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1303.7133
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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-040 LHCb-PAPER-2012-047 27 March 2013
Measurements of the branching fractions of $B^{+}\rightarrow p\bar{p}K^{+}$
decays
The LHCb collaboration†††Authors are listed on the following pages.
The branching fractions of the decay $B^{+}\rightarrow p\bar{p}K^{+}$ for
different intermediate states are measured using data, corresponding to an
integrated luminosity of $1.0\,{\rm{fb^{-1}}}$, collected by the LHCb
experiment. The total branching fraction, its charmless component
$(M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2})$ and the branching fractions via the
resonant $c\bar{c}$ states $\eta_{c}(1S)$ and $\psi(2S)$ relative to the decay
via a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ intermediate state are
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow p\bar{p}K^{+})_{\rm
total}}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,4.91\pm 0.19\,{(\rm stat)}\pm 0.14\,{(\rm
syst)},$ $\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow
p\bar{p}K^{+})_{M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}}}{{\mathcal{B}}(B^{+}\rightarrow
J/\psi K^{+}\rightarrow p\bar{p}K^{+})}=$ $\displaystyle\,2.02\pm 0.10\,{(\rm
stat)}\pm 0.08\,{(\rm syst)},$
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow\eta_{c}(1S)K^{+}\rightarrow
p\bar{p}K^{+})}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,0.578\pm 0.035\,{(\rm stat)}\pm 0.027\,{(\rm
syst)},$
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow\psi(2S)K^{+}\rightarrow
p\bar{p}K^{+})}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,0.080\pm 0.012\,{(\rm stat)}\pm 0.009\,{(\rm
syst)}.$
Upper limits on the $B^{+}$ branching fractions into the $\eta_{c}(2S)$ meson
and into the charmonium-like states $X(3872)$ and $X(3915)$ are also obtained.
Submitted to EPJ C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht9, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E. Bowen37, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15,52, A. Cook43, M. Coombes43, S. Coquereau8, G.
Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47,
C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J.
Dickens44, H. Dijkstra35, M. Dogaru26, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46,35, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, U. Eitschberger9,
R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, D. Ferguson47, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-
Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R.
Forty35, O. Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E.
Furfaro21, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y.
Gao3, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35,
R. Gauld52, E. Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V.
Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A.
Gomes2, H. Gordon52, M. Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado
Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T.
Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51,
P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P.
Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52,
M. Hoballah5, C. Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
R. Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, I. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G.
Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H.
Lu3, J. Luisier36, H. Luo47, F. Machefert7, I.V. Machikhiliyan4,28, F.
Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
R. McNulty12, B. Meadows57,52, F. Meier9, M. Meissner11, M. Merk38, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1,
M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T. Nikodem11, S. Nisar56, A.
Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pietrzyk4,
T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23,
A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A.
Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian4, J.H.
Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N.
Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S.
Ricciardi46, A. Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P. Schaack50, M.
Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B.
Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I.
Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35,
P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O.
Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, M.D.
Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci37, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, V. Syropoulos39, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14,
G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43,
A. Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12,
S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, J. Wiechczynski23, D. Wiedner11, L.
Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46,
J. Wishahi9, M. Witek23, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y.
Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The $B^{+}\rightarrow p\bar{p}K^{+}$ decay111The inclusion of charge-conjugate
modes is implied throughout the paper. offers a clean environment to study
$c\bar{c}$ states and charmonium-like mesons that decay to $p\bar{p}$ and
excited ${\bar{}\mathchar 28931\relax}$ baryons that decay to $\bar{p}K^{+}$,
and to search for glueballs or exotic states. The presence of $p\bar{p}$ in
the final state allows intermediate states of any quantum numbers to be
studied and the existence of the charged kaon in the final state significantly
enhances the signal to background ratio in the selection procedure.
Measurements of intermediate charmonium-like states, such as the $X(3872)$,
are important to clarify their nature [1, 2] and to determine their partial
width to $p\bar{p}$, which is crucial to predict the production rate of these
states in dedicated experiments [3]. BaBar and Belle have previously measured
the $B^{+}\rightarrow p\bar{p}K^{+}$ branching fraction, including
contributions from the $J/\psi$ and $\eta_{c}(1S)$ intermediate states [4, 5].
The data sample, corresponding to an integrated luminosity of
$1.0\,{\rm{fb^{-1}}}$, collected by LHCb at $\sqrt{s}=7\,{\rm{TeV}}$ allows
the study of substructures in the $B^{+}\rightarrow p\bar{p}K^{+}$ decays with
a sample ten times larger than those available at previous experiments.
In this paper we report measurements of the ratios of branching fractions
${\cal R}({\rm mode})=\frac{{\cal B}(B^{+}\rightarrow{\rm mode}\rightarrow
p\bar{p}K^{+})}{{\cal B}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})},$ (1)
where “mode” corresponds to the intermediate $\eta_{c}(1S)$, $\psi(2S)$,
$\eta_{c}(2S)$, $\chi_{c0}(1P)$, $h_{c}(1P)$, $X(3872)$ or $X(3915)$ states,
together with a kaon.
## 2 Detector and software
The LHCb detector [6] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has momentum $(p)$ resolution $\Delta p/p$ that
varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers.
The trigger [7] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage where candidates
are fully reconstructed. The hardware trigger selects hadrons with high
transverse energy in the calorimeter. The software trigger requires a two-,
three- or four-track secondary vertex with a high $p_{\rm T}$ sum of the
tracks and a significant displacement from the primary $pp$ interaction
vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter (IP)
$\chi^{2}$ with respect to the primary interaction greater than 16. The IP
$\chi^{2}$ is defined as the difference between the $\chi^{2}$ of the PV
reconstructed with and without the considered track. A multivariate algorithm
is used for the identification of secondary vertices consistent with the decay
of a $b$ hadron.
Simulated $B^{+}\rightarrow p\bar{p}K^{+}$ decays, generated uniformly in
phase space, are used to optimize the signal selection and to evaluate the
ratio of the efficiencies for each considered channel with respect to the
$J/\psi$ channel. Separate samples of $B^{+}\rightarrow J/\psi
K^{+}\rightarrow p\bar{p}K^{+}$ and
$B^{+}\rightarrow\eta_{c}(1S)K^{+}\rightarrow p\bar{p}K^{+}$ decays, generated
with the known angular distributions, are used to check the dependence of the
efficiency ratio on the angular distribution. In the simulation, $pp$
collisions are generated using Pythia 6.4 [8] with a specific LHCb
configuration [9]. Decays of hadronic particles are described by EvtGen [10]
in which final state radiation is generated by Photos [11]. The interaction of
the generated particles with the detector and its response are implemented
using the Geant4 toolkit [12, *Agostinelli:2002hh] as described in Ref. [14].
## 3 Candidate selection
Candidate $B^{+}\rightarrow p\bar{p}K^{+}$ decays are reconstructed from any
combination of three charged tracks with total charge of $+1$. The final state
particles are required to have a track fit with a $\chi^{2}/{\rm ndf}<3$ where
ndf is the number of degrees of freedom. They must also have
$p>1500\,{\rm{MeV/}}c$, $p_{\rm T}$ $>100\,{\rm{MeV}}/c$, and IP $\chi^{2}>1$
with respect to any primary vertex in the event. Particle identification (PID)
requirements, based on the RICH detector information, are applied to $p$ and
$\bar{p}$ candidates. The discriminating variables between different particle
hypotheses ($\pi$, $K$, $p$) are the differences between log-likelihood values
$\Delta$ln${\mathcal{L}}_{\alpha\beta}$ under particle hypotheses $\alpha$ and
$\beta$, respectively. The $p$ and $\bar{p}$ candidates are required to have
$\Delta$ln${\mathcal{L}}_{p\pi}>-5$. The reconstructed $B^{+}$ candidates are
required to have an invariant mass in the range $5079-5579\,{\rm{MeV/}}c^{2}$.
The asymmetric invariant mass range around the nominal $B^{+}$ mass is
designed to select also $B^{+}\rightarrow p\bar{p}\pi^{+}$ candidates without
any requirement on the PID of the kaon. The PV associated to each $B^{+}$
candidate is defined to be the one for which the $B^{+}$ candidate has the
smallest IP $\chi^{2}$. The $B^{+}$ candidate is required to have a vertex fit
with a $\chi^{2}/{\rm ndf}<12$ and a distance greater than $3\,{\rm{mm}}$, a
$\chi^{2}$ for the flight distance greater than $500$, and an IP $\chi^{2}<10$
with respect to the associated PV. The maximum distance of closest approach
between daughter tracks has to be less than $0.2\,{\rm{mm}}$. The angle
between the reconstructed momentum of the $B^{+}$ candidate and the $B^{+}$
flight direction ($\theta_{\rm fl}$) is required to have $\cos\theta_{\rm
fl}>0.99998$.
The reconstructed candidates that meet the above criteria are filtered using a
boosted decision tree (BDT) algorithm [15]. The BDT is trained with a sample
of simulated $B^{+}\rightarrow p\bar{p}K^{+}$ signal candidates and a
background sample of data candidates taken from the invariant mass sidebands
in the ranges $5080-5220\,{\rm{MeV/}}c^{2}$ and $5340-5480\,{\rm{MeV/}}c^{2}$.
The variables used by the BDT to discriminate between signal and background
candidates are: the $p_{\rm T}$ of each reconstructed track; the sum of the
daughters’ $p_{T}$; the sum of the IP $\chi^{2}$ of the three daughter tracks
with respect to the primary vertex; the IP of the daughter, with the highest
$p_{\rm T}$, with respect to the primary vertex; the number of daughters with
$p_{\rm T}$ $>900\,{\rm{GeV/}}c$; the maximum distance of closest approach
between any two of the $B^{+}$ daughter particles; the IP of the $B^{+}$
candidate with respect to the primary vertex; the distance between primary and
secondary vertices; the $\theta_{\rm fl}$ angle; the $\chi^{2}/{\rm ndf}$ of
the secondary vertex; a pointing variable defined as
$\frac{P\sin\theta}{P\sin\theta+\sum_{i}p_{\rm T,i}}$, where $P$ is the total
momentum of the three-particle final state, $\theta$ is the angle between the
direction of the sum of the daughter’s momentum and the direction of the
flight distance of the $B^{+}$ and $\sum_{i}p_{{\rm T},i}$ is the sum of the
transverse momenta of the daughters; and the log likelihood difference for
each daughter between the assumed PID hypothesis and the pion hypothesis. The
selection criterion on the BDT response (Fig. 1) is chosen in order to have a
signal to background ratio of the order of unity. This corresponds to a BDT
response value of $-0.11$. The efficiency of the BDT selection is greater than
$92\%$ with a background rejection greater than $86\%$.
Figure 1: Distribution of the BDT algorithm response evaluated for background
candidates from the data sidebands (red), and signal candidates from
simulation (blue). The black dotted line indicates the chosen BDT response
value.
Figure 2: Invariant mass distribution of a) all selected $B^{+}\rightarrow
p\bar{p}K^{+}$ candidates and b) candidates having
$M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}$. The points with error bars are the data
and the solid lines are the result of the fit. The dotted lines represent the
two Gaussian functions (red) and the dashed line the linear function (green)
used to parametrize the signal and the background, respectively. The vertical
lines indicate the signal region. The two plots below the mass distributions
show the pulls.
## 4 Signal yield determination
Figure 3: Invariant mass distribution of the $p\bar{p}$ system for
$B^{+}\rightarrow p\bar{p}K^{+}$ candidates within the $B^{+}$ mass signal
window, $|M(p\bar{p}K^{+})-M_{B^{+}}|<50\,{\rm{MeV/}}c^{2}$. The dotted lines
represent the Gaussian and Voigtian functions (red) and the dashed line the
smooth function (green) used to parametrize the signal and the background,
respectively. The bottom plot shows the pulls.
Figure 4: Invariant mass distribution of the $p\bar{p}$ system in the regions
around a) the $\eta_{c}(1S)$ and $J/\psi$ and b) the $\eta_{c}(2S)$ and
$\psi(2S)$ states. The dotted lines represent the Gaussian and the Voigtian
functions (red) and the dashed line the smooth function (green) used to
parametrize the signal and the background, respectively. The two plots below
the mass distribution show the pulls.
Figure 5: Invariant mass distribution of the $p\bar{p}$ system in the regions
around a) the $\chi_{c0}(1P)$ and $h_{c}$ and b) the $X(3872)$ and $X(3915)$
states. The dotted lines represent the Gaussian and Voigitian functions (red)
and the dashed line the smooth function (green) used to parametrize the signal
and the background, respectively. The two plots below the mass distribution
show the pulls.
The signal yield is determined from an unbinned extended maximum likelihood
fit to the invariant mass of selected $B^{+}\rightarrow p\bar{p}K^{+}$
candidates, shown in Fig. 2a). The signal component is parametrized as the sum
of two Gaussian functions with the same mean and different widths. The
background component is parametrized as a linear function. The signal yield of
the charmless component is determined by performing the same fit described
above to the sample of $B^{+}\rightarrow p\bar{p}K^{+}$ candidates with
$M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}$, shown in Fig. 2b). The $B^{+}$ mass and
widths, evaluated with the invariant mass fits to all of the $B^{+}\rightarrow
p\bar{p}K^{+}$ candidates, are compatible with the values obtained for the
charmless component.
The signal yields for the charmonium contributions,
$B^{+}\rightarrow(c\bar{c})K^{+}\rightarrow p\bar{p}K^{+}$, are determined by
fitting the $p\bar{p}$ invariant mass distribution of $B^{+}\rightarrow
p\bar{p}K^{+}$ candidates within the $B^{+}$ mass signal window,
$|M_{p\bar{p}K^{+}}-M_{B^{+}}|<50\,{\rm{MeV/}}c^{2}$. Simulations show that no
narrow structures are induced in the $p\bar{p}$ spectrum as kinematic
reflections of possible $B^{+}\rightarrow p{\it\bar{\Lambda}}\rightarrow
p\bar{p}K^{+}$ intermediate states.
An unbinned extended maximum likelihood fit to the $p\bar{p}$ invariant mass
distribution, shown in Fig. 3, is performed over the mass range
$2400-4500\,{\rm{MeV/}}c^{2}$. The signal components of the narrow resonances
$J/\psi$, $\psi(2S)$, $h_{c}(1P)$, and $X(3872)$, whose natural widths are
much smaller than the $p\bar{p}$ invariant mass resolution, are parametrized
by Gaussian functions. The signal components for the $\eta_{c}(1S)$,
$\chi_{c0}(1P)$, $\eta_{c}(2S)$, and $X(3915)$ are parametrized by Voigtian
functions.222A Voigtian function is the convolution of a Breit-Wigner function
with a Gaussian distribution. Since the $p\bar{p}$ invariant mass resolution
is approximately constant in the explored range, the resolution parameters for
all resonances, except the $\psi(2S)$, are fixed to the $J/\psi$ value
($\sigma_{J/\psi}=8.9\pm 0.2\,{\rm{MeV/}}c^{2}$). The background shape is
parametrized as $f(M)=e^{c_{1}M+c_{2}M^{2}}$ where $c_{1}$ and $c_{2}$ are fit
parameters. The $J/\psi$ and $\psi(2S)$ resolution parameters, the mass values
of the $\eta_{c}(1S)$, $J/\psi$, and $\psi(2S)$ states, and the $\eta_{c}(1S)$
natural width are left free in the fit. The masses and widths for the other
signal components are fixed to the corresponding world averages [16]. The
$p\bar{p}$ invariant mass resolution, determined by the fit to the $\psi(2S)$
is $\sigma_{\psi(2S)}=7.9\pm 1.7\,{\rm{MeV/}}c^{2}$.
The fit result is shown in Fig. 3. Figures 4 and 5 show the details of the fit
result in the regions around the $\eta_{c}(1S)$ and $J/\psi$, $\eta_{c}(2S)$
and $\psi(2S)$, $\chi_{c0}(1P)$ and $h_{c}$, and $X(3872)$ and $X(3915)$
resonances. Any bias introduced by the inaccurate description of the tails of
the $\eta_{c}(1S)$, $J/\psi$ and $\psi(2S)$ resonances is taken into account
in the systematic uncertainty evaluation.
The contribution of $c\bar{c}\rightarrow p\bar{p}$ from processes other than
$B^{+}\rightarrow p\bar{p}K^{+}$ decays, denoted as “non-signal”, is estimated
from a fit to the $p\bar{p}$ mass in the $B^{+}$ mass sidebands $5130-5180$
and $5380-5430\,{\rm{MeV/}}c^{2}$. Except for the $J/\psi$ mode, no evidence
of a non-signal contribution is found. The non-signal contribution to the
$J/\psi$ signal yield in the $B^{+}$ mass window is $43\pm 11$ candidates and
is subtracted from the number of $J/\psi$ signal candidates.
The signal yields, corrected for the non-signal contribution, are reported in
Table 1. For the intermediate charmonium states $\eta_{c}(2S)$,
$\chi_{c0}(1P)$, $h_{c}(1P)$, $X(3872)$ and $X(3915)$, there is no evidence of
signal. The $95\%\,{\rm{CL}}$ upper limits on the number of candidates are
shown in Table 1 and are determined from the likelihood profile integrating
over the nuisance parameters. Since for the $X(3872)$ the fitted signal yield
is negative, the upper limit has been calculated integrating the likelihood
only in the physical region of a signal yield greater than zero.
Table 1: Signal yields for the different channels and corresponding 95% CL upper limits for modes with less than 3$\sigma$ statistical significance. For the $J/\psi$ mode, the non-signal yield is subtracted. Uncertainties are statistical only. $B^{+}$ decay mode | Signal yield | Upper limit (95% CL)
---|---|---
$p\bar{p}K^{+}\;{\rm[total]}$ | $6951\,$$\pm$ | $\,176$ |
$p\bar{p}K^{+}\;[M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}]$ | $3238\,$$\pm$ | $\,122$ |
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $1458\,$$\pm$ | $\,42$ |
$\eta_{c}(1S)K^{+}$ | $856\,$$\pm$ | $\,46$ |
$\psi(2S)K^{+}$ | $107\,$$\pm$ | $\,16$ |
$\eta_{c}(2S)K^{+}$ | $39\,$$\pm$ | $\,15$ | $<65.4$
$\chi_{c0}(1P)K^{+}$ | $15\,$$\pm$ | $\,13$ | $<38.1$
$h_{c}(1P)K^{+}$ | $21\,$$\pm$ | $\,11$ | $<40.2$
$X(3872)K^{+}$ | $-9\,$$\pm$ | $\,8$ | $<10.3$
$X(3915)K^{+}$ | $13\,$$\pm$ | $\,17$ | $<42.1$
## 5 Efficiency determination
The ratio of branching fractions is calculated using
${\cal R}({\rm mode})=\frac{{\cal B}(B^{+}\rightarrow{\rm mode}\rightarrow
p\bar{p}K^{+})}{{\cal B}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=\frac{N_{\rm mode}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\times\frac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\epsilon_{\rm mode}},$ (2)
where $N_{\rm mode}$ and $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
are the signal yields for the given mode and the reference mode,
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\rightarrow
p\bar{p}K^{+}$, and $\epsilon_{\rm
mode}/\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the
corresponding ratio of efficiencies. The efficiency is the product of the
reconstruction, trigger, and selection efficiencies, and is estimated using
simulated data samples.
Figure 6: Efficiency as a function of $M_{p\bar{p}}$ for $B^{+}\rightarrow
p\bar{p}K^{+}$ decays. The solid line represents the linear fit to the
efficiency distribution; the dashed line is the point-by-point interpolation
used to estimate the systematic uncertainty.
Since the track multiplicity distribution for simulated events differs from
that observed in data, simulated candidates are assigned a weight so that the
weighted distribution reproduces the observed multiplicity distribution. The
distributions of $\Delta$ln${\mathcal{L}}_{K\pi}$ and
$\Delta$ln${\mathcal{L}}_{p\pi}$ for kaons and protons in data are obtained in
bins of momentum, pseudorapidity and number of tracks from control samples of
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays for kaons
and ${\it\mathchar 28931\relax}\rightarrow p\pi^{-}$ decays for protons, which
are then used on a track-by-track basis to correct the simulation. The
efficiency as a function of $M_{p\bar{p}}$ is shown in Fig. 6. A linear fit to
the efficiency distribution is performed and the efficiency ratios are
determined based on the fit result.
Table 2: Relative systematic uncertainties (in $\%$) on the relative branching
fractions from different sources. The total systematic uncertainty is
determined by adding the individual contributions in quadrature.
Source | ${\cal R}({\rm total})$ | ${\cal R}(M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2})$ | ${\cal R}(\eta_{c}(1S))$ | ${\cal R}(\psi(2S))$
---|---|---|---|---
Efficiency ratio | 0.21 | 0.5 | 3.3 | 4.8
$B^{+}$ mass fit range | 0.16 | 0.5 | $-$ | $-$
Sig. and Bkg. shape | 2.5 | 3.6 | 1.8 | 6.5
$B^{+}$ mass window | 0.6 | 0.6 | 0.9 | 3.8
Non-signal component | $-$ | $-$ | 0.4 | 5.1
Signal tail param. | 1.0 | 1.0 | 1.2 | 4.3
Total | 2.8 | 3.8 | 4.1 | 11.3
Source | ${\cal R}(\eta_{c}(2S))$ | ${\cal R}(\chi_{c0}(1P))$ | ${\cal R}(h_{c}(1P))$ | ${\cal R}(X(3872))$ | ${\cal R}(X(3915))$
---|---|---|---|---|---
Efficiency ratio | 4.4 | 2.5 | 3.4 | 6.5 | 7.0
$B^{+}$ mass fit range | $-$ | $-$ | $-$ | $-$ |
Sig. and Bkg. shape | 3.9 | 3.3 | 14.3 | 5.6 | 10.1
$B^{+}$ mass window | 11.3 | 23.6 | 23.6 | 17.5 | 7.5
Non-signal component | $-$ | $-$ | $-$ | $-$ | $-$
Signal tail param. | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Total | 12.8 | 24.0 | 27.8 | 19.5 | 15.5
## 6 Systematic uncertainties
The measurements of the relative branching fractions depend on the ratios of
signal yields and efficiencies with respect to the reference mode. Since the
final state is the same in all cases, most of the systematic uncertainties
cancel. The systematic uncertainty on the efficiency ratio, in each region of
$p\bar{p}$ invariant mass, is determined from the difference between the
efficiency ratios calculated using the solid fitted line and the dashed point-
by-point interpolation shown in Fig. 6. The uncertainty associated with the
evaluation of the $B^{+}$ signal yield has been determined by varying the fit
range by $\pm 30\,{\rm{MeV/}}c^{2}$, using a single Gaussian instead of a
double Gaussian function to model the signal PDF, and using an exponential
function to model the background. For each charmonium resonance the systematic
uncertainty on the signal yield has been investigated by varying the $B$ mass
signal window by $\pm 10\,{\rm{MeV/}}c^{2}$, the signal and background shape
parametrization and the subtraction of the $c\bar{c}$ contribution from the
continuum. The systematic uncertainty associated with the parametrization of
the signal tails of the $J/\psi$, $\eta_{c}(1S)$ and $\psi(2S)$ resonances is
taken into account by taking the difference between the number of candidates
in the observed distribution and the number of candidates calculated from the
integral of the fit function in the range $-6\sigma$ to $-2.5\sigma$. The
systematic uncertainty associated with the selection procedure is estimated by
changing the value of the BDT selection to $-0.03$, which retains $85\%$ of
the signal with a $30\%$ background, and is found to be negligible. The
contributions to the systematic uncertainties from the different sources are
listed in Table 2. The total systematic uncertainty is determined by adding
the individual contributions in quadrature.
## 7 Results
The results are summarized in Table 3 and the values of the product of
branching fractions derived from our measurement using the world average
values ${\cal B}(B^{+}\rightarrow J/\psi K^{+})=(1.013\pm 0.034)\times
10^{-3}$ and ${\cal B}(J/\psi\rightarrow p\bar{p})=(2.17\pm 0.07)\times
10^{-3}$ [16] are listed in Table 4.
Table 3: Signal yields, efficiency ratios, ratios of branching fractions and corresponding upper limits. $B^{+}\rightarrow({\rm mode})$ | Yield | $\epsilon_{\rm mode}/\epsilon_{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ | ${\cal R}({\rm mode})$ | Upper Limit
---|---|---|---|---
$\rightarrow p\bar{p}K^{+}$ | $\pm$ stat $\pm$ syst | $\pm$ syst | $\pm$ stat $\pm$ syst | $95\%$ CL
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $1458\,$$\pm$ | $\,42\,$$\pm$ | $\,24$ | $-$ | 1 | $-$
total | $6951\,$$\pm$ | $\,176\,$$\pm$ | $\,171$ | $0.970\pm 0.002$ | $4.91\,$$\pm$ | $\,0.19\,$$\pm$ | $\,0.14$ | $-$
${M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}}$ | $3238\,$$\pm$ | $\,122\,$$\pm$ | $\,121$ | $1.097\pm 0.006$ | $2.02\,$$\pm$ | $\,0.10\,$$\pm$ | $\,0.08$ | $-$
$\eta_{c}(1S)K^{+}$ | $856\,$$\pm$ | $\,46\,$$\pm$ | $\,19$ | $1.016\pm 0.034$ | $0.578\,$$\pm$ | $\,0.035\,$$\pm$ | $\,0.026$ | $-$
$\psi(2S)K^{+}$ | $107\,$$\pm$ | $\,16$$\pm$ | $\,13$ | $0.921\pm 0.044$ | $0.080\,$$\pm$ | $\,0.012\,$$\pm$ | $\,0.009$ | $-$
$\eta_{c}(2S)K^{+}$ | $39\,$$\pm$ | $\,15\,$$\pm$ | $\,5$ | $0.927\pm 0.041$ | $0.029\,$$\pm$ | $\,0.011\,$$\pm$ | $\,0.004$ | $<0.048$
$\chi_{c0}(1P)K^{+}$ | $15\,$$\pm$ | $\,13\,$$\pm$ | $\,4$ | $0.957\pm 0.024$ | $0.011\,$$\pm$ | $\,0.009\,$$\pm$ | $\,0.003$ | $<0.028$
$h_{c}(1P)K^{+}$ | $21\,$$\pm$ | $\,11\,$$\pm$ | $\,5$ | $0.943\pm 0.032$ | $0.015\,$$\pm$ | $\,0.008\,$$\pm$ | $\,0.004$ | $<0.029$
$X(3872)K^{+}$ | $-9\,$$\pm$ | $\,8\,$$\pm$ | $\,2$ | $0.896\pm 0.058$ | $-0.007\,$$\pm$ | $\,0.006\,$$\pm$ | $\,0.002$ | $<0.008$
$X(3915)K^{+}$ | $13\,$$\pm$ | $\,17\,$$\pm$ | $\,5$ | $0.890\pm 0.062$ | $0.010\,$$\pm$ | $\,0.013\,$$\pm$ | $\,0.002$ | $<0.032$
Table 4: Branching fractions for $B^{+}\rightarrow({\rm mode})\rightarrow p\bar{p}K^{+}$ derived using the world average value of the ${\cal B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$ and ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow p\bar{p})$ branching fractions [16]. For the charmonium modes we compare our values to the product of the indipendently measured branching fractions. The first uncertainties are statistical, the second systematic in the present measurement, and the third systematic from the uncertainty on the $J/\psi$ branching fraction. $B^{+}$ | ${\cal B}(B^{+}\rightarrow({\rm mode})\rightarrow p\bar{p}K^{+})$ | UL $(95\%$ CL) | Previous measurements
---|---|---|---
decay mode | ($\times 10^{6}$) | ($\times 10^{6}$) | ($\times 10^{6}$) [4, 5]
total | $10.81\,$$\pm$ | $\,0.42\,$$\pm$ | $\,0.30\,$$\pm$ | $\,0.49$ | | $10.76^{+0.36}_{-0.33}\pm 0.70$
${M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}}$ | $4.46\,$$\pm$ | $\,0.21\,$$\pm$ | $\,0.18\,$$\pm$ | $\,0.20$ | | $5.12\pm 0.31$
$\eta_{c}(1S)K^{+}$ | $1.27\,$$\pm$ | $\,0.08\,$$\pm$ | $\,0.05\,$$\pm$ | $\,0.06\,$ | | $1.54\pm 0.16$
$\psi(2S)K^{+}$ | $0.175\,$$\pm$ | $\,0.027\,$$\pm$ | $\,0.020\,$$\pm$ | $\,0.008$ | | $0.176\pm 0.012$
$\eta_{c}(2S)K^{+}$ | $0.063\,$$\pm$ | $\,0.025\,$$\pm$ | $\,0.009\,$$\pm$ | $\,0.003$ | $<0.106$ |
$\chi_{c0}(1P)K^{+}$ | $0.024\,$$\pm$ | $\,0.021\,$$\pm$ | $\,0.006\,$$\pm$ | $\,0.001$ | $<0.062$ | $0.030\pm 0.004$
$h_{c}(1P)K^{+}$ | $0.034\,$$\pm$ | $\,0.018\,$$\pm$ | $\,0.008\,$$\pm$ | $\,0.002$ | $<0.064$ |
$X(3872)K^{+}$ | $-0.015\,$$\pm$ | $\,0.013\,$$\pm$ | $\,0.003\,$$\pm$ | $\,0.001$ | $<0.017$ |
$X(3915)K^{+}$ | $0.022\,$$\pm$ | $\,0.029\,$$\pm$ | $\,0.004\,$$\pm$ | $\,0.001$ | $<0.071$ |
The branching fractions obtained are compatible with the world average values
[16]. The upper limit on
${\mathcal{B}}(B^{+}\rightarrow\chi_{c0}(1P)K^{+}\rightarrow p\bar{p}K^{+})$
is compatible with the world average
${\mathcal{B}}(B^{+}\rightarrow\chi_{c0}(1P)K^{+})\times{\mathcal{B}}(\chi_{c0}(1P)\rightarrow
p\bar{p})=(0.030\pm 0.004)\times 10^{-6}$ [16]. We combine our upper limit for
$X(3872)$ with the known value for ${\mathcal{B}}(B^{+}\rightarrow
X(3872)K^{+})\times{\mathcal{B}}(X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-})=(8.6\pm 0.8)\times 10^{-6}$ [16] to obtain the limit
$\frac{{\mathcal{B}}(X(3872)\rightarrow
p\bar{p})}{{\mathcal{B}}(X(3872)\rightarrow J/\psi\pi^{+}\pi^{-})}<2.0\times
10^{-3}.$
This limit challenges some of the predictions for the molecular
interpretations of the $X(3872)$ state and is approaching the range of
predictions for a conventional $\chi_{c1}(2P)$ state [17, 18]. Using our
result and the $\eta_{c}(2S)$ branching fraction
${\mathcal{B}}(B^{+}\rightarrow\eta_{c}(2S)K^{+})\times{\mathcal{B}}(\eta_{c}(2S)\rightarrow
K\bar{K}\pi)=(3.4\,^{+2.3}_{-1.6})\times 10^{-6}$ [16], a limit of
$\frac{{\mathcal{B}}(\eta_{c}(2S)\rightarrow
p\bar{p})}{{\mathcal{B}}(\eta_{c}(2S)\rightarrow K\bar{K}\pi)}<3.1\times
10^{-2}$
is obtained.
## 8 Summary
Based on a sample of $6951\pm 176$ $B^{+}\rightarrow p\bar{p}K^{+}$ decays
reconstructed in a data sample, corresponding to an integrated luminosity of
$1.0\,{\rm{fb^{-1}}}$, collected with the LHCb detector, the following
relative branching fractions are measured
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow p\bar{p}K^{+})_{\rm
total}}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,4.91\pm 0.19\,{(\rm stat)}\pm 0.14\,{(\rm
syst)},$ $\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow
p\bar{p}K^{+})_{M_{p\bar{p}}<2.85\,{\rm{GeV/}}c^{2}}}{{\mathcal{B}}(B^{+}\rightarrow
J/\psi K^{+}\rightarrow p\bar{p}K^{+})}=$ $\displaystyle\,2.02\pm 0.10\,{(\rm
stat)}\pm 0.08\,{(\rm syst)},$
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow\eta_{c}(1S)K^{+}\rightarrow
p\bar{p}K^{+})}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,0.578\pm 0.035\,{(\rm stat)}\pm 0.025\,{(\rm
syst)},$
$\displaystyle\frac{{\mathcal{B}}(B^{+}\rightarrow\psi(2S)K^{+}\rightarrow
p\bar{p}K^{+})}{{\mathcal{B}}(B^{+}\rightarrow J/\psi K^{+}\rightarrow
p\bar{p}K^{+})}=$ $\displaystyle\,0.080\pm 0.012\,{(\rm stat)}\pm 0.009\,{(\rm
syst)}.$
An upper limit on the ratio $\frac{{\mathcal{B}}(B^{+}\rightarrow
X(3872)K^{+}\rightarrow p\bar{p}K^{+})}{{\mathcal{B}}(B^{+}\rightarrow J/\psi
K^{+}\rightarrow p\bar{p}K^{+})}<0.017$ is obtained, from which a limit of
$\frac{{\mathcal{B}}(X(3872)\rightarrow
p\bar{p})}{{\mathcal{B}}(X(3872)\rightarrow J/\psi\pi^{+}\pi^{-})}<2.0\times
10^{-3}$
is derived.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-03-28T14:24:56 |
2024-09-04T02:49:43.576279
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D. Craik,\n S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis,\n K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De\n Silva, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H.\n Dijkstra, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A.\n Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P.\n Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton,\n F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei,\n M. Karacson, T.M. Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B.\n Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi,\n M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez,\n S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski,\n A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek,\n S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang,\n R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L.\n Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Roberta Cardinale",
"url": "https://arxiv.org/abs/1303.7133"
}
|
1303.7453
|
# Atomically Sharp 318nm Gd:AlGaN Ultraviolet Light Emitting Diodes on Si with
Low Threshold Voltage
Thomas F. Kent Department of Materials Science and Engineering, The Ohio
State University, Columbus, Ohio 43210, USA Santino D. Carnevale Department
of Materials Science and Engineering, The Ohio State University, Columbus,
Ohio 43210, USA Roberto C. Myers Department of Materials Science and
Engineering, The Ohio State University, Columbus, Ohio 43210, USA Deparment
of Electrical and Computer Engineering, The Ohio State University, Columbus,
Ohio 43210, USA
###### Abstract
Self-assembled AlxGa1-xN polarization-induced nanowire light emitting diodes
(PINLEDs) with Gd-doped AlN active regions are prepared by plasma-assisted
molecular beam epitaxy on Si substrates. Atomically sharp electroluminescence
(EL) from Gd intra-f-shell electronic transitions at 313nm and 318nm is
observed under forward biases above 5V. The intensity of the Gd 4f EL scales
linearly with current density and increases at lower temperature. The low
field excitation of Gd 4f EL in PINLEDs is contrasted with high field
excitation in metal/Gd:AlN/polarization-induced n-AlGaN devices; PINLED
devices offer over a three fold enhancement in 4f EL intensity at a given
device bias.
Keywords: AlGaN, Nanowires, Rare Earth, Electroluminescent Devices
In this letter we report on ultraviolet light emitting diodes based on self-
assembled AlGaN nanowire heterojunctions doped with Gd, which emit ultraviolet
(UV) radiation at 318nm and operate at lower device bias compared with
existing Gd electroluminescent technology. Following the realization of the
ruby laser over half a century agomaiman , many materials have been developed
as hosts for elements with optically-active energetic transitions to produce
lasers and electroluminescent devices with emission in the visible and
infrared part of the electromagnetic spectrum. In the last twenty years,
electrically driven devices utilizing rare earth phosphors in wide or medium
gap semiconductors have begun to make an appearancerack . These thin film
electroluminescent devices (TFED) have found applications as visible and IR
emitters, and with the increasing use of wide gap materials, such as GaN and
AlN are currently being developed to take of advantage of phosphors in the
blue and UV.
Figure 1: Optically active Gd3+ 4f levels in the active region of a III-N
nanowire polarization induced light emitting diode. a. Calculated device band
diagram showing location of optically active Gd ions. Insert shows device
schematic as measured. b. Device IV exhibiting turn on in forward bias at 5V
Figure 2: Optically active Gd ions in the depletion region of a polarization-
induced light emitting diode (PINLED). a. Room temperature device EL spectra
for multiple currents. b. Detail of spectral region of interest for the Gd 4f
shell EL peaks, showing first excited and second excited state transitions.
Insert shows behavior of peak intensity with increasing current density. c.
Detail of spectra region of interest for the Gd 4f EL at multiple
temperatures. Insert shows variation of peak intensity with temperature.
Design of optoelectronic devices which utilize atomic transitions in the rare
earths for EL can offer a number of advantages over EL produced from band-to-
band recombination. First, the transition energy is dictated by the energy
level scheme of the 4f orbital, which is relatively unperturbed by the
crystalline environment due to the fact that the 5d and 5s,5p orbitals extend
to further radial distances, and fill before the f shell of lower principal
quantum numberatkins . This shielding by the earlier filled 5d and 5s orbitals
causes the energies of the 4f transitions to be relatively insensitive to
crystalline imperfections, unlike transitions based on band-to-band
transitions in semiconductors. Band-to-band optical transitions are well
knowndavies02 to be sensitive to deep levels, exciton-phonon interaction, and
crystalline disorder which can lead to broadening of emission or parasitic
emission at an unintended energy. Additionally, due to the decoupling of the
4f orbital with the lattice, emission from rare earth centers is spectrally
pure with common FWHM of less than 30meV.
In recent years, the family of III-nitrides, particularly the pseudobinary
AlxGa1-xN, has become attractive as a potential host for rare earth elemental
phosphorswakahara . This material system offers a high breakdown field
(12MV/cm for AlN) and sufficiently wide bandgaps (3.4 to 6.1eV) to accommodate
phosphors with emission from the UV to IR. Equally significant is the way in
which rare earths incorporate in the wurtzite structure of GaN and AlN alloys.
Rare earth (RE) ions commonly exhibit the RE3+ ionization state, which makes
them isovalent with the Al3+ and Ga3+ cations of AlxGa1-xN. RE atoms exhibit
high solubility (in excess of 1at% has been reportedwang ) and regularly
incorporate at the cation site. Due to the wide range of energy level schemes
of the lanthanides, EL from RE centers in AlN and GaN have been reported for
Er3+ (IR, green)Lee , Tm3+(blue)Lee , Eu3+(Red)wakahara , and Gd3+(UV)kita .
Additonally, room temperature optically pumped lasing in Eu-doped GaNpark has
been successfully demonstrated.
Although most RE phosphors have been developed for optical transitions in the
visible and infrared parts of the electromagnetic spectrum, the energy scheme
of Gd3+ in wurtzite AlN offers an energy difference between ground and first
excited state of 3.90eV (318nm). The spectrally-narrow and energetically-
stable nature of the Gd3+ fluorescence emission make it a potential candidate
for spectroscopic and lithographic applications in the UV. This led to
exploration of dilutely Gd doped AlxGa1-xN in the form of fluorescencezavada
and cathodoluminescence experimentsvetter ; gruber ; kita .Although the 4f
levels in the RE3+ are typically thought not to interact with the surrounding
lattice, cathodoluminescence data for Gd:AlN thin films show phonon replica
satellite peaks of the Gd3+ 6P7/2$\rightarrow$8S7/2 (318nm) transitionsvetter
. These data suggest that the 4f electrons in Gd3+ in AlN are not completely
decoupled from the host lattice.
Although there have been a number of reportskita ; zavada ; vetter ; gruber
on the spectroscopy of Gd:AlGaN compounds, less work has focused on
development of active optoelectronic devices that utilize Gd3+ 4f transitions.
This is likely due to the difficulty of achieving electrical contact to uid-
AlN. Reportskita ; kitayama have been made of a “field emission device”
consisting of a reactive ion sputtered AlxGd1-xN film with metal contacts,
forming a MIS structure whereby a high voltage 270V to $>$1kV driven across
the device produces fluorescence of the Gd3+ ions, likely by the process of
impact excitation.
Figure 3: Optically active Gd ions in a purely unipolar metal-insulator-
semiconductor heterostructure a. Device heterostructure and calculated band
diagram. Insert shows device schematic. b. Device IV curve exhibiting
rectification and a turn on voltage above 10V. c. EL spectra showing weak EL
of the Gd 4f 6P7/2$\rightarrow$8S7/2 transition. Insert shows detail of the 4f
EL peak with an overlay of the more intense spectrum from the PINLED device
under comparable conditions.
The dominant mechanism of RE3+ 4f excitation in solids depends on the electric
field regime where the device functions. MIS devices function largely in the
high field regime, causing the dominant mechanism to be direct transfer of
kinetic energy from hot electrons to the RE3+ center by collision, known as
impact excitation. At lower fields, the excitation mechanism becomes more
complex, involving typically a multi-step, defect assisted Auger process, or
exciton localization by the RE3+ center.godlewski ; bodiou In this work, we
study two seperate heterostructures which are designed to generate EL under
low electric field conditions (pn-diode) and high electric field conditions
(Gd:AlGaN MIS structure). In the PINLED, the initially large built in electric
field in the depletion region is reduced with forward bias of the device. In
contrast, the MIS structure has initially flat bands and thus no electric
field in the active region and when biased, the electric field is increased in
the Gd doped region.
Self-assembled III-nitride nanowire heterostructures grown by plasma assisted
molecular beam epitaxy have recently gained popularity for applications
requiring high crystalline quality and largely mismatched, complex
heterostructures that would otherwise be difficult to form in thin films due
to strain considerationscarnevale1 ; carnevale2 ; carnevale3 . In addition,
they have been shown to function at high current densitiescarnevale3 .
Additionally, UV-LEDs based on III-nitride nanowire heterostructures have been
demonstrated to accommodate active regions spanning from high %Al AlGaN to
GaNcarnevale3 .
Polarization-induced nanowire diodes (PINLEDs)carnevale2 containing Gd doping
in their active region are prepared by plasma-assisted molecular beam epitaxy
on n-Si(111) in the III-limited growth regime, the details of which are
discussed elsewherecarnevale2 . These devices consist of a GaN nucleation
layer followed by a linear grade in composition from GaN to AlN over 100nm.
This is followed by an active region consisting of 2.4ML of GdN deposited
between two 5nm uid-AlN spacers. The structure is then linearly graded in
composition from AlN back to GaN. This structure forms a pn-diode as shown in
Fig. 1a, the band diagram of which is calculated with a self consistent
Schrödinger-Poisson solverbandeng , as described in refcarnevale2 . The
spontaneous polarization present in wurtzite AlxGa1-xN combined with a
gradient in composition give rise to polarization-induced hole and electron
doping, respectivelysimon . In addition to the built in polarization doping,
wires are supplementary doped with Mg and Si in p and n regions, assuming
c-axis,Ga-polar orientation of the nanowires.
From these heterostructures, electrical devices are fashioned by depositing
semitransparent 10nm/20nm Ni/Au contacts with an electron beam evaporator for
the top contact, which connect the vertical ensemble in parallel. The back
contact is fashioned by mechanically removing the nanowires adjacent to the
top contact with a diamond scribe and thermally diffusing In metal directly to
the n-Si with a soldering ion. Current-Voltage (IV) behavior of the devices
are measured with a probe station and an Agilent B1500 semiconductor parameter
analyzer. Device IV’s show rectifying behavior with device turn on at 5V, as
shown in Fig. 1b.
After the IV behavior of the devices is characterized, they are transferred to
a variable temperature UV-VIS-NIR spectroscopy system consisting of a closed
cycle ARS DMX20-OM cryostat and a Princeton instruments SP2500i 0.5m
spectrometer equipped with a Princeton instruments PIXIS100 UV-VIS-NIR CCD.
Devices are connected to a Yokogawa DC constant current source and Keithley
2700 data acquisition system. Prior to the collection of any spectra, a
background spectrum is collected with no current injection in the device.
Constant currents from 10mA to 120mA, corresponding to device current
densities from 1.32A/cm2 to 14.5A/cm2 (the current density through any given
nanowire is unknown since not all individual nanowires give ELlimbach ) are
sourced at room temperature and the resulting EL is collected through a 50mm,
f/2 uv-fused silica singlet lens, collimated and subsequently focused onto the
entrance slit of the 0.5m spectrometer.
Room temperature EL spectra, shown for multiple current densities in Fig. 2a.,
exhibit multiple emission peaks from the UV through the visible parts of the
spectrum. The sharp peak at 318nm (Fig. 2b.), corresponds to the
6P7/2$\rightarrow$8S7/2 first excited state to ground state transition of the
Gd3+ iongruber . More careful inspection of spectra around the Gd atomic line
region reveals an additional peak at the correct energy for the
6P5/2$\rightarrow$8S7/2 second excited state to ground state transitiongruber
. In addition, the intensity of these peaks scale linearly with current
density, within the range of currents investigated. The Gd emission exhibits
FWHM of 23.1meV. Additional broad EL peaks in the 400nm-700nm range are
attributed to below band gap defects in AlGaN, due to observation of identical
emission spectra in non-Gd containing devices, though the graded nature of
structure prevents precise identification of the deep levels responsible.
In order to further investigate the mechanism by which the UV emission occurs,
variable temperature EL measurements are conducted from room temperature to
30K, the results of which are shown in Fig. 2c. Spectral intensity of the main
6P7/2$\rightarrow$8S7/2 Gd 4f peak is observed to be invariant at temperatures
above 75K, below which the intensity increases dramatically (Fig. 2c.,
insert). Similar behavior has been previously observed in Eu-doped GaNnyein
as well as Er-doped InGaPneuhalfen which is attributed to thermal quenching
of the multi-step excitation mechanism of the RE3+ ion. At low temperature
another peak becomes distinct at 324nm. This peak has been previously
identified as a phonon replica of the primary 318nm peak in
cathodoluminescence experimentsvetter . From analysis of the peak positions, a
phonon mode energy of 72.2meV is measured, which is smaller than the LO phonon
energy of the surrounding AlN matrix (110meVbergman ) as well as GaN
(92meVcingolani ). This phonon energy agrees within the resolution of the
spectrometer used with results from cathodoluminescence experiments, which
report 72.9meVvetter .
Many RE electroluminescent devices rely on the impact excitation mechanism to
drive intra-f-shell EL. In the interest of investigating EL under these
conditions in nanowire based devices, a structure consisting of an n-type
nanowire graded from GaN to AlN, with a 200nm uid-AlN layer doped with Gd ions
(1E18cm-3) and capped with a small amount of n++ AlN for a top contact is
prepared. Growth conditions for the heterostructure are identical to the
n-region and depletion region of the PINLED, thus similar defect content are
expected in both structures. An identical device contact scheme to that of the
heterojunction diode is used and is shown in Fig. 3a. This device is again
rectifying, as shown in Fig. 3b., but is less conductive than the PINLED
device, producing 5.9A/cm2 compared to 11.2A/cm2 when forward biased to 15V.
At 15V it is calculated that the active region of the MIS device develops an
electric field in excess of 0.75MV/cm, where as the PINLED should have
approximately flat band conditions in the active region, due to reduction of
band bending in forward bias. The lower conductivity of the MIS devices can be
attributed to the uid-AlN center region as well as a large Schottky barrier
between the n++AlN and the Ti/Au top contact which produce additional series
resistance in the device over the PINLED. Electroluminescence spectroscopy
(Fig. 3c.) reveals a weak, but detectable peak at 318nm among a large
background of defect luminescence, indicating that some Gd ions are being
excited by hot electrons passing through the structure as well as impact
excitation of band to defect luminescence. Comparison of emission from PINLED
devices and Gd:AlGaN MIS structures (Fig. 3c., insert) shows a 372%
enhancement of intensity of the 6P7/2$\rightarrow$8S7/2 transition in the
PINLED devices over the Gd:AlGaN MIS devices at 15V bias. Additionally, no
peak corresponding to the 6P5/2$\rightarrow$8S7/2 higher order transition is
present in the emission spectra from the Gd:AlGaN MIS structure. It is noted
that due to the variety of possible mechanisms for RE 4f excitation in III-V
materials, the difference in performance between the MIS device and the PINLED
device could be affected by phenomena which are extrinsic to the E-field
regime, such as preferential interaction of one carrier type with the RE ion.
In conclusion, polarization induced light emitting diodes (PINLEDS) doped with
Gd in an AlN active region are prepared by plasma assisted molecular beam
epitaxy on n-Si substrates. These devices function at one to two orders of
magnitude lower biases than previously reported Gd:AlN electroluminescent
devices, making them attractive for low power, UV EL applications,
particularly portable devices. When forward biased, devices emit sharp peaks
at 318nm and 313nm, which correspond to the Gd intra-f-shell
6P7/2$\rightarrow$8S7/2 and 6P5/2$\rightarrow$8S7/2 transitions, respectively
and scale linearly with current density. Emission intensity is shown to be
temperature independent above 75K, below which it increases strongly. By
studying two different devices, designed to produced Gd 4f EL under both low
and high electric field conditions, we observe a significant improvement in
emission intensity for PINLED devices which function under low-field operation
conditions over hot electron MIS devices. Although this device has been
applied to Gd, it would be possible in principle to dope with any of the 4f
phosphor rare earths to achieve spectrally stable electrically driven emission
at a variety of wavelengths.
This work is supported by the Center of Emergent Materials at OSU under NSF
DMR-0820414 and National Science Foundation CAREER award (DMR-1055164). S. D.
Carnevale acknowledges support from the National Science Foundation Graduate
Research Fellowship Program (2011101708).
## References
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* (2) P. D. Rack, P. H. Holloway “The structure, device physics and material properties of thin film electroluminescent displays” Mat. Sci. and Eng., R21(19988)171-219(1998)
* (3) P. W. Atkins Molecular Quantum Mechanics 2nd Ed. Oxford, New York. 1992 pg. 223
* (4) J. H. Davies The Physics of Low-dimensional Semiconductors: An Introduction Cambridge University Press (1997)
* (5) A. Wakahara, H. Sekiguchi, H. Okada, Y. Takagi “Current status for light-emitting diode with Eu-doped GaN active layer grown by MBE” J. Lumin. (2012)
* (6) R. Wang and A. J. Steckl “Effect of growth conditions on Eu${}^{3}+$ luminescence in GaN” J. Crys. Grow. 312 (2010) 680-684
* (7) D. S. Lee and A. J. Steckl “Enhanced blue and green emission in rare-earth-doped GaN electroluminescent devices by optical photopumping” Appl. Phys. Lett. 81, 2331(2002);
* (8) T. Kita, et. al. “Narrow-band deep-ultraviolet light emitting device using Al1-xGdxN” Appl. Phys. Lett. 93, 211901 (2008)
* (9) J. H. Park. and A. J. Steckl “Laser action in Eu-doped GaN thin-film cavity at room temperature” Appl. Phys. Lett. 85, 4588 (2004)
* (10) J. M . Zavada et. al., “Ultraviolet photoluminescence from Gd-implanted AlN epilayers” Appl. Phys. Lett. 89, 152107 (2006)
* (11) U. Vetter, J. Zenneck and Hofsass “Intense ultraviolet cathodoluminescence at 318nm from Gd3+-doped AlN” Appl. Phys. Lett. 83, 2145 (2003)
* (12) J. B. Gruber, U. Vetter, H. Hofsass, B. Zandi and M. F. Reid “Spectra and energy levels of Gd3+ (4f7) in AlN” Phys. Rev. B 69, 195202 (2004)
* (13) M.A. Scarpulla, C. S. Gallinat, S. Mack, J. S. Speck, A. C. Gossard, J. Crys. Grow. 311, 1239 (2009)
* (14) S. Kitayama, et. al. “Influence of local atomic configuration in AlGdN phosphor thin films on deep ultra-violet luminescence intensity” J. Appl. Phys. 110, 093108 (2011)
* (15) M. Godlewski, M. Leskel’́a, CRC Critical Reviews in Solid State and Materials Sciences 19 (1994) 199 “Excitation and recombination processes during elecroluminescence of rare earth-activated materials”
* (16) L. Bodiou, A. Braud “Direc evidence of trap-mediated excitation in GaN:Er3+ with a two-color experiment” Appl. Phys. Lett 93, 151107 (2008)
* (17) S. D. Carnevale , J. Yang , Patrick J. Phillips , M. J. Mills, and R. C. Myers “Three-Dimensional GaN/AlN Nanowire Heterostructures by Separating Nucleation and Growth Processes” Nano. Lett.,2011, 11(2), pp 866-867
* (18) S. D. Carnevale, T. F. Kent, P. J. Phillips, M. J. Mills, S. Rajan, and R. C. Myers “Polarization-Induced pn Diodes in Wide-Band-Gap Nanowires with Ultraviolet Electroluminescence” Nano. Lett.,2012, 12(2), pp 915-920
* (19) S. D. Carnevale, C. Marginean, P. J. Phillips, T. F. Kent, A. T. M. G. Sarwar, M. J. Mills, and R. C. Myers “Coaxial nanowire resonant tunneling diodes from non-polar AlN/GaN on silicon” Appl. Phys. Lett. 100, 142115 (2012)
* (20) M. Grundman (2005) BandEng Available online: http://my.ece.ucsb.edu/mgrundmann/bandeng.htm
* (21) J. Simon, V. Protasenko, C. Lian, H. Xing, D. Jena “Polarization-induced hole doping in wide-band-gap uniaxial semiconductor heterostructures” Science 2010, 327, 60-64
* (22) F. Limbach, C. Hauswalf, J. Lähnemann, M. Wölz, O. Brandt, A. Trampert, M Hanke,et. al., Nanotechnology 23 (2012) 465301
* (23) E. Nyein, U. Hömmerich, J. Heikenfeld, D. Lee, A. J. Steckl, et. al. Appl. Phys. Lett. 82, 1655 (2003)
* (24) A. Neuhalfen, B. Wessels Appl. Phys. Lett. 60, 2657 (1992)
* (25) L. Bergman, M. Dutta, C. Balkas, R. Davis, J. Christman, et. al. J. Appl. Phys. 85, 3535 (1999)
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|
arxiv-papers
| 2013-03-29T18:24:53 |
2024-09-04T02:49:43.594766
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas F. Kent, Santino D. Carnevale, Roberto C. Myers",
"submitter": "Roberto Myers",
"url": "https://arxiv.org/abs/1303.7453"
}
|
1303.7454
|
# Constructive Interference in Linear Precoding Systems: Power Allocation and
User Selection
Dimitrios Christopoulos1, Symeon Chatzinotas1, Ioannis Krikidis2 and Björn
Ottersten1 1SnT - securityandtrust.lu, University of Luxembourg
e-mail: {dimitrios.christopoulos, symeon.chatzinotas, bjorn.ottersten}@uni.lu
2Department of Electrical and Computer Engineering, University of Cyprus
e-mail: [email protected]
###### Abstract
The exploitation of interference in a constructive manner has recently been
proposed for the downlink of multiuser, multi-antenna transmitters. This novel
linear precoding technique, herein referred to as constructive interference
zero forcing (CIZF) precoding, has exhibited substantial gains over
conventional approaches; the concept is to cancel, on a symbol-by-symbol
basis, only the interfering users that do not add to the intended signal
power. In this paper, the power allocation problem towards maximizing the
performance of a CIZF system with respect to some metric (throughput or
fairness) is investigated. What is more, it is shown that the performance of
the novel precoding scheme can be further boosted by choosing some of the
constructive multiuser interference terms in the precoder design. Finally,
motivated by the significant effect of user selection on conventional, zero
forcing (ZF) precoding, the problem of user selection for the novel precoding
method is tackled. A new iterative, low complexity algorithm for user
selection in CIZF is developed. Simulation results are provided to display the
gains of the algorithm compared to known user selection approaches.
## I Introduction
The capacity of the multiple input multiple output (MIMO) broadcast channel
(BC) can be reached by non-linear precoding methods, namely dirty paper coding
(DPC)[1]. However, linear precoding methods, like zero forcing (ZF) precoding,
can still attain the channel capacity in a multiuser environment [2, 3, 4],
while proven more realistic in terms of practical implementation. Linear
precoding techniques, especially ZF, have been extensively investigated in [5,
3] and the references therein. In these cases, ZF precoding constitutes a
simple precoder design solution. By inverting the channel, multiuser
interferences are cancelled and the precoding design problem is reduced to a
power allocation problem over new equivalent channels; hence a simple concave
optimization problem[6] needs to be solved. To maximize the throughput (sum-
rate, SR), the well known water-filling solution can be straightforwardly
applied [7]. To maximize the minimum offered rate (i.e. the fairness problem),
the problem is still convex and thus solvable[5]. The key assumption of all
the above considerations however is the assumption of Gaussian signaling.
The concept of constructive interference linear precoding, initially proposed
in [8] for code division multiple access (CDMA) systems and then extended to
apply for MIMO communications in [9], is based on the multiuser interference
cancellation concept of channel inversion. An example of the concept is
described in Fig. 1. The novelty of this precoder design lies in considering
practical constellations and allowing users that add up to the intended user’s
signal power to interfere. This is referred to as constructive interference
(CI) and it can be exploited by acknowledging each users’ channel and
modulated signal. The problem of power allocation in constructive interference
zero forcing (CIZF) precoding techniques has not been studied in existing
literature. Existing works on this topic only assumed CIZF precoding with
equal power allocation for all users[9].
Figure 1: The constructive interference (CI) concept over binary phase shift
keying (BPSK) modulations: The $k$-th user’s transmit symbol is $s_{k}=+1$.
The $i$-th user’s symbol, $s_{i}$, multiplied by the cross-correlation between
the $k$-th and the $j$-th user’s channels, $\rho_{kj}$ (see Sec. II-B), is a
vector that when added to $s_{k}$ will move the resulting vector further away
from the decision threshold (0 for BPSK). Subsequently, not cancelling this
user will benefit the $k$-th user. On the other hand, the $j$-th user is still
interfering thus needs to be cancelled by the precoding design.
Another very important aspect of linear precoding is the user selection
problem investigated in [3, 6]. ZF performance is increased when user channels
are orthogonal to each other. Under the assumption of large random user sets,
the probability of orthogonal users increases and with that the complexity of
the user selection problem. Nevertheless, simple suboptimal algorithms in the
existing literature provide substantial gains with affordable complexity.
Based on existing methods, Yoo et al [2] proposed a low complexity, iterative
user selection algorithm that allows ZF to achieve the performance of non-
linear precoding[10] as the number of available for selection users grows to
infinity.
The contribution of the present paper is twofold. Firstly, the effect of power
allocation on CIZF precoding transmitters is investigated; a design parameter
that has not been examined in existing literature. Secondly, motivated by the
fact that user selection can optimize the ZF performance, the problem of user
selection in CIZF systems is defined and solved by a low complexity algorithm.
This algorithm achieves substantial gains and approaches the performance of
the optimal user selection, as derived by full space search.
The rest of the present paper is structured as follows. The considered system
model is described with detail in Section II, where the concepts of
conventional ZF and novel CIZF are also described. Section III explores the
effects of power allocation in CI based linear precoding methods with the
support of simulation results. In Section IV, the user selection problem is
defined and solved via a novel heuristic algorithm. Conclusions are drawn in
Section V.
Notation: Throughout the paper, $\left(\cdot\right)^{\dagger}$,
$\mathfrak{Re}(\cdot)$ and $||\cdot||,$ denote the conjugate transpose, the
real part of complex elements and the Euclidean norm operations, respectively,
while $[\cdot]_{ij}$ denotes the $i,j$-th element of a matrix. The element-
wise matrix product is denoted by $\circ$. Bold face lower case characters
denote column vectors and upper case denote matrices while the operator
diag($\mathbf{x}$) produces a diagonal square matrix composed of the elements
of $\mathbf{x}$. An identity matrix of size $n$ is denoted by
$\mathbf{I}_{n}$. Upper case calligraphic characters denote sets. The
operation $\mathcal{A-B}$ is the relative complement of $\mathcal{B}$ in
$\mathcal{A}$, while $|\mathcal{A}|$, denotes the cardinality of a set.
## II System Model
A multi-user (MU) multiple input single output (MISO) network consisting of
one transmitter with $N_{t}$ antennas and $K\geq N_{t}$ single antenna
receivers, is considered. At each time, the transmitter serves exactly
$K=N_{t}$ users which are selected either randomly or based on a selection
scheme as described in Sec. IV. The received signal at the $k$-th user can be
expressed as
$y_{k}=\mathbf{h}^{{\dagger}}_{k}\mathbf{x}+n_{k},$ (1)
where $\mathbf{h}_{k}$ is an $N_{t}\times 1$ vector composed of the channel
coefficients between the $k$-th user and the $N_{t}$ antennas of the source,
$\mathbf{x}$ is an $N_{t}\times 1$ vector of transmitted symbols and $n_{k}$
is the independent identically distributed (i.i.d) zero mean Additive White
Gaussian Noise (AWGN) measured at the $k$-th user’s receive antenna. The noise
is assumed normalized, thus $\mathcal{E}\left\\{|n_{k}|^{2}\right\\}=1$. In
matrix form, this MU MISO BC reads as
$\displaystyle\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n},$ (2)
where $\mathbf{y}=[y_{1},y_{2},\ldots,y_{N_{t}}]^{\dagger}$,
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{N_{t}}]^{\dagger}$, $\mathbf{H}$ is the
$N_{t}\times N_{t}$ square matrix that contains the user complex vector
channels, i.e.
$\mathbf{H}=\left[\mathbf{h}_{1},\mathbf{h}_{2}\dots,\mathbf{h}_{N_{t}}\right]^{\dagger}$
and $\mathbf{n}=[n_{1},n_{2},\ldots,n_{N_{t}}]^{{\dagger}}$. The transmitter
linearly precodes the information symbols:
$\displaystyle\mathbf{x}=\mathbf{W}\mathbf{P}^{1/2}\mathbf{s},$ (3)
where the $N_{t}\times N_{t}$ matrix $\mathbf{W}$ is the precoding matrix,
$s_{k}$ denotes the symbol for the $k$-th destination and $\mathbf{s}=[s_{1},\
s_{2},\ldots,s_{K}]^{\dagger}$ with
$\mathbb{E}[\mathbf{s}\mathbf{s}^{\dagger}]=\mathbf{I}$ and
$\mathbf{P}^{1/2}=\operatorname{diag}([\sqrt{p_{1}},\sqrt{p_{2}}\ldots\sqrt{p_{k}}])$
is a diagonal $K\times K$ matrix composed of the transmit powers allocated to
the $k$ users111The notion of transmit power allocated to a user is explained
in [3].. For shortness and since the results can be straightforwardly
generalized for higher order constellations, in this study only real valued
signals will be assumed (binary phase shift keying-BPSK modulation), hence
$\displaystyle{s_{i}}=\pm 1,\ i=1\ldots N_{t}.$ (4)
### II-A Zero Forcing beamforming
Transmit beamforming is a multiuser precoding technique that separates user
data streams in different parallel beamforming directions[3]. A linear
precoding technique with reasonable computational complexity that still
achieves full spatial multiplexing and multiuser diversity gains, is ZF
precoding [11, 12, 3]. The ability of ZF to fully cancel out multiuser
interference makes it useful in the high Signal to Noise Ratio
($\mathrm{SNR}$) regime. However, it performs far from the optimal in the
noise limited regime. In addition, it can only simultaneously serve as many
single antenna users as the number of transmit antennas. A common solution for
the ZF precoding matrix is the pseudo-inverse of the $K\times N_{t}$ channel
matrix. Under a total power constraint, the pseudo-inverse is the optimal
solution (rather than any generalized inverse) in terms of maximum SR and
maximum fairness [5]. The precoding matrix can be expressed as
$\displaystyle\mathbf{W}=\mathbf{H}^{\dagger}\mathbf{R}^{-1}\mathbf{T},$ (5)
where we define the matrix $\mathbf{R}$ as
$\displaystyle\mathbf{R}=\mathbf{HH}^{\dagger}.$ (6)
The matrix $\mathbf{T}$ has been introduced by [9] to model the CI scheme as
explained in the next subsection. The general model described by (5), for
$\displaystyle\mathbf{T}=\mathbf{I}\circ\mathbf{R,}$ (7)
where the non zero elements of $\mathbf{T}$ are
$\tau_{kk}=\sum\mathbf{h}_{k}^{\dagger}\mathbf{h}_{k}$, will yield the
conventional ZF design. The complete cancellation of interferences in this
case, will reduce the $k$-th users received signal to
$\displaystyle
y_{k}=\mathbf{h}^{{\dagger}}_{k}\sqrt{p_{k}}\mathbf{w}_{k}s_{k}+n_{k},$ (8)
where $\mathbf{w}_{k}$ is the $k$-th column of the total precoding matrix.
Assuming uniform power allocation across users, as in [9], the total power
constraint over the transmit antennas $P_{tot}$ will yield[5]:
$\displaystyle\mathcal{E}\\{||\mathbf{x}||^{2}\\}=\operatorname{Tr}\\{\mathbf{xx}^{\dagger}\\}\leq
P_{tot}.$ (9)
From (3) and (9), the transmit power allocated to the $k$-th user becomes
$\displaystyle\sqrt{p}_{k}=\sqrt{\frac{P_{tot}}{\operatorname{Tr}\\{\mathbf{T}^{\dagger}\mathbf{R}^{-1}\mathbf{T}\\}}},\
\forall\ k.$ (10)
By examining (10), it is clear that the transmit power allocated to the $k$-th
user is a function of the precoder design. In precoding, channel inversion
i.e. projecting the actual channels on orthogonal dimensions, leads to the
reduction of each users effective channel. Subsequently, since the precoders
are not normalized, the sum of the individual powers allocated to each user
($\sqrt{p_{k}}$) is not equal to the sum of powers transmitted (9). The notion
of individual user consumption can be introduced to better explain this power
loss due to the precoding. Finally, the $k$-th user $\mathrm{SINR}$ for the ZF
precoding will read as
$\mathrm{SINR}_{k}^{\text{ZF}}=|\tau_{kk}|^{2}p_{k}.$ (11)
### II-B Constructive Interference Zero Forcing beamforming
The CIZF scheme, introduced in [9], allows the so-called constructive
multiuser interference (cross-interference) to be added to the useful signal
at each receiver. In general, given the full channel state information
available at the transmitter and acknowledging the signal constellation, the
CIZF scheme does not suppress the part of the cross-interference that is
constructive and thus increases the power of the useful signal. A simple
example of this concept is explained in Fig. 1.
As discussed with detail in [9], the symbol to symbol multiuser interference
results from the $i,j$-th element of the matrix $\mathbf{R}$ :
$\rho_{ij}=\sum_{n=1}^{N_{t}}h_{in}\cdot({h_{jn}}^{\dagger})$. In the CI
scenario, the received signal of (8) will become
$\displaystyle y_{k}=\mathbf{\tau}_{kk}\sqrt{p_{k}}s_{k}+\sum_{j\neq
k}\text{CI}_{kj}+n_{k}$ (12)
where $\text{CI}_{kj}=\tau_{ki}\sqrt{p_{j}}s_{j}$ denotes the constructive
cross-interference from the $j$-th data flow ($j$-th user) to the $k$-th user.
Subsequently, the $k$-th user’s signal to interference plus noise ratio
($\mathrm{SINR}$) will read as
$\mathrm{SINR}_{k}=\sum^{K}_{j=1}|\tau_{kj}|^{2}p_{j}.$ (13)
Let us define as
$\mathbf{G}=\text{diag}(\mathbf{s})\cdot\mathfrak{Re}(\mathbf{R})\cdot\text{diag}(\mathbf{s}),$
which yields
$\displaystyle\mathbf{G}=\begin{pmatrix}s_{1}\mathfrak{Re}(\rho_{11})s_{1}&\dots&&\\\
\vdots&\ddots&&\\\ &&s_{k}\mathfrak{Re}(\rho_{kj})s_{j}&\\\ &&&\\\
\end{pmatrix}.$ (14)
In order to indicate the cross-interference as CI, the signal constellation
needs to be accounted for. Subsequently, the terms that position the received
signal into the decision region of the transmitted symbols are beneficial and
thus not cancelled by the precoding design. For the simple case of BPSK
modulation, the cross-interference generated by the $j$-th data flow to the
$k$-th destination, is considered to be constructive when
$\displaystyle s_{k}\mathfrak{Re}(\rho_{kj})s_{j}>0,$ (15)
which can be expressed as $\mathbf{G}_{kj}>0.$ Thus the CIZF precoder is
deduced from
$\displaystyle\tau_{kk}=\rho_{kk}$ (16)
$\displaystyle\tau_{ki}=\begin{cases}\rho_{ki},&\text{If}\
[\mathbf{G}]_{kj}>0\\\ 0,&\text{elsewhere.}\end{cases}$
Therefore, the precoding matrix is computed on a symbol-by-symbol basis.
## III Power Allocation in Linear Precoding
The impact of power allocation (PA) on the CIZF has not been addressed in
existing literature on this topic [8, 9]. Therein, the problem was simplified
by a uniform power allocation assumption, as defined in (10). In general, PA
is performed to the end of maximizing some performance metric. The performance
metrics commonly addressed in literature involve either the total throughput
performance (i.e. max SR criterion) or the $\mathrm{SINR}$ level of the worst
user (i.e. max fairness criterion). Another important parameter in linear
precoding is the type of constraints that will be assumed. Usually, a total
sum power constraint simplifies the analysis and provides better results since
the available power is freely allocated across antennas. Herein, two objective
functions of the achievable user rates that ensure maximum fairness
(availability) and maximum SR (throughput), are considered. More specifically,
the optimization problem reads as
$\displaystyle\max_{\mathbf{P}\geq\mathbf{0}}f(\mathbf{P})$ (17) s.t.
$\displaystyle\sum_{i=1}^{N_{t}}\sum_{j=1}^{K}|\omega_{ij}|^{2}p_{j}\leq
P_{tot}$
where $\omega_{ij}$ is the $i,j$-th element of $\mathbf{W}$ and the objective
function $f$ is given by [5]:
$\displaystyle
f(\mathbf{P})=\begin{cases}\sum_{k}\log_{2}(1+\mathrm{SINR}_{k}),&\text{\
Throughput}\\\ \min_{k}\mathrm{SINR}_{k},&\text{\ Fairness}\end{cases}$ (18)
where $\mathrm{SINR}_{k}$ is given by (13) and $1\leq k\leq N_{t}$. Based on
(18), the objective function is concave in $\mathbf{P}\geq 0$, for both
scenarios and therefore the optimization problem is a simple concave
maximization with one linear constraint. It is worth noting that for the case
of the maximum throughput, the optimization problem can be solved using the
water-filling solution. The problem of allocating the power to the end of
maximising some system performance metric is discussed in the following
Section (III-A).
### III-A Power Allocation
An appropriate PA scheme distributes the total available power to the data
flows in a way that maximizes an objective function of the achievable rates.
In the case of conventional ZF precoding, the max throughput PA problem, under
a total power constraint, reads as in (17) with objective function
$\displaystyle
f(\mathbf{P})=\sum_{k}\log_{2}\left(1+\mathrm{SINR}_{k}^{\text{ZF}}\right),$
(19)
where the $\mathrm{SINR}_{k}^{\text{ZF}}$ is given by (11).
In order to be able to easily show the impact of PA on CIZF and maintain
concavity for the formulated optimization problems, we assume that the
equivalent CIZF channel (with the modulation-based CI) refers to Gaussian
inputs; this assumption allows to approximate the channel capacity of the
system with the simple $\log$-based Shannon expressions. It should be
clarified here, that the optimal power allocation problem for linear precoders
under the constraint of finite input alphabets is a highly complex problem.
The most recent attempt to solve it can be found in [13] where a heuristic
optimization algorithm is developed. However, in the present paper, a
preliminary study to exhibit the impact of PA on CIZF is performed, hence the
strictly optimal solution is beyond the scope of the present work.
#### III-A1 Simulation Results
The effect of power allocation in the CIZF precoding design is plotted in Fig.
2. The power allocation problem (18) has been solved using the CVX tool in
MATLAB [14]. Simulations where carried for $100$ channel instances, and for
$K=N_{t}=4$. In Fig. 2, the gain from CIZF with uniform power allocation, as
proposed in [9], over the conventional ZF is evident by comparing the dashed
lines. The novel result, depicted in Fig. 2, is that power allocation further
boosts the CIZF gain (continuous lines). More specifically, for the
conventional ZF scheme power allocation introduces approximately $1$ dB of
gain over the uniform allocation. However, when PA is applied in CIZF, then
more than $2$ dB gain can be gleaned. It is therefore concluded that power
allocation over the CIZF precoding scheme is an important aspect that
introduces significant gains. Finally, in the same figure, the realistic
region where the results apply is defined by a dashed line. This restriction
comes from the acknowledgement of BPSK modulation as a practical constellation
choice. The alleviation of this restriction via adaptive modulation methods is
part of the future extensions of this work.
Figure 2: Per user spectral efficiency of conventional ZF precoding under
uniform and optimized PA, compared to the efficiency of the novel CIZF
precoding under equivalent PA assumptions. PA optimization for max system
throughput. The modulation constrained threshold for BPSK is also plotted.
### III-B Power Constrained Transmission
In the present section, the existence of redundant CI interference terms is
discussed. As can be seen from (10) and (13), the consideration of non-zero,
off diagonal elements in the precoding matrices, i.e. CI terms, has a double
impact on each user’s $\mathrm{SINR}$ and thus the sum system capacity; from
one hand, it increases the expression $\sum_{i}|\tau_{ki}|^{2}p_{k}$ by adding
more positive terms in the summation, but on the other hand, it changes the
power allocated to each user in (9), (10). Therefore, a CI term in the CIZF is
not always beneficial for the system performance. This partially constructive
interference zero forcing (P-CIZF) scheme examines the tradeoff between the
positive and the negative impact of an CI term by searching all combinations
and selecting the most beneficial set of CI terms. The P-CIZF scheme can be
formulated as
$\displaystyle\max_{\\{\mathbf{p},\mathbf{T}^{(\mathcal{S})}\\}}f(\mathbf{p},\mathbf{T}^{(s)})$
(20) s.t.
$\displaystyle\mathbf{T}^{(\mathcal{S})}\subseteq\mathbf{T}^{(\mathcal{S}_{tot})},$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{K}|\omega_{ij}|^{2}p_{j}\leq P_{tot}.$
(21)
where
$\displaystyle\mathcal{S}\subseteq\mathcal{S}_{tot}=2^{m},$
with $m=\left|[\mathbf{G}]_{ki}>0\right|$, the number of CI terms. The above
definition means that the Partially-CIZF scheme searches all the possible
combinations ($2^{m}$) of the CI terms and holds the one that maximizes the
objective function considered. The optimization problem (20), will be solved
under uniform (10) and optimal (21) power allocation considerations.
Intuitively, the second consideration provides more flexibility in the design
but finding the strictly beneficial terms while at the same time optimizing
the PA is a highly complex procedure; for every possible combination of the CI
terms, a new convex optimization PA problem is solved.
As the purpose of this work is to demonstrate this interesting trade-off, more
practical implementation of the P-CIZF scheme are beyond the scope of this
paper.
#### III-B1 Simulation Results
In Fig. 3 the improvements of P-CIZF over the CIZF scheme are depicted. The
performance is evaluated under uniform and optimized power allocation.
Starting with the CIZF scheme under uniform power allocation, in Fig. 3,
finding the strictly beneficial CI terms provides some gains. In each point of
the figure, the percentage of the CI terms kept in the P-CIZF scheme over the
total CI terms of the CIZF scheme is also depicted. Focusing on the dotted
curves, in the uniform power allocation case, it is apparent that maintaining
approximately $88\%$ of the total number of CI terms, provides small gains.
These results exhibit the optimality of this approach, while analytical proof
of the optimality is part of future work. The continuous curves in Fig. 3
correspond to an optimal (with respect to maximizing the total SR) power
allocation assumption in P-CIZF precoding and again some gains are gleaned;
thus the strict optimality of this approach is exhibited. Since the degrees of
freedom in the precoder design are increased, less CI terms are maintained
(approximately $65\%$).
Intuitively, the above results can be explained by the fact that allowing only
strictly beneficial terms in the precoder reduces power consumption, thus
allowing for more power to be allocated to the users without exceeding the
total power constraint imposed on the transmit antennas. It should be noted
here, however, that the effects of this approach on the minimum supported rate
(fairness) are not examined in the present work. Finally, higher gains of this
approach could be gleaned over larger user sets, but by exponentially
increasing the iterations of the searching algorithm. The investigation of
such scenarios is part of future work.
Figure 3: Evaluation of the P-CIZF compared to fully CIZF under equal and
optimal power allocation. The percentages of the constructive terms maintained
in the P-CIZF scheme, for each $\mathrm{SNR}$ point are also presented.
## IV User selection
In this section, the user selection problem is formulated and a novel
selection technique based on the exploitation of CI is proposed. Considering
the decoupled nature between the user selection and the PA problem, as proven
in [3], the performance of user selection is not affected by PA. However, in
accordance with the previous results and more importantly to maximise the
overall performance of the assumed system, PA is optimized independently after
the user selection process.
### IV-A User Selection Methods
#### IV-A1 Orthogonal user selection
ZF beamforming has the potential of approaching the optimal channel capacity,
otherwise only achievable by DPC, when all the users are perfectly orthogonal
to each other, as proven by [3]. The same authors provided a heuristic low
complexity user selection algorithm, namely the semi-orthogonal user selection
($\mathrm{SUS}$) algorithm, which was proven to select the optimal users, as
the number of available users approaches infinity. However, it has never been
applied in the CIZF framework. Due to lack of space, the reader is referred
to[3, 2] for more details on this algorithm. It should be mentioned here, that
the orthogonal user selection does not take into account the CI during the
selection procedure. However, to maintain fairness in the study, after the
selection, any CI terms that exist are maintained.
#### IV-A2 Optimal Constructive Interference User Selection
The investigated precoding scheme strongly depends on the transmitted signal
(user constellation) and therefore the above conventional user selection
schemes become unsuitable. A CIZF-based user selection metric should also
consider the transmitted symbols, since the precoding matrix is defined via
the CI and affects the final performance. In order to attain an upper bound
for any CIZF-based user selection policy, all the possible combinations of
users $\mathcal{Q}$ are examined and out of them, the combination
$\mathcal{U}$ that maximizes the system throughput is chosen:
$\displaystyle\mathcal{U}=\arg_{\mathcal{U}}\max_{\mathcal{U}\in\mathcal{Q}}\sum_{m}\log_{2}\left(1+\mathrm{SINR}_{m}\right),$
(22)
where $\mathrm{SINR}_{m}$ is given by (13) under a uniform power allocation
assumption, i.e. (10). It should be stressed that this method relies on
exhaustive search over all possible combinations of users. As a result, a
searching algorithm requires $\binom{K}{N_{t}}=K!/(K-N_{t})!$ iterations in
order to decide about the optimal combination at each transmission.
Considering that user selection methods perform better as the number of users
increases[3], i.e. as $K>>N_{t}$, then the optimal solution becomes difficult
to compute. In the scope of this work, a simpler heuristic algorithm is
presented hereafter.
#### IV-A3 Semi parallel user selection
Inspired by the concept of user orthogonality, the purpose of this selection
method is dual: users with CI need to be selected and furthermore these users
need to be aligned (rather than orthogonal) so that the aggregate beneficial
receive power is increased. Following this concept the semi-parallel user
selection ($\mathrm{SPUS}$) algorithm, provided in pseudo-code in Alg. 1, has
been developed. An analytic description of the algorithm follows.
Initially, the algorithm accepts as input the CI matrix $\mathbf{G}$. The
first step is to choose the user with the larger diagonal element. For this
user, the cross-correlation elements with all other users are stored in the
buffer vector $\mathbf{c}_{(i)}$ where $i$ is the iteration counter. Also the
sets $\mathcal{S},\mathcal{T}$ that include the available and the selected
users, are initialized and updated in every iteration. Cross-correlation is
the inner product of the vector channels of two users and represents the level
of orthogonality between the users. In this scenario, the purpose is to have
as little orthogonality as possible so as to increase the received CI. In each
of the $M$ iterations, (Step 2) the user with the strongest element in
$\mathbf{c}_{(i)}$ is chosen. Then this user’s corresponding cross-
correlations with all the users is added in the buffer matrix. By adding the
cross-correlation of each selected user in the buffer matrix, a metric for the
subspace of the previously selected users is created, since the aim of the
algorithm is to find the most parallel users to the subspace spanned by the
ones already selected. This is achieved by choosing the strongest element of
the $\mathbf{c}$ vector in each iteration. Since in the previous steps no
guarantee exists that a selected user has only CI towards the selected ones,
in Step 3, the residual non-CI terms are removed from the precoding matrix.
The developed heuristic, iterative algorithm runs for exactly $K$ iterations.
#### Simulation Results
A comparison in terms of maximum SR performance of the algorithms described in
the previous section is presented in Fig. 4, where the performance of these
algorithms was studied with optimized PA to maximize the total throughput. In
this figure, the gain of the optimal user selection algorithm compared to
existing approaches is clear. The best algorithm for ZF precoding, i.e.
$\mathrm{SUS}$ [3], performs better than assuming no selection; however,
approximately $6$ dB loss is expected over the optimal CIZF selection. This
observation emanates the need for better user selection algorithms, when
exploiting the benefits of CI. Accounting also for the complexity of an
exhaustive search selection algorithm, as discussed in the previous section, a
less complex heuristic algorithm is further necessitated. In this direction,
$\mathrm{SPUS}$ has been developed. In Fig. 4, the close to optimal
performance of the developed algorithm is clear; the proposed technique
performs less than 1 dB away from the optimal selection.
Furthermore, the performance of the developed algorithm under fairness
maximization PA is examined in Fig. 5. Results indicate, that when the
$\mathrm{SPUS}$ algorithm is combined with max fairness PA optimization, the
performance degradation with respect to the optimal selection policy, is
relatively small. Therefore, the proposed algorithm, developed for maximising
the total throughput of the system, does not severely compromise the fairness
of the system. By comparing Fig. 4 and 5, it is also noted that the the
minimum user rates are not far from the average rate. This result indicates
that the variance between the user rates is kept in reasonable levels when
user selection is combined with PA to optimize the total SR. It is therefore
concluded that fairness is not severely compromised in user selection
scenarios, even when PA optimization is performed to maximize to total system
SR.
In Fig. 6 the performance of the discussed algorithms is studied with respect
the size of the available user set. The beneficial effect of the increasing
number of users is clear for all algorithms. What is more, for relatively
small user pools the performance of the algorithms is beginning to saturate
thus indicating that the main gains are gleaned for finite numbers of users,
that are in line with the dimensions of practical operating multiuser systems.
Figure 4: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a total
pool of $K=12$ users. PA has been optimized to maximise the total system
throughput.
Figure 5: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a total
pool of $K=12$ users. PA has been optimized to maximise minimum user rate.
Figure 6: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a
variable pool of users, for a fixed total transmit power of $15$ dB.
## V Conclusions and future work
The concept of constructive interference in linear precoding systems has been
examined under the framework of power optimization for the maximization of the
system performance. Results indicate that an excess of 2dB gain can be gleaned
by optimizing the power allocation with the aim of increasing the total system
throughput in constructive interference precoding systems. Moreover, as it has
been shown, the individual user power consumption of these schemes can be
further reduced, thus leading to some gains. Finally, the user selection
problem has been tackled for the novel type of precoding and a heuristic, low
complexity, iterative algorithm with close to optimal performance has been
proposed.
Future extensions of this work include the investigation of constructive
interference amongst users with different constellations in an adaptive
modulation environment where the limitations induced by these practical
constellations are alleviated.
## Acknowledgment
This work was partially supported by the National Research Fund, Luxembourg
under the project “$CO^{2}SAT:$ Cooperative & Cognitive Architectures for
Satellite Networks’.
Semi-Parallel User Selection (SPUS) algorithm
Output: $\mathbf{G}_{out}$
Input:
$\mathbf{G}=\text{diag}(\mathbf{s})\cdot\mathfrak{Re}(R)\cdot\text{diag}(\mathbf{s}),$
_Step 1: Initialization_
$\pi_{(0)}=\arg\max||\mathbf{h}_{k}||=\arg\max[\mathbf{G}]_{kk}$,
$\forall k=1,...M:\mathbf{c}_{(0)}=\mathbf{G}(\pi_{(0)},k)$,
$\mathcal{S}_{(0)}=\pi_{(0)}$
$\mathcal{T}_{(0)}=\\{1,\dots K\\}-\\{\pi_{(0)}\\}\ $ set of unprocessed
users.
for _$i=1\to M$ _ do
_Step 2: Selection_ $\pi_{(i)}=\arg\max\mathbf{c}_{(i-1)},\text{ Provided that
}\pi_{(i)}\in\ \mathcal{T}_{(i-1)}$;
$\forall k=1,...M:\
\mathbf{c}_{(i)}=\mathbf{G}(\pi_{(i-1)},k)+\mathbf{G}(\pi_{(i)},k);$
$\mathcal{T}_{(i)}=\mathcal{T}_{(i-1)}-\\{\pi_{(i)}\\};$
$\mathcal{S}_{(i)}=\mathcal{S}_{(i-1)}+\\{\pi_{(i)}\\};$
end for
_Step 3: Output_
$\mathbf{G}_{out}=\mathbf{G}(\mathcal{S}_{(M)})$;
for _$m=1\to M$_ do
for _$l=1\to M$_ do
if _$\mathbf{G}(m,k) <0$ _ then
$\mathbf{G}_{out}(m,k)=0$
end if
end for
end for
Algorithm 1 Semi-Parallel User Selection Algorithm (SPUS)
## References
* [1] M. Costa, “Writing on dirty paper,” _IEEE Trans. Inf. Theory_ , vol. 29, no. 3, pp. 439–441, 1983.
* [2] T. Yoo and A. Goldsmith, “Optimality of zero-forcing beamforming with multiuser diversity,” in _IEEE Int. Conf. on Commun. (ICC)_ , vol. 1, May 2005, pp. 542–546.
* [3] ——, “On the optimality of multi-antenna broadcast scheduling using zero-forcing beamforming,” _IEEE J. Select. Areas Commun._ , vol. 24, Mar. 2006.
* [4] B. L. Ng, J. Evans, S. Hanly, and D. Aktas, “Distributed downlink beamforming with cooperative base stations,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 12, pp. 5491 –5499, dec. 2008.
* [5] A. Wiesel, Y. C. Eldar, and S. Shamai, “Zero forcing precoding and generalized inverses,” _IEEE Trans. Signal Process._ , vol. 56, no. 9, pp. 4409–4418, Sept. 2008.
* [6] G. Dimic and N. Sidiropoulos, “On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm,” _IEEE Trans. Signal Process._ , vol. 53, no. 10, pp. 3857 – 3868, Oct. 2005.
* [7] T. Yoo and A. Goldsmith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 5, pp. 2203–2214, May 2006.
* [8] C. Masouros and E. Alsusa, “A novel transmitter-based selective-precoding technique for DS/CDMA systems,” _IEEE Signal Process. Lett._ , vol. 14, no. 9, pp. 637 –640, Sep. 2007.
* [9] ——, “Dynamic linear precoding for the exploitation of known interference in MIMO broadcast systems,” _IEEE Trans. Wireless Commun._ , vol. 8, no. 3, pp. 1396 –1404, Mar. 2009.
* [10] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 9, pp. 3936–3964, 2006.
* [11] H. Viswanathan, S. Venkatesan, and H. Huang, “Downlink capacity evaluation of cellular networks with known-interference cancellation,” _IEEE J. Select. Areas Commun._ , vol. 21, no. 5, pp. 802–811, 2003.
* [12] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 49, no. 7, pp. 1691–1706, July 2003.
* [13] Y. Wu, M. Wang, C. Xiao, Z. Ding, and X. Gao, “Linear precoding for MIMO broadcast channels with finite-alphabet constraints,” _IEEE Trans. Wireless Commun._ , vol. 11, no. 8, pp. 2906–2920, Aug. 2012.
* [14] S. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge Univ. Press, 2004.
|
arxiv-papers
| 2013-03-29T18:25:52 |
2024-09-04T02:49:43.601698
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dimitrios Christopoulos, Symeon Chatzinotas, Ioannis Krikidis and\n Bjorn Ottersten",
"submitter": "Dimitrios Christopoulos",
"url": "https://arxiv.org/abs/1303.7454"
}
|
1304.0026
|
# Socle pairings on tautological rings
Felix Janda and Aaron Pixton
###### Abstract.
We study some aspects of the $\lambda_{g}$ pairing on the tautological ring of
$M_{g}^{c}$, the moduli space of genus $g$ stable curves of compact type. We
consider pairing $\kappa$ classes with pure boundary strata, all tautological
classes supported on the boundary, or the full tautological ring. We prove
that the rank of this restricted pairing is equal in the first two cases and
has an explicit formula in terms of partitions, while in the last case the
rank increases by precisely the rank of the $\lambda_{g}\lambda_{g-1}$ pairing
on the tautological ring of $M_{g}$.
## 1\. Introduction
Let $M_{g,n}$ be the moduli space of smooth curves of genus $g$ with $n$
marked points and let $\overline{M}_{g,n}$ be the Deligne-Mumford
compactification, the moduli space of stable $n$-pointed nodal curves of
arithmetic genus $g$. Inside this, let $M_{g,n}^{c}$ be the subspace of stable
pointed curves of compact type, i.e. curves whose dual graph is a tree.
The intersection theory of these moduli spaces of curves is a subject of
fundamental importance in algebraic geometry. When studying the Chow ring
$A^{*}(\overline{M}_{g,n})$, one is naturally led to consider a subring
consisting of the classes such as the Arborella-Cornalba $kappa$ classes that
are defined via certain tautological maps between the $\overline{M}_{g,n}$.
This subring is the tautological ring $R^{*}(\overline{M}_{g.n})$.
Tautological rings $R^{*}(M_{g,n})$ and $R^{*}(M_{g,n}^{c})$ for $M_{g,n}$ and
$M^{c}_{g,n}$ can be defined by restriction. We will primarily be interested
in $R^{*}(M_{g}^{c})$, the case of compact type with no marked points.
Inside $R^{*}(M_{g,n}^{c})$ there is the subring $\kappa^{*}(M_{g,n}^{c})$
generated by the $\kappa$ classes $\kappa_{1},\kappa_{2},\ldots$. The kappa
ring $\kappa^{*}(M_{g,n}^{c})$ has been studied in detail by Pandharipande
[6]. In particular, for $n>1$ a complete description of the kappa ring is
given.
When restricted to the moduli space of smooth curves $M_{g}$, the tautological
ring $R^{*}(M_{g})$ is actually equal to the kappa ring $\kappa^{*}(M_{g})$.
This means that on $M_{g}^{c}$, any tautological class can be written as the
sum of a polynomial in the $\kappa$ classes and a class supported on the
boundary. We denote by $BR^{*}(M_{g}^{c})$ the ideal of tautological classes
supported on the boundary, so the tautological ring $R^{*}(M_{g}^{c})$ is
linearly spanned by $\kappa^{*}(M_{g}^{c})$ and $BR^{*}(M_{g}^{c})$.
A general element of $BR^{*}(M_{g}^{c})$ is a linear combination of classes
obtained by taking the pushforward of tautological classes via gluing maps
$M_{g_{1},n_{1}}^{c}\times M_{g_{2},n_{2}}^{c}\times\cdots\times
M_{g_{k},n_{k}}^{c}\to M_{g}^{c}.$
When the class $1$ is pushed forward along such a map, this construction gives
pure boundary strata. We let $PBR^{*}(M_{g}^{c})$ denote the linear subspace
of $BR^{*}(M_{g}^{c})$ generated by the pure boundary strata.
There are natural bilinear pairings
$\displaystyle R^{r}(M_{g}^{c})\times R^{2g-3-r}(M_{g}^{c})\to
R^{2g-3}(M_{g}^{c})\cong\mathbb{Q},$ $\displaystyle R^{r}(M_{g})\times
R^{g-2-r}(M_{g})\to R^{g-2}(M_{g})\cong\mathbb{Q},$
given by the product in the Chow ring and the socle evaluations. These
pairings are called the $\lambda_{g}$ and $\lambda_{g}\lambda_{g-1}$ pairings
respectively because they may be defined by integrating against these classes
in $\overline{M}_{g}$.
In this paper we will study the restriction
$\kappa^{d}(M_{g}^{c})\times R^{r}(M_{g}^{c})\to\mathbb{Q}$
of the $\lambda_{g}$-pairing for $r+d=2g-3$, for any $g\geq 2$. The following
theorems, our main results, were previously conjectured by Pandharipande.
###### Housing Theorem.
The rank of the $\lambda_{g}$-pairing of $\kappa$ classes against boundary
classes
$\kappa^{d}(M_{g}^{c})\times BR^{r}(M_{g}^{c})\to\mathbb{Q}$
equals the rank of the $\lambda_{g}$-pairing of $\kappa$ classes against pure
boundary strata
$\kappa^{d}(M_{g}^{c})\times PBR^{r}(M_{g}^{c})\to\mathbb{Q}.$
Furthermore, these ranks are equal to the number of partitions of $d$ of
length less than $r+1$ plus the number of partitions of $d$ of length $r+1$
which contain at least two even parts.
###### Rank Theorem.
The rank of the $\lambda_{g}$-pairing of $\kappa$ classes against general
tautological classes
$\kappa^{d}(M_{g}^{c})\times R^{r}(M_{g}^{c})\to\mathbb{Q}$
equals the sum of the rank of the $\lambda_{g}$-pairing of $\kappa$ classes
against boundary classes
$\kappa^{d}(M_{g}^{c})\times BR^{r}(M_{g}^{c})\to\mathbb{Q}$
and the rank of the $\lambda_{g}\lambda_{g-1}$ pairing
$\kappa^{r}(M_{g})\times\kappa^{g-2-r}(M_{g})\to\mathbb{Q}.$
These theorems will be proven by direct combinatorial analysis of the well
known formulae for calculating the integrals arising in the pairings. In
particular, we have no geometric explanation of the Rank Theorem, which
connects the compact type case and the smooth case.
### 1.1. Consequences
It has been conjectured by Faber [1] that $\kappa^{*}(M_{g})=R^{*}(M_{g})$ is
a Gorenstein ring with socle in degree $g-2$. He verified this for $g\leq 23$
by computing many relations between the $\kappa$ classes and checking that
they produced a Gorenstein ring. However, starting in genus $24$, the known
methods of producing relations have failed to give enough relations to yield a
Gorenstein ring. In fact, the known relations have all been in the span of the
Faber-Zagier (FZ) relations, and these relations produce a Gorenstein ring if
and only if $g\leq 23$.
There are therefore _mystery relations_ in $R^{*}(M_{g})$: formal polynomials
in $\kappa$ classes which pair to zero with any $\kappa$ polynomial in
$R^{*}(M_{g})$ of complementary degree but are not a linear combination of FZ
relations. If one assumes Faber’s Gorenstein conjecture then these relations
must hold in $R(M_{g})$. Since FZ relations extend to tautological relations
in $R^{*}(\overline{M}_{g})$ (this is a consequence of the proof of the FZ
relations in [7]), a possible reason for the existence of mystery relations
might be if they do not extend tautologically to $R^{*}(M_{g}^{c})$ or
$R^{*}(\overline{M}_{g})$. The Rank Theorem can be interpreted as saying that
part of the obstruction to this extension is zero: the mystery relations at
least extend to classes in the tautological ring of $M_{g}^{c}$ which pair to
zero with the $\kappa$ subring. It is an interesting question whether the
mystery relations extend to classes in the tautological ring of $M_{g}^{c}$
which are relations in the Gorenstein quotient (i.e. pair to zero with the
entire tautological ring).
In [6] Pandharipande gives a minimal set of generators of
$\kappa^{*}(M_{g,n}^{c})$ and relates higher genus relations to genus 0
relations. More precisely, he shows that there is a surjective (graded) ring
homomorphism
$\kappa^{*}(M_{0,2g+n}^{c})\stackrel{{\scriptstyle\iota_{g,n}}}{{\to}}\kappa^{*}(M_{g,n}^{c}),$
which is an isomorphism for $n\geq 1$, or in degrees up to $g-2$ when $n=0$.
The Rank Theorem gives us information about the $n=0$ case in higher degrees.
###### Theorem 1.
Let $g\geq 2$, $0\leq e\leq g-2$, and $d=g-1+e$. Let $\delta_{d}$ be the rank
of the kernel of the map from $\kappa^{d}(M_{g}^{c})$ to the Gorenstein
quotient of $R^{*}(M_{g}^{c})$. Let $\gamma_{e}$ be the rank of the space of
$\kappa$ relations of degree $e$ in the Gorenstein quotient of $R^{*}(M_{g})$.
Then the degree $d$ part of the kernel of $\iota_{g,0}$ has rank
$\gamma_{e}-\delta_{d}$.
###### Proof.
By [6], the rank of $\kappa^{d}(M_{0,2g}^{c})$ is equal to $|P(d,2g-2-d)|$,
the number of partitions of $d$ of length at most $2g-2-d$. On the other side,
the rank of $\kappa^{r}(M_{g}^{c})$ is equal to $\delta_{r}$ plus the rank of
the first pairing appearing in the Rank Theorem. The rank of the second
pairing appearing in the Rank Theorem is given by the Housing Theorem, and the
rank of the third pairing appearing in the Rank Theorem is equal to
$|P(e)|-\gamma_{e}$. Putting these pieces together gives the theorem
statement. ∎
Remark. The components $\gamma_{e}$ and $\delta_{d}$ appearing in the above
theorem both have conjectural values. The FZ relations give a prediction for
$\gamma_{e}$ (if they are the only relations in the first half of the
Gorenstein quotient and are linearly independent):
$\gamma_{e}=\begin{cases}a(3e-g-1)&\text{ if }e\leq\frac{g-2}{2}\\\
a(3(g-2-e)-g-1)&\text{ else},\end{cases}$
where $a(n)$ is the number of partitions of $n$ with no parts of sizes
$5,8,11,\ldots$.
The Gorenstein conjecture in compact type would imply that $\delta_{d}=0$,
though in fact this is a much weaker statement. Combining these predictions
gives a conjecture for all the Betti numbers of $\kappa^{*}(M_{g}^{c})$.
### 1.2. Plan of the paper
In Section 2 we review basic facts about the tautological ring. In Section 3
we prove the Housing Theorem. In Section 4 we state and prove a slightly more
explicit version of the Rank Theorem (see Theorem 2).
### Acknowledgments
The first named author wants to thank his advisor Rahul Pandharipande for the
introduction to this topic and various discussions. The beginning of Section
1.1 elaborates an email from him.
The first named author was supported by the Swiss National Science Foundation
grant SNF 200021_143274. The second named author was supported by an NDSEG
graduate fellowship.
## 2\. The tautological ring
### 2.1. Tautological Classes
The subrings $R^{*}(\overline{M}_{g.n})$ of tautological classes in the Chow
rings $A^{*}(\overline{M}_{g,n})$ are collectively defined as the smallest
subrings which are closed under pushforward by the maps forgetting markings
$\overline{M}_{g,n}\to\overline{M}_{g,n-1}$ and the gluing maps
$\overline{M}_{g_{1},n_{1}\sqcup\\{\star\\}}\times\overline{M}_{g_{2},n_{2}\sqcup\\{\bullet\\}}\to\overline{M}_{g_{1}+g_{2},n_{1}+n_{2}}$
and $\overline{M}_{g,n\sqcup\\{\star,\bullet\\}}\to\overline{M}_{g+1,n}$
defined by gluing together $\star$ and $\bullet$. It turns out that nearly all
classes on the moduli space of curves that appear naturally in geometry lie in
the tautological ring.
For each $i=1,2,\ldots,n$, there is a line bundle $\mathbb{L}_{i}$ on
$\overline{M}_{g,n}$ given by the cotangent space at the $i$th marked point.
The first Chern classes of these line bundles are denoted by
$\psi_{i}=c_{1}(\mathbb{L}_{i})\in A^{1}(\overline{M}_{g,n})$. The $\kappa$
classes are then pushforwards of powers of the $\psi$ classes:
$\kappa_{m}=\pi_{*}(\psi_{n+1}^{m+1})\in A^{m}(\overline{M}_{g,n}),$
where $\pi$ is the forgetful map $\overline{M}_{g,n+1}\to\overline{M}_{g,n}$.
It is well known (see e.g. [5]) that the $\kappa$ and $\psi$ classes combined
with pushforward by the gluing morphisms alone are sufficient to generate the
tautological rings. In other words, $R^{*}(\overline{M}_{g,n})$ is additively
generated by classes of the form
$\xi_{\Gamma}\left(\prod_{v\text{ vertex of }\Gamma}\theta_{v}\right),$
where $\Gamma$ is a stable graph expressing the data of the gluing map
$\xi_{\Gamma}:\prod_{v\text{ vertex of
}\Gamma}\overline{M}_{g(v),n(v)}\to\overline{M}_{g,n}$
and the $\theta_{v}\in R^{*}(\overline{M}_{g(v),n(v)})$ are arbitrary
monomials in the $\psi$ and $\kappa$ classes.
The tautological rings $R^{*}(M_{g,n}^{c})$ and $R^{*}(M_{g,n})$ are defined
as the image of $R^{*}(\overline{M}_{g,n})$ under restriction. In the case of
$R^{*}(M_{g,n}^{c})$, this means that the stable graph $\Gamma$ must be a
tree, while $R^{*}(M_{g,n})$ is simply the subring of polynomials in the
$\kappa$ and $\psi$ classes.
The ring $R^{*}(M_{g,n}^{c})$ has one-dimensional socle, in degree $2g-3+n$:
$R^{2g-3+n}(M_{g,n}^{c})\cong\mathbb{Q}.$
This gives a canonical (up to scaling) bilinear pairing on
$R^{*}(M_{g,n}^{c})$, which can be realized explicitly by integrating against
the Hodge class $\lambda_{g}$:
$R^{*}(M_{g,n}^{c})\times
R^{*}(M_{g,n}^{c})\to\mathbb{Q},\quad(\alpha,\beta)\mapsto\int_{\overline{M}_{g,n}}\alpha\beta\lambda_{g}.$
Here the integral is defined by taking any extensions of $\alpha$ and $\beta$
to $R^{*}(\overline{M}_{g,n})$. It is independent of which particular
extension one has chosen because $\lambda_{g}$ vanishes on the complement of
$M_{g,n}^{c}$.
The $\lambda_{g}\lambda_{g-1}$ pairing is a similar pairing for the moduli
space of smooth curves, given by
$R^{*}(M_{g})\times
R^{*}(M_{g})\to\mathbb{Q},\quad(\alpha,\beta)\mapsto\int_{\overline{M}_{g}}\alpha\beta\lambda_{g}\lambda_{g-1}.$
Notice that the $\lambda_{g}$ pairing on $R^{*}(M_{g}^{c})$ vanishes above
degree $2g-3$ whereas the $\lambda_{g}\lambda_{g-1}$ pairing on $R^{*}(M_{g})$
already vanishes above degree $g-2$.
### 2.2. Notation concerning partitions
In the following sections we will use the following notation heavily. A
_partition_ $\sigma$ is an unordered collection of natural numbers (a
multiset). We call its elements _parts_. Its _size_ is the sum of all its
parts. The _length_ $\ell(\sigma)$ of a partition $\sigma$ is the number of
parts in it. For natural numbers $n$, $r$ we denote by $P(n)$ the set of
partitions of size $n$ and by $P(n,r)$ the set of partitions of size $n$ and
length at most $r$. Furthermore, let $I(\sigma)$ be a set of $\ell(\sigma)$
elements which we will use to index the parts of $\sigma$. For example we
could take
$I(\sigma)=[\ell(\sigma)]:=\\{1,\dots,\ell(\sigma)\\}.$
For two partitions $\sigma,\tau\in P(n)$ and a map $\varphi:I(\sigma)\to
I(\tau)$ we say that $\varphi$ is a _refining function_ of $\tau$ into
$\sigma$ if for any $i\in I(\tau)$ we have
$\tau_{i}=\sum_{j\in\varphi^{-1}(i)}\sigma_{j}.$
If for given $\sigma,\tau$ there exists a refining function $\varphi$ of
$\tau$ into $\sigma$ we say that $\sigma$ is a refinement of $\tau$.
For a finite set $S$, a _set partition_ $P$ of $S$ (written $P\vdash S$) is a
set $P=\\{S_{1},\dots,S_{m}\\}$ of nonempty subsets of $S$ such that $S$ is
the disjoint union of the $S_{i}$.
For a partition $\sigma$ and a set $S$ of subsets of $I(\sigma)$ we define a
new partition $\sigma^{S}$ indexed by the elements of $S$ by setting
$(\sigma^{S})_{s}=\sum_{i\in s}\sigma_{i}$ for each $s\in S$. Usually we will
take a set partition $P$ of $I(\sigma)$ for $S$. For a subset $T\subseteq
I(\sigma)$ we define the _restriction $\sigma|_{T}$ of $\sigma$ to $T$_ by
$\sigma^{S}$, where $S$ is the set of all 1-element subsets of $T$; in other
words, $\sigma|_{T}=(\sigma_{t})_{t\in T}$.
### 2.3. Integral calculations
The basic formula for the evaluation of the integrals arising in the
$\lambda_{g}$ pairing is (see [3])
$\int_{\overline{M}_{g,n}}\prod_{i=1}^{n}\psi_{i}^{\tau_{i}}\lambda_{g}=\binom{2g-3+n}{\tau}\int_{\overline{M}_{g,1}}\psi_{1}^{2g-2}\lambda_{g},$
where $\tau_{1},\dotsc,\tau_{n}$ are nonnegative integer numbers with sum
$2g-3+n$. The formula is symmetric with respect to the sorting of the markings
and hence we only need to know the partition corresponding to $\tau$ in order
to calculate these integrals. The only thing we will need to know about the
integral on the right hand side is that it is nonzero (see [2]) since we are
only interested in the ranks of the pairing.
We will need to evaluate integrals involving $\psi$ classes as part of the
proof of the housing theorem. However our main interest lies in the
calculation of integrals involving $\kappa$ classes. Using the definition of
the $\kappa$ classes as push-forwards of powers of $\psi$ classes we can find
a nice expression for the quotients
$\vartheta(\sigma;\tau):=\left(\int_{\overline{M}_{g,\ell(\tau)}}\kappa_{\sigma}\psi^{\tau}\lambda_{g}\right)\left(\int_{\overline{M}_{g,1}}\psi_{1}^{2g-2}\lambda_{g}\right)^{-1}.$
In this equation we have used $\kappa_{\sigma}$ as an abbreviation for
$\prod_{i\in I(\sigma)}\kappa_{\sigma_{i}}$ and $\psi^{\tau}$ for $\prod_{i\in
I(\tau)}\psi_{i}^{\tau_{i}}$ indexing the $|\tau|$ marked points by the parts
of $\tau$. We will write
$\vartheta(\sigma):=\vartheta(\sigma;\emptyset)$
when we just have $\kappa$ classes and no $\psi$ classes.
###### Lemma 1.
For partitions $\sigma$ and $\tau$ such that $2g-3+\ell(\tau)=|\sigma|+|\tau|$
we have
$\vartheta(\sigma;\tau)=\sum_{P\vdash
I(\sigma)}(-1)^{|P|+\ell(\sigma)}\binom{2g-3+|P|+\ell(\tau)}{((\sigma^{P})_{i}+1)_{i\in
P},\tau}.$
###### Proof.
From the basic socle evaluation formula we see that it suffices to prove the
identity
$\kappa_{\sigma}\psi^{\tau}\lambda_{g}=\sum_{P\vdash
I(\sigma)}(-1)^{|P|+\ell(\sigma)}\pi_{*}\left(\psi^{((\sigma^{P})_{i}+1)_{i\in
P}}\psi^{\tau}\lambda_{g}\right).$
in the Chow ring $R(\overline{M}_{g,\ell(\tau)})$, where by abuse of notation
$\pi$ is the forgetful map
$\overline{M}_{g,\ell(\tau)+n}\to\overline{M}_{g,\ell(\tau)}$ for the
appropriate $n$. Since $\pi^{*}(\lambda_{g})=\lambda_{g}$, we can further
reduce to
$\kappa_{\sigma}\psi^{\tau}=\sum_{P\vdash
I(\sigma)}(-1)^{|P|+\ell(\sigma)}\pi_{*}\left(\psi^{((\sigma^{P})_{i}+1)_{i\in
P}}\psi^{\tau}\right).$
This follows from the pushforward formula
$\pi_{*}\left(\psi^{(\sigma_{i}+1)_{i\in P}}\psi^{\tau}\right)=\sum_{P\vdash
I(\sigma)}\left(\prod_{S\in P}(|S|-1)!\right)\kappa_{\sigma^{P}}\psi^{\tau}.$
and partition refinement inversion. ∎
To evaluate the more general integrals which arise when we pair $\kappa$
classes with arbitrary tautological classes, we can restrict ourselves to
pairing a $\kappa$ monomial with the additive set of generators described in
Section 2.1. In this case we have to sum over the set of possible
distributions of the $\kappa$ classes to the vertices of $\Gamma$ and then
multiply the $\lambda_{g}$ integrals at each vertex.
The $\lambda_{g}\lambda_{g-1}$ pairing formula is similar:
$\int_{\overline{M}_{g,n}}\psi^{\sigma}\lambda_{g}\lambda_{g-1}=\frac{(2g-3+\ell(\sigma))!(2g-1)!!}{(2g-1)!\prod_{i\in
I(p)}(2\sigma_{i}+1)!!}\int_{\overline{M}_{g}}\psi^{g-2}\lambda_{g}\lambda_{g-1}.$
The integral on the right hand side is known to be nonzero (see [4]). We can
calculate the $\kappa$ integrals analogously to Lemma 1.
## 3\. The Housing Theorem
### 3.1. Housing Partitions
Let us now study pairing $\kappa$ monomials of degree $d$ with pure boundary
classes via the $\lambda_{g}$ pairing. Each pure boundary stratum in
codimension $2g-3-d$ is determined by a tree $\Gamma=(V,E)$ with $|V|=2g-2-d$
vertices and $|E|=2g-3-d$ edges and a genus function $g:V\to\mathbb{Z}_{\geq
0}$ with $\sum_{v\in V}g(v)=g$. Then the class is the push-forward of $1$
along the gluing map $\xi_{\Gamma}:\prod_{v\in V}M_{g(v),n(v)}^{c}\to
M_{g}^{c}$ corresponding to the tree $\Gamma$, where $n(v)$ is the degree of
the vertex $v$. From this data we obtain a partition of
$\displaystyle\sum_{v\in V}(2g(v)-3+n(v))$
$\displaystyle=2g-3(2g-2-d)+2(2g-3-d)$ $\displaystyle=d$
by collecting the socle dimensions $2g(v)-3+n(v)$ for each vertex $v\in V$ and
throwing away the zeroes. We will call this partition the _housing data_ of
the pure boundary stratum. From the $\lambda_{g}$ formula it is easy to see
that the pairing of the $\kappa$ ring with a pure boundary stratum is
determined by its housing data.
On the other hand it is interesting to consider which partitions of $d$ can
arise as housing data corresponding to a pure boundary stratum. We will call
these partitions _housing partitions_.
###### Lemma 2.
A partition $\sigma$ of $d$ is a housing partition if and only if it either
has fewer than $2g-2-d$ parts or exactly $2g-2-d$ parts with at least two
even.
###### Proof.
Only partitions of length at most $2g-2-d$ can be housing partitions because
there are only that many vertices. Furthermore it is easy to see that no
partition of $2g-2-d$ parts with fewer than two even parts can arise since
every vertex with only one edge gives an even part (or no part if $g(v)=1$).
Now suppose $\sigma$ is a partition of $d$ with either fewer than $2g-2-d$
parts or exactly $2g-2-d$ parts with at least two even. Let $(\tau_{i})_{1\leq
i\leq 2g-2-d}$ be the tuple of nonnegative integers given by appending
$2g-2-d-\ell(\sigma)$ zeroes to $\sigma$, so the sum of the $\tau_{i}$ is $d$
and exactly $2k+2$ of the $\tau_{i}$ are even for some nonnegative integer
$k$.
Construct a tree $\Gamma$ by taking a path of $2g-2-d-k$ vertices and adding
$k$ additional leaves connected to vertices $2,3,\ldots,k+1$ along the path
respectively. Thus $\Gamma$ has $2g-2-d$ vertices, each of degree at most
three, and exactly $2k+2$ of the vertices of $\Gamma$ have odd degree. We now
choose a bijection between the $\tau_{i}$ and the vertices of $\Gamma$ such
that even $\tau_{i}$ are assigned to vertices of odd degree. We can then
assign a genus $g_{i}=(\tau_{i}+3-n_{i})/2$ to each vertex, where $n_{i}$ is
the degree of the vertex to which $\tau_{i}$ was assigned. The resulting
stable tree has housing data $\sigma$, as desired. ∎
### 3.2. Reduction to a combinatorial problem
We have already described the housing data of a pure boundary stratum. Let us
now describe a similar notion for any class in the generating set described in
Section 2.1. Such a class is given by a boundary stratum corresponding to a
tree $\Gamma=(V,E)$ and a genus assignment $g:V\to\mathbb{Z}_{\geq 0}$, along
with assignments of monomials in $\kappa$ and $\psi$ classes (of degrees
$r(v)$ and $s(v)$ respectively) to each component of the stratum. Let
$k=\sum_{v\in V}(r(v)+s(v))$; then we must have $|E|=2g-3-d-k$ edges in the
tree in order to obtain a class of degree $2g-3-d$. If this class does not
vanish by dimension reasons then we can obtain a partition $\gamma$ of
$\sum_{v\in V}(2g(v)-3+n(v)-r(v)-s(v))=\\\ 2g-3(2g-2-d-k)+2(2g-3-d-k)-k=d$
by assigning to each vertex of $V$ the number $2g(v)-3+n(v)-r(v)-s(v)$. This
is exactly the degree $d^{\prime}(v)$ such that the $\lambda_{g(v)}$ pairing
of $R^{d^{\prime}(v)}(M_{g(v),n(v)}^{c})$ with the monomial of $\psi$ and
$\kappa$ classes at $v$ is not zero for dimension reasons. Then the pairing
with the boundary class is determined by the partition $\gamma$, an assignment
of degrees $r(i)$ and $s(i)$ to the parts $i\in I(\gamma)$ and partitions
$\tau_{i}\in P(r(i))$ and $\rho_{i}\in P(s(i))$ corresponding to the $\kappa$
and $\psi$ monomials. In particular we can leave out classes which were
assigned to vertices with $2g(v)-3+n(v)-r(v)-s(v)=0$ and we do not need to
remember which node corresponds to each $\psi$. The result of the
$\lambda_{g}$ pairing of this class together with a $\kappa$ monomial
corresponding to a partition $\pi$ of $d$ is (up to scaling) given by
$\sum_{\varphi}\prod_{j\in
I(\gamma)}\vartheta\left(\pi_{\varphi^{-1}(j)},\tau_{j};\rho_{j}\right),$
where the sum runs over all refining functions $\varphi$ of $\gamma$ into
$\pi$.
When we view $\mathbb{Q}^{P(d)}$ as a ring of formal $\kappa$ polynomials,
this pairing gives linear forms
$v_{\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}\in\left(\mathbb{Q}^{P(d)}\right)^{*}$.
We notice that the formulas still make combinatorial sense even if the
$\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}$ data does not come from pairing with an
actual tautological class.
The special case where all the $r(i)$ and $s(i)$ are zero gives the pairing of
$\kappa$ classes with pure boundary classes. We get $|P(d)|$ linear forms
$M_{\lambda}$, which we normalize such that $M_{\lambda}(\lambda)=1$:
(1) $M_{\lambda}(\pi)=\frac{1}{\mathrm{Aut}(\lambda)}\sum_{\varphi}\prod_{j\in
I(\lambda)}\vartheta\left(\pi_{\varphi^{-1}(j)}\right).$
In this way we obtain a basis of $\left(\mathbb{Q}^{P(d)}\right)^{*}$. If we
sort partitions in any way such that shorter partitions come before longer
partitions, then the basis change matrix from this basis to the standard basis
is triangular with ones on the diagonal. Note that this basis uses some
partitions which are not housing partitions.
The housing theorem can now be reformulated as follows:
###### Claim.
The span of $\\{M_{\lambda}:\lambda\text{ is a housing partition}\\}$ in
$\left(\mathbb{Q}^{P(d)}\right)^{*}$ equals the span of the
$v_{\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}$ for all choices of housing data.
To prove this claim we will first express the vectors
$v_{\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}$ for any choice of housing data in
terms of the basis of $\left(\mathbb{Q}^{P(d)}\right)^{*}$ we have described
above in Section 3.3. We will then in Section 3.4 rewrite the coefficients as
counts of certain combinatorial objects. This combinatorial interpretation is
proved in Section 3.5. We conclude in Section 3.6 by showing that when
expressing vectors $v$ corresponding to actual housing data in terms of the
$M_{\lambda}$, the coefficient is zero whenever $\lambda$ is not a housing
partition.
### 3.3. A Matrix Inversion
In section 3.2 we have seen that there are formal expansions
$v_{\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=\sum_{\lambda\in
P(d)}c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}M_{\lambda}$
for some coefficients $c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}$.
We can calculate $c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}$ explicitly
by inverting the triangular matrix given by equation (1). We obtain
$\displaystyle
c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=\sum_{l=0}^{\infty}(-1)^{l}\sum_{\lambda_{0}\stackrel{{\scriptstyle\varphi_{1}}}{{\to}}\dots\stackrel{{\scriptstyle\varphi_{l}}}{{\to}}\lambda_{l}\stackrel{{\scriptstyle\varphi_{l+1}}}{{\to}}\gamma}\frac{v_{\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}(\lambda_{l})}{\prod_{i=1}^{l}|\mathrm{Aut}(\lambda_{i})|}\prod_{i=1}^{l}\prod_{j\in
I(\lambda_{i})}\vartheta\left((\lambda_{i-1})_{\varphi_{i}^{-1}(j)}\right),$
where we sum over chains $\lambda=\lambda_{0},\dotsc\lambda_{l}$ of
refinements of $\gamma$ with corresponding refinement functions $\varphi_{i}$.
In particular $c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=0$ if
$\lambda$ is not a refinement of $\gamma$.
We can reduce to the special case in which $\gamma=(d)$ is of length one by
splitting this sum based on the composition
$\varphi:=\varphi_{l+1}\circ\varphi_{l}\circ\dots\circ\varphi_{1}$ and
examining the contribution of the preimages of the $j\in I(\gamma)$. The
result is
(2)
$c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=\sum_{\varphi}\prod_{j\in
I(\gamma)}c_{\lambda_{\varphi^{-1}(j)},(\gamma_{j}),\\{\tau_{j}\\},\\{\rho_{j}\\}},$
summed over refinements $\varphi$ of $\gamma$ into $\lambda$.
When $\gamma=(d)$, we set $\tau_{1}=:\tau$ and $\rho_{1}=:\rho$ and we can
write more compactly
(3)
$c_{\lambda,(d),\\{\tau\\},\\{\rho\\}}=\sum_{l=0}^{\infty}(-1)^{l}\sum_{\lambda_{0}\stackrel{{\scriptstyle\varphi_{1}}}{{\to}}\dots\stackrel{{\scriptstyle\varphi_{l}}}{{\to}}\lambda_{l}}\frac{\vartheta(\lambda_{l},\tau;\rho)}{\prod_{i=1}^{l}|\mathrm{Aut}(\lambda_{i})|}\prod_{i=1}^{l}\prod_{j\in
I(\lambda_{i})}\vartheta\left((\lambda_{i-1})_{\varphi_{i}^{-1}(j)}\right).$
### 3.4. Interpreting the coefficients combinatorially
We will interpret the coefficients $c_{\lambda,(d),\\{\tau\\},\\{\rho\\}}$ as
counting certain permutations of symbols labeled by the parts of the
partitions $\lambda,\tau$, and $\rho$. We say that a symbol is _of kind $i$_
if it is labelled by some $i$ belonging to the disjoint union of the indexing
sets of the partitions, $I(\lambda)\sqcup I(\tau)\sqcup I(\rho)$. There will
in general be multiple symbols of a given kind.
###### Main Claim.
The coefficient $c_{\lambda,(d),\\{\tau\\},\\{\rho\\}}$ counts the number of
permutations of
* •
$\lambda_{i}+1$ symbols of kind $i$ for each $i\in I(\lambda)$,
* •
$\tau_{i}+1$ symbols of kind $i$ for each $i\in I(\tau)$, and
* •
$\rho_{i}$ symbols of kind $i$ for each $i\in I(\rho)$
such that:
1. (1)
If the last symbol of some kind $i$ is immediately followed by the first
symbol of kind $j$ with $i,j\in I(\lambda)\sqcup I(\tau)$, then we have $i<j$.
2. (2)
For $i\in I(\lambda)$ the last element of kind $i$ is not immediately followed
by a symbol of kind $j$ for any $j\in I(\lambda)$,
averaged over all total orders $<$ of $I(\lambda)\sqcup I(\tau)$ such that
elements of $I(\tau)$ are smaller than elements of $I(\lambda)$.
It follows in particular that the coefficient
$c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}$ is non-negative.
### 3.5. Proof of the main claim
#### 3.5.1. Refinements of permutations of symbols
For given natural numbers $d$, $n$ and a partition $\tau\in P(d)$ we will
study permutations of $\tau_{i}+1$ symbols of kind $i$ for $i\in I(\tau)$ and
$n$ symbols of kind $c$. (The permutations of symbols appearing in the
previous section are an instance of this.) We will need to construct refined
permutations of this type for partition refinements $\varphi:I(\sigma)\to
I(\tau)$. For this we need additional _refinement data_ : for each $i\in
I(\tau)$, let $T_{i}$ be a permutation of $\sigma_{j}+1$ symbols of kind $j$
for $j\in\varphi^{-1}(i)$. Then we can obtain a permutation $S^{\prime}$ of
$\sigma_{i}+1$ symbols of kind $i$ and $n$ symbols of kind $c$ in the
following way:
For each $i\in I(\tau)$ and each $j\in\varphi^{-1}(i)$, modify $T_{i}$ by
gluing the last symbol of kind $j$ with the immediately following symbol; the
result is a permutation $T^{\prime}_{i}$ of $\tau_{i}+1$ symbols. To construct
$S^{\prime}$ from $S$, for each $i$ we replace the symbols of kind $i$ by
$T^{\prime}_{i}$ and then remove the glue.
#### 3.5.2. Reinterpretation
We start with a combinatorial interpretation of the number
$\vartheta(\sigma;\tau)$ for partitions $\sigma$ and $\tau$.
###### Lemma 3.
Given an arbitrary total order $<$ on $I(\sigma)$, the number
$\vartheta(\sigma;\tau)$ is equal to the number of permutations of
* •
$\sigma_{i}+1$ symbols of kind $i$ for each $i\in I(\sigma)$ and
* •
$\tau_{i}$ symbols of kind $i$ for each $i\in I(\tau)$
such that the following property holds:
If the last symbol of kind $i$ is immediately followed by the first symbol of
kind $j$ for $i,j\in I(\sigma)$ then we have $i<j$.
###### Proof.
For each permutation $S$ of symbols as above, but not necessarily satifying
the property, we can assign a set partition $Q_{S}\vdash I(\sigma)$ which
measures in what ways it fails to satisfy the property: $Q_{S}$ is the finest
set partition such that if $i<j$ and the last symbol of kind $i$ is
immediately followed by the first symbol of kind $j$ in $S$, then $i$ and $j$
are in the same part of $Q_{S}$. Thus $S$ satisfies the given property if and
only if $Q_{S}$ is the set partition with all parts of size $1$.
The multinomial coefficient in the summand in the formula for
$\vartheta(\sigma;\tau)$ given by Lemma 1 corresponding to a set partition
$P\vdash I(\sigma)$ counts the number of permutations $S$ such that for
$p=\\{p_{1},\ldots,p_{k}\\}\in P$ with $p_{1}<\cdots<p_{k}$, the last element
of kind $p_{i}$ is immediately followed by the first element of kind $p_{i+1}$
in $S$ for $i=1,\cdots,k-1$. These are precisely the $S$ such that $Q_{S}$ can
be obtained by combining parts of $P$ such that the largest element in one
part is smaller than the smallest element of the other part.
This means that the contribution of a permutation with failure set partition
$Q=\\{Q_{1},\ldots,Q_{k}\\}$ to the sum in Lemma 1 is precisely
$\prod_{i=1}^{k}\sum_{j=0}^{|Q_{k}|-1}(-1)^{j}\binom{|Q_{k}|-1}{j},$
which is $1$ for $Q$ the set partition with all parts of size $1$ and $0$
otherwise. ∎
Equipped with Lemma 3, the next step is to interpret the coefficient
$c_{\lambda,(d),\\{\tau\\},\\{\rho\\}}$ as the sum of the values of a function
$f$ on the set $S_{\lambda,\tau,\rho}$ of permutations of $\lambda_{i}+1$,
$\tau_{i}+1$, $\rho_{i}$ symbols of kind $i$ for $i\in I(\lambda)$, $i\in
I(\tau)$ and $i\in I(\rho)$ respectively.
Fix
* •
a chain of partitions $\lambda_{0}=\lambda,\lambda_{1},\dots,\lambda_{l}$ with
refining maps $\varphi_{i}$,
* •
an order $<$ on $I(\lambda_{l})\sqcup I(\tau)$ such that elements of $I(\tau)$
appear before elements of $I(\lambda_{l})$,
* •
orders on $\varphi_{i}^{-1}(j)$ for $1\leq i\leq l$ and $j\in I(\lambda_{i})$.
Then we identify each $\kappa$ socle evaluation factor
$\vartheta\left((\lambda_{i-1})_{\varphi_{i}^{-1}(j)}\right)$
with the number of permutations of $(\lambda_{i-1})_{k}+1$ symbols of kind
$k\in\varphi_{i}^{-1}(j)$ such that if the last symbol of kind $k$ is
immediately followed by the first symbol of kind $k^{\prime}$, then
$k<k^{\prime}$. We interpret each such permutation as refinement data
corresponding to the refinement $\varphi_{i}$ of $\lambda_{i}$ into
$\lambda_{i+1}$.
Furthermore we interpret the factor
$\vartheta\left(\lambda_{l},\tau;\rho\right)$
as the number of permutations of $(\lambda_{l})_{k}+1$, $\tau_{k}+1$ and
$\rho_{k}$ symbols of kind $k$ with $k\in I(\lambda_{l})$, $k\in I(\tau)$ and
$k\in I(\rho)$ respectively such that if the last symbol of kind $k$ is
immediately followed by the first symbol of kind $k^{\prime}$ for
$k,k^{\prime}\in I(\lambda_{l})\sqcup I(\tau)$, then $k<k^{\prime}$. In order
to remove the dependence on the chosen orders we will average over all choices
of them.
#### 3.5.3. Simplification
Given all this data, we can build a “composite permutation” by repeatedly
refining the collection of symbols of kind $k$ with $k\in I(\lambda_{l})$
using the construction from Section 3.5.1 and keeping the order of the other
symbols intact. The result is a permutation of $\lambda_{k}+1$, $\tau_{k}+1$
and $\rho_{k}$ symbols of kind $k$ for $k\in I(\lambda)$, $k\in I(\tau)$ and
$k\in I(\rho)$ respectively. Any permutation obtained in this way has the
property that the last symbol of any kind $j\in I(\lambda)$ is not immediately
followed by the first symbol of some kind $j^{\prime}\in I(\tau)$.
For any permutation in $S_{\lambda,\tau,\rho}$ we assign a set partition
$P\vdash I(\lambda)$, which measures in what way it fails to satisfy condition
(2) in the main claim. We define $P$ to be the finest set partition such that
if the last symbol of kind $i$ is immediately followed by a symbol of kind $j$
for $i,j\in I(\lambda)$ then $i$ and $j$ lie in the same set in $P$.
Now, suppose we are given a chain of partitions
$\lambda,\lambda_{1},\dots,\lambda_{l}$ along with additional refining data
and base permutation as above, and supppose the resulting composite
permutation has failure set partition $P$ that is not the partition into one-
element sets.
By the definition of $P$, if we change the order on $I(\lambda)\sqcup I(\tau)$
such that the order on $I(\tau)$ and each element of $P$ is preserved, all the
conditions on the data are still satisfied.
On the other hand, consider the following data:
* •
the chain $\lambda,\lambda_{1},\dots,\lambda_{l},\lambda_{l}^{P}$ with
refining maps $\varphi_{i}$ as before,
* •
any refining map $\varphi^{\prime}:I(\lambda_{l})\to I(\lambda_{l}^{P})$ which
is up to an automorphism of $\lambda_{l}^{P}$ the canonical one,
* •
the orders and refining data corresponding to the $\varphi_{i}$ as before,
* •
in addition an order on each element of $P$ induced by the order on
$I(\lambda_{l})\sqcup I(\tau)$,
* •
refining data corresponding to $\varphi^{\prime}$ induced from the permutation
corresponding to $\lambda_{l}$, $\tau$ and $\rho$,
* •
any order on $I(\lambda_{l}^{P})\sqcup I(\tau)$ such that the restriction to
$I(\tau)$ is the restriction of the order on $I(\lambda_{l})\sqcup I(\tau)$
and such that elements of $I(\tau)$ appear before elements of
$I(\lambda_{l}^{P})$,
* •
permutations of $(\lambda_{l}^{P})_{i}+1$, $\tau_{i}+1$, $\rho_{i}$ symbols of
kind $i$ for $i\in P$, $i\in I(\tau)$ and $i\in I(\rho)$ respectively, defined
from the permutation corresponding to $\lambda_{l}$ by leaving out the last
symbol of any kind $i\in I(\lambda_{l})$ which is not the last one in a set of
$P$ and identifying symbols according to $P$.
It is easy to check that the refining data and the permutation still satisfy
the order conditions. Furthermore, the failure set partition of the composite
partition of this new data is the partition into one-element sets.
The original chain with additional data giving failure set partition $P$ and
the extended chain with additional data giving failure set partition the
partition into one-element sets contribute to
$c_{\lambda,(d),\\{\tau\\},\\{\rho\\}}$ in formula (3) with opposite signs,
since the extended chain is one element longer. We claim these contributions
are actually equal.
For the original chain, we have
$\frac{(\ell(\lambda_{l}))!}{\prod_{j\in P}|\varphi^{\prime-1}(j)|!}$
choices of orders on $I(\lambda)\sqcup I(\tau)$ in the above construction. For
the extended chain, we made
$|\mathrm{Aut}(\lambda_{l}^{P})|(\ell(\lambda_{l}^{P}))!$
choices in the above construction.
However, the contributions are also weighted by averaging over choices of
orders and by the coefficients in (3). For the original chain the weight is
$((\ell(\lambda_{l}))!)^{-1}$
and for the extended chain the weight is
$\left(|\mathrm{Aut}(\lambda_{l}^{P})|(\ell(\lambda_{l}^{P}))!\prod_{j\in
P}|\varphi^{\prime-1}(j)|!\right)^{-1}.$
Thus the two contributions cancel.
The only remaining contributions come when $l=0$ and $P$ is the set partition
into one-element sets. These are the permutations counted in the main claim.
### 3.6. Proof of the Housing Theorem
We begin with a simple lemma.
###### Lemma 4.
Suppose $r+s+\ell(\gamma)<\ell(\lambda)$. Then
$c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=0$.
###### Proof.
We examine the summand in formula (2) corresponding to some $\varphi$. A
factor in this summand can only be nonzero if
$r(i)+s(i)+1\geq\ell(\varphi^{-1}(i))$. Therefore each summand will vanish
unless $r+s+\ell(\gamma)\geq\ell(\lambda)$. ∎
Now let us suppose that $\gamma$, $\\{\tau_{i}\\}$, $\\{\rho_{i}\\}$ is the
housing data of a boundary class of the generating set. We need to show that
$c_{\lambda,\gamma,\\{\tau_{i}\\},\\{\rho_{i}\\}}=0$ for each $\lambda$ which
is not a housing partition.
Let us first study the case $\ell(\lambda)>2g-2-d$. Since $\gamma$ is derived
from a boundary stratum of at most codimension $2g-3-d-r-s$ (we are missing
the $\psi$ and $\kappa$ classes from the components, which do not contribute
to $\gamma$) by diminishing parts by their $\kappa$ and $\psi$ degrees, we
have the inequality $\ell(\gamma)\leq 2g-2-d-r-s$. Then by Lemma 4 we are done
in this case. The same argument settles also the case where there are
components of the boundary stratum we are considering which do not appear in
$\gamma$ and $\ell(\lambda)=2g-2-d$.
Now assume that $\ell(\lambda)=2g-2-d$ and that $\lambda$ contains no even
part. Then by the same arguments if the coefficient is nonzero, we must have
$\ell(\gamma)=2g-2-d-r-s$. Then from the proof of Lemma 4 we see that
$r(i)+s(i)=\ell(\varphi^{-1}(i))-1$ for each $i\in I(\gamma)$. This implies
$\ell(\varphi^{-1}(i))+r(i)+s(i)\equiv 1\pmod{2}$ and therefore for each $i\in
I(\gamma)$ we have $\gamma_{i}+r(i)+s(i)\equiv 1\pmod{2}$. Hence each part of
the housing data (for the underlying boundary stratum), which $\gamma$ was
obtained from by subtraction of $r(i)+s(i)$ at each part, is odd. This is a
contradiction, so the coefficient must be zero, as desired.
## 4\. The Rank Theorem
### 4.1. Reformulation
Let us first formulate a stronger version of the Rank Theorem.
###### Theorem 2.
For any $\kappa$ polynomial $F$ in degree $r:=2g-3-d$ the following two
statements are equivalent:
1. (1)
For any $\pi\in P(g-2-r)$ we have
$\int_{\overline{M}_{g}}F\kappa_{\pi}\lambda_{g}\lambda_{g-1}=0$.
2. (2)
There is a $B\in PBR^{r}(M_{g}^{c})$ such that for any $\pi^{\prime}\in
P(2g-3-r)$ we have
$\int_{\overline{M}_{g}}(F-B)\kappa_{\pi^{\prime}}\lambda_{g}=0$.
It will be convenient to show that we can replace the first condition in
Theorem 2 by
1. (3)
For any $\pi\in P(g-2-r)$ of length at most $r+1$ we have
$\int_{\overline{M}_{g}}F\kappa_{\pi}\lambda_{g}\lambda_{g-1}=0$.
Then Theorem 2 will follow from the following simple argument. Consider an $F$
satisfying the second condition and we want to show that
$\int_{\overline{M}_{g}}F\kappa_{\pi}\lambda_{g}\lambda_{g-1}=0$ for some
given $\pi\in P(g-2-r)$. Notice that then also $F\kappa_{\pi}$ satisfies the
second condition since $B\kappa_{\pi}$ lies in $BR^{g-2}(M^{c}_{g})$ and by
the housing theorem can be replaced by some $B^{\prime}\in
PBR^{g-2}(M^{c}_{g})$. We then find that
$\int_{\overline{M}_{g}}F\kappa_{\pi}\lambda_{g}\lambda_{g-1}=0$ since in this
case the length condition is trivial.
As we have seen in Section 3.5.2, not only boundary classes but also every
$\kappa$ class can be written in terms of virtual boundary strata in the
$\lambda_{g}$ pairing with the kappa ring. So the second condition in the
theorem is equivalent to the condition that only actual boundary strata are
needed in the expansion of $F$. Notice that by Lemma 4 we only need strata
corresponding to partitions of $2g-3-r$ of length at most $r+1$. However we
might need terms corresponding to partitions $2g-3-r$ of length equal to $r+1$
with only odd parts, and those are the terms we are interested in. For the
proof of the Rank Theorem we will need to understand the coefficients
corresponding to these classes better.
Observe that partitions of $2g-3-r$ of length being equal to $r+1$ with only
odd parts correspond to partitions of $g-2-r$ of length at most $r+1$. So for
any $\sigma\in P(g-2-r,r+1)$ we can look at
$\eta_{\sigma},\mu_{\sigma}\in(\mathbb{Q}^{P(r)})^{*}$ with
$\eta_{\sigma}(\tau):=c_{\lambda,\tau,\emptyset}$, where $\lambda$ is the
partition of $2g-3-r$ of length $r+1$ corresponding to $\sigma$, and
$\mu_{\sigma}(\tau)$ is up to a factor the integral
$\int_{\overline{M}_{g}}\kappa_{\sigma}\kappa_{\tau}\lambda_{g}\lambda_{g-1}$,
namely
$\mu_{\sigma}(\tau)=\sum_{P\vdash I(\sigma)\sqcup
I(\tau)}(-1)^{\ell(\sigma)+\ell(\tau)+|P|}\frac{(2g-3+|P|)!}{\prod_{i\in
P}(2(\sigma,\tau)^{P}_{i}+1)!!}.$
So what we need to show is the following:
###### Claim.
The $\mathbb{Q}$-subspaces of $(\mathbb{Q}^{P(r)})^{*}$ spanned by
$\eta_{\sigma}$ and $\mu_{\sigma}$ for $\sigma$ ranging over all partitions of
$g-2-r$ of length at most $r+1$ are equal.
Recall from Section 3.5.2 that $\eta_{\sigma}(\tau)$ is the number of all
permutations $S$ of $\lambda_{i}+1$ symbols of kind $i\in I(\lambda)$ and
$\tau_{i}+1$ symbols of kind $i\in I(\tau)$ satisfying
1. (1)
The last symbol of kind $i$ for some $i\in I(\lambda)$ is either at the end of
the sequence or immediately followed by a symbol of kind $j$ for some $j\in
I(\tau)$ which is not the first of its kind.
2. (2)
The successor of the last element of kind $i$ is not the first element of kind
$j$ for any $i,j\in I(\tau)$ with $i<j$, where we fix some order on $I(\tau)$.
Before coming to the main part of the proof we apply an invertible
transformation $\Phi$ to $(\mathbb{Q}^{P(r)})^{*}$ to simplify the definitions
of $\eta$ and $\mu$. The inverse of the transformation we want to apply sends
a linear form $\varphi^{\prime}\in(\mathbb{Q}^{P(r)})^{*}$ to a linear form
$\varphi$ defined by
$\varphi(\tau)=\sum_{P\vdash
I(\tau)}(-1)^{\ell(\tau)+|P|}\varphi^{\prime}(\tau^{P}).$
The transformation $\Phi$ defined in this way is clearly invertible. By a
similar argument as in the proof of Lemma 3, we can show that the image
$\eta^{\prime}_{\sigma}$ of $\eta_{\sigma}$ under $\Phi$ is defined in the
same way as $\eta_{\sigma}$ but leaving out Condition 2 on the permutations.
To study the action of $\Phi$ on $\mu$ we use the following lemma:
###### Lemma 5.
Let $F$ be a function $F:P(n+m)\to\mathbb{Q}$ and define for any $\sigma\in
P(n)$ functions $G_{\sigma},G^{\prime}_{\sigma}:P(m)\to\mathbb{Q}$ in terms of
$F$ by
$\displaystyle G_{\sigma}(\tau)$ $\displaystyle=\sum_{P\vdash I(\sigma)\sqcup
I(\tau)}F((\sigma\sqcup\tau)^{P})$ $\displaystyle G^{\prime}_{\sigma}(\tau)$
$\displaystyle=\sum_{\begin{subarray}{c}P\vdash I(\sigma)\sqcup I(\tau)\\\
P\text{ separates }I(\tau)\end{subarray}}F((\sigma\sqcup\tau)^{P}),$
where the second sum just runs over set partitions $P$ such that each element
of $I(\tau)$ belongs to a separate part. Then
$G_{\sigma}(\tau)=\sum_{P\vdash I(\tau)}G^{\prime}_{\sigma}(\tau^{P}).$
###### Proof.
Given set partitions $P$ of $I(\tau)$ and $Q$ of $I(\sigma)\sqcup
I(\tau^{P})$, with $Q$ separating $I(\tau^{P})$, we can alter $Q$ by replacing
each element of $I(\tau^{P})$ by the elements in the corresponding part of
$P$. Each set partition of $I(\sigma)\sqcup I(\tau)$ is obtained exactly once
by this construction. ∎
Using this lemma and keeping track of the sign factors, we have that
$\mu^{\prime}_{\sigma}(\tau)$ is
$\mu^{\prime}_{\sigma}(\tau)=\sum_{\begin{subarray}{c}P\vdash I(\sigma)\sqcup
I(\tau)\\\ P\text{ separates
}I(\tau)\end{subarray}}(-1)^{\ell(\sigma)+\ell(\tau)+|P|}\frac{(2g-3+|P|)!}{\prod_{i\in
P}(2(\sigma\sqcup\tau)^{P}_{i}+1)!!}.$
We can use the lemma again with the roles of $\sigma$ and $\tau$ interchanged
to replace the generators of the span of $\mu^{\prime}_{\sigma}$ by
$\mu^{\prime\prime}_{\sigma}$ with
(4) $\mu^{\prime\prime}_{\sigma}(\tau):=\sum_{\begin{subarray}{c}P\vdash
I(\sigma)\sqcup I(\tau)\\\ P\text{ separates }I(\tau)\\\ P\text{ separates
}I(\sigma)\end{subarray}}(-1)^{\ell(\sigma)+\ell(\tau)+|P|}\frac{(2g-3+|P|)!}{\prod_{i\in
P}(2(\sigma\sqcup\tau)^{P}_{i}+1)!!}.$
Therefore we have reduced the proof of the Rank Theorem to proving the
following claim.
###### Claim.
The $\mathbb{Q}$-subspaces of $(\mathbb{Q}^{P(r)})^{*}$ spanned by
$\eta^{\prime}_{\sigma}$ and $\mu^{\prime\prime}_{\sigma}$ for $\sigma$
ranging over all partitions of $g-2-r$ of length at most $r+1$ are equal.
### 4.2. Further strategy of proof
In order to prove the claim we will establish interpretations for
$\eta^{\prime}_{\sigma}(\tau)$ and $\mu^{\prime\prime}_{\sigma}(\tau)$ as
counts of symbols of different kinds satisfying some ordering constraints.
This enables us to find nonzero constants $F(i)$ for each $i\in I(\sigma)$
independent of $\tau$ such that
$\mu^{\prime\prime}_{\sigma}(\tau)=\sum_{P\vdash I(\sigma)}\prod_{i\in
P}F(i)\frac{\eta^{\prime}_{\sigma^{P}}(\tau)}{(r+1-|P|)!},$
giving a triangular transformation.
For the interpretations the notion of a _comb-like order_ plays an important
role. We say that symbols $i_{1}\dotsc i_{2m+1}$ are in comb-like order if we
have the relations $i_{1}<i_{3}<\dots<i_{2m+1}$ and $i_{2j}<i_{2j+1}$ for
$j\in[m]$. This is illustrated in Figure 1.
$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$$i_{2m-1}$$i_{2m}$$i_{2m+1}$ Figure 1. A
comb-like order
Note that the number of comb-like orderings of $2m+1$ symbols is
$(2m+1)!/(2m+1)!!$. More generally the number
$\frac{(2|\pi|+\ell(\pi))!}{\prod_{i\in I(\pi)}(2\pi_{i}+1)!!}$
corresponding to a partition $\pi$ counts the number of permutations of the
$2|\pi|+\ell(\pi)$ symbols $\bigcup_{i\in
I(\pi)}\\{i_{1},\dots,i_{2\pi_{i}+1}\\}$ such that symbols corresponding to
the same part of $\pi$ appear in comb-like order.
### 4.3. Combing orders
We obtain a first reinterpretation of $\eta^{\prime}_{\sigma}(\tau)$ by
numbering the symbols of equal kind:
###### Interpretation A1.
$\eta^{\prime}_{\sigma}(\tau)$ is the number of all permutations of symbols
$i_{1},\dotsc,i_{\tau_{i}+1}$ for $i\in I(\tau)$ and
$i_{1},\dotsc,i_{\lambda_{i}+1}$ for $i\in I(\lambda)$ such that for fixed
$i\in I(\tau)\sqcup I(\lambda)$ the $i_{j}$ appear in order and for all $i\in
I(\lambda)$ the symbol $i_{\lambda_{i}+1}$ is either at the end of the
sequence or immediately followed by some $j_{k}$ for $j\in I(\tau)$ and $k\neq
1$.
Since $\lambda$ has length $r+1$ and $|\tau|=r$, such a permutation gives a
bijection between the $j_{k}$ for $j\in I(\tau)$ with $k\neq 1$ and all but
one of the $i_{\lambda_{i}+1}$ for $i\in I(\lambda)$. After picking this
bijection, we can remove the $i_{\lambda_{i}+1}$.
###### Interpretation A2.
$\eta^{\prime}_{\sigma}(\tau)$ is the sum over bijections
$\varphi:I(\lambda)\to\\{i_{j}\mid i\in I(\tau),j\neq
1\\}\sqcup\\{\text{End}\\}$
of the number of permutations of symbols $i_{1},\dotsc,i_{\tau_{i}+1}$ for
$i\in I(\tau)$ and $i_{1},\dotsc,i_{\lambda_{i}}$ for $i\in I(\lambda)$ such
that symbols of the same kind appear in order and all symbols $i_{j}$ for
$i\in I(\lambda)$ appear before $\varphi(i)$ (this condition is empty if
$\varphi(i)=\text{End}$).
We can then add new symbols immediately following each $i_{\lambda_{i}}$ for
$i\in I(\lambda)$ and reindex the $i_{j}$ for $i\in I(\tau)$ to create comb-
like orderings.
###### Interpretation A3.
$\eta^{\prime}_{\sigma}(\tau)$ is the sum over bijections
$\varphi:I(\lambda)\to\\{i_{j}\mid i\in I(\tau),j\text{
even}\\}\sqcup\\{\text{End}\\}$
of the number of permutations of symbols $i_{1},\dotsc,i_{2\tau_{i}+1}$ for
$i\in I(\tau)$, $i_{1},\dotsc,i_{\lambda_{i}}$ for $i\in I(\lambda)$, and an
additional symbol End such that the $i_{j}$ for $i\in I(\tau)$ appear in comb-
like order, the $i_{j}$ for $i\in I(\lambda)$ appear in order, and
$i_{\lambda_{i}}$ for $i\in I(\lambda)$ is immediately followed by
$\varphi(i)$.
Recall that $\lambda$ is defined in terms of $\sigma$ by taking the numbers
$2\sigma_{i}+1$ for each $i\in I(\sigma)$ and adding as many ones as needed to
reach length $r+1$. There is only one symbol $i_{1}$ of kind $i$ for $i\in
I(\lambda)\setminus I(\sigma)$ in Interpretation Interpretation A3 of
$\eta^{\prime}_{\sigma}(\tau)$ and it must be immediately followed by
$\varphi(i)$. For convenience set
$(r+1-\ell(\sigma))!\cdot\eta^{\prime\prime}_{\sigma}:=\eta^{\prime}_{\sigma}$.
Removing these symbols $i_{1}$ gives an interpretation of
$\eta^{\prime\prime}_{\sigma}$.
###### Interpretation A4.
$\eta^{\prime\prime}_{\sigma}(\tau)$ is a sum over all injections
$\varphi:I(\sigma)\to\\{i_{j}\mid i\in I(\tau),j\text{
even}\\}\sqcup\\{\text{End}\\}$
of the number of permutations of symbols $i_{1},\dotsc,i_{2\tau_{i}+1}$ for
$i\in I(\tau)$, $i_{1},\dotsc,i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$, and an
additional symbol End such that the $i_{j}$ for $i\in I(\tau)$ appear in comb-
like order, the $i_{j}$ for $i\in I(\sigma)$ appear in order, and
$i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ is immediately followed by
$\varphi(i)$.
We now switch to the interpretation of $\mu^{\prime\prime}_{\sigma}(\tau)$,
which was defined in (4). The coefficient corresponding to a set partition $P$
can be interpreted as the number of permutations of symbols
$i_{1},\dotsc,i_{2(\sigma\sqcup\tau)^{P}_{i}+1}$ for $i\in
I((\sigma\sqcup\tau)^{P})$ and one additional symbol $\star$ such that all
$i_{1},\dotsc,i_{2(\sigma\sqcup\tau)^{P}_{i}+1}$ for $i\in
I((\sigma\sqcup\tau)^{P})$ appear in comb-like order.
Because of the restrictions in the sum, the parts of $P$ are either singletons
or contain exactly one element from each of $I(\sigma)$ and $I(\tau)$. This
defines a function $\psi:I(\sigma)\to I(\tau)\sqcup\\{\star\\}$, injective
when restricted to the preimage of $I(\tau)$. Interpreting the summands as
counting comb-like orders and cutting combs into two pieces for each part of
$P$ of size two gives the following:
###### Interpretation B1.
$\mu^{\prime\prime}_{\sigma}(\tau)$ is the sum over functions
$\psi:I(\sigma)\to I(\tau)\sqcup\\{\star\\}$
such that $\psi|_{\psi^{-1}(I(\tau))}$ is injective, of a sign of
$(-1)^{|\psi^{-1}(I(\tau))|}$ times the number of permutations of symbols
$i_{1},\dotsc,i_{2\tau_{i}+1}$ for $i\in I(\tau)$,
$i_{1},\dotsc,i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ and one additional
symbol $\star$ such that all $i_{1},\dotsc,i_{2\tau_{i}+1}$ for $i\in I(\tau)$
and all $i_{1},\dotsc,i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ appear in comb-
like order and such that $i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ with
$\psi(i)\neq\star$ is immediately followed by $\psi(i)_{1}$.
Now we split the set of such permutations depending on the symbols immediately
following symbols $i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$. We notice that the
signed sum exactly kills those permutations where some $i_{2\sigma_{i}+1}$ for
$i\in I(\sigma)$ is immediately followed by some $j_{1}$ for $j\in I(\tau)$
since if such a summand appears for some $\psi$ with $\psi(i)\neq j$ we must
have $\psi(i)=\star$ and we find the same summand with opposite sign in the
sum corresponding to the map $\psi^{\prime}$ defined by $\psi^{\prime}(i)=j$
and $\psi^{\prime}(k)=\psi(k)$ for $k\neq i$ and vice versa.
###### Interpretation B2.
$\mu^{\prime\prime}_{\sigma}(\tau)$ is the number of permutations of symbols
$i_{1},\dotsc,i_{2\tau_{i}+1}$ for $i\in I(\tau)$,
$i_{1},\dotsc,i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ and one additional
symbol $\star$ such that all $i_{1},\dotsc,i_{2\tau_{i}+1}$ for $i\in I(\tau)$
and all $i_{1},\dotsc,i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ appear in comb-
like order and such that $i_{2\sigma_{i}+1}$ for $i\in I(\sigma)$ is not
immediately followed by a symbol of the form $j_{1}$ with $j\in I(\tau)$.
Interpretations Interpretation A4 and Interpretation B2 are very close. The
differences between the two of them are that the $\sigma$-type symbols are in
total order rather than comb-like order in Interpretation A4 and that the
conditions on the elements immediately following the $i_{2\sigma_{i}+1}$ are
different.
We now break $\mu^{\prime\prime}_{\sigma}(\tau)$ into a sum over set
partitions $P$ of $I(\sigma)$. Given a permutation of the symbols appearing in
Interpretation Interpretation B2, define a function
$\varphi:I(\sigma)\to\\{i_{j}\mid i\in I(\tau),j\text{
even}\\}\sqcup\\{\text{End}\\}$
recursively by
$\varphi(i)=\left\\{\begin{aligned} j_{2k}&&\text{if }i_{2\sigma_{i}+1}\text{
for }i\in I(\sigma)\text{ is immediately followed by a symbol }\\\ &&\text{ of
the form }j_{2k}\text{ or }j_{2k+1}\text{ with }j\in I(\tau),\\\
\text{End}&&\text{if }i_{2\sigma_{i}+1}\text{ for }i\in I(\sigma)\text{ is
immediately followed by }\star\\\ &&\text{ or at the end of the sequence},\\\
\varphi(j)&&\text{if }i_{2\sigma_{i}+1}\text{ for }i\in I(\sigma)\text{ is
immediately followed by a symbol }\\\ &&\text{ of the form }j_{k}\text{ with
}j\in I(\sigma).\end{aligned}\right.$
Then let $P$ be the set partition of preimages under $\varphi$. We will
identify the summand of $\mu^{\prime\prime}_{\sigma}(\tau)$ corresponding to
such a set partition $P$ as $\eta^{\prime\prime}_{\sigma^{P}}(\tau)$ times a
factor depending only on $\sigma$ and $P$.
This factor is equal to
$\prod_{i\in P}F(i),$
where
$F(i)=\frac{(2\sigma^{P}_{i}+|i|+1)!}{\prod_{j\in i}(2\sigma_{j}+1)!!}.$
Here $F(i)$ should be interpreted as the number of permutations of
$2\sigma_{j}+1$ symbols of kind $j$ for each $j\in i$ and one additional
symbol End such that the symbols of each kind appear in comb-like order. If
these permutations are interpreted as refinement data, then the permutations
counted by the $P$-summand of $\mu^{\prime\prime}_{\sigma}(\tau)$ are the
refinements by this data of the permutations counted by
$\eta^{\prime\prime}_{\sigma^{P}}(\tau)$.
Thus we have the identity
$\mu^{\prime\prime}_{\sigma}=\sum_{P\vdash I(\sigma)}\prod_{i\in
P}F(i)\eta^{\prime\prime}_{\sigma^{P}}.$
This is a triangular change of basis with nonzero entries on the diagonal, so
the $\mu^{\prime\prime}$ and $\eta^{\prime\prime}$ span the same subspace in
$(\mathbb{Q}^{P(r)})^{*}$. This completes the proof of the Rank Theorem.
## References
* [1] C. Faber. A conjectural description of the tautological ring of the moduli space of curves. In Moduli of curves and abelian varieties, Aspects Math., E33, pages 109–129. Vieweg, Braunschweig, 1999.
* [2] C. Faber and R. Pandharipande. Hodge integrals and Gromov-Witten theory. Invent. Math., 139(1):173–199, 2000.
* [3] C. Faber and R. Pandharipande. Hodge integrals, partition matrices, and the $\lambda_{g}$ conjecture. Ann. of Math. (2), 157(1):97–124, 2003.
* [4] E. Getzler and R. Pandharipande. Virasoro constraints and the Chern classes of the Hodge bundle. Nuclear Phys. B, 530(3):701–714, 1998.
* [5] T. Graber and R. Pandharipande. Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J., 51(1):93–109, 2003.
* [6] R. Pandharipande. The kappa ring of the moduli of curves of compact type. Acta Math., 208(2):335–388, 2012.
* [7] R. Pandharipande and A. Pixton. Relations in the tautological ring of the moduli space of curves. arXiv:1301.4561, Jan 2013.
|
arxiv-papers
| 2013-03-29T21:05:35 |
2024-09-04T02:49:43.615152
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Felix Janda and Aaron Pixton",
"submitter": "Felix Janda",
"url": "https://arxiv.org/abs/1304.0026"
}
|
1304.0065
|
A Perturbed Sums of Squares Theorem for Polynomial Optimization and its
Applications
Masakazu Muramatsu111Department of Communication Engineering and Informatics,
The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo,
182-8585 JAPAN. [email protected] , Hayato Waki222Institute of Mathematics
for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395,
JAPAN. [email protected] and Levent Tunçel333 Department of
Combinatorics and Optimization, Faculty of Mathematics, University of
Waterloo, Waterloo, Ontario N2L 3G1 CANADA. [email protected]
Abstract
We consider a property of positive polynomials on a compact set with a small
perturbation. When applied to a Polynomial Optimization Problem (POP), the
property implies that the optimal value of the corresponding SemiDefinite
Programming (SDP) relaxation with sufficiently large relaxation order is
bounded from below by $(f^{\ast}-\epsilon)$ and from above by
$f^{\ast}+\epsilon(n+1)$, where $f^{\ast}$ is the optimal value of the POP. We
propose new SDP relaxations for POP based on modifications of existing sums-
of-squares representation theorems. An advantage of our SDP relaxations is
that in many cases they are of considerably smaller dimension than those
originally proposed by Lasserre. We present some applications and the results
of our computational experiments.
## 1 Introduction
### 1.1 Lasserre’s SDP relaxation for POP
We consider the POP:
$\mbox{ \rm minimize }\ f(x)\ \mbox{ \rm subject to }\ f_{i}(x)\geq 0\
(i=1,\ldots,m),$ (1)
where $f$, $f_{1},\ldots,f_{m}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ are
polynomials. The feasible region is denoted by
$K=\\{\,x\in\mathbb{R}^{n}\,:\,f_{j}(x)\geq 0\ (j=1,\ldots,m)\,\\}$. Then it
is easy to see that the optimal value $f^{\ast}$ can be represented as
$f^{\ast}=\sup\left\\{\,\rho\,:\,f(x)-\rho\geq 0\ (\forall x\in
K)\,\right\\}.$
First, we briefly describe the framework of the SDP relaxation method for POP
$(\ref{eq:POP0})$ proposed by Lasserre [17]. See also [25]. We denote the set
of polynomials and sums of squares by $\mathbb{R}[x]$ and $\Sigma$,
respectively. $\mathbb{R}[x]_{r}$ is the set of polynomials whose degree is
less than or equal to $r$. We let $\Sigma_{r}=\Sigma\cap\mathbb{R}[x]_{2r}$.
We define the quadratic module generated by $f_{1},\ldots,f_{m}$ as
$M(f_{1},\ldots,f_{m})=\left\\{\,\sigma_{0}+\sum_{j=1}^{m}\sigma_{j}f_{j}\,:\,\sigma_{0},\ldots,\sigma_{m}\in\Sigma\,\right\\}.$
The truncated quadratic module whose degree is less than or equal to $2r$ is
defined by
$M_{r}(f_{1},\ldots,f_{m})=\left\\{\,\sigma_{0}+\sum_{i=1}^{m}\sigma_{j}f_{j}\,:\,\sigma_{0}\in\Sigma_{r},\sigma_{j}\in\Sigma_{r_{j}}(j=1,\ldots,m)\,\right\\},$
where $r_{j}=r-\lceil\deg f_{j}/2\rceil$ for $j=1,\ldots,m$.
Replacing the condition that $f(x)-\rho$ is nonnegative by a relaxed condition
that the polynomial is contained in $M_{r}(f_{1},\ldots,f_{m})$, we obtain the
following SOS relaxation:
$\rho_{r}=\sup\left\\{\,\rho\,:\,f(x)-\rho\in
M_{r}(f_{1},\ldots,f_{m})\,\right\\}.$ (2)
Lasserre[17] showed that $\rho_{r}\rightarrow f^{\ast}$ as
$r\rightarrow\infty$ if $M(f_{1},\ldots,f_{m})$ is archimedean. See [22, 26]
for a definition of archimedean. An easy way to ensure that
$M(f_{1},\ldots,f_{m})$ is archimedean is to make sure that
$M(f_{1},\ldots,f_{m})$ contains a representation of a ball of finite (but
possibly very large) radius. In particular, we point out that when
$M(f_{1},\ldots,f_{m})$ is archimedean, $K$ is compact.
The problem $(\ref{eq:SOS1})$ can be encoded as an SDP problem. Note that we
can express a sum of squares $\sigma\in\Sigma_{r}$ by using a positive
semidefinite matrix $X\in\mathbb{S}^{s(r)}_{+}$ as
$\sigma(x)=u_{r}(x)^{T}Xu_{r}(x)$, where $s(r)={{n+r}\choose{n}}$ and
$u_{r}(x)$ is the monomial vector which contains all the monomials in $n$
variables up to and including degree $r$ with an appropriate order. By using
this relation, the containment by $M_{r}(f_{1},\ldots,f_{m})$ constraints in
$(\ref{eq:SOS1})$, i.e.,
$f-\rho=\sigma_{0}+\sum_{j=1}^{m}\sigma_{j}f_{j},$
can be transformed to linear equations involving semidefinite matrix variables
corresponding to $\sigma_{0}$ and $\sigma_{j}$’s.
Note that, in this paper, we neither assume that $K$ is compact nor that
$M(f_{1},\ldots,f_{m})$ is archimedean. Still, the framework of Lasserre’s SDP
relaxation described above can be applied to $(\ref{eq:POP0})$, although the
good theoretical convergence property may be lost.
### 1.2 Problems in the SDP relaxation for POP
Since POP is NP-hard, solving POP in practice is sometimes extremely
difficult. The SDP relaxation method described above also has some difficulty.
A major difficulty arises from the size of the SDP relaxation problem
$(\ref{eq:SOS1})$. In fact, $(\ref{eq:SOS1})$ contains ${{n+2r}\choose{n}}$
variables and $s(r)\times s(r)$ matrix. When $n$ and/or $r$ get larger,
solving $(\ref{eq:SOS1})$ can become just impossible.
To overcome this difficulty, several techniques, using sparsity of
polynomials, are proposed. See, e.g., [15, 19, 22, 23, 29]. Based on the fact
that most of the practical POPs are sparse in some sense, these techniques
exploit special sparsity structure of POPs to reduce the number of variables
and the size of the matrix variable in the SDP $(\ref{eq:SOS1})$. Recent work
in this direction, e.g., [6, 7] also exploit special structure of POPs to
solve larger sized problems. Nie and Wang [24] proposes a use of
regularization method for solving SDP relaxation problems instead of primal-
dual interior-point methods.
Another problem with the SDP relaxation is that $(\ref{eq:SOS1})$ is often
ill-posed. In [11, 31, 33], strange behaviors of SDP solvers are reported.
Among them is that an SDP solver returns an ‘optimal’ value of
$(\ref{eq:SOS1})$ which is significantly different from the true optimal value
without reporting any numerical errors. Even more strange is that the returned
value by the SDP solver is nothing but the real optimal value of the POP
$(\ref{eq:POP0})$. We refer to this as a ‘super-accurate’ property of the SDP
relaxation for POP.
### 1.3 Contribution of this paper
POP contains very hard problems as well as some easier ones. We would like an
approach which will exploit the structure in the easier instances of POP. In
the context of current paper the notion of “easiness” will be based on sums of
squares certificate and sparsity. Based on Theorems 1, 2 and its variants, we
propose new SDP relaxations. We call it Adaptive SOS relaxation in this paper.
Adaptive SOS relaxations can be interpreted as relaxations of those originally
proposed by Lasserre. As a result, the bounds generated by our approach cannot
be superior to those generated by Lasserre’s approach for the same order
relaxations. However, Adaptive SOS relaxations are of significantly smaller
dimensions (compared to Lasserre’s SDP relaxations) and as the computational
experiments in Section 3 indicate, we obtain very significant speed-up factors
and we are able to solve larger instances and higher-order SDP relaxations.
Moreover, in most cases, the amount of loss in the quality of bounds is small,
even for the same order SDP relaxations.
The rest of this paper is organized as follows. Section 2 gives our main
results and Adaptive SOS relaxation based on Theorem 1. In Section 3, we
present the results of some numerical experiments. We give a proof of Theorem
1 and some of extensions, and the related work to Theorem 1 in Section 4.
## 2 Adaptive SOS relaxation
### 2.1 Main results
We assume that there exists an optimal solution $x^{\ast}$ of
$(\ref{eq:POP0})$. Let
$\displaystyle b$ $\displaystyle=$
$\displaystyle\max\left(1,\max\\{\,|x^{\ast}_{i}|\,:\,i=1,\ldots,n\,\\}\right)$
$\displaystyle B$ $\displaystyle=$ $\displaystyle[-b,b]^{n}.$
Obviously $x^{\ast}\in B$. We define:
$\displaystyle\bar{K}$ $\displaystyle=$ $\displaystyle B\cap K$ $\displaystyle
R_{j}$ $\displaystyle=$ $\displaystyle\max\left\\{\,|f_{j}(x)|\,:\,x\in
B\,\right\\}\ (j=1,\ldots,m).$
Define also, for a positive integer $r$,
$\displaystyle\psi_{r}(x)$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{m}f_{j}(x)\left(1-\frac{f_{j}(x)}{R_{j}}\right)^{2r},$
$\displaystyle\Theta_{r}(x)$ $\displaystyle=$ $\displaystyle
1+\sum_{i=1}^{n}x_{i}^{2r},$ $\displaystyle\Theta_{r,b}(x)$ $\displaystyle=$
$\displaystyle 1+\sum_{i=1}^{n}\left(\frac{x_{i}}{b}\right)^{2r}.$
We start with the following theorem.
###### Theorem 1
Suppose that for $\rho\in\mathbb{R}$, $f(x)-\rho>0$ for every $x\in\bar{K}$,
i.e., $\rho$ is a lower bound of $f^{\ast}$.
1. i.
Then there exists $\tilde{r}\in\mathbb{N}$ such that for all $r\geq\tilde{r}$,
$f-\rho+\psi_{r}$ is positive over $B$.
2. ii.
In addition, for every $\epsilon>0$, there exists a positive integer $\hat{r}$
such that, for every $r\geq\hat{r}$,
$f-\rho+\epsilon\Theta_{r,b}+\psi_{\tilde{r}}\in\Sigma.$
Theorem 1 will be proved in Section 4 as a corollary of Theorem 5. We remark
that $\hat{r}$ depends on $\rho$ and $\epsilon$, while $\tilde{r}$ depends on
$\rho$, but not $\epsilon$. The implication of this theorem is twofold.
First, it elucidates the super-accurate property of the SDP relaxation for
POPs. Notice that by construction, $-\psi_{\tilde{r}}(x)\in
M_{\bar{r}}(f_{1},\ldots,f_{m})$ where
$\bar{r}=\tilde{r}\max_{j}(\deg(f_{j}))$. Now assume that in
$(\ref{eq:SOS1})$, $r\geq\bar{r}$. Then, for any lower bound $\bar{\rho}$ of
$f^{\ast}$, Theorem 1 means that $f-\bar{\rho}+\epsilon\Theta_{r,b}\in
M_{r}(f_{1},\ldots,f_{m})$ for arbitrarily small $\epsilon>0$ and sufficiently
large $r$.
Let us discuss this in more details. Define $\Pi$ be the set of the
polynomials such that abosolute value of each coefficient is less than or
equal to $1$. Suppose that $\bar{\rho}$ is a “close” lower bound of $f^{\ast}$
such that the system $f-\bar{\rho}+\psi_{\tilde{r}}\in\Sigma$ is infeasible.
Let us admit an error $\epsilon$ in the above system, i.e., consider
$f-\bar{\rho}+\epsilon h+\psi_{\tilde{r}}\in\Sigma,\ h\in\Pi.$ (3)
The system $(\ref{eq:h})$ restricts the amount of the infinity norm error in
the equality condition of the SDP relaxation problem to be less than or equal
to $\epsilon$. Since we can decompose $h=h_{+}-h_{-}$ where
$h_{+},h_{-}\in\Sigma\cap\Pi$, now the system $(\ref{eq:h})$ is equivalent
with:
$f-\bar{\rho}+\epsilon h_{+}+\psi_{\tilde{r}}\in\Sigma,\
h_{+}\in\Pi\cap\Sigma.$ (4)
This observation shows that $-h_{-}$ is not the direction of errors.
Furthermore, because $\Theta_{r,b}\in\Pi\cap\Sigma$, the system
$(\ref{eq:hplus})$ is feasible due to ii of Theorem 1. Therefore, if we admit
an error $\epsilon$, the system $f-\bar{\rho}+\psi_{\tilde{r}}\in\Sigma$ is
considered to be feasible, and $\bar{\rho}$ is recognized as a lower bound for
$f^{\ast}$. As a result, we may obtain $f^{\ast}$ due to the numerical errors.
On the other hand, we point out that when we do not admit an error, but are
given a direction of error $h$ implicitly by the floating point arithmetic, it
does not necessarily satisfy the left inclusion of $(\ref{eq:h})$. However,
some numerical experiments show that this is true in most cases (e.g., [31]).
The reason is not clear.
Second, we can use the result to construct new sparse SDP relaxations for POP
$(\ref{eq:POP0})$. Our SDP relaxation is weaker than Lasserre’s, but the size
of our SDP relaxation can become smaller than Lasserre’s. As a result, for
some large-scale and middle-scale POPs, our SDP relaxation can often obtain a
lower bound, while Lasserre’s cannot.
A naive idea is that we use $(\ref{eq:POP0})$ as is. Note that
$-\psi_{\tilde{r}}(x)$ contains only monomials whose exponents are contained
in
$\bigcup_{j=1}^{m}\left(\mathcal{F}_{j}+\underbrace{\tilde{\mathcal{F}}_{j}+\cdots+\tilde{\mathcal{F}}_{j}}_{2\tilde{r}}\right),$
where $\mathcal{F}_{j}$ is the support of the polynomial $f_{j}$, i.e., the
set of exponents of monomials with nonzero coefficients in $f_{j}$, and
$\tilde{\mathcal{F}}_{j}=\mathcal{F}_{j}\cup\\{0\\}$. To state the idea more
precisely, we introduce some notation. For a finite set
$\mathcal{F}\subseteq\mathbb{N}^{n}$ and a positive integer $r$, we denote
$r\mathcal{F}=\underbrace{\mathcal{F}+\cdots+\mathcal{F}}_{r}$ and
$\Sigma(\mathcal{F})=\left\\{\,\sum_{k=1}^{q}g_{k}(x)^{2}\,:\,\mbox{supp}(g_{k})\subseteq\mathcal{F}\,\right\\},$
where $\mbox{supp}(g_{k})$ is the support of $g_{k}$. Note that
$\Sigma(\mathcal{F})$ is the set of sums of squares of polynomials whose
supports are contained in $\mathcal{F}$.
Now, fix an admissible error $\epsilon>0$ and $\tilde{r}$ as in Theorem 1, and
consider:
$\hat{\rho}(\epsilon,\tilde{r},r)=\sup\left\\{\,\rho\,:\,f-\rho+\epsilon\Theta_{r,b}-\sum_{j=1}^{m}f_{j}\sigma_{j}=\sigma_{0},\sigma_{0}\in\Sigma_{r},\sigma_{j}\in\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j})\,\right\\}$
(5)
for some $r\geq\tilde{r}$. Due to Theorem 1, $(\ref{eq:sparse1})$ has a
feasible solution for all sufficiently large $r$.
###### Theorem 2
For every $\epsilon>0$, there exist $\tilde{r},r\in\mathbb{N}$ such that
$f^{*}-\epsilon\leq\hat{\rho}(\epsilon,\tilde{r},r)\leq f^{*}+\epsilon(n+1)$.
Proof : We apply Theorem 1 to POP (1) with $\rho=f^{*}-\epsilon$. Then for any
$\epsilon>0$, there exist $\hat{r},\tilde{r}\in\mathbb{N}$ such that for every
$r\geq\hat{r}$,
$f-(f^{*}-\epsilon)+\epsilon\Theta_{r,b}+\psi_{\tilde{r}}\in\Sigma$. Choose a
positive integer $r\geq\hat{r}$ which satisfies
$r\geq\max\\{\lceil\deg(f)/2\rceil,\lceil(\tilde{r}+1/2)\deg(f_{1})\rceil,\ldots,\lceil(\tilde{r}+1/2)\deg(f_{m})\rceil\\}.$
(6)
Then there exists $\tilde{\sigma}_{0}\in\Sigma_{r}$ such that
$f-(f^{*}-\epsilon)+\epsilon\Theta_{r,b}+\psi_{\tilde{r}}=\tilde{\sigma}_{0}$,
because the degree of the polynomial in the left hand side is equal to $2r$.
We denote $\tilde{\sigma}_{j}:=\left(1-f_{j}/R_{j}\right)^{2\tilde{r}}$ for
all $j$. The triplet $(f^{*}-\epsilon,\tilde{\sigma}_{0},\tilde{\sigma}_{j})$
is feasible in (5) because
$\left(1-f_{j}/R_{j}\right)^{2\tilde{r}}\in\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j})$.
Therefore, we have $f^{*}-\epsilon\leq\hat{\rho}(\epsilon,\tilde{r},r)$.
We prove that $\hat{\rho}(\epsilon,\tilde{r},r)\leq f^{*}+\epsilon(n+1)$. We
choose $r$ as in (6) and consider the following POP:
$\tilde{f}:=\inf_{x\in\mathbb{R}^{n}}\left\\{f(x)+\epsilon\Theta_{r,b}(x):f_{1}(x)\geq
0,\ldots,f_{m}(x)\geq 0\right\\}.$ (7)
Applying Lasserre’s SDP relaxation with relaxation order $r$ to (7), we obtain
the following SOS relaxation problem:
$\hat{\rho}(\epsilon,r):=\sup\left\\{\,\rho\,:\,f-\rho+\epsilon\Theta_{r,b}=\sigma_{0}+\sum_{j=1}^{m}f_{j}\sigma_{j},\sigma_{0}\in\Sigma_{r},\sigma_{j}\in\Sigma_{r_{j}}\,\right\\},$
(8)
where $r_{j}:=r-\lceil\deg(f_{j})/2\rceil$ for $j=1,\ldots,m$. Then we have
$\hat{\rho}(\epsilon,r)\geq\hat{\rho}(\epsilon,\tilde{r},r)$ because
$\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j})\subseteq\Sigma_{r_{j}}$ for all $j$.
Indeed, it follows from (6) and the definition of $r_{j}$ that
$r_{j}\geq\tilde{r}\deg(f_{j})$, and thus
$\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j})\subseteq\Sigma_{r_{j}}$.
Every optimal solution $x^{*}$ of POP (1) is feasible for (7) and its
objective value is $f^{*}+\Theta_{r,b}(x^{*})$. We have
$f^{*}+\Theta_{r,b}(x^{*})\geq\hat{\rho}(\epsilon,r)$ because (8) is the
relaxation problem of (7). In addition, it follows from $x^{*}\in B$ that
$n+1\geq\Theta_{r,b}(x^{*})$, and thus
$\hat{\rho}(\epsilon,\tilde{r},r)\leq\hat{\rho}(\epsilon,r)\leq
f^{*}+\epsilon(n+1)$. $\Box$
Lasserre [17] proved the convergence of his SDP relaxation under the
assumption that the quadratic module $M(f_{1},\ldots,f_{m})$ associated with
POP (1) is archimedean. In contrast, Theorem 2 does not require such an
assumption and ensures that we can obtain a sufficiently close approximation
to the optimal value $f^{*}$ of POP (1) by solving (5).
We delete the perturbed part $\epsilon\Theta_{r,b}(x)$ from the above sparse
relaxation $(\ref{eq:sparse1})$ in our computations, because it may be
implicitly introduced in the computation by using floating-point arithmetic.
In the above sparse relaxation $(\ref{eq:sparse1})$, we have to consider only
those positive semidefinite matrices whose rows and columns correspond to
$\tilde{r}\tilde{\mathcal{F}}_{j}$ for $f_{j}$. In contrast, in Lasserre’s SDP
relaxation, we have to consider the whole set of monomials whose degree is
less than or equal to $r_{j}$ for each polynomial $f_{j}$. Only $\sigma_{0}$
is large; it contains the set of all monomials whose degree is less than or
equal to $r$. However, since the other polynomials do not contain most of the
monomials of $\sigma_{0}$, such monomials can safely be eliminated to reduce
the size of $\sigma_{0}$ (as in [15]). As a result, our sparse relaxation
reduces the size of the matrix significantly if each $|\mathcal{F}_{j}|$ is
small enough. We note that in many of the practical cases, this in fact is
true. We will call this new relaxation Adaptive SOS relaxation in the
following.
### 2.2 Proposed approach: Adaptive SOS relaxation
An SOS relaxation (5) for POP (1) has been introduced. However, this
relaxation has some weak points. In particular, we do not know the value
$\tilde{r}$ in advance. Also, introducing small perturbation $\epsilon$
intentionally may lead numerical difficulty in solving SDP.
To overcome these difficulties, we ignore the perturbation part
$\epsilon\Theta_{r,b}(x)$ in (5) because the perturbation part may be
implicitly introduced by floating point arithmetic. In addition, we choose a
positive integer $r$ and find $\tilde{r}$ by increasing $r$. Furthermore, we
replace $\sigma_{j}\in\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j})$ by
$\sigma_{j}\in\Sigma(\tilde{r}_{j}\tilde{\mathcal{F}}_{j})$ in (5), where
$\tilde{r}_{j}$ is defined for a given integer $r$ as
$\tilde{r}_{j}=\left\lfloor\frac{r}{\deg(f_{j})}-\frac{1}{2}\right\rfloor,$
to have $\deg(f_{j}\sigma_{j})\leq 2r$ for all $j=1,\ldots,m$. Then, we obtain
the following SOS problem:
$\rho^{*}(r):=\sup_{\rho\in\mathbb{R},\sigma_{0}\in\Sigma_{r},\sigma_{j}\in\Sigma(\tilde{r}_{j}\tilde{\mathcal{F}}_{j})}\left\\{\rho:f-\rho-\sum_{j=1}^{m}f_{j}\sigma_{j}=\sigma_{0}\right\\}.$
(9)
We call (9) Adaptive SOS relaxation for POP (1). Note that we try to use
numerical errors in a positive way; even though Adaptive SOS relaxation has a
different optimal value from that of POP, we may hope that the contaminated
computation produces the correct optimal value of POP.
In general, we have
$\Sigma(\tilde{r}_{j}\tilde{\mathcal{F}}_{j})\subseteq\Sigma_{r_{j}}$ because
of $\tilde{r}_{j}\deg(f_{j})\leq r_{j}$. Recall that
$r_{j}=r-\lceil\deg(f_{j})/2\rceil$ and is used in Lasserre’s SDP relaxation
(2). This implies that Adaptive SOS relaxation is no stronger than Lasserre’s
SDP relaxation, i.e., the optimal value $\rho^{*}(r)$ is lower than or equal
to the optimal value $\rho(r)$ of Lasserre’s SDP relaxation for POP (1) for
all $r$. We further remark that $\rho^{*}(r)$ may not converge to the optimal
value $f^{*}$ of POP (1). However, we can hope for the convergence of
$\rho^{*}(r)$ to $f^{*}$ from Theorem 1 and some numerical results in [11, 31,
33].
In the rest of this subsection, we provide a property of Adaptive SOS
relaxation for the quadratic optimization problem
$\inf_{x\in\mathbb{R}^{n}}\left\\{f(x):=x^{T}P_{0}x+c_{0}^{T}x:f_{j}(x):=x^{T}P_{j}x+c_{j}^{T}x+\gamma_{j}\geq
0\ (j=1,\ldots,m)\right\\}.$ (10)
The proposition implies that we do not need to compute $\rho^{*}(r)$ for even
$r$.
###### Proposition 3
Assume that the degree $\deg(f_{j})=2$ for all $j=1,\ldots,m$ for QOP (10).
Then, the optimal value $\rho^{*}(r)$ of Adaptive SOS relaxation is equal to
$\rho^{*}(r-1)$ if $r$ is even.
Proof : It follows from definition of $\tilde{r}_{j}$ that we have
$\tilde{r}_{j}=\left\lfloor\frac{r-1}{2}\right\rfloor=\left\\{\begin{array}[]{cl}\frac{r-1}{2}&\mbox{if
}r\mbox{ is odd},\\\ \frac{r}{2}-1&\mbox{if }r\mbox{ is
even}.\end{array}\right.$
We assume that $r$ is even and give Adaptive SOS relaxation problems with
relaxation order $r$ and $r-1$:
$\displaystyle\rho^{*}(r)$ $\displaystyle=$
$\displaystyle\sup\left\\{\rho:\begin{array}[]{l}f-\rho-\displaystyle\sum_{j=1}^{m}f_{j}\sigma_{j}=\sigma_{0},\rho\in\mathbb{R},\sigma_{0}\in\Sigma_{r},\\\
\sigma_{j}\in\Sigma\left(\displaystyle\left(\frac{r}{2}-1\right)\tilde{\mathcal{F}}_{j}\right)\end{array}\right\\},$
(13) $\displaystyle\rho^{*}(r-1)$ $\displaystyle=$
$\displaystyle\sup\left\\{\rho:\begin{array}[]{l}f-\rho-\displaystyle\sum_{j=1}^{m}f_{j}\sigma_{j}=\sigma_{0},\rho\in\mathbb{R},\sigma_{0}\in\Sigma_{r-1},\\\
\sigma_{j}\in\Sigma\left(\left(\displaystyle\frac{r}{2}-1\right)\tilde{\mathcal{F}}_{j}\right)\end{array}\right\\}.$
(16)
We have $\rho^{*}(r)\geq\rho^{*}(r-1)$ for (13) and (16). All feasible
solutions $(\rho,\sigma_{0},\sigma_{j})$ of (13) satisfy the following
identity:
$f_{0}-\rho=\sigma_{0}+\sum_{j=1}^{m}\sigma_{j}f_{j}.$
Since $r$ is even, the degrees of $\sum_{j=1}^{m}\sigma_{j}(x)f_{j}(x)$ and
$f_{0}(x)-\rho$ are less than or equal to $2r-2$ and 2 respectively, and thus,
the degree of $\sigma_{0}$ is less than or equal to $2r-2$. Indeed, we can
write $\sigma_{0}(x)=\sum_{k=1}^{\ell}\left(g_{k}(x)+h_{k}(x)\right)^{2}$,
where $\deg(g_{k})\leq r-1$ and $h_{k}$ is a homogenous polynomial with degree
$r$. Then we obtain $0=\sum_{k=1}^{\ell}h_{k}^{2}(x)$, which implies $h_{k}=0$
for all $k=1,\ldots,\ell$. Therefore, all feasible solutions
$(\rho,\sigma_{0},\sigma_{j})$ in SDP relaxation problem (13) are also
feasible in SDP relaxation problem (16), and we have
$\rho^{*}(r)=\rho^{*}(r-1)$ if $r$ is even. $\Box$
## 3 Numerical Experiments
In this section, we compare Adaptive SOS relaxation with Lasserre’s SDP
relaxation and the sparse SDP relaxation using correlative sparsity proposed
in [29]. To this end, we perform some numerical experiments. We observe from
the results of our computational experiments that (i) although Adaptive SOS
relaxation is often strictly weaker than Lasserre’s, i.e., the value obtained
by Adaptive SOS relaxation is less than Lasserre’s, the difference is small in
many cases, (ii) Adaptive SOS relaxation solves at least 10 times faster than
Lasserre’s in middle to large scale problems. Therefore, we conclude that
Adaptive SOS relaxation can be more effective than Lasserre’s for large- and
middle-scale POPs. We will also observe a similar relationship against the
sparse relaxation in [29]; Adaptive SOS relaxation is weaker but much faster
than the sparse one.
We use a computer with Intel (R) Xeon (R) 2.40 GHz cpus and 24GB memory, and
MATLAB R2010a. To construct Lasserre’s [17], sparse [29] and Adaptive SOS
problems, we use SparsePOP 2.99 [30]. To solve the resulting SDP relaxation
problems, we use SeDuMi 1.3 [27] and SDPT3 4.0 [28] with the default
parameters. The default tolerances for stopping criterion of SeDuMi and SDPT3
are 1.0e-9 and 1.0e-8, respectively.
To determine whether the optimal value of an SDP relaxation problem is the
exact optimal value of a given POP or not, we use the following two criteria
$\epsilon_{\mbox{obj}}$ and $\epsilon_{\mbox{feas}}$: Let $\hat{x}$ be a
candidate of an optimal solution of the POP obtained from the SDP relaxations.
We apply a projection of the dual solution of the SDP relaxation problem onto
$\mathbb{R}^{n}$ for obtaining $\hat{x}$ in this section. See [29] for the
details. We define:
$\displaystyle\epsilon_{\mbox{obj}}$ $\displaystyle:=$
$\displaystyle\frac{|\mbox{the optimal value of the SDP
relaxation}-f(\hat{x})|}{\max\\{1,|f(\hat{x})|\\}},$
$\displaystyle\epsilon_{\mbox{feas}}$ $\displaystyle:=$
$\displaystyle\min_{k=1,\ldots,m}\\{f_{k}(\hat{x})\\}.$
If $\epsilon_{\mbox{feas}}\geq 0$, then $\hat{x}$ is feasible for the POP. In
addition, if $\epsilon_{\mbox{obj}}=0$, then $\hat{x}$ is an optimal solution
of the POP and $f(\hat{x})$ is the optimal value of the POP.
We introduce the following value to indicate the closeness between the
obtained values of Lasserre’s, sparse and Adaptive SOS relaxations.
$\mbox{Ratio}:=\frac{(\mbox{obj. val. of Lasserre's or sparse SDP relax.
})}{(\mbox{obj. val. of Adaptive SOS
relax.})}=\frac{\rho^{*}_{r}}{\rho^{*}(r)}.$ (17)
If the signs of both optimal values are the same and Ratio is sufficiently
close to 1, then the optimal value of Adaptive SOS relaxation is close to the
optimal value of Lasserre’s and sparse SDP relaxations. In general, this value
is meaningless for measuring the closeness if those signs are different or
either of values is zero. Fortunately, those values are not zero and those
signs are the same in all numerical experiments in this section.
To reduce the size of the resulting SDP relaxation problems, SparsePOP has
functions based on the methods proposed in [15, 34]. These methods are closely
related to a facial reduction algorithm proposed by Borwein and Wolkowicz [1,
2], and thus we can expect the numerical stability of the primal-dual
interior-point methods for the SDP relaxations may be improved. In this
section, except for Subsection 3.1, we apply the method proposed in [34].
For POPs which have lower and upper bounds on variables, we can strengthen the
SDP relaxations by adding valid inequalities based on these bound constraints.
In this section, we add them as in [29]. See Subsection 5.5 in [29] for the
details.
Table 1 shows the notation used in the description of numerical experiments in
the following subsections.
Table 1: Notation iter. | the number of iterations in SeDuMi and SDPT3
---|---
rowA, colA | the size of coefficient matrix $A$ in the SeDuMi input format
nnzA | the number of nonzero elements in coefficient matrix $A$ in the SeDuMi input format
SDPobj | the objective value obtained by SeDuMi for the resulting SDP relaxation problem
POPobj | the value of $f$ at a solution $\hat{x}$ retrieved by SparsePOP
#solved | the number of the POPs which are solved by SDP relaxation in 30 problems. If both $\epsilon_{\mbox{obj}}$ and $\epsilon_{\mbox{feas}}$ are smaller than 1.0e-7, we regard that the SDP relaxation attains the optimal value of the POP.
minRatio | minimum value of Ratio defined in (17) in 30 problems
aveRatio | average of Ratio defined in (17) in 30 problems
maxRatio | maximum value of Ratio defined in (17) in 30 problems
$\sec$ | cpu time consumed by SeDuMi or SDPT3 in seconds
min.t | minimum cpu time consumed by SeDuMi or SDPT3 in seconds among 30 resulting SDP relaxations
ave.t | average cpu time consumed by SeDuMi or SDPT3 in seconds among 30 resulting SDP relaxations
max.t | maximum cpu time consumed by SeDuMi or SDPT3 in seconds among 30 resulting SDP relaxations
### 3.1 Numerical results for POP whose quadratic module is non-archimedean
In this subsection, we give the following POP and apply Adaptive SOS
relaxation:
$\inf_{x,y\in\mathbb{R}}\left\\{-x-y:\begin{array}[]{l}f_{1}(x,y):=x-0.5\geq
0,\\\ f_{2}(x,y):=y-0.5\geq 0,\\\ f_{3}(x,y):=0.5-xy\geq
0\end{array}\right\\}.$ (18)
The optimal value is $-1.5$ and the solutions are $(0.5,1)$ and $(1,0.5)$. It
was proved in [26, 33] that the quadratic module associated with POP (18) is
non-archimedean and that all the resulting SDP relaxation problems are weakly
infeasible. However, the convergence of computed values of Lasserre’s SDP
relaxation for POP (18) was observed in [33].
In [33], it was shown that Lasserre’s SDP relaxation $(\ref{eq:SOS1})$ for
$(\ref{prestel-delzell})$ is weakly infeasible. Since Adaptive SOS relaxation
for $(\ref{prestel-delzell})$ has less monomials for representing
$\sigma_{j}$’s than that of Lasserre’s, the resulting SDP relaxation problems
are necessarily infeasible.
However, we expect from Thorem 2 that Adaptive SOS relaxation attains the
optimal value $-1.5$. Table 2 provides numerical results for Adaptive SOS
relaxation based on $(\ref{sosProb})$. In fact, we observe from Table 2 that
$\rho^{\ast}(r)$ obtained by SeDuMi is equal to $-1.5$ at $r=7,8,9,10$. By
SDPT3, we observe similar results.
Table 2: The approximate optimal value, cpu time, the number of iterations by SeDuMi and SDPT3 $r$ | Software | iter. | SDPobj | [$\sec$]
---|---|---|---|---
1 | SeDuMi | 46 | -5.9100801e+07 | 0.31
| SDPT3 | 37 | -1.8924840e+06 | 0.57
2 | SeDuMi | 38 | -6.8951407e+02 | 0.29
| SDPT3 | 72 | -1.1676106e+04 | 1.28
3 | SeDuMi | 32 | -4.2408507e+01 | 0.22
| SDPT3 | 77 | -2.0928888e+00 | 1.43
4 | SeDuMi | 35 | -1.2522887e+01 | 0.30
| SDPT3 | 76 | -1.8195861e+00 | 1.74
5 | SeDuMi | 32 | -3.5032311e+00 | 0.39
| SDPT3 | 86 | -1.6015287e+00 | 2.65
6 | SeDuMi | 33 | -1.8717460e+00 | 0.48
| SDPT3 | 86 | -1.5025613e+00 | 3.43
7 | SeDuMi | 17 | -1.5000064e+00 | 0.47
| SDPT3 | 21 | -1.5000022e+00 | 1.18
8 | SeDuMi | 16 | -1.5000030e+00 | 0.58
| SDPT3 | 25 | -1.5000001e+00 | 2.03
9 | SeDuMi | 15 | -1.5000023e+00 | 0.75
| SDPT3 | 21 | -1.4999912e+00 | 1.95
10 | SeDuMi | 15 | -1.5000015e+00 | 0.99
| SDPT3 | 17 | -1.5003641e+00 | 1.89
### 3.2 The difference between Lasserre’s and Adaptive SOS relaxations
In this subsection, we show a POP where Adaptive SOS relaxation converges to
the optimal value strictly slower than Lasserre’s, practically. This POP is
available at [8], whose name is “st_e08.gms”.
$\inf_{x,y\in\mathbb{R}}\left\\{2x+y:\begin{array}[]{ll}f_{1}(x,y):=xy-1/16\geq
0,&f_{2}(x,y):=x^{2}+y^{2}-1/4\geq 0,\\\ f_{3}(x,y):=x\geq
0,&f_{4}(x,y):=1-x\geq 0,\\\ f_{5}(x,y):=y\geq 0,&f_{6}(x,y):=1-y\geq
0.\end{array}\right\\}.$ (19)
The optimal value is $(3\sqrt{6}-\sqrt{2})/8\approx 0.741781958247055$ and
solution is $(x^{*},y^{*})=((\sqrt{6}-\sqrt{2})/8,(\sqrt{6}+\sqrt{2})/8)$.
Table 3: Numerical results on SDP relaxation problems in Subsection 3.2 by SeDuMi and SDPT3 | Lasserre | Adaptive SOS
---|---|---
$r$ | Software | (SDPobj, POPobj$|$ $\epsilon_{\mbox{obj}},\epsilon_{\mbox{feas}}$ $|$ [$\sec$]) | (SDPobj, POPobj$|$ $\epsilon_{\mbox{obj}},\epsilon_{\mbox{feas}}$ $|$ [$\sec$])
1 | SeDuMi | (0.00000e+00, 0.00000e+00$|$ 0.0e+00, -1.0e+00$|$ 0.02) | (0.00000e+00, 0.00000e+00$|$ 0.0e+00, -1.0e+00$|$ 0.02 )
| SDPT3 | (-1.16657e-09, 5.89142e-10$|$ 1.8e-09, -1.0e+00$|$ 0.14) | (-1.16657e-09, 5.89142e-10$|$ 1.8e-09, -1.0e+00$|$ 0.06)
2 | SeDuMi | (3.12500e-01, 3.12500e-01$|$ -9.5e-10, -8.4e-01$|$ 0.09) | (2.69356e-01, 2.69356e-01$|$ -1.7e-10, -9.3e-01$|$ 0.09)
| SDPT3 | (3.12500e-01, 3.12500e-01$|$ 2.0e-09 , -8.4e-01$|$ 0.22) | (2.69356e-01, 2.69356e-01$|$ 1.1e-09, -9.3e-01$|$ 0.21)
3 | SeDuMi | (7.41782e-01, 7.41782e-01$|$ -2.0e-11, -1.1e-09$|$ 0.15) | (3.06312e-01, 3.06312e-01$|$ -1.1e-09, -8.3e-01$|$ 0.13)
| SDPT3 | (7.41782e-01, 7.41782e-01$|$ 2.0e-08, 0.0e+00$|$ 0.26) | (3.06312e-01, 3.06312e-01$|$ 4.6e-09, -8.3e-01$|$ 0.25)
4 | SeDuMi | (7.41782e-01, 7.41782e-01$|$ 1.1e-10, -1.5e-09$|$ 0.15) | (7.29855e-01, 7.29855e-01$|$ -1.2e-07, -4.9e-02$|$ 0.24)
| SDPT3 | (7.41782e-01, 7.41782e-01$|$ 2.8e-09, 0.0e+00$|$ 0.34) | (7.29855e-01, 7.29855e-01$|$ 2.5e-08, -4.9e-02$|$ 0.36)
5 | SeDuMi | (7.41782e-01, 7.41782e-01$|$ 8.3e-11, -4.5e-10$|$ 0.19) | (7.36195e-01, 7.36194e-01$|$ -9.5e-07, -4.2e-02$|$ 0.33)
| SDPT3 | (7.41782e-01, 7.41782e-01$|$ -6.3e-10, 0.0e+00$|$ 0.72) | (7.36195e-01, 7.36195e-01$|$ 5.3e-08, -4.2e-02$|$ 0.50)
6 | SeDuMi | (7.41782e-01, 7.41782e-01$|$ 2.3e-11, -6.1e-11$|$ 0.27) | (7.41782e-01, 7.41782e-01$|$ -1.0e-09, -6.6e-09$|$ 0.20)
| SDPT3 | (7.41782e-01, 7.41782e-01$|$ 3.4e-10, 0.0e+00$|$ 1.02) | (7.41782e-01, 7.41782e-01$|$ -4.7e-11, 0.0e+00$|$ 0.98)
Table 3 show the numerical results of SDP relaxations for POP (19) by SeDuMi
and SDPT3. We observe that Lasserre’s SDP relaxation attains the optimal value
of (19) by relaxation order $r=3$, while Adaptive SOS relaxation attains it
only at the relaxation order by $r=6$.
### 3.3 Numerical results for detecting the copositivity
The symmetric matrix $A$ is said to be copositive if $x^{T}Ax\geq 0$ for all
$x\in\mathbb{R}^{n}_{+}$. We can formulate the problem for detecting whether a
given matrix is copositive, as follows:
$\inf_{x\in\mathbb{R}^{n}}\left\\{x^{T}Ax:f_{i}(x):=x_{i}\geq 0\
(i=1,\ldots,n),f_{n+1}(x):=1-\sum_{i=1}^{n}x_{i}=0,\right\\}.$ (20)
If the optimal value of this problem is nonnegative, then $A$ is copositive.
Table 4: Information on SDP relaxations problems in Subsection 3.3 by SeDuMi and SDPT3 | Lasserre | Adaptive SOS |
---|---|---|---
$n$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
5 | SeDuMi | (30 $|$ 0.14 0.18 0.50) | (30 $|$ 0.12 0.16 0.20) | (1.0, 1.0, 1.0)
| SDPT3 | (30 $|$ 0.40 0.44 0.85) | (30 $|$ 0.34 0.42 0.53) | (1.0, 1.0, 1.0)
10 | SeDuMi | (29 $|$ 0.36 0.42 0.50) | (29 $|$ 0.23 0.31 0.42) | (1.0, 1.0, 1.0)
| SDPT3 | (29 $|$ 0.73 1.00 1.48) | (30 $|$ 0.66 0.88 1.23) | (1.0, 1.0, 1.0)
15 | SeDuMi | (30 $|$ 1.59 1.99 2.52) | (30 $|$ 0.75 0.99 1.31) | (1.0, 1.0, 1.0)
| SDPT3 | (29 $|$ 2.91 3.40 4.73) | (23 $|$ 1.58 2.04 2.80) | (1.0, 1.0, 1.0)
20 | SeDuMi | (30 $|$ 10.22 14.06 19.98) | (30 $|$ 4.47 6.02 7.72) | (1.0, 1.0, 1.0)
| SDPT3 | (26 $|$ 11.40 16.23 19.73) | (1 $|$ 6.65 8.64 11.32) | (1.0, 1.0, 1.0)
25 | SeDuMi | (29 $|$ 215.94 263.88 336.96) | (29 $|$ 49.69 66.63 84.07) | (1.0, 1.0, 1.0)
| SDPT3 | (20 $|$ 51.53 64.31 77.35) | (4 $|$ 26.91 36.06 44.74) | (1.0, 1.0, 1.0)
30 | SeDuMi | (27 $|$ 1970.59 2322.30 2930.30) | (28 $|$ 1031.91 1198.05 1527.01) | (1.0, 1.0, 1.0)
| SDPT3 | (0 $|$ 136.59 401.23 1184.76) | (0 $|$ 92.96 165.22 295.23) | (0.4, 1.0, 1.6)
In this experiment, we solve 30 problems generated randomly. In particular,
the coefficients of all diagonal of $A$ are set to be $\sqrt{n}/2$ and the
other coefficients are chosen from [-1, 1] uniformly. In addition, since the
positive semidefiniteness implies the copositivity, we chose the matrices $A$
which are not positive semidefinite.
We apply Lasserre’s and Adaptive SOS relaxations with relaxation order $r=2$.
Table 4 shows the numerical results by SeDuMi and SDPT3 for (20),
respectively. We observe the following.
* •
SDPT3 fails to solve almost all problems (20), while SeDuMi solves them for
$n=20,25,30$. In particular, Adaptive SOS relaxations return the optimal
values of the original problems although it is no stronger than Lasserre’s
theoretically.
* •
SeDuMi solves Adaptive SOS relaxation problems faster than Lasserre’s because
the sizes of Adaptive SOS relaxation problems are smaller than those of
Lasserre’s.
* •
SDPT3 cannot solve any problems with $n=30$ by Lasserre’s and Adaptive SOS
relaxation although it terminates faster than SeDuMi. In particular, for
almost all SDP relaxation problems, SDPT3 returns the message “stop: progress
is bad” or “stop: progress is slow” and terminates. This means that it is
difficult for SDPT3 to solve those SDP relaxation problems numerically.
### 3.4 Numerical results for BoxQP
In this subsection, we solve BoxQP:
$\inf_{x\in\mathbb{R}^{n}}\left\\{x^{T}Qx+c^{T}x:0\leq x_{i}\leq 1\
(i=1,\ldots,n)\right\\},$ (21)
where each element in $Q\in\mathbb{S}^{n}$ and $c\in\mathbb{R}^{n}$ is chosen
from [-50, 50] uniformly. In particular, we vary the number $n$ of the
variables in (21) and the density of $Q,c$. In this subsection, we compare
Adaptive SOS relaxation based on Theorem 5 with sparse SDP relaxation [29]
instead of Lasserre’s. Indeed, when the density of $Q$ is small, the BoxQP has
sparse structure, and thus sparse SDP relaxation is more effective than
Lasserre’s.
Table 5: Information on SDP relaxation problems in Subsection 3.4 with density 0.2 by SeDuMi and SDPT3 | Sparse | Adaptive SOS |
---|---|---|---
$n$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
5 | SeDuMi | (23 $|$0.15, 0.24, 0.48) | (23 $|$ 0.14, 0.23, 0.52) | (0.00072, 12.34638, 342.39518)
| SDPT3 | (23 $|$0.20, 0.37, 2.48) | (22 $|$ 0.19, 0.26, 0.34) | (0.00072, 0.97463, 1.24265)
10 | SeDuMi | (13 $|$0.33, 0.55, 0.70) | (12 $|$ 0.28, 0.40, 0.53) | (0.97227, 0.99609, 1.00000)
| SDPT3 | (12 $|$0.28, 0.51, 0.62) | (12 $|$ 0.22, 0.29, 0.39) | (0.97227, 0.99609, 1.00000)
15 | SeDuMi | (14 $|$0.57, 0.95, 1.68) | ( 3 $|$ 0.42, 0.63, 0.85) | (0.96590, 0.99172, 1.00000)
| SDPT3 | (14 $|$0.54, 0.93, 1.22) | ( 3 $|$ 0.43, 0.57, 0.76) | (0.96590, 0.99172, 1.00000)
20 | SeDuMi | (11 $|$1.40, 2.57, 5.32) | ( 0 $|$ 0.80, 0.97, 1.27) | (0.94812, 0.98422, 0.99978)
| SDPT3 | (10 $|$1.41, 2.31, 3.55) | ( 0 $|$ 0.55, 0.69, 1.01) | (0.94812, 0.98422, 0.99978)
25 | SeDuMi | ( 7 $|$2.57, 5.15, 10.03) | ( 0 $|$ 0.95, 1.09, 1.42) | (0.94333, 0.97591, 0.99923)
| SDPT3 | ( 6 $|$4.60, 7.24, 12.46) | ( 0 $|$ 0.59, 0.85, 1.31) | (0.94333, 0.97591, 0.99923)
30 | SeDuMi | (12 $|$3.43, 15.60, 26.86) | ( 0 $|$ 1.27, 1.51, 2.02) | (0.93773, 0.97542, 0.99843)
| SDPT3 | (10 $|$8.02, 22.87, 38.42) | ( 0 $|$ 0.94, 1.33, 1.67) | (0.93773, 0.97542, 0.99843)
35 | SeDuMi | (12 $|$26.57, 67.79, 143.06) | ( 0 $|$ 1.77, 2.15, 3.33) | (0.93271, 0.97236, 0.99648)
| SDPT3 | ( 9 $|$44.14, 80.48, 135.30) | ( 0 $|$ 1.06, 1.83, 2.63) | (0.93271, 0.97236, 0.99648)
40 | SeDuMi | Not solved | ( 0 $|$ 2.47, 2.89, 3.57) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$ 2.13, 3.13, 3.87) | (–, –, –)
45 | SeDuMi | Not solved | ( 0 $|$ 3.58, 4.17, 5.51) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$ 4.12, 5.09, 6.35) | (–, –, –)
50 | SeDuMi | Not solved | ( 0 $|$ 5.30, 7.02, 9.48) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$ 5.19, 6.83, 8.34) | (–, –, –)
55 | SeDuMi | Not solved | ( 0 $|$ 8.75, 10.43, 12.23) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$ 8.31, 10.77, 13.60) | (–, –, –)
60 | SeDuMi | Not solved | ( 0 $|$ 12.21, 15.16, 19.59) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$ 12.62, 16.57, 22.44) | (–, –, –)
Table 6: Information on SDP relaxation problems in Subsection 3.4 with density 0.4 by SeDuMi and SDPT3 | Sparse | Adaptive SOS |
---|---|---|---
$n$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
5 | SeDuMi | (24 $|$0.14, 0.19, 0.28) | (22 $|$ 0.13, 0.17, 0.27) | (0.98678, 0.99849, 1.00000)
| SDPT3 | (23 $|$0.23, 0.27, 0.35) | (22 $|$ 0.17, 0.23, 0.34) | (0.98678, 0.99849, 1.00000 )
10 | SeDuMi | (19 $|$0.28, 0.49, 0.77) | ( 9 $|$ 0.25, 0.35, 0.46) | (0.95400, 0.98958, 1.00000)
| SDPT3 | (18 $|$0.32, 0.53, 0.86) | ( 7 $|$ 0.26, 0.33, 0.52) | (0.95400, 0.98958, 1.00000)
15 | SeDuMi | (13 $|$0.76, 1.21, 2.50) | ( 3 $|$ 0.46, 0.56, 0.65) | (0.95219, 0.98580, 1.00000)
| SDPT3 | (13 $|$0.84, 1.32, 2.26) | ( 3 $|$ 0.37, 0.54, 0.81) | (0.95219, 0.98580, 1.00000)
20 | SeDuMi | (11 $|$2.10, 3.51, 5.45) | ( 0 $|$ 0.70, 0.79, 0.97) | (0.94457, 0.97953, 0.99933)
| SDPT3 | (11 $|$3.22, 5.61, 8.30) | ( 0 $|$ 0.50, 0.73, 1.01) | (0.94457, 0.97953, 0.99933)
25 | SeDuMi | (11 $|$6.65, 13.88, 24.32) | ( 0 $|$ 1.02, 1.13, 1.28) | (0.92917, 0.96999, 0.99596)
| SDPT3 | (10 $|$11.48, 21.00, 30.98) | ( 0 $|$ 0.69, 1.03, 1.47) | (0.92917, 0.96999, 0.99596)
30 | SeDuMi | (14 $|$27.25, 60.67, 108.22) | ( 0 $|$ 1.31, 1.62, 2.26) | (0.92761, 0.97283, 0.99608)
| SDPT3 | (12 $|$43.33, 66.25, 95.80) | ( 0 $|$ 1.29, 1.71, 2.22) | (0.92761, 0.97283, 0.99608)
35 | SeDuMi | ( 8 $|$76.07, 328.08, 589.43) | ( 0 $|$ 2.11, 2.42, 2.95) | (0.93669, 0.96707, 0.99717)
| SDPT3 | ( 6 $|$116.23, 218.61, 322.82) | ( 0 $|$ 2.21, 2.87, 5.03) | (0.93669, 0.96707, 0.99717)
40 | SeDuMi | Not solved | ( 0 $|$3.11, 3.54, 4.69) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$3.29, 4.50, 5.39) | (–, –, –)
45 | SeDuMi | Not solved | ( 0 $|$4.99, 5.79, 7.10) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$5.43, 6.89, 8.85) | (–, –, –)
50 | SeDuMi | Not solved | ( 0 $|$7.09, 8.47, 11.58) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$9.09, 11.30, 15.02) | (–, –, –)
55 | SeDuMi | Not solved | ( 0 $|$11.84, 14.34, 17.72) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$14.09, 18.30, 22.13) | (–, –, –)
60 | SeDuMi | Not solved | ( 0 $|$19.33, 24.23, 29.13) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$19.45, 22.96, 26.65) | (–, –, –)
Table 7: Information on SDP relaxation problems in Subsection 3.4 with density 0.6 by SeDuMi and SDPT3 | Sparse | Adaptive SOS |
---|---|---|---
$n$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
5 | SeDuMi | (27 $|$0.13, 0.22, 0.54) | (25 $|$ 0.12, 0.17, 0.38) | (0.93673, 0.99543, 1.00000)
| SDPT3 | (26 $|$0.21, 0.26, 0.33) | (25 $|$ 0.18, 0.21, 0.29) | (0.93673, 0.99543, 1.00000)
10 | SeDuMi | (19 $|$0.36, 0.68, 1.26) | ( 6 $|$ 0.33, 0.48, 0.79) | (0.94709, 0.98678, 1.00000)
| SDPT3 | (18 $|$0.37, 0.48, 0.72) | ( 6 $|$ 0.25, 0.31, 0.40) | (0.94709, 0.98678, 1.00000)
15 | SeDuMi | (14 $|$0.71, 1.52, 3.70) | ( 6 $|$ 0.42, 0.61, 1.01) | (0.95463, 0.98581, 1.00000)
| SDPT3 | (14 $|$0.77, 1.33, 2.04) | ( 6 $|$ 0.34, 0.41, 0.51) | (0.95463, 0.98581, 1.00000)
20 | SeDuMi | (13 $|$1.92, 5.18, 7.99) | ( 2 $|$ 0.72, 0.91, 1.56) | (0.92378, 0.97521, 1.00000)
| SDPT3 | (11 $|$2.25, 5.54, 8.21) | ( 2 $|$ 0.52, 0.61, 0.75) | (0.92378, 0.97521, 1.00000)
25 | SeDuMi | (15 $|$9.56, 29.31, 57.08) | ( 0 $|$ 1.03, 1.24, 1.94) | (0.92768, 0.96827, 0.99715)
| SDPT3 | (12 $|$15.55, 26.06, 40.61) | ( 0 $|$ 0.75, 0.93, 1.19) | (0.92768, 0.96827, 0.99715)
30 | SeDuMi | (11 $|$50.72, 168.53, 368.04) | ( 0 $|$ 1.56, 1.97, 2.99) | (0.93048, 0.96888, 0.99470)
| SDPT3 | ( 9 $|$42.25, 90.31, 140.94) | ( 0 $|$ 1.27, 1.50, 2.10) | (0.93048, 0.96888, 0.99470)
35 | SeDuMi | (12 $|$510.67, 964.20, 1489.56) | ( 0 $|$ 2.52, 3.11, 4.27) | (0.90892, 0.95875, 0.99301)
| SDPT3 | (11 $|$217.87, 303.90, 366.57) | ( 0 $|$ 2.16, 2.55, 3.09) | (0.90892, 0.95875, 0.99301)
40 | SeDuMi | Not solved | ( 0 $|$3.77, 4.34, 5.77) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$3.37, 4.24, 5.12) | (–, –, –)
45 | SeDuMi | Not solved | ( 0 $|$6.08, 6.91, 8.33) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$5.63, 7.07, 9.33) | (–, –, –)
50 | SeDuMi | Not solved | ( 0 $|$8.97, 10.66, 12.82) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$8.87, 10.59, 11.84) | (–, –, –)
55 | SeDuMi | Not solved | ( 0 $|$13.95, 17.13, 20.71) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$10.26, 13.64, 20.92) | (–, –, –)
60 | SeDuMi | Not solved | ( 0 $|$21.94, 25.42, 30.36) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$15.48, 19.66, 27.26) | (–, –, –)
Table 8: Information on SDP relaxation problems in Subsection 3.4 with density 0.8 by SeDuMi and SDPT3 | Sparse | Adaptive SOS |
---|---|---|---
$n$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
5 | SeDuMi | (25 $|$0.15, 0.19, 0.34) | (22 $|$ 0.13, 0.17, 0.24) | (0.94896, 0.99548, 1.00000)
| SDPT3 | (25 $|$0.22, 0.27, 0.37) | (22 $|$ 0.18, 0.22, 0.29) | (0.94896, 0.99548, 1.00000)
10 | SeDuMi | (20 $|$0.36, 0.54, 0.80) | (11 $|$ 0.26, 0.37, 0.52) | (0.96388, 0.99365, 1.00000)
| SDPT3 | (20 $|$0.40, 0.62, 0.99) | (10 $|$ 0.29, 0.39, 0.59) | (0.96388, 0.99365, 1.00000)
15 | SeDuMi | (14 $|$0.93, 1.67, 2.93) | ( 1 $|$ 0.50, 0.59, 0.71) | (0.94514, 0.98537, 1.00000)
| SDPT3 | (12 $|$1.29, 1.85, 2.63) | ( 1 $|$ 0.42, 0.51, 0.71) | (0.94514, 0.98537, 1.00000)
20 | SeDuMi | (14 $|$2.51, 5.22, 8.98) | ( 2 $|$ 0.66, 0.85, 1.15) | (0.95261, 0.98061, 1.00000)
| SDPT3 | (12 $|$4.50, 6.70, 9.35) | ( 2 $|$ 0.56, 0.76, 1.13) | (0.95261, 0.98061, 1.00000)
25 | SeDuMi | (10 $|$10.64, 23.57, 56.02) | ( 0 $|$ 1.13, 1.25, 1.52) | (0.95060, 0.97500, 0.99997)
| SDPT3 | (10 $|$14.13, 26.81, 44.75) | ( 0 $|$ 0.87, 1.11, 1.66) | (0.95060, 0.97500, 0.99997)
30 | SeDuMi | (11 $|$42.70, 156.60, 507.20) | ( 0 $|$ 1.68, 1.89, 2.18) | (0.94199, 0.96738, 0.99484)
| SDPT3 | ( 9 $|$53.52, 104.12, 173.49) | ( 0 $|$ 1.43, 1.88, 2.49) | (0.94199, 0.96738, 0.99484 )
35 | SeDuMi | (15 $|$185.51, 1000.24, 2158.08) | ( 0 $|$ 2.66, 2.89, 3.15) | (0.92313, 0.96254, 0.99485)
| SDPT3 | (12 $|$157.31, 337.69, 508.43) | ( 0 $|$ 2.52, 2.99, 3.60) | (0.92313, 0.96258, 0.99485)
40 | SeDuMi | Not solved | ( 0 $|$4.45, 4.89, 6.34) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$4.11, 5.22, 6.66) | (–, –, –)
45 | SeDuMi | Not solved | ( 0 $|$6.52, 7.63, 8.86) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$7.00, 8.05, 9.51) | (–, –, –)
50 | SeDuMi | Not solved | ( 0 $|$10.45, 11.70, 13.89) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$10.57, 12.65, 15.41) | (–, –, –)
55 | SeDuMi | Not solved | ( 0 $|$15.96, 19.55, 24.40) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$11.84, 16.07, 21.26) | (–, –, –)
60 | SeDuMi | Not solved | ( 0 $|$26.31, 32.04, 36.89) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$17.69, 22.33, 27.93) | (–, –, –)
70 | SeDuMi | Not solved | ( 0 $|$69.62, 91.01, 123.14) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$26.30, 34.00, 45.75) | (–, –, –)
80 | SeDuMi | Not solved | ( 0 $|$182.40, 218.82, 268.42) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$46.87, 52.48, 59.51) | (–, –, –)
90 | SeDuMi | Not solved | ( 0 $|$406.85, 478.44, 619.49) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$77.36, 91.34, 107.29) | (–, –, –)
100 | SeDuMi | Not solved | ( 0 $|$844.15, 943.74, 1138.27) | (–, –, –)
| SDPT3 | Not solved | ( 0 $|$130.50, 148.36, 172.25) | (–, –, –)
We observe the following from Table 5.
* •
Sparse SDP relaxation obtains the optimal solution for some BoxQPs, while
Adaptive SOS relaxation cannot.
* •
Adaptive SOS relaxation solves the resulting SDP problems approximately 10
$\sim$ 30 times faster than Lasserre’s.
* •
The values obtained by Adaptive SOS relaxation are within 10% of Sparse SDP
relaxation, except for $n=5$.
### 3.5 Numerical results for Bilinear matrix inequality eigenvalue problems
In this subsection, we solve the binary matrix inequality eigenvalue problems.
$\inf_{s\in\mathbb{R},x\in\mathbb{R}^{n},y\in\mathbb{R}^{m}}\left\\{\,s\,:\,sI_{k}-B_{k}(x,y)\in\mathbb{S}_{+}^{k},x\in[0,1]^{n},y\in[0,1]^{m}\,\right\\},$
(22)
where we define for $k\in\mathbb{N}$, $x\in\mathbb{R}^{n}$ and
$y\in\mathbb{R}^{m}$:
$B_{k}(x,y)=\sum_{i=1}^{n}\sum_{j=1}^{m}B_{ij}x_{i}y_{j}+\sum_{i=1}^{n}B_{i0}x_{i}+\sum_{j=1}^{m}B_{0j}y_{j}+B_{00},$
where $B_{ij}(i=0,\ldots,n,j=0,\ldots,m)$ are $k\times k$ symmetric matrices.
In this numerical experiment, each element of $B_{ij}$ is chosen from $[-1,1]$
uniformly. (22) is the problem of minimizing the maximum eigenvalue of
$B_{k}(x,y)$ keeping $B_{k}(x,y)$ positive semidefinite.
We apply Lasserre and Adaptive SOS relaxations with relaxation order $r=3$.
Tables 9 shows the numerical results for BMIEP (22) with $k=5,10$ by SeDuMi
and SDPT3, respectively.
Table 9: Information on SDP relaxation problems in Subsection 3.5 by SeDuMi and SDPT3 | Lasserre | Adaptive SOS |
---|---|---|---
$(n,m,k)$ | Software | (#solved $|$ min.t, ave.t, max.t) | (#solved $|$ min.t, ave.t, max.t) | (minR, aveR, maxR)
(1, 1, 5) | SeDuMi | (21 $|$ 0.11, 0.16, 0.25) | (16 $|$ 0.10, 0.17, 0.29) | (1.00000, 1.00103, 1.02352)
| SDPT3 | (21 $|$ 0.28, 0.36, 0.42) | (16 $|$ 0.26, 0.32, 0.41) | (1.00000, 1.00103, 1.02352)
(1, 1, 10) | SeDuMi | (20 $|$ 0.12, 0.16, 0.21) | (18 $|$ 0.11, 0.16, 0.30) | (1.00000, 1.00018, 1.00450)
| SDPT3 | (20 $|$ 0.32, 0.37, 0.47) | (18 $|$ 0.26, 0.33, 0.44) | (1.00000, 1.00018, 1.00450)
(3, 3, 5) | SeDuMi | (3 $|$ 1.95, 3.49, 4.74) | (1 $|$ 0.43, 0.63, 0.81) | (0.878394, 1.01520, 1.20254)
| SDPT3 | (3 $|$ 4.81, 7.12, 8.77) | (1 $|$ 0.67, 0.97, 1.14) | (0.878394, 1.01520, 1.20254 )
(3, 3, 10) | SeDuMi | (0 $|$ 2.46, 3.89, 4.77) | (0 $|$ 0.54, 0.69, 0.93) | (1.00000, 1.00407, 1.01243)
| SDPT3 | (0 $|$ 5.51, 7.63, 8.95) | (0 $|$ 0.88, 1.04, 1.16) | (1.00000, 1.00407, 1.01243 )
(5, 5, 5) | SeDuMi | (0 $|$ 219.93, 350.02, 545.81) | (0 $|$ 8.25, 10.99, 14.08) | (0.649823, 1.04081, 1.26310)
| SDPT3 | (0 $|$ 160.89, 247.24, 298.97) | (0 $|$ 4.45, 5.50, 6.97) | (0.649823, 1.04081, 1.26310)
(5, 5, 10) | SeDuMi | (0 $|$ 285.21, 420.27, 509.31) | (0 $|$ 7.96, 10.53, 15.04) | (1.00000, 1.01445, 1.02818)
| SDPT3 | (0 $|$ 217.48, 276.67, 309.27) | (0 $|$ 4.34, 5.37, 6.66) | (1.00000, 1.01445, 1.02818)
We observe the following:
* •
SDPT3 solves SDP relaxation problems faster than SeDuMi for $(n,m)=(5,5)$.
* •
Adaptive SOS relaxation can solve the resulting SDP problems faster than
Lasserre’s. In particular, SDPT3 works efficiently for Adaptive SOS relaxation
for BMIEP (22).
## 4 Extensions
In this section, we give three extensions of Theorem 1 and present some
related work to Theorem 1.
### 4.1 Sums of squares of rational polynomials
We can extend part i. of Theorem 1 with sums of squares of rational
polynomials. We assume that for all $j=1,\ldots,m$, there exists
$g_{j}\in\mathbb{R}[x]$ such that $|f_{j}(x)|\leq g_{j}(x)$ and $g_{j}(x)\neq
0$ for all $x\in B$. We define
$\tilde{\psi}_{r}(x)=-\sum_{j=1}^{m}f_{j}(x)\left(1-\frac{f_{j}(x)}{g_{j}(x)}\right)^{2r}$
for all $r\in\mathbb{N}$. Then, we can prove the following corollary by using
almost the same arguments as Theorem 1.
###### Corollary 4
Suppose that for $\rho\in\mathbb{R}$, $f(x)-\rho>0$ for every $x\in\bar{K}$,
i.e., $\rho$ is an lower bound of $f^{\ast}$. Then there exists
$\tilde{r}\in\mathbb{N}$ such that for all $r\geq\tilde{r}$,
$f-\rho+\tilde{\psi}_{r}$ is positive over $B$.
It is difficult to apply Corollary 4 to the framework of SDP relaxations,
because we deal with rational polynomials in $\tilde{\psi}_{r}$. However, we
may be able to reduce the degrees of sums of squares in $\tilde{\psi}_{r}$ by
using Corollary 4. For instance, we consider $f_{1}(x)=1-x^{4}$ and
$B=[-1,1]$. Choose $g_{1}(x)=2(1+x^{2})$. Then $g_{1}$ dominates $|f_{1}|$
over $B$, i.e., $|f_{1}(x)|\leq g_{1}(x)$ for all $x\in B$. We have
$\tilde{\psi}_{r}(x)=-(1-x^{4})\left(1-\frac{1-x^{4}}{2(1+x^{2})}\right)^{2r}=-(1-x^{4})\left(1-\frac{1-x^{2}}{2}\right)^{2r},$
and the degree of $\tilde{\psi}$ in Corollary 4 is $4r$, while the degree of
$\psi$ in Theorem 1 is $8r$.
### 4.2 Extension to POP with correlative sparsity
In [29], the authors introduced the notion of correlative sparsity for POP
(1), and proposed a sparse SDP relaxation that exploits the correlative
sparsity. They then demonstrated that the sparse SDP relaxation outperforms
Lasserre’s SDP relaxation. The sparse SDP relaxation is implemented in [30]
and its source code is freely available.
We give some of the definition of the correlative sparsity for POP (1). For
this, we use an $n\times n$ symbolic symmetric matrix $R$, whose elements are
either $0$ or $\star$ representing a nonzero value. We assign either $0$ or
$\star$ as follows:
$R_{k,\ell}=\left\\{\begin{array}[]{ll}\star&\mbox{if }k=\ell,\\\
\star&\mbox{if }\alpha_{k}\geq 1\mbox{ and }\alpha_{\ell}\geq 1\mbox{ for some
}\alpha\in\mathcal{F},\\\ \star&\mbox{if }x_{k}\mbox{ and }x_{\ell}\mbox{ are
involved in the polynomial }f_{j}\mbox{ for some }j=1,\ldots,m,\\\
0&\mbox{o.w.}\end{array}\right.$
POP (1) is said to be correlatively sparse if the matrix $R$ is sparse.
We give some of the details of the sparse SDP relaxation proposed in [29] for
the sake of completeness. We construct an undirected graph $G=(V,E)$ from $R$.
Here $V:=\\{1,\ldots,n\\}$ and $E:=\\{(k,\ell):R_{k,\ell}=\star\\}$. After
applying the chordal extension to $G=(V,E)$, we generate all maximal cliques
$C_{1},\ldots,C_{p}$ of the extension $G=(V,\tilde{E})$ with
$E\subseteq\tilde{E}$. See [5, 29] and references therein for the details of
the construction of the chordal extension. For a finite set
$C\subseteq\mathbb{N}$, $x_{C}$ denotes the subvector which consists of
$x_{i}\ (i\in C)$. For all $f_{1},\ldots,f_{m}$ in POP (1), $F_{j}$ denotes
the set of indices whose variables are involved in $f_{j}$, i.e.,
$F_{j}:=\\{i\in\\{1,\ldots,n\\}:\alpha_{i}\geq 1\ \mbox{for some
}\alpha\in\mathcal{F}_{j}\\}$. For a finite set $C\subseteq\mathbb{N}$, the
sets $\Sigma_{r,C}$ and $\Sigma_{\infty,C}$ denote the subsets of $\Sigma_{r}$
as follows:
$\displaystyle\Sigma_{r,C}$ $\displaystyle:=$
$\displaystyle\left\\{\sum_{k=1}^{q}g_{k}(x)^{2}:\forall
k=1,\ldots,q,g_{k}\in\mathbb{R}[x_{C}]_{r}\right\\},$
$\displaystyle\Sigma_{\infty,C}$ $\displaystyle:=$
$\displaystyle\bigcup_{r\geq 0}\Sigma_{r,C}.$
Note that if $C=\\{1,\ldots,n\\}$, then we have $\Sigma_{r,C}=\Sigma_{r}$ and
$\Sigma_{\infty,C}=\Sigma$. The sparse SDP relaxation problem with relaxation
order $r$ for POP (1) is obtained from the following SOS relaxation problem:
$\rho_{r}^{\mbox{\scriptsize
sparse}}:=\sup\left\\{\rho:\begin{array}[]{l}f-\rho=\sum_{h=1}^{p}\sigma_{0,h}+\sum_{j=1}^{m}\sigma_{j}f_{j},\\\
\sigma_{0,h}\in\Sigma_{r,C_{h}}\
(h=1,\ldots,p),\sigma_{j}\in\Sigma_{r_{j},D_{j}}\
(j=1,\ldots,m)\end{array}\right\\},$ (23)
where $D_{j}$ is the union of some of the maximal cliques $C_{1},\ldots,C_{p}$
such that $F_{j}\subseteq C_{h}$ and $r_{j}=r-\lceil\deg(f_{j})/2\rceil$ for
$j=1,\ldots,m$.
It should be noted that other sparse SDP relaxations are proposed in [9, 19,
22] and the asymptotic convergence is proved. In contrast, the convergence of
the sparse SDP relaxation (23) is not shown in [29].
We give an extension of Theorem 1 to POP with correlative sparsity. If
$C_{1},\ldots,C_{p}\subseteq\\{1,\ldots,n\\}$ satisfy the following property,
we refer this property as the running intersection property (RIP):
$\forall h\in\\{1,\ldots,p-1\\},\exists t\in\\{1,\ldots,p\\}\mbox{ such that
}C_{h+1}\cap(C_{1}\cup\cdots\cup C_{h})\subsetneq C_{t}.$
For $C_{1},\ldots,C_{p}\subseteq\\{1,\ldots,n\\}$, we define sets
$J_{1},\ldots,J_{p}$ as follows:
$J_{h}:=\left\\{j\in\\{1,\ldots,m\\}:f_{j}\in\mathbb{R}[x_{C_{h}}]\right\\}.$
Clearly, we have $\cup_{h=1}^{p}J_{h}=\\{1,\ldots,m\\}$. In addition, we
define
$\displaystyle\psi_{r,h}(x)$ $\displaystyle:=$ $\displaystyle-\sum_{j\in
J_{h}}f_{j}(x)\left(1-\frac{f_{j}(x)}{R_{j}}\right)^{2r},$
$\displaystyle\Theta_{r,h,b}(x)$ $\displaystyle:=$ $\displaystyle 1+\sum_{i\in
C_{h}}\left(\frac{x_{i}}{b}\right)^{2r}$
for $h=1,\ldots,p$.
Using a proof similar to the one for the theorem on convergence of the sparse
SDP relaxation given in [9], we can establish the correlatively sparse case of
Theorem 1. Indeed, we can obtain the theorem by using [9, Lemma 4] and Theorem
1.
###### Theorem 5
Assume that nonempty sets $C_{1},\ldots,C_{p}\subseteq\\{1,\ldots,n\\}$
satisfy (RIP) and we can decompose $f$ into $f=\hat{f}_{1}+\cdots+\hat{f}_{p}$
with $\hat{f}_{h}\in\mathbb{R}[x_{C_{h}}]\ (h=1,\ldots,p)$. Under the
assumptions of Theorem 1, there exists $\tilde{r}\in\mathbb{N}$ such that for
all $r\geq\tilde{r}$, $f-\rho+\sum_{h=1}^{p}\psi_{r,h}$ is positive over
$B=[-b,b]^{n}$. In addition, for every $\epsilon>0$, there exists
$\hat{r}\in\mathbb{N}$ such that for all $r\geq\hat{r}$,
$f-\rho+\epsilon\sum_{h=1}^{p}\Theta_{r,h,b}+\sum_{h=1}^{p}\psi_{\tilde{r},h}\in\Sigma_{\infty,C_{1}}+\cdots+\Sigma_{\infty,C_{p}}.$
(24)
Note that if $p=1$, i.e., $C_{1}=\\{1,\ldots,n\\}$, then we have
$\psi_{r,1}=\psi_{r}$ and $\Theta_{r,1,b}=\Theta_{r,b}$, and thus Theorem 5 is
reduced to Theorem 1. Therefore, we will concentrate our effort to prove
Theorem 5 in the following. In addition, we remark that it would follow from
[9, Theorem 5] that $(\ref{eq:sparsemain})$ holds without the polynomial
$\epsilon\sum_{h=1}^{p}\Theta_{r,h,b}$ if we assume that all quadratic modules
generated by $f_{j}\ (j\in C_{h})$ for all $h=1,\ldots,p$ are archimedean.
To prove Theorem 5, we use Lemma 4 in [9] and Corollary 3.3 of [21].
###### Lemma 6
(modified version of [9, Lemma 4]) Assume that we decompose $f$ into
$f=\hat{f}_{1}+\cdots+\hat{f}_{p}$ with $\hat{f}_{h}\in\mathbb{R}[x_{C_{h}}]$
and $f>0$ on $K$. Then, for any bounded set $B\subseteq\mathbb{R}^{n}$, there
exist $\tilde{r}\in\mathbb{N}$ and $g_{h}\in\mathbb{R}[x_{C_{h}}]$ with
$g_{h}>0$ on $B$ such that for every $r\geq\tilde{r}$,
$f=-\sum_{h=1}^{p}\psi_{r,h}+\sum_{h=1}^{p}g_{h}.$
###### Remark 7
The original statement in [9, Lemma 4] is slightly different from Lemma 6. In
[9, Lemma 4], it is proved that there exists $\lambda\in(0,1]$,
$\tilde{r}\in\mathbb{N}$ and $g_{h}\in\mathbb{R}[x_{C_{h}}]$ with $g_{h}>0$ on
$B$ such that
$f=\sum_{h=1}^{p}\sum_{j\in J_{h}}\left(1-\lambda
f_{j}\right)^{2\tilde{r}}f_{j}+\sum_{h=1}^{p}g_{h}.$
In Appendix A, we establish the correctness of Lemma 6 by using [9, Lemma 4].
###### Lemma 8
(Corollary 3.3 of [21]) Let $f\in\mathbb{R}[x]$ be a polynomial nonnegative on
$[-1,1]^{n}$. For arbitrary $\epsilon>0$, there exists some $\hat{r}$ such
that for every $r\geq\hat{r}$, the polynomial $(f+\epsilon\Theta_{r})$ is a
SOS.
Proof of Theorem 5 : We may choose $[-b,b]^{n}$ as $B$ in Lemma 6. It follows
from the assumption in Theorem 5 that we can decompose $f-\rho$ into
$(\hat{f}_{1}-\rho)+\hat{f}_{2}+\cdots+\hat{f}_{p}$. Since
$\hat{f}_{1}-\rho\in\mathbb{R}[x_{C_{1}}]$, it follows from Lemma 6 that there
exists $\tilde{r}\in\mathbb{N}$ and $g_{h}\in\mathbb{R}[x_{C_{h}}]$ with
$g_{h}>0$ on $B$ such that for every $r\geq\tilde{r}$,
$f-\rho=(\hat{f}_{1}-\rho)+\hat{f}_{2}+\cdots+\hat{f}_{p}=-\sum_{h=1}^{p}\psi_{r,h}+\sum_{h=1}^{p}g_{h}.$
Therefore, the polynomial $f-\rho+\sum_{h=1}^{p}\psi_{r,h}$ is positive on $B$
for all $r\geq\tilde{r}$.
For simplicity, we fix $h$ and define $C_{h}=\\{c_{1},\ldots,c_{k}\\}$. Then,
$g_{h}$ consists of the $k$ variables $x_{c_{1}},\ldots,x_{c_{k}}$. Since
$g_{h}>0$ on $B$, it is also positive on
$B^{\prime}:=\\{(x_{c_{1}},\ldots,x_{c_{k}}):-b\leq x_{c_{j}}\leq b\
(j=1,\ldots,k)\\}$. We define $\hat{g}_{h}(y)=g_{h}(by)$. Since $g_{h}$ is
positive on $B^{\prime}$,
$\hat{g}_{h}\in\mathbb{R}[y_{c_{1}},\ldots,y_{c_{k}}]$ is also positive on the
set $\\{(y_{c_{1}},\ldots,y_{c_{k}}):-1\leq y_{c_{j}}\leq 1\
(j=1,\ldots,k)\\}$. Applying Lemma 8 to $\hat{g}_{h}$, for all $\epsilon>0$,
there exists $\hat{r}_{h}\in\mathbb{N}$ such that for every
$r\geq\hat{r}_{h}$,
$\hat{g}_{h}(y_{c_{1}},\ldots,y_{c_{k}})+\epsilon\sum_{i=1}^{k}y_{c_{i}}^{2r}=\sigma_{h}(y_{c_{1}},\ldots,y_{c_{k}})$
for some $\sigma_{h}\in\Sigma_{\infty,C_{h}}$. Substituting
$x_{c_{1}}=by_{c_{1}},\ldots,x_{c_{k}}=by_{c_{k}}$, we obtain
$g_{h}+\epsilon\Theta_{r,h,b}\in\Sigma_{\infty,C_{h}}.$
We fix $\epsilon>0$. Applying the above discussion to all $h=1,\ldots,p$, we
obtain the numbers $\hat{r}_{1},\ldots,\hat{r}_{p}$. We denote the maximum
over $\hat{r}_{1},\ldots,\hat{r}_{p}$ by $\hat{r}$. Then, we have
$f-\rho+\epsilon\sum_{h=1}^{p}\Theta_{r,h,b}+\sum_{h=1}^{p}\psi_{\tilde{r},h}\in\Sigma_{\infty,C_{1}}+\cdots+\Sigma_{\infty,C_{p}}$
for every $r\geq\hat{r}$. $\Box$
### 4.3 Extension to POP with symmetric cones
In this subsection, we extend Theorem 1 to POP over symmetric cones, i.e.,
$f^{*}:=\inf_{x\in\mathbb{R}^{n}}\left\\{f(x):G(x)\in\mathcal{E}_{+}\right\\},$
(25)
where $f\in\mathbb{R}[x]$, $\mathcal{E}_{+}$ is a symmetric cone associated
with an $N$-dimensional Euclidean Jordan algebra $\mathcal{E}$, and $G$ is
$\mathcal{E}$-valued polynomial in $x$. The feasible region $K$ of POP (25) is
$\\{x\in\mathbb{R}^{n}:G(x)\in\mathcal{E}_{+}\\}$. Note that if $\mathcal{E}$
is $\mathbb{R}^{m}$ and $\mathcal{E}_{+}$ is the nonnegative orthant
$\mathbb{R}^{m}_{+}$, then (25) is identical to (1). In addition,
$\mathbb{S}^{n}_{+}$, the cone of $n\times n$ symmetric positive semidefinite
matrices, is a symmetric cone, the bilinear matrix inequalities can be
formulated as (25).
To construct $\psi_{r}$ for (25), we introduce some notation and symbols. The
Jordan product and inner product of $x,y\in\mathcal{E}$ are denoted by,
respectively, $x\circ y$ and $x\bullet y$. Let $e$ be the identity element in
the Jordan algebra $\mathcal{E}$. For any $x\in\mathcal{E}$, we have $e\circ
x=x\circ e=x$. We can define eigenvalues for all elements in the Jordan
algebra $\mathcal{E}$, generalizing those for Hermitian matrices. See [4] for
the details. We construct $\psi_{r}$ for (25) as follows:
$\displaystyle M$ $\displaystyle:=$ $\displaystyle\sup\left\\{\mbox{maximum
absolute eigenvalue of }G(x):x\in\bar{K}\right\\},$ $\displaystyle\psi_{r}(x)$
$\displaystyle:=$
$\displaystyle-G(x)\bullet\left(e-\frac{G(x)}{M}\right)^{2r},$ (26)
where we define $x^{k}:=x^{k-1}\circ x$ for $k\in\mathbb{N}$ and
$x\in\mathcal{E}$.
Lemma 4 in [16] shows that $\psi_{r}$ defined in (26) has the same properties
as $\psi_{r}$ in Theorem 1.
###### Theorem 9
For a given $\rho$, suppose that $f(x)-\rho>0$ for every $x\in\bar{K}$. Then,
there exists $\tilde{r}\in\mathbb{N}$ such that for all $r\geq\tilde{r}$,
$f-\rho+\psi_{r}$ is positive over $B$. Moreover, for any $\epsilon>0$, there
exists $\hat{r}\in\mathbb{N}$ such that for every $r\geq\hat{r}$,
$f-\rho+\epsilon\Theta_{r,b}+\psi_{\tilde{r}}\in\Sigma.$
### 4.4 Another perturbed sums of squares theorem
In this subsection, we present another perturbed sums of squares theorem for
POP (1) which is obtained by combining results in [14, 18].
To use the result in [14], we introduce some notation and symbols. We assume
that $K\subseteq B:=[-b,b]^{n}$. We choose $\gamma\geq 1$ such that for all
$j=0,1,\ldots,m$,
$\displaystyle|f_{j}(x)/\gamma|$ $\displaystyle\leq$ $\displaystyle 1\mbox{ if
}\|x\|_{\infty}\leq\sqrt{2}b,$ $\displaystyle|f_{j}(x)/\gamma|$
$\displaystyle\leq$ $\displaystyle\|x/b\|_{\infty}^{d}\mbox{ if
}\|x\|_{\infty}\geq\sqrt{2}b,$
where $f_{0}$ denotes the objective function $f$ in POP (1), and
$d=\max\\{\deg(f),\deg(f_{1}),\ldots,\deg(f_{m})\\}$. For $r\in\mathbb{N}$, we
define
$\displaystyle\psi_{r}(x)$ $\displaystyle:=$
$\displaystyle-\sum_{j=1}^{m}\left(1-\frac{f_{j}(x)}{\gamma}\right)^{2r}f_{j}(x),$
$\displaystyle\phi_{r,b}(x)$ $\displaystyle:=$
$\displaystyle-\frac{(m+2)\gamma}{b^{2}}\sum_{i=1}^{n}\left(\frac{x_{i}}{b}\right)^{2d(r+1)}(b^{2}-x_{i}^{2}).$
From (a), (b) and (c) of Lemma 3.2 in [14], we obtain the following result:
###### Proposition 10
Assume that the feasible region $K$ of POP (1) is contained in $B=[-b,b]^{n}$.
In addition, we assume that for $\rho\in\mathbb{R}$, we have $f-\rho>0$ over
$K$. Then there exists $\tilde{r}\in\mathbb{N}$ such that for all
$r\geq\tilde{r}$, $(f-\rho+\psi_{r}+\phi_{r,b})$ is positive over
$\mathbb{R}^{n}$.
We remark that we do not need to impose the assumption on the compactness of
$K$ in Proposition 10. Indeed, we can drop it by replacing $K$ by $\bar{K}$
defined in Subsection 2.1 as in Theorem 1.
Next, we describe a result from [18] which is useful in deriving another
perturbed sums of squares theorem.
###### Theorem 11
((iii) of Theorem 4.1 in [18]) Let $f\in\mathbb{R}[x]$ be a nonnegative
polynomial. Then for every $\epsilon>0$, there exists $\hat{r}\in\mathbb{N}$
such that for all $r\geq\hat{r}$,
$f+\epsilon\theta_{r}\in\Sigma,$
where $\theta_{r}(x):=\sum_{i=1}^{n}\sum_{k=0}^{r}(x_{i}^{2k}/k!)$.
By incorporating Proposition 10 with Theorem 11, we obtain yet another
perturbation theorem.
###### Theorem 12
We assume that for $\rho\in\mathbb{R}$, we have $f-\rho>0$ over $K$. Then we
have
1. i.
there exists $\tilde{r}\in\mathbb{N}$ such that for all $r\geq\tilde{r}$,
$(f-\rho+\psi_{r}+\phi_{r,b})$ is positive over $\mathbb{R}^{n}$;
2. ii.
moreover, for every $\epsilon>0$, there exists $\hat{r}\in\mathbb{N}$ such
that for all $r\geq\hat{r}$,
$(f-\rho+\psi_{\tilde{r}}+\phi_{\tilde{r},b}+\epsilon\theta_{r})\in\Sigma.$
We give an SDP relaxation analogous to (5), based on Theorem 12, as follows:
$\eta(\epsilon,\tilde{r},r):=\sup\left\\{\eta:\begin{array}[]{l}f-\eta+\epsilon\theta_{r}-\displaystyle\sum_{j=1}^{m}f_{j}\sigma_{j}-\displaystyle\sum_{i=1}^{n}(b^{2}-x_{i}^{2})\mu_{i}=\sigma_{0},\\\
\sigma_{0}\in\Sigma_{r},\sigma_{j}\in\Sigma(\tilde{r}\tilde{\mathcal{F}}_{j}),\mu_{i}\in\Sigma(\\{d(\tilde{r}+1)e_{i}\\})\end{array}\right\\},$
(27)
for some $r\geq\tilde{r}$, where $e_{i}$ is the $i$th standard unit vector in
$\mathbb{R}^{n}$. One of the differences between (5) and (27) is that (27) has
$n$ SOS variables $\mu_{1},\ldots,\mu_{n}$. These variables correspond to
nonnegative variables in the SDP formulation, but not positive semidefinite
matrices, since these consist of a single monomial. On the other hand, it is
difficult to estimate $\tilde{r}$ in the SDP relaxations (5) and (27), and
thus we could not compare the size and the quality of the optimal value of (5)
with (27) so far.
We obtain a result similar technique to Theorem 2. We omit the proof because
we obtain the inequalities by applying a proof similar to that of Theorem 2.
###### Theorem 13
For every $\epsilon>0$, there exists $r,\tilde{r}\in\mathbb{N}$ such that
$f^{*}-\epsilon\leq\eta(\epsilon,\tilde{r},r)\leq f^{*}+\epsilon ne^{b^{2}}$.
## 5 Concluding Remarks
We mention other research related to our work related to Theorem 1. A common
element in all of these approaches is to use perturbations
$\epsilon\theta_{r}(x)$ or $\epsilon\Theta_{r}(x)$ for finding an approximate
solution of a given POP.
In [10, 12], the authors added $\epsilon\Theta_{r}(x)$ to the objective
function of a given unconstrained POP and used algebraic techniques to find a
solution. In [13], the following equality constraints were added in the
perturbed unconstrained POP and Lasserre’s SDP relaxation was applied to the
new POP:
$\frac{\partial f_{0}}{\partial x_{i}}+2r\epsilon x_{i}^{2r-1}=0\
(i=1,\ldots,n).$
Lasserre in [20] proposed an SDP relaxation via $\theta_{r}(x)$ defined in
Theorem 11 and a perturbation theorem for semi-algebraic set defined by
equality constraints $g_{k}(x)=0$ $(k=1,\ldots,m)$. The SDP relaxation can be
applied to the following equality constrained POP:
$\inf_{x\in\mathbb{R}^{n}}\left\\{f_{0}(x):g_{k}(x)=0\
(k=1,\ldots,m)\right\\};$ (28)
To obtain the SDP relaxations, $\epsilon\theta_{r}(x)$ is added to the
objective function in POP (28) and the equality constraints in POP (28) is
replaced by $g_{k}^{2}(x)\leq 0$. In the resulting SDP relaxations,
$\theta_{r}(x)$ is explicitly introduced and variables associated with
constraints $g_{k}^{2}(x)\leq 0$ are not positive semidefinite matrices, but
nonnegative variables.
In this paper, we present a perturbed SOS theorem (Theorem 1) and its
extensions, and propose a new sparse relaxation called Adaptive SOS
relaxation. During the course of the paper, we have shed some light on why
Lasserre’s SDP relaxation calculates the optimal value of POP even if its SDP
relaxation has a different optimal value. The numerical experiments clearly
show that Adaptive SOS relaxation is promising, justifying the need for future
research in this direction.
Of course, if the original POP is dense, i.e., $\tilde{F}_{j}$ contains many
elements for almost all $j$, then the proposed relaxation has little effect in
reducing the SDP relaxation. However, in real applications, such cases seem
rare.
In the numerical experiments, we sometimes observe that the behaviors of
SeDuMi and SDPT3 are very different each other. See, for example, Table 4. In
the column of Adaptive SOS, SeDuMi solved significantly fewer problems than
SDPT3. On the other hand, there are several cases where SeDuMi outperforms
SDPT3. For such an example, see the sparse relaxation column of Table 7. This
is why we present the results of both solvers in every table. In solving a
real problem, one should be very careful in choosing the appropriate SDP
solver for the problem at hand.
## Acknowledgements
The first author was supported by in part by a Grant-in-Aid for Scientific
Research (C) 19560063. The second author was supported in part bya Grant-in-
Aid for Young Scientists (B) 22740056. The third author was supported in part
by a Discovery Grant from NSERC, a research grant from University of Waterloo
and by ONR research grant N00014-12-10049.
## Appendix A A proof of Lemma 6
As we have already mentioned in Remark 7, Lemma 6 is slightly different from
the original one in [9, Lemma 4]. To show the correctness of Lemma 6, we use
the following lemma:
###### Lemma 14
([9, Lemma 3]) Let $B\subseteq\mathbb{R}^{n}$ be a compact set. Assume that
nonempty sets $C_{1},\ldots,C_{p}\subseteq\\{1,\ldots,n\\}$ satisfy (RIP) and
we can decompose $f$ into $f=\hat{f}_{1}+\cdots+\hat{f}_{p}$ with
$\hat{f}_{h}\in\mathbb{R}[x_{C_{h}}]\ (h=1,\ldots,p)$. In addition, suppose
that $f>0$ on $B$. Then there exists $g_{h}\in\mathbb{R}[x_{C_{h}}]$ with
$g_{h}>0$ on $B$ such that
$f=g_{1}+\cdots+g_{p}.$
We can prove Lemma 6 in a manner similar to [9, Lemma 4]. We define
$F_{r}:\mathbb{R}^{n}\to\mathbb{R}$ as follows:
$F_{r}=f-\sum_{h=1}^{p}\psi_{r,h}.$
We recall that $\psi_{r,h}=\sum_{j\in C_{h}}(1-f_{j}/R_{j})^{2r}f_{j}$ for all
$h=1,\ldots,p$ and $r\in\mathbb{N}$, and that $R_{j}$ is the maximum value of
$|f_{j}|$ on $B$ for all $j=1,\ldots,m$. It follows from the definitions of
$\psi_{r,h}$ and $R_{j}$ that we have $\psi_{r,h}\geq\psi_{r+1,h}$ on $B$ for
all $h=1,\ldots,p$ and $r\in\mathbb{N}$, and thus we have $F_{r}\leq F_{r+1}$
on $B$. In addition, we can prove that (i) on $B\cap K$, $F_{r}\to f$ as
$r\to\infty$, and (ii) on $B\setminus K$, $F_{r}\to\infty$ as $r\to\infty$.
Since $B$ is compact, it follows from (i), (ii) and the positiveness of $f$ on
$B$ that there exists $\tilde{r}\in\mathbb{N}$ such that for every
$r\geq\tilde{r}$, $F_{r}>0$ on $B$. Applying Lemma 14 to $F_{r}$, we obtain
the desired result.
## References
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|
arxiv-papers
| 2013-03-30T03:59:23 |
2024-09-04T02:49:43.627702
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Masakazu Muramatsu and Hayato Waki and Levent Tuncel",
"submitter": "Hayato Waki",
"url": "https://arxiv.org/abs/1304.0065"
}
|
1304.0067
|
# On an Operator Preserving Inequalities between Polynomials
N. A. Rather, Suhail Gulzar
Department of Mathematics, University of Kashmir, Srinagar 190006, India
[email protected], [email protected]
Abstract. Let $\mathscr{P}_{n}$ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$ and $\mathcal{B}_{n}$ a family
of operators that maps $\mathscr{P}_{n}$ into itself. In this paper, we
consider a problem of investigating the dependence of
$\left|B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}B[P\circ\rho](z)\right|$
on the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real
or complex numbers $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq
1,|\beta|\leq 1,R>r\geq k,$ $\sigma(z)=Rz,$ $\rho(z)=rz$ and establish certain
sharp operator preserving inequalities between polynomials, from which a
variety of interesting results follow as special cases.
Keywords: Polynomials; Inequalities in the complex domain;
$\mathcal{B}_{n}$-operator.
2000 AMS Subject Classification: 30A10; 30D15; 41A17
## 1\. Introduction
Let $\mathscr{P}_{n}$ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$. A famous result known as
Bernstein’s inequality (for reference, see [8, p.531], [10, p.508] or [11]
states that if $P\in\mathscr{P}_{n}$, then
(1) $\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq
n\underset{\left|z\right|=1}{Max}\left|P(z)\right|,$
whereas concerning the maximum modulus of $P(z)$ on the circle
$\left|z\right|=R>1$, we have
(2) $\underset{\left|z\right|=R}{Max}\left|P(z)\right|\leq
R^{n}\underset{\left|z\right|=1}{Max}\left|P(z)\right|,\,\,\,R\geq 1.$
(for reference, see [7, p.442] or [8, vol.I, p.137] ).
If we restrict ourselves to the class of polynomials $P\in\mathscr{P}_{n}$
having no zero in $|z|<1$, then inequalities (1) and (2) can be respectively
replaced by
(3)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq\frac{n}{2}\underset{\left|z\right|=1}{Max}\left|P(z)\right|,$
and
(4)
$\underset{\left|z\right|=R}{Max}\left|P(z)\right|\leq\frac{R^{n}+1}{2}\underset{\left|z\right|=1}{Max}\left|P(z)\right|,\,\,\,R\geq
1.$
Inequality (3) was conjectured by Erdös and later verified by Lax [5], whereas
inequality (4) is due to Ankey and Ravilin [1]. Aziz and Dawood [2] further
improved inequalities (3) and (4) under the same hypothesis and proved that,
(5)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq\frac{n}{2}\left\\{\underset{\left|z\right|=1}{Max}\left|P(z)\right|-\underset{\left|z\right|=1}{Min}\left|P(z)\right|\right\\},$
(6)
$\underset{\left|z\right|=R}{Max}\left|P(z)\right|\leq\frac{R^{n}+1}{2}\underset{\left|z\right|=1}{Max}\left|P(z)\right|-\frac{R^{n}-1}{2}\underset{\left|z\right|=1}{Min}\left|P(z)\right|,\,\,\,R\geq
1.$
As a compact generalization of Inequalities (1) and (2), Aziz and Rather [3]
have shown that if $P\in\mathscr{P}_{n}$ then for $\alpha,\beta\in\mathbb{C}$
with $|\alpha|\leq 1,$ $|\beta|\leq 1,$ $R>1$ and $|z|\geq 1,$
$\displaystyle\bigg{|}P(Rz)-\alpha P(z)$
$\displaystyle+\beta\left\\{\left(\frac{R+1}{2}\right)^{n}-|\alpha|\right\\}P(z)\bigg{|}$
(7)
$\displaystyle\leq|z|^{n}\left|R^{n}-\alpha+\beta\left\\{\left(\frac{R+1}{2}\right)^{n}-|\alpha|\right\\}\right|\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$
The result is sharp and equality in (1\. Introduction) holds for the
polynomial $P(z)=az^{n},$ $a\neq 0.$
As a corresponding compact generalization of Inequalities (3) and (4), they
[3] have also shown that if $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish
in $|z|<1,$ then for all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq
1,|\beta|\leq 1,$ $R>1$ and $|z|\geq 1,$
$\displaystyle\Bigg{|}P(Rz)-\alpha P(z)+\beta$
$\displaystyle\left\\{\left(\frac{R+1}{2}\right)^{n}-|\alpha|\right\\}P(z)\Bigg{|}$
$\displaystyle\leq\frac{1}{2}$
$\displaystyle\Bigg{[}\left|R^{n}-\alpha+\beta\left\\{\left(\frac{R+1}{2}\right)^{n}-|\alpha|\right\\}\right||z|^{n}$
(8)
$\displaystyle+\left|1-\alpha+\beta\left\\{\left(\frac{R+1}{2}\right)^{n}-|\alpha|\right\\}\right|\Bigg{]}\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$
The result is best possible and equality in (1\. Introduction) holds for
$P(z)=az^{n}+b,$ $|a|=|b|.$
Q. I. Rahman [9] (see also Rahman and Schmeisser [10, p. 538]) introduced a
class $\mathcal{B}_{n}$ of operators $B$ that carries a polynomial
$P\in\mathscr{P}_{n}$ into
(9)
$B[P](z)=\lambda_{0}P(z)+\lambda_{1}\left(\dfrac{nz}{2}\right)\dfrac{P^{\prime}(z)}{1!}+\lambda_{2}\left(\dfrac{nz}{2}\right)^{2}\dfrac{P^{\prime\prime}(z)}{2!},$
where $\lambda_{0},\lambda_{1}$ and $\lambda_{2}$ are such that all the zeros
of
(10) $U(z)=\lambda_{0}+n\lambda_{1}z+\frac{n(n-1)}{2}\lambda_{2}z^{2}$
lie in half plane $|z|\leq\left|z-n/2\right|.$
As a generalization of the inequalities (1) and (3), Q. I. Rahman [9,
inequalities 5.2 and 5.3] proved that if $P\in\mathscr{P}_{n},$ then
(11)
$|B[P](z)|\leq|B[z^{n}]|\underset{|z|=1}{Max}|P(z)|,\,\,\,\,\,\,\textnormal{for}\,\,\,\,\,\,\,|z|\geq
1,$
and if $P\in\mathscr{P}_{n},$ $P(z)\neq 0$ in $|z|<1,$ then
(12)
$|B[P](z)|\leq\dfrac{1}{2}\left\\{|B[z^{n}]|+|\lambda_{0}|\right\\}\underset{|z|=1}{Max}|P(z)|,\,\,\,\,\,\,\textnormal{for}\,\,\,\,\,\,\,|z|\geq
1,$
where $B\in\mathcal{B}_{n}.$
In this paper, we denote for any complex functions
$P,\,\rho:\mathbb{C}\rightarrow\mathbb{C}$ the composite function of $P$ and
$\rho$, defined by
$\left(P\circ\rho\right)(z)=P\left(\rho(z)\right)\,\,(z\in\mathbb{C}),$ as
$P\circ\rho$.
## 1\. Preliminaries
For the proof of our results, we need the following Lemmas.
###### Lemma 1.1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ have all its zeros in $\left|z\right|\leq
k$ where $k\geq 0$, then for every $R\geq r,$ $Rr\geq k^{2}$ and
$\left|z\right|=1$, we have
$\left|P(Rz)\right|\geq\left(\frac{R+k}{r+k}\right)^{n}\left|P(rz)\right|.$
The above is due to Aziz and Zargar [4]. The next lemma follows from Corollary
$18.3$ of [6, p. 86].
###### Lemma 1.2.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has all zeros in $|z|\leq k,$ where $k>0$
then all the zeros of $B[P](z)$ also lie in $|z|\leq k.$
###### Lemma 1.3.
If $P\in\mathscr{P}_{n}$ and $P(z)$ have no zero in $\left|z\right|<k,$ where
$k>0,$ then for all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1$ , $R>r\geq k$ and $|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (13)
$\displaystyle\leq
k^{n}\left|B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\right|$
where $Q(z)=z^{n}\overline{P(1/\overline{z})},$ $\sigma(z)=Rz,$ $\rho(z)=rz,$
$\tau(z)=Rz/k^{2},$ $\eta(z)=rz/k^{2}$ and
(14)
$\Phi_{k}(R,r,\alpha,\beta)=\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}-\alpha.$
###### Proof.
By hypothesis, the polynomial $P(z)$ does not vanish in $|z|<k.$ Therefore,
all the zeros of polynomial $Q(z/k^{2})$ lie in $|z|<k$. As
$|k^{n}Q(z/k^{2})|=|P(z)|\,\,\,\,\textrm{for}\,\,\,\,|z|=k,$
applying Theorem 2.1 to $P(z)$ with $F(z)$ replaced by $k^{n}Q(z/k^{2}),$ we
get for arbitrary real or complex numbers $\alpha,\beta$ with $|\alpha|\leq
1,$ $|\beta|\leq 1,$ $R>r\geq k$ and $|z|\geq 1,$
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$
$\displaystyle\leq
k^{n}\left|B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\right|,$
This proves Lemma 1.3.
∎
###### Lemma 1.4.
If $P\in\mathscr{P}_{n}$ and $Q(z)=z^{n}\overline{P(1/\overline{z})}$ then for
$\alpha,\beta\in\mathbb{C}$ ,with $|\alpha|\leq 1,|\beta|\leq 1,R>r\geq k$,
$k\leq 1$ and $|z|\geq 1$,
$\displaystyle\Big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\Big{|}$
$\displaystyle+k^{n}$
$\displaystyle\Big{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)]\Big{|}$
(15)
$\displaystyle\leq\left\\{|\lambda_{0}|\big{|}1+\Phi_{k}(R,r,\alpha,\beta)\big{|}+\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right|\right\\}\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz,$ $\tau(z)=Rz/k^{2},$ $\eta(z)=rz/k^{2}$ and
$\Phi_{k}(R,r,\alpha,\beta)$ is given by (14).
###### Proof.
Let $M=Max_{\left|z\right|=k}\left|P(z)\right|,$ then by Rouche’s theorem, the
polynomial $F(z)=P(z)-\mu M$ does not vanish in $|z|<k$ for every
$\mu\in\mathbb{C}$ with $|\mu|>1.$ Applying Lemma 1.3 to polynomial $F(z)$, we
get for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ and
$|z|\geq 1$,
$\left|B[F\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[F\circ\rho](z)\right|\leq
k^{n}\left|B[H\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[H\circ\eta](z)\right|,$
where $H(z)=z^{n}\overline{F(1/\overline{z})}=Q(z)-\overline{\mu}Mz^{n}$.
Replacing $F(z)$ by $P(z)-\mu M$ and $H(z)$ by $Q(z)-\overline{\mu}Mz^{n},$ we
have for $|\alpha|\leq 1,|\beta|\leq 1$ and $|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)$
$\displaystyle
B[P\circ\rho](z)-\mu\lambda_{0}\left(1+\Phi_{k}(R,r,\alpha,\beta)\right)M\big{|}$
$\displaystyle\leq k^{n}\Bigg{|}B[Q\circ\tau](z)]$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)]$ (16)
$\displaystyle-\frac{\overline{\mu}}{k^{2n}}\left(R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right)MB[z^{n}]\Bigg{|}$
where $Q(z)=z^{n}\overline{P(1/\overline{z})}$.
Now choosing argument of $\mu$ in the right hand side of inequality (1) such
that
$\displaystyle k^{n}$
$\displaystyle\bigg{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)-\frac{\overline{\mu}}{k^{2n}}\left(R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right)MB[z^{n}]\bigg{|}$
$\displaystyle=\frac{|\overline{\mu}|}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right||B[z^{n}]|M-k^{n}\left|B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)]\right|$
which is possible by applying Corollary 2.3 to polynomial $Q(z/k^{2})$ and
using the fact $Max_{\left|z\right|=k}\left|Q(z/k^{2})\right|$ $=M/k^{n}$, we
get for $|\alpha|\leq 1,|\beta|\leq 1$ and $|z|\geq 1$,
$\displaystyle\big{|}$ $\displaystyle
B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}-|\mu\lambda_{0}|\big{|}\left(1+\Phi_{k}(R,r,\alpha,\beta)\right)M\big{|}$
$\displaystyle\leq\frac{|\overline{\mu}|}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right||B[z^{n}]|M-k^{n}\left|B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)]\right|$
Equivalently for $|\alpha|\leq 1,|\beta|\leq 1$ and $|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma](z)$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}+k^{n}\left|B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\right|$
$\displaystyle\leq|\mu|\left\\{|\lambda_{0}|\big{|}1+\Phi_{k}(R,r,\alpha,\beta)\big{|}+\frac{1}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right||B[z^{n}]|\right\\}M$
Letting $|\mu|\rightarrow 1$ , we get the conclusion of Lemma 1.4 and this
completes proof of Lemma 1.4. ∎
## 2\. Main results
###### Theorem 2.1.
If $F\in\mathscr{P}_{n}$ and $F(z)$ has all its zeros in the disk
$\left|z\right|\leq k$ where $k>0$ and $P(z)$ is a polynomial of degree at
most n such that
$\left|P(z)\right|\leq\left|F(z)\right|\,\,\,for\,\,\,|z|=k,$
then for $\left|\alpha\right|\leq 1,\left|\beta\right|\leq 1$, $R>r\geq k$ and
$|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (17)
$\displaystyle\leq\left|B[F\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[F\circ\rho](z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz$ and
(18)
$\Phi_{k}(R,r,\alpha,\beta)=\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}-\alpha.$
The result is best possible and the equality holds for the polynomial
$P(z)=e^{i\gamma}F(z)$ where $\gamma\in\mathbb{R}.$
###### Proof of Theorem 2.1.
Since polynomial $F(z)$ of degree $n$ has all its zeros in $|z|\leq k$ and
$P(z)$ is a polynomial of degree at most $n$ such that
(19) $|P(z)|\leq|F(z)|\,\,\,\,\textrm{for}\,\,\,\,|z|=k,$
therefore, if $F(z)$ has a zero of multiplicity $s$ at $z=ke^{i\theta_{0}},$
$0\leq\theta_{0}<2\pi,$ then $P(z)$ has a zero of multiplicity at least $s$ at
$z=ke^{i\theta_{0}}$. If $P(z)/F(z)$ is a constant, then inequality (2.1) is
obvious. We now assume that $P(z)/F(z)$ is not a constant, so that by the
maximum modulus principle, it follows that
$|P(z)|<|F(z)|\,\,\,\textrm{for}\,\,|z|>k\,\,.$
Suppose $F(z)$ has $m$ zeros on $|z|=k$ where $0\leq m<n$, so that we can
write
$F(z)=F_{1}(z)F_{2}(z)$
where $F_{1}(z)$ is a polynomial of degree $m$ whose all zeros lie on $|z|=k$
and $F_{2}(z)$ is a polynomial of degree exactly $n-m$ having all its zeros in
$|z|<k$. This implies with the help of inequality (19) that
$P(z)=P_{1}(z)F_{1}(z)$
where $P_{1}(z)$ is a polynomial of degree at most $n-m$. Again, from
inequality (19), we have
$|P_{1}(z)|\leq|F_{2}(z)|\,\,\,for\,\,|z|=k\,$
where $F_{2}(z)\neq 0\,\,for\,\,|z|=k$. Therefore for every real or complex
number $\lambda$ with $|\lambda|>1$, a direct application of Rouche’s theorem
shows that the zeros of the polynomial $P_{1}(z)-\lambda F_{2}(z)$ of degree
$n-m\geq 1$ lie in $|z|<k$ hence the polynomial
$G(z)=F_{1}(z)\left(P_{1}(z)-\lambda F_{2}(z)\right)=P(z)-\lambda F(z)$
has all its zeros in $|z|\leq k$ with at least one zero in $|z|<k$, so that we
can write
$G(z)=(z-te^{i\delta})H(z)$
where $t<k$ and $H(z)$ is a polynomial of degree $n-1$ having all its zeros in
$|z|\leq k$. Applying Lemma 1.1 to the polynomial $H(z)$, we obtain for every
$R>r\geq k$ and $0\leq\theta<2\pi$,
$\displaystyle|G(Re^{i\theta})|=$
$\displaystyle|Re^{i\theta}-te^{i\delta}||H(Re^{i\theta})|$
$\displaystyle\geq$
$\displaystyle|Re^{i\theta}-te^{i\delta}|\left(\frac{R+k}{k+r}\right)^{n-1}|H(re^{i\theta})|,$
$\displaystyle=$
$\displaystyle\left(\frac{R+k}{k+r}\right)^{n-1}\frac{|Re^{i\theta}-te^{i\delta}|}{|re^{i\theta}-te^{i\delta}|}|(re^{i\theta}-te^{i\delta})H(re^{i\theta})|,$
$\displaystyle\geq$
$\displaystyle\left(\frac{R+k}{k+r}\right)^{n-1}\left(\frac{R+t}{r+t}\right)|G(re^{i\theta})|.$
This implies for $R>r\geq k$ and $0\leq\theta<2\pi$,
(20)
$\left(\frac{r+t}{R+t}\right)|G(Re^{i\theta})|\geq\left(\frac{R+k}{k+r}\right)^{n-1}|G(re^{i\theta})|.$
Since $R>r\geq k$ so that $G(Re^{i\theta})\neq 0$ for $0\leq\theta<2\pi$ and
$\frac{r+k}{k+R}>\frac{r+t}{R+t}$, from inequality (20), we obtain
(21)
$|G(Re^{i\theta})|>\left(\frac{R+k}{k+r}\right)^{n}|G(re^{i\theta})|,\,\,\,\,\,\,R>r\geq
k\,\,\,\,\textrm{and}\,\,\,\,\,0\leq\theta<2\pi.$
Equivalently,
$|G(Rz)|>\left(\frac{R+k}{k+r}\right)^{n}|G(rz)|$
for $|z|=1$ and $R>r\geq k$. Hence for every real or complex number $\alpha$
with $|\alpha|\leq 1$ and $R>r\geq k,$ we have
(22) $\displaystyle\left|G(Rz)-\alpha G(rz)\right|$
$\displaystyle\geq\left|G(Rz)\right|-|\alpha|\left|G(rz)\right|$
$\displaystyle>\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}|G(rz)|,\,\,\,\textrm{for}\,\,\,|z|=1.$
Also, inequality (21) can be written in the form
(23) $|G(re^{i\theta})|<\left(\frac{k+r}{R+k}\right)^{n}|G(Re^{i\theta})|$
for every $R>r\geq k$ and $0\leq\theta<2\pi.$ Since $G(Re^{i\theta})\neq 0$
and $\left(\frac{k+r}{R+k}\right)^{n}<1$, from inequality (23), we obtain for
$0\leq\theta<2\pi$ and $R>r\geq k$,
$|G(re^{i\theta})|<|G(Re^{i\theta})|.$
That is,
$|G(rz)|<|G(Rz)|\,\,\,\textrm{for}\,\,\,\,|z|=1.$
Since all the zeros of $G(Rz)$ lie in $|z|\leq(k/R)<1$, a direct application
of Rouche’s theorem shows that the polynomial $G(Rz)-\alpha G(rz)$ has all its
zeros in $|z|<1$ for every real or complex number $\alpha$ with $|\alpha|\leq
1$. Applying Rouche’s theorem again, it follows from (22) that for arbitrary
real or complex numbers $\alpha,\beta$ with $|\alpha|\leq 1,|\beta|\leq 1$ and
$R>r\geq k$, all the zeros of the polynomial
$\displaystyle T(z)=$ $\displaystyle G(Rz)-\alpha
G(rz)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}G(rz)$
$\displaystyle=$ $\displaystyle\left[P(Rz)-\alpha
P(rz)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}P(rz)\right]$
$\displaystyle\,\,\,\,-\lambda\left[F(Rz)-\alpha
F(rz)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}F(rz)\right]$
lie in $|z|<1$.
Applying Lemma 1.3 to the polynomial $T(z)$ and noting that $B$ is a linear
operator, it follows that all the zeros of polynomial
$\displaystyle B[T](z)=$ $\displaystyle\left[B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}B[P\circ\rho](z)\right]$
$\displaystyle\,\,\,-\lambda\left[B[F\circ\sigma](z)-\alpha
B[F\circ\rho](z)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}[F\circ\rho](z)\right]$
lie in $|z|<1.$ This implies
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (24)
$\displaystyle\leq\left|B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\right|,$
for $|z|\geq 1$ and $R>r\geq k$. If inequality (2) is not true, then there a
point $z=z_{0}$ with $|z_{0}|\geq 1$ such that
$\displaystyle\big{|}\big{\\{}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{\\}}_{z=z_{0}}\big{|}$
$\displaystyle\geq\left|\left\\{B[F\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[F\circ\rho](z)\right\\}_{z=z_{0}}\right|,$
But all the zeros of $F(Rz)$ lie in $|z|<(k/R)<1$, therefore, it follows (as
in case of $G(z)$) that all the zeros of $F(Rz)-\alpha
F(rz)+\beta\left\\{\left(\frac{R+k}{k+r}\right)^{n}-|\alpha|\right\\}F(rz)$
lie in $\left|z\right|<1$. Hence, by Lemma 1.3,
$\left\\{B[F\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[F\circ\rho](z)\right\\}_{z=z_{0}}\neq
0$
with $|z_{0}|\geq 1$.We take
$\lambda=\dfrac{\left\\{B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\right\\}_{z=z_{0}}}{\left\\{B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\right\\}_{z=z_{0}}},$
then $\lambda$ is a well defined real or complex number with $|\lambda|>1$ and
with this choice of $\lambda$, we obtain $\\{B[T](z)\\}_{z=z_{0}}=0$ where
$|z_{0}|\geq 1$. This contradicts the fact that all the zeros of $B[T(z)]$ lie
in $|z|<1$. Thus (2) holds for $|\alpha|\leq 1$, $|\beta|\leq 1$, $|z|\geq 1$,
and $R>r\geq k.$
∎
For $\alpha=0$ in Theorem 2.1, we obtain the following.
###### Corollary 2.2.
If $F\in\mathscr{P}_{n}$ and $F(z)$ has all its zeros in the disk
$\left|z\right|\leq k,$ where $k>0$ and $P(z)$ is a polynomial of degree at
most n such that
$\left|P(z)\right|\leq\left|F(z)\right|\,\,\,for\,\,\,|z|=k,$
then for $\left|\beta\right|\leq 1$, $R>r\geq k$ and $|z|\geq 1$,
$\displaystyle\bigg{|}B[P\circ\sigma](z)+$
$\displaystyle\beta\left(\frac{R+k}{k+r}\right)^{n}B[P\circ\rho](z)\bigg{|}$
(25)
$\displaystyle\leq\left|B[F\circ\sigma](z)+\beta\left(\frac{R+k}{k+r}\right)^{n}B[F\circ\rho](z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz.$ The result is sharp, and the equality
holds for the polynomial $P(z)=e^{i\gamma}F(z)$ where $\gamma\in\mathbb{R}.$
If we choose $F(z)=z^{n}M/k^{n}$, where
$M=Max_{\left|z\right|=k}\left|P(z)\right|$ in Theorem 2.1, we get the
following result.
###### Corollary 2.3.
If $P\in\mathscr{P}_{n}$ then for $\alpha,\beta\in\mathbb{C}$ with
$|\alpha|\leq 1,$ $|\beta|\leq 1,$ $R>r\geq k>0$ and $|z|=1,$
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (26)
$\displaystyle\leq\frac{1}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right||B[z^{n}]|\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz$ and $\Phi_{k}(R,r,\alpha,\beta)$ is given
by (18). The result is best possible and equality in (2.3) holds for
$P(z)=az^{n},$ $a\neq 0.$
Next, we take $P(z)=z^{n}m/k^{n}$, where
$m=Min_{\left|z\right|=k}\left|P(z)\right|$ in Theorem 2.1, we get the
following result.
###### Corollary 2.4.
If $F\in\mathscr{P}_{n}$ and $F(z)$ have all its zeros in the disk $|z|\leq
k,$ where $k>0$ then for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1,$ $R>r\geq k>0$
$\displaystyle\underset{|z|=1}{Min}\big{|}B[F\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[F\circ\rho](z)\big{|}$ (27)
$\displaystyle\geq\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right|\underset{\left|z\right|=k}{Min}\left|P(z)\right|,$
where $\sigma(z)=Rz,$ $\rho(z)=rz$ and $\Phi_{k}(R,r,\alpha,\beta)$ is given
by (18). The result is Sharp.
If we take $\beta=0$ in (2.3), we get the following result.
###### Corollary 2.5.
If $P\in\mathscr{P}_{n}$ then for $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$
$R>r\geq k>0$ and $|z|\geq 1,$
(28) $\left|B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)\right|\leq\frac{1}{k^{n}}\left|R^{n}-\alpha
r^{n}\right||B[z^{n}]|\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz.$ The result is best possible as shown by
$P(z)=az^{n},a\neq 0.$
For polynomials $P\in\mathscr{P}_{n}$ having no zero in $|z|<k$, we establish
the following result which leads to a compact generalization of inequality
(3),(4),(1\. Introduction) and (12).
###### Theorem 2.6.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in the disk $|z|<k,$ where
$k\leq 1,$ then for all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1$ , $R>r\geq k>0$ and $|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (29)
$\displaystyle\leq\frac{1}{2}\bigg{[}\frac{|B[z^{n}]|}{k^{{}^{n}}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}+\left|1+\Phi_{k}(R,r,\alpha,\beta)\right||\lambda_{0}|\bigg{]}\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
where $\sigma(z)=Rz,$ $\rho(z)=rz$ and $\Phi_{k}(R,r,\alpha,\beta)$ is given
by (18).
###### Proof of Theorem 2.6.
Since $P(z)$ does not vanish in $|z|<k,\,\,k\leq 1$, by Lemma 1.3, we have for
all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,\,\,|\beta|\leq 1,$ $R>1$
and $|z|\geq 1,$
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$ (30)
$\displaystyle\leq
k^{n}\left|B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\right|,$
where$\sigma(z)=Rz,$ $\rho(z)=rz,$ $\tau(z)=Rz/k^{2},$ $\eta(z)=rz/k^{2}$ and
$Q(z)=z^{n}\overline{P(1/\overline{z})}.$ Inequality (2) in conjunction with
Lemma 1.4 gives for all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq
1,\,\,|\beta|\leq 1,$ $R>r\geq k$ and $|z|\geq 1,$
$\displaystyle 2\big{|}$ $\displaystyle
B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$
$\displaystyle\leq\big{|}B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}+k^{n}\left|B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\right|$
$\displaystyle\leq\left\\{|\lambda_{0}|\big{|}1+\Phi_{k}(R,r,\alpha,\beta)\big{|}+\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right|\right\\}|\underset{\left|z\right|=k}{Max}\left|P(z)\right|.$
This completes the proof of Theorem 2.6.
∎
We finally prove the following result, which is the refinement of Theorem 2.6.
###### Theorem 2.7.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in the disk $|z|<k,$ where
$k\leq 1,$ then for all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1$ , $R>r\geq k>0$ and $|z|=1$,
$\displaystyle\bigg{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\Bigg{[}\bigg{\\{}\frac{|B[z^{n}]|}{k^{{}^{n}}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right|+\left|1+\Phi_{k}(R,r,\alpha,\beta)\right||\lambda_{0}|\bigg{\\}}\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
(31)
$\displaystyle-\bigg{\\{}\frac{|B[z^{n}]|}{k^{{}^{n}}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right|-\left|1+\Phi_{k}(R,r,\alpha,\beta)\right||\lambda_{0}|\bigg{\\}}\underset{\left|z\right|=k}{Min}\left|P(z)\right|\Bigg{]}$
where $\sigma(z)=Rz,$ $\rho(z)=rz$ and $\Phi_{k}(R,r,\alpha,\beta)$ is given
by (18).
###### Proof of Theorem 2.7.
Let $m=Min_{\left|z\right|=k}\left|P(z)\right|.$ If $P(z)$ has a zero on
$|z|=k,$ then the result follows from Theorem 2.6. We assume that $P(z)$ has
all its zeros in $|z|>k$ where $k\leq 1$ so that $m>0$. Now for every $\delta$
with $|\delta|<1$, it follows by Rouche’s theorem $h(z)=P(z)-\delta m$ does
not vanish in $|z|<k$. Applying Lemma 1.3 to the polynomial $h(z),$ we get for
all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$, $R>r\geq
k$ and $|z|\geq 1$
$\left|B[h\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)B[h\circ\rho](z)\right|\leq
k^{n}\left|B[q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[q\circ\eta](z)]\right|,$
where $\sigma(z)=Rz,$ $\rho(z)=rz,$ $\tau(z)=Rz/k^{2},$ $\eta(z)=rz/k^{2}$ and
$q(z)=z^{n}\overline{h(1/\overline{z})}=z^{n}\overline{P(1/\overline{z})}-\overline{\delta}mz^{n}$.
Equivalently,
$\displaystyle\big{|}B[P\circ\sigma](z)+$
$\displaystyle\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)-\delta\lambda_{0}\left(1+\Phi_{k}(R,r,\alpha,\beta)\right)m\big{|}$
$\displaystyle\leq$ $\displaystyle
k^{n}\bigg{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)$ (32)
$\displaystyle\,\,\,\,-\frac{\overline{\delta}}{k^{2n}}\left(R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right)mB[z^{n}]\bigg{|}$
where $Q(z)=z^{n}\overline{P(1/\overline{z})}.$ Since all the zeros of
$Q(z/k^{2})$ lie in $|z|\leq k,$ $k\leq 1$ by Corollary 2.4 applied to
$Q(z/k^{2})$, we have for $R>1$ and $|z|=1,$
$\displaystyle\big{|}B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)$
$\displaystyle B[Q\circ\eta](z)]\big{|}$
$\displaystyle\geq\frac{1}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|\underset{|z|=k}{Min}Q(z/k^{2})$
(33)
$\displaystyle=\frac{1}{k^{2n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|m.$
Now, choosing the argument of $\delta$ on the right hand side of inequality
(2) such that
$\displaystyle k^{n}$
$\displaystyle\bigg{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)-\frac{\overline{\delta}}{k^{2n}}\left(R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right)mB[z^{n}]\bigg{|}$
$\displaystyle=$ $\displaystyle
k^{n}\big{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\big{|}-\frac{1}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|m.$
for $|z|=1,$ which is possible by inequality (2). We get for $|z|=1$ ,
$\displaystyle\big{|}B[P\circ\sigma](z)+\Phi_{k}(R,r,\alpha,\beta)$
$\displaystyle
B[P\circ\rho](z)\big{|}-|\delta||\lambda_{0}||1+\Phi_{k}(R,r,\alpha,\beta)\big{|}m$
$\displaystyle\leq k^{n}\big{|}B[Q\circ\tau](z)$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\big{|}$ (34)
$\displaystyle\,\,\,\,-\frac{|\delta|}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|m.$
Equivalently for $|z|=1,R>r\geq k$, we have
$\displaystyle\big{|}B[P\circ\sigma](z)$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}-k^{n}\big{|}B[Q\circ\tau](z)]+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)]\big{|}$
(35)
$\displaystyle\leq|\delta|\left\\{|\lambda_{0}||1+\Phi_{k}(R,r,\alpha,\beta)\big{|}-\frac{1}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|\right\\}m.$
Letting $|\delta|\rightarrow 1$ in inequality (2), we obtain for all
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1,R>r\geq k$ and
$|z|=1,$
$\displaystyle\big{|}B[P\circ\sigma](z)$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}-k^{n}\big{|}B[Q\circ\tau](z)+\Phi_{k}(R,r,\alpha,\beta)B[Q\circ\eta](z)\big{|}$
(36)
$\displaystyle\leq\left\\{|\lambda_{0}||1+\Phi_{k}(R,r,\alpha,\beta)\big{|}-\frac{1}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|\right\\}m.$
Inequality (2) in conjunction with Lemma 1.4 gives for all
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$ $|\beta|\leq 1,R>1$ and
$|z|=1,$
$\displaystyle 2\big{|}B[P\circ\sigma](z)$
$\displaystyle+\Phi_{k}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{|}$
$\displaystyle\leq$
$\displaystyle\left\\{|\lambda_{0}|\big{|}1+\Phi_{k}(R,r,\alpha,\beta)\big{|}+\frac{1}{k^{n}}\left|R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\right||B[z^{n}]|\right\\}|\underset{|z|=k}{Max}|P(z)|$
$\displaystyle+\left\\{|\lambda_{0}||1+\Phi_{k}(R,r,\alpha,\beta)\big{|}-\frac{1}{k^{n}}\big{|}R^{n}+r^{n}\Phi_{k}(R,r,\alpha,\beta)\big{|}|B[z^{n}]|\right\\}\underset{|z|=k}{Min}|P(z)|.$
which is equivalent to inequality (2.7) and thus completes the proof of
theorem 2.7.
∎
If we take $\alpha=0,$ we get the following.
###### Corollary 2.8.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $|z|<k$ where $k\leq
1,$ then for all $\beta\in\mathbb{C}$ with $|\beta|\leq 1$ , $R>r\geq k$ and
$|z|=1$,
$\displaystyle\bigg{|}B[P\circ\sigma](z)$
$\displaystyle+\beta\left(\frac{R+k}{k+r}\right)^{n}B[P\circ\rho](z)\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\Bigg{[}\Bigg{\\{}\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}+r^{n}\beta\left(\frac{R+k}{k+1}\right)^{n}\right|+\left|1+\beta\left(\frac{R+k}{k+1}\right)^{n}\right||\lambda_{0}|\bigg{\\}}\underset{\left|z\right|=k}{Max}\left|B[P](z)\right|$
(37)
$\displaystyle-\bigg{\\{}\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}+r^{n}\beta\left(\frac{R+k}{k+1}\right)^{n}\right|-\left|1+\beta\left(\frac{R+k}{k+1}\right)^{n}\right||\lambda_{0}|\bigg{\\}}\underset{\left|z\right|=k}{Min}\left|B[P](z)\right|\Bigg{]}$
where $\sigma(z)=Rz$ and $\rho(z)=rz.$
For $\beta=0,$ Theorem 2.6 reduces to the following result.
###### Corollary 2.9.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $|z|<k$ where $k\leq
1,$ then for all $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>r\geq k$ and
$|z|=1$,
$\displaystyle\left|B[P\circ\sigma](z)-\alpha B[P\circ\rho](z)\right|$
$\displaystyle\leq\frac{1}{2}\Bigg{[}\left\\{\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}-\alpha
r^{n}\right|+\left|1-\alpha\right||\lambda_{0}|\right\\}\underset{\left|z\right|=k}{Max}\left|P(z)\right|$
(38) $\displaystyle-\left\\{\frac{|B[z^{n}]|}{k^{n}}\left|R^{n}-\alpha
r^{n}\right|-\left|1-\alpha\right||\lambda_{0}|\right\\}\underset{\left|z\right|=k}{Min}\left|P(z)\right|\Bigg{]}$
where $\sigma(z)=Rz$ and $\rho(z)=rz.$ The result is sharp and extremal
polynomial is $P(z)=az^{n}+b,|a|=|b|\neq 0.$
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|
arxiv-papers
| 2013-03-30T05:01:38 |
2024-09-04T02:49:43.640328
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. A. Rather and Suhail Gulzar",
"submitter": "Suhail Gulzar Mattoo Suhail Gulzar",
"url": "https://arxiv.org/abs/1304.0067"
}
|
1304.0138
|
# Differential equations and logarithmic intertwining operators for strongly
graded vertex algebras
Jinwei Yang
###### Abstract
We derive certain systems of differential equations for matrix elements of
products and iterates of logarithmic intertwining operators among strongly
graded generalized modules for a strongly graded conformal vertex algebra
under suitable assumptions. Using these systems of differential equations, we
verify the convergence and extension property needed in the logarithmic tensor
category theory for such strongly graded generalized modules developed by
Huang, Lepowsky and Zhang.
## 1 Introduction
In the present paper, we generalize the arguments in [H] and [HLZ] to prove
that for a strongly graded conformal vertex algebra $V$, matrix elements of
products and iterates of logarithmic intertwining operators among triples of
strongly graded generalized $V$-modules under suitable assumptions satisfy
certain systems of differential equations and that the prescribed singular
points are regular. Using these differential equations, we verify the
convergence and extension property needed in the theory of logarithmic tensor
categories for strongly graded generalized $V$-modules in [HLZ]. Consequently,
under certain assumptions on the strongly graded generalized modules for a
strongly graded conformal vertex algebra $V$, we obtain a natural structure of
braided tensor category on the category of strongly graded generalized
$V$-modules using the main result of [HLZ].
The notion of strongly graded conformal vertex algebra and the notion of its
strongly graded module were introduced in [HLZ] as natural concepts from which
the theory of logarithmic tensor categories was developed. A strongly
$A$-graded conformal vertex algebra $V$ (respectively, a strongly
$\tilde{A}$-graded $V$-module) is a vertex algebra (respectively, a
$V$-module), with a weight-grading provided by a conformal vector in $V$ (an
$L(0)$-eigenspace decomposition), and with a second, compatible grading by an
abelian group $A$ (respectively, an abelian group $\tilde{A}$ containing $A$
as its subgroup), satisfying certain grading restriction conditions. One
important source of examples of strongly graded conformal vertex algebras and
modules comes from the vertex algebras and modules associated with not
necessarily positive definite even lattices. In particular, the tensor
products of vertex operator algebras and the vertex algebras associated with
even lattices are strongly graded conformal vertex algebras (see [Y]). In
[B1], Borcherds used the vertex algebra associated with the self-dual
Lorentzian lattice of rank $2$ and its tensor product with $V^{\natural}$ to
construct the “Monster” Lie algebra.
It was proved in [H] that if every module $W$ for a vertex operator algebra
$V=\coprod_{n\in\mathbb{Z}}V_{(n)}$ satisfies the $C_{1}$-cofiniteness
condition, that is, dim $W/C_{1}(W)<\infty$, where $C_{1}(W)$ is the subspace
of $W$ spanned by elements of the form $u_{-1}w$ for $u\in
V_{+}=\coprod_{n>0}V_{(n)}$ and $w\in W$, then matrix elements of products and
iterates of intertwining operators among triples of $V$-modules satisfy
certain systems of differential equations. Moreover, for prescribed singular
points, there exist such systems of differential equations such that the
prescribed singular points are regular. In Section $11$ of [HLZ] (Part VII),
using the same argument as in [H], certain systems of differential equations
were derived for matrix elements of products and iterates of logarithmic
intertwining operators among triples of generalized $V$-modules. In this
paper, we prove similar, more general results for matrix elements of products
and iterates of logarithmic intertwining operators among triples of strongly
graded generalized modules for a strongly graded vertex algebra.
In the present paper, we generalize the $C_{1}$-cofiniteness condition for
generalized modules for a vertex operator algebra to a $C_{1}$-cofiniteness
condition for strongly graded generalized modules for a strongly graded vertex
algebra. That is, every strongly graded generalized $\tilde{A}$-module $W$ for
a strongly $A$-graded vertex algebra $V$ satisfies the condition dim
$W^{(\beta)}/(C_{1}(W))^{(\beta)}<\infty$ where $C_{1}(W)$ is the subspace of
$W$ spanned by elements of the form $u_{-1}w$ for $u\in
V_{+}=\coprod_{n>0}V_{(n)}$ and $w\in W$, $W^{(\beta)}$ and
$(C_{1}(W))^{(\beta)}$ are $\tilde{A}$-homogeneous subspace of $W$ and
$C_{1}(W)$ with $\tilde{A}$-grading $\beta$ for $\beta\in\tilde{A}$.
Furthermore, let $V_{0}$ be a strongly graded vertex subalgebra of $V$, the
$C_{1}$-cofiniteness condition for $W$ as a $V_{0}$-module implies the
$C_{1}$-cofiniteness condition for $W$ as a $V$-module. In particular, the
case that $W$ satisfies $C_{1}$-cofiniteness condition as a module for
$V^{(0)}$—the $A$-homogeneous subspace of $V$ with $A$-weight $0$—is the same
as the case that $W$ satisfies $C_{1}$-cofiniteness condition as a vertex
operator algebra module.
The key step in deriving systems of differential equations in [H] is to
construct a finitely generated $R=\mathbb{C}[z_{1}^{\pm 1},z_{2}^{\pm
1},(z_{1}-z_{2})^{-1}]$-module that is a quotient module of the tensor product
of $R$ and a quadruple of modules for a vertex operator algebra. However, for
a strongly graded conformal vertex algebra, the quotient module constructed in
the same way is not finitely generated since there are infinitely many
$\tilde{A}$-homogeneous subspaces in the strongly graded generalized modules.
In order to obtain a finitely generated quotient module, we assume that fusion
rules for triples of certain $\tilde{A}$-homogeneous subspaces of strongly
graded generalized $V$-modules viewed as $V^{(0)}$-modules are zero for all
but finitely many triples of such $\tilde{A}$-homogeneous subspaces.
Under the assumption on the fusion rules for triples of certain
$\tilde{A}$-homogeneous subspaces and the $C_{1}$-cofiniteness condition for
the strongly graded generalized modules, we construct a natural map from a
finitely generated $R$-module to the set of matrix elements of products and
iterates of logarithmic intertwining operators among triples of strongly
graded generalized $V$-modules. The images of certain elements under this map
provide systems of differential equations for the matrix elements of products
and iterates of logarithmic intertwining operators, as a consequence of the
$L(-1)$-derivative property for the logarithmic intertwining operators.
Moreover, for any prescribed singular point, we derive certain systems of
differential equations such that this prescribed singular point is regular.
Using these systems of differential equations, we verify the convergence and
extension property needed in the construction of associativity isomorphism for
the logarithmic tensor category structure developed in [HLZ]. Consequently, if
all the assumptions mentioned above are satisfied, we obtain a braided tensor
category structure on the category of strongly graded generalized $V$-modules.
The present paper is organized as follows: In section $2$, we recall the
definitions and some basic properties of strongly graded vertex algebras and
their strongly graded generalized modules. The $C_{1}$-cofiniteness condition
for strongly graded generalized modules is introduced in section $3$ and the
definitions of logarithmic intertwining operators among strongly graded
generalized modules is recalled in section $4$. The existence of systems of
differential equations and the existence of systems with regular prescribed
singular points are established in section $5$ and $6$, respectively. In
section $7$, we prove the convergence and extension property for products and
iterates of logarithmic intertwining operators among strongly graded
generalized modules for a strongly graded vertex algebra. Consequently, we
obtain the braided tensor category structure on the category of strongly
graded generalized modules generalizing the results in [HLZ].
Acknowledgements I would like to thank Professor Yi-Zhi Huang and Professor
James Lepowsky for helpful discussions and suggestions.
## 2 Strongly graded vertex algebras and their modules
In this section, we recall the basic definitions from [HLZ] (cf. [Y]).
###### Definition 2.1
A conformal vertex algebra is a ${\mathbb{Z}}$-graded vector space
$V=\coprod_{n\in{\mathbb{Z}}}V_{(n)}$
equipped with a linear map:
$\displaystyle V$ $\displaystyle\to$ $\displaystyle({\rm End}\;V)[[x,x^{-1}]]$
$\displaystyle v$ $\displaystyle\mapsto$ $\displaystyle
Y(v,x)={\displaystyle\sum_{n\in{\mathbb{Z}}}}v_{n}x^{-n-1},$
and equipped also with two distinguished vectors: vacuum vector ${\bf 1}\in
V_{(0)}$ and conformal vector $\omega\in V_{(2)}$, satisfying the following
conditions for $u,v\in V$:
* •
the lower truncation condition:
$u_{n}v=0\;\;\mbox{\; for}\;\;n\mbox{ \;sufficiently\; large};$
* •
the vacuum property:
$Y({\bf 1},x)=1_{V};$
* •
the creation property:
$Y(v,x){\bf 1}\in V[[x]]\;\;\mbox{ and }\;\lim_{x\rightarrow 0}Y(v,x){\bf
1}=v;$
* •
the Jacobi identity (the main axiom):
$\displaystyle
x_{0}^{-1}\delta\bigg{(}{\displaystyle\frac{x_{1}-x_{2}}{x_{0}}}\bigg{)}Y(u,x_{1})Y(v,x_{2})-x_{0}^{-1}\delta\bigg{(}{\displaystyle\frac{x_{2}-x_{1}}{-x_{0}}}\bigg{)}Y(v,x_{2})Y(u,x_{1})$
$\displaystyle=x_{2}^{-1}\delta\bigg{(}{\displaystyle\frac{x_{1}-x_{0}}{x_{2}}}\bigg{)}Y(Y(u,x_{0})v,x_{2});$
* •
the Virasoro algebra relations:
$[L(m),L(n)]=(m-n)L(m+n)+{\displaystyle\frac{1}{12}}(m^{3}-m)\delta_{n+m,0}c$
for $m,n\in{\mathbb{Z}}$, where
$L(n)=\omega_{n+1}\;\;\mbox{ for
}\;n\in{\mathbb{Z}},\;\;\mbox{i.e.},\;\;Y(\omega,x)=\sum_{n\in{\mathbb{Z}}}L(n)x^{-n-2},$
$c\in{\mathbb{C}}\;\;\;(\mbox{central charge of}\;V);$
satisfying the $L(-1)$-derivative property:
${\displaystyle\frac{d}{dx}}Y(v,x)=Y(L(-1)v,x);$
and
$L(0)v=nv=(\mbox{\rm wt}\ v)v\;\;\mbox{ for }\;n\in{\mathbb{Z}}\;\mbox{ and
}\;v\in V_{(n)}.$
This completes the definition of the notion of conformal vertex algebra. We
will denote such a conformal vertex algebra by $(V,Y,{\bf 1},\omega)$.
###### Definition 2.2
Given a conformal vertex algebra $(V,Y,{\bf 1},\omega)$, a module for $V$ is a
${\mathbb{C}}$-graded vector space
$W=\coprod_{n\in{\mathbb{C}}}W_{(n)}$ (2.1)
equipped with a linear map
$\displaystyle V$ $\displaystyle\rightarrow$ $\displaystyle(\mbox{End}\
W)[[x,x^{-1}]]$ $\displaystyle v$ $\displaystyle\mapsto$ $\displaystyle
Y(v,x)=\sum_{n\in{\mathbb{Z}}}v_{n}x^{-n-1}$
such that the following conditions are satisfied:
* •
the lower truncation condition: for $v\in V$ and $w\in W$,
$v_{n}w=0\;\;\mbox{ for }\;n\;\mbox{ sufficiently large};$
* •
the vacuum property:
$Y(\mbox{\bf 1},x)=1_{W};$
* •
the Jacobi identity for vertex operators on $W$: for $u,v\in V$,
$\displaystyle{\displaystyle x^{-1}_{0}\delta\bigg{(}{x_{1}-x_{2}\over
x_{0}}\bigg{)}Y(u,x_{1})Y(v,x_{2})-x^{-1}_{0}\delta\bigg{(}{x_{2}-x_{1}\over-
x_{0}}\bigg{)}Y(v,x_{2})Y(u,x_{1})}$
$\displaystyle{\displaystyle=x^{-1}_{2}\delta\bigg{(}{x_{1}-x_{0}\over
x_{2}}\bigg{)}Y(Y(u,x_{0})v,x_{2})};$
* •
the Virasoro algebra relations on $W$ with scalar $c$ equal to the central
charge of $V$:
$[L(m),L(n)]=(m-n)L(m+n)+{\displaystyle\frac{1}{12}}(m^{3}-m)\delta_{n+m,0}c$
for $m,n\in{\mathbb{Z}}$, where
$L(n)=\omega_{n+1}\;\;\mbox{ for }n\in{\mathbb{Z}},\;\;{\rm
i.e.},\;\;Y(\omega,x)=\sum_{n\in{\mathbb{Z}}}L(n)x^{-n-2};$
satisfying the $L(-1)$-derivative property
$\displaystyle\frac{d}{dx}Y(v,x)=Y(L(-1)v,x);$
and
$(L(0)-n)w=0\;\;\mbox{ for }\;n\in{\mathbb{C}}\;\mbox{ and }\;w\in W_{(n)}.$
(2.2)
This completes the definition of the notion of module for a conformal vertex
algebra.
###### Definition 2.3
A generalized module for a conformal vertex algebra is defined in the same way
as a module for a conformal vertex algebra except that in the grading (2.1),
each space $W_{(n)}$ is replaced by $W_{[n]}$, where $W_{[n]}$ is the
generalized $L(0)$-eigenspace corresponding to the generalized eigenvalue
$n\in\mathbb{C}$; that is, (2.1) and (2.2) in the definition are replaced by
$W=\coprod_{n\in{\mathbb{C}}}W_{[n]}$
and
${\rm for}\ n\in\mathbb{C}\ {\rm and}\ w\in W_{[n]},\ (L(0)-n)^{k}w=0,\ {\rm
for}\ k\in\mathbb{N}\ {\rm sufficiently\ large},$
respectively. For $w\in W_{[n]}$, we still write wt $w=n$ for the generalized
weight of $w$.
###### Definition 2.4
Let $A$ be an abelian group. A conformal vertex algebra
$V=\coprod_{n\in{\mathbb{Z}}}V_{(n)}$
is said to be strongly graded with respect to $A$ (or strongly $A$-graded, or
just strongly graded if the abelian group $A$ is understood) if it is equipped
with a second gradation, by $A$,
$V=\coprod_{\alpha\in A}V^{(\alpha)},$
such that the following conditions are satisfied: the two gradations are
compatible, that is,
$V^{(\alpha)}=\coprod_{n\in{\mathbb{Z}}}V^{(\alpha)}_{(n)},\;\;\mbox{where}\;V^{(\alpha)}_{(n)}=V_{(n)}\cap
V^{(\alpha)}\;\mbox{ for any }\;\alpha\in A;$
for any $\alpha,\beta\in A$ and $n\in{\mathbb{Z}}$,
$\displaystyle V^{(\alpha)}_{(n)}=0\;\;\mbox{ for }\;n\;\mbox{ sufficiently
negative};$ $\displaystyle\dim V^{(\alpha)}_{(n)}<\infty;$ $\displaystyle{\bf
1}\in V^{(0)}_{(0)};\;\;\;\;\omega\in V^{(0)}_{(2)};$ $\displaystyle
v_{l}V^{(\beta)}\subset V^{(\alpha+\beta)}\;\;\mbox{ for any }\;v\in
V^{(\alpha)},\;l\in{\mathbb{Z}}.$
This completes the definition of the notion of strongly $A$-graded conformal
vertex algebra.
For modules for a strongly graded algebra we will also have a second grading
by an abelian group, and it is natural to allow this group to be larger than
the second grading group $A$ for the algebra. (Note that this already occurs
for the first grading group, which is ${\mathbb{Z}}$ for algebras and
${\mathbb{C}}$ for modules.)
###### Definition 2.5
Let $A$ be an abelian group and $V$ a strongly $A$-graded conformal vertex
algebra. Let $\tilde{A}$ be an abelian group containing $A$ as a subgroup. A
$V$-module (respectively, generalized $V$-module)
$W=\coprod_{n\in{\mathbb{C}}}W_{(n)}\;\;(\mbox{respectively,
}\;W^{(\beta)}=\coprod_{n\in{\mathbb{C}}}W_{[n]})$
is said to be strongly graded with respect to $\tilde{A}$ (or strongly
$\tilde{A}$-graded, or just strongly graded) if the abelian group $\tilde{A}$
is understood) if it is equipped with a second gradation, by $\tilde{A}$,
$W=\coprod_{\beta\in\tilde{A}}W^{(\beta)},$
such that the following conditions are satisfied: the two gradations are
compatible, that is, for any $\beta\in\tilde{A}$,
$W^{(\beta)}=\coprod_{n\in{\mathbb{C}}}W^{(\beta)}_{(n)},\;\;\mbox{where
}\;W^{(\beta)}_{(n)}=W_{(n)}\cap W^{(\beta)}$ $(\mbox{respectively,
}\;W^{(\beta)}=\coprod_{n\in{\mathbb{C}}}W^{(\beta)}_{[n]},\;\;\mbox{where
}\;W^{(\beta)}_{[n]}=W_{[n]}\cap W^{(\beta)});$
for any $\alpha\in A$, $\beta\in\tilde{A}$ and $n\in{\mathbb{C}}$,
$\displaystyle W^{(\beta)}_{(n+k)}=0\;\;(\mbox{respectively,
}\;W^{(\beta)}_{[n+k]}=0)\;\;\mbox{ for }\;k\in{\mathbb{Z}}\;\mbox{
sufficiently negative};$ (2.3) $\displaystyle\dim
W^{(\beta)}_{(n)}<\infty\;\;(\mbox{respectively, }\;\dim
W^{(\beta)}_{[n]}<\infty);$ $\displaystyle v_{l}W^{(\beta)}\subset
W^{(\alpha+\beta)}\;\;\mbox{ for any }\;v\in V^{(\alpha)},\;l\in{\mathbb{Z}}.$
A strongly $\tilde{A}$-graded (generalized) $V$-module $W$ is said to be lower
bounded if instead of (2.3), it satisfies the stronger condition that for any
$\beta\in\tilde{A}$,
$W_{(n)}^{(\beta)}=0\;\;(\mbox{respectively,}\;W_{[n]}^{(\beta)}=0)\;\;\mbox{for}\;\;n\in\mathbb{C}\;\;\mbox{and}\;\;\mathfrak{R}(n)\;\;\mbox{sufficiently
negative}.$
This completes the definition of the notion of strongly $\tilde{A}$-graded
generalized module for a strongly $A$-graded conformal vertex algebra.
###### Remark 2.6
In the strongly graded case, subalgebras (submodules) are vertex subalgebras
(submodules) that are strongly graded; algebra and module homomorphisms are of
course understood to preserve the grading by $A$ or $\tilde{A}$.
With the strong gradedness condition on a (generalized) module, we can now
define the corresponding notion of contragredient module.
###### Definition 2.7
Let $W=\coprod_{\beta\in\tilde{A},n\in\mathbb{C}}W_{[n]}^{(\beta)}$ be a
strongly $\tilde{A}$-graded generalized module for a strongly $A$-graded
conformal vertex algebra. For each $\beta\in\tilde{A}$ and $n\in\mathbb{C}$,
let us identify $(W_{[n]}^{(\beta)})^{*}$ with the subspace of $W^{*}$
consisting of the linear function on $W$ vanishing on each
$W_{[n]}^{(\gamma)}$ with $\gamma\neq\beta$ or $m\neq n$. We define
$W^{\prime}$ to be the $(\tilde{A}\times\mathbb{C})$-graded vector subspaces
of $W^{*}$ given by
$W^{\prime}=\coprod_{\beta\in\tilde{A},n\in\mathbb{C}}(W^{\prime})_{[n]}^{(\beta)},\;\;\mbox{where}\;\;(W^{\prime})_{[n]}^{(\beta)}=(W_{[n]}^{(-\beta)})^{*}.$
The adjoint vertex operators $Y^{\prime}(v,z)\;(v\in V)$ on $W^{\prime}$ is
defined in the same way as vertex operator algebra in section 5.2 in [FHL]
(see Section $2$ of [HLZ]). The pair $(W^{\prime},Y^{\prime})$ carries a
strongly graded module structure as follow:
###### Proposition 2.8
Let $\tilde{A}$ be an abelian group containing $A$ as a subgroup and $V$ a
strongly $A$-graded conformal vertex algebra. Let $(W,Y)$ be a strongly
$\tilde{A}$-graded $V$-module (respectively, generalized $V$-module). Then the
pair $(W^{\prime},Y^{\prime})$ carries a strongly $\tilde{A}$-graded
$V$-module (respectively, generalized $V$-module) structure. If $W$ is lower
bounded, so is $W^{\prime}$.
###### Definition 2.9
The pair $(W^{\prime},Y^{\prime})$ is called the contragredient module of
$(W,Y)$.
###### Example 2.10
Note that the notion of conformal vertex algebra strongly graded with respect
to the trivial group is exactly the notion of vertex operator algebra. Let $V$
be a vertex operator algebra, viewed (equivalently) as a conformal vertex
algebra strongly graded with respect to the trivial group. Then the
$V$-modules that are strongly graded with respect to the trivial group (in the
sense of Definition 2.5) are exactly the ($\mathbb{C}$-graded) modules for $V$
as a vertex operator algebra, with the grading restrictions as follows: For
$n\in\mathbb{C}$,
$W_{(n+k)}=0\;\;\mbox{ for }\;k\in{\mathbb{Z}}\;\mbox{ sufficiently negative}$
and
$\dim W_{(n)}<\infty.$
###### Example 2.11
An important source of examples of strongly graded conformal vertex algebras
and modules comes from the vertex algebras and modules associated with even
lattices. We recall the following construction from [FLM]. Let $L$ be an even
lattice, i.e., a finite-rank free abelian group equipped with a nondegenerate
symmetric bilinear form $\langle\cdot,\cdot\rangle$, not necessarily positive
definite, such that $\langle\alpha,\alpha\rangle\in 2{\mathbb{Z}}$ for all
$\alpha\in L$. Let $\mathfrak{h}=L\otimes_{\mathbb{Z}}\mathbb{C}$. Then
$\mathfrak{h}$ is a vector space with a nonsingular bilinear form
$\langle\cdot,\cdot\rangle$, extended from $L$. We form a Heisenberg algebra
$\widehat{\mathfrak{h}}_{\mathbb{Z}}=\coprod_{n\in\mathbb{Z},\ n\neq
0}\mathfrak{h}\otimes t^{n}\oplus\mathbb{C}c.$
Let $(\widehat{L},\bar{}\ )$ be a central extension of $L$ by a finite cyclic
group $\langle\kappa\;|\;\kappa^{s}=1\rangle$. Fix a primitive $s$th root of
unity, say $\omega$, and define the faithful character
$\chi:\langle\kappa\rangle\rightarrow\mathbb{C}^{*}$
by the condition
$\chi(\kappa)=\omega.$
Denote by $\mathbb{C}_{\chi}$ the one-dimensional space $\mathbb{C}$ viewed as
a $\langle\kappa\rangle$-module on which $\langle\kappa\rangle$ acts according
to $\chi$:
$\kappa\cdot 1=\omega,$
and denote by $\mathbb{C}\\{L\\}$ the induced $\widehat{L}$-module
$\mathbb{C}\\{L\\}={\rm
Ind}^{\widehat{L}}_{\langle\kappa\rangle}\mathbb{C}_{\chi}=\mathbb{C}[\widehat{L}]\otimes_{\mathbb{C}[\langle\kappa\rangle]}\mathbb{C}_{\chi}.$
Then
$V_{L}=S(\widehat{\mathfrak{h}}_{\mathbb{Z}}^{-})\otimes\mathbb{C}\\{L\\}$
has a natural structure of conformal vertex algebra; see [B1] and Chapter 8 of
[FLM]. For $\alpha\in L$, choose an $a\in\widehat{L}$ such that
$\bar{a}=\alpha$. Define
$\iota(a)=a\otimes 1\in\mathbb{C}\\{L\\}$
and
$V_{L}^{(\alpha)}=\mbox{span}\;\\{h_{1}(-n_{1})\cdots
h_{k}(-n_{k})\otimes\iota(a)\\},$
where $h_{1},\dots,h_{k}\in\mathfrak{h}$, $n_{1},\dots,n_{k}>0$, and where
$h(n)$ is the natural operator associated with $h\otimes t^{n}$ via the
$\hat{\mathfrak{h}}_{\mathbb{Z}}$-module structure of $V_{L}$. Then $V_{L}$ is
equipped with a natural second grading given by $L$ itself. Also for
$n\in\mathbb{Z}$, we have
$(V_{L})_{(n)}^{(\alpha)}={\rm span}\ \\{h_{1}(-n_{1})\cdots
h_{k}(-n_{k})\otimes\iota(a)|\
\bar{a}=\alpha,\sum_{i=1}^{k}n_{i}+\frac{1}{2}\langle\alpha,\alpha\rangle=n\\},$
making $V_{L}$ a strongly $L$-graded conformal vertex algebra in the sense of
definition 2.4. When the form $\langle\cdot,\cdot\rangle$ on $L$ is also
positive definite, then $V_{L}$ is a vertex operator algebra, that is, as in
example 2.10, $V_{L}$ is a strongly $A$-graded conformal vertex algebra for
$A$ the trivial group. In general, a conformal vertex algebra may be strongly
graded for several choinces of $A$.
Any sublattice $M$ of the “dual lattice” $L^{\circ}$ of $L$ containing $L$
gives rise to a strongly $M$-graded module for the strongly $L$-graded
conformal vertex algebra (see Chapter 8 of [FLM]; cf. [LL]). In fact, any
irreducible (generalized) $V_{L}$-module is equivalent to a $V_{L}$-module of
the form $V_{L+\beta}\subset V_{L^{\circ}}$ for some $\beta\in L^{\circ}$ and
any (generalized) $V_{L}$-module $W$ is equivalent to a direct sum of
irreducible $V_{L}$-modules. i.e.,
$W=\coprod_{\gamma_{i}\in L^{\circ},\ i=1,\dots,n}V_{\gamma_{i}+L},$
where $\gamma_{i}$’s are arbitrary elements of $L^{\circ}$, and
$n\in\mathbb{N}$ (see [D], [DLM]; cf. [LL]).
###### Definition 2.12
Let $V$ be a strongly $A$-graded conformal vertex algebra. The subspaces
$V_{(n)}^{(\alpha)}$ for $n\in\mathbb{Z}$, $\alpha\in A$ are called the doubly
homogeneous subspaces of $V$. The elements in $V_{(n)}^{(\alpha)}$ are called
doubly homogeneous elements. Similar definitions can be used for
$W^{(\beta)}_{(n)}$ (respectively, $W^{(\beta)}_{[n]}$) in the strongly graded
(generalized) module $W$.
###### Notation 2.13
Let $v$ be a doubly homogeneous element of $V$. Let wt $v_{n}$,
$n\in\mathbb{Z}$, refer to the weight of $v_{n}$ as an operator acting on $W$,
and let $A$-wt $v_{n}$ refer to the $A$-weight of $v_{n}$ on $W$. Similarly,
let $w$ be a doubly homogeneous element of $W$. We use wt $w$ to denote the
weight of $w$ and $\tilde{A}$-wt $w$ to denote the $\tilde{A}$-grading of $w$.
###### Lemma 2.14
Let $v\in V^{(\alpha)}_{(n)}$, for $n\in\mathbb{Z}$, $\alpha\in A$. Then for
$m\in\mathbb{Z}$, wt$\ v_{m}$ = $n-m-1$ and $A$-wt$\ v_{m}=\alpha$.
Proof. The first equation is standard from the theory of graded conformal
vertex algebras and the second follows easily from the definitions.
## 3 $C_{1}$-cofiniteness condition
In this section, we will let $V$ denote a strongly $A$-graded conformal vertex
algebra and let $W$ denote a strongly $\tilde{A}$-graded lower bounded
(generalized) $V$-module, where $A$, $\tilde{A}$ are abelian groups such that
$A\subset\tilde{A}$.
In the following definition, we generalize the $C_{1}$-cofiniteness condition
for the (generalized) modules for a vertex operator algebra to a
$C_{1}$-cofiniteness condition for the strongly graded (generalized) modules
for a strongly graded conformal vertex algebra.
###### Definition 3.1
Let $C_{1}(W)$ be the subspace of $W$ spanned by elements of the form
$u_{-1}w$ for
$u\in V_{+}=\coprod_{n>0}V_{(n)}$
and $w\in W$. The $\tilde{A}$-grading on $W$ induces a $\tilde{A}$-grading on
$W/C_{1}(W)$ with
$(W/C_{1}(W))^{(\beta)}=W^{(\beta)}/(C_{1}(W))^{(\beta)}.$
If dim $(W/C_{1}(W))^{(\beta)}<\infty$ for $\beta\in\tilde{A}$, we say that
$W$ is $C_{1}$-cofinite or $W$ satisfies the $C_{1}$-cofiniteness condition.
###### Remark 3.2
Let $V_{0}$ be a conformal vertex subalgebra of $V$ strongly graded with
respect to $A_{0}\subset A$. We can also define $C_{1}$-cofiniteness condition
for $W$ as a strongly graded (generalized) $V_{0}$-module. If $W$ is
$C_{1}$-cofinite as a strongly graded (generalized) $V_{0}$-module. Then $W$
is $C_{1}$-cofinite as a strongly graded (generalized) $V$-module.
###### Example 3.3
Let $V_{L}$ be the conformal vertex algebra associated with a nondegenerate
even lattice $L$ and let $W$ be a (generalized) $V_{L}$-module as in Example
2.11. Then the strongly graded (generalized) $V_{L}$-module $W$ satisfies the
$C_{1}$-cofiniteness condition as a $V_{L}^{(0)}$-module. Thus $W$ is also
$C_{1}$-cofinite as a strongly graded $V_{L}$-module.
## 4 Logarithmic intertwining operators
Throughout this paper, we shall use $x,x_{0},x_{1},x_{2},\dots$ to denote
commuting formal variables and $z,z_{0},z_{1},z_{2},\dots$ to denote complex
variables or complex numbers. We first recall the following definitions from
[HLZ].
###### Definition 4.1
Let $(W_{1},Y_{1})$, $(W_{2},Y_{2})$ and $(W_{3},Y_{3})$ be generalized
modules for a conformal vertex algebra $V$. A logarithmic intertwining
operator of type ${W_{3}\choose W_{1}\,W_{2}}$ is a linear map
${\cal Y}(\cdot,x)\cdot:W_{1}\otimes W_{2}\to W_{3}[\log x]\\{x\\},$ (4.1)
or equivalently,
$w_{(1)}\otimes w_{(2)}\mapsto{\cal
Y}(w_{(1)},x)w_{(2)}=\sum_{n\in{\mathbb{C}}}\sum_{k\in{\mathbb{N}}}{w_{(1)}}_{n;\,k}^{\cal
Y}w_{(2)}x^{-n-1}(\log x)^{k}\in W_{3}[\log x]\\{x\\}$ (4.2)
for all $w_{(1)}\in W_{1}$ and $w_{(2)}\in W_{2}$, such that the following
conditions are satisfied: the lower truncation condition: for any $w_{(1)}\in
W_{1}$, $w_{(2)}\in W_{2}$ and $n\in{\mathbb{C}}$,
${w_{(1)}}_{n+m;\,k}^{\cal Y}w_{(2)}=0\;\;\mbox{ for
}\;m\in{\mathbb{N}}\;\mbox{ sufficiently large,\, independently of}\;k;$ (4.3)
the Jacobi identity:
$\displaystyle\displaystyle x^{-1}_{0}\delta\bigg{(}{x_{1}-x_{2}\over
x_{0}}\bigg{)}Y_{3}(v,x_{1}){\cal Y}(w_{(1)},x_{2})w_{(2)}$ (4.4)
$\displaystyle\hskip 20.00003pt-x^{-1}_{0}\delta\bigg{(}{x_{2}-x_{1}\over-
x_{0}}\bigg{)}{\cal Y}(w_{(1)},x_{2})Y_{2}(v,x_{1})w_{(2)}$
$\displaystyle{\displaystyle=x^{-1}_{2}\delta\bigg{(}{x_{1}-x_{0}\over
x_{2}}\bigg{)}{\cal Y}(Y_{1}(v,x_{0})w_{(1)},x_{2})w_{(2)}}$
for $v\in V$, $w_{(1)}\in W_{1}$ and $w_{(2)}\in W_{2}$ (note that the first
term on the left-hand side is meaningful because of (4.3)); the
$L(-1)$-derivative property: for any $w_{(1)}\in W_{1}$,
${\cal Y}(L(-1)w_{(1)},x)=\frac{d}{dx}{\cal Y}(w_{(1)},x).$ (4.5)
###### Definition 4.2
In the setting of Definition 4.1, suppose in addition that $V$ and $W_{1}$,
$W_{2}$ and $W_{3}$ are strongly graded. A logarithmic intertwining operator
$\cal{Y}$ as in Definition 4.1 is a grading-compatible logarithmic
intertwining operator if for $\beta,\gamma\in\tilde{A}$ and $w_{1}\in
W_{1}^{(\beta)}$, $w_{2}\in W_{2}^{(\gamma)}$, $n\in\mathbb{C}$ and
$k\in\mathbb{N}$, we have
$(w_{1})_{n;k}w_{2}\in W_{3}^{(\beta+\gamma)}.$
###### Definition 4.3
In the setting of Definition 4.2, the grading-compatible logarithmic
intertwining operators of a fixed type ${W_{3}\choose W_{1}\,W_{2}}$ form a
vector space, which we denote by $\mathcal{V}_{W_{1}W_{2}}^{W_{3}}$. We call
the dimension of $\mathcal{V}_{W_{1}W_{2}}^{W_{3}}$ the fusion rule for
$W_{1}$, $W_{2}$ and $W_{3}$ and denote it by $N_{W_{1}W_{2}}^{W_{3}}$.
We shall use the following two sets in the next section: For
$\beta_{i}\in\tilde{A}$, $i=1,2,3$, set
$\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}=(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})\times(\beta_{3}+A_{0}).$
For any strongly $\tilde{A}$-graded generalized $V$-modules $W_{i}$
$(i=0,1,\dots,4)$ and any logarithmic intertwining operators $\mathcal{Y}_{1}$
and $\mathcal{Y}_{2}$ of type ${W_{0}^{\prime}\choose W_{1}\,W_{4}}$ and
${W_{4}\choose W_{2}\,W_{3}}$, respectively, set
$\displaystyle
I^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}=\left\\{(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}\;\middle|\begin{array}[]{ccc}&\mbox{if
there exist}\;\;\;w_{i}\in
W_{i}^{(\widetilde{\beta_{i}})}\;(i=1,2,3)\;\;\mbox{such that}\\\
&\;\;\;\mathcal{Y}_{1}(w_{1},x_{1})\mathcal{Y}_{2}(w_{2},x_{2})w_{3}\neq 0\\\
\end{array}\right\\}.$
For brevity, we will use $I^{(\beta_{1},\beta_{2},\beta_{3})}$ to denote the
set $I^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ in
the rest of this paper.
###### Lemma 4.4
Let $V$ be a strongly $A$-graded vertex algebra with a vertex subalgebra
$V_{0}$ strongly graded with respect to $A_{0}\subset A$. Suppose that every
strongly graded $V$-module satisfies $C_{1}$-cofiniteness condition as a
$V_{0}$-module. Also suppose that for any two fixed elements $\beta_{1}$ and
$\beta_{2}$ in $\tilde{A}$ and any triple of strongly graded generalized
$V$-modules $M_{1}$, $M_{2}$ and $M_{3}$, the fusion rule
$N_{M_{1}^{(\widetilde{\beta_{1}})}M_{2}^{(\widetilde{\beta_{2}})}}^{M_{3}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}})}}\neq
0$
for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$.
Then the set $I^{(\beta_{1},\beta_{2},\beta_{3})}$ defined above is a finite
set.
Proof. Since for the triple of strongly graded generalized modules
$(W_{1},W_{2},W_{3})$, the fusion rules
$N_{W_{1}^{(\widetilde{\beta_{1}})}W_{2}^{(\widetilde{\beta_{2}})}}^{W_{3}^{(\widetilde{\beta}_{1}+\widetilde{\beta}_{2})}}\neq
0$ for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$,
the logarithmic intertwining operator $\mathcal{Y}_{2}(w_{2},x_{2})w_{3}$,
where $w_{2}\in W_{2}^{(\widetilde{\beta_{2}})}$ and $w_{3}\in
W_{3}^{(\widetilde{\beta_{3}})}$, have to be $0$ except for finitely many
pairs
$(\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in(\beta_{2}+A_{0})\times(\beta_{3}+A_{0})$,
and then there are only finitely many triples
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}$
such that the products of logarithmic intertwining operators
$\displaystyle\mathcal{Y}_{1}(w_{1},x_{1})\mathcal{Y}_{2}(w_{2},x_{2})w_{3}\neq
0,$
where $w_{1}\in W_{1}^{(\widetilde{\beta_{1}})}$, $w_{2}\in
W_{2}^{(\widetilde{\beta_{2}})}$ and $w_{3}\in
W_{3}^{(\widetilde{\beta_{3}})}$. Thus the set
$I^{(\beta_{1},\beta_{2},\beta_{3})}$ is a finite set.
###### Remark 4.5
In the case that $A_{0}$ is a finite subgroup of $A$, the assumption in Lemma
4.4 holds automatically.
## 5 Differential equations
In the rest of this paper, we assume that $V$ is a strongly $A$-graded vertex
algebra with a vertex subalgebra $V_{0}$ strongly graded with respect to
$A_{0}\subset A$, and assume that every strongly graded (generalized)
$V$-module is $\mathbb{R}$-graded, lower bounded and satisfies
$C_{1}$-cofiniteness condition as a $V_{0}$-module.
Let $W_{i}$ be strongly $\tilde{A}$-graded generalized $V$-modules for
$i=0,1,\dots,4$ and let $\mathcal{Y}_{1}$ and $\mathcal{Y}_{2}$ be logarithmic
intertwining operators of type ${W_{0}^{\prime}\choose W_{1}\,W_{4}}$ and
${W_{4}\choose W_{2}\,W_{3}}$, respectively. Let
$\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}$ and
$I^{(\beta_{1},\beta_{2},\beta_{3})}$ be the two sets defined in the previous
section.
Let $R=\mathbb{C}[z_{1}^{\pm 1},z_{2}^{\pm 1},(z_{1}-z_{2})^{-1}]$,
$\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ be three fixed elements in
$\tilde{A}$. Set
$\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}=\coprod_{(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}}R\otimes
W_{0}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}})}\otimes
W_{1}^{\widetilde{(\beta_{1})}}\otimes W_{2}^{\widetilde{(\beta_{2})}}\otimes
W_{3}^{\widetilde{(\beta_{3})}}$
and
$T^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}=\coprod_{(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in
I^{(\beta_{1},\beta_{2},\beta_{3})}}R\otimes
W_{0}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}})}\otimes
W_{1}^{\widetilde{(\beta_{1})}}\otimes W_{2}^{\widetilde{(\beta_{2})}}\otimes
W_{3}^{\widetilde{(\beta_{3})}}.$
Then $\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$ and
$T^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ have
natural $R$-module structures. For convenience, in the rest of this paper, we
will use $T^{(\beta_{1},\beta_{2},\beta_{3})}$ to denote
$T^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$.
For simplicity, we shall omit one tensor symbol to write
$f(z_{1},z_{2})\otimes w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$ as
$f(z_{1},z_{2})w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$ in
$\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$ and
$T^{(\beta_{1},\beta_{2},\beta_{3})}$. For a strongly $\tilde{A}$-graded
generalized $V$-module $W$, let $(W^{\prime},Y^{\prime})$ be the
contragredient module of $W$ (recall definition 2.9). In particular, for $u\in
V$ and $n\in\mathbb{Z}$, we have the operators $u_{n}$ on $W^{\prime}$. Let
$u_{n}^{*}:W\rightarrow W$ be the adjoint of $u_{n}:W^{\prime}\rightarrow
W^{\prime}$. Note that since wt $u_{n}=$ wt $u-n-1$, we have wt
$u_{n}^{*}=-$wt $u+n+1$. Also, $A$-wt $u_{n}^{*}$ $=$ $-(A$-wt $u_{n})$.
Let
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}$
and let
$\widetilde{\beta_{0}}=\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}}$.
For $u\in(V_{0})_{+}$ and $w_{i}\in W_{i}^{(\widetilde{\beta_{i}})}$
$(i=0,1,2,3)$, let $J^{(\beta_{1},\beta_{2},\beta_{3})}$ be the submodule of
$\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$ generated by elements of the
form
$\displaystyle\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle\;\;\;\;\;\;\;\;=\sum_{k\geq 0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-z_{1})^{k}u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}-w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes w_{1}\otimes
u_{k}w_{2}\otimes w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)(-z_{1})^{-1-k}w_{0}\otimes
w_{1}\otimes w_{2}\otimes u_{k}w_{3},$
$\displaystyle\;\;\;\;\;\;\mathcal{B}(u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle\;\;\;\;\;\;\;\;=\sum_{k\geq 0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-z_{2})^{k}u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes u_{k}w_{1}\otimes
w_{2}\otimes w_{3}-w_{0}\otimes w_{1}\otimes u_{-1}w_{2}\otimes w_{3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)(-z_{2})^{-1-k}w_{0}\otimes
w_{1}\otimes w_{2}\otimes u_{k}w_{3},$
$\displaystyle\mathcal{C}(u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle\;\;\;\;\;\;\;\;=u_{-1}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}-\sum_{k\geq 0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)z_{1}^{-1-k}w_{0}\otimes u_{k}w_{1}\otimes w_{2}\otimes
w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)z_{2}^{-1-k}w_{0}\otimes
w_{1}\otimes u_{k}w_{2}\otimes w_{3}-w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{-1}w_{3},$
$\displaystyle\;\;\;\;\;\mathcal{D}(u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle\;\;\;\;\;\;\;\;=u_{-1}w_{0}\otimes w_{1}\otimes w_{2}\otimes
w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)z_{1}^{k+1}w_{0}\otimes
e^{z_{1}^{-1}L(1)}(-z_{1}^{2})^{L(0)}u_{k}(-z_{1}^{-2})^{L(0)}e^{-z_{1}^{-1}L(1)}w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)z_{2}^{k+1}w_{0}\otimes
w_{1}\otimes
e^{z_{2}^{-1}L(1)}(-z_{2}^{2})^{L(0)}u_{k}(-z_{2}^{-2})^{L(0)}e^{-z_{2}^{-1}L(1)}w_{2}\otimes
w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-w_{0}\otimes w_{1}\otimes
w_{2}\otimes u_{-1}^{*}w_{3}.$
We shall also need a submodule
$S^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ of
$\tilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$ generated by elements of the
form
$w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$
for $w_{i}\in W_{i}^{(\widetilde{\beta_{i}})}(i=0,1,2,3)$,
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in\tilde{I}^{(\beta_{1},\beta_{2},\beta_{3})}\setminus
I^{(\beta_{1},\beta_{2},\beta_{3})}$. For simplicity, we denote
$S^{(\beta_{1},\beta_{2},\beta_{3})}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ by
$S^{(\beta_{1},\beta_{2},\beta_{3})}$.
###### Lemma 5.1
Let $\beta_{i}\in\tilde{A}$. Then
$\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}=T^{(\beta_{1},\beta_{2},\beta_{3})}\oplus
S^{(\beta_{1},\beta_{2},\beta_{3})}.$
We shall find an $R$-submodule of
$\tilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$ such that its complement in
$T^{(\beta_{1},\beta_{2},\beta_{3})}$ is finitely generated. For this purpose,
we use the following $R$-submodule of
$\tilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$:
$\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}=J^{(\beta_{1},\beta_{2},\beta_{3})}\oplus
S^{(\beta_{1},\beta_{2},\beta_{3})}.$
For $r\in R$, we can define the $R$-submodules
$T^{(\beta_{1},\beta_{2},\beta_{3})}_{(r)}$,
$F_{r}(T^{(\beta_{1},\beta_{2},\beta_{3})})$ and
$F_{r}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ as in [H]. Note that
$F_{r}(T^{(\beta_{1},\beta_{2},\beta_{3})})$ is a finitely generated
$R$-module since $I^{(\beta_{1},\beta_{2},\beta_{3})}$ is a finite set by
Lemma 4.4.
###### Proposition 5.2
Let $W_{i}$ be strongly $\tilde{A}$-graded generalized $V$-modules and let
$\beta_{i}\in\tilde{A}$ for $i=0,1,2,3$. Then there exists $M\in\mathbb{Z}$
such that for any $r\in\mathbb{R}$,
$F_{r}(T^{(\beta_{1},\beta_{2},\beta_{3})})\subset
F_{r}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
In particular,
$T^{(\beta_{1},\beta_{2},\beta_{3})}\subset\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Proof. For $\widetilde{\beta_{i}}\in\tilde{A}$, let $\widetilde{\beta_{0}}$
denote $\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}}$ and
let $(C_{1}(W_{i}))^{(\widetilde{\beta_{i}})}$ be the subspace of $W_{i}$
spanned by elements of the form $u_{-1}w_{i}\in
W_{i}^{(\widetilde{\beta_{i}})}$, where
$u\in(V_{0})_{+}=\coprod_{n>0}(V_{0})_{(n)}.$
Since dim
$W_{i}^{(\widetilde{\beta_{i}})}/(C_{1}(W_{i}))^{(\widetilde{\beta_{i}})}<\infty$
for $i=0,1,2,3$, there exists $M\in\mathbb{Z}$ such that
$\displaystyle\coprod_{n>M}T^{(\beta_{1},\beta_{2},\beta_{3})}_{(n)}\subset\coprod_{(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in
I^{(\beta_{1},\beta_{2},\beta_{3})}}$ $\displaystyle
R((C_{1}(W_{0}))^{(\widetilde{\beta_{0}})}\otimes
W_{1}^{(\widetilde{\beta_{1}})}\otimes W_{2}^{(\widetilde{\beta_{2}})}\otimes
W_{3}^{(\widetilde{\beta_{3}})})$ $\displaystyle+$ $\displaystyle
R(W_{0}^{(\widetilde{\beta_{0}})}\otimes(C_{1}(W_{1}))^{(\widetilde{\beta_{1}})}\otimes
W_{2}^{(\widetilde{\beta_{2}})}\otimes W_{3}^{(\widetilde{\beta_{3}})})$
$\displaystyle+$ $\displaystyle R(W_{0}^{(\widetilde{\beta_{0}})}\otimes
W_{1}^{(\widetilde{\beta_{1}})}\otimes(C_{1}(W_{2}))^{(\widetilde{\beta_{2}})}\otimes
W_{3}^{(\widetilde{\beta_{3}})})$ $\displaystyle+$ $\displaystyle
R(W_{0}^{(\widetilde{\beta_{0}})}\otimes
W_{1}^{(\widetilde{\beta_{1}})}\otimes
W_{2}^{(\widetilde{\beta_{2}})}\otimes(C_{1}(W_{3}))^{(\widetilde{\beta_{3}})}).$
We use induction on $r\in\mathbb{R}$. If $r$ is equal to $M$,
$F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})\subset
F_{M}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Now we assume that $F_{r}(T^{(\beta_{1},\beta_{2},\beta_{3})})\subset
F_{r}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$
for $r<s$ where $s>M$. We want to show that any homogeneous element of
$T^{(\beta_{1},\beta_{2},\beta_{3})}_{(s)}$ can be written as a sum of an
element of $F_{s}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ and an element
of $F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$. Since $s>M$, by (5), any
element of $T^{(\beta_{1},\beta_{2},\beta_{3})}_{(s)}$ is an element of the
right hand side of (5). We shall discuss only the case that this element is in
$R(W_{0}^{(\widetilde{\beta_{0}})}\otimes(C_{1}(W_{1}))^{(\widetilde{\beta_{1}})}\otimes
W_{2}^{(\widetilde{\beta_{2}})}\otimes W_{3}^{(\widetilde{\beta_{3}})})$; the
other cases are completely similar.
We need only discuss elements of the form $w_{0}\otimes u_{-1}w_{1}\otimes
w_{2}\otimes w_{3}$, where $w_{i}\in W_{i}^{(\widetilde{\beta_{i}})}$ for
$i=0,2,3$, $u_{-1}w_{1}\in(C_{1}(W_{1}))^{(\widetilde{\beta_{1}})}$ and
$u\in(V_{0})_{+}$. We see from Lemma 5.1 that the elements
$u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$, $w_{0}\otimes
w_{1}\otimes u_{k}w_{2}\otimes w_{3}$ and $w_{0}\otimes w_{1}\otimes
w_{2}\otimes u_{k}w_{3}$ for $k\geq 0$ are either in
$S^{(\beta_{1},\beta_{2},\beta_{3})}$ or in
$T^{(\beta_{1},\beta_{2},\beta_{3})}$. By assumption, the weight of
$w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$ is $s$, then the weight
of $u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}$, $w_{0}\otimes
w_{1}\otimes u_{k}w_{2}\otimes w_{3}$ and $w_{0}\otimes w_{1}\otimes
w_{2}\otimes u_{k}w_{3}$ for $k\geq 0$, are all less than $s$. Thus these
elements either lie in $F_{s}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ or
in $F_{s-1}(T^{(\beta_{1},\beta_{2},\beta_{3})})$. Also, since
$\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3})\in
F_{s}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$, we see that
$\displaystyle w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$
$\displaystyle\;\;\;\;\;\;\;\;=\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3})+\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-z_{1})^{k}u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes w_{1}\otimes
u_{k}w_{2}\otimes w_{3}$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)(-z_{1})^{-1-k}w_{0}\otimes
w_{1}\otimes w_{2}\otimes u_{k}w_{3}$
can be written as a sum of an element of
$F_{s}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ and elements of
$F_{s-1}(T^{(\beta_{1},\beta_{2},\beta_{3})})$. Thus by the induction
assumption, the element $w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$
can be written as a sum of an element of
$F_{s}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ and an element of
$F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Now we have
$\displaystyle T^{(\beta_{1},\beta_{2},\beta_{3})}$
$\displaystyle=\coprod_{r\in\mathbb{R}}F_{r}(T^{(\beta_{1},\beta_{2},\beta_{3})})$
$\displaystyle\subset\coprod_{r\in\mathbb{R}}F_{r}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$
$\displaystyle=\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})}).$
We immediately obtain the following:
###### Corollary 5.3
The quotient $R$-module
$T^{(\beta_{1},\beta_{2},\beta_{3})}/T^{(\beta_{1},\beta_{2},\beta_{3})}\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
is finitely generated.
Proof. We have the following R-module isomorphism:
$T^{(\beta_{1},\beta_{2},\beta_{3})}/T^{(\beta_{1},\beta_{2},\beta_{3})}\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}\simeq
T^{(\beta_{1},\beta_{2},\beta_{3})}+\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}/\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}.$
By the previous Proposition, the R-module
$T^{(\beta_{1},\beta_{2},\beta_{3})}+\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}/\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
is a submodule of
$\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})/\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}\simeq
F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})/F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})},$
which is finitely generated.
For an element $\mathcal{W}\in T^{(\beta_{1},\beta_{2},\beta_{3})}$, we shall
use $[\mathcal{W}]$ to denote the equivalence class in
$T^{(\beta_{1},\beta_{2},\beta_{3})}/T^{(\beta_{1},\beta_{2},\beta_{3})}\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
containing $\mathcal{W}$. We also have:
###### Corollary 5.4
Let $W_{i}$ be strongly $\tilde{A}$-graded generalized $V$-modules for
$i=0,1,2,3$. For any $\tilde{A}$-homogeneous elements $w_{i}\in W_{i}$
$(i=0,1,2,3)$, let $M_{1}$ and $M_{2}$ be the $R$-submodules of
$T^{(\beta_{1},\beta_{2},\beta_{3})}/T^{(\beta_{1},\beta_{2},\beta_{3})}\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
generated by $[w_{0}\otimes L(-1)^{j}w_{1}\otimes w_{2}\otimes w_{3}]$, $j\geq
0$, and by $[w_{0}\otimes w_{1}\otimes L(-1)^{j}w_{2}\otimes w_{3}]$, $j\geq
0$, respectively. Then $M_{1}$, $M_{2}$ are finitely generated. In particular,
for any $\tilde{A}$-homogeneous elements $w_{i}\in W_{i}$ $(i=0,1,2,3)$, there
exist $a_{k}(z_{1},z_{2})$, $b_{l}(z_{1},z_{2})\in R$ for $k=1,\dots,m$ and
$l=1,\dots,n$ such that
$\displaystyle[w_{0}\otimes L(-1)^{m}w_{1}\otimes w_{2}\otimes
w_{3}]+a_{1}(z_{1},z_{2})[w_{0}\otimes L(-1)^{m-1}w_{1}\otimes w_{2}\otimes
w_{3}]$ $\displaystyle+\cdots+a_{m}(z_{1},z_{2})[w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}]=0,$ (5.15)
$\displaystyle[w_{0}\otimes w_{1}\otimes L(-1)^{n}w_{2}\otimes
w_{3}]+b_{1}(z_{1},z_{2})[w_{0}\otimes w_{1}\otimes L(-1)^{n-1}w_{2}\otimes
w_{3}]$ $\displaystyle+\cdots+b_{n}(z_{1},z_{2})[w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}]=0.$ (5.16)
Now we establish the existence of systems of differential equations:
###### Theorem 5.5
Let $V$ be a strongly $A$-graded vertex algebra with a vertex subalgebra
$V_{0}$ strongly graded with respect to $A_{0}\subset A$. Suppose that every
strongly graded $V$-module satisfies $C_{1}$-cofiniteness condition as a
$V_{0}$-module. Also suppose that for any two fixed elements $\beta_{1}$ and
$\beta_{2}$ in $\tilde{A}$ and any triple of strongly graded generalized
$V$-modules $M_{1}$, $M_{2}$ and $M_{3}$, the fusion rule
$N_{M_{1}^{(\widetilde{\beta_{1}})}M_{2}^{(\widetilde{\beta_{2}})}}^{M_{3}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}})}}\neq
0$
for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$.
Let $W_{i}$ be strongly $\tilde{A}$-graded generalized $V$-modules for
$i=0,1,2,3,4$ and let $\mathcal{Y}_{1}$ and $\mathcal{Y}_{2}$ be logarithmic
intertwining operators of type ${W_{0}^{\prime}\choose W_{1}\,W_{4}}$,
${W_{4}\choose W_{2}\,W_{3}}$. Then for any $\tilde{A}$-homogeneous elements
$w_{i}\in W_{i}$ ($i=0,1,2,3$), there exist
$a_{k}(z_{1},z_{2}),b_{l}(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm},(z_{1}-z_{2})^{-1}]$
for $k=1,\dots,m$ and $l=1,\dots,n$ such that the series
$\langle
w_{0},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle,$
(5.17)
satisfying the expansions of the system of differential equations
$\frac{\partial^{m}\varphi}{\partial
z_{1}^{m}}+a_{1}(z_{1},z_{2})\frac{\partial^{m-1}\varphi}{\partial
z_{1}^{m-1}}+\cdots+a_{m}(z_{1},z_{2})\varphi=0,$ (5.18)
$\frac{\partial^{n}\varphi}{\partial
z_{2}^{n}}+b_{1}(z_{1},z_{2})\frac{\partial^{n-1}\varphi}{\partial
z_{2}^{n-1}}+\cdots+b_{n}(z_{1},z_{2})\varphi=0$ (5.19)
in the region $|z_{1}|>|z_{2}|>0$.
Proof. The proof is similar to the proof of Theorem 1.4 in [H] except the
difference on the $R$-module $\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$. We
sketch the proof as follows:
Let $\Delta={\rm wt}\ w_{0}-{\rm wt}\ w_{1}-{\rm wt}\ w_{2}-{\rm wt}\ w_{3}$.
For $(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in
I^{(\beta_{1},\beta_{2},\beta_{3})}$, let
$\widetilde{\beta_{0}}=\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}}$.
Let $\mathbb{C}(\\{x\\})$ be the space of all series of the form
$\sum_{n\in\mathbb{R}}a_{n}x^{n}$ for $n\in\mathbb{R}$ such that $a_{n}=0$
when the real part of $n$ is sufficiently negative.
Consider the map
$\displaystyle\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}:T^{(\beta_{1},\beta_{2},\beta_{3})}\longrightarrow
z_{1}^{\Delta}\mathbb{C}(\\{z_{2}/z_{1}\\})[z_{1}^{\pm 1},z_{2}^{\pm 1}]$
defined by
$\displaystyle\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}(f(z_{1},z_{2})w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3})$
$\displaystyle=\iota_{|z_{1}|>|z_{2}|>0}(f(z_{1},z_{2}))\langle
w_{0},\mathcal{Y}_{1}(w_{1},z_{1})\mathcal{Y}_{2}(w_{2},z_{2})w_{3}\rangle,$
where
$\displaystyle\iota_{|z_{1}|>|z_{2}|>0}:R$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{C}[[z_{2}/z_{1}]][z_{1}^{\pm 1},z_{2}^{\pm 1}]$
is the map expanding elements of $R$ as series in the regions
$|z_{1}|>|z_{2}|>0$.
Using the Jacobi identity for the logarithmic intertwining operators, we have
that elements of $J^{(\beta_{1},\beta_{2},\beta_{3})}$ are in the kernel of
$\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$. The elements of
$S^{(\beta_{1},\beta_{2},\beta_{3})}$ are in the kernel by the construction of
the set $I^{(\beta_{1},\beta_{2},\beta_{3})}$. From Lemma 5.1, we have
$\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})=0.$
Thus the map $\phi_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ induces a map
$\displaystyle\bar{\phi}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}:T^{(\beta_{1},\beta_{2},\beta_{3})}/T^{(\beta_{1},\beta_{2},\beta_{3})}\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}\longrightarrow
z_{1}^{\Delta}\mathbb{C}(\\{z_{2}/z_{1}\\})[z_{1}^{\pm 1},z_{2}^{\pm 1}].$
Applying $\bar{\phi}_{\mathcal{Y}_{1},\mathcal{Y}_{2}}$ to (5.4) and (5.4) and
then use the $L(-1)$-derivative property for logarithmic intertwining
operators, we see that (5.17) indeed satisfies the expansions of the system of
differential equations in the regions $|z_{1}|>|z_{2}|>0$.
###### Remark 5.6
Note that in the theorems above, $a_{k}(z_{1};z_{2})$ for $k=1,\dots,m-1$ and
$b_{l}(z_{1};z_{2})$ for $l=1,\dots,l-1$, and consequently the corresponding
system, depend on the logarithmic intertwining operators $\mathcal{Y}_{1}$,
$\mathcal{Y}_{2}$.
The following result can be proved in the same method, so we omit the proof.
###### Theorem 5.7
Let $V$ be a strongly $A$-graded vertex algebra with a vertex subalgebra
$V_{0}$ strongly graded with respect to $A_{0}\subset A$. Suppose that every
strongly graded $V$-module satisfies $C_{1}$-cofiniteness condition as a
$V_{0}$-module. Also suppose that for any two fixed elements $\beta_{1}$ and
$\beta_{2}$ in $\tilde{A}$ and any triple of strongly graded generalized
$V$-modules $M_{1}$, $M_{2}$ and $M_{3}$, the fusion rules
$N_{M_{1}^{(\widetilde{\beta_{1}})}M_{2}^{(\widetilde{\beta_{2}})}}^{M_{3}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}})}}\neq
0$
for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$.
Let $W_{i}$ be strongly $\tilde{A}$-graded generalized $V$-modules for
$i=0,\dots,n+1$. For any generalized V-modules
$\widetilde{W_{1}},\dots,\widetilde{W_{n-1}}$, let
$\mathcal{Y}_{1},\mathcal{Y}_{2},\dots,\mathcal{Y}_{n-1},\mathcal{Y}_{n}$
be logarithmic intertwining operators of types
${W_{0}\choose W_{1}\,\widetilde{W_{1}}},{\widetilde{W_{1}}\choose
W_{2}\,\widetilde{W_{2}}},\dots,{\widetilde{W_{n-2}}\choose
W_{n-1}\,\widetilde{W_{n-1}}},{\widetilde{W_{n-1}}\choose W_{n}\,W_{n+1}},$
respectively. Then for any $\tilde{A}$-homogeneous elements
$w_{(0)}^{\prime}\in W_{0}^{\prime}$, $w_{(1)}\in W_{1},\dots,w_{(n+1)}\in
W_{n+1}$, there exist
$a_{k,l}(z_{1},\dots,z_{n})\in\mathbb{C}[z_{1}^{\pm 1},\dots,z_{n}^{\pm
1},(z_{1}-z_{2})^{-1},(z_{1}-z_{3})^{-1},\dots,(z_{n-1}-z_{n})^{-1}]$
for $k=1,\dots,m$ and $l=1,\dots,n$ such that the series
$\langle
w_{(0)}^{\prime},\mathcal{Y}_{1}(w_{(1)},z_{1})\cdots\mathcal{Y}_{n}(w_{(n)},z_{n})w_{(n+1)}\rangle$
satisfies the system of differential equations
$\frac{\partial^{m}\varphi}{\partial
z_{l}^{m}}+\sum_{k=1}^{m}a_{k,l}(z_{1},\dots,z_{n})\frac{\partial^{m-k}\varphi}{\partial
z_{l}^{m-k}}=0,\ \ l=1,\dots,n$ (5.20)
in the region $|z_{1}|>\cdots>|z_{n}|>0$.
###### Remark 5.8
Under the same condition as in the Theorem 5.5, it follows from the same
argument in this section that matrix elements of iterates of logarithmic
intertwining operators
$\langle
w_{(0)}^{\prime},\mathcal{Y}_{1}(\mathcal{Y}_{2}(w_{1},z_{1}-z_{2}),z_{2})w_{2}\rangle$
(5.21)
also satisfy the expansions of the system of differential equations of the
form (5.18) and (5.19) in the region $|z_{2}|>|z_{1}-z_{2}|>0$.
###### Example 5.9
Let $V_{L}$ be the conformal vertex algebra associated with a nondegenerate
even lattice $L$. Then any strongly graded generalized $V_{L}$-module $W$ (in
this example, all the generalized modules are modules) satisfies the
assumption in Theorem 5.5 and the series (5.17), (5.21) satisfies the
expansions of the system of differential equations (5.18) and (5.19) in the
regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}-z_{2}|>0$, respectively.
## 6 The regularity of the singular points
We first recall the definition for regular singular points for a system of
differential equations given in [K]. For the system of differential equations
of form (5.20), a singular point
$z_{0}=(z_{0}^{(1)},\dots,z_{0}^{(n)})$
is an isolated singular point of the coefficient matrix
$a_{k,l}(z_{1},\dots,z_{n})\in\mathbb{C}[z_{1}^{\pm 1},\dots,z_{n}^{\pm
1},(z_{1}-z_{2})^{-1},(z_{1}-z_{3})^{-1},\dots,(z_{n-1}-z_{n})^{-1}]$
for $k=1,\dots,m$ and $l=1,\dots,n$. For
$s=(s_{1},\dots,s_{n})\in\mathbb{Z}_{+}^{n}$, set
$|s|=\sum_{i=0}^{n}s_{i}$
and
$(\log(z-z_{0}))^{s}=(\log(z_{1}-z_{0}^{(1)}))^{s_{1}}\cdots(\log(z_{n}-z_{0}^{(n)}))^{s_{n}}.$
For $t=(t^{(1)},\dots,t^{(n)})\in\mathbb{C}^{n}$, set
$(z-z_{0})^{t}=(z_{1}-z_{0}^{(1)})^{t^{(1)}}\cdots(z_{n}-z_{0}^{(n)})^{t^{(n)}}.$
A singular point $z_{0}$ for the system of differential equations of form
(5.20) is regular if every solution in the punctured disc $(D^{\times})^{n}$
$0<|z_{i}-z_{0}^{(i)}|<a_{i}$
with some $a_{i}\in\mathbb{R}_{+}$ $(i=1,\dots,n)$ is of the form
$\varphi(z)=\sum_{i=1}^{r}\sum_{|m|<M}(z-z_{0})^{t_{i}}(\log(z-z_{0}))^{m}f_{t_{i},m}(z-z_{0})$
with $M,r\in\mathbb{Z}_{+}$ and each $f_{t_{i},m}(z-z_{0})$ holomorphic in
$(D^{\times})^{n}$. Theorem B.16 in [K] gives a sufficient condition for a
singular point of a system of differential equations to be regular.
As in [H], for $r\in\mathbb{R}$, we define the $R$-modules
$F_{r}^{(z_{1}=z_{2})}(R)$,
$F_{r}^{(z_{1}=z_{2})}(T^{(\beta_{1},\beta_{2},\beta_{3})})$ and
$F_{r}^{(z_{1}=z_{2})}(\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})})$,
which provide filtration associated to the singular point $z_{1}=z_{2}$ on
$R$, $R$-modules $T^{(\beta_{1},\beta_{2},\beta_{3})}$ and
$\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})}$, respectively.
For convenience, we shall use $\widetilde{\beta_{0}}$ to denote
$\widetilde{\beta_{1}}+\widetilde{\beta_{2}}+\widetilde{\beta_{3}}$ for
$\widetilde{\beta_{i}}\in\beta_{i}+A_{0}$ $(i=1,2,3)$. We shall also consider
the ring $\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$ and the
$\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$-module
$(T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})}=\coprod_{(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in
I^{(\beta_{1},\beta_{2},\beta_{3})}}\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]\otimes
W_{0}^{(\widetilde{\beta_{0}})}\otimes W_{1}^{(\widetilde{\beta_{1}})}\otimes
W_{2}^{(\widetilde{\beta_{2}})}\otimes W_{3}^{(\widetilde{\beta_{3}})}.$
Let $(T^{(\beta_{1},\beta_{2},\beta_{3})})_{(r)}^{(z_{1}=z_{2})}$ be the space
of elements of $(T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})}$ of
weight $r$ for $r\in\mathbb{R}$. Let
$F_{r}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})=\coprod_{s\leq
r}(T^{(\beta_{1},\beta_{2},\beta_{3})})_{(s)}^{(z_{1}=z_{2})}$. These
subspaces give a filtration of
$(T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})}$ in the following
sense: $F_{r}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})\subset
F_{s}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})$ for $r\leq s$
and
$(T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})}=\coprod_{r\in\mathbb{R}}F_{r}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})$.
Let
$F_{r}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})=F_{r}^{(z_{1}=z_{2})}(\widetilde{T}^{(\beta_{1},\beta_{2},\beta_{3})})\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
for $r\in\mathbb{R}$. We have the following lemma:
###### Lemma 6.1
For any $r\in\mathbb{R}$,
$F_{r}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})\subset
F_{r}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Proof. The proof is similar to the proof of Proposition 5.2 except some slight
differences. We discuss elements of the form $w_{0}\otimes u_{-1}w_{1}\otimes
w_{2}\otimes w_{3}$ with weight $s$, where $w_{i}\in
W_{i}^{(\widetilde{\beta_{i}})}$ for $i=0,1,2,3$ and $u\in(V_{0})_{+}$. By
definition of the element $\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3})$ in the
$R$-submodule $\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$, we have
$\displaystyle w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$
$\displaystyle\;\;\;\;\;\;\;\;\;=\sum_{k\geq 0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-z_{1})^{k}u_{-1-k}^{*}w_{0}\otimes w_{1}\otimes
w_{2}\otimes w_{3}-\mathcal{A}(u,w_{0},w_{1},w_{2},w_{3})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\
k\end{array}\right)(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes w_{1}\otimes
u_{k}w_{2}\otimes w_{3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\sum_{k\geq
0}\left(\begin{array}[]{c}-1\\\ k\end{array}\right)(-z_{1})^{-1-k}w_{0}\otimes
w_{1}\otimes w_{2}\otimes u_{k}w_{3}.$
We know from Lemma 5.1 that the elements $u_{-1-k}^{*}w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3}$, $w_{0}\otimes w_{1}\otimes u_{k}w_{2}\otimes
w_{3}$ and $w_{0}\otimes w_{1}\otimes w_{2}\otimes u_{k}w_{3}$ for $k\geq 0$
are either in
$S^{(\beta_{1},\beta_{2},\beta_{3})}\subset\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
or in $T^{(\beta_{1},\beta_{2},\beta_{3})}$ with weights less than the weight
of $w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$.
In the first case, since elements of the form $w_{0}\otimes w_{1}\otimes
u_{k}w_{2}\otimes w_{3}$ are in $F_{s-k-1}^{(z_{1}=z_{2})}(\tilde{J})$,
$(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes w_{1}\otimes u_{k}w_{2}\otimes w_{3}\in
F_{s}^{(z_{1}=z_{2})}(\tilde{J})$. Thus in this case, $w_{0}\otimes
u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$ is an element of
$F_{s}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$.
In the second case, by induction assumption, $u_{-1-k}^{*}w_{0}\otimes
w_{1}\otimes w_{2}\otimes w_{3}$, $w_{0}\otimes w_{1}\otimes w_{2}\otimes
u_{k}w_{3}\in
F_{s}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$
and $w_{0}\otimes w_{1}\otimes u_{k}w_{2}\otimes w_{3}\in
F_{s-k-1}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Hence the element $(-(z_{1}-z_{2}))^{-1-k}w_{0}\otimes w_{1}\otimes
u_{k}w_{2}\otimes w_{3}\in
F_{s}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Thus in this case, $w_{0}\otimes u_{-1}w_{1}\otimes w_{2}\otimes w_{3}$ can be
written as a sum of an element of
$F_{s}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})$ and an
element of $F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Using Lemma 6.1, we get the following refinement of proposition 5.2:
###### Proposition 6.2
For any $r\in\mathbb{R}$,
$F_{r}^{(z_{1}=z_{2})}(T^{(\beta_{1},\beta_{2},\beta_{3})})\subset
F_{r}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
In particular,
$F_{r}^{(z_{1}=z_{2})}(T^{(\beta_{1},\beta_{2},\beta_{3})})=F_{r}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})\cap
T^{(\beta_{1},\beta_{2},\beta_{3})}+F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$.
Proof. It is a consequence of the decomposition:
$F_{r}^{(z_{1}=z_{2})}(T^{(\beta_{1},\beta_{2},\beta_{3})})=\coprod_{i=0}^{r}(z_{1}-z_{2})^{-i}F_{r-i}((T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})})$
and Lemma 6.1.
Let $w_{i}\in W_{i}^{(\widetilde{\beta_{i}})}$ for $i=0,1,2,3$ and
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}},\widetilde{\beta_{3}})\in
I^{(\beta_{1},\beta_{2},\beta_{3})}$. Then by Proposition 6.2,
$w_{0}\otimes w_{1}\otimes w_{2}\otimes w_{3}=\mathcal{W}_{1}+\mathcal{W}_{2}$
where $\mathcal{W}_{1}\in
F_{\sigma}^{(z_{1}=z_{2})}(\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})})\cap
T^{(\beta_{1},\beta_{2},\beta_{3})}=F_{\sigma}^{(z_{1}=z_{2})}(T^{(\beta_{1},\beta_{2},\beta_{3})})\cap\tilde{J}^{(\beta_{1},\beta_{2},\beta_{3})}$
and $\mathcal{W}_{2}\in F_{M}(T^{(\beta_{1},\beta_{2},\beta_{3})})$. Using the
same proof as Lemma 2.2 in [H], we have the following lemma:
###### Lemma 6.3
For any $s\in[0,1)$, there exist $S\in\mathbb{R}$ such that
$s+S\in\mathbb{Z}_{+}$ and for any $w_{i}\in W_{i}$, $i=0,1,2,3$, satisfying
$\sigma\in s+\mathbb{Z}$,
$(z_{1}-z_{2})^{\sigma+S}\mathcal{W}_{2}\in(T^{(\beta_{1},\beta_{2},\beta_{3})})^{(z_{1}=z_{2})}$.
###### Theorem 6.4
Let $V$ be a strongly $A$-graded vertex algebra with a vertex subalgebra
$V_{0}$ strongly graded with respect to $A_{0}\subset A$. Suppose that every
strongly graded $V$-module satisfies $C_{1}$-cofiniteness condition as a
$V_{0}$-module. Also suppose that for any two fixed elements $\beta_{1}$ and
$\beta_{2}$ in $\tilde{A}$ and any triple of strongly graded generalized
$V$-modules $M_{1}$, $M_{2}$ and $M_{3}$, the fusion rule
$N_{M_{1}^{(\widetilde{\beta_{1}})}M_{2}^{(\widetilde{\beta_{2}})}}^{M_{3}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}})}}\neq
0$
for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$.
Let $W_{i}$, $w_{i}\in W_{i}$ for $i=0,1,2,3,4$, $\mathcal{Y}_{1}$ and
$\mathcal{Y}_{2}$ be the same as in Theorem 5.5. For any possible singular
point of the form $(z_{1}=0,z_{2}=0,z_{1}=\infty,z_{2}=\infty,z_{1}=z_{2})$,
$z_{1}^{-1}(z_{1}-z_{2})=0$, or $z_{2}^{-1}(z_{1}-z_{2})=0$, there exist
$a_{k}(z_{1},z_{2}),b_{l}(z_{1},z_{2})\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm},(z_{1}-z_{2})^{-1}]$
for $k=1,\dots,m$ and $l=1,\dots,n$, such that this singular point of the
system (5.18) and (5.19) satisfied by (5.17) is regular.
Proof. The proof is the same as the proof of Theorem 2.3 in [H] except that we
use Proposition 6.2 and Lemma 6.3 here.
We can prove the following theorem using the same method, so we omit the proof
here.
###### Theorem 6.5
For any set of possible singular points of the system (5.20) in Theorem 5.7 of
the form $z_{i}=0$ or $z_{i}=\infty$ for some $i$ or $z_{i}=z_{j}$ for some
$i\neq j$, the $a_{k,l}(z_{1},\dots,z_{n})$ in Theorem 5.7 can be chosen for
$k=1,\dots,m$ and $l=1,\dots,n$ so that these singular points are regular.
## 7 Braided tensor category structure
In the logarithmic tensor category theory developed in [HLZ], the convergence
and expansion property for the logarithmic intertwining operators are needed
in the construction of the associativity isomorphism. In this section, we will
recall the definition of convergence and expansion property for products and
iterates of logarithmic intertwining operators and then follow [HLZ] to give
sufficient conditions for a category to have these properties.
Throughout this section, we will let $\mathcal{M}_{sg}$ (respectively,
$\mathcal{GM}_{sg}$) denote the category of the strongly $\tilde{A}$-graded
(respectively, generalized) $V$-modules. We are going to study the subcategory
$\mathcal{C}$ of $\mathcal{M}_{sg}$ (respectively, $\mathcal{GM}_{sg}$)
satisfying the following assumptions.
###### Assumption 7.1
We shall assume the following:
* •
$A$ is an abelian group and $\tilde{A}$ is an abelian group containing $A$ as
a subgroup.
* •
$V$ is a strongly $A$-graded conformal vertex algebra with a strongly
$A_{0}\subset A$-graded vertex subalgebra $V_{0}$ and $V$ is an object of
$\cal{C}$ as a $V$-module.
* •
All (generalized) $V$-modules are lower bounded, satisfy the
$C_{1}$-cofiniteness condition as $V_{0}$-modules and for any two fixed
elements $\beta_{1}$ and $\beta_{2}$ in $\tilde{A}$ and any triple of strongly
graded generalized $V$-modules $M_{1}$, $M_{2}$ and $M_{3}$, the fusion rule
$N_{M_{1}^{(\widetilde{\beta_{1}})}M_{2}^{(\widetilde{\beta_{2}})}}^{M_{3}^{(\widetilde{\beta_{1}}+\widetilde{\beta_{2}})}}\neq
0$
for only finitely many pairs
$(\widetilde{\beta_{1}},\widetilde{\beta_{2}})\in(\beta_{1}+A_{0})\times(\beta_{2}+A_{0})$.
* •
For any object of $\cal{C}$, the (generalized) weights are real numbers and in
addition there exist $K\in\mathbb{Z}$ such that $(L(0)-L(0)_{s})^{K}=0$ on the
generalized module.
* •
$\cal{C}$ is closed under images, under the contragredient functor, under
taking finite direct sums.
Given objects $W_{1},W_{2},W_{3},W_{4},M_{1}$ and $M_{2}$ of the category
$\cal{C}$, let $\mathcal{Y}_{1},\mathcal{Y}_{2},\mathcal{Y}^{1}$ and
$\mathcal{Y}^{2}$ be logarithmic intertwining operators of types
${W_{4}\choose W_{1}\,M_{1}}$, ${M_{1}\choose W_{2}\,W_{3}}$, ${W_{4}\choose
M_{2}\,W_{3}}$ and ${M_{2}\choose W_{1}\,W_{2}}$, respectively. We recall the
following definitions and theorems from Section $11$ in [HLZ] (part VII):
Convergence and extension property for products For any $\beta\in\tilde{A}$,
there exists an integer $N_{\beta}$ depending only on $\mathcal{Y}_{1}$ and
$\mathcal{Y}_{2}$ and $\beta$, and for any doubly homogeneous elements
$w_{(1)}\in(W_{1})^{(\beta_{1})}$ and $w_{(2)}\in(W_{2})^{(\beta_{2})}$
$(\beta_{1},\beta_{2}\in\tilde{A})$ and any $w_{(3)}\in W_{3}$ and
$w_{(4)}^{\prime}\in W_{4}^{\prime}$ such that
$\beta_{1}+\beta_{2}=-\beta,$
there exist $M\in\mathbb{N}$, $r_{k},s_{k}\in\mathbb{R}$,
$i_{k},j_{k}\in\mathbb{N}$, $k=1,\dots,M$, and analytic functions $f_{k}(z)$
on $|z|<1$, $k=1,\dots,M$, satisfying
$\mbox{\rm wt}\ w_{(1)}+\mbox{\rm wt}\ w_{(2)}+s_{k}>N_{\beta},\ k=1,\dots,M,$
such that
$\langle
w_{(4)}^{\prime},\mathcal{Y}_{1}(w_{(1)},x_{1})\mathcal{Y}_{2}(w_{(2)},x_{2})w_{(3)}\rangle_{W_{4}}|_{x_{1}=z_{1},\
x_{2}=z_{2}}$
is absolutely convergent when $|z_{1}|>|z_{2}|>0$ and can be analytically
extended to the multivalued analytic function
$\sum_{k=1}^{M}z_{2}^{r_{k}}(z_{1}-z_{2})^{s_{k}}(\log
z_{2})^{i_{k}}(\log(z_{1}-z_{2}))^{j_{k}}f_{k}(\frac{z_{1}-z_{2}}{z_{2}})$
(here $\log(z_{1}-z_{2})$ and $\log z_{2}$, and in particular, the powers of
the variables, mean the multivalued functions, not the particular branch we
have been using) in the region $|z_{2}|>|z_{1}-z_{2}|>0$.
Convergence and extension property without logarithms for products When
$i_{k}=j_{k}=0$ for $k=1,\dots,M$, we call the property above the convergence
and extension property without logarithms for products.
Convergence and extension property for iterates For any $\beta\in\tilde{A}$,
there exists an integer $\tilde{N_{\beta}}$ depending only on
$\mathcal{Y}^{1}$ and $\mathcal{Y}^{2}$ and $\beta$, and for any doubly
homogeneous elements $w_{(1)}\in(W_{1})^{(\beta_{1})}$ and
$w_{(2)}\in(W_{2})^{(\beta_{2})}$ $(\beta_{1},\beta_{2}\in\tilde{A})$ and any
$w_{(3)}\in W_{3}$ and $w_{(4)}^{\prime}\in W_{4}^{\prime}$ such that
$\beta_{1}+\beta_{2}=-\beta,$
there exist $\tilde{M}\in\mathbb{N}$,
$\tilde{r_{k}},\tilde{s_{k}}\in\mathbb{R}$,
$\tilde{i_{k}},\tilde{j_{k}}\in\mathbb{N}$, $k=1,\dots,\tilde{M}$, and
analytic functions $\tilde{f_{k}}(z)$ on $|z|<1$, $k=1,\dots,M$, satisfying
$\mbox{\rm wt}\ w_{(1)}+\mbox{\rm wt}\
w_{(2)}+\tilde{s_{k}}>\tilde{N_{\beta}},\ k=1,\dots,\tilde{M},$
such that
$\langle
w_{(0)}^{\prime},\mathcal{Y}_{1}(\mathcal{Y}_{2}(w_{(1)},x_{0})w_{(2)},x_{2})w_{(3)}\rangle_{W_{4}}|_{x_{0}=z_{1}-z_{2},\
x_{2}=z_{2}}$
is absolutely convergent when $|z_{2}|>|z_{1}-z_{2}|>0$ and can be
analytically extended to the multivalued analytic function
$\sum_{k=1}^{\tilde{M}}z_{1}^{\tilde{r_{k}}}z_{2}^{\tilde{s_{k}}}(\log
z_{1})^{\tilde{i_{k}}}(\log
z_{2})^{\tilde{j_{k}}}\tilde{f_{k}}(\frac{z_{2}}{z_{1}})$
(here $\log z_{1}$ and $\log z_{2}$, and in particular, the powers of the
variables, mean the multivalued functions, not the particular branch we have
been using) in the region $|z_{1}|>|z_{2}|>0$.
Convergence and extension property without logarithmic for iterates When
$i_{k}=j_{k}=0$ for $k=1,\dots,M$, we call the property above the convergence
and extension property without logarithms for iterates.
If the convergence and extension property (with or without logarithms) for
products holds for any objects $W_{1},W_{2},W_{3},W_{4}$ and $M_{1}$ of
$\cal{C}$ and any logarithmic intertwining operators $\mathcal{Y}_{1}$ and
$\mathcal{Y}_{2}$ of the types as above, we say that the convergence and
extension property for products holds in $\cal{C}$. We similarly define the
meaning of the phrase the convergence and extension property for iterates
holds in $\cal{C}$.
The following theorem generalizes Theorem $11.8$ in [HLZ] to the strongly
graded generalized modules for a strongly graded conformal vertex algebra:
###### Theorem 7.2
Let $V$ be a strongly graded conformal vertex algebra. Then
* 1.
The convergence and extension properties for products and iterates hold in
$\cal{C}$. If $\cal{C}$ is in $\mathcal{M}_{sg}$ and if every object of
$\cal{C}$ is a direct sum of irreducible objects of $\cal{C}$ and there are
only finitely many irreducible objects of $\cal{C}$ (up to equivalence), then
the convergence and extension properties without logarithms for products and
iterates hold in $\cal{C}$.
* 2.
For any $n\in\mathbb{Z}_{+}$, any objects $W_{1},\dots,W_{n+1}$ and
$\widetilde{W_{1}},\dots,\widetilde{W_{n-1}}$ of $\cal{C}$, any logarithmic
intertwining operators
$\mathcal{Y}_{1},\mathcal{Y}_{2},\dots,\mathcal{Y}_{n-1},\mathcal{Y}_{n}$
of types
${W_{0}\choose W_{1}\,\widetilde{W_{1}}},{\widetilde{W_{1}}\choose
W_{2}\,\widetilde{W_{2}}},\dots,{\widetilde{W_{n-2}}\choose
W_{n-1}\,\widetilde{W_{n-1}}},{\widetilde{W_{n-1}}\choose W_{n}\,W_{n+1}},$
respectively, and any $w_{(0)}^{\prime}\in W_{0}^{\prime}$, $w_{(1)}\in
W_{1},\dots,W_{(n+1)}\in W_{n+1}$, the series
$\langle
w_{(0)}^{\prime},\mathcal{Y}_{1}(w_{(1)},z_{1})\cdots\mathcal{Y}_{n}(w_{(n)},z_{n})w_{(n+1)}\rangle$
is absolutely convergent in the region $|z_{1}|>\cdots>|z_{n}|>0$ and its sum
can be analytically extended to a multivalued analytic function on the region
given by $z_{1}\neq 0$, $i=1,\dots,n$, $z_{i}\neq z_{j}$, $i\neq j$, such that
for any set of possible singular points with either $z_{i}=0,z_{i}=\infty$ or
$z_{i}=z_{j}$ for $i\neq j$, this multivalued analytic function can be
expanded near the singularity as a series having the same form as the
expansion near the singular points of a solution of a system of differential
equations with regular singular points.
Proof. The first statement in the first part and the statement in the second
part of the theorem follow directly from Theorem 5.7 and Theorem 6.5 and the
theorem of differential equations with regular singular points. The second
statement in the first part can be proved using the same method in [H].
In order to construct braided tensor category on the category of strongly
graded generalized $V$-modules, we need the following assumption on $\cal{C}$
(see Assumption 10.1, Theorem 11.4 of [HLZ]).
###### Assumption 7.3
Suppose the following two conditions are satisfied:
* 1.
$\mathcal{C}$ is closed under $P(z)$-tensor products for some
$z\in\mathbb{C}^{\times}$.
* 2.
Every finite-generated lower bounded doubly graded generalized $V$-module is
an object of $\mathcal{C}$.
###### Conjecture 7.4
We conjectured that the category of certain strongly graded generalized
$V$-modules satisfying the first condition in Assumption 7.3. The case for the
vertex operator algebra was proved in [H1].
Under Assumption 7.1 and Assumption 7.3 on the category
$\mathcal{C}\subset\mathcal{GM}_{sg}$, we generalize the main result (Theorem
12.15) of [HLZ] to the category of strongly graded generalized modules for a
strongly graded vertex algebra:
###### Theorem 7.5
Let $V$ be a strongly graded conformal vertex algebra. Then the category
$\mathcal{C}$, equipped with the tensor product bifunctor $\boxtimes$, the
unit object $V$, the braiding isomorphism $\mathcal{R}$, the associativity
isomorphism $\mathcal{A}$, and the left and right unit isomorphisms $l$ and
$r$ in [HLZ], is an additive braided tensor category.
In the case that $\cal{C}$ is an abelian category, we have:
###### Corollary 7.6
If the category $\cal{C}$ is an abelian category, then $\mathcal{C}$, equipped
with the tensor product bifunctor $\boxtimes$, the unit object $V$, the
braiding isomorphism $\mathcal{R}$, the associativity isomorphism
$\mathcal{A}$, and the left and right unit isomorphisms $l$ and $r$ in [HLZ],
is a braided tensor category.
## REFERENCES
* [B1] R. E. Borcherds, Monstrous moonshine and the monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.
* [B2] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.
* [D] C. Dong, Vertex algebras associated with even lattices, J. of Algebra 161 (1993), 245–265.
* [DLM] C. Dong, H. Li and G. Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), 148–166.
* [FHL] I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, Amer. Math. Soc., Providence, 1993, no. 494 (preprint, 1989).
* [FLM] I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.
* [H] Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005), 375–400.
* [H1] Y.-Z. Huang, Cofiniteness conditions, projective covers and the logarithmic tensor product theory, J. Pure Appl. Alg. 213 (2009), 458–475.
* [HLZ] Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, Parts I - VIII, arXiv:1012.4193, arXiv:1012.4196, arXiv:1012.4197, arXiv:1012.4198, arXiv:1012.4199, arXiv:1012.4202, arXiv:1110.1929, arXiv:1110.1931.
* [K] A. W. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, Princeton, New Jersey, 1986.
* [LL] J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhäuser, Boston, 2003.
* [Y] J. Yang, Tensor products of strongly graded vertex algebras and their modules, J. Pure Appl. Alg. 217 (2013), 348–363.
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd.,
Piscataway, NJ 08854-8019
E-mail address: [email protected]
|
arxiv-papers
| 2013-03-30T21:17:25 |
2024-09-04T02:49:43.654839
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jinwei Yang",
"submitter": "Jinwei Yang",
"url": "https://arxiv.org/abs/1304.0138"
}
|
1304.0292
|
# Semiconcave functions in Alexandrov’s geometry
Anton Petrunin111Supported in part by the National Science Foundation under
grant # DMS-0406482.
###### Abstract
The following is a compilation of some techniques in Alexandrov’s geometry
which are directly connected to convexity.
## 0 Introduction
This paper is not about results, it is about available techniques in
Alexandrov’s geometry which are linked to semiconcave functions. We consider
only spaces with lower curvature bound, but most techniques described here
also work for upper curvature bound and even in more general settings.
Many proofs are omitted, I include only those which necessary for a continuous
story and some easy ones. The proof of the existence of quasigeodesics is
included in appendix A (otherwise it would never be published).
I did not bother with rewriting basics of Alexandrov’s geometry but I did
change notation, so it does not fit exactly in any introduction. I tried to
make it possible to read starting from any place. As a result the dependence
of statements is not linear, some results in the very beginning depend on
those in the very end and vice versa (but there should not be any cycle).
Here is a list of available introductions to Alexandrov’s geometry:
1. $\diamond$
[BGP] and its extension [Perelman 1991] is the first introduction to
Alexandrov’s geometry. I use it as the main reference.
Some parts of it are not easy to read. In the English translation of [BGP]
there were invented some militaristic terms, which no one ever used, mainly
_burst point_ should be _strained point_ and _explosion_ should be _collection
of strainers_.
2. $\diamond$
[Shiohama] intoduction to Alexandrov’s geometry, designed to be reader
friendly.
3. $\diamond$
[Plaut 2002] A survey in Alexandrov’s geometry written for topologists. The
first 8 sections can be used as an introduction. The material covered in my
paper is closely related to sections 7–10 of this survey.
4. $\diamond$
[BBI, Chapter 10] is yet an other reader friendly introduction.
I want to thank Karsten Grove for making me write this paper, Stephanie
Alexander, Richard Bishop, Sergei Buyalo, Vitali Kapovitch, Alexander Lytchak
and Conrad Plaut for many useful discussions during its preparation and
correction of mistakes, Irina Pugach for correcting my English.
###### Contents
1. 0 Introduction
1. 0.1 Notation and conventions
2. 1 Semi-concave functions.
1. 1.1 Definitions
2. 1.2 Variations of definition.
3. 1.3 Differential
3. 2 Gradient curves.
1. 2.1 Definition and main properties
2. 2.2 Gradient flow
3. 2.3 Applications
4. 3 Gradient exponent
1. 3.2 Spherical and hyperbolic gradient exponents
2. 3.3 Applications
5. 4 Extremal subsets
1. 4.1 Definition and properties.
2. 4.2 Applications
6. 5 Quasigeodesics
1. 5.1 Definition and properties
2. 5.2 Applications.
7. 6 Simple functions
1. 6.2 Smoothing trick.
8. 7 Controlled concavity
1. 7.2 General definition.
2. 7.3 Applications
9. 8 Tight maps
1. 8.2 Applications.
10. 9 Please deform an Alexandrov’s space.
11. A Existence of quasigeodesics
1. A.0 Step 0: Monotonic curves
2. A.1 Step 1: Convex curves.
3. A.2 Step 2: Pre-quasigeodesics
4. A.3 Step 3: Quasigeodesics
5. A.4 Quasigeodesics in extremal subsets.
### 0.1 Notation and conventions
1. $\diamond$
By $\text{{\nnn Alex}}^{m}(\kappa)$ we will denote the class of
$m$-dimensional Alexandrov’s spaces with curvature $\geqslant\kappa$. In this
notation we may omit $\kappa$ and $m$, but if not stated otherwise we assume
that dimension is finite.
2. $\diamond$
Gromov–Hausdorff convergence is understood with fixed sequence of
approximations. I.e. once we write
$X_{n}\buildrel\mathrm{GH}\over{\longrightarrow}X$ that means that we fixed a
sequence of Hausdorff approximations $f_{n}\colon X_{n}\to X$ (or equivalently
$g_{n}\colon X\to X_{n}$).
This makes possible to talk about limit points in $X$ for a sequence $x_{n}\in
X_{n}$, limit of functions $f_{n}\colon X_{n}\to\mathbb{R}$, Hausdorff limit
of subsets $S_{n}\subset X_{n}$ as well as weak limit of measures $\mu_{n}$ on
$X_{n}$.
3. $\diamond$
regular fiber — see page 32
4. $\diamond$
$\measuredangle xyz$ — angle at $y$ in a geodesic triangle $\triangle
xyz\subset A$
5. $\diamond$
$\measuredangle(\xi,\eta)$ — an angle between two directions
$\xi,\eta\in\Sigma_{p}$
6. $\diamond$
$\tilde{\measuredangle}_{\kappa}xyz$ — a comparison angle, i.e. angle of the
model triangle $\tilde{\triangle}xyz$ in $\hbox{\tencyr L}_{\kappa}$ at $y$.
7. $\diamond$
$\tilde{\measuredangle}_{\kappa}(a,b,c)$ — an angle opposite $b$ of a triangle
in $\hbox{\tencyr L}_{\kappa}$ with sides $a,b$ and $c$. In case $a+b<c$ or
$b+c<a$ we assume $\tilde{\measuredangle}_{\kappa}(a,b,c)=0$.
8. $\diamond$
$\uparrow_{p}^{q}$ — a direction at $p$ of a minimazing geodesic from $p$ to
$q$
9. $\diamond$
$\Uparrow_{p}^{q}$ — the set of all directions at $p$ of minimizing geodesics
from $p$ to $q$
10. $\diamond$
$A$ — usually an Alexandrov’s space
11. $\diamond$
$\operatorname{argmax}$ — see page 8
12. $\diamond$
$\partial A$ — boundary of $A$
13. $\diamond$
$\operatorname{dist}_{x}(y)=|xy|$ — distance between $x$ and $y$
14. $\diamond$
$d_{p}f$ — differential of $f$ at $p$, see page 1.3.1
15. $\diamond$
$\operatorname{gexp}_{p}$ — see section 3
16. $\diamond$
$\operatorname{gexp}_{p}(\kappa;v)$ — see section 3.2
17. $\diamond$
$\gamma^{\pm}$ — right/left tangent vector, see 2.1
18. $\diamond$
$\hbox{\tencyr L}_{\kappa}$ — model plane see page 4
19. $\diamond$
$\hbox{\tencyr L}_{\kappa}^{+}$ — model halfplane see page 3.3
20. $\diamond$
$\hbox{\tencyr L}_{\kappa}^{m}$ — model $m$-space, see page 7
21. $\diamond$
$\log_{p}$ — see page 1.3.2
22. $\diamond$
$\nabla_{p}f$ — gradient of $f$ at $p$, see definition 1.3.2
23. $\diamond$
$\rho_{\kappa}$ — see page 1.2.
24. $\diamond$
$\Sigma(X)$ — the spherical suspension over $X$ see [BGP, 4.3.1], in [Plaut
2002, 89] and [Berestovskii] it is called _spherical cone_.
25. $\diamond$
$\sigma_{\kappa}$ — see footnote 16 on page 16.
26. $\diamond$
$T_{p}=T_{p}A$ — tangent cone at $p\in A$, see page 1.3.
27. $\diamond$
$T_{p}E$ — see page 4.1.5
28. $\diamond$
$\Sigma_{p}=\Sigma_{p}A$ — see footnote 5 on page 5.
29. $\diamond$
$\Sigma_{p}E$ — see page 29
30. $\diamond$
$f^{\pm}$ — see page 2.1
## 1 Semi-concave functions.
### 1.1 Definitions
###### 1.1.1.
Definition for a space without boundary. Let $A\in\text{{\nnn Alex}}$,
$\partial A=\varnothing$ and $\Omega\subset A$ be an open subset.
A locally Lipschitz function $f\colon\Omega\rightarrow\mathbb{R}$ is called
$\lambda$-_concave_ if for any unit-speed geodesic $\gamma$ in $\Omega$, the
function
$f\circ\gamma(t)-\tfrac{\lambda}{2}{\cdot}t^{2}$
is concave.
If $A$ is an Alexandrov’s space with non-empty boundary222Boundary of
Alexandrov’s space is defined in [BGP, 7.19]., then its doubling333i.e. two
copies of $A$ glued along their boundaries. $\tilde{A}$ is also an
Alexandrov’s space (see [Perelman 1991, 5.2]) and
$\partial\tilde{A}=\varnothing$.
Set $\mathtt{p}\colon\tilde{A}\to A$ to be the canonical map.
###### 1.1.2.
Definition for a space with boundary. Let $A\in\text{{\nnn Alex}}$, $\partial
A\not=\varnothing$ and $\Omega\subset A$ be an open subset.
A locally Lipschitz function $f\colon\Omega\rightarrow\mathbb{R}$ is called
$\lambda$-_concave_ if $f\circ\mathtt{p}$ is $\lambda$-concave in
$\mathtt{p}^{-1}(\Omega)\subset\tilde{A}$.
Remark. Note that the restriction of a linear function on $\mathbb{R}^{n}$ to
a ball is not $0$-concave in this sense.
### 1.2 Variations of definition.
A function $f\colon A\to\mathbb{R}$ is called semiconcave if for any point
$x\in A$ there is a neighborhood $\Omega_{x}\ni x$ and $\lambda\in\mathbb{R}$
such that the restriction $f|_{\Omega_{x}}$ is $\lambda$-concave.
Let $\varphi\colon\mathbb{R}\to\mathbb{R}$ be a continuous function. A
function $f\colon A\to\mathbb{R}$ is called _$\varphi(f)$ -concave_ if for any
point $x\in A$ and any $\varepsilon>0$ there is a neighborhood $\Omega_{x}\ni
x$ such that $f|_{\Omega_{x}}$ is $(\varphi\circ f(x)+\varepsilon)$-concave
For the Alexandrov’s spaces with curvature $\geqslant\kappa$, it is natural to
consider the class of $(1-\kappa{\cdot}f)$-concave functions. The advantage of
such functions comes from the fact that on the model space444 i.e. the simply
connected 2-manifold of constant curvature $\kappa$ (the Russian L is for
Lobachevsky) $\hbox{\tencyr L}_{\kappa}$, one can construct model
$(1-\kappa{\cdot}f)$-concave functions which are equally concave in all
directions at any fixed point. The most important example of
$(1-\kappa{\cdot}f)$-concave function is
$\rho_{\kappa}\circ\operatorname{dist}_{x}$, where
$\operatorname{dist}_{x}(y)=|xy|$ denotes distance function from $x$ to $y$
and
$\rho_{\kappa}(x)=\left[\begin{matrix}\tfrac{1}{\kappa}{\cdot}(1-\cos(x{\cdot}\sqrt{\kappa}))&{\text{if
}}&\kappa>0\\\ x^{2}/2&{\text{if }}&\kappa=0\\\
\tfrac{1}{\kappa}{\cdot}(\operatorname{ch}(x{\cdot}\sqrt{-\kappa})-1)&{\text{if
}}&k<0\end{matrix}\right.$
In the above definition of $\lambda$-concave function one can exchange
Lipschitz continuity for usual continuity. Then it will define the same set of
functions, see corollary 3.3.2.
### 1.3 Differential
Given a point $p$ in an Alexandrov’s space $A$, we denote by $T_{p}=T_{p}A$
the tangent cone at $p$.
For an Alexandrov’s space, the tangent cone can be defined in two equivalent
ways (see [BGP, 7.8.1]):
1. $\diamond$
As a cone over space of directions at a point and
2. $\diamond$
As a limit of rescalings of the Alexandrov’s space, i.e.:
Given $s>0$, we denote the space $(A,s{\cdot}d)$ by $s{\cdot}A$, where $d$
denotes the metric of an Alexandrov’s space $A$, i.e. $A=(A,d)$. Let
$i_{s}\colon s{\cdot}A\to A$ be the canonical map. The limit of
$(s{\cdot}A,p)$ for $s\to\infty$ is the tangent cone $(T_{p},o_{p})$ at $p$
with marked origin $o_{p}$.
###### 1.3.1.
Definition. Let $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ be an open
subset.
For any function $f\colon\Omega\rightarrow\mathbb{R}$ the function
$d_{p}f\colon T_{p}\rightarrow\mathbb{R}$, $p\in\Omega$ defined by
$d_{p}f=\lim_{s\to\infty}s{\cdot}(f\circ i_{s}-f(p)),\ \ f\circ i_{s}\colon
s{\cdot}A\to\mathbb{R}$
is called the _differential_ of $f$ at $p$.
It is easy to see that the differential $d_{p}f$ is well defined for any
semiconcave function $f$. Moreover, $d_{p}f$ is a concave function on the
tangent cone $T_{p}$ which is positively homogeneous, i.e. $d_{p}f(r\cdot
v)=r\cdot d_{p}f(v)$ for $r\geqslant 0$.
#### Gradient.
With a slight abuse of notation, we will call elements of the tangent cone
$T_{p}$ the “tangent vectors” at $p$. The origin $o=o_{p}$ of $T_{p}$ plays
the role of a “zero vector”. For a tangent vector $v$ at $p$ we define its
absolute value $|v|$ as the distance $|ov|$ in $T_{p}$. For two tangent
vectors $u$ and $v$ at $p$ we can define their “scalar product”
$\langle
u,v\rangle\buildrel\mathrm{def}\over{=}(|u|^{2}+|v|^{2}-|uv|^{2})/2=|u|\cdot|v|\cdot\cos\alpha,$
where $\alpha=\measuredangle uov=\tilde{\measuredangle}_{0}uov$ in $T_{p}$.
It is easy to see that for any $u\in T_{p}$, the function $x\mapsto-\langle
u,x\rangle$ on $T_{p}$ is concave.
###### 1.3.2.
Definition. Let $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ be an open
subset. Given a $\lambda$-concave function $f\colon\Omega\to\mathbb{R}$, a
vector $g\in T_{p}$ is called a _gradient_ of $f$ at $p\in\Omega$ (in short:
$g=\nabla_{p}f$) if
(i) $d_{p}f(x)\leqslant\langle g,x\rangle\ \hbox{for any}\ x\in T_{p}$, and
(ii) $d_{p}f(g)=\langle g,g\rangle.$
It is easy to see that any $\lambda$-concave function
$f\colon\Omega\to\mathbb{R}$ has a uniquely defined gradient vector field.
Moreover, if $d_{p}f(x)\leqslant 0$ for all $x\in T_{p}$, then
$\nabla_{p}f=o_{p}$; otherwise,
$\nabla_{p}f=d_{p}f(\xi_{\max})\cdot\xi_{\max}$
where $\xi_{\max}\in\Sigma_{p}$555By $\Sigma_{p}\subset T_{p}$ we denote the
set of unit vectors, which we also call directions at $p$. The space
$(\Sigma_{p},\measuredangle)$ with angle metric is an Alexandrov’s space with
curvature $\geqslant 1$. $(\Sigma_{p},\measuredangle)$ it is also path-
isometric to the subset $\Sigma_{p}\subset T_{p}$. is the (necessarily unique)
unit vector for which the function $d_{p}f$ attains its maximum.
For two points $p,q\in A$ we denote by $\uparrow_{p}^{q}\,\in\Sigma_{p}$ a
direction of a minimizing geodesic from $p$ to $q$. Set
$\log_{p}q=|pq|\cdot\mskip-3.0mu\mskip-3.0mu\uparrow_{p}^{q}\in T_{p}$. In
general, $\uparrow_{p}^{q}$ and $\log_{p}q$ are not uniquely defined.
The following inequalities describe an important property of the “gradient
vector field” which will be used throughout this paper.
$p$$q$$\uparrow_{p}^{q}$$\uparrow_{q}^{p}$$\ell$$\nabla_{p}f$$\nabla_{q}f$
###### 1.3.3.
Lemma. Let $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ be an open subset,
$f\colon\Omega\to\mathbb{R}$ be a $\lambda$-concave function. Assume all
minimizing geodesics between $p$ and $q$ belong to $\Omega$, set $\ell=|pq|$.
Then
$\langle\uparrow_{p}^{q},\nabla_{p}f\rangle\geqslant{\\{f(q)-f(p)-\tfrac{\lambda}{2}{\cdot}\ell^{2}\\}}/{\ell},$
and in particular
$\langle\uparrow_{p}^{q},\nabla_{p}f\rangle+\langle\uparrow_{q}^{p},\nabla_{q}f\rangle\geqslant-\lambda{\cdot}\ell.$
Proof. Let $\gamma\colon[0,\ell]\to\Omega$ be a unit-speed minimizing geodesic
from $p$ to $q$, so
$\gamma(0)=p,\ \ \gamma(\ell)=q,\ \ \gamma^{+}(0)=\uparrow_{p}^{q}.$
From definition 1.3.2 and the $\lambda$-concavity of $f$ we get
$\displaystyle\langle\uparrow_{p}^{q},\nabla_{p}f\rangle$
$\displaystyle=\langle\gamma^{+}(0),\nabla_{p}f\rangle\geqslant$
$\displaystyle\geqslant d_{p}f(\gamma^{+}(0))=$
$\displaystyle=(f\circ\gamma)^{+}(0)\geqslant$
$\displaystyle\geqslant\frac{f\circ\gamma(\ell)-f\circ\gamma(0)-\tfrac{\lambda}{2}{\cdot}\ell^{2}}{\ell}$
and the first inequality follows (for definition of $\gamma^{+}$ and
$(f\circ\gamma)^{+}$ see 2.1).
The second inequality is just a sum of two of the first type. ∎
###### 1.3.4.
Lemma. Let $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$.
Let $f_{n}\colon A_{n}\to\mathbb{R}$ be a sequence of $\lambda$-concave
functions and $f_{n}\to f\colon A\to\mathbb{R}$.
Let $x_{n}\in A_{n}$ and $x_{n}\to x\in A$.
Then
$|\nabla_{x}f|\leqslant\liminf_{n\to\infty}|\nabla_{x_{n}}f_{n}|.$
In particular we have lower-semicontinuity of the function
$x\mapsto|\nabla_{x}f|$:
###### 1.3.5.
Corollary. Let $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ be an open
subset.
If $f\colon\Omega\to\mathbb{R}$ is a semiconcave function then the function
$x\mapsto|\nabla_{x}f|$
is lower-semicontinuos, i.e. for any sequence $x_{n}\to x\in\Omega$, we have
$|\nabla_{x}f|\leqslant\liminf_{n\to\infty}|\nabla_{x_{n}}f|.$
Proof of lemma 1.3.4. Fix an $\varepsilon>0$ and choose $q$ near $p$ such that
$\frac{f(q)-f(p)}{|pq|}>|\nabla_{p}f|-\varepsilon.$
Now choose $q_{n}\in A_{n}$ such that $q_{n}\to q$. If $|pq|$ is sufficiently
small and $n$ is sufficiently large, the $\lambda$-concavity of $f_{n}$ then
implies that
$\liminf_{n\to\infty}d_{p_{n}}f_{n}(\uparrow_{p_{n}}^{q_{n}})\geqslant|\nabla_{p}f|-2{\cdot}\varepsilon.$
Hence,
$\liminf_{n\to\infty}|\nabla_{p_{n}}f_{n}|\geqslant|\nabla_{p}f|-2{\cdot}\varepsilon\
\ \text{for any}\ \ \varepsilon>0$
and therefore
$\liminf_{n\to\infty}|\nabla_{p_{n}}f_{n}|\geqslant|\nabla_{p}f|.$
∎
#### Supporting and polar vectors.
###### 1.3.6.
Definition. Assume $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ is an open
subset, $p\in\Omega$, let $f\colon\Omega\to\mathbb{R}$ be a semiconcave
function.
A vector $s\in T_{p}$ is called a _supporting vector_ of $f$ at $p$ if
$d_{p}f(x)\leqslant-\langle s,x\rangle\ \ \hbox{for any}\ \ x\in T_{p}$
The set of supporting vectors is not empty, i.e.
###### 1.3.7.
Lemma. Assume $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ is an open
subset, $f\colon\Omega\to\mathbb{R}$ is a semiconcave function, $p\in\Omega$.
Then set of supporting vectors of $f$ at $p$ form a non-empty convex subset of
$T_{p}$.
Proof. Convexity of the set of supporting vectors follows from concavity of
the function $x\to-\langle u,x\rangle$ on $T_{p}$. To show existence, consider
a minimum point $\xi_{\min}\in\Sigma_{p}$ of the function
$d_{p}f|_{\Sigma_{p}}$. We will show that the vector
$s=\left[-d_{p}f(\xi_{\min})\right]\cdot\xi_{\min}$
is a supporting vector for $f$ at $p$. Assume that we know the existence of
supporting vectors in dimension $<m$. Applying it to $d_{p}f|_{\Sigma_{p}}$ at
$\xi_{\min}$, we get $d_{\xi_{\min}}(d_{p}f|_{\Sigma_{p}})\equiv 0$.
Therefore, since $d_{p}f|_{\Sigma_{p}}$ is $(-d_{p}f)$-concave (see section
1.2) for any $\eta\in\Sigma_{p}$ we have
$d_{p}f(\eta)\leqslant
d_{p}f(\xi_{\min})\cdot\cos\measuredangle(\xi_{\min},\eta)$
hence the result. ∎
In particular, it follows that if the space of directions $\Sigma_{p}$ has a
diameter666We always consider $\Sigma_{p}$ with angle metric.
$\leqslant\tfrac{\pi}{2}$ then $\nabla_{p}f=o$ for any $\lambda$-concave
function $f$.
Clearly, for any vector $s$, supporting $f$ at $p$ we have
$|s|\geqslant|\nabla_{p}f|.$
###### 1.3.8.
Definition. Two vectors $u,v\in T_{p}$ are called _polar_ if for any vector
$x\in T_{p}$ we have
$\langle u,x\rangle+\langle v,x\rangle\geqslant 0.$
More generally, a vector $u\in T_{p}$ is called polar to a set of vectors
$\mathcal{V}\subset T_{p}$ if
$\langle u,x\rangle+\sup_{v\in\mathcal{V}}\langle v,x\rangle\geqslant 0.$
Note that if $u,v\in T_{p}$ are polar to each other then
$d_{p}f(u)+d_{p}f(v)\leqslant 0.$ $None$
Indeed, if $s$ is a supporting vector then
$d_{p}f(u)+d_{p}f(v)\leqslant-\langle s,u\rangle-\langle s,v\rangle\leqslant
0.$
Similarly, if $u$ is polar to a set $\mathcal{V}$ then
$d_{p}f(u)+\inf_{v\in\mathcal{V}}\\{d_{p}f(v)\\}\leqslant 0.$ $None$
Examples of pairs of polar vectors.
1. (i)
If two vectors $u,v\in T_{p}$ are _antipodal_ , i.e. $|u|=|v|$ and
$\measuredangle uo_{p}v=\pi$ then they are polar to each other.
In general, if $|u|=|v|$ then they are polar if and only if for any $x\in
T_{p}$ we have $\measuredangle uo_{p}x+\measuredangle xo_{p}v\leqslant\pi$.
2. (ii)
If $\uparrow_{q}^{p}$ is uniquely defined then $\uparrow_{q}^{p}$ is polar to
$\nabla_{q}\operatorname{dist}_{p}$.
More generally, if $\Uparrow_{p}^{q}\subset\Sigma_{p}$ denotes the set of all
directions from $p$ to $q$ then $\nabla_{q}\operatorname{dist}_{p}$ is polar
to the set $\Uparrow_{q}^{p}$.
Both statement follow from the identity
$d_{q}(v)=\min_{\xi\in\Uparrow_{q}^{p}}\\{-\langle\xi,v\rangle\\}$
and the definition of gradient (see 1.3.2).
Given a vector $v\in T_{p}$, applying above property (ii) to the function
$\operatorname{dist}_{v}\colon T_{p}\to\nobreak\mathbb{R}$ we get that
$\nabla_{o}f_{v}$ is polar to $\uparrow_{o}^{v}$. Since there is a natural
isometry $T_{o}T_{p}\to T_{p}$ we have
###### 1.3.9.
Lemma. Given any vector $v\in T_{p}$ there is a polar vector $v^{*}\in T_{p}$.
Moreover, one can assume that $|v^{*}|\leqslant|v|$
In A.3.2 using quasigeodesics we will show that in fact one can assume
$|v^{*}|=|v|$
## 2 Gradient curves.
The technique of gradient curves was influenced by Sharafutdinov’s retraction,
see [Sharafutdinov]. These curves were designed to simplify Perelman’s proof
of existence of quasigeodesics. However, it turned out that gradient curves
themselves provide a superior tool, which is in fact almost universal in
Alexandrov’s geometry. Unlike most of Alexandrov’s techniques, gradient curves
work equally well for infinitely dimensional Alexandrov’s spaces (the proof
requires some quasifications, but essentially is the same), for spaces with
curvature bounded above and for locally compact spaces with well defined
tangent cone at each point, see [Lytchak]. It was pointed out to me that some
traces of these properties can be found even in general metric spaces see
[AGS].
### 2.1 Definition and main properties
Given a curve $\gamma(t)$ in an Alexandrov’s space $A$, we denote by
$\gamma^{+}(t)$ the right, and by $\gamma^{-}(t)$ the left, tangent vectors to
$\gamma(t)$, where, respectively,
$\gamma^{\pm}(t)\in T_{\gamma(t)},\ \ \ \gamma^{\pm}(t)=\lim_{\varepsilon\to
0+}\frac{\log_{\gamma(t)}\gamma(t\pm\varepsilon)}{\varepsilon}.$
This sign convention is not quite standard; in particular, for a function
$f\colon\mathbb{R}\to\mathbb{R}$, its right derivative is equal to $f^{+}$ and
its left derivative is equal to $-f^{-}(t)$. For example
$\ \ \text{if}\ \ f(t)=t\ \ \text{then}\ \ f^{+}(t)\equiv 1\ \ \text{and}\ \
f^{-}(t)\equiv-1.$
###### 2.1.1.
Definition. Let $A\in\text{{\nnn Alex}}$ and $f\colon A\to\mathbb{R}$ be a
semiconcave function.
A curve $\alpha(t)$ is called $f$-_gradient curve_ if for any $t$
$\alpha^{+}(t)=\nabla_{\alpha(t)}f.$
###### 2.1.2.
Proposition. Given a $\lambda$-concave function $f$ on an Alexandrov’s space
$A$ and a point $p\in A$ there is a unique gradient curve
$\alpha\colon[0,\infty)\rightarrow A$ such that $\alpha(0)=p$.
The gradient curve can be constructed as a limit of broken geodesics, made up
of short segments with directions close to the gradient. Convergence,
uniqueness, follow from lemma 1.3.3, while corollary 1.3.5 guarantees that the
limit is indeed a gradient curve.
#### Distance estimates.
###### 2.1.3.
Lemma. Let $A\in\text{{\nnn Alex}}$ and $f\colon A\to\mathbb{R}$ be a
$\lambda$-concave function and $\alpha(t)$ be an $f$-gradient curve.
Assume $\bar{\alpha}(s)$ is the reparametrization of $\alpha(t)$ by arclength.
Then $f\circ\bar{\alpha}$ is $\lambda$-concave.
Proof. For $s>s_{0}$,
$\displaystyle(f\circ\bar{\alpha})^{+}(s_{0})$
$\displaystyle=|\nabla_{\bar{\alpha}(s_{0})}f|\geqslant$
$\displaystyle\geqslant
d_{\bar{\alpha}(s_{0})}f\left(\uparrow_{\bar{\alpha}(s_{0})}^{\bar{\alpha}(s)}\right)\geqslant$
$\displaystyle\geqslant\frac{f(\bar{\alpha}(s))-f(\bar{\alpha}(s_{0}))-\tfrac{\lambda}{2}{\cdot}|\bar{\alpha}(s)\,\bar{\alpha}(s_{0})|^{2}}{|\bar{\alpha}(s)\,\bar{\alpha}(s_{0})|}.$
Therefore, since
$s-s_{0}\geqslant|\bar{\alpha}(s)\,\bar{\alpha}(s_{0})|=s-s_{0}-o(s-s_{0})$,
we have
$(f\circ\bar{\alpha})^{+}(s_{0})\geqslant\frac{f(\bar{\alpha}(s))-f(\bar{\alpha}(s_{0}))-\tfrac{\lambda}{2}{\cdot}(s-s_{0})^{2}}{s-s_{0}}+o(s-s_{0})$
i.e. $f\circ\bar{\alpha}$ is $\lambda$-concave.∎
The following lemma states that there is a nice parametrization of a gradient
curve (by $\vartheta_{\lambda}$) which makes them behave as a geodesic in some
respects.
###### 2.1.4.
Lemma. Let $A\in\text{{\nnn Alex}}$, $f\colon A\to\mathbb{R}$ be a
$\lambda$-concave function and $\alpha,\beta\colon[0,\infty)\to A$ be two
$f$-gradient curves with $\alpha(0)=p$, $\beta(0)=q$.
Then
1. (i)
for any $t\geqslant 0$,
$|\alpha(t)\beta(t)|\leqslant e^{\lambda{\cdot}t}|pq|$
2. (ii)
for any $t\geqslant 0$,
$|\alpha(t)q|^{2}\leqslant|pq|^{2}+\left\\{2{\cdot}f(p)-2{\cdot}f(q)+\lambda{\cdot}|pq|^{2}\right\\}\cdot\vartheta_{\lambda}(t)+|\nabla_{p}f|^{2}\cdot\vartheta^{2}_{\lambda}(t),$
where
$\vartheta_{\lambda}(t)=\int_{0}^{t}e^{\lambda{\cdot}t}\cdot
dt=\left[\begin{matrix}t&\text{if}&\lambda=0\\\
\frac{e^{\lambda{\cdot}t}-1}{\lambda}&\text{if}&\lambda\not=0\end{matrix}\right.$
3. (iii)
if $t_{p}\geqslant t_{q}\geqslant 0$ then
$|\alpha(t_{p})\beta(t_{q})|^{2}\leqslant
e^{2\lambda{\cdot}t_{q}}\bigl{[}|pq|^{2}+$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\left\\{2{\cdot}f(p)-2{\cdot}f(q)+\lambda{\cdot}|pq|^{2}\right\\}\cdot\vartheta_{\lambda}(t_{p}-t_{q})+$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+|\nabla_{p}f|^{2}\cdot\vartheta^{2}_{\lambda}(t_{p}-t_{q})\bigr{]}.$
In case $\lambda>0$, this lemma can also be reformulated in a geometer-
friendly way:
2.1.4${}^{\prime}\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu$. Lemma. Let $\alpha$,
$\beta$, $p$ and $q$ be as in lemma 2.1.4 and $\lambda>0$. Consider points
$\tilde{o},\tilde{p},\tilde{q}\subset\mathbb{R}^{2}$ defined by the following:
$|\tilde{p}\tilde{q}|=|pq|,\ \
\lambda{\cdot}|\tilde{o}\tilde{p}|=|\nabla_{p}f|,$
$\tfrac{\lambda}{2}{\cdot}\left(|\tilde{o}\tilde{q}|^{2}-|\tilde{o}\tilde{p}|^{2}\right)=f(q)-f(p)$
Let $\tilde{\alpha}(t)$ and $\tilde{\beta}(t)$ be
$\left({\frac{\lambda}{2}{\cdot}\operatorname{dist}^{2}_{\tilde{o}}}\right)$-gradient
curves in $\mathbb{R}^{2}$ with $\tilde{\alpha}(0)=\tilde{p}$,
$\tilde{\beta}(0)=\tilde{q}$. Then,
1. (i)
$|\alpha(t)q|\leqslant|\tilde{\alpha}(t)\tilde{q}|$ for any $t>0$
2. (ii)
$|\alpha(t)\beta(t)|\leqslant|\tilde{\alpha}(t)\tilde{\beta}(t)|$
3. (iii)
if $t_{p}\geqslant t_{q}$ then
$|\alpha(t_{p})\beta(t_{q})|\leqslant|\tilde{\alpha}(t_{p})\tilde{\beta}(t_{q})|$
$p$$q$$\alpha(t)$$\beta$
Proof. (ii). If $\lambda=0$, from lemma 2.1.3 it follows that777For
$\lambda\not=0$ it will be
$f\circ\alpha(t)-f\circ\alpha(0)\leqslant\left|\nabla_{\bar{\alpha}(0)}f\right|^{2}\cdot[\vartheta_{\lambda}(t)+\tfrac{\lambda}{2}{\cdot}\vartheta_{\lambda}^{2}(t)]$.
$f\circ\alpha(t)-f\circ\alpha(0)\leqslant\left|\nabla_{\bar{\alpha}(0)}f\right|^{2}\cdot
t.$
Therefore from lemma 1.3.3, setting $\ell=\ell(t)=|q\alpha(t)|$, we get888For
$\lambda\not=0$ it will be
$\left({\ell^{2}}/2\right)^{\prime}-\tfrac{\lambda}{2}{\cdot}\ell^{2}\leqslant
f(p)-f(q)+\left|\nabla_{p}f\right|^{2}\cdot[\vartheta_{\lambda}(t)+\tfrac{\lambda}{2}{\cdot}\vartheta_{\lambda}^{2}(t)]$.
$\left({\ell^{2}}/2\right)^{\prime}\leqslant
f(p)-f(q)+\left|\nabla_{p}f\right|^{2}\cdot t,$
hence the result.
(i) follows from the second inequality in lemma 1.3.3;
(iii) follows from (i) and (ii). ∎
#### Passage to the limit.
The next lemma states that gradient curves behave nicely with Gromov–Hausdorff
convergence, i.e. a limit of gradient curves is a gradient curve for the limit
function.
###### 2.1.5.
Lemma. Let $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$, $A_{n}\ni p_{n}\to p\in A$.
Let $f_{n}\colon A_{n}\to\mathbb{R}$ be a sequence of $\lambda$-concave
functions and $f_{n}\to f\colon A\to\mathbb{R}$.
Let $\alpha_{n}\colon[0,\infty)\to A_{n}$ be the sequence of $f_{n}$-gradient
curves with $\alpha_{n}(0)=p_{n}$ and let $\alpha\colon[0,\infty)\to A$ be the
$f$-gradient curve with $\alpha(0)=p$.
Then $\alpha_{n}\to\alpha$ as $n\to\infty$.
Proof. Let $\bar{\alpha}_{n}(s)$ denote the reparametrization of
$\alpha_{n}(t)$ by arc length. Since all $\bar{\alpha}_{n}$ are $1$-Lipschitz,
we can choose a partial limit, say $\bar{\alpha}(s)$ in $A$. Note that we may
assume that $f$ has no critical points and so $d(f\circ\bar{\alpha})\not=0$.
Otherwise consider instead the sequence $A^{\prime}_{n}=A_{n}\times\mathbb{R}$
with $f^{\prime}_{n}(a\times x)=f_{n}(a)+x$.
Clearly, $\bar{\alpha}$ is also 1-Lipschitz and hence, by Lemma 1.3.4,
$\displaystyle\lim_{n\to\infty}f_{n}\circ\bar{\alpha}_{n}|_{a}^{b}$
$\displaystyle=\lim_{n\to\infty}\int_{a}^{b}|\nabla_{\bar{\alpha}_{n}(s)}f_{n}|\cdot
ds\geqslant$
$\displaystyle\geqslant\int_{a}^{b}|\nabla_{\bar{\alpha}(s)}f|\cdot
ds\geqslant$
$\displaystyle\geqslant\int_{a}^{b}d_{\bar{\alpha}(s)}f(\bar{\alpha}^{+}(s))\cdot
ds=$ $\displaystyle=f\circ\bar{\alpha}|_{a}^{b},$
where $\bar{\alpha}^{+}(s)$ denotes any partial limit of
$\log_{\bar{\alpha}(s)}\bar{\alpha}(s+\varepsilon)/\varepsilon$,
$\varepsilon\to 0+$.
On the other hand, since $\bar{\alpha}_{n}\to\bar{\alpha}$ and $f_{n}\to f$ we
have $f_{n}\circ\bar{\alpha}_{n}|_{a}^{b}\to f\circ\bar{\alpha}|_{a}^{b}$,
i.e. equality holds in both of these inequalities. Hence
$|\nabla_{\bar{\alpha}(s)}f|=\lim_{n\to\infty}|\nabla_{\bar{\alpha}_{n}(s)}f_{n}|,\
\ \ |\bar{\alpha}^{+}(s)|=1\ \ \ \text{a.e.}$
and the directions of $\bar{\alpha}^{+}(s)$ and $\nabla_{\bar{\alpha}(s)}f$
coincide almost everywhere.
This implies that $\bar{\alpha}(s)$ is a gradient curve reparametrized by arc
length. It only remains to show that the original parameter $t_{n}(s)$ of
$\alpha_{n}$ converges to the original parameter $t(s)$ of $\alpha$.
Notice that $|\nabla_{\bar{\alpha}_{n}(s)}f_{n}|\cdot dt_{n}=ds$ or
$dt_{n}/ds=ds/d(f_{n}\circ\bar{\alpha}_{n})$. Likewise,
$dt/ds=ds/d(f\circ\bar{\alpha})$. Then the convergence $t_{n}\to t$ follows
from the $\lambda$-concavity of $f_{n}\circ\bar{\alpha}_{n}$ (see Lemma 2.1.3)
and the convergence $f_{n}\circ\bar{\alpha}_{n}\to f\circ\bar{\alpha}.$∎
### 2.2 Gradient flow
Let $f$ be a semi-concave function on an Alexandrov’s space $A$. We define the
$f$-_gradient flow_ to be the one parameter family of maps
$\Phi^{t}_{f}\colon A\to A,\ \ \Phi^{t}_{f}(p)=\alpha_{p}(t),$
where $t\geqslant 0$ and $\alpha_{p}\colon[0,\infty)\to A$ is the $f$-gradient
curve which starts at $p$ (i.e. $\alpha_{p}(0)=p$). 999In general the domain
of definition of $\Phi^{t}_{f}$ can be smaller than $A$, but it is defined on
all $A$ for a reasonable type of function, say for $\lambda$-concave and for
$(1-\kappa{\cdot}f)$-concave functions. Obviously
$\Phi^{t+\tau}_{f}=\Phi^{t}_{f}\circ\Phi^{\tau}_{f}.$
This map has the following main properties:
1. 1.
$\Phi^{t}_{f}$ is locally Lipschitz (in the domain of definition). Moreover,
if $f$ is $\lambda$-concave then it is $e^{\lambda{\cdot}t}$-Lipschitz.
This follows from lemma 2.1.4(i).
2. 2.
Gradient flow is stable under Gromov–Hausdorff convergence, namely:
If $A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$, $f_{n}\colon
A_{n}\to\mathbb{R}$ is a sequence of $\lambda$-concave functions which
converges to $f\colon A\to\mathbb{R}$ then $\Phi_{f_{n}}^{t}\colon A_{n}\to
A_{n}$ converges pointwise to $\Phi_{f}^{t}\colon A\to A$.
This follows from lemma 2.1.5.
3. 3.
For any $x\in A$ and all sufficiently small $t\geqslant 0$, there is $y\in A$
so that $\Phi_{f}^{t}(y)=x$.
For spaces without boundary this follows from [Grove–Petersen 1993, lemma 1].
For spaces with boundary one should consider its doubling.
Gradient flow can be used to deform a mapping with target in $A$. For example,
if $X$ is a metric space, then given a Lipschitz map $F\colon X\to A$ and a
positive Lipschitz function $\tau\colon X\to\mathbb{R}_{+}$ one can consider
the map $F^{\prime}$ called _gradient deformation_ of $F$ which is defined by
$F^{\prime}(x)=\Phi_{f}^{\tau(x)}\circ F(x),\ \ \ F^{\prime}\colon X\to A.$
From lemma 2.1.4 it is easy to see that the _dilation_ 101010i.e. its optimal
Lipschitz constant. of $F^{\prime}$ can be estimated in terms of $\lambda$,
$\sup_{x}\tau(x)$, dilation of $F$ and the Lipschitz constants of $f$ and
$\tau$.
Here is an optimal estimate for the length element of a curve which follows
from lemma 2.1.4:
###### 2.2.1.
Lemma. Let $A\in\text{{\nnn Alex}}$. Let $\gamma_{0}(s)$ be a curve in $A$
parametrized by arc-length, $f\colon A\to\mathbb{R}$ be a $\lambda$-concave
function, and $\tau(s)$ be a non-negative Lipschitz function. Consider the
curve
$\gamma_{1}(s)=\Phi^{\tau(s)}_{f}\circ\gamma_{0}(s).$
If $\sigma=\sigma(s)$ is its arc-length parameter then
$d\sigma^{2}\leqslant e^{2\lambda\tau}\left[ds^{2}+2\cdot
d(f\circ\gamma_{0})d\tau+|\nabla_{\gamma_{0}(s)}f|^{2}\cdot d\tau^{2}\right]$
### 2.3 Applications
Gradient flow gives a simple proof to the following result which generalizes a
key lemma in [Liberman]. This generalization was first obtained in
[Perelman–Petrunin 1993, 5.3], a simplified proof was given in [Petrunin 1997,
1.1]. See sections 4 and 5 for definition of extremal subset and
quasigeodesic.
###### 2.3.1.
Generalized Lieberman’s Lemma. Any unit-speed geodesic for the induced
intrinsic metric on an extremal subset is a quasigeodesic in the ambient
Alexandrov’s space.
Proof. Let $\gamma\colon[a,b]\to E$ be a unit-speed minimizing geodesic in an
extremal subset $E\subset A$ and $f$ be a $\lambda$-concave function defined
in a neighborhood of $\gamma$. Assume $f\circ\gamma$ is not $\lambda$-concave,
then there is a non-negative Lipschitz function $\tau$ with support in $(a,b)$
such that
$\int\limits_{a}^{b}\left[(f\circ\gamma)^{\prime}\tau^{\prime}+\lambda\tau\right]\cdot
ds<0$
Then as follows from lemma 2.2.1, for small $t\geqslant 0$
$\gamma_{t}(s)=\Phi^{t\cdot\tau(s)}_{f}\circ\gamma_{0}(s)$
gives a length-contracting homotopy of curves relative to ends and according
to definition 4.1.1, it stays in $E$ — this is a contradiction.∎
The fact that gradient flow is stable with respect to collapsing has the
following useful consequence: Let $M_{n}$ be a collapsing sequence of
Riemannian manifolds with curvature $\geqslant\kappa$ and
$M_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$. For a regular point $p$
let us denote by $F_{n}(p)$ the _regular fiber_ 111111see footnote 32 on page
32 over $p$, it is well defined for all large $n$. Let $f\colon
A\to\mathbb{R}$ be a $\lambda$-concave function. If $\alpha(t)$ is an
$f$-gradient curve in $A$ which passes only through regular points, then for
any $t_{0}<t_{1}$ there is a homotopy equivalence $F_{n}(\alpha(t_{0}))\to
F_{n}(\alpha(t_{1}))$ with dilation $\approx e^{\lambda{\cdot}(t_{1}-t_{0})}$.
This observation was used in [KPT] to prove some properties of almost
nonnegatively curved manifolds. In particular, it gave simplified proofs of
the results in [Fukaya–Yamaguchi]):
###### 2.3.2.
Nilpotency theorem. Let $M$ be a closed almost nonnegatively curved manifold.
Then a finite cover of $M$ is a _nilpotent space_ , i.e. its fundamental group
is nilpotent and it acts nilpotently on higher homotopy groups.
###### 2.3.3.
Theorem. Let $M$ be an almost nonnegatively curved $m$-manifold. Then
$\pi_{1}(M)$ is $\operatorname{Const}(m)$-nilpotent, i.e., $\pi_{1}(M)$
contains a nilpotent subgroup of index at most $\operatorname{Const}(m)$.
Gradient flow also gives an alternative proof of the homotopy lifting theorem
4.2.3. To explain the idea let us start with definition:
Given a topological space $X$, a map $F\colon X\to A$, a finite sequence of
$\lambda$-concave functions $\\{f_{i}\\}$ on $A$ and continuous functions
$\tau_{i}\colon X\to\mathbb{R}_{+}$ one can consider a composition of gradient
deformations (see 2.2)
$F^{\prime}(x)=\Phi_{f_{N}}^{\tau_{N}(x)}\circ\cdots\circ\Phi_{f_{2}}^{\tau_{2}(x)}\circ\Phi_{f_{1}}^{\tau_{1}(x)}\circ
F(x),\ \ \ F^{\prime}\colon X\to A,$
which we also call gradient deformation of $F$.
Let us define gradient homotopy to be a gradient deformation of trivial
homotopy
$F\colon[0,1]\times X\to A,\ \ \ F_{t}(x)=F_{0}(x)$
with the functions
$\tau_{i}\colon[0,1]\times X\to\mathbb{R}_{+}\ \ \ \text{such that}\ \ \
\tau_{i}(0,x)\equiv 0.$
If $Y\subset X$, then to define _gradient homotopy relative to_ $Y$ we assume
in addition
$\tau_{i}(t,y)=0\ \ \ \text{for any }\ \ \ y\in Y,\ \ t\in[0,1].$
Then theorem 4.2.3 follows from lemma 2.1.5 and the following lemma:
###### 2.3.4.
Lemma [Petrunin-GH]. Let $A$ be an Alexandrov’s space without proper extremal
subsets and $K$ be a finite simplicial complex. Then, given $\varepsilon>0$,
for any homotopy
$F_{t}\colon K\to A,\ \ t\in[0,1]$
one can construct an $\varepsilon$-close gradient homotopy
$G_{t}\colon K\to A$
such that $G_{0}\equiv F_{0}$.
## 3 Gradient exponent
One of the technical difficulties in Alexandrov’s geometry comes from
nonextendability of geodesics. In particular, the exponential map,
$\exp_{p}\colon T_{p}\to A$, if defined the usual way, can be undefined in an
arbitrary small neighborhood of origin. Here we construct its analog, the
_gradient exponential map_ $\operatorname{gexp}_{p}\colon T_{p}\to A$, which
practically solves this problem. It has many important properties of the
ordinary exponential map, and is even “better” in certain respects.
Let $A$ be an Alexandrov’s space and $p\in A$, consider the function
$f=\nobreak\operatorname{dist}_{p}^{2}/2$. Recall that $i_{s}\colon
s{\cdot}A\to A$ denotes canonical maps (see page 2). Consider the one
parameter family of maps
$\Phi^{t}_{f}\circ i_{e^{t}}\colon e^{t}{\cdot}A\to A\ \ \ \text{as}\ \ \
t\to\infty\ \ \ \text{so}\ \ \
(e^{t}{\cdot}A,p)\buildrel\mathrm{GH}\over{\longrightarrow}(T_{p},o_{p})$
where $\Phi^{t}_{f}$ denotes gradient flow (see section 2.2). Let us define
the gradient exponential map as the limit
$\operatorname{gexp}_{p}\colon T_{p}A\to A,\ \ \
\operatorname{gexp}_{p}=\lim_{t\to\infty}\Phi^{t}_{f}\circ i_{e^{t}}.$
Existence and uniqueness of gradient exponential. If $A$ is an Alexandrov’s
space with curvature $\geqslant 0$, then $f$ is $1$-concave, and from lemma
2.1.4, $\Phi^{t}_{f}$ is an $e^{t}$-Lipschitz and therefore compositions
$\Phi^{t}_{f}\circ i_{e^{t}}\colon e^{t}{\cdot}A\to A$ are _short_ 121212i.e.
maps with Lipschitz constant 1.. Hence a partial limit
$\operatorname{gexp}_{p}\colon T_{p}A\to A$ exists, and it is a short
map.131313For general lower curvature bound, $f$ is only
$(1+O(r^{2}))$-concave in the ball $B_{r}(p)$. Therefore $\Phi^{1}_{f}\colon
B_{r/e}(p)\to B_{r}(p)$ is $e(1+O(r^{2}))$-Lipschitz. By taking compositions
of these maps for different $r$ we get that $\Phi^{N}_{f}\colon
B_{r/e^{N}}(p)\to B_{r}(p)$ is $e^{N}(1+O(r^{2}))$-Lipschitz. Obviously, the
same is true for any $t\geqslant 0$, i.e. $\Phi^{t}_{f}\colon
B_{r/e^{t}}(p)\to B_{r}(p)$ is $e^{t}(1+O(r^{2}))$-Lipschitz, or
$\Phi^{t}_{f}\circ i_{e^{t}}\colon e^{t}{\cdot}A\to A$ is
$(1+O(r^{2}))$-Lipschitz on $B_{r}(p)\subset e^{t}{\cdot}A$. This is
sufficient for existence of partial limit $\operatorname{gexp}_{p}\colon
T_{p}A\to A$, which turns out to be $(1+O(r^{2}))$-Lipschitz on a central ball
of radius $r$ in $T_{p}$.
Clearly for any partial limit we have
$\Phi^{t}_{f}\circ\operatorname{gexp}_{p}(v)=\operatorname{gexp}_{p}(e^{t}\cdot
v)$ $None$
and since $\Phi^{t}$ is $e^{t}$-Lipschitz, it follows that
$\operatorname{gexp}_{p}$ is uniquely defined.
###### 3.1.1.
Property. If $E\in A$ is an extremal subset, $p\in E$ and $\xi\in\Sigma_{p}E$
then $\operatorname{gexp}_{p}(t\cdot\xi)\in E$ for any $t\geqslant 0$.
It follows from above and from definition of extremal subset (4.1.1).
Radial curves. From identity $(*)$, it follows that for any
$\xi\in\Sigma_{p}$, curve
$\alpha_{\xi}\colon t\mapsto\operatorname{gexp}_{p}(t\cdot\xi)$
satisfies the following differention equation
$\alpha_{\xi}^{+}(t)=\frac{|p\,\alpha_{\xi}(t)|}{t}{\cdot}\nabla_{\alpha_{\xi}(t)}\operatorname{dist}_{p}\
\ \ \text{for all}\ \ t>0\ \ \ \text{and}\ \ \ \alpha_{\xi}^{+}(0)=\xi$ $None$
We will call such a curve radial curve from $p$ in the direction $\xi$. From
above, such radial curve exists and is unique in any direction.
Clearly, for any radial curve from $p$, $|p\alpha_{\xi}(t)|\leqslant t$; and
if this inequality is exact for some $t_{0}$ then
$\alpha_{\xi}\colon[0,t_{0}]\to A$ is a unit-speed minimizing geodesic
starting at $p$ in the direction $\xi\in\Sigma_{p}$. In other words,
$\operatorname{gexp}_{p}\circ\log_{p}=\operatorname{id}_{A}.$
Next lemma gives a comparison inequality for radial curves.
###### 3.1.2.
Lemma. Let $A\in\text{{\nnn Alex}}$, $f\colon A\to\mathbb{R}$ be a
$\lambda$-concave function $\lambda\geqslant 0$ then for any $p\in A$ and
$\xi\in\Sigma_{p}$
$f\circ\operatorname{gexp}_{p}(t\cdot\xi)\leqslant f(p)+t\cdot
d_{p}f(\xi)+t^{2}{\cdot}\tfrac{\lambda}{2}.$
Moreover, the function
$\vartheta(t)=\\{f\circ\operatorname{gexp}_{p}(t\cdot\xi)-f(p)-t^{2}\cdot\tfrac{\lambda}{2}\\}/t$
is non-increasing.
In particular, applying this lemma for $f=\operatorname{dist}_{q}^{2}/2$ we
get
###### 3.1.3.
Corollary. If $A\in\text{{\nnn Alex}}(0)$ then for any $p,q,\in A$ and
$\xi\in\Sigma_{p}$,
$\tilde{\measuredangle}_{0}(t,|\operatorname{gexp}_{p}(t{\cdot}\xi)q|,|pq|)$
is non-increasing in $t$.151515$\tilde{\measuredangle}_{\kappa}(a,b,c)$
denotes angle opposite to $b$ in a triangle with sides $a,b,c$ in
$\hbox{\tencyr L}_{\kappa}$. In particular,
$\tilde{\measuredangle}_{0}(t,|\operatorname{gexp}_{p}(t{\cdot}\xi)\,q|,|pq|)\leqslant\measuredangle(\xi,\uparrow_{p}^{q}).$
In 3.2 you can find a version of this corollary for arbitrary lower curvature
bound.
Proof of lemma 3.1.2. Recall that $\nabla_{q}\operatorname{dist}_{p}$ is polar
to the set $\Uparrow_{q}^{p}\subset T_{q}$ (see example (ii) on page ii). In
particular, from inequality $(**)$ on page 1.3,
$d_{q}f(\nabla_{q}\operatorname{dist}_{p})+\inf_{\zeta\in\Uparrow_{q}^{p}}\\{d_{q}f(\zeta)\\}\leqslant
0$
On the other hand, since $f$ is $\lambda$-concave,
$d_{q}f(\zeta)\geqslant\frac{f(p)-f(q)-\lambda{\cdot}|pq|^{2}/2}{|pq|}\ \
\text{for any}\ \ \zeta\in\Uparrow_{q}^{p},$
therefore
$d_{q}f(\nabla_{q}\operatorname{dist}_{p})\leqslant\frac{f(q)-f(p)+\tfrac{\lambda}{2}{\cdot}|pq|^{2}}{|pq|}.$
Set $\alpha_{\xi}(t)=\operatorname{gexp}(t\cdot\xi)$, $q=\alpha_{\xi}(t_{0})$,
then
$\alpha^{+}_{\xi}(t_{0})=\tfrac{|pq|}{t}{\cdot}\nabla_{q}\operatorname{dist}_{p}$
as in $(\diamond)$. Therefore,
$\displaystyle(f\circ\alpha_{\xi})^{+}(t_{0})$
$\displaystyle=d_{q}f(\alpha^{+}_{\xi}(t_{0}))\leqslant$
$\displaystyle\leqslant\frac{|pq|}{t_{0}}\cdot\left[\frac{f(q)-f(p)+\tfrac{\lambda}{2}{\cdot}|pq|^{2}}{|pq|}\right]=$
$\displaystyle=\frac{f(q)-f(p)+\tfrac{\lambda}{2}{\cdot}|pq|^{2}}{t_{0}}\leqslant$
since $|pq|\leqslant t_{0}$ and $\lambda\geqslant 0$,
$\displaystyle\leqslant\frac{f(q)-f(p)+\tfrac{\lambda}{2}{\cdot}t^{2}_{0}}{t_{0}}=$
$\displaystyle=\frac{f(\alpha_{\xi}(t_{0}))-f(p)+\tfrac{\lambda}{2}{\cdot}t^{2}_{0}}{t_{0}}.$
Substituting this inequality in the expression for derivative of $\vartheta$,
$\vartheta^{+}(t_{0})=\frac{(f\circ\alpha_{\xi})^{+}(t)}{t_{0}}-\frac{f\circ\operatorname{gexp}_{p}(t_{0}\cdot\xi)-f(p)}{t_{0}^{2}}-\tfrac{\lambda}{2},$
we get $\vartheta^{+}\leqslant 0$, i.e. $\vartheta$ is non-increasing.
Clearly, $\vartheta(0)=d_{p}f(\xi)$ and so the first statement follows.∎
### 3.2 Spherical and hyperbolic gradient exponents
The gradient exponent described above is sufficient for most applications. It
works perfectly for non-negatively curved Alexandrov’s spaces and where one
does not care for the actual lower curvature bound. However, for fine analysis
on spaces with curvature $\geqslant\kappa$, there is a better analog of this
map, which we denote $\operatorname{gexp}_{p}(\kappa;v)$;
$\operatorname{gexp}_{p}(0;v)=\operatorname{gexp}_{p}(v)$.
In addition to case $\kappa=0$, it is enough to consider only two cases:
$\kappa=\pm 1$, the rest can be obtained by rescalings. We will define two
maps: $\operatorname{gexp}_{p}(-1,*)$ and $\operatorname{gexp}_{p}(1,*)$, and
list their properties, leaving calculations to the reader. These properties
are analogous to the following properties of the ordinary gradient exponent:
1. $\diamond$
if $A\in\text{{\nnn Alex}}({0})$, then $\operatorname{gexp}_{p}\colon T_{p}\to
A$ is distance non-increasing.
Moreover, for any $q\in A$, the angle
$\tilde{\measuredangle}_{0}(t,|\operatorname{gexp}_{p}(t\cdot\xi)\,q|,|pq|)$
is non-increasing in $t$ (see corollary 3.1.3). In particular
$\tilde{\measuredangle}_{0}(t,|\operatorname{gexp}_{p}(t\cdot\xi)\,q|,|pq|)\leqslant\measuredangle(\xi,\uparrow_{p}^{q}).$
###### 3.2.1.
Case $\kappa=-1$.
The hyperbolic radial curves are defined by the following differential
equation
$\alpha^{+}_{\xi}(t)=\frac{\operatorname{th}|p\alpha_{\xi}(t)|}{\operatorname{th}t}\cdot\nabla_{\alpha_{\xi}(t)}\operatorname{dist}_{p}\
\ \ \text{and}\ \ \ \alpha^{+}_{\xi}(0)=\xi.$
These radial curves are defined for all $t\in[0,\infty)$. Let us define
$\operatorname{gexp}_{p}(-1;t\cdot\xi)=\alpha_{\xi}(t).$
This map is defined on tangent cone $T_{p}$. Let us equip the tangent cone
with a hyperbolic metric $\mathfrak{h}(u,v)$ defined by the hyperbolic rule of
cosines
$\operatorname{ch}(\mathfrak{h}(u,v))=\operatorname{ch}|u|\cdot\operatorname{ch}|v|-\operatorname{sh}|u|\cdot\operatorname{sh}|v|\cdot\cos\alpha,$
where $u,v\in T_{p}$ and $\alpha=\measuredangle uo_{p}v$.
$(T_{p},\mathfrak{h})\in\text{{\nnn Alex}}(-1)$, this is a so called _elliptic
cone_ over $\Sigma_{p}$; see [BGP, 4.3.2], [Alexander–Bishop 2004]. Here are
the main properties of $\operatorname{gexp}(-1;*)$:
1. $\diamond$
if $A\in\text{{\nnn Alex}}(-1)$, then
$\operatorname{gexp}(-1;*)\colon(T_{p},\mathfrak{h})\to A$ is distance non-
increasing.
Moreover, the function
$t\mapsto\tilde{\measuredangle}_{-1}(t,|\operatorname{gexp}(-1;t\cdot\xi)\,q|,|pq|)$
is non-increasing in $t$. In particular for any $t>0$,
$\tilde{\measuredangle}_{-1}(t,|\operatorname{gexp}(-1;t\cdot\xi)\,q|,|pq|)\leqslant\measuredangle(\xi,\uparrow_{p}^{q}).$
###### 3.2.2.
Case $\kappa=1$.
For unit tanget vector $\xi\in\Sigma_{p}$, the spherical radial curve is
defined to satisfy the following identity:
$\alpha^{+}_{\xi}(t)=\frac{\operatorname{tg}|p\alpha_{\xi}(t)|}{\operatorname{tg}t}\cdot\nabla_{\alpha_{\xi}(t)}\operatorname{dist}_{p}\
\ \ \text{and}\ \ \ \alpha^{+}_{\xi}(0)=\xi.$
These radial curves are defined for all $t\in[0,\tfrac{\pi}{2}]$. Let us
define the spherical gradient exponential map by
$\operatorname{gexp}_{p}(1;t\cdot\xi)=\alpha_{\xi}(t).$
This map is well defined on $\bar{B}_{\pi/2}(o_{p})\subset T_{p}$. Let us
equip $\bar{B}_{\pi/2}(o_{p})$ with a spherical distance $\mathfrak{s}(u,v)$
defined by the spherical rule of cosines
$\cos(\mathfrak{s}(u,v))=\cos|u||\cdot\cos|v|+\sin|u||\cdot\sin|v||\cdot\cos\alpha,$
where $u,v\in B_{\pi}(o_{p})\subset T_{p}$ and $\alpha=\measuredangle
uo_{p}v$. $(\bar{B}_{\pi}(o_{p}),\mathfrak{s})\in\text{{\nnn Alex}}(1)$, this
is isometric to _spherical suspension_ $\Sigma(\Sigma_{p})$, see [BGP, 4.3.1],
[Alexander–Bishop 2004]. Here are the main properties of
$\operatorname{gexp}(1;*)$:
1. $\diamond$
If $A\in\text{{\nnn Alex}}(1)$ then
$\operatorname{gexp}_{p}(1,*)\colon(\bar{B}_{\pi/2}(o_{p}),\mathfrak{s})\to A$
is distance non-increasing.
Moreover, if $|pq|\leqslant\tfrac{\pi}{2}$, then function
$t\mapsto\tilde{\measuredangle}_{1}(t,|\operatorname{gexp}_{p}(1;t\cdot\xi)\,q|,|pq|)$
is non-increasing in $t$. In particular, for any $t>0$
$\tilde{\measuredangle}_{1}(t,|\operatorname{gexp}_{p}(1;t\cdot\xi)\,q|,|pq|)\leqslant\measuredangle(\xi,\uparrow_{p}^{q}).$
### 3.3 Applications
One of the main applications of gradient exponent and radial curves is the
proof of existence of quasigeodesics; see property 4 page 4 and appendix A for
the proof.
An infinite-dimensional generalization of gradient exponent was introduced by
Perelman to make the last step in the proof of equality of Hausdorff and
topological dimension for Alexandrov’s spaces, see [Perelman–Petrunin QG,
A.4]. According to [Plaut 1996] (or [Plaut 2002, 151]), if
$\operatorname{dim}_{H}A\geqslant m$, then there is a point $p\in A$, the
tangent cone of which contains a subcone $W\subset T_{p}$ isometric to
Euclidean $m$-space. Then infinite-dimensional analogs of properties in
section 3.2 ensure that image $\operatorname{gexp}_{p}(W)$ has topological
dimension $\geqslant m$ and therefore $\operatorname{dim}A\geqslant m$.
The following statement has been proven in [Perelman 1991], then its
formulation was made more exact in [Alexander–Bishop 2003]. Here we give a
simplified proof with the use of a gradient exponent.
###### 3.3.1.
Theorem. Let $A\in\text{{\nnn Alex}}(\kappa)$ and $\partial
A\not=\varnothing$; then the function
$f=\nobreak\sigma_{\kappa}\circ\operatorname{dist}_{\partial
A}$161616$\sigma_{\kappa}\colon\mathbb{R}\to\mathbb{R}$ is defined by
$\sigma_{\kappa}(x)=\sum_{n=0}^{\infty}\frac{(-\kappa)^{n}}{(2n+1)!}{\cdot}x^{2n+1}=\left[\begin{matrix}{\frac{1}{\sqrt{\kappa}}{\cdot}\sin({x{\cdot}\sqrt{\kappa}})}&{\hbox{if}\
\kappa>0}\\\ {x}&{\hbox{if}\ \kappa=0}\\\
{\frac{1}{\sqrt{-\kappa}}{\cdot}\operatorname{sh}({x{\cdot}\sqrt{-\kappa}})}&{\hbox{if}\
\kappa<0}\\\ \end{matrix}\right..$ is $(-\kappa{\cdot}f)$-concave in
$\Omega=A\backslash\partial A$.171717Note that by definition 1.1.2, $f$ is not
semiconcave in $A$.
In particular,
1. (i)
if $\kappa=0$, $\operatorname{dist}_{\partial A}$ is concave in $\Omega$;
2. (ii)
if $\kappa>0$, the level sets $L_{x}=\operatorname{dist}^{-1}_{\partial
A}(x)\subset A$, $x>0$ are strictly concave hypersurfaces.
$\tilde{\gamma}(0)$$\tilde{\gamma}(\tau)$$\alpha$$\tilde{\beta}$$\tilde{p}$$\tilde{q}$$\partial\hbox{\tencyr
L}^{+}_{\kappa}$
Proof. We have to show that for any unit-speed geodesic $\gamma$, the function
$f\circ\gamma$ is $(-\kappa{\cdot}f)$-concave; i.e. for any $t_{0}$,
$(f\circ\gamma)^{\prime\prime}(t_{0})\leqslant-\kappa{\cdot}f\circ\gamma(t_{0})$
_in a barrier sense_ 181818For a continuous function $f$,
$f^{\prime\prime}(t_{0})\leqslant c$ _in a barrier sense_ means that there is
a smooth function $\bar{f}$ such that $f\leqslant\bar{f}$,
$f(t_{0})=\bar{f}(t_{0})$ and $\bar{f}^{\prime\prime}(t_{0})\leqslant c$.
Without loss of generality we can assume $t_{0}=0$.
Direct calculations show that the statement is true for $A=\hbox{\tencyr
L}_{\kappa}^{+}$, the halfspace of the model space $\hbox{\tencyr
L}_{\kappa}$.
Let $p\in\partial A$ be a closest point to $\gamma(0)$ and
$\alpha=\measuredangle(\gamma^{+}(0),\uparrow_{\gamma(0)}^{p})$.
Consider the following picture in the model halfspace $\hbox{\tencyr
L}_{\kappa}^{+}$: Take a point $\tilde{p}\in\partial\hbox{\tencyr
L}_{\kappa}^{+}$ and consider the geodesic $\tilde{\gamma}$ in $\hbox{\tencyr
L}_{\kappa}^{+}$ such that
$|\gamma(0)p|=|\tilde{\gamma}(0)\tilde{p}|=|\tilde{\gamma}(0)\,\partial\hbox{\tencyr
L}^{+}_{\kappa}|,$
so $\tilde{p}$ is the closest point to $\tilde{\gamma}(0)$ on the
boundary191919in case $\kappa>0$ it is possible only if
$|\gamma(0)p|\leqslant\frac{\pi}{2{\cdot}\sqrt{\kappa}}$, but this is always
the case since otherwise any small variation of $p$ in $\partial A$ decreases
distance $|\gamma(0)p|$. and
$\measuredangle(\tilde{\gamma}^{+}(0),\uparrow_{\tilde{\gamma}(0)}^{\tilde{p}})=\alpha.$
Then it is enough to show that
$\operatorname{dist}_{\partial
A}\gamma(\tau)\leqslant\operatorname{dist}_{\partial\hbox{\sevencyr
L}_{\kappa}^{+}}\tilde{\gamma}(\tau)+o(\tau^{2}).$
Set
$\beta(\tau)=\measuredangle\gamma(0)\,p\,\gamma(\tau)$
and
$\tilde{\beta}(\tau)=\measuredangle\tilde{\gamma}(0)\,\tilde{p}\,\tilde{\gamma}(\tau).$
From the comparison inequalities
$|p\gamma(\tau)|\leqslant|\tilde{p}\tilde{\gamma}(\tau)|$
and
$\vartheta(\tau)=\max\left\\{0,\,\tilde{\beta}(\tau)-\beta(\tau)\right\\}=o(\tau).$
$None$
Note that the tangent cone at $p$ splits: $T_{p}A=\mathbb{R}_{+}\times
T_{p}\partial A$.202020This follows from the fact that $p$ lies on a shortest
path between two preimages of $\gamma(0)$ in the doubling $\tilde{A}$ of $A$,
see [BGP, 7.15]. Therefore we can represent $v=\log_{p}\gamma(\tau)\in T_{p}A$
as $v=(s,w)\in\mathbb{R}_{+}\times T_{p}\partial A$. Let
$\tilde{q}=\tilde{q}(\tau)\in\partial\hbox{\tencyr L}_{\kappa}$ be the closest
point to $\tilde{\gamma}(\tau)$, so
$\displaystyle\measuredangle(\uparrow_{p}^{\gamma(\tau)},w)$
$\displaystyle=\tfrac{\pi}{2}-\beta(\tau)\leqslant$
$\displaystyle\leqslant\tfrac{\pi}{2}-\tilde{\beta}(\tau)-\vartheta(\tau)=$
$\displaystyle=\measuredangle\tilde{\gamma}(\tau)\tilde{p}\tilde{q}+o(\tau).$
Set
$q=\operatorname{gexp}_{p}\left(\kappa;|\tilde{p}\tilde{q}|\frac{w}{|w|}\right)$.212121
Alternatively, one can set $q=\gamma(|\tilde{p}\tilde{q}|)$, where $\gamma$ is
a quasigeodesic in $\partial A$ starting at $p$ in direction
$\frac{w}{|w|}\in\Sigma_{p}$ (it exists by second part of property 4 on page
4). Since gradient curves preserve extremal subsets $q\in\partial A$ (see
property 3.1.1 on page 3.1.1). Clearly $|\tilde{p}\tilde{q}|=O(\tau)$,
therefore applying the comparison from section 3.2 (or Corollary 3.1.3 if
$\kappa=0$) together with $(*)$, we get
$\displaystyle\operatorname{dist}_{\partial A}\gamma(\tau)$
$\displaystyle\leqslant|q\gamma(\tau)|\leqslant$
$\displaystyle\leqslant|\tilde{q}\tilde{\gamma}(\tau)|+O\left(|\tilde{p}\tilde{q}|\cdot\vartheta(\tau)\right)=$
$\displaystyle=\operatorname{dist}_{\partial\hbox{\sevencyr
L}_{\kappa}^{+}}\tilde{\gamma}(\tau)+o(\tau^{2}).$
∎
The following corollary implies that the Lipschitz condition in the definition
of convex function 1.1.2– 1.1.1 can be relaxed to usual continuity.
###### 3.3.2.
Corollary. Let $A\in\text{{\nnn Alex}}$, $\partial A=\varnothing$,
$\lambda\in\mathbb{R}$ and $\Omega\subset A$ be open.
Assume $f\colon\Omega\to\mathbb{R}$ is a continuous function such that for any
unit-speed geodesic $\gamma$ in $\Omega$ we have that the function
$t\mapsto f\circ\gamma-\tfrac{\lambda}{2}{\cdot}t^{2}$
is concave; then $f$ is locally Lipschitz.
In particular, $f$ is $\lambda$-concave in the sense of definition 1.1.2.
Proof. Assume $f$ is not Lipschitz at $p\in\Omega$. Without loss of generality
we can assume that $\Omega$ is convex222222Otherwise, pass to a small convex
neighborhood of $p$ which exists by by corollary 7.1.2. and
$\lambda<0$232323Otherwise, add a very concave (Lipschitz) function which
exists by theorem 7.1.1. Then, since $f$ is continuous, sub-graph
$X_{f}=\\{(x,y)\in\bar{\Omega}\times\mathbb{R}|y\leqslant f(x)\\}$
is closed convex subset of $A\times\mathbb{R}$, therefore it forms an
Alexandrov’s space.
Since $f$ is not Lipschitz at $p$, there is a sequence of pairs of points
$(p_{n},q_{n})$ in $A$, such that
$p_{n},q_{n}\to p\ \ \ \text{and}\ \ \
\frac{f(p_{n})-f(q_{n})}{|p_{n}q_{n}|}\to+\infty.$
Consider a sequence of radial curves $\alpha_{n}$ in $X_{f}$ which extend
shortest paths from $(p_{n},f(p_{n}))$ to $(q_{n},f(q_{n}))$. Since the
boundary $\partial X_{f}\subset X_{f}$ is an extremal subset, we have
$\alpha_{n}(t)\in\partial X_{f}$ for all
$\displaystyle t$ $\displaystyle\geqslant\ell_{n}=$
$\displaystyle=|(p_{n},f(p_{n}))(q_{n},f(q_{n}))|=$
$\displaystyle=\sqrt{|p_{n}q_{n}|^{2}+(f(p_{n})-f(q_{n}))^{2}}.$
Clearly, the function $h\colon X_{f}\to\mathbb{R}$, $h\colon(x,y)\mapsto y$ is
concave. Therefore, from 3.1.2, there is a sequence $t_{n}>\ell_{n}$, so
$\alpha_{n}(t_{n})\to(p,f(p)-1)$. Therefore, $(p,f(p)-1)\in\partial X_{f}$
thus $p\in\partial A$, i.e. $\partial A\not=\varnothing$, a contradiction. ∎
###### 3.3.3.
Corollary. Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$, $m\geqslant 2$ and
$\gamma$ be a unit-speed curve in $A$ which has a convex $\kappa$-developing
with respect to any point. Then $\gamma$ is a quasigeodesic, i.e. for any
$\lambda$-concave function $f$, function $f\circ\gamma$ is $\lambda$-concave.
Proof. Let us first note that in the proof of theorem 3.3.1 we used only two
properties of curve $\gamma$: $|\gamma^{\pm}|=1$ and the convexity of the
$\kappa$-development of $\gamma$ with respect to $p$.
Assume $\kappa=\lambda=0$ then sub-graph of $f$
$X_{f}=\\{(x,y)\in A\times\mathbb{R}\ |\ y\leqslant f(x)\\}$
is a closed convex subset, therefore it forms an Alexandrov’s space.
Applying the above remark, we get that if $\gamma$ is a unit-speed curve in
$X_{f}\backslash\partial X_{f}$ with convex $0$-developing with respect to any
point then $\operatorname{dist}_{\partial X_{f}}\circ\gamma$ is concave.
Hence, for any $\varepsilon>0$, the function $f_{\varepsilon}$, which has the
level set $\operatorname{dist}_{\partial
X_{f}}^{-1}(\varepsilon)\subset\mathbb{R}\times A$ like the graph, has a
concave restriction to any curve $\gamma$ in $A$ with a convex $0$-developing
with respect to any point in $A\backslash\gamma$. Clearly, $f_{\varepsilon}\to
f$ as $\varepsilon\to 0$, hence $f\circ\gamma$ is concave.
For $\lambda$-concave function the set $X_{f}$ is no longer convex, but it
becomes convex if one changes metric on $A\times\mathbb{R}$ to _parabolic
cone_ 242424 i.e. warped-product
$\mathbb{R}\times_{\exp(\operatorname{Const}{\cdot}t)}A$, which is an
Alexandrov’s space, see [BGP, 4.3.3], [Alexander–Bishop 2004] and then one can
repeat the same arguments.∎
Remark One can also get this corollary from the following lemma:
###### 3.3.4.
Lemma. Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$, $\Omega$ be an open subset of
$A$ and $f\colon\Omega\to\mathbb{R}$ be a $\lambda$-concave $L$-Lipschitz
function. Then function
$f_{\varepsilon}(y)=\min_{x\in\Omega}\\{f(x)+\tfrac{1}{\varepsilon}{\cdot}|xy|^{2}\\}$
is $(\lambda+\delta)$-concave in the domain of definition252525i.e. at the set
where the minimum is defined. for some262626this function
$\delta(L,\lambda,\kappa,\varepsilon)$ is achieved for the model space
$\Lambda_{\kappa}$ $\delta=\delta(L,\lambda,\kappa,\varepsilon)$, $\delta\to
0$ as $\varepsilon\to 0$.
Moreover, if $m\geqslant 2$ and $\gamma$ is a unit-speed curve in $A$ with
$\kappa$-convex developing with respect to any point then
$f_{\varepsilon}\circ\gamma$ is also $(\lambda+\delta)$-concave.
Proof. It is analogous to theorem 3.3.1. We only indicate it in the simplest
case, $\kappa=\lambda=0$. In this case $\delta$ can be taken to be $0$.
Let $\gamma$ be a unit-speed geodesic (or it satisfies the last condition in
the lemma). It is enough to show that for any $t_{0}$
$(f_{\varepsilon}\circ\gamma)^{\prime\prime}(t_{0})\leqslant 0$
in a barrier sense.
Let $y=\gamma(t_{0})$ and $x\in\Omega$ be a point for which
$f_{\varepsilon}(y)=f(x)+\tfrac{1}{\varepsilon}{\cdot}|xy|^{2}$. The tangent
cone $T_{x}$ splits in direction $\uparrow_{y}^{x}$, i.e. there is an isometry
$T_{x}\to\mathbb{R}\times\operatorname{Cone}$ such that
$\uparrow_{x}^{y}\mapsto(1,o)$, where $o\in\operatorname{Cone}$ is its origin.
Let
$\log_{x}\gamma(t)=(a(t),v(t))\in\mathbb{R}\times\operatorname{Cone}=T_{x}.$
Consider vector
$w(t)=(a(t)-|xy|,v(t))\in\mathbb{R}\times\operatorname{Cone}=T_{x}.$
Clearly $|w(t)|\geqslant|x\gamma(t)|$. Set
$x(t)=\operatorname{gexp}_{y}(w(t))$ then lemma 3.1.2 gives an estimate for
$f\circ x(t))$ while corollary 3.1.3 gives an estimate for
$|\gamma(t)x(t)|^{2}$. Hence the result. ∎
Here is yet another illustration for the use of gradient exponents. At first
sight it seems very simple, but the proof is not quite obvious. In fact, I did
not find any proof of this without applying the gradient exponent.
###### 3.3.5.
Lytchak’s problem. Let $A\in\text{{\nnn Alex}}^{m}(1)$. Show that
$\operatorname{vol}_{m-1}\partial A\leqslant\operatorname{vol}_{m-1}S^{m-1}$
where $\partial A$ denotes the boundary of $A$ and $S^{m-1}$ the unit
$(m-1)$-sphere.
The problem would have followed from conjecture 9.1.1 (that boundary of an
Alexandrov’s space is an Alexandrov’s space), but before this conjecture has
been proven, any partial result is of some interest. Among other corollaries
of conjecture 9.1.1, it is expected that if $A\in\text{{\nnn Alex}}(1)$ then
$\partial A$, equipped with induced intrinsic metric, admits a noncontracting
map to $S^{m-1}$. In particular, its intrinsic diameter is at most $\pi$, and
perimeter of any triangle in $\partial A$ is at most $2\pi$. This does not
follow from the proof below, since in general
$\operatorname{gexp}_{z}(1;\partial B_{\pi/2}(o_{z}))\not\subset\partial A$,
i.e. $\operatorname{gexp}_{z}(1;\partial B_{\pi/2}(o_{z}))$ might have some
creases left inside of $A$, which might be used as a shortcut for curves with
ends in $\partial A$.
Let us first prepare a proposition:
###### 3.3.6.
Proposition. The inverse of the gradient exponential map
$\operatorname{gexp}^{-1}_{p}(\kappa;*)$ is uniquely defined inside any
minimizing geodesic starting at $p$.
Proof. Let $\gamma\colon[0,t_{0}]\to A$ be a unit-speed minimizing geodesic,
$\gamma(0)=p$, $\gamma(t_{0})=q$. From the angle comparison we get that
$|\nabla_{x}\operatorname{dist}_{p}|\geqslant-\cos\tilde{\measuredangle}_{\kappa}pxq$.
Therefore, for any $\zeta$ we have
$|p\alpha_{\zeta}(t)|^{+}_{t}\geqslant-|\alpha^{+}_{\zeta}(t)|{\cdot}\cos\tilde{\measuredangle}_{\kappa}p\,\alpha_{\zeta}(t)\,q\
\ \text{and}\ \
|\alpha_{\zeta}(t)q|^{+}_{t}\geqslant-|\alpha^{+}_{\zeta}(t)|.$
Therefore, $\tilde{\measuredangle}_{\kappa}p\,q\,\alpha_{\zeta}(t)$ is
nondecreasing in $t$, hence the result. ∎
Proof of 3.3.5. Let $z\in A$ be the point at maximal distance from $\partial
A$, in particular it realizes maximum of
$f=\sigma_{1}\circ\operatorname{dist}_{\partial
A}=\sin\circ\operatorname{dist}_{\partial A}$. From theorem 3.3.1, $f$ is
$(-f)$-concave and $f(z)\leqslant 1$.
Note that $A\subset\bar{B}_{\pi/2}(z)$, otherwise if $y\in A$ with
$|yz|>\tfrac{\pi}{2}$, then since $f$ is $(-f)$-concave and $f(y)\geqslant 0$,
we have $df(\uparrow_{z}^{y})>0$; i.e., $z$ is not a maximum of $f$.
From this it follows that gradient exponent
$\operatorname{gexp}_{z}(1;*)\colon(\bar{B}_{\pi/2}(o_{z}),\mathfrak{s})\to A$
is a short onto map.
Moreover,
$\partial A\subset\operatorname{gexp}_{z}(\partial B_{\pi/2}(o_{z})).$
Indeed, $\operatorname{gexp}$ gives a homotopy equivalence $\partial
B_{\pi/2}(o_{z})\to A\backslash\\{z\\}$. Clearly,
$\Sigma_{z}=\nobreak\partial(B_{\pi/2}(o_{z}),\mathfrak{s})$ has no boundary,
therefore $H_{m-1}(\partial A,\mathbb{Z}_{2})\not=0$, see [Grove–Petersen
1993, lemma 1]. Hence for any point $x\in\partial A$, any minimizing geodesic
$zx$ must have a point of the image $\operatorname{gexp}(1;\partial
B_{\pi/2}(o))$ but, as it is shown in proposition 3.3.6, it can only be its
end $x$.
Now since
$\operatorname{gexp}_{z}(1;*)\colon(\bar{B}_{\pi/2}(o_{z}),\mathfrak{s})\to A$
is short and $(\partial B_{\pi/2}(o),\mathfrak{s})$ is isometric to
$\Sigma_{z}A$ we get $\operatorname{vol}\partial
A\leqslant\operatorname{vol}\Sigma_{z}A$ and clearly,
$\operatorname{vol}\Sigma_{z}A\leqslant\operatorname{vol}S^{m-1}$.∎
## 4 Extremal subsets
Imagine that you want to move a heavy box inside an empty room by pushing it
around. If the box is located in the middle of the room, you can push it in
any direction. But once it is pushed against a wall you can not push it back
to the center; and once it is pushed into a corner you cannot push it anywhere
anymore. The same is true if one tries to move a point in an Alexandrov’s
space by pushing it along a gradient flow, but the role of walls and corners
is played by extremal subsets.
Extremal subsets first appeared in the study of their special case — the
boundary of an Alexandrov’s space; introduced in [Perelman–Petrunin 1993], and
were studied further in [Petrunin 1997], [Perelman 1997].
An Alexandrov’s space without extremal subsets resembles a very non-smooth
Riemannian manifold. The presence of extremal subsets makes it behave as
something new and maybe intersting; it gives an interesting additional
combinatoric structure which reflects geometry and topology of the space
itself, as well as of nearby spaces.
### 4.1 Definition and properties.
It is best to define extremal subsets as “ideals” of the gradient flow, i.e.
###### 4.1.1.
Definition. Let $A\in\text{{\nnn Alex}}$.
$E\subset A$ is an _extremal subset_ , if for any semiconcave function $f$ on
$A$, $t\geqslant 0$ and $x\in E$, we have $\Phi_{f}^{t}(x)\in E$.
Recall that $\Phi^{t}_{f}$ denotes the $f$-gradient flow for time $t$, see
2.2. Here is a quick corollary of this definition:
1. 1.
Extremal subsets are closed. Moreover:
1. (i)
For any point $p\in A$, there is an $\varepsilon>0$, such that if an extremal
subset intersects $\varepsilon$-neighborhood of $p$ then it contains $p$.
2. (ii)
On each extremal subset the intrinsic metric is locally finite.
These properties follow from the fact that the gradient flow for a
$\lambda$-concave function with $d_{p}f|_{\Sigma_{p}}<0$ pushes a small ball
$B_{\varepsilon}(p)$ to $p$ in time proportionate to $\varepsilon$.
Examples.
1. (i)
An Alexandrov’s space itself, as well as the empty set, forms an extremal
subsets.
2. (ii)
A point $p\in A$ forms a one-point extremal subset if its space of directions
$\Sigma_{p}$ has a diameter $\leqslant\tfrac{\pi}{2}$
3. (iii)
If one takes a subset of points of an Alexandrov’s space with tangent cones
homeomorphic272727Equivalently, with homeomorphic small spherical
neigborhoods. The equivalence follows from Perelman’s stability theorem. to
each other then its closure282828As well as the closure of its connected
component. forms an extremal subset.
In particular, if in this construction we take points with tangent cone
homeomorphic to $\mathbb{R}_{+}\times\mathbb{R}^{m-1}$ then we get the
boundary of an Alexandrov’s space.
This follows from theorem 4.1.2 and the Morse lemma (property 7 page 7).
4. (iv)
Let $A/G$ be a factor of an Alexandrov’s space by an isometry group, and
$S_{H}\subset A$ be the set of points with stabilizer conjugate to a subgroup
$H\subset G$ (or its connected component). Then the closure of the projection
of $S_{H}$ in $A/G$ forms an extremal subset.
For example: A cube can be presented as a quotient of a flat torus by a
discrete isometry group, and each face of the cube forms an extremal subset.
The following theorem gives an equivalence of our definition of extremal
subset and the definition given in [Perelman–Petrunin 1993]:
###### 4.1.2.
Theorem. A closed subset $E$ in an Alexandrov’s space $A$ is extremal if and
only if for any $q\in A\backslash E$, the following condition is fulfilled:
If $\operatorname{dist}_{q}$ has a local minimum on $E$ at a point $p$, then
$p$ is a critical point of $\operatorname{dist}_{q}$ on $A$, i.e.,
$\nabla_{p}\operatorname{dist}_{q}=o_{p}$.
Proof. For the “only if” part, note that if $p\in E$ is not a critical point
of $\operatorname{dist}_{q}$, then one can find a point $x$ close to $p$ so
that $\uparrow_{p}^{x}$ is uniquely defined and close to the direction of
$\nabla_{p}\operatorname{dist}_{q}$, so
$d_{p}\operatorname{dist}_{q}(\uparrow_{p}^{x})>0$. Since
$\nabla_{p}\operatorname{dist}_{x}$ is polar to $\uparrow_{p}^{x}$ (see page
1.3) we get
$(d_{p}\operatorname{dist}_{q})(\nabla_{p}\operatorname{dist}_{x})<0,$
see inequality 1.3.8 on page 1.3.8. Hence, the gradient flow
$\Phi_{\operatorname{dist}_{x}}^{t}$ pushes the point $p$ closer to $q$, which
contradicts the fact that $p$ is a minimum point $\operatorname{dist}_{q}$ on
$E$.
To prove the “if” part, it is enough to show that if $F\subset A$ satisfies
the condition of the theorem, then for any $p\in F$, and any semiconcave
function $f$, either $\nabla_{p}f=o_{p}$ or
$\tfrac{\nabla_{p}f}{|\nabla_{p}f|}\in\Sigma_{p}F$. If so, an $f$-gradient
curve can be obtained as a limit of broken lines with vertexes on $F$, and
from uniqueness, any gradient curve which starts at $F$ lives in $F$.
Let us use induction on $\operatorname{dim}A$. Note that if $F\subset A$
satisfies the condition, then the same is true for
$\Sigma_{p}F\subset\Sigma_{p}$, for any $p\in F$. Then using the inductive
hypothesis we get that $\Sigma_{p}F\subset\Sigma_{p}$ is an extremal subset.
If $p$ is isolated, then clearly
$\operatorname{diam}\Sigma_{p}\leqslant\tfrac{\pi}{2}$ and therefore
$\nabla_{p}f=o$, so we can assume $\Sigma_{p}F\not=\varnothing$.
Note that $d_{p}f$ is $(-d_{p}f)$-concave on $\Sigma_{p}$ (see 1.2, page 1.2).
Take $\xi=\tfrac{\nabla_{p}f}{|\nabla_{p}f|}$, so $\xi\in\Sigma_{p}$ is the
maximal point of $d_{p}f$. Let $\eta\in\Sigma_{p}F$ be a direction closest to
$\xi$, then $\measuredangle(\xi,\eta)\leqslant\tfrac{\pi}{2}$; otherwise $F$
would not satisfy the condition in the theorem for a point $q$ with
$\uparrow_{p}^{q}\,\approx\xi$. Hence, since $\Sigma_{p}F\subset\Sigma_{p}$ is
an extremal subset, $\nabla_{\eta}(d_{p}f)\in\Sigma_{\eta}\Sigma_{p}F$ and
therefore
$(d_{\eta}d_{p}f)(\uparrow_{\eta}^{\xi})\leqslant\langle\nabla_{\eta}d_{p}f,\,\uparrow_{\eta}^{\xi}\rangle\leqslant
0.$
Hence, $d_{p}f(\eta)\geqslant d_{p}f(\xi)$, and therefore $\xi=\eta$, i.e.
$\tfrac{\nabla_{p}f}{|\nabla_{p}f|}\in\Sigma_{p}F$. ∎
From this theorem it follows that in the definition of extremal subset
(4.1.1), one has to check only squares of distance functions. Namely: Let
$A\in\text{{\nnn Alex}}$, then $E\subset A$ is an extremal subset, if for any
point $p\in A$, and any $x\in E$, we have
$\Phi_{\operatorname{dist}_{p}^{2}}^{t}(x)\in E$ for any $t\geqslant 0$.
In particular, applying lemma 2.1.5 we get
###### 4.1.3.
Lemma. The limit of extremal subsets is an extremal subset.
Namely, if $A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ and $E_{n}\subset A_{n}$ is
a sequence of extremal subsets such that $E_{n}\to E\subset A$ then $E$ is an
extremal subset of $A$.
The following is yet another important technical lemma:
###### 4.1.4.
Lemma. [Perelman–Petrunin 1993, 3.1(2)] Let $A\in\text{{\nnn Alex}}$ be
compact, then there is $\varepsilon>0$ such that $\operatorname{dist}_{E}$ has
no critical values in $(0,\varepsilon)$. Moreover,
$|\nabla_{x}\operatorname{dist}_{E}|>\varepsilon\ \ \text{if}\ \
0<\operatorname{dist}_{E}(x)<\varepsilon.$
For a non-compact $A$, the same is true for the restriction
$\operatorname{dist}_{E}|_{\Omega}$ to any bounded open $\Omega\subset A$.
Proof. Follows from lemma 4.1.5 and theorem 4.1.2.∎
###### 4.1.5.
Lemma about an obtuse angle. Given $v>0$, $r>0$, $\kappa\in\mathbb{R}$ and
$m\in\mathbb{N}$, there is $\varepsilon=\varepsilon(v,r,\kappa,m)>0$ such that
if $A\in\text{{\nnn Alex}}^{m}(\kappa)$, $p\in A$,
$\operatorname{vol}_{m}B_{r}(p)>v$, then for any two points $x,y\in B_{r}(p)$,
$|xy|<\varepsilon$ there is point $z\in B_{r}(p)$ such that $\measuredangle
zxy>\tfrac{\pi}{2}+\varepsilon$ or $\measuredangle
zyx>\tfrac{\pi}{2}+\varepsilon$.
The proof is based on a volume comparison for $\log_{x}\colon A\to T_{x}$
similar to [Grove–Petersen 1988, lemma 1.3].
Note that the tangent cone $T_{p}E$ of an extremal subset $E\subset A$ is well
defined; i.e. for any $p\in E$, subsets $s\cdot E$ in $(s{\cdot}A,p)$ converge
to a subcone of $T_{p}E\subset T_{p}A$ as $s\to\infty$. Indeed, assume
$E\subset A$ is an extremal subset and $p\in E$. For any
$\xi\in\Sigma_{p}E$292929For a closed subset $X\subset A$, and $p\in X$,
$\Sigma_{p}X\subset\Sigma_{p}$ denotes the set of tangent directions to $X$ at
$p$, i.e. the set of limits of $\uparrow_{p}^{q_{n}}$ for $q_{n}\to p$,
$q_{n}\in X$., the radial curve $\operatorname{gexp}(t\cdot\xi)$ lies in
$E$.303030that follows from the fact that the curves
$t\mapsto\operatorname{gexp}(t\,\cdot\mskip-3.0mu\uparrow_{p}^{q_{n}})$
starting with $q_{n}$ belong to $E$ and their converge to
$\operatorname{gexp}(t\cdot\xi)$ In particular, there is a curve which goes in
any tangent direction of $E$. Therefore, as $s\to\infty$, $(s\cdot E\subset
s{\cdot}A,p)$ converges to a subcone $T_{p}E\subset T_{p}A$, which is simply
cone over $\Sigma_{p}E$ (see also [Perelman–Petrunin 1993, 3.3])
Next we list some properties of tangent cones of extremal subsets:
1. 2.
A closed subset $E\subset A$ is extremal if and only if the following
condition is fulfilled:
1. $\diamond$
At any point $p\in E$, its tangent cone $T_{p}E\subset T_{p}A$ is well
defined, and it is an extremal subset of the tangent cone $T_{p}A$; compare
[Perelman–Petrunin 1993, 1.4].
(Here is an equivalent formulation in terms of the space of directions: For
any $p\in E$, either (a) $\Sigma_{p}E=\varnothing$ and
$\operatorname{diam}\Sigma_{p}\leqslant\tfrac{\pi}{2}$ or (b)
$\Sigma_{p}E=\\{\xi\\}$ is one point extremal subset and
$\bar{B}_{\pi/2}(\xi)=\Sigma_{p}$ or (c) $\Sigma_{p}E$ is extremal subset of
$\Sigma_{p}$ with at least two points.)
$T_{p}E$ is extremal as a limit of extremal subsets, see lemma 4.1.3. On the
other hand for any semiconcave function $f$ and $p\in E$, the differential
$d_{p}f\colon T_{p}\to\mathbb{R}$ is concave and since $T_{p}E\subset T_{p}$
is extremal we have $\nabla_{p}f\in\nobreak T_{p}E$. I.e. gradient curves can
be approximated by broken geodesics with vertices on $E$, see page 2.1.2.
2. 3.
[Perelman–Petrunin 1993, 3.4–5] If $E$ and $F$ are extremal subsets then so
are
1. (i)
$E\cap F$ and for any $p\in E\cap F$ we have $T_{p}(E\cup
F)=T_{p}E\cup\Sigma_{p}F$
2. (ii)
$E\cup F$ and for any $p\in E\cup F$ we have $T_{p}(E\cap
F)=T_{p}E\cap\Sigma_{p}F$
3. (iii)
$\overline{E\backslash F}$ and for any $p\in\overline{E\backslash F}$ we have
$T_{p}(\overline{E\backslash F})=\overline{T_{p}E\backslash T_{p}F}$
In particular, if $T_{p}E=T_{p}F$ then $E$ and $F$ coincide in a neighborhood
of $p$.
The properties (i) and (ii) are obvious. The property (iii) follows from
property 2 and lemma 4.1.4.
We continue with properties of the intrinsic metric of extremal subsets:
1. 4.
[Perelman–Petrunin 1993, 3.2(3)] Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$ and
$E\subset A$ be an extremal subset. Then the induced metric of $E$ is locally
bi-Lipschitz equivalent to its induced intrinsic metric. Moreover, the local
Lipschitz constant at point $p\in E$ can be expressed in terms of $m$,
$\kappa$ and volume of a ball $v=\operatorname{vol}B_{r}(p)$ for some (any)
$r>0$.
From lemma 4.1.5, it follows that for two sufficiently close points $x,y\in E$
near $p$ there is a point $z$ so that
$\langle\nabla_{x}\operatorname{dist}_{z},\uparrow_{x}^{y}\rangle>\varepsilon$
or
$\langle\nabla_{y}\operatorname{dist}_{z},\uparrow_{y}^{x}\rangle>\varepsilon$.
Then, for the corresponding point, say $x$, the gradient curve
$t\to\Phi^{t}_{\operatorname{dist}_{z}}(x)$ lies in $E$, it is 1-Lipschitz and
the distance $|\Phi^{t}_{\operatorname{dist}_{z}}(x)\,y|$ is decreasing with
the speed of at least $\varepsilon$. Hence the result.
2. 5.
Let $A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ without collapse (i.e.
$\operatorname{dim}A=m$) and $E_{n}\subset A_{n}$ be extremal subsets. Assume
$E_{n}\to E\subset A$ as subsets. Then
1. (i)
[Kapovitch 2007, 9.1] For all large $n$, there is a homeomorphism of pairs
$(A_{n},E_{n})\to(A,E)$. In particular, for all large $n$, $E_{n}$ is
homeomorphic to $E$,
2. (ii)
[Petrunin 1997, 1.2] $E_{n}\buildrel\mathrm{GH}\over{\longrightarrow}E$ as
length metric spaces (with the intrinsic metrics induced from $A_{n}$ and
$A$).
The first property is a coproduct of the proof of Perelman’s stability
theorem. The proof of the second is an application of quasigeodesics.
3. 6.
[Petrunin 1997, 1.4]The first variation formula. Assume $A\in\text{{\nnn
Alex}}$ and $E\subset A$ is an extremal subset, let us denote by
$|\mskip-3.0mu**|_{E}$ its intrinsic metric. Let $p,q\in E$ and $\alpha(t)$ be
a curve in $E$ starting from $p$ in direction $\alpha^{+}(0)\in\Sigma_{p}E$.
Then
$|\alpha(t)\,q|_{E}=|pq|_{E}-\cos\varphi\cdot t+o(t).$
where $\varphi$ is the minimal (intrinsic) distance in $\Sigma_{p}E$ between
$\alpha^{+}(0)$ and a direction of a shortest path in $E$ from $p$ to $q$ (if
$\varphi>\pi$, we assume $\cos\varphi=-1$).
4. 7.
Generalized Lieberman’s Lemma. Any minimizing geodesic for the induced
intrinsic metric on an extremal subset is a quasigeodesic in the ambient
space.
See 2.3.1 for the proof and discussion.
Let us denote by $\operatorname{Ext}(x)$ the minimal extremal subset which
contains a point $x\in A$. Extremal subsets which can be obtained this way
will be called _primitive_. Set
$\operatorname{Ext}^{\circ}(x)=\\{y\in\operatorname{Ext}(x)|\operatorname{Ext}(y)=\operatorname{Ext}(x)\\};$
let us call $\operatorname{Ext}^{\circ}(x)$ the _main part_ of
$\operatorname{Ext}(x)$. $\operatorname{Ext}^{\circ}(x)$ is the same as
$\operatorname{Ext}(x)$ with its proper extremal subsets removed. From
property 3iii on page 3iii, $\operatorname{Ext}^{\circ}(x)$ is open and
everywhere dense in $\operatorname{Ext}(x)$. Clearly the main parts of
primitive extremal subsets form a disjoint covering of $M$.
1. 8.
[Perelman–Petrunin 1993, 3.8] Stratification. The main part of a primitive
extremal subset is a topological manifold. In particular, the main parts of
primitive extremal subsets stratify Alexandrov’s space into topological
manifolds.
This follows from theorem 4.1.2 and the Morse lemma (property 7 page 7); see
also example iii, page iii.
### 4.2 Applications
The notion of extremal subsets is used to make more precise formulations. Here
is the simplest example, a version of the radius sphere theorem:
###### 4.2.1.
Theorem. Let $A\in\text{{\nnn Alex}}^{m}(1)$,
$\operatorname{diam}A>\tfrac{\pi}{2}$ and $A$ have no extremal subsets. Then
$A$ is homeomorphic to a sphere.
From lemma 5.2.1 and theorem 4.1.2, we have $A\in\text{{\nnn Alex}}(1)$,
$\operatorname{rad}A>\tfrac{\pi}{2}$ implies that $A$ has no extremal subsets.
I.e. this theorem does indeed generalize the radius sphere theorem 5.2.2(ii).
Proof. Assume $p,q\in A$ realize the diameter of $A$. Since $A$ has no
extremal subsets, from example iii, page iii, it follows that a small
spherical neighborhood of $p\in A$ is homeomorphic to $\mathbb{R}^{m}$. From
angle comparison, $\operatorname{dist}_{p}$ has only two critical points $p$
and $q$. Therefore, this theorem follows from the Morse lemma (property 7 page
7) applied to $\operatorname{dist}_{p}$. ∎
The main result of such type is the result in [Perelman 1997]. It roughly
states that a collapsing to a compact space without proper extremal subsets
carries a natural Serre bundle structure.
This theorem is analogous to the following:
###### 4.2.2.
Yamaguchi’s fibration theorem [Yamaguchi]. Let $A_{n}\in\text{{\nnn
Alex}}^{m}(\kappa)$ and $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}M$,
$M$ be a Riemannian manifold.
Then there is a sequence of locally trivial fiber bundles $\sigma_{n}\colon
A_{n}\to M$. Moreover, $\sigma_{n}$ can be chosen to be _almost submetries_
313131i.e. Lipshitz and co-Lipschitz with constants almost 1. and the
diameters of its fibers converge to $0$.
The conclusion in Perelman’s theorem is weaker, but on the other hand it is
just as good for practical purposes. In addition it is sharp, i.e. there are
examples of a collapse to spaces with extremal subsets which do not have the
homotopy lifting property. Here is a source of examples: take a compact
Riemannian manifold $M$ with an isometric and non-free action by a compact
connected Lie group $G$, then $(M\times\varepsilon
G)/G\buildrel\mathrm{GH}\over{\longrightarrow}M/G$ as $\varepsilon\to 0$ and
since the curvature of $G$ is non-negative, by O’Naill’s formula, we get that
the curvature of $(M\times\varepsilon G)/G$ is uniformly bounded below.
###### 4.2.3.
Homotopy lifting theorem. Let
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$, $A_{n}\in\text{{\nnn
Alex}}^{m}(\kappa)$, $A$ be compact without proper extremal subsets and $K$ be
a finite simplicial complex.
Then, given a homotopy
$F_{t}\colon K\to A,\ \ t\in[0,1]$
and a sequence of maps $G_{0;n}\colon K\to A_{n}$ such that $G_{0,n}\to F_{0}$
as $n\to\infty$ one can extend $G_{0;n}$ by homotopies
$G_{t;n}\colon K\to A$
such that $G_{t;n}\to F_{t}$ as $n\to\infty$.
An alternative proof is based on Lemma 2.3.4.
###### 4.2.4.
Remark. As a corollary of this theorem one obtains that for all large $n$ it
is possible to write a homotopy exact sequence:
$\cdots\pi_{k}(F_{n})\longrightarrow\pi_{k}(A_{n})\longrightarrow\pi_{k}(A)\longrightarrow\pi_{k-1}(F_{n})\cdots,$
where the space $F_{n}$ can be obtained the following way: Take a point $p\in
A$, and fix $\varepsilon>0$ so that $\operatorname{dist}_{p}\colon
A\to\mathbb{R}$ has no critical values in the interval
$(0,2{\cdot}\varepsilon)$. Consider a sequence of points $A_{n}\ni p_{n}\to p$
and take $F_{n}=B_{\varepsilon}(p_{n})\subset A_{n}$. In particular, if $p$ is
a regular point then for large $n$, $F_{n}$ is homotopy equivalent to a
_regular fiber over $p$_323232 It is constructed the following way: take a
distance chart $G\colon B_{2{\cdot}\varepsilon}(p)\to\mathbb{R}^{k}$,
$k=\operatorname{dim}A$ around $p\in A$ and lift it to $A_{n}$. It defines a
map $G_{n}\colon B_{\varepsilon}(p_{n})\to\mathbb{R}^{k}$. Then take
$F_{n}=G_{n}^{-1}\circ G(p)$ for large $n$. If $A_{n}$ are Riemannian then
$F_{n}$ are manifolds and they do not depend on $p$ up to a homeomorphism.
Moreover, $F_{n}$ are almost non-negatively curved in a generalized sense; see
[KPT, definition 1.4]..
Next we give two corollaries of the above remark. The last assertion of the
following theorem was conjectured in [Shioya] and was proved in [Mendonça].
###### 4.2.5.
Theorem [Perelman 1997, 3.1]. Let $M$ be a complete noncompact Riemannian
manifold of nonnegative sectional curvature. Assume that its asymptotic cone
$\operatorname{Cone}_{\infty}(M)$ has no proper extremal subsets, then $M$
splits isometrically into the product $L\times N$, where $L$ is a compact
Riemannian manifold and $N$ is a non-compact Riemannian manifold of the same
dimension as $\operatorname{Cone}_{\infty}(M)$.
In particular, the same conclusion holds if radius of the ideal boundary of
$M$ is at least $\tfrac{\pi}{2}$.
The proof is a direct application of theorem 4.2.3 and remark 4.2.4 for
collapsing
$\varepsilon{\cdot}M\buildrel\mathrm{GH}\over{\longrightarrow}\operatorname{Cone}_{\infty}(M),\
\ \text{as}\ \ \varepsilon\to 0.$
###### 4.2.6.
Theorem [Perelman 1997, 3.2]. Let $A_{n}\in\text{{\nnn Alex}}^{m}(1)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ be a collapsing sequence
(i.e. $m>\operatorname{dim}A$), then $\operatorname{Cone}(A)$ has proper
extremal subsets. In particular, $\operatorname{rad}A\leqslant\tfrac{\pi}{2}$.
The last assertion of this theorem (in a stronger form) has been proven in
[Grove–Petersen 1993, 3(3)].
The proof is a direct application of theorem 4.2.3 and remark 4.2.4 for
collapsing of spherical suspensions
$\Sigma(A_{n})\buildrel\mathrm{GH}\over{\longrightarrow}\Sigma(A),\ \
n\to\infty.$
## 5 Quasigeodesics
The class of quasigeodesics333333It should be noted that the class of
quasigeodesics described here has nothing to do with the Gromov’s
quasigeodesics in $\delta$-hyperbolic spaces. generalizes the class of
geodesics to nonsmooth metric spaces. It was first introduced in [Alexandrov
1945] for $2$-dimensional convex hypersurfaces in the Euclidean space, as the
curves which “turn” right and left simultaneously. They were studied further
in [Alexandrov–Burago], [Pogorelov], [Milka 1971] and was generalized to
surfaces with bounded integral curvature [Alexandrov 1949] and to
multidimensional polyhedral spaces [Milka 1968], [Milka 1969]. For multi-
dimensional Alexandrov’s spaces they were introduced in the author’s master
thesis; in print they appear first in [Perelman–Petrunin QG].
In Alexandrov’s spaces, quasigeodesics behave more naturally than geodesics,
mainly:
1. $\diamond$
There is a quasigeodesic starting in any direction from any point;
2. $\diamond$
The limit of quasigeodesics is a quasigeodesic.
Quasigeodesics have beauty on their own, but also due to the generalized
Lieberman lemma (2.3.1), they are very useful in the study of intrinsic metric
of extremal subsets, in particular the boundary of Alexandrov’s space.
Since quasigeodesics behave almost as geodesics, they are often used instead
of geodesics in the situations when there is no geodesic in a given direction.
In most of these applications one can instead use the radial curves of
gradient exponent, see section 3; a good example is the proof of theorem
3.3.1, see footnote 21, page 21. In this type of argument, radial curves could
be considered as a simpler and superior tool since they can be defined in a
more general setting, in particular, for infinitely dimensional Alexandrov’s
spaces.
### 5.1 Definition and properties
In section 1, we defined $\lambda$-concave functions as those locally
Lipschitz functions whose restriction to any unit-speed minimizing geodesic is
$\lambda$-concave. Now consider a curve $\gamma$ in an Alexandrov’s space such
that restriction of any $\lambda$-concave function to $\gamma$ is
$\lambda$-concave. It is easy to see that for any Riemannian manifold $\gamma$
has to be a unit-speed geodesic. In a general Alexandrov’s space $\gamma$
should only be a quasigeodesic.
###### 5.1.1.
Definition. A curve $\gamma$ in an Alexandrov’s space is called
_quasigeodesic_ if for any $\lambda\in\mathbb{R}$, given a $\lambda$-concave
function $f$ we have that $f\circ\gamma$ is $\lambda$-concave.
Although this definition works for any metric space, it is only reasonable to
apply it for the spaces where we have $\lambda$-concave functions, but not all
functions are $\lambda$-concave, and Alexandrov’s spaces seem to be the
perfect choice.
The following is a list of corollaries from this definition:
1. 1.
Quasigeodesics are unit-speed curves. I.e., if $\gamma(t)$ is a quasigeodesic
then for any $t_{0}$ we have
$\lim_{t\to t_{0}}\frac{|\gamma(t)\gamma(t_{0})|}{|t-t_{0}|}=1.$
To prove that quasigeodesic $\gamma$ is $1$-Lipschitz at some $t=t_{0}$, it is
enough to apply the definition for $f=\operatorname{dist}_{\gamma(t_{0})}^{2}$
and use the fact that in any Alexandrov’s space $\operatorname{dist}_{p}^{2}$
is $(2+O(r^{2}))$-concave in $B_{r}(p)$. The lower bound is more complicated,
see theorem 7.3.3.
2. 2.
For any quasigeodesic the right and left tangent vectors $\gamma^{+}$,
$\gamma^{-}$ are uniquely defined unit vectors.
To prove, take a partial limits $\xi^{\pm}\in T_{\gamma(t_{0})}$ for
$\frac{\log_{\gamma(t_{0})}\gamma(t_{0}\pm\tau)}{\tau},\ \ \text{as}\ \
\tau\to 0+$
It exists since quasigeodesics are 1-Lipschitz (see the previous property).
For any semiconcave function $f$, $(f\circ\gamma)^{\pm}$ are well defined,
therefore
$(f\circ\gamma)^{\pm}(t_{0})=d_{\gamma(t_{0})}f(\xi^{\pm}).$
Taking $f=\operatorname{dist}_{q}^{2}$ for different $q\in A$, one can see
that $\xi^{\pm}$ is defined uniquely by this identity, and therefore
$\gamma^{\pm}(t_{0})=\xi^{\pm}$.
3. 3.
Generalized Lieberman’s Lemma. Any unit-speed geodesic for the induced
intrinsic metric on an extremal subset is a quasigeodesic in the ambient
Alexandrov’s space.
See 2.3.1 for the proof and discussion.
4. 4.
For any point $x\in A$, and any direction $\xi\in\Sigma_{x}$ there is a
quasigeodesic $\gamma\colon\mathbb{R}\to A$ such that $\gamma(0)=x$ and
$\gamma^{+}(0)=\xi$.
Moreover, if $E\subset A$ is an extremal subset and $x\in E$,
$\xi\in\Sigma_{x}E$, then $\gamma$ can be chosen to lie completely in $E$.
The proof is quite long, it is given in appendix A.
Applying the definition locally, we get that if $f$ is a
$(1-\kappa{\cdot}f)$-concave function then $f\circ\gamma$ is
$(1-\kappa{\cdot}f\circ\gamma)$-concave (see section 1.2). In particular, if
$A$ is an Alexandrov’s space with curvature $\geqslant\kappa$, $p\in A$ and
$h_{p}(t)=\rho_{\kappa}\circ\operatorname{dist}_{p}\circ\gamma(t)$343434Function
$\rho_{\kappa}\colon\mathbb{R}\to\mathbb{R}$ is defined on page 1.2 then we
have the following inequality in the _barrier sense_
$h_{p}^{\prime\prime}\leqslant 1-\kappa{\cdot}h_{p}.$
This inequality can be reformulated in an equivalent way: Let $A\in\text{{\nnn
Alex}}^{m}(\kappa)$, $p\in A$ and $\gamma$ be a quasigeodesic, then function
$t\mapsto\tilde{\measuredangle}_{\kappa}(|\gamma(0)p|,|\gamma(t)p|,t)$
is decreasing for any $t>0$ (if $\kappa>0$ then one has to assume
$t\leqslant\pi/\sqrt{\kappa}$).
In particular,
$\tilde{\measuredangle}_{\kappa}(|\gamma(0)p|,|\gamma(t)p|,t)\leqslant\measuredangle(\uparrow_{\gamma(0)}^{p},\gamma^{+}(0))$
for any $t>0$ (if $\kappa>0$ then in addition $t\leqslant\pi/\sqrt{\kappa}$).
It also can be reformulated more geometrically using the notion of developing
(see below):
Any quasigeodesic in an Alexandrov’s space with curvature $\geqslant\kappa$,
has a convex $\kappa$-developing with respect to any point.
###### 5.1.2.
Definition of developing [Alexandrov 1957]. Fix a real $\kappa$.
Let $X$ be a metric space, $\gamma\colon[a,b]\to X$ be a 1-Lipschitz curve and
$p\in X\backslash\gamma$. If $\kappa>0$, assume in addition that
$|p\gamma(t)|<\pi/\sqrt{\kappa}$ for all $t\in[a,b]$.
Then there exists a unique (up to rotation) curve
$\tilde{\gamma}\colon[a,b]\to\hbox{\tencyr L}_{\kappa}$, parametrized by the
arclength, and such that $|o\tilde{\gamma}(t)|=|p\gamma(t)|$ for all $t$ and
some fixed $o\in\hbox{\tencyr L}_{\kappa}$, and the segment
$o\tilde{\gamma}(t)$ turns clockwise as $t$ increases (this is easy to prove).
Such a curve $\tilde{\gamma}$ is called the _$\kappa$ -development of $\gamma$
with respect to $p$_.
The development $\tilde{\gamma}$ is called _convex_ if for every $t\in(a,b)$,
for sufficiently small $\tau>0$ the curvilinear triangle, bounded by the
segments $o\tilde{\gamma}(t\pm\tau)$ and the arc
$\tilde{\gamma}|_{t-\tau,t+\tau}$, is convex.
In [Milka 1971], it has been proven that the developing of a quasigeodesic on
a convex surface is convex.
1. 5.
Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$, $m>1$353535This condition is only
needed to ensure that the set $A\backslash\gamma$ is everywhere dense.. A
curve $\gamma$ in $A$ is a quasigeodesic if and only if it is parametrized by
arc-length and one of the following properties is fulfilled:
1. (i)
For any point $p\in A\backslash\gamma$ the $\kappa$-developing of $\gamma$
with respect to $p$ is convex.
2. (ii)
For any point $p\in A$, if
$h_{p}(t)=\rho_{\kappa}\circ\operatorname{dist}_{p}\circ\gamma(t)$, then we
have the following inequality in _a barrier sense_
$h_{p}^{\prime\prime}\leqslant 1-\kappa{\cdot}h_{p}.$
3. (iii)
Function
$t\mapsto\tilde{\measuredangle}_{\kappa}(|\gamma(0)p|,|\gamma(t)p|,t)$
is decreasing for any $t>0$ (if $\kappa>0$ then in addition
$t\leqslant\pi/\sqrt{\kappa}$).
4. (iv)
The inequality
$\measuredangle(\uparrow_{\gamma(0)}^{p},\gamma^{+}(0))\geqslant\tilde{\measuredangle}_{\kappa}(|\gamma(0)p|,|\gamma(t)p|,t)$
holds for all small $t>0$.
The “only if” part has already been proven above, and the “if” part follows
from corollary 3.3.3
2. 6.
A pointwise limit of quasigeodesics is a quasigeodesic. More generally:
Assume $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$, $\operatorname{dim}A=m$ (i.e. it is
not a collapse).
Let $\gamma_{n}\colon[a,b]\to A_{n}$ be a sequence of quasigeodesics which
converges pointwise to a curve $\gamma\colon[a,b]\to A$. Then $\gamma$ is a
quasigeodesic.
As it follows from lemma 7.2.3, the statement in the definition is correct for
any $\lambda$-concave function $f$ which has controlled convexity type
$(\lambda,\kappa)$. I.e. $\gamma$ satisfies the property 7.3.4. In particular,
the $\kappa$-developing of $\gamma$ with respect to any point $p\in A$ is
convex, and as it is noted in remark 7.3.5, $\gamma$ is a unit-speed curve.
Therefore, from corollary 3.3.3 we get that it is a quasigeodesic.
Here is a list of open problems on quasigeodesics:
1. (i)
Is there an analog of the Liouvile theorem for “quasigeodesic flow”?
2. (ii)
Is it true that any finite quasigeodesic has bounded variation of turn?
or
Is it possible to approximate any finite quasigeodesic by sequence of broken
lines with bounded variation of turn?
3. (iii)
Is it true that in an Alexandrov’s space without boundary there is an
infinitely long geodesic?
As it was noted by A. Lytchak, the first and last questions can be reduced to
the following: Assume $A$ is a compact Alexandrov’s $m$-space without
boundary. Let us set $V(r)=\int_{A}\operatorname{vol}_{m}(B_{r}(x))$, then
$V(r)=\operatorname{vol}_{m}(A)\omega_{m}r^{m}+o(r^{m+1})$. The technique of
_tight maps_ makes it possible to prove only that
$V(r)=\nobreak\operatorname{vol}_{m}(A)\omega_{m}r^{m}+O(r^{m+1})$. Note that
if $A$ is a Riemannian manifold with boundary then
$V(r)=\operatorname{vol}_{m}(A)\omega_{m}r^{m}+\operatorname{vol}_{m-1}(\partial
A)\omega^{\prime}_{m}r^{m+1}+o(r^{m+1})$.
### 5.2 Applications.
The quasigeodesics is the main technical tool in the questions linked to the
intrinsic metric of extremal subsets, in particular the boundary of
Alexandrov’s space. The main examples are the proofs of convergence of
intrinsic metric of extremal subsets and the first variation formula (see
properties 5ii and 6, on page 5ii).
Below we give a couple of simpler examples:
###### 5.2.1.
Lemma. Let $A\in\text{{\nnn Alex}}^{m}(1)$ and
$\operatorname{rad}A>\tfrac{\pi}{2}$. Then for any $p\in A$ the space of
directions $\Sigma_{p}$ has radius $>\tfrac{\pi}{2}$.
Proof. Assume that $\Sigma_{p}$ has radius $\leq\tfrac{\pi}{2}$, and let
$\xi\in\Sigma_{p}$ be a direction, such that
$\bar{B}_{\xi}(\tfrac{\pi}{2})=\Sigma_{p}$. Consider a quasigeodesic $\gamma$
starting at $p$ in direction $\xi$.
Then for $q=\gamma(\tfrac{\pi}{2})$ we have $\bar{B}_{q}(\tfrac{\pi}{2})=A$.
Indeed, for any point $x\in A$ we have
$\measuredangle(\xi,\uparrow_{p}^{x})\leqslant\tfrac{\pi}{2}$. Therefore, by
the comparison inequality (property 5iv, page 5iv),
$|xq|\leqslant\tfrac{\pi}{2}$. This contradicts our assumption that
$\operatorname{rad}A>\tfrac{\pi}{2}$. ∎
###### 5.2.2.
Corollary. Let $A\in\text{{\nnn Alex}}^{m}(1)$ and
$\operatorname{rad}A>\tfrac{\pi}{2}$ then
1. (i)
$A$ has no extremal subsets.
2. (ii)
[Grove–Petersen 1993](radius sphere theorem) $A$ is homeomorphic to an
$m$-sphere.
Yet another proof of the radius sphere theorem follows immediately from
[Perelman–Petrunin 1993, 1.2, 1.4.1]; theorem 4.2.1 gives a slight
generalization.
Proof. Part (i) is obvious.
Part (ii): From lemma 5.2.1, $\operatorname{rad}\Sigma_{p}>\tfrac{\pi}{2}$.
Since $\operatorname{dim}\Sigma_{p}<m$, by the induction hypothesis we have
$\Sigma_{p}\simeq S^{m-1}$. Now the Morse lemma (see property 7, page 7) for
$\operatorname{dist}_{p}\colon A\to\mathbb{R}$ gives that
$A\simeq\Sigma(\Sigma_{p})\simeq S^{m}$, here $\Sigma(\Sigma_{p})$ denotes a
spherical suspension over $\Sigma_{p}$.∎
## 6 Simple functions
This is a short technical section. Here we introduce _simple functions_ , a
subclass of semiconcave functions which on one hand includes all functions we
need and in addition is liftable; i.e. for any such function one can construct
a nearby function on a nearby space with “similar” properties.
Our definition of simple function is a modification of two different
definitions of so called “admissible functions” given in [Perelman 1993, 3.2]
and [Kapovitch 2007, 5.1].
###### 6.1.1.
Definition Let $A\in\text{{\nnn Alex}}$, a function $f\colon A\to\mathbb{R}$
is called _simple_ if there is a finite set of points $\\{q_{i}\\}_{i=1}^{N}$
and a semiconcave function $\Theta\colon\mathbb{R}^{N}\to\mathbb{R}$ which is
non-decreasing in each argument such that
$f(x)=\Theta(\operatorname{dist}_{q_{1}}^{2},\operatorname{dist}_{q_{2}}^{2},\dots,\operatorname{dist}_{q_{N}}^{2})$
It is straightforward to check that simple functions are semiconcave. Class of
simple functions is closed under summation, multiplication by a positive
constant363636as well as multiplication by positive simple functions and
taking the minimum.
In addition this class is liftable; i.e. given a converging sequence of
Alexandrov’s spaces $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ and a
simple function $f\colon A\to\mathbb{R}$ there is a way to construct a
sequence of functions $f_{n}\colon A_{n}\to\mathbb{R}$ such that $f_{n}\to f$.
Namely, for each $q_{i}$ take a sequence $A_{n}\ni q_{i,n}\to q_{i}\in A$ and
consider function $f_{n}\colon A_{n}\to\mathbb{R}$ defined by
$f_{n}=\Theta(\operatorname{dist}_{q_{1,n}}^{2},\operatorname{dist}_{q_{2,n}}^{2},\dots,\operatorname{dist}_{q_{N,n}}^{2}).$
### 6.2 Smoothing trick.
Here we present a trick which is very useful for doing local analysis in
Alexandrov’s spaces, it was introduced in [Otsu–Shioya, section 5].
Consider function
$\widetilde{\operatorname{dist}}_{p}=\oint\limits_{B_{\varepsilon}(p)}\operatorname{dist}_{x}\cdot
dx.$
In this notation, we do not specify $\varepsilon$ assuming it to be very
small. It is easy to see that $\widetilde{\operatorname{dist}}_{p}$ is
semiconcave.
Note that
$d_{y}\widetilde{\operatorname{dist}}_{p}=\oint\limits_{B_{\varepsilon}(p)}d_{y}\operatorname{dist}_{x}\cdot
dx.$
If $y\in A$ is regular, i.e. $T_{y}$ is isometric to Euclidean space, then for
almost all $x\in B_{\varepsilon}(p)$ the differential
$d_{y}\operatorname{dist}_{x}\colon T_{y}\to\mathbb{R}$ is a linear function.
Therefore $\widetilde{\operatorname{dist}}_{p}$ is differentiable at every
regular point, i.e.
$d_{y}\widetilde{\operatorname{dist}}_{p}\colon T_{y}\to\mathbb{R}$
is a linear function for any regular $y\in A$.
The same trick can be applied to any simple function
$f(x)=\Theta(\operatorname{dist}_{q_{1}}^{2},\operatorname{dist}_{q_{2}}^{2},\dots,\operatorname{dist}_{q_{N}}^{2}).$
This way we obtain function
$\tilde{f}(x)=\oint_{B_{\varepsilon}(q_{1})\times
B_{\varepsilon}(q_{2})\times\cdots\times
B_{\varepsilon}(q_{N})}\Theta(\operatorname{dist}_{x_{1}}^{2},\operatorname{dist}_{x_{2}}^{2},\dots,\operatorname{dist}_{x_{N}}^{2})\cdot
dx_{1}\cdot dx_{2}\cdots dx_{N},$
which is differentiable at every regular point, i.e. if $T_{y}$ is isometric
to the Euclidean space then
$d_{y}\tilde{f}\colon T_{y}\to\mathbb{R}$
is a linear function.
## 7 Controlled concavity
In this and the next sections we introduce a couple of techniques which use
comparison of $m$-dimensional Alexandrov’s space with a model space of the
same dimension $\hbox{\tencyr L}_{\kappa}^{m}$ (i.e. simply connected
Riemannian manifold with constant curvature $\kappa$). These techniques were
introduced in [Perelman 1993] and [Perelman-DC].
We start with the local existence of a strictly concave function on an
Alexandrov’s space.
###### 7.1.1.
Theorem [Perelman 1993, 3.6]. Let $A\in\text{{\nnn Alex}}$.
For any point $p\in A$ there is a strictly concave function $f$ defined in an
open neighborhood of $p$.
Moreover, given $v\in T_{p}$, the differential, $d_{p}f(x)$, can be chosen
arbitrarily close to $x\mapsto-\langle v,x\rangle$
$q$$\gamma(t)$$\alpha(t)$
Proof. Consider the real function
$\varphi_{r,c}(x)=(x-r)-c{(x-r)^{2}}/r,$
so we have
$\varphi_{r,c}(r)=0,\ \ \varphi_{r,c}^{\prime}(r)=1\ \
\varphi_{r,c}^{\prime\prime}(r)=-{2c}/{r}.$
Let $\gamma$ be a unit-speed geodesic, fix a point $q$ and set
$\alpha(t)=\measuredangle(\gamma^{+}(t),\uparrow_{\gamma(t)}^{q}).$
If $r>0$ is sufficiently small and $|q\gamma(t)|$ is sufficiently close to
$r$, then direct calculations show that
$(\varphi_{r,c}\circ\operatorname{dist}_{q}\circ\gamma)^{\prime\prime}(t)\leqslant\frac{3-c\cdot\cos^{2}\alpha(t)}{r}.$
Now, assume $\\{q_{i}\\}$, $i=\\{1,..,N\\}$ is a finite set of points such
that $|pq_{i}|=r$ for any $i$. For $x\in A$ and $\xi_{x}\in\Sigma_{x}$, set
$\alpha_{i}(\xi_{x})=\measuredangle(\xi_{x},\uparrow_{p}^{q_{i}})$. Assume we
have a collection $\\{q_{i}\\}$ such that for any $x\in B_{\varepsilon}(p)$
and $\xi_{x}\in\Sigma_{x}$ we have
$\max_{i}\\{|\alpha_{i}(\xi_{x})-\tfrac{\pi}{2}|\\}\geqslant\varepsilon>0$.
Then taking in the above inequality $c>3N/\cos^{2}\varepsilon$, we get that
the function
$f=\sum_{i}\varphi_{r,c}\circ\operatorname{dist}_{q_{i}}$
is strictly concave in $B_{\varepsilon^{\prime}}(p)$ for some positive
$\varepsilon^{\prime}<\varepsilon$.
To construct the needed collection $\\{q_{i}\\}$, note that for small $r>0$
one can construct $N_{\delta}\geqslant\operatorname{Const}/\delta^{(m-1)}$
points $\\{q_{i}\\}$ such that $|pq_{i}|=r$ and
$\tilde{\measuredangle}_{\kappa}q_{i}pq_{j}>\delta$ (here
$\operatorname{Const}=\operatorname{Const}(\Sigma_{p})>0$). On the other hand,
the set of directions which is orthogonal to a given direction is smaller than
$S^{m-2}$ and therefore contains at most
$\operatorname{Const}(m)/\delta^{(m-2)}$ directions with angles at least
$\delta$. Therefore, for small enough $\delta>0$, $\\{q_{i}\\}$ forms the
needed collection.
If $r$ is small enough, points $q_{i}$ can be chosen so that all directions
$\uparrow_{p}^{q_{i}}$ will be $\varepsilon$-close to a given direction $\xi$
and therefore the second property follows. ∎
Note that in the theorem 7.1.1 (as well as in theorem 7.2.2), the function $f$
can be chosen to have maximum value $0$ at $p$, $f(p)=0$ and with $d_{p}f(x)$
arbitrary close to $-|x|$. It can be constructed by taking the minimum of the
functions in these theorems.
In particular it follows that
###### 7.1.2.
Claim. For any point of an Alexandrov’s space there is an arbitrary small
closed convex neighborhood.
By rescaling and passing to the limit one can even estimate the size of the
convex hull in an Alexandrov’s space in terms of the volume of a ball
containing it:
###### 7.1.3.
Lemma on strictly concave convex hulls [Perelman–Petrunin 1993, 4.3]. For any
$v>0$, $r>0$ and $\kappa\in\mathbb{R}$, $m\in\mathbb{N}$ there is
$\varepsilon>0$ such that, if $A\in\text{{\nnn Alex}}^{m}(\kappa)$ and
$\operatorname{vol}B_{r}(p)\geqslant v$ then for any $\rho<\varepsilon\cdot
r$,
$\operatorname{diam}\operatorname{Conv}B_{\rho}(p)\leqslant\rho/\varepsilon.$
In particular, for any compact Alexandrov’s $A$ space there is
$\operatorname{Const}\in\mathbb{R}$ such that for any subset $X\subset A$
$\operatorname{diam}\left(\operatorname{Conv}X\right)\leqslant\operatorname{Const}\cdot\operatorname{diam}X.$
### 7.2 General definition.
The above construction can be generalized and optimized in many ways to fit
particular needs. Here we introduce one such variation which is not the most
general, but general enough to work in most applications.
Let $A$ be an Alexandrov’s space and $f\colon A\to\mathbb{R}$,
$f=\Theta(\operatorname{dist}^{2}_{q_{1}},\operatorname{dist}^{2}_{q_{2}},\dots,\operatorname{dist}^{2}_{q_{N}})$
be a _simple function_ (see section 6). If $A$ is $m$-dimensional, we say that
such a function $f$ has _controlled concavity of type_ $(\lambda,\kappa)$ at
$p\in A$, if for any $\varepsilon>0$ there is $\delta>0$, such that for any
collection of points $\\{\tilde{p},\tilde{q}_{i}\\}$ in the _model
$m$-space_373737i.e. a simply connected $m$-manifold with constant curvature
$\kappa$. $\hbox{\tencyr L}_{\kappa}^{m}$ satisfying
$|\tilde{q}_{i}\tilde{q}_{j}|>|q_{i}q_{j}|-\delta\ \ \text{and}\ \
\bigl{|}|\tilde{p}\tilde{q}_{i}|-|pq_{i}|\bigr{|}<\delta\ \ \text{for all}\ \
i,j,$
we have that the function $\tilde{f}\colon\hbox{\tencyr
L}_{\kappa}^{m}\to\mathbb{R}$ defined by
$\tilde{f}=\Theta(\operatorname{dist}^{2}_{\tilde{q}_{1}},\operatorname{dist}^{2}_{\tilde{q}_{2}},..,\operatorname{dist}^{2}_{\tilde{q}_{n}})$
is $(\lambda-\varepsilon)$-concave in a small neighborhood of $\tilde{p}$.
The following lemma states that the conrolled concavity is stronger than the
usual concavity.
###### 7.2.1.
Lemma. Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$.
If a simple function
$f=\Theta(\operatorname{dist}^{2}_{q_{1}},\operatorname{dist}^{2}_{q_{2}},..,\operatorname{dist}^{2}_{q_{N}}),\
\ f\colon A\to\mathbb{R}$
has a conrolled concavity type $(\lambda,\kappa)$ at each point $p\in\Omega$,
then $f$ is $\lambda$-concave in $\Omega$.
The proof is just a direct calculation similar to that in the proof of 7.1.1.
Note also, that the function constructed in the proof of theorem 7.1.1 has
controlled concavity. In fact from the same proof follows:
###### 7.2.2.
Existence. Let $A\in\text{{\nnn Alex}}$, $p\in A$,
$\lambda,\kappa\in\mathbb{R}$. Then there is a function $f$ of controlled
concavity $(\lambda,\kappa)$ at $p$.
Moreover, given $v\in T_{p}$, the function $f$ can be chosen so that its
differential $d_{p}f(x)$ will be arbitrary close to $x\mapsto-\langle
v,x\rangle$.
Since functions with a conrolled concavity are simple they admit liftings, and
from the definition it is clear that these liftings also have controlled
concavity of the same type, i.e.
###### 7.2.3.
Concavity of lifting. Let $A\in\text{{\nnn Alex}}^{m}$.
Assume a simple function
$f\colon A\to\mathbb{R},\ \
f=\Theta(\operatorname{dist}^{2}_{q_{1}},\operatorname{dist}^{2}_{q_{2}},..,\operatorname{dist}^{2}_{q_{N}})$
has controlled concavity type $(\lambda,\kappa)$ at $p$.
Let $A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ (so, no collapse) and
$\\{p_{n}\\},\\{q_{i,n}\\}\in A_{n}$ be sequences of points such that
$p_{n}\to p\in A$ and $q_{i,n}\to q_{i}\in A$ for each $i$.
Then for all large $n$, the liftings of $f$,
$f_{n}\colon A_{n}\to\mathbb{R},\ \
f_{n}=\Theta(\operatorname{dist}^{2}_{q_{1,n}},\operatorname{dist}^{2}_{q_{2,n}},..,\operatorname{dist}^{2}_{q_{N,n}})$
have controlled concavity type $(\lambda,\kappa)$ at $p_{n}$.
In other words, if $f\colon A\to\mathbb{R}$ has controlled concavity type
$(\lambda,\kappa)$ at all points of some open set $\Omega\subset A$, then
$f_{n}\colon A_{n}\to\mathbb{R}$ have controlled concavity type
$(\lambda,\kappa)$ at all points of some sequence of open sets
$\Omega_{n}\subset A_{n}$, such that $\Omega_{n}$ complement-converges to
$\Omega$ (i.e. $A_{n}\backslash\Omega_{n}\to A\backslash\Omega$ in Hausdorff
sense).
### 7.3 Applications
As was already noted, in the theorems 7.1.1 and 7.2.2, the function $f$ can be
chosen to have a maximum value $0$ at $p$, and with $d_{p}f(x)$ arbitrary
close to $-|x|$. This observation was used in [Kapovitch 2002] to solve the
second part of problem 32 from [Petersen 1996]:
###### 7.3.1.
Petersen’s problem. Let $A$ be a smoothable Alexandrov’s $m$-space, i.e. there
is a sequence of Riemannian $m$-manifolds $M_{n}$ with curvature
$\geqslant\kappa$ such that
$M_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$.
Prove that the space of directions $\Sigma_{x}A$ for any point $x\in A$ is
homeomorphic to the standard sphere.
Note that Perelman’s stability theorem (see [Perelman 1991], [Kapovitch 2007])
only gives that $\Sigma_{x}A$ has to be homotopically equivalent to the
standard sphere.
Sketch of the proof: Fix a big negative $\lambda$ and construct a function
$f\colon A\to\mathbb{R}$ with $d_{p}f(x)\approx-|x|$ and controlled concavity
of type $(\lambda,\kappa)$. From 7.2.1, the liftings $f_{n}\colon
M_{n}\to\mathbb{R}$ of $f$ (see 7.2.3) are strictly concave for large $n$. Let
us slightly smooth the functions $f_{n}$ keeping them strictly concave. Then
the level sets $f^{-1}_{n}(a)$, for values of $a$, which are little below the
maximum of $f_{n}$, have strictly positive curvature and are diffeomorphic to
the standard sphere383838Since $f$ has only one critical value above $a$ and
it is a local maximum..
Let us denote by $p_{n}\in M_{n}$ a maximum point of $f_{n}$. Then it is not
hard to choose a sequence $\\{a_{n}\\}$ and a sequence of rescalings
$\\{s_{n}\\}$ so that
$(s_{n}M_{n},p_{n})\buildrel\mathrm{GH}\over{\longrightarrow}(T_{p},o_{p})$
and $s_{n}\cdot f^{-1}_{n}(a_{n})\subset s_{n}M_{n}$ converge to a convex
hypersurface $S$ close to $\Sigma_{p}\subset T_{p}$. Then, from Perelman’s
stability theorem, it follows that $S$ and therefore $\Sigma_{p}$ is
homeomorphic to the standard sphere. ∎
Remark. From this proof it follows that $\Sigma_{p}$ is itself smoothable.
Moreover, there is a non-collapsing sequence of Riemannian metrics $g_{n}$ on
$S^{m-1}$ such that
$(S^{m-1},g_{n})\buildrel\mathrm{GH}\over{\longrightarrow}\Sigma_{p}$. This
observation makes possible to proof a similar statement for iterated spaces of
directions of smoothable Alexandrov space.
In the case of collapsing, the liftings $f_{n}$ of a function $f$ with
controlled concavity type do not have the same controlled concavity type.
Nevertheless, the liftings are semiconcave and moreover, as was noted in
[Kapovitch 2005], if $M_{n}$ is a sequence of $(m+k)$-dimensional Riemannian
manifolds with curvature $\geqslant\kappa$,
$M_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$, $\operatorname{dim}A=m$,
then one has a good control over the sum of $k+1$ maximal eigenvalues of their
Hessians. In particular, a construction as in the proof of theorem 7.1.1 gives
a strictly concave function on $A$ for which the liftings $f_{n}$ on $A_{n}$
have Morse index $\leqslant k$. It follows that one can retract an
$\varepsilon$-neighborhood of $p_{n}$ to a $k$-dimensional CW-complex393939it
is unknown whether it could be retracted to an $k$-submanifold. If true, it
would give some interesting applications, where $p_{n}\in A_{n}$ is a maximum
point of $f_{n}$ and $\varepsilon$ does not depend on $n$. This observation
gives a lower bound for the _codimension of a collapse_ 404040in our case, it
is $k$; the difference between the dimension of spaces from the collapsing
sequence and the dimension of the limit space to particular spaces. For
example, for any lower curvature bound $\kappa$, the codimension of a collapse
to $\Sigma(\mathbb{H}\mathrm{P}^{m})$414141i.e. a spherical suspension over
$\mathbb{H}\mathrm{P}^{m}$ is at least 3, and for
$\Sigma(Ca\mskip-3.0mu\operatorname{P}^{2})$ is at least 8 (it is expected to
be $\infty$). In addition, it yields the following theorem, which seems to be
the only sphere theorem which does not assume positiveness of curvature.
###### 7.3.2.
Funny sphere theorem. If a $4{\cdot}(m+1)$-dimensional Riemannian manifold $M$
with sectional curvature $\geqslant\kappa$ is sufficiently close424242i.e.
$\varepsilon$-close for some $\varepsilon=\varepsilon(\kappa,m)$ to
$\Sigma(\mathbb{H}\mathrm{P}^{m})$, then it is homeomorphic to a sphere.
The controlled concavity also gives a short proof of the following result:
###### 7.3.3.
Theorem. Any quasigeodesic is a unit-speed curve.
Proof. To prove that a quasigeodesic $\gamma$ is $1$-Lipschitz at some
$t=t_{0}$, it is enough to apply the definition for
$f=\operatorname{dist}_{\gamma(t_{0})}^{2}$ and use the fact that in any
Alexandrov’s space $\operatorname{dist}_{p}^{2}$ is $(2+O(r^{2}))$-concave in
$B_{r}(p)$.
Note that if $A_{n},A\in\text{{\nnn Alex}}^{m}(\kappa)$,
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$ without collapse, and
$\gamma_{n}$ in $A_{n}$ is a sequence of quasigeodesics which converges to a
curve $\gamma$ in $A$, then $\gamma$ has the following property434343from
statement 6, page 6, we that $\gamma$ is a quasigeodesic, but its proof is
based on this theorem:
###### 7.3.4.
Property. For any function $f$ on $A$ with controlled concavity type
$(\lambda,\kappa)$ we have that $f\circ\gamma$ is $\lambda$-concave.
If $\gamma$ is a quasigeodesic in $A$ with $\gamma(0)=p$, then the curves
$\gamma(t/s)$ are quasigeodesics in $s{\cdot}A$. Therefore, as $s\to\infty$,
the limit curve
$\gamma_{\infty}(t)=\left[\begin{matrix}|t|\cdot\gamma^{+}(0)&\text{if}\ \
t\geqslant 0\\\ |t|\cdot\gamma^{-}(0)&\text{if}\ \ t<0\\\ \end{matrix}\right.$
in $T_{p}$ has the above property. By a construction similar444444Setting
$v=\gamma^{\pm}(0)\in T_{p}$ and $w=2\gamma^{\pm}(0)$, this function can be
presented as a sum
$f=A(\varphi_{r,c}\circ\operatorname{dist}_{o}+\varphi_{r,c}\circ\operatorname{dist}_{w})+B\sum_{i}\varphi_{r^{\prime},c^{\prime}}\circ\operatorname{dist}_{q_{i}},$
for appropriately chosen positive reals $A,\ B,\ r,\ r^{\prime},\ c,\
c^{\prime}$ and a collection of points $q_{i}$ such that, $\measuredangle
opq_{i}=\tilde{\measuredangle}_{0}opq_{i}=\tfrac{\pi}{2}$, $|pq_{i}|=r$ . to
theorem 7.1.1, for any $\varepsilon>0$ there is a function $f$ of controlled
concavity type $(-2+\varepsilon,-\varepsilon)$ on a neighborhood of
$\gamma^{\pm}\in T_{p}$ such that
$f(t\cdot\gamma^{\pm})=-(t-1)^{2}+o((t-1)^{2}).$
Applying the property above we get $|\gamma^{\pm}(0)|\geqslant 1$. ∎
###### 7.3.5.
Remark. Note that we have proven a slightly stronger statement; namely, if a
curve $\gamma$ satisfies the property 7.3.4 then it is a unit-speed curve.
###### 7.3.6.
Question. Is it true that for any point $p\in A$ and any $\varepsilon>0$,
there is a $(-2+\varepsilon)$-concave function $f_{p}$ defined in a
neighborhood of $p$, such that $f_{p}(p)=0$ and
$f_{p}\geqslant-\operatorname{dist}_{p}^{2}$?
Existence of a such function would be a useful technical tool. In particular,
it would allow for an easier proof of the above theorem.
## 8 Tight maps
The tight maps considered in this section give a more flexible version of
distance charts.
Similar maps (so called _regular maps_) were used in [Perelman 1991] [Perelman
1993], and then they were modified to nearly this form in [Perelman-DC]. This
technique is also useful for Alexandrov’s spaces with upper curvature bound,
see [Lytchak–Nagano].
###### 8.1.1.
Definition. Let $A\in\text{{\nnn Alex}}^{m}$ and $\Omega\subset A$ be an open
subset. A collection of semiconcave functions $f_{0},f_{1},\dots,f_{\ell}$ on
$A$ is called _tight in $\Omega$_ if
$\sup_{x\in\Omega,\,i\not=j}\\{d_{x}f_{i}(\nabla_{x}f_{j})\\}<0.$
In this case the map
$F\colon\Omega\to\mathbb{R}^{\ell+1},\ \ F\colon
x\mapsto(f_{0}(x),f_{1}(x),\dots,f_{\ell}(x))$
is called _tight_.
A point $x\in\Omega$ is called a _critical point_ of $F$ if
$\min_{i}\\{d_{x}f_{i}\\}\leqslant 0$, otherwise the point $x$ is called
_regular_.
###### 8.1.2.
Main example. If $A\in\text{{\nnn Alex}}^{m}(\kappa)$ and
$a_{0},a_{1},\dots,a_{\ell},p\in A$ such that
$\tilde{\measuredangle}_{\kappa}a_{i}pa_{j}>\tfrac{\pi}{2}\ \ \text{for all}\
\ i\not=j$
then the map $x\mapsto(|a_{0}x|,|a_{1}x|,\dots,|a_{\ell}x|)$ is tight in a
neighborhood of $p$.
The inequality in the definition follows from inequality $(**)$ on page 1.3
and a subsequent to it example (ii).
This example can be made slightly more general. Let $f_{0},f_{1},...,f_{\ell}$
be a collection of simple functions
$f_{i}=\Theta_{i}(\operatorname{dist}_{a_{1,i}}^{2},\operatorname{dist}_{a_{2,i}x}^{2},\dots,\operatorname{dist}_{a_{n_{i},i}x}^{2})$
and the sets of points $K_{i}=\\{a_{k,i}\\}$ satisfy the following inequality
$\tilde{\measuredangle}_{\kappa}xpy>\tfrac{\pi}{2}\ \ \text{for any}\ \ x\in
K_{i},\ \ y\in K_{j},\ \ i\not=j.$
Then the map $x\mapsto(f_{0}(x),f_{1}(x),...,f_{\ell}(x))$ is tight in a
neighborhood of $p$. We will call such a map a _simple tight map_.
Yet further generalization is given in the property 1 below.
The maps described in this example have an important property, they are
liftable and their lifts are tight. Namely, given a converging sequence
$A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$, $A_{n}\in\text{{\nnn
Alex}}^{m}(\kappa)$ and a simple tight map $F\colon A\to\mathbb{R}^{\ell+1}$
around $p\in A$, the construction in section 6 gives simple tight maps
$F_{n}\colon A_{n}\to\mathbb{R}^{\ell}$ for large $n$, $F_{n}\to F$.
I was unable to prove that tightness is a stable property in a sense
formulated in the question below. It is not really important for the theory
since all maps which appear naturally are simple (or, in the worst case they
are as in the generalization and as in the property 1). However, for the
beauty of the theory it would be nice to have a positive answer to the
following question.
###### 8.1.3.
Question. Assume $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$, $f,g\colon A\to\mathbb{R}$ is a
tight collection around $p$ and $f_{n},g_{n}\colon A_{n}\to\mathbb{R}$,
$f_{n}\to f$, $g_{n}\to g$ are two sequences of $\lambda$-concave functions
and $A_{n}\ni p_{n}\to p\in A$. Is it true that for all large $n$, the
collection $f_{n},g_{n}$ must be tight around $p_{n}$?
If not, can one modify the definition of tightness so that
1. (i)
it would be stable in the above sense,
2. (ii)
the definition would make sense for all semiconcave functions
3. (iii)
the maps described in the main example above are tight?
Let us list some properties of tight maps with sketches of proofs:
1. 1.
Let $x\mapsto(f_{0}(x),f_{1}(x),...,f_{\ell}(x))$ be a tight map in an open
subset $\Omega\subset A$, then there is $\varepsilon>0$ such that if
$g_{0},g_{1},...,g_{n}$ is a collection of $\epsilon$-Lipschitz semiconcave
functions in $\Omega$ then the map
$x\mapsto(f_{0}(x)+g_{0}(x),f_{1}(x)+g_{1}(x),...,f_{\ell}(x)+g_{\ell}(x))$
is also tight in $\Omega$.
2. 2.
The set of regular points of a tight map is open.
Indeed, let $x\in\Omega$ be a regular point of tight map
$F=(f_{0},f_{1},\dots,f_{\ell})$. Take real $\lambda$ so that all $f_{i}$ are
$\lambda$-concave in a neighborhood of $x$. Take a point $p$ sufficiently
close to $x$ such that $d_{x}f_{i}(\uparrow_{x}^{p})>0$ and moreover
$f_{i}(p)-f_{i}(x)>\tfrac{\lambda}{2}{\cdot}|xp|^{2}$ for each $i$. Then, from
$\lambda$-concavity of $f_{i}$, there is a small neighborhood $\Omega_{x}\ni
x$ such that for any $y\in\Omega_{x}$ and $i$ we have
$d_{y}f_{i}(\uparrow_{y}^{p})\geqslant\varepsilon$ for some fixed
$\varepsilon>0$.
3. 3.
If one removes one function from a tight collection (in $\Omega$) then (for
the corresponding map) all points of $\Omega$ become regular. In other words,
the projection of a tight map $F$ to any coordinate hyperplane is a tight map
with all regular points (in $\Omega$).
This follows from the property 3 on page 3 applied to the flow for the removed
$f_{i}$.
4. 4.
The converse also holds, i.e. if $F$ is regular at $x$ then one can find a
semiconcave function $g$ such that map $z\mapsto(F(z),g(z))$ is tight in a
neighborhood of $x$. Moreover, $g$ can be chosen to have an arbitrary
controlled concavity type.
Indeed, one can take $g=\operatorname{dist}_{p}$, where $p$ as in the property
2. Then we have
$d_{x}g(v)=-\max_{\xi\in\Uparrow_{x}^{p}}\\{\langle\xi,v\rangle\\}$
and therefore
$d_{x}g(\nabla_{x}f_{i})=-\max_{\xi\in\Uparrow_{x}^{p}}\\{\langle\xi,\nabla_{x}f_{i}\rangle\\}\leqslant-\max_{\xi\in\Uparrow_{x}^{p}}\\{d_{x}f(\xi)\\}\leqslant-\varepsilon.$
On the other hand, from inequality $(**)$ on page 1.3 and example (ii)
subsequent to it, we have
$d_{x}f_{i}(\nabla_{x}g)+\min_{\xi\in\Uparrow_{x}^{p}}\\{d_{x}f_{i}(\xi)\\}\leqslant
0.$
The last statement follows from the construction in theorem 7.1.1.
5. 5.
A tight map is open and even _co-Lipschitz_ 454545A map $F\colon X\to Y$
between metric spaces is called $L$-co-Lipschitz in $\Omega\subset X$ if for
any ball $B_{r}(x)\subset\Omega$ we have $F(B_{r}(x))\supset B_{r/L}(F(x))$ in
$Y$ in a neighborhood of any regular point.
This follows from lemma 8.1.4.
6. 6.
Let $A\in\text{{\nnn Alex}}$, $\Omega\subset A$ be an open subset. If
$F\colon\Omega\to\mathbb{R}^{\ell+1}$ is tight then
$\ell\leqslant\operatorname{dim}A$.
Follows from the properties 3 and 5.
7. 7.
Morse lemma. A tight map admits a local splitting in a neighborhood of its
regular point, and a proper everywhere regular tight map is a locally trivial
fiber bundle. Namely
1. (i)
If $F\colon\Omega\to\mathbb{R}^{\ell+1}$ is a tight map and $p\in\Omega$ is a
regular point, then there is a neighborhood $\Omega\supset\Omega_{p}\ni p$ and
homeomorphism
$h\colon\Upsilon\times F(\Omega_{p})\to\Omega_{p},$
such that $F\circ h$ coincides with the projection to the second coordinate
$\Upsilon\times F(\Omega_{p})\to F(\Omega_{p})$.
2. (ii)
If $F\colon\Omega\to\Delta\subset\mathbb{R}^{\ell+1}$ is a proper tight map
and all points in $\Delta\subset\nobreak\mathbb{R}^{\ell+1}$ are regular
values of $F$, then $F$ is a locally trivial fiber bundle.
The proof is a backward induction on $\ell$, see [Perelman 1993, 1.4],
[Perelman 1991, 1.4.1] or [Kapovitch 2007, 6.7].
The following lemma is an analog of lemmas [Perelman 1993, 2.3] and [Perelman-
DC, 2.2].
###### 8.1.4.
Lemma. Let $x$ be a regular point of a tight map
$F\colon x\mapsto(f_{0}(x),f_{1}(x),\dots,f_{\ell}(x)).$
Then there is $\varepsilon>0$ and a neighborhood $\Omega_{x}\ni x$ such that
for any $y\in\Omega_{x}$ and $i\in\\{0,1,\dots,\ell\\}$ there is a unit vector
$w_{i}\in\Sigma_{x}$ such that $d_{x}f_{i}(w_{i})\geqslant\varepsilon$ and
$d_{x}f_{j}(w_{i})=0$ for all $j\not=i$.
Moreover, if $E\subset A$ is an extremal subset and $y\in E$ then $w_{i}$ can
be chosen in $\Sigma_{y}E$.
Proof. Take $p$ as in the property 2 page 2. Then we can find a neighborhood
$\Omega_{x}\ni x$ and $\varepsilon>0$ so that for any $y\in\Omega_{x}$
1. (i)
$d_{y}f_{i}(\uparrow_{y}^{p})>\varepsilon$ for each $i$;
2. (ii)
$-d_{y}f_{i}(\nabla_{y}f_{j})>\varepsilon.$ for all $i\not=j$.
Note that if $\alpha(t)$ is an $f_{i}$-gradient curve in $\Omega_{x}$ then
$(f_{i}\circ\alpha)^{+}>0\ \ \text{and}\ \
(f_{j}\circ\alpha)^{+}\leqslant-\varepsilon\ \ \text{for any}\ \ j\not=i.$
Applying lemma 2.1.5 for
$(s{\cdot}A,y)\buildrel\mathrm{GH}\over{\longrightarrow}T_{y}$,
$s\cdot[f_{i}-f_{i}(y)]\to d_{y}f_{i}$, we get the same inequalities for
$d_{y}f_{i}$-gradient curves on $T_{y}$, i.e. if $\beta(t)$ is an
$d_{y}f_{i}$-gradient curve in $T_{y}$ then
$(d_{y}f_{i}\circ\beta)^{+}>0\ \ \text{and}\ \
(d_{y}f_{j}\circ\beta)^{+}\leqslant-\varepsilon\ \ \text{for any}\ \ j\not=i.$
Moreover, $d_{y}f_{i}(v)>0$ implies
$\langle\nabla_{v}(d_{y}f_{i}),\uparrow_{v}^{o}\rangle<0$, therefore in this
case $|\beta(t)|^{+}>\nobreak 0$.
Take $w_{0}\in T_{y}$ to be a maximum point for $d_{y}f_{0}$ on the set
$\\{v\in T_{y}|f_{i}(v)\geqslant 0,|v|\leqslant 1\\}.$
Then
$d_{y}f_{0}(w_{0})\geqslant d_{y}f_{0}(\uparrow_{y}^{p})>\varepsilon.$
Assume for some $j\not=0$ we have $f_{j}(w_{0})>0$. Then
$\min_{i\not=j}\\{d_{w_{0}}d_{y}f_{i},d_{w_{0}}\nu\\}\leqslant 0,$
where the function $\nu$ is defined by $\nu\colon v\mapsto-|v|$; this is a
concave function on $T_{y}$. Therefore, if $\beta_{j}(t)$ is a
$d_{y}f_{j}$-gradient curve with an end464646it does exist by property 3 on
page 3 point at $w_{0}$, then moving along $\beta_{j}$ from $w_{0}$ backwards
decreases only $d_{y}f_{j}$, and increases the other $d_{y}f_{i}$ and $\nu$ in
the first order; this is a contradiction.
To prove the last statement it is enough to show that $w_{0}\in T_{y}E$, which
follows since $T_{y}E\subset T_{y}$ is an extremal subset (see property 2 on
page 2). ∎
###### 8.1.5.
Main theorem. Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$, $\Omega\subset A$ be
the interior of a compact convex subset, and
$F\colon\Omega\to\mathbb{R}^{\ell+1},\ \ F\colon
x\mapsto(f_{0}(x),f_{1}(x),\dots,f_{\ell}(x))$
be a tight map. Assume all $f_{i}$ are strictly concave. Then
1. (i)
the set of critical points of $F$ in $\Omega$ forms an $\ell$-submanifold $M$
2. (ii)
$F\colon M\to\mathbb{R}^{\ell+1}$ is an embedding.
3. (iii)
$F(M)\subset\mathbb{R}^{\ell+1}$ is a convex hypersurface which lies in the
boundary of $F(\Omega)$474747In fact $F(M)=\partial F(\Omega)\cap F(\Omega)$..
###### 8.1.6.
Remark. The condition that all $f_{i}$ are strictly concave seems to be very
restrictive, but that is not really so; if $x$ is a regular point of a tight
map $F$ then, using properties 1 and 4 on page 1, one can find $\varepsilon>0$
and $g$ such that
$F^{\prime}\colon y\mapsto(f_{0}(y)+\varepsilon
g(y),\dots,f_{\ell}(y)+\varepsilon g(y),g(y))$
is tight in a small neighborhood of $x$ and all its coordinate functions are
strictly concave. In particular, in a neighborhood of $x$ we have
$F=L\circ F^{\prime}$
where $L\colon\mathbb{R}^{\ell+2}\to\mathbb{R}^{\ell+1}$ is linear.
###### 8.1.7.
Corollary. In the assumptions of theorem 8.1.5, if in addition $m=\ell$ then
$M=\Omega$, $F(\Omega)$ is a convex hypersurface in $\mathbb{R}^{m+1}$ and
$F\colon\Omega\to\mathbb{R}^{m+1}$ is a locally bi-Lipschitz embedding.
Moreover, each projection of $F$ to a coordinate hyperplane is a locally bi-
Lipschitz homeomorphism.
Proof of theorem 8.1.5. Let $\gamma\colon[0,s]\to A$ be a minimal unit-speed
geodesic connecting $x,y\in\Omega$, so $s=|xy|$. Consider a straight segment
$\bar{\gamma}$ connecting $F(x)$ and $F(y)$:
$\bar{\gamma}\colon[0,s]\to\mathbb{R}^{\ell+1},\ \
\bar{\gamma}(t)=F(x)+\tfrac{t}{s}\cdot\left[F(y)-F(x)\right].$
Each function $f_{i}\circ\gamma$ is concave, therefore all coordinates of
$F\circ\gamma(t)-\bar{\gamma}(t)$
are non-negative. This implies that the Minkowski sum484848equivalently
$Q=\\{(x_{0},x_{1},\dots,x_{\ell})\in\mathbb{R}^{\ell+1}|\exists(y_{0},y_{1},\dots,y_{\ell})\in
F(\Omega)\forall i\ x_{i}\leqslant y_{i}\\}$.
$Q=F(\Omega)+(\mathbb{R}_{-})^{\ell+1}$
is a convex set.
Let $x_{0}\in\Omega$ be a critical point of $F$. Since
$\min_{i}\\{d_{x_{0}}f_{i}\\}\leqslant 0$, at least one of coordinates of
$F(x)$ is smaller than the corresponding coordinate of $F(x_{0})$ for any
$x\in\Omega$. In particular, $F$ sends its critical point to the boundary of
$Q$.
Consider map
$G\colon\mathbb{R}^{\ell+1}\to A,\ \
G\colon(y_{0},y_{1},\dots,y_{\ell})\mapsto\operatorname{argmax}\\{\min_{i}\\{f_{i}-y_{i}\\}\\}$
where $\operatorname{argmax}\\{f\\}$ denotes a maximum point of $f$. The
function $\min_{i}\\{f_{i}-y_{i}\\}$ is strictly concave; therefore
$\operatorname{argmax}\\{\min_{i}\\{f_{i}-y_{i}\\}\\}$ is uniquely defined and
$G$ is continuous in the domain of definition.494949We do not need it, but
clearly $G(y_{0},y_{1},\dots,y_{\ell})=G(y_{0}+h,y_{1}+h,\dots,y_{\ell}+h)$
for any $h\in\mathbb{R}$. The image of $G$ coincides with the set of critical
points of $F$ and moreover $G\circ F|_{M}=\operatorname{id}_{M}$. Therefore
$F|_{M}$ is a homeomorphism505050In general, $G$ is not Lipschitz (even on
$F(M)$); even in the case when all functions $f_{i}$ are $(-1)$-concave it is
only possible to prove that $G$ is Hölder continuous of class
$C^{0;\frac{1}{2}}$. (In fact the statement in [Perelman 1991], page 20, lines
23–25 is wrong but the proposition 3.5 is still OK.). ∎
Proof of corollary 8.1.7. It only remains to show that $F$ is locally bi-
Lipschitz.
Note that for any point $x\in\Omega$, one can find $\varepsilon>0$ and a
neighborhood $\Omega_{x}\ni x$, so that for any direction $\xi\in\Sigma_{y}$,
$y\in\Omega_{x}$ one can choose $f_{i}$, $i\in\nobreak\\{0,1,\dots,m\\}$, such
that $d_{x}f_{i}(\xi)\leqslant-\varepsilon$. Otherwise, by a slight
perturbation515151as in the property 1 on page 1 of collection $\\{f_{i}\\}$
we get a map $F\colon A^{m}\to\mathbb{R}^{m+1}$ regular at $y$, which
contradicts property 5.
Therefore applying it for $\xi=\uparrow_{z}^{y}$ and $\uparrow_{y}^{z}$,
$z,y\in\Omega$, we get two values $i,j$ such that
$f_{i}(y)-f_{i}(z)\geqslant\varepsilon{\cdot}|yz|\ \ \text{and}\ \
f_{j}(z)-f_{j}(y)\geqslant\varepsilon{\cdot}|yz|.$
Therefore $F$ is bi-Lipschits.
Clearly $i\not=j$ and therefore at least one of them is not zero. Hence the
projection map $F^{\prime}\colon x\mapsto(f_{1}(x),\dots,f_{m}(x))$ is also
locally bi-Lipschitz. ∎
### 8.2 Applications.
One series of applications of tight maps is Morse theory for Alexandrov’s
spaces, it is based on the main theorem 8.1.5. It includes Morse lemma
(property 7 page 7) and
1. $\diamond$
Local structure theorem [Perelman 1993]. Any small spherical neighborhood of a
point in an Alexandrov’s space is homeomorphic to a cone over its boundary.
2. $\diamond$
Stability theorem [Perelman 1991]. For any compact $A\in\text{{\nnn
Alex}}^{m}(\kappa)$ there is $\varepsilon>0$ such that if
$A^{\prime}\in\text{{\nnn Alex}}^{m}(\kappa)$ is $\varepsilon$-close to $A$
then $A$ and $A^{\prime}$ are homeomorphic.
The other series is the regularity results on an Alexandrov’s space. These
results obtained in [Perelman-DC] are improvements of earlier results in
[Otsu–Shioya], [Otsu]. It use mainly the corollary 8.1.7 and the smoothing
trick; see subsection 6.2.
1. $\diamond$
Components of metric tensor of an Alexandrov’s space in a chart are continuous
at each regular point525252i.e. at each point with Euclidean tangent space.
Moreover they have bounded variation and are differentiable almost everywhere.
2. $\diamond$
The Christoffel symbols in a chart are well defined as signed Radon measures.
3. $\diamond$
Hessian of a semiconcave function on an Alexandrov’s space is defined almost
everywhere. I.e. if $f\colon\Omega\to\mathbb{R}$ is a semiconcave function,
then for almost any $x_{0}\in\Omega$ there is a symmetric bi-linear form
$\operatorname{Hess}_{f}$ such that
$f(x)=f(x_{0})+d_{x_{0}}f(v)+\operatorname{Hess}_{f}(v,v)+o(|v|^{2}),$
where $v=\log_{x_{0}}x$. Moreover, $\operatorname{Hess}_{f}$ can be calculated
using standard formulas in the above chart.
Here is yet another, completely Riemannian application. This statement has
been proven by Perelman, a sketch of its proof is included in an appendix to
[Petrunin 2003]. The proof is based on the following observation: if $\Omega$
is an open subset of a Riemannian manifold and
$F\colon\Omega\to\mathbb{R}^{\ell+1}$ is a tight map with strictly concave
coordinate functions, then its level sets $F^{-1}(x)$ inherit the lower
curvature bound.
1. $\diamond$
_Continuity of the integral of scalar curvature._ Given a compact Riemannian
manifold $M$, let us define $\mathcal{F}(M)=\int_{M}\operatorname{Sc}$. Then
$\mathcal{F}$ is continuous on the space of Riemannian $m$-dimensional
manifolds with uniform lower curvature and upper diameter bounds.535353In fact
$\mathcal{F}$ is also bounded on the set of Riemannian $m$-dimensional
manifolds with uniform lower curvature, this is proved in [Petrunin 2007] by a
similar method.
## 9 Please deform an Alexandrov’s space.
In this section we discuss a number of related open problems. They seem to be
very hard, but I think it is worth to write them down just to indicate the
border between known and unknown things.
The main problem in Alexandrov’s geometry is to find a way to vary
Alexandrov’s space, or simply to find a nearby Alexandrov’s space to a given
Alexandrov’s space. Lack of such variation procedure makes it impossible to
use Alexandrov’s geometry in the way it was designed to be used:
For example, assume you want to solve the Hopf conjecture545454i.e. you want
to find out if $S^{2}\times S^{2}$ carries a metric with positive sectional
curvature.. Assume it is wrong, then there is a volume maximizing Alexandrov’s
metrics $d$ on $S^{2}\times S^{2}$ with curvature $\geqslant 1$555555There is
no reason to believe that this metric $d$ is Riemannian, but from Gromov’s
compactness theorem such Alexandrov’s metric should exist.. Provided we have a
procedure to vary $d$ while keeping its curvature $\geqslant 1$, we could find
some special properties of $d$ and in ideal situation show that $d$ does not
exist.
Unfortunately, at the moment, except for boring rescaling, there is no
variation procedure available. The following conjecture (if true) would give
such a procedure. Although it will not be sufficient to solve the Hopf
conjecture, it will give some nontrivial information about the critical
Alexandrov’s metric.
###### 9.1.1.
Conjecture. The boundary of an Alexandrov’s space equipped with induced
intrinsic metric is an Alexandrov’s space with the same lower curvature bound.
This also can be reformulated as:
9.1.1${}^{\prime}\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu$. Conjecture. Let $A$ be
an Alexandrov space without boundary. Then a convex hypersurface in $A$
equipped with induced intrinsic metric is an Alexandrov’s space with the same
lower curvature bound.
This conjecture, if true, would give a variation procedure. For example if $A$
is a non-negatively curved Alexandrov’s space and $f\colon A\to\mathbb{R}$ is
concave (so $A$ is necessarily open) then for any $t$ the graph
$A_{t}=\\{(x,t{\cdot}f(x))\in A\times\mathbb{R}\\}$
with induced intrinsic metric would be an Alexandrov’s space. Clearly
$A_{t}\buildrel\mathrm{GH}\over{\longrightarrow}A$ as $t\to 0$. An analogous
construction exists for semiconcave functions on closed manifolds, but one has
to take a _parabolic cone_ 565656see footnote 24 on page 24 instead of the
product.
It seems to be hopeless to attack this problem with purely synthetic methods.
In fact, so far, even for a convex hypersurface in a Riemannian manifold,
there is only one proof available (see [Buyalo]) which uses smoothing and the
Gauss formula575757In fact in this paper the curvature bound is not optimal,
but the statement follows from nearly the same idea; see [AKP].. There is one
beautiful synthetic proof (see [Milka 1979]) for a convex surface in the
Euclidian space, but this proof heavily relies on Euclidean structure and it
seems impossible to generalize it even to the Riemannian case.
There is a chance of attacking this problem by proving a type of the Gauss
formula for Alexandrov’s spaces. One has to start with defining a curvature
tensor of Alexandrov’s spaces (it should be a measure-valued tensor field),
then prove that the constructed tensor is really responsible for the geometry
of the space. Such things were already done in the two-dimensional case and
for spaces with bilaterly bounded curvature, see [Reshetnyak] and [Nikolaev]
respectively. So far the best results in this direction are given in
[Perelman-DC], see also section 8.2 for more details. This approach, if works,
would give something really new in the area.
Almost everything that is known so far about the intrinsic metric of a
boundary is also known for the intrinsic metric of a general extremal subset.
In [Perelman–Petrunin 1993], it was conjectured that an analog of conjecture
9.1.1 is true for any _primitive extremal subset_ , but it turned out to be
wrong; a simple example was constructed in [Petrunin 1997]. All such examples
appear when codimension of extremal subset is $\geqslant 3$. So it still might
be true that
###### 9.1.2.
Conjecture. Let $A\in\text{{\nnn Alex}}(\kappa)$, $E\subset A$ be a primitive
extremal subset and $\operatorname{codim}E=2$ then $E$ equipped with induced
intrinsic metric belongs to $\text{{\nnn Alex}}(\kappa)$
The following question is closely related to conjecture 9.1.1.
###### 9.1.3.
Question. Assume $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$, $\operatorname{dim}A=m$ (i.e. it is
not a collapse).
Let $f$ be a $\lambda$-concave function of an Alexandrov’s space $A$. Is it
always possible to find a sequence of $\lambda$-concave functions $f_{n}\colon
A_{n}\to\mathbb{R}$ which converges to $f\colon A\to\mathbb{R}$?
Here is an equivalent formulation:
9.1.3.′ Question. Assume $A_{n}\buildrel\mathrm{GH}\over{\longrightarrow}A$,
$A_{n}\in\text{{\nnn Alex}}^{m}(\kappa)$, $\operatorname{dim}A=m$ (i.e. it is
not a collapse) and $\partial A=\varnothing$.
Let $S\subset A$ be a convex hypersurface. Is it always possible to find a
sequence of convex hypersurfaces $S_{n}\subset A_{n}$ which converges to $S$?
If true, this would give a proof of conjecture 9.1.1 for the case of a
_smoothable Alexandrov’s space_ (see page 7.3.1).
In most of (possible) applications, Alexandrov’s spaces appear as limits of
Riemannian manifolds of the same dimension. Therefore, even in this reduced
generality, a positive answer would mean enough.
The question of whether an Alexandrov space is smoothable is also far from
being solved. From Perelamn’s stability theorem, if an Alexandrov’s space has
topological singularities then it is not smoothable. Moreover, from [Kapovitch
2002] one has that any space of directions of a smoothable Alexandrov’s space
is homeomorphic to the sphere. Except for the 2-dimensional case, it is only
known that any polyhedral metric of non-negative curvature on a 3-manifold is
smoothable (see [Matveev–Shevchishin]). There is yet no procedure of smoothing
an Alexandrov’s space even in a neighborhood of a regular point.
Maybe a more interesting question is whether smoothing is unique up to a
diffeomorphism. If the answer is positive it would imply in particular that
any Riemannian manifold with curvature $\geqslant 1$ and
$\operatorname{diam}>\tfrac{\pi}{2}$ is diffeomorphic(!) to the standard
sphere, see [Grove–Wilhelm] for details. Again, from Perelman’s stability
theorem ([Perelman 1991]), it follows that any two smoothings must be
homeomorphic. In fact it seems likely that any two smoothings are PL-
homeomorphic; see [Kapovitch 2007, question 1.3] and discussion right before
it. It seems that today there is no technique which might approach the general
uniqueness problem (so maybe one should try to construct a counterexample).
One may also ask similar questions in the collapsing case. In [PWZ] there were
constructed Alexandrov’s spaces with curvature $\geqslant 1$ which can not be
presented as a limit of an (even collapsing) sequence of Riemannian manifolds
with curvature $\geqslant\kappa>\tfrac{1}{4}$. In [Kapovitch 2005] there were
found some lower bounds for codimension of collapse with arbitrary lower
curvature bound to some special Alexandrov’s spaces, see section 7.3 for more
discussion. It is expected that the same spaces (for example, the spherical
suspension over the Cayley plane) can not be approximated by sequence of
Riemannian manifolds of any fixed dimension and any fixed lower curvature
bound, but so far this question remains open.
## Appendix A Existence of quasigeodesics
This appendix is devoted to the proof of property 4 on page 4, i.e.
###### A.0.1.
Existence theorem. Let $A\in\text{{\nnn Alex}}^{m}$, then for any point $x\in
A$, and any direction $\xi\in\Sigma_{x}$ there is a quasigeodesic
$\gamma\colon\mathbb{R}\to A$ such that $\gamma(0)=x$ and $\gamma^{+}(0)=\xi$.
Moreover if $E\subset A$ is an extremal subset and $x\in E$,
$\xi\in\Sigma_{x}E$ then $\gamma$ can be chosen to lie completely in $E$.
The proof is quite long; it was obtained by Perelman around 1992; here we
present a simplified proof similar to [Perelman–Petrunin QG] which is based on
the gradient flow technique. We include a complete proof here, since otherwise
it would never be published.
Quasigeodesics will be constructed in three big steps.
A.1
Monotonic curves $\longrightarrow$ convex curves.
A.2
Convex curves $\longrightarrow$ pre-quasigeodesics.
A.3
Pre-quasigeodesics $\longrightarrow$ quasigeodesics.
In each step, we construct a better type of curves from a given type of curves
by an extending-and-chopping procedure and then passing to a limit. The last
part is most complicated.
The second part of the theorem is proved in the subsection A.4.
### A.0 Step 0: Monotonic curves
As a starting point we use radial curves, which do exist for any initial data
(see section 3), and by lemma 3.1.2 are monotonic in the sense of the
following definition:
###### A.0.1.
Definition. A curve $\alpha(t)$ in an Alexandrov’s space $A$ is called
_monotonic_ with respect to a parameter value $t_{0}$ if for any
$\lambda$-concave function $f$, $\lambda\geqslant 0$, we have that function
$t\mapsto\frac{f\circ\alpha(t+t_{0})-f\circ\alpha(t_{0})-\tfrac{\lambda}{2}{\cdot}t^{2}}{t}$
is non-increasing for $t>0$.
Here is a construction which gives a new monotonic curve out of two. It will
be used in the next section to construct _convex curves_.
###### A.0.2.
Extention. Let $A\in\text{{\nnn Alex}}$, $\alpha_{1}[a,\infty)\to A$ and
$\alpha_{2}\colon[b,\infty)\to A$ be two monotonic curves with respect to $a$
and $b$ respectively.
Assume
$a\leqslant b,\ \ \alpha_{1}(b)=\alpha_{2}(b)\ \ \text{and}\ \
\alpha^{+}_{1}(b)=\alpha^{+}_{2}(b).$
Then its joint
$\beta\colon[a,\infty)\to A,\ \
\beta(t)=\left[\begin{matrix}\alpha_{1}(t)&\text{if}&t<b\\\
\alpha_{2}(t)&\text{if}&t\geqslant b\end{matrix}\right.$
is monotonic with respect to $a$ and $b$.
Proof. It is enough to show that
$t\mapsto\frac{f\circ\alpha_{2}(t+a)-f\circ\alpha_{1}(a)-\tfrac{\lambda}{2}{\cdot}t^{2}}{t}$
is non-increasing for $t\geqslant b-a$. By simple algebra, it follows from the
following two facts:
1. $\diamond$
$\alpha_{2}$ is monotonic and therefore
$t\mapsto\frac{f\circ\alpha_{2}(t+b)-f\circ\alpha_{2}(b)-\tfrac{\lambda}{2}{\cdot}t^{2}}{t}$
is non-increasing for $t>0$.
2. $\diamond$
From monotonicity of $\alpha_{1}$,
$\displaystyle(f\circ\alpha_{2})^{+}(b)$
$\displaystyle=d_{\alpha_{1}(b)}f(\alpha_{1}^{+}(b))=$
$\displaystyle=(f\circ\alpha_{1})^{+}(b)\leqslant$
$\displaystyle\leqslant\frac{f\circ\alpha_{1}(b)+f\circ\alpha_{1}(a)-\tfrac{\lambda}{2}(b-a)^{2}}{b-a}.$
∎
### A.1 Step 1: Convex curves.
In this step we construct _convex curves_ with arbitrary initial data.
###### A.1.1.
Definition. A curve $\beta\colon[0,\infty)\to A$ is called _convex_ if for any
$\lambda$-concave function $f$, $\lambda\geqslant 0$, we have that function
$t\mapsto f\circ\beta(t)-\tfrac{\lambda}{2}{\cdot}t^{2}$
is concave.
Properties of convex curves. Convex curves have the following properties; the
proofs are either trivial or the same as for quasigeodesics:
1. 1.
A curve is convex if and only if it is monotonic with respect to any value of
parameter.
2. 2.
Convex curves are $1$-Lipschitz.
3. 3.
Convex curves have uniquely defined right and left tangent vectors.
4. 4.
A limit of convex curves is convex and the natural parameter converges to the
natural parmeter of the limit curves (the proof the last statement is based on
the same idea as theorem 7.3.3).
The next is a construction similar to A.0.2 which gives a new convex curve out
of two. It will be used in the next section to construct _pre-quasigeodesics_.
###### A.1.2.
Extention. Let $A\in\text{{\nnn Alex}}$, $\beta_{1}\colon[a,\infty)\to A$ and
$\beta_{2}\colon[b,\infty)\to A$ be two convex curves. Assume
$a\leqslant b,\ \ \beta_{1}(b)=\beta_{2}(b)\ \ \text{and}\ \
\beta^{+}_{1}(b)=\beta^{+}_{2}(b)$
then its joint
$\gamma\colon[a,\infty)\to A,\ \
\gamma(t)=\left[\begin{matrix}\beta_{1}(t)&\text{if}&t\leqslant b\\\
\beta_{2}(t)&\text{if}&t\geqslant b\end{matrix}\right.$
is a convex curve.
Proof. Follows immidetely from A.0.2 and property 1 above.
###### A.1.3.
Existence. Let $A\in\text{{\nnn Alex}}$, $x\in A$ and $\xi\in\Sigma_{x}$. Then
there is a convex curve $\beta_{\xi}\colon[0,\infty)\to A$ such that
$\beta_{\xi}(0)=x$ and $\beta_{\xi}^{+}(0)=\xi$.
Proof. For $v\in T_{x}A$, consider the radial curve
$\alpha_{v}(t)=\operatorname{gexp}_{x}(tv)$
According to lemma 3.1.2 if $|v|=1$ then $\alpha_{v}$ is $1$-Lipschitz and
monotonic. Moreover, straightforward calculations show that the same is true
for $|v|\leqslant 1$.
Fix $\varepsilon>0$. Given a direction $\xi\in\Sigma_{x}$, let us consider the
following recursively defined sequence of radial curves $\alpha_{v_{n}}(t)$
such that $v_{0}=\xi$ and $v_{n}=\alpha^{+}_{v_{n-1}}(\varepsilon)$. Then
consider their joint
$\beta_{\xi,\varepsilon}(t)=\alpha_{v_{\lfloor
t/\varepsilon\rfloor}}(t-\varepsilon\lfloor t/\varepsilon\rfloor).$
Applying an extension procedure A.0.2 we get that
$\beta_{\xi,\varepsilon}\colon[0,\infty)\to A$ is monotonic with respect to
any $t=n{\cdot}\varepsilon$.
By property 1 on page 1, passing to a partial limit
$\beta_{\xi,\varepsilon}\to\beta_{\xi}$ as $\varepsilon\to 0$ we get a convex
curve $\beta_{\xi}\colon[0,\infty)\to A$.
It only remains to show that $\beta_{\xi}^{+}(0)=\xi$.
Since $\beta_{\xi}$ is convex, its right tangent vector is well defined and
$|\beta^{+}_{\xi}(0)|\leqslant\nobreak 1$585858see properties 3 and 2, page 3.
On the other hand, since $\beta_{\xi,\varepsilon}$ are monotonic with respect
to $0$, for any semiconcave function $f$ we have
$\displaystyle d_{x}f(\beta^{+}_{\xi}(0))$
$\displaystyle=(f\circ\beta_{\xi})^{+}(0)\leqslant$
$\displaystyle\leqslant\lim_{\varepsilon_{i}\to
0}(f\circ\beta_{\xi,\varepsilon})^{+}(0)=$ $\displaystyle=d_{x}f(\xi).$
Substituting in this inequality $f=\operatorname{dist}_{y}$ with
$\measuredangle(\uparrow_{x}^{y},\xi)<\varepsilon$, we get
$\langle\beta^{+}_{\xi}(0),\uparrow_{x}^{y}\rangle>1-\varepsilon$
for any $\varepsilon>0$. Together with $|\beta^{+}_{\xi}(0)|\leqslant 1$
(property 2 on page 2), it implies that
$\beta^{+}(0)=\xi.$
∎
### A.2 Step 2: Pre-quasigeodesics
In this step we construct a _pre-quasigeodesic_ with arbitrary initial data.
###### A.2.1.
Definition. A convex curve $\gamma\colon[a,b)\rightarrow A$ is called a pre-
quasigeodesic if for any $s\in[a,b)$ such that ${|\gamma^{+}(s)|}>0$, the
curve $\gamma^{s}$ defined by
$\gamma^{s}(t)=\gamma\left(s+\frac{t}{|\gamma^{+}(s)|}\right)$
is convex for $t\geqslant 0$, and if ${|\gamma^{+}(s)|}=0$ then
$\gamma(t)=\gamma(s)$ for all $t\geqslant s$.
Let us first define entropy of pre-quasigeodesic, which measures “how far” a
given pre-quasigeodesic is from being a quasigeodesic.
###### A.2.2.
Definition. Let $\gamma$ be a pre-quasigeodesic in an Alexandrov’s space.
The _entropy_ of $\gamma$, $\mu_{\gamma}$ is the measure on the set of
parameters defined by
$\mu_{\gamma}((a,b))=\ln|\gamma^{+}(a)|-\ln|\gamma^{-}(b)|.$
Here are its main properties:
1. 1.
The entropy of a pre-quasigeodesic $\gamma$ is zero if and only if $\gamma$ is
a quasigedesic.
2. 2.
For a converging sequence of pre-quasigeodesics $\gamma_{n}\to\gamma$, the
entropy of the limit is a weak limit of entropies,
$\mu_{\gamma_{n}}\rightharpoonup\mu_{\gamma}$.
It follows from property 4 on page 4.
The next statement is similar to A.0.2 and A.1.2; it makes a new pre-
quasigeodesic out of two. It will be used in the next section to construct
_quasigeodesics_.
###### A.2.3.
Extention. Let $A\in\text{{\nnn Alex}}$, $\gamma_{1}\colon[a,\infty)\to A$ and
$\gamma_{2}\colon[b,\infty)\to A$ be two pre-quasigeodesics. Assume
$a\leqslant b,\ \ \gamma_{1}(b)=\gamma_{2}(b),\ \ \gamma^{-}_{1}(b)\ \
\text{is polar to}\ \ \gamma^{+}_{2}(b)\ \ \text{and}\ \
|\gamma^{+}_{2}(b)|\leqslant|\gamma^{-}_{1}(b)|$
then its joint
$\gamma\colon[a,\infty)\to A,\ \
\gamma(t)=\left[\begin{matrix}\gamma_{1}(t)&\text{if}&t\leqslant b\\\
\gamma_{2}(t)&\text{if}&t\geqslant b\end{matrix}\right.$
is a pre-quasigeodesic. Moreover, its entropy is defined by
$\mu_{\gamma}|_{(a,b)}=\mu_{\gamma_{1}},\ \
\mu_{\gamma}|_{(b,c)}=\mu_{\gamma_{2}}\ \ \text{and}\ \
\mu_{\gamma}(\\{b\\})=\ln|\gamma^{+}(b)|-\ln|\gamma^{-}(b)|.$
Proof. The same as for A.0.2. ∎
###### A.2.4.
Existence. Let $A\in\text{{\nnn Alex}}$, $x\in A$ and $\xi\in\Sigma_{x}$. Then
there is a pre-quasigeodesic $\gamma\colon[0,\infty)\to A$ such that
$\gamma(0)=x$ and $\gamma^{+}(0)=\xi$.
Proof. Let us choose for each point $x\in A$ and each direction
$\xi\in\Sigma_{x}$ a convex curve $\beta_{\xi}\colon[0,\infty)\to A$ such that
$\beta_{\xi}(0)=x$, $\beta_{\xi}^{+}(0)=\xi$. If $v=r\xi$, then set
$\beta_{v}(t)=\beta_{\xi}(rt).$
Clearly $\beta_{v}$ is convex if $0\leqslant r\leqslant 1$.
Let us construct a convex curve
$\gamma_{\varepsilon}\colon[0,\infty)\rightarrow M$ such that there is a
representation of $[0,\infty)$ as a countable union of disjoint half-open
intervals $[a_{i},\bar{a}_{i})$, such that
$|\bar{a}_{i}-a_{i}|\leqslant\varepsilon$ and for any
$t\in[a_{i},\bar{a}_{i})$ we have
$|\gamma_{\varepsilon}^{+}(a_{i})|\geqslant|\gamma_{\varepsilon}^{+}(t)|\geqslant(1-\varepsilon)\cdot|\gamma_{\varepsilon}^{+}(a_{i})|.$
$None$
Moreover, for each $i$, the curve
$\gamma_{\varepsilon}^{a_{i}}\colon[0,\infty)\to A$,
$\gamma_{\varepsilon}^{a_{i}}(t)=\gamma_{\varepsilon}\left({a_{i}}+\frac{t}{|\gamma_{\varepsilon}^{+}({a_{i}})|}\right)$
is also convex.
Assume we already can construct $\gamma_{\varepsilon}$ in the interval
$[0,t_{\max})$, and cannot do it any further. Since $\gamma_{\varepsilon}$ is
1-Lipschitz, we can extend it continuously to $[0,t_{\max}]$. Use lemma 1.3.9
to construct a vector $v^{*}$ polar to $\gamma^{-}_{\varepsilon}(t_{\max})$
with $|v^{*}|\leqslant|\gamma^{-}_{\varepsilon}(t_{\max})|$. Consider the
joint of $\gamma_{\varepsilon}$ with a short half-open segment of $\beta_{v}$,
a longer curve with the desired property. This is a contradiction.
Let $\gamma$ be a partial limit of $\gamma_{\varepsilon}$ as $\varepsilon\to
0$. From property 4 on page 4, we get that for almost all $t$ we have
$|\gamma^{+}(t)|=\lim|\gamma_{\varepsilon_{n}}^{+}(t)|$. Combining this with
inequality $(*)$ shows that for any $a\geqslant 0$
$\gamma^{a}(t)=\gamma\left({a}+\frac{t}{|\gamma^{+}({a})|}\right)$
is convex. ∎
### A.3 Step 3: Quasigeodesics
We will construct quasigeodesics in an $m$-dimensional Alexandrov’s space,
assuming we already have such a construction in all dimensions $<m$. This
construction is much easier for the case of an Alexandrov’s space with only
$\delta$-strained points; in this case we construct a sequence of special pre-
quasigeodesics only by extending/chopping procedures (see below) and then pass
to the limit. In a general Alexandrov’s space we argue by contradiction, we
assume that $\Omega$ is a maximal open set such that for any initial data one
can construct an $\Omega$-quasigeodesic (i.e. a pre-quasigeodesic with zero
entropy on $\Omega$, see A.2.2), and arrive at a contradiction with the
assumption $\Omega\not=A$.
The following extention and chopping procedures are essential in the
construction:
###### A.3.1.
Extention procedure. Given a pre-quasigeodesic $\gamma\colon[0,t_{\max})\to A$
we can extend it as a pre-quasigeodesic $\gamma\colon[0,\infty)\to A$ so that
$\mu_{\gamma}(\\{t_{\max}\\})=0.$
Proof. Let us set $\gamma(t_{\max})$ to be the limit of $\gamma(t)$ as $t\to
t_{\max}$ (it exists since pre-quasigeodesics are Lipschitz).
From Milka’s lemma A.3.2, we can construct a vector $\gamma^{+}(t_{\max})$
which is polar to $\gamma^{-}(t_{\max})$ and such that
$|\gamma^{+}(t_{\max})|=|\gamma^{-}(t_{\max})|$. Then extend $\gamma$ by a
pre-quasigeodesic in the direction $\gamma^{+}(t_{\max})$. By A.2.3, we get
$\mu_{\gamma}\\{t_{\max}\\}=\ln|\gamma^{+}(t_{\max})|-\ln|\gamma^{-}(t_{\max})|=0.$
∎
###### A.3.2.
Milka’s lemma (existence of the polar direction). For any unit vector
$\xi\in\Sigma_{p}$ there is a polar unit vector $\xi^{*}$, i.e.
$\xi^{*}\in\Sigma_{p}$ such that
$\langle\xi,v\rangle+\langle\xi^{*},v\rangle\geqslant 0$
for any $v\in T_{p}$.
The proof is taken from [Milka 1968]. That is the only instance where we use
existence of quasigeodesics in lower dimensional spaces.
Proof. Since $\Sigma_{p}$ is an Alexandrov’s $(m-1)$-space with curvature
$\geqslant 1$, given $\xi\in\Sigma_{p}$ we can construct a quasigeodesic in
$\Sigma_{p}$ of length $\pi$, starting at $\xi$; the comparison inequality
(theorem 5(5iv)) implies that the second endpoint $\xi^{*}$ of this
quasigeodesic satisfies
$|\xi\,\eta|_{\Sigma_{q}}+|\eta\,\xi^{*}|_{\Sigma_{q}}=\measuredangle(\xi,\eta)+\measuredangle(\eta,\xi^{*})\leq\pi\
\ \text{for all}\ \ \eta\in\Sigma_{p},$
which is equivalent to the statement that $\xi$ and $\xi^{*}$ are polar in
$T_{p}$. ∎
###### A.3.3.
Chopping procedure. Given a pre-quasigeodesic $\gamma\colon[0,\infty)\to A$,
for any $t\geqslant 0$ and $\varepsilon>0$ there is $\bar{t}>t$ such that
$\mu_{\gamma}\left((t,\bar{t})\right)<\varepsilon[\vartheta+\bar{t}-t],\ \
\bar{t}-t<\varepsilon,\ \ \vartheta<\varepsilon,$
where
$\vartheta=\vartheta(t,\bar{t})=\measuredangle\left(\gamma^{+}(t),\uparrow_{\gamma(t)}^{\gamma(\bar{t})}\right).$
$\gamma(t)$$\gamma(\bar{t})$$\vartheta$$\gamma$
Proof. For all sufficiently small $\tau>0$ we have
$\vartheta(t,t+\tau)<\varepsilon$
and from convexity of $\gamma^{t}$ it follows that
$\mu\left((t,t+\tau/3)\right)<C{\cdot}\vartheta^{2}(t,t+\tau).$
The following exercise completes the proof. ∎
###### A.3.4.
Exercise. Let the functions $h,g\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ be such
that for any sufficiently small $s$,
$h(s/3)\leqslant g^{2}(s),\ s\leqslant g(s)\ \hbox{and}\ \lim_{s\to 0}g(s)=0.$
Show that for any $\varepsilon>0$ there is $s>0$ such that
$h(s)<10{\cdot}g^{2}(s)\ \hbox{and}\ g(s)\leqslant\varepsilon.$
Construction in the $\mathbf{\delta}$-strained case. From the extension
procedure, it is sufficient to construct a quasigeodesic $\gamma\colon[0,T)\to
A$ with any given initial data $\gamma^{+}(0)=\xi\in\Sigma_{p}$ for some
positive $T=T(p)$.
The plan: Given $\varepsilon>0$, we first construct a pre-quasigeodesic
$\gamma_{\varepsilon}\colon[0,T)\to A,\ \ \ \gamma_{\varepsilon}^{+}(0)=\xi$
such that one can present $[0,T)$ as a countable union of disjoint half-open
intervals $[a_{i},\bar{a}_{i})$ with the following property ($\vartheta$ is
defined in the chopping procedure A.3.3):
$\mu\left([a_{i},\bar{a}_{i})\right)<\varepsilon{\cdot}\vartheta(a_{i},\bar{a}_{i}),\
\ \bar{a}_{i}-a_{i}<\varepsilon,\ \ \vartheta(a_{i},\bar{a}_{i})<\varepsilon.$
$None$
Then we show that the entropies $\mu_{\gamma_{\varepsilon}}([0,T))\to 0$ as
$\varepsilon\to 0$ and passing to a partial limit of $\gamma_{\varepsilon}$ as
$\varepsilon\to 0$ we get a quasigeodesic.
Existence of $\gamma_{\varepsilon}$: Assume that we already can construct
$\gamma_{\varepsilon}$ on an interval $[0,t_{\max})$, $t_{\max}<T$ and cannot
construct it any further, then applying the extension procedure A.3.1 for
$\gamma_{\varepsilon}\colon[0,t_{\max})\to A$ and then chopping it (A.3.3)
starting from $t_{\max}$, we get a longer curve with the desired property;
that is a contradiction.
Vanishing entropy: From $(\star)$ we have that
$\mu_{\gamma_{\varepsilon}}([0,T))<\varepsilon\cdot\left[T+\sum_{i}\vartheta(a_{i},\bar{a}_{i})\right].$
Therefore, to show that $\mu_{\gamma_{\varepsilon}}([0,T))\to 0$, it only
remains to show that $\sum_{i}\vartheta(a_{i},\bar{a}_{i})$ is bounded above
by a constant independent of $\varepsilon$.
That will be the only instance, where we apply that $p$ is $\delta$-strained
for a small enough $\delta$.
It is easy to see that there is $\varepsilon=\varepsilon(\delta)\to 0$ as
$\delta\to 0$ and $T=T(p)>0$ such that there is a finite collection of points
$\\{q_{k}\\}$ which satisfy the following property: for any $x\in B_{T}(p)$
and $\xi\in\Sigma_{x}$ there is $q_{k}$ such that
$\measuredangle(\xi,\uparrow_{x}^{q_{k}})<\varepsilon$. Moreover, we can
assume $\operatorname{dist}_{q_{k}}$ is $\lambda$-concave in $B_{T}(p)$ for
some $\lambda>0$.
Note that for any convex curve $\gamma\colon[0,T)\to B_{T}(p)\subset A$, the
measures $\chi_{k}$ on $[0,T)$, defined by
$\chi_{k}((a,b))=(\operatorname{dist}_{q_{k}}\circ\gamma)^{-}(b)-(\operatorname{dist}_{q_{k}}\circ\gamma)^{+}(a)+\lambda{\cdot}(b-a),$
are positive and their total mass is bounded by $\lambda T+2$ (this follows
from the fact that $\operatorname{dist}_{q_{k}}$ is $\lambda$-concave and
1-Lipschitz).
Let $x\in B_{T}(p)$, and $\delta$ be small enough. Then for any two directions
$\xi,\nu\in\nobreak\Sigma_{x}$ there is $q_{k}$ which satisfies the following
property:
$\tfrac{1}{10}{\cdot}\measuredangle_{x}(\xi,\nu)\leqslant
d_{x}\operatorname{dist}_{q_{k}}(\xi)-d_{x}\operatorname{dist}_{q_{k}}(\nu)\ \
\ \text{and}\ \ \ d_{x}\operatorname{dist}_{q_{k}}(\nu)\geqslant 0.$ $None$
Substituting in this inequality
$\xi=\gamma^{+}(a_{i})/|\gamma^{+}(a_{i})|,\ \ \
\nu=\uparrow_{\gamma(a_{i})}^{\gamma(\bar{a}_{i})},$
and applying lemma A.3.5, we get
$\vartheta(a_{i},\bar{a}_{i})=\measuredangle(\xi,\nu)\leqslant
10{\cdot}\sum_{n}\chi_{k}([a_{i},\bar{a}_{i})).$
Therefore
$\sum_{i}\vartheta(a_{i},\bar{a}_{i})\leqslant 10{\cdot}N{\cdot}(\lambda
T+2),$
where $N$ is the number of points in the collection $\\{q_{k}\\}$. ∎
###### A.3.5.
Lemma. Let $A\in\text{{\nnn Alex}}$, $\gamma\colon[0,t]\to A$ be a convex
curve $|\gamma^{+}(0)|=1$ and $f$ be a $\lambda$-concave function,
$\lambda\geqslant 0$. Set $p=\gamma(0)$, $q=\gamma(t)$, $\xi=(\gamma)^{+}(0)$
and $\nu=\uparrow_{p}^{q}$. Then
$d_{p}f(\xi)-d_{p}f(\nu)\leqslant(f\circ\gamma)^{+}(0)-(f\circ\gamma)^{-}(t)+\lambda{\cdot}t,$
provided that $d_{p}f(\nu)\geqslant 0$.
$p$$q$$\xi$$\nu$$\gamma$
Proof. Clearly,
$f(q)\leq
f(p)+d_{p}f(\nu){\cdot}|pq|+\tfrac{\lambda}{2}{\cdot}|pq|^{2}\leqslant
f(p)+d_{p}f(\nu)t+\tfrac{\lambda}{2}{\cdot}t^{2}.$
On the other hand,
$f(p)\leqslant
f(q)-(f\circ\gamma)^{-}(t){\cdot}t+\tfrac{\lambda}{2}{\cdot}t^{2}.$
Clearly, $d_{p}f(\xi)=(f\circ\gamma)^{+}(0)$, whence the result. ∎
What to do now? We have just finished the proof for the case, where all points
of $A$ are $\delta$-strained. From this proof it follows that if we denote by
$\Omega_{\delta}$ the subset of all $\delta$-strained points of $A$ (which is
an open everywhere dense set, see [BGP, 5.9]), then for any initial data one
can construct a pre-quasigeodesic $\gamma$ such that
$\mu_{\gamma}(\gamma^{-1}(\Omega_{\delta}))=0$. Assume $A$ has no boundary;
set $\mathfrak{C}=A\backslash\Omega_{\delta}$. In this case it seems unlikely
that we hit $\mathfrak{C}$ by shooting a pre-quasigeodesic in a generic
direction. If we could prove that it almost never happens, then we obtain
existence of quasigeodesics in all directions as the limits of quasigeodesics
in generic directions (see property 6 on page 6) and passing to doubling in
case $\partial A\not=\varnothing$. Unfortunately, we do not have any tools so
far to prove such a thing595959It might be possible if we would have an analog
of the Liouvile theorem for “pre-quasigeodesic flow”. Instead we generalize
inequality $(*)$.
###### A.3.6.
The $\mathbf{(*)}$ inequality. Let $A\in\text{{\nnn Alex}}^{m}(\kappa)$ and
$\mathfrak{C}\subset A$ be a closed subset. Let $p\in\mathfrak{C}$ be a point
with $\delta$-maximal $\operatorname{vol}_{m-1}\Sigma_{p}$, i.e.
$\operatorname{vol}_{m-1}\Sigma_{p}+\delta>\inf_{x\in\mathfrak{C}}\\{\operatorname{vol}_{m-1}\Sigma_{p}\\}.$
Then, if $\delta$ is small enough, there is a finite set of points
$\\{q_{i}\\}$ and $\varepsilon>0$, such that for any
$x\in\mathfrak{C}\cap\bar{B}_{\varepsilon}(p)$ and any pair of directions
$\xi\in\Sigma_{x}\mathfrak{C}$606060$\Sigma_{x}\mathfrak{C}$ is defined on
page 29. and $\nu\in\Sigma_{x}$ we can choose $q_{i}$ so that
$\tfrac{1}{10}{\cdot}\measuredangle_{x}(\xi,\nu)\leqslant
d_{x}\operatorname{dist}_{q_{i}}(\xi)-d_{x}\operatorname{dist}_{q_{i}}(\nu)\ \
\ \text{and}\ \ \ d_{x}\operatorname{dist}_{q_{k}}(\nu)\geqslant 0.$
Proof. We can choose $\varepsilon>0$ so small that for any
$x\in\bar{B}_{\varepsilon}(p)$, $\Sigma_{x}$ is almost bigger than
$\Sigma_{p}$.616161i.e. for small $\delta>0$ there is a map
$f\colon\Sigma_{p}\to\Sigma_{x}$ such that $|f(x)f(y)|>|xy|-\delta$. Since
$\operatorname{vol}_{m-1}\Sigma_{p}$ is almost maximal we get that for any
$x\in\mathfrak{C}\cap\bar{B}_{\varepsilon}(p)$, $\Sigma_{x}$ is almost
isometric to $\Sigma_{p}$. In particular, if one takes a set $\\{q_{i}\\}$ so
that directions $\uparrow_{p}^{q_{i}}$ form a sufficiently dense set and
$\measuredangle q_{i}pq_{j}\approx\tilde{\measuredangle}_{\kappa}q_{i}pq_{j}$,
then directions $\uparrow_{x}^{q_{i}}$ will form a sufficiently dense set in
$\Sigma_{x}$ for all $x\in\mathfrak{C}\cap\bar{B}_{\varepsilon}(p)$.
Note that for any $x\in\mathfrak{C}\cap\bar{B}_{\varepsilon}(p)$ and
$\xi\in\Sigma_{x}\mathfrak{C}$, there is an almost isometry
$\Sigma_{x}\to\Sigma(\Sigma_{\xi}\Sigma_{x})$ such that $\xi$ goes to north
pole of the spherical suspension
$\Sigma(\Sigma_{\xi}\Sigma_{x})=\Sigma_{\xi}T_{x}$. 626262Otherwise, taking a
point $y\in\mathfrak{C}$, close to $x$ in direction $\xi$ we would get that
$\operatorname{vol}_{m-1}\Sigma_{y}$ is essentially bigger than
$\operatorname{vol}_{m-1}\Sigma_{x}$, which is impossible since both are
almost equal to $\operatorname{vol}_{m-1}\Sigma_{p}$.
Using these two properties, we can find $q_{i}$ so that
$\uparrow^{\nu}_{\xi}\approx\uparrow_{\xi}^{\uparrow_{x}^{q_{i}}}$ in
$\Sigma_{\nu}(\Sigma_{x}A)$ and
$\measuredangle(\xi,\uparrow_{x}^{q_{i}})>\tfrac{\pi}{2}$, hence the statement
follows. ∎
Now we are ready to finish construction in the general case. Let us define a
subtype of pre-quasigeodesics:
###### A.3.7.
Definition. Let $A\in\text{{\nnn Alex}}$ and $\Omega\subset A$ be an open
subset. A pre-quasigeodesic $\gamma\colon[0,T)\to A$ is called
$\Omega$-quasigeodesic if its entropy vanishes on $\Omega$, i.e.
$\mu_{\gamma}(\gamma^{-1}(\Omega))=0$
From property 2 on page 2, it follows that the limit of
$\Omega$-quasigeodesics is a $\Omega$-quasigeodesic. Moreover, if for any
initial data we can construct an $\Omega$-quasigeodesic and an
$\Omega^{\prime}$-quasigeodesic, then it is possible to construct an
$\Omega\cup\Omega^{\prime}$-quasigeodesic for any initial data; for
$\Upsilon\Subset\Omega\cup\Omega^{\prime}$, $\Upsilon$-quasigeodesic can be
constructed by joining together pieces of $\Omega$ and
$\Omega^{\prime}$-quasigeodesics and $\Omega\cup\Omega^{\prime}$-quasigeodesic
can be constructed as a limit of $\Upsilon_{n}$-quasigeodesics as
$\Upsilon_{n}\to\Omega\cup\Omega^{\prime}$.
Let us denote by $\Omega$ the maximal open set such that for any initial data
one can construct an $\Omega$-quasigeodesic. We have to show then that
$\Omega=A$.
Let $\mathfrak{C}=A\backslash\Omega$, and let $p\in\mathfrak{C}$ be the point
with almost maximal $\operatorname{vol}_{m-1}\Sigma_{p}$. We will arrive to a
contradiction by constructing a $B_{\varepsilon}(p)\cup\Omega$-quasigeodesic
for any initial data.
Choose a finite set of points $q_{i}$ as in A.3.6. Given $\varepsilon>0$, it
is enough to construct an $\Omega$-quasigeodesic
$\gamma_{\varepsilon}\colon[0,T)\to A$, for some fixed $T>0$ with the given
initial data $x\in\bar{B}_{\varepsilon}(p)$, $\xi\in\Sigma_{x}$, such that the
entropies $\mu_{\gamma_{\varepsilon}}((0,T))\to 0$ as $\varepsilon\to 0$.
The $\Omega$-quasigeodesic $\gamma_{\varepsilon}$ which we are going to
construct will have the following property: one can present $[0,T)$ as a
countable union of disjoint half-open intervals $[a_{i},\bar{a}_{i})$ such
that
$\text{if}\ \ \
\frac{\gamma^{+}(a_{i})}{|\gamma^{+}(a_{i})|}\in\Sigma_{\gamma(a_{i})}\mathfrak{C}\
\ \ \text{then}\ \ \
\mu_{\gamma}([a_{i},\bar{a}_{i}))\leqslant\varepsilon{\cdot}\vartheta(a_{i},\bar{a}_{i})$
and
$\text{if}\ \ \
\frac{\gamma^{+}(a_{i})}{|\gamma^{+}(a_{i})|}\not\in\Sigma_{\gamma(a_{i})}\mathfrak{C}\
\ \ \text{then}\ \ \ \mu_{\gamma}([a_{i},\bar{a}_{i}))=0$
Existence of $\gamma_{\varepsilon}$ is being proved the same way as in the
$\delta$-strained case, with the use of one additional observation: if
$\frac{\gamma^{+}(t_{\max})}{|\gamma^{+}(t_{\max})|}\not\in\Sigma_{\gamma(a_{i})}\mathfrak{C}$
then any $\Omega$-quasigeodesic in this direction has zero entropy for a short
time.
Then, just as in the $\delta$-strained case, applying inequality A.3.6 we get
that $\mu_{\gamma_{\varepsilon}}(0,T)\to 0$ as $\varepsilon\to 0$. Therefore,
passing to a partial limit $\gamma_{\varepsilon}\to\gamma$ gives a
$B_{\varepsilon}(p)\cup\Omega$-quasigeodesic $\gamma\colon[0,T)\to A$ for any
initial data in $B_{\varepsilon}(p)$. ∎
### A.4 Quasigeodesics in extremal subsets.
The second part of theorem A.0.1 follows from the above construction, but we
have to modify Milka’s lemma A.3.2:
###### A.4.1.
Extremal Milka’s lemma. Let $E\subset T_{p}$ be an extremal subset of a
tangent cone then for any vector $v\in E$ there is a polar vector $v^{*}\in E$
such that $|v|=|v^{*}|$.
Proof. Set $X=E\cap\Sigma_{p}$. If $\Sigma_{\xi}X\not=\varnothing$ then the
proof is the same as for the standard Milka’s lemma; it is enough to choose a
direction in $\Sigma_{\xi}X$ and shoot a quasigedesic $\gamma$ of length $\pi$
in this direction such that $\gamma\subset X$ ($\gamma$ exists from the
induction hypothesis).
If $X=\\{\xi\\}$ then from the extremality of $E$ we have
$B_{\pi/2}(\xi)=\Sigma_{p}$. Therefore $\xi$ is polar to itself.
Otherwise, if $\Sigma_{\xi}X=\varnothing$ and $X$ contains at least two
points, choose $\xi^{*}$ to be closest point in $X\backslash\xi$ from $\xi$.
Since $X\subset\Sigma_{p}$ is extremal we have that for any
$\eta\in\Sigma_{p}$
$\measuredangle_{\Sigma_{p}}\eta\xi^{*}\xi\leqslant\tfrac{\pi}{2}$ and since
$\Sigma_{\xi}X=\varnothing$ we have
$\measuredangle_{\Sigma_{p}}\eta\xi\xi^{*}\leqslant\tfrac{\pi}{2}$. Therefore,
from triangle comparison we have
$|\xi\eta|_{\Sigma_{p}}+|\eta\xi^{*}|_{\Sigma_{p}}=\measuredangle(\xi,\eta)+\measuredangle(\eta,\xi^{*})\leqslant\pi$
∎
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* [Shiohama] Shiohama, Katsuhiro An introduction to the geometry of Alexandrov spaces. Lecture Notes Series, 8. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. ii+78 pp.
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|
arxiv-papers
| 2013-04-01T03:56:21 |
2024-09-04T02:49:43.674368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1304.0292"
}
|
1304.0293
|
KEK-TH-1616
# What can we learn from the 126 GeV Higgs boson
for the Planck scale physics ?
\- Hierarchy problem and the stability of the vacuum - 111Contribution to
SCGT12 ”KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC
Perspective”, 4-7 Dec. 2012, Nagoya University and HPNP2013 ”Toyama
International Workshop on Higgs as a Probe of New Physics 2013”, 13–16, Feb.
2013, Toyama University
Satoshi Iso Theory Center, Institute of Particles and Nuclear Studies
High Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba, Ibaraki, Japan 305-0801
###### Abstract
The discovery of the Higgs particle at around 126 GeV has given us a big hint
towards the origin of the Higgs potential. The running quartic self-coupling
decreases and crosses zero somewhere in the very high energy scale. It is
usually considered as a signal of the instability of the standard model (SM)
vacuum, but it can also indicate a link between the physics in the electroweak
scale and the Planck scale. Furthermore, the LHC experiments as well as the
flavor physics experiments give strong constraints on the physics beyond the
SM. It urges us to reconsider the widely taken approach to the physics beyond
the SM (BSM), namely the approach based on the gauge unification below the
Planck scale and the resulting hierarchy problem. Motivated by the recent
experiments, we first revisit the hierarchy problem and consider an
alternative appoach based on a classical conformality of the SM without the
Higgs mass term.
In this talk, I review our recent proposal of a B-L extension of the SM with a
flat Higgs potential at the Planck scale IsoOrikasa ; IOO . This model can be
an alternative solution to the hierarchy problem as well as being
phenomenologically viable to explain the neutrino oscillations and the baryon
asymmetry of the universe. With an assumption that the Higgs has a flat
potential at the Planck scale, we show that the B-L symmetry is radiatively
broken at the TeV scale via the Coleman-Weinberg mechanism, and it triggers
the electroweak symmetry breaking through a radiatively generated scalar
mixing. The ratio of these two breaking scales is dynamically determined by
the B-L gauge coupling.
## I Central dogma of particle physics
In the LHC era, we acquired various hints towards the physics beyond the SM.
The first hint is of course the mass of the recently discovered Higgs-like
particle. The value of 126 GeV is quite interesting because it is close to the
boarder of the stability bound. Given the vev of the Higgs at 246 GeV, its
mass gives an information of the curvature of the potential at the minimum. As
the mass 126 GeV is smaller than 246 GeV, the Higgs potential is rather
shallow and unstable against the radiative corrections. The (in)stability of
the SM vacuum can be investigated explicitly by looking at the running
bahaviour of the quartic coupling. The beta function of the quartic Higgs
coupling $\lambda_{H}$ is given by
$\displaystyle\beta_{H}=\frac{1}{16\pi^{2}}\left(24\lambda_{H}^{2}-6Y_{t}^{4}+\frac{9}{8}g^{4}+\frac{3}{8}g_{Y}^{4}\right).$
(1)
$Y_{t}$ is the top Yukawa coupling and $g,g_{Y}$ are $SU(2)_{L},U(1)_{Y}$
gauge couplings. It is either positive or negative whether the negative
contribution by the large top Yukawa coupling is compensated by the gauge
couplings and the Higgs quartic coupling. The corresponding quartic coupling
to the 126 GeV Higgs boson does not suffice to compensate it and the beta
function is negative. So the running quartic coupling crosses zero somewhere
at the UV energy scale. It is very suggestive that the observed value of the
Higgs mass is close to the stability bound up to the Planck scale Espinosa
$\displaystyle
M_{h}[\mbox{GeV}]>129.2+1.8\left(\frac{M_{t}[\mbox{GeV}]-173.2}{0.9}\right)-0.5\left(\frac{\alpha_{s}(M_{Z})-0.1184}{0.0007}\right)\pm
1.0_{th}.$ (2)
When the Higgs mass is lighter than the above bound, new physcis must appear
below the Planck scale. But if it lies just on the border of the stability
bound, it gives a big hint to the origin of the Higgs potential at the Planck
scaleFP1 .
Another important information is that the LHC results are almost consistent
with the SM. Furhtermore the precision experiments of the flavor physics,
Babar, Belle and LHCb, gave stringent constraints on the physics beyond the
SM. Of course, in spite of the above rather unexpectedly good agreement with
the SM, there exist phenomena which cannot be explained within the SM.
Nuetrino oscillation requires the dimension 5 operator $l\phi l\phi$ and a new
scale beyond the SM must be introduced. The baryon asymmetry of the universe
also requires an additional source of the CP violation. The SM anyway needs to
be extended to explain these phenomana.
The most common appoach to go beyond the SM is based on a unification of the
gauge couplings below the Planck scale, i.e. the GUT scale. Then we need a
natural explanation why the electroweak scale is much smaller than the GUT
scale. In order to solve the hierarchical structure of the scales, the
supersymmetry is introduced. Here I call the sequence of ideas from GUT to the
hierarchy problem and the low energy supersymmetry the central dogma of
particle physics. In addition to solving the hierarchy problem, it can improve
the gauge coupling unification as well as providing candidates of the dark
matter particles. But as the bonus we get or as the price we pay, it predicts
many new particles at the TeV scale and the recent experiments have given
strong constraints on the models with low energy supersymmetry.
In such circumstances, it may be a good time to reconsider the central dogma
of particle physics. In this note, we take an approach to the hierarchy
problem suggested by Bardeen. In the next section, we interpret the Bardeen’s
argument in terms of the renormalization group. If we adopt the argument, the
most natural mechanism to break the electreweak symmetry is the Coleman-
Weinberg (CW) mechanism. But we know that the CW mechanism does not work
within the SM because of the large top Yukawa coupling, so we need to extend
the SM. In section 3, we introduce our model, a classically conformal $B-L$
extension of the SM and then discuss the dynamics of the model.
## II Bardeen’s argument of the hierarchy problem
We pay a special attention to the almost scale invariance of the SM. At the
classical level, the SM Lagrangian is conformal invariant except for the Higgs
mass term. Bardeen argued Bardeen that once the classical conformal
invariance and its minimal violation by quantum anomalies are imposed on the
SM, it may be free from the quadratic divergences.
Bardeen’s argument on the hierarchy problem may be interpreted as follows
AokiIso . We classify divergences of the scalar mass term in the SM into the
following 3 classes,
* •
quadratic divergences: $\Lambda^{2}$
* •
logarithmic divergences with a small coefficient: $m^{2}\log(\Lambda/\mu)$
* •
logarithmic divergences with a large coefficient: $M^{2}\log(\Lambda/\mu)$
The logarithmic divergences are operative both in the UV and the IR. In
particular, it controls a running of coupling constants and is observable. On
the other hand, the quadratic divergence can be always removed by a
subtraction. Once subtracted, it no longer appears in observable quantities.
In this sense, it gives a boundary condition of a quantity in the IR theory at
the UV energy scale where the IR theory is connected with a UV completion
theory. Indeed, the RGE of a Higgs mass term $m^{2}$ in the SM
$\displaystyle V(H)=-m^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}$ (3)
is approximately given by
$\displaystyle\frac{dm^{2}}{dt}=\frac{m^{2}}{16\pi^{2}}\left(12\lambda_{H}+6Y_{t}^{2}-\frac{9}{2}g^{2}-\frac{3}{2}g_{Y}^{2}\right).$
(4)
The quadratic divergence is subtracted by a boundary condition either at the
IR or UV scale. Once the initial condition of the RGE is given at the UV
scale, it is no longer operative in the IR. The RGE shows that the mass term
$m^{2}$ is multiplicatively renormalized. If it is zero at a UV scale
$M_{UV}$, it continues to be zero at lower energy scales. In this sense, the
quadratic divergence is not the issue in the IR effective theory, but the
issue in the UV completion theory. Hence if the SM (and its extension at the
TeV scale) is directly connected with a UV completion theory at the Planck
scale physics, the hierarchy problem turns out to be a problem of the boundary
condition at the UV scale. If the UV completion theory is an ordinary field
theory, it will be difficult to protect the masslessness of the scalar
particle against radiative corrections by massive particles of the UV scale
unless we introduce, e.g. the low-energy supersymmetry. But in the string
theory, symmetry is sometimes enhanced on a moduli space and massless scalars
can survive even without supersymmetry. Also discrete symmetry like T-duality,
which is invisible in the low energy effective theory, may prohibit a
generation of potential at the string scale.
The multiplicative renormalization of the Higgs mass term is violated by a
mixing with a massive field in the loop. If the massive field aquires its mass
in a different mechanism with the EWSB, the Higgs mass has a logarithmic
divergence
$\displaystyle\delta
m^{2}\sim\frac{\lambda^{2}M^{2}}{16\pi^{2}}\log(\Lambda^{2}/m^{2})$ (5)
which modifies the RGE as
$\displaystyle\frac{dm^{2}}{dt}=\frac{m^{2}}{16\pi^{2}}\left(12\lambda_{H}+6Y_{t}^{2}-\frac{9}{2}g^{2}-\frac{3}{2}g_{Y}^{2}\right)+\frac{M^{2}}{8\pi^{2}}\lambda^{2}.$
(6)
The last term corresponds to the logarithmic divergence with a large
coefficient. The coefficient $M^{2}$ has nothing to do with the mass of the
Higgs $m^{2}$, and it violates the multiplicativity of the Higgs mass. Thus
the hierarchy problem, namely the stability of the EWSB scale, is caused by
such a mixing of relevant operators (mass terms) with hierarchical energy
scales $m\ll M$. In the Bardeen’s argument, he also imposes an absence of
intermediate scales above the EW scale. The logarithmic divergence with a
large coefficient (5) is sometimes confused with the quadratic divergence, but
if the UV completion theory is something like a string theory, they should be
distinguished.
From the above considerations, the hierarchy problem can be solved by imposing
the following two different conditions;
* •
Correct boundary condition at the UV (Planck) scale $M_{pl}$
* •
Absence of mixings in intermediate scales below $M_{pl}$
The first condition subtracts the quadratic divergence at the Planck scale. It
must be solved in the UV completion theory such as the string theory. The most
natural boundary condition is that scalar fields which appear in the low
energy physics are massless at the Planck scale. On the other hand, the second
condition assures the absence of logarithmic divergences with large
coefficients. Even if the scalars are massless at the Planck scale, they
receive large radiative corrections from the mixing with other relevant
operators. Without a cancellation mechanism like the supersymmetry, we need to
impose an absence of intermediate scales between EW (or TeV) and Planck
scales. Hence all symmetries are broken either at the Planck scale or near the
EW scale. Especially, the breakings of the supersymmetry or the grand
unification of gauge coupling should occur at the Planck scale. This second
condition is also emphasized in the Bardeen’s argument Bardeen . In such a
scenario, Planck scale physics is directly connected with the electroweak
physics shapo .
Hence a natural boundary condition of the mass term at the UV cut-off scale,
e.g. $M_{Pl}$, is
$\displaystyle m^{2}(M_{Pl})=0.$ (7)
This is the condition of the classical conformality of the BSM. The condition
(7) must be justified in the UV completion theory, and from the low energy
effective theory point of view, it is just imposed as a boundary condition
222The condition (7) may look similar to the Veltman condtion Veltman , but
they are conceptually different at all. In the Veltman condition, the
quadratic divergence is considered to be cancelled between various
contributions of bosons and fermions. Such a cancellation occurs in a very
special situation of the IR physics. On the contrary, the condition (7) is
independent of the matter content in the IR, and robust against a change of
scales. .
## III Flat potential at the Planck scale
The condition (7) restricts the form of the Higgs potential as
$\displaystyle V(H)=\lambda_{H}(H^{\dagger}H)^{2}.$ (8)
Here $\lambda_{H}$ is the running coupling and the RG improved effecitve
potential is given by making the coupling $\lambda_{H}(H)$ field dependent.
The mass term is not generated even in the IR as discussed in the previous
section once the boundary condtion (7) is imposed at the boundary with the UV
completion theory. The mass of the Higgs at 126 GeV suggests that the running
coupling becomes asymptotically vanishing near the Planck scale. The current
bounds (2) is a bit heavier than the experimental data, but in this note, we
assume that the Higgs quartic coupling vanishes at the Planck scale. Hence
$\displaystyle V(H)=0\mbox{ at the Planck scale}.$ (9)
The condition may connect the SM in the IR with the string theory in the UV.
Now we have to solve two problems. The first is whether we can construct a
phenomenologically viable model starting from the condtion of the flatness of
the Higgs potential (9), and the second is to derive such a boundary condtion
from the UV completion theory such as a string theory. Supersymmetry or grand
unification, if exists, are broken at the Planck scale. In the following we
focus on the first problem by proposing a B-L extension of the SM with a flat
potential at the Planck scale. The second issue is left for future
investigations.
Since the IR theory is assumed to have the boundary condition (9), the
electroweak symmetry breaking should occur radiatively, namely the Coleman-
Weinberg mechanism. However, it is now well-known that the CW mechanism cannot
occur within the SM because of the large top-Yukawa coupling. Indeed, the CW
mechanism is realized only when the beta-function of the quartic scalar
coupling is positive and the running quartic coupling crosses zero somewhere
in the IR. But as we saw, the beta function of the quartic Higgs coupling is
positive in the SM and its behavior is opposite to the CW mechanism. Hence, in
order to realize the EWSB, we need an additional sector in which the symmetry
is broken radiatively by the CW mechanism and whose symmetry breaking triggers
the EWSB. In the next section, we introduce our model, namely a B-L extension
of the SM with a flat potential at the Planck scale.
## IV B-L extension of the SM with flat potential at Planck
The idea to utilize the CW mechanism to solve the hierarchy problem was first
modelled by Meissner and Nicolai MaNi (see also Dias ). In addition to the SM
particles, they introduced right-handed neutrinos and a SM singlet scalar
$\Phi$. Inspired by the work, we proposed a minimal phenomenologically viable
model IOO . It is the minimal B-L model B-L , but with a classical
conformality. The model is similar to the one proposed by Meissner and Nicolai
MaNi , but the difference is whether the B-L symmetry is gauged or not. In a
recent paper we further showed that by imposing the flatness (9) of the Higgs
potential at the Planck scale the B-L breaking scale is related with the EWSB
scale. The ratio of two scales is dynamically determined by the B-L gauge
coupling and the B-L breaking scale is naturally constrained to be around TeV
scale B-L2 for a not so small B-L gauge coupling.
Besides the SM particles the model consists of the B-L gauge field with the
gauge coupling $g_{B-L}$, right-handed nuetrinos $\nu_{R}^{i}$ ($i=1,2,3$
denotes the generation index) and a SM singlet complex scalar field $\Phi$
with two units of the B-L charge. The model is anomaly free. The Lagrangian
contains Majorana Yukawa coupling $\sim
Y_{N}^{i}\Phi\bar{\nu}_{R}^{ic}\nu_{R}^{i}$, and the see-saw mechanism gives
masses to the left-handed neutrinos once the scalar $\Phi$ acquires vev.
## V Symmetry breakings of B-L and EW
Since the B-L gauge symmetry is broken by the CW mechanism, the breaking scale
is correlated with the quartic coupling $\lambda_{\Phi}$ at the UV scale. Its
running is described by
$\frac{d\lambda_{\Phi}}{dt}=\frac{1}{16\pi^{2}}\left(20\lambda_{\Phi}^{2}-\frac{1}{2}Tr\left[Y_{N}^{4}\right]+96g_{B-L}^{4}+\cdots\right).$
(10)
If the Mayorana Yukawa coupling is not so large, the beta function is
positive. The typical behavior of the running $\lambda_{\phi}$ is drawn in
Fig. 1. It crosses zero at a lower energy scale $M_{0}$, then the B-L symmetry
is broken at $M_{B-L}\sim M_{0}\exp(-1/4)$ through the CW mechanismIsoOrikasa
.
Figure 1: RG evolution of the self-coupling $\lambda_{\phi}$ of a SM singlet
scalar $\phi$. Since the $\beta$ function is positive, the running coupling
crosses zero at a lower energy scale.
As shown in the paper IOO , the ratio of the scalar boson mass to the B-L
gauge boson mass is given by
$\displaystyle\left(\frac{m_{\Phi}}{m_{Z^{\prime}}}\right)^{2}\sim\frac{6}{\pi}\alpha_{B-L}\ll
1.$ (11)
The condition that the B-L gauge coupling does not diverge up to the Planck
scale requires $\alpha_{B-L}<0.015$ at $M_{B-L}$. Hence the scalar boson
becomes lighter than the B-L gauge boson, $m^{2}_{\Phi}<0.03\
m^{2}_{Z^{\prime}}$. Such a very light scalar boson is a general prediction of
the CW mechanism.
The EWSB is triggered by the B-L breaking. The flatness condition (9) of the
Higgs potential predicts an absence of the scalar mixing at the Planck scale.
Hence B-L and EW sectors are decoupled each other in the UV. But since the
matter fields are coupled to both $U(1)_{Y}$ and $U(1)_{B-L}$, these two
sectors become mixed through the $U(1)$-mixing. As a result, the scalar mixing
term $\lambda_{mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)$ appears in the IR. It
is interesting that a very small negative mixing $\lambda_{mix}$ is always
generated irrespective of the details of other parameters once we assume the
flatness condition $\lambda_{mix}(M_{pl})=0$. By solving the RGE IsoOrikasa
we showed that the scalar mixing term is dynamically generated around
$\lambda_{mix}\sim-4\times 10^{-4}$. If the $\Phi$ field acquires a VEV
$\langle\Phi\rangle=M_{B-L}$, the mixing term
$\lambda_{mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)$ gives an effective mass term
of the Higgs field. Since the coefficient $\lambda_{mix}$ is negative, the
EWSB is triggered and the Higgs VEV is given by
$\displaystyle v=\langle
H\rangle=\sqrt{\frac{|\lambda_{mix}|}{\lambda_{H}}}M_{B-L}$ (12)
This gives a ratio between the EWSB scale to the B-L symmetry breaking scale.
The scalar mixing is determined in terms of the gauge couplings, so the ratio
of two breaking scales is also determined dynacamilly in terms of the gauge
coupling $g_{B-L}$.
## VI Model predictions
The dyanamics of the model is controlled by two parameters, $g_{B-L}$ and
$\lambda_{\Phi}$, which determines the two breaking scales of B-L and EW.
Figure 2: Model prediction is drawn in the black line (from top left to down
right). The $B-L$ gauge coupling $\alpha_{B-L}$ and the gauge boson mass
$m_{Z^{\prime}}$ are related because of the flat potential assumption at the
Planck scale. The left side of the most left solid line in blue has been
already excluded by the LEP experiment. The left of the dashed line can be
explored in the 5-$\sigma$ significance at the LHC with $\sqrt{s}$=14 TeV and
an integrated luminosity 100 fb-1. The left of the most right solid line (in
red) can be explored at the ILC with $\sqrt{s}$=1 TeV, assuming 1% accuracy.
The experimental input $v=246$ GeV gives a relation between these two and the
dynamics of the model is essentially described by a single parameter. The
figure 2 shows the prediction of our model. The vertical axis is the strength
of $\alpha_{B-L}$ and the horizontal axis is the mass of the B-L gauge boson.
The black line (from top left to down right) shows the prediction of our
model. If an extra $U(1)$ gauge boson and a SM singlet scalar are found in the
future, the prediction of our model is the mass relation (11), e.g.,
$m_{\phi}\sim 0.1\ m_{Z^{\prime}}$ for $\alpha_{B-L}\sim 0.005$. The CW
mechanism in the B-L sector predicts a much lighter SM singlet Higgs boson
than the extra $U(1)$ gauge boson. It is different from the ordinary TeV scale
B-L model where the symmetry is broken by a negative squared mass term.
Nuetrino oscillation is realized by the type I see-saw mechanism with small
neutrino Yukawa couplings. Baryon number asymmetry of the universe may be
generated through the TeV scale leptogenesis with almost degenerate Majorana
massesIOO3 . Furhter phenomenological issues such as $U(1)$ mixing or the
lepton number violation at the TeV scale are discussed in a separate paper.
## References
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* (5) W. A. Bardeen, FERMILAB-CONF-95-391-T
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* (10) R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980); R. E. Marshak and R. N. Mohapatra, Phys. Lett. B 91, 222 (1980); C. Wetterich, Nucl. Phys. B 187, 343 (1981); A. Masiero, J. F. Nieves and T. Yanagida, Phys. Lett. B 116, 11 (1982); R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 27, 254 (1983); W. Buchmuller, C. Greub and P. Minkowski, Phys. Lett. B 267, 395 (1991).
* (11) S. Khalil, J. Phys. G 35, 055001 (2008) M. Abbas and S. Khalil, JHEP 056, 804 (2008) S. Khalil and A. Masiero, Phys. Lett. B 665, 374 (2008) L. Basso, A. Belyaev, S. Moretti and C. H. Shepherd-Themistocleous, Phys. Rev. D 80, 055030 (2009) L. Basso, A. Belyaev, S. Moretti, G. M. Pruna and C. H. Shepherd-Themistocleous, Eur. Phys. J. C 71, 1613 (2011)
* (12) S. Iso, N. Okada and Y. Orikasa, Phys. Rev. D 83, 093011 (2011) N. Okada, Y. Orikasa and T. Yamada, Phys. Rev. D 86, 076003 (2012)
* (13) M. J. G. Veltman, Acta Phys. Polon. B 12, 437 (1981). See also a recent attempt, Y. Hamada, H. Kawai, K. -y. Oda Phys. Rev. D 87 (2013) 053009
|
arxiv-papers
| 2013-04-01T04:09:43 |
2024-09-04T02:49:43.695317
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Satoshi Iso",
"submitter": "Satoshi Iso",
"url": "https://arxiv.org/abs/1304.0293"
}
|
1304.0338
|
11institutetext: University of Bucharest 11email: [email protected]
# Existence of equilibrium for multiobjective games in abstract convex spaces
Monica Patriche University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, 010014 Bucharest, Romania
###### Abstract
In this paper we use the minimax inequalities obtained by S. Park (2011) to
prove the existence of weighted Nash equilibria and Pareto Nash equilibria of
a multiobjective game defined on abstract convex spaces.
###### Keywords:
minimax inequality, weighted Nash equilibria, Pareto Nash equilibria,
multiobjective game, abstract convex space.
2000 Mathematics Subject Classification: 47H10, 55M20, 91B50.
## 1 INTRODUCTION
Recently, in [8], S. Park introduced a new concept of abstract convex space
and several classes of correspondences having the KKM property. With this new
concept, the KKM type correspondences were used to obtain coincidence
theorems, fixed point theorems and minimax inequalities. S. Park generalizes
and unifies most of important results in the KKM theory on G-convex spaces,
H-spaces, and convex spaces (for example, see [8]-[13]).
For the history of KKM literature, we must remind Ky Fan [3], who extended the
original KKM theorem to arbitrarily topological vector space. The property of
close-valuedness of related KKM correspondences was replaced with more general
concepts. In [7], Luc and al. have introduced the concept of intersectionally
closed-valued correspondences and in [13], S. Park has obtained new KKM type
theorems for this kind of KKM correspondences.
In this paper we use the minimax inequalities obtained by S. Park in [13] to
prove the existence of weighted Nash equilibria and Pareto Nash Equilibria of
a multiobjective game defined on abstract convex spaces. For the history of
minimax theorems, I also must remind the name of Ky Fan (see [4]). Among the
authors who studied the existence of Pareto equilibria in game theory with
vector payoffs, I emphasize S. Chebbi [2], W. K. Kim [5], W. K. Kim, X. P.
Ding [6], H. Yu [16], J. Yu, G. X.-Z Yuan [17], X. Z. Yuan, E. Tarafdar [18].
A reference work is the paper of M. Zeleny [19]. The approaches of above-
mentioned authors deal with the Ky Fan minimax inequality, quasi-equilibrium
theorems or quasi-variational inequalities. We must mention the papers of P.
Borm, F. Megen, S. Tijs [1], who introduced the concept of perfectness for
multicriteria games and M. Voorneveld, S. Grahn, M. Dufwenberg [14], who
studied the existence of ideal equilibria. Ather authors, as H. Yu (see [16]),
obtained the existence of a solution of multiobjective games by using new
concepts of continuity and convexity.
The paper is organised as follows: In section 2, some notation, terminological
convention, basic definitions and results about abstract convex spaces and
minimax inequalities are given. Section 3 introduces the model, that is, a
multiobjective game defined on an abstract convex space and the concept of
weight Nash equilibrium. Section 4 contains existence results for weight Nash
equilibrium and Pareto Nash equilibrium.
## 2 ABSTRACT CONVEX SPACES AND MINIMAX INEQUALITIES
Let $A$ be a subset of a topological space $X$. $2^{A}$ denotes the family of
all subsets of $A$. $\overline{A}$ denotes the closure of $A$ in $X$ and
int$A$ denotes the interiorof $A$. If $A$ is a subset of a vector space, co$A$
denotes the convex hull of $A$. If $F$, $G:$ $X\rightarrow 2^{Y}$ are
correspondences, then co$G$, cl $G$, $G\cap F$ $:$ $X\rightarrow 2^{Y}$ are
correspondences defined by $($co$G)(x)=$co$G(x)$, $($cl$G)(x)=$cl$G(x)$ and
$(G\cap F)(x)=G(x)\cap F(x)$ for each $x\in X$, respectively. The graph of
$F:X\rightarrow 2^{Y}$ is the set Gr$(F)=\\{(x,y)\in X\times Y\mid y\in
F(x)\\}$ and $F^{-}:Y\rightarrow 2^{X}$ is defined by $F^{-}(y)=\\{x\in X:y\in
F(x)\\}$ for $y\in Y.$ Let $\tciFourier(A)$ be the set of all nonempty finite
subsets of a set $A.\vskip 6.0pt plus 2.0pt minus 2.0pt$
For the reader’s convenience, we review a few basic definitions and results
from abstract convex spaces.
Definition 1 [13]. Let $X$ be a topological space, $D$ be a nonempty set and
let $\Gamma:\tciFourier(D)\rightarrow 2^{X}$ be a correspondence with nonempty
values $\Gamma_{A}=\Gamma(A)$ for $A\in\tciFourier(D).$ The family
$(X,D;\Gamma)$ is called an abstract convex space.
Definition 2 [13]. For a nonempty subset $D^{\prime}$ of $D$, we define the
$\Gamma$-convex hull of $D^{\prime}$, denoted by co${}_{\Gamma}D^{\prime}$, as
co${}_{\Gamma}D^{\prime}$=$\cup\\{\Gamma_{A}:A\in\tciFourier(D^{\prime})\\}\subset
X.$
Definition 3 [13]. Given an abstract convex space ($X,D,\Gamma$), a nonempty
subset $Y$ of $X$ is called to be a $\Gamma$-convex subset of ($X,D,\Gamma$)
relative to $D^{\prime}$ if for any $A\in\tciFourier$($D^{\prime}$), we have
$\Gamma_{A}\subset Y$, that is, co${}_{\Gamma}D^{\prime}\subset Y.$
Definition 4 [13]. When $D\subset X$ in ($X,D,\Gamma$), a subset $Y$ of $X$ is
said to be $\Gamma$-convex if co${}_{\Gamma}(Y\cap D)\subset Y;$ in other
words, $Y$ is $\Gamma$-convex relative to $D^{\prime}=Y\cap D.$ In case $X=D,$
let $(X,\Gamma)=(X,X,\Gamma).$
Definition 5 [13]. The abstract convex space ($X,D,\Gamma$) is called compact
if $X$ is compact.
We have abstract convex subspaces as the following simple observation.
###### Proposition 1
For an abstract convex space ($X,D,\Gamma$) and a nonempty subset $D^{\prime}$
of $D$, let $Y$ be a $\Gamma$-convex subset of $X$ relative to $D^{\prime}$
and $\Gamma^{\prime}:\tciFourier$($D^{\prime}$)$\rightarrow 2^{Y}$ a
correspondence defined by
$\Gamma_{A}^{\prime}=\Gamma_{A}\subset X$ for $A\in\tciFourier$($D^{\prime}$).
Then ($Y,D^{\prime},\Gamma^{\prime}$) itself is an abstract convex space
called a subspace relative to $D^{\prime}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
The following result is known.
###### Lemma 1 (12)
Let ($X_{i},D_{i},\Gamma_{i}$)i∈I be any family of abstract convex spaces. Let
$X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i}$ be equipped with the
product topology and $D=\mathop{\textstyle\prod}\nolimits_{i\in I}D_{i}$. For
each $i\in I$, let $\pi_{i}:D\rightarrow D_{i}$ be the projection. For each
$A\in\tciFourier$($D$), define
$\Gamma(A)=\mathop{\textstyle\prod}\nolimits_{i\in I}\Gamma_{i}(\pi_{i}(A))$.
Then ($X,D,\Gamma$) is an abstract convex space.
Definition 6 [13]. Let ($X,D,\Gamma$) be an abstract convex space. Then
$F:D\rightarrow 2^{X}$ is called a KKM correspondence if it satisfies
$\Gamma_{A}\subset F(A):=\cup_{y\in A}F(y)$ for all $A\in\tciFourier$($D$).
Definition 7 [13]. The partial KKM principle for an abstract convex space
($X,D,\Gamma$) is the statement that, for any closed-valued KKM correspondence
$F:D\rightarrow 2^{X}$, the family $\\{F(z)\\}_{z\in D}$ has the finite
intersection property. The KKM principle is the statement that the same
property also holds for any open-valued KKM correspondence.
An abstract convex space is called a KKM space if it satisfies the KKM
principle.
###### Proposition 2
Let ($X,D,\Gamma$) be an abstract convex space and
($X,D^{\prime},\Gamma^{\prime}$) a subspace. If ($X,D,\Gamma$) satisfies the
partial KKM principle, then so does ($X$, $D^{\prime},\Gamma^{\prime}$).
Let ($X,D,\Gamma$) be an abstract convex space.
Definition 8 [13]. The function $f:X\rightarrow\overline{\mathbb{R}}$ is said
to be quasiconcave (resp. quasiconvex) if $\\{x\in X:f(x)>r\\}$ (resp.,
$\\{x\in X:f(x)<r\\}$ is $\Gamma$-convex for each
$r\in\overline{\mathbb{R}}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
In [7], Luc and al. have introduced the concept of intersectionally closed-
valued correspondences.
Definition 9. Let $F:D\rightarrow 2^{X}$ be a correspondence.
(i) [7] $F$ is intersectionally closed-valued if $\cap_{z\in
D}\overline{F(z)}=\overline{\cap_{z\in D}F(z)};$
(ii) $F$ is transfer closed-valued if $\cap_{z\in D}\overline{F(z)}=\cap_{z\in
D}F(z);$
(iii) [7] $F$ is unionly open-valued if Int$\cup_{z\in D}F(z)=\cup_{z\in
D}$Int$F(z);$
(iv) $F$ is transfer open-valued if $\cup_{z\in D}F(z)=\cup_{z\in
D}$Int$F(z);$
Luc at al. [7] noted that (ii)$\Rightarrow$(i$).$
###### Proposition 3 (7)
The correspondence $F$ is intersectionally closed-valued (resp. transfer
closed-valued) if only if its complement $F^{C}$ is unionly open-valued (resp.
transfer open-valued).
Definition 10 [13]. Let$Y$ be a subset of $X.$
(i) $Y$ is said to be intersectionally closed (resp. transfer closed) if there
is an intersectionally (resp., transfer) closed-valued correspondence
$F:D\rightarrow 2^{X}$ such that $Y=F(z)$ for some $z\in D.$
(ii) $Y$ is said to be unionly open (resp. transfer open) if there is an
unionly (resp., transfer) open-valued correspondence $F:D\rightarrow 2^{X}$
such that $Y=F(z)$ for some $z\in D.\vskip 6.0pt plus 2.0pt minus 2.0pt$
S. Park gives in [13] the concept of generally lower (resp. upper)
semicontinuous function.
Definition 11 [13]. The function $f:D\times X\rightarrow\overline{\mathbb{R}}$
is said to be generally lower (resp. upper) semicontinuous (g.l.s.c.) (resp.
g.u.s.c.) on $X$ whenever, for each $z\in D,$ $\\{y\in X:f(z,y)\leq r\\}$
(resp., $\\{y\in X:f(z,y)\geq r\\})$ is intersectionally closed for each
$r\in\overline{\mathbb{R}}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
The aim of this paper is to prove the existence of a weighted Nash equilibrium
for a multicriteria game defined in the framework of abstract convex spaces.
For our purpose, we need the following theorem (variant of Theorem 6.3 in
[13]).
###### Theorem 2.1
(Minimax inequality, [13]). Let ($X,D=X,\Gamma$) an abstract convex space
satisfying the partial KKM principle, $f,g:X\times
X\rightarrow\overline{\mathbb{R}}$ extended real-valued functions and
$\gamma\in\overline{\mathbb{R}}$ such that
(i) for each $x\in X,$ $g(x,x)\leq\gamma;$
(ii) for each $y\in X,$ $F(y)=\\{x\in X:f(x,y)\leq\gamma\\}$ is
intersectionally closed (respectiv, transfer closed);
(iii) for each $x\in X,$ co${}_{\Gamma}\\{y\in
X:f(x,y)>\gamma\\}\subset\\{y\in X:g(x,y)>\gamma\\};$
(iv) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:
there exists a nonempty compact subset $K$ of $X$ such that either
(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or
(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
X^{\prime}}\overline{F(y)}\neq\phi.$
Then
1) there exists a $x_{0}\in X$ $($resp., $x_{0}\in K)$ such that
$f(x_{0},y)\leq\gamma$ for all $y\in X;$
2) if $\gamma:=\sup_{x\in X}g(x,x)$, then we have
inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$
For the case when $X=D$ (we are concerned with compact abstract spaces
($X,\Gamma$) satisfying the partial KKM principle), we have the following
variants of the corollaries stated in [13].
###### Corollary 1 (13)
Let $f,$ $g:X\times X\rightarrow\mathbb{R}$ be real-valued functions and
$\gamma\in\mathbb{R}$ such that
(i) for each $x,y\in X,$ $f(x,y)\leq g(x,y)$ and $g(x,x)\leq\gamma;$
(ii) for each $y\in X,$ $\\{x\in X:f(x,y)>\gamma\\}$ is unionly open in $X$;
(iii) for each $x\in X,$ $\\{y\in X:g(x,y)>\gamma\\}$ is $\Gamma$-convex on
$X;$
Then
1) there exists a $x_{0}\in X$ such that $f(x_{0},y)\leq\gamma$ for all $y\in
X;$
2) if $\gamma:=\sup_{x\in X}g(x,x)$, then we have
inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$
###### Corollary 2 (13)
Let $f,$ $g:X\times X\rightarrow\mathbb{R}$ be functions such that
(i) for each $x,y\in X,$ $f(x,y)\leq g(x,y)$ and $g(x,x)\leq\gamma;$
(ii) for each $y\in X,$ $f(\cdot,y)$ is g.l.s.c on $X$;
(iii) for each $x\in X,$ $f(x,\cdot)$ is quasiconcave on $X;$
Then we have
inf${}_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}g(x,x).\vskip 6.0pt plus
2.0pt minus 2.0pt$
## 3 MULTIOBJECTIVE GAMES
Now we consider the multicriteria game (or multiobjective game) in its
strategic form. Let $I$ be a finite set of players and for each $i\in I,$ let
$X_{i}$ be the set of strategies such that
$X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i}$ and ($X_{i}$,
$D_{i},\Gamma_{i}$ for each $i\in I$) is an abstract convex space with
$D_{i}\subset X_{i}$. Let $T^{i}:X\rightarrow 2^{\mathbb{R}^{k_{i}}}$, where
$k_{i}\in\mathbb{N}$, which is called the payoff function (or called
multicriteria). From Lemma 1, we also have that $(X,D,\Gamma)$ is an abstract
convex space, where $X=\mathop{\textstyle\prod}\nolimits_{i\in I}X_{i},$
$D=\mathop{\textstyle\prod}\nolimits_{i\in I}D_{i}$ and
$\Gamma(A)=\mathop{\textstyle\prod}\nolimits_{i\in I}\Gamma_{i}(\pi_{i}(A))$
for each $A\in\tciFourier$($D$).
Definition 12. The family $G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ is
called multicriteria game.
If an action $x:=(x_{1},x_{2},...,x_{n})$ is played, each player $i$ is trying
to find his/her payoff function
$T^{i}(x):=(T_{1}^{i}(x),...,T_{k_{i}}^{i}(x)),$ which consists of
noncommensurable outcomes. We assume that each player is trying to minimize
his/her own payoff according with his/her preferences.
In order to introduce the equilibrium concepts of a multicriteria game, we
need several necessary notation.
Notation. We shall denote by
$\mathbb{R}_{+}^{m}:=\\{u=(u_{1},u_{2},...u_{m})\in\mathbb{R}^{m}:u_{j}\geq 0$
$\forall j=1,2,...,m\\}$ and
int$\mathbb{R}_{+}^{m}:=\\{u=(u_{1},u_{2},...u_{m})\in\mathbb{R}^{m}:u_{j}>0$
$\forall j=1,2,...,m\\}$
the non-negative othant of $\mathbb{R}^{m}$ and respective the non-empty
interior of $\mathbb{R}_{+}^{m}$ with the topology induced in terms of
convergence of vector with respect to the Euclidian metric.
Notation. For each $i\in I,$ denote
$X_{-i}:=\mathop{\textstyle\prod}\nolimits_{j\in I\setminus\\{i\\}}X_{j}.$ If
$x=(x_{1},x_{2},...,x_{n})\in X,$ we denote
$x_{-i}=(x_{1},...,x_{i-1},x_{i+1},...,x_{n})\in X_{-i}.$ If $x_{i}\in X_{i}$
and $x_{-i}\in X_{-i}$, we shall use the notation
$(x_{-i},x_{i})=(x_{1},...,x_{i-1},x_{i},x_{i+1},...,x_{n})=x\in X.$
Notation. For each $u,v\in\mathbb{R}^{m}$, $u\cdot v$ denote the standard
Euclidian inner product.
Let $\widehat{x}=(\widehat{x}_{1},\widehat{x}_{2},...,\widehat{x}_{n})\in X.$
Now we have the following definitions.
Definition 13. A strategy $\widehat{x}_{i}\in X_{i}$ of player $i$ is said to
be a Pareto efficient strategy (resp., a weak Pareto efficient strategy) with
respect to $\widehat{x}\in X$ of the multiobjective game
$G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if there is no strategy
$x_{i}\in X_{i}$ such that
$T^{i}(\widehat{x})-T^{i}(\widehat{x}_{-i},x_{i})\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$
(resp.,
$T^{i}(\widehat{x})-T^{i}(\widehat{x}_{-i},x_{i})\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}).\vskip
6.0pt plus 2.0pt minus 2.0pt$
###### Remark 1
Each Pareto equilibrium is a weak Pareto equilibrium, but the converse is not
always true.
Definition 14. A strategy $\widehat{x}\in X$ is said to be a Pareto
equilibrium (resp., a weak Pareto equilibrium) of the multiobjective game
$G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if for each player $i\in I$,
$\widehat{x}_{i}\in X_{i}$ is a Pareto efficient strategy (resp., a weak
Pareto efficient strategy) with respect to $\widehat{x}.\vskip 6.0pt plus
2.0pt minus 2.0pt$
Definition 15. A strategy $\widehat{x}\in X$ is said to be a weighted Nash
equilibrium with respect to the weighted vector $W=(W_{i})_{i\in I}$ with
$W_{i}=(W_{i,1},W_{i,2},...,W_{i,k_{i}})\in\mathbb{R}_{+}^{k_{i}}$ of the
multiobjective game $G=((X_{i},D_{i},\Gamma_{i}),T^{i})_{i\in I}$ if for each
player $i\in I$, we have
(i) $W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\};$
(ii) $W_{i}\cdot T^{i}(\widehat{x})\leq W_{i}\cdot
T^{i}(\widehat{x}_{-i},x_{i}),$ $\forall x_{i}\in X_{i},$where $\cdot$ denotes
the inner product in $\mathbb{R}^{k_{i}}.$
###### Remark 2
In particular, if $W_{i}\in\mathbb{R}_{+}^{k_{i}}$ with
$\mathop{\textstyle\sum}\nolimits_{j=1}^{k_{i}}W_{i,j}=1$ for each $i\in I,$
then the strategy $\widehat{x}\in X$ is said to be a normalized weighted Nash
equilibrium with respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$
## 4 EXISTENCE OF WEIGHTED NASH EQUILIBRIUM AND PARETO NASH EQUILIBRIUM
Now, as an application of Theorem 1, we have the following existence theorem
of weighted Nash equilibria for multiobjective games.
###### Theorem 4.1
Let $I$ be a finite set of indices, let ($X_{i},D_{i}=X_{i},\Gamma_{i}$)i∈I be
any finite family of abstract convex spaces such that the product space
($X,\Gamma$) satisfies the partial KKM principle. If there is a weighted
vector $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:
(i) for each $y\in X,$ $F(y)=\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
0\\}$ is intersectionally closed (respectiv, transfer closed);
(ii) there exists $g:X\times X\rightarrow\overline{\mathbb{R}}$ extended real-
valued function such that for each $x\in X,$ $g(x,x)\leq 0$ and for each $x\in
X,$ co${}_{\Gamma}\\{y\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}\subset\\{y\in
X:g(x,y)>0\\};$
(iii) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:
there exists a nonempty compact subset $K$ of $X$ such that either
(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or
(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
x^{\prime}}\overline{F(y)}\neq\phi;$
then there exists $\widehat{x}\in K$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Proof. Define the function $f:X\times X\rightarrow\mathbb{R}$ by
$f(x,y)=\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i})),$
$(x,y)\in X\times X.$ By Theorem 1, we have that inf${}_{x\in X}\sup_{y\in
X}f(x,y)\leq\sup_{x\in X}g(x,x)=0.$ It follows that there exists an
$\widehat{x}\in K$ such that $f(\widehat{x},y)\leq 0$ for any $y\in X.$ That
is
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i})\leq
0$ for any $y\in X.$ For any given $i\in I$ and any given $y_{i}\in X_{i},$
let $y=(\widehat{x}_{-i},y_{i}).$ Then we have
$W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))=$
$=\mathop{\textstyle\sum}\nolimits_{j=1}^{n}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))-\mathop{\textstyle\sum}\nolimits_{j\neq
i}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))$
$=\mathop{\textstyle\sum}\nolimits_{j=1}^{n}W_{j}\cdot(T^{j}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))\leq
0.$
Therefore, we have
$W_{i}\cdot(T^{i}(\widehat{x}_{-i},\widehat{x}_{i})-T^{i}(\widehat{x}_{-i},y_{i}))$
$\leq 0$ for each $i\in I$ and $y_{i}\in X_{i},$ that is $\widehat{x}\in K$ is
a weighted Nash equilibrium of the game $G$ with respect to $W.\vskip 6.0pt
plus 2.0pt minus 2.0pt$
We obtain the following corollaries for the compact games when $X=D$.
###### Corollary 3
Let $I$ be a finite set of indices, let ($X_{i},\Gamma_{i}$)i∈I be any finite
family of abstract convex spaces such that the product space ($X,\Gamma$)
satisfies the partial KKM principle. If there is a weighted vector
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:
(i) there exists $g:X\times X\rightarrow R$ such that for each $x,y\in X,$
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
g(x,y)$ and $g(x,x)\leq 0;$
(ii) for each $y\in X,$ $\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}$
is unionly open in $X$;
(iii) for each $x\in X,$ $\\{y\in X:g(x,y)>0\\}$ is $\Gamma$-convex on $X;$
then there exists $\widehat{x}\in X$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$
###### Corollary 4
Let $I$ be a finite set of indices, let ($X_{i},\Gamma_{i}$)i∈I be any finite
family of abstract convex spaces such that the product space ($X,\Gamma$)
satisfies the partial KKM principle. If there is a weighted vector
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:
(i) there exists $g:X\times X\rightarrow R$ such that for each $x,y\in X,$
$\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
g(x,y);$
(ii) for each fixed $y\in X,$ the function
$x\rightarrow\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))$
is g.l.s.c on $X$;
(iii) for each fixed $x\in X,$ the function
$y\rightarrow\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))$
is quasiconcave on $X$;
then there exists $\widehat{x}\in X$ such that $\widehat{x}$ is a weighted
Nash equilibria of the game $G=((X_{i},\Gamma_{i}),T^{i})_{i\in I}$ with
respect to $W.\vskip 6.0pt plus 2.0pt minus 2.0pt$
In order to prove an existence theorem of Pareto equilibria for multiobjective
games, we need the following lemma.
###### Lemma 2 (15)
Each normalized weighted Nash equilibrium $\widehat{x}\in X$ with a weight
$W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ (resp.,
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\})$ and
$\mathop{\textstyle\sum}\nolimits_{j=1}^{k_{i}}W_{i,j}=1$ for each $i\in I,$
for a multiobjective game $G=(X_{i},T^{i})_{i\in I}$ is a weak Pareto
equilibrium (resp. a Pareto equilibrium) of the game $G.$
###### Remark 3
The conclusion of Lemma2 still holds if $\widehat{x}\in X$ is a weighted Nash
equilibrium with a weight $W=(W_{1},W_{2},...,W_{n})$,
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I$(resp.,
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I)$ of the
game $G.$
###### Remark 4
A Pareto equilibrium of $G$ is not necessarily a weighted Nash equilibrium of
the game $G.\vskip 6.0pt plus 2.0pt minus 2.0pt$
###### Theorem 4.2
Let $I$ be a finite set of indices, let ($X_{i},D_{i}=X_{i},\Gamma_{i}$)i∈I be
any finite family of abstract convex spaces such that the product space
($X,\Gamma$) satisfies the partial KKM principle. If there is a weighted
vector $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ such that the followings are
satisfied:
(i) for each $y\in X,$ $F(y)=\\{x\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))\leq
0\\}$ is intersectionally closed (respectiv, transfer closed);
(ii) there exists $g:X\times X\rightarrow\overline{\mathbb{R}}$ extended real-
valued function such that for each $x\in X,$ $g(x,x)\leq 0$ and for each $x\in
X,$ co${}_{\Gamma}\\{y\in
X:\mathop{\textstyle\sum}\nolimits_{i=1}^{n}W_{i}\cdot(T^{i}(x_{-i},x_{i})-T^{i}(x_{-i},y_{i}))>0\\}\subset\\{y\in
X:g(x,y)>0\\};$
(iii) the correspondence $F:X\rightarrow 2^{X}$ satisfies the following
condition:
there exists a nonempty compact subset $K$ of $X$ such that either
(a) $K\supset\cap\\{\overline{F(y)}:y\in M\\}$ for some
$M\in\tciFourier$($X$); or
(b) for each $N\in\tciFourier$($X$), there exists a compact $\Gamma$-convex
subset $L_{N}$ of $X$ relative to some $X^{\prime}\subset X$ such that
$N\subset X^{\prime}$ and $K\supset L_{N}\cap\cap_{y\in
x^{\prime}}\overline{F(y)}\neq\phi;$
then there exists $\widehat{x}\in K$ such that $\widehat{x}$ is a weak Pareto
equilibrium of the game $G=((X_{i},D_{i}=X_{i},\Gamma_{i}),T^{i})_{i\in I}.$
In addition, if $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in$int$R_{+}^{k_{i}}\backslash\\{0\\}$ for $i\in I,$ then $G$ has at
least a Pareto equilibrium point $\widehat{x}\in X.$
Proof. By Theorem 2, $G$ has at least weighted Nash equilibrium point
$\widehat{x}\in K$ with respect of the weighted vector $W.$ Lemma 2 and Remark
3 shows that $\widehat{x}$ is also a weak Pareto equilibrium point of $G,$ and
a Pareto equilibrium point of $G$ if $W=(W_{1},W_{2},...,W_{n})$ with
$W_{i}\in$int$\mathbb{R}_{+}^{k_{i}}\backslash\\{0\\}$ for each $i\in I.$
Acknowledgment: This work was supported by the strategic grant
POSDRU/89/1.5/S/58852, Project ”Postdoctoral programme for training scientific
researchers” cofinanced by the European Social Found within the Sectorial
Operational Program Human Resources Development 2007-2013.
The author thanks to Professor João Paulo Costa from the University of Coimbra
for the fruitfull discussions and for the hospitality he proved during the
visit to his departament.
## References
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* (3) K. Fan, A generalization of Tyhonoff’s fixed point theorem. Math. Ann. 142 (1961), 305-310.
* (4) K. Fan, A minimax inequality and applications in: O. Shisha (Ed.), Inequalities III, Academic Press, New York, 1972, pp. 103-113.
* (5) W. K. Kim, Weight Nash equilibria for generalized multiobjective games. J. Chungcheong Math. Soc. 13 (2000), 1, 13-20.
* (6) W. K. Kim, X. P. Ding, On generalized weight Nash equilibria for generalized multiobjective games. J. Korean Math. Soc. 40 (2003), 5, 883-899.
* (7) D. T. Luc, E. Sarabi and A. Soubeyran, Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364 (2010), 544-555.
* (8) S. Park, On generalizations of the KKM principle on abstract convex spaces. Nonlinear Anal. Forum 11 (2006), 1, 67–77.
* (9) S. Park, Elements of the KKM theory on abstract convex spaces. J. Korean Math. Soc. 45 (2008), 1, 1–27.
* (10) S. Park, Generalizations of the Nash EquilibriumTheorem in the KKM Theory. Fixed Point Theory Appl. doi:10.1155/2010/234706
* (11) S. Pak, The KKM principle in abstract convex spaces: equivalent formulations and applications. Nonlinear Anal. 73 (2010), 1028-1042.
* (12) S. Park, Generalizations of the Nash Equilibrium Theorem in the KKM Theory. Fixed Point Theory Appl., doi:10.1155/2010/234706.
* (13) S. Park, New generalizations of basic theorems in the KKM theory. Nonlinear Anal. 74 (2011), 3000-3010.
* (14) M. Voorneveld, S. Grahn, M. Dufwenberg, Ideal equilibria in noncooperative multicriteria games. Math. Meth. Oper. Res. 52 (2000), 65-77.
* (15) S. Y. Wang, Existence of a Pareto equilibrium, J. Optim. Theory Appl. 95 (1997), 373-384.
* (16) H. Yu, Weak Pareto Equilibria for Multiobjective Constrained games. Appl. Math. Let. 16 (2003), 773-776.
* (17) J. Yu, G. X.-Z Yuan, The study of Pareto Equilibria for Multiobjective games by fixed point and Ky Fan Minimax Inequality methods. Computers Math. Applic. 35, (1998), 9, 17-24.
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|
arxiv-papers
| 2013-04-01T11:42:57 |
2024-09-04T02:49:43.703590
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1304.0338"
}
|
1304.0339
|
# Minimax Theorems for Set-valued Maps Without Continuity Assumptions
Monica Patriche
Abstract. We introduce several classes of set-valued maps with generalized
convexity and we obtain minimax theorems for set-valued maps which satisfy the
introduced properties and which are not continuous. Our method consists of the
use
of a fixed-point theorem for weakly naturally quasi-concave set-valued maps
defined
on a simplex in a topological vector space or of a constant selection of
quasi-convex
set-valued maps.
Key Words. minimax theorems, fixed point theorem, weakly naturally quasi-
concave set-valued map, $S$-transfer $\mu$-convex set-valued map, transfer
properly $S-$quasi-convex, weakly $z-$convex set-valued map.
2010 Mathematics Subject Classification: 49J35, 90C47.
1\. Introduction
The classical Ky Fan inequalities [4], [5], [6] are an undeniably important
tool in the study of many important results concerning the variational
inequalities, game theory, mathematical economics, control theory and fixed-
point theory. e.g., see [1], [2], [7], [8], [10], [13]-[17], [19]-[21], [23],
[25]-[32] and the references therein. Within recent years, many
generalizations have been successfully obtained and here we must emphasize Ky
Fan’s study of minimax theorems for vector-valued mappings and for set-valued
maps. We refer the reader, for instance, to Li and Wang [15], Luo [19], Zhang
and Li [31], [32], Zhang, Cheng and Li [30]. In [20], Nessah and Tian search
the condition concerning the existence of solution of minimax inequalities for
real-valued mappings, without convexity and compactness assumptions. They
define the local dominatedness property and prove that it is necessary and
further, under some mild continuity condition, sufficient for the existence of
equilibrium in minimax inequalities. This type of characterization of the
solution for minimax theorems leads us to the question whether similar results
can be obtained, but, by keeping the convexity assumptions and by giving up
the continuity ones over the set-valued maps.
We are introduced into the extremely limited literature concerning the minimax
theorems for set-valued maps with the opportunity to see the things from a new
perspective and to propose coherent answers to the problem of the solution
existence. Our results could be particularly designed to identify new methods
of proof for this kind of problems and to assess whether the convexity
framework can be adapted to set-valued maps with two variables and whether
classes of weakened convexity can be implemented, particularly by relying on a
mechanism which takes into accont the behaviour of the maps in the points
where their values contain or not maximal (resp. minimal) elements of certain
sets of type $\mathop{\textstyle\bigcup}\limits_{y\in X}F(x,y)$ or
$\mathop{\textstyle\bigcup}\limits_{x\in X}F(x,y).$
In this paper, we study vector minimax inequalities for set-valued maps. We
give up the condition of continuity of the set-valued maps and, instead, we
work with some new classes of generalized convexity which we introduce:
$S$-transfer $\mu$-convexity, transfer properly $S-$quasi-convexity and weakly
$z-$convexity. In order to prove our results, we construct a constant
selection for a quasi-convex correspondence and we use the fixed point theorem
for weakly naturally quasi-concave set-valued maps defined on a simplex in a
topological vector space (see [22]).
The article is organized as follows. In Section 2, we introduce notations and
preliminary results. In Section 3, the convex-type properties for set-valued
maps are defined and some exemples are given as well. In Section 4, we obtain
two types of Ky Fan minimax inequalities for set-valued maps. We also provide
some examples to illustrate our results. Concluding remarks are presented in
Section 5.
2\. Preliminaries and Notation
We shall use the following notations and definitions:
Let $A$ be a subset of a topological space $X.$ $2^{A}$ denotes the family of
all subsets of $A$ and $\overline{A}$ denotes the closure of $A$ in $X$. If
$A$ is a subset of a vector space, co$A$ denotes the convex hull of $A$. If
$F$, $G:X\rightrightarrows Z$ are set-valued maps, then co $G$, $G\cap
F:X\rightrightarrows Z$ are set-valued maps defined by $($co $G)(x):=$co
$G(x)$ and $(G\cap F)(x):=G(x)\cap F(x)$ for each $x\in X$, respectively.
In this paper, we will consider $E$ and $Z$ to be real Hausdorff topological
vector spaces and we will assume that $S$ is a pointed closed convex cone in
$Z$ with its interior int$S\neq\emptyset.$
Definition 2.1 (see [11]). Let $A\subset Z$ be a non-empty subset.
(i) A point $z\in A$ is said to be a minimal point of $A$ iff
$A\cap(z-S)=\\{z\\},$ and Min$A$ denotes the set of all minimal points of $A.$
(ii) A point $z\in A$ is said to be a weakly minimal point of $A$ iff
$A\cap(z-$int$S)=\emptyset,$ and Min${}_{w}A$ denotes the set of all weakly
minimal points of $A.$
(iii) A point $z\in A$ is said to be a maximal point of $A$ iff
$A\cap(z+S)=\\{z\\},$ and Max$A$ denotes the set of all maximal points of $A.$
(iv) A point $z\in A$ is said to be a weakly maximal point of $A$ iff
$A\cap(z+$int$S)=\emptyset,$ and Max${}_{w}A$ denotes the set of all weakly
maximal points of $A.$
It is easy to check that Min$A\subset$Min${}_{w}A$ and
Max$A\subset$Max${}_{w}A.$
Lemma 2.1 (see [7])Let $A\subset Z$ be a non-empty compact subset. Then, (i)
Min$A\neq\emptyset;$ (ii) $A\subset$Min$A+S;$ (iii)
$A\subset$Min${}_{w}A+$int$S\cup\\{0_{F}\\};$ (iv)Max$A\neq\emptyset;$ (v)
$A\subset$Max$A-S;$ (vi) $A\subset$Max${}_{w}A-$int$S\cup\\{0_{F}\\}.$
Notation. If $X$ and $Y$ are sets and $F:X\times X\rightrightarrows Y$ is a
set-valued map, we will denote $F(x,X)=\mathop{\textstyle\bigcup}\limits_{y\in
X}F(x,y)$ and $F(X,y)=\mathop{\textstyle\bigcup}\limits_{x\in X}F(x,y).$
We present the following types of generalized convex mappings and set-valued
maps.
Definition 2.2 Let $X$ be a non-empty convex subset of a topological vector
space $E,$ $Z$ a real topological vector space and $S$ a pointed closed convex
cone in $Z$ with its interior int$S\neq\emptyset.$ Let $F:X\rightrightarrows
Z$ be a set-valued map with non-empty values.
(i) $F$ is said to be (in the sense of [12 , Definition 3.6]) type-(iii)
properly $S-$quasi-convex on $X$ (see [9]), iff for any $x_{1},x_{2}\in X$ and
$\lambda\in[0,1],$ either $F(x_{1})\subset F(\lambda
x_{1}+(1-\lambda)x_{2})+S$ or $F(x_{2})\subset F(\lambda
x_{1}+(1-\lambda)x_{2})+S.$
(ii) $F$ is said to be (in the sense of [12 , Definition 3.6]) type-(v)
properly $S-$quasi-convex on $X$ (see [9]), iff for any $x_{1},x_{2}\in X$ and
$\lambda\in[0,1],$ either $F(\lambda x_{1}+(1-\lambda)x_{2})\subset
F(x_{1})-S$ or $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-S.$
If $-F$ is a type-(iii) [resp. type-(v)] $S-$properly quasiconvex set-valued
map, then, $F$ is said be type-(iii) [resp. type-(v)] $S-$properly quasi-
concave, which is equivalent to type-(iii) [resp. type-(v)] $(-S)$-properly
quasi-convex set valued map.
(iii) $F:X\rightrightarrows Y$ is said to be (in the sense of [12 , Definition
3.6]) type-(iii) _naturally S-quasi-convex_ on $X$, iff for any
$x_{1}$,$x_{2}\in X$ and $\lambda\in[0,1],$ co$(F(x_{1})\cup F(x_{2}))\subset$
$F(\lambda x_{1}+(1-\lambda)x_{2})+S$.
iv) $F:X\rightrightarrows Y$ is said to be (in the sense of [12 , Definition
3.6]) type-(v) _naturally S-quasi-convex_ on $X$, iff for any
$x_{1}$,$x_{2}\in X$ and $\lambda\in[0,1],$ $F(\lambda
x_{1}+(1-\lambda)x_{2})\subset$co$(F(x_{1})\cup F(x_{2}))-S$.
$F$ is said to be type-(iii) [resp. type-(v)] naturally $S-$quasi-concave on
$X$, iff $-F$ is type-(iii) [resp. type-(v)] naturally $S-$quasi-convex on
$X.$
(v) $F:X\rightrightarrows Y$ is said to be _S-quasi-convex_ on $X$ (see [24]),
iff for any $x_{1}$,$x_{2}\in X$ and $\lambda\in[0,1],$
($F(x_{1})+S)\cap(F(x_{2})+S)\subset$ $F(\lambda x_{1}+(1-\lambda)x_{2})+S$.
(vi) $F$ is quasi-convex $X$ [24] iff, for each $n$ and for every
$x_{1},x_{2},...,x_{n}\in X,$
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$
$\mathop{\textstyle\bigcap}\limits_{i=1}^{n}F(x_{i})\subset
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}).$
$F$ is said to be quasi-concave on $X$, iff $-F$ is quasi-convex on $X.$
Definition 2.3 (see [26]) Let $X$ be a non-empty convex subset of a
topological vector space $E$, let $Y$ be a subset of a topological vector
space $Z$ and $S$ a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ A vector-valued mapping $f:X\rightarrow Y$ is said to be
natural __ $S-$quasi-convex on $X$ iff $f(\lambda
x_{1}+(1-\lambda)x_{2})\in$co$\\{f(x_{1}),f(x_{2})\\}-S$ for every
$x_{1},x_{2}\in X$ and $\lambda\in[0,1].$ This condition is equivalent with
the following condition: there exists $\mu\in[0,1]$ such that $f(\lambda
x_{1}+(1-\lambda)x_{2})\leq_{S}\mu f(x_{1})+(1-\mu)f(x_{2}),$ where
$x\leq_{S}y$ $\Leftrightarrow$ $y-x\in S.$
A vector-valued mapping f is said to be _natural_ $S-$_quasi-concave_ on $X$
if $-f$ is natural quasi $S-$convex on $X$.
Notation. We will denote by $\Delta_{n-1}$ the standard (n-1)-dimensional
simplex in $R^{n},$ that is
$\Delta_{n-1}=\left\\{(\lambda_{1},\lambda_{2},...,\lambda_{n})\in
R^{n}:\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}=1\text{
and }\lambda_{i}\geqslant 0,i=1,2,...,n\right\\}.$
In this paper, we will also use the following notation:
$C^{\ast}(\Delta_{n-1})=\\{g=(g_{1},g_{2},...,g_{n}):\Delta_{n-1}\rightarrow\Delta_{n-1}$
where $g_{i}$ is continuous, $g_{i}(1)=1$ and $g_{i}(0)=0$ for each
$i\in\\{1,2,...,n\\}\\}$
Definition 2.4 (see [3]) Let $X$ be a non-empty convex subset of a topological
vector space $E$ and $Y$ a non-empty subset of $E$. The set-valued map
$F:X\rightrightarrows Y$ is said to have _weakly convex graph_ (in short, it
is a WCG correspondence) if, for each $n\in N$ and for each finite set
$\\{x_{1},x_{2},...,x_{n}\\}\subset X$, there exists $y_{i}\in F(x_{i})$,
$(i=1,2,...,n)$ such that
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1.1)\ \ \ \ \
$co$(\\{(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\\})\subset$Gr$(F)$
The relation (1.1) is equivalent to
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1.2)\ \ \ \
\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}y_{i}\in
F(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}x_{i})\ \ \ \
\ \ \ (\forall(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}).$
In [22] we introduced the concept of weakly naturally quasi-concave set-valued
map.
Definition 2.5 (see [22])Let $X$ be a non-empty convex subset of a topological
vector space $E$ and $Y$ a non-empty subset of a topological vector space $Z$.
The set-valued map $F:X\rightrightarrows Y$ is said to be _weakly naturally
quasi-concave (WNQ)_ iff, for each $n$ and for each finite set
$\\{x_{1},x_{2},...,x_{n}\\}\subset X$, there exists $y_{i}\in F(x_{i})$,
$(i=1,2,...,n)$ and $g\in C^{\ast}(\Delta_{n-1})$ such that
$\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}g_{i}(\lambda_{i})y_{i}\in
F(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}x_{i})$ for
every $(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}.$
Remark 2.1 If $g_{i}(\lambda_{i})=\lambda_{i}$ for each $i\in(1,2,...,n)$ and
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$ we get a set-
valued map with weakly convex graph, as it is defined by Ding and He Yiran in
[3]. In the same time, the weakly naturally quasi-concavity is a weakening of
the notion of naturally S-quasi-concavity with $S=\\{0\\}.$
Remark 2.2 If $F$ is a single-valued mapping, then, it must be natural
$S$-quasiconcave for $S=\\{0\\}.$
Example 2.1 (see [22]) Let $F:[0,4]\rightrightarrows[-2,2]$ be defined by
$F(x)=\left\\{\begin{array}[]{c}[0,2]\text{ if }x\in[0,2);\\\ [-2,0]\text{ \ \
if \ }x=2;\\\ (0,2]\text{ if }x\in(2,4].\end{array}\right.$
$F$ is neither upper semicontinuous, nor lower semicontinuous in $2.$ $F$ has
not either got a weakly convex graph, since, if we consider $n=2,$ $x_{1}=1$
and $x_{2}=3,$ we have that co$\\{(1,y_{1}),(3,y_{2})\\}\nsubseteq$Gr$F,$ for
every $y_{1}\in F(x_{1}),y_{2}\in F(x_{2}).$ We notice that $F$ is not
naturally $\\{0\\}-$quasi-concave, but it is weakly naturally quasi-concave.
We proved in [22] the following fixed point theorem.
Theorem 2.1 (see [22])Let $Y$ be a non-empty subset of a topological vector
space $E$ and $K$ be a $(n-1)$\- dimensional simplex in $E$. Let
$F:K\rightrightarrows Y$ be an weakly naturally quasi-concave set-valued map
and $s:Y\rightarrow K$ be a continuous function. Then, there exists
$x^{\ast}\in K$ such that $x^{\ast}\in s\circ F(x^{\ast})$.
3\. Set-valued Maps with Generalized Convexity
In this section, we introduce several classes of cone convexity in order to
generalize the requirements for results concerning minimax inequalities.
Concerning the minimax problems we consider in this paper, we must underline
the behaviour importance of the set-valued maps $F(\cdot,\cdot):X\times
X\rightarrow Y$ in the points where their values contain or not maximal (resp.
minimal) elements of the certain sets of type
$\mathop{\textstyle\bigcup}\limits_{y\in X}F(x,y)$ or
$\mathop{\textstyle\bigcup}\limits_{x\in X}F(x,y)$. We obtain the new
definitions through transferring the convexity properties of the maps from a
variable to another and by taking into consideration the maximal (resp.
minimal) elements. The reasons for our conception of generalized convex set-
valued maps come from the motivating work in the framework of minimax theory,
where the new properties prove to be necessary in order to obtain results by
giving up the continuity assumptions.
We firstly define the $S-$transfer $\mu-$convexity.
Definition 3.1 Let $X$ be a convex set of a topological vector space $E,$ let
$Y$ be a non-empty set in the topological vector space $Z$ and let $F:X\times
X\rightrightarrows Y$ be a set valued map with non-empty values. $F$ is called
$S-$transfer type-(v) $\mu-$convex in the first argument on $X\times X$ iff,
for each $n\in N$, $x_{1},x_{2},...,x_{n}\in X$ and $z\in X,$ we have that,
for each $i\in\\{1,2,...,n\\},$ there exists
$z_{i}=z_{i}(x_{1},x_{2},...,x_{n},z)\in X$ such that:
i)
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y))\subset F(x_{i},z_{i})-S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y)\neq\emptyset$ or,
ii)
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y))\subset F(x_{i},z_{i})-$int$S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y)=\emptyset.$
$F$ is called $S-$transfer type-(v) $\mu-$concave in the first argument on
$X\times X$ if $-F$ is $S-$transfer type-(v) $\mu-$convex in the first
argument on $X\times X.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Remark 3.1 We can similarily define the $S-$transfer type-(iii) $\mu-$convex
set-valued maps.
Example 3.1 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[-1,y]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,y]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We will prove that $F$ is $S-$transfer type-(v) $\mu-$convex in the first
argument.
Let $x_{1},x_{2},...,x_{n}\in X$ and $z\in Y.$ For each $i\in\\{1,2,...,n\\},$
$\mathop{\textstyle\bigcup}_{y\in X}F(x_{i},y)=[-1,1].$
Moreover, by computing, we obtain Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y)=\\{1\\}$ and
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)=\left\\{\begin{array}[]{c}[-1,z]\text{
\ \ \ \ \ \ \ if \ \ \ \ \ \
}0\leq\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}\leq z\leq 1;\\\
[-\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z]\text{ if }0\leq
z<\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}\leq 1.\end{array}\right.$
For each $i\in\\{1,2,...,n\\},$ there exists $z_{i}\in Y,$
$z_{i}\geq\max\\{z,x_{i}\\},$ so that $F(x_{i},z_{i})=[-1,z_{i}]$ and then:
i) if $z=1,$
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y)=F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap\\{1\\}\neq\emptyset$
and $F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\subset
F(x_{i},z_{i})-S$ or
ii) if $z<1,$
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y)=F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap\\{1\\}=\emptyset$
and $F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\subset
F(x_{i},z_{i})-$int$S$
Remark 3.2 The $S-$transfer type-(v) $\mu-$convexity in the first argument is
implied by the following property, which we call $\alpha:$
$(\alpha):$ For each $x\in X,$ $A_{x}=\cup_{y\in X}F(x,y)$ is compact and
there exists $z_{x}\in Z$ such that $z_{x}\in$Max$\cup_{y\in X}F(x,y)$ and
$\cup_{y\in X}F(x,y)\subset z_{x}-S.$
We note that according to Lemma 2.1, $\cup_{y\in
X}F(x,y)\subset$Max$\cup_{y\in X}F(x,y)-S.$
The $S-$transfer type-(v) $\mu-$concavity in the second argument is implied by
the following property $\alpha^{\prime}:$
$(\alpha^{\prime}):$ For each $y\in X,$ $A_{y}=\cup_{x\in X}F(x,y)$ is compact
and there exists $z_{y}\in Z$ such that $z_{y}\in$Max$\cup_{x\in X}F(x,y)$ and
$\cup_{x\in X}F(x,y)\subset z_{y}+S.$
The set valued map from Example 3.1 verifies the property $\alpha$.
The condition $\alpha$ is not fulfilled in the next example.
Example 3.2 Let $S((0,0),x)=\\{(u,v)\in[-1,1]\times[-1,1]:u^{2}+v^{2}\leq
x^{2}\\},$
$S_{+}((0,0),x)=\\{(u,v)\in[0,1]\times[-1,1]:u^{2}+v^{2}\leq x^{2}\\}$ and
$S_{-}((0,0),x)=\\{(u,v)\in[-1,0]\times[-1,1]:u^{2}+v^{2}\leq x^{2}\\}.$
Let us define $F:[0,1]\times[0,1]\rightrightarrows[-1,1]\times[-1,1]$ by
$F(x,y)=\left\\{\begin{array}[]{c}S((0,0),1)\text{ \ \ \ \ if \ \ \ \
}x=1\text{ \ \ \ \ and }y\in[0,1].\\\ S_{+}((0,0),x)\text{ \ if }0<x<1\text{ \
and \ }x\leq y\leq 1;\\\ S_{-}((0,0),x)\text{ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \
\ \ \ \ \ \ }0<y<x<1;\\\ \\{(0,0)\\}\text{ \ \ \ \ \ \ if }x=0\text{ \ \ \ \ \
and \ \ \ \ \ }y\in[0,1].\end{array}\right.$
$F$ is $R_{+}^{2}$-transfer type-(v) $\mu$ convex in the first argument.
The Definition 3.1 can be weakened in the following way.
Definition 3.2 Let $X$ be a convex set of a topological vector space $E,$ let
$Y$ be a non-empty set in the topological vector space $Z$ and let $F:X\times
X\rightrightarrows Y$ be a set-valued map with non-empty values. $F$ is called
$S-$transfer weakly type-(v) $\mu-$convex in the first argument on $X\times X$
iff, for each $n\in N$, $x_{1},x_{2},...,x_{n}\in X$ and $z\in X,$ we have
that, there exist $i_{0}\in\\{1,2,...,n\\}$ and
$z_{i_{0}}=z_{i_{0}}(x_{1},x_{2},...,x_{n},z)\in X$ such that:
i)
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i_{0}},y))\subset F(x_{i_{0}},z_{i_{0}})-S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y)\neq\emptyset$ or,
ii)
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i_{0}},y))\subset F(x_{i_{0}},z_{i_{0}})-$int$S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y)=\emptyset.$
$F$ is called $S-$transfer weakly type-(v) $\mu-$concave in the first argument
on $X\times X$ if $-F$ is $S-$transfer weakly type-(v) $\mu-$convex in the
first argument on $X\times X.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Remark 3.3.We can similarily define the $S$-transfer weakly type-(iii) $\mu$
convex set-valued maps.
Remark 3.4. If $F:X\times X\rightarrow Z$ is type-(v) properly $S-$quasi-
convex in the first argument, then, $F$ is $S$-transfer weakly type-(v) $\mu$
convex in the first argument.
Indeed, let $x_{1},x_{2},...,x_{n}\in X$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$. We have
that $F(\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}x_{i},y)\subset
F(x_{i_{0}},y)-S$ for each $\lambda\in\Delta_{n-1}$, $y\in X$ and an idex
$i_{0}\in\\{1,2,...,n\\}.$ Then, for each $z\in X,$ there exists $z_{i_{0}}=z$
such that
$F(\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}\limits_{z\in
X}F(x_{i_{0}},z))\subset F(x_{i_{0}},z_{i_{0}})-S.$
Consequently, the notion of $S-$transfer weakly type-(v) $\mu-$convexity is
weaker than the type-(v) properly $S-$quasi-convexity and, in certain cases,
it is implied by the property $\alpha.$
Example 3.3 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,y]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,y]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
$F$ is $S-$transfer weakly type-(v) $\mu-$convex in the first argument.
Now, we are introducing a similar definition for single valued mappings.
Definition 3.3 Let $X$ be a convex set of a topological vector space $E$ and
let $Y$ be a non-empty set in the topological vector space $Z.$
The mapping $f:X\times X\rightarrow Y$ is called $S-$transfer $\mu-$convex in
the first argument on $X\times X$ iff, for each $n\in N$,
$x_{1},x_{2},...,x_{n}\in X$ and $z\in X,$ we have that, for each
$i\in\\{1,2,...,n\\},$ there exists $z_{i}=z_{i}(x_{1},x_{2},...,x_{n},z)\in
X$ such that, if
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in\mathop{\textstyle\bigcup}_{y\in
X}f(x_{i},y)$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$ the
following condition is fulfilled:
i) $f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in
f(x_{i},z_{i})-S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in$Max${}_{w}(\mathop{\textstyle\bigcup}_{y\in
X}f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y))$ or,
ii) $f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in
f(x_{i},z_{i})-$int$S$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ with the
property that
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\notin$Max${}_{w}(\mathop{\textstyle\bigcup}_{y\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},y)).$
The mapping $f$ is called $S-$transfer $\mu-$concave in the first argument on
$X\times X$ iff $-f$ is $S-$transfer $\mu-$convex in the first argument on
$X\times X.$
Example 3.4 Let $X=[0,1],$ $Y=[-1,0],$ $S=[0,\infty)$ and $f:X\times
X\rightarrow Y$ be defined by $f(x,y)=\left\\{\begin{array}[]{c}1\text{ if
}0\leq x\leq y\leq 1;\\\ x\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We will prove that $f$ is $S-$transfer $\mu-$convex in the first argument.
Let $x_{1},x_{2},...,x_{n},z\in X.$ For each $i\in\\{1,2,...,n\\},$
$\mathop{\textstyle\bigcup}_{y\in X}f(x_{i},y)=\\{x_{i},1\\},$ Maxw
$\mathop{\textstyle\bigcup}_{y\in X}f(x_{i},y)=\\{1\\}$ and we have that, for
each $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$ if
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in\\{x_{i},1\\},$
there exists $z_{i}\in Y,$ $z_{i}\geq\max\\{z,x_{i}\\},$ so that
$f(x_{i},z_{i})=1,$ and then:
i) if $z=1$ and $f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)=1$ for
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$ we have that
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in f(x_{i},z_{i})-S$
or,
ii) if $z<1$ and $f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\neq
1$ for $(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1},$ we have
that $f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\in
f(x_{i},z_{i})-$int$S.\vskip 6.0pt plus 2.0pt minus 2.0pt$
The next notion is stronger than the properly $S-$quasi-convexity and it is
adapted for set-valued maps with two variables. We consider pairs of points in
the product space $X\times X$. We keep constant one component and we consider
any convex combination of the other ones. By comparing the images of $F$ in
all these pairs of points, we obtain the following definition.
Definition 3.4 Let $X$ be a non-empty convex subset of a topological vector
space $E,$ $Z$ a real topological vector space and $S$ a pointed closed convex
cone in $Z$ with its interior int$S\neq\emptyset.$ Let $F:X\rightrightarrows
Z$ be a set-valued map with non-empty values.
(i) $F$ is said to be type-(iii) pair properly $S-$quasi-convex on $X\times X$
in the first argument, iff, for any $(x_{1},y_{1}),(x_{2},y_{2})\in X\times X$
and $\lambda\in[0,1],$ either $F(x_{1},y_{1})\subset F(\lambda
x_{1}+(1-\lambda)x_{2},y_{1})+S$ or $F(x_{2},y_{2})\subset F(\lambda
x_{1}+(1-\lambda)x_{2},y_{2})+S.$
(ii) $F$ is said to be type-(v) pair properly $S-$quasi-convex on $X\times X$
in the first argument, iff, for any $(x_{1},y_{1}),(x_{2},y_{2})\in X\times X$
and $\lambda\in[0,1],$ either $F(\lambda x_{1}+(1-\lambda)x_{2},y_{1})\subset
F(x_{1},y_{1})-S$ or $F(\lambda x_{1}+(1-\lambda)x_{2},y_{2})\subset
F(x_{2},y_{2})-S.$
(iii) $F$ is said to be type-(iii) [resp. type-(v)] pair properly $S-$quasi-
concave on $X$ in the first argument, iff, $-F$ is type-(iii) [resp. type-(v)]
pair properly $S-$quasi-convex in the first argument on $X.$
(iv) $F$ is said to be pair properly quasi-convex iff for any
$(x_{1},y_{1}),(x_{2},y_{2})\in X\times X$ and $\lambda\in[0,1],$ either
$F(x_{1},y_{1})\subset F(\lambda x_{1}+(1-\lambda)x_{2},y_{1})$ or
$F(x_{2},y_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2},y_{2}).$
$\mathit{F}$ is said to be pair properly quasi-concave if $-F$ is pair
properly $S-$quasi-convex.
Example 3.5 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[-1,1]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,1]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
$F$ is type-(iii) pair properly quasi-concave in the second argument on $X.$
Remark 3.5. $S-$transfer $\mu-$convexity does not imply pair properly
$S-$quasi-convexity. The set valued map from Example 3.2 is
$R_{+}^{2}-$transfer type-(v) $\mu-$convex in the first argument, but it is
not type-(v) pair properly $R_{+}^{2}-$quasi-convex in the first argument.
If we consider $(x_{1},y_{1})=(\frac{1}{15},\frac{9}{10}),$
$(x_{2},y_{2})=(\frac{1}{4},\frac{1}{5})$ and
$x_{0}=\frac{1}{5}\in$co$\\{x_{1},x_{2}\\},$ then,
$F(x_{1},y_{1})=S_{+}((0,0),\frac{1}{15}),$
$F(x_{2},y_{2})=S_{-}((0,0),\frac{1}{4}),$
$F(x_{0},y_{1})=S_{+}((0,0),\frac{1}{5})$ and
$F(x_{0},y_{2})=S_{+}((0,0),\frac{1}{5}).$ It follows that neither
$F(x_{0},y_{1})\subset F(x_{1},y_{1})-R_{+}^{2},$ nor $F(x_{0},y_{2})\subset
F(x_{2},y_{2})-R_{+}^{2}$ and then, $F$ is not type-(v) pair properly
$R_{+}^{2}-$quasi-convex in the first argument.
Conversely, the pair properly $S-$quasi-convexity does not imply $S-$transfer
$\mu-$convexity. The following example is concludent in this respect.
Example 3.6. For each $(x,y)\in[0,1]\times[0,1],$ let us define
$S((0,y),x)=\\{(u,v)\in R^{2}\times R^{2}:u^{2}+(v-y)^{2}\leq x^{2}\\}$ and
$S((y,0),x)=\\{(u,v)\in\in R^{2}\times R^{2}:(u-y)^{2}+v^{2}\leq x^{2}\\}.$
Let $S=R_{+}^{2}$ and $F:[0,1]\times[0,1]\rightrightarrows[-2,2]\times[-2,2]$
be defined by
$F(x,y)=\left\\{\begin{array}[]{c}S((0,y),x)\text{ \ \ \ \ if \ \
}(x,y)\in[0,1]\times([0,1]\cap Q);\\\ S((y,0),x)\text{ if
}(x,y)\in[0,1]\times([0,1]\cap(R\backslash Q)).\end{array}\right.$
The set valued map $F$ is type-(v) pair properly $R_{+}^{2}-$quasi-convex in
the first argument, but it is not $R_{+}^{2}-$transfer type-(v) $\mu-$convex
in the first argument.
Indeed, let us consider first $(x_{1},y_{1})$ and $(x_{2},y_{2})\in[0,1].$
Without loss of generalization, we can assume that $x_{1}\leq x(\lambda)\leq
x_{2}$ for each $\lambda\in[0,1]$, where $x(\lambda)=\lambda
x_{1}+(1-\lambda)x_{2}.$ Consequently, $F(x(\lambda),y_{2})\subset
F(x_{2},y_{2})-S$ and $F$ is type-(v) pair properly $R_{+}^{2}-$quasi-convex
in the first argument.
In order to prove the second assertion, let us consider $x_{1},x_{2}\in[0,1]$
and
$x(\lambda)=\lambda x_{1}+(1-\lambda)x_{2},$ where $\lambda\in[0,1].$
For $i=1,2$ and $y=0$ the following equality holds:
$F(x(\lambda),0)\cap\mathop{\textstyle\bigcup}\limits_{y\in[0,1]}F(x_{i},y)=$
$(F(x(\lambda),0)\cap\mathop{\textstyle\bigcup}\limits_{y\in[0,1]\cap
Q}F(x_{i},y))\cup(F(x(\lambda),0)\cap\mathop{\textstyle\bigcup}\limits_{y\in[0,1]\cap(R\backslash
Q)}F(x_{i},y))$
and there is not any $z_{i}\in[0,1]$ such that
$F(x(\lambda),0)\cap\mathop{\textstyle\bigcup}\limits_{y\in[0,1]}F(x_{i},y)\subset
F(x_{i},z_{i})-R_{+}^{2}.$
We conclude that $F$ is not $R_{+}^{2}-$transfer type-(v) $\mu-$convex in the
first argument.
For single valued mappings, the next definition is proposed.
Definition 3.5 Let $X$ be a nonempty convex subset of a topological vector
space $E,$ $Z$ a real topological vector space and $S$ a pointed closed convex
cone in $Z$ with its interior int$S\neq\emptyset.$ Let $f:X\rightarrow Z$ be a
set-valued map with non-empty values.
(i) $f$ is said to be pair properly $S-$quasi-convex on $X\times X$ in the
first argument, iff, for any $(x_{1},y_{1}),(x_{2},y_{2})\in X\times X$ and
$\lambda\in[0,1],$ either $f(x_{1},y_{1})\subset f(\lambda
x_{1}+(1-\lambda)x_{2},y_{1})+S$ or $f(x_{2},y_{2})\subset f(\lambda
x_{1}+(1-\lambda)x_{2},y_{2})+S.$
$f$ is said to be pair properly $S-$quasi-concave in the first argument on
$X\times X$, iff $-f$ is properly $S-$quasi-convex in the first argument on
$X\times X.$
The usual naturally $S-$quasi-convexity requirement in the minimax
inequalities for set-valued maps can be weakened. In the definition we propose
below, we take into consideration the bahaviour of the set-valued maps in the
points where their values do not contain minimal (resp. maximal) points of
some certain sets of $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ or
$\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ types.
Definition 3.6 Let $X$ be a convex set of a topological vector space $E,$ let
$Y$ be a non-empty set in the topological vector space $Z$ and let $F:X\times
X\rightrightarrows Y$ be a set-valued map with non-empty values.
i) $F$ is called transfer type-(iii) properly $S-$quasi-convex in the first
argument on $X\times X$ iff, for each elements $x_{1},x_{2},z\in X,$
$\lambda\in(0,1)$ and $i\in\\{1,2\\}$, the following condition is fulfilled:
$F(\lambda
x_{1}+(1-\lambda)x_{2},z)\cap$Min${}_{w}(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y))=\emptyset$ implies that $F(x_{i},z)\subset F(\lambda
x_{1}+(1-\lambda)x_{2},z)+S$.
ii) $F$ is called transfer type-(v) properly $S-$quasi-convex in the first
argument on $X\times X$ iff, for each elements $x_{1},x_{2},z\in X,$
$\lambda\in(0,1)$ and $i\in\\{1,2\\}$, the following condition is fulfilled:
$F(\lambda
x_{1}+(1-\lambda)x_{2},z)\cap$Min${}_{w}(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y))=\emptyset$ implies $F(\lambda x_{1}+(1-\lambda)x_{2},z)\subset
F(x_{i},z)-S$.
$F$ is called transfer type-(iii) [resp.type-(v)] properly $S-$quasi-concave
in the first argument on $X\times X$ if $-F$ is transfer type-(iii)
[resp.type-(v)] properly $S-$quasi-convex in the first argument on $X\times
X.$
Remark 3.6. If $F(\cdot,y)$ is naturally $S-$quasi-convex for each $y\in X,$
then, $F$ is transfer properly $S-$quasi-convex in the first argument on
$X\times X.$
Remark 7. If $F$ is transfer properly $S-$quasi-convex in the first argument
on $X\times X,$ then, $F$ is $S-$transfer weakly $\mu-$convex in the first
argument.
Conversely, it is not true. The set-valued map $F$ defined in Example 3.2 is
$S-$transfer weakly (type-v) $\mu-$convex in the first argument, but it is not
type-(v) transfer properly $S-$quasi-convex.
Example 3.7 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,y]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,y]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We prove that $F(\cdot,y)$ is type-(iii) naturally $S-$quasi-concave on $X$
(and then, $F$ is transfer type-(iii) properly $S-$quasi-concave in the first
argument on $X\times X$).
Let $y\in[0,1]$ be fixed, $x_{1},x_{2}\in[0,1]$, $\lambda\in[0,1]$ and
$x(\lambda)=\lambda x_{1}+(1-\lambda)x_{2}$.
1) If $x_{1}\geq x_{2}\geq y,$ then, $F(x_{1},y)=[-x_{1},y],$
$F(x_{2},y)=[-x_{2},y]$, $F(x(\lambda),y)=[-x(\lambda),y]$ and
co{$F(x_{1},y),F(x_{2},y)\\}=[-x_{1},y]\subset[-x(\lambda),y]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
2) if $x_{1}\leq x_{2}\leq y,$ then, $F(x_{1},y)=[0,y],$ $F(x_{2},y)=[0,y]$,
$F(x(\lambda),y)=[0,y]$ and
co{$F(x_{1},y),F(x_{2},y)\\}=[0,y]\subset[0,y]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
3) if $x_{1}\geq y\geq x_{2},$ then, $F(x_{1},y)=[-x_{1},y]$,
$F(x_{2},y)=[0,y]$ and
co{$F(x_{1},y),F(x_{2},y)\\}=[-x_{1},y];$
if $x_{1}\geq x(\lambda)\geq y\geq x_{2},$ then,
$F(x(\lambda),y)=[-x(\lambda),y]$ and
co{$F(x_{1},y),F(x_{2},y)\\}=[-x_{1},y]\subset[-x(\lambda),y]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
if $x_{1}\geq y\geq x(\lambda)\geq x_{2},$ then, $F(x(\lambda),y)=[0,y]$ and
co{$F(x_{1},y),F(x_{2},y)\\}=[-x_{1},y]\subset[0,y]-[0,\infty)=F(x(\lambda),y)-[0,\infty).$
The usual properly $S-$quasi-convexity assumption in the minimax theorems with
set-valued maps can be also generalized. In order to obtain necessary
conditions in our results, we introduce the following definitions.
Definition 3.7 Let $X$ be a convex set of a topological vector space $E,$ let
$Y$ be a non-empty set in the topological vector space $Z$ and let $F:X\times
X\rightrightarrows Y$ be a set-valued map with non-empty values. $F$ satisfies
the condition $\gamma$ on $X\times X$ iff:
$(\gamma)$ there exist $n\in N,$
$(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\in X\times X$,
$y^{\ast}\in$co$\\{x_{1},x_{2},...,x_{n}\\}$ such that $F(x_{i},y_{i})\subset
F(x_{i},y^{\ast})-S$ and $F(x_{i},y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i},z)\neq\emptyset$ for each $i\in\\{1,2,...,n\\}$.
Example 3.8 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,y]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,y]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We prove that $F(x,\cdot)$ satisfies the condition $\gamma.$ In fact, there
exist $(x_{1},y_{1})=(0,1),$ $(x_{2},y_{2})=(1,1)\in X\times X$ such that
$F(x_{i},y_{i})\cap$Max${}_{w}\cup_{z\in X}F(x_{i},z)\neq\emptyset,$ $i=1,2.$
There also exists $y^{\ast}=1\in$co$\\{x_{1},x_{2}\\}$ such that
$[0,1]=F(x_{1},y_{1})\subset F(x_{1},y^{\ast})-[0,\infty)$ and
$[0,1]=F(x_{2},y_{2})\subset F(x_{2},y^{\ast})-[0,\infty).$
Definition 3.8 Let $X$ be a convex set of a topological vector space $E,$ let
$Y$ be a non-empty set in the topological vector space $Z$ and let $F:X\times
X\rightrightarrows Y$ be a set-valued map with non-empty values. $F$ satisfies
the condition $\gamma^{\prime}$ on $X\times X$ iff:
$(\gamma^{\prime})$ there exist $n\in N,$
$(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\in X\times X$ and
$x^{\ast}\in$co$\\{y_{1},y_{2},...,y_{n}\\}$ such that $F(x_{i},y_{i})\subset
F(x_{i},y^{\ast})+S$ and $F(x_{i},y_{i})\cap$Min${}_{w}\cup_{x\in
X}F(x,y_{i})\neq\emptyset$ for each $i\in\\{1,2,...,n\\}$.
4\. Minimax Theorems for Set-valued Maps without Continuity
In this section, we establish some generalized Ky Fan minimax inequalities.
Firstly, we are proving the following lemma, which is comparable with Lemma
3.1 in [32], but our result does not involve continuity assumptions. Instead,
we use several generalized convexity properties for set-valued maps introduced
in Section 3.
Lemma 4.1 will be used to prove the minimax Theorem 4.1.
Lemma 4.1 Let $X$ be a (n-1) dimensional simplex of a Hausdorff topological
vector space $E,$ $Y$ a compact set in the Hausdorff topological vector space
$Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with non-empty values.
(i) Let us suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a
compact set for each $x\in X$. If $F$ is $S-$transfer type-(v) $\mu-$convex in
the first argument on $X\times X$, $F$ is type-(iii) pair properly quasi-
concave in the second argument on $X\times X$ and $F(\cdot,y)$ is type-(iii)
naturally $S-$quasi-concave on $X$ for each $y\in X,$ then, there exists
$x^{\ast}\in X$ such that
$F(x^{\ast},x^{\ast})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x^{\ast},y)\neq\emptyset.$
(ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If $F$ is transfer type-(v) $\mu-$concave in the second
argument on $X\times X,$ $F$ is type-(iii) pair properly quasi-convex in the
first argument on $X\mathit{\times X}$ and $F(x,\cdot)$ is type-(iii)
naturally $S-$quasi-convex on $X$ for each $x\in X,$ then, there exists
$y^{\ast}\in X$ such that $F(y^{\ast},y^{\ast})\cap$Min${}_{w}\cup_{x\in
X}F(x,y^{\ast})\neq\emptyset.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Proof. (i) Let us define the set-valued map $T:X\rightrightarrows X$ by
$T(x)=\\{y\in X:F(x,y)\cap$Max${}_{w}\cup_{z\in X}F(x,z)\neq\emptyset\\}$ for
each $x\in X.$
We claim that $T$ is non-empty valued. Indeed, since $\cup_{z\in X}F(x,z)$ is
a compact set for each $x\in X,$ according to Lemma 2.1, Max${}_{w}\cup_{z\in
X}F(x,z)\neq\emptyset.$ For each $x\in X,$ let $z_{x}\in$Max${}_{w}\cup_{z\in
X}F(x,z).$ Then, there exists $y_{x}\in X$ such that $z_{x}\in F(x,y_{x}).$ It
is clear that $y_{x}\in T(x)=\\{y\in X:F(x,y)\cap$Max${}_{w}\cup_{z\in
X}F(x,z)\\}$ and, consequently, $T(x)\neq\emptyset$ for each $x\in X.$
Further,we will prove that $T$ is weakly naturally quasi-concave.
Let $x_{1},x_{2},...,x_{n}\in X.$ For each $i\in 1,...,n,$ there exists
$y_{i}\in T(x_{i})$, that is
$F(x_{i},y_{i})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i},z)\neq\emptyset.$
By contrary, we assume that $T$ is not weakly naturally quasi-concave. Then,
for each $g\in C^{\ast}(\Delta_{n-1})$, there exists
$\lambda^{g}=(\lambda_{1}^{g},\lambda_{2}^{g},...,\lambda_{n}^{g})\in\Delta_{n-1}$
such that $\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i}\notin
T(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i}),$ relation which is
equivalent with the following one:
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},z)=\emptyset.$
Since the set-valued map $F$ is $S-$transfer type-(v) $\mu-$convex in the
first argument and
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},X)=\emptyset$,
it follows that, for each $i\in\\{1,2,...,n\\},$ there exists the element
$z_{i_{0}}\in X$ such that the following relation is fulfilled:
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i},z))\subset F(x_{i},z_{i_{0}})-$int$S.$
Let $t_{i}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i},z))$ and $u_{i}\in F(x_{i},z_{i_{o}})$ such that
$t_{i}=u_{i}-s_{i},$ $s_{i}\in$int$S.$ It follows that
$u_{i}\in\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i},z)\cap\\{t_{i}+$int$S\\}\neq\emptyset,$ that is $t_{i}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i},z))$ implies the fact that $t_{i}\notin$Max${}_{w}\cup_{z\in
X}F(x_{i},z).$ Consequently, we have that, for each index
$i\in\\{1,2,...,n\\},$
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i},z)=\emptyset.$
We claim that
$F(x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i},z)=\emptyset$ for each $i\in\\{1,2,...,n\\}.$ Indeed, if, by
contrary, we assume that there exists $i_{0}\in\\{1,2,...,n\\}$ and $t\in
F(x_{i_{0}},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})$
such that $t\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z),$ then, it is true that
$t\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})-S$
(1) and $t\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z)$ (2).
According to (1), we have $t=t^{\prime}-s_{0},$ where $t^{\prime}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})$
and $s_{0}\in S,$ therefore $t^{\prime}=t+s_{0}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i}).$
According to the relation (2), $\cup_{z\in
X}F(x_{i_{0}},z)\cap\\{t+$int$S\\}=\emptyset.$ Consequently,
$t^{\prime}+s\notin\cup_{z\in X}F(x_{i_{0}},z)$ if $s\in$int$S$ (we take into
account that $t^{\prime}+s=t+(s_{0}+s)\in t+$int$S).$ Then, $\cup_{z\in
X}F(x_{i_{0}},z)\cap\\{t^{\prime}+$int$S\\}=\emptyset,$ which implies
$t^{\prime}\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z).$
Thus, we have that $t^{\prime}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{g}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z),$ which is a contradiction. It remains that
$F(x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}(\lambda_{i}^{g})y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i},z)=\emptyset$ for each $i\in\\{1,2,...,n\\}.$
Since $F$ is type-(iii) pair properly quasi-concave in the second argument on
$X\times X,$ there exists $j\in\\{1,2,...,n\\}$ such that
$F(x_{j},y_{j})\cap$Max${}_{w}\cup_{z\in X}F(x_{j},z)=\emptyset,$ which
contradicts the assumption about $(x_{j},y_{j})$. According toTheorem 2.1,
there exists $x^{\ast}\in T(x^{\ast}),$ that is,
$F(x^{\ast},x^{\ast})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x^{\ast},y)\neq\emptyset.$
(ii) Let us define the set-valued map $Q:X\rightrightarrows X$ by
$Q(y)=\\{x\in X:F(x,y)\cap$Min${}_{w}\cup_{x\in X}F(x,y)\neq\emptyset\\}$ for
each $y\in X.$
Further, the proof follows a similar line as above and we conclude that there
exists $y^{\ast}\in Q(y^{\ast}),$ that is,
$F(y^{\ast},y^{\ast})\cap$Min${}_{w}\cup_{x\in X}F(x,y^{\ast})\neq\emptyset.$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \square$
Remark 4.1. The $S-$transfer type-(v) $\mu-$convexity of $F$ in the first
argument on $X\times X$ is verified by all real-valued set valued maps which
fulfill the property that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a
compact set for each $x\in X$. This fact is a consequence of Remark 3.1.
As a first application of the previous lemma, we obtain the following result,
which differs from Theorem 3.1 in [32] becose we only take into consideration
the hypothesis which concern convexity properties of set-valued maps. No form
of continuity is assumed.
Theorem 4.1 Let $X$ be a (n-1) dimensional simplex of a Hausdorff topological
vector space $E,Y$ be a compact set in a Hausdorff topological vector space
$Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with non-empty values.
i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a compact set
for each $x\in X$. If the set-valued map $F$ is $S-$transfer type-(v)
$\mu-$convex in the first argument on $X\times X$, type-(iii) pair properly
quasi-concave in the second argument on $X\times X$ and $F(\cdot,y)$ is
type-(iii) naturally $S-$quasi-concave on $X$ for each $y\in X,$ then, there
exist the elements $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If the set-valued map $F$ is $S-$transfer type-(v)
$\mu-$concave in the second argument on $X\times X,$ type-(iii) pair properly
quasi-convex in the first argument on $X\times X$ and $F(x,\cdot)$ is
type-(iii) naturally $S-$quasi-convex on $X$ for each $x\in X,$ then, there
exist the elements $z_{1}\in$Min$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$ such that
$z_{1}\in z_{2}-S.$
Proof. i) According to Lemma 4.1, there exists $x^{\ast}\in X$ such that
$F(x^{\ast},x^{\ast})\cap$
Max${}_{w}\cup_{y\in X}F(x^{\ast},y)\neq\emptyset.$
We have $F(x^{\ast},x^{\ast})\subset\overline{\cup_{x\in X}F(x,x)}$ and,
according to Lemma 2.1, it follows that $\overline{\cup_{x\in
X}F(x,x)}\subset$Max $\overline{\cup_{x\in X}F(x,x)}-S,$ so that,
$F(x^{\ast},x^{\ast})\subset$Max$\overline{\cup_{x\in X}F(x,x)}-S.$
On the other hand, Max${}_{w}\cup_{y\in
X}F(x^{\ast},y)\subset\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ and,
according to Lemma 2.1, it follows that $\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}\subset$Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}+S,$ so that, Max${}_{w}\cup_{y\in
X}F(x^{\ast},y)\subset$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}+S.$
Hence, for every $u\in F(x^{\ast},x^{\ast})$ and $v\in$Max${}_{w}\cup_{y\in
X}F(x^{\ast},y),$ there exist the elements $z_{1}\in$Max$\overline{\cup_{x\in
X}F(x,x)}$ and $z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$
such that $u\in z_{1}-S$ and $v\in z_{2}+S.$ If we take $u=v,$ we have
$z_{1}\in z_{2}+S.$
ii) According to Lemma 4.1, there exists $y^{\ast}\in X$ such that
$F(y^{\ast},y^{\ast})\cap$
Min${}_{w}\cup_{x\in X}F(x,y^{\ast})\neq\emptyset.$
We have $F(y^{\ast},y^{\ast})\subset\overline{\cup_{x\in X}F(x,x)}$ and,
according to Lemma 2.1, it follows that $\overline{\cup_{x\in
X}F(x,x)}\subset$Min $\overline{\cup_{x\in X}F(x,x)}+S,$ so that,
$F(y^{\ast},y^{\ast})\subset$Min$\overline{\cup_{x\in X}F(x,x)}+S.$
On the other hand, Min${}_{w}\cup_{x\in
X}F(x,y^{\ast})\subset\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$ and,
according to Lemma 2.1, it follows that $\overline{\cup_{y\in
X}\text{Min}_{w}F(X,y)}\subset$Max$\overline{\cup_{y\in
X}\text{Min}_{w}F(X,y)}-S,$ consequently, Min${}_{w}\cup_{x\in
X}F(x,y^{\ast})\subset$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}-S.$
Hence, for every $u\in F(y^{\ast},y^{\ast})$ and $v\in$Min${}_{w}\cup_{x\in
X}F(x,y^{\ast}),$ there exist the elements $z_{1}\in$Min$\overline{\cup_{x\in
X}F(x,x)}$ and $z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$
such that $u\in z_{1}+S$ and $v\in z_{2}-S.$ If we take $u=v,$ we have
$z_{1}\in z_{2}-S.$ $\square$
An important version of Theorem 4.1 is obtained in the case when the set-
valued map has the property $\alpha$ (resp.$\alpha^{\prime}).$
Theorem 4.2 Let $X$ be a (n-1) dimensional simplex of a Hausdorff topological
vector space $E,Y$ be a compact set in a Hausdorff topological vector space
$Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with nonempty values.
i) Suppose that $F$ satisfies the property $\alpha.$ If $F$ is type-(iii) pair
properly quasi-concave in the second argument on $X\times X$ and $F(\cdot,y)$
is type-(iii) naturally $S-$quasi-concave on $X$ for each $y\in X,$ then,
there exist the elements $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $F$ satisfies the property $\alpha^{\prime}.$ If $F$ is
type-(iii) pair properly quasi-convex in the first argument on $X\times X$ and
$F(x,\cdot)$ is type-(iii) naturally $S-$quasi-convex on $X$ for each $x\in
X,$ then, there exist the elements $z_{1}\in$Min$\overline{\cup_{x\in
X}F(x,x)}$ and $z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$
such that $z_{1}\in z_{2}-S.$
Example 4.1 Let $S=-R_{+}^{2}$, and for each $x\in[0,1],$ let
$S^{\ast}((0,0),x)=\\{(u,v)\in[0,1]\times[0,1]:u^{2}\times v^{2}\leq x^{2}\\}$
and $F:[0,1]\times[0,1]\rightrightarrows[0,1]\times[0,1]$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}\\{(0,0)\\}\text{ \ \ \ \ for each\ \ \
}0\leq x\leq y\leq 1;\\\ S^{\ast}((0,0),x)\text{ for each }0\leq y<x\leq
1.\end{array}\right.$
We notice that $F$ is not continuous on $X.$
a) $F$ is $-R_{+}^{2}-$transfer type-(v) $\mu-$convex in the first argument.
Let $x_{1},x_{2},...,x_{n}\in[0,1]$ and $z\in[0,1].$ For each
$i\in\\{1,2,...,n\\},$ there exists
$z_{i}=z_{i}(x_{1},x_{2},...,x_{n},z)\geq\max_{i=1,2,...,n}x_{i}\in[0,1]$ such
that $F(x_{i},z_{i})=\\{(0,0)\\}$ for each $i=1,2,...n$ and then,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},z)\cap(\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i},y))\subset\\{(0,0)\\}-(-R_{+}^{2})$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$.
It follows that $F$ is $-R_{+}^{2}-$transfer type-(v) $\mu-$convex in the
first argument on $[0,1]\times[0,1].$
b) $F$ is type-(iii) pair properly $-R_{+}^{2}-$quasiconcave in the second
argument on $[0,1]\times[0,1].$
Let us consider $(x_{1},y_{1}),$ $(x_{2},y_{2})\in[0,1]\times[0,1]$ and let us
assume, without loss of generalization, that $y_{1}\leq y(\lambda)\leq y_{2}$
for each $\lambda\in[0,1],$ where $y(\lambda)=\lambda y_{1}+(1-\lambda)y_{2}.$
$F(x_{1},y_{1})=\left\\{\begin{array}[]{c}\\{(0,0)\\}\text{ \ \ \ \ \ \ \ \ \
\ \ \ \ \ for each\ \ \ \ \ \ \ \ \ \ \ }0\leq x_{1}\leq y_{1}\leq 1;\\\
S^{\ast}((0,0),x_{1})\text{ \ \ \ \ \ \ \ \ for each \ \ \ \ \ \ \ \ }0\leq
y_{1}<x_{1}\leq 1,\end{array}\right.$
$F(x_{2},y_{2})=\left\\{\begin{array}[]{c}\\{(0,0)\\}\text{ \ \ \ for each\ \
\ \ \ \ }0\leq x_{2}\leq y_{2}\leq 1;\\\ S^{\ast}((0,0),x_{2})\text{ \ for
each }0\leq y_{2}<x_{2}\leq 1\end{array}\right.$,
$F(x_{1},y(\lambda))=\left\\{\begin{array}[]{c}\\{(0,0)\\}\text{\ \ \ \ \ for
each\ \ \ \ \ }0\leq x_{1}\leq y(\lambda)\leq 1;\\\
S^{\ast}((0,0),x_{1})\text{ for each }0\leq y(\lambda)<x_{1}\leq
1;\end{array}\right.$
$F(x_{2},y(\lambda))=\left\\{\begin{array}[]{c}\\{(0,0)\\}\text{ \ \ \ \ for
each\ \ \ \ }0\leq x_{2}\leq y(\lambda)\leq 1;\\\ S^{\ast}((0,0),x_{2})\text{
for each }0\leq y(\lambda)<x_{2}\leq 1.\end{array}\right.$
b1) If $x_{1}\leq y_{1}\leq y(\lambda),$ then, $F(x_{1},y_{1})=\\{(0,0)\\},$
$F(x_{1},y(\lambda))=\\{(0,0)\\};$
b2) if $y_{1}\leq y(\lambda)<x_{1},$ then,
$F(x_{1},y_{1})=S^{\ast}((0,0),x_{1}),$
$F(x_{1},y(\lambda))=S^{\ast}((0,0),x_{1});$
b3) if $y_{1}<x_{1}\leq y(\lambda),$ then,
$F(x_{1},y_{1})=S^{\ast}((0,0),x_{1}),$ $F(x_{1},y(\lambda))=\\{(0,0)\\}.$
Then, $F(x_{1},y_{1})\subset F(x_{1},y(\lambda))-(-R_{+}^{2})$ for each
$\lambda\in[0,1].$
c) We prove that $F(\cdot,y)$ is type-(iii) naturally
$-R_{+}^{2}-$quasiconcave on $[0,1]$ for each $y\in[0,1].$
Let $y\in[0,1]$ be fixed, $x_{1},x_{2}\in[0,1]$, $\lambda\in[0,1]$ and
$x(\lambda)=\lambda x_{1}+(1-\lambda)x_{2}$.
c1) If $x_{1}\geq x_{2}>y,$ $F(x_{1},y)=S^{\ast}((0,0),x_{1}),$
$F(x_{2},y)=S^{\ast}((0,0),x_{2})$,
$F(x(\lambda),y)=S^{\ast}((0,0),x(\lambda))$ and
co$\\{F(x_{1},y)\cup F(x_{2},y)\\}=S^{\ast}((0,0),x_{1})\subset
S^{\ast}((0,0),x(\lambda))-(-R_{+}^{2})=F(x(\lambda),y)-(-R_{+}^{2});$
c2) if $x_{1}\leq x_{2}\leq y,$ $F(x_{1},y)=\\{(0,0)\\},$
$F(x_{2},y)=\\{(0,0)\\}$, $F(x(\lambda),y)=\\{(0,0)\\}$ and
co$\\{F(x_{1},y)\cup F(x_{2},y)\\}=\\{(0,0)\\}\subset
F(x(\lambda),y)-(-R_{+}^{2});$
c3) if $x_{1}>y\geq x_{2},$ then, $F(x_{1},y)=S^{\ast}((0,0),x_{1}),$
$F(x_{2},y)=\\{(0,0)\\}$ and
co$\\{F(x_{1},y)\cup F(x_{2},y)\\}=S^{\ast}((0,0),x_{1});$
if $x_{1}\geq x(\lambda)>y\geq x_{2},$ then,
$F(x(\lambda),y)=S^{\ast}((0,0),x(\lambda))$ and
co$\\{F(x_{1},y)\cup F(x_{2},y)\\}=S^{\ast}((0,0),x_{1})\subset
F(x(\lambda),y)-(-R_{+}^{2});$
if $x_{1}>y\geq x(\lambda)\geq x_{2},$ then, $F(x(\lambda),y)=\\{(0,0)\\}$ and
co$\\{F(x_{1},y)\cup
F(x_{2},y)\\}=S^{\ast}((0,0),x_{1})\subset\\{(0,0)\\}-(-R_{+}^{2})=F(x(\lambda),y)-(-R_{+}^{2}).$
The following equality is true:
$\cup_{y\in X}F(x,y)=S^{\ast}((0,0),x)$ and, consequently, $\cup_{y\in
X}F(x,y)$ is a compact set.
All the assumptions of Theorem 4.2 are fulfilled, then, there exist the
elements $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+(-R_{+}^{2}).$
In our case, $\cup_{x\in X}F(x,x)=\\{(0,0)\\},$ Max$\overline{\cup_{x\in
X}F(x,x)}=\\{(0,0)\\},$ Max${}_{w}F(x,X)=\\{(0,0)\\}$ and
Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}=\\{(0,0)\\}.$ Then, taking
$z_{1}=(0,0)$ and $z_{2}=(0,0),$ we have that $z_{1}\in z_{2}+(-R_{+}^{2}).$
Considering Remark 4.2, we obtain the following result as a consequence of
Theorem 4.2, for the real-valued maps case.
Corollary 4.1 Let $X$ be a (n-1) dimensional simplex of a Hausdorff
topological vector space $E,Y$ a compact set in $\mathit{R}$ and let $S$ be a
pointed closed convex cone in $R$ with its interior int$S\neq\emptyset.$ Let
$F:X\times X\rightrightarrows Y$ be a set-valued map with non-empty values.
i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a compact set
for each $x\in X$. If $F$ is type-(iii) pair properly quasi-concave in the
second argument on $X\times X$ and $F(\cdot,y)$ is type-(iii) naturally
$S-$quasi-concave on $X$ for each $y\in X,$ then, there exist
$z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If $F$ is type-(iii) pair properly quasi-convex in the
first argument on $X\times X$ and $F(x,\cdot)$ is type-(iii) naturally
$S-$quasi-convex on $X$ for each $x\in X,$ then, there exist
$z_{1}\in$Min$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$ such that
$z_{1}\in z_{2}-S.$
Example 4.2 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[-1,1]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,1]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We notice that $F$ is not continuous on $X$ and it is $S-$transfer type-(v)
$\mu-$convex in the first argument.
a) In Example 3.5 we have seen that $F$ is type-(iii) pair properly
quasiconcave in the second argument on $X\times X.$
b) We prove that $F(\cdot,y)$ is type-(iii) naturally $S-$quasiconcave on $X$
for each $y\in X.$
Let $y\in[0,1]$ be fixed, $x_{1},x_{2}\in[0,1]$, $\lambda\in[0,1]$ and
$x(\lambda)=\lambda x_{1}+(1-\lambda)x_{2}$.
b1) If $x_{1}\geq x_{2}\geq y,$ $F(x_{1},y)=[-x_{1},1],$
$F(x_{2},y)=[-x_{2},1]$, $F(x(\lambda),y)=[-x(\lambda),1]$
and co$\\{F(x_{1},y)\cup
F(x_{2},y)\\}=[-x_{1},1]\subset[-x(\lambda),1]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
b2) if $x_{1}\leq x_{2}\leq y,$ $F(x_{1},y)=[-1,1],$ $F(x_{2},y)=[-1,1]$,
$F(x(\lambda),y)=[-1,1]$ and
co$\\{F(x_{1},y)\cup
F(x_{2},y)\\}=[-1,1]\subset[-1,1]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
b3) if $x_{1}\geq y\geq x_{2},$ then, $F(x_{1},y)=[-x_{1},1],$
$F(x_{2},y)=[-1,1]$ and
co$\\{F(x_{1},y)\cup F(x_{2},y)\\}=[-1,1];$
if $x_{1}\geq x(\lambda)\geq y\geq x_{2},$ then,
$F(x(\lambda),y)=[-x(\lambda),1]$ and
co$\\{F(x_{1},y)\cup
F(x_{2},y)\\}=[-1,1]\subset[-x(\lambda),1]-[0,\infty)=F(x(\lambda),y)-[0,\infty);$
if $x_{1}\geq y\geq x(\lambda)\geq x_{2},$ then, $F(x(\lambda),y)=[-1,1]$ and
co$\\{F(x_{1},y)\cup
F(x_{2},y)\\}=[-1,1]\subset[-1,1]-[0,\infty)=F(x(\lambda),y)-[0,\infty).$
The following equalities are true:
$\cup_{y\in X}F(x,y)=\cup_{y<x}[-x,1]\cup\cup_{y\geq
x}[-1,1]=[-x,1]\cup[-1,1]=[-1,1]$ and, consequently, $\cup_{y\in X}F(x,y)$ is
a compact set.
All the assumptions of Corollary 4.1 are fulfilled, then, there exist the
elements $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
In our case, $\cup_{x\in X}F(x,x)=[-1,1],$ Max$\overline{\cup_{x\in
X}F(x,x)}=\\{1\\},$ Max${}_{w}F(x,X)=\\{1\\}$ and Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}=\\{1\\}.$ Then, taking $z_{1}=1$ and $z_{2}=1,$ we
have that $z_{1}\in z_{2}+S.$
The next corollary is a particular case of Theorem 4.1.
Corollary 4.2 Let $X$ be a (n-1) dimensional simplex of a Hausdorff
topological vector space $E,$ $Y$ be a compact set in a Hausdorff topological
vector space $Z$ and let $S$ be a pointed closed convex cone in $Z$ with its
interior int$S\neq\emptyset.$ Let $f:X\times X\rightarrow Y$ be a vector-
valued mapping.
i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}f(x,y)$ is a compact set
for each $x\in X$. If the mapping $f$ is $S-$transfer $\mu-$convex in the
first argument on $X\times X$, pair properly quasi-concave in the second
argument on $X\mathit{\times X}$ and $f(\cdot,y)$ is naturally $S-$quasi-
concave on $X$ for each $y\in X,$ then, there exist
$z_{1}\in$Max$\overline{\cup_{x\in X}f(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}f(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}f(x,y)$ is a compact set
for each $y\in X$. If the mapping $f$ is $S$ transfer $\mu-$concave in the
second argument on $X\times X$, pair properly quasi-convex in the first
argument on $X\times X$ and $f(x,\cdot)$ is naturally $S$ quasi-convex on $X$
for each $x\in X,$ then, there exist $z_{1}\in$Min$\overline{\cup_{x\in
X}f(x,x)}$ and $z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}f(X,y)}$such
that $z_{1}\in z_{2}-S.$
We search to weaken the assumptions from Lemma 4.1, especially the
$S-$transfer $\mu-$convexity (resp. $S-$transfer $\mu-$concavity) and the
naturally $S-$quasi-concavity (resp. naturally $S-$quasi-convexity), but
another proving method needs to be used: we build a constant selection for a
set-valued map. This change requires a new condition instead of pair quasi-
convexity (resp. pair quasi-concavity), a condition we called $\gamma$ (resp.
$\gamma^{\prime}$). Under the condition $\gamma$ (resp.$\gamma^{\prime}$), the
assumption of transfer properly $S-$quasi-concavity (resp. transfer properly
$S-$quasi-convexity) proves to be necessary. The next Lemma is the key used in
order to obtain Theorem 4.3.
Lemma 4.2 Let $X$ be a convex set in a Hausdorff topological vector space $E,$
$Y$ a compact set in the Hausdorff topological vector space $Z$ and let $S$ a
pointed closed convex cone in $Z$ with its interior int$S\neq\emptyset.$ Let
$F:X\times X\rightrightarrows Y$ be a set-valued map with non-empty values.
(i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a compact set
for each $x\in X$. If $F$ is $S-$transfer weakly type-(v) $\mu-$convex in the
first argument on $X\times X$, transfer type-(iii) properly $S-$quasi-concave
in the first argument on $X\times X$ and satisfies the condition $\gamma,$
then there exists $x^{\ast}\in X$ such that
$F(x^{\ast},x^{\ast})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x^{\ast},y)\neq\emptyset.$
(ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If $F$ is transfer weakly type-(v) $\mu-$concave in the
second argument on $X\times X$, transfer type-(iii) properly $S-$quasi-convex
in the second argument on $X\times X,$ and satisfies the condition
$\gamma^{\prime}$, then, there exists $y^{\ast}\in X$ such that
$F(y^{\ast},y^{\ast})\cap$Min${}_{w}\cup_{x\in
X}F(x,y^{\ast})\neq\emptyset.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Proof. Let us define the set-valued map $T:X\rightrightarrows X$ by
$T(x)=\\{y\in X:F(x,y)\cap$Max${}_{w}\cup_{z\in X}F(x,z)\neq\emptyset\\}$ for
each $x\in X.$
We claim that $T$ is non-empty valued. Indeed, since $\cup_{z\in X}F(x,z)$ is
a compact set for each $x\in X,$ by Lemma 2.1, Max${}_{w}\cup_{z\in
X}F(x,z)\neq\emptyset.$ For each $x\in X,$ let $z_{x}\in$Max${}_{w}\cup_{z\in
X}F(x,z).$ Then, there exists $y_{x}\in X$ such that $z_{x}\in F(x,y_{x}).$ It
is clear that, $y_{x}\in T(x)=\\{y\in X:F(x,y)\cap$Max${}_{w}\cup_{z\in
X}F(x,z)\neq\emptyset\\}$ and consequently, $T(x)\neq\emptyset$ for each $x\in
X.$
Since $F$ satisfies the condition $\gamma$, there exist $n\in N,$
$(x_{1},y_{1}),(x_{2},y_{2})...,(x_{n},y_{n})\in X\times X$ and
$y^{\ast}\in$co$\\{x_{i}:i=1,2,...,n\\}$ such that $F(x_{i},y_{i})\subset
F(x_{i},y^{\ast})-S$ and $F(x_{i},y_{i})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i},z)\neq\emptyset$ for each $i\in\\{1,2,...,n\\}.$
Let us fix $i_{0}\in\\{1,2,...,n\\}.$ There exists $t_{i_{0}}\in
F(x_{i_{0}},y_{i_{0}})\cap$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z).$ This means
that $t_{i_{0}}\in F(x_{i_{0}},y_{i_{0}})$ and $\cup_{z\in
X}F(x_{i_{0}},z)\cap(t_{i_{0}}+$int$S)=\emptyset.$ There exists
$t_{i_{0}}^{\prime}\in F(x_{i_{0}},y^{\ast})$ and $s_{i_{0}}\in S$ such that
$t_{i_{0}}^{\prime}=t_{i_{0}}+s_{i_{0}}.$ Therefore,
$t_{i_{0}}^{\prime}\in\cup_{z\in X}F(x_{i_{0}},z)$ and, for each
$s^{\prime}\in$int$S,$ $(t_{i_{0}}^{\prime}+s^{\prime})\cap\cup_{z\in
X}F(x_{i_{0}},z)=(t_{i_{0}}+s_{i_{0}}+s^{\prime})\cap\cup_{z\in
X}F(x_{i_{0}},z)=\emptyset.$ It follows that
$(t_{i_{0}}^{\prime}+$int$S)\cap\cup_{z\in X}F(x_{i_{0}},z)=\emptyset$, and,
since $t_{i_{0}}^{\prime}\in\cup_{z\in X}F(x_{i_{0}},z),$ we have that
$t_{i_{0}}\in F(x_{i_{0}},y^{\ast})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z).$ We showed that $y^{\ast}\in T(x_{i_{0}}),$ and, since
$i_{0}$ is arbitrary and $y^{\ast}\in$co$\\{x_{i}:i=1,2,...,n\\}$, then,
$y^{\ast}\in\mathop{\textstyle\bigcap}\limits_{i=1}^{n}T(x_{i})\cap$co$\\{x_{i}:i=1,2,...,n\\}.$
Hence, $\mathop{\textstyle\bigcap}\limits_{i=1}^{n}T(x_{i})$ is non-empty.
Further, we will prove that $T$ is quasi-convex. By contrary, we assume that
$T$ is not quasi-convex. Then, suppose that there exists
$z^{\ast}\in\mathop{\textstyle\bigcap}\limits_{i=1}^{n}T(x_{i})$ and
$\lambda^{\ast}\in\Delta_{n-1}$ such that $z^{\ast}\notin
T(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i}),$ that is
$F(x_{i},z^{\ast})\cap$Max${}_{w}\cup_{z\in X}F(x_{i},z)\neq\emptyset$ for
each $i\in\\{1,2,...,n\\}$ and
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap$Max${}_{w}\cup_{z\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z)=\emptyset.$
Since $F$ is $S-$transfer weakly type-(v) $\mu-$convex in the first argument
and we also have
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{z\in
X}F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z)=\emptyset$, it
follows that, there exists $i_{0}\in I$ and $z_{i_{0}}\in X$ such that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i_{0}},z))\subset F(x_{i_{0}},z_{i_{0}})-$int$S.$
Let $t\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i_{0}},z))$ and let $u_{i_{0}}\in F(x_{i_{0}},z_{i_{o}})$ such that
$t=u_{i_{0}}-s_{i_{0}},$ $s_{i_{0}}\in$int$S.$ Since $t\in
F(x_{i_{0}},z_{i_{0}})-$int$S,$ it follows that
$u_{i_{0}}\in\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i_{0}},z)\cap\\{t+$int$S\\}\neq\emptyset,$ that is $t\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap(\mathop{\textstyle\bigcup}_{z\in
X}F(x_{i_{0}},z))$ implies the fact that $t\notin$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z).$
Consequently,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z)=\emptyset.$
We claim that $F(x_{i_{0}},z^{\ast})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z)=\emptyset.$ On the contrary, we assumethat there exists $t\in
F(x_{i_{0}},z^{\ast})$ such that $t\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z).$
Since $F$ is transfer type-(iii) properly $S-$quasi-concave in the first
argument, then, $t\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})-S.$ We
have $t=t^{\prime}-s_{0},$ where $t^{\prime}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})$ and
$s_{0}\in S,$ therefore $t^{\prime}=t+s_{0}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})$. Since
$t\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z),$
$F(x_{i_{0}},z)\cap\\{t+$int$S\\}=\emptyset.$ For each $s\in$int$S,$
$t^{\prime}+s=t+s_{0}+s\in t+$int$S$, which implies $t^{\prime}+s\notin
F(x_{i_{0}},z)$, that is,
$F(x_{i_{0}},z)\cap\\{t^{\prime}+$int$S\\}=\emptyset,$ or, equivalently,
$t^{\prime}\in$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z).$ We obtained
$t^{\prime}\in
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z^{\ast})\cap$Max${}_{w}\cup_{z\in
X}F(x_{i_{0}},z),$ which is a contradiction. It remains that
$F(x_{i_{0}},z^{\ast})\cap$Max${}_{w}\cup_{z\in X}F(x_{i_{0}},z)=\emptyset,$
and then, $z^{\ast}\notin T(x_{i_{0}}),$ which contradicts
$z^{\ast}\in\mathop{\textstyle\bigcap}\limits_{i=1}^{n}T(x_{i})$. Therefore,
$T$ is quasi-convex.
We proved that there exist the elements $x^{\ast},x_{1},x_{2},...,x_{n}\in X$
such that
$x^{\ast}\in\mathop{\textstyle\bigcap}\limits_{i=1}^{n}T(x_{i})\cap$co$\\{x_{i}:i=1,2,...,n\\}\subset
T(x)$ for each $x\in$co$\\{x_{i}:i=1,2,..,n\\}$, then, $x^{\ast}\in
T(x^{\ast}),$ that is,
$F(x^{\ast},x^{\ast})\cap$Max${}_{w}\mathop{\textstyle\bigcup}_{y\in
X}F(x^{\ast},y)\neq\emptyset.$
(ii) Let us define the set-valued map $Q:X\rightrightarrows X$ by
$Q(y)=\\{x\in X:F(x,y)\cap$Min${}_{w}\cup_{x\in X}F(x,y)\neq\emptyset\\}$ for
each $y\in X.$
Further, the proof follows a similar line as above and we conclude that there
exists $y^{\ast}\in Q(y^{\ast}),$ that is,
$F(y^{\ast},y^{\ast})\cap$Min${}_{w}\mathop{\textstyle\bigcup}_{x\in
X}F(x,y^{\ast})\neq\emptyset.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \square\vskip 6.0pt plus 2.0pt minus
2.0pt$
Remark 4.2. The transfer type-(iii) properly $S-$quasiconcavity in the first
argument of $F$ is a necessary condition for Lemma 4.2 i). In the following
example, we have that $F$ satisfies the condition $\gamma$, it is not transfer
type-(iii) properly $S-$quasiconcave in the first argument and the conclusion
of Lemma 4.2 i) is not fulfilled.
Let $X=[0,1],$ $Y=[0,1]$ and $F:X\times X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,1]\text{ if
}(x,y)\in[\frac{1}{4},\frac{3}{4}]\times\\{1\\}\cup([0,\frac{1}{4}]\cup[\frac{3}{4},1])\times\\{\frac{1}{2}\\};\\\
\\{(0\\}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise.}\end{array}\right.\vskip 6.0pt plus
2.0pt minus 2.0pt$
Remark 4.3. The two assumptions from Lemma 4.2 i), namely, the $S-$transfer
weakly type-(v) $\mu-$convexity in the first argument and the transfer
type-(iii) properly $S-$quasiconcavity of $F$ in the first argument on
$X\times X,$ imply the following:
for each $x_{1},x_{2},...,x_{n}\in X$ and $z\in X$ , there exists
$\lambda^{\ast}\in\Delta_{n-1},$ $i_{o}\in\\{1,2,...n\\}$ and $z_{i_{0}}\in X$
such that if
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z)\cap$Max${}_{y}\mathop{\textstyle\bigcup}_{y\in
X}F(x_{i_{0}},y)=\emptyset,$ it follows that
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z)\cap\mathop{\textstyle\bigcup}\limits_{y\in
X}F(x_{i_{0}},y)\subset
F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}^{\ast}x_{i},z_{i_{0}}).\vskip
6.0pt plus 2.0pt minus 2.0pt$
As a first application of the previous lemma, we obtain the following result,
which differs from Theorem 3.1 in [32] by the fact that the continuity
assumptions are dropped.
Theorem 4.3 Let $X$ be a convex set be in a Hausdorff topological vector space
$E,$ $Y$ a compact set in a Hausdorff topological vector space $Z$ and let $S$
be a pointed closed convex cone in $Z$ with its interior int$S\neq\emptyset.$
Let $F:X\times X\rightrightarrows Y$ be a set-valued map with non-empty
values.
i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a compact set
for each $x\in X$. If $F$ is $S-$transfer weakly type-(v) $\mu-$convex in the
first argument on $X\times X$, transfer type-(iii) properly $S-$quasi-concave
in the first argument on $X\times X$ and satisfies the condition $\gamma,$
then, there exist $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If $F$ is transfer weakly type-(v) $\mu-$concave in the
second argument on $X\times X$, transfer type-(iii) properly $S-$quasi-convex
in the second argument on $X\times X$ and satisfies the condition
$\gamma^{\prime},$ then there exist $z_{1}\in$Min$\overline{\cup_{x\in
X}F(x,x)}$ and $z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)\text{
}}$such that $z_{1}\in z_{2}-S.$
Proof. i) According to Lemma 4.2, in the case i) there exists $x^{\ast}\in X$
such that $F(x^{\ast},x^{\ast})\cap$Max${}_{w}\cup_{y\in
X}F(x^{\ast},y)\neq\emptyset$ and in the case ii), there exists $y^{\ast}\in
X$ such that $F(y^{\ast},y^{\ast})\cap$Min${}_{w}\cup_{x\in
X}F(x,y^{\ast})\neq\emptyset.$
Further, the proof is similar to the proof of Theorem 4.1. $\square$
If $F$ satisfies the property $\alpha$ (resp. $\alpha^{\prime}),$ we obtain
the following variant of Theorem 4.3.
Theorem 4.4 Let $X$ be a convex set be in a Hausdorff topological vector space
$E,$ $Y$ be a compact set in a Hausdorff topological vector space $Z$ and let
$S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with non-empty values.
i) Suppose that $F$ satisfies the property $\alpha$. If $F$ is transfer
type-(iii) properly $S-$quasi-concave in the first argument on $X\times X$ and
also satisfies the condition $\gamma,$ then, there exist
$z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that that $F$ satisfies the property $\alpha^{\prime}$. If $F$ is
transfer type-(iii) properly $S-$quasi-convex in the second argument on
$X\times X$ and also satisfies the condition $\gamma^{\prime},$ then there
exist $z_{1}\in$Min$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)\text{ }}$such that
$z_{1}\in z_{2}-S.$
We obtain the following corollary of Theorem 4.4, for the case of the real-
valued maps.
Corollary 4.3 Let $X$ be a convex set in a Hausdorff topological vector space
$E,Y$ be a compact set in $\mathit{R}$ and let $S$ be a pointed closed convex
cone in $R$ with its interior int$S\neq\emptyset.$ Let $F:X\times
X\rightrightarrows Y$ be a set-valued map with non-empty values.
i) Suppose that $\mathop{\textstyle\bigcup}_{y\in X}F(x,y)$ is a compact set
for each $x\in X$. If $F$ is transfer type-(iii) properly $S-$quasi-concave in
the first argument on $X\times X$ and satisfies the condition $\gamma,$ then,
there exist $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
ii) Suppose that $\mathop{\textstyle\bigcup}_{x\in X}F(x,y)$ is a compact set
for each $y\in X$. If $F$ is transfer type-(iii) properly $S-$quasi-convex in
the second argument on $X\times X$ and satisfies the condition
$\gamma^{\prime},$ then, there exist $z_{1}\in$Min$\overline{\cup_{x\in
X}F(x,x)}$ and $z_{2}\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$
such that $z_{1}\in z_{2}-S.$
Example 4.3 Let $X=[0,1],$ $Y=[-1,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,y]\text{ if }0\leq x\leq y\leq 1;\\\
[-x,y]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We notice that $F$ is not continuous on $X.$
According to Examples 3.7 and 3.8, $F$ is transfer type-(iii) properly
$S-$quasi-concave in the first argument on $X\times X$ and it has the property
$\gamma.$
All the assumptions of Corollary 4.3 are fulfilled, then, there exists the
elements $z_{1}\in$Max$\overline{\cup_{x\in X}F(x,x)}$ and
$z_{2}\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ such that
$z_{1}\in z_{2}+S.$
It is also true that:
$\cup_{x\in X}F(x,x)=\cup_{x\in X}[0,x]=[0,1];$ Max$\overline{\cup_{x\in
X}F(x,x)}=\\{1\\};$ Max${}_{w}F(x,X)=\\{1\\}$ and Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}=\\{1\\}.$
Then, taking $z_{1}=1$ and $z_{2}=1,$ we have $z_{1}\in z_{2}+S.$
We introduce the following definition which concerns the convexity properties
of set-valued maps with two variables. It will be used to obtain different
minimax inequalities.
Definition 4.1 Let $X$ be a (n-1) dimensional simplex of a Hausdorff
topological vector space $E,Y$ a subset of a Hausdorff topological vector
space $Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set valued map
with nonempty values.
$F$ is weakly $z-$convex on $X$ for $z\in A\subseteq Z$, iff for each $z\in A$
and $x_{1},...,x_{n}\in X,$ there exist $y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in
X$ and $g^{z}\in C^{\ast}(\Delta_{n-1})$ such that
$F(x_{i},y_{i}^{z})\cap(z+S)\neq\emptyset$ for each $i\in\\{1,2,...,n\\}$
implies
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\cap(z+S)\neq\emptyset$
for each $(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}.$
Example 4.4 Let $X=[0,1],$ $Y=[0,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,x]\text{ if }0\leq x\leq y\leq 1;\\\
[0,1]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
For each $z\in[0,1)$ and $x_{1},x_{2},...,x_{n}\in X,$ there exists
$y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in X$ with $0\leq x_{i}\leq y_{i}^{z}$ for
each $i\in\\{1,2,...,n\\},$ such that
$F(x_{i},y_{i}^{z})\cap(z+S)=[0,x_{i}]\cap[z,\infty)\neq\emptyset$. It follows
that $z\leq$min${}_{i=1,...,n}\\{x_{i}\\}.$ Consequently,
$z\leq\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}$ and $0\leq
x_{i}\leq\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z}$ for
each $i\in\\{1,2,...,n\\}$, $g^{z}\in C^{\ast}(\Delta_{n-1})$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$. Then,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})=[0,\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}].$
Hence,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\cap(z+S)=[0,\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}]\cap[z,\infty)\neq\emptyset.$
For $z=1$ and for any $x_{1},x_{2},...,x_{n}\in X,$ there exists
$y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in X$ with $0\leq y_{i}^{z}<x_{i}$ for each
$i\in\\{1,2,...,n\\},$ such that
$F(x_{i},y_{i}^{z})\cap(z+S)=[0,1]\cap[1,\infty)\neq\emptyset$ for each
$i\in\\{1,2,...,n\\}.$ We have that
$0\leq\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z}<x_{i}$
for each $i\in\\{1,2,...,n\\}$, $g_{i}^{z}\in C^{\ast}(\Delta_{n-1})$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ and then,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})=[0,1].$
Therefore,
$F(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\cap(z+S)=[0,1]\cap[1,\infty)\neq\emptyset.\vskip
6.0pt plus 2.0pt minus 2.0pt$
If $f$ is a mapping, we obtain the following definition.
Definition 4.2 Let $X$ be a (n-1) dimensional simplex of a Hausdorff
topological vector space $E,Y$ a subset of a Hausdorff topological vector
space $Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $f:X\times X\rightarrow Y$ be a mapping.
$f$ is weakly $z$-convex on $X$ for $z\in A\subseteq Z$, iff for each $z\in A$
and $x_{1},...,x_{n}\in X,$ there exist $y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in
X$, $g^{z}\in C^{\ast}(\Delta_{n-1})$ such that $f(x_{i},y_{i}^{z})\in z+S$
for each $i\in\\{1,2,...,n\\}$ implies
$f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\in
z+S$ for each $(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}.$
Example 4.5 Let $X=[0,1],$ $Y=[0,1]\times[0,1],$ $S=R_{+}^{2}$ and $f:X\times
X\rightarrow Y$ be defined by $f(x,y)=\left\\{\begin{array}[]{c}(x,y)\text{ if
}0\leq x\leq y\leq 1;\\\ (1,y)\text{ if }0\leq y<x\leq 1.\end{array}\right.$
For each $z=(z^{\prime},z^{\prime\prime})\in[0,1)\times[0,1]$ and
$x_{1},x_{2},...,x_{n}\in X,$ there exists $y_{1}^{z},y_{2}^{z},...,$
$y_{n}^{z}\in X$ with $0\leq x_{i}\leq y_{i}^{z}$ for each
$i\in\\{1,2,...,n\\}$ such that
$(x_{i},y_{i}^{z})=f(x_{i},y_{i}^{z})\in(z+S)=[z^{\prime},\infty)\times[z^{\prime\prime},\infty).$
It follows that $z^{\prime}\leq$min${}_{i=1,...,n}\\{x_{i}\\}.$ Consequently,
$z^{\prime}\leq\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i}$ and $0\leq
x_{i}\leq\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z}$ for
each $i\in\\{1,2,...,n\\}$, $g^{z}\in C^{\ast}(\Delta_{n-1})$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ and then,
$(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})=f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\in(z+S)=[z^{\prime},\infty)\times[z^{\prime\prime},\infty).$
For $z=(1,y)$ with $y\in[0,1)$ and for any $x_{1},x_{2},...,x_{n}\in X,$ there
exists $y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in X$ with $0\leq y_{i}^{z}<x_{i}$
for each $i\in\\{1,2,...,n\\}$ such that
$(1,y_{i}^{z})=f(x_{i},y_{i}^{z})\in(z+S)=[1,\infty)\times[y,\infty).$ We have
that
$0\leq\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z}<x_{i}$
for each $i\in\\{1,2,...,n\\}$, $g^{z}\in C^{\ast}(\Delta_{n-1})$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$ and then,
$(1,\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})=f(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i},\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z})\in(z+S)=[1,\infty)\times[y,\infty).\vskip
6.0pt plus 2.0pt minus 2.0pt$
Theorem 4.5 is a minimax theorem in which the set-valued map satisfies the
above defined property.
Theorem 4.5 Let $X$ be a (n-1) dimensional simplex of a Hausdorff topological
vector space $E,$ $Y$ a compact set in a Hausdorff topological vector space
$Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with non-empty values such that $\cup_{x\in X}F(x,x)$ and $\cup_{y\in
X}F(x,y)$ are compact sets for each $x\in X$. Suppose the following conditions
are fulfilled:
(i) $\mathit{F}$ is weakly $z-$convex for each $z\in$Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)};$
(ii) for each $x\in X,$ Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}\subset F(x,X)-S.$
Then, Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}\subset$Max$\cup_{x\in
X}F(x,x)-S.$
Proof. According to assumptions and Lemma 2.1, Max${}_{w}F(x,X)\neq\emptyset$
for each $x\in X$ and Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}\neq\emptyset.$
Let $z\in$Min$\overline{\cup_{x\in X}\text{Max}_{w}F(x,X)}$ and let us define
the set-valued map $T:X\rightrightarrows X$ by $T(x)=\\{y\in
X:F(x,y)\cap(z+S)\neq\emptyset\\}$ for each $x\in X.$ According to assumption
(ii), it follows that $T(x)$ is nonempty for each $x\in X.$
According to Assumption (i), we have that $T$ is weakly naturally quasi-
convex: for any $x_{1},x_{2},...,x_{n}\in X$ and $z\in Y,$ there exist
$y_{1}^{z},y_{2}^{z},...,y_{n}^{z}\in X$, $g^{z}\in C^{\ast}(\Delta_{n-1}),$
such that, if $y_{i}\in T(x_{i})$ for each $i\in\\{1,2,...,n\\},$ then,
$\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})y_{i}^{z}\in
T(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}x_{i})$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}.$
Therefore, according to the fixed point Theorem 2.1, there exists $x^{\ast}\in
T(x^{\ast}),$ that is, $F(x^{\ast},x^{\ast})\cap(z+S)\neq\emptyset.$ Then,
according to Lemma 2.1, we have $z\in F(x^{\ast},x^{\ast})-S\subset\cup_{x\in
X}F(x,x)-S\subset$Max$\cup_{x\in X}F(x,x)-S.$ $\square$
Example 4.6 Let $X=[0,1],$ $Y=[0,1],$ $S=[0,\infty)$ and $F:X\times
X\rightrightarrows Y$ be defined by
$F(x,y)=\left\\{\begin{array}[]{c}[0,x]\text{ if }0\leq x\leq y\leq 1;\\\
[0,1]\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We saw in Example 4.2 that $F$ is weakly $z-$convex for each $z\in Z.$
Further, we have that, $\cup_{x\in X}F(x,x)=\cup_{x\in X}[0,x]=[0,1]$ and for
each $x\in X,$ $\cup_{y\in X}F(x,y)=[0,1]$, so that, $\cup_{x\in X}F(x,x)$ and
$\cup_{y\in X}F(x,y)$ are compact sets, for each $x\in X.$
It is also true that Max${}_{w}F(x,X)=\\{1\\}$ and Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}=\\{1\\}.$
$F(x,X)-S=(-\infty,1]$ and then, for each $x\in X,$ Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}$
$\subset F(x,X)-S.$ All the assumptions of Theorem 4.4 are fulffiled.
Then, $\\{1\\}=$Min$\overline{\cup_{x\in
X}\text{Max}_{w}F(x,X)}\subset$Max$\cup_{x\in X}F(x,x)-S=(-\infty,1].\vskip
6.0pt plus 2.0pt minus 2.0pt$
The next corollary is obtained by considering single valued mappings, as a
particular case, in Theorem 4.5.
Corollary 4.4 Let $X$ be an (n-1) dimensional simplex of a Hausdorff
topological vector space $E,Y$ a compact set in a Hausdorff topological vector
space $Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $f:X\times X\rightarrow Y$ be a mapping such that
$\cup_{y\in X}f(x,y)$ and $\cup_{x\in X}f(x,x)$ are compact sets for each
$x\in X$. Suppose the following conditions are fulfilled:
(i) $\mathit{f}$ is weakly $z-$convex for each $z\in$Min$\overline{\cup_{x\in
X}\text{Max}_{w}f(x,X)};$
(ii) for each $x\in X,$ Min$\overline{\cup_{x\in
X}\text{Max}_{w}f(x,X)}\subset f(x,X)-S.$
Then, Min$\overline{\cup_{x\in X}\text{Max}_{w}f(x,X)}\subset$Max$\cup_{x\in
X}f(x,x)-S.$
Example 4.7 Let $X=[0,1],$ $Y=[0,1]\times[0,1],$ $S=IR_{+}^{2}$ and $f:X\times
X\rightarrow Y$ be defined by $f(x,y)=\left\\{\begin{array}[]{c}(x,y)\text{ if
}0\leq x\leq y\leq 1;\\\ (1,1)\text{ if }0\leq y<x\leq 1.\end{array}\right.$
We notice that $f$ is not continuous. The mapping $f$ is weakly $z-$convex for
each $z\in Y$. According to the definition of $f$, $\cup_{y\in
X}f(x,y)=\\{x\\}\times[x,1]\cup\\{(1,1)\\}$ and $\cup_{x\in
X}f(x,x)=\\{(x,x):x\in[0,1]\\},$ which are compact sets.
The following equalities take place:
Max${}_{w}\cup_{y\in X}f(x,y)=\\{1\\}\times[0,1]\cup[0,1]\times\\{1\\}$ and
Min$\overline{\cup_{x\in
X}\text{Max}_{w}f(x,X)}=\\{1\\}\times[0,1]\cup[0,1]\times\\{1\\}.$
Finally, we have that, for each $x\in X,$ Min$\overline{\cup_{x\in
X}\text{Max}_{w}f(x,X)}\subset f(x,X)-S$ and then, all the assumptions of the
Corrollary are satisfied.
Hence, Min$\overline{\cup_{x\in X}\text{Max}_{w}f(x,X)}\subset$Max$\cup_{x\in
X}f(x,x)-S.$
Another result is obtained in the same context of Theorem 4.5.
Theorem 4.6 Let $X$ be an (n-1) dimensional simplex of a Hausdorff topological
vector space $E,Y$ a compact set in a Hausdorff topological vector space $Z$
and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $F:X\times X\rightrightarrows Y$ be a set-valued map
with non-empty values such that $\cup_{x\in X}F(x,x)$ and $\cup_{y\in
X}F(x,y)$ are compact sets for each $x\in X$. Suppose the following conditions
are fulfilled:
(i) $\mathit{F}$ is weakly $z-$convex for each $z\in$Max$\overline{\cup_{y\in
X}\text{Min}_{w}F(X,y)};$
(ii) for each $x\in X,$ Max$\overline{\cup_{y\in
X}\text{Min}_{w}F(X,y)}\subset F(X,y)+S.$
Then, Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}\subset$Min$\cup_{x\in
X}F(x,x)+S.$
Proof. According to the assumptions and Lemma 2.1,
Min${}_{w}F(X,y)\neq\emptyset$ for each $y\in X$ and Max$\overline{\cup_{y\in
X}\text{Min}_{w}F(X,y)}.$
Let $z\in$Max$\overline{\cup_{y\in X}\text{Min}_{w}F(X,y)}$ and let us define
the set-valued map $Q:X\rightrightarrows X$ by $Q(y)=\\{x\in
X:F(x,y)\cap(z-S)\neq\emptyset\\}$ for each $y\in X.$ According to the
assumption (ii), it follows that $Q(y)$ is non-empty for each $y\in X.$
Accordint to the Assumption (i), we have that $Q$ is weakly naturally quasi-
convex: for any $y_{1},y_{2},...,y_{n}\in X$ and $z\in Y,$ there exist
$x_{1}^{z},x_{2}^{z},...,x_{n}^{z}\in X$ and $g^{z}\in
C^{\ast}(\Delta_{n-1}),$ such that, if $x_{i}\in Q(y_{i})$ for each
$i\in\\{1,2,...,n\\},$ then,
$\mathop{\textstyle\sum}_{i=1}^{n}g_{i}^{z}(\lambda_{i})x_{i}^{z}\in
Q(\mathop{\textstyle\sum}_{i=1}^{n}\lambda_{i}y_{i})$ for each
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}.$
Therefore, according to the fixed point Theorem 2.1, there exists $y^{\ast}\in
Q(y^{\ast}),$ that is, $F(y^{\ast},y^{\ast})\cap(z-S)\neq\emptyset.$ According
to Lemma 2.1, we have that $z\in F(y^{\ast},y^{\ast})+S\subset\cup_{x\in
X}F(x,x)+S\subset$Min$\cup_{x\in X}F(x,x)+S.$ $\square$
The last result from this paper is stated now.
Corollary 4.5 Let $X$ be an (n-1) dimensional simplex of a Hausdorff
topological vector space $E,Y$ a compact set in a Hausdorff topological vector
space $Z$ and let $S$ be a pointed closed convex cone in $Z$ with its interior
int$S\neq\emptyset.$ Let $f:X\times X\rightarrow Y$ be a mapping such that
$\cup_{y\in X}f(x,y)$ and $\cup_{x\in X}f(x,x)$ are compact sets for each
$x\in X$. Let us suppose that the following conditions are fulfilled:
(i) $\mathit{f}$ is weakly $z-$convex for each $z\in$Max$\overline{\cup_{y\in
X}\text{Min}_{w}f(X,y)};$
(ii) for each $x\in X,$ Max$\overline{\cup_{y\in
X}\text{Min}_{w}f(X,y)}\subset f(X,y)+S.$
Then, Max$\overline{\cup_{y\in X}\text{Min}_{w}f(X,y)}\subset$Min$\cup_{x\in
X}f(x,x)+S.$
Concluding Remarks
We have proven the existence of equilibria in minimax inequalities without
assuming any form of continuity of functions or set-valued maps. New
conditions of convexity have been introduced. The main tools to prove our
results have been a fixed-point theorem for weakly naturally quasi-concave set
valued maps and a constant selection for quasi-convex set-valued maps. Several
examples have been provided in order to illustrate our results.
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|
arxiv-papers
| 2013-04-01T11:46:10 |
2024-09-04T02:49:43.711071
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1304.0339"
}
|
1304.0375
|
11institutetext: University of Bucharest 11email: [email protected]
# Existence of equilibrium for an abstract economy with private information
and a countable space of actions
Monica Patriche University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, 010014 Bucharest, Romania
###### Abstract
We define the model of an abstract economy with private information and a
countable set of actions. We generalize the H. Yu and Z. Zhang’s model (2007),
considering that each agent is characterised by a preference correspondence
instead of having an utility function. We establish two different equilibrium
existence results.
###### Keywords:
private information, upper semicontinuous correspondences, abstract economy,
equilibrium.
2000 Mathematics Subject Classification: 47H10, 55M20, 91B50.
## 1 INTRODUCTION
We define the model of an abstract economy with private information and a
countable set of actions. The preference correspondences need not to be
represented by utility functions. The equilibrium concept is an extension of
the deterministic equilibrium. We present the H. Yu and Z. Zhang’s model in
[18], in which the agents maximize their expected utilities. Our model is a
generalization of H. Yu and Z. Zhang’s one.
A purpose in this paper is to prove the existence of equilibrium for an
abstract economy with private information and a countable set of actions. The
assumptions on correspondences refer to upper semicontinuity and
measurability.
The existence of pure strategy equilibrium for a game with finitely many
players, finite action space and diffuse and independent private information
was first proved by Radner and Rosenthal [15]. This result was extended by
Khan and Sun [10] to the case of a finite game with diffuse and independent
private information and with countable compact metric spaces as their action
spaces. These authores have shown in [8] that Radner and Rosenthal’s result
can not be extended to a general action space. We quote the papers of M.A.
Khan, K. Rath, Y. Sun [7], [8], [9] and M.A. Khan, Y. Sun [11], [12],
concerning this subject of research. In [19], H. Yu and Z. Zhang showed the
existence of pure strategy equilibrium for games with countable complete
metric spaces and worked with compact-valued correspondences. They relied on
the Bollobás and Varopulos’s extension [2] of the marriage lemma to construct
a theory of the distribution of a correspondence from an atomless probability
space to a countable complete metric space. They also studied the case of the
game with a continuum of players.
The classical model of Nash [14] was generalized by many authors. Models were
proposed in his pioneering works by Debreu [3] or later by A. Borglin and H.
Keiding [2], Shafer and Sonnenschein [16], Yannelis and Prahbakar [18].
Yannelis and Prahbakar developed new tehniques of work for showing the
existence of equilibrium. That is the reason for what we defined a new model
that can be integrated in this direction of development of the game theory. We
use the fixed point method of finding the equilibrium, precisely we use Ky Fan
fixed point theorem for upper semicontinuous correspondences.
The paper is organised as follows: In section 2, some notation and
terminological convention are given. In section 3, H. Yu and Z. Zhang’s
expected utility model with a finite number of agents and private information
and their main result in [19] are presented. Section 4 introduces our model,
that is, an abstract economy with private information and a countable space of
actions. Section 5 contains existence results for upper semicontinuous
correspondences.
## 2 PRELIMINARIES AND NOTATION
Throughout this paper, we shall use the following notations and definitions:
Let $A$ be a subset of a topological space $X$.
1. 1.
$\tciFourier(A)$ denotes the family of all non-empty finite subsets of $A$.
2. 2.
$2^{A}$ denotes the family of all subsets of $A$.
3. 3.
cl $A$ denotes the closure of $A$ in $X$.
4. 4.
If $A$ is a subset of a vector space, co$A$ denotes the convex hull of $A$.
5. 5.
If $F$, $G:$ $X\rightarrow 2^{Y}$ are correspondences, then co$G$, cl $G$,
$G\cap F$ $:$ $X\rightarrow 2^{Y}$ are correspondences defined by
$($co$G)(x)=$co$G(x)$, $($cl$G)(x)=$cl$G(x)$ and $(G\cap F)(x)=G(x)\cap F(x)$
for each $x\in X$, respectively.
Definition 1. Each correspondence $F:$ $X\rightarrow 2^{Y}$ has two natural
inverses:
1. 1.
the upper inverse $F^{u}$ (also called the strong inverse) of a subset $A$ of
$Y$ is defined by $F^{u}(A)=\left\\{x\in A:F(x)\subset A\right\\}.$
2. 2.
the lower inverse $F^{l}$ (also called the weak inverse) of a subset $A$ of
$Y$ is defined by $F^{l}(A)=\left\\{x\in A:F(x)\cap
A\not=\emptyset\right\\}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Definition 2. Let $X$, $Y$ be topological spaces and $F:X\rightarrow 2^{Y}$ be
a correspondence.
1\. $F$ is said to be upper semicontinuous if for each $x\in X$ and each open
set $V$ in $Y$ with $F(x)\subset V$, there exists an open neighborhood $U$ of
$x$ in $X$ such that $F(y)\subset V$ for each $y\in U$.
2\. $F$ is said to be lower semicontinuous (l.s.c) if for each $x\in X$ and
each open set $V$ in $Y$ with $F(x)\cap V\neq\emptyset$, there exists an open
neighbourhood $U$ of $x$ in $X$ such that $F(y)\cap V\neq\emptyset$ for each
$y\in U$.
Lemma 1 [20]. Let $X$ and $Y$ be two topological spaces and let $A$ be a
closed (resp. open) subset of $X.$ Suppose $F_{1}:X\rightarrow 2^{Y}$,
$F_{2}:X\rightarrow 2^{Y}$ are lower semicontinuous (resp. upper
semicontinuous) such that $F_{2}(x)\subset F_{1}(x)$ for all $x\in A.$ Then
the correspondence $F:X\rightarrow 2^{Y}$ defined by
$\mathit{F(x)=}\left\\{\begin{array}[]{c}F_{1}(x)\text{, \ \ \ \ \ \ \ if
}x\notin A\text{, }\\\ F_{2}(x)\text{, \ \ \ \ \ \ \ \ \ if }x\in
A\end{array}\right.$
is also lower semicontinuous (resp. upper semicontinuous).
Definition 3 Let $(T$, $\mathcal{T})$ be a measurable space, $Y$ a topological
space and $F:T\rightarrow 2^{Y}$ a corespondence.
1. 1.
$F$ is weakly measurable if $F^{l}(A)\in\mathcal{T}$ for each open subset $A$
of $Y;$
2. 2.
$F$ is measurable if $F^{l}(A)\in\mathcal{T}$ for each closed subset $A$ of
$Y.$
Remark. Let $(T$, $\mathcal{T})$ be a measurable space, $Y$ a countable set
and $F:T\rightarrow 2^{Y}$ a corespondence. Then $F$ is measurable if for each
$y\in Y,F^{-1}(y)=\left\\{t\in T:y\in F(t)\right\\}$ is
$\mathcal{T-}$measurable.
Lemma 2 [1]. For a correspondence $F:T\rightarrow 2^{Y}$ from a measurable
space into a metrizable space we have the following:
1. 1.
If $F$ is measurable, then it is also weakly measurable;
2. 2.
If $F$ is compact valued and weakly measurable, it is measurable.
Definition 4 [19]. Let $Y$ be a countable complete metric space, $(T$,
$\mathcal{T}$, $\lambda)$ an atomless probability space and $F:T\rightarrow
2^{Y}$ a measurable corespondence. The function $f:T\rightarrow Y$ is said to
be a selection of $F$ if $f(t)\in F(t)$ for $\lambda-$almost $t\in T.$ Denote
$\mathcal{D}_{F}=\left\\{\lambda f^{-1}:f\text{ is a measurable selection of
}F\right\\}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Lemma 3 [19]. Let $Y$ be a countable complete metric space, $(T$,
$\mathcal{T}$, $\lambda)$ an atomless probability space and $F:T\rightarrow
2^{Y}$ a measurable corespondence. Then $\mathcal{D}_{F}$ is nonempty and
convex in the space $\mathcal{M}(Y)$ \- the space of probability measure on
$Y$, equipped with the topology of weak convergence.
Lemma 4 [19]. Let $Y$ be a countable complete metric space, and $(T$,
$\mathcal{T}$, $\lambda)$ be an atomless probability space and $F:T\rightarrow
2^{Y}$ be a measurable corespondence. If $F$ is compact valued, then
$\mathcal{D}_{F}$ is compact in $\mathcal{M}(Y).\vskip 6.0pt plus 2.0pt minus
2.0pt$
Lemma 5 [19]. Let $X$ be a metric space, $(T$, $\mathcal{T}$, $\lambda)$ be an
atomless probability space, $Y$ be a countable complete metric space and
$F:T\times X\rightarrow 2^{Y}$ a correspondence. Assume that for any fixed $x$
in $X$, $F(\cdot,x)$ (also denoted by $F_{x})$ is a compact-valued measurable
correspondence, and for each fixed $t\in T,$ $F(t,\cdot)$ is upper
semicontinuous on $X$. Also, assume that there exists a compact valued
corespondence $H:T\times X\rightarrow 2^{Y}$ such that $F(t,x)\subset H(t)$
for all $t$ and $x$. Then $\mathcal{D}_{F_{x}}$ is upper semicontinuous on
$X.$
Theorem 1 (Kuratowski-Ryll-Nardzewski Selection Theorem) [1]. A weakly
measurable correspondence with nonempty closed values from a measurable space
into a Polish space admits a measurable selector.
## 3 A Nash Equilibrium Existence Theorem
We present Yu and Zhang’s model of a finite game with private information. In
this model it is assigned to each agent a private information related to his
action and payoff described by the random mappings $\tau_{i}$ and $\chi_{i},$
mappings defined on $(\Omega,\mathcal{F})=\underset{i\in
I}{(\prod}(Z_{i},X_{i}),\underset{i\in
I}{\prod}(\mathcal{Z}_{i},\mathcal{X}_{i})),$ where $(X_{i}$,
$\mathcal{X}_{i})$ and $(Z_{i},\mathcal{Z}_{i})$ are measurable spaces. For a
point $\omega=(z_{1},x_{1},...,z_{n},x_{n})\in\Omega,$ $\tau_{i}$ and
$\chi_{i}$ are the coordinate projections
$\tau_{i}(\omega)=z_{i}$, $\chi_{i}(\omega)=x_{i}.$ Each player $i$ in $I$
first observes the realization, say $z_{i}\in Z_{i},$ of the random element
$\tau_{i}(\omega),$ then chooses his own action from a nonempty compact subset
$D_{i}(z_{i})$ of a countable complete metric space $A_{i},$ with
$D_{i}(\cdot)$ measurable. The payoff of each player $i$ is given by the the
utility function $u_{i}:A\times X_{i}\rightarrow\mathbb{R},$ where
$A=\underset{j\in I}{\prod}A_{j}$ is the set of of all combinations of all
players’ moves. Let $\mu$ be a probability measure on $\Omega.$ It is assumed
the following uniform integrability condition (UI):
(UI) For every $i\in I,$ there is a real-valued integrable function $h_{i}:$
$\Omega\rightarrow\mathbb{R}$ such that $\mu-$almost all $\omega\in\Omega,$
$\mid u_{i}(a,\chi_{i}(\omega))\mid\leq h_{i}(\omega)$ holds for all $a\in A.$
Definition 5 [19]. A finite game with private information is a family
$\Gamma=(I,((Z_{i},\mathcal{Z}_{i}),(X_{i},\mathcal{X}_{i}),(A_{i},D_{i}),u_{i})_{i\in
I},\mu)$.
For each $i\in I,$ let meas($Z_{i},D_{i})$ be the set of measurable mappings
$f$ from $Z_{i}$ to $A_{i}$ such that $f(z_{i})\in D_{i}(z_{i})$ for each
$z_{i}\in Z_{i}.$ An element $g_{i}$ of meas($Z_{i},A_{i})$ is called a pure
strategy for player $i.$ A pure strategy profile $g$ is an n-vector function
$(g_{1},g_{2},...,g_{n})$ that specifies a pure strategy for each player.
Definition 6 [19]. For a pure strategy profile $g=(g_{1},g_{2},...,g_{n}),$
the expected payoff for player $i$ is
$U_{i}(g)=\mathop{\textstyle\int}\limits_{\omega\in\Omega}u_{i}(g_{1}(\tau_{1}(\omega)),...,g_{n}(\tau_{n}(\omega)),\chi_{i}(\omega))\mu
d(\omega).\vskip 6.0pt plus 2.0pt minus 2.0pt$
Definition 7 [19]. A Nash equilibrium in pure strategies is defined as a pure
strategy profile $(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ such that for
each player $i\in I$
$U_{i}(g^{\ast})\geq U_{i}(g_{i},g_{-i}^{\ast})$ for all
$g_{i}\in$Meas$(Z_{i},D_{i}).\vskip 6.0pt plus 2.0pt minus 2.0pt$
The following theorem is the main result of Zhang in [19].
Theorem 2 Suppose that for every player $i,$ the compact valued $D_{i}$
corespondence is measurable, and
a) the distribution $\mu\tau_{i}^{-1}$ of $\tau_{i}$ is an atomless measure;
b) the random elements $\left\\{\tau_{j}:j\not=i\right\\}$ together with the
random element $\xi_{i}\equiv(\tau_{i},\chi_{i})$ form a mutually independent
set;
c) for any fixed $x_{i}\in X_{i},$ $u_{i}(\cdot,x_{i})$ is a continuous
function on $A;$ for any fixed $a\in A,$ $u_{i}(a,\cdot)$ is a measurable
function on $(X_{i},\mathcal{X}_{i});$
d) the uniform integrability condition (UI) holds.
Then the game $\Gamma$ has a Nash equilibrium in pure strategies.
## 4 The Model of an abstract economy with private information
In this section we define a model of abstract economy with private information
and a countable set of actions. We also prove the existence of equilibrium of
abstract economies.
Let $I$ be a nonempty and finite set (the set of agents). For each $i\in I$,
the space of actions, $A_{i}$ is a countable complete metric space and
$(Z_{i},\mathcal{Z}_{i})$ is measurable space. Let $(\Omega,\mathcal{F})$ be
the product measurable space$,\underset{i\in I}{(\prod}Z_{i},\underset{i\in
I}{\prod}\mathcal{Z}_{i})$, and $\mu$ a probability measure on
$(\Omega,\mathcal{F}).$ For a point $\omega=(z_{1},...,z_{n})\in\Omega,$
define the coordinate projections
$\tau_{i}(\omega)=z_{i}.$
The random mapping $\tau_{i}(\omega)$ is interpreted as player i’s private
information related to his action.
For each $i\in I,$ we also denote by meas($Z_{i},A_{i})$ the set of measurable
mappings $f$ from $Z_{i}$ to $A_{i}.$ An element $g_{i}$ of
meas($Z_{i},A_{i})$ is called a pure strategy for player $i.$ A pure strategy
profile $g$ is an n-vector function $(g_{1},g_{2},...,g_{n})$ that specifies a
pure strategy for each player.
We suppose that there exists a correspondence $D_{i}:Z_{i}\rightarrow
2^{A_{i}}$ such that each agent $i$ can choose an action from
$D_{i}(z_{i})\subset A_{i}$ for each $z_{i}\in Z_{i}.$
Let $D_{\mathcal{D}_{i}}$ be the set
$\left\\{(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}\text{ is a measurable selection of
}D_{i}\right\\}.$
For each $i\in I,$ let the constraint correspondence be
$\alpha_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}D_{D_{i}}\rightarrow 2^{A_{i}}$, such that
$\alpha_{i}(z_{i},(\mu\tau_{1}^{-1})g_{1}^{-1},(\mu\tau_{2}^{-1})g_{2}^{-1},...,(\mu\tau_{n}^{-1})g_{n}^{-1})\subset
A_{i}$ and the preference correspondence is
$P_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in I}D_{D_{i}}\rightarrow
2^{A_{i}}$, such that $P_{i}(z_{i},(\mu\tau_{1}^{-1})g_{1}^{-1},$
$(\mu\tau_{2}^{-1})g_{2}^{-1},...,(\mu\tau_{n}^{-1})g_{n}^{-1})\subset A_{i}.$
Definition 8. An abstract economy (or a generalized game) with private
information and a countable space of actions is defined as
$\Gamma=(I,((Z_{i},\mathcal{Z}_{i}),\newline (A_{i},\alpha_{i},P_{i}))_{i\in
I},\mu)$.
Definition 9. An equilibrium for $\Gamma$ is defined as a strategy profile
$(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(Z_{i},D_{i})$ such that for each $i\in I:$
1)
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})$
for each $z_{i}\in Z_{i};$
2)
$\alpha_{i}(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})\cap$
$P_{i}(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})=\phi$
for each $z_{i}\in Z_{i}.$
.
## 5 Existence of equilibrium for abstract economies with private information
We state some new equilibrium existence theorems for abstract economies.
Theorem 3 is an existence theorem of equilibrium for an abstract economy with
upper semicontinuous correspondences $\alpha_{i}$ and $P_{i}.$
Theorem 3. Let
$\Gamma=(I,((Z_{i},\mathcal{Z}_{i}),(A_{i},\alpha_{i},P_{i}))_{i\in I},\mu)$
be an abstract economy with private information and a countable space of
action, where $I$ is a finite index set such that for each $i\in I:$
a) $A_{i}$ is a countable complete metric space and $(Z_{i},\mathcal{Z}_{i})$
is a measurable space; $(\Omega,F)$ is the product measurable space
$\underset{i\in I}{(\prod}(Z_{i},\mathcal{Z}_{i}))$ and $\mu$ an atomless
probability measure on $(\Omega,F);$
b) the correspondence $D_{i}:Z_{i}\rightarrow 2^{A_{i}}$ is measurable with
compact values;
c) the correspondence
$\alpha_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ is measurable with respect to
$z_{i}$ and, for all $z_{i}\in Z_{i},$
$\alpha_{i}(z_{i},\cdot,\cdot,...,\cdot)$ is upper semicontinuous with
nonempty, compact values;
d) the correspondence $P_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ is measurable with respect to
$z_{i}$ and, for all $z_{i}\in Z_{i},$ $P_{i}(z_{i},\cdot,\cdot,...,\cdot)$ is
upper semicontinuous with nonempty, compact values;
e) for each $z_{i}\in Z_{i}$ and each
$(g_{1},g_{2},...,g_{n})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(Z_{i},A_{i}),$
$g_{i}(z_{i})\not\in
P_{i}(z_{i},(\mu\tau_{1}^{-1})g_{1}^{-1},(\mu\tau_{2}^{-1})g_{2}^{-1},...,(\mu\tau_{n}^{-1})g_{n}^{-1});$
f) the set $U_{i}:=$
$\left\\{(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in
Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}:(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\emptyset\right\\}$ is
open for each $z_{i}\in Z_{i}$.
Then there exists
$(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(Z_{i},A_{i})$ an equilibrium for $\Gamma.$
Proof. By Lemma 3, $D_{D_{i}}$ is nonempty and convex. By Lemma 4, $D_{D_{i}}$
is compact. For each $i\in I$ the set
$U_{i}:=\left\\{(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in
Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}:(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\emptyset\right\\}$ is
open
and we define $F_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ by
$F_{i}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\left\\{\begin{array}[]{c}(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\text{ if
}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\not\in U_{i},\\\
\alpha_{i}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\text{ if
}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in U_{i}.\end{array}\right.$
Then the correspondence $F_{i}$ has nonempty, compact values and is measurable
with respect to $z_{i}$ and upper semicontinuous with respect to
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}.$
We denote $\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})=$
=$\left\\{(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}\text{ is a measurable selection
of }F_{i}(\cdot,\lambda_{1},\lambda_{2},...,\lambda_{n})\right\\}.$ Then:
i) $\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is nonempty
because there exists a measurable selection from the correspondence $F_{i}$ by
Kuratowski-Ryll-Nardewski Selection Theorem.
ii) $\ \mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is convex
and compact by Lemma 3 and Lemma 4.
We define $\Phi:$ $\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}},$
$\Phi(\lambda_{1},\lambda_{2},...,\lambda_{n})=\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...$
$,\lambda_{n}).$
The set $\mathop{\textstyle\prod}\limits_{i\in I}\mathcal{D}_{D_{i}}$ is
nonempty, compact and convex. By Lemma 5 the correspondence
$\mathcal{D}_{F_{i}}$ is upper semicontinuous. Then the correspondence $\Phi$
is upper semicontinuous and has nonempty compact and convex values. By Ky Fan
fixed point Theorem, we know that there exists a fixed point
$(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in\Phi(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast}).$
In particular, for each player $i,$
$\lambda_{i}^{\ast}\in\mathcal{D}_{F_{i}}(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast}).$
Therefore, for each player $i,$ there exists
$g_{i}^{\ast}\in$Meas$(Z_{i},A_{i})$ such that $g_{i}^{\ast}$ is a selection
of $F_{i}(\cdot,\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
and $(\mu\tau_{i}^{-1})(g_{i}^{\ast})^{-1}=\lambda_{i}^{\ast}.$
We prove that $(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ is an equilibrium
for $\Gamma.$ For each $i\in I,$ because $g_{i}^{\ast}$ is a selection of
$F_{i}(\cdot,\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$,
it follows that $g_{i}^{\ast}(z_{i})\in(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$ if
$(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\not\in
U_{i}$ or
$g_{i}^{\ast}(z_{i})\in\alpha_{i}^{{}^{\prime}}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
if $(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in
U_{i}.$
By the assumption d) it follows that $g_{i}^{\ast}(z_{i})\not\in
P_{i}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$ for
each $z_{i}\in Z_{i}.$ Then
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
and $(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in
U_{i}.$ This is equivalent with the fact that
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,$
$(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})$ and $(\alpha_{i}\cap
P_{i})(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})=\emptyset$
for each $z_{i}\in Z_{i}.$ Consequently,
$(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ is an equilibrium for
$\Gamma.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Theorem 4. Let
$\Gamma=(I,((Z_{i},\mathcal{Z}_{i}),(A_{i},\alpha_{i},P_{i}))_{i\in I},\mu)$.
be an abstract economy with private information and a countable space of
action, where $I$ is a finite index set such that for each $i\in I,$
a) $A_{i}$ is a countable complete metric space and $(Z_{i}$,
$\mathcal{Z}_{i})$ is a measurable space; $(\Omega,\mathcal{F})$ is the
product measurable space $\underset{i\in I}{\prod}(Z_{i},\mathcal{Z}_{i})$ and
$\mu$ an atomless probability measure on $(\Omega,\mathcal{F});$
b) the correspondence $D_{i}:Z_{i}\rightarrow 2^{A_{i}}$ is measurable with
compact values; let $D_{D_{i}}$ be the set
$\left\\{(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}\text{ {is a measurable selection
of} }D_{i}\right\\};$
c) the correspondence
$\alpha_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ is measurable with respect to
$z_{i}$ and, for all $z_{i}\in Z_{i},$
$\alpha_{i}(z_{i},\cdot,\cdot,...,\cdot)$ is upper semicontinuous with
nonempty, compact values;
d) there exists a selector
$G_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ for $(\alpha_{i}\cap
P_{i}):Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ such that
$G_{i}(z_{i},(\mu\tau_{1}^{-1})g_{1}^{-1},(\mu\tau_{2}^{-1})g_{2}^{-1},...,(\mu\tau_{n}^{-1})g_{n}^{-1})$
is measurable with respect to $z_{i}$ and, for all $z_{i}\in Z_{i},$
$G_{i}(z_{i},\cdot,\cdot,...,\cdot)$ is upper semicontinuous with nonempty,
compact values;
e) for each $z_{i}\in Z_{i}$ and each
$(g_{1},g_{2},...,g_{n})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(Z_{i},A_{i}),$
$g_{i}(z_{i})\not\in
G_{i}(z_{i},(\mu\tau_{1}^{-1})g_{1}^{-1},(\mu\tau_{2}^{-1})g_{2}^{-1},...,(\mu\tau_{n}^{-1})g_{n}^{-1});$
f) the set
$U_{i}:=\left\\{(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in
Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}:(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\emptyset\right\\}$ is
open for each $z_{i}\in Z_{i}$.
Then there exists
$(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(Z_{i},A_{i})$ an equilibrium for $\Gamma.$
Proof. By Lemma 3, $D_{D_{i}}$ is nonempty and convex. By Lemma 4,
$\mathcal{D}_{D_{i}}$ is compact. For each $i\in I$ the set
$U_{i}:=\left\\{(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in
Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}:(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\emptyset\right\\}$ is
open
and we define $F_{i}:Z_{i}\times\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{A_{i}}$ by
$F_{i}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})=\left\\{\begin{array}[]{c}G_{i}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\text{
if }(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\not\in U_{i},\\\
\alpha_{i}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\text{ if
}(z_{i},\lambda_{1},\lambda_{2},...,\lambda_{n})\in U_{i}.\end{array}\right.$
Then the correspondence $F_{i}$ has nonempty, compact values and is measurable
with respect to $z_{i}$ and upper semicontinuous with respect to
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}.$
We denote $\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})=$
$=\left\\{(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}\text{ is a measurable selection
of }F_{i}(\cdot,\lambda_{1},\lambda_{2},...,\lambda_{n})\right\\}.$ Then:
i) $\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is nonempty
because there exists a measurable selection from the correspondence $F_{i}$ by
Kuratowski-Ryll-Nardewski Selection Theorem.
ii) $\ \mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n})$ is convex
and compact by Lemma 3 and Lemma 4.
We define $\Phi:$ $\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}\rightarrow 2^{\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{D_{i}}},$
$\Phi(\lambda_{1},\lambda_{2},...,\lambda_{n})=\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{F_{i}}(\lambda_{1},\lambda_{2},...,\lambda_{n}).$
The set $\mathop{\textstyle\prod}\limits_{i\in I}\mathcal{D}_{D_{i}}$ is
nonempty, compact convex. By Lemma 5 the correspondence $\mathcal{D}_{F_{i}}$
is upper semicontinuous. Then the correspondence $\Phi$ is upper
semicontinuous and has nonempty, compact and convex values. By Ky Fan fixed
point theorem, we know that there exists a fixed point
$(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in\Phi(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast}).$
In particular, for each player $i,$
$\lambda_{i}^{\ast}\in\mathcal{D}_{F_{i}}(\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast}).$
Therefore, for each player $i,$ there exists
$g_{i}^{\ast}\in$Meas$(Z_{i},A_{i})$ such that $g_{i}^{\ast}$ is a selection
of $F_{i}(\cdot,\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
and $(\mu\tau_{i}^{-1})(g_{i}^{\ast})^{-1}=\lambda_{i}^{\ast}.$
We prove that $(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ is an equilibrium
for $\Gamma.$ For each $i\in I,$ because $g_{i}^{\ast}$ is a selection of
$F_{i}(\cdot,\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$,
it follows that $g_{i}^{\ast}(z_{i})\in(\alpha_{i}\cap
P_{i})(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$ if
$(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\not\in
U_{i}$ or
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
if $(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in
U_{i}.$
By the hypothesis d), it follows that $g_{i}^{\ast}(z_{i})\not\in
G_{i}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$ for
each $z_{i}\in Z_{i}.$ Then
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})$
and $(z_{i},\lambda_{1}^{\ast},\lambda_{2}^{\ast},...,\lambda_{n}^{\ast})\in
U_{i}.$ This is equivalent with the fact that
$g_{i}^{\ast}(z_{i})\in\alpha_{i}(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...\newline
,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})$ and $(\alpha_{i}\cap
P_{i})(z_{i},(\mu\tau_{1}^{-1})(g_{1}^{\ast})^{-1},(\mu\tau_{2}^{-1})(g_{2}^{\ast})^{-1},...,(\mu\tau_{n}^{-1})(g_{n}^{\ast})^{-1})=\emptyset$
for each $z_{i}\in Z_{i}.$ Consequently,
$(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ is an equilibrium for
$\Gamma.\vskip 6.0pt plus 2.0pt minus 2.0pt$
## References
* (1) C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer-Verlag, Berlin, 1994.
* (2) B. Bollobas and N. T. Varopoulos, Representation of systems of measurable sets. Mathematical Proceeding of the Cambridge Philosophical Society 78 (1975), 323-325.
* (3) A. Borglin and H. Keiding, Existence of equilibrium actions and of equilibrium: A note on the ’new’ existence theorem, J. Math. Econom. 3 (1976), 313-316.
* (4) G. Debreu, A social equilibrium existence theorem. Proc. Nat. Acad. Sci. 38 (1952), 886-893.
* (5) K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121-126.
* (6) H. Fu, Y. Sun, N. C. Yannelis and N. C. Zhang, Pure strategy equilibria in games with private and public information, Journal of Mathematical Economics (Published online), 2006.
* (7) M.A. Khan, K. Rath, Y. Sun, On the existence of pure strategy equilibria in games with a continuum of players, Journal of Economic Theory 76 (1997), 13-46.
* (8) M.A. Khan, K. Rath, Y. Sun, On a private information game without pure strategy equilibria, Journal of Mathematical Economics 31 (1999), 341-359.
* (9) M.A. Khan, K. Rath, Y. Sun, The Dvoretzky-Wald-Wolfowitz Theorem and purification in atomless finite-action games. International Journal of Game Theory 34 (2006), 91-104.
* (10) M.A. Khan, Y. Sun, Pure strategies in games with private information. Journal of Mathematical Economics 24(1995), 633-653.
* (11) M.A. Khan, Y. Sun, Integrals of set valued functions with a countable range. Mathematics of Operations Research 21(1996), 946-954.
* (12) M.A. Khan, Y. Sun, Non-cooperative games on hyperfinite Loeb spaces. Journal of Mathematical Economics 31 (1999), 455-492.
* (13) E. Klein and A. C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley&Sons, Inc., New York, 1984.
* (14) J. F. Nash, Non-cooperative games. Ann. Math. 54 (1951), 286-295.
* (15) R. Radner, R. W. Rosenthal, Private information and pure-strategy equilibria. Mathematics of Operations Research 7(1982), 401-409.
* (16) W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, Journal of Mathematical Economics 2 (1975), 345-348.
* (17) Y. Sun, Distributional properties of correspondences on Loeb Spaces, Journal of Functional Analysis 139(1996), 68-93.
* (18) N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibrium in linear topological spaces, J. Math. Econom. 12 (1983), 233-245.
* (19) H. Yu and Z. Zhang, Pure strategy equilibria in games with countable actions, Journal of Mathematical Economics 43 (2007), 192-200.
* (20) X. Z. Yuan, The Study of Minimax Inequalities and Applications to Economies and Variational Inequalities, Memoirs of the American Society, 132, (1988).
|
arxiv-papers
| 2013-04-01T15:22:49 |
2024-09-04T02:49:43.722625
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1304.0375"
}
|
1304.0444
|
# Zygmund-type inequalities for an operator preserving inequalities between
polynomials
Nisar. A. Rather, Suhail Gulzar and K.A. Thakur
###### Abstract.
In this paper, we present certain new $L_{p}$ inequalities for
$\mathcal{B}_{n}$-operators which include some known polynomial inequalities
as special cases.
00footnotetext: AMS Mathematics Subject Classification(2010): 26D10,
41A17.00footnotetext: Keywords: $L^{p}$ inequalities,
$\mathcal{B}_{n}$-operators, polynomials.
P.G.Department of Mathematics, Kashmir University, Hazratbal,
Srinagar-190006, India
e-mail: [email protected]
*Department of Mathematics, Degree College Ganderbal,
Ganderbal Kashmir- India
## 1\. Introduction and statement of results
Let $\mathscr{P}_{n}$ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$. For $P\in\mathscr{P}_{n}$,
define
$\left\|P(z)\right\|_{0}:=\exp\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|P(e^{i\theta})\right|d\theta\right\\},$
$\left\|P(z)\right\|_{p}:=\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}\right\\}^{1/p},\,\,1\leq
p<\infty,$
$\left\|P(z)\right\|_{\infty}:=\underset{\left|z\right|=1}{\max}\left|P(z)\right|,\quad
m(P,k):=\underset{\left|z\right|=k}{\min}\left|P(z)\right|,\,k>0$
and denote for any complex function $\psi:\mathbb{C}\rightarrow\mathbb{C}$ the
composite function of $P$ and $\psi$, defined by
$\left(P\circ\psi\right)(z):=P\left(\psi(z)\right)\,\,\,\,(z\in\mathbb{C})$,
as $P\circ\psi$.
If $P\in\mathscr{P}_{n}$, then
(1.1) $\left\|P^{\prime}(z)\right\|_{p}\leq
n\left\|P(z)\right\|_{p},\,\,\,\,\,\,p\geq 1$
and
(1.2) $\left\|P(Rz)\right\|_{p}\leq
R^{n}\left\|P(z)\right\|_{p},\,\,\,R>1,\,\,\,\,p>0,$
Inequality (1.1) was found out by Zygmund [20] whereas inequality (1.2) is a
simple consequence of a result of Hardy [8]. Arestov [2] proved that (1.1)
remains true for $0<p<1$ as well. For $p=\infty$, the inequality (1.1) is due
to Bernstein (for reference, see [11, 15, 18]) whereas the case $p=\infty$ of
inequality (1.2) is a simple consequence of the maximum modulus principle (
see [11, 12, 15]). Both the inequalities (1.1) and (1.2) can be sharpened if
we restrict ourselves to the class of polynomials having no zero in
$\left|z\right|<1.$ In fact, if $P\in\mathscr{P}_{n}$ and $P(z)\neq 0$ in
$\left|z\right|<1$, then inequalities (1.1) and (1.2) can be respectively
replaced by
(1.3) $\left\|P^{\prime}(z)\right\|_{p}\leq
n\frac{\left\|P(z)\right\|_{p}}{\left\|1+z\right\|_{p}},\,\,\,\,p\geq 0$
and
(1.4)
$\left\|P(Rz)\right\|_{p}\leq\frac{\left\|R^{n}z+1\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p},\,\,\,R>1,\,\,\,p>0.$
Inequality (1.3) is due to De-Bruijn [7](see also [3]) for $p\geq 1$. Rahman
and Schmeisser [1] extended it for $0<p<1,$ whereas the inequality (1.4) was
proved by Boas and Rahman [6] for $p\geq 1$ and later it was extended for
$0<p<1$ by Rahman and Schmeisser [14]. For $p=\infty$, the inequality (1.3)
was conjectured by Erdös and later verified by Lax [9] whereas inequality
(1.4) was proved by Ankeny and Rivlin [1].
As a compact generalization of inequalities (1.3) and (1.5), Aziz and Rather
[5] proved that if $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in
$|z|<1,$ then for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1,$ $R>r\geq 1$ and $p>0$,
(1.5)
$\left\|P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\right\|_{p}\leq\dfrac{C_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p}$
where
(1.6)
$C_{p}=\left\|\left(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\right)z+(1+\phi_{n}(R,r,\alpha,\beta))\right\|_{p}$
and
(1.7)
$\phi_{n}\left(R,r,\alpha,\beta\right)_{n}=\beta\left\\{\left(\dfrac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}-\alpha.$
If we take $\beta=0,\,\alpha=1$ and $r=1$ in (1.5) and divide two sides of
(1.5) by $R-1$ then make $R\rightarrow 1,$ we obtain inequality (1.3). Whereas
inequality (1.4) is obtained from (1.5) by taking $\alpha=\beta=0.$
Rahman [13] (see also Rahman and Schmeisser [15, p. 538]) introduced a class
$\mathcal{B}_{n}$ of operators $B$ that maps $P\in\mathscr{P}_{n}$ into
itself. That is, the operator $B$ carries $P\in\mathscr{P}_{n}$ into
(1.8)
$B[P](z):=\lambda_{0}P(z)+\lambda_{1}\left(\frac{nz}{2}\right)\frac{P^{\prime}(z)}{1!}+\lambda_{2}\left(\frac{nz}{2}\right)^{2}\frac{P^{\prime\prime}(z)}{2!}$
where $\lambda_{0},\lambda_{1}$ and $\lambda_{2}$ are such that all the zeros
of
(1.9)
$u(z):=\lambda_{0}+C(n,1)\lambda_{1}z+C(n,2)\lambda_{2}z^{2},\,\,\,C(n,r)=n!/r!(n-r)!,$
lie in the half plane
(1.10) $|z|\leq|z-n/2|$
and proved that if $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in
$|z|<1$, then
(1.11)
$\left|B[P\circ\sigma](z)\right|\leq\frac{1}{2}\left\\{R^{n}\left|\Lambda_{n}\right|+\left|\lambda_{0}\right|\right\\}\left\|P(z)\right\|_{\infty}\,\,\,\mbox{}for\,\mbox{}\,\,\,|z|=1,$
(see [13, Inequalities (5.2) and (5.3)]) where $\sigma(z)=Rz$, $R\geq 1$ and
(1.12)
$\Lambda_{n}:=\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}+\lambda_{2}\frac{n^{3}(n-1)}{8}.$
As an extension of inequality (1.11) to $L_{p}$-norm, recently W.M. Shah and
A. Liman [19] while seeking the desired extension, they [19, Theorem 2] have
made an incomplete attempt by claiming to have proved that if
$P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$, then
for each $R\geq 1$ and $p\geq 1$,
(1.13)
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\frac{R^{n}|\Lambda_{n}|+|\lambda_{0}|}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}.$
where $B\in B_{n}$ and $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by 1.12.
Rather and Shah [16] pointed an error in the proof of (1.13), they not only
provided a correct proof but also extended it for $0\leq p<1$ as well. They
proved:
###### Theorem A.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$0\leq p<\infty$ and $R>1,$
(1.14)
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\dfrac{\left\|R^{n}\Lambda_{n}z+\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
$B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by (1.12).
The result is sharp as shown by $P(z)=az^{n}+b,$ $|a|=|b|=1.$
Recently, Rather and Suhail Gulzar [16] obtained the following result which is
a generalization of Theorem A.
###### Theorem B.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $0\leq p<\infty$ and $R>1,$
(1.15) $\displaystyle\left\|B[P\circ\sigma](z)-\alpha
B[P](z)\right\|_{p}\leq\dfrac{\left\|(R^{n}-\alpha)\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
where $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by
(1.12). The result is best possible and equality in (1.15) holds for
$P(z)=az^{n}+b,$ $|a|=|b|=1.$
If we take $\alpha=0$ in Theorem B, we obtain Theorem A.
In this paper, we investigating the dependence of
$\left\|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right\|_{p}$
on $\left\|P(z)\right\|_{p}$ for $\alpha$, $\beta\in\mathbb{C}$ with
$|\alpha|\leq 1$, $|\beta|\leq 1$, $R>r\geq 1$, $0\leq p<\infty$,
$\sigma(z):=Rz$, $\rho(z):=rz$ and $\phi_{n}\left(R,r,\alpha,\beta\right)$ is
given by (1.7) and establish certain generalized $L_{p}$-mean extensions of
the inequality (1.11) for $0\leq p<\infty$ and also a generalization of (1.5).
In this direction, we first present the following result which is a compact
generalization of the inequalities (1.3), (1.4), (1.5) and (1.11) for $0\leq
p<1$ as well.
###### Theorem 1.1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1$, $R>r\geq 1$ and $0\leq p<\infty$,
$\displaystyle\left\|B[P\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](z)\right\|_{p}$
(1.16)
$\displaystyle\qquad\qquad\qquad\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)\Lambda_{n}z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$
where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by(1.7) and (1.12)
respectively. The result is best possible and equality in (1.1) holds for
$P(z)=az^{n}+b,|a|=|b|\neq 0$
###### Remark 1.1.
If we take $\lambda_{1}=\lambda_{2}=0$ in (1.1), we obtain inequality (1.5).
For $\beta=0,$ inequality (1.1) reduces the following result.
###### Corollary 1.1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for every real or complex number $\alpha$ with $|\alpha|\leq 1$, $R>r\geq
1$ and $0\leq p<\infty$,
(1.17) $\left\|B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)\right\|_{p}\leq\frac{\left\|(R^{n}-\alpha
r^{n})\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$
where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$ and $\Lambda_{n}$
is defined by (1.12). The result is best possible and equality in (1.17) holds
for $P(z)=az^{n}+b,\,\,\,|a|=|b|\neq 0$.
###### Remark 1.2.
For taking $\alpha=0$ in (1.17), we obtain Theorem (A) and for $r=1$ in
(1.17), we get Theorem B.
Instead of proving Theorem 1.1, we prove the following more general result
which includes Theorem 1.1 as a special case.
###### Theorem 1.2.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta,\delta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1,$ $|\delta|\leq 1,$ $R>r\geq 1$ and $0\leq p<\infty$,
$\displaystyle\Bigg{\|}B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)$
$\displaystyle+\delta$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\|}_{p}$
(1.18)
$\displaystyle\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)\Lambda_{n}z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$
where $B\in B_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by(1.7) and (1.12)
respectively. The result is best possible and equality in (1.1) holds for
$P(z)=az^{n}+b,|a|=|b|\neq 0.$
###### Remark 1.3.
For $\delta=0$ in (1.2), we get Theorem 1.1.
Next, corollary which is a generalization of (1.5) follows by taking
$\lambda_{1}=\lambda_{2}=0$ in (1.2).
###### Corollary 1.2.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta,\delta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1,$ $|\delta|\leq 1,$ $R>r\geq 1$ and $0\leq p<\infty$,
$\displaystyle\Bigg{\|}P(Rz)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)$
$\displaystyle+\delta\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||-|1+\phi_{n}\left(R,r,\alpha,\beta\right)|\Big{)}m}{2}\Bigg{\|}_{p}$
(1.19)
$\displaystyle\qquad\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$
where $\phi_{n}\left(R,r,\alpha,\beta\right)$ is defined by(1.7). The result
is best possible and equality in (1.2) holds for $P(z)=az^{n}+b,|a|=|b|\neq
0.$
## 2\. Lemmas
For the proofs of these theorems, we need the following lemmas. The first
Lemma is easy to prove.
###### Lemma 2.1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has all its zeros in $\left|z\right|\leq
1$, then for every $R\geq r\geq 1$ and $\left|z\right|=1$,
$\left|P(Rz)\right|\geq\left(\frac{R+1}{r+1}\right)^{n}\left|P(rz)\right|.$
The following Lemma follows from [10, Corollary 18.3, p. 65].
###### Lemma 2.2.
If all the zeros of polynomial $P\in\mathscr{P}_{n}$ lie in
$\left|z\right|\leq 1$, then all the zeros of the polynomial $B[P](z)$ also
lie in $\left|z\right|\leq 1$.
###### Lemma 2.3.
If $F\in\mathscr{P}_{n}$ has all its zeros in $\left|z\right|\leq 1$ and
$P(z)$ is a polynomial of degree at most $n$ such that
$|P(z)|\leq|F(z)|\,\,\,\textrm{for}\,\,\,|z|=1,$
then for every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1$, $|\beta|\leq
1$, $R\geq r\geq 1$, and $|z|\geq 1$,
(2.1)
$|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|\leq|B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$
where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.
###### Proof.
Since the polynomial $F(z)$ of degree $n$ has all its zeros in $|z|\leq 1$ and
$P(z)$ is a polynomial of degree at most $n$ such that
(2.2) $|P(z)|\leq|F(z)|\,\,\,\,\textrm{for}\,\,\,\,|z|=1,$
therefore, if $F(z)$ has a zero of multiplicity $s$ at $z=e^{i\theta_{0}}$,
then $P(z)$ has a zero of multiplicity at least $s$ at $z=e^{i\theta_{0}}$. If
$P(z)/F(z)$ is a constant, then the inequality (2.1) is obvious. We now assume
that $P(z)/F(z)$ is not a constant, so that by the maximum modulus principle,
it follows that
$|P(z)|<|F(z)|\,\,\,\textrm{for}\,\,|z|>1\,\,.$
Suppose $F(z)$ has $m$ zeros on $|z|=1$ where $0\leq m\leq n$, so that we can
write
$F(z)=F_{1}(z)F_{2}(z)$
where $F_{1}(z)$ is a polynomial of degree $m$ whose all zeros lie on $|z|=1$
and $F_{2}(z)$ is a polynomial of degree exactly $n-m$ having all its zeros in
$|z|<1$. This implies with the help of inequality (2.2) that
$P(z)=P_{1}(z)F_{1}(z)$
where $P_{1}(z)$ is a polynomial of degree at most $n-m$. Now, from inequality
(2.2), we get
$|P_{1}(z)|\leq|F_{2}(z)|\,\,\,\textrm{for}\,\,|z|=1\,$
where $F_{2}(z)\neq 0\,\,for\,\,|z|=1$. Therefore for every
$\lambda\in\mathbb{C}$ with $|\lambda|>1$, a direct application of Rouche’s
theorem shows that the zeros of the polynomial $P_{1}(z)-\lambda F_{2}(z)$ of
degree $n-m\geq 1$ lie in $|z|<1$. Hence the polynomial
$f(z)=F_{1}(z)\left(P_{1}(z)-\lambda F_{2}(z)\right)=P(z)-\lambda F(z)$
has all its zeros in $|z|\leq 1$ with at least one zero in $|z|<1$, so that we
can write
$f(z)=(z-te^{i\delta})H(z)$
where $t<1$ and $H(z)$ is a polynomial of degree $n-1$ having all its zeros in
$|z|\leq 1$. Applying Lemma 2.1 to the polynomial $f(z)$ with $k=1$, we obtain
for every $R>r\geq 1$ and $0\leq\theta<2\pi$,
$\displaystyle|f(Re^{i\theta})|=$
$\displaystyle|Re^{i\theta}-te^{i\delta}||H(Re^{i\theta})|$
$\displaystyle\geq$
$\displaystyle|Re^{i\theta}-te^{i\delta}|\left(\frac{R+1}{r+1}\right)^{n-1}|H(re^{i\theta})|$
$\displaystyle=$
$\displaystyle\left(\frac{R+1}{r+1}\right)^{n-1}\frac{|Re^{i\theta}-te^{i\delta}|}{|re^{i\theta}-te^{i\delta}|}|(re^{i\theta}-te^{i\delta})H(re^{i\theta})|$
$\displaystyle\geq$
$\displaystyle\left(\frac{R+1}{r+1}\right)^{n-1}\left(\frac{R+t}{r+t}\right)|f(re^{i\theta})|.$
This implies for $R>r\geq 1$ and $0\leq\theta<2\pi$,
(2.3)
$\left(\frac{r+t}{R+t}\right)|f(Re^{i\theta})|\geq\left(\frac{R+1}{r+1}\right)^{n-1}|f(re^{i\theta})|.$
Since $R>r\geq 1>t$ so that $f(Re^{i\theta})\neq 0$ for $0\leq\theta<2\pi$ and
$\frac{1+r}{1+R}>\frac{r+t}{R+t}$, from inequality (2.3), we obtain $R>r\geq
1$ and $0\leq\theta<2\pi$,
(2.4) $|f(Re^{i\theta})|>\left(\frac{R+1}{r+1}\right)^{n}|f(re^{i\theta})|.$
Equivalently,
$|f(Rz)|>\left(\frac{R+1}{r+1}\right)^{n}|f(rz)|$
for $|z|=1$ and $R>r\geq 1$. Hence for every $\alpha\in\mathbb{C}$ with
$|\alpha|\leq 1$ and $R>r\geq 1,$ we have
$\displaystyle|f(Rz)-\alpha f(rz)|$ $\displaystyle\geq|f(Rz)|-|\alpha||f(rz)|$
(2.5)
$\displaystyle>\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}|f(rz)|,\,\,\,\,\,|z|=1.$
Also, inequality (2.4) can be written in the form
(2.6) $|f(re^{i\theta})|<\left(\frac{r+1}{R+1}\right)^{n}|f(Re^{i\theta})|$
for every $R>r\geq 1$ and $0\leq\theta<2\pi.$ Since $f(Re^{i\theta})\neq 0$
and $\left(\frac{r+1}{R+1}\right)^{n}<1$, from inequality (2.6), we obtain for
$0\leq\theta<2\pi$ and $R>r\geq 1$,
$|f(re^{i\theta})|<|f(Re^{i\theta})|.$
Equivalently,
$|f(rz)|<|f(Rz)|\,\,\,\textrm{for}\,\,\,\,|z|=1.$
Since all the zeros of $f(Rz)$ lie in $|z|\leq(1/R)<1$, a direct application
of Rouche’s theorem shows that the polynomial $f(Rz)-\alpha f(rz)$ has all its
zeros in $|z|<1$ for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1$.
Applying Rouche’s theorem again, it follows from (2.4) that for
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq
1$, all the zeros of the polynomial
$\displaystyle T(z)=$ $\displaystyle f(Rz)-\alpha
f(rz)+\beta\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}f(rz)$
$\displaystyle=f(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)f(rz)$
$\displaystyle=\big{(}P(Rz)-\lambda
F(Rz)\big{)}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}P(rz)-\lambda
F(rz)\big{)}$
$\displaystyle=\big{(}P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\big{)}-\lambda\big{(}F(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)F(rz)\big{)}$
lie in $|z|<1$ for every $\lambda\in\mathbb{C}$ with $|\lambda|>1$. Using
Lemma 2.2 and the fact that $B$ is a linear operator, we conclude that all the
zeros of polynomial
$\displaystyle W(z)$ $\displaystyle=B[T](z)$
$\displaystyle=(B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\lambda(B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z))$
also lie in $|z|<1$ for every $\lambda$ with $|\lambda|>1$. This implies
(2.7)
$|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|\leq|B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$
for $|z|\geq 1$ and $R>r\geq 1$. If inequality (2.7) is not true, then exist a
point $z=z_{0}$ with $|z_{0}|\geq 1$ such that
$|B[P\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z_{0})|>|B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})|.$
But all the zeros of $F(Rz)$ lie in $|z|<1$, therefore, it follows (as in case
of $f(z)$) that all the zeros of
$F(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)F(rz)$ lie in $|z|<1$. Hence by
Lemma 2.2, all the zeros of
$B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)$
also lie in $|z|<1$, which shows that
$B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})\neq
0.$
We take
$\lambda=\frac{B[P\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z_{0})}{B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})},$
then $\lambda$ is a well defined real or complex number with $|\lambda|>1$ and
with this choice of $\lambda$, we obtain $W(z_{0})=0$. This contradicts the
fact that all the zeros of $W(z)$ lie in $|z|<1$. Thus (2.7) holds and this
completes the proof of Lemma 2.3. ∎
###### Lemma 2.4.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has all its zeros in $|z|\leq 1,$ then for
every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ and
$|z|\geq 1,$
(2.8)
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$
where $m={\min}_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$
$\sigma(z)=Rz,\,\rho(z)=rz,$ $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.
###### Proof.
By hypothesis, all the zeros of $P(z)$ lie in $|z|\leq 1$ and
$m|z|^{n}\leq|P(z)|\,\,\,\,\textnormal{for}\,\,\,\,|z|=1.$
We first show that the polynomial $g(z)=P(z)-\lambda mz^{n}$ has all its zeros
in $|z|\leq 1$ for every $\lambda\in\mathbb{C}$ with $|\lambda|<1.$ This is
obvious if $m=0,$ that is if $P(z)$ has a zero on $|z|=1.$ Henceforth, we
assume $P(z)$ has all its zeros in $|z|<1,$ then $m>0$ and it follows by
Rouche’s theorem that the polynomial $g(z)$ has all its zeros in $|z|<1$ for
every $\lambda\in\mathbb{C}$ with $|\lambda|<1.$ Proceeding similarly as in
the proof of Lemma 2.3, we obtain that for $\alpha,\beta\in\mathbb{C}$ with
$|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq 1$, all the zeros of the
polynomial
$\displaystyle H(z)=$ $\displaystyle g(Rz)-\alpha
g(rz)+\beta\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}g(rz)$
$\displaystyle=g(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)g(rz)$
$\displaystyle=\big{(}P(Rz)-\lambda
R^{n}z^{n}m\big{)}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}P(rz)-\lambda
r^{n}z^{n}m\big{)}$
$\displaystyle=\big{(}P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\big{)}-\lambda\big{(}R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\big{)}mz^{n}$
lie in $|z|<1.$ Applying Lemma 2.1 to $H(z)$ and noting that $B$ is a linear
operator, it follows that all the zeros of polynomial
$\displaystyle B[H](z)=$
$\displaystyle\left\\{B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right\\}$
(2.9)
$\displaystyle\qquad\qquad\qquad\qquad-\lambda\big{(}R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\big{)}mB[z^{n}]$
lie in $|z|<1.$ This gives
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
(2.10)
$\displaystyle\qquad\qquad\qquad\qquad\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m\,\,\,\,\,\textnormal{for}\,\,\,\,\,|z|\geq
1.$
If (2) is not true, then there is point $w$ with $|w|\geq 1$ such that
(2.11)
$\displaystyle\left|B[P\circ\sigma](w)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](w)\right|<|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||w|^{n}m.$
We choose
$\lambda=\dfrac{B[P\circ\sigma](w)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](w)}{R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||w|^{n}m.},$
then clearly $|\lambda|<1$ and with this choice of $\lambda,$ from (2), we get
$B[H](w)=0$ with $|w|\geq 1.$ This is clearly a contradiction to the fact that
all the zeros of $H(z)$ lie in $|z|<1.$ Thus for every
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$ $|\beta|\leq 1,$
$\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$
for $|z|\geq 1$ and $R>r\geq 1.$ ∎
###### Lemma 2.5.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq
1,R>r\geq 1$ and $|z|\geq 1,$
$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$ (2.12)
$\displaystyle\leq|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
where $P^{*}(z):=z^{n}\overline{P(1/\overline{z})}$, $B\in\mathcal{B}_{n}$,
$\sigma(z):=Rz$, $\rho(z):=rz$, and $\phi_{n}\left(R,r,\alpha,\beta\right)$ is
defined by (1.7).
###### Proof.
By hypothesis the polynomial $P(z)$ of degree $n$ does not vanish in $|z|<1$,
therefore, all the zeros of the polynomial
$P^{*}(z)=z^{n}\overline{P(1/\overline{z})}$ of degree $n$ lie in $|z|\leq 1$.
Applying Lemma 2.3 with $F(z)$ replaced by $P^{*}(z)$, it follows that
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\leq\left|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)\right|$
for $|z|\geq 1,|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq 1$. This proves the
Lemma 2.5. ∎
###### Lemma 2.6.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has no zero in $\left|z\right|<1,$ then
for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>r\geq 1$ and
$|z|\geq 1$,
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\leq\left|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)\right|$
(2.13)
$\displaystyle\qquad\quad-\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m,$
where $P^{\star}(z)=z^{n}\overline{P(1/\overline{z})},$
$m={\min}_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz,$ $\rho(z)=rz,$
$\Lambda_{n}$ and $\phi_{n}\left(R,r,\alpha,\beta\right)$ are given by (1.12)
and (1.7) respectively.
###### Proof.
By hypothesis $P(z)$ has all its zeros in $|z|\geq 1$ and
(2.14) $\displaystyle m\leq|P(z)|\,\,\,\textnormal{for}\,\,\,\,|z|=1.$
We show $F(z)=P(z)+\lambda m$ does not vanish in $|z|<1$ for every
$\lambda\in\mathbb{C}$ with $|\lambda|<1.$ This is obvious if $m=0$ that is,
if $P(z)$ has a zero on $|z|=1.$ So we assume all the zeros of $P(z)$ lie in
$|z|>1,$ then $m>0$ and by the maximum modulus principle, it follows from
(2.14),
(2.15) $\displaystyle m<|P(z)|\,\,\,\textnormal{for}\,\,\,|z|<1.$
Now if $F(z)=P(z)+\lambda m=0$ for some $z_{0}$ with $|z_{0}|<1,$ then
$\displaystyle P(z_{0})+\lambda m=0$
This implies
(2.16) $\displaystyle|P(z_{0})|=|\lambda|m\leq
m,\,\,\,\textnormal{for}\,\,\,|z_{0}|<1$
which is clearly contradiction to (2.15). Thus the polynomial $F(z)$ does not
vanish in $|z|<1$ for every $\lambda$ with $|\lambda|<1.$ Applying Lemma 2.3
to the polynomial $F(z),$ we get
$\displaystyle|B[F\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$
$\displaystyle\leq|B[F^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F^{\star}\circ\rho](z)|$
for $|z|=1$ and $R>r\geq 1.$ Replacing $F(z)$ by $P(z)+\lambda m,$ we obtain
$\displaystyle|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)$
$\displaystyle
B[P\circ\rho](z)+\lambda(1+\phi_{n}\left(R,r,\alpha,\beta\right))\lambda_{0}m|$
$\displaystyle\leq|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)$
(2.17)
$\displaystyle\qquad\qquad\qquad+\bar{\lambda}(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n})\Lambda_{n}z^{n}m|$
Now choosing the argument of $\lambda$ in the right hand side of (2) such that
$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}$
$\displaystyle\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)+\bar{\lambda}(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n})\Lambda_{n}z^{n}m|$
$\displaystyle=|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
$\displaystyle\qquad\qquad\qquad-|\bar{\lambda}||R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$
for $|z|=1,$which is possible by Lemma 2.4,we get
$\displaystyle|B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|-|\lambda||1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|m$
$\displaystyle\leq$
$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
$\displaystyle\qquad\qquad\qquad-|\lambda||R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$
Equivalently,
$\displaystyle|B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$
$\displaystyle\leq$
$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
(2.18)
$\displaystyle\quad-|\lambda|\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m.$
Letting $|\lambda|\rightarrow 1$ in (2) we obtain inequality (2.6) and this
completes the proof of Lemma 2.6. ∎
Next we describe a result of Arestov [2].
For
$\gamma=\left(\gamma_{0},\gamma_{1},\cdots,\gamma_{n}\right)\in\mathbb{C}^{n+1}\,\,\,\,\mbox{}\,\,\,\text{and}\,\,\,P(z)=\sum_{j=0}^{n}a_{j}z^{j}$,
we define
$C_{\gamma}P(z)=\sum_{j=0}^{n}\gamma_{j}a_{j}z^{j}.$
The operator $C_{\gamma}$ is said to be admissible if it preserves one of the
following properties:
1. (i)
$P(z)$ has all its zeros in $\left\\{z\in\mathbb{C}:|z|\leq 1\right\\}$,
2. (ii)
$P(z)$ has all its zeros in $\left\\{z\in\mathbb{C}:|z|\geq 1\right\\}$.
The result of Arestov may now be stated as follows.
###### Lemma 2.7.
[2, Theorem 2] Let $\phi(x)=\psi(\log x)$ where $\psi$ is a convex non-
decreasing function on $\mathbb{R}$. Then for all $P\in\mathscr{P}_{n}$ and
each admissible operator $\Lambda_{\gamma}$,
$\int_{0}^{2\pi}\phi\left(|C_{\gamma}P(e^{i\theta})|\right)d\theta\leq\int_{0}^{2\pi}\phi\left(c(\gamma,n)|P(e^{i\theta})|\right)d\theta$
where $c(\gamma,n)=\max\left(|\gamma_{0}|,|\gamma_{n}|\right)$.
In particular Lemma 2.7 applies with $\phi:x\rightarrow x^{p}$ for every
$p\in(0,\infty)$ and $\phi:x\rightarrow\log x$ as well. Therefore, we have for
$0\leq p<\infty$,
(2.19)
$\left\\{\int_{0}^{2\pi}\phi\left(|C_{\gamma}P(e^{i\theta})|^{p}\right)d\theta\right\\}^{1/p}\leq
c(\gamma,n)\left\\{\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta\right\\}^{1/p}.$
From Lemma 2.7, we deduce the following result.
###### Lemma 2.8.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for each $p>0$, $R>1$ and $\eta$ real, $0\leq\eta<2\pi$,
$\displaystyle\int_{0}^{2\pi}|\big{(}B[P\circ\sigma](e^{i\theta})$
$\displaystyle+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{)}e^{i\eta}$
$\displaystyle+\big{(}B[P^{*}\circ\sigma]^{*}(e^{i\theta})+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(e^{i\theta})\big{)}|^{p}d\theta$
$\displaystyle\leq|(R^{n}+\phi_{n}(R,r,$
$\displaystyle\alpha,\beta)r^{n})\Lambda_{n}e^{i\eta}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}|^{p}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$,
$B[P^{*}\circ\sigma]^{*}(z):=(B[P^{*}\circ\sigma](z))^{*}$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.
###### Proof.
Since $P(z)$ does not vanish in $\left|z\right|<1$ and
$P^{*}(z)=z^{n}\overline{P(1/\bar{z})}$, by Lemma 2.5, we have for $R>r\geq
1$,
$\displaystyle|B[P\circ\sigma](z)+\phi_{n}$
$\displaystyle\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$ (2.20)
$\displaystyle\leq|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
Also, since
$P^{*}(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P^{*}(rz)=R^{n}z^{n}\overline{P(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}z^{n}\overline{P(1/r\bar{z})}$,
therefore,
$\displaystyle
B[P^{*}\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P^{*}\circ\rho](z)$
$\displaystyle=\lambda_{0}\big{(}R^{n}z^{n}\overline{P(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}z^{n}\overline{P(1/r\bar{z})}\big{)}+\lambda_{1}\left(\frac{nz}{2}\right)\Big{(}nR^{n}z^{n-1}\overline{P(1/R\bar{z})}$
$\displaystyle\quad-R^{n-1}z^{n-2}\overline{P^{\prime}(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}nr^{n}z^{n-1}\overline{P(1/r\bar{z})}-r^{n-1}z^{n-2}\overline{P^{\prime}(1/r\bar{z})}\big{)}\Big{)}$
$\displaystyle\quad+\frac{\lambda_{2}}{2!}\left(\frac{nz}{2}\right)^{2}\Big{(}n(n-1)R^{n}z^{n-2}\overline{P(1/R\bar{z})}-2(n-1)R^{n-1}z^{n-3}\overline{P^{\prime}(1/R\bar{z})}$
$\displaystyle\quad+R^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}n(n-1)r^{n}z^{n-2}\overline{P(1/r\bar{z})}$
$\displaystyle\quad-2(n-1)r^{n-1}z^{n-3}\overline{P^{\prime}(1/r\bar{z})}+r^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/r\bar{z})}\big{)}\Big{)}$
and hence,
$\displaystyle B[P^{*}\circ\sigma]^{*}$
$\displaystyle(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)$
$\displaystyle=$
$\displaystyle\big{(}B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)\big{)}^{*}$
$\displaystyle=$
$\displaystyle\left(\bar{\lambda_{0}}+\bar{\lambda_{1}}\frac{n^{2}}{2}+\bar{\lambda_{2}}\frac{n^{3}(n-1)}{8}\right)\left(R^{n}P(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n}P(z/r)\right)$
$\displaystyle-\left(\bar{\lambda_{1}}\frac{n}{2}+\bar{\lambda_{2}}\frac{n^{2}(n-1)}{4}\right)\Big{(}R^{n-1}zP^{\prime}(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n-1}zP^{\prime}(z/r)\Big{)}$
(2.21)
$\displaystyle+\bar{\lambda_{2}}\frac{n^{2}}{8}\Big{(}R^{n-2}z^{2}P^{\prime\prime}(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n-2}z^{2}P^{\prime\prime}(z/r)\Big{)}.$
Also, for $|z|=1$
$\displaystyle|B[P^{*}\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
$\displaystyle=|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|.$
Using this in (2), we get for $|z|=1$ and $R>r\geq 1$,
$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$
$\displaystyle\leq|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|.$
Since all the zeros of $P^{*}(z)$ lie in $|z|\leq 1$, as before, all the zeros
of $P^{*}(Rz)+\phi_{n}(R,r,\alpha,\beta)P^{*}(rz)$ lie in $|z|<1$ for all real
or complex numbers $\alpha,\beta$ with $|\alpha|\leq 1$, $|\beta|\leq 1$ and
$R>r\geq 1$. Hence by Lemma 2.2, all the zeros of
$B[P^{*}\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P^{*}\circ\rho](z)$ lie in
$|z|<1$, therefore, all the zeros of
$B[P^{*}\circ\sigma]^{*}(z)+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(z)$
lie in $|z|>1$. Hence by the maximum modulus principle,
$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
(2.22)
$\displaystyle<|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|\quad\textrm{for}\quad|z|<1.$
A direct application of Rouche’s theorem shows that
$\displaystyle C_{\gamma}P(z)=$
$\displaystyle\big{(}B[P\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{)}e^{i\eta}$
$\displaystyle+\big{(}B[P^{*}\circ\sigma]^{*}(z)+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(z)\big{)}$
$\displaystyle=$
$\displaystyle\left\\{(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\eta}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\right\\}a_{n}z^{n}$
$\displaystyle+\cdots+\left\\{(R^{n}+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})r^{n})\bar{\Lambda_{n}}+e^{i\eta}(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}\right\\}a_{0}$
does not vanish in $|z|<1$. Therefore, $C_{\gamma}$ is an admissible operator.
Applying (2.19) of Lemma 2.7, the desired result follows immediately for each
$p>0$. ∎
We also need the following lemma [4].
###### Lemma 2.9.
If $A,B,C$ are non-negative real numbers such that $B+C\leq A,$ then for each
real number $\gamma,$
$|(A-C)e^{i\gamma}+(B+C)|\leq|Ae^{i\gamma}+B|.$
## 3\. Proof of the Theorems
###### Proof of Theorem 1.2.
By hypothesis $P(z)$ does not vanish in $|z|<1,$ therefore by Lemma 2.6, we
have
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\leq\left|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)\right|$
(3.1)
$\displaystyle\qquad\quad-\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m,$
for $|z|=1,$ $|\alpha|\leq 1$ and $R>r\geq 1$ where
$P^{\star}(z)=z^{n}\overline{P(1/\overline{z})}.$ Since
$B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)$
is the conjugate of
$B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)$
and
$\displaystyle|B[P^{\star}\circ\sigma]^{\star}(z)$
$\displaystyle+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)|$
$\displaystyle=|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
Thus (3) can be written as
$\displaystyle\big{|}B$
$\displaystyle[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\big{|}$
$\displaystyle\qquad\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}$
$\displaystyle\leq\left|B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)\right|$
(3.2)
$\displaystyle\qquad\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2},$
for $|z|=1.$ Taking
$A=\left|B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)\right|$
$B=\big{|}B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\big{|},$
and
$C=\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}$
in Lemma 2.9 and noting by (3) that
$B+C\leq A-C\leq A,$
we get for every real $\gamma$,
$\displaystyle\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}.$
This implies for each $p>0,$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
(3.3)
$\displaystyle\quad\quad\qquad\qquad\quad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\theta.$
Integrating both sides of (3) with respect to $\gamma$ from $0$ to $2\pi,$ we
get with the help of Lemma 2.8 for each $p>0,$
$\displaystyle\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta
d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Big{|}\big{|}B[$
$\displaystyle
P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\theta
d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}\Big{|}$
$\displaystyle\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\gamma\Bigg{\\}}\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\Big{(}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\Big{)}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\Big{(}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\Big{)}\Bigg{|}^{p}d\gamma\Bigg{\\}}\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\Big{(}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\Big{)}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\Big{(}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\Big{)}\Bigg{|}^{p}d\theta\Bigg{\\}}\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma$
(3.4)
$\displaystyle\qquad\qquad\qquad\qquad\times\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
Now it can be easily verified that for every real number $\gamma$ and $s\geq
1$,
$\left|s+e^{i\alpha}\right|\geq\left|1+e^{i\alpha}\right|.$
This implies for each $p>0$,
(3.5)
$\int_{0}^{2\pi}\left|s+e^{i\gamma}\right|^{p}d\gamma\geq\int_{0}^{2\pi}\left|1+e^{i\gamma}\right|^{p}d\gamma.$
If
$\displaystyle\big{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\neq
0,$
we take
$s=\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}},$
then by (3), $s\geq 1$ and we get with the help of (3.5),
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}$
$\displaystyle\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\quad\quad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}\Bigg{|}^{p}d\gamma$
$\displaystyle=\Bigg{|}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}$
(3.10) $\displaystyle\times\int\limits_{0}^{2\pi}$
$\displaystyle\left|e^{i\gamma}+\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}\right|^{p}d\gamma$
$\displaystyle=\Bigg{|}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}$
(3.15) $\displaystyle\times\int\limits_{0}^{2\pi}$
$\displaystyle\left|e^{i\gamma}+\left|\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}\right|^{p}\,\right|d\gamma$
$\displaystyle\geq$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
(3.16) $\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma.$
For
$\displaystyle\big{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\neq
0,$
then (3) is trivially true. Using this in (3), we conclude for every
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ $R>r\geq 1$
and $p>0$,
$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
This gives for every $\delta,\alpha,\beta$ with $|\delta|\leq 1,$
$|\alpha|\leq 1,$ $|\beta|\leq 1,$ $R>r\geq 1$ and $\gamma$ real
$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})$
$\displaystyle+\delta$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
(3.17)
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
Since
$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}|e^{i\gamma}+|(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}|e^{i\gamma}+|(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
(3.18)
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
the desired result follows immediately by combining (3) and (3). This
completes the proof of Theorem 1.2 for $p>0$. To establish this result for
$p=0$, we simply let $p\rightarrow 0+$.
∎
## References
* [1] N.C. Ankeny and T.J.Rivli, On a theorm of S.Bernstein, Pacific J. Math., 5(1955), 849 - 852.
* [2] V.V. Arestov, On integral inequalities for trigonometric polynimials and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981),3-22[in Russian]. English translation; Math.USSR-Izv.,18 (1982), 1-17.
* [3] A. Aziz, A new proof and a generalization of a theorem of De Bruijn, proc. Amer Math. Soc., 106(1989), 345-350.
* [4] A. Aziz and N.A. Rather, $L^{p}$ inequalities for polynomials,Glas. Math., 32 (1997), 39-43.
* [5] A. Aziz and N.A. Rather, Some new generalizations of Zygmund-type inequalities for polynomials, Math. Ineq. Appl., 15(2012), 469-486.
* [6] R.P. Boas, Jr. and Q.I. Rahman , $L^{p}$ inequalities for polynomials and entire functions, Arch. Rational Mech. Anal., 11(1962),34-39.
* [7] N.G. Bruijn, Inequalities concerning polynomials in the complex domain, Nederal. Akad.Wetensch. Proc.,50(1947), 1265-1272.
* [8] G.H. Hardy, The mean value of the modulus of an analytic functions, Proc. London Math. Soc., 14(1915), 269-277.
* [9] P.D. Lax, Proof of a conjecture of P.Erdös on the derivative of a polynomial, Bull. Amer. Math.Soc.,50(1944), 509-513.
* [10] M. Marden, Geometry of polynomials, Math. Surveys, No.3, Amer. Math.Soc. Providence, RI, 1949.
* [11] G.V. Milovanovic, D.S. Mitrinovic and Th.M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore,(1994).
* [12] G. Polya and G. Szegö, Aufgaben und lehrsätze aus der analysis, Springer-Verlag, Berlin (1925).
* [13] Q.I. Rahman, Functions of exponential type, Trans. Amer. Math. Soc., 135(1969), 295-309.
* [14] Q.I. Rahman and G. Schmeisser, $L^{p}$ inequalities for polynomials, J. Approx. Theory, 53(1988), 26-32.
* [15] Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York, 2002.
* [16] N.A. Rather and M.A. Shah, On an operator preserving $L_{p}$ inequalities between polynomials, 399 (2013), 422-432.
* [17] N.A. Rather and Suhail Gulzar, Integral mean estimates for an operator preserving inequalities between polynomials, J. Inequal. Spec. Funct., 3 (2012), 24 - 41.
* [18] A.C. Schaffer, Inequalities of A.Markov and S.Bernstein for polynomials and related functions, Bull.Amer. Math. Soc., 47(1941), 565-579.
* [19] W.M.Shah and A.Liman, Integral estimates for the family of B-operators, Operators and Matrices, 5(2011), 79-87.
* [20] A. Zygmund, A remark on conjugate series, Proc. London Math. Soc., 34(1932),292-400.
|
arxiv-papers
| 2013-04-01T13:44:11 |
2024-09-04T02:49:43.731526
|
{
"license": "Public Domain",
"authors": "Nisar. A. Rather, Suhail Gulzar, K. A. Thakur",
"submitter": "Suhail Gulzar Mattoo",
"url": "https://arxiv.org/abs/1304.0444"
}
|
1304.0600
|
Software for creating pictures in the LaTeX environment
R. V. Bezhencev, [email protected]
###### Abstract
To create a text with graphic instructions for output pictures into LaTeX
document, we offer software that allows us to build a picture in WIZIWIG mode
and for setting the text with these graphical instructions.
Keywords: LaTeX, TeX, GUI, drawing.
## Introduction
As we know, for drawing picture in LaTeX environment, user has to write
commands, which contain itself a set of primitives, which together completed
drawing. How creating LaTeX-integrated graphics and animations wrote Francesc
Sunol [1].
About drawing problems in LaTeX and motivation don’t integrated final image is
well described in the thesis Jie Xiao [2]. Since the process of creating
images in the LaTeX environment is not a WYSIWYG (“What You See Is what You
Get”), and reduced to manual writing graphics output commands in the TeX
language, the user has only to imagine how it will look finished drawing, and
approximately select control points.
This paper describes the software developed by the author of PaintTeX,
designed to solve this problem. It was developed in C and WinAPI, using the
methods of multi-threading, which guarantees the performance of the program.
To simplify the creation of drawings by other authors also develops software
Graphviz [2] for drawing graphs, Drawlets [2] for drawing arbitrary graphics
and FeynEdit [3] and JaxoDraw [4] for drawing Feynman diagrams.
## 1 Output line segment
To display the line segment or vector in the user text in addition to the
coordinates of reference point, it is necessary to specify a slope angle with
a width to height ratio. In the TeX language command output segment is as
follows:
\put(60,50){\line(1,-2){20}}
where (60,50) - the coordinates of the start point of the segment, (1,-2) \-
angle as the ratio of length to height, 20 - the length of the projection on
the axis $OX$. Values in a proportion of given inclination should not exceed 6
in absolute value of segments, 4 of vectors, and don’t have common divisors
other than 1. Details can be found in the books of [5] and [6].
Created by the author software PaintTeX provides WYSIWYG interface for drawing
images using primitives, and then converts each primitive in the appropriate
command output graphics TeX language. For example, to draw the image shown in
picture 1, you need a long time to calculate the coordinates of control points
and other parameters of output commands for each graphic primitive, or fit
them around. This picture was painted in the program Paint TeX, the output
code is as follows:
\begin{picture}(215,283)
\qbezier(99,172)(105,172)(112,172)
\qbezier(112,172)(108,174)(105,175)
\qbezier(112,172)(108,171)(105,169)
\qbezier(63,193)(82,170)(102,147)
\qbezier(102,147)(100,151)(99,155)
\qbezier(102,147)(98,149)(95,151)
\qbezier(0,14)(111,14)(209,14)
\qbezier(209,14)(205,16)(202,17)
\qbezier(209,14)(205,13)(202,11)
\qbezier(168,22)(89,129)(8,234)
\qbezier(8,22)(93,22)(168,22)
\qbezier(8,234)(8,128)(8,22)
\qbezier(0,13)(0,145)(0,276)
\qbezier(0,276)(-1,273)(-3,269)
\qbezier(0,276)(1,273)(3,270)
\put(64,192){\circle{38}}
\put(101,160){V}
\put(8,2){O}
\put(10,283){Y}
\put(215,0){X}
\end{picture}
VOYX
Picture 1 - Example of a picture in LaTeX
Let us consider PaintTeX in action. The user selects the desired primitive and
draws it, pointing coordinates of the reference points on which the program
draws the primitive and stores them in memory. When user save a drawing
program inserts into the file text of outputting commands of the primitive
with stored coordinates of reference points. Complex primitives are displayed
in the form of a composition of simpler primitives, for example, vector - is
three straight lines, connections of the ends at the one point, which forms
the arrow, and rectangle - 4 straight. This method can display a myriad of
shapes, including three-dimensional.
## 2 How the program works
The principle of a program under development is as follows. When the program
starts, a window appears with menus, toolbar and drawing area. The user
selects the primitive on the toolbar, and then sets the coordinates of the
mouse control points. Control points are stored in an instance of the class
selected shape and it draws primitive. When you select “Save”, the program
saves output commands primitives in results file, inserting the necessary
parameters (coordinates, radius) from the coordinates of the control points.
For each primitive in the program is allocated class object of the primitive.
At the current stage of development, there are 7 classes: VETREX, LINE, LABEL,
BIZE, SQVR, CIRKLE, FISH. Each of these classes is a child of the base FIGURE
class. FIGURE class content that:
class FIGURE
{
public:
POINT *pt;
FIGURE *nextFig;
Ψvirtual void print() = 0;
};
Due to the mechanism of inheritance, each child class inherits from a base
pointer types POINT, FIGURE and virtual function print (). When you create a
primitive, start initialization function, which converts a pointer *pt to the
array points, required for a given dimension of the primitive. That is, if you
create line segment, in the constructor LINE works command pt = new POINT[2],
and if the rectangle - pt = new POINT[4] in the constructor SQVR. Pointer
*nextFig serves to form a stack of primitives. Through the mechanism of
inheritance it can point to any class of the primitive.
Each description of classes of primitives in their own redefined output
function print(). This function writhen the primitive drawing commands into a
text file, from which a set of commands, and then you can copy in the article
and compile LaTeX tools. In each class, this function outputs in the file own
command and parameters, contained in the selected object. Below, for example,
is the content of a class of primitives “label”:
class LABEL : public FIGURE
{
public:
Ψchar *lab;
Ψvoid ini (int x, int y, char *str, int len, HDC hdc)
Ψ{
ΨΨpt = new POINT;
ΨΨpt[0].x = x;
ΨΨpt[0].y = y;
ΨΨlab = new char[len+1];
ΨΨstrcpy(lab, str);
ΨΨTextOutA(hdc, pt->x, pt->y, lab, strlen(lab));
Ψ}
ΨLABEL (){}
Ψvoid print()
Ψ{
Ψofile << "\\put(" << pt[0].x - Canv_left << ","Ψ
<< Canv_top - pt[0].y << "){" << lab << "}" << endl;
Ψ}
}*label;
This class contains a pointer *lab, which is converted to a string for storing
text of the label, the initialization function, which stores the data in the
structure and draws the text, the function print(), a transformative figure in
the commands of graphics output with the crop, and a pointer *label,
responsible for work stack. Function to create a primitive “label” is as
follows:
LABEL *new_label(int x, int y, char *str, int len, HDC hdc)
{
ΨLABEL *label_new = new LABEL;
Ψlabel_new->ini(x, y, str, len, hdc);
Ψif (!labelcount++) label_new->nextFig = 0;
Ψelse label_new->nextFig = label;
Ψreturn label_new;
}
When the user draws a primitive, in this case, the label, the function of
creating passed the coordinates to reference point, text string, the length of
the string and device handle, which will be drawn text. Since for each new
primitive memory is allocated dynamically, it have to use for initialize the
initialization function, not the designer.
When you save commands, the program for each class of primitive creates a
separate thread. Each thread runs a function that using mutex synchronizes the
output of each command. Declaration of the function follows:
void save(FIGURE *curfig, int counter)
As you can see, the argument * curfig - stack pointer primitives, and counter
- their total number. Through the mechanism of inheritance, each class
primitive is a class of FIGURE, which means for synchronous output primitives
of any class is sufficient to use a single function. So, thanks to the virtual
function print (), with curfig-¿ print (); You can access the output function
of each primitive, and the program will know what kind of entity it is
necessary to bring in a file.
Mathematical models have been taken from the book “ Mathematical Foundations
of Computer Graphics” cite momg. For example, a Bezier curve - parametric
curve given by the expression
$B(t)=\sum_{i=0}^{n}P_{i}b_{i,n}(t),0<t<1$
Where $P_{i}$ \- function of the components of the reference peaks, and
$b_{i,n}(t)$ \- basic functions of a Bezier curve, also called the Bernstein
polynomials.
$b_{i,n}(t)=\left(n\atop i\right)t^{i}(1-t)^{n-i}$
Where $\left(n\atop i\right)=\frac{n!}{I!(Ni)!}$ \- Number of combinations of
$n$ on $i$, where $n$ \- polynomial degree, $i$ \- number of reference peaks.
Since the syntax TeX can display curves only by three points, the formula for
the output has been simplified.
X = (1 - t)*(1 - t) * pt[0].x + 2*t*(1-t)*pt[1].x + t*t*pt[2].x;
Y = (1 - t)*(1 - t) * pt[0].y + 2*t*(1-t)*pt[1].y + t*t*pt[2].y;
Where pt[0].x, pt[1].x, pt[2].x - control points along the axis of $OX$, and
pt[0].y, pt[1].y, pt[2].y - coordinates of the reference points on the axis
$OY$. In the construction of the curve, the program increments t = t + 0.01
finds points on the curve, and then joins them small segments.
## 3 The problems in during the implementation the software
While working on the software adds the following problem. Since the values are
responsible for the slope in the primitive “segment” and “vector” must be
integers, and their number is very limited, and then the slope of the
primitive there is limited number of angles. A forthcoming software user draws
segments and vectors by specifying the coordinates of the starting and ending
point. Convert their coordinates in the output instruction in the TeX language
was not possible, so to print straight lines, it was decided to use Bezier
curves, defining the beginning of a line, a middle and an end. Since the
withdrawal of the Bezier curves does not specify a value for the slope and
length of the projection, the curves can be output through the straight
segments and vectors at any angle.
Just had a problem with the definition of the figure. In the LaTeX drawing
area is defined manually, and the user, as well as entities that also have to
pick up some, determining what sizes will be drawing. Thanks to the automatic
cutting PaintTeX defines the boundaries of the rectangle (canvas), which was
painted the image and crop a picture to fit your needs, inserting the
appropriate parameters in the command beginpicture().
Another problem - work with coordinates Windows and LaTeX. As the starting
point coordinates in Windows is the upper left edge of the window, and in the
LaTeX bottom left, when converting images to files saved coordinates Windows,
and then compile the image look in the mirror image vertically. Now PaintTeX
while saving the figure takes into account this nuance.
## References
* [1] F. Sunol, Tools for creating LaTeX-integrated graphics and animations under GNU/Linux. - The PracTeX Journal, N 1 (2010), P. 1-12.
* [2] X. Jie,Extending Two Drawing Frameworks to Create : Presented in partial Fulfillment of the requirements for the degree of master of computer science, Concordia University Monreal, Quebec, Canada, 2005. - 151 p.
* [3] T. Hahn, P. Lang, “FeynEdit - a tool for drawing Feynman diagrams”, Munich, 2007. 9p. Preprint, arXiv:0711.1345v1 [hep-ph], Cornell Univ. http://arxiv.org/abs/0711.1345v1
* [4] D. Binosi, J. Colins, C. Kaufhold, L. Theussl, “JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes.”, Comput. Phys. Commun. 2008. 17p. Preprint, arXiv:0811.4113v1 [hep-ph], Cornell Univ. http://arxiv.org/abs/0811.4113v1
* [5] S. M. Lvovsky, Typesetting set in the LaTeX system, St. Petersburg: Piter, 2003. - 448 p.
* [6] D. E. Knuth, The TeXbook, part A series Computers and Typesetting. - Addison-Wesley, 1994, 640 p.
* [7] J. A. Adams, D. F. Rogers, Computer-aided Heat Transfer Analysis, McGraw, N. Y., 1973. - 604 p.
|
arxiv-papers
| 2013-04-02T11:55:33 |
2024-09-04T02:49:43.745700
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bezhentcev Roman Vadimovich",
"submitter": "Roman Bezhencev Vadimovich",
"url": "https://arxiv.org/abs/1304.0600"
}
|
1304.0670
|
# Prospects for Localization of Gravitational Wave Transients by the Advanced
LIGO and Advanced Virgo Observatories
J. Aasi1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
J. Abadie1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
B. P. Abbott1 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R. Abbott1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA T. D. Abbott2 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Abernathy3 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA T. Accadia4 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA F. Acernese5ac 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA C. Adams6 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA T. Adams7 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. Addesso8 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. X. Adhikari1
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C.
Affeldt9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA M. Agathos11a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA O. D. Aguiar12 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA P. Ajith1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA B. Allen9,13,10 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. Allocca14ac 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA E. Amador Ceron13 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. Amariutei15
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S. B.
Anderson1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
W. G. Anderson13 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA K. Arai1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. C. Araya1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA C. Arceneaux16 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Ast9,10 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. M. Aston6 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA P. Astone17a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA D. Atkinson18 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA P. Aufmuth10,9
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C.
Aulbert9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Austin1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA B. E. Aylott19 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Babak20 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA P. Baker21 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA G. Ballardin22 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA S. Ballmer23 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA Y. Bao15 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. C. Barayoga1
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Barker18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
F. Barone5ac 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA B. Barr3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Barsotti24 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Barsuglia25 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. A. Barton18 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA I. Bartos26 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA R. Bassiri3,27 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Bastarrika3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Basti14ab
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Batch18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
J. Bauchrowitz9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA Th. S. Bauer11a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Bebronne4 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA B. Behnke20 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA M. Bejger28c 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M.G. Beker11a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. S. Bell3 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA C. Bell3 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA G. Bergmann9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J. M.
Berliner18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. Bertolini9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J. Betzwieser6 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA N. Beveridge3 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA P. T. Beyersdorf29 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA T. Bhadbade27 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA I. A. Bilenko30
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G.
Billingsley1 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J. Birch6 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Biscans24 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Bitossi14a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. A. Bizouard31a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. Black1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. K. Blackburn1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA L. Blackburn32 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. Blair33 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA B. Bland18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Blom11a 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
O. Bock9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA T. P. Bodiya24 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. Bogan9,10 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA C. Bond19 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA F. Bondu34b 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA L. Bonelli14ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. Bonnand35 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA R. Bork1 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA M. Born9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA V. Boschi14a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Bose36 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA L. Bosi37a 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA B. Bouhou25
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Bowers2 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. Bradaschia14a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA P. R. Brady13 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. B. Braginsky30 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Branchesi38ab 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. E. Brau39 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. Breyer9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA T.
Briant40 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. O. Bridges6 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA A. Brillet34a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Brinkmann9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. Brisson31a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Britzger9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. F. Brooks1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. A. Brown23
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. D.
Brown19 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
F. Brueckner19 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA K. Buckland1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA T. Bulik28b 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA H. J. Bulten11ab 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Buonanno41 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. Burguet–Castell42 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA D. Buskulic4 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. Buy25 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. L. Byer27
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA L.
Cadonati43 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
G. Cagnoli35,44 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA E. Calloni5ab 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J. B. Camp32 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA P. Campsie3 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA K. Cannon45 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. Canuel22 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. Cao46 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. D. Capano41
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Carbognani22 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Carbone19 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Caride47 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA A. D. Castiglia48 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Caudill13 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Cavaglià16 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA F. Cavalier31a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. Cavalieri22
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G.
Cella14a 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. Cepeda1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. Cesarini49a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA T. Chalermsongsak1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA S. Chao101 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA P. Charlton50 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. Chassande-Mottin25 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA X. Chen33 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA Y. Chen51 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. Chincarini52 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Chiummo22
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA H. S.
Cho53 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Chow54 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA N.
Christensen55 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA Q. Chu33 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. S. Y. Chua54 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. T. Y. Chung56 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA G. Ciani15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA F. Clara18 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. E. Clark27 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. A. Clark43 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA F. Cleva34a 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA E. Coccia49ab
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA P.-F.
Cohadon40 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. N. Colacino14ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Colla17ab 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA M. Colombini17b 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Constancio Jr.12 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. Conte17ab 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. Cook18 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA T. R. Corbitt2 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Cordier29 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA N. Cornish21
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Corsi103 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. A. Costa2,12 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Coughlin57 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J.-P. Coulon34a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Countryman26 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. Couvares23 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. M. Coward33
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Cowart6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. C. Coyne1 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA K. Craig3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J. D. E. Creighton13 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA T. D. Creighton44 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. Cumming3 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA L. Cunningham3 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA E. Cuoco22 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA K. Dahl9,10 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA M. Damjanic9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S. L.
Danilishin33 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. D’Antonio49a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA K. Danzmann9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. Dattilo22 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA B. Daudert1 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA H. Daveloza44 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Davier31a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA G. S. Davies3
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA E. J.
Daw58 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA T.
Dayanga36 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. De Rosa5ab 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA G. Debreczeni59 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J. Degallaix35 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA W. Del Pozzo11a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. Deleeuw15 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA T. Denker10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA T. Dent9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA V. Dergachev1
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA R.
DeRosa2 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. DeSalvo8 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Dhurandhar60 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. Di Fiore5a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Di Lieto14ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA I. Di Palma9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. Di Virgilio14a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Díaz44 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Dietz4,16
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Donovan24 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
K. L. Dooley9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Doravari1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA M. Drago61ab 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Drasco20 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. W. P. Drever62 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. C. Driggers1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA Z. Du46 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J.-C. Dumas33
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S.
Dwyer24 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Eberle9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA M. Edwards7 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Effler2 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA P. Ehrens1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. S. Eikenberry15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Endrőczi59 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Engel1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Essick24 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA T. Etzel1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA K. Evans3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Evans24 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA T. Evans6 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA M.
Factourovich26 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA V. Fafone49ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Fairhurst7 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA Q. Fang33 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA B. F. Farr63 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA W. Farr63 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Favata13 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. Fazi63 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA H. Fehrmann9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA D. Feldbaum15 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA I. Ferrante14ab
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Ferrini22 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
F. Fidecaro14ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. S. Finn64 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA I. Fiori22 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA R. P. Fisher23 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. Flaminio35 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA S. Foley24 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA E. Forsi6 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA L. A. Forte5a
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA N.
Fotopoulos1 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J.-D. Fournier34a 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA J. Franc35 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Franco31a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Frasca17ab 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA F. Frasconi14a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Frede9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M. A.
Frei48 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA Z.
Frei65 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Freise19 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Frey39 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. T. Fricke9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA D. Friedrich9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA P. Fritschel24 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA V. V. Frolov6 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M.-K. Fujimoto66 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA P. J. Fulda15
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M. Fyffe6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J. Gair57
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Galimberti35 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Gammaitoni37ab 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA J. Garcia18 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA F. Garufi5ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. E. Gáspár59 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA N. Gehrels32 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA G. Gelencser65
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G.
Gemme52 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. Genin22 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. Gennai14a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Á. Gergely67 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Ghosh36 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA J. A. Giaime2,6 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Giampanis13 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA K. D. Giardina6 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. Giazotto14a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Gil-Casanova42
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C. Gill3
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Gleason15 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. Goetz9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA G. González2 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA N. Gordon3 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA M. L. Gorodetsky30 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Gossan51 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Goßler9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Gouaty4 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. Graef9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA P. B.
Graff32 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
M. Granata35 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA A. Grant3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Gras24 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA C. Gray18 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R. J. S. Greenhalgh68 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. M. Gretarsson69 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA C. Griffo70 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA H. Grote9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA K. Grover19 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Grunewald20
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G. M.
Guidi38ab 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. Guido6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. K. Gustafson1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA R. Gustafson47 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA D. Hammer13 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Hammond3 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA J. Hanks18 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA C. Hanna71 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. Hanson6 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA K. Haris72
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Harms62 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
G. M. Harry73 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA I. W. Harry23 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA E. D. Harstad39 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. T. Hartman15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA K. Haughian3 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA K. Hayama66 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. Heefner1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Heidmann40
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M. C.
Heintze6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
H. Heitmann34a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA P. Hello31a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA G. Hemming22 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA M. A. Hendry3 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA I. S. Heng3 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. W. Heptonstall1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Heurs9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Hewitson9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S. Hild3
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. Hoak43
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA K. A.
Hodge1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA K.
Holt6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Holtrop74 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Hong51 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
S. Hooper33 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J. Hough3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA E. J. Howell33 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA V. Huang101 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA E. A. Huerta23 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA B. Hughey69 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. H. Huttner3 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Huynh13 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA T. Huynh–Dinh6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. R.
Ingram18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Inta54 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Isogai24 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA A. Ivanov1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA B. R. Iyer75 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA K. Izumi66 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Jacobson1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. James1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA H. Jang76 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA Y. J. Jang63 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA P. Jaranowski28d 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA E. Jesse69 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA W. W. Johnson2
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Jones18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. I. Jones77 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R. Jones3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R.J.G. Jonker11a 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. Ju33 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA P. Kalmus1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA V. Kalogera63 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Kandhasamy78 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Kang76 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. B. Kanner32 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Kasprzack22,31a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. Kasturi79
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA E.
Katsavounidis24 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA W. Katzman6 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA H. Kaufer9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA K. Kawabe18 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Kawamura66 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA F. Kawazoe9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. Keitel9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Kelley23 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
W. Kells1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. G. Keppel9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Khalaidovski9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA F. Y. Khalili30 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. A. Khazanov80 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. K. Kim76 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. Kim76 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA K. Kim81 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA N. Kim27 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA Y. M. Kim53 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA P. J. King1
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. L.
Kinzel6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
J. S. Kissel24 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Klimenko15 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J. Kline13 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA K. Kokeyama2 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. Kondrashov1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Koranda13 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA W. Z. Korth1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA I. Kowalska28b 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. Kozak1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. Kozameh82
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Kremin78 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
V. Kringel9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA B. Krishnan10,9 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Królak28ae 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. Kucharczyk27 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Kuehn9,10 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA P. Kumar23 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Kumar3 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. J. Kuper70 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. Kurdyumov27
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA P. Kwee24
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Landry18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
B. Lantz27 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
P. D. Lasky56 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA C. Lawrie3 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Lazzarini1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Le Roux6 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA P. Leaci20 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA C. H. Lee53 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA H. K. Lee81 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA H. M. Lee83 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA J. Lee70 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA J. R. Leong9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA N.
Leroy31a 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
N. Letendre4 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA B. Levine18 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA V. Lhuillier18 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA T. G. F. Li11a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. C. Lin27 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA V. Litvine1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA Y. Liu46 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA Z. Liu15 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA N. A. Lockerbie84 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. Lodhia19 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA K. Loew69 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA J. Logue3 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA A. L.
Lombardi41 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
M. Lorenzini49ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA V. Loriette31b 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Lormand6 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Losurdo38a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. Lough23 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Lubinski18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA H.
Lück9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. P. Lundgren9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J. Macarthur3 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA E. Macdonald7 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA B. Machenschalk9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. MacInnis24 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. M. Macleod7
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Magana-Sandoval70 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Mageswaran1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA K. Mailand1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. Majorana17a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA I. Maksimovic31b 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA V. Malvezzi49a
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA N. Man34a
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G.
Manca20 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
I. Mandel19 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA V. Mandic78 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Mantovani14a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA F. Marchesoni37ac 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA F. Marion4 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA S. Márka26 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA Z. Márka26 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Markosyan27
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA E. Maros1
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Marque22 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
F. Martelli38ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA I. W. Martin3 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA R. M. Martin15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA D. Martonov1 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA J. N. Marx1 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA K. Mason24 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Masserot4
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Matichard24 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Matone26 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA R. A. Matzner85 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA N. Mavalvala24 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. May15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Mazzolo9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA K. McAuley29 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. McCarthy18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. E.
McClelland54 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. C. McGuire86 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA G. McIntyre1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA J. McIver43 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA G. D. Meadors47 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Mehmet9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. Meidam11a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA T. Meier10,9
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Melatos56 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
G. Mendell18 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R. A. Mercer13 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Meshkov1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA C. Messenger7 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. S. Meyer6 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA H. Miao51 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA C. Michel35 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA L. Milano5ab 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. Miller54 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA Y. Minenkov49a
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C. M. F.
Mingarelli19 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Mitra60 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA V. P. Mitrofanov30 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA G. Mitselmakher15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. Mittleman24 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. Moe13 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Mohan22 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. R. P.
Mohapatra23,48 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA F. Mokler10,9 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA D. Moraru18 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA G. Moreno18 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA N. Morgado35 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA T. Mori66 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. R. Morriss44 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA K. Mossavi9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA B. Mours4 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. M.
Mow–Lowry9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA C. L. Mueller15 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA G. Mueller15 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA S. Mukherjee44 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Mullavey2,54 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. Munch87 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. Murphy26 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. G. Murray3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Mytidis15
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D. Nanda
Kumar15 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Nash1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
L. Naticchioni17ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA R. Nayak88 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA V. Necula15 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA I. Neri37ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Newton3 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA T. Nguyen54 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA E. Nishida66 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. Nishizawa66
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A. Nitz23
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Nocera22 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. Nolting6 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA M. E. Normandin44 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA L. Nuttall7 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA E. Ochsner13 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. O’Dell68 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA E. Oelker24 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA G. H. Ogin1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. J. Oh89 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA S. H. Oh89
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F. Ohme7
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA P.
Oppermann9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA B. O’Reilly6 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA R. O’Shaughnessy13 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. Osthelder1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA C. D. Ott51 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. J. Ottaway87 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. S. Ottens15 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. Ou101 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA H. Overmier6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA B. J.
Owen64 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C.
Padilla70 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. Page19 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. Pai72 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
L. Palladino49ac 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. Palomba17a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA Y. Pan41 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. Pankow13 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA F. Paoletti14a,22 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Paoletti14ac 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. A. Papa20,13
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA H.
Paris18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
M. Parisi5ab 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA W. Parkinson90 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Pasqualetti22 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA R. Passaquieti14ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA D. Passuello14a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Pedraza1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Penn79 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. Peralta20
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Perreca23 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
M. Phelps1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
M. Pichot34a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA M. Pickenpack9,10 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA F. Piergiovanni38ab 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. Pierro8 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA L. Pinard35 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA I. M. Pinto8 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Pitkin3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA H. J. Pletsch9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA R.
Poggiani14ab 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J. Pöld9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA F. Postiglione91 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. Poux1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA V. Predoi7 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA T. Prestegard78 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA L. R. Price1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Prijatelj9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S.
Privitera1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
G. A. Prodi61ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. G. Prokhorov30 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA O. Puncken44 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Punturo37a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. Puppo17a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA V. Quetschke44
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA E.
Quintero1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Quitzow-James39 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA F. J. Raab18 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA D. S. Rabeling11ab 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA I. Rácz59 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA H. Radkins18 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA P. Raffai26 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA S. Raja92 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA M. Rakhmanov44 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA C. Ramet6 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. Rapagnani17ab 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA V. Raymond1 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA V. Re49ab 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA C. M. Reed18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA T. Reed93
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA T.
Regimbau34a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Reid94 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA D. H. Reitze1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA F. Ricci17ab 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA R. Riesen6 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA K. Riles47 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Roberts27 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA N. A. Robertson1,3 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA F. Robinet31a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA E. L. Robinson20
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Rocchi49a 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
S. Roddy6 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. Rodriguez63 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Rodriguez85 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Rodruck18 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA L. Rolland4 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J. G. Rollins1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. D. Romano44 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Romano5ac 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J. H. Romie6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Rosińska28cf 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA C. Röver9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA S. Rowan3 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA A. Rüdiger9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA P. Ruggi22 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA K. Ryan18 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA F. Salemi9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA L. Sammut56 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA V. Sandberg18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Sanders47 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
S. Sankar24 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA V. Sannibale1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. Santamaría1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA I. Santiago-Prieto3 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA E. Saracco35 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA B. Sassolas35 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. S. Sathyaprakash7 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA P. R. Saulson23
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA R. L.
Savage18 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Schilling9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA R. Schnabel9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA R. M. S. Schofield39 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA D. Schuette9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA B. Schulz9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA B. F. Schutz20,7
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA P.
Schwinberg18 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA J. Scott3 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. M. Scott54 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA F. Seifert1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA D. Sellers6 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. S. Sengupta95 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA D. Sentenac22 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. Sergeev80 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA D. A. Shaddock54
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S. Shah96
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Shaltev9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA Z. Shao1 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA B. Shapiro27 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA P. Shawhan41 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA D. H. Shoemaker24 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA T. L Sidery19 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA X. Siemens13 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. Sigg18 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA D. Simakov9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. Singer1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA L. Singer1 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA A. M. Sintes42
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA G. R.
Skelton13 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
B. J. J. Slagmolen54 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J. Slutsky9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J. R. Smith70 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. R. Smith1 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA R. J. E. Smith19 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA N. D. Smith-Lefebvre1 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA E. J. Son89
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA B.
Sorazu3 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Souradeep60 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Sperandio49ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. Stefszky54 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA E. Steinert18 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. Steinlechner9,10 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA S. Steinlechner9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Steplewski36
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Stevens63 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
A. Stochino54 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA R. Stone44 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA K. A. Strain3 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. E. Strigin30 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. S. Stroeer44 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA R. Sturani38ab 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA A. L. Stuver6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA T. Z.
Summerscales97 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. Susmithan33 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA P. J. Sutton7 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA B. Swinkels22 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA G. Szeifert65 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Tacca22 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA L. Taffarello61c
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA D.
Talukder39 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
D. B. Tanner15 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. P. Tarabrin9,10 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA R. Taylor1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. P. M. ter Braack11a 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA M. Thomas6 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA P. Thomas18 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA K. A. Thorne6
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA K. S.
Thorne51 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. Thrane1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
V. Tiwari15 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA K. V. Tokmakov84 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. Tomlinson58 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Toncelli14ab 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. Tonelli14ab 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA O. Torre14ac 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA C. V. Torres44
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA C. I.
Torrie1,3 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
E. Tournefier4 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA F. Travasso37ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA G. Traylor6 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA M. Tse26 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA D. Ugolini98 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. S. Unnikrishnan99 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA H. Vahlbruch10,9 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA G. Vajente14ab 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Vallisneri51
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J. F. J.
van den Brand11ab 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. Van Den Broeck11a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. van der Putten11a 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA M. V. van der Sluys63 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA A. A. van Veggel3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Vass1 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Vasuth59 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA R. Vaulin24
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Vavoulidis31a 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA A. Vecchio19 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA G. Vedovato61c 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J. Veitch7 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA K. Venkateswara100 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA D. Verkindt4 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA S. Verma33 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA F. Vetrano38ab
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A.
Viceré38ab 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Vincent-Finley86 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA J.-Y. Vinet34a 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA S. Vitale24 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Vitale11a 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA T. Vo18 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA H. Vocca37a 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA C. Vorvick18 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA W. D. Vousden19
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA S. P.
Vyatchanin30 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA A. Wade54 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Wade13 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA M. Wade13 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA S. J. Waldman24 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA L. Wallace1 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA Y. Wan46 1LIGO - California Institute of Technology, Pasadena,
CA 91125, USA J. Wang101 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA M. Wang19 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA X. Wang46 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA A. Wanner9,10 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. L. Ward25,54 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Was9,10,31a 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA M. Weinert9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA A. J.
Weinstein1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
R. Weiss24 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
T. Welborn6 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA L. Wen33 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA P. Wessels9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. West23 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA T. Westphal9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA K. Wette9,10 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA J. T. Whelan48 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA D. J. White58 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA B. F. Whiting15
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA K.
Wiesner9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA C. Wilkinson18 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA P. A. Willems1 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA L. Williams15 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA R. Williams1 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA T. Williams90 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA J. L. Willis102 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA B. Willke9,10
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA M.
Wimmer9,10 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
L. Winkelmann9,10 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA W. Winkler9,10 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA C. C. Wipf24 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA A. G. Wiseman13 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA H. Wittel9,10 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA G. Woan3 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA R. Wooley6 1LIGO
- California Institute of Technology, Pasadena, CA 91125, USA J. Worden18
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA J.
Yablon63 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
I. Yakushin6 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA H. Yamamoto1 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA C. C. Yancey41 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA H. Yang51 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA D. Yeaton-Massey1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA S. Yoshida90 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA H. Yum63 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA M. Yvert4 1LIGO - California Institute
of Technology, Pasadena, CA 91125, USA A. Zadrożny28e 1LIGO - California
Institute of Technology, Pasadena, CA 91125, USA M. Zanolin69 1LIGO -
California Institute of Technology, Pasadena, CA 91125, USA J.-P. Zendri61c
1LIGO - California Institute of Technology, Pasadena, CA 91125, USA F.
Zhang24 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
L. Zhang1 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
C. Zhao33 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
H. Zhu64 1LIGO - California Institute of Technology, Pasadena, CA 91125, USA
X. J. Zhu33 1LIGO - California Institute of Technology, Pasadena, CA 91125,
USA N. Zotov93 1LIGO - California Institute of Technology, Pasadena, CA
91125, USA M. E. Zucker24 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA J. Zweizig1 1LIGO - California Institute of
Technology, Pasadena, CA 91125, USA (The LIGO Scientific Collaboration and
the Virgo Collaboration) 1LIGO - California Institute of Technology,
Pasadena, CA 91125, USA
###### Abstract
We present a possible observing scenario for the Advanced LIGO and Advanced
Virgo gravitational wave detectors over the next decade, with the intention of
providing information to the astronomy community to facilitate planning for
multi-messenger astronomy with gravitational waves. We determine the expected
sensitivity of the network to transient gravitational-wave signals, and study
the capability of the network to determine the sky location of the source. For
concreteness, we focus primarily on gravitational-wave signals from the
inspiral of binary neutron star (BNS) systems, as the source considered likely
to be the most common for detection and also promising for multimessenger
astronomy. We find that confident detections will likely require at least 2
detectors operating with BNS sensitive ranges of at least 100 Mpc, while
ranges approaching 200 Mpc should give at least $\sim$1 BNS detection per year
even under pessimistic predictions of signal rates. The ability to localize
the source of the detected signals depends on the geographical distribution of
the detectors and their relative sensitivity, and can be as large as thousands
of square degrees with only 2 sensitive detectors operating. Determining the
sky position of a significant fraction of detected signals to areas of 5 deg2
to 20 deg2 will require at least 3 detectors of sensitivity within a factor of
$\sim$ 2 of each other and with a broad frequency bandwidth. Should one of the
LIGO detectors be relocated in India as expected, many gravitational-wave
signals will be localized to a few square degrees by gravitational-wave
observations alone.
## 1 Introduction
Advanced LIGO (aLIGO) [1] and Advanced Virgo (AdV) [2, 3] are kilometer-scale
gravitational wave (GW) detectors that are expected to yield direct
observations of gravitational waves. In this document we describe the
currently projected schedule, sensitivity, and sky localization accuracy for
the GW detector network. The purpose of this document is to provide
information to the astronomy community to assist in the formulation of plans
for the upcoming era of GW observations. In particular, we intend this
document to provide the information required for assessing the features of
programs for joint observation of GW events using electromagnetic, neutrino,
or other observing facilities.
The full science of aLIGO and AdV is broad [4], and is not covered in this
document. We concentrate solely on candidate GW transient signals. We place
particular emphasis on the coalescence of neutron-star binary systems, which
are the GW source with the most reliable predictions on the prospects of
detection.
Although our collaborations have amassed a great deal of experience with GW
detectors and analysis, it is still very difficult to make predictions for
both improvements in search methods and for the rate of progress for detectors
which are not yet fully installed or operational. We stress that the scenarios
of LIGO and Virgo detector sensitivity evolution and observing times given
here represent our best estimates at present. They should not be considered as
fixed or firm commitments. As the detectors’ construction and commissioning
progresses, we intend to release updates versions of this document.
## 2 Commissioning and Observing Phases
We divide the roadmap for the aLIGO and AdV observatories into three phases:
1. 1.
Construction includes the installation and testing of the detectors. This
phase ends with acceptance of the detectors. Acceptance means that the
interferometers can lock for periods of hours: light is resonant in the arms
of the interferometer with _no guaranteed gravitational-wave sensitivity._
Construction will likely involve several short engineering runs with no
expected astrophysical output as the detectors progress towards acceptance.
2. 2.
Commissioning will take the detectors from their configuration at acceptance
through progressively better sensitivity to the ultimate second-generation
detector sensitivity. Engineering and science runs in the commissioning phase
will allow us to understand our detectors and analyses in an observational
mode. It is expected that science runs will produce astrophysical results,
including upper limits on the rate of sources and quite possibly the first
detections of GWs. During this phase, exchange of GW candidates with partners
outside the LSC and Virgo collaborations will be governed by memoranda of
understanding (MOUs) [5].
3. 3.
Observing runs begin when the detectors are at a sensitivity which makes
detections likely. We anticipate that there will be a gradual transition from
the commissioning to the observing phases. If it has not happened previously,
the first few GW signals will be observed and the LSC and Virgo will be
engaged in a long-term campaign to observe the GW sky. After the first four
detections [5] we expect free exchange of GW event candidates with the
astronomical community and the maturation of GW astronomy.
The progress in sensitivity as a function of time will affect the duration of
the runs that we plan at any stage, as we strive to minimize the time to the
first gravitational wave detection. Commissioning is a complex process which
involves both scheduled improvements to the detectors and tackling unexpected
new problems. While our experience makes us cautiously optimistic regarding
the schedule for the advanced detectors, we note that we are targeting an
order of magnitude improvement in sensitivity relative to the previous
generation of detectors over a much wider frequency band. Consequently it is
not possible to make concrete predictions for sensitivity as a function of
time. We can, however, use our previous experience as a guide to plausible
scenarios for the detector operational states that will allow us to reach the
desired sensitivity. Unexpected problems could slow down the commissioning,
but there is also the possibility that progress may happen faster than
predicted here. As the detectors begin to be commissioned, information on the
cost in time and benefit in sensitivity will become more apparent and drive
the schedule of runs. More information on event rates, including the first
detection, will also very likely change the schedule and duration of runs. In
section 2.1 we present the commissioning plans for the aLIGO and AdV
detectors. A summary of expected science runs is in section 2.2.
### 2.1 Commissioning and Observing Roadmap
The anticipated strain sensitivity evolution for aLIGO and AdV is shown in
Fig. 1. A standard figure of merit for the sensitivity of an interferometer is
the binary neutron star (BNS) range: the volume- and orientation-averaged
distance at which a compact binary coalescence consisting of two
$\mathrm{1.4\,M_{\odot}}$ neutron stars gives a matched filter signal-to-noise
ratio of 8 in a single detector [6]111 Another often quoted number is the BNS
_horizon_ —the distance at which an optimally oriented and located BNS system
would be observed with a signal to noise ratio of 8. The horizon is a factor
of 2.26 larger than the range. . The BNS ranges for the various stages of
aLIGO and AdV expected evolution are also provided in Fig. 1.
Figure 1: aLIGO (left) and AdV (right) target strain sensitivity as a
function of frequency. The average distance to which binary neutron star (BNS)
signals could be seen is given in Mpc. Current notions of the progression of
sensitivity are given for early, middle, and late commissioning phases, as
well as the final design sensitivity target and the BNS-optimized sensitivity.
While both dates and sensitivity curves are subject to change, the overall
progression represents our best current estimates.
The installation of aLIGO is well underway. The plan calls for three identical
4 km interferometers, referred to as H1, H2, and L1. In 2011, the LIGO Lab and
IndIGO consortium in India proposed installing one of the aLIGO Hanford
detectors, H2, at a new observatory in India (LIGO-India). As of early 2013
LIGO Laboratory has begun preparing the H2 interferometer for shipment to
India. Funding for the Indian portion of LIGO-India is in the final stages of
consideration by the Indian government.
The first aLIGO science run is expected in 2015. It will be of order three
months in duration, and will involve the H1 and L1 detectors (assuming H2 is
placed in storage for LIGO-India). The detectors will _not_ be at full design
sensitivity; we anticipate a possible BNS range of 40 – 80 Mpc. Subsequent
science runs will have increasing duration and sensitivity. We aim for a BNS
range of 80 – 170 Mpc over 2016–18, with science runs of several months.
Assuming that no unexpected obstacles are encountered, the aLIGO detectors are
expected to achieve a 200 Mpc BNS range circa 2019. After the first observing
runs, circa 2020, it might be desirable to optimize the detector sensitivity
for a specific class of astrophysical signals, such as BNSs. The BNS range may
then become 215 Mpc. The sensitivity for each of these stages is shown in Fig.
1.
Because of the planning for the installation of one of the LIGO detectors in
India, the installation of the H2 detector has been deferred. This detector
will be reconfigured to be identical to H1 and L1 and will be installed in
India once the LIGO-India Observatory is complete. The final schedule will be
adopted once final funding approvals are granted. It is expected that the site
development would start in 2014, with installation of the detector beginning
in 2018. Assuming no unexpected problems, first runs are anticipated circa
2020 and design sensitivity at the same level as the H1 and L1 detectors is
anticipated for no earlier than 2022.
The commissioning timeline for AdV [3] is still being defined, but it is
anticipated that in 2015 AdV might join the LIGO detectors in their first
science run depending on the sensitivity attained. Following an early step
with sensitivity corresponding to a BNS range of 20 – 60 Mpc, commissioning is
expected to bring AdV to a 60 – 85 Mpc in 2017–18. A configuration upgrade at
this point will allow the range to increase to approximately 65 – 115 Mpc in
2018–20. The final design sensitivity, with a BNS range of 130 Mpc, is
anticipated circa 2021. The corresponding BNS-optimised range would be 145
Mpc. The sensitivity curves for the various AdV configurations are shown in
Fig. 1.
The GEO600 [7] detector will likely be operational in the early to middle
phase of the AdV and aLIGO science runs, i.e. from 2015–2017. The sensitivity
that potentially can be achieved by GEO in this timeframe is similar to the
AdV sensitivity of the early and mid scenarios at frequencies around 1 kHz and
above. Around 100 Hz GEO will be at least 10 times less sensitive than the
early AdV and aLIGO detectors.
Japan has recently begun the construction of an advanced detector, KAGRA [8].
KAGRA is designed to have a BNS range comparable to AdV at final sensitivity.
While we do not consider KAGRA in this document, we note that the addition of
KAGRA to the worldwide GW detector network will improve both sky coverage and
localization capabilities beyond those envisioned here.
### 2.2 Estimated observing schedule
Keeping in mind the mentioned important caveats about commissioning affecting
the scheduling and length of science runs, the following is a plausible
scenario for the operation of the LIGO-Virgo network over the next decade:
* •
2015: A 3 month run with the two-detector H1L1 network at early aLIGO
sensitivity (40 – 80 Mpc BNS range). Virgo in commissioning at $\sim$ 20 Mpc
with a chance to join the run.
* •
2016–17: A 6 month run with H1L1 at 80 – 120 Mpc and Virgo at 20 – 60 Mpc.
* •
2017–18: A 9 month run with H1L1 at 120 – 170 Mpc and Virgo at 60 – 85 Mpc.
* •
2019+: Three-detector network with H1L1 at full sensitivity of 200 Mpc and V1
at 65 – 130 Mpc.
* •
2022+: Four-detector H1L1V1+LIGO-India network at full sensitivity (aLIGO at
200 Mpc, AdV at 130 Mpc).
The observational implications of this scenario are discussed in section 4.
## 3 Searches for gravitational-wave transients
Data from gravitational wave detectors are searched for many types of possible
signals [4]. Here we focus on signals from compact binary coalescences (CBC),
including BNS systems, and on generic transient or burst signals. See [9, 10,
11] for recent observational results from LIGO and Virgo for such systems.
The gravitational waveform from a binary neutron star coalescence is well
modelled and matched filtering can be used to search for signals and measure
the system parameters. For systems containing black holes, or in which the
component spin is significant, uncertainties in the waveform model can reduce
the sensitivity of the search. Searches for bursts make few assumptions on the
signal morphology, using time-frequency decompositions to identify
statistically significant excess power transients in the data. Burst searches
generally perform best for short-duration signals ($\lesssim$1 s); their
astrophysical targets include core-collapse supernovae, magnetar flares, black
hole binary coalescence, cosmic string cusps, and possibly as-yet-unknown
systems.
In the era of advanced detectors, the LSC and Virgo will search in near real-
time for CBC and burst signals for the purpose of rapidly identifying event
candidates. A prompt notice of a potential GW transient by LIGO-Virgo might
enable followup observations in the electromagnetic spectrum. A first followup
program including low-latency analysis, event candidate selection, position
reconstruction and the sending of alerts to several observing partners
(optical, X-ray, and radio) was implemented and exercised during the 2009–2010
LIGO-Virgo science run [12, 13, 14]. Latencies of less than 1 hour were
achieved and we expect to improve this in the advanced detector era. Increased
detection confidence, improved sky localization, and identification of host
galaxy and redshift are just some of the benefits of joint GW-electromagnetic
observations. With this in mind, we focus on two points of particular
relevance for followup of GW events: the source localization afforded by a GW
network and the relationship between signal significance (or false alarm rate)
and localization.
### 3.1 Localization
The aLIGO-AdV network will determine the sky position of a GW transient source
mainly by triangulation using the observed time delays between sites [15, 16].
The effective single-site timing accuracy is approximately
$\sigma_{t}=\frac{1}{2\pi\rho\sigma_{f}}\,,$ (1)
where $\rho$ is the signal-to-noise ratio in the given detector and
$\sigma_{f}$ is the effective bandwidth of the signal in the detector,
typically of order $100$ Hz. Thus a typical timing accuracy is on the order of
$10^{-4}$ s (about $1/100$ of the light travel time between sites). This sets
the localization scale. Equation (1) ignores many other relevant issues such
as uncertainty in the emitted gravitational waveform, instrumental calibration
accuracies, and correlation of sky location with other binary parameters [15,
17, 18, 19, 20, 21]. While many of these will affect the measurement of the
time of arrival in individual detectors, such factors are largely common
between two similar detectors, so the time difference between the two
detectors is relatively uncorrelated with these “nuisance” parameters. The
triangulation approach therefore provides a good leading order estimate to
localizations.
Source localization using only timing for a 2-site network yields an annulus
on the sky; see Fig. 2. Additional information such as signal amplitude, spin,
and precessional effects can sometimes resolve this to only parts of the
annulus, but even then sources will only be localized to regions of hundreds
to thousands of square degrees. For three detectors, the time delays restrict
the source to two sky regions whose locations are mirror images in the plane
formed by the three detectors. It is often possible to eliminate one of these
regions by requiring consistent amplitudes in all detectors. For signals just
above the detection threshold, this typically yields regions with areas of
several tens of square degrees. If there is significant difference in
sensitivity between detectors, the source is less well localized and we may be
left with the majority of the annulus on the sky determined by the two most
sensitive detectors. With four or more detectors, timing information alone is
sufficient to localize to a single sky region, and the additional baselines
help to limit the region to under 10 square degrees for some signals.
Figure 2: Source localization by triangulation for the aLIGO-AdV network. The
locus of constant time delay (with associated timing uncertainty) between two
detectors forms an annulus on the sky concentric about the baseline between
the two sites. For three detectors, these annuli may intersect in two
locations. One is centered on the true source direction, $S$, while the other
($S^{\prime}$) is its mirror image with respect to the geometrical plane
passing through the three sites. For four or more detectors there is a unique
intersection region of all of the annuli. Figure adapted from [22].
From (1), it follows that the linear size of the localization ellipse scales
inversely with the signal to noise ratio (SNR) of the signal and the frequency
bandwidth of the signal in the detector. For GWs that sweep across the band of
the detector, such as binary merger signals, the effective bandwidth is $\sim
100$ Hz, determined by the most sensitive frequencies of the detector. For
shorter transients the bandwidth $\sigma_{f}$ depends on the specific signal.
For example, GWs emitted by various processes in core-collapse supernovae are
anticipated to have relatively large bandwidths, between 150-500 Hz [23, 24,
25, 26], largely independent of detector configuration. By contrast, the sky
localization region for narrowband burst signals may consist of multiple
disconnected regions; see for example [27, 12].
Finally, we note that some GW searches are triggered by electromagnetic
observations, and in these cases localization information is known a priori.
For example, in GW searches triggered by gamma-ray bursts [10] the triggering
satellite provides the localization. The rapid identification of a GW
counterpart to such a trigger could prompt further followups by other
observatories. This is of particular relevance to binary mergers, which are
considered the likely progenitors of most short gamma-ray bursts. It is
therefore important to have high-energy satellites operating during the
advanced detector era.
Finally, it is also worth noting that all GW data are stored permanently, so
that it is possible to perform retroactive analyses at any time.
### 3.2 Detection and False Alarm Rates
The rate of BNS coalescences is uncertain, but is currently predicted to lie
between $10^{-8}-10^{-5}$ Mpc-3 yr-1 [28]. This corresponds to between 0.4 and
400 signals above SNR 8 per year of observation for a single aLIGO detector at
final sensitivity [28]. The predicted observable rates for NS-BH and BBH are
similar. Expected rates for other transient sources are lower and/or less well
constrained.
The rate of false alarm triggers above a given SNR will depend critically upon
the data quality of the advanced detectors; non-stationary transients or
glitches will produce an elevated background of loud triggers. For low-mass
binary coalescence searches, the waveforms are well modelled and signal
consistency tests reduce the background significantly. For burst sources which
are not well modelled, or which spend only a short time in the detectors’
sensitive band, it is more difficult to distinguish between the signal and a
glitch, and so a reduction of the false alarm rate comes at a higher cost in
terms of reduced detection efficiency.
Figure 3: False alarm rate versus detection statistic for CBC and burst
searches on 2009-2010 LIGO-Virgo data. Left: Cumulative rate of background
events for the CBC search, as a function of the threshold ranking statistic
$\rho_{c}$ [9]. Right: Cumulative rate of background events for the burst
search, as a function of the coherent network amplitude $\eta$ [11]. In the
large-amplitude limit $\eta$ is related to the combined SNR by
$\rho_{c}\sim\sqrt{2K}\eta$, where $K$ is the number of detectors. The burst
events are divided into two sets based on their central frequency.
Figure 3 shows the noise background as a function of detection statistic for
the low-mass binary coalescence and burst searches with the 2009–2010 LIGO-
Virgo data [9, 11]. For binary mergers, the background rate decreases by a
factor of $\sim$100 for every unit increase in combined SNR $\rho_{c}$, with
no evidence of a tail even at low false alarm rates. Here, $\rho_{c}$ is a
combined, re-weighted SNR. The re-weighting is designed to reduce the SNR of
glitches while leaving signals largely unaffected. Consequently, for a signal
$\rho_{c}$ is essentially the root-sum-square of the SNRs in the individual
detectors.
We conservatively estimate a $\rho_{c}$ threshold of 12 is required for a
false rate below $\sim$ $10^{-2}$ yr-1 in aLIGO-AdV, where we have taken into
account trials factors due to the increase in the number of template waveforms
required to search the advanced detector data. In future sections, we quote
results for this threshold. A combined SNR of 12 corresponds to a single
detector SNR of 8.5 in each of two detectors or 7 in three detectors. At this
threshold we estimate approximately a quarter of detected signals can be
localized with 90% containment to areas of 20 deg2 or less by the H1L1V1
network at design sensitivity; see the 2019$+$ epoch in Table 1 for details.
For a background rate of 1 yr-1 (100 yr-1) the threshold $\rho_{c}$ decreases
by about 10% (20%), the number of signals above threshold increases by about
30% (90%), and the area localization for these low-threshold signals is
degraded by approximately 20% (60%).
Imperfections in the data can have a greater effect on the burst search. At
frequencies above 200 Hz the rate of background events falls off steeply as a
function of amplitude. At lower frequencies, however, the data often exhibit a
significant tail of loud background events that are not removed by multi-
detector consistency tests. While the extent of these tails varies, when
present they typically begin at rates of approximately 1 yr-1, hindering the
confident detection of low-frequency gravitational-wave transients. Although
the advanced detectors are designed with many technical improvements, we must
anticipate that burst searches will likely still have to deal with such tails
in some cases, particularly at low frequencies. The unambiguous observation of
an electromagnetic counterpart could increase the detection confidence in
these cases.
A study [27] of the localization capability of the burst search for the aLIGO-
AdV network using a variety of waveform morphologies finds that at an SNR of
$\rho_{c}\simeq 17$ (false rate of $\lesssim 0.1$ yr-1 from Fig. 3) the
typical error box area for 50% (90%) containment is approximately 40 deg2 (400
deg2). The median 50% containment area increases to 100 deg2 at
$\rho_{c}\simeq 12$, and drops to approximately 16 deg2 at $\rho_{c}\simeq
25$. These results are broadly consistent with a study of two burst detection
algorithms using real LIGO-Virgo data from 2009 [12], which shows that for
signals near the nominal search threshold (coherent network amplitude
$\eta\gtrsim 6$, corresponding $\rho_{c}\gtrsim 15$ [11]) median containment
regions are typically between 30 deg2 and 200 deg2, dropping to approximately
10 deg2 at large amplitudes. See Fig. 4 for an example.
Figure 4: (left) Plot of typical uncertainty region sizes for the burst
search, as a function of GW strain amplitude at Earth, for a mix of ad hoc
Gaussian, sine-Gaussian, and broadband white-noise burst waveforms [12]. The
“searched area” is the area of the skymap with a likelihood value greater than
the likelihood value at the true source location. The solid line represents
the median (50%) performance, while the upper and lower limits of the shaded
area show the 75% and 25% quartile values. The detection threshold of
$\eta~{}\simeq 6$ corresponds to signal root-sum-square amplitudes
($h_{\mathrm{rss}}^{2}=\int[h_{+}^{2}+h_{\times}^{2}]dt$) of approximately
$h_{\text{rss}}\sim 0.5\times 10^{-21}\,\text{Hz}^{-1/2}$ to $\sim 2\times
10^{-21}\,\text{Hz}^{-1/2}$ [11], depending on signal frequency. Median
uncertainty regions at these amplitudes are typically between 30 deg2 and 200
deg2. (right) Typical uncertainty region sizes for two specific signal models:
short-duration Gaussian-modulated sinusoids (sine-Gaussians) with central
frequency 153 Hz or 1053 Hz and bandwidths of 17 Hz or 117 Hz. The larger-
bandwidth signal is more precisely localized, as expected from the discussion
in Sect. 3.1. See [12] for more details.
## 4 Observing Scenario
In this section we estimate the sensitivity, possible number of detections,
and localization capability for each of the observing scenarios laid out in
section 2.2. We discuss each future science run in turn and also summarize the
results in Table 1.
We estimate the expected number of binary neutron star coalescence detections
using both the lower and upper estimates on the BNS source rate density,
$10^{-8}-10^{-5}$ Mpc-3 yr-1 [28]. Similar estimates may be made for neutron
star – black hole (NS-BH) binaries using the fact that the NS-BH range is
approximately a factor of 2 larger222This assumes a black hole mass of
$10\,M_{\odot}$. than the BNS range, though the uncertainty in the NS-BH
source rate density is slightly larger [28]. We assume a nominal $\rho_{c}$
threshold of 12, at which the expected false alarm rate is $10^{-2}$ yr-1.
However, such a stringent threshold may not be appropriate for selecting
candidates triggers for electromagnetic followup. For example, selecting CBC
candidates at thresholds corresponding to a higher background rate of 1 yr-1
(100 yr-1) would increase the number of true signals subject to
electromagnetic followup by about 30% (90%). The area localization for these
low-threshold signals is only fractionally worse than for the high-threshold
population – by approximately 20% (60%). The localization of NS-BH signals is
expected to be similar to that of BNS signals.
For typical burst sources the GW waveform is not well known. However, the
performance of burst searches is largely independent of the detailed waveform
morphology [11], allowing us to quote an approximate sensitive range
determined by the total energy $E_{\mathrm{GW}}$ emitted in GWs, the central
frequency $f_{0}$ of the burst, the detector noise spectrum $S(f_{0})$, and
the single-detector SNR threshold $\rho_{\mathrm{det}}$ [29]:
$D\simeq\left(\frac{G}{2\pi^{2}c^{3}}\frac{E_{\mathrm{GW}}}{S(f_{0})f_{0}^{2}\rho_{\mathrm{det}}^{2}}\right)^{\frac{1}{2}}\,.$
In this document we quote ranges using $E_{\mathrm{GW}}=10^{-2}M_{\odot}c^{2}$
and $f_{0}=150$ Hz. We note that $E_{\mathrm{GW}}=10^{-2}M_{\odot}c^{2}$ is an
optimistic value for GW emission by various processes (see e.g. [10]); for
other values the distance reach scales as $E_{\mathrm{GW}}^{1/2}$. We use a
single-detector SNR threshold of 8, corresponding to a typical $\rho_{c}\simeq
12$ and false alarm rates of $\sim$0.3 yr-1. Due to the tail of the low-
frequency background-rate-vs.-amplitude distribution in Fig. 3, we see that
varying the selection threshold from a background of $0.1$ yr-1
($\rho_{c}\gtrsim 15$) to even 3 yr-1 ($\rho_{c}\gtrsim 10$) would increase
the number of true signals selected for electromagnetic followup by a factor
$(15/10)^{3}\sim 3$, though the area localization for low-SNR bursts may be
particularly challenging.
The run durations discussed below are in calendar time. Based on prior
experience, we can reasonably expect a duty cycle of $\sim$80% for each
instrument after a few science runs. Assuming downtime periods are
uncorrelated among detectors, this means 50% coincidence time in a 3-detector
network. Our estimates of expected number of detections account for these duty
cycles. They also account for the uncertainty in the detector sensitive ranges
as indicated in Fig. 1.
### 4.1 2015 run: aLIGO 40 – 80 Mpc, AdV 20 Mpc
This is envisioned as the first advanced detector science run, lasting three
months. The aLIGO sensitivity is expected to be similar to the “early” curve
in Fig. 1, with a BNS range of 40 – 80 Mpc and a burst range of 40 – 60 Mpc.
The Virgo detector will be in commissioning, but may join the run with a
$\sim$ 20 Mpc BNS range.
A three month run gives a BNS search volume333 The search volume is ${4\over
3}\pi R^{3}\times T$, where $R$ is the range and $T$ the observing time. of
$(0.4-3)\times 10^{5}$ Mpc3 yr at the confident detection threshold of
$\rho_{c}=12$. We therefore expect $0.0004-3$ BNS detections. A detection is
likely only if the most optimistic astrophysical rates hold.
With the 2-detector H1-L1 network any detected events would not be well
localized, and even if AdV joins the run this will continue to be the case due
to its lower sensitivity. Follow-up observations of a GW signal would
therefore likely rely on localizations provided by another instrument, such as
a gamma-ray burst satellite.
### 4.2 2016–17 run: aLIGO 80 – 120 Mpc, AdV 20 – 60 Mpc
This is envisioned to be a six month run with three detectors. The aLIGO
performance is expected to be similar to the “mid” curve in Fig. 1, with a BNS
range of 80 – 120 Mpc and a burst range of 60 – 75 Mpc. The AdV range may be
similar to the “early” curve, approximately 20 – 60 Mpc for BNS and 20 – 40
Mpc for bursts. This gives a BNS search volume of $(0.6-2)\times 10^{6}$ Mpc3
yr, and an expected number of $0.006-20$ BNS detections. Source localization
for various points in the sky for CBC signals for the 3-detector network is
illustrated in Fig. 5.
### 4.3 2017–18 run: aLIGO 120 – 170 Mpc, AdV 60 – 85 Mpc
This is envisioned to be a nine month run with three detectors. The aLIGO
(AdV) sensitivity will be similar to the “late” (“mid”) curve of Fig. 1, with
BNS ranges of 120 – 170 Mpc and 60 – 85 Mpc respectively and burst ranges of
75 – 90 Mpc and 40 – 50 Mpc respectively. This gives a BNS search volume of
$(3-10)\times 10^{6}$ Mpc3 yr, and an expected $0.04-100$ BNS detections.
Source localization for CBC signals is illustrated in Fig. 5. While the
greater range compared to the 2016–17 run increases the expected number of
detections, the detector bandwidths are marginally smaller. This slightly
degrades the localization capability for a source at a fixed signal-to-noise
ratio.
### 4.4 2019+ run: aLIGO 200 Mpc, AdV 65 – 130 Mpc
At this point we anticipate extended runs with the detectors at or near design
sensitivity. The aLIGO detectors are expected to have a sensitivity curve
similar to the “design (2019)” curve of Fig. 1. AdV may be operating similarly
to the “late” curve, eventually reaching the “design” sensitivity c.2021. This
gives a per-year BNS search volume of $2\times 10^{7}$ Mpc3 yr, giving an
expected (0.2 - 200) confident BNS detections annually. Source localization
for CBC signals is illustrated in Fig. 5. The fraction of signals localized to
areas of a few tens of square degrees is greatly increased compared to
previous runs. This is due to the much larger detector bandwidths,
particularly for AdV; see Fig. 1.
### 4.5 2022+ run: aLIGO (including India) 200 Mpc, AdV 130 Mpc
The four-site network incorporating LIGO-India at design sensitivity will have
both improved sensitivity and better localization capabilities. The per-year
BNS search volume increases to $4\times 10^{7}$ Mpc3 yr, giving an expected
$0.4-400$ BNS detections annually. Source localization is illustrated in Fig.
5. The addition of a fourth detector site allows for good source localization
over the whole sky.
Figure 5: Network sensitivity and localization accuracy for face-on BNS systems with advanced detector networks. The ellipses show 90% confidence localization areas, and the red crosses show regions of the sky where the signal would not be confidently detected. The top two plots show the localization expected for a BNS system at 80 Mpc by the HLV network in the 2016–17 run (left) and 2017–18 run (right). The bottom two plots show the localization expected for a BNS system at 160 Mpc by the HLV network in the 2019+ run (left) and by the HILV network in 2022+ with all detectors at final design sensitivity (right). The inclusion of a fourth site in India provides good localization over the whole sky. | Estimated | $E_{\mathrm{GW}}=10^{-2}M_{\odot}c^{2}$ | | Number | % BNS Localized
---|---|---|---|---|---
| Run | Burst Range (Mpc) | BNS Range (Mpc) | of BNS | within
Epoch | Duration | LIGO | Virgo | LIGO | Virgo | Detections | $5\deg^{2}$ | $20\deg^{2}$
2015 | 3 months | 40 – 60 | – | 40 – 80 | – | 0.0004 – 3 | – | –
2016–17 | 6 months | 60 – 75 | 20 – 40 | 80 – 120 | 20 – 60 | 0.006 – 20 | 2 | 5 – 12
2017–18 | 9 months | 75 – 90 | 40 – 50 | 120 – 170 | 60 – 85 | 0.04 – 100 | 1 – 2 | 10 – 12
2019+ | (per year) | 105 | 40 – 80 | 200 | 65 – 130 | 0.2 – 200 | 3 – 8 | 8 – 28
2022+ (India) | (per year) | 105 | 80 | 200 | 130 | 0.4 – 400 | 17 | 48
Table 1: Summary of a plausible observing schedule, expected sensitivities,
and source localization with the advanced LIGO and Virgo detectors, which will
be strongly dependent on the detectors’ commissioning progress. The burst
ranges assume standard-candle emission of $10^{-2}M_{\odot}c^{2}$ in GWs at
150 Hz and scale as $E_{\mathrm{GW}}^{1/2}$. The burst and binary neutron star
(BNS) ranges and the BNS localizations reflect the uncertainty in the detector
noise spectra shown in Fig. 1. The BNS detection numbers also account for the
uncertainty in the BNS source rate density [28], and are computed assuming a
false alarm rate of $10^{-2}$ yr-1. Burst localizations are expected to be
broadly similar to those for BNS systems, but will vary depending on the
signal bandwidth. Localization and detection numbers assume an 80% duty cycle
for each instrument.
## 5 Conclusions
We have presented a possible observing scenario for the Advanced LIGO and
Advanced Virgo network of gravitational wave detectors, with emphasis on the
expected sensitivities and sky localization accuracies. This network is
expected to begin operations in 2015. Unless the most optimistic astrophysical
rates hold, two or more detectors with an average range of at least 100 Mpc
and with a run of several months will be required for detection.
Electromagnetic followup of GW candidates may help confirm GW candidates that
would not be confidently identified from GW observations alone. However, such
follow-ups would need to deal with large position uncertainties, with areas of
many tens to thousands of square degrees. This is likely to remain the
situation until late in the decade. Optimizing the EM follow-up and source
identification is an outstanding research topic. Triggering of focused
searches in GW data by EM-detected events can also help in recovering
otherwise hidden GW signals.
Networks with at least 2 detectors with sensitivities of the order of 200 Mpc
are expected to yield detections with a year of observation based purely on GW
data even under pessimistic predictions of signal rates. Sky localization will
continue to be poor until a third detector reaches a sensitivity within a
factor of $\sim$ 2 of the others and with a broad frequency bandwidth. With a
four-site detector network at final design sensitivity, we may expect a
significant fraction of GW signals to be localized to as well as a few square
degrees by GW observations alone.
The purpose of this document is to provide information to the astronomy
community to facilitate planning for multi-messenger astronomy with advanced
gravitational-wave detectors. While the scenarios described here are our best
current projections, they will likely evolve as detector installation and
commissioning progresses. We will therefore update this document regularly.
The authors gratefully acknowledge the support of the United States National
Science Foundation for the construction and operation of the LIGO Laboratory,
the Science and Technology Facilities Council of the United Kingdom, the Max-
Planck-Society, and the State of Niedersachsen/Germany for support of the
construction and operation of the GEO600 detector, and the Italian Istituto
Nazionale di Fisica Nucleare and the French Centre National de la Recherche
Scientifique for the construction and operation of the Virgo detector. The
authors also gratefully acknowledge the support of the research by these
agencies and by the Australian Research Council, the International Science
Linkages program of the Commonwealth of Australia, the Council of Scientific
and Industrial Research of India, the Istituto Nazionale di Fisica Nucleare of
Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria
d’Economia Hisenda i Innovació of the Govern de les Illes Balears, the
Foundation for Fundamental Research on Matter supported by the Netherlands
Organisation for Scientific Research, the Polish Ministry of Science and
Higher Education, the FOCUS Programme of Foundation for Polish Science, the
Royal Society, the Scottish Funding Council, the Scottish Universities Physics
Alliance, The National Aeronautics and Space Administration, the Carnegie
Trust, the Leverhulme Trust, the David and Lucile Packard Foundation, the
Research Corporation, and the Alfred P. Sloan Foundation. This document has
been assigned LIGO Document number P1200087, Virgo Document number
VIR-0288A-12.
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|
arxiv-papers
| 2013-04-02T15:40:39 |
2024-09-04T02:49:43.754218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "The LIGO Scientific Collaboration, the Virgo Collaboration, and the\n KAGRA Collaboration: B. P. Abbott, R. Abbott, T. D. Abbott, S. Abraham, F.\n Acernese, K. Ackley, C. Adams, V. B. Adya, C. Affeldt, M. Agathos, K.\n Agatsuma, N. Aggarwal, O. D. Aguiar, L. Aiello, A. Ain, P. Ajith, T. Akutsu,\n G. Allen, A. Allocca, M. A. Aloy, P. A. Altin, A. Amato, A. Ananyeva, S. B.\n Anderson, W. G. Anderson, M. Ando, S. V. Angelova, S. Antier, S. Appert, K.\n Arai, Koya Arai, Y. Arai, S. Araki, A. Araya, M. C. Araya, J. S. Areeda, M.\n Ar\\`ene, N. Aritomi, N. Arnaud, K. G. Arun, S. Ascenzi, G. Ashton, Y. Aso, S.\n M. Aston, P. Astone, F. Aubin, P. Aufmuth, K. AultONeal, C. Austin, V.\n Avendano, A. Avila-Alvarez, S. Babak, P. Bacon, F. Badaracco, M. K. M. Bader,\n S. W. Bae, Y. B. Bae, L. Baiotti, R. Bajpai, P. T. Baker, F. Baldaccini, G.\n Ballardin, S. W. Ballmer, S. Banagiri, J. C. Barayoga, S. E. Barclay, B. C.\n Barish, D. Barker, K. Barkett, S. Barnum, F. Barone, B. Barr, L. Barsotti, M.\n Barsuglia, D. Barta, J. Bartlett, M. A. Barton, I. Bartos, R. Bassiri, A.\n Basti, M. Bawaj, J. C. Bayley, M. Bazzan, B. B\\'ecsy, M. Bejger, I.\n Belahcene, A. S. Bell, D. Beniwal, B. K. Berger, G. Bergmann, S. Bernuzzi, J.\n J. Bero, C. P. L. Berry, D. Bersanetti, A. Bertolini, J. Betzwieser, R.\n Bhandare, J. Bidler, I. A. Bilenko, S. A. Bilgili, G. Billingsley, J. Birch,\n R. Birney, O. Birnholtz, S. Biscans, S. Biscoveanu, A. Bisht, M. Bitossi, M.\n A. Bizouard, J. K. Blackburn, C. D. Blair, D. G. Blair, R. M. Blair, S.\n Bloemen, N. Bode, M. Boer, Y. Boetzel, G. Bogaert, F. Bondu, E. Bonilla, R.\n Bonnand, P. Booker, B. A. Boom, C. D. Booth, R. Bork, V. Boschi, S. Bose, K.\n Bossie, V. Bossilkov, J. Bosveld, Y. Bouffanais, A. Bozzi, C. Bradaschia, P.\n R. Brady, A. Bramley, M. Branchesi, J. E. Brau, T. Briant, J. H. Briggs, F.\n Brighenti, A. Brillet, M. Brinkmann, V. Brisson, P. Brockill, A. F. Brooks,\n D. A. Brown, D. D. Brown, S. Brunett, A. Buikema, T. Bulik, H. J. Bulten, A.\n Buonanno, D. Buskulic, C. Buy, R. L. Byer, M. Cabero, L. Cadonati, G.\n Cagnoli, C. Cahillane, J. Calder\\'on Bustillo, T. A. Callister, E. Calloni,\n J. B. Camp, W. A. Campbell, M. Canepa, K. Cannon, K. C. Cannon, H. Cao, J.\n Cao, E. Capocasa, F. Carbognani, S. Caride, M. F. Carney, G. Carullo, J.\n Casanueva Diaz, C. Casentini, S. Caudill, M. Cavagli\\`a, F. Cavalier, R.\n Cavalieri, G. Cella, P. Cerd\\'a-Dur\\'an, G. Cerretani, E. Cesarini, O.\n Chaibi, K. Chakravarti, S. J. Chamberlin, M. Chan, M. L. Chan, S. Chao, P.\n Charlton, E. A. Chase, E. Chassande-Mottin, D. Chatterjee, M. Chaturvedi, K.\n Chatziioannou, B. D. Cheeseboro, C. S. Chen, H. Y. Chen, K. H. Chen, X. Chen,\n Y. Chen, Y. R. Chen, H.-P. Cheng, C. K. Cheong, H. Y. Chia, A. Chincarini, A.\n Chiummo, G. Cho, H. S. Cho, M. Cho, N. Christensen, H. Y. Chu, Q. Chu, Y. K.\n Chu, S. Chua, K. W. Chung, S. Chung, G. Ciani, A. A. Ciobanu, R. Ciolfi, F.\n Cipriano, A. Cirone, F. Clara, J. A. Clark, P. Clearwater, F. Cleva, C.\n Cocchieri, E. Coccia, P.-F. Cohadon, D. Cohen, R. Colgan, M. Colleoni, C. G.\n Collette, C. Collins, L. R. Cominsky, M. Constancio Jr., L. Conti, S. J.\n Cooper, P. Corban, T. R. Corbitt, I. Cordero-Carri\\'on, K. R. Corley, N.\n Cornish, A. Corsi, S. Cortese, C. A. Costa, R. Cotesta, M. W. Coughlin, S. B.\n Coughlin, J.-P. Coulon, S. T. Countryman, P. Couvares, P. B. Covas, E. E.\n Cowan, D. M. Coward, M. J. Cowart, D. C. Coyne, R. Coyne, J. D. E. Creighton,\n T. D. Creighton, J. Cripe, M. Croquette, S. G. Crowder, T. J. Cullen, A.\n Cumming, L. Cunningham, E. Cuoco, T. Dal Canton, G. D\\'alya, S. L.\n Danilishin, S. D'Antonio, K. Danzmann, A. Dasgupta, C. F. Da Silva Costa, L.\n E. H. Datrier, V. Dattilo, I. Dave, M. Davier, D. Davis, E. J. Daw, D. DeBra,\n M. Deenadayalan, J. Degallaix, M. De Laurentis, S. Del\\'eglise, W. Del Pozzo,\n L. M. DeMarchi, N. Demos, T. Dent, R. De Pietri, J. Derby, R. De Rosa, C. De\n Rossi, R. DeSalvo, O. de Varona, S. Dhurandhar, M. C. D\\'iaz, T. Dietrich, L.\n Di Fiore, M. Di Giovanni, T. Di Girolamo, A. Di Lieto, B. Ding, S. Di Pace,\n I. Di Palma, F. Di Renzo, A. Dmitriev, Z. Doctor, K. Doi, F. Donovan, K. L.\n Dooley, S. Doravari, I. Dorrington, T. P. Downes, M. Drago, J. C. Driggers,\n Z. Du, J.-G. Ducoin, P. Dupej, S. E. Dwyer, P. J. Easter, T. B. Edo, M. C.\n Edwards, A. Effler, S. Eguchi, P. Ehrens, J. Eichholz, S. S. Eikenberry, M.\n Eisenmann, R. A. Eisenstein, Y. Enomoto, R. C. Essick, H. Estelles, D.\n Estevez, Z. B. Etienne, T. Etzel, M. Evans, T. M. Evans, V. Fafone, H. Fair,\n S. Fairhurst, X. Fan, S. Farinon, B. Farr, W. M. Farr, E. J. Fauchon-Jones,\n M. Favata, M. Fays, M. Fazio, C. Fee, J. Feicht, M. M. Fejer, F. Feng, A.\n Fernandez-Galiana, I. Ferrante, E. C. Ferreira, T. A. Ferreira, F. Ferrini,\n F. Fidecaro, I. Fiori, D. Fiorucci, M. Fishbach, R. P. Fisher, J. M. Fishner,\n M. Fitz-Axen, R. Flaminio, M. Fletcher, E. Flynn, H. Fong, J. A. Font, P. W.\n F. Forsyth, J.-D. Fournier, S. Frasca, F. Frasconi, Z. Frei, A. Freise, R.\n Frey, V. Frey, P. Fritschel, V. V. Frolov, Y. Fujii, M. Fukunaga, M.\n Fukushima, P. Fulda, M. Fyffe, H. A. Gabbard, B. U. Gadre, S. M. Gaebel, J.\n R. Gair, L. Gammaitoni, M. R. Ganija, S. G. Gaonkar, A. Garcia, C.\n Garc\\'ia-Quir\\'os, F. Garufi, B. Gateley, S. Gaudio, G. Gaur, V. Gayathri, G.\n G. Ge, G. Gemme, E. Genin, A. Gennai, D. George, J. George, L. Gergely, V.\n Germain, S. Ghonge, Abhirup Ghosh, Archisman Ghosh, S. Ghosh, B. Giacomazzo,\n J. A. Giaime, K. D. Giardina, A. Giazotto, K. Gill, G. Giordano, L. Glover,\n P. Godwin, E. Goetz, R. Goetz, B. Goncharov, G. Gonz\\'alez, J. M. Gonzalez\n Castro, A. Gopakumar, M. L. Gorodetsky, S. E. Gossan, M. Gosselin, R. Gouaty,\n A. Grado, C. Graef, M. Granata, A. Grant, S. Gras, P. Grassia, C. Gray, R.\n Gray, G. Greco, A. C. Green, R. Green, E. M. Gretarsson, P. Groot, H. Grote,\n S. Grunewald, P. Gruning, G. M. Guidi, H. K. Gulati, Y. Guo, A. Gupta, M. K.\n Gupta, E. K. Gustafson, R. Gustafson, L. Haegel, A. Hagiwara, S. Haino, O.\n Halim, B. R. Hall, E. D. Hall, E. Z. Hamilton, G. Hammond, M. Haney, M. M.\n Hanke, J. Hanks, C. Hanna, M. D. Hannam, O. A. Hannuksela, J. Hanson, T.\n Hardwick, K. Haris, J. Harms, G. M. Harry, I. W. Harry, K. Hasegawa, C.-J.\n Haster, K. Haughian, H. Hayakawa, K. Hayama, F. J. Hayes, J. Healy, A.\n Heidmann, M. C. Heintze, H. Heitmann, P. Hello, G. Hemming, M. Hendry, I. S.\n Heng, J. Hennig, A. W. Heptonstall, M. Heurs, S. Hild, Y. Himemoto, T.\n Hinderer, Y. Hiranuma, N. Hirata, E. Hirose, D. Hoak, S. Hochheim, D. Hofman,\n A. M. Holgado, N. A. Holland, K. Holt, D. E. Holz, Z. Hong, P. Hopkins, C.\n Horst, J. Hough, E. J. Howell, C. G. Hoy, A. Hreibi, B. H. Hsieh, G. Z.\n Huang, P. W. Huang, Y. J. Huang, E. A. Huerta, D. Huet, B. Hughey, M. Hulko,\n S. Husa, S. H. Huttner, T. Huynh-Dinh, B. Idzkowski, A. Iess, B. Ikenoue, S.\n Imam, K. Inayoshi, C. Ingram, Y. Inoue, R. Inta, G. Intini, K. Ioka, B.\n Irwin, H. N. Isa, J.-M. Isac, M. Isi, Y. Itoh, B. R. Iyer, K. Izumi, T.\n Jacqmin, S. J. Jadhav, K. Jani, N. N. Janthalur, P. Jaranowski, A. C.\n Jenkins, J. Jiang, D. S. Johnson, A. W. Jones, D. I. Jones, R. Jones, R. J.\n G. Jonker, L. Ju, K. Jung, P. Jung, J. Junker, T. Kajita, C. V. Kalaghatgi,\n V. Kalogera, B. Kamai, M. Kamiizumi, N. Kanda, S. Kandhasamy, G. W. Kang, J.\n B. Kanner, S. J. Kapadia, S. Karki, K. S. Karvinen, R. Kashyap, M. Kasprzack,\n S. Katsanevas, E. Katsavounidis, W. Katzman, S. Kaufer, K. Kawabe, K.\n Kawaguchi, N. Kawai, T. Kawasaki, N. V. Keerthana, F. K\\'ef\\'elian, D.\n Keitel, R. Kennedy, J. S. Key, F. Y. Khalili, H. Khan, I. Khan, S. Khan, Z.\n Khan, E. A. Khazanov, M. Khursheed, N. Kijbunchoo, Chunglee Kim, C. Kim, J.\n C. Kim, J. Kim, K. Kim, W. Kim, W. S. Kim, Y.-M. Kim, C. Kimball, N. Kimura,\n E. J. King, P. J. King, M. Kinley-Hanlon, R. Kirchhoff, J. S. Kissel, N.\n Kita, H. Kitazawa, L. Kleybolte, J. H. Klika, S. Klimenko, T. D. Knowles, P.\n Koch, S. M. Koehlenbeck, G. Koekoek, Y. Kojima, K. Kokeyama, S. Koley, K.\n Komori, V. Kondrashov, A. K. H. Kong, A. Kontos, N. Koper, M. Korobko, W. Z.\n Korth, K. Kotake, I. Kowalska, D. B. Kozak, C. Kozakai, R. Kozu, V. Kringel,\n N. Krishnendu, A. Kr\\'olak, G. Kuehn, A. Kumar, P. Kumar, Rahul Kumar, R.\n Kumar, S. Kumar, J. Kume, C. M. Kuo, H. S. Kuo, L. Kuo, S. Kuroyanagi, K.\n Kusayanagi, A. Kutynia, K. Kwak, S. Kwang, B. D. Lackey, K. H. Lai, T. L.\n Lam, M. Landry, B. B. Lane, R. N. Lang, J. Lange, B. Lantz, R. K. Lanza, A.\n Lartaux-Vollard, P. D. Lasky, M. Laxen, A. Lazzarini, C. Lazzaro, P. Leaci,\n S. Leavey, Y. K. Lecoeuche, C. H. Lee, H. K. Lee, H. M. Lee, H. W. Lee, J.\n Lee, K. Lee, R. K. Lee, J. Lehmann, A. Lenon, M. Leonardi, N. Leroy, N.\n Letendre, Y. Levin, J. Li, K. J. L. Li, T. G. F. Li, X. Li, C. Y. Lin, F.\n Lin, F. L. Lin, L. C. C. Lin, F. Linde, S. D. Linker, T. B. Littenberg, G. C.\n Liu, J. Liu, X. Liu, R. K. L. Lo, N. A. Lockerbie, L. T. London, A. Longo, M.\n Lorenzini, V. Loriette, M. Lormand, G. Losurdo, J. D. Lough, C. O. Lousto, G.\n Lovelace, M. E. Lower, H. L\\\"uck, D. Lumaca, A. P. Lundgren, L. W. Luo, R.\n Lynch, Y. Ma, R. Macas, S. Macfoy, M. MacInnis, D. M. Macleod, A. Macquet, F.\n Maga\\~na-Sandoval, L. Maga\\~na Zertuche, R. M. Magee, E. Majorana, I.\n Maksimovic, A. Malik, N. Man, V. Mandic, V. Mangano, G. L. Mansell, M.\n Manske, M. Mantovani, F. Marchesoni, M. Marchio, F. Marion, S. M\\'arka, Z.\n M\\'arka, C. Markakis, A. S. Markosyan, A. Markowitz, E. Maros, A. Marquina,\n S. Marsat, F. Martelli, I. W. Martin, R. M. Martin, D. V. Martynov, K. Mason,\n E. Massera, A. Masserot, T. J. Massinger, M. Masso-Reid, S. Mastrogiovanni,\n A. Matas, F. Matichard, L. Matone, N. Mavalvala, N. Mazumder, J. J. McCann,\n R. McCarthy, D. E. McClelland, S. McCormick, L. McCuller, S. C. McGuire, J.\n McIver, D. J. McManus, T. McRae, S. T. McWilliams, D. Meacher, G. D. Meadors,\n M. Mehmet, A. K. Mehta, J. Meidam, A. Melatos, G. Mendell, R. A. Mercer, L.\n Mereni, E. L. Merilh, M. Merzougui, S. Meshkov, C. Messenger, C. Messick, R.\n Metzdorff, P. M. Meyers, H. Miao, C. Michel, Y. Michimura, H. Middleton, E.\n E. Mikhailov, L. Milano, A. L. Miller, A. Miller, M. Millhouse, J. C. Mills,\n M. C. Milovich-Goff, O. Minazzoli, Y. Minenkov, N. Mio, A. Mishkin, C.\n Mishra, T. Mistry, S. Mitra, V. P. Mitrofanov, G. Mitselmakher, R. Mittleman,\n O. Miyakawa, A. Miyamoto, Y. Miyazaki, K. Miyo, S. Miyoki, G. Mo, D. Moffa,\n K. Mogushi, S. R. P. Mohapatra, M. Montani, C. J. Moore, D. Moraru, G.\n Moreno, S. Morisaki, Y. Moriwaki, B. Mours, C. M. Mow-Lowry, Arunava\n Mukherjee, D. Mukherjee, S. Mukherjee, N. Mukund, A. Mullavey, J. Munch, E.\n A. Mu\\~niz, M. Muratore, P. G. Murray, K. Nagano, S. Nagano, A. Nagar, K.\n Nakamura, H. Nakano, M. Nakano, R. Nakashima, I. Nardecchia, T. Narikawa, L.\n Naticchioni, R. K. Nayak, R. Negishi, J. Neilson, G. Nelemans, T. J. N.\n Nelson, M. Nery, A. Neunzert, K. Y. Ng, S. Ng, P. Nguyen, W. T. Ni, D.\n Nichols, A. Nishizawa, S. Nissanke, F. Nocera, C. North, L. K. Nuttall, M.\n Obergaulinger, J. Oberling, B. D. O'Brien, Y. Obuchi, G. D. O'Dea, W. Ogaki,\n G. H. Ogin, J. J. Oh, S. H. Oh, M. Ohashi, N. Ohishi, M. Ohkawa, F. Ohme, H.\n Ohta, M. A. Okada, K. Okutomi, M. Oliver, K. Oohara, C. P. Ooi, P. Oppermann,\n Richard J. Oram, B. O'Reilly, R. G. Ormiston, L. F. Ortega, R. O'Shaughnessy,\n S. Oshino, S. Ossokine, D. J. Ottaway, H. Overmier, B. J. Owen, A. E. Pace,\n G. Pagano, M. A. Page, A. Pai, S. A. Pai, J. R. Palamos, O. Palashov, C.\n Palomba, A. Pal-Singh, Huang-Wei Pan, K. C. Pan, B. Pang, H. F. Pang, P. T.\n H. Pang, C. Pankow, F. Pannarale, B. C. Pant, F. Paoletti, A. Paoli, M. A.\n Papa, A. Parida, J. Park, W. Parker, D. Pascucci, A. Pasqualetti, R.\n Passaquieti, D. Passuello, M. Patil, B. Patricelli, B. L. Pearlstone, C.\n Pedersen, M. Pedraza, R. Pedurand, A. Pele, F. E. Pe\\~na Arellano, S. Penn,\n C. J. Perez, A. Perreca, H. P. Pfeiffer, M. 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"submitter": "LVK Publication",
"url": "https://arxiv.org/abs/1304.0670"
}
|
1304.0716
|
11institutetext: University of Bucharest 11email: [email protected]
# Fixed point theorems for nonconvex valued correspondences and applications
in game theory
Monica Patriche University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, 010014 Bucharest, Romania
###### Abstract
In this paper, we introduce several types of correspondences: weakly naturally
quasiconvex, *-weakly naturally quasiconvex, weakly biconvex and
correspondences with *–weakly convex graph and we prove some fixed point
theorems for these kinds of correspondences. As a consequence, using a version
of W. K. Kim’s quasi-point theorem, we obtain the existence of equilibria for
a quasi-game.
###### Keywords:
Fixed point theorem, correspondences with *–weakly convex graph, weakly
naturally quasiconvex correspondences, quasi game, quasi-quilibrium.
2010 Mathematics Subject Classification: 47H10, 91A47, 91A80.
## 1 Introduction
The aim of this paper is to prove some fixed points theorems for
correspondences which are not continuous or convex valued and to give
applications in game theory.
The significance of equilibrium theory stems from the fact that it develops
important tools (as fixed point and selection theorems) to prove the existence
of equilibrium for different types of games. In 1950, J. F. Nash [15] first
proved a theorem of equilibrium existence for games where the player’s
preferences were representable by continuous quasi-concave utilities. G.
Debreu’s works on the existence of equilibrium in a generalized N-person game
or on an abstract economy [6] were extended by several authors. In [16] W.
Shafer and H. Sonnenschein proved the existence of equilibrium of an economy
with finite dimensional commodity space and irreflexive preferences
represented as correspondences with open graph. N. C. Yannelis and N. D.
Prahbakar [19] developed new techniques based on selection theorems and fixed-
point theorems. Their main result concerns the existence of equilibrium when
the constraint and preference correspondences have open lower sections. They
worked within different frameworks (countable infinite number of agents,
infinite dimensional strategy spaces). K. J. Arrow and G. Debreu proved the
existence of Walrasian equilibrium in [3]. In [20], X. Z. Yuan proposed a
model of abstract economy more general than that introduced by Borglin and
Keing in [4].
Within the last years, a lot of authors generalized the classical model of
abstract economy. For example, K. Vind [18] defined the social system with
coordination, X. Z. Yuan [20] proposed the model of the general abstract
economy. Motivated by the fact that any preference of a real agent could be
unstable by the fuzziness of consumers’ behaviour or market situation, W. K.
Kim and K. K. Tan [12] defined the generalized abstract economies. Also W. K.
Kim [13] obtained a generalization of the quasi fixed-point theorem due to I.
Lefebvre [14], and as an application, he proved an existence theorem of
equilibrium for a generalized quasi-game with infinite number of agents. W. K.
Kim’s result concerns generalized quasi-games where the strategy sets are
metrizable subsets in locally convex linear topological spaces.
Biconvexity was studied by R. Aumann, S. Hart in [2] and J. Gorski, F.
Pfeuffer and K. Klamroth in [10].
An open problem of the fixed point theory is to prove the existence of fixed
points for correspondences without continuity or convex values. X. Ding and He
Yiran introduced in [7] the correspondences with weakly convex graph to prove
a fixed point theorem. The result concerning the existence of the affine
selection on a special type of sets (simplex) proves to be redundant, since a
correspondence $T:X\rightarrow 2^{Y}$ has an affine selection if and only if
it has a weakly convex graph. This result is stronger than it needs in order
to obtain a fixed point theorem. We try to weaken these conditions by defining
several types of correspondences which are not continuous or convex valued:
weakly naturally quasiconvex, *-weakly naturally quasiconvex, correspondences
with *–weakly convex graph and weakly biconvex correspondences. We prove fixed
point theorems for these kinds of correspondences and using a version of W. K.
Kim’s quasi-point theorem, we prove the existence of equilibria for a quasi-
game. We use the continuous selection technique introduced by N. C. Yannelis
and N. D. Prahbakar in [19].
The paper is organized in the following way: Section 2 contains preliminaries
and notation. The fixed point theorems are given in Section 3 and the
equilibrium theorems are stated in Section 4.
## 2 Preliminaries and notation
Throughout this paper, we shall use the following notations and definitions:
Let $A$ be a subset of a topological space $X$.
1\. 2A denotes the family of all subsets of $A$.
2\. cl $A$ denotes the closure of $A$ in $X$.
3\. If $A$ is a subset of a vector space, co$A$ denotes the convex hull of
$A$.
4\. If $F$, $T:$ $A\rightarrow 2^{X}$ are correspondences, then co$T$, cl $T$,
$T\cap F$ $:$ $A\rightarrow 2^{X}$ are correspondences defined by
$($co$T)(x)=$co$T(x)$, $($cl$T)(x)=$cl$T(x)$ and $(T\cap F)(x)=T(x)\cap F(x)$
for each $x\in A$, respectively.
5\. The graph of $T:X\rightarrow 2^{Y}$ is the set Gr$(T)=\\{(x,y)\in X\times
Y\mid y\in T(x)\\}$
6\. The correspondence $\overline{T}$ is defined by $\overline{T}(x)=\\{y\in
Y:(x,y)\in$clX×YGr$T\\}$ (the set clX×YGr$(T)$ is called the adherence of the
graph of T).
It is easy to see that cl$T(x)\subset\overline{T}(x)$ for each $x\in X.\vskip
6.0pt plus 2.0pt minus 2.0pt$
###### Definition 1
Let $X$, $Y$ be topological spaces and $T:X\rightarrow 2^{Y}$ be a
correspondence. $T$ is said to be upper semicontinuous if for each $x\in X$
and each open set $V$ in $Y$ with $T(x)\subset V$, there exists an open
neighborhood $U$ of $x$ in $X$ such that $T(x)\subset V$ for each $y\in U$.
Let $X\subset E_{1}$ and $Y\subset E_{2}$ be two nonempty convex sets,
$E_{1},E_{2}$ be topological vector spaces and let $B\subset X\times Y.$
###### Definition 2 (2)
The set $B\subset X\times Y$ is called a biconvex set on $X\times Y$ if the
section $B_{x}=\left\\{y\in Y:(x,y)\in B\right\\}$ is convex for every $x\in
X$ and the section $B_{y}=\left\\{x\in X:(x,y)\in B\right\\}$ is convex for
every $y\in Y.\vskip 6.0pt plus 2.0pt minus 2.0pt$
###### Definition 3 (2)
Let $(x_{i},y_{i})\in X\times Y$ for $i=1,2,...n.$ A convex combination
$(x,y)=\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}(x_{i},y_{i})$,
(with $\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}=1,$
$\lambda_{i}\geq 0$ $i=1,2,...,n$) is called biconvex combination if
$x_{1}=x_{2}=...=x_{n}=x$ or $y_{1}=y_{2}=...=y_{n}=y.\vskip 6.0pt plus 2.0pt
minus 2.0pt$
###### Theorem 2.1
(Aumann and Hart [2]). A set $B\subseteq X\times Y$ is biconvex if and only if
$B$ contains all biconvex combinations of its elements.
###### Definition 4 (2)
Let $A\subseteq X\times Y$ be a given set. The set
$H:=\mathop{\textstyle\bigcap}\\{A_{I}:A\subseteq A_{I},$
$A_{I}$ is biconvex} is called biconvex hull of $A$ and is denoted
biconv$(A).\vskip 6.0pt plus 2.0pt minus 2.0pt$
###### Theorem 2.2
(Aumann and Hart [2]). The biconvex hull of a set $A$ is biconvex.
Furthermore, it is the smallest biconvex set (in the sens of set inclusion),
which contains $A.\vskip 6.0pt plus 2.0pt minus 2.0pt$
###### Lemma 1
(Gorski, Pfeuffer and Klamroth [10]). Let $A\subseteq X\times Y$ be a given
set. Then biconv$(A)\subseteq$conv$(A).\vskip 6.0pt plus 2.0pt minus 2.0pt$
NOTATION. We denote the standard $(n-1)$\- dimensional simplex by
$\Delta_{n-1}=\\{(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\mathbb{R}^{n}:\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}=1,\lambda_{i}\geqslant
0,i=1,2,...,n\\}$.
## 3 Selection theorems and fixed point theorems
An open problem of the fixed point theory is to prove the existence of fixed
points for correspondences without continuity or convex values. In this
section we introduce some types of correspondences which are not continuous or
convex valued and prove selection theorems and fixed point theorems.
First, we introduce the concept of weakly naturally quasiconvex
correspondence.
###### Definition 5
Let $X,Y$ be nonempty convex subsets of topological vector spaces $E,$
respectively $F$. The correspondence $T:X\longrightarrow 2^{Y}$ is said to be
_weakly naturally quasiconvex (WNQ)_ if for each $n\in\mathbb{N}$ and for each
finite set $\\{x_{1},x_{2},...,x_{n}\\}\subset X$, there exists $y_{i}\in
T(x_{i})$ , $(i=1,2,...,n)$ and
$g=(g_{1},g_{2},...,g_{n}):\Delta_{n-1}\rightarrow\Delta_{n-1}$ a bijective
function depending on $x_{1},x_{2},...,x_{n}$ with $g_{i}$ continuous,
$g_{i}(1)=1,$ $g_{i}(0)=0$ for each $i=1,2,...n$, such that for every
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$, there exists $y\in
T(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}x_{i})$ and
$y=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}g_{i}(\lambda_{i})y_{i}.\vskip
6.0pt plus 2.0pt minus 2.0pt$
###### Remark 1
A weakly naturally quasiconvex correspondence may not be continuous or convex
valued.
We give an economic interpretation of the weakly naturally quasiconvex
correspondences.
We consider an abstract economy $\Gamma=(X_{i},A_{i},P_{i})_{i\in I}$ with $I$
\- the set of agents. Each agent can choose a strategy from the set $X_{i}$
and has a preferrence correspondence
$P_{i}:X=\mathop{\textstyle\prod}\limits_{i\in I}X_{i}\rightarrow 2^{X_{i}}$
and a constraint correspondence $A_{i}:X=\mathop{\textstyle\prod}\limits_{i\in
I}X_{i}\rightarrow 2^{X_{i}}.$ The traditional approach considers that the
preferrence of agent $i$ is characterized by a binary relation $\succeq_{i}$
on the set $X_{i}.$ A real valued function $u_{i}:X\rightarrow\mathbb{R}$ that
satisfies $x\succeq_{i}y$ $\Leftrightarrow$ $u_{i}(x)\geq u_{i}(y)$ is called
an utility function of the preferrence $\succeq_{i}.$ The relation between the
utility function $u_{i}$ and the preferrence correspondence $P_{i}$, for each
agent $i$ is:
$P_{i}(x)=\left\\{y_{i}\in X_{i}:u_{i}(x,y_{i})>u_{i}(x,x_{i})\right\\},$
where, in this case, $u_{i}:X\times X_{i}\rightarrow\mathbb{R}.$
The aim of the equilibrium theory is to maximize each agent’s utility on a
strategy set.
For the case that, for each index $i$, $P_{i}$ is a weakly naturally
quasiconvex correspondence, the interpretation is the following: for all
certain amounts $x^{1},x^{2},...x^{n}\in X,$ the agent $i$ with the
correspondence $P_{i}$ will always prefer $y_{i},$ the weighted average of
some quantities $y_{i}^{k}\in P_{i}(x^{k}),$ $k=1,n.$ This implies that there
exist $y_{i}^{1}\in P_{i}(x^{1}),y_{i}^{2}\in P_{i}(x^{2}),...,y_{i}^{n}\in
P_{i}(x^{n})$ and
$g=(g_{1},g_{2},...,g_{n}):\Delta_{n-1}\rightarrow\Delta_{n-1}$ a bijective
function depending on $x^{1},x^{2},...,x^{n}$ with $g_{i}$ continuous for each
$i=1,2,...,n,$ such that, for each $\lambda\in\Delta_{n-1},$ there exists
$y_{i}=\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}g_{i}(\lambda_{k})y_{i}^{k}$
and $y_{i}\in
P_{i}(\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}\lambda_{k}x^{k})$
(i.e., if there exist utility functions $u_{i}:X\times
X_{i}\rightarrow\mathbb{R}$ such that, if
$u_{i}(x^{k},y_{i}^{k})>u_{i}(x^{k},x_{i}^{k})$ for every
$k\in\\{1,2,...n\\},$ we have that $y_{i}\in
A_{i}(\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}\lambda_{k}x^{k})$
and
$u_{i}(\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}\lambda_{k}x^{k},y_{i})>u_{i}(\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}\lambda_{k}x^{k},(\overset{n}{\underset{k=1}{\mathop{\textstyle\sum}}}\lambda_{k}x^{k})_{i}).$
###### Theorem 3.1
(selection theorem). Let $K$ be a simplex in a topological vector space $F$
and $Y$ be a non-empty convex subset of a topological vector space $E.$ Let
$T:K\rightarrow 2^{Y}$ be a weakly naturally quasiconvex correspondence. Then,
$T$ has a continuous selection on $K$.
Proof. Assume that $K$ is a simplex, i.e., the convex hull of an affinely
independent set $\\{a_{1},a_{2},...,a_{n}\\}.$ Since $T$ is weakly naturally
quasiconvex, there exist $b_{i}\in T(a_{i})$, $(i=1,2,...,n)$ and
$g=(g_{1},g_{2},...,g_{n}):\Delta_{n-1}\rightarrow\Delta_{n-1}$ a bijective
function with $g_{i}$ continuous for each $i=1,2,...,n$, such that for every
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$, there exists $y\in
T(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}a_{i})$ with
$y=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}g_{i}(\lambda_{i})y_{i}.$
Since $K$ is a $(n-1)$-dimensional simplex with the vertices
$a_{1},...,a_{n},$ there exists unique continuous functions
$\lambda_{i}:K\rightarrow\mathbb{R},$ $i=1,2,...,n$ such that for each $x\in
K,$ we have
$(\lambda_{1}(x),\lambda_{2}(x),...,\lambda_{n}(x))\in\Delta_{n-1}$ and
$x=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x)a_{i}.$
Let’s define $f:K\rightarrow Y$ by
$f(a_{i})=b_{i}$ $(i=1,...,n)$ and
$f(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}a_{i})=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}g_{i}(\lambda_{i})b_{i}\in
T(x).$
We show that $f$ is continuous.
Let $(x_{m})_{m\in N}$ be a sequence which converges to $x_{0}\in K,$ where
$x_{m}=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x_{m})a_{i}$
and $x_{0}=$
$\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x_{0})a_{i}.$
By the continuity of $\lambda_{i},$ it follows that for each $i=1,2,...,n$,
$\lambda_{i}(x_{m})\rightarrow\lambda_{i}(x_{0})$ as $m\rightarrow\infty.$
Since $\ g_{1},g_{2},...,g_{n}$ are continuous, we have
$g_{i}(\lambda_{i}(x_{m}))\rightarrow g_{i}(\lambda_{i}(x_{0}))$ as
$m\rightarrow\infty.$ Hence, $f(x_{m})\rightarrow f(x_{0})$ as
$m\rightarrow\infty,$ i.e. $f$ is continuous.
We proved that $T$ has a continuous selection on $K.\vskip 6.0pt plus 2.0pt
minus 2.0pt$
By Brouwer’s fixed point theorem, we obtain the following fixed point theorem
for weakly naturally quasiconvex correspondences.
###### Theorem 3.2
Let $K$ be a simplex in a topological vector space $F.$ Let $T:K\rightarrow
2^{K}$ be a weakly naturally quasiconvex correspondence. Then, $T$ has a fixed
point in $K$.
Proof. By Theorem 3, $T$ has a continuous selection on $K,$ $f:K\rightarrow
K.$
Since $f$ has a fixed point $x^{\ast}\in K,$ we have that
$x^{\ast}=f(x^{\ast})\in T(x^{\ast}).$
NOTATION. For the correspondence $T:X\rightarrow 2^{Y}$ and for the set $V\in
Y,$ we denote $T_{V}$ the correspondence $T_{V}:X\rightarrow 2^{Y}$, defined
by $T_{V}(x)=(T(x)+V)\cap Y$ for each $x\in X.$
If $Y=K,$ we obtain the following fixed point theorem:
###### Theorem 3.3
Let $K$ be a simplex in a topological vector space $F$ and let $T:K\rightarrow
2^{K}$ be a correspondence. Assume that for each neighborhood $V$ of the
origin in $F$, there is $T^{V}:K\rightarrow 2^{K}$ a weakly naturally
quasiconvex correspondence such that Gr$T^{V}\subset$clGr$T_{V}$. Then there
exists a point $x^{\ast}\in K$ such that $x^{\ast}\in\overline{T}(x^{\ast}).$
To prove Theorem 5, we need the following lemma from [20].
###### Lemma 2 (20)
Let $X$ be a topological space, $Y$ be a non-empty subset of a topological
vector space E, ß be a base of the neighborhoods of $0$ in $E$ and
$T:X\rightarrow 2^{Y}.$ If $x^{\ast}\in X$ and $\widehat{y}\in Y$ are such
that $\widehat{y}\in\cap_{V\in\text{\ss}}\overline{T_{V}}(x^{\ast}),$ then
$\widehat{y}\in\overline{T}(x^{\ast}),$ where $\overline{T}$ $:X\rightarrow
2^{Y}$ is defined by $\overline{T}(x)=\\{y\in Y:(x,y)\in$clX×YGr$T\\}.\vskip
6.0pt plus 2.0pt minus 2.0pt$
Proof of Theorem 5. Let ß denote the family of all neighborhoods of zero in
$F.$ Let $V\in$ß. By the fixed point theorem 4, it follows that for each
neighborhood $V$ of the origin in $Y,$ there exists $x_{V}^{\ast}\in
T^{V}(x_{V}^{\ast})\subset(T(x_{V}^{\ast})+V)\cap K.$
For each $V\in\text{\ss,\ we \ define }Q_{V}=\\{x\in K:$ $x\in(T(x)+V)\cap
K\\}.$
$Q_{V}$ is nonempty since $x_{V}^{\ast}\in Q_{V},$ then cl$Q_{V}$ is nonempty.
We prove that the family $\\{$cl$Q_{V}:V\in\text{\ss}\\}$ has the finite
intersection property.
Let $\\{V^{(1)},V^{(2)},...V^{(n)}\\}$ be any finite set of $\text{\ss}.$ Let
$V=\underset{k=1}{\overset{n}{\cap}}V^{(k)}$, then $V\in\text{\ss}$. Clearly
$Q_{V}\subset\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}$ so that
$\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}\neq\emptyset.$ Then
$\underset{k=1}{\overset{n}{\cap}}$cl$Q_{V^{(k)}}\neq\emptyset.$
Since $K$ is compact and the family $\\{$cl$Q_{V}:V\in\text{\ss}\\}$ has the
finite intersection property, we have that
$\cap\\{$cl$Q_{V}:V\in\text{\ss}\\}\neq\emptyset.$ Take any
$x^{\ast}\in\cap\\{$cl$Q_{V}:V\in$ß$\\},$ then for each $V\in\text{\ss},$
$x^{\ast}\in$cl$\left\\{x^{\ast}\in K:x^{\ast}\in(T(x^{\ast})+V)\cap
K\right\\}$. Hence $(x^{\ast},x^{\ast})\in$clGr($(T(x)+V)\cap K)$ for each
$V\in$ß. By Lemma 2 __ we have that __ $x^{\ast}\in\overline{T}(x^{\ast}),$
i.e. $x^{\ast}$ is a fixed point for $\overline{T}$ $\Box$
The weakly convex correspondences are defined in [7].
###### Definition 6
[7]. Let $X$ and $Y$ be nonempty convex subsets of a topological vector space
$E.$ The correspondence $T:X\longrightarrow 2^{Y}$ is said to have _weakly
convex graph_ (in short it is a WCG correspondence), if for each finite set
$\\{x_{1},x_{2},...,x_{n}\\}\subset X$, there exists $y_{i}\in T(x_{i})$,
$(i=1,2,...,n)$, such that
(1) co$(\\{(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\\})\subset$Gr$(T)$
The relation (1) is equivalent to
(2) $\ \ \ \
\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}y_{i}\in
T(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}x_{i})$
$(\forall(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}).$
It is clear that if either Gr$(T)$ is convex, or $\
\mathop{\textstyle\bigcap}\\{T(x):x\in X\\}\neq\emptyset$, then $T$ has a
weakly convex graph.
Remark 4\. Let $T:X\rightarrow 2^{Y}$ be a WCG correspondence and $X_{0}$ be a
non-empty convex subset of $X$. Then, the restriction of $T$ on $X_{0}$,
$T_{\mid X_{0}}:X_{0}\rightarrow 2^{Y}$ is a WCG correspondence, too.
Now we introduce the following definition.
###### Definition 7
Let $E$, $F$ be topological vector spaces, $X$ and $Y$ be nonempty convex
subsets of $E,$ respectively $F$ and $T:X\rightarrow 2^{Y}$ be a
correspondence. $T$ is said to have a *-weakly convex graph if for each
neighborhood $V$ of the origin in $F,$ the correspondence $T_{V}:X\rightarrow
2^{Y},$ defined by $T_{V}(x)=(T(x)+V)\cap Y$ for each $x\in X$ has an weakly
convex graph.
The next theorem (its proof follows the same lines as that of Theorem 5) is a
fixed point result for a correspondence with *-weakly convex graph.
###### Theorem 3.4
Let $K$ be a simplex in a topological vector space $F.$ Let $T:K\rightarrow
2^{K}$ be a correspondence with *-weakly convex graph. Then, there exists a
point $x^{\ast}\in K$ such that $x^{\ast}\in\overline{T}(x^{\ast}).$
We get the following corollary.
###### Corollary 1
Let $K$ be a simplex in a topological vector space $F.$ Let $S,T:K\rightarrow
2^{K}$ be two correspondences with the following conditions:
(i) for each $x\in K,$ $\overline{S}(x)\subset T(x)$ and $S(x)\neq\emptyset,$
(ii) $S$ has *-weakly convex graph.
Then, there exists a point $x^{\ast}\in K$ such that $x^{\ast}\in
T(x^{\ast}).\vskip 6.0pt plus 2.0pt minus 2.0pt$
Now, we introduce the concept of *-weakly naturally quasiconvex
correspondence.
###### Definition 8
Let $E$, $F$ be topological vector spaces, $X$ and $Y$ be nonempty convex
subsets of $E,$ respectively $F$ and $T:X\rightarrow 2^{Y}$ be a
correspondence. $T$ is said to be *-weakly naturally quasiconvex if for each
neighborhood $V$ of the origin in $F,$ the corespondence $T_{V}:X\rightarrow
2^{Y},$ defined by $T_{V}(x)=(T(x)+V)\cap Y$ for each $x\in X$ is weakly
naturally quasiconvex.
Theorem 7 is a fixed point theorem for *-weakly naturally quasiconvex
correspondences.
###### Theorem 3.5
Let $K$ be a non-empty simplex in a topological vector space $F.$ Let
$T:K\rightarrow 2^{K}$ be a *-weakly naturally quasiconvex correspondence.
Then there exists a point $x^{\ast}\in K$ such that
$x^{\ast}\in\overline{T}(x^{\ast}).$
Proof. Let ß denote the family of all neighborhoods of zero in $F$ and let
$V\in$ß. The corespondence $T_{V}:K\rightarrow 2^{K},$ defined by
$T_{V}(x)=(T(x)+V)\cap K$ for each $x\in K$ is *-weakly naturally quasiconvex.
Then there exists a continuous selection $f_{V}:K\rightarrow K$ such that
$f_{V}(x)\in T_{V}(x).$ The proof follows the same line as in Theorem 5.$\Box$
Now we introduce the following definition.
###### Definition 9
Let $B\subset X\times Y$ be a biconvex set, $Z$ a nonempty convex subset of a
topological vector space $F$ and $T:B\rightarrow 2^{Z}$ a correspondence. $T$
is called weakly biconvex if for each finite set
$\\{(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\\}\subset B$, there exists
$z_{i}\in T(x_{i},y_{i})$, $(i=1,2,...,n)$ such that for every biconvex
combination
$(x,y)=\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}(x_{i},y_{i})\in B$
(with $\mathop{\textstyle\sum}\limits_{i=1}^{n}\lambda_{i}=1,$
$\lambda_{i}\geq 0$ $i=1,2,...,n$), there exists $y^{\prime}\in
T(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x_{i},y_{i}))$
and
$y^{\prime}=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}z_{i}.\vskip
6.0pt plus 2.0pt minus 2.0pt$
We state the following selection theorem for weakly biconvex correspondences.
###### Theorem 3.6
(selection theorem). Let $Y$ be a non-empty convex subset of a topological
vector space $F$ and $K\subset E_{1}\times E_{2},$ where $E_{1},E_{2}$ are
topological vector spaces. Suppose that $K$ is the biconvex hull of
$\\{(a_{1},b_{1}),(a_{2},b_{2}),...,(a_{n},b_{n})\\}\subset E_{1}\times
E_{2}$. Let $T:K\rightarrow 2^{Y}$ be a weakly biconvex correspondence. Then,
$T$ has a continuous selection on $K$.
Proof. Since $T$ is weakly biconvex, there exists $c_{i}\in T(a_{i},b_{i})$,
$(i=1,2,...,n),$ such that for every
$(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\Delta_{n-1}$, there exists $z\in
T(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(a_{i},b_{i}))$
with $z=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}z_{i}.$
Since $K$ is biconvex hull of $(a_{1},b_{1}),...,(a_{n},b_{n}),$ there exist
unique continuous functions $\lambda_{i}:K\rightarrow\mathbb{R},$
$i=1,2,...,n$ such that for each $(x,y)\in K,$ we have
$(\lambda_{1}(x,y),\lambda_{2}(x,y),...,\lambda_{n}(x,y))\in\Delta_{n-1}$ and
$(x,y)=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x,y)(a_{i},b_{i}).$
Define $f:K\rightarrow 2^{Y}$ by
$f(a_{i},b_{i})=c_{i}$ $(i=1,...,n)$ and
$f(\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(a_{i},b_{i}))=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}c_{i}\in
T(x,y).$
We show that $f$ is continuous.
Let $(x_{m},y_{m})_{m\in N}$ be a sequence which converges to $x_{0}\in K,$
where
$(x_{m},y_{m})=\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x_{m},y_{m})(a_{i},b_{i})$
implies $a_{1}=a_{2}=...=a_{n}=a$ or $b_{1}=b_{2}=...=b_{n}=b$ and
$(x_{0},y_{0})=$
$\overset{n}{\underset{i=1}{\mathop{\textstyle\sum}}}\lambda_{i}(x_{0})(a_{i},b_{i})$
with $a_{1}=a_{2}=...=a_{n}=a$ or $b_{1}=b_{2}=...=b_{n}=b.$ By the continuity
of $\lambda_{i},$ it follows that for each $i=1,2,...,n$,
$\lambda_{i}(x_{m},y_{m})\rightarrow\lambda_{i}(x_{0},y_{0})$ as
$m\rightarrow\infty.$ Hence $f(x_{m},y_{m})\rightarrow f(x_{0},y_{0})$ as
$m\rightarrow\infty,$ i.e. $f$ is continuous.
We proved that $T$ has a continuous selection on $K.$
In order to prove the existence theorems of equilibria for a generalized
quasi-game, we need the following version of Kim’s quasi fixed-point theorem:
###### Theorem 3.7
Let $I$ and $J$ be any (possible uncountable) index sets. For each $i\in I$
and $j\in J$, let $X_{i}$ and $Y_{j}$ be non-empty compact convex subsets of
Hausdorff locally convex spaces $E_{i}$ and respectively $F_{j}$.
Let $X:=\prod X_{i}$, $Y:=\underset{i\in I}{\prod}Y_{j}$ and $Z:=X\times Y$.
For each $i\in I$ let $\Phi_{i}:Z\rightarrow 2^{X_{i}}$ be a correspondence
such that the set $W_{i}=\left\\{(x,y)\in Z\text{
}\mid\Phi_{i}(x,y)\neq\emptyset\right\\}$ is open and $\Phi_{i}$ has a
continuous selection fi on $W_{i}$.
For each j$\in J$ let $\Psi_{j}:Z\rightarrow 2^{Y_{j}}$ be an upper
semicontinuous correspondence with non-empty closed convex values.
Then there exists a point $(x^{\ast},y^{\ast})\in Z$ such that for each $i\in
I$, either $\Phi_{i}(x^{\ast},y^{\ast})=\emptyset$ or
$\overset{\\_}{x}_{i}\in\Phi_{i}(x^{\ast},y^{\ast})$, and for each $j\in J$,
$y_{j}^{\ast}\in\Psi_{j}(x^{\ast},y^{\ast})$.
Proof. We first endow $\underset{i\in I}{\prod}E_{i}$ and $\underset{j\in
J}{\prod}F_{j}$ with the product topologies; and then $\underset{i\in
I}{\prod}E_{i}\times$ $\underset{j\in J}{\prod}F_{j}$ is also a locally convex
Hausdorff topological vector space.
For each $i\in I$, we define a correspondence
$\Phi_{i}^{{}^{\prime}}:Z\rightarrow 2^{X_{i}}$ by
$\Phi_{i}^{{}^{\prime}}(x,y):=\left\\{\begin{array}[]{c}\\{f_{i}(x,y)\\}\text{,
if }(x,y)\in W_{i}\text{, }\\\ X_{i}\text{, \qquad if }(x,y)\notin
W_{i}\text{;}\end{array}\right.$
Then for each $(x,y)\in Z$, $\Phi_{i}^{{}^{\prime}}(x,y)$ is a non-empty
closed convex subset of $X_{i}$. Also, $\Phi_{i}^{{}^{\prime}}$ is an upper
semicontinuous correspondence on $Z$. In fact, for each proper open subset $V$
of $X_{i}$, we have
$U:=\\{(x,y)\in Z$ $\mid$ $\Phi_{i}^{{}^{\prime}}(x,y)\subset V\\}$
=$\\{(x,y)\in W_{i}$ $\mid$ $\Phi_{i}^{{}^{\prime}}(x,y)\subset V\cup(x,y)\in
Z\setminus W_{i}$ $\mid$ $\Phi_{i}^{{}^{\prime}}(x,y)\subset V\\}$
…=$\left\\{(x,y)\in W_{i}\text{ }\mid\text{ }f_{i}(x,y)\in
V\right\\}\cup\left\\{(x,y)\in Z\setminus W_{i}\text{ }\mid\text{
}X_{i}\subset V\right\\}$
=$\left\\{(x,y)\in W_{i}\text{ }\mid\text{ }f_{i}(x,y)\in
V\right\\}=f_{i}^{-1}(V)\cap W_{i}$.
Since $W_{i}$ is open and $f_{i}$ is a continuous map on $W_{i},U$ is open,
and hence $\Phi_{i}^{{}^{\prime}}$ is upper semicontinuous on $Z$.
Finally, we define a correspondence $\Phi:Z\rightarrow 2^{Z}$ by
$\Phi(x,y):=\underset{i\in
I}{\prod}\Phi_{i}^{{}^{\prime}}(x,y)\times\underset{j\in
J}{\prod}\Psi_{j}(x,y)$ for each $(x,y)\in Z$.
Then, by Lemma 3 in [8], $\Phi$ is an upper semicontinuous correspondence such
that each $\Phi(x,y)$ is non-empty closed convex. Therefore, by the Fan-
Glicksebrg fixed point theorem [9] there exists a fixed point
$(x^{\ast}$,$y^{\ast})\in Z$ such that $(x^{\ast},y^{\ast})$ $\in\Phi(x,y)$,
i.e., for each $i\in I$, $x_{i}^{\ast}\in\Phi_{i}^{{}^{\prime}}(x,y)$, and for
each $j\in J$, $y_{j}^{\ast}\in\Psi_{j}(x,y)$. If $(x^{\ast},y^{\ast})\in
W_{i}$ for some $i\in I$, then
$x_{i}^{\ast}=f_{i}(x^{\ast},y^{\ast})\in\Phi_{i}(x^{\ast},y^{\ast})$; and if
$(x^{\ast},y^{\ast})\notin W_{i}$ for some $i\in I$, then
$\Phi_{i}(x^{\ast},y^{\ast})=\emptyset$. Therefore, we have that for each
$i\in I$, either $\Phi_{i}(x^{\ast},y^{\ast})=\emptyset$ or
$x_{i}^{\ast}\in\Phi_{i}(x^{\ast},y^{\ast})$. Also, for each $j\in J$, we
already have $y_{j}^{\ast}\in\Psi_{j}(x^{\ast},y^{\ast})$. This completes the
proofs. $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$
We have the following corollary.
###### Corollary 2
Let $I$ and $J$ be any (possible uncountable) index sets. For each $i\in I$
and $j\in J$, let $X_{i}$ and $Y_{j}$ be non-empty compact convex subsets of
Hausdorff locally convex spaces $E_{i}$ and respectivelly $F_{j}$.
Let $X:=\prod X_{i}$, $Y:=\underset{i\in I}{\prod}Y_{j}$ and $Z:=X\times Y$.
For each $i\in I$ let $S_{i}:Z\rightarrow 2^{X_{i}}$ be a correspondence such
that the set $W_{i}=\left\\{(x,y)\in Z\text{ }\mid
S_{i}(x,y)\neq\emptyset\right\\}$ is the interior of the biconvex hull of
$\\{(a_{1},b_{1}),(a_{2},b_{2}),...,(a_{n},b_{n})\\}\subset Z$ and $S_{i}$ is
weakly biconvex on $W_{i}$.
For each $\mathit{j}\in J$ let $T_{j}:Z\rightarrow 2^{Y_{j}}$ be an upper
semicontinuous correspondence with non-empty closed convex values.
Then there exists a point $(x^{\ast},y^{\ast})\in Z$ such that for each $i\in
I$, either $S_{i}(x^{\ast},y^{\ast})=\emptyset$ or $x_{i}^{\ast}\in
S_{i}(x^{\ast},y^{\ast})$, and for each $j\in J$, $y_{j}^{\ast}\in
T_{j}(x^{\ast},y^{\ast})$.
## 4 Applications in the equilibrium theory
In this paper, we study the following model of a generalized quasi-game.
###### Definition 10
Let $I$ be a nonempty set (the set of agents). For each $i\in I$, let $X_{i}$
be a non-empty topological vector space representing the set of actions and
define $X:=\underset{i\in I}{\prod}X_{i}$; let $A_{i}$, $B_{i}:X\times
X\rightarrow 2^{X_{i}}$ be the constraint correspondences and $P_{i}:X\times
X\rightarrow 2^{X_{i}}$ the preference correspondence. A generalized quasi-
game $\Gamma=(X_{i},A_{i},B_{i},P_{i})_{i\in I}$ is defined as a family of
ordered quadruples $(X_{i},A_{i},B_{i},P_{i})$.
In particular, when $I$=$\left\\{1\text{, }2\text{...}n\right\\}$, $\Gamma$ is
called n-person quasi-game.
###### Definition 11
An equilibrium for $\Gamma$ is defined as a point $(x^{\ast},y^{\ast})\in
X\times X$ such that for each $i\in I$,
$y_{i}^{\ast}\in$cl$B_{i}(x^{\ast},y^{\ast})$ and
$A_{i}(x^{\ast},y^{\ast})\cap P_{i}(x^{\ast},y^{\ast})=\emptyset$.
If $A_{i}(x,y)=B_{i}(x,y)$ for each $(x,y)\in X\times X$ and $i\in I$, this
model coincides with that one introduced by W. K. Kim [13].
If, in addition, for each $i\in I$, $\ A_{i},P_{i}$ are constant with respect
to the first argument, this model coincides with the classical one of the
abstract economy and the definition of equilibrium is that given in [4].
In this work, Kim established an existence result for a generalized quasi-game
with a possibly uncountable set of agents, in a locally convex Hausdorff
topological vector space.
Here is his result:
###### Theorem 4.1 (11)
Let $\Gamma=(X_{i},A_{i},B_{i},P_{i})_{i\in I}$ be a generalized quasi-game,
where $I$ is a (possibly uncountable) set of agents such that for each $i\in
I$ :
(1) $X_{i}$ is a non-empty compact convex subset of a Hausdorff locally convex
space $E_{i}$ and denote $X:=\underset{i\in I}{\prod}X_{i}$ and $Z:=X\times
X$;
(2) The correspondence $A_{i}:X\times X\rightarrow 2^{X_{i}}$ is upper
semicontinuous such that $A_{i}(x,y)$ is a non-empty convex subset of $X_{i}$
for each $(x,y)\in Z$;
(3) $A_{i}^{-1}\left(x_{i}\right)$ is (possibly empty) open for each $x_{i}\in
X_{i}$;
(4) the correspondence $P_{i}:Z\rightarrow 2^{X_{i}}$ is such that $(A_{i}\cap
P_{i})^{-1}\left(x_{i}\right)$ is (possibly empty) open for each $x_{i}\in
X_{i}$;
(5) the set $W_{i}$ $:$ $=\left\\{(x,y)\in Z\text{ }\mid\text{
}\left(A_{i}\cap P_{i}\right)(x,y)\neq\emptyset\right\\}$ is perfectly normal;
(6) for each $(x,y)\in W_{i}$, $x_{i}\notin coP_{i}(x,y)$.
Then there exists an equilibrium point $(x^{\ast},y^{\ast})\in X\times X$ for
$\Gamma$, $i.e$., for each $i\in I$,
$y_{i}^{\ast}\in$cl$A_{i}(x^{\ast},y^{\ast})$ and
$A_{i}(x^{\ast},y^{\ast})\cap P_{i}(x^{\ast},y^{\ast})=\emptyset$.
As application of the selection theorems from section 3, we state a theorem on
the existence of the equilibrium for a generalized quasi-game.
###### Theorem 4.2
Let $\Gamma=(X_{i},A_{i},B_{i},P_{i})_{i\in I}$ be a generalized quasi-game
where $I$ is a (possibly uncountable) set of agents such that for each $i\in
I:$
(1) $X_{i}$ is a non-empty compact convex set in a Hausdorff locally convex
space $E_{i}$ and denote $X:=\underset{i\in I}{\prod}X_{i}$ and $Z:=X\times
X$;
(2) The correspondence $B_{i}:Z\rightarrow 2^{X_{i}}$ is non-empty, convex
valued such that for each ($x,y)\in Z$, $A_{i}(x,y)\subset B_{i}(x,y)\ $and
cl$B_{i}$ is upper semicontinuous;
(3) the correspondence $A_{i}\cap P_{i}:W_{i}\rightarrow 2^{X_{i}}$ is weakly
naturally quasiconvex;
(4) the set $W_{i}:$ $=\left\\{(x,y)\in Z\text{ / }\left(A_{i}\cap
P_{i}\right)(x,y)\neq\emptyset\right\\}$ is open and cl$W_{i}$ is a $(n-1)$
dimensional simplex in $Z$;
(5) for each $(x,y)\in W_{i},$ $x_{i}\notin P_{i}(x,y)$.
Then there exists an equilibrium point $(x^{\ast},y^{\ast})\in Z$ for
$\Gamma$,$\ i.e.$, for each $i\in I$,
$y_{i}^{\ast}\in$cl$B_{i}(x^{\ast},y^{\ast})$ and
$A_{i}(x^{\ast},y^{\ast})\cap P_{i}(x^{\ast},y^{\ast})=\emptyset$.
Proof. For each $i\in I$, we define $\Phi_{i}:Z\rightarrow 2^{X_{i}}$ by
$\Phi_{i}(x,y)=\left\\{\begin{array}[]{c}(A_{i}\cap P_{i})(x,y)\text{, if
}(x,y)\in W_{i}\text{, }\\\ \emptyset\text{, \qquad\qquad\qquad\ \ if
}(x,y)\notin W_{i}\text{;}\end{array}\right.$
By applying Theorem 3 to the restrictions $A_{i}\cap P_{i}$ on $W_{i}$, we can
obtain that there exists a continuous selection $f_{i}:W_{i}\rightarrow X_{i}$
such that $f_{i}(x,y)\in(A_{i}\cap P_{i})(x,y)$ for each $(x,y)\in W_{i}$.
For each $j\in I$, we define $\Psi_{j}:Z\rightarrow 2^{X_{i}}$, by
$\Psi_{j}(x,y)=$cl$B_{j}(x,y)$ for each $(x,y)\in Z$.
Then $\Psi_{j}$ is an upper semicontinuous correspondence and $\Psi_{j}(x,y)$
is a non-empty, convex, closed subset of $X_{j}$ for each $(x,y)\in Z$.
By Theorem 9, it follows that there exists $(x^{\ast},y^{\ast})\in Z$ such
that for each $i\in I$, either $\Phi_{i}(x^{\ast},y^{\ast})=\emptyset$ or
$x_{i}^{\ast}\in\Phi_{i}(x^{\ast},y^{\ast})$ and for each $j\in J$,
$y_{j}^{\ast}\in\Psi_{j}(x^{\ast},y^{\ast})$.
If $x_{i}^{\ast}\in\Phi_{i}(x^{\ast},y^{\ast})$ for some $i\in I$, then
$x_{i}^{\ast}\in\Phi_{i}(x^{\ast},y^{\ast})=(A_{i}\cap
P_{i})(x^{\ast},y^{\ast})\subset P_{i}(x^{\ast},y^{\ast})$ which contradicts
the assumption (5).
Therefore, for each $i\in I$, $\Phi_{i}(x,y)=\emptyset$ and then
$(x^{\ast},y^{\ast})\notin W_{i}$. Hence, $(A_{i}\cap
P_{i})(x^{\ast},y^{\ast})=\emptyset$ and for each $i\in I$,
$y^{\ast}\in\Psi_{i}(x^{\ast},y^{\ast})=$cl$B_{i}(x^{\ast},y^{\ast})$. $\Box$
By using a similar type of proof and Theorem 8, we obtain Theorem 12.
###### Theorem 4.3
Let $\Gamma=(X_{i},A_{i},B_{i},P_{i})_{i\in I}$ be a generalized quasi-game
where $I$ is a (possibly uncountable) set of agents such that for each $i\in
I:$
(1) $X_{i}$ is a non-empty compact convex set in a Hausdorff locally convex
space $E_{i}$ and denote $X:=\underset{i\in I}{\prod}X_{i}$ and $Z:=X\times
X$;
(2) The correspondence $B_{i}:Z\rightarrow 2^{X_{i}}$ is non-empty, convex
valued such that for each ($x,y)\in Z$, $A_{i}(x,y)\subset B_{i}(x,y)\ $and
cl$B_{i}$ is upper semicontinuous;
(3) $A_{i}\cap P_{i}$ is a weakly biconvex correspondence on $W_{i}$;
(4) the set $W_{i}:$ $=\left\\{(x,y)\in Z\text{ / }\left(A_{i}\cap
P_{i}\right)(x,y)\neq\emptyset\right\\}$ is the interior of the biconvex hull
of $\\{(a_{1},b_{1}),(a_{2},b_{2}),...,(a_{n},b_{n})\\}\subset Z$;
(5) for each $(x,y)\in W_{i},$ $x_{i}\notin$co$P_{i}(x,y)$.
Then there exists an equilibrium point $(x^{\ast},y^{\ast})\in Z$ for
$\Gamma$,$\ i.e.$, for each $i\in I$,
$y_{i}^{\ast}\in$cl$B_{i}(x^{\ast},y^{\ast})$ and
$A_{i}(x^{\ast},y^{\ast})\cap P_{i}(x^{\ast},y^{\ast})=\emptyset$.
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* (20) X. Z. Yuan, The Study of Minimax inequalities and Applications to Economies and Variational inequalities. Memoirs of the American Society 132, 625, (1988).
|
arxiv-papers
| 2013-04-02T18:05:12 |
2024-09-04T02:49:43.770061
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1304.0716"
}
|
1304.0718
|
# A Peer-based Model of Fat-tailed Outcomes
Ben Klemens
US Census Bureau
[email protected] paper originated as work done at Caltech, under
the guidance of Matt Jackson, Kim Border, and Peter Bossaerts. The agent-based
modeling work was done at the Brookings Institution’s Center for Social and
Economic Dynamics, and the author thanks the CSED’s members for their support.
Thanks also to Josh Tokle and Taniecea Arceneaux of the United States Census
Bureau.
###### Abstract
It is well known that the distribution of returns from various financial
instruments are leptokurtic, meaning that the distributions have “fatter
tails” than a Normal distribution, and have skew toward zero. This paper
presents a graceful micro-level explanation for such fat-tailed outcomes,
using agents whose private valuations have Normally-distributed errors, but
whose utility function includes a term for the percentage of others who also
buy.
## 1 Introduction
Many researchers have pointed out that day-to-day returns on equities have
“fat tails,” in the sense that extreme events happen much more frequently than
would be predicted by a Normal distribution, and have skew toward zero,
meaning that extreme negative returns are more likely than extreme positive
returns. This has been re-verified by many of the sources listed below. The
fat tails of actual equity return distributions is far from academic trivia:
if extreme events are more likely than predicted by a Normal distribution,
models based on Normally-distributed returns can systematically under-predict
risk.
Here, I present an explanation for the non-Normality of equity returns using a
micro-level model where agents observe and emulate the behavior of others.
There are several reasons for rational agents to take note of the actions of
other rational agents; the model here is agnostic as to which best describes
real-world agents, but given some motivation to emulate others, I show that
the wider-than-Normal distribution of equity returns follows.
From the tulip bubble of 1637 to the housing bubble of 2007, herding behavior
has been used to explain extreme market movements (Mackay, 1841; Schiller,
2008). Most of the literature discussed below focuses on models where the herd
almost always leads itself to an extreme outcome, where goods are blockbusters
or flops. Typically, agents in these models have private information or
preferences that are easily drowned out by observing the behavior of others
(and in some cases they have no private information at all). Conversely, the
model here shows that when agents have an evaluation strategy that is a mix of
both private preferences and public actions or information, then outcome
distributions look much like that of day-to-day equity returns: they may have
kurtosis and skew that are arbitrarily large, but they remain unimodal. As the
individual utility function is adjusted so that private information is of
little value, the model outcomes replicate the blockbusters, flops, and market
bifurcations in the literature.
Section 2 will give a quick overview of the mostly empirical literature that
has demonstrated that equity returns are fat-tailed, and that equity traders
(and those who advise equity traders) demonstrate emulative behavior. Epstein
and Axtell (1996, p 20, emphasis in original.) wrote “Perhaps one day people
will interpret the question ‘Can you explain it?’ as asking ‘Can you grow
it?”’ Section 3 will demonstrate that once we take emulative behavior as
given, it is easy to grow fat-tailed outcomes. Section 4 concludes, pointing
out that, because situations where outcomes are fat-tailed but not entirely
off the charts are common, we may be able to use emulative preferences to
explain more than they have been used for in the past.
## 2 Literature
This section gives an overview of two threads of the economics literature that
do not quite meet. The first is an overview of the existing literature on the
distribution of equity returns; the second is a survey of the situations
posited in the finance literature where individuals gain utility from
emulating others.
### 2.1 Fitting non-Normal distributions
The second central moment, also known as the variance, is defined as:
$\mu_{2}=\sigma^{2}=\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)dx,$
where $x\in{\mathbb{R}}$ is a random variable, $f(x)$ is the probability
distribution function (PDF) on $x$, and $\mu$ is the mean of $x$
$\left(\int_{-\infty}^{\infty}xf(x)dx\right)$.
One could similarly define the third and fourth central moments:
$\displaystyle\mu_{3}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}(x-\mu)^{3}f(x)dx,\hbox{ and}$
$\displaystyle\mu_{4}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}(x-\mu)^{4}f(x)dx.$
Depending on the author, the skew is sometimes the third central moment,
${\cal{S}}\equiv\mu_{3}$, and sometimes ${\cal{S}}\equiv\mu_{3}/\sigma^{3}$.
The kurtosis may be $\kappa\equiv\mu_{4}$, $\kappa\equiv\mu_{4}/\sigma^{4}$,
or $\kappa\equiv\mu_{4}/\sigma^{4}-3$. In this paper, I will use
$\displaystyle{\cal{S}}$ $\displaystyle\equiv$ $\displaystyle\mu_{3}\hbox{,
and}$ $\displaystyle\kappa$ $\displaystyle\equiv$
$\displaystyle\mu_{4}/\sigma^{4}.$
I will refer to $\kappa$ as normalized kurtosis to remind the reader that it
is divided by variance squared.
The more elaborate normalizations make it easy to compare these moments to a
Normal distribution, because for a Normal distribution with mean $\mu$ and
standard deviation $\sigma$, $\mu_{4}/\sigma^{4}=3$. A Normal distribution is
symmetric and therefore has zero skew (whether normalized or not). One can use
these facts to check empirical distributions for deviations from the Normal.
Fama (1965) ran such a test on equity returns, and found that they were
leptokurtic, meaning that $\mu_{4}\gg 3\sigma^{4}$, and were skewed. However,
he is not the first to notice these features—Mandelbrot (1963, footnote 3)
traces awareness of the non-Normality of return distributions as far back as
1915. Many of the papers cited in the following few paragraphs reproduce the
results using their own data sets. Bakshi et al. (2003) gathered data on
several index and equity returns, and (with few exceptions) found a skew
toward zero (i.e., negative skew, meaning that extreme downward events are
more likely than extreme upward events).
Most of the explanations for the deviation from the Normal have focused on
finding a closed-form PDF that better fits the data. Mandelbrot (1963) showed
that a stable Paretian (aka symmetric-stable) distribution fit better than the
Normal. Blattberg and Gonedes (1974) showed that a renormalized Student’s $t$
distribution fit better than a symmetric-stable distribution. Kon (1984) found
that a mixture of Normal distributions fit better than a Student’s $t$. The
mixture model produces an output distribution by summing a first Normal
distribution, ${\cal N}(\mu_{1}$, $\sigma_{1})$, with an independent second
Normal distribution, ${\cal N}(\mu_{2}$, $\sigma_{2})$. Depending on the
values of the five input parameters (two means, two standard deviations, and a
mixing parameter), the distribution produced by summing the two can take on a
wide range of mean, standard deviation, skew, and kurtosis.
The mixture model raises a few critiques. Kon found that the sum of two
distributions satisfactorily matches only about half of the equity return
distributions he tests. Others require as many as four input distributions—and
thus eleven input parameters—to explain the four moments of the distribution
to be matched. Barbieria et al. (2010, pp 1095–96) tested a set of four broad
equity indices (MSCI’s USA, Europe, UK, and Japan indices) against a
comparable model claiming Normality with variances changing over time, and
rejected the model for all four indices.
As with all of the distribution models, the use of a sum of several
distributions raises the question of how the given distributions go beyond
being a good fit to being a valid explanation of market behavior. After all,
one could fit a Fourier sequence to a data series to arbitrary precision, but
it is not necessarily an explanation of market behavior. This brings us to the
second thread of the literature, covering the micro-level behavior of market
actors.
### 2.2 Emulation
The literature provides many rational motivations for emulating others,
variously termed herding, information cascades, network effects, peer effects,
spillovers—not to mention simple questions of fashion. This section provides a
sample of some of the theoretical results for such models, and a discussion of
herding in the finance context. None of these models were written with the
stated intention of describing an observed leptokurtic distribution, but this
section will calculate the kurtosis of the output distributions implied by
some of these models to see how they fare.
#### The restaurant problem
Among the most common of the models where agents emulate others are the
herding or information cascade models, e.g. Banerjee (1992) or Bikhchandani et
al. (1992). In these models, agents use the prior choices of other agents as
information when making decisions.
A sequence of agents chooses to eat at restaurant $A$ or $B$. The first will
use its private information to choose. The second will use its private
information, plus the information revealed by the observable choice made by
the first agent. The third agent will add to its private information the
information provided by observing where the first two entrants are eating.
Thus, if the first two agents are eating at restaurant $A$, the third may
ignore a preference for restaurant $B$ and eat at $A$. Once the preponderance
of prior choices leans toward restaurant $A$, we can expect that all future
arrivals will choose it as well. The next day, both restaurants start off
empty again, and early arrivals in the sequence might have private information
that restaurant $B$ is better, so subsequent arrivals would all go to
restaurant $B$.
Network externalities are a property of goods where consumption by others
increases the utility of the good, such as a social networking web site whose
utility depends on how many others are also subscribed, computer equipment
that needs to interoperate with others’ equipment, or coordination problems
like the choice of whether to drive on the right or left side of the road. The
typical analysis (e.g., that of Choi (1997)) matches that of the restaurant
problem.
Both the information and the direct utility stories can be shown to produce a
bifurcated distribution of results with probability one: over many days,
restaurant $A$ will show either about 0% attendance or about 100% attendance
every day. Many goods show such a blockbuster/flop dichotomy, such as movies
(de Vaney and Walls, 1996).
But for our purposes, a sharply bimodal distribution is not desirable. First,
one would be hard-pressed to find an equity whose returns are truly bimodal.
More importantly, such a bifurcated outcome distribution is typically
platykurtic, the opposite of the leptokurtosis we seek. Consider an ideal
bimodal distribution with density $r\in(0,1)$ at $a$ and density $1-r$ at $b$
(for any values of $a,b\in{\mathbb{R}}$, $a\neq b$). The distribution has
normalized kurtosis equal to
$\frac{1}{r-r^{2}}-3.$
For a symmetric distribution, $r=0.5$, the normalized kurtosis is one, and it
remains less than three for any $r\in(.211,.789)$. Thus, a model that predicts
a bifurcated distribution can only show a large fourth moment if the
distribution is lopsided, which is not sustainable for equity returns.
#### Distribution models
Brock and Durlauf (2001) specify a model similar to the one presented here. In
the first round, a prior percentage of actors is given, and people act iff
that percentage would be large enough to give them a positive utility from
acting. In subsequent rounds, individuals use the percentage of people who
chose to act in the prior round to decide whether to act or not.
The specific details of Brock and Durlauf’s assumptions lead to two possible
outcomes. One is a bifurcation, much like the outcomes for the restaurant
problem models above. The other, due to the specific form of the assumptions,
is that the output distribution is the input distribution transformed via the
hyperbolic tangent. The $\tanh$ transformation reduces the normalized
kurtosis, and is therefore inappropriate for deriving leptokurtic equity
returns.
Glaeser et al. (1996) point out that the more people emulate others, the more
likely are extreme outcomes, which they measure via “excess variance.” They do
this via a Binomial model: if being the victim of a crime is a draw from a
Bernoulli trial with probability $p$, then the mean of $n$ such trials is
$np$, and the variance is $np(1-p)$. Thus, given $n$ and the sample mean (or
equivalently, $n$ and $p$) we can solve for the expected variance, and if the
observed variance is significantly greater, then we can reject the hypothesis
of independent Bernoulli trials. However, this process says nothing about
whether the observed victimization rates are Normally distributed or not:
excess variance is not excess kurtosis or skew.
#### Finance
Within the theoretical finance literature, papers abound regarding herding
behavior (e.g., Grossman (1976, 1981), Radner (1979), Choi (1997), Minehart
and Scotchmer (1999)), although they concern themselves not with explaining
herding, but with the information aggregation issues entailed by herding. Many
stories regarding the emulation of others apply to the situation of the
rational, self-interested manager of an asset portfolio:
* •
Pricing is partly based on the value of the underlying asset and partly on
what others are willing to pay for the asset. At the extreme, people will buy
a stock which pays zero dividends only if they are confident that there are
other people who will also buy the stock; as more people are willing to buy,
the value of the stock to any individual rises.
* •
It has long been a lament of the fund manager that if the herd does badly but
he breaks even, he sees little benefit; but if the herd does well and he
breaks even, then he gets fired. Therefore, behaving like others may
explicitly appear in a risk-averse fund manager’s utility function.
* •
Since an undercapitalized company is likely to fail, the success of a public
offering may depend on how well-subscribed it is, providing another
justification for putting the behavior of others in the fund manager’s utility
function.
* •
If a large number of banks take simultaneous large losses, then they may be
bailed out; since a bail-out is unlikely if only one bank takes a loss, this
may also serve as an incentive for financiers to take risks together.
* •
Simply following the herd: “[…] elements such as fashion and sense of honour
affected the banks’ decision to take part in a syndicated loan. Banks are
certainly not insensitive to prevailing trends, and if it is ‘the in thing’ to
take part in syndicated loans[…], people sometimes consent too readily.”
(Jepma et al., 1996, p 337)
The model of this paper is a reduced form model which simply assumes that a
financier’s expected utility from an action is increasing with the percentage
of other people acting. I make no effort to explain which of the above
motivations are present at any time, but assert that given these effects, the
model below is applicable.
Empirical studies of analyst recommendations find that they do indeed herd.
For example, Graham (1999) finds evidence of herding among investment
newsletter recommendations, and finds that the more reputable ones are more
likely to herd. Meanwhile, Hong et al. (2000) finds evidence of herding among
investment analysts, and finds that inexperienced analysts are “more likely to
be terminated for bold forecasts that deviate from consensus,” and therefore
less reputable analysts are more likely to herd. Welch (2000) finds that an
analyst recommendation has a strong impact on the next two recommendations for
the same security by other analysts, and that this effect is uncorrelated with
whether the recommendations prove to be correct or not. Although these papers
disagree in the details, they all find empirical evidence that analysts are
inclined to behave like other analysts (and therefore the people who listen to
analysts are likely to also behave alike), so the model below is apropos.
## 3 The model
One run of the model below finds an output equilibrium demand given an input
distribution of individual preferences. Repeating a single run thousands of
times gives a distribution of equilibrium outcomes, which will have large
kurtosis and skew under certain conditions.
One run of the model consists of a plurality of agents (the simulations below
use 10,000), each privately deciding whether to purchase a good. Each has an
individual taste for consuming, $t\in{\mathbb{R}}$, where $t\sim{\cal
N}(\epsilon,1)$ and $\epsilon$ is a small non-negative offset, fixed at zero
or 0.05 in the simulations to follow.
Let the proportion of the population consuming be $k\in[0,1]$, and let the
desire to emulate others be represented by a coefficient
$\alpha\in[0,\infty)$. Then the utility from consuming is
$U_{c}=t+\alpha k.$ (1)
The utility from not consuming is
$U_{nc}=\alpha(1-k).$ (2)
That is, agents who do not consume get utility from emulating the $1-k$ agents
who also do not consume, but have a taste for non-consumption normalized to
zero. One can show that this normalization is without loss of generality.
Agents consume iff $U_{c}>U_{nc}$.
A Bayesian Nash equilibrium is a set of acting agents, comprising the
proportion $k_{a}$ percent of the population, where all acting agents have
$U_{c}>U_{nc}$ given $k_{a}$ percent acting, and all agents outside the acting
set have $U_{c}\leq U_{nc}$ given $k_{a}$ percent acting.
It can be shown that, given the assumptions here, the game has a cutoff-type
equilibrium, where there is a cutoff value $T$ such that every agent with
private tastes greater than $T$ acts and every agent with $t\leq T$ does not
act. An agent with private taste $t$ equal to the cutoff $T$ will have
$U_{c}=U_{nc}$.
One could embed this model of the distribution of demand into a larger model,
such as a simple supply-demand model where supply remains fixed and demand
shifts as per the model here, and prices thus vary with demand. To maintain
focus on the core concept, this paper will cover only the core model
describing the distribution of $\hat{k}$.
### 3.1 Implementation
Recall the restaurant problem, where we measured the turnout to restaurant $A$
every day for a few weeks or months. Each day gave us another draw of diners
from the population, and it was the aggregate of turnouts over several days
that added up to the bimodal distribution. Similarly, the literature on
equities did not claim that if we surveyed willingness to pay by all members
of the market at some instant in time, the distribution would be leptokurtic;
rather, the claim is that every day there is a new distribution of willingness
to pay, which produces a single outcome for the day, and tallying those
outcomes over time generates a leptokurtic distribution. This model draws a
sample distribution (which one could think of as today’s market, and which
will be close to a Normal distribution), finds the equilibrium value
$\hat{k}$, and then repeats until there are enough samples of $\hat{k}$ that
we can estimate the moments of $\hat{k}$’s distribution.
We must first solve for the equilibrium percent acting for a single run.
Briefly switching from the equilibrium percent acting $k$ to the equilibrium
cutoff taste $T$, one can find the equilibrium for a single distribution by
finding the value of $T$ such that an agent with that value is indifferent
between action and inaction, given that the cutoff is at that value (that is,
$U_{c}$ in Equation 1 equals $U_{nc}$ in Equation 2). Write the proportion not
acting given cutoff $T$ as $\hbox{CDF}(T)$ (i.e. the cumulative distribution
function of the empirical distribution of tastes up to the cutoff $T$); then
any value of $T$ that satisfies
$T=\alpha(1-2\hbox{CDF}(T))$ (3)
is an equilibrium.
There are typically no closed-form solutions for $T$, so the work will require
a numeric search.
I use an agent-based simulation to organize the draws. For each step, the
simulation draws 10,000 agents from the fixed distribution, then the
simulation algorithm solves for equilibrium via tatônnement, as detailed
below. The equilibrium reached via market simulation is a Bayesian Nash
equilibrium as in Equation 3. There are other search strategies for finding
the equilibrium given the draws of $t$, but the agent-based model has the
advantages of always finding the equilibrium and providing a realistic story
of what happens in the market.
Repeating the process for thousands of draws from the fixed distribution,
starting each simulation with a new set of random draws of tastes $t$ from the
same distribution, will produce a distribution of the statistics $\hat{T}$ and
$\hat{k}$, including multiple modes when there are multiple equilibria.
The algorithm for a single run of the simulation is displayed in Figure 1. In
each step, agents consume or do not based on the value of $k$ from the last
step, and the process repeats until the value of $k$ no longer changes. The
output of the process is the equilibrium value of $\hat{T}$ and the
equilibrium percent acting $\hat{k}$.
With a sufficiently large number of runs (in the simulations here, 20,000), it
is possible to calculate the moments ${\cal{S}}(\hat{k})$ and
$\kappa(\hat{k})$.
–Fix $N$ and $\epsilon$.
–Generate a new population of agents:
Hey.–Set the initial value of $k=\frac{1}{2}$.
Hey.–For each agent:
Hey.Hey.–Draw a taste $t$ from a ${\cal N}(\epsilon,1)$ distribution.
–While $k$ this period is not equal to $k$ last period:
Hey.–For each agent:
Hey.Hey.–Consume iff $U_{k}\geq U_{nk}$.
Hey.–Recalculate $k$.
–Record the equilibrium percent acting $k$.
Figure 1: The algorithm for finding the equilibrium level of consumption for
one run.
The code itself is a short script written in C using the open source Apophenia
library (Klemens, 2008), and is available upon request.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 $\alpha=1$ $\alpha=1.3$ $\alpha=1.6$
Figure 2: Above, three distributions of the equilibrium percent acting $k$,
for 20,000 runs with $\alpha$ equal to 1 (unimodal), 1.3 (bimodal with modes
near 0.3 and 0.7), and 1.6 (bimodal with modes near 0.1 and 0.9). Below, a
full sequence of such distributions, for $\alpha=0.5$ in front up to
$\alpha=2$ at the back. Vertical axis is the percent of runs (out of 20,000
per $\alpha$) whose equilibrium is in the given histogram bin. The three
slices in the 2-D plot are indicated by a line on the floor of the 3-D plot.
1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5
Normalized kurtosis ($\kappa/\sigma^{4}$)
Emulation parameter ($\alpha$)
Figure 3: The normalized kurtosis reveals the narrow range of transition from
Normal-type distribution ($\kappa/\sigma^{4}=3$) to bimodal-type distribution
($\kappa/\sigma^{4}=1$).
### 3.2 Results
It is instructive to begin with the symmetric case, where $\epsilon=0$, so
agents’ private tastes are drawn from a ${\cal N}(0,1)$ distribution.
Figure 2 shows a sequence of distributions of the equilibrium percent acting
$k$, from the distribution given $\alpha=1$ up to the distribution for
$\alpha=2.5$, with distributions for three specific values of $\alpha$
highlighted. Small values of $\alpha$ (where utility is mostly private
valuation) result in a Normal output distribution of prices, while large
values of $\alpha$ (where utility is mostly public) give a coordination-game
style bifurcation.
As $\alpha$ goes from the Normal range to the bifurcated range, there is a
small range of $\alpha$ where the transition occurs, and the distribution is
neither fully bifurcated nor Normal.
At large $\alpha$, the value of $k$ between the sink that sends the simulation
to the lower equilibrium and the sink that sends the simulation to the higher
equilibrium (near 0.5) is an unstable equilibrium; in theory it occurs with
probability zero, but in a finite simulation it occurs with small
probability.222The figures are the aggregate of 20,000 runs of the simulation.
If an equilibrium was reached even once, then it appears as a mark in the 3-D
plot. The 2-D plots have lower resolution, and unlikely events may blend with
the axes. Below, we will see that these distributions with a small middle mode
behave like a bifurcated distribution, so I will refer to them as such.
The small transition range is especially clear when we look at the normalized
kurtosis of each $\alpha$’s distribution, which is not at all a uniform shift.
As in Figure 3, the normalized kurtosis is consistently three for small values
of $\alpha$ (as for a Normal distribution), is consistently one for large
values of $\alpha$ (as for a symmetric bimodal distribution), and has a quick
period of transition between $\alpha\approx 1$ and $\alpha\approx 1.4$.333The
units on $\alpha$ are utils per percent acting, so exact values of $\alpha$
are basically meaningless. Rescaling $t$ (by changing its variance) would
produce entirely different values of $\alpha$, but the qualitative effects
described here would still hold.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9 1 $\alpha=1$ $\alpha=1.3$ $\alpha=1.6$
Figure 4: Two views of the $\alpha$-to-cutoff-frequency relation. PDFs of
cutoffs for three given levels of $\alpha$ (1=unimodal near center,
1.3=unimodal to right, 1.6=bimodal) are displayed in 2-D form at top. At
bottom is a series of PDFs for a range of values of $\alpha$ from $\alpha=0.5$
at front to $\alpha=2$ at rear.
0 50 100 150 200 250 0 0.5 1 1.5 2 2.5
Normalized kurtosis ($\kappa/\sigma^{4}$)
Emulation parameter ($\alpha$)
-12 -10 -8 -6 -4 -2 0 2 0 0.5 1 1.5 2 2.5
Normalized skew (${\cal{S}}/\sigma^{3}$)
Emulation parameter ($\alpha$)
Figure 5: The relationship between $n$ (on the horizontal axis) and
$\kappa/\sigma^{4}$ (on the vertical axis, top) or ${\cal{S}}/\sigma^{3}$ (on
the vertical axis, bottom).
Figure 4 shows the sequence of distributions where $\epsilon=0.05$. For
$\alpha\approx 0$, the distribution is roughly equivalent to the $\epsilon=0$
situation but shifted upward slightly; for $\alpha\approx 2$ and above, where
the outcome distribution is bifurcated, the slight shift in the distribution’s
center causes positive outcomes to be more likely than negative outcomes.
However, between these two outcomes lies a range of $\alpha$ where the
$\epsilon=0$ case would have led to a bifurcation, but the lower tail of the
distribution is suppressed because the nobody-acts equilibrium is not
feasible. In this range, we have an asymmetric but unimodal distribution.
Figure 5 plots normalized kurtosis for each $\alpha$’s distribution. The
neighborhood of $\alpha\approx 1.3$ is again salient, because the normalized
kurtosis in that range is an order of magnitude larger than three. The model’s
exceptional success in generating a leptokurtic outcome makes the plot’s
vertical scale rather large, so it may be difficult to discern that the
kurtosis up to the peak is three, and after the peak is one, as in the
$\epsilon=0$ case.
The bottom plot of Figure 5 shows that normalized skew follows the same story
relative to $\alpha$ as did kurtosis: it spikes around 1.3, where the
distribution of equilibrium percent acting has heavily negative skew.
Thus, given a realistic value of $\epsilon$ (i.e., anything but exactly zero),
and a value of $\alpha$ that is not too small to be equivalent to the private
preferences case and not too large to be equivalent to the full herding case,
the distribution of outcomes is unimodal, leptokurtic, and has a negative
skew.
## 4 Conclusion
There are several explanations for why rational agents would choose to emulate
others, all of which advise that a utility function meant to describe a trader
in the finance markets should include a term for the desire to emulate others.
Meanwhile, we know that equity return distributions show certain consistent
deviations from the Normal distribution implied by naïve application of a
Central Limit Theorem. Adding a term for the emulation of others to individual
utilities produces aggregate outcome distributions that show these same
deviations from Normal: extreme outcomes happen more often, and do so
asymmetrically.
However, the story is not quite as simple as saying that people tend to
imitate others. The type of distribution observed in equity returns appears in
a middle-ground between two extreme types of utility function. With $\alpha$
small, the distribution of cutoffs is more-or-less that of a situation of
purely private utility. With $\alpha$ large, the distribution follows the
story of agents that simply follow the herd. But between these two situations,
there is a transition range where the distribution of cutoffs has the desired
characteristics of being unimodal, having large kurtosis, and skew toward
zero. Thus, the model explains this type of distribution via an interplay
between private and emulative utility.
This paper has shown that peer effects can generate leptokurtic outcomes under
certain conditions. This creates the possibility that an observed leptokurtic
distribution can be explained by peer effects. For example, Jones et al.
(2003) found leptokurtic outcomes in Congressional actions such as budget
allocations; I suggest in this paper that a model of Congressional
representatives who emulate each other can generate such an outcome
distribution. When outcomes have a blockbuster/flop bimodality, there is
little doubt that peer effects are at play, but the model here shows that even
more subtle outcomes, with unimodal distributions but fat tails, may also be
the result of agents who gain direct or indirect utility from emulating each
other.
## References
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|
arxiv-papers
| 2013-04-02T18:07:18 |
2024-09-04T02:49:43.778256
|
{
"license": "Public Domain",
"authors": "Ben Klemens",
"submitter": "Ben Klemens",
"url": "https://arxiv.org/abs/1304.0718"
}
|
1304.0753
|
# A generalization of the Shafer-Fink inequality
Jacopo D’Aurizio
Università di Pisa
[email protected]
(02-03-2013)
In this article we will prove some generalizations and extensions of the
Shafer-Fink ([3]) double inequality for the arctangent function:
###### Theorem 1.
For any positive real number $x$,
$\frac{3x}{1+2\sqrt{1+x^{2}}}<\arctan x<\frac{\pi x}{1+2\sqrt{1+x^{2}}}$
holds.
###### Proof.
Following the lines of ([2]), we consider the substitution $x=\tan\theta$,
that gives the following, equivalent form of the inequality:
$\forall\theta\in
I=(0,\pi/2),\qquad\theta(\cos\theta+2)-\pi\sin\theta<0<\theta(\cos\theta+2)-3\sin\theta.$
If now we set
$f_{K}(\theta)=(\cos\theta+2)-K\,\frac{\sin\theta}{\theta}$
we have:
$\theta^{2}\,\frac{df_{K}}{d\theta}=(K-\theta^{2})\sin\theta-K\theta\cos\theta.$
Since for any $\theta\in I$ we have:
$\frac{\theta}{\tan\theta}<1-\frac{\theta^{2}}{3}<1-\frac{\theta^{2}}{\pi},$
$f_{3}(\theta)$ ed $f_{\pi}(\theta)$ are both non-decreasing on $I$, in virtue
of $\frac{df_{K}}{d\theta}\geq 0$; moreover, $f_{K}^{\prime}(0)=0$ and
$f_{K}^{\prime}$ cannot be zero on $I$. Since:
$f_{3}(0)=0,\quad f_{3}(\pi/2)>0,\quad f_{\pi}(0)<0,\quad f_{\pi}(\pi/2)=0,$
the claim follows. ∎
We give now a different proof of this inequality, that relies on the bisection
formula for the cotangent function and the associated Weierstrass product.
From the logarithmic derivative of the Weierstrass product for the sine
function we know that for any $x\in[0,\pi/2]$
$f(x)=x\cot x=1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)}{\pi^{2k}}\,x^{2k}$
holds. Since $f(x)$ is an even function, there exists a suitable linear
combination $g_{1}(x)$ of $f(x)$ and $f(x/2)$ that satisfies:
$g_{1}(x)=A_{0}f(x)+A_{1}f(x/2)=1-\sum_{k\geq 2}C^{(1)}_{k}\,x^{2k}.$
With the choices $A_{0}=-\frac{1}{3},A_{1}=\frac{4}{3}$ the previous identity
holds, and, for any $k\geq 2$:
$C^{(1)}_{k}=\left(A_{0}+\frac{A_{1}}{4^{k}}\right)\frac{\zeta(2k)}{\pi^{2k}}<0,$
so $g_{1}(x)$ is an increasing and convex function over $I=[0,\pi/2]$. From
that,
$\forall x\in I,\quad\left(-\frac{1}{3}x\cot
x+\frac{2}{3}x\cot\frac{x}{2}\right)\in[g_{1}(0),g_{1}(\pi/2)]=[1,\pi/3]$
follows. If now we consider the bisection formula for the cotangent function:
$\cot\frac{x}{2}=\cot x+\sqrt{1+\cot^{2}x}$
we have a different proof of the Shafer-Fink inequality.
We consider now $g_{2}(x)$ as a linear combination of $f(x),f(x/2)$ and
$f(x/4)$ such that:
$g_{2}(x)=A_{0}f(x)+A_{1}f(x/2)+A_{2}f(x/4)=1-\sum_{k\geq
3}C^{(2)}_{k}\,x^{2k}.$
From the annihilation of the coefficient of $x^{2}$ in the RHS we deduce the
constraint $A_{0}+A_{1}\cdot\frac{1}{4}+A_{2}\cdot\frac{1}{16}=0$, and from
the annihilation of the coefficient of $x^{4}$ we deduce the constraint
$A_{0}+A_{1}\cdot\frac{1}{16}+A_{2}\cdot\frac{1}{256}=0$. If we take
$p_{2}(x)=A_{0}+A_{1}x+A_{2}x^{2}$, such constraints translate into
$p_{2}(1/4)=p_{2}(1/16)=0$, from which:
$p_{2}(x)=K_{2}\left(x-\frac{1}{4}\right)\left(x-\frac{1}{16}\right),$
with $K_{2}=(1-1/4)^{-1}\cdot(1-1/16)^{-1}$ in order to grant
$A_{0}+A_{1}+A_{2}=p_{2}(1)=1$.
Since $C^{(2)}_{k}=\frac{\zeta(2k)}{\pi^{2k}}p_{2}(4^{-k})$, all the non-zero
coefficients of the Taylor series of $g_{2}(x)$, except (at most) the first
one, have the same sign, so $g_{2}(x)$ is a monotonic function over $I$. In
particular:
$\displaystyle\forall x\in
I,\qquad\frac{\pi(3+8\sqrt{2})}{45}=g_{2}(\pi/2)\leq g_{2}(x)$
$\displaystyle=\frac{1}{45}\left(f(x)-20f(x/2)+64f(x/4)\right)$
$\displaystyle=\frac{x}{45}\left(\cot x-10\cot(x/2)+16\cot(x/4)\right)\leq 1,$
from which we get:
$\pi(3+8\sqrt{2})\leq x\left(\cot x-10\cot(x/2)+16\cot(x/4)\right)\leq 45.$
By using twice the bisection formula for the cotangent, we have the following
strengthening of the Shafer-Fink inequality:
###### Theorem 2 (D’Aurizio).
For any positive real number $x$
$\pi(3+8\sqrt{2})\cdot f(x)<\arctan x<45\cdot f(x)$
holds, where:
$f(x)=\frac{x}{7+6\,\sqrt{1+x^{2}}+16\sqrt{2}\,\sqrt{1+x^{2}+\sqrt{1+x^{2}}}}.$
The same approach leads to an arbitrary strengthening of the Shafer-Fink
inequality:
###### Theorem 3 (D’Aurizio).
For any positive real number $x$ and for any positive natural number $n$,
once defined:
$f(x)=x\cot x=1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)}{\pi^{2k}}\,x^{2k},$
$p_{n}(x)=\prod_{k=1}^{n}\frac{(4^{k}x-1)}{(4^{k}-1)}=A_{0}+A_{1}x+\ldots+A_{n}x^{n},$
$g_{n}(x)=\sum_{k=0}^{n}A_{k}\,f(2^{-k}x)=x\sum_{k=0}^{n}\frac{A_{k}}{2^{k}}\,\cot(2^{-k}x),$
$e_{j}(x_{1},\ldots,x_{k})=\sum_{sym}x_{1}\cdot\ldots\cdot x_{j},$
$L_{0}(x)=1,\qquad L_{n+1}(x)=L_{n}(x)+\sqrt{x^{2}+L_{n}(x)^{2}},$
we have:
$K_{low}\cdot a_{n}(x)<\arctan(x)<K_{high}\cdot a_{n}(x),$
where $K_{low}=\min(g_{n}(0),g_{n}(\pi/2))$,
$K_{high}=\max(g_{n}(0),g_{n}(\pi/2))$ and:
$a_{n}(x)=x\cdot\left(\sum_{j=0}^{n}(-1)^{n-j}\cdot L_{j}(x)\cdot 2^{j}\cdot
e_{j}(1,4,\ldots,4^{n-1})\right)^{-1}.$
Moreover, $K_{high}-K_{low}<\frac{1}{4^{n}}$.
###### Proof.
By taking
$p_{n}(x)=\prod_{k=1}^{n}\frac{(4^{k}x-1)}{(4^{k}-1)}=A_{0}+A_{1}x+\ldots+A_{n}x^{n}$
we have $p_{n}(1)=1$ and $p_{n}(4^{-j})=0$ for every $j\in[1,n]$. In
particular, the Taylor series of
$g_{n}(x)=\sum_{k=0}^{n}A_{k}\,f(2^{-k}x)=x\sum_{k=0}^{n}\frac{A_{k}}{2^{k}}\,\cot(2^{-k}x).$
is equal to:
$1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)p_{n}(4^{-k})}{\pi^{2k}}\,x^{2k}=1-2\sum_{k>n}C^{(k)}_{n}\,x^{2k},$
and all the $C^{(k)}_{n}$ with $k>n$ have the same sign, so $g_{n}(x)$ is
monotonic over $[0,\pi/2]$, with $g_{n}(0)=1$.
In particular, we have:
$\forall x\in[0,\pi/2],\qquad
x\cdot\sum_{j=0}^{n}(-1)^{n-j}\cot\left(\frac{x}{2^{j}}\right)2^{j}\,e_{j}(1,4,\ldots,4^{n-1})\leq\prod_{k=1}^{n}(4^{k}-1),$
where $e_{j}$ is the $j$-th elementary symmetric function. Since for any $m>n$
we have $|p_{n}(4^{-m})|<1$,
$\left|g_{n}(\pi/2)-g_{n}(0)\right|\leq\sum_{k>n}\frac{\zeta(2k)}{4^{k}}<\frac{1}{4^{n}}.$
holds. ∎
We give now another upper bound for the arctangent function that does not
belong to the last family of inequalities, but that strenghtens the inequality
$\arctan x<\frac{\pi x}{1+2\sqrt{1+x^{2}}}$, too.
###### Theorem 4.
For any positive real number $x$
$\arctan x<\frac{\pi x}{\frac{4}{\pi}+\sqrt{2}\sqrt{1+x^{2}+x\sqrt{1+x^{2}}}}$
holds.
###### Proof.
By using the substitution $x=\tan\theta$, it is sufficient to prove that for
any $\theta\in I=[0,\pi/2]$ we have:
$\theta\leq\frac{\pi\sin\theta}{\frac{4}{\pi}\cos\theta+\sqrt{2+2\sin\theta}},$
that is also equivalent, up to the change of variable $\theta=\pi/2-\phi$, to
the inequality:
$\frac{\pi}{2}-\phi\leq\frac{\pi\cos\phi}{\frac{4}{\pi}\sin\phi+2\cos(\phi/2)},$
or the inequality:
$\frac{\cos\phi}{1-\frac{2\phi}{\pi}}\geq\cos(\phi/2)\left(\frac{4}{\pi}\sin(\phi/2)+1\right).$
In order to prove the latter it is sufficient to prove:
$\frac{\cos\phi}{1-\frac{2\phi}{\pi}}\geq\cos(\phi/2)\left(1+\frac{2\phi}{\pi}\right),$
or:
$\frac{\cos\phi}{1-\frac{4\phi^{2}}{\pi^{2}}}\geq\cos(\phi/2).$
By considering the Weierstrass product for the cosine function we may rewrite
the last line in the form:
$\prod_{k=1}^{+\infty}\left(1-\frac{4x^{2}}{(2k+1)^{2}\pi^{2}}\right)\geq\prod_{k=1}^{+\infty}\left(1-\frac{x^{2}}{(2k-1)^{2}\pi^{2}}\right).$
By considering the Taylor series of the logarithm of both sides, we simply
have to prove:
$\forall
m\in\mathbb{N}_{0},\qquad(4^{m}-1)\zeta(2m)-4^{m}-(1-4^{-m})\zeta(2m)\leq 0,$
that is a consequence of:
$\forall m\in\mathbb{N}_{0},\qquad\zeta(2m)\leq\frac{4^{m}+1}{4^{m}-1},$
implied by:
$\forall m\in\mathbb{N}_{0},\qquad(4^{m}-1)(\zeta(2m)-1)\leq 2.$
An upper bound for the LHS is the series:
$1+\sum_{k=1}^{+\infty}\left(\frac{4}{(2k+1)^{2}}\right)^{m},$
whose value decreases as $m$ increases; so we have:
$(4^{m}-1)(\zeta(2m)-1)\leq
1+\sum_{k=1}^{+\infty}\frac{4}{(2k+1)^{2}}=3\zeta(2)-3,$
and the RHS is less than $2$ since $\pi^{2}<10$ holds. ∎
Now we make a step back into the general setting of double inequalities for
the arctangent function.
###### Lemma 1.
If $f(u),g(u)$ are a couple of real functions such that, for any $u\in[0,1]$,
$f(u)\leq\arctan u\leq g(u)$
holds, then:
$2\cdot f\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\leq\arctan x\leq 2\cdot
g\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)$
holds for any $x\in\mathbb{R}^{+}$.
###### Proof.
In virtue of the angle bisector theorem,
$\arctan t=2\arctan\left(\frac{t}{1+\sqrt{1+t^{2}}}\right)$
for any $t\geq 0$, so if the first inequality holds for any $\theta=\arctan u$
in the range $[0,\pi/4]$, the second inequality holds for any $\theta=\arctan
x$ in the range $[0,\pi/2]$. ∎
The last lemma gives a third way to prove the Shafer-Fink inequality. By
direct inspection of the Taylor series of $\frac{\arctan u}{u}$, it is easy to
show that $(3+u^{2})\frac{\arctan u}{u}$ is an increasing function over
$[0,1]$, so:
$\frac{3u}{3+u^{2}}\leq\arctan u\leq\frac{\pi u}{3+u^{2}},$
and it is sufficient to use the substitution $u=\frac{x}{1+\sqrt{1+x^{2}}}$ to
give another proof of the Shafer-Fink inequality.
###### Lemma 2.
If an approximation $f(u)$ of the arctangent function satisfies:
$\|f(u)-\arctan(u)\|_{\mathbb{R}^{+}}=\sup_{u\in\mathbb{R}^{+}}|f(u)-\arctan(u)|=C_{\infty},$
then
$\left\|2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)-\arctan(u)\right\|_{\mathbb{R}^{+}}=2\cdot\|f(u)-\arctan(u)\|_{(0,1)}=2\cdot
C_{1},$
and, for any $t\in(0,1)$,
$\left\|2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)-\arctan(u)\right\|_{(0,t)}=2\cdot\|f(u)-\arctan(u)\|_{\left(0,\frac{2t}{1-t^{2}}\right)}.$
This simple consequence of the previous lemma tell us the fact that any
algebraic approximation of the arctangent function in a right neighbourhood of
zero can be “lifted” to an algebraic approximation over the whole
$\mathbb{R}^{+}$, through the iteration of the map
$f(u)\quad\longrightarrow\quad 2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right).$
For example, if we consider the Lagrange interpolation polynomial for the
arctangent function with respect to the points
$(0,\tan(\pi/8)=\sqrt{2}-1,\tan(\pi/4)=1)$
$p(x)=\frac{\pi}{4}\cdot\frac{x(x-\sqrt{2}+1)}{2-\sqrt{2}}+\frac{\pi}{8}\cdot\frac{x(x-1)}{(\sqrt{2}-1)(\sqrt{2}-2)},$
we have
$\|p(x)-\arctan x\|_{(0,1)}<\frac{1}{230},$
so, by considering $2\cdot p\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)$:
###### Theorem 5.
For any non negative real number $x$, the absolute difference between
$\arctan(x)$ and
$\frac{\pi
x\left(\left(4+\sqrt{2}\right)\left(1+\sqrt{1+x^{2}}\right)-\sqrt{2}\,x\right)}{8\left(1+\sqrt{1+x^{2}}\right)^{2}}$
is less than $\frac{1}{115}$.
Another way to produce really effective approximation is to use the Chebyshev
expansion for the arctangent function:
###### Lemma 3.
The sequence of functions:
$f_{n}(x)=2\sum_{k=0}^{n}\frac{(-1)^{k}}{(2k+1)(1+\sqrt{2})^{2k+1}}\;T_{2k+1}(x),$
where $T_{k}(x)$ is the $k$-th Chebyshev polynomial of the first kind, gives a
uniform approximation of the arctangent function over the interval $[0,1]$:
$\|\arctan x-f_{n}(x)\|_{[0,1]}\leq\frac{1}{(1+\sqrt{2})^{2n+3}}.$
Moreover,
$\arctan(mx)=2\sum_{k=0}^{+\infty}\frac{(-1)^{k}}{(2k+1)}\left(\frac{m}{1+\sqrt{1+m^{2}}}\right)^{2k+1}\,T_{2k+1}(x)$
holds for any $x\in(-1,1)$ and for any $m\in\mathbb{N}_{0}$.
###### Theorem 6.
For any $n\in\mathbb{N}_{0}$ and for any $x\in\mathbb{R}$
$\left|\;\arctan
x-4\sum_{k=0}^{n}\frac{(-1)^{k}}{(2k+1)(1+\sqrt{2})^{2k+1}}\;T_{2k+1}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\right|\leq\frac{1}{\left(3+2\sqrt{2}\right)^{n}}.$
Still another way is to use the continued fraction representation for the
arctangent funtion:
$\arctan
z=\frac{z}{1+\frac{z^{2}}{3+\frac{4z^{2}}{5+\frac{9z^{2}}{7+\frac{16z^{2}}{9+\frac{25z^{2}}{11+\ldots}}}}}},$
from which we get a sequence of approximations for $\arctan x$ over $[0,1]$:
$\left\\{\begin{array}[]{cll}K_{1}(x)&=\displaystyle\frac{x}{1+x^{2}/3},&\\\\[11.38092pt]
K_{2}(x)&=\displaystyle\frac{x}{1+x^{2}/(3+4x^{2}/5)}&=\displaystyle\frac{x(15+4x^{2})}{15+9x^{2}},\\\\[11.38092pt]
K_{3}(x)&=\displaystyle\frac{x}{1+x^{2}/(3+4x^{2}/(5+9x^{2}/7))}&=\displaystyle\frac{5x\left(21+11x^{2}\right)}{105+90x^{2}+9x^{4}}\\\\[11.38092pt]
\ldots&&\end{array}\right.$
that satisfy:
$\|\arctan x-K_{n}(x)\|_{[0,1]}\leq\frac{1}{2\cdot 4^{n}},$
so:
$\left\|\arctan
x-K_{n}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\right\|_{\mathbb{R}}\leq\frac{1}{4^{n}},$
with an error term that is the same achieved by $a_{n}(x)$, defined as in
Theorem (3).
By using the Taylor series for the arctangent function with respect to the
point $x=1$ one has:
$\arctan{x}=\frac{\pi}{4}-\sum_{j=0}^{+\infty}\left(-\frac{(1-x)^{4}}{4}\right)^{j}\cdot\left(\frac{(1-x)}{2(4j+1)}+\frac{(1-x)^{2}}{2(4j+2)}+\frac{(1-x)^{3}}{4(4j+3)}\right).$
By plugging in $x=2/3$ we have:
$\arctan\frac{1}{5}=\sum_{j=0}^{+\infty}\left(-\frac{1}{324}\right)^{j}\cdot\left(\frac{1}{6(4j+1)}+\frac{1}{18(4j+2)}+\frac{1}{108(4j+3)}\right),$
and by plugging in $x=119/120$ we have:
$\arctan\frac{1}{239}=\sum_{j=0}^{+\infty}\left(-\frac{1}{829440000}\right)^{j}\cdot\left(\frac{1}{240(4j+1)}+\frac{1}{28800(4j+2)}+\frac{1}{6912000(4j+3)}\right).$
The Machin Formula
$\frac{\pi}{4}=4\arctan\frac{1}{5}+\arctan\frac{1}{239}$
give us the possibility to exhibit a good approximation for $\pi$:
$\displaystyle\pi=8\;$
$\displaystyle\sum_{j=0}^{+\infty}\left(-\frac{1}{324}\right)^{j}\cdot\left(\frac{1}{3(4j+1)}+\frac{1}{9(4j+2)}+\frac{1}{54(4j+3)}\right)+$
$\displaystyle+$
$\displaystyle\sum_{j=0}^{+\infty}\left(-\frac{1}{829440000}\right)^{j}\cdot\left(\frac{1}{60(4j+1)}+\frac{1}{7200(4j+2)}+\frac{1}{1728000(4j+3)}\right).$
In the same fashion, we have that:
$\arctan\frac{1}{2z-1}=\sum_{j=0}^{+\infty}\left(-\frac{1}{4z^{4}}\right)^{j}\cdot\left(\frac{1}{2z(4j+1)}+\frac{1}{2z^{2}(4j+2)}+\frac{1}{4z^{3}(4j+3)}\right)$
holds for any $z\geq 1$, and the truncated series gives a better and better
approximation as $z$ goes to infinity. By a change of variable, the same is
true for:
$\arctan\frac{1}{t}=\sum_{j=0}^{+\infty}\left(-\frac{4}{(t+1)^{4}}\right)^{j}\cdot\left(\frac{1}{(t+1)(4j+1)}+\frac{2}{(t+1)^{2}(4j+2)}+\frac{2}{(t+1)^{3}(4j+3)}\right),$
and:
$\arctan
u=\sum_{j=0}^{+\infty}\left(-\frac{4u^{4}}{(u+1)^{4}}\right)^{j}\cdot\left(\frac{u}{(u+1)(4j+1)}+\frac{2u^{2}}{(u+1)^{2}(4j+2)}+\frac{2u^{3}}{(u+1)^{3}(4j+3)}\right)$
holds for any $u\in[0,1]$. By taking:
$s_{n}(u)=\sum_{j=0}^{n}\left(-\frac{4u^{4}}{(u+1)^{4}}\right)^{j}\cdot\left(\frac{u}{(u+1)(4j+1)}+\frac{2u^{2}}{(u+1)^{2}(4j+2)}+\frac{2u^{3}}{(u+1)^{3}(4j+3)}\right)$
we have that:
$\left|\arctan u-s_{n}(u)\right|\leq\left(\frac{\sqrt{2}\,u}{u+1}\right)^{4n}$
for any $u\in[0,1]$, with $s_{n}$ being an upper bound for $\arctan u$ over
$[0,1]$ for any even $n$ and a lower bound for any odd $n$. If we consider:
$t_{n}(u)=\frac{\pi}{4}-s_{n}\left(\frac{1-u}{1+u}\right)=\frac{\pi}{4}-\sum_{j=0}^{n}\left(-\frac{(1-u)^{4}}{4}\right)^{j}\cdot\left(\frac{1-u}{2(4j+1)}+\frac{(1-u)^{2}}{2(4j+2)}+\frac{(1-u)^{3}}{4(4j+3)}\right),$
then $t_{n}$ is a lower/upper bound for the arctangent function over $[0,1]$
if and only if $s_{n}$ is a lower/upper bound, and:
$\left|\arctan u-t_{n}(u)\right|\leq\left(\frac{1-u}{\sqrt{2}}\right)^{4n}$
holds. Any convex combination of $s_{n}$ and $t_{n}$ is still a lower/upper
bound - by taking:
$w_{n}(u)=\frac{u^{4n+4}\cdot t_{n}(u)+(1-u)^{4n+4}\cdot
s_{n}(u)}{u^{4n+4}+(1-u)^{4n+4}}$
we can perform a reduction of the uniform error, since:
$\left|w_{n}(u)-\arctan u\right|\leq\frac{1}{20^{n}}$
and the error function goes very fast to zero when $u$ approaches $0$ or $1$.
This gives that
$w_{n}\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)$
is an especially good lower/upper bound for the arctangent function when $u$
is close to $0$ or much bigger than $1$, achieving the same uniform error term
with respect to the generalized Shafer-Fink inequality or the continued
fraction expansion.
## References
* [1] A.M. Fink: Two inequalities, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 6 (1995), 48–49.
(http://pefmath.etf.bg.ac.yu/)
* [2] Feng Qi, Shi-Qin Zhang, Bai-Ni Guo: Sharpening and generalizations of Shafer’s inequality
for the arc tangent function, Journal of Inequalities and Applications 2009
(2009).
* [3] R. E. Shafer, E 1867, Amer. Math. Monthly 73 (1966), no. 3, 309.
* [4] D. S. Mitrinović, Elementary Inequalities, Groningen, 1964.
|
arxiv-papers
| 2013-03-10T12:49:02 |
2024-09-04T02:49:43.786861
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jacopo D'Aurizio",
"submitter": "Jacopo D'Aurizio",
"url": "https://arxiv.org/abs/1304.0753"
}
|
1304.0806
|
# $IFP$-intuitionistic fuzzy soft set theory and its applications
Faruk Karaaslan [email protected] Naim Çağman
[email protected] Şaban Yılmaz [email protected] Department of
Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University, 60250
Tokat, Turkey
###### Abstract
In this work, we present definition of intuitionistic fuzzy parameterized
$(IFP)$ intuitionistic fuzzy soft set and its operations. Then we define
$IFP$-aggregation operator to form $IFP$-intuitionistic fuzzy soft-decision-
making method which allows constructing more efficient decision processes.
###### keywords:
Soft set, fuzzy set, intuitionistic fuzzy set, intuitionistic fuzzy soft set,
intuitionistic fuzzy parameterized intuitionistic fuzz soft set, aggregation
operator.
## 1 Introduction
Problems in economy, engineering, environmental science and social science and
many fields involve data that contain uncertainties. This problems may not be
successfully modeled by existing methods in classical mathematics because of
various types of uncertainties. There are some well known mathematical
theories for dealing with uncertainties such as; fuzzy set theory [17], soft
set theory [15], intuitionistic fuzzy set theory [1], fuzzy soft set theory
[12] and so on.
In 1999, Molodtsov [15] firstly introduced the soft set theory as a general
mathematical tool for dealing with uncertainty and vagueness. Since then some
authors studied on the operations of soft sets [2, 3, 14].
Many interesting results of soft set theory have been studied by embedding the
ideas of fuzzy sets. Fuzzy Soft sets [5, 4, 9, 12, 16], intuitionistic fuzzy
soft sets [10, 11, 13]. Firstly, fuzzy parameterized soft set and fuzzy
parameterized fuzzy soft set and their operations are introduced Çag̃man et
al. in [4, 5]. Intuitionistic fuzzy parameterized soft set is define by
Çag̃man and Deli [7] and intuitionistic fuzzy parameterized fuzzy soft set and
its operations are introduced by Çag̃man and Karaaslan [8].
In this paper, firstly we present preliminaries and then we introduce
intuitionistic fuzzy parameterized intuitionistic fuzzy soft set and their
properties. We also define $IFP$-aggregation operator to form
$IFP-$intuitionistc fuzzy soft decision making method that allows constructing
more efficient decision processes. We finally present examples which shows
that the methods can be successfully applied to many problems that contain
uncertainties.
## 2 Preliminary
In this section, we present definitions and some results of soft set, fuzzy
set, fuzzy soft set, intuitionistic fuzzy set and intuitionistic fuzzy soft
set theory that can be found details [1, 3, 12, 5, 11, 14, 15, 17].
Throughout this subsection $U$ refers to an initial universe, $E$ is a set of
parameters, $P(U)$ is the power set of $U$.
###### Definition 1
[3] Let $U$ be an initial universe, $P(U)$ be the power set of $U$, $E$ is the
set of all parameter and $A\subseteq E$. Then, a soft set $F_{A}$ on the
universe $U$ is defined by a function $f_{A}$ representing a mapping
$f_{A}:E\rightarrow P(U)\textrm{ such that }f_{A}(x)=\emptyset\textrm{ if
}x\notin A$
Here $f_{A}$ is called approximate function of soft set $F_{A}$, and the value
$f_{A}(x)$ is a set called $x$-element of the soft set for all $x\in E$. It is
worth nothing that the set $f_{A}(x)$ may be arbitrary. Some of them may be
empty, some may have nonempty intersection. Thus, a soft set $F_{A}$ over $U$
can be represented by the set of ordered pairs
$F_{A}=\\{(x,f_{A}):x\in E,f_{A}(x)\in P(U)\\}$
Note that the set of all soft sets over $U$ will be denoted by $S(U)$.
###### Definition 2
[17] Let $U$ be a universe. A fuzzy set $X$ over $U$ is a set defined by a
function $\mu_{X}$ representing a mapping
$\mu_{X}:U\rightarrow[0,1]$
Here, $\mu_{X}$ called membership function of $X$ and the value $\mu_{X}(u)$
is called the grade of membership of $u\in U$. The value represents the degree
of $u$ belonging to fuzzy set $X$. Thus, a fuzzy set $X$ over $U$ can be
represented as follows,
$X=\\{(u,\mu_{X}(u)):u\in U,\mu_{X}(u)\in[0,1]\\}$
Note that the set of all the fuzzy sets over $U$ will be denoted by $F(U)$.
###### Definition 3
[1] An intuitionistic fuzzy set $(IFS)$ $X$ in U is defined as an object of
the following form
$X=\\{(x,\mu_{X}(u),\nu_{X}(u)):u\in U\\},$
where the functions $\mu_{X}:U\to[0,1]\textrm{ and }\nu_{X}:U\to[0,1]$ define
the degree of membership and the degree of non-membership of the element $u\in
U$, respectively, and for every $u\in U$,
$0\leq\mu_{X}(u)+\nu_{X}(u)\leq 1.$
In addition for all $u\in U$, $U=\\{(u,1,0):u\in
U\\}\,\,,\,\emptyset=\\{(u,0,1):u\in U\\}$ are intuitionistic fuzzy universal
and intuitionistic fuzzy empty set, respectively.
###### Theorem 1
[1] Let $X$ and $Y$ be two intuitionistic fuzzy sets. Then,
1. i.
$X\subseteq Y\Leftrightarrow\forall u\in
U,\mu_{X}(u)\leq\mu_{Y}(u),\nu_{X}(u)\geq\nu_{Y}(u)$
2. ii.
$X\cap
Y=\\{(u,min\\{\mu_{X}(u),\mu_{Y}(u)\\},max\\{\nu_{X}(u),\nu_{Y}(u)\\}):u\in
U\\}$
3. iii.
$X\cup
Y=\\{(u,max\\{\mu_{X}(u),\mu_{Y}(u)\\},min\\{\nu_{X}(u),\nu_{Y}(u)\\}):u\in
U\\}.$
4. iv.
$X^{c}=\\{(u,\nu_{X}(u),\mu_{X}(u)):u\in U\\}.$
Note that the set of all the fuzzy sets over $U$ will be denoted by
$\mathcal{IF}(U)$.
###### Definition 4
[6] Let $U$ be an initial universe, $\mathcal{IF}(U)$ be the set of all
intuitionistic fuzzy sets over $U$, E be a set of all parameters and
$A\subseteq E$. Then, an intuitionistic fuzzy soft set (IFS-set) $\gamma_{A}$
over U is a function from $E$ into $\mathcal{IF}(U)$.
Where, the value $\gamma_{A}(x)$ is an intuitionistic fuzzy set over U. That
is,
$\gamma_{A}(x)=\\{(u,\overline{\gamma}_{A(x)}(u),\underline{\gamma}_{A(x)}(u)):x\in
E,u\in U)\\}$, where $\overline{\gamma}_{A(x)}(u)$ and
$\underline{\gamma}_{A(x)}(u)$ are the membership and non-membership degrees
of $u$ to the parameter x, respectively.
Note that, the set of all intuitionistic fuzzy soft sets over $U$ is denoted
by $\mathcal{IFS}(U)$.
###### Definition 5
[6] Let $A,B\subseteq E$, $\gamma_{A}$ and $\gamma_{B}$ be two IFS-sets. Then,
$\gamma_{A}$ is said to be an intuitionistic fuzzy soft subset of $\gamma_{B}$
if
(1) $A\subseteq B$ and
(2) $\gamma_{A}(x)$ is an intuitionistic fuzzy subset of $\gamma_{B}(x)$
$\forall x\in A$.
This relationship is denoted by $\gamma_{A}\tilde{\subseteq}\gamma_{B}$.
Similarly, $\gamma_{A}$ is said to be an intuitionistic fuzzy soft superset of
$\gamma_{B}$, if $\gamma_{B}$ is an intuitionistic fuzzy soft subset of
$\gamma_{A}$ and denoted by $\gamma_{A}\tilde{\supseteq}\gamma_{B}$.
###### Definition 6
[6] Let $\gamma_{A}$ and $\gamma_{B}$ be two intuitionistic fuzzy soft sets
over U. Then, $\gamma_{A}$ and $\gamma_{B}$ are said to be intuitionistic
fuzzy soft equal if and only if $\gamma_{A}$ is an intuitionistic fuzzy soft
subset of $\gamma_{B}$ and $\gamma_{B}$ is an intuitionistic fuzzy soft subset
of $\gamma_{A}$, and written by $\gamma_{A}$=$\gamma_{B}$.
###### Definition 7
[6] Let $\gamma_{A}$ be an IFS-set over $\mathcal{IF}(U)$. If
$\gamma_{A}(x)=\emptyset$ for all $x\in E$, then $\gamma_{A}$ is called empty
IFS-set and denoted by $\gamma_{\phi}$.
###### Definition 8
[6] Let $\gamma_{A}$ be an IFS-set over $\mathcal{IF}(U)$. If
$\gamma_{A}(x)=\\{(u,1,0):\forall u\in U\\}$ for all $x\in A$, then
$\gamma_{A}$ is called A-universal IFS-set and denoted by $\gamma_{\hat{A}}$.
If A=E, then the A-universal IFS-set is called universal IFS-set and denoted
by $\gamma_{\hat{E}}$.
###### Definition 9
[6] Let $\gamma_{A}$ and $\gamma_{B}$ be two IFS-sets over $\mathcal{IF}(U)$.
Union of $\gamma_{A}$ and $\gamma_{B}$, denoted by
$\gamma_{A}\tilde{\cup}\gamma_{B}$, and is defined by
$\gamma_{A}\tilde{\cup}\gamma_{B}=\\{(x,\gamma_{A\tilde{\cup}B}(x)):x\in E\\}$
where
$\gamma_{A\tilde{\cup}B}(x)=\\{(u,max\\{\overline{\gamma}_{A(x)}(u),\overline{\gamma}_{B(x)}(u)\\},min\\{\underline{\gamma}_{A(x)}(u),\underline{\gamma}_{B(x)}(u)\\}):u\in
U\\}.$
###### Definition 10
[6] Let $\gamma_{A}$ and $\gamma_{B}$ be two IFS-set over $\mathcal{IF}(U)$.
Intersection of $\gamma_{A}$ and $\gamma_{B}$, denoted by
$\gamma_{A}\tilde{\cap}\gamma_{B}$, and is defined by
$\gamma_{A}\tilde{\cap}\gamma_{B}=\\{(x,\gamma_{A\tilde{\cap}B}(x)):x\in E\\}$
where
$\gamma_{A\tilde{\cap}B}(x)=\\{(u,min\\{\overline{\gamma}_{A(x)}(u),\overline{\gamma}_{B(x)}(u)\\},max\\{\underline{\gamma}_{A(x)}(u),\underline{\gamma}_{B(x)}(u)\\}):u\in
U\\}.$
###### Definition 11
[6] Let $\gamma_{A}$ be an IFS-set over $\mathcal{IF}(U)$. Complement of
$\gamma_{A}$, denoted by $\gamma_{A}^{c}$, and is defined by
$\gamma_{A}^{c}=\\{(x,\gamma_{A^{c}}(x)):x\in E\\}$
where $\gamma_{A^{c}}(x)=\gamma_{A}^{c}(x)$ is the complement of
intuitionistic fuzzy set $\gamma_{A}(x)$, defined by
$\gamma_{A}^{c}(x)=\\{(u,\underline{\gamma}_{A(x)}(u),\overline{\gamma}_{A(x)}(u)):u\in
U\\}$
for all $x\in E$
###### Definition 12
[6]Let $\gamma_{A}$ and $\gamma_{B}$ be two IFS-set over $\mathcal{IF}(U)$.
$\wedge-$product of $\gamma_{A}$ and $\gamma_{B}$, denoted by
$\gamma_{A}\wedge\gamma_{B}$, and is defined by
$\gamma_{A}\wedge\gamma_{B}=\\{((x,y),\gamma_{A\wedge B}(x,y)):(x,y)\in
E\times E\\}$
where
$\gamma_{A\wedge
B}(x,y))=\\{(u,min(\mu_{\gamma_{A(x)}}(u),\mu_{\gamma_{B(x)}}(u))),max(\nu_{\gamma_{A(x)}}(u),\nu_{\gamma_{B(x)}}(u))):u\in
U\\}$
for all $x,y\in E$
###### Definition 13
[6]Let $\gamma_{A}$ and $\gamma_{B}$ be two IFS-set over $\mathcal{IF}(U)$.
$\vee-$product of $\gamma_{A}$ and $\gamma_{B}$, denoted by
$\gamma_{A}\vee\gamma_{B}$, and is defined by
$\gamma_{A}\vee\gamma_{B}=\\{((x,y),\gamma_{A\vee B}(x,y)):(x,y)\in E\times
E\\}$
where
$\gamma_{A\vee
B}(x,y))=\\{(u,max(\mu_{\gamma_{A(x)}}(u),\mu_{\gamma_{B(x)}}(u))),min(\nu_{\gamma_{A(x)}}(u),\nu_{\gamma_{B(x)}}(u))):u\in
U\\}$
for all $x,y\in E$
###### Definition 14
[5] Let $U$ be an initial universe, $E$ be the set of all parameters and $X$
be a fuzzy set over $E$ with the membership function
$\mu_{X}:E\rightarrow[0,1]$ and $\gamma_{X}(x)$ be a fuzzy set over $U$ for
all $x\in E$. Then, and $fpfs-$set $\Gamma_{X}$ over $U$ is a set defined by a
function $\gamma_{X}(x)$ representing a mapping
$\gamma_{X}:E\rightarrow F(U)\textrm{ such that
}\gamma_{X}(x)=\emptyset\textrm{ if }\mu_{X}(x)=0$
Here, $\gamma_{X}$ is called fuzzy approximate function of ($fpfs$-set)
$\Gamma_{X}$, and the value $\gamma_{X}(x)$ is a fuzzy set called $x-$element
of the $fpfs-$set for all $x\in E$. Thus, an $fpfs-$set $\Gamma_{X}$ over $U$
can be represented by the set of ordered pairs
$\Gamma_{X}=\\{(\mu_{X}(x)/x,\gamma_{X}(x)):x\in E,\gamma_{X}(x)\in
F(U),\mu_{X}(x)\in[0,1]\\},$
It must be noted that the sets of all $fpfs$-sets over $U$ will be denoted by
$FPFS(U)$.
###### Definition 15
[7] Let $U$ be an initial universe, $P(U)$ be the power set of $U$, $E$ is the
set of all parameters and $X$ be a intuitionistic fuzzy set over E with the
membership function $\mu_{X}:E\rightarrow[0,1]$ and non-membership function
$\nu_{X}:E\rightarrow[0,1]$. Then, an $ifps-$set $F_{X}$ over $U$ is a set
defined by a function $f_{X}$ representing a mapping
$f_{X}:E\rightarrow P(U)\textrm{ such that }f_{X}(x)=\emptyset\textrm{ if
}\mu_{x}=0,\nu_{x}=1$
Here, $f_{X}$ is called approximate function of the $ifps$-set $F_{X}$, and
the value $f_{X}(x)$ is a set called $x-$element of $ifps$-set for all $x\in
E$. Thus, $ifps-$set $F_{X}$ over $U$ can be represented by the set of ordered
pairs
$F_{X}=\\{((\mu_{X}(x),\nu_{X}(x))/x,f_{X}(x)):x\in E,f_{X}(x)\in
P(U),\mu_{X}(x),\nu_{X}(x)\in[0,1]\\}$
###### Definition 16
[8] Let $U$ be an initial universe, $P(U)$ be the power set of $U$, $E$ is the
set of all parameters and $X$ be a intuitionistic fuzzy set over E with the
membership function $\mu_{X}:E\rightarrow[0,1]$ and non-membership function
$\nu_{X}:E\rightarrow[0,1]$. Then, an $ifps-$set $F_{X}$ over $U$ is a set
defined by a function $f_{X}$ representing a mapping
$f_{X}:E\rightarrow P(U)\textrm{ such that }f_{X}(x)=\emptyset\textrm{ if
}\mu_{x}=0,\nu_{x}=1$
Here, $f_{X}$ is called approximate function of the $ifps$-set $F_{X}$, and
the value $f_{X}(x)$ is a set called $x-$element of $ifps$-set for all $x\in
E$. Thus, $ifps-$set $F_{X}$ over $U$ can be represented by the set of ordered
pairs
$F_{X}=\\{((\mu_{X}(x),\nu_{X}(x))/x,f_{X}(x)):x\in E,f_{X}(x)\in
P(U),\mu_{X}(x),\nu_{X}(x)\in[0,1]\\}$
## 3 $IFP-$intuitionistic fuzzy soft sets
In this section, we define intuitionistic fuzzy parameterized intuitionistic
fuzzy soft sets and their operations with examples.
Throughout this work, we use $\Omega_{X},\Omega_{Y},\Omega_{Z},...,$etc. for
$\Omega$-sets and $\omega_{X},\omega_{Y},\omega_{Z},...,$etc. for their
intuitionistic fuzzy approximate function, respectively.
###### Definition 17
Let $U$ be an initial universe, $E$ be the set of all parameters and $X$ be an
intuitionistic fuzzy set over $E$ with the membership function
$\mu_{X}:E\rightarrow[0,1]$ and non-membership function
$\nu_{X}:E\rightarrow[0,1]$ and $\omega_{X}$ is an intuitionistic fuzzy set
over $U$ for all $x\in E$. Then, an $\Omega$-set $\Omega_{X}$ over
$\mathcal{IF}(U)$ is a set defined by a function $\omega_{X}(x)$ representing
a mapping
$\omega_{X}:E\rightarrow\mathcal{IF}(U)\textrm{ such that
}\omega_{X}(x)=\emptyset\textrm{ if }x\not\in X$
Here, $\omega_{X}$ is called intuitionistic fuzzy approximation of
$\Omega$-set $\Omega_{X}$. $\omega_{X}(x)$ is an intuitionistic fuzzy set
called $x$-element of the $\Omega$-set for all $x\in E$. Thus, an $\Omega$-set
$\Omega_{X}$ over $U$ can be represented by the set of ordered pairs
$\Omega_{X}=\\{((\mu_{X}(x),\nu_{X}(x))/x,\omega_{X}(x)):x\in E,u\in
U,\omega_{X}(x)\in\mathcal{IF}(U)\\}$
Note that, If $\mu_{X}(x)=0,\nu_{X}(x)=1$ and $\omega_{X}(x)=\emptyset$, we
don’t display such elements in the set. Also, it must be noted that the sets
of all $\Omega$-sets over $\mathcal{IF}(U)$ will be denoted by $\Omega(U)$.
###### Example 1
Assume that $U=\\{u_{1},u_{2}\,u_{3},u_{4},u_{5}\\}$ is an universal set and
$E=\\{x_{1},x_{2},x_{3},x_{4},x_{5}\\}$ a set of parameters. If
$X=\\{(0.5,0.2)/x_{1},(0.6,0.3)/x_{3},(1.0,0.0)/x_{4}\\}$
$\omega_{X}(x_{1})=\\{(0.7,0.2)/u_{1},(0.5,0.4)/u_{4}\\},\\\
\omega_{X}(x_{2})=\emptyset,\\\
\omega_{X}(x_{3})=\\{(0.4,0.3)/u_{2},(0.8,0.1)/u_{3},(0.6,0.3)/u_{5}\\}$
$\omega_{X}(x_{4})=U$,
then the $\Omega_{X}$ is written as follow
$\begin{array}[]{rcl}\Omega_{X}&=&\\{((0.5,0.2)/x_{1},\\{(0.7,0.2)/u_{1},(0.5,0.4)/u_{4}\\}),\\\
&&((0.6,0.3)/x_{3},\\{(0.4,0.3)/u_{2},(0.8,0.1)/u_{3},(0.6,0.3)/u_{5}\\}),\\\
&&((1.0,0.0)/x_{4},U)\\}\end{array}$
###### Definition 18
Let $\Omega_{X}\in\Omega$(U). If $\omega_{X}(x)=\emptyset$ for all $x\in E$,
then $\Omega_{X}$ is called an $X$-empty $\Omega$-set, denoted by
$\Omega_{\emptyset_{X}}$.
If $X=\emptyset,$ then the $X-$empty $\Omega$-set $(\Omega_{\emptyset_{X}})$
is called empty $\Omega$-set, denoted by $\Omega_{\emptyset}$. Here,
$\emptyset$ mean that intuitionistic fuzzy empty set.
###### Definition 19
Let $\Omega_{X}\in\Omega$(U). If $\mu_{X}(x)=1$, $\nu_{X}(x)=0$ and
$\omega_{X}(x)=U$ for all $x\in X$, then $\Omega_{X}$ is called X-universal
$\Omega$-set, denoted by $\Omega_{\tilde{X}}$.
If $X$ is equal to intuitionistic fuzzy universal set over $E$, then the
$X$-universal $\Omega$-set is called universal $\Omega$-set, denoted by
$\Omega_{\tilde{E}}$. Here, $U$ mean that intuitionistic fuzzy universal set.
###### Example 2
Let $U=\\{u_{1},u_{2},u_{3},u_{4}\\}$ be a universal set and
$E=\\{x_{1},x_{2},x_{3},x_{4}\\}$ be a set of parameters. If
$\begin{array}[]{rcl}X&=&\\{(0.2,0,5)/x_{2},(0.5,0,3)/x_{3},(1.0,0)/x_{4}\\}\textrm{
and }\\\ \omega_{X}(x_{1})&=&\emptyset,\\\
\omega_{X}(x_{2})&=&\\{(0.5,0.4)/u_{1},(0.7,0.3)/u_{5}\\},\\\
\omega_{X}(x_{3})&=&\emptyset,\\\ \omega_{X}(x_{4})&=&U,\end{array}$
then the $\Omega$-set $\Omega_{X}$ is written by
$\Omega_{X}=\\{((0.2,0,5)/x_{2},\\{(0.5,0.4)/u_{1},(0,1.0)/u_{2},(0,1.0)/u_{3},\\\
(0.7,0.3)/u_{4}\\}),((0,5.0,3)/x_{3},\emptyset),((1.0,0)/x_{4},U)\\}$.
If $Y=\\{(1.0,0)/x_{1},(0.7,0.2)/x_{4}\\}$ and $\omega_{Y}(x_{1})=\emptyset$,
$\omega_{Y}(x_{4})=\emptyset$ then the $\Omega$-set $\Omega_{Y}$ is an
$Y$-empty $\Omega$-set, i.e., $\Omega_{Y}=\Omega_{\Phi_{Y}}$.
If $Z=\\{(1.0,0)/x_{1},(1.0,0)/x_{2}\\}$, $\omega_{Z}(x_{1})=U$, and
$\omega_{Z}(x_{2})=U$, then the $\Omega$-set $\Omega_{Z}$ is $Z$-universal
$\Omega$-set, i.e., $\Omega_{Z}=\Omega_{\tilde{Z}}$.
If $X=E$ and $\omega_{X}(x_{i})=U$ for all $x_{i}\in E$, where $i=1,2,3,4$,
then the $\Omega$-set $\Omega_{X}$ is a universal $\Omega$-set, i.e.,
$\Omega_{X}=\Omega_{\tilde{E}}$.
###### Definition 20
Let $\Omega_{X},\Omega_{Y}\in\Omega$(U). Then, $\Omega_{X}$ is an
$\Omega$-subset of $\Omega_{Y}$, denoted by
$\Omega_{X}\widetilde{\subseteq}\Omega_{Y}$, if
$\mu_{X}(x)\leq\mu_{Y}(x),\nu_{X}(x)\geq\nu_{X}(x)\textrm{ and
}\omega_{X}(x)\subseteq\omega_{Y}(x)$ for all $x\in E$.
###### Proposition 1
Let $\Omega_{X},\Omega_{Y}\in\Omega$(U). Then,
(i)
$\Omega_{X}\widetilde{\subseteq}\Omega_{\tilde{E}}$
(ii)
$\Omega_{\Phi_{X}}\widetilde{\subseteq}\Omega_{X}$
(iii)
$\Omega_{\Phi}\widetilde{\subseteq}\Omega_{X}$
(iv)
$\Omega_{X}\widetilde{\subseteq}\Omega_{X}$
(v)
$\Omega_{X}\widetilde{\subseteq}\Omega_{Y}$ and
$\Omega_{Y}\widetilde{\subseteq}\Omega_{Z}\Rightarrow\Omega_{X}\widetilde{\subseteq}\Omega_{Z}$
###### Proof 1
They can be proved easily by using the fuzzy approximate and membership
functions of the $\Omega$-sets.
###### Definition 21
Let $\Omega_{X},\Omega_{Y}\in\Omega(U)$. Then, $\Omega_{X}$ and $\Omega_{Y}$
are $\Omega-$equal, written as $\Omega_{X}=\Omega_{Y}$, if and only if
$\mu_{X}(x)=\mu_{Y}(x)$, $\nu_{X}(x)=\nu_{Y}(x)$ and
$\omega_{X}(x)=\omega_{Y}(x)$ for all $x\in E$.
###### Proposition 2
Let $\Omega_{X},\Omega_{Y},\Omega_{Z}\in\Omega$(U). Then,
(i)
$\Omega_{X}=\Omega_{Y}$ and
$\Omega_{Y}=\Omega_{Z}\Leftrightarrow\Omega_{X}=\Omega_{Z}$
(ii)
$\Omega_{X}\widetilde{\subseteq}\Omega_{Y}$ and
$\Omega_{Y}\widetilde{\subseteq}\Omega_{X}\Leftrightarrow\Omega_{X}=\Omega_{Y}$
###### Definition 22
Let $\Omega_{X}\in\Omega$(U). Then the complement of $\Omega_{X}$, denoted by
$\Omega_{X}^{\tilde{c}}$, is defined by
$\Omega^{\tilde{c}}_{X}=\\{((\nu_{X}(x),\mu_{X}(x))/x,\omega^{c}_{X}(x)):x\in
E,\omega^{c}_{X}(x)\in\mathcal{IF}(U)\\}$
where $\omega_{X}^{c}(x)$ is complement of the intuitionistic fuzzy set
$\omega_{X}(x)$, that is, $\omega^{c}_{X}(x)=\omega_{X^{c}}(x)$ for every
$x\in E$.
###### Proposition 3
Let $\Omega_{X}\in\Omega$(U). Then,
(i)
$(\Omega_{X}^{\tilde{c}})^{\tilde{c}}=\Omega_{X}$
(ii)
$\Omega_{\Phi}^{\tilde{c}}=\Omega_{\tilde{E}}$
###### Proof 2
By using the intuitionistic fuzzy approximate, membership functions and
nonmembership functions of the $\Omega$-sets, the proof is straightforward.
###### Definition 23
Let $\Omega_{X},\Omega_{Y}\in\Omega(U)$. Then, union of $\Omega_{X}$ and
$\Omega_{Y}$, denoted by $\Omega_{X}\widetilde{\cup}\Omega_{Y}$, is defined by
$\Omega_{X}\cup\Omega_{Y}=\\{((\mu_{X\widetilde{\cup}Y}(x),\nu_{X\widetilde{\cap}Y}(x))/x,\omega_{X\widetilde{\cup}Y}(x)):x\in
E\\}.$
Here,
$\mu_{X\widetilde{\cup}Y}(x)=\max\\{\mu_{X}(x),\mu_{Y}(x)\\},\nu_{X\widetilde{\cup}Y}(x)=\min\\{\nu_{X}(x),\nu_{Y}(x)\\}\textrm{
and }$
$\omega_{X\widetilde{\cup}Y}(x)=\omega_{X}(x)\cup\omega_{Y}(x),\textrm{for all
}x\in E.$
Note that here $\omega_{X}(x)$ and $\omega_{Y}(x)$ are intuitionistic fuzzy
sets. Thus, in operations of between $\omega_{X}(x)$ and $\omega_{Y}(x)$, we
use the operations of intuitionistic fuzzy sets.
###### Proposition 4
Let $\Omega_{X},\Omega_{Y},\Omega_{Z}\in\Omega(U)$. Then,
(i)
$\Omega_{X}\widetilde{\cup}\Omega_{X}=\Omega_{X}$
(ii)
$\Omega_{X}\widetilde{\cup}\Omega_{\Phi}=\Omega_{X}$
(iii)
$\Omega_{X}\widetilde{\cup}\Omega_{\tilde{E}}=\Omega_{\tilde{E}}$
(iv)
$\Omega_{X}\widetilde{\cup}\Omega_{Y}=\Omega_{Y}\widetilde{\cup}\Omega_{X}$
(v)
$(\Omega_{X}\widetilde{\cup}\Omega_{Y})\widetilde{\cup}\Omega_{Z}=\Omega_{X}\widetilde{\cup}(\Omega_{Y}\widetilde{\cup}\Omega_{Z})$
###### Proof 3
The proofs can be easily obtained from Definition 23.
###### Definition 24
Let $\Omega_{X},\Omega_{Y}\in\Omega(U)$. Then, intersection of $\Omega_{X}$
and $\Omega_{Y}$, denoted by $\Omega_{X}\widetilde{\cap}\Omega_{Y}$, is
defined by
$\Omega_{X}\cap\Omega_{Y}=\\{((\mu_{X\widetilde{\cap}Y}(x),\nu_{X\widetilde{\cup}Y}(x))/x,\omega_{X\widetilde{\cap}Y}(x)):x\in
E\\}.$
Here,
$\mu_{X\widetilde{\cap}Y}(x)=\min\\{\mu_{X}(x),\mu_{Y}(x)\\},\nu_{X\widetilde{\cap}Y}(x)=\max\\{\nu_{X}(x),\nu_{Y}(x)\\}$
and
$\omega_{X\widetilde{\cap}Y}(x)=\omega_{X}(x)\cap\omega_{Y}(x)\,\textrm{for
all }x\in E.$
Note that here $\omega_{X}(x)$ and $\omega_{Y}(x)$ are intuitionistic fuzzy
sets. Thus, in operations of between $\omega_{X}(x)$ and $\omega_{Y}(x)$, we
use the operations of intuitionistic fuzzy sets.
###### Proposition 5
Let $\Omega_{X},\Omega_{Y},\Omega_{Z}\in\Omega(U)$. Then,
(i)
$\Omega_{X}\widetilde{\cap}\Omega_{X}=\Omega_{X}$
(ii)
$\Omega_{X}\widetilde{\cap}\Omega_{\Phi}=\Omega_{\Phi}$
(iii)
$\Omega_{X}\widetilde{\cap}\Omega_{\tilde{E}}=\Omega_{X}$
(iv)
$\Omega_{X}\widetilde{\cap}\Omega_{Y}=\Omega_{Y}\widetilde{\cap}\Omega_{X}$
(v)
$(\Omega_{X}\widetilde{\cap}\Omega_{Y})\widetilde{\cap}\Omega_{Z}=\Omega_{X}\widetilde{\cap}(\Omega_{Y}\widetilde{\cap}\Omega_{Z})$
###### Proof 4
The proofs can be easily obtained from Definition 24.
###### Remark 1
Let $\Omega_{X}\in\Omega(U)$. If $\Omega_{X}\neq\Omega_{\Phi}$ or
$\Omega_{X}\neq\Omega_{\tilde{E}}$, then
$\Omega_{X}\widetilde{\cup}\Omega_{X}^{\tilde{c}}\neq\Omega_{\tilde{E}}$ and
$\Omega_{X}\widetilde{\cap}\Omega_{X}^{\tilde{c}}\neq\Omega_{\Phi}$.
###### Proposition 6
Let $\Omega_{X},\Omega_{Y}\in\Omega(U)$. Then De Morgan’s laws are valid
(i)
$(\Omega_{X}\widetilde{\cup}\Omega_{Y})^{\tilde{c}}=\Omega_{X}^{\tilde{c}}\widetilde{\cap}\Omega_{Y}^{\tilde{c}}$
(ii)
$(\Omega_{X}\widetilde{\cap}\Omega_{Y})^{\tilde{c}}=\Omega_{X}^{\tilde{c}}\widetilde{\cup}\Omega_{Y}^{\tilde{c}}$
###### Proof 5
Firstly, for all $x\in E$,
$\begin{array}[]{lll}i.\omega_{(X\widetilde{\cup}Y)^{\tilde{c}}}(x)&=&\omega_{X\widetilde{\cup}Y}^{c}(x)\\\
&=&(\omega_{X}(x)\cup\omega_{Y}(x))^{c}\\\
&=&(\omega_{X}(x))^{c}\cap(\omega_{Y}(x))^{c}\\\
&=&\omega^{c}_{X}(x)\cap\omega^{c}_{Y}(x)\\\
&=&\omega_{X^{\tilde{c}}}(x)\cap\omega_{Y^{\tilde{c}}}(x)\\\
&=&\omega_{X^{\tilde{c}}\widetilde{\cap}Y^{\tilde{c}}}(x).\end{array}$
and
$\begin{array}[]{rcl}\Omega_{X}\widetilde{\cup}\Omega_{Y}&=&\\{(\max\\{\mu_{X}(x),\mu_{Y}(x)\\},\min\\{\nu_{X}(x),\nu_{Y}(x)))/x,\omega_{X\tilde{\cup}Y}(x)\\\
&&:x\in E\\}\\\
(\Omega_{X}\widetilde{\cup}\Omega_{Y})^{\tilde{c}}&=&\\{((\min\\{\nu_{X}(x),\nu_{Y}(x)\\},\max\\{\mu_{X}(x),\mu_{Y}(x)\\})/x,\omega_{(X\tilde{\cup}Y)^{c}}(x)):\\\
&&x\in E\\}\\\
&=&\\{((\min\\{\nu_{X}(x),\nu_{Y}(x)\\},\max\\{\mu_{X}(x),\mu_{Y}(x)\\})/x,\omega_{X^{c}\tilde{\cap}Y^{c}}(x)):\\\
&&x\in E\\}\\\ &=&\\{((\nu_{X}(x),\mu_{X}(x))/x,\omega_{X^{c}}(x)):x\in
E\\}\\\ &\cap&\\{((\nu_{Y}(x),\mu_{Y}(x)))/x,\omega_{Y^{c}}(x)):x\in E\\}\\\
&=&\Omega_{X}^{\tilde{c}}\widetilde{\cap}\Omega_{Y}^{\tilde{c}}\end{array}$
The proof of _ii._ can be made similarly.
###### Proposition 7
Let $\Omega_{X},\Omega_{Y},\Omega_{Z}\in\Omega(U)$. Then,
(i)
$\Omega_{X}\widetilde{\cup}(\Omega_{Y}\widetilde{\cap}\Omega_{Z})=(\Omega_{X}\widetilde{\cup}\Omega_{Y})\widetilde{\cap}(\Omega_{X}\widetilde{\cup}\Omega_{Z})$
(ii)
$\Omega_{X}\widetilde{\cap}(\Omega_{Y}\widetilde{\cup}\Omega_{Z})=(\Omega_{X}\widetilde{\cap}\Omega_{Y})\widetilde{\cup}(\Omega_{X}\widetilde{\cap}\Omega_{Z})$
###### Proof 6
For all $x\in E$,
$\begin{array}[]{lll}i.\,\,\,\mu_{X\widetilde{\cup}(Y\widetilde{\cap}Z)}(x)&=&\max\\{\mu_{X}(x),\mu_{Y\widetilde{\cap}Z}(x)\\}\\\
&=&\max\\{\mu_{X}(x),\min\\{\mu_{Y}(x),\mu_{Z}(x)\\}\\}\\\
&=&\min\\{\max\\{\mu_{X}(x),\mu_{Y}(x)\\},\max\\{\mu_{X}(x),\mu_{Z}(x)\\}\\}\\\
&=&\min\\{\mu_{X\widetilde{\cup}Y}(x),\mu_{X\widetilde{\cup}Z}(x)\\}\\\
&=&\mu_{(X\widetilde{\cup}Y)\widetilde{\cap}(X\widetilde{\cup}Z)}(x)\\\
\textrm{ and }\\\
\nu_{X\widetilde{\cup}(Y\widetilde{\cap}Z)}(x)&=&\min\\{\nu_{X}(x),\nu_{Y\widetilde{\cap}Z}(x)\\}\\\
&=&\min\\{\nu_{X}(x),\max\\{\nu_{Y}(x),\nu_{Z}(x)\\}\\}\\\
&=&\min\\{\max\\{\nu_{X}(x),\nu_{Y}(x)\\},\max\\{\nu_{X}(x),\nu_{Z}(x)\\}\\}\\\
&=&\min\\{\nu_{X\widetilde{\cup}Y}(x),\nu_{X\widetilde{\cup}Z}(x)\\}\\\
&=&\nu_{(X\widetilde{\cup}Y)\widetilde{\cap}(X\widetilde{\cup}Z)}(x)\end{array}$
$\begin{array}[]{lll}\omega_{X\widetilde{\cup}(Y\widetilde{\cap}Z)}(x)&=&\omega_{X}(x)\cup\omega_{Y\widetilde{\cap}Z}(x)\\\
&=&\omega_{X}(x)\cup(\omega_{Y}(x)\cap\omega_{Z}(x))\\\
&=&(\omega_{X}(x)\cup\omega_{Y}(x))\cap(\omega_{X}(x)\cup\omega_{Z}(x))\\\
&=&\omega_{X\widetilde{\cup}Y}(x)\cap\omega_{X\widetilde{\cup}Z}(x)\\\
&=&\omega_{(X\widetilde{\cup}Y)\widetilde{\cap}(X\widetilde{\cup}Z)}(x)\end{array}$
The proof of _ii._ can be made in a similar way.
## 4 Decision making method
The approximate function of an $\Omega$-set is intuitionistic fuzzy set. The
$\Omega_{agg}$ on the intuitionistic fuzzy sets is an operation by which
several approximate functions of an $\Omega$-set are combined to produce a
single intuitionistic fuzzy set that is the aggregate intuitionistic fuzzy set
of the $\Omega$-set. Once an aggregate intuitionistic fuzzy set has been
arrived at, it may be necessary to choose the best single crisp alternative
from this set. Therefore, we can construct a decision-making method by the
following algorithm.
_Step 1._
Construct an $\Omega$-set $\Omega_{X}$ over $U$,
_Step 2._
Find the aggregate intuitionistic fuzzy set $\Omega_{X}^{*}$ of $\Omega_{X}$,
_Step 3._
Find $max(u)=max\\{\mu_{\Omega_{X}^{*}}(u):u\in U\\}$ and
$min(v)=min\\{\nu_{\Omega_{X}^{*}}(v):v\in U\\}$
_Step 4._
Find $\alpha\in[0,1]$ such that $(max(u),\alpha)/u\in\Omega^{*}_{X}$ and
$\beta\in[0,1]$ such that $(\beta,min(v))/v\in\Omega^{*}_{X}$
_Step 5._
Find $\frac{max(u)}{max(u)+\alpha}=\alpha^{\prime}$ and
$\frac{\beta}{min(v)+\beta}=\beta^{\prime}$
_Step 6._
Opportune element of $U$ is denoted by $Opp(u)$ and it is chosen as follow
$Opp(u)$=$\left\\{\begin{array}[]{c}u,if\,\,\alpha^{\prime}>\beta^{\prime}\\\
v,if\,\,\beta^{\prime}<\alpha^{\prime}\end{array}\right.$
###### Example 3
In this example, we present an application for the $\Omega$-decision-making
method.
Let us assume that a company wants to fill a position. There are five
candidates who form the set of alternatives,
$U=\\{u_{1},u_{2},u_{3},u_{4},u_{5}\\}$. The choosing committee consider a set
of parameters, $E=\\{x_{1},x_{2},x_{3},x_{4}\\}$. For $i=1,2,3,4,5$, the
parameters $x_{i}$ stand for ”experience”, ”computer knowledge”, ”young age”
and ”good speaking”, respectively.
After a serious discussion each candidate is evaluated from point of view of
the goals and the constraint according to a chosen subset
$X=\\{(0.7,0.2)/x_{2},(0.8,0,2)/x_{3},(0.6,0.3)/x_{4}\\}$ of $E$. Finally, the
committee constructs the following $\Omega$-set over $U$.
Step 1: Let the constructed $\Omega$-set, $\Omega_{X}$, be as follows,
$\begin{array}[]{rl}\Omega_{X}=&\bigg{\\{}(0.7,0.2)/x_{2},\\{(0.4,0.3)/u_{1},(0.7,0.3)/u_{2},(0.6,0.2)/u_{3},(0.1,0.5)/u_{4},\\\
&(0.9,0.1)/u_{5}\\}),((0.8,0.2)/x_{3},\\{0.8,0.1)/u_{1},(0.8,0.2)/u_{2},(0.5,0.3)/u_{3},\\\
&(0.7,0.3)/u_{4}\\}),((0.6,0.3)/x_{4},\\{(0.5,0.5)/u_{1},(0.6,0.1)/u_{3},(0.3,0,6)/u_{5}\\})\bigg{\\}}\\\
\end{array}$
Step 2: The aggregate intuitionistic fuzzy set can be found as,
$\begin{array}[]{rcl}\Omega_{X}^{*}&=&\\{(0.318,0.057)/u_{1},(0.248,0.100)/u_{2},(0.295,0.033)/u_{3},(0.158,0.115)/u_{4},\\\
&&(0.203,0.100)/u_{5}\\}\end{array}$
Step 3:$max(u)=0.318$ and $min(v)=0.033$
Step 4: $(0.318,0.057)/u_{1}\in\Omega_{X}^{*}$ and
$(0.295,0.033)/u_{3}\in\Omega_{X}^{*}$
Step 5: $\alpha^{\prime}=\frac{0.318}{0.318+0.057}=0.848$ and
$\beta^{\prime}=\frac{0.295}{0.295+0.033}=0.899$
Step 6:Since $\alpha^{\prime}<\beta^{\prime}$, $Opp(u)=u_{3}$.
Note that, although membership degree of $u_{1}$ is bigger than $u_{3}$,
opportune element of $U$ is $u_{3}$. This example show how the effect on
decision making of non-membership degrees of elements.
## 5 Conclusion
In this paper, firstly we have defined $IFP$-intuitionistic fuzzy soft sets
and their operations. Then we have presented a decision making methods on the
$IFP$-intuitionistic fuzzy soft set theory. Finally, we have provided an
example that demonstrating that this method can successfully work. It can be
applied to problems of many fields that contain uncertainty.
## References
* [1] Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Set Syst 20:87-96
* [2] M.I. Ali, F. Feng, X. Liu, W.K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) 1547-1553.
* [3] N. C̣ağman , S. Enginoğlu (2010) Soft set theory and uni-int decision making. Eur J Oper Res 207:848-855
* [4] ̣Cağman N, Enginoğlu S, C̣ıtak F (2011) Fuzzy Soft Set Theory and Its Applications. Iran J Fuzzy Syst 8(3):137-147
* [5] N. Çağman, F C̣ıtak , S. Enginoğlu, Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal of Fuzzy System 1/1 (2010) 21-35.
* [6] N. Çağman, S. Karataş, Intuitionistic fuzzy soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems DOI:10.3233/IFS-2012-0601.
* [7] N. Çağman, I.Deli (2012) Intuitionistic fuzzy parametrized soft set theory and its decision making, Submitted.
* [8] N. Çağman, F.Karaaslan (2012) $IFP-$fuzzy soft set theory and its applications, Submitted.
* [9] Feng F, Li C, Davvaz B, Ali MI (2010) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14(9):899-911
* [10] Jiang Y, Tang Y, Chen Q, Liu H, Tang J (2010) Interval-valued intuitionistic fuzzy soft sets and their properties. Comput Math Appl 60(3):906-918
* [11] Jiang Y, Tang Y, Chen Q (2011) An adjustable approach to intuitionistic fuzzy soft sets based decision making. Appl Math Model 35:824-836
* [12] Maji PK, Biswas R, Roy AR (2001) Fuzzy Soft Sets. J Fuzzy Math 9(3):589-602
* [13] Maji PK, Biswas R, Roy AR (2001) Intuitionistic fuzzy soft sets. J Fuzzy Math 9(3):677-692
* [14] P.K. Maji, R. Biswas and A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562.
* [15] D.A. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999) 19-31.
* [16] Majumdar P, Samanta SK (2010) Generalised fuzzy soft sets. Comput Math Appl 59:1425-1432
* [17] Zadeh LA (1965) Fuzzy Sets. Inform Control 8:338-353
|
arxiv-papers
| 2013-04-02T22:10:00 |
2024-09-04T02:49:43.794294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Faruk Karaaslan, Naim Cagman, Saban Yilmaz",
"submitter": "Faruk Karaaslan",
"url": "https://arxiv.org/abs/1304.0806"
}
|
1304.0809
|
[to be supplied]
DRAFT New Equations for Neutral Terms
Guillaume Allais and Conor McBride University of Strathclyde
{guillaume.allais, conor.mcbride}@strath.ac.uk Pierre Boutillier PPS - Paris
Diderot [email protected]
# New Equations for Neutral Terms
A Sound and Complete Decision Procedure, Formalized
(2013)
###### Abstract
The definitional equality of an intensional type theory is its test of type
compatibility. Today’s systems rely on ordinary evaluation semantics to
compare expressions in types, frustrating users with type errors arising when
evaluation fails to identify two ‘obviously’ equal terms. If only the machine
could decide a richer theory! We propose a way to decide theories which
supplement evaluation with ‘$\nu$-rules’, rearranging the neutral parts of
normal forms, and report a successful initial experiment.
We study a simple $\lambda$-calculus with primitive fold, map and append
operations on lists and develop in Agda a sound and complete decision
procedure for an equational theory enriched with monoid, functor and fusion
laws.
###### keywords:
Normalization by Evaluation, Logical Relations, Simply-Typed Lambda Calculus,
Map Fusion
††conference: ICFP ’13 September 25–27, 2013, Boston
## 1 Introduction
The programmer working in intensional type theory is no stranger to ‘obviously
true’ equations she wishes held _definitionally_ for her program to typecheck
without having to chase down ill-typed terms and brutally coerce them. In this
article, we present one way to relax definitional equality, thus accommodating
some of her longings. We distinguish three types of fundamental relations
between terms.
The first denotes computational rules: it is untyped, _oriented_ and denoted
by $\leadsto$ in its one step version or $\leadsto^{\star}$ when the reflexive
transitive congruence closure is considered. In Table 1, we introduce a few
such rules which correspond to the equations the programmer writes to define
functions. They are referred to as $\delta$ (for _definitions_) and $\iota$
(for pattern-matching on _inductive_ data) rules and hold computationally just
like the more common $\beta$-rule.
map : (a $\rightarrow$ b) $\rightarrow$ list a $\rightarrow$ list bmap f []
$\leadsto$ []map f (x :: xs) $\leadsto$ f x :: map f xs(++) : list a
$\rightarrow$ list a $\rightarrow$ list a[] ++ ys $\leadsto$ ysx :: xs ++ ys
$\leadsto$ x :: (xs ++ ys)fold : (a $\rightarrow$ b $\rightarrow$ b)
$\rightarrow$ b $\rightarrow$ list a $\rightarrow$ bfold c n [] $\leadsto$
nfold c n (x :: xs) $\leadsto$ c x (fold c n xs)
Table 1: $\delta\iota$-rules - computational
The second is the judgmental equality ($\equiv$): it is typed, tractable for a
machine to decide and typically includes $\eta$-rules for negative types
therefore internalizing some kind of _extensionality_. Table 2 presents such
rules, explaining that some types have unique constructors which the
programmer can demand. They are well supported in e.g. Epigram Chapman et al.
[2005] and Agda Norell [2008] both for functions and records but still lacking
for records in Coq INRIA .
$\Gamma\ \vdash$ f $\equiv$ $\lambda$ x. f x : a $\to$ b$\Gamma\ \vdash$ p
$\equiv$ ($\operatorname{\pi_{1}}$ p , $\operatorname{\pi_{2}}$ p) : a *
b$\Gamma\ \vdash$ u $\equiv$ () : 1
Table 2: $\eta$-rules - canonicity
The third is the propositional equality ($=$): this lets us state and give
evidence for equations on open terms which may not be identified judgmentally.
Table 3 shows a kit for building computationally inert _neutral_ terms growing
layers of thwarted progress around a variable which we dub the ‘nut’, together
with some equations on neutral terms which held only propositionally – until
now. This paper shows how to extend the judgmental equality with these new
‘$\nu$-rules’. We gain, for example, that map swap . map swap $\equiv$ id,
where swap swaps the elements of a pair.
x a $\operatorname{\pi_{1}}$ $\operatorname{\pi_{2}}$ ++ ys map f fold n c
xs ++ [] | = | xs
---|---|---
(xs ++ ys) ++ zs | = | xs ++ (ys ++ zs)
map id xs | = | xs
map f (map g xs) | = | map (f . g) xs
map f (xs ++ ys) | = | map f xs ++ map f ys
fold c n (map f xs) | = | fold (c . f) n xs
fold c n (xs ++ ys) | = | fold c (fold c n ys) xs
Table 3: $\nu$-rules
A $\nu$-rule is an equation between neutral terms with the same nut which
holds just by structural induction on the nut, with $\beta\delta\iota$
reducing subgoals to inductive hypotheses – the classic proof pattern of Boyer
and Moore Boyer and Moore [1975]. Consequently, we need only use $\nu$-rules
to standardize neutral terms after ordinary evaluation stops. This
separability makes implementation easy, but the proof of its completeness
correspondingly difficult. Here, we report a successful experiment in
formalizing a modified normalization by evaluation proof for simply-typed
$\lambda$-calculus with list primitives and the $\nu$-rules above.
#### Contents
We define the terms of the theory and deliver a sound and complete
normalization algorithm in Sections 2 to 5. We then explain how this promising
experiment can be scaled up to type theory (Section 6) thus suggesting that
other frustrating equations of a similar character may soon come within our
grasp (Section 7).
## 2 Our Experimental Setting
In a dependently-typed setting, one has to deal with issues unrelated to the
matter at hand: Danielsson’s formalization of a Type Theory as an inductive-
recursive family uses a non strictly positive datatype Danielsson [2007], Abel
et al. Abel et al. [2007a] resort to recursive domain equations together with
logical relations proving them meaningful, McBride’s proposition McBride
[2010] is only able to steal the judgmental equality of the implementation
language and Chapman’s big step formulation is not proven terminating Chapman
[2009].
We propose a preliminary experiment on a calculus for which the formalization
in Agda is tractable: we are interested in the modifications to be made to an
existing implementation in order to get a complete procedure for the extended
equational theory. We developed the algorithm during Boutillier’s internship
at Strathclyde Boutillier [2009]; Allais completed the formalized meta-theory.
#### Types
The set of types is parametrized by a finite set of base types
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{1}},\dots,{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{n}}$
it can build upon. These unanalysed base types give us a simple way to model
expressions exhibiting some parametric polymorphism.
$\sigma,\tau,\dots\operatorname{::=~{}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}\mid{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}\mid\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau\mid\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau\mid{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$
###### Remark 2.1.
In the Agda implementation this indexing by a finite set of base types is
modelled by defining a nat-indexed family
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{type}}_{n}$
with a constructor
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}}$
taking a natural number $k$ bounded by $n$ (an element of
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Fin}}~{}n$)
to refer to the $k^{th}$ base type.
#### Terms
Terms follow the grammar presented below and the typing rules described in
Figure 1 where contexts are just snoc lists of variable names together with
their type.
$\displaystyle t,u,\dots$
$\displaystyle\operatorname{::=~{}}x\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.t\mid
t\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}u\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\mid
t\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}u\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}t\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}t\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
$\displaystyle\mid
hd\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}tl\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,xs)\mid
xs\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}ys\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(c,n,xs)$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip 22.74313pt\vbox{\vbox{}\hbox{\hskip-22.74313pt\hbox{\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{base}}}\colon\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\varepsilon}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\varepsilon}}$}}}}}}$ $\displaystyle\displaystyle{\hbox{\hskip 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72.73763pt\vbox{\hbox{\hskip-72.73763pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}xs~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$}\hskip 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50.06451pt\vbox{\vbox{}\hbox{\hskip-50.0645pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}xs\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}ys~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$}}}}}}$ $\displaystyle\displaystyle{\hbox{\hskip 70.40549pt\vbox{\hbox{\hskip-70.40549pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}f~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau$}\hskip 20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}xs~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$}}}}\vbox{}}}\over\hbox{\hskip 51.57487pt\vbox{\vbox{}\hbox{\hskip-51.57486pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,xs)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\tau$}}}}}}$ $\displaystyle\displaystyle{\hbox{\hskip 107.85016pt\vbox{\hbox{\hskip-107.85014pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}c~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau$}\hskip 20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}n~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}\tau$}\hskip 20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}xs~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$}}}}}\vbox{}}}\over\hbox{\hskip 44.14426pt\vbox{\vbox{}\hbox{\hskip-44.14424pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(c,n,xs)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}\tau$}}}}}}$
---|---
Figure 1: Context inclusion and typing rules
For sake of clarity in the formalization, we quote the constructors of our
object language, making a clear distinction from the corresponding features of
the host language, Agda, where we use the standard ‘typed de Bruijn index’
representation of well-typed terms de Bruijn [1972]; Altenkirch and Reus
[1999] to eliminate junk from consideration. In our treatment here, we always
assume freshness of the variables introduced by $\lambda$-abstractions. And we
do not artificially separate well-typed terms and typing derivations; in other
words we will use alternatively
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\colon}\sigma$
and
$t\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
to denote the same objects.
#### Weakening
The notion of context inclusion gives rise to a weakening operation
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{\\_}}$
which can be viewed as the action on morphisms of the functor
$\\_~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
from the category of contexts and their inclusions to the category of well-
typed terms and functions between them. It is defined inductively (cf. Figure
1) rather than as a function transporting membership predicates from one
context to its extension in order to avoid having to use an extensionality
axiom to prove two context inclusion proofs to be the same. This more
intensional presentation can already be found under the name order preserving
embeddings in Chapman’s thesis Chapman [2009].
#### From types to contexts
We can lift the notion of well-typed terms
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
to whole parallel substitutions. For any two contexts named $\Gamma$ and
$\Delta$, the well-typed parallel substitution from $\Gamma$ to $\Delta$ is
defined by:
$\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash^{s}}~{}\Gamma=\left\\{\begin{array}[]{l@{\text{
if }\Gamma=~}l}\top\hfil\text{ if }\Gamma=~&\varepsilon\\\
\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash^{s}}~{}\Gamma^{\prime}\times\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma\hfil\text{
if
}\Gamma=~&\Gamma^{\prime}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\cdot}~{}(x:\sigma)\end{array}\right.$
We write $t[\rho]$ for the application of the parallel substitution
$\rho\colon\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash^{s}}~{}\Gamma$
to the term
$t\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
yielding a term of type
$\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$.
###### Remark 2.3.
All the notions described in this document can be lifted in a pointwise
fashion to either contexts when they are defined on types or parallel
substitutions when they deal with terms. We will assume these extensions
defined and casually use the same name (augmented with: s) for the extension
and the original concept.
#### Judgmental Equality
The equational theory of the calculus, denoted
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv_{\operatorname{\beta\delta\iota\eta\nu}{}}}}$,
is quite naturally the congruence closure of the
$\operatorname{\beta\delta\iota\eta\nu}$-rules described earlier where
reductions under $\lambda$-abstraction are allowed. In this paper, we also
mention the relation
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}$
where the rules presented earlier are all considered with a left to right
orientation (except for the identity laws for the list functor and the list
monoid) thus inducing a notion of _reduction_. The soundness theorem proves
that not only is the term produced by our normalization procedure related to
the source one but it is a reduct of it.
One easy sanity check we recommend before starting to work on the meta-theory
was to give a shallow embedding of the calculus in a pre-existing sound type
theory and to show that the reduction relation is compatible with the
propositional equality in this theory. We used Agda extended with a postulate
stating extensional equality for non-dependent functions in our formalization.
Once the reader is convinced that no silly mistakes were made in the
equational theory, she can start the implementation.
## 3 Reduction Machinery
When looking in details at different accounts of normalization by evaluation
Berger and Schwichtenberg [1991]; Coquand and Dybjer [1997]; Coquand [2002];
Ahman [2012], the reader should be able to detect that there are two phases in
the process: firstly the evaluation function building elements of the model
from well-typed terms performs $\beta\delta\iota$-reductions and does not
reduce under $\lambda$-abstractions effectively building closures – using the
$\lambda$-abstractions of the host language – when encountering one. Secondly
the quoting machinery extracting terms from the model performs
$\eta$-expansions where needed which will cause the closures to be reduced and
new computations to be started. This two-step process was already more or less
present in Berger and Schwichtenberg’s original paper Berger and
Schwichtenberg [1991]:
> Obviously each term in $\beta$-normal form may be transformed into long
> $\beta$-normal form by suitable $\eta$-expansions. Therefore each term $r$
> may be transformed into a unique long $\beta$-normal form $r^{\star}$ by
> $\beta$-conversion and $\eta$-expansions.
Building on this ascertainment, we construct a three (rather than two) staged
process successively performing $\beta\delta\iota$, $\eta$ and finally $\nu$
reductions whilst always potentially calling back a procedure from a preceding
stage to reduce further non-normal terms appearing when e.g. going under
$\lambda$-abstractions during $\eta$-expansion, distributing a map over an
append, etc.
### 3.1 The Three Stages of Standardization
The normalization and standardization process goes through three successive
stages whence the need to define three different subsets of terms of our
calculus. They have to be understood simply as syntactic category restricting
the shape of terms typed in the same way as the ones in the original languages
except for the few extra constructors for which we explicitly detail what they
mean.
###### Remark 3.1.
It should be noted that the two last steps never reduce a term to a
constructor-headed one for datatypes (lists in our setting). In particular,
the last step only rearranges stuck terms to produce terms which are
themselves stuck. In other words: if a term (a list in our case) is
convertible to a constructor headed term (be it either nil or cons), then it
is reduced to it by the first step of the reduction.
###### Example 3.2.
We will consider the normalization of
$(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)$
of type
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\varepsilon}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}})~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}})}$
as a running example demonstrating the successive steps.
#### Untyped $\beta\iota$-reductions
The first intermediate language we are going to encounter is composed of weak-
head $\beta\delta\iota$-normal expressions i.e. we never reduce under a
lambda, this role being assigned to the $\eta$-expansion routine. Having
$\lambda$-closures as first-class values is one of the characteristics of this
approach.
$\displaystyle m\operatorname{::=~{}}$ $\displaystyle x\mid
m\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}w\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}m\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}m\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(w_{1},w_{2},m)$
$\displaystyle\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(w,m)\mid
m\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}w$
$\displaystyle w\operatorname{::=~{}}$ $\displaystyle
m\mid{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda[}\rho{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}]}x.t\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\mid
w_{1}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}w_{2}\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\mid
w_{1}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}w_{2}$
$\displaystyle\rho\operatorname{::=~{}}$
$\displaystyle\varepsilon\mid\rho,x\mapsto w$ Figure 2: Weak-head normal forms
These values are computed using a simple off the shelf environment machine
which returns a constructor when facing one; stores the evaluation environment
in a $\lambda$-closure when evaluating a term starting with a
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}$;
and calls an helper function (e.g.
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\$\$}}}}$,
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\pi_{1}}}}}$,
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\pi_{2}}}}}$,
etc.) on the recursively evaluated subterms when uncovering an eliminator.
These helper functions either return a neutral if the interesting subterm was
stuck or perform the elimination which may start new computations (e.g. in the
application case). We call
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-
norm}}}$ this evaluation function.
###### Remark 3.3.
This reduction step is absolutely type-agnostic and could therefore be
performed on terms devoid of any type information as in e.g. Coq where
conversion is untyped. Keeping and propagating _some_ types (e.g. the codomain
of the function in a map) is nonetheless needed to be able to infer back the
type of the whole expression which is crucial in the following steps.
###### Example 3.4.
The untyped evaluation reduces our simple example
$(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)$
to the usual identity function:
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda[}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{tt}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}]}x.x$.
#### Type-directed $\eta$-expansion
Then an $\eta$-expansion step kicks in and produces $\eta$-long values in a
type-directed way. It insists that the only neutrals worthy of being
considered normal forms are the ones of the base type. It also carves out the
subset of stuck lists in a separate syntactic category $l$ thus preparing for
the last step which will leave most of the rest of the language untouched.
$\displaystyle n\operatorname{::=~{}}$ $\displaystyle x\mid
n\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}v\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}n\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}n\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(v_{1},v_{2},l)$
$\displaystyle v\operatorname{::=~{}}$ $\displaystyle
n_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}\mid
l\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.v\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\mid
v_{1}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}v_{2}\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\mid
v_{1}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}v_{2}$
$\displaystyle l\operatorname{::=~{}}$ $\displaystyle
n_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}\mid\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(v,l)\mid
l\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}v$
Figure 3: $\eta$-long values
The $\eta$-expansion of product and function type actually calls back the
subroutines for $\beta\delta\iota$-rules projecting components out of pairs or
performing function application – here to the variable newly introduced. This
step is the only one requiring a name generator which allows us to avoid
threading such an artifact along the whole reduction machinery. We call
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}$
the main function performing this step and present it in Figure 4.
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{list}}}$
and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{neut}}}$
are two trivial auxiliary functions going structurally through either lists or
neutral terms and calling
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}$
whenever necessary.
$\begin{array}[]{@{\etanorm(}l@{)~t~=~}l}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}(\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}\hfil)~{}t~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{neut}}}t\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}(\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma\hfil)~{}t~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{list}}}~{}\sigma~{}t\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}(\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}\hfil)~{}t~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}(\lx@intercol\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau\hfil)~{}t~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}\sigma~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\pi_{1}}}}}t)\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}\tau~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\pi_{2}}}}}t)\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}(\lx@intercol\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau\hfil)~{}t~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}\tau(t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\$\$}}}}x))\end{array}$
Figure 4: From weak-head normal forms to $\eta$-long ones
###### Example 3.5.
The $\eta$-expansion of the evaluated form
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda[}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{tt}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}]}x.x$
of type
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\varepsilon}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}})~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}})}$
proceeds in multiple steps.
* •
The arrow type forces us to introduce a $\lambda$-abstraction:
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}))~{}(({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda[}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{tt}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}]}x.x)\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\$\$}}}}x)$.
* •
Now,
$({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda[}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{tt}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}]}x.x)\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\$\$}}}}x$
trivially reduces to $x$, a neutral of list type, left unmodified by
$\eta$-expansion. Hence the $\eta$-long form:
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x$.
#### $\nu$-rules reorganizing neutrals
Standard forms have a very specific shape due to the fact that we now
completely internalize the $\nu$-rules. The new constructor
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\\_{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\\_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\\_$
– referred to as “mapp” – has the obvious semantics that it represents the
concatenation of a stuck map and a list.
Figure 5: StandardForms
The standard lists $s$ are produced by flattening the stuck map / append trees
present in $l$ after the end of the previous procedure whilst the fold / map
and fold / append fusion rules are applied in order to compute folds further
and reach the point where a stuck fold is stuck on a _real_ neutral lists.
These reductions are computed by the mutually defined
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}$,
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
neut}}}$ and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}$ respectively turning $\eta$-long normals, neutrals and lists into
elements of the corresponding standard classes.
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}$ and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
neut}}}$ are mostly structural except for the few cases described in Figure 6.
We define
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}$
as being the composition of
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}$
and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}$ whilst
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}$
is the composition of
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-
norm}}}$ and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}$.
As one can see below, $\nu$-rules can restart computations in subterms by
invoking subroutines of the evaluation function
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-
norm}}}$. Formally proving the termination of the whole process is therefore
highly non-trivial.
$\displaystyle\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma)\mathit{xs_{ne}}$
$\displaystyle=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}\mathit{xs}$
$\displaystyle\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
neut}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}c~{}n~{}\mathit{xs})$
$\displaystyle=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
fold}}}c~{}n~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}\mathit{xs})$
$\begin{array}[]{@{\nflist~}l@{~=~}l}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}~{}\lx@intercol\mathit{xs_{ne}}\hfil~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x){~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}f\mathit{xs})\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
map}}}~{}f~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}\mathit{xs})\\\
\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}~{}\lx@intercol(\mathit{xs}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{ys})\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-\operatorname{++}}}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
list}}}\mathit{xs})(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}~{}\\_~{}\mathit{ys})\end{array}$
$\begin{array}[]{ll@{~=~}l}\lx@intercol\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
fold}}}c~{}n~{}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}ys)=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}~{}\mathit{cf}~{}\mathit{ih}~{}\mathit{xs}\hfil\lx@intercol\\\
\text{
where}&\mathit{cf}\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}~{}\\_~{}(c\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\circ\circ}}}}f)\\\
&\mathit{ih}\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}~{}\\_~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-
fold}}}~{}c~{}n~{}\mathit{ys})\\\ \\\
\lx@intercol\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
map}}}f~{}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}g{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}ys)=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}fg{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}fys\hfil\lx@intercol\\\
\text{
where}&\mathit{fg}\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}~{}\\_~{}(f\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{\circ\circ}}}}g))\\\
&\mathit{fys}\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}~{}\\_~{}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-
map}}}~{}f~{}\mathit{ys})\\\ \\\
\lx@intercol\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-\operatorname{++}}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}ys)~{}\mathit{zs}=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}xs{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}yzs\hfil\lx@intercol\\\
\text{
where}&\mathit{yzs}\hfil~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{standard}}}~{}\\_~{}(ys\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wh-\operatorname{++}}}}zs)\end{array}$
Figure 6: From $\eta$-long values to standard ones
###### Example 3.6.
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{nf-
norm}}}$ does not touch the $\lambda$-abstraction but expands the neutral $x$
of type
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}})$
to
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\texttt{id}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}x{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
where id is the normal form of the identity function on
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}$.
We leave it to the reader to check that:
id
$\displaystyle=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}})~{}p$
id
$\displaystyle=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}})~{}p$
$\displaystyle=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}p)\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\eta\mathtt{norm}}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}~{}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p))$
$\displaystyle=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p)$
Hence the final standard form of
$(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x)$:
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p){~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}x{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
The grammar of standard terms explicitly defines a hierarchy between stuck
functions: appends are forbidden to appear inside maps and both of them have
better not be found sitting in a fold. It is but one way to guarantee the
existence of standard forms and future extensions hopefully allowing the
programmer to add the $\nu$-rules she fancies holding definitionally will have
to make sure –for completeness’ sake– that such standard forms exist.
## 4 Formalization of the Procedure
What we are interested in here is to demonstrate the decidability of the
equational theory’s extension rather than explaining how to prove termination
of a big step semantics in Agda and rely on functional induction to prove the
different properties. The reader keen on learning about the latter should
refer to James Chapman’s thesis Chapman [2009] where he describes a principled
solution to proving termination of big step semantics for various calculi. We,
on the other hand, will focus on the former: we opted for a version of the
algorithm based, in the tradition of normalization by evaluation, on a model
construction which basically collapses the layered stages but is trivially
terminating by a structural argument.
#### Type directed partial evaluation
(or normalization by evaluation) is a way to compute the canonical forms by
using the evaluation mechanism of the host language whilst exploiting the
available type information to retrieve terms from the semantical objects. It
was introduced by Berger and Schwichtenberg Berger and Schwichtenberg [1991]
in order to have an efficient normalization procedure for Minlog. It has since
been largely studied in different settings:
Danvy’s lecture notes Danvy [1999] review its foundations and presents its
applications as a technique to get rid of static redexes when compiling a
program. It also discusses various refinements of the naïve approach such as
the introduction of let bindings to preserve a call-by-value semantics or the
addition of extra reduction rules111E.g. $n+0\leadsto n$ in a calculus where
$\\_+\\_$ is defined by case analysis on the first argument and this
expression is therefore stuck. to get cleaner code generated. Our $\nu$-rules
are somehow reminiscent of this approach.
T. Coquand and Dybjer Coquand and Dybjer [1997] introduced a glued model
recording the partial application of combinators in order to be able to build
the reification procedure for a combinatorial logic. In this case the naïve
approach is indeed problematic given that the SK structure is lost when
interpreting the terms in the naïve model and is impossible to get back. This
was of great use in the design of a model outside the scope of this paper
computing only weak-head normal forms Allais [2012].
C. Coquand Coquand [2002] showed in great details how to implement and prove
sound and complete an extension of the usual algorithm to a simply-typed
lambda calculus with explicit substitutions. This development guided our
correctness proofs.
More recently Abel et al. Abel et al. [2007a, b] built extensions able to deal
with a variety of type theories. Last but not least Ahman and Staton Ahman
[2012]; Ahman and Sam [2013] explained how to treat calculi equipped with
algebraic effects which can be seen as an extension of the calculus of Watkins
et al. Watkins et al. [2003] extending judgmental equality with equations for
concurrency or Filinski’s computational $\lambda$-calculus. Filinski [2001]
###### Remark 4.1.
We will call
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{nf}}}~{}\sigma$
the typing derivations restricted to standard values as per the previous
section’s definitions and
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma$
the corresponding ones for standard neutrals. Standard list will be silently
embedded in standard values: the separation of $s$ and $v$ is an important
vestige of the syntactic category $l$ of stuck lists but inlining it in the
grammar yields exactly the same set of terms.
###### Remark 4.2.
Following Agda’s color scheme, function names and type constructors will be
typeset in blue, constructors will appear in green and variables will be left
black.
###### Definition 4.3 (Model).
The model is defined by induction on the type using an auxiliary inductive
definition parametric in its arguments –which guarantees that the definition
is strictly positive therefore meaningful– to give a semantical account of
lists. One should remember that the calculus enjoys $\eta$-rules for unit,
product and arrow types; therefore the semantical counterpart of terms with
such types need not be more complex than unit, pairs and actual function
spaces.
$\begin{array}[]{@{\NBE(\Gamma,~}l@{~)~}cl}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol\\_\hfil~{})~{}&:&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{type}}_{n}\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}\hfil~{})~{}&=&\top\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}\hfil~{})~{}&=&\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau\hfil~{})~{}&=&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)\times{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\tau)\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau\hfil~{})~{}&=&\forall\Delta,\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\tau)\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}(\Gamma,~{}\lx@intercol{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma\hfil~{})~{}&=&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(.~{},\sigma))\\\
\end{array}$
Standardization may trigger new reductions and we have therefore the
obligation to somehow store the computational power of the functions part of
stuck maps. This is a bit tricky because the domain type of such functions is
nowhere related to the overall type of the expression meaning that no
induction hypothesis can be used. Luckily these new computations are only ever
provoked by neutral terms: they come from function compositions caused by map
or map-fold fusions.
$\displaystyle\displaystyle{\hbox{\hskip
109.29079pt\vbox{\hbox{\hskip-109.29077pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Con}}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{type}}_{n})$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\sigma\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{type}}_{n}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathtt{M}_{\sigma}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Con}}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{type}}_{n})\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}$}}}}}\vbox{}}}\over\hbox{\hskip
31.67926pt\vbox{\vbox{}\hbox{\hskip-31.67924pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
28.62373pt\vbox{\vbox{}\hbox{\hskip-28.62373pt\hbox{\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}:{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
62.23764pt\vbox{\hbox{\hskip-62.23764pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{HD}\colon\mathtt{M}_{\sigma}(\Gamma)$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{TL}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}}}}\vbox{}}}\over\hbox{\hskip
46.25925pt\vbox{\vbox{}\hbox{\hskip-46.25923pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
153.78053pt\vbox{\hbox{\hskip-153.78053pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{F}\colon\forall\Delta,\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta\rightarrow\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\tau\rightarrow\mathtt{M}_{\sigma}(\Delta)$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{xs}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\tau}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
\mathit{YS}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}}}}}\vbox{}}}\over\hbox{\hskip
72.742pt\vbox{\vbox{}\hbox{\hskip-72.742pt\hbox{\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\mathit{F}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}}}}}}$
###### Remark 4.4.
One should notice the Kripke flavour of the interpretation of function types.
It is exactly what is needed to write down a weakening operation thus giving
the entire model a Kripke-like structure.
###### Definition 4.5 (Reify and reflect).
Mutually defined processes allow normal forms
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{nf}}}~{}\sigma$
to be extracted from elements of the model
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
whilst neutral forms
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma$
can be turned into elements of the model.
###### Proof 4.6.
Both
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)\rightarrow\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{nf}}}~{}\sigma$
and
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
are defined by induction on their type index $\sigma$.
#### Unit, base and product types
The unit case is trivial: the reification process returns
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}$
while the reflection one produces the only inhabitant of $\top$. The base type
case is solved by the embedding of neutrals into normals on one hand and by
the identity function on the other hand. The product case is simply discharged
by invoking the induction hypotheses: the reification is the pairing of the
reifications of the subterms while the reflection is the reflection of the
$\eta$-expansion of the stuck term. We can now focus on the more subtle cases.
#### Arrow type
The function case is obtained by $\eta$-expansion both at the term level (the
normal form will start with a
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}$)
and the semantical level (the object will be a function). It is here that the
fact that the definitions are mutual is really important.
$\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau}F$
$\displaystyle\overset{\text{def}}{=}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\tau}F(\\_,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}x)$
$\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau}f$
$\displaystyle\overset{\text{def}}{=}\lambda\Delta~{}inc~{}x.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\tau}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc}}(f)\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}x)$
#### Lists
The list case is dealt with by recursion on the semantical list for the
reification process and a simple injection for the reflection case. We write
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}$
and
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow\uparrow}_{\sigma}$
for the helper functions performing reification and reflection on lists of
type
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$.
$\begin{array}[]{@{\listreify}l@{~\eqdef~}l}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}\lx@intercol\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\hfil~{}\overset{\text{def}}{=}~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}\lx@intercol\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL}\hfil~{}\overset{\text{def}}{=}~{}&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}\mathit{TL}\\\
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}\lx@intercol\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS}\hfil~{}\overset{\text{def}}{=}~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}f(x){~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{\sigma}\mathit{YS}\end{array}$
This injection corresponds to applying the identity functor and monoid law.
Indeed
$\lambda\Delta\\_.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}$
denotes the identity function and has the appropriate type
$\forall\Delta,\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta\rightarrow\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)$
to fit in the semantical list mapp constructor.
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow\uparrow}_{\sigma}xs~{}\overset{\text{def}}{=}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\lambda\Delta\\_.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
###### Example 4.7.
of $\eta\nu$-expansions provoked by the reflect / reify functions: for
$\mathit{xs}$ a neutral list of type
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}})$,
we get an expanded version by drowning it in the model and reifying it back:
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow\downarrow}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow\uparrow}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}\mathit{xs})=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p){~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
This showcases the $\eta$-expansion of unit, products and functions as well as
the use of the identity laws mentioned during the definition of
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow\uparrow}_{\sigma}$.
Proving that every term can be normalized now amounts to proving the existence
of an evaluation function producing a term $T$ of the model
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)$
given a well-typed term $t$ of the language
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
and a semantical environment
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)$.
Indeed the definition of the reflection function
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}$
together with the existence of environment weakenings give us the necessary
machinery to produce a diagonal semantical environment
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Gamma,\Gamma)$
which could then be fed to such an evaluation function.
In order to keep the development tidy and have a more modular proof of
correctness, it is wise to give this evaluation function as much structure as
possible. This is done through a multitude of helper functions explaining what
the semantical counterparts of the usual combinators of the calculus (except
for lambda which, integrating a weakening to give the model its Kripke
structure, is a bit special) ought to look like.
$\begin{array}[]{l@{~\vappend\mathit{ZS}=\,}l}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\hfil~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}&\mathit{ZS}\\\ \mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL}\hfil~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}&\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}(\mathit{TL}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}\mathit{ZS})\\\ \operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}F{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS}\hfil~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}\mathit{F}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}(\mathit{YS}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}\mathit{ZS})\end{array}$ $\begin{array}[]{@{\vmap~\mathit{F}~}l@{~=~}l}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}~{}\mathit{F}~{}\lx@intercol\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\hfil~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}~{}\mathit{F}~{}\lx@intercol(\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL})\hfil~{}=~{}&F(\\_,\mathit{HD})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}F\,\mathit{TL}\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}~{}\mathit{F}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}G{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS})\hfil~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}F\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\circ}}G{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}F\,\mathit{YS}\\\ \lx@intercol\text{where}~{}F~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\circ}}~{}G=\lambda E~{}\mathit{inc}~{}t.F(\mathit{inc},G(\mathit{inc},t))\hfil\lx@intercol\end{array}$ $\begin{array}[]{l@{~=~}l}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}~{}~{}C~{}N~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\hfil~{}=~{}&N\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}~{}~{}C~{}N~{}(\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL})\hfil~{}=~{}&C(\\_,\mathit{HD},\\_,\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}~{}C~{}N~{}\mathit{TL})\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}_{\tau}C~{}N~{}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}F{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS})\hfil~{}=~{}&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\tau}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(c,n,\mathit{xs})\\\ \lx@intercol\begin{array}[]{ll@{~=~}l}\text{where}&c\hfil~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}y.{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\tau}C(\\_,F(\\_,x)),\\_,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\tau}y)\\\ &n\hfil~{}=~{}&{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\tau}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}~{}C~{}N~{}\mathit{YS}\end{array}\hfil\lx@intercol\end{array}$ | $\begin{array}[]{@{\termeval~}l@{~R~=~}l}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol x\hfil~{}R~{}=~{}&R(x)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.t)\hfil~{}R~{}=~{}&\lambda E~{}\mathit{inc}~{}S.\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}t~{}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}^{s}}_{\mathit{inc}}(R),x\mapsto S)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(f\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}x)\hfil~{}R~{}=~{}&(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}f~{}R)(\\_,\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}x~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}})\hfil~{}R~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{tt}}}\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(a\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}b)\hfil~{}R~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}a~{}R,\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}b~{}R\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}t)\hfil~{}R~{}=~{}&\operatorname{\pi_{1}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}t~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}t)\hfil~{}R~{}=~{}&\operatorname{\pi_{2}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}t~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}})\hfil~{}R~{}=~{}&\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\mathit{hd}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{tl})\hfil~{}R~{}=~{}&(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{hd}~{}R)\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{tl}~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\mathit{xs}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{ys})\hfil~{}R~{}=~{}&(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{xs}~{}R)\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{ys}~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,\mathit{xs}))\hfil~{}R~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}f~{}R)(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{xs}~{}R)\\\ \operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\lx@intercol(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}(c,n,\mathit{xs}))\hfil~{}R~{}=~{}&\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}c~{}R)(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}n~{}R)(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}~{}\mathit{xs}~{}R)\end{array}$
---|---
Figure 7: Evaluation function and semantical counterparts of list primitives
###### Theorem 4.8 (Evaluation function).
Given a term in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
and a semantical environment in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)$,
one can build a semantical object in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)$.
###### Proof 4.9.
A simple induction on the term to be evaluated using the semantical
counterparts of the calculus’ combinators to assemble semantical objects
obtained by induction hypotheses discharges most of the goals. See Figure 7
for the details of the code.
In the lambda case, we have the body of the lambda $b$ in
$\Gamma\cdot\sigma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\tau$,
an evaluation environment $R$ in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)$
and we are given a context $E$, a proof inc that
$\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}E$
and an object $S$ living in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(E,\sigma)$.
By combining $S$ and a weakening of $R$ along inc, we get an evaluation
environment of type
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(E,\Gamma\cdot\sigma)$
which is just what we needed to conclude by using the
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(E,\tau)$
delivered by the induction hypothesis on $b$.
###### Remark 4.10.
Unlike traditional normalization by evaluation, reflection and reification are
used when defining the interpretation of terms in the model. This is made
necessary by the presence of syntactical artifacts (stuck lists) in the mapp
constructor. Growing the spine of stuck eliminators calls for the reification
of these eliminators’ parameters and the reflection of the whole stuck
expression to re-inject it in the model.
This kind of patterns also appeared in the glueing construction introduced by
Coquand and Dybjer in their account of normalization by evaluation for the
simply-typed SK-calculus Coquand and Dybjer [1997] and can be observed in
other variants of normalization by evaluation deciding more exotic equational
theories e.g. having $\beta$-reduction but no $\eta$-rules for the simply-
typed $\lambda$-calculus Allais [2013].
###### Remark 4.11.
The only place where type information is needed is when reorganizing neutrals
following $\nu$-rules e.g. in the semantical fold. The evaluation function is
therefore faithful to the staged evaluation approach. The model is indeed
related to the algorithm presented earlier on in section 3.1: we _describe_
all the computations eagerly for Agda to see the termination argument but a
subtle evaluation strategy applied to the produced code could reclaim the
behaviour of the layered approach. It would have to form lambda closures in
the arrow case, fire eagerly only the reductions eliminating constructors in
the
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{map}}}$,
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{++}}}$
and
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}\mathtt{fold}}}$
helper functions thus postponing the execution of the code corresponding to
$\eta\nu$-rules to reification time.
###### Corollary 4.12.
There is a normalization function
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}$
turning terms in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
into normal forms in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{nf}}}~{}\sigma$.
###### Proof 4.13.
Given $t$ a term of type
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
and
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\texttt{id}}^{s}}$
the function turning a context $\Gamma$ into the corresponding diagonal
semantical environment
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Gamma,\Gamma)$,
the normalization procedure is given by the composition of evaluation and
reification:
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}t\overset{\text{def}}{=}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}(\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}(t,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\texttt{id}}^{s}}\Gamma))$
## 5 Correctness
The typing information provided by the implementation language guarantees that
the procedure computes terms in normal forms from its inputs and that they
have the same type. This is undoubtedly a good thing to know but does not
forbid all the potentially harmful behaviours: the empty list is a type
correct normal form for any input of type list but it certainly is not a
satisfactory answer with respect to
$\operatorname{\beta\delta\iota\eta\nu}$-equality. Hence the need for a
soundness and a completeness theorem tightening the specification of the
procedure.
The meta-theory is an ad-hoc extension of the techniques already well
explained by Catarina Coquand Coquand [2002] in her presentation of a simply-
typed lambda calculus with explicit substitutions (but no data). Soundness is
achieved through a simple logical relation while completeness needs two
mutually defined notions explaining what it means for elements of
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}$
to be semantically equal and to behave uniformly on extensionally equal terms.
The reader should think of these logical relations as specifying requirements
for a characterization (being equal, being uniform) to be true of an element
at some type. The natural deduction style presentation of these recursive
functions should then be quite natural for her: read in a bottom-top fashion,
they express that the (dependent) conjunction of the hypotheses – the empty
conjunction being $\top$– is the requirement for the goal to hold. Hence
leading to a natural interpretation:
$\displaystyle\displaystyle{\hbox{\hskip
31.72469pt\vbox{\hbox{\hskip-31.72469pt\hbox{\hbox{$\displaystyle\displaystyle
A$}\hskip 20.00003pt\hbox{\hbox{$\displaystyle\displaystyle B$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
C$}}}}}\vbox{}}}\over\hbox{\hskip
9.60419pt\vbox{\vbox{}\hbox{\hskip-9.60419pt\hbox{\hbox{$\displaystyle\displaystyle
F(t)$}}}}}}$ $\displaystyle\displaystyle{\huge\leadsto}$
$\displaystyle\displaystyle F(t)=A\times B\times C$
### 5.1 Soundness
Soundness amounts to re-building the propositional part of the reducibility
candidate argument Girard [2006] which has been erased to get the bare bones
model. The logical relation
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$
relates a semantical object $T$ in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
and a term $t$ in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
which is morally the source of the semantical object.
###### Definition 5.1 (Logical Relation for Soundness).
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$
is defined by induction on the type $\sigma$ plus an appropriate inductive
definition for the list case
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\mathit{xs}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\mathit{XS}$.
Here are the formation rules of these types.
$\displaystyle\displaystyle{\hbox{\hskip
48.21074pt\vbox{\hbox{\hskip-48.21074pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{t}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{T}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$}}}}\vbox{}}}\over\hbox{\hskip
45.00826pt\vbox{\vbox{}\hbox{\hskip-45.00824pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
151.88522pt\vbox{\hbox{\hskip-151.88521pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{xs}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{XS}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma})$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_\colon\forall\Gamma,\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma\rightarrow\mathtt{M}_{\sigma}\Gamma\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}$}}}}}\vbox{}}}\over\hbox{\hskip
81.7673pt\vbox{\vbox{}\hbox{\hskip-81.7673pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\mathit{xs}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\mathit{XS}\colon{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Set}}$}}}}}}$
###### Remark 5.2.
It should be no surprise to the now experienced reader that the inductive
definition of the logical relation for
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$
is parametrized by $\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_$, the logical
relation for elements of type $\sigma$ which will be lifted to lists, simply
to avoid positivity problems. It is ultimately instantiated with the logical
relation taken at type $\sigma$.
She will also have noticed that the uses of both
$\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\mathcal{M}}$
and
$\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\mathcal{L}}$
on the left of
$\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\ni}$
are but syntactical artifacts to hint at the connection with the model
definition. Hence the different arity in the case of the logical relation for
lists.
#### Unit, base and product types
The unit and base type cases are, as expected, the simplest ones and the
product case is not very much more exciting:
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
36.59569pt\vbox{\vbox{}\hbox{\hskip-36.59567pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
17.01454pt\vbox{\hbox{\hskip-17.01453pt\hbox{\hbox{$\displaystyle\displaystyle
t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}T$}}}\vbox{}}}\over\hbox{\hskip
38.84001pt\vbox{\vbox{}\hbox{\hskip-38.84001pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
174.90216pt\vbox{\hbox{\hskip-174.90216pt\hbox{\hbox{$\displaystyle\displaystyle
a\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
b\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\tau$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}a\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}b$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}a~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}A$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}b~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}B$}}}}}}}\vbox{}}}\over\hbox{\hskip
53.58994pt\vbox{\vbox{}\hbox{\hskip-53.58994pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}A,B$}}}}}}$
#### Arrow type
Function types on the other hand give rise to a Kripke-like structure in two
ways: in addition to the quantification on all possible future context which
we need to match the model construction, there is also a quantification on all
possible source term reducing to the current one.
$\displaystyle\displaystyle{\hbox{\hskip
202.61156pt\vbox{\hbox{\hskip-202.61156pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(inc\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)~{}x~{}X,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}x~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}X\rightarrow$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\forall
t,t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc}}f\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}x\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}F(\mathit{inc},X)$}}}}\vbox{}}}\over\hbox{\hskip
47.27573pt\vbox{\vbox{}\hbox{\hskip-47.27573pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}f~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}F$}}}}}}$
#### Lists
The cases for nil and cons are simply saying that the source term indeed
reduces to a term with the corresponding head-constructors and that the
eventual subterms are also related to the sub-objects.
$\displaystyle\displaystyle{\hbox{\hskip
18.67601pt\vbox{\hbox{\hskip-18.676pt\hbox{\hbox{$\displaystyle\displaystyle
t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$}}}\vbox{}}}\over\hbox{\hskip
69.37839pt\vbox{\vbox{}\hbox{\hskip-69.37839pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
152.70146pt\vbox{\hbox{\hskip-152.70145pt\hbox{\hbox{$\displaystyle\displaystyle
t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}\mathit{hd}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{tl}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathtt{M}_{\sigma}~{}\mathit{hd}~{}\lightning~{}\mathit{HD}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\mathit{tl}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\mathit{TL}$}}}}}\vbox{}}}\over\hbox{\hskip
87.0139pt\vbox{\vbox{}\hbox{\hskip-87.01389pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL}$}}}}}}$
The mapp case is a bit more complex. The source term is expected to reduce to
a term with the same canonical shape and then we expect the semantical
function to behave like the one discovered.
$\displaystyle\displaystyle{\hbox{\hskip
185.01125pt\vbox{\hbox{\hskip-56.15881pt\hbox{\hbox{$\displaystyle\displaystyle
t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,\mathit{xs})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{ys}$}}}\vbox{\hbox{\hskip-185.01125pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(\mathit{inc}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)~{}t\rightarrow\mathtt{M}_{\sigma}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc}}(f)\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}t~{}\lightning~{}F(\mathit{inc},t)$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\mathit{ys}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\mathit{YS}$}}}}\vbox{}}}}\over\hbox{\hskip
114.14249pt\vbox{\vbox{}\hbox{\hskip-114.14249pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{L}}}(\Gamma,\sigma,\mathtt{M}_{\sigma},{\mathtt{M}_{\sigma}~{}\\_~{}\lightning~{}\\_})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}{\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}F{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}\mathit{xs}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\mathit{YS}}$}}}}}}$
The first thing to notice is that whenever two objects are related by this
logical relation then the property of interest holds true i.e. the semantical
object indeed is a reduct of the source term. This result which mentions the
reifying function has to be proven together with the corresponding one about
the mutually defined reflection function.
###### Definition 5.3 (Pointwise extension).
We denote by
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\\_,\\_)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\\_~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning\lightning}~{}\\_$
the pointwise extension of the soundness logical relation to parallel
substitutions and semantical environments.
###### Lemma 5.4.
Reflect and reify are compatible with this logical relation in the sense that:
1. 1.
If $t_{\mathit{ne}}$ is a neutral
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma$
then
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t_{\mathit{ne}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}t_{\mathit{ne}}$.
2. 2.
If $t$ and $T$ are such that
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$
then
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}T$
The Kripke-style structure we mentioned during the definition of the logical
relation adds just what is need to have it closed under anti-reductions of the
source term:
###### Proposition 5.5.
For all $s$ and $t$ in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$,
if
$s\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}t$
then for all $T$ such that
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$,
it is also true that
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}s~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$
The proof of soundness then mainly involves showing that the semantical
counterparts of the language’s combinators we defined during the model
construction are compatible with the logical relation. Namely that e.g. if
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}f~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}F$
and
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}xs~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}XS$
hold then it is also true that:
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\tau})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,xs)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(F,XS)$.
###### Theorem 5.6.
Given a term
$t\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$,
a parallel substitution
$\rho\colon\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash^{s}}~{}\Gamma$
and an evaluation environment $R$ such that $\rho$ and $R$ are related
(${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\rho~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning\lightning}~{}R$
holds), the evaluation of $t$ in $R$ is related to $t[\rho]$:
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}{t[\rho]}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}(t,R)$
###### Proof 5.7.
The theorem is proved by structural induction on the shape of the typing
derivation of $t$. The variable case is trivially discharged by using the
proof of
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}\rho~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning\lightning}~{}R$.
All the other cases – except for the lambda one – can be solved by combining
induction hypotheses with the appropriate lemma proving that the corresponding
semantical combinator respects the logical relation.
In the case where
$t=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.b$,
we are given a context $E$ together with a proof $inc$ that it is an extension
of $\Delta$, a term $u$ and an object $U$ which are related
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(E,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}u~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}U$
and, finally, a term $s\colon
E~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\tau$
which reduces to
$(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.b)[\rho]\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\$}}}u$.
First of all, we should notice that
$s\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}b[\rho,x\mapsto
u]$ and therefore that to prove
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(E,\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}s~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$
it is enough to prove that
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(E,\tau)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}{b[\rho,x\mapsto
u]}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}T$.
And we get just that by using the induction hypothesis with the related
parallel substitution $\rho^{\prime}$ and evaluation environment $R^{\prime}$
obtained by the combination of the weakening of $\rho$ (resp. $R$) along $inc$
with $u$ (resp. $U$).
###### Corollary 5.8.
A term $t$ reduces to the normal form produced by the normalization by
evaluation procedure:
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}t$.
And if two terms $t$ and $u$ have the same normal form up-to
$\alpha$-equivalence then they are indeed related:
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv_{\operatorname{\beta\delta\iota\eta\nu}{}}}}u$.
###### Proof 5.9.
The identity parallel substitution is related to the diagonal evaluation
environment and
$t[{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{id}}_{\Gamma}]$
is equal to $t$ hence, by the previous theorem,
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\ni}~{}t~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\lightning}~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}(t,{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{id}}_{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}~{}\Gamma})$
and then
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}^{*}}}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}t$.
### 5.2 Completeness
Completeness can be summed up by the fact that the interpretation of
$\operatorname{\beta\delta\iota\eta\nu}$-convertible elements produces
semantical objects behaving similarly. This notion of similar behaviour is
formalized as _semantic equality_ where, in the function case, we expect both
sides to agree on any _uniform_ input rather than any element of the model. As
usual the list case is dealt with by using an auxiliary definition parametric
in its ”interesting” arguments.
###### Definition 5.10.
The semantic equality of two elements $T,U$ of
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
is written
$T~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}U$
while
$T\in{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
being uniform is written
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}T$.
They are both mutually defined by induction on the index $\sigma$ in Figure 8.
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
17.59418pt\vbox{\vbox{}\hbox{\hskip-17.59416pt\hbox{\hbox{$\displaystyle\displaystyle
T~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}}~{}U$}}}}}}$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
15.99799pt\vbox{\vbox{}\hbox{\hskip-15.99797pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}}~{}T$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
14.24196pt\vbox{\hbox{\hskip-14.24194pt\hbox{\hbox{$\displaystyle\displaystyle
T=U$}}}\vbox{}}}\over\hbox{\hskip
18.86867pt\vbox{\vbox{}\hbox{\hskip-18.86865pt\hbox{\hbox{$\displaystyle\displaystyle
T~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}~{}U$}}}}}}$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
17.27248pt\vbox{\vbox{}\hbox{\hskip-17.27246pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}}~{}T$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
43.26479pt\vbox{\hbox{\hskip-43.26479pt\hbox{\hbox{$\displaystyle\displaystyle
A~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}C$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
B~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\tau}~{}D$}}}}\vbox{}}}\over\hbox{\hskip
43.43292pt\vbox{\vbox{}\hbox{\hskip-43.4329pt\hbox{\hbox{$\displaystyle\displaystyle(A,B)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau}~{}(C,D)$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
41.44739pt\vbox{\hbox{\hskip-41.44737pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}A$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\tau}~{}B$}}}}\vbox{}}}\over\hbox{\hskip
31.7527pt\vbox{\vbox{}\hbox{\hskip-31.75269pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}\tau}~{}(A,B)$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
137.49814pt\vbox{\hbox{\hskip-137.49812pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(inc:\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)(S:{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma))\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}S\rightarrow
F(\mathit{inc},S)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\tau}~{}G(\mathit{inc},S)$}}}\vbox{}}}\over\hbox{\hskip
22.54855pt\vbox{\vbox{}\hbox{\hskip-22.54854pt\hbox{\hbox{$\displaystyle\displaystyle
F~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau}~{}G$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
161.31305pt\vbox{\hbox{\hskip-88.94633pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(inc:\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta),{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}S\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\tau}~{}F(inc,S)$}}}\vbox{\hbox{\hskip-161.31305pt\hbox{\hbox{$\displaystyle\displaystyle$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(\mathit{inc}:\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}S_{1}\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}S_{2}\rightarrow
S_{1}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}S_{2}\rightarrow
F(\mathit{inc},S_{1})~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\tau}~{}F(\mathit{inc},S_{2})$}}}}\vbox{\hbox{\hskip-126.03639pt\hbox{\hbox{$\displaystyle\displaystyle\forall\mathit{inc_{1}},\mathit{inc_{2}}\rightarrow{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}S\rightarrow{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc_{1}}}~{}F(\mathit{inc_{2}},S)}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\tau}~{}{F(\mathit{inc_{2}}\cdot\mathit{inc_{1}},{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc_{1}}}~{}S)}$}}}\vbox{}}}}}\over\hbox{\hskip
20.98013pt\vbox{\vbox{}\hbox{\hskip-20.98012pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}\tau}~{}F$}}}}}}$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
26.04106pt\vbox{\vbox{}\hbox{\hskip-26.04106pt\hbox{\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}\colon\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}^{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{`list}}}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{}\over\hbox{\hskip
22.54068pt\vbox{\vbox{}\hbox{\hskip-22.54066pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
66.81468pt\vbox{\hbox{\hskip-66.81467pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{hd}\colon\mathit{EQ_{\sigma}}(X,Y)$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{tl}\colon\mathit{XS}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}^{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{`list}}}~{}\mathit{YS}$}}}}\vbox{}}}\over\hbox{\hskip
67.91281pt\vbox{\vbox{}\hbox{\hskip-67.9128pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{hd}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{tl}\colon
X\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{XS}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}^{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{`list}}}~{}Y\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{YS}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
54.41675pt\vbox{\hbox{\hskip-54.41675pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}\mathit{HD}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}~{}\mathit{TL}$}}}}\vbox{}}}\over\hbox{\hskip
40.17618pt\vbox{\vbox{}\hbox{\hskip-40.17618pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}~{}\mathit{HD}\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`::}}~{}}\mathit{TL}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
198.94366pt\vbox{\hbox{\hskip-198.94366pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{xs}\colon\mathit{xs_{1}}\equiv\mathit{xs_{2}}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\mathit{YS}\colon\mathit{YS_{1}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}^{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{`list}}}~{}\mathit{YS_{2}}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle
F\colon\forall\Delta(\mathit{inc}:\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)(t:\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\tau),\mathit{EQ_{\sigma}}(F_{1}(\mathit{inc},t),F_{2}(\mathit{inc},t))$}}}}}\vbox{}}}\over\hbox{\hskip
165.68811pt\vbox{\vbox{}\hbox{\hskip-165.68811pt\hbox{\hbox{\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(F,\mathit{xs})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{YS}\colon$}}\hskip
20.00003pt\hbox{$\displaystyle\displaystyle\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(F_{1},\mathit{xs_{1}})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{YS_{1}}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}^{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{`list}}}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(F_{2},\mathit{xs_{2}})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{YS_{2}}$}}}}}}$
$\displaystyle\displaystyle{\hbox{\hskip
141.59784pt\vbox{\hbox{\hskip-92.36194pt\hbox{\hbox{$\displaystyle\displaystyle\forall\Delta(\mathit{inc}\colon\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\subseteq}~{}\Delta)(t\colon\Delta~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\tau),{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}F(\mathit{inc},t)$}}}\vbox{\hbox{\hskip-141.59784pt\hbox{\hbox{$\displaystyle\displaystyle\forall\mathit{inc_{1}},\mathit{inc_{2}},t,{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc_{1}}}~{}F(\mathit{inc_{2}},t)}~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}{F(\mathit{inc_{2}}\cdot\mathit{inc_{1}},{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{wk}}_{\mathit{inc_{1}}}~{}t)}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}~{}\mathit{YS}$}}}}\vbox{}}}}\over\hbox{\hskip
62.58598pt\vbox{\vbox{}\hbox{\hskip-62.58597pt\hbox{\hbox{$\displaystyle\displaystyle{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma}~{}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(F,\mathit{xs})\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\mathit{YS}$}}}}}}$
Figure 8: Semantic equality and uniformity of objects in the model
Quite unsurprisingly, the unit case is of no interest: all the semantical
units are equivalent and uniform. Semantic equality for elements with base
types is up-to $\alpha$-equivalence: inhabitants are just bits of data
(neutrals) which can be compared in a purely syntactical fashion because we
use nameless terms. They are always uniform.
In the product case, the semantical objects are actual pairs and the
definition just forces the properties to hold for each one of the pair’s
components.
The function type case is a bit more hairy. While extensionality on uniform
arguments is simple to state, uniformity has to enforce a lot of invariants:
application of uniform objects should yield a uniform object, application of
extensionally equal uniform objects should yield extensionally equal objects
and weakening and application should commute (up to extensionality).
In the
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}\sigma$
case, extensional equality is an inductive set basically building the
(semantical) diagonal relation on lists of the same type. It is parametrized
by a relation $EQ_{\sigma}$ on terms of type
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Delta,\sigma)$
(for any context $\Delta$) which is, in the practical case instantiated with
$\\_~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}\\_$
as one would expect. Uniformity is, on the other hand, defined by recursion on
the semantical list. It could very well be defined as being parametric in
something behaving like
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}\\_$
but this is not necessary: there are no positivity problems! It is therefore
probably better to stick to a lighter presentation here. The empty list indeed
is uniform. A constructor-headed list is said to be uniform if its head of
type
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}}(\Gamma,\sigma)$
is uniform and its tail also is uniform. The criterion for a stuck list is a
bit more involved. Mimicking the definition of uniformity for functions, there
are two requirements on the stuck map: applying it to a neutral yields a
uniform element of the model and application and weakening commute. Lastly the
second argument of the stuck append should be uniform too.
###### Remark 5.11.
The careful reader will already have noticed that this defines a family of
equivalence relations; we will not make explicit use of reflexivity, symmetry
and transitivity in the paper but it is fundamental in the formalization.
Recall that the completeness theorem was presented as expressing the fact that
elements equivalent with respect to the reduction relation were interpreted as
semantical objects behaving similarly. For this approach to make sense,
knowing that two semantical objects are extensionally equal should immediately
imply that their respective reifications are syntactically equal. Which is the
case.
###### Lemma 5.12.
Reification, reflection and weakenings are compatible with the notions of
extensional equality and uniformity.
1. 1.
If
$T~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}U$
then
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}T={\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\downarrow}_{\sigma}U$
2. 2.
If $t_{ne}$ is a neutral
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash_{\mathit{ne}}}~{}\sigma$
then
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathtt{Uni}}_{\sigma}~{}({\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\uparrow}_{\sigma}t_{ne})$
3. 3.
Weakening and reification commute for uniform objects
Now that we know that all the theorem proving ahead of us will not be
meaningless, we can start actually tackling completeness. When applying an
extensional function, it is always required to prove that the argument is
uniform. Being able to certify the uniformity of the evaluation of a term is
therefore of the utmost importance.
###### Lemma 5.13.
Evaluation preserves properties of the evaluation environment.
1. 1.
Evaluation in uniform environments produces uniform values
2. 2.
Evaluation in semantically equivalent environments produces semantically
equivalent values
3. 3.
Weakening the evaluation of a term is equivalent to evaluating this term in a
weakened environment
###### Theorem 5.14.
If $s$ and $t$ are two terms in
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$
such that
$s\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\leadsto_{\operatorname{\beta\delta\iota\eta\nu}}}}t$
and if $R$ is a uniform environment in
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\mathcal{M}}^{s}}(\Delta,\Gamma)$
then
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}(s,R)~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv}_{\sigma}~{}\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}(t,R)$.
###### Proof 5.15.
One proceeds by induction on the proof that $s$ reduces to $t$.
#### Structural rules
Structural rules can be discharged by combining induction hypotheses and
reflexivity proofs using previously proved lemma such as the fact that
evaluation in uniform environments yields uniform elements for the structural
rule for the argument part of application.
#### $\beta\iota$-rules
Each one the $\iota$ rules holds by reflexivity of the extensional equality,
indeed evaluation realizes these computation rules syntactically. The case of
the $\beta$ rule is slightly more complicated. Given a function
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.b$
and its argument $u$, one starts by proving that the diagonal semantical
environment extended with the evaluation of $u$ in $R$ is extensionally equal
to the evaluation in $R$ of the diagonal substitution extended with $u$.
Thence, knowing that the evaluations of a term in two extensionally equal
environments are extensionally equal, one can see that the evaluation of the
redex is related to the evaluation of the body in an environment corresponding
to the evaluation of the substitution generated when firing the redex.
Finally, the fact that
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}eval}}$
and substitution commute (up-to-extensionality) lets us conclude.
#### $\eta\nu$-rules
definitely are the most work-intensive ones: except for the ones for product
and unit types which can be discharged by reflexivity of the semantic
equality, all of them need at least a little bit of theorem proving to go
through. It is possible to prove the map-id, map-append, append-nil,
associativity of append and various fusion rules by induction on the ‘nut’ for
uniform values. Solving the goals is then just a matter of combining the right
auxiliary lemma with facts proved earlier on, typically the uniformity of
semantical object obtained by evaluating a term in a uniform environment.
###### Corollary 5.16 (Completeness).
For all terms $t$ and $u$ of type
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$,
if
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv_{\operatorname{\beta\delta\iota\eta\nu}{}}}}u$
then
$\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}t=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}u$.
###### Proof 5.17.
Reflection produces uniform values and uniformity is preserved through
weakening hence the fact that the trivial diagonal environment is uniform.
Combined with iterations of the previous lemma along the proof that
$t\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\equiv_{\operatorname{\beta\delta\iota\eta\nu}{}}}}u$,
we get that the respective evaluations of $t$ and $u$ are extensionally equal
which we have proved to be enough to get syntactically equal reifications.
###### Corollary 5.18.
The equational theory enriched with $\nu$-rules is decidable.
###### Proof 5.19.
Given terms $t$ and $u$ of the same type
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}\sigma$,
we can get two normal forms
$t_{nf}=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}t$
and
$u_{nf}=\operatorname{{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}norm}}u$
and test them for equality up-to $\alpha$-conversion (which is a simple
syntactic check in our nameless representation in Agda).
If $t_{nf}=u_{nf}$ then the soundness result allows us to conclude that $t$
and $u$ are convertible terms.
If $t_{nf}\neq u_{nf}$ then $t$ and $u$ are not convertible. Indeed, if they
were then the completeness result guarantees us that $t_{nf}$ and $u_{nf}$
would be equal which leads to a contradiction.
###### Example 5.20.
of terms which are identified thanks to the internalization of the
$\nu$-rules.
1. 1.
In a context with two functions $f$ and $g$ of type
$\sigma~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`1}}$,
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(f,xs)$
and
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(g,xs)$
both normalize to
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}\\_.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\langle\rangle}}},xs)\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
and are therefore declared equal.
2. 2.
At type
$\Gamma~{}{\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}\vdash}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{l}}})~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\rightarrow}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`list}}~{}({\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{k}}~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\times}~{}{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\alpha}_{l}}})}$,
the terms
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.xs$
and
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(swap,\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(swap,xs))$
where $swap$ is the function
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}p)$
swapping the order of a pair’s elements are convertible with normal form
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}xs.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}p.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}p\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}p),xs)\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$.
## 6 Scaling up to Type Theory
Now that we know for sure that the judgmental equality can be safely extended
with some $\nu$-rules, we are ready to tackle more complex type theories. We
have already experimented with extending our simply-typed setting to a
universe of polynomial datatypes with map and fold. We have to identify which
parts of the setting are key to the success of this technique and how to
enforce that the generalized version still has good properties.
#### Types
Arrow types will be replaced by $\Pi$-types and product types by
$\Sigma$-types but the basic machinery of evaluation and type-directed
$\eta$-expansion work in much the same way.
In Type Theory, it is not quite enough to be able to decide the judgmental
equality. Pollack’s PhD thesis (Pollack [1994], Section 5.3.1), taught us how
to turn the typing relation with a conversion rule into a syntax-directed
typechecking algorithm by relying on ordinary evaluation (cf. the application
typing rule in Figure 9). It is therefore quite crucial for ensuring the
reusability of previous typechecking algorithms to be able to guarantee that
ordinary evaluation is complete for uncovering constructor-headed terms i.e.
$\Gamma\vdash t\equiv C~{}\vec{t_{i}}\colon T$ should imply that
$t\leadsto^{\star}C~{}\vec{t_{i}}^{\prime}$. This can be enforced by making
sure that candidates for $\nu$-rules are only reorganizing spines of stuck
eliminators and are absolutely never emitting new constructors.
$\displaystyle\displaystyle{\hbox{\hskip
135.91005pt\vbox{\hbox{\hskip-135.91003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma\vdash
f\colon F\qquad F\leadsto^{\star}(x:S)\rightarrow T$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma\vdash s\colon
S^{\prime}$}\hskip
20.00003pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma\vdash S\equiv
S^{\prime}\colon\mathtt{Set}$}}}}}\vbox{}}}\over\hbox{\hskip
34.77243pt\vbox{\vbox{}\hbox{\hskip-34.77243pt\hbox{\hbox{$\displaystyle\displaystyle\Gamma\vdash
fs\colon T[s/x]$}}}}}}$ Figure 9: Syntax-directed typing rule for application,
Pollack Pollack [1994]
#### $\eta$-rules
A Type Theory does not need to have judgmental $\eta$-rules for the
$\nu$-rules to make sense. However this partially defeats the purpose of this
extension: without $\eta$-rules for products we fail to identify the silly
identity on lists of products map swap . map swap with the more traditional
one $\lambda x.x$ because
$f_{1}=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.x$
is different from
$f_{2}=\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.(\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{1}}}}}x\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`,}}~{}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`\operatorname{\pi_{2}}}}}x)$
when both terms would reduce respectively to
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f_{1}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}x{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$
and
$\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`}\lambda}}x.\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\langle}f_{2}{~{}\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12},~{}}x{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\rangle\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}}\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`[]}}}$.
So close yet so far away!
#### Defined symbols
In this presentation, a handful of functions are built-in rather than user-
defined. This will probably be one of the biggest changes when moving to a
usable Type Theory. We can enforce that functions defined by pattern-matching
have a fixed arity and are always fully applied at that arity. Such a function
is stuck if it is strict in a neutral argument. Some type theories reduce
pattern matching to the primitive elimination operator for each datatype. To
apply $\nu$-rules, we need to detect which stuck eliminators correspond to
which stuck pattern matches. This is the same problem as producing readable
output from normalizing open terms, and it has already been solved by the
‘labelled type’ translation used in Epigram, which effectively inserts
documentation of stuck pattern matches into spines of stuck eliminators
McBride and McKinna [2004].
#### Criteria for $\nu$-rules
Working in a setting where the datatypes are given by a universe Chapman et
al. [2010], we should at least expect that built-in generic operators, e.g.
map, have associated $\nu$-rules. However, it is clearly desirable to allow
the programmer to propose $\nu$-rules for programs of her own construction.
How will the machine check that proposed $\nu$-rules keep evaluation canonical
and judgmental equality consistent and decidable? We have already seen that
$\nu$-rules must avoid to emit new constructors; this can be summed up by the
mantra: “A $\nu$-rule may restart computation _within_ its contractum but
_never_ in its enclosing context”.
The candidates for $\nu$-rules should hold trivially by a Boyer-Moore style
induction; in other words, the $\beta\delta\iota-\nu$ critical pairs should be
convergent. This tells us that these rules are consistent and can be delayed
until after evaluation.
Obviously, the $\nu-\nu$ critical pairs should also be convergent. These three
criteria are all easy to check provided that $\nu$-reductions give rise to a
terminating term rewrite system.
This termination requirement is the last criterion. As a first instance, a
rather conservative approach could be to ask the user for a linear order on
defined symbols which we would lift to expressions by using the lexicographic
ordering of the encountered defined symbols starting from the “nut” and going
outwards. If this ordering is compatible with a left to right orientation of
the $\nu$-rules she wants to hold, then it is terminating. In the set of
$\nu$-rules used as an example in this paper, the simple ordering
$\operatorname{~{}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`++}}~{}}>\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`map}}}>\operatorname{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\mathtt{`fold}}}$
is compatible with the rules.
## 7 Further Opportunities for $\nu$-Rules
We were motivated to develop a proof technique for extending definitional
equality with $\nu$-rules because there are many opportunities where we might
profit by doing so. Let us set out a prospectus.
#### Reflexive coercion for type-based equality.
Altenkirch, McBride and W. Swierstra developed a propositional equality for
intensional type theory Altenkirch et al. [2007] which differs from the usual
inductive definition ($\mathtt{refl~{}a~{}:~{}a~{}=~{}a}$) in that its main
eliminator
$\displaystyle\displaystyle{\hbox{\hskip
60.25812pt\vbox{\hbox{\hskip-60.2581pt\hbox{\hbox{$\displaystyle\displaystyle
S,T:\mathtt{Set}\quad Q:S=T\quad s:S$}}}\vbox{}}}\over\hbox{\hskip
33.82764pt\vbox{\vbox{}\hbox{\hskip-33.82764pt\hbox{\hbox{$\displaystyle\displaystyle
s[Q:S=T\rangle:T$}}}}}}$
computes by structural recursion first on the _types_ $S$ and $T$, and then
(where appropriate) on $s$, rather than by pattern matching on the proof $Q$.
Equality is still reflexive, so evaluation can leave us with terms
$n[\mathtt{refl}\>n:N=N\rangle:N$ where $n$ is a neutral term in a neutral
type $N$. It is clearly a nuisance that this term does not compute to $n$, as
would happen if the eliminator matched on the proof. The fix is to add a
$\nu$-rule which discards coercions whenever it is type-safe to do so:
$\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$s$}}}$[Q:S=T\rangle$}}}=\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$s$}}}\qquad\mbox{if}\;S\equiv
T:\mathtt{Set}$
It is easy to check that adding this rule for neutral terms makes it
admissible for all terms, and hence that we need add it not to evaluation, but
only to the reification process which follows, just as with the $\nu$-rules in
this paper. There, as here, this spares the evaluation process from decisions
which involve $\eta$-expansion and thus require a name supply. The $\nu$-rule
thus gives us a non-disruptive means to respect the full computational
behaviour of inductive equality in the observational setting.
#### Functor laws.
Barral and Soloviev give a treatment of functor laws for parametrized
inductive datatypes by modifying the $\iota$-rules of their underlying type
theory Barral and Soloviev [2006]. We should very much hope to achieve the
same result, as we did here in the special case of lists, just by adding
$\nu$-rules. Our preliminary experiments McBride [2010] suggest that we can
implement functor laws once and for all in a type theory whose datatypes are
given once and for all by a syntactic encoding of strictly positive functors,
as Dagand and colleagues propose Chapman et al. [2010]; Dagand and McBride
[2012]. Moreover, Luo and Adams have shown Luo and Adams [2008] that
structural subtyping for inductive types can be reified by a coherent system
of implicit coercions if functor laws hold definitionally.
#### Monad laws.
Watkins et al. give a definitional treatment of monad laws in order to achieve
an adequate representation of concurrent processes encapsulated monadically in
a logical framework Watkins et al. [2003]. For straightforward free monads, an
experimental extension of Epigram (by Norell, as it happens) McBride [2010]
suggests that we may readily allow $\nu$-rules:
$\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$t$}}}>\\!\\!>\\!\\!=\mathtt{return}$}}}=\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$t$}}}\qquad\framebox{{\raisebox{0.0pt}[7.22743pt][2.8903pt]{$\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$(\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$t$}}}>\\!\\!>\\!\\!=\sigma)$}}}>\\!\\!>\\!\\!=\rho$}}}=\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$t$}}}>\\!\\!>\\!\\!=((>\\!\\!>\\!\\!=\sigma)\cdot\rho)$}}}$
Atkey’s Foveran system uses a similar normalization method for free monad laws
Atkey [2011], again for an encoded universe of underlying functors.
#### Decomposing functors.
Dagand and colleagues further note that their syntax of descriptions for
indexed functors is, by virtue of being a syntax, itself presentable as the
free monad of a functor. The description decoder
$\mathtt{Decode}:\mathtt{IDesc}\>I\to(I\to\mathtt{Set})\to\mathtt{Set}$
is structurally recursive in the description and lifts pointwise to an
interpretation of substitutions in the $\mathtt{IDesc}$ monad
$\begin{array}[]{l}\llbracket\\_\rrbracket:(O\to\mathtt{IDesc}\>I)\;\;\to\;\;(I\to\mathtt{Set})\to(O\to\mathtt{Set})\\\
\llbracket\sigma\rrbracket\>X\>o=\mathtt{Decode}\>(\sigma\>o)\>X\end{array}$
as indexed functors with a ‘map’ operation satisfying functor laws. However,
not only does this interpretation _deliver_ functors, it is _itself_ a
contravariant functor: the identity substitution yields the identity functor
just by $\beta\delta\iota$, but we may also interpret Kleisli composition as
reverse functor composition
$\llbracket(>\\!\\!>\\!\\!=\sigma)\cdot\rho\rrbracket=\llbracket\rho\rrbracket\cdot\llbracket\sigma\rrbracket$
by means of a $\nu$-rule
$\framebox{{\raisebox{0.0pt}[7.22743pt][2.8903pt]{Decode~{}\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$(\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$D$}}}>\\!\\!>\\!\\!=\sigma)$}}}~{}$X$}}}=\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{Decode~{}\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$D$}}}~{}$(\llbracket\sigma\rrbracket\>X)$}}}$
taking each $D$ to be some $\rho\>o$. If we want to do a ‘scrap your
boilerplate’ style traversal of some described container-like structure, we
need merely exhibit the decomposition of the description as some
$(>\\!\\!>\\!\\!=\sigma)\cdot\rho$, where $\rho$ describes the invariant
superstructures and $\sigma$ the modified substructures, then invoke the
functoriality of $\llbracket\rho\rrbracket$. This $\nu$-rule thus lets us
expose functoriality over substructures not anticipated by explicit
parametrization in datatype declarations. We thus recover the kind of ad-hoc
data traversal popularized by Lämmel and Peyton Jones Lämmel and Jones [2003]
by static structural means.
#### Universe embeddings.
A type theory with inductive-recursive definitions is powerful enough to
encode universes of dependent types by giving a datatype of codes _in tandem_
with their interpretations Dybjer and Setzer [1999], the paradigmatic example
being
$\begin{array}[]{@{}l@{\;}|@{\;}l@{}}\mathtt{U}_{1}:\mathtt{Set}&\mathtt{El}_{1}:\mathtt{U}_{1}\to\mathtt{Set}\\\
\mathtt{`Pi}_{1}:(S:\mathtt{U}_{1})\to&\mathtt{El}_{1}\>(\mathtt{`Pi}_{1}\>S\>T)=\\\
\qquad\qquad(\mathtt{El}_{1}\>S\to\mathtt{U}_{1})\to\mathtt{U}_{1}&\;\;(s:\mathtt{El}_{1}\>S)\to\mathtt{El}_{1}\>(T\>s)\\\
\vdots&\vdots\end{array}$
Palmgren Palmgren [1998] suggests that one way to model a cumulative hierarchy
of such universes is to give each a code in the next, so
$\begin{array}[]{@{}l@{\;}|@{\;}l@{}}\mathtt{U}_{2}:\mathtt{Set}&\mathtt{El}_{2}:\mathtt{U}_{2}\to\mathtt{Set}\\\
\mathtt{`U}_{1}:\mathtt{U}_{2}&\mathtt{El}_{2}\>\mathtt{`U}_{1}=\mathtt{U}_{1}\\\
\mathtt{`Pi}_{2}:(S:\mathtt{U}_{2})\to&\mathtt{El}_{2}\>(\mathtt{`Pi}_{2}\>S\>T)=\\\
\qquad\qquad(\mathtt{El}_{2}\>S\to\mathtt{U}_{2})\to\mathtt{U}_{2}&\;\;(s:\mathtt{El}_{2}\>S)\to\mathtt{El}_{2}\>(T\>s)\\\
\vdots&\vdots\end{array}$
and then define an embedding recursively
$\begin{array}[]{l}\uparrow:\mathtt{U}_{1}\to\mathtt{U}_{2}\\\
\uparrow(\mathtt{`Pi}_{1}\>S\>T)=\mathtt{`Pi}_{2}\>(\uparrow S)\>(\lambda
s.\>\uparrow(T\>s))\end{array}$
but a small frustration with this proposal is that $s$ is abstracted at type
$\mathtt{El}_{2}\>(\uparrow S))$, but used at type $\mathtt{El}_{1}\>S$, and
these two types are not definitionally equal for an abstract $S$. One
workaround is to make $\uparrow$ a constructor of $\mathtt{U}_{2}$, at the
cost of some redundancy of representation, but now we might also consider
fixing the discrepancy with a $\nu$-rule
$\framebox{{\raisebox{0.0pt}[7.22743pt][2.8903pt]{$\mathtt{El}_{2}\>\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$(\uparrow\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$S$}}})$}}})$}}}=\framebox{{\raisebox{0.0pt}[5.78172pt][1.4457pt]{$\mathtt{El}_{1}\>\framebox{{\raisebox{0.0pt}[4.33601pt][0.0pt]{$S$}}}$}}}$
This is peculiar for our examples thus far, in that the $\nu$-rule is needed
even to typecheck the $\delta\iota$-rules for $\uparrow$, reflecting the fact
that $\uparrow$ should not be any old function from $\mathtt{U}_{1}$ to
$\mathtt{U}_{2}$, but rather one which preserves the meanings given by
$\mathtt{El}_{1}$ and $\mathtt{El}_{2}$. In effect, the $\nu$-rule is
expressing the coherence property of a richer notion of morphism. It is
inviting to wonder what other notions of coherence we might enable and enforce
by checking that $\nu$-rules hold of the operations we implement.
#### Non-examples.
A key characteristic of a $\nu$-rule is that it is a nut-preserving
rearrangement of neutral term layers. Whilst this is good for associativity
and sometimes for distributivity, it is perfectly useless for commutativity.
Suppose $+$ for natural numbers is recursive on its first argument, and
observe that rewriting $x+y$ to $y+x$ when $x$ is neutral will not result in a
neutral term unless $y$ is also neutral. Less ambitious rules such as
$x+\mathtt{suc}\>y=\mathtt{suc}\>(x+y)$ and $x*0=0$ similarly make neutral
terms come unstuck, and so cannot be postponed until reification if we want to
be sure that evaluation suffices to show whether any expression in a datatype
can be put into constructor-headed form. Walukiewicz-Chrzaszcz has proposed a
more invasive adoption of rewriting for Coq, necessitating a modified
evaluator, but incorporating rules which can expose constructors Walukiewicz-
Chrzaszcz [2003]. Her untyped rewriting approach sits awkwardly with
$\eta$-laws, but we can find a more carefully structured compromise.
## 8 Discussion
We fully expect to scale this technology up to type theory. Abel and Dybjer
(with Aehlig Abel et al. [2007a] and T. Coquand Abel et al. [2007b]) have
already given normalization by evaluation algorithms which we plan to adapt.
Finding good criteria for checking that candidate $\nu$-rules can safely be
added is of the utmost importance. We want to let the programmer negotiate the
new $\nu$-rules she wants, as long as the machine can check that they yield a
notion of standard form and lift from neutral terms to all terms by the prior
equational theory.
It is also interesting to try to integrate $\nu$-rules with more practical
presentations of normalization. For instance Grégoire and Leroy’s conversion
by compilation to a bytecode machine derived from Ocaml’s ZAM Grégoire and
Leroy [2002] decides $\eta$ by expansion only when provoked by a $\lambda$:
such laziness is desirable when possible but causes trouble with $\eta$-rules
for unit types and may conceal the potential to apply $\nu$-rules. Hereditary
substitution Watkins et al. [2003], formalized by Abel Abel [2009] and by
Keller and Altenkirch Keller and Altenkirch [2010], may be easier to adapt.
## Acknowledgements
We would like to thank the anonymous reviewers for their helpful comments and
suggestions as well as Stevan Andjelkovic for carefully reading our draft.
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|
arxiv-papers
| 2013-04-02T22:55:32 |
2024-09-04T02:49:43.808325
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guillaume Allais, Pierre Boutillier, Conor McBride",
"submitter": "Guillaume Allais",
"url": "https://arxiv.org/abs/1304.0809"
}
|
1304.0899
|
# Ligth-flavour identified charged-hadron production in pp and Pb–Pb
collisions at the LHC
Roberto Preghenella for the ALICE Collaboration Centro Studi e Ricerche e
Museo Storico della Fisica “Enrico Fermi”, Rome, Italy
Sezione INFN, Bologna, Italy [email protected]
###### Abstract
Thanks to the unique detector design adopted to fulfill tracking and particle-
identification (PID) requirements (e.g. low momentum cut-off and low material
budget), the ALICE experiment provides significant information about hadron
production both in pp and Pb–Pb collisions. In particular, the $p_{\rm
T}$-differential and integrated production yields of identified particles play
a key role in the study of the collective and thermal properties of the matter
formed in high-energy heavy-ion collisions. Furthermore, the production of
high-$p_{\rm T}$ particles provides insights into the property of the hot
medium created in such collisions and the in-medium energy-loss mechanisms.
Transverse momentum spectra of $\pi^{\pm}$, K±, p and $\bar{\rm p}$ are
measured at mid-rapidity ($\left|y\right|~{}<~{}0.5$) over a wide momentum
range, from $\sim$ 100 MeV/$c$ up to $\sim$ 20 GeV/$c$. The measurements are
performed exploiting the d$E$/d$x$ in silicon and gas, the time-of-flight and
the ring-imaging Cherenkov particle-identification techniques, which will be
briefly reviewed in this report. The current results on light-flavour charged-
hadron production will be presented for pp collisions at $\sqrt{s}$ = 0.9,
2.76 and 7 TeV and for Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.
Integrated production yields, transverse momentum spectra and particle ratios
in pp are discussed as a function of the collision energy and compared to
previous experiments and commonly-used Monte Carlo models. Pb–Pb collisions at
the LHC feature the highest radial flow ever observed and an unexpectedly low
p/$\pi$ production ratio. The results are presented as a function of collision
centrality and compared to RHIC data in Au–Au collisions at $\sqrt{s_{\rm
NN}}$ = 200 GeV and predictions from thermal and hydrodynamic models. The
nuclear modification factor ($R_{\rm AA}$) of identified hadrons will also be
discussed and compared to unidentified charged particles and theoretical
predictions. This is observed to be identical for all particle species at
high-$p_{\rm T}$.
## 1 Introduction
ALICE (A Large Ion Collider Experiment) is a general-purpose heavy-ion
detector at the CERN LHC (Large Hadron Collider). It has been designed in
order to fulfill the requirements to track and identify particles from very
low ($\sim$100 MeV/$c$) up to quite high ($\sim$100 GeV/$c$) transverse
momenta in an environment with large charged-particle multiplicities as in the
case of central lead-lead (Pb–Pb) collisions at the LHC.
;
Figure 1: Schematic layout of the ALICE detector with its main subsystems.
The ALICE experiment, shown in Figure 1, consists of a central-barrel detector
and several forward detector systems. The central system covers the mid-
rapidity region ($\left|\eta\right|\leq$ 0.9) over the full azimuthal angle.
It is installed inside a large solenoidal magnet providing a moderate magnetic
field of 0.5 T. It includes a six-layer high-resolution inner-tracking system
(ITS), a large-volume time-projection chamber (TPC) and electron and charged-
hadron identification detectors which exploit transition-radiation (TRD) and
time-of-flight (TOF) techniques, respectively. Small-area systems for
high-$p_{\rm T}$ particle-identification (HMPID), photon and neutral-meson
measurements (PHOS) and jet reconstruction (EMCal) complement the central
barrel. Thanks to these unique features the experiment is able to identify
hadrons in a wide momentum range by employing different detection systems and
techniques, as discussed in Section 2. The detectors which cover larger
rapidity regions include a single-arm muon spectrometer covering the
pseudorapidity range -4.0 $\leq\eta\leq$ -2.4 and several smaller detectors
(VZERO, TZERO, FMD, ZDC, and PMD) for triggering, multiplicity measurements
and centrality determination. A detailed description of the ALICE detector
layout and of its subsystems can be found in [1].
Since November 2009 when the first collisions at the LHC occurred, the ALICE
experiment has collected proton-proton data at several centre-of-mass energies
($\sqrt{s}$ = 0.9, 2.76, 7 and 8 TeV). During the first two LHC heavy-ion
runs, in 2010 and 2011, the detector recorded Pb–Pb collisions at a centre-of-
mass energy per nucleon pair of $\sqrt{s_{\rm NN}}$ = 2.76 TeV and could
profit from of an integrated luminosity of about 10 $\mu\rm b^{-1}$ and 100
$\mu\rm b^{-1}$, respectively. It is worth stressing the outstanding
performance of the LHC complex: the instant luminosity exceeded $10^{26}\rm
cm^{-2}s^{-1}$ in the second run, higher than the design value. Proton-lead
(p–Pb) collision data were also collected with the ALICE detector. This
occurred during a short run performed in September 2012 in preparation for the
main p–Pb run at the beginning of 2013. Eight pairs of bunches collided in the
ALICE interaction region, providing a luminosity of about $8\times 10^{25}\rm
cm^{-2}s^{-1}$. The configuration (4 TeV protons colliding with fully stripped
208Pb ions at $82\times 4$ TeV) resulted in interactions at $\sqrt{s_{\rm
NN}}$ = 5.02 TeV in the nucleon-nucleon centre-of-mass system, which moves
with a rapidity of $\Delta y_{\rm NN}$ = 0.465.
## 2 Particle identification
Figure 2: Energy loss d$E$/d$x$ in the ITS (top-left) and in the TPC (top-
right). The continuous curves represent the Bethe-Bloch parametrization.
(bottom-left) particle velocity $\beta$ measured with TOF as a function of
momentum. (bottom-right) Cherenkov angle measured in the HMPID as a function
of the track momentum. Figure 3: Transverse DCA ($DCA_{xy}$) of protons in the
range between 0.6 GeV/$c$ and 0.65 GeV/$c$ in 0-5% most central Pb–Pb
collisions together with the Monte Carlo templates which are fitted to the
data.
In this section the main particle-identification (PID) detectors relevant to
the analyses presented in this paper are briefly discussed. A detailed review
of the ALICE experiment and of its PID capabilities can be found in [2]. The
ITS is the innermost detector system, a six-layer silicon detector located at
radii between 4 and 43 cm. Four of the six layers provide specific energy loss
d$E$/d$x$ measurements and are used for particle identification in the non-
relativistic ($1/\beta^{2}$) region. By using the ITS as a standalone tracker
it is possible to reconstruct and identify low-momentum particles (below 200
MeV/c) not reaching the main tracking systems (Figure 2 top-left). The TPC is
the main central-barrel tracking detector of ALICE. It provides three-
dimensional hit information and specific energy-loss measurements with up to
159 samples. With the measured particle momentum and $\langle$d$E$/d$x\rangle$
the particle type can be determined by comparing the measurements with the
Bethe-Bloch expectation (Figure 2 top-right). The TOF detector is a large-area
array of Multigap Resistive Plate Chambers (MRPC) and covers the central
pseudorapidity region ($\left|\eta\right|<$ 0.9, full azimuth). Particle
identification is performed by matching momentum and trajectory-length
measurements performed by the tracking system with the time-of-flight
information provided by the TOF system. The overall time-of-flight resolution
is measured to be about 85 ps in Pb–Pb collisions (about 120 ps in pp
collisions) and it is determined by the time resolution of the detector itself
and by the start-time resolution (Figure 2 bottom-left). The HMPID detector
consists of seven identical proximity focusing RICH (Ring Imaging Cherenkov)
counters. Photon detection is performed using the proportional multiwire
chambers coupled to pad-segmented CsI photocathode. Particle identification is
obtained by means of the measurement of the Cherenkov angle allowing the
separation of pions and kaons between 1 GeV/$c$ and 3 GeV/$c$ and protons from
1.5 GeV/$c$ up to 5 GeV/$c$ (Figure 2).
The transverse momentum spectra of primary $\rm\pi^{\pm}$, $\rm K^{\pm}$, $\rm
p$ and $\rm\bar{p}$ are measured at mid-rapidity ($\left|y\right|~{}<~{}0.5$)
combining the techniques and detectors described above. Primary particles are
defined as prompt particles produced in the collision and all decay daughters,
except products from weak decays of strange particles. The contribution from
the feed-down of weakly-decaying particles to $\rm\pi^{\pm}$, $\rm p$ and
$\rm\bar{p}$ and from protons emitted from secondary interactions with
material are subtracted by fitting the data using Monte Carlo templates of the
DCA111Distance of Closest Approach to the reconstructed primary vertex.
distributions (Figure 3). Particles can also be identified in ALICE via their
characteristics decay topology or invariant mass. This, combined with the
direct identification of the decay daughters, allows one to reconstruct
weakly-decaying particles and hadronic resonances with a good signal-to-
background ratio.
## 3 Light-flavour hadron production
Figure 4: Transverse momentum distributions of the sum of positive and
negative pions, kaons and protons for central Pb–Pb collisions. The results
are compared to RHIC data and hydrodynamic models.
Figure 5: (left) The thermal fit of ALICE data showing data and model for the
best fit. (right) Mid-rapidity particle ratios compared to RHIC results and
predictions from thermal models for central Pb–Pb collisions at the LHC.
Figure 6: $\rm K/\pi$ (top) and $\rm p/\pi$ production ratio in pp collisions
compared with PYTHIA Monte Carlo predictions and NLO calculations.
Figure 7: Charged pion, kaon and (anti)proton nuclear modification factor $\rm
R_{AA}$ as a function of $p_{\rm T}$ for Pb–Pb collisions at $\sqrt{s_{\rm
NN}}$ = 2.76 TeV for centrality 0-5% (left) and 20-40% (right). Statistical
(vertical error bars) and systematic (gray and colored boxes) are shown for
the charged pion RAA. The gray boxes contains the common systematic error
related to pp normalization to INEL and $\rm N_{coll}$. Figure 8: Nuclear
modification factor $\rm R_{AA}$ of charged pions compared to the $\rm R_{AA}$
of unidentified charged particles as a function of $p_{\rm T}$ for different
centrality classes. Statistical (vertical error bars) and systematic (gray and
colored boxes) errors are shown for the charged pions. The colored boxes
contain the common systematic uncertainty related to the number of binary
collisions and the pp normalization to INEL. Only statistical errors are shown
for the unidentified charged $\rm R_{AA}$. Figure 9: Proton-to-pion ratio in
Pb–Pb collisions compared with pp results at 2.76 TeV in several centrality
bins. In Pb–Pb the statistical and systematic uncertainties are displayed as
error bars and gray bands, respectively. In pp the systematic uncertainty is
displayed in black squares.
Figure 10: (left) Proton-to-pion ratio in Pb–Pb collisions in several
centrality bins compared with pp results at 2.76 TeV. (right) Proton-to-pion
ratio in central Pb–Pb collisions compared with models.
ALICE has measured the production yields of primary charged pions, kaon and
(anti)protons in a wide momentum range and in several colliding systems. The
measurements have been performed in proton-proton collisions at several
centre-of-mass energies ($\sqrt{s}$ = 0.9, 2.76 and 7 TeV) and in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV as a function of collision
centrality.
In Pb–Pb collisions the transverse momentum $p_{\rm T}$ distributions and
yields are compared to previous results at RHIC and expectations from
hydrodynamic and thermal model. The results obtained for central Pb–Pb
collisions are shown in Figure 4 and 5. The spectral shapes are harder than
those observed at RHIC, indicating an increase of the radial flow velocity
with the centre-of-mass energy. The radial flow at the LHC is found to be
about 10% higher than at RHIC energy. The hydrodynamic models shown in Figure
4 give for central collisions a fair description of the data, describing the
experimental spectra within ∼20%. This supports a hydrodynamic interpretation
of the transverse momentum distribution in central collisions at the LHC.
Further details of this anaysis and the models used to compare with the data
can be found in [5].
While the K/$\pi$ integrated production ratio is observed to be in line with
lower energy measurements and predictions from the thermal model, both the
p/$\pi$ and the $\Lambda/\pi$ ratios are lower than those at RHIC and
significantly lower (a factor $\sim$ 1.5 and 1.35, respectively) than
predictions. A possible explanation of these deviations from the thermal-model
predictions may be re-interactions in the hadronic phase due to large cross
sections for antibaryon-baryon annihilation [6, 7, 8].
The $p_{\rm T}$-dependent yield of charged kaons and protons normalized to
charged pions are shown in Figure 6 for pp collisions at $\sqrt{s}$ = 2.76 and
7 TeV where they are compared with theoretical model predictions. Within the
experimental uncertainties no energy dependence is observed in the data. The
observed production ratios are not reproduced by NLO calculations [9]. PYTHIA
Monte Carlo generator [10] underpredicts the proton-to-pion ratio at
intermediate $p_{\rm T}$.
Pion, kaon and (anti)proton production in Pb–Pb collisions have been compared
to that in pp collisions and all show a suppression pattern which is similar
to that of inclusive charged hadrons at high momenta ($p_{\rm T}$ above
$\simeq$ 10 GeV/$c$), as shown in Figure 7 and 8 [11]. This seems to suggest
that the dense medium formed in Pb–Pb collisions does not affect the
fragmentation. A similar conclusion can be drawn from the proton-to-pion ratio
measured in Pb–Pb collisions (Figure 9 and 10): for intermediate momenta (3–7
GeV/$c$) it exibits a relatively strong enhancement, by a factor of 3 compared
to that in pp collisions at $p_{\rm T}\approx$ 3 GeV/$c$ and returns to the
value in pp collisions at higher momenta ($p_{\rm T}$ above $\simeq$ 10
GeV/$c$) [11]. A similar observation was also reported for the
$\rm\Lambda/K^{0}_{s}$ ratio and possible explanations have been proposed
which include particle production via quark recombination [12].
## 4 Transverse momentum distribution in proton-lead collisions
Particle production in proton-lead (p–Pb) collisions allows one to study and
understand QCD at low parton fractional momentum $x$ and high gluon density.
It is moreover expected to be sensitive to nuclear effects in the initial
state. For this reason p–Pb measurements provide an essential reference tool
to discriminate between initial and final state effects and they are crucial
for the studies and the understanding of deconfined matter created in nucleus-
nucleus collisions.
Figure 11: The nuclear modification factor of charged particles as a function
of transverse momentum in NSD p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV compared to measurements in central (0-5%) and peripheral (70-80%) Pb–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.
The measurement of the transverse momentum $p_{\rm T}$ distributions of
charged particle in p–Pb collisions were reported already in [13]. It was
previously shown that the production of charged hadrons in central Pb–Pb
collisions at the LHC is strongly suppressed [14, 15]. The suppression remains
substantial up to 100 GeV/$c$ and is also seen in reconstructed jets [16].
Proton-lead collisions provide a control experiment to clearly establish
whether the initial state of the colliding nuclei plays a role in the observed
high-$p_{\rm T}$ hadron production in Pb–Pb collisions. In order to quantify
nuclear effects, the $p_{\rm T}$-differential yield relative to the proton-
proton reference, the so-called nuclear modification factor, is calculated.
The nuclear modification factor is expected to be unity for hard processes
which exhibit binary collision scaling. This has been recently confirmed in
Pb–Pb collisions at the LHC by the measurements of direct photon [17], Z0 [18]
and W± [19], observables which are not affected by hot QCD matter. In Figure
11 the measurement of the nuclear modification factor in p–Pb collisions RpPb
is compared to that in central (0-5% centrality) and peripheral (70–80%) Pb–Pb
collisions RPbPb. RpPb is observed to be consistent with unity for transverse
momenta higher than about 2 GeV/$c$. This important measurement demonstrates
that the strong suppression observed in central Pb–Pb collisions at the LHC is
not due to an initial-state effect, but it is a final state effect related to
the hot matter created in high-energy heavy-ion reactions.
## 5 Summary and conclusions
ALICE has obtained so far a wealth of physics results both from the analysis
of proton-proton collision data and from the first two LHC heavy-ion runs.
The transverse momentum spectra of $\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$
have been measured with ALICE in several colliding systems and energies at the
LHC demonstrating the excellent PID capabilities of the experiment. Data in pp
collisions show no evident $\sqrt{s}$ dependence in hadron production ratios.
In Pb–Pb collisions $\bar{p}/\pi^{-}$ integrated ratio is significantly lower
than statistical model predictions with a chemical freeze-out temperature
$T_{ch}\simeq 160-170$ MeV. The average transverse momenta and the transverse
momentum spectra indicate a $\sim$10% stronger radial flow than at RHIC
energies.
In the intermediate transverse momentum $p_{\rm T}$ region, an enhancement of
the baryon-to-meson ratio is observed. The maximum of the ratio is shifted to
higher $p_{\rm T}$ with respect to RHIC measurements. The results of the
measurements of charged pion, kaon, protons and antiproton production at high
$p_{\rm T}$ were also presented. The nuclear modification factors for these
species are similar in magnitude, suggesting that the medium does not
significantly affect fragmentation.
First results from a short pilot run with proton-lead beams have been also
reported. The measurement of the charged-particle transverse momentum $p_{\rm
T}$ spectra and nuclear modification factor in p–Pb collisions at
$\sqrt{s_{\rm NN}}$ = 5.02 TeV, covering 0.5 $<p_{\rm T}<$ 20 GeV/$c$, show a
nuclear modification factor consistent with unity for $p_{\rm T}>$ 2 GeV/c.
This measurement indicates that the strong suppression of hadron production at
high $p_{\rm T}$ observed at the LHC in Pb–Pb collisions is not due to an
initial-state effect, but is the fingerprint of jet quenching in hot QCD
matter.
## References
## References
* [1] K. Aamodt et al. [ALICE Collaboration], “The ALICE experiment at the CERN LHC”, JINST 3 (2008) S08002.
* [2] G. Alessandro, (Ed.) et al. [ALICE Collaboration], “ALICE: Physics performance report, volume II”, J. Phys. G 32 (2006) 1295.
* [3] B. Abelev et al. [ALICE Collaboration], “Pseudorapidity density of charged particles $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV”, arXiv:1210.3615 [nucl-ex].
* [4] B. Alver, M. Baker, C. Loizides and P. Steinberg, “The PHOBOS Glauber Monte Carlo”, arXiv:0805.4411 [nucl-ex].
* [5] B. Abelev et al. [ALICE Collaboration], “Pion, Kaon, and Proton Production in Central Pb–Pb Collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Rev. Lett. 109 (2012) 252301 [arXiv:1208.1974 [hep-ex]].
* [6] J. Steinheimer, J. Aichelin and M. Bleicher, “Non-thermal $p/\pi$ ratio at LHC as a consequence of hadronic final state interactions”, arXiv:1203.5302 [nucl-th].
* [7] F. Becattini, M. Bleicher, T. Kollegger, M. Mitrovski, T. Schuster and R. Stock, “Hadronization and Hadronic Freeze-Out in Relativistic Nuclear Collisions”, Phys. Rev. C 85 (2012) 044921 [arXiv:1201.6349 [nucl-th]].
* [8] Y. Pan and S. Pratt, “Baryon Annihilation in Heavy Ion Collisions”, arXiv:1210.1577 [nucl-th].
* [9] R. Sassot, P. Zurita and M. Stratmann, “Inclusive Hadron Production in the CERN-LHC Era”, Phys. Rev. D 82 (2010) 074011 [arXiv:1008.0540 [hep-ph]].
* [10] P. Z. Skands, Phys. Rev. D 82 (2010) 074018 [arXiv:1005.3457 [hep-ph]].
* [11] A. Ortiz Velasquez [ALICE Collaboration], “Production of pions, kaons and protons at high $p_{T}$ in $\sqrt{s_{NN}}=2.76$ TeV Pb-Pb collisions”, arXiv:1210.6995 [hep-ex].
* [12] R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, “Hadronization in heavy ion collisions: Recombination and fragmentation of partons”, Phys. Rev. Lett. 90 (2003) 202303 [nucl-th/0301087].
* [13] B. Abelev et al. [ALICE Collaboration], “Transverse Momentum Distribution and Nuclear Modification Factor of Charged Particles in $p$-Pb Collisions at $\sqrt{s_{NN}}=5.02$ TeV”, arXiv:1210.4520 [nucl-ex].
* [14] K. Aamodt et al. [ALICE Collaboration], “Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb–Pb Collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Lett. B 696 (2011) 30 [arXiv:1012.1004 [nucl-ex]].
* [15] B. Abelev et al. [ALICE Collaboration], “Centrality Dependence of Charged Particle Production at Large Transverse Momentum in Pb–Pb Collisions at $\sqrt{s_{\rm{NN}}}=2.76$ TeV”, [arXiv:1208.2711 [hep-ex]].
* [16] G. Aad et al. [ATLAS Collaboration], “Measurement of the jet radius and transverse momentum dependence of inclusive jet suppression in lead-lead collisions at $\sqrt{s_{NN}}=2.76$ TeV with the ATLAS detector”, arXiv:1208.1967 [hep-ex].
* [17] S. Chatrchyan et al. [CMS Collaboration], “Measurement of isolated photon production in $pp$ and PbPb collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Lett. B 710 (2012) 256 [arXiv:1201.3093 [nucl-ex]].
* [18] S. Chatrchyan et al. [CMS Collaboration], “Study of Z boson production in PbPb collisions at nucleon-nucleon centre of mass energy = 2.76 TeV”, Phys. Rev. Lett. 106 (2011) 212301 [arXiv:1102.5435 [nucl-ex]].
* [19] S. Chatrchyan et al. [CMS Collaboration], “Study of $W$ boson production in PbPb and $pp$ collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Lett. B 715 (2012) 66 [arXiv:1205.6334 [nucl-ex]].
|
arxiv-papers
| 2013-04-03T10:10:33 |
2024-09-04T02:49:43.825827
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roberto Preghenella (for the ALICE Collaboration)",
"submitter": "Roberto Preghenella",
"url": "https://arxiv.org/abs/1304.0899"
}
|
1304.1000
|
# Passages in Graphs
W.M.P. van der Aalst Department of Mathematics and Computer Science,
Technische Universiteit Eindhoven, The Netherlands.
BPM Discipline, Queensland University of Technology, GPO Box 2434, Brisbane
QLD 4001, Australia.
WWW: www.vdaalst.com, E-mail: [email protected]
###### Abstract
Directed graphs can be partitioned in so-called _passages_. A passage $P$ is a
set of edges such that any two edges sharing the same initial vertex or
sharing the same terminal vertex are both inside $P$ or are both outside of
$P$. Passages were first identified in the context of process mining where
they are used to successfully decompose process discovery and conformance
checking problems. In this article, we examine the properties of passages. We
will show that passages are closed under set operators such as union,
intersection and difference. Moreover, any passage is composed of so-called
minimal passages. These properties can be exploited when decomposing graph-
based analysis and computation problems.
###### keywords:
Directed graphs , Process modeling , Decomposition
††journal:
## 1 Introduction
Recently, the notion of _passages_ was introduced in the context of process
mining [2]. There it was used to decompose process discovery and conformance
checking problems [1]. Any directed graph can be partitioned into a collection
of non-overlapping passages. Analysis can be done per passage and the results
can be combined easily, e.g., for conformance checking a process model can be
decomposed into process fragments using passages and traces in the event log
fit the overall model if and only if they fit all process fragments.
As shown in this article, passages have various elegant problems. Although the
notion of passages is very simple, we could not find this graph notion in
existing literature on (directed) graphs [3, 6]. Classical graph partitioning
approaches [7, 8] decompose the vertices of a graph rather than the edges,
i.e., the goal there is to decompose the graph in smaller components of
similar size that have few connecting edges. Some of these notions have been
extended to vertex-cut graph partitioning [5, 9]. However, these existing
notions are not applicable in our problem setting where components need to
behave synchronously and splits and joins cannot be partitioned. We use
passages to _decompose a graph into sets of edges such that all edges sharing
an initial vertex or terminal vertex are in the same set_. To the best of our
knowledge, the notion of passages has not been studied before. However, we
believe that this notion can be applied in various domains (other than process
mining). Therefore, we elaborate on the foundational properties of passages.
The remainder is organized as follows. In Section 2 we define the notion of
passages, provide alternative characterizations, and discuss elementary
properties. Section 3 shows that any graph can be partitioned into passages
and that any passage is composed of so-called minimal passages. Section 4
introduces passage graphs visualizing the relations between passages. Graphs
may be partitioned in different ways. Therefore, Section 5 discusses the
quality of passage partitionings. Section 6 concludes this article.
## 2 Defining Passages
Passages are defined on directed graphs, simply referred to as graphs.
###### Definition 1 (Graph)
A (directed) graph is a pair $G=(V,E)$ composed of a set of vertices $V$ and a
set of edges $E\subseteq V\times V$.
Figure 1: Graph $G_{1}$ with 9 vertices, 12 edges, and 32 passages.
A passage is a set of edges such that any two edges sharing the same initial
vertex (tail) or sharing the same terminal vertex (head) are both inside or
both outside of the passage. For example, $\\{(a,b),(a,c)\\}$ is a passage in
graph $G_{1}$ shown in Figure 1 because there are no other edges having $a$ as
initial vertex or $b$ or $c$ as terminal vertex.
###### Definition 2 (Passage)
Let $G=(V,E)$ be a graph. $P\subseteq E$ is a passage if for any $(x,y)\in P$
and $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq E$:
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P$. $\mathit{pas}(G)$ is the set
of all passages of $G$.
Figure 2 shows 7 of the 32 passages of graph $G_{1}$ shown in Figure 1.
$P_{2}=\\{(b,e),(b,f),(c,f),(c,d),(d,d),(d,f)\\}$ is a passage as there are no
other edges having $b$, $c$, or $d$ as initial vertex or $d$, $e$, or $f$ as
terminal vertex. Figure 2 does not show the two trivial passages: $\emptyset$
(no edges) and $E$ (all edges).
Figure 2: Seven example passages of graph $G_{1}$ shown in Figure 1.
###### Lemma 1 (Trivial Passages)
Let $G=(V,E)$ be a graph. The _empty passage_ $\emptyset$ and the _full
passage_ $E$ are trivial passages of $G$. Formally:
$\\{\emptyset,E\\}\subseteq\mathit{pas}(G)$ for any $G$.
Some of the passages in Figure 2 are overlapping: $P_{6}=P_{3}\cup P_{4}\cup
P_{5}$ and $P_{7}=P_{1}\cup P_{3}\cup P_{4}$. To combine passages into new
passages and to reason about the properties of passages we define the
following notations.
###### Definition 3 (Passage Operators)
Let $G=(V,E)$ be a graph with $P,P_{1},P_{2}\subseteq E$. $P_{1}\cup P_{2}$,
$P_{1}\cap P_{2}$, $P_{1}\setminus P_{2}$, $P_{1}=P_{2}$, $P_{1}\neq P_{2}$,
$P_{1}\subseteq P_{2}$, and $P_{1}\subset P_{2}$ are defined as usual.
$\pi_{1}(P)=\\{x\mid(x,y)\in P\\}$ are the initial vertices of $P$,
$\pi_{2}(P)=\\{y\mid(x,y)\in P\\}$ are the terminal vertices of $P$,
$P_{1}\\#P_{2}$ if and only if $P_{1}\cap P_{2}=\emptyset$,
$P_{1}\triangleright P_{2}$ if and only if
$\pi_{2}(P_{1})\cap\pi_{1}(P_{2})\neq\emptyset$.
Note that $d$ is both an initial and terminal vertex of $P_{2}$ in Figure 2:
$\pi_{1}(P_{2})=\\{b,c,d\\}$ and $\pi_{2}(P_{2})=\\{d,e,f\\}$. $P_{5}\\#P_{7}$
because $P_{5}\cap P_{7}=\emptyset$. $P_{4}\triangleright P_{5}$ because
$\pi_{2}(P_{4})\cap\pi_{1}(P_{5})=\\{h\\}\neq\emptyset$.
The union, intersection and difference of passages yield passages. For
example, $P_{7}=P_{1}\cup P_{3}\cup P_{4}$ is a passage composed of three
smaller passages. $P_{5}=P_{6}\setminus P_{7}$ and $P_{6}\cap P_{7}=P_{3}\cup
P_{4}$ are passages.
###### Lemma 2 (Passages Are Closed under $\cup$, $\cap$ and $\setminus$)
Let $G=(V,E)$ be a graph. If $P_{1},P_{2}\in\mathit{pas}(G)$ are two passages,
then $P_{1}\cup P_{2}$, $P_{1}\cap P_{2}$, and $P_{1}\setminus P_{2}$ are also
passages.
###### Proof 1
Let $P_{1},P_{2}\in\mathit{pas}(G)$, $(x,y)\in P_{1}\cup P_{2}$, and
$\\{(x,y^{\prime}),\allowbreak(x^{\prime},y)\\}\subseteq E$. We need to show
that $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}\cup P_{2}$. If
$(x,y)\in P_{1}$, then $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq
P_{1}\subseteq P_{1}\cup P_{2}$. If $(x,y)\in P_{2}$, then
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{2}\subseteq P_{1}\cup P_{2}$.
Let $P_{1},P_{2}\in\mathit{pas}(G)$, $(x,y)\in P_{1}\cap P_{2}$, and
$\\{(x,y^{\prime}),\allowbreak(x^{\prime},y)\\}\subseteq E$. We need to show
that $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}\cap P_{2}$. Since
$(x,y)\in P_{1}$, $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}$. Since
$(x,y)\in P_{2}$, $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{2}$. Hence,
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}\cap P_{2}$.
Let $P_{1},P_{2}\in\mathit{pas}(G)$, $(x,y)\in P_{1}\setminus P_{2}$, and
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq E$. We need to show that
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}\setminus P_{2}$. Since
$(x,y)\in P_{1}$, $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq P_{1}$. Since
$(x,y)\not\in P_{2}$, $\\{(x,y^{\prime}),(x^{\prime},y)\\}\cap
P_{2}=\emptyset$. Hence, $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq
P_{1}\setminus P_{2}$.
A passage is fully characterized by both the set of initial vertices and the
set of terminal vertices. Therefore, the following properties hold.
###### Lemma 3 (Passage Properties)
Let $G=(V,E)$ be a graph. For any $P_{1},P_{2}\in\mathit{pas}(G)$:
* 1.
$\pi_{1}(P_{1})=\pi_{1}(P_{2})\ \Leftrightarrow\ P_{1}=P_{2}\ \Leftrightarrow\
\pi_{2}(P_{1})=\pi_{2}(P_{2})$,
* 2.
$P_{1}\\#P_{2}\ \Leftrightarrow\ \pi_{1}(P_{1})\cap\pi_{1}(P_{2})=\emptyset$,
and
* 3.
$P_{1}\\#P_{2}\ \Leftrightarrow\ \pi_{2}(P_{1})\cap\pi_{2}(P_{2})=\emptyset$.
###### Proof 2
$X=\pi_{1}(P)$ implies $P=\\{(x,y)\in E\mid x\in X\\}$ (definition of
passages). Hence, $\pi_{1}(P_{1})=\pi_{1}(P_{2})\ \Rightarrow\ P_{1}=P_{2}$
(because a passage $P$ is fully determined by $\pi_{1}(P)$). The other
direction ($\Leftarrow$) holds trivially. A passage $P$ is also fully
determined by $\pi_{2}(P)$. Hence, $\pi_{2}(P_{1})=\pi_{2}(P_{2})\
\Rightarrow\ P_{1}=P_{2}$. Again the other direction ($\Leftarrow$) holds
trivially.
The second property follows from the observation that two passages share an
edge if and only if the initial vertices overlap. If two passages share an
edge $(x,y)$, they also share initial vertex $x$. If two passage share initial
vertex $x$, then they also share some edges $(x,y)$.
Due to symmetry, the same holds for the third property.
The following lemma shows that a passage can be viewed as a fixpoint:
$P=((\pi_{1}(P)\times V)\cup(V\times\pi_{2}(P)))\cap E$. This property will be
used to construct minimal passages.
###### Lemma 4 (Another Passage Characterization)
Let $G=(V,E)$ be a graph. $P\subseteq E$ is a passage if and only if
$P=((\pi_{1}(P)\times V)\cup(V\times\pi_{2}(P)))\cap E$.
###### Proof 3
Suppose $P$ is a passage: it is fully characterized by $\pi_{1}(P)$ and
$\pi_{2}(P)$. Take all edges leaving from $\pi_{1}(P)$: $P=(\pi_{1}(P)\times
V)\cap E$. Take all edges entering $\pi_{2}(P)$: $P=(V\times\pi_{2}(P))\cap
E$. Hence, $P=(\pi_{1}(P)\times V)\cap E=(V\times\pi_{2}(P))\cap E$. So,
$P=((\pi_{1}(P)\times V)\cup(V\times\pi_{2}(P)))\cap E$.
Suppose $P=((\pi_{1}(P)\times V)\cup(V\times\pi_{2}(P)))\cap E$. Let $(x,y)\in
P$ and $\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq E$. Clearly,
$(x,y^{\prime})\in(\pi_{1}(P)\times V)\cap E$ and
$(x^{\prime},y)\in(V\times\pi_{2}(P))\cap E$. Hence,
$\\{(x,y^{\prime}),(x^{\prime},y)\\}\subseteq((\pi_{1}(P)\times
V)\cup(V\times\pi_{2}(P)))\cap E=P$.
## 3 Passage Partitioning
After introducing the notion of passages and their properties, we now show
that graph can be _partitioned_ using passages. For example, the set of
passages $\\{P_{1},P_{2},P_{3},P_{4},\allowbreak P_{5}\\}$ in Figure 2
partitions $G_{1}$. Other passage partitionings for graph $G_{1}$ are
$\\{P_{2},P_{5},P_{7}\\}$ and $\\{P_{1},P_{2},P_{6}\\}$.
###### Definition 4 (Passage Partitioning)
Let $G=(V,E)$ be a graph. ${\cal
P}=\\{P_{1},P_{2},\ldots,P_{n}\\}\subseteq\mathit{pas}(G)\setminus\\{\emptyset\\}$
is a _passage partitioning_ if and only if $\bigcup{\cal P}=E$ and
$\forall_{1\leq i<j\leq n}\ \allowbreak P_{i}\\#P_{j}$.
Any passage partitioning ${\cal P}$ defines an equivalence relation on the set
of edges. For $e_{1},e_{2}\in E$, $e_{1}\sim_{{\cal P}}e_{2}$ if there exists
a $P\in{\cal P}$ with $\\{e_{1},e_{2}\\}\subseteq P$.
###### Lemma 5 (Equivalence Relation)
Let $G\allowbreak=(V,E)$ be a graph with passage partitioning ${\cal P}$.
$\sim_{\cal P}$ defines an equivalence relation.
###### Proof 4
We need to prove that $\sim_{\cal P}$ is reflexive, symmetric, and transitive.
Let $e,e^{\prime},e^{\prime\prime}\in E$. Clearly, $e\sim_{\cal P}e$ because
$e\in E=\bigcup{\cal P}$ (${\cal P}$ is a passage partitioning). Hence, there
must be a $P\in{\cal P}$ with $e\in{\cal P}$ (reflexivity). If $e\sim_{\cal
P}e^{\prime}$, then $e^{\prime}\sim_{\cal P}e$ (symmetry). If $e\sim_{\cal
P}e^{\prime}$ and $e^{\prime}\sim_{\cal P}e^{\prime\prime}$, then there must
be a $P\in{\cal P}$ such that $\\{e_{1},e_{2},e_{3}\\}\subseteq P$. Hence,
$e\sim_{\cal P}e^{\prime\prime}$ (transitivity).
Any graph has a passage partitioning, e.g., $\\{E\\}$ is always a valid
passage partitioning. However, to decompose analysis one is typically
interested in partitioning the graph in as many passages as possible.
Therefore, we introduce the notion of a _minimal_ passage. Passage $P_{6}$ in
Figure 2 is not minimal because it contains smaller non-empty passages:
$P_{3}$, $P_{4}$, and $P_{5}$. Passage $P_{7}$ is also not minimal. Only the
first five passages in Figure 2 ($P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$ and
$P_{5}$) are minimal.
###### Definition 5 (Minimal Passages)
Let $G=(V,E)$ be a graph and $P\in\mathit{pas}(G)$ a passage. $P$ is minimal
if and only if there is no non-empty passage
$P^{\prime}\in\mathit{pas}(G)\setminus\\{\emptyset\\}$ such that
$P^{\prime}\subset P$. $\mathit{pas}_{\mathit{min}}(G)$ is the set of all non-
empty minimal passages.
Two different minimal passages cannot share the same edge. Otherwise, the
difference between both passages would yield a smaller non-empty minimal
passage. Hence, an edge can be used to uniquely identify a minimal passage.
The fixpoint characterization given in Lemma 4 suggests an iterative procedure
that starts with a single edge. In each iteration edges are added that must be
part of the same minimal passage. As shown this procedure can be used to
determine all minimal passages.
###### Lemma 6 (Constructing Minimal Passages)
Let $G\allowbreak=(V,E)$ be a graph. For any $(x,y)\in E$, there exists
precisely one minimal passage $P_{(x,y)}\in\mathit{pas}_{\mathit{min}}(G)$
such that $(x,y)\in P_{(x,y)}$.
###### Proof 5
Initially, set $P:=\\{(x,y)\\}$. Extend $P$ as follows: $P:=((\pi_{1}(P)\times
V)\cup(V\times\pi_{2}(P)))\cap E$. Repeat extending $P$ until it does not
change anymore. Finally, return $P_{(x,y)}=P$. The procedure ends because the
number of edges is finite. If $P=((\pi_{1}(P)\times
V)\cup(V\times\pi_{2}(P)))\cap E$ (i.e., $P$ does not change anymore), then
$P$ is indeed a passage (see Lemma 4). $P$ is minimal because no unnecessary
edges are added: if $(x,y)\in P$, then any edge starting in $x$ or ending in
$y$ has to be included.
To prove the latter one can also consider all passages ${\cal
P}=\\{P_{1},P_{2},\ldots,P_{n}\\}$ that contain $(x,y)$. The intersection of
all such passages $\bigcap{\cal P}$ contains edge $(x,y)$ and is again a
passage because of Lemma 2. Hence, $\bigcap{\cal P}=P_{(x,y)}$.
The construction described in the proof can be used compute all minimal
passages and is quadratic in the number of edges.
$\mathit{pas}_{\mathit{min}}(G_{1})=\\{P_{1},P_{2},P_{3},P_{4},P_{5}\\}$ for
the graph shown in Figure 1. This is also a passage partitioning. (Note that
the construction in Lemma 6 is similar to the computation of so-called
clusters in a Petri net [4].)
###### Theorem 1 (Minimal Passage Partitioning)
Let $G\allowbreak=\allowbreak(V,E)$ be a graph.
$\mathit{pas}_{\mathit{min}}(G)$ is a passage partitioning.
###### Proof 6
Let $\mathit{pas}_{\mathit{min}}(G)=\\{P_{1},P_{2},\ldots,P_{n}\\}$. Clearly,
$\\{P_{1},\allowbreak
P_{2},\ldots,P_{n}\\}\subseteq\mathit{pas}(G)\setminus\\{\emptyset\\}$,
$\bigcup_{1\leq i\leq n}P_{i}=E$ and $\forall_{1\leq i<j\leq n}\ \allowbreak
P_{i}\\#P_{j}$ (follows from Lemma 6).
Figure 3 shows a larger graph $G_{2}=(V_{2},E_{2})$ with
$V_{2}=\\{a,b,\ldots,o\\}$ and $E_{2}=\\{(a,b),(b,e),\ldots,(n,o)\\}$. The
figure also shows six passages. These form a passage partitioning. Each edge
has a number that refers to the corresponding passage, e.g., edge $(h,k)$ is
part of passage $P_{4}$. Passages are shown as rectangles and vertices are put
on the boundaries of at most two passages. Vertex $a$ in Figure 3 is on the
boundary of $P_{1}$ because $(a,b)\in P_{1}$. Vertex $b$ is on the boundary of
$P_{1}$ and $P_{2}$ because $(a,b)\in P_{1}$ and $(b,e)\in P_{2}$. $G_{2}$ has
no isolated vertices, so all vertices are on the boundary of at least one
passage.
Figure 3: A passage partitioning for graph $G_{2}$.
The passage partitioning shown in Figure 3 is not composed of minimal passages
as is indicated by the two dashed lines. Both $P_{1}$ and $P_{6}$ are not
minimal. $P_{1}$ can be split into minimal passages $P_{1a}=\\{(a,b)\\}$ and
$P_{1b}=\\{(c,d)\\}$. $P_{6}$ can be split into minimal passages
$P_{6a}=\\{(m,l)\\}$ and $P_{6b}=\\{(n,o),(n,m)\\}$. In fact, as shown next,
any passage can be decomposed into minimal non-empty passages.
###### Theorem 2 (Composing Minimal Passages)
Let $G=(V,E)$ be a graph. For any passage $P\in\mathit{pas}(G)$ there is a set
of minimal non-empty passages
$\\{P_{1},P_{2},\ldots,P_{n}\\}\subseteq\mathit{pas}_{\mathit{min}}(G)$ such
that $\bigcup_{1\leq i\leq n}P_{i}=P$ and $\forall_{1\leq i<j\leq n}\
P_{i}\\#P_{j}$.
###### Proof 7
Let $\\{P_{1},P_{2},\ldots,P_{n}\\}=\\{P_{(x,y)}\mid(x,y)\in P\\}$. These
passages are minimal (Lemma 6) and also cover all edges in $P$. Moreover, two
different minimal passages cannot share edges.
A graph without edges has only one passage. Hence, if $E=\emptyset$, then
$\mathit{pas}(G)=\\{\emptyset\\}$ (just one passage),
$\mathit{pas}_{\mathit{min}}(G)=\emptyset$ (no minimal non-empty passages),
and $\emptyset$ is the only passage partitioning. If $E\neq\emptyset$, then
there is always a trivial singleton passage partitioning $\\{E\\}$ and a
minimal passage partitioning $\mathit{pas}_{\mathit{min}}(G)$ (but there may
be many more).
###### Lemma 7 (Number of Passages)
Let $G=(V,E)$ be a graph with $k=|\mathit{pas}_{\mathit{min}}(G)|$ minimal
non-empty passages. There are $2^{k}$ passages and $B_{k}$ passage
partitionings.111$B_{k}$ is the $k$-th Bell number (the number of partitions
of a set of size $k$), e.g., $B_{3}=5$, $B_{4}=15$, and $B_{5}=52$ [10]. For
any passage partitioning $\\{P_{1},P_{2},\ldots,P_{n}\\}$ of $G$: $n\leq
k\leq|E|$.
###### Proof 8
Any passage can be composed of minimal non-empty passages. Hence, there are
$2^{k}$ passages. $B_{k}$ is the number of partitions of a set with $k$
members, thus corresponding to the number of passage partitionings.
If there are no edges, there are no minimal non-empty passages ($k=0$) and
there is only one possible passage partitioning: $\emptyset$. Hence, $n=0$. If
$E\neq\emptyset$, then $\mathit{pas}_{\mathit{min}}(G)$ is the most refined
passage partitioning. There are at most $|E|$ minimal non-empty passages as
they cannot share edges. Hence, $n\leq k\leq|E|$. Note that $n\geq 1$ if
$E\neq\emptyset$.
Graph $G_{2}$ in Figure 3 has $2^{8}=256$ passages and $B_{8}=4140$ passage
partitionings.
## 4 Passage Graphs
Passage partitionings can be visualized using _passage graphs_. To relate
passages, we first define the input/output vertices of a passage.
###### Definition 6 (Input and Output Vertices)
Let $G=(V,E)$ be a graph and $P\in\mathit{pas}(G)$ a passage.
$\mathit{in}(P)=\pi_{1}(P)\setminus\pi_{2}(P)$ are the input vertices of $P$,
$\mathit{out}(P)=\pi_{2}(P)\setminus\pi_{1}(P)$ are the output vertices of
$P$, and $\mathit{io}(P)=\pi_{1}(P)\cap\pi_{2}(P)$ are the input/output
vertices of $P$.
Note the difference between input, output, and input/output vertices on the
one hand and the initial and terminal vertices of a passage on the other hand.
Given a passage partitioning, there are five types of vertices: isolated
vertices, input vertices, output vertices, connecting vertices, and local
vertices.
###### Definition 7 (Five Types of Vertices)
Let $G=(V,E)$ be a graph and ${\cal P}=\\{P_{1},P_{2},\ldots,P_{n}\\}$ a
passage partitioning. $V_{\mathit{iso}}=V\setminus(\pi_{1}(E)\cup\pi_{2}(E))$
are the isolated vertices of ${\cal P}$,
$V_{\mathit{in}}=\pi_{1}(E)\setminus\pi_{2}(E)$ are the input vertices of
${\cal P}$, $V_{\mathit{out}}=\pi_{2}(E)\setminus\pi_{1}(E)$ are the output
vertices of ${\cal P}$, $V_{\mathit{con}}=\bigcup_{i\neq
j}\pi_{2}(P_{i})\cap\pi_{1}(P_{j})$ are the connecting vertices of ${\cal P}$,
$V_{\mathit{loc}}=\bigcup_{i}\pi_{1}(P_{i})\cap\pi_{2}(P_{i})$ are the local
vertices of ${\cal P}$.
Note that $V=V_{\mathit{iso}}\cup V_{\mathit{in}}\cup V_{\mathit{out}}\cup
V_{\mathit{con}}\cup V_{\mathit{loc}}$ and the five sets are pairwise
disjoint, i.e., they partition $V$. In the passage partitioning shown in
Figure 3: $a$ is the only input vertex, $k$ and $o$ are output vertices, and
$e$, $i$ and $m$ are local vertices. All other vertices are connecting
vertices.
###### Definition 8 (Passage Graph)
Let $G=(V,E)$ be a graph and ${\cal P}=\\{P_{1},P_{2},\ldots,P_{n}\\}$ a
passage partitioning. $({\cal P},\\{(P,P^{\prime})\in{\cal P}\times{\cal
P}\mid P\triangleright P^{\prime}\\})$ is corresponding passage graph .
Figure 4 shows a passage graph. The graph shows the relationships among
passages and can be used to partition the vertices $V$ into
$V_{\mathit{iso}}\cup V_{\mathit{in}}\cup V_{\mathit{out}}\cup
V_{\mathit{con}}\cup V_{\mathit{loc}}$.
Figure 4: Passage graph based on the passage partitioning shown in Figure 3.
## 5 Quality of a Passage Partitioning
Passages can be used to decompose analysis problems (e.g., conformance
checking and process discovery [2]). In the extreme case, there is just one
minimal passage covering all edges in the graph. In this case, the graph
cannot be decomposed. Ideally, we would like to use a passage partitioning
${\cal P}=\\{P_{1},P_{2},\ldots,P_{n}\\}$ that is accurate and that has only
small passages. One could aim at as many passages as possible in order to
minimize the average size per passage: $\mathit{av}({\cal P})=\frac{|E|}{n}$
per passage. One can also aim at minimizing the size of the biggest passage
(i.e., $\mathit{big}({\cal P})=\mathit{max}_{1\leq i\leq n}\ |P_{i}|$) because
the biggest passage often takes most of the computation time.
To have smaller passages, one may need to abstract from edges that are less
important. To reason about such “approximate passages” we define the input as
$G_{\pi}=(V,\pi)$ with vertices $V$ and weight function $\pi\in(V\times
V)\rightarrow[-1,1]$. Given two vertices $x,y\in V$: $\pi(x,y)$ is “weight” of
the possible edge connecting $x$ and $y$. If $\pi(x,y)>0$, then it is more
likely than unlikely that there is an edge connecting $x$ and $y$. If
$\pi(x,y)<0$, then it is more unlikely than likely that there is an edge
connecting $x$ and $y$. One can view $\frac{\pi(x,y)+1}{2}$ as the
“probability” that there is such an edge. The penalty for leaving out an edge
$(x,y)$ with $\pi(x,y)=0.99$ is much bigger than leaving out an edge
$(x^{\prime},y^{\prime})$ with $\pi(x^{\prime},y^{\prime})=0.15$. The accuracy
of a passage partitioning ${\cal P}=\\{P_{1},P_{2},\ldots,P_{n}\\}$ with
$E=\cup_{1\leq i\leq n}\ P_{i}$ for input $G_{\pi}=(V,\pi)$ can be defined as
$\mathit{acc}({\cal P})=\frac{\sum_{(x,y)\in
E}\pi(x,y)}{\mathit{max}_{E^{\prime}\subseteq V\times V}\sum_{(x,y)\in
E^{\prime}}\pi(x,y)}$. If $\mathit{acc}({\cal P})=1$, then all edges having a
positive weight are included in some passage and none of edges having a
negative weight are included. Often there is a trade-off between higher
accuracy and smaller passages, e.g., discarding a potential edge having a low
weight may allow for splitting a large passage into two smaller ones. Just
like in traditional graph partitioning [7, 8], one can look for the passage
partitioning that maximizes $\mathit{acc}({\cal P})$ provided that
$\mathit{av}({\cal P})\leq\tau_{\mathit{av}}$ and/or $\mathit{big}({\cal
P})\leq\tau_{\mathit{big}}$, where $\tau_{\mathit{av}}$ and
$\tau_{\mathit{big}}$ are suitably chosen thresholds. Whether one needs to
resort to approximate passages depends on the domain, e.g., when discovering
process models from event logs causalities tend to be uncertain and including
all potential causalities results in Spaghetti-like graphs [1], therefore
approximate passages are quite useful.
## 6 Conclusion
In this article we introduced the new notion of passages. Passages have been
shown to be useful in the domain of process mining. Given their properties and
possible applications in other domains, we examined passages in detail.
Passages are closed under the standard set operators (union, difference, and
intersection). A graph can be partitioned into components based on its minimal
passages and any passage is composed of minimal passages. The theory of
passages can be extended to deal with approximate passages. We plan to examine
these in the context of process mining, but are also looking for applications
of passage partitionings in other domains (e.g., distributed enactment and
verification).
## References
* [1] W.M.P. van der Aalst. Process Mining: Discovery, Conformance and Enhancement of Business Processes. Springer-Verlag, Berlin, 2011.
* [2] W.M.P. van der Aalst. Decomposing Process Mining Problems Using Passages. In S. Haddad and L. Pomello, editors, Applications and Theory of Petri Nets 2012, volume 7347 of Lecture Notes in Computer Science, pages 72–91. Springer-Verlag, Berlin, 2012.
* [3] J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications (Second Edition). Springer-Verlag, Berlin, 2009.
* [4] J. Desel and J. Esparza. Free Choice Petri Nets, volume 40 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 1995.
* [5] U. Feige, M. Hajiaghayi, and J. Lee. Improved Approximation Algorithms for Minimum-Weight Vertex Separators. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 563–572. ACM, New York, 2005.
* [6] J.L. Gross and J. Yellen. Handbook of Graph Theory. CRC Press, 2004.
* [7] G. Karpis and V. Kumar. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1998.
* [8] B.W. Kernighan and S. Lin. An Efficient Heuristic Procedure for Partitioning Graphs. The Bell Systems Technical Journal, 49(2), 1970.
* [9] M. Kim and K. Candan. SBV-Cut: Vertex-Cut Based Graph Partitioning Using Structural Balance Vertices. Data and Knowledge Engineering, 72:285–303, 2012.
* [10] N.J.A. Sloane. Bell Numbers. In Encyclopedia of Mathematics. Kluwer Academic Publishers, 2002\. http://www.encyclopediaofmath.org/index.php?title=Bell_numbers&oldid=14335.
|
arxiv-papers
| 2013-04-03T16:07:46 |
2024-09-04T02:49:43.837324
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wil van der Aalst",
"submitter": "Wil van der Aalst",
"url": "https://arxiv.org/abs/1304.1000"
}
|
1304.1024
|
# Silicon spin chains at finite temperature: dynamics of Si(553)-Au
Steven C. Erwin [email protected] Center for Computational Materials
Science, Naval Research Laboratory, Washington, DC 20375 P.C. Snijders
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37830
###### Abstract
When gold is deposited on Si(553), the surface self-assembles to form a
periodic array of steps with nearly perfect structural order. In scanning
tunneling microscopy these steps resemble quasi-one-dimensional atomic chains.
At temperatures below $\sim$50 K the chains develop a tripled periodicity. We
recently predicted, on the basis of density-functional theory calculations at
$T=0$, that this tripled periodicity arises from the complete polarization of
the electron spin on every third silicon atom along the step; in the ground
state these linear chains of silicon spins are antiferromagnetically ordered.
Here we explore, using ab-initio molecular dynamics and kinetic Monte Carlo
simulations, the behavior of silicon spin chains on Si(553)-Au at finite
temperature. Thermodynamic phase transitions at $T>0$ in one-dimensional
systems are prohibited by the Mermin-Wagner theorem. Nevertheless we find that
a surprisingly sharp onset occurs upon cooling—at about 30 K for perfect
surfaces and at higher temperature for surfaces with defects—to a well-ordered
phase with tripled periodicity, in good agreement with experiment.
## I Introduction
Linear atomic chains of metal atoms on semiconductor surfaces offer, in
principle, the physical realization of phenomena predicted theoretically for
one-dimensional model systems. In practice, however, unanticipated
interactions can often complicate the picture and lead to behavior not easily
explained by simple models. In this article we demonstrate theoretically how
the complex interactions among polarized electron spins in silicon surface
states determine the observed behavior of a well-studied atomic chain system,
Si(553)-Au, over a wide range of temperatures. The methods developed here and
the resulting predictions—which are qualitatively and quantitatively
consistent with experimental observations—are also likely to apply more
broadly to other vicinal Si/Au chain systems, such as Si(557)-Au.
The Si(553)-Au surface was first investigated in Ref. Crain _et al._ , 2003,
an experimental study which established that the electronic band dispersion
and fermi surface were indeed those of a nearly one-dimensional metal. Since
then, numerous lines of research have emerged. Efforts to determine the basic
atomic structure of the surface have been based on data from diffraction
experimentsGhose _et al._ (2005); Takayama _et al._ (2009); Voegeli _et
al._ (2010) and on the results of theoretical total-energy
calculations.Riikonen and Sanchez-Portal (2005, 2006, 2008); Krawiec (2010)
These were greatly aided by the first definitive determination of the coverage
of Au atoms on Si(553)-Au.Barke _et al._ (2009) Other investigations have
explored the properties of finite-length chainsCrain and Pierce (2005); Crain
_et al._ (2006) as well as various native defectsOkino _et al._ (2007a) and
foreign adsorbates Ryang _et al._ (2007); Okino _et al._ (2007b); Ahn _et
al._ (2008); Kang _et al._ (2009a, a, b); Nita _et al._ (2011); Krawiec and
Jalochowski (2013) on the nominally clean Si(553)-Au surface.
One particularly interesting line of research has focused on the collective
behavior in Si(553)-Au that emerges at low temperature. Ideal one-dimensional
metals with partially filled bands exhibit a broken symmetry at low
temperature, namely a charge-density wave arising from the Peierls
instability. Indeed, broken symmetries in Si(553)-Au were observed using
scanning tunneling microscopy (STM) in Refs. Ahn _et al._ , 2005 and Snijders
_et al._ , 2006. Images acquired at room temperature showed alternating bright
and dim rows with unit periodicity $a_{0}$ along the rows. Below $\sim$50 K
these rows separately developed higher-order periodicity: a tripled period
(3$a_{0}$) along the bright rows and a doubled period (2$a_{0}$) along the dim
rows. Subsequent review articles have discussed possible explanations for
these observations.Snijders and Weitering (2010); Hasegawa (2010)
Notwithstanding the fact that Peierls instabilities lead to higher-order
periodicity, a completely different theoretical explanation for the coexisting
triple and double periodicities in Si(553)-Au was proposed in Ref. Erwin and
Himpsel, 2010. The key idea, which was based on the results of density-
functional theory (DFT) calculations, was that the ground state of Si(553)-Au
is spin polarized. In the DFT ground state, the silicon atoms that comprise
the steps on this vicinal surface have dangling bonds, every third of which is
occupied by a single fully polarized electron while the other two are doubly
occupied. The bright rows seen in empty-state STM images arise from these
step-edge silicon atoms. At low temperature the $3a_{0}$ peaks that appear in
this row are from the spin-polarized atoms, which have precisely this
periodicity. The DFT ground state also reveals a period doubling within the
row of Au atoms. Both of these higher-order periodicities disappear if spin
polarization is suppressed in the calculation, providing compelling evidence
that spin polarization is the primary mechanism underlying the observed
symmetry breaking in Si(553)-Au.
Experiments were subsequently carried out to look for a spectroscopic
signature of this predicted spin-polarized ground state. The DFT calculations
showed that an unoccupied state should exist 0.5 eV above the fermi level and
be localized at the polarized silicon atoms.Erwin and Himpsel (2010) The
existence and spectral and spatial location of this state was indeed confirmed
by two-photon photoemission Biedermann _et al._ (2012) and by scanning
tunneling spectroscopy.Snijders _et al._ (2012)
The predictions of Ref. Erwin and Himpsel, 2010 only addressed the zero-
temperature ground state of Si(553)-Au. Left unanswered in that work was the
question of how the broken-symmetry ground state evolves to have normal
$a_{0}$ periodicity above $\sim$50 K. This article addresses that question
from a theoretical and computational perspective. Although it may seem obvious
that thermal fluctuations are important, the nature of these fluctuations
turns out to be unexpectedly subtle. Nevertheless, we derive here a number of
detailed qualitative as well as quantitative predictions that can easily be
tested experimentally. The results of these tests will furnish additional
evidence for evaluating the validity of the basic mechanism proposed in Ref.
Erwin and Himpsel, 2010.
## II Ground state configuration
The physical and magnetic structure of Si(553)-Au in its ground state were
first proposed and discussed in Ref. Erwin and Himpsel, 2010 and for reference
are reproduced in Fig. 1. This is a stepped surface consisting of (111)
terraces and bilayer steps, and is stabilized by Au atoms that substitute for
Si atoms in the surface layer of the terrace. The steps themselves consist of
Si atoms organized into a thin graphitic strip of honeycomb hexagons (the
green atoms in Fig. 1).
Figure 1: (Color online) (a,b) Perspective and top views of Si(553)-Au in its
electronic ground state. Yellow atoms are Au, all others are Si. The Au atoms
are embedded in flat terraces, which are separated by steps consisting of Si
atoms arranged as a honeycomb chain (green). Every third Si atom (red, blue)
at the step has a spin magnetic moment of one Bohr magneton ($S=1/2$, arrows)
from the complete polarization of the electron occupying the dangling-bond
orbital. The sign of the polarization (red vs. blue) alternates along the
step. The six atoms in the outlined box are the focus of the ab-initio
molecular dynamics discussed in Sec. III.
The surface electronic structure of Si(553)-Au has two main contributions. The
first consists of two intense quasi-1D parabolic electron bands centered at
the boundary of the surface Brillouin zone. These “Au bands” arise from the
bonding and antibonding combinations of Au 6$s$ and subsurface Si orbitals
(purple atoms). The bonding Au band is approximately half-filled and the
antibonding band approximately one-fourth filled.
The second contribution arises from the very edge of the Si honeycomb chain,
which consists of threefold-coordinated Si atoms. The unpassivated $sp^{3}$
orbitals of these atoms can in principle be occupied by zero, one, or two
electrons. The Si atoms themselves supply, on average, one electron per
orbital. The step edge does not necessarily maintain this average occupancy,
because electronic charge can also be transferred to or from the Au bands.
Indeed, DFT calculations predict that the lowest energy configuration has one
electron in every third orbital (red and blue atoms in Fig. 1) and double
occupancy everywhere else (green atoms). The singly occupied orbitals are
completely spin-polarized and hence have local spin moments of 1 bohr magneton
each. Physically, these atoms relax slightly downward, by 0.3 Å, compared to
their nonpolarized neighbors. The sign of the spins alternates along the step
edge, with antiferromagnetic order favored by 15 meV per spin relative to
ferromagnetic order. Therefore the magnetic periodicity is 6$a_{0}$, where
$a_{0}$ is the Si surface lattice constant. This is also the smallest period
that allows for the coexistence of 3$a_{0}$ spacing of the spins and 2$a_{0}$
spacing (period doubling) within the Au chain. This coexistence was first
observed in STM experimentsAhn _et al._ (2005); Snijders _et al._ (2006) and
emerges naturally in DFT calculations—but only when the spin degree of freedom
is unconstrained.Erwin and Himpsel (2010)
## III Finite temperature dynamics
The remainder of this article explores excitations of Si(553)-Au from its
ground state due to finite temperature. Two main theoretical tools were used:
ab-initio molecular dynamics (MD) and kinetic Monte Carlo (kMC) simulations.
The first was used to identify the most important low-energy activated
processes and to determine their activation barriers. Because of the
complexity of the system only small time scales (tens of ps) and a small
(1$\times$6) simulation cell could be addressed using ab-initio MD. To reach
much longer time scales (tens of ns) and larger system sizes (up to 128 spins)
we constructed a one-dimensional kMC model based on the processes and barriers
determined from ab-initio MD. In particular, the kMC model allowed us to
investigate finite-temperature behavior in the presence of pinning
defects—providing useful insight into temperature-dependent results from
scanning probe experiments, where defects often play a critical role.
Two simplifying assumptions were used throughout this work. (1) Electronic
excitations were not considered, and consequently the system stays on the
Born-Oppenheimer surface. This assumption is reasonable in view of the modest
temperatures—room temperature and lower—considered here. (2) Spin flips were
not allowed. Although, as we will see below, the spins can diffuse among the
Si step-edge atoms, their signs and ordering remained that of the original
antiferromagnetic ordering. Although a different initial spin ordering might
affect some details of the simulation, the overall qualitative findings would
be very similar.
### III.1 Ab-initio molecular dynamics
The MD simulations were performed using the same basic geometry and
computational parameters described in Ref. Erwin and Himpsel, 2010. The
Si(553)-Au surface was represented by six layers of Si plus the reconstructed
top surface layer and a vacuum region of 10 Å. All atoms were free to move
during the simulation except the bottom Si layer and its passivating hydrogen
layer. Total energies and forces were calculated within the generalized-
gradient approximation of Perdew, Burke, and Ernzerhof to DFT using projector-
augmented wave potentials, as implemented in VASP.Kresse and Hafner (1993);
Kresse and Furthmüller (1996); Blochl (1994); Kresse and Joubert (1999) The
plane-wave cutoff was 200 eV and only the $\Gamma$ point was used. The
dynamics simulations were performed in the canonical ensemble using a Nosé
thermostat and a time step of 3 fs. Five temperatures, equally spaced in
$1/T$, were used (57, 67, 80, 110, 133 K). For each temperature a
thermalization run of 10 ps was first performed, followed by a dynamics run of
20 ps.
Figure 2 shows the resulting atomic trajectories during the entire run of 104
MD time steps for the lowest temperature studied, 57 K. The six curves are for
the six Si step-edge atoms in the outlined box of Fig. 1(b). The upper and
lower panels show the relative heights of the atoms and their local spin
moments, respectively. After thermalization was achieved the system settled
into its ground state configuration with two spin-polarized atoms (red and
blue) sitting $\sim$0.3 Å lower than their four non-polarized neighbors.
Figure 2: (Color online) (a) Ab-initio molecular dynamics trajectories of the
six Si step-edge atoms outlined in Fig. 1, at 57 K. Red and blue curves denote
the red and blue atoms, which are initially spin-polarized. Other colors
(magenta, cyan, dark green, light green) denote initially non-polarized atoms.
Thermalization is completed by about 10 ps. Upper panel: height of each atom,
relative to the average height of nonpolarized atoms. Lower panel: local spin
moment of each atom. (b) Expanded view of two spin hops occurring at 15.44 ps
(from the red atom to the cyan atom) and at 15.71 ps (from the blue atom to
the magenta atom).
The expanded view in Fig. 2(b) focuses on two events that occurred between 15
and 16 ps. At 15.44 ps the magnitude of the moment on the spin-up red atom
went rapidly to zero while, concurrently, a spin-up moment rapidly developed
on the neighboring cyan atom. At essentially the same time the height of the
red atom increased by 0.3 Å to that of a non-polarized atom, while the cyan
atom moved down by the same amount. In summary, the spin-up moment that was
localized on the red atom hopped to one of its neighbors.
Very soon after, a second hop occurred at 15.71 ps. This hop was made by the
other spin (with the opposite sign) which moved from the blue atom to the
magenta atom. It is not a coincidence that this hop occurred so soon after the
first. The first hop changed the minimum spacing between spins from 3$a_{0}$
to 2$a_{0}$, incurring an energy penalty (discussed in detail below). This
increase in energy in turn reduced the barrier for any hop that restores the
spacing to its optimal value. The cyan and magenta atoms are indeed separated
by 3$a_{0}$, and thus after two rapid spin hops the system was restored to an
equivalent ground state configuration, in which it remained for the rest of
the simulation.
The very small number of hops observed at 57 K makes it clear that ab-initio
MD simulations of Si(553)-Au at still lower temperatures, where many of the
relevant experiments are conducted, are not feasible. Instead we turn to
higher temperatures and ask how the frequency of hopping events depends on
temperature. This information will be useful in Sec. III.2 for calibrating and
validating our kMC model in a temperature range accessible to both methods.
Figure 3 shows the resulting time-averaged hopping rate for a single spin,
versus inverse temperature. The rates for low temperatures have large
statistical uncertainties (not shown) and hence it is reasonable to describe
these results by a simple linear Arrhenius fit, as shown. The attempt
frequency, $2.0\times 10^{13}$ s-1, is on the order of a surface vibrational
frequency, as expected. The activation energy, 12 meV, represents a
characteristic average of the individual barriers for spin hops weighted by
their relative probability of occurrence.
Figure 3: Temperature dependence of the hopping rate for Si spins along the
Si(553)-Au step edge. The rates are time averages extracted from ab-initio
molecular dynamics (MD) trajectories, and are compared to rates from kinetic
Monte Carlo (kMC) simulations. The linear fits describe Arrhenius behavior.
The fit to ab-initio MD rates (thick line) gives a pre-exponential factor
$2.0\times 10^{13}$ s-1 and activation barrier 12 meV. For kMC rates (thin
line) the values are $2.2\times 10^{13}$ s-1 and 13 meV.
### III.2 Kinetic Monte Carlo model
To construct the kMC model one first needs to enumerate all the relevant spin
hops and their individual rates. We used DFT results obtained from the full
Si(553)-Au system for this task. In the spirit of simplicity we constructed
the kMC model itself to be strictly one-dimensional, with an arbitrarily large
unit cell and periodic boundary conditions. Thus the kMC simulations inherit
much of the accuracy of the DFT calculations but make the additional
approximation that spin hops along different step edges are independent.
Figure 4 shows the DFT potential energy surface for the spin hop observed in
Fig. 2(b) at 15.44 ps into the MD simulation. Because the spins and the
heights of the atoms are tightly linked, the reaction coordinate $x$ is
approximately given by the relative heights $h$ of the red and cyan atoms,
$x\approx[1-(h_{\rm cyan}-h_{\rm red})/\Delta h]/2,$ (1)
where $\Delta h=0.3$ Å is the equilibrium height difference between spin-
polarized and non-polarized atoms. To definitively determine the detailed
reaction pathway and potential energy surface we used the nudged elastic-band
method.
Figure 4: DFT potential energy surface for a single spin hopping from the red
atom to the neighboring cyan atom. The initial state (0) is the ground state
depicted in Fig. 1. The final state (1) is the metastable state, fully
relaxed, that exists between 15.44 and 15.71 ps in the MD simulation of Fig.
2. The activation barrier is 30 meV for the forward reaction and 5 meV for the
reverse reaction. Atom colors correspond to the trajectories in Fig. 2.
This potential energy surface confirms the assertion, made in Sec. III.1, that
the red-to-cyan (forward) spin hop incurs an energy penalty that leads to a
smaller barrier for the cyan-to-red (reverse) hop. Specifically, the
activation barrier for the forward hop is 30 meV, the resulting energy penalty
is 25 meV, and the barrier for the reverse hop is 5 meV. These two types of
hops, and their calculated barriers, are two of the three fundamental
processes included in our kMC model.
For convenience we define here a more compact notation for enumerating the
different types of spin hops. Careful examination of the ab-initio MD
trajectories reveals that all spin hops were to an adjacent site; there were
no double hops. Hence we can label every hop as either leftward ($\leftarrow$)
or rightward ($\rightarrow$). We assume that the barrier for a spin hop
depends only on the spin’s immediate environment, that is, on the distances
$ma_{0}$ and $na_{0}$ to the left and right neighboring spins, respectively,
measured before making the hop. Using this notation we can express the
barriers for the two hops shown in Fig. 4 as $b(3,3,\leftarrow)=30$ meV and
$b(2,4,\rightarrow)=5$ meV, where the two numerical arguments denote $m$ and
$n$, respectively.
The third important hop we considered occurs when $m+n=5$, rather than 6 as
depicted in Fig. 4. From DFT nudged elastic-band calculations we find
$b(3,2,\leftarrow)$ = $b(2,3,\rightarrow)=14$ meV (the barriers are equal by
symmetry). As expected from the distances to the neighboring spins, this
barrier is in between the previous two.
The MD trajectories also show that two spins never occupy adjacent sites.
Because of this, our enumeration of the possible spin hops is already complete
for all cases with $m+n\leq 6$. (It is worth noting that the configuration in
which a spin has both neighbors at $2a_{0}$ is allowed, but because its
adjacent sites cannot be occupied this spin cannot hop until one of its
neighbors does.) The cases with $m+n\geq 7$ are difficult to treat within DFT
but occur more rarely and thus are less important. For this reason we treated
the effect of neighbors beyond $3a_{0}$ as negligible, used the barrier of 30
meV for any hop that brings a spin within $2a_{0}$ of its neighbor, and
assigned a single (arbitrary) barrier of 10 meV to hops that maintain larger
separations than this. This completes our enumeration.
To finish the construction of the kMC model, we assumed that the rates for all
allowed spin hops are given by $r=a\exp(-b/kT)$, where $a$ is a common
prefactor and $b=b(m,n,\leftarrow)$ and $b(m,n,\rightarrow)$ are the DFT
barriers.
To determine the optimal value of $a$ and compare the predictions of the kMC
model to the ab-initio MD results, we applied the model to the system
discussed in Sec. III.1—two spin-polarized atoms in a six-atom unit cell with
periodic boundary conditions. The resulting kMC spin hopping rates obtained
using $a=6\times 10^{12}$ s-1 are plotted in Fig. 3 for direct comparison with
the rates from ab-initio MD. The kMC rates have negligible statistical errors
and it is clear that a simple Arrhenius fit describes them very well.
Moreover, the fitted attempt frequency, $2.2\times 10^{13}$ s-1, and
activation energy, 12 meV, are within 10% of the MD values. This confirms that
the kMC model accurately reproduces the ab-initio results within the
temperature range considered.
### III.3 Spins at finite temperature near a defect
In real systems, the behavior of collective phenomena is often controlled by
defects that pin the phase of a low-temperature state having broken symmetry.
On the Si(553)-Au surface, a variety of defects—missing atoms, absorbates,
etc.—have been observed to act as pinning sites that locally stabilize the
1$\times$3 ground state.Snijders _et al._ (2006); Kang _et al._ (2009b);
Hasegawa (2010); Shin _et al._ (2012) The important role played by such
pinning defects motivates our first application of the kMC model.
We prepared a system consisting of 128 independent spins, with periodic
boundary conditions, initially arranged in the antiferromagnetic ground state
with uniform 3$a_{0}$ spacing. One of the spins (at position 0) was pinned in
place throughout the simulation, thus representing a generic immobile defect.
The spins were allowed to hop stochastically among the 3$\times$128 lattice
sites according to probabilities defined by the hopping rates $r$.
Figure 5 displays the resulting trajectories at 300 K of all the spins over
the first 10 ns of the simulation. For clarity every tenth trajectory trace is
colored. As the system evolved, each spin explored a region of the lattice
around its initial position. These explorations were relatively small for
spins near the pinning defect and became progressively larger for spins
farther away.
Figure 5: (Color online) Kinetic Monte Carlo trajectories of 128 spins at 300
K in the presence of a pinning defect at the origin. Every tenth trace is
colored for clarity. Right panel: histogram of the positions occupied by each
spin, weighted by the time spent there, obtained over a simulation time of 1
$\mu$s. The heavy curve is the envelope function $d^{-2/3}$ describing the
decay of histogram heights with distance $d$ from the pinning defect.
The right panel in Fig. 5 examines this thermally induced wandering in greater
detail. For each of the 128 spins a histogram was made representing the
position of that spin at 300 K. At this temperature a simulation time of 1
$\mu$s was sufficient to obtain the steady-state distribution. It is readily
apparent from examining the colored histograms that each is well described by
a gaussian function centered on the spin’s initial position. Thus each of
these gaussians is entirely specified by its variance $\sigma^{2}$, whose
value depends on the distance $d$ to the pinning defect. To deduce this
dependence we first note that the area under each gaussian is by construction
the same. Hence the height of each gaussian is proportional to $1/\sigma$. We
find empirically that the dependence of these heights on distance is given
with excellent accuracy as $d^{-2/3}$. An envelope function with this
dependence is shown on the histogram plot as a heavy black curve. From this
dependence we thus deduce that the thermally induced widths $w$, defined here
as $2\sigma$, increase with distance from a pinning defect as $w\sim d^{2/3}$.
Now we move on to explore how temperature affects the thermal wandering of
spins near a pinning defect. We repeated the kMC simulation and analysis in
Fig. 5 for a series of temperatures between 10 and 300 K. We focus on the
variation of the thermal widths $w$ as a function of temperature $T$.
Figure 6: (Color online) Temperature dependence of the thermal widths $w(T)$
of every tenth spin, in the presence of a pinning defect. Labels indicate the
spin’s distance from the defect, in units of 3$a_{0}$. Colors correspond to
the trajectories in Fig. 5. The light gray shaded area is bounded by
logarithmic fits to the lower (red) and upper (orange) data points. The
characteristic temperatures $T_{0}$, $T_{1}$, and $T_{3}$ describe different
criteria by which thermal wandering of spins is expected to be either
eliminated or suppressed; see discussion in text.
Figure 6 summarizes the resulting temperature dependence. The six datasets
show $w(T)$ for every tenth spin of the 128-spin simulation cell. For
reference, the six values at $T=300$ K correspond to the six gaussian widths
in the upper half of the histogram panel in Fig. 5. We find empirically that
the dependence of each dataset on temperature is close to logarithmic, as
shown by the light gray shaded area. This implies that the dependence of the
thermal widths on distance and temperature can be separated and written as
$w(d,T)=w_{0}(d)\ln(T/T_{0}),$ (2)
where $w_{0}(d)\sim d^{2/3}$ and the characteristic temperature $T_{0}$ has
the fitted value 21 K. All thermal wandering is, by definition, completely
eliminated at $T_{0}$. But two less restrictive criteria may be more relevant
for interpreting the experimentally observed transition to the period-tripled
ground state. At $T_{1}=27$ K the thermal widths for all 128 spins become
smaller than the width of a single lattice site. Hence, below this temperature
the spins will in effect be frozen into place on every third lattice site. At
still higher temperature, $T_{3}=42$ K, all thermal widths are less than or
equal to the average spacing (three lattice sites) between spins. Hence the
spins will first become distinguishable as the system is cooled below this
temperature.
To generalize this result and make predictions that can be tested by
experiment, we first assume that real systems can be characterized by a known
average concentration $c$ of pinning defects. Because the defects are
distributed in 1D, the characteristic distance $d$ from a spin to the nearest
defect scales as $1/c$. By inverting Eq. 2 we then immediately obtain a simple
result: the temperatures $T_{1,3}$ at which all spins in the system become
either frozen into place or distinguishable will scale as
$T_{1,3}\sim\exp(c^{2/3})$. Thus we predict this scaling to describe the
temperature at which the period-tripled ground state of Si(553)-Au is first
observed.
By inserting into this qualitative relationship the appropriate constants
obtained from the kMC simulations, we derive a quantitative prediction for the
freezing temperature,
$T_{1}=T_{0}\exp(k\,c^{2/3}),$ (3)
where $k=11.8$ is a dimensionless constant and $c$ is expressed in the
dimensionless units of defects per lattice site. The corresponding equation
for $T_{3}$ can be obtained from Eq. 3 by multiplying the argument of the
exponential by three. A useful guide to understanding the importance of
defects in Si(553)-Au is provided by linearizing Eq. 3 around a physically
plausible value (10-2) for the defect concentration. This leads to the result
that a change in the defect concentration will raise the freezing temperature
by $\Delta T_{1}=\gamma\Delta c$, with proportionality constant $\gamma=1330$
K. Thus, for samples with approximately one defect every 100 lattice sites, a
doubling of this concentration will increase the freezing temperature by 13 K.
### III.4 Spins at finite temperature in the absence of defects
Although a system completely free of defects is obviously unrealistic, the
behavior of such an idealized system nevertheless offers complementary insight
into the thermal wandering of spins when the concentration of defects is very
low. As we show below, despite the absence of defects, the statistical
behavior of the spins still exhibits a sudden and qualitative change at about
30 K.
Figure 7: Trajectory of a single spin at 100 K in the absence of pinning
defects. Gray curve shows the theoretical average displacement versus time for
a isolated random walker in one dimension, $\langle d\rangle\propto\sqrt{t}$.
We constructed a periodic system of 64 spins similar to that described in Sec.
III.3, but now without a pinning defect. Thus each spin executed a random walk
in 1D. Figure 7 shows a typical trajectory at 100 K for one of the 64 spins.
Despite the stochastic nature of this single trajectory, it is already
plausible that the average displacement $\langle d\rangle$ depends on the time
$t$ according to $\langle d\rangle\propto\sqrt{t}$, which is the well-known
result for a single unbiased random walker in one dimension.
Figure 8: Average displacement versus time of a single spin in the absence of
defects, at the indicated temperatures. Circles are statistical averages over
2000 kMC simulations. Straight lines are fits to $t^{H}$, where $H$ is the
Hurst exponent. The dotted line indicates $H$=1/2.
To analyze this behavior more systematically, we performed many independent
kMC simulations and computed the average displacements as a function of time.
Figure 8 shows these averages on a log-log scale for a range of temperatures.
At high temperatures we indeed obtain the behavior $\langle d\rangle\propto
t^{1/2}$, which is indicated by the dotted line. This behavior persists until
the temperature reaches the range 30–35 K, where it still exhibits power-law
behavior $\langle d\rangle\propto t^{H}$ but with a progressively larger
exponent $H>1/2$.
Figure 9: Hurst exponent versus temperature for spins in the absence of a
defect. The solid curve is a fit to a generalized susceptibility with a
characteristic temperature $T_{c}=28$ K.
In general, a system for which the long-time displacements are characterized
by a Hurst exponent $H=1/2$ is said to be uncorrelated, while $H>1/2$
indicates that long-time correlations are present.Rangarajan and Ding (2000)
Figure 9 shows the Hurst exponent $H$, obtained by fitting the time-averaged
displacements, as a function of temperature. Above $\sim$40 K the system
displays uncorrelated behavior. Below this temperature we observe the rapid
onset of correlated behavior. To quantify the temperature of this transition
we fit the Hurst exponents to a generalized susceptibility of the form
$H=(1/2)/[1-(T_{c}/T)^{\nu}]$ and obtain a characteristic temperature
$T_{c}=28$ K. Hence as the clean system is cooled toward $T_{c}$ the random
walks described by individual spins rapidly lose their independent character.
This transition occurs with a characteristic temperature comparable to that
obtained, $T_{0}=21$ K, by extrapolating from the behavior in the presence of
defects. Thus these two complementary approaches lead to qualitatively as well
as quantitatively similar conclusions.
## IV Discussion and conclusions
A guiding principle in one-dimensional physics is provided by the Mermin-
Wagner theorem, which states that phase transitions cannot occur above $T=0$
if the interactions are short-ranged.Mermin and Wagner (1966) Exploring the
different manifestations of this theorem in real systems can yield new and
unanticipated insights. For example, a previous publication by one of us
demonstrated theoretically that when the interactions are $not$ short-ranged,
as for $1/r$ Coulomb interactions, then a well-defined thermodynamic phase
transition can indeed occur—and likely does occur for a system of Ba
adsorbates on Si(111).Erwin and Hellberg (2005); Bruch (2005) In the present
article the new insights into 1D physics are of a different kind: we have
shown that even when the interactions are short-ranged, and the system purely
1D, a well-ordered phase with the clear signature of a broken symmetry can
form well above $T=0$. Moreover, an approximate transition temperature can be
readily identified using realistic simulations and straightforward statistical
analysis.
For Si(553)-Au the precise nature of the interacting entities is unexpected
and somewhat subtle: our MD simulations showed that they are neither simple
vibrations of atoms, nor spins on a simple fixed lattice, but rather a tightly
coupled combination of both. In this sense a description of Si(553)-Au based
on spin polarons is appropriate. Our kMC simulations showed that as the
temperature of the system is raised, the ground state $3a_{0}$ crystal formed
by these polarons melts at a temperature we estimate to be $\sim$30 K for
perfectly clean systems, and higher for systems with pinning defects. A direct
experimental test of this description is afforded by Eq. 3, which predicts how
the transition temperature varies with the concentration of defects.
It is important to acknowledge some limitations of this work. Because our
focus has been on the behavior of Si(553)-Au at low temperature, we have
assumed that the spin-polarized silicon states remain polarized at higher
temperatures. Our preliminary calculations indicate that this assumption is
completely justified at the temperatures of interest here. However, at much
higher temperatures the thermally induced vibrations of the step edge atoms
increasingly render the system nonpolarized for part of the time. For example,
at room temperature the average spin moment is reduced to roughly 2/3 of its
low-temperature value of 1 bohr magneton. Future theoretical investigations
into the behavior of Si(553)-Au near room temperature will have to account for
this thermal suppression of the spin polarization.
Finally, as mentioned in Sec. III, we have throughout assumed for simplicity
that the ordering of the spins remains antiferromagnetic, as in the ground
state. Preliminary calculations show that the barriers for spin hopping depend
quantitatively, although not qualitatively, on the signs of the neighboring
spins. A generalization of our kMC model that includes spin flips would be a
very interesting direction to pursue, but we anticipate that the qualitative
conclusions drawn here—as well as the overall consistency between our findings
and those of existing experiments—would be largely unchanged.
###### Acknowledgements.
Many discussions with F.J. Himpsel are gratefully acknowledged. This work was
supported by the Office of Naval Research (SCE) and the Department of Energy,
Basic Energy Sciences, Materials Sciences and Engineering Division (PCS).
Computations were performed at the DoD Major Shared Resource Centers at AFRL
and ERDC.
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|
arxiv-papers
| 2013-04-03T17:35:51 |
2024-09-04T02:49:43.845139
|
{
"license": "Public Domain",
"authors": "Steven C. Erwin and P.C. Snijders",
"submitter": "Steven C. Erwin",
"url": "https://arxiv.org/abs/1304.1024"
}
|
1304.1033
|
11institutetext: 1 University of Vigo 11email: [email protected]
2 University of Bucharest 11email: [email protected]
# A Fixed Point Theorem and Equilibria of Abstract Economies with w-Upper
Semicontinuous Set-Valued Maps
Carlos Hervés-Beloso1 and Monica Patriche2 1 RGEA, Facultad de Económicas,
Universidad de Vigo, Campus Universitario, E-36310 Vigo, Spain
2 University of Bucharest, Faculty of Mathematics and Computer Science, 14
Academiei Street, 010014 Bucharest, Romania
###### Abstract
We introduce the notion of w-upper semicontinuous set valued maps
###### Keywords:
Fixed point theorem, w-upper semicontinuous set valued maps,
2010 Mathematics Subject Classification. 47H10, 91A47, 91A80.
1. Introduction
The pioneer work of Nash [11] first proved a theorem of equilibrium existence
for games where the player’s payoffs are represented by continuous quasi-
concave utilities. Arrow and Debreu used the work by Nash to prove the
existence of equilibrium in a generalized N-person game or on abstract economy
[7] which implies the Walrasian equilibrium existence [2]. These ideas were
extended by various authors in several ways. In [16], Shafer and Sonnenschein
proved the existence of equilibrium of an economy with finite dimensional
commodity space and irreflexive preferences represented as set valued maps
with open graph. Yannelis and Prahbakar [22] developed new techniques based on
selection theorems and fixed-point theorems. Their main result concerns the
existence of equilibrium when the constraint and preference set valued maps
have open lower sections. They work within different frameworks (countable
infinite number of agents, infinite dimensional strategy spaces).
Borglin and Keiding [3] used new concepts of K.F.-set valued maps and KF-
majorized set valued maps for their existence results . The concept of KF-
majorized set valued maps was extended by Yannelis and Prabhakar [22] to
L-majorized set valued maps. In [23], Yuan proposed a more general model of
abstract economy than the one introduced by Borglin and Keing in [3], in the
sense that the constraint mapping was split into two parts $A$ and $B.$ This
is due to the ”small” constraint set valued map $A$ which could not have
enough fixed points even though the ”big” constraint set valued map $B$ could.
Most existence theorems of equilibrium deal with preference set valued maps
which have lower open sections or are majorized by set valued maps with lower
open sections. In the last few years, some existence results were obtained for
lower semicontinuous and upper semicontinuous set valued maps. Some recent
results concerning upper semicontinuous set valued maps and fixed points can
be found in [1], [4], [18], [19], [20], [24]. New results on equilibrium
existence in games are given in [10], [12], [13], [17].
In this paper, we define two types of set valued maps: w-upper semicontinuous
set valued maps and set valued maps that have e-USS-property. We prove a fixed
point theorem for w-upper semicontinuous set valued maps. This result is a Wu
like result [20] and generalizes the Himmelberg’s fixed point theorem in [9].
We use this theorem for proving our first theorem of equilibrium existence for
abstract economies having w-upper semicontinuous constraint and preference set
valued maps. On the other hand, we use a technique of approximation to prove
an equilibrium existence theorem for set valued maps having e-USS-property.
The paper is organized in the following way: Section 2 contains preliminaries
and notations. The fixed point theorem is presented in Section 3 and the
equilibrium theorems are stated in Section 4.
2. Preliminaries and Notation
Throughout this paper, we shall use the following notations and definitions:
Let $A$ be a subset of a topological space $X$. $2^{A}$ denotes the family of
all subsets of $A$. cl$A$ denotes the closure of $A$ in $X$. If $A$ is a
subset of a vector space, co$A$ denotes the convex hull of $A$. If $F$, $G:$
$X\rightrightarrows Y$ are set valued maps, then conv $G$, cl $G$, $G\cap F$
$:$ $X\rightrightarrows Y$ are set valued maps defined by $($conv $G)(x)=$conv
$G(x)$, $($cl $G)(x)=$cl $G(x)$ and $(G\cap F)(x)=G(x)\cap F(x)$ for each
$x\in X$, respectively. The graph of $T:X\rightrightarrows Y$ is the set Gr
$(T)=\\{(x,y)\in X\times Y\mid y\in T(x)\\}.$
The set valued map $\overline{T}$ is defined by $\overline{T}(x):=\\{y\in
Y:(x,y)\in$clX×Y Gr $T\\}$ (the set clX×Y Gr $(T)$ is called the adherence of
the graph of $T$). It is easy to see that cl $T(x)\subset\overline{T}(x)$ for
each $x\in X.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Let $X$, $Y$ be topological spaces and $T:X\rightrightarrows Y$ be a set
valued map. $T$ is said to be upper semicontinuous if for each $x\in X$ and
each open set $V$ in $Y$ with $T(x)\subset V$, there exists an open
neighborhood $U$ of $x$ in $X$ such that $T(y)\subset V$ for each $y\in U$.
$T$ is said to be almost upper semicontinuous if for each $x\in X$ and each
open set $V$ in $Y$ with $T(x)\subset V$, there exists an open neighborhood
$U$ of $x$ in $X$ such that $T(y)\subset$cl $V$ for each $y\in U$.
Lemma 2.1(Lemma 3.2, pag. 94 in [25]) Let $X$ be a topological space, $Y$ be a
topological linear space, and let $S:X\rightrightarrows Y$ be an upper
semicontinuous set valued map with compact values. Assume that the set
$C\subset Y$ is closed and $K\subset Y$ is compact. Then $T:X\rightrightarrows
Y$ defined by $T(x)=(S(x)+C)\cap K$ for all $x\in X$ is upper semicontinuous.
Lemma 2.2 is a version of Lemma 1.1 in [21] ( for $D=Y,$ we obtain Lemma 1.1
in [21]). For the reader’s convenience, we include its proof below.
Lemma 2.2 Let $X$ be a topological space, $Y$ be a nonempty subset of a
locally convex topological vector space $E$ and $T:X\rightrightarrows Y$ be a
set valued map. Let ß be a basis of neighbourhoods of $0$ in $E$ consisting of
open absolutely convex symmetric sets. Let $D$ be a compact subset of $Y$. If
for each $V\in$ß, the set valued map $T^{V}:X\rightrightarrows Y$ is defined
by $T^{V}(x)=(T(x)+V)\cap D$ for each $x\in X,$ then
$\cap_{V\in\text{\ss}}\overline{T^{V}}(x)\subseteq\overline{T}(x)$ for every
$x\in X.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Proof Let $x$ and $y$ be such that
$y\in\cap_{V\in\text{\ss}}\overline{T^{V}}(x)$ and suppose, by way of
contradiction, that $y\notin\overline{T}(x).$ This means that $(x,y)\notin$cl
Gr $T,$ so that there exists an open neighborhood $U$ of $x$ and $V\in$ß such
that:
$(U\times(y+V))\cap$Gr $T=\emptyset.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ (1)$
Choose $W\in$ß such that $W-W\subseteq V$ (e.g. $W=\frac{1}{2}V)$. Since $y\in
T^{W}(x)$, then $(x,y)\in$cl Gr $T^{W},$ so that
$(U\times(y+W))\cap\text{Gr }T^{W}\neq\emptyset.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
There are some $x^{\prime}\in U$ and $w^{\prime}\in W$ such that
$(x^{\prime},y+w^{\prime})\in$Gr $T^{W},$ i.e. $y+w^{\prime}\in
T^{W}(x^{\prime}).$ Then, $y+w^{\prime}\in D$ and
$y+w^{\prime}=y^{\prime}+w^{{}^{\prime\prime}}$ for some $y^{\prime}\in
T(x^{\prime})$ and $w^{{}^{\prime\prime}}\in W.$ Hence,
$y^{\prime}=y+(w^{\prime}-w^{{}^{\prime\prime}})\in y+(W-W)\subseteq y+V,$ so
that $T(x^{\prime})\cap(y+V)\neq\emptyset.$ Since $x^{\prime}\in U,$ this
means that $(U\times(y+V))\cap$Gr $T\neq\emptyset,$ contradicting (1). $\ \ \
\ \ \ \ \ \ \ \square$
We introduce the following definitions.
Let $X$ be a topological space, $Y$ be a nonempty subset of a topological
vector space $E$ and $D$ be a subset of $Y$.
Definition 2.1 The set valued map $T:X\rightrightarrows Y$ is said to be
w-upper semicontinuous (weakly upper semicontinuous) with respect to the set
$D$ if there exists a basis ß of open symmetric neighborhoods of $0$ in $E$
such that, for each $V\in$ß, the set valued map $T^{V}$ is upper
semicontinuous.
Definition 2.2 The set valued map $T:X\rightrightarrows Y$ is said to be
almost w-upper semicontinuous (almost weakly upper semicontinuous) with
respect to the set $D$ if there exists a basis ß of open symmetric
neighborhoods of $0$ in $E$ such that, for each $V\in$ß, the set valued map
$\overline{T^{V}}$ is upper semicontinuous.
Example 2.1 Let $T_{1}:(0,2)\rightrightarrows(0,2)$ be defined by
$T_{1}(x):=\left\\{\begin{array}[]{c}(0,1)\text{ if }x\in(0,1];\\\ [1,2)\text{
if x}\in(1,2).\end{array}\right.$
$T_{1}$ and $T_{1}\cap\\{1\\}:=\left\\{\begin{array}[]{c}\phi\text{ \ if \
}x\in(0,1];\\\ \\{1\\}\text{ if x}\in(1,2)\end{array}\right.$ are not upper
semicontinuous on $(0,2).$
Let $D:=\\{1\\}$ and let $V:=(-\varepsilon,\varepsilon),$ $\varepsilon>0,$ be
an open symmetric neighbourhood of $0$ in $IR.$ Then, it results that
for $\varepsilon>0,$
$T_{1}(x)+(-\varepsilon,\varepsilon):=\left\\{\begin{array}[]{c}(-\varepsilon,1+\varepsilon)\text{
\ \ if \ \ \ }x\in(0,1];\\\ (1-\varepsilon,2+\varepsilon)\text{\ if \ \ \ \ \
}x\in(1,2);\end{array}\right.$
$T_{1}^{V}(x):=(T_{1}(x)+(-\varepsilon,\varepsilon))\cap\\{1\\}=\\{1\\}$ for
any $x\in(0,2).$
$\overline{T_{1}^{V}}(x)=\\{1\\}$ for $x\in(0,2).$
For each $V=(-\varepsilon,\varepsilon)$ with $\varepsilon>0,$ the set valued
maps $T_{1}^{V}$ and $\overline{T_{1}^{V}}$ are upper semicontinuous and
$\overline{T_{1}^{V}}$ has nonempty values. We conclude that $T_{1}$ is
w-upper semicontinuous with respect to $D$ and it is also almost w-upper
semicontinuous with respect to $D.$
We also define the dual w-upper semicontinuity with respect to a compact set.
Definition 2.3 Let $T_{1},T_{2}:X\rightrightarrows Y$ be set valued maps. The
pair $(T_{1},T_{2})$ is said to be dual almost w-upper semicontinuous (dual
almost weakly upper semicontinuous) with respect to the set $D$ if there
exists a basis ß of open symmetric neighborhoods of $0$ in $E$ such that, for
each $V\in$ß, the set valued map $\overline{T_{(1,2)}^{V}}:X\rightrightarrows
D$ is lower semicontinuous, where $T_{(1,2)}^{V}:X\rightrightarrows D$ is
defined by $T_{(1,2)}^{V}(x):=(T_{1}(x)+V)\cap T_{2}(x)\cap D$ for each $x\in
X$.
Example 2.2 Let $\ D:=[1,2],$ $T_{1}:(0,2)\rightrightarrows[1,4]$ be the set
valued map defined by
$T_{1}(x):=\left\\{\begin{array}[]{c}[2-x,2],\text{ if }x\in(0,1);\\\
\\{4\\}\text{ \ \ \ \ \ \ if \ \ \ \ \ \ \ }x=1;\\\ [1,2]\text{ \ \ \ if \ \ \
}x\in(1,2).\end{array}\right.$
and $T_{2}:(0,2)\rightrightarrows[2,3]$ be the set valued map defined by
$T_{2}(x):=\left\\{\begin{array}[]{c}[2,3],\text{ if }x\in(0,1];\\\
\\{2\\}\text{ \ if \ \ }x\in(1,2);\end{array}\right..$
The set valued map $T_{1}$ is not upper semicontinuous on $(0,2)$.
For $\varepsilon\in(0,2],$ $(T_{1}(x)+(-\varepsilon,\varepsilon))\cap D\cap
T_{2}(x)=\left\\{\begin{array}[]{c}\\{2\\}\text{ if }x\in(0,1)\cup(1,2);\\\
\phi\text{ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ \ }x=1.\end{array}\right.$
For $\varepsilon\in(2,\infty),$ $(T_{1}(x)+(-\varepsilon,\varepsilon))\cap
D\cap T_{2}(x)=\\{2\\}$ for each $x\in(0,2).$
Then, we have that for each $\varepsilon>0,$
$\overline{T_{(1,2)}^{V}}(x)=\\{2\\}$ for each $x\in[0,2]$ and the set valued
map $\overline{T_{(1,2)}^{V}}$ is upper semicontinuous and has nonempty
values.
We conclude that the pair $(T_{1},T_{2})$ is dual almost w-upper
semicontinuous with respect to $D.$
3. A New Fixed Point Theorem
We obtain the following fixed point theorem which generalizes Himmelberg’s
fixed point theorem in [9]:
Theorem 3.1 Let $I$ be an index set. For each $i\in I,$ let $X_{i}$ be a
nonempty convex subset of a Hausdorff locally convex topological vector space
$E_{i}$, $D_{i}$ be a nonempty compact convex subset of $X_{i}$ and
$S_{i},T_{i}:X:=\mathop{\textstyle\prod}\limits_{i\in I}X_{i}\rightrightarrows
X_{i}$ be two set valued maps with the following conditions:
1) for each $x\in X$, $\overline{S}_{i}(x)\subseteq T_{i}(x)$;
2) $S_{i}$ is almost w-upper semicontinuous with respect to $D_{i}$ and
$\overline{S_{i}^{V_{i}}}$ is convex nonempty valued for each absolutely
convex symmetric neighborhood $V_{i}$ of $0$ in $E_{i}$.
Then there exists $x^{\ast}\in D:=\mathop{\textstyle\prod}\limits_{i\in
I}D_{i}$ such that $x_{i}^{\ast}\in T_{i}(x^{\ast})$ for each $i\in I.\vskip
6.0pt plus 2.0pt minus 2.0pt$
Proof Since $D_{i}$ is compact, $D:=\mathop{\textstyle\prod}\limits_{i\in
I}D_{i}$ is also compact in $X.$ For each $i\in I,$ let ßi be a basis of open
absolutely convex symmetric neighborhoods of zero in $E_{i}$ and let
ß=$\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i}.$ For each system of
neighborhoods $V=(V_{i})_{i\in I}\in\mathop{\textstyle\prod}\limits_{i\in
I}$ß${}_{i},$ let’s define the set valued maps
$S_{i}^{V_{i}}:X\rightrightarrows D_{i},$ by
$S_{i}^{V_{i}}(x)=(S_{i}(x)+V_{i})\cap D_{i}$, $x\in X,$ $i\in I.$ By
assumption 2) each $\overline{S_{i}^{V_{i}}}$ is u.s.c with nonempty closed
convex values. Let’s define $S^{V}:X\rightrightarrows D$ by
$S^{V}(x)=\mathop{\textstyle\prod}\limits_{i\in I}\overline{S_{i}^{V_{i}}}(x)$
for each $x\in D.$ The set valued map $S^{V}$ is upper semicontinuous with
closed convex values. Therefore, according to Himmelberg’s fixed point theorem
[9], there exists $x_{V}^{\ast}=\mathop{\textstyle\prod}\limits_{i\in
I}(x_{V}^{\ast})_{i}\in D$ such that $x_{V}^{\ast}\in S^{V}(x_{V}^{\ast}).$ It
follows that $(x_{V}^{\ast})_{i}\in\overline{S_{i}^{V_{i}}}(x_{V}^{\ast})$ for
each $i\in I.$
For each $V=(V_{i})_{i\in I}\in$ß, let’s define $Q_{V}=\cap_{i\in I}\\{x\in
D:$ $x_{i}\in\overline{S_{i}^{V_{i}}}(x)\\}.$
$Q_{V}$ is nonempty since $x_{V}^{\ast}\in Q_{V},$ then $Q_{V}$ is nonempty
and closed.
We prove that the family $\\{Q_{V}:V\in\text{\ss}\\}$ has the finite
intersection property.
Let $\\{V^{(1)},V^{(2)},...,V^{(n)}\\}$ be any finite set of ß and let
$V^{(k)}=\underset{i\in I}{\mathop{\textstyle\prod}}V_{i}^{(k)}$, $k=1,...,n.$
For each $i\in I$, let $V_{i}=\underset{k=1}{\overset{n}{\cap}}V_{i}^{(k)}$,
then $V_{i}\in\text{\ss}_{i};$ thus $V=\underset{i\in
I}{\mathop{\textstyle\prod}}V_{i}\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}.$ Clearly
$Q_{V}\subseteq\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}$ so that
$\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}\neq\emptyset.$
Since $D$ is compact and the family $\\{Q_{V}:V\in\text{\ss}\\}$ has the
finite intersection property, we have that
$\cap\\{Q_{V}:V\in\text{\ss}\\}\neq\emptyset.$ Take any
$x^{\ast}\in\cap\\{Q_{V}:V\in$ß$\\},$ then for each $V_{i}\in\text{\ss}_{i},$
$x_{i}^{\ast}\in\overline{S_{i}^{V_{i}}}(x^{\ast})$. According to Lemma 2.2,__
we have that __ $x_{i}^{\ast}\in\overline{S_{i}}(x^{\ast}),$ for each $i\in
I,$ therefore $x_{i}^{\ast}\in T(x^{\ast}).$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \square$
If $\left|I\right|=1$ we get the result below.
Corollary 3.1 Let $X$ be a nonempty subset of a Hausdorff locally convex
topological vector space $F,$ $D$ be a nonempty compact convex subset of $X$
and $S,T:X\rightrightarrows X$ be two set valued maps with the following
conditions:
1) for each $x\in X,$ $\overline{S}(x)\subseteq T(x)$ and $S(x)\neq\emptyset,$
2) $S$ is almost w-upper semicontinuous with respect to $D$ and
$\overline{S^{V}}$ is convex valued for each open absolutely convex symmetric
neighborhood $V$ of $0$ in $E$.
Then, there exists a point $x^{\ast}\in D$ such that $x^{\ast}\in
T(x^{\ast}).$
In the particular case that the set valued map $S=T$ the following result
stands.
Corollary 3.2 Let $X$ be a nonempty subset of a Hausdorff locally convex
topological vector space $F,$ $D$ be a nonempty compact convex subset of $X$
and $T:X\rightrightarrows X$ be an almost w- upper semicontinuous set valued
map with respect to $D$ and $\overline{T^{V}}$ is convex valued for each open
absolutely convex symmetric neighborhood $V$ of $0$ in $E$. Then, there exists
a point $x^{\ast}\in D$ such that $x^{\ast}\in\overline{T}(x^{\ast}).\vskip
6.0pt plus 2.0pt minus 2.0pt$
4\. Application in the Equilibrium Theory
4.1 The Model of an Abstract Economy
We will consider further Yuan’s model of an abstract economy (see [23]). Let
$I$ be a nonempty set (the set of agents). For each $i\in I$, let $X_{i}$ be a
non-empty subset of a topological vector space representing the agent’s $i$
set of actions and define $X:=\underset{i\in I}{\prod}X_{i}$; let $A_{i}$,
$B_{i}:X\rightrightarrows X_{i}$ be the constraint set valued maps and $P_{i}$
the preference set valued map.
Definition 4.1 [23] An abstract economy
$\Gamma=(X_{i},A_{i},P_{i},B_{i})_{i\in I}$ is defined as a family of ordered
quadruples $(X_{i},A_{i},P_{i},B_{i})$.
Definition 4.2 [23] An equilibrium for $\Gamma$ is defined as a point
$x^{\ast}\in X$ such that for each $i\in I$,
$x_{i}^{\ast}\in\overline{B}_{i}(x^{\ast})$ and $A_{i}(x^{\ast})\cap
P_{i}(x^{\ast})=\emptyset$.
Remark 4.1 When, for each $i\in I$, $A_{i}(x)=B_{i}(x)$ for all $x\in X,$ the
abstract economy model coincides with the classical one introduced by Borglin
and Keiding in [3]. If in addition,
$\overline{B}_{i}(x^{\ast})=$cl${}_{X_{i}}$ $B_{i}(x^{\ast})$ for each $x\in
X,$ which is the case where $B_{i}$ has a closed graph in $X\times X_{i}$, the
definition of equilibrium coincides with the one used by Yannelis and
Prabhakar in [22].
Remark 4.2 If the preference set valued map $P_{i}$ is defined by using a
utility function $u_{i}$, that is $P_{i}(x)=\\{y\in
X_{i}:u_{i}(y)>u_{i}(x_{i})\\},$ the irreflexibility condition
$x_{i}\notin\overline{P_{i}}\left(x\right),$ which appears among the
hypothesis of the existence equilibrium theorems, may fail. A case in which
this condition is verified, is when $P_{i}$ is an order interval preference.
Order interval preferences are studied, for instance, in Chateauneuf [5].
These preference relations $\prec$ (on $X$) are representable, if two real
valued functions $u$ and $v$ on $X$ exist and are such that: $x\prec y$
$\Leftrightarrow$ $u(x)<v(y)$. If a representation of the preference relation
$\prec_{i}$ exist, we can define the preference set valued map $P_{i}$ by
$P_{i}(x)=\\{y\in X_{i}:v_{i}(y)>u_{i}(x_{i})\\}$ and the condition
$x_{i}\notin\overline{P_{i}}\left(x\right)$ can be fulfilled.
4.2 Examples of Abstract Economies with Two Constraint Set Valued Maps
A first example of an abstract economy with two constraint set valued maps is
the one associated to the model proposed by Radner [14] and his followers.
This is a model of a pure exchange economy with two periods, present and
future, and uncertainty on the state of nature in the future. There is a
finite number $n$ of agents and a finite number $m$ of possible future states
of nature. Let $I=\\{1,2,...,n\\}$ be the set of agents and
$\Omega=\\{s_{1},s_{2},...,s_{m}\\}$ the set of the future states of the
nature. Each agent has his own private, and tipically incomplete, information
about the future state of nature. For each agent $i$ in $I$, the initial
private information is a partition on $\Omega$ , induced by a signal
$\pi_{i}:\Omega\rightarrow Y_{i}$. In Radner [14], agents make decisions today
without knowing the future state of nature tomorrow. The initial agent’s
information is kept fixed and their consumption plans need to be made
compatible with their information, in the sense that their consumption must be
the same in states that they do not distinguish. In a different framework,
Radner [15] consider the notion of rational expectation equilibrium. In this
model, agents are able to forecast the future equilibrium price. Consequently,
their initial information is updated with a signal given by the future
equilibrium prices $p$ and a more refined partition of $\Omega$ is obtained as
the joint of the initial information and the information generated by
$\widehat{\pi_{i}}(p):\Omega\rightarrow\Delta$, defined by
$\widehat{\pi_{i}}(p)(s)=p_{i}(s)$, where $\Delta$ be the normalized set of
prices. Here, we consider a model in which agents may be able to learn from
market signals. These market signal are summarized by equilibrium prices that
may not be fully revealing. Without loss of generality, let denote the joint
of the initial information and the information generated by prices by
$\widehat{\pi_{i}}$.
For agent $i$ in $I$, the consumption plan in the first period will be denoted
by $x_{0}^{i}\in IR_{+}^{l}$ and in the second period, for each state $s_{j},$
$j=1,2,...,m$, it will be denoted by $x_{j}^{i}\in IR_{+}^{l}.$ A bundle for
agent $i$ is $\ x^{i}=(x_{0}^{i},x_{1}^{i},x_{2}^{i},...,x_{m}^{i}).$ Let
$X_{i}=IR_{+}^{ml+1}.$ Each agent has a preference set valued map
$Q_{i}^{\prime}:\mathop{\textstyle\prod}\limits_{i\in I}X_{i}\rightrightarrows
X_{i}$ and an initial endowment
$e^{i}=(e_{0}^{i},e_{1}^{i},e_{2}^{i},...,e_{m}^{i})\in X_{i}.$
Definition 4.3 A pure exchange economy with assymmetric information is the
family $\mathcal{E=}(I,\Omega,\widehat{\pi}_{i},Q_{i}^{\prime},e^{i})_{i\in
I}.$
Definition 4.4 An allocation for the economy $\mathcal{E}$ is $x=(x^{i})_{i\in
I}.$ The allocation is called phisically feasible if
$\mathop{\textstyle\sum}\limits_{i\in
I}x^{i}\leq\mathop{\textstyle\sum}\limits_{i\in I}e^{i}$ and informationally
feasible for each agent $i$ if
$\widehat{\pi}_{i}(p)(s)=\widehat{\pi}_{i}(p)(s^{\prime})$ implies
$x_{s}^{i}=x_{s^{\prime}}^{i}.$
Let $p_{0}$ be the price in the first period, for the second period let
$p_{j}$ be the price in the state $j,$ $j=1,2,...,m$ and let
$p=(p_{0},p_{1},...,p_{m}).$ Let $\Delta$ be the normalized set of prices.
Without loss of generality, we assume that $p$ belongs to $\Delta$.
The budget set valued map of agent $i$ is $B_{i}:\Delta\rightrightarrows
IR_{+}^{lm+1},$ defined by $B_{i}(p)=\\{x^{i}\in
IR_{+}^{lm+1}:px^{i}<pe^{i}\\}.$
The information set valued map of agent $i$ is $I_{i}:\Delta\rightrightarrows
IR_{+}^{lm+1},$ defined by $I_{i}(p)=\\{x^{i}\in
IR_{+}^{lm+1}:x_{s}^{i}=x_{s^{\prime}}^{i}$ if
$\widehat{\pi}_{i}(p)(s)=\widehat{\pi}_{i}(p)(s^{\prime})\\}.$
Definition 4.5 The pair $(x^{\ast},p^{\ast})\in IR_{+}^{n(lm+1)}\times\Delta$
is an equilibrium for the asymmetrically informed economy $\mathcal{E}$ if
1) $\mathop{\textstyle\sum}\limits_{i\in
I}(x^{\ast})^{i}\leq\mathop{\textstyle\sum}\limits_{i\in I}e^{i}$
and for each $i\in I,$
2)
$(x^{\ast})^{i}\in\overline{I_{i}}(p^{\ast})\cap\overline{B_{i}}(p^{\ast});$
3) $y^{i}\in Q_{i}^{\prime}(x^{\ast})\cap I_{i}(p^{\ast})$ implies that
$y^{i}\notin B_{i}(p^{\ast}).\vskip 6.0pt plus 2.0pt minus 2.0pt$
Let $X:=\mathop{\textstyle\prod}\limits_{i\in I}X_{i}\times\Delta,$ where for
$i\in I,$ $X_{i}=IR_{+}^{lm+1}$ is the consumption set of agent $i.$ Let’s
define the following set valued maps:
-for each $i\in I,$ $Q_{i}:X\rightrightarrows X_{i}$ is the preference set valued map defined by $Q_{i}(x,p)=Q_{i}^{\prime}(x)$ for each $(x,p)\in X;$
\- $Q_{n+1}:X\rightrightarrows\Delta$ is the preference set valued map defined
by $Q_{n+1}(x,p):=\\{q\in\Delta:q(\mathop{\textstyle\sum}\limits_{i\in
I}(x^{i}-e^{i}))>p(\mathop{\textstyle\sum}\limits_{i\in I}(x^{i}-e^{i}))\\}$
for each $(x,p)\in X;$
-for $i\in I,$ $A_{i}:X\rightrightarrows 2^{X_{i}}$ is defined by $A_{i}(x,p):=\\{y^{i}\in IR_{+}^{lm+1}:py^{i}<pe^{i}\\}$ for each $(x,p)\in X;$
-$A_{n+1}:X\rightrightarrows\Delta$ is defined by $A_{n+1}(x,p):=\Delta$ for each $(x,p)\in X;$
-for $i\in I,$ $I_{i}:X\rightrightarrows X_{i}$ is defined by $I_{i}(x,p):=\\{y^{i}\in IR_{+}^{lm+1}:y_{s}^{i}=y_{s^{\prime}}^{i}$ if $\widehat{\pi}_{i}(p)(s)=\widehat{\pi}_{i}(p)(s^{\prime})\\}$ for each $(x,p)\in X;$
\- $I_{n+1}:X\rightrightarrows\Delta$ is defined by $I_{n+1}(x,p):=\Delta$ for
each $(x,p)\in X;\vskip 6.0pt plus 2.0pt minus 2.0pt$
Definition 4.6 The abstract economy associated to the model of the pure
exchange economy with assymmetric information is
$\Gamma=(X_{i},A_{i},P_{i},B_{i})_{i\in\\{1,2,...,n+1\\}},$ where:
-for $i\in I,$ $X_{i}:=IR_{+}^{lm+1}$ is the consumption set of agent $i$ and let $X:=\mathop{\textstyle\prod}\limits_{i\in I}X_{i}\times\Delta;$
-$P_{i}:X\rightrightarrows X_{i}$ $(i\in I)$ and $P_{n+1}:X\rightrightarrows\Delta$ are the preference set valued maps defined by $P_{i}(x,p)=Q_{i}(x,p)\cap I_{i}(x,p)$ for each $(x,p)\in X$ and $i\in\\{1,2,...,n+1\\};$
-$A_{i}:X\rightrightarrows X_{i}$ $(i\in I)$ and $A_{n+1}:X\rightrightarrows\Delta$ are the constraint set valued maps defined above;
-$B_{i}:X\rightrightarrows X_{i}$ $(i\in I)$ and $B_{n+1}:X\rightrightarrows\Delta$ are the constraint set valued maps defined by $B_{i}(x,p):=A_{i}(x,p)\cap I_{i}(x,p)$ for each $(x,p)\in X$ and $i\in\\{1,2,...,n+1\\}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Remark 4.3 We note that $A_{i}(x,p)\cap P_{i}(x,p)\subseteq B_{i}(x,p)$ for
each $(x,p)\in X$ and for each $i\in\\{1,2,...,n+1\\}.$
Proposition 4.1 An equilibrium for the associated abstract economy $\Gamma$ is
an equilibrium of the economy with assymmetric information $E$.
Proof Let $(x^{\ast},p^{\ast})$ be an equilibrium for $\Gamma.$
1) For each $i\in\\{1,2,...,n\\},$ we have that
$(x^{\ast})^{i}\in\overline{B_{i}}(x^{\ast},p^{\ast})=\overline{(A_{i}\cap
I_{i})}(x^{\ast},p^{\ast})$ and then, by definition of $A_{i}$ and $I_{i},$
$(x^{\ast})^{i}\in\overline{(I_{i}\cap B_{i})}(p^{\ast})$;
2) $p^{\ast}\in\overline{B_{n+1}}(x^{\ast},p^{\ast})=\Delta;$
3) for each $i\in\\{1,2,...,n\\},$ we have that $A_{i}(x^{\ast},p^{\ast})\cap
P_{i}(x^{\ast},p^{\ast})=\phi$, which implies that if $y^{i}\in
P_{i}(x^{\ast},p^{\ast})=Q_{i}(x^{\ast},p^{\ast})\cap
I_{i}(x^{\ast},p^{\ast}),$ then $y^{i}\notin A_{i}(x^{\ast},p^{\ast});$ This
means that $y^{i}\in Q_{i}^{\prime}(x^{\ast})\cap I_{i}(p^{\ast})$ implies
that $y^{i}\notin B_{i}(p^{\ast});$
4) we have that $A_{n+1}(x^{\ast},p^{\ast})\cap
P_{n+1}(x^{\ast},p^{\ast})=\phi$, which is equivalent with
$\\{q\in\Delta:q(\mathop{\textstyle\sum}\limits_{i\in
I}((x^{\ast})^{i}-e^{i}))>p^{\ast}(\mathop{\textstyle\sum}\limits_{i\in
I}((x^{\ast})^{i}-e^{i}))\\}\cap\Delta=\phi.$ This fact implies that
$q(\mathop{\textstyle\sum}\limits_{i\in I}((x^{\ast})^{i}-e^{i}))\leq
p^{\ast}(\mathop{\textstyle\sum}\limits_{i\in I}((x^{\ast})^{i}-e^{i}))\leq 0$
for all $q\in\Delta.$ If we choose $q$ as a vector of the canonical basis of
$IR^{ml+1},$ that is $q_{j}=1$ and $q_{i}=0$ for $i\neq j,$ where
$i,j\in\\{1,2,...,ml+1\\},$ we obtain that
$\mathop{\textstyle\sum}\limits_{i\in
I}(x^{\ast})^{i}\leq\mathop{\textstyle\sum}\limits_{i\in I}e^{i}.$ $\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \square\vskip 6.0pt plus 2.0pt minus 2.0pt$
The second example is the abstract economy associated to an exchange economy
with two constraint set valued maps, the first one being the budget set valued
map and the second one being the consumption set that depends on prices.
The third example follows the idea of an exchange economy which has, beyond
the budget set valued map, a second constraint set valued map $G_{i},$ defined
by the delivery conditions as stated in the paper by Correia-da-Silva and
Herves-Beloso [6].
Let’s assume that the set of the states of nature is $\Omega=\\{1,2,...,m\\}$,
the future prices are $p_{1},p_{2},...,p_{m}\in IR_{+}^{l}$ and that each
agent $i$ has a signal $f_{i}:\Omega\rightarrow Y_{i}$ such that
$f_{i}(s)=f_{i}(s^{\prime})$ if $s$ and $s^{\prime}$ are states that cannot be
distinguished. The agent $i$ chooses a portfolio $y(s)$ in the following way:
$p_{s}y(s)\leq p_{s}y(s^{\prime})$ for all $s^{\prime}$ such that
$f_{i}(s^{\prime})=f_{i}(s).$
The set valued map $G_{i}:X\times\Delta\rightrightarrows IR^{lm}$ is defined
by $G_{i}(x,p)=\\{y\in IR^{lm}:p_{s}y(s)\leq p_{s}y(s^{\prime})$ for all
$s^{\prime}$ such that $f_{i}(s^{\prime})=f_{i}(s)\\}.\vskip 6.0pt plus 2.0pt
minus 2.0pt$
4.3 The Existence of Equilibria in Locally Convex Spaces
As an application of the fixed point Theorem 3.1, we have the following
result.
Theorem 4.1 Let $\Gamma=\left\\{X_{i},A_{i},B_{i},P_{i}\right\\}_{i\in I}$ be
an abstract economy such that for each $i\in I,$ the following conditions are
fulfilled:
1) $X_{i}$ is a nonempty convex subset of a Hausdorff locally convex
topological vector space $E_{i}\,$ and $D_{i}$ is a nonempty compact convex
subset of $X_{i}$;
2) for each $x\in X:=\prod\limits_{i\in I}X_{i},$ $A_{i}\left(x\right)$ and
$P_{i}(x)$ are convex, $B_{i}\left(x\right)$ is nonempty, convex and
$A_{i}\left(x\right)\cap P_{i}(x)\subset B_{i}(x)$;
3) $W_{i}:=\left\\{x\in X:A_{i}\left(x\right)\cap
P_{i}\left(x\right)\neq\emptyset\right\\}$ is open in $X$.
4) $H_{i}:X\rightrightarrows X_{i}$ defined by
$H_{i}\left(x\right):=A_{i}(x)\cap P_{i}\left(x\right)$ for each $x\in X$ is
almost w-upper semicontinuous with respect to $D_{i}$ on $W_{i}$ and
$\overline{H_{i}^{V_{i}}}$ is convex nonempty valued for each open absolutely
convex symmetric neighborhood $V_{i}$ of $0$ in $E_{i}$;
5) $B_{i}:X\rightrightarrows X_{i}$ is almost w-upper semicontinuous with
respect to $D_{i}$ and $\overline{B_{i}^{V_{i}}}$ is convex nonempty valued
for each open absolutely convex symmetric neighborhood $V_{i}$ of $0$ in
$E_{i}$;
6) for each $x\in X$ , $x_{i}\notin\overline{\mathit{(}A_{i}\cap
P_{i})}\left(x\right)$;
Then there exists $x^{\ast}\in D=$ $\prod\limits_{i\in I}D_{i}$ such that
$x_{i}^{\ast}\in\overline{B}_{i}\left(x^{\ast}\right)$ and $(A_{i}\cap
P_{i})(x^{\ast})=\emptyset$ for each $i\in I.\vskip 6.0pt plus 2.0pt minus
2.0pt$
Proof Let $i\in I.$ By condition (3) we know that $W_{i}$ is open in $X.$
Let’s define $T_{i}:X\rightrightarrows X_{i}$ by
$T_{i}\left(x\right):=\left\\{\begin{array}[]{c}A_{i}\left(x\right)\cap
P_{i}\left(x\right),\text{ if }x\in W_{i},\\\ B_{i}\left(x\right),\text{ \ \ \
\ \ \ \ \ \ \ \ if }x\notin W_{i}\end{array}\right.$ for each $x\in X.$
Then $T_{i}:X\rightrightarrows X_{i}$ is a set valued map with nonempty convex
values. We shall prove that $T_{i}:X\rightrightarrows D_{i}$ is almost w-upper
semicontinuous with respect to $D_{i}$. Let ßi be a basis of open absolutely
convex symmetric neighborhoods of $0$ in $E_{i}$ and let
ß=$\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i}.$
For each $V=(V_{i})_{i\in I}\in\mathop{\textstyle\prod}\limits_{i\in
I}$ß${}_{i},$ for each $x\in X,$ let for each $i\in I$
$B^{V_{i}}(x):=(B_{i}\left(x\right)+V_{i})\cap D_{i}$,
$F^{V_{i}}(x):=((A_{i}\left(x\right)\cap P_{i}\left(x\right))+V_{i})\cap
D_{i}$ and
$T_{i}^{V_{i}}(x)"=\left\\{\begin{array}[]{c}F^{V_{i}}(x),\text{ if }x\in
W_{i},\\\ B^{V_{i}}(x),\text{\ if }x\notin W_{i}.\end{array}\right.$
For each open set $V_{i}^{\prime}$ in $D_{i}$, the set
$\left\\{x\in X:\overline{T_{i}^{V_{i}}}\left(x\right)\subset
V_{i}^{\prime}\right\\}=$
$=\left\\{x\in W_{i}:\overline{F^{V_{i}}}(x)\subset
V_{i}^{\prime}\right\\}\cup\left\\{x\in X\smallsetminus
W_{i}:\overline{B^{V_{i}}}(x)\subset V_{i}^{{}^{\prime}}\right\\}$
$=\left\\{x\in W_{i}:\overline{F^{V_{i}}}(x)\subset
V_{i}^{{}^{\prime}}\right\\}\cup\left\\{x\in X:\overline{B^{V_{i}}}(x)\subset
V_{i}^{\prime}\right\\}.$
According to condition (4), the set $\left\\{x\in
W_{i}:\overline{F^{V_{i}}}(x)\subset V_{i}^{\prime}\right\\}$ is open in $X$.
The set $\left\\{x\in X:\overline{B^{V_{i}}}(x)\subset
V_{i}^{\prime}\right\\}$ is open in $X$ because $\overline{B^{V_{i}}}$ is
upper semicontinuous.
Therefore, the set $\left\\{x\in
X:\overline{T_{i}^{V_{i}}}\left(x\right)\subset V_{i}^{\prime}\right\\}$ is
open in $X.$ It shows that $\overline{T_{i}^{V_{i}}}:X\rightrightarrows D_{i}$
is upper semicontinuous. According to Theorem 3.1, there exists $x^{\ast}\in
D=$ $\prod\limits_{i\in I}D_{i}$ such that
$x^{\ast}\in\overline{T}_{i}\left(x^{\ast}\right),$ for each $i\in I.$ By
condition (5) we have that
$x_{i}^{\ast}\in\overline{B}_{i}\left(x^{\ast}\right)$ and $(A_{i}\cap
P_{i})(x^{\ast})=\emptyset$ for each $i\in I.$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \square\vskip 6.0pt plus 2.0pt minus 2.0pt$
Example 4.1 Let $\Gamma=\left\\{X_{i},A_{i},B_{i},P_{i}\right\\}_{i\in I}$ be
an abstract economy, where $I=\\{1,2,...,n\\},$ $X_{i}:=[0,4]$ be a compact
convex choice set, $D_{i}:=[0,2]$ for each $i\in I$ and $X:=\prod\limits_{i\in
I}X_{i}$.
Let the set valued maps $A_{i},B_{i},P_{i}:X\rightrightarrows X_{i}$ be
defined as follows:
for each $(x_{1},x_{2},...,x_{n})\in X,$
$A_{i}(x):=\left\\{\begin{array}[]{c}[1-x_{i},2]\text{ if
}x\in(0,\frac{1}{2})^{n};\\\ [1-x_{i},2)\text{ if }x\in[\frac{1}{2},1)^{n};\\\
[3,4]\text{ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ }x=0;\\\ [0,\frac{1}{2}]\text{,
\ \ \ \ \ \ \ \ \ \ \ \ otherwise;}\end{array}\right.$
$P_{i}(x):=\left\\{\begin{array}[]{c}[\frac{3}{2},2+x_{i}]\text{ if
}x\in[0,1)^{n};\\\ [1,2]\text{, \ \ \ \ \ \ \ \ \ \ \ \ \ \
otherwise;}\end{array}\right.$
$B_{i}(x):=\left\\{\begin{array}[]{c}[0,2]\text{ if }x\in[0,1);\\\ [3,4]\text{
\ \ \ if \ \ \ }x=0;\\\ [0,2)\text{,\ \ \ \ \ otherwise.}\end{array}\right.$
The set valued maps $A_{i},B_{i},P_{i}$ are not upper semicontinuous on $X.$
$A_{i}(x)\cap P_{i}(x):=\left\\{\begin{array}[]{c}[\frac{3}{2},2]\text{ if
}x\in(0,\frac{1}{2})^{n};\\\ [\frac{3}{2},2)\text{ if
}x\in[\frac{1}{2},1)^{n};\\\ \phi\text{, \ \ \ \ \ \ \ \ \ \ \ \ \ \
otherwise.}\end{array}\right.$
$W_{i}:=\left\\{x\in X:A_{i}\left(x\right)\cap
P_{i}\left(x\right)\neq\emptyset\right\\}=(0,1)^{n}$ is open in $X.$
$\overline{\mathit{(}A_{i}\cap
P_{i})}\left(x\right):=\left\\{\begin{array}[]{c}[\frac{3}{2},2]\text{ if
}x\in[0,1]^{n};\\\ \phi\text{, \ \ \ \ \ \ \ \ \ \ \
otherwise.}\end{array}\right.$
We notice that for each $x\in X$ , $x_{i}\notin\overline{\mathit{(}A_{i}\cap
P_{i})}\left(x\right).$
We shall prove that $B_{i}$ and $\mathit{(}A_{i}\cap P_{i})_{W_{i}}$ are
almost w-upper semicontinuous with respect to $D_{i}=[0,2].$
On $W_{i},$
$\mathit{(}A_{i}\cap
P_{i})\left(x\right):=\left\\{\begin{array}[]{c}[\frac{3}{2},2]\text{ if
}x\in(0,\frac{1}{2})^{n};\\\ [\frac{3}{2},2)\text{ if
}x\in[\frac{1}{2},1)^{n};\end{array}\right.,$
$\mathit{(}A_{i}\cap
P_{i})\left(x\right)+(-\varepsilon,\varepsilon)=(\frac{3}{2}-\varepsilon,2+\varepsilon)$
if $x\in(0,1)^{n};$
Let $\mathit{(}A_{i}\cap P_{i})^{V}\left(x\right)=(\mathit{(}A_{i}\cap
P_{i})\left(x\right)+(-\varepsilon,\varepsilon))\cap[0,2],$ where
$V=(-\varepsilon,\varepsilon).$
Then,
if $\varepsilon\in(0,\frac{3}{2}],$
$\mathit{(}A_{i}\cap P_{i})^{V}\left(x\right)=(\frac{3}{2}-\varepsilon,2]$ if
$x\in(0,1)^{n};$
if $\varepsilon>\frac{3}{2},$
$\mathit{(}A_{i}\cap P_{i})^{V}\left(x\right)=[0,2]$ if $x\in(0,1)^{n};$
Hence, for each $V=(-\varepsilon,\varepsilon),$ $\overline{\mathit{(}A_{i}\cap
P_{i})^{V}}_{W_{i}}$ is upper semicontinuous and has nonempty values.
$B_{i}\left(x\right)+(-\varepsilon,\varepsilon)=\left\\{\begin{array}[]{c}(-\varepsilon,\text{
}2+\varepsilon)\text{ if },\text{ }x\in(0,1)^{n};\\\ (3-\varepsilon,\text{
}4+\varepsilon)\text{\ \ \ if \ \ }x=0;\\\ (-\varepsilon,\text{
}2+\varepsilon)\text{ \ \ \ \ \ \ otherwise.}\end{array}\right.$
Let
$B_{i}{}^{V}\left(x\right)=(B_{i}\left(x\right)+(-\varepsilon,\varepsilon))\cap[0,2],$
where $V=(-\varepsilon,\varepsilon).$
Then,
if $\varepsilon\in(0,1],$
$B_{i}{}^{V}\left(x\right)=\left\\{\begin{array}[]{c}\phi\text{ \ \ \ \ \ if \
}x=0;\\\ [0,2]\text{\ otherwise;}\end{array}\right.$
if $\varepsilon\in(1,3],$
$B_{i}{}^{V}\left(x\right)=\left\\{\begin{array}[]{c}[0,2]\text{ if
}x\in[0,1)^{n};\\\ (3-\varepsilon,2]\text{ if }x=0;\\\ [0,2],\text{ \ \ \ \ \
\ otherwise.}\end{array}\right.$
and if $\varepsilon>3,$ $B_{i}{}^{V}\left(x\right)=[0,2]$ if $x\in X.$
Then, for each $V=(-\varepsilon,\varepsilon),$ $\overline{B_{i}^{V}}$ is upper
semicontinuous and has nonempty values.
Therefore, all hypotheses of Theorem 4.1 are satisfied, so that there exist
equilibrium points. For example,
$x^{\ast}=\\{\frac{3}{2},\frac{3}{2},...,\frac{3}{2}\\}\in X$ verifies
$x_{i}^{\ast}\in\overline{B}_{i}\left(x^{\ast}\right)$ and $(A_{i}\cap
P_{i})(x^{\ast})=\emptyset.$
Theorem 4.2 deals with abstract economies which have dual w-upper
semicontinuous pairs of set valued maps.
Theorem 4.2 Let $\Gamma=\left\\{X_{i},A_{i},B_{i},P_{i}\right\\}_{i\in I}$ be
an abstract economy such that for each $i\in I,$ the following conditions are
fulfilled:
1) $X_{i}$ is a nonempty convex subset of a Hausdorff locally convex
topological vector space $E_{i}\,$and $D_{i}$ is a nonempty compact convex
subset of $X_{i}$;
2) for each $x\in X:=\prod\limits_{i\in I}X_{i},$ $P_{i}(x)\subset D_{i},$
$A_{i}\left(x\right)\cap P_{i}\left(x\right)\subset B_{i}\left(x\right)$ and
$B_{i}\left(x\right)$ is nonempty;
3) the set $W_{i}:=\left\\{x\in X:A_{i}\left(x\right)\cap
P_{i}\left(x\right)\neq\emptyset\right\\}$ is open in $X$;
4) the pair $(A_{i\mid\text{cl}W_{i}},P_{i\mid\text{cl}W_{i}})$ is dual almost
w-upper semicontinuous with respect to $D_{i}$, $B_{i}:X\rightrightarrows
X_{i}$ is almost w-upper semicontinuous with respect to $D_{i}$;
5) if $T_{i,V_{i}}:X\rightrightarrows X_{i}$ is defined by
$T_{i,V_{i}}(x):=(A_{i}(x)+V_{i})\cap D_{i}\cap P_{i}(x)$ for each $x\in X,$
then the set valued maps $\overline{B_{i}^{V_{i}}}$ and
$\overline{T_{i,V_{i}}}$ are nonempty convex valued for each open absolutely
convex symmetric neighborhood $V_{i}$ of $0$ in $E_{i}$;
6) for each $x\in X$ , $x_{i}\notin\overline{P}_{i}\left(x\right)$;
Then, there exists $x^{\ast}\in D:=$ $\prod\limits_{i\in I}D_{i}$ such that
$x_{i}^{\ast}\in\overline{B}_{i}\left(x^{\ast}\right)$ and
$A_{i}\left(x^{\ast}\right)\cap P_{i}\left(x^{\ast}\right)=\emptyset$ for all
$i\in I.\vskip 6.0pt plus 2.0pt minus 2.0pt$
Proof For each $i\in I,$ let ßi denote the family of all open absolutely
convex symmetric neighborhoods of zero in $E_{i}$ and let
ß$=\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i}.$ For each
$V=\mathop{\textstyle\prod}\limits_{i\in
I}V_{i}\in\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i},$ for each $i\in
I,$ let
$B^{V_{i}}(x):=(B_{i}\left(x\right)+V_{i})\cap D_{i}$ for each $x\in X$ and
$S_{i}^{V_{i}}\left(x\right):=\left\\{\begin{array}[]{c}T_{i,V_{i}}(x),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if }x\in W_{i},\\\ B_{i}^{V_{i}}(x),\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if }x\notin W_{i},\end{array}\right.$
$\overline{S_{i}^{V_{i}}}$ has closed values. Next, we shall prove that
$\overline{S_{i}^{V_{i}}}:X\rightrightarrows D_{i}$ is upper semicontinuous.
For each open set $V^{\prime}$ in $D_{i}$, the set
$\left\\{x\in X:\overline{S_{i}^{V_{i}}}\left(x\right)\subset
V^{\prime}\right\\}=$
$=\left\\{x\in W_{i}:\overline{T_{i,V_{i}}}(x)\subset
V^{\prime}\right\\}\cup\left\\{x\in X\smallsetminus
W_{i}:\overline{B_{i}^{V_{i}}}(x)\subset V^{\prime}\right\\}$
=$\left\\{x\in W_{i}:\overline{T_{i,V_{i}}}(x)\subset
V^{\prime}\right\\}\cup\left\\{x\in X:\overline{B_{i}^{V_{i}}}(x)\subset
V^{\prime}\right\\}.$
We know that the set valued map $\overline{T_{i,V_{i}}}(x)_{\mid W_{i}}:$
$W_{i}\rightrightarrows D_{i}$ is upper semicontinuous. The set $\left\\{x\in
W_{i}:\overline{T_{i,V_{i}}}(x)\subset V^{\prime}\right\\}$ is open in $X.$
Since $\overline{B_{i}^{V_{i}}}(x):X\rightrightarrows D_{i}$ is upper
semicontinuous, the set $\\{x\in X:\overline{B_{i}^{V_{i}}}(x)\\}\subset
V^{\prime}$ is open in $X$ and therefore, the set $\left\\{x\in
X:\overline{S_{i}^{V_{i}}}\left(x\right)\subset V^{\prime}\right\\}$ is open
in $X$. It proves that $\overline{S_{i}^{V_{i}}}:X\rightrightarrows D_{i}$ is
upper semicontinuous. According to Himmelberg’s Theorem, applied for the set
valued maps $\overline{S_{i}^{V_{i}}},$ there exists a point $x_{V}^{\ast}\in
D=$ $\prod\limits_{i\in I}D_{i}$ such that $(x_{V}^{\ast})_{i}\in
S_{i}^{V_{i}}\left(x_{V}^{\ast}\right)$ for each $i\in I.$ By condition (5),
we have that
$(x_{V}^{\ast})_{i}\notin\overline{P_{i}}\left(x_{V}^{\ast}\right),$ hence,
$(x_{V}^{\ast})_{i}\notin\overline{A_{i}^{V_{i}}}\left(x_{V}^{\ast}\right)\cap\overline{P_{i}}\left(x_{V}^{\ast}\right)$.
We also have that cl Gr $(T_{i,V_{i}})\subseteq$ cl Gr $(A_{i}^{V_{i}})\cap$cl
Gr $P_{i}.$ Then $\overline{T_{i,V_{i}}}(x)\subseteq$
$\overline{A_{i}^{V_{i}}}(x)\cap\overline{P_{i}}\left(x\right)$ for each $x\in
X.$ It follows that
$(x_{V}^{\ast})_{i}\notin\overline{T_{i,V_{i}}}(x_{V}^{\ast}).$ Therefore,
$(x_{V}^{\ast})_{i}\in\overline{B^{V_{i}}}\left(x_{V}^{\ast}\right).$
For each $V=(V_{i})_{i\in I}\in\mathop{\textstyle\prod}\limits_{i\in
I}$ß${}_{i},$ let’s define $Q_{V}=\cap_{i\in I}\\{x\in
D:x\in\overline{B^{V_{i}}}\left(x\right)$ and $A_{i}\left(x\right)\cap
P_{i}\left(x\right)=\emptyset\\}.$
$Q_{V}$ is nonempty since $x_{V}^{\ast}\in Q_{V},$ and it is a closed subset
of $D$ according to (3). Then, $Q_{V}$ is nonempty and compact.
Let ß=$\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i}.$ We prove that the
family $\\{Q_{V}:V\in\text{\ss}\\}$ has the finite intersection property.
Let $\\{V^{(1)},V^{(2)},...,V^{(n)}\\}$ be any finite set of ß and let
$V^{(k)}=\underset{i\in I}{\mathop{\textstyle\prod}}V_{i}^{(k)}{}_{i\in I}$,
$k=1,...,n.$ For each $i\in I$, let
$V_{i}=\underset{k=1}{\overset{n}{\cap}}V_{i}^{(k)}$, then
$V_{i}\in\text{\ss}_{i};$ thus $V\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}.$ Clearly
$Q_{V}\subset\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}$ so that
$\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}\neq\emptyset.$
Since $D$ is compact and the family $\\{Q_{V}:V\in\text{\ss}\\}$ has the
finite intersection property, we have that
$\cap\\{Q_{V}:V\in\text{\ss}\\}\neq\emptyset.$ Take any
$x^{\ast}\in\cap\\{Q_{V}:V\in$ß$\\},$ then for each $V\in\text{\ss},$
$x^{\ast}\in\cap_{i\in I}\left\\{x^{\ast}\in
D:x_{i}^{\ast}\in\overline{B^{V_{i}}}\left(x\right)\text{ and
}A_{i}\left(x\right)\cap P_{i}\left(x\right)=\emptyset)\right\\}.$
Hence, $x_{i}^{\ast}\in\overline{B^{V_{i}}}\left(x^{\ast}\right)$ for each
$V\in$ß and for each $i\in I.$ According to Lemma __ 2.2,__ we have that __
$x_{i}^{\ast}\in\overline{B_{i}}(x^{\ast})$ and $(A_{i}\cap
P_{i})(x^{\ast})=\emptyset$ for each $i\in I.$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \square$
We now introduce the following concept, which also generalizes the concept of
lower semicontinuous set valued maps.
Definition 4.7 Let $X$ be a non-empty convex subset of a topological linear
space $E$, $Y$ be a non-empty set in a topological space and $K\subseteq
X\times Y.$
The set valued map $T:X\times Y\rightrightarrows X$ has the e-USCS-property
(e-upper semicontinuous selection property) on $K,$ if for each absolutely
convex neighborhood $V$ of zero in $E,$ there exists an upper semicontinuous
set valued map with convex values $S^{V}:X\times Y\rightrightarrows X$ such
that $S^{V}(x,y)\subset T(x,y)+V$ and $x\notin$cl $S^{V}(x,y)$ for every
$(x,y)\in K$.
The following theorem is an equilibrium existence result for economies with
constraint set valued maps having e-USCS-property.
Theorem 4.3 Let $\Gamma=(X_{i},A_{i},P_{i},B_{i})_{i\in I}$ be an abstract
economy, where $I$ is a (possibly uncountable) set of agents such that for
each $i\in I:$
(1) $X_{i}$ is a non-empty compact convex set in a locally convex space
$E_{i}$;
(2) cl $B_{i}$ is upper semicontinuous with non-empty convex values;
(3) the set $W_{i}:$ $=\left\\{x\in X\text{ / }\left(A_{i}\cap
P_{i}\right)(x)\neq\emptyset\right\\}$ is open;
(3) cl $(A_{i}\cap P_{i})$ has the e-USCS-property on $W_{i}$.
Then there exists an equilibrium point $x^{\ast}\in X$ for $\Gamma$,$\ i.e.$,
for each $i\in I$, $x_{i}^{\ast}\in\overline{B}_{i}(x^{\ast})$ and
$A_{i}(x^{\ast})\cap P_{i}(x^{\ast})=\emptyset$.
Proof For each $i\in I$, let ßi denote the family of all open convex
neighborhoods of zero in $E_{i}.$ Let $V=(V_{i})_{i\in
I}\in\mathop{\textstyle\prod}\limits_{i\in I}$ß${}_{i}.$ Since cl $(A_{i}\cap
P_{i})$ has the e-USCS-property on $W_{i}$, it follows that there exists an
upper semicontinuous set valued map $F_{i}^{V_{i}}:X\rightrightarrows X_{i}$
such that $F_{i}^{V_{i}}(x)\subset$cl $(A_{i}\cap P_{i})(x)+V_{i}$ and
$x_{i}\notin$cl $F_{i}^{V_{i}}(x)$ for each $x\in W_{i}$.
Define the set valued map $T_{i}^{V_{i}}:X\rightrightarrows X_{i}$, by
$T_{i}^{V_{i}}(x):=\left\\{\begin{array}[]{c}\text{cl
}\\{F_{i}^{V_{i}}(x)\\}\text{, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
if }x\in W_{i}\text{, }\\\ \text{cl }(B_{i}(x)+V_{i})\cap X_{i}\text{, \ \ \ \
\ \ \ \ \ if }x\notin W_{i}\text{;}\end{array}\right.$
$B_{i}^{V_{i}}:X\rightrightarrows X_{i},$ $B_{i}^{V_{i}}(x):=$cl
$(B_{i}(x)+V_{i})\cap X_{i}=($cl $B_{i}(x)+$cl $V_{i})\cap X_{i}$ is upper
semicontinuous by Lemma 2.1 _._
Let $U$ be an open subset of $\ X_{i}$, then
$U^{{}^{\prime}}:=\\{x\in X$ $\mid T_{i}^{V_{i}}(x)\subset U\\}$
=$\\{x\in W_{i}$ $\mid T_{i}^{V_{i}}(x)\subset U\\}\cup\\{x\in X\setminus
W_{i}$ $\mid$ $T_{i}^{V_{i}}(x)\subset U\\}$
=$\left\\{x\in W_{i}\text{ }\mid\text{cl }F_{i}^{V_{i}}(x)\subset
U\right\\}\cup\left\\{x\in X\mid\text{ }(\text{cl
}B_{i}(x)+\overline{V_{i}})\cap X_{i}\subset U\right\\}$
$U^{{}^{\prime}}$ is an open set, because $W_{i}$ is open, $\left\\{x\in
W_{i}\text{ }\mid\text{cl }F_{i}^{V_{i}}(x)\subset U\right\\}$ is open since
cl$F_{i}^{V_{i}}(x)$ is an upper semicontinuous map on $W_{i}.$ We have also
that the set $\left\\{x\in X\mid\text{ }(\text{cl }B_{i}(x)+\text{cl
}V_{i})\cap X_{i}\subset U\right\\}$ is open since $($cl $B_{i}+$cl
$V_{i})\cap X_{i}$ is u.s.c. Then $T_{i}^{V_{i}}$ is upper semicontinuous on
$X$ and has closed convex values.
Define $T^{V}:X\rightrightarrows X$ by $T^{V}(x):=\underset{i\in
I}{\prod}T_{i}^{V_{i}}(x)$ for each $x\in X$.
$T^{V}$ is an upper semicontinuous set valued map and has also non-empty
convex closed values.
Since $X$ is a compact convex set, by Fan’s fixed-point theorem [8], there
exists $x_{V}^{\ast}\in X$ such that $x_{V}^{\ast}\in T^{V}(x_{V}^{\ast})$,
i.e., for each $i\in I$, $(x_{V}^{\ast})_{i}\in T_{i}^{V_{i}}(x_{V}^{\ast})$.
If $x_{V}^{\ast}\in W_{i},$ $(x_{V}^{\ast})_{i}\in$cl
$F_{i}^{V_{i}}(x_{V}^{\ast})$, which is a contradiction.
Hence, $(x_{V}^{\ast})_{i}\in$cl $(B_{i}(x_{V}^{\ast})+V_{i})\cap X_{i}$ and
$(A_{i}\cap P_{i})(x_{V}^{\ast})=\emptyset,$ i.e. $x_{V}^{\ast}\in Q_{V}$
where
$Q_{V}=\cap_{i\in I}\\{x\in X:$ $x_{i}\in$cl $(B_{i}(x)+V_{i})\cap X_{i}$ and
$(A_{i}\cap P_{i})(x)=\emptyset\\}.$
Since $W_{i}$ is open, $Q_{V}$ is the intersection of non-empty closed sets,
therefore it is non-empty, closed in $X$.
We prove that the family $\\{Q_{V}:V\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}\\}$ has the finite intersection
property.
Let $\\{V^{(1)},V^{(2)},...,V^{(n)}\\}$ be any finite set of $\underset{i\in
I}{\mathop{\textstyle\prod}\text{\ss}_{i}}$ and let
$V^{(k)}=(V_{i}^{(k)})_{i\in I}$, $k=1,...n.$ For each $i\in I$, let
$V_{i}=\underset{k=1}{\overset{n}{\cap}}V_{i}^{(k)}$, then
$V_{i}\in\text{\ss}_{i};$ thus $V=(V_{i})_{i\in I}\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}.$ Clearly
$Q_{V}\subset\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}$ so that
$\underset{k=1}{\overset{n}{\cap}}Q_{V^{(k)}}\neq\emptyset.$
Since $X$ is compact and the family $\\{Q_{V}:V\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}\\}$ has the finite intersection
property, we have that $\cap\\{Q_{V}:V\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}\\}\neq\emptyset.$ Take any
$x^{\ast}\in\cap\\{Q_{V}:V\in\underset{i\in
I}{\mathop{\textstyle\prod}}\text{\ss}_{i}\\},$ then for each $i\in I$ and
each $V_{i}\in\text{\ss}_{i},$ $x_{i}^{\ast}\in$cl$(B_{i}(x^{\ast})+V_{i})\cap
X_{i}$ and $(A_{i}\cap P_{i})(x^{\ast})=\emptyset;$ but then
$x_{i}^{\ast}\in\overline{B}_{i}(x^{\ast})$ from Lemma 2.2 __ and $(A_{i}\cap
P_{i})(x^{\ast})=\emptyset$ for each $i\in I$ so that $x^{\ast}$ is an
equilibrium point of $\Gamma$ in X. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \square$
5\. Concluding Remarks
We proved a fixed point theorem for the w-upper semicontinuous set valued
maps. We also obtained results concerning the existence of equilibria for
Yuan’s model of an abstract economy without continuity assumptions.
The first author thanks the support by Research Grants ECO2012-38860-C02-02
(Ministerio de Economia y Competitividad), RGEA and 10PXIB300141PR (Xunta de
Galicia and FEDER).
References
## References
* (1) R. P. Agarwal, O’Regan, 2001. A Note on Equilibria for abstract economies, Mathematical and Computer Modelling 34, 331-343.
* (2) K. J. Arrow, G. Debreu, 1954. Existence of an Equilibrium for a Competitive Economy. Econometrica 22, 265-290.
* (3) A. Borglin and H. Keiding, 1976. Existence of equilibrium action and of equilibrium:A note on the ”new” existence theorem. J. Math. Econom. 3, 313-316.
* (4) S.-Y. Chang, 2010. Inequalities and Nash equilibria. Nonlinear Analysis: Theory, Methods & Applications, Vol. 73, 9, 2933-2940.
* (5) A. Chateauneuf, 1987. Continuous representation of a preference relation on a connected topological space, Journal of Mathematical Economics 16, 139-146.
* (6) J. Correia-da-Silva and C. Hervés-Beloso, 2012. General equilibrium in economies with uncertain delivery, Economic Theory, 51 3,, 729-755.
* (7) G. Debreu, 1952. A social equilibrium existence theorem. Proc. Nat. Acad. Sci. 38, 886-893.
* (8) K. Fan,1961. A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142, 305-310.
* (9) C. J. Himmelberg, 1972. Fixed points of compact multifunctions. J. Math. Anal. Appl. 38, 205-207.
* (10) A. A. Kulkarni, V. Shanbhag, Revisiting Generalized Nash Games and Variational Inequalities. Journal of Optimization Theory and Applications, 154 (1), 175-186 (2012)
* (11) J.F. Nash, 1951. Non-cooperative games. Ann. Math. 54, 286-295.
* (12) R. Nessah, G. Tian, Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions. J.O.T.A., 2012 DOI 10.1007/s10957-012-0176-5
* (13) M. Patriche, Equilibrium in games and competitive economies.The Publishing House of the Romanian Academy (2011)
* (14) R. Radner, 1968. Competitive Equilibrium Under Uncertainty, Econometrica, 36 1, 31-58.
* (15) R. Radner, 1979. Rational Expectations Equilibrium: Generic Existence and the Information Revealed by Prices. Econometrica, 47 3, 655-678.
* (16) W. Shafer, H. Sonnenschein, 1975. Equilibrium in abstract economies without ordered preferences. J. Math. Econom. 2, 345-348.
* (17) A. Stefanescu, M. Ferrara, M. V. Stefanescu, Equilibria of the Games in Choice Form. Journal of Optimization Theory and Applications, 155 (3), 1060-1072 (2012)
* (18) K.K. Tan, Z. Wu, 1998. A Note on Abstract Economies with upper semicontinuous correspondence. Appl. Math. Letters. 5 Vol. 11, 21-22.
* (19) L. A. Tuan, G. M. Lee, P. H. Sach, 2010. Upper semicontinuity in a parametric general variational problem and application. Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, 3-4, 1500-1513.
* (20) X. Wu, 1997. A new fixed point theorem and its applications, Proc. Amer. Math. Soc. 125, 1779-1783.
* (21) X. Wu, 20001. Equilibria of lower semicontinuous games, Computer Math. Appl. 42, 13-22.
* (22) N. C. Yannelis and N. D. Prabhakar, 1983. Existence of maximal elements and equilibrium in linear topological spaces. J. Math. Econom. 12, 233-245.
* (23) X. Z. Yuan, 1998. The Study of Minimax inequalities and Applications to Economies and Variational inequalities. Memoirs of the American Society 132, 625.
* (24) X. Z. Yuan, E. Taradfar, 1999. Maximal elements and equilibria of generalized games for U-majorized and condensing correspondences, Internat. J. Math. Sci. 1, vol 22, 179-189.
* (25) X. Zheng, 1997. Approximate selection theorems and their applications. J. Math. An. Appl. 212, 88-97.
|
arxiv-papers
| 2013-04-03T17:52:09 |
2024-09-04T02:49:43.854129
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Herv\\'es-Beloso and Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1304.1033"
}
|
1304.1062
|
# Heegaard Floer homology and rational cuspidal curves
Maciej Borodzik Institute of Mathematics, University of Warsaw, ul. Banacha
2, 02-097 Warsaw, Poland [email protected] and Charles Livingston
Department of Mathematics, Indiana University, Bloomington, IN 47405
[email protected]
###### Abstract.
We apply the methods of Heegaard Floer homology to identify topological
properties of complex curves in $\mathbb{C}P^{2}$. As one application, we
resolve an open conjecture that constrains the Alexander polynomial of the
link of the singular point of the curve in the case that there is exactly one
singular point, having connected link, and the curve is of genus 0.
Generalizations apply in the case of multiple singular points.
###### Key words and phrases:
rational cuspidal curve, $d$–invariant, surgery, semigroup density
###### 2010 Mathematics Subject Classification:
primary: 14H50, secondary: 14B05, 57M25, 57R58
The first author was supported by Polish OPUS grant No 2012/05/B/ST1/03195
The second author was supported by National Science Foundation Grant 1007196.
## 1\. Introduction
We consider irreducible algebraic curves $C\subset\mathbb{C}P^{2}$. Such a
curve has a finite set of singular points, $\\{z_{i}\\}_{i=1}^{n}$; a
neighborhood of each intersects $C$ in a cone on a link $L_{i}\subset S^{3}$.
A fundamental question asks what possible configurations of links
$\\{L_{i}\\}$ arise in this way. In this generality the problem is fairly
intractable and research has focused on a restricted case, in which each
$L_{i}$ is connected, and thus a knot $K_{i}$, and $C$ is a rational curve,
meaning that there is a rational surjective map $\mathbb{C}P^{1}\to C$. Such a
curve is called rational cuspidal. Being rational cuspidal is equivalent to
$C$ being homeomorphic to $S^{2}$.
Our results apply in the case of multiple singular points, but the following
statement gives an indication of the nature of the results and their
consequences.
###### Theorem 1.1.
Suppose that $C$ is a rational cuspidal curve of degree $d$ with one singular
point, a cone on the knot $K$, and the Alexander polynomial of $K$ is expanded
at $t=1$ to be
$\Delta_{K}(t)=1+\frac{(d-1)(d-2)}{2}(t-1)+(t-1)^{2}\sum_{l}k_{l}t^{l}$. Then
for all $j,0\leq j\leq d-3$, $k_{d(d-j-3)}=(j-1)(j-2)/2$.
There are three facets to the work here:
1. (1)
We begin with a basic observation that a neighborhood $Y$ of $C$ is built from
the 4–ball by attaching a 2–handle along the knot $K=\\#K_{i}$ with framing
$d^{2}$, where $d$ is the degree of the curve. Thus, its boundary,
$S^{3}_{d^{2}}(K)$, bounds the rational homology ball
$\mathbb{C}P^{2}\setminus Y$. From this, it follows that the Heegaard Floer
correction term satisfies $d(S^{3}_{d^{2}}(K),{{\mathfrak{s}}}_{m})=0$ if
$d|m$, for properly enumerated Spincstructures ${{\mathfrak{s}}}_{m}$.
2. (2)
Because each $K_{i}$ is an algebraic knot (in particular an $L$–space knot),
the Heegaard Floer complex $\operatorname{\it CFK}^{\infty}(S^{3},K_{i})$ is
determined by the Alexander polynomial of $K_{i}$, and thus the complex
$\operatorname{\it CFK}^{\infty}(S^{3},K)$ and the $d$–invariants are also
determined by the Alexander polynomials of the $K_{i}$.
3. (3)
The constraints that arise on the Alexander polynomials, although initially
appearing quite intricate, can be reinterpreted in compact form using
semigroups of singular points. In this way, we can relate these constraints to
well-known conjectures.
### 1.1. The conjecture of Fernández de Bobadilla, Luengo, Melle-Hernandez
and Némethi
In [5] the following conjecture was proposed. It was also verified for all
known examples of rational cuspidal curves.
###### Conjecture 1.2 ([5]).
Suppose that the rational cuspidal curve $C$ of degree $d$ has critical points
$z_{1},\dots,z_{n}$. Let $K_{1},\dots,K_{n}$ be the corresponding links of
singular points and let $\Delta_{1},\dots,\Delta_{n}$ be their Alexander
polynomials. Let $\Delta=\Delta_{1}\cdot\ldots\cdot\Delta_{n}$, expanded as
$\Delta(t)=1+\frac{(d-1)(d-2)}{2}(t-1)+(t-1)^{2}\sum_{j=0}^{2g-2}k_{l}t^{l}.$
Then for any $j=0,\dots,d-3$, $k_{d(d-j-3)}\leq(j+1)(j+2)/2$, with equality
for $n=1$.
We remark that the case $n=1$ of the conjecture is Theorem 1.1. We will prove
this result in Section 4.4. Later we will also prove an alternative
generalization of Theorem 1.1 for the case $n>1$, stated as Theorem 5.4, which
is the main result of the present article. The advantage of this formulation
over the original conjecture lies in the fact that it gives precise values of
the coefficients $k_{d(d-j-3)}$. Theorem 6.5 provides an equivalent statement
of Theorem 5.4.
###### Acknowledgements.
The authors are grateful to Matt Hedden, Jen Hom and András Némethi for
fruitful discussions. The first author wants to thank Indiana University for
hospitality.
## 2\. Background: Algebraic Geometry and Rational Cuspidal Curves
In this section we will present some of the general theory of rational
cuspidal curves. Section 2.1 includes basic information about singular points
of plane curves. In Section 2.2 we discuss the semigroup of a singular point
and its connections to the Alexander polynomial of the link. We shall use
results from this section later in the article to simplify the equalities that
we obtain. In Section 2.3 we describe results from [5] to give some flavor of
the theory. In Section 2.4 we provide a rough sketch of some methods used to
study rational cuspidal curves. We refer to [13] for an excellent and fairly
up-to-date survey of results on rational cuspidal curves.
### 2.1. Singular points and algebraic curves
For a general introduction and references to this subsection, we refer to [3,
7], or to [12, Section 10] for a more topological approach. In this article we
will be considering algebraic curves embedded in $\mathbb{C}P^{2}$. Thus we
will use the word _curve_ to refer to a zero set of an irreducible homogeneous
polynomial $F$ of degree $d$. The _degree_ of the curve is the degree of the
corresponding polynomial.
Let $C$ be a curve. A point $z\in C$ is called _singular_ if the gradient of
$F$ vanishes at $z$. Singular points of irreducible curves in
$\mathbb{C}P^{2}$ are always isolated. Given a singular point and a
sufficiently small ball $B\subset\mathbb{C}P^{2}$ around $z$, we call
$K=C\cap\partial B$ the _link_ of the singular point. The singular point is
called _cuspidal_ or _unibranched_ if $K$ is a knot, that is a link with one
component, or equivalently, if there is an analytic map $\psi$ from a disk in
$\mathbb{C}$ onto $C\cap B$.
Unless specified otherwise, all singular points are assumed to be cuspidal.
Two unibranched singular points are called _topologically equivalent_ if the
links of these singular points are isotopic; see for instance [7, Definition
I.3.30] for more details. A unibranched singular point is topologically
equivalent to one for which the local parametrization $\psi$ is given in local
coordinates $(x,y)$ on $B$ by $t\mapsto(x(t),y(t))$, where $x(t)=t^{p}$,
$y(t)=t^{q_{1}}+\ldots+t^{q_{n}}$ for some positive integers
$p,q_{1},\ldots,q_{n}$ satisfying $p<q_{1}<q_{2}<\ldots<q_{n}$. Furthermore,
if we set $D_{i}=\gcd(p,q_{1},\ldots,q_{i})$, then $D_{i}$ does not divide
$q_{i+1}$ and $D_{n}=1$. The sequence $(p;q_{1},\ldots,q_{n})$ is called the
_characteristic sequence_ of the singular point and $p$ is called the
_multiplicity_. Sometimes $n$ is referred to as the _number of Puiseux pairs_
, a notion which comes from an alternative way of encoding the sequence
$(p;q_{1},\ldots,q_{n})$. We will say that a singular point is of type
$(p;q_{1},\ldots,q_{n})$ if it has a presentation of this sort in local
coefficients.
The link of a singular point with a characteristic sequence
$(p;q_{1},\ldots,q_{n})$ is an $(n-1)$–fold iterate of a torus knot
$T(p^{\prime},q^{\prime})$, where $p^{\prime}=p/D_{1}$ and
$q^{\prime}=q_{1}/D_{1}$; see for example [3, Sections 8.3 and 8.5] or [26,
Chapter 5.2]. In particular, if $n=1$, the link is a torus knot $T(p,q_{1})$.
In all cases, the genus of the link is equal to $\mu/2=\delta$, where $\mu$ is
the Milnor number and $\delta$ is the so-called $\delta$–invariant of the
singular point, see [7, page 205], or [12, Section 10]. The genus is also
equal to half the degree of the Alexander polynomial of the link of the
singular point. The Milnor number can be computed from the following formula,
see [12, Remark 10.10]:
$\mu=(p-1)(q_{1}-1)+\sum_{i=2}^{n}(D_{i}-1)(q_{i}-q_{i-1}).$
Suppose that $C$ is a degree $d$ curve with singular points
$z_{1},\ldots,z_{n}$ (and $L_{1},\ldots,L_{n}$ are their links). The genus
formula, due to Serre (see [12, Property 10.4]) states that the genus of $C$
is equal to
$g(C)=\frac{1}{2}(d-1)(d-2)-\sum_{i=1}^{n}\delta_{i}.$
If all the critical points are cuspidal, we have $\delta_{i}=g(L_{i})$, so the
above formula can be written as
(2.1) $g(C)=\frac{1}{2}(d-1)(d-2)-\sum_{i=1}^{n}g(L_{i}).$
In particular, $C$ is rational cuspidal (that is, it is a homeomorphic image
of a sphere) if and only $\sum g(L_{i})=\frac{1}{2}(d-1)(d-2)$.
### 2.2. Semigroup of a singular point
The notion of the semigroup associated to a singular point is a central notion
in the subject, although in the present work we use only the language of
semigroups, not the algebraic aspects. We refer to [26, Chapter 4] or [7, page
214] for details and proofs. Suppose that $z$ is a cuspidal singular point of
a curve $C$ and $B$ is a sufficiently small ball around $z$. Let
$\psi(t)=(x(t),y(t))$ be a local parametrization of $C\cap B$ near $z$; see
Section 2.1. For any polynomial $G(x,y)$ we look at the order at $0$ of an
analytic map $t\mapsto G(x(t),y(t))\in\mathbb{C}$. Let $S$ be the set
integers, which can be realized as the order for some $G$. Then $S$ is clearly
a semigroup of $\mathbb{Z}_{\geq 0}$. We call it the _semigroup of the
singular point_. The semigroup can be computed from the characteristic
sequence, for example for a sequence $(p;q_{1})$, $S$ is generated by $p$ and
$q_{1}$. The _gap sequence_ , $G:=\mathbb{Z}_{\geq 0}\setminus S$, has
precisely $\mu/2$ elements and the largest one is $\mu-1$, where $\mu$ is the
Milnor number.
We now assume that $K$ is the link of the singular point $z$. Explicit
computations of the Alexander polynomial of $K$ show that it is of the form
(2.2) $\Delta_{K}(t)=\sum_{i=0}^{2m}(-1)^{i}t^{n_{i}},$
where $n_{i}$ form an increasing sequence with $n_{0}=0$ and $n_{2m}=2g$,
twice the genus of $K$.
Expanding $t^{n_{2i}}-t^{n_{2i-1}}$ as
$(t-1)(t^{n_{2i}-1}+t^{n_{2i}-2}+\ldots+t^{n_{2i-1}})$ yields
(2.3) $\Delta_{K}(t)=1+(t-1)\sum_{j=1}^{k}t^{g_{j}},$
for some finite sequence $0<g_{1}<\ldots<g_{k}$. We have the following result
(see [26, Exercise 5.7.7]).
###### Lemma 2.4.
The sequence $g_{1},\ldots,g_{k}$ is the gap sequence of the semigroup of the
singular point. In particular $k=\\#G=\mu/2$, where $\mu$ is the Milnor
number, so $\\#G$ is the genus.
Writing $t^{g_{j}}$ as $(t-1)(t^{g_{j}-1}+t^{g_{j}-2}+\ldots+t+1)+1$ in (2.3)
yields the following formula
(2.5) $\Delta_{K}(t)=1+(t-1)g(K)+(t-1)^{2}\sum_{j=0}^{\mu-2}k_{j}t^{j},$
where $k_{j}=\\#\\{m>j\colon m\not\in S\\}$.
We shall use the following definition.
###### Definition 2.6.
For any finite increasing sequence of positive integers $G$, we define
(2.7) $I_{G}(m)=\\#\\{k\in G\cup\mathbb{Z}_{<0}\colon k\geq m\\},$
where $\mathbb{Z}_{<0}$ is the set of the negative integers. We shall call
$I_{G}$ the _gap function_ , because in most applications $G$ will be a gap
sequence of some semigroup.
###### Remark 2.8.
We point out that for $j=0,\ldots,\mu-2$, we have $I_{G}(j+1)=k_{j}$, where
the $k_{j}$ are as in (2.5).
###### Example 2.9.
Consider the knot $T(3,7)$. Its Alexander polynomial is
$\displaystyle\frac{(t^{21}-1)(t-1)}{(t^{3}-1)(t^{7}-1)}=$ $\displaystyle\
1-t+t^{3}-t^{4}+t^{6}-t^{8}+t^{9}-t^{11}+t^{12}$ $\displaystyle=$
$\displaystyle\ 1+(t-1)(t+t^{2}+t^{4}+t^{5}+t^{8}+t^{11})$ $\displaystyle=$
$\displaystyle\ 1+6(t-1)+$ $\displaystyle+(t-1)^{2}$
$\displaystyle\left(6+5t+4t^{2}+4t^{3}+3t^{4}+2t^{5}+2t^{6}+2t^{7}+t^{8}+t^{9}+t^{10}\right).$
The semigroup is $(0,3,6,7,9,10,12,13,14,\dots)$. The gap sequence is
$1,2,4,5,8,11$.
###### Remark 2.10.
The passage from (2.2) through (2.3) to (2.5) is just an algebraic
manipulation, and thus it applies to any knot whose Alexander polynomial has
form (2.2). In particular, according to [21, Theorem 1.2] it applies to any
$L$–space knot. In this setting we will also call the sequence
$g_{1},\dots,g_{k}$ the _gap sequence_ of the knot and denote it by $G_{K}$;
we will write $I_{K}(m)$ for the gap function relative to $G_{K}$. Even though
the complement $\mathbb{Z}_{\geq 0}\setminus G_{K}$ is not always a semigroup,
we still have $\\#G_{K}=\frac{1}{2}\deg\Delta_{K}$. This property follows
immediately from the symmetry of the Alexander polynomial.
### 2.3. Rational cuspidal curves with one cusp
The classification of rational cuspidal curves is a challenging old problem,
with some conjectures (like the Coolidge–Nagata conjecture [4, 14]) remaining
open for many decades. The classification of curves with a unique critical
point is far from being accomplished; the special case when the unique
singular point has only one Puiseux term (its link is a torus knot) is
complete [5], but even in this basic case, the proof is quite difficult.
To give some indication of the situation, consider two families of rational
cuspidal curves. The first one, written in projective coordinates on
$\mathbb{C}P^{2}$ as $x^{d}+y^{d-1}z=0$ for $d>1$, the other one is
$(zy-x^{2})^{d/2}-xy^{d-1}=0$ for $d$ even and $d>1$. These are of degree $d$.
Both families have a unique singular point, in the first case it is of type
$(d-1;d)$, in the second of type $(d/2;2d-1)$. In both cases, the Milnor
number is $(d-1)(d-2)$, so the curves are rational. An explicit
parametrization can be easily given as well.
There also exist more complicated examples. For instance, Orevkov [18]
constructed rational cuspidal curves of degree $\phi_{j}$ having a single
singular point of type $(\phi_{j-2};\phi_{j+2})$, where $j$ is odd and $j>5$.
Here the $\phi_{j}$ are the Fibonacci numbers, $\phi_{0}=0$, $\phi_{1}=1$,
$\phi_{j+2}=\phi_{j+1}+\phi_{j}$. As an example, there exists a rational
cuspidal curve of degree $13$ with a single singular point of type $(5;34)$.
Orevkov’s construction is inductive and by no means trivial. Another family
found by Orevkov are rational cuspidal curves of degree $\phi_{j-1}^{2}-1$
having a single singular point of type $(\phi_{j-2}^{2};\phi_{j}^{2})$, for
$j>5$, odd.
The main result of [5] is that apart of these four families of rational
cuspidal curves, there are only two sporadic curves with a unique singular
point having one Puiseux pair, one of degree $8$, the other of degree $16$.
### 2.4. Constraints on rational cuspidal curves.
Here we review some constraints for rational cuspidal curves. We refer to [13]
for more details and references. The article [5] shows how these constraints
can be used in practice. The fundamental constraint is given by (2.1). Next,
Matsuoka and Sakai [11] proved that if $(p_{1};q_{11},\ldots,q_{1k_{1}})$,
…,$(p_{n};q_{n1},\ldots,q_{nk_{n}})$ are the only singular points occurring on
a rational cuspidal curve of degree $d$ with $p_{1}\geq\ldots\geq p_{n}$, then
$p_{1}>d/3$. Later, Orevkov [18] improved this to
$\alpha(p_{1}+1)+1/\sqrt{5}>d$, where $\alpha=(3+\sqrt{5})/2\sim 2.61$ and
showed that this inequality is asymptotically optimal (it is related to the
curves described in Section 2.1). Both proofs use very deep algebro-geometric
tools. We reprove the result of [11] in Proposition 6.7 below.
Another obstruction comes from the semicontinuity of the spectrum, a concept
that arises from Hodge Theory. Even a rough definition of the spectrum of a
singular point is beyond the scope of this article. We refer to [1, Chapter
14] for a definition of the spectrum and to [5] for illustrations of its use.
We point out that recently (see [2]) a tight relation has been drawn between
the spectrum of a singular point and the Tristram–Levine signatures of its
link. In general, semicontinuity of the spectrum is a very strong tool, but it
is also very difficult to apply.
Using tools from algebraic geometry, such as the Hodge Index Theorem, Tono in
[25] proved that any rational cuspidal curve can have at most eight singular
points. An old conjecture is that a rational cuspidal curve can have at most
$4$ singular points; see [22] for a precise statement.
In [6] a completely new approach was proposed, motivated by a conjecture on
Seiberg–Witten invariants of links of surface singularities made by Némethi
and Nicolaescu; see [16]. Specifically, Conjecture 1.2 in the present article
arises from these considerations. Another reference for the general conjecture
on Seiberg–Witten invariants is [15].
## 3\. Topology, algebraic topology, and Spinc structures
Let $C\subset\mathbb{C}P^{2}$ be a rational cuspidal curve. Let $d$ be its
degree and $z_{1},\ldots,z_{n}$ be its singular points. We let $Y$ be a closed
manifold regular neighborhood of $C$, let $M=\partial Y$, and
$W=\overline{\mathbb{C}P^{2}-Y}$.
### 3.1. Topological descriptions of $Y$ and $M$
The neighborhood $Y$ of $C$ can be built in three steps. First, disk
neighborhoods of the $z_{i}$ are selected. Then neighborhoods of $N-1$
embedded arcs on $C$ are adjoined, yielding a 4–ball. Finally, the remainder
of $C$ is a disk, so its neighborhood forms a 2–handle attached to the 4–ball.
Thus, $Y$ is a 4–ball with a 2–handle attached. The attaching curve is easily
seen to be $K=\\#K_{i}$. Finally, since the self-intersection of $C$ is
$d^{2}$, the framing of the attaching map is $d^{2}$. In particular,
$M=S^{3}_{d^{2}}(K)$.
One quickly computes that $H_{2}(\mathbb{C}P^{2},C)=\mathbb{Z}_{d}$, and
$H_{4}(\mathbb{C}P^{2},C)=\mathbb{Z}$, with the remaining homology groups 0.
Using excision, we see that the groups $H_{i}(W,M)$ are the same. Via
Lefschetz duality and the universal coefficient theorem we find that
$H_{0}(W)=\mathbb{Z}$, $H_{1}(W)=\mathbb{Z}_{d}$ and all the other groups are
0. Finally, the long exact sequence of the pair $(W,M)$ yields
$0\to H_{2}(W,M)\to H_{1}(M)\to H_{1}(W)\to 0$
which in this case is
$0\to\mathbb{Z}_{d}\to\mathbb{Z}_{d^{2}}\to\mathbb{Z}_{d}\to 0.$
This is realized geometrically by letting the generator of $H_{2}(W,M)$ be
${\it H}\cap W$, where $\it H\subset\mathbb{C}P^{2}$ is a generic line. Its
boundary is algebraically $d$ copies of the meridian of the attaching curve
$K$ in the 2–handle decomposition of $Y$.
Taking duals we see that the map $H^{2}(W)\to H^{2}(M)$, which maps
$\mathbb{Z}_{d}\to\mathbb{Z}_{d^{2}}$, takes the canonical generator to $d$
times the dual to the meridian in $M=S^{3}_{d^{2}}(K)$.
### 3.2. Spinc structures
For any space $X$ there is a transitive action of $H^{2}(X)$ on Spinc($X$).
Thus, $W$ has $d$ Spincstructures and $M$ has $d^{2}$ such structures.
Since $\mathbb{C}P^{2}$ has a Spincstructure with first Chern class a dual to
the class of the line, its restriction to $W$ is a structure whose restriction
to $M$ has first Chern class equal to $d$ times the dual to the meridian.
For a cohomology class $z\in H^{2}(X)$ and a Spincstructure
${{\mathfrak{s}}}$, one has
$c_{1}(z\cdot{{\mathfrak{s}}})-c_{1}({{\mathfrak{s}}})=2z$. Thus for each
$k\in\mathbb{Z}$, there is a Spincstructure on $M$ which extends to $W$ having
first Chern class of the form $d+2kd$. Notice that for $d$ odd, all
$md\in\mathbb{Z}_{d^{2}}$ for $m\in\mathbb{Z}$ occur as first Chern classes of
Spincstructures that extend over $W$, but for $d$ even, only elements of the
form $md$ with $m$ odd occur. (Thus, for $d$ even, there are $d$ extending
structures, but only $d/2$ first Chern classes that occur.)
According to [20, Section 3.4], the Spincstructures on $M$ have an enumeration
${{\mathfrak{s}}}_{m}$, for $m\in[-d^{2}/2,d^{2}/2]$, which can be defined via
the manifold $Y$. Specifically, ${{\mathfrak{s}}}_{m}$ is defined to be the
restriction to $M$ of the Spincstructure on $Y$, ${{\mathfrak{t}}}_{m}$, with
the property that $\left<c_{1}({{\mathfrak{t}}}_{m}),C\right>+d^{2}=2m$. We
point out that if $d$ is even, ${{\mathfrak{s}}}_{d^{2}/2}$ and
${{\mathfrak{s}}}_{-d^{2}/2}$ denote the same structure; compare Remark 4.5
below.
It now follows from our previous observations that the structures
${{\mathfrak{s}}}_{m}$ that extend to $W$ are those with $m=kd$ for some
integer $k$, $-d/2\leq k\leq d/2$ if $d$ is odd. If $d$ is even, then those
that extend have $m=kd/2$ for some odd $k$, $-d\leq k\leq d$. For future
reference, we summarize this with the following lemma.
###### Lemma 3.1.
If $W^{4}=\overline{\mathbb{C}P^{2}-Y}$ where $Y$ is a neighborhood of a
rational cuspidal curve $C$ of degree $d$ (as constructed above), then the
Spincstructure ${{\mathfrak{s}}}_{m}$ on $\partial W^{4}$ extends to $W^{4}$
if $m=kd$ for some integer $k$, $-d/2\leq k\leq d/2$ if $d$ is odd. If $d$ is
even, then those that extend have $m=kd/2$ for some odd $k$, $-d\leq k\leq d$.
Here ${{\mathfrak{s}}}_{m}$ is the Spincstructure on $\partial W$ which
extends to a structure ${{\mathfrak{t}}}$ on $Y$ satisfying
$\left<c_{1}({{\mathfrak{t}}}_{m}),C\right>+d^{2}=2m$.
## 4\. Heegaard Floer theory
Heegaard Floer theory [19] associates to a 3–manifold $M$ with Spincstructure
${{\mathfrak{s}}}$, a filtered, graded chain complex
$CF^{\infty}(M,{{\mathfrak{s}}})$ over the field $\mathbb{Z}_{2}$. A
fundamental invariant of the pair $(M,{{\mathfrak{s}}})$, the correction term
or $d$–invariant, $d(M,{{\mathfrak{s}}})\in\mathbb{Q}$, is determined by
$CF^{\infty}(M,{{\mathfrak{s}}})$. The manifold $M$ is called an $L$–space if
certain associated homology groups are of rank one [21].
A knot $K$ in $M$ provides a second filtration on
$CF^{\infty}(M,{{\mathfrak{s}}})$ [19]. In particular, for $K\subset S^{3}$
there is a bifiltered graded chain complex $\operatorname{\it
CFK}^{\infty}(K)$ over the field $\mathbb{Z}_{2}$. It is known that for
algebraic knots the complex is determined by the Alexander polynomial of $K$.
More generally, this holds for any knot upon which some surgery yields an
$L$–space; these knots are called $L$–space knots.
The Heegaard Floer invariants of surgery on $K$, in particular the
$d$–invariants of $S^{3}_{q}(K)$, are determined by this complex, and for
$q>2(\textrm{genus}(K))$ the computation of $d(S^{3}_{q}(K),{{\mathfrak{s}}})$
from $CFK^{\infty}(K)$ is particularly simple. In this section we will
illustrate the general theory, leaving the details to references such as [9,
10].
### 4.1. $\operatorname{\it CFK}^{\infty}(K)$ for $K$ an algebraic knot
Figure 1 is a schematic illustration of a finite complex over
$\mathbb{Z}_{2}$. Each dot represents a generator and the arrows indicate
boundary maps. Abstractly it is of the form
$0\to\mathbb{Z}_{2}^{4}\to\mathbb{Z}_{2}^{5}\to 0$ with homology
$\mathbb{Z}_{2}$. The complex is bifiltered with the horizontal and vertical
coordinates representing the filtrations levels of the generators. We will
refer to the two filtrations levels as the $(i,j)$–filtrations levels. The
complex has an absolute grading which is not indicated in the diagram; the
generator at filtration level $(0,6)$ has grading 0 and the boundary map
lowers the grading by 1. Thus, there are five generators at grading level 0
and four at grading level one. We call the first set of generators type A and
the second type B.
We will refer to a complex such as this as a staircase complex of length $n$,
$\operatorname{St}(v)$, where $v$ is a $(n-1)$–tuple of positive integers
designating the length of the segments starting at the top left and moving to
the bottom right in alternating right and downward steps. Furthermore we
require that the top left vertex lies on the vertical axis and the bottom
right vertex lies on the horizontal axis. Thus, the illustration is of
$\operatorname{St}(1,2,1,2,2,1,2,1)$. The absolute grading of
$\operatorname{St}(v)$ is defined by setting the grading of the top left
generator to be equal to $0$ and the boundary map to lower the grading by $1$.
The vertices of $\operatorname{St}(K)$ will be denoted
$\operatorname{Vert}(St(K))$. We shall write
$\operatorname{Vert}_{A}(\operatorname{St}(K))$ to denote the set of type A
vertices and write $\operatorname{Vert}_{B}(\operatorname{St}(K))$ for the set
of vertices of type B.
If $K$ is a knot admitting an $L$–space surgery, in particular an algebraic
knot (see [8]), then it has Alexander polynomial of the form
$\Delta_{K}(t)=\sum_{i=0}^{2m}(-1)t^{n_{i}}$. To such a knot we associate a
staircase complex, $\operatorname{St}(K)=\operatorname{St}(n_{i+1}-n_{i})$,
where $i$ runs from 0 to $2m-1$. As an example, the torus knot $T(3,7)$ has
Alexander polynomial $1-t+t^{3}-t^{4}+t^{6}-t^{8}+t^{9}-t^{11}+t^{12}$. The
corresponding staircase complex is $\operatorname{St}(1,2,1,2,2,1,2,1)$.
0.6[subgriddiv=1,gridcolor=gray](-1,-1)(-1,-1)(7,7) Figure 1. The staircase
complex $\operatorname{St}(K)$ for the torus knot $T(3,7)$.
Given any finitely generated bifiltered complex $S$, one can form a larger
complex $S\otimes\mathbb{Z}_{2}[U,U^{-1}]$, with differentials defined by
$\partial(x\otimes U^{i})=(\partial x)\otimes U^{i}$. It is graded by
$gr(x\otimes U^{k})=gr(x)-2k$. Similarly, if $x$ is at filtration level
$(i,j)$, then $x\otimes U^{i}$ is at filtration level $(i-k,j-k)$. If $K$
admits an $L$–space surgery, then
$\operatorname{St}(K)\otimes\mathbb{Z}_{2}[U,U^{-1}]$ is isomorphic to
$\operatorname{\it CFK}^{\infty}(K)$. Figure 2 illustrates a portion of
$\operatorname{St}(T(3,7))\otimes\mathbb{Z}_{2}[U,U^{-1}]$; that is, a portion
of the Heegaard Floer complex $\operatorname{\it CFK}^{\infty}(T(3,7))$.
0.6[subgriddiv=1,gridcolor=gray](-1,-1)(-1,-1)(7,7) Figure 2. A portion of
$\operatorname{\it CFK}^{\infty}(T(3,7))$.
### 4.2. $d$–invariants from $\operatorname{\it CFK}^{\infty}(K)$.
We will not present the general definition of the $d$–invariant of a
3–manifold with Spincstructure; details can be found in [19]. However, in the
case that a 3–manifold is of the form $S^{3}_{q}(K)$ where $q\geq
2($genus($K$)), there is a simple algorithm (originating from [20, Section 4],
we use the approach of [9, 10]) to determine this invariant from
$\operatorname{\it CFK}^{\infty}(K)$.
If $m$ satisfies $-d/2\leq m\leq d/2$, one can form the quotient complex
$\operatorname{\it CFK}^{\infty}(K)/\operatorname{\it
CFK}^{\infty}(K)\\{i<0,j<m\\}.$
We let $d_{m}$ denote the least grading in which this complex has a nontrivial
homology class, say $[z]$, where $[z]$ must satisfy the added constraint that
for all $i>0$, $[z]=U^{i}[z_{i}]$ for some homology class $[z_{i}]$ of grading
$d_{m}+2i$.
In [20, Theorem 4.4], we find the following result.
###### Theorem 4.1.
For the Spincstructure ${{\mathfrak{s}}}_{m}$,
$d(S^{3}_{q}(K),{{\mathfrak{s}}}_{m})=d_{m}+\frac{(-2m+q)^{2}-q}{4q}$.
### 4.3. From staircase complexes to the $d$–invariants
Let us now define a distance function for a staircase complex by the formula
$J_{K}(m)=\min_{(v_{1},v_{2})\in\operatorname{Vert}(\operatorname{St}(K))}\max(v_{1},v_{2}-m),$
where $v_{1},v_{2}$ are coordinates of the vertex $v$. Observe that the
minimum can always be taken with respect to the set of vertices of type A. The
function $J_{K}(m)$ represents the greatest $r$ such that the region $\\{i\leq
0,j\leq m\\}$ intersects $\operatorname{St}(K)\otimes U^{r}$ nontrivially. It
is immediately clear that $J_{K}(m)$ is a non-increasing function. It is also
immediate that for $m\geq g$ we have $J_{K}(m)=0$.
0.6[subgriddiv=1,gridcolor=gray](0,0)(-7,-7)(7,7) Figure 3. The function
$J(m)$ for the knot $T(3,7)$. When $(0,m)$ lies on the dashed vertical
intervals, the function $J(m)$ is constant; when it is on solid vertical
intervals the function $J(m)$ is decreasing. The dashed lines connecting
vertices to points on the vertical axis indicate how the ends of dashed and
solid intervals are constructed.
For the sake of the next lemma we define $n_{-1}=-\infty$.
###### Lemma 4.2.
Suppose $m\leq g$. We have $J_{K}(m+1)-J_{K}(m)=-1$ if $n_{2i-1}-g\leq
m<n_{2i}-g$ for some $i$, and $J_{K}(m+1)=J_{K}(m)$ otherwise.
###### Proof.
The proof is purely combinatorial. We order the type A vertices of
$\operatorname{St}(K)$ so that the first coordinate is increasing, and we
denote these vertices $v_{0},\ldots,v_{k}$. For example, for
$\operatorname{St}(T(3,7))$ as depicted on Figure 1, we have $v_{0}=(0,6)$,
$v_{1}=(1,4)$, $v_{2}=(2,2)$, $v_{3}=(4,1)$ and $v_{4}=(6,0)$. We denote by
$(v_{i1},v_{i2})$ the coordinates of the vertex $v_{i}$.
A verification of the two following facts is straightforward:
(4.3) $\begin{split}\max(v_{i1},v_{i2}-m)&=v_{i1}\textrm{ if and only if
$m\geq v_{i1}-v_{i2}$}\\\
\max(v_{i1},v_{i2}-m)&\geq\max(v_{i-1,1},v_{i-1,2}-m)\textrm{ if and only if
$m\leq v_{i1}-v_{i-1,2}$}.\end{split}$
By the definition of the staircase complex we also have
$v_{i1}-v_{i2}=n_{2i}-g$ and $v_{i1}-v_{i-1,2}=n_{2i-1}-g$. The second
equation of (4.3) yields
$J_{K}(m)=\max(v_{i1},v_{i2}-m)\text{ if and only if
}m\in[n_{2i-1},n_{2i+1}].$
Then the first equation of (4.3) allows to compute the difference
$J_{K}(m+1)-J_{K}(m)$. ∎
The relationship between $J_{K}$ and the $d$–invariant is given by the next
result.
###### Proposition 4.4.
Let $K$ be an algebraic knot, let $q>2g(K)$, and let $m\in[-q/2,q/2]$ be an
integer. Then
$d(S^{3}_{q}(K),\mathfrak{s}_{m})=\frac{(-2m+q)^{2}-q}{4q}-2J(m).$
###### Proof.
Denote by $S_{i}$ the subcomplex $\operatorname{St}(K)\otimes U^{i}$ in
$\operatorname{\it CFK}^{\infty}(K)$. The result depends on understanding the
homology of the image of $S_{i}$ in $\operatorname{\it
CFK}^{\infty}(K)/\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$. Because of
the added constraint (see the paragraph before Theorem 4.1), we only have to
look at the homology classes supported on images of the type A vertices.
Notice that if $i>J_{K}(m)$, then at least one of the type A vertices is in
$\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$. But all the type A vertices
are homologous in $S_{i}$, and since these generate $H_{0}(S_{i})$, the
homology of the image in the quotient is 0. On the other hand, if $i\leq
J_{K}(m)$, then none of the vertices of $S_{i}$ are in $\operatorname{\it
CFK}^{\infty}(K)\\{i<0,j<m\\}$ and thus the homology of $S_{i}$ survives in
the quotient.
It follows that the least grading of a nontrivial class in the quotient arises
from the $U^{J_{K}(m)}$ translate of one of type A vertices of
$S_{0}=\operatorname{St}(K)$. Since $U$ lowers grading by 2, the grading is
$-2J_{K}(m)$. The result follows by applying the shift described in Theorem
4.1. ∎
###### Remark 4.5.
Notice that in the case that $q$ is even, the integer values $m=-q/2$ and
$m=q/2$ are both in the given interval. One easily checks that Proposition 4.4
yields the same value at these two endpoints.
We now relate the $J$ function to the semigroup of the singular point. Let
$I_{K}$ be the gap function as in Definition 2.6 and Remark 2.10.
###### Proposition 4.6.
If $K$ is the link of an algebraic singular point, then for $-g\leq m\leq g$
$J_{K}(m)=I_{K}(m+g)$.
###### Proof.
In Section 2.2 we described the gap sequence in terms of the exponents
$n_{i}$. It follows immediately that the growth properties of $I_{K}(m+g)$ are
identical to those of $J_{K}(m)$, as described in Lemma 4.2. Furthermore,
since the largest element in the gap sequence is $2g-1$, we have
$I_{K}(2g)=J_{K}(g)=0$. ∎
### 4.4. Proof of Theorem 1.1
According to Lemma 3.1, the Spincstructures on $S^{3}_{d^{2}}(K)$ that extend
to the complement $W$ of a neighborhood of $C$ are precisely those
${{\mathfrak{s}}}_{m}$ where $m=kd$ for some $k$, where $-d/2\leq k\leq d/2$;
here $k\in\mathbb{Z}$ if $d$ odd, and $k\in\mathbb{Z}+\frac{1}{2}$ if $d$ is
even. Since $W$ is a rational homology sphere, by [19, Proposition 9.9] the
associated $d$–invariants are 0, so by Proposition 4.4, letting $q=d^{2}$ and
$m=kd$, we have
$2J_{K}(kd)=\frac{(-2kd+d^{2})^{2}-d^{2}}{4d^{2}}.$
By Proposition 4.6 we can replace this with
$8I_{G_{K}}(kd+g)=(d-2k-1)(d-2k+1).$
Now $g=d(\frac{d-3}{2})+1$, so by substituting $j=k+\frac{d-3}{2}$ we obtain
$8I_{K}(jd+1)=4(d-j+1)(d-j+2)$
and $j\in[-3/2,\ldots,d-3/2]$ is an integer regardless of the parity of $d$.
The proof is accomplished by recalling that $k_{jd}=I_{K}(jd+1)$, see Remark
2.8.
## 5\. Constraints on general rational cuspidal curves
### 5.1. Products of staircase complexes and the $d$–invariants
In the case that there is more than one cusp, the previous approach continues
to apply, except the knot $K$ is now a connected sum of algebraic knots.
For the connected sum $K=\\#K_{i}$, the complex $\operatorname{\it
CFK}^{\infty}(K)$ is the tensor product of the $\operatorname{\it
CFK}^{\infty}(K_{i})$. To analyze this, we consider the tensor product of the
staircase complexes $\operatorname{St}(K_{i})$. Although this is not a
staircase complex, the homology is still $\mathbb{Z}_{2}$, supported at
grading level 0. For the tensor product we shall denote by
$\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\ldots\otimes\operatorname{St}(K_{n}))$
the set of vertices of the corresponding complex. These are of the form
$v_{1}+\ldots+v_{n}$, where $v_{j}\in\operatorname{Vert}(K_{j})$,
$j=1,\ldots,n$.
Any element of the form $a_{1q_{1}}\otimes a_{2q_{2}}\otimes\cdots\otimes
a_{nq_{n}}$ represents a generator of the homology of the tensor product,
where the $a_{iq_{i}}$ are vertices of type A taken from each
$\operatorname{St}(K_{i})$. Furthermore, if the translated subcomplex
$\text{St}(K)\otimes U^{i}\subset\text{St}(K)\otimes\mathbb{Z}_{2}[U,U^{-1}]$
intersects $\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$ nontrivially,
then the intersection contains one of these generators. Thus, the previous
argument applies to prove the following.
###### Proposition 5.1.
Let $q>2g-1$, where $g=g(K)$ and $m\in[-q/2,q/2]$. Then we have
$d(S^{3}_{q}(K),\mathfrak{s}_{m})=-2J_{K}(m)+\frac{(-2m+q)^{2}-q}{4q},$
where $J_{K}(m)$ is the minimum of $\max(\alpha,\beta-m)$ over all elements of
form $a_{1q_{1}}\otimes a_{2q_{2}}\otimes\ldots\otimes a_{nq_{n}}$, where
$(\alpha,\beta)$, is the filtration level of the corresponding element.
Since the $d$–invariants vanish for all Spincstructures that extend to $W$, we
have:
###### Theorem 5.2.
If $C$ is a rational cuspidal curve of degree $d$ with singular points $K_{i}$
and $K=\\#K_{i}$, then for all $k$ in the range $[-d/2,d/2]$, with
$k\in\mathbb{Z}$ for $d$ odd and $k\in\mathbb{Z}+\frac{1}{2}$ for $d$ even:
$J_{K}(kd)=\frac{(d-2k-1)(d-2k+1)}{8}.$
###### Proof.
We have from the vanishing of the $d$–invariants,
$d(S^{3}_{d^{2}}(K),{{\mathfrak{s}}}_{m})$ (for $m=kd$) the condition
$J_{K}(m)=\frac{(-2m+d^{2})^{2}-d^{2}}{8d^{2}}.$
The result then follows by substituting $m=kd$ and performing algebraic
simplifications. ∎
### 5.2. Restatement in terms of $I_{K_{i}}(m)$.
We now wish to restate Theorem 5.2 in terms of the coefficients of the
Alexander polynomial, properly expanded. As before, for the gap sequence for
the knot $K_{i}$, denoted $G_{K_{i}}$, let
$I_{i}(s)=\\#\\{k\geq s\colon k\in G_{K_{i}}\cup\mathbb{Z}_{<0}\\}.$
For two functions $I,I^{\prime}\colon\mathbb{Z}\to\mathbb{Z}$ bounded below we
define the following operation
(5.3) $I\diamond I^{\prime}(s)=\min_{m\in\mathbb{Z}}I(m)+I^{\prime}(s-m).$
As pointed out to us by Krzysztof Oleszkiewicz, in real analysis this
operation is sometimes called the _infimum convolution_.
The following is the main result of this article.
###### Theorem 5.4.
Let $C$ be a rational cuspidal curve of degree $d$. Let $I_{1},\dots,I_{n}$ be
the gap functions associated to each singular point on $C$. Then for any
$j\in\\{-1,0,\ldots,d-2\\}$ we have
$I_{1}\diamond I_{2}\diamond\ldots\diamond
I_{n}(jd+1)=\frac{(j-d+1)(j-d+2)}{2}.$
###### Remark 5.5.
* •
For $j=-1$, the left hand side is $d(d-1)/2=d-1+(d-1)(d-2)/2$. The meaning of
the equality is that $\sum\\#G_{j}=(d-1)(d-2)/2$ which follows from (2.1) and
Lemma 2.4. Thus, the case $j=-1$ does not provide any new constraints.
Likewise, for $j=d-2$ both sides are equal to $0$.
* •
We refer to Section 6.2 for a reformulation of Theorem 5.4.
* •
We do not know if Theorem 5.4 settles Conjecture 1.2 for $n>1$. The passage
between the two formulations appears to be more complicated; see [17,
Proposition 7.1.3] and the example in Section 6.1.
Theorem 5.4 is an immediate consequence of the arguments in Section 4.4
together with the following proposition.
###### Proposition 5.6.
As in (5.3), let $I_{K}$ be given by $I_{1}\diamond\ldots\diamond I_{n}$, for
the gap functions $I_{1},\ldots,I_{n}$. Then $J_{K}(m)=I_{K}(m+g)$.
###### Proof.
The proof follows by induction over $n$. For $n=1$, the statement is
equivalent to Proposition 4.6. Suppose we have proved it for $n-1$. Let
$K^{\prime}=K_{1}\\#\ldots\\#K_{n-1}$ and let $J_{K^{\prime}}(m)$ be the
corresponding $J$ function. Let us consider a vertex
$v\in\operatorname{Vert}(\operatorname{St}_{1}(K)\otimes\ldots\otimes\operatorname{St}_{n}(K))$.
We can write this as $v^{\prime}+v_{n}$, where
$v^{\prime}\in\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\cdots\otimes\operatorname{St}(K_{n-1}))$
and $v_{n}\in\operatorname{Vert}(\operatorname{St}(K_{n}))$. We write the
coordinates of the vertices as $(v_{1},v_{2})$,
$(v^{\prime}_{1},v^{\prime}_{2})$ and $(v_{n1},v_{n2})$, respectively. We have
$v_{1}=v^{\prime}_{1}+v_{n1}$, $v_{2}=v^{\prime}_{2}+v_{n2}$. We shall need
the following lemma.
###### Lemma 5.7.
For any four integers $x,y,z,w$ we have
$\max(x+y,z+w)=\min_{k\in\mathbb{Z}}\left(\max(x,z-k)+\max(y,w+k)\right).$
###### Proof of Lemma 5.7.
The direction ‘$\leq$’ is trivial. The equality is attained at $k=z-x$. ∎
_Continuation of the proof of Proposition 5.6._
Applying Lemma 5.7 to $v_{1}^{\prime},v_{2}^{\prime},v_{n1},v_{n2}-m$ and
taking the minimum over all vertices $v$ we obtain
$J_{K}(m)=\min_{v\in\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\ldots\otimes\operatorname{St}(K_{n}))}\max(v_{1},v_{2}-m)=\\\
\min_{v^{\prime}\in\operatorname{Vert}^{\prime}}\min_{v_{n}\in\operatorname{Vert}_{n}}\min_{k\in\mathbb{Z}}\left(\max(v^{\prime}_{1},v_{2}^{\prime}-k)+\max(v_{n1},v_{n2}+k-m)\right),$
where we denote
$\operatorname{Vert}^{\prime}=\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\cdots\otimes\operatorname{St}(K_{n-1}))$
and $\operatorname{Vert}_{n}=\operatorname{Vert}(\operatorname{St}(K_{n}))$.
The last expression is clearly
$\min_{k\in\mathbb{Z}}J_{K^{\prime}}(k)+J_{K_{n}}(m-k)$. By the induction
assumption this is equal to
$\min_{k\in\mathbb{Z}}I_{K^{\prime}}(k+g^{\prime})+I_{K_{n}}(m-k+g_{n})=I_{K}(m+g),$
where $g^{\prime}=g(K^{\prime})$ and $g_{n}=g(K_{n})$ are the genera, and we
use the fact that $g=g^{\prime}+g_{n}$. ∎
## 6\. Examples and applications
### 6.1. A certain curve of degree $6$
As described, for instance, in [5, Section 2.3, Table 1], there exists an
algebraic curve of degree $6$ with two singular points, the links of which are
$K=T(4,5)$ and $K^{\prime}=T(2,9)$. The values of $I_{K}(m)$ for
$m\in\\{0,\ldots,11\\}$ are $\\{6,6,5,4,3,3,3,2,1,1,1,1\\}$. The values of
$I_{K^{\prime}}(m)$ for $m\in\\{0,\ldots,7\\}$ are $\\{4,4,3,3,2,2,1,1\\}$. We
readily get
$I\diamond I^{\prime}(1)=10,\ I\diamond I^{\prime}(7)=6,\ I\diamond
I^{\prime}(13)=3,\ I\diamond I^{\prime}(19)=1,$
exactly as predicted by Theorem 5.4.
On the other hand, the computations in [5] confirm Conjecture 1.2 but we
sometimes have an inequality. For example $k_{6}=5$, whereas Conjecture 1.2
states $k_{6}\leq 6$. This shows that Theorem 5.4 is indeed more precise.
### 6.2. Reformulations of Theorem 5.4
Theorem 5.4 was formulated in a way that fits best with its theoretical
underpinnings. In some applications, it is advantageous to reformulate the
result in terms of the function counting semigroup elements in the interval
$[0,k]$. To this end, we introduce some notation.
Recall that for a semigroup $S\subset\mathbb{Z}_{\geq 0}$, the gap sequence of
$G$ is $\mathbb{Z}_{\geq 0}\setminus S$. We put $g=\\#G$ and for $m\geq 0$ we
define
(6.1) $R(m)=\\#\\{j\in S\colon j\in[0,m)\\}.$
###### Lemma 6.2.
For $m\geq 0$, $R(m)$ is related to the gap function $I(m)$ (see (2.7)) by the
following relation:
(6.3) $R(m)=m-g+I(m).$
###### Proof.
Let us consider an auxiliary function $K(m)=\\#\\{j\in[0,m):j\in G\\}$. Then
$K(m)=g-I(m)$. Now $R(m)+K(m)=m$, which completes the proof. ∎
We extend $R(m)$ by (6.3) for all $m\in\mathbb{Z}$. We remark that $R(m)=m-g$
for $m>\sup G$ and $R(m)=0$ for $m<0$. In particular, $R$ is a non-negative,
non-decreasing function.
We have the following result.
###### Lemma 6.4.
Let $I_{1},\dots,I_{n}$ be the gap functions corresponding to the semigroups
$S_{1},\ldots,S_{n}$. Let $g_{1},\dots,g_{n}$ be given by
$g_{j}=\\#{\mathbb{Z}_{\geq 0}\setminus S_{j}}$. Let $R_{1},\ldots,R_{n}$ be
as in (6.1). Then
$R_{1}\diamond R_{2}\diamond\ldots\diamond
R_{n}(m)=m-g+I_{1}\diamond\ldots\diamond I_{n}(m),$
where $g=g_{1}+\ldots+g_{n}$.
###### Proof.
To simplify the notation, we assume that $n=2$; the general case follows by
induction. We have
$\displaystyle R_{1}\diamond R_{2}(m)=$
$\displaystyle\min_{k\in\mathbb{Z}}R_{1}(k)+R_{2}(m-k)=$
$\displaystyle=\min_{k\in\mathbb{Z}}(k-g_{1}+I_{1}(k)+m-k-g_{2}+I_{2}(m-k))=$
$\displaystyle=m-g_{1}-g_{2}+I_{1}\diamond I_{2}(m).$
∎
Now we can reformulate Theorem 5.4:
###### Theorem 6.5.
For any rational cuspidal curve of degree $d$ with singular points
$z_{1},\dots,z_{n}$, and for $R_{1},\dots,R_{n}$ the functions as defined in
(6.1), one has that for any $j=\\{-1,\ldots,d-2\\}$,
$R_{1}\diamond R_{2}\diamond\ldots\diamond R_{n}(jd+1)=\frac{(j+1)(j+2)}{2}.$
This formulation follows from Theorem 5.4 by an easy algebraic manipulation
together with the observation that by (2.1) and Lemma 2.4, the quantity $g$
from Lemma 6.4 is given by $\frac{(d-1)(d-2)}{2}$.
The formula bears strong resemblance to [5, Proposition 2], but in that
article only the ‘$\geq$’ part is proved and an equality in case $n=1$ is
conjectured.
###### Remark 6.6.
Observe that by definition
$R_{1}\diamond\ldots\diamond
R_{n}(k)=\min_{\begin{subarray}{c}k_{1},\ldots,k_{n}\in\mathbb{Z}\\\
k_{1}+\ldots+k_{n}=k\end{subarray}}R_{1}(k_{1})+\ldots+R_{n}(k_{n}).$
Since for negative values $R_{j}(k)=0$ and $R_{j}$ is non-decreasing on
$[0,\infty)$, the minimum will always be achieved for
$k_{1},\ldots,k_{n}\geq-1$.
### 6.3. Applications
From Theorem 6.5 we can deduce many general estimates for rational cuspidal
curves. Throughout this subsection we shall be assuming that $C$ has degree
$d$, its singular points are $z_{1},\ldots,z_{n}$, the semigroups are
$S_{1},\ldots,S_{n}$, and the corresponding $R$–functions are
$R_{1},\ldots,R_{n}$. Moreover, we assume that the characteristic sequence of
the singular point $z_{i}$ is $(p_{i};q_{i1},\ldots,q_{ik_{i}})$. We order the
singular points so that that $p_{1}\geq p_{2}\geq\ldots\geq p_{n}$.
We can immediately prove the result of Matsuoka–Sakai, [11], following the
ideas in [5, Section 3.5.1].
###### Proposition 6.7.
We have $p_{1}>d/3$.
###### Proof.
Suppose $3p_{1}\leq d$. It follows that for any $j$, $3p_{j}\leq d$. Let us
choose $k_{1},\ldots,k_{n}\geq-1$ such that $\sum k_{j}=d+1$. For any $j$, the
elements $0,p_{j},2p_{j},\ldots$ all belong to the $S_{j}$. The function
$R_{j}(k_{j})$ counts elements in $S_{j}$ strictly smaller than $k_{j}$, hence
for any $\varepsilon>0$ we have
$R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{k_{j}-\varepsilon}{p_{j}}.$
Using $3p_{j}\leq d$ we rewrite this as $R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{3k_{j}-3\varepsilon}{d}$. Since
$\varepsilon>0$ is arbitrary, setting $\delta_{j}=1$ if $d|3k_{j}$, and $0$
otherwise, we write
$R_{j}(k_{j})\geq 1+\genfrac{\lfloor}{\rfloor}{}{1}{3k_{j}}{d}-\delta_{j}.$
We get
(6.8) $\sum_{j\colon
d|3k_{j}}R_{j}(k_{j})\geq\genfrac{\lfloor}{\rfloor}{}{1}{\sum 3k_{j}}{d}.$
Using the fact that
$\genfrac{\lfloor}{\rfloor}{}{1}{a}{d}+\genfrac{\lfloor}{\rfloor}{}{1}{b}{d}\geq\genfrac{\lfloor}{\rfloor}{}{1}{a+b}{d}-1$
for any $a,b\in\mathbb{Z}$, we estimate the other terms:
(6.9) $\sum_{j\colon d\not\;|\,3k_{j}}R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{3\sum k_{j}}{d}.$
Since $\sum k_{j}=d+1$, there must be at least one $j$ for which $d$ does not
divide $3k_{j}$. Hence adding (6.8) to (6.9) we obtain
$R_{1}(k_{1})+\ldots+R_{n}(k_{n})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{\sum_{j=1}^{n}3k_{j}}{d}=1+\genfrac{\lfloor}{\rfloor}{}{1}{3d+3}{d}=4.$
This contradicts Theorem 6.5 for $j=1$, and the contradiction concludes the
proof. ∎
We also have the following simple result.
###### Proposition 6.10.
Suppose that $p_{1}>\frac{d+n-1}{2}$. Then $q_{11}<d+n-1$.
###### Proof.
Suppose that $p_{1}>\frac{d+n-1}{2}$ and $q_{11}>d+n-1$. It follows that
$R_{1}(d+n)=2$. But then we choose $k_{1}=d+n$, $k_{2}=\ldots=k_{n}=-1$ and we
get $\sum_{j=1}^{n}R_{j}(k_{j})=2$, hence
$R_{1}\diamond R_{2}\diamond\ldots\diamond R_{n}(d+1)\leq 2$
contradicting Theorem 6.5. ∎
### 6.4. Some examples and statistics
We will now present some examples and statistics, where we compare our new
criterion with the semicontinuity of the spectrum as used in [5, Property
$(SS_{l})$] and the Orevkov criterion [18, Corollary 2.2]. It will turn out
that the semigroup distribution property is quite strong and closely related
to the semicontinuity of the spectrum, but they are not the same. There are
cases which pass one criterion and fail to another. Checking the semigroup
property is definitely a much faster task than comparing spectra; refer to [6,
Section 3.6] for more examples.
###### Example 6.11.
Among the 1,920,593 cuspidal singular points with Milnor number of the form
$(d-1)(d-2)$ for $d$ ranging between $8$ and $64$, there are only 481 that
pass the semigroup distribution criterion, that is Theorem 1.1. All of these
pass the Orevkov criterion $\overline{M}<3d-4$. Of those 481, we compute that
475 satisfy the semicontinuity of the spectrum condition and 6 them fail the
condition; these are: $(8;28,45)$, $(12;18,49)$, $(16;56,76,85)$,
$(24;36,78,91)$, $(24;84,112,125)$, $(36;54,114,133)$.
###### Remark 6.12.
The computations in Example 6.11 were made on a PC computer during one
afternoon. Applying the spectrum criteria for all these cases would take much
longer. The computations for degrees between $12$ and $30$ is approximately
$15$ times faster for semigroups; the difference seems to grow with the
degree. The reason is that even though the spectrum can be given explicitly
from the characteristic sequence (see [24]), it is a set of fractional numbers
and the algorithm is complicated.
###### Example 6.13.
There are $28$ cuspidal singular points with Milnor number equal to
$110=(12-1)(12-2)$. We ask, which of these singular points can possibly occur
as a unique singular point on a degree $12$ rational curve? We apply the
semigroup distribution criterion. Only 8 singular points pass the criterion,
as is seen on Table 1.
(3;56) | fails at $j=1$ | (6;9,44) | fails at $j=1$ | (8;12,14,41) | fails at $j=3$
---|---|---|---|---|---
(4;6,101) | fails at $j=1$ | (6;10,75) | fails at $j=1$ | (8;12,18,33) | fails at $j=4$
(4;10,93) | fails at $j=1$ | (6;14,59) | fails at $j=2$ | (8;12,22,25) | passes
(4;14,85) | fails at $j=1$ | (6;15,35) | fails at $j=2$ | (8;12,23) | passes
(4;18,77) | fails at $j=1$ | (6;16,51) | fails at $j=2$ | (8;14,33) | fails at $j=1$
(4;22,69) | fails at $j=1$ | (6;20,35) | fails at $j=4$ | (9;12,23) | passes
(4;26,61) | fails at $j=1$ | (6;21,26) | passes | (10;12,23) | passes
(4;30,53) | fails at $j=1$ | (6;22,27) | passes | (11;12) | passes
(4;34,45) | fails at $j=1$ | (6;23) | passes | |
(6;8,83) | fails at $j=1$ | (8;10,57) | fails at $j=2$ | |
Table 1. Semigroup property for cuspidal singular points with Milnor number
$12$. If a cuspidal singular point fails the semigroup criterion, we indicate
the first $j$ for which $I(12j+1)\neq\frac{(j-d+1)(j-d+2)}{2}$.
Among the curves in Table 1, all those that are obstructed by the semigroup
distribution, are also obstructed by the semicontinuity of the spectrum. The
spectrum also obstructs the case of $(8;12,23)$.
###### Example 6.14.
There are 2330 pairs $(a,b)$ of coprime integers, such that $(a-1)(b-1)$ is of
form $(d-1)(d-2)$ for $d=5,\ldots,200$. Again we ask if there exists a degree
$d$ rational cuspidal curve having a single singular point with characteristic
sequence $(a;b)$. Among these 2330 cases, precisely 302 satisfy the semigroup
distribution property. Out of these 302 cases, only one, namely $(2;13)$, does
not appear on the list from [5]; see Section 2.3 for the list. It is therefore
very likely that the semigroup distribution property alone is strong enough to
obtain the classification of [5].
###### Example 6.15.
In Table 2 we present all the cuspidal points with Milnor number
$(30-1)(30-2)$ that satisfy the semicontinuity of the spectrum. Out of these,
all but the three ($(18;42,65)$, $(18;42,64,69)$ and $(18;42,63,48)$) satisfy
the semigroup property. All three fail the semigroup property for $j=1$. In
particular, for these three cases the semigroup property obstructs the cases
which pass the semicontinuity of the spectrum criterion.
(15; 55, 69) | (18;42,64,69) | (20; 30, 59) | (25; 30, 59)
---|---|---|---
(15; 57, 71) | (18;42,63,68) | (24; 30, 57, 62) | (27; 30, 59)
(15;59) | (20; 30,55,64) | (24;30,58,63) | (28; 30,59)
(18;42,65) | (20; 30,58,67) | (24; 30,59) | (29; 30)
Table 2. Cuspidal singular points with Milnor number $752$ satisfying the
semicontinuity of the spectrum criterion.
###### Example 6.16.
The configuration of five critical points $(2;3)$, $(2;3)$, $(2;5)$, $(5;7)$
and $(5;11)$ passes the semigroup, the spectrum and the Orevkov criterion for
a degree $10$ curve. In other words, none of the aforementioned criteria
obstructs the existence of such curve. We point out that it is conjectured
(see [13, 22]) that a rational cuspidal curve can have at most $4$ singular
points. In other words, these three criteria alone are insufficient to prove
that conjecture.
## References
* [1] V.I. Arnold, A.N. Varchenko, S.M. Gussein–Zade, Singularities of differentiable mappings. II., “Nauka”, Moscow, 1984.
* [2] M. Borodzik, A. Némethi, Spectrum of plane curves via knot theory, J. London Math. Soc. 86 (2012), 87–110.
* [3] E. Brieskorn, H. Knörrer, Plane Algebraic Curves, Birkhäuser, Basel–Boston–Stuttgart, 1986.
* [4] J. Coolidge, _A treatise on plane algebraic curves_ , Oxford Univ. Press, Oxford, 1928.
* [5] J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi, _Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair_ , Proceedings of Sao Carlos Workshop 2004 Real and Complex Singularities, Series Trends in Mathematics, Birkhäuser 2007, 31–46.
* [6] J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi, _On rational cuspidal projective plane curves_ , Proc. of London Math. Soc., 92 (2006), 99–138.
* [7] G. M. Greuel, C. Lossen, E. Shustin, _Introduction to singularities and deformations_ , Springer Monographs in Mathematics. Springer, Berlin, 2007.
* [8] M. Hedden, _On knot Floer homology and cabling. II_ , Int. Math. Res. Not. 2009, No. 12, 2248–2274.
* [9] S. Hancock, J. Hom, M. Newmann, _On the knot Floer filtration of the concordance group_ , preprint 2012, arxiv:1210.4193.
* [10] M. Hedden, C. Livingston, D. Ruberman, _Topologically slice knots with nontrivial Alexander polynomial_ , Adv. Math. 231 (2012), 913–939.
* [11] T. Matsuoka, F. Sakai, _The degree of rational cuspidal curves_ , Math. Ann. 285 (1989), 233–247.
* [12] J. Milnor, _Singular points of complex hypersurfaces_ , Annals of Mathematics Studies. 61, Princeton University Press and the University of Tokyo Press, Princeton, NJ, 1968.
* [13] T. K. Moe, _Rational cuspidal curves_ , Master Thesis, University of Oslo 2008, permanent link at University of Oslo: https://www.duo.uio.no/handle/123456789/10759.
* [14] M. Nagata, _On rational surfaces. I: Irreducible curves of arithmetic genus 0 or 1_ , Mem. Coll. Sci., Univ. Kyoto, Ser. A 32 (1960), 351–370.
* [15] A. Némethi, _Lattice cohomology of normal surface singularities_ , Publ. RIMS. Kyoto Univ., 44 (2008), 507–543.
* [16] A. Némethi, L. Nicolaescu, _Seiberg-Witten invariants and surface singularities: Splicings and cyclic covers_ , Selecta Math., New series, Vol. 11 (2005), 399–451.
* [17] A Némethi, F. Róman, _The lattice cohomology of $S^{3}_{−d}(K)$_ in: Zeta functions in algebra and geometry, 261–292, Contemp. Math., 566, Amer. Math. Soc., Providence, RI, 2012.
* [18] S. Orevkov, _On rational cuspidal curves. I. Sharp estimates for degree via multiplicity_ , Math. Ann. 324 (2002), 657–673.
* [19] P. Ozsváth, Z. Szabó, _Absolutely graded Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary_ , Adv. Math. 173 (2003), 179–261.
* [20] P. Ozsváth, Z. Szabó, _Holomorphic disks and knot invariants_ , Adv. Math. 186 (2004), 58–116.
* [21] P. Ozsváth, Z. Szabó, _On knot Floer homology and lens space surgeries_ , Topology 44 (2005), 1281–1300.
* [22] J. Piontkowski, _On the number of cusps of rational cuspidal plane curves_ , Exp. Math. 16, no. 2 (2007), 251–255.
* [23] J. Rasmussen, _Floer homology and knot complements_ , Harvard thesis, 2003, available at arXiv:math/0306378.
* [24] M. Saito, _Exponents and Newton polyhedra of isolated hypersurface singularities_ , Math. Ann. 281 (1988), 411–417.
* [25] K. Tono, _On the number of cusps of cuspidal plane curves_ , Math. Nachr. 278 (2005), 216–221.
* [26] C. T. C. Wall, Singular Points of Plane Curves London Mathematical Society Student Texts, 63. Cambridge University Press, Cambridge, 2004.
|
arxiv-papers
| 2013-04-03T19:14:35 |
2024-09-04T02:49:43.866537
|
{
"license": "Public Domain",
"authors": "Maciej Borodzik, Charles Livingston",
"submitter": "Maciej Borodzik",
"url": "https://arxiv.org/abs/1304.1062"
}
|
1304.1403
|
# On the effect of rearrangement on complex interpolation for families of
Banach spaces
Yanqi QIU
###### Abstract.
We give a new proof to show that the complex interpolation for families of
Banach spaces is not stable under rearrangement of the given family on the
boundary, although, by a result due to Coifman, Cwikel, Rochberg, Sagher and
Weiss, it is stable when the latter family takes only 2 values. The non-
stability for families taking 3 values was first obtained by Cwikel and
Janson. Our method links this problem to the theory of matrix-valued Toeplitz
operator and we are able to characterize all the transformations on
$\mathbb{T}$ that are invariant for complex interpolation at 0, they are
precisely the origin-preserving inner functions.
2010 Mathematics Subject Classification: 46B70, 46M35
Key words: complex interpolation method for families, rearrangement, matrix
valued outer functions, Toeplitz operator, duality
## Introduction
This paper is a remark on the theory of complex interpolation for families of
Banach spaces, developed by Coifman, Cwikel, Rochberg, Sagher and Weiss in
[CCRSW82]. To avoid technical difficulties, we will concentrate on finite
dimensional spaces.
Let $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ be the unit disc with boundary
$\mathbb{T}=\partial\mathbb{D}$. The normalised Lebesgue measure on
$\mathbb{T}$ is denoted by $m$. By an interpolation family, we mean a
measurable family of complex $N$-dimensional normed spaces
$\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$, i.e., $E_{\gamma}$ is $\mathbb{C}^{N}$
equipped with norm $\|\cdot\|_{\gamma}$ and for each $x\in\mathbb{C}^{N}$, the
function $\gamma\mapsto\|x\|_{\gamma}$ defined on $\mathbb{T}$ is measurable.
We should also assume that $\int\log^{+}\|x\|_{\gamma}dm(\gamma)<\infty$ for
any $x\in\mathbb{C}^{N}$. By definition, the interpolated space at 0 is
$E[0]:=H^{\infty}(\mathbb{T};\\{E_{\gamma}\\})/zH^{\infty}(\mathbb{T};\\{E_{\gamma}\\}).$
That is, for all $x\in\mathbb{C}^{N}$,
$\|x\|_{E[0]}=\inf\Big{\\{}\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|f(\gamma)\|_{E_{\gamma}}\Big{|}f:\mathbb{T}\rightarrow\mathbb{C}^{N}\text{
analytic},f(0)=x\Big{\\}}.$
More generally, for any $z\in\mathbb{D}$, the interpolated space at $z$ for
the family $\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$ is denoted by $E[z]$ or
$\\{E_{\gamma}:\gamma\in\mathbb{T}\\}[z]$ whose norm is defined as follows.
For any $x\in\mathbb{C}^{N}$,
$\|x\|_{E[z]}:=\inf\Big{\\{}\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|f(\gamma)\|_{E_{\gamma}}\Big{|}f:\mathbb{T}\rightarrow\mathbb{C}^{N}\text{
analytic},f(z)=x\Big{\\}}.$
It is known (cf. [CCRSW82, Prop. 2.4]) that in the above definition, instead
of using $\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|f(\gamma)\|_{E_{\gamma}}$, we can use
$\Big{(}\int\|f(\gamma)\|_{E_{\gamma}}^{p}\,P_{z}(d\gamma)\Big{)}^{1/p}$ for
$0<p<\infty$ or
$\exp\Big{(}\int\log\|f(\gamma)\|_{E_{\gamma}}\,P_{z}(d\gamma)\Big{)}$ without
changing the norm on $E[z]$. Here $P_{z}(d\gamma)$ is the harmonic measure on
$\mathbb{T}$ associated to $z$.
The goal of this paper is to investigate when the norm of the space $E[0]$ is
invariant under a (measure preserving) rearrangement of the family
$\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$. A trivial example of such a
rearrangement is a rotation on $\mathbb{T}$. But, as we will see, there are
non trivial instances of this phenomenon. In particular, we recall the
following well-known result.
###### Theorem 0.1.
[CCRSW82, Cor. 5.1] If $X_{\gamma}=Z_{0}$ for all $\gamma\in\Gamma_{0}$ and
$X_{\gamma}=Z_{1}$ for all $\gamma\in\Gamma_{1}$, where $\Gamma_{0}$ and
$\Gamma_{1}$ are disjoint measurable sets whose union is $\mathbb{T}$, then
$X[0]=(Z_{0},Z_{1})_{\theta}$, where $\theta=m(\Gamma_{1})$ and
$(Z_{0},Z_{1})_{\theta}$ is the classical complex interpolation space for the
pair $(Z_{0},Z_{1})$.
The key fact behind this theorem is the existence for any measurable partition
$\Gamma_{0}\cup\Gamma_{1}$ of the unit circle of an origin-preserving inner
function taking $\Gamma_{0}$ to an arc of length $2\pi m(\Gamma_{0})$ and
$\Gamma_{1}$ to the complementary arc of length. For details, see the
appendix. More generally, complex interpolation at 0 is stable under the
rearrangements given by any inner function vanishing at 0.
###### Proposition 0.2.
Let $\varphi:\mathbb{D}\rightarrow\mathbb{C}$ be an inner function vanishing
at 0. Its boundary value is denoted again by
$\varphi:\mathbb{T}\rightarrow\mathbb{T}$. Then for any interpolation family
$\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$, the canonical identity:
$Id:\\{E_{\gamma}:\gamma\in\mathbb{T}\\}[0]\rightarrow\\{E_{\varphi(\gamma)}:\gamma\in\mathbb{T}\\}[0]$
is isometric.
###### Proof.
The proof is routine, for details, see the last step in the proof of Theorem
0.1 in the appendix. ∎
Theorem 0.1 shows in particular that in the 2-valued case, the complex
interpolation is stable under rearrangement (the reader is referred to Lemma
5.1 for the detail). We show that in the general case, this is not the case.
We learnt from the referee that this result was previously obtained by Cwikel
and Janson in [Cwikel-Janson] with a different method, the statement is at the
bottom of page 214, the proof is from page 278 to page 283.
Our method is simpler and it also yields a characterization of all the
transformations on $\mathbb{T}$ that are invariant for complex interpolation
at 0, they are precisely the inner functions vanishing at 0. In other words,
the converse of Proposition 0.2 holds.
Here is how the paper is organised.
In §1, we recall a result from Helson and Lowdenslager’s papers [HL58, HL61]
on the matrix-valued outer function $F_{W}:\mathbb{D}\rightarrow M_{N}$
associated to a given matrix weight $W:\mathbb{T}\rightarrow M_{N}$. This
result allows us to give an approximation formula for $|F_{W}(0)|^{2}$ when
$W$ is a small perturbation of the constant weight $I$, where $I$ is the
identity matrix in $M_{N}$.
In §2, we study the interpolation families consisting of distorted Hilbert
spaces (i.e., $\mathbb{C}^{N}$ equipped with norms
$\|x\|_{\gamma}=\|W(\gamma)^{1/2}x\|_{\ell_{2}^{N}}$ for a.e.
$\gamma\in\mathbb{T}$). We produce an explicit example of such a family for
which complex interpolation at 0 is not stable under rearrangement.
Our main results are given in §3, where we study some interpolation families
consisting of 3 distorted Hilbert spaces. It is shown that in this restricted
case, the complex interpolation at 0 is already non-stable under
rearrangement. One advantage of our method is that we are able to characterize
all the transformations on $\mathbb{T}$ that are invariant for complex
interpolation at 0, they are precisely the inner functions
$\Theta:\mathbb{T}\rightarrow\mathbb{T}$ such that
$\Theta(0)=\widehat{\Theta}(0)=0$.
§4 is mainly devoted to the stability of complex interpolation under
rearrangement for families of compatible Banach lattices. We also exhibit a
rather surprising non-stability example of interpolation family taking values
in $\\{X,\overline{X},X^{*},\overline{X}^{*}\\}$.
Finally, in the Appendix, we reformulate the argument of [CCRSW82] to prove
Theorem 0.1, the proof somewhat explains why the 3-valued case is different
from the 2-valued case.
## 1\. An approximation formula
In this section, we first recall some results from [HL58, §5] and [HL61, §10,
§11, §12] in the forms that will be convenient for us, and then deduce from
them a useful formula.
Let $W:\mathbb{T}\rightarrow M_{N}$ be a measurable positive semi-definite
$N\times N$-matrix valued function such that $\text{tr}(W)$ is integrable.
Such a function should be considered as a matrix weight. Without mentioning,
all matrix weights in this paper satisfy: There exist $c,C>0$ such that
(1) $\displaystyle cI\leq W(\gamma)\leq CI\quad\text{ for
}a.e.\,\gamma\in\mathbb{T};$
where $I$ is the identity matrix in $M_{N}$. For such a matrix weight, let
$L^{2}_{W}=L^{2}(\mathbb{T},W;S_{2}^{N})$ be the set of functions
$f:\mathbb{T}\rightarrow M_{N}$ for which
$\|f\|_{L_{W}^{2}}^{2}=\int\text{tr}\Big{(}f(\gamma)^{*}W(\gamma)f(\gamma)\Big{)}dm(\gamma)<\infty.$
Clearly, $L_{W}^{2}$ is a Hilbert space.
We will consider two subspaces $H^{2}(W)\subset L_{W}^{2}$ and
$H^{2}_{0}(W)\subset L_{W}^{2}$ defined as follows:
$H^{2}(W)=\\{f\in L_{W}^{2}|\hat{f}(n)=0,\forall n<0\\},$
$H_{0}^{2}(W)=\\{f\in L_{W}^{2}|\hat{f}(n)=0,\forall n\leq 0\\}.$
Given the assumption (1) on $W$, the identity map
$Id:L_{2}(\mathbb{T};S_{2}^{N})\rightarrow L_{W}^{2}$ is an isomorphism, more
precisely,
(2) $\displaystyle
c^{1/2}\|f\|_{L^{2}(\mathbb{T};S_{2}^{N})}\leq\|f\|_{L^{2}_{W}}\leq
C^{1/2}\|f\|_{L^{2}(\mathbb{T};S_{2}^{N})}.$
In particular, $H^{2}_{0}(\mathbb{T};S_{2}^{N})$ and $H^{2}_{0}(W)$ are set
theoretically identical but equipped with equivalent norms.
In the sequel, any element $F\in H^{2}(\mathbb{T};S_{2}^{N})$ will be
identified with its holomorphic extension on $\mathbb{D}$, in particular,
$F(0)=\widehat{F}(0)$, the 0-th Fourier coefficient.
We recall the following theorem (a restricted form) of Helson and Lowdenslager
from [HL58] and [Hel64]. We denote by $S_{2}^{N}$ the spaces of $N\times N$
complex matrices equipped with the Hilbert-Schmidt norm.
###### Theorem 1.1 (Helson-Lowdenslager).
Assume $W$ a matrix weight satisfying the assumption (1). Then there exists
$F\in H^{2}(\mathbb{T};S_{2}^{N})$ such that
* •
$F(\gamma)^{*}F(\gamma)=W(\gamma)$ for a.e. $\gamma\in\mathbb{T}$.
* •
$F$ is a right outer function, that is, $F\cdot H^{2}(\mathbb{T};S_{2}^{N})$
is dense in $H^{2}(\mathbb{T};S_{2}^{N})$.
Let $\Phi$ be the orthogonal projection of the constant function $I$ to the
subspace $H^{2}(W)\ominus H_{0}^{2}(W)\subset L_{W}^{2}$, i.e.,
$\Phi=P_{H^{2}(W)\ominus H_{0}^{2}(W)}(I),$ then
(3) $\displaystyle\Phi(\gamma)^{*}W(\gamma)\Phi(\gamma)=|F(0)|^{2}\,\text{ for
}a.e.\,\gamma\in\mathbb{T}.$
Moreover, $\Phi$ and $F$ and both invertible.
If $F$ and $G$ are two (right) outer functions such that
$F(\gamma)^{*}F(\gamma)=G(\gamma)^{*}G(\gamma)=W(\gamma)\,\text{ for
}a.e.\,\gamma\in\mathbb{T},$
then there is a constant unitary matrix $U\in\mathscr{U}(N)$ such that
$F(z)=UG(z)$ for all $z\in\mathbb{D}$. In particular, $|F(0)|^{2}=|G(0)|^{2}$
is uniquely determined by $W$, as shown by the equation (3). Within all
possible such outer functions, there is a unique one such that $F(0)$ is
positive, we will denote it by $F_{W}$.
Let $\Psi=P_{H^{2}_{0}(W)}(I)$, where the orthogonal projection
$P_{H_{0}^{2}(W)}$ is defined on the space $L_{W}^{2}$. Clearly, we have
(4) $\displaystyle\Phi=I-\Psi.$
We have already known that set theoretically,
$H^{2}_{0}(W)=H^{2}_{0}(\mathbb{T};S_{2}^{N})$, and they are equipped with
equivalent norms, thus we have a Fourier series for $\Psi\in
H_{0}^{2}(W)=H_{0}^{2}(\mathbb{T};S_{2}^{N})$:
$\Psi=\sum_{n\geq 1}\widehat{\Psi}(n)\gamma^{n};$
where the convergence is in $H^{2}_{0}(\mathbb{T};S_{2}^{N})$ and hence in
$H^{2}_{0}(W)$.
By definition, $\Psi$ is characterized as follows. For any $A\in M_{N}$ and
any $n\geq 1$, we have $\langle\Psi,\gamma^{n}A\rangle_{L_{W}^{2}}=\langle
I,\gamma^{n}A\rangle_{L_{W}^{2}},$ i.e.,
$\int\text{tr}(\gamma^{-n}A^{*}W\Psi)dm(\gamma)=\int\text{tr}(\gamma^{-n}A^{*}W)dm(\gamma).$
Or equivalently,
(5) $\displaystyle\int\gamma^{-n}W\Psi
dm(\gamma)=\int\gamma^{-n}Wdm(\gamma),\text{ for }n\geq 1.$
We denote by $P_{+}$ the orthogonal projection of $L^{2}(\mathbb{T})$ onto the
subspace $H^{2}_{0}(\mathbb{T})$. The generalized projection $P_{+}\otimes
I_{X}$ on $L_{p}(\mathbb{T};X)$ for $1<p<\infty$ will still be denoted by
$P_{+}$ . Note that $P_{+}$ is slightly different to the usual Riesz
projection, the latter is defined as the orthogonal projection onto
$H^{2}(\mathbb{T})$. Similarly, we denote by $P_{-}$ the orthogonal projection
onto $\overline{H_{0}^{2}(\mathbb{T})}$ and also its generalisation on
$L_{p}(\mathbb{T};X)$ when it is bounded. With this notation, the equation
system (5) is equivalent to
(6) $\displaystyle P_{+}(W\Psi)=P_{+}(W).$
Key observation:
If $W$ is a perturbation of identity, that is, if there exists a measurable
function $\Delta:\mathbb{T}\rightarrow M_{N}$ such that
$\Delta(\gamma)^{*}=\Delta(\gamma)\,\text{ for }a.e.\gamma\in\mathbb{T}\text{
and }\|\Delta\|_{L_{\infty}(\mathbb{T};M_{N})}<1$
and
$W=I+\Delta;$
then the equation (6) has the form
(7) $\displaystyle\Psi+P_{+}(\Delta\Psi)=P_{+}(\Delta).$
The above equation can be solved using a Taylor series.
To make the last sentence in the preceding observation rigorous, we introduce
the following Toeplitz type operator:
$T_{\Delta}:H_{0}^{2}(\mathbb{T};S_{2}^{N})\xrightarrow{L_{\Delta}}L^{2}(\mathbb{T};S_{2}^{N})\xrightarrow{P_{+}}H_{0}^{2}(\mathbb{T};S_{2}^{N});$
where $L_{\Delta}:H_{0}^{2}(\mathbb{T};S_{2}^{N})\rightarrow
L^{2}(\mathbb{T};S_{2}^{N})$ is the left multiplication by $\Delta$ on the
subspace $H_{0}^{2}(\mathbb{T};S_{2}^{N})$. More precisely,
$(L_{\Delta}f)(\gamma)=\Delta(\gamma)f(\gamma)\text{ for any }f\in
L^{2}(\mathbb{T};S_{2}^{N}).$
Clearly, we have
$\|T_{\Delta}\|\leq\|\Delta\|_{L_{\infty}(\mathbb{T};M_{N})}<1.$
The term $P_{+}(\Delta)$ in equation (7) should be treated as an element in
$H_{0}^{2}(\mathbb{T};S_{2}^{N})$, then the equation (7) has the form
(8) $\displaystyle(Id+T_{\Delta})(\Psi)=P_{+}(\Delta).$
Since $\|T_{\Delta}\|<1$, the operator $Id+T_{\Delta}$ is invertible. Thus
equation (8) has a unique solution $\Psi\in
H_{0}^{2}(\mathbb{T};S_{2}^{N})=H^{2}_{0}(W)$ given by the formula:
(9) $\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(Id+T_{\Delta})^{-1}(P_{+}(\Delta))=\sum_{n=0}^{\infty}(-1)^{n}T_{\Delta}^{n}(P_{+}(\Delta));$
where $T^{0}_{\Delta}(P_{+}(\Delta))=P_{+}(\Delta)$, and the convergence is
understood in the space $H^{2}_{0}(\mathbb{T};S_{2}^{N})$. Combining equations
(3), (4) and (9), we deduce the following formula:
$\displaystyle|F_{I+\Delta}(0)|^{2}=$
$\displaystyle\Big{[}I-\sum_{n=0}^{\infty}(-1)^{n}T_{\Delta}^{n}(P_{+}(\Delta))\Big{]}^{*}(I+\Delta)\times$
$\displaystyle\times\Big{[}I-\sum_{n=0}^{\infty}(-1)^{n}T_{\Delta}^{n}(P_{+}(\Delta))\Big{]}.$
We summarize the above discussion in the following:
###### Proposition 1.2.
Let $\Delta:\mathbb{T}\rightarrow M_{N}$ be a measurable bounded selfadjoint
function such that $\|\Delta\|_{L_{\infty}(\mathbb{T};M_{N})}<1$. Let
$\varepsilon\in[0,1]$, then we have
$\displaystyle|F_{I+\varepsilon\Delta}(0)|^{2}=$
$\displaystyle\Big{[}I-\sum_{n=0}^{\infty}(-1)^{n}\varepsilon^{n+1}T_{\Delta}^{n}(P_{+}(\Delta))\Big{]}^{*}(I+\varepsilon\Delta)\times$
$\displaystyle\times\Big{[}I-\sum_{n=0}^{\infty}(-1)^{n}\varepsilon^{n+1}T_{\Delta}^{n}(P_{+}(\Delta))\Big{]}.$
In particular, we have
(10)
$\displaystyle|F_{I+\varepsilon\Delta}(0)|^{2}=I+\varepsilon\widehat{\Delta}(0)-\varepsilon^{2}\sum_{n\geq
1}|\widehat{\Delta}(n)|^{2}+\mathcal{O}(\varepsilon^{3}),\text{ as
}\varepsilon\to 0^{+}.$
###### Proof.
It suffices to prove the approximation identity (10). We have
(11)
$\displaystyle\begin{split}|F_{I+\varepsilon\Delta}(0)|^{2}=&\Big{[}I-\varepsilon
P_{+}(\Delta)+\varepsilon^{2}T_{\Delta}(P_{+}(\Delta))+\mathcal{O}(\varepsilon^{3})\Big{]}^{*}(I+\varepsilon\Delta)\times\\\
&\times\Big{[}I-\varepsilon
P_{+}(\Delta)+\varepsilon^{2}T_{\Delta}(P_{+}(\Delta))+\mathcal{O}(\varepsilon^{3})\Big{]}\\\
=&I+\varepsilon R_{1}+\varepsilon^{2}R_{2}+\mathcal{O}(\varepsilon^{3}),\text{
as }\varepsilon\to 0^{+};\end{split}$
where
$R_{1}=\Delta-P_{+}(\Delta)-P_{+}(\Delta)^{*},$
$R_{2}=P_{+}(\Delta)^{*}P_{+}(\Delta)-\Delta
P_{+}(\Delta)-P_{+}(\Delta)^{*}\Delta+T_{\Delta}(P_{+}(\Delta))+T_{\Delta}(P_{+}(\Delta))^{*}.$
For $R_{1}$, we note that since $\Delta$ is selfadjoint,
$P_{-}(\Delta)=P_{+}(\Delta)^{*}$ and hence
(12)
$\displaystyle\Delta=P_{+}(\Delta)+P_{+}(\Delta)^{*}+\widehat{\Delta}(0).$
Thus
$R_{1}=\widehat{\Delta}(0).$
For $R_{2}$, we note that since the left hand side of equation (11) is
independent of $\gamma\in\mathbb{T}$, the right hand side should also be
independent of $\gamma$, hence $R_{2}$ must be independent of $\gamma$, it
follows that
$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle\int
R_{2}(\gamma)dm(\gamma)$ $\displaystyle=$
$\displaystyle\int\Big{(}P_{+}(\Delta)^{*}P_{+}(\Delta)-\Delta
P_{+}(\Delta)-P_{+}(\Delta)^{*}\Delta\Big{)}dm(\gamma)$ $\displaystyle=$
$\displaystyle-\sum_{n\geq
1}\widehat{\Delta}(n)^{*}\widehat{\Delta}(n)=-\sum_{n\geq
1}|\widehat{\Delta}(n)|^{2}.$
∎
## 2\. Interpolation Families in the Continuous Case
To any invertible matrix $A\in GL_{N}(\mathbb{C})$ is associated a Hilbertian
norm $\|\cdot\|_{A}$ on $\mathbb{C}^{N}$, which is defined as follows:
$\|x\|_{A}=\|Ax\|_{\ell_{2}^{N}},\text{ for any }x\in\mathbb{C}^{N};$
where $\ell_{2}^{N}$ denotes the space $\mathbb{C}^{N}$ with the usual
Euclidean norm. Let us denote $\ell_{A}^{2}:=(\mathbb{C}^{N},\|\cdot\|_{A}).$
We have the following elementary properties:
* •
Let $A,B\in GL_{N}(\mathbb{C})$, then they define the same norm on
$\mathbb{C}^{N}$ if and only if $|A|=|B|$. Thus, if $U\in\mathscr{U}(N)$ is a
$N\times N$ unitary matrix, then $\|\cdot\|_{UA}=\|\cdot\|_{A}$.
* •
We define a pairing $(x,y)=\sum_{n=1}^{N}x_{n}y_{n}$ for any
$x,y\in\mathbb{C}^{N}$, then under this pairing, we have the canonical
isometries:
$(\ell_{A}^{2})^{*}=\ell_{A^{-T}}^{2};$
where $A^{-T}$ is the inverse of the tranpose matrix $A^{T}$.
* •
We have the following canonical isometries:
$\overline{\ell_{A}^{2}}=\ell_{\overline{A}}^{2}\,\text{ and
}\,\overline{\ell_{A}^{2}}^{*}=\ell_{(A^{*})^{-1}}^{2}.$
Here we recall that, for a complex Banach space $X$, its complex conjugate
$\overline{X}$ is defined to be the space consists of the same element of $X$,
but with scalar multiplication
$\lambda\cdot v=\bar{\lambda}v,\text{ for }\lambda\in\mathbb{C},v\in X.$
Consider an $N\times N$-matrix weight $W$. To such a weight is associated an
interpolation family
$\\{\ell^{2}_{w(\gamma)}:\gamma\in\mathbb{T}\\},\text{ where
}w(\gamma)=\sqrt{W(\gamma)}.$
The following elementary proposition will be used frequently:
###### Proposition 2.1.
For interpolation family $\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$ with
$E_{\gamma}=\ell_{w(\gamma)}^{2}$, we have $E[0]=\ell^{2}_{F(0)}$, that is,
$\|x\|_{E[0]}=\|F(0)x\|_{\ell_{2}^{N}},\,\text{ for all }x\in\mathbb{C}^{N};$
where $F(z)$ is any right outer function associated to the weight $W$.
###### Proof.
By the definition of right outer function associated to the weight $W$,
(13) $\displaystyle F(\gamma)^{*}F(\gamma)=W(\gamma)\,\text{ for
}a.e.\,\gamma\in\mathbb{T}.$
For any $x\in\mathbb{C}^{N}$, define an analytic function
$f_{x}:\mathbb{D}\rightarrow\mathbb{C}^{N}$ by
$f_{x}(z)=F(z)^{-1}F(0)x,$
then $f_{x}(0)=x$ and for a.e. $\gamma\in\mathbb{T}$,
$\displaystyle\|f_{x}(\gamma)\|_{w(\gamma)}^{2}$ $\displaystyle=$
$\displaystyle\langle W(\gamma)F(\gamma)^{-1}F(0)x,F(\gamma)^{-1}F(0)x\rangle$
$\displaystyle=$ $\displaystyle\langle
F(\gamma)^{*}F(\gamma)F(\gamma)^{-1}F(0)x,F(\gamma)^{-1}F(0)x\rangle$
$\displaystyle=$ $\displaystyle\|F(0)x\|_{\ell_{2}^{N}}^{2}.$
This shows that
$\|f_{x}\|_{H^{\infty}(\mathbb{T};\\{E_{\gamma}\\})}\leq\|F(0)x\|_{\ell_{2}^{N}}$,
whence
$\|x\|_{E[0]}\leq\|F(0)x\|_{\ell_{2}^{N}}=\|x\|_{\ell_{F(0)}^{2}}.$
The converse inequality will be given by duality, it suffices to show that
$\|x\|_{E[0]^{*}}\leq\|x\|_{\ell_{F(0)^{-T}}^{2}}=\|x\|_{(\ell_{F(0)}^{2})^{*}}.$
Consider the dual interpolation family
$\\{E_{\gamma}^{*}:\gamma\in\mathbb{T}\\}=\\{\ell_{w(\gamma)^{-T}}^{2}:\gamma\in\mathbb{T}\\},$
which is naturally given by the weight
$W(\gamma)^{-T}=(w(\gamma)^{-T})^{*}w(\gamma)^{-T}$. By [CCRSW82, Th. 2.12],
we have a canonical isometry
$\\{E_{\gamma}^{*}:\gamma\in\mathbb{T}\\}[0]=E[0]^{*}.$
The identity (13) implies
$(F(\gamma)^{-T})^{*}F(\gamma)^{-T}=W(\gamma)^{-T}\,\text{ for
}a.e.\,\gamma\in\mathbb{T}.$
Thus $F(z)^{-T}$ is the right outer function associated to the weight
$W(\gamma)^{-T}$. Then the same argument as above yields that
$\|x\|_{E[0]^{*}}\leq\|x\|_{\ell_{F(0)^{-T}}^{2}}=\|x\|_{(\ell_{F(0)}^{2})^{*}}.$
∎
###### Remark 2.2.
More generally, assume that $X$ is a (finite dimensional) normed space such
that $M_{N}\subset End(X)$ and $\|u\cdot x\|_{X}=\|x\|_{X}$ for any
$u\in\mathscr{U}(N)$. For instance $X=S_{p}^{N}\,(1\leq p\leq\infty)$ and
$M_{N}$ acts on $S_{p}^{N}$ by the usual left multiplications of matrices.
Consider the interpolation family $E_{\gamma}=(X,\|\cdot\|_{X;\,A(\gamma)})$
with $\|x\|_{X;\,A(\gamma)}=\|A(\gamma)\cdot x\|_{X}$ for any
$\gamma\in\mathbb{T}$, then $E[0]=(X,\|\cdot\|_{B(0)})$ with
$\|x\|_{B(0)}=\|B(0)\cdot x\|_{X}$, where $B(z)$ is any right outer function
associated to the matrix weight $A(\gamma)^{*}A(\gamma)$.
The following result is probably known to the experts of prediction theory,
since we do not find it in the literature, we include its proof.
###### Proposition 2.3.
The function $\\{W(\gamma):\gamma\in\mathbb{T}\\}\mapsto F_{W}(0)$ or
equivalently $\\{W(\gamma):\gamma\in\mathbb{T}\\}\mapsto|F(0)|^{2}$ is not
stable under rearrangement. More precisely, there exists a family
$\\{W(\gamma):\gamma\in\mathbb{T}\\}$ and a measure preserving mapping
$S:\mathbb{T}\rightarrow\mathbb{T}$, such that
$F_{W}(0)\neq F_{W\circ S}(0).$
Before we proceed to the proof of the proposition, let us mention that if the
weight $W(\gamma)$ takes only 2 distinct values, i.e., if $W(\gamma)=A_{0}$
for $\gamma\in\Gamma_{0}$ and $W(\gamma)=A_{1}$ for $\gamma\in\Gamma_{1}$ with
$\mathbb{T}=\Gamma_{0}\cup\Gamma_{1}$ a measurable partition, then a detailed
computation shows that we have
$F_{W}(0)^{2}=A_{0}^{1/2}(A_{0}^{-1/2}A_{1}A_{0}^{-1/2})^{m(\Gamma_{1})}A_{0}^{1/2}=A_{1}^{1/2}(A_{1}^{-1/2}A_{0}A_{1}^{-1/2})^{m(\Gamma_{0})}A_{1}^{1/2}.$
In particular, $F_{W}(0)=F_{W\circ M}(0)$ for any measure preserving mapping
$M:\mathbb{T}\rightarrow\mathbb{T}$. Of course, this can be viewed as a
special case of Theorem 0.1. The fact that we can calculate $F_{W}(0)$
efficiently in the above situation is due to the fundamental fact that two
quadratic forms can always be simultaneously diagonalized.
###### Proof.
Fix $r>0$, define two $M_{2}$-valued bounded analytic functions
$F_{1},F_{2}:\mathbb{D}\rightarrow M_{2}$ by
$F_{1}(z)=\left[\begin{array}[]{cc}(1+r^{2})^{1/4}&r(1+r^{2})^{-1/4}z\\\
0&(1+r^{2})^{-1/4}\end{array}\right],$
$F_{2}(z)=\left[\begin{array}[]{cc}(1+r^{2})^{-1/4}&0\\\
r(1+r^{2})^{-1/4}z&(1+r^{2})^{1/4}\end{array}\right].$
Note that they are both outer since $z\rightarrow F_{1}(z)^{-1}$ and
$z\rightarrow F_{2}(z)^{-1}$ are bounded on $\mathbb{D}$. By a direct
computation,
$F_{1}(e^{i\theta})^{*}F_{1}(e^{i\theta})=W_{1}(e^{i\theta})=\left[\begin{array}[]{cc}(1+r^{2})^{1/2}&re^{i\theta}\\\
re^{-i\theta}&(1+r^{2})^{1/2}\end{array}\right],$
$F_{2}(e^{i\theta})^{*}F_{2}(e^{i\theta})=W_{2}(e^{i\theta})=\left[\begin{array}[]{cc}(1+r^{2})^{1/2}&re^{-i\theta}\\\
re^{i\theta}&(1+r^{2})^{1/2}\end{array}\right].$
If we define $S:\mathbb{T}\rightarrow\mathbb{T}$ by
$S(\gamma)=\overline{\gamma}$, then $S$ is measure preserving and
$W_{2}=W_{1}\circ S$. By noting that $F_{1}(0)$ and $F_{2}(0)$ are positive,
we have $F_{1}=F_{W_{1}}$ and $F_{2}=F_{W_{2}}=F_{W_{1}\circ S}$. However,
$F_{W_{1}\circ S}(0)=F_{2}(0)\neq F_{1}(0)=F_{W_{1}}(0)$. ∎
We denote
$W^{(r)}(e^{i\theta}):=\left[\begin{array}[]{cc}(1+r^{2})^{1/2}&re^{i\theta}\\\
re^{-i\theta}&(1+r^{2})^{1/2}\end{array}\right],$
and let $w^{(r)}(\gamma)=\sqrt{W^{(r)}(\gamma)}$. The notation
$S:\mathbb{T}\rightarrow\mathbb{T}$ will be reserved for the complex
conjugation mapping.
An immediate consequence of Propositions 2.1 and 2.3 is the following:
###### Corollary 2.4.
The interpolation family $\\{\widetilde{E}^{(r)}_{\gamma}=\ell_{(w^{(r)}\circ
S)(\gamma)}^{2}:\gamma\in\mathbb{T}\\}$ is a rearrangement of the family
$\\{E^{(r)}_{\gamma}=\ell^{2}_{w^{(r)}(\gamma)}:\gamma\in\mathbb{T}\\}$. The
identity mapping $Id:\widetilde{E}^{(r)}[0]\rightarrow E^{(r)}[0]$ has norm
$\|Id:\widetilde{E}^{(r)}[0]\rightarrow E^{(r)}[0]\|=(1+r^{2})^{1/2}.$
###### Proof.
Indeed, we have:
$\displaystyle\|Id:\widetilde{E}^{(r)}[0]\rightarrow E^{(r)}[0]\|=\sup_{x\neq
0}\frac{\|F_{W^{(r)}}(0)x\|_{\ell_{2}^{2}}}{\|F_{W^{(r)}\circ
S}(0)x\|_{\ell_{2}^{2}}}$ $\displaystyle=\|F_{W^{(r)}}(0)F_{W^{(r)}\circ
S}(0)^{-1}\|_{M_{2}}=(1+r^{2})^{1/2}.$
∎
###### Remark 2.5.
By Corollary 2.4 and a suitable discretization argument, we can show that if
$J_{k}=\Big{\\{}e^{i\theta}:\frac{(k-1)\pi}{4}\leq\theta<\frac{k\pi}{4}\Big{\\}},$
for $1\leq k\leq 8$, and let $\gamma_{k}\in J_{k}$ be the center point on
$J_{k}$, then the interpolation families
$B^{(r_{0})}_{\gamma}=\ell^{2}_{w^{(r_{0})}(\gamma_{k})}$ if $\gamma\in J_{k}$
and
$\widetilde{B}^{(r_{0})}_{\gamma}=\ell^{2}_{w^{(r_{0})}(\bar{\gamma}_{k})}$ if
$\gamma\in J_{k}$ for $r_{0}=\sqrt{2+2\sqrt{2}}$ give different interpolation
space at 0, i.e.,
$\|Id:\widetilde{B}^{(r_{0})}[0]\rightarrow B^{(r_{0})}[0]\|>1.$
We omit its proof, because in the next section, we give a better result by
using the formula obtained in §1.
## 3\. Interpolation for three Hilbert spaces
In this section, we will show that complex interpolation is not stable even
for a familiy taking only 3 distinct Hilbertian spaces. The starting point of
this section is Proposition 1. Our proof is somewhat abstract, but it explains
why the 3-valued case becomes different from the 2-valued case, the idea used
in the proof will be applied further to get a characterization of measurable
transformations on $\mathbb{T}$ that perserve complex interpolation at 0.
###### Theorem 3.1.
There are two different measurable partitions of the unit circle:
$\mathbb{T}=S_{1}\cup S_{2}\cup S_{3}=S_{1}^{\prime}\cup S_{2}^{\prime}\cup
S_{3}^{\prime},\,\,m(S_{k})=m(S_{k}^{\prime}),\text{ for }k=1,2,3,$
and three constant selfadjoint matrices $\Delta_{k}\in M_{2}$ for $k=1,2,3$,
such that if we let
$\Delta=\Delta_{1}1_{S_{1}}+\Delta_{2}1_{S_{2}}+\Delta_{3}1_{S_{3}}\text{ and
}\Delta^{\prime}=\Delta_{1}1_{S_{1}^{\prime}}+\Delta_{2}1_{S_{2}^{\prime}}+\Delta_{3}1_{S_{3}^{\prime}},$
then
$\sum_{n\geq 1}|\widehat{\Delta}(n)|^{2}\neq\sum_{n\geq
1}|\widehat{\Delta^{\prime}}(n)|^{2}.$
Before turning to the proof of the above theorem, we state our main result.
###### Corollary 3.2.
Let $\Delta,\Delta^{\prime}$ be as in Theorem 3.1. For
$0<\varepsilon<\frac{1}{\,\,\|\Delta\|_{\infty}}$, we define two matrix
weights which are perturbation of identity:
$W_{\varepsilon}=I+\varepsilon\Delta,\,\,W_{\varepsilon}^{\prime}=I+\varepsilon\Delta^{\prime}.$
Denote $w_{\varepsilon}$ and $w_{\varepsilon}^{\prime}$ the square root of
$W_{\varepsilon}$ and $W^{\prime}_{\varepsilon}$ respectively. Then there
exists $\varepsilon_{0}<1$ such that whenever $0<\varepsilon<\varepsilon_{0}$,
we have
$|F_{W_{\varepsilon}}(0)|^{2}\neq|F_{W_{\varepsilon}^{\prime}}(0)|^{2}.$
Thus, whenever $0<\varepsilon<\varepsilon_{0}$, the following two
interpolation families
$\\{\ell^{2}_{w_{\varepsilon}(\gamma)}:\gamma\in\mathbb{T}\\},\quad\\{\ell^{2}_{w^{\prime}_{\varepsilon}(\gamma)}:\gamma\in\mathbb{T}\\}$
have the same distribution and take only 3 distinct values. However, the
interpolation spaces at 0 given by these two families are different:
$\ell^{2}_{F_{W_{\varepsilon}(0)}}\neq\ell^{2}_{F_{W^{\prime}_{\varepsilon}(0)}}.$
###### Proof.
This is an immediate corollary of Proposition 1.2 and Theorem 3.1. The last
assertion follows from Proposition 2.1. ∎
###### Remark 3.3.
We verify that in the two main cases where the interpolation is stable under
rearrangement, the function $\Delta\mapsto\sum_{n\geq 1}|\hat{\Delta}(n)|^{2}$
is stable under rearrangement. Note first that we have the following matrix
identity:
$\sum_{n\geq 1}|\widehat{\Delta}(n)|^{2}=\int|P_{+}(\Delta)|^{2}dm.$
* •
2-valued case: If $\Delta$ is a 2-valued selfadjoint function, i.e, there is a
measurable subset $A\subset\mathbb{T}$ and two selfadjoint matrices
$\Delta_{1},\Delta_{2}\in M_{N}$, such that
$\Delta=\Delta_{1}1_{A}+\Delta_{2}1_{A^{c}}$ then
$\displaystyle\sum_{n\geq 1}|\widehat{\Delta}(n)|^{2}=$
$\displaystyle\int|P_{+}(\Delta)|^{2}dm=\int\Big{|}P_{+}\Big{(}(\Delta_{1}-\Delta_{2})1_{A}+\Delta_{2}\Big{)}\Big{|}^{2}dm$
$\displaystyle=$
$\displaystyle|\Delta_{1}-\Delta_{2}|^{2}\int|P_{+}(1_{A})|^{2}dm$
$\displaystyle=$
$\displaystyle\frac{m(A)-m(A)^{2}}{2}|\Delta_{1}-\Delta_{2}|^{2},$
which depends on the measure of $A$ but not the other structure of $A$.
More generally, we note in passing that for any real valued $f$ in
$L_{2}(\mathbb{T})$ the expression $2\|P_{+}(f)\|_{2}^{2}=2\sum_{n\geq
1}|\widehat{f}(n)|^{2}$ coincides with the variance of $f$.
* •
Rearrangement under inner functions: Let
$\varphi:\mathbb{T}\rightarrow\mathbb{T}$ be the boundary value of an origin-
preserving inner function. Assume $\Delta:\mathbb{T}\rightarrow M_{N}$
selfadjoint. Note that $P_{+}(\Delta\circ\varphi)=P_{+}(\Delta)\circ\varphi$
and that $\varphi$ preserves the measure $m$. Hence
$\displaystyle\sum_{n\geq 1}|\widehat{(\Delta\circ\varphi)}(n)|^{2}=$
$\displaystyle\int|P_{+}(\Delta\circ\varphi)|^{2}dm=\int|P_{+}(\Delta)\circ\varphi|^{2}dm$
$\displaystyle=$ $\displaystyle\int|P_{+}(\Delta)|^{2}dm=\sum_{n\geq
1}|\widehat{\Delta}(n)|^{2}.$
###### Proof of Theorem 3.1.
Assume by contradiction that for any pair of 3-valued selfadjoint functions
$\Delta$ and $\Delta^{\prime}$ as in the statement of Theorem 3.1, we have
(14) $\displaystyle\sum_{n\geq 1}|\widehat{\Delta}(n)|^{2}=\sum_{n\geq
1}|\widehat{\Delta^{\prime}}(n)|^{2}.$
We make the following reduction.
Step 1:
The above assumption implies that for any pair of functions,
$\Delta,\Delta^{\prime}$ taking values in the same set of three matrices and
having identical distribution, the equation (14) holds as well. Indeed, given
such a pair, we can consider the pair of selfadjoint functions which are still
3-valued:
$\gamma\rightarrow\left[\begin{array}[]{cc}0&\Delta(\gamma)^{*}\\\
\Delta(\gamma)&0\end{array}\right]\text{ and
}\gamma\rightarrow\left[\begin{array}[]{cc}0&\Delta^{\prime}(\gamma)^{*}\\\
\Delta^{\prime}(\gamma)&0\end{array}\right].$
Then the square of the $n$-th Fourier coefficient becomes
$\left[\begin{array}[]{cc}|\widehat{\Delta}(n)|^{2}&0\\\
0&|\widehat{\Delta^{*}}(n)|^{2}\end{array}\right]\text{ and
}\left[\begin{array}[]{cc}|\widehat{\Delta^{\prime}}(n)|^{2}&0\\\
0&|\widehat{\Delta^{\prime*}}(n)|^{2}\end{array}\right]$
respectively. The block (1, 1)-terms then give the desired equation.
Step 2:
If we take $N=1$ in the above step, then the conclusion is that for any pair
of 3-valued scalar functions $f,f^{\prime}\in L_{\infty}(\mathbb{T})$ such
that $f\stackrel{{\scriptstyle d}}{{=}}f^{\prime}$, we have $\sum_{n\geq
1}|\widehat{f}(n)|^{2}=\sum_{n\geq 1}|\widehat{f^{\prime}}(n)|^{2}$, or
equivalently,
$\|P_{+}(f)\|_{2}^{2}=\|P_{+}(f^{\prime})\|_{2}^{2}.$
Consequence I:
Under the above assumption, if $(A_{1},A_{2})$ is a pair of two disjoint
measurable subsets of $\mathbb{T}$, and $(A_{1}^{\prime},A_{2}^{\prime})$ is
another such pair such that $m(A_{1})=m(A_{1}^{\prime})$ and
$m(A_{2})=m(A_{2}^{\prime})$, then
(15) $\displaystyle\langle
P_{+}(1_{A_{1}}),P_{+}(1_{A_{2}})\rangle_{L^{2}(\mathbb{T})}=\langle
P_{+}(1_{A^{\prime}_{1}}),P_{+}(1_{A^{\prime}_{2}})\rangle_{L^{2}(\mathbb{T})}$
Indeed, if we define $A_{3}:=\mathbb{T}\setminus(A_{1}\cup A_{2})$ and
$A^{\prime}_{3}:=\mathbb{T}\setminus(A^{\prime}_{1}\cup A^{\prime}_{2})$. For
any $\alpha\in\mathbb{C},\alpha\neq 0,1$, consider
$f_{\alpha}=\alpha 1_{A_{1}}+1_{A_{2}}+0\times 1_{A_{3}},\quad
f^{\prime}_{\alpha}=\alpha 1_{A^{\prime}_{1}}+1_{A^{\prime}_{2}}+0\times
1_{A^{\prime}_{3}},$
then $f_{\alpha}$ and $f^{\prime}_{\alpha}$ are two functions taking exactly 3
values 0, 1, $\alpha$ and $f_{\alpha}\stackrel{{\scriptstyle
d}}{{=}}f^{\prime}_{\alpha}$. Hence by the assumption, we have
(16) $\displaystyle\|\alpha
P_{+}(1_{A_{1}})+P_{+}(1_{A_{2}})\|_{2}^{2}=\|\alpha
P_{+}(1_{A^{\prime}_{1}})+P_{+}(1_{A^{\prime}_{2}})\|_{2}^{2},\text{ for any
}\alpha\in\mathbb{C}.$
Note that for any measurable set $A$, since $1_{A}$ is real,
(17) $\displaystyle\|P_{+}(1_{A})\|_{2}^{2}=\frac{m(A)-m(A)^{2}}{2}$
Taking this in consideration, the equation (16) implies that
$\Re\Big{(}\alpha\langle
P_{+}(1_{A_{1}}),P_{+}(1_{A_{2}})\rangle\Big{)}=\Re\Big{(}\alpha\langle
P_{+}(1_{A^{\prime}_{1}}),P_{+}(1_{A^{\prime}_{2}})\rangle\Big{)},\text{ for
any }\alpha\in\mathbb{C},$
hence the equation (15) holds.
Step 3:
We can deduce from our assumption the following consequence.
Consequence II:
For any pair of scalar functions $f,f^{\prime}\in L_{\infty}(\mathbb{T})$
(without the assumption that they are both 3-valued), such that
$f\stackrel{{\scriptstyle d}}{{=}}f^{\prime}$ , we have
$\|P_{+}(f)\|_{2}=\|P_{+}(f^{\prime})\|_{2}.$
Indeed, if
$f=\sum_{k=1}^{n}f_{k}1_{A_{k}},\quad
f^{\prime}=\sum_{k=1}^{n}f_{k}1_{A^{\prime}_{k}},$
where $(A_{k})_{k=1}^{n}$ are disjoint subsets of $\mathbb{T}$, so is
$(A^{\prime}_{k})_{k=1}^{n}$, moreover $m(A_{k})=m(A_{k}^{\prime})$. By (15)
and (17), we have
$\displaystyle\|P_{+}(f)\|_{2}^{2}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}|f_{k}|^{2}\cdot\|P_{+}(1_{A_{k}})\|_{2}^{2}+\sum_{1\leq
k\neq l\leq n}f_{k}f_{l}\langle P_{+}(1_{A_{1}}),P_{+}(1_{A_{2}})\rangle$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}|f_{k}|^{2}\cdot\|P_{+}(1_{A^{\prime}_{k}})\|_{2}^{2}+\sum_{1\leq
k\neq l\leq n}f_{k}f_{l}\langle
P_{+}(1_{A^{\prime}_{1}}),P_{+}(1_{A^{\prime}_{2}})\rangle$ $\displaystyle=$
$\displaystyle\|P_{+}(f^{\prime})\|_{2}^{2}.$
Then by an approximation argument, more precisely, by using the fact that two
functions $f,f^{\prime}\in L^{2}(\mathbb{T})$ such that
$f\stackrel{{\scriptstyle d}}{{=}}f$ can be approximated in
$L^{2}(\mathbb{T})$ by two sequences of simple functions $(g_{n})$ and
$(g_{n}^{\prime})$ such that $g_{n}\stackrel{{\scriptstyle
d}}{{=}}g_{n}^{\prime}$, we can extend the above equality for pairs of
equidistributed simple functions to the general equidistributed pairs of
functions, as stated in Consequence II.
Step 4:
Now if we take $f,f^{\prime}\in L_{\infty}(\mathbb{T})$ to be
$f(\gamma)=\gamma$ and $f^{\prime}(\gamma)=\overline{\gamma}$, then
$f\stackrel{{\scriptstyle d}}{{=}}f^{\prime}$, but we have
$\|P_{+}(f)\|_{2}=1\neq 0=\|P_{+}(f^{\prime})\|_{2},$ which contradicts
Consequence II. This completes the proof.
∎
Define
$T_{k}:=\Big{\\{}e^{i\theta}|\frac{2(k-1)\pi}{3}\leq\theta<\frac{2k\pi}{3}\Big{\\}}\text{
for }k=1,2,3.$
We claim that in Theorem 3.1 and hence in Corollary 3.2, we can take for
example
$S_{1}=S_{1}^{\prime}=T_{1},\quad S_{2}=S_{3}^{\prime}=T_{2},\quad
S_{3}=S_{2}^{\prime}=T_{3}.$
Indeed, by the proof of Theorem 3.1, here we only need to show that
$\displaystyle\langle P_{+}(1_{T_{1}}),P_{+}(1_{T_{2}})\rangle\neq\langle
P_{+}(1_{T_{1}}),P_{+}(1_{T_{3}})\rangle.$
Since $1_{T_{1}}(\gamma)=1_{T_{3}}(e^{-i\frac{2\pi}{3}}\gamma)$ and
$1_{T_{2}}(\gamma)=1_{T_{1}}(e^{-i\frac{2\pi}{3}}\gamma)$, we have
$\langle P_{+}(1_{T_{1}}),P_{+}(1_{T_{3}})\rangle=\langle
P_{+}(1_{T_{2}}),P_{+}(1_{T_{1}})\rangle.$
Thus we only need to show that
(18) $\displaystyle\Im\Big{(}\langle
P_{+}(1_{T_{1}}),P_{+}(1_{T_{2}})\rangle\Big{)}\neq 0.$
Note that
$\Im\Big{(}\widehat{1_{T_{1}}}(n)\overline{\widehat{1_{T_{2}}}(n)}\Big{)}=\frac{\sin\frac{2\pi}{3}(1-\cos\frac{2\pi}{3})}{2\pi^{2}n^{2}}\times\left\\{\begin{array}[]{cl}0,&\text{
if }n\equiv 0\mod 3;\\\ 1,&\text{ if }n\equiv 1\mod 3;\\\ -1,&\text{ if
}n\equiv 2\mod 3.\end{array}\right.$
Hence
$\displaystyle\Im\Big{(}\langle
P_{+}(1_{T_{1}}),P_{+}(1_{T_{2}})\rangle\Big{)}$ $\displaystyle=$
$\displaystyle\frac{3\sin\frac{2\pi}{3}(1-\cos\frac{2\pi}{3})}{2\pi^{2}}\sum_{k=0}^{\infty}\frac{2k+1}{(3k+1)^{2}(3k+2)^{2}},$
which is non-zero, as we expected.
The same idea as in the proof of Theorem 3.1 yields the following
characterization: combining with Proposition 0.2, we have characterized all
measurable transformations on $\mathbb{T}$ that preserve complex interpolation
at 0. At this stage, the proof is quite direct.
###### Theorem 3.4.
Let $\Theta:\mathbb{T}\rightarrow\mathbb{T}$ be a measurable transformation.
If for any interpolation family $\\{E_{\gamma};\gamma\in\mathbb{T}\\}$, we
have
$\\{E_{\gamma}:\gamma\in\mathbb{T}\\}[0]=\\{E_{\Theta(\gamma)}:\gamma\in\mathbb{T}\\}[0],$
then $\Theta$ is an inner function and $\hat{\Theta}(0)=0$.
###### Remark 3.5.
The main point of Theorem 3.4 is to characterize all the transformations which
preserve the interpolation spaces at origin.
###### Proof.
It suffices to show that $\Theta\in H_{0}^{\infty}(\mathbb{T})$, since by
definition $\Theta(\gamma)$ has modulus 1 for $a.e.\,\gamma\in\mathbb{T}$. By
Propositions 1.2, 2.1 and similar arguments in the proof of Theorem 3.1, we
have
(19) $\displaystyle\|P_{+}(f\circ\Theta)\|_{2}=\|P_{+}(f)\|_{2},\text{ for any
scalar function }f\in L_{\infty}(\mathbb{T}).$
Now take $f(\gamma)=\overline{\gamma}$, we have
$\|P_{+}(\overline{\Theta})\|_{2}=\|P_{+}(\overline{\gamma})\|_{2}=0$, which
implies that $\overline{\Theta}\in\overline{H^{\infty}(\mathbb{T})}$ and hence
$\Theta\in H^{\infty}(\mathbb{T})$. Then we can write
$\Theta=\widehat{\Theta}(0)+P_{+}(\Theta)$. In (19), if we take
$f(\gamma)=\gamma$, then $\|P_{+}(\Theta)\|_{2}=\|P_{+}(\gamma)\|_{2}=1$. Note
that
$1=\|\Theta\|_{2}^{2}=|\widehat{\Theta}(0)|^{2}+\|P_{+}(\Theta)\|_{2}^{2},$
whence $\widehat{\Theta}(0)=0$. This completes the proof. ∎
## 4\. Some related comments
Recall that an $N$-dimensional (complex) Banach space $\mathscr{L}$ is called
a (complex) Banach lattice with respect to a fixed basis
$(e_{1},\cdots,e_{N})$ of $\mathscr{L}$ if it satisfies the lattice axiom: For
any $x_{k},y_{k}\in\mathbb{C}$ such that $|x_{k}|\leq|y_{k}|$ for all $1\leq
k\leq N$,
$\|\sum_{k=1}^{N}x_{k}e_{k}\|_{\mathscr{L}}\leq\|\sum_{k=1}^{N}y_{k}e_{k}\|_{\mathscr{L}}.$
Thus in particular,
$\|\sum_{k=1}^{N}x_{k}e_{k}\|_{\mathscr{L}}=\|\sum_{k=1}^{N}|x_{k}|e_{k}\|_{\mathscr{L}}.$
The above fixed basis $(e_{1},\cdots,e_{N})$ will be called a lattice-basis of
$\mathscr{L}$. Such a Banach lattice $\mathscr{L}$ will be viewed as function
spaces over the $N$-point set $[N]=\\{1,\cdots,N\\}$ in such a way that
$e_{k}$ corresponds to the Dirac function at the point $k$. Thus for
$x,y\in\mathscr{L}$, we can write $|x|\leq|y|$ if $|x_{k}|\leq|y_{k}|$ for all
$1\leq k\leq N$, and $\log|x|=\sum_{k=1}^{N}\log|x_{k}|e_{k}$, suppose that
$x_{k}\neq 0$ for all $1\leq k\leq N$.
We will call
$\\{\mathscr{L}_{\gamma}=(\mathbb{C}^{N},\|\cdot\|_{\gamma}):\gamma\in\mathbb{T}\\}$
a family of compatible Banach lattices, if there is an algebraic basis
$(e_{1},\cdots,e_{N})$ of $\mathbb{C}^{N}$ which is simultaneously a lattice-
basis of $\mathscr{L}_{\gamma}$ for a.e. $\gamma\in\mathbb{T}$ and such that
(20) $\displaystyle 0<\underset{\gamma\in\mathbb{T}}{\text{ess inf
}}\|e_{k}\|_{\gamma}\leq\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|e_{k}\|_{\gamma}<\infty\text{ for all }1\leq k\leq N.$
In the sequel, the notation
$\\{\mathscr{L}_{\gamma}=(\mathbb{C}^{N},\|\cdot\|_{\gamma}):\gamma\in\mathbb{T}\\}$
is reserved for a family of compatible Banach lattices with respect to the
canonical basis of $\mathbb{C}^{N}$.
Complex interpolation at 0 for families of compatible Banach lattices is
stable under any rearrangement. The proof of the following proposition is
standard.
###### Proposition 4.1.
If
$\\{\mathscr{L}_{\gamma}=(\mathbb{C}^{N},\|\cdot\|_{\gamma}):\gamma\in\mathbb{T}\\}$
be an interpolation family of compatible Banach lattices, then
(21)
$\displaystyle\log\|x\|_{\mathscr{L}[0]}=\inf\int\log\|f(\gamma)\|_{\gamma}\,dm(\gamma),$
where the infimum runs over the set of all measurable coordinate bounded
functions $f:\mathbb{T}\rightarrow\mathbb{C}^{N}$, i.e.,
$f_{k}:\mathbb{T}\rightarrow\mathbb{C}$ is bounded for all $1\leq k\leq N$
such that $($ by convention $\log 0:=-\infty$ $)$
$\log|x|\leq\int\log|f(\gamma)|\,dm(\gamma).$
In particular, if $M:\mathbb{T}\rightarrow\mathbb{T}$ is measure preserving
and let
$\\{\widetilde{\mathscr{L}}_{\gamma}=\mathscr{L}_{M(\gamma)}:\gamma\in\mathbb{T}\\}$,
then
$Id:\mathscr{L}[0]\rightarrow\widetilde{\mathscr{L}}[0]$
is isometric.
###### Proof.
It suffices to show (21). Assume that $x\in\mathbb{C}^{N}$ and
$\|x\|_{\mathscr{L}[0]}<\lambda$. Without loss of generality, we can assume
$x_{k}\neq 0$ for all $1\leq k\leq N$. By the definition of $\mathscr{L}[0]$
there exists an analytic function
$f=(f_{1},\cdots,f_{N}):\mathbb{D}\rightarrow\mathbb{C}^{N}$ such that
$f(0)=x\text{ and }\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|f(\gamma)\|_{\gamma}<\lambda.$
By (20), this implies in particular that $f$ is coordinate bounded. Since
$z\mapsto\log|f_{k}(z)|$ is subharmonic, we have
$\log|x_{k}|=\log|f_{k}(0)|\leq\int\log|f_{k}(\gamma)|dm(\gamma),\text{ for
}1\leq k\leq N.$
Hence $\log|x|\leq\int\log|f(\gamma)|dm(\gamma)$. Obviously,
$\int\log\|f(\gamma)\|_{\gamma}dm(\gamma)<\log\lambda$, whence
$\inf\int\log\|f(\gamma)\|_{\gamma}\,dm(\gamma)\leq\log\|x\|_{\mathscr{L}[0]}.$
Conversely, assume that $x\in\mathbb{C}^{N}$ and $x_{k}\neq 0$ for all $1\leq
k\leq N$ and let $f:\mathbb{D}\rightarrow\mathbb{C}^{N}$ be any coordinate
bounded analytic function such that
$\log|x|\leq\int\log|f(\gamma)|dm(\gamma)$. Then by (20),
$\underset{\gamma\in\mathbb{T}}{\text{ess sup }}\|f(\gamma)\|_{\gamma}<\infty$
and there exists $y\in\mathbb{C}^{N}$ such that
(22) $\displaystyle|x|\leq|y|\text{ and
}\log|y|=\int\log|f(\gamma)|dm(\gamma).$
Define $u(\gamma):=\log|f(\gamma)|$. By assumption, $x_{k}\neq 0$ and $f_{k}$
is bounded, hence $\log|f_{k}|\in L_{1}(\mathbb{T})$, so we can define the
Hilbert transform of $u_{k}$. Let $\tilde{u}(\gamma)$ be the Hilbert transform
of $u(\gamma)$ and define $g(\gamma)=e^{u(\gamma)+i\tilde{u}(\gamma)}$. Then
$g_{k}(\gamma)=e^{u_{k}(\gamma)+i\tilde{u}_{k}(\gamma)}$ is the boundary value
of an outer function, hence
$\log|g_{k}(0)|=\int\log|g_{k}(\gamma)|dm(t)=\int
u_{k}(\gamma)dm(\gamma)=\log|y_{k}|.$
Thus $|y|=|g(0)|$. By [CCRSW82, Prop. 2.4], we have
$\displaystyle\|y\|_{\mathscr{L}[0]}$ $\displaystyle=$
$\displaystyle\|g(0)\|_{\mathscr{L}[0]}\leq\exp\Big{(}\int\log\|g(\gamma)\|_{\gamma}\,dm(\gamma)\Big{)}$
$\displaystyle=$
$\displaystyle\exp\Big{(}\int\log\|f(\gamma)\|_{\gamma}\,dm(t)\Big{)}.$
It is easy to see that $\mathscr{L}[0]$ is a Banach lattice and by (22),
$\|x\|_{\mathscr{L}[0]}\leq\|y\|_{\mathscr{L}[0]}.$
Thus
$\log\|x\|_{\mathscr{L}[0]}\leq\log\|y\|_{\mathscr{L}[0]}\leq\int\log\|f(\gamma)\|_{\gamma}\,dm(\gamma).$
This proves the converse inequality. ∎
###### Remark 4.2.
The preceding result should be compared with [CCRSW82, Cor. 5.2], where it is
shown that
$\Big{\\{}L^{p_{\gamma}}(X,\Sigma,\mu):\gamma\in\mathbb{T}\Big{\\}}[z]=L^{p_{z}}(X,\Sigma,\mu),$
where $1/p_{z}=\int(1/p_{\gamma})P_{z}(d\gamma)$.
###### Definition 4.3.
Let $\mathscr{L}=(\mathbb{C}^{N},\|\cdot\|_{\mathscr{L}})$ be a symmetric
Banach lattices, we define $S_{\mathscr{L}}$ to be the space of $N\times N$
matrices equipped with the norm :
$\|A\|_{S_{\mathscr{L}}}=\|(s_{1}(A),\cdots,s_{N}(A))\|_{\mathscr{L}},$
where $s_{1}(A),\cdots,S_{N}(A)$ are singular numbers of the matrix $A$.
If the Banach lattices $\mathscr{L}_{\gamma}$ considered above are all
symmetric, i.e., for any permutation $\sigma\in\mathfrak{S}_{N}$ and any
$x_{k}\in\mathbb{C}$,
$\|\sum_{k=1}^{N}x_{k}e_{\sigma(k)}\|_{\mathscr{L}_{\gamma}}=\|\sum_{k=1}^{N}x_{k}e_{k}\|_{\mathscr{L}_{\gamma}},$
then to each $\mathscr{L}_{\gamma}$ is associated a Schatten type space
$S_{\mathscr{L}_{\gamma}}=(M_{N},\|\cdot\|_{S_{\mathscr{L}_{\gamma}}})$.
The following proposition is classical (c.f. [Pie71]), we omit its proof.
###### Proposition 4.4.
Let
$\\{\mathscr{L}_{\gamma}=(\mathbb{C}^{N},\|\cdot\|_{\gamma}):\gamma\in\mathbb{T}\\}$
be an interpolation family of compatible symmetric Banach lattices and
consider the associated interpolation family:
$\\{S_{\mathscr{L}_{\gamma}}=(M_{N},\|\cdot\|_{S_{\mathscr{L}_{\gamma}}}):\gamma\in\mathbb{T}\\}.$
Then for any $z\in\mathbb{D}$, we have the following isometric identification
$Id:S_{\mathscr{L}[z]}\rightarrow\\{S_{\mathscr{L}_{\gamma}}\\}[z].$
Combining Propositions 4.1 and 4.4, we have the following:
###### Corollary 4.5.
Consider the interpolation family
$\\{S_{\mathscr{L}_{\gamma}}:\gamma\in\mathbb{T}\\}.$ Let
$M:\mathbb{T}\rightarrow\mathbb{T}$ be measure preserving and let
$\\{\widetilde{S}_{\mathscr{L}_{\gamma}}=S_{\mathscr{L}_{M(\gamma)}}:\gamma\in\mathbb{T}\\}$,
then
$Id:\\{S_{\mathscr{L}_{\gamma}}\\}[0]\rightarrow\\{\widetilde{S}_{\mathscr{L}_{\gamma}}\\}[0]$
is isometric.
The following proposition is related to our problem, see the discussion after
it.
###### Proposition 4.6.
Let $\\{E_{\gamma}:\gamma\in\mathbb{T}\\}$ be an interpolation family of
$N$-dimensional spaces such that there exist $c,C>0$, for any
$x\in\mathbb{C}^{N}$,
$c\cdot\min_{k}|x_{k}|\leq\|x\|_{\gamma}\leq C\cdot\max_{k}|x_{k}|\text{ for
}a.e.\,\gamma\in\mathbb{T}.$
Assume that $Id:E_{\bar{\gamma}}\rightarrow\overline{E_{\gamma}}^{*}$ is
isometric for $a.e.\,\gamma\in\mathbb{T}$. Then
$E[\zeta]=\ell_{2}^{N},\text{ for any }\zeta\in(-1,1).$
###### Proof.
Fix $\zeta\in(-1,1)$. For any $x\in\mathbb{C}^{N}$. Given any analytic
function $f:\mathbb{D}\rightarrow\mathbb{C}^{N}$ such that $f(\zeta)=x$ and
$\|f\|_{H^{\infty}(\\{E_{\gamma}\\})}<\infty$. Since $\zeta=\bar{\zeta}$, we
have $f(\zeta)=f(\bar{\zeta})=x$. The assumption on the interpolation family
implies that the function $z\mapsto\langle f(z),f(\bar{z})\rangle$ is bounded
analytic, hence
$\displaystyle\log\|x\|_{\ell_{2}^{N}}^{2}$ $\displaystyle=$
$\displaystyle\log|\langle f(\zeta),f(\bar{\zeta})\rangle|\leq\int\log|\langle
f(\gamma),f(\bar{\gamma})\rangle|P_{\zeta}(d\gamma)$ $\displaystyle\leq$
$\displaystyle\int\log\Big{(}\|f(\gamma)\|_{E_{\gamma}}\|f(\bar{\gamma})\|_{\overline{E_{\gamma}^{*}}}\Big{)}P_{\zeta}(d\gamma)$
$\displaystyle=$
$\displaystyle\int\log\Big{(}\|f(\gamma)\|_{E_{\gamma}}\|f(\bar{\gamma})\|_{E_{\bar{\gamma}}}\Big{)}P_{\zeta}(d\gamma)$
$\displaystyle\leq$
$\displaystyle\log\Big{(}\|f\|_{H^{\infty}(\\{E_{\gamma}\\})}^{2}\Big{)}.$
Hence $\|x\|_{\ell_{2}^{N}}\leq\|f\|_{H^{\infty}(\\{E_{\gamma}\\})}.$ It
follows that
$\|x\|_{\ell_{2}^{N}}\leq\|x\|_{E[\zeta]}.$
By duality, this inequality also holds in the dual case, hence we must have
$\|x\|_{\ell_{2}^{N}}=\|x\|_{E[\zeta]}.$ ∎
Let $Q_{j}$ be the open arc of $\mathbb{T}$ in the $j$-th quadrant, i.e.,
$Q_{j}=\Big{\\{}e^{i\theta}:\frac{(k-1)\pi}{2}<\theta<\frac{k\pi}{2}\Big{\\}}\text{
for }1\leq j\leq 4.$
Suppose that $X$ and $Y$ are $N$-dimensional, define two interpolation
families $\\{Z_{\gamma}:\gamma\in\mathbb{T}\\}$ and
$\\{\widetilde{Z}_{\gamma}:\gamma\in\mathbb{T}\\}$ by letting
$Z_{\gamma}=\left\\{\begin{array}[]{cl}X,&\gamma\in Q_{1}\\\ Y,&\gamma\in
Q_{2}\\\ \overline{Y^{*}},&\gamma\in Q_{3}\\\ \overline{X^{*}},&\gamma\in
Q_{4}\end{array},\right.\quad\widetilde{Z}_{\gamma}=\left\\{\begin{array}[]{cl}X,&\gamma\in
Q_{1}\\\ Y,&\gamma\in Q_{2}\\\ \overline{X^{*}},&\gamma\in Q_{3}\\\
\overline{Y^{*}},&\gamma\in Q_{4}\end{array}.\right.$
By Proposition 4.6, $Z[0]=\ell_{2}^{N}$. For suitable choices of $X$ and $Y$,
we could have $\widetilde{Z}[0]\neq\ell_{2}^{N}$. More precisely, we have the
following proposition.
###### Proposition 4.7.
For any $\alpha\in\mathbb{T}$, define a $2\times 2$ selfadjoint matrix
$\delta_{\alpha}:=\left[\begin{array}[]{cc}0&\overline{\alpha}\\\
\alpha&0\end{array}\right].$
For $0<\varepsilon<1$, let
$w^{\alpha,\varepsilon}=(I+\varepsilon\delta_{\alpha})^{1/2}$ and
$X=\ell^{2}_{w^{\alpha,\varepsilon}}$. Consider the weight
$W^{\alpha,\varepsilon}$ and the interpolation family generated by it as
follows:
$W^{\alpha,\varepsilon}(\gamma)=\left\\{\begin{array}[]{cl}I+\varepsilon\delta_{\alpha},&\gamma\in
Q_{1}\\\ (I+\varepsilon\overline{\delta_{\alpha}})^{-1},&\gamma\in Q_{2}\\\
(I+\varepsilon\delta_{\alpha})^{-1},&\gamma\in Q_{3}\\\
I+\varepsilon\overline{\delta_{\alpha}},&\gamma\in
Q_{4}\end{array};\right.\quad\widetilde{Z^{\alpha,\varepsilon}_{\gamma}}=\left\\{\begin{array}[]{cl}X,&\gamma\in
Q_{1}\\\ X^{*},&\gamma\in Q_{2}\\\ \overline{X}^{*},&\gamma\in Q_{3}\\\
\overline{X},&\gamma\in Q_{4}\end{array}.\right.$
There exists $\alpha\in\mathbb{T}$ and $0<\varepsilon_{0}<1$, such that if
$0<\varepsilon<\varepsilon_{0}$ then
$\widetilde{Z^{\alpha,\varepsilon}}[0]\neq\ell_{2}^{N}.$
###### Proof.
We have
$W^{\alpha,\varepsilon}(\gamma)=\left\\{\begin{array}[]{cl}I+\varepsilon\delta_{\alpha},&\gamma\in
Q_{1}\\\
I-\varepsilon\overline{\delta_{\alpha}}+\varepsilon^{2}I+\mathcal{O}(\varepsilon^{3}),&\gamma\in
Q_{2}\\\
I-\varepsilon\delta_{\alpha}+\varepsilon^{2}I+\mathcal{O}(\varepsilon^{3}),&\gamma\in
Q_{3}\\\ I+\varepsilon\overline{\delta_{\alpha}},&\gamma\in
Q_{4}\end{array}.\right.$
Applying a slightly modified variant of the approximation equation (10), we
have
$\displaystyle|F_{W^{\alpha,\varepsilon}}(0)|^{2}$ $\displaystyle=$
$\displaystyle
I+\frac{\varepsilon^{2}I}{2}-\varepsilon^{2}\left[\begin{array}[]{cc}\|P_{+}(h_{\alpha})\|_{2}^{2}&0\\\
0&\|P_{+}(\overline{h_{\alpha}})\|_{2}^{2}\end{array}\right]+\mathcal{O}(\varepsilon^{3});$
where $h_{\alpha}=\alpha 1_{Q_{1}}-\overline{\alpha}1_{Q_{2}}-\alpha
1_{Q_{3}}+\overline{\alpha}1_{Q_{4}}.$
Assume by contradiction that
$\widetilde{Z^{\alpha,\varepsilon}}[0]=\ell_{2}^{N}$ for any
$\alpha\in\mathbb{T}$ and small $\varepsilon$. Then we must have
$\|P_{+}(h_{\alpha})\|_{2}^{2}=\frac{1}{2}$ for any $\alpha\in\mathbb{T}$. In
particular,
$\alpha\mapsto\|P_{+}(h_{\alpha})\|_{2}^{2}\text{ is a constant function on
$\mathbb{T}$}.$
It follows that the following function is a constant function:
$\displaystyle C(\alpha)$ $\displaystyle=$ $\displaystyle\Re\langle\alpha
P_{+}(1_{Q_{1}}),-\overline{\alpha}P_{+}(1_{Q_{2}})\rangle+\Re\langle\alpha
P_{+}(1_{Q_{1}}),\overline{\alpha}P_{+}(1_{Q_{4}})\rangle$ $\displaystyle+$
$\displaystyle\Re\langle-\overline{\alpha}P_{+}(1_{Q_{2}}),-\alpha
P_{+}(1_{Q_{3}})\rangle+\Re\langle-\alpha
P_{+}(1_{Q_{3}}),\overline{\alpha}P_{+}(1_{Q_{4}})\rangle.$
Clearly, by translation invariance of Haar measure, we have
$\langle P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle=\langle
P_{+}(1_{Q_{2}}),P_{+}(1_{Q_{3}})\rangle=\langle
P_{+}(1_{Q_{3}}),P_{+}(1_{Q_{4}})\rangle,$ $\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{4}})=\langle
P_{+}(1_{Q_{2}}),P_{+}(1_{Q_{1}})\rangle,$
hence
$C(\alpha)=-\Re\Big{\\{}2\alpha^{2}\Big{(}\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle-\overline{\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle}\Big{)}\Big{\\}}.$
Then $\alpha\mapsto C(\alpha)$ is constant function if and only if
$\langle P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle-\overline{\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle}=0,$
which is equivalent to
(24) $\displaystyle\Im\Big{(}\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle\Big{)}=0.$
By a similar computation as in the proof of inequality (18), we have
$\Im\Big{(}\langle
P_{+}(1_{Q_{1}}),P_{+}(1_{Q_{2}})\rangle\Big{)}=\frac{4}{\pi^{2}}\sum_{k=0}^{\infty}\frac{2k+1}{(4k+1)^{2}(4k+3)^{2}},$
this contradicts (24), and hence completes the proof. ∎
## 5\. Appendix
Here we reformulate the argument of [CCRSW82] to emphasize the crucial role
played by a certain inner function associated to the measurable partition of
the unit circle in proving Theorem 0.1. It follows from the preceding that the
analogous inner function for a measurable partition into 3 subsets does not
exist.
###### Lemma 5.1.
Suppose that $\Gamma_{0}\cup\Gamma_{1}$ is a measurable partition of
$\mathbb{T}$. Then there exists an inner function $\varphi$ such that
$\varphi(0)=0$. And $\varphi(\Gamma_{0})\cup\varphi(\Gamma_{1})$ is a
partition of $\mathbb{T}$ into two disjoint arcs (up to negligible sets).
Moreover,
(25) $\displaystyle m(\varphi(\Gamma_{0}))=m(\Gamma_{0})\textit{ and
}m(\varphi(\Gamma_{1}))=m(\Gamma_{1}).$
###### Proof.
Since any origin-preserving inner function $\varphi$ preserves the measure $m$
on $\mathbb{T}$ (indeed note
$\int_{\mathbb{T}}\varphi(\gamma)^{n}dm(\gamma)=\int_{\mathbb{T}}\gamma^{n}dm(\gamma)\,\forall
n\in\mathbb{Z}$), it suffices to show the existence of an inner function
satisfying the partition condition.
Let $v=1_{\Gamma_{1}}:\mathbb{T}\rightarrow\mathbb{R}$ be the characteristic
function of $\Gamma_{1}$, its harmonic extension on $\mathbb{D}$ will also be
denoted by $v$. Note that $0<v(z)<1$ for any $z\in\mathbb{D}$. Let $\tilde{v}$
be the harmonic conjugate of $v$ and define $\psi=v+i\tilde{v}$ on
$\mathbb{D}$. Then $\psi$ is an analytic map from $\mathbb{D}$ to
$\mathcal{S}:=\\{z\in\mathbb{C}:0<\Re(z)<1\\}$ and has non-tangential limit
$\psi(\gamma)=v(\gamma)+i\tilde{v}(\gamma)$, a.e. $\gamma\in\mathbb{T}$. Thus
$\psi(\Gamma_{0})\subset\partial_{0}\text{ and
}\psi(\Gamma_{1})\subset\partial_{1},$
where $\partial_{0}=\\{z\in\mathbb{C}:\Re(z)=0\\}$ and
$\partial_{1}=\\{z\in\mathbb{C}:\Re(z)=1\\}$. Let
$\tau:\mathcal{S}\rightarrow\mathbb{D}$ be a Riemann conformal mapping such
that $\tau(\psi(0))=0$. Note that $\tau(\partial_{0})$ and
$\tau(\partial_{1})$ are disjoint open arcs of $\mathbb{T}$. Define
$\varphi=\tau\circ\psi:\mathbb{D}\rightarrow\mathbb{D}.$ Then $\varphi$ is an
inner function such that $\varphi(0)=0$. We have
$\varphi(\Gamma_{0})\subset\tau(\partial_{0})\text{ and
}\varphi(\Gamma_{1})\subset\tau(\partial_{1}).$
Hence $m(\varphi(\Gamma_{0}))\leq m(\tau(\partial_{0}))$ and
$m(\varphi(\Gamma_{1}))\leq m(\tau(\partial_{1}))$. Since $\varphi$ preserves
the measure $m$, we have
$1=m(\varphi(\Gamma_{0}))+m(\varphi(\Gamma_{1}))\leq
m(\tau(\partial_{0}))+m(\tau(\partial_{1}))=1.$
Thus up to negligible sets, we have
$\varphi(\Gamma_{0})=\tau(\partial_{0})\text{ and
}\varphi(\Gamma_{1})=\tau(\partial_{1}).$
∎
###### Proof of Theorem 0.1.
Suppose $\Gamma_{0}\cup\Gamma_{1}$ is a measurable partition of the circle and
let the interpolation family $\\{X_{\gamma}:\gamma\in\mathbb{T}\\}$ be such
that
$X_{\gamma}=Z_{0}\text{ for all
}\gamma\in\Gamma_{0},\,\,X_{\gamma}=Z_{1}\text{ for all }\gamma\in\Gamma_{1}.$
By Lemma 5.1, we can find an inner function $\varphi$ such that $\varphi(0)=0$
and $\varphi(\Gamma_{0})=J_{0},\,\,\varphi(\Gamma_{1})=J_{1}$ up to negligible
sets, where $J_{0}\cup J_{1}$ is a partition of the circle into disjoint arcs.
Consider the interpolation family of spaces
$\\{\widetilde{X}_{\gamma}:\gamma\in\mathbb{T}\\}$ such that
$\widetilde{X}_{\gamma}=Z_{0}\text{ for all }\gamma\in
J_{0},\,\,\widetilde{X}_{\gamma}=Z_{1}\text{ for all }\gamma\in J_{1}.$
Then by a conformal mapping, it is easy to see
(26)
$\displaystyle\widetilde{X}[0]=(Z_{0},Z_{1})_{\theta},\,\,\theta=m(J_{1})=m(\Gamma_{1}).$
We have $\widetilde{X}_{\varphi(\gamma)}=X_{\gamma}$ for a.e.
$\gamma\in\mathbb{T}$. If $x\in\mathbb{C}^{N}$ is such that
$\|x\|_{\widetilde{X}[0]}<1$, then by definition, there exists an analytic
function $f:\mathbb{T}\rightarrow\mathbb{C}^{N}$ such that $f(0)=x$ and
$\underset{t\in\mathbb{T}}{\text{ess sup
}}\|f(\gamma)\|_{\widetilde{X}_{\gamma}}<1$. Thus
$\underset{\gamma\in\mathbb{T}}{\text{ess sup
}}\|(f\circ\varphi)(\gamma)\|_{X_{t}}=\underset{\gamma\in\mathbb{T}}{\text{ess
sup
}}\|(f\circ\varphi)(\gamma)\|_{\widetilde{X}_{\varphi(\gamma)}}=\underset{\gamma\in\mathbb{T}}{\text{ess
sup }}\|f(\gamma)\|_{\widetilde{X}_{\gamma}}<1.$
Since $(f\circ\varphi)(0)=f(0)=x$, the above inequality shows that
$\|x\|_{X[0]}<1.$ By homogeneity, $\|x\|_{X[0]}\leq\|x\|_{\widetilde{X}[0]}.$
But if we consider the dual of the above interpolation family, then we get the
same inequality, hence we must have
(27) $\displaystyle\|x\|_{X[0]}=\|x\|_{\widetilde{X}[0]}.$
By (27) and (26), we have
$X[0]=(Z_{0},Z_{1})_{\theta},\,\,\theta=m(\Gamma_{1}).$
∎
By definition, a space is arcwise $\theta$-Hilbertian if it can be obtained by
complex interpolation of a family of spaces on the circle such that on an arc,
the spaces are Hilbertian.
###### Remark 5.2 (Communicated by Gilles Pisier).
The preceding argument also shows that, as conjectured in [Pis10], of which we
use the terminology, any $\theta$-Hilbertian Banach space is automatically
arcwise $\theta$-Hilbertian, at least under suitable assumptions on the dual
spaces, that are automatic in the finite dimensional case. We merely indicate
the argument in the latter case. Consider a measurable partition
$\Gamma_{0}\cup\Gamma_{1}$ of the unit circle with $m(\Gamma_{1})=\theta$ and
a family of $n$-dimensional spaces $\\{E_{\gamma}\mid\gamma\in\partial D\\}$
such that $E_{\gamma}=\ell_{2}^{n}$ for any $\gamma\in\Gamma_{1}$ but
$E_{\gamma}$ is arbitrary for $\gamma\in\Gamma_{0}$. If $\varphi$ is the inner
function appearing in Lemma 5.1, and if we set
$F_{\gamma}=E_{\varphi(\gamma)}$ then the identity map $Id:E[0]\to F[0]$ is
clearly contractive and $F[0]$ is arcwise $\theta$-Hilbertian. Applying this
to the dual family $\\{E_{\gamma}^{*}\\}$ in place of $\\{E_{\gamma}\\}$ and
using the duality theorem from [CCRSW82, Th. 2.12 ]) we find that $Id:\
{E[0]}^{*}\to{F[0]}^{*}$ is also contractive, and hence is isometric. This
shows that $E[0]$ is arcwise $\theta$-Hilbertian.
## Acknowledgements
The author is grateful to Gilles Pisier for stimulating discussions and
valuable suggestions, he would like to thank the referee for careful reading
of the manuscript. The author was partially supported by the ANR grant
2011-BS01-00801 and the A*MIDEX grant.
## References
* [CCRSW79] R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, _Complex interpolation for families of Banach spaces_ , Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 269–282. MR 545314 (81a:46082)
* [CCRSW82] R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, _A theory of complex interpolation for families of Banach spaces_ , Adv. in Math. 43 (1982), no. 3, 203–229. MR 648799 (83j:46084)
* [Hel64] Henry Helson, _Lectures on invariant subspaces_ , Academic Press, New York (1964).
* [HL58] Henry Helson and David Lowdenslager, _Prediction theory and Fourier series in several variables_ , Acta Math. 99 (1958), 165–202. MR 0097688 (20 #4155)
* [HL61] by same author, _Prediction theory and Fourier series in several variables. II_ , Acta Math. 106 (1961), 175–213. MR 0176287 (31 #562)
* [Pie71] Albrecht Pietsch, _Interpolationsfunktoren, Folgenideale und Operatorenideale_ , Czechoslovak Math. J. 21(96) (1971), 644–652. MR 0293410 (45 #2487)
* [Pis10] Gilles Pisier, _Complex interpolation between Hilbert, Banach and operator spaces_ , Mem. Amer. Math. Soc. 208 (2010), no. 978, vi+78. MR 2732331 (2011k:46024)
Yanqi QIU
|
arxiv-papers
| 2013-04-04T15:39:42 |
2024-09-04T02:49:43.883985
|
{
"license": "Public Domain",
"authors": "Yanqi Qiu",
"submitter": "Yanqi Qiu",
"url": "https://arxiv.org/abs/1304.1403"
}
|
1304.1411
|
# RITA: An Index-Tuning Advisor for Replicated Databases
Quoc Trung Tran
UC Santa Cruz [email protected] Ivo Jimenez
UC Santa Cruz [email protected] Rui Wang
UC Santa Cruz [email protected] Neoklis Polyzotis
UC Santa Cruz [email protected] Anastasia Ailamaki
École Polytechnique F́edérale de Lausanne [email protected]
###### Abstract
Given a replicated database, a divergent design tunes the indexes in each
replica differently in order to specialize it for a specific subset of the
workload. This specialization brings significant performance gains compared to
the common practice of having the same indexes in all replicas, but requires
the development of new tuning tools for database administrators. In this paper
we introduce RITA (Replication-aware Index Tuning Advisor), a novel divergent-
tuning advisor that offers several essential features not found in existing
tools: it generates robust divergent designs that allow the system to adapt
gracefully to replica failures; it computes designs that spread the load
evenly among specialized replicas, both during normal operation and when
replicas fail; it monitors the workload online in order to detect changes that
require a recomputation of the divergent design; and, it offers suggestions to
elastically reconfigure the system (by adding/removing replicas or
adding/dropping indexes) to respond to workload changes. The key technical
innovation behind RITA is showing that the problem of selecting an optimal
design can be formulated as a Binary Integer Program (BIP). The BIP has a
relatively small number of variables, which makes it feasible to solve it
efficiently using any off-the-shelf linear-optimization software. Experimental
results demonstrate that RITA computes better divergent designs compared to
existing tools, offers more features, and has fast execution times.
## 1 Introduction
Database replication is used heavily in distributed systems and database-as-a-
service platforms (e.g., Amazon’s Relational Database Service [1] or Microsoft
SQL Azure [2]), to increase availability and to improve performance through
parallel processing. The database is typically replicated across several
nodes, and replicas are kept synchronized (eagerly or lazily) when updates
occur so that incoming queries can be evaluated on any replica.
_Divergent-design tuning_ [5] represents a new paradigm to tune workload
performance over a replicated database. A divergent design leverages
replication as follows: it specializes each replica to a specific subset of
the workload by installing indexes that are particularly beneficial for the
corresponding workload statements. Thus, queries can be evaluated more
efficiently by being routed to a specialized replica. As shown in a previous
study, a divergent design brings significant performance improvements when
compared to a uniform design that uses the same indexes in all replicas:
queries are executed faster due to replica specialization (up to 2x
improvement on standard benchmarks), but updates as well become significantly
more efficient (more than 2x improvement) since fewer indexes need to be
installed per replica.
To reap the benefits of divergent designs in practice, DB administrators need
new index-tuning advisors that are replication-aware. The original study [5]
introduces an advisor called DivgDesign, which creates specialized designs per
replica but has severe limitations that restrict its usefulness in practice.
Firstly, DivgDesign assumes that replicas are always operational. Replica
failures, however, are common in real systems, and the resulting workload
redistribution may cause queries to be routed to low-performing replicas, with
predictably negative effects on the overall system performance. An effective
advisor should generate robust divergent designs that allow the system to
adapt gracefully to replica failures. Secondly, DivgDesign ignores the effect
of specialization to each replica’s load, and can therefore incur a skewed
load distribution in the system. Our experiments suggest that DivgDesign can
cause certain replicas to be twice as loaded as others. A good advisor should
take the replica load into account, and generate divergent designs that
provide the benefits of specialization while maintaining a balanced load
distribution. Lastly, DivgDesign targets a static system where the database
workload and the number of replicas are assumed to remain unchanged. A
replicated database system, however, is typically volatile: the workload may
change over time, and in response the DBA may wish to elastically reconfigure
the system by expanding or shrinking the set of replicas and by incrementally
adding or dropping indexes at different replicas. A replication-aware advisor
should alert the DBA when a workload change necessitates retuning the
divergent design, and also help the DBA evaluate options for changing the
design.
The limitations of DivgDesign stem from the fact that it internally employs a
conventional index-tuning advisor, e.g., DB2’s db2advis or the index advisor
of MS SQL Server, which is not suitable for modeling and solving the
aforementioned issues. Modifying DivgDesign to address its limitations would
require a non-trivial redesign of the advisor. A more general question is
whether it is even feasible to reap the performance benefits demonstrated in
[5] and at the same time maintain a balanced load and the ability to adapt
gracefully to failures. Our work shows that this is indeed feasible but
requires the development of a new type of index-tuning advisor that is
replication-aware.
Contributions. In this paper, we introduce a novel index advisor termed RITA
(Replication-aware Index Tuning Advisor) that provides DBAs with a powerful
tool for divergent index tuning. Instead of relying on conventional techniques
for index tuning, RITA is a new type of index advisor that is designed from
the ground up to take into account replication and the unique characteristics
of divergent designs. RITA’s foundation is a novel reduction of the problem of
divergent design tuning to Binary Integer Programming (BIP). The BIP
formulation allows RITA to employ an off-the-shelf linear optimization solver
to compute near-optimal designs that satisfy complex constraints (e.g., even
load distribution or robustness to failures). Compared to DivgDesign, RITA
offers richer tuning functionality and is able to compute divergent designs
that result in significantly better performance.
More concretely, the contributions of our work can be summarized as follows:
(1.) To make divergent designs suitable for the characteristics of real-world
systems, we introduce a generalized version of the problem of divergent design
tuning that has two important features: it takes into account the probability
of replica failures and their effect on workload performance; and, it allows
for an expanded class of constraints on the computed divergent design and in
particular constraints on global system-properties, e.g., maintaining an even
load distribution (Section 3).
(2.) We prove that, under realistic assumptions about the underlying system,
the generalized tuning problem can be formulated as a compact Binary Integer
Program (BIP), i.e., a linear-optimization problem with a relatively small
number of binary variables. The implication is that we can use an off-the-
shelf solver to efficiently compute a (near-)optimal divergent design that
also satisfies any given constraints (Section 4).
(3.) We propose RITA as a new index-tuning tool that leverages the previous
theoretical result to implement a unique set of features. RITA allows the DBA
to initially tune the divergent design of the system using a training
workload. Subsequently, RITA continuously analyzes the incoming workload and
alerts the DBA if a retuning of the divergent design could lead to substantial
performance improvements. The DBA can then examine how to elastically adapt
the divergent design to the changed workload, e.g., by expanding/shrinking the
set of replicas, incrementally adding/removing indexes, or changing how
queries are distributed across replicas. Internally, RITA translates the DBA’s
requests to BIPs that are solved efficiently by a linear-optimization solver.
In fact, RITA often returns its answers in seconds, thus facilitating an
exploratory approach to index tuning (Section 5).
(4.) We perform an extensive experimental study to validate the effectiveness
of RITA as a tuning advisor. The results show that the designs computed by
RITA can improve system performance by up to a factor of four compared to the
standard uniform design that places the same indexes on all replicas.
Moreover, RITA outperforms DivgDesign by up to a factor of three in terms of
the performance of the computed divergent designs, while supporting a larger
class of constraints (Section 6).
Overall, RITA provides a positive answer to our previously stated question: a
divergent design can bring significant performance benefits while maintaining
important properties such as a balanced load distribution and tolerance to
failures. Consequently, divergent design advisors can be practically employed
on real systems and guide further development of tuning tools. The underlying
theoretical results (problem definition and BIP formulation) are also
significant, as they expand on the previous work on single-system tuning [6]
and demonstrate a wider applicability of Binary Integer Programming to index-
tuning problems.
## 2 Related work
Index tuning. There has been a long line of research studies on the problem
of tuning the index configuration of a single DBMS (e.g., [3, 6, 18]). These
methods analyze a representative workload and recommend an index configuration
that optimizes the evaluation of the workload according to the optimizer’s
estimates. A recent study [6] has introduced the COPHY index advisor that
outperforms state-of-the-art commercial and research techniques by up to an
order of magnitude in terms of both solution quality and total execution time.
Both RITA and COPHY leverage the same underlying principle of linear
composability, which we will define and discuss extensively in Section 4.1, in
order to cast the index-tuning problem as a compact, efficiently-solvable
Binary Integer Program (BIP). However, COPHY targets the conventional index-
tuning problem where the goal is to compute a single index configuration for a
single-node system. This problem scenario is much simpler than what we
consider in our work, where there are several nodes in the system, each can
carry a different index configuration, queries have to be distributed in a
balanced fashion and the system must recover gracefully from failures.
Leveraging the principle of linear composability in this generalized problem
scenario is one of the key contributions of our work.
Physical data organization on replicas. Previous works also considered the
idea of diverging the physical organization of replicated data. The technique
of Fractured Mirrors [12] builds a mirrored database that stores its base data
in a different physical organizations on disk (specifically, in a row-based
and a column-based organization). Similarly, Distorted Mirrors [14] presents
logically but not physically identical mirror disks for replicated data.
However, they do not consider how to tune the indexes for each mirror.
There are recent works [9, 8] that explore different physical designs for
different replicas in Hadoop context. Specifically, TROJAN HDFS [9] organizes
each physical replica of an HDFS block in a different data layout, where each
data layout corresponds to a different vertical partitioning. Likewise, HAIL
[8] organizes each physical replica of an HDFS block in a different sorted
order, which essentially amounts to exactly one clustered index per replica.
Our work differs from these works in several aspects. First, these works do
not consider the problem of spreading the load evenly among specialized
replicas that we are considering. Second, performance is very sensitive to
failures, because the tuning options considered by these papers lead to
replicas that are highly specialized for subsets of the workload. When a
replica fails, the corresponding queries will be rerouted to replicas with
little provision to handle the rerouted workload, and hence performance may
suffer. In contrast, we focus on divergent designs that directly take into
account the possibility of replicas failing, thus offering more stable
performance when one or more replicas become unavailable. Third, [8] creates
one index per replica which restricts the extent to which we can tune each
replica to the workload. Our methods do not have any such built-in limitations
and are only restricted by configurable constraints on the materialized
indexes (e.g., total space consumption, or total maintenance cost). Fourth,
our work can return a set of possible designs that represent trade-off points
within a multi-dimensional space, e.g., between workload evaluation cost and
design-materialization cost. These works do not support this functionality.
The original study on divergent-design tuning [5] introduced the DivgDesign
advisor which is the direct competitor to our proposed RITA advisor. However,
as we discussed in Section 1, DivgDesign is fundamentally limited by the
functionality of the underlying single-system advisor, and cannot support many
essential tuning functionalities as RITA.
## 3 Divergent Design Tuning: Problem Statement
In this section, we formalize the problem of divergent design tuning. The
problem statement borrows several concepts from the original problem statement
in [5] but also provides a non-trivial generalization. A comparison to the
original study appears at the end of the section.
### 3.1 Basic Definitions
We consider a database comprising tables $T_{1},\dots,T_{n}$. An _index
configuration_ $X$ is a set of indexes defined over the database tables. We
assume that $X$ is a subset of a universe of candidate indexes ${\cal S}={\cal
S}_{1}\ \cup\ \cdots\ \cup\ {\cal S}_{n}$, where ${\cal S}_{i}$ represents the
set of candidate indexes on table $T_{i}$. Each ${\cal S}_{i}$ represents a
very large set of indexes and can be derived manually by the DBA or by mining
the query logs. We do not place any limitations on the indexes regarding their
type or the type or count of attributes that they cover, except that each
index in $X$ is defined on exactly one table (i.e., no join indexes).
We use ${\mathit{cost}}(q,X)$ to denote the cost of evaluating query $q$
assuming that $X$ is materialized. The cost function can be evaluated
efficiently in modern systems (i.e., without materializing $X$) using a _what-
if optimizer_ [4]. We define ${\mathit{cost}}(u,X)$ similarly for an update
statement $u$, except that in this case we also consider the overhead of
maintaining the indexes in $X$ due to the update. Following common practice
[13, 6], we break the execution of $u$ into two orthogonal components: (1) a
query shell $q_{sel}$ that selects the tuples to be updated, and (2) an update
shell that performs the actual update on base tables and also updates any
affected materialized indexes. Hence, the total cost of an update statement
can be expressed as $cost(u,X)=cost(q_{sel},X)+\sum_{a\in X}ucost(u,a)+c_{u}$,
where $ucost(u,a)$ is the cost to update index $a$ with the effects of the
update and can be estimated again using the what-if optimizer. The constant
$c_{u}$ is simply the cost to update the base data which does not depend on
$X$.
We consider a database that is fully replicated in $N$ nodes, i.e., each node
$i\in[1,N]$ holds a full copy of the database. The replicas are kept
synchronized by forwarding each database update to all replicas (lazily or
eagerly). At the same time, a query can be evaluated by any replica. Since we
are dealing with a multi-node system, we have to take into account the
possibility of replicas failing. We use $\alpha$ to denote the probability of
at least one replica failing. Setting this parameter can be done once in the
beginning to the best of the DBA’s ability and then it can be updated with
easy statistics as the system is used (you adjust it based on the failure rate
you see). To simplify further notation, we will assume that at most one
replica can fail at any point in time. The extension to multiple replicas
failing together is straightforward for our problem.
We define $W=Q\cup U$ as a workload comprising a set $Q$ of query statements
and a set $U$ of update statements. Workload $W$ serves as the representative
workload for tuning the system. As is typical in these cases, we also define a
weight function $f:W\rightarrow\Re$ such that $f(x)$ corresponds to the
importance of query or update statement $x$ in $W$. The input workload and
associated weights can be hand-crafted by the DBA or they can be obtained
automatically, e.g., by analyzing the query logs of the database system.
### 3.2 Problem Statement
At a high level, a divergent design allows each replica to have a different
index configuration, tailored to a particular subset of the workload. To
evaluate the query workload $Q$, an ideal strategy would route each $q\in Q$
to the replica that minimizes the execution cost for $q$. However, this ideal
routing may not be feasible for several reasons, e.g., the replica may not be
reachable or may be overloaded. Hence, the idea is to have several low-cost
replicas for $q$, so as to provide some flexibility for query evaluation. For
this purpose, we introduce a parameter $m\in[1,N]$, which we term routing
multiplicity factor. Informally, for every query $q\in Q$, a divergent design
specifies a set of $m$ low-cost replicas that $q$ can be routed to. The value
of $m$ is assumed to be set by the administrator who is responsible for tuning
the system: $m=1$ leads to a design that favors specialization; $m=N$ provides
for maximum flexibility; $1<m<N$ achieves some trade-off between the two
extremes.
Formally, we define a divergent design as a pair $(\mathbf{I},\mathbf{h})$.
The first component $\mathbf{I}=(\mathit{I}_{1},\dots,\mathit{I}_{N})$ is an
$N$-tuple, where $I_{r}$ is the index configuration of replica $r\in[1,N]$.
The second component
$\mathbf{h}=(\mathit{h}_{0},\mathit{h}_{1},\cdots,\mathit{h}_{N})$ is a
$(N+1)$-tuple of routing functions. Specifically, $\mathit{h}_{0}()$ is a
function over queries such that $\mathit{h}_{0}(q)$ specifies the set of $m$
replicas to which $q$ can be routed when all replicas are operational (i.e.,
there are no failures). Intuitively, $\mathit{h}_{0}(q)$ indicates the
replicas that can evaluate $q$ at low cost while respecting other constraints
(e.g., bounding load skew among replicas, which we discus later), and is meant
to serve as a hint to the runtime query scheduler. Therefore, a key
requirement is that $\mathit{h}_{0}()$ can be evaluated on any query $q$ and
not just the queries in the training workload. The remaining functions
$\mathit{h}_{1},\dots,\mathit{h}_{N}$ have a similar functionality but cover
the case when replicas fail: $\mathit{h}_{j}()$, for $j\in[1,N]$, specifies
how to route each query when replica $j$ has failed and is not reachable.
Notice that in this case there may be fewer than $m$ replicas in
$\mathit{h}_{j}(q)$ for any $q\in Q$ if the DBA has originally specified
$m=N$.
In order to quantify the goodness of a divergent design, we first use a metric
that captures the performance of the workload under the normal operation when
no running replica fails as follows.
$\displaystyle\mathit{TotalCost}(\mathbf{I},\mathbf{h})$ $\displaystyle=$
$\displaystyle\sum_{q\in
Q}\sum_{r\in\mathit{h}_{0}(q)}\frac{f(q)}{m}{\mathit{cost}}(q,\mathit{I}_{r})+$
$\displaystyle\sum_{u\in
U}\sum_{i\in[1,N]}f(u){\mathit{cost}}(u,\mathit{I}_{i})$
The second term simply captures the cost to propagate each update $u\in U$ to
each replica in the system. The first summation captures the cost to evaluate
the query workload $Q$. We assume that $q$ is routed uniformly among its $m$
replicas in $\mathit{h}_{0}(q)$, and hence the weight of $q$ is scaled by
$1/m$ for each replica. The intuition behind the
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$ metric is that it captures the
ability of the divergent design to achieve both replica specialization and
flexibility in load balancing with respect to $m$.
To capture the case of failures, we define
$\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$ as the performance of the
workload when replica $j\in[1,N]$ fails:
$\displaystyle\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$ $\displaystyle=$
$\displaystyle\sum_{q\in
Q}\sum_{r\in\mathit{h}_{j}(q)}\frac{f(q)}{\max\\{m,N-1\\}}{\mathit{cost}}(q,\mathit{I}_{r})+$
$\displaystyle\sum_{u\in
U}\sum_{i\in\\{1,\cdots,N\\}-\\{j\\}}f(u){\mathit{cost}}(u,\mathit{I}_{i})$
The expression for $\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$ is similar
to $\mathit{TotalCost}(\mathbf{I},\mathbf{h})$, except that, since replica $j$
is unavailable, the update cost on replica $j$ is discarded and routing
function $\mathit{h}_{j}$ is used instead of $\mathit{h}_{0}$.
We quantify the goodness of a divergent design $(\mathbf{I},\mathbf{h})$ based
on the expected cost of the workload, denoted as
$\mathit{ExpTotalCost}(\mathbf{I},\mathbf{h})$, by combining
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$ and
$\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$ weighted appropriately. Recall
that $\alpha$ is a DBA-specified probability that a failure will occur. It
follows that $(1-\alpha)$ is the probability that all replicas are operational
and hence the performance of the workload is computed by
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$. Conversely, the probability of a
specific replica $j$ failing is $\alpha/N$, assuming that all replicas can
fail independently with the same probability. In that case, the cost of
workload evaluation is $\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$. Putting
everything together, we obtain the following definition for the expected
workload cost:
$\displaystyle\mathit{ExpTotalCost}(\mathbf{I},\mathbf{h})$ $\displaystyle=$
$\displaystyle(1-\alpha)\cdot\mathit{TotalCost}(\mathbf{I},\mathbf{h})+$
$\displaystyle\sum_{j\in[1,N]}\frac{\alpha}{N}\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$
Our assumption so far is that at most one replica can be inoperational at any
point in time. The extension to concurrent failures is straightforward. All
that is needed is extending $\mathbf{h}$ with routing functions for
combinations of failed replicas, and then extending the expression of
$\mathit{ExpTotalCost}(\mathbf{I},\mathbf{h})$ with the corresponding cost
terms and associated probabilities.
We are now ready to formally define the problem of Divergent Design Tuning,
referred to as DDT.
###### Problem 1
(Divergent Design Tuning - DDT) We are given a replicated database with $N$
replicas, a workload $W=Q\ \cup\ U$, a candidate index-set ${\cal S}$, a set
of constraints $C$, a routing multiplicity factor $m$, and a probability of
failure $\alpha$. The goal is to compute a divergent design
$(\mathbf{I},\mathbf{h})$ that employs indexes in ${\cal S}$, satisfies the
constraints in $C$, and $\mathit{ExpTotalCost}(\mathbf{I},\mathbf{h})$ is
minimal among all feasible divergent designs. $\Box$
Constraints in DDT. The set of constraints $C$ enables the DBA to control the
space of divergent designs considered by the advisor. An _intra-replica_
constraint specifies some desired property that is local to a replica.
Examples include the following:
* •
The size of $I_{j}$ in $\mathbf{I}$ is within a storage-space budget.
* •
Indexes in $I_{j}$ must have specific properties, e.g., no index can be more
than 5-columns wide, or the count of multi-key indexes is below a limit.
* •
The cost to update the indexes in $I_{j}$ is below a threshold.
Conversely, an _inter-replica_ constraint specifies some property that
involves all the replicas. Examples include the following:
* •
If $(\mathbf{I}_{c},\mathbf{h}_{c})$ represents the current divergent design
of the system, then $\mathit{ExpTotalCost}(\mathbf{I},\mathbf{h})$ must
improve on $\mathit{ExpTotalCost}(\mathbf{I}_{c},\mathbf{h}_{c})$ by at least
some percentage.
* •
The total cost to materialize $(\mathbf{I},\mathbf{h})$ (i.e., to build each
$I_{j}$ in each replica) must be below some threshold.
* •
The load skew among replicas must be below some threshold. (We discuss this
constraint in more detail shortly.)
We will formalize later the precise class of constraints $C$ that we can
support in RITA. The goal is to provide support for a large class of practical
constraints, while retaining the ability to find effective designs
efficiently.
Bounding load skew is a particularly important inter-replica constraint that
we examine in our work. The replica-specialization imposed by a divergent
design means that each replica may receive a different subset of the workload,
and hence a different load. The $\mathit{ExpTotalCost}()$ metric does not take
into account these different loads, which means that minimizing workload cost
may actually lead to a high skew in terms of load distribution. Our
experiments verify this conjecture, showing that an optimal divergent design
in terms of $\mathit{ExpTotalCost}()$ can cause loads at different replicas to
differ by up to a factor of two. This situation, which is clearly detrimental
for good performance in a distributed setting, can be avoided by including in
$C$ a constraint on the load skew among replicas. More concretely, the load of
replica $j$ under normal operation can be computed as:
$\mathit{load}(\mathbf{I},\mathbf{h},j)=\sum_{q\in Q\land\
j\in\mathit{h}_{0}(q)}\frac{f(q)}{m}{\mathit{cost}}(q,\mathit{I}_{j})+\sum_{u\in
U}f(u){\mathit{cost}}(u,\mathit{I}_{j})$
We say that design $(\mathbf{I},\mathbf{h})$ has _load skew_ ${{\tau}\geq 0}$
if and only if
$\mathit{load}(\mathbf{I},\mathbf{h},r)\leq(1+{\tau})\cdot\mathit{load}(\mathbf{I},\mathbf{h},j)$
for any $1\leq r\neq j\leq N$. A low value is desirable for ${\tau}$, as it
implies that $(\mathbf{I},\mathbf{h})$ keeps the different replicas relatively
balanced.
We can define a load-skew constraint for the case of failures in exactly the
same way. Specifically, we define $\mathit{fload}(\mathbf{I},\mathbf{h},j,f)$
as the load of replica $j$ when replica $f$ fails. The formula of
$\mathit{fload}(\mathbf{I},\mathbf{h},j,f)$ is similar to that of
$\mathit{load}(\mathbf{I},\mathbf{h},j)$ except that $\mathit{h}_{0}$ is
replaced by $\mathit{h}_{f}$. The constraint then specifies that
$\mathit{fload}(\mathbf{I},j,f)\leq(1+{\tau}^{\prime})\mathit{fload}(\mathbf{I},\mathbf{h},r,f)$
for any valid choice of $j,r,f$ and a skew factor ${\tau}^{\prime}\geq 0$.
It is straightforward to verify that zero skew is always possible by assigning
the same index configuration to each replica. One may ask whether there is a
tradeoff between specialization (and hence overall performance) and a low skew
factor. One of the contributions of our work is to show that this is _not_ the
case, i.e., it is possible to compute divergent designs that exhibit both good
performance and a low skew factor.
Theoretical Analysis. Computing the optimal divergent design implies computing
a partitioning of the workload to replicas and an optimal index configuration
per replica. Not surprisingly, the problem is computationally hard, as
formalized in the following theorem. The proof is provided in Appendix A.
###### Theorem 1
It is not possible to compute an optimal solution to ${\it DDT}$ in polynomial
time unless $P=NP$.
### 3.3 Comparison to Original Study [5]
The formulation of DDT expands on the original problem statement in [5] in
several non-trivial ways. First, DDT incorporates the expected cost under the
case of failures into the objective function, whereas failures were completely
ignored in [5]. Second, our formulation allows a much richer set of
constraints $C$ compared to the original study which considered solely intra-
replica constraints. As discussed earlier, the omission of such constraints
may lead to divergent designs with undesirable effects on the overall system,
e.g., the load skew issue that we discussed earlier. Finally, the original
problem statement imposed a restriction for $\mathit{h}_{0}(q)$ to correspond
to the $m$ replicas with the least evaluation cost for $q$, that is, $\forall
q\in Q$ and $\forall i,j\in[1,N]$ such that $i\in\mathit{h}_{0}(q)$ and
$j\notin\mathit{h}_{0}(q)$ it must be that $cost(q,\mathit{I}_{i})\leq
cost(q,\mathit{I}_{j})$. We remove this restriction in our formulation in
order to explore a larger space of divergent designs, which is particularly
important in light of the richer class of constraints that we consider.
## 4 Divergent Design Tuning as Binary Integer Programming
In this section, we show that the problem of Divergent Design Tuning (${\it
DDT}$) can be reduced to a Binary Integer Program (BIP) that contains a
relatively small number of variables. The implication is that we can leverage
several decades of research in linear-optimization solvers in order to
efficiently compute near-optimal divergent designs. Reliance on these off-the-
shelf solvers brings other important benefits as well, e.g., simpler
implementation and higher portability of the index advisor, or the ability to
operate in “any-time” mode where the DBA can interrupt the tuning session at
any point in time and obtain the best design computed thus far. We discuss
these features in more detail in Section 5, when we describe the architecture
of RITA.
The remainder of the section presents the technical details of the reduction.
We first review some basic concepts for _fast what-if optimization_ , which
forms the basis for the development of our results. We then present the
reduction for a simple variant of ${\it DDT}$ and then generalize to the full
problem statement.
### 4.1 Fast What-If Optimization
What-if optimization is a principled method to estimate ${\mathit{cost}}(q,X)$
and ${\mathit{cost}}(u,X)$ for any $q\in Q$, $u\in U$ and index set $X$, but
it remains an expensive operation that can easily become the bottleneck in any
index-tuning tool. To mitigate the high overhead of what-if optimization,
recent studies have developed two techniques for fast what-if optimization,
termed INUM [11] and C-PQO [3] respectively, that can be used as drop-in
replacements for a what-if optimizer. In what follows, we focus on INUM but
note that the same principles apply for C-PQO.
We first introduce some necessary notation. A configuration $A\subseteq{\cal
S}$ is called atomic [11] if $A$ contains at most one index from each ${\cal
S}_{i}$. We represent $A$ as a vector with $n$ elements, where $A[i]$ is an
index from ${\cal S}_{i}$ or a symbol $\mbox{\small\rm SCAN}_{\small i}$
indicating that no index of ${\cal S}_{i}$ is selected. For an arbitrary index
set $X$, we use $atom(X)$ to denote the set of atomic configurations in $X$.
To simplify presentation, we assume that a query $q$ references a specific
table $T_{i}$ with at most one tuple variable. The extension to the general
case is straightforward at the expense of complicated notation.
For each query $q$, INUM makes a few carefully selected calls to the what-if
optimizer in order to compute a set of _template plans_ , denoted as
$\mathit{TPlans}(q)$. A template plan $p\in\mathit{TPlans}(q)$ is a physical
plan for $q$ except that all access methods (i.e., the leaf nodes of the plan)
are substituted by “slots”. Given a template $p\in\mathit{TPlans}(q)$ and an
atomic index configuration $A$, we can instantiate a concrete physical
execution plan by instantiating each slot with the corresponding index in $A$,
or a sequential scan if $A$ does not prescribe an index for the corresponding
relation. Figure 1 shows an example of this process for a simple query over
three tables $T_{1}$, $T_{2}$, and $T_{3}$, and an atomic configuration that
specifies an index on $T_{1}$ and another index on $T_{3}$. Each template is
also associated with an internal plan cost, which is the sum of the costs of
the operators in this plan except the access methods. Given an atomic
configuration $A$, the cost of the instantiated plan, denoted as
${\mathit{cost}}(p,A)$, is the sum of the internal plan cost and the cost of
the instantiated access methods.
The intuition is that $\mathit{TPlans}(q)$ represents the possibilities for
the optimal plan of $q$ depending on the set of materialized indexes. Hence,
given a hypothetical index configuration $X$, INUM estimates
${\mathit{cost}}(q,X)$ as the minimum ${\mathit{cost}}(p,A)$ over
$p\in\mathit{TPlans}(q)$ and $A\in\mathit{Atom}(X)$. Note that a slot in $p$
may have restrictions on its sorted order, e.g., the template plan in Figure 1
prescribes that the slot for $T_{1}$ must be accessed in sorted order of
attribute $x$. If $A$ does not provide a suitable access method that respects
this sorted order, then ${\mathit{cost}}(p,A)$ is set to $\infty$. INUM
guarantees that there is at least one plan $p$ in $\mathit{TPlans}(q)$ such
that ${\mathit{cost}}(p,A)<\infty$ for any $A\in\mathit{Atom}(X)$. As shown in
the original study [11], INUM provides an accurate approximation for the
purpose of index tuning, and is orders-of-magnitude faster compared to
conventional what-if optimization.
Figure 1: Example of template plans and instantiated plans. The configuration
$A$ has the following contents: $A[1]=a$, an index with key $T_{1}.x$;
$A[2]=\mbox{\small\rm SCAN}_{\small 2}$; $A[3]=b$, an index with key
$(T_{2}.x,T_{2}.w)$ [6]
Linear composability. The approximation provided by INUM and C-PQO can be
formalized in terms of a property that is termed linear composability in [6].
###### Definition 1 (Linear composability [6])
Function ${\mathit{cost}}()$ is linearly composable for a select-statement $q$
if there exists a set of identifiers $K_{q}$ and constants $\beta_{p}$ and
$\gamma_{pa}$ for $p\in K_{q}$, $a\in{\cal S}\cup\\{\mbox{\small\rm
SCAN}_{\small 1}\\}\cup\cdots\cup\\{\mbox{\small\rm SCAN}_{\small n}\\}$ such
that:
$cost(q,X)=min\\{\beta_{p}+\sum_{a\in A}\gamma_{pa},p\in
K_{q},A\in\mathit{Atom}(X)\\}$
for any configuration X. Function $cost()$ is linearly composable for an
update-statement $q$ if it is linearly composable for its query shell. $\Box$
It has been shown in [6] that both INUM and C-PQO compute a cost function that
is linearly composable. For INUM, $K_{q}=\mathit{TPlans}(q)$ and each $p$
corresponds to a distinct template plan in $\mathit{TPlans}(q)$. Here, we use
$\mathit{TPlans}(q)$ for the set of identifiers and overload
$p\in\mathit{TPlans}(q)$ to represent an identifier. In turn, the expression
$\beta_{p}+\sum\gamma_{pa}$ corresponds to $cost(p,A)$, where $\beta_{p}$
denotes the internal plan cost of $p$, and $\gamma_{pa}$ is the cost of
implementing the corresponding slot in $p$ using index $a$. (The slot covers
the relation on which the index is defined.) Note that linear composability
does not imply a linear cost model for the query optimizer – non-linearities
are simply hidden inside the constants $\beta_{qp}$.
For the remainder of the paper, we assume that ${\mathit{cost}}(q,X)$ is
computed by either INUM or C-PQO (for the purpose of fast what-if
optimization) and hence respects linear composability.
### 4.2 Basic DDT
In this subsection, we discuss how to reduce ${\it DDT}$ to a compact BIP for
the case when $\alpha=0$, $C=\emptyset$ (i.e., no failures and no constraints)
and the workload comprises solely queries, i.e., $W=Q$. This reduction forms
the basis for generalizing to the full problem statement, which we discuss
later.
Minimize:
$\hat{\mathit{TotalCost}}(\mathbf{I},\mathbf{h})=\hat{\mathit{QueryCost}}(\mathbf{I},\mathbf{h})\boxed{+\hat{\mathit{UpdateCost}}(\mathbf{I},\mathbf{h})}$,
where:
$\hat{\mathit{QueryCost}}(\mathbf{I},\mathbf{h})=\sum_{q\in
Q}\sum_{r\in[1,N]}\frac{f(q)}{m}{\hat{{\mathit{cost}}}}(q,r)$
$\displaystyle\hat{\mathit{UpdateCost}}(\mathbf{I},\mathbf{h})$
$\displaystyle=$ $\displaystyle\sum_{q\in
Q_{upd}}\sum_{r\in[1,N]}f(q){\hat{{\mathit{cost}}}}(q,r)$ $\displaystyle+$
$\displaystyle\sum_{u\in U}\sum_{r\in[1,N]}f(u)s^{r}_{a}\cdot ucost(u,a)$
${\hat{{\mathit{cost}}}}(q,r)=\sum_{p\in\mathit{TPlans}(q)}\beta_{p}y^{r}_{p}+\sum_{\begin{subarray}{c}p\in\mathit{TPlans}(q)\\\
a\in{\cal S}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
1}\\}\cup\cdots\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
n}\\}\end{subarray}}\gamma_{pa}x^{r}_{pa},\begin{subarray}{c}\forall
r\in[1,N],\\\ \forall q\in Q\cup Q_{upd}\end{subarray}$ (1)
such that:
$\sum_{r\in[1,N]}t^{r}_{q}=m,\forall q\in Q$ (2)
$\sum_{r\in[1,N]}t^{r}_{q}=N,\forall q\in Q_{upd}$ (3)
$\sum_{p\in\mathit{TPlans}(q)}y^{r}_{p}=t^{r}_{q},\ \ \ \forall q\in Q\cup
Q_{upd}$ (4)
$s^{r}_{a}\geq x^{r}_{pa},\ \ \ \forall q\in Q\cup
Q_{upd},p\in\mathit{TPlans}(q),\ a\in{\cal S}$ (5)
$\sum_{a\in{\cal S}_{i}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
i}\\}}x^{r}_{pa}=y^{r}_{p},\ \ \begin{subarray}{c}\forall q\in Q\cup
Q_{upd},p\in\mathit{TPlans}(q),\\\ i\in[1,n],\ T_{i}\mbox{ is referenced in
q}\end{subarray}$ (6)
Figure 2: The BIP for Divergent Design Tuning.
BIP formulation. At a high level, we are given an instance of ${\it DDT}$ and
we wish to construct a BIP whose solution provides an optimal divergent
design. This reduction will hinge upon the linear composability property,
i.e., we assume that each query $q\in W$ has been preprocessed with INUM and
therefore we can approximate cost$(q,X)$ for any $X\subseteq{\cal S}$ as
expressed in Definition 1.
Figure 2 shows the constructed BIP. (Ignore for now the boxed expressions.) In
what follows, we will explain the different components of the BIP and also
formally state its correctness. The BIP uses two sets of binary variables to
encode the choice for a divergent design $(\mathbf{I},\mathbf{h})$:
* •
Variable $s^{r}_{a}$ is set to $1$ if and only if index $a$ is part of the
index design $I_{r}$ on replica $r$. In other words, $I_{r}=\\{a\ |\
s^{r}_{a}=1\\}$.
* •
Variable $t^{r}_{q}$ is set to $1$ if and only if query $q$ is routed to
replica $r$, i.e., $r\in\mathit{h}_{0}(q)$. (Recall that we ignore failures
for now.) In other words, $\mathit{h}_{0}(q)=\\{r\ |\ t^{r}_{q}=1\\}$.
Under our assumption of using fast what-if optimization, the cost of a query
$q$ in some replica $r$ can be expressed as
${\mathit{cost}}(q,I_{r})={\mathit{cost}}(p^{\prime},A^{\prime})$ for some
choice of $p^{\prime}\in\mathit{TPlans}(q)$ and an atomic configuration
$A^{\prime}\in\mathit{Atom}(I_{r})$. To encode these two choices, we introduce
two different sets of binary variables:
* •
Variable $x^{r}_{pa}$, where $p$ is a template in $\mathit{TPlans}(q)$ and $a$
is an index in ${\cal S}$$\cup\\{\mbox{\small\rm SCAN}_{\small
1}\\}\cup\cdots\cup\\{\mbox{\small\rm SCAN}_{\small n}\\}$, is equal to $1$ if
and only if $p=p^{\prime}$ and $a\in A^{\prime}$.
* •
Variable $y^{r}_{p}=1$ if and only if $p=p^{\prime}$.
The BIP specifies several constraints that govern the valid value assignments
to the aforementioned variables:
* •
Constraint (2) specifies that query $q$ must be routed to exactly $m$
replicas.
* •
Constraint (4) specifies that there must be exactly one variable $y^{r}_{p}$
set to $1$ if $t^{r}_{q}=1$, i.e., exactly one template $p$ chosen for
computing ${\mathit{cost}}(q,I_{r})$ if $q$ is routed to $r$. Conversely,
$y^{r}_{p}=0$ for all templates $p$ if $t^{r}_{q}=0$.
* •
Constraint (5) specifies that an index $a$ can be used in instantiating a
template $p$ at replica $r$ only if it appears in the corresponding design
$I_{r}$.
* •
Constraint (6) specifies that if $y^{r}_{p}=1$, i.e., $p$ is used to compute
${\mathit{cost}}(q,I_{r})$, then there must be exactly one access method $a$
per slot such that $x^{r}_{pa}=1$. Essentially, the choices of $a$ for which
$x^{r}_{pa}=1$ must correspond to an atomic configuration. Conversely,
$x^{r}_{pa}=0$ for all $a$ if $y^{r}_{p}=0$.
Given these variables, we can express ${\mathit{cost}}(q,I_{r})$ as in
Equation 1 in Figure 2. The equation is a restatement of linear composability
(Definition 1) by translating the minimization to a guarded summation using
the binary variables $y^{r}_{p}$ and $x^{r}_{pa}$. Specifically, if
$t^{r}_{q}=1$, then constraint (4) forces the solver to pick exactly one $p$
such that $y^{r}_{p}=1$, and constraint (6) forces setting $x^{r}_{pa}=1$ for
the same choice of $p$ and corresponding to an atomic configuration. Hence,
minimizing the expression in Equation 1 corresponds to computing
${\mathit{cost}}(q,I_{r})$. Otherwise, if $t^{r}_{q}=0$, then the same
constraints force ${\mathit{cost}}(q,I_{r})=0$. In turn, it follows that the
objective function of the BIP corresponds to
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$.
Handling update statements. The total cost to execute update statements,
$\mathit{UpdateCost}(\mathbf{I},\mathbf{h})$, includes two terms, as shown in
the second boxed expression in Figure 2. Here, $Q_{upd}$ denotes the set of
all the query-shells, each of which corresponds to each update statement in
$U$. The first component of $\mathit{UpdateCost}()$ is the total cost to
evaluate every query-shell in $Q_{upd}$ at every replica. This component is
expressed as the summation of ${\hat{{\mathit{cost}}}}(q,r)$ for all
$q_{sel}\in Q_{upd}$ and $r\in[1,N]$ in our BIP. Since each query-shell needs
to be routed to all replicas, we impose the constraint (3).
The second component of $\mathit{UpdateCost}()$ is the total cost to update
the affected indexes. Using variable $s_{a}^{r}$ that tracks the selection of
an index at replica $r$ in the recommended configuration, the cost of updating
an index $a$ at replica $r$ given the presence of an update statement $u$ is
computed as the product of $s^{r}_{a}$ and $ucost(u,a)$.
Correctness. Up to this point, we argued informally about the correctness of
the BIP. The following theorem formally states this property. The proof is
given in Appendix B.
###### Theorem 2
A solution to the BIP in Figure 2 corresponds to the optimal divergent design
for ${\it DDT}$ when $\alpha=0$ and $C=\emptyset$.
As stated repeatedly, the key property of the BIP is that it contains a
relatively small number of variables and constraints, which means that a BIP-
solver is likely to find a good solution efficiently. Formally:
###### Corollary 1
The number of variables and constraints in the BIP shown in Figure 2 is in the
order of $O(N|W||{\cal S}|)$.
In fact, it is possible to eliminate some variables and constraints from the
BIP while maintaining its correctness. We do not show this extension since it
does not change the order of magnitude for the variable count but it makes the
BIP less readable and harder to explain.
### 4.3 Factoring Failures
To extend the BIP to the case when $\alpha>0$ (i.e., failures are possible),
we first introduce additional variables $t^{r,j}_{q}$, $y^{r,j}_{p}$ and
$x^{r,j}_{pa}$, for $j\in[1,N]$. These variables have the same meaning as
their counterparts in Figure 2, except that they refer to the case where
replica $j$ fails. For instance, $t^{r,j}_{q}=1$ if and only if $q$ is routed
to replica $r$ when $j$ fails, i.e., $\mathit{h}_{j}(q)=\\{r\ |\
t^{r,j}_{q}=1\\}$. We augment the BIP with the corresponding constraints as
well. For instance, we add the constraint $\sum_{r\neq
j}t^{r,j}_{q}=\max\\{N-1,m\\}$, $\forall q\in Q,j\in[1,N]$ to express the fact
that function $\mathit{h}_{j}()$ must respect the routing-multiplicity factor
$m$. Finally, we change the objective function to $\mathit{ExpTotalCost}()$,
which is already linear, and express each term
$\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)$ as a summation that involves
the new variables.
The complete details for this extension, including the proof of correctness,
can be found in Appendix C. We should mention that this extension increases
the number of variables and constraints by a factor of $N$ to
$O(N^{2}|W||{\cal S}|)$, since it becomes necessary to reason about the
failure of every replica $j\in[1,N]$.
### 4.4 Adding Constraints
In this subsection, we discuss how to extend the BIP when $C\neq\emptyset$,
i.e., the DBA specifies constraints for the divergent design.
Obviously, we can attach to the BIP any type of linear constraint. As it turns
out, linear constraints can capture a surprisingly large class of practical
constraints. In what follows, we present three examples of how to translate
common constraints to linear expressions that be directly added to the BIP.
Space budget. Let $size(a)$ denote the estimated size of an index $a$, and $b$
be the storage budget at each replica. Using the variable $s_{a}^{r}$ that
tracks the selection of an index at replica $r$ in the recommended
configuration, the storage constraint can be encoded as: $\sum_{a\in{\cal
S}}s^{r}_{a}size(a)\leq b,\ \forall r\in[1,N]$. In general, variables
$s^{r}_{a}$ can be used to express several types of intra-replica constraints
that involve the selected indexes, e.g., bound the total number of multi-key
indexes per replica, or bound the total update cost for the indexes in each
replica.
Bounding load-skew. Recall that $\mathit{load}(\mathbf{I},\mathbf{h},j)$
captures the total load of replica $j$ under a divergent design
$(\mathbf{I},\mathbf{h})$. The load-skew constraint specifies that
$\mathit{load}(\mathbf{I},\mathbf{h},j)\leq(1+\tau)\mathit{load}(\mathbf{I},\mathbf{h},r)$,
for any $r\neq j$, where $\tau$ is the load-skew factor provided by the DBA.
It is straightforward to translate the constraint between two specific
replicas $j$ and $r$ into a linear inequality, by using variables $x^{r}_{pa}$
and $y^{r}_{p}$ to rewrite the corresponding $\mathit{load}()$ terms as linear
sums. Specifically, $\mathit{load}(\mathbf{I},\mathbf{h},j)$ can be expressed
as a linear sum similarly to $\hat{\mathit{TotalCost}}()$ in Figure 2, except
that we only consider replica $j$ and the queries for which
$j\in\mathit{h}_{0}(q)$, and the same goes for expressing
$\mathit{load}(\mathbf{I},\mathbf{h},r)$.
Based on this translation, we can add $N(N-1)$ constraints to the BIP, one for
each possible choice of $j$ and $r$. We can actually do better, by observing
that we can sort replicas in ascending order of their load, and then impose a
single load-skew constraint between the first and last replica. By virtue of
the sorted order, the constraint will be satisfied by any other pair of
replicas. Specifically, we add the following two constraints to the BIP:
$\displaystyle\mathit{load}(\mathbf{I},\mathbf{h},i)\leq\mathit{load}(\mathbf{I},\mathbf{h},i+1),\
\forall i\in[1,N-1]$ (7)
$\displaystyle\mathit{load}(\mathbf{I},\mathbf{h},N)\leq(1+{\tau})\cdot\mathit{load}(\mathbf{I},\mathbf{h},1)$
(8)
This approach requires only $N$ constraints and is thus far more effective.
The final step requires adding another set of constraints on
${\hat{{\mathit{cost}}}}(q,I_{r})$. This is a subtle technical point that
concerns the correctness of the reduction when the constraints are infeasible.
More concretely, the solver may assign variables $y^{r}_{p}$ and $x^{r}_{pa}$
for some query $q$ so that constraints (7)–(8) are satisfied even though this
assignment does not correspond to the optimal cost ${\mathit{cost}}(q,I_{r})$.
To avoid this situation, we introduce another set of variables that are
isomorphic to $x^{r}_{pa}$ and are used to force a cost-optimal selection for
$y^{r}_{p}$ and $x^{r}_{pa}$. The details are given in Appendix D.1, but the
upshot is that we need to add $O(N|W||{\cal S}|)$ additional constraints.
We have also developed an approximate scheme to handle load-skew constraints
in the BIP. The approximate scheme allows the BIP to be solved considerably
faster, but the compromise is that the resulting divergent design may not be
optimal. However, our experimental results (see Section 6) suggest that the
loss in quality is not substantial. The details of the approximate scheme can
be found in Appendix D.2
Materialization cost constraint. This constraint specifies that the total
cost to materialize $(\mathbf{I},\mathbf{h})$ must be below some threshold
${\mathit{C_{m}}}$. The materialization cost is computed with respect to the
current design $(\mathbf{I}_{c},\mathbf{h}_{c})$ and takes into account the
cost to scale up or down the current number of replicas, and the cost to
create additional indexes or drop redundant indexes in each replica.
We first consider the case when the number of replicas remains unchanged
between $(\mathbf{I},\mathbf{h})$ and $(\mathbf{I}_{c},\mathbf{h}_{c})$. Let
us consider a specific replica $r$ and the new design $I_{r}\in\mathbf{I}$.
Let $I^{c}_{r}\in\mathbf{I}_{c}$ denote the previous design. Clearly, we need
to create every index in $I_{r}-I^{c}_{r}$ and to delete every index in
$I^{c}_{r}-I_{r}$. Assuming that $\mathit{ccost}(a)$ and $\mathit{dcost}(a)$
denote the cost to create and drop index $a$ respectively, we can express the
reconfiguration cost for replica $r$ as $\sum_{a\not\in
I^{c}_{r}}s^{r}_{a}\mathit{ccost}(a)+\sum_{a\in
I^{c}_{r}}(1-s^{r}_{a})\mathit{dcost}(a)$. If each replica can install indexes
in parallel, then the materialization cost constraint can be expressed as:
$\sum_{a\in{\cal S}\wedge a\not\in
I^{c}_{r}}s^{r}_{a}\mathit{ccost}(a)+\sum_{a\in{\cal S}\wedge a\in
I^{c}_{r}}(1-s^{r}_{a})\mathit{dcost}(a)\leq{\mathit{C_{m}}},\forall
r\in[1,N]$
We can also express a single constraint on the aggregate materialization cost
by summing the per-replica costs.
We next consider the case when the DBA wants to shrink the number of replicas
to be $N_{d}<N$. In this case, the BIP solver should try to find which
replicas to maintain and how to adjust their index configurations so that the
total materialization cost remains below threshold. For this purpose, we
introduce $N$ new binary variables $z^{r}$ with $r\in[1,N]$ associated with
each replica $r$, where $z^{r}=1$ if replica $r$ is kept in the new divergent
design, and $z^{r}=0$ otherwise. The materialization cost can be computed in a
similar way as discussed above, except that we need to add the following two
additional constraints to the BIP.
$\displaystyle t^{r}_{q}\leq z^{r},\forall q\in Q\cup Q_{upd},r\in[1,N]$ (9a)
$\displaystyle\sum_{r\in[1,N]}z^{r}=N_{d}$ (9b)
The first constraint ensures that we can route queries only to live replicas.
The second simply restricts the number of live replicas to the desired number.
Lastly, we consider the case when the DBA wants to expand the number of
replicas to be $N_{d}>N$. The set of constraints in the BIP can be re-used
except that all the variables are defined according to $N_{d}$ replicas
(instead of $N$ replicas as before). The materialization cost can also be
computed in a similar way. In addition, we also take into account the cost to
deploy the database in new replicas, which appear as constants in the total
cost to materialize a design in a new replica.
### 4.5 Routing Queries
Recall that a divergent design $(\mathbf{I},\mathbf{h})$ includes both the
index-sets for different replicas and the routing functions
$\mathit{h}_{0}(),\mathit{h}_{1}(),\dots,\mathit{h}_{N}()$. These functions
are used at runtime, after the divergent design has been materialized, to
route queries to different specialized replicas. A solution to the BIP
determines how to compute these functions for a training query $q$ in $Q$,
based on the variables $t^{r}_{q}$ and $t^{r,j}_{q}$. Here, we describe how to
compute these functions for any query $q^{\prime}$ that is not part of the
training workload. We focus on the computation of $\mathit{h}_{0}(q^{\prime})$
but our techniques readily extend to the other functions.
Our first approach is inspired by the original problem statement of the tuning
problem [5] and computes $\mathit{h}_{0}(q^{\prime})$ as the $m$ replicas with
the lowest evaluation cost for $q^{\prime}$. Normally this requires $N$ what-
if optimizations for $q^{\prime}$, but we can leverage again fast what-if
optimization in order to achieve the same result more efficiently.
Specifically, we first compute $\mathit{TPlans}(q^{\prime})$ (which requires a
few calls to the what-if optimizer) and then formulate a BIP that computes the
top $m$ replicas for $q^{\prime}$.
Our second approach tries to match more closely the revised problem statement,
where a query is not necessarily routed to its top $m$ replicas. Our approach
is to match $q^{\prime}$ to its most “similar” query $q$ in the training
workload $Q$, and then to set $\mathit{h}_{0}(q^{\prime})=\mathit{h}_{0}(q)$.
The intuition is that the two queries would affect the divergent design
similarly if they were both included in the training workload. We can use
several ways to assess similarity, but we found that fast what-if
optimizations provides again a nice solution. Specifically, we compute again
$\mathit{TPlans}(q^{\prime})$ and then quickly find the optimal plan for
$q^{\prime}$ in each replica. We then form a vector $v_{q^{\prime}}$ where the
$i$-th element is the set of indexes in the optimal plan of $q^{\prime}$ at
replica $i$. We can compute a similar vector for $v_{q}$ and then compute the
similarity between $q^{\prime}$ and $q$ as the similarity between the
corresponding vectors111Any vector-similarity metric will do. We first convert
$v_{q^{\prime}}$ $v_{q}$ to binary vectors indicating which indexes are used
at each replica and then use a cosine-similarity metric.. The intuition is
that $q^{\prime}$ is similar to $q$ if in each replica they use similar sets
of indexes. We can refine this approach further by taking into account the
top-$2$ plans for each query, but our empirical results suggest that the
simple approach works quite well.
## 5 RITA: Architecture and Functionality
Figure 3: The architecture of RITA.
In this section we describe the architecture and the functionality of RITA,
our proposed index-tuning advisor. RITA builds on the reduction presented in
the previous section in order to offer a rich set of features.
Figure 3 shows the architecture of RITA. It comprises two main modules: the
online monitor, which continuously analyzes the workload in order to detect
changes and opportunities for retuning; and the recommender, which is invoked
by the DBA in order to run a tuning session. As we will see later, both
modules solve a variant of the ${\it DDT}$ problem in order to perform their
function. Also, both modules make use of the reduction we presented in the
previous section in order to solve the respective tuning problems. For this
purpose, they employ an off-the-shelf BIP solver. The remaining sections
discuss the two modules in more detail.
### 5.1 Online Monitor
The online monitor maintains a divergent design $(\mathbf{I}^{\mbox{\tiny\rm
slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$ that is continuously re-computed
based on the latest queries in the workload. Concretely, the monitor maintains
a sliding window over the current workload (the length of the window is a
parameter defined by the DBA) and then solves ${\it DDT}$ using the sliding
window as the training workload. Each new statement in the running workload
causes an update of the window and a re-computation of
$(\mathbf{I}^{\mbox{\tiny\rm slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$.
Once computed, the up-to-date design $(\mathbf{I}^{\mbox{\tiny\rm
slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$ is compared against the current
design $(\mathbf{I}^{\mbox{\tiny\rm curr}},\mathbf{h}^{\mbox{\tiny\rm curr}})$
of the system, using the $\mathit{ExpTotalCost}()$ metric of each design on
the workload in the sliding window. The module outputs the difference between
the two as the performance improvement if $(\mathbf{I}^{\mbox{\tiny\rm
slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$ were materialized. This output,
which is essentially a time series since $(\mathbf{I}^{\mbox{\tiny\rm
slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$ is being continuously updated,
can inform the DBA about the need to retune the system.
Clearly, it is important for the online monitor to maintain
$(\mathbf{I}^{\mbox{\tiny\rm slide}},\mathbf{h}^{\mbox{\tiny\rm slide}})$ up-
to-date with the latest statements in the workload. For this purpose, the
online monitor solves a bare-bones variant of ${\it DDT}$ that assumes
$\alpha=0$ (i.e., no failures) and does not employ any constraints except
perhaps very basic ones (e.g., a space budget per replica). Beyond being fast
to solve, this formulation also reflects the best-case potential to improve
performance, which again can inform the DBA about the need to retune the
system. RITA allows the DBA to impose additional constraints inside the online
monitor at the expense of taking longer to update the output of the online
monitor.
### 5.2 Recommender
The DBA invokes the recommender module to run a tuning session, for the
purpose of tuning the initial divergent design or retuning the current design
when the workload changes. The DBA provides an instance of the ${\it DDT}$
problem, e.g., a training workload, the parameter $\alpha$ and several
constraints, and the recommender returns the corresponding (near-)optimal
divergent design. The recommender leverages the BIP-based formulation of ${\it
DDT}$ in order to compute its output efficiently.
If desired by the DBA, the recommender can also return a set of possible
designs that represent trade-off points within a multi-dimensional space. For
example, suppose that the DBA specifies the workload-evaluation cost and the
materialization cost of each design as the two dimensions of this space. We
expect that a design with a higher materialization cost will have more
indexes, and hence will have a lower workload-evaluation cost. The recommender
formulates a BIP to compute an optimal divergent design that does not bound
the materialization cost. The solution provides an upper bound on
materialization cost, henceforth denoted as ${\mathit{C_{m}}}$. Subsequently,
the recommender formulates several tuning BIPs where each BIP puts a different
threshold on the materialization cost based on ${\mathit{C_{m}}}$ and some
factor (e.g., materialization cost should not exceed
$.5\times{\mathit{C_{m}}}$). The thresholds for these Pareto-optimal designs
can be predefined or chosen based on more involved strategies such as the
Chord algorithm [7]. An important point is that the successive BIPs are
essentially identical except for the modified constraint on the
materialization cost, which enables the BIP solver to work fast by reusing
previous computations.
The DBA can also add other parameters into this exploration. For example,
adding the number of replicas as another parameter will cause the recommender
to use the same process to generate designs for the hypothetical scenarios of
expanding/shrinking the set of replicas. The final output can inform the DBA
about the trade-off between workload-evaluation cost and design-
materialization cost, and how it is affected by the number of replicas.
Besides being able to perform tuning sessions efficiently, RITA’s recommender
module gains two important features through its reliance on a BIP solver.
* •
Fast refinement. As mentioned earlier, the BIP solver can reuse computation if
the current BIP is sufficiently similar to previously solved BIPs. RITA takes
advantage of this feature to offer fast refinement of the solution for small
changes to the input. E.g., the optimal divergent design can be updated very
efficiently if the DBA wishes to change the set of candidate indexes or impose
additional inter-replica constraints.
* •
Early termination. In the course of solving a BIP, the solver maintains the
currently-best solution along with a bound on its suboptimality. This
information can be leveraged by RITA to support early termination based on
time or quality. For instance, the DBA may instruct the recommender to return
the first solution that is within 5% of the optimal, which can reduce
substantially the total running time without compromising performance for the
output divergent design. Or, the DBA may ask for the best solution that can be
computed within a specific time interval.
## 6 Experimental Study
This section presents the results of the experimental study that we conducted
in order to evaluate the effectiveness of RITA. In what follows, we first
discuss the experimental methodology and then present the findings of the
experiments.
Parameter | Values
---|---
Number of replicas ($N$) | $2$, 3, $4$, $5$
Routing multiplicity ($m$) | $1$, 2, $3$
Space budget ($b$) | 0.25$\times$, 0.5$\times$, 1.0$\times$, INF
Prob. of failure ($\alpha$) | 0.0, $0.1$, $0.2$, $0.3$, $0.4$
Load skew (${\tau}$) | $1.3$, $1.5$, $1.7$, $1.9$, $2.1$, INF
Percentage-update ($\mathit{p_{upd}}$) | $10^{-5}$, $10^{-4}$, ${\textbf{10}}^{\textbf{-3}}$, $10^{-2}$
Sliding window ($w$) | $40$, 60, $80$, $100$
Table 1: Experimental parameters (default in bold). | |
---|---|---
Figure 4: Varying space budget on $\mathit{TPCDS\mbox{-}query}$, $\alpha=0$, ${\tau}=+\infty$. | Figure 5: Varying number of replicas on $\mathit{TPCDS\mbox{-}mix}$, $\alpha=0$, ${\tau}=+\infty$. | Figure 6: Constraint the update cost on $\mathit{TPCDS\mbox{-}mix}$, $\alpha=0$, ${\tau}=+\infty$.
### 6.1 Methodology
Advisors. Our experiments use a prototype implementation of RITA written in
Java. The prototype employs CPLEX v12.3 as the off-the-shelf BIP solver, and a
custom implementation of INUM for fast what-if optimization. The database
system in our experiments is the freely available IBM DB2 Express-C. The CPLEX
solver is tuned to return the first solution that is within 5% of the optimal.
In all experiments, we use ${p_{\mbox{\tiny\rm RITA}}}$ to denote the
divergent design computed by RITA.
We compare RITA against the heuristic advisor DivgDesign that was introduced
in the original study of divergent designs [5]. DivgDesign employs IBM’s
physical design advisor internally. Similar to [5], we run DivgDesign five
times and output the lowest-cost design out of all the independent runs. We
denote this final design as ${p_{\mbox{\tiny\rm DD}}}$. We note that the
comparison against DivgDesign concerns only a restricted definition of the
general tuning problem, since DivgDesign supports only a space budget
constraint and does not take into account replica failures.
We also include in the comparison the common practice of using the same index
configuration with each replica. The identical configuration is computed by
invoking the DB2 index-tuning advisor on the whole workload. We use
${p_{\mbox{\tiny\rm UNIF}}}$ to refer to the resulting design.
Data Sets and Workloads. We use a 100GB TPC-DS database [15] for our
experiments, along with three different workloads, namely
$\mathit{TPCDS\mbox{-}query}$, $\mathit{TPCDS\mbox{-}mix}$ and
$\mathit{TPCDS\mbox{-}dyn}$. $\mathit{TPCDS\mbox{-}query}$ comprises 40
complex TPC-DS benchmark queries that are currently supported by our INUM
implementation [16]. $\mathit{TPCDS\mbox{-}mix}$ adds INSERT statements that
model updates to the base data. $\mathit{TPCDS\mbox{-}dyn}$ models a workload
of 600 queries that goes through three phases, each phase corresponding to a
specific distribution of the queries that appear in
$\mathit{TPCDS\mbox{-}query}$. The first phase corresponds mostly to queries
of low execution cost222The execution cost is measured with respect to the
optimal index-set for each query returned by the DB2 advisor., then the
distribution is inverted for the second phase, and reverts back to the
starting distribution in the first phase.
In all cases, the weight for each query is set to one, whereas the update of
each INSERT statement is determined as the product of the cardinality of the
corresponding relation and a _percentage-update_ parameter
($\mathit{p_{upd}}$). This parameter allows us to simulate different volumes
of updates when we test the advisors.
Candidate Index Generation. Recall from Section 3 that the DDT problem assumes
that a set of candidate indexes ${\cal S}$ is provided as input. There are
many methods for generating ${\cal S}$ based on the database and
representative workload. In our setting, we use DB2’s service to select the
optimal indexes per query (without any space constraints) and then perform a
union of the returned index-sets. The resulting index-set, which is optimal
for the workload in the absence of constraints and update statements, contains
$103$ candidate indexes and has a total size of $265$GB.
Experimental Parameters. Our experiments vary the following parameters: the
number of replicas $N$, the per-replica space budget $b$, the probability of
failure $\alpha$, the load-skew factor ${\tau}$, the percentage of updates in
the workload $\mathit{p_{upd}}$ (for $\mathit{TPCDS\mbox{-}mix}$), and the
size of the sliding window $w$ for online monitoring. The routing multiplicity
factor ($m$) is set to be $\lceil N/2\rceil$. We report the additional
experimental results when varying $m$ in Appendix D.3. Table 1 shows the
parameter values tested in our experiments. Note that the storage space budget
is measured as a multiple of the base data size, i.e., given TPCDS $100$ GB
base data size, a space budget of $0.5\times$ indicates a $50$ GB storage
space budget.
Metrics. We use $\mathit{ExpTotalCost}()$ to measure the performance of a
divergent design. To allow meaningful comparisons among the designs generated
by different advisors, we compute this metric for a specific design by
invoking DB2’s what-if optimizer for all the required cost factors. This
methodology, which is consistent with previous studies on physical design
tuning, allows us to gauge the effectiveness of the divergent design in
isolation from any estimation errors in the optimizer’s cost models. In some
cases, we also report the performance improvement of ${p_{\mbox{\tiny\rm
RITA}}}$ over ${p_{\mbox{\tiny\rm DD}}}$ and ${p_{\mbox{\tiny\rm UNIF}}}$,
where the performance improvement of a design $X$ over a design $Y$ is
computed as $1-\mathit{ExpTotalCost}(X)/\mathit{ExpTotalCost}(Y)$. We also
report the time that is taken to execute the index advisor for the
corresponding divergent design.
Testing Platform. All measurements are taken on a machine running 64-bit
Ubuntu OS with four-core Intel(R) Core(TM) i7 CPU Q820 @1.73GHz CPU and 4GB
RAM.
### 6.2 Results
Basic Tuning Problem. We first consider a basic case of DDT when $\alpha=0$
and ${\tau}=+\infty$, i.e., no failures occur and there is no constraint on
load skew. There is a single constraint on the divergent design which is the
per-replica space budget. This setting corresponds essentially to the original
problem statement in [5].
We begin with a set of experiments that evaluates the performance of RITA and
the competitor advisors on the query-only workload
$\mathit{TPCDS\mbox{-}query}$. In this case indexes can only bring benefit to
queries, and hence the only restraint in materializing indexes comes from any
constraints. Figure 6 shows the performance of the divergent designs computed
by RITA, DivgDesign, and Unif, as we vary the space budget parameter. (All
other parameters are set to their default values according to Table 1.) The
results show that RITA consistently outperforms the other two competitors for
a wide range of space budgets. The improvement is up to 75% over
${p_{\mbox{\tiny\rm UNIF}}}$ and up to 67% for ${p_{\mbox{\tiny\rm DD}}}$,
i.e., the performance of ${p_{\mbox{\tiny\rm RITA}}}$ is 4$\times$ better than
${p_{\mbox{\tiny\rm UNIF}}}$ and is 3$\times$ better than ${p_{\mbox{\tiny\rm
DD}}}$. Another way to view these results is that RITA can make much more
effective usage of the aggregate disk space for indexes. For instance,
${p_{\mbox{\tiny\rm RITA}}}$ at $b=0.25\times$ matches the performance of
${p_{\mbox{\tiny\rm DD}}}$ at $b=1.0\times$, i.e., with four times as much
space for indexes. In all cases, RITA’s better performance can be attributed
to the fact that it searches a considerably larger space of possible designs,
through the reduction to a BIP. As the space budget increases, the performance
of ${p_{\mbox{\tiny\rm RITA}}}$, ${p_{\mbox{\tiny\rm DD}}}$ and
${p_{\mbox{\tiny\rm UNIF}}}$ converge as all beneficial indexes can be
materialized in every design.
We next examine the performance of RITA and the competitor advisors on a
workload of queries and updates. Figure 6 reports the performance of
${p_{\mbox{\tiny\rm RITA}}}$, ${p_{\mbox{\tiny\rm DD}}}$ and
${p_{\mbox{\tiny\rm UNIF}}}$ for the workload $\mathit{TPCDS\mbox{-}mix}$, as
we vary the number of replicas in the system. We chose this parameter as
updates have to be routed to all replicas and hence it controls directly the
total cost of updates. We observe that the improvement of RITA over Unif is in
the order of $50\%$ and the improvement of RITA over DivgDesign is $38\%$. Not
surprisingly, the improvements increase with the number of replicas. The
reason is that RITA is able to find designs with much fewer indexes per
replica compared to ${p_{\mbox{\tiny\rm UNIF}}}$ and ${p_{\mbox{\tiny\rm
DD}}}$, which contributes to a lower update cost. For instance for $N=3$ and
$b=0.5\times$, the number of indexes per replica of ${p_{\mbox{\tiny\rm
RITA}}}$ is $(44,44,31)$ compared to $(70,70,70)$ for ${p_{\mbox{\tiny\rm
UNIF}}}$ and $(46,50,53)$ for ${p_{\mbox{\tiny\rm DD}}}$. We conducted similar
experiments with different weights for the update statements and observed
similar trends.
| |
---|---|---
Figure 7: Varying probability of failure on $\mathit{TPCDS\mbox{-}query}$, $\alpha\geq 0$, ${\tau}=+\infty$. | Figure 8: Varying load skew on $\mathit{TPCDS\mbox{-}query}$, $\alpha\geq 0$, ${\tau}<+\infty$. | Figure 9: Routing queries
The next experiment examines how RITA’s advanced functionality can control
even further the cost of updates. Instead of having RITA minimize the combined
cost of queries and updates, we instruct the advisor to perform the following
constrained optimization: minimize query cost such that update cost is at most
$x\%$ of the update cost of a uniform design. Essentially, the desire is to
make updates much faster compared to the uniform design, and also try to get
some benefits for query processing. This changed optimization requires minimal
changes to the underlying BIP: the objective function includes only the cost
of evaluating queries, and the constraints include an additional linear
constraint on the total update cost based on the update cost of the uniform
design (which can be treated as a constant). The ease by which we can support
this advanced functionality reflects the power of expressing ${\it DDT}$ as a
BIP.
Figure 6 depicts the cost of the query workload under ${p_{\mbox{\tiny\rm
RITA}}}$ as we vary the factor that bounds the update cost relative to
${p_{\mbox{\tiny\rm UNIF}}}$. For comparison we also show the cost of the
query workload for ${p_{\mbox{\tiny\rm UNIF}}}$. The results show clearly that
the designs computed by RITA can improve performance dramatically even in this
scenario. As a concrete data point, when the bounding factor is set to $0.4$,
${p_{\mbox{\tiny\rm RITA}}}$ makes query evaluation more than 2$\times$
cheaper compared to ${p_{\mbox{\tiny\rm UNIF}}}$ and incurs an update cost
that is less than half the update cost of ${p_{\mbox{\tiny\rm UNIF}}}$.
Overall, our results demonstrate that RITA clearly outperforms its competitors
on the basic definition of the divergent-design tuning problem. From this
point onward, we will evaluate RITA’s effectiveness with respect to the
generalized version of the problem (i.e., including failures and a richer set
of constraints). In the interest of space, we present results with query-only
workloads, as the trends were very similar when we experimented with mixed
workloads.
Factoring Failures. We first evaluate how well RITA can tailor the divergent
design in order to account for possible failures, as captured by the failure
probability $\alpha$.
Figure 13 shows the $\mathit{ExpTotalCost}()$ metric for ${p_{\mbox{\tiny\rm
RITA}}}$, ${p_{\mbox{\tiny\rm DD}}}$ and ${p_{\mbox{\tiny\rm UNIF}}}$ as we
vary the probability of failure $\alpha$. There are two interesting take-away
points from the results. The first is that ${p_{\mbox{\tiny\rm RITA}}}$ has a
relatively stable performance as we vary $\alpha$. Essentially, we can reap
the benefits of divergent designs even when there is an increased probability
of failure in the system, as long as there is a judicious specialization for
each replica and a controlled strategy to redistribute the workload (two
things that RITA clearly achieves). The second interesting point is that the
gap between ${p_{\mbox{\tiny\rm RITA}}}$ and ${p_{\mbox{\tiny\rm DD}}}$
increases with $\alpha$. Basically, ${p_{\mbox{\tiny\rm DD}}}$ ignores the
possibility of failures (i.e., it always assumes that $\alpha=0$) and hence
the computed design ${p_{\mbox{\tiny\rm DD}}}$ cannot handle effectively a
redistribution of the workload when a replica becomes unavailable. As a side
note, the cost of ${p_{\mbox{\tiny\rm UNIF}}}$ is unchanged for different
values of $\alpha$, since each query has the same cost under
${p_{\mbox{\tiny\rm UNIF}}}$ on all replicas, and hence a redistribution of
the workload does not change the total cost.
Bounding Load Skew. We next study how RITA handles a (inter-replica)
constraint on load skew. Recall that the constraint has the following form:
for any two replicas, their load should not differ by a factor of more than
$1+\tau$, where $\tau\geq 0$ is the load-skew parameter. A balanced load
distribution is important for good performance in a distributed system and
hence we are interested in small values for $\tau$. The ability to satisfy
such constraints is part of RITA’s novel functionality.
Figure 9 shows the performance of ${p_{\mbox{\tiny\rm RITA}}}$,
${p_{\mbox{\tiny\rm DD}}}$ and ${p_{\mbox{\tiny\rm UNIF}}}$ as we vary
parameter ${\tau}$ that bounds the load skew (recall that ${\tau}=0$ implies
no skew). We report two sets of results for RITA corresponding to $\alpha=0$
(no failures) and $\alpha=0.1$ (10% chance that one replica will fail)
respectively, in order to examine the interplay between $\alpha$ and ${\tau}$.
Note that we report the results for the greedy version of RITA, which are
identical to the exact solution of the constraint. The chart shows a single
point corresponding to ${p_{\mbox{\tiny\rm DD}}}$, given that it is not
possible to constrain load skew within DivgDesign. As shown,
${p_{\mbox{\tiny\rm DD}}}$ has a significant load skew of up to a $2x$
difference between replicas. This magnitude of skew limits severely the
ability of the system to maintain a balanced load and to route queries
effectively. In contrast, RITA is able to compute designs that maintain a low
expected cost (up to 4$\times$ better than Unif) and also satisfy the bound on
load skew. These savings are not affected by the value of $\alpha$–RITA is
again able to make a judicious choice for the divergent design in order to
satisfy all constraints and handle failures. Note that the uniform design
trivially satisfies the load-skew constraint for all values of ${\tau}$ as
every replica has the same design and hence the system can be perfectly
balanced.
| | $\alpha=0$ | $\alpha>0$
---|---|---
${\tau}=+\infty$ | $4$ | $60$
${\tau}<+\infty$ | $9$ | $84$
| | $\alpha=0$ | $\alpha>0$
---|---|---
${\tau}=+\infty$ | $20$ | $120$
${\tau}<+\infty$ | $30$ | $146$
(a) Workload $\mathit{TPCDS\mbox{-}query}$ | (b) Workload $\mathit{TPCDS\mbox{-}dyn}$
Table 2: The average running time of RITA (in seconds)
Running Time. Given an instance of the basic DDT problem ($\alpha=0$,
${\tau}=+\infty$), RITA spends $180$ seconds to initialize INUM, a step that
is dependent solely on the input workload, and then requires only four seconds
to formulate and solve the resulting BIP. An important point is that the
initialization step can be reused for free if the workload remains unchanged,
e.g., if the DBA runs several tuning sessions using the same workload but
different constraints each time. Each subsequent tuning session can thus be
executed in the order of a few seconds, offering an almost interactive
response to the DBA.
Table 2(a) shows the running time for RITA on $\mathit{TPCDS\mbox{-}query}$
workload as we vary the load-skew factor and the probability of failure, two
parameters that correspond to novel features of our generalized tuning
problem. Note that the time to initialize INUM remains the same as before and
is excluded from all the cells of the table. Clearly, the new features
complicate the tuning problem and hence have an impact on running time. Still,
even for the most complex combination (${\tau}>0$ and $\alpha>0$) RITA has a
reasonable running time of at most $84$ seconds. Moreover, as noted in Section
5, RITA can always be invoked with a time threshold and return the best design
that has been identified within the allotted time.
Table 2(b) shows the same details about the running time of RITA on
$\mathit{TPCDS\mbox{-}dyn}$ workload, consisting of $600$ queries. RITA also
runs efficiently for this large workload.
| |
---|---|---
Figure 10: Online monitoring | Figure 11: Elasticity retuning, varying number of replicas and materialization costs | Figure 12: Elasticity retuning, varying routing multiplicity factor and materialization costs
Routing. The next set of experiments examines the effectiveness of the routing
scheme we introduced in Section 4.5, which determines how to route unseen
queries (i.e., queries not in $W$ for which the routing functions
$\mathit{h}_{j}$ cannot be applied) to “good” specialized replicas.
Our test methodology splits $\mathit{TPCDS\mbox{-}query}$ into two
(sub)workloads: (1) a training workload that plays the role of $W$ and
consists of $30$ randomly-chosen queries of $\mathit{TPCDS\mbox{-}query}$, and
(2) a testing workload that plays the role of the unseen queries and consists
of the remaining $10$ queries. We compute a divergent design
${p_{\mbox{\tiny\rm RITA}}}$ for the training workload, and route the queries
in $\mathit{TPCDS\mbox{-}query}$ (including both seen and unseen queries)
assuming ${p_{\mbox{\tiny\rm RITA}}}$ is deployed. For comparison, we apply
the same methodology to the uniform design: we first derive
${p_{\mbox{\tiny\rm UNIF}}}$ for the training workload and then route the
queries in $\mathit{TPCDS\mbox{-}query}$ workload in round-robin fashion. We
repeat this experiment for ten independent runs, where each run involves a
different random split of the workload.
Figure 9 shows the expected cost of the workload for ${p_{\mbox{\tiny\rm
RITA}}}$ and ${p_{\mbox{\tiny\rm UNIF}}}$ for each run. The results show that
RITA outperforms Unif consistently, even though replica specialization does
not take into account the unseen queries. The improvements vary across
different runs depending on the choice of the workload split, but overall we
can reap the benefits of divergent designs even with incomplete knowledge of
the workload.
Online Monitoring. The aforementioned routing scheme can help the system cope
with unseen queries, but at some point it may become necessary to retune the
divergent design if the actual workload is substantially different than the
training workload. The next experiment evaluates the online-monitoring module
inside RITA which is designed for the task of detecting workload changes.
We assume that the system receives the dynamic workload
$\mathit{TPCDS\mbox{-}dyn}$, which shifts to a different query distribution
after query 200 and then shifts back to the original distribution at query
400. Initially, the system is equipped with a divergent design
$p_{\mbox{\tiny\rm RITA}}^{\mbox{\tiny\rm curr}}$ that is tuned with a
training workload from the first query distribution. The monitoring module
continuously computes a divergent design $p_{\mbox{\tiny\rm
RITA}}^{\mbox{\tiny\rm slide}}$ based on a sliding window of the last 60
queries in the workload, and outputs the improvement on
$\mathit{ExpTotalCost}()$ if $p_{\mbox{\tiny\rm RITA}}^{\mbox{\tiny\rm
slide}}$ were used instead of $p_{\mbox{\tiny\rm RITA}}^{\mbox{\tiny\rm
curr}}$.
Figure 12 shows the monitoring statistics produced by the online-monitoring
module of RITA for the $\mathit{TPCDS\mbox{-}dyn}$ workload. Matching our
intuition, the output shows that $p_{\mbox{\tiny\rm RITA}}^{\mbox{\tiny\rm
slide}}$ has small improvements for the first 200 queries (around 30%), since
the current design $p_{\mbox{\tiny\rm RITA}}^{\mbox{\tiny\rm curr}}$ is
already tuned for the particular phase of the workload. However, as soon as
the workload shifts to a different distribution, the output shows a
considerable improvement of more than 60%. This can be viewed as a strong
indication that a retuning of the system can yield significant performance
improvements. The spike tapers off close to query $450$, since in this
experiment the workload shifts back to its previous distribution and hence
there is no benefit to changing the current design.
RITA requires $1.2$ seconds on average to analyze each new query in this
workload, and can thus generate an output that accurately reflects the actual
workload. We conducted similar experiments with different values of the length
of the sliding window and observed similar results. For instance, RITA takes
at most $3$ seconds when the sliding window is set to $100$ statements.
Elastic Retuning. After observing the monitoring output, the DBA can invoke
the recommender module to examine different recommendations for retuning the
system in an elastic fashion. The next set of experiments evaluate how fast
RITA can generate these recommendations and also their quality.
We employ a scenario that builds on the previous experiment on online
monitoring. Specifically, we assume that the DBA invokes the recommender using
the sliding window of 60 queries that corresponds to the spike in Figure 12.
Moreover, the DBA specifies two dimensions of interest with respect to a new
divergent design: the workload-evaluation cost and the cost of materializing
the design. Also, the DBA wants to study the effect of shrinking and expanding
the number of replicas. We assume that the DBA sets the probability of failure
($\alpha$) to be $0$ in order to allow RITA to execute fast and generate the
output in a timely fashion. After inspecting the output, the DBA may invoke
another (more expensive) tuning session for a specific choice of replicas (or
routing multiplicity factor) and reconfiguration cost, and a non-zero
$\alpha$. Our results in Figure 13 show that RITA can compute a divergent
design that matches the same level of performance as the case for $\alpha=0$.
Figures 12 shows the output of the recommender based on our testing scenario.
Each point $(x,y)$ on the chart corresponds to a divergent design that
requires $x$ cost units to materialize and whose $\mathit{ExpTotalCost}()$ is
equal to $y$. The three curves labeled $N=z$, $z\in\\{2,3,4\\}$, represent
divergent designs that employ $z$ replicas. We assume that $N=3$ is the
current setting in the system, and hence $N=2$ (resp. $N=4$) represents
dropping (resp. adding) a replica. The chart also shows the
$\mathit{ExpTotalCost}()$ metric of the current design, for comparison. As
shown, there are several options to significantly improve (by up to $7\times$)
the performance of the current design. Moreover, the DBA obtains the following
valuable information: there is a least materialization cost in order to get
some improvement; designs that require more than 160 units of materialization
cost offer diminishing returns for $N=3$ and $N=4$; and there is not much
benefit to increasing the number of replicas, since $N=3$ and $N=4$ have
virtually identical performance. Based on these data points, the DBA can make
an informed decision about how to retune the divergent design in the system.
RITA requires a total of $20$ seconds to generate the points in the chart.
Note that the recommender does not have to initialize INUM for the training
workload, as this initialization has already been performed inside the
monitoring module. This short computation time facilitates an exploratory
approach to index tuning.
We employ another scenario that is similar to the previous one except that we
assume the DBA wants to study the effect of using different values for the
routing multiplicity factor, while keeping the number of replicas unchanged.
Figure 12 shows the output of the recommender based on the above testing
scenario. The three curves labeled $m=z$, $z\in\\{1,2,3\\}$, represent
divergent designs that have the routing multiplicity factor $z$ (We assume
that $m=2$ is the current setting in the system). We observe that designs that
require more than 80 units of materialization cost when routing queries for
$m=2$ has slightly better performance when routing queries for $m=1$. This
result indicates that we can obtain designs with some flexibility in routing
queries (i.e., $m=2$) and without sacrifying much in terms of performance as
designs that have the most specialization (i.e., $m=1$). RITA requires a total
of $10$ seconds to generate the points in the chart.
## 7 Conclusion
In this paper, we introduced RITA, a novel index tuning advisor for replicated
databases, that provides DBAs with a powerful tool for divergent index tuning.
The key technical contribution of RITA is a reduction of the problem to a
compact binary integer program, which enables the efficient computation of a
(near-)optimal divergent design using mature, off-the-shelf software for
linear optimization. Our experimental studies demonstrate that, compared to
state-of-the-art solutions, RITA offers richer tuning functionality and is
able to compute divergent designs that result in significantly better
performance.
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## Appendix A Proving Theorem 1
We reduce the original problem studied in [5] to ${\it DDT}$ by proving their
equivalence when $\alpha=0$ and $C$ contains solely a space-budget constraint
per replica. Since the original problem is NP-Hard, the same follows for ${\it
DDT}$. The result in Lemma 1 (See below) is the key to prove their
equivalence. It is important to note from Section 3.3 that in the general
setting of DDT, $\mathit{h}_{0}(q)$ might not correspond to the $m$ replicas
with the least evaluation cost for $q$.
###### Lemma 1
In the problem setting of ${\it DDT}$ when $\alpha=0$ and $C$ contains solely
a space-budget constraint per replica, $\mathit{h}_{0}(q)$ corresponds to the
$m$ replicas with the least evaluation cost for $q$.
We prove Lemma 1 using contradiction. Assume that for some query $q$, there
exist two replicas $r_{1}$ and $r_{2}$ such that $r_{1}\in\mathit{h}_{0}(q)$,
$r_{2}\not\in\mathit{h}_{0}(q)$ and $cost(q,I_{r_{1}})>cost(q,I_{r_{2}})$. We
then derive another routing function $\mathbf{h}^{\prime}$ that is similar to
$\mathbf{h}$ except that $\mathit{h}_{0}^{\prime}$ is slightly modified as
follows:
$\mathit{h}_{0}^{\prime}(q)=\mathit{h}_{0}(q)\cup\\{r_{2}\\}-\\{r_{1}\\}$.
Clearly,
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})>\mathit{TotalCost}(\mathbf{I},\mathbf{h}^{\prime})$.
This contradicts to the requirement to minimize
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$ in the problem setting of DDT.
## Appendix B Proving Theorem 2
We prove the theorem in two steps. First, we show that every divergent design
$(\mathbf{I},\mathbf{h})$ corresponds to a value-assignment ${\bf v}$ for
variables in the BIP such that ${\bf v}$ satisfies the constraints (Lemma 2).
This property guarantees that the solution space of the BIP contains all
possible solutions for the divergent design tuning problem. Subsequently, we
prove that the optimal assignment ${\bf v}^{*}$ corresponds to a divergent
design. Combining these two results, we can then conclude the correctness of
the theorem (Lemma 3).
To simplify the presentation and without loss of generality, we prove the
theorem for the basic DDT when $\alpha=0$, $C=\emptyset$ and the workload
comprises solely queries, i.e., $W=Q$.
Given a valid-assignment ${\bf v}$, we use $BIPcost({\bf v})$ to denote the
value of the objective function of the BIP under the assignment ${\bf v}$.
###### Lemma 2
For any divergent physical design $(\mathbf{I},\mathbf{h})$, there is an
assignment ${\bf v}$ s.t.
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})=BIPcost({\bf v})$.
###### Lemma 3
Let ${\bf v^{*}}$ denote the solution to the BIP problem. Then,
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})=BIPcost({\bf v^{*}})$, where
$(\mathbf{I},\mathbf{h})$ is the divergent design derived from ${\bf v^{*}}$.
### B.1 Proof of Lemma 2
Given a divergent design $(\mathbf{I},\mathbf{h})$ and for every query $q\in
Q$, using the linear decomposability property, we can express the cost of $q$
at replica $r\in\mathit{h}_{0}(q)$ as:
$cost(q,\mathit{I}_{r})=\beta_{p}+\sum_{i\in[1,n],a=Y[i]}\gamma_{pa}$
for some choice of $p=p^{r}\in\mathit{TPlans}(q)$ and
$Y=Y^{p,r}\in\mathit{Atom}(\mathit{I}_{r})$. We assign the values for
variables as follows.
* •
${\bf v}(t^{r}_{q})=1$ if $r\in\mathit{h}_{0}(q)$,
* •
${\bf v}(y^{r}_{p})=1$ if $p=p^{r}$, $r\in\mathit{h}_{0}(q)$,
* •
${\bf v}(x^{r}_{pa})=1$ if $p=p^{r}$, $r\in\mathit{h}_{0}(q)$ and
$a=Y^{p,r}[i]$, $i\in[1,n]$,
* •
${\bf v}(s^{r}_{a})=1$ if $a\in I_{r}$, $r\in[1,N]$, and
* •
The other cases of variables are assigned value $0$
We observe that under this assignment, all constraints in the BIP are
satisfied. For instance, since ${\bf v}(t^{r}_{q})=1$ when
$r\in\mathit{h}_{0}(q)$ and $\mathit{h}_{0}(q)$ has $m$ values, it can be
immediately derived that $\sum_{r\in[1,N]}t^{r}_{q}=m$, i.e., constraint (2)
is satisfied.
By eliminating terms with value $0$, we obtain the following results.
$BIPCost({\bf v})=\sum_{q\in
Q}\sum_{r\in\mathit{h}_{0}(q)}\frac{f(q)}{m}{\hat{{\mathit{cost}}}}(q,r)$
${\hat{{\mathit{cost}}}}(q,r)=\beta_{p}+\sum_{i\in[1,n],a=Y[i]}\gamma_{pa},\mbox{
for }r\in\mathit{h}_{0}(q),p=p^{r},Y=Y^{p,r}\in\mathit{Atom}(\mathit{I}_{r})$
Thus, $BIPCost({\bf v})=\mathit{TotalCost}(\mathbf{I},\mathbf{h})$.
### B.2 Proof of Lemma 3
The following arguments are derived based on the assumption that ${\bf v^{*}}$
satisfies the BIP formulation.
First, based on (2), we derive that for every query $q$, there exists a set
$S_{q}=\\{r\ |\ r\in[1,N]\\}$ and $|S_{q}|=m$ such that ${\bf
v^{*}}(t^{r}_{q})=1$ iff $r\in S_{q}$.
Second, based on (4), we derive that for every query $q$ and every $r\in
S_{q}$, there exists exactly one plan $p=p^{r}\in\mathit{TPlans}(q)$ such that
${\bf v^{*}}(y^{r}_{p})=1$.
Third, based on (6), there exists an atomic configuration $Y^{p,r}$, $r\in
S_{q}$, $p=p^{r}$ that corresponds to the assignments for ${\bf
v}(x^{r}_{pa})$.
Finally, we prove that $p^{r}$ and $Y^{p,r}$, $r\in S_{q}$, correspond to the
choice of plan $p$ and atomic configuration $Y$ that yields the minimum value
of $cost(q,\mathit{I}_{r})$, by using contradiction. Combining these results,
we conclude that $BIPCost({\bf
v^{*}})=\mathit{TotalCost}(\mathbf{I},\mathbf{h})$.
Suppose that there exists a different choice $p^{c}\in\mathit{TPlans}(q)$ and
$Y^{c}\in\mathit{Atom}(\mathit{I}_{r})$, $r\in S_{q}$, such that
$cost(q,p^{c},Y^{c})<cost(q,p^{r},Y^{p,r}$). Here, we use $cost(q,p,Y)$ denote
the cost of $q$ using the template plan $p$ and the atomic configuration $Y$.
We can now derive an alternative assignment ${\bf v^{c}}$ that is similar to
${\bf v^{*}}$ except the followings:
* •
Variables corresponding to $p^{r}$ and $Y^{p,r}$ are assigned value $0$, and
* •
${\bf v}(y^{r}_{p})=1$ if $p=p^{c}$, $r\in S_{q}$, and
* •
${\bf v}(x^{r}_{pa})=1$, if $p=p^{c}$, $r\in S_{q}$ and $a=Y^{c}[i]$,
$i\in[1,n]$.
We observe that ${\bf v^{c}}$ is a valid constraint-assignment for the
formulated BIP. However, since $BIPcost({\bf v^{c}})<BIPcost({\bf v^{*}})$,
this contradicts our assumption about the optimality of ${\bf v^{*}}$.
## Appendix C Factoring Failures
In this section, we present the full details of how RITA integrates failures
into the BIP.
Under our assumption of using fast what-if optimization, the cost of a query
$q$ in some replica $r$ can be expressed as
${\mathit{cost}}(q,I_{r})={\mathit{cost}}(p^{\prime},A^{\prime})$ for some
choice of $p^{\prime}\in\mathit{TPlans}(q)$ and an atomic configuration
$A^{\prime}\in\mathit{Atom}(I_{r})$ We introduce the following additional
variables.
* •
$t^{r,j}_{q}=1$ if and only if $q$ is routed to replica $r$ when $j$ fails,
i.e., $\mathit{h}_{j}(q)=\\{r\ |\ t^{r,j}_{q}=1\\}$
* •
$x^{r,j}_{pa}=1$ if and only if $q$ is routed to replica $r$ when $j$ fails,
$p=p^{\prime}$ and $a\in A^{\prime}$.
* •
$y^{r,j}_{p}=1$ if and only if $q$ is routed to replica $r$ when $j$ fails,
$p=p^{\prime}$.
We also need to add a new set of constraints, as given in Figure 13. These
constraints are very similar to their counterparts in Figure 2. The
correctness of the BIP is proven in the same way as presented in Appendix B.
$\mathit{FTotalCost}(\mathbf{I},\mathbf{h},j)=\sum_{q\in
Q}\sum_{r\in[1,N]\wedge r\neq
j}\frac{f(q)}{\max{m,N-1}}{\hat{{\mathit{cost}}}}(q,r,j)$
${\hat{{\mathit{cost}}}}(q,r,j)=\sum_{p\in\mathit{TPlans}(q)}\beta_{p}y^{r,j}_{p}+\sum_{\begin{subarray}{c}p\in\mathit{TPlans}(q)\\\
a\in{\cal S}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
1}\\}\cup\cdots\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
n}\\}\end{subarray}}\gamma_{pa}x^{r,j}_{pa},\begin{subarray}{c}\forall
r\in[1,N],\\\ \forall q\in Q\cup Q_{upd}\end{subarray}$ (10)
such that:
$\sum_{r\in[1,N]}t^{r,j}_{q}=\max\\{N-1,m\\},\forall q\in Q$ (11)
$\sum_{p\in\mathit{TPlans}(q)}y^{r,j}_{p}=t^{r,j}_{q},\ \ \ \forall q\in Q\cup
Q_{upd}$ (12)
$s^{r}_{a}\geq x^{r,j}_{pa},\ \ \ \forall q\in Q\cup
Q_{upd},p\in\mathit{TPlans}(q),\ a\in{\cal S}$ (13)
$\sum_{a\in{\cal S}_{i}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
i}\\}}x^{r,j}_{pa}=y^{r,j}_{p},\ \ \begin{subarray}{c}\forall q\in Q\cup
Q_{upd},p\in\mathit{TPlans}(q),\\\ i\in[1,n],\ T_{i}\mbox{ is referenced in
q}\end{subarray}$ (14)
Figure 13: Augmented BIP to handle failures.
## Appendix D Bounding Load-skew
### D.1 Additional Constraints for Exact Solution
This section presents the set of constraints that RITA formulates in order to
ensure the optimality of ${\hat{{\mathit{cost}}}}(q,r)$ with the presence of
bounding load-skew constraints.
RITA introduces a new cost formula $\mathit{cost^{opt}}(q,r)=cost(q,I_{r})$
for $r\in[1,N]$. The formula of $\mathit{cost^{opt}}(q,r)$ is very similar to
$cost(q,r)$; the variables $\mathit{yo}^{r}_{p}$ (resp.
$\mathit{xo}^{r}_{pa}$) have the same meaning with $y^{r}_{p}$ (resp.
$x^{r}_{pa}$). The main difference is that for $r\not\in\mathit{h}_{0}(q)$, we
have ${\hat{{\mathit{cost}}}}(q,r)=0$ whereas
$\mathit{cost^{opt}}(q,r)=cost(q,I_{r})>0$. The atomic constraint in (16) are
somehow similar to the atomic constraints on $cost(q,r)$. Note that in (16a),
the constraint requires exactly one template plan to be chosen to compute
$\mathit{cost^{opt}}(q,r)$ in order for this value corresponds to the query
execution cost of $q$ on replica $r$.
$\mathit{cost^{opt}}(q,r)=\sum_{p\in\mathit{TPlans}(q)}\beta_{p}\mathit{yo}^{r}_{p}+\sum_{\begin{subarray}{c}p\in\mathit{TPlans}(q)\\\
a\in{\cal S}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
1}\\}\cup\cdots\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
n}\\}\end{subarray}}\gamma_{pa}\mathit{xo}^{r}_{pa},\begin{subarray}{c}\forall
q\in Q\cup Q_{upd}\\\ \forall r\in[1,N]\end{subarray}$ (15)
$\displaystyle\sum_{p\in\mathit{TPlans}(q)}\mathit{yo}^{r}_{p}$
$\displaystyle=1$ (16a) $\displaystyle\sum_{a\in{\cal
S}_{i}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny i}\\}}\mathit{xo}^{r}_{pa}$
$\displaystyle=\mathit{yo}^{r}_{p},\ \ \begin{subarray}{c}\forall
p\in\mathit{TPlans}(q)\\\ \forall i\in[1,n]\wedge\ T_{i}\mbox{ is referenced
in q}\end{subarray}$ (16b)
$\mathit{cost^{opt}}(q,r)\leq\beta_{p}+\sum_{\begin{subarray}{c}i\in[1,n]\\\
a\in{\cal S}\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
1}\\}\cup\cdots\cup\\{\mbox{\tiny\rm SCAN}_{\tiny
n}\\}\end{subarray}}\gamma_{pa}u^{r}_{pa},\ \ \forall p\in\mathit{TPlans}(q)$
(17)
$\displaystyle\sum_{a\in{\cal S}_{i}\cup I_{\emptyset}}u^{r}_{pa}$
$\displaystyle=1,\ \ \begin{subarray}{c}\forall t\in[1,K_{q}],\\\ \forall
i\in[1,n]\ \wedge\ T_{i}\mbox{ is referenced in q}\end{subarray}$ (18a)
$\displaystyle u^{r}_{pa}$ $\displaystyle\leq s^{r}_{a},\forall
p\in\mathit{TPlans}(q)\wedge a\in{\cal S}$ (18b)
$\sum_{b\in{\cal S}_{i}\cup I_{\emptyset}\ \wedge\
\gamma_{pa}\geq\gamma_{pb}}u^{r}_{pb}\geq s^{r}_{a},\ \ \forall
p\in\mathit{TPlans}(q),i\in[1,n],a\in{\cal S}_{i}$ (19)
Figure 14: Query-Optimal Constraints
To establish the optimal cost constraints, we use the following alternative
way to compute $cost(q,X)$. For each internal plan cost $\beta_{p}$,
$p\in\mathit{TPlans}(q)$, we first derive a “local” optimal cost, referred to
as $C^{local}_{t}$, which is the smallest cost that can be obtained by
“plugging” all possible atomic configurations $A\in\mathit{Atom}(X)$ into the
slot of the template plan of $\beta_{p}$. Essentially,
$C^{local}_{t}=\beta_{p}+I^{local}_{p}$, where $I^{local}_{p}$ is the smallest
value of the total access cost using some atomic-configuration
$A\in\mathit{Atom}(X)$ to plug into the template plan of $\beta_{p}$. To
obtain $I^{local}_{p}$, for each slot in the internal plan of $\beta_{p}$, we
enumerate all possible indexes in $X$ that can be “plugged” into, and find the
one that yields the smallest access cost to sum up into $I^{local}_{p}$.
Lastly, $cost_{q}(X)$ is then obtained as the smallest value among the derived
$C^{local}_{p}$ with $p\in\mathit{TPlans}(q)$.
The right hand-side of (17) is the formula of $C^{local}_{p}$. Here, we
introduce variables $u^{r}_{pa}$; where $u^{r}_{pa}=1$ iff the index $a$ is
used at slot $i$ in the template plan $\beta_{p}$ to compute $C^{local}_{p}$.
For $C^{local}_{p}$ to correspond to some atomic configuration, we impose the
constraint in (18a).
Furthermore, an index $a$ can be used in $C^{opt}_{p}$ if and only if $a$ is
recommended at replica $r$ (constraint (18b)).
The constraint (19) ensures that the candidate index with the smallest access
cost is selected to plug into each slot of $\beta_{t}$ in computing
$I^{local}_{t}$.
### D.2 Greedy Approach
This section presents our proposal of a greedy scheme that trade-offs the
quality of the design for the efficiency.
First, we derive an optimal design $(\mathbf{I}_{opt},\mathbf{h}_{opt})$
assuming there is no load imbalance constraint and the probability of failure
is $0$. We then compute an approximation factor
$\beta=\frac{{\tau}-1}{1+(N-1){\tau}}$. and add the following constraint into
the BIP.
$\mathit{load}(\mathbf{I},\mathbf{h},r)\leq\frac{(1+\beta)\mathit{TotalCost}(\mathbf{I}_{opt},\mathbf{h}_{opt})}{N},\forall
r\in[1,N]$ (20)
This constraint is an easy constraint, as its right handside is a constant. We
prove that if the BIP solver can find a solution for the modified BIP, the
returned solution is a valid solution and has
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})$ bounded as the following theorem
shows.
###### Theorem 3
The divergent design returned by the greedy solution satisfies all constraints
in DDT problem and has
$\mathit{TotalCost}(\mathbf{I},\mathbf{h})\leq(1+\beta)\mathit{TotalCost}(\mathbf{I}_{opt},\mathbf{h}_{opt})$.
$\Box$
###### Proof D.4.
We overload $I_{opt}$ (resp. $\mathbf{I}$) to refer to the total cost of the
design $I_{opt}$ (resp. $\mathbf{I}$) as well.
The maximum load of a replica in $\mathbf{I}$ is $\frac{(1+\beta)I_{opt}}{N}$
(due to the constraint 20). By summing up the load of all replicas in
$\mathbf{I}$, we obtain: $\mathbf{I}\leq(1+\beta)I_{opt}$. Therefore,
$\mathbf{I}$ differs from $I_{opt}$ by an approximation ratio $(1+\beta)$. All
remaining issue is to prove that $\mathbf{I}$ satisfies the load-imbalance
constraint.
Without loss of generality, assume that $load(1,\mathbf{I})\leq
load(j,\mathbf{I})$, $\forall j\in[2,N]$.
Since $\mathbf{I}$ is load-imbalance, we can derive the followings:
$\displaystyle\mathbf{I}=\sum_{j\in[2,N]}\mathit{load}(j,\mathbf{I})+\mathit{load}(1,\mathbf{I})$
(21a) $\displaystyle\frac{(1+\beta)I_{opt}}{N}\geq\mathit{load}(j,\mathbf{I})$
(21b)
$\displaystyle\frac{(N-1)}{N}(1+\beta)I_{opt}+load(1,\mathbf{I})\geq\mathbf{I}$
(21c) $\displaystyle\mathbf{I}\geq I_{opt}$ (21d) $\displaystyle
load(1,\mathbf{I})\geq\left(1-\frac{(N-1)}{N}(1+\alpha)\right)I_{opt}$ (21e)
The maximum load in $\mathbf{I}$ is $\frac{1}{N}(1+\alpha)I_{opt}$ and the
minimum load is $\left(1-\frac{(N-1)}{N}(1+\alpha)\right)I_{opt}$. Therefore,
the load-imbalance factor of $\mathbf{I}$ is $\frac{1+\beta}{1-(N-1)\alpha}$.
By replacing the value of $\beta$, we obtain the load-imbalance factor
${\tau}$.
Note that this greedy scheme does not encounter the aforementioned problem
with ${\hat{{\mathit{cost}}}}(q,r)$ not to be equal to $cost(q,I_{r})$.
Informally, the reason is due to the fact that the right hand-side of the
inequality constraint in (20) is a constant.
### D.3 Additional Experimental Results
This section presents the comparison between RITA, DivgDesign and Unif when we
vary the routing multiplicity factor. Figure 15 presents one representative
result when we vary this factor on $\mathit{TPCDS\mbox{-}query}$ workload with
$b=0.5\times$ and $N=3$.
As expected, when the value of $m$ increases, the total cost of
${p_{\mbox{\tiny\rm RITA}}}$ and ${p_{\mbox{\tiny\rm DD}}}$ increase, since
queries need to be sent to more places. Note that the cost of
${p_{\mbox{\tiny\rm UNIF}}}$ remains the same, as all replicas have the same
index configuration under UNIF design. Also, when $m=N$, the total costs of
${p_{\mbox{\tiny\rm DD}}}$ and ${p_{\mbox{\tiny\rm UNIF}}}$ are the same,
since DivgDesign needs to send every query to every replica, and it uses the
same black-box design advisor as Unif to compute the recommended index-set at
each replica.
We observe that in all cases, RITA significantly outperforms DivgDesign and
Unif. The reason can be again attributed to the fact that RITA searches a
considerably larger space of possible designs.
Figure 15: Varying the routing multiplicity factor on
$\mathit{TPCDS\mbox{-}query}$ workload
|
arxiv-papers
| 2013-04-04T15:54:48 |
2024-09-04T02:49:43.896098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quoc Trung Tran, Ivo Jimenez, Rui Wang, Neoklis Polyzotis, Anastasia\n Ailamaki",
"submitter": "Quoc Trung Tran",
"url": "https://arxiv.org/abs/1304.1411"
}
|
1304.1829
|
ON DOUBLE 3-TERM ARITHMETIC PROGRESSIONS
Tom Brown
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada
[email protected]
Veselin Jungić
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada
[email protected]
Andrew Poelstra
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada
[email protected]
Abstract
In this note we are interested in the problem of whether or not every
increasing sequence of positive integers $x_{1}x_{2}x_{3}\cdots$ with bounded
gaps must contain a double 3-term arithmetic progression, i.e., three terms
$x_{i}$, $x_{j}$, and $x_{k}$ such that $i+k=2j$ and $x_{i}+x_{k}=2x_{j}$. We
consider a few variations of the problem, discuss some related properties of
double arithmetic progressions, and present several results obtained by using
RamseyScript, a high-level scripting language.
## 1 Introduction
In 1987, Tom Brown and Allen Freedman ended their paper titled Arithmetic
progressions in lacunary sets [Brown and Freedman 1987] with the following
conjecture.
###### Conjecture 1.
Let $(x_{i})_{i\geq 1}$ be a sequence of positive integers with $1\leq
x_{i}\leq K$. Then there are two consecutive intervals of positive integers
$I,J$ of the same length, with $\sum_{i\in I}x_{i}=\sum_{j\in J}x_{j}$.
Equivalently, if $a_{1}<a_{2}<\cdots$ satisfy $a_{n+1}-a_{n}\leq K$, for all
$n$, then there exist $i<j<k$ such that $i+k=2j$ and $a_{i}+a_{k}=2a_{j}$.
If true, Conjecture 1 would imply that if the sum of the reciprocals of a set
$A=\\{a_{1}<a_{2}<a_{3}<\cdots\\}$ of positive integers diverges, and
$a_{n+1}-a_{n}\rightarrow\infty$ as $n\rightarrow\infty$, and there exists $K$
such that $a_{i+1}-a_{i}\leq a_{j+1}-a_{j}+K$ for all $1\leq i\leq j,$ then
$A$ contains a 3-term arithmetic progression. This is a special case of the
famous Erdős conjecture that if the sum of the reciprocals of a set $A$ of
positive integers diverges, then $A$ contains arbitrarily long arithmetic
progressions.
Conjecture 1 is a well-known open question in combinatorics of words and it is
usually stated in the following form:
> Must every infinite word on a finite alphabet consisting of positive
> integers contain an additive square, i.e., two adjacent blocks of the same
> length and the same sum?
The answer is trivially yes in case the alphabet has size at most 3. For more
on this question see, for example, [Au et al. 2011, Freedman 2013+], [Grytczuk
2008]. Also see [Halbeisen and Hungerb$\ddot{\text{u}}$hler 2000], [Pirillo
and Varricchio 1994] and [Richomme et al. 2011].
We mention two relatively recent positive results. Freedman [Freedman 2013+]
has shown that if $a<b<c<d$ satisfy the Sidon equation $a+d=b+c$, then every
word on $\\{a,b,c,d\\}$ of length 61 contains an additive square. His proof is
a clever reduction of the general problem to several cases that are then
checked by computer.
Ardal, Brown, Jungić, and Sahasrabudhe [Ardal et al. 2012] proved that if an
infinite word $\omega=a_{1}a_{2}a_{3}\cdots$ has the property that there is a
constant $M$, such that for each positive integer $n$ the number of possible
sums of $n$ consecutive terms in $\omega$ does not exceed $M$, then for each
positive integer $k$ there is a $k$-term arithmetic progression
$\\{m+id:i=0,\cdots,k-1\\}$ such that
$\sum_{i=m+1}^{m+d}a_{i}=\sum_{i=m+d+1}^{m+2d}a_{i}=\cdots=\sum_{i=m+(k-2)d+1}^{m+(k-1)d}a_{i}.$
The proof of this fact is based on van der Waerden’s theorem [van der Waerden
1927].
This note is inspired by the second statement in Conjecture 1. Before
restating this part of the conjecture we introduce the following terms.
We say that an increasing sequence of positive integers
$a_{1},a_{2},a_{3},\dots$ has bounded gaps if there is a constant $K$ such
that
$a_{n+1}-a_{n}\leq K$
for all positive integers $n$.
We say that an increasing sequence of positive integers
$a_{1},a_{2},a_{3},\dots$ contains a double $k$-term arithmetic progression if
there are $p_{1}<p_{2}<\cdots<p_{k}$ such that both
$\\{a_{p_{1}},a_{p_{2}},\dots,a_{p_{k}}\\}$ and
$\\{p_{1},p_{2},\dots,p_{k}\\}$ are arithmetic progressions.
###### Problem 1.
Does every increasing sequence of positive integers with bounded gaps contain
a double 3-term arithmetic progression?
It is straightforward to check that Problem 1 is equivalent to the question
above concerning additive squares: Given positive integers $K$ and
$a_{1}<a_{2}<a_{3}<\cdots$, with $a_{i+1}-a_{i}\leq K$ for all $i\geq 1,$ let
$x_{i}=a_{i+1}-a_{i},\ i\geq 1.$ Then $x_{1}x_{2}x_{3}\cdots$ is an infinite
word on a finite alphabet of positive integers. Given an infinite word
$x_{1}x_{2}x_{3}\cdots$ on a finite alphabet of positive integers, define
$a_{1},a_{2},a_{3},\ldots$ recursively by
$a_{1}\in\mathbb{N},a_{i+1}=x_{i}+a_{i},\ i\geq 1.$ Then
$a_{1}<a_{2}<a_{3}<\cdots$, and $a_{i+1}-a_{i}\leq K$ for some $K$ and all
$i\geq 1.$ In both cases, an additive square in $x_{1}x_{2}x_{3}\cdots$
corresponds exactly to a double 3-term arithmetic progression in
$a_{1}<a_{2}<a_{3}<\cdots$.
The existence of an infinite word on four integers with no additive cubes,
i.e., with no three consecutive blocks of the same length and the same sum,
established by Cassaigne, Currie, Schaeffer, and Shallit [Cassaigne et al.
2013+], translates into the fact that there is an increasing sequence of
positive integers with bounded gaps with no double 4-term arithmetic
progression.
But what about a double variation on van der Waerden’s theorem?
###### Problem 2.
If the set of positive integers is finitely colored, must there exist a color
class, say $A=\\{a_{1}<a_{2}<a_{3}<\cdots\\}$ for which there exist $i<j<k$
with $a_{i}+a_{k}=2a_{j}$ and $i+k=2j$?
We have just seen that an affirmative answer to Problem 1 gives an affirmative
answer to the question concerning additive squares. It is also true that an
affirmative answer to Problem 1 implies an affirmative answer to Problem 2.
###### Proposition 1.
Assume that every increasing sequence of positive integers
$x_{1}x_{2}x_{3}\cdots$ with bounded gaps contains a double 3-term arithmetic
progression. Then if the set of positive integers is finitely colored, there
must exist a color class, say $A=\\{a_{1}<a_{2}<a_{3}<\cdots\\}$, which
contains a double 3-term arithmetic progression.
###### Proof.
We use induction on the number of colors, denoted by $r$. For $r=1$ the
conclusion trivially follows. Now assume that for every $r$-coloring of
$\mathbb{N}$ there exists a color class which contains a double 3-term
arithmetic progression. By the Compactness Principle there exists
$M\in\mathbb{N}$ such that every $r$-coloring of $[1,M]$ (or of any translate
of $[1,M]$) yields a monochromatic double 3-term arithmetic progression.
Assume now that there is an $(r+1)$-coloring of $\mathbb{N}$ for which there
does not exist a monochromatic double 3-term arithmetic progression. Let the
$(r+1)$st color class be $C(r+1)=\\{x_{1}<x_{2}<\cdots\\}.$ By the induction
hypothesis on $r$ colo rs, $C(r+1)$ is infinite. By the assumption that every
increasing sequence of positive integers $x_{1}x_{2}x_{3}\cdots$ with bounded
gaps contains a double 3-term arithmetic progression, $C(r+1)$ does not have
bounded gaps. In particular, there is $p\geq 1$ such that $x_{p+1}-x_{p}\geq
M+2.$ But then the interval $[x_{p}+1,x_{p+1}-1]$ contains a translate of
$[1,M]$ and is colored with only $r$ colors, so that $[x_{p}+1,x_{p+1}-1]$
does contain a monochromatic double 3-term arithmetic progression. This
contradiction completes the proof. ∎
More generally, if the set of positive integers is finitely colored and if
each color class is regarded as an increasing sequence, must there be a
monochromatic double $k$-term arithmetic progression, for a given positive
integer $k$? What if the gaps between consecutive elements colored with same
color are pre-prescribed, say at most 4 for the first color, at most 6 for the
second color, and at most 8 for the third color, and so on?
In the spirit of van der Waerden’s numbers $w(r,k)$ [Graham et al. 1990] we
define the following.
###### Definition 1.
For given positive integers $r$ and $k$ greater than 1, let $w^{\ast}(r,k)$ be
the least integer, if it exists, such that for any $r$-coloring of the
interval $[1,w^{\ast}(r,k)]$ there is a monochromatic double $k$-term
arithmetic progression.
For given positive numbers $r$, $k$, $a_{1},a_{2},\ldots,a_{r}$ let
$w^{\ast}(k;a_{1},a_{2},\ldots,a_{r})$ be the least integer, if it exists,
such that for any $r$-coloring of the interval
$[1,w^{\ast}(k;a_{1},a_{2},\ldots,a_{r})]=A_{1}\cup A_{2}\cup\cdots\cup A_{r}$
such that for each $i$ the gap between any two consecutive elements in $A_{i}$
is not greater than $a_{i}$, there is a monochromatic double $k$-term
arithmetic progression.
We will show that $w^{\ast}(2,3)$ is relatively simple to obtain. We will give
lower bounds for $w^{\ast}(3,3)$ and $w^{\ast}(4,2)$ and a table with values
of $w^{\ast}(3;a_{1},a_{2},a_{3})$ for various triples $(a_{1},a_{2},a_{3})$
and propose a related conjecture.
We will share with the reader some insights related to the general question
about the existence of double 3-term arithmetic progressions in increasing
sequences with bounded gaps.
Finally, we will describe RamseyScript, a high-level scripting language
developed by the third author that was used to obtain the colorings and bounds
that we have established.
## 2 $w^{\ast}(r,3)$
Now we look more closely at $w^{\ast}(r,3),$ the least integer, if it exists,
such that for every $r$-coloring of the interval $[1,w^{\ast}(r,3)]$ there is
a monochromatic double 3-term arithmetic progression.
Suppose that $w^{\ast}(r,3)$ does not exist for some $r$, but
$w^{\ast}(r-1,3)$ does exist. Then, by the Compactness Principle, there is a
coloring of the positive integers with $r$ colours, say with colour classes
$A_{1},A_{2},\ldots,A_{r}$, such that no colour class contains a double 3-term
arithmetic progression. Then (a) $A_{1}$ contains no double 3-term arithmetic
progression, (b) $A_{1}$ has bounded gaps because $w^{\ast}(r-1,3)$ exists,
and (c) $A_{1}$ is infinite, because $w^{\ast}(r-1,3)$ exists.
Let $d_{1},d_{2},\ldots$ be the sequence of consecutive differences of the
sequence $A_{1}$. That is, if $A_{1}=\\{a_{1},a_{2},a_{3},\ldots\\}$ then
$d_{n}=a_{n}-a_{n-1}$, $n\geq 1$. Then the sequence $d_{1},d_{2},\ldots$ is a
sequence on a finite set of integers which does not contain any additive
square.
Thus if there exists $r$ such that $w^{\ast}(r,3)$ does not exist, then there
exists a sequence on a finite set of integers which does not contain an
additive square.
It is conceivable that proving that $w^{\ast}(r,3)$ does not exist for all $r$
(if this is true!) is easier than directly proving the existence of a sequence
on a finite set of integers with no additive square.
###### Theorem 1.
$w^{\ast}(2,3)=17$.
###### Proof.
Color $[1,m]$ with two colors, with no monochromatic double 3-term arithmetic
progressions. Then the first color class must have gaps of either 1, 2, or 3.
Thus the sequence of gaps of the first color class is a sequence of 1s, 2s,
and 3s, and this sequence must have length at most 7, otherwise there is an
additive square, which would give a double 3-term arithmetic progression in
the first color class. Hence, the first colour class can contain at most 8
elements (only 7 consecutive differences) and similarly for the second colour
class. This shows that $w^{\ast}(2,3)\leq 8+8+1=17$. On the other hand, the
following 2-coloring of $[1,16]$ has no monochromatic double 3-term arithmetic
progression:
$0010110100101101.$
Hence $w^{\ast}(2,3)=17$. ∎
###### Theorem 2.
$w^{\ast}(3,3)\geq 414$.
The following 3-coloring of $[1,413]$ avoids monochromatic 3-term double
arithmetic progressions:
$\begin{array}[]{l}0101102210100201200100221221010010220010112011211202210112122112202210\\\
0110010220201122022002202001012212112122001001120121100110020022002110\\\
2001101001121120210020011210201121122112122010110100110102201220201221\\\
1210021122112122112200110011212200202202001212212112212200110010110012\\\
0211212200220100112202200220200122102212211211002101220022001001100221\\\
211010010110020022110010110010221211020220200220221001122011211.\\\
\end{array}$
This coloring is the result of about 8 trillion iterations of RamseyScript,
using the Western Canada Research Grid111http://www.westgrid.ca. We started
with a seed 3-coloring of the interval $[1,61]$ and searched the entire space
of extensions. Figure 1 gives the number of double 3-AP free extensions of the
seed coloring versus their lengths.
Figure 1: Number of double 3-AP free extensions versus length
To get more information about $w^{\ast}(3,3)$ we define $w^{\ast}(3,3;d)$ to
be the smallest $m$ such that whenever $[1,m]$ is 3-coloured so that each
colour class has maximum gap at most $d$, then there is a monochromatic double
3-term arithmetic progression. Our goal was to compute $w^{\ast}(3;3;d)$ for
small values of $d$. (See Table 1.)
| | $w^{\ast}(3,3;d)$
---|---|---
Max gap $d$ | 2 | 11
3 | 22
4 | 39
5 | 100
6 | $>152$
7 | ?
Table 1: Known Values of $w^{\ast}(3,3;d)$
We note that $w^{\ast}(3,3;d)$ is already difficult to compute when $d$ is
much smaller than $w^{\ast}(2,3)=17$. (In a 3-coloring containing no
monochromatic double 3-term arithmetic progression the maximum gap size of any
color class is 17.)
Freedman [Freedman 2013+] showed that there were 16 double 3-AP free 51-term
sequences having the maximum gap of at most $4$. The fact that
$w^{\ast}(3,3;4)=39$ is an interesting contrast, and shows that considering a
single sequence instead of partitioning an interval of positive integers into
three sequences is somewhat less restrictive.
###### Theorem 3.
$w^{\ast}(2,4)\geq 30830$.
Starting with the seed 2-coloring
$[1,10]=\\{1,4,6,7\\}\cup\\{2,3,5,8,9,10\\}$, after $2\cdot 10^{8}$ iterations
RamseyScript produced a double 4-AP free 2-coloring of the interval
$[1,30829]$ that is available at the web page people.math.sfu.ca/
vjungic/Double/w-4-2.txt.
## 3 $w^{\ast}(3;a,b,c)$ and $w^{\ast}(k;a,b)$
Recall that $w^{\ast}(3;a,b,c)$ is the least number such that every 3-coloring
of$[1,w^{\ast}(3;a,b,c)]$, with gap sizes on the three colors restricted to
$a$, $b$, and $c$, respectively, has a monochromatic double 3-term arithmetic
progression. Similarly, $w^{\ast}(k;a,b)$ is the least number such that every
2-coloring of $[1,w^{\ast}(k;a,b)]$, with gap sizes on the two colors
restricted to $a$ and $b$, respectively, has a monochromatic double $k$-term
arithmetic progression.
Table 2 shows values of $w^{\ast}(3;a,b,c)$ for some small values of $a$, $b$,
and $c$. Table 3 shows values of $w^{\ast}(k;a,b)$ for some small values of
$a$, $b$, and $k$.
| | Max Green Gaps
---|---|---
| | 3 | 4 | 5 | 6 | 7+
Max Blue Gaps | 3 | 22 | | | |
4 | 31 | 31 | | |
5 | 33 | 38 | 43 | |
6 | 33 | 41 | 44 | 45 |
7 | 33 | 41 | 46 | 46 | 46
8+ | 33 | 41 | 46 | 46 | 47
Max Red Gap 3
| | Max Green
---|---|---
| | 5 | 6 | 7 | 8+
Max Blue | 5 | 100 | | |
6 | $>113$ | $>133$ | |
7 | ? | ? | ? |
8+ | ? | ? | ? | ?
Max Red Gap 5
| | Max Green Gaps
---|---|---
| | 4 | 5 | 6 | 7 | 8 | 9+
Max Blue Gaps | 4 | 39 | | | | |
5 | 49 | 63 | | | |
6 | 56 | 79 | 91 | | |
7 | 76 | 96 | $>$105 | $>$121 | |
8 | 81 | 96 | $>$114 | $>$131 | $>$131 |
9 | 81 | 96 | $>$114 | $>$133 | $>$133 | $>$133
10 | 81 | 96 | $>$114 | $>$133 | $>$135 | $>$135
11+ | 81 | 97 | $>$114 | $>$133 | $>$135 | $>$135
Max Red Gap 4
Table 2: Known Values and Bounds for $w^{\ast}(3;a,b,c)$
| | Red
---|---|---
| | 2 | 3
Blue | 2 | 7 |
3 | 11 | 17
Double 3-AP’s
| | Red
---|---|---
| | 2 | 3 | 4+
Blue | 2 | 11 | |
3 | 22 | $>176$ |
4+ | 22 | $>2690$ | $>3573$
Double 4-AP’s
| | Red
---|---|---
| | 2 | 3 | 4 | 5+
Blue | 2 | 15 | | |
3 | 37 | $>131000$ | |
4 | $>25503$ | ? | ? |
| 5+ | $>33366$ | ? | ? | ?
Double 5-AP’s
Table 3: Known Values and Bounds for $w^{\ast}(k;a,b)$
Based on this evidence, we propose the following conjecture.
###### Conjecture 2.
The number $w^{*}(3,3)$ exists. The number $w^{*}(2,4)$ does not exist.
Our guess would be that $w^{*}(3,3)<500$. Also we recall that $w^{*}(2,3)=17$
and $w^{\ast}(2,4)\geq 30830$.
## 4 Double 3-term Arithmetic Progressions in Increasing Sequences of
Positive Integers
In this section, we return to Problem 1: the existence of double 3-term
arithmetic progressions in infinite sequences of positive integers with
bounded gaps.
We remind the reader of the meaning of the following terms from combinatorics
of words.
An infinite word on a finite subset $S$ of $\mathbb{Z}$, called the alphabet,
is defined as a map $\omega:\mathbb{N}\to S$ and is usually written as
$\omega=x_{1}x_{2}\cdots,$ with $x_{i}\in S$, $i\in\mathbb{N}$. For
$n\in\mathbb{N}$, a factor $B$ of the infinite word $\omega$ of length $n=|B|$
is the image of a set of $n$ consecutive positive integers by $\omega$,
$B=\omega(\\{i,i+1,\cdots,i+n-1\\})=x_{i}x_{i+1}\cdots x_{i+n-1}$. The sum of
the factor $B$ is $\sum B=x_{i}+x_{i+1}+\cdots+x_{i+n-1}$. A factor
$B=\omega(\\{1,2,\cdots,n\\})=x_{1}x_{2}\cdots x_{n}$ is called a prefix of
$\omega$.
###### Theorem 4.
The following statements are equivalent:
* (1)
For all $k>1$, every infinite word on $\\{1,2,\cdots,k\\}$ has two adjacent
factors with equal length and equal sum.
* (1a)
For all $k>1$, there exists $R=R(k)$ such that every word on
$\\{1,2,\cdots,k\\}$ of length $R$ has two adjacent factors with equal length
and equal sum.
* (2)
For all $n>1$, if $x_{1}<x_{2}<x_{3}<\cdots$ is an infinite sequence of
positive integers such that $x_{i+1}-x_{i}\leq n$ for all $i>1$, then there
exist $1\leq i<j<k$ such that $x_{i}+x_{k}=2x_{j}$ and $i+k=2j$.
* (2a)
For all $n>1$, there exists $S=S(n)$ such that if
$x_{1}<x_{2}<x_{3}<\cdots<x_{S}$ are positive integers with $x_{i+1}-x_{i}\leq
n$ whenever $1\leq i\leq S-1$, then there exist $1\leq i<j<k\leq S$ such that
$x_{i}+x_{k}=2x_{j}$ and $i+k=2j$.
* (3)
For all $t>1$, if $\mathbb{N}=A_{1}\cup A_{2}\cup\cdots\cup A_{t}$, then there
exists $q$, $1\leq q\leq t$, such that if $A_{q}=\\{x_{1}<x_{2}<\cdots\\}$,
there are $1\leq i<j<k$ such that $x_{i}+x_{k}=2x_{j}$ and $i+k=2j$.
* (3a)
For all $t>1$, there exists $T=T(t)$ such that for all $a>1$, if
$\\{a,a+1,\cdots,a+T-1\\}=A_{1}\cup A_{2}\cup\cdots\cup A_{t}$, then there
exists $q$, $1\leq q\leq t$, such that if
$A_{q}=\\{x_{1}<x_{2}<\cdots<x_{p}\\}$, there are $1\leq i<j<k$ such that
$x_{1}+x_{k}=2x_{j}$ and $i+k=2j$.
###### Remark 1.
Note that in (3) and (3a) the statements concern coverings (by not necessarily
disjoint sets) and not partitions (colorings). This turns out to be essential,
since if we used colorings in (3) and (3a) (call these new statements (3’) and
(3a’)), then (3’) would not imply (2), although (2) would still imply (3a’).
This can be seen from the proofs below.
###### Remark 2.
In each case $i=1,2,3$, the statement (ia) is the finite form of the statement
(i).
###### Proof.
We start by proving that (2) implies (2a). (The proof that (1) implies (1a)
follows the same form, and is a little more routine.)
Suppose that (2a) is false. Then there exists $n$ such that for all $S>1$
there are $x_{1}<x_{2}<x_{3}<\cdots<x_{S}$, with $x_{i+1}-x_{i}\leq n$
whenever $1\leq i\leq S-1$, such that there do not exist $1\leq i<j<k\leq S$
such that $x_{i}+x_{k}=2x_{j}$ and $i+k=2j$. Replace
$x_{1}<x_{2}<x_{3}<\cdots<x_{S}$ by its characteristic binary word (of length
$x_{S}$)
$B_{S}=b_{1}b_{2}b_{3}\cdots b_{x_{S}}$
defined by $b_{i}=1$ if $i$ is in $\\{x_{1},x_{2},x_{3},\ldots,x_{S}\\}$, and
$b_{i}=0$ otherwise. Let $H$ be the (infinite) collection of binary words
obtained in this way. Note that if $B_{S}$ is in $H$, then consecutive 1s in
$B_{S}$ are separated by at most $n-1$ 0s.
Now construct, inductively, an infinite binary word $w$ such that each prefix
of $w$ is a prefix of infinitely many words $B_{S}$ in $H$ in the following
way. Let $w_{1}$ be a prefix of an infinite set $H_{1}$ of words in $H$. Let
$w_{1}w_{2}$ be a prefix of an infinite set $H_{2}$ of words in $H_{1}$. And
so on. Set $w=w_{1}w_{2}\cdots$ .
Define $x_{1}<x_{2}<x_{3}<\cdots$ so that $w$ is the characteristic word of
$x_{1}<x_{2}<x_{3}<\cdots$ and note that $x_{i+1}-x_{i}\leq n$ for all $i>1$.
Now it follows that there cannot exist $1\leq i<j<k$ with $x_{1}+x_{k}=2x_{j}$
and $i+k=2j$. (For these $i,j,k$ would occur inside some prefix of $w$. But
that prefix is itself a prefix of some word $B_{S}=b_{1}b_{2}b_{3}\cdots
b_{S}$, where there do not exist such $i,j,k$.) Thus if (2a) is false, (2) is
false.
Next we prove that (3) implies (3a). Suppose that (3a) is false. Then there
exists $t$ such that for all $T$ there is, without loss of generality, a
covering $\\{1,2,\ldots,T\\}=A_{1}\cup A_{2}\cup\cdots\cup A_{t}$ such that
there does not exist $q$ with $A_{q}=\\{x_{1}<x_{2}<\cdots<x_{p}\\}$ and
$i<j<k$ with $x_{1}+x_{k}=2x_{j}$ and $i+k=2j$. Represent the cover
$\\{1,2,\ldots,T\\}=A_{1}\cup A_{2}\cup\cdots\cup A_{t}$ by a word
$B_{T}=b_{1}b_{2}b_{3}\cdots b_{T}$ on the alphabet consisting of the non-
empty subsets of $\\{1,2,\ldots,t\\}$. Here for each $i$, $1\leq i\leq T$,
$b_{i}=\\{\mbox{the set of }p,1\leq p\leq t,\mbox{ such that }i\mbox{ is in
}A_{p}\\}$. Let $H$ be the set of all words $B_{T}$ obtained in this way.
Construct an infinite word $w=w_{1}w_{2}w_{3}\dots$ on the alphabet consisting
of the non-empty subsets of $\\{1,2,\ldots,t\\}$, such that each prefix of $w$
is a prefix of infinitely many of the words $B_{T}$ in $H$. Thus $w$
represents a cover $\mathbb{N}=A_{1}\cup A_{2}\cup\cdots\cup A_{t}$, where
$A_{i}=\\{j\geq 1\mbox{ such that $i$ is in }w_{j}\\}$, $1\leq i\leq t$, for
which there does not exist $i$, $A_{i}=\\{x_{1}<x_{2}<\cdots\\}$, with $1\leq
i<j<k$ such that $x_{1}+x_{k}=2x_{j}$ and $i+k=2j$, contradicting (3).
It is not difficult to show that (1) is equivalent to (2), that (1) is
equivalent to (1a), that (2a) implies (2), and that (3a) implies (3). We have
shown that (2) implies (2a) and that (3) implies (3a).
The final steps are:
Proof that (3) implies (2). If $n$ and $A_{0}=\\{x_{1}<x_{2}<x_{3}<\cdots\\}$
are given, with $x_{i+1}-x_{i}\leq n$ for all $i>1$, let $A_{i}=A_{0}+i$,
$0\leq i\leq n-1$. Then $\mathbb{N}=A_{0}\cup A_{1}\cup\cdots\cup A_{n-1}$,
and now (3) implies (2).
Proof that (2) implies (3) and (3a). Assume (2), and use induction on $t$ to
show that (3) and (3a) are true for $t$. (Note that (3) for a given value of
$t$ is equivalent to (3a) for the same value of $t$.) For $t=1$ this is
trivial. Fix $t\geq 1,$ assume (3) and (3a) for this $t$, and let
$\mathbb{N}=A_{1}\cup A_{2}\cup\cdots\cup A_{t+1}$. If $A_{t+1}$ is finite we
are done by the induction hypothesis on (3). If $A_{t+1}$ has bounded gaps, we
are done by (2). In the remaining case, there are arbitrarily long intervals
which are subsets of $A_{1}\cup A_{2}\cup\cdots\cup A_{t}$, and we are done by
the induction hypothesis on (3a).
∎
###### Remark 3.
If true, perhaps (3a) can be proved by a method such as van der Waerden’s
proof that any finite coloring of $\mathbb{N}$ has a monochromatic 3-AP.
Here is another remark on double 3-term arithmetic progressions.
###### Theorem 5.
The following two statements are equivalent:
* (1)
For all $n\geq 1$, every infinite sequence of positive integers
$x_{1}<x_{2}<\cdots$ such that $x_{i+1}-x_{i}\leq n$ contains a double 3-term
arithmetic progression.
* (2)
For all $n\geq 1$, every infinite sequence of positive integers
$x_{1}<x_{2}<\cdots$ such that $x_{i+1}-x_{i}\leq n$ contains a double 3-term
arithmetic progression $x_{i},x_{j},x_{k}$ with the property that $j-i=k-j\geq
m$ for any fixed $m\in\mathbb{N}$.
###### Proof.
Certainly (2) implies (1). We prove that (1) implies (2).
Let $n$ and $m$ be given positive integers. Let $X=\\{x_{1}<x_{2}<\cdots\\}$
be an infinite sequence with gaps from $\\{1,\ldots,n\\}$. For
$j\in\mathbb{N}$ we define $y_{j}=x_{jm+1}-x_{(j-1)m+1}$. Note that $m\leq
y_{j}\leq nm$. Next we define an increasing sequence
$Z=\\{z_{1}~{}<~{}z_{2}~{}<~{}\cdots\\}$ with gaps from
$\\{m,m+1,\ldots,nm\\}$ by
$z_{i}=\sum_{j=1}^{i}y_{j}=\sum_{j=1}^{i}x_{jm+1}-\sum_{j=0}^{i-1}x_{jm+1}.$
By (1) the sequence $Z$ contains a double 3-term arithmetic progression
$z_{p},z_{q},z_{r}$ with
$z_{r}-z_{q}=z_{q}-z_{p}\mbox{ and }p+r=2q.$
It follows that
$\sum_{j=q+1}^{r}x_{jm+1}-\sum_{j=q}^{r-1}x_{jm+1}=\sum_{j=p+1}^{q}x_{jm+1}-\sum_{j=p}^{q-1}x_{jm+1}$
and
$x_{rm+1}-x_{qm+1}=x_{qm+1}-x_{pm+1}.$
From
$(pm+1)+(rm+1)=m(p+r)+2=2mq+2=2(mq+1)$
we conclude that $x_{pm+1},x_{qm+1},x_{rm+1}$ form a double 3-term arithmetic
progression with
$rm+1-(qm+1)=(r-q)m\geq m.$
Since $m$ and $X$ are arbitrary, we conclude that (2) holds.
∎
We wonder if one could get some intuitive “evidence” that it is easier to show
that $w^{\ast}(3,3)$ exists than it is to show that every increasing sequence
with gaps from $\\{1,2,3,\ldots,17\\}$ has a double 3-term arithmetic
progression. The “17” is chosen because in a 3-coloring of $[1,m]$ which has
no monochromatic double 3-AP, the gaps between elements of this color class
are colored with 2 colors, and $w^{\ast}(2,3)=17$.
RamseyScript was used for search of an increasing sequence with gaps from
$\\{1,2,3,\ldots,17\\}$ with no double 3-term arithmetic progressions. The
first search produced a sequence of the length 2207. The histogram with the
distribution of gaps in this sequence is given on Figure 2.
Figure 2: Histogram of Gaps in a 2207-term Double 3-AP Free Sequence
In another attempt we changed the order of gaps in the search, taking
$[16,12,11,17,10,14,15,8,5,3,6,4,2,1,13,7,9]$
instead of $[1,2,\cdots,17]$. RamseyScript produced a 5234-term double 3-AP
free sequence. The corresponding histogram of gaps is given on Figure 3.
Figure 3: Histogram of Gaps in a 5234-term Double 3-AP Free Sequence
Here are a few conclusion that one can make from this experiment.
1. 1.
Initial choices of the order of gaps matter very much when constructing a
double 3-AP free sequence, because we cannot backtrack in a reasonable (human)
timespan at these lengths.
2. 2.
We do not really know anything about how long a sequence there will be.
3. 3.
The search space is very big. Table 4 gives the recursion tree size vs.
maximum sequence length considered.
Max. Seq. Length | Size of Search
---|---
0 | 1
1 | 18
2 | 307
3 | 4931
4 | 78915
5 | 1216147
6 | 18695275
7 | 278661995
8 | ????
Table 4: Recursion Tree Size vs. Maximum Sequence Length
## 5 RamseyScript
To handle the volume and variety of computation required by this project and
related ones, we use the utility RamseyScript, developed by the third author,
which provides a high-level scripting language. In creating RamseyScript, we
had two goals:
* -
To provide a uniform framework for Ramsey-type computational problems (which
despite being minor variations of each other, are traditionally handled by _ad
hoc_ academic code).
* -
To provide a correct and efficient means to actually carry out these
computations.
To achieve these goals, RamseyScript appears to the user as a declarative
scripting language which is used to define a backtracking algorithm to be run.
It exposes three main abstractions: search space, filters and targets.
The _search space_ is a set of objects — typically $r$-colorings of the
natural numbers or sequences of positive integers — which can be generated
recursively and checked to satisfy certain conditions, such as being
squarefree or containing no monochromatic progressions.
The conditions to be checked are specified as _filters_. Typically when
extending RamseyScript to handle a new type of problem, only a new filter
needs to be written. This saves development time and effort compared to
writing a new program, while also making available additional features, e.g.
for splitting the problem across a computing cluster.
Finally, _targets_ describe the information that should be shown to the user.
The default target, max-length, informs the user of the largest object in the
search space which passed the filters.
With these parameters set, RamseyScript then runs a standard backtracking
algorithm, which essentially runs as follows:
1. 1.
Start with some element $x$ in the search space. For example, $x$ might be the
trivial coloring of the empty interval.
2. 2.
Check that $x$ passes each filter. If not, skip steps 3 and 4.
3. 3.
Check each target against $x$ (e.g., is $x$ the longest coloring obtained so
far?).
4. 4.
For each possible extension $\hat{x}$ of $x$, repeat step 2. For example, if
$x$ is the interval $[1,n]$ and the search space is the set of $r$-colorings,
then the possible extensions of $x$ are the $r$ colorings of $[1,n+1]$ which
match $x$ on the first $n$ elements.
5. 5.
Output the current state of all targets.
Here is an example script to demonstrate these ideas and syntax:
# Output a brief description
echo Find the longest interval [1, n] that cannot be 4-colored
echo without a monochromatic 3-AP or a rainbow 4-AP.
# Set up environment
set n-colors 4
set ap-length 3
# Choose filters
filter no-n-aps
filter no-rainbow-aps
# Use the default target (max-length)
# Backtrack on the space of 4-colorings
search colorings
Its output is
find the longest interval [1, n] that cannot be 4_colored
without a monochromatic 3_ap or a rainbow 4_ap.
Added filter ‘‘no-3-aps’’.
Added filter ‘‘no-rainbow-aps’’.
#### Starting coloring search ####
Targets: Ψmax-length
Filters: Ψno-rainbow-aps no-3-aps
Dump data: Ψ
Seed:ΨΨ[[] [] [] []]
Max. coloring (len 56): [[removed due to length]]
Time taken: 7s. Iterations: 4546107
#### Done. ####
RamseyScript has many options to control the backtracking algorithm and its
output. For full details see the README, available alongside its source code
at https://www.github.com/apoelstra/RamseyScript. It is licensed under the
Creative Commons 0 public domain dedication license.
Acknowledgement. The authors would like to acknowledge the IRMACS Centre at
Simon Fraser University for its support.
## References
* [Ardal et al. 2012] H. Ardal, T. Brown, V. Jungić, and J. Sahasrabudhe, On additive and abelian complexity in infinite words, Integers, Electron. J. Combin. Number Theory 12 (2012) A21.
* [Au et al. 2011] Yu-Hin Au, Aaron Robertson, and Jeffrey Shallit, Van der Waerden’s theorem and avoidability in words, INTEGERS: Elect. J. Combin. Number Theory 11 #A6 (electronic), 2011.
* [Brown and Freedman 1987] T.C. Brown and A.R. Freedman, Arithmetic progressions in lacunary sets, Rocky Mountain J. Math. 17, Number 3 (1987), 587–596.
* [Cassaigne et al. 2013+] Julien Cassaigne, James D. Currie, Luke Schaeffer, and Jeffrey Shallit, Avoiding three consecutive blocks of the same size and same sum, arXiv:1106.5204.
* [Freedman 2013+] Allen R. Freedman, Sequences on sets of four numbers, to appear in INTEGERS: Elect. J. Combin. Number Theory.
* [Graham et al. 1990] R. Graham, B. Rothschild, and J. H. Spencer, Ramsey Theory (2nd ed.), New York: John Wiley and Sons, 1990.
* [Grytczuk 2008] Jaroslaw Grytczuk, Thue type problems for graphs, points, and numbers, Discrete Math. 308, 4419–4429, 2008.
* [Halbeisen and Hungerb$\ddot{\text{u}}$hler 2000] L. Halbeisen and N. Hungerb$\ddot{\text{u}}$hler, An application of van der Waerden’s theorem in additive number theory, INTEGERS: Elect. J. Combin. Number Theory 0 # A7 (electronic), 2000.
* [Pirillo and Varricchio 1994] G. Pirillo and S. Varricchio, On uniformly repetitive semigroups, Semigroup Forum 49, 125–129, 1994.
* [Richomme et al. 2011] Gwénaël Richomme, Kalle Saari, and Luca Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. Soc. 83(1), 79–95, 2011.
* [van der Waerden 1927] B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15, 212–216, 1927.
|
arxiv-papers
| 2013-04-05T22:02:39 |
2024-09-04T02:49:43.921771
|
{
"license": "Public Domain",
"authors": "Tom Brown, Veselin Jungi\\'c, Andrew Poelstra",
"submitter": "Andrew Poelstra",
"url": "https://arxiv.org/abs/1304.1829"
}
|
1304.1844
|
# The causal meaning of Fisher’s average effect
James J. Lee1∗
Carson C. Chow1
1Laboratory of Biological Modeling
National Institute of Diabetes and Digestive and Kidney Diseases
National Institutes of Health
Bethesda, MD 20892, USA
∗To whom correspondence should be addressed; E-mail: [email protected]
RESEARCH PAPER
RUNNING HEAD: Causal meaning of average effect
Summary
In order to formulate the Fundamental Theorem of Natural Selection, Fisher
defined the _average excess_ and _average effect_ of a gene substitution.
Finding these notions to be somewhat opaque, some authors have recommended
reformulating Fisher’s ideas in terms of covariance and regression, which are
classical concepts of statistics. We argue that Fisher intended his two
averages to express a distinction between correlation and causation. On this
view the average effect is a specific weighted average of the actual
phenotypic changes that result from physically changing the allelic states of
homologous genes. We show that the statistical and causal conceptions of the
average effect, perceived as inconsistent by Falconer, can be reconciled if
certain relationships between the genotype frequencies and non-additive
residuals are conserved. There are certain theory-internal considerations
favoring Fisher’s original formulation in terms of causality; for example, the
frequency-weighted mean of the average effects equaling zero at each locus
becomes a derivable consequence rather than an arbitrary constraint. More
broadly, Fisher’s distinction between correlation and causation is of critical
importance to gene-trait mapping studies and the foundations of evolutionary
biology.
Keywords: quantitative genetics, causality, confounding, selection bias,
natural selection
## 1\. Introduction
Darwin perceived that hereditary variation in fitness leads to an increase in
adaptive complexity. In an attempt to provide a Mendelian and mathematical
formulation of this profound insight, Fisher expounded the Fundamental Theorem
of Natural Selection (FTNS), which in a modern paraphrase states that the
partial increase in population mean fitness ascribable solely to changes in
allele frequencies by natural selection is equal to the additive genetic
variance in fitness [1, 33, 51, 13, 15, 16, 30, 9, 10, 36, 45]. In the
discrete-time formulation of the FTNS, the additive genetic variance is
proportional to this partial increase, as it must be divided by the mean
fitness.
In his exposition of the FTNS, Fisher took some pains to define the concepts
of _average excess_ and _average effect_. In his own words,
> Let us now consider the manner in which any quantitative individual
> measurement, such as human stature, may depend upon the individual genetic
> constitution. We may imagine, in respect of any pair of alternative
> [alleles], the population divided into two portions, each comprising one
> homozygous type together with half of the heterozygotes, which must be
> divided equally between the two portions. The difference in average stature
> between these two groups may then be termed the _average excess_ (in
> stature) associated with the gene substitution in question…. [21, p. 30,
> emphasis added]
In contrast,
> [b]y whatever rules …the frequency of different gene combinations, may be
> governed, the substitution of a small proportion of the genes of one
> [allelic] kind by the genes of another will produce a definite proportional
> effect upon the average stature. The amount of the difference produced, on
> the average, in the total stature of the population, for each such gene
> substitution, may be termed the _average effect_ of such substitution, in
> contra-distinction to the average excess as defined above…. [21, p. 31,
> emphasis added]
>
> It is natural to conceive [of the average effect] as the actual increase in
> the total of the measurements of a population, when without change in the
> environment, or the mating system, the gene substitution is _experimentally_
> brought about, as it might be by mutation. [24, p. 373, emphasis added]
This paper addresses a puzzle raised by [18] in his brilliant explication of
Fisher’s two genetic averages. Falconer assumed that what Fisher meant by the
quoted definition of the average effect was as follows. We randomly sample a
zygote immediately after fertilization but before the onset of any
developmental events. If the zygote’s genotype contains a gene of a certain
allelic type, say $\mathcal{A}_{1}$, we change it to $\mathcal{A}_{2}$. This
experimental intervention may lead to a value of the focal phenotype at the
time of measurement that differs from what it would have been if the
intervention had not been performed. Falconer reasoned that the expected
magnitude of this difference corresponds to Fisher’s verbal definition of the
average effect.
Falconer then showed that [24]’s (fisher:1941) now widely accepted
mathematical definition of the average effect—the partial regression
coefficient of gene count in the linear regression of the phenotype on all
loci in the genome—does not generally coincide with the definition in terms of
experimental gene substitutions performed at random. Falconer expressed
surprise at the apparent invalidity of the latter definition, given that
“Fisher uses the imaginary replacement of one allele by another as a verbal
description to introduce the idea of average effect, and it seems to have been
seen by him as the basis for the concept” (p. 334).
Falconer correctly perceived the importance of experimental intervention to
Fisher’s conception of the average effect. Indeed, Fisher did not even bother
to spell out his regression definition in the first edition of _The Genetical
Theory of Natural Selection_. Furthermore, to any reader familiar with
Fisher’s work on experimental design and his controversial stance on the
tobacco-cancer connection, the quotations given above must bring into mind his
repeated admonition that an observed _excess_ in the average measurement of
one group over another can always be interpreted as the causal _effect_ of the
factor distinguishing the groups under the following circumstance: the
allotment of members to groups has been randomized in a controlled experiment
[23, 27]. This preoccupation with causation is one of the stark contrasts
between Fisher and his nemesis Karl Pearson; contrary to the intellectual
fashion of the Edwardian era, Fisher did not regard causality as a meaningless
concept. In the inaugural issue of the journal _Philosophy of Science_ , the
word _cause_ and its derivatives appear in [22] no fewer than seventy times.
Over much resistance by seasoned experimenters [3], Fisher advocated
randomization in experimental design for the precise purpose of distinguishing
causation from spurious correlations brought about by confounding variables.
There is thus compelling reason to believe that the notion of experimental
control revealing causation is critical to the proper interpretation of the
average effect.
We argue that a more nuanced reading of Fisher’s writings can bring his
experimental and regression definitions of the average effect into full
agreement in certain special cases. We then provide reasons to favor the
experimental definition in more general situations. A striking disadvantage of
the regression definition is that its use invalidates the FTNS if some of the
variance in fitness has environmental causes.
For simplicity our main text mostly follows [18] in treating the case of a
single locus with two alleles. We provide the generalization to multiple
alleles and loci in two of the later sections. Some interesting new concepts
do arise in this generalization, but the central ideas can be conveyed without
multilocus notation, which seems inevitably to be either cumbersome or opaque.
## 2\. A Notation for Causal Notions
A formal symbolic language to distinguish causal relations from merely
correlational ones, such as the counterfactual notation of [42] and [52], was
not to our knowledge ever adopted by Fisher. This is despite the fact that he
frequently wrote about this distinction. Although such formalisms lack the
elegance of Fisher’s prose, adopting the appropriate formalism is an aid to
understanding.
For this purpose we adopt the $do$ operator of [46]. We are to interpret an
expression such as $\mathbb{E}[Y\,|\,do(x)]$ to mean the expectation of $Y$
given that the random variable $X$ has been _experimentally fixed_ to the
value $x$. The contrast between conditional quantities containing the $do$
symbol and traditional conditional quantities is evident in the expressions
$\mathbb{P}(mud\,|\,rain)\geq\mathbb{P}(mud)\quad\textrm{and}\quad\mathbb{P}(rain\,|\,mud)\geq\mathbb{P}(rain)$
(1)
and
$\mathbb{P}[mud\,|\,do(rain)]\geq\mathbb{P}(mud)\quad\textrm{and}\quad\mathbb{P}[rain\,|\,do(mud)]=\mathbb{P}(rain).$
(2)
(1) indicates that we are more likely to find mud if we have already observed
rain. Because co-occurrence is symmetric, it also becomes more likely that it
has rained if we have already observed mud. On the other hand, (2) symbolizes
the much stronger and asymmetrical assertion that rain causes mud and not
_vice versa_ ; muddying up the backyard with a garden hose will not make it
rain.
This notation and its associated machinery may be of some benefit in the
burgeoning field of genome-wide association studies (GWAS), where it is
important to single out genetic variants with a causal effect on a given
phenotype from markers that are merely associated with the phenotype for other
reasons, including linkage disequilibrium (LD) with a nearby causal variant
[57]. Letting $Y$ denote the phenotype of interest, we can say that a genetic
variant is a causal variant if the equality
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]=\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$
(3)
does not hold. The expectation is taken over the space of all possible
multilocus genotypes and environments. Note that the equality does in fact
hold for a non-causal marker locus in LD with a causal locus. If we could
experimentally mutate a randomly chosen zygote’s genotype at a biologically
inert marker locus immediately before the onset of development, we would not
expect any ensuing change in the phenotype.
The $do$ notation is more than a convenient means of fixing ideas. The
treatise of [47] grounds this symbol in a rich syntax and semantics. From one
point of view, the work of Pearl can be regarded as a vast generalization of
[58]’s (wright:1968) path analysis.
For simplicity we will speak of events in the life cycle such as
fertilization, development, and phenotypic measurement as if all individuals
experienced each such event at the same time—a convention that is appropriate
for an organism with a life cycle consisting of discrete and non-overlapping
generations. We can then speak of selecting one zygote for an experimental
treatment from all those zygotes making up the current generation. Our
discussion also applies, however, to organisms with a life cycle consisting of
continuous and overlapping generations. In this case a quantity such as
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$ is to be interpreted as
the present phenotypic value that a randomly selected organism would have been
expected to obtain if its genotype could have been converted to
$\mathcal{A}_{1}\mathcal{A}_{1}$ immediately after its own fertilization.
Fisher’s own writings suggest the importance of counterfactual thinking. In a
summary of his work on the correlations between relatives, he wrote: “[I]t
should be clearly understood what we mean by a _cause of variability_. If we
say, ‘This boy has grown tall because he has been well fed,’ we are not merely
tracing out cause and effect in an individual instance; we are suggesting that
he might quite probably have been worse fed, and that in this case he would
have been shorter” [20, p. 214, emphasis in original]. The $do$ operator bears
both interventional and counterfactual interpretations. If necessary, each
organism can be weighted by reproductive value.
## 3\. Falconer’s Interpretation of the Experimental Average Effect
We can use the $do$ operator to symbolize the gene substitutions in Fisher’s
thought experiment. Here we use it to review Falconer’s understanding of this
experiment for a single biallelic locus. We first note that if genotypic and
environmental causes of phenotypic variation act additively and independently,
then quantities such as $\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})$ are
precisely equal to $\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$ at
the single causal locus. Until we say otherwise, we assume the stochastic
independence of genotypes and environments.
Following the notation of [19], we let $P$, $2Q$, and $R$ denote the
respective frequencies of the genotypes $\mathcal{A}_{1}\mathcal{A}_{1}$,
$\mathcal{A}_{1}\mathcal{A}_{2}$, and $\mathcal{A}_{2}\mathcal{A}_{2}$. Given
that a zygote’s genotype is $\mathcal{A}_{1}\mathcal{A}_{1}$, we write the
expected phenotypic effect of changing a gene’s allelic type from
$\mathcal{A}_{1}$ to $\mathcal{A}_{2}$ as
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2}),\mathcal{A}_{1}\mathcal{A}_{1}]-\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}).$
(4)
There is no contradiction in conditioning on both the observation of
$\mathcal{A}_{1}\mathcal{A}_{1}$ and the experimental setting of the genotype
to $\mathcal{A}_{1}\mathcal{A}_{2}$. This simply means that instead of
performing the experiment on a zygote sampled at random from the entire
population, we perform it specifically on a zygote that would otherwise have
borne the genotype $\mathcal{A}_{1}\mathcal{A}_{1}$. Similarly, we define
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2}),\mathcal{A}_{1}\mathcal{A}_{2}]-\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}).$
(5)
The problem with identifying _the_ effect of a gene substitution—as in
identifying the effect of an alteration to any nonlinear causal system—is that
the expected change depends on the context. In other words (4) and (5) are not
equal in general. Falconer supposed that Fisher arrived at the “average
effect” of substituting $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$ by averaging
(4) and (5) in the following way. We sample a zygote at random and then select
one of its genes at random. If the chosen gene is of allelic type
$\mathcal{A}_{2}$, we leave it alone. If the chosen gene is of type
$\mathcal{A}_{1}$, we change it to $\mathcal{A}_{2}$. The expected phenotypic
effect of the gene substitutions performed under this scheme is thus
$\frac{P(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2})+Q(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2})}{P+Q}.$
(6)
Falconer pointed out that (6) does not agree with the regression definition of
the average effect that [24] gave in an article criticizing Sewall Wright for
conflating the average excess and average effect. This article required
explicit expressions for the two genetic averages in traditional notation, and
Fisher obtained an expression for the average effect adequate for
demonstrating its distinctness from the average excess by minimizing the sum
of squares
$P[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})-\nu+\alpha]^{2}+2Q[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})-\nu]^{2}+R[\mathbb{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})-\nu-\alpha]^{2},$
(7)
where $\nu$ is the regression constant. Using a notation that generalizes to a
locus with more than two alleles, we can express this sum of squares
equivalently as
$P[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})-\mu-2\alpha_{1}]^{2}+2Q[\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})-\mu-\alpha_{1}-\alpha_{2}]^{2}\\\
+R[\mathbb{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})-\mu-2\alpha_{2}]^{2}\quad\textrm{where
$\mu=\mathbb{E}(Y)$}.$ (8)
In the definition (7), then, the average effect $\alpha$ is the slope in the
regression of the phenotype on gene count. $\alpha_{1}$ and $\alpha_{2}$ in
(8) are the average effects of the two alleles individually—a notion to which
we will return. For now we simply note that $\alpha$ will turn out to equal
$\alpha_{2}-\alpha_{1}$ in magnitude. There is some ambiguity in the
literature over whether the outcome variable in the regression should be
defined, as in (8), with the subtraction of the unconditional phenotypic mean
[28, 51, 16]. However, this choice simply adds a constant term to the average
effects of the individual alleles, and this term disappears in the biallelic
average effect $\alpha=\alpha_{2}-\alpha_{1}$. In our later discussion of
individual average effects, we will give a compelling reason to favor the mean
subtraction.
Perhaps frustrated by Fisher’s concise style, Falconer concluded his article
by approvingly quoting [51]’s (price:1972) remark that Fisher’s ideas can be
translated into well-understood concepts such as covariance and regression
without dealing with his “special” notions of average excess and average
effect.
In the following we show that the two definitions of the average effect can be
reconciled, in the case of genotype-environment independence, for a specific
weighting of the two possible substitutions. However, if such independence
fails to hold, it is not possible to dispense with Fisher’s “special”
definition in terms of experimental gene substitutions.
## 4\. Fisher’s Experimental Average Effect
Fisher conditioned the gene substitutions in his hypothetical experiment on
the “rules” by which “the frequency of different gene combinations may be
governed.” It is this difficult subtlety that Falconer did not take into
account. In _The Genetical Theory_ Fisher’s wording seems to imply that it is
only the mating scheme that determines how different alleles combine to form
whole-genome genotypes. Later he acknowledged that other factors also
influence the departure of genotype frequencies from random combination of
genes, explicitly mentioning “the partial isolation of sections of the
population” [24, p. 54]. The implication for the experimental gene
substitutions is that they must be carried out in a manner that does not
disturb the arrangement of alleles into genotypes called for by the
population’s rules of formation.
The three genotype frequencies sum to unity, as do the frequencies of the two
alleles. Thus, given the frequency of one allele, one more parameter is
required to specify the genotype frequencies. There appears to be complete
freedom in the choice of this parameter. For example, one possibility is
Wright’s inbreeding coefficient $F$ [7]. As we later show, if we require the
experimental average effect to coincide with the regression average effect in
the case of genotype-environment independence, then we must choose the
parameter to be $\lambda=Q^{2}/(PR)$, the ratio of the squared (ordered)
heterozygote frequency to the product of the homozygote frequencies. $\lambda$
can be written in the symmetrical form
$\frac{\mathbb{P}(\mathcal{A}_{2}\,|\,\mathcal{A}_{1})}{\mathbb{P}(\mathcal{A}_{1}\,|\,\mathcal{A}_{1})}\cdot\frac{\mathbb{P}(\mathcal{A}_{1}\,|\,\mathcal{A}_{2})}{\mathbb{P}(\mathcal{A}_{2}\,|\,\mathcal{A}_{2})},$
and it attains the constant value of unity if the population mates randomly, a
fact first noted by [31].
Let $p=Q+R$ denote the frequency of $\mathcal{A}_{2}$, and write the
population mean of $Y$ as a function of allele frequency and the rules of
combination, $\mu(p,\lambda)$. We now show that the expression
$\mu(p+dp,\lambda)-\mu(p,\lambda)$ is proportional to the average effect,
$\alpha$, obtained from regression equation (7). In other words the ratio
$\lambda$ must be kept constant under this manipulation, _whatever_ the
population’s rules of formation have determined this ratio to be, in order for
the experimental gene substitutions to yield what Fisher intended by the
average effect.
The population mean is given by the expression
$\mu=P\,\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]+2Q\,\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]+R\,\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})].$
(9)
The average effect is then proportional to the change of $\mu$ with respect to
$p$ while holding $\lambda$ constant. We can increase $p$ by carrying out
either the intervention $\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$ or $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$. As detailed in the Appendix,
upon noting that the differential of $Q^{2}=\lambda PR$ for constant $\lambda$
yields the differential equation
$\frac{dP}{P}+\frac{dR}{R}=\frac{2dQ}{Q},$ (10)
we find that Fisher’s average effect is
$\alpha=\frac{c_{1}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2})+c_{2}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2})}{c_{1}+c_{2}},$
(11)
where the weights are
$\displaystyle c_{1}$ $\displaystyle=P(Q+R),$ $\displaystyle c_{2}$
$\displaystyle=R(P+Q).$
Let us recapitulate the meaning of (11). Immediately after fertilization we
take a random sample of the zygotes bearing the genotype
$\mathcal{A}_{1}\mathcal{A}_{1}$. We then randomly assign some of these
zygotes to the “treatment,” which consists of changing the allelic type of a
gene from $\mathcal{A}_{1}$ to $\mathcal{A}_{2}$. The expected difference in
phenotype between treatments and controls at the time of measurement is the
causal effect of the gene substitution. We perform the analogous experiment to
determine the causal effect of changing $\mathcal{A}_{1}\mathcal{A}_{2}$ to
$\mathcal{A}_{2}\mathcal{A}_{2}$. The weighted average of the two causal
effects—where the weights $c_{1}$ and $c_{2}$ are chosen so as to preserve
$\lambda$ if the two types of gene substitutions are applied to the population
in the ratio $c_{1}/c_{2}$—is the average effect of gene substitution holding
constant the rules governing the frequencies of the different genotypes.
Now that the average effect has been defined in (11), we can apply it to an
example of a population changing in mean phenotypic value under a sequence of
gene substitutions (Table 1). This example may be seen as a numerical
counterpart to the diagrammatic illustration by [10]. Suppose that the effect
of changing an $\mathcal{A}_{1}\mathcal{A}_{1}$ individual to
$\mathcal{A}_{1}\mathcal{A}_{2}$ is 3 phenotypic units, whereas the effect of
changing $\mathcal{A}_{1}\mathcal{A}_{2}$ to $\mathcal{A}_{2}\mathcal{A}_{2}$
is $-2$. Suppose also that the numbers of the genotypes
$\mathcal{A}_{1}\mathcal{A}_{1}$, $\mathcal{A}_{1}\mathcal{A}_{2}$, and
$\mathcal{A}_{2}\mathcal{A}_{2}$ in this population are 40, 40, and 20
respectively. These genotype frequencies imply that $(c_{1},c_{2})$ is
proportional to $(4,3)$. Table 1 shows how the average phenotypic change and
$\lambda$ are affected by each step in a sequence of gene substitutions
leading to an increase in $p$ but tending to keep $\lambda$ constant. The
first column gives the gene substitution. In this sequence the two types of
substitution alternate, but this is not an essential feature. The second
column gives the numbers of the genotypes after the gene substitution. The
third column gives the cumulative change in the total phenotypic measurements
(the mean phenotype times the population size) divided by the number of gene
substitutions. The fourth column gives the new value of $\lambda$ after the
gene substitution.
Table 1: _Sequence of experimental gene substitutions yielding the average effect._ experimental change | genotype numbers | $\frac{\Delta(\mu N)}{\textrm{number of changes}}$ | $\lambda$
---|---|---|---
— | 40, 40, 20 | — | 1/2
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 39, 41, 20 | 3 | .5387821
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 39, 40, 21 | 1/2 | .4884005
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 38, 41, 21 | 4/3 | .5266291
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 38, 40, 22 | 1/2 | .4784689
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 37, 41, 22 | 1 | .5162776
$\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2}$ | 37, 40, 23 | 1/2 | .4700353
$\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$ | 36, 41, 23 | 6/7 | .5075483
It is readily confirmed that the final value of $\lambda$ is the closest to
the starting value of $1/2$ that can be achieved with 7 gene substitutions. If
we take population size to infinity, we can make the discrepancy between the
original and new values of $\lambda$ as small as we please.
In the special case of genotype-environment independence considered so far,
where equalities such as
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
always hold, Fisher’s experimental and regression definitions of the average
effect coincide for constant $\lambda$. In the example above, after assigning
each genotype an expected phenotypic value consistent with the magnitudes of
the experimental effects, it is easily verified that the slope in the least-
squares regression of phenotypic value on $\mathcal{A}_{2}$ gene count is 6/7.
## 5\. Gene-Environment Correlation and Interaction
As a preliminary matter, we note that any variable along a causal path (in the
sense of Wright and Pearl) from genotype to phenotype must not be counted as
environmental. For example, if dairy consumption affects stature, it is
tempting to regard dairy consumption as an environmental (non-genotypic)
variable with respect to stature. But if genetic variation affects lactose
tolerance and thus the amount of milk consumed, assigning the effect of dairy
consumption on stature to the environment ignores the fact that the path
_genotype_ $\rightarrow$ _lactose tolerance_ $\rightarrow$ _dairy consumption_
$\rightarrow$ _stature_ ultimately begins with a genetic variable. This
subtlety may have been among the reasons why Fisher favored “speaking of the
residue as non-genetic, rather than environmental …” [2, p. 260]
It is worth asking whether Fisher intended the average effect to be defined in
the event that genotypic and environmental causes are either dependent or non-
additive. In many places he certainly assumed or argued for independence and
additivity [19, 24, 26, 29], and it has been asserted that Fisher’s
biometrical theory is meaningless if these conditions are not met [56, _e.g._
,].
As [51] has pointed out, Fisher’s exposition in _The Genetical Theory_ leaves
much to be desired. A close reading of this text and Fisher’s other writings,
however, turns up many reasons to suspect that Fisher regarded independence
and additivity as reasonable specifications for certain demonstrations and not
as strictly necessary conditions for the average effect to be defined.
1. 1.
In the discussion of the average effect in _The Genetical Theory_ , Fisher did
not explicitly refer to his other work where he made special assumptions
regarding the environment.
2. 2.
The average effect is a key concept in the FTNS, which Fisher regarded as an
exact and rigorous statement. One would like to believe that Fisher, having
been trained in mathematical physics, would not have compared the FTNS to the
second law of thermodynamics if the FTNS depended on assumptions regarding the
environment that must always be approximations at best.
3. 3.
We can read that “[t]he genetic variance as here defined is only a portion of
the variance determined genotypically, and this will differ from, and usually
be somewhat less than, the total variance to be observed” [21, p. 34]. The
genotypic variance is greater than the total variance only if “good” genotypes
tend to be found in “bad” environments, and thus Fisher was clearly allowing
for the possibility of dependence.
4. 4.
In a letter to J. A. Fraser Roberts, Fisher wrote that
> [t]here is one point in which Hogben and his associates are riding for a
> fall, and that is in making a great song about the possible, but unproved,
> importance of non-linear interactions between hereditary and environmental
> factors…. What they do not see is that we ordinarily count as genetic only
> such part of the genetic effect as may be included in a linear formula and
> that we make a present to the environmentalists of such variation due to the
> combined action of genetic and environmental factors as is not expressible
> in such a formula. [2, p. 260]
These remarks clearly show that Fisher did not regard genotype-environment
interaction as an obstacle to defining the average effect.
Emboldened by this evidence regarding the intended generality of the average
effect, we extend our treatment to encompass gene-environment correlation and
interaction.
We first suppose that genotypic and environmental causes act additively but
are not independent. Additivity means that the experimental effect of a gene
substitution remains the same regardless of the environment in which the
experiment is carried out; varying the environment simply raises or lowers the
expected phenotypic values of all three genotypes by the same amount. For
instance,
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i}=\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{j}$
(12)
for any choice of environments $\mathcal{E}_{i}$ and $\mathcal{E}_{j}$. In
this case all of the discussion in previous sections continues to apply
_except_ for the equivalence of the experimental and regression average
effects. If some genotypes are more frequently found in favorable environments
for phenotypic development, then the regression of phenotypic value on gene
count does not have a simple genetic interpretation.
Non-additivity means that at least one equality of the kind in (12) does not
hold. The precise magnitude of the expected change upon an experimental gene
substitution now depends on some aspect of the environment that the
manipulated zygote will experience between the onset of development and the
time of measurement. This case is problematic because now a quantity such as
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$
is not necessarily equal to $\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{1}\mathcal{A}_{1}$,
since the genotypes $\mathcal{A}_{1}\mathcal{A}_{1}$ and
$\mathcal{A}_{1}\mathcal{A}_{2}$ may tend to be found in different
environments. This difficulty can be overcome by redefining expressions such
as $\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}$
so that each symbolizes a difference between experimental treatments rather
than a difference between a treatment and an unperturbed control group. For
example, (4) would become
$\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2}=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]-\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})].$
Seeking an equivalent generalization that retains the interventional form of
(4) and (5), however, sheds substantially greater light on the problem.
Before taking up the issue of gene-environment interaction, it is helpful to
review Fisher’s motivation for holding $\lambda$ constant as a means to
address gene-gene interaction. In order to formulate the FTNS, Fisher wished
to quantify the causal effect of changing allele frequency while holding the
environment constant. In his view the way in which alleles combine to form
genotypes, as parameterized by $\lambda$, should be regarded as part of the
environment. Although this choice may initially seem eccentric, because
fitness differences among genotypes will typically change both $p$ and
$\lambda$, it becomes reasonable when we realize that $\lambda$ may also
change as a result of extrinsic events such as the formation or dissolution of
geographical hindrances to random mating.
There is an analogy here to Fisher’s analysis of covariance to separate the
direct and indirect effects of a given experimental manipulation on a focal
outcome. For instance, in an experiment to determine whether a given
fertilizer affects the purity of sugar extracted from sugar-beets, the
experimenter may already know that the fertilizer affects the weight of the
beet roots, which in turn affects sugar purity [29, pp. 283–284]. The
experimenter may wish to know whether the fertilizer affects sugar purity
through a direct causal path, _fertilizer_ $\rightarrow$ _sugar purity_ ,
distinct from the indirect path _fertilizer_ $\rightarrow$ _root weight_
$\rightarrow$ _sugar purity_. In certain cases adjustment for root weight by
analysis of covariance yields the target quantity: the amount by which sugar
purity would change upon application of the fertilizer, if root weight could
be experimentally clamped to the value that it would have obtained in the
control condition. Similarly, while gene substitutions that are not
deliberately balanced as in (11) will typically change both $p$ and $\lambda$,
we can still mathematically define an average effect stipulating that
$\lambda$ remains clamped to a constant value. This point of view is similar
to one expressed by [45].
Once we regard any change in how alleles are arranged into genotypes as
environmentally caused, it perhaps becomes obvious that we should regard
certain changes in the allotment of genotypes to environments as such. After
all, a redistribution among environments might lead to changes in the
phenotypic means of the genotypes. Such changes in the genotype-phenotype
mapping, when caused by extrinsic events such as climate change, are readily
classified as environmental in nature. This consideration suggests that the
gene substitutions defining the average effect in the presence of genotype-
environment interaction should be balanced in such a way that the phenotypic
means of the genotypes remain constant.
Since equalities such as
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
do not hold when genotypes and environments are also dependent, there is
ambiguity in what is meant by holding constant the phenotypic means. We first
consider holding constant the _observed_ means. If the environments
interacting with genotypes can be classified discretely, then we can write an
equation like
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=\sum_{i}\mathbb{P}(\mathcal{E}_{i}\,|\,\mathcal{A}_{1}\mathcal{A}_{1})\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})$
(13)
for each genotype. Because genotypes and environments exhaust all possible
causes of phenotypic variation,
$\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})$ is
equivalent to
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1}),do(\mathcal{E}_{i})]$. In
a sense even the expectation operator is unnecessary because $Y$ is a
deterministic function when both genotype and environment are specified.
Constancy of observed means requires constancy of the conditional
probabilities taking the form
$\mathbb{P}(\mathcal{E}_{i}\,|\,\mathcal{A}_{1}\mathcal{A}_{1})$. A candidate
definition for the average effect is then
$2\alpha
dp=\mu[p+dp,\lambda,\mathbb{P}(\mathcal{E}_{1}\,|\,\mathcal{A}_{1}\mathcal{A}_{1}),\ldots,\mathbb{P}(\mathcal{E}_{n}\,|\,\mathcal{A}_{2}\mathcal{A}_{2})]\\\
-\mu[p,\lambda,\mathbb{P}(\mathcal{E}_{1}\,|\,\mathcal{A}_{1}\mathcal{A}_{1}),\ldots,\mathbb{P}(\mathcal{E}_{n}\,|\,\mathcal{A}_{2}\mathcal{A}_{2})].$
The problem with this candidate definition, however, is that it can lead to a
nonzero average effect even if in each environment neither gene substitution
has a causal effect. This is because preserving a genotype’s conditional
probabilities of being found in the various environments may require that some
gene substitutions be accompanied by the placement of the manipulated organism
in a different environment; the resulting change in phenotype may then be
entirely the result of the environmental change.
If we instead consider holding constant the _experimental_ means, then we
obtain
$\displaystyle\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$
$\displaystyle=\sum_{i}\mathbb{P}[\mathcal{E}_{i}\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]\,\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1}),\mathcal{E}_{i}]$
$\displaystyle=\sum_{i}\mathbb{P}(\mathcal{E}_{i})\mathbb{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i}).$
(14)
The left-hand side is the expected phenotypic value upon sampling a zygote at
random and, if its genotype is not $\mathcal{A}_{1}\mathcal{A}_{1}$, making it
so. Since changing the genotype of a zygote cannot affect its environment, we
have
$\mathbb{P}[\mathcal{E}_{i}\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]=\mathbb{P}(\mathcal{E}_{i})$
for each $i$ and thus a justification of the second line. Therefore preserving
the experimental means only requires a constant marginal distribution of
environmental states. Of course, we can always abide by this constraint if we
never foster any manipulated organism in a different environment. This ensures
that a nonzero average effect is indeed an average of genetic effects, at
least one of which would turn out to be nonzero under experimental control.
Hence a natural definition of the average effect in the presence of genotype-
environment interaction is
$\alpha=\frac{\sum_{i}c_{1,i}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1}\rightarrow\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})+c_{2,i}(\Delta
Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2}\rightarrow\mathcal{A}_{2}\mathcal{A}_{2},\mathcal{E}_{i})}{\sum_{i}c_{1,i}+c_{2,i}},$
(15)
where
$\displaystyle c_{1,i}=c_{1}\mathbb{P}(\mathcal{E}_{i}),$ $\displaystyle
c_{2,i}=c_{2}\mathbb{P}(\mathcal{E}_{i}).$
## 6\. Average Effects of Individual Alleles
We will now explain how the experimental average effect of an individual
allele may be defined for a locus with any number of alleles. Since there are
${n\choose 2}$ possible gene substitutions at a locus with $n$ alleles, we can
no longer speak of a single average effect in the case of $n>2$, and thus an
extension of this kind is plainly necessary. In the second edition of _The
Genetical Theory_ , we can read that “[w]ith multiple allelomorphism it is
convenient to define [the average effect of an allele] by the effect of
substituting any chosen gene for a random selection of the genes homologous
with it” [28, p. 35]. This definition can be explicated with respect to a
given allele, say $\mathcal{A}_{1}$, as follows. Immediately after
fertilization but before the onset of any developmental events, we select the
allelic type of a gene to be changed into $\mathcal{A}_{1}$ in such a way that
the probabilities of selection are equal to the allele frequencies. That is,
if the vector of allele frequencies is $(p_{1},\ldots,p_{n})$, then the gene
to be changed is $\mathcal{A}_{1}$ with probability $p_{1}$, $\mathcal{A}_{2}$
with probability $p_{2}$, and so on. If the gene to be changed happens to be
$\mathcal{A}_{1}$ itself, then the $\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}$ change will have no phenotypic consequence. For all changes
other than the null change, the choice of the undisturbed gene in the genotype
is made in such a way that the population’s rules of genotype formation are
preserved. If genotypes and environments are both dependent and interacting,
then the marginal distribution of environmental states must be considered as
in (15). The expected change in the phenotype of the manipulated organism is
then $\alpha_{1}$, the average effect of $\mathcal{A}_{1}$.
From this definition we can derive some important consequences. Let $N_{k}$
stand for the number of $\mathcal{A}_{k}$ genes in the population. The total
number of genes is $\sum_{k=1}^{n}N_{k}=N$. Among the $n$ experiments defining
the individual average effects, choose one to perform with a probability equal
to its corresponding allele frequency. The expected vector of allele
frequencies following the randomly chosen experiment is then
$\sum_{k=1}^{n}\frac{N_{k}}{N}\left\\{\sum_{\ell=1}^{n}\frac{N_{\ell}}{N}\left[\left(\frac{N_{1}}{N},\ldots,\frac{N_{n}}{N}\right)+\frac{1}{N}\mathbf{e}_{k}-\frac{1}{N}\mathbf{e}_{\ell}\right]\right\\},$
(16)
where $\mathbf{e}_{k}$ is the vector of length $n$ with element unity at
position $k$ and zeroes elsewhere. After some algebra we find that the first
element of the expected vector is $N_{1}\left(\sum
N_{k}\right)^{2}/N^{3}=p_{1}$, the second is $N_{2}\left(\sum
N_{k}\right)^{2}/N^{3}=p_{2}$, and so on. The expected outcome of the randomly
chosen experiment is a population with exactly the same allele frequencies,
rules of genotype formation, and phenotypic mean. We have thus proved that the
experimental average effects satisfy
$\sum_{k=1}^{n}p_{k}\alpha_{k}=0.$ (17)
With the generalization of the experimental average effect given in the next
section, (17) holds at any one of arbitrarily many multiallelic loci. In the
case of a single locus, (17) holds for the regression average effects in (8)
[16], and agreement of the regression and experimental average effects thus
requires the mean subtraction in that expression.
Let us apply the definition of the individual average effect to the biallelic
example in Table 1. There are initially 120 $\mathcal{A}_{2}$ genes in this
population of 200 total genes. If we perform the experiment defining
$\alpha_{1}$, then with probability .40 the population gene numbers remain at
$(80,120)$ and with probability .60 the numbers become $(81,119)$. In the
event of a non-null substitution, with probability $4/7$ (given by
$\frac{c_{1}}{c_{1}+c_{2}}$) the change is $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{1}\mathcal{A}_{1}$ and with probability $3/7$
(given by $\frac{c_{2}}{c_{1}+c_{2}}$) it is $\mathcal{A}_{2}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{1}\mathcal{A}_{2}$. The expected outcome of the
experiment is thus a population with gene numbers $(80.6,119.4)$ and, up to
the limits of finite size, the same value of $\lambda$. Using simple
probability calculus, we can calculate that the numerical value of
$\alpha_{1}$ is $-18/35$.
In summary, the experiment defining $\alpha_{1}$ will lead to the null
substitution $\mathcal{A}_{1}$ $\rightarrow$ $\mathcal{A}_{1}$ with
probability $p_{1}$ (in which case the causal effect is zero) and to the
substitution $\mathcal{A}_{2}$ $\rightarrow$ $\mathcal{A}_{1}$ with
probability $p_{2}$ (in which case the effect is equal in magnitude to the
average effect of gene substitution with respect to the entire locus).
Therefore $\alpha_{1}$ must be equal to $(p_{1})(0)+(p_{2})(-\alpha)$, and
from this we can use $p_{1}\alpha_{1}+p_{2}\alpha_{2}=0$ to derive
$\alpha=\alpha_{2}-\alpha_{1}$ algebraically. The meaning of this relation
among the three average effects is as follows. The expected outcome of the
experiment defining $\alpha_{2}$ is a population with gene numbers
$(79.6,120.4)$ and nearly the same value of $\lambda$. Now suppose that we
perform the “opposite” of the experiment defining $\alpha_{1}$, on average
reducing the number of $\mathcal{A}_{1}$ genes rather than increasing them. We
compose this experiment with the one defining $\mathcal{A}_{2}$, which in our
example has a numerical value of $12/35$. The population is thus expected to
proceed through the sequence $(80,120)$ $\rightarrow$ $(79.4,120.6)$
$\rightarrow$ $(79,121)$, preserving $\lambda$ at each step. The final state
is precisely the one expected upon performing the experiment defining
$\alpha$, the average effect of gene substitution for the entire locus. We can
see in what sense the average effect of gene substitution ($6/7$) is equal to
the effect of removing one gene ($18/35$) and then replacing it with another
($12/35$).
## 7\. Average Effects in the Case of Multiple Loci
In the case of a single locus with two alleles, we can just as well define the
average effect of gene substitution as
$\alpha=\frac{1}{2}\frac{\partial\mu(p,\lambda)}{\partial p},$ (18)
where $\mu$ is defined as in (9). From this starting point, we can derive the
equivalence of the regression (7) and experimental (11) definitions in the
case of genotype-environment independence. (18) fills the lacuna in Wright’s
casual use of the expression
$\frac{d\overline{W}}{dp},$
to which [24] strongly objected. The explicit dependence of $\mu$ on
$\lambda$, a measure of departure from random combination of genes, meets the
criticism that “the numerator involves the average of [the phenotype] for a
number of different genotypes …exceeding the number of gene frequencies $p$ on
which their frequencies are taken to depend” (p. 57).
It is interesting that the only genetic condition governing the gene
substitutions defining the average effect for a single biallelic locus is the
constancy of $\lambda$, a parameter that depends on the genotype frequencies
but not the genotypic means. One might have thought that these means,
appearing as they do in (7), must play some role in the weighting of the two
possible gene substitutions. It is then natural to ask whether the
generalization to multiple loci retains the appealing feature that constancy
of appropriately quantified departures from Hardy-Weinberg and linkage
disequilibrium is sufficient—without any additional information regarding the
genotypic means—for an experimental average effect to agree with its
corresponding partial regression coefficient. According to our analysis in the
Appendix, the multilocus average effects do not in fact retain this feature.
That is, we would like to define the multilocus average effect of allele
$i_{k}$ at locus $k$, $\mathcal{A}^{(k)}_{i_{k}}$, as
$\alpha^{(k)}_{i_{k}}=\frac{1}{2}\frac{\partial\mu(\mathbf{p},\bm{\lambda})}{\partial
p^{(k)}_{i_{k}}},$ (19)
where $\mathbf{p}$ is now a vector of allele frequencies at several loci,
$p^{(k)}_{i_{k}}$ being the element corresponding to
$\mathcal{A}^{(k)}_{i_{k}}$, and $\bm{\lambda}$ is a vector of whatever
measures of departure from random combination are preserved under the
appropriately balanced gene substitutions. However, as will be demonstrated,
such a mean-invariant description of the average effects does not seem to
exist.
To set up a weaker definition of the multilocus average effects, we require
some additional definitions and notational conventions. Suppose that there are
$L$ causal loci, in the sense of (3), affecting the focal phenotype. Suppose
also that there are $n_{\ell}$ alleles $\mathcal{A}^{(\ell)}_{i_{\ell}}$
$(i_{\ell}=1,\ldots,n_{\ell})$ at locus $\ell$. We have already stipulated
that $p^{(\ell)}_{i_{\ell}}$ is the frequency of allele
$\mathcal{A}^{(\ell)}_{i_{\ell}}$. Put $i=(i_{1},\ldots,i_{L})$ and denote the
gamete $\mathcal{A}^{(1)}_{i_{1}}\cdots\mathcal{A}^{(L)}_{i_{L}}$ by the
multi-index $i$. In addition, denote the frequency of the ordered multilocus
genotype containing gametes $i$ and $j$ as $P_{ij}$.
Define the _coefficient of departure from random combination_ ,
$\theta_{ij}=\frac{P_{ij}}{\prod_{k}p^{(k)}_{i_{k}}p^{(k)}_{j_{k}}},$ (20)
as the ratio of the (ordered) whole-genome genotype $ij$ to the products of
its constituent allele frequencies. The $\theta_{ij}$ are thus measures of
both Hardy-Weinberg and linkage disequilibrium; they are all equal to unity if
and only if the rules of genotype formation call for the random combination of
all genes. Special cases of this coefficient were introduced by [33], although
[41] has pointed out that some of Kimura’s expressions employing these
coefficients are incorrect. To capture how the experimental gene substitutions
defining the average effects change the departures from random combination,
let
$\mathring{\theta}_{ij}=\frac{\Delta
P_{ij}}{P_{ij}}-\sum_{k}\left(\frac{\Delta
p^{(k)}_{i_{k}}}{p^{(k)}_{i_{k}}}+\frac{\Delta
p^{(k)}_{j_{k}}}{p^{(k)}_{j_{k}}}\right)$ (21)
denote the relative change in $\theta_{ij}$. In the limit of infinitesimal
changes, this is equivalent to the logarithmic derivative of $\theta_{ij}$.
Now the experimenter must ascertain the mean of each whole-genome genotype by
experimental control and then fit the equation
$\mathbb{E}[Y\,|\,do(ij)]=\mu+\alpha_{ij}+\varepsilon_{ij},\quad\textrm{where
$\alpha_{ij}=\alpha_{i}+\alpha_{j}$,
$\alpha_{i}=\sum_{k=1}^{L}\alpha^{(k)}_{i_{k}}$},$ (22)
to the treatment means thus obtained. The $\alpha^{(k)}_{i_{k}}$ are the
average effects of the individual alleles. The residuals $\varepsilon_{ij}$
will reflect both dominance and epistasis, and in the general case it does not
seem profitable to separate the two in the manner that [33] attempted. The
fitting is accomplished by seeking the vector of average effects,
$\bm{\alpha}$, that minimizes the sum of squares
$\sum_{i,j}P_{ij}\varepsilon_{ij}^{2}.$ (23)
Whereas the minimization defines the $\varepsilon_{ij}$ uniquely, the
$\alpha^{(k)}_{i_{k}}$ are so far defined only up to a constant term in the
sense that one constant may be added to the average effects at one locus and
the same constant subtracted from the average effects at another locus without
changing the minimum sum of squares [16]. The experimental average effect of a
given allele, however, is obviously not defined only up to a constant term but
rather must be equal to the precise number determined by the experiment of
replacing a random homologous gene with a gene of the given allelic kind. In
the Appendix we show that performing a non-null substitution in this
experiment, in a manner preserving the rules of genotype formation, amounts to
weighting the possible gene substitutions such that the scalar quantity
$\overline{\varepsilon\,\mathring{\theta}}=\sum_{i,j}P_{ij}\varepsilon_{ij}\mathring{\theta}_{ij}$
(24)
is equal to zero. Another way to phrase this key result is that the vanishing
of $\overline{\varepsilon\,\mathring{\theta}}$ is a necessary and sufficient
condition for the regression and experimental average effects to coincide in
the case of genotype-environment independence. [33] showed that constancy of
$\lambda$ suffices for $\overline{\varepsilon\,\mathring{\theta}}$ to vanish
in the case of a single biallelic locus; it is worth mentioning that even in
this simplest possible case there do not generally exist changes in the
genotype frequencies such that each individual $\mathring{\theta}_{ij}$
vanishes.
Our theoretical experimenter can of course perform all $\sum_{k=1}^{L}n_{k}$
experiments to determine the unique values of the elements in the vector
$\bm{\alpha}$. However, given our demonstration that the mean of the
experimental average effects at any given locus is equal to zero, it suffices
to impose (17) for each locus as a constraint on the minimization of (23). The
proof of (17) is still valid for each of multiple loci because the vanishing
of $\overline{\varepsilon\,\mathring{\theta}}$ along each possible branch of
the random experiment implies that the expected change in phenotypic mean must
be equal to
$2\sum_{i_{k}}^{n_{k}}\mathbb{E}\left(\Delta
p^{(k)}_{i_{k}}\right)\alpha^{(k)}_{i_{k}},$ (25)
and since the expected outcome of the experiment is a population with the same
allele frequencies, (17) is assured.
The vanishing of $\overline{\varepsilon\,\mathring{\theta}}$ preserves the
population’s rules of genotype formation in the following sense. Although the
number of parameters required to describe departure from random combination of
genes increases very rapidly with the number of alleles and loci, (24) implies
it is not necessary for each and every such parameter to stay constant. It is
enough, roughly speaking, for the average change in these parameters to equal
zero. $\overline{\varepsilon\,\mathring{\theta}}$ is similar in form to the
weighted average of the relative changes in the departures from random
combination, those genotypes with large non-additive residuals being weighted
more heavily.
The expression
$\alpha^{(k)}_{i_{k}}=\frac{1}{2}\left(\frac{\partial}{\partial
p^{(k)}_{i_{k}}}\sum_{i,j}P_{ij}\mathbb{E}[Y\,|\,do(ij)]\right)_{\overline{\varepsilon\,\mathring{\theta}}=0}$
(26)
may therefore serve as the definition of the experimental average effect in
the case of multiple loci.
Let us recapitulate the meaning of (26). Our variable of interest is the
population average of the experimentally determined phenotypic means of the
genotypes. If genotypes and environments are dependent, this variable is not
the same as the population mean $\mathbb{E}(Y)$. Partial differentiation with
respect to the frequency of allele $\mathcal{A}^{(k)}_{i_{k}}$ indicates that
we examine how our variable of interest responds to the replacement of a small
number of randomly chosen homologous genes with genes of the given allelic
kind. The constraint on the partial derivative indicates that we consider only
those counterfactual populations that can be reached from the original
population by experimental replacements that result in the vanishing of (24).
The factor of $\frac{1}{2}$ is owed to diploidy.
It may seem from the form of the constrained derivative that this definition
contains an element of circularity, since the $\varepsilon_{ij}$ are defined
relative to the average effects in (22). Any such concern should be dispelled
by the fact that (26) fully encodes our argument from (22) to (25), which
provides an unambiguous sequence of instructions for the theoretical
experimenter to follow. The Appendix provides some numerical examples.
## 8\. Average Effects and Natural Selection
At this point the reader may be questioning the need for defining the average
effect in terms of causality, as might be revealed by experimentally
controlled gene substitutions. Modern texts give only the regression
definition [38, 5], and those who are accustomed to these accounts may resist
the new notation and new way of thinking.
We have already given one strong motivation to adopt the criterion of
sensitivity to experimental manipulation: the need to distinguish a causal
variant from the non-causal markers in LD with it. Another motivation is that
dependence of genotypes and environments is a frequent occurrence. For
instance, a major concern in GWAS is ensuring that discovered associations are
not attributable to population stratification, which is essentially a form of
confounding. A well-known apocryphal example is the “chopstick gene.” A
geneticist performing a GWAS of chopstick skill in a large sample containing
both Europeans and East Asians will undoubtedly find many marker loci failing
to satisfy the equality
$E(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1})=E(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2})=E(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2})$
(27)
even if, unbeknownst to the geneticist, the corresponding equality (3) is
obeyed at all loci linked to the statistically significant markers. This is
because the Europeans and East Asians differ both in allele frequencies at
these loci and in the prevalence of chopstick use; the latter difference
presumably has arisen for reasons having nothing to do with genetics. A
regression of the observed phenotypic values on gene count will nevertheless
lead to a nonzero “average effect” in violation of both Fisher’s verbal
definition and common sense.
GWAS investigators attempt to control confounding by including all other
genotyped markers in the regression. Since the number of genotyped markers
typically exceeds the sample size, techniques such as principal components and
mixed linear modeling are typically employed [49, 59]. The reason for the
frequent effectiveness of these techniques is that genomic background become
an extremely good proxy for the subpopulation to which a given sample member
belongs as the number of loci grows large [11]. However, one can construct
examples where partialing out other loci fails to deal with confounding [39],
and in any case a theoretical definition whose usefulness depends on
contingent quantities such as genome size and genetic diversity is inherently
unattractive.
Perhaps the most conspicuous failure of the regression definition occurs in
the very situation that motivated Fisher to define the average effect. This is
when the phenotype is fitness itself. In this case the regression average
effect will generically fail to be proportional to the partial change in
genetic mean per change in allele frequency _even if_ the genotypic and
environmental causes of fitness variation are additive and initially
independent.
A simple simulation will bear out this perhaps surprising claim. The simulated
organism follows a life cycle consisting of non-overlapping generations. The
population size is 20,000. Fitness is determined by a single locus and the
environment; the frequency of $\mathcal{A}_{2}$ is initially 1/2, and the
population mated at random in the previous generation. The genotypic
fitnesses—the values of
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$,
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]$,
$\mathbb{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$—are .4, .5, and .6
respectively. We determine the phenotypic fitness of each individual in the
following way. Immediately after fertilization but before the onset of
viability selection, an environmental disturbance of .3 in absolute value is
added to each individual’s genotypic fitness. Positive and negative
disturbances are equally probable. This scheme ensures that genotypes and
environments are independent at this time.
Whether an individual withstands viability selection to mate with a random
fellow survivor is determined by a discrete approximation of an exponential
process. We stipulate ten discrete time intervals between fertilization and
reproduction, each of which an individual survives with a probability chosen
so that the probability of surviving all ten intervals is equal to the
individual’s phenotypic fitness. By dividing the time between fertilization
and mating into more intervals, we could more closely approach a true
continuous-time model, where the logarithm of phenotypic fitness would be
similar to the Malthusian parameter. Ten intervals, however, suffice to make
the point at issue.
Table 2: _Evolutionary change across time intervals in a simulated organism._ | $\beta$ | $\Delta p$ | $\Delta A$
---|---|---|---
fertilization | .100 | $9.33\times 10^{-3}$ | $1.87\times 10^{-3}$
time 1 | .091 | $8.07\times 10^{-3}$ | $1.61\times 10^{-3}$
time 2 | .084 | $5.86\times 10^{-3}$ | $1.17\times 10^{-3}$
time 3 | .079 | $6.11\times 10^{-3}$ | $1.22\times 10^{-3}$
time 4 | .073 | $5.08\times 10^{-3}$ | $1.02\times 10^{-3}$
time 5 | .069 | $4.72\times 10^{-3}$ | $9.45\times 10^{-4}$
time 6 | .065 | $4.10\times 10^{-3}$ | $8.20\times 10^{-4}$
time 7 | .062 | $3.47\times 10^{-3}$ | $6.95\times 10^{-4}$
time 8 | .060 | $3.00\times 10^{-3}$ | $5.91\times 10^{-4}$
time 9 | .059 | $1.28\times 10^{-3}$ | $2.57\times 10^{-4}$
time 10 | .060 | — | —
Table 2 shows the evolution of this population from fertilization to mating.
The first column gives the time interval. The second column gives the
regression average effect—the slope in the regression of phenotypic values on
$\mathcal{A}_{2}$ gene count among those individuals alive at the beginning of
the time interval; $\beta$ is the conventional notation for a regression
coefficient. The third column gives the change in $\mathcal{A}_{2}$ frequency
from the beginning of the current time interval to the beginning of the next.
The fourth column gives the change in the mean genotypic fitness from the
beginning of the current time interval to the beginning of the next. Because
the effect of substituting $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$ does not
depend on the allelic type of the undisturbed gene, the experimental average
effect is of course .10. In this case of additive gene action, the genotypic
value is the same as the “breeding” or “additive genetic” value, which is now
often denoted by the symbol $A$.
Immediately after fertilization, the regression and experimental average
effects coincide, as expected from the fact that genetic values and
environmental disturbances are initially independent. The change in mean
genetic value from fertilization to the beginning of the first time interval
is equal to two times the experimental average effect times the change in
allele frequency. The relation $\Delta A=2\alpha\Delta p$ in fact holds for
each transition from one time interval to the next. The relation $\Delta
A=2\beta\Delta p$, however, does not hold for any transition besides the
first. Note the decline in $\beta$, far greater and more systematic than can
be explained by sampling fluctuations, with the passage of time.
What explains the increasing discrepancy between $\alpha$ and $\beta$? This is
an example of what some methodologists call _selection bias_ [47, _e.g._ ,].
Suppose that intelligence and athletic ability are uncorrelated in the
population at large. However, if we limit our observations to the students
attending a university that uses both of these attributes as admissions
criteria, then we will find that intelligence and athleticism are negatively
correlated. If we learn that a student at this university is academically
undistinguished, then it becomes more probable that the student is a good
athlete. Otherwise the student would likely not have been admitted.
Similarly, if there is some relation between fitness at different points of
the lifespan, then with the passage of time the genetic and environmental
causes of fitness will tend to become correlated even if they were initially
independent. If we learn that a particular survivor of a rigorous selection
scheme has an unfit genotype, then it becomes more probable that the organism
has benefited from a favorable environment. This same principle explains why
selection tends to induce deviations from Hardy-Weinberg and linkage
equilibrium [4, 41, 5, 15]; if we find that a survivor has an unfit gene at
one genomic position, it becomes more probable that the survivor bears fit
genes at other positions. As stated previously, the dependence of genotypes
and environment leads to a divergence between the experimental and regression
average effects, and the latter then has no straightforward genetic
interpretation.
It is important to note that our example does not necessarily impugn the
validity of the FTNS, under the regression definition of the average effect,
with respect to organisms living in discrete time. This is because in this
model the FTNS has come to be interpreted as concerning the change in mean
breeding value between generations, and the correctness of the FTNS is
preserved when the mean is measured upon fertilization and the regression
average effect is measured at the beginning of the parental generation.
However, because our model places deaths along a temporal dimension between
birth and mating, it should properly be classified as a continuous-time model.
The FTNS is intended to apply at every point in continuous time, and therefore
our argument for the experimental definition of the average effect retains its
full force for organisms following such a life cycle.
Fisher knew that selection bias with respect to the outcome variable prevents
regression coefficients from being interpretable. In _Statistical Methods for
Research Workers_ , he pointed out that the application of a selection process
to the outcome variable will change the regression line [29, p. 130]. It is
thus rather curious that Fisher never mentioned this principle in connection
with natural selection, a form of selection bias that is always and everywhere
operating.
The regression definition is made viable by stipulating the use of “true” or
“intrinsic” phenotypic measurements as the outcome variable rather than the
actual measurements. This approach, which we adopt in the Appendix, may be
natural and inevitable in the case of multiple loci. Because of the need to
know the residuals in the multilocus case, it does not seem possible to banish
the concept of least-squares linear regression from the theory of average
effects. The concepts of regression and causality need to work together.
Needless to say, the notion of causality remains an essential partner in this
collaboration. A definition calling for the regression of “true” phenotypic
measurements on gene content really amounts to replacing the observed
phenotypic means of the three genotypes in (7) with the experimental means,
which requires the same $do$ operator incorporated in (11) and (15). The
instance of $do$ in (26) actually covers two points where we must invoke
experimental control: once in the determination of the genotypic means,
breeding values, and non-additive residuals, and again in the replacement of
randomly chosen homologous genes to resolve the non-uniqueness of the
individual average effects. To capture what Fisher intended by the average
effect in a formal and transparent way, we cannot easily avoid a special
notation for singling out causal relations from merely correlational ones.
## 9\. Discussion
[18] had the good sense to intuit that sensitivity to physical change was
important to Fisher’s conception of the average effect. Indeed, among all
twentieth-century scientists, Fisher might have been the one most likely to
incorporate the distinction between an observed excess and a causal effect
into a formal theory. The discrepancy that Falconer thought he had uncovered
between Fisher’s regression and experimental definitions of the average effect
can be reconciled, in the case of genotype-environment independence, by using
a specific weighted average of the two possible gene substitutions rather than
a naive average. If the phenotype is affected by one biallelic locus, the
weights are chosen so that a population subject to gene substitutions in
numbers proportional to the weights retains the same value of
$\lambda=Q^{2}/(PR)$, a parameter describing the way in which alleles are
combined into genotypes. If genotypes and environments interact non-
additively, then the gene substitutions must also be balanced with respect to
the marginal distribution of environmental states. This balancing has the
desirable property of preserving the experimentally ascertained phenotypic
means of the genotypes. In the case of multiple loci, there is no longer a
fixed parameterization of genotype formation to which the weightings of the
gene substitutions must conform, but in a loose sense the changes in the
departures from random combination must average out to zero. These
restrictions are requirements for a change in allele frequency “without change
in the environment, or in the mating system [rules of genotype combination].”
When genotypes and environments are dependent—which must always be the case,
even if only slightly, as a result of natural selection—the experimental
definition is to be preferred.
[24] gave one reason why a definition based on experimental gene substitutions
may be inferior to one based on passive observations of a static population
(although later in this paper he reverted to the language of gene
substitutions). He pointed out that changes in the frequencies of the
different genotypes may feed back to change the phenotypic means themselves.
He gave the example of experimental gene substitutions increasing milk yield,
which lead to females in the next generation who can leverage their superior
nourishment to provide even more milk to their own offspring. Fisher wished to
discount such knock-on effects—presumably because they are too complex to form
general rules about them. These knock-on effects can be positive or negative.
When fitnesses are frequency-dependent, the knock-on effects of naturally
selected changes in allele frequencies can steadily decrease the mean fitness
of the population [44]. The approach of a female-skewed sex ratio to a stable
fifty-fifty equilibrium in a polygynous species can be an example of precisely
this phenomenon ([pp. 141-143]fisher:1930:gtns[p. 232]bennett:1983). Therefore
Fisher consigned changes in the genotype-phenotype mapping—the
$\mathbb{E}[Y\,|\,do(ij)]$—brought about by gene substitutions with all other
possible such changes, including those brought about by unpredictable changes
in climate, predators, parasites, and so on. Our preferred resolution of the
dilemma raised by the cascade of additional phenotypic changes that may be
initiated by a physical gene substitution is to stipulate the constancy of (4)
and (5), for instance, in the experimental definition of the average effect.
That is, the average effect is calculated on the assumption that the
prevailing genotype-phenotype mapping will not itself change as a result of
the gene substitutions. This is equivalent to the _stable unit treatment value
assumption_ (SUTVA) in the Neyman-Rubin counterfactual framework.
SUTVA may often have a reasonable interpretation. For example, in the cases of
fecundity selection and frequency-dependent fitnesses of game-theoretic
strategies, we may interpret each causal effect as the expected phenotypic
change upon placing a manipulated organism in a virtual environment containing
the same mixture of types constituting the undisturbed population. In any
event finding an interpretation of SUTVA may not be important in most
biological situations, so long as any frequency-dependent changes ensuing from
the experimental manipulation of a few individuals can be neglected in a
theoretically infinite population.
It is the constancy of the $\mathbb{E}[Y\,|\,do(ij)]$ rather than the
constancy of the corresponding observed phenotypic means that is satisfied by
the gene substitutions defining the average effect in the case of genotype-
environment dependence and interaction. This striking fact further affirms the
priority of causal quantities over observables that may have no causal
interpretation.
A renewed understanding of the average effect is especially timely given the
enablement of GWAS by modern technology and the upsurge of research into the
inheritance of fitness in human populations [54]. The findings of the [12]
indicate that the fine-mapping of the variants with nonzero experimental
average effects responsible for a given association signal may turn out to be
less onerous than was once supposed. However, care is needed as researchers
isolate variants with ever smaller average effects, which will be difficult to
distinguish from spurious signals generated by subtle confounding or selection
bias.
An appealing feature of GWAS is the availability of a complementary study
design, pioneered by [53], that offers nearly the entirety of the benefits
inhering in experimental control. According to Mendel’s laws, a parent passes
on a randomly chosen gene from each of its homologous pairs to a given
offspring. Given the applicability of Mendel’s laws, we can then treat the
genotype of an offspring given the parental genotypes much like a treatment in
a randomized experiment. It follows that a significant association between
transmission of a particular allele and the focal phenotype cannot be the
result of confounding; in the absence of selection bias, the only feasible
explanation is linkage with a locus where the average effect is nonzero.
Fisher himself noted this feature of family-based studies:
> Genetics is indeed in a peculiarly favoured condition in that Providence has
> shielded the geneticist from many of the difficulties of a reliably
> controlled comparison. The different genotypes possible from the same mating
> have been beautifully randomized by the meiotic process. A more perfect
> control of conditions is scarcely possible, than that of different genotypes
> appearing in the same litter. [25, p. 7]
Family-based studies have successfully been used to replicate findings from
studies of nominally unrelated individuals [35, 55], and this is another way
in which the thought experiments defining the average effect are becoming less
like _Gedanken_ and more like routine empirical operations. We note that when
[53] introduced their family-based test, their null hypothesis was no linkage
with a causal locus despite the presence of population association. This test
and its variants have since often been used to test the null hypothesis that
there is neither linkage nor association. We anticipate that there will be a
trend back toward the original form of the test. Because parent-offspring
trios and sets of siblings can be difficult to recruit and require more
genotyping, investigators find it convenient to test for population
association in large samples of unrelated individuals. Those markers showing
evidence of association can then be interrogated, however, for linkage with
loci where there are nonzero average effects. The follow-up cohorts of
families will typically be much smaller and less likely to yield genome-wide
significant $p$-values, but it will be reasonable to require less stringent
evidence or merely overall sign agreement greatly exceeding 50 percent. This
procedure can provide a check on whether the association stage is producing an
acceptably low rate of false positives with respect to the causal hypothesis
of a nonzero average effect—which, of course, is not strictly the same as the
statistical hypothesis of a nonzero partial regression coefficient.
We note that family-based studies are not immune to selection bias intervening
between fertilization and the time of measurement, which may rise to an
appreciable level in studies of phenotypes strongly affecting fitness. This
may be a challenge for gene-trait mapping studies conducted in the near
future.
It may be tempting to define the average effect in terms of a hypothetical
family-based study. However, whereas rejecting the null hypothesis of a zero
average effect requires only the assumptions of Mendel’s laws, effect
estimation requires additional assumptions and thus does not seem particularly
suited for a theoretical definition after all [17].
Finally, we comment on the role of the average effect in the FTNS. We write
the breeding (additive genetic) value of a given individual as
$A=\sum_{\ell=1}^{L}\sum_{i_{\ell}=1}^{n_{\ell}}\chi\left(\mathcal{A}^{(\ell)}_{i_{\ell}}\right)\alpha^{(\ell)}_{i_{\ell}},$
(28)
where $\chi(\cdot)$ is a function giving the number of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ genes (0, 1, or 2) present in the
individual’s genotype. The variance in breeding values, $\textrm{Var}(A)$, is
now called the _additive genetic variance_ , and the ratio
$\textrm{Var}(A)/\textrm{Var}(Y)$ the _heritability in the narrow sense_. It
is important to keep in mind that these breeding values are linear functions
of _experimental_ average effects; we are building up a predicted value for a
given individual from the causal effects of the genes present in the genotype.
The FTNS states that the partial change in mean fitness attributable to
changes in allele frequencies caused by natural selection is proportional to
the additive genetic variance in fitness, which can be shown to equal
$2\sum_{\ell=1}^{L}\sum_{i_{\ell}=1}^{n_{\ell}}p^{(\ell)}_{i_{\ell}}a^{(\ell)}_{i_{\ell}}\alpha^{(\ell)}_{i_{\ell}},$
(29)
where the meaning of $a^{(\ell)}_{i_{\ell}}$ is as follows. If genotypes and
environments are independent, then this quantity is the average excess of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$, which is usually defined as the difference
in mean fitness between the bearers of the given allele and the entire
population. (29) is invariably derived under the assumption that genotypes and
environments are independent. Because under our definitions the values of the
experimental average effects do not depend on the extent of genotype-
environment dependence, it follows that the breeding values and hence the
additive genetic variance are also insensitive to genotype-environment
dependence. The equality of (29) with $\textrm{Var}(A)$ is thus fully valid in
our account—given the following modification regarding
$a^{(\ell)}_{i_{\ell}}$.
If genotypes and environments are not independent, $a^{(\ell)}_{i_{\ell}}$ in
(29) is not exactly the same as the average excess defined by [28, p. 35 ]. It
is rather the average excess that _would_ be observed if genotypes were
distributed randomly among environments. In other words each
$a^{(\ell)}_{i_{\ell}}$ only reflects confounding with other genetic loci and
not with environmental causes. To repeat, this is a consequence of the fact
that our experimental average effects—and hence all quantities derived from
them, including the additive genetic variance—are sensitive only to the
marginal distribution of environmental states. Every factor in (29), including
the $a^{(\ell)}_{i_{\ell}}$, must therefore be equal to whatever they would be
under genotype-environment independence, the standard setting in which (29) is
calculated. If the “full” average excesses were substituted into (29), then
the expression would no longer be interpretable as a variance; it could then
possibly be negative.
It is well known that the change in the frequency of
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ is proportional to the product of
$p^{(\ell)}_{i_{\ell}}$ and the actual difference in mean fitness between the
bearers of the given allele and the entire population [50, _e.g._ ,]. From the
fact that the difference is not necessarily equal to our
$a^{(\ell)}_{i_{\ell}}$, we learn that there is partition of the total change
in allele frequency between the change caused by natural selection and the
change attributable to how genotypes are distributed across environments
varying in severity. This partition is in the same spirit of Fisher’s
conditions discussed previously. Like changes in the rules of genotype
formation or the $\mathbb{E}[Y\,|\,do(ij)]$, deviations from genotype-
environment independence cannot generally lead to an increase in fitness, and
indeed the example set out in Table 2 demonstrates that the dependence induced
by natural selection itself tends to retard the frequency increase of the
superior allele.
Each increment of naturally selected change in allele frequency is a direct
cause of a change in the mean fitness equal to $2\alpha^{(\ell)}_{i_{\ell}}$.
Any discrepancy between the total change and this partial change, summed over
all loci and alleles, is owed to indirect effects acting through changes in
the rules of genotype formation, the distribution of environmental states, or
some other determinant of fitness. This completes the FTNS: the _increase_ in
the mean fitness of a population caused exclusively by the effect of natural
selection on allele frequencies—setting aside those _changes_ in fitness
(which can be positive or negative) ascribable to other causes—is equal to the
additive genetic variance in fitness.
Fisher’s contributions to biology and applied mathematics were of course
numerous and profound. Judging from his writing in _The Genetical Theory_ ,
however, we surmise that he considered the FTNS to be the most important of
his achievements. The FTNS quantifies Darwin’s notion of hereditary variation
in fitness leading to adaptation and provides a deeper understanding of it. It
is interesting that (29), Fisher’s “supreme law of the biological sciences,”
explicitly encodes a distinction between an observed excess and a causal
effect, the same distinction that animated his work on experimental design,
which [43] praised as the greatest of Fisher’s contributions to statistics.
The FTNS was thus another blow struck by Fisher against his scientific
adversary Karl Pearson, who believed it was possible both to study evolution
mathematically and to discard the notion of causality. If causality appears
inevitably in the formulation of a phenomenon as fundamental as evolution by
natural selection, then it surely cannot be a dispensable “fetish amidst the
inscrutable arcana of modern science” [48, p. xii].
## Acknowledgements
We thank A. W. F. Edwards for sharing his unpublished work and his
correspondence with the late Douglas Falconer, Sabin Lessard for helpfully
answering one of our queries, and the reviewers for comments and suggestions
that have greatly improved this paper. This work was supported by the
Intramural Research Program of the NIDDK, NIH.
## Appendix
Here we explicitly derive the conditions under which the regression and
experimental definitions of average effect are equivalent. We assume that the
equivalence can always be secured in a meaningful way, either because
genotypes and environments are independent or because the regression has been
performed on the experimental genotypic means rather than the observed
genotypic means. We will often refer to an experimental average effect in the
sense of an arbitrary linear combination of relevant causal effects
(differences between genotypic means) and narrow down our reference to
particular linear combinations as the given argument proceeds. We first treat
the case of a single biallelic locus, which is of special interest because it
is possible here to find explicit expressions for the weights $c_{1}$ and
$c_{2}$ in (11).
Let $i$ stand for $\textrm{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{1})]$, $j$
for $\textrm{E}[Y\,|\,do(\mathcal{A}_{1}\mathcal{A}_{2})]$, and $k$ for
$\textrm{E}[Y\,|\,do(\mathcal{A}_{2}\mathcal{A}_{2})]$. This notation is
similar to that of [19, 24]. By using the $do$ symbol, however, our argument
below is meaningful even if genotypes and environments are dependent and non-
additive.
To minimize the sum of squares
$P(i-\nu+\alpha)^{2}+2Q(j-\nu)^{2}+R[k-\nu-\alpha]^{2},$
we take partial derivatives with respect to $\nu$ and $\alpha$ and set them
equal to zero. Solving the two resulting equations gives
$\alpha=\frac{P(Q+R)(j-i)+R(P+Q)(k-j)}{PQ+QR+2PR},$ (A1)
which can easily be recognized as equivalent to (11) in the case that
genotypes and environments act additively. Using (5\. Gene-Environment
Correlation and Interaction) to expand each experimental mean, we find that
the numerator of (A1) becomes
$c_{1}\left[\sum_{i}\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})-\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{1},\mathcal{E}_{i})\right]\\\
+c_{2}\left[\sum_{i}\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{2}\mathcal{A}_{2},\mathcal{E}_{i})-\textrm{Pr}(\mathcal{E}_{i})\textrm{E}(Y\,|\,\mathcal{A}_{1}\mathcal{A}_{2},\mathcal{E}_{i})\right],$
(A2)
which means that (A1) is also equivalent to (15).
Now consider the change in the mean phenotype caused by experimental gene
substitutions. The contribution to the population mean phenotype by the
experimental means of the genotypes is given by
$\mu=iP+2jQ+kR,$ (A3)
and the change in the population mean upon effecting the gene substitutions is
$d\mu=idP+2jdQ+kdR.$ (A4)
The changes $dP,dQ,dR$ have two degrees of freedom. To express the changes in
terms of a single change $dp$, we must obtain another condition, which can be
expressed without loss of generality as $f(P,Q,R)=0$. [24] gave the condition
that $\lambda=Q^{2}/(PR)$ remains constant, but his concise argument has
puzzled many commentators.
It turns out that Fisher set $d\mu=idP+2jdQ+kdR$ equal to $2\alpha dp$ and
equated the coefficients of $i,j,k$ [8], which yields
$\displaystyle dP$ $\displaystyle=-2P(Q+R)dp/S,$ $\displaystyle dQ$
$\displaystyle=Q(P-R)dp/S,$ $\displaystyle dR$ $\displaystyle=2R(P+Q)dp/S,$
(A5)
where $S=P(Q+R)+R(P+Q)$. The function $f$ satisfies the differential equation
$\frac{\partial f}{\partial P}dP+\frac{\partial f}{\partial
Q}dQ+\frac{\partial f}{\partial R}dR=0.$ (A6)
Inserting (A5) into (A6) gives
$-2P(Q+R)\frac{\partial f}{\partial P}+Q(P-R)\frac{\partial f}{\partial
Q}+2R(P+Q)\frac{\partial f}{\partial R}=0.$ (A7)
Now $(-2P(Q+R),Q(P-R),2R(P+Q))$ and $(\frac{\partial f}{\partial
p},\frac{\partial f}{\partial Q},\frac{\partial f}{\partial R})$ can be
regarded as two orthogonal vectors in three-space. We want the second
condition to be independent of the conservation of probability condition and
not to be the trivial zero vector. By inspection, we see that a solution is
given by
$\displaystyle\frac{\partial f}{\partial P}$ $\displaystyle=\frac{\phi}{P},$
$\displaystyle\frac{\partial f}{\partial Q}$ $\displaystyle=\frac{-2\phi}{Q},$
$\displaystyle\frac{\partial f}{\partial R}$ $\displaystyle=\frac{\phi}{R},$
(A8)
where $\phi$ is an arbitrary function of $P,Q,R$. A simple solution is given
by setting $\phi$ equal to the constant $a$, whereupon (A8) can be integrated
to obtain
$f=-2a\ln Q+a\ln P+a\ln R+a\ln\lambda,$ (A9)
which gives the condition $Q^{2a}=(\lambda PR)^{a}$. $a=1$ gives the condition
expressed in terms of the classic Fisher parameter. Conversely, if we let
$\phi=PRQ^{-2}$ then we get $f=PRQ^{-2}-(1/\lambda)$, which also gives the
Fisher parameter.
Taking the partial second derivatives gives the compatibility conditions that
$\phi$ must satisfy:
$\displaystyle\frac{1}{P}\frac{\partial\phi}{\partial R}$
$\displaystyle=\frac{1}{R}\frac{\partial\phi}{\partial P},$
$\displaystyle\frac{1}{P}\frac{\partial\phi}{\partial Q}$
$\displaystyle=\frac{-2}{Q}\frac{\partial\phi}{\partial P},$
$\displaystyle\frac{1}{R}\frac{\partial\phi}{\partial Q}$
$\displaystyle=\frac{-2}{Q}\frac{\partial\phi}{\partial R}.$ (A10)
Hence, any differentiable function of $PRQ^{-2}$ is a solution. This then
implies that $f$ can be any differentiable function of $PRQ^{-2}$ as well.
This shows that the average phenotypic increment caused by a number of
experimental gene substitutions is the same as the slope in the regression of
the phenotype on the experimental genotypic means if the substitutions are
performed in a background where any function of $PRQ^{-2}$ is held constant,
with $\lambda$ being the simplest one.
We now treat a phenotype affected by an arbitrary number of multiallelic loci.
As shown in Section 7, the experimentally determined phenotypic means of the
whole-genome genotypes can be expressed as
$\mathbb{E}[Y\,|\,do(ij)]=\mu+\alpha_{ij}+\varepsilon_{ij}.$
In the remainder we abbreviate $\mathbb{E}[Y\,|\,do(ij)]$ as $G_{ij}$ and set
$a_{ij}=G_{ij}-\mu$, which obeys the condition $\sum_{i,j}P_{ij}a_{ij}=0$.
The average effects can be written as
$\alpha_{ij}=\alpha_{i}+\alpha_{j}=\sum_{\ell}(\alpha_{i_{\ell}}^{(\ell)}+\alpha_{j_{\ell}}^{(\ell)})$
and are obtained by minimizing
$\sum_{i,j}P_{ij}(a_{ij}-\alpha_{ij})^{2}.$ (A11)
The minimum obeys the condition
$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{i\nmid
i_{k}}\sum_{j}P_{ij}\alpha_{ij}=\sum_{i\nmid
i_{k}}\sum_{j}P_{ij}\sum_{\ell}\left(\alpha_{i_{\ell}}^{(\ell)}+\alpha_{j_{\ell}}^{(\ell)}\right),$
(A12)
where
$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{i\nmid i_{k}}\sum_{j}a_{ij}P_{ij}$ (A13)
defines the average excesses. A sum running over $i\nmid i_{k}$ should be
understood as a sum over all multi-indices $i$ where the $k$th element is
fixed to $i_{k}$. These relations imply that $\sum_{ij}P_{ij}\alpha_{ij}=0$,
which also implies that $\sum_{ij}P_{ij}\varepsilon_{ij}=0$.
Equation (A12) can be rewritten as
$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\left(p_{i_{k}}^{(k)}+q_{i_{k}i_{k}}^{(kk)}\right)\alpha_{i_{k}}^{(k)}+\sum_{\ell\neq
k}\sum_{i_{\ell}}p_{i_{k}i_{\ell}}^{(k\ell)}\alpha_{i_{\ell}}^{(\ell)}+\sum_{\ell}\sum_{j_{\ell}\neq
i_{k}}q_{i_{k}j_{\ell}}^{(k\ell)}\alpha_{j_{\ell}}^{(\ell)}\\\
\equiv\sum_{j_{\ell}}H^{(k\ell)}_{i_{k}j_{\ell}}\alpha_{j_{\ell}}^{(\ell)},$
(A14)
where
$p^{(k\ell)}_{i_{k}i_{\ell}}=\sum_{i\nmid i_{k},i_{\ell}}\sum_{j}P_{ij}$
denotes the frequency of gametes that carry $\mathcal{A}^{(k)}_{i_{k}}$ and
$\mathcal{A}^{(\ell)}_{i_{\ell}}$ and
$q^{(k\ell)}_{i_{k}j_{\ell}}=\sum_{i\nmid i_{k}}\sum_{j\nmid j_{\ell}}P_{ij}$
denotes the frequency of all multilocus genotypes that carry
$\mathcal{A}^{(k)}_{i_{k}}$ and $\mathcal{A}^{(\ell)}_{j_{\ell}}$ on different
chromosomes. The matrix $\mathsf{H}$ in (A14) is constructed as follows. Let
$\mathbf{p}$ denote the vector of allele frequencies, $\mathbf{a}$ the vector
of average excesses, and $\bm{\alpha}$ the vector of average effects. These
vectors have length $\sum_{k}^{L}n_{k}$, and their elements are ordered by
locus. We can then define
$\mathsf{H}=\mathsf{D}+\mathsf{P}+\mathsf{Q},$ (A15)
where $\mathsf{D}$ is the diagonal matrix with the components of $\mathbf{p}$
on the diagonal, $\mathsf{P}$ is the matrix with entries
$p^{(k\ell)}_{i_{k}i_{\ell}}$ if $k\neq\ell$ and 0 otherwise, and $\mathsf{Q}$
is the matrix with entries $q^{(k\ell)}_{i_{k}j_{\ell}}$ [14, 6]. We will use
the notation $\mathbf{p}\cdot\mathbf{a}$ to designate the component-wise
product of the vectors $\mathbf{p}$ and $\mathbf{a}$, i.e.,
$(\mathbf{p}\cdot\mathbf{a})_{i}=\mathbf{p}_{i}\mathbf{a}_{i}$. (A14) can thus
be rewritten again as
$\bm{\alpha}=\mathsf{H}^{-1}(\mathbf{p}\cdot\mathbf{a})$ (A16)
subject to suitable constraints on $\bm{\alpha}$. We will shortly see that
these constraints turn out to be (17) for each locus. Given our ordering
convention, the element $H^{(k\ell)}_{i_{k}j_{\ell}}$ lies in the row of
$\mathsf{H}$ corresponding to allele $\mathcal{A}^{(k)}_{i_{k}}$ and the
column corresponding to $\mathcal{A}^{(\ell)}_{i_{\ell}}$.
The total change in $\mu$ is
$d\mu=\sum_{i_{k}}\sum_{i,j}G_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}=\sum_{i_{k}}\sum_{i,j}(\mu+\alpha_{ij}+\varepsilon_{ij})\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}\\\
=\sum_{i_{k}}\sum_{i,j}(\alpha_{ij}+\varepsilon_{ij})\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}$ (A17)
upon performing a number of experimental gene substitutions at locus $k$.
Agreement of the experimental and regression average effects implies that this
change must equal the change predictable from the breeding values,
$d\mu=\sum_{i_{k}}\sum_{i,j}\alpha_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)},$ (A18)
which implies in turn that
$\sum_{i,j}\sum_{i_{k}}\varepsilon_{ij}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}dp_{i_{k}}^{(k)}=0$ (A19)
is a necessary and sufficient condition for the experimental and regression
average effects to coincide. The bald statement that the changes in genotype
frequencies must somehow nullify the non-additive residuals, however, is not
very revealing. We can render (A19) into a more insightful form by noting that
$\sum_{i,j}P_{ij}\sum_{\ell=1}^{L}\left(\frac{\partial
p_{i_{\ell}}}{p_{i_{\ell}}}+\frac{\partial
p_{j_{\ell}}}{p_{j_{\ell}}}\right)\varepsilon_{ij}=0$ (A20)
because the sum over $\ell$ is a constant determined by the experimenter.
Using this, from (A19) we obtain
$\sum_{ij}P_{ij}\left[\frac{1}{P_{ij}}\frac{\partial P_{ij}}{\partial
p^{(k)}_{i_{k}}}-\sum_{\ell}\left(\frac{\partial
p_{i_{\ell}}}{p_{i_{\ell}}}+\frac{\partial
p_{j_{\ell}}}{p_{j_{\ell}}}\right)\right]\varepsilon_{ij}=0,$ (A21)
which leads to (24). This argument, which simplifies one given by [36], can be
used to construct a variety of quantities measuring departures from random
combination. The $\theta_{ij}$ appear to be the simplest such quantities.
The criterion (A21) does not pick out a unique weighting of the possible gene
substitutions for a given genetic architecture. It would be of great
significance if a subset of the possible weights could be characterized in a
manner that does not depend on the non-additive residuals. We have done this
for a single biallelic locus, where the subset contains the singleton
weighting of the two possible gene substitutions that conserves $\lambda$. If
a general procedure for constructing such a residual-free characterization for
any number of loci exists, then the following argument should be able to find
it.
The contribution of the experimental genotypic means to the population mean is
$\mu=\sum_{i,j}G_{ij}P_{ij}.$ (A22)
The definition of the experimental average effect can be written as
$\alpha_{i_{k}}^{(k)}=\frac{1}{2}\frac{\partial\mu}{\partial
p_{i_{k}}^{(k)}}.$ (A23)
Imposing constancy of the experimental means, we can write the change in the
population mean due to a change in frequency of allele
$\mathcal{A}^{(k)}_{i_{k}}$ as
$\frac{\partial}{\partial p^{(k)}_{i_{k}}}\mu=\sum_{i,j}G_{ij}\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}=\sum_{i,j}(G_{ij}-\mu)\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}}=\sum_{i,j}a_{ij}\frac{\partial
P_{ij}}{\partial p^{(k)}_{i_{k}}},$ (A24)
using the fact that $\sum_{i,j}\frac{\partial}{\partial
p^{(k)}_{i_{k}}}P_{ij}=0$. The indeterminacy in the partial derivatives with
respect to allele frequency will be resolved by the properties of
$\bm{\lambda}$ in (19) that emerge from the subsequent analysis.
Substituting (A24) and (A23) into (A14) using (A13) gives the condition
$\sum_{i\nmid
i_{k}}\sum_{j}a_{ij}P_{ij}=\frac{1}{2}\sum_{j}\sum_{j_{\ell}}H^{(k\ell)}_{i_{k}j_{\ell}}\sum_{m,n}a_{mn}\frac{\partial
P_{mn}}{\partial p^{(\ell)}_{j_{\ell}}}$ (A25)
for each $i_{k}$ and $k$. Closed-form solutions of these partial differential
equations will not exist in general. However, using symmetry conditions and
properties of $\mathsf{H}$, we may infer some necessary conditions on the
genotype frequencies that must be satisfied.
We first note that the image space of $\mathsf{H}$ contains all permissible
vectors of allele-frequency changes [37]. Since $\mathsf{H}$ is invertible on
its image space, we may operate on (A25) by the inverse of $\mathsf{H}$ (which
we call $\mathsf{J}$) and thereby separate the PDE system into a set of $\sum
n_{\ell}$ ordinary differential equations, which we denote by
$2\sum_{i_{k}}J_{j_{k}i_{k}}p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}=\sum_{m,n}a_{mn}\frac{\partial
P_{mn}}{\partial p^{(k)}_{j_{k}}}.$ (A26)
We may now select any row of (A26), expand the
$p_{i_{k}}^{(k)}a_{i_{k}}^{(k)}$ in terms of the $a_{mn}$, and equate the LHS
and RHS coefficients of $a_{mn}$. This will result in a set of $\prod
n_{\ell}\times\prod n_{\ell}$ ordinary differential equations of the form
$\frac{1}{P_{mn}}\frac{\partial P_{mn}}{\partial
p^{(k)}_{j_{k}}}=\phi_{mn}(j_{k}),$ (A27)
where $\phi_{mn}(j_{k})$ is some linear combination of the elements of the
vector $J_{j_{\ell}=j_{k},i_{\ell}}$. From this point the
$a_{mn}=\alpha_{mn}+\varepsilon_{mn}$ no longer appear in the argument, and it
follows that we must be finding properties of a solution that depends on
neither the breeding values nor the non-additive residuals.
Conserved quantities imposed by (A25), which can be used to form elements of
$\bm{\lambda}$, can be constructed by taking linear combinations of the ODEs
such that
$\sum_{m,n}\sigma_{mn}\frac{1}{P_{mn}}\frac{\partial P_{mn}}{\partial
p^{(k)}_{j_{k}}}=0,$ (A28)
from which we obtain conserved measures of departure from random combination
assuming the form
$\frac{\prod_{\\{\sigma>0\\}}P_{\alpha\beta}}{\prod_{\\{\sigma<0\\}}P_{\gamma\delta}}=\lambda_{\sigma},$
(A29)
where $\sigma_{mn}$ is some set of coefficients that are positive, zero, or
negative. These conserved quantities will form a set of necessary conditions
for the equivalence of the experimental and regression definitions of the
average effects.
Note that the coefficients of $a_{mn}$ on the LHS of (A26) are grouped
according to the $a_{i_{k}}^{(k)}$. Thus all of the $a_{mn}$ expressed in a
given $a_{i_{k}}^{(k)}$ will have the same coefficient (one of the elements of
$\mathsf{J}$). We can thus construct conserved measures of Hardy-Weinberg and
linkage disequilibrium without an explicit calculation of $\mathsf{J}$ because
we know which sets of coefficients are equal.
Our first numerical example is of a single locus with three alleles (Table
A1). The case of a single locus with any number of alleles was analytically
treated by [32]. The equating of coefficients along the $i$th row of (A26)
leads to the matrix of equations
$\begin{pmatrix}J_{i1}=\dfrac{1}{P_{11}}\dfrac{\partial P_{11}}{\partial
p_{i}}&J_{i1}=\dfrac{1}{P_{12}}\dfrac{\partial P_{12}}{\partial
p_{i}}&J_{i2}=\dfrac{1}{P_{21}}\dfrac{\partial P_{21}}{\partial p_{i}}\\\
&&\\\ J_{i2}=\dfrac{1}{P_{22}}\dfrac{\partial P_{22}}{\partial
p_{i}}&J_{i1}=\dfrac{1}{P_{13}}\dfrac{\partial P_{13}}{\partial
p_{i}}&J_{i3}=\dfrac{1}{P_{31}}\dfrac{\partial P_{31}}{\partial p_{i}}\\\
&&\\\ J_{i3}=\dfrac{1}{P_{33}}\dfrac{\partial P_{33}}{\partial
p_{i}}&J_{i2}=\dfrac{1}{P_{23}}\dfrac{\partial P_{23}}{\partial
p_{i}}&J_{i3}=\dfrac{1}{P_{32}}\dfrac{\partial P_{32}}{\partial p_{i}}\\\
\end{pmatrix}$ (A30)
for allele $i$. The notation $P_{ij}$ now means the ordered genotype with
alleles $i$ and $j$. This matrix gives a set of nine conditions plus
conservation of probability that must be satisfied to ensure the equality of
(A25). However, given that there are only six unique genotypes, these
conditions are overdetermined and will not necessarily be solvable. We can
attempt to formulate a solvable set by combining these conditions. We can see
that the second and third elements in a given row of this matrix must equal
the sum of the elements in the first column corresponding to the homozygous
bearers of the relevant alleles. For example,
$\frac{1}{P_{12}}\frac{\partial P_{12}}{\partial
p_{i}}+\frac{1}{P_{21}}\frac{\partial P_{21}}{\partial
p_{i}}=\frac{1}{P_{11}}\frac{\partial P_{11}}{\partial
p_{i}}+\frac{1}{P_{22}}\frac{\partial P_{22}}{\partial p_{i}}=J_{i1}+J_{i2},$
(A31)
and these equations lead collectively to the three conserved measures of
Hardy-Weinberg disequilibrium
$\lambda_{12}=\frac{P_{12}^{2}}{P_{11}P_{22}},\quad\lambda_{13}=\frac{P_{13}^{2}}{P_{11}P_{33}},\quad\lambda_{23}=\frac{P_{23}^{2}}{P_{22}P_{33}}.$
(A32)
Two of the allele frequencies and these three conserved quantities appear to
be a complete specification of the six genotype frequencies. By the implicit
function theorem, invertibility of the Jacobian at any solution ($p_{1}$,
$p_{2}$, $\lambda_{12}$, $\lambda_{13}$, $\lambda_{23}$) specifying a valid
vector of genotype frequencies ensures that there are unique solutions for
small perturbations of the allele frequencies. Numerical testing suggests that
invertibility of the Jacobian is a generic property of this five-dimensional
system.
Table A1: A trait affected by a single triallelic locus. genotype | $\mathbb{E}[Y\,|\,do(\cdot)]$ | frequency | $\varepsilon$
---|---|---|---
$\mathcal{A}_{1}\mathcal{A}_{1}$ | 10 | .2 | $-.3402778$
$\mathcal{A}_{2}\mathcal{A}_{2}$ | 13 | .2 | .2152778
$\mathcal{A}_{3}\mathcal{A}_{3}$ | 12 | .2 | $-.6875$
$\mathcal{A}_{1}\mathcal{A}_{2}$ | 11 | .2 | $-.5625$
$\mathcal{A}_{1}\mathcal{A}_{3}$ | 14 | .1 | 2.4861111
$\mathcal{A}_{2}\mathcal{A}_{3}$ | 13 | .1 | .2638889
Given the numerical values in Table A1, what is the experimental average
effect of substituting $\mathcal{A}_{2}$ for $\mathcal{A}_{1}$? There are
three ways in which this gene substitution can be brought about:
$\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$, $\mathcal{A}_{1}\mathcal{A}_{2}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$, and
$\mathcal{A}_{1}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{3}$. The causal effects of these three
substitutions are 1, 2, and $-1$ respectively.
We first attempt to satisfy the weaker criterion that (24) is equal to zero by
determining which weighted average of the first two substitutions yields the
smallest absolute value of $\overline{\varepsilon\,\mathring{\theta}}$. To
calculate a discrete approximation of the $\mathring{\theta}_{ij}$, we use a
population size of 10,000. We examine all integer weights such that the
weights sum to 90. There are 91 such weighted averages, and it turns out that
the weights $(70,20)$ yield the minimum. In fact, the absolute value of
$\overline{\varepsilon\,\mathring{\theta}}$ yielded by these weights is
roughly $1.5\times 10^{-16}$, which is nearly within machine error of zero.
The 90 other weighted averages lead to absolute values of
$\overline{\varepsilon\,\mathring{\theta}}$ exceeding $1\times 10^{-4}$.
These weights lead to an experimental average effect, $\alpha_{2}-\alpha_{1}$,
equaling 11/9. In the case of a single locus, the regression average effects
(which we now denote by $\beta$) do not require the imposition of (17) to be
identified, and the calculations yielding the values of the $\varepsilon_{ij}$
in Table A1 also give us ($-0.7798611$, 0.4423611, 0.39375) as the numerical
value of ($\beta_{1}$, $\beta_{2}$, $\beta_{3}$). It appears that
$\beta_{2}-\beta_{1}$ is exactly equal to 11/9.
We can use a different pair of substitutions, say
$\mathcal{A}_{1}\mathcal{A}_{2}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{2}$ and $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{3}$, to yield the experimental
average effect $\alpha_{2}-\alpha_{1}$. We examine all integer weightings of
these two substitutions such that the weights sum to 270. It turns out that
the weighting $(200,70)$ yields the minimum. The absolute value of
$\overline{\varepsilon\,\mathring{\theta}}$ yielded by these weights is
roughly $4\times 10^{-16}$, again nearly within machine error of zero, whereas
the 270 other weighted averages all lead to absolute values of
$\overline{\varepsilon\,\mathring{\theta}}$ exceeding $3\times 10^{-4}$. These
minimizing weights again lead to an experimental average effect of 11/9. It is
rather interesting that the neighboring weights $(199,71)$ and $(201,69)$ lead
to such higher values of $\overline{\varepsilon\,\mathring{\theta}}$ despite
the numerical closeness of these weighted averages and the fineness of our
discretization. In fact, we have chosen to present this example because of
this phenomenon, which we conjecture to be related to the fact that the
$\alpha_{2}-\alpha_{1}$ happens to be rational and thus exactly equal to some
integer-weighted average of the causal effects.
Evidently it should not be possible to obtain a valid average effect by using
only the substitutions $\mathcal{A}_{1}\mathcal{A}_{1}$ $\rightarrow$
$\mathcal{A}_{1}\mathcal{A}_{2}$ and $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{3}$. Examining all integer weights
summing to 1000, we find that $\overline{\varepsilon\,\mathring{\theta}}$
declines linearly from $(0,1000)$ to $(1000,0)$; the absolute minimum of
$\overline{\varepsilon\,\mathring{\theta}}$ is thus attained at a boundary,
and it is not especially small ($\sim 2\times 10^{-2}$).
We examine whether our conception of individual average effects is valid.
Using the method of minimizing $\overline{\varepsilon\,\mathring{\theta}}$, we
find that $\alpha_{2}-\alpha_{3}$ is approximately .049. According to our
notion of substituting $\mathcal{A}_{2}$ for a random homologous gene,
$\alpha_{2}$ must be equal to
$p_{1}(\alpha_{2}-\alpha_{1})+p_{3}(\alpha_{2}-\alpha_{3})$. In our example
($p_{1}$, $p_{2}$, $p_{3}$) happens to be (.35, .35, .30), which leads to
$.4425$ as the approximate numerical value of $\alpha_{2}$. This is in good
agreement with $\beta_{2}$. Continuing this exercise, we can satisfy ourselves
that ($\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$) and ($\beta_{1}$, $\beta_{2}$,
$\beta_{3}$) are equal.
We now attempt to satisfy the stronger criterion that the quantities in (A32)
remain constant. The numerical value of ($p_{1}$, $p_{2}$, $\lambda_{12}$,
$\lambda_{13}$, $\lambda_{23}$) is (35/100, 35/100, 1/4, 1/16, 1/16), and a
perturbation of ($-1/1000$, $1/1000$, 0, 0, 0) leads to a numerical solution
that specifies another valid vector of genotype frequencies. The weighting of
the possible gene substitutions satisfying the changes in genotype frequencies
is typically not unique. In a population of size $10^{8}$, one permissible
vector of weights for our example can be reasonably well approximated by
${\mathcal{A}_{1}\mathcal{A}_{1}}$${\mathcal{A}_{1}\mathcal{A}_{2}}$${\mathcal{A}_{2}\mathcal{A}_{2}}$${\mathcal{A}_{3}\mathcal{A}_{3}}$${\mathcal{A}_{1}\mathcal{A}_{3}}$${\mathcal{A}_{2}\mathcal{A}_{3}}$88,82188,951622,2226
(A33)
where the label of each arrow indicates how many gene substitutions of that
kind are to be performed. Notice that there are 12 gene substitutions
involving a genotype containing the allele $\mathcal{A}_{3}$. For each
$\mathcal{A}_{3}$ gene created by $\mathcal{A}_{1}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{3}\mathcal{A}_{3}$, another $\mathcal{A}_{3}$ is
destroyed by $\mathcal{A}_{2}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{2}\mathcal{A}_{2}$, and the net result is the same frequency of
$\mathcal{A}_{3}$. These 12 substitutions turn out to be a way of decreasing
the number of $\mathcal{A}_{1}$ genes and increasing the number of
$\mathcal{A}_{2}$ without directly converting one to the other. We might as
well pair each $\mathcal{A}_{1}\mathcal{A}_{3}$ $\rightarrow$
$\mathcal{A}_{3}\mathcal{A}_{3}$ with $\mathcal{A}_{2}\mathcal{A}_{3}$
$\rightarrow$ $\mathcal{A}_{2}\mathcal{A}_{2}$, treating each such pair as a
single substitution. The weighted average of the gene substitutions is then
$\frac{88,821(1)+88,951(2)+22,222(-1)+6(-2+0)}{88,821+88,951+22,222+6},$
which diverges from 11/9 at the fourth decimal place.
We now apply our argument to the case of two biallelic loci. Here we will
encounter a contradiction.
The equating of coefficients along the row of (A26) corresponding to allele
$\mathcal{A}^{(k)}_{i_{k}}$ now leads to the matrix of equations
$\begin{pmatrix}J_{i1}+J_{i3}=\frac{1}{P_{11,11}}\frac{\partial
P_{11,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,11}}\frac{\partial
P_{12,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,11}}\frac{\partial
P_{21,11}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,11}}\frac{\partial
P_{22,11}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,12}}\frac{\partial P_{11,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,12}}\frac{\partial
P_{12,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,12}}\frac{\partial
P_{21,12}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,12}}\frac{\partial
P_{22,12}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,21}}\frac{\partial P_{11,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,21}}\frac{\partial
P_{12,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,21}}\frac{\partial
P_{21,21}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,21}}\frac{\partial
P_{22,21}}{\partial p^{(k)}_{i_{k}}}\\\
J_{i1}+J_{i3}=\frac{1}{P_{11,22}}\frac{\partial P_{11,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i1}+J_{i4}=\frac{1}{P_{12,22}}\frac{\partial
P_{12,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i3}=\frac{1}{P_{21,22}}\frac{\partial
P_{21,22}}{\partial
p^{(k)}_{i_{k}}}&J_{i2}+J_{i4}=\frac{1}{P_{22,22}}\frac{\partial
P_{22,22}}{\partial p^{(k)}_{i_{k}}}\\\ \end{pmatrix}$ (A34)
plus conservation of probability that must be satisfied to ensure the equality
of (A25). An argument analogous to the one below (A30) shows that six
quantities of the form
$\lambda_{ij}=\frac{P_{ij}^{2}}{P_{ii}P_{jj}}$ (A35)
must be conserved. If we do not assume that the double heterozygotes are
phenotypically equivalent, then these six measures of Hardy-Weinberg
disequilibrium, the allele frequencies at the two loci, and conservation of
probability leave one more condition to specify ten genotype frequencies.
Rearrange each element of (A34) to put the genotype frequency on one side and
form the four column sums. Each such sum is the marginal frequency of a
gamete. For example, we have
$P_{11}=(J_{i1}+J_{i3})^{-1}\sum_{11,j}\frac{\partial P_{11,j}}{\partial
p^{(k)}_{i_{k}}},$ (A36)
which implies that
$J_{i1}+J_{i3}=\frac{1}{P_{11}}\frac{\partial P_{11}}{\partial
p^{(k)}_{i_{k}}}.$ (A37)
Combining all columns, we get
$\frac{\partial P_{11}}{\partial p^{(k)}_{i_{k}}}+\frac{\partial
P_{22}}{\partial p^{(k)}_{i_{k}}}-\frac{\partial P_{12}}{\partial
p^{(k)}_{i_{k}}}-\frac{\partial P_{21}}{\partial p^{(k)}_{i_{k}}}=0,$ (A38)
which yields the condition that
$\zeta=\frac{P_{11}P_{22}}{P_{12}P_{21}}$ (A39)
remains constant. $\zeta$ is the measure introduced by [34], and the multi-
index notation immediately reveals that it is equal to unity in linkage
equilibrium.
The equality of the regression and experimental average effects for constant
$\bm{\lambda}=(\lambda_{11,21},\ldots,\lambda_{12,22},\zeta)$ appears to
conflict with the result of [40] that the stipulation of $\Delta\zeta=0$ and
random mating to reset the $\lambda_{ij}$ to unities among zygotes does not
lead to the change in the mean phenotype equaling the summed products of
average effects and changes in allele frequencies (in the case that the
phenotype is fitness). Our next numerical example shows that we have indeed
reached a contradiction (Table A2).
Table A2: A trait affected by two biallelic loci. genotype | $\mathbb{E}[Y\,|\,do(\cdot)]$ | frequency | $\varepsilon$
---|---|---|---
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ | 17 | .054 | 5.0100265
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ | 12 | .036 | $-1.438691$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}$ | 13 | .257 | $-.8874187$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 14 | .140 | $-.3345667$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 18 | .080 | $-.7832893$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 10 | .039 | $-4.7832893$
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 16 | .066 | 4.7679882
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}$ | 15 | .041 | $-.6791599$
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{1}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 11 | .029 | $-3.2178824$
$\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}/\mathcal{A}^{(1)}_{2}\mathcal{A}^{(2)}_{2}$ | 20 | .258 | .4233950
Numerical testing suggests that invertibility of the Jacobian is also a
generic property of the nine-dimensional system ($p^{(1)}$, $p^{(2)}$,
$\lambda_{11,21}$, …, $\lambda_{12,22}$, $\zeta$). We numerically update the
vector of genotype frequencies in Table A2 by increasing the frequency of
allele $\mathcal{A}^{(1)}_{2}$ by $10^{-6}$. The regression average effect at
locus 1, as determined by the Levenberg-Marquardt algorithm, is approximately
2.4934. However, when we multiply this by two times $10^{-6}$, the result does
not closely agree with $G_{ij}\Delta P_{ij}$. The discrepancy is close to 12
percent and does not diminish as $\Delta p^{(1)}$ is made smaller. We conclude
that we have falsified our initial assumption that a residual-free description
of the average effects always exists.
Sampling vectors of initial genotype frequencies from the Dirichlet
distribution, we find that the changes implied by constancy of $\bm{\lambda}$
in the case of two biallelic loci do not typically produce such a large
discrepancy. The error is usually less than 7 percent. This suggests to us
that there may exist a subset of weights, distinguished by the changes in the
departures from random combination all being “small” in some sense, that can
be mathematically described. We leave this issue to future research.
The vanishing of $\overline{\varepsilon\,\mathring{\theta}}$ is still an
applicable criterion. For example, the genotype
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{2}$/
$\mathcal{A}^{(1)}_{1}\mathcal{A}^{(2)}_{1}$ can be transformed into either
double heterozygote, depending on whether the left or right gene at locus 1 is
the target of the substitution. In one case the causal effect is 6, and in the
other it is $-2$. Among all integer weightings of these two substitutions
summing to 1000, the weights (562, 438) yield the minimum. The corresponding
weighted average of the causal effects, $\alpha^{(1)}_{2}-\alpha^{(1)}_{1}$,
equals 2.496 and is also the closest to
$\beta^{(1)}_{2}-\beta^{(1)}_{1}\approx 2.493$ that can be obtained given our
discretization. The replacement of randomly chosen homologous genes can now be
used to determine ($\alpha^{(1)}_{1}$, $\alpha^{(1)}_{2}$).
## References
* [1] J Henry Bennett “Population genetics and natural selection” In _Genetica_ 28.1, 1956, pp. 297–307
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|
arxiv-papers
| 2013-04-06T02:42:08 |
2024-09-04T02:49:43.931090
|
{
"license": "Public Domain",
"authors": "James J. Lee and Carson C. Chow",
"submitter": "James Lee",
"url": "https://arxiv.org/abs/1304.1844"
}
|
1304.1851
|
# Managing Interference Correlation Through Random Medium Access
Yi Zhong, Wenyi Zhang, _Senior Member, IEEE_ and Martin Haenggi, _Senior
Member, IEEE_ Y. Zhong and W. Zhang are with Department of Electronic
Engineering and Information Science, University of Science and Technology of
China, Hefei 230027, China (email: [email protected],
[email protected]). M. Haenggi is with Department of Electrical
Engineering, University of Notre Dame, Notre Dame, IN 46556, USA (email:
[email protected]). The research has been supported by the National Basic
Research Program of China (973 Program) through grant 2012CB316004, National
Natural Science Foundation of China through grant 61071095, MIIT of China
through grant 2011ZX03001-006-01, and by the US NSF through grant CCF 1216407.
###### Abstract
The capacity of wireless networks is fundamentally limited by interference.
However, little research has focused on the interference correlation, which
may greatly increase the local delay (namely the number of time slots required
for a node to successfully transmit a packet). This paper focuses on the
question whether increasing randomness in the MAC, specifically frequency-
hopping multiple access (FHMA) and ALOHA, helps to reduce the effect of
interference correlation. We derive closed-form results for the mean and
variance of the local delay for the two MAC protocols and evaluate the optimal
parameters that minimize the mean local delay. Based on the optimal
parameters, we identify two operating regimes, the correlation-limited regime
and the bandwidth-limited regime. Our results reveal that while the mean local
delays for FHMA with $N$ sub-bands and for ALOHA with transmit probability $p$
essentially coincide when $p=\frac{1}{N}$, a fundamental discrepancy exists
between their variances. We also discuss implications from the analysis,
including an interesting mean delay-jitter tradeoff, and convenient bounds on
the tail probability of the local delay, which shed useful insights into
system design.
###### Index Terms:
ALOHA, frequency-hopping, interference correlation, local delay, Poisson point
process, stochastic geometry.
## I Introduction
### I-A Motivation
A main limitation to the capacity of wireless communication systems is
interference, which depends upon a number of factors, including the locations
of interfering transmitters. The issue of interference has been studied
extensively in the literature; however, much less attention has been paid to
the topic of interference correlation until recently. Interference correlation
generally captures the fact that the interference created by interfering
transmitters is a correlated stochastic process both spatially and temporally.
It is well recognized that correlated fading reduces the performance gain in
multi-antenna communications [1]. Likewise, it has recently been also proved
that interference correlation decreases the diversity gain [2][3].
Interference correlation partially comes come from correlated channel
attenuation, like correlated fading and shadowing, but more importantly, such
correlation stems from the spatial distribution of transmitters and the MAC
protocols since they determine the locations and the active pattern of the
interferers, which then determine the structure of the interference. The lines
of recent research can be divided into three categories based on different
configurations for the receiver:
* •
Correlation between different time slots: Assume that the receiver is equipped
with a single antenna. This line of research explored the interference
correlation at the same receiver between different time slots. Related works
include [4, 5, 6, 3].
* •
Correlation between different receive antennas: Assume that the receiver is
equipped with co-located multiple antennas. The correlation between different
antennas exists because the interferences received by different antennas come
from the same source of transmitters. Related works include [2].
* •
Correlation between different receivers: This refers to the interference
correlation between different receivers which are separated (a few wavelengths
apart). Since the network may make use of relay and cooperative transmission,
it is necessary to consider this type of interference correlation for an
accurate analysis. Related works include [5].
In this work, we focus on the interference correlation between different time
slots at the same receiver, i.e., the temporal correlation. The interference
power constitutes a stochastic process, wherein the randomness comes from
three sources: the spatial distribution of nodes, the fading and the MAC. The
interferences at two different time slots are correlated because they come
from correlated sets of transmitters and the fading, shadowing and traffic may
also be correlated. In this paper, we only focus on the correlation caused by
the spatial distribution of transmitters and the MAC, assuming that fading and
shadowing are independent. This type of correlation brings about the fact that
if transmission fails in a previous time slot, there is a significant
probability that the subsequent transmission will also fail in the next few
time slots [3][5]. Thus a simple retransmission mechanism may not be an
effective method. The most direct impact of this type of correlation is the
increase of the local delay. Local delay is defined as the number of time
slots required by a node to successfully transmit a packet to its next-hop
node111The definition of local delay in our work is consistent with [7]. In
some other works, like [6], the local delay denotes the _mean_ number of time
slots required to successfully transmit a packet..
As a motivating example, consider a spatial network without mobility or fading
and without a MAC coordinating. Hence the interference power experienced by a
receiver remains fixed for all time slots; it is a randomly variable uniquely
determined by the spatial distribution of nodes. The local delay, as a random
variable, in that extreme case is two-valued: either one frame (good
realization of the spatial distribution of nodes) or infinite (bad realization
of the spatial distribution of nodes). In this case, the transmission success
events are fully correlated (one success implies success in each time slot,
and vice versa), and the mean local delay is infinite.
In view of this, we consider some forms of man-made randomization by
introducing MAC dynamics to reduce the interference correlation. The following
analysis will be carried out in parallel under two different kinds of MAC
protocols:
* •
FHMA (Frequency-hopping multiple access): FHMA is implemented by simply
dividing the entire frequency band into $N$ sub-bands and letting each
transmitter independently choose a sub-band uniformly randomly in each time
slot. We focus on slow frequency-hopping, i.e., hopping at the time scale of a
time slot, not at the time scale of a symbol. There are three benefits by
splitting the entire frequency bands into sub-bands. First and foremost, it
increases the uncertainty in the active pattern of interfering nodes, thereby
reducing the effect of interference correlation. Second, the interference for
a given transmission is also reduced because the intensity of the interfering
transmitters are scaled by $\frac{1}{N}$. Third, the noise power is also
scaled by $\frac{1}{N}$ since each transmission occurs in a narrow sub-band.
Meanwhile, on the other side, splitting into sub-bands scales down the rate.
* •
ALOHA: In ALOHA, if a packet is to be transmitted during a time slot, the
packet will only be transmitted with a certain probability using the entire
frequency band. Decreasing the transmit probability increases the uncertainty
in the active pattern of interfering nodes and reduces the interference, while
the noise power will not be reduced. Meanwhile, transmitting probabilistically
scales down the rate.
Since FHMA is often viewed as a spread-spectrum technique, we briefly comment
on DS-CDMA. For synchronous orthogonal CDMA like those using Walsh codes, a
receiver can in theory completely reject arbitrarily strong signals from
interfering transmitters using different spreading sequences; thus, only those
transmitters using the same spreading sequence as the desired link will cause
interference. If the spreading sequence is randomly chosen for each
transmission, the analysis and results of the local delay are exactly the same
as that for FHMA. For asynchronous CDMA using pseudo-noise (PN) sequences, the
interference comes from all transmitters and is usually approximated as
Gaussian noise in the literature. The works in [8] and [9] have discussed the
difference between asynchronous CDMA and FHMA in terms of outage probability
and throughput. In asynchronous CDMA, although the desired signal is increased
by the processing gain, the interference still comes from all transmitters.
Therefore, the analysis of the local delay is similar as that for FHMA with
$N=1$, i.e., no bandwidth splitting is employed. We will show that in this
case the distribution of the local delay has a heavy tail, which results in an
infinite mean local delay.
### I-B Related Works
Recently, the tools from stochastic geometry [10] have been used extensively
in modeling and analysis of wireless communication systems; see, e.g., [11,
12, 13, 14] and references therein. This mathematical framework permits the
derivation of closed-form results for various system metrics and makes it
possible to evaluate the interference correlation. A number of works
considering the related problems are as follows. In [5] the authors evaluated
the spatio-temporal correlation coefficient of the interference and the joint
probability of success in ALOHA networks, and in [4] the authors calculated
the correlation coefficient of interference under different assumptions of
dependence. The framework for the analysis of the local delay was provided in
[6][7][15][16], where different scenarios were considered and it was observed
that the mean local delay may be infinite under certain system parameters. The
work in [17] extended the results to the case of finite mobility. In [18], a
new model, which characterizes different degrees of temporal dependence, was
proposed to evaluate the local delay by using joint interference statistics.
In [19], the optimal power control policies for different fading statistics
were proposed to minimize the mean local delay. All the above works are based
on the Poisson point process (PPP) model, while the work in [20] analyzed the
local delay in clustered networks.
### I-C Contributions
In this work, we focus on the question that whether increasing randomness in
the MAC helps reduce the local delay. We apply the so-called Poisson bipolar
model (see [13, Sec. 5.3]), and derive the mean and variance of the local
delay under FHMA and ALOHA. Based on the mean and variance of the local delay
we have derived, we explore the essential difference between the two MAC
protocols. We also evaluate the optimal number of sub-bands for FHMA and the
optimal transmit probability for ALOHA that minimize the mean local delay. The
issue of optimizing the number of sub-bands was also considered in [21], where
the optimal number of sub-bands is derived to maximize the number of
concurrent transmissions. However, such outage-based framework used in [21]
cannot capture the effects of correlated interference. In the last part of our
work, we evaluate the mean delay-jitter tradeoff and the bounds on the tail
probability of the local delay, both of which are critical issues for the
system design.
Our results reveal that the means of the local delay of the two protocols,
FHMA and ALOHA, coincide when the number of sub-bands $N$ in FHMA is equal to
the reciprocal of the transmit probability $p$ in ALOHA (with thermal noise
ignored). However, the variances of the local delay for the two protocols are
drastically different: when $p=\frac{1}{N}$ and $N\rightarrow\infty$, the
variance in FHMA converges to a constant which is typically small, while in
ALOHA the variance scales as $\Theta(N^{2})$. Moreover, we calculate bounds on
the complementary cumulative distribution function (ccdf) of the local delay
when no MAC dynamic is introduced. In that case, the distribution of the local
delay has a heavy tail, which results in an infinite mean local delay. By
employing the MAC randomness of either FHMA or ALOHA, the ccdf of the local
delay will decay fast, and the mean local delay will then be finite. This
observation reveals the underlying mechanism why even such simple MAC
protocols can greatly reduce the local delay.
The remaining part of this paper is organized as follows. Section II describes
the network model and the MAC protocols. Section III then establishes the main
analytical results of this paper, including the mean and variance of the local
delay for FHMA and for ALOHA. Section IV evaluates the optimal number of sub-
bands for FHMA and the optimal transmit probability for ALOHA that minimize
the mean local delay. Section V evaluates the optimal SINR threshold that
minimizes the mean local delay. Section VI presents the mean delay-jitter
tradeoff and the bounds on the tail probability of the local delay, and
Section VII offers the concluding remarks.
## II System Model
### II-A Network Model
To obtain the most essential features, we consider the widely used Poisson
bipolar model. In this model, the locations of the transmitters are modeled as
a PPP $\Phi=\\{x_{i}\\}\subset\mathbb{R}^{d}$ of intensity $\lambda$. Each
transmitter is associated with one receiver which is at a fixed distance
$r_{0}$ to the corresponding transmitter. In the analysis, we will condition
on a particular desired transmitter $x_{0}\in\Phi$, and denote by
$r_{0}=|x_{0}|$ the distance from this transmitter to the origin where the
receiver resides. Such conditioning is equivalent to adding the point $x_{0}$
to the PPP and guarantees that the link between $x_{0}$ and the origin is a
typical link, in the sense that this link behaves statistically the same as
all other links (see [13, Ch. 8]).
Figure 1: Spatial distribution of different network entities.
We assume that the time is divided into discrete slots with equal duration.
Each transmission attempt occupies one time slot, and if a transmission fails
in a certain time slot, a retransmission will be conducted. The local delay is
defined as the number of time slots until a packet is successfully received
[6][7]. In this paper, we assume fully backlogged nodes so that whenever a
node is scheduled to access the channel it always has data to transmit. The
local delay is thus basically the transmission delay, but not the queueing
delay.
For the propagation model, we consider the common path loss $l(r)=\kappa
r^{-\alpha}$, where $\alpha$ is the path loss exponent and $\kappa$ is a
constant. We will further discuss the effect of bounded path loss
$l(r)=\kappa(r^{\alpha}+\varepsilon)^{-1}$ in the subsection III-E. We assume
that the power fading coefficients are spatially and temporally independent
with exponential distribution of unit mean (i.e., Rayleigh fading), and let
$h_{k,x}$ be the fading coefficient between transmitter $x$ and the considered
receiver located at origin $o$ in time slot $k$. Without loss of generality,
we assume that all transmitters transmit at a normalized power level of unity.
This constant power assumption is consistent with the bipolar network model,
in which all link distances are identical. The thermal noise is assumed to be
white Gaussian with power spectral density $N_{r}$. To simplify the notations,
we introduce the normalized noise power spectral density as
$N_{0}=N_{r}/\kappa$.
We assume that the SINR threshold model is applied. That is, for each time-
frequency resource block, as long as the SINR is above a threshold $\theta$,
it can be successfully used for information transmission at spectral
efficiency $\log_{2}(1+\theta)$ bits per second per Hz. We also assume that a
packet of a fixed size needs exactly one time slot to be transmitted if it is
allocated the entire frequency band $W$ under SINR threshold $\theta$ and
successfully transmitted in that time slot. In that way, in the FHMA case if
the entire frequency band is split into $N$ sub-bands, a packet will need $N$
successful time slots. Meanwhile, in the ALOHA case, each active transmission
will make use of the entire frequency band; thus, only one successful time
slot is needed. Notice that the local delay is measured by the number of time
slots. Since different system configurations may apply different durations of
time slot, we should normalize the local delay so that the actual delays of
different system configurations can be compared fairly. The duration of each
time slot is proportional to $\frac{1}{\log_{2}(1+\theta)}$ because the size
of a packet is fixed and the spectral efficiency is proportional to
$\log_{2}(1+\theta)$. Therefore, when comparing the actual delays under
different SINR thresholds $\theta$, we normalize the local delay by
$\frac{1}{\log_{2}(1+\theta)}$ as the metric.
In static or moderately mobile network, the locations of the transmitters
during all time slots are deemed to be correlated, resulting in the temporal
interference correlation. This type of correlation decreases the successful
probability for retransmissions if the first transmission attempt failed, thus
increasing the local delay. In order to reduce the effect of interference
correlation, we study two kinds of MAC randomness described as follows.
### II-B FHMA
In the FHMA case, we assume that the total frequency band $W$ is divided into
$N$ sub-bands and each transmitter chooses a sub-band uniformly randomly,
independently of the location and the time slot (i.e., memoryless both
spatially and temporally). Let $s\in\mathbb{S}=\\{1,2,\cdots,N\\}$ be the sub-
band index, and let $\mathcal{S}_{k}(x)\in\mathbb{S}$ denote the index of the
sub-band used by node $x\in\Phi$ in time slot $k$. With these notations, the
interference at the typical receiver located at the origin $o$ in time slot
$k$ is given by
$I_{k}=\sum_{x\in\Phi\backslash\\{x_{0}\\}}h_{k,x}\kappa|x|^{-\alpha}\mathbf{1}(\mathcal{S}_{k}(x)=\mathcal{S}_{k}(x_{0})),$
(1)
where $\mathbf{1}(\cdot)$ is the indicator function and $|x|$ denotes the
distance between $x$ and the origin $o$. Note that the exclusion of $x_{0}$
from the sum over the point process does not imply that $x_{0}\notin\Phi$, but
it ensures that when we condition on $x_{0}\in\Phi$, the power received from
this node is not counted as interference.
Besides reducing the interference and breaking the correlation, introducing
FHMA has the additional benefit that the noise power decreases from $W\kappa
N_{0}$ to $\frac{W}{N}\kappa N_{0}$. By taking this noise scaling into
consideration, we obtain the SINR of the typical receiver in time slot $k$ as
$\displaystyle\mathrm{SINR}_{k}=\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad$
$\displaystyle\frac{h_{k,x_{0}}r_{0}^{-\alpha}}{\frac{WN_{0}}{N}+\sum_{x\in\Phi\backslash\\{x_{0}\\}}h_{k,x}|x|^{-\alpha}\mathbf{1}(\mathcal{S}_{k}(x)=\mathcal{S}_{k}(x_{0}))}.$
(2)
### II-C ALOHA
In the ALOHA case, let $\Phi_{k}$ be the transmitting set in time slot $k$.
The interference at the typical receiver located at the origin $o$ in time
slot $k$ is
$I_{k}=\sum_{x\in\Phi\backslash\\{x_{0}\\}}h_{k,x}\kappa|x|^{-\alpha}\mathbf{1}(x\in\Phi_{k}).$
(3)
Unlike FHMA, the noise scaling effect does not exist for ALOHA since the
entire frequency band is used for each transmission. The SINR of the typical
receiver in time slot $k$ is
$\mathrm{SINR}_{k}=\frac{h_{k,x_{0}}r_{0}^{-\alpha}}{WN_{0}+\sum_{x\in\Phi\backslash\\{x_{0}\\}}h_{k,x}|x|^{-\alpha}\mathbf{1}(x\in\Phi_{k})}.$
(4)
## III Mean and Variance of the Local Delay
In this section, we derive the mean and variance of the local delay for FHMA
and for ALOHA respectively.
### III-A FHMA
#### III-A1 Mean local delay
The following theorem gives the mean local delay in FHMA networks.
###### Theorem 1
In FHMA with $N$ sub-bands, the mean local delay is
$\displaystyle D(N)$ $\displaystyle=$ $\displaystyle
N\exp\left(\frac{A}{(N-1)^{1-\delta}N^{\delta}}+\frac{B}{N}\right),$ (5)
where $A=\lambda c_{d}r_{0}^{d}\theta^{\delta}C(\delta)$, $B=\theta
r_{0}^{\alpha}WN_{0}$, $\delta=d/\alpha$,
$C(\delta)=1/{\mathrm{sinc}(\delta)}$, and $c_{d}=|b(o,1)|$ is the volume of
the $d$-dimensional unit ball222Since the equation (5) has implied that
$D(1)=\infty$, without loss of generality, we can regard the domain of $D(N)$
as $N\geqslant 1$ with $D(1)=\infty$..
###### Proof:
Let $\mathcal{C}_{\Phi}$ be the event that a transmission succeeds conditioned
on the PPP $\Phi$. The probability for successful transmission given $\Phi$ is
the same for each time slot. Our analysis below is conditioned on $\Phi$
having a point at $x_{0}$. This means that the probability measure of the
point process is the Palm probability $\mathbb{P}^{x_{0}}$ (see Ch. 8 in
[13]). Correspondingly, the expectation, denoted by $\mathbb{E}^{x_{0}}$, is
taken with respect to the measure $\mathbb{P}^{x_{0}}$. With this notation, by
setting the SINR threshold to be $\theta$, we denote the probability of
successful transmission conditioned on $\Phi$ as
$\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})=\mathbb{P}^{x_{0}}(\mathrm{SINR}_{k}>\theta\mid\Phi)$,
which can be evaluated as
$\displaystyle\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})=\mathbb{P}^{x_{0}}(\mathrm{SINR}_{k}>\theta\mid\Phi)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathbb{P}^{x_{0}}\Big{(}h_{k,x_{0}}r_{0}^{-\alpha}>\theta\Big{(}\frac{W}{N}N_{0}+I_{k}\Big{)}\mid\Phi\Big{)}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}\Big{(}\exp\Big{(}-\theta
r_{0}^{\alpha}\Big{(}\frac{W}{N}N_{0}+I_{k}\Big{)}\Big{)}\mid\Phi\Big{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}\Big{(}\exp\Big{(}-\theta
r_{0}^{\alpha}\frac{W}{N}N_{0}-$
$\displaystyle\sum_{x\in\Phi\backslash\\{x_{0}\\}}\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\mathbf{1}(\mathcal{S}_{k}(x)=\mathcal{S}_{k}(x_{0}))\Big{)}\mid\Phi\Big{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\exp\Big{(}-\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}\Big{)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\mathbb{E}^{x_{0}}\left(\exp\left(-\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\mathbf{1}(\mathcal{S}_{k}(x)=\mathcal{S}_{k}(x_{0}))\right)\mid\Phi\right)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\exp\Big{(}-\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}\Big{)}$
$\displaystyle\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\Big{(}\frac{1}{N}\mathbb{E}^{x_{0}}\left(\exp\left(-\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\right)\mid\Phi\right)+\frac{N-1}{N}\Big{)}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(b)}}{{=}}\\!\\!\\!$
$\displaystyle\exp\Big{(}-\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}\Big{)}\\!\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\\!\Big{(}\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}\Big{)}.$
In steps $(a)$ and $(b)$ of the derivation above, we have applied the property
that the fading coefficients $h_{k,x}$ are i.i.d. random variables with
exponential distribution of unit mean. The number of time slots needed until a
successful time slot appears, denoted by $\Delta$, is a random variable called
_delay till success_ (DTS) [19]. Conditioned upon $\Phi$, the success events
in different time slots are independent with probability
$\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})$; therefore, the DTS with given
$\Phi$, denoted by $\Delta_{\Phi}$, is a random variable with geometric
distribution given by
$\mathbb{P}^{x_{0}}\left(\Delta_{\Phi}=k\right)=\left(1-\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})\right)^{k-1}\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}).$
(7)
The conditional expectation of $\Delta_{\Phi}$ is taken w.r.t. the fading and
the MAC, given by
$\mathbb{E}^{x_{0}}\left(\Delta_{\Phi}\right)=1/\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})$.
Noticing that a packet will need $N$ successful time slots to finish
transmission in FHMA, the mean local delay can be evaluated as
$\displaystyle D(N)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\mathbb{E}^{x_{0}}\left(\Delta\right)$ (8) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle
N\mathbb{E}^{x_{0}}_{\Phi}\left(\mathbb{E}^{x_{0}}\left(\Delta_{\Phi}\right)\right)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\mathbb{E}^{x_{0}}_{\Phi}\Big{(}\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\Big{)}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!\\!\\!$
$\displaystyle N\exp\Big{(}\frac{\theta r_{0}^{\alpha}WN_{0}}{N}\Big{)}$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\bigg{(}\frac{1}{\prod_{x\in\Phi\backslash\\{x_{0}\\}}\left(\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}\right)}\bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle N\exp\left(\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}\right)$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\bigg{(}\prod_{x\in\Phi\backslash\\{x_{0}\\}}\frac{1}{\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}}\bigg{)}.$
where $(a)$ follows from (LABEL:equ:succ). By applying the probability
generating functional (PGFL) of the PPP, we obtain
$\displaystyle D(N)$ (9) $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\left(\frac{\theta r_{0}^{\alpha}WN_{0}}{N}\right)$
$\displaystyle\exp\bigg{(}-\lambda\int_{\mathbb{R}^{d}}\bigg{(}1-\frac{1}{\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}}\bigg{)}\mathrm{d}x\bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle N\exp\bigg{(}\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}+\lambda
c_{d}d\int_{0}^{\infty}\frac{r^{d-1}}{\frac{N}{\theta
r_{0}^{\alpha}}r^{\alpha}+N-1}\mathrm{d}r\bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle N\exp\bigg{(}\frac{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)}{(N-1)^{1-\delta}N^{\delta}}+\frac{\theta
r_{0}^{\alpha}WN_{0}}{N}\bigg{)}.$
where $\delta=d/\alpha$, $C(\delta)$ is given by
$C(\delta)=\Gamma\left(1+\delta\right)\Gamma\left(1-\delta\right)=\frac{1}{\mathrm{sinc}(\delta)}$,
and $c_{d}=|b(o,1)|$ is the volume of the $d$-dimensional unit ball. ∎
The result in Theorem 1 is closed-form and easy to evaluate and interpret. The
value of $A$ is determined by the interference and that of $B$ is due to the
thermal noise. From (5), we have
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!D(N)=N\exp\bigg{(}A\bigg{(}1-\frac{1}{N}\bigg{)}^{\delta-1}\frac{1}{N}+\frac{B}{N}\bigg{)}$
(10) $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\bigg{(}A\bigg{(}1-\frac{\delta-1}{N}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}\bigg{)}\frac{1}{N}+\frac{B}{N}\bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\bigg{(}\frac{A+B}{N}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}\bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N+A+B+O\bigg{(}\frac{1}{N}\bigg{)}.$
The result shows that when $N$ is large, the mean local delay increases
linearly with $N$. Since $D(1)$ is infinity, there exists an optimal number of
sub-bands $N_{\mathrm{opt}}$ that minimizes the mean local delay. Inspecting
$D(N)$, we see that there are two effects by splitting the entire frequency
band into $N$ sub-bands: first, the mean local delay $D(N)$ tends to decrease
due to the reduced interference correlation; second, $D(N)$ tends to increase
since the number of time slots needed becomes $N$ times larger. In view of
this, we introduce two regimes, _correlation-limited_ regime and _bandwidth-
limited_ regime. For $N<N_{\mathrm{opt}}$, the first effect outweighs the
second one, and the network operates in the correlation-limited regime. For
$N>N_{\mathrm{opt}}$, it is the opposite and the network operates in the
bandwidth-limited regime.
In the above, we have derived results under the assumption that the frequency
allocation is dynamic (i.e., the sub-bands are allocated randomly and
independently in each time slot). Alternatively, one could consider the case
where the frequency allocation is static over time. That case is exactly the
same as the case where no frequency splitting is applied, with the only
difference that the intensity of the interfering transmitters is scaled down
to $\lambda/N$. The mean local delay in that case is also infinite. This fact
explains that even though frequency splitting is introduced, if the sub-bands
are not reallocated randomly temporally, the mean local delay will still be
infinite. This is a nontrivial observation since it reveals that the reduction
of the mean local delay by introducing FHMA does not come from reducing the
interference or the thermal noise, but mainly comes from reducing the
interference correlation.
Based on Theorem 1, we show how the normalized mean local delay
$\frac{D(N)}{\log_{2}(1+\theta)}$ varies with $N$ numerically. As for the
parameters, we ignore the thermal noise ($N_{0}=0$) and set the intensity of
transmitters as $\lambda=0.01\mathrm{m}^{-2}$ by default, which means that the
coverage area of each transmitter is $100\mathrm{m}^{2}$ on average,
reasonable for a typical deployment of WLAN. The path loss exponent is set as
$\alpha=4$ by default, and the distance between the receiver and the typical
desired transmitter is $r_{0}=5\mathrm{m}$. Let $\theta$ be the outage
threshold for SINR. The relationship between $\frac{D(N)}{\log_{2}(1+\theta)}$
and $N$ is depicted in Fig. 2.
By changing the values of $\alpha$ and $\lambda$ respectively, we get the
curves in Fig. 2. Comparing the curves in Fig. 2(a) with those in Fig. 2(b)
and Fig. 2(c), we observe that the optimal number of sub-bands increases when
$\alpha$ decreases or when $\lambda$ increases. This observation is consistent
with the intuition: Smaller $\alpha$ implies that the signal strength decays
more slowly with distance, and larger $\lambda$ implies that more transmitters
exist in the same region, so in both cases more interference is created.
Therefore, the entire frequency band should be divided into more sub-bands,
namely larger $N_{\mathrm{opt}}$, to reduce the interference and interference
correlation.
(a) $\lambda=0.01$ and $\alpha=4$.
(b) $\lambda=0.01$ and $\alpha=3$.
(c) $\lambda=0.04$ and $\alpha=4$.
Figure 2: The normalized mean local delay $\frac{D(N)}{\log_{2}(1+\theta)}$ as
a function of the number of sub-bands $N$, when $d=2$, $r=5$m, and thermal
noise ignored.
#### III-A2 Variance of the local delay
The mean local delay discussed above has characterized the mean number of time
slots needed until a packet is successfully transmitted. In order to better
understand the distribution of the local delay, we also derive its variance.
The following theorem gives the variance of the local delay for FHMA.
###### Theorem 2
In FHMA with $N$ sub-bands, the variance of the local delay is
$V(N)=N\left(N+1\right)\exp\bigg{(}\frac{(2N-1-\delta)A}{N^{\delta}(N-1)^{2-\delta}}+\frac{2B}{N}\bigg{)}\\\
$ $\qquad\qquad-D(N)-D^{2}(N).$ (11)
###### Proof:
In order to transmit a packet in FHMA, $N$ successful transmissions are
needed. Letting $\Delta_{i}$ $(1\leq i\leq N)$ be the DTS of the $i$th
transmission, we get the local delay of a packet as
$\sum_{i=1}^{N}\Delta_{i}$. For $1\leq i,j\leq N$ and $i\neq j$, $\Delta_{i}$
and $\Delta_{j}$ are dependent because the interference of the $i$th
transmission and that of the $j$th transmission are correlated. However, if we
condition on $\Phi$, $\\{\Delta_{i}\\}$ are i.i.d. random variables with
geometric distribution given by (7). With these notations, we obtain the
variance of the local delay as
$\displaystyle\\!\\!\\!\\!\\!\\!V(N)=\mathbb{E}^{x_{0}}\left(\left(\sum_{i=1}^{N}\Delta_{i}\right)^{2}\right)-\left(\mathbb{E}^{x_{0}}\left(\sum_{i=1}^{N}\Delta_{i}\right)\right)^{2}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}\left(\sum_{i=1}^{N}\Delta_{i}^{2}+\sum_{\stackrel{{\scriptstyle
i,j=1}}{{i\neq
j}}}^{N}2\Delta_{i}\Delta_{j}\right)-\left(\sum_{i=1}^{N}\mathbb{E}^{x_{0}}\left(\Delta_{i}\right)\right)^{2}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!\\!\\!$
$\displaystyle\sum_{i=1}^{N}\mathbb{E}^{x_{0}}\left(\Delta_{i}^{2}\right)+\sum_{\stackrel{{\scriptstyle
i,j=1}}{{i\neq
j}}}^{N}2\mathbb{E}^{x_{0}}\left(\Delta_{i}\Delta_{j}\right)-D^{2}(N),$
where $(a)$ follows from the definition of the mean local delay. By applying
the total expectation formula, we have
$\displaystyle\\!\\!\\!V(N)\\!\\!\\!$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\\!\\!\\!\sum_{i=1}^{N}\mathbb{E}^{x_{0}}_{\Phi}\left(\mathbb{E}^{x_{0}}\left(\Delta_{i}^{2}\mid\Phi\right)\right)$
(12) $\displaystyle+\sum_{\stackrel{{\scriptstyle i,j=1}}{{i\neq
j}}}^{N}2\mathbb{E}^{x_{0}}_{\Phi}\left(\mathbb{E}^{x_{0}}\left(\Delta_{i}\Delta_{j}\mid\Phi\right)\right)-D^{2}(N)$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$
$\displaystyle\\!\\!\\!N\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{2-\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)$
$\displaystyle+N\left(N-1\right)\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)-D^{2}(N)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\\!\\!\\!N\left(N+1\right)\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)$
$\displaystyle-N\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\right)-D^{2}(N)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\\!\\!\\!N\left(N+1\right)\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)$
$\displaystyle-D(N)-D^{2}(N),$
where $(b)$ follows from the second moment of the geometrically distributed
random variable.
From (LABEL:equ:succ), we have
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\exp\left(\frac{2\theta
r_{0}^{\alpha}WN_{0}}{N}\right)$
$\displaystyle\quad\mathbb{E}^{x_{0}}_{\Phi}\Bigg{(}\frac{1}{\prod_{x\in\Phi\backslash\\{x_{0}\\}}\left(\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}\right)^{2}}\Bigg{)}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(c)}}{{=}}\\!\\!\\!$
$\displaystyle\exp\Bigg{(}\frac{2\theta r_{0}^{\alpha}WN_{0}}{N}$
$\displaystyle\quad-\lambda\int_{\mathbb{R}^{d}}\Bigg{(}1-\frac{1}{\left(\frac{1}{N}\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+\frac{N-1}{N}\right)^{2}}\Bigg{)}\mathrm{d}x\Bigg{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\exp\Big{(}\frac{2\theta
r_{0}^{\alpha}WN_{0}}{N}$ $\displaystyle\\!\\!\\!-\lambda
c_{d}d\int_{0}^{\infty}\left(1-\frac{N^{2}(1+\theta
r_{0}^{\alpha}r^{-\alpha})^{2}}{\left(N+(N-1)\theta
r_{0}^{\alpha}r^{-\alpha}\right)^{2}}\right)r^{d-1}\mathrm{d}r\Big{)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\exp\left(\frac{2\theta
r_{0}^{\alpha}WN_{0}}{N}+\frac{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)(2N-1-\delta)}{N^{\delta}(N-1)^{2-\delta}}\right),$
where $(c)$ follows by applying the PGFL of the PPP. Plugging
(LABEL:equ:quadprob) into (12), we get the variance of the local delay as in
Theorem 2. ∎
### III-B ALOHA
The fundamental difference between FHMA and ALOHA is that if a packet is to be
transmitted during a time slot, in FHMA the packet will be surely transmitted
by randomly choosing a sub-band, while in ALOHA the packet will only be
transmitted with a given probability. Similar to the analysis of FHMA, we also
assume that a packet needs exactly one time slot if it is allocated the entire
frequency band $W$ under SINR threshold $\theta$ and successfully transmitted
in that time slot. We assume that each node transmits with probability $p$ in
each time slot and if it transmits, it will make use of the entire frequency
band. In that way, only one successful time slot is needed to transmit a
packet, and the local delay is the DTS of one transmission, denoted by
$\Delta$.
#### III-B1 Mean local delay
The following theorem gives the mean local delay for ALOHA.
###### Theorem 3
In ALOHA with transmit probability $p$, the mean local delay is
$\displaystyle\widetilde{D}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{p}\exp\bigg{(}\frac{pA}{(1-p)^{1-\delta}}+B\bigg{)}.$
(14)
###### Proof:
In each time slot, a packet will be transmitted with probability $p$ and the
transmission will be successful with probability
$\mathbb{P}^{x_{0}}(\mathrm{SINR}_{k}>\theta\mid\Phi)$ conditioned upon
$\Phi$. Therefore, similar to the derivation of (LABEL:equ:succ), the
probability for successfully transmitting a packet conditioned upon $\Phi$ in
a time slot is
$\displaystyle\\!\\!\\!\\!\\!\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})\stackrel{{\scriptstyle(a)}}{{=}}p\mathbb{P}^{x_{0}}(\mathrm{SINR}_{k}>\theta\mid\Phi)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
p\mathbb{P}^{x_{0}}\left(h_{k,x_{0}}r_{0}^{-\alpha}>\theta\left({W}N_{0}+I_{k}\right)\mid\Phi\right)$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(b)}}{{=}}\\!\\!\\!\\!$
$\displaystyle p\mathbb{E}^{x_{0}}\left(\exp\left(-\theta
r_{0}^{\alpha}\left({W}N_{0}+I_{k}\right)\right)\mid\Phi\right)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
p\mathbb{E}^{x_{0}}\Big{(}\exp\Big{(}-\theta r_{0}^{\alpha}{W}N_{0}$
$\displaystyle-\\!\\!\sum_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\mathbf{1}(x\in\Phi_{k})\Big{)}\mid\Phi\Big{)}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle p\exp\left(-{\theta
r_{0}^{\alpha}WN_{0}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\mathbb{E}^{x_{0}}\left(\exp\left(-\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\mathbf{1}(x\in\Phi_{k})\right)\mid\Phi\right)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle p\exp\left(-{\theta
r_{0}^{\alpha}WN_{0}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\Big{(}p\mathbb{E}^{x_{0}}\Big{(}\exp\big{(}-\theta
r_{0}^{\alpha}h_{k,x}|x|^{-\alpha}\big{)}\mid\Phi\Big{)}+1-p\Big{)}$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(c)}}{{=}}\\!\\!\\!\\!$
$\displaystyle p\exp\big{(}-\theta
r_{0}^{\alpha}WN_{0}\big{)}\\!\\!\\!\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\\!\\!\Big{(}\frac{p}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+1-p\Big{)}.$
where $(a)$ is because a transmission occurs with probability $p$, and $(b)$
and $(c)$ follows because the fading coefficients $h_{k,x}$ are i.i.d. random
variables with exponential distribution of unit mean. Then, the mean local
delay for ALOHA is given by
$\displaystyle\widetilde{D}(p)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\Big{(}\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\Big{)}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{p}\exp\Big{(}\theta r_{0}^{\alpha}WN_{0}$
$\displaystyle-\lambda\int_{\mathbb{R}^{d}}\Big{(}1-\frac{1}{\frac{p}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+1-p}\Big{)}\mathrm{d}x\Big{)}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{p}\exp\left(\frac{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)p}{(1-p)^{1-\delta}}+\theta
r_{0}^{\alpha}WN_{0}\right).$
Applying the definition of $A$ and $B$ in Theorem 1, we obtain the result in
Theorem 3. ∎
#### III-B2 Variance of the local delay
The variance of the local delay in ALOHA is given by the following theorem.
###### Theorem 4
In ALOHA with transmit probability $p$, the variance of the local delay is
$\displaystyle\widetilde{V}(p)$ $\displaystyle=$
$\displaystyle\frac{2}{p^{2}}\exp\Big{(}\frac{(2-p-\delta
p)pA}{(1-p)^{2-\delta}}+2B\Big{)}$ (17)
$\displaystyle\qquad-\widetilde{D}(p)-\widetilde{D}^{2}(p).$
###### Proof:
In the ALOHA case, in order to transmit a packet, one successful transmission
is needed. The variance of local delay for ALOHA is thus
$\displaystyle\widetilde{V}(p)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}\left(\Delta^{2}\right)-\left(\mathbb{E}^{x_{0}}\left(\Delta\right)\right)^{2}$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\left(\mathbb{E}^{x_{0}}\left(\Delta^{2}|\Phi\right)\right)-\widetilde{D}^{2}(p)$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(b)}}{{=}}\\!\\!\\!\\!$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\bigg{(}\frac{2-\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\bigg{)}-\widetilde{D}^{2}(p)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
2\mathbb{E}^{x_{0}}_{\Phi}\bigg{(}\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\bigg{)}-\widetilde{D}(p)-\widetilde{D}^{2}(p),$
where $(a)$ follows from the total expectation formula, and $(b)$ follows from
the second moment of the geometrically distributed random variable. From
(LABEL:equ:succ_aloha), we have
$\displaystyle\\!\\!\\!\\!\\!\\!\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{(\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi}))^{2}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!\\!$ $\displaystyle
p^{-2}\exp\big{(}{2\theta r_{0}^{\alpha}WN_{0}}\big{)}$
$\displaystyle\mathbb{E}^{x_{0}}_{\Phi}\bigg{(}\frac{1}{\prod_{x\in\Phi\backslash\\{x_{0}\\}}\big{(}p\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+1-p\big{)}^{2}}\bigg{)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\stackrel{{\scriptstyle(c)}}{{=}}\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle p^{-2}\exp\bigg{(}{2\theta r_{0}^{\alpha}WN_{0}}$
$\displaystyle-\lambda\int_{\mathbb{R}^{d}}\bigg{(}1-\frac{1}{\big{(}p\frac{1}{1+\theta
r_{0}^{\alpha}|x|^{-\alpha}}+1-p\big{)}^{2}}\bigg{)}\mathrm{d}x\bigg{)}$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{1}{p^{2}}\exp\left(2\theta
r_{0}^{\alpha}WN_{0}+\frac{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)(2-p-\delta
p)p}{(1-p)^{2-\delta}}\right),$
where $(c)$ follows from the probability generating functional (PGFL) of the
PPP. Plugging (LABEL:equ:quadprob_aloha) into (LABEL:equ:vartmp_aloha) and
applying the definition of $A$ and $B$ in Theorem 1, we get the variance of
the local delay in Theorem 4. ∎
### III-C Comparison Between FHMA and ALOHA
#### III-C1 Mean local delay
The mean local delay in FHMA is given by $D(N)$ in (5), and that in ALOHA is
given by $\widetilde{D}(p)$ in (14). In ALOHA, if the transmit probability $p$
is set as $\frac{1}{N}$, by comparing $\widetilde{D}(\frac{1}{N})$ to the
result of FHMA, $D(N)$ given in (5), we observe that the only difference lies
in the thermal noise term. In FHMA, the entire frequency band is divided into
a number of sub-bands, thus reducing the noise power. However, with ALOHA, the
noise scaling effect does not exist since the entire frequency band is used
for transmission. The mean local delays of the two schemes are the same if we
ignore the thermal noise term and set $p=\frac{1}{N}$.
#### III-C2 Variance of the local delay
Comparing (11) and (17), we observe that even if the noise is ignored and the
transmit probability is set as $p=\frac{1}{N}$ in ALOHA, there is still an
significant difference between the variances of the two schemes when $N>1$: in
FHMA, a factor $N\left(N+1\right)$ exists in the first term, while in ALOHA
the factor is $2N^{2}$. When $N>1$, we have $V(N)<\widetilde{V}(\frac{1}{N})$
and this illustrates that the variance of the local delay for FHMA is less
than that for ALOHA (see Fig. 3). We further observe from Fig. 3 that when
$N\rightarrow\infty$, the variance for FHMA stabilizes at a typically small
value, while for ALOHA, the variance increases quickly with $N$. To understand
the limiting characteristics quantitatively, we evaluate how the variances of
the local delay scale with $N$ in the following proposition.
###### Proposition 1
For FHMA with number of sub-bands $N>1$, the variance of the local delay is
$\displaystyle V(N)=(2-\delta)A+B+O\bigg{(}\frac{1}{N}\bigg{)}=\Theta(1).$
(20)
For ALOHA with transmit probability $p=\frac{1}{N}<1$, the variance of the
local delay is
$\displaystyle\widetilde{V}(\frac{1}{N})=\Theta(N^{2}).$ (21)
###### Proof:
For FHMA with number of sub-bands $N>1$, from (5), we have
$\displaystyle D(N)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\bigg{(}\frac{A+B}{N}+\frac{(1-\delta)A}{N^{2}}+O\bigg{(}\frac{1}{N^{3}}\bigg{)}\bigg{)}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!$ $\displaystyle N+(A+B)$
$\displaystyle\\!\\!\\!\\!+\bigg{(}\frac{(A+B)^{2}}{2}+(1-\delta)A\bigg{)}\frac{1}{N}+O\bigg{(}\frac{1}{N^{2}}\bigg{)}.$
Then, $D^{2}(N)$ is given by
$\displaystyle D^{2}(N)$ $\displaystyle=$ $\displaystyle
N^{2}+2(A+B)N+2(A+B)^{2}$ (23)
$\displaystyle+2(1-\delta)A+O\left(\frac{1}{N}\right).$
From (11), we have the variance of the local delay as
$\displaystyle V(N)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N(N+1)\exp\bigg{(}\frac{2B}{N}+\frac{2A}{N}-\frac{3(\delta-1)A}{N^{2}}$ (24)
$\displaystyle+O\bigg{(}\frac{1}{N^{3}}\bigg{)}\bigg{)}-D(N)-D^{2}(N)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!$ $\displaystyle N^{2}+(2A+2B+1)N+2(A+B)$
$\displaystyle+2(A+B)^{2}-3(\delta-1)A$
$\displaystyle-D(N)-D^{2}(N)+O\left(\frac{1}{N}\right).$
Plugging (LABEL:equ:var_lim1) and (23) into (24), we get the result in (20).
The derivations of the limiting for ALOHA is similar, and we omit the details
of that proof. ∎
Figure 3: Normalized variances of the local delay for FHMA,
$\frac{V(N)}{(\log_{2}(1+\theta))^{2}}$, and for ALOHA,
$\frac{\widetilde{V}(\frac{1}{N})}{(\log_{2}(1+\theta))^{2}}$, as a function
of the number of sub-bands $N$, when $d=2$, $\lambda=0.01\mathrm{m}^{-2}$,
$\alpha=4$, $r=5$m, and thermal noise ignored.
### III-D Finiteness of the Mean Local Delay
To understand why the mean local delay goes to infinity when $N$ is set to
one, let us consider the expression for the mean local delay
$D=N\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\right)$
in FHMA, where $\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}$ is a random
variable with support set $(1,+\infty)$ because it is the reciprocal of the
successful transmit probability conditioned upon the PPP $\Phi$. When $N=1$,
the expectation
$\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\right)$
is infinity because the ccdf of
$\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}$ has a heavy tail. To show
the heavy tail behavior, let us derive a lower bound for the ccdf of the local
delay when $N=1$. Ignoring the thermal noise term in (LABEL:equ:succ), for
$N=1$ and any $t\in(1,+\infty)$, we have
$\displaystyle\\!\\!\mathbb{P}^{x_{0}}\bigg{(}\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}>t\bigg{)}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{P}^{x_{0}}\bigg{(}\\!\\!\\!\prod_{x\in\Phi\backslash\\{x_{0}\\}}\\!\\!\\!\\!\\!\big{(}1+\theta
r_{0}^{\alpha}|x|^{-\alpha}\big{)}>t\bigg{)}$ (25)
$\displaystyle\\!\\!\\!\\!>\\!\\!\\!\\!$
$\displaystyle\mathbb{P}^{x_{0}}\left(1+\theta
r_{0}^{\alpha}|x_{\mathrm{min}}|^{-\alpha}>t\right)$
$\displaystyle\\!\\!\\!\\!>\\!\\!\\!\\!$
$\displaystyle\mathbb{P}^{x_{0}}\left(\theta
r_{0}^{\alpha}|x_{\mathrm{min}}|^{-\alpha}>t\right)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{P}^{x_{0}}\left(|x_{\mathrm{min}}|<\theta^{\frac{1}{\alpha}}t^{-\frac{1}{\alpha}}r_{0}\right),$
where $x_{\mathrm{min}}=\mathrm{argmin}_{x\in\Phi\setminus\\{x_{0}\\}}|x|$ is
the nearest interfering transmitter to the receiver. The distance between the
receiver and its nearest interfering transmitter has cumulative distribution
function (cdf) as $b(r)=1-\exp(-c_{d}\lambda r^{d})$. Substituting this cdf
into (25) and letting $\delta=\frac{2}{\alpha}$, we get
$\displaystyle\\!\\!\\!\\!\mathbb{P}^{x_{0}}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}>t\right)$
$\displaystyle\\!\\!\\!\\!>\\!\\!\\!\\!$ $\displaystyle
1-\exp\left(-c_{d}\lambda\theta^{\delta}t^{-\delta}r_{0}^{d}\right)\qquad$
(26) $\displaystyle\\!\\!\\!\\!\sim\\!\\!\\!\\!$
$\displaystyle\frac{c_{d}\lambda\theta^{\delta}r_{0}^{d}}{t^{\delta}},\quad
t\rightarrow\infty.$ (27)
The function $g(t)=1-\exp\left(-C_{0}t^{-\delta}\right)$, where
$C_{0}=c_{d}\lambda\theta^{\delta}r_{0}^{d}$, gives a lower bound for the ccdf
of the local delay when $N=1$ (see Fig. 4).
By the identity of $\mathbb{E}(X)=\int_{0}^{\infty}\mathbb{P}(X>t)\mathrm{d}t$
for any non-negative random variable $X$ and the inequality
$e^{-x}<1-x+\frac{x^{2}}{2}$ for $x>0$, we have
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\mathbb{E}^{x_{0}}_{\Phi}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}\right)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\int_{0}^{\infty}\left(\mathbb{P}^{x_{0}}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}>t\right)\right)\mathrm{d}t$
(28) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1+\int_{1}^{\infty}\left(\mathbb{P}^{x_{0}}\left(\frac{1}{\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})}>t\right)\right)\mathrm{d}t$
$\displaystyle\\!\\!\\!\\!>\\!\\!\\!\\!$ $\displaystyle
1+\int_{1}^{\infty}\left(1-\exp\left(-C_{0}t^{-\delta}\right)\right)\mathrm{d}t$
$\displaystyle\\!\\!\\!\\!>\\!\\!\\!\\!$ $\displaystyle
1+\int_{1}^{\infty}\left(C_{0}t^{-\delta}-\frac{1}{2}C_{0}^{2}t^{-2\delta}\right)\mathrm{d}t$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle\infty.$
Figure 4: Lower bound for the ccdf of the local delay, given by (26), when
$N=1$ in the 2-dimensional case ($d=2$). The intensity of transmitters is
$\lambda=0.01\mathrm{m}^{-2}$ and the path loss exponent is $\alpha=4$.
When FHMA is applied with $N>1$, there is an additional term $\frac{N-1}{N}$
in the success probability given by (LABEL:equ:succ), which prevents
$\mathbb{P}^{x_{0}}(\mathcal{C}_{\Phi})$ from getting too small when $|x|$
approaches zero. It can be interpreted intuitively that, although there are
some interfering transmitters very close to the receiver, the application of
FHMA guarantees that there is always a relatively large probability that those
transmitters do not continuously cause interference to the receiver.
### III-E The effect of bounded path loss function
In the discussion above, we have considered the unbounded path loss function
$l(r)=\kappa r^{-\alpha}$. Though the unbounded path loss function is an
idealized model, it gives an effective approximation to the actual path loss
and results in concise results [22, 23]. In this subsection, we compare the
results derived under the unbounded path loss function to that under the
bounded path loss function $l(r)=\kappa(r^{\alpha}+\varepsilon)^{-1}$, where
$\varepsilon>0$. The unbounded path loss function is the limiting case of the
bounded path loss function as $\varepsilon\rightarrow 0$.
Without loss of generality, we take FHMA as an example. By replacing
$l(r)=\kappa r^{-\alpha}$ with $l(r)=\kappa(r^{\alpha}+\varepsilon)^{-1}$ in
the derivations of Theorem 1 and Theorem 2, we obtain the mean and variance of
the local delay under the assumption of bounded path loss function as follows.
$\displaystyle
D_{\varepsilon}(N)=N\exp\bigg{(}\frac{(r_{0}^{\alpha}+\varepsilon)\theta
WN_{0}}{N}$ $\displaystyle+\frac{\lambda
c_{d}(r_{0}^{\alpha}+\varepsilon)\theta
C(\delta)}{(N\varepsilon+(N-1)\theta(r_{0}^{\alpha}+\varepsilon))^{1-\delta}N^{\delta}}\bigg{)},$
(29)
$\displaystyle V_{\varepsilon}(N)=N\left(N+1\right)\exp\bigg{(}\frac{2\theta
r_{0}^{\alpha}WN_{0}}{N}$
$\displaystyle+\frac{(2N\varepsilon+(2N-1-\delta)\theta(r_{0}^{\alpha}+\varepsilon))\lambda
c_{d}(r_{0}^{\alpha}+\varepsilon)\theta
C(\delta)}{N^{\delta}(N\varepsilon+(N-1)\theta(r_{0}^{\alpha}+\varepsilon))^{2-\delta}}\bigg{)}$
$\qquad\qquad-D_{\varepsilon}(N)-D_{\varepsilon}^{2}(N).$ (30)
When $N=1$, both $D_{\varepsilon}(1)$ and $V_{\varepsilon}(1)$ are finite if
$\varepsilon>0$. Setting $\varepsilon=0$ reproduces the results for the
unbounded model, as expected. As can be seen, the difference between the
results for the unbounded model and the bounded one decreases with increasing
$r_{0}$ or decreasing $\theta$. In order to evaluate the difference, we set
$\varepsilon$ as the typical value $\kappa$ (i.e., the path loss becomes
$l(r)=\kappa(r^{\alpha}+\kappa)^{-1}$) such that the received power never
exceeds the transmitted one without fading. The value of $\kappa$ is the path
loss at $1$m TX-RX separation, which is rather small, typically like $-30$dB
[24, Ch. 3]. Therefore, as $\varepsilon=\kappa\rightarrow 0$, the mean local
delay for $N=1$ is approximated as
$D_{\varepsilon}(1)\sim\exp\left(\theta r_{0}^{\alpha}WN_{0}+\frac{\lambda
c_{d}r_{0}^{\alpha}\theta
C(\delta)}{\varepsilon^{1-\delta}}\right),\quad\varepsilon\rightarrow 0.$ (31)
It is observed that $D_{\varepsilon}(1)$ increases exponentially with respect
to $1/\epsilon^{1-\delta}$ as $\varepsilon\rightarrow 0$. For the realistic
bounded path loss model, in which $\varepsilon$ is rather small, the mean
local delay when $N=1$ is finite though extremely large. Thus, we can conclude
that for the realistic bounded path loss model, when $N>2$ the boundedness of
the path loss has only negligible effect on the mean and the variance of the
local delay; when $N=1$ the mean local delay is extremely large and thus can
be considered as infinity for practical purposes.
## IV Optimal Parameters To Minimize Mean Local Delay
In this section, we analyze the optimal number of sub-bands in FHMA and
optimal transmit probability in ALOHA to minimize the mean local delay.
Deriving the optimal parameters is difficult, and the results may not be
compact; thus, we resort to deriving tight bounds for the optimal values.
### IV-A FHMA
In the derivation, we relax $N$ to be continuous and subsequently take the
actual optimal number to be a nearby integer. The following theorem gives the
bounds of the optimal number of sub-bands.
###### Theorem 5
The bounds of the optimal number of sub-bands that minimizes the mean local
delay are given by
$\displaystyle\\!\\!\\!\\!N_{\mathrm{opt}}\in[\lfloor t_{0}\rfloor,\lceil
t_{0}\rceil+2],\quad t_{0}=\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta r_{0}^{\alpha}WN_{0}.$
###### Proof:
Based on the result of (5), we get the derivative of the mean local delay
$D^{\prime}(N)$ when $N>1$ as follows
$D^{\prime}(N)=f(N)\exp\big{(}\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)(N-1)^{\delta-1}N^{-\delta}$
$\qquad+\theta r_{0}^{\alpha}WN_{0}N^{-1}\big{)},$ (33)
where
$f(N)=1-\frac{1}{N}\bigg{(}\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)\bigg{(}\frac{N}{N-1}\bigg{)}^{1-\delta}\frac{N-\delta}{N-1}$
$+\theta r_{0}^{\alpha}WN_{0}\bigg{)}.$ (34)
We observe that $f(N)$ is strictly monotonically increasing in $N$; this means
that there is only one optimal value $N_{\mathrm{opt}}$ that satisfies
$D^{\prime}(N_{\mathrm{opt}})=0$, which is given by $f(N_{\mathrm{opt}})=0$.
From $f(N_{\mathrm{opt}})=0$, we get
$\displaystyle N_{\mathrm{opt}}$ $\displaystyle=$ $\displaystyle\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)\left(\frac{N_{\mathrm{opt}}}{N_{\mathrm{opt}}-1}\right)^{2-\delta}\frac{N_{\mathrm{opt}}-\delta}{N_{\mathrm{opt}}}$
(35) $\displaystyle+\theta r_{0}^{\alpha}WN_{0}.$
Since $N_{\mathrm{opt}}/(N_{\mathrm{opt}}-1)>1$ and $0<\delta<1$, we have
$\displaystyle N_{\mathrm{opt}}$ $\displaystyle>$ $\displaystyle\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)\left(\frac{N_{\mathrm{opt}}}{N_{\mathrm{opt}}-1}\right)^{2-\delta}\frac{N_{\mathrm{opt}}-1}{N_{\mathrm{opt}}}$
(36) $\displaystyle+\theta r_{0}^{\alpha}WN_{0}$ $\displaystyle>$
$\displaystyle\lambda c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta
r_{0}^{\alpha}WN_{0},$
This gives a lower bound for the optimal value $N_{\mathrm{opt}}$. Next we
derive an upper bound for $N_{\mathrm{opt}}$. From (35), we have
$\displaystyle N_{\mathrm{opt}}$ $\displaystyle<$ $\displaystyle\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)\left(\frac{N_{\mathrm{opt}}}{N_{\mathrm{opt}}-1}\right)^{2}+\theta
r_{0}^{\alpha}WN_{0}$ $\displaystyle<$ $\displaystyle\left(\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta
r_{0}^{\alpha}WN_{0}\right)\left(\frac{N_{\mathrm{opt}}}{N_{\mathrm{opt}}-1}\right)^{2}.$
Then, we have
$\displaystyle\frac{(N_{\mathrm{opt}}-1)^{2}}{N_{\mathrm{opt}}}$
$\displaystyle<$ $\displaystyle\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta r_{0}^{\alpha}WN_{0}.$
$\displaystyle N_{\mathrm{opt}}-2+\frac{1}{N_{\mathrm{opt}}}$ $\displaystyle<$
$\displaystyle\lambda c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta
r_{0}^{\alpha}WN_{0}.$ $\displaystyle N_{\mathrm{opt}}<$ $\displaystyle\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+\theta r_{0}^{\alpha}WN_{0}+2.$ (37)
Combining (36) and (37) and noting that $N$ is an integer, we get the bounds
of $N_{\mathrm{opt}}$. ∎
The bounds given here are rather simple and tight. If frequency splitting is
not applied (the case of $N=1$), the mean local delay will surely be infinite.
To guarantee a finite mean local delay, the value of $N$ should be at least
two. It is valuable to investigate for which range of parameters the optimal
value $N_{\mathrm{opt}}$ will be two. The following corollary gives such a
condition.
###### Corollary 1
If the intensity of transmitters $\lambda$ satisfies the following inequality,
$\displaystyle\lambda$ $\displaystyle<$
$\displaystyle\frac{\ln\frac{3}{2}-\frac{1}{6}\theta
r_{0}^{\alpha}WN_{0}}{c_{d}r_{0}^{d}\theta^{\delta}C(\delta)(2^{-\delta}-2^{\delta-1}3^{-\delta})},$
(38)
the optimal number of sub-bands $N_{\mathrm{opt}}$ that minimizes the mean
local delay will be two.
###### Proof:
Since we have proved that mean local delay $D(N)$ is a function that first
decreases and then increases with $N$, the condition for $N_{\mathrm{opt}}=2$
is $D(2)<D(3)$. By substituting the expression of mean local delay (5) into
$D(2)<D(3)$, we get the condition in the corollary. Notice that the right side
of the inequality in (38) may be negative, in which case the condition for
$\lambda$ cannot be satisfied and then the optimal number of sub-bands will
never be two. ∎
In Fig. 5, we plot the optimal number of sub-bands $N_{\mathrm{opt}}$ and its
bounds given in (LABEL:equ:thmbounds) as a function of the path loss exponent
for different $\theta$. The optimal number $N_{\mathrm{opt}}$ is obtained by
numerical calculation of the solution of the equation (35). We observe from
Fig. 5 that the bounds are quite tight and give excellent approximation of the
value $N_{\mathrm{opt}}$. This figure also shows that the optimal value
$N_{\mathrm{opt}}$ decreases with increasing path loss exponent, which
verifies our aforementioned discussion regarding Fig. 2. The curves show that
when the path loss exponent is fixed, the optimal number $N_{\mathrm{opt}}$ is
an increasing function of the SINR threshold $\theta$, which can be also
perceived from the expression (LABEL:equ:thmbounds). This is reasonable since
with larger SINR threshold, the condition for successful transmission becomes
harsher, and more sub-bands are needed to meet the stronger requirements. In
Fig. 6, we plot the minimum value of the normalized mean local delay when the
optimum number of sub-bands $N_{\mathrm{opt}}$ is used. We observe from Fig. 6
that there are intersection points between different curves, implying that the
choice of SINR threshold $\theta$ has direct impact on the mean local delay.
In Section V, we will try to obtain the optimal SINR threshold.
Figure 5: Optimal number of sub-bands $N_{\mathrm{opt}}$ and its bounds
$(t_{0},t_{0}+2)$ as a function of the path loss exponent for varying
$\alpha$. Figure 6: Minimum of the normalized mean local delay
$\frac{D(N_{\mathrm{opt}})}{\log_{2}(1+\theta)}$ as a function of the path
loss exponent $\alpha$.
### IV-B ALOHA
Since the expressions of the mean local delays for ALOHA and for FHMA are the
same when $p=1/N$ and thermal noise ignored, we give the optimal transmit
probability in the following theorem directly and omit the proof.
###### Theorem 6
The bounds of the optimal transmit probability which minimizes the mean local
delay is
$\displaystyle p_{\mathrm{opt}}\in\left(\frac{1}{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)+2},\frac{1}{\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)}\right).$ (39)
## V Optimal SINR Threshold $\theta$
In the discussion above, we have already derived tight bounds for the optimal
number of sub-bands $N_{\mathrm{opt}}$ and the optimal transmit probability
$p_{\mathrm{opt}}$ to minimize the mean local delay when the SINR threshold
$\theta$ is fixed. The following analysis will focus on deriving the optimal
threshold $\theta_{\mathrm{opt}}$ or its bounds when the number of sub-bands
$N$ or the transmit probability $p$ is fixed. However, as mentioned in Section
II, the duration of each time slot is proportional to
$\frac{1}{\log_{2}(1+\theta)}$. In order to characterize the actual delay, we
slightly modify the optimization objective as the normalized mean local delay,
i.e., $\frac{D(N)}{\log_{2}(1+\theta)}$ for FHMA and
$\frac{\widetilde{D}(p)}{\log_{2}(1+\theta)}$ for ALOHA. In the following
analysis, we consider two asymptotic regimes: the interference-limited regime
and the noise-limited regime. The interference-limited regime is typically
encountered in cellular radio systems like CDMA networks, where the
interference dominates over the thermal noise. The noise-limited regime is
appropriate if the distance between concurrent transmitters is much larger
than the distance of the typical link, in which case the interference in the
network is negligible.
### V-A FHMA
The following theorem gives the optimal threshold $\theta_{\mathrm{opt}}$ and
its bounds for FHMA in the interference-limited regime and noise-limited
regime respectively.
###### Theorem 7
In the noise-limited regime, the optimal threshold $\theta_{\mathrm{opt}}$
that minimizes the normalized mean local delay
$\frac{D(N)}{\log_{2}(1+\theta)}$ for FHMA is given by
$\displaystyle\theta_{\mathrm{opt}}$ $\displaystyle=$
$\displaystyle\exp\left(\mathcal{W}\left(\frac{N}{r_{0}^{\alpha}WN_{0}}\right)\right)-1,$
(40)
where $\mathcal{W}(z)$ is the Lambert $\mathcal{W}$ function which solves
$\mathcal{W}(z)e^{\mathcal{W}(z)}=z$. In the interference-limited regime, the
bounds of $\theta_{\mathrm{opt}}$ are given by
$\displaystyle\theta_{\mathrm{opt}}\in\left(b_{0}^{-1/(\delta+1)}-1,b_{0}^{-1/\delta}\right),$
$\displaystyle b_{0}=\lambda c_{d}r_{0}^{d}\delta
C(\delta)(N-1)^{\delta-1}N^{-\delta}.$ (41)
###### Proof:
The derivative of the normalized mean local delay with respective to $\theta$
is
$\displaystyle\frac{\partial\left(\frac{D(N)}{\log_{2}(1+\theta)}\right)}{\partial\theta}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{h(\theta)\theta^{\delta-1}N}{\log_{2}(1+\theta)}\exp\big{(}\theta
r_{0}^{\alpha}WN_{0}N^{-1}$ (42)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\lambda
c_{d}r_{0}^{d}\theta^{\delta}C(\delta)(N-1)^{\delta-1}N^{-\delta}\big{)},$
where $h(\theta)$ is as follows
$\displaystyle h(\theta)=\lambda c_{d}r_{0}^{d}\delta
C(\delta)(N-1)^{\delta-1}N^{-\delta}$
$\displaystyle+r_{0}^{\alpha}WN_{0}N^{-1}\theta^{1-\delta}-\frac{1}{\theta^{\delta-1}(1+\theta)\ln(1+\theta)}.$
(43)
Next, we prove that $h(\theta)$ is a strictly increasing function of $\theta$,
then we show that the equation $h(\theta)=0$ has a unique solution. Let
$l(\theta)=\theta^{\delta-1}(1+\theta)\ln(1+\theta)$ and the derivative of
$l(\theta)$ is as follows
$\displaystyle l^{\prime}(\theta)$ $\displaystyle=$
$\displaystyle((\delta-1)\theta^{\delta-2}+\delta\theta^{\delta-1})\ln(1+\theta)+\theta^{\delta-1}$
$\displaystyle>$
$\displaystyle((\delta-1)\theta^{\delta-2}+\delta\theta^{\delta-1})\theta+\theta^{\delta-1}$
$\displaystyle>$ $\displaystyle 0.$
Thus, $l(\theta)$ is a strictly increasing function of $\theta$. This implies
that $h(\theta)$ is also a strictly increasing function of $\theta$. Since
$\lim_{\theta\rightarrow 0^{+}}h(\theta)=-\infty$ and
$\lim_{\theta\rightarrow\infty}h(\theta)=+\infty$, the equation $h(\theta)=0$
has a unique solution $\theta_{\mathrm{opt}}$ that minimizes the mean local
delay.
In the noise-limited regime, the equation $h(\theta)=0$ has the form
$\displaystyle
r_{0}^{\alpha}WN_{0}N^{-1}\theta^{1-\delta}-\frac{1}{\theta^{\delta-1}(1+\theta)\ln(1+\theta)}=0.\quad$
(44)
Solving this equation we obtain (40).
In the interference-limited regime, the noise is ignored, and $h(\theta)=0$
has the form
$\displaystyle\lambda c_{d}r_{0}^{d}\delta
C(\delta)(N-1)^{\delta-1}N^{-\delta}$
$\displaystyle-\frac{1}{\theta^{\delta-1}(1+\theta)\ln(1+\theta)}=0.$ (45)
A closed-form solution for the above equation does not exist. By applying the
inequalities $\frac{\theta}{1+\theta}<\ln(1+\theta)<\theta$ to (45), we get
the following inequalities
$\displaystyle\frac{1}{(1+\theta)^{\delta+1}}<\frac{1}{\theta^{\delta}(1+\theta)}$
$\displaystyle<\lambda c_{d}r_{0}^{d}\delta
C(\delta)(N-1)^{\delta-1}N^{-\delta}<\frac{1}{\theta^{\delta}}.$ (46)
From the above inequalities, we get the bounds in (41) ∎
### V-B ALOHA
Based on the similarity between the expressions of the normalized mean local
delays for ALOHA and for FHMA, we obtain the optimal threshold for ALOHA
directly.
###### Theorem 8
In noise-limited regime, the optimal threshold $\theta_{\mathrm{opt}}$ that
minimizes the normalized mean local delay for ALOHA is as follows
$\displaystyle\theta_{\mathrm{opt}}$ $\displaystyle=$
$\displaystyle\mathcal{W}\left(\exp\left(\frac{1}{r_{0}^{\alpha}WN_{0}}\right)\right)-1.$
(47)
In interference-limited regime, the bounds of $\theta_{\mathrm{opt}}$ are
given by
$\displaystyle\theta_{\mathrm{opt}}\in\left(b_{0}^{-1/(\delta+1)}-1,b_{0}^{-1/\delta}\right),$
$\displaystyle b_{0}=\lambda c_{d}r_{0}^{d}\delta C(\delta)p(1-p)^{\delta-1}.$
(48)
## VI Design Insights
### VI-A Mean Delay-Jitter Tradeoff
The jitter of delay, typically characterized by the packet delay variation, is
defined in [25] and [26]. In the system design, the delay variation is an
important measure that characterizes the fluctuation of delay [27]. For
interactive real-time applications, e.g., VoIP, large delay variance can be a
serious issue. To the best of our knowledge, the variance of local delay has
not been explored in the existing work. The optimal value of $N$ that
minimizes the mean local delay is often not the one that minimizes the
variance; thus there is a tradeoff between the mean and the variance of the
local delay. Fig. 7(a) and Fig. 7(b) visualize the relationship between the
mean and the variance of the normalized local delay for FHMA and for ALOHA
respectively. From Fig. 7(a) we observe that in FHMA the favorable operating
point has a reasonably wide tuning range because the variance stabilizes fast
as $N$ increases. In contrast, we observe from Fig. 7(b) that in ALOHA the
curves turn sharply.
(a) FHMA
(b) ALOHA
Figure 7: Mean delay-jitter tradeoff.
### VI-B Tail Probability of the Local Delay
The tail probability is an important measure of the system performance since
one may require (as a QoS constraint) that the probability that the local
delay exceeds a certain threshold is less than a predefined value. Based on
the mean and variance we have derived and by applying the one-tailed
Chebyshev’s inequality, we obtain an upper bound for the tail probability.
For example, in FHMA with $N>1$, letting $X=\sum_{i=1}^{N}\Delta_{i}$ be the
local delay, the tail probability is upper bounded as follows
$\\!\\!\\!\\!\mathbb{P}\\{X>T_{0}\\}\leq\frac{V(N)}{V(N)+(T_{0}-D(N))^{2}},\quad\mathrm{for}\quad
T_{0}>D(N).\\\ $ (49)
For example, if we let the design requirement be that the probability that the
local delay exceeds $10$ is less than $5\%$. Then, when the threshold of the
local delay is fixed as $T_{0}=10$, the upper bound of the tail probability
given by (49) with varying $N$ is shown in Fig. 8. We observe that in order to
achieve the probability $5\%$, the number of sub-bands $N$ for the case when
$\theta=1,10,100$ should be chosen larger than $15,50,95$ respectively.
The bounds based on Chebyshev’s inequality will typically not provide the
tightest bounds. However, it generally cannot be improved if only the mean and
variance are available. On the other hand, if further statistical information
is provided, a number of methods may be developed to improve the sharpness of
the bounds, for example, through the use of semivariances if some samples are
available, or through the use of Bhattacharyya’s inequality or large-
deviations based inequalities if higher moments or even the moment generating
functions are available.
Figure 8: Upper bounds of the tail probability of local delay given by (49) as
a function of the number of sub-bands $N$ when fixing $T_{0}=10$ in FHMA.
## VII Conclusions
In this work, we studied the problem of reducing the effect of interference
correlation by introducing MAC dynamics. We derived the mean and variance of
the local delay and evaluated how the interference correlation can be reduced
by FHMA and ALOHA. We also evaluated the optimal number of sub-bands in FHMA
and the optimal transmit probability in ALOHA that minimize the mean local
delay.
The results reveal that there exist two operation regimes for the network, the
correlation-limited regime and bandwidth-limited regime, which are separated
by the optimal number of sub-bands in FHMA and the optimal transmit
probability in ALOHA. If no MAC dynamics is employed, the local delay has a
heavy tail distribution which results in infinite mean local delay; meanwhile,
employing FHMA and ALOHA will greatly decrease the mean local delay. By
comparing the results of FHMA and ALOHA, we observed that while the mean local
delays of the two protocols are the same for certain parameters, the variances
are rather different. According to the results established herein, FHMA
outperforms ALOHA if implementation costs like overhead are not taken into
consideration; however, when considering the implementation costs, the
overhead for FHMA may be much higher than ALOHA because in FHMA each
transmitter should inform the corresponding receiver which sub-band to listen
on.
## Acknowledgement
The authors wish to thank the anonymous reviewers for their constructive
comments.
## References
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* [2] M. Haenggi, “Diversity Loss due to Interference Correlation,” _IEEE Communications Letters_ , vol. 16, no. 10, pp. 1600–1603, Oct. 2012.
* [3] M. Haenggi and R. Smarandache, “Diversity Polynomials for the Analysis of Temporal Correlations in Wireless Networks,” _IEEE Transactions on Wireless Communications_ , 2013, accepted. Available at http://www.nd.edu/~mhaenggi/pubs/twc14.pdf.
* [4] U. Schilcher, C. Bettstetter, and G. Brandner, “Temporal Correlation of Interference in Wireless Networks with Rayleigh Block Fading,” _IEEE Transactions on Mobile Computing_ , vol. 11, no. 12, pp. 2109–2120, 2012.
* [5] R. Ganti and M. Haenggi, “Spatial and temporal correlation of the interference in ALOHA ad hoc networks,” _IEEE Communications Letters_ , vol. 13, no. 9, pp. 631–633, 2009.
* [6] M. Haenggi, “The Local Delay in Poisson Networks,” _IEEE Transactions on Information Theory_ , vol. 59, no. 3, pp. 1788–1802, Mar. 2013\.
* [7] F. Baccelli and B. Błaszczyszyn, “A new phase transitions for local delays in MANETs,” in _Proceedings IEEE INFOCOM_ , 2010, pp. 1–9.
* [8] J. G. Andrews, S. Weber, and M. Haenggi, “Ad Hoc Networks: To Spread or not to Spread?” _IEEE Communications Magazine_ , vol. 45, no. 12, pp. 84–91, Dec. 2007.
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* [10] D. Stoyan, W. Kendall, J. Mecke, and L. Ruschendorf, _Stochastic geometry and its applications, 2nd Edition_. Wiley New York, 1987.
* [11] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 27, no. 7, pp. 1029–1046, 2009.
* [12] M. Haenggi and R. Ganti, _Interference in large wireless networks_. Now Publishers Inc, 2009, vol. 3, no. 2.
* [13] M. Haenggi, _Stochastic Geometry for Wireless Networks_. Cambridge University Press, 2012.
* [14] F. Baccelli and B. Błaszczyszyn, _Stochastic Geometry and Wireless Networks, Volume II: Applications. Foundations and Trends in Networking. Now Publishers_ , 2009.
* [15] M. Haenggi, “Local Delay in Poisson Networks with and without Interference,” in _Allerton Conference on Communication, Control and Computing_ , Sep. 2010.
* [16] ——, “Local Delay in Static and Highly Mobile Poisson Networks with ALOHA,” in _Proc. IEEE International Conference on Communications (ICC’10)_ , Cape Town, South Africa, May 2010.
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* [22] H. Inaltekin, S. B. Wicker, M. Chiang, and H. V. Poor, “On unbounded path-loss models: effects of singularity on wireless network performance,” _IEEE Journal on Selected Areas in Communications_ , vol. 27, no. 7, pp. 1078–1092, 2009\.
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* [25] C. Demichelis and P. Chimento, “IP packet delay variation metric for IP performance metrics (IPPM),” _RFC 3393_ , Nov. 2002.
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* [27] D. Verma, H. Zhang, and D. Ferrari, “Delay jitter control for real-time communication in a packet switching network,” in _IEEE Conference on Communications Software, 1991, ’Communications for Distributed Applications and Systems’, Proceedings of TRICOMM ’91._ , 1991, pp. 35–43.
| Yi Zhong Yi Zhong received his B.S. degree in Electronic Engineering from
University of Science and Technology of China (USTC) in 2010. He is now a
Ph.D. student in Electronic Engineering at USTC, Hefei, China. From August to
December 2012, he was a visiting student in Prof. Martin Haenggi’s group at
University of Notre Dame. From July to October 2013, he worked as an intern in
Qualcomm, Corporate Research and Development, Beijing. His research interests
include heterogeneous and femtocell-overlaid cellular networks, wireless ad
hoc networks, stochastic geometry and point process theory.
---|---
| Wenyi Zhang Wenyi Zhang (S’00, M’07, SM 11) received the B.E. degree in
automation from Tsinghua University, Beijing, China, in 2001, and the M.S. and
Ph.D. degrees in electrical engineering both from the University of Notre
Dame, Notre Dame, IN, in 2003 and 2006, respectively. He was affiliated with
University of Southern California as a Postdoctoral Research Associate, and
with the Qualcomm Corporate Research and Development, Qualcomm Incorporated.
He is currently on the faculty of Department of Electronic Engineering and
Information Science, University of Science and Technology of China. His
research interests include wireless communications and networking, information
theory, and statistical signal processing.
---|---
| Martin Haenggi Martin Haenggi (S-95, M-99, SM-04) is a Professor of
Electrical Engineering and a Concurrent Professor of Applied and Computational
Mathematics and Statistics at the University of Notre Dame, Indiana, USA. He
received the Dipl.-Ing. (M.Sc.) and Dr.sc.techn. (Ph.D.) degrees in electrical
engineering from the Swiss Federal Institute of Technology in Zurich (ETH) in
1995 and 1999, respectively. After a postdoctoral year at the University of
California in Berkeley, he joined the University of Notre Dame in 2001. In
2007-2008, he spent a Sabbatical Year at the University of California at San
Diego (UCSD). For both his M.Sc. and Ph.D. theses, he was awarded the ETH
medal, and he received a CAREER award from the U.S. National Science
Foundation in 2005 and the 2010 IEEE Communications Society Best Tutorial
Paper award. He served an Associate Editor of the Elsevier Journal of Ad Hoc
Networks from 2005-2008, of the IEEE Transactions on Mobile Computing (TMC)
from 2008-2011, and of the ACM Transactions on Sensor Networks from 2009-2011,
as a Guest Editor for the IEEE Journal on Selected Areas in Communications in
2008-2009 and the IEEE Transactions on Vehicular Technology in 2012-2013, and
as a Steering Committee Member for the TMC. Presently he is the chair of the
Executive Editorial Committee of the IEEE Transactions on Wireless
Communications. He also served as a Distinguished Lecturer for the IEEE
Circuits and Systems Society in 2005-2006, as a TPC Co-chair of the
Communication Theory Symposium of the 2012 IEEE International Conference on
Communications (ICC’12), and as a General Co-chair of the 2009 International
Workshop on Spatial Stochastic Models for Wireless Networks (SpaSWiN’09) and
the 2012 DIMACS Workshop on Connectivity and Resilience of Large-Scale
Networks, and as the Keynote Speaker of SpaSWiN’13. He is a co-author of the
monograph ”Interference in Large Wireless Networks” (NOW Publishers, 2009) and
the author of the textbook ”Stochastic Geometry for Wireless Networks”
(Cambridge University Press, 2012). His scientific interests include
networking and wireless communications, with an emphasis on ad hoc, cognitive,
cellular, sensor, and mesh networks.
---|---
|
arxiv-papers
| 2013-04-06T03:56:14 |
2024-09-04T02:49:43.948479
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi Zhong, Wenyi Zhang, Martin Haenggi",
"submitter": "Yi Zhong",
"url": "https://arxiv.org/abs/1304.1851"
}
|
1304.1867
|
# Geometrical scaling in high energy collisions
and its breaking ††thanks: Presented at the Conference Excited QCD,
Bjelasnica, Sarajevo, Feb. 3 – 9, 2013.
Michal Praszalowicz M. Smoluchowski Institute of Physics, Jagellonian
University, Reymonta 4, 30-059 Krakow, Poland
###### Abstract
We analyze geometrical scaling (GS) in Deep Inelstic Scattering at HERA and in
pp collisions at the LHC energies and in NA61/SHINE experiment. We argue that
GS is working up to relatively large Bjorken $x\sim 0.1$. This allows to study
GS in negative pion multiplicity $p_{\rm T}$ distributions at NA61/SHINE
energies where clear sign of scaling violations is seen with growing rapidity
when one of the colliding partons has Bjorekn $x\geq 0.1$.
13.85.Ni,12.38.Lg
## 1 Introduction
In this short note, following Refs. [1]–[5] where also an extensive list of
references can be found, we will focus on the scaling law, called geometrical
scaling (GS), which has been introduced in the context of DIS [6]. Recently it
has been shown that GS is also exhibited by the $p_{\text{T}}$ spectra at the
LHC [1]–[3]. An onset of GS in heavy ion collisions at RHIC energies has been
reported in Ref. [3]. At low Bjorken $x<x_{\mathrm{max}}$ proton is
characterized by an intermediate energy scale $Q_{\text{s}}(x)$ – called
saturation scale [7, 8] – defined as the border line between dense and dilute
gluonic systems within a proton (for review see _e.g._ Refs. [9, 10]). For the
present study, however, the details of saturation are not of primary interest,
it is the very existence of $Q_{\text{s}}(x)$ which is of importance.
Here we present analysis of three different pieces of data which exhibit both
emergence and violation of geometrical scaling. In Sect. 2 we briefly describe
the method used to assess the existence of GS. Secondly, in Sect. 3 we
describe our recent analysis [4] of combined HERA data [11] where it has been
shown that GS in DIS works very well up to relatively large
$x_{\text{max}}\sim 0.1$ (see also [12]). Next, in Sect. 4, on the example of
the CMS $p_{\rm T}$ spectra in central rapidity [13], we show that GS can be
extended to hadronic collisions. For particles produced at non-zero
rapidities, one (larger) Bjorken $x=x_{1}$ may leave the domain of GS, _i.e._
$x_{1}>x_{\text{max}}$, and violation of GS should appear. In Sect. 5 we
present analysis of very recent pp data from NA61/SHINE experiment at CERN
[14] and show that GS is indeed violated once rapidity is increased. We
conclude in Sect. 6.
## 2 Method of ratios
Geometrical scaling hypothesis means that some observable $\sigma$ that in
principle depends on two independent kinematical variables, say $x$ and
$Q^{2}$, in fact depends only on a specific combination of them denoted as
$\tau$:
$\sigma(x,Q^{2})=F(\tau)/{Q_{0}^{2}}.$ (1)
Here function $F$ in Eq. (1) is a dimensionless function of scaling variable
$\tau=Q^{2}/Q_{\text{s}}^{2}(x).$ (2)
and
$Q_{\text{s}}^{2}(x)=Q_{0}^{2}\left({x}/{x_{0}}\right)^{-\lambda}$ (3)
is the saturation scale. Here $Q_{0}$ and $x_{0}$ are free parameters which
can be extracted from the data within some specific model for $\sigma$, and
exponent $\lambda$ is a dynamical quantity of the order of $\lambda\sim 0.3$.
Throughout this paper we shall test the hypothesis whether different pieces of
data can be described by formula (1) with constant $\lambda$, and what is the
kinematical range where GS is working satisfactorily.
In view of Eq. (1) observables $\sigma(x_{i},Q^{2})$ for different $x_{i}$’s
should fall on one universal curve, if evaluated not in terms of $Q^{2}$ but
in terms of $\tau$. This means in turn that ratios
$R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})=\frac{\sigma(x_{i},\tau(x_{i},Q_{k}^{2};\lambda))}{\sigma(x_{\text{ref}},\tau(x_{\text{ref}},Q_{k,\text{ref}}^{2};\lambda))}$
(4)
should be equal to unity independently of $\tau$. Here for some $x_{\rm ref}$
we pick up all $x_{i}<x_{\rm ref}$ which have at least two overlapping points
in $Q^{2}$.
For $\lambda\neq 0$ points of the same $Q^{2}$ but different $x$’s correspond
in general to different $\tau$’s. Therefore one has to interpolate
$\sigma(x_{\text{ref}},\tau(x_{\text{ref}},Q^{2};\lambda))$ to
$Q_{k,\text{ref}}^{2}$ such that
$\tau(x_{\text{ref}},Q_{k,\text{ref}}^{2};\lambda)=\tau_{k}$. This procedure
is described in detail in Refs. [4].
By tuning $\lambda$ one can make
$R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})\rightarrow 1$ for all $\tau_{k}$.
In order to find optimal value $\lambda_{\rm min}$ that minimizes deviations
of ratios (4) from unity we form the chi-square measure
$\chi_{x_{i},x_{\text{ref}}}^{2}(\lambda)=\frac{1}{N_{x_{i},x_{\text{ref}}}-1}{\displaystyle\sum\limits_{k\in
x_{i}}}\frac{\left(R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})-1\right)^{2}}{\Delta
R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})^{2}}$ (5)
where the sum over $k$ extends over all points of given $x_{i}$ that have
overlap with $x_{\text{ref}}$, and ${N_{x_{i},x_{\text{ref}}}}$ is a number of
such points.
## 3 Deep Inelastic Scattering at HERA
In the case of DIS the relevant scaling observable is $\gamma^{\ast}p$ cross
section and variable $x$ is simply Bjorken $x$. In Fig. 1 we present 3-d plot
of $\lambda_{\min}({x,x_{\rm ref}})$ which has been found by minimizing (5).
Figure 1: Three dimensional plot of
$\lambda_{\mathrm{min}}(x,x_{\mathrm{ref}})$ obtained by minimization of Eq.
(5).
Qualitatively, GS is given by the independence of $\lambda_{\text{min}}$ on
Bjorken $x$ and by the requirement that the pertinent value of
$\chi_{x,x_{\text{ref}}}^{2}(\lambda_{\text{min}})$ should be small (for the
discussion of the latter see Refs. [4]). We see from Fig. 1 that the stability
corner of $\lambda_{\text{min}}$ extends up to $x_{\text{ref}}\lesssim 0.1$,
which is well above the original expectations. In Ref. [4] we have shown that:
$\lambda=0.32-0.34\,\,\,\,\,{\rm for}\,\,\,\,\,x\leq 0.08.$ (6)
## 4 Central rapidity $p_{\rm T}$ spectra at the LHC
Figure 2: Ratios of CMS $p_{\mathrm{T}}$ spectra [13] at 7 TeV to 0.9 (blue
circles) and 2.36 TeV (red triangles) plotted as functions of $p_{\mathrm{T}}$
(left) and scaling variable $\sqrt{\tau}$ (right) for $\lambda=0.27$.
In hadronic collisions at c.m. energy $W=\sqrt{s}$ particles are produced in
the scattering process of two patrons carrying Bjorken $x$’s
$x_{1,2}=e^{\pm y}\,p_{\text{T}}/W.$ (7)
For central rapidities $x=x_{1}\sim x_{2}$. It has been shown that in this
case charged particle multiplicity spectra exhibit GS [1]
$\left.\frac{dN}{dyd^{2}p_{\text{T}}}\right|_{y\simeq
0}=\frac{1}{Q_{0}^{2}}F(\tau)$ (8)
where $F$ is a universal dimensionless function of the scaling variable
$\tau=p_{\text{T}}^{2}/Q_{\text{s}}^{2}(x)=p_{\text{T}}^{2}/Q_{0}^{2}\,\left(p_{\rm
T}/(x_{0}\sqrt{s})\right)^{\lambda}.$ (9)
Therfore the scaling observable is $\sigma(W,p_{\rm
T}^{2})={dN}/{dyd^{2}p_{\text{T}}}$ and the method of ratios is applied to the
multiplicity distributions at different energies ($W_{i}$ taking over the role
of $x_{i}$ in Eq. (4)). For $W_{\rm ref}$ we take the highest LHC energy of 7
TeV. Therefore one can form two ratios $R_{W_{i},W_{\rm ref}}$ with
$W_{1}=2.36$ and $W_{2}=0.9$ TeV. These ratios are plotted in Fig. 2 for the
CMS single non-diffractive spectra for $\lambda=0$ and for $\lambda=0.27$,
which minimizes (5) in this case. We see that original ratios plotted in terms
of $p_{\text{T}}$ range from 1.5 to 7, whereas plotted in terms of
$\sqrt{\tau}$ they are well concentrated around unity. The optimal exponent
$\lambda$ is, however, smaller than in the case of DIS. Why this so, remains
to be understood.
## 5 Violation of geometrical scaling in forward rapidity region
For $y>0$ two Bjorken $x$’s can be quite different: $x_{1}>x_{2}$. Therefore
looking at the spectra with increasing $y$ one can eventually reach
$x_{1}>x_{\mathrm{max}}$ and GS violation should be seen. To this end we shall
use pp data from NA61/SHINE experiment at CERN [14] at different rapidities
$y=0.1-3.5$ and at five different energies
$W_{1,\ldots,5}=17.28,\;12.36,\;8.77,\;7.75$, and $6.28$ GeV.
Figure 3: Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for the lowest
rapidity $y=0.1$: a) for $\lambda=0$ when $\sqrt{\tau}=p_{\mathrm{T}}$ and b)
for $\lambda=0.27$ which corresponds to GS.
Figure 4: Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for $\lambda=0.27$ and
for different rapidities a) $y=0.7$ and b) $y=1.3$. With increase of rapidity,
gradual closure of the GS window can be seen.
In Fig. 3 we plot ratios $R_{1i}=R_{W_{1},W_{i}}$ (4) for $\pi^{-}$ spectra in
central rapidity for $\lambda=0$ and 0.27. For $y=0.1$ the GS region extends
towards the smallest energy because $x_{\rm max}$ is as large as 0.08.
However, the quality of GS is the worst for the lowest energy $W_{5}$. By
increasing $y$ some points fall outside the GS window because $x_{1}\geq
x_{\rm max}$, and finally for $y\geq 1.7$ no GS should be present in
NA61/SHINE data. This is illustrated nicely in Fig. 4.
## 6 Conclusions
We have shown that GS in DIS works well up to rather large Bjorken $x$’s with
exponent $\lambda=0.32-0.34$. In pp collisions at the LHC energies in central
rapidity GS is seen in the charged particle multiplicity spectra, however,
$\lambda=0.27$ in this case. By changing rapidity one can force one of the
Bjorken $x$’s of colliding patrons to exceed $x_{\rm max}$ and GS violation is
expected. Such behavior is indeed observed in the NA61/SHINE pp data.
The author wants to thank M. Gazdzicki and Sz. Pulawski for the access to the
NA61/SHINE data and to T. Stebel for collaboration and remarks. Many thanks
are due to the organizers of this successful series of conferences. This work
was supported by the Polish NCN grant 2011/01/B/ST2/00492.
## References
* [1] L. McLerran and M. Praszalowicz, Acta Phys. Polon. B 41 (2010) 1917 and Acta Phys. Polon. B 42 (2011) 99.
* [2] M. Praszalowicz, Phys. Rev. Lett. 106 (2011) 142002.
* [3] M. Praszalowicz, Acta Phys. Polon. B 42 (2011) 1557 and arXiv:1205.4538 [hep-ph].
* [4] M. Praszalowicz and T. Stebel, JHEP 1303, 090 (2013) and arXiv:1302.4227 [hep-ph], to be published in JHEP.
* [5] M. Praszalowicz, arXiv:1301.4647 [hep-ph], to be published in Phys. Rev. D.
* [6] A. M. Stasto, K. J. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. 86, 596 (2001).
* [7] L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rept. 100 (1983) 1;
A. H. Mueller and J-W. Qiu, Nucl. Phys. 268 (1986) 427; A. H. Mueller, Nucl.
Phys. B558 (1999) 285.
* [8] K. J. Golec-Biernat and M. Wüsthoff, Phys. Rev. D 59 (1998) 014017 and Phys. Rev. D 60 (1999) 114023.
* [9] A. H. Mueller, _Parton Saturation: An Overview_ , arXiv:hep-ph/0111244.
* [10] L. McLerran, Acta Phys. Pol. B 41, 2799 (2010).
* [11] C. Adloff et al. [H1 Collaboration], Eur. Phys. J. C 21 (2001) 33; S. Chekanov et al. [ZEUS Collaboration], Eur. Phys. J. C 21 (2001) 443.
* [12] F. Caola, S. Forte and J. Rojo, Nucl. Phys. A 854, 32 (2011).
* [13] V. Khachatryan et al. [CMS Collaboration], JHEP 1002 (2010) 041 and Phys. Rev. Lett. 105 (2010) 022002 and JHEP 1101 (2011) 079.
* [14] N. Abgrall et al. [NA61/SHINE Collaboration], Report from the NA61/SHINE experiment at the CERN SPS CERN-SPSC-2012-029, SPSC-SR-107;
A. Aduszkiewicz, Ph.D. Thesis in prepartation, University of Warsaw, 2013;
Sz. Pulawski, talk at 9th Polish Workshop on Relativistic Heavy-Ion
Collisions, Kraków, November 2012 and private communication.
|
arxiv-papers
| 2013-04-06T09:03:58 |
2024-09-04T02:49:43.960210
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michal Praszalowicz",
"submitter": "Michal Praszalowicz",
"url": "https://arxiv.org/abs/1304.1867"
}
|
1304.2314
|
arxiv-papers
| 2013-04-08T19:00:01 |
2024-09-04T02:49:43.981261
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Morrell",
"submitter": "Thomas Morrell",
"url": "https://arxiv.org/abs/1304.2314"
}
|
|
1304.2389
|
11institutetext: A. Ptok 22institutetext: Institute of Physics, University of
Silesia, 40-007 Katowice, Poland, 22email: [email protected] 33institutetext: D.
Crivelli 44institutetext: Institute of Physics, University of Silesia, 40-007
Katowice, Poland, 44email: [email protected]
# The Fulde-Ferrell-Larkin-Ovchinnikov state in pnictides
Andrzej Ptok Dawid Crivelli
(Received: date / Accepted: date)
###### Abstract
Fe-based superconductors (FeSC) exhibit all the properties of systems that
allow the formation of a superconducting phase with oscillating order
parameter, called the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase. By the
analysis of the Cooper pair susceptibility in two-band FeSC, such systems are
shown to support the existence of a FFLO phase, regardless of the exhibited
order parameter symmetry. We also show the state with nonzero Cooper pair
momentum, in superconducting FeSC with $\sim\cos(k_{x})\cdot\cos(k_{y})$
symmetry, to be the ground state of the system in a certain parameter range.
###### Keywords:
FFLO pnictides
###### pacs:
74.20.-z 74.70.Xa 74.81.-g
††journal: Journal of Low Temperature Physics
## 1 Introduction
At low temperatures the orbital pair breaking effects are smaller in magnitude
than the Pauli paramagnetic effect, so that superconductivity survives up to
the Pauli limit – a phase with oscillating order parameter (called the
Fulde–Ferrell–Larkin–Ovchinnikov phase or FFLO in short) FFLO can be more
stable than a phase with a constant order parameter (the
Bardeen–Cooper–Schrieffer phase, or BCS in short). In this case, Cooper pairs
may be formed with non-zero total momentum between Zeeman-split parts of the
Fermi surface.
Properties of this phase have been usually evaluated in tight-binding models
of one-band systems. tbmodel However the latest experimental fflo.fesc and
theoretical fflo.fesc.th works suggest we can expect the existence of the
FFLO phase in multi-band Fe-based superconductors (FeSC). It follows from the
fact that they possess properties close to heavy fermions systems,
matsuda.shimahara.07 for which strong experimental evidence suggest the
existence of said phase. fflo.hf Both kinds of systems are multi-layered,
clean and have a relatively high Maki parameter.
In this paper, making use of the Cooper pair susceptibility and the the
minimization of free energy of the system, we discuss the possible appearance
of the FFLO phase in pnictides. In Section 2 we describe the selected model of
FeSC, in Section 3 we present our methods. In Section 4 we illustrate and
discuss our numerical results. We summarize the results in Section 5.
## 2 Theoretical model
The FeSC system is described using a two-orbital per site model, with
hybridization between the $d_{xz}$ and $d_{yz}$ orbitals. We adopt the band
structure proposed in Ref. raghu.qi.08 and assume that the external magnetic
field is parallel to the plane. The Hamiltonian of the system in momentum
space takes the following form:
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{k}}\sigma}\sum_{\alpha\beta}(T^{\alpha\beta}_{\bm{k}}-(\mu+\sigma
h)\delta_{\alpha\beta})c_{\alpha{\bm{k}}\sigma}^{\dagger}c_{\beta{\bm{k}}\sigma}$
(1) $\displaystyle T^{11}_{\bm{k}}$ $\displaystyle=$
$\displaystyle-2\left(t_{1}\cos(k_{x})+t_{2}\cos(k_{y})\right)-4t_{3}\cos(k_{x})\cos(k_{y}),$
$\displaystyle T^{22}_{\bm{k}}$ $\displaystyle=$
$\displaystyle-2\left(t_{2}\cos(k_{x})+t_{1}\cos(k_{y})\right)-4t_{3}\cos(k_{x})\cos(k_{y}),$
$\displaystyle T^{12}_{\bm{k}}$ $\displaystyle=$ $\displaystyle
T^{21}_{\bm{k}}=-4t_{4}\sin(k_{x})\sin(k_{y}),$
where $c_{\alpha{\bm{k}}\sigma}^{\dagger}$ ($c_{\alpha{\bm{k}}\sigma}$) is the
creation (annihilation) operator of a particle with momentum ${\bm{k}}$ and
spin $\sigma$ in the orbital $\alpha$. $T^{\alpha\beta}_{{\bm{k}}\sigma}$ is
the kinetic energy term of a particle with momentum ${\bm{k}}$ changing the
orbital from $\beta$ to $\alpha$, $\mu$ is the chemical potential and $h$ is
the external magnetic field. The hoppings have magnitudes:
$(t_{1},t_{2},t_{3},t_{4})=(-1.0,1.3,-0.85,-0.85)$, in units of $|t_{1}|$. At
half-filling, a configurations with two electrons per site requires
$\mu=1.54|t_{1}|$. Our choice of the parameter set is motivated by the fact
that it reproduces the same Fermi surface structure as the local-density
approximation calculations of band structure. fermisurface
By diagonalizing the above Hamiltonian, one obtains
$\displaystyle H^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\sum_{\varepsilon{\bm{k}}\sigma}E_{\varepsilon{\bm{k}}\sigma}d_{\varepsilon{\bm{k}}\sigma}^{\dagger}d_{\varepsilon{\bm{k}}\sigma}$
(2)
with eigenvalues
$E_{\varepsilon{\bm{k}}\sigma}=E_{\varepsilon{\bm{k}}}-(\mu+\sigma h)$, where:
$E_{\pm,{\bm{k}}}=\frac{T_{\bm{k}}^{11}+T_{\bm{k}}^{22}}{2}\pm\sqrt{\left(\frac{T_{\bm{k}}^{11}-T_{\bm{k}}^{22}}{2}\right)^{2}+\left(T_{\bm{k}}^{12}\right)^{2}},$
(3)
$d_{\varepsilon{\bm{k}}\sigma}^{\dagger}$ is a new fermion quasi-particle
operator in the band $\varepsilon=\pm$. In this case we have two Fermi
surfaces (Fig. 4.a) – giving an electron-like band ($\varepsilon=+$) and hole-
like band ($\varepsilon=-$).
## 3 Methods
Figure 1: (Color online) The vector ${\bm{\delta}}$ defines the pairing
between sites $i$ and $i+{\bm{\delta}}$ for different symmetries of the order
parameter. Colors and symbols correspond to the sign of the order parameter
for a given direction in real space. For s-wave symmetry the pairing is
between two electrons on the same site of the lattice, while for other
symmetries it is between two other sites (nearest neighbors or next nearest
neighbors). In contrast to $d$ type symmetries, $s$ type symmetries do not
change sign depending on the direction.
We introduce a superconducting pairing between the long-lived quasi-particles
in bands $\varepsilon=\pm$. linder.sudbo.09 To determine the possibility of
formation of the FFLO phase, we turn our attention to the static Cooper pairs
susceptibility:
$\displaystyle\chi_{\varepsilon}^{\Delta}({\bm{q}})$ $\displaystyle\equiv$
$\displaystyle\lim_{\omega\rightarrow
0}\frac{-1}{N}\sum_{{\bm{i}}{\bm{j}}}\exp(i{\bm{q}}\cdot({\bm{i}}-{\bm{j}}))\langle\langle\widehat{\Delta}_{\varepsilon{\bm{i}}}|\widehat{\Delta}_{\varepsilon{\bm{j}}}^{\dagger}\rangle\rangle^{r},$
(4)
where $\langle\langle\ldots\rangle\rangle^{r}$ is the retarded Green’s
function and
$\widehat{\Delta}_{\varepsilon{\bm{i}}}=\sum_{\bm{j}}\vartheta({\bm{j}}-{\bm{i}})d_{\varepsilon{\bm{i}}\uparrow}d_{\varepsilon{\bm{j}}\downarrow}$
is the OP in band $\varepsilon$. The operator $d_{\varepsilon{\bm{i}}\sigma}$
in real space corresponds to the operator $d_{\varepsilon{\bm{k}}\sigma}$ in
momentum space. The Factor $\vartheta({\bm{j}}-{\bm{i}})$ defines the OP
symmetries (Fig. 1) – for example for $d_{x^{2}-y^{2}}$-wave pairing,
$\vartheta({\bm{\delta}})$ is equal to $1$ ($-1$) for
${\bm{\delta}}=\pm\hat{x}$ ($\pm\hat{y}$) and zero otherwise. In momentum
space:
$\displaystyle\chi_{\varepsilon}^{\Delta}({\bm{q}})=\lim_{\omega\rightarrow
0}\frac{-1}{N}\sum_{{\bm{k}}{\bm{l}}}\eta(-{\bm{k}}-{\bm{q}})\eta({\bm{l}}){\bm{G}}_{\varepsilon}({\bm{k}},{\bm{l}},{\bm{q}},\omega),$
(5)
$\displaystyle{\bm{G}}_{\varepsilon}({\bm{k}},{\bm{l}},{\bm{q}},\omega)=\langle\langle
d_{\varepsilon{\bm{k}}\uparrow}d_{\varepsilon,-{\bm{k}}-{\bm{q}}\downarrow}|d_{\varepsilon,-{\bm{l}}-{\bm{q}}\downarrow}^{\dagger}d_{\varepsilon{\bm{l}}\uparrow}^{\dagger}\rangle\rangle^{r}=\delta_{{\bm{k}}{\bm{l}}}\frac{f(-E_{\varepsilon{\bm{k}}\uparrow})-f(E_{\varepsilon,-{\bm{k}}-{\bm{q}}\downarrow})}{\omega-
E_{\varepsilon{\bm{k}}\uparrow}-E_{\varepsilon,-{\bm{k}}-{\bm{q}}\downarrow}},$
where $\eta({\bm{k}})$ is the structure factor:
$\displaystyle\eta({\bm{k}})=\left\\{\begin{array}[]{cc}1&$for s-wave$\\\
2\left(\cos(k_{x})+\cos(k_{y})\right)&$for $s_{x^{2}+y^{2}}$-wave$,\\\
4\cos(k_{x})\cos(k_{y})&$for $s_{x^{2}y^{2}}(s_{\pm})$-wave$,\\\
2\left(\cos(k_{x})-\cos(k_{y})\right)&$for $d_{x^{2}-y^{2}}$-wave$,\\\
4\sin(k_{x})\sin(k_{y})&$for $d_{x^{2}y^{2}}$-wave$,\end{array}\right.$ (12)
corresponding to the type of symmetry of the OP.
We investigate the tendency to form the FFLO phase in the system, using the
static Cooper pairs susceptibility $\chi_{\varepsilon}^{\Delta}({\bm{q}})$. In
magnetic fields of the order of the Pauli limit, when the critical FFLO field
($h_{c}^{FFLO}$) is bigger than the corresponding BCS field ($h_{c}^{BCS}$),
the FFLO phase is favored. In such case, the divergence of this function for
some ${\bm{q}}\neq 0$ may imply a second-order transition to the FFLO state of
corresponding symmetry from the normal phase. mierzejewski.ptok.09 The
location of the maximum of the response function
$\chi_{\varepsilon}^{\Delta}({\bm{q}})$ matches the preferred momentum of the
Cooper pairs in the system described by the Hamiltonian (2) in magnetic field
$h$. This method allows to establish the propensity to form the
superconducting phase (with non-zero momentum of the Cooper pairs) without
specifying the mechanisms responsible for the ordered phases with given
symmetry. Additionally we obtain the change in the pair susceptibilities
$\delta\chi_{\varepsilon}^{\Delta}({\bm{q}})=\chi_{\varepsilon}^{\Delta}({\bm{q}})-\bar{\chi}_{\varepsilon}^{\Delta}({\bm{q}})$
due to the external magnetic field ($\chi_{\varepsilon}^{\Delta}({\bm{q}})$
with the field, $\bar{\chi}_{\varepsilon}^{\Delta}({\bm{q}})$ without
respectively).
It should be noted that the divergence of the Cooper-pair susceptibility is
neither a sufficient condition nor evidence for the transition to the FFLO
state. In order for this to happen the system energy $\Omega({\bm{q}})$ should
attain its minimum at a nonzero Cooper pair momentum ${\bm{q}}$ in a magnetic
field $h>h_{c}^{BCS}$, equivalent to the condition $h_{c}^{FFLO}>h_{c}^{BCS}$.
To check this, we effectively describe superconductivity in the FFLO phase by
the Hamiltonian:
$H_{SC}=\sum_{\varepsilon{\bm{k}}}\left(\Delta_{\varepsilon{\bm{k}}}d_{\varepsilon{\bm{k}}\uparrow}d_{\varepsilon,-{\bm{k}}+{\bm{q}}_{\varepsilon}\downarrow}+H.c.\right),$
(13)
where $\Delta_{\varepsilon{\bm{k}}}=\Delta_{\varepsilon}\eta({\bm{k}})$ is the
amplitude of the OP for Cooper pairs with total momentum
${\bm{q}}_{\varepsilon}$ (in band $\varepsilon$ with symmetry described by
$\eta({\bm{k}})$). As we see, in the operator basis
$d_{\varepsilon{\bm{k}}\sigma}$ the total Hamiltonian
$H=H^{\prime}_{0}+H_{SC}$ formally describes a system with two independent
bands. Using the Bogoliubov transformation we can find the eigenvalues of $H$:
$\displaystyle\lambda_{\varepsilon{\bm{k}}}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{E_{\varepsilon{\bm{k}}\uparrow}-E_{\varepsilon,-{\bm{k}}+{\bm{q}}\downarrow}}{2}\pm\sqrt{\left(\frac{E_{\varepsilon{\bm{k}}\uparrow}+E_{\varepsilon,-{\bm{k}}+{\bm{q}}\downarrow}}{2}\right)^{2}+|\Delta_{\varepsilon{\bm{k}}}|^{2}}.$
(14)
The free energy is given by:
$\displaystyle\Omega=-kT\sum_{\alpha\in\pm}\sum_{\varepsilon{\bm{k}}}\ln\left(1+\exp(-\beta\lambda_{\varepsilon{\bm{k}}}^{\alpha})\right)+\sum_{\varepsilon{\bm{k}}}\left(E_{\varepsilon{\bm{k}}\downarrow}-\frac{2|\Delta_{\varepsilon}|^{2}}{V_{\varepsilon}}\right),$
(15)
where $V_{\varepsilon}$ is the interaction intensity in band $\varepsilon$.
The global ground state for fixed $h$ and $T$ is found by minimizing the free
energy w.r.t. the OPs and ${\bm{q}}$.
## 4 Numerical results and discussion
Numerical calculations were carried out for a square lattice $N_{x}\times
N_{y}=600\times 600$ with periodic boundary conditions, for
$kT=10^{-5}|t_{1}|$. As a first step the static Cooper pairs susceptibility
$\chi_{\varepsilon}^{\Delta}({\bm{q}})$ was calculated in magnetic field
$h=0.25|t_{1}|$. Then the free energy $\Omega({\bm{q}})$ of the
superconducting system was evaluated for magnetic fields near the Pauli limit
$h_{P}\simeq 0.25|t_{1}|$.
### The static Cooper pairs susceptibility.
Assuming different symmetries $\eta({\bm{k}})$ of the superconducting OP in
bands $\varepsilon=\pm$, we characterized the Cooper pair susceptibility –
Fig. 2. For every OP symmetry, in the band $\varepsilon=-$ the static Cooper
pairs susceptibility $\chi_{-}^{\Delta}({\bm{q}})$ takes its maximum for
${\bm{q}}\neq 0$. Conversely in the band $\varepsilon=+$, with
$s_{x^{2}+y^{2}}$ and $d_{x^{2}y^{2}}$ symmetry of the OP, there is a strong
tendency to form a BCS phase (maximum $\chi_{+}^{\Delta}({\bm{q}})$ for
${\bm{q}}=0$). When $h_{c}^{FFLO}>h_{c}^{BCS}$ this can be a sign of the
existence of the FFLO phase in the band $\varepsilon=-$, while the band
$\varepsilon=+$ is in the normal state. Numerical data for both d-wave type
symmetries in band $\varepsilon=-$ is less clear cut, as the maximum
$\chi({\bm{q}})$ is only slightly greater than $\chi({\bm{0}})$.
Figure 2: (Color online) The static Cooper pairs susceptibility
$\chi_{\varepsilon}^{\Delta}({\bm{q}})$ in the presence of the external
magnetic field $h=0.25|t_{1}|$ and $kT=10^{-5}|t_{1}|$ for different
symmetries. Figure 3: (Color online) Change in the static Cooper pairs
susceptibility $\delta\chi_{\varepsilon}^{\Delta}({\bm{q}})$ (for data
presented in Fig. 2)
There is a clear preference in case of $\varepsilon=-$ for much smaller
momenta than in band $\varepsilon=+$, due to the relative width of the bands.
Cooper pair momenta depend on the split in the Fermi surface, caused by the
external magnetic field, which is larger for the broader band $\varepsilon=+$.
Additionally the presence of a magnetic field causes a dampening in each case
of the response function near zero momentum (Fig. 3 – in blue). Nonetheless
larger momenta are unaffected and increasing (in red).
The behaviour of the response function $\chi_{\varepsilon}^{\Delta}({\bm{q}})$
shows that multi-band systems have the characteristics typical of one-band
systems – Cooper pairs in the FFLO phase possessing momentum along the
principal directions of the system are preferred, fflo.onedirection – for
example in directions $[\pm 1,0]$ and $[\pm 1,1]$ for s-wave and
$d_{x^{2}-y^{2}}$-wave symmetry respectively in band $\varepsilon=+$ (Fig. 2).
### Minimization of free energy.
Theoretical results indicate the presence of
$s_{x^{2}y^{2}}\sim\cos(k_{x})\cdot\cos(k_{y})$ (also called $s_{\pm}$)
pairing symmetry in FeSC . op.symmetry In this case the OP exhibits a sign
reversal between the hole pockets and electron pockets (Fig. 4.a). Taking this
into account, in this paragraph only consider $s_{x^{2}y^{2}}$ symmetry.
$V_{\varepsilon}$ was taken such that the Pauli limit was of the order
$h_{P}\simeq 0.25|t_{1}|$ ($V_{+}=-0.74|t_{1}|$ and $V_{-}=-1.56|t_{1}|$).
Figure 4: (Color online) Detailed study of the minimal two-band model
describing iron-base superconductors with $s_{x^{2}y^{2}}(s_{\pm})$-wave
symmetry proposed by Ref. raghu.qi.08 . (Panel a) Fermi surfaces (solid line)
for $\mu=1.54|t_{1}|$. The background color describes the sign of the OP (red
for $\eta({\bm{k}})>0$ and blue for $\eta({\bm{k}})<0$). (Panel b) The free
energy $\Omega({\bm{q}})$ in the two bands $\varepsilon=\pm$, for different
values of the Cooper pair momentum ${\bm{q}}$, showing the location of the
minima and indicating the existence of different phases. Results for
$h=0.25|t_{1}|$ and temperature $kT=10^{-5}|t_{1}|$.
The study of the free energy $\Omega({\bm{q}})$ for the BCS state
(${\bm{q}}=0$) w.r.t. magnetic fields $h\simeq h_{P}$, showed that phase
transitions in both bands are first-order for all symmetries, except for
$s_{x^{2}+y^{2}}$ and $d_{x^{2}y^{2}}$ which are second-order. Only the
minimization of $\Omega({\bm{q}})$ w.r.t. ${\bm{q}}$, allows to check whether
the system exhibits a FFLO phase. Varying ${\bm{q}}\in FBZ$ in case of
$s_{x^{2}y^{2}}$ pairing showed that the band $\varepsilon=+$ undergoes a
transition from BCS to the normal state and the band $\varepsilon=-$ from BCS
to FFLO state (Fig. 4.b). Further increasing the magnetic field, the FFLO
phase persists in $\varepsilon=-$. It should be pointed out that in this band
exist four equivalent Cooper pair momenta $(\pm q,0)$ and $(0,\pm q)$, in
agreement with the static Cooper pairs susceptibility results, and also with
previous works. fflo.onedirection Moreover, it is reasonable to expect that
the phase with an OP given by the superposition of plane waves with said
momenta would be energetically favored by the system. moreq
## 5 Summary
FeSC exhibit many characteristic features of systems in which we can expect
the existence of the FFLO phase. Using a minimal two-band model for FeSC, we
conducted a numerical study of FFLO phase in multi-band systems. The static
Cooper pair susceptibility suggests that we can expect the system to prefer
the state with nonzero Cooper pair momenta (the FFLO phase) regardless of the
OP symmetry, when $h_{c}^{FFLO}>h_{c}^{BCS}$. Moreover, the ground state of
the system with $s_{x^{2}y^{2}}\sim\cos(k_{x})\cdot\cos(k_{y})$ symmetry OP,
can be the state with nonzero Cooper pair momentum for magnetic fields near
the Pauli limit.
###### Acknowledgements.
D.C. acknowledges a scholarship from the TWING project, co-funded by the
European Social Fund.
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|
arxiv-papers
| 2013-04-08T12:49:59 |
2024-09-04T02:49:43.987098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrzej Ptok and Dawid Crivelli",
"submitter": "Andrzej Ptok",
"url": "https://arxiv.org/abs/1304.2389"
}
|
1304.2449
|
# On nonlinear Schrödinger equations with random potentials: existence and
probabilistic properties
Leandro Cioletti
Departamento de Matemática, UnB, 70910-900 Brasília, Brazil.
E-mail:[email protected]
Lucas C. F. Ferreira
Universidade Estadual de Campinas, IMECC - Departamento de Matemática,
Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil.
E-mail:[email protected]
Marcelo Furtado
Departamento de Matemática, UnB, 70910-900 Brasília, Brazil.
E-mail:[email protected]
###### Abstract
In this paper we are concerned with nonlinear Schrödinger equations with
random potentials. Our class includes continuum and discrete potentials.
Conditions on the potential $V_{\omega}$ are found for existence of solutions
almost sure $\omega$. We study probabilistic properties like central limit
theorem and law of larger numbers for the obtained solutions by independent
ensembles. We also give estimates on the expected value for the
$L^{\infty}$-norm of the solution showing how it depends on the size of the
potential.
AMS 2000 subject classification: 47B80, 60H25, 35J60, 35R60, 82B44, 47H10
Keywords: Random potentials; Random nonlinear equations; Schrödinger operators
## 1 Introduction
A class of models that appears naturally in a wide number of phenomena are the
random differential equations. This occurs because randomness is a powerful
tool and concept to control complex systems involving a large number of
variables and particles. The basic idea is describe complex systems by means
of their statistical properties. Another kind of phenomena are those governed
by quantum mechanics and uncertainty principle. In this direction, we have
Schrödinger equations, and their random versions, which are core in the study
of condensed matter.
In this paper we are concerned with a random version of the nonlinear
Schrödinger equation
$ih\dfrac{\partial\psi}{\partial
t}=-h^{2}\Delta\psi+V(x)\psi-|\psi|^{p-1}\psi,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{n},$ (1.1)
where $t\in\mathbb{R}$, $n\geq 3$, $p>1$, $h$ is the Planck constant and $i$
is the imaginary unit. When looking for standing wave solutions, namely those
which have the special form $\psi(x,t):=e^{-i\frac{E}{h}t}u(x)$, with
$E\in\mathbb{R}$, we are leading to solve the following stationary equation
$-\Delta u+V(x)u=|u|^{p-1}u,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N}.$
From the physical viewpoint, the function $V$ is the potential energy, and
therefore the force acting on the system is given by $F(x)=-\nabla V(x)$. In
the deterministic case, there are many papers concerning existence,
multiplicity and qualitative properties for the solution of the above equation
(see [19, 11, 2, 1] and references therein).
The main interest of this paper is to study situations where the potential $V$
is not deterministic. Worth to mention that during the last thirty years,
random Schrödinger operators, which originated in condensed matter physics,
have been studied intensively by physicists and mathematicians. The theory is
at the crossroads of a number of mathematical fields: the theory of operators,
partial differential equations, the theory of probabilities and also
stochastic process. This paper aims to prove the existence and probabilistic
properties of bounded solutions for the random equation
$\left\\{\begin{array}[]{rcll}-\Delta
u+V_{\omega}(x)u&=&b(x)u|u|^{p-1}+g(x),&\text{if}\ x\in U;\\\\[4.26773pt]
u&=&0,&\text{if}\ x\in\partial U,\end{array}\right.$ (1.2)
where $V_{\omega}$ is a random variable, $U\subset\mathbb{R}^{n}$ is a bounded
domain and the terms $b,\,g\in L^{\infty}(U)$ are deterministic. In fact, the
boundedness of $U$ is not essential and could be circumvented by working in
weighted $L^{\infty}$-spaces or Lebesgue spaces $L^{s}(\mathbb{R}^{n})$ with
$s\neq\infty$ (see [13, 14]). However, here this condition will simplify
matters a bit. The random potential $V_{\omega}$ is constructed via a
convolution with a realization of a random variable valued in the finite
random measure space. Precisely, given a continuous function
$f:\mathbb{R}^{N}\rightarrow\mathbb{R}$ we consider
$V_{\omega}(x):=\int_{U}f(x-y)\,d{\mu_{\omega}}(y)$ (1.3)
where $\mu_{\omega}$ is a $\mathcal{M}(U)$-valued random variable.
We present here some examples of (1.3) that have been treated in the
literature (see e.g. the review [17]). We first consider a model of an
unordered alloy, that is, a mixture of several materials with atoms located at
lattice positions. If we assume that the type of atom at the lattice
$i\in\mathbb{Z}^{n}$ is random we are leading to consider the following type
of potential
$V_{\omega}(x)=\sum_{i\in\mathbb{Z}^{n}}q_{i}(\omega)f(x-i),$ (1.4)
where the random variables $q_{i}$ describe the charge of the atom at the
position $i$ of the lattice. Other example can be obtained if we consider
materials like glass or rubber, where the position of the atoms of the
material are located at random points $\eta_{i}$ in space. By normalizing the
charge of the atoms, the suggested potential is formally
$V_{\omega}(x)=\sum_{i\in\mathbb{Z}^{n}}f(x-\eta_{i}(\omega)),$ (1.5)
where the $\eta_{i}(\omega)$ are random variables which localize the atoms in
the spaces.
The class of potentials allowed here is sufficient large to consider many
known models. For example, the case of glass considered in (1.5) can be
obtained if we take the random point measure
$\mu_{\omega}=\sum_{i}\delta_{\eta_{i}(\omega)}$. Actually, for this choice of
the measure we have that
$\sum_{i\in\mathbb{Z}^{n}\cap
U}f(x-\eta_{i}(\omega))=\int_{U}f(x-\eta)d\mu_{\omega}(\eta).$ (1.6)
Also, a combination of potentials like (1.4) and (1.5), namely
$\Sigma_{{}_{i\in\mathbb{Z}^{n}\cap U}}q_{i}(\omega)f(x-\eta_{i}(\omega))$
(see [8]), is also covered by (1.3) with
$\mu_{\omega}=\Sigma_{{}_{i\in\mathbb{Z}^{n}\cap
U}}q_{i}(\omega)\delta_{\eta_{i}(\omega)}$. It is not difficult to see that we
can also consider other models like, e.g., the Poisson model (see [17] for
more examples).
The models (1.4) and (1.5) correspond to discrete measures $\mu_{\omega}$ and
results for them about localization, spectral properties or decays can be
found in [5, 8, 15, 17, 20]. For Schrödinger equations defined in a lattice,
that is $x\in\mathbb{Z}^{n}$, we refer the reader to [4, 6]. Considering a
random time-dependent potential for (1.1), the authors of [3] studied
asymptotic behavior of solutions by showing convergence for stochastic
Gaussian limits when the two-point correlation function of the potential is
rapidly decaying. Still for time random potentials, scaling limits for
parabolic waves in random media were investigated in [12]. Despite important
progress in the last years, there is still a lack of result for random
equations, including Schrödinger ones (see [7]), mainly with respect to the
continuum case which seems to be harder-to-treating. Another type of random
equations are the parabolic ones, for which we refer the works [9, 10] and
their references.
In this paper we find conditions on the potential $V_{\omega}$ for the
nonlinear equation (1.2) having solutions almost sure $\omega.$ The solution
are understood in an integral sense coming from Green functions. From Theorem
3.5 we see how the expected value of the $L^{\infty}$-norm of solutions
depends on the size of potential. Our results also cover continuum random
potential like, among others, the examples given in Remark 3.3 and Theorem
3.4. Moreover, we study probabilistic properties like central limit theorem
and law of larger numbers for the obtained solutions by independent ensembles.
It is worthwhile to mention that, when dealing with the random variable
$\omega\mapsto u(x,\omega)$ which maps an element of $\Omega$ in the solution
of (1.2) associated with the random potential $V_{\omega}$, we need to extend
some known concepts of real random variables for that taking values in a more
general Banach space. We refer to Section 2 for more details.
As a further comment, we observe that the random potentials considered in this
paper are built from a very general probability space. In this setting does
not always make sense to ask what is the probability that the problem (1.2)
has an unique solution in $L^{\infty}(U)$. In order to give some sense to this
question we should restrict ourself to probability spaces
$(\Omega,\mathcal{F},\mathbb{P})$ and random potentials $V$ where the set
$\\{\omega\in\Omega:\ \text{the problem \eqref{eqdif-est1} has a unique
solution in}\ L^{\infty}(U)\\}$
is an event (measurable). Working in such probability spaces Theorem 3.2 give
us immediately a lower bound for the probability that the non-linear problem
(1.2) has a unique solution.
The manuscript is organized as follows. In the next section, we introduce some
notations, basic definitions and give some properties for an integral operator
associated with the random potential $V_{\omega}.$ The results are stated and
proved in Section 3.
## 2 Preliminaries and notation
Throughout this paper $(\Omega,\mathcal{F},\mathbb{P})$ denotes a given
complete probability space. If $(E,\mathcal{E})$ is a measurable space, any
$(\mathcal{F},\mathcal{E})$-measurable function $X:\Omega\rightarrow E$ will
be called a $E$-valued random variable. We use the abbreviation a.s. for
almost surely or almost sure.
Let $U\subset\mathbb{R}^{n}$ be a bounded domain. We adopt the standard
notation $\mathcal{M}(U)$ to denote the set of all Random measures over $U$
having finite variation and we call $\mathscr{B}(\mathcal{M}(U))$ the
$\sigma$-algebra of the borelians of $\mathcal{M}(U)$ generated by the total
variation norm. The space of all bounded continuous real-valued functions
defined on $U$ will be denoted by $BC(U)$. Since $BC(U)$ is a metric space
with the supremum norm, when we refer to a $BC(U)$-valued random variable, the
$\sigma$-algebra we are considering is always the one generated by the
borelians. Similarly to a $\mathcal{X}$-valued Borel random variable
$X:\Omega\rightarrow\mathcal{X},$ where $\mathcal{X}$ is an arbitrary metric
space.
The random potentials considered here are the $BC(U)$-valued random variables
defined as follows. Take any random variable
$X:\Omega\rightarrow\mathcal{M}(U)$ (which is simply a random measure in
$\mathcal{M}(U)$) and a fixed function $f\in BC(\mathbb{R}^{n})$. Then, for
$\mu_{\omega}=X(\omega)$, the function $V:\Omega\rightarrow BC(U)$ defined by
$V_{\omega}(x):=\int_{U}f(x-y)\,d\mu_{\omega}(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U,$
is a $BC(U)$-valued random variable that will be called a random potential. To
see that $V$ is a well-defined $BC(U)$-valued random variable, is enough to
consider the mapping $T_{f}:\mathcal{M}(U)\rightarrow BC(U)$ given by
$T_{f}(\mu)(x)=\int_{U}f(x-y)\,d\mu(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U,$
and to observe that $V=T_{f}\circ X$. In fact, if we denote by
$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
the total variation of the measure $\mu$, the inequality
$\|T_{f}(\mu)\|_{\infty}:=\sup_{x\in
U}|T_{f}(\mu)(x)|\leq\left(\sup_{x\in\mathbb{R}^{n}}\left|f(x)\right|\right)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
(2.1)
implies that $T_{f}$ is a continuous and Borel measurable function. Since $V$
is a composition of two Borel measurable functions, $V$ is a $BC(U)$-valued
random variable.
As usual, if $(U,\mathscr{B},\mu)$ is a measure space, we define
$\|f\|_{L^{\infty}(U,d\mu)}=\inf\left\\{a\geq
0:\mu(\\{x:|f(x)|>a\\})=0\right\\}$
and the space $L^{\infty}(U,\mathscr{B}(U),\mu)$ as being the set
$\\{f:U\rightarrow\mathbb{R}:f\ \text{is Borel measurable and}\
\|f\|_{L^{\infty}(U,d\mu)}<\infty\\}.$
When $d\mu=dx$ is the Lebesgue measure in $U\subset\mathbb{R}^{n},$ we simply
denote $L^{\infty}(U)=L^{\infty}(U,\mathscr{B}(U),dx)$. Although we are
assuming that $f\in BC(\mathbb{R}^{n})$, most of the results presented here
are also valid if we suppose only the weaker condition
$f\in\cap_{\mu\in\mathcal{M}(U-U)}L^{\infty}(U-U,\mathscr{B}(U-U),\mu)$.
In order to state some convergence results obtained in this paper we need to
use the notion of Bochner integrals. Let
$(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ be a Banach space and
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If
$X:\Omega\rightarrow\mathcal{X}$ is a $\mathcal{X}$-valued Borel random
variable such that $X=Y$ a.s. in $\Omega,$ where
$Y:\Omega\rightarrow\mathcal{X}$ is a $\mathcal{X}$-valued Borel random
variable with $Y(\Omega)\subset\mathcal{X}$ separable, and
$\int_{\Omega}\|X(\omega)\|_{\mathcal{X}}\,d\mathbb{P}(\omega)<\infty,$
then there exist a unique element $\mathbb{E}[X]\in\mathcal{X}$ with the
property
$\ell(\mathbb{E}[X])=\int_{\Omega}\ell(X(\omega))\,d\mathbb{P}(\omega)$
for all $\ell\in\mathcal{X}^{\ast}$, where $\mathcal{X}^{\ast}$ is the dual of
$\mathcal{X}$. Following the standard notation we write
$\mathbb{E}[X]=\int_{\Omega}X(\omega)\,d\mathbb{P}(\omega).$
We call $\mathbb{E}[X]$ the Bochner integral of $X$ with respect to
$\mathbb{P}$. More details about the existence and some properties of this
integral can be found in [16, 18]. For these $\mathcal{X}$-valued random
variables we define the convergence in probability similarly to the real-
valued case, that is, if $\\{X_{j}\\}$ is a sequence of $\mathcal{X}$-valued
random variable we say that $X_{j}$ converges to a $\mathcal{X}$-valued random
variable $X$ in probability if for all $\varepsilon>0$, we have
$\lim_{j\rightarrow\infty}\mathbb{P}(\\{\omega\in\Omega:\|X_{j}(\omega)-X(\omega)\|_{\mathcal{X}}\geq\varepsilon\\})=0.$
(2.2)
When $X$ is real-valued random variable, we use the usual notation and denote
the expected value of $X$ and its variance by
$\mathbb{E}[X]:=\int_{\Omega}X(\omega)\,d\mathbb{P}(\omega)\text{ \ \ and \
Var}\,X:=\mathbb{E}[(\mathbb{E}[X]-X)^{2}],$
respectively. For the both senses of expectation presented above we also use
the notation
$\mathbb{E}_{A}[X]=\int_{A}X(\omega)\,d\mathbb{P}(\omega),$
whenever $A\subset\Omega$ is measurable and the right-hand-side of the
expression makes sense.
Let $X$ and $Y$ be two $E$-valued random variable in the same probability
space. We say that they are identically distributed if for all
$A\in\mathcal{E}$ we have $\mathbb{P}(X^{-1}(A))=\mathbb{P}(Y^{-1}(A))$. Now
we introduce the notion of independence. Given a finite set of random
variables $X_{1},\ldots X_{j}$ we say they are independent if for all
$A_{i}\in\mathcal{E},1\leq i\leq j$, we have
$\mathbb{P}(\cap_{i=1}^{j}X_{i}\in A_{i})=\prod_{i=1}^{j}\mathbb{P}(X_{i}\in
A_{i}).$
Finally a sequence of random variables $\\{X_{1},X_{2}\ldots\\}$ is said
independent if all finite collection of this sequence form a set of
independent random variables. If $X_{1},X_{2},\ldots$ is a sequence of
independent and identically distributed random variables we say that
$X_{1},X_{2},\ldots$ are i.i.d. random variables.
## 3 Main results and proofs
Let $G$ be the Green function of the laplacian operator $-\Delta$ in the
bounded domain $U\subset\mathbb{R}^{n}$ with $n\geq 3.$ It is known that, for
all $x,\,y\in U$, there holds
$0\leq
G(x,y)\leq\frac{1}{n\alpha_{n}(n-2)}\frac{1}{|x-y|^{n-2}},\leavevmode\nobreak\
$
where $\alpha_{n}$ stands for the volume of the unit ball in $\mathbb{R}^{n}$.
Hence, if we denote by $d_{U}$ the the diameter of $U$, namely
$d_{U}:=\sup_{x_{1},\,x_{2}\in U}{|}x_{1}-x_{2}{|},$
and $B_{d_{U}}(x)=\\{x\in\mathbb{R}^{n};\left|x\right|<d_{U}\\},$ a
straightforward calculation provides
$\begin{array}[]{lcl}\displaystyle\int_{U}G(x,y)dy&\leq&\dfrac{1}{n\alpha_{n}(n-2)}\displaystyle\int_{B_{d_{U}}(x)}\dfrac{1}{|x-y|^{n-2}}dy\vspace{0.2cm}\\\
&=&\dfrac{1}{n\alpha_{n}(n-2)}\dfrac{n\alpha_{n}d_{U}^{2}}{2}=\dfrac{d_{U}^{2}}{2(n-2)},\end{array}$
(3.1)
for all $x\in U$. From now on we write only $l_{0}=l_{0}(n,U)$ to denote the
following quantity
$l_{0}:=\dfrac{d_{U}^{2}}{2(n-2)}.$ (3.2)
Inequality (3.1) implies that is well defined the map
$H:L^{\infty}(U)\rightarrow L^{\infty}(U)$ given by
$H(\varphi)(x):=\int_{U}G(x,y)\varphi(y)dy,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in U.$
More specifically, for any $\varphi\in L^{\infty}(U)$, there holds
$|H(\varphi)(x)|\leq\int_{U}G(x,y)|\varphi(y)|dy\leq\|\varphi\|_{\infty}\int_{U}G(x,y)dy$
and therefore
$\|H(\varphi)\|_{\infty}\leq l_{0}\|\varphi\|_{\infty}.$ (3.3)
Standard calculations show that the problem (1.2) is formally equivalent to
the integral equation
$u(x)=H(g)+H(V_{\omega}u)+H(bu|u|^{p-1}).$ (3.4)
In what follows we make suitable estimates on the terms of the integral
equation in order to be able to apply a fixed point argument. We first set
$\mathcal{X}:=L^{\infty}(U)$ and define, for any fixed $\omega\in\Omega$, the
linear function $T:\mathcal{X}\rightarrow\mathcal{X}$ by
$T(u):=H(V_{\omega}u),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\,u\in\mathcal{X}.$
It follows from (3.3) and (2.1) that, for any $u\in\mathcal{X}$, there holds
$\|T(u)\|_{\infty}\leq l_{0}\|V_{\omega}u\|_{\infty}\leq
l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
}\|u\|_{\infty},$ (3.5)
and therefore
$\|T\|_{\infty}\leq
l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}.$
For the nonlinear term we define $B:\mathcal{X}\rightarrow\mathcal{X}$ by
setting
$B(u):=H(b|u|^{p-1}u),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\,u\in\mathcal{X}.$
If $a_{1},\,a_{2}\in\mathbb{R}$ there holds
$\left|a_{1}|a_{1}|^{p-1}-a_{2}|a_{2}|^{p-1}\right|\leq
p|a_{1}-a_{2}|\left(|a_{1}|^{p-1}-|a_{2}|^{p-1}\right),$
and therefore it follows that
$\|b(\cdot)\left(u|u|^{p-1}-\tilde{u}|\tilde{u}|^{p-1}\right)\|_{\infty}\leq\|b\|_{\infty}\|u-\tilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}-\|\tilde{u}\|_{\infty}^{p-1}\right).$
This inequality and the same argument used in (3.5) imply that
$\|B(u)-B(\tilde{u})\|_{\infty}\leq
l_{0}p\|b\|_{\infty}\|u-\tilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}-\|\tilde{u}\|_{\infty}^{p-1}\right),$
(3.6)
for any $u,\,\tilde{u}\in L^{\infty}(U)$.
All together, the above estimates enable us to solve the random equation (1.2)
as follows.
###### Proposition 3.1.
Given $f,\,b,g\in L^{\infty}(U)$ and $\omega\in\Omega$, we consider the
potential $V_{\omega}$ induced by the random measure
$\mu_{\omega}:=X(\omega)$. Let $l_{0}$ be the quantity introduced in (3.2) and
set
$\tau_{\omega}:=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
\ \ and}\leavevmode\nobreak\ \leavevmode\nobreak\ K:=l_{0}p\|b\|_{\infty}.$
(3.7)
If $\varepsilon>0$ and $\omega\in\Omega$ are such that
$0\leq\tau_{\omega}<1,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}+\tau_{\omega}<1,$
(3.8)
and $\|g\|_{\infty}\leq\varepsilon/l_{0}$, then the equation (1.2) has a
unique integral solution
$u_{\omega}=u(\cdot,\omega)\in L^{\infty}(U)\text{ such that
}\|u_{\omega}\|_{\infty}\leq\frac{2\varepsilon}{1-\tau_{\omega}}.$ (3.9)
Proof. For each $\omega\in\Omega,$ we consider the closed ball
$\mathcal{B}_{\varepsilon}=\left\\{u\in
L^{\infty}(U);\|u\|_{\infty}\leq\frac{2\varepsilon}{(1-\tau_{\omega})}\right\\}$
endowed with the metric $d(u,v):=\|u-v\|_{\infty}.$ We are going to show that
the map
$\Phi(u):=H(g)+H(V_{\omega}u)+H(bu\left|u\right|^{p-1})=H(g)+T(u)+B(u)$ (3.10)
is a contraction on the complete metric space $(\mathcal{B}_{\varepsilon},d).$
Using the estimates (3.3), (3.5), and (3.6) with $\tilde{u}=0,$ we obtain
$\displaystyle\left\|\Phi(u)\right\|_{\infty}$
$\displaystyle\leq\|H(g)\|_{\infty}+\|T(u)\|_{\infty}+\|B(u)\|_{\infty}$
$\displaystyle\leq
l_{0}\left\|g\right\|_{\infty}+\tau_{\omega}\|u\|_{\infty}+K\|u\|_{\infty}^{p}$
$\displaystyle\leq\varepsilon+\tau_{\omega}\frac{2\varepsilon}{1-\tau_{\omega}}+\frac{2^{p}K\varepsilon^{p}}{(1-\tau_{\omega})^{p}}$
$\displaystyle=\left(1+\tau_{\omega}+\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}\right)\frac{\varepsilon}{1-\tau_{\omega}}$
for all $u\in\mathcal{B}_{\varepsilon}$ and $\omega\in\Omega$. Hence, it
follows from (3.8) that
$\left\|\Phi(u)\right\|_{\infty}\leq\frac{2\varepsilon}{1-\tau_{\omega}}.$
This shows that $\Phi$ maps $\mathcal{B}_{\varepsilon}$ into
$\mathcal{B}_{\varepsilon}$.
For any $u,\widetilde{u}\in\mathcal{B}_{\varepsilon},$ it follows from (3.5)
and (3.6) that
$\displaystyle\|\Phi(u)-\Phi(\widetilde{u})\|_{\infty}$
$\displaystyle=\|T(u-\widetilde{u})\|_{\infty}+\|B(u)-B(\widetilde{u})\|_{\infty}$
$\displaystyle\leq\tau_{\omega}\|u-\widetilde{u}\|_{\infty}+K\|u-\widetilde{u}\|_{\infty}\left(\|u\|_{\infty}^{p-1}+\|\widetilde{u}\|_{\infty}^{p-1}\right)$
$\displaystyle\leq\left(\tau_{\omega}+\frac{2^{p}K\varepsilon^{p-1}}{(1-\tau_{\omega})^{p-1}}\right)\|u-\widetilde{u}\|_{\infty}.$
Recalling (3.8), the above estimate implies that the map $\Phi$ is a
contraction. The Banach fixed point theorem assures that there is a unique
solution $u$ for the integral equation (3.4) such that
$\|u\|_{\infty}\leq(2\varepsilon)/(1-\tau_{\omega}).$
The next results are related to the randomness introduced by the random
potential $V$ and the existence and uniqueness of solutions for the problem
(1.2). Roughly speaking, we first obtain the probability of (1.2) having a
solution given by the method discussed above. In the sequel we study two
important limit theorems in probability theory, namely, the central limit
theorem and the law of large numbers for a sequence of random potentials.
###### Theorem 3.2.
Let $\nu$ be the probability measure induced on $\mathbb{R}$ by the random
variable
$\omega\mapsto\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\,.$
Let $g\in L^{\infty}(U)$ be such that
$\left\|g\right\|_{\infty}<\frac{1}{l_{0}}(\frac{1}{2^{p}K})^{\frac{1}{p-1}}$,
where $K=l_{0}p\|b\|_{\infty}$. Choose $0<c_{0}<1$and set
$\varepsilon_{0}:=\left(\frac{(1-c_{0})^{p}}{2^{p}K}\right)^{\frac{1}{p-1}}.$
Let $\mathcal{A}$ be the set of $\omega\in\Omega$ such that (1.2) has a unique
solution $u(\cdot,\omega)$ given by Proposition 3.1 with
$\varepsilon=\varepsilon_{0}$. The set $\mathcal{A}$ is called the admissible
one for the random variable $X.$
* (i)
The set $\mathcal{A}$ is $\mathcal{F}$-measurable and the probability of (1.2)
having a solution is
$\mathbb{P(\mathcal{A})}=\nu\left(\left[0,\frac{1}{l_{0}\|f\|_{\infty}}\right)\right).$
* (ii)
Let $u_{\omega},\,\tilde{u}_{\omega}$ be two solutions of (1.2) corresponding,
respectively, to $\mu_{\omega},g,\mathcal{A}$ and
$\tilde{\mu}_{\omega},\tilde{g},\widetilde{\mathcal{A}}$. Assume that
$\mathcal{A\cap}\widetilde{\mathcal{A}}\neq\varnothing$ and define, for
$\omega\in\mathcal{A\cap}\widetilde{\mathcal{A}}$,
$\eta_{\omega}:=l_{0}\|f\|_{\infty}\max\\{\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt},\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\widetilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\\}.$
We have that
$\|u(\cdot,\omega)-\tilde{u}(\cdot,\omega)\|_{\infty}\leq\frac{l_{0}\left(\|g-\tilde{g}\|_{\infty}+\dfrac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\right)}{1-\eta_{\omega}-\dfrac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}}$
(3.11)
for all $\omega\in\mathcal{A\cap}\widetilde{\mathcal{A}}$.
* (iii)
The map $\ \mathcal{U}:\mathcal{A}\rightarrow L^{\infty}(U)$ given by
$\mathcal{U}(\omega):=u(\cdot,\omega)$ is a random variable and there holds
$\|u(\cdot,\omega)\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\tau_{\omega}}=2\varepsilon_{0}\sum_{j=0}^{\infty}\tau_{\omega}^{j},$
(3.12)
for all $\omega\in\mathcal{A}.$
Proof. We first notice that the choice of $\varepsilon_{0}$ implies that
$\|g\|_{\infty}\leq\varepsilon_{0}/l_{0}.$ Moreover, $\omega\in\mathcal{A}$ if
only if
$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
verifies (3.8) with $\varepsilon=\varepsilon_{0}.$ Then, if
$Y(\omega)=\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,X(\omega)\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,=\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt},$
it follows that
$\mathcal{A}=\left\\{Y\in\left[0,\frac{1}{l_{0}\|f\|_{\infty}}\right)\right\\}$
is measurable and
$\begin{array}[]{lcl}\mathbb{P}(\mathcal{A})&=&\mathbb{P}\left(Y\in\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right)=\mathbb{P}_{Y}\left(\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right)\vspace{0.2cm}\\\
&=&\nu\left(\left[0,\dfrac{1}{l_{0}\|f\|_{\infty}}\right)\right).\end{array}$
This establishes (i).
Now we deal with item (ii). Firstly, observe that
$\eta_{\omega}=\max\\{\tau_{\omega},\widetilde{\tau}_{\omega}\\},$ where
$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\text{
and
}\widetilde{\tau}_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}.$
Subtracting the integral equations verified by $u_{\omega}$ and
$\,\tilde{u}_{\omega},$ and afterwards computing $\|\cdot\|_{\infty}$, we
obtain
$\begin{array}[]{lcl}\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}&\leq&\left\|H(g-\tilde{g})\right\|_{\infty}+\left\|H(V_{\omega}(u-\tilde{u}_{\omega}))\right\|_{\infty}\vspace{0.2cm}\\\
&&+\|H((V_{\omega}-\widetilde{V}_{\omega})\tilde{u}_{\omega})\|_{\infty}\vspace{0.2cm}\\\
&&+\left\|H(b\left(u_{\omega}|u_{\omega}|^{p-1}-\tilde{u}_{\omega}|\tilde{u}_{\omega}|^{p-1}\right))\right\|_{\infty}\vspace{0.2cm}\\\
&\leq&l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}\vspace{0.2cm}\\\
&&+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\|\tilde{u}_{\omega}\|_{\infty}\vspace{0.2cm}\\\
&&+l_{0}p\|b\|_{\infty}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}(\|u_{\omega}\|_{\infty}^{p-1}-\|\tilde{u}_{\omega}\|_{\infty}^{p-1}).\end{array}$
It follows from (3.9) that
$\|u_{\omega}\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\tau_{\omega}}\leq\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\text{
and
}\|\tilde{u}\|_{\infty}\leq\frac{2\varepsilon_{0}}{1-\widetilde{\tau}_{\omega}}\leq\frac{2\varepsilon_{0}}{1-\eta_{\omega}}.$
The two above expressions give us
$\displaystyle\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}$
$\displaystyle\leq$ $\displaystyle
l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty}$
$\displaystyle+\,l_{0}\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}+\frac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}\|u_{\omega}-\tilde{u}_{\omega}\|_{\infty}$
$\displaystyle=$ $\displaystyle
l_{0}\|g-\tilde{g}\|_{\infty}+l_{0}\frac{2\varepsilon_{0}}{1-\eta_{\omega}}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}-\tilde{\mu}_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
$\displaystyle+\,\left[\eta_{\omega}+\frac{2^{p}K\varepsilon_{0}^{p-1}}{(1-\eta_{\omega})^{p-1}}\right]\left\|u_{\omega}-\tilde{u}_{\omega}\right\|_{\infty},$
which yields (3.11).
Taking $\mu_{\omega},\tilde{\mu}_{\omega}$ independent of $\omega,$ i.e.
$\mu_{\omega}=\mu$ and $\tilde{\mu}_{\omega}=\tilde{\mu},$ for all
$\omega\in\Omega,$ we see from (3.7) and (3.11) that the data-map solution
$\mathcal{L}(\mu,g)=u$ is continuous from
$\left\\{(\mu,g)\in\mathcal{M}(U)\times L^{\infty}(U);\text{
}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\frac{1}{l_{0}\|f\|_{\infty}},\left\|g\right\|_{\infty}<\frac{1}{l_{0}}\left(\frac{1}{2^{p}K}\right)^{\frac{1}{p-1}}\right\\}\text{to
}L^{\infty}(U),$ (3.13)
where $u$ is the deterministic solution of (1.2) corresponding to the data
$(\mu,g).$ From this, and because $X|_{\mathcal{A}}$ given by
$X(\omega)=\mu_{\omega}$ is measurable, it follows that the composition
$\mathcal{U}(\omega)=\mathcal{L}(\mu_{\omega},g)=\mathcal{L}(X(\omega),g)\,$
from $\mathcal{A}$ to $L^{\infty}(U)$ is measurable.
In view of the series $\frac{1}{1-z}=\sum_{j=0}^{\infty}z^{j}$ for
$\left|z\right|<1,$ we finish by observing that (3.12) follows at once from
(3.9) with $\varepsilon=\varepsilon_{0}$ and $\omega\in\mathcal{A}.$
###### Remark 3.3.
Here we give examples of random potentials for which there exists solution
almost surely in $\Omega$. The first setting occurs if we suppose that the
measure $\nu$ has compact support contained in the interval $[0,a]$, with
$a<\frac{1}{l_{0}\|f\|_{\infty}}$. In this case it follows from the first item
of the above theorem that $\mathbb{P(\mathcal{A})}=1$, i.e., the solution
exists almost surely in $\Omega.$ Secondly, we take
$\\{\mu_{j}\\}_{j\in\mathbb{N}}$ a sequence in $\mathcal{M}(U)$ and let
$\\{a_{j}(\omega)\\}_{j\in\mathbb{N}}$ be a sequence of random variables from
$\Omega$ to $\mathbb{R}.$ Consider the random variable $\mu_{\omega}$ defined
by
$\mu_{\omega}=\sum_{j=1}^{\infty}a_{j}(\omega)\mu_{j}.$
For $q>1,$ suppose that
$|a_{j}(\omega)|<\frac{(\sum_{k=1}^{\infty}\frac{1}{k^{q}})^{-1}}{l_{0}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{j}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\|f\|_{\infty}}\cdot\frac{1}{j^{q}}\text{
a.s. in }\Omega,$
for all $j\in\mathbb{N}.$ Then
$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\leq\sum_{j=1}^{\infty}|a_{j}(\omega)|\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{j}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,<\frac{1}{l_{0}\|f\|_{\infty}}\text{
a.s. in }\Omega,$
and Theorem 3.2 assures that there is an integral solution for (1.2) a.s. in
$\Omega.$
In the sequel we show how the Borel-Cantelli’s Lemma can be used to give a
sufficient condition for the existence of solution a.s. in $\Omega$.
###### Theorem 3.4.
Let $\\{\mu_{j}\\}_{j\in\mathbb{N}}$ be a sequence in $\mathcal{M}(U)$ and let
$\\{a_{j}(\omega)\\}_{j\in\mathbb{N}}$ be a sequence of random variables from
$\Omega$ to $\mathbb{R}.$ Assume that the following series is convergent in
$\mathcal{M}(U)$
$\mu_{\omega}=\sum_{j=1}^{\infty}a_{j}(\omega)\mu_{j}.$
For any $k\in\mathbb{N}$ define
$S_{k}(\omega)=\sum_{j=1}^{k}a_{j}(\omega)\mu_{j}$
and
$L_{k}=\\{\omega\in\Omega:\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\geq\tilde{c}\\}$,
with $0<\tilde{c}<1/(l_{0}\|f\|_{\infty})$. If
$\sum_{k=1}^{\infty}\mathbb{P}(L_{k})<\infty$
then there is an integral solution for (1.2) almost surely in $\Omega$.
Proof. By the Borel-Cantelli’s Lemma we get that $\mathbb{P}(\limsup
L_{k})=0$, that is,
$\mathbb{P}\left(\cup_{j=1}^{\infty}\cap_{k=j}^{\infty}\\{\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\tilde{c}\\}\right)=1$
It follows that, for almost sure $\omega,$ there is $j_{0}=$ $j_{0}(\omega)$
such that for all $j>j_{0}$, we have
$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{k}(\omega)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\tilde{c}.$
Therefore by taking the limit when $k$ goes to infinity, we obtain
$\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}=\lim_{j\rightarrow\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,S_{j}(\omega)\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\leq\tilde{c}<\frac{1}{l_{0}\|f\|_{\infty}}\quad\mbox{a.s.
in }\Omega.$
This inequality and Theorem 3.2 imply that there is an integral solution
$u(x,\omega)$ for (1.2) almost surely in $\Omega$.
A straightforward calculation shows that in general
$\mathbb{E}_{\Omega}(u(x,\omega))$ does not satisfies the equation (1.2), even
if we replace the random potential by its mean. However, we are able to obtain
some information on the average and moments of the random solution
$u_{\omega}$ previously obtained. It is worthwhile to mention that, when
dealing with the random variable $\omega\mapsto u_{\omega}$, the expectation
has to be understood in the Bochner sense (see Section 2). Note also that a
solution $u_{\omega}\in L^{\infty}(U)$ for (3.4) in fact belongs to the
separable subspace $C(\overline{U}).$
###### Theorem 3.5.
Under hypotheses of Theorem 3.2 let us denote by
$u_{\omega}(x)=u(x,\omega)\in\mathcal{A}$ the solution of (1.2). Let
$m\in\mathbb{N}$ and suppose that
$\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}(l_{0}\|f\|_{\infty})^{j}\
\mathbb{E}_{\mathcal{A}}[\
\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}^{\,j}\,]<+\infty.$
(3.14)
Then $\mathbb{E}_{\mathcal{A}}[|u|^{m}(x,\omega)]\in L^{\infty}(U)$ and
$\mathbb{E}_{\mathcal{A}}\left[\left\||u|^{m}(\cdot,\omega)\right\|_{L^{\infty}(U)}\right]<\infty.$
(3.15)
In particular, $\mathbb{E}_{\mathcal{A}}[u(x,\omega)]\in L^{\infty}(U).$
Proof. It follows from (3.12) that
$\||u|^{m}(\cdot,\omega)\|_{L^{\infty}(U)}\leq\|u(\cdot,\omega)\|_{L^{\infty}(U)}^{m}\leq\frac{(2\varepsilon_{0})^{m}}{(1-\tau_{\omega})^{m}}.$
(3.16)
Recalling that
$\tau_{\omega}=l_{0}\|f\|_{\infty}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}$
and computing $\mathbb{E}_{\mathcal{A}}$ in (3.16), we obtain
$\displaystyle\left\|\mathbb{E}_{\mathcal{A}}\left[|u|^{m}(x,\omega)\right]\right\|_{L^{\infty}(U)}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\mathcal{A}}\left[\left\||u|^{m}(x,\omega)\right\|_{L^{\infty}(U)}\right]$
$\displaystyle\leq$
$\displaystyle(2\varepsilon_{0})^{m}\mathbb{E}_{\mathcal{A}}\left[\left(1+\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}\tau_{\omega}^{j}\right)\right]$
By using the linearity of the expectation and definition of $\tau_{\omega}$ we
get the following upper bound for the right hand side above
$(2\varepsilon_{0})^{m}+(2\varepsilon_{0})^{m}\sum_{j=1}^{\infty}\frac{(m+j-1)!}{(m-1)!j!}\left(l_{0}\|f\|_{\infty}\right)^{j}\mathbb{E}_{\mathcal{A}}\left[\
\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}^{\,j}\
\right],$
which is finite due to (3.14). The last assertion of the statement follows
from (3.15) with $m=1$ and the easy estimate
$\left\|\mathbb{E}_{\mathcal{A}}\left[u(x,\omega)\right]\right\|_{L^{\infty}(U)}\leq\mathbb{E}_{\mathcal{A}}\left[\left\||u|(x,\omega)\right\|_{L^{\infty}(U)}\right].$
### 3.1 Classical Probability Limit Theorems
We start this section by recalling basic background concerning to some main
limit theorems in probability. A real-valued random variable
$X:\Omega\to\mathbb{R}$ in a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ has standard normal distribution, notation
$X\sim N(0,1)$, if for all $x\in\mathbb{R}$ its cumulative distribution
function verifies
$\mathbb{P}(X\leq
x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{1}{2}t^{2}}dt.$
A sequence of real-valued random variable $\\{Y_{j}\\}_{j\in\mathbb{N}}$ in a
probability space $(\Omega,\mathcal{F},\mathbb{P})$ is said to converge in
distribution to a standard normal random variable, notation $Y_{j}\to N(0,1)$,
if for all $x\in\mathbb{R}$ we have
$\lim_{j\to\infty}\mathbb{P}(Y_{j}\leq
x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{1}{2}t^{2}}dt.$
In the sequel we show versions of the central limit theorem and a weak law of
large numbers for the random $L^{\infty}(U)$-solutions obtained in Section 2.
###### Theorem 3.6.
Let $\\{X_{j}\\}_{j\in\mathbb{N}}$ be an independent identically distributed
(i.i.d.) sequence of random variables $X_{j}:\Omega\rightarrow\mathcal{M}(U)$.
Assume that the admissible set $\mathcal{A}_{j}=\Omega$ for all $j$, and let
$u_{j}(\cdot,\omega)\in L^{\infty}(U)$ be the solution given by Theorem 3.2
with respect to $X_{j}(\omega)=\mu_{\omega,j}$ and $g.$ We have that
$\\{Z_{j}\\}_{j\in\mathbb{N}}$ given by
$Z_{j}(\omega):=\|u_{j}(\cdot,\omega)\|_{\infty}$ is a i.i.d. sequence of
random variables, and if
$m=\mathbb{E}[\|u_{j}(\cdot,\omega)\|_{\infty}]<\infty$ and
$\sigma^{2}:=\text{Var}\,Z_{j}<\infty$ then following holds as
$k\rightarrow+\infty$
$\sum_{j=1}^{k}\frac{(Z_{j}-m)}{\sigma\sqrt{k}}\rightarrow N(0,1).$
Proof. Recall the data-solution map $\mathcal{L}(\mu,g)$ defined in the proof
of Theorem 3.2 (see (3.13)). Fixed $g$ such that
$\left\|g\right\|_{\infty}<\frac{1}{l_{0}}(\frac{1}{2^{p}K})^{\frac{1}{p-1}},$
consider
$S_{g}(\mu)=\mathcal{L}(\mu,g)$ (3.17)
defined from $D$ to $L^{\infty}(U),$ where
$D=\left\\{\mu\in\mathcal{M}(U):\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}<\frac{1}{l_{0}\|f\|_{\infty}}\right\\}.$
Since $\|\cdot\|_{\infty}$ is continuous from $L^{\infty}(U)$ to $\mathbb{R}$
and
$Z_{j}(\omega)=\|u_{j}(\cdot,\omega)\|_{\infty}=\|S_{g}\circ
X_{j}(\omega)\|_{\infty},$
we get that $\\{Z_{j}\\}_{j\in\mathbb{N}}$ is a i.i.d. sequence. The
convergence stated in the theorem follows from the central limit theorem.
###### Theorem 3.7.
Let $\\{X_{j}\\}_{j\in\mathbb{N}}$ be an independent sequence of random
variables $X_{j}:\Omega\rightarrow\mathcal{M}(U)$. Assume that the admissible
set $\mathcal{A}_{j}=\Omega$ for all $j$, and let $u_{j}(\cdot,\omega)\in
L^{\infty}(U)$ be the solution given by Theorem 3.2 with respect to
$X_{j}(\omega)=\mu_{\omega,j}$ and $g.$ If $X_{j}\rightarrow X$ a.s. and
$L=\sup_{j\in\mathbb{N}}\left(\mathrm{ess}\sup_{\omega\in\Omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega,j}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt}\right)<\frac{1}{l_{0}\|f\|_{\infty}},$
(3.18)
then
$\sum_{j=1}^{k}\frac{u_{j}(x,\omega)-\mathbb{E}_{\Omega}[u_{j}(x,\omega)]}{k}\rightarrow
0$ (3.19)
and
$\sum_{j=1}^{k}\frac{\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]}{k}\rightarrow
0,$ (3.20)
when $k\rightarrow\infty$, where the convergence in (3.19) and (3.20) are in
probability sense (see (2.2)).
Proof. Notice that $X_{j}\rightarrow X$ a.s. is equivalent to
$\mu_{\omega,j}\rightarrow\mu_{\omega}=X(\omega)$ in $\mathcal{M}(U)$ almost
surely. From this and the continuity of data-solution map
$\mathcal{L}(\cdot,\cdot)$ (see (3.13)), it follows that
$\|u_{j}(\cdot,\omega)-u(\cdot,\omega)\|_{\infty}=\|\mathcal{L}(\mu_{\omega,j},g)-\mathcal{L}(\mu,g)\|_{\infty}\rightarrow
0,$
when $j\rightarrow\infty$. Recalling (3.12) and afterwards using (3.18), we
obtain
$\displaystyle\|u_{j}(\cdot,\omega)\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\frac{2\varepsilon_{0}}{1-l_{0}\|f\|_{\infty}(\mathrm{ess}\sup_{\omega\in\Omega}\rule[-2.27621pt]{1.13809pt}{9.6739pt}\,\mu_{\omega},_{j}\,\rule[-2.27621pt]{1.13809pt}{9.6739pt})}$
(3.21) $\displaystyle\leq$
$\displaystyle\frac{2\varepsilon_{0}}{1-L}=Q_{0},\text{ a.s. in }\Omega.$
Since $X_{j}$’s are independent, it follows that
$\\{Y_{j}\\}_{j\in\mathbb{N}}$ defined by
$Y_{j}=\left\|u_{j}(\cdot,\omega)\right\|_{\infty}=\|S_{g}\circ
X_{j}(\omega)\|_{\infty}$ are also independent, where $S_{g}$ is as in (3.17).
So, from Chebyshev’s inequality and the independence of
$\\{Y_{j}\\}_{j\in\mathbb{N}}$, we have that
$\displaystyle\mathbb{P}\left(\left|k^{-1}\sum_{j=1}^{k}(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}])\right|\geq\delta\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{(k\delta)^{2}}\mathbb{E}_{\Omega}\left[\left|\sum_{j=1}^{k}\left(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]\text{
}\right)\right|^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{(k\delta)^{2}}\sum_{j=1}^{k}\mathbb{E}_{\Omega}\left[\left|\left(\|u_{j}(\cdot,\omega)\|_{\infty}-\mathbb{E}_{\Omega}[\|u_{j}(\cdot,\omega)\|_{\infty}]\text{
}\right)\right|^{2}\right]$ $\displaystyle\leq$
$\displaystyle\frac{1}{(k\delta)^{2}}\sum_{j=1}^{k}\mathbb{E}_{\Omega}\left[\left|2Q_{0}\right|^{2}\right]\leq\frac{4Q_{0}^{2}}{\delta^{2}}\frac{1}{k},$
where we have used (3.21). Letting $k\rightarrow+\infty$ in the above
expression we get (3.20). The convergence (3.19) can be proved with similar
arguments.
## Acknowledgments
L.C.F. Ferreira was supported by FAPESP-SP and CNPq, Brazil. M. Furtado was
supported by CNPq, Brazil.
## References
* [1] A. Ambrosetti, A. Malchiodi and S. Secchi, _Multiplicity results for some nonlinear Schrödinger equations with potentials_ , Arch. Rational Mech. Anal. 159 (2001), 253–271.
* [2] A. Ambrosetti, M. Badiale and S. Cingolani, _Semiclassical states of nonlinear Schrödinger equations_ , Arch. Rational Mech. Anal. 140 (1997), 285-300.
* [3] G. Bal, T. Komorowski and L. Ryzhik: Asymptotic of the Solutions of the Random Schrödinger Equation. Arch. Rational Mech. Anal. 200, 613-664 (2011).
* [4] J. Bourgain: Nonlinear Schrödinger Equation With a Random Potential. Illinois J. math. 50, 183-188 (2006).
* [5] J. Bourgain and C. Kenig: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. math. 161, 389-426 (2005).
* [6] J. Bourgain and W.-M. Wang: Quasi-periodic solutions of nonlinear random Schrödinger equations. J. Eur. Math Soc. 10, 1-45 (2008).
* [7] R. Carmona, J. Lacroix, Spectral theory of random Schrödinger operators. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990.
* [8] J.-M. Combes and P.D. Hislop: Localization for Some Continuous, Random Hamiltonians in $d$-dimensions. J. Funct. Anal. 124, 149-180 (1994).
* [9] J. G. Conlon and A. Naddaf: Green’s Functions for Elliptic and Parabolic Equations with Random Coefficients. New York J. Math. 6, 153-225 (2000).
* [10] D. A. Dawson and M. Kouritzin: Invariance Principles for Parabolic Equations with Random Coefficients. J. Funct. Anal. 149, 377-414 (1997).
* [11] M. Del Pino and P. Felmer, _Local Mountain Pass for semilinear elliptic problems in unbounded domains_ , Calc. Var. Partial Differential Equations 4 (1996), 121–137.
* [12] A. Fannjiang: Self-Averaging Scaling Limits for Random Parabolic Waves. Arch. Rational Mech. Anal. 175, 343-387 (2008).
* [13] L.C.F. Ferreira and M. Montenegro: Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials. J. Differential Equations 250, 2045-2063 (2011).
* [14] L.C.F. Ferreira, E.S. Medeiros and M. Montenegro: A class of elliptic equations in anisotropic spaces, to appear in Annali di Matematica Pura ed Applicata doi:10.1007/s10231-011-0236-8 (2012).
* [15] F. Germinet, A. Klein and J. Schenker: Dynamical delocalization in random Landau Hamiltonians. Ann. of Math. 166, 215-244 (2007).
* [16] E. Hille and R.S. Phillips: Functional Analysis and Semigroups. Amer. Math Soc. Colloquium Publ. 31 Amer. Math. Soc., Providence, Rhode (1957).
* [17] W. Kirsch: An invitation to random Schrödinger operators. In Random Schrödinger operators, volume 25 of Panor. Synthèses, Soc. Math. France, Paris, 1-119 (2008).
* [18] K.R. Parthasarathy: Probability Measures on Metric Spaces. Academic Press (1967).
* [19] P.H. Rabinowitz, _On a class of nonlinear Schrödinger equations_ , Z. Angew Math. Phys. 43 (1992), 270-291.
* [20] O. Safronov: Absolutely continuous spectrum of one random elliptic operator. J. Funct. Anal. 255, 755-767 (2008).
|
arxiv-papers
| 2013-04-09T02:32:15 |
2024-09-04T02:49:43.997347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leandro Cioletti, Lucas C. F. Ferreira and Marcelo Furtado",
"submitter": "Leandro Cioletti",
"url": "https://arxiv.org/abs/1304.2449"
}
|
1304.2451
|
# Multihomogeneous Normed Algebras and
Polynomial Identities
Leandro Cioletti
José Antônio Freitas
Departamento de Matemática,
Universidade de Brasília,
70910-900 Brasília, DF, Brazil
Dimas José Gonçalves
Departamento de Matemática,
Universidade Federal de São Carlos,
13565-905 São Carlos, SP, Brazil [email protected]@mat.unb.br;
Partially supported by grant from CNPq No. 478318/2010-3 [email protected];
Partially supported by grant from CNPq No. 478318/2010-3
###### Abstract
In this paper we consider PI-algebras $A$ over $\mathbb{R}$ or $\mathbb{C}$.
It is well known that in general such algebras are not normed algebras. In
fact, there is a nilpontent commutative algebra which is not a normed algebra,
see [1]. Here we address the question of whether it is possible to find a
normed PI-algebra $B$ with the same polynomial identities as $A$, and
moreover, whether there is some Banach PI-algebra with this property. Our main
theorem provides an affirmative answer for this question and moreover we also
show the existence of a Banach Algebra with the same polynomial identities as
$A$. As a byproduct we prove that if $A$ is a normed PI-algebra and its
completion is nil, then $A$ is nilpotent. By introducing the concept of
multihomogeneous norm we obtain as an application of our main results that if
$F\langle X\rangle$ is multihomogeneus normed algebra and $A$ is a PI-algebra
such that the completion of the quotient space $F\langle X\rangle/Id(A)$ is
nil, then $A$ is nilpotent. Both applications are extensions of the study
initiated in [4].
Key words: PI-Algebras, Normed Algebras, Banach Algebras.
2010 Mathematics Subject Classification: 16R10, 16R40, 46H10.
## 1 Introduction
We begin this article by stating precisely some of our main results and then
we proceed to introduce the concept of multihomogeneous norm. In order to be
concise and objective, we will skip the precise definition of some of the
basic concepts needed here, such as PI-algebra and Normed algebra. These,
together with other additional background concepts, will appear in detail in
Section 2. The proofs of the results stated here are found in the last
section.
Let $F$ be the field $\mathbb{R}$ or $\mathbb{C}$ and $F\langle X\rangle$ the
free non-unitary associative algebra, freely generated over $F$ by the
infinite set $X=\\{x_{1},x_{2},\ldots\\}$. All the algebras considered in this
paper will be non-unitary, associative and over the field $F$. Thus for
convenience we will only use the term algebra. A good example to keep in mind
is the algebra $F\langle X\rangle$.
For a normed algebra $A$ we write $C(A)$ to denote its completion. If $A$ is a
PI-algebra then we denote by $Id(A)$ the set of all polynomial identities of
$A$.
The statement of our first result is:
###### Proposition 1.
If $A$ is a normed PI-algebra, then $Id(A)=Id(C(A))$.
In other words this proposition tell us that every normed PI-algebra $A$ has
the same polynomial identities that some Banach PI-algebra. Since not all PI-
algebras are normed PI-algebras, see for example [1], a natural question to
ask is: given a PI-algebra $A$, is there some Banach PI-algebra $B$ with the
same polynomial identities of $A$ ? As we said before we give an affirmative
answer for this question and we also show how to construct such algebra $B$.
To explain the construction we introduce some definitions.
Let $f=f(x_{1},\dots,x_{n})\in F\langle X\rangle$ be a polynomial, which will
be written as
$f=\displaystyle\sum_{d_{1}\geq 0,\dots,d_{n}\geq 0}f^{(d_{1},\dots,d_{n})}\
,$
where $f^{(d_{1},\dots,d_{n})}=f^{(d_{1},\dots,d_{n})}(x_{1},\dots,x_{n})$ is
the multihomogeneous component of $f$ with multidegree $(d_{1},\ldots,d_{n})$.
###### Definition 2.
A norm $||\cdot||$ in $F\langle X\rangle$ is called multihomogenous if
$||f^{(d_{1},\ldots,d_{n})}||\leq||f||$
for all $f=f(x_{1},\ldots,x_{n})\in F\langle X\rangle$ and all
$d=(d_{1},\ldots,d_{n})$. If $F\langle X\rangle$ is a normed algebra with
respect to a multihomegeneous norm, then we say that $F\langle X\rangle$ is a
MN-algebra.
An example of MN-algebras can be obtained as follows. Take
$f=\sum_{m}\alpha_{m}m$, where $\alpha_{m}\in F$ and $m$ is a monomial, then
$F\langle X\rangle$ with the norm
$||f||=\sum_{m}|\alpha_{m}|.$
is a MN-algebra.
In the sequel, we prove that if $F\langle X\rangle$ is a MN-algebra and if $A$
is a PI-algebra, then $Id(A)$ is a closed ideal of $F\langle X\rangle$. Thus
the multihomogeneous norm in $F\langle X\rangle$ induces a norm in the
quotient algebra $F\langle X\rangle/Id(A)$ by
$||f+Id(A)||=\mbox{inf}\\{||f+g||\,:\,g\in Id(A)\\},$
where $f\in F\langle X\rangle$. We remark that the quotient $F\langle
X\rangle/Id(A)$ is a normed algebra with this norm and using this fact we
obtain our second main result:
###### Theorem 3.
Let $F\langle X\rangle$ be a MN-algebra. If $A$ is a PI-algebra, then
$Id(A)=Id\left(C\left(\frac{F\langle X\rangle}{Id(A)}\right)\right).$
In particular, a PI-algebra has the same polinomial identities that some
Banach PI-algebra. As an application, we obtain similar results as in the
Grabiner’s paper. In [4] the author proves the following:
###### Theorem 4.
Let $A$ be a Banach algebra. If $A$ is nil then $A$ is nilpotent.
In the above theorem, the algebra $A$ is required to be a Banach algebra. Here
we investigate when the nilpotency of an algebra $A$ can be obtained by
hypothesis imposed on $C(A)$. In this direction our first result is
###### Corollary 5.
Let $A$ be a normed PI-algebra. If $C(A)$ is nil, then $A$ is nilpotent.
Our second result relates the nilpotency of $A$ to the completion of certain
quotient space related to the polynomial identities of $A$. To be more precise
we prove the following:
###### Corollary 6.
Let $F\langle X\rangle$ be a MN-algebra and let $A$ be a PI-algebra. If
$C\left(\frac{F\langle X\rangle}{Id(A)}\right)$
is nil, then $A$ is nilpotent.
## 2 Banach and PI-Algebras
An algebra $A$ is said to be normed if it satisfies the followings properties:
* a)
$A$ has a norm $\|\cdot\|$;
* b)
$\|ab\|\leq\|a\|\|b\|$ for all $a,b\in A$.
A normed algebra $A$ is called Banach algebra if $A$ is a complete normed
space. It’s well known that every normed algebra $A$ is contained in some
Banach algebra $C(A)$ such that $A$ is dense in $C(A)$. This algebra is the
completion of $A$. Here we recall its construction: we first define the
relation $\sim$ in the set of all Cauchy sequences of $A$ by
$(a_{n})_{n}\sim(b_{n})_{n}\Longleftrightarrow\lim_{n\rightarrow\infty}\|a_{n}-b_{n}\|=0.$
Denote by $C(A)$ the set of all equivalence classes. If $(a_{n})_{n}$ is a
Cauchy sequence in $A$, we denote by $[(a_{n})_{n}]$ its equivalence class.
The algebraic operations in $C(A)$ are defined as usual:
$[(a_{n})_{n}]+[(b_{n})_{n}]=[(a_{n}+b_{n})_{n}]$, for any $\lambda\in F$, we
put $\lambda[(a_{n})_{n}]=[(\lambda a_{n})_{n}]$ and
$[(a_{n})_{n}][(b_{n})_{n}]=[(a_{n}b_{n})_{n}]$. Endowed with these operations
and with the norm
$\|[(a_{n})_{n}]\|=\lim_{n\rightarrow\infty}\|a_{n}\|$
we have that $C(A)$ is a Banach algebra. Note that an element $a\in A$ is
identified with the class $[(a)_{n}]\in C(A)$ of the constant sequence equal
to $a$. For more details see [6, 5].
Let $F\langle X\rangle$ be the free non-unitary associative algebra, freely
generated over $F$ by the infinite set $X=\\{x_{1},x_{2},\ldots\\}$. The
elements of $F\langle X\rangle$ are called polynomials and a polynomial of the
kind $x_{i_{1}}x_{i_{1}}\ldots x_{i_{n}}$ is called monomial.
A polynomial $f(x_{1},\dots,x_{m})\in F\langle X\rangle$ is called a
polynomial identity for an algebra $A$ if $f(a_{1},\dots,a_{m})=0$, for all
$a_{1},\ldots,a_{m}\in A$. We denote by $Id(A)$ the set of all polynomial
identities of $A$. If $Id(A)\neq\\{0\\}$ then we say that $A$ is a PI-algebra.
The set $Id(A)$ is an ideal in $F\langle X\rangle$ and has the property
$f(g_{1},\ldots,g_{m})\in Id(A)$ for all $f(x_{1},\ldots,x_{m})\in Id(A)$ and
$g_{1},\ldots,g_{m}\in F\langle X\rangle$. Thus we say that $Id(A)$ is a
T-ideal. For details, see [2, 3].
Let $F\langle X\rangle^{(d_{1},\ldots,d_{m})}$ be the vector subspace of
$F\langle X\rangle$ spanned by all monomials $u=x_{j_{1}}\ldots x_{j_{t}}$,
where the variable $x_{i}$ appears $d_{i}$ times in $u$ for all
$i=1,\ldots,m$. If $f(x_{1},\ldots,x_{m})\in F\langle
X\rangle^{(d_{1},\ldots,d_{m})}$ then we say that $f$ is multihomegenous of
multidegree $(d_{1},\ldots,d_{m})$. Note that if $f=f(x_{1},\dots,x_{m})\in
F\langle X\rangle$, we can always write
$f=\displaystyle\sum_{d_{1}\geq 0,\dots,d_{m}\geq 0}f^{(d_{1},\dots,d_{m})}$
where $f^{(d_{1},\dots,d_{m})}\in F\langle X\rangle^{(d_{1},\dots,d_{m})}$.
The polynomials $f^{(d_{1},\ldots,d_{m})}$ are called the multihomogenous
components of $f$.
Now we recall an important result that will be used in the next section.
###### Theorem 7.
If $f=f(x_{1},\dots,x_{m})$ is a polynomial identity for an algebra $A$, then
every multihomogeneous component of $f$ is a polynomial identity for $A$.
###### Proof.
See [3], Theorem 1.3.2. ∎
We call the reader’s attention to the fact that the above result holds for
every infinite field $F$, but it is not always true for finite fields.
## 3 Proofs of the Main Results
In this section we give the proofs of the main results of the paper.
###### Proof.
(Proposition 1) Since $A\subseteq C(A)$, it follows that $Id(C(A))\subseteq
Id(A)$. Let $f(x_{1},\ldots,x_{m})\in Id(A)$ and $a_{1},\ldots,a_{m}\in C(A)$.
By the construction of $C(A)$ we have
$f(a_{1},\ldots,a_{m})=f([(a_{1n})_{n}],\ldots,[(a_{mn})_{n}])=[(f(a_{1n},\ldots,a_{mn}))_{n}]=[(0)_{n}].$
The last identity implies that $f\in Id(C(A))$. Therefore $Id(A)\subseteq
Id(C(A))$. ∎
Let $g(x_{1},\ldots,x_{t})$ a polynomial and let $g^{(d_{1},\ldots,d_{t})}$
its multihomogeneous component of multidegree $(d_{1},\ldots,d_{t})$. If $m<t$
and $d_{m+1}=d_{m+2}=\ldots=d_{t}=0$, then we write
$g^{(d_{1},\ldots,d_{m},\ldots,d_{t})}=g^{(d_{1},\ldots,d_{m})}.$
If $t<m$, then we write
$g^{(d_{1},\ldots,d_{t})}=g^{(d_{1},\ldots,d_{t},0,\ldots,0)},$
where the number of zeros is $m-t$. With this convention we have the following
lemma:
###### Lemma 8.
Let $F\langle X\rangle$ be a MN-algebra and let $f=f(x_{1},\ldots,x_{m})$ be a
polynomial. If $(f_{n})_{n}$ is a sequence in $F\langle X\rangle$ such that
$f_{n}\to f$, then
$f_{n}^{(d_{1},\ldots,d_{m})}\to f^{(d_{1},\ldots,d_{m})}$
for all multidegree $d=(d_{1},\ldots,d_{m})$.
###### Proof.
Once $\|\cdot\|$ is a multihomogeneous norm we have the following inequality
$\|f_{n}^{(d_{1},\ldots,d_{m})}-f^{(d_{1},\ldots,d_{m})}\|\leq\|f_{n}-f\|.$
Since $f_{n}\to f$ it follows that $f_{n}^{(d_{1},\ldots,d_{m})}\to
f^{(d_{1},\ldots,d_{m})}$. ∎
###### Proposition 9.
Let $A$ be a PI-algebra. If $F\langle X\rangle$ is a MN-algebra, then $Id(A)$
is a closed ideal.
###### Proof.
Let $(f_{n})_{n}\in Id(A)$ be a sequence of polynomials such that $f_{n}\to
f$. We want to prove that $f\in Id(A)$. Write
$f=f(x_{1},\ldots,x_{m})=\sum_{(d_{1},\dots,d_{m})}f^{(d_{1},\dots,d_{m})}.$
By the Lemma 8, we have that $f_{n}^{(d_{1},\dots,d_{m})}\to
f^{(d_{1},\dots,d_{m})}$ for all multidegree $(d_{1},\dots,d_{m})$. Note that
$f_{n}^{(d_{1},\dots,d_{m})}\in F\langle X\rangle^{(d_{1},\dots,d_{m})}\cap
Id(A)$ by the Theorem 7.
Since $F\langle X\rangle^{(d_{1},\dots,d_{m})}$ is a finite-dimensional vector
space follows that
$F\langle X\rangle^{(d_{1},\dots,d_{m})}\cap Id(A)$
has also finite dimension. Since every finite-dimensional space is closed in
the norm topology, we have that $F\langle X\rangle^{(d_{1},\dots,d_{m})}\cap
Id(A)$ is closed. Thus
$f^{(d_{1},\dots,d_{m})}\in F\langle X\rangle^{(d_{1},\dots,d_{m})}\cap Id(A)$
and therefore $f\in Id(A)$. ∎
If $A$ is a PI-algebra and $F\langle X\rangle$ is MN-algebra, then by the
above proposition we can define a norm in the quotient algebra $F\langle
X\rangle/Id(A)$:
$\|f+Id(A)\|=\mbox{inf}\\{\|f+g\|\,:\,g\in Id(A)\\},$
where $f\in F\langle X\rangle$. With this norm we have that $F\langle
X\rangle/Id(A)$ is a normed algebra. So we can see that the Theorem 3
describes the polynomial identities of the completion of this quotient
algebra. Now we proceed to its proof.
###### Proof.
(Theorem 3) By a classical result in PI-Algebra we have
$Id(A)=Id\left(\frac{F\langle X\rangle}{Id(A)}\right),$
see [3]. So the proof of the theorem follows immediately from Proposition 1. ∎
###### Proof.
(Corollary 5) If $C(A)$ is nil, then by Theorem 4, we have that $C(A)$ is
nilpotent. Thus $x_{1}x_{2}\ldots x_{n}$ is a polynomial identity of $C(A)$
for some $n$. Since by Proposition 1 we have $Id(A)=Id(C(A))$, follows that
$A$ is nilpotent. ∎
###### Proof.
(Corollary 6) Let $B=F\langle X\rangle/Id(A)$. If $C(B)$ is nil, then by
Theorem 4 we have that $C(B)$ is nilpotent. Thus $x_{1}x_{2}\ldots x_{n}$ is
polynomial identity of $C(B)$ for some $n$. Since by Theorem 3 we have
$Id(A)=Id(C(B))$, follows that $A$ is nilpotent. ∎
## References
* [1] H. G. Dales, Norming Nil Algebras, Proc. Amer. Math. Soc., 83, Number 1, 71–74, 1981.
* [2] V. Drensky, Free algebras and PI-algebras, Graduate Course in Algebra, Springer, Singapore, 1999.
* [3] A. Giambruno, M. Zaicev, Polynomial identities and asymptotic methods, Math. Surveys Monographs 122, AMS, Providence, RI, 2005\.
* [4] S. Grabine, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc., 21, 510, 1969.
* [5] I. Kaplansky, Set Theory and Metric Spaces, Allyn and Bacon Series in Advanced Mathematics, Boston, Mass., 1972.
* [6] M. A. Naimark, Normed algebra, Wolters-Noordhoff, 3 edition, 1972.
|
arxiv-papers
| 2013-04-09T02:41:43 |
2024-09-04T02:49:44.005883
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leandro Cioletti, Jos\\'e Ant\\^onio Freitas and Dimas Jos\\'e\n Gon\\c{c}alves",
"submitter": "Leandro Cioletti",
"url": "https://arxiv.org/abs/1304.2451"
}
|
1304.2472
|
# Dualities for absolute zeta functions and multiple gamma functions
Nobushige Kurokawa Nobushige Kurokawara
Tokyo Institute of Technology [email protected] and Hiroyuki Ochiai
Hiroyuki Ochiai
Kyushu University [email protected]
###### Abstract.
We study absolute zeta functions from the view point of a canonical
normalization. We introduce the absolute Hurwitz zeta function for the
normalization. In particular, we show that the theory of multiple gamma and
sine functions gives good normalizations in cases related to the Kurokawa
tensor product. In these cases, the functional equation of the absolute zeta
function turns out to be equivalent to the simplicity of the associated non-
classical multiple sine function of negative degree.
###### Key words and phrases:
absolute zeta function, multiple gamma function, multiple sine function
###### 2000 Mathematics Subject Classification:
Primary 11M06
## 1\. Introduction
The absolute zeta function of a scheme $X$ over ${\mathbb{F}_{1}}$ was first
studied by Soulé [S] as a “limit of $p\to 1$” of the (congruence) zeta
function over ${\mathbb{F}_{p}}$: see Kurokawa [K2] and Deitmar [D] also.
Then, Connes and Consani [CC1] [CC2] investigated the absolute zeta function
as the following integral
$\zeta_{X}(s)=\exp\left(\mbox{$\displaystyle\int_{1}^{\infty}\frac{N_{X}(u)}{u^{s+1}\log
u}du$}\right),$
where
$N_{X}(u)=\left|X(\mathbb{F}_{1^{u-1}})\right|$
is a suitably interpolated “counting function.” Here we must pay attention to
the needed normalization for the integral near $u=1$: see [CC1] [CC2] for a
discussion. In [CC1, Theorem 4.13] [CC2, Theorem 4.3] Connes and Consani
calculated $\zeta_{X}(s)$ for Noetherian schemes via the Kurokawa tensor
product of [K1].
Our purpose is to introduce the absolute Hurwitz zeta function
$Z_{X}(w;s)=\frac{1}{\Gamma(w)}\int_{1}^{\infty}\frac{N_{X}(u)}{u^{s+1}(\log
u)^{1-w}}du$
to get the canonical normalization:
$\zeta_{X}(s)=\exp\left(\left.\frac{\partial}{\partial
w}Z_{X}(w;s)\right|_{w=0}\right).$
This normalization is essentially due to Riemann (1859) and it is used in the
theory of multiple gamma and sine function as follows.
For each integer $r\geq 1$, the $r$-ple Hurwitz zeta function $\zeta_{r}(w;x)$
is defined in $\operatorname{Re}(w)>r$ as
$\zeta_{r}(w;x)=\sum_{n=0}^{\infty}{}_{r}H_{n}(n+x)^{-w}$ (1)
where ${}_{r}H_{n}={{n+r-1}\choose{n}}$.
The analytic continuation of $\zeta_{r}(w;x)$ to all $w\in\mathbb{C}$ is
obtained via the integral representation of Riemann
$\displaystyle\zeta_{r}(w;x)$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{0}^{\infty}(1-e^{-t})^{-r}e^{-xt}t^{w-1}dt$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(1-u^{-1})^{-r}u^{x-1}(\log
u)^{w-1}du$
by treating the integral around $u=1$ in the usual way.
Thus, by using such analytic continuation we get the $r$-ple gamma function
$\Gamma_{r}(x)=\exp\left(\left.\frac{\partial}{\partial
w}\zeta_{r}(w;x)\right|_{w=0}\right)$
and the $r$-ple sine function
$S_{r}(x)=\Gamma_{r}(x)^{-1}\Gamma_{r}(r-x)^{(-1)^{r}}.$
We refer to Barnes [B] (1904) and Kurokawa-Koyama [KK] (2003) for details,
where more general multiple gamma functions and multiple sine functions were
treated respectively.
We report three results in this introduction. First, for a function
$N:(1,\infty)\rightarrow\mathbb{C}$ we use
$Z_{N}(w;s)=\frac{1}{\Gamma(w)}\int_{1}^{\infty}N(u)u^{-s-1}(\log u)^{w-1}du$
and
$\zeta_{N}(s)=\exp\left(\left.\frac{\partial}{\partial
w}Z_{N}(w;s)\right|_{w=0}\right)$
also.
###### Theorem A.
Let $N(u)=\displaystyle\sum_{\alpha}m(\alpha)u^{\alpha}$ be a finite sum.
Then:
* (1)
$Z_{N}(w;s)=\displaystyle\sum_{\alpha}m(\alpha)(s-\alpha)^{-w}$.
* (2)
$\zeta_{N}(s)=\displaystyle\prod_{\alpha}(s-\alpha)^{-m(\alpha)}$.
This result is applicable to calculate many examples (see [K2]) of absolute
zeta functions under our canonical normalization. We note two simple examples.
###### Example 1.
Let $X=\operatorname{Spec}{\mathbb{F}_{1}}$. Then
$\displaystyle N_{X}(u)=1,$ $\displaystyle Z_{X}(w;s)=s^{-w},$
$\displaystyle\zeta_{X}(s)=1/s.$
###### Example 2.
Let $X=\mathbb{SL}(2)$. Then
$\displaystyle N_{X}(u)=u^{3}-u,$ $\displaystyle
Z_{X}(w;s)=(s-3)^{-w}(s-1)^{-w},$ $\displaystyle\zeta_{X}(s)=(s-1)/(s-3).$
Now the following result shows a functoriality.
###### Theorem B.
* (1)
For $N_{1},N_{2}:(1,\infty)\rightarrow\mathbb{C}$ let
$(N_{1}\oplus N_{2})(u)=N_{1}(u)+N_{2}(u).$
Then
$Z_{N_{1}\oplus N_{2}}(w;s)=Z_{N_{1}}(w;s)+Z_{N_{2}}(w;s)$
and
$\zeta_{N_{1}\oplus N_{2}}(s)=\zeta_{N_{1}}(s)\zeta_{N_{2}}(s).$
* (2)
Let
$N_{i}(u)=\sum_{\alpha_{i}}m_{i}(\alpha_{i})u^{\alpha_{i}}$
for $i=1,2$. Suppose that both are finite sums. Put
$(N_{1}\otimes N_{2})(u)=N_{1}(u)N_{2}(u).$
Then
$\displaystyle Z_{N_{1}\otimes N_{2}}(w;s)$
$\displaystyle=\sum_{\alpha_{1},,\alpha_{2}}m_{1}(\alpha_{1})m_{2}(\alpha_{2})(s-(\alpha_{1}+\alpha_{2}))^{-w}$
and
$\zeta_{N_{1}\otimes
N_{2}}(s)=\prod_{\alpha_{1},\alpha_{2}}(s-(\alpha_{1}+\alpha_{2}))^{-m_{1}(\alpha_{2})m_{2}(\alpha_{2})}.$
This tensor product is essentially the Kurokawa tensor product originated in
[K1] (see [M], [CC1] and [CC2]) when $\alpha_{j}$’s are real. We remark that
for general $N_{j}$’s (“infinite sums” or “generalized functions”) we must
resolve various difficulties.
For the next result we notice that our construction of $\zeta_{r}(w;x)$,
$\Gamma_{r}(x)$ and $S_{r}(x)$ is valid for negative $r$ also (see the later
explanation).
###### Theorem C.
Let $r$ be a positive integer. Then
* (1)
$\displaystyle Z_{\mathbb{G}_{\rm m}^{\otimes r}}(w;s)=\zeta_{-r}(w;s-r)$.
* (2)
$\displaystyle\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\Gamma_{-r}(s-r)$
$\displaystyle=\prod_{j=1}^{r}(s-j)^{(-1)^{r-j-1}{r\choose j}}$
$\displaystyle=\left((1-1/s)^{\otimes r}\right)^{-1}$,
where $\otimes r$ is the Kurokawa tensor product.
* (3)
We have the functional equation
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\zeta_{\mathbb{G}_{\rm m}^{\otimes
r}}(r-s)^{(-1)^{r}},$
which is equivalent to $S_{-r}(x)=1$.
Our result would suggest that
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\Gamma_{-r}(s-r)$
holds for $r<0$ also with the functional equation $s\leftrightarrow-r-s$. For
example
$\zeta_{\mathbb{G}_{\rm
m}^{\otimes-1}}(s)=\Gamma_{1}(s+1)=\frac{\Gamma(s+1)}{\sqrt{2\pi}}$
and the functional equation $s\leftrightarrow 1-s$ is the reflection formula
of Euler:
$\Gamma_{1}(s+1)\Gamma(2-s)=S_{1}(s+1)^{-1}=-\frac{1}{2\sin(\pi s)}.$
We remark that Manin [M, §1.7] indicated an idea to consider the gamma
function as the zeta function of the “dual infinite dimensional projective
space over ${\mathbb{F}_{1}}$.”
## 2\. Multiple gamma functions and multiple sine functions
We recall the construction of the multiple Hurwitz zeta function:
$\displaystyle\zeta_{r}(w;x)$ $\displaystyle=\sum_{n=0}^{\infty}{n+r-1\choose
n}(n+x)^{-w}$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{0}^{\infty}(1-e^{-t})^{-r}e^{-xt}t^{w-1}dt$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(1-u^{-1})^{-r}u^{-x-1}(\log
u)^{w-1}du.$
This definition is valid for any $r\in\mathbb{R}$ with sufficiently large
$\operatorname{Re}(x)$ and $\operatorname{Re}(w)$, so we have the analytic
continuation to all $w\in\mathbb{C}$ via the usual method. Thus, we get
$\Gamma_{r}(x)=\exp\left(\left.\frac{\partial}{\partial
w}\zeta_{r}(w;x)\right|_{w=0}\right)$
and
$S_{r}(x)=\Gamma_{r}(x)^{-1}\Gamma_{r}(r-x)^{(-1)^{r}}$
for any $r\in\mathbb{R}$ (or $r\in\mathbb{Z}$ at least without ambiguity of
the meaning of $(-1)^{r}$). For readers interested in the theory of $r<0$, we
refer to [KO].
###### Theorem 1.
Let $r$ be a negative integer. Then
* (1)
$\displaystyle\Gamma_{r}(x)=\prod_{n=0}^{-r}(x+n)^{(-1)^{n+1}{-r\choose n}}$.
* (2)
$S_{r}(x)=1$.
###### Proof.
We have
$\displaystyle\zeta_{r}(w;x)$ $\displaystyle=\sum_{n=0}^{\infty}{n+r-1\choose
n}(n+x)^{-w}$ $\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}{-r\choose
n}(n+x)^{-w}.$
Hence
$\displaystyle\Gamma_{r}(x)$
$\displaystyle=\exp\left(\sum_{n=0}^{-r}(-1)^{n+1}{-r\choose
n}\log(n+x)\right)$ $\displaystyle=\prod_{n=0}^{-r}(n+x)^{(-1)^{n+1}{-r\choose
n}}.$
Next,
$\displaystyle S_{r}(x)=\Gamma_{r}(x)^{-1}\Gamma_{r}(r-x)^{(-1)^{r}}$
$\displaystyle=\prod_{n=0}^{-r}(n+x)^{(-1)^{n}{-r\choose
n}}\times\prod_{n=0}^{-r}(n+r-x)^{(-1)^{n-r+1}{-r\choose n}}$
$\displaystyle=\prod_{n=0}^{-r}(n+x)^{(-1)^{n}{-r\choose n}}$
$\displaystyle\qquad\times\prod_{n=0}^{-r}((-r-n)+x)^{(-1)^{(-r-n)+1}{-r\choose
n}},$
where we used
$\sum_{n=0}^{-r}(-1)^{n}{-r\choose n}=0.$
Hence
$\displaystyle S_{r}(x)$
$\displaystyle=\prod_{n=0}^{-r}(n+x)^{(-1)^{n}{-r\choose
n}}\times\prod_{n=0}^{-r}(n+x)^{(-1)^{n+1}{-r\choose n}}$ $\displaystyle=1.$
∎
This result can be generalized to the multi-period case
$\underline{\omega}=(\omega_{1},\ldots,\omega_{r})$ with
$\omega_{1},\ldots,\omega_{r}>0$ as follows, where the above case is contained
as $\underline{\omega}=(1,\ldots,1)$. Put
$\displaystyle\zeta_{-r}(w;x,\underline{\omega})$
$\displaystyle\qquad=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
r}(-1)^{k}(x+\omega_{i_{1}}+\cdots+\omega_{i_{k}})^{-w},$
$\displaystyle\Gamma_{-r}(w,\underline{\omega})=\exp\left(\left.\frac{\partial}{\partial
w}\zeta_{-r}(w;x,\underline{\omega})\right|_{w=0}\right),$ and $\displaystyle
S_{-r}(x,\underline{\omega})=\Gamma_{-r}(x,\underline{\omega})^{-1}$
$\displaystyle\quad\qquad\times\Gamma_{-r}(-(\omega_{1}+\cdots+\omega_{r})-x,\underline{\omega})^{(-1)^{r}}.$
Then we have (see [KO] for more generalizations also)
$\displaystyle\zeta_{-r}(w;x,\underline{\omega})$
$\displaystyle\quad=\frac{1}{\Gamma(w)}\int_{0}^{w}(1-e^{-t\omega_{1}})\cdots(1-e^{-t\omega_{r}})e^{-xt}t^{w-1}dt,$
$\displaystyle\Gamma_{-r}(x,\underline{\omega})=\prod_{1\leq
i_{1}<\cdots<i_{k}\leq r}(x+\omega_{i_{1}}+\cdots+\omega_{i_{k}})^{(-1)^{k}},$
and $\displaystyle S_{-r}(x,\underline{\omega})=1.$
For example, we get
$\zeta_{\mathbb{SL}(2)}(s)=\Gamma_{-1}(s-3,2)=\frac{s-1}{s-3}.$
More generally:
$\displaystyle\zeta_{\mathbb{SL}(r)}(s)=\Gamma_{-(r-1)}(s-(r^{2}-1),(2,3,\cdots,r))$
and
$\displaystyle\zeta_{\mathbb{GL}(r)}(s)=\Gamma_{-r}(s-r^{2},(1,2,3,\cdots,r)),$
where $\left\\{\begin{array}[]{l}r-1=\operatorname{rank}\mathbb{SL}(r)\\\
r^{2}-1=\dim\mathbb{SL}(r)\end{array}\right.$ and
$\left\\{\begin{array}[]{l}r=\operatorname{rank}\mathbb{GL}(r)\\\
r^{2}=\dim\mathbb{GL}(r).\end{array}\right.$ We obtain the functional
equations
$\displaystyle\zeta_{\mathbb{SL}(r)}(s)=\zeta_{\mathbb{SL}(r)}(r(3r-1)/2-1-s)^{(-1)^{r-1}},$
and
$\displaystyle\zeta_{\mathbb{GL}(r)}(s)=\zeta_{\mathbb{GL}(r)}(r(3r-1)/2-s)^{(-1)^{r}}$
from the triviality of the multiple sine function of negative order exactly
similar to Theorem C.
###### Theorem 2.
Let $r$ be a negative real number. Then:
* (1)
$\zeta_{r}(m;x)=0$ for each integer $m$ satisfying $r<m\leq 0$.
* (2)
$\displaystyle\Gamma_{r}(x)=\exp\left(\int_{1}^{\infty}(1-u^{-1})^{-r}u^{-x-1}(\log
u)^{-1}du\right)$ for $\operatorname{Re}(x)>0$.
###### Example 3.
$\zeta_{-3}(w;x)=x^{-w}-3(x+1)^{-w}+3(x+2)^{-w}-(x+3)^{-w}$
and
$\zeta_{-3}(0;x)=\zeta_{-3}(-1;x)=\zeta_{-3}(-2;x)=0.$
Notice that $\zeta_{-3}(-3;x)=-6$. (In general $\zeta_{-m}(-m;x)=(-1)^{m}m!$
for integers $m\geq 0$.
###### Example 4.
$\zeta_{-\frac{1}{2}}(w;x)=x^{-w}-\sum_{n=1}^{\infty}\frac{{2n\choose
n}}{(2n-1)4^{n}}(n+x)^{-w}$
and
$\zeta_{-\frac{1}{2}}(0;x)=1-\sum_{n=1}^{\infty}\frac{{2n\choose
n}}{(2n-1)4^{n}}=0,$
that is
$\sum_{n=1}^{\infty}\frac{{2n\choose n}}{(2n-1)4^{n}}=1.$
###### Proof.
The fact (1) follows from the integral representation
$\zeta_{r}(w;x)=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(1-u^{-1})^{-r}u^{-x-1}(\log
u)^{w-1}du,$
since this integral converges for $\operatorname{Re}(w)>-r$ when
$\operatorname{Re}(x)>0$, and $1/\Gamma(w)$ has zeros at $w=0,-1,\ldots,r+1$.
Similarly, (2) is seen by looking at $w=0$. ∎
## 3\. Proof of Theorem A
For a function $N:(1,\infty)\rightarrow\mathbb{C}$ we defined
$Z_{N}(w;s)=\frac{1}{\Gamma(w)}\int_{1}^{\infty}N(u)u^{-s-1}(\log u)^{w-1}du$
and
$\zeta_{N}(s)=\exp\left(\left.\frac{\partial}{\partial
w}Z_{N}(w;s)\right|_{w=0}\right).$
We calculate these functions in the case of a finite sum
$N(u)=\sum_{\alpha}m(\alpha)u^{\alpha}.$
It is sufficient to calculate the following monomial case.
###### Lemma.
Let $N(u)=u^{\alpha}$, then
$Z_{N}(w;s)=(s-\alpha)^{-w}$
and
$\zeta_{N}(s)=\frac{1}{s-\alpha}.$
###### Proof.
$\displaystyle Z_{N}(w;s)$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}u^{\alpha-s-1}(\log
u)^{w-1}du$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{0}^{\infty}e^{-(s-\alpha)t}t^{w-1}dt$
$\displaystyle=(s-\alpha)^{-w}.$
Hence
$\left.\frac{\partial}{\partial w}Z_{N}(w,s)\right|_{w=0}=-\log(s-\alpha)$
and
$\zeta_{N}(s)=\frac{1}{s-\alpha}.$
∎
## 4\. Proof of Theorem B
(1) Since
$\displaystyle Z_{N_{1}\oplus N_{2}}(w;s)$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(N_{1}\oplus
N_{2})(u)u^{-s-1}(\log u)^{w-1}du$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(N_{1}(u)+N_{2}(u))u^{-s-1}(\log
u)^{w-1}du$ $\displaystyle=Z_{N_{1}}(w;s)+Z_{N_{2}}(w;s),$
we have
$\zeta_{N_{1}\oplus N_{2}}(s)=\zeta_{N_{1}}(s)\zeta_{N_{2}}(s).$
(2) From
$\displaystyle(N_{1}\oplus N_{2})(u)$ $\displaystyle=N_{1}(u)N_{2}(u)$
$\displaystyle=(\sum_{\alpha_{1}}m_{1}(\alpha_{1})u^{\alpha_{1}})(\sum_{\alpha_{2}}m_{2}(\alpha_{2})u^{\alpha_{2}})$
$\displaystyle=\sum_{\alpha_{1},\alpha_{2}}m_{1}(\alpha_{1})m_{2}(\alpha_{2})u^{\alpha_{1}+\alpha_{2}},$
we have
$\displaystyle Z_{N_{1}\otimes N_{2}}(w;s)$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(N_{1}\otimes
N_{2})(u)u^{-s-1}(\log u)^{w-1}du$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}\Big{(}\sum_{\alpha_{1},\alpha_{2}}m_{1}(\alpha_{1})m_{2}(\alpha_{2})u^{\alpha_{1}+\alpha_{2}}\Big{)}$
$\displaystyle\hskip 113.81102pt\times u^{-s-1}(\log u)^{w-1}du$
$\displaystyle=\sum_{\alpha_{1},\alpha_{2}}m_{1}(\alpha_{1})m_{2}(\alpha_{2})(s-(\alpha_{1}+\alpha_{2}))^{-w}.$
Hence
$\zeta_{N_{1}\otimes
N_{2}}(s)=\prod_{\alpha_{1},\alpha_{2}}(s-(\alpha_{1}+\alpha_{2}))^{-m_{1}(\alpha_{1})m_{2}(\alpha_{2})}.\qed$
## 5\. Absolute zeta functions
###### Theorem 3.
Let $r$ be a positive integer. Then
* (1)
$\displaystyle\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\Gamma_{-r}(s-r)$.
* (2)
$\displaystyle\zeta_{\mathbb{G}_{\rm m}^{\otimes
r}}(s)=\exp\Big{(}\int_{1}^{\infty}N_{\mathbb{G}_{\rm m}^{\otimes
r}}(u)u^{-s-1}(\log u)^{-1}du\Big{)}.$
###### Proof.
(1) Since
$N_{\mathbb{G}_{\rm m}^{\otimes r}}(u)=(u-1)^{r},$
we have
$\displaystyle Z_{\mathbb{G}_{\rm m}^{\otimes r}}(w;s)$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(u-1)^{r}u^{-s-1}(\log
u)^{w-1}du$
$\displaystyle=\frac{1}{\Gamma(w)}\int_{1}^{\infty}(1-u^{-1})^{r}u^{-s+r-1}(\log
u)^{w-1}du$ $\displaystyle=\zeta_{-r}(w;s-r).$
Thus,
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\Gamma_{-r}(s-r).$
(2) This follows from (1) and Theorem 2(2). ∎
We notice that Theorem 1 and Theorem 3(1) imply Theorem C(1)(2).
## 6\. Functional equations
###### Theorem 4.
Let $r$ be a positive integer. Then
$\frac{\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)}{\zeta_{\mathbb{G}_{\rm
m}^{\otimes r}}(r-s)^{(-1)^{r}}}=S_{-r}(s-r)^{-1}.$
###### Proof.
From Theorem 3(1), we have
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\Gamma_{-r}(s-r)$
and
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(r-s)=\Gamma_{-r}(-s).$
Hence,
$\displaystyle\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)\zeta_{\mathbb{G}_{\rm
m}^{\otimes r}}(r-s)^{(-1)^{r+1}}$
$\displaystyle=\Gamma_{-r}(s-r)\Gamma_{-r}(-s)^{(-1)^{r+1}}$
$\displaystyle=S_{-r}(s-r)^{-1}.$
∎
We remark that we have the functional equation
$\zeta_{\mathbb{G}_{\rm m}^{\otimes r}}(s)=\zeta_{\mathbb{G}_{\rm m}^{\otimes
r}}(r-s)^{(-1)^{r}}$
from Theorem 1(2) and we know that it is equivalent to $S_{-r}(x)=1$. Thus we
have Theorem C(3).
## References
* [B] E.W. Barnes, On the theory of the multiple gamma functions. Trans. Cambridge Philos. Soc. 19 (1904) 374–425.
* [CC1] A. Connes and C. Consani, Schemes over ${\mathbb{F}}_{1}$ and zeta functions. Compositio Mathematica 146 (2010) 1383–1415.
* [CC2] A. Connes and C. Consani, Characteristic one, entropy and the absolute point. In ”Noncommutative Geometry, Arithmetic, and Related Topics, Proceedings of the JAMI Conference 2009”, Johns Hopkins University Press (2011) 75–140.
* [D] A. Deitmar, Remarks on zeta functions and $K$-theory over ${\mathbf{F}}_{1}$, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006) 141–146.
* [K1] N. Kurokawa, Multiple zeta functions: an example. In “Zeta Functions in Geometry” (Tokyo 1990), Adv. Stud. Pure Math. 21, Kinokuniya, Tokyo, 1992, 219–226.
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|
arxiv-papers
| 2013-04-09T07:07:28 |
2024-09-04T02:49:44.011819
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nobushige Kurokawa and Hiroyuki Ochiai",
"submitter": "Hiroyuki Ochiai",
"url": "https://arxiv.org/abs/1304.2472"
}
|
1304.2591
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-058 LHCb-PAPER-2013-009 May $27$, $2013$
Limits on neutral Higgs boson production in the forward region in $pp$
collisions at $\sqrt{s}={7~{}\mathrm{TeV}}$
The LHCb collaboration†††Authors are listed on the following pages.
Limits on the cross-section times branching fraction for neutral Higgs bosons,
produced in $pp$ collisions at ${\sqrt{s}=7~{}\mathrm{Te\kern-1.00006ptV}}$,
and decaying to two tau leptons with pseudorapidities between $2.0$ and $4.5$,
are presented. The result is based on a dataset, corresponding to an
integrated luminosity of $1.0~{}\mathrm{fb}^{-1}$, collected with the LHCb
detector. Candidates are identified by reconstructing final states with two
muons, a muon and an electron, a muon and a hadron, or an electron and a
hadron. A model independent upper limit at the ${95\%}$ confidence level is
set on a neutral Higgs boson cross-section times branching fraction. It varies
from ${8.6~{}\mathrm{pb}}$ for a Higgs boson mass of
${90~{}\mathrm{Ge\kern-1.00006ptV}}$ to ${0.7~{}\mathrm{pb}}$ for a Higgs
boson mass of ${250~{}\mathrm{Ge\kern-1.00006ptV}}$, and is compared to the
Standard Model expectation. An upper limit on ${\tan\beta}$ in the Minimal
Supersymmetric Model is set in the ${m_{h^{0}}^{\mathrm{max}}}$ scenario. It
ranges from $34$ for a $C\\!P$-odd Higgs boson mass of
${90~{}\mathrm{Ge\kern-1.00006ptV}}$ to $70$ for a pseudo-scalar Higgs boson
mass of ${140~{}\mathrm{Ge\kern-1.00006ptV}}$.
Published in JHEP Vol. 2013, Number 5 (2013), 132, DOI 10.1007/JHEP05(2013)132
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen59, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso56, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso57, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani12, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk56, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia56, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton56, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D. Craik47, S. Cunliffe52, R. Currie49,
C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis59, I. De Bonis4, K. De
Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De Paula2, W. De
Silva59, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, D.
Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M.
Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A.
Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C.
Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F.
Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini56, Y. Gao3, J. Garofoli56, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel57, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O. Grünberg58, B.
Gui56, E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou56, G.
Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T. Hampson45, S. Hansmann-
Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann58, J. He37,
V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4,
W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D.
Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11,
E. Jans40, P. Jaton38, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, B.
Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R.
Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I.
Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L.
Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9,
M. Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, R. McNulty12, A. Mcnab53, B. Meadows59,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez57, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain56, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal56, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan56, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A.
Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W.
Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42,
N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki56, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff59, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone56,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß58, H. Voss10, R. Waldi58, R. Wallace12,
S. Wandernoth11, J. Wang56, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing56, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang56, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH University of Science and Technology, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56Syracuse University, Syracuse, NY, United States
57Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
58Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
59University of Cincinnati, Cincinnati, OH, United States, associated to 56
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The discovery of a boson with a mass of about
${125~{}\mathrm{Ge\kern-1.00006ptV}}$ by the ATLAS [1] and CMS [2]
collaborations requires further investigations to confirm whether its
properties are compatible with a Standard Model (SM) Higgs boson or if it is
better described by theories beyond the SM, such as supersymmetry. The ATLAS
and CMS measurements have been made at central values of pseudorapidity,
$\eta$; investigations in the forward region can be provided by the LHCb
experiment, which is fully instrumented between ${2<\eta<5}$. Both
measurements of cross-sections and branching fractions allow different models
to be tested. In this paper, model-independent limits on the Higgs boson‡‡‡The
symbol ${\Phi^{0}}$ is used throughout to indicate any neutral Higgs boson.
Additionally, charge conjugation is implied and the speed of light is taken as
$1$. cross-section times branching fraction into two tau leptons are presented
for the forward region and compared to SM Higgs boson predictions. Model-
dependent limits for the Minimal Supersymmetric Model (MSSM) Higgs bosons, in
the scenario where the lightest supersymmetric Higgs boson mass is maximal
(${m_{h^{0}}^{\mathrm{max}}}$) [3], are also given for the ratio between up-
and down-type Higgs vacuum expectation values (${\tan\beta}$) as a function of
the $C\\!P$-odd Higgs boson (${A^{0}}$) mass.
## 2 Detector and datasets
The LHCb detector [4] is a single-arm forward spectrometer. The components of
particular relevance for this analysis are a high-precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$ interaction
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about ${4~{}\mathrm{Tm}}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. Photon, electron and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and pre-shower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The trigger [5] consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage, which
applies a full event reconstruction.
Simulated data samples are used to calculate signal and background
contributions, determine efficiencies, and estimate systematic uncertainties.
Each sample was generated as described in Ref. [6], with Pythia $6.4$ [7]
using the CTEQ$6$L$1$ leading-order PDF set [8] and passed through a Geant4
[9, *GeantB] based simulation of the detector [11]. The LHCb reconstruction
software [12] was used to perform trigger emulation and full event
reconstruction.
The dataset used for this analysis is identical to that described in our
previous measurement of the $Z$ cross-section using tau final states [13],
which corresponded to an integrated luminosity of ${1028\pm
36~{}\mathrm{pb}^{-1}}$, taken at a centre-of-mass energy of
${7~{}\mathrm{Te\kern-1.00006ptV}}$. The ${Z\to{\tau\tau}}$ decays are
identified in five categories: ${\tau_{\mu}\tau_{\mu}}$,
${\tau_{\mu}\tau_{e}}$, ${\tau_{e}\tau_{\mu}}$, ${\tau_{\mu}\tau_{h}}$ and
${\tau_{e}\tau_{h}}$, defined so as to be exclusive, where the subscripts
indicate tau decays containing a muon ($\mu$), electron ($e$), or hadron ($h$)
and the ordering specifies the first and second tau decay product on which
different requirements are applied. The first tau decay product is required to
have transverse momentum, $p_{\mathrm{T}}$, above
${20~{}\mathrm{Ge\kern-1.00006ptV}}$ and the second to have
${p_{\mathrm{T}}>5~{}\mathrm{Ge\kern-1.00006ptV}}$. Both tracks are required
to have pseudorapidities between $2.0$ and $4.5$, to be isolated with little
surrounding activity, to be approximately back-to-back in the azimuthal
coordinate, and their combined invariant mass must be greater than
${20~{}\mathrm{Ge\kern-1.00006ptV}}$. The tracks in the
${\tau_{\mu}\tau_{\mu}}$, ${\tau_{\mu}\tau_{h}}$, and ${\tau_{e}\tau_{h}}$
categories are required to be displaced from the primary vertex. Additionally,
the ${\tau_{\mu}\tau_{\mu}}$ category requires a difference between the
$p_{\mathrm{T}}$ of the two tracks and excludes di-muon invariant masses
between $80$ and ${100~{}\mathrm{Ge\kern-1.00006ptV}}$, to suppress the direct
decays of $Z$ bosons into two muons. Full details on the selection criteria
can be found in Ref. [13].
The invariant mass distribution of the two final state particles for the
selected ${{\Phi^{0}}\to{\tau\tau}}$ candidates is plotted in Fig. 1 for each
of the five categories separately and combined together. No candidates are
observed with a mass above ${120~{}\mathrm{Ge\kern-1.00006ptV}}$. The
distributions of Fig. 1 differ from those of Ref. [13] as the simulated mass
shapes are calibrated to correct for differences between data and simulation,
and the $Z\to{\tau\tau}$ distributions are normalised to theory.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1: Invariant mass distributions for LABEL:sub@fig:mumu
${\tau_{\mu}\tau_{\mu}}$, LABEL:sub@fig:mue ${\tau_{\mu}\tau_{e}}$,
LABEL:sub@fig:emu ${\tau_{e}\tau_{\mu}}$, LABEL:sub@fig:muh
${\tau_{\mu}\tau_{h}}$, LABEL:sub@fig:eh ${\tau_{e}\tau_{h}}$, and
LABEL:sub@fig:all all candidates. The ${Z\to{\tau\tau}}$ background (solid
red) is normalised to the theoretical expectation. The $\mathrm{QCD}$
(horizontal green), electroweak (vertical blue), and $Z$ (solid cyan)
backgrounds are estimated from data. The ${t\bar{t}}$ (vertical orange) and
${WW}$ (horizontal magenta) backgrounds are estimated from simulation and
generally not visible. The contribution that would be expected from an MSSM
signal for ${M_{A^{0}}=125~{}\mathrm{Ge\kern-0.90005ptV}}$ and
${\tan\beta=60}$ is shown in solid green.
Six background components are considered: ${Z\to{\tau\tau}}$; hadronic
processes (QCD); electroweak (EWK), where one $\tau$ decay product candidate
originates from a $W$ or $Z$ boson and the other comes from the underlying
event; ${t\bar{t}}$; ${WW}$; and ${Z\to{\ell\ell}}$ where ${\ell\ell}$
indicates electrons or muons originating from a leptonic $Z$ decay.
All backgrounds, except ${Z\to{\tau\tau}}$, have been estimated in Ref. [13].
The distribution and normalisation of QCD background events is found from data
using same-sign events. The electroweak invariant mass distribution is taken
from simulation and normalised using data. The small contributions from
${t\bar{t}}$ and ${WW}$ production are taken from simulation, while the
${Z\to{\ell\ell}}$ invariant mass shape and normalisation are determined from
data.
The invariant mass distributions for ${{\Phi^{0}}\to{\tau\tau}}$ and
${Z\to{\tau\tau}}$ decays are evaluated from simulation where the mass
resolution has been calibrated using the ${Z\to{\mu\mu}}$ invariant mass peak.
Each event is re-weighted by a factor
${(\sigma\times{\varepsilon})/(\sigma_{\mathrm{sim}}\times{\varepsilon_{\mathrm{sim}}})}$,
which provides a negligible correction in comparison to the mass resolution
calibration. The efficiency, ${\varepsilon}$, for triggering, reconstructing
and selecting candidates has been evaluated as a function of momentum and
pseudorapidity using data-driven techniques and is described in Ref. [13],
while ${\varepsilon_{\mathrm{sim}}}$ is the corresponding efficiency in
simulation. The cross-section for the process in simulation is represented by
$\sigma_{\mathrm{sim}}$, while $\sigma$ is the theoretical cross-section. The
${Z\to{\tau\tau}}$ sample is normalised using the cross-section calculated
with Dynnlo [14] using the MSTW08 PDF set [15]. The
${{\Phi^{0}}\to{\tau\tau}}$ signal distribution is found from simulated gluon-
fusion events. The signal samples were generated in mass steps of
${10~{}\mathrm{Ge\kern-1.00006ptV}}$ from ${90~{}\mathrm{Ge\kern-1.00006ptV}}$
to $250~{}\mathrm{Ge\kern-1.00006ptV}$. For both the SM and MSSM Higgs bosons,
the normalisation of the signal uses the theoretical calculations described
below.
The SM cross-sections, using the recommendations of Refs. [16] and [17], are
calculated at ${\sqrt{s}=7~{}\mathrm{Te\kern-1.00006ptV}}$ with the program
dfg [18] in the complex-pole scheme at next-to-next-to-leading log in QCD
contributions and next-to-leading order (NLO) in electroweak contributions.
The large parameter space in the MSSM necessitates the use of benchmark
scenarios [3]. Only the ${m_{h^{0}}^{\mathrm{max}}}$ scenario is considered
for comparison with previous results. Both gluon-fusion and associated
${b\bar{b}}$ production mechanisms are considered; the former is calculated at
NLO in QCD using Higlu [19] with the top-loop corrected to NNLO using ggh@nnlo
[20], while the latter is calculated at NNLO in QCD using bbh@nnlo [21] with
the five flavour scheme. For both SM and MSSM Higgs bosons, the branching
fractions are calculated using FeynHiggs [22] at the two-loop level.
The expected distributions of background events are displayed in Fig. 1 and
the estimated numbers of events with their associated systematic
uncertainties, as well as the observed numbers of candidates from data, are
given in Table 1. The systematic uncertainty on the ${Z\to{\tau\tau}}$
background is dominated by the statistical uncertainty on the data-driven
determination of the efficiency; the other background uncertainties are
described in Ref. [13].
Table 1: Estimated number of events for each background component and their sum, together with the observed number of candidates and the expected number of SM signal events for $M_{H}=125~{}\mathrm{Ge\kern-0.90005ptV}$, separated by analysis category. | ${\tau_{\mu}\tau_{\mu}}$ | ${\tau_{\mu}\tau_{e}}$ | ${\tau_{e}\tau_{\mu}}$ | ${\tau_{\mu}\tau_{h}}$ | ${\tau_{e}\tau_{h}}$
---|---|---|---|---|---
${Z\to{\tau\tau}}$ | $79.8\,\pm$ | $5.6$ | $288.2\,\pm$ | $26.2$ | $115.8\,\pm$ | $12.7$ | $146.1\,\pm$ | $9.7$ | $\phantom{1}62.1\,\pm$ | $8.0$
$\mathrm{QCD}$ | $11.7\,\pm$ | $3.4$ | $72.4\,\pm$ | $2.2$ | $54.0\,\pm$ | $3.0$ | $41.9\,\pm$ | $0.5$ | $24.5\,\pm$ | $0.6$
$\mathrm{EWK}$ | $0.0\,\pm$ | $3.5$ | $40.3\,\pm$ | $4.3$ | $0.0\,\pm$ | $1.3$ | $10.8\,\pm$ | $0.5$ | $9.3\,\pm$ | $0.5$
${t\bar{t}}$ | $<0.1\,\pm$ | $0.1$ | $3.6\,\pm$ | $0.4$ | $1.0\,\pm$ | $0.1$ | $<0.1\,\pm$ | $0.1$ | $0.7\,\pm$ | $0.4$
${WW}$ | $<0.1\,\pm$ | $0.1$ | $13.3\,\pm$ | $1.2$ | $1.6\,\pm$ | $0.2$ | $0.2\,\pm$ | $0.1$ | $<0.1\,\pm$ | $0.1$
${Z\to{\ell\ell}}$ | $29.8\,\pm$ | $7.0$ | $-$ | $-$ | $0.4\,\pm$ | $0.1$ | $2.0\,\pm$ | $0.2$
${\rm Total}$ | $121.4\,\pm$ | $10.2$ | $417.9\,\pm$ | $26.7$ | $172.4\,\pm$ | $13.1$ | $199.3\,\pm$ | $\phantom{1}9.7$ | $98.7\,\pm$ | $\phantom{1}8.0$
${\rm Observed}$ | 124 | 421 | 155 | 189 | 101
${\rm SM~{}Higgs\times 100}$ | $3.9\,\pm$ | $0.5$ | $11.9\,\pm$ | $1.6$ | $3.8\,\pm$ | $0.5$ | $9.7\,\pm$ | $1.3$ | $4.2\,\pm$ | $0.6$
## 3 Results
Limits for model independent and MSSM Higgs boson production are calculated
using the method of Ref. [23] with ${{\mathrm{CL_{s}}}=95\%}$ and the test
statistic of Eq. $14$ from Ref. [24]. The test statistic is defined using the
profile extended-likelihood ratio of the distributions in Fig. 1, where the
systematic uncertainties in Table 1 and the uncertainty on the simulated
invariant mass shapes have been incorporated using normally distributed
nuisance parameters. The uncertainty for the invariant mass shape is
determined from the momentum resolution calibration for simulation, while the
primary normalisation uncertainties are from luminosity determination and the
electron reconstruction efficiency. The distribution of this test statistic is
assumed to follow the result of Wilks [25]; this assumption has been validated
using a simple likelihood ratio. The expected limits have been determined
using Asimov datasets [24].
Figure 2: Model independent combined limit on cross-section by branching
fraction for a Higgs boson decaying to two tau leptons at ${95\%}$
${\mathrm{CL_{s}}}$ as a function of $M_{\Phi^{0}}$ is given on the left. The
background only expected limit (dashed red) and ${\pm 1\sigma}$ (green) and
${\pm 2\sigma}$ (yellow) bands are compared with the observed limit (solid
black) and the expected SM theory (dotted black) with uncertainty (grey). The
combined MSSM ${95\%}$ ${\mathrm{CL_{s}}}$ upper limit on ${\tan\beta}$ as a
function of $M_{A^{0}}$ is given on the right and compared to ATLAS (dotted
maroon and dot-dashed magenta), CMS (dot-dot-dashed blue and dot-dot-dot-
dashed cyan), and LEP (hatched orange) results.
The upper limit on the cross-section times branching fraction of a model
independent Higgs boson decaying to two tau leptons with ${2.0<\eta<4.5}$ is
plotted on the left of Fig. 2 as a function of the Higgs boson mass. The
upper-limit on ${\tan\beta}$ for the production of neutral MSSM Higgs bosons,
as a function of the $C\\!P$-odd Higgs boson mass, $M_{A^{0}}$, is provided in
the right plot of Fig. 2. Previously published exclusion limits from ATLAS
[26, 27], CMS [28, 29], and LEP [30] are provided for comparison.
## 4 Conclusions
A model independent search for a Higgs boson decaying to two tau leptons with
pseudorapidities between $2.0$ and $4.5$ gives an upper bound, at the $95\%$
confidence level, on the cross-section times branching fraction of
$8.6~{}\mathrm{pb}$ for a Higgs boson mass of
${90~{}\mathrm{Ge\kern-1.00006ptV}}$ with the bound decreasing smoothly to
$0.7~{}\mathrm{pb}$ for a Higgs boson mass of
${250~{}\mathrm{Ge\kern-1.00006ptV}}$.
Limits on a MSSM Higgs bosons have been set in the
${m_{h^{0}}^{\mathrm{max}}}$ scenario. Values above ${\tan\beta}$ ranging from
$34$ to $70$ are excluded over the $C\\!P$-odd MSSM Higgs boson mass range of
$90$ to $140~{}\mathrm{Ge\kern-1.00006ptV}$. For
${M_{A^{0}}<110~{}\mathrm{Ge\kern-1.00006ptV}}$, these are comparable to the
limits obtained by ATLAS and CMS using the $2010$ data sets but are
considerably less stringent than the ATLAS and CMS results using $2011$ data.
The forthcoming running of the LHC should allow the boson, observed by ATLAS
and CMS, to be seen in the LHCb detector through a combination of channels and
should provide complementary information on its properties.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-09T13:51:39 |
2024-09-04T02:49:44.028673
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto,\n H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D.\n Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen,\n E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P.\n Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik,\n P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina,\n I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, B. Leverington,\n Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, A. Mcnab, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G. Punzi,\n W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, E.\n Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E. Santovetti, M.\n Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack,\n M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F.\n Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson,\n A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Philip Ilten",
"url": "https://arxiv.org/abs/1304.2591"
}
|
1304.2600
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-055 LHCb-PAPER-2013-002 May 22, 2013
Measurement of $C\\!P$ violation and the $B^{0}_{s}$ meson decay width
difference with $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and
$B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The time-dependent $C\\!P$ asymmetry in $B^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ decays is measured using $pp$ collision data at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $1.0$$\mbox{\,fb}^{-1}$, collected with the LHCb detector. The
decay time distribution is characterised by the decay widths
$\Gamma_{\mathrm{L}}$ and $\Gamma_{\mathrm{H}}$ of the light and heavy mass
eigenstates of the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system and by a
$C\\!P$-violating phase $\phi_{s}$. In a sample of 27 617
$B^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$ decays, where the dominant
contribution comes from $B^{0}_{s}\rightarrow J/\psi\phi$ decays, these
parameters are measured to be $\phi_{s}=0.07\pm 0.09\text{(stat)}\pm
0.01\text{(syst)}\ \text{rad}$,
$\Gamma_{s}\equiv(\Gamma_{\mathrm{L}}+\Gamma_{\mathrm{H}})/2=0.663\pm
0.005\text{(stat)}\pm 0.006\text{(syst)}\ {\rm\,ps^{-1}}$ and
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{L}}-\Gamma_{\mathrm{H}}=0.100\pm
0.016\text{(stat)}\pm 0.003\text{(syst)}\ {\rm\,ps^{-1}}$, corresponding to
the single most precise determination of $\phi_{s}$, $\Delta\Gamma_{s}$ and
$\Gamma_{s}$. The result of performing a combined analysis with
$B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays gives $\phi_{s}=0.01\pm
0.07\text{(stat)}\pm 0.01\text{(syst)}\ \text{rad}$, $\Gamma_{s}=0.661\pm
0.004\text{(stat)}\pm 0.006\text{(syst)}\ {\rm\,ps^{-1}}$ and
$\Delta\Gamma_{s}=0.106\pm 0.011\text{(stat)}\pm 0.007\text{(syst)}\
{\rm\,ps^{-1}}$. All measurements are in agreement with the Standard Model
predictions.
Submitted to Phys. Rev. D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso57,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso58, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani12, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A. Cardini15,
H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51, M.
Cattaneo37, Ch. Cauet9, M. Charles54, Ph. Charpentier37, P. Chen3,38, N.
Chiapolini39, M. Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L.
Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J.
Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15, A.
Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan38,
D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di
Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil
Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, A. Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S.
Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L.
Eklund50,37, I. El Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C.
Färber11, G. Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49,
V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32,
M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,
A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J.
Garofoli57, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R.
Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V.
Gibson46, V.V. Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A.
Gomes2, H. Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, O. Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, B. Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37,
T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18,37, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, B.
Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B.
Liu3, G. Liu37, J. von Loeben20, S. Lohn37, J.H. Lopes2, E. Lopez Asamar35, N.
Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7,
I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d,
G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, L. Martin54, A. Martín Sánchez7, M. Martinelli40, D. Martinez
Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, R. McNulty12,
A. Mcnab53, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren35,o, B.K. Pal57,
A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, V.
Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K.
Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The interference between $B^{0}_{s}$ meson decay amplitudes to $C\\!P$
eigenstates ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ directly or via
mixing gives rise to a measurable $C\\!P$-violating phase $\phi_{s}$. In the
Standard Model (SM), for $b\rightarrow c\overline{c}s$ transitions and
ignoring subleading penguin contributions, this phase is predicted to be
$-2\beta_{s}$, where
$\beta_{s}=\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)$ and $V_{ij}$
are elements of the CKM quark flavour mixing matrix [1, *Cabibbo:1963yz]. The
indirect determination via global fits to experimental data gives
$2\beta_{s}=0.0364\pm 0.0016\rm\,rad$ [3]. This precise indirect determination
within the SM makes the measurement of $\phi_{s}$ interesting since new
physics (NP) processes could modify the phase if new particles were to
contribute to the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ box diagrams [4, 5] shown in
Fig. 1.
Direct measurements of $\phi_{s}$ using
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays have been reported
previously. In the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channel, the decay width difference of the light (L) and heavy (H) $B^{0}_{s}$
mass eigenstates,
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{L}}-\Gamma_{\mathrm{H}}$, and the
average $B^{0}_{s}$-decay width,
$\Gamma_{s}=(\Gamma_{\mathrm{L}}+\Gamma_{\mathrm{H}})/2$ are also measured.
The measurements of $\phi_{s}$ and $\Delta\Gamma_{s}$ are shown in Table 1.
This paper extends previous LHCb measurements in the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ [6]
and $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ [7] channels. In the previous
analysis of $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays, the invariant mass of the $K^{+}K^{-}$ system was limited
to $\pm 12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the $\phi(1020)$
mass [8], which selected predominately resonant P-wave $\phi\rightarrow
K^{+}K^{-}$ events, although a small S-wave $K^{+}K^{-}$ component was also
present. In this analysis the $K^{+}K^{-}$ mass range is extended to $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the notation
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
is used to include explicitly both P- and S-wave decays [9]. In both channels
additional same-side flavour tagging information is used. The data were
obtained from $pp$ collisions collected by the LHCb experiment at a centre-of-
mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ during 2011, corresponding to
an integrated luminosity of $1.0\mbox{\,fb}^{-1}$.
Table 1: Results for $\phi_{s}$ and $\Delta\Gamma_{s}$ from different experiments. The first uncertainty is statistical and the second is systematic (apart from the D0 result, for which the uncertainties are combined). The CDF confidence level (CL) range quoted is that consistent with other experimental measurements of $\phi_{s}$. Experiment | Dataset [$\mbox{\,fb}^{-1}$ ] | Ref. | $\phi_{s}$[$\rm\,rad$ ] | $\Delta\Gamma_{s}$[${\rm\,ps^{-1}}$ ]
---|---|---|---|---
LHCb ($B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$) | $0.4$ | [6] | $0.15\pm 0.18\pm 0.06$ | $0.123\pm 0.029\pm 0.011$
LHCb ($B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$) | $1.0$ | [7] | $-0.019\,^{+0.173+0.004}_{-0.174-0.003}$ | –
LHCb (combined) | $0.4$+$1.0$ | [7] | $0.06\pm 0.12\pm 0.06$ | –
ATLAS | $4.9$ | [10] | $0.22\pm 0.41\pm 0.10$ | $0.053\pm 0.021\pm 0.010$
CMS | $5.0$ | [11] | – | $0.048\pm 0.024\pm 0.003$
D0 | $8.0$ | [12] | $-0.55\,^{+0.38}_{-0.36}$ | $0.163\,^{+0.065}_{-0.064}$
CDF | $9.6$ | [13] | $[-0.60,\,0.12]$ at 68% CL | $0.068\pm 0.026\pm 0.009$
This paper is organised as follows. Section 2 presents the phenomenological
aspects related to the measurement. Section 3 presents the LHCb detector. In
Sect. 4 the selection of $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ candidates
is described. Section 5 deals with decay time resolution, Sect. 6 with the
decay time and angular acceptance effects and Sect. 7 with flavour tagging.
The maximum likelihood fit is explained in Sect. 8. The results and systematic
uncertainties for the $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ channel are
given in Sections 9 and 10, the results for the $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ channel are given in Sect. 11 and finally the combined
results are presented in Sect. 12. Charge conjugation is implied throughout
the paper.
## 2 Phenomenology
Figure 1: Feynman diagrams for $B^{0}_{s}$–$\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ mixing, within the SM.
\begin{overpic}[scale={1},clip={true},trim=91.04881pt 512.1496pt 0.0pt
142.26378pt]{final_figs/tree_penguin.pdf} \put(18.0,0.0){(a) tree}
\put(57.0,0.0){(b) penguin} \end{overpic} Figure 2: Feynman diagrams
contributing to the decay $B^{0}_{s}\rightarrow J/\psi h^{+}h^{-}$ within the
SM, where $h=\pi,K$.
The $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ decay proceeds predominantly via
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ with
the $\phi$ meson subsequently decaying to $K^{+}K^{-}$. In this case there are
two intermediate vector particles and the $K^{+}K^{-}$ pair is in a P-wave
configuration. The final state is then a superposition of $C\\!P$-even and
$C\\!P$-odd states depending upon the relative orbital angular momentum of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and the $\phi$. The
phenomenological aspects of this process are described in many articles, e.g.,
Refs. [14, 15]. The main Feynman diagrams for $B_{s}^{0}\rightarrow J/\psi
K^{+}K^{-}$ decays are shown in Fig. 2. The effects induced by the sub-leading
penguin contributions are discussed, e.g., in Ref. [16]. The same final state
can also be produced with $K^{+}K^{-}$ pairs in an S-wave configuration [17].
This S-wave final state is $C\\!P$-odd. The measurement of $\phi_{s}$ requires
the $C\\!P$-even and $C\\!P$-odd components to be disentangled by analysing
the distribution of the reconstructed decay angles of the final-state
particles.
In contrast to Ref. [6], this analysis uses the decay angles defined in the
helicity basis as this simplifies the angular description of the background
and acceptance. The helicity angles are denoted by
$\Omega=(\cos\theta_{K},\cos\theta_{\mu},\varphi_{h})$ and their definition is
shown in Fig. 3. The polar angle $\theta_{K}$ ($\theta_{\mu}$) is the angle
between the $K^{+}$ ($\mu^{+}$) momentum and the direction opposite to the
$B^{0}_{s}$ momentum in the $K^{+}K^{-}$ ($\mu^{+}\mu^{-}$) centre-of-mass
system. The azimuthal angle between the $K^{+}K^{-}$ and $\mu^{+}\mu^{-}$
decay planes is $\varphi_{h}$. This angle is defined by a rotation from the
$K^{-}$ side of the $K^{+}K^{-}$ plane to the $\mu^{+}$ side of the
$\mu^{+}\mu^{-}$ plane. The rotation is positive in the $\mu^{+}\mu^{-}$
direction in the $B^{0}_{s}$ rest frame. A definition of the angles in terms
of the particle momenta is given in Appendix A.
Figure 3: Definition of helicity angles as discussed in the text.
The decay can be decomposed into four time-dependent complex amplitudes,
$A_{i}(t)$. Three of these arise in the P-wave decay and correspond to the
relative orientation of the linear polarisation vectors of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\phi$ mesons, where
$i\in\\{0,\parallel,\perp\\}$ and refers to the longitudinal, transverse-
parallel and transverse-perpendicular orientations, respectively. The single
$K^{+}K^{-}$ S-wave amplitude is denoted by $A_{\rm S}(t)$.
The distribution of the decay time and angles for a $B^{0}_{s}$ meson produced
at time $t=0$ is described by a sum of ten terms, corresponding to the four
polarisation amplitudes and their interference terms. Each of these is given
by the product of a time-dependent function and an angular function [14]
$\frac{\mathrm{d}^{4}\Gamma(B^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})}{\mathrm{d}t\;\mathrm{d}\Omega}\;\propto\;\sum^{10}_{k=1}\>h_{k}(t)\>f_{k}(\Omega)\,.$
(1)
The time-dependent functions $h_{k}(t)$ can be written as
$h_{k}(t)\;=\;N_{k}e^{-\Gamma_{s}t}\>[a_{k}\cosh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)+b_{k}\sinh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)\\\
+c_{k}\cos(\Delta m_{s}t)\,+d_{k}\sin(\Delta m_{s}t)],$ (2)
where $\Delta m_{s}{}$ is the mass difference between the heavy and light
$B^{0}_{s}$ mass eigenstates. The expressions for the $f_{k}(\Omega)$ and the
coefficients of Eq. 2 are given in Table 2 [18, 19]. The coefficients $N_{k}$
are expressed in terms of the $A_{i}(t)$ at $t=0$, from now on denoted as
$A_{i}$. The amplitudes are parameterised by $|A_{i}|e^{i\delta_{i}}$ with the
conventions $\delta_{0}=0$ and $|A_{0}|^{2}+|A_{\|}|^{2}+|A_{\perp}|^{2}=1$.
The S-wave fraction is defined as $F_{\text{S}}=|A_{\rm
S}|^{2}/(|A_{0}|^{2}+|A_{\|}|^{2}+|A_{\perp}|^{2}+|A_{\rm S}|^{2})=|A_{\rm
S}|^{2}/(|A_{\rm S}|^{2}+1)$.
Table 2: Definition of angular and time-dependent functions.
$\begin{array}[]{c|c|c|c|c|c|c}k&f_{k}(\theta_{\mu},\theta_{K},\varphi_{h})&N_{k}&a_{k}&b_{k}&c_{k}&d_{k}\\\
\hline\cr\rule{0.0pt}{8.53581pt}1&2\cos^{2}\theta_{K}\sin^{2}\theta_{\mu}&|A_{0}|^{2}&1&D&C&-S\\\
2&\sin^{2}\theta_{K}\left(1-\sin^{2}\theta_{\mu}\cos^{2}\varphi_{h}\right)&|A_{\|}|^{2}&1&D&C&-S\\\
3&\sin^{2}\theta_{K}\left(1-\sin^{2}\theta_{\mu}\sin^{2}\varphi_{h}\right)&|A_{\perp}|^{2}&1&-D&C&S\\\
4&\sin^{2}\theta_{K}\sin^{2}\theta_{\mu}\sin
2\varphi_{h}&|A_{\|}A_{\perp}|&C\sin(\delta_{\perp}-\delta_{\parallel})&S\cos(\delta_{\perp}-\delta_{\parallel})&\sin(\delta_{\perp}-\delta_{\parallel})&D\cos(\delta_{\perp}-\delta_{\parallel})\\\
5&\tfrac{1}{2}\sqrt{2}\sin 2\theta_{K}\sin
2\theta_{\mu}\cos\varphi_{h}&|A_{0}A_{\|}|&\cos(\delta_{\parallel}-\delta_{0})&D\cos(\delta_{\parallel}-\delta_{0})&C\cos(\delta_{\parallel}-\delta_{0})&-S\cos(\delta_{\parallel}-\delta_{0})\\\
6&-\frac{1}{2}\sqrt{2}\sin 2\theta_{K}\sin
2\theta_{\mu}\sin\varphi_{h}&|A_{0}A_{\perp}|&C\sin(\delta_{\perp}-\delta_{0})&S\cos(\delta_{\perp}-\delta_{0})&\sin(\delta_{\perp}-\delta_{0})&D\cos(\delta_{\perp}-\delta_{0})\\\
7&\tfrac{2}{3}\sin^{2}\theta_{\mu}&|A_{\rm S}|^{2}&1&-D&C&S\\\
8&\tfrac{1}{3}\sqrt{6}\sin\theta_{K}\sin 2\theta_{\mu}\cos\varphi_{h}&|A_{\rm
S}A_{\|}|&C\cos(\delta_{\parallel}-\delta_{\rm
S})&S\sin(\delta_{\parallel}-\delta_{\rm
S})&\cos(\delta_{\parallel}-\delta_{\rm
S})&D\sin(\delta_{\parallel}-\delta_{\rm S})\\\
9&-\tfrac{1}{3}\sqrt{6}\sin\theta_{K}\sin 2\theta_{\mu}\sin\varphi_{h}&|A_{\rm
S}A_{\perp}|&\sin(\delta_{\perp}-\delta_{\rm
S})&-D\sin(\delta_{\perp}-\delta_{\rm S})&C\sin(\delta_{\perp}-\delta_{\rm
S})&S\sin(\delta_{\perp}-\delta_{\rm S})\\\
10&\tfrac{4}{3}\sqrt{3}\cos\theta_{K}\sin^{2}\theta_{\mu}&|A_{\rm
S}A_{0}|&C\cos(\delta_{0}-\delta_{\rm S})&S\sin(\delta_{0}-\delta_{\rm
S})&\cos(\delta_{0}-\delta_{\rm S})&D\sin(\delta_{0}-\delta_{\rm S})\\\
\end{array}$
For the coefficients $a_{k},\ldots,d_{k}$, three $C\\!P$ violating observables
are introduced
$C\;\equiv\;\frac{1-|\lambda|^{2}}{1+|\lambda|^{2}}\;,\qquad
S\;\equiv\;\frac{2\Im(\lambda)}{1+|\lambda|^{2}}\;,\qquad
D\;\equiv\;-\frac{2\Re(\lambda)}{1+|\lambda|^{2}}\;,$ (3)
where the parameter $\lambda$ is defined below. These definitions for $S$ and
$C$ correspond to those adopted by HFAG [20] and the sign of $D$ is chosen
such that it is equivalent to the symbol $A^{\Delta\Gamma}_{f}$ used in Ref.
[20]. The $C\\!P$-violating phase $\phi_{s}$ is defined by
$\phi_{s}\equiv-\arg(\lambda)$ and hence $S$ and $D$ can be written as
$S\;\equiv\;-\frac{2|\lambda|\sin{\phi_{s}}}{1+|\lambda|^{2}}\;,\qquad
D\;\equiv\;-\frac{2|\lambda|\cos{\phi_{s}}}{1+|\lambda|^{2}}\;.$ (4)
The parameter $\lambda$ describes $C\\!P$ violation in the interference
between mixing and decay, and is derived from the $C\\!P$-violating parameter
[21] associated with each polarisation state $i$
$\lambda_{i}\;\equiv\;\frac{q}{p}\;\frac{\bar{A}_{i}}{A_{i}},$ (5)
where $A_{i}$ ($\bar{A}_{i}$) is the amplitude for a $B_{s}^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$) meson to decay to final state
$i$ and the complex parameters $p=\langle B_{s}^{0}|B_{L}\rangle$ and
$q=\langle\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}|B_{L}\rangle$
describe the relation between mass and flavour eigenstates. The polarisation
states $i$ have $C\\!P$ eigenvalue $\eta_{i}=+1\>\ \text{for
$i\in\\{0,\parallel\\}$}$ and $\eta_{i}=-1\>\text{for $i\in\\{\perp,{\rm
S}\\}$}$. Assuming that any possible $C\\!P$ violation in the decay is the
same for all amplitudes, then the product $\eta_{i}\bar{A}_{i}/A_{i}$ is
independent of $i$. The polarisation-independent $C\\!P$-violating parameter
$\lambda$ is then defined such that $\lambda_{i}=\eta_{i}\lambda$. The
differential decay rate for a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson produced at time $t=0$
can be obtained by changing the sign of $c_{k}$ and $d_{k}$ and by including a
relative factor $|p/q|^{2}$.
The expressions are invariant under the transformation
$(\phi_{s},\Delta\Gamma_{s},\delta_{0},\delta_{\parallel},\delta_{\perp},\delta_{\rm
S})\longmapsto(\pi-\phi_{s},-\Delta\Gamma_{s},-\delta_{0},-\delta_{\parallel},\pi-\delta_{\perp},-\delta_{\rm
S})\>,$ (6)
which gives rise to a two-fold ambiguity in the results.
In the selected $\pi^{+}\pi^{-}$ invariant mass range the $C\\!P$-odd fraction
of $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays is greater than 97.7% at
95% confidence level (CL) as described in Ref. [22]. As a consequence, no
angular analysis of the decay products is required and the differential decay
rate can be simplified to
$\frac{\mathrm{d}\Gamma(B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-})}{\mathrm{d}t}\;\propto\;h_{7}(t).$ (7)
## 3 Detector
The LHCb detector [23] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons are
identified using two ring-imaging Cherenkov detectors [24]. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage which applies a full event
reconstruction [25].
Simulated $pp$ collisions are generated using Pythia 6.4 [26] with a specific
LHCb configuration [27]. Decays of hadronic particles are described by EvtGen
[28] in which final state radiation is generated using Photos [29]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [30, *Agostinelli:2002hh] as described in
Ref. [32].
## 4 Selection of $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ candidates
The reconstruction of $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ candidates
proceeds using the decays $J\\!/\\!\psi\rightarrow\mu^{+}\mu^{-}$ combined
with a pair of oppositely charged kaons. Events are first required to pass a
hardware trigger [25], which selects events containing muon or hadron
candidates with high transverse momentum ($p_{\rm T}$). The subsequent
software trigger [25] is composed of two stages, the first of which performs a
partial event reconstruction. Two types of first-stage software trigger are
employed. For the first type, events are required to have two well-identified
oppositely-charged muons with invariant mass larger than
$2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. This trigger has an almost
uniform acceptance as a function of decay time and will be referred to as
unbiased. For the second type there must be at least one muon (one
high-$p_{\rm T}$ track) with transverse momentum larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
($1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and impact parameter larger than
100$\,\upmu\rm m$ with respect to the PV. This trigger introduces a non-
trivial acceptance as a function of decay time and will be referred to as
biased. The second stage of the trigger performs a full event reconstruction
and only retains events containing a $\mu^{+}\mu^{-}$ pair with invariant mass
within $120{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $J\\!/\\!\psi$
mass [8] and which form a vertex that is significantly displaced from the PV,
introducing another small decay time biasing effect.
The final $B^{0}_{s}$ candidate selection is performed by applying kinematic
and particle identification criteria to the final-state tracks. The
$J\\!/\\!\psi$ meson candidates are formed from two oppositely-charged
particles, originating from a common vertex, which have been identified as
muons and which have $p_{\rm T}$ larger than
500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The invariant mass of the
$\mu^{+}\mu^{-}$ pair, $m(\mu^{+}\mu^{-})$, must be in the range
$[3030,3150]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. During subsequent steps
of the selection, $m(\mu^{+}\mu^{-})$ is constrained to the $J\\!/\\!\psi$
mass [8].
The $K^{+}K^{-}$ candidates are formed from two oppositely-charged particles
that have been identified as kaons and which originate from a common vertex.
The $K^{+}K^{-}$ pair is required to have a $p_{\rm T}$ larger than
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass of the
$K^{+}K^{-}$ pair, $m(K^{+}K^{-})$, must be in the range
$[990,1050]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $B^{0}_{s}$ candidates are reconstructed by combining the $J\\!/\\!\psi$
candidate with the $K^{+}K^{-}$ pair, requiring their invariant mass
$m(J\\!/\\!\psi K^{+}K^{-})$ to be in the range
$[5200,5550]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The decay time, $t$, of
the $B^{0}_{s}$ candidate is calculated from a vertex and kinematic fit that
constrains the $B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ candidate to
originate from its associated PV [33]. The $\chi^{2}$ of the fit (which has 7
degrees of freedom) is required to be less than 35. Multiple $B^{0}_{s}$
candidates are found in less than $1\%$ of events; in these cases the
candidate with the smallest $\chi^{2}$ is chosen. $B^{0}_{s}$ candidates are
required to have decay time in the range $[0.3,14.0]{\rm\,ps}$; the lower
bound on the decay time suppresses a large fraction of the prompt
combinatorial background whilst having a negligible effect on the sensitivity
to $\phi_{s}$. The kinematic fit evaluates an estimated decay time
uncertainty, $\sigma_{t}$. Candidates with $\sigma_{t}$ larger than
0.12${\rm\,ps}$ are removed from the event sample.
Figure 4: Invariant mass distribution of the selected $B^{0}_{s}\rightarrow
J\\!/\\!\psi K^{+}K^{-}$ candidates. The mass of the $\mu^{+}\mu^{-}$ pair is
constrained to the $J\\!/\\!\psi$ mass [8]. Curves for the fitted
contributions from signal (dotted red), background (dotted green) and their
combination (solid blue) are overlaid.
\begin{overpic}[trim=36.98857pt 31.29802pt 22.76219pt
71.13188pt,clip={true},width=223.0721pt]{final_figs/mumuMassLin.pdf}
\put(26.0,46.0){(a)} \end{overpic}\begin{overpic}[trim=36.98857pt 31.29802pt
22.76219pt 71.13188pt,clip={true},width=223.0721pt]{final_figs/KKMassLin.pdf}
\put(26.0,46.0){(b)} \end{overpic}
Figure 5: Background subtracted invariant mass distributions of the (a)
$\mu^{+}\mu^{-}$ and (b) $K^{+}K^{-}$ systems in the selected sample of
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ candidates. The solid blue line
represents the fit to the data points described in the text.
Figure 4 shows the $m(J\\!/\\!\psi K^{+}K^{-})$ distribution for events
originating from both the unbiased and biased triggers, along with
corresponding projection of an unbinned maximum log-likelihood fit to the
sample. The probability density function (PDF) used for the fit is composed of
the sum of two Gaussian functions with a common mean and separate widths and
an exponential function for the combinatorial background. In total, after the
trigger and full offline selection requirements, there are $27\,617\pm 115$
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ signal events found by the fit.
Of these, $23\,502\pm 107$ were selected by the unbiased trigger and $4115\pm
43$ were exclusively selected by the biased trigger. The uncertainties quoted
here come from propagating the uncertainty on the signal fraction evaluated by
the fit.
Figure 5 shows the invariant mass of the $\mu^{+}\mu^{-}$ and $K^{+}K^{-}$
pairs satisfying the selection requirements. The background has been
subtracted using the sPlot [34] technique with $m(J\\!/\\!\psi K^{+}K^{-})$ as
the discriminating variable. In both cases fits are also shown. For the di-
muon system the fit model is a double Crystal Ball shape [35]. For the di-kaon
system the total fit model is the sum of a relativistic P-wave Breit-Wigner
distribution convolved with a Gaussian function to model the dominant $\phi$
meson peak and a polynomial function to describe the small $K^{+}K^{-}$ S-wave
component.
## 5 Decay time resolution
If the decay time resolution is not negligibly small compared to the
$B^{0}_{s}$ meson oscillation period $2\pi/\Delta m_{s}\approx 350$ fs, it
affects the measurement of the oscillation amplitude, and thereby $\phi_{s}$.
For a given decay time resolution, $\sigma_{t}$, the dilution of the amplitude
can be expressed as ${\cal D}=\exp(-\sigma_{t}^{2}\Delta m_{s}^{2}/2)$ [36].
The relative systematic uncertainty on the dilution directly translates into a
relative systematic uncertainty on $\phi_{s}$.
For each reconstructed candidate, $\sigma_{t}$ is estimated by the vertex fit
with which the decay time is calculated. The signal distribution of
$\sigma_{t}$ is shown in Fig. 6 where the sPlot technique is used to subtract
the background. To account for the fact that track parameter resolutions are
not perfectly calibrated and that the resolution function is not Gaussian, a
triple Gaussian resolution model is constructed
$R(t;\sigma_{t})\;=\;\sum_{i=1}^{3}\>\frac{f_{i}}{\sqrt{2\pi}r_{i}\sigma_{t}}\>\exp\left[-\frac{(t-d)^{2}}{2r_{i}^{2}\sigma_{t}^{2}}\right],$
(8)
where $d$ is a common small offset of a few fs, $r_{i}$ are event-independent
resolution scale factors and $f_{i}$ is the fraction of each Gaussian
component, normalised such that $\sum f_{i}=1$.
Figure 6: Decay time resolution, $\sigma_{t}$, for selected
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ signal events. The curve shows
a fit to the data of the sum of two gamma distributions with a common mean.
\begin{overpic}[width=227.62204pt]{final_figs/prescaled_dg_de_gausswpv_floating_zoom_linear.pdf}
\end{overpic}
\begin{overpic}[width=227.62204pt]{final_figs/prescaled_dg_de_gausswpv_floating.pdf}
\end{overpic}
Figure 7: Decay time distribution of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ candidates. The curve
(solid blue) is the decay time model convolved with a Gaussian resolution
model. The decay time model consists of a delta function for the prompt
component and two exponential functions with different decay constants, which
represent the $B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ signal and long-
lived background, respectively. The decay constants are determined from the
fit. The same dataset is shown in both plots, on different scales.
The scale factors are estimated from a sample of prompt
$\mu^{+}\mu^{-}K^{+}K^{-}$ combinations that pass the same selection criteria
as the signal except for those that affect the decay time distribution. This
sample consists primarily of prompt combinations that have a true decay time
of zero. Consequently, the shape of the decay time distribution close to zero
is representative of the resolution function itself.
Prompt combinations for which the muon pair originates from a real
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson have a better resolution
than those with random muon pairs. Furthermore, fully simulated events confirm
that the resolution evaluated using prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
with two random kaons is more representative for the resolution of $B_{s}^{0}$
signal decays than the purely combinatorial background. Consequently, in the
data only ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ events are
used to estimate the resolution function. These are isolated using the sPlot
method to subtract the $\mu^{+}\mu^{-}$ combinatorial background.
The background subtracted decay time distribution for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ candidates is shown
in Fig. 7 using linear and logarithmic scales. The distribution is
characterised by a prompt peak and a tail due to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from $B$ decays. The
resolution model parameters are determined by fitting the distribution with a
decay time model that consists of a prompt peak and two exponential functions,
convolved with the resolution model given in Eq. 8.
The per-event resolution receives contributions both from the vertex
resolution and from the momentum resolution. The latter contribution is
proportional to the decay time and cannot be calibrated with the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ control sample. When
using a scale factor for the resolution there is an assumption that the vertex
contribution and the momentum contribution have a common scale. This
assumption is tested in simulations and a systematic uncertainty is assigned.
The effective dilution of the resolution function is calculated by taking its
Fourier transform calculated at frequency $\Delta m_{s}$ [36]
${\cal D}\;=\;\int_{-\infty}^{\infty}{\rm d}t\;\cos(\Delta
m_{s}t)\>R(t;\sigma_{t}).$ (9)
Taking into account the distribution of the per-event resolution, the
effective dilution for the calibrated resolution model is $0.72\pm 0.02$. This
dilution corresponds to an effective single Gaussian resolution of
approximately 45 fs. The systematic uncertainty accounts for uncertainties due
to the momentum resolution scale and other differences between the control
sample and signal decays. It is derived from simulations.
The sample used to extract the physics parameters of interest consists only of
events with $t>0.3$ ps. The observed decay time distribution of these events
is not sensitive to details of the resolution function. Therefore, in order to
simplify the fit procedure the resolution function for the final fit
(described in Sect. 8) is modelled with a single Gaussian distribution with a
resolution scale factor, $r_{t}$, chosen such that its effective dilution
corresponds to that of the multiple Gaussian model. This scale factor is
$r_{t}=1.45\pm 0.06$.
## 6 Acceptance
There are two distinct decay time acceptance effects that influence the
$B^{0}_{s}$ decay time distribution. First, there is a decrease in
reconstruction efficiency for tracks with a large impact parameter with
respect to the beam line. This effect is present both in the trigger and the
offline reconstruction, and translates to a decrease in the $B^{0}_{s}$ meson
reconstruction efficiency as a function of its decay time. This decrease is
parameterised by a linear acceptance function
$\varepsilon_{t}(t)\propto(1+\beta t)$, which multiplies the time dependent
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ PDF described below. The
parameterisation is determined using a control sample of $B^{\pm}\rightarrow
J\\!/\\!\psi K^{\pm}$ events from data and simulated $B^{0}_{s}\\!\rightarrow
J\\!/\\!\psi\phi$ events, leading to $\beta=(-8.3\pm 4.0)\times
10^{-3}{\rm\,ps^{-1}}$. The uncertainty directly translates to a $4.0\times
10^{-3}{\rm\,ps^{-1}}$ systematic uncertainty on $\Gamma_{s}$.
Secondly, a non-trivial decay time acceptance is introduced by the trigger
selection. Binned functional descriptions of the acceptance for the unbiased
and biased triggers are obtained from the data by exploiting the sample of
$B^{0}_{s}$ candidates that are also selected by a trigger that has no decay
time bias, but was only used for a fraction of the recorded data. Figure 8
shows the corresponding acceptance functions that are included in the fit
described in Sect. 8.
\begin{overpic}[width=227.62204pt]{final_figs/Bs_HltPropertimeAcceptance_PhiMassWindow30MeV_NextBestPVCut_Data_40bins_AlmostUnbiased.pdf}
\put(66.0,34.0){(a)} \end{overpic}
\begin{overpic}[width=227.62204pt]{final_figs/Bs_HltPropertimeAcceptance_PhiMassWindow30MeV_NextBestPVCut_Data_40bins_ExclusivelyBiased.pdf}
\put(66.0,34.0){(b)} \end{overpic}
Figure 8: $B^{0}_{s}$ decay time trigger-acceptance functions obtained from
data. The unbiased trigger category is shown on (a) an absolute scale and (b)
the biased trigger category on an arbitrary scale.
The acceptance as a function of the decay angles is not uniform due to the
forward geometry of LHCb and the requirements placed upon the momenta of the
final-state particles. The three-dimensional acceptance function,
$\varepsilon_{\Omega}$, is determined using simulated events which are
subjected to the same trigger and selection criteria as the data. Figure 9
shows the angular efficiency as a function of each decay angle, integrated
over the other angles. The relative acceptances vary by up to 20% peak-to-
peak. The dominant effect in $\cos\theta_{\mu}$ is due to the $p_{\rm T}$ cuts
applied to the muons.
The acceptance is included in the unbinned maximum log-likelihood fitting
procedure to signal weighted distributions (described in Sect. 8). Since only
a PDF to describe the signal is required, the acceptance function needs to be
included only in the normalisation of the PDF through the ten integrals
$\int\mathrm{d}\Omega\,\varepsilon_{\Omega}(\Omega)\,f_{k}(\Omega)$. The
acceptance factors for each event $i$, $\varepsilon_{\Omega}(\Omega_{i})$,
appear only as a constant sum of logarithms and may be ignored in the
likelihood maximisation. The ten integrals are determined from the fully
simulated events using the procedure described in Ref. [37].
\begin{overpic}[trim=128.0374pt 36.98857pt 54.06023pt
91.04881pt,clip={true},width=145.68143pt]{final_figs/angEffIntBinsCtk}
\put(70.0,67.0){\small(a)} \end{overpic}\begin{overpic}[trim=128.0374pt
36.98857pt 54.06023pt
91.04881pt,clip={true},width=145.68143pt]{final_figs/angEffIntBinsCtl}\put(70.0,67.0){\small(b)}
\end{overpic}\begin{overpic}[trim=128.0374pt 36.98857pt 54.06023pt
91.04881pt,clip={true},width=145.68143pt]{final_figs/angEffIntBinsPhi}
\put(70.0,67.0){\small(c)} \end{overpic}
Figure 9: Angular acceptance function evaluated with simulated
$B^{0}_{s}\\!\rightarrow J\\!/\\!\psi\phi$ events, scaled by the mean
acceptance. The acceptance is shown as a function of (a) $\cos\theta_{K}$, (b)
$\cos\theta_{\mu}$ and (c) $\varphi_{h}$, where in all cases the acceptance is
integrated over the other two angles. The points are obtained by summing the
inverse values of the underlying physics PDF for simulated events and the
curves represent a polynomial parameterisation of the acceptance.
## 7 Tagging the $B^{0}_{s}$ flavour at production
Each reconstructed candidate is identified by flavour tagging algorithms as
either a $B^{0}_{s}$ meson ($q=+1$) or a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson ($q=-1$) at production.
If the algorithms are unable to make a decision, the candidate is untagged
($q=0$).
The tagging decision, $q$, is based upon both opposite-side and same-side
tagging algorithms. The opposite-side (OS) tagger relies on the pair
production of $b$ and $\overline{}b$ quarks and infers the flavour of the
signal $B^{0}_{s}$ meson from identification of the flavour of the other
$b$-hadron. The OS tagger uses the charge of the lepton ($\mu$, $e$) from
semileptonic $b$ decays, the charge of the kaon from the $b\rightarrow
c\rightarrow s$ decay chain and the charge of the inclusive secondary vertex
reconstructed from $b$-hadron decay products. The same-side kaon (SSK) tagger
exploits the hadronization process of the $\overline{b}$($b$) quark forming
the signal $B_{s}^{0}$($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$)
meson. In events with a $B^{0}_{s}$ candidate, the fragmentation of a
$\overline{}b$ quark can lead to an extra $\overline{}s$ quark being available
to form a hadron, often leading to a charged kaon. This kaon is correlated to
the signal $B^{0}_{s}$ in phase space and the sign of the charge identifies
its initial flavour.
The probability that the tagging determination is wrong (estimated wrong-tag
probability, $\eta$) is based upon the output of a neural network trained on
simulated events. It is subsequently calibrated with data in order to relate
it to the true wrong-tag probability of the event, $\omega$, as described
below.
The tagging decision and estimated wrong-tag probability are used event-by-
event in order to maximise the tagging power, ${\varepsilon_{\rm tag}}{\cal
D}^{2}$, which represents the effective reduction of the signal sample size
due to imperfect tagging. In this expression $\varepsilon_{\rm tag}$ is the
tagging efficiency, i.e., the fraction of events that are assigned a non-zero
value of $q$, and ${\cal D}=1-2\omega$ is the dilution.
### 7.1 Opposite side tagging
The OS tagging algorithms and the procedure used to optimise and calibrate
them are described in Ref. [38]. In this paper the same approach is used,
updated to use the full 2011 data set.
Calibration of the estimated wrong-tag probability, $\eta$, is performed using
approximately 250 000 $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ events selected from data. The values of $q$ and $\eta$ measured
by the OS taggers are compared to the known flavour, which is determined by
the charge of the final state kaon. Figure 10 shows the average wrong tag
probability in the $B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}$ control channel in bins of $\eta$. For calibration purposes a
linear relation is assumed
$\displaystyle\omega(\eta)$ $\displaystyle=$ $\displaystyle p_{0}+\frac{\Delta
p_{0}}{2}+p_{1}(\eta-\langle\eta\rangle)\,,$ (10)
$\displaystyle\overline{}\omega(\eta)$ $\displaystyle=$ $\displaystyle
p_{0}-\frac{\Delta p_{0}}{2}+p_{1}(\eta-\langle\eta\rangle)\,,$
where $\omega(\eta)$ and $\overline{}\omega(\eta)$ are the calibrated
probabilities for wrong-tag assignment for $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ mesons, respectively. This
parametrisation is chosen to minimise the correlation between the parameters
$p_{0}$ and $p_{1}$. The resulting values of the calibration parameters
$p_{0}$, $p_{1}$, $\Delta p_{0}$ and $\langle\eta\rangle$ (the mean value of
$\eta$ in the sample) are given in Table 3. The systematic uncertainties for
$p_{0}$ and $p_{1}$ are determined by comparing the tagging performance for
different decay channels, comparing different data taking periods and by
modifying the assumptions of the fit model. The asymmetry parameter $\Delta
p_{0}$ is obtained by performing the calibration separately for $B^{+}$ and
$B^{-}$ decays. No significant difference of the tagging efficiency or of
$p_{1}$ is measured ($\Delta{\varepsilon_{\rm tag}}=(0.00\pm 0.10)$%, $\Delta
p_{1}=0.06\pm 0.04$). Figure 10 shows the relation between $\omega$ and $\eta$
for the full data sample.
The overall effective OS tagging power for $B^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ candidates is ${{\varepsilon_{\rm tag}}{\cal D}^{2}}=(2.29\pm
0.06)$%, with an efficiency of ${\varepsilon_{\rm tag}}=(33.00\pm 0.28)$% and
an effective average wrong-tag probability of $(36.83\pm 0.15)$% (statistical
uncertainties only).
Figure 10: Average measured wrong-tag probability ($\omega$) versus estimated wrong-tag probability ($\eta$) calibrated on $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal events for the OS tagging combinations for the background subtracted events in the signal mass window. Points with errors are data, the red curve represents the result of the wrong-tag probability calibration, corresponding to the parameters of Table 3. Table 3: Calibration parameters ($p_{0}$, $p_{1}$,$\langle\eta\rangle$ and $\Delta p_{0}$) corresponding to the OS and SSK taggers. The uncertainties are statistical and systematic, respectively, except for $\Delta p_{0}$ where they have been added in quadrature. Calibration | $p_{0}$ | $p_{1}$ | $\langle\eta\rangle$ | $\Delta p_{0}$
---|---|---|---|---
OS | $0.392\pm 0.002\pm 0.008$ | $1.000\pm 0.020\pm 0.012$ | $0.392$ | +$0.011\pm 0.003$
SSK | $0.350\pm 0.015\pm 0.007$ | $1.000\pm 0.160\pm 0.020$ | $0.350$ | $-0.019\pm 0.005$
### 7.2 Same side kaon tagging
One of the improvements introduced in this analysis compared to Ref. [6] is
the use of the SSK tagger. The SSK tagging algorithm was developed using large
samples of simulated $B^{0}_{s}$ decays to $D^{-}_{s}\pi^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and is documented in Ref.
[39]. The algorithm preferentially selects kaons originating from the
fragmentation of the signal $B^{0}_{s}$ meson, and rejects particles that
originate either from the opposite-side $B$ decay or the underlying event. For
the optimisation, approximately 26 000 $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ data events are used. The same fit procedure employed to
determine the $B^{0}_{s}$ mixing frequency $\Delta m_{s}$ [40] is used to
maximise the effective tagging power ${\varepsilon_{\rm tag}}{\cal D}^{2}$.
The calibration was also performed using $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ events and assuming the same linear relation given by Eq.
10. The resulting values of the calibration parameters ($p_{0},p_{1},\Delta
p_{0}$) are given in the second row of Table 3. In contrast to the OS tagging
case, it is more challenging to measure $p_{0}$ and $p_{1}$ separately for
true $B$ or $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mesons at
production using $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ events. Therefore,
assuming that any tagging asymmetry is caused by the difference in interaction
with matter of $K^{+}$ and $K^{-}$, $\Delta p_{0}$ is estimated using
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$, where the
$p$ and $p_{\rm T}$ distributions of the OS tagged kaons are first reweighted
to match those of SSK tagged kaons from a large sample of fully simulated
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ events.
The effective SSK tagging power for $B^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
events is ${{\varepsilon_{\rm tag}}{\cal D}^{2}}=(0.89\pm 0.17)$% and the
tagging efficiency is ${\varepsilon_{\rm tag}}=(10.26\pm 0.18)$% (statistical
uncertainties only).
### 7.3 Combination of OS and SSK tagging
Only a small fraction of tagged events are tagged by both the OS and the SSK
algorithms. The algorithms are uncorrelated as they select mutually exclusive
charged particles, either in terms of the impact parameter significance with
respect to the PV, or in terms of the particle identification requirements.
The two tagging results are combined taking into account both decisions and
their corresponding estimate of $\eta$. The combined estimated wrong-tag
probability and the corresponding uncertainties are obtained by combining the
individual calibrations for the OS and SSK tagging and propagating their
uncertainties according to the procedure defined in Ref. [38]. To simplify the
fit implementation, the statistical and systematic uncertainties on the
combined wrong-tag probability are assumed to be the same for all of these
events. They are defined by the average values of the corresponding
distributions computed event-by-event. The effective tagging power for these
OS+SSK tagged events is ${{\varepsilon_{\rm tag}}{\cal D}^{2}}=(0.51\pm
0.03)$%, and the tagging efficiency is ${\varepsilon_{\rm tag}}=(3.90\pm
0.11)$%.
### 7.4 Overall tagging performance
The overall effective tagging power obtained by combining all three categories
is ${{\varepsilon_{\rm tag}}{\cal D}^{2}}=(3.13\pm 0.12\pm 0.20)$%, the
tagging efficiency is ${\varepsilon_{\rm tag}}=(39.36\pm 0.32)$% and the
wrong-tag probability is $\omega=35.9$%. Figure 11 shows the distributions of
the estimated wrong-tag probability $\eta$ of the $B^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ signal events obtained with the sPlot technique using
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-})$ as the
discriminating variable.
\begin{overpic}[angle={0},width=204.85844pt]{final_figs/eta_OS.pdf}
\put(30.0,55.0){(a)}
\end{overpic}\begin{overpic}[angle={0},width=204.85844pt]{final_figs/eta_SSK.pdf}
\put(30.0,55.0){(b)} \end{overpic}
Figure 11: Distributions of the estimated wrong-tag probability, $\eta$, of
the $B^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$ signal events obtained using the
sPlot method on the $J\\!/\\!\psi K^{+}K^{-}$ invariant mass distribution.
Both the (a) OS-only and (b) SSK-only tagging categories are shown.
## 8 Maximum likelihood fit procedure
Each event is given a signal weight, $W_{i}$, using the sPlot [34] method with
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-})$ as the
discriminating variable. A weighted fit is then performed using a signal-only
PDF, denoted by ${\cal S}$, the details of which are described below. The
joint negative log likelihood, ${\cal L}$ constructed as
$-\ln{\cal L}=-\alpha\sum_{\mathrm{events}\;i}{W_{i}\ln{{\cal S}}},$ (11)
is minimised in the fit, where the factor
$\alpha=\sum_{i}W_{i}/\sum_{i}W_{i}^{2}$ is used to include the effect of the
weights in the determination of the uncertainties [41].
### 8.1 The mass model used for weighting
The signal mass distribution, ${\cal
S}_{m}(m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-});m_{B^{0}_{s}},\sigma_{m},r_{21},f_{1})$, is modelled by a
double Gaussian function. The free parameters in the fit are the common mean,
$m_{B^{0}_{s}}$, the width of the narrower Gaussian function, $\sigma_{m}$,
the ratio of the second to the first Gaussian width, $r_{21}$, and the
fraction of the first Gaussian, $f_{1}$.
The background mass distribution, ${\cal
B}_{m}(m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}))$ is modelled
by an exponential function. The full PDF is then constructed as
${\cal P}_{m}=f_{s}\;{\cal S}_{m}+(1-f_{s})\;{\cal B}_{m},$ (12)
where $f_{s}$ is the signal fraction. Fig. 4 shows the result of fitting this
model to the selected candidates.
### 8.2 Dividing the data into bins of $m(K^{+}K^{-})$
The events selected for this analysis are within the $m(K^{+}K^{-})$ range
$[990,1050]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The data are divided
into six independent sets, where the boundaries are given in Table 4. Binning
the data this way leads to an improvement in statistical precision by
separating events with different signal fractions and the analysis becomes
insensitive to correction factors which must be applied to each of the three
S-wave interference terms in the differential decay rate ($f_{8},f_{9},f_{10}$
in Table 2). These terms are required to account for an averaging effect
resulting from the variation within each bin of the S-wave line-shape (assumed
to be approximately uniform) relative to that of the P-wave (a relativistic
Breit-Wigner function). In each bin, the correction factors are calculated by
integrating the product of $p$ with $s^{*}$ which appears in the interference
terms between the P- and S-wave, where $p$ and $s$ are the normalised
$m(K^{+}K^{-})$ lineshapes and ∗ is the complex conjugation operator,
$\int_{m^{L}}^{m^{H}}{ps^{*}}\>\>{\rm d}m(K^{+}K^{-})=C_{\rm
SP}e^{-i\theta_{\rm SP}},$ (13)
where $[m^{L},m^{H}]$ denotes the boundaries of the $m(K^{+}K^{-})$ bin,
$C_{\rm SP}$ is the correction factor and $\theta_{\rm SP}$ is absorbed in the
measurements of $\delta_{\rm S}-\delta_{\perp}$. The $C_{\rm SP}$ correction
factors are given in Table 4. By using several bins these factors are close to
one, whereas if only a single bin were used the correction would differ
substantially from one. The effect of these factors on the fit results is very
small and is discussed further in Sect. 10, where a different S-wave lineshape
is considered. Binning the data in $m(K^{+}K^{-})$ allows a repetition of the
procedure described in Ref. [42] to resolve the ambiguous solution described
in Sect. 1 by inspecting the trend in the phase difference between the S- and
P-wave components.
Table 4: Bins of $m(K^{+}K^{-})$ used in the analysis and the $C_{\rm SP}$ correction factors for the S-wave interference term, assuming a uniform distribution of non-resonant $K^{+}K^{-}$ contribution and a non-relativistic Breit-Wigner shape for the decays via the $\phi$ resonance. $m(K^{+}K^{-})$ bin [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | $C_{\rm SP}$
---|---
0990 – 1008 | 0.966
1008 – 1016 | 0.956
1016 – 1020 | 0.926
1020 – 1024 | 0.926
1024 – 1032 | 0.956
1032 – 1050 | 0.966
The weights, $W_{i}$, are determined by performing a simultaneous fit to the
$m(J\\!/\\!\psi K^{+}K^{-})$ distribution in each of the $m(K^{+}K^{-})$ bins,
using a common set of signal mass parameters and six independent background
mass parameters. This fit is performed for $m(J\\!/\\!\psi K^{+}K^{-})$ in the
range $[5200,5550]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the results
for the signal mass parameters are shown in Table 5.
Table 5: Parameters of the common signal fit to the $m(J\\!/\\!\psi K^{+}K^{-})$ distribution in data. Parameter | Value
---|---
$m_{B^{0}_{s}}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | $5368.22\phantom{0}\pm 0.05$
$\sigma_{m}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | $\phantom{000}6.08\phantom{0}\pm 0.13$
$f_{1}$ | $\phantom{0000}0.760\pm 0.035$
$r_{21}$ | $\phantom{000}2.07\phantom{0}\pm 0.09$
### 8.3 The signal PDF
The physics parameters of interest in this analysis are $\Gamma_{s}$,
$\Delta\Gamma_{s}$, $|A_{0}|^{2}$, $|A_{\perp}|^{2}$, $F_{\text{S}}$,
$\delta_{\parallel}$, $\delta_{\perp}$, $\delta_{\rm S}$, $\phi_{s}$,
$|\lambda|$ and $\Delta m_{s}$, all of which are defined in Sect. 2. The
signal PDF, ${\cal S}$, is a function of the decay time, $t$, and angles,
$\Omega$, and is conditional upon the estimated wrong-tag probability for the
event, $\eta$, and the estimate of the decay time resolution for the event,
$\sigma_{t}$. The data are separated into disjoint sets corresponding to each
of the possible tagging decisions $q\in\\{-1,0,+1\\}$ and the unbiased and
biased trigger samples. A separate signal PDF, ${\cal
S}_{q}(t,\Omega|\sigma_{t},\eta;Z,N)$, is constructed for each event set,
where $Z$ represents the physics parameters and $N$ represents nuisance
parameters described above.
The ${\cal S}_{q}$ are constructed from the differential decay rates of
$B^{0}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons
described in Sect. 2. Denoting
$\frac{\mathrm{d}^{4}\Gamma(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}KK)}{\mathrm{d}t\;\mathrm{d}\Omega}$ by $X$ and
$\frac{\mathrm{d}^{4}\Gamma(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}KK)}{\mathrm{d}t\;\mathrm{d}\Omega}$ by $\overline{X}$, then
${\cal S}_{q}=\frac{s_{q}}{\int s_{q}\>\mathrm{d}t\;\mathrm{d}\Omega},\\\ $
(14)
where
$\displaystyle s_{+1}$ $\displaystyle=$
$\displaystyle\Big{[}[\>(1-\omega)\;X(t,\Omega;Z)+\bar{\omega}\;\overline{X}(t,\Omega;Z)\>]\otimes
R(t;\sigma_{t})\Big{]}\;\varepsilon_{t}(t)\;\varepsilon_{\Omega}(\Omega),$
$\displaystyle s_{-1}$ $\displaystyle=$
$\displaystyle\Big{[}[\>\omega\;X(t,\Omega;Z)+(1-\bar{\omega})\;\overline{X}(t,\Omega;Z)\>]\otimes
R(t;\sigma_{t})\Big{]}\;\varepsilon_{t}(t)\;\varepsilon_{\Omega}(\Omega),$
(15) $\displaystyle s_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}[\>X(t,\Omega;Z)+\overline{X}(t,\Omega;Z)\>]\otimes
R(t;\sigma_{t})\Big{]}\;\varepsilon_{t}(t)\;\varepsilon_{\Omega}(\Omega).\ $
Asymmetries in the tagging efficiencies and relative magnitudes of the
production rates for $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons, as well as the factor
$|p/q|^{2}$ are not included in the model. Sensitivity to these effects is
reduced by the use of separately normalised PDFs for each of the tagging
decisions and any residual effect is shown to be negligible.
All physics parameters are free in the fit apart from $\Delta m_{s}$, which is
constrained to the value measured by LHCb of $17.63\pm 0.11{\rm\,ps^{-1}}$
[40]. The parameter $\delta_{\rm S}-\delta_{\perp}$ is used in the
minimisation instead of $\delta_{\rm S}$ as there is a large (90%) correlation
between $\delta_{\rm S}$ and $\delta_{\perp}$.
In these expressions the terms $\omega$ and $\overline{\omega}$ represent the
wrong-tag probabilities for a candidate produced as a genuine $B^{0}_{s}$ or
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson, respectively,
and are a function of $\eta$ and the (nuisance) calibration parameters
$(p_{1},p_{0},\langle\eta\rangle,\Delta p_{0})$ as given in Eq. 10. The
calibration parameters are given in Table 3 and are all included in the fit
via Gaussian constraints with widths equal to their uncertainties.
The expressions are convolved with the decay time resolution function,
$R(t;\sigma_{t})$ (Sect. 5). The scale factor parameter, $r_{t}$, is included
in the fit with its value constrained by a Gaussian constraint with width
equal to its uncertainty. The $\varepsilon_{t}(t)$ and
$\varepsilon_{\Omega}(\Omega)$ terms are the decay time acceptance and decay-
angle acceptance, respectively. The two different trigger samples have
different decay time acceptance functions. These are described in Sect. 6.
Since this weighted fit uses only a signal PDF there is no need to include the
distributions of either the estimated wrong tag probability, $\eta$, or the
decay time resolution for each event, $\sigma_{t}$. The physics parameter
estimation is then performed by a simultaneous fit to the weighted data in
each of the $m(K^{+}K^{-})$ bins for each of the two trigger samples. All
parameters are common, except for the S-wave fraction $F_{\rm S}$ and the
phase difference $\delta_{\rm S}-\delta_{\perp}$, which are independent
parameters for each range.
## 9 Results for $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ decays
The results of the fit for the principal physics parameters are given in Table
6 for the solution with $\Delta\Gamma_{s}>0$, showing both the statistical and
the total systematic uncertainties described in Sect. 10.
The statistical correlation matrix is shown in Table 7. The projections of the
decay time and angular distributions are shown in Fig. 12. It was verified
that the observed uncertainties are compatible with the expected
sensitivities, by generating and fitting to a large number of simulated
experiments.
Figure 13 shows the 68%, 90% and 95% CL contours obtained from the two-
dimensional profile likelihood ratio in the ($\Delta\Gamma_{s}$, $\phi_{s}$)
plane, corresponding to decreases in the log-likelihood of 1.15, 2.30 and 3.00
respectively. Only statistical uncertainties are included. The SM expectation
[43, *Badin:2007bv, *Lenz:2011ti] is shown.
The results for the S-wave parameters are shown in Table 8. The likelihood
profiles for these parameters are non-parabolic and are asymmetric. Therefore
the 68% CL intervals obtained from the likelihood profiles, corresponding to a
decrease of 0.5 in the log-likelihood, are reported. The variation of
$\delta_{\rm S}-\delta_{\perp}$ with $m(K^{+}K^{-})$ is shown in Fig. 14. The
decreasing trend confirms that expected for the physical solution with
$\phi_{s}$ close to zero, as found in Ref. [42].
All results have been checked by splitting the dataset into sub-samples to
compare different data taking periods, magnet polarities, $B^{0}_{s}$-tags and
trigger categories. In all cases the results are consistent between the
independent sub-samples. The measurements of $\phi_{s}$, $\Delta\Gamma_{s}$
and $\Gamma_{s}$ are the most precise to date. Both $\Delta\Gamma_{s}$ and
$\phi_{s}$ agree well with the SM expectation [3, 43].
These data also allow an independent measurement of $\Delta m_{s}$ without
constraining it to the value reported in Ref. [40]. This is possible because
there are several terms in the differential decay rate of Eq. 1, principally
$h_{4}$ and $h_{6}$, which contain sinusoidal terms in $\Delta m_{s}t$ that
are not multiplied by $\sin\phi_{s}$. Figure 15 shows the likelihood profile
as a function of $\Delta m_{s}$ from a fit to the data where $\Delta m_{s}$ is
not constrained. The result of the fit gives
$\Delta m_{s}=17.70\pm 0.10\;\text{(stat)}\pm
0.01\;\text{(syst)${\rm\,ps^{-1}}$},$
which is consistent with other measurements [40, 46, 47, 48].
Table 6: Results of the maximum likelihood fit for the principal physics
parameters. The first uncertainty is statistical and the second is systematic.
The value of $\Delta m_{s}$ was constrained to the measurement reported in
Ref. [40]. The evaluation of the systematic uncertainties is described in
Sect. 10.
Parameter Value $\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] $0.663\pm 0.005\pm 0.006$
$\Delta\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] $0.100\pm 0.016\pm 0.003$
$|A_{\perp}|^{2}$ $0.249\pm 0.009\pm 0.006$ $|A_{0}|^{2}$ $0.521\pm 0.006\pm
0.010$ $\delta_{\parallel}$ [rad] $3.30\,^{+0.13}_{-0.21}\pm 0.08$
$\delta_{\perp}$ [rad] $3.07\pm 0.22\pm 0.08$ $\phi_{s}$ [rad] $0.07\pm
0.09\pm 0.01$ $|\lambda|$ $0.94\pm 0.03\pm 0.02$
Table 7: Correlation matrix for the principal physics parameters. | $\Gamma_{s}$ | $\Delta\Gamma_{s}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\parallel}$ | $\delta_{\perp}$ | $\phi_{s}$ | $|\lambda|$
---|---|---|---|---|---|---|---|---
| [${\rm\,ps^{-1}}$ ] | [${\rm\,ps^{-1}}$ ] | | | [rad] | [rad] | [rad] |
$\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | $\phantom{+}1.00$ | ${-0.39}$ | $\phantom{+}{0.37}$ | ${-0.27}$ | $-0.09$ | $-0.03$ | $\phantom{+}0.06$ | $\phantom{+}0.03$
$\Delta\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | | $\phantom{+}1.00$ | ${-0.68}$ | $\phantom{+}{0.63}$ | $\phantom{+}0.03$ | $\phantom{+}0.04$ | $-0.04$ | $\phantom{+}0.00$
$|A_{\perp}|^{2}$ | | | $\phantom{+}1.00$ | ${-0.58}$ | ${-0.28}$ | $-0.09$ | $\phantom{+}0.08$ | $-0.04$
$|A_{0}|^{2}$ | | | | $\phantom{+}1.00$ | $-0.02$ | $-0.00$ | $-0.05$ | $\phantom{+}0.02$
$\delta_{\parallel}$ [rad] | | | | | $\phantom{+}1.00$ | $\phantom{+}{0.32}$ | $-0.03$ | $\phantom{+}0.05$
$\delta_{\perp}$ [rad] | | | | | | $\phantom{+}1.00$ | $\phantom{+}{0.28}$ | $\phantom{+}0.00$
$\phi_{s}$ [rad] | | | | | | | $\phantom{+}1.00$ | $\phantom{+}0.04$
$|\lambda|$ | | | | | | | | $\phantom{+}1.00$
Figure 12: Decay-time and helicity-angle distributions for $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ decays (data points) with the one-dimensional projections of the PDF at the maximal likelihood point. The solid blue line shows the total signal contribution, which is composed of $C\\!P$-even (long-dashed red), $C\\!P$-odd (short-dashed green) and S-wave (dotted-dashed purple) contributions. Figure 13: Two-dimensional profile likelihood in the ($\Delta\Gamma_{s}$, $\phi_{s}$) plane for the $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ dataset. Only the statistical uncertainty is included. The SM expectation of $\Delta\Gamma_{s}=0.087\pm 0.021{\rm\,ps^{-1}}$ and $\phi_{s}=-0.036\pm 0.002\rm\,rad$ is shown as the black point with error bar [3, 43]. Figure 14: Variation of $\delta_{\rm S}-\delta_{\perp}$ with $m(K^{+}K^{-})$ where the uncertainties are the quadrature sum of the statistical and systematic uncertainties in each bin. The decreasing phase trend (blue circles) corresponds to the physical solution with $\phi_{s}$ close to zero and $\Delta\Gamma_{s}>0$. The ambiguous solution is also shown. Table 8: Results of the maximum likelihood fit for the S-wave parameters, with asymmetric statistical and symmetric systematic uncertainties. The evaluation of the systematic uncertainties is described in Sect. 10. $m(K^{+}K^{-})$ bin [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ] | Parameter | Value | $\sigma_{\text{stat}}$ (asymmetric) | $\sigma_{\text{syst}}$
---|---|---|---|---
$\phantom{0}990-1008$ | $F_{\text{S}}$ | 0.227 | ${+0.081},{-0.073}$ | 0.020
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | 1.31 | $+0.78,-0.49$ | 0.09
$1008-1016$ | $F_{\text{S}}$ | 0.067 | $+0.030,-0.027$ | 0.009
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | 0.77 | $+0.38,-0.23$ | 0.08
$1016-1020$ | $F_{\text{S}}$ | 0.008 | $+0.014,-0.007$ | 0.005
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | 0.51 | $+1.40,-0.30$ | 0.20
$1020-1024$ | $F_{\text{S}}$ | 0.016 | $+0.012,-0.009$ | 0.006
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | $-0.51$ | $+0.21,-0.35$ | 0.15
$1024-1032$ | $F_{\text{S}}$ | 0.055 | $+0.027,-0.025$ | 0.008
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | $-0.46$ | $+0.18,-0.26$ | 0.05
$1032-1050$ | $F_{\text{S}}$ | 0.167 | $+0.043,-0.042$ | 0.021
| $\delta_{\rm S}-\delta_{\perp}$ [rad] | $-0.65$ | $+0.18,-0.22$ | 0.06
Figure 15: Profile likelihood for $\Delta m_{s}$ from a fit where $\Delta
m_{s}$ is unconstrained.
## 10 Systematic uncertainties for $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$
decays
The parameters $\Delta m_{s}$, the tagging calibration parameters, and the
event-by-event proper time scaling factor, $r_{t}$, are all allowed to vary
within their uncertainties in the fit. Therefore the systematic uncertainties
from these sources are included in the statistical uncertainty on the physics
parameters. The remaining systematic effects are discussed below and
summarised in Tables 9, 10 and 11.
The parameters of the $m(J\\!/\\!\psi K^{+}K^{-})$ fit model are varied within
their uncertainties and a new set of event weights are calculated. Repeating
the full decay time and angular fit using the new weights gives negligible
differences with respect to the results of the nominal fit. The assumption
that $m(J\\!/\\!\psi K^{+}K^{-})$ is independent of the decay time and angle
variables is tested by re-evaluating the weights in bins of the decay time and
angles. Repeating the full fit with the modified weights gives new estimates
of the physics parameter values in each bin. The total systematic uncertainty
is computed from the square root of the sum of the individual variances,
weighted by the number of signal events in each bin in cases where a
significant difference is observed.
Using simulated events, the only identified peaking background is from
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*}(892)^{0}$
events where the pion from the $K^{*}(892)^{0}$ decay is misidentified as a
kaon. The fraction of this contribution was estimated from the simulation to
be at most 1.5% for $m(J\\!/\\!\psi K^{+}K^{-})$ in the range
$[5200,5550]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The effect of this
background (which is not included in the PDF modelling) was estimated by
embedding the simulated $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$ events in the signal sample and repeating the fit. The
resulting variations are taken as systematic uncertainties. The contribution
of $B^{0}_{s}$ mesons coming from the decay of $B_{c}^{+}$ mesons is estimated
to be negligible.
Since the angular acceptance function, $\varepsilon_{\Omega}$, is determined
from simulated events, it is important that the simulation gives a good
description of the dependence of final-state particle efficiencies on their
kinematic properties. Figure 16 shows significant discrepancies between
simulated $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ events and selected
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
data events where the background has been subtracted. To account for this
difference the simulated events are re-weighted such that the kaon momentum
distribution matches the data (re-weighting the muon momentum has negligible
effect). A systematic uncertainty is estimated by determining
$\varepsilon_{\Omega}$ after this re-weighting and repeating the fit. The
changes observed in physics parameters are taken as systematic uncertainties.
A systematic uncertainty is included which arises from the limited size of the
simulated data sample used to determine $\varepsilon_{\Omega}$.
\begin{overpic}[width=186.65173pt]{final_figs/pk_paper.pdf}
\put(64.0,41.0){(a)}
\end{overpic}\begin{overpic}[width=186.65173pt]{final_figs/pmu_paper.pdf}
\put(64.0,41.0){(b)} \end{overpic}
Figure 16: Background-subtracted (a) kaon and (b) muon momentum distributions
for $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ signal events in data compared to simulated
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ signal
events. The distributions are normalised to the same area. A larger deviation
is visible for kaons.
The lower decay time acceptance is included in the PDF using the binned
functions described in Sect. 6. A systematic uncertainty is determined by
repeating the fits with the bin values varied randomly within their
statistical precision. The standard deviation of the distribution of central
values obtained for each fit parameter is then assigned as the systematic
uncertainty. The slope of the acceptance correction at large lifetimes is
$\beta=(-8.3\pm 4.0)\times 10^{-3}{\rm\,ps^{-1}}$. This leads to a $4.0\times
10^{-3}{\rm\,ps^{-1}}$ systematic uncertainty on $\Gamma_{s}$.
The uncertainty on the LHCb length scale is estimated to be at most 0.020%,
which translates directly in an uncertainty on $\Gamma_{s}$ and
$\Delta\Gamma_{s}$ of 0.020% with other parameters being unaffected. The
momentum scale uncertainty is at most 0.022%. As it affects both the
reconstructed momentum and mass of the $B^{0}_{s}$ meson, it cancels to a
large extent and the resulting effect on $\Gamma_{s}$ and $\Delta\Gamma_{s}$
is negligible.
The $C_{\rm SP}$ factors (Table 4) used in the nominal fit assume a non-
resonant shape for the S-wave contribution. As a cross-check the factors are
re-evaluated assuming a Flatté shape [49] and the fit is repeated. There is a
negligible effect on all physics parameters except $\delta_{\rm
S}-\delta_{\perp}$. A small shift (approximately 10% of the statistical
uncertainty) is observed in $\delta_{\rm S}-\delta_{\perp}$ in each bin of
$m(K^{+}K^{-})$, and is assigned as a systematic uncertainty.
A possible bias of the fitting procedure is investigated by generating and
fitting many simplified simulated experiments of equivalent size to the data
sample. The resulting biases are small, and those which are not compatible
with zero within three standard deviations are quoted as systematic
uncertainties.
The small offset, $d$, in the decay time resolution model was set to zero
during the fitting procedure. A corresponding systematic uncertainty was
evaluated using simulated experiments and found to be negligible for all
parameters apart from $\phi_{s}$ and $\delta_{\perp}$.
A measurement of the asymmetry that results from $C\\!P$ violation in the
interference between $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing and decay is
potentially affected by $C\\!P$ violation in the mixing, direct $C\\!P$
violation in the decay, production asymmetry and tagging asymmetry. In the
previous analysis [6] an explicit systematic uncertainty was included to
account for this. In this analysis the fit parameter $|\lambda|$ is added,
separate tagging calibrations are used for $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decisions, as well as separate
normalisations of the PDF for each tagging decision. Any residual effects due
to tagging efficiency asymmetry and production asymmetry are shown to be
negligible through simulation studies.
The measurement of $\Delta m_{s}$ determined from these data alone without
applying a constraint has been reported in Sect. 9. The dominant sources of
systematic uncertainty come from the knowledge of the LHCb length and momentum
scales. No significant systematic effect is observed after varying the decay
time and angular acceptances and the decay time resolution. Adding all
contributions in quadrature gives a total systematic uncertainty of $\pm
0.01{\rm\,ps^{-1}}$.
Table 9: Statistical and systematic uncertainties. Source | $\Gamma_{s}$ | $\Delta\Gamma_{s}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\parallel}$ | $\delta_{\perp}$ | $\phi_{s}$ | $|\lambda|$
---|---|---|---|---|---|---|---|---
| [ps-1] | [ps-1] | | | [rad] | [rad] | [rad] |
Stat. uncertainty | 0.0048 | 0.016 | 0.0086 | 0.0061 | ${}^{+0.13}_{-0.21}$ | 0.22 | 0.091 | 0.031
Background subtraction | 0.0041 | 0.002 | – | 0.0031 | 0.03 | 0.02 | 0.003 | 0.003
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ background | – | 0.001 | 0.0030 | 0.0001 | 0.01 | 0.02 | 0.004 | 0.005
Ang. acc. reweighting | 0.0007 | – | 0.0052 | 0.0091 | 0.07 | 0.05 | 0.003 | 0.020
Ang. acc. statistical | 0.0002 | – | 0.0020 | 0.0010 | 0.03 | 0.04 | 0.007 | 0.006
Lower decay time acc. model | 0.0023 | 0.002 | – | – | – | – | – | –
Upper decay time acc. model | 0.0040 | – | – | – | – | – | – | –
Length and mom. scales | 0.0002 | – | – | – | – | – | – | –
Fit bias | – | – | 0.0010 | – | – | – | – | –
Decay time resolution offset | – | – | – | – | – | 0.04 | 0.006 | –
Quadratic sum of syst. | 0.0063 | 0.003 | 0.0064 | 0.0097 | 0.08 | 0.08 | 0.011 | 0.022
Total uncertainties | 0.0079 | 0.016 | 0.0107 | 0.0114 | ${}^{+0.15}_{-0.23}$ | 0.23 | 0.092 | 0.038
Table 10: Statistical and systematic uncertainties for S-wave fractions in bins of $m(K^{+}K^{-})$. Source | bin 1 | bin 2 | bin 3 | bin 4 | bin 5 | bin 6
---|---|---|---|---|---|---
| $F_{\rm S}$ | $F_{\rm S}$ | $F_{\rm S}$ | $F_{\rm S}$ | $F_{\rm S}$ | $F_{\rm S}$
Stat. uncertainty | ${}^{+0.081}_{-0.073}$ | ${}^{+0.030}_{-0.027}$ | ${}^{+0.014}_{-0.007}$ | ${}^{+0.012}_{-0.009}$ | ${}^{+0.027}_{-0.025}$ | ${}^{+0.043}_{-0.042}$
Background subtraction | 0.014 | 0.003 | 0.001 | 0.002 | 0.004 | 0.006
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ background | 0.010 | 0.006 | 0.001 | 0.001 | 0.002 | 0.018
Angular acc. reweighting | 0.004 | 0.006 | 0.004 | 0.005 | 0.006 | 0.007
Angular acc. statistical | 0.003 | 0.003 | 0.002 | 0.001 | 0.003 | 0.004
Fit bias | 0.009 | – | 0.002 | 0.002 | 0.001 | 0.001
Quadratic sum of syst. | 0.020 | 0.009 | 0.005 | 0.006 | 0.008 | 0.021
Total uncertainties | ${}^{+0.083}_{-0.076}$ | ${}^{+0.031}_{-0.029}$ | ${}^{+0.015}_{-0.009}$ | ${}^{+0.013}_{-0.011}$ | ${}^{+0.028}_{-0.026}$ | ${}^{+0.048}_{-0.047}$
Table 11: Statistical and systematic uncertainties for S-wave phases in bins of $m(K^{+}K^{-})$. Source | bin 1 | bin 2 | bin 3 | bin 4 | bin 5 | bin 6
---|---|---|---|---|---|---
| $\delta_{\rm S}-\delta_{\perp}$ | $\delta_{\rm S}-\delta_{\perp}$ | $\delta_{\rm S}-\delta_{\perp}$ | $\delta_{\rm S}-\delta_{\perp}$ | $\delta_{\rm S}-\delta_{\perp}$ | $\delta_{\rm S}-\delta_{\perp}$
| [rad] | [rad] | [rad] | [rad] | [rad] | [rad]
Stat. uncertainty | ${}^{+0.78}_{-0.49}$ | ${}^{+0.38}_{-0.23}$ | ${}^{+1.40}_{-0.30}$ | ${}^{+0.21}_{-0.35}$ | ${}^{+0.18}_{-0.26}$ | ${}^{+0.18}_{-0.22}$
Background subtraction | 0.03 | 0.02 | – | 0.03 | 0.01 | 0.01
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ background | 0.08 | 0.04 | 0.08 | 0.01 | 0.01 | 0.05
Angular acc. reweighting | 0.02 | 0.03 | 0.12 | 0.13 | 0.03 | 0.01
Angular acc. statistical | 0.033 | 0.023 | 0.067 | 0.036 | 0.019 | 0.015
Fit bias | 0.005 | 0.043 | 0.112 | 0.049 | 0.022 | 0.016
$C_{SP}$ factors | 0.007 | 0.028 | 0.049 | 0.025 | 0.021 | 0.020
Quadratic sum of syst. | 0.09 | 0.08 | 0.20 | 0.15 | 0.05 | 0.06
Total uncertainties | ${}^{+0.79}_{-0.50}$ | ${}^{+0.39}_{-0.24}$ | ${}^{+1.41}_{-0.36}$ | ${}^{+0.26}_{-0.38}$ | ${}^{+0.19}_{-0.26}$ | ${}^{+0.19}_{-0.23}$
## 11 Results for $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays
The $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ analysis used in this paper is
unchanged with respect to Ref. [7] except for:
1. 1.
the inclusion of the same-side kaon tagger in the same manner as has already
been described for the $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ sample. This
increases the number of tagged signal candidates to 2146 OS-only, 497 SSK-only
and 293 overlapped events compared to 2445 in Ref. [7]. The overall tagging
efficiency is $(39.5\pm 0.7)\%$ and the tagging power increases from $(2.43\pm
0.08\pm 0.26)\%$ to $(3.37\pm 0.12\pm 0.27)\%$;
2. 2.
an updated decay time acceptance model. For this, the decay channel
$B^{0}\rightarrow J\\!/\\!\psi K^{*}(892)^{0}$, which has a well known
lifetime, is used to calibrate the decay time acceptance, and simulated events
are used to determine a small relative correction between the acceptances for
the $B^{0}\rightarrow J/\psi K^{*}(892)^{0}$ and $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decays;
3. 3.
use of the updated values of $\Gamma_{s}$ and $\Delta\Gamma_{s}$ from the
$B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ analysis presented in this paper as
constraints in the fit for $\phi_{s}$.
The measurement of $\phi_{s}$ using only the $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ events is
$\phi_{s}=-0.14\,^{+0.17}_{-0.16}\pm 0.01\rm\,rad,$
where the systematic uncertainty is obtained in the same way as described in
Ref. [7]. The decay time resolution in this channel is approximately $40$ fs
and its effect is included in the systematic uncertainty.
In addition, the effective lifetime $\tau^{\rm
eff}_{B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}$ is measured by fitting a single exponential function to
the $B^{0}_{s}$ decay time distribution with no external constraints on
$\Gamma_{s}$ and $\Delta\Gamma_{s}$ applied. The result is
$\tau^{\rm eff}_{B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}=1.652\pm 0.024\ (\mathrm{stat})\pm 0.024\
(\mathrm{syst}){\rm\,ps}.$
This is equivalent to a decay width of
$\Gamma^{\rm eff}_{B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}=0.605\pm 0.009\ (\mathrm{stat})\pm 0.009\
(\mathrm{syst}){\rm\,ps^{-1}},$
which, in the limit $\phi_{s}=0$ and $|\lambda|=1$, corresponds to
$\Gamma_{\rm H}$. This result supersedes that reported in Ref. [50]. The
uncertainty on the $B^{0}$ lifetime [8] used to calibrate the decay time
acceptance is included in the statistical uncertainty. The remaining
systematic uncertainty is evaluated by changing the background model and
assigning half of the relative change between the fit results with and without
the decay time acceptance correction included, leading to uncertainties of
$0.011{\rm\,ps}$ and $0.021{\rm\,ps}$, respectively. The total systematic
uncertainty obtained by adding the two contributions in quadrature is
$0.024{\rm\,ps}$.
## 12 Combined results for $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and
$B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ datasets
This section presents the results from a simultaneous fit to both
$B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ datasets. The joint log-likelihood is minimised with the
common parameters being $\Gamma_{s}$, $\Delta\Gamma_{s}$, $\phi_{s}$,
$|\lambda|$, $\Delta m_{s}$ and the tagging calibration parameters. The
combined results are given in Table 12. The correlation matrix for the
principal parameters is given in Table 13.
For all parameters, except $\Gamma_{s}$ and $\Delta\Gamma_{s}$, the same
systematic uncertainties as presented for the stand-alone
$B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ analysis are assigned. For
$\Gamma_{s}$ and $\Delta\Gamma_{s}$ additional systematic uncertainties of
$0.001{\rm\,ps^{-1}}$ and $0.006{\rm\,ps^{-1}}$ respectively are included, due
to the $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ background model and decay
time acceptance variations described above.
Table 12: Results of combined fit to the $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ datasets. The first uncertainty is statistical and the second is systematic. Parameter | Value
---|---
$\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | $0.661\pm 0.004\pm 0.006$
$\Delta\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | $0.106\pm 0.011\pm 0.007$
$|A_{\perp}|^{2}$ | $0.246\pm 0.007\pm 0.006$
$|A_{0}|^{2}$ | $0.523\pm 0.005\pm 0.010$
$\delta_{\parallel}$ [rad] | $3.32\,^{+0.13}_{-0.21}\pm 0.08$
$\delta_{\perp}$ [rad] | $3.04\pm 0.20\pm 0.08$
$\phi_{s}$ [rad] | $0.01\pm 0.07\pm 0.01$
$|\lambda|$ | $0.93\pm 0.03\pm 0.02$
Table 13: Correlation matrix for statistical uncertainties on combined results. | $\Gamma_{s}$ | $\Delta\Gamma_{s}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\parallel}$ | $\delta_{\perp}$ | $\phi_{s}$ | $|\lambda|$
---|---|---|---|---|---|---|---|---
| [${\rm\,ps^{-1}}$ ] | [${\rm\,ps^{-1}}$ ] | | | [rad] | [rad] | [rad] |
$\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | $\phantom{+}1.00$ | $\phantom{+}0.10$ | $\phantom{+}0.08$ | $\phantom{+}0.03$ | $-0.08$ | $-0.04$ | $\phantom{+}0.01$ | $\phantom{+}0.00$
$\Delta\Gamma_{s}$ [${\rm\,ps^{-1}}$ ] | | $\phantom{+}1.00$ | ${-0.49}$ | $\phantom{+}{0.47}$ | $\phantom{+}0.00$ | $\phantom{+}0.00$ | $\phantom{+}0.00$ | $-0.01$
$|A_{\perp}|^{2}$ | | | $\phantom{+}1.00$ | ${-0.40}$ | ${-0.37}$ | $-0.14$ | $\phantom{+}0.02$ | $-0.05$
$|A_{0}|^{2}$ | | | | $\phantom{+}1.00$ | $-0.05$ | $-0.03$ | $-0.01$ | $\phantom{+}0.01$
$\delta_{\parallel}$ [rad] | | | | | $\phantom{+}1.00$ | $\phantom{+}{0.39}$ | $-0.01$ | $\phantom{+}0.13$
$\delta_{\perp}$ [rad] | | | | | | $\phantom{+}1.00$ | $\phantom{+}0.21$ | $\phantom{+}0.03$
$\phi_{s}$ [rad] | | | | | | | $\phantom{+}1.00$ | $\phantom{+}0.06$
$|\lambda|$ | | | | | | | | $\phantom{+}1.00$
## 13 Conclusion
A sample of $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$,
corresponding to an integrated luminosity of $1.0$$\mbox{\,fb}^{-1}$,
collected with the LHCb detector is used to select $27\,617\pm 115$
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ events in a $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around the $\phi(1020)$
meson mass [8]. The effective tagging efficiency from the opposite-side (same-
side kaon) tagger is ${\varepsilon_{\rm eff}=2.29\pm 0.22}$% ($0.89\pm
0.18$%). A combination of data and simulation based techniques are used to
correct for detector efficiencies. These data have been analysed in six bins
of $m(K^{+}K^{-})$, allowing the resolution of two symmetric solutions,
leading to the single most precise measurements of $\phi_{s}$, $\Gamma_{s}$
and $\Delta\Gamma_{s}$
$\begin{array}[]{ccllllllll}\phi_{s}&\;=&0.07&\pm&0.09&\text{(stat)}&\pm&0.01&\text{(syst)}&\text{rad},\\\
\Gamma_{s}&\;=&0.663&\pm&0.005&\text{(stat)}&\pm&0.006&\text{(syst)}&{\rm\,ps^{-1}},\rule{0.0pt}{14.22636pt}\\\
\Delta\Gamma_{s}&\;=&0.100&\pm&0.016&\text{(stat)}&\pm&0.003&\text{(syst)}&{\rm\,ps^{-1}}.\rule{0.0pt}{14.22636pt}\\\
\end{array}$
The $B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ events also allow an
independent determination of ${\Delta m_{s}=17.70\pm 0.10\pm
0.01{\rm\,ps^{-1}}}$.
The time-dependent $C\\!P$-asymmetry measurement using $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ events from Ref. [7] is updated to include same-side
kaon tagger information. The result of performing a combined fit using both
$B^{0}_{s}\rightarrow J\\!/\\!\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ events gives
$\begin{array}[]{ccllllllll}\phi_{s}&\;=&0.01&\pm&0.07&\text{(stat)}&\pm&0.01&\text{(syst)}&\text{rad},\\\
\Gamma_{s}&\;=&0.661&\pm&0.004&\text{(stat)}&\pm&0.006&\text{(syst)}&{\rm\,ps^{-1}},\rule{0.0pt}{14.22636pt}\\\
\Delta\Gamma_{s}&\;=&0.106&\pm&0.011&\text{(stat)}&\pm&0.007&\text{(syst)}&{\rm\,ps^{-1}}.\rule{0.0pt}{14.22636pt}\\\
\end{array}$
The measurements of $\phi_{s}$, $\Delta\Gamma_{s}$ and $\Gamma_{s}$ are the
most precise to date and are in agreement with SM predictions [3, 43,
*Badin:2007bv, *Lenz:2011ti]. All measurements using $B^{0}_{s}\rightarrow
J\\!/\\!\psi K^{+}K^{-}$ decays supersede our previous measurements reported
in Ref. [6], and all measurements using $B^{0}_{s}\rightarrow
J\\!/\\!\psi\pi^{+}\pi^{-}$ decays supersede our previous measurements
reported in Ref. [7]. The $B^{0}_{s}\rightarrow J\\!/\\!\psi\pi^{+}\pi^{-}$
effective lifetime measurement supersedes that reported in Ref. [50]. The
combined results reported in Ref. [7] are superseded by those reported here.
Since the combined results for $\Gamma_{s}$ and $\Delta\Gamma_{s}$ include all
lifetime information from both channels they should not be used in conjunction
with the ${B^{0}_{s}\rightarrow J\\!/\\!\psi\pi^{+}\pi^{-}}$ effective
lifetime measurement.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
## Appendix A Definition of helicity decay angles
The helicity angles can be defined in terms of the momenta of the decay
particles. The momentum of particle $a$ in the centre-of-mass system of $S$ is
denoted by $\vec{p}_{a}^{\;S}$. With this convention, unit vectors are defined
along the helicity axis in the three centre-of-mass systems and the two unit
normal vectors of the $K^{+}K^{-}$ and $\mu^{+}\mu^{-}$ decay planes as
$\begin{gathered}\hat{e}_{z}^{\,KK\mu\mu}=+\frac{\vec{p}_{\mu^{+}}^{\;KK\mu\mu}+\vec{p}_{\mu^{-}}^{\;KK\mu\mu}}{|\vec{p}_{\mu^{+}}^{\;KK\mu\mu}+\vec{p}_{\mu^{-}}^{\;KK\mu\mu}|},\qquad\hat{e}_{z}^{\,KK}=-\frac{\vec{p}_{\mu^{+}}^{\;KK}+\vec{p}_{\mu^{-}}^{\;KK}}{|\vec{p}_{\mu^{+}}^{\;KK}+\vec{p}_{\mu^{-}}^{\;KK}|},\qquad\hat{e}_{z}^{\,\mu\mu}=-\frac{\vec{p}_{K^{+}}^{\;\mu\mu}+\vec{p}_{K^{-}}^{\;\mu\mu}}{|\vec{p}_{K^{+}}^{\;\mu\mu}+\vec{p}_{K^{-}}^{\;\mu\mu}|},\\\
\hat{n}_{KK}=\frac{\vec{p}_{K^{+}}^{\;KK\mu\mu}\times\vec{p}_{K^{-}}^{\;KK\mu\mu}}{|\vec{p}_{K^{+}}^{\;KK\mu\mu}\times\vec{p}_{K^{-}}^{\;KK\mu\mu}|},\qquad\qquad\hat{n}_{\mu\mu}=\frac{\vec{p}_{\mu^{+}}^{\;KK\mu\mu}\times\vec{p}_{\mu^{-}}^{\;KK\mu\mu}}{|\vec{p}_{\mu^{+}}^{\;KK\mu\mu}\times\vec{p}_{\mu^{-}}^{\;KK\mu\mu}|}.\end{gathered}$
(16)
The helicity angles are defined in terms of these vectors as
$\displaystyle\cos\theta_{K}$
$\displaystyle=\frac{\vec{p}_{K^{+}}^{\;KK}}{|\vec{p}_{K^{+}}^{\;KK}|}\cdot\hat{e}_{z}^{\,KK},$
$\displaystyle\qquad\quad\cos\theta_{\mu}$
$\displaystyle=\frac{\vec{p}_{\mu^{+}}^{\;\mu\mu}}{|\vec{p}_{\mu^{+}}^{\;\mu\mu}|}\cdot\hat{e}_{z}^{\,\mu\mu},$
(17) $\displaystyle\cos\varphi_{h}$
$\displaystyle=\hat{n}_{KK}\cdot\hat{n}_{\mu\mu},$
$\displaystyle\sin\varphi_{h}$
$\displaystyle=\left(\hat{n}_{KK}\times\hat{n}_{\mu\mu}\right)\cdot\hat{e}_{z}^{\,KK\mu\mu}.$
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|
arxiv-papers
| 2013-04-09T14:08:07 |
2024-09-04T02:49:44.039041
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson,\n K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C.\n Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy,\n F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost,\n M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R. Kenyon, U. Kerzel,\n T. Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, B. Leverington, Y. Li, L. Li Gioi, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, S. Lohn, J.H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H.\n Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A.\n Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos,\n D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E.\n Maurice, A. Mazurov, J. McCarthy, R. McNulty, A. Mcnab, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, M.J. Morello, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, R. 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Williams, F.F. Wilson,\n J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Greig Cowan Dr",
"url": "https://arxiv.org/abs/1304.2600"
}
|
1304.2669
|
# On Normal forms for Levi-flat hypersurfaces with an isolated line
singularity
Arturo Fernández Pérez Departamento de Matemática, Universidade Federal de
Minas Gerais, UFMG Av. Antônio Carlos, 6627 C.P. 702, 30123-970 - Belo
Horizonte - MG, Brazil. [email protected]
###### Abstract.
We prove the existence of normal forms for some local real-analytic Levi-flat
hypersurfaces with an isolated line singularity. We also give sufficient
conditions for that a Levi-flat hypersurface with a complex line as
singularity to be a pullback of a real-analytic curve in $\mathbb{C}$ via a
holomorphic function.
###### Key words and phrases:
Levi-flat hypersurfaces - Holomorphic foliations
###### 2010 Mathematics Subject Classification:
Primary 32V40 - 32S65
## 1\. Introduction
Let $M\subset U\subset\mathbb{C}^{n}$ be a real-analytic hypersurface, where
$U$ is an open set and denote by $M^{*}$ the regular part, that is, near each
point $p\in M^{*}$, the variety $M$ is a manifold of real codimension one. For
each $p\in M^{*}$, there is a unique complex hyperplane $L_{p}$ contained in
the tangent space $T_{p}M^{*}$, and consequently defines a real-analytic
distribution $p\mapsto L_{p}$ of complex hyperplanes in $T_{p}M^{*}$, the so-
called Levi distribution. We say that $M$ is Levi-flat, if the Levi
distribution is integrable in sense of Frobenius. The foliation defined by
this distribution is called Levi-foliation. The local structure near regular
points is very well understood, according to E. Cartan, around each $p\in
M^{*}$ we can find local holomorphic coordinates $z_{1},\ldots,z_{n}$ such
that $M^{*}=\\{\mathcal{R}e(z_{n})=0\\}$, and consequently the leaves of Levi-
foliation are imaginary levels of $z_{n}$. The singular case was studied by
Burns-Gong [2], The authors classified singular Levi-flat hypersurfaces in
$\mathbb{C}^{n}$ with quadratic singularities and also proved the existence of
a normal form, in the case of generic (Morse) singularities. In [4], Cerveau-
Lins Neto have proved that a local real-analytic Levi-flat hypersurface $M$
with a sufficiently small singular set is given by the zeros of the real part
of a holomorphic function.
The aim of this paper is to prove the existence of some normal forms for local
real-analytic Levi-flat hypersurfaces defined by the vanishing of real part of
holomorphic functions with an isolated line singularity (for short: ILS). In
particular, we establish an analogous result like in Singularity Theory for
germs of holomorphic functions.
The main motivation for this work is a result due to Dirk Siersma, who
introduced in [14] the class of germs of holomorphic functions with an ILS.
More precisely, let
$\mathcal{O}_{n+1}:=\\{f:(\mathbb{C}^{n+1},0)\rightarrow\mathbb{C}\\}$ be the
ring of germs of holomorphic functions and let $m$ be its maximal ideal. If
$(x,y)=(x,y_{1},\ldots,y_{n})$ denote the coordinates in $\mathbb{C}^{n+1}$
and consider the line $L:=\\{y_{1}=\ldots=y_{n}=0\\}$, let
$I:=(y_{1},\ldots,y_{n})\subset\mathcal{O}_{n+1}$ be its ideal and denote by
$\mathcal{D}_{I}$ the group of local analytic isomorphisms
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ for which
$\varphi(L)=L$. Then $\mathcal{D}_{I}$ acts on $I^{2}$ and for $f\in I^{2}$,
the tangent space of (the orbit of) $f$ with respect to this action is the
ideal defined by
$\tau(f):=m.\frac{\partial{f}}{\partial{x}}+I.\frac{\partial{f}}{\partial{y}}$
and the codimension of (the orbit) of $f$ is
$c(f):=\dim_{\mathbb{C}}\frac{I^{2}}{\tau(f)}.$
A line singularity is a germ $f\in I^{2}$. An ILS is a line singularity $f$
such that $c(f)<\infty$. Geometrically, $f\in I^{2}$ is an ILS if and only if
the singular locus of $f$ is $L$ and for every $x\neq 0$, the germ of (a
representative of) $f$ at $(x,0)\in L$ is equivalent to
$y^{2}_{1}+\ldots+y^{2}_{n}$. In a certain sense ILS are the first
generalization of isolated singularities. D. Siersma proved the following
result. (The topology on $\mathcal{O}_{n+1}$ is introduced as in [5, p. 145]).
###### Theorem 1.1.
A germ $f\in I^{2}$ is $D_{I}$-simple (i.e. $c(f)<\infty$ and $f$ has a
neighborhood in $I^{2}$ which intersects only a finite number of
$D_{I}$-orbits) if and only if $f$ is $D_{I}$-equivalent to one the germs in
the following table
Type | Normal form | Conditions
---|---|---
$A_{\infty}$ | $y_{1}^{2}+y_{2}^{2}+\ldots+y^{2}_{n}$ |
$D_{\infty}$ | $xy^{2}_{1}+y^{2}_{2}+\ldots+y^{2}_{n}$ |
$J_{k,\infty}$ | $x^{k}y^{2}_{1}+y_{1}^{3}+y^{2}_{2}+\ldots+y_{n}^{2}$ | $k\geq 2$
$T_{\infty,k,2}$ | $x^{2}y^{2}_{1}+y_{1}^{k}+y_{2}^{2}+\ldots+y_{n}^{2}$ | $k\geq 4$
$Z_{k,\infty}$ | $xy_{1}^{3}+x^{k+2}y_{1}^{2}+y_{2}^{2}+\ldots+y_{n}^{2}$ | $k\geq 1$
$W_{1,\infty}$ | $x^{3}y_{1}^{2}+y_{1}^{4}+y_{2}^{2}+\ldots+y_{n}^{2}$ |
$T_{\infty,q,r}$ | $xy_{1}y_{2}+y_{1}^{q}+y_{2}^{r}+y^{2}_{3}\ldots+y_{n}^{2}$ | $q\geq r\geq 3$
$Q_{k,\infty}$ | $x^{k}y_{1}^{2}+y_{1}^{3}+xy_{2}^{2}+y^{2}_{3}\ldots+y_{n}^{2}$ | $k\geq 2$
$S_{1,\infty}$ | $x^{2}y_{1}^{2}+y_{1}^{2}y_{2}+y_{3}^{2}+\ldots+y_{n}^{2}$ |
Table 1. Isolated Line singularities
The singularities in Theorem 1.1 are analogous of the $A$-$D$-$E$
singularities due to Arnold [1]. A new characterization of simple ILS have
been proved by A. Zaharia [15]. We prove the existence of normal forms for
Levi-flat hypersurfaces with an ILS.
###### Theorem 1.
Let $M=\\{F=0\\}$ be a germ of an irreducible real-analytic hypersurface on
$(\mathbb{C}^{n+1},0)$, $n\geq 3$. Suppose that
1. (1)
$F(x,y)=\mathcal{R}e(P(x,y))+H(x,y),$ where $P(x,y)$ is one of the germs of
the Table 1.
2. (2)
$M=\\{F=0\\}$ is Levi-flat.
3. (3)
$H(x,0)=0$ for all $x\in(\mathbb{C},0)$ and $j_{0}^{k}(H)=0$, for $k=\deg(P)$.
Then there exists a biholomorphism
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ preserving $L$
such that
$\varphi(M)=\\{\mathcal{R}e(P(x,y))=0\\}.$
This result is a Siersma’s type Theorem for singular Levi-flat hypersurfaces.
We remark that the function $H$ is of course restricted by the assumption that
$M$ is Levi flat. Now, if $\varphi(M)=\\{\mathcal{R}e(P(x,y))=0\\}$, where $P$
is a germ with an ILS at $L$ then $\textsf{Sing}(M)=L$. In other words, $M$ is
a Levi-flat hypersurface with an ILS at $L$. If $P(x,y)$ is the germ
$A_{\infty}$, we prove that Theorem 1 is true in the case $n=2$.
###### Theorem 2.
Let $M=\\{F=0\\}$ be a germ of an irreducible real-analytic Levi-flat
hypersurface on $(\mathbb{C}^{3},0)$. Suppose that $F$ is defined by
$F(x,y)=\mathcal{R}e(y_{1}^{2}+y_{2}^{2})+H(x,y),$
where $H$ is a germ of real-analytic function such that $H(x,0)=0$ and
$j_{0}^{k}(H)=0$ for $k=2$. Then there exists a biholomorphism
$\varphi:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{3},0)$ preserving $L$ such
that $\varphi(M)=\\{\mathcal{R}e(y_{1}^{2}+y_{2}^{2})=0\\}$.
The above result should be compared to [2, Theorem 1.1]. This result can be
viewed as a Morse’s Lemma for Levi-flat hypersurfaces with an ILS at $L$. The
problem of normal forms of Levi-flat hypersurfaces in $\mathbb{C}^{3}$ with an
ILS seems difficult in the other cases. To prove these results we use
techniques of holomorphic foliations developed in [4] and [6]. Another normal
forms of singular Levi-flat hypersurfaces have been obtained in [2], [7] and
[9].
This paper is organized as follows: In Section 2, we recall some definitions
and known results about Levi-flat and holomorphic foliations. Section 3 is
devoted to prove Theorem 1. In Section 4, we prove Theorem 2. Finally, in
Section 5, using holomorphic foliations, we give sufficient conditions for
that a Levi-flat hypersurface with a complex line as singularity to be a
pullback of a real-analytic curve in $\mathbb{C}$ via a holomorphic function,
(see Theorem 5.7).
## 2\. Levi-flat hypersurfaces and Foliations
In this section we works with germs at $0\in\mathbb{C}^{n+1}$ of irreducible
real-analytic hypersurfaces and of codimension one holomorphic foliations. Let
$M=\\{F=0\\}$, where $F:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{R},0)$ is a
germ of an irreducible real-analytic function, and
$M^{*}:=\\{F=0\\}\backslash\\{dF=0\\}$. Let us define the singular set of $M$
(or “set of critical points” of $M$) by
$\textsf{Sing}(M):=\\{F=0\\}\cap\\{dF=0\\}.$ (2.1)
Note that $\textsf{Sing}(M)$ contains all points $q\in M$ such that $M$ is
smooth at $q$, but the codimension of $M$ at $q$ is at least two. In general
the singular set of a real-analytic subvariety $M$ in a complex manifold is
defined as the set of points near which $M$ is not a real-analytic submanifold
(of any dimension) and “in general” has structure of a semianalytic set; see
for instance, [11]. In this paper, we work with $\textsf{Sing}(M)$ as defined
in (2.1). We recall that (in this case) the Levi distribution $L$ on $M^{*}$
is defined by
$\displaystyle L_{p}:=ker(\partial{F}(p))\subset
T_{p}M^{*}=ker(dF(p)),\,\,\,\,\text{for any}\,\,p\in M^{*}.$ (2.2)
Let us suppose that $M$ is Levi-flat, this implies that $M^{*}$ is foliated by
complex codimension one holomorphic submanifolds immersed on $M^{*}$.
Note that the Levi distribution $L$ on $M^{*}$ can be defined by the real-
analytic 1-form $\eta=i(\partial{F}-\bar{\partial}F)$, which is called the
Levi 1-form of $F$. It is well known that the integrability condition of $L$
is equivalent to equation
$(\partial{F}-\bar{\partial}F)\wedge\partial\bar{\partial}F|_{M^{*}}=0.$
Let us consider the series Taylor of $F$ at $0\in\mathbb{C}^{n+1}$,
$F(x,y)=\sum_{i,\mu,j,\nu}F_{i\mu j\nu}x^{i}y^{\mu}\bar{x}^{j}\bar{y}^{\nu}$
where $\bar{F}_{i\mu j\nu}=F_{j\nu i\mu}$; $i,j\in\mathbb{N}$,
$\mu=(\mu_{1},\ldots,\mu_{n})$, $\nu=(\nu_{1},\ldots,\nu_{n})$,
$(x,y)\in\mathbb{C}\times\mathbb{C}^{n}$, $y^{\mu}=y_{1}^{\mu_{1}}\ldots
y_{n}^{\mu_{n}}$ and
$\bar{y}^{\nu}=\bar{y}_{1}^{\nu_{1}}\ldots\bar{y}_{n}^{\nu_{n}}$. The
complexification $F_{\mathbb{C}}\in\mathcal{O}_{2n+2}$ of $F$ is defined by
the serie
$F_{\mathbb{C}}(x,y,z,w)=\sum_{i,\mu,j,\nu}F_{i\mu
j\nu}x^{i}y^{\mu}z^{j}w^{\nu},$
where $z\in\mathbb{C}$, $w=(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}$ and
$w^{\nu}=w_{1}^{\nu_{1}}\ldots w_{n}^{\nu_{n}}$. Notice that
$F(x,y)=F_{\mathbb{C}}(x,y,\bar{x},\bar{y})$. The complexification
$M_{\mathbb{C}}$ of $M$ is defined as $M_{\mathbb{C}}:=\\{F_{\mathbb{C}}=0\\}$
and defines a complex subvariety in $\mathbb{C}^{2n+2}$, its regular part is
$M^{*}_{\mathbb{C}}:=M_{\mathbb{C}}\backslash\\{dF_{\mathbb{C}}=0\\}$. Now,
assume that $M$ is Levi-flat. Then the integrability condition of
$\eta=i(\partial{F}-\bar{\partial}F)|_{M^{*}}$
implies that $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ is integrable, where
$\eta_{\mathbb{C}}:=i[(\partial_{x}F_{\mathbb{C}}+\partial_{y}F_{\mathbb{C}})-(\partial_{z}F_{\mathbb{C}}+\partial_{w}F_{\mathbb{C}})].$
Therefore $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ defines a codimension one
holomorphic foliation $\mathcal{L}_{\mathbb{C}}$ on $M^{*}_{\mathbb{C}}$ that
will be called the complexification of $\mathcal{L}$.
Let
$W:=M_{\mathbb{C}}^{*}\backslash\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$
and denote by $L_{\zeta}$ the leaf of $\mathcal{L}_{\mathbb{C}}$ through
$\zeta$, where $\zeta\in W$. The next results will be used several times along
of the paper.
###### Lemma 2.1 (Cerveau-Lins Neto [4]).
For any $\zeta\in W$, the leaf $L_{\zeta}$ of $\mathcal{L}_{\mathbb{C}}$
through $\zeta$ is closed in $M_{\mathbb{C}}^{*}$.
###### Definition 2.2.
The algebraic dimension of $\textsf{Sing}(M)$ is the complex dimension of the
singular set of $M_{\mathbb{C}}$.
The following result will be used enunciated in the context of Levi-flat
hypersurfaces in $\mathbb{C}^{n+1}$.
###### Theorem 2.3 (Cerveau-Lins Neto [4]).
Let $M=\\{F=0\\}$ be a germ of an irreducible analytic Levi-flat hypersurface
at $0\in\mathbb{C}^{n+1}$, $n\geq{2}$, with Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$. Assume that the algebraic dimension of
$\textsf{Sing}(M)\leq 2n-2$. Then there exists a unique germ at
$0\in\mathbb{C}^{n+1}$ of holomorphic codimension one foliation
$\mathcal{F}_{M}$ tangent to $M$, if one of the following conditions is
fulfilled:
1. (1)
$n\geq 3$ and
$cod_{M_{\mathbb{C}}^{*}}(\textsf{Sing}(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
3$.
2. (2)
$n\geq 2$,
$cod_{M_{\mathbb{C}}^{*}}(\textsf{Sing}(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
2$ and $\mathcal{L}_{\mathbb{C}}$ admits a non-constant holomorphic first
integral.
Moreover, in both cases the foliation $\mathcal{F}_{M}$ admits a non-constant
holomorphic first integral $f$ such that $M=\\{\mathcal{R}e(f)=0\\}$.
## 3\. Proof of Theorem 1
We write
$F(x,y)=\mathcal{R}e(P(x,y_{1},\ldots,y_{n}))+H(x,y_{1},\ldots,y_{n}),$
where $P(x,y_{1},\ldots,y_{n})$ is one of the polynomials of the Table 1,
$H:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{R},0)$ is a germ of real-analytic
function such that $H(x,0)=0$ for all $x\in(\mathbb{C},0)$ and
$j_{0}^{k}(H)=0$, for $k=\deg(P)$. The complexification of $F$ is given by
$F_{\mathbb{C}}(x,y,z,w)=\frac{1}{2}P(x,y)+\frac{1}{2}P(z,w)+H_{\mathbb{C}}(x,y,z,w),$
thus
$M_{\mathbb{C}}=\\{F_{\mathbb{C}}(x,y,z,w)=0\\}\subset(\mathbb{C}^{2n+2},0)$,
where $z\in\mathbb{C}$ and $w=(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}$.
Since $P(x,y)$ has an ILS at $L$, we get
$\textsf{Sing}(M_{\mathbb{C}})=\\{y=w=0\\}\simeq\mathbb{C}^{2}$. In
particular, the algebraic dimension of $\textsf{Sing}(M)$ is $2$. On the other
hand, the complexification of $\eta=i(\partial{F}-\bar{\partial}F)$ is
$\eta_{\mathbb{C}}:=i[(\partial_{x}F_{\mathbb{C}}+\partial_{y}F_{\mathbb{C}})-(\partial_{z}F_{\mathbb{C}}+\partial_{w}F_{\mathbb{C}})].$
Recall that $\eta|_{M^{*}}$ and $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$
define $\mathcal{L}$ and $\mathcal{L}_{\mathbb{C}}$ respectively. Now we
compute $\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$. We can write
$dF_{\mathbb{C}}=\alpha+\beta$, with
$\alpha:=\frac{\partial{F_{\mathbb{C}}}}{\partial{x}}dx+\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{y}_{j}}dy_{j}=\frac{1}{2}\frac{\partial{P}}{\partial{x}}(x,y)dx+\frac{1}{2}\sum_{j=1}^{n}\frac{\partial{P}}{\partial{y}_{j}}(x,y)dy_{j}+\theta_{1}$
and
$\beta:=\frac{\partial{F_{\mathbb{C}}}}{\partial{z}}dz+\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}=\frac{1}{2}\frac{\partial{P}}{\partial{z}}(z,w)dz+\frac{1}{2}\sum_{j=1}^{n}\frac{\partial{P}}{\partial{w}_{j}}(z,w)dw_{j}+\theta_{2}$
where
$\theta_{1}=\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}dx+\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}$
and
$\theta_{2}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}}dz+\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}$.
Note that $\eta_{\mathbb{C}}=i(\alpha-\beta)$, and so
$\displaystyle\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=(\eta_{\mathbb{C}}+idF_{\mathbb{C}})|_{M^{*}_{\mathbb{C}}}=2i\alpha|_{M^{*}_{\mathbb{C}}}=-2i\beta|_{M^{*}_{\mathbb{C}}}.$
(3.1)
In particular, $\alpha|_{M^{*}_{\mathbb{C}}}$ and
$\beta|_{M^{*}_{\mathbb{C}}}$ define $\mathcal{L}_{\mathbb{C}}$. Therefore
$\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ can be split in two
parts. In fact, let $M_{1}:=\\{(x,y,z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{z}}\neq 0$ or
$\frac{\partial{F}_{\mathbb{C}}}{\partial{w_{j}}}\neq 0$ for some
$j=1,\ldots,n\\}$ and $M_{2}:=\\{(x,y,z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{x}}\neq 0$ or
$\frac{\partial{F}_{\mathbb{C}}}{\partial{z_{j}}}\neq 0$ for some
$j=1,\ldots,n\\}$, then $M_{\mathbb{C}}=M_{1}\cup M_{2}$. If we denote by
$A_{0}=\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}$,
$A_{j}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}$ for all $1\leq j\leq
n$ and by $B_{0}=\frac{\partial{H_{\mathbb{C}}}}{\partial{z}}$,
$B_{j}=\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}$ for all $1\leq j\leq
n$, we obtain that
$\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=X_{1}\cup X_{2}$,
where
$X_{1}:=M_{1}\cap\\{\frac{\partial{P}}{\partial{x}}(x,y)+A_{0}=\frac{\partial{P}}{\partial{y}_{1}}(x,y)+A_{1}=\ldots=\frac{\partial{P}}{\partial{y}_{n}}(x,y)+A_{n}=0\\}$
and
$X_{2}:=M_{2}\cap\\{\frac{\partial{P}}{\partial{z}}(z,w)+B_{0}=\frac{\partial{P}}{\partial{w}_{1}}(z,w)+B_{1}=\ldots=\frac{\partial{P}}{\partial{w}_{n}}(z,w)+B_{n}=0\\}.$
Since $P$ is a polynomial with an ILS at $L=\\{y=0\\}$, we conclude that
$cod_{M^{*}_{\mathbb{C}}}\textsf{Sing}(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=n.$
By hypothesis $n\geq 3$, then it follows from Theorem 2.3, part $(1)$ that
there exists a germ $f\in\mathcal{O}_{n+1}$ such that the holomorphic
foliation $\mathcal{F}$ defined by $df=0$ is tangent to $M$. Moreover
$M=\\{\mathcal{R}e(f)=0\\}$. Note that if
$M=\\{\mathcal{R}e(f)=0\\}=\\{F=0\\}$, with $F$ an irreducible germ, we must
have that $\mathcal{R}e(f)=U\cdot F$, where $U$ is a germ of real-analytic
function with $U(0)\neq 0$. Without loss of generality, we can assume that
$U(0)=1$. In particular, $\mathcal{R}e(f)=U\cdot F$ implies that $f=P+h.o.t$.
According to Theorem 1.1, there exists a biholomorphism
$\varphi:(\mathbb{C}^{n+1},0)\rightarrow(\mathbb{C}^{n+1},0)$ preserving $L$
such that $f\circ\varphi^{-1}=P$, ($f$ is $D_{I}$-equivalent to $P$, because
$f$ is a germ with ILS at $L$). Therefore,
$\varphi(M)=\\{\mathcal{R}e(P)=0\\}$ and the proof ends.
## 4\. Proof of Theorem 2
The idea is to use Theorem 2.3, part (2). In order to prove our result in the
case $n=2$, we are going to prove that $\mathcal{L}_{\mathbb{C}}$ has a non-
constant holomorphic first integral.
We begin by a blow-up along
$C:=\\{y_{1}=y_{2}=w_{1}=w_{2}=0\\}\simeq\mathbb{C}^{2}\subset\mathbb{C}^{6}$.
Let $F(x,y_{1},y_{2})=\mathcal{R}e(y^{2}_{1}+y^{2}_{2})+H$ and $M=\\{F=0\\}$
Levi-flat. Its complexification can be written as
$F_{\mathbb{C}}(x,y_{1},y_{2},z,w_{1},w_{2})=\frac{1}{2}(y^{2}_{1}+y^{2}_{2})+\frac{1}{2}(w^{2}_{1}+w^{2}_{2})+H_{\mathbb{C}}(x,y_{1},y_{2},z,w_{1},w_{2}).$
Note that
$\textsf{Sing}(M_{\mathbb{C}})=\\{y=w=0\\}=C.$
Let $E$ be the exceptional divisor of the blow-up
$\pi:\tilde{\mathbb{C}}^{6}\rightarrow\mathbb{C}^{6}$ along $C$. Denote by
$\tilde{M}_{\mathbb{C}}:=\overline{\pi^{-1}(M_{\mathbb{C}}\setminus\\{C\\})}\subset\tilde{\mathbb{C}}^{6}$
the strict transform of $M_{\mathbb{C}}$ via $\pi$ and by
$\tilde{\mathcal{F}}:=\pi^{*}(\mathcal{L}_{\mathbb{C}})$ the foliation on
$\tilde{M}_{\mathbb{C}}$.
Now, we consider an especial situation. Suppose that $\tilde{M}_{\mathbb{C}}$
is smooth and set $\tilde{C}:=\tilde{M}_{\mathbb{C}}\cap E$. Moreover, assume
that $\tilde{C}$ is invariant by $\tilde{\mathcal{F}}$. Take
$S=\tilde{C}\setminus\textsf{Sing}\tilde{\mathcal{F}}$, then $S$ is a smooth
leaf of $\tilde{\mathcal{F}}$. Pick $p_{0}\in S$ and a transverse section
$\sum$ through $p_{0}$. Let $G\subset\operatorname{Diff}(\sum,p_{0})$ be the
holonomy group of the leaf $S$ of $\tilde{\mathcal{F}}$. Since
$\textsf{dim}(\sum)=1$, we can assume that
$G\subset\operatorname{Diff}(\sum,0)$. We state a fundamental lemma.
###### Lemma 4.1 (Fernández-Pérez [9]).
In the above situation, suppose that the following properties are verified:
1. (1)
For any $p\in S\backslash\textsf{Sing}{(\tilde{\mathcal{F}})}$ the leaf
$L_{p}$ of $\tilde{\mathcal{F}}$ through $p$ is closed in $S$.
2. (2)
$g^{\prime}(0)$ is a primitive root of unity, for all $g\in
G\backslash\\{id\\}$.
Then $\mathcal{L}_{\mathbb{C}}$ admits a non-constant holomorphic first
integral.
###### Proof.
Let $G^{\prime}=\\{g^{\prime}(0)/g\in G\\}$ and consider the homomorphism
$\phi:G\rightarrow G^{\prime}$ defined by $\phi(g)=g^{\prime}(0)$. We claim
that $\phi$ is injective. In fact, assume that $\phi(g)=1$ and suppose by
contradiction that $g\neq id$. In this case $g(z)=z+az^{r+1}+\ldots$, where
$a\neq 0$. According to [12], the pseudo-orbits of this transformation
accumulate at $0\in(\sum,0)$, contradicting the fact that the leaves of
$\tilde{\mathcal{F}}$ are closed and so the assertion is proved. Now, it
suffices to prove that any element $g\in G$ has finite order (cf. [13]). In
fact, $\phi(g)=g^{\prime}(0)$ is a root of unity thus $g$ has finite order
because $\phi$ is injective. Hence, all transformations of $G$ have finite
order and $G$ is linearizable.
This implies that there is a coordinate system $w$ on $(\sum,0)$ such that
$G=\langle w\rightarrow\lambda w\rangle$, where $\lambda$ is a
$d^{th}$-primitive root of unity (cf. [13]). In particular, $\psi(w)=w^{d}$ is
a first integral of $G$, that is $\psi\circ g=\psi$ for any $g\in G$.
Let $\Gamma$ be the union of the separatrices of $\mathcal{L}_{\mathbb{C}}$
through $0\in\mathbb{C}^{6}$ and $\tilde{\Gamma}$ be its strict transform
under $\pi$. The first integral $\psi$ can be extended to a first integral
$\varphi:\tilde{M}_{\mathbb{C}}\backslash\tilde{\Gamma}\rightarrow\mathbb{C}$
by setting
$\varphi(q)=\psi(\tilde{L}_{q}\cap\sum),$
where $\tilde{L}_{p}$ denotes the leaf of $\tilde{\mathcal{F}}$ through $q$.
Since $\psi$ is bounded (in a compact neighborhood of $0\in\sum$), so is
$\varphi$. It follows from Riemann extension theorem that $\varphi$ can be
extended holomorphically to $\tilde{\Gamma}$ with $\varphi(\tilde{\Gamma})=0$.
This provides the first integral of $\mathcal{L}_{\mathbb{C}}$. ∎
The rest of the proof is devoted to prove that we are indeed in the conditions
of Lemma 4.1. It is follows from Lemma 2.1 that the leaves of
$\mathcal{L}_{\mathbb{C}}$ are closed. Therefore, we need to prove that each
generator of the holonomy group $G$ of $\tilde{\mathcal{F}}$ with respect to
$S$ has finite order.
Consider for instance the chart $(U_{1},(x,t,s,z,u,v))$ of
$\tilde{\mathbb{C}}^{6}$ where
$\pi(x,t,s,z,u,v)=(x,tu,su,z,u,vu)=(x,y_{1},y_{2},z,w_{1},w_{2}).$
We have
$\tilde{M}_{\mathbb{C}}\cap U_{1}=\\{(x,t,s,z,u,v)\in
U_{1}|1+t^{2}+s^{2}+v^{2}+uH_{1}(x,t,s,z,u,v)=0\\},$
where $H_{1}=H(x,ut,us,z,u,uv)/u^{3}$ and this fact imply that
$E\cap\tilde{M}_{\mathbb{C}}\cap U_{1}=\\{(x,t,s,z,u,v)\in
U_{1}|1+t^{2}+s^{2}+v^{2}=u=0\\}.$
It is not difficult to see that these complex subvarieties are smooth. Now,
let us describe the foliation $\tilde{\mathcal{F}}$ on $U_{1}$. In fact, note
that the foliation $\mathcal{L}_{\mathbb{C}}$ is defined by
$\alpha|_{M^{*}_{\mathbb{C}}}=0$, where
$\alpha=\frac{1}{2}\frac{\partial{P}}{\partial{x}}dx+\frac{1}{2}\frac{\partial{P}}{\partial{y_{1}}}dy_{1}+\frac{1}{2}\frac{\partial{P}}{\partial{y_{2}}}dy_{2}+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H}_{\mathbb{C}}}{\partial{y_{j}}}dy_{j}.$
It follows that
$\alpha=y_{1}dy_{1}+y_{2}dy_{2}+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H}_{\mathbb{C}}}{\partial{y_{j}}}dy_{j}$,
then $\tilde{\mathcal{F}}|_{U_{1}}$ is defined by
$\tilde{\alpha}|_{\tilde{M}_{\mathbb{C}}\cap U_{1}}=0$, where
$\displaystyle\tilde{\alpha}=(t^{2}+s^{2})du+utdt+usds+u\tilde{\theta},$ (4.1)
and
$\tilde{\theta}=\frac{\pi^{*}(\frac{\partial{H_{\mathbb{C}}}}{\partial{x}}dx+\sum^{2}_{j=1}\frac{\partial{H_{\mathbb{C}}}}{\partial{y_{j}}}dy_{j})}{u^{2}}.$
Therefore, the singular set of $\tilde{\mathcal{F}}|_{U_{1}}$ is given by
$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}=\\{u=t+is=0\\}\cup\\{u=t-is=0\\}.$
On the other hand, note that the exceptional divisor $E$ is invariant by
$\tilde{\mathcal{F}}$ and the intersection with
$\textsf{Sing}\widetilde{\mathcal{F}}$ is
$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}\cap
E=\\{u=t+is=v^{2}+1=0\\}\cup\\{u=t-is=v^{2}+1=0\\}.$
In particular,
$S:=(E\cap\tilde{M}_{\mathbb{C}})\backslash\textsf{Sing}\widetilde{\mathcal{L}}_{\mathbb{C}}$
is a leaf of $\widetilde{\mathcal{F}}$. We calculate the generators of the
holonomy group $G$ of the leaf $S$. We work in the chart $U_{1}$, because of
the symmetry of the variables in the definition of the variety
$\tilde{M}_{\mathbb{C}}$.
Pick $p_{0}=(0,1,0,0,0,0)\in S\cap U_{1}$ and a transversal
$\sum=\\{(0,1,0,0,\lambda,0)|\lambda\in\mathbb{C}\\}$ parameterized by
$\lambda$ at $p_{0}$. We have that
$\textsf{Sing}\tilde{\mathcal{F}}|_{U_{1}}\cap
E=\\{u=t+is=v^{2}+1=0\\}\cup\\{u=t-is=v^{2}+1=0\\}.$
For each $j=1,2$; let $\rho_{j}$ be a $2^{td}$-primitive root of $-1$. The
fundamental group $\pi_{1}(S,p_{0})$ can be written in terms of generators as
$\pi_{1}(S,p_{0})=\langle\gamma_{j},\delta_{j}\rangle_{1\leq j\leq 2},$
where for each $j=1,2$; $\gamma_{j}$ are loops that turn around
$\\{u=t+is=v-\rho_{j}=0\\}$ and $\delta_{j}$ are loops that turns around
$\\{u=t-is=v-\rho_{j}=0\\}$. Therefore, $G=\langle f_{j},g_{j}\rangle_{1\leq
j\leq 2}$, where $f_{j}$ and $g_{j}$ correspond to $[\gamma_{j}]$ and
$[\delta_{j}]$, respectively. We get from (4.1) that
$f^{\prime}_{j}(0)=e^{-\pi i}$ and $g^{\prime}_{j}(0)=e^{-\pi i}$ for all
$1\leq j\leq 2$. The proof of the theorem is complete.
## 5\. Levi-flat hypersurfaces with a complex line as singularity
In this section, we work with the system of coordinates
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$. The canonical local models examples
of Levi-flat hypersurfaces $M$ in $\mathbb{C}^{3}$ such that
$\textsf{Sing}(M)=L=\\{z_{1}=z_{2}=0\\}$ are
$\\{\mathcal{R}e(z^{2}_{1}+z^{2}_{2})=0\\}$ and
$\\{z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}=0\\}$.
Recently, Burns and Gong [2] classified, up to local biholomorphism, all germs
of quadratic Levi-flat hypersurfaces. Namely, up to biholomorphism, there is
only five models:
Type | Normal form | Singular set
---|---|---
$Q_{0,2k}$ | $\mathcal{R}e(z_{1}^{2}+z_{2}^{2}+\ldots+z^{2}_{k})$ | $\mathbb{C}^{n-k}$
$Q_{1,1}$ | $z^{2}_{1}+2z^{2}_{1}\bar{z}_{1}+z^{2}_{1}$ | empty
$Q^{\lambda}_{1,2}$ | $z^{2}_{1}+2\lambda z^{2}_{1}\bar{z}_{1}+z^{2}_{1}$ | $\mathbb{C}^{n-1}$
$Q_{2,2}$ | $(z_{1}+\bar{z}_{1})(z_{2}+\bar{z}_{2})$ | $\mathbb{R}^{2}\times\mathbb{C}^{n-2}$
$Q_{2,4}$ | $z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}$ | $\mathbb{C}^{n-2}$
Table 2. Levi-flat quadrics
We address the problem of provide conditions to characterize singular Levi-
flat hypersurfaces with a complex line as singularity. Using the
classification due to Burns and Gong [2], it is not hard to prove the
following proposition.
###### Proposition 5.1.
Suppose that $M$ is a quadratic real-analytic Levi-flat hypersurface in
$\mathbb{C}^{n}$, $n\geq 3$ such that
$\textsf{Sing}(M)=\\{z_{1}=z_{2}=\ldots=z_{n-1}=0\\}$. Then
1. (1)
If $n=3$, $M$ is biholomorphically equivalent to $Q_{0,2}$ or $Q_{2,4}$.
2. (2)
If $n\geq 4$, $M$ is biholomorphically equivalent to $Q_{0,2(n-1)}$.
###### Proof.
To prove part (1), observe that only there are two models of $M$ which admits
$\textsf{Sing}(M)=\\{z_{1}=z_{2}=0\\}$ as singularity, $Q_{0,2}$ or $Q_{2,4}$.
Now to prove part (2), note that if $n\geq 4$, the real hypersurface
$\\{z_{1}\bar{z}_{2}-\bar{z}_{1}z_{2}=0\\}$ has a complex subvariety of
dimension $n-2$ as singularity. It is follows that $M$ is biholomorphically
equivalent to $Q_{0,2(n-1)}$. ∎
In order to obtain a characterization, we define the Segre varieties
associated to real-analytic hypersurfaces. Let $M$ be a real-analytic
hypersurface defined by $\\{F=0\\}$. Fix $p\in M$, the Segre variety
associated to $M$ at $p$ is the complex variety in $(\mathbb{C}^{n},p)$
defined by
$Q_{p}:=\\{z\in(\mathbb{C}^{n},p):F_{\mathbb{C}}(z,\bar{p})=0\\}.$ (5.1)
Now assume that $M$ is Levi-flat and denote by $L_{p}$ the leaf of
$\mathcal{L}$ through $p\in M^{*}$. We denote by $Q^{\prime}_{p}$ the union of
all branches of $Q_{p}$ which are contained in $M$. Observe that
$Q^{\prime}_{p}$ could be the empty set when $p\in\textsf{Sing}(M)$.
Otherwise, it is a complex variety of pure dimension $n-1$.
The following result is classical, we proved it here for completeness.
###### Proposition 5.2.
In above situation, $L_{p}$ is an irreducible component of $(Q_{p},p)$ and
$Q^{\prime}_{p}=L_{p}$.
###### Proof.
Since $p\in M^{*}$, E. Cartan’s theorem assures that there exists a
holomorphic coordinate system such that near of $p$, $M$ is given by
$\\{\mathcal{R}e(z_{n})=0\\}$ and $p$ is the origin. In this coordinates
system the foliation $\mathcal{L}$ is defined by $dz_{n}|_{M^{*}}=0$. In
particular, $L_{0}=\\{z_{n}=0\\}$ and obviously $\\{z_{n}=0\\}$ is a branch of
$Q_{0}$. Furthermore, $L_{0}$ is the unique germ of complex variety of pure
dimension $n-1$ at $0$ which is contained in $M$. Hence
$Q^{\prime}_{0}=L_{0}$. ∎
Let $p\in\textsf{Sing}(M)$, we say that $p$ is a Segre degenerate singularity
if $Q_{p}$ has dimension $n$, that is, $Q_{p}=(\mathbb{C}^{n},p)$. Otherwise,
we say that $p$ is a Segre nondegenerate singularity.
Suppose that $M$ is defined by $\\{F=0\\}$ in a neighborhood of $p$, observe
that $p$ is a degenerate singularity of $M$ if $z\longmapsto
F_{\mathbb{C}}(z,\bar{p})$ is identically zero.
###### Remark 5.3.
If $V$ is a germ of complex variety of dimension $n-1$ contained in $M$ then
for $p\in V$, we have $(V,p)\subset(Q_{p},p)$. In particular, if there exists
distinct infinitely many complex varieties of dimension $n-1$ through $p\in M$
then $p$ is a Segre degenerate singularity.
To continuation, we consider a germ at $0\in\mathbb{C}^{n}$ of a codimension
one singular holomorphic foliation $\mathcal{F}$.
###### Definition 5.4.
We say that $\mathcal{F}$ and $M$ are tangent, if the leaves of the Levi
foliation $\mathcal{L}$ on $M$ are also leaves of $\mathcal{F}$.
###### Definition 5.5.
A meromorphic (holomorphic) function $h$ is called a meromorphic (holomorphic)
first integral for $\mathcal{F}$ if its indeterminacy (zeros) set is contained
in $\textsf{Sing}(\mathcal{F})$ and its level hypersurfaces contain the leaves
of $\mathcal{F}$.
Recently, Cerveau and Lins Neto proved the following result.
###### Theorem 5.6 (Cerveau-Lins Neto [4]).
Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{3}$, of
holomorphic codimension one foliation tangent to a germ of an irreducible real
analytic hypersurface $M$. Then $\mathcal{F}$ has a non-constant meromorphic
first integral.
In our context, we prove the following result.
###### Theorem 5.7.
Let $M$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$ of an irreducible real-
analytic Levi-flat hypersurfaces such that
$\textsf{Sing}(M)=L:=\\{z_{1}=z_{2}=\ldots=z_{n-1}=0\\}$. Suppose that:
1. (1)
Every point in $\textsf{Sing}(M)$ is a Segre nondegenerate singularity.
2. (2)
The Levi-foliation $\mathcal{L}$ on $M^{*}$ extends to a holomorphic foliation
$\mathcal{F}$ in some neighborhood of $M$.
Then there exists $f\in\mathcal{O}_{n}$ and a real-analytic curve
$\gamma\subset\mathbb{C}$ such that $M=f^{-1}(\gamma)$.
###### Proof.
Since the Levi-foliation $\mathcal{L}$ on $M^{*}$ extends to a holomorphic
foliation $\mathcal{F}$, we can apply directly Theorem 5.6, this means that
$\mathcal{F}$ has a non-constant meromorphic first integral $f=g/h$, where $g$
and $h$ are relatively prime. We asserts that $f$ is holomorphic. In fact, if
$f$ is purely meromorphic, we have that for all $\zeta\in\mathbb{C}$, the
complex hypersurfaces $V_{\zeta}=\\{g(z)-\zeta h(z)=0\\}$ contains leaves of
$\mathcal{F}$. In particular, $M$ contains an infinitely many of hypersurfaces
$V_{\zeta}$, because $M$ is closed and $\mathcal{F}$ is tangent to $M$. Set
$\Lambda:=\\{\zeta\in\mathbb{C}:V_{\zeta}\subset M\\}$. Note also that the
foliation $\mathcal{F}$ is singular at $L$, so that
$\mathcal{I}_{f}:=\\{h=g=0\\}$ the indeterminacy set of $f$ intersect $L$.
Therefore, we have a point $q$ at $\mathcal{I}_{f}\cap L$ which would be a
Segre degenerate singularity, because $q\in V_{\zeta}$, for all
$\zeta\in\Lambda$. It is a contradiction and the assertion is proved.
The foliation $\mathcal{F}$ is defined by $df=0$, $f\in\mathcal{O}_{n}$ and is
tangent to $M$. Without loss generality, we can assume that $f$ is an
irreducible germ in $\mathcal{O}_{n}$. According to a remark of Brunella [3,
pg. 8], there exists a real-analytic curve $\gamma\subset\mathbb{C}$ through
the origin such that $M=f^{-1}(\gamma)$. ∎
###### Remark 5.8.
In [11], J. Lebl gave conditions for the Levi-foliation on $M^{*}$ does
extended to a holomorphic foliation. One could be considered these hypothesis
and establish a theorem more refined. Note also that if $\textsf{Sing}(M)$ is
a germ of smooth complex curve, it is possible adapted the proof of Theorem
5.7. In general, the holomorphic extension problem for the Levi-foliation of a
Levi-flat real-analytic hypersurface remains open and is of independent
interest, for more details see [8].
Acknowledgments.– This work was partially supported by PRPq - Universidade
Federal de Minas Gerias UFMG 2013 and FAPEMIG APQ-00371-13. I would like to
thank Maurício Corrêa JR for his comments and suggestions, and the referee for
pointing out corrections.
## References
* [1] V.I. Arnold: Normal forms of functions near degenerate critical points, the Weyl groups $A_{k},D_{k},E_{k}$ and Lagrangian singularities. Funkcional. Anal. i Priložen. (6), no. 4, pp. 3-25, (1972).
* [2] D. Burns, X. Gong: Singular Levi-flat real analytic hypersurfaces. Amer. J. Math. 121, no. 1, pp. 23-53, $(1999)$.
* [3] M. Brunella: Some remarks on meromorphic first integrals. Enseign. Math. (2), 58 (3-4): 315-324, $(2012)$.
* [4] D. Cerveau, A. Lins Neto: Local Levi-Flat hypersurfaces invariants by a codimension one holomorphic foliation. Amer. J. Math. 133, no. 3, pp. 677-716, $(2011)$.
* [5] A. H. Durfee: Fifteen characterizations of rational double points and simple critical points. Enseign. Math. 25, pp. 131-163, (1979).
* [6] A. Fernández-Pérez: Singular Levi-flat hypersurfaces. An approach through holomorphic foliations. Ph. D. Thesis IMPA - Brazil (2010).
* [7] A. Fernández-Pérez: On normal forms of singular Levi-flat real analytic hypersurfaces. Bull. Braz. Math. Soc. 42 (1), pp. 75-85, (2011).
* [8] A. Fernández-Pérez: On Levi-flat hypersurfaces with generic real singularities. J. Geom. Anal. v. 23, pp. 2020-2033, (2013).
* [9] A. Fernández-Pérez: Normal forms of Levi-flat hypersurfaces with Arnold type singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (2014, to appear) doi:10.2422/2036-2145.201112_003
* [10] J. Lebl: Algebraic Levi-flat hypervarieties in complex projective space. J. Geom. Anal. 22 (2), 410-432, (2012).
* [11] J. Lebl: Singular set of a Levi-flat hypersurface is Levi-flat. Math. Ann. 355. no. 3. pp. 1177-1199, (2013).
* [12] F. Loray: Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Avaliable in http://hal.archives-ouvertures.fr/ccsd-00016434
* [13] J.F. Mattei, R. Moussu: Holonomie et intégrales premières. Ann. Ec. Norm. Sup. 13, pp. 469-523, $(1980)$.
* [14] D. Siersma: Isolated line singularity. Proc. of Symposia in Pure Math. (2) 40, pp. 485-496, $(1983)$.
* [15] A. Zaharia: Characterizations of simple isolated line singularities. Canad. Math. Bull. Vol. 42 (4), pp. 499-506, (1999).
|
arxiv-papers
| 2013-04-09T17:23:13 |
2024-09-04T02:49:44.056904
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arturo Fern\\'andez-P\\'erez",
"submitter": "Arturo Fernandez",
"url": "https://arxiv.org/abs/1304.2669"
}
|
1304.2780
|
# Direct Ultraviolet Imaging
and Spectroscopy of Betelgeuse
A. K. Dupree and R. P. Stefanik Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138 USA
###### Abstract
Direct images of Betelgeuse were obtained over a span of 4 years with the
Faint Object Camera on the Hubble Space Telescope. These images reveal the
extended ultraviolet continuum emission ($\sim$2 times the optical diameter),
the varying overall ultraviolet flux levels and a pattern of bright surface
continuum features that change in position and appearance over several months
or less. Concurrent photometry and radial velocity measures support the model
of a pulsating star, first discovered in the ultraviolet from IUE. Spatially
resolved HST spectroscopy reveals a larger extention in chromospheric
emissions of Mg II as well as the rotation of the supergiant. Changing
localized subsonic flows occur in the low chromosphere that can cover a
substantial fraction of the stellar disk and may initiate the mass outflow.
††slugcomment: European Astronomical Society Publication Series, 2013,
Eds. P. Kervella, Th. Le Bertre & G. Perrin, in press
## 1 Introduction
Alpha Orionis (Betelgeuse) has been long and well-studied with a variety of
ground and space-based techniques as a prototypical supergiant. Ultraviolet
observations have been particularly useful because they probe the very outer
layers of this star and can pinpoint the onset of outflowing material and
indicate the driving mechanisms behind the mass loss from the star (Dupree
2010). In fact the monitoring of the flux from Betelgeuse shows that its
photometric behavior and its ’spottedness’ differ from the signals of magnetic
activity found in the Sun and active cool stars. The long-lived IUE satellite
clearly demonstrated the presence of periodic fluctuations and a traveling
disturbance in the outer atmosphere (Dupree et al. 1987). And the Hubble Space
Telescope with its Faint Object Camera acquired the first direct image of a
star other than the Sun (Gilliland & Dupree 1996) which in concert with
ground-based photometry and spectroscopy and further ultraviolet imaging
reveals the extent and characteristics of the supergiant’s variability.
Figure 1: Panels showing the V magnitude from the AAVSO photoelectric database
(Henden 2012), the UV continuum and the Mg II k-line flux measured from IUE
spectra, and the radial velocity. In the bottom panel, measurements denoted by
the plus-symbol (+) are taken from Smith et al. (1989); the open diamonds
($\diamond$) represent measures from Oak Ridge Observatory. The radial
velocity measures are generally made from photospheric neutral metal lines. A
discontinuity occurs between the two sets of radial velocity measures; this
discontinuity appears to be real, since no cause has been identified.
## 2 Spatially Unresolved Photometry and Spectroscopy
Over a time span of $\sim$ 16 years, photometry in the V-band and the
ultraviolet continuum ($\lambda\lambda$ 2950–3050Å), the Mg II k-line emission
flux, and the radial velocity are shown in Fig. 1. For $\sim$3 years (JD
6000-7400, mod JD2440000), the chromosphere displayed a brightness fluctuation
with a period of 420 days (Dupree et al. 1987) which later became
substantially weaker, and then disappeared entirely. The appearance of a
period suggests that the global brightness variations do not arise from the
appearance of convection cells on the surface which would not be expected to
be periodic. A period of $\sim$400 d is consistent with models of fundamental
pulsation modes of the star (Lovy et al. 1984, Stothers 2010). From Fig. 1, it
is apparent that the continuum brightenings are correlated with the high
chromospheric (Mg II) brightenings. This is also a clue that Betelgeuse’s
behavior is not the result of magnetically active ’star spots’. Cool stars
with magnetic activity are well-known to show an anti-correlation between
photometric brightenings and chromospheric activity. When classical star spots
are present, they are cooler and photometrically ’dark’, the continuum flux
decreases, and chromospheric emission lines become stronger as a result of
magnetically-associated chromospheric heating. The V-magnitude decreases
during three instances (JD7600, JD9100, JD9800, mod JD2440000) where the UV
continuum measures, observed simultaneously with V-band photometry exhibits a
decreased flux. Additionally, the flux modulation in Betelgeuse is substantial
… about a factor of 2 in the lines and continuum … and such an excursion
surpasses that found in low gravity magnetically active stars such as RS Cvn
binaries.
An indication of the presence of a travelling wave in the atmosphere comes
from measures of the B-magnitude variations and the flux in the chromospheric
lines reported in Dupree et al. (1987). On several occasions during the time
when the 420-day period was evident, the B magnitude became faint and then
recovered while, after a delay of 55 days, the Mg II h-line flux became faint
and subsequently recovered. If a propagating wave caused the decrease in
emission measure, followed by an increase, and this wave travelled at $\sim$2
km s-1, it would cover a reasonable distance of 0.1 R⋆ in 55 days, using the
larger distance of Betelgeuse (191 pc) suggested by Harper et al. (2008).
Observation of the variation of the Mg II h and k lines suggests a lag of
$\sim$70 days beween the h and k line variations. Because the opacity in the
k-line is larger than the h-line, the k-line is formed further out in the
atmosphere and the lag in flux variation between this two lines again is
consistent with a propagating disturbance.
The radial velocity measures might offer additional information. The values
displayed in Fig. 1 (lower panel) suggest a long-term variability ($\sim$13
yr) on which shorter variations ($\sim$400 d) are superposed. In a Cepheid
star, the light maximum is close to but does not always coincide with the
maximum velocity of approach (cf. Robinson & Hoffleit 1932; Bersier 2002). One
CS Mira star, R CMi, has shown light maximum after maximum velocity infall
(Jorissen 2004, and Lion et al. 2013). Detailed study shows the velocity
pattern in Miras is complex and varies with the line diagnostic (Hinkle et al.
1982). Inspection of Fig. 1 shows an inconsistent pattern at many light maxima
in Betelgeuse. For instance, the V-magnitude displays brightenings at JD8600,
JD 10800, JD 11200 (mod JD2440000) coincident with the radial velocity
corresponding to a local infall maximum - not subsequent to the infall maximum
as seen in a Mira variable. However, the data at JD9800 exhibit a minimum
V-magnitude brightness, yet the radial velocity measures signal maximum
inflow. Gaps in the observational measures obviously can compromise
conclusions here. The spatially resolved observations discussed later in this
paper appear to offer an explanation of the radial velocity behavior,
suggesting that a clean interpretation and seeking similarity with other
pulsating stars remains challenging.
In sum, the spatially unresolved measures suggest that pulsation phenomena
could dominate the photometric variability of Betelgeuse, but the details do
not consistently replicate in detail what is found in globally pulsating
Cepheid stars or in Mira giants.
## 3 Direct UV Imaging of Betelgeuse
The Faint Object Camera on HST was used (Gilliland & Dupree 1996) to image
Betelgeuse directly in the ultraviolet continuum ($\lambda$2550Å). This first
direct image of the surface of a star other than the Sun provided about 10
resolution elements on the ultraviolet disk (38 mas point spread function)
which has a diameter about 2.2 times larger than the optical diameter. This
image revealed a bright spot in the SW quadrant of the star which comprised
10% of the area and 20% of the flux from Betelgeuse at that time.
Subsequently, we followed up with similar ultraviolet images spanning 4.1
years. All of the images were obtained with a combination of filters: a
medium-band filter (F253M) was crossed with a second UV filter, F220W, and 4
magnitudes of neutral-density filter inserted also. These images are shown in
Fig. 2 where different scalings are used in the upper and lower panel set.
Each of these images contains the comparison star, HZ 4, taken during the
first visit of HST ($t=0$) demonstrating the extended nature of Betelgeuse in
the ultraviolet.
When the images are scaled to the same exposure time (Fig. 2, top panels), it
is obvious that the total ultraviolet flux not surprisingly varies on a time
scale of months. The lower panels in Fig. 2 show the same images scaled to the
brightest pixel. This figure reveals the changing brightness pattern across
the stellar chromosphere. The single bright area found at $t=0$ becomes
smaller and fainter over the next 2.6 yr, then appears to move to the north,
and becomes greatly extended in the 3.5 yr observation, approximately
’circling’ the spot in the original ($t=0$) image, before fading in the final
image at 4.1 yr. The bright spots seem to stay in approximately the same
position on the star. We find no large excursion to the stellar limb in the
position of the bright spot.
Characteristics of the UV images are shown in Fig. 3 with respect to the V
magnitude and the radial velocity. The mean UV flux from the HST images (third
panel) generally tracks the V magnitude. In fact, the faintest excursion in
magnitude is in harmony with the lowest value of the UV flux, and the times of
brightest optical magnitude generally agree with the brightest UV flux. The
relation beween UV flux and radial velocity is not so clear. The highest
infall velocity occurs twice during the HST observations, and at these times
the
Figure 2: Top 2 panels: UV images scaled to the same exposure time, 3559 s,
which corresponds to the longest summed exposure. Lower 2 panels: UV (F253M)
dithered image scaled to the brightest pixel.
mean UV flux is first at a minimum and then at a maximum value.
A Voigt profile was fit to the UV continuum images and the full width at half
maximum (FWHM) is also shown in Fig. 3 (bottom panel). There does not appear
to be a relationship between the diameter of the UV image and the photospheric
radial velocity. During the time span of the HST images, the V magnitude
displays a period of 366 days whereas the period found for the radial velocity
variation is 440 days. These periods were derived using the Lomb-Scargle
technique for irregularly-spaced data after removing a linear trend (Horne &
Baliunas 1986). The absence of correlations may be understood from the results
of the spatially resolved spectra discussed below.
Figure 3: These panels relate the magnitude and metal-line radial velocity to
the HST measures of flux and stellar diameter. The broken lines are meant to
guide the eye. Long term trends have been removed with a second-order
polynomial from the V magnitude and radial velocity values. Figure 4: UV image
of Betelgeuse with the sense of rotation shown as a wire frame (Uitenbroek et
al. 1998).
At the same time of the first UV image, spatially resolved UV spectra were
obtained (Uitenbroek et al. 1998) in the near ultraviolet region that included
both the resonance Mg II emission lines and several photospheric absorption
lines. The absorption lines from the spatial scan NW to SE across the stellar
disk displayed a systematic shift from negative to positive velocities with a
total amplitude $\sim$10 km s-1. This behavior is interpreted as due to the
rotation of the star. Uitenbroek et al. proposed that the bright spot
coincided with the pole of rotation (which is also consistent with the
measures of the angle of highest polarization), making the inclination of the
rotation axis of the star 20∘ from the line of sight (see Fig. 4). This
inclination, coupled with the measured 5 km s-1 rotation velocity suggests
that the deprojected radial velocity is 14.6 km s-1, and the rotational period
is 25.5 yr at a distance of 191 pc. Thus it appears plausible that the bright
spots shown in Fig. 2 emerge preferentially around the pole of the star.
Figure 5: The flow velocity in the low chromosphere, as inferred from modeling
of the Si I line, $\lambda$2516, observed with STIS (the 7 STIS aperture
positions are marked by broken lines) at time $\Delta t=4.1yr$. The length and
direction of the arrows indicate the magnitude of the velocity and the
direction of flow (Lobel & Dupree 2001).
## 4 Spatially Resolved Spectroscopy
STIS, the Space Telescope Imaging Spectrograph on HST possesses a narrow
aperture (25 $\times$ 100 mas) which offers true spatial resolution for the
ultraviolet emission lines of Betelgeuse since they have a diameter of
$\sim$270 mas or larger. In addition, line profiles of several neutral and
singly ionized species that occur in the near-ultraviolet exhibit centrally-
reversed emission which serves as a diagnostic of mass motions in the
atmosphere. Lobel & Dupree (2001) detected changes in many profiles with
spatial position on the disk that indicated both outflowing and inflowing
chromospheric material with velocities $\sim$2 km s-1. These velocities change
with position on the disk and also with time. Detailed non-LTE models in
spherical geometry were constructed to match the profiles of many lines
including Fe I, Fe II, Si I, and Al II. The 4 spectroscopic observations
spanned 1.3 yr with sampling of about 0.3 yr. These began at $t=2.8$ yr
corresponding to the times in Fig. 2. Beginning at $t=2.8$ yr, the flow
pattern in the low chromosphere changed from a global decelerating inflow to
outflow in one quadrant and subsequently the outflow extended to almost the
whole stellar hemisphere (Fig. 5). The spatially resolved spectroscopy reveals
that the outer atmosphere of Betelgeuse does not behave in a global fashion
but that asymmetric time dependent dynamics are present.
## 5 Conclusions
When observed as a star, the photosphere and chromosphere of Betelgeuse are
subject to a semi-periodic travelling oscillation with a period of $\sim$400
days. This period can be coherent for $\sim$4 years.
Spatially resolved imaging shows that the image size in the near ultraviolet
continuum ($\lambda$ 2550Å) exceeds the optical diameter (taken as 55 mas) by
about a factor of 2.2, and the chromospheric Mg II lines extend even further …
to a diameter $\sim$4 times that of the optical. Bright regions occur on the
ultraviolet disk that change in position and strength over a period of months.
They appear to be localized around the rotational pole of the star. Spatially
resolved spectroscopy demonstrates that the low chromosphere does not behave
uniformly, but that the dynamics are complex. We have discovered gradually
changing inflow and outflow patterns suggesting asymmetric mass motions. Such
behavior obviously complicates the interpretation of the spatially unresolved
radial velocity measures.
We acknowledge with thanks the variable star observations from the AAVSO
International Database contributed by observers worldwide and used in this
research.
## References
* (1)
* (2) Bersier, D. 2002, ApJS, 140, 265
* (3)
* (4) Dupree, A. K. 2010, in Physics of Sun and Star Spots, ed. D. P. Choudhary & K. G. Strassmeier, Proc IAU Symp. 273, 188
* (5)
* (6) Dupree, A. K., Baliunas, S. L., Guinan, E. F., Hartmann, L., Nassiopoulos, G. E., & Sonneborn, G. 1987, ApJ, 317, L85
* (7) Gilliland, R. L., & Dupree, A. K. 1996, ApJ, 463, L29
* (8) Harper, G. M., Brown, A., & Guinan, E. F. 2008, AJ, 135, 1430
* (9)
* (10) Henden, A. A., 2012, Observations from the AAVSO International Database, private communication
* (11)
* (12) Hinkle, K. H., Hall, D. N. B., & Ridgway, S. T. 1982, ApJ, 252, 697
* (13)
* (14) Horne, J. H., & Baliunas, S. L. 1986, ApJ., 302, 757
* (15)
* (16) Jorissen, A. 2004, in Asymptotic Giant Branch Stars, ed. H. Habing & H. Olofsson, (Berlin: Springer), p. 461
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* (18) Lion, S., Van Eck, S., Chiavassa, A., Plez, B., & Jorissen, A. 2013, this volume
* (19)
* (20) Lobel, A., & Dupree, A. K. 2001, ApJ, 558, 815
* (21)
* (22) Lovy, D., Maeder, A., Noels, A. & Gabriel, M. 1984, A&A, 133, 307
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* (24) Robinson, L. V., & Hoffleit, D. 1932, Harvard College Observatory Bulletin No. 888, 12.
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* (26) Smith, M. A., Patten, B. M., & Goldberg, L. 1989, AJ, 98, 2233
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* (29) Uitenbroek, H., Dupree, A. K., & Gilliland, R. L. 1998, AJ, 116, 2501
|
arxiv-papers
| 2013-04-09T20:01:08 |
2024-09-04T02:49:44.069362
|
{
"license": "Public Domain",
"authors": "A. K. Dupree and R. P. Stefanik",
"submitter": "Andrea Dupree",
"url": "https://arxiv.org/abs/1304.2780"
}
|
1304.2960
|
# Standardized network reconstruction of E. coli metabolism
Kieran Smallbone
_Manchester Centre for Integrative Systems Biology_
_131 Princess Street, Manchester M1 7DN, UK_
[email protected]
###### Abstract
We have created a genome-scale network reconstruction of Escherichia coli
metabolism. Existing reconstructions were improved in terms of annotation
standards, to facilitate their subsequent use in dynamic modelling. The
resultant network is available from EcoliNet (http://ecoli.sf.net/).
## EcoliNet
The structure of metabolic networks can be determined by a reconstruction
approach, using data from genome annotation, metabolic databases and chemical
databases [1]. We built upon an existing reconstruction of the metabolic
network of E. coli that was based on genomic and literature data (known as
iJO1366, [2]). This model contains 1366 genes, 2251 metabolic reactions, and
1136 unique metabolites. Comparison to experimental data sets shows that it
makes accurate phenotypic predictions of growth on different substrates and
for gene knockout strains [2].
iJO1366 suffers from the use of non-standard names and is not annotated with
methods that are machine-readable. The model was thus updated according to
existing community-driven annotation standards [3]. The reconstruction is
described and made available in Systems Biology Markup Language (SBML)
(http://sbml.org/, [4]), an established community XML format for the mark-up
of biochemical models that is understood by a large number of software
applications. The network is available from EcoliNet (http://ecoli.sf.net/).
### Annotation
The highly-annotated network is primarily assembled and provided as an SBML
file. Specific model entities, such as species or reactions, are annotated
using ontological terms. These annotations, encoded using the resource
description framework (RDF) [5], provide the facility to assign definitive
terms to individual components, allowing software to identify such components
unambiguously and thus link model components to existing data resources [6].
Minimum Information Requested in the Annotation of Models (MIRIAM, [7])
–compliant annotations have been used to identify components unambiguously by
associating them with one or more terms from publicly available databases
registered in MIRIAM resources [8]. Thus this network is entirely traceable
and is presented in a computational framework.
Nine different databases are used to annotate entities in the network (see
Table 1). The Systems Biology Ontology (SBO) [9] is also used to semantically
discriminate between entity types. Eight different SBO terms are used to
annotate entities in the network (see Table 2).
example | identifier | database
---|---|---
EcoliNet | 562 | taxonomy
EcoliNet | 21988831 | pubmed
cytoplasm | GO:0005737 | obo.go
(-)-ureidoglycolate | C00603 | kegg.compound
(-)-ureidoglycolate | CHEBI:57296 | chebi
glgB | eco:b3432 | kegg.genes
glgB | P07762 | uniprot
1,4-alpha-glucan branching enzyme | 2.4.1.18 | ec-code
2-dehydro-3-deoxygalactonokinase | 1555810845 | isbn
Table 1: MIRIAM annotations used in the model. example | SBO term | interpretation
---|---|---
cytoplasm | 290 | compartment
(-)-ureidoglycolate | 247 | metabolite
tRNA (Glu) | 250 | ribonucleic acid
glgB | 252 | enzyme
1,4-alpha-glucan branching enzyme | 176 | biochemical reaction
1,4-alpha-glucan transport | 185 | transport reaction
biomass objective function | 397 | modelling reaction
glgB $\rightarrow$ 1,4-alpha-glucan branching enzyme | 460 | catalyst
Table 2: SBO terms used in the model.
### Use
We maintain the distinction between the E. coli GEnome scale Network
REconstruction (GENRE) [10] and its derived GEnome scale Model (GEM) [11].
This is important to differentiate between the established biochemical
knowledge included in a GENRE and the modelling assumptions required for
analysis or simulation with a GEM. A GENRE serves as a structured knowledge
base of established biochemical facts, while a GEM is a model which
supplements the established biochemical information with additional
(potentially hypothetical) information to enable computational simulation and
analysis [12]. Reactions added to the GEM include the biomass objective
function – a sink representing cellular growth – and hypothetical
transporters.
Three versions of the network are made available:
* •
<organism>_<version>.xml, a GEM for use in flux analyses, provided in Flux
Balance Constraints (FBC) format [13]
* •
<organism>_<version>_cobra.xml, the same GEM network, provided in Cobra format
[14]
* •
<organism>_<version>_recon.xml, a GENRE containing only reactions for which
there is experimental evidence
## YeastNet
YeastNet is an annotated metabolic network of Saccharomyces cerevisiae S288c
that is periodically updated by a team of collaborators from various research
groups. It started on the shoulders of previous reconstructions of the yeast
metabolic network that were published separately (iLL672 [15] and iMM904
[16]). However, due to the different approaches utilised, those earlier
reconstructions had a significant number of differences. A community effort in
2007 resulted in a consensus network representation of yeast metabolism,
reconciling the earlier results.
As of December 2012, six versions of the network have been released (see Table
3).
version | date | publications
---|---|---
1 | February 2008 | [3]
2 | June 2009 | –
3 | October 2009 | –
4 | March 2010 | [17]
5 | September 2011 | [12]
6 | December 2012 | –
Table 3: Development of YeastNet
The EcoliNet and YeastNet networks are structured identically to facilitate
comparative studies. YeastNet is available from http://yeast.sf.net/.
#### Acknowledgements
This work is deliverable 4.1 of the EU FP7 (KBBE) grant 289434 “BioPreDyn: New
Bioinformatics Methods and Tools for Data-Driven Predictive Dynamic Modelling
in Biotechnological Applications”.
## References
* [1] Palsson BØ, Thiele I: A protocol for generating a high-quality genome-scale metabolic reconstruction. Nature Protoc 2010, 5:91–121. doi:10.1038/nprot.2009.203
* [2] Orth JD, Conrad TM, Na J, Lerman JA, Nam H, Feist AM, Palsson BØ: A comprehensive genome-scale reconstruction of Escherichia coli metabolism – 2011. Mol Syst Biol 2011, 7:535. doi:10.1038/msb.2011.65
* [3] Herrgård MJ, Swainston N, Dobson P, Dunn WB, Arga KY, Arvas M, Blüthgen N, Borger S, Costenoble R, Heinemann M, Hucka M, Le Novére N, Li P, Liebermeister W, Mo M, Oliveira AP, Petranovic D, Pettifer S, Simeonidis E, Smallbone K, Spasić I, Weichart D, Brent R, Broomhead DS, Westerhoff HV, Kırdar B, Penttilä M, Klipp E, Palsson BØ, Sauer U, Oliver SG, Mendes P, Nielsen J, Kell DB: A consensus yeast metabolic network obtained from a community approach to systems biology. Nature Biotechnol 2008, 26:1155–1160. doi:10.1038/nbt1492
* [4] Hucka M, Finney A, Sauro H, Bolouri H, Doyle J, Kitano H, Arkin A, Bornstein B, Bray D, Cornish-Bowden A, Cuellar A, Dronov S, Gilles E, Ginkel M, Gor V, Goryanin I, Hedley W, Hodgman T, Hofmeyr J,Hunter P, Juty N, Kasberger J, Kremling A, Kummer U, Le Novère N, Loew L, Lucio D, Mendes P, Minch E, Mjolsness E, Nakayama Y, Nelson M, Nielsen P, Sakurada T, Schaff J, Shapiro B, Shimizu T, Spence H, Stelling J, Takahashi K, Tomita M, Wagner J, Wang J: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 2003, 19:524–531. doi:10.1093/bioinformatics/btg015
* [5] Wang XS, Gorlitsky R, Almeida JS: From XML to RDF: how semantic web technologies will change the design of ‘omic’ standards. Nature Biotechnol 2005, 23:1099–1103. doi:10.1038/nbt1139
* [6] Kell DB, Mendes P: The markup is the model: reasoning about systems biology models in the Semantic Web era. J Theor Biol 2008, 252:538–543. doi:10.1016/j.jtbi.2007.10.023
* [7] Le Novére N, Finney A, Hucka M, Bhalla US, Campagne F, Collado-Vides J, Crampin EJ, Halstead M, Klipp E, Mendes P, Nielsen P, Sauro H, Shapiro B, Snoep JL, Spence HD, Wanner BL: Minimum information requested in the annotation of biochemical models (MIRIAM). Nature Biotechnol 2005, 23:1509–1515. doi:10.1038/nbt1156
* [8] Laibe C, Le Novére N: MIRIAM resources: tools to generate and resolve robust cross-references in Systems Biology. BMC Syst Biol 2008, 252:538–543. doi:10.1186/1752-0509-1-58
* [9] Courtot M., Juty N., Knüpfer C., Waltemath D., Zhukova A., Dr ger A., Dumontier M., Finney A., Golebiewski M., Hastings J., Hoops S., Keating S., Kell D.B., Kerrien S., Lawson J., Lister A., Lu J., Machne R., Mendes P., Pocock M., Rodriguez N., Villeger A., Wilkinson D.J., Wimalaratne S., Laibe C., Hucka M., Le Novére N.: Controlled vocabularies and semantics in systems biology.. Mol Syst Biol 2011, 7:-543. doi:10.1038/msb.2011.77
* [10] Price ND, Reed JL, Palsson BØ: Genome-scale models of microbial cells: evaluating the consequences of constraints. Nat Rev Microbiol 2004, 2:886–897. doi:10.1038/nrmicro1023
* [11] Feist AM, Herrgård MJ, Thiele I, Reed JL, Palsson BØ: Reconstruction of biochemical networks in microorganisms. Nat Rev Microbiol 2008, 7:129–143. doi:10.1038/nrmicro1949
* [12] Heavner BD, Smallbone K, Barker B, Mendes P, Walker LP: Yeast 5 – an expanded reconstruction of the Saccharomyces cerevisiae metabolic network. BMC Syst Biol 2012, 6:55. doi:10.1186/1752-0509-6-55
* [13] Olivier BG, Bergmann FT: Flux Balance Constraints, Version 1 Release 1. Available from COMBINE. 2013\.
* [14] Schellenberger J, Que R, Fleming RM, Thiele I, Orth JD, Feist AM, Zielinski DC, Bordbar A, Lewis NE, Rahmanian S, Kang J, Hyduke DR, Palsson BØ: Quantitative prediction of cellular metabolism with constraint-based models: the COBRA Toolbox v2.0. Nat Protoc 2011, 6:1290–1307. doi:10.1038/nprot.2011.308.4
* [15] Kuepfer L, Sauer U, Blank LM: Metabolic functions of duplicate genes in Saccharomyces cerevisiae. Genome Res 2005, 15:1421–1430. doi:10.1101/gr.3992505
* [16] Mo ML, Palsson BØ, Herrgård MJ: Connecting extracellular metabolomic measurements to intracellular flux states in yeast. BMC Syst Biol 2009, 3:37. doi:10.1186/1752-0509-3-37
* [17] Dobson PD, Jameson D, Simeonidis E, Lanthaler K, Pir P, Lu C, Swainston N, Dunn WB, Fisher P, Hull D, Brown M, Oshota O, Stanford NJ, Kell DB, King RD, Oliver SG, Stevens RD, Mendes P: Further developments towards a genome-scale metabolic model of yeast. BMC Syst Biol 2010, 4:145. doi:10.1186/1752-0509-4-145
|
arxiv-papers
| 2013-04-09T09:07:13 |
2024-09-04T02:49:44.082145
|
{
"license": "Public Domain",
"authors": "Kieran Smallbone",
"submitter": "Kieran Smallbone",
"url": "https://arxiv.org/abs/1304.2960"
}
|
1304.3010
|
# Characterizing the Life Cycle of Online News Stories
Using Social Media Reactions
Carlos Castillo
Mohammed El-Haddad
Jürgen Pfeffer
Matt Stempeck
Qatar Computing Research Institute Doha, Qatar [email protected] Al Jazeera
Doha, Qatar mohammed.haddad
@aljazeera.net Carnegie Mellon University Pittsburgh, USA
[email protected] MIT Center for Civic Media Cambridge, USA
[email protected]
###### Abstract
This paper presents a study of the life cycle of news articles posted online.
We describe the interplay between website visitation patterns and social media
reactions to news content. We show that we can use this hybrid observation
method to characterize distinct classes of articles. We also find that social
media reactions can help predict future visitation patterns early and
accurately.
We validate our methods using qualitative analysis as well as quantitative
analysis on data from a large international news network, for a set of
articles generating more than 3,000,000 visits and 200,000 social media
reactions. We show that it is possible to model accurately the overall traffic
articles will ultimately receive by observing the first ten to twenty minutes
of social media reactions. Achieving the same prediction accuracy with visits
alone would require to wait for three hours of data. We also describe
significant improvements on the accuracy of the early prediction of shelf-life
for news stories.
###### category:
H.4.m Information Systems Applications Miscellaneous
###### keywords:
Web analytics; predictive web analytics; online news
;
## 1 Introduction
Traditional newspapers have been in decline in recent years in terms of
readership and revenue; in comparison, digital online news have been steadily
increasing according to both
metrics.111http://stateofthemedia.org/2012/overview-4/key-findings/
Recent surveys have shown that about half of the population of the US gets
their news online, and about one third goes online every day for
news.222http://www.people-press.org/2012/09/27/section-2-online-and-digital-
news-2/
The study of patterns of consumption of online news has attracted considerable
attention from the research community for over a decade. This research started
with the analysis of access patterns to websites, and has expanded to include
topics such as new engagement metrics, personalized news recommendations and
summaries, etc. (see Section 2 for an overview).
One line of research looks at consumption and interaction patterns as a single
time series and attempts several prediction tasks on it. For example,
predicting total comments from early comments [18, 28], total visits from
early visits [16], etc. More recent works incorporate attributes from each
specific article (e.g. topic, source, etc.) into the prediction [4].
We adopt a novel approach, in which we integrate different types of
interactions of users with an online news article including visits, social
media reactions, and search/referrals. We evaluate our methods on data from Al
Jazeera English, a large international news network, deeply characterizing
different classes of articles, and predicting their total number of page views
and their effective shelf-life (the effective shelf-life of an article is the
time span during which it receives most of its visits).
The characterization and prediction of user behavior around news articles is
valuable for a news organization, as it allows them (i) to gain a better
understanding of how people consume different types of news online; (ii) to
deliver more relevant and engaging content in a proactive manner; and (iii) to
improve the allocation of resources to developing stories over their life
cycle.
Our contributions. In this paper we present a qualitative and quantitative
analysis of the life cycle of online news stories. Our main contributions are
the following:
* •
We find that social media reactions can contribute substantially to the
understanding of visitation patterns in online news.
* •
We characterize two fundamental classes of news stories: breaking news and in-
depth articles, and describe the differences in users’ behavior around them.
* •
We describe classes of short-term audience response profiles to news articles
in terms of visits and social media reactions (decreasing, steady, increasing,
and rebounding).
* •
We improve significantly the accuracy and timeliness of predictive models of
total visits and shelf-life of articles, by incorporating social media
reactions.
The remainder of this paper is organized as follows. Section 2 provides an
overview of previous works related to ours. Section 3 introduces our data
collection and defines the concepts and variables we use. The main results of
our paper are presented as descriptive and predictive analysis in the two
following sections: Section 4 describes user behavior with respect to
different classes of articles, and Section 5 demonstrates the importance of
incorporating social media information into the predictive modeling of visits.
The last section concludes the paper.
## 2 Related work
One of the earliest published studies of user behavior in online news was
conducted by Aikat aikat_1998_news, who studied the web sites of two large
newspapers from November 1995 to May 1997. This work describes many of the
patterns still seen in news sites today: visits occur mostly during weekdays
and working hours; readers “skim” pages for information so dwell times tend to
be short, and there are clear traffic “bursts” that can be attributed to
specific news developments.
With the advent in recent years of what can be considered as new forms of
journalism (blogs) and new propagation mechanisms for news (micro-blogs and
online social networking sites), the volume of research publications in this
area has increased considerably. In this section we overview a few previous
works closely related to ours, but our coverage is by no means complete.
Behavioral-driven article classification. Previous works including [8, 19]
that have studied online activities around online resources (e.g. visiting,
voting, sharing, etc.), have consistently identified broad classes of temporal
patterns. These classes can be generally characterized, first, by the presence
or absence of a clear “peak” of activity; and second, by the amount of
activity before and after the peak.
Crane and Sornette crane_2008_response describe classes of visitation patterns
to online videos, and present models that are consistent with propagation
phenomena in social networks. Lehmann et al. lehmann_2012_dynamical extend
these classes by observing that for Twitter “hashtags” (user-defined topics)
the distributions of activity in different periods (before/during/after)
induce distinct clusters of activity that can be interpreted considering the
semantics of each hashtag. Romero et al. romero_2011_differences describe how
manually-assigned classes of hashtags are related to different shapes of the
exposure curve: the probability that a user will propagate some information
(“retweet” in the case of Twitter) after being exposed to the information by a
certain number of her neighbors.
Yang and Leskovec yang_2011_patterns describe six classes of temporal shapes
of attention. Attention is measured in terms of the number of appearances of a
given phrase (of a variation of it) corresponding to an event. The patterns
describe the distribution of attention over time, as well as the ordering in
which different types of media (professional blogs, news agencies, etc.)
“break” the story.
In general, previous works have established that the evolution of the
popularity of different on-line items depends on their class. Figueiredo et
al. figueiredo_2011_tube describe how YouTube videos that are posted to a
“top” page on the website, and videos that are making use of professionally
produced content, are different from randomly-chosen videos in terms of their
visit patterns.
Recently, researchers at URL shortening service Bit.ly [6] described how an
article’s half-life (see definition in Section 3) is affected by topics,
extending a previous observation than in general there are some topics that
are more time-sensitive than others [12]. For instance, business-related
articles have on average a longer half-life, while articles related to
politics/celebrities/entertainment have an intermediate one. Sports-related
articles have in comparison a shorter half-life. Previously, Bit.ly
researchers [5] have shown that this half-life is also affected by the social
media platform where the link is first posted (e.g. links on Facebook were
longer-lived than links on Twitter).
Table 1: Selected references on predictive modeling of user behavior, sorted by publication year. Reference | Collection | Input / Output
---|---|---
Tatar et al. tatar_2001_predicting | 20
Minutes | input: publication hour, number of comments after a short time, section; output: total number of comments
Brody, Harnad, and Carr brody_2006_citations | arXiv pre-prints | input: short-term article downloads; output: long-term article citations
Lee, Moon, and Salamatian lee_2010_popularity | DPReviews / Myspace | input: time to first-comment, inter comment arrival stats; output: time to last comment
Lerman and Hogg lerman_2010_news | Digg | input: visits; output: parameters of models that consider examination and promotion patterns
Kim, Kim, and Cho kim_2011_temperature | Blogs | input: clicks on first 30 minutes; output: clicks until end of lifetime
Yu, Chen, and Kwok yu_2011_predicting | Facebook pages | input: content and media type; output: number of FB likes/shares of each post
Lakkaraju and Ajmera lakkaraju_2011_attention | Facebook pages | input: text- and other characteristics of the posting and the page; output: number of FB likes/shares of each post
Szabo and Huberman szabo2012predicting | YouTube and Digg | input: views (Y), votes (D) in first 10d (Y), 2h (D); output: total number of views/votes
Bandari, Asur, and Huberman bandari2012pulse | News
aggregator | input: text analysis incl. topics, named entities, subjectivity, etc., source popularity; output: tweet count
Ruan et al. ruan_2012_prediction | Tweets | input: topics, past tweets, content features, user features, etc.; output: tweet count for a given topic
Pinto, Almeida, and Gonçalves pinto_2013_predicting | YouTube | input: time series of views in first 7 days; output: number of views after 30 days
Ahmed et al. ahmed_2013_predicting | YouTube, Digg, Vimeo | input: views (Y, V) and votes (D) over time; output: predict future popularity
We deepen and complement previous works on behavioral-driven characterization
of online content, by describing the life-cycle of online news articles
considering their visitation patterns as well as their social media reactions.
Prediction of users’ activity. The prediction of the volume of user activities
with respect to on-line content items has attracted a considerable amount of
research. This is attested by a number of papers, some of which are outlined
in Table 1. Another active topic that is closely related, but different, is
that of predicting real-world variables such as sales or profits using social
media signals (e.g. [13] and many others).
Over the years, the models used to predict user behavior in social media have
increased in complexity. For instance, Bandari et al. bandari2012pulse and
Ruan et al. ruan_2012_prediction incorporate into their models features
extracted from the content of the articles, such as topics. Yin et al.
yin_2012_straw study voting behavior over on-line contents and describe a
model that considers that users are divided into two populations: a group that
follows the majority opinion, and a group that does not. Myers et al.
myers_2012_external study models that describe user activity in terms of
information propagations, including the presence of external influences, e.g.
traditional media sources that can reach vast audiences, such as television
networks. Huang et al. huang_2012_predicting consider an online model of
social activities that evolves over time as more information becomes
available.
In contrast with previous works, we focus on the dynamic relation between
social media reactions and visits over time, and show that both are useful to
understand the differences among classes of articles and to predict future
visit patterns.
Analysis of news visits and social media responses. Dezso et al.
dezso_2006_dynamics analyze the visits to a large news portal in Hungary. One
aspect they study which is closely related to our work is the half-life of
articles, which is shown to be distributed according to a power-law across a
broad range, with a mean of 36 hours. Agarwal et al. agarwal_2012_multi study
the actions users perform after reading an article, which include printing,
commenting, rating, and sharing through e-mail or social media. Their focus is
on performing personalized recommendations, but they also uncover that article
topics have an effect on the probability of each action, with a division
between articles users read privately and articles they share publicly: “Users
tend to share articles that earn them social prestige and credit but they do
not mind clicking and reading some salacious news occasionally in private.”
Social media reactions to traditional news media can vary not only in volume
but also qualitatively. Hu et al. hu_2011_event record tweets during the
broadcast of a speech of the US President. They observe that many tweets refer
to the speech in general, except for certain topics which are discussed in
more detail.
Finally, social media optimization company SocialFlow describes in a
whitepaper [22] a comparative study of social media responses to several large
media outlets: Al Jazeera, BBC News, CNN, The Economist, Fox News and The New
York Times. Among other findings, they note that the probability that a user
clicks on a tweet is higher for The Economist ($\approx 19\%$) than for Fox
News ($\approx 16\%$), Al Jazeera ($\approx 11\%$) or The New York Times
($\approx 4\%$). However, followers of Al Jazeera are almost twice as likely
to retweet article links than followers of the other channels.
In contrast with previous works, we consider jointly traffic to the website
and social media reactions, as both constitute acts in which users engage with
the news content. Additionally, we quantify the richness of Twitter messages
over time measuring entropy and counting unique tweets, and show that these
variables are key to more accurate predictions of future visits.
## 3 Context and dataset
In this section we provide some context to our research and describe the
dataset that will be used on the remainder of the paper.
### 3.1 Traditional news and social media
Our dataset is provided by Al Jazeera English,333http://www.aljazeera.com/ a
well-established news organization that reaches hundreds of millions of
viewers through its TV channel. Their website is divided into five major
sections: News, In-Depth, Programmes, Sports, and Weather – plus a collection
of blogs, which is outside the scope of this study. Approximately 40
editors/producers work on the areas of News, In-Depth, and Programmes.
The editors of Al Jazeera English maintain Facebook and Twitter444Currently
Facebook and Twitter are the two most frequent sources of social media
referrals to Al Jazeera. Reddit appears in a third place but only through few
articles having extremely high visibility. accounts (we call them “corporate
accounts” in the rest of the paper) and use them actively to announce their
content. This seems to be a standard practice adopted by all major media
organizations in recent years. Each account (facebook.com/aljazeera and
@AJEnglish) has over 1.5 million followers as of May 2013. Using these
accounts, articles in the News section are shared immediately after being
posted online. Articles on the In-Depth and Programmes sections are shared
throughout the day with the goal of maximizing audience reach across multiple
time zones.
The corporate social media accounts re-share articles at different times of
the day, sometimes up to 4 times, on a schedule determined by editors’
judgment and designed to increase user engagement. Close attention is paid to
the wording of the items posted in social media, including aspects such as
their length and the use of hashtags in the case of Twitter. Editors use a
variety of online tools to obtain low-latency analytics of traffic and social
media, and to decide which hashtags and keywords to use in their postings.
More than half of the visitors to the Al Jazeera English website are from the
USA, the United Kingdom, or Canada. According to an online survey taken by Al
Jazeera English in 2011 ($n=4,500$), 18% of respondents said they used
Twitter, 42% Facebook, and 12% both.
Social media interactions and traffic to the website can complement or
substitute each other. Most frequently, they complement each other: people
click on the shared content and visit the website. Sometimes, the social media
share can be a substitute for a visit to the article, such as when a video can
be viewed directly on the social media site, or when the social media content
itself delivers enough information to satisfy users without requiring them to
click through to the full article.
For instance, the news “Pakistan’s Malala now able to stand in UK” (19 Oct
2012) generated an unusually large number of shares on Facebook, but
comparatively little traffic on the website. At the time, the student-activist
was being treated from nearly-fatal wounds received ten days before, and it is
likely that users who were following the story just wanted to express their
relief or satisfaction at her recovery.
In summary, for Al Jazeera and for most large news organizations, social media
is important both because it attracts more visitors to their website than any
other external referrer, as well as because it provides more platforms in
which to have an audience. Hence, many news organizations adopt an active role
in social media in order to increase this positive effect.
Figure 1: Overall visits to articles. Our study considers all articles posted
from Oct. 8th to Oct 29th, 2012 (the graph extends until Nov. 6th).
### 3.2 Data collection
We focus on a period of three weeks between October 8th, 2012, and October
29th, 2012. The choice of this period is not random: it was a relatively
stable period of traffic, only exhibiting a relatively minor peak on October
29th due to Hurricane Sandy. Figure 1 depicts the frequency of visits to all
the articles in our dataset during the observation period.
The data collection is done via a “beacon” embedded in all article pages; this
produces events that are processed using Apache
S4,555http://incubator.apache.org/s4/ a high-performance system for online
processing, which is used to collect and aggregate the visits with a 1-minute
granularity. For efficiency reasons, only articles obtaining at least 5 visits
in a 10-hour window are monitored. The collected data is stored using a
Cassandra666http://cassandra.apache.org/ NoSQL database.
Our system also collects messages from Facebook (using the Facebook Query
Language API) and Twitter (using their Search API). Both platforms have strict
limitations on polling frequencies, which impose a trade-off between the
number of articles we can monitor and the frequency with which we monitor
them. To obtain more accurate results for popular articles, and after
experimenting with different settings, we decided to poll social media
reactions for articles that are within the list of the 30 most visited
articles during each five-minute data collection window. We remark that this
list varies considerably over time.
We selected a uniform random sample of articles whose first visit was recorded
during the observation period, and kept only those accumulating at least 100
visits during their first week after publication. A total of 606 articles was
included; this covers over 3.6 million visits and at least 235,000 social
media reactions. Table 2 presents some summary statistics on this dataset.
Table 2: Summary statistics of our dataset. | Total | Article avg.
---|---|---
Number of articles | 606 | -
Visits after 1 hour | 260 K | 430
Visits after 1 day | 2.5 M | 4,273
Visits after 7 days | 3.6 M | 5,971
Facebook shares | 155 K | 256
Tweets | 80 K | 133
Tweet entropy | | 5.6 bits
Fraction of unique tweets | | 19.9 %
Fraction of corporate retweets | | 36.8 %
### 3.3 Metrics
For each article we collected a number of metrics regarding user visits and
social media reactions. First, we observed at a granularity of one minute the
number of visits (page views) to each article, and the URL of the previous
page seen by the users before reaching an article (referral). We bucketed the
latter into four classes:
* •
internal links, mostly from the home page of the website: these are the
majority of the traffic sources and comprise 70% of the visits;
* •
external links from other sources including social media sites, news
aggregators, and others: 14%;
* •
direct links, which have an empty referral and correspond
mostly777http://www.theatlantic.com/technology/archive/2012/10/dark-social-we-
have-the-whole-history-of-the-web-wrong/263523/ to people sharing news through
instant messaging, e-mail, or other non-web application: 11%; and
* •
search referrals, basically links from organic search results: 5%.
We remark that this distribution of referrals corresponds to the articles in
our sample, which do not include the homepage of the website, section index
pages, or older articles. If we take those into account, the numbers are
different, e.g. the search referrals account for 30% of the visits.
We also collected periodically the number of times an article has been shared
on Facebook, and the content of any Twitter message containing the URL of the
article, or a variant of the URL produced by a URL shortening service. We used
this data to compute the following variables:
* •
Number of Facebook shares per minute (interpolated).
* •
Number of tweets per minute.
* •
Number of unique tweets per minute. A tweet is deemed unique if its edit
distance with all previous tweets pointing to the same article (after
discarding shortened URLs and “retweet” prefixes) is more than 10 characters.
* •
Tweet vocabulary entropy. To compute this, at any given point in time we
create a document by concatenating all the tweets received up to that time.
Then, we compute the entropy of the distribution of terms in that document.
* •
Number of corporate retweets per minute. A tweet is a “corporate retweet” if
it includes “RT @AJEnglish” or “RT @AJELive” in its text. A tweet can be both
corporate retweet and unique, as users are free to edit the retweet before
posting it.
* •
Number of followers, friends (followees) and statuses of each of the users
posting a tweet.
Table 3: Top 10 most frequent words (stemmed and lowercased) in article titles in the “News” and “In-Depth” sections. Words that appear in both lists are italicized. Top words (News, $n=322$) | Top words (In-Depth, $n=139$)
---|---
Word | News | In-Depth | Word | News | In-Depth
us | 34 | 17 | us | 34 | 17
kill | 21 | 1 | pictur | 0 | 10
attack | 19 | 0 | obama | 6 | 6
syria | 15 | 4 | interact | 0 | 6
dead | 15 | 1 | america | 0 | 6
protest | 13 | 0 | muslim | 0 | 5
rebel | 12 | 2 | syrian | 11 | 4
vote | 11 | 3 | syria | 15 | 4
syrian | 11 | 4 | presid | 4 | 4
pakistan | 10 | 2 | polici | 2 | 4
## 4 Behavioral-Driven Classes
In this section we describe classes of articles according to patterns of user
behavior.
### 4.1 News vs In-Depth
We observe that articles in the two larger sections of the Al Jazeera English
website trigger distinct user behavior patterns: visits and social media
reactions on articles in the News section (322 articles in our sample) are
different from the ones on articles in the In-Depth section (139 articles).
Titles. Table 3 includes the most frequent words in titles of articles in
these two sections, after converting to lowercase and applying Porter’s
stemmer.888http://snowball.tartarus.org/algorithms/porter/stemmer.html While
in our sample the US and Syria appear prominently in both sections, articles
in the News section include several violent acts, while articles in the In-
Depth section are dominated by photos and political analysis. A chi-squared
test comparing the entire distributions shows $p<10^{-13}$, rejecting the
hypothesis that they are equal.
Figure 2: Visits per minute (left y-axis) as well as Tweets and Facebook
shares per minute (right y-axis) for the first 12 hours. For visits, the
shaded area covers 50% of the data (quantiles 0.25 to 0.75). Top: average for
a News item. Bottom: average for an In-Depth item.
Visits. Figure 2 (top) depicts the average time series of some variables for
articles in the News section. Time is expressed in hours-equivalent, which are
hours corrected by the seasonality (day-night, weekday-weekend) of traffic on
the website, as in [27]. Initially there are a number of visits and activity
on Twitter and Facebook, that decays rapidly after a short time. This is often
the pattern in news media as observed e.g. by [9, 21] and others. After a few
hours, a large amount of visits can be explained by “internal traffic”, i.e.
visitors arriving from the homepage of the site. For most articles, once the
news article is displaced from the homepage by more recent items, its traffic
slows down considerably.
The profile of visits to In-Depth articles can be more complex. Figure 2
(bottom) depicts the average series for these articles. We can observe that a
sustained level of visits is observed during several hours, as the contents of
these articles are not as time-sensitive as those of the News section. We
remark that in both cases (News and In-Depth) there is considerable
variability from one article to another.
Figure 3: Visits in the first hour versus visits on the first week for
articles in the two largest categories. A simple function that assumes that
visits after seven days are a multiple of visits after one hour has been
included, by performing a least-squares fit in the central portion of each
distribution. Figure 4: Differences in the distribution of Facebook vs Twitter
shares. On average the ratio of Facebook shares to tweets is 1.9:1 (1.6:1 for
News, 2.7:1 for In-Depth). The result of a least-squares fit in the central
portion of each distribution is included. Figure 5: Differences in the
distribution of the fraction of unique tweets. In both cases, Twitter activity
is dominated by re-tweets or repetitions of the same tweets, but In-Depth
articles attract more unique tweets. Figure 6: Differences in the distribution
of fraction of corporate retweets. In-Depth articles have a larger share of
re-tweets from the @AJEnglish and @AJELive accounts.
News items compared to In-Depth items have a more intense first hour, as can
be seen in Figure 3. For News, visits in one hour are roughly $1/12$th of the
visits in the first week, while for In-Depth they are on average around
$1/29$th. The two groups are similar to “promoted” (homepage) and “not-
promoted” stories in Digg as observed in [27].
This difference in behavior can to some extent be explained by the design of
the website. News articles are displayed more prominently on the home page,
with the most salient location being typically used by a news item; however,
In-Depth articles are also visible across the website, including a prominent
slot on the top right corner of every page. Additionally to the differences in
social media sharing that we discuss next, we observe that long-lived News
articles (in terms of effective shelf-life as defined in Section 5.2) tend to
include analysis that would actually make them fit for the In-Depth section.
Indeed, the top-3 longer lived News articles in our observation period are
“Profile: Malala Yousafzai” (Oct 10th, 2012), “Syrian rebels in uneasy
alliances” (Oct 25th, 2012), and “Malala is the daughter of Pakistan” (Oct
13th, 2012); their contents, while motivated by specific news events such as
the Syrian conflict and the shooting of a school girl, do not describe the
events but rather the context in which they are taking place.
Social media. On average the ratio of Facebook shares to tweets per article is
1.9:1, which is to some extent consistent with the survey described in Section
3.1 that indicated that there were twice as many website visitors using
Facebook as there were Twitter users. Additionally, In-Depth articles are
shared more on Facebook given the same level of activity on Twitter, as shown
in Figure 4. On average News articles have 1.6 Facebook shares per tweet,
while In-Depth articles have 2.7.
As shown in Figure 5 there is also a difference in the number of unique
tweets. On average, 17% of the tweets about News articles are unique, versus
25% of the tweets about In-Depth articles. This means that a majority of users
do not change the content of the tweets when clicking on the “tweet” button
next to the articles, or when retweeting from another Twitter user.
There is also a difference in the number of corporate retweets, as shown in
Figure 6. On average, 27% of tweets about News articles are corporate
retweets, compared to 44% of tweets about In-Depth articles. This means that
for In-Depth articles a larger share of Twitter activity can be attributed to
users who are followers of @AJEnglish or @AJELive, and thus are probably more
engaged with these Twitter accounts.
Anecdotally, we know that editors spend more time crafting tweets to promote
In-Depth articles than News articles, given that the former are not as time
sensitive as the latter. In the case of News, the headline is often posted
without modifications to Twitter, which may produce a comparatively less
appealing tweet.
### 4.2 Analysis of news articles
We observe that News articles have attention profiles that are quite
predictable, while In-Depth and other article categories show significantly
more variability. We focus on the first 12 hours-equivalent after publication
of each article in the News section, and observe the time series of data from
all sources including internal links, external links, search engines, and
social media (similarly to the time series shown in Figure 2, but for each
individual article instead of as an average). We then classify articles into
several classes based on visit patterns that are apparent from these
observations, starting with the largest class (“decreasing”) and following
with the other classes. The classification is done by the authors seeking
consensus and discussing borderline cases.
At a high level, the classes of articles in our News sample can be roughly
described by an “80:10:10 rule”. The traffic to $\sim$80% of the articles
decreases monotonically during the first 12 hours, the traffic to $\sim$10%
does not decrease, and the traffic to the remaining $\sim$10% decreases first,
but then rebounds. Articles on each are listed in Appendix A.
Next we provide a brief description of each class and examples of stories
appearing in each of them. We remark that in this work we do not attempt to
provide a comprehensive content-based typology for news articles within each
attention profile.
Decreasing (78%). The largest article class represents about 78% of the sample
set. Articles in this class demonstrate an initial spike in visits following
article publication, followed by a rather consistent drop in the number of
visits, either immediately (244 articles), or after a short delay (7
articles).
Delayed onset traffic decreases have been observed before, such as in [21]
with respect to the shooting in Aurora, Colorado, in 2012. This attention
pattern can often be attributed to breaking news that resonates with readers
located in a time zone that is off-peak when the article is first posted, such
as when that portion of the audience is mostly asleep. A story about Hurricane
Sandy’s movement up the East Coast of the United States, for example, sees an
initially sharp visit growth that begins to decline as the East Coast retires
for the evening.
The predominance of this class of article indicates that while news itself
occurs, and can even be covered, at a constant rate, in most cases readers
will only be interested on a news article for a brief period of time after its
publication.
Steady or Increasing (12%) Roughly 9% of the sample’s articles retain
relatively constant visitor rates during their first 12 hours. Compared to
news categories with very short shelf-lives, such as sports news, these
articles are remarkably consistent. In this subset of news articles, dramatic
news and emotional stories appear to garner Facebook shares and, often as a
result, extended shelf-lives.
In the U.S., multiple articles on Obama and Romney’s sharp-tongued
presidential debate drive consistent Facebook and Twitter responses for a
relatively long period of time following the articles’ publication. A poll on
racism in the US has similar staying power and Facebook sharing. In Central
Asia, the Taliban attack on Pakistani schoolgirl Malala appears in a number of
these articles, where consistent Facebook sharing buoys the article traffic
beyond average shelf-life. In Europe, furor over a seismology scandal is
posted to Facebook, while in the Middle East, atrocities in the war in Syria
and violence between Israel and Hamas also generate hours of steady traffic.
Africa sees a new prime minister in Libya, the police shooting of 34 striking
miners in South Africa, and a bomb attack on a church in Nigeria, all of which
see sustained traffic thanks in part to significant Facebook and Twitter
sharing many hours after their initial publication.
Stories in this group were mostly developing stories and many of them had
regular updates. One such example is the story about Malala for which Al
Jazeera sent a correspondent to the Swat Valley. Being a complex region to
cover, a series of news articles and feature stories were written. In
addition, Al Jazeera reached out to the Reddit community for a Q&A session
which topped the “Ask me anything (AmA)”
section999http://www.reddit.com/r/IAmA/comments/11p6q3/i_am_asad_hashim_journalist_for_the_al_jazeera/.
section.
A relatively small number of articles (3% of our sample) buck the usual trend
and see increased page traffic as time passes after their publication, rather
than a decline. To the extent that these articles can be generalized, they
resemble the class of articles detailed above. Some of these articles were
also updated with supporting content. For at least half of them, web producers
added video packages after publication, which may explain to some extent the
increase in visits.
Rebounding (10%). About 10% of the articles in our sample initially exhibit a
decline in visits/minute, until a point where such decline is reversed. This
“rebound” occurs either because of internal or external links.
In the case of internal traffic, the traffic patterns behind these rebounding
articles sometimes reflect the common newsroom practice of linking to previous
coverage in more recent articles. This practice provides additional background
context to readers just arriving at the story, but also helps news
organizations extract additional value from articles that are otherwise
statistically becoming valueless. Stories that required a significant
investment of resources to produce are also promoted more heavily than regular
articles. We can see that in these cases, these internal links do indeed
deliver readers to articles whose shelf-lives have nearly expired, when
measured by homepage and social media traffic.
The articles that rebound as a result of external traffic are beneficiaries of
attention directed from outside of the news organization (e.g. a social
networking site, the website of another news network, etc.). Typically each
observed burst in external web traffic can be tracked to a single source.
Breaking stories can also gain visits as ongoing developments drive
significant additional interest. This phenomenon is evidenced, for instance,
by three rebounding articles tracking Hurricane Sandy’s descent upon the
United States.
In general, we see that when News articles cover topics that stray from “hard
news”, the article’s attention profile reflects the increased variability seen
in the In-Depth pieces. For example, some articles ostensibly cover specific
actualities, but also bridge into long-standing issues: in the U.S.,
“Immigrant family in pursuit of the American Dream” and “Living the modern
American Dream” stoke passions around immigration.
The sometimes blurry line between reporting on immediate actualities and
longer-term trends like immigration is an area of tension in journalism, one
identified by Galtung and Ruge when they asked “how do ’events’ become
’news”’? [11].
## 5 Improving Traffic Predictions
Using Social Media Data
An increased amount of social media reactions is often correlated with more
traffic to online articles. This is particularly marked in the case of non-
decreasing and rebounding News articles, as well as In-Depth articles whose
visitation patterns are more varied and less predictable than regular
(decreasing) News articles. In this section, we combine social media reactions
with early visitation measures to provide improved predictions of (i) the
volume of visits to an article after 7 days from its publication and (ii) the
effective shelf-life of articles, i.e. the time during which they will receive
most of their visits. We begin by fitting models to our sample data, and then
explore the practicality of this approach for new data.
### 5.1 Modeling visiting volume
Our first goal is to determine to what extent social media reactions can
improve the prediction of the overall popularity (total number of visits) of
an article. The dependent variable that we want to describe with our models is
the total number of visits after 7 days ($v7d$). We use a straightforward
approach to answer this question—linear regression models. We include the
following variables (described in Section 3.3) as observed at the time at
which the prediction is performed: number of visits ($v$), number of visits
from link referrals ($vr$) and from “direct” traffic from e-mail/IM ($vd$),
shares on Facebook ($f$), Twitter ($t$), mean number of followers of people
sharing on Twitter (foll), entropy of tweets ($ent$), number and fraction of
unique tweets ($uni$, $unip$) and fraction of corporate retweets ($cp$). We
use a linear regression model that includes all first-order effects as well as
second order interactions. We included second-order interactions because of
the interdependency of the variables (e.g. an article with more visits is more
likely to have more social media reactions):
$lm(vis7d\sim v)$
$lm(vis7d\sim(v+vr+vd+f+t+\textrm{\em{foll}}+ent+uni+unip+cp)^{2})$.
Figure 7: Proportion of explained variance ($r^{2}$) for the prediction of total volume of visits, for News and In-depth articles. Table 4: Modeling visiting volume after 7 days: Significance levels for regression models after 20 minutes. Variable | In-depth | News
---|---|---
Facebook shares | 0.0349 | * | 0.0204 | *
Twitter tweets | 0.0026 | ** | $<$0.0001 | ***
Twitter entropy | $<$0.0001 | *** | 0.0003 | ***
Twitter avg. followers | $<$0.0001 | *** | |
Volume of unique tweets | | | $<$0.0001 | ***
Unique tweets % | | | $<$0.0001 | ***
Corporate retweets % | 0.0092 | ** | |
The distribution of visits to articles is log-normal distributed in our data,
consistently with previous works [29, 4]. We log-transform ${log(x+1)}$ the
visits as well as the volume of social media reactions. For t=5, 10, 15, …we
calculate the proportion of the explained variance of these two linear models.
The result is shown in Figure 7.
It takes about 3 hours to be able to explain $>0.6$ of the variance for In-
Depth articles, and the additional variables are profitable from the first
minutes. After 10-20 minutes we observe the largest difference in our
regression models (+0.5 in terms of $r^{2}$).
We take a closer look at the model variables after 20 minutes to identify the
sources of this improvement. For this purpose we stepwise fit the model
variables by AIC (Akaike information criterion) as implemented in stats.step
in R. Table 4 shows the reliability of the Social media variables to serve as
good predictor for the volume of visits after 7 days.
The fraction of traffic from different sources does not appear to be a
reliable predictor when all variables are used for the model; when we reduce
the model to exclusively these two variables, the traffic from e-mail/IM is a
more reliable predictor than the traffic from external links.
Social media variables, particularly the number of tweets and the entropy of
the vocabulary used in them, seem to be reliable predictors for both In-Depth
and News articles. The number of followers of people posting an article on
Twitter together with the fraction of corporate retweets seem to be
particularly important for In-Depth articles. A possible interpretation is
that the response to these articles has a larger component driven by
influential accounts and the actions of Al Jazeera editors. In contrast, the
number and fraction of unique tweets can be used for the prediction of traffic
to News articles. Consequently, a rich online discussion around a breaking
within its first minutes is a signal of potentially high and sustained user
interest.
Figure 8: Distribution of effective shelf-life. Table 5: Modeling effective shelf-life: Significance levels for regression models after 20 minutes. Variable | In-depth | News
---|---|---
Visits $R^{2}$ | 0.0005 | | 0.0921 |
Social media $R^{2}$ | 0.4457 | | 0.2193 |
Social media $R^{2}$ adjusted | 0.2274 | | 0.1505 |
Twitter tweets | 0.0138 | * | 0.0061 | **
Twitter entropy | 0.0027 | ** | 0.0024 | **
Twitter avg. followers | | | 0.0001 | ***
Volume of unique tweets | 0.0026 | ** | |
Unique tweets % | 0.0190 | * | 0.0445 | *
Corporate retweets | 0.0001 | *** | |
Traffic from e-mail/IM | 0.0482 | * | |
### 5.2 Modeling shelf-life
We define the effective shelf-life $\tau_{\ell}$ of an article as the time
passed between its first visit and the time at which it has received a
fraction $\ell$ of the visits it will ever receive. In this work we set
$\ell=0.90$, but similar values (e.g. $0.85$, or $0.95$) yield similar results
to the ones presented here. When $\ell=0.50$ this is equivalent to half-life
[5, 6].
Given that our observation period is finite, we use a seven-day observation
period as a proxy for the total number of visits the articles will ever
receive, as for basically all the articles in our sample, there is little
activity after 3 or 4 days. This is consistent with the experience of Al
Jazeera editors and with observations in previous works (e.g. [30]). We remark
however that there are rare cases where an article is “re-born” after weeks,
for instance when it provides background information for a new development.
The distribution of the shelf-life for both classes is depicted in Figure 8.
As observed in the qualitative analysis, the average shelf-life of In-Depth
articles, 2 days and 9 hours, is longer than the one of News articles, 1 day
and 16 hours. Their average half-lives are respectively 20 hours and 8 hours
(both are shorter than the 36 hours observed by [9]).
We observe that the effective shelf-life of all articles is independent from
their total number of visits after 7 days (Pearson’s correlation $r=-0.03$).
This will lead to low accuracy when predicting based solely on visits. For the
predictive task the linear regression model setup is analogous to the one used
to model visiting volume; this time the dependent variable is $\tau_{90}$. Our
focus is again on the variables after 20 minutes. Running the first regression
model (only visits) for this time period reveals differences for News and In-
Depth stories (Table 5). For News stories, at least 9.2% of effective shelf-
life variance can be described, while visits show no predictive information
for In-Depth stories. Including social media variables changes this picture
dramatically. Especially for In-Depth stories, a significant part of the
variance can now be described. Stepwise fitting of the social media models
shows that the number of Facebook shares and the traffic from external links
are no reliable predictors for effective shelf-life. In contrast, all Tweet
variables reach significant levels. For In-Depth articles corporate retweets
and traffic from e-mail/IM also serve as reliable predictors.
In a nutshell, using social media variables to model effective shelf-life of
stories can increase the accuracy of early prediction significantly. This is a
very promising result for future research given that we describe the effective
shelf-life pattern with data from one single time point without the use of
time series or other elaborate models.
### 5.3 Online predictions
Figure 9: Screenshot of http://fast.qcri.org/ depicting predictions for four
articles. Green bars indicate number of pageviews so far, gray bars indicate
predictions. Exact numbers are business-sensitive so they are omitted.
A live system implementing these ideas on data from the Al Jazeera English
website is available online.101010http://fast.qcri.org/ This allows us to
further test the effectiveness of our methods in an online setting, in
addition to the off-line tests we have described so far. Figure 9 shows a
screenshot of this application.
The system collects data for all articles irrespective of their section, and
produces predictions for all articles in the News section using one set of
models, and for all the remaining articles (In-Depth, Videos, Programmes,
etc.) using another set. In each model set there are models that are executed
1 hour, 6 hours, 12 hours and 24 hours after an article is published. The
target variable in this live system is page-views after 3 days. Every 24
hours, it re-trains the models by adding the articles that have passed the
3-day deadline to the training set. After an initial warm up period of 3
weeks, we monitored all 350 article URLs published during a period of 1 week
in July 2013 and kept all the predictions done by the system.
First, we evaluated the coverage of our system, which as explained in Section
3.2 is designed to focus on the top 30 most visited URLs in every 5-minute
period. In practice, we produce predictions within 6 hours for 194 (55%) of
the articles seen. Taking as comparison Google Analytics, which is also used
by this website,111111http://analytics.google.com/ we observe that this covers
65% of the page-views to Al Jazeera English articles. We remark that this
partial coverage is not an intrinsic aspect of the system, but a limitation of
using public (instead of paid) access to Twitter’s API.
Second, we evaluated the quality of the predictions. In order to do so, we
store the predictions done by the different models. Conceptually, each of the
350 articles is in the testing set, and the training set is composed of all
the articles published in the period of 1 to 4 weeks before its publication
day. Figure 10 compares the actual number of page-views with predictions done
1 hour and 6 hours after an article is published. The quality of the
predictions after one hour is similar to off-line tests ($r^{2}=0.72$) even
when we are mixing articles from other sections; 134 articles (38%) not in
News or In-Depth – but we also remark the predictive horizon of the live
system is shorter (3 days vs 7 days). Predictions after 6 hours have
$r^{2}=0.85$. The location of the best trade-off between timeliness and
accuracy in the range of 1 to 6 hours is an important problem, which requires
understanding how editors react to the predictions and use them in practice.
Figure 10: Predictions of visits after 3 days using the online system across
all articles. Left: predictions 1 hour after publication. Right: predictions 6
hours after publication.
## 6 Conclusions
Main findings. By adopting an integrated view of users’ behavior, we have
observed that there are two classes of articles that generate qualitatively
and quantitatively different responses from readers. News articles describing
breaking news events tend to decay in attention shortly after they are
published and thus have a shorter shelf-life. These articles also have more
repetitive social media reactions, as most users simply repeat the news
headlines without commenting on them. In-Depth items portraying or analyzing a
topic tend to exhibit a longer shelf-life and a richer social media response,
including more content-rich tweets in terms of vocabulary entropy and fraction
of unique tweets, and more shares on Facebook for the same level of tweets.
By going deeper into the first few hours after publication of News articles,
we found three distinctive response patterns in a roughly 80:10:10 proportion:
decreasing traffic, steady or increasing traffic, and rebounding traffic. We
found that there can be multiple causes for non-decreasing traffic, including
the addition of new content to articles, social media reactions, and other
types of referrals.
We have shown that social media signals can improve by a large margin the
accuracy of predictions of future visits, as well as the accuracy of
predictions of article shelf-life. In particular for In-Depth articles which
exhibit more complex visit patterns over time, we have found that
incorporating measures of the quantity and variety of social media reactions
can lead to substantial gains in terms of prediction accuracy.
Practical significance. From the perspective of a news provider, while no
automatic system can replace editorial judgment, understanding and predicting
the life cycle of stories has three main benefits:
* •
In the case of News stories, knowing how the audience is interacting with an
article is not just “nice to have”, but increasingly a critical component in
delivering timely and relevant content to an ever growing online audience.
* •
For In-Depth stories, which operate on a slower news cycle, knowing when to
allocate additional time and resources can significantly improve the news
planning process. This is particularly useful for an emerging class of news
programmes that combine live online discussions with more traditional TV
coverage.
* •
To a web producer, an article with a longer shelf-life means judicious time
can be spent preparing backgrounder pieces which are valuable in providing
context to a story. From a reach perspective, articles with steady or
increasing levels of traffic translate into higher user engagement.
Our work depends on having access to a large repository of social media
reactions. As more people get into social media (e.g. Twitter), this line of
work will become more relevant and will be able to produce even higher quality
predictions.
Limitations and future work. We combine findings from computer science,
journalism, and media studies. The research presented here is more difficult
to execute than the traditional single-discipline study, but we expect
interdisciplinary work on this area to become increasingly common as our media
and technology continue to converge.
Our data gathering system collects only aggregate information and does not
attempt to link actions across sessions or users across platforms; this
prevents us from separating post-read from pre-read sharing, an important
distinction explored in [1]. Another limitation of our work is that we used
data from a single website, and we are in the process of gathering data from
other sources in order to strengthen our claims. We also used a manual process
for categorization of the article classes
(decreasing/steady/increasing/rebounding), and we did not attempt a
comprehensive content-based classification of articles inside each class.
In this work, we used linear models and did not attempt anything more
sophisticated. We do not claim that our models are the more accurate that can
be built using this data, but used them to demonstrate in a clear way the
importance of social media signals for the predictive tasks we undertake.
Better models are definitively possible, and may yield even larger gains in
accuracy when incorporating social media signals.
We also used a data-driven approach in which shelf-life is derived from
observations. Alternatively, shelf-life can be derived by fitting a visitation
curve produced by a parametrized model [29]. This may lead to an improvement
in the prediction accuracy.
Reproducibility. The data sample used for this study, including feature
vectors and the categorization of articles done during the qualitative
analysis, is available for research purposes upon request. A live demo is
available at http://fast.qcri.org/
Acknowledgments. The authors wish to thank Al Jazeera English for the data
used for this study, Kiran Garimella from QCRI for his work in the live
system, Janette Lehmann from Yahoo! Research for her valuable help and
comments on an early version of this manuscript; Michael K. Martin and Ju-Sung
Lee from Carnegie Mellon University for insightful discussions on regression
models; and Edward Schiappa for his feedback on the methodology for the
analysis of article classes.
Key references: [7, 27]
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## Appendix A Example articles
List of articles in the non-majority classes described in the qualitative
assessment of Section 4.2. The data sample is available for research purposes
upon request.
Delayed decreasing:
Hurricane Sandy moves up US Atlantic coast - Americas
Skydiver lands safely after record jump - Americas
Third-party candidates spar in US debate - Americas
Arrests by French police foiled ’bomb plot’ - Europe
Scotland’s independence referendum signed - Europe
Rival protesters clash in Egypt’s capital - Middle East
Syria opposition ’captures’ Assad soldiers - Middle East
Steady:
Bomb attack hits northern Nigerian church - Africa
Libya assembly elects new prime minister - Africa
Police admit ’overreacting’ at Marikana - Africa
Marking the Cuban missile crisis - Americas
Obama and Romney face off in final debate - Americas
Obama and Romney meet in combative debate - Americas
Poll finds fresh increase in US racism - Americas
US exports to Iran soar despite sanctions - Americas
Asad Hashim: Ask Me Anything on Malala - Central & South Asia
Clerics declare Malala shooting ’un-Islamic’ - Central & South Asia
India suspends Kingfisher licence - Central & South Asia
Pakistani schoolgirl Malala arrives to UK - Central & South Asia
Profile: Malala Yousafzai - Central & South Asia
Teenage rights activist shot in Pakistan - Central & South Asia
Italian seismologists could face jail term - Europe
Karadzic to begin Srebrenica defence at Hague - Europe
Russia says fighters killed in North Caucasus - Europe
Scientists found guilty in Italy quake trial - Europe
Bomb blast hits Damascus’ Old City - Middle East
Fatah claims victory in West Bank poll - Middle East
Fighting dims hopes for Syria Eid truce - Middle East
Hariri calls on Lebanese to attend funeral - Middle East
Israel strikes Gaza after Hamas retaliation - Middle East
Marginalisation of disabled people in Egypt - Middle East
Palestinians vote in municipal elections - Middle East
Rights group says Syria used cluster bombs - Middle East
Syrian children killed in Idlib air raids - Middle East
US and EU urge political stability in Lebanon - Middle East
Increasing:
Colombia and FARC rebels launch negotiations - Americas
Immigrant family in pursuit of American Dream - Americas
Living the modern ’American Dream’ - Americas
Man charged over attempted US bank bomb plot - Americas
Minors flee Central American violence - Americas
Anti-austerity protests erupt in Athens - Europe
Lithuanians vote out austerity government - Europe
Scientists await verdict in Italy quake trial - Europe
Assault on Yemen base blamed on al-Qaeda - Middle East
Qatari emir in historic Gaza visit - Middle East
Rebounding:
African and EU leaders to hold Mali summit - Africa
Evidence of mass murder after Gaddafi’s death - Africa
Nigerian soldiers kill dozens of civilians - Africa
State-linked Libyan militias shell Bani Walid - Africa
Tunisia clash leaves opposition official dead - Africa
UN urges military action plan for Mali - Africa
Wounded Mauritania president flown to Paris - Africa
Argentine crew to vacate ship seized in Ghana - Americas
Armstrong ’unaffected’ by doping report - Americas
Biden and Ryan set for crucial VP debate - Americas
Brazil forces set for raid on Rio slums - Americas
Candidates spar in US vice president debate - Americas
Cuba’s Castro appears in public - Americas
First planet with four suns discovered - Americas
Forecasters predict ’serious’ Hurricane Sandy - Americas
Hurricane Sandy approaches eastern US - Americas
Tsunami warning for Hawaii lifted - Americas
US deficit tops $1 trillion for fourth year - Americas
US East Coast prepares for Hurricane Sandy - Americas
Dozens dead in Afghanistan Eid suicide blast - Central & South Asia
Pakistan court probes bartering of girls - Central & South Asia
Pakistan teen activist in critical condition - Central & South Asia
Berlusconi vows to remain in political arena - Europe
Boxer a big hit as Ukraine readies for vote - Europe
EU leaders agree on banking supervisor - Europe
Germany’s Merkel reassures Greece - Europe
Merkel arrives in Greece amid tight security - Europe
Russia demands Turkey explain intercepted jet - Europe
Russian opposition aide arrested - Europe
Baghdad area hit by more deadly Eid attacks - Middle East
Eid truce awaits Syrian government response - Middle East
Kuwait police fire tear gas at protesters - Middle East
Syrian forces continue to shell Aleppo - Middle East
|
arxiv-papers
| 2013-04-10T16:04:42 |
2024-09-04T02:49:44.089700
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Castillo, Mohammed El-Haddad, J\\\"urgen Pfeffer and Matt\n Stempeck",
"submitter": "Carlos Castillo",
"url": "https://arxiv.org/abs/1304.3010"
}
|
1304.3035
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-048 LHCb-PAPER-2013-005 April 10, 2013
Measurement of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction
at low dilepton mass
The LHCb collaboration†††Authors are listed on the following pages.
The branching fraction of the rare decay $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ in the dilepton mass region from 30 to 1000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ has been measured by the LHCb
experiment, using $pp$ collision data, corresponding to an integrated
luminosity of 1.0 $\mbox{\,fb}^{-1}$, at a centre-of-mass energy of 7
$\mathrm{\,Te\kern-1.00006ptV}$. The decay mode
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ is utilized as a normalization channel. The
branching fraction ${{\cal B}(B^{0}\rightarrow K^{*0}e^{+}e^{-}})$ is measured
to be
${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7},$
where the first error is statistical, the second is systematic, and the third
comes from the uncertainties on the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow e^{+}e^{-}$
branching fractions.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk57, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
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Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De Paula2, W. De
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Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M.
Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A.
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Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C.
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P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
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Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
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Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
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T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, L. Martin54, A. Martín Sánchez7, M. Martinelli40,
D. Martinez Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, R.
McNulty12, A. Mcnab53, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40,
D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53,
P. Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
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Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
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Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
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Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,
P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T.
Skwarnicki57, N.A. Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P.
Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49,
P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone57, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The $b\\!\rightarrow s\gamma$ transition proceeds through flavour changing
neutral currents, and thus is sensitive to the effects of physics beyond the
Standard Model (BSM). Although the branching fraction of the
$B^{0}\\!\rightarrow K^{*0}\gamma$ decay has been measured [1, 2, 3] to be
consistent with the Standard Model (SM) prediction [4], BSM effects could
still be present and detectable through more detailed studies of the decay
process. In particular, in the SM the photon helicity is predominantly left-
handed, with a small right-handed current arising from long distance effects
and from the non-zero value of the ratio of the $s$-quark mass to the
$b$-quark mass. Information on the photon polarisation can be obtained with an
angular analysis of the $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ decay
($\ell=e,\mu$) in the low dilepton invariant mass squared ($q^{2}$) region
where the photon contribution dominates. The inclusion of charge-conjugate
modes is implied throughout the paper. The low $q^{2}$ region also has the
benefit of reduced theoretical uncertainties due to long distance
contributions compared to the full $q^{2}$ region [5]. The more precise SM
prediction allows for increased sensitivity to contributions from BSM. In the
low $q^{2}$ interval there is a contribution from $B^{0}\rightarrow
K^{*0}V(V\rightarrow\ell^{+}\ell^{-})$ where $V$ is one of the vector
resonances $\rho$, $\omega$ or $\phi$; however this contribution has been
calculated to be at most 1% [6]. The diagrams contributing to the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay are shown in Fig. 1.
Figure 1: Dominant Standard Model diagrams contributing to the decay
${B^{0}\rightarrow K^{*0}e^{+}e^{-}}$.
With the LHCb detector, the $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$
analysis can be carried out using either muons [7] or electrons.
Experimentally, the decay with muons in the final state produces a much higher
yield per unit integrated luminosity than electrons, primarily due to the
clean trigger signature. In addition, the much smaller bremsstrahlung
radiation leads to better momentum resolution, allowing a more efficient
selection. On the other hand, the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay
probes lower dilepton invariant masses, thus providing greater sensitivity to
the photon polarisation [5]. Furthermore, the formalism is greatly simplified
due to the negligible lepton mass [8]. It is therefore interesting to carry
out an angular analysis of the decay $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ in
the region where the dilepton mass is less than
1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The lower limit is set to
30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ since below this value the
sensitivity for the angular analysis decreases because of a degradation in the
precision of the orientation of the $e^{+}e^{-}$ decay plane due to multiple
scattering. Furthermore, the contamination from the $B^{0}\\!\rightarrow
K^{*0}\gamma$ decay, with the photon converting into an $e^{+}e^{-}$ pair in
the detector material, increases significantly as $q^{2}\rightarrow 0$.
The first step towards performing the angular analysis is to measure the
branching fraction in this very low dilepton invariant mass region. Indeed,
even if there is no doubt about the existence of this decay, no clear
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal has been observed in this region
and therefore the partial branching fraction is unknown. The only experiments
to have observed $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ to date are BaBar [9]
and Belle [10], which have collected about 30 $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ events each in the region
$q^{2}<2{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, summing over electron
and muon final states.
## 2 The LHCb detector, dataset and analysis strategy
The study reported here is based on $pp$ collision data, corresponding to an
integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$, collected at the Large Hadron
Collider (LHC) with the LHCb detector [11] at a centre-of-mass energy of 7 TeV
during 2011. The LHCb detector is a single-arm forward spectrometer covering
the pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. It includes a high precision tracking system
consisting of a silicon-strip vertex detector (VELO) surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $(\Delta p/p)$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad (SPD) and preshower (PS)
detectors, an electromagnetic calorimeter (ECAL) and a hadronic calorimeter.
Muons are identified by a system composed of alternating layers of iron and
multiwire proportional chambers. The trigger [12] consists of a hardware
stage, based on information from the calorimeter and muon systems, followed by
a software stage which applies a full event reconstruction.
For signal candidates to be considered in this analysis, at least one of the
electrons from the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay must pass the
hardware electron trigger, or the hardware trigger must be satisfied
independently of any of the daughters of the signal $B^{0}$ candidate (usually
triggering on the other $b$-hadron in the event). The hardware electron
trigger requires the presence of an ECAL cluster with a transverse energy
greater than 2.5 GeV. An energy deposit is also required in one of the PS
cells in front of the ECAL cluster, where the threshold corresponds to the
energy that would be deposited by the passage of five minimum ionising
particles. Finally, at least one SPD hit is required among the SPD cells in
front of the cluster. The software trigger requires a two-, three- or four-
track secondary vertex with a high sum of the $p_{\rm T}$ of the tracks and a
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP $\chi^{2}$ with respect
to the primary interaction greater than 16. The IP $\chi^{2}$ is defined as
the difference between the $\chi^{2}$ of the PV reconstructed with and without
the considered track. A multivariate algorithm is used for the identification
of secondary vertices consistent with the decay of a $b$-hadron.
The strategy of the analysis is to measure a ratio of branching fractions in
which most of the potentially large systematic uncertainties cancel. The decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ is used as normalization mode, since it has the same
final state as the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay and has a well
measured branching fraction [13, 14], approximately 300 times larger than
${\cal B}(B^{0}\\!\rightarrow K^{*0}e^{+}e^{-})$ in the $e^{+}e^{-}$ invariant
mass range 30 to 1000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Selection
efficiencies are determined using data whenever possible, otherwise simulation
is used, with the events weighted to match the relevant distributions in data.
The $pp$ collisions are generated using Pythia 6.4 [15] with a specific LHCb
configuration [16]. Hadron decays are described by EvtGen [17] in which final
state radiation is generated using Photos [18]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].
## 3 Selection and backgrounds
The candidate selection is divided into three steps: a loose selection, a
multivariate algorithm to suppress the combinatorial background, and
additional selection criteria to remove specific backgrounds.
Candidate $K^{*0}$ mesons are reconstructed in the $K^{*0}\rightarrow
K^{+}\pi^{-}$ mode. The $p_{\rm T}$ of the charged $K$ ($\pi$) mesons must be
larger than 400 (300) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Particle
identification (PID) information is used to distinguish charged pions from
kaons [22]. The difference between the logarithms of the likelihoods of the
kaon and pion hypotheses is required to be larger than 0 for kaons and smaller
than 5 for pions; the combined efficiency of these cuts is 88%. Candidates
with a $K^{+}\pi^{-}$ invariant mass within 130
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $K^{*0}$ mass and a
good quality vertex fit are retained for further analysis. To remove
background from $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})\phi$ and $B^{0}_{s}\\!\rightarrow\phi e^{+}e^{-}$ decays,
where one of the kaons is misidentified as a pion, the mass computed under the
$K^{+}K^{-}$ hypothesis is required to be larger than 1040
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Bremsstrahlung radiation, if not accounted for, would worsen the $B^{0}$ mass
resolution. If the radiation occurs downstream of the dipole magnet the
momentum of the electron is correctly measured and the photon energy is
deposited in the same calorimeter cell as the electron. In contrast, if
photons are emitted upstream of the magnet, the measured electron momentum
will be that after photon emission, and the measured $B^{0}$ mass will be
degraded. In general, these bremsstrahlung photons will deposit their energy
in different calorimeter cells than the electron. In both cases, the ratio of
the energy detected in the ECAL to the momentum measured by the tracking
system, an important variable in identifying electrons, is unbiased. To
improve the momentum reconstruction, a dedicated bremsstrahlung recovery
procedure is used, correcting the measured electron momentum by the
bremsstrahlung photon energy. As there is little material within the magnet,
the bremsstrahlung photons are searched for among neutral clusters with an
energy larger than 75 $\mathrm{\,Me\kern-1.00006ptV}$ in a well defined
position given by the electron track extrapolation from before the magnet.
Oppositely-charged electron pairs with an electron $p_{\rm T}$ larger than 350
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and a good quality vertex are used to
form $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ candidates. The $e^{+}e^{-}$ invariant mass is
required to be in the range 30 – 1000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or 2400 – 3400
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the two decay modes,
respectively. Candidate $K^{*0}$ mesons and $e^{+}e^{-}$ pairs are combined to
form $B^{0}$ candidates which are required to have a good-quality vertex. For
each $B^{0}$ candidate, the production vertex is assigned to be that with the
smallest IP $\chi^{2}$. The $B^{0}$ candidate is also required to have a
direction that is consistent with coming from the PV as well as a
reconstructed decay point that is significantly separated from the PV.
In order to maximize the signal efficiency while still reducing the high level
of combinatorial background, a multivariate analysis, based on a Boosted
Decision Tree (BDT) [23, *Roe] with the AdaBoost algorithm [25], is used. The
signal training sample is $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ simulated
data. The background training sample is taken from the upper sideband
($m_{B^{0}}>5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) from half of the
data sample. The variables used in the BDT are the $p_{\rm T}$, the IP and
track $\chi^{2}$ of the final state particles; the $K^{*0}$ candidate
invariant mass, the vertex $\chi^{2}$ and flight distance $\chi^{2}$ (from the
PV) of the $K^{*0}$ and $e^{+}e^{-}$ candidates; the $B^{0}$ $p_{\rm T}$, its
vertex $\chi^{2}$, flight distance $\chi^{2}$ and IP $\chi^{2}$, and the angle
between the $B^{0}$ momentum direction and its direction of flight from the
PV. A comparison of the BDT output for the data and the simulation for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays is shown in Fig. 2. The candidates for this
test are reconstructed using a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mass constraint and the background is statistically subtracted using the sPlot
technique [26] based on a fit to the $B^{0}$ invariant mass spectrum. The
agreement between data and simulation confirms a proper modelling of the
relevant variables. The optimal cut value on the BDT response is chosen by
considering the combinatorial background yield ($b$) on the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ invariant mass distribution outside the
signal region111The signal region is defined as $\pm
300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $B^{0}$ mass.
and evaluating the signal yield ($s$) using the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ simulation assuming a visible $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ branching fraction of $2.7\times 10^{-7}$. The quantity
$s/\sqrt{s+b}$ serves as an optimisation metric, for which the optimal BDT cut
is 0.96. The signal efficiency of this cut is about 93% while the background
is reduced by two orders of magnitude.
Figure 2: Output of the BDT for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ data (points) and simulation (red line).
After applying the BDT selection, specific backgrounds from decays that have
the same visible final state particles as the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ signal remain. Since some of these backgrounds have larger
branching fractions, additional requirements are applied to the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ candidates.
A large non-peaking background comes from the $B^{0}\\!\rightarrow
D^{-}e^{+}\nu$ decay, with ${D^{-}\rightarrow e^{-}{\overline{\nu}}K^{*0}}$.
The branching fraction for this channel is about five orders of magnitude
larger than that of the signal. When the neutrinos have low energies, the
signal selections are ineffective at rejecting this background. Therefore, the
$K^{*0}e^{-}$ invariant mass is required to be larger than 1900
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is 97% efficient on signal
decays. Another important source of background comes from the
$B^{0}\\!\rightarrow K^{*0}\gamma$ decay, where the photon converts into an
$e^{+}e^{-}$ pair. In LHCb, approximately 40% of the photons convert before
the calorimeter, and although only about 10% are reconstructed as an
$e^{+}e^{-}$ pair, the resulting mass of the $B^{0}$ candidate peaks in the
signal region. This background is suppressed by a factor 23 after the
selection cuts (including the 30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
minimum requirement on the $e^{+}e^{-}$ invariant mass). The fact that signal
$e^{+}e^{-}$ pairs are produced at the $B^{0}$ decay point, whereas conversion
electrons are produced in the VELO detector material, is exploited to further
suppress this background. The difference in the $z$ coordinates, $\Delta z$,
between the first VELO hit and the expected position of the first hit,
assuming the electron was produced at the $K^{*0}$ vertex, should satisfy
$|\Delta z|<30$ mm. In addition, we require that the calculated uncertainty on
the $z$-position of the $e^{+}e^{-}$ vertex be less than 30 mm, since a large
uncertainty makes it difficult to determine if the $e^{+}e^{-}$ pair
originates from the same vertex as the $K^{*0}$ meson, or from a point inside
the detector material. These two additional requirements reject about 2/3 of
the remaining $B^{0}\\!\rightarrow K^{*0}\gamma$ background, while retaining
about 90% of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal. After applying
these cuts, the $B^{0}\\!\rightarrow K^{*0}\gamma$ contamination under the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal peak is estimated to be $(10\pm
3)\%$ of the expected signal yield.
Other specific backgrounds have been studied using either simulated data or
analytical calculations and include the decays $B\rightarrow
K^{\ast}\eta,K^{\ast}\eta^{\prime},K^{\ast}\pi^{0}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{\ast}\gamma$, where
$\mathchar 28931\relax^{\ast}$ represents a high mass resonance decaying into
a proton and a charged kaon. The main source of background is found to be the
$B\rightarrow K^{\ast}\eta$ mode, followed by a Dalitz decay
($\eta\rightarrow\gamma e^{+}e^{-}$). These events form an almost flat
background in the mass range
$4300-5250{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. None of these backgrounds
contribute significantly in the $B^{0}$ mass region, and therefore are not
specifically modelled in the mass fits described later.
More generally, partially reconstructed backgrounds arise from $B$ decays with
one or more decay products in addition to a $K^{*0}$ meson and an $e^{+}e^{-}$
pair. In the case of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay, there are two sources for these partially
reconstructed events: those from the hadronic part, such as events with higher
$K^{*}$ resonances (partially reconstructed hadronic background), and those
from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ part (partially
reconstructed $J/\psi$ background), such as events coming from $\psi{(2S)}$
decays. For the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay mode, only the
partially reconstructed hadronic background has to be considered.
## 4 Fitting procedure
Since the signal resolution, type and rate of backgrounds depend on whether
the hardware trigger was caused by a signal electron or by other activity in
the event, the data sample is divided into two mutually exclusive categories:
events triggered by an extra particle $(e,\gamma,h,\mu)$ excluding the four
final state particles (called HWTIS, since they are triggered independently of
the signal) and events for which one of the electrons from the $B^{0}$ decay
satisfies the hardware electron trigger (HWElectron). Events satisfying both
requirements (20%) are assigned to the HWTIS category. The numbers of
reconstructed signal candidates are determined from unbinned maximum
likelihood fits to their mass distributions separately for each trigger
category. The mass distribution of each category is fitted to a sum of
probability density functions (PDFs) modelling the different components.
1. 1.
The signal is described by the sum of two Crystal Ball functions [27] (CB)
sharing all their parameters but with different widths.
2. 2.
The combinatorial background is described by an exponential function.
3. 3.
The shapes of the partially reconstructed hadronic and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ backgrounds are described by
non-parametric PDFs [28] determined from fully simulated events.
The signal shape parameters are fixed to the values obtained from simulation,
unless otherwise specified.
There are seven free parameters for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit for each trigger category. These include the
peak value of the $B^{0}$ candidate mass, a scaling factor applied to the
widths of the CB functions to take into account small differences between
simulation and data, and the exponent of the combinatorial background. The
remaining four free parameters are the yields for each fit component. The
invariant mass distributions together with the PDFs resulting from the fit are
shown in Fig. 3. The number of signal events in each category is summarized in
Table 1.
A fit to the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ candidates is then
performed, with several parameters fixed to the values found from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. These fixed parameters are the scaling factor
applied to the widths of the CB functions, the peak value of the $B^{0}$
candidate mass and the ratio of the partially reconstructed hadronic
background to the signal yield. The $B^{0}\\!\rightarrow K^{*0}\gamma$ yield
is fixed in the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ mass fit using the
fitted $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ signal yield, the ratio of efficiencies of the
$B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ modes, and the ratio of branching fractions ${\cal
B}(B^{0}\\!\rightarrow K^{*0}\gamma)/{\cal
B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0})$. Hence there are three free parameters for the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ fit for each trigger category: the
exponent and yield of the combinatorial background and the signal yield. The
invariant mass distributions together with the PDFs resulting from the fit are
shown in Fig. 4. The signal yield in each trigger category is summarized in
Table 1. The probability of the background fluctuating to obtain the observed
signal corresponds to 4.1 standard deviations for the HWElectron category and
2.4 standard deviations for the HWTIS category, as determined from the change
in the value of twice the natural logarithm of the likelihood of the fit with
and without signal. Combining the two results, the statistical significance of
the signal corresponds to 4.8 standard deviations.
Figure 3: Invariant mass distributions for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay mode for the (left) HWElectron and (right)
HWTIS trigger categories. The dashed line is the signal PDF, the light grey
area corresponds to the combinatorial background, the medium grey area is the
partially reconstructed hadronic background and the dark grey area is the
partially reconstructed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
background component.
Figure 4: Invariant mass distributions for the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay mode for the (left) HWElectron and (right) HWTIS trigger categories. The dashed line is the signal PDF, the light grey area corresponds to the combinatorial background, the medium grey area is the partially reconstructed hadronic background and the black area is the $B^{0}\\!\rightarrow K^{*0}\gamma$ component. Table 1: Signal yields with their statistical uncertainties. Trigger category | $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(e^{+}e^{-})K^{*0}$ | $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
---|---|---
HWElectron | $5082\pm 104$ | $15.0\,^{+5.1}_{-4.5}$
HWTIS | $4305\pm 101$ | $14.1\,^{+7.0}_{-6.3}$
## 5 Results
The $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction is calculated in
each trigger category using the measured signal yields and the ratio of
efficiencies
$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=$
$\displaystyle\frac{N(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})}{N(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0})}\times r_{\rm sel}\times r_{\rm PID}\times r_{\rm
HW}$ $\displaystyle\times{\cal
B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})\times{\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow e^{+}e^{-}),$
where the ratio of efficiencies is sub-divided into the contributions arising
from the selection requirements (including acceptance effects, but excluding
PID), $r_{\rm sel}$, the PID requirements $r_{\rm PID}$ and the trigger
requirements $r_{\rm HW}$. The values of $r_{\rm sel}$ are determined using
simulated data, while $r_{\rm PID}$ and $r_{\rm HW}$ are obtained directly
from calibration data samples: ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow e^{+}e^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$ from
$D^{*+}$ decays for $r_{\rm PID}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays for $r_{\rm HW}$. The values are summarized
in Table 2. The only ratio that is inconsistent with unity is the hardware
trigger efficiency due to the different mean electron $p_{\rm T}$ for the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays.
The branching fraction for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decay
mode is taken from Ref. [14] and a correction factor of 1.02 has been applied
to take into account the difference in the $K\pi$ invariant mass range used,
and therefore the different S-wave contributions.
Table 2: Ratios of efficiencies used for the measurement of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction. The ratio $r_{\rm HW}$ for the HWTIS trigger category is assumed to be equal to unity. The uncertainties are the total ones and are discussed in Sec. 6. | HWElectron category | HWTIS category
---|---|---
$r_{\rm sel}$ | $1.03\pm 0.02$ | $1.03\pm 0.02$
$r_{\rm PID}$ | $1.01\pm 0.02$ | $1.03\pm 0.02$
$r_{\rm HW}$ | $1.35\pm 0.03$ | 1
The $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction, for each
trigger category, is measured to be
$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\text{HWElectron}}$
$\displaystyle=$ $\displaystyle(3.3\,^{+1.1}_{-1.0})\times 10^{-7}$
$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\text{HWTIS}}$
$\displaystyle=$ $\displaystyle(2.8\,^{+1.4}_{-1.2})\times 10^{-7},$
where the uncertainties are statistical only.
## 6 Systematic uncertainties
Several sources of systematic uncertainty are considered, affecting either the
determination of the number of signal events or the computation of the
efficiencies. They are summarized in Table 3.
The ratio of trigger efficiencies is determined using a
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ calibration sample from data, which is reweighted
using the $p_{\rm T}$ of the triggering electron in order to model properly
the kinematical properties of the two decays. The uncertainties due to the
limited size of the calibration samples are propagated to get the related
systematic uncertainty shown in Table 2.
The PID calibration introduces a systematic uncertainty on the calculated PID
efficiencies as given in Table 2. For the kaon and pion candidates this
systematic uncertainty is estimated by comparing, in simulated events, the
results obtained using a $D^{*+}$ calibration sample to the true simulated PID
performance. For the $e^{+}e^{-}$ candidates, the systematic uncertainty is
assessed ignoring the $p_{\rm T}$ dependence of the electron identification.
The resulting effect is limited by the fact that the kinematic differences
between the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ and the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
decays are small once the full selection chain is applied.
The fit procedure is validated with pseudo-experiments. Samples are generated
with different fractions or shapes for the partially reconstructed hadronic
background, or different values for the fixed signal parameters and are then
fitted with the standard PDFs. The corresponding systematic uncertainty is
estimated from the bias in the results obtained by performing the fits
described above. The resulting deviations from zero of each variation are
added in quadrature to get the total systematic uncertainty due to the fitting
procedure. The parameters of the signal shape are varied within their
statistical uncertainties as obtained from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. An alternate signal shape, obtained by studying
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ signal decays in data both with and without a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass constraint is also tried;
the difference in the yields from that obtained using the nominal signal shape
is taken as an additional source of uncertainty. The ratio of the partially
reconstructed hadronic background to the signal yield is assumed to be
identical to that determined from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. The systematic uncertainty linked to this
hypothesis is evaluated by varying the ratio by $\pm 50\%$. The fraction of
partially reconstructed hadronic background thus determined is in agreement
within errors with the one found in $B^{0}\\!\rightarrow K^{*0}\gamma$ decays
[29]. The shape of the partially reconstructed background used in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ and the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ fits
are the same. The related systematic uncertainty has been evaluated using an
alternative shape obtained from charmless $b$-hadron decays. The
$B^{0}\\!\rightarrow K^{*0}\gamma$ contamination in the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ signal sample is $1.2\pm 0.4$ and $1.5\pm 0.5$ events for
the HWElectron and HWTIS signal samples, respectively. Combining the
systematic uncertainties in quadrature, the branching fractions are found to
be
$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\rm
HWElectron}$ $\displaystyle=$ $\displaystyle(3.3\,^{+1.1\mbox{
}+0.2}_{-1.0\mbox{ }-0.3}\pm 0.2)\times 10^{-7}$ $\displaystyle{\cal
B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\rm
HWTIS}$ $\displaystyle=$ $\displaystyle(2.8\,^{+1.4\mbox{ }+0.2}_{-1.2\mbox{
}-0.3}\pm 0.2)\times 10^{-7},$
where the first error is statistical, the second systematic, and the third
comes from the uncertainties on the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow e^{+}e^{-}$
branching fractions [13, 14]. The branching ratios are combined assuming all
the systematic uncertainties to be fully correlated between the two trigger
categories except those related to the size of the simulation samples. The
combined branching ratio is found to be
${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7}.$
Table 3: Absolute systematic uncertainties on the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching ratio (in $10^{-7}$) . Source | HWElectron category | HWTIS category
---|---|---
Simulation sample statistics | 0.06 | 0.05
Trigger efficiency | 0.07 | -
PID efficiency | 0.08 | 0.10
Fit procedure | ${}^{+0.09}_{-0.22}$ | ${}^{+0.07}_{-0.23}$
$B^{0}\\!\rightarrow K^{*0}\gamma$ contamination | 0.08 | 0.08
Total | ${}^{+0.17}_{-0.26}$ | ${}^{+0.16}_{-0.27}$
## 7 Summary
Using $pp$ collision data corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in 2011 at a centre-of-
mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$, a sample of approximately 30
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ events, in the dilepton mass range 30
to 1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, has been observed. The
probability of the background to fluctuate upward to form the signal
corresponds to 4.6 standard deviations including systematic uncertainties. The
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay mode is utilized as a normalization channel,
and the branching fraction $\cal B$($B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$) is
measured to be
${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7}.$
This result can be compared to theoretical predictions. A simplified formula
suggested in Ref. [5] takes into account only the photon diagrams of Fig. 1.
When evaluated in the 30 to 1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$e^{+}e^{-}$ invariant mass interval using $\cal B$($B^{0}\\!\rightarrow
K^{*0}\gamma$) [1, 2, 3], it predicts a $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
branching fraction of $2.35\times 10^{-7}$. A full calculation has been
recently performed [30] and the numerical result for the $e^{+}e^{-}$
invariant mass interval of interest is $(2.43^{+0.66}_{-0.47})\times 10^{-7}$.
The consistency between the two values reflects the photon pole dominance. The
result presented here is in good agreement with both predictions.
Using the full LHCb data sample obtained in 2011 – 2012 it will be possible to
do an angular analysis. The measurement of the $A_{\mathrm{T}}^{2}$ parameter
[8] thus obtained, is sensitive to the existence of right handed currents in
the virtual loops in diagrams similar to those of Fig. 1. For this purpose,
the analysis of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay is
complementary to that of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ mode.
Indeed, it is predominantly sensitive to a modification of $\mathcal{C}_{7}$
(the so-called $\mathcal{C}_{7}^{{}^{\prime}}$ terms) while, because of the
higher $q^{2}$ in the decay, the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
$A_{\mathrm{T}}^{2}$ parameter has a larger possible contribution from the
$\mathcal{C}_{9}^{{}^{\prime}}$ terms [31].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-10T17:42:51 |
2024-09-04T02:49:44.102935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M.\n De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff,\n L. Del Buono, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen,\n E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P.\n Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik,\n P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina,\n I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki,\n J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev,\n Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, R. McNulty, A.\n Mcnab, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello,\n R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B.\n Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K.\n Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak,\n S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas,\n J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch,\n A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Marie-Helene Schune",
"url": "https://arxiv.org/abs/1304.3035"
}
|
1304.3054
|
# Friedmann equations from emergence of cosmic space
Ahmad Sheykhi111 [email protected] Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Center for Excellence in Astronomy and Astrophysics (CEAA-RIAAM) Maragha, P.
O. Box 55134-441, Iran
###### Abstract
Padmanabhan [arXiv:1206.4916] argues that the cosmic acceleration can be
understood from the perspective that spacetime dynamics is an emergence
phenomena. By calculating the difference between the surface degrees of
freedom and the bulk degrees of freedom in a region of space, he also arrived
at Friedmann equation in flat universe. In this paper, by modification his
proposal, we are able to derive the Friedmann equation of the Friedmann-
Robertson-Walker (FRW) Universe with any spatial curvature. We also extend the
study to higher dimensional spacetime and derive successfully the Friedmann
equations not only in Einstein gravity, but also in Gauss-Bonnet and more
general Lovelock gravity with any spacial curvature. This is the first
derivation of Friedmann equations in these gravity theories in a nonflat FRW
Universe by using the novel idea proposed by Padmanabhan. Our study indicates
that the approach presented here is enough powerful and further supports the
viability of the Padmanabhan’s perspective of emergence gravity.
PACS number:04.20.Cv, 04.50.-h, 04.70.Dy
## I Introduction
Physicists have been speculating on the nature and origin of gravity for a
long time. Newton believed that gravity is just a force like other forces of
the nature and does not affect on the space. This was a general belief until
Einstein presented his theory of general relativity in $1915$. According to
Einstein’s theory, gravity is just the spacetime curvature. In this new
picture, the matter field tells space (geometry) how to curve, and the
geometry tells matter how to move. Also, according to the equivalence
principle of general relativity, gravity is just the dynamics of spacetime.
This implies that gravity is an emergent phenomenon.
In $1970^{\prime}s$ thermodynamics of black holes were studied. According to
laws of black holes mechanics, a black hole can be regarded as a
thermodynamical system which has temperature proportional to its surface
gravity and an entropy proportional to its horizon area. This indicates that
geometrical quantities such as horizon area and surface gravity are closely
related to the thermodynamic quantities like temperature and entropy. Are
there a direct connection between gravitational field equations describing the
geometry of spacetime and the first law of thermodynamics? Jacobson Jac was
indeed the first who answered this question by disclosing that the Einstein
field equations can be derived by applying the Clausius relation $\delta
Q=T\delta S$ on the horizon of spacetime, here $\delta S$ is the change in the
entropy and $\delta Q$ and $T$ are, respectively, the energy flux across the
horizon and the Unruh temperature seen by an accelerating observer just inside
the horizon.
The next great step toward understanding the nature of gravity put forwarded
by Verlinde Ver in $2010$ who claimed that gravity is not a fundamental
interaction but should be interpreted as an entropic force caused by changes
of entropy associated with the information on the holographic screen. Applying
the first principles, namely the holographic principle and the equipartition
law of energy, Verlinde derived the Newton’s law of gravitation, the Poisson
equation and in the relativistic regime the Einstein field equations. Although
in Pad0 Padmanabhan observed that the equipartition law of energy for the
horizon degrees of freedom combining with the thermodynamic relation $S=E/2T$,
leads to the Newton’s law of gravity, however, the idea that gravity is not a
fundamental force and can be interpreted as the entopic force was first
pointed by Verlinde Ver . Following Ver , some attempts have been done to
investigate the entropic origin of gravity in different setups (see Cai4 ;
Other ; newref ; sheyECFE ; Ling ; Modesto ; Yi ; Sheykhi2 and references
therein). Nevertheless, there are some critical comments on Verlinde’s
proposal crit . Strong criticism against the entropic origin of gravity was
presented by Visser Vis who claimed that the interpretation of gravity as an
entropic force is untenable. According to Visser arguments Vis , if one would
like to reformulate classical Newtonian gravity in terms of an entropic force,
then the fact that Newtonian gravity is described by a conservative force
places significant constraints on the form of the entropy and temperature
functions.
Although Verlinde’s proposal has changed our understanding on the origin and
nature of gravity, but it considers the gravitational field equations as the
equations of emergent phenomenon and leave the spacetime as a background
geometric which has already exist. Is it possible to regard the spacetime
itself as an emergent structure? Recently, by calculating the difference
between the surface degrees of freedom and the bulk degrees of freedom in a
region of space, Padmanabhan Pad1 argued that spacetime dynamics can be
emerged. As a result, he is able to explain the origin of the acceleration of
the universe expansion from his new perspective Pad1 . According to
Padmanabhan, the spatial expansion of our universe can be regarded as the
consequence of emergence of space and the cosmic space is emergent as the
cosmic time progresses. Using this new idea, Padmanabhan Pad1 derived the
Friedmann equation of a flat FRW Universe. Following Pad1 , Cai obtained the
Friedmann equation of a higher dimensional FRW Universe in Einstein, Gauss-
Bonnet and Lovelock theory Cai1 . Similar derivation were also made by the
authors of Yang . Instead of modifying the number of degrees of freedom on the
holographic surface of the Hubble sphere, and the volume increase, the authors
of Yang , assumed that $(dV/dt)$ is proportional to a function $f(\triangle
N)$. Here $\triangle N=N_{\mathrm{sur}}-N_{\mathrm{bulk}}$, where
$N_{\mathrm{sur}}$ is the number of degrees of freedom on the boundary and
$N_{\mathrm{bulk}}$ is the number of degrees of freedom in the bulk. When the
volume of the spacetime is constant, the function $f(\triangle N)$ is equal to
zero. It is worth mentioning that the authors of Cai1 ; Yang only derived the
Friedmann equations of the spatially flat FRW Universe in Gauss-Bonnet and
Lovelock gravities, and failed to arrive at Friedmann equations with any
spacial curvature in these gravity theories. For this purpose, they proposed
the Hawking temperature associated with the Hubble horizon to be $T=H/2\pi$
and the volume of the universe is $V=4\pi H^{-3}/3$.
In this paper, by modifying the original proposal of Padmanabhan Pad1 , we are
able to derive the Friedmann equation of the FRW Universe with any spacial
curvature. Note that in a nonflat universe, the Hawking temperature and the
volume are usually taken as $T=1/2\pi\tilde{r}_{A}$ and
$V=4\pi\tilde{r}^{3}_{A}/3$, respectively, where $\tilde{r}_{A}$ is the
apparent horizon radius CaiKim . We also generalize the study to the higher
dimensional spacetime and higher order gravities, and derive the corresponding
dynamical equations governing the evolution of the universe with any spacial
curvature not only in Einstein gravity, but also in Gauss-Bonnet and more
general lovelock gravity. For consistency, in all cases we set the integration
constant equal to zero. In the next section we extract the Friedmann equation
by properly modifying the proposal of Pad1 . In section III, we extend our
study to higher order gravity theory in arbitrary dimension. We summarize our
results in section IV.
## II Friedmann equation in 4D Einstein gravity
We assume the background spacetime is spatially homogeneous and isotropic
which is described by the line element
$ds^{2}={h}_{ab}dx^{a}dx^{b}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric is
$h_{ab}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of space
with $k=0,1,-1$ corresponding to flat, closed, and open universes,
respectively. The dynamical apparent horizon, a marginally trapped surface
with vanishing expansion, is determined by the relation
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$. For a dynamical
spacetime, the apparent horizon has been argued to be a causal horizon and is
associated with the gravitational entropy and surface gravity Bak . A simple
calculation gives the apparent horizon radius for the FRW Universe as Hay
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}},$ (2)
where $H=\dot{a}/a$ is the Hubble parameter. It is widely accepted that the
apparent horizon is a suitable boundary of our universe, from thermodynamic
viewpoint, for which all laws of thermodynamics are hold on it.
Thermodynamical properties of the apparent horizon has been studied in
different setups SheyW1 ; Wang2 ; Shey3 . Following Pad1 , we assume the
number of degrees of freedom on the spherical surface of apparent horizon with
radius $\tilde{r}_{A}$ is proportional to its area and is given by
$N_{\mathrm{sur}}=4S=\frac{4\pi\tilde{r}^{2}_{A}}{L_{p}^{2}},$ (3)
where $L_{p}$ is the Planck length, $A=4\pi\tilde{r}^{2}_{A}$ represents the
area of the apparent horizon and $S$ is the entropy which obeys the area law.
Assume the temperature associated with the apparent horizon is the Hawking
temperature CaiKim
$T=\frac{1}{2\pi\tilde{r}_{A}},$ (4)
and the energy contained inside the sphere with volume
$V=4\pi\tilde{r}^{3}_{A}/3$ is the Komar energy
$E_{\mathrm{Komar}}=|(\rho+3p)|V.$ (5)
According to the equipartition law of energy, the bulk degrees of freedom obey
$N_{\mathrm{bulk}}=\frac{2|E_{\mathrm{Komar}}|}{T}.$ (6)
Through this paper we set $k_{B}=1=c=\hbar$ for simplicity. The novel idea of
Padmanabhan is that the cosmic expansion, conceptually equivalent to the
emergence of space, is being driven towards holographic equipartition, and the
basic law governing the emergence of space must relate the emergence of space
to the difference between the number of degrees of freedom in the holographic
surface and the one in the emerged bulk Pad1 . He proposed that in an
infinitesimal interval $dt$ of cosmic time, the increase $dV$ of the cosmic
volume, in flat universe, is given by
$\frac{dV}{dt}=L_{p}^{2}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).$ (7)
In general, one may expect ${dV}/{dt}$ to be some function of
$(N_{\mathrm{sur}}-N_{\mathrm{bulk}})$ which vanishes when the latter does. In
this case one may regard Eq. (7) as a Taylor series expansion of this function
truncated at the first order Pad1 . This approach was studied recently Yang .
Motivated by (7), we propose the volume increase, in a nonflat FRW Universe,
is still proportional to the difference between the number of degrees of
freedom on the apparent horizon and in the bulk, but the function of
proportionality is not just a constant, and it equals to the ratio of the
apparent horizon and Hubble radius. Therefore we write down
$\frac{dV}{dt}=L_{p}^{2}\frac{\tilde{r}_{A}}{H^{-1}}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).$
(8)
It is well known that for pure de Sitter spacetime the number of degrees of
freedom in a bulk and the number of degrees of freedom on the boundary surface
are equal, namely $N_{\rm sur}=N_{\rm bulk}$ Pad1 . Since our universe, is not
exactly de Sitter but it is asymptotically de Sitter, thus for our universe,
Padmanabhan proposed Pad1
$\frac{dV}{dt}\propto(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).$ (9)
In order to arrive at the desired dynamical equations for the FRW Universe, in
Einstein gravity, he assumed the constant of proportionality to be
$L_{p}^{2}$. For a nonflat Universe and other gravity theories, the assumption
(7) does not work and we found out that it should be modified as in Eq. (8).
One may regard the assumption (8) to the status of a postulate and verify
whether it can lead to the correct Friedmann equations describing the
evolution of the Universe. In this paper, we will show that with this
modification, we are able to extract the Friedmann equations with any spacial
curvature in Einstein, Gauss-Bonnet and more general Lovelock gravity. This
may justify the correctness of our assumption in (8). For spatially flat
universe, $\tilde{r}_{A}=H^{-1}$, and one recovers the proposal (7).
Taking the time derivative of the cosmic volume $V=4\pi\tilde{r}^{3}_{A}/3$,
we have
$\frac{dV}{dt}=4\pi\tilde{r}^{2}_{A}\dot{\tilde{r}}_{A}.$ (10)
Substituting the cosmic volume $V$ and the temperature (4) in Eq. (6), we find
the numbers of degrees of freedom in the bulk as
$N_{\mathrm{bulk}}=-\frac{16\pi^{2}}{3}(\rho+3p)\tilde{r}_{A}^{4}.$ (11)
In order to have $N_{\rm bulk}>0$, we take $\rho+3p<0$ Pad1 . Substituting
Eqs. (3), (10), (11) into (8), we arrive at
$4\pi\tilde{r}^{2}_{A}\dot{\tilde{r}}_{A}=L_{p}^{2}\frac{\tilde{r}_{A}}{H^{-1}}\left[\frac{4\pi\tilde{r}^{2}_{A}}{L_{p}^{2}}+\frac{16\pi^{2}}{3}(\rho+3p)\tilde{r}^{4}_{A}\right].$
(12)
Rearranging the terms we obtain
$4\pi\tilde{r}^{2}_{A}\left(\dot{\tilde{r}}_{A}H^{-1}-\tilde{r}_{A}\right)=\frac{16\pi^{2}L_{p}^{2}}{3}(\rho+3p)\tilde{r}^{5}_{A},$
(13)
which can be simplified as
$\tilde{r}^{-3}_{A}(\dot{\tilde{r}}_{A}H^{-1}-\tilde{r}_{A})=\frac{4\pi
L_{p}^{2}}{3}\left[3(\rho+p)-2\rho\right].$ (14)
Using the continuity equation, $\dot{\rho}+3H(\rho+p)=0,$ we reach
$\tilde{r}^{-3}_{A}(\dot{\tilde{r}}_{A}H^{-1}-\tilde{r}_{A})=-\frac{4\pi
L_{p}^{2}}{3}\left[\dot{\rho}H^{-1}+2\rho\right].$ (15)
Multiplying the both hand side of (15) by factor $2\dot{a}a$, and using the
fact that $H^{-1}=a/\dot{a}$, we get
$2\dot{a}a\tilde{r}^{-2}_{A}-2a^{2}\dot{\tilde{r}}_{A}\tilde{r}^{-3}_{A}=\frac{8\pi
L_{p}^{2}}{3}\left[\dot{\rho}a^{2}+2\rho\dot{a}a\right].$ (16)
The above equation can be further rewritten as
$\frac{d}{dt}\left(a^{2}\tilde{r}_{A}^{-2}\right)=\frac{d}{dt}\left[a^{2}\left(H^{2}+\frac{k}{a^{2}}\right)\right]=\frac{8\pi
L_{p}^{2}}{3}\frac{d}{dt}(\rho a^{2}),$ (17)
where we have also used relation (2). Integrating, we obtain
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi L_{p}^{2}}{3}\rho,$ (18)
where we have set the integration constant equal to zero. In this way we
derive the Friedmann equation of the FRW Universe with any spacial curvature,
by calculating the difference between the number of degrees of freedom in the
bulk and on the apparent horizon. Let us stress here the difference between
our derivation and ones presented in Cai1 ; Yang . The authors of Cai1 ; Yang
arrived at (18), by using proposal Pad1 given by Eq. (7), and interpreting
the integration constant as the special curvature, while we arrive at the same
result by modifying the proposal of Pad1 in the form of (8), and setting the
integration constant equal to zero.
## III Friedmann equation in Gauss-Bonnet and Lovelock gravity
In this section, we apply the approach developed in the previous section to
derive the Friedmann equations in Gauss-Bonnet and more general Lovelock
gravity with any spacial curvature. This is the first derivation of Friedmann
equations in these gravity theories in a nonflat FRW Universe by using the
novel idea presented in Pad1 . We first extend the approach of the previous
section to the $(n+1)$-dimensional spacetime. In this case the number of
degrees of freedom on the apparent horizon turn out to be Cai1
$N_{\mathrm{sur}}=\alpha\frac{A}{L_{p}^{2}},$ (19)
where $A=n\Omega_{n}\tilde{r}^{n-1}_{A}$ and $\alpha=(n-1)/2(n-2)$, with
$\Omega_{n}$ is the volume of an unit $n$-sphere. We also modify our proposal
in (8) a little as
$\alpha\frac{dV}{dt}=L_{p}^{n-1}\frac{\tilde{r}_{A}}{H^{-1}}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}),$
(20)
where the volume of the $n$-sphere is $V=\Omega_{n}\tilde{r}^{n}_{A}$. The
bulk Komar energy in $(n+1)$-dimensions is given by Cai2
$E_{\rm Komar}=\frac{(n-2)\rho+np}{n-2}V,$ (21)
and hence the bulk degrees of freedom is obtained as
$N_{\rm bulk}=-4\pi\Omega_{n}\tilde{r}^{n+1}_{A}\frac{(n-2)\rho+np}{n-2},$
(22)
where we take $(n-2)\rho+np<0$ in order to have $N_{\rm bulk}>0$ Pad1 .
Substituting Eqs. (19) and (22) in relation (20), one gets
$\tilde{r}^{-2}_{A}-\dot{\tilde{r}}_{A}H^{-1}\tilde{r}^{-3}_{A}=-\frac{8\pi
L_{p}^{n-1}}{n(n-1)}[(n-2)\rho+np].$ (23)
Multiplying the both hand side by factor $2\dot{a}a$, after using the
continuity equation in $(n+1)$-dimensions as
$\dot{\rho}+nH(\rho+p)=0,$ (24)
we arrive at
$\frac{d}{dt}\left[a^{2}\left(H^{2}+\frac{k}{a^{2}}\right)\right]=\frac{16\pi
L_{p}^{n-1}}{n(n-1)}\frac{d}{dt}(\rho a^{2}).$ (25)
Integrating, we find
$H^{2}+\frac{k}{a^{2}}=\frac{16\pi L_{p}^{n-1}}{n(n-1)}\rho,$ (26)
where we have set the integration constant equal to zero. This is the
Friedmann equation of $(n+1)$-dimensional FRW Universe with any spacial
curvature CaiKim .
Up to now we only considered Einstein gravity, and derive the corresponding
Friedmann equations in a Universe with spacial curvature. Now we want to see
whether the above procedure works or not in other gravity theories such as the
Gauss-Bonnet and more general Lovelock gravity. Lovelock gravity is the most
general lagrangian which keeps the field equations of motion for the metric of
second order, as the pure Einstein-Hilbert action Lov . Let us first consider
the Gauss-Bonnet theory. The key point which should be noticed here is that in
Gauss-Bonnet gravity the entropy of the holographic screen does not obey the
area law. Static black hole solutions of Gauss-Bonnet gravity have been found
and their thermodynamics have been investigated in ample details Bou ; caigb .
The entropy of the static spherically symmetric black hole in Gauss-Bonnet
theory has the following expression caigb
$S=\frac{A_{+}}{4L_{p}^{n-1}}\left[1+\frac{n-1}{n-3}\frac{2\tilde{\alpha}}{r_{+}^{2}}\right],$
(27)
where $A_{+}=n\Omega_{n}r^{n-1}_{+}$ is the horizon area and $r_{+}$ is the
horizon radius. In the above expression $\tilde{\alpha}=(n-2)(n-3)\alpha$,
where $\alpha$ is the Gauss-Bonnet coefficient which is positive Bou . For
$n=3$ we have $\tilde{\alpha}=0$, thus the Gauss-Bonnet correction term
contributes only for $n\geq 4$. We assume the entropy expression (27) also
holds for the apparent horizon of the FRW Universe in Gauss-Bonnet gravity.
The only change we need to apply is the replacement of the horizon radius
$r_{+}$ with the apparent horizon radius $\tilde{r}_{A}$, namely
$S=\frac{A}{4L_{p}^{n-1}}\left[1+\frac{n-1}{n-3}\frac{2\tilde{\alpha}}{\tilde{r}_{A}^{2}}\right],$
(28)
where $A=n\Omega_{n}\tilde{r}_{A}^{n-1}$ is the apparent horizon area. We
define the effective area of the holographic surface corresponding to the
entropy (28) as
$\displaystyle\widetilde{A}=n\Omega_{n}\tilde{r}_{A}^{n-1}\left[1+\frac{n-1}{n-3}\frac{2\tilde{\alpha}}{\tilde{r}_{A}^{2}}\right].$
(29)
Now we calculate the increasing in the effective volume as
$\displaystyle\frac{d\widetilde{V}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{A}}{(n-1)}\frac{d\widetilde{A}}{dt}=n\Omega_{n}\dot{\tilde{r}}_{A}\tilde{r}^{n-1}_{A}(1+2\tilde{\alpha}\tilde{r}^{-2}_{A})$
(30) $\displaystyle=$
$\displaystyle-\frac{n\Omega_{n}\tilde{r}^{n+2}_{A}}{2}\frac{d}{dt}\left(\tilde{r}^{-2}_{A}+\tilde{\alpha}\tilde{r}^{-4}_{A}\right).$
(31)
Inspired by (31), we propose that the number of degrees of freedom on the
apparent horizon, in Gauss-Bonnet gravity, is given by
$N_{\mathrm{sur}}=\frac{\alpha
n\Omega_{n}\tilde{r}^{n+1}_{A}}{L_{p}^{n-1}}\left(\tilde{r}^{-2}_{A}+\tilde{\alpha}\tilde{r}^{-4}_{A}\right).$
(32)
The bulk degrees of freedom is still given by (22). Inserting Eqs. (22), (30)
and (32) in relation (20), with replacing $V\rightarrow\tilde{V}$, we obtain
$\displaystyle(\tilde{r}^{-2}_{A}+\tilde{\alpha}\tilde{r}^{-4}_{A})-\dot{\tilde{r}}_{A}H^{-1}\tilde{r}^{-3}_{A}(1+2\tilde{\alpha}\tilde{r}^{-2}_{A})$
(33) $\displaystyle=$ $\displaystyle-\frac{8\pi
L_{p}^{n-1}}{n(n-1)}[(n-2)\rho+np].$ (34)
Multiplying the both hand side of (34) by factor $2\dot{a}a$, with help of
continuity equation (24) and relation (2), we get
$\frac{d}{dt}\Bigg{\\{}a^{2}\left[H^{2}+\frac{k}{a^{2}}+\tilde{\alpha}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}\right]\Bigg{\\}}=\frac{16\pi
L_{p}^{n-1}}{n(n-1)}\frac{d}{dt}(\rho a^{2}).$ (35)
Integrating, we find
$H^{2}+\frac{k}{a^{2}}+\tilde{\alpha}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}=\frac{16\pi
L_{p}^{n-1}}{n(n-1)}\rho,$ (36)
where again we have set the integration constant equal to zero. This is
indeed, the corresponding Friedmann equation of the FRW Universe with any
spacial curvature in Gauss-Bonnet gravity CaiKim . Note that the authors of
Refs. Cai1 ; Yang could derive the above equation only in a flat FRW
Universe, while we derive it with arbitrary spacial curvature. This may show
the viability of our proposal (20).
Finally, we consider the more general Lovelock gravity. The entropy of the
spherically symmetric black hole solutions in Lovelock theory can be expressed
as caiLo
$S=\frac{A_{+}}{4L_{p}^{n-1}}\sum_{i=1}^{m}\frac{i(n-1)}{(n-2i+1)}{\hat{c}_{i}}{{r}_{+}}^{2-2i},$
(37)
where $m=[n/2]$ and the coefficients ${\hat{c}_{i}}$ are given by
${\hat{c}_{0}}=\frac{{c_{0}}}{n(n-1)},\ \ {\hat{c}_{1}}=1,\ \
{\hat{c}_{i}}=c_{i}\prod_{j=3}^{2m}(n+1-j)\ \ i>1.$ (38)
We further assume the entropy expression (37) are valid for a FRW Universe
bounded by the apparent horizon in the Lovelock gravity provided we replace
the horizon radius $r_{+}$ with the apparent horizon radius $\tilde{r}_{A}$,
namely
$S=\frac{A}{4L_{p}^{n-1}}\sum_{i=1}^{m}\frac{i(n-1)}{(n-2i+1)}{\hat{c}_{i}}{\tilde{r}_{A}}^{2-2i}.$
(39)
It is easy to show that, the first term in the above expression leads to the
well known area law. The second term yields the apparent horizon entropy in
Gauss-Bonnet gravity. We suppose from the entropy expression that the
effective area of the apparent horizon in Lovelock gravity is given by
$\displaystyle\widetilde{A}=n\Omega_{n}\tilde{r}^{n-1}_{A}\sum_{i=1}^{m}\frac{i(n-1)}{(n-2i+1)}{\hat{c}_{i}}{\tilde{r}_{A}}^{2-2i},$
(40)
and the increase of the effective volume is then given by
$\displaystyle\frac{d\widetilde{V}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{A}}{(n-1)}\frac{d\widetilde{A}}{dt}=n\Omega_{n}\tilde{r}^{n+1}_{A}\left(\sum_{i=1}^{m}i\hat{c}_{i}{\tilde{r}_{A}}^{-2i}\right)\dot{\tilde{r}}_{A}$
(41) $\displaystyle=$
$\displaystyle-\frac{n\Omega_{n}\tilde{r}^{n+2}_{A}}{2}\frac{d}{dt}\left(\sum_{i=1}^{m}\hat{c}_{i}{\tilde{r}_{A}}^{-2i}\right).$
(42)
In this case, we assume from (42) that the number of degrees of freedom on the
apparent horizon, in Lovelock gravity, is
$N_{\mathrm{sur}}=\frac{\alpha
n\Omega_{n}}{L_{p}^{n-1}}\tilde{r}^{n+1}_{A}\sum_{i=1}^{m}\hat{c}_{i}{\tilde{r}_{A}}^{-2i}.$
(43)
Substituting (22), (41) and (43) into (20), we reach
$\displaystyle\sum_{i=1}^{m}\hat{c}_{i}{\tilde{r}_{A}}^{-2i}-\dot{\tilde{r}}_{A}H^{-1}\sum_{i=1}^{m}i\hat{c}_{i}{\tilde{r}_{A}}^{-2i-1}$
(44) $\displaystyle=$ $\displaystyle-\frac{8\pi
L_{p}^{n-1}}{n(n-1)}[(n-2)\rho+np].$ (45)
Multiplying the both hand side by factor $2\dot{a}a$, after using the
continuity equation (24) as well as definition (2), we obtain
$\frac{d}{dt}\left[a^{2}\sum_{i=1}^{m}\hat{c}_{i}\left(H^{2}+\frac{k}{a^{2}}\right)^{i}\right]=\frac{16\pi
L_{p}^{n-1}}{n(n-1)}\frac{d}{dt}(\rho a^{2}).$ (46)
After integrating and setting the constant of integration equal to zero, we
find the corresponding Friedmann equation of the FRW Universe with any spacial
curvature in Lovelock gravity,
$\sum_{i=1}^{m}\hat{c}_{i}\left(H^{2}+\frac{k}{a^{2}}\right)^{i}=\frac{16\pi
L_{p}^{n-1}}{n(n-1)}\rho.$ (47)
This is exactly the result obtained in CaiKim by applying the first law of
thermodynamics on the apparent horizon of the FRW Universe in Lovelock
gravity. Here we arrived at the same result by using quite different approach.
This indicates that, given the entropy expression at hand, one is able to
reproduce the corresponding dynamical equation with any spacial curvature, by
applying the proposal (20).
## IV Summary and discussion
We have investigated the novel idea recently proposed by Padmanabhan Pad1 ,
which states that the emergence of space and Universe expansion can be
understood by calculating the difference between the number of degrees of
freedom on the Hubble horizon and the one in the emerged bulk. Applying this
idea to a flat FRW Universe with Hubble horizon, he derived the dynamical
equation describing the evolution of the Universe Pad1 . In this paper, by
properly modification his idea, we derived the Friedmann equation of a FRW
Universe with any spacial curvature. Our approach not only works in Einstein
gravity, but also works very well in Gauss-Bonnet and more general Lovelock
gravity. The key assumption here is that in a nonflat Universe, the volume
increase, is still proportional to the difference between the number of
degrees of freedom on the apparent horizon and in the bulk, but the function
of proportionality is not just the constant $L_{p}^{2}$, instead it equals to
the ratio of the apparent horizon radius and the Hubble radius, i.e.,
$L_{p}^{2}\tilde{r}_{A}/H^{-1}$.
It is important to note that Padmanabhan’s proposal (7) can lead to the
Friedmann equation with spacial curvature only in Einstein gravity Cai1 ; Yang
. The main result of the present work is that the modified proposal (8) can
lead to the Friedmann equations of the FRW Universe with any spacial curvature
in higher order gravity theories. Indeed, while the authors of Cai1 ; Yang
interpreted the integration constant as the spatial curvature $k$ in Einstein
gravity, they failed to interpret the constant of integration as the spatial
curvature in the cases of Gauss-Bonnet and Lovelock gravities. This is due to
the fact that, in Einstein gravity, the de Sitter Universe can be described
either by $k=0$ or $k=1$. As a result, in Gauss-Bonnet and Lovelock gravity,
with proposal (7), they could only derive the Friedmann equations of the flat
Universe.
In summary, given the entropy expression at hand, one is able to reproduce the
corresponding dynamical equation of the FRW Universe with any spacial
curvature, by calculating the difference between the horizon degrees of
freedom and the bulk degrees of freedom in a region of space and applying the
proposal (8). The results obtained in this paper together with those of Cai1 ;
Yang further support the new proposal of Padmanabhan Pad1 and its
modification as (8) and show that this approach is powerful enough to apply
for deriving the dynamical equations describing the evolution of the Universe
in other gravity theories with any spacial curvature.
###### Acknowledgements.
I thank the referee for constructive comments which helped me to improve the
paper significantly. I also thank the Research Council of Shiraz University.
This work has been supported financially by Center for Excellence in Astronomy
and Astrophysics of IRAN (CEAAI-RIAAM).
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|
arxiv-papers
| 2013-04-10T19:01:19 |
2024-09-04T02:49:44.114397
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1304.3054"
}
|
1304.3057
|
# Rotating black strings in $f(R)$-Maxwell theory
A. Sheykhi1,[email protected], S. Salarpour 3 and Y. Bahrampour
[email protected] 1 Physics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
3 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
4 Department of Mathematics, Shahid Bahonar University, P.O. Box 76175,
Kerman, Iran
###### Abstract
In general, the field equations of $f(R)$ theory coupled to a matter field are
very complicated and hence it is not easy to find exact analytical solutions.
However, if one considers traceless energy-momentum tensor for the matter
source as well as constant scalar curvature, one can derive some exact
analytical solutions from $f(R)$ theory coupled to a matter field. In this
paper, by assuming constant curvature scalar, we construct a class of charged
rotating black string solutions in $f(R)$-Maxwell theory. We study the
physical properties and obtain the conserved quantities of the solutions. The
conserved and thermodynamic quantities computed here depend on function
$f^{\prime}(R_{0})$ and differ completely from those of Einstein theory in AdS
spaces. Besides, unlike Einstein gravity, the entropy does not obey the area
law. We also investigate the validity of the first law of thermodynamics as
well as the stability analysis in the canonical ensemble, and show that the
black string solutions are always thermodynamically stable in $f(R)$-Maxwell
theory with constant curvature scalar. Finally, we extend the study to the
case where the Ricci scalar is not a constant and in particular $R=R(r)$. In
this case, by using the Lagrangian multipliers method, we derive an analytical
black string solution from $f(R)$ gravity and reconstructed the function
$R(r)$. We find that this class of solutions has an additional logaritmic term
in the metric function which incorporates the effect of the $f(R)$ theory in
the solutions.
Keywords: modified gravity; string; thermodynamics.
## I Introduction
There has been considerable attentions in the past years in modified gravity
theories, specially $f(R)$ theory which is one of the encouraging candidates
for explaining the current accelerating of the universe expansion Odin ; Capo
(see also Anto for a comprehensive review on $f(R)$ theories). In fact $f(R)$
theories can be regarded as the simplest extension of general relativity. Many
$f(R)$ models have passed all the available experimental tests and fit the
cosmological data. To prevent a ghost state, $f^{\prime}(R)>0$ for $R\geq
R_{0}$ is required Nun ; Fara . $f^{\prime\prime}(R)>0$ for $R\geq R_{0}$, is
needed to avoid the negative mass squared of a scalar-field degree of freedom
(tachyon) Anto . $f(R)\rightarrow R-2\Lambda$ for $R\geq R_{0}$, is required
for the presence of the matter era and for consistency with local gravity
constraints Anto . It was shown that $f(R)$ theories can be considered as
general relativity with an additional scalar field that provide new insight in
the two cases of Brans-Dicke theory with ${\omega}_{0}=0$ and
${\omega}_{0}=-3/2$ Soti .
There have been a lot of works in the literature attempting to construct
static and stationary black hole solutions in $f(R)$ gravity theories. One may
expect that some signatures of black holes in $f(R)$ theories will be in
disagreement with the expected physical results of Einstein’s gravity. In Cruz
the authors studied general solutions in $f(R)$ theory using a perturbation
approach around the Einstein-Hilbert action. In Psal black hole solutions
were found by adding dynamical vector and tensor degrees of freedom to the
Einstein-Hilbert action. Also, the transition from neutron stars to a strong
scalar- field state in $f(R)$ gravity has been studied in Nova . Physical
properties of the matter forming an accretion disk in the spherically
symmetric background in $f(R)$ theories were explored in Pun . In Ref. Lobo
the construction of traversable wormhole geometries was discussed in $f(R)$
gravity. The Schwarzschild-de Sitter black hole like solutions of $f(R)$
gravity were obtained for a positively constant and a non-constant curvature
scalar in Cogn and Sebas , respectively. A black hole solution was obtained
from $f(R)$ theories by requiring the negative constant curvature scalar Cruz
. If $1+f^{\prime}(R_{0})>0$, this black hole is similar to the Schwarzschild-
AdS (SAdS) black hole. It was argued that $f(R)$ and SAdS black holes have no
big difference in thermodynamic quantities when using the Euclidean action
approach and replacing the Newtonian constant $G$ by $G_{\rm
eff}=G/(1+f^{\prime}(R_{0}))$ Cruz . It is also interesting to study black
hole solutions in $f(R)$ theory coupled to a matter field. In general, the
field equations of $f(R)$ theory coupled to the matter field are very
complicated and hence it is not easy to find exact analytical solutions. In
order to construct the constant curvature scalar black hole solutions from
$f(R)$ gravity coupled to the matter, the trace of its energy-momentum tensor
$T_{\mu\nu}$ should be zero Moon . Two examples for the traceless $T_{\mu\nu}$
are Maxwell and Yang-Mills fields which were studied in Moon ; Habib .
Thermodynamics and properties of these solutions were also studied in ample
details Moon . It was found that these solutions are similar to the Reissner-
Nordström–AdS (RNAdS) black hole when making appropriate replacements Moon .
The Kerr-Newman black hole solutions with non-zero constant scalar curvature
in $f(R)$-Maxwell theory, their thermodynamics, as well as their local and
global stability were also studied in Alex .
In this paper we would like to continue the investigation on the $f(R)$ black
holes, by constructing a new class of charged rotating black string solutions
in $R+f(R)$-Maxwell theory with constant curvature scalar. The traceless
property of the energy-momentum tensor of the Maxwell field plays a crucial
role in our derivation. With assumptions $R_{0}<0$ and $1+f^{\prime}(R_{0})>0$
our solution is similar to charged black string solution in AdS space with
suitable replacing the parameters. We will also suggest the suitable
counterterm which removes the divergences of the action. We calculate the
conserved and thermodynamic quantities of these black strings by using the
counterterm method. We obtain a Smarr-type formula for the mass of the black
string and check the validity of the first law of thermodynamics. We perform
the stability analysis in the canonical ensemble and show that the black
strings are always thermodynamically stable in $f(R)$-Maxwell theory with
constant curvature scalar. Finally, we extend the study to the case where the
Ricci scalar is not constant and in particular $R=R(r)$ and derive an
analytical black string solution.
## II Field Equations and solutions
We start from the four-dimensional $R+f(R)$ theory coupled to the Maxwell
field
$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(R+f(R)-F_{\mu\nu}F^{\mu\nu}\right)-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-h}\Theta(h),$
(1)
where ${R}$ is the Ricci scalar curvature,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, and $A_{\mu}$ is the electromagnetic potential.
The last term in Eq. (1) is the Gibbons-Hawking boundary term. It is required
for the variational principle to be well-defined. The factor $\Theta$
represents the trace of the extrinsic curvature for the boundary
${\partial\mathcal{M}}$ and $h$ is the induced metric on the boundary. The
equations of motion can be obtained by varying the action (1) with respect to
the gravitational field $g_{\mu\nu}$ and the gauge field $A_{\mu}$ which
yields the following field equations
$\displaystyle{R}_{\mu\nu}\left(1+f^{\prime}(R)\right)-\frac{1}{2}g_{\mu\nu}(R+f(R))+\left(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}\right)f^{\prime}(R)=8\pi
T_{\mu\nu},$ (2) $\displaystyle\nabla_{\mu}F^{\mu\nu}=0,$ (3)
with the energy-momentum tensor
$T_{\mu\nu}=\frac{1}{4\pi}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{4}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right).$ (4)
The above energy-momentum tensor is traceless in four dimension, i. e.,
$T^{\mu}_{\text{ }\ \mu}=0$. As we mentioned already this property plays an
important role in our derivation. In Eq. (2) the “prime” denotes
differentiation with respect to curvature scalar $R$. Assuming the constant
curvature scalar $R=R_{0}$, the trace of Eq. (2) yields
$\displaystyle
R_{0}\left(1+f^{\prime}(R_{0})\right)-2\left(R_{0}+f(R_{0})\right)=0,$ (5)
Solving the above equation for negative $R_{0}$, gives
$\displaystyle R_{0}=\frac{2f(R_{0})}{f^{\prime}(R_{0})-1}\equiv
4{\Lambda_{\rm f}}<0.$ (6)
Substituting the above relation into Eq. (2), we obtain the following equation
for Ricci tensor
${R}_{\mu\nu}=\frac{1}{2}g_{\mu\nu}\left(\frac{f(R_{0})}{f^{\prime}(R_{0})-1}\right)+\frac{2}{1+f^{\prime}(R_{0})}T_{\mu\nu}.$
(7)
Now, we want to construct charged rotating black string solutions of the field
equations (2) and (3) and investigate their properties. We are looking for the
four-dimensional rotating solution with cylindrical or toroidal horizons. The
metric which describes such a spacetime can be written in the following form
Lem ; shey
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-N(r)\left(\Xi dt-
ad\phi\right)^{2}+r^{2}\left(\frac{a}{l^{2}}dt-\Xi
d\phi\right)^{2}+\frac{dr^{2}}{N(r)}+\frac{r^{2}}{l^{2}}dz^{2},$
$\displaystyle\Xi^{2}$ $\displaystyle=$ $\displaystyle 1+\frac{a^{2}}{l^{2}},$
(8)
where $a$ is the rotation parameter. The function $N(r)$ should be determined
and $l$ has the dimension of length which is related to the constant
$\Lambda_{\rm f}$ by the relation $l^{2}=-3/\Lambda_{\rm f}$. The two
dimensional space, $t$=constant and $r$ =constant, can be (i) the flat torus
model $T^{2}$ with topology $S^{1}\times S^{1}$, and $0\leq\phi<2\pi$, $0\leq
z<2\pi l$, (ii) the standard cylindrical model with topology $R\times S^{1}$,
and $0\leq\phi<2\pi$, $-\infty<z<\infty$, and (iii) the infinite plane $R^{2}$
with $-\infty<\phi<\infty$ and $-\infty<z<\infty$. We will focus upon (i) and
(ii). The Maxwell equation (3) can be integrated immediately to give
$\displaystyle F_{tr}$ $\displaystyle=$ $\displaystyle\frac{q\Xi}{r^{2}},$
$\displaystyle F_{\phi r}$ $\displaystyle=$
$\displaystyle-\frac{a}{\Xi}F_{tr},$ (9)
where $q$ is the charge parameter of the black string. Substituting the
Maxwell fields (II) as well as the metric (II) in the field equation (2) with
constant curvature, the non-vanishing independent components of the field
equations for $a=0$ reduce to
$\displaystyle\left(1+{\it
f^{\prime}(R_{0})}\right)\left(2r^{4}\frac{d^{2}N(r)}{dr^{2}}+4r^{3}\frac{dN(r)}{dr}+R_{0}r^{4}\right)-4q^{2}=0,$
(10) $\displaystyle\left(1+{\it
f^{\prime}(R_{0})}\right)\left(4r^{3}\frac{dN(r)}{dr}+4r^{2}N(r)+R_{0}r^{4}\right)+4q^{2}=0.$
(11)
One can easily show that the above equations have the following solution
$N(r)=-\frac{2m}{r}+\frac{q^{2}}{(1+f^{\prime}(R_{0}))r^{2}}-\frac{R_{0}}{12}r^{2},$
(12)
where $m$ is an integration constant which is related to the mass of the
string. One can also check that these solutions satisfy equations (2)-(3) in
the rotating case where $a\neq 0$. It is apparent that this spacetime is
similar with asymptotically AdS black string. Indeed, with the following
replacement
$\displaystyle\frac{q^{2}}{\left(1+f^{\prime}(R_{0})\right)}\rightarrow Q^{2}$
(13) $\displaystyle\frac{R_{0}}{4}\rightarrow\Lambda$ (14)
the solution reduces to the asymptotically AdS charged black string for
$\Lambda=-3/l^{2}$ Lem . Next we study the physical properties of the
solutions. The Kretschmann scalar for this solution is given by
$\displaystyle
R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}=\frac{8}{3r^{8}(1+f^{\prime}(R_{0}))^{2}}\left[r^{2}(\frac{1}{16}{R_{0}}^{2}r^{6}+18m^{2})(1+f^{\prime}(R_{0}))^{2}-36mrq^{2}(1+f^{\prime}(R_{0}))+21q^{4}\right].$
(15)
Figure 1: The function $N(r)$ versus $r$ for $m=2$, $f^{\prime}(R_{0})=2$ and
$q=1$. $R_{0}=12$ (bold line) and $R_{0}=-12$ (continuous line).
Figure 2: The function $N(r)$ versus $r$ for $m=2$, $q=1$ and $R_{0}=-12$.
$f^{\prime}(R_{0})=0.5$ (bold line), $f^{\prime}(R_{0})=0$ (continuous line)
and $f^{\prime}(R_{0})=-0.5$ (dashed line).
Figure 3: The function $N(r)$ versus $r$ for $m=2$, $f^{\prime}(R_{0})=-2$ and
$q=1$. $R_{0}=12$ (bold line) and $R_{0}=-12$ (continuous line).
When $r\rightarrow 0$, the dominant term in the Kretschmann scalar is
${56q^{4}}/[(1+f^{\prime}(R_{0}))^{2}r^{8}]$. Therefore we have an essential
singularity located at $r=0$. The Kretschmann scalar also approaches
${R_{0}}^{2}/6$ as $r\rightarrow\infty$. As one can see from Eq. (12), the
solution is ill-defined for $f^{\prime}(R_{0})=-1$. The cases with
$f^{\prime}(R_{0})>-1$ and $f^{\prime}(R_{0})<-1$ should be considered
separately. In the first case where $f^{\prime}(R_{0})>-1$, there exist a
cosmological horizon for $R_{0}>0$, while there is no cosmological horizons if
$R_{0}<0$ (see fig. 1). Indeed, for $1+f^{\prime}(R_{0})>0$ and $R_{0}<0$ the
black string can have two inner and outer horizons provided the parameters of
the solutions are chosen suitably (see fig. 2). In the latter case
($f^{\prime}(R_{0})<-1$), the signature of the spacetime changes and the
conserved quantities such as mass and angular momenta become negative, as we
will see in the next section, thus this is not a physical case and we rule it
out from our consideration (see fig. 3 ).
## III Conserved and Thermodynamic quantities
Next, we calculate the conserved quantities of the solutions by using the
counterterm method inspired by (A)dS/CFT correspondence Mal . The spacetimes
under consideration in this paper has zero curvature boundary,
$R_{abcd}(h)=0$, and therefore the counterterm for the stress energy tensor
should be proportional to $h^{ab}$. We find the suitable counterterm which
removes the divergences of the action in the form
$I_{\rm
ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-h}\sqrt{-\frac{R_{0}}{3}},$
(16)
where $R_{0}<0$. Having the total finite action $I=I_{G}+I_{\mathrm{ct}}$ at
hand, one can use the quasilocal definition to construct a divergence free
stress-energy tensor BY . Thus the finite stress-energy tensor can be written
as
$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta
h^{ab}-\sqrt{-\frac{R_{0}}{3}}h^{ab}\right].$ (17)
The first two terms in Eq. (17) are the variation of the action (1) with
respect to $h_{ab}$, and the last term is the variation of the boundary
counterterm (16) with respect to $h_{ab}$. To compute the conserved charges of
the spacetime, one should choose a spacelike surface $\mathcal{B}$ in
$\partial\mathcal{M}$ with metric $\sigma_{ij}$, and write the boundary metric
in ADM (Arnowitt-Deser-Misner) form:
$h_{ab}dx^{a}dx^{a}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$
where the coordinates $\varphi^{i}$ are the angular variables parameterizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (17) can be written as
$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(18)
where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. The first Killing vector of the
spacetime is $\xi=\partial/\partial t$, and therefore its associated conserved
charge of the string is the mass per unit volume. A simple calculation gives
$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b}=\frac{(3\Xi^{2}-1)m}{8\pi
l}\left[1+f^{\prime}(R_{0})\right].$ (19)
The second conserved quantity is the angular momentum per unit volume
associated with the rotational Killing vectors
$\varsigma=\partial/\partial\phi$ which can be calculated as
$\displaystyle
J=\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}=\frac{3\Xi
m\sqrt{\Xi^{2}-1}}{8\pi}\left[1+f^{\prime}(R_{0})\right].$ (20)
For $a=0$ ($\Xi=1$), the angular momentum per unit volume vanishes, and
therefore $a$ is the rotational parameters of the spacetime. Next we calculate
the entropy of the black string. Let us first give a brief discussion
regarding the entropy of the black hole solutions in $f(R)$ gravity. To this
aim, we follow the arguments presented in Brevik . If one use the Noether
charge method for evaluating the entropy associated with black hole solutions
in $f(R)$ theory with constant curvature, one finds Cogn
${S}=\frac{A}{4G}f^{\prime}(R_{0}),$ (21)
where $A=4\pi r_{+}^{2}$ is the horizon area. As a result, in $f(R)$ gravity,
the entropy does not obey the area law and one obtains a modification of the “
area law”. Motivated by the above argument, for the rotating black string
solution in $R+f(R)$ gravity, we find the entropy per unit length of the
string as
${S}=\frac{r_{+}^{2}\Xi}{4l}\left[1+f^{\prime}(R_{0})\right].$ (22)
Then we obtain the temperature and angular velocity of the horizon by analytic
continuation of the metric. Although our solution is not static, the Killing
vector
$\chi=\partial_{t}+\Omega\partial_{\phi}$ (23)
is the null generator of the event horizon where $\Omega$ is the angular
velocity of the outer horizon. The analytical continuation of the Lorentzian
metric by $t\rightarrow i\tau$ and $a\rightarrow ia$ yields the Euclidean
section, whose regularity at $r=r_{+}$ requires that we should identify
$\tau\sim\tau+\beta_{+}$ and $\phi\sim\phi+i\beta_{+}\Omega_{+}$ where
$\beta_{+}$ and $\Omega_{+}$ are the inverse Hawking temperature and the
angular velocity of the horizon. We find
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\Xi}\left(\frac{dN(r)}{dr}\right)_{r=r_{+}}=-\frac{\left[R_{0}r_{+}^{4}(1+f^{\prime}(R_{0}))+4q^{2}\right]}{16\pi\Xi[1+f^{\prime}(R_{0})]r_{+}^{3}},$
(24) $\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\frac{a}{\Xi
l^{2}},$ (25)
where we have used equation $N(r_{+})=0$ for omitting the mass parameter $m$
from temperature expression. Since $1+f^{\prime}(R_{0})>0$, therefore the
temperature is non negative provided
$\displaystyle R_{0}r_{+}^{4}(1+f^{\prime}(R_{0}))\leq-4q^{2}\rightarrow
R_{0}\leq-\frac{4q^{2}}{r_{+}^{4}(1+f^{\prime}(R_{0}))},$ (26)
where the equality holds for extremal black string with zero temperature. The
next quantity we are going to calculate is the electric charge of the string.
To determine the electric field we should consider the projections of the
electromagnetic field tensor on special hypersurface. The normal vectors to
such hypersurface are
$u^{0}=\frac{1}{N},\text{ \ }u^{r}=0,\text{ \ }u^{i}=-\frac{V^{i}}{N},$ (27)
where $N$ and $V^{i}$ are the lapse function and shift vector. Then the
electric field is $E^{\mu}=g^{\mu\rho}F_{\rho\nu}u^{\nu}$, and the electric
charge per unit length of the string can be found by calculating the flux of
the electric field at infinity,
${Q}=\frac{\Xi q}{4\pi l\sqrt{1+f^{\prime}(R_{0})}}.$ (28)
The electric potential $U$, measured at infinity with respect to the horizon,
is defined by Cal
$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(29)
where $\chi$ is the null generator of the event horizon given in Eq. (23). One
can easily obtain the electric potential as
$U=\frac{q}{\Xi r_{+}}\sqrt{1+f^{\prime}(R_{0})}.$ (30)
Then, we consider the first law of thermodynamics for the black string. In
order to do this, we obtain the mass $M$ as a function of extensive quantities
$S$, ${J}$ and $Q$. Using the expression for the mass, the angular momenta,
the entropy and the charge given in Eqs. (19), (20), (22) and (28) and the
fact that $N(r_{+})=0$, one can obtain a Smarr-type formula as
$M(S,J,Q)=\frac{J(3Z-1)}{3l\sqrt{Z(Z-1)}},$ (31)
where $Z=\Xi^{2}$ is the positive real root of the following equation:
$\displaystyle\frac{3\sqrt{Z-1}\pi^{2}Q^{2}l^{2}[1+f^{\prime}(R_{0})]^{2}-2J\pi\sqrt{\sqrt{Z}(1+f^{\prime}(R_{0}))Sl}+3S^{2}\sqrt{Z-1}}{2\pi\sqrt{\sqrt{Z}(1+f^{\prime}(R_{0}))Sl}}=0.$
(32)
One may then regard the parameters $S$, ${J}$ and $Q$ as a complete set of
extensive parameters for the mass $M(S,{J},Q)$ and define the intensive
parameters conjugate to $S$, ${J}$ and $Q$. These quantities are the
temperature, the angular velocities and the electric potential
$\displaystyle T$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial S}\right)_{J,Q}=-\frac{J\left\\{\left[\pi
Ql(1+f^{\prime}(R_{0}))\right]^{2}-3S^{2}\right\\}}{3Sl\sqrt{Z(Z-1)}\left\\{\left[\pi
Ql(1+f^{\prime}(R_{0}))\right]^{2}+S^{2}\right\\}},$ (33)
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial J}\right)_{S,Q}$ (34) $\displaystyle=$
$\displaystyle\frac{3\left(3Z-1\right)\left\\{\left[\pi
Ql(1+f^{\prime}(R_{0}))\right]^{2}+S^{2}\right\\}-4\pi
J\sqrt{\sqrt{Z}(Z-1)(1+f^{\prime}(R_{0}))Sl}}{9l\sqrt{Z(Z-1)}\left\\{\left[\pi
Ql(1+f^{\prime}(R_{0}))\right]^{2}+S^{2}\right\\}},$ $\displaystyle U$
$\displaystyle=$ $\displaystyle\left(\frac{\partial M}{\partial
Q}\right)_{S,J}=\frac{4\pi^{2}QlJ[1+f^{\prime}(R_{0})]^{2}}{3\sqrt{Z(Z-1)}\left\\{\left[\pi
Ql(1+f^{\prime}(R_{0}))\right]^{2}+S^{2}\right\\}}.$ (35)
Numerical calculations show that the intensive quantities calculated by Eqs.
(33)-(35) coincide with Eqs. (24), (25) and (30), respectively. Thus, these
thermodynamics quantities satisfy the first law of thermodynamics
$dM=TdS+\Omega d{J}+Ud{Q}.$ (36)
## IV Thermal Stability of black string
Figure 4: The function $(\partial^{2}M/\partial S^{2})_{J,Q}$ versus $q$ for
$l=1$, $\Xi=1.25$, $r_{+}=0.7$ and $R_{0}=-12$. $f^{\prime}(R_{0})=0$ (bold
line), $f^{\prime}(R_{0})=1$ (continuous line) and $f^{\prime}(R_{0})=2$
(dashed line).
Figure 5: The function $(\partial^{2}M/\partial S^{2})_{J,Q}$ versus $r_{+}$
for $l=1$, $\Xi=1.25$, $R_{0}=-12$ and $f^{\prime}(R_{0})=1$. $q=0.5$ (bold
line), $q=1$ (continuous line) and $q=1.5$ (dashed line).
Figure 6: The function $(\partial^{2}M/\partial S^{2})_{J,Q}$ versus $q$ for
$l=1$, $f^{\prime}(R_{0})=1$ and $R_{0}=-12$. $\Xi=1.25$, (bold line),
$\Xi=1.75$, (continuous line) and $\Xi=2.25$, (dashed line).
Finally, we investigate the thermal stability of rotating black string
solutions in $f(R)$ gravity coupled to a matter field. The stability of a
thermodynamic system with respect to small variations of the thermodynamic
coordinates is usually performed by analyzing the behavior of the entropy
$S(M,J,Q)$ around the equilibrium. The local stability in any ensemble
requires that $S(M,{J},Q)$ be a convex function of the extensive variables or
its Legendre transformation must be a concave function of the intensive
variables. The stability can also be studied by the behavior of the energy
$M(S,J,Q)$ which should be a convex function of its extensive variables. Thus,
the local stability can in principle be carried out by finding the determinant
of the Hessian matrix of $M(S,{J},Q)$ with respect to its extensive variables
$X_{i}$, $\mathbf{H}_{X_{i}X_{j}}^{M}=[\partial^{2}M/\partial X_{i}\partial
X_{j}]$ Cal ; Gub . In our case the mass $M$ is a function of entropy, angular
momenta, and charge. The number of thermodynamic variables depends on the
ensemble that is used. In the canonical ensemble, the charge and the angular
momenta are fixed parameters, and therefore the positivity of the
$(\partial^{2}M/\partial S^{2})_{{J},Q}$ is sufficient to ensure local
stability. We find that the black string solutions are always thermally stable
independent of the value of the parameters $q$ and $\Xi$. We have shown the
behavior of $(\partial^{2}M/\partial S^{2})_{{J},Q}$ as a function $q$ and
$r_{+}$ for different value of $\Xi$ and $f^{\prime}(R_{0})$ in figures 4-6.
These figures show that the black string solutions in $f(R)$-Maxwell theory
with constant curvature scalar are always thermally stable.
## V Solution with non constant Ricci scalar
In this section we would like to extend the study to the case where the Ricci
scalar is not a constant, instead we reconstruct it as $R=R(r)$ as a result of
our calculations. In this case, we find out that we can obtain solution only
in the absence of the matter field. As we mentioned in the introduction, in
general, the field equations of $f(R)$ theory coupled to the matter field are
very complicated and hence it is not easy to find exact analytical solutions.
Thus we only consider the uncharged black string solution. We start with the
following action
$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}f(R)\,,$ (37)
where $f(R)$ is a generic function of the Ricci scalar $R$. We also modify our
metric (II) a bit as follow
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N(r)\mathrm{e}^{2\alpha(r)}\left(\Xi dt-
ad\phi\right)^{2}+r^{2}\left(\frac{a}{l^{2}}dt-\Xi
d\phi\right)^{2}+\frac{dr^{2}}{N(r)}+\frac{r^{2}}{l^{2}}dz^{2},$ (38)
where we have added an additional function $\alpha(r)$ in the metric
coefficients. For simplicity we only consider the non-rotating black string
with $a=0$, thus the above metric reduces to
$\displaystyle
ds^{2}=-N(r)\mathrm{e}^{2\alpha(r)}dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}d\phi^{2}+\frac{r^{2}}{l^{2}}dz^{2}.$
(39)
The scalar curvature for metric (39) reads
$\displaystyle R$ $\displaystyle=$
$\displaystyle-3\,{\frac{dN\left(r\right)}{dr}}{\frac{d\alpha\left(r\right)}{dr}}-2\,N\left(r\right)\left[{\frac{d}{dr}}\alpha\left(r\right)\right]^{2}-{\frac{d^{2}N\left(r\right)}{d{r}^{2}}}-2\,N\left(r\right){\frac{d^{2}\alpha\left(r\right)}{d{r}^{2}}}$
(40)
$\displaystyle-\frac{4}{r}\frac{dN\left(r\right)}{dr}-\frac{4N\left(r\right)}{r}\frac{d\alpha\left(r\right)}{dr}-\frac{2N\left(r\right)}{{r}^{2}}\,.$
We use the Lagrangian multipliers method. In the framework of Friedmann-
Robertson-Walker universe this method was studied in vile ; Capozziello ;
Monica , while for static spherically symmetric black hole solutions it was
investigated in Sebas ; Capozziello2 . In this approach one may consider the
scalar curvature $R$ as independent Lagrangian coordinates in addition to the
functions $\alpha(r)$ and $N(r)$, which appear from the metric line element.
Introducing the Lagrangian multipliers $\lambda$, after using (40), the action
(37) can be written
$\displaystyle S$ $\displaystyle\equiv$ $\displaystyle\frac{1}{16\pi}\int
dt\int
d{r}\left(e^{\alpha(r)}r^{2}\right)\left\\{f(R)-\lambda\left\\{R+\left\\{3\,\left[{\frac{d}{dr}}N\left(r\right)\right]{\frac{d}{dr}}\alpha\left(r\right)\right.\right.\right.$
(41)
$\displaystyle+2\,N\left(r\right)\left[{\frac{d}{dr}}\alpha\left(r\right)\right]^{2}+{\frac{d^{2}N\left(r\right)}{d{r}^{2}}}+2\,N\left(r\right)\frac{d^{2}\alpha\left(r\right)}{d{r}^{2}}+\frac{4}{r}\frac{dN\left(r\right)}{dr}$
$\displaystyle+\frac{4N\left(r\right)}{r}\frac{d\alpha\left(r\right)}{dr}+{\frac{2N\left(r\right)}{{r}^{2}}}\left.\left.\left.\right\\}\right\\}\right\\}\,.$
Varying the above action with respect to $R$, one gets
$\lambda=f^{\prime}(R),$ (42)
where the prime denotes the derivative with respect to the scalar curvature
$R$. Substituting this value and integrating by part, the Lagrangian takes the
form
$\displaystyle L(\alpha,d\alpha/dr,N,dN/dr,R,dR/dr)$ $\displaystyle=$
$\displaystyle
e^{\alpha}\left\\{r^{2}\left[f(R)-Rf^{\prime}(R)\right]-2f^{\prime}(R)\left(r\frac{dN(r)}{dr}+N(r)\right)\right.$
(43)
$\displaystyle+\left.f^{\prime\prime}(R)\frac{dR}{dr}r^{2}\left(\frac{dN(r)}{dr}+2N(r)\frac{d\alpha(r)}{dr}\right)\right\\}\,.$
Making the variation with respect to $\alpha$, one gets the first equation of
motion
$\displaystyle\frac{Rf^{\prime}(R)-f(R)}{f^{\prime}(R)}+\frac{2}{r^{2}}\left[N(r)+r\frac{dN(r)}{dr}\right]$
(44)
$\displaystyle+\frac{2N(r)f^{\prime\prime}(R)}{f^{\prime}(R)}\left[\frac{d^{2}R}{dr^{2}}+\left(\frac{dN(r)/dr}{2N(r)}\right)\frac{dR}{dr}+\frac{f^{\prime\prime\prime}(R)}{f^{\prime\prime}(R)}\left(\frac{dR}{dr}\right)^{2}\right]=0\,.$
The variation with respect to $N(r)$ leads the second equation of motion
$\left[\frac{d\alpha(r)}{dr}\left(\frac{f^{\prime\prime}(R)}{f^{\prime}(R)}\frac{dR}{dr}\right)-\frac{f^{\prime\prime}(R)}{f^{\prime}(R)}\frac{d^{2}R}{dr^{2}}-\frac{f^{\prime\prime\prime}(R)}{f^{\prime}(R)}\left(\frac{dR}{dr}\right)^{2}\right]=0\,,$
(45)
while by making the variation with respect to $R$, we recover Eq. (40). Given
$f(R)$, together with equation (40), the above equations form a system of
three differential equations for the three unknown quantities $\alpha(r),N(r)$
and $R(r)$. We would like to note that one advantage of this approach is that
$\alpha$ does not appear in Eq.(44). In what follow, we will find exact
solutions of the above system of differential equations.
In the special case of constant curvature $R=R_{0}$ and $\alpha=\rm constant$,
it is easy to show that the only solution of Eqs. (40) and(44) is the
Schwarzshild de Sitter black string solution with flat horizon,
$N(r)=-\frac{2m}{r}-\frac{\Lambda}{3}r^{2},$ (46)
where $m$ is a constant of integration which can be interpreted as the mass
parameter of the black string and we have defined $2\Lambda\equiv
R_{0}-f(R_{0})/f^{\prime}(R_{0})$ and $R_{0}=4\Lambda$. Notice that in the
absence of the matter field, $q=0$, solution (12) coincides with the result
obtained in (46), as expected.
Next, we consider the case of non constant Ricci curvature, but still with
$\alpha=\rm constant$. From Eq. (45) we have
$f^{\prime\prime\prime}\left(\frac{dR}{dr}\right)^{2}+f^{\prime\prime}\left(\frac{d^{2}R}{dr^{2}}\right)=\frac{d^{2}}{dr^{2}}f^{\prime}(R)=0,$
(47)
which has the following solution,
$f^{\prime}(R)=mr+n,$ (48)
where $m$ and $n$ are two integration constants. Given the explicit form of
$R$, we may find $r$ as a function of Ricci scalar and reconstruct
$f^{\prime}(R)$ realizing such solution. From Eq. (40) with constant $\alpha$,
one gets
$R=-{\frac{d^{2}N\left(r\right)}{d{r}^{2}}}-\frac{4}{r}\frac{dN\left(r\right)}{dr}-\,{\frac{2N\left(r\right)}{{r}^{2}}}.$
(49)
Using the fact that $(f^{\prime\prime}(R))dR/dr=df^{\prime}(R)/dr=m$ and
$df(R)/dr=f^{\prime}(R)dR/dr$, and multiplying Eq. (44) by $f^{\prime}(R)$, we
arrive at
$-\frac{d^{2}N(r)}{dr^{2}}\left(m+\frac{n}{r}\right)+\frac{4mN(r)}{r^{2}}+\frac{2nN(r)}{r^{3}}-\frac{m}{r}\frac{dN(r)}{dr}=0\,.$
(50)
When $m=0$, the solution of the above equation is ones obtained in (46). For
$n=0$, the general solution is
$N(r)=-C_{1}r^{2}+\frac{C_{2}}{r^{2}}.$ (51)
Substituting in Eq. (49) we again arrive at constant Ricci scalar,
$R=12C_{1}$. Although in this case $f^{\prime}(R)=df(R)/dR=mr$ is not a
constant, but still we have $df(R)/dr=0$, which implies that $f(R)=\rm
constant$. Next we look for the most general solution of Eq.(50) with $n\neq
0$ and $m\neq 0$. Solving (50), we find
$\displaystyle
N\left(r\right)=-C_{1}{r}^{2}+\frac{C_{2}}{r}\left[2\,{n}^{3}-3mn^{2}r+6\,{m}^{2}{r}^{2}n-6{m}^{3}r^{3}\ln\left(m+\frac{n}{r}\right)\right],$
(52)
where $C_{1}$ and $C_{2}$ are two arbitrary constants. Given solution (52) one
can basically construct $f(R)$ by using Eqs. (48) and (49). In order to
simplify the above solution, we choose $n=1$ and $C_{2}=-1/m$,
$N\left(r\right)=3-\frac{2}{mr}-6\,{mr}-C_{1}{r}^{2}+6{m}^{2}r^{2}\ln\left(m+\frac{1}{r}\right).$
(53)
For this general case the Ricci scalar becomes
${R(r)}=12C_{1}-72m^{2}\ln\left(m+\frac{1}{r}\right)+\frac{6\left(12m^{3}r^{3}+18m^{2}r^{2}+4mr-1\right)}{r^{2}(mr+1)^{2}}.$
(54)
which is clearly not a constant. Now we want to reconstruct the corresponding
$f(R)$ theory. From Eq.(48), for $n=1$ one has
$f^{\prime}(R)=\frac{df(R)}{dR}=\frac{df(R)}{dr}\frac{dr}{dR}=mr+1.$ (55)
Integrating (55), by using (54), we get
$f[R(r)]=-36m^{2}\ln\left(m+\frac{1}{r}\right)+\frac{3\left(12m^{3}r^{3}+18m^{2}r^{2}+4mr-2\right)}{r^{2}(ar+1)^{2}}.$
(56)
Combining Eqs. (54), (55) and (56), one gets the following differential
equation for function $f(R)$,
$\frac{3m^{2}}{[f^{\prime}(R)]^{2}[f^{\prime}(R)-1]^{2}}+f(R)-\frac{R}{2}+6C_{1}=0.$
(57)
This equation has a simple solution as
$f(R)=\frac{R}{2}-6C_{1}-48m^{2},$ (58)
but it has also another complicated solution which we have not presented it
here. Thus we have found a black string solution in $f(R)$ gravity with non
constant Ricci scalar. This approach also leads to construct Ricci scalar as a
function of $r$, as given in Eq. (54). The obtained solutions in this section
differ from that presented in Capozziello2 for axially symmetric solutions in
$f(R)$ gravity. It is worth mentioning that metric (38) has a good property
for which its static and rotating solutions coincide and so in this section we
only study the static case. Following the approach of this section, one can
easily check that solution (52) can be deduced for rotating case where $a\neq
0$. Besides, in this section we only considered the case with $\alpha=\rm
constant$, and derived the metric function (52) as well as $R(r)$. The study
can also be generalized to the case where $\alpha=\alpha(r)$. We leave it and
also thermodynamic considerations of the obtained solution in this section,
for future investigations.
## VI Conclusions
In order to obtain the constant curvature black hole solution in $f(R)$
gravity theory coupled to a matter field, the trace of the energy-momentum
tensor of the matter field should be zero Moon . Since the energy-momentum
tensor of Maxwell field is traceless in four dimensions, therefore spherically
symmetric black hole solutions from $f(R)$ theory coupled to Maxwell field was
derived in four dimensional spacetime Moon .
In this paper we continued the study by constructing a new class of charged
rotating solutions in $f(R)$-Maxwell theory with constant curvature scalar.
This class of solutions describe the four dimensional charged rotation black
string with cylindrical or toroidal horizons with zero curvature boundary.
These solutions are similar to asymptotically AdS black string of Einstein-
Maxwell gravity with suitably replacement of the parameters. However, the
solution presented in this paper has at least two differences from AdS black
string solutions of Einstein-Maxwell gravity. First, the conserved and
thermodynamic quantities computed here depend on function $f^{\prime}(R_{0})$
and differ completely from those of Einstein theory in AdS spaces. Clearly the
presence of the general function $f^{\prime}(R_{0})$ changes the physical
values of conserved and thermodynamic quantities. Second, unlike Einstein
gravity, the entropy does not obey the area law for black string solutions in
$f(R)$-Maxwell theory as one can see from Eq (22). We studied the physical
properties of the solutions and found a suitable conterterm which removes the
divergence of the action. We obtained mass and angular momenta of the string
through the use of conterterm method. We also derived the entropy of the black
string in $f(R)$ gravity which has a modification from the area law. We
obtained a Smarr-type formula for the mass namely $M(S,J,Q)$ and checked that
the obtained conserved and thermodynamic quantities satisfy the first law of
black hole thermodynamics. Finally, we explored the thermal stability of the
solutions in the canonical ensemble and showed that the black strings derived
from $f(R)$\- Maxwell theory are always thermally stable. This is commensurate
with the fact that there is no Hawking-Page phase transition for black objects
with zero curvature horizon Wit .
We also extend the study to the case where the Ricci scalar is not a constant.
For this purpose we used the Lagrangian multipliers method and found an exact
black string solution in $f(R)$ gravity. In this approach one may consider the
scalar curvature $R(r)$ as an independent Lagrangian coordinates in addition
to the metric functions and deduce $R(r)$ as a solution of the field
equations. We found the explicit form of $R(r)$ as well as the metric function
which has a logaritmic term. It is worth noting that since for the non
constant Ricci scalar, the field equations of $f(R)$ theory coupled to the
matter field become very complicated, in this case, we could only derived
analytical solution in the absence of the matter field.
###### Acknowledgements.
We thank the referee for constructive comments which helped us to improve the
paper significantly. We also grateful to S. H. Hendi for useful comments and
helpful discussions. The work of A. Sheykhi has been supported financially by
Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran. A.
Sheykhi also thank from the Research Council of Shiraz University.
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|
arxiv-papers
| 2013-04-10T19:14:32 |
2024-09-04T02:49:44.122032
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Salarpour, A. Sheykhi and Y. Bahrampour",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1304.3057"
}
|
1304.3139
|
# The Complexity of Approximating Vertex Expansion
Anand Louis
Georgia Tech
[email protected] Supported by National Science Foundation awards AF-0915903
and AF-0910584. Prasad Raghavendra
UC Berkeley
[email protected] Supported by NSF Career Award and Alfred. P. Sloan
Fellowship Santosh Vempala 11footnotemark: 1
Georgia Tech
[email protected]
We study the complexity of approximating the vertex expansion of graphs
$G=(V,E)$, defined as
$\phi^{\sf V}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{S\subset
V}n\cdot\frac{\left\lvert N(S)\right\rvert}{\left\lvert
S\right\rvert\left\lvert V\setminus S\right\rvert}.$
We give a simple polynomial-time algorithm for finding a subset with vertex
expansion $\mathcal{O}\left(\sqrt{\phi^{\sf V}\log d}\right)$ where $d$ is the
maximum degree of the graph. Our main result is an asymptotically matching
lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to
find a subset with expansion less than $C\sqrt{\phi^{\sf V}\log d}$ for an
absolute constant $C$. In particular, this implies for all constant
$\varepsilon>0$, it is ${\sf SSE}$-hard to distinguish whether the vertex
expansion $<\varepsilon$ or at least an absolute constant. The analogous
threshold for edge expansion is $\sqrt{\phi}$ with no dependence on the degree
(Here $\phi$ denotes the optimal edge expansion). Thus our results suggest
that vertex expansion is harder to approximate than edge expansion. In
particular, while Cheeger’s algorithm can certify constant edge expansion, it
is SSE-hard to certify constant vertex expansion in graphs.
Our proof is via a reduction from the Unique Games instance obtained from the
SSE hypothesis to the vertex expansion problem. It involves the definition of
a smoother intermediate problem we call Balanced Analytic Vertex Expansion
which is representative of both the vertex expansion and the conductance of
the graph. Both reductions (from the UGC instance to this problem and from
this problem to vertex expansion) use novel proof ideas.
## 1 Introduction
Vertex expansion is an important parameter associated with a graph, one that
has played a major role in both algorithms and complexity. Given a graph
$G=(V,E)$, the vertex expansion of a set $S\subseteq V$ of vertices is defined
as
$\phi^{\sf V}(S)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\lvert
V\right\rvert\cdot\frac{\left\lvert N(S)\right\rvert}{\left\lvert
S\right\rvert\left\lvert V\setminus S\right\rvert}$
Here $N(S)$ denotes the outer boundary of the set $S$, i.e. $N(S)=\left\\{i\in
V\backslash S|\exists u\in S\textrm{ such that }\left\\{u,v\right\\}\in
E\right\\}$. The vertex expansion of the graph is given by $\phi^{\sf
V}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{S\subset V}\phi^{\sf V}(S)$.
The problem of computing $\phi^{\sf V}$ is a major primitive for many graph
algorithms specifically for those that are based on the divide and conquer
paradigm [LR99]. It is NP-hard to compute the vertex expansion $\phi^{\sf V}$
of a graph exactly. In this work, we study the approximability of vertex
expansion $\phi^{\sf V}$ of a graph.
A closely related notion to vertex expansion is that of edge expansion. The
edge expansion of a set $S$ is defined as
$\phi(S)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\mu(E(S,\bar{S}))}{\mu(S)}$
and the edge expansion of the graph is $\phi=\min_{S\subset V}\phi(S)$. Graph
expansion problems have received much attention over the past decades, with
applications to many algorithmic problems, to the construction of pseudorandom
objects and more recenlty due to their connection to the unique games
conjecture.
The problem of approximating edge or vertex expansion can be studied at
various regimes of parameters of interest. Perhaps the simplest possible
version of the problem is to distinguish whether a given graph is an expander.
Fix an absolute constant $\delta_{0}$. A graph is a $\delta_{0}$-vertex (edge)
expander if its vertex (edge) expansion is at least $\delta_{0}$. The problem
of recognizing a vertex expander can be stated as follows:
###### Problem 1.1.
Given a graph $G$, distinguish between the following two cases
(Non-Expander) the vertex expansion is $<\varepsilon$
(Expander) the vertex expansion is $>\delta_{0}$ for some absolute constant
$\delta_{0}$.
Similarly, one can define the problem of recognizing an edge expander graph.
Notice that if there is some sufficiently small absolute constant
$\varepsilon$ (depending on $\delta_{0}$), for which the above problem is
easy, then we could argue that it is easy to “recognize” a vertex expander.
For the edge case, the Cheeger’s inequality yields an algorithm to recognize
an edge expander. In fact, it is possible to distinguish a $\delta_{0}$ edge
expander graph, from a graph whose edge expansion is $<\delta_{0}^{2}/2$, by
just computing the second eigenvalue of the graph Laplacian.
It is natural to ask if there is an efficient algorithm with an analogous
guarantee for vertex expansion. More precisely, is there some sufficiently
small $\varepsilon$ (an arbitrary function of $\delta_{0}$), so that one can
efficiently distinguish between a graph with vertex expansion $>\delta_{0}$
from one with vertex expansion $<\varepsilon$. In this work, we show a
hardness result suggesting that there is no efficient algorithm to recognize
vertex expanders. More precisely, our main result is a hardness for the
problem of approximating vertex expansion in graphs of bounded degree $d$. The
hardness result shows that the approximability of vertex expansion degrades
with the degree, and therefore the problem of recognizing expanders is hard
for sufficiently large degree. Furthermore, we exhibit an approximation
algorithm for vertex expansion whose guarantee matches the hardness result up
to constant factors.
#### Related Work
The first approximation for conductance was obtained by discrete analogues of
the Cheeger inequality shown by Alon-Milman [AM85] and Alon [Alo86].
Specifically, Cheeger’s inequality relates the conductance $\phi$ to the
second eigenvalue of the adjacency matrix of the graph – an efficiently
computable quantity. This yields an approximation algorithm for $\phi$, one
that is used heavily in practice for graph partitioning. However, the
approximation for $\phi$ obtained via Cheeger’s inequality is poor in terms of
a approximation ratio, especially when the value of $\phi$ is small. An
$\mathcal{O}\left(\log n\right)$ approximation algorithm for $\phi$ was
obtained by Leighton and Rao [LR99]. Later work by Linial et al. [LLR95] and
Aumann and Rabani [AR98] established a strong connection between the Sparsest
Cut problem and the theory of metric spaces, in turn spurring a large and rich
body of literature. The current best algorithm for the problem is an
$O(\sqrt{\log n})$ approximation for due to Arora et al. [ARV04] using
semidefinite programming techniques.
Ambühl, Mastrolilli and Svensson [AMS07] showed that $\phi^{\sf V}$ and $\phi$
have no PTAS assuming that SAT does not have sub-exponential time algorithms.
The current best approximation factor for $\phi^{\sf V}$ is
$\mathcal{O}\left(\sqrt{\log n}\right)$ obtained using a convex relaxation
[FHL08]. Beyond this, the situation is much less clear for the approximability
of vertex expansion. Applying Cheeger’s method leads to a bound of
$\mathcal{O}\left(\sqrt{d\operatorname{{\sf OPT}}}\right)$ [Alo86] where $d$
is the maximum degree of the input graph.
#### Small Set Expansion Hypothesis
A more refined measure of the edge expansion of a graph is its expansion
profile. Specifically, for a graph $G$ the expansion profile is given by the
curve
$\phi(\delta)=\min_{\mu(S)\leqslant\delta}\phi(S)\qquad\qquad\forall\delta\in[0,\nicefrac{{1}}{{2}}]\,.$
The problem of approximating the expansion profile has received much less
attention, and is seemingly far less tractable. In summary, the current state-
of-the-art algorithms for approximating the expansion profile of a graph are
still far from satisfactory. Specifically, the following hypothesis is
consistent with the known algorithms for approximating expansion profile.
###### Hypothesis (Small-Set Expansion Hypothesis, [RS10]).
For every constant $\eta>0$, there exists sufficiently small $\delta>0$ such
that given a graph $G$ it is NP-hard to distinguish the cases,
Yes:
there exists a vertex set $S$ with volume $\mu(S)=\delta$ and expansion
$\phi(S)\leqslant\eta$,
No:
all vertex sets $S$ with volume $\mu(S)=\delta$ have expansion
$\phi(S)\geqslant 1-\eta$.
Apart from being a natural optimization problem, the Small-Set Expansion
problem is closely tied to the Unique Games Conjecture. Recent work by
Raghavendra-Steurer [RS10] established reduction from the Small-Set Expansion
problem to the well known Unique Games problem, thereby showing that Small-Set
Expansion Hypothesis implies the Unique Games Conjecture. This result suggests
that the problem of approximating expansion of small sets lies at the
combinatorial heart of the Unique Games problem.
In a breakthrough work, Arora, Barak, and Steurer [ABS10] showed that the
problem $\textsc{Small-Set Expansion}(\eta,\delta)$ admits a subexponential
algorithm, namely an algorithm that runs in time $\exp(n^{\eta}/\delta)$.
However, such an algorithm does not refute the hypothesis that the problem
$\textsc{Small-Set Expansion}(\eta,\delta)$ might be hard for every constant
$\eta>0$ and sufficiently small $\delta>0$.
The Unique Games Conjecture is not known to imply hardness results for
problems closely tied to graph expansion such as Balanced Separator. The
reason being that the hard instances of these problems are required to have
certain global structure namely expansion. Gadget reductions from a unique
games instance preserve the global properties of the unique games instance
such as lack of expansion. Therefore, showing hardness for graph expansion
problems often required a stronger version of the Expanding Unique Games,
where the instance is guaranteed to have good expansion. To this end, several
such variants of the conjecture for expanding graphs have been defined in
literature, some of which turned out to be false [AKK+08]. The Small-Set
Expansion Hypothesis could possibly serve as a natural unified assumption that
yields all the implications of expanding unique games and, in addition, also
hardness results for other fundamental problems such as Balanced Separator. In
fact, Raghavendra, Steurer and Tulsiani [RST12] show that the the SSE
hypothesis implies that the Cheeger’s algorithm yields the best approximation
for the balanced separator problem.
#### Formal Statement of Results
Our first result is a simple polynomial-time algorithm to obtain a subset of
vertices $S$ whose vertex expansion is at most
$\mathcal{O}\left(\sqrt{\phi^{\sf V}\log d}\right)$. Here $d$ is the largest
vertex degree of $G$. The algorithm is based on a Poincairé-type graph
parameter called $\lambda_{\infty}$ defined by Bobkov, Houdré and Tetali
[BHT00], which approximates $\phi^{\sf V}$. While $\lambda_{\infty}$ also
appears to be hard to compute, its natural SDP relaxation gives a bound that
is within $\mathcal{O}\left(\log d\right)$, as observed by Steurer and Tetali
[ST12], which inspires our first Theorem.
###### Theorem 1.2.
There exists a polynomial time algorithm which given a graph $G=(V,E)$ having
vertex degrees at most $d$, outputs a set $S\subset V$, such that $\phi^{\sf
V}(S)=\mathcal{O}\left(\sqrt{\phi^{\sf V}_{G}\log d}\right)$.
It is natural to ask if one can prove better inapproximability results for
vertex expansion than those that follow from the inapproximability results for
edge expansion. Indeed, the best one could hope for would be a lower bound
matching the upper bound in the above theorem. Our main result is a reduction
from SSE to the problem of distinguishing between the case when vertex
expansion of the graph is at most $\varepsilon$ and the case when the vertex
expansion is at least $\Omega(\sqrt{\varepsilon\log d})$. This immediately
implies that it is SSE-hard to find a subset of vertex expansion less than
$C\sqrt{\phi^{\sf V}\log d}$ for some constant $C$. To the best of our
knowledge, our work is the first evidence that vertex expansion might be
harder to approximate than edge expansion. More formally, we state our main
theorem below.
###### Theorem 1.3.
For every $\eta>0$, there exists an absolute constant $C$ such that
$\forall\varepsilon>0$ it is SSE-hard to distinguish between the following two
cases for a given graph $G=(V,E)$ with maximum degree $d\geqslant
100/\varepsilon$.
Yes
: There exists a set $S\subset V$ of size $\left\lvert
S\right\rvert\leqslant\left\lvert V\right\rvert/2$ such that
$\phi^{\sf V}(S)\leqslant\varepsilon$
No
: For all sets $S\subset V$,
$\phi^{\sf V}(S)\geqslant\min\left\\{10^{-10},C\sqrt{\varepsilon\log
d}\right\\}-\eta$
By a suitable choice of parameters in the above theorem, we obtain the main
theorem of this work, Theorem 1.4.
###### Theorem 1.4.
There exists an absolute constant $\delta_{0}>0$ such that for every constant
$\varepsilon>0$ the following holds: Given a graph $G=(V,E)$, it is SSE-hard
to distinguish between the following two cases:
Yes
: There exists a set $S\subset V$ of size $\left\lvert
S\right\rvert\leqslant\left\lvert V\right\rvert/2$ such that $\phi^{\sf
V}(S)\leqslant\varepsilon$
No
: ($G$ is a vertex expander with constant expansion) For all sets $S\subset
V$, $\phi^{\sf V}(S)\geqslant\delta_{0}$
In particular, the above result implies that it is SSE-hard to certify that a
graph is a vertex expander with constant expansion. This is in contrast to the
case of edge expansion, where the Cheeger’s inequality can be used to certify
that a graph has constant edge expansion.
At the risk of being redundant, we note that our main theorem implies that any
algorithm that outputs a set having vertex expansion less than
$C\sqrt{\phi^{\sf V}\log d}$ will disprove the SSE hypothesis; alternatively,
to improve on the bound of $\mathcal{O}\left(\sqrt{\phi^{\sf V}\log
d}\right)$, one has to disprove the SSE hypothesis. From an algorithmic
standpoint, we believe that Theorem 1.4 exposes a clean algorithmic challenge
of recognizing a vertex expander – a challenging problem that is not only
interesting on its own right, but whose resolution would probably lead to a
significant advance in approximation algorithms.
At a high level, the proof is as follows. We introduce the notion of Balanced
Analytic Vertex Expansion for Markov chains. This quantity can be thought of
as a ${\sf CSP}$ on $(d+1)$-tuples of vertices. We show a reduction from
Balanced Analytic Vertex Expansion of a Markov chain, say $H$, to vertex
expansion of a graph, say $H_{1}$ (Section 7). Our reduction is generic and
works for any Markov chain $H$. Surprisingly, the ${\sf CSP}$-like nature of
Balanced Analytic Vertex Expansion makes it amenable to a reduction from
Small-Set Expansion (Section 6). We construct a gadget for this reduction and
study its embedding into the Gaussian graph to analyze its soundness (Section
4 and Section 5). The gadget involves a sampling procedure to generate a
bounded-degree graph.
## 2 Proof Overview
Figure 1: Reduction from SSE to Vertex Expansion
#### Balanced Analytic Vertex Expansion
To exhibit a hardness result, we begin by defining a combinatorial
optimization problem related to the problem of approximating vertex expansion
in graphs having largest degree $d$. This problem referred to as Balanced
Analytic Vertex Expansion can be motivated as follows.
Fix a graph $G=(V,E)$ and a subset of vertices $S\subset V$. For any vertex
$v\in V$, $v$ is on the boundary of the set $S$ if and only if $\max_{u\in
N(v)}\left\lvert\varmathbb{I}_{S}\left[u\right]-\varmathbb{I}_{S}\left[v\right]\right\rvert=1$,
where $N(v)$ denotes the neighbourhood of vertex $v$. In particular, the
fraction of vertices on the boundary of $S$ is given by
$\operatorname*{\varmathbb{E}}_{v}\max_{u\in
N(v)}\left\lvert\varmathbb{I}_{S}\left[u\right]-\varmathbb{I}_{S}\left[v\right]\right\rvert$.
The symmetric vertex expansion of the set $S\subseteq V$ is given by,
$n\cdot\frac{\left\lvert N(S)\cup N(V\backslash S)\right\rvert}{\left\lvert
S\right\rvert\left\lvert V\backslash
S\right\rvert}=\frac{\operatorname*{\varmathbb{E}}_{v}\max_{u\in
N(v)}\left\lvert\varmathbb{I}_{S}\left[u\right]-\varmathbb{I}_{S}\left[v\right]\right\rvert}{\operatorname*{\varmathbb{E}}_{u,v}\left\lvert\varmathbb{I}_{S}\left[u\right]-\varmathbb{I}_{S}\left[v\right]\right\rvert}\,.$
Note that for a degree $d$ graph, each of the terms in the numerator is
maximization over the $d$ edges incident at the vertex. The formal definition
of Balanced Analytic Vertex Expansion is as shown below.
###### Definition 2.1.
An instance of Balanced Analytic Vertex Expansion, denoted by
$(V,\mathcal{P})$, consists of a set of variables $V$ and a probability
distribution $\mathcal{P}$ over $(d+1)$-tuples in $V^{d+1}$. The probability
distribution $\mathcal{P}$ satisfies the condition that all its $d+1$ marginal
distributions are the same (denoted by $\mu$). The goal is to solve the
following optimization problem
$\Phi({V,\mathcal{P}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{F:V\to\left\\{0,1\right\\}|\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert\geqslant\frac{1}{100}}\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(Y_{i})-F(X)\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert}$
For constant $d$, this could be thought of as a constraint satisfaction
problem (CSP) of arity $d+1$. Every $d$-regular graph $G$ has an associated
instance of Balanced Analytic Vertex Expansion whose value corresponds to the
vertex expansion of $G$. Conversly, we exhibit a reduction from Balanced
Analytic Vertex Expansion to problem of approximating vertex expansion in a
graph of degree $\operatorname{{\sf poly}}(d)$ (Section 7 for details).
#### Dictatorship Testing Gadget
As with most hardness results obtained via the label cover or the unique games
problem, central to our reduction is an appropriate dictatorship testing
gadget.
Simply put, a dictatorship testing gadget for Balanced Analytic Vertex
Expansion is an instance $\mathcal{H}^{R}$ of the problem such that, on one
hand there exists the so-called dictator assignments with value $\varepsilon$,
while every assignment far from every dictator incurs a cost of at least
$\Omega(\sqrt{\varepsilon\log d})$.
The construction of the dictatorship testing gadget is as follows. Let $H$ be
a Markov chain on vertices $\\{s,t,t^{\prime},s^{\prime}\\}$ connected to form
a path of length three. The transition probabilities of the Markov chain
$\mathcal{H}$ are so chosen to ensure that if $\mu_{H}$ is the stationary
distribution of $H$ then $\mu_{H}(t)=\mu_{H}(t^{\prime})=\varepsilon/2$ and
$\mu_{H}(s)=\mu_{H}(s^{\prime})=(1-\varepsilon)/2$. In particular, $H$ has a
vertex separator $\\{t,t^{\prime}\\}$ whose weight under the stationary
distribution is only $\varepsilon$.
The dictatorship testing gadget is over the product Markov chain $H^{R}$ for
some large constant $R$. The constraints $\mathcal{P}$ of the dictatorship
testing gadget $H^{R}$ are given by the following sampling procedure,
* –
Sample $x\in H^{R}$ from the stationary distribution of the chain.
* –
Sample $d$-neighbours $y_{1},\ldots,y_{d}\in H^{R}$ of $x$ independently from
the transition probabilities of the chain $H^{R}$. Output the tuple
$(x,y_{1},\ldots,y_{d})$.
For every $i\in[R]$, the $i^{th}$ dictator solution to the above described
gadget is given by the following function,
$F(x)=\begin{cases}1&\text{ if }x_{i}\in\\{s,t\\}\\\ 0&\text{
otherwise}\end{cases}$
It is easy to see that for each constraint
$(x,y_{1},\ldots,y_{d})\sim\mathcal{P}$, $\max_{j}\left\lvert
F(x)-F(y_{j})\right\rvert=0$ unless $x_{i}=t$ or $x_{i}=t^{\prime}$. Since $x$
is sampled from the stationary distribution for $\mu_{H}$,
$x_{i}\in\\{t,t^{\prime}\\}$ happens with probability $\varepsilon$. Therefore
the expected cost incurred by the $i^{th}$ dictator assignment is at most
$\varepsilon$.
#### Soundness Analysis of the Gadget
The soundness property desired of the dictatorship testing gadget can be
stated in terms of influences. Specifically, given an assignment
$F:V(H)^{R}\to[0,1]$, the influence of the $i^{th}$ coordinate is given by
$\operatorname{{\sf
Inf}}_{i}[F]=\operatorname*{\varmathbb{E}}_{x_{[R]\backslash
i}}\mathsf{Var}_{x_{i}}[F(x)]$, i.e., the expected variance of the function
after fixing all but the $i^{th}$ coordinate randomly. Henceforth, we will
refer to a function $F:H^{R}\to[0,1]$ as far from every dictator if the
influence of all of its coordinates are small (say $<\tau$).
We show that the dictatorship testing gadget $H^{R}$ described above satisfies
the following soundness – for every function $F$ that is far from every
dictator, the cost of $F$ is at least $\Omega(\sqrt{\varepsilon\log d})$. To
this end, we appeal to the invariance principle to translate the cost incurred
to a corresponding isoperimetric problem on the Gaussian space. More
precisely, given a function $F:H^{R}\to[0,1]$, we express it as a polynomial
in the eigenfunctions over $H$. We carefully construct a Gaussian ensemble
with the same moments up to order two, as the eigenfunctions at the query
points $(x,y_{1},\ldots,y_{d})\in\mathcal{P}$. By appealing to the invariance
principle for low degree polynomials, this translates in to the following
isoperimetric question over Gaussian space $\mathcal{G}$.,
Suppose we have a subset $S\subseteq\mathcal{G}$ of the $n$-dimensional
Gaussian space. Consider the following experiment:
* –
Sample a point $z\in\mathcal{G}$ the Gaussian space.
* –
Pick $d$ independent perturbations
$z^{\prime}_{1},z^{\prime}_{2},\ldots,z^{\prime}_{d}$ of the point $z$ by
$\varepsilon$-noise.
* –
Output $1$ if at least one of the edges $(z,z^{\prime}_{i})$ crosses the cut
$(S,\bar{S})$ of the Gaussian space.
Among all subsets $S$ of the Gaussian space with a given volume, which set has
the least expected output in the above experiment? The answer to this
isoperimetric question corresponds to the soundness of the dictatorship test.
A halfspace of volume $\frac{1}{2}$ has an expected output of
$\sqrt{\varepsilon\log d}$ in the above experiment. We show that among all
subsets of constant volume, halfspaces acheive the least expected output
value.
This isoperimetric theorem proven in Section 4 yields the desired
$\Omega(\sqrt{\varepsilon\log d})$ bound for the soundness of the dictatorship
test constructed via the Markov chain $H$. Here the noise rate of
$\varepsilon$ arises from the fact that all the eigenfunctions of the Markov
chain $H$ have an eigenvalue smaller than $1-\varepsilon$. The details of the
argument based on invariance principle is presented in Section 5
We show a $\Omega(\sqrt{\varepsilon\log d})$ lower bound for the isoperimetric
problem on the Gaussian space. The proof of this isoperimetric inequality is
included in Section 4
We would like to point out here that the traditional noisy cube gadget does
not suffice for our application. This is because in the noisy cube gadget
while the dictator solutions have an edge expansion of $\varepsilon$ they have
a vertex expansion of $\varepsilon d$, yielding a much worse value than the
soundness.
#### Reduction from Small-Set Expansion problem
Gadget reductions from the Unique Games problem cannot be used towards proving
a hardness result for edge or vertex expansion problems. This is because if
the underlying instance of Unique Games has a small vertex separator, then the
graph produced via a gadget reduction would also have small vertex expansion.
Therefore, we appeal to a reduction from the Small-Set Expansion problem
(Section 6 for details).
Raghavendra et al. [RST12] show optimal inapproximability results for the
Balanced separator problem using a reduction from the Small-Set Expansion
problem. While the overall approach of our reduction is similar to theirs, the
details are subtle. Unlike hardness reductions from unique games, the
reductions for expansion-type problems starting from Small-Set Expansion are
not very well understood. For instance, the work of Raghavendra and Tan [RT12]
gives a dictatorship testing gadget for the Max-Bisection problem, but a
Small-Set Expansion based hardness for Max-Bisection still remains open.
### 2.1 Notation
We use $\mu_{G}$ to denote a probability distribution on vertices of the graph
$G$. We drop the subscript $G$, when the graph is clear from the context. For
a set of vertices $S$, we define $\mu(S)=\int_{x\in S}\mu(x)$. We use
$\mu_{|S}$ to denote the distribution $\mu$ restricted to the set $S\subset
V(G)$. For the sake of simplicity, we sometimes say that vertex $v\in V(G)$
has weight $w(v)$, in which case we define $\mu(v)=w(v)/\sum_{u\in V}w(u)$. We
denote the weight of a set $S\subseteq V$ by $w(S)$. We denote the degree of a
vertex $v$ by $\deg(v)$. We denote the neighborhood of $S$ in $G$ by
$N_{G}(S)$, i.e.
$N_{G}(S)=\\{v\in\bar{S}|\exists u\in S\textrm{ such that
}\left\\{u,v\right\\}\in E(G)\\}\,.$
We drop the subscript $G$ when the graph is clear from the context.
### 2.2 Organization
We begin with some definitions and the statements of the SSEhypotheses in
Section 3. In Section A, we show that the computation of vertex expansion and
symmetric vertex expansion is equivalent upto constant factors. We prove a new
Gaussian isoperimetry results in Section 4 that we use in our soundness
analysis. In Section 5 we show the construction of our main gadget and analyze
its soundness and completeness using Balanced Analytic Vertex Expansion as the
test function. We show a reduction from a reduction from Balanced Analytic
Vertex Expansion to vertex expansion in Section 7. In Section 6, we use this
gadget to show a reduction SSE to Balanced Analytic Vertex Expansion. Finally,
in Section 8, we show how to put all the reductions togethor to get optimal
SSE-hardness for vertex expansion.
Complimenting our lower bound, we give an algorithm that outputs a set having
vertex expansion at most $\mathcal{O}\left(\sqrt{\phi^{\sf V}\log d}\right)$
in Section 9.
## 3 Preliminaries
#### Symmetric Vertex Expansion
For our proofs, the notion of Symmetric Vertex Expansion is useful.
###### Definition 3.1.
Given a graph $G=(V,E)$, we define the the symmetric vertex expansion of a set
$S\subset V$ as follows.
$\Phi^{\sf
V}_{G}(S)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}n\cdot\frac{\left\lvert
N_{G}(S)\cup N_{G}(V\backslash S)\right\rvert}{\left\lvert
S\right\rvert\left\lvert V\backslash{S}\right\rvert}$
#### Balanced Vertex Expansion
We define the balanced vertex expansion of a graph as follows.
###### Definition 3.2.
Given a graph $G$ and balance parameter $b$, we define the $b$-balanced vertex
expansion of $G$ as follows.
$\phi^{\sf
V,bal}_{b}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{S:\left\lvert
S\right\rvert\left\lvert V\backslash S\right\rvert\geqslant bn^{2}}\phi^{\sf
V}(S).$
and
$\Phi^{\sf
V,bal}_{b}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{S:\left\lvert
S\right\rvert\left\lvert V\backslash S\right\rvert\geqslant bn^{2}}\Phi^{\sf
V}(S).$
We define $\phi^{\sf V,bal}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\phi^{\sf
V,bal}_{1/100}$ and $\Phi^{\sf
V,bal}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Phi^{\sf V,bal}_{1/100}$.
#### Analytic Vertex Expansion
Our reduction from SSE to vertex expansion goes via an intermediate problem
that we call $d$-Balanced Analytic Vertex Expansion. We define the notion of
$d$-Balanced Analytic Vertex Expansion as follows.
###### Definition 3.3.
An instance of $d$-Balanced Analytic Vertex Expansion, denoted by
$(V,\mathcal{P})$, consists of a set of variables $V$ and a probability
distribution $\mathcal{P}$ over $(d+1)$-tuples in $V^{d+1}$. The probability
distribution $\mathcal{P}$ satisfies the condition that all its $d+1$ marginal
distributions are the same (denoted by $\mu$). The $d$-Balanced Analytic
Vertex Expansion under a function $F:V\to\left\\{0,1\right\\}$ is defined as
$\Phi({V,\mathcal{P}})(F)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(Y_{i})-F(X)\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert}\,.$
The $d$-Balanced Analytic Vertex Expansion of $(V,\mathcal{P})$ is defined as
$\Phi({V,\mathcal{P}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{F:V\to\left\\{0,1\right\\}|\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert\geqslant\frac{1}{100}}\Phi({V,\mathcal{P}})(F).$
When drop the degree $d$ from the notation, when it is clear from the context.
For an instance $(V,\mathcal{P})$ of Balanced Analytic Vertex Expansion and an
assignment $F:V\to\left\\{0,1\right\\}$ define
$\operatorname{val}_{\mathcal{P}}(F)=\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(Y_{i})-F(X)\right\rvert.$
#### Gaussian Graph
Recall that two standard normal random variables $X,Y$ are said to be
$\alpha$-correlated if there exists an independent standard normal random
variable $Z$ such that $Y=\alpha X+\sqrt{1-\alpha^{2}}Z$.
###### Definition 3.4.
The Gaussian Graph $\mathcal{G}_{\Lambda,\Sigma}$ is a complete weighted graph
on the vertex set $V(\mathcal{G}_{\Lambda,\Sigma})=\varmathbb R^{n}$. The
weight of the edge between two vertices $u,v\in
V(\mathcal{G}_{\Lambda,\Sigma})$ is given by
$w(\left\\{u,v\right\\})=\operatorname*{\varmathbb{P}}\left[X=u\textrm{ and
}Y=v\right]$
where $Y\sim\mathcal{N}(\Lambda X,\Sigma)$, where $\Lambda$ is a diagonal
matrix such that $\left\lVert\Lambda\right\rVert\leqslant 1$ and
$\Sigma\succeq\varepsilon I$ is a diagonal matrix.
###### Remark 3.5.
Note that for any two non-empty disjoint sets $S_{1},S_{2}\subset
V(\mathcal{G}_{\Lambda,\Sigma})$, the total weight of the edges between
$S_{1}$ and $S_{2}$ can be non-zero even though every single edge in the
$\mathcal{G}_{\Lambda,\Sigma}$ has weight zero.
###### Definition 3.6.
We say that a family of graphs $\mathcal{G}_{d}$ is $\Theta(d)$-regular, if
there exist absolute constants $c_{1},c_{2}\in\varmathbb R^{+}$ such that for
every $G\in\mathcal{G}_{d}$, all vertices $i\in V(G)$ have
$c_{1}d\leqslant\deg(i)\leqslant c_{2}d$.
We now formalize our notion of hardness.
###### Definition 3.7.
A constrained minimization problem $\mathcal{A}$ with its optimal value
denoted by $\operatorname{val}(\mathcal{A})$ is said to be c-vs-s hard if it
is SSE-hard to distinguish between the following two cases.
Yes:
$\operatorname{val}(\mathcal{A})\leqslant c\,.$
No:
$\operatorname{val}(\mathcal{A})\geqslant s\,.$
#### Variance
For a random variable $X$, define the variance and $\ell_{1}$-variance as
follows,
$\mathsf{Var}[X]=\operatorname*{\varmathbb{E}}_{X_{1},X_{2}}[(X_{1}-X_{2})^{2}]\qquad\mathsf{Var}_{1}[X]=\operatorname*{\varmathbb{E}}_{X_{1},X_{2}}[|X_{1}-X_{2}|]$
where $X_{1},X_{2}$ are two independent samples of $X$.
#### Small-Set Expansion Hypothesis
###### Problem 3.8 (Small-Set Expansion $(\gamma,\delta)$).
Given a regular graph $G=(V,E)$, distinguish between the following two cases:
Yes:
There exists a non-expanding set $S\subset V$ with $\mu(S)=\delta$ and
$\Phi_{G}(S)\leqslant\gamma$.
No:
All sets $S\subset V$ with $\mu(S)=\delta$ are highly expanding having
$\Phi_{G}(S)\geqslant 1-\gamma$.
###### Hypothesis 3.9 (Hardness of approximating Small-Set Expansion).
For all $\gamma>0$, there exists $\delta>0$ such that the promise problem
Small-Set Expansion ($\gamma,\delta$) is NP-hard.
For the proofs, it shall be more convenient to use the following version of
the Small-Set Expansion problem, in which we high expansion is guaranteed not
only for sets of measure $\delta$, but also within an arbitrary multiplicative
factor of $\delta$.
###### Problem 3.10 (Small-Set Expansion $(\gamma,\delta,M)$).
Given a regular graph $G=(V,E)$, distinguish between the following two cases:
Yes:
There exists a non-expanding set $S\subset V$ with $\mu(S)=\delta$ and
$\Phi_{G}(S)\leqslant\gamma$.
No:
All sets $S\subset V$ with $\mu(S)\in\left(\tfrac{\delta}{M},M\delta\right)$
have $\Phi_{G}(S)\geqslant 1-\gamma$.
The following stronger hypothesis was shown to be equivalent to Small-Set
Expansion Hypothesis in [RST12].
###### Hypothesis 3.11 (Hardness of approximating Small-Set Expansion).
For all $\gamma>0$ and $M\geqslant 1$, there exists $\delta>0$ such that the
promise problem Small-Set Expansion ($\gamma,\delta,M$) is NP-hard.
## 4 Isoperimetry of the Gaussian Graph
In this section we bound the Balanced Analytic Vertex Expansion of the
Gaussian graph. For the Gaussian Graph, we define the canonical probability
distribution on $V^{d+1}$ as follows. The marginal distribution along any
component $X$ or $Y_{i}$ is the standard Gaussian distribution in $\varmathbb
R^{n}$, denoted here by $\mu=\mathcal{N}(0,1)^{n}$.
$\mathcal{P}_{\mathcal{G}_{\Lambda,\Sigma}}(X,Y_{1},\ldots,Y_{d})=\frac{\Pi_{i=1}^{d}w(X,Y_{i})}{\mu(X)^{d-1}}=\mu(X)\Pi_{i=1}^{d}\operatorname*{\varmathbb{P}}\left[Y=Y_{i}\right].$
Here, random variable $Y$ is sampled from $\mathcal{N}(\Lambda X,\Sigma)$.
###### Theorem 4.1.
For any closed set $S\subset ofV(\mathcal{G}_{\Lambda,\Sigma})$ with $\Lambda$
a diagonal matrix satisfying $\left\lVert\Lambda\right\rVert\leqslant 1$, and
$\Sigma$ a diagonal matrix satisfying $\Sigma\succeq\varepsilon I$, we have
$\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{\mathcal{G}_{\Lambda,\Sigma}}}\max_{i}\left\lvert\varmathbb{I}_{S}\left[X\right]-\varmathbb{I}_{S}\left[Y_{i}\right]\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert\varmathbb{I}_{S}\left[X\right]-\varmathbb{I}_{S}\left[Y\right]\right\rvert}=\frac{\operatorname*{\varmathbb{E}}_{X\sim\mu}\operatorname*{\varmathbb{E}}_{Y_{1},\ldots
Y_{d}\sim\mathcal{N}(\Lambda
X,\Sigma)}\max_{i}\left\lvert\varmathbb{I}_{S}\left[X\right]-\varmathbb{I}_{S}\left[Y_{i}\right]\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert\varmathbb{I}_{S}\left[X\right]-\varmathbb{I}_{S}\left[Y\right]\right\rvert}\geqslant
c\sqrt{\varepsilon\log d}$
for some absolute constant $c$.
###### Lemma 4.2.
Let $u,v\in\varmathbb R^{n}$ satisfy $\left\lvert
u-v\right\rvert\leqslant\sqrt{\varepsilon\log d}$. Let $\Lambda$ be a diagonal
matrix satisfying $\left\lVert\Lambda\right\rVert\leqslant 1$, and let
$\Sigma$ a diagonal matrix satisfying $\Sigma\succeq\varepsilon I$. Let
$P_{u},P_{v}$ be the distributions $\mathcal{N}(\Lambda u,\Sigma)$ and
$\mathcal{N}(\Lambda v,\Sigma)$ respectively. Then,
$d_{\sf TV}(P_{u},P_{v})\leqslant 1-\frac{1}{d}.$
###### Proof.
First, we note that that for the purpose of estimating their total variation
distance, we can view $P_{u},P_{v}$ as one-dimensional Gaussians along the
line $\Lambda u-\Lambda v$. Since $\left\lVert\Lambda\right\rVert\leqslant 1$,
$\left\lVert\Lambda u-\Lambda v\right\rVert\leqslant\left\lVert
u-v\right\rVert\leqslant\sqrt{\varepsilon\log d}\,.$
Wlog, we may take $\Lambda u=0$ and $\Lambda v=\sqrt{\varepsilon\log d}$.
Next, by the definition of total variation distance,
$\displaystyle d_{\sf TV}(P_{u},P_{v})$ $\displaystyle=$
$\displaystyle\int_{x:P_{v}(x)\geqslant P_{u}(x)}|P_{v}(x)-P_{u}(x)|dx$
$\displaystyle=$ $\displaystyle\int_{\Lambda
v/2}^{\infty}(P_{v}(x)-P_{u}(x))dx$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\varepsilon}}\int_{\Lambda
v/2}^{\infty}e^{-\frac{\|x-\Lambda
v\|^{2}}{2\varepsilon}}\,dx-\frac{1}{\sqrt{2\pi\varepsilon}}\int_{\Lambda
v/2}^{\infty}e^{-\frac{\|x\|^{2}}{2\varepsilon}}\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\varepsilon}}\int_{-\Lambda v/2}^{\Lambda
v/2}e^{-\frac{\|x\|^{2}}{2\varepsilon}}\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\sqrt{\log d}/2}^{\sqrt{\log
d}/2}e^{-\frac{\|x\|^{2}}{2}}\,dx$ $\displaystyle=$ $\displaystyle
1-2\cdot\frac{1}{\sqrt{2\pi}}\int_{\sqrt{\log
d}/2}^{\infty}e^{-\frac{\|x\|^{2}}{2}}\,dx$ $\displaystyle<$ $\displaystyle
1-\frac{1}{d}.$
where the last step uses a standard bound on the Gaussian tail.
∎
###### Proof of Theorem 4.1..
Let $\mu_{X}$ denote the Gaussian distribution $\mathcal{N}(\Lambda
X,\Sigma)$. Then the LHS is:
$\int_{\varmathbb R^{n}\setminus
S}\left(1-(1-\mu_{X}(S))^{d}\right)\,d\mu(X)+\int_{S}\left(1-(1-\mu_{X}(\varmathbb
R^{n}\setminus S))^{d}\right)\,d\mu(X).$
To bound this, we will restrict ourselves to points $X$ for which the
$\mu_{X}$ measure of the complementary set is at least $1/d$. Roughly
speaking, these will be points near the boundary of $S$. Define:
$S_{1}=\left\\{x\in S\,:\,\mu_{X}(\varmathbb R^{n}\setminus
S)<\frac{1}{2d}\right\\},\ S_{2}=\left\\{x\in\varmathbb R^{n}\setminus
S\,:\,\mu_{X}(S)<\frac{1}{2d}\right\\}$
and
$S_{3}=\varmathbb R^{n}\setminus S_{1}\setminus S_{2}.$
For $u\in\varmathbb R^{n}$, let $P_{u}$ be the distribution
$\mathcal{N}(\Lambda u,\Sigma)$. For any $u\in S_{1},v\in S_{2}$, we have
$d_{\sf TV}(P_{u},P_{v})>1-\frac{1}{2d}-\frac{1}{2d}=1-\frac{1}{d}.$
Therefore, by Lemma 4.2, $\|u-v\|>\sqrt{\varepsilon\log d}$, i.e.,
$d(S_{1},S_{2})>\sqrt{\varepsilon\log d}$. Next we bound the measure of
$S_{3}$. We can assume wlog that $\mu(S)\leqslant\mu(\varmathbb R^{n}\setminus
S)$ and $\mu(S_{1})\geqslant\mu(S)/2$ (else $\mu(S_{3})\geqslant\mu(S)/2$ and
we are done). Applying the isoperimetric inequaity for Gaussian space [Bor75,
ST78], for subsets at this distance,
$\mu(S_{3})\geqslant\sqrt{\frac{2}{\pi}}\sqrt{\varepsilon\log
d}\cdot\mu(S_{1})\mu(S_{2})\geqslant\sqrt{\frac{\varepsilon\log
d}{2\pi}}\cdot\mu(S)\mu(\varmathbb R^{n}\setminus S).$
We are now ready to complete the proof.
$\displaystyle\frac{1}{2}\left(\int_{\varmathbb R^{n}\setminus
S}(1-(1-\mu_{X}(S))^{d})\,d\mu(X)+\int_{S}(1-(1-\mu_{X}(\varmathbb
R^{n}\setminus S))\,d\mu(X)\right)$ $\displaystyle\geqslant$
$\displaystyle\frac{1}{2}\left(\int_{X\in\varmathbb R^{n}\setminus
S,\mu_{X}(S)\geqslant 1/d}(1-(1-\mu_{X}(S))^{d})\,d\mu(X)+\int_{X\in
S,\mu_{X}(\varmathbb R^{n}\setminus S)\geqslant 1/d}(1-(1-\mu_{X}(\varmathbb
R^{n}\setminus S))\,d\mu(X)\right)$ $\displaystyle\geqslant$
$\displaystyle\frac{e-1}{2e}\left(\int_{X\in\varmathbb R^{n}\setminus
S,\mu_{X}(S)\geqslant 1/d}\,d\mu(X)+\int_{X\in S,\mu_{X}(\varmathbb
R^{n}\setminus X)\geqslant 1/d}\,d\mu(X)\right)$ $\displaystyle\geqslant$
$\displaystyle\frac{e-1}{2e}\mu(S_{3})$ $\displaystyle\geqslant$
$\displaystyle c\sqrt{\varepsilon\log d}\cdot\mu(S)\mu(\varmathbb
R^{n}\setminus S).$
∎
We prove the following Theorem which helps us to bound the isoperimetry of the
Gaussian graph for over all functions over the range $[0,1]$.
###### Theorem 4.3.
Given an instance $(V,\mathcal{P})$ and a function $F:V\to[0,1]$, there exists
a function $F^{\prime}:V\to\left\\{0,1\right\\}$, such that
$\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert}\geqslant\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F^{\prime}(X)-F^{\prime}(Y_{i})\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F^{\prime}(X)-F^{\prime}(Y)\right\rvert}$
###### Proof.
For every $r\in[0,1]$, we define $F_{r}:V\to\left\\{0,1\right\\}$ as follows.
$F_{r}(X)=\begin{cases}1&F(X)\geqslant r\\\ 0&F(X)<r\end{cases}$
Clearly,
$F(X)=\int_{0}^{1}F_{r}(X)dr\,.$
Now, observe that if $F(X)-F(Y)\geqslant 0$ then $F_{r}(X)-F_{r}(Y)\geqslant
0\ \forall r\in[0,1]$ and similiarly, if $F(X)-F(Y)<0$ then
$F_{r}(X)-F_{r}(Y)\leqslant 0\ \forall r\in[0,1]$. Therefore,
$\left\lvert
F(X)-F(Y)\right\rvert=\left\lvert\int_{0}^{1}\left(F_{r}(X)-F_{r}(Y)\right)dr\right\rvert=\int_{0}^{1}\left\lvert
F_{r}(X)-F_{r}(Y)\right\rvert dr\,.$
Also, observe that if $\left\lvert
F(X)-F(Y_{1})\right\rvert\geqslant\left\lvert F(Y_{i})-F(X)\right\rvert$ then
$\left\lvert F_{r}(X)-F_{r}(Y_{1})\right\rvert\geqslant\left\lvert
F_{r}(Y_{i})-F_{r}(X)\right\rvert\ \forall r\in[0,1]$
Therefore,
$\displaystyle\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert}$ $\displaystyle=$
$\displaystyle\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\int_{0}^{1}\left\lvert
F_{r}(X)-F_{r}(Y_{i})\right\rvert
dr}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\int_{0}^{1}\left\lvert
F_{r}(X)-F_{r}(Y)\right\rvert dr}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{1}\left(\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F_{r}(X)-F_{r}(Y_{i})\right\rvert\right)dr}{\int_{0}^{1}\left(\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F_{r}(X)-F_{r}(Y)\right\rvert\right)dr}$ $\displaystyle\geqslant$
$\displaystyle\min_{r\in[0,1]}\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F_{r}(X)-F_{r}(Y_{i})\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F_{r}(X)-F_{r}(Y)\right\rvert}$
Let $r^{\prime}$ be the value of $r$ which minimizes the expression above.
Taking $F^{\prime}$ to be $F_{r^{\prime}}$ finishes the proof.
∎
###### Corollary 4.4 (Corollary to Theorem 4.1 and Theorem 4.3).
Let $F:V(\mathcal{G}_{\Lambda,\Sigma})\to[0,1]$ be any function. Then, for
some absolute constant $c$,
$\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{\mathcal{G}_{\Lambda,\Sigma}}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu}\left\lvert
F(X)-F(Y)\right\rvert}\geqslant c\sqrt{\varepsilon\log d}\,.$
## 5 Dictatorship Testing Gadget
In this section we initiate the construction of the dictatorship testing
gadget for reduction from SSE.
Overall, the dictatorship testing gadget is obtained by picking an
appropriately chosen constant sized Markov-chain $H$, and considering the
product Markov chain $H^{R}$. Formally, given a Markov chain $H$, define an
instance of Balanced Analytic Vertex Expansion with vertices as $V_{H}$ and
the constraints given by the following canonical probability distribution over
$V_{H}^{d+1}$.
* –
Sample $X\sim\mu_{H}$, the stationary distribution of the Markov chain
$V_{H}$.
* –
Sample $Y_{1},\ldots,Y_{d}$ independently from the neighbours of $X$ in
$V_{H}$
For our application, we use a specific Markov chain $H$ on four vertices.
Define a Markov chain $H$ on $V_{H}=\\{s,t,t^{\prime},s^{\prime}\\}$ as
follows,$p(s|s)=p(s^{\prime}|s^{\prime})=1-\frac{\varepsilon}{1-2\varepsilon}$,
$p(t|s)=p(t^{\prime}|s^{\prime})=\frac{\varepsilon}{1-2\varepsilon}$,
$p(s|t)=p(s^{\prime}|t^{\prime})=\frac{1}{2}$ and
$p(t^{\prime}|t)=p(t|t^{\prime})=\frac{1}{2}$. It is easy to see that the
stationary distribution of the Markov chain $H$ over $V_{H}$ is given by,
$\mu_{H}(s)=\mu_{H}(s^{\prime})=\frac{1}{2}-\varepsilon\qquad\qquad\mu_{H}(t)=\mu_{H}(t^{\prime})=\varepsilon$
From this Markov chain, construct a dictatorship testing gadget
$(V_{H}^{R},\mathcal{P}_{H}^{R})$ as described above. We begin by showing that
this dictatorship testing gadget has small vertex separators corresponding to
dictator functions.
###### Proposition 5.1 (Completeness).
For each $i\in[R]$, the $i^{th}$-dictator set defined as $F(x)=1$ if
$x_{i}\in\\{s,t\\}$ and $0$ otherwise satisfies,
$\mathsf{Var}_{1}[F]=\frac{1}{2}\qquad\text{ and
}\qquad\operatorname{val}_{\mathcal{P}_{H^{R}}}(F)\leqslant 2\varepsilon$
###### Proof.
Clearly,
$\operatorname*{\varmathbb{E}}_{X,Y\sim\mu_{H}}\left\lvert
F(X)-F(Y)\right\rvert=\frac{1}{2}$
Observe that for any choice of
$(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}$, $\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert$ is non-zero if and only if either $x_{i}=t$ or
$x_{i}=t^{\prime}$. Therefore we have,
$\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert\leqslant\operatorname*{\varmathbb{P}}[x_{i}\in\\{t,t^{\prime}\\}])=2\varepsilon\,,$
which concludes the proof. ∎
### 5.1 Soundness
We will show a general soundness claim that holds for dictatorship testing
gadgets $(V(H^{R}),\mathcal{P}_{H^{R}})$ constructed out of arbitrary Markov
chains $H$ with a given spectral gap. Towards formally stating the soundness
claim, we recall some background and notation about polynomials over the
product Markov chain $H^{R}$.
### 5.2 Polynomials over $H^{R}$
In this section, we recall how functions over the product Markov chain $H^{R}$
can be written as multilinear polynomials over the eigenfunctions of $H$.
Let $e_{0},e_{1},\ldots,e_{n}:V(H)\to\varmathbb R$ be an orthonormal basis of
eigenvectors of $H$ and let $\lambda_{0},\ldots,\lambda_{n}$ be the
corresponding eigenvalues. Here $e_{0}=1$ is the constant function whose
eigenvalue $\lambda_{0}=1$. Clearly $e_{0},\ldots,e_{n}$ form an orthonormal
basis for the vector space of functions from $V(H)$ to $\varmathbb R$.
It is easy to see that the eigenvectors of the product chain $H^{R}$ are given
by products of $e_{0},\dots,e_{n}$. Specifically, the eigenvectors of $H^{R}$
are indexed by $\sigma\in[n]^{R}$ as follows,
$e_{\sigma}(x)=\prod_{i=1}^{R}e_{\sigma_{i}}(x_{i})$
Every function $f:H^{R}\to\varmathbb R$ can be written in this orthonormal
basis $f(x)=\sum_{\sigma\in[n]^{R}}\hat{f}_{\sigma}e_{\sigma}(x)$. For a
multi-index $\sigma\in[n]^{R}$, the function $e_{\sigma}$ is a monomial of
degree $|\sigma|=|\\{i|\sigma_{i}\neq 0\\}|$.
For a polynomial $Q=\sum_{\sigma}\hat{Q}_{\sigma}e_{\sigma}$, the polynomial
$Q^{>p}$ denotes the projection on to degrees higher than $p$, i.e.,
$Q^{>p}=\sum_{\sigma,|\sigma|>p}\hat{Q}_{\sigma}e_{\sigma}$. The influences of
a polynomial $Q=\sum_{\sigma}\hat{Q}_{\sigma}$ are defined as,
$\operatorname{{\sf Inf}}_{i}(Q)=\sum_{\sigma:\sigma_{i}\neq
0}\hat{Q}_{\sigma}^{2}$
The above notions can be naturally extended to vectors of multilinear
polynomials $Q=(Q_{0},Q_{1},\ldots,Q_{d})$.
Note that every real-valued function on the vertices $V(H)$ of a Markov chain
$H$ can be thought of as a random variable. For each $i>0$, the random
variable $e_{i}(x)$ has mean zero and variance $1$. The same holds for all
$e_{\sigma}(x)$ for all $|\sigma|\neq 0$. For a function
$Q:V(H^{R})\to\varmathbb R$ (or equivalently a polynomial), $\mathsf{Var}[Q]$
denotes the variance of the random variable $Q(x)$ for a random $x$ from
stationary distribution of $H^{R}$. It is an easy computation to check that
this is given by,
$\mathsf{Var}[Q]=\sum_{\sigma:|\sigma|\neq 0}\hat{Q}_{\sigma}^{2}$
We will make use of the following Invariance Principle due to Isaksson and
Mossel [IM12].
###### Theorem 5.2 ([IM12]).
Let $X=(X_{1},\ldots,X_{n})$ be an independent sequence of ensembles, such
that
$\operatorname*{\varmathbb{P}}\left[X_{i}=x\right]\geqslant\alpha>0,\forall
i,x$. Let $Q$ be a $d$-dimensional multilinear polynomial such that
$\mathsf{Var}(Q_{j}(X))\leqslant 1$,
$\mathsf{Var}(Q_{j}^{>p})\leqslant(1-\varepsilon\eta)^{2p}$ and
$\operatorname{{\sf Inf}}_{i}(Q_{j})\leqslant\tau$ where
$p=\frac{1}{18}\log(1/\tau)/\log(1/\alpha)$. Finally, let $\psi:\varmathbb
R^{k}\to\varmathbb R$ be Lipschitz continuous. Then,
$\left\lvert\operatorname*{\varmathbb{E}}\left[\psi(Q(X))\right]-\operatorname*{\varmathbb{E}}\left[\psi(Q(Z))\right]\right\rvert=\mathcal{O}\left(\tau^{\frac{\varepsilon\eta}{18}/\log\frac{1}{\alpha}}\right)$
where $Z$ is an independent sequence of Gaussian ensembles with the same
covariance structure as $X$.
### 5.3 Noise Operator
We define a noise operator $\Gamma_{1-\eta}$ on functions on the Markov chain
$H$ as follows :
$\Gamma_{1-\eta}F(X)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(1-\eta)F(X)+\eta\operatorname*{\varmathbb{E}}_{Y\sim
X}F(Y)$
for every function $F:H\to\varmathbb R$. Similarly, one can define the noise
operator $\Gamma_{1-\eta}$ on functions over $H^{R}$.
Applying the noise operator $\Gamma_{1-\eta}$ on a function $F$, smoothens the
function or makes it closer to a low-degree polynomial. This resulting
function $\Gamma_{1-\eta}F$ is close to a low-degree polynomial, and therefore
is amenable to applying an invariance principle. Formally, one can show the
following decay of coefficients of high degree for $\Gamma_{1-\eta}F$. We
defer the proof to the Appendix (Lemma C.1).
###### Lemma 5.3.
(Decay of High degree Coefficients) Let $Q_{j}$ be the multi-linear polynomial
representation of $\Gamma_{1-\eta}F(X)$, and let $\varepsilon$ be the spectral
gap of the Markov chain $H$. Then,
$\mathsf{Var}(Q_{j}^{>p})\leqslant(1-\varepsilon\eta)^{2p}$
Furthermore, on applying the noise operator $\Gamma_{1-\eta}$, the resulting
function $\Gamma_{1-\eta}F$ can have a bounded number of influential
coordinates as shown by the following lemma.
###### Lemma 5.4.
(Sum of Influences Lemma) If the spectral gap of a Markov chain is at least
$\varepsilon$ then for any function $F:V_{H}^{R}\to\varmathbb R$,
$\sum_{i\in[R]}\operatorname{{\sf
Inf}}_{i}(\Gamma_{1-\eta}F)\leqslant\frac{1}{\eta\varepsilon}\mathsf{Var}[F]$
###### Proof.
By suitable normalization, we may assume without loss of generality that
$\mathsf{Var}[F]=1$. If $Q$ denotes the multilinear representation of
$\Gamma_{1-\eta}F$, then the sum of influences can be written as,
$\displaystyle\sum_{i\in[R]}\operatorname{{\sf Inf}}_{i}(\Gamma_{1-\eta}F)$
$\displaystyle\leqslant\sum_{|\sigma|\neq 0}|\sigma|\hat{Q}_{\sigma}^{2}$
$\displaystyle\leqslant\sum_{|\sigma|\neq
0}|\sigma|(1-\eta\varepsilon)^{2|\sigma|}\hat{F}_{\sigma}^{2}$
$\displaystyle\leqslant\left(\max_{k\in\varmathbb
N}k(1-\eta\varepsilon)^{2k}\right)\sum_{|\sigma|\neq
0}\hat{F}_{\sigma}^{2}<\frac{1}{\eta\varepsilon}$
where we used the fact that the function $h(t)=t(1-\eta\varepsilon)^{2t}$
achieves its maximum value at $t=-\frac{1}{2}\ln(1-\eta\varepsilon)$. ∎
### 5.4 Soundness Claim
Now we are ready to formally state our soundness claim for a dictatorship test
gadget constructed out of a Markov chain.
###### Proposition 5.5 (Soundness).
For all $\varepsilon,\eta,\alpha,\tau>0$ the following holds. Let $H$ be a
finite Markov-chain with a spectral gap of at least $\varepsilon$, and the
probability of every state under stationary distribution is $\geqslant\alpha$.
Let $F:V(H^{R})\to\left\\{0,1\right\\}$ be a function such that
$\max_{i\in[R]}\operatorname{{\sf Inf}}_{i}(\Gamma_{1-\eta}F)\leqslant\tau$.
Then we have
$\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}[\max_{i}\left\lvert
F(Y_{i})-F(X)\right\rvert]\geqslant\Omega(\sqrt{\varepsilon\log
d})\operatorname*{\varmathbb{E}}_{X,Y\sim\mu_{H^{R}}}\left\lvert
F(X)-F(Y)\right\rvert-O(\eta)-\tau^{\Omega(\varepsilon\eta/\log(1/\alpha))}$
For the sake of brevity, we define ${\sf
soundness}(V(H^{R}),\mathcal{P}_{H^{R}})$ to be the following :
###### Definition 5.6.
${\sf
soundness}(V(H^{R}),\mathcal{P}_{H^{R}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{F:\max_{i\in[R]}\operatorname{{\sf
Inf}}_{i}(F)\leqslant\tau}\frac{\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}[\max_{i}\left\lvert
F(Y_{i})-F(X)\right\rvert]}{\operatorname*{\varmathbb{E}}_{X,Y\sim\mu_{H^{R}})}\left\lvert
F(X)-F(Y)\right\rvert}$
In the rest of the section, we will present a proof of Proposition 5.5. First,
we construct gaussian random variables with moments matching the eigenvectors
of the chain $H$.
#### Gaussian Ensembles
Let $Q=(Q_{0},Q_{1},\ldots,Q_{d})$ be the multi-linear polynomial
representation of the vector-valued function
$\left(\Gamma_{1-\eta}F(X),\Gamma_{1-\eta}F(Y_{1}),\ldots,\Gamma_{1-\eta}F(Y_{d})\right)$.
Let $E$ denote the ensemble of $nd$ random variables
$(e_{0}(X),e_{1}(X),\ldots,e_{n}(X)),(e_{0}(Y_{1}),\ldots,e_{n}(Y_{1})),\ldots,(e_{0}(Y_{d}),\ldots,e_{n}(Y_{d}))$.
Let $E_{1},\ldots,E_{R}$ be $R$ independent copies of the ensemble $E$.
Clearly, the polynomial $Q$ can be thought of as a polynomial over
$E_{1},\ldots,E_{R}$. For each random variable $x$ in $E_{1},\ldots,E_{R}$ and
a value $\beta$ in its support,
$\operatorname*{\varmathbb{P}}\left[x=\beta\right]$ is at least the minimum
probability of a vertex in $H$ under its stationary distribution.
This polynomial $Q$ satisfies the requirements of Theorem 5.2 because on the
one hand, the influences of $F$ are $\leqslant\tau$ and on the other by Lemma
5.3, $\mathsf{Var}(Q^{\geqslant p})\leqslant(1-\varepsilon\eta)^{2p}$. Now we
will apply the invariance principle to relate the soundness to the
corresponding quantity on the gaussian graph, and then appeal to the
isoperimetric result on the Gaussian graph (Theorem 4.1).
The invariance principle translates the polynomial
$(Q_{0}(X),Q_{1}(Y_{1}),\ldots Q_{d}(Y_{d}))$ on the sequence of independent
ensembles $E_{1},\dots,E_{R}$, to a polynomial on a corresponding sequence of
gaussian ensembles with the same moments up to degree two.
Consider the ensemble $E$. For each $i\neq 0$, the expectation
$\operatorname*{\varmathbb{E}}[e_{i}(X)]=\operatorname*{\varmathbb{E}}[e_{i}(Y_{1})=0]=\ldots\operatorname*{\varmathbb{E}}[e_{i}(Y_{d})]=0$.
For each $i\neq j$, it is easy to see that,
$\operatorname*{\varmathbb{E}}[e_{i}(X)e_{j}(X)]=\operatorname*{\varmathbb{E}}[e_{i}(Y_{1})e_{j}(Y_{1})]=\ldots\operatorname*{\varmathbb{E}}[e_{i}(Y_{d})e_{j}(Y_{d}]=0$.
Moreover,
$\operatorname*{\varmathbb{E}}[e_{i}(X)e_{j}(Y_{a})]=\operatorname*{\varmathbb{E}}[e_{i}(Y_{a})e_{j}(Y_{b})]=0$
whenever $i\neq j$ and all $a,b\in\\{1,\ldots d\\}$. The only non-trivial
correlations are $\operatorname*{\varmathbb{E}}[e_{i}(X)e_{i}(Y_{a})]$ and
$\operatorname*{\varmathbb{E}}[e_{i}(Y_{a})e_{i}(Y_{b})]$ for all $i\in[n]$
and $a,b\in[d]$. It is easy to check that
$\operatorname*{\varmathbb{E}}[e_{i}(X)e_{i}(Y_{a})]=\lambda_{i}\qquad\qquad\operatorname*{\varmathbb{E}}[e_{i}(Y_{a})e_{i}(Y_{b})]=\lambda_{i}^{2}$
From the above discussion, we see that the following gaussian ensemble
$z=(z_{X},z_{Y_{1}},\ldots,z_{Y_{d}})$ has the same covariance as the ensemble
$E$.
1. 1.
Sample $z_{X}$ and $n$-dimensional Gaussian random vector.
2. 2.
Sample $z_{Y_{1}},\ldots,z_{Y_{d}}\in\varmathbb R^{n}$ i.i.d as follows : The
$i^{th}$ coordinate of each $z_{Y_{a}}$ is sampled from
$\lambda_{i}z_{X}(i)+\sqrt{1-\lambda_{i}^{2}}\xi_{a,i}$ where $\xi_{a,i}$ is a
Gaussian random variable independent of $z_{X}$ and all other $\xi_{a,i}$.
Let $Z_{X},Z_{Y_{1}},\ldots,Z_{Y_{d}}\in\varmathbb R^{nR}$ be the ensemble
obtained by $R$ independent samples from $z_{X},z_{Y_{1}},\ldots,z_{Y_{d}}$.
Let $\Sigma$ denote the $nR\times nR$ diagonal matrix whose entries are
$1-\lambda^{2}_{1},\ldots,1-\lambda^{2}_{n}$ repeated $R$ times. Since the
spectral gap of $H$ is $\varepsilon$, we have that $1-\lambda^{2}_{i}\geqslant
2\varepsilon-\varepsilon^{2}>\varepsilon$ for all $i\in\\{1,\ldots,n\\}$.
Therefore, we have $\Sigma>\varepsilon I$.
#### Proof of soundness
Now we return to the proof of the main soundness claim for the dictatorship
testing gadget ($V(H^{R})$, $\mathcal{P}_{\mathcal{H}^{R}}$) constructed out
an arbitrary Markov chain.
###### Proof of Proposition 5.5.
Let $Q=(Q_{0},Q_{1},\ldots,Q_{d})$ be the multi-linear polynomial
representation of the vector-valued function
$\left(\Gamma_{1-\eta}F(X),\Gamma_{1-\eta}F(Y_{1}),\ldots,\Gamma_{1-\eta}F(Y_{d})\right)$.
Define a function $s:\varmathbb R\to\varmathbb R$ as follows
$s(x)=\begin{cases}0&\text{ if }x<0\\\ x&\text{ if }x\in[0,1]\\\ 1&\text{ if
}x>1\end{cases}$
Define a function $\Psi:\varmathbb R^{d+1}\to\varmathbb R$ as,
$\Psi(x,y_{1},\ldots,y_{d})=\max_{i}|s(y_{i})-s(x)|$. Clearly, $\Psi$ is a
Lipshitz function with a constant of 1.
Using the fact that $F$ is bounded in $[0,1]$,
$\displaystyle\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}\left\lvert
F(X)-F(Y_{a})\right\rvert\geqslant\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}\left\lvert\Gamma_{1-\eta}F(X)-\Gamma_{1-\eta}F(Y_{a})\right\rvert-2\eta$
(5.1)
Furthermore, since $\Gamma_{1-\eta}F$ is also bounded in $[0,1]$, we have
$s(\Gamma_{1-\eta}F)=\Gamma_{1-\eta}F$. Therefore,
$\displaystyle\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}\left\lvert\Gamma_{1-\eta}F(X)-\Gamma_{1-\eta}F(Y_{a})\right\rvert=\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}\left\lvert
s\left(\Gamma_{1-\eta}F(X)\right)-s\left(\Gamma_{1-\eta}F(Y_{a})\right)\right\rvert$
(5.2)
Apply the invariance principle to the polynomial
$Q=\left(\Gamma_{1-\eta}F,\Gamma_{1-\eta}F,\ldots,\Gamma_{1-\eta}F\right)$ and
Lipshitz function $\Psi$. By invariance principle Theorem 5.2, we get
$\displaystyle\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}$
$\displaystyle\left\lvert
s\left(\Gamma_{1-\eta}F(X)\right)-s\left(\Gamma_{1-\eta}F(Y_{a})\right)\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{(Z_{X},Z_{Y_{1}},\ldots,Z_{Y_{d}})\sim\mathcal{P}_{\mathcal{G}_{\Lambda,\Sigma}}}\max_{a}\left\lvert
s\left(\Gamma_{1-\eta}F(Z_{X})\right)-s\left(\Gamma_{1-\eta}F(Z_{Y_{a}})\right)\right\rvert-\tau^{\Omega(\varepsilon\eta/\log(1/\alpha))}$
(5.3)
Observe that $s\circ(\Gamma_{1-\eta}F)$ is bounded in $[0,1]$ even over the
gaussian space. Hence, by using the isoperimetric result on gaussian graphs
(Corollary 4.4), we know that
$\displaystyle\operatorname*{\varmathbb{E}}_{(Z_{X},Z_{Y_{1}},\ldots,Z_{Y_{d}})\sim\mathcal{P}_{\mathcal{G}_{\Lambda,\Sigma}}}\max_{a}\left\lvert
s\left(\Gamma_{1-\eta}F(Z_{X})\right)-s\left(\Gamma_{1-\eta}F(Z_{Y_{a}})\right)\right\rvert\geqslant
c\sqrt{\varepsilon\log
d}\operatorname*{\varmathbb{E}}_{Z_{X},Z_{Y}\sim\mu_{\mathcal{G}_{\Lambda,\Sigma}}}\left\lvert
s\left(\Gamma_{1-\eta}F(Z_{X})\right)-s\left(\Gamma_{1-\eta}F(Z_{Y})\right)\right\rvert$
(5.4)
Now we apply the invariance principle on the polynomial
$(\Gamma_{1-\eta}F,\Gamma_{1-\eta}F)$ and the functional $\Psi:\varmathbb
R^{2}\to\varmathbb R$ given by $\Psi(a,b)=|s(a)-s(b)|$. This yields,
$\displaystyle\operatorname*{\varmathbb{E}}_{Z_{X},Z_{Y}\sim\mu_{\mathcal{G}_{\Lambda,\Sigma}}}\left\lvert
s\left(\Gamma_{1-\eta}F(Z_{X})\right)-s\left(\Gamma_{1-\eta}F(Z_{Y})\right)\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{X,Y\sim\mu(H^{R})}\left\lvert
s\left(\Gamma_{1-\eta}F(X)\right)-s\left(\Gamma_{1-\eta}F(Y)\right)\right\rvert-\tau^{\Omega(\varepsilon\eta/\log(1/\alpha))}$
(5.5)
Over $H^{R}$, the function $\Gamma_{1-\eta}F$ is bounded in $[0,1]$, which
implies that $s(\Gamma_{1-\eta}F(X))=\Gamma_{1-\eta}F(X)$ and
$\Gamma_{1-\eta}F(X)\geqslant F(X)-\eta$.
$\displaystyle\operatorname*{\varmathbb{E}}_{X,Y\sim\mu(H^{R})}\left\lvert
s\left(\Gamma_{1-\eta}F(X)\right)-s\left(\Gamma_{1-\eta}F(Y)\right)\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{X,Y\sim\mu(H^{R})}\left\lvert
F(X)-F(Y)\right\rvert-2\eta$ (5.6)
From equations (5.1) to (5.6) we get,
$\displaystyle\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}_{H^{R}}}\max_{a}\left\lvert
F(X)-F(Y_{a})\right\rvert$ $\displaystyle\geqslant\Omega(\sqrt{\varepsilon\log
d})\operatorname*{\varmathbb{E}}_{X,Y\sim\mu(H^{R})}\left\lvert
F(X)-F(Y)\right\rvert-4\eta-\tau^{\Omega(\varepsilon\eta/\log(1/\alpha))}$
∎
## 6 Hardness Reduction from SSE
In this section we will present a reduction from Small-Set Expansion problem
to Balanced Analytic Vertex Expansion problem.
Let $G=(V,E)$ be an instance of Small-Set Expansion $(\gamma,\delta,M)$.
Starting with the instance $G=(V,E)$ of $\textsc{Small-Set
Expansion}(\gamma,\delta,M)$, our reduction produces an instance
$(\mathcal{V}^{\prime},\mathcal{P}^{\prime})$ of Balanced Analytic Vertex
Expansion.
To describe our reduction, let us fix some notation. For a set $A$, let
${A}^{\\{R\\}}$ denote the set of all multisets with $R$ elements from $A$.
Let $G_{\eta}=(1-\eta)G+\eta K_{V}$ where $K_{V}$ denotes the complete graph
on the set of vertices $V$. For an integer $R$, define $G_{\eta}^{\otimes R}$
to be the product graph $G_{\eta}^{\otimes R}=(G_{\eta})^{R}$.
Define a Markov chain $H$ on $V_{H}=\\{s,t,t^{\prime},s^{\prime}\\}$ as
follows,$p(s|s)=p(s^{\prime}|s^{\prime})=1-\frac{\varepsilon}{1-2\varepsilon}$,
$p(t|s)=p(t^{\prime}|s^{\prime})=\frac{\varepsilon}{1-2\varepsilon}$,
$p(s|t)=p(s^{\prime}|t^{\prime})=\frac{1}{2}$ and
$p(t^{\prime}|t)=p(t|t^{\prime})=\frac{1}{2}$. It is easy to see that the
stationary distribution of the Markov chain $H$ over $V_{H}$ is given by,
$\mu_{H}(s)=\mu_{H}(s^{\prime})=\frac{1}{2}-\varepsilon\qquad\qquad\mu_{H}(t)=\mu_{H}(t^{\prime})=\varepsilon$
The reduction consists of two steps. First, we construct an “unfolded”
instance $(\mathcal{V},\mathcal{P})$ of the Balanced Analytic Vertex
Expansion, then we merge vertices of $(\mathcal{V},\mathcal{P})$ to create the
final output instance $(\mathcal{V}^{\prime},\mathcal{P}^{\prime})$. The
details of the reduction are presented below.
Reduction Input: A graph $G=(V,E)$ \- an instance of $\textsc{Small-Set
Expansion}(\gamma,\delta,M)$. Parameters: $R=\frac{1}{\delta}$, $\varepsilon$
Unfolded instance $(\mathcal{V},\mathcal{P})$ Set $\mathcal{V}=(V\times
V_{H})^{R}$. The probability distribution $\mu$ on $\mathcal{V}$ is given by
$(\mu_{V}\times\mu_{H})^{R}$. The probability distribution $\mathcal{P}$ is
given by the following sampling procedure. 1. Sample a random vertex $A\in
V^{R}$. 2. Sample $d+1$ random neighbors $B,C_{1},\ldots,C_{d}\sim
G_{\eta}^{\otimes R}(A)$ of the vertex $A$ in the tensor-product graph
$G_{\eta}^{\otimes R}$. 3. Sample $x\in V_{H}^{R}$ from the product
distribution $\mu^{R}$. 4. Independently sample $d$ neighbours
$y^{(1)},\ldots,y^{(d)}$ of $x$ in the Markov chain $H^{R}$, i.e.,
$y^{(i)}\sim\mu_{H}^{R}(x)$. 5. Output
$\left((B,x),(C_{1},y_{1}),\ldots,(C_{d},y_{d})\right)$ Folded Instance
$(\mathcal{V}^{\prime},\mathcal{P}^{\prime})$ Fix
$\mathcal{V}^{\prime}=(V\times\\{s,t\\})^{\\{R\\}}$. Define a projection map
$\Pi:\mathcal{V}\to\mathcal{V}^{\prime}$ as follows:
$\Pi(A,x)=\\{(a_{i},x_{i})|x_{i}\in\\{s,t\\}\\}$ for each
$(A,x)=\left((a_{1},x_{1}),(a_{2},x_{2}),\ldots,(a_{R},x_{R})\right)$ in
$(V\times\\{s,t\\})^{\\{R\\}}$. Let $\mu^{\prime}$ be the probability
distribution on $\mathcal{V}^{\prime}$ obtained by projection of probability
distribution $\mu$ on $\mathcal{V}$. Similarly, the probability distribution
$\mathcal{P}^{\prime}$ on $(\mathcal{V}^{\prime})^{d+1}$ by applying the
projection $\Pi$ to the probability distribution $\mathcal{P}$.
Observe that each of the queries $\Pi(B,x)$ and
$\\{\Pi(C_{i},y_{i})\\}_{i=1}^{d}$ are distributed according to $\mu^{\prime}$
on $\mathcal{V}^{\prime}$. Let
$F^{\prime}\colon\mathcal{V}^{\prime}\to\\{0,1\\}$ denote the indicator
function of a subset for the instance. Let us suppose that
$\operatorname*{\varmathbb{E}}_{X,Y\sim\mathcal{V}}\left[|F^{\prime}(X)-F^{\prime}(Y)|\right]\geqslant\frac{1}{10}$
For the whole reduction, we fix $\eta=\varepsilon/(100d)$. We will restrict
$\gamma<\varepsilon/(100d)$. We will fix its value later.
###### Theorem 6.1.
(Completeness) Suppose there exists a set $S\subset V$ such that
$\operatorname{{\sf vol}}(S)=\delta$ and $\Phi(S)\leqslant\gamma$ then there
exists $F^{\prime}:\mathcal{V}^{\prime}\to\\{0,1\\}$ such that,
$\operatorname*{\varmathbb{E}}_{X,Y\sim\mathcal{V}^{\prime}}\left[|F^{\prime}(X)-F^{\prime}(Y)|\right]\geqslant\frac{1}{10}$
and,
$\operatorname*{\varmathbb{E}}_{X,Y_{1},\ldots,Y_{d}\sim\mathcal{P}}\left[\max_{i}|F^{\prime}(X)-F^{\prime}(Y_{i})|\right]\leqslant
2\varepsilon+\mathcal{O}\left(d(\eta+\gamma)\right)\leqslant 4\varepsilon$
###### Proof.
Define $F:\mathcal{V}\to\\{0,1\\}$ as follows:
$F(A,x)=\begin{cases}1&\text{ if }|\Pi(A,x)\cap(S\times\\{s,t\\})|=1\\\
0&\text{ otherwise}\end{cases}$
Observe that by definition of $F$, the value of $F(A,x)$ only depends on
$\Pi(A,x)$. So the function $F$ naturally defines a map
$F^{\prime}:\mathcal{V}^{\prime}\to\\{0,1\\}$. Therefore we can write,
$\displaystyle\operatorname*{\varmathbb{P}}\left[F(A,x)=1\right]$
$\displaystyle=\sum_{i\in[R]}\operatorname*{\varmathbb{P}}\left[x_{i}\in\\{s,t\\}\right]\operatorname*{\varmathbb{P}}\left[\\{a_{1},\ldots,a_{R}\\}\cap
S=\\{a_{i}\\}|x_{i}\in\\{s,t\\}\right]$ $\displaystyle\geqslant
R\cdot\frac{1}{2}\cdot\frac{1}{R}\cdot\left(1-\frac{1}{R}\right)^{R-1}\geqslant\frac{1}{10}$
and,
$\operatorname*{\varmathbb{P}}\left[F(A,x)=1\right]=\operatorname*{\varmathbb{P}}\left[|\Pi(A,x)\cap(S\times\\{s,t\\})|=1\right]\leqslant\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}\left[|\Pi(A,x)\cap(S\times\\{s,t\\})|\right]=R\cdot\frac{1}{2}\cdot\frac{|S|}{|V|}\leqslant\frac{1}{2}$
The above bounds on $\operatorname*{\varmathbb{P}}\left[F(A,x)=1\right]$ along
with the fact that $F$ takes values only in $\\{0,1\\}$, we get that
$\operatorname*{\varmathbb{E}}_{X,Y\sim\mathcal{V}^{\prime}}\left\lvert
F^{\prime}(X)-F^{\prime}(Y)\right\rvert=\operatorname*{\varmathbb{E}}_{(A,x),(B,y)\sim\mathcal{V}}{|F(A,x)-F(B,y)|}\geqslant\frac{1}{10}$
Suppose we sample $A\in V^{R}$ and $B,C_{1},\ldots,C_{d}$ independently from
$G_{\eta}^{\otimes R}(A)$. Let us denote $A=(a_{1},\ldots,a_{R})$,
$B=(b_{1},\ldots,b_{R})$, $C_{i}=(c_{i1},\ldots,c_{iR})$ for all $i\in[d]$.
Note that,
$\displaystyle\operatorname*{\varmathbb{P}}\left[\exists i\in[R]\text{ such
that }\left\lvert\\{a_{i},b_{i}\\}\cap S\right\rvert=1\right]$
$\displaystyle\leqslant\sum_{i\in[R]}(1-\eta)\operatorname*{\varmathbb{P}}\left[(a_{i},b_{i})\in
E[S,\bar{S}]\right]+\eta\operatorname*{\varmathbb{P}}\left[(a_{i},b_{i})\in
S\times\bar{S}\right]$ $\displaystyle\leqslant R(\operatorname{{\sf
vol}}(S)\Phi(S)+2\eta\operatorname{{\sf vol}}(S))\leqslant 2(\gamma+\eta)\,.$
Similarly, for each $j\in[d]$,
$\operatorname*{\varmathbb{P}}\left[\exists i\in[R]||\\{a_{i},c_{ji}\\}\cap
S|=1\right]\leqslant\sum_{i\in[R]}\operatorname*{\varmathbb{P}}\left[(a_{i},c_{ji})\in
E[S,\bar{S}]\right]\leqslant R\operatorname{{\sf vol}}(S)\Phi(S)\leqslant
2(\gamma+\eta)\,.$
By a union bound, with probability at least $1-2(d+1)(\gamma+\eta)$ we have
that none of the edges $\\{(a_{i},b_{i})\\}_{i\in[R]}$ and
$\\{(a_{i},c_{ji})\\}_{j\in[d],i\in[R]}$ cross the cut $(S,\bar{S})$.
Conditioned on the above event, we claim that if
$(B,x)\cap\left(S\times\\{t,t^{\prime}\\}\right)=\emptyset$ then
$\max_{i}|F(B,x)-F(C_{i},y_{i})|=0$. First, if
$(B,x)\cap\left(S\times\\{t,t^{\prime}\\}\right)=\emptyset$ then for each
$b_{i}\in S$ the corresponding $x_{i}\in\\{s,s^{\prime}\\}$. In particular,
this implies that for each $b_{i}\in S$, either all of the pairs
$(b_{i},x_{i}),\\{(c_{ji},y_{ji})\\}_{j\in[d]}$ are either in
$S\times\\{s,t\\}$ or $S\times\\{s^{\prime},t^{\prime}\\}$, thereby ensuring
that $\max_{i}|F(B,x)-F(C_{i},y_{i})|=0$.
From the above discussion we conclude,
$\displaystyle\operatorname*{\varmathbb{E}}_{(B,x),(C_{1},y_{1}),\ldots,(C_{d},y_{d})\sim\mathcal{P}}\left[\max_{i}|F(B,x)-F(C_{i},y_{i})|\right]$
$\displaystyle\leqslant\operatorname*{\varmathbb{P}}\left[|(B,x)\cap\left(S\times\\{t,t^{\prime}\\}\right)|\geqslant
1\right]+2(d+1)(\gamma+\eta)$
$\displaystyle\leqslant\operatorname*{\varmathbb{E}}\left[|(B,x)\cap\left(S\times\\{t,t^{\prime}\\}\right)|\right]+2(d+1)(\gamma+\eta)$
$\displaystyle=R\cdot\operatorname{{\sf
vol}}(S)\cdot\varepsilon+2(d+1)(\gamma+\eta)=\varepsilon+2(d+1)(\gamma+\eta)$
∎
Let $F^{\prime}:\mathcal{V}^{\prime}\to\\{0,1\\}$ be a subset of the instance
$(\mathcal{V}^{\prime},\mathcal{P}^{\prime})$. Let us define the following
notation.
$\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}^{\prime}}\left[\max_{i\in[d]}\left\lvert
F^{\prime}(X)-F^{\prime}(Y_{i})\right\rvert\right]\qquad\mathsf{Var}_{1}[F^{\prime}]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname*{\varmathbb{E}}_{X,Y\sim\mathcal{V}^{\prime}}\left\lvert
F^{\prime}(X)-F^{\prime}(Y)\right\rvert$
We define the functions $F:\mathcal{V}\to[0,1]$ and
$f_{A},g_{A}:V_{H}^{R}\to[0,1]$ for each $A\in V^{R}$ as follows.
$F(A,x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}F^{\prime}(\Pi(A,x))\qquad
f_{A}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}F(A,x)\qquad\qquad
g_{A}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname*{\varmathbb{E}}_{B\sim
G_{\eta}^{\otimes R}(A)}F(B,x)$
###### Lemma 6.2.
$\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\geqslant\operatorname*{\varmathbb{E}}_{A\in
V^{R}}\operatorname{val}_{\mu_{H}^{R}}(g_{A})$
###### Proof.
$\displaystyle\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
$\displaystyle=\operatorname{val}_{\mathcal{P}}(F)$
$\displaystyle=\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y_{1},\ldots,y_{d}\sim\mu^{R}_{H}(x)}\operatorname*{\varmathbb{E}}_{B,C_{1},\ldots,C_{d}\sim
G_{\gamma}^{\otimes R}(A)}\max_{i}\left\lvert
F(B,x)-F(C_{i},y_{i})\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y_{1},\ldots,y_{d}\sim\mu^{R}_{H}(x)}\max_{i}\left\lvert\operatorname*{\varmathbb{E}}_{B\sim
G_{\gamma}^{\otimes R}(A)}F(B,x)-\operatorname*{\varmathbb{E}}_{C_{i}\sim
G_{\gamma}^{\otimes R}(A)}F(C_{i},y_{i})\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y_{1},\ldots,y_{d}\sim\mu^{R}_{H}(x)}\max_{i}\left\lvert
g_{A}(x)-g_{A}(y_{i})\right\rvert$
$\displaystyle=\operatorname*{\varmathbb{E}}_{A\in
V^{R}}\operatorname{val}_{\mu_{H}^{R}}(g_{A})$
∎
###### Lemma 6.3.
$\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}g_{A}(x)^{2}\geqslant\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F^{2}(A,x)-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
###### Proof.
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}g_{A}(x)^{2}$
$\displaystyle=\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}F(B,x)F(C,x)$
$\displaystyle=\frac{1}{2}\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}F^{2}(B,x)+F^{2}(C,x)-(F(B,x)-F(C,x))^{2}$
$\displaystyle=\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}F^{2}(A,x)-\frac{1}{2}\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}(F(B,x)-F(C,x))^{2}$ (6.1)
where in the last step we used the fact that $B,C$ have the same distribution
as $A\sim V^{R}$. Since the function $F$ is bounded in $[0,1]$, we have
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}(F(B,x)-F(C,x))^{2}$
$\displaystyle\leqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}\left\lvert F(B,x)-F(C,x)\right\rvert$ (6.2)
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim V^{R}}$
$\displaystyle\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}\left\lvert F(B,x)-F(C,x)\right\rvert$
$\displaystyle\leqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y\sim\mu^{R}_{H}(x)}\operatorname*{\varmathbb{E}}_{B,C,D\sim
G_{\eta}^{\otimes R}(A)}\left\lvert F(B,x)-F(D,y)\right\rvert+\left\lvert
F(C,x)-F(D,y)\right\rvert$
$\displaystyle=2\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y\sim\mu^{R}_{H}(x)}\operatorname*{\varmathbb{E}}_{B,D\sim
G_{\eta}^{\otimes R}(A)}\left\lvert
F(B,x)-F(D,y)\right\rvert\quad\text{(because (B,D), (C,D) \text{ have same
distribution} )}$ $\displaystyle\leqslant
2\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\sim\mu_{H}^{R}}\operatorname*{\varmathbb{E}}_{y_{1},\ldots,y_{d}\sim\mu^{R}_{H}(x)}\operatorname*{\varmathbb{E}}_{B,D_{1},\ldots,D_{d}\sim
G_{\eta}^{\otimes R}(A)}\max_{i}\left\lvert F(B,x)-F(D_{i},y_{i})\right\rvert$
$\displaystyle=2\operatorname{val}_{\mathcal{P}}(F)=2\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
(6.3)
Equations (6.1), (6.2) and (6.3) yield the desired result. ∎
###### Lemma 6.4.
$\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\mathsf{Var}_{1}[g_{A}]=\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}\left\lvert
g_{A}(x)-g_{A}(y)\right\rvert\geqslant\frac{1}{2}(\mathsf{Var}_{1}[F])^{2}-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
###### Proof.
Since the function $g_{A}$ is bounded in $[0,1]$ we can write
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}\left\lvert
g_{A}(x)-g_{A}(y)\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}\left(g_{A}(x)-g_{A}(y)\right)^{2}$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\in\mu_{H}^{R}}g^{2}_{A}(x)-\operatorname*{\varmathbb{E}}_{A}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}g_{A}(x)g_{A}(y)$
(6.4)
In the above expression there are two terms. From Lemma 6.3, we already know
that
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\in\mu_{H}^{R}}g^{2}_{A}(x)\geqslant\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F^{2}(A,x)-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
(6.5)
Let us expand out the other term in the expression.
$\displaystyle\operatorname*{\varmathbb{E}}_{A}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}g_{A}(x)g_{A}(y)$
$\displaystyle=\operatorname*{\varmathbb{E}}_{A}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes
R}(A)}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}F^{\prime}(\Pi(B,x))F^{\prime}(\Pi(C,y))$
(6.6)
Now consider the following graph $\mathcal{H}$ on $\mathcal{V}^{\prime}$
defined by the following edge sampling procedure.
* –
Sample $A\in V^{R}$, and $x,y\in\mu_{H}^{R}$.
* –
Sample independently $B\sim G_{\eta}^{\otimes R}(A)$ and $C\sim
G_{\eta}^{\otimes R}(A)$
* –
Output the edge $\Pi(B,x)$ and $\Pi(C,y)$
Let $\lambda$ denote the second eigenvalue of the adjacency matrix of the
graph $\mathcal{H}$.
$\displaystyle\operatorname*{\varmathbb{E}}_{A}\operatorname*{\varmathbb{E}}_{B,C\sim
G_{\eta}^{\otimes R}(A)}$
$\displaystyle\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}F^{\prime}(\Pi(B,x))F^{\prime}(\Pi(C,y))=\langle
F^{\prime},\mathcal{H}F^{\prime}\rangle$
$\displaystyle\leqslant\left(\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F^{\prime}(\Pi(A,x))\right)^{2}+\lambda\left(\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}\left(F^{\prime}(\Pi(A,x))\right)^{2}-(\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F^{\prime}(\Pi(A,x)))^{2}\right)$
$\displaystyle=\lambda\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F(A,x)^{2}+(1-\lambda)(\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F(A,x))^{2}\quad\text{(because
$F^{\prime}(\Pi(A,x))=F(A,x)$)}$
Using the above inequality with equations (6.4), (6.5), (6.6) we can derive
the following,
$\displaystyle\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}\left\lvert
g_{A}(x)-g_{A}(y)\right\rvert$
$\displaystyle\geqslant\operatorname*{\varmathbb{E}}_{A\sim
V^{R}}\operatorname*{\varmathbb{E}}_{x\in\mu_{H}^{R}}g^{2}_{A}(x)-\operatorname*{\varmathbb{E}}_{A}\operatorname*{\varmathbb{E}}_{x,y\in\mu_{H}^{R}}g_{A}(x)g_{A}(y)$
$\displaystyle\geqslant(1-\lambda)\left[\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F^{2}(A,x)-(\operatorname*{\varmathbb{E}}_{(A,x)\sim\mathcal{V}}F(A,x))^{2}\right]-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
$\displaystyle\geqslant(1-\lambda)\mathsf{Var}[F]-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})$
$\displaystyle\geqslant(1-\lambda)(\mathsf{Var}_{1}[F])^{2}-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\quad\text{(because
$\mathsf{Var}[F]>\mathsf{Var}_{1}[F]^{2}$ for all $F$)}$
To finish the argument, we need to bound the second eigenvalue $\lambda$ for
the graph $\mathcal{H}$. Here we will present a simple argument showing that
the second eigenvalue $\lambda$ for the graph $\mathcal{H}$ is strictly less
than $\frac{1}{2}$. Let us restate the procedure to sample edges from
$\mathcal{H}$ slightly differently.
* –
Define a map $\mathcal{M}:V\times
V_{H}\to(V\cup\perp)\times(V_{H}\cup\\{\perp\\})$ as follows,
$\mathcal{M}(b,x)=(b,x)$ if $x\in\\{s,t\\}$ and
$\mathcal{M}(b,x)=(\perp,\perp)$ otherwise. Let
$\Pi^{\prime}:((V\cup\perp)\times(V_{H}\cup\perp))^{R}\to(V\times\\{s,t\\})^{\\{R\\}}$
denote the following map.
$\Pi^{\prime}(B^{\prime},x^{\prime})=\\{(b^{\prime}_{i},x^{\prime}_{i})|x_{i}\in\\{s,t\\}\\}$
* –
Sample $A\in V^{R}$ and $x,y\in\mu_{H}^{R}$
* –
Sample independently $B=(b_{1},\ldots,b_{R})\sim G_{\eta}^{\otimes R}(A)$ and
$C=(c_{1},\ldots,c_{R})\sim G_{\eta}^{\otimes R}(A)$.
* –
Let
$\mathcal{M}(B,x),\mathcal{M}(C,y)\in\left((V\cup\\{\perp\\})\times(V_{H}\cup\\{\perp\\})\right)^{R}$
be obtained by applying $\mathcal{M}$ to each coordinate of $(B,x)$ and
$(C,y)$.
* –
Output an edge between
$(\Pi^{\prime}(\mathcal{M}(B,x)),\Pi^{\prime}(\mathcal{M}(C,y)))$.
It is easy to see that the above procedure also samples the edges of
$\mathcal{H}$ from the same distribution as earlier. Note that $\Pi^{\prime}$
is a projection from $((V\cup\perp)\times(V_{H}\cup\perp))^{R}$ to
$(V\times\\{s,t\\})^{\\{R\\}}$. Therefore, the second eigenvalue of the graph
$\mathcal{H}$ is upper bounded by the second eigenvalue of the graph on
$((V\cup\perp)\times(V_{H}\cup\\{\perp\\}))^{R}$ defined by
$\mathcal{M}(B,x)\sim\mathcal{M}(C,y)$. Let $\mathcal{H}_{1}$ denote the graph
defined by the edges $\mathcal{M}(B,x)\sim\mathcal{M}(C,y)$. Observe that the
coordinates of $\mathcal{H}_{1}$ are independent, i.e.,
$\mathcal{H}_{1}=\mathcal{H}_{2}^{R}$ for a graph $\mathcal{H}_{2}$
corresponding to each coordinate of $\mathcal{M}(B,x)$ and $\mathcal{M}(C,y)$.
Therefore, the second eigenvalue of $\mathcal{H}_{1}$ is at most the second
eigenvalue of $\mathcal{H}_{2}$. The Markov chain $\mathcal{H}_{2}$ on
$(V\cup\\{\perp\\})\times(V_{H}\cup\perp)$ is defined as follows,
* –
Sample $a\in V$ and two neighbors $b\sim G_{\eta}(a)$ and $c\sim G_{\eta}(a)$.
* –
Sample $x,y\in V_{H}$ independently from the distribution $\mu_{H}$.
* –
Output an edge between $\mathcal{M}(b,x)$ $\mathcal{M}(c,y)$.
Notice that in the Markov chain $\mathcal{H}_{2}$, for every choice of
$\mathcal{M}(b,x)$ in $(V\cup\\{\perp\\})\times(V_{H}\cup\perp)$, with
probability at least $\frac{1}{2}$, the other endpoint
$\mathcal{M}(c,y)=(\perp,\perp)$. Therefore, the second eigenvalue of
$\mathcal{H}_{2}$ is at most $\frac{1}{2}$, giving a bound of $\frac{1}{2}$ on
the second eigen value of $\mathcal{H}$. ∎
Now we restate a claim from [RST12] that will be useful for our our soundness
proof.
###### Theorem 6.5.
(Restatment of Lemma 6.11 from [RST12]) Let $G$ be a graph with a vertex set
$V$. Let a distribution on pairs of tuples $(A,B)$ be defined by $A\sim
V^{R}$, $B\sim G_{\eta}^{\otimes R}(A)$. Let $\ell:V^{R}\to[R]$ be a labelling
such that over the choice of random tuples and two random permutations
$\pi_{A},\pi_{B}$
$\operatorname*{\varmathbb{P}}_{A\sim V^{R},B\sim G_{\eta}^{\otimes
R}(A)}\operatorname*{\varmathbb{P}}_{\pi_{A},\pi_{B}}\left\\{\pi_{A}^{-1}\left(\ell(\pi_{A}(A))\right)=\pi_{B}^{-1}\left(\ell(\pi_{B}(B))\right)\right\\}\geqslant\zeta$
Then there exists a set $S\subset V$ with $\operatorname{{\sf
vol}}(S)\in\left[\frac{\zeta}{16R},\frac{3}{\eta R}\right]$ satisfying
$\Phi(S)\leqslant 1-\zeta/16$.
The following lemma asserts that if the graph $G$ is a $NO$-instance of Small-
Set Expansion ($\gamma$, $\delta$,$M$) then for almost all $A\in V^{R}$ the
functions have no influential coordinates.
###### Lemma 6.6.
Fix $\delta=1/R$. Suppose for all sets $S\subset V$ with $\operatorname{{\sf
vol}}(S)\in\left(\delta/M,M\delta\right)$ , $\Phi(S)\geqslant 1-\gamma$ then
for all $\tau>0$,
$\operatorname*{\varmathbb{P}}_{A\sim V^{R}}\left[\exists
i\mid\operatorname{{\sf
Inf}}_{i}[\Gamma_{1-\eta}g_{A}]\geqslant\tau\right]\leqslant\frac{1000}{\tau^{3}\varepsilon^{2}\eta^{2}}\cdot\max(1/M,\gamma)$
###### Proof.
For each $A\in V^{R}$, let $L_{A}=\left\\{i\in[R]\mid\operatorname{{\sf
Inf}}_{i}(\Gamma_{1-\eta}f_{A})>\tau/2\right\\}$ and
$L^{\prime}_{A}=\left\\{i\in[R]\mid\operatorname{{\sf
Inf}}_{i}(\Gamma_{1-\eta}g_{A})>\tau\right\\}$. Call a vertex $A\in V^{R}$ to
be good if $L^{\prime}_{A}\neq\emptyset$. By Lemma 5.4, the sum of influences
of $\Gamma_{1-\eta}g_{A}$ is at most
$\frac{1}{\varepsilon\eta}\mathsf{Var}[g_{A}]\leqslant\frac{1}{\varepsilon\eta}$.
Therefore, the cardinality of $L^{\prime}_{A}$ is upper bounded by
$|L^{\prime}_{A}|\leqslant\frac{2}{\tau\varepsilon\eta}$. Similarly, the
cardinality of $L_{A}$ is upper bounded by
$|L_{A}|\leqslant\frac{1}{\tau\varepsilon\eta}$.
The lemma asserts that at most a
$\frac{1000}{\tau^{3}\eta^{2}\varepsilon^{2}}\cdot\max(1/M,\gamma)$ fraction
of vertices are good. For the sake of contradiction, assume that
$\operatorname*{\varmathbb{P}}_{A\in
V^{R}}\left[L^{\prime}_{A}\neq\emptyset\right]\geqslant
1000\max(1/M,\gamma)/\tau^{2}\varepsilon^{2}\eta^{2}$.
Define a labelling $\ell:V^{R}\to[R]$ as follows: for each $A\in V^{R}$, with
probability $\frac{1}{2}$ choose a random coordinate in $L_{A}$ and with
probability $\nicefrac{{1}}{{2}}$, choose a random coordinate in
$L^{\prime}_{A}$. If the sets $L_{A},L^{\prime}_{A}$ are empty, then we choose
a uniformly random coordinate in $[R]$.
Observe that for each $A\in V^{R}$, the function $g_{A}$ is the average over
bounded functions $f_{B}\colon V_{H}^{R}\to[0,1]$, where $B\sim
G^{R}_{\eta}(A)$. Fix a vertex $A\in V^{R}$ such that
$L^{\prime}_{A}\neq\emptyset$ and a coordinate $i\in L^{\prime}_{A}$. In
particular, we have that $\operatorname{{\sf
Inf}}_{i}[\Gamma_{1-\eta}g_{A}]\geqslant\tau$. Using convexity of influences,
this implies that,
$E_{B\sim G^{\otimes R}_{\eta}(A)}\operatorname{{\sf
Inf}}_{i}[\Gamma_{1-\eta}f_{B}]\geqslant\tau\,.$
Specifically, this implies that for at least a $\frac{\tau}{2}$ fraction of
the neighbours $B\sim G^{R}_{\eta}(A)$, the influence of the $i^{th}$
coordinate on $f_{B}$ is at least $\frac{\tau}{2}$. Hence, if
$L^{\prime}_{A}\neq\emptyset$ then for at least a $\tau/2$ fraction of
neighbours $B\sim G^{\otimes R}_{\eta}(A)$ we have $L^{\prime}_{A}\cap
L_{B}\neq\emptyset$.
By definition of the functions $f_{A},g_{A}$, it is clear that for every
permutation $\pi:[R]\to[R]$, $f_{A}(\pi(x))=f_{\pi(A)}(x)$ and
$g_{A}(\pi(x))=g_{\pi(A)}(x)$. Therefore, for every permutation
$\pi:[R]\to[R]$ and $A\in V^{R}$,
$L_{A}=\pi^{-1}(L_{\pi(A)})\qquad\text{ and
}L^{\prime}_{A}=\pi^{-1}(L^{\prime}_{\pi(A)})$
From the above discussion, for every good vertex $A\in V^{R}$, for at least a
$\tau/2$ fraction of the vertices $B\sim G^{\otimes R}_{\eta}(A)$, and every
pair of permutations $\pi_{A},\pi_{B}:[R]\to[R]$, we have
$\pi^{-1}_{A}(L^{\prime}_{\pi_{A}(A)})\cap\pi^{-1}_{B}(L_{\pi_{B}(B)})\neq\emptyset$.
This implies that,
$\displaystyle\operatorname*{\varmathbb{P}}_{A\sim V^{R},B\sim
G_{\eta}^{\otimes
R}(A)}\operatorname*{\varmathbb{P}}_{\pi_{A},\pi_{B}}\left\\{\pi_{A}^{-1}\left(\ell(\pi_{A}(A))\right)=\pi_{B}^{-1}\left(\ell(\pi_{B}(B))\right)\right\\}$
$\displaystyle\geqslant\operatorname*{\varmathbb{P}}_{A\sim
V^{R}}[L^{\prime}_{A}\neq\emptyset]\cdot\operatorname*{\varmathbb{P}}_{B\sim
G^{\otimes R}_{\eta}(A)}[L^{\prime}_{A}\cap
L_{B}\neq\emptyset|L^{\prime}_{A}\neq\emptyset]\cdot\operatorname*{\varmathbb{P}}\left[\pi_{A}^{-1}(\ell(\pi_{A}(A)))=\pi_{B}^{-1}(\ell(\pi_{B}(B)))\mid
L^{\prime}_{A}\cap L_{B}\neq\emptyset\right]$
$\displaystyle\geqslant\operatorname*{\varmathbb{P}}_{A\sim
V^{R}}[L^{\prime}_{A}\neq\emptyset]\cdot\left(\frac{\tau}{2}\right)\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{|L^{\prime}_{A}|}\frac{1}{|L_{B}|}$
$\displaystyle\geqslant\operatorname*{\varmathbb{P}}_{A\sim
V^{R}}[L^{\prime}_{A}\neq\emptyset]\cdot\left(\frac{\tau}{2}\right)\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\left(\frac{\tau\eta\varepsilon}{2}\right)^{2}$
$\displaystyle\geqslant 16\max(\nicefrac{{1}}{{M}},\gamma)$
By Theorem 6.5, this implies that there exists a set $S\subset V$ with
$\operatorname{{\sf vol}}(S)\in[\frac{1}{MR},\frac{3}{\eta R}]$ satisfying
$\Phi(S)\leqslant 1-\gamma$. A contradiction. ∎
Finally, we are ready to show the soundness of the reduction.
###### Theorem 6.7.
(Soundness) For all $\varepsilon,d$ there exists choice of $M$ and
$\gamma,\eta$ such that the following holds. Suppose for all sets $S\subset V$
with $\operatorname{{\sf vol}}(S)\in\left(\delta/M,M\delta\right)$ ,
$\Phi(S)\geqslant 1-\eta$, then for all
$F^{\prime}:\mathcal{V}^{\prime}\to[0,1]$ such that
$\mathsf{Var}_{1}[F^{\prime}]\geqslant\frac{1}{10}$, we have
$\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\geqslant\Omega(\sqrt{\varepsilon\log{d}})$
###### Proof.
Recall that we had fixed $\eta=\varepsilon/(100d)$. We will choose $\tau$ to
small enough so that the error term in the soundness of dictatorship test
(Proposition 5.5) is smaller than $\varepsilon$. Since the least probability
of any vertex in Markov chain $H$ is $\varepsilon$, setting
$\tau=\varepsilon^{1/\varepsilon^{3}}$ would suffice.
First, we know that if $G$ is a $NO$-instance of Small-Set Expansion
($\gamma,\delta,M$) then for almost all $A\in V^{R}$, the function $g_{A}$ has
no influential coordinates. Formally, by Lemma 6.6, we will have
$\operatorname*{\varmathbb{P}}_{A\sim V^{R}}\left[\exists
i\mid\operatorname{{\sf
Inf}}_{i}[\Gamma_{1-\eta}g_{A}]\geqslant\tau\right]\leqslant\frac{1000}{\tau^{3}\eta^{2}}\cdot\max(1/M,\gamma)\,.$
For an appropriate choice of $M,\gamma$, the above inequality implies that for
all but an $\varepsilon$-fraction of vertices $A\in V^{R}$, the function
$g_{A}$ will have no influential coordinates.
Without loss of generality, we may assume that
$\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\leqslant\sqrt{\varepsilon\log
d}$, else we would be done. Applying Lemma 6.4, we get that
$\operatorname*{\varmathbb{E}}_{A\in
V^{R}}\mathsf{Var}_{1}[g_{A}]\geqslant(\mathsf{Var}_{1}[F])^{2}-\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\geqslant\frac{1}{200}$.
This implies that for at least a $\frac{1}{400}$ fraction of $A\in V^{R}$,
$\mathsf{Var}_{1}[g_{A}]\geqslant 1/400$. Hence for at least an
$1/400-\varepsilon$ fraction of vertices $A\in V^{R}$ we have,
$\mathsf{Var}_{1}[g_{A}]\geqslant\frac{1}{400}\qquad\text{ and
}\qquad\max_{i}\operatorname{{\sf
Inf}}_{i}(\Gamma_{1-\eta}(g_{A}))\leqslant\tau$
By appealing to the soundness of the gadget (Proposition 5.5), for every such
vertex $A\in V^{R}$,
$\operatorname{val}_{\mu_{H}^{R}}(g_{A})\geqslant\Omega(\sqrt{\varepsilon\log{d}})-O(\varepsilon)=\Omega(\sqrt{\varepsilon\log{d}})$.
Finally, by applying Lemma 6.2, we get the desired conclusion.
$\operatorname{val}_{\mathcal{P}^{\prime}}(F^{\prime})\geqslant\operatorname*{\varmathbb{E}}_{A\in
V^{R}}\operatorname{val}_{\mu_{H}^{R}}(g_{A})\geqslant\Omega(\sqrt{\varepsilon\log{d}})$
∎
## 7 Reduction from Analytic $d$-Vertex Expansion to Vertex Expansion
###### Theorem 7.1.
A c-vs-s hardness for $d$-Balanced Analytic Vertex Expansion implies a 4
c-vs-s/16 hardness for balanced symmetric-vertex expansion on graphs of degree
at most $D$, where $D=\max\left\\{100d/s,2\log(1/c)\right\\}$.
At a high level, the proof of Theorem 7.1 has two steps.
1. 1.
We show that a c-vs-s hardness for Balanced Analytic Vertex Expansion. implies
a 2 c-vs-s/4 hardness for instances of Balanced Analytic Vertex Expansion
having uniform distribution (Proposition 7.2).
2. 2.
We show that a c-vs-s hardness for instances of $d$-Balanced Analytic Vertex
Expansion having uniform stationary distribution implies a 2 c-vs-s/2 hardness
for balanced symmetric-vertex expansion on $\Theta(D)$-regular graphs.
(Proposition 7.5).
###### Proposition 7.2.
A c-vs-s hardness for Balanced Analytic Vertex Expansion. implies a 2 c-vs-s/4
hardness for instances of Balanced Analytic Vertex Expansion having uniform
distribution.
###### Proof.
Let $(V,\mathcal{P})$ be an instance of Balanced Analytic Vertex Expansion. We
construct an instance $(V^{\prime},\mathcal{P}^{\prime})$ as follows. Let
$T=2n^{2}$. We first delete all vertices $i$ from $V$ which have
$\mu(i)<1/2n^{2}$, i.e. $V\leftarrow V\backslash\left\\{i\in
V:\mu(i)<1/2n^{2}\right\\}$. Note that after this operation, the total weight
of the remaining vertices is still at least $1-1/2n$ and the Balanced Analytic
Vertex Expansion can increase or decrease by at most a factor of $2$. Next for
each $i$, we introduce introduce $\lceil\mu(i)T\rceil$ copies of vertex $i$.
We will call these vertices the cloud for vertex $i$ and index them as $(i,a)$
for $a\in[\mu(i)T]$.
We set the probability mass of each $(d+1)$-tuple
$((i,a),(j_{1},b_{1})\ldots,(j_{d},b_{d}))$ as follows :
$\mathcal{P}^{\prime}((i,a),(j_{1},b_{1})\ldots,(j_{d},b_{d}))=\frac{\mathcal{P}(i,j_{1},\ldots,j_{d})}{(\mu(i)T)\cdot\Pi_{\ell=1}^{d}(\mu(j_{\ell})T)}$
It is easy to see that $\mu^{\prime}(i,a)=1/T$ for all vertices $(i,a)\in
V^{\prime}$. The analytic $d$-vertex expansion under a function $F$ is given
by,
$\frac{\operatorname*{\varmathbb{E}}_{((i,a),(j_{1},b_{1})\ldots,(j_{d},b_{d}))\sim\mathcal{P}^{\prime}}\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell},b_{\ell})\right\rvert}{\operatorname*{\varmathbb{E}}_{(i,a),(j,b)\sim\mu^{\prime}}\left\lvert
F(i,a)-F(j,b)\right\rvert}$
where $X=(i,a)$ and $Y_{\ell}=(j,b)$ which are sampled as follows:
1. 1.
Sample a $(d+1)$-tuple $(i,j_{1},\ldots,j_{d})$ from $\mathcal{P}$.
2. 2.
Sample $a$ uniformly at random from ${1,\ldots,\mu(i)T}$.
3. 3.
Sample $b_{\ell}$ uniformly at random from
$\left\\{1,\ldots,\mu(j_{\ell})T\right\\}$ for each $\ell\in[d]$.
#### Completeness
Suppose, $\Phi({V,\mathcal{P}})\leqslant c$. Let $f$ be the corresponding cut
function. The function $f:V\to\left\\{0,1\right\\}$ can be trivially extended
to a function $F:V^{\prime}\to\left\\{0,1\right\\}$ thereby certifying that
$\Phi({V^{\prime},\mathcal{P}^{\prime}})\leqslant 2c$.
#### Soundness
Suppose $\Phi({V,\mathcal{P}})\geqslant s$. Let
$F:V^{\prime}\to\left\\{0,1\right\\}$ be any balanced function. By convexity
of absolute value function we get
$\operatorname*{\varmathbb{E}}_{((i,a),(j_{1},b_{1}),\ldots,(j_{d},b_{d}))\sim\mathcal{P}^{\prime}}\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell,b_{\ell}})\right\rvert\geqslant\operatorname*{\varmathbb{E}}_{(i,j_{1},\ldots,j_{d})\sim\mathcal{P}}\max_{\ell}\left\lvert\operatorname*{\varmathbb{E}}_{a}F(i,a)-\operatorname*{\varmathbb{E}}_{\ell}F(j_{\ell},b_{\ell})\right\rvert.$
So if we define $f(i)=E_{a}F(i,a)$, the numerator for analytic $d$-vertex
expansion in $(V,\mathcal{P})$ for $f$ is only lower than the corresponding
numerator for $F$ in $(V^{\prime},\mathcal{P}^{\prime})$. We need to lower
bound the denominator, $\operatorname*{\varmathbb{E}}_{i,j\sim\mu}\left\lvert
f(i)-f(j)\right\rvert$. The requisite lower bound follows from the following
two lemmas.
###### Lemma 7.3.
$\operatorname*{\varmathbb{E}}_{i,j\sim\mu}\left\lvert
f(i)-f(j)\right\rvert\geqslant\operatorname*{\varmathbb{E}}_{(i,a),(j,b)\sim\mu^{\prime}}\left\lvert
F(i,a)-F(j,b)\right\rvert-\operatorname*{\varmathbb{E}}_{(i,a),(i,b)\sim\mu^{\prime}}\left\lvert
F(i,a)-F(i,b)\right\rvert$
###### Proof.
The Lemma follows directly from the following two inequalities.
$\operatorname*{\varmathbb{E}}_{(i,a),(j,b)}\left\lvert
F(i,a)-F(j,b)\right\rvert\leqslant\operatorname*{\varmathbb{E}}_{(i,a)}\left\lvert
F(i,a)-f(i)\right\rvert+\operatorname*{\varmathbb{E}}_{(j,b)}\left\lvert
F(j,b)-f(j)\right\rvert+\operatorname*{\varmathbb{E}}_{i,j}\left\lvert
f(i)-f(j)\right\rvert\qquad\textrm{(Triangle Inequality)}$
and
$\operatorname*{\varmathbb{E}}_{i,a}\left\lvert
F(i,a)-f(i)\right\rvert\leqslant\operatorname*{\varmathbb{E}}_{i,a,b}\left\lvert
F(i,a)-F(i,b)\right\rvert$
∎
###### Lemma 7.4.
$\operatorname*{\varmathbb{E}}_{i,a,b}\left\lvert
F(i,a)-F(i,b)\right\rvert\leqslant
2\operatorname{val}_{\mathcal{P}^{\prime}}(F)=2\operatorname*{\varmathbb{E}}_{(i,a),(j_{1},c_{1}),\ldots(j_{d},c_{d})\sim\mathcal{P}^{\prime}}\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell},c_{\ell})\right\rvert$
###### Proof.
Sample $(i,j_{1},\ldots,j_{d})\sim\mathcal{P}$. For any neighbour $(j,c)$ of
$(i,a),(i,b)$, using the Triangle Inequality we have
$\left\lvert F(i,a)-F(i,b)\right\rvert\leqslant\left\lvert
F(i,a)-F(j,c)\right\rvert+\left\lvert F(j,c)-F(i,b)\right\rvert$
Therefore,
$\displaystyle\left\lvert F(i,a)-F(i,b)\right\rvert$ $\displaystyle\leqslant$
$\displaystyle\frac{\sum_{\ell}\left\lvert
F(i,a)-F(j_{\ell},c_{\ell})\right\rvert+\sum_{\ell}\left\lvert
F(i,b)-F(j_{\ell},c_{\ell})\right\rvert}{d}$ $\displaystyle\leqslant$
$\displaystyle\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell},c_{\ell})\right\rvert+\max_{\ell}\left\lvert
F(i,b)-F(j_{\ell},c_{\ell})\right\rvert$
Taking expectations over the uniformly random choice of $a$ and $b$ from the
cloud of $i$,
$\operatorname*{\varmathbb{E}}_{(i,a),(i,b)}\left\lvert
F(i,a)-F(i,b)\right\rvert\leqslant
2\operatorname*{\varmathbb{E}}_{((i,a),(j_{1},b_{1}),\ldots,(j_{d},b_{d}))\sim\mathcal{P}^{\prime}}\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell},c_{\ell})\right\rvert$
∎
Lemma 7.3 and Lemma 7.4 together show that
$\operatorname*{\varmathbb{E}}_{i,j}\left\lvert
f(i)-f(j)\right\rvert\geqslant\frac{\operatorname*{\varmathbb{E}}_{(i,a),(j,b)}\left\lvert
F(i,a)-F(j,b)\right\rvert}{2}.$
as long as the value
$\operatorname{val}_{\mathcal{P}^{\prime}}(F)<\mathsf{Var}_{1}[F]/4$.
Therefore, for any $F:V^{\prime}\to\left\\{0,1\right\\}$,
$\frac{\operatorname*{\varmathbb{E}}_{((i,a),(j_{1},b_{1})\ldots,(j_{d},b_{d}))\sim\mathcal{P}^{\prime}}\max_{\ell}\left\lvert
F(i,a)-F(j_{\ell},b_{\ell})\right\rvert}{\operatorname*{\varmathbb{E}}_{(i,a),(j,b)\sim\mu^{\prime}}\left\lvert
F(i,a)-F(j,b)\right\rvert}\geqslant\frac{s}{4}\,.$
Theorem 4.3 shows that the minimum value of Balanced Analytic Vertex Expansion
is obtained by boolean functions. Therefore,
$\Phi({V^{\prime},\mathcal{P}^{\prime}})\geqslant s/4$. ∎
###### Proposition 7.5.
A c-vs-s hardness for instances of $d$-Balanced Analytic Vertex Expansion
having uniform stationary distribution implies a 2 c-vs-s/4 hardness for
balanced symmetric-vertex expansion on $\Theta(D)$-regular graphs. Here
$D\geqslant\max\left\\{100d/s,2\log(1/c)\right\\}$.
###### Proof.
Let $(V^{\prime},\mathcal{P}^{\prime})$ be an instance of $d$-Balanced
Analytic Vertex Expansion. We construct a graph $G$ from
$(V^{\prime},\mathcal{P}^{\prime})$ as follows. We initially set
$V(G)=V^{\prime}$. For each vertex $X$ we pick $D$ neighbors by sampling $D/d$
tuples from the marginal distribution of $\mathcal{P}^{\prime}$ on tuples
containing $X$ in the first coordinate.
Let $\deg(i)$ denote the degree of vertex $i$, i.e. the number of vertices
adjacent to vertex $i$ in $G$. It is easy to see that $\deg(i)\geqslant D$ and
$\operatorname*{\varmathbb{E}}\left[\deg(i)\right]=2D\ \forall i\in V(G)$. Let
$L=\left\\{i\in V(G)|\deg(i)>4D\right\\}$. Using Hoeffding’s Inequality, we
get a tight concentration for $\deg(i)$ around $2D$.
$\operatorname*{\varmathbb{P}}\left[\deg(i)>4D\right]\leqslant e^{-D}\,.$
Therefore, $\operatorname*{\varmathbb{E}}\left[\left\lvert
L\right\rvert\right]<n/e^{D}$. We delete these vertices from $G$, i.e.
$V(G)\leftarrow V(G)\backslash L$. With constant probability, all remaining
vertices will have their degrees in the range $[D/2,4D]$. Also, the vertex
expansion of every set will decrease by at most an additive $1/e^{D}$.
#### Completeness
Let $\Phi({V^{\prime},\mathcal{P}^{\prime}})\leqslant c$ and let
$F:V^{\prime}\to\left\\{0,1\right\\}$ be the function corresponding to
$\Phi({V^{\prime},\mathcal{P}^{\prime}})$. Let the set $S$ be the support of
the function $F$. Clearly, the set $S$ is balanced. Therefore, with constant
probability, we have
$\Phi^{\sf V}(G)\leqslant\Phi^{\sf
V}_{G}(S)\leqslant\Phi({V^{\prime},\mathcal{P}^{\prime}})+1/e^{D}\leqslant
2c\,.$
#### Soundness
Suppose $\Phi({V^{\prime},\mathcal{P}^{\prime}})\geqslant s$. Let
$F:V^{\prime}\to\left\\{0,1\right\\}$ be any balanced function.
Since the max is larger than the average, we get
$\operatorname*{\varmathbb{E}}_{X}\max_{Y_{i}\in N_{G}(X)}\left\lvert
F(X)-F(Y_{i})\right\rvert\geqslant\frac{d}{D}\sum_{j=1}^{D/d}\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert$
By Hoeffding’s inequality, we get
$\displaystyle\operatorname*{\varmathbb{P}}\left[\left(\operatorname*{\varmathbb{E}}_{X}\max_{Y_{i}\in
N(X)}\left\lvert F(X)-F(Y_{i})\right\rvert\right)<s/4\right]$
$\displaystyle\leqslant$
$\displaystyle\operatorname*{\varmathbb{P}}\left[\left(\frac{d}{D}\sum_{j=1}^{D/d}\operatorname*{\varmathbb{E}}_{(X,Y_{1},\ldots,Y_{d})\sim\mathcal{P}}\max_{i}\left\lvert
F(X)-F(Y_{i})\right\rvert\right)<s/4\right]$ $\displaystyle\leqslant$
$\displaystyle\exp\left(-n(sD/d)^{2}\right)$
Here, the last inequality follows from Hoeffding’s inequality over the index
$X$. There are at most $2^{n}$ boolean functions on $V$. Therefore, using a
union bound on all those functions we get,
$\operatorname*{\varmathbb{P}}\left[\Phi^{\sf V}(G)\geqslant
s/4\right]\geqslant 1-2^{n}\exp\left(-n(sD/d)^{2}\right).$
Since $D>d/s$, we get that with probability $1-o(1)$, $\Phi^{\sf
V}(G)\geqslant s/4$.
∎
###### Proof of Theorem 7.1.
Theorem 7.1 follows directly from Proposition 7.2 and Proposition 7.5. ∎
## 8 Hardness of Vertex Expansion
We are now ready to prove Theorem 1.3. We restate the Theorem below.
###### Theorem 8.1.
For every $\eta>0$, there exists an absolute constant $C$ such that
$\forall\varepsilon>0$ it is SSE-hard to distinguish between the following two
cases for a given graph $G=(V,E)$ with maximum degree $d\geqslant
100/\varepsilon$.
Yes
: There exists a set $S\subset V$ of size $\left\lvert
S\right\rvert\leqslant\left\lvert V\right\rvert/2$ such that
$\phi^{\sf V}(S)\leqslant\varepsilon$
No
: For all sets $S\subset V$,
$\phi^{\sf V}(S)\geqslant\min\left\\{10^{-10},C\sqrt{\varepsilon\log
d}\right\\}-\eta$
###### Proof.
From Theorem 6.1 and Theorem 6.7 we get that for an instance of Balanced
Analytic Vertex Expansion $(V,\mathcal{P})$, it is SSE-hard to distinguish
between the following two cases cases:
Yes
:
$\Phi({V,\mathcal{P}})\leqslant\varepsilon$
No
:
$\Phi({V,\mathcal{P}})\geqslant\min\left\\{10^{-4},c_{1}\sqrt{\varepsilon\log
d}\right\\}-\eta$
Now from Theorem 7.1 we get that for a graph $G$, it is SSE-hard to
distinguish between the following two cases cases:
Yes
:
$\Phi^{\sf V,bal}\leqslant\varepsilon$
No
:
$\Phi^{\sf V,bal}\geqslant\min\left\\{10^{-6},c_{2}\sqrt{\varepsilon\log
d}\right\\}-\eta$
We use a standard reduction from Balanced vertex expansion to vertex
expansion. For the sake of completeness we give a proof of this reduction in
Lemma B.2. Using this reduction, we get that for a graph $G$, it is SSE-hard
to distinguish between the following two cases cases:
Yes
:
$\Phi^{\sf V}\leqslant\varepsilon$
No
:
$\Phi^{\sf V}\geqslant\min\left\\{10^{-8},c_{3}\sqrt{\varepsilon\log
d}\right\\}-\eta$
Finally, using the computational equivalence of vertex expansion and symmetric
vertex expansion (Theorem A.1), we get that for a graph $G$, it is SSE-hard to
distinguish between the following two cases cases:
Yes
:
$\phi^{\sf V}\leqslant\varepsilon$
No
:
$\phi^{\sf V}\geqslant\min\left\\{10^{-10},C\sqrt{\varepsilon\log
d}\right\\}-\eta$
This completes the proof of the theorem.
∎
## 9 An Optimal Algorithm for vertex expansion
In this section we give a simple polynomial time algorithm which outputs a set
$S$ whose vertex expansion is at most $\mathcal{O}\left(\sqrt{\phi^{\sf V}\log
d}\right)$. We restate Theorem 1.2.
###### Theorem 9.1.
There exists a polynomial time algorithm which given a graph $G=(V,E)$ having
vertex degrees at most $d$, outputs a set $S\subset V$, such that $\phi^{\sf
V}(S)=\mathcal{O}\left(\sqrt{\phi^{\sf V}\log d}\right)$.
For an undirected graph $G$, Bobkov et al. [BHT00] define $\lambda_{\infty}$
as follows.
$\lambda_{\infty}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min_{x}\frac{\sum_{i}\max_{j\sim
i}(x_{i}-x_{j})^{2}}{\sum_{i}x_{i}^{2}-\frac{1}{n}(\sum_{i}x_{i})^{2}}$
They also prove the following Theorem.
###### Theorem 9.2 ([BHT00]).
For any unweighted, undirected graph $G$, we have
$\frac{\lambda_{\infty}}{2}\leqslant\phi^{\sf
V}\leqslant\sqrt{2\lambda_{\infty}}$
Consider the following SDP relaxation of $\lambda_{\infty}$.
###### SDP 9.3.
$\displaystyle{\sf
SDPval}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min\sum_{i\in}\alpha_{i}$
subject to: $\displaystyle\left\lVert v_{j}-v_{i}\right\rVert^{2}$
$\displaystyle\leqslant$ $\displaystyle\alpha_{i}\qquad\forall i\in V\textrm{
and }\forall j\sim i$ $\displaystyle\sum_{i}\left\lVert
v_{i}\right\rVert^{2}-\frac{1}{n}\left\lVert\sum_{i}v_{i}\right\rVert^{2}$
$\displaystyle=$ $\displaystyle 1$
It’s easy to see that this is a relaxation for $\lambda_{\infty}$. We present
a simple randomized rounding of this SDP which, with constant probability,
outputs a set with vertex expansion at most $C\sqrt{\phi^{\sf V}\log d}$ for
some absolute constant $C$.
###### Algorithm 9.4.
– Input : A graph $G=(V,E)$ – Output : A set $S$ with vertex expansion at
most $576\sqrt{{\sf SDPval}\log d}$ (with constant probability). 1. Compute
graph $G^{\prime}$ as in Theorem A.2, let $n=\left\lvert
V(G^{\prime})\right\rvert$. 2. Solve SDP 9.3 for graph $G^{\prime}$. 3. Pick a
random Gaussian vector $g\sim N(0,1)^{n}$. 4. For each $i\in[n]$, define
$x_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\langle v_{i},g\rangle$. 5.
Sort the $x_{i}$’s in decreasing order $x_{i_{1}}\geqslant
x_{i_{2}}\geqslant\ldots x_{i_{n}}$. Let $S_{j}$ denote the set of the first
$j$ vertices appearing in the sorted order. Let $l$ be the index such that
$l={\sf argmin}_{1\leqslant j\leqslant n/2}\Phi^{\sf V}(S_{j})\,.$ 6. Output
the set corresponding to $S_{l}$ in $G$.
We first prove a technical lemma which shows that we can a recover a a set
with small vertex expansion from a good linear-ordering (Step $3$ in Algorithm
9.4).
###### Lemma 9.5.
For any $y_{1},y_{2},\ldots,y_{n}\in\varmathbb R^{+}\cup\left\\{0\right\\}$,
let $Y\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[y_{1}y_{2}\ldots y_{n}]^{T}$
and $\alpha\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\sum_{i}\max_{j\sim
i}|y_{j}-y_{i}|}{\sum_{i}y_{i}}$. Then $\exists S\subseteq\operatorname{\sf
supp}(Y)$ such that $\phi^{\sf V}(S)\leqslant\alpha$. Morover, such a set can
be computed in polynomial time.
###### Proof.
W.l.o.g we may assume that $y_{1}\geqslant y_{2}\geqslant\ldots\geqslant
y_{n}\geqslant 0$. Then
$\frac{\sum_{i}\max_{j\sim i,j<i}(y_{j}-y_{i})}{\sum_{i}y_{i}}\leqslant\alpha$
(9.1)
and
$\frac{\sum_{i}\max_{j\sim i,j>i}(y_{i}-y_{j})}{\sum_{i}y_{i}}\leqslant\alpha$
(9.2)
Let $i_{\max}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\textrm{\sf
argmax}_{i}y_{i}>0$, i.e. $i_{\max}$ be the largest index such that
$y_{i_{\max}}>0$. Let
$S_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\\{y_{1},\ldots,y_{i}\right\\}$.
Suppose $\forall i<i_{\max}$ $N^{v}(S_{i})>\alpha|S_{i}|$.
Now, from Inequality 9.2,
$\alpha\geqslant\frac{\sum_{i}\max_{j\sim
i,j<i}(y_{j}-y_{i})}{\sum_{i}y_{i}}=\frac{\sum_{i}\max_{j\sim
i,j<i}\sum_{l=j}^{l=i-1}(y_{l}-y_{l+1})}{\sum_{i}y_{i}}=\frac{\sum_{i}(y_{i}-y_{i+1})|N(S_{i})|}{\sum_{i}y_{i}}>\alpha\frac{\sum_{i}(y_{i}-y_{i+1})|S_{i}|}{\sum_{i}y_{i}}=\alpha$
Thus we get $\alpha>\alpha$ which is a contradition. Therefore, $\exists
i\leqslant i_{\max}$ such that $\phi^{\sf V}(S_{i})\leqslant\alpha$. ∎
Next we show a $\lambda_{\infty}$-like bound for the $x_{i}$’s.
###### Lemma 9.6.
Let $x_{1},\ldots,x_{n}$ be as defined in Algorithm 9.4. Then, with constant
probability, we have
$\frac{\sum_{i}\max_{j\sim
i}(x_{i}-x_{j})^{2}}{\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}}\leqslant
96\ {\sf SDPval}\log d.$
###### Proof.
We will make use of the following fact that is part of the folkore about
Gaussian random variables. For the sake of completeness, we prove this Fact in
Appendix B (Fact B.3).
###### Fact 9.7.
Let $Y_{1},Y_{2},\ldots,Y_{d}$ be $d$ normal random variables with mean $0$
and variance at most $\sigma^{2}$. Let $Y$ be the random variable defined as
$Y\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\left\\{Y_{i}|i\in[d]\right\\}$.
Then
$\operatorname*{\varmathbb{E}}\left[Y\right]\leqslant 2\sigma\sqrt{\log d}$
Now using this fact we get,
$\operatorname*{\varmathbb{E}}\left[\max_{j\sim
i}(x_{j}-x_{j})^{2}\right]=\operatorname*{\varmathbb{E}}\left[\max_{j\sim
i}\langle v_{i}-v_{j},g\rangle^{2}\right]\leqslant 2\max_{j\sim i}\left\lVert
v_{j}-v_{i}\right\rVert^{2}\log d.$
Therefore, $\operatorname*{\varmathbb{E}}\left[\sum_{i}\max_{j\sim
i}(x_{j}-x_{j})^{2}\right]\leqslant 2\ {\sf SDPval}\log d$. Using Markov’s
Inequality we get
$\operatorname*{\varmathbb{P}}\left[\sum_{i}\max_{j\sim
i}(x_{j}-x_{j})^{2}>48\ {\sf SDPval}\log d\right]\leqslant\frac{1}{24}$ (9.3)
For the denominator, using linearity of expectation, we get
$\operatorname*{\varmathbb{E}}\left[\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}\right]=\sum_{i}\left\lVert
v_{i}\right\rVert^{2}-\frac{1}{n}\left\lVert\sum_{i}v_{i}\right\rVert^{2}.$
Also recall that the denominator can be re-written as
$\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}=\frac{1}{n}\sum_{i,j}(x_{i}-x_{j})^{2}\,,$
which is a sum of squares of gaussians. Now applying Lemma 9.8 to the
denominator we conclude
$\operatorname*{\varmathbb{P}}\left[\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}\geqslant\frac{1}{2}\right]\geqslant\frac{1}{12}.$
(9.4)
Using (9.3) and (9.4) we get that
$\operatorname*{\varmathbb{P}}\left[\frac{\sum_{i}\max_{j\sim
i}(x_{i}-x_{j})^{2}}{\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}}\leqslant
96\ {\sf SDPval}\log d\right]>\frac{1}{24}.$
∎
###### Lemma 9.8.
Suppose $z_{1},\ldots,z_{m}$ are gaussian random variables (not necessarily
independent) such $\operatorname*{\varmathbb{E}}[\sum_{i}z_{i}^{2}]=1$ then
$\operatorname*{\varmathbb{P}}\left[\sum_{i}z_{i}^{2}\geqslant\frac{1}{2}\right]\geqslant\frac{1}{12}$
###### Proof.
We will bound the variance of the random variable $R=\sum_{i}z_{i}^{2}$ as
follows,
$\displaystyle\operatorname*{\varmathbb{E}}[R^{2}]$
$\displaystyle=\sum_{i,j}E[z_{i}^{2}z_{j}^{2}]$
$\displaystyle\leqslant\sum_{i,j}\left(E[z_{i}^{4}]\right)^{\frac{1}{2}}\left(E[z_{j}^{4}]\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{i,j}3E[z_{i}^{2}]E[z_{j}^{2}]\qquad\textrm{ (Using
}\operatorname*{\varmathbb{E}}[g^{4}]=3\operatorname*{\varmathbb{E}}[g^{2}]\textrm{
for gaussians )}$ $\displaystyle=3\left(\sum_{i}E[z_{i}^{2}]\right)^{2}=3$
By the Paley-Zygmund inequality,
$\operatorname*{\varmathbb{P}}\left[R\geqslant\frac{1}{2}\operatorname*{\varmathbb{E}}[R]\right]\geqslant\left(\frac{1}{2}\right)^{2}\frac{(\operatorname*{\varmathbb{E}}[R])^{2}}{\operatorname*{\varmathbb{E}}[R^{2}]}\geqslant\frac{1}{12}\,.$
∎
We are now ready to complete the proof of Theorem 1.2.
###### Proof of Theorem 1.2.
Let the $x_{i}$’s be as defined in Algorithm 9.4. W.l.o.g, we may assume
that111For any $x\in\varmathbb R$,
$x^{+}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\left\\{x,0\right\\}$.
$\left\lvert\operatorname{\sf
supp}(x^{+})\right\rvert<\left\lvert\operatorname{\sf
supp}(x^{-})\right\rvert$. For each $i\in[n]$, we define $y_{i}=x_{i}^{+}$.
Lemma 9.6 shows that with constant probability we have
$\frac{\sum_{i}\max_{j\sim
i}(x_{i}-x_{j})^{2}}{\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}}\leqslant
96\ {\sf SDPval}\log d.$
We need to show that
$\frac{\sum_{i}\max_{j\sim i}\left\lvert
y_{i}^{2}-y_{j}^{2}\right\rvert}{\sum_{i}y_{i}^{2}-\frac{1}{n}\left(\sum_{i}y_{i}\right)^{2}}\leqslant
6\sqrt{\frac{\sum_{i}\max_{j\sim
i}(x_{i}-x_{j})^{2}}{\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}}}.$
This fact is proved in [BHT00]. For the sake of completeness, we give a proof
of this fact in Appendix B (Lemma B.1). Using Lemma 9.6, we get
$\frac{\sum_{i}\max_{j\sim i}\left\lvert
y_{i}^{2}-y_{j}^{2}\right\rvert}{\sum_{i}y_{i}^{2}-\frac{1}{n}\left(\sum_{i}y_{i}\right)^{2}}\leqslant
576\sqrt{{\sf SDPval}\log d}.$
From Lemma 9.5 we get that the set output in Step $3$ of Algorithm 9.4 has
vertex expansion at most $576\sqrt{{\sf SDPval}\log d}$. ∎
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## Appendix A Reduction between Vertex Expansion and Symmetric Vertex
Expansion
In this section we show that the computation of the vertex expansion is
essentially equivalent to the computation of symmetric vertex expansion.
Formally, we prove the following theorems.
###### Theorem A.1.
Given a graph $G=(V,E)$, there exists a graph $H$ such that $\max_{i\in
V(H)}\deg(i)\leqslant\left(\max_{i\in V(G)}\deg(i)\right)^{2}+\max_{i\in
V(G)}\deg(i)$ such that
$\Phi^{\sf V}(G)\leqslant\phi^{\sf V}(H)\leqslant\frac{\Phi^{\sf
V}(G)}{1-\Phi^{\sf V}(G)}\,.$
###### Proof.
Let $G^{2}$ denote the graph on $V(G)$ that corresponds to two hops in the
graph $G$. Formally,
$\left\\{u,v\right\\}\in E(G^{2})\iff\exists w\in V(G),(u,w)\in E(G)\textrm{
and }(w,v)\in E(G)\,.$
Let $H=G\cup G^{2}$, i.e., $V(H)=V(G)$ and $E(H)=E(G)\cup E(G^{2})$.
Let $S\subset V(G)$ be a set with small symmetric vertex expansion $\Phi^{\sf
V}(S)=\varepsilon$. Let $S^{\prime}=S-N_{G}(\bar{S})$ be the set of vertices
obtained from $S$ by deleting it’s internal boundary. It is easy to see that
$N_{H}(S^{\prime})=N_{G}(S)\cup N_{G}(\bar{S})\,.$
Moreover, since $N_{G}(\bar{S})\leqslant\Phi^{\sf V}(S)w(S)$ we have
$w(S^{\prime})\geqslant w(S)(1-\Phi^{\sf V}_{G}(S))$. Hence the vertex
expansion of the set $S^{\prime}$ is upper-bounded by,
$\phi^{\sf V}_{H}(S^{\prime})\leqslant\frac{\Phi^{\sf V}_{G}(S)}{1-\Phi^{\sf
V}_{G}(S)}\,.$
Conversely, suppose $T\subset V(H)$ be a set with small vertex expansion
$\phi^{\sf V}_{H}(T)=\varepsilon$. Consider the set $T^{\prime}=T\cup
N_{G}(T)$. Observe that the internal boundary of $T^{\prime}$ in the graph $G$
is given by $N_{G}(\bar{T}^{\prime})=N_{G}(T)$. Further the external boundary
of $T^{\prime}$ is given by $N_{G}(T^{\prime})=N_{G}(N_{G}(T))=N_{G^{2}}(T)$.
Therefore, we have
$N_{G}(T^{\prime})\cup N_{G}(\bar{T}^{\prime})=N_{G}(T)\cup
N_{G^{2}}(T)=N_{H}(T)\,.$
Further since $w(T^{\prime})\geqslant w(T)$, we have $\Phi^{\sf
V}_{G}(T^{\prime})\leqslant\phi^{\sf V}_{H}(T)$.
This completes the proof of the Theorem.
∎
###### Theorem A.2.
Given a graph $G$, there exists a graph $G^{\prime}$ such that $\max_{i\in
V(G)}\deg(i)=\max_{i\in V(G^{\prime})}\deg(i)$ and $\phi^{\sf
V}(G)=\Theta(\Phi^{\sf V}(G^{\prime}))$. Moreover, such a $G^{\prime}$ can be
computed in time polynomial in the size of $G$.
###### Proof.
Given graph $G$, we construct $G^{\prime}$ as follows. We start with
$V(G^{\prime})=V(G)\cup E(G)$, i.e., $G^{\prime}$ has a vertex for each vertex
in $G$ and for each edge in $G$. For each edge $\left\\{u,v\right\\}\in E(G)$,
we add edges $\left\\{u,\left\\{u,v\right\\}\right\\}$ and
$\left\\{v,\left\\{u,v\right\\}\right\\}$ in $G^{\prime}$. For a vertex $i\in
V(G)\cap V(G^{\prime})$, we set its weight to be $w(i)$. For a vertex
$\left\\{u,v\right\\}\in E(G)\cap V(G^{\prime})$, we set its weight to be
$\min\left\\{w(u)/\deg(u),w(v)/\deg(v)\right\\}$.
It is easy to see that $G^{\prime}$ can be computed in time polynomial in the
size of $G$, and that $\max_{i\in V(G)}\deg(i)=\max_{i\in
V(G^{\prime})}\deg(i)$.
We first show that $\phi^{\sf V}(G)\geqslant\Phi^{\sf V}(G^{\prime})/2$. Let
$S\subset V(G)$ be the set having the least vertex expansion in $G$. Let
$S^{\prime}=S\cup\left\\{\left\\{u,v\right\\}|\left\\{u,v\right\\}\in
E(G)\textrm{ and }u\in S\textrm{ or }v\in S\right\\}\,.$
By construction, we have $w(S)\leqslant w(S^{\prime})$,
$N_{G}(S)=N_{G^{\prime}}(S^{\prime})$ and
$w(N_{G^{\prime}}(\bar{S}^{\prime}))\leqslant\sum_{u\in
N_{G^{\prime}}(S^{\prime})}\deg(u)\frac{w(u)}{\deg(u)}\leqslant
w(N_{G^{\prime}}(S^{\prime}))\,.$
Therefore,
$\Phi^{\sf V}(G^{\prime})\leqslant\Phi^{\sf
V}_{G^{\prime}}(S^{\prime})=\frac{w(N_{G^{\prime}}(S^{\prime}))+w(N_{G^{\prime}}(\bar{S}^{\prime}))}{w(S^{\prime})}\leqslant\frac{2w(N_{G}(S))}{w(S)}=2\phi^{\sf
V}_{G}(S)=2\phi^{\sf V}(G)\,.$
Now, let $S^{\prime}\subset V(G^{\prime})$ be the set having the least value
of $\Phi^{\sf V}_{G^{\prime}}(S^{\prime})$ and let $\varepsilon=\Phi^{\sf
V}_{G^{\prime}}(S^{\prime})$. We construct the set $S$ as follows. We let
$S_{1}=S^{\prime}\backslash N_{G^{\prime}}(\bar{S}^{\prime})$, i.e. we obtain
$S_{1}$ from $S^{\prime}$ by deleting it’s internal boundary. Next we set
$S=S_{1}\cap V(G)$. More formally, we let $S$ be the following set.
$S=\left\\{v\in S^{\prime}\cap V(G)|v\notin
N_{G^{\prime}}(\bar{S}^{\prime})\right\\}\,.$
By construction, we get that $N_{G}(S)\subseteq N_{G^{\prime}}(S^{\prime})\cup
N_{G^{\prime}}(\bar{S}^{\prime})$. Now, the internal boundary of $S^{\prime}$
has weight at most $\varepsilon w(S^{\prime})$. Therefore, we have
$w(S_{1})\geqslant(1-\varepsilon)w(S^{\prime})\,.$
We need a lower bound on the weight of the set $S$ we constructed. To this
end, we make the following observation. For each vertex
$\left\\{u,v\right\\}\in S_{1}\cap E(G)$, $u$ or $v$ also has to be in $S_{1}$
(If not, then deleting $\left\\{u,v\right\\}$ from $S^{\prime}$ will result in
a decrease in the vertex expansion thereby contradicting the optimality of the
choice of the set $S^{\prime}$). Therefore, we have the following
$\sum_{\left\\{u,v\right\\}\in S_{1}\cap
E(G)}w(\left\\{u,v\right\\})=\sum_{\left\\{u,v\right\\}\in S_{1}\cap
E(G)}\min\left\\{\frac{w(u)}{\deg(u)},\frac{w(u)}{\deg(u)}\right\\}\leqslant\sum_{u\in
S_{1}\cap V(G)}w(u)=w(S)\,.$
Therefore,
$w(S)\geqslant\frac{w(S_{1})}{2}\geqslant(1-\varepsilon)\frac{w(S^{\prime})}{2}$
Therefore, we have
$\phi^{\sf V}(G)\leqslant\phi^{\sf
V}_{G}(S)=\frac{w(N_{G}(S))}{w(S)}\leqslant\frac{w(N_{G^{\prime}}(S^{\prime})\cup
N_{G^{\prime}}(\bar{S}^{\prime})}{(1-\varepsilon)w(S^{\prime})/2}=4\Phi^{\sf
V}_{G^{\prime}}(S^{\prime})=4\Phi^{\sf V}(G^{\prime})\,.$
Putting these two together, we have
$\frac{\phi^{\sf V}(G)}{2}\leqslant\Phi^{\sf V}(G^{\prime})\leqslant
4\phi^{\sf V}(G)\,.$
∎
## Appendix B Omitted Proofs
###### Lemma B.1 ([BHT00]).
Let $z_{1},\ldots,z_{n}\in R$ and let
$x_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}z_{i}^{+}$. Then
$\frac{\sum_{i}\max_{j\sim i}\left\lvert
x_{i}^{2}-x_{j}^{2}\right\rvert}{\sum_{i}x_{i}^{2}-\frac{1}{n}\left(\sum_{i}x_{i}\right)^{2}}\leqslant
6\sqrt{\frac{\sum_{i}\max_{j\sim
i}(z_{i}-z_{j})^{2}}{\sum_{i}z_{i}^{2}-\frac{1}{n}\left(\sum_{i}z_{i}\right)^{2}}}.$
###### Proof.
W.l.o.g we may assume that $\left\lvert\operatorname{\sf
supp}(Z^{+})\right\rvert=\left\lvert\operatorname{\sf
supp}(Z^{-})\right\rvert=\lceil n/2\rceil$ and that $z_{1}\geqslant
z_{2}\geqslant\ldots\geqslant z_{n}$.
Note that for any $i\in[n]$, we have $\max_{j\sim
i,\&j<i}(z_{j}^{+}-z_{i}^{+})^{2}+\max_{j\sim
i,\&j>i}(z_{j}^{-}-z_{i}^{-})^{2}\leqslant 2\max_{j\sim i}(z_{j}-z_{i})^{2}$.
Now,
$\displaystyle\frac{\sum_{i}\max_{j\sim
i}(z_{j}-z_{i})^{2}}{\sum_{i}z_{i}^{2}}$ $\displaystyle\geqslant$
$\displaystyle\frac{\sum_{i}\max_{j<i\&j\sim
i}(z_{j}^{+}-z_{i}^{+})^{2}+\sum_{i}\max_{j>i\&j\sim
i}(z_{j}^{-}-z_{i}^{-})^{2}}{2\left(\sum_{i\in\operatorname{\sf
supp}(Z^{+})}z_{i}^{2}+\sum_{i\in\operatorname{\sf
supp}(Z^{-})}z_{i}^{2}\right)}$ $\displaystyle\geqslant$
$\displaystyle\min\left\\{\frac{\sum_{i}\max_{j<i\&j\sim
i}(z_{j}^{+}-z_{i}^{+})^{2}}{2\sum_{i\in\operatorname{\sf
supp}(Z^{+})}z_{i}^{2}},\frac{\sum_{i}\max_{j>i\&j\sim
i}(z_{j}^{-}-z_{i}^{-})^{2}}{2\sum_{i\in\operatorname{\sf
supp}(Z^{-})}z_{i}^{2}}\right\\}$
W.l.o.g we may assume that
$\frac{\sum_{i}\max_{j<i\&j\sim
i}(z_{j}^{+}-z_{i}^{+})^{2}}{\sum_{i\in\operatorname{\sf
supp}(Z^{+})}z_{i}^{2}}\leqslant\frac{\sum_{i}\max_{j>i\&j\sim
i}(z_{j}^{-}-z_{i}^{-})^{2}}{\sum_{i\in\operatorname{\sf
supp}(Z^{-})}z_{i}^{2}}$
$\frac{\sum_{i}\max_{j\sim i}(x_{j}-x_{i})^{2}}{\sum_{i}x_{i}^{2}}\leqslant
2\frac{\sum_{i}\max_{j\sim i}(z_{j}-z_{i})^{2}}{\sum_{i}z_{i}^{2}}$
We have
$\displaystyle\max_{j\sim i,j<i}(x_{j}^{2}-x_{i}^{2})$ $\displaystyle=$
$\displaystyle\max_{j\sim i,j<i}(x_{j}-x_{i})(x_{j}+x_{i})$
$\displaystyle\leqslant$ $\displaystyle\max_{j\sim
i,j<i}\left((x_{j}-x_{i})^{2}+2x_{i}(x_{j}-x_{i})\right)$
$\displaystyle\leqslant$ $\displaystyle\max_{j\sim
i,j<i}(x_{j}-x_{i})^{2}+2x_{i}\max_{j\sim i,j<i}(x_{j}-x_{i})$
$\displaystyle\leqslant$ $\displaystyle\sum_{i}\max_{j\sim
i,j<i}(x_{j}-x_{i})^{2}+2\sqrt{\sum_{i}x_{i}^{2}}\sqrt{\max_{j\sim
i,j<i}(x_{j}-x_{i})^{2}}\qquad\textrm{ Cauchy-Schwarz}$ $\displaystyle=$
$\displaystyle\lambda_{\infty}\sum_{i}x_{i}^{2}+2\sqrt{\lambda_{\infty}}\sum_{i}x_{i}^{2}$
Thus we have
$\frac{\sum_{i}\max_{j\sim
i,j<i}(x_{j}^{2}-x_{i}^{2})}{\sum_{i}x_{i}^{2}}\leqslant
6\sqrt{\frac{\sum_{i}\max_{j\sim i}(z_{j}-z_{i})^{2}}{\sum_{i}z_{i}^{2}}}$
∎
###### Lemma B.2.
A c-vs-s hardness for $b$-Balanced-vertex expansion implies a 2 c-vs-s/2
hardness for vertex expansion.
###### Proof.
Fix a graph $G=(V,E)$.
#### Completeness
If $G$ has Balanced-vertex expansion at most $c$, then clearly its vertex
expansion is also at most $c$.
#### Soundness
Suppose we have a polynomial time algorithm that outputs a set $S$ having
$\phi^{\sf V}(S)\leqslant s$ whenever $G$ has a set $S^{\prime}$ having
$\phi^{\sf V}(S^{\prime})\leqslant 2c$. Then this algorithm can be used as an
oracle to find a balanced set of vertex expansion at most $s$. This would
contradict the hardness of Balanced-vertex expansion.
First we find a set, say $T$, having $\phi^{\sf V}(T)\leqslant s$. If we are
unable to find such a $T$, we stop. If we find such a set $T$ and $T$ has
balance at least $b$, then we stop. Else, we delete the vertices in $T$ from
$G$ and repeat. We continue until the number of deleted vertices first exceeds
a $b/2$ fraction of the vertices.
If the process deletes less than $b/2$ fraction of the vertices, then the
remaining graph (which has at least $(1-b/2)n$ vertices) has conductance $2c$,
and thus in the original graph every $b$-balanced cut has conductance at least
$c$. This is a contradiction !
If the process deletes between $b/2$ and $1/2$ of the nodes, then the union of
the deleted sets gives a set $T^{\prime}$ with $\phi^{\sf
V}(T^{\prime})\leqslant s$ and balance of $T^{\prime}$ at least $b/2$. ∎
###### Fact B.3.
Let $Y_{1},Y_{2},\ldots,Y_{d}$ be $d$ standard normal random variables. Let
$Y$ be the random variable defined as
$Y\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\max\left\\{Y_{i}|i\in[d]\right\\}$.
Then
$\operatorname*{\varmathbb{E}}\left[Y^{2}\right]\leqslant 4\log
d\qquad\textrm{ and
}\qquad\operatorname*{\varmathbb{E}}\left[Y\right]\leqslant 2\sqrt{\log d}\,.$
###### Proof.
For any $Z_{1},\ldots,Z_{d}\in\varmathbb R$ and any $p\in\varmathbb Z^{+}$, we
have $\max_{i}\left\lvert
Z_{i}\right\rvert\leqslant(\sum_{i}Z_{i}^{p})^{\frac{1}{p}}$. Now
$Y^{2}=(\max_{i}X_{i})^{2}\leqslant\max_{i}X_{i}^{2}$.
$\displaystyle\operatorname*{\varmathbb{E}}\left[Y^{2}\right]$
$\displaystyle\leqslant$
$\displaystyle\operatorname*{\varmathbb{E}}\left[\left(\sum_{i}X_{i}^{2p}\right)^{\frac{1}{p}}\right]\leqslant\left(\operatorname*{\varmathbb{E}}\left[\sum_{i}X_{i}^{2p}\right]\right)^{\frac{1}{p}}\quad\textrm{
( Jensen's Inequality )}$ $\displaystyle\leqslant$
$\displaystyle\left(\sum_{i}\left(\operatorname*{\varmathbb{E}}\left[X_{i}^{2}\right]\right)\frac{(2p)!}{(p)!2^{p}}\right)^{\frac{1}{p}}\leqslant
2pd^{\frac{1}{p}}\quad\textrm{(using $(2p)!/p!\leqslant(2p)^{p}$ )}$
Picking $p=\log d$ gives
$\operatorname*{\varmathbb{E}}\left[Y^{2}\right]\leqslant 2e\log d$.
Therefore
$\operatorname*{\varmathbb{E}}\left[Y\right]\leqslant\sqrt{\operatorname*{\varmathbb{E}}\left[Y^{2}\right]}\leqslant\sqrt{2e\log
d}$.
∎
## Appendix C Noise Operators
Let $H$ be a Markov chain and let $F:V(H^{k})\to\left\\{0,1\right\\}$ be any
boolean function. In this section we prove some basic properties of
$\Gamma_{1-\eta}F$. We restate the definition of our Noise Operator
$\Gamma_{1-\eta}$.
$\Gamma_{1-\eta}F(X)=(1-\eta)F(X)+\eta\operatorname*{\varmathbb{E}}_{Y\sim
X}F(Y)$
The Fourier expansion of the function $F$ is
$F=\sum_{\sigma}\hat{f}_{\sigma}e_{\sigma}$ where
$\left\\{e_{\sigma}\right\\}$ is the set of eigenvectors of $H^{k}$. It is
easy to see that $e_{\sigma}=e_{\sigma_{1}}\otimes\ldots\otimes
e_{\sigma_{k}}$, where the $\left\\{e_{\sigma_{i}}\right\\}$ are the
eigenvectors of $H$.
###### Lemma C.1.
(Decay of High degree Coefficients) Let $Q_{j}$ be the multi-linear polynomial
representation of
$\left\lvert\Gamma_{1-\eta}F(X)-\Gamma_{1-\eta}F(Y_{j})\right\rvert$. Then,
$\mathsf{Var}(Q_{j}^{>p})\leqslant(1-\varepsilon\eta)^{2p}$
###### Proof.
$\displaystyle\Gamma_{1-\eta}F(X)$ $\displaystyle=$
$\displaystyle(1-\eta)F(X)+\eta\operatorname*{\varmathbb{E}}_{Y\sim X}F(Y)$
$\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}\operatorname*{\varmathbb{E}}\left[e_{\sigma}(X)+\operatorname*{\varmathbb{E}}_{Y\sim
X}F(Y)\right]$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}\Pi_{i\in\sigma}\left((1-\eta)e_{\sigma_{i}}(X_{i})+\operatorname*{\varmathbb{E}}_{Y_{i}\sim
X_{i}}e_{\sigma_{i}}(Y_{i})\right)$
We bound the second moment of $\Gamma_{1-\eta}F$ as follows
$\displaystyle\operatorname*{\varmathbb{E}}_{X}\left(\Gamma_{1-\eta}F(X)\right)^{2}$
$\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}^{2}\operatorname*{\varmathbb{E}}_{X}\Pi_{i\in\sigma}\left((1-\eta)e_{\sigma_{i}}(X_{i})+\eta\operatorname*{\varmathbb{E}}_{Y_{i}\sim
X_{i}}e_{\sigma_{i}}(Y_{i})\right)^{2}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}^{2}\Pi_{i\in\sigma}\left((1-\eta)^{2}\operatorname*{\varmathbb{E}}_{X_{i}}e_{\sigma_{i}}(X_{i})^{2}+\eta^{2}\operatorname*{\varmathbb{E}}_{X_{i}}\left(\operatorname*{\varmathbb{E}}_{Y_{i}\sim
X_{i}}e_{\sigma_{i}}(Y_{i})\right)^{2}+2\eta(1-\eta)\operatorname*{\varmathbb{E}}_{X_{i}}\operatorname*{\varmathbb{E}}_{Y_{i}\sim
X_{i}}e_{\sigma_{i}}(X_{i})e_{\sigma_{i}}(Y_{i})\right)^{2}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}^{2}\Pi_{i\in\sigma}\left((1-\eta)^{2}+\eta^{2}\lambda_{i}^{2}+2\eta(1-\eta)\lambda_{i}\right)$
$\displaystyle=$
$\displaystyle\sum_{\sigma}\hat{f}_{\sigma}^{2}\Pi_{i\in\sigma}\left(1-\eta+\eta\lambda_{i}\right)^{2}$
Therefore,
$\displaystyle\mathsf{Var}(Q_{j}^{>p})$ $\displaystyle\leqslant$
$\displaystyle
4\sum_{\sigma:\left\lvert\sigma\right\rvert>p}\hat{f}_{\sigma}^{2}\Pi_{i\in\sigma}\left(1-\eta+\eta\lambda_{i}\right)^{2}$
$\displaystyle\leqslant$
$\displaystyle\sum_{\sigma:\left\lvert\sigma\right\rvert>p}\hat{f}_{\sigma}^{2}\left(1-\varepsilon\eta\right)^{2\left\lvert\sigma\right\rvert}$
$\displaystyle\leqslant$ $\displaystyle(1-\varepsilon\eta)^{2p}$
Here the second inequality follows from the fact that all non-trivial
eigenvalues of $H$ are at most $1-\varepsilon$ and the third inequality
follows Parseval’s indentity. ∎
|
arxiv-papers
| 2013-04-10T20:31:28 |
2024-09-04T02:49:44.134215
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anand Louis, Prasad Raghavendra, Santosh Vempala",
"submitter": "Anand Louis",
"url": "https://arxiv.org/abs/1304.3139"
}
|
1304.3146
|
# Standardized network reconstruction of CHO cell metabolism
Kieran Smallbone
_Manchester Centre for Integrative Systems Biology_
_131 Princess Street, Manchester M1 7DN, UK_
[email protected]
###### Abstract
We have created a genome-scale network reconstruction of chinese hamster ovary
(CHO) cell metabolism. Existing reconstructions were improved in terms of
annotation standards, to facilitate their subsequent use in dynamic modelling.
The resultant network is available from ChoNet (http://cho.sf.net/).
## ChoNet
The structure of metabolic networks can be determined by a reconstruction
approach, using data from genome annotation, metabolic databases and chemical
databases [1]. We built upon an existing reconstruction of the metabolic
network of CHO cells that was based on genomic and literature data (Selvarasu
et al. [2]). This model contains 1065 genes, 1545 metabolic reactions, and
1218 unique metabolites. Use of in silico modelling allows characterisation
internal metabolic behaviour during growth and non-growth phases [2].
Selvarasu et al. suffers from the use of non-standard names and is not
annotated with methods that are machine-readable. The model was thus updated
according to existing community-driven annotation standards [3]. The
reconstruction is described and made available in Systems Biology Markup
Language (SBML) (http://sbml.org/, [4]), an established community XML format
for the mark-up of biochemical models that is understood by a large number of
software applications. The network is available from ChoNet
(http://cho.sf.net/). As supplied, the model has an optimal growth rate of
0.0257 flux units.
### Annotation
The highly-annotated network is primarily assembled and provided as an SBML
file. Specific model entities, such as species or reactions, are annotated
using ontological terms. These annotations, encoded using the resource
description framework (RDF) [5], provide the facility to assign definitive
terms to individual components, allowing software to identify such components
unambiguously and thus link model components to existing data resources [6].
Minimum Information Requested in the Annotation of Models (MIRIAM, [7])
–compliant annotations have been used to identify components unambiguously by
associating them with one or more terms from publicly available databases
registered in MIRIAM resources [8]. Thus this network is entirely traceable
and is presented in a computational framework.
Six different databases are used to annotate entities in the network (see
Table 1). The Systems Biology Ontology (SBO) [9] is also used to semantically
discriminate between entity types. Five different SBO terms are used to
annotate entities in the network (see Table 2).
example | identifier | database
---|---|---
ChoNet | 10029 | taxonomy
ChoNet | 22252269 | pubmed
cytosol | GO:0005737 | obo.go
N-methylhistamine | CHEBI:29009 | chebi
1-oxidoreductase | 1.1.99.1 | ec-code
1-oxidoreductase | 218865 | ncbigene
Table 1: MIRIAM annotations used in the model. example | SBO term | interpretation
---|---|---
cytosol | 290 | compartment
N-methylhistamine | 247 | metabolite
N-methylhistamine | 176 | biochemical reaction
AATRA20 | 185 | transport reaction
biomass objective function | 397 | modelling reaction
Table 2: SBO terms used in the model.
### Use
We maintain the distinction between the CHO cell GEnome scale Network
REconstruction (GENRE) [10] and its derived GEnome scale Model (GEM) [11].
This is important to differentiate between the established biochemical
knowledge included in a GENRE and the modelling assumptions required for
analysis or simulation with a GEM. A GENRE serves as a structured knowledge
base of established biochemical facts, while a GEM is a model which
supplements the established biochemical information with additional
(potentially hypothetical) information to enable computational simulation and
analysis [12]. Reactions added to the GEM include the biomass objective
function – a sink representing cellular growth – and hypothetical
transporters.
Three versions of the network are made available:
* •
<organism>_<version>.xml, a GEM for use in flux analyses, provided in Flux
Balance Constraints (FBC) format [13]
* •
<organism>_<version>_cobra.xml, the same GEM network, provided in Cobra format
[14]
* •
<organism>_<version>_recon.xml, a GENRE containing only reactions for which
there is experimental evidence
## EcoliNet and YeastNet
EcoliNet (http://ecoli.sf.net/) and YeastNet (http://yeast.sf.net/) are
annotated metabolic network of Escherichia coli and Saccharomyces cerevisiae
S288c, respectively, that are periodically updated by a team of collaborators
from various research groups. The three networks are structured identically to
facilitate comparative studies.
#### Acknowledgements
This work is deliverable 4.3 of the EU FP7 (KBBE) grant 289434 “BioPreDyn: New
Bioinformatics Methods and Tools for Data-Driven Predictive Dynamic Modelling
in Biotechnological Applications”.
## References
* [1] Palsson BØ, Thiele I: A protocol for generating a high-quality genome-scale metabolic reconstruction. Nature Protoc 2010, 5:91–121. doi:10.1038/nprot.2009.203
* [2] Selvarasu S, Ho YS, Chong WP, Wong NS, Yusufi FN, Lee YY, Yap MG, Lee DY: Combined in silico modeling and metabolomics analysis to characterize fed-batch CHO cell culture. Biotechnol Bioeng 2012, 109:1415–1429. doi:10.1002/bit.24445
* [3] Herrgård MJ, Swainston N, Dobson P, Dunn WB, Arga KY, Arvas M, Blüthgen N, Borger S, Costenoble R, Heinemann M, Hucka M, Le Novére N, Li P, Liebermeister W, Mo M, Oliveira AP, Petranovic D, Pettifer S, Simeonidis E, Smallbone K, Spasić I, Weichart D, Brent R, Broomhead DS, Westerhoff HV, Kırdar B, Penttilä M, Klipp E, Palsson BØ, Sauer U, Oliver SG, Mendes P, Nielsen J, Kell DB: A consensus yeast metabolic network obtained from a community approach to systems biology. Nature Biotechnol 2008, 26:1155–1160. doi:10.1038/nbt1492
* [4] Hucka M, Finney A, Sauro H, Bolouri H, Doyle J, Kitano H, Arkin A, Bornstein B, Bray D, Cornish-Bowden A, Cuellar A, Dronov S, Gilles E, Ginkel M, Gor V, Goryanin I, Hedley W, Hodgman T, Hofmeyr J,Hunter P, Juty N, Kasberger J, Kremling A, Kummer U, Le Novère N, Loew L, Lucio D, Mendes P, Minch E, Mjolsness E, Nakayama Y, Nelson M, Nielsen P, Sakurada T, Schaff J, Shapiro B, Shimizu T, Spence H, Stelling J, Takahashi K, Tomita M, Wagner J, Wang J: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 2003, 19:524–531. doi:10.1093/bioinformatics/btg015
* [5] Wang XS, Gorlitsky R, Almeida JS: From XML to RDF: how semantic web technologies will change the design of ‘omic’ standards. Nature Biotechnol 2005, 23:1099–1103. doi:10.1038/nbt1139
* [6] Kell DB, Mendes P: The markup is the model: reasoning about systems biology models in the Semantic Web era. J Theor Biol 2008, 252:538–543. doi:10.1016/j.jtbi.2007.10.023
* [7] Le Novére N, Finney A, Hucka M, Bhalla US, Campagne F, Collado-Vides J, Crampin EJ, Halstead M, Klipp E, Mendes P, Nielsen P, Sauro H, Shapiro B, Snoep JL, Spence HD, Wanner BL: Minimum information requested in the annotation of biochemical models (MIRIAM). Nature Biotechnol 2005, 23:1509–1515. doi:10.1038/nbt1156
* [8] Laibe C, Le Novére N: MIRIAM resources: tools to generate and resolve robust cross-references in Systems Biology. BMC Syst Biol 2008, 252:538–543. doi:10.1186/1752-0509-1-58
* [9] Courtot M., Juty N., Knüpfer C., Waltemath D., Zhukova A., Dr ger A., Dumontier M., Finney A., Golebiewski M., Hastings J., Hoops S., Keating S., Kell D.B., Kerrien S., Lawson J., Lister A., Lu J., Machne R., Mendes P., Pocock M., Rodriguez N., Villeger A., Wilkinson D.J., Wimalaratne S., Laibe C., Hucka M., Le Novére N.: Controlled vocabularies and semantics in systems biology.. Mol Syst Biol 2011, 7:-543. doi:10.1038/msb.2011.77
* [10] Price ND, Reed JL, Palsson BØ: Genome-scale models of microbial cells: evaluating the consequences of constraints. Nat Rev Microbiol 2004, 2:886–897. doi:10.1038/nrmicro1023
* [11] Feist AM, Herrgård MJ, Thiele I, Reed JL, Palsson BØ: Reconstruction of biochemical networks in microorganisms. Nat Rev Microbiol 2008, 7:129–143. doi:10.1038/nrmicro1949
* [12] Heavner BD, Smallbone K, Barker B, Mendes P, Walker LP: Yeast 5 – an expanded reconstruction of the Saccharomyces cerevisiae metabolic network. BMC Syst Biol 2012, 6:55. doi:10.1186/1752-0509-6-55
* [13] Olivier BG, Bergmann FT: Flux Balance Constraints, Version 1 Release 1. Available from COMBINE. 2013\.
* [14] Schellenberger J, Que R, Fleming RM, Thiele I, Orth JD, Feist AM, Zielinski DC, Bordbar A, Lewis NE, Rahmanian S, Kang J, Hyduke DR, Palsson BØ: Quantitative prediction of cellular metabolism with constraint-based models: the COBRA Toolbox v2.0. Nat Protoc 2011, 6:1290–1307. doi:10.1038/nprot.2011.308.4
|
arxiv-papers
| 2013-04-09T09:09:16 |
2024-09-04T02:49:44.150948
|
{
"license": "Public Domain",
"authors": "Kieran Smallbone",
"submitter": "Kieran Smallbone",
"url": "https://arxiv.org/abs/1304.3146"
}
|
1304.3192
|
11institutetext: Y. Guo 22institutetext: College of Electronic Science and
Engineering,
National University of Defense Technology,
Changsha, Hunan, P.R.China
22email: [email protected] 33institutetext: Y. Guo 44institutetext: F.
Sohel 55institutetext: M. Bennamoun 66institutetext: School of Computer
Science and Software Engineering,
The University of Western Australia,
Perth, Australia 77institutetext: M. Lu 88institutetext: J. Wan
99institutetext: College of Electronic Science and Engineering,
National University of Defense Technology
# Rotational Projection Statistics for 3D Local Surface Description and Object
Recognition ††thanks: This research is supported by a China Scholarship
Council (CSC) scholarship and Australian Research Council grants (DE120102960,
DP110102166).
Yulan Guo Ferdous Sohel Mohammed Bennamoun Min Lu Jianwei Wan
(Received: date / Accepted: date)
###### Abstract
Recognizing 3D objects in the presence of noise, varying mesh resolution,
occlusion and clutter is a very challenging task. This paper presents a novel
method named Rotational Projection Statistics (RoPS). It has three major
modules: Local Reference Frame (LRF) definition, RoPS feature description and
3D object recognition. We propose a novel technique to define the LRF by
calculating the scatter matrix of all points lying on the local surface. RoPS
feature descriptors are obtained by rotationally projecting the neighboring
points of a feature point onto 2D planes and calculating a set of statistics
(including low-order central moments and entropy) of the distribution of these
projected points. Using the proposed LRF and RoPS descriptor, we present a
hierarchical 3D object recognition algorithm. The performance of the proposed
LRF, RoPS descriptor and object recognition algorithm was rigorously tested on
a number of popular and publicly available datasets. Our proposed techniques
exhibited superior performance compared to existing techniques. We also showed
that our method is robust with respect to noise and varying mesh resolution.
Our RoPS based algorithm achieved recognition rates of 100%, 98.9%, 95.4% and
96.0% respectively when tested on the Bologna, UWA, Queen’s and Ca’ Foscari
Venezia Datasets.
###### Keywords:
Surface descriptor Local feature Local reference frame 3D representation
Feature matching 3D object recognition
††journal: International Journal of Computer Vision
## 1 Introduction
Object recognition is an active research area in computer vision with numerous
applications including navigation, surveillance, automation, biometrics,
surgery and education (Guo et al., 2013c; Johnson and Hebert, 1999; Lei et
al., 2013; Tombari et al., 2010). The aim of object recognition is to
correctly identify the objects that are present in a scene and recover their
poses (i.e., position and orientation) (Mian et al., 2006b). Beyond object
recognition from 2D images (Brown and Lowe, 2003; Lowe, 2004; Mikolajczyk and
Schmid, 2004), 3D object recognition has been extensively investigated during
the last two decades due to the availability of low cost scanners and high
speed computing devices (Mamic and Bennamoun, 2002). However, recognizing
objects from range images in the presence of noise, varying mesh resolution,
occlusion and clutter is still a challenging task.
Existing algorithms for 3D object recognition can broadly be classified into
two categories, i.e., global feature based and local feature based algorithms
(Bayramoglu and Alatan, 2010; Castellani et al., 2008). The global feature
based algorithms construct a set of features which encode the geometric
properties of the entire 3D object. Examples of these algorithms include the
geometric 3D moments (Paquet et al., 2000), shape distribution (Osada et al.,
2002) and spherical harmonics (Funkhouser et al., 2003). However, these
algorithms require complete 3D models and are therefore sensitive to occlusion
and clutter (Bayramoglu and Alatan, 2010). In contrast, the local feature
based algorithms define a set of features which encode the characteristics of
the local neighborhood of feature points. The local feature based algorithms
are robust to occlusion and clutter. They are therefore even suitable to
recognize partially visible objects in a cluttered scene (Petrelli and Di
Stefano, 2011).
A number of local feature based 3D object recognition algorithms have been
proposed in the literature, including point signature based (Chua and Jarvis,
1997), spin image based (Johnson and Hebert, 1999), tensor based (Mian et al.,
2006b) and Exponential Map (EM) based (Bariya et al., 2012) algorithms. Most
of these algorithms follow a paradigm that has three phases, i.e., feature
matching, hypothesis generation and verification, and pose refinement (Taati
and Greenspan, 2011). Among these phases, feature matching plays a critical
role since it directly affects the effectiveness and efficiency of the two
subsequent phases (Taati and Greenspan, 2011).
Descriptiveness and robustness of a feature descriptor are crucial for
accurate feature matching (Bariya and Nishino, 2010). The feature descriptors
should be highly descriptive to ensure an accurate and efficient object
recognition. That is because the accuracy of feature matching directly
influences the quality of the estimated transformation which is used to align
the model to the scene, as well as the computational time required for
verification and refinement (Taati and Greenspan, 2011). Moreover, the feature
descriptors should be robust to a set of nuisances, including noise, varying
mesh resolution, clutter, occlusion, holes and topology changes (Bronstein et
al., 2010a; Boyer et al., 2011).
A number of local feature descriptors exist in literature (Section 2.1). These
descriptors can be divided into two broad categories based on whether they use
a Local Reference Frame (LRF) or not. Feature descriptors without any LRF use
a histogram or the statistics of the local geometric information (e.g.,
normal, curvature) to form a feature descriptor (Section 2.1.1). Examples of
this category include surface signature (Yamany and Farag, 2002), Local
Surface Patch (LSP) (Chen and Bhanu, 2007) and THRIFT (Flint et al., 2007). In
contrast, feature descriptors with LRF encode the spatial distribution and/or
geometric information of the neighboring points with respect to the defined
LRF (Section 2.1.2). Examples include spin image (Johnson and Hebert, 1999),
Intrinsic Shape Signatures (ISS) (Zhong, 2009) and MeshHOG (Zaharescu et al.,
2012). However, most of the existing feature descriptors still suffer from
either low descriptiveness or weak robustness (Bariya et al., 2012).
In this paper we present a highly descriptive and robust feature descriptor
together with an efficient 3D object recognition algorithm. This paper first
proposes a unique, repeatable and robust LRF for both local feature
description and object recognition (Section 3). The LRF is constructed by
performing an eigenvalue decomposition on the scatter matrix of all the points
lying on the local surface together with a sign disambiguation technique. A
novel feature descriptor, namely Rotational Projection Statistics (RoPS), is
then presented (Section 4). RoPS exhibits both high discriminative power and
strong robustness to noise, varying mesh resolution and a set of deformations.
The RoPS feature descriptor is generated by rotationally projecting the
neighboring points onto three local coordinate planes and calculating several
statistics (e.g, central moment and entropy) of the distribution matrices of
the projected points. Finally, this paper presents a novel hierarchical 3D
object recognition algorithm based on the proposed LRF and RoPS feature
descriptor (Section 6). Comparative experiments on four popular datasets were
performed to demonstrate the superiority of the proposed method (Section 7).
The rest of this paper is organized as follows. Section 2 provides a brief
literature review of local surface feature descriptors and 3D object
recognition algorithms. Section 3 introduces a novel technique for LRF
definition. Section 4 describes our proposed RoPS method for local surface
feature description. Section 5 presents the evaluation results of the RoPS
descriptor on two datasets. Section 6 introduces a RoPS based hierarchical
algorithm for 3D object recognition. Section 7 presents the results and
analysis of our 3D object recognition experiments on four datasets. Section 8
concludes this paper.
## 2 Related Work
This section presents a brief overview of the existing main methods for local
surface feature description and local feature based 3D object recognition.
### 2.1 Local Surface Feature Description
#### 2.1.1 Features without LRF
Stein and Medioni (1992) proposed a splash feature by recording the
relationship between the normals of the geodesic neighboring points and the
feature point. This relationship is then encoded into a 3D vector and finally
transformed into curvatures and torsion angles. Hetzel et al. (2001)
constructed a set of features by generating histograms using depth values,
surface normals, shape indices and their combinations. Results show that the
surface normal and shape index exhibit high discrimination capabilities.
Yamany and Farag (2002) introduced a surface signature by encoding the surface
curvature information into a 2D histogram. This method can be used to estimate
scaling transformations as well as recognizing objects in 3D scenes. Chen and
Bhanu (2007) proposed a LSP feature that encodes the shape indices and normal
deviations of the neighboring points. Flint et al. (2008) introduced a THRIFT
feature by calculating a weighted histogram of the deviation angles between
the normals of the neighboring points and the feature point. Taati et al.
(2007) considered the selection of a good local surface feature for 3D object
recognition as an optimization problem and proposed a set of Variable-
Dimensional Local Shape Descriptors (VD-LSD). However, the process of
selecting an optimized subset of VD-LSDs for a specific object is very time
consuming (Taati and Greenspan, 2011). Kokkinos et al. (2012) proposed a
generalization of 2D shape context feature (Belongie et al., 2002) to curved
surfaces, namely Intrinsic Shape Context (ISC). The ISC is a meta-descriptor
which can be applied to any photometric or geometric field defined on a
surface.
Without LRF, most of these methods generate a feature descriptor by
accumulating certain geometric attributes (e.g., normal, curvature) into a
histogram. Since most of the 3D spatial information is discarded during the
process of histogramming, the descriptiveness of the features without LRF is
limited (Tombari et al., 2010).
#### 2.1.2 Features with LRF
Chua and Jarvis (1997) proposed a point signature by using the distances from
the neighboring points to their corresponding projections on a fitted plane.
One merit of the point signature is that no surface derivative is required.
One of its limitations relate to the fact that the reference direction may not
be unique. It is also sensitive to mesh resolution (Mian et al., 2010).
Johnson and Hebert (1998) used the surface normal as a reference axis and
proposed a spin image representation by spinning a 2D image about the normal
of a feature point and summing up the number of points falling into the bins
of that image. The spin image is one of the most cited methods. But its
descriptiveness is relatively low and it is also sensitive to mesh resolution
(Zhong, 2009). Frome et al. (2004) also used the normal vector as a reference
axis and generated a 3D Shape Context (3DSC) by counting the weighted number
of points falling in the neighboring 3D spherical space. However, a reference
axis is not a complete reference frame and there is an uncertainty in the
rotation around the normal (Petrelli and Di Stefano, 2011).
Sun and Abidi (2001) introduced an LRF by using the normal of a feature point
and an arbitrarily chosen neighboring point. Based on the LRF, they proposed a
descriptor named point’s fingerprint by projecting the geodesic circles onto
the tangent plane. It was reported that their approach outperforms the 2D
histogram based methods. One major limitation of this method is that their LRF
is not unique (Tombari et al., 2010). Mian et al. (2006b) proposed a tensor
representation by defining an LRF for a pair of oriented points and encoding
the intersected surface area into a multidimensional table. This
representation is robust to noise, occlusion and clutter. However, a pair of
points are required to define an LRF, which causes a combinatorial explosion
(Zhong, 2009). Novatnack and Nishino (2008) used the surface normal and a
projected eigenvector on the tangent plane to define an LRF. They proposed an
EM descriptor by encoding the surface normals of the neighboring points into a
2D domain. The effectiveness of exploiting geometric scale variability in the
EM descriptor has been demonstrated. Zhong (2009) introduced an LRF by
calculating the eigenvectors of the scatter matrix of the neighboring points
of a feature point, and proposed an ISS feature by recording the point
distribution in the spherical angular space. Since the sign of the LRF is not
defined unambiguously, four feature descriptors can be generated from a single
feature point. Mian et al. (2010) proposed a keypoint detection method and
used a similar LRF to Zhong (2009) for their feature description. Tombari et
al. (2010) analyzed the strong impact of LRF on the performance of feature
descriptors and introduced a unique and unambiguous LRF by performing an
eigenvalue decomposition on the scatter matrix of the neighboring points and
using a sign disambiguation technique. Based on the proposed LRF, they
introduced a feature descriptor called Signature of Histograms of OrienTations
(SHOT). SHOT is very robust to noise, but sensitive to mesh resolution
variation. Petrelli and Di Stefano (2011) proposed a novel LRF which aimed to
estimate a repeatable LRF at the border of a range image. Zaharescu et al.
(2012) proposed a MeshHOG feature by first projecting the gradient vectors
onto three planes defined by an LRF and then calculating a two-level histogram
of these vectors.
However, none of the existing LRF definition techniques is simultaneously
unique, unambiguous, and robust to noise and mesh resolution. Besides, most of
the existing feature descriptors suffer from a number of limitations,
including a low robustness and discriminating power (Bariya et al., 2012).
### 2.2 3D Object Recognition
Most of the existing algorithms for local feature based 3D object recognition
follow a three-phase paradigm including feature matching, hypothesis
generation and verification, and pose refinement (Taati and Greenspan, 2011).
Stein and Medioni (1992) used the splash features to represent the objects and
generated hypotheses by using a set of triplets of feature correspondences.
These hypotheses are then grouped into clusters using geometric constraints.
They are finally verified through a least square calculation. Chua and Jarvis
(1997) used point signatures of a scene to match them against those of their
models. The rigid transformation between the scene and a candidate model was
then calculated using three pairs of corresponding points. Its ability to
recognize objects in both single-object and multi-object scenes has been
demonstrated. However, verifying each triplet of feature correspondences is
very time consuming. Johnson and Hebert (1999) generated point correspondences
by matching the spin images of the scene with the spin images of the models.
These point correspondences are first grouped using geometric consistency. The
groups are then used to calculate rigid transformations, which are finally be
verified. This algorithm is robust to clutter and occlusion, and capable to
recognize objects in complicated real scenes. Yamany and Farag (2002) used
surface signatures as feature descriptors and adopted a similar strategy to
Johnson and Hebert (1999) for object recognition. Mian et al. (2006b) obtained
feature correspondences and model hypothesis by matching the tensor
representations of the scene with those of the models. The hypothesis model is
then transformed to the scene and finally verified using the Iterative Closest
Point (ICP) algorithm (Besl and McKay, 1992). Experimental results revealed
that it is superior in terms of recognition rate and efficiency compared to
the spin image based algorithm. Mian et al. (2010) also developed a 3D object
recognition algorithm based on keypoint matching. This algorithm can be used
to recognize objects at different and unknown scales. Taati and Greenspan
(2011) developed a 3D object recognition algorithm based on their proposed VD-
LSD feature descriptors. The optimal VD-LSD descriptor is selected based on
the geometry of the objects and the characteristics of the range sensors.
Bariya et al. (2012) introduced a 3D object recognition algorithm based on the
EM feature descriptor and a constrained interpretation tree.
There are some algorithms in the literature which do not follow the
aforementioned three-phase paradigm. For example, Frome et al. (2004)
performed 3D object recognition using the sum of the distances between the
scene features (i.e. 3DSC) and their corresponding model features. This
algorithm is efficient. However, it is not able to segment the recognized
object from a scene, and its effectiveness on real data has not been
demonstrated. Shang and Greenspan (2010) proposed a Potential Well Space
Embedding (PWSE) algorithm for real-time 3D object recognition in sparse range
images. It cannot however handle clutter and therefore requires the objects to
be segmented a priori from the scene.
None of the existing object recognition algorithms has explicitly explored the
use of LRF to boost the performance of the recognition. Moreover, most of
these algorithms require three pairs of feature correspondences to establish a
transformation between a model and a scene. This not only increases the run
time due to the combinatorial explosion of the matching pairs, but also
decreases the precision of the estimated transformation (since the chance to
find three correct feature correspondences is much lower compared to finding
only one correct correspondence).
### 2.3 Paper Contributions
This paper is an extended version of (Guo et al., 2013a, b). It has three
major contributions, which are summarized as follows.
i) We introduce a unique, unambiguous and robust 3D LRF using all the points
lying on the local surface rather than just the mesh vertices. Therefore, our
proposed LRF is more robust to noise and varying mesh resolution. We also use
a novel sign disambiguation technique, our proposed LRF is therefore unique
and unambiguous. This LRF offers a solid foundation for effective and robust
feature description and object recognition.
ii) We introduce a highly descriptive and robust RoPS feature descriptor. RoPS
is generated by rotationally projecting the neighboring points onto three
coordinate planes and encoding the rich information of the point distribution
into a set of statistics. The proposed RoPS descriptor has been evaluated on
two datasets. Experimental results show that RoPS achieved a high power of
descriptiveness. It is shown to be robust to a number of deformations
including noise, varying mesh resolution, rotation, holes and topology
changes. (see Section 5 for details) .
iii) We introduce an efficient hierarchical 3D object recognition algorithm
based on the LRF and RoPS feature descriptor. One major advantage of our
algorithm is, a single correct feature correspondence is sufficient for object
recognition. Moreover, by integrating our robust LRF, the proposed object
recognition algorithm can work with any of the existing feature descriptors
(e.g., spin image) in the literature. Rigorous evaluations of the proposed 3D
object recognition algorithm were conducted on four different popular
datasets. Experimental results show that our algorithm achieved high
recognition rates, good efficiency and strong robustness to different
nuisances. It consistently resulted in the best recognition results on the
four datasets.
## 3 Local Reference Frame
A unique, repeatable and robust LRF is important for both effective and
efficient feature description and 3D object recognition. Advantages of such an
LRF are many fold. First, the repeatability of an LRF directly affects the
descriptiveness and robustness of the feature descriptor, i.e., an LRF with a
low repeatability will result in a poor performance of feature matching
(Petrelli and Di Stefano, 2011). Second, compared with the methods which
associate multiple descriptors to a single feature point (e.g., ISS (Zhong,
2009)), a unique LRF can help to improve both the precision and the efficiency
of feature matching (Tombari et al., 2010). Third, a robust 3D LRF helps to
boost the performance of 3D object recognition.
We propose a novel LRF by fully employing the point localization information
of the local surface. The three axes for the LRF are determined by performing
an eigenvalue decomposition on the scatter matrix of all points lying on the
local surface. The sign of each axis is disambiguated by aligning the
direction to the majority of the point scatter.
### 3.1 Coordinate Axis Construction
Given a feature point $\boldsymbol{p}$ and a support radius $r$, the local
surface mesh $S$ which contains $N$ triangles and $M$ vertices, is cropped
from the range image using a sphere of radius $r$ centered at
$\boldsymbol{p}$. For the $i$th triangle with vertices $\boldsymbol{p}_{i1}$,
$\boldsymbol{p}_{i2}$ and $\boldsymbol{p}_{i3}$, a point lying within the
triangle can be represented as:
$\boldsymbol{p}_{i}\left(s,t\right)=\boldsymbol{p}_{i1}+s(\boldsymbol{p}_{i2}-\boldsymbol{p}_{i1})+t\left(\boldsymbol{p}_{i3}-\boldsymbol{p}_{i1}\right),$
(1)
where $0\leq s,t\leq 1$, and $s+t\leq 1$, as illustrated in Fig. 1.
Figure 1: An illustration of a triangle mesh and a point lying on the surface.
An arbitrary point within a triangle can be represented by the triangle’s
vertices.
(a) Armadillo (b) Asia Dragon (c) Bunny (d) Dragon
(e) Happy Buddha (f) Thai Statue
Figure 2: The six models of the Tuning Dataset.
The scatter matrix $\mathbf{C}_{i}$ of all the points lying within the $i$th
triangle can be calculated as:
$\mathbf{C}_{i}=\frac{\int_{0}^{1}\int_{0}^{1-s}\left(\boldsymbol{p}_{i}\left(s,t\right)-\boldsymbol{p}\right)\left(\boldsymbol{p}_{i}\left(s,t\right)-\boldsymbol{p}\right)^{\textrm{T}}dtds}{\int_{0}^{1}\int_{0}^{1-s}dtds}.$
(2)
Using Eq. 1, the scatter matrix $\mathbf{C}_{i}$ be can expressed as:
$\displaystyle\mathbf{C}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\sum_{j=1}^{3}\sum_{k=1}^{3}\left(\boldsymbol{p}_{ij}-\boldsymbol{p}\right)\left(\boldsymbol{p}_{ik}-\boldsymbol{p}\right)^{\textrm{T}}$
(3)
$\displaystyle+\frac{1}{12}\sum_{j=1}^{3}\left(\boldsymbol{p}_{ij}-\boldsymbol{p}\right)\left(\boldsymbol{p}_{ij}-\boldsymbol{p}\right)^{\textrm{T}}.$
The overall scatter matrix $\mathbf{C}$ of the local surface S is calculated
as the weighted sum of the scatter matrices of all the triangles, that is:
$\mathbf{C}=\sum_{i=1}^{N}w_{i1}w_{i2}\mathbf{C}_{i},$ (4)
where $N$ is the number of triangles in the local surface $S$. Here, $w_{i1}$
is the ratio between the area of the $i$th triangle and the total area of the
local surface $S$, that is:
$w_{i1}=\frac{\left|\left(\boldsymbol{p}_{i2}-\boldsymbol{p}_{i1}\right)\times\left(\boldsymbol{p}_{i3}-\boldsymbol{p}_{i1}\right)\right|}{\sum_{i=1}^{N}\left|\left(\boldsymbol{p}_{i2}-\boldsymbol{p}_{i1}\right)\times\left(\boldsymbol{p}_{i3}-\boldsymbol{p}_{i1}\right)\right|},$
(5)
where $\times$ denotes the cross product.
$w_{i2}$ is a weight that is related to the distance from the feature point to
the centroid of the $i$th triangle, that is:
$w_{i2}=\left(r-\left|\boldsymbol{p}-\frac{\boldsymbol{p}_{i1}+\boldsymbol{p}_{i2}+\boldsymbol{p}_{i3}}{3}\right|\right)^{2}.$
(6)
Note that, the first weight $w_{i1}$ is expected to improve the robustness of
LRF to varying mesh resolutions, since a compensation with respect to the
triangle area is incorporated through this weighting. The second weight
$w_{i2}$ is expected to improve the robustness of LRF to occlusion and
clutter, since distant points will contribute less to the overall scatter
matrix.
We then perform an eigenvalue decomposition on the overall scatter matrix
$\mathbf{C}$, that is:
$\mathbf{C}\mathbf{V}=\mathbf{EV},$ (7)
where $\mathbf{E}$ is a diagonal matrix of the eigenvalues
$\left\\{\lambda_{1},\lambda_{2},\lambda_{3}\right\\}$ of the matrix
$\mathbf{C}$, and $\mathbf{V}$ contains three orthogonal eigenvectors
$\left\\{\boldsymbol{v}_{1},\boldsymbol{v}_{2},\boldsymbol{v}_{3}\right\\}$
that are in the order of decreasing magnitude of their associated eigenvalues.
The three eigenvectors offer a basis for LRF definition. However, the signs of
these vectors are numerical accidents and are not repeatable between different
trials even on the same surface (Bro et al., 2008; Tombari et al., 2010). We
therefore propose a novel sign disambiguation technique which is described in
the next subsection.
It is worth noting that, although some existing techniques also use the idea
of eigenvalue decomposition to construct the LRF (e.g., (Mian et al., 2010;
Tombari et al., 2010; Zhong, 2009)), they calculate the scatter matrix using
just the mesh vertices. Instead, our technique employs all the points in the
local surface and, is therefore more robust compared to exiting techniques (as
demonstrated in Section 3.3).
### 3.2 Sign Disambiguation
In order to eliminate the sign ambiguity of the LRF, each eigenvector should
point in the major direction of the scatter vectors (which start from the
feature point and point in the direction of the points lying on the local
surface). Therefore, the sign of each eigenvector is determined from the sign
of the inner product of the eigenvector and the scatter vectors. Specifically,
the unambiguous vector $\widetilde{\boldsymbol{v}_{1}}$ is defined as:
$\widetilde{\boldsymbol{v}_{1}}=\boldsymbol{v}_{1}\cdot\textrm{sign}\left(h\right),$
(8)
where $\mathrm{sign\left(\cdot\right)}$ denotes the signum function that
extracts the sign of a real number, and $h$ is calculated as:
$\displaystyle h$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}w_{i1}w_{i2}\left(\int_{0}^{1}\int_{0}^{1-s}\left(\boldsymbol{p}_{i}\left(s,t\right)-\boldsymbol{p}\right)\boldsymbol{v}_{1}dtds\right)$
(9) $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}w_{i1}w_{i2}\left(\frac{1}{6}\sum_{j=1}^{3}\left(\boldsymbol{p}_{ij}-\boldsymbol{p}\right)\boldsymbol{v}_{1}\right).$
Similarly, the unambiguous vector $\widetilde{\boldsymbol{v}_{3}}$ is defined
as:
$\widetilde{\boldsymbol{v}_{3}}=\boldsymbol{v}_{3}\cdot\textrm{sign}\left(\sum_{i=1}^{N}w_{i1}w_{i2}\left(\frac{1}{6}\sum_{j=1}^{3}\left(\boldsymbol{p}_{ij}-\boldsymbol{p}\right)\boldsymbol{v}_{3}\right)\right).$
(10)
Given two unambiguous vectors $\widetilde{\boldsymbol{v}_{1}}$ and
$\widetilde{\boldsymbol{v}_{3}}$, $\widetilde{\boldsymbol{v}_{2}}$ is defined
as $\widetilde{\boldsymbol{v}_{3}}\times\widetilde{\boldsymbol{v}_{1}}$.
Therefore, a unique and unambiguous 3D LRF for feature point $\boldsymbol{p}$
is finally defined. Here, $\boldsymbol{p}$ is the origin, and
$\widetilde{\boldsymbol{v}_{1}}$, $\widetilde{\boldsymbol{v}_{2}}$ and
$\widetilde{\boldsymbol{v}_{3}}$ are the $x$, $y$ and $z$ axes respectively.
With this LRF, a unique, pose invariant and highly discriminative local
feature descriptor can now be generated.
### 3.3 Performance of the Proposed LRF
To evaluate the repeatability and robustness of our proposed LRF, we
calculated the LRF errors between the corresponding points in the scenes and
models. The six models (i.e., “Armadillo”, “Asia Dragon”, “Bunny”, “Dragon”,
“Happy Buddha” and “Thai Statue”) used in this experiment were taken from the
Stanford 3D Scanning Repository (Curless and Levoy, 1996). They are shown in
Fig. 2. The six scenes were created by resampling the models down to
$\nicefrac{{1}}{{2}}$ of their original mesh resolution and then adding
Gaussian noise with a standard deviation of 0.1 mesh resolution (mr) to the
data. We refer to this dataset as the “Tuning Dataset” in the rest of this
paper.
We randomly selected 1000 points in each model and we refer to these points as
feature points. We then obtained the corresponding points in the scene by
searching the points with the smallest distances to the feature points in the
model. For each point pair
$\left(\boldsymbol{p}_{Si},\boldsymbol{p}_{Mi}\right)$, we calculated the LRFs
for both points, denoted as $\mathbf{L}_{Si}$ and $\mathbf{L}_{Mi}$,
respectively. Using the similar criterion as in (Mian et al., 2006a), the
error between two LRFs of the $i$th point pair can be calculated by:
$\epsilon_{i}=\arccos\left(\frac{\textrm{trace}\left(\mathbf{L}_{Si}\mathbf{L}_{Mi}^{-1}\right)-1}{2}\right)\frac{180}{\pi},$
(11)
where $\epsilon_{i}$ represents the amount of rotation error between two LRFs
and is zero in the case of no error.
Our proposed LRF technique was tested on the Tuning Dataset with comparison to
several existing techniques, e.g., proposed by Novatnack and Nishino (2008),
Mian et al. (2010), Tombari et al. (2010), and Petrelli and Di Stefano (2011).
We tested each LRF technique five times by randomly selecting 1000 different
point pairs each time. The overall LRF errors of each technique are shown in
Fig. 3 as a histogram. Ideally, all of the LRF errors should lie around the
zero value (in the first bin of the histogram). It is clear that our proposed
technique performed best, with 83.5% of the point pairs having LRF errors less
than 10 degrees. Whereas the second best one (i.e., proposed by Petrelli and
Di Stefano (2011)) secured only 43.2% of the point pairs with LRF errors less
than 10 degrees. Other techniques only had around 40% point pairs with LRF
errors less than 10 degrees. These results clearly indicate that our proposed
LRF is more repeatable and more robust than the state-of-the-art in the
presence of noise and mesh resolution variation.
In order to further assess the influence of a weighting strategy, we used a
distance weight
$w_{i3}=r-\left|\boldsymbol{p}-\frac{\boldsymbol{p}_{i1}+\boldsymbol{p}_{i2}+\boldsymbol{p}_{i3}}{3}\right|$
(following the approach of (Tombari et al., 2010)) to replace the weights
$w_{i1}$ and $w_{i2}$ in Equations 4, 9 and 10, resulting in a modified LRF.
The histogram of LRF errors of the modified technique is shown in Fig. 3. The
performance of the modified LRF decreased significantly compared to the
original proposed LRF. This observation reveals that the weighting strategy
using both quadratic distance weight $w_{i2}$ and area weight $w_{i1}$
produced more robust results compared to those using only a linear distance
weight $w_{i3}$.
Fig. 3 shows that part of the LRF errors of each technique are larger than 80
degrees. This is mainly due to the presence of local symmetrical surfaces
(e.g., flat or spherical surfaces) in the scenes. For a local symmetrical
surface, there is an inherent sign ambiguity of its LRF because the
distribution of points is almost the same in all directions. In order to deal
with this case, we adopt a feature point selection technique which uses the
ratio of eigenvalues to avoid local symmetrical surfaces (see Section 6.2).
Once an LRF is determined, the next step is to define a local surface
descriptor. In the next section, we propose a novel RoPS descriptor.
## 4 Local Surface Description
A local surface descriptor needs to be invariant to rotation and robust to
noise, varying mesh resolution, occlusion, clutter and other nuisances. In
this section, we propose a novel local surface feature descriptor namely RoPS
by performing local surface rotation, neighboring points projection and
statistics calculation.
### 4.1 RoPS Feature Descriptor
An illustrative example of the overall RoPS method is given in Fig. 4. From a
range image/model, a local surface is selected for a feature point
$\boldsymbol{p}$ given a support radius $r$. Figures 4(a) and (b) respectively
show a model and a local surface. We already have defined the LRF for
$\boldsymbol{p}$ and the vertices of the triangles in the local surface $S$
constitute a pointcloud
$\mathbf{Q}=\left\\{\boldsymbol{q}_{1},\boldsymbol{q}_{2},\ldots,\boldsymbol{q}_{M}\right\\}$.
The pointcloud
$\mathbf{Q}=\left\\{\boldsymbol{q}_{1},\boldsymbol{q}_{2},\ldots,\boldsymbol{q}_{M}\right\\}$
is then transformed with respect to the LRF in order to achieve rotation
invariance, resulting in a transformed pointcloud
$\mathbf{Q}^{\prime}=\left\\{\boldsymbol{q}_{1}^{\prime},\boldsymbol{q}_{2}^{\prime},\ldots,\boldsymbol{q}_{M}^{\prime}\right\\}$.
We then follow a number of steps which are described as follows.
Figure 3: Histogram of the LRF errors for the six scenes and models of the
Tuning Dataset. Our proposed technique outperformed the existing techniques by
a large margin. (Figure best seen in color.) Figure 4: An illustration of the
generation of a RoPS feature descriptor for one rotation. (a) The Armadillo
model and the local surface around a feature point. (b) The local surface is
cropped and transformed in the LRF. (c) The local surface is rotated around a
coordinate axis. (d) The neighboring points are projected onto three 2D
planes. (e) A distribution matrix is obtained for each plane by partitioning
the 2D plane into bins and counting up the number of points falling into each
bin. The dark color indicates a large number. (f) Each distribution matrix is
then encoded into several statistics. (g) The statistics from three
distribution matrices are concatenated to form a sub-feature descriptor for
one rotation. (Figure best seen in color.)
First, the pointcloud is rotated around the $x$ axis by an angle $\theta_{k}$,
resulting in a rotated pointcloud
$\mathbf{Q}^{\prime}\left(\theta_{k}\right)$, as shown in Fig. 4(c). This
pointcloud $\mathbf{Q}^{\prime}\left(\theta_{k}\right)$ is then projected onto
three coordinate planes (i.e., the $xy$, $xz$ and $yz$ planes) to obtain three
projected pointclouds
$\widetilde{\mathbf{Q}^{\prime}}_{i}\left(\theta_{k}\right),i=1,2,3$. Note
that, the projection offers a means to describe the 3D local surface in a
concise and efficient manner. That is because 2D projections clearly preserve
a certain amount of unique 3D geometric information of the local surface from
that particular viewpoint.
Next, for each projected pointcloud
$\widetilde{\mathbf{Q}^{\prime}}_{i}\left(\theta_{k}\right)$, a 2D bounding
rectangle is obtained, which is subsequently divided into $L\times L$ bins, as
shown in Fig. 4(d). The number of points falling into each bin is then counted
to yield an $L\times L$ matrix $\mathbf{D}$, as shown in Fig. 4(e). We refer
to the matrix $\mathbf{D}$ as a “distribution matrix” since it represents the
2D distribution of the neighboring points. The distribution matrix
$\mathbf{D}$ is further normalized such that the sum of all bins is equal to
one in order to achieve invariance to variations in mesh resolution.
The information in the distribution matrix $\mathbf{D}$ is further condensed
in order to achieve computational and storage efficiency. In this paper, a set
of statistics is extracted from the distribution matrix $\mathbf{D}$,
including central moments (Demi et al., 2000; Hu, 1962) and Shannon entropy
(Shannon, 1948). The central moments are utilized for their mathematical
simplicity and rich descriptiveness (Hu, 1962), while Shannon entropy is
selected for its strong power to measure the information contained in a
probability distribution (Shannon, 1948).
The central moment $\mu_{mn}$ of order $m+n$ of matrix $\mathbf{D}$ is defined
as:
$\mu_{mn}=\sum_{i=1}^{L}\sum_{j=1}^{L}\left(i-\bar{i}\right)^{m}\left(j-\bar{j}\right)^{n}\mathbf{D}\left(i,j\right),$
(12)
where
$\bar{i}=\sum_{i=1}^{L}\sum_{j=1}^{L}i\mathbf{D}\left(i,j\right),$ (13)
and
$\bar{j}=\sum_{i=1}^{L}\sum_{j=1}^{L}j\mathbf{D}\left(i,j\right).$ (14)
The Shannon entropy $e$ is calculated as:
$e=-\sum_{i=1}^{L}\sum_{j=1}^{L}\mathbf{D}\left(i,j\right)\log\left(\mathbf{D}\left(i,j\right)\right).$
(15)
Theoretically, a complete set of central moments can be used to uniquely
describe the information contained in a matrix (Hu, 1962). However in
practice, only a small subset of the central moments can sufficiently
represent the distribution matrix $\mathbf{D}$. These selected central moments
together with the Shannon entropy are then used to form a statistics vector,
as shown in Fig. 4(f). The three statistics vectors from the $xy$, $xz$ and
$yz$ planes are then concatenated to form a sub-feature
$\boldsymbol{f}_{x}\left(\theta_{k}\right)$. Note that
$\boldsymbol{f}_{x}\left(\theta_{k}\right)$ denotes the total statistics for
the $k$th rotation around the $x$ axis, as shown in Fig. 4(g).
In order to encode the “complete” information of the local surface, the
pointcloud $\mathbf{Q}^{\prime}$ is rotated around the $x$ axis by a set of
angles $\left\\{\theta_{k}\right\\},k=1,2,\ldots,T$, resulting in a set of
sub-features
$\left\\{\boldsymbol{f}_{x}\left(\theta_{k}\right)\right\\},k=1,2,\ldots,T$.
Further, $\mathbf{Q}^{\prime}$ is rotated by a set of angles around the $y$
axis and a set of sub-features
$\left\\{\boldsymbol{f}_{y}\left(\theta_{k}\right)\right\\},k=1,2,\ldots,T$ is
calculated. Finally, $\mathbf{Q}^{\prime}$ is rotated by a set of angles
around the $z$ axis and a set of sub-features
$\left\\{\boldsymbol{f}_{z}\left(\theta_{k}\right)\right\\},k=1,2,\ldots,T$ is
calculated. The overall feature descriptor is then generated by concatenating
the sub-features of all the rotations into a vector, that is:
$\boldsymbol{f}=\left\\{\boldsymbol{f}_{x}\left(\theta_{k}\right),\boldsymbol{f}_{y}\left(\theta_{k}\right),\boldsymbol{f}_{z}\left(\theta_{k}\right)\right\\},k=1,2,\ldots,T.$
(16)
It is expected that the RoPS descriptor would be highly discriminative (as
demonstrated in Section 5) since it encodes the geometric information of a
local surface from a set of viewpoints. Note that, some existing view-based
methods can be found in the literature, such as (Yamauchi et al., 2006),
(Ohbuchi et al., 2008) and (Atmosukarto and Shapiro, 2010). However, these
methods are based on global features and originate from the 3D shape retrieval
area. They are, however, not suitable for 3D object recognition due to their
sensitivity to occlusion and clutter.
Other related methods, however, include the spin image (Johnson and Hebert,
1999) and snapshot (Malassiotis and Strintzis, 2007) descriptors. A spin image
is generated by projecting a local surface onto a 2D plane using a cylindrical
parametrization. Similarly, a snapshot is obtained by rendering a local
surface from the viewpoint which is perpendicular to the surface. Our RoPS
differs from these methods in several aspects. First, RoPS represents a local
surface from a set of viewpoints rather than just one view (as in the case of
spin image and snapshot). Second, RoPS is associated with a unique and
unambiguous LRF, and it is invariant to rotation. In contrast, spin image
discards cylindrical angular information and snapshot is prone to rotation.
Third, RoPS is more compact than spin image and snapshot since RoPS further
encodes 2D matrices with a set of statistics. The typical lengths of RoPS,
spin image and snapshot are 135, 225 and 1600, respectively (see Table 2,
(Johnson and Hebert, 1999) and (Malassiotis and Strintzis, 2007)).
### 4.2 RoPS Generation Parameters
The RoPS feature descriptor has four parameters: i) the combination of
statistics, ii) the number of partition bins $L$, iii) the number of rotations
$T$ around each coordinate axis, and iv) the support radius $r$. The
performance of RoPS descriptor against different settings of these parameters
was tested on the Tuning Dataset using the criterion of Recall vs 1-Precision
Curve (RP Curve).
RP Curve is one of the most popular criteria used for the assessment of a
feature descriptor (Flint et al., 2008; Hou and Qin, 2010; Ke and Sukthankar,
2004; Mikolajczyk and Schmid, 2005). It is calculated as follows: given a
scene, a model and the ground truth transformation, a scene feature is matched
against all model features to find the closest feature. If the ratio between
the smallest distance and the second smallest one is less than a threshold,
then the scene feature and the closest model feature are considered a match.
Further, a match is considered a true positive only if the distance between
the physical locations of the two features is sufficiently small, otherwise it
is considered a false positive. Therefore, recall is defined as:
$\textrm{recall}=\frac{\textrm{the number of true positives}}{\textrm{total
number of positives}}.$ (17)
1-precision is defined as:
$\textrm{1-precision}=\frac{\textrm{the number of false
positives}}{\textrm{total number of matches}}.$ (18)
By varying the threshold, a RP Curve can be generated. Ideally, a RP Curve
would fall in the top left corner of the plot, which means that the feature
obtains both high recall and precision.
(a) (b) (c) (d)
Figure 5: Effect of the RoPS generation parameters. (a) Different combinations
of statistics. (b) The number of partition bins $L$. There is a twin plot in
(b), where the right plot is a magnified version of the region indicated by
the rectangle in the left plot. (c) The number of rotations $T$. There is a
twin plot in (c), where the right plot is a magnified version of the region
indicated by the rectangle in the left plot. (d) The support radius $r$. (We
chose the No.6 combination of the statistics and set $L=5$, $T=3$ and $r=15$mr
in this paper as a tradeoff between effectiveness and efficiency. Figure best
seen in color.)
#### 4.2.1 The Combination of Statistics
The selection of the subset of statistics plays an important role in the
generation of a RoPS feature descriptor. It determines not only the capability
for encapsulating the information in a distribution matrix but also the size
of a feature vector. We considered eight combinations of statistics (a number
of low-order moments and entropy), as listed in Table 1, and tested the
performance for each combination in the terms of RP Curve. The other three
parameters were set constant as $L=5$, $T=3$ and $r=15$mr. It is worth noting
that the zeroth-order central moment $\mu_{00}$ and the first-order central
moments $\mu_{01}$ and $\mu_{10}$ were excluded from the combinations of the
statistics. Because these moments are constant (i.e., $\mu_{00}=1$,
$\mu_{01}=0$ and $\mu_{10}=0$) and therefore contain no information of the
local surface. Our experimental results are shown in Fig. 5(a).
Table 1: Different combinations of the statistics. No. | Combination of the statistics
---|---
1 | $\mu_{02},\mu_{11},\mu_{20}$
2 | $\mu_{02},\mu_{11},\mu_{20}$,$\mu_{03},\mu_{12},\mu_{21},\mu_{30}$
3 | $\mu_{02},\mu_{11},\mu_{20}$,$\mu_{03},\mu_{12},\mu_{21},\mu_{30}$,$\mu_{04},\mu_{13},\mu_{22},\mu_{31},\mu_{40}$
4 | $\mu_{02},\mu_{11},\mu_{20}$,$\mu_{03},\mu_{12},\mu_{21},\mu_{30}$,$\mu_{04},\mu_{13},\mu_{22},\mu_{31},\mu_{40},e$
5 | $\mu_{11},\mu_{21},\mu_{12},\mu_{22}$
6 | $\mu_{11},\mu_{21},\mu_{12},\mu_{22},e$
7 | $\mu_{11},\mu_{21},\mu_{12},\mu_{22},\mu_{31},\mu_{13}$
8 | $\mu_{11},\mu_{21},\mu_{12},\mu_{22},\mu_{31},\mu_{13},e$
It is clear that the No.6 combination achieved the best performance, followed
by the No.5 combination. While the No.3, No.4 and No.8 combinations obtained
comparable performance, with recall being a little lower than the No.6
combination. The superior performance of the No.6 combination is due to the
facts that, first, the low-order moments $\mu_{11},\mu_{21},\mu_{12},\mu_{22}$
and entropy $e$ contain the most meaningful and significant information of the
distribution matrix. Consequently, the descriptiveness of these statistics is
sufficiently high. Second, the low-order moments are more robust to noise and
varying mesh resolution compared to the high-order moments. Beyond the high
precision and recall, the size of the No.6 combination is also small, which
means that the calculation and matching of feature descriptors can be
performed efficiently. Therefore, the No.6 combination, i.e.,
$\left\\{\mu_{11},\mu_{21},\mu_{12},\mu_{22},e\right\\}$, was selected to
represent the information in a distribution matrix and to form the RoPS
descriptor.
#### 4.2.2 The Number of Partition Bins
The number of partition bins $L$ is another important parameter in the RoPS
generation. It determines both the descriptiveness and robustness of a
descriptor. That is, a dense partition of the projected points offers more
details about the point distribution, it however increases the sensitivity to
noise and varying mesh resolution. We tested the performance of RoPS
descriptor on the Tuning Dataset with respect to a number of partition bin,
while the two other parameters were set to $T=3$ and $r=15$mr. The
experimental results are shown in Fig. 5(b) as a twin plot, where the right
plot is a magnified version of the region indicated by the rectangle in the
left plot.
The plot shows that the performance of RoPS descriptor improved as the number
of partition bins increased from 3 to 5. This is because more details about
the point distribution were encoded into the feature descriptor. However, for
a number of partition bins larger than 5, the performance degraded as the
number of partition bins increased. This is due to the reason that a dense
partition makes the distribution matrix more susceptible to the variation of
spatial position of the neighboring points. It can therefore be inferred that
5 is the most suitable number of partitions as a tradeoff between the
descriptiveness and the robustness to noise and varying mesh resolution. We
therefore used $L=5$ in this paper.
#### 4.2.3 The Numbers of Rotations
The number of rotations $T$ determines the “completeness” when describing the
local surface using a RoPS feature descriptor. That is, increasing the number
of rotations means that more information of the local surface are encoded into
the overall feature descriptor. We tested the performance of the RoPS feature
descriptor with respect to a varying number of rotations while keeping the
other parameters constant (i.e., $r=15$mr). The results are given in Fig. 5(c)
as a twin plot, where the right plot is a magnified version of the region
indicated by the rectangle in the left plot.
It was found that as the number of rotations increased, the descriptiveness of
the RoPS increased, resulting in an improvement of the matching performance
(which confirmed our assumption). Specifically, the performance of the RoPS
descriptor improved significantly as the number of rotations increased from 1
to 2, as shown in the left plot of Fig. 5(c). The performance then improved
slightly as the number of rotations increased from 2 to 6, as indicated in the
magnified version shown in the right plot of Fig. 5(c). In fact, there was no
notable difference between the performance with respect to the number of
rotations of 3 and 6. That is because almost all the information of the local
surface is encoded in the feature descriptor by rotating the neighboring
points 3 times around each axis. Therefore, increasing the number of rotations
any further will not necessarily add any significant information to the
feature descriptor. Moreover, increasing the number of rotations will cost
more computational and memory resources. We therefore, set the number of
rotations to be 3 in this paper.
#### 4.2.4 The Support Radius
The support radius $r$ determines the amount of surface that is encoded by the
RoPS feature descriptor. The value of $r$ can be chosen depending on how local
the feature should be, and a tradeoff lies between the feature’s
descriptiveness and robustness to occlusion. That is, a large support radius
enables the RoPS descriptor to encapsulate more information of the object and
therefore provides more descriptiveness. On the other hand, a large support
radius increases the sensitivity to occlusion and clutter. We tested the
performance of the RoPS feature descriptor with respect to varying support
radius while keeping the other parameters fixed. The results are given in Fig.
5(d).
The results show that the recall and precision performance of the RoPS feature
descriptor improved steadily as the support radius increased from 5mr (mr =
mesh resolution) to 25mr. Specifically, there was a significant improvement of
the matching performance as the support radius increased from 5mr to 10mr,
this is because a radius of 5mr is too small to contain sufficient
discriminating information of the underlying surface. The RoPS feature
descriptor achieved good results with a support radius of 15mr, achieving a
high precision of about 0.9 and a high recall of about 0.9. Although the
performance of RoPS feature descriptor further improved slightly as the
support radius was increased to 25mr, the performance deteriorated sharply
when the support radius was set to 30mr. We choose to set the support radius
to 15mr in the paper to maintain a strong robustness to occlusion and clutter.
An illustration is shown in Fig. 6. The range image contains two objects in
the presence of occlusion and clutter, and a feature point is selected near
the tail of the chicken. The red, green and blue spheres, respectively
represent the support regions with radius of 25 mr, 15mr and 5mr for the
feature point. As the radius increases from 5mr to 25 mr, points on the
surface within the support region are more likely to be missing due to
occlusion, and points from other objects (e.g., T-rex on the right) are more
likely to be included in the support region due to clutter. Therefore, the
resulting feature descriptor is more likely to be affected by occlusion and
clutter.
Figure 6: An illustration of the descriptor’s robustness to occlusion and
clutter with respect to varying support radius. The red, green and blue
spheres respectively represent the support regions with radius of 25 mr, 15mr
and 5mr for a feature point. (Figure best seen in color.)
Note that, several adaptive-scale keypoint detection methods have been
proposed for the purpose of determining the support radius based on the
inherent scale of a feature point (Tombari et al., 2013). However, we simply
adopt a fixed support radius since our focus is on feature description and
object recognition rather than keypoint detection. Moreover, our proposed RoPS
descriptor has been demonstrated to achieve an even better performance
compared to the methods with adaptive-scale keypoint detection (e.g., EM
matching and keypoint matching), as analyzed in Section 7.
## 5 Performance of the RoPS Descriptor
The descriptiveness and robustness of our proposed RoPS feature descriptor was
first evaluated on the Bologna Dataset (Tombari et al., 2010) with respect to
different levels of noise, varying mesh resolution and their combinations. It
was also evaluated on the PHOTOMESH Dataset (Zaharescu et al., 2012) with
respect to 13 transformations. In these experiments, the RoPS was compared to
several state-of-the-art feature descriptors.
### 5.1 Performance on The Bologna Dataset
#### 5.1.1 Dataset and Parameter Setting
The Bologna Dataset used in this paper comprises six models and 45 scenes. The
six models (i.e., “Armadillo”, “Asia Dragon”, “Bunny”, “Dragon”, “Happy
Buddha” and “Thai Statue”) were taken from the Stanford 3D Scanning
Repository. They are shown in Fig. 2. Each scene was synthetically generated
by randomly rotating and translating three to five models in order to create
clutter and pose variances. As a result, the ground truth rotations and
translations between each model and its instances in the scenes were known a
priori during the process of construction. An example scene is shown in Fig.
7.
Figure 7: A scene on the Bologna Dataset.
The performance of each feature descriptor was assessed using the criterion of
RP Curve (as detailed in Section 4.2). We compared our RoPS feature descriptor
with five state-of-the-art feature descriptors, including spin image (Johnson
and Hebert, 1999), normal histogram (NormHist) (Hetzel et al., 2001), LSP
(Chen and Bhanu, 2007), THRIFT (Flint et al., 2007) and SHOT (Tombari et al.,
2010). The support radius $r$ for all methods was set to be 15mr as a
compromise between the descriptiveness and the robustness to occlusion. The
parameters for generating all these feature descriptors were tuned by
optimizing the performance in terms of RP Curve on the Tuning Dataset. The
tuned parameter settings for all feature descriptors are presented in Table 2.
Table 2: Tuned parameter settings for six feature descriptors. | Support Radius | Dimensionality | Length
---|---|---|---
Spin image | 15mr | 15*15 | 225
NormHist | 15mr | 15*15 | 225
LSP | 15mr | 15*15 | 225
THRIFT | 15mr | 32*1 | 32
SHOT | 15mr | 8*2*2*10 | 320
RoPS | 15mr | 3*3*3*5 | 135
In order to avoid the impact of the keypoint detection method on feature’s
descriptiveness, we randomly selected 1000 feature points from each model, and
extracted their corresponding points from the scene. We then employed the
methods listed in Table 2 to extract feature descriptors for these feature
points. Finally, we calculated a RP Curve for each feature descriptor to
evaluate the performance.
#### 5.1.2 Robustness to Noise
(a) Noise free (b) Noise with a standard deviation of 0.1mr (c) Noise with a
standard deviation of 0.2mr (d) Noise with a standard deviation of 0.3mr
(e) Noise with a standard deviation of 0.4mr (f) Noise with a standard
deviation of 0.5mr
Figure 8: Recall vs 1-Precision curves in the presence of noise. (Figure best
seen in color.)
(a) $\nicefrac{{1}}{{2}}$ mesh decimation (b) $\nicefrac{{1}}{{4}}$ mesh
decimation (c) $\nicefrac{{1}}{{8}}$ mesh decimation (d) $\nicefrac{{1}}{{2}}$
mesh decimation and 0.1mr Gaussian noise
Figure 9: Recall vs 1-Precision curves with respect to mesh resolution.
(Figure best seen in color.)
In order to evaluate the robustness of these feature descriptors to noise, we
added a Gaussian noise with increasing standard deviation of 0.1mr, 0.2mr,
0.3mr, 0.4mr and 0.5mr to the scene data. The RP Curves under different levels
of noise are presented in Fig. 8.
We made a number of observations. i) These feature descriptors achieved
comparable performance on noise free data, with high recall together with high
precision, as shown in Fig. 8(a).
ii) With noise, our proposed RoPS feature descriptor achieved the best
performance in most cases, and is followed by SHOT. Specifically, the
performance of RoPS is better than SHOT under a low-level noise with a
standard deviation of 0.1mr, as shown in Fig. 8(b). As the standard deviation
of the noise increased to 0.2mr and 0.3mr, SHOT performed slightly better than
RoPS, as indicated in Figures 8(c) and (d). However, the performance of our
proposed RoPS was significantly better than SHOT under high levels of noise,
e.g., with a noise deviation larger than 0.3mr, as shown in Figures 8(e) and
(f). It can be inferred that RoPS is very robust to noise, particularly in the
case of scenes with a high level of noise.
iii) As the noise level increased, the performance of LSP and THRIFT
deteriorated sharply, as shown in Figures 8(b-e). THRIFT failed to work even
under a low-level of noise with a standard deviation of 0.1mr. This result is
also consistent with the conclusion given in (Flint et al., 2008). Although
NormHist and spin image worked relatively well under low- and medium-level
noise with a standard deviation less than 0.2mr, they failed completely under
noise with a large standard deviation. The sensitivity of spin image,
NormHist, THR-IFT and LSP to noise is due to the fact that, they rely on
surface normals to generate their feature descriptors. Since the calculation
of surface normal includes a process of differentiation, it is very
susceptible to noise.
iv) The strong robustness of our RoPS feature descriptor to noise can be
explained by at least three facts. First, RoPS encodes the “complete”
information of the local surface from various viewpoints through rotation and
therefore, encodes more information than the existing methods. Second, RoPS
only uses the low-order moments of the distribution matrices to form its
feature descriptor and is therefore less affected by noise. Third, our
proposed unique, unambiguous and stable LRF also helps to increase the
descriptiveness and robustness of the RoPS feature descriptor.
#### 5.1.3 Robustness to Varying Mesh Resolution
In order to evaluate the robustness of these feature descriptors to varying
mesh resolution, we resampled the noise free scene meshes to
$\nicefrac{{1}}{{2}}$, $\nicefrac{{1}}{{4}}$ and $\nicefrac{{1}}{{8}}$ of
their original mesh resolution. The RP Curves under different levels of mesh
decimation are presented in Figures 9(a-c).
It was found that our proposed RoPS feature descriptor outperformed all the
other descriptors by a large margin under all levels of mesh decimation. It is
also notable that the performance of our RoPS feature descriptor with
$\nicefrac{{1}}{{8}}$ of original mesh resolution was even comparable to the
best results given by the existing feature descriptors with
$\nicefrac{{1}}{{2}}$ of original mesh resolution. Specifically, RoPS obtained
a precision more than 0.7 and a recall more than 0.7 with
$\nicefrac{{1}}{{8}}$ of original mesh resolution, whereas spin image obtained
a precision around 0.8 and a recall around 0.8 with $\nicefrac{{1}}{{2}}$ of
original mesh resolution, as shown in Figures 9(a) and (c). This indicated
that our RoPS feature descriptor is very robust to varying mesh resolution.
The strong robustness of RoPS to varying mesh resolution is due to at least
two factors. First, the LRF of RoPS is derived by calculating the scatter
matrix of all the points lying on the local surface rather than just the
vertices, which makes RoPS robust to different mesh resolution. Second, the 2D
projection planes are sparsely partitioned and only the low-order moments are
used to form the feature descriptor, which further improves the robustness of
our method to mesh resolution.
#### 5.1.4 Robustness to Combined Noise and Mesh Decimation
In order to further test the robustness of these feature descriptors to
combined noise and mesh decimation, we resampled the scene meshes down to
$\nicefrac{{1}}{{2}}$ of their original mesh resolution and added a Gaussian
random noise with a standard deviation of 0.1mr to the scenes. The resulting
RP Curves are presented in Fig. 9(d).
As shown in Fig. 9(d), RoPS significantly outperformed the other methods in
the scenes with both noise and mesh decimation, obtaining a high precision of
about 0.9 and a high recall of about 0.9. It is followed by NormHist, SHOT,
spin image and LSP, while THRIFT failed to work.
As summarized in Table 2, the RoPS feature descriptor length is 135, while the
others such as spin image, NormHist, LSP and SHOT are 225, 225, 225 and 320,
respectively. So RoPS is more compact and therefore more efficient for feature
matching compared to these methods. Note that, although the length of THRIFT
is smaller than RoPS, THRIFT’s performance in terms of recall and precision
results is surpassed by our RoPS feature descriptor by a large margin.
### 5.2 Performance on The PHOTOMESH Dataset
The PHOTOMESH Dataset contains three null shapes. Two of the null shapes were
obtained with multi-view stereo reconstruction algorithms, and the other one
was generated with a modeling program. 13 transformations were applied to each
shape. The transformations include color noise, color shot noise, geometry
noise, geometry shot noise, rotation, scale, local scale, sampling, hole,
micro-hole, topology changes and isometry. Each transformation has five
different levels of strength.
To make a rigorous comparison with (Zaharescu et al., 2012), we set the
support radius $r$ to $\sqrt{\nicefrac{{\alpha_{r}A_{M}}}{{\pi}}}$, where
$A_{M}$ is the total area of a mesh, and $\alpha_{r}$ is 2%. RoPS feature
descriptors were calculated at all points of the shapes, without any feature
detection. We used the average normalized $L_{2}$ distance between the feature
descriptors of corresponding points to measure the quality of a feature
descriptor, as in (Zaharescu et al., 2012). The experimental results of the
RoPS descriptor are shown in Table 3. For comparison, the results of the
MeshHOG descriptor (Gaussian curvature) without and with MeshDOG are also
reported in Tables 4 and 5, respectively.
The RoPS descriptor was clearly invariant to color noise and color shot noise.
Because the geometric information used in RoPS cannot be affected by color
deformations. RoPS was also invariant to rotation and scale, which means that
it was invariant to rigid transformations.
The RoPS descriptor turned out to be very robust to geometry noise, geometry
shot noise, local scale, holes, micro-holes, topology and isometry with noise.
The average normalized $L_{2}$ distances for all these transformations were no
more than 0.06, even under the highest level of transformations. The biggest
challenge for RoPS descriptor was sampling. The average normalized $L_{2}$
distance increased from 0.01 to 0.06 as the strength level changed from 1 to
5. However, RoPS was more robust to sampling than MeshHOG. As shown in Tables
3 and 4, the average normalized $L_{2}$ distance of RoPS with a strength level
of 5 was even smaller than that of MeshHOG with a strength level of 1, i.e.,
0.02 and 0.04, respectively. Overall, the average normalized $L_{2}$ distances
of RoPS descriptor were much smaller under all strength levels of all
transformations compared to MeshHOG.
Table 3: Robustness of RoPS descriptor. | Strength
---|---
Transform. | 1 | $\leq$2 | $\leq$3 | $\leq$4 | $\leq$5
Color Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Color Shot Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Geometry Noise | 0.01 | 0.01 | 0.01 | 0.02 | 0.02
Geometry Shot Noise | 0.01 | 0.01 | 0.02 | 0.03 | 0.05
Rotation | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Scale | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Local Scale | 0.01 | 0.01 | 0.02 | 0.02 | 0.02
Sampling | 0.01 | 0.02 | 0.04 | 0.05 | 0.06
Holes | 0.01 | 0.01 | 0.01 | 0.01 | 0.02
Marco-Holes | 0.00 | 0.01 | 0.01 | 0.01 | 0.01
Topology | 0.01 | 0.01 | 0.02 | 0.02 | 0.03
Isometry + Noise | 0.02 | 0.02 | 0.01 | 0.02 | 0.02
Average | 0.00 | 0.01 | 0.01 | 0.02 | 0.02
Table 4: Robustness of MeshHOG (Gaussian curvature) without MeshDOG detector. | Strength
---|---
Transform. | 1 | $\leq$2 | $\leq$3 | $\leq$4 | $\leq$5
Color Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Color Shot Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Geometry Noise | 0.07 | 0.08 | 0.09 | 0.10 | 0.11
Geometry Shot Noise | 0.02 | 0.03 | 0.05 | 0.06 | 0.09
Rotation | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Scale | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Local Scale | 0.06 | 0.07 | 0.08 | 0.09 | 0.10
Sampling | 0.10 | 0.12 | 0.13 | 0.13 | 0.13
Holes | 0.01 | 0.02 | 0.04 | 0.03 | 0.05
Marco-Holes | 0.01 | 0.01 | 0.03 | 0.04 | 0.04
Topology | 0.07 | 0.10 | 0.11 | 0.11 | 0.12
Isometry + Noise | 0.08 | 0.08 | 0.08 | 0.09 | 0.09
Average | 0.04 | 0.04 | 0.05 | 0.06 | 0.06
## 6 3D Object Recognition Algorithm
So far we have developed a novel LRF and a RoPS feature descriptor. In this
section, we propose a new hierarchical 3D object recognition algorithm based
on the LRF and RoPS descriptor. Our 3D object recognition algorithm consists
of four major modules, i.e., model representation, candidate model generation,
transformation hypothesis generation, verification and segmentation. A flow
chart illustration of the algorithm is given in Fig. 10.
Table 5: Robustness of MeshHOG (Gaussian curvature) with MeshDOG detector. | Strength
---|---
Transform. | 1 | $\leq$2 | $\leq$3 | $\leq$4 | $\leq$5
Color Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Color Shot Noise | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Geometry Noise | 0.26 | 0.29 | 0.31 | 0.33 | 0.34
Geometry Shot Noise | 0.04 | 0.09 | 0.14 | 0.21 | 0.29
Rotation | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Scale | 0.01 | 0.01 | 0.01 | 0.01 | 0.00
Local Scale | 0.21 | 0.25 | 0.28 | 0.30 | 0.31
Sampling | 0.31 | 0.34 | 0.34 | 0.36 | 0.36
Holes | 0.02 | 0.02 | 0.07 | 0.07 | 0.07
Marco-Holes | 0.01 | 0.01 | 0.07 | 0.07 | 0.08
Topology | 0.13 | 0.20 | 0.22 | 0.25 | 0.28
Isometry + Noise | 0.23 | 0.24 | 0.22 | 0.25 | 0.25
Average | 0.10 | 0.12 | 0.14 | 0.15 | 0.17
Figure 10: Flow chart of the 3D object recognition algorithm. The module of
model representation is performed offline, and the other modules are operated
online.
### 6.1 Model Representation
We first construct a model library for the 3D objects that we are interested
in. Given a model $\mathsf{\mathscr{\mathcal{M}}}$, $N_{m}$ seed points are
evenly selected from the model pointcloud. Since the feature descriptors of
closely located feature points may be similar (since they represent more or
less the same local surface), a resolution control strategy (Zhong, 2009) is
further enforced on these seed points to extract the final feature points. For
each feature point $\boldsymbol{p}_{m}$, the LRF $\mathbf{F}_{m}$ and the
feature descriptor (e.g., our RoPS descriptor) $\boldsymbol{f}_{m}$ are
calculated. The point position $\boldsymbol{p}_{m}$, LRF $\mathbf{F}_{m}$ and
feature descriptor $\boldsymbol{f}_{m}$ of all the feature points are then
stored in a library for object recognition.
In order to speed up the process of feature matching during online
recognition, the local feature descriptors from all models are indexed using a
$k$-d tree method (Bentley, 1975). Note that, the model feature calculation
and indexing can be performed offline, while the following modules are
operated online.
### 6.2 Candidate Model Generation
The input scene $\mathcal{S}$ is first decimated, which results in a low
resolution mesh $\mathcal{S}^{\prime}$. The vertices of $\mathcal{S}$ which
are nearest to the vertices of $\mathcal{S}^{\prime}$ are selected as seed
points (following a similar approach of (Mian et al., 2006b)). Next, a
resolution control strategy (Zhong, 2009) is enforced on these seed points to
prune out redundant seed points. A boundary checking strategy (Mian et al.,
2010) is also applied to the seed points to eliminate the boundary points of
the range image. Further, since the LRF of a point can be ambiguous when two
eigenvalues of the overall scatter matrix of the underlying local surface (see
Eq. 4) are equal, we impose a constraint on the ratios of the eigenvalues
$\nicefrac{{\lambda_{1}}}{{\lambda_{2}}}>\tau_{\lambda}$ to exclude seed
points with symmetrical local surfaces, as in (Zhong, 2009; Mian et al.,
2010). The remaining seed points are considered feature points. It is worth
noting that, the feature point detection and LRF calculation procedures can be
performed simultaneously. Given the LRF $\mathbf{F}_{s}$ of a feature point
$\boldsymbol{p}_{s}$, its feature descriptor $\boldsymbol{f}_{s}$ is
subsequently calculated.
The scene features are exactly matched against all model features in the
library using the previously constructed $k$-d tree. If the ratio between the
smallest distance and the second smallest one is less than a threshold
$\tau_{f}$, the scene feature and its closest model feature are considered a
feature correspondence. Each feature correspondence votes for a model. These
models which have received votes from feature correspondences are considered
candidate models. They are then ranked according to the number of votes
received. With this ranked models, the subsequent steps (Sections 6.3 and 6.4)
can be performed from the most likely candidate model.
### 6.3 Transformation hypothesis Generation
For a feature correspondence which votes for the model
$\mathsf{\mathscr{\mathcal{M}}}$, a rigid transformation is calculated by
aligning the LRF of the model feature to the LRF of the scene feature.
Specifically, given the LRF $\mathbf{F}_{s}$ and the point position
$\boldsymbol{p}_{s}$ of a scene feature, the LRF $\mathbf{F}_{m}$ and the
point position $\boldsymbol{p}_{m}$ of a corresponding model feature, the
rigid transformation can be estimated by:
$\mathbf{R}=\mathbf{F}_{s}^{\mathrm{T}}\mathbf{F}_{m},$ (19)
$\boldsymbol{t}=\boldsymbol{p}_{s}-\boldsymbol{p}_{m}\mathbf{R},$ (20)
where $\mathbf{R}$ is the rotation matrix and $\boldsymbol{t}$ is the
translation vector of the rigid transformation. It is worth noting that a
transformation can be estimated from a single feature correspondence using our
RoPS feature descriptor. This is a major advantage of our algorithm compared
with most of the existing algorithms (e.g., splash, point signatures and spin
image based methods) which require at least three correspondences to calculate
a transformation (Johnson and Hebert, 1999). Our algorithm not only eliminates
the combinatorial explosion of feature correspondences but also improves the
reliability of the estimated transformation.
As all the plausible transformations
$\left(\mathbf{R}_{i},\boldsymbol{t}_{i}\right),i=1,2,\cdots,N_{t}$ between
the scene $\mathcal{S}$ and the model $\mathsf{\mathscr{\mathcal{M}}}$ are
calculated, these transformations are then grouped into several clusters.
Specifically, for each plausible transformation, its rotation matrix
$\mathbf{R}_{i}$ is first converted into three Euler angles which form a
vector $\boldsymbol{u}_{i}$. In this manner, the difference between any two
rotation matrices can be measured by the Euclidean distance between their
corresponding Euler angles. These transformations whose Euler angles are
around $\boldsymbol{u}_{i}$ (with distances less than $\tau_{a}$) and
translations are around $\boldsymbol{t}_{i}$ (with distances less than
$\tau_{t}$) are grouped into a cluster $\mathcal{C}_{i}$. Therefore, each
plausible transformation $\left(\mathbf{R}_{i},\boldsymbol{t}_{i}\right)$
results in a cluster $\mathcal{C}_{i}$. The cluster center
$\left(\mathbf{R}_{c},\boldsymbol{t}_{c}\right)$ of $\mathcal{C}_{i}$ is
calculated as the average rotation and translation in that cluster. Next, a
confidence score $s_{c}$ for each cluster is calculated as:
$s_{c}=\frac{n_{f}}{d},$ (21)
where $n_{f}$ is the number of feature correspondences in the cluster, and $d$
is the average distance between the scene features and their corresponding
model features which fall within the cluster. These clusters are sorted
according to their confidence scores, the ones with confidence scores smaller
than half of the maximum score are first pruned out. We then select the valid
clusters from these remaining clusters, starting from the highest scored one
and discarding the nearby clusters whose distances to these selected clusters
are small (using $\tau_{a}$ and $\tau_{t}$). $\tau_{a}$ and $\tau_{t}$ are
empirically set to 0.2 and 30mr throughout this paper. These selected clusters
are then allowed to proceed to the final verification and segmentation stage
(Section 6.4).
### 6.4 Verification and Segmentation
Given a scene $\mathcal{S}$, a candidate model
$\mathsf{\mathscr{\mathcal{M}}}$ and a transformation hypothesis
$\left(\mathbf{R}_{c},\boldsymbol{t}_{c}\right)$, the model
$\mathsf{\mathscr{\mathcal{M}}}$ is first transformed to the scene
$\mathcal{S}$ by using the transformation hypothesis
$\left(\mathbf{R}_{c},\boldsymbol{t}_{c}\right)$. This transformation is
further refined using the ICP algorithm (Besl and McKay, 1992), resulting in a
residual error $\varepsilon$. After ICP refinement, the visible proportion
$\alpha$ is calculated as:
$\alpha=\frac{n_{c}}{n_{s}},$ (22)
where $n_{c}$ is the number of corresponding points between the scene
$\mathcal{S}$ and the model $\mathsf{\mathscr{\mathcal{M}}}$, $n_{s}$ is the
total number of points in the scene $\mathcal{S}$. Here, a scene point and a
transformed model point are considered corresponding if their distance is less
than twice the model resolution (Mian et al., 2006b).
The candidate model $\mathsf{\mathscr{\mathcal{M}}}$ and the transformation
hypothesis $\left(\mathbf{R}_{c},\boldsymbol{t}_{c}\right)$ are accepted as
being correct only if the residual error $\varepsilon$ is smaller than a
threshold $\tau_{\varepsilon}$ and the proportion $\alpha$ is larger than a
threshold $\tau_{\alpha}$. However, it is hard to determine the thresholds.
Because selecting strict thresholds will reject correct hypotheses which are
highly occluded in the scene, while selecting loose thresholds will produce
many false positives. In this paper, a flexible thresholding scheme is
developed. To deal with a highly occluded but well aligned object, we select a
small error threshold $\tau_{\varepsilon 1}$ together with a small proportion
threshold $\tau_{\alpha 1}$. Meanwhile, in order to increase the tolerance to
the residual error which resulted from an inaccurate estimation of the
transformation, we select a relatively larger error threshold
$\tau_{\varepsilon 2}$ together with a larger proportion threshold
$\tau_{\alpha 2}$. We chose these thresholds empirically and set them as
$\tau_{\varepsilon 1}=0.75\textrm{mr}$, $\tau_{\varepsilon 2}=1.5\textrm{mr}$,
$\tau_{\alpha 1}=0.04$ and $\tau_{\alpha 2}=0.2$ throughout the paper.
Therefore, once $\varepsilon<\tau_{\varepsilon 1}$ but $\alpha>\tau_{\alpha
1}$, or $\varepsilon<\tau_{\varepsilon 2}$ but $\alpha>\tau_{\alpha 2}$, the
candidate model $\mathsf{\mathscr{\mathcal{M}}}$ and the transformation
hypothesis $\left(\mathbf{R}_{c},\boldsymbol{t}_{c}\right)$ are accepted, the
scene points which correspond to this model are removed from the scene.
Otherwise, this transformation hypothesis is rejected and the next
transformation hypothesis is verified by turn. If no transformation hypothesis
results in an accurate alignment, we conclude that the model
$\mathsf{\mathscr{\mathcal{M}}}$ is not present in the scene $\mathcal{S}$.
While if more than one transformation hypotheses are accepted, it means that
multiple instances of the model $\mathsf{\mathscr{\mathcal{M}}}$ are present
in the scene $\mathcal{S}$.
Once all the transformation hypotheses for a candidate model
$\mathsf{\mathscr{\mathcal{M}}}$ are tested, the object recognition algorithm
then proceeds to the next candidate model. This process continues until either
all the candidate models have been verified or there are too few points left
in the scene for recognition.
## 7 Performance of 3D Object Recognition
The effectiveness of our proposed RoPS based 3D object recognition algorithm
was evaluated by a set of experiments on four datasets, including the Bologna
Dataset (Tombari et al., 2010), the UWA Dataset (Mian et al., 2006b), the
Queen’s Dataset (Taati and Greenspan, 2011) and the Ca’ Foscari Venezia
Dataset (Rodolà et al., 2012). These four datasets are amongst the most
popular datasets publicly available, containing multiple objects in each scene
in the presence of occlusion and clutter.
(a) Recognition rates in the presence of noise (b) Recognition rates with
respect to varying mesh resolution
Figure 11: Recognition rates on the Bologna Dataset. (Figure best seen in
color.)
### 7.1 Recognition Results on The Bologna Dataset
We used the Bologna Dataset to evaluate the effectiveness of our proposed RoPS
based 3D object recognition algorithm. We specifically focused on the
performance with respect to noise and varying mesh resolution. We also aimed
to demonstrate the capability of our 3D object recognition algorithm to
integrate the existing feature descriptors without LRF.
(a) Chef (b) Chicken (c) Parasaurolophus (d) Rhino
(e) T-Rex
Figure 12: The five models of the UWA Dataset.
(a) The first sample scene (b) Our recognition result (c) The second sample
scene (d) Our recognition result
Figure 13: Two sample scenes and our recognition results on the UWA Dataset.
The correctly recognized objects have been superimposed by their 3D complete
models from the library. All objects were correctly recognized except for the
T-Rex in (d). (Figure best seen in color.)
We used our RoPS together with the five feature descriptors (as detailed in
Section 5.1.1) to perform object recognition. For feature descriptors that do
not have a dedicated LRF, e.g., spin image, NormHist, LSP and THRIFT, the LRFs
were defined using our proposed technique. The average number of detected
feature points in an unsampled scene and a model were 985 and 1000,
respectively.
In order to evaluate the performance of the 3D object recognition algorithms
on noisy data, we added a Gaussian noise with increasing standard deviation of
0.1mr, 0.2mr, 0.3mr, 0.4mr and 0.5mr to each scene data, the average
recognition rates of the six algorithms on the 45 scenes are shown in Fig.
11(a). It can be seen that both RoPS and SHOT based algorithms achieved the
best results, with recognition rates of 100% under all levels of noise. Spin
image and NormHist based algorithms achieved recognition rates higher than 97%
under low-level noise with deviations less than 0.1mr. However, their
performance deteriorated sharply as the noise increased. While LSP and THRIFT
based algorithms were very sensitive to noise.
In order to evaluate the effectiveness of the 3D object recognition algorithms
with respect to varying mesh resolution, the 45 noise free scenes were
resampled to $\nicefrac{{1}}{{2}}$, $\nicefrac{{1}}{{4}}$ and
$\nicefrac{{1}}{{8}}$ of their original mesh resolution. The average
recognition rates on the 45 scenes with respect to different mesh resolutions
are given in Fig. 11(b). It is shown that RoPS based algorithm achieved the
best performance, obtaining 100% recognition rate under all levels of mesh
decimation. It was followed by NormHist and spin image based algorithms. That
is, they obtained recognition rates of 97.8% and 91.1% respectively in scenes
with $\nicefrac{{1}}{{8}}$ of original mesh resolution.
### 7.2 Recognition Results on The UWA Dataset
The UWA Dataset contains five 3D models and 50 real scenes. The scenes were
generated by randomly placing four or five real objects together in a scene
and scanned from a single viewpoint using a Minolta Vivid 910 scanner. An
illustration of the five models is given in Fig. 12, and two sample scenes are
shown in Figures 13(a) and (c).
For the sake of consistency in comparison, RoPS based 3D object recognition
experiments were performed on the same data as Mian et al. (2006b) and Bariya
et al. (2012). Besides, the Rhino model was excluded from the recognition
results, since it contained large holes and cannot be recognized by the spin
image based algorithm in any of the scenes. Comparison was performed with a
number of state-of-the-art algorithms, such as tensor (Mian et al., 2006b),
spin image (Mian et al., 2006b), keypoint (Mian et al., 2010), VD-LSD (Taati
and Greenspan, 2011) and EM based (Bariya et al., 2012) algorithms. Comparison
results are shown in Fig. 14 with respect to varying levels of occlusion. The
average number of detected feature points in a scene and a model were 2259 and
4247, respectively.
Occlusion is defined according to Johnson and Hebert (1999) as:
$\textrm{occlusion}=\frac{\textrm{model surface patch area in
scene}}{\textrm{total model surface area}}.$ (23)
The ground truth occlusion values were automatically calculated for the
correctly recognized objects and manually calculated for the objects which
were not correctly recognized. As shown in Fig. 14, our RoPS based algorithm
outperformed all the existing algorithms. It achieved a recognition rate of
100% with up to 80% occlusion, and a recognition rate of 93.1% even under 85%
occlusion. The average recognition rate of our RoPS based algorithm was 98.8%,
while the average recognition rate of spin image, tensor and EM based
algorithms were 87.8%, 96.6% and 97.5% respectively, with up to 84% occlusion.
The overall average recognition rate of our RoPS based algorithm was 98.9%.
Moreover, no false positive occurred in the experiments when using our RoPS
based algorithm, and only two out of the total 188 objects in the 50 scenes
was not correctly recognized. These results confirm that our RoPS based
algorithm is able to recognize objects in complex scenes in the presence of
significant clutter, occlusion and mesh resolution variation.
Figure 14: Recognition rates on the UWA Dataset. (Figure best seen in color.)
Two sample scenes and their corresponding recognition results are shown in
Fig. 13. All objects were correctly recognized and their poses were accurately
recovered except for the T-Rex in Fig. 13(d). The reason for the failure in
Fig. 13(d) relates to the excessive occlusion of the T-Rex. It is highly
occluded and the visible surface is sparsely distributed in several parts of
the body rather than in a single area. Therefore, almost no reliable feature
could be extracted from the object.
(a) Angle (b) Big Bird (c) Gnome (d) Kid
(e) Zoe
Figure 15: The five models in the Queen’s Dataset.
(a) The first sample scene (b) Our recognition result (c) The second sample
scene (d) Our recognition result
Figure 16: Two sample scenes and our recognition results on the Queen’s
dataset. The correctly recognized objects have been superimposed by their 3D
complete models from the library. All objects were correctly recognized except
for the Angle in (d). (Figure best seen in color.)
Note that, although we used a fixed support radius (i.e., $r$ = 15mr) for
feature description throughout this paper, the proposed algorithm is generic,
and different adaptive-scale keypoint detection methods can be seamlessly
integrated within our RoPS descriptor. In order to further demonstrate the
generic nature of our algorithm, we generated RoPS descriptors using the
support radii estimated by the adaptive-scale method in (Mian et al., 2010).
The recognition result is shown in Fig. 14. The recognition performance of the
adaptive-scale RoPS based algorithm was better than that reported in (Mian et
al., 2010), which means that our RoPS descriptor was more descriptive than the
descriptor used in (Mian et al., 2010). It is also observed that the
performance of adaptive-scale RoPS was marginally worse than the fixed-scale
counterpart. This is because the errors of scale estimation adversely affected
the performance of feature matching, and ultimately object recognition. That
is, the corresponding points in a scene and model may have different estimated
scales due to the estimation errors. As reported in (Tombari et al., 2013),
the scale repeatability of the adaptive-scale detector in (Mian et al., 2010)
were less than 85% and 60% on the Retrieval dataset and Random Views dataset,
respectively.
### 7.3 Recognition Results on The Queen’s Dataset
The Queen’s Dataset contains five models and 80 real scenes. The 80 scenes
were generated by randomly placing one, three, four or five of the models in a
scene and scanned from a single viewpoint using a LIDAR sensor. The five
models were generated by merging several range images of a single object.
Since all scenes and models were represented in the form of pointclouds, we
first converted them into triangular meshes in order to calculate the LRFs
using our proposed technique. A scene pointcloud was converted by mapping the
3D pointcloud onto the 2D retina plane of the sensor and performing a 2D
Delaunay triangulation over the mapped points. The 2D points and triangles
were then mapped back to the 3D space, resulting in a triangular mesh. A model
pointcloud was converted into a triangular mesh using the Marching Cubes
algorithm (Guennebaud and Gross, 2007). An illustration of the five models is
given in Fig. 15, and two sample scenes are shown in Figures 16(a) and (c).
Table 6: Recognition rates (%) on the Queen’s Dataset. The results of the tests on the full dataset containing 80 scenes are shown in parentheses. The others were tested on a subset dataset which contains 55 scenes. ‘NA’ indicates that the corresponding item is not available. The best results are in bold fonts. Method | Angel | Big Bird | Gnome | Kid | Zoe | Average
---|---|---|---|---|---|---
RoPS | 97.4 (97.9) | 100.0 (100.0) | 97.4 (97.9) | 94.9 (95.8) | 87.2 (85.4) | 95.4 (95.4)
EM | NA (77.1) | NA (87.5) | NA (87.5) | NA (83.3) | NA (76.6) | 81.9 (82.4)
VD-LSD(SQ) | 89.7 | 100.0 | 70.5 | 84.6 | 71.8 | 83.8
VD-LSD(VQ) | 56.4 | 97.4 | 69.2 | 51.3 | 64.1 | 67.7
3DSC | 53.8 | 84.6 | 61.5 | 53.8 | 56.4 | 62.1
Spin image (impr.) | 53.8 | 84.6 | 38.5 | 51.3 | 41.0 | 53.8
Spin image (orig.) | 15.4 | 64.1 | 25.6 | 43.6 | 28.2 | 35.4
Spin image spherical (impr.) | 53.8 | 74.4 | 38.5 | 61.5 | 43.6 | 54.4
Spin image spherical (orig.) | 12.8 | 61.5 | 30.8 | 43.6 | 30.8 | 35.9
First, we performed object recognition using our RoPS based algorithm on the
full dataset which contains 80 real scenes. The average number of detected
feature points in a scene and a model were 3296 and 4993, respectively. The
results are shown in parentheses in Table 6, with a comparison to the results
given by Bariya et al. (2012). It can be seen that the average recognition
rate of our algorithm is 95.4%, in contrast, the average recognition rate of
the EM based algorithm is 82.4%. These results indicate that our algorithm is
superior to the EM based algorithm although a complicated keypoint detection
and scale selection strategy has been adopted by the EM based algorithm.
To make a direct comparison with the results given by Taati and Greenspan
(2011), we performed our RoPS based 3D object recognition on the same subset
dataset which contains 55 scenes. The results are given in Table 6, with
comparisons to the results provided by two variants of VD-LSD, 3DSC and four
variants of spin image. As shown in Table. 6, our average recognition rate was
95.4%, while the second best result achieved by VD-LSD (SQ) was 83.8%. The
RoPS based algorithm achieved the best recognition rates for all the five
models. More than 97% of the instances of Angle, Big Bird and Gnome were
correctly recognized. Although RoPS’s recognition rate for Zoe was relatively
low (i.e., 87.2%), it still outperformed the existing algorithms by a large
margin, since the second best result achieved by VD-LSD (SQ) was 71.8%. Fig.
16 shows two sample scenes and our recognition results on the Queen’s Dataset.
It can be seen that our RoPS based algorithm was able to recognize objects
with large amounts of occlusion and clutter.
Note that, the Queen’s Dataset is more challenging than the UWA Dataset since
the former is more noisy and the points are not uniformly distributed. That is
the reason why the spin image based algorithm had a significant drop in the
recognition performance when tested on the two datasets. Specifically, the
average recognition rate of spin image based algorithm on the UWA Dataset was
87.8% while the best result on the Queen’s Dataset was only 54.4%. Similarly,
a notable decrease of performance can also be found for the EM based
algorithm, with 97.5% recognition rate for the UWA Dataset and 81.9%
recognition rate for the Queen’s Dataset. However, our RoPS based algorithm
was consistently effective and robust to different kinds of variations
(including noise, varying mesh resolution and occlusion), it outperformed the
existing algorithms and achieved comparable results in both datasets,
obtaining a recognition rate of 98.9% on the UWA Dataset and 95.4% on the
Queen’s Dataset.
We also performed a timing experiment to measure the average processing time
to recognize each object in the scene. The experiment was conducted on a
computer with a 3.16 GHz Intel Core2 Duo CPU and a 4GB RAM. The code was
implemented in MATLAB without using any program optimization or parallel
computing technique. The average computational time to detect feature points
and calculate LRFs was 42.6s. The average computational time to generate RoPS
descriptors was 7.2s. Feature matching consumed 46.6s, while the computational
time for the transformation hypothesis generation was negligible. Finally,
verification and segmentation cost 57.4s in average.
### 7.4 Recognition Results on The Ca’ Foscari Venezia Dataset
This dataset is composed of 20 models and 150 scenes. Each scene contains 3 to
5 objects in the presence of occlusion and clutter. Totally, there are 497
object instances in all scenes. This dataset has been released just recently.
It is the largest available 3D object recognition dataset. It is also more
challenging than many other datasets, containing several models with large
flat and featureless areas, and several models which are very similar in shape
(Rodolà et al., 2012).
Table 7: Precision and recall values on the Ca’ Foscari Venezia Dataset. The best results are in bold fonts. | | Armadillo | Bunny | Cat1 | Centaur1 | Chef | Chicken | Dog7 | Dragon | Face
---|---|---|---|---|---|---|---|---|---|---
Precision | RoPS | 97 | 100 | 100 | 100 | 100 | 97 | 100 | 100 | 100
Game-theoretic | 100 | 100 | 78 | 96 | 93 | 93 | 95 | 100 | 91
Recall | RoPS | 100 | 100 | 44 | 100 | 100 | 100 | 91 | 100 | 100
Game-theoretic | 97 | 97 | 82 | 100 | 100 | 100 | 86 | 89 | 95
| | Ganesha | Gorilla0 | Horse7 | Lioness13 | Para | Rhino | T-Rex | Victoria3 | Wolf2
Precision | RoPS | 100 | 100 | 100 | 100 | 97 | 96 | 100 | 100 | 100
Game-theoretic | 89 | 95 | 97 | 88 | 97 | 91 | 97 | 83 | 82
Recall | RoPS | 100 | 100 | 100 | 100 | 97 | 100 | 100 | 95 | 100
Game-theoretic | 100 | 91 | 100 | 100 | 94 | 91 | 97 | 83 | 95
The precision and recall values of RoPS based algorithm on this dataset is
shown in Table 7, the results as reported in (Rodolà et al., 2012) are also
reported for comparison. As in (Rodolà et al., 2012), two out of the 20 models
were left out from the recognition tests and used as clutter. The average
number of detected feature points in a scene and a model were 2210 and 5000,
respectively. The RoPS based algorithm achieved better precision results
compared to (Rodolà et al., 2012). The average precision of RoPS based
algorithm was 99%, which was higher than (Rodolà et al., 2012) by a margin of
6%. Besides, the precision values of 14 individual models were as high as
100%.
The average recall of RoPS based algorithm was 96%, in contrast, the average
recall of (Rodolà et al., 2012) was 95%. Moreover, RoPS based algorithm
achieved equal or better recall values on 17 individual models out of the 18
models. Note that, SHOT descriptors and a game-theoretic framework is used in
(Rodolà et al., 2012) for 3D object recognition. It is observed that our RoPS
based algorithm performed better than SHOT based algorithm on this Dataset.
In summary, the superior performance of our RoPS based 3D object recognition
algorithm is due to several reasons. First, the highly descriptiveness and
strong robustness of our RoPS feature descriptor improve the accuracy of
feature matching and therefore boost the performance of 3D object recognition.
Second, the unique, repeatable and robust LRF enables the estimation of a
rigid transformation from a single feature correspondence, which therefore
reduces the errors of transformation hypotheses. This is because the
probability of selecting only one correct feature correspondence is much
higher than the probability of selecting three correct feature
correspondences. Moreover, our proposed hierarchical object recognition
algorithm enables object recognition to be performed in an effective and
efficient manner.
## 8 Conclusion
In this paper, we proposed a novel RoPS feature descriptor for 3D local
surface description, and a new hierarchical RoPS based algorithm for 3D object
recognition. The RoPS feature descriptor is generated by rotationally
projecting the neighboring points around a feature point onto three coordinate
planes and calculating the statistics of the distribution of the projected
points. We also proposed a novel LRF by calculating the scatter matrix of all
points lying on the local surface rather than just mesh vertices. The unique
and highly repeatable LRF facilitates the effectiveness and robustness of the
RoPS descriptor.
We performed a set of experiments to assess our RoPS feature descriptor with
respect to a set of different nuisances including noise, varying mesh
resolution and holes. Comparative experimental results show that our RoPS
descriptor outperforms the state-of-the-art methods, obtaining high
descriptiveness and strong robustness to noise, varying mesh resolution and
other deformations.
Moreover, we performed extensive experiments for 3D object recognition in
complex scenes in the presence of noise, varying mesh resolution, clutter and
occlusion. Experimental results on the Bologna Dataset show that our RoPS
based algorithm is very effective and robust to noise and mesh resolution
variation. Experimental results on the UWA Dataset show that RoPS based
algorithm is very robust to occlusion and outperforms existing algorithms. The
recognition results achieved on the Queen’s Dataset show that our algorithm
outperforms the state-of-the-art algorithms by a large margin. The RoPS based
algorithm was further tested on the largest available 3D object recognition
dataset (i.e., the Ca’ Foscari Venezia Dataset), reporting superior results.
Overall, our algorithm has achieved significant improvements over the existing
3D object recognition algorithms when tested on the same dataset.
Interesting future research directions include the extension of the proposed
RoPS feature to encode both geometric and photometric information. Integrating
geometric and photometric cues would be beneficial for the recognition of 3D
objects with poor geometric but rich photometric features (e.g., a flat or
spherical surface). Another direction is to adopt our RoPS descriptors to
perform 3D shape retrieval on a large scale 3D shape corpus, e.g., the SHREC
Datasets (Bronstein et al., 2010b).
###### Acknowledgements.
The authors would like to acknowledge the following institutions. Stanford
University for providing the 3D models; Bologna University for providing the
3D scenes; INRIA for providing the PHOTOMESH Dataset; Queen’s University for
providing the 3D models and scenes; Università Ca’ Foscari Venezia for
providing the 3D models and scenes. The authors also acknowledge A. Zaharescu
from Aimetis Corporation for the results on the PHOTOMESH Dataset shown in
Tables 3 and 4.
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|
arxiv-papers
| 2013-04-11T04:26:52 |
2024-09-04T02:49:44.161535
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yulan Guo, Ferdous Sohel, Mohammed Bennamoun, Min Lu, Jianwei Wan",
"submitter": "Yulan Guo",
"url": "https://arxiv.org/abs/1304.3192"
}
|
1304.3244
|
# RX J1301.9+2747: A Highly Variable Seyfert Galaxy with Extremely Soft X-ray
Emission
Luming Sun11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei, Anhui 230026; [email protected], [email protected],
[email protected] , Xinwen Shu11affiliation: CAS Key Laboratory for Research
in Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui 230026; [email protected],
[email protected], [email protected] , and Tinggui Wang11affiliation:
CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei, Anhui 230026;
[email protected], [email protected], [email protected]
###### Abstract
In this paper we present a temporal and spectral analysis of X-ray data from
the XMM-Newton and Chandra observations of the ultrasoft and variable Seyfert
galaxy RX J1301.9+2747. In both observations the source clearly displays two
distinct states in the X-ray band, a long quiescent state and a short flare
(or eruptive) state which differs in count rates by a factor of 5–7. The
transition from quiescent to flare state occurs in 1–2 ks. We have observed
that the quiescent state spectrum is unprecedentedly steep with a photon index
$\Gamma\sim 7.1$, and the spectrum of the flare state is flatter with
$\Gamma\sim 4.4$. X-rays above 2 keV were not significantly detected in either
state. In the quiescent state, the spectrum appears to be dominated by a black
body component of temperature about $\sim$30–40 eV, which is comparable to the
expected maximum effective temperature from the inner accretion disk. The
quiescent state however, requires an additional steep power-law, presumably
arising from the Comptonization by transient heated electrons. Optical
spectrum from the Sloan Digital Sky Survey shows Seyfert-like narrow lines for
RX J1301.9+2747, while the HST imaging reveals a central point source for the
object. In order to precisely determine the hard X-ray component, future
longer X-ray observations are required. This will help constrain the accretion
disk model for RX J1301.9+2747, and shed new light into the characteristics of
the corona and accretion flows around black holes.
###### Subject headings:
accretion, accretion disks — galaxies: active — galaxies: individual (RX
J1301.9+2747) — X-rays: galaxies
## 1\. Introduction
Active galactic nuclei (AGNs) are thought to be powered by supermassive black
holes of $M_{\rm BH}\sim 10^{6}-10^{9}~{}M_{\odot}$ accreting the surrounding
gas (see Rees, 1984, for a review). They are also considered to be scaled-up
versions of Galactic black hole binaries (BHBs, $M_{\rm BH}\sim
10~{}M_{\odot}$, McHardy et al. (2006) and references therein). The rapid
X-ray variability is one example of the similarities between these two types
of systems (Gierlinski et al., 2008). In Seyfert galaxies, variations of the
X-ray continuum emission over a timescale from minutes to hours have been
reported (e.g. Ulrich et al., 1997; Boller et al., 1997; Ponti et al., 2012),
however, persistent giant and rapid variability appears to be fairly rare and
its origin is still poorly understood.
Soft X-ray excesses above an extrapolation of the underlying hard X-ray power-
law is commonly observed in Type 1 AGNs and radio quiet quasars (Piconcelli et
al., 2005; Bianchi et al., 2009). The origin of this additional component is
not clear, and may be the high-energy tail of the AGN accretion disk emission
(e.g. Grupe et al., 1995). The problem using this explanation is that the
temperatures of the soft X-ray excesses appear to fall within a narrow range
(kT $\sim$ 0.1–0.2 keV) from a sample of AGNs containing a large range of
black hole (BH) mass, which is difficult to explain using the standard
accretion disk models (e.g. Gierlinski & Done, 2004; Crummy et al., 2006).
So far there has been no convincing evidence for the presence of direct
accretion disk emission seen in the X-ray spectra of AGNs. Yuan et al. (2010)
reported a luminous ultra-soft excess in the narrow line Seyfert 1 galaxy
(NLS1) J1633+4718 from archival ROSAT spectra, and found a lowest soft excess
temperature of 32 eV among AGNs. This characteristic of the soft excess is
likely an observational signature for the accretion disk emission. However,
the blackbody nature of this emission needs to be tested further, utilizing
higher quality X-ray data, as the ROSAT spectra ($\sim$ 0.1–2.4 keV) are less
sensitive to constrain the harder power-law emission above $\sim$2 keV.
Recently, Terashima et al. (2012) reported the discovery of a candidate
’ultrasoft’ AGN, whose X-ray spectrum can be represented $purely$ by a soft
thermal component with a blackbody temperature of $kT\sim$0.13–0.15 keV, by
analog with the accretion disk dominated spectrum typically seen in the
high/soft state of BHBs. Additionally, the soft X-ray emission obtained shows
spectral variability consistent with being caused by strong Comptonization.
Interestingly, the object was later optically confirmed to be a Type 2 AGN (Ho
et al., 2012) with a central BH mass as small as $10^{5}M_{\odot}$. However,
in their work they did not test in detail the possibility of the accretion
disk emission as the origin for the soft excess.
In this paper, we report results of new XMM-Newton and Chandra observations of
RX 1301.9+2747 at $z$=0.0237 (hereafter J1302), a highly variable and ultra-
soft X-ray source in a post-starburst galaxy (Dewangan et al., 2000). Our
detailed analysis of the optical spectrum from the SDSS revealed that it is a
Seyfert galaxy. The ultrasoft X-ray emission of J1302 was confirmed in the new
X-ray observations. In particular, we found unusual giant flares in both XMM-
Newton and Chandra light curves, accompanied by spectra hardening during the
flare state. In the quiescent state, the spectrum appears to be dominated by a
thermal blackbody component, whose temperature is comparable to the predicted
maximum accretion disk temperature. Throughout this paper, we assume a
cosmology with $H_{0}$ = 0.71, $\Omega_{M}$ = 0.27, $\Omega_{\Lambda}$ = 0.73.
## 2\. Data analysis and result
### 2.1. Optical Spectrum
J1302 was spectroscopically observed by the SDSS in March 2007. Figure 1 shows
the rest-frame spectrum for J1302 (black line), which is dominated by the
starlight of the host galaxy. To subtract the starlight and the nuclear
continuum, we followed the recipe described in detail in Dong et al. (2005).
As seen in Figure 1, the galaxy starlight model (green line) gives a very good
fit to the optical continuum ($\chi^{2}$/dof = 3648/3208). After the
subtraction of stellar absorption lines, we fitted the emission-line spectrum,
represented by a blue line, by using Gaussians to derive the parameters of the
emission lines. [OIII] $\lambda\lambda$4959, 5007, [NII] $\lambda\lambda$6548,
6583, H$\alpha$ and [SII] $\lambda\lambda$6717, 6731 emission lines are
clearly detected with S/N $>$ 5, while H$\beta$ line is only weakly
distinguishable with S/N $\sim$ 1.4. The right panel in Fig. 1 displays the
emission-line spectra, alongside the best fit Gaussian models. There is no
apparent broad component of H$\alpha$ line, and a narrow Gaussian with a line
width of FWHM $\sim$ 240km s-1 can provide good fit. In order to verify the
absence of the broad H$\alpha$ line, we add an additional Gaussian to the
narrow H$\alpha$, with the width fixed at 2000 km s-1. The flux is allowed to
vary in the fitting process. We found that the fitting was marginally improved
by adding this component, and the S/N for the H$\alpha$ broad component is
only $\sim$0.5. This suggests that the broad H$\alpha$ line, if there is any,
is extremely weak in J1302. The ratios of the narrow lines [OIII]
$\lambda$5007/H$\beta\,>$ 4.8 (using 3$\sigma$ upper limit of H$\beta$ line
flux), and [NII] $\lambda$6583/H$\alpha$ = 2.3, place J1302 into the Seyfert
regime on the BPT diagram of Kewley et al. (2006). The flux ratios of
[SII]/H$\alpha$ and [OI]/H$\alpha$ are 0.56 and 0.32, respectively, further
strengthening the Seyfert nature of J1302 according to line ratio diagnostic
diagrams of Kewley et al. (2006).
### 2.2. X-ray Observations
J1302 was observed by XMM-Newton EPIC cameras in December 2000 with a total
exposure time of 29 ks. It was detected $\sim$7.3 arcmin away from the center
of the field of view in the XMM-Newton imaging of the Coma cluster (ObsID
0124710801). The XMM-Newton data were reprocessed with the Science Analysis
Software version 11.0.0, using the calibration files as of December 2011. We
used principally the PN data, which have much higher sensitivity, using the
MOS data only to check for consistency. Spectra and light curves of source
were extracted from a circular region with a radius of 40$\arcsec$ centered at
the source position for both PN and MOS cameras. Background spectra were made
from source-free areas on the same chip using four circular regions identical
to the source region. The epochs of high background events were examined and
excluded by using the light curves in the energy band above 12 keV.
The Chandra pointing observation of J1302 was taken in June 2009 for about 5
ks. The data were processed with CIAO (version 4.3) and CALDB (version 4.4.1),
following the standard criteria. Fig. 2 shows the Chandra X-ray contours of
J1302, overlaid on the HST image in the B band. It is clear from the figure
that the X-ray emission of J1302 is point-like. The center for the X-ray
source is coincident with the optical nucleus of the galaxy (with a positional
offset of $\sim$0.1$\arcsec$). Given the subarcsecond spatial resolution of
Chandra, we conclude that most of the X-ray emission, if not all, comes from
the nuclear region of the galaxy, likely related to the AGN.
### 2.3. X-ray light curve
Light curves from the XMM-Newton and Chandra observations are shown in Fig. 3.
The source exhibits large-amplitude count rate variations in both
observations. It can be seen from the PN light curve that there is a giant
flare with count rates rising by a factor of 5 times the average value, having
a duration of $\sim$2 ks. The X-ray flux then declines to a relatively steady
state. Such X-ray flare is confirmed by the MOS light curves, in which a
possible decline of another flare is also recorded at the beginning of the MOS
observation. The time interval of the two flares is about 17 ks.
Interestingly, similar flare is seen in the Chandra light curve, with count
rates increasing by a factor of 7 within $\sim$1 ks. The similar amplitudes of
flares in the XMM-Newton and Chandra observations which span $\sim$ 9 years
suggest that the flare behaviour in J1302 seems persistent on time scale of
$\sim$ decade. The spectral variability during the flare will be explored in
detail in Section 2.4.
### 2.4. X-ray Spectra
As both XMM-Newton and Chandra observations show peculiar flare behaviours, we
attempt to quantify the spectral variability during flares by dividing the
data into high and low flux intervals, using count rate thresholds of 0.35
counts s-1 for XMM-PN and 0.08 counts s-1 for Chandra, respectively. For
simplicity, we classify the data above the count rate thresholds as belonging
to the flare state, and that which falls below, to the quiescent state. The
spectra data were grouped in the following manner: data from the XMM-Newton
had at least 20 counts per bin to ensure the $\chi^{2}$ statistics, and the
Chandra data had at least 3 counts per bin and utilized the $C$-statistics
which was adopted for minimization. Spectral fitting was performed using the
XSPEC (Version 12.6) and limited to the 0.2–2 keV range for XMM-Newton, since
the emission is background dominated above that energy range. The Chandra data
was fitted in the energy range between 0.3 and 3 keV. Throughout the model
fittings, the Galactic column density was considered and fixed at $N_{\rm
H}^{\rm Gal}=0.75\times 10^{20}$ cm-2 (Kalberla et al., 2005).
Spectral variability is clearly present between the two states with the source
being much harder in the flare state. To illustrate the differences in the
spectral slope between the two states, we show the spectra together with an
absorbed power-law model in Figure 4. While the fit is generally acceptable
for the two states, as confirmed by the reduced $\chi^{2}$ value (see Table
1), the photon indices obtained from a power-law fit are, however, extremely
steep with $\Gamma$ = $4.4^{+0.5}_{-0.4}$ for $N_{H}=4.3^{+2.2}_{-1.8}\times
10^{20}$ cm-2, and $\Gamma$ = $7.1^{+0.9}_{-0.7}$ for
$N_{H}=3.6^{+1.9}_{-1.6}\times 10^{20}$ cm-2, for the XMM flare and quiescent
state, respectively. The photon indices belong to the steepest values obtained
from AGN X-ray spectra. For comparison, the mean photon index in the 0.2–2.0
keV band is $\sim 2.9$ for a sample of soft X-ray selected AGNs observed with
the ROSAT RASS (Grupe et al., 2010). The result of this comparison indicates
that the X-ray spectrum of J1302 is extremely soft compared to other AGNs. The
absorption-corrected luminosity in the 0.5–2.0 keV range for this simple
power-law model is $6.7\times 10^{41}$ and $2.8\times 10^{40}$ ${\rm erg\
s^{-1}}$ for the XMM flare and quiescent state, respectively.
In order to further investigate the spectral variability in J1302, we then
attempted to fit the spectra with a blackbody (bbody in XSPEC) or Multiple
Color Disk model (MCD, diskbb in XSPEC), and a thermally Comptonized disk
model (compTT in XSPEC, Titarchuk, 1994), both of these alternative models
have been used to fit the spectra of Galactic BHBs (e.g., Done et al., 2007),
and the soft X-ray excess emission in AGNs (e.g., Porquet et al., 2004;
Patrick et al., 2012) . The diskbb model integrates over the surface of
accretion disk to form a multicolor blackbody spectrum, and compTT is an
analytic model that self-consistently calculates the spectrum produced by the
Comptonization of soft seed photons in a hot corona above the accretion disk.
The physical parameters of the compTT model are: the soft photon temperature
($kT_{0}$), the temperature of the Comptonizing electrons ($kT_{e}$), the
plasma scattering optical depth ($\tau$). For our fitting with the compTT
model, a disk geometry was assumed for the comptonizing region, and the seed
photons were assumed to follow Wien’s law with a temperature of 22 eV (the
expected disk temperature in section 3.2). Because the temperature and optical
depth of the Comptonizing plasma are strongly coupled (both are equally
involved in shaping the spectrum) and thus cannot be constrained
simultaneously, we fixed the plasma temperature at 20 keV and obtained
constraints on the optical depth111 Leaving the plasma temperature as a free
parameter yields $kT_{e}=21(<27)$ keV and $\tau<1.3$ for the XMM quiescent
state spectrum, but both parameters cannot be constrained by the data during
the flare.. The single Comptonized model yields consistent fitting results
with the previous simple power-law model for the spectrum at both states. The
Compton optical depth is $\tau=0.16^{+0.07}_{-0.05}$ and $\tau<0.03$ for the
flare and quiescent state, respectively. For the spectrum in the XMM-Newton
flare state, a multicolor-disk blackbody gives equivalent fit, which is
statistically better than the simple blackbody emission.
The spectrum at the XMM-Newton quiescent state, however, shows an excess of
emission at energies above $\sim$ 0.7 keV when fitted with a thermal model
(bbody or diskbb). The addition of a power-law to the model improves the fit
with very high statistical significance ($\chi^{2}$ decreased by 13.9 for two
extra parameters, at a 99.98% level according to $F$-test). The power-law
component contributes $\sim$15% of the total luminosity in the 0.3–2 keV band.
In this case, we obtain an effective blackbody temperature of $kT_{\rm
BB}=43^{+6}_{-3}$ eV, comparable to the seed photon temperature assumed in the
compTT model. Although with large uncertainties, the additional power-law
component is relatively steep with photon index $\Gamma=4.3^{+1.6}_{-1.9}$,
and it is close to what is observed in the flare state. The spectral fitting
results for the PN data, alongside the observed flux and intrinsic luminosity
in the 0.5–2 keV range for each model, are shown in Table 1. Note that we used
the same models to fit the MOS data and found that the results agree well with
the PN data.
The Chandra spectra were fitted with the same models used in the XMM-Newton
observation and the results are listed in Table 1. During the first run we
found that the photon indices derived from the simplest power-law model are
systematically flatter than the values for the XMM-Newton data. The spectrum
during the flare can be well fitted by a power-law model ($C$/dof=22.5/22). On
the other hand, a power-law model is not sufficient to fit the data at the
Chandra quiescent state. The addition of a soft thermal component with respect
to the power-law model improves the fit significantly ($C$ value decreases by
$\sim$9 for two extra parameters, corresponding to a significance level of
98.7%). The resulting best-fit photon index is flatter, with
$\Gamma=3.5^{+0.8}_{-1.0}$. Similarly, we obtain an effective disk
temperature, when fitted with a blackbody, of $kT_{\rm BB}=29^{+19}_{-16}$ eV.
This value is slightly lower than the blackbody temperature for the XMM-Newton
quiescent state, but the parameter is loosely constrained due to the poor
statistics of the Chandra data. The power-law component in this case
contributes $\sim 40\%$ of the total luminosity in the 0.3–2 keV, indicating a
possible change of the power-law or the blackbody emission between Chandra and
XMM-Newton observations.
Although a Comptonized model as opposed to a power-law model also provides a
good fit for the Chandra flare state spectrum, it is not sufficient to fit the
data at the quiescent state. The addition of an extra hard power-law component
is needed at a significance level of 98.7% according to $F$-test. The best-
fitted optical depth $\tau$ for the compTT model is not significantly
different from that obtained with the XMM-Newton data. The unabsorbed
luminosity in the 0.5–2 keV band, based on the best-fit power-law model and
the diskbb+power-law model for the Chandra flare and quiescent state, is
$5.1\times 10^{41}$ and $4.4\times 10^{40}$ erg s-1, respectively.
## 3\. Discussion
### 3.1. AGN characteristics in RX J1301.9+2747
X-ray observation with Chandra , which has superb spatial resolution of $\sim
0.5\arcsec$, revealed the presence of an AGN in J1302: the center of the
bright unresolved X-ray emission coinciding with the optical point-like
nucleus of the galaxy (with a position offset of $\sim 0.1\arcsec$). Both the
Chandra and XMM-Newton observations show that it has Seyfert-like X-ray
luminosity of $\sim 10^{41}$ erg s-1 in the energy range of 0.5–2 keV ( $\sim
10^{42}$ erg s-1 in the 0.2–2 keV band ), and rapid X-ray variability down to
a time-scale $\sim$ 1 ks. Additionally, the optical spectrum of J1302 displays
Seyfert-like narrow emission line ratios. All these observational facts point
to the presence of an AGN (Seyfert nucleus) in this galaxy. This can be
further supported by the point-like appearance of a nuclear source from the
HST imaging observations with $\sim 0.1\arcsec$ resolution (Caldwell et al.,
1999).
Based on ROSAT X-ray observations, Dewangan et al. (2000) argued for the
presence of an AGN in J1302, a view that is consistent with ours on the basis
of the new observations. However, based on their observed optical spectrum,
Dewangan et al. conclude that the galaxy nucleus is more like a LINER. In this
paper, we have carefully modeled the host galaxy’s starlight, especially the
stellar absorption features, and subtracted them from the new SDSS spectrum,
enabling us to accurately measure the weak AGN emission lines in J1302 (e.g.,
Dong et al., 2005), thus more resolutely confirming the Seyfert nature based
on the line ratio diagnostics.
The lack of detectable broad permitted lines prevents us from estimating the
central BH mass of J1302 using conventional linewidth-luminosity-mass scaling
relation. In the SDSS spectral fitting, we found the strongest narrow line,
[OIII]$\lambda$5007, is marginally resolved with Gaussian $\sigma=58\pm 9$ km
s-1 (after correcting for the instrument resolution).Using the width of the
[OIII]$\lambda$5007 line as a proxy for the stellar velocity dispersion of the
host galaxy, we obtained a BH mass of $M_{\rm BH}=8\times 10^{5}M_{\odot}$
with an intrinsic scatter of 0.5 dex (e.g., Xiao et al., 2011).
The bolometric luminosity for J1302 can be estimated from the optical
continuum luminosity. We retrieved the high resolution HST/WFPC2 images of
J1302 in the $B$ (F450W filter) and $I$ (F814W filter) passbands from the HST
archive. The HST observations (dataset U39D0301M–U39D0304M) were made in July
1997, with two 600 s exposures in the $B$ band and two 400 s exposures in the
$I$ band, respectively. The images were processed using the standard HST
pipeline routines in
IRAF/STSDAS222http://www.stsci.edu/institute/software_hardware/stsdas. We then
performed two-dimensional profile decompositions of this galaxy with the code
GALFIT (version 3.0, Peng et al. 2010). Our model consists of an exponential
disk component, a bulge component, and an unresolved central point source for
the nuclear AGN emission. In our GALFIT modeling, the point-spread function
was generated by the $\tt{Tiny\,Tim}$ software (Krist et al., 1995). Note that
the bulge for this galaxy displays a box/peanut shape, which is commonly seen
in edge-on barred disk galaxy, consequently we added a boxiness parameter to
the bulge profile in GALFIT. The model generally matches the data well
($\chi^{2}_{\nu}\sim 1.02$). The inferred flux for the central point source in
the $B$-band is $M_{B}$ = -15.8, corresponding to a nuclear luminosity of $\nu
L_{\nu B}$ = $8\times 10^{41}$ erg s-1. For comparison, the B-band bulge
luminosity of this galaxy derived from the GALFIT decomposition is LB,bulge
$\sim 6\times 10^{42}$ erg s-1. If indeed the nuclear emission comes from
AGN333Note that we have not accounted for the dust extinction in estimating
the luminosity., we estimate the bolometric luminosity to be $1\times 10^{43}$
${\rm erg\ s^{-1}}$ using the $B$-band luminosity for the central point source
by adopting a bolometric correction of 13 (Marconi et al., 2004). For a BH
mass of $8\times 10^{5}M_{\odot}$, the accretion rate in Eddington unit is
$L/L_{\rm{EDD}}\sim$ 0.1, which suggests J1302 is accreting at high Eddington
ratio.
### 3.2. Extremely Soft X-ray Emission
One of the remarkable features of J1302 is the extreme softness of the X-ray
spectra. The best-fitted power-law index for the spectrum in the XMM-Newton
quiescent state ($\Gamma\sim 7$) is one of the steepest soft X-ray photon
indices among AGNs (e.g., Grupe et al., 1995; Boller et al., 2011).
Understanding the origin of the ultrasoft X-ray emission will help to pin down
the nature of the source. Some AGNs such as NLS1s can be very soft (e.g.,
Boller et al., 1996; Middleton et al., 2007), showing a strong soft X-ray
excess over an underlying power-law component. The strength of soft excess can
be quantitatively described as the ratio of flux at 0.5 keV to the power law
extrapolation of the fitting to the spectrum above 2 keV (Middleton et al.,
2007). Since no significant hard X-ray emission above $\sim$2 keV was detected
in the XMM observation of J1302, we estimated 90% confidence upper limit on
the count rates in 2–7 keV, $\sim 6.7\times 10^{-4}$ counts s-1, and converted
it to an upper limit on the extrapolated flux at 0.5 keV. XSPEC simulations of
models with power-law $\Gamma$ = 2, 2.5 and 3 show that the lower limits on
the ratios defined above are 37, 15 and 6.1, respectively. Note that the
ratios for a sample of NLS1s (Middleton et al., 2007) are usually found to be
less than 10. Thus, the soft emission relative to that at the hard X-rays in
J1302 is extremely strong compared to other AGNs.
The origin of soft excess is still unclear. The spectrum at the XMM quiescent
state can be fitted well by a Comptonized model and a blackbody plus power-law
emission equally. Interestingly, the fitted blackbody temperature ($kT_{\rm
BB}=43^{+6}_{-3}$ eV) is much lower than the canonical values of $\sim$
0.1–0.2 keV found for AGNs (Crummy et al., 2006). Standard accretion disk
models (Shakura & Sunyaev, 1973) give a maximum effective temperature of the
accreting material $kT_{\rm max}\sim$ $11.5(\dot{m}/M_{8})^{1/4}$ eV, where
$\dot{m}$ is mass accretion rate in Eddington unit and $M_{8}$ = $M_{\rm
BH}/10^{8}M_{\odot}$. Using $M_{\rm BH}=8\times 10^{5}M_{\odot}$ and
$\dot{m}\sim 0.1$ for J1302, we obtained $kT_{\rm max}\sim$22 eV, which is
comparable to the fitted blackbody temperature. Note that though with larger
uncertainties, the lower fitted temperature for the Chandra quiescent state
data ($kT_{\rm BB}=29^{+19}_{-16}$ eV) is more compatible with the predicted
maximum disk temperature. Therefore, the ultrasoft X-ray emission in J1302 may
be connected with the direct thermal emission from the accretion disk. Another
constraint on the X-ray spectra due to thermal disk emission can come from the
optical/UV data. The optical $B$-band flux of nuclear point source from the
HST observation is, however, much higher than the extrapolated MCD flux in the
optical ($M_{B}\sim-12.7$), about one order of magnitude difference. The
difference cannot be explained by the contamination from nuclear star
clusters, as they have typical absolute $I$-band magnitudes between $-14$ and
$-10$ (Böker et al., 2002), much lower than the observed $I$-band flux of
J1302 nucleus ($M_{I}\sim-16.8$, obtained with the same GALFIT decomposition
of the HST/WFPC2 image as detailed in Section 3.1). To further investigate
this difference we need a more realistic disk spectral modeling, and fit to a
broader band data, which is beyond the scope of this paper.
### 3.3. Unusual X-ray variability
Another unusual feature of J1302 is that it clearly shows two distinct states
in the X-ray, a flare (or eruptive) state and a quiescent state. The
amplitudes of the flare in the Chandra and XMM-Newton light curves look very
similar, with the count rates increased by a factor of 5–7 within $\sim$1–2
ks. Both the XMM-Newton observation in 2000 and the Chandra observation in
2009 detected rapid flares, suggesting that the flare itself appears
repetitive and occurs very frequently in the object. In fact, ROSAT PSPC
observations also found that the object is highly variable and demonstrates a
rapid flare event in light curve lasting $\sim$1.3 ks (Dewangan et al., 2000).
The rapid energetic flare in J1302 is fairly rare among Seyfert galaxies and
quasars, although some extreme variations have also been found in objects such
as IRAS 13224-3809 (Boller et al., 1997) and PKS 0558-504 (Wang et al., 2001).
As discussed by Wang et al. (2001), a magnetic coronal model in which
electrons in the corona are continuously heated by magnetic reconnection, can
produce rapid energetic X-ray flare. A realistic physical model to explain a
rapid energetic flare in AGNs, and the comparison with Galactic BHBs are
beyond the scope of this work and will be presented elsewhere.
The spectral variability is clearly present between the two flux states with
the source being much harder in the flare state. This behavior is markedly
contrast to what is commonly seen in AGNs and BHBs (e.g., Markowitz & Edelson,
2004; Done et al., 2007). Clearly the spectral variability cannot be explained
by any simple spectral model. One possibility is that the spectrum in the
quiescent state is represented by a relatively stable thermal emission from
accretion disk, and the spectral variability is caused by the flux variations
of a second additional component such as strongly Comptonized power-law
emission. This perhaps explains why the photon index for the power-law
emission in the XMM-Newton flare state (which dominates the spectrum) is
consistent with the additional power-law component in the quiescent state,
whose strength is relatively weak. In this case, the underlying power-law
emission is still steep ($\Gamma\sim$4) compared to other AGNs, the nature of
which remains understood. Note that similar spectral change has also been seen
in the Chandra observation of J1302, though the spectra statistics for the
data is low. Longer X-ray observations are required to examine the presence of
persistent flares and to investigate the origin of spectral variability during
flare periods. This can in turn shed new lights on the characteristics of
corona and accretion flows around BHs.
This research made use of the HEASARC online data archive services, supported
by NASA/GSFC. We would like to thank the anonymous referee for his/her helpful
comments to improve the content of the paper. We are also grateful to G.
Mckenzie for offering language assistance. This work was supported by Chinese
NSF through grant 11103017, 11233002, and National Basic Research Program
2013CB834905. X.S. acknowledges the support from the Fundamental Research
Funds for the Central Universities (grant Nos. WK2030220004, WK2030220005).
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Table 1Spectral fitting results for the XMM-Newton and Chandra observation at
different states
XMM-PN flare state
---
wabs*modela | $\rm{N_{H}}$ | $\Gamma$ | kT | $\tau$b | $\chi^{2}$/d.o.f. | F0.5-2keVc | L0.5-2keVc |
| ($10^{20}\,\rm{cm}^{-2}$) | | (eV) | | | ($10^{-14}$ erg s-1 cm-2) | ($10^{40}$ erg s-1) |
powl | $4.3^{+2.2}_{-1.8}$ | $4.4^{+0.5}_{-0.4}$ | | | 40.4/28 | 40 | 67 |
bbody | 0.75(fixed)d | | $99^{+6}_{-5}$ | | 55.5/29 | 38 | 54 |
diskbb | 0.75(fixed) | | $134^{+9}_{-9}$ | | 43.7/29 | 39 | 55 |
compTTe | $4.3^{+2.2}_{-1.8}$ | | | $0.16^{+0.07}_{-0.05}$ | 40.3/28 | 40 | 67 |
XMM-PN quiescent state
powl | $3.6^{+1.9}_{-1.6}$ | $7.1^{+0.9}_{-0.7}$ | | | 27.9/40 | 1.6 | 2.8 |
bbody+powl | 0.75(fixed) | $4.3^{+1.6}_{-1.9}$ | $43^{+6}_{-3}$ | | 26.3/39 | 1.7 | 2.5 |
diskbb+powl | 0.75(fixed) | $3.7^{+2.0}_{-2.0}$ | $52^{+5}_{-5}$ | | 25.9/39 | 1.7 | 2.6 |
compTT | $2.9^{+1.4}_{-1.3}$ | | | 0.016($<$0.03) | 27.7/40 | 1.6 | 2.7 |
Chandra flare state
wabs*model | $\rm{N_{H}}$ | $\Gamma$ | kT | $\tau$ | $C$/d.o.f. | f0.5-2keV | Lintr,0.5-2keV |
| ($10^{20}\,\rm{cm}^{-2}$) | | (eV) | | | ($10^{-14}$ erg s-1 cm-2) | ($10^{40}$ erg s-1) |
powl | 3.7($<$11) | $3.2^{+1.1}_{-0.6}$ | | | 22.5/22 | 34 | 51 |
bbody | 0.75(fixed) | | $195^{+24}_{-20}$ | | 27.6/23 | 35 | 47 |
diskbb | 0.75(fixed) | | $275^{+51}_{-38}$ | | 22.5/23 | 35 | 47 |
compTT | 3($<$20) | | | $0.44^{+0.28}_{-0.28}$ | 22.5/22 | 34 | 50 |
Chandra quiescent state
powl | 0.75(fixed) | $4.5^{+0.6}_{-0.6}$ | | | 16.5/14 | 3.7 | 5.3 |
bbody+powl | 0.75(fixed) | $3.5^{+0.8}_{-1.0}$ | $29^{+19}_{-16}$ | | 8.0/12 | 3.2 | 4.5 |
diskbb+powl | 0.75(fixed) | $3.5^{+0.8}_{-1.0}$ | $32^{+25}_{-18}$ | | 8.0/12 | 3.2 | 4.4 |
compTT+powl | 0.75(fixed) | 2.8($<$3.9) | | 0.01($<$0.09)f | 8.1/12 | 3.1 | 4.3 |
* a
Spectral model (as given in XSPEC) multiplied by a neutral absorption (wabs)
with column density $\rm{N_{H}}$. wabs is the photo-electric absorption model
using Wisconsin–Morrison & McCammon (1983) cross-sections.
* b
Plasma optical depth in the CompTT model.
* c
$F_{\rm 0.5-2keV}$ is the observed 0.5–2 keV flux in units of $10^{-14}$ ${\rm
erg\ cm^{-2}\ s^{-1}}$ . $L_{\rm 0.5-2keV}$ is the unabsorbed luminosity in
the energy range of 0.5–2 keV, in units of $10^{40}$ ${\rm erg\ s^{-1}}$ .
* d
The column density was fixed to Galactic value $\rm{N_{H}^{Gal}}$ if the
fitting yields a $\rm{N_{H}}<$$\rm{N_{H}^{Gal}}$.
* e
The spectrum of the seed photons is assumed to be Wien law with a temperature
of 22 eV (see Section 3.2 for details). We fixed the plasma temperature at 20
keV and obtained constraints on the optical depth.
* f
Pegged at the minimum value allowed in XSPEC.
Figure 1.— Illustration of the continuum and emission-line fittings of the
SDSS spectrum. Left panel: observed spectrum (black), stellar continuum model
(green) and residual (blue) which is used to fit the emission lines. Right
panel: a zoomed-in view of the emission-line profile fitting for the
H$\alpha$+[NII] and the [SII] doublet lines. The inset in the left panel shows
a zoomed-in view of the H$\beta$ and [OIII] region. Gaussian line models are
plotted in cyan line, and the residual is shown in the lower panel. In the
emission-line fits, the line ratios of [OIII] and [NII] doublet lines are
fixed to the theoretical values. Figure 2.— Contours of the Chandra image
(green) overlaid with the HST $B$-band image. The direction of north is up and
east is left. White plus marks the position of the central point source from
the HST imaging. For comparison, the center of the radio emission obtained
with the Very Large Array (Miller et al., 2009) is shown in cyan cross. Figure
3.— XMM-Newton and Chandra light curves for J1302. Top panel: XMM-PN and the
summed MOS1+MOS2 light curves, with a time bin size of 100 s. The PN and MOS
background light curves are also shown for comparison (lower panel). Bottom
panel: Chandra light curve with the same bin size as XMM-Newton. The dotted
lines represent the count rate thresholds dividing the data into flare and
quiescent state, which are 0.35 counts s-1 for XMM-PN, and 0.08 counts s-1 for
Chandra , respectively.
Figure 4.— Top panel: Spectra of the XMM-Newton flare state (black) and the
quiescent state (red) for J1302. Only the PN data are shown for clarity. The
solid lines are the simple power-law model fits for both states, and the
corresponding residuals are shown in the lower panels. Bottom panel: as top
panel, but showing for the Chandra data.
|
arxiv-papers
| 2013-04-11T10:05:26 |
2024-09-04T02:49:44.176952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luming Sun (USTC), Xinwen Shu (USTC) and Tinggui Wang (USTC)",
"submitter": "Xinwen Shu",
"url": "https://arxiv.org/abs/1304.3244"
}
|
1304.3249
|
J.Y. Moyen and P. Parisen Toldin
# A polytime complexity analyser for Probabilistic Polynomial Time over
imperative stack programs.
J.Y. Moyen LIPN, UMR 7030, CNRS, Universitè Paris 13 F-93430 Villetaneuse,
France. [email protected] P. Parisen Toldin Dipartimento
di Scienze dell’Informazione, Università di Bologna Équipe FOCUS, INRIA Sophia
Antipolis Mura Anteo Zamboni 7, 40127 Bologna, Italy. [email protected]
###### Abstract.
We present ${\mathbf{iSAPP}}$ (Imperative Static Analyser for Probabilistic
Polynomial Time), a complexity verifier tool that is sound and extensionally
complete for the Probabilistic Polynomial Time ($\mathbf{PP}$) complexity
class. ${\mathbf{iSAPP}}$ works on an imperative programming language for
stack machines. The certificate of polynomiality can be built in polytime,
with respect to the number of stacks used.
###### keywords:
ICC, Probabilistic Polytime, Static verifier
###### 1991 Mathematics Subject Classification:
Theory, Verification
## 1\. Introduction
One of the crucial problem in program analysis is to understand how much time
it takes a program to complete its run. Having a bound on running time or on
space consumption is really useful, specially in fields of information
technology working with limited computing power. Solving this problem for
every program is well known to be undecidable. The best we can do is to create
an analyser for a particular complexity class able to say “yes”, “no”, or
“don’t know”. Creating such an analyser can be quite easy: the one saying
every time “don’t know” is a static complexity analyser. The most important
thing is to create one that answers “don’t know” the minimum number of time as
possible.
We try to combine this problem with techniques derived from Implicit
Computational Complexity (ICC). Such research field combines computational
complexity with mathematical logic, in order to give machine independent
characterisations of complexity classes. ICC has been successfully applied to
various complexity classes such as $\mathbf{FP}$ [2, 11, 4], $\mathbf{PSPACE}$
[12], $\mathbf{LOGSPACE}$ [8].
ICC systems usually work by restricting the constructions allowed in a
program. This _de facto_ creates a small programming language whose programs
all share a given complexity property (such as computing in polynomial time).
ICC systems are normally extensionally complete: for each function computable
within the given complexity bound, there exists one program in the system
computing this function. They also aim at intentional completeness: each
program computing within the bound should be recognised by the system. Full
intentional completeness, however, is undecidable and ICC systems try to
capture as many programs as possible (that is, answer “don’t know” as little
time as possible).
Having an ICC system characterising a complexity class $\mathcal{C}$ is a good
starting point for developing a static complexity analyser. There is a large
literature on static analysers for complexity bounds. We develop an analysis
recalling methods from [9, 3, 10]. Comparatively to these approaches our
system works with a more concrete language of stacks, where variables,
constants and commands are defined; we are also sound and complete with
respect to the Probabilistic Polynomial time complexity class ($\mathbf{PP}$)
[7].
We introduce a probabilistic variation of the Loop language. Randomised
computations are nowadays widely used and most of efficient algorithms are
written using stochastic information. There are several probabilistic
complexity classes and $\mathbf{BPP}$ (which stands for Bounded-error
Probabilistic Polytime) [7] is considered close to the informal notion of
feasibility. Our work would be a first step into the direction of being able
to capture real feasible programs solving problems in $\mathbf{BPP}$
($\mathbf{BPP}\subseteq\mathbf{PP}$) [7].
Similar work has been done in [6] with characterisation of complexity class
$\mathbf{PP}$; This work gives a characterisation of complexity class
$\mathbf{PP}$ by using a functional language with safe recursion as in
Bellantoni and Cook [2].
Our system is called ${\mathbf{iSAPP}}$, which stands for _Imperative Static
Analyser for Probabilistic Polynomial Time_. It works on a prototype of
imperative programming language based on the Loop language [14]. The main
purpose of this paper is to present a minimal probabilistic polytime certifier
for imperative programming languages.
Following ideas from [9, 3] we “type” commands with matrices, while we do not
type expressions since they have constant size. The underlying idea is that
these matrices express a series of polynomials bounding the size of stacks,
with respect to their input size. The algebra on which these matrices and
vectors are based is a finite (more or less tropical) semi-ring.
## 2\. Stacks machines
We study _stacks machines_ , a generalisation of the classical counters
machines. Informally, a stacks machine work with _letters_ belonging to a
finite alphabet and _stacks_ of letters. Letters can be manipulated with
_operators_. Typical alphabet include the binary alphabet $\\{0,1\\}$ or the
set long int of 64 bits integers. On the later, typical operators are $+$ or
$*$.
Each machine has a finite number of registers that may hold letters and a
finite number of stacks that may hold stacks. Tests can be made either on
registers and letters (with boolean operators) or to check whether a given
stack holds the empty stack. There are only bounded (for) loops which are
controlled by the size of a given stack. That is, it is more alike a foreach
(element in the stack) loop. Since there are only bounded loops (and no
while), this _de facto_ limits the language to primitive recursive functions.
In this way, stack machines are a generalisation of the classical Loop
language [14]. Since our analysis is compositional, we add also functions to
the language; their certificates can be computed separately and plugged in the
right place when a call is performed.
### 2.1. Syntax and Semantics
We denote $\langle\rangle$ the empty stack and $\langle a_{1}\ldots
a_{n}\rangle$ the stack with $n$ elements and $a_{1}$ at top.
###### Definition 2.1.
A _stacks machine_ consists in:
* •
a finite alphabet $\Sigma=\\{a_{1},\ldots,a_{n}\\}$ containing at least two
values $\mathsf{true}$ and $\mathsf{false}$;
* •
a finite set of operators, $\mathtt{op}_{i}$, of type $\Sigma^{n}\to\Sigma$,
containing at least a 0-ary operator $\mathtt{rand}$, operators whose co-
domain is $\\{\mathsf{true},\mathsf{false}\\}$ are _predicates_ noted
$\mathtt{op?}$;
* •
a finite set of registers $\mathbf{r}$ and stacks, $\mathtt{S}_{j}$ (the empty
stack is noted $\langle\rangle$);
* •
and a program written in the following syntax:
$\displaystyle b\in\mathtt{BooleanExp}::=$
$\displaystyle\mathsf{true}\,|\,\mathsf{false}\,|\,\mathtt{op?}(e_{1},\ldots,e_{n})\,|\,\mathtt{rand}()\,|\,\mathtt{isempty?}(\mathtt{S})$
$\displaystyle e\in\mathtt{Expressions}::=$ $\displaystyle
c\,|\,\mathbf{r}\,|\,\mathtt{op}(e_{1},\ldots,e_{n})\,|\,\mathtt{top}({\mathtt{S}})$
$\displaystyle C\in\mathtt{Commands}::=$
$\displaystyle\textbf{skip}\,|\,\mathbf{r}:=e\,|\,\mathtt{S}_{1}:=\mathtt{S}_{2}\,|\,\mathtt{S}:=\langle
c_{1}\ldots
c_{n}\rangle\,|\,\mathtt{S}_{k}:=\textbf{call}(f,\mathtt{S}_{1}\ldots\mathtt{S}_{n})$
$\displaystyle\,|\,\textbf{pop}({\mathtt{S}})\,|\,\textbf{push}(e,\mathtt{S})\,|\,C;C\,|\,\textbf{If
}\,b\,\textbf{ Then }\,C\,\textbf{ Else
}\,C\,|\,\textbf{loop}\,{\mathtt{S}}\,\\{{C}\\}$ $\displaystyle
f\in\mathtt{Functions}::=$ $\displaystyle\textbf{def }{f}\textbf{ in
}{(\mathtt{S}_{1}\ldots\mathtt{S}_{n})}\,\,\\{{C}\\}\,\,\textbf{out}{(\mathtt{S}_{j})}$
Note that registers may not appear directly in booleans expressions to avoid
dealing with the way non-booleans values are interpreted as booleans. However,
it is easy to define a unary predicate which, _e.g._ sends $\mathsf{true}$ to
$\mathsf{true}$ and every other letter to $\mathsf{false}$ to explicitly
handle this.
Expressions always return letters (content of registers) while commands modify
the state but do not return any value. $\mathtt{top}({\ })$ does not destruct
the stack but simply returns its top element while $\textbf{pop}({\ })$ remove
the top element from the stack but does not return anything. It is also
possible to assign constant stack to a stack.
The $\mathtt{isempty?}($) predicate returns $\mathsf{true}$ if and only if the
stack given in argument holds the empty stack and $\mathsf{false}$ otherwise.
The $\textbf{loop}\,{\mathtt{S}}\,\\{{C}\\}$ commands executes $C$ as many
time as the size of $\mathtt{S}$. Moreover, $\mathtt{S}$ may not appear in
$C$. It is, however, possible to make a copy beforehand if the content is
needed within the loop. Finally, we give the possibility to have function
call. The command $\textbf{call}(f,\mathtt{S}_{1}\ldots\mathtt{S}_{n})$ call
the function $f$ passing the actual arguments
$\mathtt{S}_{1}\ldots\mathtt{S}_{n}$ and finally return the result stored in
the stack $\mathtt{S}_{j}$.
### 2.2. Complexity
The set of operators is not specified and may vary from one stacks machine to
another (together with the alphabet). This allows for a wide variety of
settings parametrised by these. Typical alphabets are the binary one
($\\{\mathsf{true},\mathsf{false}\\}$), together with classical boolean
operators (not, and, …); or the set long int of 64 bits integers with a large
number of operators such as +, *, <, …Since there is only a finite number of
letters and operators all have the alphabet as domain and co-domain, there is
only a finite number of operators at each arity. So, without going deep into
details, it makes sense to consider that each operator take a constant time to
be computed. More precisely, each operator can be computed within a time
bounded by a constant. Typically, on long int, + can be computed in 64
elementary (binary) additions and * takes a bit more operations but is still
done in bounded time.
Thus, in order to simplify the study, we consider that operators are computed
in constant time and we do not need to take individual operators into account
when bounding complexity. It is sufficient to consider the number of
operators.
The only thing that is unbounded is the size (length) of stacks. Thus, if one
want, _e.g._ to handle large integers (larger than the size of the alphabet),
one has to encode them within stacks. The most obvious ways being the unary
representation (a number is represented by the size of a stack) and the binary
one (a stack of 0 and 1 is interpreted as a binary number with least
significant bit on top). Obviously, any other base can be use. In each case,
addition (and multiplication) has to be defined for this representation of
“large integers” with the tools given by the language (loops). Of course,
encoding unbounded value is crucial in order to simulate arbitrary Turing
Machines (or even simply Ptime ones) and is thus required for the completeness
part of the result.
Note that copying a whole stack as a single instruction is a bit unrealistic
as it would rather takes time proportional to the size of the stack. However,
since each stack will individually be bounded in size by a polynomial, this
does not hampers the polynomiality of the program. A clever implementation of
stacks with pointers (_i.e._ as lists) will also allow copy of a whole stack
to be implemented as copy of a single pointer, an easy operation.
Since the language only provides bounded loops whose number of execution can
be (dynamically) known before executing them, only primitive recursive
functions may be computed. This may look like a big restriction but actually
is quite common within classical ICC results on Ptime. Notably, Cobham [5] or
Bellantoni and Cook [2] both work on restrictions of the primitive recursion
scheme; Bonfante, Marion and Moyen [4] split the size analysis (quasi-
interpretation) from a termination analysis (termination ordering) which also
characterise only primitive recursive programs; and lastly Jones and
Kristiansen [9], on which this work is directly based, use the Loop language
which also allows only primitive recursion.
Since loops are bounded by the size of stacks, it is sufficient to bound the
size of stacks in order to bound the time complexity of the program. Indeed,
if each stack has a size smaller than $p$ and the program has never more than
$k$ nested loops, then its runtime cannot be larger than $p^{k}$. Similarly,
in the original $mwp$ calculus of Jones and Kristiansen, it was sufficient to
bound the value of stacks in order to bound the runtime of programs (for the
same reasons). Note that to have a large number of iterations, one first has
to create a stack of large size, that is when bounding the number of
iterations stacks are considered _de facto_ as unary numbers.
For each stack, we keep the dependencies it has from the other stacks. For
example, after a copy ($\mathtt{S}_{1}:=\mathtt{S}_{2}$), the size of
$\mathtt{S}_{1}$ is the same as the size of $\mathtt{S}_{2}$. Keeping precise
dependencies is not manageable, so we only keep the _shape_ of the dependence
(_e.g._ the degree with which it appear in a polynomial). These shapes are
collected in a vector (for each stack) and combining all of them gives a
matrix certificate expressing the size of the output stacks relatively to the
size of the input stacks. The matrix calculus we obtain for the certificates
is compositional. This allows for a modular approach of building certificates.
## 3\. Algebra
Before going deeply in explaining our system, we need to present the algebra
on which it is based. ${\mathbf{iSAPP}}$ is based on a finite algebra of
values. The set of scalars is $\mathtt{Values}=\\{0,L,A,M\\}$ and these are
ordered in the following way $0<L<A<M$. The idea behind these elements is to
express how the value of stacks influences the result of an expression. $0$
expresses no-dependency between stack and result; $L$ (stands for “Linear”)
expresses that the result linearly depends with coefficient $1$ from this
stack. $A$ (stands for “Additive”) expresses the idea of generic affine
dependency. $M$ (stands for “Multiplicative”) expresses the idea of generic
polynomial dependency.
We define sum, multiplication and union in our algebra as expressed in Table
1. The reader will immediately notice that $L+L$ gives $A$, while $L\cup L$
gives $L$ The operator $\cup$ works as a maximum.
$\times$ | 0 | L | A | M
---|---|---|---|---
0 | 0 | 0 | 0 | 0
L | 0 | L | A | M
A | 0 | A | A | M
M | 0 | M | M | M
+ | 0 | L | A | M
---|---|---|---|---
0 | 0 | L | A | M
L | L | A | A | M
A | A | A | A | M
M | M | M | M | M
$\cup$ | 0 | L | A | M
---|---|---|---|---
0 | 0 | L | A | M
L | L | L | A | M
A | A | A | A | M
M | M | M | M | M
Table 1. Multiplication, addition and union of values
Over this semi-ring we create a module of matrices, where values are elements
of $\mathtt{Values}$. We define a partial order $\leq$ between matrices of the
same size as component wise ordering. Particular matrices are $\mathbf{0}$,
the one filled with all $0$, and $\mathbf{I}$, the identity matrix, where
elements of the main diagonal are $L$ and all the others are $0$. If
$\textit{v}\in\mathtt{Values}$, a particular vector is
$\mathbf{{V}}^{\textit{v}}_{i}$ that is a column vector full of zeros and
having v at $i$-th row. Multiplication and addition between matrices work as
usual111That is:
$(\mathbf{A}+\mathbf{B})_{i,j}=\mathbf{A}_{i,j}+\mathbf{B}_{i,j}$ and
$(\mathbf{A}\times\mathbf{B})_{i,j}=\sum\mathbf{A}_{i,k}\times\mathbf{B}_{k,j}$
and we define point-wise union between matrices:
$(\mathbf{A}\cup\mathbf{B})_{i,j}=\mathbf{A}_{i,j}\cup\mathbf{B}_{i,j}$.
Notice that $\mathbf{A}\cup\mathbf{B}\leq\mathbf{A}+\mathbf{B}$. As usual,
multiplication between a value and a matrix corresponds to multiplying every
element of the matrix by that value.
We can now move on and present some new operators and properties of matrices.
Given a column vector ${V}$ of dimension $n$, a matrix $\mathbf{A}$ of
dimension $n\times m$ an index $i$ ($i\leq m$), we indicate with
${\mathbf{A}}\xleftarrow{i}{{V}}$ a substitution of the $i$-th column of the
matrix $\mathbf{A}$ with the vector ${V}$.
Next, we need a closure operator. The “union closure” is the union of all
powers of the matrix: ${\mathbf{A}}^{\cup}=\bigcup_{i\geq 0}\mathbf{A}^{i}$.
It is always defined because the set of possible matrices is finite. We will
need also a “merge down” operator. Its use is to propagate the influence of
some stacks to some other and it is used to correctly detect the influence of
stacks controlling loops onto stacks modified within the loop (hence, we can
also call it “loop correction”). The last row and column of the matrix is
treated differently because it will be use to handle constants and not stacks.
In the following, $n$ is size of the vector, $k<n$ and $j<n$.
* •
$({{V}}^{\downarrow{k,n}})_{i}={V}_{i}$
* •
$({{V}}^{\downarrow{k,j}})_{k}=\begin{cases}M&\text{ if $\exists p<n,p\neq k$
such that ${V}_{p}\neq 0$}\\\ 0&\text{ otherwise and ${V}_{n}=0$}\\\ L&\text{
otherwise and ${V}_{n}=L$}\\\ A&\text{ otherwise and ${V}_{n}\geq A$}\\\
{V}_{k}&\text{ otherwise}\\\ \end{cases}$
* •
$({{V}}^{\downarrow{k,j}})_{i}=\begin{cases}0&\text{ if $i=n$ }\\\ M&\text{ if
$i\neq j$, ${V}_{i}\neq 0$ and ${V}_{j}\neq 0$}\\\ {V}_{i}&\text{
otherwise}\end{cases}$
In the following we will use a slightly different notation. Given a matrix
$\mathbf{A}$ and an index $k$, ${\mathbf{A}}^{\downarrow{k}}$ is the matrix
obtained by applying the previous definition of merge down on each column of
$\mathbf{A}$. Formally, if ${V}$ is the $j$-th column of $\mathbf{A}$, then
$j$-th column of ${\mathbf{A}}^{\downarrow{k}}$ is ${{V}}^{\downarrow{k,j}}$.
Finally, the last operator that we are going to introduce is the “re-ordering”
operator. Given a vector ${V}$ we write
$\left[\stackrel{{\scriptstyle{V}}}{{1\rightarrow i,\ldots,n\rightarrow
j}}\right]$ to indicate that the result is a vector whose rows are permuted.
The first raw goes in the $i$-th position and so on till the $n$-th to the
$j$-th. In order to use a short notation, if a row is flowing in its same
position, then we don’t explicit it. Formally, if
${U}=\left[\stackrel{{\scriptstyle{V}}}{{1\rightarrow i,\ldots,n\rightarrow
j}}\right]$, then: ${U}_{p}=\sum_{k}{V}_{k}\,|\,k\rightarrow p$.
So, in case two or more rows clash on the same final row, we perform a sum
between the values. This operator is used for certificate the function calls.
Indeed we have to connect the formal parameters with the actual parameters.
Therefore, we have to permute the result of the function in order to keep
track where the actual parameters has been substituted in place of the formal
parameters.
## 4\. Multipolynomials and abstraction
We can now proceed and introduce another fundamental concept for
${\mathbf{iSAPP}}$: multipolynomials. This concept was firstly presented in
[13]. A multipolynomial represents real bounds and its abstraction is a
matrix. In the following, we assume that every polynomial have positive
coefficients and it is in the canonical form.
First, we need to introduce some operator working on polynomials.
###### Definition 4.1 (Union of polynomial).
Be $\mathtt{{p}}$, $\mathtt{{q}}$ the canonical form of the polynomials $p$,
$q$ and let $r,s$ polynomials, $\alpha,\beta$ natural numbers, we define the
operator $({p}\oplus{q})$ over polynomials in the following way:
$({p}\oplus{q})=\begin{cases}\max{(\alpha,\beta)}+({r}\oplus{s})&\text{if
$\mathtt{{p}}=\alpha+r$ and $\mathtt{{q}}=\beta+s$.}\\\
\max{(\alpha,\beta)}\cdot X_{i}+({r}\oplus{s})&\text{otherwise if
$\mathtt{{p}}=\alpha X_{i}+r$ and $\mathtt{{q}}=\beta X_{i}+s$.}\\\
\mathtt{{p}}+\mathtt{{q}}&\text{otherwise}\end{cases}\ $
Let’s see some example. Suppose we have these two polynomials:
${X_{1}}+2{X_{2}}+3{X_{4}}^{2}{X_{5}}$ and
${X_{1}}+3{X_{2}}+3{X_{4}}^{2}{X_{5}}+{X_{6}}$. Call them, respectively $p$
and $q$. We have that $({p}\oplus{q})$ is
${X_{1}}+3{X_{2}}+3{X_{4}}^{2}{X_{5}}+{X_{6}}$.
First we need to introduce the concept of abstraction of polynomial.
Abstraction gives a vector representing the shape of our polynomial and how
variables appear inside it.
###### Definition 4.2 (Abstraction of polynomial).
Let $p(\overline{X})$ a polynomial over $n$ variables,
$\lceil{{p(\overline{X})}}\rceil$ is a column vector of size $n+1$ such that:
* •
If $p(\overline{X})$ is a constant $c>1$, then $\lceil{p(\overline{X})}\rceil$
is $\mathbf{V}^{\mathbf{A}}_{n}$
* •
Otherwise if $p(\overline{X})$ is a constant $0$ or $1$, then
$\lceil{p(\overline{X})}\rceil$ is respectively $\mathbf{V}^{\mathbf{0}}_{n}$
or $\mathbf{V}^{\mathbf{I}}_{n}$.
* •
Otherwise if $p(\overline{X})$ is ${X}_{i}$, then
$\lceil{p(\overline{X})}\rceil$ is $\mathbf{V}^{\mathbf{I}}_{i}$.
* •
Otherwise if $p(\overline{X})$ is $\alpha{X}_{i}$ (for some constant
$\alpha>1$), then $\lceil{p(\overline{X})}\rceil$ is
$\mathbf{V}^{\mathbf{A}}_{i}$.
* •
Otherwise if $p(\overline{X})$ is $q(\overline{X})+r(\overline{X})$, then
$\lceil{p(\overline{X})}\rceil$ is
$\lceil{q(\overline{X})}\rceil+\lceil{r(\overline{X})}\rceil$.
* •
Otherwise, $p(\overline{X})$ is $q(\overline{X})\cdot r(\overline{X})$, then
$\lceil{p(\overline{X})}\rceil$ is $M\cdot\lceil{q(\overline{{X}})}\rceil\cup
M\cdot\lceil{r(\overline{{X}})}\rceil$.
Size of vectors is $n+1$ because $n$ cells are needed for keeping track of $n$
different variables and the last cell is the one associated to constants. We
can now introduce multipolynomials and their abstraction.
###### Definition 4.3 (Multipolynomials).
A multipolynomial is a tuple of polynomials. Formally
$P=(p_{1},\ldots,p_{n})$, where each $p_{i}$ is a polynomial.
In the following, in order to refere to a particular polynomial of a
multipolynomial we will use an index. So, $P_{i}$ refers to the $i$-th
polynomial of $P$. Now that we have introduced the definition of
multipolynomials, we can go on and present two foundamental operation on them:
sum and composition.
###### Definition 4.4 (Sum of multipolynomials).
Given two multipolynomials $P$ and $Q$ over the same set of variables, we
define addition in the following way: $(P\oplus
Q)_{i}=({P_{i}}\oplus{Q_{i}})$.
###### Definition 4.5 (Composition of multipolynomial).
Given two multipolynomials $P$ and $Q$ over the same set of variables, the
composition of two multipolynomials is defined as the composition component-
wise of each polynomial. Formally we define composition in the following way:
$(P\cdot Q)=Q_{1}\cdot P_{1},\ldots,Q_{n}\cdot P_{n}$.
Abstracting a multipolynomial naturally gives a matrix where each column is
the abstraction of one of the polynomials.
###### Definition 4.6.
Let $P$ be a multipolynomial, its abstraction $\lceil{P}\rceil$ is a matrix
where the $i$-th column is the vector $\lceil{P_{i}}\rceil$.
In the following, we use polynomials to bound size of single stacks. Since
handling polynomials is too hard (_i.e._ undecidable), we only keep their
abstraction. Similarly, we use multipolynomials to bound the size of all the
stacks of a program at once. Again, rather than handling the multipolynomials,
we only work with their abstractions.
## 5\. Typing and certification
We presented all the ingredients of ${\mathbf{iSAPP}}$ and we are ready to
introduce certifying rules. Certifying rules, in figure 1, associate at every
command a matrix. We suppose to have $n-1$ stacks. Notice how expressions are
not typed; indeed, we don’t need to type them because their size is fixed.
$n>1$ (Const-A) $\vdash\mathtt{S}:=\langle c_{1}\ldots
c_{n}\rangle:{\mathbf{I}}\xleftarrow{i}{\mathbf{V}^{\mathbf{A}}_{n}}$
(Const-L)
$\vdash\mathtt{S}:=\langle{c_{1}}\rangle:{\mathbf{I}}\xleftarrow{i}{\mathbf{V}^{\mathbf{I}}_{n}}$
(Const-0)
$\vdash\mathtt{S}:=\langle{}\rangle:{\mathbf{I}}\xleftarrow{i}{\mathbf{V}^{\mathbf{0}}_{n}}$
(Axiom-Reg) $\vdash\mathbf{r}:=e:\mathbf{I}$ (Push)
$\vdash\textbf{push}(e,\mathtt{S}_{i}):{\mathbf{I}}\xleftarrow{i}{(\mathbf{V}^{\mathbf{I}}_{n}+\mathbf{V}^{\mathbf{I}}_{i})}$
$\vdash C_{1}:\mathbf{A}$ $\vdash C_{2}:\mathbf{B}$ (Concat) $\vdash
C_{1};C_{2}:\mathbf{A}\times\mathbf{B}$ $\vdash C_{1}:\mathbf{A}$ $\forall
i,{({\mathbf{A}}^{\cup})}_{i,i}<A$ (Loop)
$\vdash\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\}:{({\mathbf{A}}^{\cup})}^{\downarrow{k}}$
$\vdash C:\mathbf{A}$ $\mathbf{A}\leq\mathbf{B}$ (Subtyp) $\vdash
C:\mathbf{B}$ (Asgn)
$\vdash\mathtt{S}_{i}:=\mathtt{S}_{j}:{\mathbf{I}}\xleftarrow{i}{\mathbf{V}^{\mathbf{I}}_{j}}$
$b_{1}\in\mathtt{BooleanExp}$ $\vdash C_{1}:\mathbf{A}$ $\vdash
C_{2}:\mathbf{B}$ (IfThen) $\vdash\textbf{If }\,b_{1}\,\textbf{ Then
}\,C_{1}\,\textbf{ Else }\,C_{2}:\mathbf{A}\cup\mathbf{B}$ $\vdash
C:\mathbf{A}$ (Fun) $\textbf{def }{f}\textbf{ in
}{(\mathtt{S}_{1}\ldots\mathtt{S}_{n})}\,\,\\{{C}\\}\,\,\textbf{out}{(\mathtt{S}_{j})}:\mathbf{A}$
(Skip) $\vdash\textbf{skip}:\mathbf{I}$ (Pop)
$\vdash\textbf{pop}({\mathtt{S}}):\mathbf{I}$ $\textbf{def }{f}\textbf{ in
}{(\mathtt{S}_{1}\ldots\mathtt{S}_{n})}\,\,\\{{C}\\}\,\,\textbf{out}{(\mathtt{S}_{j})}:\mathbf{A}$
(FunCall)
${\mathtt{S}_{i}}:=\textbf{call}(f,\mathtt{S}_{k},\ldots,\mathtt{S}_{p}):{\mathbf{I}}\xleftarrow{k}{\left[\stackrel{{\scriptstyle\mathbf{A}_{j}}}{{(1\rightarrow
k,\ldots,n\rightarrow p)}}\right]}$
Figure 1. Typing rules for commands and functions
These matrices tell us about the behaviour of a command and functions. We can
think about them as certificates. Certificates for commands tell us about the
correlation between input and output stacks. Each column gives the bound of
one output stack while each row corresponds to one input stack. Last row and
column handle constants.
As example, command (Skip) tells us that no stack is changed. Concatenation of
commands (Concat) tells us how to find a certificate for a series of commands.
The intrinsic meaning of matrix multiplication is to “connect” output of the
first certificate with input of the second. In this way we rewrite outputs of
the second certificate respect to inputs of the first one. Notice how the rule
for (Push) does not have any hypothesis. Indeed, this command just increase by
$+1$ (a constant) the size of the stack $\mathtt{S}_{i}$. When there is a
test, taking the union (_i.e._ maximum) of the certificates means taking the
worst possible case between the two branches. The most interesting type rule
is the one concerning the (Loop) command. The right premise acts as a guard:
an $A$ on the diagonal means that there is a stack $\mathtt{S}_{i}$ such that
iterating the loop a certain number of time results in (the size of)
$\mathtt{S}_{i}$ depending affinely of itself, _e.g._
$|\mathtt{S}_{i}|=2\times|\mathtt{S}_{i}|$. Obviously, iterating this loop may
create an exponential growth, so we stop the analysis immediately. Next, the
union closure used as a certificate corresponds to a worst case scenario. We
can’t know if the loop will be executed 0, 1, 2, …times each corresponding to
certificates $\mathbf{A}^{0},\mathbf{A}^{1},\mathbf{A}^{2},\ldots$ Thus we
assume the worst and take the union of these, that is the union closure.
Finally, the loop correction (merge down) is here to take into account the
fact that the result will also depends on the size of the stack controlling
the loop (_i.e._ the index $k$ is the number of the variable $S_{k}$
controlling the loop).
Before start to prove the main theorems, let present some examples using the
commands $\textbf{call}()$, $\textbf{loop}\,{}\,\\{{}\\}$. In the following we
will use integer number like $0,1,2,\ldots$ intending a constant list of size
$0,1,2,\ldots$. This should help the reader.
###### Example 5.1 (Addition).
We are going to present the function $+$ (a shortcut for the following
function). We can check that the analysis of this function is exactly the one
expected. The size of the result is the sum of the sizes of the two stacks.
def addition in ($S_{1},S_{2}$){
$S_{3}:=S_{2}$
loop ($S_{2}$){
push($\mathtt{top}({S_{3}}),S_{1}$)
pop($S_{3}$)
}
}out($S_{1}$)
The associate matrix of this function is exactly what we are expecting.
Indeed, the matrix is the following one: $\begin{bmatrix}L&0&0&0\\\ L&L&L&0\\\
0&0&0&0\\\ 0&0&0&L\\\ \end{bmatrix}$
###### Example 5.2 (Multiplication).
In the following we present a way to type multiplication between a number and
a variable. In the following $S_{2}$ is multiplied by $n$ and the result is
stored in $S_{1}$.
$S_{1}:=0$
loop ($S_{2}$){
$S_{1}=S_{1}+n$
}
typed with $\begin{bmatrix}0&0&0\\\ A&L&0\\\ 0&0&L\\\ \end{bmatrix}$
###### Example 5.3 (Multiplication).
In this example we show how to type a multiplication between two variables.
def multiplication in ($S_{1},S_{2}$){
$S_{3}:=0$
loop ($S_{2}$){
$S_{3}:=S_{1}+S_{3}$
}
}out($S_{1}$)
The loop is typed with the matrix $\begin{bmatrix}L&0&M&0\\\ 0&L&M&0\\\
0&0&L&0\\\ 0&0&0&L\\\ \end{bmatrix}$.
So, the entire function is typed with $\begin{bmatrix}L&0&M&0\\\ 0&L&M&0\\\
0&0&0&0\\\ 0&0&0&L\\\ \end{bmatrix}$, as is was expected.
###### Example 5.4 (Subtraction).
In this example we show how to type the subtraction between two variables.
def subtraction in ($S_{1},S_{2}$){
loop ($S_{2}$){
$\textbf{pop}({S_{1}})$
}
}out($S_{1}$)
The function is typed with the identity matrix $\mathbf{I}$, since the
$\textbf{pop}({})$ command is typed with the identity.
## 6\. Semantics
Semantics of the programs generated by the grammar in def 2.1 is the usual and
expected one. In the following we are using $\sigma$ as the state function
associating to each variable a stack and to each register a letter. Semantics
for boolean value is labelled with probability, while semantics for
expressions ($\rightarrow_{a}$) is not carrying anything. In figure 2 is shown
the semantic for booleans and expressions. Most of boolean operator have
probability $1$ and operator rand reduced to $\mathsf{true}$ or
$\mathsf{false}$ with probability $\frac{1}{2}$. Notice how there is no
semantic associated to operators $op?()$ and $op()$. Of course, their
semantics depends on how they will be implemented.
Since semantics for boolean is labelled with probability, also semantics of
commands ($\rightarrow_{c}^{{\alpha}}$) is labelled with a probability, It
tells us the probability to reach a particularly final state after having
execute a command from a initial state.
if $\sigma(S)=\langle\rangle$ $\langle
S,\sigma\rangle\rightarrow_{b}^{{1}}\mathsf{true}$ if
$\sigma(S)!=\langle\rangle$ $\langle
S,\sigma\rangle\rightarrow_{b}^{{1}}\mathsf{false}$
$\langle\mathtt{rand},\sigma\rangle\rightarrow_{b}^{{1/2}}\mathsf{true}$
$\langle\mathtt{rand},\sigma\rangle\rightarrow_{b}^{{1/2}}\mathsf{false}$
$\langle\mathbf{r},\sigma\rangle\rightarrow_{a}\sigma(\mathbf{r})$ if
$\sigma(S)=\langle c_{1}\ldots c_{n}\rangle$
$\langle\mathtt{top}({S}),\sigma\rangle\rightarrow_{a}c_{1}$
Figure 2. Semantics of booleans and expressions
In figure 3 are presented the semantics for commands. Since a compile time all
the functions definitions can be collected, we suppose that exists a set of
defined function called $\mathtt{DefinedFunctions}$ where all the functions
defined belong.
$\langle\textbf{skip},\sigma\rangle\rightarrow_{c}^{{1}}\sigma$
$\sigma(S)=\langle<c_{1},c_{2},\ldots,c_{n}>\rangle$
$\langle\textbf{pop}({S}),\sigma\rangle\rightarrow_{c}^{{1}}\sigma[S/\langle
c_{2},\ldots,c_{n}\rangle]$ $\sigma(S)=\langle<c_{1},\ldots,c_{n}>\rangle$
$\langle\textbf{push}(e,S),\sigma\rangle\rightarrow_{c}^{{1}}\sigma[S/\langle
e,c_{1},\ldots,c_{n}\rangle]$
$\langle\mathbf{r}:=e_{1},\sigma\rangle\rightarrow_{c}^{{1}}\sigma[\mathbf{r}/e_{1}]$
$\langle
S_{1}:=S_{2},\sigma\rangle\rightarrow_{c}^{{1}}\sigma[S_{1}/\sigma(S_{2})]$
$\langle S_{1}:=\langle
c_{1},\ldots,c_{n}\rangle,\sigma\rangle\rightarrow_{c}^{{1}}\sigma[S_{1}/\langle
c_{1},\ldots,c_{n}\rangle]$ $\textbf{def }{myfun}\textbf{ in
}{(S_{1},\ldots,S_{n})}\,\,\\{{C}\\}\,\,\textbf{out}{(S_{m})}\in\mathtt{DefinedFunctions}$
$\langle
C,\sigma[S_{1},\ldots,S_{n}/\overline{S}]\rangle\rightarrow_{c}^{{\alpha}}\sigma_{1}$
$\langle
S_{p}:=\textbf{call}(myfun,\overline{S}),\sigma\rangle\rightarrow_{c}^{{\alpha}}\sigma_{1}$
$\langle C_{1},\sigma_{1}\rangle\rightarrow_{c}^{{\alpha}}\sigma_{2}$ $\langle
C_{2},\sigma_{2}\rangle\rightarrow_{c}^{{\beta}}\sigma_{3}$ $\langle
C_{1};C_{2},\sigma\rangle\rightarrow_{c}^{{\alpha\beta}}\sigma_{3}$ $\langle
b_{1},\sigma\rangle\rightarrow_{b}^{{\alpha}}\mathsf{true}$ $\langle
C_{1},\sigma\rangle\rightarrow_{c}^{{\beta}}\sigma_{1}$ $\langle\textbf{If
}\,b_{1}\,\textbf{ Then }\,C_{1}\,\textbf{ Else
}\,C_{2},\sigma\rangle\rightarrow_{c}^{{\alpha\beta}}\sigma_{1}$ $\langle
S_{k},\sigma\rangle\rightarrow_{a}\langle\rangle$
$\langle\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\},\sigma\rangle\rightarrow_{c}^{{1}}\sigma$
$\langle b_{1},\sigma\rangle\rightarrow_{b}^{{\alpha}}\mathsf{false}$ $\langle
C_{2},\sigma\rangle\rightarrow_{c}^{{\beta}}\sigma_{1}$ $\langle\textbf{If
}\,b_{1}\,\textbf{ Then }\,C_{1}\,\textbf{ Else
}\,C_{2},\sigma\rangle\rightarrow_{c}^{{\alpha\beta}}\sigma_{1}$ $\langle
S_{k},\sigma\rangle\rightarrow_{a}\langle c_{1}\ldots c_{n}\rangle$ $\langle
C_{1},\sigma\rangle\rightarrow_{c}^{{\alpha_{1}}}\sigma_{1}$ $\ldots$ $\langle
C_{1},\sigma_{n-1}\rangle\rightarrow_{c}^{{\alpha_{n}}}\sigma_{n}$
$\langle\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\},\sigma\rangle\rightarrow_{c}^{{\Pi\alpha_{i}}}\sigma_{n}$
Figure 3. semantics of commands
Since ${\mathbf{iSAPP}}$ is working on stochastic computations, in order to
reach soundness and completeness respect to $\mathbf{PP}$, we need to define a
semantics for distribution of final states. We need to introduce some more
definitions. Let $\mathscr{D}$ be a distribution of probabilities over states.
Formally, $\mathscr{D}$ is a function whose type is
$(\mathtt{Stacks}\rightarrow\mathtt{Values})\rightarrow\alpha$. Sometimes we
will use the following notation
$\mathscr{D}=\\{\sigma_{1}^{\alpha_{1}},\ldots,\sigma_{n}^{\alpha_{n}}\\}$
indicating that probability of $\sigma_{i}$ is $\alpha_{i}$.
We can so define semantics for distribution; the most important rules are
shown in Figure 4. Since semantics for some commands computes with probability
equal to $1$, the correspondent rule for distributions is not presented.
Unions of distributions and multiplication between real number and a
distribution have the natural meaning. Notice also how all the final
distributions are normalized distributions.
$\langle C_{1},\sigma\rangle\rightarrow_{\mathscr{D}}\mathscr{D}$
$\forall\sigma_{i}\in\mathscr{D}.\langle
C_{2},\sigma_{i}\rangle\rightarrow_{\mathscr{D}}\mathscr{E}_{i}$ $\langle
C_{1};C_{2},\sigma\rangle\rightarrow_{\mathscr{D}}\bigcup_{i}\mathscr{D}(\sigma_{i})\cdot\mathscr{E}_{i}$
$\langle S_{k},\sigma\rangle\rightarrow_{a}0$
$\langle\textbf{loop}\,{S_{k}}\,\\{{C}\\},\sigma\rangle\rightarrow_{\mathscr{D}}\\{\sigma^{1}\\}$
$\langle S_{k},\sigma\rangle\rightarrow_{a}\langle c_{1}\ldots c_{n}\rangle$
$\langle\,\overbrace{C;C;\ldots;C}^{n},\sigma\rangle\rightarrow_{\mathscr{D}}\mathscr{E}$
$\langle\textbf{loop}\,{S_{k}}\,\\{{C}\\},\sigma\rangle\rightarrow_{\mathscr{D}}\mathscr{E}$
$\langle b,\sigma\rangle\rightarrow_{b}^{{\alpha}}\mathsf{true}$ $\langle
C_{1},\sigma\rangle\rightarrow_{\mathscr{D}}\mathscr{D}$ $\langle
C_{2},\sigma\rangle\rightarrow_{\mathscr{D}}\mathscr{E}$ $\langle\textbf{If
}\,b\,\textbf{ Then }\,C_{1}\,\textbf{ Else
}\,C_{2},\sigma\rangle\rightarrow_{\mathscr{D}}(\alpha\cdot\mathscr{D})\cup((1-\alpha)\cdot\mathscr{E})$
Figure 4. Distributions of output states
Here we can present our first result.
###### Theorem 6.1.
A command $C$ in a state $\sigma_{1}$ reduce to another state $\sigma_{2}$
with probability equal to $\mathscr{D}(\sigma_{2})$, where $\mathscr{D}$ is
the distribution of probabilities over states such that $\langle
C,\sigma_{1}\rangle\rightarrow_{\mathscr{D}}\mathscr{D}$.
Proof is done by structural induction on derivation tree. It is quite easy to
check that this property holds, as the rules in Figure 4 are showing us
exactly this statement. The reader should also not be surprised by this
property. Indeed, we are not considering just one possible derivation from
$\langle C_{1},\sigma_{1}\rangle$ to $\sigma_{2}$, but all the ones going from
the first to the latter.
## 7\. Soundness
The language recognised by ${\mathbf{iSAPP}}$ is an imperative language where
the iteration schemata is restricted and the size of objects (here, stacks) is
bounded. These are ingredients of a lot of well known ICC polytime systems.
There is no surprise that every program certified by ${\mathbf{iSAPP}}$ runs
in probabilistic polytime.
Now we can start to present theorems and lemmas of our system. First we will
focus on multipolynomial properties in order to show that the behaviour of
these algebraic constructor is similar to the behaviour of matrices in our
system. Finally we will link these things together to get polytime bound for
${\mathbf{iSAPP}}$. Here are two fundamental lemmas. Their proofs are
straightforward.
###### Lemma 7.1.
Let $p$ and $q$ two positive polynomials, then it holds that $\lceil{p\oplus
q}\rceil=\lceil{p}\rceil\cup\lceil{q}\rceil$.
###### Proof 7.2.
by induction on the size of the two polynomials. By definition the union
between two polynomial is defined in 4.1 as the maximum of the comparable
monomials. Let’s analyze the different cases:
* •
If $p=c_{1}+r$ and $q=c_{2}+s$. By induction, $\lceil{r\oplus
s}\rceil=\lceil{r}\rceil\cup\lceil{s}\rceil$. By definition 4.1, $p\oplus q$
is $\max{(\alpha,\beta)}+({r}\oplus{s})$ and so, by definition 4.2, the
abstraction is defined as
$\lceil{\max{(\alpha,\beta)}}\rceil+\lceil{({r}\oplus{s})}\rceil$. By using
induction hypothesis we get
$\lceil{\max{(\alpha,\beta)}}\rceil+\lceil{r}\rceil\cup\lceil{s}\rceil$. It’s
clear that $\lceil{\max{(\alpha,\beta)}}\rceil$ is equal to
$\max{(\lceil{\alpha}\rceil,\lceil{\beta}\rceil)}$, since the abstraction take
in account the value of the constants. This is, by definition, the union of
the two abstracted polynomials. We get, so
$(\lceil{\alpha}\rceil\cup\lceil{\beta}\rceil)+(\lceil{r}\rceil\cup\lceil{s}\rceil)$.
Notice how the abstractions of the two constants are two column vectors having
$0$ everywhere except for the last row. We can so rewrite the previous
equation as
$(\lceil{\alpha}\rceil+\lceil{r}\rceil)\cup(\lceil{\beta}\rceil+\lceil{s}\rceil)$,
that is the thesis.
* •
If $\mathtt{{p}}=\alpha X_{i}+r$ and $\mathtt{{q}}=\beta X_{i}+s$. This case
is very similar to the previous one.
* •
The last case is where the two polynomials are not comparable. In this case,
the union is defined as $p+q$. There are two cases:
* –
If some variables are present just in one polynomial and not in the other one,
then the correspondent rows, for each single variable, is not influenced by
the abstraction of the polynomial in which the variable does not appear.
* –
If some variables are present in both. In this case it means that the
variables appear in at least one monomial with grade gretar than one or in a
monomial having more than one variable. In both cases the associated
abstracted value for both is $M$.
The thesis holds. This concludes the proof.
###### Lemma 7.3.
Let $P$ and $Q$ two positive multipolynomials, then it holds that
$\lceil{({P}\oplus{Q})}\rceil=\lceil{P}\rceil\cup\lceil{Q}\rceil$.
###### Proof 7.4.
By definition of sum between multipolynomials 4.1 we know that sum is defined
componentwise, $(P\oplus Q)_{i}=({P_{i}}\oplus{Q_{i}})$. By lemma 7.1 we prove
the theorem.
###### Lemma 7.5.
Let $P$ and $Q$ two positive multipolynomials (over $n$ variables) in
canonical form, then it holds that $\lceil{P\cdot
Q}\rceil\leq\lceil{Q}\rceil\times\lceil{P}\rceil$
###### Proof 7.6.
We will consider the element in position $i,j$ and so we have:
$(\lceil{Q}\rceil\times\lceil{P}\rceil)_{i,j}=\sum_{k}{\lceil{Q}\rceil_{i,k}\times\lceil{P}\rceil_{k,j}}$.
We can start by making some algebraic passages:
$\lceil{P\cdot
Q}\rceil_{i,j}=\lceil{P(Q_{1},\ldots,Q_{n})}\rceil_{i,j}=\lceil{P_{j}(Q_{1},\ldots,Q_{n})}\rceil_{i}$
The equality holds because we are considering the element in the $j$-th
column. Since we are interested at the element in position $i$-th we have to
understand how the variable $X_{i}$ (or constant) in each $Q_{k}$ is
substituted.
* •
Case where $i=n+1$.
* –
If none of the polynomials $Q_{k}$ has a constant inside, then the proof is
evident, since the only possible constant appearing in the result is the
possible constant appearing in $P_{j}$. Recall that the element in position
$(n+1,n+1)$ is $L$ by definition, so
$\sum_{k}{\lceil{Q}\rceil_{i,k}\times\lceil{P}\rceil_{k,j}}$ contain at least
$\lceil{P}\rceil_{n+1,i}$.
* –
Otherwise some constants appear in some $Q_{k}$. This means that the expected
abstraction for the element at position $(n+1,j)$ may be $A$ or $L$. If $L$ is
the result, then is clear that and equality holds, since it means that the
constant is $1$. The inequality hold if the expected result is $A$, since that
on the right side we have to perform the following sum:
$\sum_{k}{\lceil{Q}\rceil_{i,k}\times\lceil{P}\rceil_{k,j}}$ and we could find
an $A$ or $M$ value.
* •
Case where $i<n+1$. In this case we are considering how the variable $X_{i}$
appears. We have four possibilities:
* –
If $X_{i}$ does not appear in any $Q_{k}$ polynomials. In this case the
expected abstract value is $0$. Is easy to check that this holds, since on the
left side of the inequality we get $0$ and on the right side we get
$\sum_{k}{\lceil{Q}\rceil_{i,k}\times\lceil{P}\rceil_{k,j}}$ that is $0$,
since all $\lceil{Q}\rceil_{i,k}$ are $0$.
* –
In the following we will consider that $X_{i}$ appears in some $Q_{k}$
polynomials. Call them $\overline{Q}_{X_{i}}$. If some of the polynomials
where $X_{i}$ appears is substituted in some monomial of $P_{j}$ of shape as
$\alpha X_{p}\cdot q(\overline{X})$ in place of some $X_{p}$, then for sure on
the right side of the inequality we will get a value $M$. On the right side,
considering $\sum_{k}{\lceil{Q}\rceil_{i,k}\times\lceil{P}\rceil_{k,j}}$ we
will multiply for sure an $M$ value with the abstracted value for $X_{i}$ of
the $\overline{Q}_{X_{i}}$ where it appears. The result is so for sure $M$.
* –
Otherwise, if some of the polynomials where $X_{i}$ appears is substituted in
some monomial of $P_{j}$ of shape as $\alpha X_{p}$ ($\alpha>1$), then the
expected abstract value depends on how $X_{i}$ appears in
$\overline{Q}_{X_{i}}$. For all the three possible cases of $X_{i}$ in
$\overline{Q}_{X_{i}}$ the abstracted value obtained on the left side is equal
to the value obtained on the right side.
* –
Otherwise, $X_{i}$ appears is substituted in some monomial of $P_{j}$ of shape
as $X_{p}$; then the substitution gives in output exactly the
$\overline{Q}_{X_{i}}$ substituted. The equality holds because on the right
side we are going to multiply by $L$ the abstracted value found for each
$Q_{k}$.
This concludes the proof.
Let’s now present the results about the probabilistic polytime soundness. The
following theorem tell us that at each step of execution of a program, size of
variables are polynomially correlated with size of variables in input.
###### Theorem 7.7.
Given a command $C$ well typed in ${\mathbf{iSAPP}}$ with matrix $\mathbf{A}$,
such that $\langle C,\sigma_{1}\rangle\rightarrow_{c}^{{\alpha}}\sigma_{2}$ we
get that exists a multipolynomial $P$ such that for all stacks $S_{i}$ we have
that $|\sigma_{2}(S_{i})|\leq
P_{i}(|\sigma_{1}(S_{1})|,\ldots,|\sigma_{1}(S_{n})|)$ and $\lceil{P}\rceil$
is $\mathbf{A}$.
###### Proof 7.8.
By structural induction on typing tree. We will present just the most
important cases.
* •
If the last rule is (Const-0), it means that we have only one stack and its
size is $0$. The relative vector in the matrix is a $\mathbf{V}^{\mathbf{0}}$.
We can choose the constant polynomial $0$, whose abstraction is exactly
$\mathbf{V}^{\mathbf{0}}$. The polynomial $0$ bounds the size of the stack.
* •
If the last rule is one of the following (Skip), (Const-A), (Const-L), (Axiom-
Reg), then the proof is trivial.
* •
If the last rule is (Push), then we know that the size of of the stack $S_{i}$
has been increased by $1$. The associated vector is a column vector having $L$
on the $i$-th row and $L$ on the last line. The correspondent polynomial,
$X_{i}+1$, is the correct bounding polynomial for the $i$-th stack.
* •
If the last rule is (Subtyp), then by induction on the hypothesis we can
easily find a new polynomial bound.
* •
If the last rule is (Asgn), then we know that the size of the $i$-th stack is
equal to the size of the stack $j$-th. So, the polynomial bounding the size of
the $i$-th stack uses at least two variables and the correct one is
$P_{i}(X_{i},X_{j})=X_{j}$.
* •
If the last rule is (IfThen), then by applying induction hypothesis on the two
premises we have multipolynomial bounds $Q$, $R$ such that
$\lceil{Q}\rceil=\mathbf{A}$ and $\lceil{R}\rceil=\mathbf{B}$. By lemma 7.3 we
get the thesis.
* •
If the last rule is (Fun), then by applying the induction hypothesis on the
premise we directly prove the thesis.
* •
If the last rule is (FunCall), then by applying the induction hypothesis on
the premise we have a polynomial bound $Q$ such that
$\lceil{Q}\rceil=\mathbf{A}$. For all the stacks different from the $i$-th,
the bounding polynomial is trivial, while for the stack $S_{i}$ depends on the
result of the function call.
The function return the value stored in the $j$-th stack used inside the
function. According to the actual parameters, the actual polynomial bound is
different from the one retrieving by applying the induction hypothesis.
* •
If last rule is (Concat), then by lemma 7.5 we can easily conclude the thesis.
* •
If last rule is (Loop), we are in the following case; so, $\mathbf{A}$ is
${({\mathbf{B}}^{\cup})}^{\downarrow{k}}$. The typing and the associated
semantic are the following:
$\vdash C_{1}:\mathbf{B}$ $\forall i,{({\mathbf{B}}^{\cup})}_{i,i}<A$ (Loop)
$\vdash\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\}:{({\mathbf{B}}^{\cup})}^{\downarrow{k}}$
$\langle S_{k},\sigma\rangle\rightarrow_{a}\langle c_{1}\ldots c_{n}\rangle$
$\langle C_{1},\sigma\rangle\rightarrow_{c}^{{\alpha_{1}}}\sigma_{1}$ $\ldots$
$\langle C_{1},\sigma_{n-1}\rangle\rightarrow_{c}^{{\alpha_{n}}}\sigma_{n}$
$\langle\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\},\sigma\rangle\rightarrow_{c}^{{\Pi\alpha_{i}}}\sigma_{n}$
We consider just the case where $n>0$, since the other one is trivial. By
induction on the premise we have a multipolynomial $P$ bound for command
$C_{1}$ such that its abstraction is $\mathbf{B}$. If $P$ is a bound for
$C_{1}$, then $P\cdot P$ is a bound for $C_{1};C_{1}$ and $(P\cdot P)\cdot P$
is a bound for $C_{1};C_{1};C_{1}$ and so on. All of these are multipolynomial
because we are composing multipolynomials with multipolynomials.
By lemma 7.5 and knowing that $\lceil{P}\rceil$ is $\mathbf{B}$ we can easily
deduce to have a multipolynomial bound for every iteration of command $C_{1}$.
In particularly by lemma 7.3 we can easily sum up everything and find out a
multipolynomial $Q$ such that $\lceil{Q}\rceil$ is ${\mathbf{B}}^{\cup}$. This
means that further iterations of sum of powers of $P$ will not change the
abstraction of the result.
So, for every iteration of command $C_{1}$ we have a multipolynomial bound
whose abstraction cannot be greater than ${\mathbf{B}}^{\cup}$. So, we study
the worst case; we analyse the matrix ${\mathbf{B}}^{\cup}$.
Side condition on (Loop) rule tells us to check elements on the main diagonal.
Recall that by definition of union closure, elements on the main diagonal are
supposed to be greater then $0$. We required also to be less then $A$. Let’s
analyse all the possibilities of an element in position $i,i$:
* –
Value $0$ means no dependencies. If value is $L$ it means that $Q_{i}$
concrete bound for such column has shape $S_{i}+r(\overline{S})$, where
$S_{i}$ does not appear in $r(\overline{S})$. Iteration of such assignment
gives us polynomial bound increment of the value of variable $S_{i}$.
* –
If value is $A$ could means that $Q_{i}$ concrete bound for such column has
shape $\alpha S_{i}+r(\overline{S})$ (for some $\alpha>1$), where $S_{i}$ does
not appear in $r(\overline{S})$. Iteration of such assignment lead us to
exponential blow up on the size of $S_{i}$.
* –
Otherwise value is $M$. This case is worse than the previous one. It’s evident
that we could have exponential blow up on the size of $S_{i}$.
The abstract bound ${\mathbf{B}}$ is still not a correct abstract bound for
the loop because loop iteration depends on some variable $S_{k}$. We need to
adjust our bound in order to keep track of the influence of variable $S_{k}$
on loop iteration.
We take multipolynomial $Q$ because we know that further iterations of the
algorithm explained before will not change its abstraction $\lceil{Q}\rceil$.
Looking at $i$-th polynomial of multipolynomial $Q$ we could have three
different cases. We behave in the following way:
* –
The polynomial has shape $S_{i}+p(\overline{S})$. In this case we multiply the
polynomial $p$ by $S_{k}$ because this is the result of iteration. We
substitute the $i$-th polynomial with the canonical form of polynomial
$S_{i}+p(\overline{S})\cdot S_{k}$.
* –
The polynomial has shape $S_{i}+\alpha$, for some constant $\alpha$. In this
case we substitute with $S_{i}+\alpha\cdot S_{k}$.
* –
The polynomial has shape $S_{i}$ or $S_{i}$ does not appear in the polynomial.
We leave as is.
In this way we generate a new multipolynomial, call it $R$. The reader should
easily check that these new multipolynomial expresses a good bound of
iterating $Q$ a number of times equal to $S_{k}$. Should also be quite easy to
check that $\lceil{R}\rceil$ is exactly
${({\mathbf{B}}^{\cup})}^{\downarrow{k}}$. This concludes the proof.
Polynomial bound on size of stacks is not enough; we should also prove
polynomiality of number of steps. Since all the programs generated by the
language terminate and all the stacks are polynomially bounded in their size,
the theorem follows straightforward.
###### Theorem 7.9.
Let $C$ be a command well typed in ${\mathbf{iSAPP}}$ and
$\sigma_{1},\sigma_{n}$ state functions. If $\pi:\langle
C_{1},\sigma_{0}\rangle\rightarrow_{c}^{{\alpha}}\sigma_{n}$, then there is a
polynomial $p$ such that $|\pi|$ is bounded by
$p(\sum_{i}|\sigma_{0}(S_{i})|)$.
###### Proof 7.10.
By induction on the associated semantic proof tree.
### 7.1. Probabilistic Polynomial Soundness
Nothing has been said about probabilistic polynomial soundness. Theorems 7.7
and 7.9 tell us just about polytime soundness. Probabilistic part is now
introduced. We will prove probabilistic polynomial soundness following idea in
[6], by using “representability by majority”.
###### Definition 7.11 (Representability by majority).
Let $\overline{\sigma_{0}}[S/n]$ define as $\forall S,\sigma_{0}(S)=n$. Then
$C$ is said to _represent-by-majority_ a language $L\subseteq\mathbb{N}$ iff:
1. (1)
If $n\in L$ and $\langle
C,\overline{\sigma_{0}}[S/n]\rangle\rightarrow_{\mathscr{D}}\mathscr{D}$, then
$\mathscr{D}(\sigma_{0})\geq\sum_{m>0}\mathscr{D}(\sigma_{m})$;
2. (2)
If $n\notin L$ and $\langle
C,\overline{\sigma_{0}}[S/n]\rangle\rightarrow_{\mathscr{D}}\mathscr{D}$, then
$\sum_{m>0}\mathscr{D}(\sigma_{m})>\mathscr{D}(\sigma_{0})$.
Observe that every command $C$ in ${\mathbf{iSAPP}}$ represents by majority a
language as defined in 7.11. In literature [1] is well known that we can
define $\mathbf{PP}$ by majority. We say that the probability error should be
at most $\frac{1}{2}$ when we are considering string in the language and
strictly smaller than $\frac{1}{2}$ when the string is not in the language. So
we can easily conclude that ${\mathbf{iSAPP}}$ is sound also respect to
probabilistic polytime.
## 8\. Probabilistic Polynomial Completeness
There are several way to demonstrate completeness respect to some complexity
class. We will show that by using language recognised by our system we are
able to encode Probabilistic Turing Machines (PTM). We will are not able to
encode all possible PTMs but all the ones with particularly shape. This lead
us to reach extensional completeness. For every problem in $\mathbf{PP}$ there
is at least an algorithm solving that problem that is recognised by
${\mathbf{iSAPP}}$.
A Probabilistic Turing Machine [7] can be seen as non deterministic TM with
one tape where at each iteration are able to flip a coin and choose between
two possible transition functions to apply.
In order to encode Probabilistic Turing Machines we will proceed with the
following steps:
* •
We show that we are able to encode polynomials. In this way we are able to
encode the polynomial representing the number of steps required by the machine
to complete.
* •
We encode the input tape of the machine.
* •
We show how to encode the transition $\delta$ function.
* •
We put all together and we have an encoding of a PTM running in polytime.
Should be quite obvious that we can encode polynomials in ${\mathbf{iSAPP}}$.
Grammar and examples 5.1, 5.2, 5.3, 5.4 give us how encode polynomials.
We need to encode the tape of our PTMs. We subdivide our tape in three sub-
tapes. The left part $\mathbf{tape}_{l}$, the head $\mathbf{tape}_{h}$ and the
right part $\mathbf{tape}_{r}$. $\mathbf{tape}_{r}$ is encoded right to left,
while the left part is encoded as usual left to right.
Let’s move on and present the encoding of transition function of PTMs.
Transition function of PTMs, denoted with $\delta$, is a relation
$\delta\subseteq(Q\times\Sigma)\times(Q\times\Sigma\times\\{\leftarrow,\downarrow,\rightarrow\\})$.
Given an input state and a symbol it may give in output more tuples of state,
a symbol and a direction of the head (left, no movement, right).
In the following we are going to present two procedures to encode movements of
the head. It is really important to pay attention on how we encode this
operations. Recall that a PTM loops the $\delta$ function and our system
requires that the matrix certifying/typing the loop needs to have values of
the diagonal less than $A$.
###### Definition 8.1 (Move head to right).
Moving head to right means to concatenate the bit pointed by the head to the
left part of the tape; therefore we need to retrieve the first bit of the
right part of the tape and associate it to the head. Procedure is presented as
algorithm 1; call it MoveToRight().
$\textbf{push}(\mathtt{top}({\mathbf{tape}_{h}}),\mathbf{tape}_{l})$
$\mathbf{tape}_{h}:=\langle\rangle$
$\textbf{push}(\mathtt{top}({\mathbf{tape}_{r}}),\mathbf{tape}_{h})$
$\textbf{pop}({\mathbf{tape}_{r}})$
Using typing rules we are able to type the algorithm with the following
matrix:
$\begin{bmatrix}L&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&L&0&0\\\ 0&0&0&L&0\\\
L&A&0&0&L\\\ \end{bmatrix}$
Algorithm 1 Move head to right
The first column of the matrix represents dependencies for variables
$\mathbf{tape}_{l}$, the second represents $\mathbf{tape}_{h}$, third is
$\mathbf{tape}_{r}$, forth is $\mathbf{M_{state}}$ and finally recall that
last column is for constants. In the following, columns of matrices are
ordered in this way.
Similarly we can encode the procedure for moving the head to left and the
possibility of not moving at all, that is a skip command. So, the $\delta$
function is then encoded in the standard way by having nested If-Then-Else
commands, checking the value of $\mathtt{rand}$, the state, the symbol on a
tape and performing the right procedure.
if $\mathtt{rand}$ then
if ${\textbf{equal?}(\mathbf{M_{state}},1)}$ then
else
if ${\textbf{equal?}(\mathbf{M_{state}},2)}$ then
$\cdots$
else
$\cdots$
end if
end if
else
$\cdots$
end if
The prototype is created by nesting If-Then-Else commands and checking the
state of the machine. for each branch, then, an operation of moving the head
is performed. Notice that since three possible operations could be performed,
all the nested If-Then-Else are typed with the following matrix:
$\begin{bmatrix}L&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&L&0&0\\\ 0&0&0&L&0\\\
L&A&L&0&L\\\ \end{bmatrix}$
Algorithm 2 Prototype of encoded $\delta$ function
Finally, we have to put the encoded $\delta$-function inside a loop. The
machine runs in a polynomial number of steps. Since the encoded
$\delta$-function is typed with the matrix presented in Alg. 2, we can easily
see that the union closure of that matrix fits the constraints of the typing
rule of $\textbf{loop}\,{}\,\\{{}\\}$. We can therefore conclude that we can
encode Probabilistic Turing Machine working in polytime.
## 9\. Polynomiality
In this last session we will discuss why ${\mathbf{iSAPP}}$ is a feasible
analyser. We already shown that is sound, so is able to understand whenever a
program does not run in Probabilistic Polynomial Time. Moreover, is also
complete, respect to $\mathbf{PP}$; this means that ${\mathbf{iSAPP}}$ is able
to recognise a lot of programs. At least one for each problem in
$\mathbf{PP}$. The final question has to do with the efficiency of our system:
“how much time does it take ${\mathbf{iSAPP}}$ to check a program?”. Can be
shown that ${\mathbf{iSAPP}}$ is running in polytime respect to the number of
variables used.
Since the typing rules are deterministic, the key problems lays on the rule
(Loop).
$\vdash C_{1}:\mathbf{A}$ $\forall i,{({\mathbf{A}}^{\cup})}_{i,i}<A$ (Loop)
$\vdash\textbf{loop}\,{S_{k}}\,\\{{C_{1}}\\}:{({\mathbf{A}}^{\cup})}^{\downarrow{k}}$
It is not trivial to understand how much it takes a union closure to be
performed. While all the typing rules for all the other commands and
expressions are trivial, the one for loop needs some more explanations. By
definition, ${\mathbf{A}}^{\cup}$ is defined as $\cup_{i}\mathbf{A}^{i}$.
Every matrix could be seen as adjacency matrix of a graph.
As example, the following matrix $\mathbf{A}$: $\begin{bmatrix}L&0&0&0\\\
L&L&L&0\\\ L&0&0&0\\\ 0&M&L&L\\\ \end{bmatrix}$ has its own representation in
the graph on the right side.
Figure 5. Example of graph representing a matrix in ${\mathbf{iSAPP}}$
In the example in Figure 5 we can easily check that $C$ flows in $S_{1}$ with
$M$ in one step. So, $\mathbf{A}^{2}$ have $M$ in position $(4,1)$, $(4,2)$,
$(4,3)$. Indeed, by using the rule of our algebra we can see how dependencies
flows in the graph.
How many unions have to be performed in order to calculate
$\cup_{i}\mathbf{A}^{i}$? In order to answer to this question, we can prove
the following theorem.
###### Theorem 9.1 (Polynomiality).
Given a squared matrix $\mathbf{A}$ of size $n$ and
$\mathbf{B}=\bigcup_{i}{\mathbf{A}^{i}}$, we get that
$\mathbf{B}=\bigcup_{i<n^{2}}\mathbf{A}^{i}$. Union closure can be calculated
by considering just the first $n^{2}$ matrix power.
###### Proof 9.2.
Here is the scratch of the proof. Since the matrix is an encoding of a flow
graph, we can see the matrix as a graph of dependencies between stacks size.
Recall that the union is component-wise, so we can focus on a singular element
of a matrix. Given two nodes $S_{1}$ and $S_{2}$ of our graph, let’s check all
the possibilities:
* •
The expected value is $M$. If so, after no more than $n$ iteration of
$\mathbf{A}$ we should have found it. If not, there are no possibilities to
have $M$ in that position. After $n$ iteration, the information has flown
through all the nodes.
* •
The expected value is $A$. We need to iterate more than $n$ times. Indeed $A$
value can be found also by adding $L+L$. In the flow-graph relation, this
means finding two distinct paths from node $S_{1}$ to $S_{2}$. This can be
easily done by encoding two paths in one. By generating all the possible pairs
of nodes, we can easily see that the number of steps to find, if exists, two
distinct paths takes $n^{2}$ number of steps (number of all pairs).
* •
If after $n^{2}$ steps no $M$ or $A$ value has been found, the maximum value
found is the correct one. Indeed, if no dependence has been found or if just a
linear dependence has been found, no further iteration could change the final
value.
## 10\. Conclusions
We presented an ICC system characterising the class $\mathbf{PP}$. There are
several improvements respect to the known systems in literature. We can
catalogue them in two sets. First, we extend the known system to probabilistic
computations, being able to characterise $\mathbf{PP}$. Since the typing
requires polynomial time, it is feasible to use ${\mathbf{iSAPP}}$ as a static
analyser for complexity. The typing/certificate gives also information about
the polynomial bound. On the other hand, respect to sequential computations,
we presented a finer analysis. ${\mathbf{iSAPP}}$ works over a concrete
language and takes care of constants and function calls. For all of these
reasons, we are able to show a program that cannot be typed correctly by
Kristiansen and Jones [9].
loop ($S_{1}$){
$S_{2}:=0*S_{2}$
}
That is typed with the identity matrix
${\mathbf{I}}\xleftarrow{2}{\mathbf{V}^{\mathbf{0}}}$. For multiplication we
use the implementation in def 5.2.
Algorithm 3 Example of recognised program
Since every constant is abstracted as a variable in [9], they cannot for sure
recognise that this program runs in polytime and for this reason this program
should be rejected. Once abstracted it is impossible to know the value of the
constant. Of course, everything depends on how the abstraction is made. In
general, every program which deals with constants could appear problematic in
[9] [3]; At least, for a lot of programs, their bounds are bigger. Moreover,
as they wrote in [3]: “Note that no procedure for inferring complexity will be
complete for $L_{concrete}$”, while our procedure is sound and complete for
our concrete language.
Finally we would like to point out some future direction:
* •
Integrating the analysis with new features in order to capture more programs.
* •
Apply our analysis to a more generic imperative programming language.
* •
Extending the algebra in such way that the associated certificates would tell
more detailed information about the polynomial bounding the complexity.
* •
Make a finer analysis in order to be sound and complete for $\mathbf{BPP}$.
## References
* [1] Sanjeev Arora and Boaz Barak. Computational Complexity, A Modern Approach. Cambridge University Press, 2009.
* [2] Stephen Bellantoni and Stephen A. Cook. A new recursion-theoretic characterization of the polytime functions. Computational Complexity, 2:97–110, 1992.
* [3] Amir M. Ben-Amram, Neil D. Jones, and Lars Kristiansen. Linear, polynomial or exponential? complexity inference in polynomial time. In Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms, CiE ’08, pages 67–76, Berlin, Heidelberg, 2008\. Springer-Verlag.
* [4] G. Bonfante, J.-Y. Marion, and J.-Y. Moyen. Quasi-interpretations a way to control resources. Theoretical Computer Science, 412(25):2776 – 2796, 2011.
* [5] Alan Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Logic, Methodology and Philosophy of Science, proceedings of the second International Congress, held in Jerusalem, 1964, Amsterdam, 1965. North-Holland.
* [6] Ugo Dal Lago and Paolo Parisen Toldin. A higher-order characterization of probabilistic polynomial time. In R. Peña, M. van Eekelen, and O. Shkaravska, editors, Proceedings of $2^{nd}$ International Workshop on Foundational and Practical Aspects of Resource Analysis, FOPARA 2011, volume 7177 of LNCS. Springer, 2011. To be appeared in.
* [7] John Gill. Computational complexity of probabilistic turing machines. SIAM J. Comput., 6(4):675–695, 1977.
* [8] Neil D. Jones. Logspace and ptime characterized by programming languages. Theoretical Computer Science, 228:151–174, October 1999.
* [9] Neil D. Jones and Lars Kristiansen. A flow calculus of mwp-bounds for complexity analysis. ACM Trans. Comput. Logic, 10(4):28:1–28:41, August 2009.
* [10] Lars Kristiansen and Neil D. Jones. The flow of data and the complexity of algorithms. In Proceedings of the First international conference on Computability in Europe: new Computational Paradigms, CiE’05, pages 263–274, Berlin, Heidelberg, 2005. Springer-Verlag.
* [11] Daniel Leivant. Stratified functional programs and computational complexity. In Principles of Programming Languages, 20th International Symposium, Proceedings, pages 325–333. ACM, 1993.
* [12] Daniel Leivant and Jean-Yves Marion. Ramified recurrence and computational complexity II: Substitution and poly-space. In Leszek Pacholski and Jerzy Tiuryn, editors, Computer Science Logic, 9th International Workshop, Proceedings, volume 933 of LNCS, pages 486–500. 1995.
* [13] R. Metnani and J.-Y. Moyen. Equivalence between the $mwp$ and Quasi-Interpretations analysis. In J.-Y. Marion, editor, DICE’11, April 2011.
* [14] Albert R. Meyer and Dennis M. Ritchie. The complexity of loop programs. In Proceedings of the 1967 22nd national conference, ACM ’67, pages 465–469, New York, NY, USA, 1967. ACM.
|
arxiv-papers
| 2013-04-11T10:20:24 |
2024-09-04T02:49:44.185718
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jean-Yves Moyen, Paolo Parisen Toldin",
"submitter": "Paolo Parisen Toldin",
"url": "https://arxiv.org/abs/1304.3249"
}
|
1304.3269
|
††thanks: School of Mathematics and Maxwell Institute of Mathematical
Sciences, James Clerk Maxwell Building, Kings Buildings, University of
Edinburgh, Edinburgh, EH9 3JZ, UK
# Robust and efficient configurational molecular sampling via Langevin
Dynamics
Benedict Leimkuhler Charles Matthews [email protected] School of
Mathematics and Maxwell Institute of Mathematical Sciences, James Clerk
Maxwell Building, Kings Buildings, University of Edinburgh, Edinburgh, EH9
3JZ, UK
###### Abstract
A wide variety of numerical methods are evaluated and compared for solving the
stochastic differential equations encountered in molecular dynamics. The
methods are based on the application of deterministic impulses, drifts, and
Brownian motions in some combination. The Baker-Campbell-Hausdorff expansion
is used to study sampling accuracy following recent work by the authors, which
allows determination of the stepsize-dependent bias in configurational
averaging. For harmonic oscillators, configurational averaging is exact for
certain schemes, which may result in improved performance in the modelling of
biomolecules where bond stretches play a prominent role. For general systems,
an optimal method can be identified that has very low bias compared to
alternatives. In simulations of the alanine dipeptide reported here (both
solvated and unsolvated), higher accuracy is obtained without loss of
computational efficiency, while allowing large timestep, and with no
impairment of the conformational exploration rate (the effective diffusion
rate observed in simulation). The optimal scheme is a uniformly better
performing algorithm for molecular sampling, with overall efficiency
improvements of 25% or more in practical timestep size achievable in vacuum,
and with reductions in the error of configurational averages of a factor of
ten or more attainable in solvated simulations at large timestep.
Langevin dynamics, configurational molecular sampling, stochastic molecular
dynamics, long term averaging, symplectic methods
## I Introduction
One of the major challenges in understanding matter at the molecular scale is
the problem of thermodynamic sampling: the calculation of averages with
respect to the canonical (Gibbs-Boltzmann) distribution. In many cases the aim
is to sample configurational quantities only, and this is the focus of this
article. Given the classical molecular potential energy function
$U:\mathbb{R}^{3N}\rightarrow\mathbb{R}$, the configurational canonical
density is
$\bar{\rho}_{\beta}(q)=Z^{-1}e^{-\beta U(q)},$
where $\beta^{-1}=k_{B}T$ where $k_{B}$ is Boltzmann’s constant, $T$ is
temperature, and $Z$ is a normalization constant so that $\bar{\rho}_{\beta}$
has unit integral over the entire configuration space. In using molecular
dynamics to sample the phase space according to the canonical distribution,
the formulation employed may not be ergodic (meaning that it may not sample
the entire phase space) and, moreover, the design of the time-discretization
methods typically distorts the equilibrium distribution. Using a stochastic
differential equation model such as Langevin dynamics, which introduces random
perturbations into each force component, we can overcome the first of these
problems, as the formulation is well known to be ergodic. As an illustration
of the effect of step size error, see Figure 1, where the potential energy in
simulations of alanine dipeptide is shown to be corrupted by a popular
Langevin discretization.
Figure 1: The computed potential energy distribution is shown for the method
of Brünger, Brooks and Karplus applied to a single alanine dipeptide protein
at 300K in a vacuum using the CHARMM22 forcefield; the energy distribution
becomes distorted as the step size increases.
Given that the majority of computational work in any MD algorithm lies in the
force calculation, most of the existing methods in common use have been
designed to require only one force evaluation per timestep. For timestepping
methods that accurately sample the canonical distribution, the available
timescales for simulation are restricted by the problem itself (e.g. the
heights of barriers, or entropic properties of the landscape). In designing a
new molecular dynamics algorithm the goal is to enlarge the usable timestep in
order to allow finite trajectories to access a larger portion of the phase
space. The drawback of working in the high-timestep regime is that for long-
time simulations the computed probability distribution is a perturbation of
$\bar{\rho}_{\beta}$, dependent on the step size, leading to a distortion in
calculated averages. The simulator must choose the step size sufficiently
small enough to avoid corruption in averages, but still large enough to ensure
a thorough exploration of configuration space.
The potential energy function in molecular dynamics determines the maximum
allowable step size. For a harmonic oscillator with frequency $\omega$ the
Verlet method has a stability restriction of $\delta t\leq 2/\omega$ 1. Most
numerical methods (including ones constructed for Langevin dynamics) suffer
from a similar limitation in the maximum timestep size driven by the presence
of stiff oscillatory solution components. However, well before reaching the
stability threshold, averages may be severely corrupted, introducing
artificial–and, often, severe–step size restriction. By removing or reducing
this bias, it becomes possible to substantially increase the timestep, with a
direct impact on the efficiency of simulation. Given the explosion in the use
of molecular dynamics in chemistry, physics, engineering and biology, it is
worth noting that where molecular dynamics is used for round-the-clock
configurational sampling calculations, a quantifiable improvement in method
efficiency (or timestep size) directly translates to a reduction in machine
costs, a reduction in energy costs, and, often, a reduction in delays to
publication.
With regard to the error in averages, it is usually assumed that the error due
to having insufficient samples dominates the timestep-dependent discretization
error, but this is not typically the case at large step size, as we
demonstrate in numerical experiments (see Section V). To dramatize the role of
discretization error, we show in Figure 2 the example of the configurational
distribution for a simple two-basin model solved using two different numerical
methods. The figure illustrates that where step size error is substantial,
crucial features of the landscape such as the heights of free energy barriers,
may be completely altered. Moreover, it is entirely possible for the relative
heights of different barriers to be altered in such a way that one transition
becomes more prevalent than another.
Figure 2: The configurational density function $\bar{\rho}_{\beta}(q)$ is
shown for a planar two-basin potential, computed using two different Langevin
dynamics methods at various (stable) timesteps increasing from left to right.
The color indicates the value of the computed probability density: from high
(red) to low (blue) over the unit square. The methods have the same
computational cost (in terms of force evaluations), but give different results
at large timesteps. Details on this computation are given in Appendix A.
Theoretical analysis of the error in the invariant distribution can be
performed for harmonic oscillators without great difficulty (see Section III),
but it can also be carried out for general nonlinear problems. This is most
straightforward in the case of splitting methods. In this article we draw on
principles of geometric integration, building on our understanding of
splitting methods from the deterministic setting 2. Splitting methods for
Langevin dynamics have been considered in the past 4, 3, 5, 7, 8, 9, 10, 6 but
a wide variety of schemes can be constructed by splitting and until now the
rational basis for selecting one scheme over another has been absent. Drawing
on the work of Talay and Tubaro 11, we have studied the generator of the
numerical method directly by examining expansions for the invariant measure of
the Langevin dynamics scheme 12; this investigation lead to the concept of the
associated density $\hat{\rho}$ of the numerical method:
$\hat{\rho}(q,p)\propto\exp\left(-\beta\left[H(q,p)+\delta
t^{2}f_{2}(q,p)+\delta t^{4}f_{4}(q,p)+\ldots\right]\right).$ (1)
This expansion allows different methods to be compared on a rational basis (as
concerns the effect of discretization error). In the past, for deterministic
methods, this type of analysis has also been used for the correction of
averages 13, 14. In typical cases which would be relevant for molecular
simulation, the error introduced in averages using such methods would be
second order in the timestep (i.e. would go to zero quadratically as the step
size is reduced).
For a particular ordering of the building blocks of the numerical method, a
“superconvergence” (cancellation) property can be obtained in the high
friction limit, meaning that in fully resolved molecular dynamics simulations
and after integrating out with respect to momenta, the leading term in the
expansion vanishes 12. This theoretical convergence result was until now only
studied for relatively simple model problems. Moreover the crucial question of
the step size (stability) threshold of the different schemes (as well as the
overall efficiency of the various methods) cannot be addressed using the
asymptotic technique since it provides information only about the small step
size limit ($\delta t\rightarrow 0$).
Another important issue raised by practitioners concerns the fact that such a
superconvergent method, relying on large friction, might not be useful in
realistic settings since it is known that large friction can reduce the
diffusion rate. The problem is complicated by a number of issues: both
friction coefficient and step size affect the long-term averaging error
differently for different methods, and the friction coefficient (and, in
principle, the step size) may affect the diffusion rates differently for
different methods. Performance is further dependent on the type of problem
under study. Thus there is a need for careful study of the methods in the
context of systems relevant for molecular dynamics (for example, containing
both steep potentials such as Lennard-Jones and stiff bonds). In this article
we address both issues: we consider large step size and modest values of the
friction coefficient, using numerical experiments to carefully examine the
relative performance of a large number of different methods.
In recent years, there has been widespread interest in multiscale methods for
enhanced sampling 15, 16, 17, 18 and such methods likely offer the best
approach to bridging the timescale gap. We observe that work on enhanced
numerical schemes for molecular dynamics remains essential as it plays a
crucial underpinning role in all the enhanced sampling approaches. Improved
trajectory generation efficiency (e.g. allowing the use of a larger basic
timestep in simulation) thus has a knock-on effect on the efficiency of all
the methods that rely on such trajectories. While relative improvements of a
few percentages in efficiency can already warrant a minor change in software
implementation, our analysis points to a more dramatic (even qualitative)
difference among various methods leading to prospects for much greater
efficiencies by selecting a suitable method. These observations are verified
in model biomolecular simulations.
Hybrid Monte-Carlo 19, 20, and other schemes based on Metropolis correction
21, are not discussed here, although these could be used in conjunction with
several of the methods implemented. The improvement in thermodynamic sampling
obtained through the use of more accurate Langevin integrators may, in some
cases, provide an alternative to Metropolis-based correction in the practical
setting. All methods under discussion require one force evaluation per
iteration, and hence have practically of the same computational cost.
This article proceeds as follows. In Section II we introduce Langevin dynamics
in the context of configurational sampling and describe our method for
examining the long-time behavior of averages under discretization of the
stochastic differential equations (SDEs). Section III discusses the harmonic
model problem, showing that for some particular schemes, the configurational
sampling can be exact; this has implications for molecular simulations
involving stiff harmonic bonds. Section IV addresses the errors obtained from
computed averages in more general systems. Section V contains numerical
experiments comparing various methods, both for one degree of freedom systems
and for solvated and unsolvated alanine dipeptide, through implementation of
the schemes in a version of NAMD 22. It is our contention that the numerical
experiments of Section V provide strong evidence for rejecting many of the
schemes in common use for stochastic molecular dynamics and favor the optimal
BAOAB scheme of 12.
## II Background
In this article we focus on Langevin dynamics,
$\displaystyle{\rm d}q$ $\displaystyle=$ $\displaystyle M^{-1}p\,{\rm d}t,$
(2) $\displaystyle{\rm d}p$ $\displaystyle=$ $\displaystyle-\nabla U(q)\,{\rm
d}t-\gamma p\,{\rm d}t+\sigma M^{1/2}\,{\rm d}W,$ (3)
where $q,p\in\mathbb{R}^{3N}$ are vectors of instantaneous position and
momenta respectively, $W=W(t)$ is a vector of $3N$ independent Wiener
processes, $\gamma>0$ is a free (scalar) parameter and $M$ is a constant
diagonal mass matrix. By choosing $\sigma=\sqrt{2\gamma\beta^{-1}}$ it is
possible to show that the unique probability distribution sampled by the
dynamics is the canonical (Gibbs-Boltzmann) density, defined as
$\rho_{\beta}(q,p)=\Omega^{-1}e^{-\beta H(q,p)},$ (4)
for total system energy (Hamiltonian) $H(q,p)=p^{T}M^{-1}p/2+U(q)$, and
normalization constant $\Omega^{-1}$ ensuring the integral is unity. We
consider numerical methods designed to integrate (2–3), primarily for the
purpose of generating trajectories that sample $\rho_{\beta}.$ Such
trajectories are often used as a means for calculating expectations of
functions purely of the position $q$, and as such the dynamical fidelity of
computed trajectories is of minor importance compared to the behavior of
averages in the large-time limit. For such an observable $\phi$, we write the
expectation as
$\mathbb{E}\left[\phi(q)\right]=\Omega^{-1}\int\int\phi(q)\rho_{\beta}(q,p)\,{\rm
d}q\,{\rm d}p=Z^{-1}\int\phi(q)\bar{\rho}_{\beta}(q)\,{\rm
d}q=\lim_{T\rightarrow\infty}T^{-1}\int_{0}^{T}\phi(q(t)){\rm d}t,$
where the ergodicity of Langevin dynamics ensures a sampling of the desired
probability distribution, and hence the ability to equate the long-time
average along a trajectory with the corresponding spatial average. The
challenge comes in integrating (2–3) effectively, and with minimal
computational cost.
Given a general potential energy function $U(q)$, we cannot integrate exactly
and must evolve the dynamics by discretizing in time. Advancing the state
requires the use of a numerical method which aims to approximate the exact
evolution. A distribution of initial conditions $\rho$ propagated using a
second-order numerical method will evolve according to the equation
$\frac{\partial\rho}{\partial t}=\hat{\cal L}^{*}\rho,$
where $\hat{\cal L}^{*}$ may be expressed in the series expansion
$\hat{{\cal L}}^{*}={\cal L}^{*}_{\text{LD}}+\delta t^{2}{\cal
L}^{*}_{2}+{\cal O}(\delta t^{3}),$ (5)
where ${\cal L}^{*}_{\text{LD}}$ is the operator associated to the exact
propagation under Langevin dynamics and $\delta t$ is the step size. The
invariant (long-time) distribution sampled by the scheme can in principle be
obtained by solving the partial differential equation (PDE) $\hat{{\cal
L}}^{*}\hat{\rho}=0$, assuming that the perturbed operator $\hat{{\cal
L}}^{*}$ is known.
Splitting methods4, 5, 6 offer a simple way of integrating the Langevin
dynamics equations; the right hand side of (2–3) is divided into pieces, eg.
$\dot{z}=f=f_{1}+f_{2}$, with each piece is solved exactly in sequence. Recent
work 12 has shown that numerical methods derived from additive splittings of
the vector field enable relatively simple computation of a method’s
characteristic operator. The order of integration, and the choice of the
splitting will define the method.
For example, one may break Langevin dynamics into three pieces:
$\left[\\!\\!\begin{array}[]{c}{\rm d}q\\\ {\rm
d}p\end{array}\\!\\!\right]=\underbrace{\left[\\!\\!\begin{array}[]{c}M^{-1}p\\\
0\end{array}\\!\\!\right]{\rm d}t}_{\rm
A}+\underbrace{\left[\\!\\!\begin{array}[]{c}0\\\ -\nabla
U(q)\end{array}\\!\\!\right]{\rm d}t}_{\rm
B}+\underbrace{\left[\\!\\!\begin{array}[]{c}0\\\ -\gamma p{\rm d}t+\sigma
M^{1/2}{\rm d}W\end{array}\\!\\!\right],}_{\rm O}$ (6)
which are labelled $A$, $B$ and $O$. Each of the three pieces may be solved
“exactly”: $A$ and $B$ correspond to a linear drift and kick when taken
individually, while the $O$ piece is associated to an Ornstein-Uhlenbeck
equation with “exact” solution
$\displaystyle q(t)$ $\displaystyle=q(0),$ $\displaystyle p(t)$
$\displaystyle=e^{-\gamma
t}p(0)+\frac{\sigma}{\sqrt{2\gamma}}\sqrt{1-e^{-2\gamma t}}M^{1/2}R_{t},$
where $R_{t}\sim{\cal N}(0,1)$ is a vector of uncorrelated noise processes.
(By “exact” we mean that this random map generates the probability
distribution $\rho(q,p,t)$ defined by the solutions of the Ornstein-Uhlenbeck
equation.)
Given the pieces of the splitting we code a method by giving the sequence of
integration, from left to right. The string “ABO” represents the method
obtained by solving first the “A” part for a timestep $\delta t$, then the “B”
part, and finally the “O” part of the system. Where a symbol is repeated, as
in “BAOAB,” there could be ambiguity in this representation but we will assume
that the method is symmetric so that all “A” and “B” parts in BAOAB are
integrated for a half timestep. Additionally, the methods of Bussi and
Parrinello 23 (OBABO) as well as gla-1 (BAO) and gla-2 (BABO) of Bou-Rabee and
Owhadi 7, are equivalent to splitting methods using these pieces.
An alternate splitting formulation8
$\left[\\!\\!\begin{array}[]{c}{\rm d}q\\\ {\rm
d}p\end{array}\\!\\!\right]=\underbrace{\left[\\!\\!\begin{array}[]{c}M^{-1}p\\\
0\end{array}\\!\\!\right]{\rm d}t}_{\rm
A}+\underbrace{\left[\\!\\!\begin{array}[]{c}0\\\ -\nabla U(q){\rm d}t-\gamma
p{\rm d}t+\sigma M^{1/2}{\rm d}W\end{array}\\!\\!\right],}_{\rm S}$ (7)
has been used to define two methods: stochastic position verlet (ASA) and
stochastic velocity verlet (SAS).
A technique is outlined in Section IV to calculate the operators $\hat{{\cal
L}}^{*}$ of such splitting methods. For methods not derived from splitting the
vector field, it can be more difficult to obtain their operators and examine
their behavior. We compare the configurational sampling for a number of
popular schemes in this article, not limiting the scope only to splitting
methods.
We will consider both ABOBA and BAOAB methods12, the Bussi/Parrinello
method23, as well as the Stochastic Position Verlet method8. The Langevin
Impulse (LI) method9, the BBK method 24 and the method of van Gunsteren and
Berendsen (VGB)25 will also be compared, as these are frequently found in
commercial software packages (for example, NAMD and GROMACS). Two first-order
methods, Ermak-McCammon (EM)26 and Ermak-Buckholtz (EB)27 will also be
considered in numerical experiments, for completeness, with each method
described in Appendix B.
## III Performance of Langevin algorithms applied to the harmonic oscillator.
We begin by considering the harmonic model problem. Harmonic oscillators are
useful in the molecular simulation setting not only because they allow
analytical determination of effective distributions, but also because they can
be seen to be relevant to understanding the timestep limiting features of
models for crystalline solids and biomolecules.
The one-dimensional harmonic oscillator $U(q)=Kq^{2}/2$, $q\in\mathbb{R}$ and
$K>0$, is a standard test case for Langevin dynamics numerical methods, as
many issues of stability and timestep in molecular dynamics simulations arise
due to harmonic potentials used to model covalent bonds. For such a simple
system we may explicitly write one iteration of a general numerical method
evolving the dynamics as 28
$\left[\begin{array}[]{c}q_{n+1}\\\
p_{n+1}\end{array}\right]\leftarrow\Psi\left[\begin{array}[]{c}q_{n}\\\
p_{n}\end{array}\right]+\mu_{n},$
where $\Psi$ is a matrix depending only on the step size $\delta t$, the
friction coefficient $\gamma$, the particle mass $M$ and the spring constant
$K$, while $\mu_{n}$ is a vector of stochastic processes.
Let $\Psi=(\psi_{ij})$ and denote the components of $\mu_{n}$ by $\mu_{n,j}$
where $i,j\in\\{1,2\\}$. Taking products of the update equations, we obtain
$\displaystyle q_{n+1}^{2}$
$\displaystyle=\psi_{11}^{2}q_{n}^{2}+\psi_{12}^{2}p_{n}^{2}+\mu_{n,1}^{2}+2\psi_{11}\psi_{12}q_{n}p_{n}+2\psi_{11}\mu_{n,1}q_{n}+2\psi_{12}\mu_{n,1}p_{n},$
(8) $\displaystyle p_{n+1}^{2}$
$\displaystyle=\psi_{21}^{2}q_{n}^{2}+\psi_{22}^{2}p_{n}^{2}+\mu_{n,2}^{2}+2\psi_{21}\psi_{22}q_{n}p_{n}+2\psi_{21}\mu_{n,2}q_{n}+2\psi_{22}\mu_{n,2}p_{n},$
(9) $\displaystyle q_{n+1}p_{n+1}$
$\displaystyle=\psi_{11}\psi_{21}q_{n}^{2}+\psi_{12}\psi_{22}p_{n}^{2}+\mu_{n,1}\mu_{n,2}$
$\displaystyle\quad+(\psi_{11}\psi_{22}+\psi_{12}\psi_{21})q_{n}p_{n}+(\psi_{11}\mu_{n,2}+\psi_{21}\mu_{n,1})q_{n}+(\psi_{22}\mu_{n,1}+\psi_{12}\mu_{n,2})p_{n}.$
(10)
We then take expectations of (8-10) in the limit $n\rightarrow\infty$, giving
simultaneous equations
$\displaystyle\langle q^{2}\rangle$ $\displaystyle=\psi_{11}^{2}\langle
q^{2}\rangle+\psi_{12}^{2}\langle
p^{2}\rangle+\langle\hat{\mu}_{1}^{2}\rangle+2\psi_{11}\psi_{12}\langle
qp\rangle+2\psi_{11}\langle\hat{\mu}_{1}q\rangle+2\psi_{12}\langle\hat{\mu}_{1}p\rangle,$
(11) $\displaystyle\langle p^{2}\rangle$ $\displaystyle=\psi_{21}^{2}\langle
q^{2}\rangle+\psi_{22}^{2}\langle
p^{2}\rangle+\langle\hat{\mu}_{2}^{2}\rangle+2\psi_{21}\psi_{22}\langle
qp\rangle+2\psi_{21}\langle\hat{\mu}_{2}q\rangle+2\psi_{22}\langle\hat{\mu}_{2}p\rangle,$
(12) $\displaystyle\langle qp\rangle$ $\displaystyle=\psi_{11}\psi_{21}\langle
q^{2}\rangle+\psi_{12}\psi_{22}\langle
p^{2}\rangle+\langle\hat{\mu}_{1}\hat{\mu}_{2}\rangle+(\psi_{11}\psi_{22}+\psi_{12}\psi_{21})\langle
qp\rangle$
$\displaystyle\quad+\psi_{11}\langle\hat{\mu}_{2}q\rangle+\psi_{21}\langle\hat{\mu}_{1}q\rangle+\psi_{22}\langle\hat{\mu}_{1}p\rangle+\psi_{12}\langle\hat{\mu}_{2}p\rangle,$
(13)
where we use notation $\langle x\rangle=\mathbb{E}[x_{n}]$ for
$x\in\\{q,p\\}$, and $\langle\hat{\mu}_{i}\rangle=\mathbb{E}[\mu_{n,i}]$. The
value of $\langle\hat{\mu}_{i}x\rangle=\mathbb{E}[\mu_{n,i}\,x_{n}]$ will
ultimately depend on the “memory” of a scheme’s stochastic process
${\mu}_{n}$, and can be found by computing $x_{n+1}\,\mu_{n+1,i}$ and taking
expectations, yielding an expression involving
$\mathbb{E}[\mu_{n,i}\mu_{n-1,i}]$ .
We can hence solve the linear system (11-13) to find the error in long-time
averages for a method, and its behavior under changes in $\delta t$ and
$\gamma$, relative to the spring constant $K$, by comparing the numerical and
analytic averages, the latter given as
$\left[\begin{array}[]{c}\langle q^{2}\rangle^{*}\\\ \langle
p^{2}\rangle^{*}\\\ \langle qp\rangle^{*}\\\
\end{array}\right]=\left[\begin{array}[]{c}K^{-1}\beta^{-1}\\\ M\beta^{-1}\\\
0\\\ \end{array}\right].$
For the BAOAB and ABOBA methods, the numerical values (in the long-time limit)
are
$\left[\begin{array}[]{c}\langle q^{2}\rangle^{\text{(BAOAB)}}\\\ \langle
p^{2}\rangle^{\text{(BAOAB)}}\\\ \langle qp\rangle^{\text{(BAOAB)}}\\\
\end{array}\right]=\left[\begin{array}[]{c}K^{-1}\beta^{-1}\\\
M\beta^{-1}\left(1-\frac{\delta t^{2}K}{4M}\right)\\\ 0\\\
\end{array}\right],\qquad\left[\begin{array}[]{c}\langle
q^{2}\rangle^{\text{(ABOBA)}}\\\ \langle p^{2}\rangle^{\text{(ABOBA)}}\\\
\langle qp\rangle^{\text{(ABOBA)}}\\\
\end{array}\right]=\left[\begin{array}[]{c}K^{-1}\beta^{-1}\\\
M\beta^{-1}\left(1-\frac{\delta t^{2}K}{4M}\right)^{-1}\\\ 0\\\
\end{array}\right],$
surprisingly giving exact values for the configurational average. Both schemes
yield the same friction-independent upper-bound on the step size of $\delta
t_{\text{max}}=2\sqrt{M/K}$: the (determinisitic) Verlet step size threshold.
This implies that we can choose any timestep below this limit and still
achieve perfect sampling of $\langle q^{2}\rangle$, up to sampling error. This
behavior is atypical of Langevin dynamics algorithms, for example comparing
the Bussi/Parrinello and BBK schemes we find
$\left[\begin{array}[]{c}\langle q^{2}\rangle^{\text{(BP)}}\\\ \langle
p^{2}\rangle^{\text{(BP)}}\\\ \langle qp\rangle^{\text{(BP)}}\\\
\end{array}\right]=\left[\begin{array}[]{c}K^{-1}\beta^{-1}\left(1-\frac{\delta
t^{2}K}{4M}\right)^{-1}\\\ M\beta^{-1}\\\ 0\\\
\end{array}\right],\qquad\left[\begin{array}[]{c}\langle
q^{2}\rangle^{\text{(BBK)}}\\\ \langle p^{2}\rangle^{\text{(BBK)}}\\\ \langle
qp\rangle^{\text{(BBK)}}\\\
\end{array}\right]=\left[\begin{array}[]{c}K^{-1}\beta^{-1}\left(1-\frac{\delta
t^{2}K}{4M}\right)^{-1}\\\ M\beta^{-1}\left(1+\frac{\gamma\delta
t}{2}\right)^{-1}\\\ 0\\\ \end{array}\right],$
giving identical second-order errors in configurational averages for this
system, with the same value of $\delta t_{\text{max}}.$ The BBK scheme has a
first order error in $\langle p^{2}\rangle$ that is also friction-dependent.
The computed configurational averages for each scheme are shown in Table 1,
while Figure 3 shows the result of computing the value of $\langle
q^{2}\rangle$ numerically, using a fixed total number of steps and varying the
step size. Three distinct regimes can be seen: the first-order “Ermak”
methods, second-order methods and the exact methods (where any error comes
solely from sampling error, rather than discretization error).
We find that the method of van Gunsteren and Berendsen 25 is in fact 2nd order
accurate for configurational sampling, not 3rd order as reported by those
authors; we attribute this to a different notion of accuracy being used in
that article.
Scheme | $\langle q^{2}\rangle$ | Scheme | $\langle q^{2}\rangle$
---|---|---|---
Exact | $K^{-1}\beta^{-1}$ | SPV | $K^{-1}\beta^{-1}\left(\gamma\,\delta t\frac{1-e^{-2\gamma\delta t}}{2\left(1-e^{-\gamma\delta t}\right)^{2}}\right)$
BAOAB | $K^{-1}\beta^{-1}$ | LI | $K^{-1}\beta^{-1}-\frac{\delta t^{2}}{12M\beta}+O\left(\delta t^{4}\right)$
ABOBA | $K^{-1}\beta^{-1}$ | VGB | $K^{-1}\beta^{-1}+\frac{\gamma^{2}M-2K}{24M\beta K}\delta t^{2}+O\left(\delta t^{4}\right)$
BBK | $K^{-1}\beta^{-1}\left(1-\frac{\delta t^{2}K}{4M}\right)^{-1}$ | EM | $K^{-1}\beta^{-1}\left(1-\frac{\delta t\,K}{2\gamma M}\right)^{-1}$
BP | $K^{-1}\beta^{-1}\left(1-\frac{\delta t^{2}K}{4M}\right)^{-1}$ | EB | $K^{-1}\beta^{-1}+\frac{\delta t}{2\gamma M\beta}+O\left(\delta t^{2}\right)$
Table 1: The expected long-time computed average of $q^{2}$ using each
Langevin dynamics scheme, for the 1D harmonic oscillator $U(q)=Kq^{2}/2$. For
brevity, some results are shown as leading order series in $\delta t.$ Figure
3: The numerically-computed average error in $\langle q^{2}\rangle$, using the
1D harmonic oscillator with a given Langevin Dynamics method. Computation was
fixed at $10^{7}$ total force evaluations, with $M=\gamma=\beta=K=1$. The
results are averaged over 2000 independent repeat runs, with error bars
included to give the standard deviation in these results. The exactness
property of the ABOBA and BAOAB schemes result in the error decreasing as step
size increases due to sampling error.
## IV Error analysis for general systems
Repeating the analysis in Section III for a more general $U(q)$ in a higher-
dimensional setting is a challenging task for complicated schemes, but we do
have a recently developed framework for carrying out the calculations. As we
have noted previously we may define the invariant distribution of a second-
order method $\hat{\rho}$ as the solution of the partial differential equation
$\hat{{\cal L}}^{*}\hat{\rho}=0.$
Expanding the operator in a perturbation series in terms of the step size
$\delta t$ and combining equations (1) and (5) yields
$\left({{\cal L}}^{*}_{\rm LD}+\delta t^{2}\hat{{\cal
L}}^{*}_{2}+\ldots\right)\rho_{\beta}\left(1-\delta t^{2}\beta
f_{2}+\ldots\right)=0.$
Equating powers of the step size, we see that the order $0$ terms match
automatically, leaving the leading order perturbation equation to be
${{\cal L}}^{*}_{\rm LD}\left(\rho_{\beta}\,f_{2}\right)=\beta^{-1}\hat{{\cal
L}}^{*}_{2}\rho_{\beta}.$ (14)
By the hypoelliptic property of the exact operator29, the unique solution to
${{\cal L}}^{*}_{\rm LD}\phi=0$ is $\phi\propto\rho_{\beta}$, hence the
homogenous solution to (14) is simply $f_{2}=c$, a constant. Therefore we need
only find a particular solution $f_{2}(q,p)$ solving (14) in order to find the
leading order error in the long-time distribution $\hat{\rho}$. Once the
perturbation is known, averages may be rebiased accordingly13, in effect
increasing the order of the method. Of course $f_{2}$ may itself be a costly
function to evaluate, involving a combination of high order derivatives of
$U(q)$, and in the general case this is likely to lead to an inefficient
method.
For the ABOBA and BAOAB methods, the right hand side of (14) is
$\displaystyle-\beta^{-1}\hat{{\cal L}}^{*({\rm ABOBA})}_{2}\rho_{\beta}$
$\displaystyle=\frac{\gamma\rho_{\beta}}{4\beta}\left(\Delta_{q}M^{-1}U(q)-\beta
p^{T}M^{-2}U^{\prime\prime}(q)p\right)+\frac{\rho_{\beta}}{4}p^{T}M^{-2}U^{\prime\prime}(q)\nabla_{q}U(q)$
$\displaystyle\quad-\frac{\rho_{\beta}}{24}p^{T}\nabla_{q}p^{T}U^{\prime\prime}(q)M^{-3}p,$
$\displaystyle-\beta^{-1}\hat{{\cal L}}^{*({\rm BAOAB})}_{2}\rho_{\beta}$
$\displaystyle=-\frac{\gamma\rho_{\beta}}{4\beta}\left(\Delta_{q}M^{-1}U(q)-\beta
p^{T}M^{-2}U^{\prime\prime}(q)p\right)-\frac{\rho_{\beta}}{4}p^{T}M^{-2}U^{\prime\prime}(q)\nabla_{q}U(q)$
$\displaystyle\quad+\frac{\rho_{\beta}}{12}p^{T}\nabla_{q}p^{T}U^{\prime\prime}(q)M^{-3}p,$
where $U^{\prime\prime}(q):=\nabla_{q}\nabla_{q}^{T}U(q)$ is the hessian
matrix of mixed partial derivatives.
Being able to write down equation (14) relies on the calculation of
$\hat{{\cal L}}^{*}_{2}$, which involves the computation of a scheme’s
perturbed operator (5), characterizing the evolution of a density of points in
the phase space. This can be a challenge in itself, though for methods that
derive from splitting the vector field, this operator is easily computed12
using successive applications of the Baker-Campbell-Hausdorff (BCH) formula
for products of exponentials 30.
As an example, consider the stochastic position verlet (SPV) method, using
splitting pieces defined in equation (7). The infinitesimal generator
associated to each part in (7) is given by
${\cal L}^{*}_{A}\phi=-M^{-1}p\cdot\nabla_{q}\phi,\hskip 14.45377pt{\cal
L}^{*}_{S}\phi=\nabla_{q}U(q)\cdot\nabla_{p}\phi+\gamma\nabla_{p}\left(\phi
p\right)+\frac{\sigma^{2}}{2}M\Delta_{p}\phi.$
Note that the exact operator ${\cal L}^{*}_{\rm LD}={\cal L}^{*}_{A}+{\cal
L}^{*}_{S}.$ Using the notation from Section II, we code this numerical method
as “ASA”. The characteristic evolution operator ${\cal L}^{*(SPV)}$ is then
computed using
$\exp\left(\delta t{\cal L}^{*(SPV)}\right)=\exp\left(\left(\delta
t/2\right)\,{\cal L}^{*}_{A}\right)\exp\left({\delta t}{\cal
L}^{*}_{S}\right)\exp\left(\left(\delta t/2\right)\,{\cal L}^{*}_{A}\right),$
where the BCH formula can be used to simplify the products of exponentials:
$\exp(t{\cal L}_{1}^{*})\exp(t{\cal L}_{2}^{*})=\exp\left(t\left({\cal
L}_{1}^{*}+{\cal L}_{2}^{*}\right)+\frac{t^{2}}{2}\left[{\cal L}_{1}^{*},{\cal
L}_{2}^{*}\right]+\frac{t^{3}}{12}\left([{\cal L}_{1}^{*},[{\cal
L}_{1}^{*},{\cal L}_{2}^{*}]]-[{\cal L}_{2}^{*},[{\cal L}_{2}^{*},{\cal
L}_{1}^{*}]]\right)+O(t^{4})\right),$
and $[{\cal L}_{1}^{*},{\cal L}_{2}^{*}]={\cal L}_{1}^{*}{\cal
L}_{2}^{*}-{\cal L}_{2}^{*}{\cal L}_{1}^{*}$ is the commutator of ${\cal
L}_{1}^{*}$ and ${\cal L}_{2}^{*}$.
By splitting the vector field (2-3), and choosing a preferred integration
sequence, one can easily create and analyse a multitude of Langevin dynamics
splitting methods using this technique; though it is perhaps surprising how
small and subtle changes to the order of each piece’s integration can yield
vastly different average behavior in the long-time limit. This effect is most
easily apparent if an asymmetry is created when two adjacent letters are
swapped in a symmetric method. Symmetry ensures that the order of a method is
at least two (by the Jacobi identity), while destroying this property could
hamper stability as well as the order of the method.
Once the right hand side of (14) has been computed, solving to find the
invariant density is an involving task and we do not pursue this here for the
methods described above. In the case of the ABOBA and BAOAB methods, solutions
can be obtained as doubly asymptotic expansions in both $\delta t$ and the
reciprocal friction coefficient $\gamma^{-1}$; moreover a superconvergence
property can be demonstrated for BAOAB configurational averages implying 4th
order accuracy12. It is interesting to note that in the case of ABOBA, no such
cancellation occurs, even though the methods have right hand sides that are
apparently similar in the leading term.
The fundamental limitation of the asymptotic approach is that it remains to
determine in which regime the theoretically obtained features of the perturbed
distribution are manifest in simulation. Large friction coefficient is known
to reduce sampling efficiency, so we would need to work with modest values of
$\gamma$, potentially invalidating the superconvergence property. Likewise the
crucial issue in many cases is the size of the allowable timestep for
simulation, not the asymptotic error behavior for small step size. These
complexities must be addressed using computer experiment.
## V Numerical results
One of the most important features of a numerical method for ergodic dynamics
(such as Langevin dynamics) is its preservation of the theoretical global
phase space exploration rate. The spectral properties of the operator ${\cal
L}^{*}_{\text{LD}}$ guarantee that we will explore the entire phase space
(ergodicity), while the relatively small perturbations to the operator induced
by numerical discretization are hoped not to significantly alter the rate of
search. Ultimately, pushing the timestep up is the only way to breach
timescale gaps, although this comes at the cost of corruption to the long-time
averages.
The self-diffusion coefficient gives a metric quantifying the diffusion rate.
It is often used as a way to compare the rate of phase space exploration
between methods, and typically calculated using the integral of the velocity
auto-correlation function. However, arbitrary methods can be constructed to
artificially scale the velocity auto-correlation function, hence giving
inaccurate diffusion constants.
Indeed, calculating the temperature of the system from an average of kinetic
energy by
$3Nk_{B}T=\mathbb{E}\left[p^{T}M^{-1}p\right],$ (15)
gives a similar problem. Alternative functions, including functions of $q$
only, can be obtained whose averages are proportional to the system
temperature31, 32, 33. Such “configurational temperature” observables are
normally based on the periodic forces of the system in order to work in
periodic boundary conditions; in the droplet simulations reported below (i.e.
without boundary conditions) we used the simpler expression:
$3Nk_{B}T=\mathbb{E}\left[q\cdot\nabla U(q)\right].$ (16)
Were one able to solve the dynamics exactly, the kinetic and configurational
temperatures would of course be equal. However, using a numerical method in
the large-timestep regime we instead sample expectations with respect to the
perturbed density $\hat{\rho}$, that may introduce discrepancies between
configurational and kinetic temperatures. It is our view that a configuration-
based temperature calculation is normally more useful and relevant for
assessing the quality of configurational sampling methods.
In a similar way, the speed of exploration of the space should not be
determined solely from functions of momentum, but should rely on actual
barrier crossing rates or times to reach some target region of phase space.
### V.1 One-dimensional double well
The advantages of performing tests initially on a simple model are that (i)
the exact solution is known (or can be numerically integrated to arbitrary
precision), while (ii) the model’s simplicity allows us to perform exhaustive
computation to refine results and determine asymptotic properties. Here we use
the algorithms to integrate Langevin dynamics for a one-dimensional model with
potential function $U(q)=(q^{2}-1)^{2}+q$, a double-well. This example is
well-studied as an approximation for modelling a dual state system. We use
unit mass, friction and temperature, and test a range of step sizes, beginning
at $\delta t=0.2$ and increasing by $5\%$ until we reach a step size where all
of the methods are no longer stable. We run $500$ independent experiments for
each step size, with computation fixed at $10^{9}$ iterations for each
realisation.
The error in configurational distribution is estimated by dividing the
interval $[-2,2]$ into 16 equal bins, and calculating the observed
configurational density in each bin for every computed trajectory. The error
in the observed densities for each bin are calculated by comparing the
absolute difference between the observed and exact expected densities (the
latter obtained using a high-order numerical solver). The overall error in the
configurational density is calculated from the root mean squared value of
these errors. Additionally, we calculate the observed kinetic temperature (15)
and configurational temperature (16) for each method. The results are shown in
Figure 4.
For the configurational distribution, all the methods shown give a second-
order relation in the step size, in contrast to Figure 3. Notably, the BAOAB
and ABOBA methods are no longer exact for this anharmonic model, while the two
first-order methods do not appear at all, as neither of them is stable in this
region. Of the methods that are stable, the BAOAB method gives both the
largest usable timestep and the smallest maximum error in the configurational
distribution for any given timestep. A sample plot of the computed
distribution for all schemes at $\delta t=0.25$ is also given in Figure 4d, it
is clear from this that the BAOAB scheme performs exceptionally well.
Figure 4: The BAOAB scheme is shown to significantly reduce configurational
discretization errors, which distort averages in Langevin Dynamics, for the
one-dimensional double-well system with potential energy function
$U(q)=(q^{2}-1)^{2}+q$. Errors are computed from the average of 500
independent trajectories, with $10^{9}$ total iterations per trajectory. The
step sizes tested began at $\delta t=0.2$ and were increased by $5\%$
incrementally until all schemes became unstable. The relative error in the
temperature (computed by averaging the momenta) is given in (a), with the
scheme of Bussi/Parrinello giving a high order relationship with the step
size. This is contrasted in (c) where the temperature is calculated using the
instantaneous system position; here it is instead the BAOAB scheme that gives
a high order result. The error in configurational distribution (calculated as
a root mean squared deviance in the histogram bins) is shown in (b), with a
sample computed distribution given in (d) for $\delta t=0.25.$ The exact
distribution is shown as a dashed line, with the inset magnifying the density
of the deepest well.
Of particular note is the apparent lack of direct correspondence between the
errors in configurational temperature and kinetic temperature. The
Bussi/Parrinello scheme is shown to preserve the kinetic temperature to a very
high degree, with less than a $1\%$ error for $\delta t<0.25,$ while, at the
same step size, the BAOAB integrator gives more than $10\%$ discrepancy in
kinetic temperature. However, these results are inverted when looking at
configurational sampling accuracy (and configurational temperature). Clearly,
if maintaining the configurational averages is the goal, estimating the
fidelity of the calculation by relying on the kinetic temperature is a risky
strategy. A much more reliable approach is to make use of the configurational
temperature, although even this does not give the complete story, since in the
example at hand the configurational temperature scales with a high power of
the step size, while configurational sampling error declines as the second
power of $\delta t$. The second order behavior does not contradict our
previous observations12 as we are here far from the large $\gamma$ limit, but
does indicate that the superconvergence property is not the key feature at
play in the setting of this model problem.
### V.2 Alanine dipeptide
In our next experiments, we studied the alanine dipeptide molecule, a classic
test case for molecular dynamics. We compare computed averages for solvated
and unsolvated alanine dipeptide using the BAOAB, ABOBA, van Gunsteren and
Berendsen, Bussi and Parrinello, Langevin Impulse, Stochastic Position Verlet
(SPV) and the Brünger/Brooks/Karplus (BBK) schemes. We obtained poor results
using the first-order schemes and therefore did not consider them here. To
provide a means of calculating basline values, we use the stochastic position
verlet (SPV) method with a small step size, for which the discretization error
is essentially negligible.
We implement each of the methods in the NAMD lite package22, and observe the
effect of discretization error (if any) on computed configurational averages.
The CHARMM22 forcefield was used to compute force interactions.
#### V.2.1 Unsolvated
We simulate the alanine dipeptide molecule (22 atoms) in vacuum at 300K for
2.5ns for multiple different step sizes and friction constants, to observe how
different simulation parameters affect computed averages. Parameters for each
run were taken from a $50\times 50$ grid, with each point on the grid
corresponding to a $(\delta t,\gamma)$ parameter set for a simulation. The
parameters for the bottom-left point on this grid are $\delta t_{1}=1$fs,
$\gamma_{1}=10^{-2}/$ps, where each grid point moving upward gives a $20.7\%$
increase in the friction value used, while each grid point moving right gives
a $2.46\%$ increase in step size. These ratios were chosen so as to give a
broad range of parameter sets to test over, while ensuring that the range was
not so wide as to yield a large number of unsuitable parameter sets (for
example, using an unstable step size) leading to wasted computation. All the
schemes were unstable for the maximum step size tested ($\delta
t_{50}=3.29$fs).
Figure 5: Results from $2.5$ns simulations of alanine dipeptide in vacuum at
the given step size (horizontal) and friction (vertical). Pixels are colored
according to relative errors for each simulation, with white pixels indicating
instability.
The results of the simulations for each scheme are given in Figure 5, where we
color points on the $50\times 50$ grid of parameter sets to indicate the
results from that respective simulation. Relative errors are calculated in the
average total potential energy and the average total bond energies, where the
“exact” comparison value is taken from averaging ten 2.5ns runs using the SPV
scheme at $\delta t=0.25$fs, where it is expected that discretization error is
not significant.
With such a small simulation we would perhaps expect to see a very “noisy”
result: high variances due to the sampling error vastly outweighing the
discretization error. But in fact the discretization error dominates and is
observable at step sizes significantly below the stability threshold.
White pixels in the grids in Figure 5 represent a method’s instability,
showing that in general there is a small stability threshold increase for the
large-friction case. There is no significant increase in this threshold
between the methods however, with BAOAB, VGB and LI schemes giving a marginal
increase over the others.
One salient feature of the results of Figure 5, is that for the BAOAB scheme
there is consistently less than a $1\%$ error in the computed configurational
temperature (for moderate friction) across all step sizes, even the largest
stable timesteps tested. The relative errors obtained were so small that no
discernable trend (with step size) can be shown, due to the sampling error,
whereas the other schemes tested show an error consistent with second-order
schemes (an example is given in Appendix C).
Self-diffusion coefficients are calculated from integrating the computed
velocity autocorrelation function, where a history is kept of the velocities
for 1ps. The values plotted in Figure 6 show that changing the step size
within the indicated range using any of the schemes has only a very slight
effect on the diffusion coefficient, while increasing the friction can
dramatically reduce it. Examining the graphs, we settle on $\gamma=1/$ps as
the largest value of $\gamma$ for which the diffusion coefficient is
unperturbed for all the schemes. It is interesting that larger damping
parameters do not substantially improve numerical stability for any of the
methods, except in an extreme case for the VGB method ($\gamma\approx 100/$ps,
where the diffusion constant is drastically reduced).
Numerical values for the computed average potential energy are given in Table
2 for varying step size at $\gamma=1/$ps.
Figure 6: The number of barrier recrossings (top) and the diffusion coefficients (bottom) are shown for each simulation, the latter in $\text{m}^{2}/$s and computed by integrating the velocity autocorrelation function over an interval of 1ps. As expected, the computed coefficients do not vary significantly between methods, though changing the friction above 1/ps has a substantial effect. Scheme | Average total potential energy (kcal/mol) | Total number of observed recrossings
---|---|---
$\delta t=1.5$fs | $\delta t=2$fs | $\delta t=2.5$fs | $\delta t=3$fs | $\delta t=1.5$fs | $\delta t=2$fs | $\delta t=2.5$fs | $\delta t=3$fs
BAOAB | $1.65\pm 0.04$ | $1.67\pm 0.05$ | $1.68\pm 0.03$ | $1.70\pm 0.05$ | $821\pm 17$ | $814\pm 25$ | $823\pm 21$ | $798\pm 11$
ABOBA | $1.69\pm 0.05$ | $1.70\pm 0.03$ | $1.75\pm 0.03$ | $1.87\pm 0.04$ | $823\pm 17$ | $829\pm 20$ | $826\pm 23$ | $833\pm 26$
SPV | $1.70\pm 0.13$ | $1.70\pm 0.05$ | $1.74\pm 0.06$ | $1.88\pm 0.03$ | $811\pm 56$ | $822\pm 22$ | $838\pm 26$ | $835\pm 22$
VGB | $1.89\pm 0.03$ | $2.14\pm 0.05$ | $2.65\pm 0.06$ | $4.27\pm 0.04$ | $802\pm 19$ | $804\pm 28$ | $812\pm 16$ | $813\pm 17$
LI | $2.05\pm 0.05$ | $2.42\pm 0.04$ | $3.21\pm 0.06$ | $5.84\pm 0.06$ | $837\pm 33$ | $819\pm 19$ | $822\pm 24$ | $821\pm 27$
BP | $2.75\pm 0.05$ | $3.91\pm 0.06$ | $6.19\pm 0.05$ | $14.0\pm 0.18$ | $828\pm 27$ | $801\pm 19$ | $821\pm 21$ | $818\pm 41$
BBK | $2.78\pm 0.03$ | $3.89\pm 0.05$ | $6.21\pm 0.06$ | $13.9\pm 0.10$ | $826\pm 18$ | $825\pm 24$ | $834\pm 19$ | $835\pm 16$
_Baseline_ | $1.66\pm 0.04$ | $808\pm 28$
Table 2: Numerical results for ten $2.5$ns simulations of unsolvated alanine
dipeptide, with friction set to $\gamma=1/$ps. The mean and standard
deviations of all simulations are given. The baseline comparison run was
completed by averaging ten $2.5$ns simulations using $\delta t=0.25$fs with
the SPV scheme. Sampling error will play a large role in the determination of
these averages, but it is clear that the BAOAB scheme outperforms the others
by a significant margin. The number of observed recrossings is obtained by
counting the number of times the central dihedral angles in the alanine
dipeptide model hop between their two configurations.
If we accept, say, a 5% error tolerance for the average potential energy, we
see that BAOAB admits a usable step size of up to $3$fs, whereas the ABOBA and
SPV schemes are restricted to a neighborhood of $2$fs, with the usable
timestep threshold for other methods well below $1.5$fs.
#### V.2.2 Solvated
We immerse the alanine dipeptide molecule in a sphere of TIP3P water (10A
radius, total system is 424 atoms) and equilibrate for 1ns at 300K to generate
an initial configuration. We then run simulations using each scheme considered
in the unsolvated case, using a 10A cutoff for electrostatics and van der
Waals potentials. The value of friction was fixed at $1/$ps, with runs
performed with increasing step size. Initial timesteps were $\delta t=2$fs,
with subsequent simulations increasing the step size by $5\%$, until reaching
a step size where all of the methods fail. Each simulation was performed for
$5$ns at $T=300$K using spherical (harmonically restrained) boundary
conditions. Although the particular boundary conditions may not be
representative of all biomolecular simulations, we contend that the crucial
features of numerical stability and relative method performance are unaffected
by the particular choice.
The results are given in Figure 7. Compared to the scheme of Bussi and
Parrinello, the relative error in average total potential energy using the
BAOAB scheme is seen to be smaller by two orders of magnitude when computed at
a step size around $\delta t=2.5$fs. The surprising downward trend of the
error using the BAOAB scheme could be indicative of higher-order terms
dominating in the error expansion, showing that our asymptotic approach does
not give definitive answers about bevavior in the large step size regime. The
analytic results obtained for the discretization error are understood only for
$\delta t\rightarrow 0$. More detailed analytical investigation of this
phenomenon is beyond the scope of this article.
Figure 7: Numerical results from 5ns runs of alanine dipeptide solvated in a
10A sphere of TIP3P water are shown, using the given algorithms. Errors are
computed against a baseline solution averaged from ten 5ns simulations using
the SPV scheme at $\delta t=0.5$fs. Results from a single run are shown for
each scheme except in the case of the BAOAB method. The BAOAB scheme shows an
order of magnitude improvement in the error in computed average total
potential energy; because of the small absolute errors, we exhibit the means
and standard deviations from 10 runs for each step size used. The lack of an
observed trend line for BAOAB suggests that the discretization error is being
dominated by the sampling error.
The breakdown of results for all energy contributions is given in Appendix D.
It is clear from these results that the average bond energy is a crucial
component in explaining the results of Figure 7. The average bond energy
computed using the BAOAB scheme gives a flat profile with respect to the
timestep increasing, whereas many other methods demonstrate an extreme drift
approaching the stability threshold, causing a large error in the average
total potential energy. The averages of other energies do not significantly
contribute to the observed errors.
The contribution of error coming from the restraining boundary condition
energy was extremely small, suggesting that the properties of the bulk water
in the model are responsible for the differences in efficiency seen here.
Hence we would expect the obseved corruption of averages to be generalizable
to any simulations involving other boundary conditions, or other simulations
involving water.
## VI Conclusion
We have studied a total of nine different integration methods for the Langevin
dynamics equations, including popular schemes that are in widespread use for
molecular sampling. We have seen that some of these can be derived as
splitting methods and in a few cases the perturbation of the invariant
distribution has been determined in some regime (for example, the small step
size, large friction limits). It is also possible to solve for the error in
averages as a function of step size in the case of a harmonic oscillator,
which we believe has direct relevance for biomolecular modelling where the
bond stretches are modelled as harmonic restraints. Harmonic models are also
likely to relate well to simulations of crystalline materials34. Our analyses
show that a particular ordering of the building blocks of a splitting method,
the BAOAB integrator12, provides exact configurational averages for the
harmonic oscillator and 4th order accurate configurational averages for a
general nonlinear model in the large friction limit. We have examined the
performance of this method in relation to other schemes for toy models and for
small biomolecular models both with and without solvent, with the observation
that the analytical results on the error in distribution are highly correlated
to their performance in practice. In particular, the BAOAB method performs
very differently than the other methods in practical simulations, giving much
higher accuracies (particularly for the configurational temperature) up to the
Verlet stability threshold.
A surprising observation is that discretization error, not sampling error,
dominates in the simulations we performed, which involved a common small
biomolecular test system and a time interval of only a few nanoseconds.
Let us put the numerical results into perspective. Our simulations explore
only a few of the available quantities that might be relevant for modelling.
It is interesting that all of the second order schemes tested provided
reasonable transition rates (in terms of barrier crossings), and so would give
a similar rate of exploration of the phase space. Since molecular dynamics is
often used for phase space exploration and supplemented by other techniques
for precise averaging, the methods may have some utility regardless of the
fact that they provide in some cases very poor approximation of
configurational averages. It is also possible for a scheme to accurately
resolve one quantity but not another (for example, in the case of unsolvated
alanine dipeptide, the scheme of van Gunsteren and Berendsen gives reasonably
good configurational temperatures but poor average potential energies).
The ABOBA and SPV methods perform very similarly, and reasonably well, both
for energy calculations and in terms of configurational temperature in all of
the numerical experiments performed with alanine dipeptide. The similarity
between these methods is a consequence of the fact that both are drift-kick-
drift style algorithms, with a subtle difference in their “kick” updates: the
ABOBA scheme solves (3) in a leapfrog manner by splitting off the force from
the Ornstein-Uhlenbeck stochastic term, where as the SPV scheme solves
equation (3) exactly for constant position $q$. One may expect that an exact
solve would provide the method better properties, but in practice the leapfrog
splitting in ABOBA is advantageous in the high friction regime. For large
$\gamma$, the result of solving exactly means that the momentum update becomes
dominated by the noise, shrinking the contribution from the force term. The
advantage of splitting up (3) is that this isolates the force from the noise,
integrating it separately, making ABOBA (and indeed other schemes using the
same splitting strategy, such as BAOAB and Bussi and Parrinello) effective for
any value of $\gamma\geq 0$.
Using quantities based on the momenta to estimate temperature or diffusion
constants is called into question; certainly the connection between the
accuracy of the kinetic energy average and the accuracy of other more directly
relevant quantities is weak. The kinetic temperature measure is an accurate
approximation of the true temperature in the case of the VGB scheme, but this
same method gives relatively poor potential energy averages.
We conclude by emphasizing that for bond energy and total potential energy
averages, in vacuum simulation, the BAOAB method performs better than the
other methods and is substantially better at large step sizes, giving a larger
useful range of step size by a factor of at least 25% with an order of
magnitude smaller errors at large step size. The differences are magnified
still further when configurational temperatures are compared.
###### Acknowledgements.
We thank David Hardy (University of Illinois) for his support with the
modification of the NAMD package. We also appreciate the support of the
Lorentz Center (Leiden, NL) and the programme on “Modelling the Dynamics of
Complex Molecular Systems” which supported the authors and provided valuable
interactions during the preparation of the article. This work has made use of
the resources provided by the Edinburgh Compute and Data Facility
(http://www.ecdf.ed.ac.uk/). The ECDF is partially supported by the eDIKT
initiative (http://www.edikt.org.uk). We further acknowledge the support of
the Engineering and Physical Sciences Research Council which has funded this
work as part of the Numerical Algorithms and Intelligent Software Centre under
Grant EP/G036136/1.
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## Appendix A Implementation details of Figure 2
We consider a single particle confined to the plane, with instantaneous
horizontal and vertical position denoted $x,y\in\mathbb{R}$ respectively. The
particle feels a force with respect to the potential energy function
$U(x,y)=\frac{1+5\Big{(}1-\exp\Big{(}-250\left(y-0.25\right)^{2}\Big{)}\Big{)}\,\exp\Big{(}-100x^{2}\Big{)}}{\left(\sqrt{x^{2}+y^{2}}-1\right)^{2}}+\frac{4\,\exp\Big{(}-20x^{2}\Big{)}}{5},$
qualitatively giving an energy surface with a narrow transition pathway
between two basins. We seek to sample the canonical distribution
$\bar{\rho}_{\beta}(x,y)$ using Langevin dynamics, comparing the computed
distributions given by the BAOAB and Bussi/Parrinello algorithms (given in the
following section) at varying stepsizes.
Figure 2 corresponds to numerical experiments using fixed friction constant
$\gamma=10$ and temperature $\beta=1$. From left to right, the stepsizes
tested were $\delta t=\left[0.005,0.021,0.024,0.027,0.03\right]$. Each image
is a two dimensional histogram over the unit square centered on the origin,
with 100 equally spaced bins in both directions. The computed density shown
for each timestep is averaged from 64 runs of $10^{8}$ steps.
## Appendix B Numerical methods
We present the numerical methods used in this article, assuming timestep
$\delta t$, friction constant $\gamma$ and temperature $T$ with diagonal mass
matrix $M$ and position and momentum vectors $q,p$ respectively. The force is
$F(q):=-\nabla U(q)$ and $k_{B}$ is Boltzmann’s constant. $R_{n}$ is a
$3N$-vector of independent, identically distributed normal random numbers with
zero mean and unit variance. Where $J>1$ random numbers are required per
degree of freedom, multiple independent (uncorrelated) random vectors are
denoted $R^{(j)}_{n},j=1,\ldots,J$ in the schemes.
BAOAB
Note: This scheme is available in recent versions of NAMD (after Jan 2013) by
including options ‘ _langevin on_ ’ and ‘ _langevinBAOAB on_ ’ in the input
parameter file.
$\displaystyle p_{n+1/3}$ $\displaystyle=p_{n}+\frac{\delta t}{2}F(q_{n}),$
$\displaystyle q_{n+1/2}$ $\displaystyle=q_{n}+\frac{\delta
t}{2}M^{-1}p_{n+1/3},$ $\displaystyle p_{n+2/3}$
$\displaystyle=e^{-\gamma\delta
t}p_{n+1/3}+\sqrt{k_{B}T\left(1-e^{-2\gamma\delta t}\right)}M^{1/2}R_{n},$
$\displaystyle q_{n+1}$ $\displaystyle=q_{n+1/2}+\frac{\delta
t}{2}M^{-1}p_{n+2/3},$ $\displaystyle p_{n+1}$
$\displaystyle=p_{n+2/3}+\frac{\delta t}{2}F(q_{n+1})$
ABOBA
$\displaystyle q_{n+1/2}$ $\displaystyle=q_{n}+\frac{\delta t}{2}M^{-1}p_{n},$
$\displaystyle p_{n+1/3}$ $\displaystyle=p_{n}+\frac{\delta
t}{2}F(q_{n+1/2}),$ $\displaystyle p_{n+2/3}$ $\displaystyle=e^{-\gamma\delta
t}p_{n+1/3}+\sqrt{k_{B}T\left(1-e^{-2\gamma\delta t}\right)}M^{1/2}R_{n},$
$\displaystyle p_{n+1}$ $\displaystyle=p_{n+2/3}+\frac{\delta
t}{2}F(q_{n+1/2}),$ $\displaystyle q_{n+1}$
$\displaystyle=q_{n+1/2}+\frac{\delta t}{2}M^{-1}p_{n+1}$
Van Gunsteren/Berendsen (VGB)
We must initialize the vector $X$,
$X_{1}=\kappa_{4}M^{-1/2}R^{(3)}_{0},$
and then iterate
$\displaystyle V_{n+1}$ $\displaystyle=\kappa_{1}M^{-1/2}R^{(1)}_{n},$
$\displaystyle\hat{V}_{n+1}$
$\displaystyle=\kappa_{2}X_{n}+\kappa_{3}M^{-1/2}R^{(2)}_{n},$ $\displaystyle
p_{n+1}$ $\displaystyle=e^{-\gamma\delta t}p_{n}+\frac{1-e^{-\gamma\delta
t}}{\gamma}F(q_{n})+M\left(V_{n+1}-e^{-\gamma\delta t}\hat{V}_{n+1}\right),$
$\displaystyle X_{n+1}$ $\displaystyle=\kappa_{4}M^{-1/2}R^{(3)}_{n},$
$\displaystyle\hat{X}_{n+1}$
$\displaystyle=\kappa_{5}V_{n+1}+\kappa_{6}M^{-1/2}R^{(4)}_{n},$
$\displaystyle q_{n+1}$ $\displaystyle=\frac{e^{\gamma\delta
t/2}-e^{-\gamma\delta t/2}}{\gamma}M^{-1}p_{n+1}+X_{n+1}-\hat{X}_{n+1},$
where we use $3N$–vectors $X,\hat{X},V,\hat{V}.$ Constants $\kappa_{i}$ are
given as
$\displaystyle\kappa_{1}$ $\displaystyle=\sqrt{k_{B}T\left(1-e^{-\gamma\delta
t}\right)},$ $\displaystyle\kappa_{2}$ $\displaystyle=\frac{2\gamma-\gamma
e^{\gamma\delta t/2}-\gamma e^{-\gamma\delta t/2}}{\gamma\delta
t-3+e^{-\gamma\delta t}\left(4e^{\gamma\delta t/2}-1\right)},$
$\displaystyle\kappa_{3}$ $\displaystyle=\sqrt{k_{B}T}\sqrt{\frac{\gamma\delta
t\left(e^{\gamma\delta t}-1\right)-4\left(e^{\gamma\delta
t/2}-1\right)^{2}}{\gamma\delta t-3+e^{-\gamma\delta t}\left(4e^{\gamma\delta
t/2}-1\right)}},$ $\displaystyle\kappa_{4}$
$\displaystyle=\gamma^{-1}\sqrt{k_{B}T}\sqrt{\gamma\delta t-3+e^{-\gamma\delta
t}\left(4e^{\gamma\delta t/2}-1\right)},$ $\displaystyle\kappa_{5}$
$\displaystyle=\gamma^{-1}\left(\frac{2-e^{\gamma\delta t}-e^{-\gamma\delta
t}}{e^{-2\gamma\delta t}-1}\right),$ $\displaystyle\kappa_{6}$
$\displaystyle=\gamma^{-1}\sqrt{k_{B}T}\sqrt{\frac{\gamma\delta
t\left(e^{-\gamma\delta t}-1\right)+4\left(e^{-\gamma\delta
t/2}-1\right)^{2}}{e^{-\gamma\delta t}-1}}.$
Stochastic Position Verlet (SPV)
$\displaystyle q_{n+1/2}$ $\displaystyle=q_{n}+\frac{\delta t}{2}M^{-1}p_{n},$
$\displaystyle p_{n+1}$ $\displaystyle=e^{-\gamma\delta
t}p_{n}+\frac{1-e^{-\gamma\delta
t}}{\gamma}F(q_{n+1/2})+\sqrt{k_{B}T\left(1-e^{-2\gamma\delta
t}\right)}M^{1/2}R_{n},$ $\displaystyle q_{n+1}$
$\displaystyle=q_{n+1/2}+\frac{\delta t}{2}M^{-1}p_{n+1}$
Bussi/Parrinello (BP)
$\displaystyle p_{n+1/4}$ $\displaystyle=e^{-\gamma\delta
t/2}p_{n}+\sqrt{k_{B}T\left(1-e^{-\gamma\delta t}\right)}M^{1/2}R^{(1)}_{n},$
$\displaystyle p_{n+2/4}$ $\displaystyle=p_{n+1/4}+\frac{\delta
t}{2}F(q_{n}),$ $\displaystyle q_{n+1}$ $\displaystyle=q_{n}+{\delta
t}M^{-1}p_{n+2/4},$ $\displaystyle p_{n+3/4}$
$\displaystyle=p_{n+2/4}+\frac{\delta t}{2}F(q_{n+1}),$ $\displaystyle
p_{n+1}$ $\displaystyle=e^{-\gamma\delta
t/2}p_{n+3/4}+\sqrt{k_{B}T\left(1-e^{-\gamma\delta
t}\right)}M^{1/2}R^{(2)}_{n}$
Langevin Impulse (LI)
We use the algorithm designed for configurational sampling; a correction term
is given in 9 to improve the sampling of momenta, though this has no effect on
configurational averages. We must initialize the $3N$–vector $Z$,
$Z_{1}=M^{1/2}\left(\alpha_{0}R_{0}+\hat{\alpha}R_{1}\right).$
and then iterate
$\displaystyle p_{n+1/4}$ $\displaystyle=p_{n}+\omega\delta tF(q_{n}),$
$\displaystyle p_{n+2/4}$ $\displaystyle=e^{-\gamma\delta
t/2}\left(p_{n+1/4}+\omega Z_{n}\right),$ $\displaystyle q_{n+1}$
$\displaystyle=q_{n}+\frac{1-e^{-\gamma\delta t}}{\gamma e^{-\gamma\delta
t/2}}M^{-1}p_{n+2/4},$ $\displaystyle Z_{n+1}$
$\displaystyle=M^{1/2}\left(\alpha R_{n}+\hat{\alpha}R_{n+1}\right),$
$\displaystyle p_{n+3/4}$ $\displaystyle=e^{-\gamma\delta
t/2}p_{n+2/4}+\hat{\omega}Z_{n+1},$ $\displaystyle p_{n+1}$
$\displaystyle=p_{n+3/4}+\hat{\omega}F(q_{n+1}),$
where
$\displaystyle\omega=\frac{e^{-\gamma\delta t}+\gamma\delta t-1}{\gamma\delta
t\left(1-e^{-\gamma\delta t}\right)},\qquad\hat{\omega}=1-\omega,$
$\displaystyle a=k_{B}T\left(2\omega^{2}\gamma\delta
t+\omega-\hat{\omega}\right),$ $\displaystyle
b=k_{B}T\left(2\omega\hat{\omega}\gamma\delta t+\hat{\omega}-\omega\right),$
$\displaystyle c=k_{B}T\left(2\hat{\omega}^{2}\gamma\delta
t+\omega-\hat{\omega}\right),$
$\displaystyle\alpha=2^{-1/2}\sqrt{c+a+\sqrt{(c+a)^{2}-4b^{2}}},$
$\displaystyle\hat{\alpha}=2^{-1/2}\sqrt{c+a-\sqrt{(c+a)^{2}-4b^{2}}},$
$\displaystyle\alpha_{0}=\sqrt{\alpha^{2}-c}.$
Brünger/Brooks/Karplus (BBK)
$\displaystyle p_{n+1/2}$ $\displaystyle=\left(1-\frac{\gamma\delta
t}{2}\right)p_{n}+\frac{\delta t}{2}F(q_{n})+\frac{1}{2}\sqrt{2\gamma
k_{B}T\delta t}M^{1/2}{R_{n}},$ $\displaystyle q_{n+1}$
$\displaystyle=q_{n}+{\delta t}M^{-1}p_{n+1/2},$ $\displaystyle p_{n+1}$
$\displaystyle=\left(1+\frac{\gamma\delta
t}{2}\right)^{-1}\left(p_{n+1/2}+\frac{\delta
t}{2}F(q_{n+1})+\frac{1}{2}\sqrt{2\gamma k_{B}T\delta
t}M^{1/2}R_{n+1}\right),$
Ermak/McCammon (EM)
As we consider only the scalar friction case, the update scheme for the
position reduces to the Euler-Maruyama algorithm, with rescaled timestep.The
update scheme for the momenta is unused in our numerical experiments, but can
be found in 26. We iterate
$\displaystyle q_{n+1}=q_{n}+\frac{\delta
tD}{k_{B}T}M^{-1}F(q_{n})+\sqrt{2D\delta t}M^{-1/2}R_{n},$
where
$D=k_{B}T/\gamma.$
Ermak/Buckholtz (EB)
$\displaystyle p_{n+1}$ $\displaystyle=e^{-\gamma\delta
t}p_{n}+\frac{1-e^{-\gamma\delta
t}}{\gamma}F(q_{n})+\sqrt{k_{B}T\left(1-e^{-\gamma\delta
t}\right)}M^{1/2}R^{(1)}_{n},$ $\displaystyle q_{n+1/3}$
$\displaystyle=q_{n}+M^{-1}\left(p_{n}+p_{n+1}-2\gamma^{-1}F(q_{n})\right)\frac{1-e^{-\gamma\delta
t}}{\gamma\left(1+e^{-\gamma\delta t}\right)},$ $\displaystyle q_{n+2/3}$
$\displaystyle=q_{n+1/3}+\gamma^{-1}\delta tM^{-1}F(q_{n}),$ $\displaystyle
q_{n+1}$ $\displaystyle=q_{n+2/3}+\gamma^{-1}\sqrt{2k_{B}T\left(\gamma\delta
t-2\frac{1-e^{-\gamma\delta t}}{1+e^{-\gamma\delta
t}}\right)}M^{-1/2}R^{(2)}_{n}$
## Appendix C Second order behavior of schemes
We demonstrate the results for the relative error in configurational
temperature, for simulations of alanine dipeptide in a vacuum at fixed
friction $\gamma=1.94/$fs. This is equivalent to plotting a horizontal cross-
section of the results given in Figure 5, at the corresponding friction value.
We find that in all but one scheme a clear second order trend is visible; the
exception is BAOAB which has much higher accuracy than the other methods and
for which a trend line could not be resolved; we conjecture that for BAOAB the
order of accuracy of the configurational temperature is substantially higher
than two (consistent with our observations for one degree-of-freedom
anharmonic cases).
## Appendix D Solvated results
The breakdown of computed average energies are given for each method. The
black dashed line marks the baseline solution for comparison.
|
arxiv-papers
| 2013-04-11T12:23:34 |
2024-09-04T02:49:44.200828
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Benedict Leimkuhler and Charles Matthews",
"submitter": "Charles Matthews",
"url": "https://arxiv.org/abs/1304.3269"
}
|
1304.3277
|
# Derivation of capture cross section from quasielastic excitation function
V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, and P.R.S.Gomes3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, 0025
Yerevan, Armenia
3Instituto de Fisica, Universidade Federal Fluminense, Av. Litorânea, s/n,
Niterói, R.J. 24210-340, Brazil
###### Abstract
The relationship between the quasielastic excitation function and the capture
cross section is derived. The quasielastic data is shown to be a useful tool
to extract the capture cross sections and the angular momenta of the captured
systems for the reactions 16O+144,154Sm,208Pb, 20Ne+208Pb, and 32S+90,96Zr at
near and above the Coulomb barrier energies.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: capture cross section, quasielastic excitation function, cold
fusion reactions
## I Introduction
The partial capture cross section is one of the important ingredients to
calculate and predict the production cross sections of exotic and superheavy
nuclei in the cold, hot, and sub-barrier astrophysical fusion reactions.
Therefore, more experimental and theoretical studies of the capture process
are required. There is a relationship between the capture and the quasielastic
scattering processes because of the conservation of the reaction flux
PRSGomes1 ; PRSGomes3 . Any loss from the quasielastic channel directly
contributes to the capture and vise versa. The quasielastic measurements are
usually not as complex as the direct capture (fusion) measurements. Thus, the
quasielastic data are suited for the extraction of the capture probabilities
and of the capture cross sections.
The paper is organized in the following way. In Sec. II we derive the formulas
for the extraction of the capture cross section and of the angular momentum of
the captured system by employing the experimental quasielastic excitation
function. In Sec. III, using these formulas, we extract the capture cross
sections and the angular momenta of the captured systems and compare with
those of direct measurements. Using the available experimental quasielasic
data, we predict the capture cross sections for the cold fusion reactions. In
Sec. IV the paper is summarized.
## II Relationship between capture and quasielastic scattering
The expression
$P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)=1$ (1)
connecting the quasielastic (reflection) $P_{qe}$ and the capture
(transmission) $P_{cap}$ probabilities follows from the conservation of the
reaction flux PRSGomes1 ; PRSGomes3 . Thus, one can extract the capture
probability $P_{cap}(E_{\rm c.m.},J=0)$ at $J=0$ from the experimental
quasielastic probability $P_{qe}(E_{\rm c.m.},J=0)$:
$P_{cap}(E_{\rm c.m.},J=0)=1-P_{qe}(E_{\rm c.m.},J=0)=1-d\sigma_{qe}(E_{\rm
c.m.})/d\sigma_{Ru}(E_{\rm c.m.}).$ (2)
Here, the quasielastic probability PRSGomes1 ; Timmers ; Timmers2 ; Zhang
$\displaystyle P_{qe}(E_{\rm c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$ (3)
for angular momentum $J=0$ is given by the ratio of the quasielastic
differential cross section and Rutherford differential cross section at 180
degrees. Further, one can approximate the $J$ dependence of the capture
probability $P_{cap}(E_{\rm c.m.},J)$ at a given energy $E_{\rm c.m.}$ by
shifting the energy Bala :
$\displaystyle P_{cap}(E_{\rm c.m.},J)\approx P_{cap}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0)=1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0),$ (4)
where $\Lambda=J(J+1)$, $R_{b}=R_{b}(J=0)$ is the position of the Coulomb
barrier at $J=0$. Then, we extract the capture cross section
$\sigma_{cap}(E_{\rm c.m.})$ from the experimental quasielastic probabilities
$P_{qe}$:
$\displaystyle\sigma_{cap}(E_{\rm c.m.})=\sum_{J=0}^{J_{cr}}\sigma_{\rm
cap}(E_{\rm
c.m.},J)=\pi\lambdabar^{2}\sum_{J=0}^{J_{cr}}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0)],$ (5)
where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie
wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$
is the nucleon mass), and at given bombarding energy $E_{\rm c.m.}$ the
summation is over the possible values of angular momentum $J$ from $J=0$ to
the critical angular momentum $J=J_{cr}$. For values $J$ greater than
$J_{cr}$, the potential pocket in the nucleus-nucleus interaction potential
vanishes and the capture is not occur. To calculate the critical angular
momentum $J_{cr}$ and the position $R_{b}$ of the Coulomb barrier, we use the
nucleus-nucleus interaction potential $V(R,J)$ of Ref. Pot . For the nuclear
part of the nucleus-nucleus potential, the double-folding formalism with the
Skyrme-type density-dependent effective nucleon-nucleon interaction is
employed Pot .
If one sets $R_{b}(J)\approx R_{b}$ in Eq. (5) for approximating the $J$-wave
penetrability by the $s$-wave penetrability at a shifted energy, one obtains
only the leading term in the series expansion in $\Lambda$. The next term in
this expansion can be easily calculated in the same way as in Ref. Bala
[$R_{b}(J)\approx R_{b}-\frac{\hbar^{2}\Lambda}{\mu\alpha R_{b}^{3}}$,
$V_{b}(J)\approx V_{b}+\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}}+\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha R_{b}^{6}}$,
$\alpha=-\partial^{2}V(R,J=0)/\partial R^{2}|_{R=R_{b}}=\mu\omega_{b}^{2}$,
$\omega_{b}=\omega_{b}(J=0)$ is the curvature of the $s$-wave potential
barrier with the height $V_{b}=V_{b}(J=0)=V(R=R_{b},J=0)$]:
$\displaystyle P_{cap}(E_{\rm c.m.},J)\approx P_{cap}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha R_{b}^{6}},J=0).$ (6)
With this improved expression for the $P_{cap}$, we obtain
$\displaystyle\sigma_{cap}(E_{\rm
c.m.})=\pi\lambdabar^{2}\sum_{J=0}^{J_{cr}}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}},J=0)][1-\frac{2\hbar^{2}\Lambda}{\mu^{2}\omega_{b}^{2}R_{b}^{4}}].$
(7)
Converting the sum over $J$ into an integral and changing variables to
$E=E_{\rm c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$ in Eq. (7), we obtain
the following simple expression:
$\displaystyle\sigma_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm
c.m.}}dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}],$ (8)
which relates the capture cross section with quasielastic excitation function.
Note that $\Lambda$ is not a small parameter, there is a natural cutoff
$\Lambda_{cr}=J_{cr}(J_{cr}+1)$ in this parameter. Because of this cutoff, the
second term $\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$ in Eq. (6) is always
larger than the third one $\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha
R_{b}^{6}}$ Bala . By using the experimental quasielastic probabilities
$P_{qe}(E_{\rm c.m.},J=0)$ and Eq. (8) one can obtain the capture cross
sections.
For the systems with $Z_{1}\times Z_{2}<2000$, the critical angular momentum
$J_{cr}$ is large enough and Eqs. (7) and (8) can be approximated with a good
accuracy as:
$\displaystyle\sigma_{cap}(E_{\rm
c.m.})\approx\pi\lambdabar^{2}\sum_{J=0}^{\infty}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}},J=0)][1-\frac{2\hbar^{2}\Lambda}{\mu^{2}\omega_{b}^{2}R_{b}^{4}}]$
(9)
and
$\displaystyle\sigma_{cap}(E_{\rm c.m.})\approx\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{0}^{E_{\rm
c.m.}}dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}].$ (10)
Following the procedure of Ref. Bala and using the extracted $\sigma_{cap}$
and the experimental $P_{qe}$, one can find the average angular momentum
$\displaystyle<J>=\frac{\pi R_{b}^{2}}{E_{\rm c.m.}\sigma_{cap}(E_{\rm
c.m.})}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu
R_{b}^{2}}}^{E_{\rm c.m.}}$ $\displaystyle
dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{5(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (11) $\displaystyle\times[(\frac{2\mu
R_{b}^{2}}{\hbar^{2}}(E_{\rm c.m.}-E)+\frac{1}{4})^{1/2}-\frac{1}{2}]$
and the second moment of the angular momentum
$\displaystyle<J(J+1)>=\frac{2\pi\mu R_{b}^{4}}{\hbar^{2}E_{\rm
c.m.}\sigma_{cap}(E_{\rm c.m.})}\int_{E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}$
$\displaystyle dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{6(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (12) $\displaystyle\times[E_{\rm
c.m.}-E]$
of the captured system.
## III Results of calculations
For the verification of our method of the extraction of $\sigma_{cap}$,
firstly we compare the extracted capture cross sections with experimental one.
In Figs. 1 and 2 one can see a good agreement between the extracted and
directly measured capture cross sections for the reactions 16O + 120Sn, 18O +
124Sn, 16O + 208Pb, and 16O + 144Sm at energies above the Coulomb barrier. The
results on the sub-barrier energy region are discussed later on. To extract
the capture cross section, we use both Eq. (8) (solid lines) and Eq. (10)
(dotted lines). The used values of critical angular momentum are $J_{cr}$=54,
56, 57, and 62 for the reactions 16O + 120Sn, 18O + 124Sn, 16O + 144Sm, and
16O + 208Pb, respectively. The difference between the results of Eqs. (8) and
(10) is less than 5$\%$ at the highest energies. At low energies, Eqs. (8) and
(10) lead to the same values of $\sigma_{cap}$. The factor $1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}$ in Eqs. (8) and (10) very weakly
influences the results of the calculations for the systems and energies
considered. Hence, one can say that for the relatively light systems the
proposed method of extracting the capture cross section is model independent
(particular, independent on the potential used).
Figure 1: The extracted capture cross sections for the reactions 16O + 120Sn
(a) and 18O + 124Sn (b) by employing Eq. (8) (solid line) and Eq. (10) (dotted
line). These lines are almost coincide. The used experimental quasielastic
data are from Ref. Sinha . The experimental capture (fusion) data (symbols)
are from Refs. Sinha ; JACOBS . Figure 2: The same as in Fig. 1, but for the
reactions 16O + 208Pb(a),144Sm(b). The used experimental quasielastic data are
from Refs. Timmers2 ; Timmers . For the 16O + 208Pb reaction, the experimental
capture (fusion) data are from Refs. Pbcap (open squares), Pbcap1 (open
circles), Pbcap2 (closed stars), and Pbcap3 (closed triangles). For the 16O
+ 144Sm reaction, the experimental capture (fusion) data are from Refs. SmCap1
(closed squares) and SmCap2 (open squares).
One can see that the used formulas are suitable not only for almost spherical
nuclei (Figs. 1 and 2), but also for the reactions with strongly deformed
target- or projectile-nucleus (Figs. 3 and 4). The deformation effect is
effectively contained in the experimental $P_{qe}$. $J_{cr}=58$, 68, 74, and
76 for the reactions 16O+154Sm, 32S+90Zr, 32S+96Zr, and 20Ne+208Pb,
respectively. The results obtained by employing the formula (10) are almost
the same and not presented in Figs. 3 and 4.
Figure 3: The same as in Fig. 1, but for the reactions 20Ne + 208Pb and 16O +
154Sm. The used experimental quasielastic data are from Refs. Piasecki ;
Timmers . The experimental capture (fusion) data (symbols) are from Refs.
SmCap2 ; Piasecki . For the 16O + 154Sm reaction, the dashed line is obtained
from the shift of the solid line by 1.7 MeV higher energies. Figure 4: The
same as in Fig. 1, but for the reactions 32S + 90Zr (a) and 32S + 96Zr (b).
For the 32S+90Zr reaction, we show the extracted capture cross sections,
increasing the experimental $P_{qe}$ by 1% (dashed line), 2% (dotted line),
and 3% (dash-dotted line). The used experimental quasielastic data are from
Ref. Zhang3 . The experimental capture (fusion) data (symbols) are from Ref.
ZhangS32Zn9096 . For the 32S + 96Zr reaction, the energy scale for the
extracted capture cross sections is adjusted to that of direct measurements.
For the reactions 16O+154Sm and 32S+96Zr, the extracted capture cross sections
are shifted in energy by 1.7 and 1.9 MeV, respectively, with respect to the
measured capture data. This could be the result of different energy
calibrations in the experiments on the capture measurement and on the
quasielastic scattering. Because of the lack of systematics in these energy
shifts, their origin remains unclear and we adjust the Coulomb barriers in the
extracted capture cross sections to the values following the experiments.
Note that the extracted and experimental capture cross sections deviate from
each other in the reactions 16O+208Pb, 16O+144Sm, and 32S+90Zr at energies
below the Coulomb barrier. Probably this deviation is a reason for the large
discrepancies in the diffuseness parameter extracted from the analyses of the
quasielastic scattering and fusion (capture) at deep sub-barrier energies. One
of the possible reasons for the overestimation of the capture cross section
from the quasielastic data at sub-barrier energies is the underestimation of
the total reaction differential cross section taken as the Rutherford
differential cross section. Indeed, for the 32S+90Zr reaction, the increase of
$P_{qe}$ within 2–3% is in order to obtain the agreement between the extracted
and measured capture cross sections at the sub-barrier energies [Fig. 4(a)].
One can use Eq. (8) and available experimental quasielasic data Ikezoe to
predict the capture cross sections for the reactions 48Ti,54Cr,56Fe,64Ni,70Zn
+ 208Pb, using $J_{cr}=78$, 74, 58, 51, 31, respectively. The extracted
capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ as a function of $E_{\rm
c.m.}$ are presented in Fig. 5 (a).
Figure 5: (a) The extracted capture cross sections employing Eq. (8) (solid
line) and Eq. (10) (dotted line) for the reactions 48Ti,54Cr,56Fe,64Ni,70Zn +
208Pb. The used experimental quasielastic data are from Ref. Ikezoe . (b) The
extracted values of the maximal angular momenta vs. energy for the above
mentioned reactions. The solid and dotted lines show the results of
calculations of $J_{max}$ by using the extracted capture cross sections
calculated with Eqs. (8) and (10), respectively.
The formulas (8) and (10) give almost the same capture cross sections for
reactions 48Ti,54Cr + 208Pb at energies under consideration. Thus, for these
systems, the values of $J_{cr}$ are relatively large and the account of
$J_{cr}$ does not affect the results. However, for heavier systems with
smaller $J_{cr}$ (the smaller potential pockets in the nucleus-nucleus
interaction potentials), the deviation between the results obtained with Eqs.
(8) and (10) increases strongly with the factor $Z_{1}\times Z_{2}$. The
$\sigma_{cap}$, calculated with the finite value of critical angular momentum,
decreases with increasing Coulomb repulsion in the system. One can try to
check experimentally these predictions of $\sigma_{cap}(E_{\rm c.m.})$ by the
direct measurement of the capture cross sections. Note that the values of the
extracted capture cross sections for the 48Ti + 208Pb system are close to
those found in the experiments 50Ti + 208Pb Naik ; Clerc . However, for the
64Ni + 208Pb system, there are strong deviations in the energy between the
extracted and experimental Bock capture cross sections.
By using the extracted $\sigma_{cap}(E_{\rm c.m.})$ and the sharp-cutoff
approximation, one can determine the maximal angular momentum $J_{max}$ in the
captured system as a function of the bombarding energies:
$\displaystyle J_{max}=[2\mu E_{\rm c.m.}\sigma_{cap}(E_{\rm
c.m.})/(\pi\hbar^{2})]^{1/2}-1.$ (13)
The extracted $J_{max}$ for the cold fusion reactions are shown in Fig. 5(b).
For the system 70Zn + 208Pb, the small depth of the potential pocket in the
nucleus-nucleus interaction potential leads to the decrease of $J_{max}$ by
the factor about of 2.4 at highest energy considered (about of 17 MeV above
the Coulomb barrier).
In the reactions with weakly bound nuclei one can extract the capture cross
section by employing the conservation of the reaction flux PRSGomes1 ; Nash ;
Be9Pb ; Li6Pb
$\displaystyle P_{cap}(E_{\rm c.m.},J=0)=1-[P_{qe}(E_{\rm
c.m.},J=0)+P_{BU}(E_{\rm c.m.},J=0)]$ (14)
and the measured probabilities of the quasielastic scattering ($P_{qe}(E_{\rm
c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$) and of the breakup ($P_{BU}(E_{\rm
c.m.},J=0)=d\sigma_{BU}/d\sigma_{Ru}$) which are defined as the differential
cross sections ratios between quasielastic scattering, breakup reaction and
the Rutherford scattering at backward angle. As seen in Fig. 6, the extracted
capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ (solid line) for the
6Li+208Pb reaction are rather close to those found in the direct measurements
Li6Pbcap at energies above the Coulomb barrier.
Figure 6: (Colour online) The extracted capture cross sections
$\sigma_{cap}(E_{\rm c.m.})$ (solid line) and $\sigma^{noBU}_{cap}(E_{\rm
c.m.})$ (dotted line) for the 6Li+208Pb reaction. The used experimental
quasielastic and quasielastic plus breakup data are from Ref. Li6Pb . The
experimental capture cross sections (solid squares) are from Refs. Li6Pbcap .
The energy scale for the extracted capture cross sections is adjusted to that
of direct measurements.
It looks that at energies near and below the Coulomb barrier the extracted
$\sigma_{cap}(E_{\rm c.m.})$ deviates from the direct measurements. It is
similarly possible to calculate the capture excitation function
$\displaystyle\sigma^{noBU}_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm
c.m.}}dEP^{nBU}_{cap}(E,J=0)[1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (15)
in the absence of the breakup process (Fig. 6, dotted line) by using the
following formula for the capture probability in this case Nash :
$\displaystyle P^{nBU}_{cap}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm
c.m.},J=0)}{1-P_{BU}(E_{\rm c.m.},J=0)}.$ (16)
By employing the measured excitation functions $P_{qe}$ and $P_{BU}$ at
backward angle Li6Pb , Eqs. (8), (15), and the formula
$\displaystyle<P_{BU}>(E_{\rm c.m.})=1-\frac{\sigma_{cap}(E_{\rm
c.m.})}{\sigma^{noBU}_{cap}(E_{\rm c.m.})},$ (17)
we extract the mean breakup probability $<P_{BU}>(E_{\rm c.m.})$ averaged over
all partial waves $J$ (Fig. 7).
Figure 7: The extracted mean breakup probability $<P_{BU}>(E_{\rm c.m.})$
[Eq. (14)] as a function of bombarding energy $E_{\rm c.m.}$ for the 6Li+208Pb
reaction. The used experimental quasielastic and quasielastic plus breakup
data are from Ref. Li6Pb .
The value of $<P_{BU}>$ has a maximum at $E_{\rm c.m.}-V_{b}\approx 4$ MeV
($<P_{BU}>$=0.26) and slightly (sharply) decreases with increasing
(decreasing) $E_{\rm c.m.}$. The experimental breakup excitation function at
backward angle has the similar energy behavior Li6Pb . By comparing the
calculated capture cross sections in the absence of breakup and experimental
capture (complete fusion) data, the opposite energy trend is found in Ref.
Nash , where $<P_{BU}>$ has a minimum at $E_{\rm c.m.}-V_{b}\approx 2$ MeV
($<P_{BU}>$=0.34) and globally increases in both sides from this minimum. It
is also shown in Refs. Nash ; PRSGomes4 that there are no systematic trends
of breakup in the complete fusion reactions with the light projectiles 9Be,
6,7,9Li, and 6,8He at near-barrier energies. Thus, by employing the
experimental quasielastic backscattering, one can obtain the additional
information about the breakup process.
Figure 8: The extracted $<J>$ and $<J^{2}>$ for the reactions 16O + 208Pb (a)
and 16O + 154Sm (b) by employing Eqs. (11) and (12). The used experimental
quasielastic data are from Ref. Timmers2 . The experimental data of $<J^{2}>$
and $<J>$ are from Refs. Vand (open squares) and Gil ; Vand2 (open squares
and circles), respectively. Figure 9: The extracted $<J>$ for the reactions
32S + 96Zr (a) and 16O + 120Sn (b) by employing Eq. (11). The used
experimental quasielastic data are from Refs. Zhang3 ; Sinha .
By using the Eqs. (11) and (12) and experimental $P_{qe}$, we extract $<J>$
and $<J^{2}>$ of the captured system for the reactions 16O + 154Sm and 16O +
208Pb, respectively (Fig. 8). The agreements with the results of direct
measurements of the $\gamma-$multiplicities in the corresponding complete
fusion reactions are quite good. For the 16O + 208Pb reaction at sub-barrier
energies, the difference between the extracted and experimental angular
momenta is related with the deviation of the extracted capture excitation
function from the experimental one (see Fig. 2). In Fig. 9 we present the
predictions of $<J>$ for the reactions 16O + 120Sn and 32S + 96Zr.
## IV Summary
We realized that the found relationship between the quasielastic excitation
function and capture cross sections is working well, and the quasielastic
technique could be an important and simple tool in the study of the capture
(fusion) research, especially, in the cold and hot fusion reactions and in the
breakup reactions at energies near and above the Coulomb barrier. Employing
the quasielastic data, one can also extract the moments of the angular
momentum of the captured system.
We thank S. Heinz, S. Hofmann and H.Q. Zhang for fruitful discussions and
suggestions. We are grateful to H. Ikezoe, C.J. Lin, E. Piasecki, and H.Q.
Zhang for providing us their experimental data. This work was supported by
DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish -
JINR(Dubna) Cooperation Programmes are gratefully acknowledged. P.R.S.G.
acknowledges the partial financial support from CNPq and FAPERJ.
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|
arxiv-papers
| 2013-04-11T12:43:07 |
2024-09-04T02:49:44.212519
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, and P.R.S.Gomes",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1304.3277"
}
|
1304.3483
|
We give a new probabilistic algorithm for interpolating a “sparse”
polynomial $f$ given by a straight-line program. Our algorithm
constructs an approximation $f^*$ of $f$, such that $f-f^*$
probably has at most half the number of terms of $f$, then
recurses on the difference $f-f^*$. Our approach builds on previous
work by [Garg and Schost, 2009], and [Giesbrecht and Roche, 2011], and is asymptotically
more efficient in terms of the total cost of the probes required
than previous methods, in many cases.
§ INTRODUCTION
We consider the problem of interpolating a sparse, univariate
\[
f = c_1 z^{e_1} + c_2 z^{e_2} + \cdots + c_t z^{e_t} \in\R[z]
\]
of degree $d$ with $t$ non-zero coefficients $c_1,\ldots,c_t$ (where
$t$ is called the sparsity of $f$) over a ring
$\R$. More formally, we are given a straight-line program that
evaluates $f$ at any point, as well as bounds $D \geq d$ and $T \geq
t$. The straight-line program is a simple but useful abstraction of a
computer program without branches, but our interpolation algorithm
will work in more common settings of “black box” sampling of $f$.
We summarize our final result as follows.
Let $f \in \R[z]$, where $\R$ is any ring. Given any straight-line
program of length $L$ that computes $f$, and bounds $T$ and $D$ for
the sparsity and degree of $f$, one can find all coefficients and
exponents of $f$ using $\softO(L T\log^3 D + LT\log D\log
(1/\mu))$[4] ring operations in $\R$, plus a similar
number of bit operations. The algorithm is probabilistic of the
Monte Carlo type: it can generate random bits at unit cost and on
any invocation returns the correct answer with probability greater
than $1-\mu$, for a user-supplied tolerance $\mu>0$.
[4]For summary convenience we use soft-Oh notation: for
functions $\phi,\psi\in\RR_{>0}\to\RR_{>0}$ we say
$\phi\in\softO(\psi)$ if and only if $\phi\in
\O(\psi(\log\psi)^c)$ for some constant $c\geq 0$.
§.§ The straight-line program model and interpolation
Straight-line programs are a useful model of computation, both as a
theoretical construct and from a more practical point of view; see,
e.g., <cit.>. Our interpolation algorithms work more
generally for $N$-variate sparse polynomials $f\in\R[z_1,\ldots,z_N]$
given by a straight-line program $\SLP_f$ defined as follows.
$\SLP_f$ takes an input $(a_1, \dots, a_N) \in
\R^N$ of length $N$, and produces a vector $b \in \R^L$ via a series
of $L$ instructions $\Gamma_i : 1 \leq i \leq L$ of the form
\[
\Gamma_i =
\begin{cases}
\gamma_i & \longleftarrow \alpha_1 \star \alpha_2, ~~\emph{or} \\
\gamma_i & \longleftarrow \delta \in \R ~~~~\emph{(i.e., a constant
from $\R$)},
\end{cases}
\]
where $\star$ is a ring operation `$+$', `$-$', or `$\times$',
and either $\alpha_\ell \in \{ a_j \}_{1 \leq j \leq n}$ or
$\alpha_\ell \in \{ \gamma_k \}_{1 \leq k < i}$ for $\ell=1,2$.
When we say $\SLP_f$ computes
$f$, we mean $\SLP_f$ sets $\gamma_L$ to $f( a_1, \dots, a_N)\in\R$.
To interpolate an $N$-variate polynomial $f\in\R[z_1,\ldots,z_N]$, we
apply a Kronecker substitution, and interpolate
\[
\fhat(z) = f\left(z, z^{(D+1)}, z^{(D+1)^2}, \dots,
z^{(D+1)^{N-1}} \right)\in\R[z].
\]
While this certainly increases the degree, $f$ and $\fhat$ have the
same number of non-zero terms, and $f$ can be easily recovered from
$\fhat$. This reduces the problem of interpolating the $N$-variate
polynomial $f$ of partial degree at most $D$ to interpolating a
univariate polynomial $\fhat$ of degree at most $(D+1)^N$. For the
remainder of this paper we thus assume $f$ is univariate.
It will also be necessary to evaluate our polynomial $f\in\R[z]$, or
rather our straight-line program $\SLP_f$ for $f$, in an extension ring of
$\R$. Precisely, we want to evaluate $f$ at symbolic $\ell$th roots
of unity for various choices of $\ell$, or algebraically, in
$\R[z]/(z^{\ell}-1)$. This may be regarded as transforming our
straight-line program by substituting operations in $\R$ with
operations in $\R[z]/(z^\ell-1)$, where each element is represented by
a polynomial in $\R[z]$ of degree less than $\ell$. Each instruction
$\Upsilon_i$ in the transformed branching program now potentially
requires $\M( \ell)$ operations in $\R$, where $\M(\ell)$ is the
number of operations in $\R$ and bit operations needed to multiply two
degree-$\ell$ polynomials over the base ring $\R$. By [Cantor and Kaltofen, 1991],
we may assume $\M(\ell) = \O(\ell \log \ell \log\log \ell)$.
Each evaluation of our straight-line program for $f$ in
$\R[z]/(z^\ell-1)$ is called a probe of degree $\ell$. Thus,
the cost of a degree-$\ell$ probe to $\SLP_f$ is $\softO( L \ell )$ operations
in $\R$, and similarly many bit operations.
This is easily connected to the more “classical” view of sparse
interpolation, in which probes are simply evaluations of a “black-box”
polynomial at a single point (and we do not have any
representation for how $f$ is calculated). Each probe in the
straight-line program model can
be thought of as evaluating $f$ at all $\ell$th roots of unity
in the classical model. Since we charge $\M(\ell)=\softO(\ell)$
operations in $\R$ for a degree $\ell$ probe in the straight-line
program model, i.e., about $\ell$ times
as much as a single black-box probe, this is consistent with the costs in a
classical model. We note that algorithms for sparse interpolation
presented below could be stated in this classical model, though we
find the straight-line program model convenient and will continue with
it throughout this paper.
§.§ Previous work
Straight-line programs, or equivalently algebraic circuits,
are important both as a computational model and as a data structure
for polynomial computation. Their rich history includes both
algorithmic advances and practical implementations
[Kaltofen, 1989, Sturtivant and Zhang, 1990, Bruno et al., 2002].
One can naively interpolate a polynomial $f\in\R[z]$ given by a straight-line
using a dense method, with $D$ probes of degree $1$. Prony's
[de Prony, 1795] interpolation algorithm — see [Ben-Or and Tiwari, 1988, Kaltofen et al., 1990, Giesbrecht et al., 2009] — is a sparse interpolation method that uses
evaluations at only $2T$ powers of a root of unity whose order is
greater than $D$. However, in the straight-line program model for a
general ring, this would require evaluating at a symbolic $D$th root
of unity, which would use at least $\Omega(D)$ ring operations and
defeat the benefit of sparsity. Problems with Prony's algorithm are
also seen in the classical model in that the underlying base ring $\R$
must also support an efficient discrete logarithm algorithm on entries
of high multiplicative order (which, for example, is not feasible
over large finite fields).
We mention two algorithms specifically intended for straight-line
§.§.§ The Garg-Schost deterministic algorithm.
[Garg and Schost, 2009] describe a novel deterministic algorithm for interpolating a
multivariate polynomial $f$ given by a straight-line program. Their
algorithm entails constructing an integer symmetric polynomial with
roots at the exponents of $f$:
\[
\chi = \prod_{i=1}^{t}(y-e_i) \in\ZZ[y],
\]
which is then factored to obtain the exponents $e_i$.
Their algorithm first finds a good prime: a prime $p$ for which
the terms of $f$ remain distinct when reduced modulo $z^p-1$. We call
such an image $f \bmod (z^p-1)$ a good image. Such an image
gives us the values $e_i \bmod p$ and hence $\chi(y) \bmod p$.
For $f = z^{33}+z^3$, $5$ is not a good prime because $f \bmod
(z^5-1) = 2z^3$. We say $z^{33}$ and $z^{3}$ collide modulo
$z^5-1$. $7$ is a good prime, as the image $f(z) \bmod (z^7-1) =
z^5+z^3$ has as many terms as $f(z)$ does.
In order to guarantee that we have a good prime, the algorithm
requires that we construct the images $f \bmod (z^p-1)$ for the first
$N$ primes, where $N$ is roughly $\softO( T^2\log D)$. A good prime
will be a prime $p$ for which the image $f \bmod (z^p-1)$ has
maximally many terms, which will be exactly $t$. Once we know we have
a good image we can discard the images $f \bmod (z^q-1)$ for
bad primes $q$, i.e. images with fewer than $t$ terms. We use
the remaining images to construct $\chi(y) = \prod_{i=1}^{t}(y-e_i)
\in \mathbb{Z}[y]$ by way of Chinese remaindering on the images
$\chi(y) \bmod p$.
We factor $\chi(y)$ to obtain the exponents $e_i$, after which we
directly obtain the corresponding coefficients $c_i$ directly from a
good image.
The algorithm of [Garg and Schost, 2009] can be made faster, albeit Monte Carlo,
using the following number-theoretic fact.
[Giesbrecht and Roche, 2011]
Let $f \in \mathcal{R}[z]$ be a polynomial with at most $T$ terms
and degree at most $D$. Let $\lambda = \max(21, \lceil
\tfrac{5}{3}T(T-1)\log D \rceil )$. A prime $p$ chosen at random in
the range $[ \lambda, 2\lambda]$ is a good prime for $f(z)$ with
probability at least $\tfrac{1}{2}$.
Thus, in order to find a good image with probability at least
$1-\varepsilon$, we can inspect images $f \bmod (z^p-1)$ for $\lceil
\log 1/ \varepsilon \rceil$ primes $p$ chosen at random in $[ \lambda,
2\lambda ]$. As the height of $\chi(y)$ can be roughly as large as
$D^T$, we still require some $\mathcal{O}^{\sim}(T \log D)$ probes to
construct $\chi(y)$.
§.§.§ The “diversified” interpolation algorithm.
[Giesbrecht and Roche, 2011] obtain better performance by way of
diversification. A polynomial $f$ is said to be diverse
if its coefficients $c_i$ are pairwise distinct. The authors show
that, for $f$ over a finite field or $\CC$ and for appropriate random
choices of $\alpha$, $f(\alpha z)$ is diverse with probability at
least $\tfrac{1}{2}$. They then try to interpolate the diversified
polynomial $f(\alpha z)$.
Once we have $t$ with high probability, we look at images $f(\alpha z)
\bmod (z^p-1)$ for primes $p$ in $[\lambda, 2\lambda]$, discarding bad
images. As $f(\alpha z)$ is diverse, we can recognize which terms in
different good images are images of the same term. Thus, as all the
$e_i$ are at most $D$, we can get all the exponents $e_i$ by looking
at some $\softO(\log D)$ good images of $f$.
§.§ Deterministic zero testing
Both the Monte Carlo algorithms of [Garg and Schost, 2009] and [Giesbrecht and Roche, 2011]
can be made Las Vegas (i.e., no possibility of erroneous output, but unbounded
worst-case running time) by way of
deterministic zero-testing. Given a polynomial $f$ represented by a
straight-line program, each of these algorithms produces a polynomial
$f^*$ that is probably $f$.
[[Bläser et al., 2009]; Lemma 13]
Let $\R$ be an integral domain, and suppose $f = f^* \bmod (z^p-1)$ for
$T\log D$ primes. Then $f = f^*$.
Thus, testing the correctness of the output of a Monte Carlo algorithm
requires some $\softO( T\log D )$ probes of degree at most
$\softO(T\log D)$. This cost does not dominate the cost of either
Monte Carlo algorithm. We note that this deterministic zero test can
dominate the cost of the interpolation algorithm presented in this
paper if $T$ is asymptotically dominated by $\log D$.
§.§ Summary of results
We state as a theorem the number and degree of probes required by our
new algorithm presented in this paper.
Let $f \in \R[z]$, where $\R$ is a ring. Given a straight-line
program for $f$, one can find all coefficients and exponents of $f$
with probability at least $1-\mu$ using $\softO\left( \log T(\log D
+ \log \tfrac{1}{\mu}) \right)$ probes of degree at most $\O( T
\log^2 D )$.
A “soft-Oh” comparison of interpolation algorithms for straight-line programs
Probes Probe degree Cost of probes Type
Dense $D$ $1$ $LD$ deterministic
Garg & Schost $T^2\log D$ $T^2\log D$ $LT^3\log^2 D$ deterministic
*Las Vegas G & S $T\log D$ $T^2\log D$ $LT^3\log^2 D$ Las Vegas
*Diversified $\log D$ $T^2\log D$ $LT^2\log^2 D$ Las Vegas
$\dagger$Recursive $\log T\log D$ $T \log^2 D$ $LT\log^3 D$ Monte Carlo
5l 0pt12pt*Average # of probes given; $\dagger$ for a fixed probability of failure $\mu$
Table <ref> gives a rough comparison of known algorithms.
Our recursive algorithm improves by a factor of $T/\log D$
over the Giesbrecht-Roche diversification algorithm — ignoring
“soft” multiplicative factors of $(\log(T/\log D))^{O(1)}$ — and as such is
better suited for moderate values of $T$. Our algorithm recursively interpolates
a series of polynomials of decreasing sparsity. An advantage of this
method is that, when we cross a threshold where $\log D$ begins to dominate
$T$, we can merely call the Monte Carlo diversification algorithm instead.
§ A RECURSIVE ALGORITHM FOR INTERPOLATING $F$
Entering each recursive step in our algorithm we have our polynomial
$f$ represented by a straight-line program, and an explicit sparse
polynomial $f^*$ “approximating” $f$, that is, whose terms mostly appear in
the sparse representation of $f$.
At each recursive step we try to interpolate the difference $g=f-f^*$.
To begin with, $f^*$ is initialized to zero.
We first find an “ok” prime $p$ which separates most of the terms of
$g$. We then use that prime $p$ to build a approximation $f^{**}$,
containing most of the terms of $g$, plus possibly some additional
“deceptive” terms. The polynomial $f^{**}$ is constructed such that
$g=f-f^*$ has, with high probability, at most $T/2$ terms. We then
recursively interpolate the difference $g-f^{**}$.
Producing images $f^* \bmod (z^\ell-1)$ is straightforward, we merely
reduce the exponents of terms of $f^*$ modulo $\ell$. We assume $g$
has a sparsity bound $T_g \leq T$.
§.§ A weaker notion of “good" primes
To interpolate a polynomial $g$, the sparse interpolation algorithm
described by [Giesbrecht and Roche, 2011] requires a good prime $p$ which
keeps the exponents of $g$ distinct modulo $p$. That is, $g \bmod
(z^p-1)$ has the same number of terms as $g$. We define a weaker notion
of a good prime, an ok prime, which separates most of the
terms of $g$. To that end we measure, for fixed $g$ and prime $p$,
how well $p$ separates the terms of $g$.
Fix a polynomial $g = \sum_{i=1}^t c_iz^{e_i}\in\R[z]$ with non-zero
$c_1,\ldots,c_t\in\R$, where $e_i < e_j$ for $i<j$, we say
$c_iz^{e_i}$ and $c_jz^{e_j}$, $i \neq j$, collide modulo
$z^p-1$ if $e_i\equiv e_j \bmod p$. We call any term $c_iz^{e_i}$ of $f$
which collides with any other term of $f$ a colliding term of $f$
$z^p-1$. We let $\COL{p} \in [0, t]$ denote the number of
colliding terms of $g$ modulo $z^p-1$.
For the polynomial $g= 1+z^5+z^7+z^{10}$, $\COL{2} = 4$, since $1$
collides with $z^{10}$ and $z^5$ collides with $z^7$ modulo $z^2-1$.
Similarly, $\COL{5} = 2$, since $z^5$ collides with $z^{10}$ modulo
We say $c_iz^{e_i}$ and $c_jz^{e_j}$ collide modulo $z^p-1$ because
both terms have the same exponent once reduced modulo $z^p-1$. All
other terms of $g$ we will call non-colliding terms modulo
In the sparse interpolation algorithm of [Giesbrecht and Roche, 2011], one chooses
a $\lambda \in \mathbb{Z}_{>0}$ such that the probability of a prime
$p \in [ \lambda, 2\lambda ]$, chosen at random and having $\COL{p}=0$,
is at least $\tfrac{1}{2}$. However, in order to guarantee that we
find such a prime with high probability, we need to choose $\lambda
\in \O( T^2\log D)$.
In this paper we will search over a range of smaller primes, while
allowing for a reasonable number of collisions. We try to pick
$\lambda$ such that
\begin{equation*}
\Pr\left( \COL{p} \geq \gamma \text{ for a random prime } p
\in [\lambda, 2\lambda] \right) < 1/2,
\end{equation*}
for a parameter $\gamma$ to be determined.
Let $g \in \R[z]$ be a polynomial with $t \leq T$ terms and degree
at most $d \leq D$. Suppose we are given $T$ and $D$, and let
$\lambda = \max\left(21, \left\lceil \tfrac{10T(T-1)\ln(D)}{3
\gamma}\right\rceil\right)$. Let $p$ be a prime chosen at
random in the range $\lambda, \dots, 2\lambda$. Then $\COL{p} \geq
\gamma$ with probability less than $\tfrac{1}{2}$.
The proof follows similarly to the proof of Lemma 2.1 in
[Giesbrecht and Roche, 2011].
Let $B$ be the set of unfavourable primes for which $\COL{p} \geq
\gamma$ terms collide modulo $z^p-1$, and denote the size of $B$ by
$\#B$. As every colliding term collides with at least one other
term modulo $z^p-1$, we know $p^{\COL{p}}$ divides $\prod_{1 \leq i
\neq j \leq t}(e_i - e_j)$. Thus, as $\COL{p} \geq \gamma$ for
$p \in B$,
\[
\lambda^{\#B \gamma} \leq \prod_{p \in B}p^\gamma \leq \prod_{1 \leq
i \neq j < t}(e_i - e_j) \leq d^{t(t-1)} \leq D^{T(T-1)}.
\]
Solving the inequality for $\#B$ gives us
\[
\#B \leq \frac{ T(T-1)\ln(D) }{ \ln( \lambda )\gamma}.
\]
The total number of primes in $[ \lambda, 2\lambda]$ is greater than
$3\lambda/(5\ln(\lambda))$ for $\lambda \geq 21$ by Corollary 3 to
Theorem 2 of [Rosser and Schoenfeld, 1962]. From our definition of $\lambda$ we have
\[
\frac{3\lambda}{5\ln(\lambda)} > \frac{2T(T-1)\ln(D)}{\ln(\lambda)\gamma} \geq 2\# B,
\]
completing the proof.
§.§.§ Relating the sparsity of $g \bmod (z^p-1)$ with
Suppose we choose $\lambda$ according to Lemma <ref>, and
make $k$ probes to compute $g \bmod (z^{p_1}-1), \dots, g \bmod
(z^{p_k}-1)$. One of the primes $p_i$ will yield an image with fewer
than $\gamma$ colliding terms (i.e. $\COL{p_i} < \gamma$) with
probability at least $1-2^{-k}$. Unfortunately, we do not know which
prime $p$ maximizes $\COL{p}$. A good heuristic might be to select
the prime $p$ for which $g \bmod (z^{p}-1)$ has maximally many terms.
However, this does not necessarily minimize $\COL{p}$. Consider the
following example.
\begin{equation*}
g = 1 + z + z^4 - 2z^{13}.
\end{equation*}
We have
\[
g \bmod (z^2-1) = 2 - z, ~~\mbox{and}~~~ g \bmod (z^3-1) = 1.
\]
While $g \bmod (z^2-1)$ has more terms than $g \bmod (z^3-1)$,
we see that $\COL{2} = 4$ is larger than $\COL{3} = 3$.
While we cannot determine the prime $p$ for which $g \bmod (z^p-1)$
has maximally many non-colliding terms, we show that choosing the
prime $p$ which maximizes the number of terms in $g \bmod (z^p-1)$ is,
in fact, a reasonable strategy.
We would like to find a precise relationship between $\COL{p}$, the
number of terms of $g$ that collide in the image $g \bmod (z^p-1)$,
and the sparsity $s$ of $g \bmod (z^p-1)$.
Suppose that $g$ has $t$ terms, and $g \bmod (z^p-1)$ has $s \leq t$
terms. Then $t-s \leq \COL{p} \leq 2(t-s)$.
To prove the lower bound, note that $t-\COL{p}$ terms of $g$ will
not collide modulo $z^p-1$, and so $g \bmod (z^p-1)$ has sparsity $s$ at
least $t-\COL{p}$.
We now prove the upper bound. Towards a contradiction, suppose that
$\COL{p} > 2(t-s)$. There are $\COL{p}$ terms of $g$ that collide modulo
$z^p-1$. Let $h$ be the $\COL{p}$-sparse polynomial
comprised of those terms of $g$.
As each term of $h$ collides with at
least one other term of $h$, $h \bmod (z^p-1)$ has sparsity at most
$\COL{p}/2$. Since none of the terms of $g-h$ collide modulo
$z^p-1$, $(g-h) \bmod (z^p-1)$ has sparsity exactly $t-\COL{p}$. It
follows that $g \bmod (z^p-1)$ has sparsity at most
$t-\COL{p}+\COL{p}/2=t-\COL{p}/2$. That is, $s \leq t-\COL{p}/2$,
and so $\COL{p} \leq 2(t-s)$.
Suppose $g$ has sparsity $t$, $g \bmod (z^q-1)$ has sparsity $s_q$, and
$g \bmod (z^p-1)$
has sparsity $s_p \geq s_q$. Then $\COL{p} \leq 2\COL{q}$.
\[
\begin{array}{rll}
\COL{p} & \leq 2(t-s_p)~~ & \emph{by the second inequality
of Lemma \ref{lem:chooseMostTerms},} \\
& \leq 2(t-s_q) & \emph{since $s_p \geq s_q$,} \\
& \leq 2\COL{q} & \emph{by the first inequality of Lemma
%\ref{lem:chooseMostTerms}.} \hbox to 0pt{\hspace*{10pt}\qed}
\ref{lem:chooseMostTerms}.} \hspace*{20pt}\qed
\end{array}
\]
Suppose then that we have computed $g \bmod (z^p-1)$, for $p$
belonging to some set of primes $S$, and the minimum value of
$\COL{p}$, $p \in S$, is less than $\gamma$. Then a prime $p^* \in S$
for which $g \bmod (z^{p^*}-1)$ has maximally many terms satisfies
$\COL{p^*} < 2\gamma$. We will call such a prime $p^*$ an ok
We then choose $\gamma = wT$ for an appropriate proportion $w \in
(0,1)$. We show that setting $w = 3/16$ allows that each recursive
call reduces the sparsity of the subsequent polynomial by at least
half. This would make $\lambda = \lceil \tfrac{10}{3w}(T-1)\ln(D)
\rceil = \lceil \tfrac{160}{9}(T-1)\ln(D) \rceil$. As per Lemma
<ref>, in order to guarantee with probability
$1-\varepsilon$ that we have come across a prime $p$ such that
$\COL{p} \leq \gamma$, we will need to perform $\lceil \log
1/\varepsilon \rceil$ probes of degree $\mathcal{O}(T\log D)$.
Procedure <ref> summarizes how we find an ok
FindOkPrime($\SLP_f, f^*, T_g, D, \varepsilon$)
* $\SLP_f$, a straight-line program that computes a polynomial $f$
* $f^*$, a current approximation to $f$
* $T_g$ and $D$, bounds on the sparsity and degree of $g=f-f^*$
* $\varepsilon$, a bound on the probability of failure
With probability at least $1-\varepsilon$, we return an “ok prime”
for $g=f-f^*$
$\lambda \longleftarrow \max\left( 21, \left\lceil \tfrac{160}{9}(T_g-1)\ln D
\right\rceil\right)$
$\left({\tt max\_sparsity}, p\right) \longleftarrow (0,0)$
$i \longleftarrow 1$ $\lceil \log 1/\varepsilon \rceil$
$p' \longleftarrow $ a random prime in $[ \lambda, 2\lambda]$
# of terms of $(f - f^*) \bmod (z^{p'}-1) \geq {\tt max\_sparsity}$
${\tt max\_sparsity} \longleftarrow $ # of terms of $(f - f^*) \bmod
$p \longleftarrow p'$
A practical application would probably choose random primes by selecting random integer values in $[\lambda, 2\lambda]$ and then applying probabilistic primality testing. In order to ensure deterministic worst-case run-time, we could pick random primes in the range $[\lambda, 2\lambda]$ by using a sieve method to pre-compute all the primes up to $2\lambda$.
§.§ Generating an approximation $f^{**}$ of
We suppose now that we have, with probability at least $1 -
\varepsilon$, an ok prime $p$; i.e., a prime $p$ such that $\COL{p}
\leq 2wT$ for a suitable proportion $w$. We now use this ok prime $p$
to construct a polynomial $f^{**}$ containing most of the terms
of $g=f-f^*$.
For a set of coprime moduli $\mathcal{Q} = \{q_1, \dots, q_k\}$
satisfying $\prod_{i=1}^k q_i > D$, we will compute $g \bmod
(z^{pq_i}-1)$ for $1 \leq i \leq k$. Here we make no requirement that
the $q_i$ be prime. We merely require that the $q_i$ are pairwise
We choose the $q_i$ as follows: denoting the $i\th$ prime by $p_i$, we
set $q_i = p_i^{\lfloor \log_{p_i} x \rfloor}$, for an appropriate
choice of $x$. That is, we let $q_i$ be the greatest power of the
$i\th$ prime that is no more than $x$. For $p_i \leq x$, we have $q_i
\geq x/p_i$ and $q_i \geq p_i$. Either $x/p_i$ or $p_i$ is at least
$\sqrt{x}$, and so $q_i\ge \sqrt{x}$ as well.
By Corollary 1 of Theorem 2 in [Rosser and Schoenfeld, 1962], there are more than
$x/\ln x$ primes less than or equal to $x$ for $x \geq 17$. Therefore
\begin{equation*}
\prod_{p_i \leq x}q_i \geq \left(\sqrt{x}\right)^{x/\ln x}.
\end{equation*}
As we want this product to exceed $D$, it suffices that
\begin{align*}
\ln D &< \ln\left( \left(\sqrt{x}\right) ^{x/\ln x}\right) = x/2.
\end{align*}
Thus, if we choose $x \geq \max( 2\ln(D), 17)$ and $k = \lceil x/\ln x
\rceil$, then $\prod_{i=1}^k q_i$ will exceed $D$. This means $q_i
\in \O(\log D)$ and $pq_i \in \O( T\log^2 D)$. The number of probes
in this step is $k \in \O( \log(D)/\log\log(D))$. Since we will use
the same set of moduli $\mathcal{Q} = \{q_1, \dots, q_k\}$ in every
recursive call, we can pre-compute $\mathcal{Q}$ prior to the first
recursive call.
We now describe how to use the images $g \bmod (z^{pq_i}-1)$ to
construct a polynomial $f^{**}$ such that $g-f^{**}$ is at most
If $cz^e$ is a term of $g$ that does not collide with any other terms
modulo $z^p-1$, then it certainly will not collide with other terms
modulo $z^{pq}-1$ for any natural number $q$. Similarly, if
$c^*z^{{e^*} \bmod p}$ appears in $g \bmod (z^p-1)$ and there exists a
unique term $c^*z^{{e^*} \bmod pq_i}$ appearing in $g \bmod
(z^{pq_i}-1)$ for $i=1, 2, \dots, k$, then $c^*z^{e^*}$ is potentially
a term of $g$. Note that $c^*z^{e^*}$ is not necessarily a
term of $g$: consider the following example.
\begin{equation*}
g(z) = 1+z+z^2+z^3 + z^{11+4} - z^{14\cdot 11+4} - z^{15\cdot 11 + 4},
\end{equation*}
with hard sparsity bound $T_g=7$ and degree bound $D=170$ and let
$p=11$. We have
\begin{equation*}
g(z) \bmod (z^{11}-1) = 1+z+z^2+z^3-z^4.
\end{equation*}
As $\deg(g)=170<2\cdot 3\cdot 5\cdot 7=210$, it suffices to make the
probes $g \bmod z^{11q}-1$ for $q=2,3,5,7$. Probing our remainder
black-box polynomial, we have
\begin{align*}
g \bmod (z^{22}-1) &= 1+z+z^2+z^3-z^{15},\\
g \bmod (z^{33}-1) &= 1+z+z^2+z^3-z^{26},\\
g \bmod (z^{55}-1) &= 1+z+z^2+z^3-z^{48},\\
g \bmod (z^{77}-1) &= 1+z+z^2+z^3-z^{15}.
\end{align*}
In each of the images $g \bmod z^{pq}-1$, there is a unique term
whose degree is congruent to one of $e=0,1,2,3,4$ modulo $p$. Four
of these terms correspond to the terms $1,z,z^2,z^3$ appearing in
$g$. Whereas the remaining term has degree $e$ satisfying $e = 1
\bmod 2$, $e = 2 \bmod 3$, $e = 3 \bmod 5$, and $e = 1 \bmod 7$. By
Chinese remaindering on the exponents, this gives a term $-z^{113}$
not appearing in $g$.
ConstructApproximation($\SLP_f, f^*, D, p,
\mathcal{Q}$)
* $\SLP_f$, a straight-line program that computes a polynomial
* $f^*$, a current approximation to $f$
* $D$ a bound on the degree of $g=f-f^*$
* $p$, an ok prime for $g$ (with high probability)
* $\mathcal{Q}$, a set of co-prime moduli whose product
exceeds $D$
A polynomial $f^{**}$ such that, if $p$ is an ok prime,
$g-f^{**}$ has sparsity at most $\lfloor T_g/2
\rfloor$, where $g$ has at most $T_g$ terms.
Collect images of $g$
$\mathcal{E} \longleftarrow $ set of exponents of terms in $(f-f^*) \bmod (z^p-1)$
$q \in \mathcal{Q}$
$h \longleftarrow (f-f^*) \bmod (z^{pq}-1)$
each term $cz^e$ in $h$
$E_{(e \bmod p), q}$ is already initialized $\mathcal{E}
\longleftarrow \mathcal{E}/\{e \bmod p\}$
$E_{(e \bmod p), q} \longleftarrow e \bmod q$
Construct terms of new approximation of $g$, $f^{**}$
$f^{**} \longleftarrow 0$
$e_p \in \mathcal{E}$
$e \longleftarrow $ least nonnegative solution to $\{e = E_{e_p, q} \bmod q $ $|$ $q \in \mathcal{Q}\}$
$c \longleftarrow $ coefficient of $z^{e_p}$ term in $(f-f^*) \bmod (z^p-1)$
$e \leq D$ $f^{**} \longleftarrow f^{**} + cz^e$
Let $c^*z^{e^*}$, $e^* \leq D$ be a monomial such that $c^*z^{e^*
\bmod p}$ appears in $g \bmod z^p-1$, and $c^*z^{e^* \bmod pq_i}$
is the unique term of degree congruent to $e^*$ modulo $p$ appearing
in $g \bmod (z^{pq_i}-1)$ for each modulus $q_i$. If $c^*z^{e^*}$
is not a term of $g$ we call it a deceptive term.
Fortunately, we can detect a collision comprised of only two terms.
Namely, if $c_1z^{e_1} + c_2z^{e_2}$ collide, there will exist a $q_i$
such that $q_i \nmid (e_1-e_2)$. That is, $g \bmod (z^{pq_i}-1)$ will
have two terms whose degree is congruent to $e_1 \bmod p$. Once we
observe that, we know the term $(c_1+c_2)z^{e_1 \bmod p}$ appearing in
$g \bmod (z^p-1)$ was not a distinct term, and we can ignore exponents
of the congruence class $e_1 \bmod p$ in subsequent images $g \bmod
Thus, supposing $g \bmod (z^p-1)$ has at most $2\gamma$ colliding
terms and at least $t-2\gamma$ non-colliding terms, $f^{**}$ will have
the $t-2\gamma$ non-colliding terms of $g$, plus potentially an
additional $\tfrac{2}{3}\gamma$ deceptive terms produced by the
colliding terms of $g$. In any case, $g-f^{**}$ has sparsity at most
$\tfrac{8}{3}\gamma$. Choosing $\gamma = \tfrac{3}{16}T_g$ guarantees
that $g-f^{**}$ has sparsity at most $T_g/2$. This would make
$\lambda = \lceil \tfrac{160}{9}(T_g-1)\ln(D) \rceil$.
Procedure <ref> gives a pseudocode
description of how we construct $f^{**}$.
If we find a prospective term in our new approximation $f^{**}$ has
degree greater than $D$, then we know that term must have been a
deceptive term and discard it. There are other obvious things we can
do to recognize deceptive terms which we exclude here. For instance,
we should check that all terms from images modulo $z^{pq}-1$ whose
degrees agree modulo $p$ share the same coefficient.
§.§ Recursively interpolating $f-f^*$
Interpolate($\SLP_f, T, D, \mu$)
* $\SLP_f$, a straight-line program that computes a polynomial $f$
* $T$ and $D$, bounds on the sparsity and degree of $f$, respectively
* $\mu$, an upper bound on the probability of failure
With probability at least $1-\mu$, we return $f$
$x \longleftarrow \max( 2\ln(D), 17)$
$\mathcal{Q} \longleftarrow \{ p^{\lfloor \log_p x \rfloor} : p \text{ is prime}, p \leq x \}$
<ref>$(\SLP_f, 0, T, D, \mathcal{Q}, \mu/(\log T + 1) )$
InterpolateRecurse($\SLP_f, f^*, T_g, D, \mathcal{Q}, \varepsilon$ )
* $\SLP_f$, a straight-line program that computes a polynomial $f$
* $f^*$, a current approximation to $f$
* $T_g$ and $D$, bounds on the sparsity and degree of $g=f-f^*$, respectively
* $\mathcal{Q}$, a set of coprime moduli whose product is at least $D$
* $\varepsilon$, a bound on the probability of failure at one recursive step
With probability at least $1 -\mu$, the algorithm outputs $f$
$T_g = 0$ $f^*$
$p \longleftarrow {\tt\ref{proc:FindOkPrime}}(\SLP_f, f^*, T_g, D, \varepsilon)$
$f^{**} \longleftarrow {\tt\ref{proc:ConstructApproximation}}(\SLP_f, f^*, D,
p, \mathcal{Q})$
<ref>$(\SLP_f, f^*+f^{**}, \lfloor T_g/2 \rfloor, D, \mathcal{Q}, \varepsilon ) $
Once we have constructed $f^{**}$, we refine our approximation $f^*$
by adding $f^{**}$ to it, giving us a new difference $g=f-f^*$
containing at most half the terms of the previous polynomial $g$. We
recursively interpolate our new polynomial $g$. With an updated
sparsity bound $\lfloor T_g/2 \rfloor$, we update the values of
$\gamma$ and $\lambda$ and perform the steps of Sections <ref>
and <ref>. We recurse in this fashion $\log T$
times. Thus, the total number of probes becomes
\[
\O\left( \log T ( \tfrac{\log D}{\log\log D} + \log(1/\varepsilon)) \right),
\]
of degree at most $\O( T\log^2 D)$.
Note now that in order for this method to work we need that, at every
recursive call, we in fact get a good prime, otherwise our sparsity bound
on the subsequent difference of polynomials could be incorrect. At
every stage we succeed with probability $1-\varepsilon$, thus the
probability of failure is $1-(1-\varepsilon)^{\lceil \log T \rceil}$.
This is less than $\lceil \log T \rceil \varepsilon$. If we want to
succeed with probability $\mu$, then we can choose $\varepsilon =
\tfrac{ \mu }{ \log T + 1} \in \O( \tfrac{ \mu }{ \log T
<ref> pre-computes our set of moduli $\mathcal{Q}$, then makes the first recursive call to <ref>, which subsequently calls itself.
§.§ A cost analysis
We analyse the cost of our algorithm, thereby proving Theorems
<ref> and <ref>.
§.§.§ Pre-computation.
Using the wheel sieve [Pritchard, 1982], we can compute the set of primes
up to $x \in \O( \log D)$ in $\softO( \log D )$ bit operations. From
this set of primes we obtain $\Q$ by computing $p^{\lfloor
\log_p x \rfloor}$ for $p \leq \sqrt{x}$ by way of
squaring-and-multiplying. For each such prime, this costs $\softO( \log x
)$ bit operations, so the total cost of computing $\Q$ is $\softO(
\log D)$.
§.§.§ Finding ok primes.
In one recursive call, we will look at some $\log 1/\varepsilon =
\O(\log 1/\mu \log\log T)$ primes in the range $[ \lambda, 2\lambda]$
in order to find an ok prime. Any practical implementation would
select such primes by using probabilistic primality testing on random
integer values in the range $[ \lambda, 2\lambda]$; however, the
probabilistic analysis of such an approach, in the context of our
interpolation algorithm, becomes somewhat ungainly. We merely note
here that we could instead pre-compute primes up to our initial value
of $\lambda \in \O(T\log D)$ in $\softO( T\log
D)$ bit operations by way of the wheel sieve.
Each prime $p$ is of order $T\log D$, and so, per our discussion in
Section <ref>, each probe costs $\softO(LT\log D)$ ring
operations and similarly many bit operations. Considering the
$\O(\log T)$ recursive calls, this totals $\softO( LT\log D\log 1/\mu )$
ring and bit operations.
§.§.§ Constructing the new approximation $f^{**}$.
Constructing $f^{**}$ requires $\softO( \log D )$ probes of degree
$\softO( T\log^2 D )$. This costs $\softO( LT \log^3 D)$ ring and bit
operations. Performing these probes at each $\O( \log T)$ recursive
call introduces an additional factor of $\log T$, which does not
affect the “soft-Oh” complexity. This step dominates the cost of the
Building a term $cz^e$ of $f^{**}$ amounts to solving a set of congruences.
By Theorem 5.8 of [Gathen and Gerhard, 2003], this requires some $\O( \log^2 D )$ word
operations. Thus the total cost of Chinese remaindering to construct $f^{**}$
becomes $\O( T\log^2 D)$. Again, the additional $\log T$ factor due to the
recursive calls does not affect the stated complexity.
§ CONCLUSIONS
We have presented a recursive algorithm for interpolating a polynomial
$f$ given by a straight-line program, using probes of smaller degree
than in previously known methods. We achieve this by looking for
“ok” primes which separate most of the terms of $f$, as opposed to
“good” primes which separate all of the terms of $f$. As is seen in
Table <ref>, our algorithm is an improvement over previous
algorithms for moderate values of $T$.
This work suggests a number of problems for future work. We believe
our algorithms have the potential for good numerical stability, and
could improve on [Giesbrecht and Roche, 2011]'s [Giesbrecht and Roche, 2011] work
on numerical interpolation of sparse complex polynomials, hopefully
capitalizing on the lower degree probes. Our Monte Carlo algorithms
are now more efficient than the best known algorithms for polynomial
identity testing, and hence these cannot be used to make them error
free. We would ideally like to expedite polynomial identity testing of
straight-line programs, the best known methods currently due to
[Bläser et al., 2009]. Finally, we believe there is still room for
improvement in sparse interpolation algorithms. The vector of
exponents of $f$ comprises some $T\log D$ bits. Assuming no
collisions, a degree-$\ell$ probe gives us some $t\log \ell$ bits of
information about these exponents. One might hope, aside from some
seemingly rare degenerate cases, that $\log D$ probes of degree $T\log
D$ should be sufficient to interpolate $f$.
§ ACKNOWLEDGEMENTS
We would like to thank Reinhold Burger and Colton Pauderis for their feedback
on a draft of this
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|
arxiv-papers
| 2013-04-11T20:43:48 |
2024-09-04T02:49:44.223659
|
{
"license": "Public Domain",
"authors": "Andrew Arnold, Mark Giesbrecht, Daniel S. Roche",
"submitter": "Daniel Roche",
"url": "https://arxiv.org/abs/1304.3483"
}
|
1304.3485
|
# Random Lattice Gauge Theories and Differential Forms111A modified version of
this manuscript appears in ISRN Mathematical Physics, vol. 2013, 487270
(2013). doi:10.1155/2013/487270
F. L. Teixeira ElectroScience Laboratory, Department of Electrical and
Computer Engineering
The Ohio State University, Columbus Ohio 43212, USA.
###### Abstract
We provide a brief overview on the application of the exterior calculus of
differential forms to the ab initio formulation of field theories based upon
random simplicial lattices. In this framework, discrete analogues of the
exterior derivative and the Hodge star operator are employed for the
factorization of discrete field equations into a purely combinatorial (metric-
free) part and a metric-dependent part. The Hodge star duality (isomorphism)
is invoked to motivate the use of primal and dual lattices (a dual cell
complex). The natural role of Whitney forms in the construction of discrete
Hodge star operators is stressed.
differential forms, discretization, electrodynamics, exterior calculus, finite
differences, finite elements, lattice field theory
###### pacs:
02.70.Bf, 02.70.Dh, 03.50.De, 11.15Ha, 41.20.-q
††preprint: ArXiv v.2
## I Introduction
The need to formulate field theories on a lattice (mesh, grid) arises from two
main reasons, which may occur simultaneously or not. First, the lattice
provides a natural ‘regularization’ of divergences in lieu of renormalization
techniques Montvay . Such regularization does not need to be viewed as an ad
hoc step, but instead as a natural consequence of assuming the field theory to
be, at some fundamental level, an effective (‘low’-energy) description Zee .
Second, the lattice provides a direct route to compute, in a non-perturbative
fashion, quantities of interest by numerical simulations. Nontrivial domains
and complex boundary conditions can then be easily treated as well Chew
,sanmartin ,Bretones ,TeixeiraAP08 . For these, the use of irregular
(‘random’) lattices are often of interest to gain geometrical flexibility.
Irregular lattices are also of interest as a means to provide a potentially
faster convergence to the continuum limit, near-isotropic lattice dispersion
properties, and better ‘conservation’ of some (e.g., long-range translational
and rotational) symmetries Christ ,Bolander . In some cases, irregular
lattices are useful for universality tests as well Drouffe ,Gockeler .
Lattice theories are typically developed by taking the counterpart continuum
theory as starting point and then applying discretization techniques whereby
derivatives are approximated by finite-differences or some constraints are
enforced on the functional space of admissible solutions to be spanned by a
finite set of ‘basis’ functions (e.g., ‘Galerkin methods’ such as spectral
elements and finite elements). These discretization strategies have proved
very useful in many settings; however, they often produce difficulties in the
case of irregular (‘random’) lattices. Among such difficulties are ($i$)
numerical instabilities in marching-on-time algorithms (regardless of the time
integration method used), ($ii$) convergence problems in algorithms relying on
iterative linear solvers, and ($iii$) spurious (‘ghost’) modes and/or
extraneous degrees of freedom. These problems often (but not always) appear
associated with highly skewed or obtuse lattice elements, or at the boundary
between heterogeneous (hybrid) lattices subcomponent, comprising overlapped
domains or “mesh-stitching” interfaces, for example. Clearly, such
difficulties put a constraint on the geometric flexibility that irregular
lattices are intended for, and may require stringent (and computationally
demanding) mesh quality controls. These difficulties also impact the ability
to utilize ‘mesh refinement’ strategies based on a priori error estimates. The
reasons behind these difficulties can be traced to an inconsistent rendering
of the differential calculus and degrees of freedom on the lattice. A rough
classification of those inconsistencies is provided in Appendix D.
The objective of this work is to provide a brief overview on the application
of exterior calculus of differential forms to the ab initio formulation of
gauge field theories on irregular simplicial (or ‘random’) lattices
KomorowskiBAPS75 ,DodziukAJM76 ,SorkinJMP75 ,weingarten77 ,MullerAM78
,BecherZPC82 ,Rabin ,Bossavit98 ,BossavitEJM91 ,AdamsArXiv , MattiussiJCP97
,KettunenMAGN98 ,MineP16 ,KettunenMAGN99 ,SenAdamsPRE00 ,ShapiroMCS00
,KotiugaPIER01 ,TeixeiraPIER01 ,KettunenPIER01 ,MineP31 ,Wise . In the
exterior calculus framework, the lattice is treated as a cell complex (in the
parlance of algebraic topology Schwarz94 ) instead of simply a collection of
discrete points, and dynamic fields are represented by means of discrete
differential forms (cochains) of various degrees KettunenPIER01 ,MineP25
,MineP27 ,Wise . This prescription provides a basis for developing a
consistent ‘discrete calculus’ on irregular lattices, and discrete analogues
to partial differential equations that better adheres to the underlying
physics.
This topic intersects many disparate application areas. For concreteness, we
use classical electrodynamics in 3+1 dimensions as a basic model. Although
some familiarity with the exterior calculus of differential forms is assumed
Bossavit98 ,BossavitEJM91 ,Whitney57 ,Misner ,DeschampsIEEE82 ,KotiugaJAP93
,MineP15 ,MineP17 ,MineP29 , the discussion is mostly kept at a tutorial
level. Finally, we stress that this is a review paper and no claim of
originality is intended.
## II Pre-metric lattice equations
Let us denote the space of differential $p$-forms on a smooth connected
manifold $\Omega$ as $\Lambda^{p}(\Omega)$. From a geometric perspective, a
differential $p$-form $\alpha^{p}\in\Lambda^{p}(\Omega)$ can be viewed as an
oriented $p$-dimensional density, or an object naturally associated with
$p$-dimensional domains of integration $U_{p}$ such that the lattice
contraction (‘pairing’) below:
$\left<U_{p}\,,\alpha^{p}\right>\doteq\int_{U_{p}}\alpha^{p}$ (1)
gives a real number (in our context) for each choice of $U_{p}$ MineP16 . On a
lattice $\mathcal{K}$, $U_{p}$ is restricted to be a union of elements from
the finite set of $p$-dimensional $N_{p}$ oriented lattice elements, which we
denote $\Gamma_{p}(\mathcal{K})=\\{\sigma_{p,i}\,,i=1,\ldots,N_{p}\\}$. These
are collective called ‘$p$-chains’. In four-dimensions for example, they
correspond to the possible unions of elements from the set of vertices (nodes)
$\sigma_{0}$, edges (‘links’) $\sigma_{1}$, facets (‘plaquettes’)
$\sigma_{2}$, volume cells (‘voxels’) $\sigma_{3}$, and hypervolume cells
$\sigma_{4}$, for $p=1,\ldots,4$, respectively. In the discrete setting, the
degrees of freedom are reduced to the set of pairings (1) on each one of the
lattice elements.
On the lattice, the pairing above can be understood as a map
${\mathcal{R}}^{p}:\Lambda^{p}(\Omega)\rightarrow\Gamma^{p}(\mathcal{K})$ such
that
${\mathcal{R}}^{p}(\alpha_{p})=\left<\sigma_{p,i}\,,\alpha^{p}\right>\doteq\int_{\sigma_{p,i}}\alpha^{p}$
(2)
defines its action on the basis of $p$-chains. Note that we use
$\Gamma^{p}(\mathcal{K})$ to denote the space dual to
$\Gamma_{p}(\mathcal{K})$, i.e. the space $p$-cochains. The latter can be
viewed as the space of ‘discrete differential forms’. Because of this, and
with some abuse of language, we use the terminology ‘differential forms’ and
‘cochains’ interchangeably to denote the same objects in what follows. The map
${\mathcal{R}}^{p}$ is called the de Rham map MineP16 .
The basic differential operator of exterior calculus is the exterior
derivative $d$, applicable to any number of dimensions. The discretization of
$d$ on a general irregular lattice can be effected by a straightforward
application of the generalized Stokes’ theorem MineP16
$\int_{\sigma_{p+1}}d\,\alpha^{p}=\int_{\partial\sigma_{p+1}}\alpha^{p}$ (3)
with $p=0,\ldots,3$ in $n=4$. In the above, $\partial$ is the boundary
operator, which simply maps a $p$-dimensional lattice element to the set of
$(p-1)$-dimensional lattice elements that comprise its boundary, preserving
orientation. This theorem sets $\partial$ as the formal adjoint of $d$ in
terms of the pairing given in (1), that is
$\left<\sigma_{p+1},d\alpha^{p}\right>=\left<\partial\sigma_{p+1},\alpha^{p}\right>$.
Computationally, the boundary operator can be implemented by means of
incidence matrices MineP16 ,MineP31 ,Guth such that
$\partial\,\sigma_{p+1,i}=\sum_{j}C_{ij}^{p}\,\sigma_{p,j}$ (4)
where the indices $i$ and $j$ run over all $(p+1)$\- and $p$-dimensional
lattice elements, respectively. The incidence matrix entries are such that
$C_{ij}^{p}\in\\{-1,0,1\\}$ for all $p$, with sign determined by the relative
orientation of lattice elements $i$ and $j$. The restriction to this set of
integer values reflects the ‘metric-free’ nature of the exterior derivative:
only information about element connectivity, that is, the combinatorial
aspects of the lattice, is involved here. It turns out that the metric is
fully encoded by Hodge star operators, the discretization of which will be
discussed further down below.
Using eqs. (3) and (4), one can write
$\int_{\sigma_{p+1,i}}d\,\alpha^{p}=\sum_{j}C_{ij}^{p}\int_{\sigma_{p,j}}\alpha^{p}$
(5)
for all $i$, so that the derivative operation is replaced by a proper sum over
$j$. On the lattice, the nilpotency of the operators
${\partial}\circ{\partial}=d\circ d=0$ Kheyfets is recovered by the
constraint MineP16
$\sum_{k}C_{ik}^{p+1}\,C_{kj}^{p}=0$ (6)
for all $i$ and $j$.
## III Example: Lattice electrodynamics
We write Maxwell’s equations in a four-dimensional Lorentzian manifold
$\Omega$ as Misner
$dF=0$ (7) $dG=*{\mathcal{J}}$ (8)
where $d$ is the four-dimensional exterior derivative, $F$ and $G$ are the so-
called Faraday and Maxwell 2-forms, respectively, and $*{\mathcal{J}}$ is the
charge-current density 3-form. The Hodge star operator $*$ is an isomorphism
that maps $p$-forms to $(4-p)$-forms, and more generally $p$ forms to $(n-p)$
forms in a $n$-dimensional manifold, and, as mentioned before, depends on the
metric of $\Omega$ MineP16 ,KettunenMAGN99 ,Misner ,DeschampsIEEE82
,HiptmairNM01 ,MineP26 ,MineP28 ,MineP30 . The above equations are
complemented by the relation $G=*F$, which indicates that $F$ and $G$ are
‘Hodge duals’ of each other.
### III.1 Primal and dual lattices
Since $F$ and $G$ are 2-forms, they should be discretized as 2-cochains
residing on plaquettes (2-chains) of the 4-dimensional lattice; however, it is
important to recognize that these two forms are of different types: $F$ is a
‘ordinary’ (or ‘non-twisted’) differential form, whereas $G$ (as well as
$*{\mathcal{J}}$) is a ‘twisted’ (or ‘odd’) differential form burke . The
basic difference here has to do with orientation: ordinary forms have internal
orientation whereas twisted forms have external orientation MattiussiJCP97
,MineP16 ,burke ,tonti76 ,TontiPIER01 . These two types of orientations
exhibit different symmetries under reflection, a distinction akin to that
between proper (or polar) tensors and pseudo (or axial) tensors. Only twisted
forms admit integration in non-orientable manifolds. These two types of forms
are associated with two distinct ‘cell complexes’ (lattices), each one
inheriting the corresponding orientation: the ordinary form $F$ is associated
with the set of plaquettes $\Gamma_{2}$ on the ‘ordinary cell complex’
$\mathcal{K}$, thus belonging to $\Gamma^{2}(\mathcal{K})$, while the twisted
forms $G$ and $*{\mathcal{J}}$ are associated with the set of plaquettes
${\tilde{\Gamma}}_{2}$ on the ‘twisted cell complex’ $\tilde{\mathcal{K}}$
MineP16 ,TeixeiraPIER01 ,TontiPIER01 ,TontiRAL72 , thus belonging to
$\Gamma^{2}(\tilde{\mathcal{K}})$. Consequently, we also have two sets of
incidence matrices $C_{ij}^{p}$ and $\tilde{C}_{ij}^{p}$, one for each
lattice. It is convenient to denote $\mathcal{K}$ as the ‘primal lattice’ and
$\tilde{\mathcal{K}}$ as the ‘dual lattice’ MineP16 .
As detailed further below, these two lattices become intertwined by the Hodge
duality $F=*G$. The need for dual lattices can be motivated from a
combinatorial standpoint AdamsArXiv ,SenAdamsPRE00 or from a computational
standpoint (to provide higher-order convergence to the continuum, for example)
YeeAP69 ,Taflove95 ,NicolaidesSIAM97 . The importance of a primal/dual lattice
setup for the discretization of the Hodge star operator in the context of
field theories was first recognized in AdamsArXiv , where it was shown that
such setup is also crucial for correctly reproducing topological invariants in
the discrete setting.
### III.2 3+1 theory
At this point, it is suitable to degeometrize time and treat it simply as a
parameter. This corresponds to the majority of low-energy applications
involving Maxwell’s equations, in which one is interested in predicting the
field evolution along different spatial slices for a given set of initial and
boundary conditions. In this case, we still use the symbols $\mathcal{K}$ and
$\tilde{\mathcal{K}}$ for the primal and dual lattices, but they now refer to
three-dimensional spatial lattices. Similarly, $\Omega$ now refers to a three-
dimensional Euclidean manifold . In such a 3+1 setting, one can decompose $F$
and $G$ as
$F=E\wedge dt+B$ (9) $G=D-H\wedge dt$ (10)
and the source density as
$*{\mathcal{J}}=-J\wedge dt+\rho$ (11)
where $\wedge$ is the wedge product, $E$ and $H$ are the electric intensity
and magnetic intensity 1-forms on $\Gamma_{1}$ and ${\tilde{\Gamma}}_{1}$
respectively, $D$ and $B$ are the electric flux and magnetic flux 2-forms on
${\tilde{\Gamma}}_{2}$ and $\Gamma_{2}$ respectively, $J$ is the electric
current density 2-form on ${\tilde{\Gamma}_{2}}$ , and $\rho$ is the electric
charge density 3-form on ${\tilde{\Gamma}_{3}}$ (corresponding assignments for
the 2+1 and 1+1 cases are provided in MineP27 ). As a result, Maxwell’s
equations reduce to
$dE=-\partial_{t}B$ (12) $dH=\partial_{t}D+J$ (13)
representing Faraday’s and Ampere’s law, respectively. Here, $d$ stands for
the 3-dimensional spatial exterior derivative. Note that both eqs. (12) and
(13) are metric-free. They are supplemented by Hodge star relations given by
$D=\star_{\epsilon}E$ (14) $H=\star_{\mu^{-1}}B$ (15)
now involving two Hodge star maps in three-dimensional space:
$\star_{\epsilon}:\Lambda^{1}(\Omega)\rightarrow\Lambda^{2}(\Omega)$ and
$\star_{\mu^{-1}}:\Lambda^{2}(\Omega)\rightarrow\Lambda^{1}(\Omega)$. On the
lattice, we have the corresponding discrete counterparts:
$[\star_{\epsilon}]:\Gamma^{1}(\mathcal{K})\rightarrow\Gamma^{2}(\tilde{\mathcal{K}})$
and
$[\star_{\mu^{-1}}]:\Gamma^{2}(\mathcal{K})\rightarrow\Gamma^{1}(\tilde{\mathcal{K}})$.
The subscripts $\epsilon$ and $\mu$ in $\star_{\epsilon}$ and
$\star_{\mu^{-1}}$ serve to indicate that these operators also incorporate
macroscopic constitutive material properties through the local permittivity
and permeability values codecasa07 (we assume dispersionless media for
simplicity). In Riemannian manifolds (and in particular, Euclidean space) and
reciprocal media, these two Hodge star operators are symmetric and positive-
definite Auchmann .
In what follows, we employ the following short-hand notation for cochains:
$\left<\sigma_{1,i},E\right>=E_{i}$,
$\left<{\tilde{\sigma}}_{1,i},H\right>=H_{i}$,
$\left<\tilde{\sigma}_{2,i},D\right>=D_{i}$,
$\left<\sigma_{2,i},B\right>=B_{i}$,
$\left<\tilde{\sigma}_{2,i},J\right>=J_{i}$, and
$\left<\tilde{\sigma}_{3,i},\rho\right>=\rho_{i}$, where the indices run over
the respective basis of $p$-chains in either $\mathcal{K}$ or
${\tilde{\mathcal{K}}}$, $p=1,2,3$. With the exception of Appendix A, we
restrict ourselves to the 3+1 setting throughout the remainder of this paper.
## IV Casting the metric on a lattice
### IV.1 Whitney forms
The Whitney map
${\mathcal{W}}:\Gamma^{p}(\mathcal{K})\rightarrow\Lambda^{p}(\Omega)$ is the
right-inverse of the de Rham map (2), that is,
${\mathcal{R}}\circ{\mathcal{W}}=\mathcal{I}$, where $\mathcal{I}$ is the
identity operator. In simplicial lattices, this morphism can be constructed
using the so-called Whitney forms MullerAM78 ,MineP16 ,KotiugaJAP93 ,MineP26
,BossavitIEE88 ,kotiugabook ,bossavit05 ,bohethesis ,HiptmairPIER01 ,rapetti
,kangas07 which are basic interpolants from cochains to differential forms
Whitney57 (other interpolants are also possible Buffa11 ,Back12 ). By
definition, all cell elements of a simplicial lattice are simplices, i.e.,
cells whose boundaries are the union of a minimal number of lower-dimensional
cells. In other words, $0$-simplices are nodes, 1-simplices are links,
2-simplices are triangles, 3-simplices are tetrahedra, and so on. Note that if
the primal lattice is simplicial, the dual lattice is not MineP25 . For a
$p$-simplex $\sigma_{p,i}$, the (lowest-order) Whitney form is given by
$\omega^{p}[\sigma_{p,i}]\doteq
p!\sum_{j=0}^{p}(-1)^{i}\lambda_{i,j}d\lambda_{i,0}\wedge d\lambda_{i,1}\cdots
d\lambda_{i,j-1}\wedge d\lambda_{i,j+1}\cdots d\lambda_{i,p}$ (16)
where $\lambda_{i,j}$, $j=0,\ldots,p$, are the barycentric coordinates
associated to $\sigma_{p,i}$. In the case of a $0$-simplex (node), (16)
reduces to $\omega^{0}[\sigma_{0,i}]=\lambda_{i}$.
From its definition, it is clear that Whitney forms have compact support.
Among its important structural properties are:
$\left<\sigma_{p,i},\omega^{p}[\sigma_{p,j}]\right>=\int_{\sigma_{p,i}}\omega^{p}[\sigma_{p,j}]=\delta_{ij}$
(17)
where $\delta_{ij}$ is the Kronecker delta, which is simply a restatement of
${\mathcal{R}}\circ{\mathcal{W}}=\mathcal{I}$, and
$\omega^{p}[\partial^{T}\sigma_{p-1,i}]=d\left(\omega^{p-1}[\sigma_{p-1,i}]\right)$
(18)
where $\partial^{T}$ is the coboundary operator kotiugabook , consistent with
the generalized Stokes’ theorem. Further structural properties are provided in
bossavit05 ,bohethesis . Higher-order version of Whitney forms also exist
HiptmairPIER01 ,rapetti . The key result
${\mathcal{W}}\circ{\mathcal{R}}\rightarrow\mathcal{I}$ holds in the limit of
zero lattice spacing. This is discussed, together with other related
convergence results in various contexts, in MullerAM78 ,Whitney57 ,albeverio90
,albeverio95 ,wilson07 ,wilson11 ,halvorsen12 .
Using the short-hand $\omega^{p}[\sigma_{p,i}]=\omega^{p}_{i}$, we can write
the following expansions for $E$ and $B$ in a irregular simplicial lattice, in
terms of its cochain representations:
$E=\sum_{i}E_{i}\,\omega^{1}_{i}$ (19) $B=\sum_{i}B_{i}\,\omega^{2}_{i}$ (20)
where the sums run over all primal lattice edges and faces, respectively.
One could argue that Whitney forms are continuum objects that should have no
fundamental place on a truly discrete theory. In our view, this is only
partially true. In many applications (see, for example, the discussion on
space-charge effects below), it is less natural to consider the lattice as
endowed with some a priori discrete metric structure than it is to consider it
instead as embedded in an underlying continuum (say, Euclidean) manifold with
metric and hence inheriting all metric properties from it. In the latter case,
Whitney forms provide the standard route to incorporate metric information
into the discrete Hodge star operators, as described next.
### IV.2 Discrete Hodge star operator
In a source-free media, we can write the Hamiltonian as
${\mathcal{H}}=\frac{1}{2}\int_{\Omega}\left(E\wedge D+H\wedge
B\right)=\int_{\Omega}\left(E\wedge\star_{\epsilon}E+\star_{\mu^{-1}}B\wedge
B\right)$ (21)
Using eqs. (19) and (20), the lattice Hamiltonian assumes the expected
quadratic form:
${\mathcal{H}}=\sum_{i}\sum_{j}E_{i}\,[\star_{\epsilon}]_{ij}\,E_{j}+\sum_{i}\sum_{j}B_{i}\,[\star_{\mu^{-1}}]_{ij}\,B_{j}$
(22)
where we immediately identify the symmetric positive definite matrices
$[\star_{\epsilon}]_{ij}=\int_{\Omega}\omega^{1}_{i}\wedge\star_{\epsilon}\omega^{1}_{j}$
(23)
$[\star_{\mu^{-1}}]_{ij}=\int_{\Omega}\left(\star_{\mu^{-1}}\omega^{2}_{i}\right)\wedge\omega^{2}_{j}$
(24)
as the discrete realization of the Hodge star operator(s) on a simplicial
lattice KettunenMAGN99 ,bossavitjapan so that
$D_{i}=\sum_{j}[\star_{\epsilon}]_{ij}E_{j}$ (25)
$H_{i}=\sum_{j}[\star_{\mu^{-1}}]_{ij}B_{j}.$ (26)
From the above, the Hamiltonian can be also expressed as
${\mathcal{H}}=\sum_{i}E_{i}\,D_{i}+\sum_{i}H_{i}\,B_{i}$ (27)
### IV.3 Symplectic structure and dynamic degrees of freedom
The Hodge star matrices $[\star_{\epsilon}]$ and $[\star_{\mu^{-1}}]$ have
different sizes. The number of elements in $[\star_{\epsilon}]$ is equal to
$N_{1}\times N_{1}$, whereas the number of elements in $[\star_{\mu}^{-1}]$ is
equal to $N_{2}\times N_{2}$. In other words,
$\Theta(E)=\Theta(D)\neq\Theta(B)=\Theta(H)$, where $\Theta$ denotes the
number of (discrete) degrees of freedom in the corresponding field.
One important property of a Hamiltonian system is its symplectic character,
associated with area preservation in phase space. The symplectic character of
the Hamiltonian in principle would require a canonical pair such as $E,B$ to
have identical number of degrees of freedom. This apparent contradiction can
be explained by the fact that Maxwell’s equations (12) and (13) can be thought
as a constrained dynamic system (by the divergence conditions) so that, even
though $\Theta(E)\neq\Theta(B)$, we still have $\Theta^{d}(E)=\Theta^{d}(B)$,
where $\Theta^{d}$ denotes the number of dynamic degrees of freedom. This is
discussed further below in Section VI, in connection with the discrete Hodge
decomposition on a lattice.
## V Semi-discrete equations
### V.1 Local and ultra-local lattice coupling
By using a contraction in the form of (2) on both sides of (12) with every
face $\sigma_{2,j}$ of $\mathcal{K}$, and using the fact that
$\left<\sigma_{2,j},\omega^{2}_{i}\right>=\left<\sigma_{1,j},\omega^{1}_{i}\right>=\delta_{ij}$
from (17), we get
$\left<\sigma_{2,j},\partial_{t}B\right>=\partial_{t}\sum_{i}B_{i}\left<\sigma_{2,j},\omega^{2}_{i}\right>=\partial_{t}B_{j}$
(28)
and
$\left<\sigma_{2,j},dE\right>=\left<\partial\sigma_{2,j},E\right>=\sum_{i}E_{i}\sum_{k}C^{1}_{jk}\left<\sigma_{1,k},\omega^{1}_{i}\right>=\sum_{i}C^{1}_{ji}\,E_{i}$
(29)
so that
$-\partial_{t}B_{i}=\sum_{j}C^{1}_{ij}\,E_{j}$ (30)
where the index $i$ runs over all faces of the primal lattice. On the dual
lattice ${\tilde{\mathcal{K}}}$, we can similarly contract both sides of eq.
(13) with every dual face ${\tilde{\sigma}}_{2,j}$ to get
$\partial_{t}D_{i}=\sum_{j}{\tilde{C}}^{1}_{ij}\,H_{j}$ (31)
where now the index $i$ runs over all faces of the dual lattice. Using eqs.
(25) and (26) and the fact that, in three-dimensions
${\tilde{C}}^{1}_{ij}=C^{1}_{ji}$ MineP16 (up to possible boundary terms
ignored here), we can write the last equation in terms of primal lattice
quantities as
$\partial_{t}\sum_{j}[\star_{\epsilon}]_{ij}\,E_{j}=\sum_{j}C^{1}_{ji}\sum_{k}[\star_{\mu^{-1}}]_{jk}B_{k}$
(32)
or, by using the inverse Hodge star matrix $[\star_{\epsilon}]^{-1}_{ij}$, as
$\partial_{t}E_{i}=\sum_{j}\Upsilon_{ij}B_{j}$ (33)
with
$\Upsilon_{ij}\doteq\sum_{k}\sum_{l}[\star_{\epsilon}]^{-1}_{ik}\,C^{1}_{lk}\,[\star_{\mu^{-1}}]_{lj}$
(34)
The matrix $[\Upsilon]$ can be viewed as the discrete realization, for $p=2$,
of the codifferential operator $\delta=(-1)^{p}*^{-1}d\,*$ that maps $p$-forms
to $(n-p)$-forms DeschampsIEEE82 .
Since the continuum operators $\star_{\epsilon}$ and $\star_{\mu^{-1}}$ are
local burke and, as seen, Whitney forms (16) have local support, it follows
that the matrices $[\star_{\epsilon}]$ and $[\star_{\mu^{-1}}]$ are sparse,
indicative of an ultra-local coupling (in the terminology of Katz1998 ). In
contrast, the numerical inverse $[\star_{\epsilon}]^{-1}$ used in eq. (34) is,
in general, not sparse so that the field coupling between distant elements is
nonzero. The lack of sparsity is a potential bottleneck in practical
simulations. However, because the coupling strength in this case decays
exponentially MineP31 ,MineP28 , we can still say (using again the terminology
of Katz1998 ) that the resulting discrete operator encoded by the matrix in
(34) is local. In practical terms, the exponential decay allows one to set a
cutoff on the nonzero elements of $[\star_{\epsilon}]$, based on element
magnitudes or on the sparsity pattern of the original matrix
$[\star_{\epsilon}]$, to build a sparse approximate inverse for
$[\star_{\epsilon}]$ and hence recover back an ultra-local representation for
$\star_{\epsilon}^{-1}$ MineP31 ,MineP27a . The sparsity pattern of
$[\star_{\epsilon}]$ encodes the nearest-neighbor edge information of the mesh
and, consequently, the sparsity pattern of $[\star_{\epsilon}]^{k}$ likewise
encodes successive ‘$k$-level’ neighbors. The latter sparsity patterns can be
used to build, quite efficiently, sparse approximations for
$[\star_{\epsilon}]^{-1}$, as detailed in MineP31 . Once such sparse
representations are obtained, eqs. (30) and (33) can be used in tandem to
construct a marching-on-time algorithm (see Appendix E (a), for example) with
a sparse structure and hence amenable for large-scale problems.
### V.2 Barycentric dual and barycentric decomposition lattices
An alternative approach, aimed at constructing a sparse discrete Hodge star
for $\star_{\epsilon}^{-1}$ directly from the dual lattice geometry is
described in TeixeiraPIER01 , based on earlier ideas exposed in AdamsArXiv
,SenAdamsPRE00 ,AdamsPRL97 . This approach is based on the fact that both
primal $\mathcal{K}$ and dual $\tilde{\mathcal{K}}$ lattices can be decomposed
into a third (underlying) lattice $\widehat{\mathcal{K}}$ by means of a
barycentric decomposition, see SenAdamsPRE00 . The dual lattice
$\tilde{\mathcal{K}}$ in this case is called the barycentric dual lattice
TeixeiraPIER01 ,AdamsPRL97 and the underlying lattice $\widehat{\mathcal{K}}$
is called the barycentric decomposition lattice. Importantly,
$\widehat{\mathcal{K}}$ is simplicial and hence admits Whitney forms built on
it using (16). Whitney forms on $\widehat{\mathcal{K}}$ can be used as
building blocks to construct (dual) Whitney forms on the (non-simplicial)
$\tilde{\mathcal{K}}$, and from that, a sparse inverse discrete Hodge star
$[\star_{\epsilon}^{-1}]$ using integrals akin to (23) and (24). An explicit
derivation of such dual lattice Whitney forms is provided in buffa .
Furthermore, a recent comprehensive survey of this and other approaches based
on dual lattices to construct discrete sparse inverse Hodge stars is provided
in GilletteCAD11 . A comparison between the properties of a barycentric dual
and a circumcentric dual is considered in Calcagni , where it is verified that
the former induces a (discrete) Laplacian with better properties (in
particular, positivity).
The barycentric dual lattice has the important property below associated with
Whitney forms:
$\left<\tilde{\sigma}_{(n-p),i},\star\omega^{p}[\sigma_{p,j}]\right>=\int_{\tilde{\sigma}_{(n-p),i}}\star\omega^{p}[\sigma_{p,j}]=\delta_{ij}$
(35)
where $\star$ stands for the spatial Hodge star operator (distilled from
constitutive material properties), and $\tilde{\sigma}_{(n-p),i}$ is the dual
element to $\sigma_{p,i}$ on the barycentric dual lattice. The operator
$\star$ is such that
$\int_{\Omega}\omega^{p}\wedge\star\omega^{p}=\int_{\Omega}|\omega|^{2}dv$
(36)
where $|\omega|^{2}$ is the two-norm of $\omega^{p}$ and $dv$ is the volume
element.
The identity (35) plays the role of structural property (17), on the dual
lattice side. We stress that identity (35) is a distinctively characteristic
feature of the barycentric dual lattice not shared by other geometrical
constructions for the dual lattice. In other words, compatibility with Whitney
forms via (35) naturally forces one to choose the dual lattice to be the
barycentric dual.
From the above, one can also define a (Hodge) duality operator directly on the
space of chains, that is
$\star_{K}:\Gamma_{p}(\mathcal{K})\mapsto\Gamma_{n-p}(\tilde{\mathcal{K}})$
with $\star_{K}(\sigma_{p,i})=\tilde{\sigma}_{(n-p),i}$ and
$\star_{\tilde{K}}:\Gamma_{p}(\tilde{\mathcal{K}})\mapsto\Gamma_{n-p}(\mathcal{K})$
with $\star_{K}(\tilde{\sigma}_{p,i})=\tilde{\sigma}_{(n-p),i}$, so that
$\star_{K}\star_{\tilde{K}}=\star_{\tilde{K}}\star_{K}=1$. This construction
is detailed in SenAdamsPRE00 .
### V.3 Galerkin duality
Even though we have chosen to assign $E$ and $B$ to the primal (simplicial)
lattice, and consequently $D$, $H$, $J$, and $\rho$ to the dual (non-
simplicial) lattice, the reverse is equally possible. In this case, the fields
$D$, $H$ become associated to a simplicial lattice and hence can be expressed
in terms of Whitney forms; the expressions dual to (19) and (20) are now
$H=\sum_{i}H_{i}\,\omega^{1}_{i}$ (37) $D=\sum_{i}D_{i}\,\omega^{2}_{i}$ (38)
with sums running over primal edges and primal faces, respectively, and where
$E_{i}=\sum_{j}[\star_{\epsilon^{-1}}]_{ij}D_{j}$ (39)
$B_{i}=\sum_{j}[\star_{\mu}]_{ij}H_{j}$ (40)
with
$[\star_{\epsilon}^{-1}]_{ij}=\int_{\Omega}\left(\star_{\epsilon^{-1}}\omega^{2}_{i}\right)\wedge\omega^{2}_{j}$
(41)
$[\star_{\mu}]_{ij}=\int_{\Omega}\omega^{1}_{i}\wedge\star_{\mu}\omega^{1}_{j}$
(42)
and the two Hodge star maps now used are such that, in the continuum,
$\star_{\epsilon}^{-1}:\Lambda^{2}(\Omega)\rightarrow\Lambda^{1}(\Omega)$ and
$\star_{\mu}:\Lambda^{1}(\Omega)\rightarrow\Lambda^{2}(\Omega)$, and, on the
lattice,
$[\star_{\epsilon}^{-1}]:\Gamma^{2}(\mathcal{K})\rightarrow\Gamma^{1}(\tilde{\mathcal{K}})$
and
$[\star_{\mu}]:\Gamma^{1}(\mathcal{K})\rightarrow\Gamma^{2}(\tilde{\mathcal{K}})$.
This alternate choice entails a duality between these two formulations, dubbed
‘Galerkin duality’. This is explored in more detail in MineP28 .
## VI Discrete Hodge decomposition and Euler’s formula
For any $p$-form $\alpha^{p}$, we can write
$\alpha^{p}=d\zeta^{p-1}+\delta\beta^{p+1}+\chi^{p},$ (43)
where $\chi^{p}$ is a harmonic form MineP25 . This Hodge decomposition is
unique. In the particular case of the $1$-form $E$, we have
$E=d\phi+\delta A+\chi,$ (44)
where $\phi$ is a $0$-form and $A$ is a $2$-form, with $d\phi$ representing
the static field, $\delta A$ the dynamic field, and $\chi$ the harmonic field
component (if any). In a contractible domain, $\chi$ is identically zero and
the Hodge decomposition simplifies to
$E=d\phi+\delta A.$ (45)
more usually known as Helmholtz decomposition in three-dimensions.
In the discrete setting, the degrees of freedom of $\phi$ are associated to
the nodes of the primal lattice. Likewise, the degrees of freedom of $A$ are
associated to the facets of the primal lattice. Consequently, we have from
(45) that
$\displaystyle\Theta^{d}\left(E\right)$ $\displaystyle=$ $\displaystyle
N_{E}^{h}-N_{V}^{h}$ (46) $\displaystyle=$
$\displaystyle\left(N_{E}-N_{E}^{b}\right)-\left(N_{V}-N_{V}^{b}\right)$
$\displaystyle=$ $\displaystyle N_{E}-N_{V},$
where $N_{V}$ is the number of primal nodes, $N_{E}$ the number of primal
edges, and $N_{F}$ the number of primal facets, with superscript $b$ standing
for boundary (fixed) elements and $h$ for interior (free) elements.
On the other hand, once we identify the lattice as a network of (in general)
polyhedra, we can apply Euler’s polyhedron formula on the primal lattice to
obtain MineP28
$N_{V}-N_{E}=1-N_{F}+N_{P},$ (47)
where $N_{P}$ represents the number of volume cells comprising the primal
lattice. A similar Euler’s polyhedron formula applies to the (closed, two-
dimensional) boundary of the primal lattice
$N_{V}^{b}-N_{E}^{b}=2-N_{F}^{b},$ (48)
Combining Eq. (47) and (48), we have
$\left(N_{E}-N_{E}^{b}\right)-\left(N_{V}-N_{V}^{b}\right)=\left(N_{F}-N_{F}^{b}\right)-\left(N_{P}-1\right).$
(49)
From the Hodge decomposition (45), we see that $\Theta^{d}\left(E\right)$ is
$\displaystyle\Theta^{d}\left(E\right)$ $\displaystyle=$ $\displaystyle
N_{E}^{in}-N_{V}^{in}$ (50) $\displaystyle=$
$\displaystyle\left(N_{E}-N_{E}^{b}\right)-\left(N_{V}-N_{V}^{b}\right).$
Note that the divergence free condition $dB=0$ produces one constraint on the
2-form $B$ for each volume element. This constraint also span the whole
lattice boundary. The total number of the constrains for $B$ is therefore
$\left(N_{P}-1\right).$ Consequently, we have
$\displaystyle\Theta^{d}\left(B\right)$ $\displaystyle=$ $\displaystyle
N_{F}^{in}-\left(N_{P}-1\right)$ (51) $\displaystyle=$
$\displaystyle\left(N_{F}-N_{F}^{b}\right)-\left(N_{P}-1\right)$
so that
$\Theta^{d}\left(B\right)=\Theta^{d}\left(E\right).$ (52)
This discussion can be generalized to lattices on non-contractible domains
with any number of holes (genus), where the identity
$\Theta^{d}\left(B\right)=\Theta^{d}\left(E\right)$ is also satisfied MineP25
. Moreover, from Hodge star isomorphism, we have
$\Theta^{d}\left(D\right)=\Theta^{d}\left(E\right)$ and
$\Theta^{d}\left(H\right)=\Theta^{d}\left(B\right)$.
In general, we can trace a direct correspondence between quantities in the
Euler’s polyhedron formula to the quantities in the Hodge decomposition
formula. For example, each term in the two-dimensional Euler’s formula
$N_{E}=N_{V}+\left(N_{F}-1\right)+g$ is associated to a corresponding term in
$E=d\phi+\delta A+\chi$; that is, the number of edges $N_{E}$ corresponds to
the dimension of the space of lattice $1$-forms $E$, which is the sum of the
number of nodes $N_{V}$ (dimension of the space of discrete $0$-forms $\phi$),
the number of faces $\left(N_{F}-1\right)$ (dimension of the space of discrete
$2$-forms $A$), and the number of holes $g$ (dimension of the space of
harmonic forms $\chi$). A similar correspondence can be traced on a three-
dimensional lattice MineP25 . This correspondence provides a physical picture
to Euler’s formula and a geometric interpretation to the Hodge decomposition.
Acknowledgments
The author thanks Weng C. Chew, Burkay Donderici, Bo He, Joonshik Kim, and
David H. Adams for technical discussions.
APPENDIX A: Differential forms and lattice fermions
Differential $p$-forms can be viewed as antisymmetric covariant tensor fields
on rank $p$. Therefore, the ingredients discussed above are applicable to any
antisymmetric tensor field theory, including (pure) non-Abelian theories
AdamsPRL97 . However, for (Dirac) fermion fields the situation is different
and, at first, it would seem unclear how differential forms could be used to
describe spinors. Nevertheless, a useful connection can indeed be established
Montvay ,BecherZPC82 ,Graf . To briefly address this point, let us consider
next the lattice transcription of the (one-flavor) Dirac equation. Needless to
say, the topic of lattice fermions is vast and we cannot do full justice to it
here; we only focus here on the aspects more germane to our main discussion.
In this Appendix, we work on Euclidean spacetime with $\hbar=c=1$ and adopt
the repeated index summation convention with $\mu$, $\nu$ as coordinate
indices, where $x$ is a point in four-dimensional space.
It is well known that fermion fields defy a lattice description with local
coupling that gives the correct energy spectrum in the limit of zero lattice
spacing and the correct chiral invariance AdamsPRD05 . This is formally stated
by the no-go theorem of Nielsen-Ninomiya Friedan and is associated to the
well-known ‘fermion-doubling’ problem Herbut . A perhaps less known fact is
that it is possible to arrive at a ‘geometrical’ interpretation of the source
of this difficulty by considering the ‘generalization’ of the Dirac equation
$(\gamma^{\mu}\partial_{\mu}+m)\psi(x)=0$ given by the Dirac-Kähler equation
$(d-\delta)\Psi(x)=-m\Psi(x)$ (53)
The square of the Dirac-Kähler operator can be viewed as the counterpart of
the Dirac operator in the sense that
$(d-\delta)^{2}=-(d\delta+\delta d)=-\Box$ (54)
recovers the Laplacian operator in the same fashion as the Dirac operator
squared does, that is
$(\gamma^{\mu}\partial_{\mu})^{2}=-\partial_{\mu}\partial^{\mu}=-\Box$, where
$\gamma^{\mu}$ represents Euclidean gamma matrices.
The Dirac-Kähler equation admits a direct transcription on the lattice because
both the exterior derivative $d$ and the codifferential $\delta$ can be simply
replaced by its lattice analogues, as discussed before. However, for the Dirac
equation the analogy has to further involve the relationship between the
4-component spinor field $\psi$ and the object $\Psi$. This relationship was
first established in BecherZPC82 ,Rabin for hypercubic lattices and later
extended to non-hypercubic lattices in Gockeler ,Raszillier . The analysis of
BecherZPC82 and Rabin has shown that $\Psi$ can be represented by a
16-component complex-valued inhomogeneous differential form:
$\Psi(x)=\sum_{p=0}^{4}\alpha^{p}(x)$ (55)
where $\alpha^{0}(x)$ is a (1-component) scalar function of position or
0-form, $\alpha^{1}(x)=\alpha^{1}_{\mu}(x)dx^{\mu}$ is a (4-component) 1-form,
and likewise for $p=2,3,4$ representing $2$-, $3$-, and $4$-forms with $6$-,
$4$-, and $1$-components respectively. By employing the following Clifford
algebra product
$dx^{\mu}\vee dx^{\nu}=g^{\mu\nu}+dx^{\mu}\wedge dx^{\nu}$ (56)
as using the anti-commutative property of the exterior product $\wedge$, we
have
$dx^{\mu}\vee dx^{\nu}+dx^{\nu}\vee dx^{\mu}=2g^{\mu\nu}$ (57)
which exactly matches the anticommutator result of the $\gamma^{\mu}$
matrices, $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}$.
This suggests that $dx^{\mu}$ plays the role of the $\gamma^{\mu}$ matrix in
the space of inhomogeneous differential forms with Clifford product kanamori ,
that is
$\gamma^{\mu}\partial_{\mu}\mapsto dx^{\mu}\vee\partial_{\mu}$ (58)
keeping in mind that while $\gamma^{\mu}\partial_{\mu}$ acts on spinors,
whereas $dx^{\mu}\vee\partial_{\mu}=(d-\delta)$ acts on inhomogeneous
differential forms. This analysis leads to a ‘geometrical’ interpretation of
the popular Kogut-Susskind staggered lattice fermions Susskind ,MineP1
because the latter can be made identical to lattice Dirac-Kähler fermions
after a simple relabeling of variables Rabin .
The 16-component object $\Psi$ can be viewed as a $4\times 4$ matrix that
produces a four-fold degeneracy with respect to the Dirac equation for $\psi$.
This degeneracy is actually not a problem in the continuum because there is a
well-defined procedure to extract the 4-components of $\psi$ from those of
$\Psi$ BecherZPC82 ,Rabin whereby the 16 scalar equations encoded by (53) all
reduce to the same copy of the four equations encoded by the standard Dirac
equation. This procedure is performed by a set of ‘projection operators’ that
form a group BecherZPC82 ,Benn . On the lattice, however, the operators $d$
and $\partial$, as well as $*$ (which plays a role on the space of
inhomogeneous differential forms $\Psi$ analogous to that of $\gamma^{5}$ on
the space of spinors $\psi$ Beauce ), behave in such a way that their action
leads to lattice translations. This is because cochains with different $p$
necessarily live on different lattice elements and also because $*$ is a map
between different lattice elements. As a consequence, the product operation of
such ‘group’ is not closed anymore. This nonclosure also stems from the fact
that the lattice operators $d$ and $\delta$ do not satisfy Leibnitz’s rule
kanamori . Because of this, the degeneracy of the Dirac equation on the
lattice is present at a more fundamental level and is harder to extricate
using the Dirac-Kähler description than the analogous degeneracy in the
continuum. In this regard, a new approach to identify the extraneous degrees
of freedom away from the continuum was recently described in AdamsPRL10 . In
addition, a split-operator approach to solve Dirac equation based on the
methods of characteristics that purports to avoid fermion doubling while
maintaining chiral symmetry on the lattice was very recently put forth in
Fillion . This approach preserves the linearity of the dispersion relation by
a splitting of the original problem into a series of one-dimensional problems
and the use of a upwind scheme with a Courant-Friedrichs-Lewy (CFL) number
equal to one, which provides an exact time-evolution (i.e. with no numerical
dispersion effects) along each reduced one-dimensional problem. The main
(practical) obstacle in this case is the need to use very small lattice
elements.
APPENDIX B: Absorbing boundary conditions
In many wave scattering simulations, the presence of long-range interactions
with slow (algebraic) decay, together with practical limitations in computer
memory resources, implies that open-space problems necessitate the use of
special techniques to suppress finite volume effects and emulate, for example,
the Sommerfeld radiation condition at infinity. Perfectly matched layers (PML)
are absorbing boundary conditions commonly used for this purpose Berenger1994
,Chew_Weedon1994 ,TeixeiraMGWL1997a ,CollinoSIAM1998 . In the continuum limit,
the PML provides a reflectionless absorption of outgoing waves, in such a way
that when the PML is used to truncate a computational lattice, finite volume
effects such as spurious reflections from the outer boundary are exponentially
suppressed. When first introduced in the literature Berenger1994 , the PML
relied upon the use of matched artificial electric and magnetic conductivities
in Maxwell’s equations and of a splitting of each vector field component into
two subcomponents. Because of this, the resulting fields inside the PML layer
are rendered ‘non-Maxwellian’. The PML concept was later shown to be
equivalent in the Fourier domain ($\partial_{t}\rightarrow-i\omega$) to a
complex coordinate stretching of the coordinate space (or an analytic
continuation to a complex-valued coordinate space) Chew_Weedon1994
,TeixeiraMGWL1997a ,CollinoSIAM1998 and, as such, applicable to any linear
wave phenomena.
Inside the PML, the (local) spatial coordinate $\zeta$ along the outward
normal direction to each lattice boundary point is complexified as
$\zeta\rightarrow\tilde{\zeta}=\int_{0}^{\zeta}s_{\zeta}(\zeta^{\prime})d\zeta^{\prime}$
(59)
where $s_{\zeta}$ is the so-called complex stretching variable written as
$s_{\zeta}(\zeta,\omega)=a_{\zeta}(\zeta)+i\Omega_{\zeta}(\zeta)/\omega$ with
$a_{\zeta}\geq 1$ and $\Omega_{\zeta}\geq 0$ (profile functions). The first
inequality ensures that evanescent waves will have a faster exponential decay
in the PML region, and the second inequality ensures that propagating waves
will decay exponentially along $\zeta$ inside the PML. As opposed to some
other lattice truncation techniques, the PML preserves the locality of the
underlying differential operators and hence retains the sparsity of the
formulation.
For Maxwell’s equations, the PML can also be effected by means of artificial
material tensors (Maxwellian PML) Sacks1995 . In three-dimensions, the
Maxwellian PML can be represented as a media with anisotropic permittivity and
permeability tensors exhibiting stratification along the normal to the
boundary $S$ that parametrizes the lattice truncation boundary. The PML
tensors properties depend on the local geometry via the two principal
curvatures of $S$ TeixeiraMGWL1997b ,TeixeiraMOTL1998 ,dondericiAP08 . The
boundary surface $S$ is assumed (constructed) as doubly differentiable with
non-negative radii of curvature, otherwise dynamic instabilities ensue during
a marching-on-time evolution MineP18 .
From (59), the PML also admits a straightforward interpretation as a
complexification of the metric MineP17 ,MineP19 . As a result, the use of
differential forms readily unifies the Maxwellian and non-Maxwellian PML
formulations because the metric is explicitly factored out into the Hodge star
operators—any transformation the metric corresponds, dually, to a
transformation on the Hodge star operators that can be mimicked by modified
constitutive relations MineP15 . In the differential forms framework, the PML
is obtained by a mapping on the Hodge star operators:
$\star_{\epsilon}\rightarrow\tilde{\star}_{\epsilon}$ and
$\star_{\mu^{-1}}\rightarrow\tilde{\star}_{\mu^{-1}}$ induced by the
complexification of the metric. The resulting differential forms inside the
PML, $\tilde{E},\tilde{D},\tilde{H},\tilde{B}$ therefore obey ‘modified’ Hodge
relations $\tilde{D}=\tilde{\star}_{\epsilon}\tilde{E}$ and
$\tilde{B}=\tilde{\star}_{\mu^{-1}}\tilde{H},$ but identical pre-metric
equations (12) and (13). In other words, (12) and (13) are invariant under the
transformation (59) MineP17 ,MineP19 .
APPENDIX C: Implementation of space charge effects
In many applications related to plasma physics or electronic devices, it is
necessary to include space charges (uncompensated charge effects) into lattice
models of macroscopic Maxwell’s equations. This is typically done by
representing the charged plasma media using particle-in-cell (PIC) methods
that track the individual particles on the lattice Hockney ,esirkepov ,ok06b .
The field/charge interaction is then modeled by ($i$) interpolating lattice
fields (cochains) to particle positions (gather step), ($ii$) advancing
particle positions and velocities in time using equations of motion, and
($iii$) interpolating back charge densities and currents onto the lattice as
cochains (scatter step). In general, the ‘particles’ do not need to be actual
individual particles, but can be a collection thereof (‘macro-particles’). To
put it simply, incorporation of space charges requires two extra steps during
the field update in any marching-on-time algorithm, which transfer information
from the instantaneous field distribution to the particle kinematic update and
vice-versa. Conventionally, this information transfer relies on spatial
interpolations that often violate the charge continuity equation and, as a
result, lead to spurious charge deposition on the lattice nodes. On regular
lattices, this problem can be corrected, for example, using approaches that
either subtract a static solution (charges) from the electric field solution
(Boris/DADI correction) or directly subtract the residual error on the Gauss
law (Langdon-Marder correction) at each time step Mardahl . On irregular
lattices, additional degrees of freedom can be added as coupled elliptical
constraints to produce a augmented Lagrange multiplier system Assous . All
these approaches necessitate changes on the original equations, while still
allowing for small violations on charge conservation. In contrast, Whitney
forms provide a direct route to construct gather and scatter steps that
satisfy charge conservation exactly even on unstructured lattices candel09
,squire12 , as explained next. To conform to the vast majority of the plasma
and electronic devices literature, we once more restrict ourselves here to the
3+1 setting (although a four-dimensional analysis in Minkowski space would
have provided a more succinct discussion).
For the gather step, Whitney forms can be used to directly compute
(interpolate) the fields at any location from the knowledge of its cochain
values, such as in (19) and (20) for example. For the scatter step, charge
movement can be modeled as the Hodge-dual of the current 2-form $J$, that is,
as the 1-form $\star J$ which can be expanded in terms of Whitney 1-forms on
the primal lattice. Here, $\star$ represents again the spatial Hodge star in
three-dimensions distilled from macroscopic constitutive properties. The
Hodge-dual current associated to an individual point charge can be expressed
as $\star J=qv^{\flat}$, where $q$ is the charge value, $v$ is the associated
velocity vector, and $\flat$ is the ‘flat’ operator or index-lowering
canonical isomorphism that maps a vector to a 1-form, given by the Euclidean
metric. Similarly, point charges can be encoded as the Hodge-dual of the
charge density 3-form $\rho$, that is, as the 0-form $\star\rho$, which can be
expanded in terms of Whitney 0-forms on the primal lattice. These two Whitney
maps are linked in such a way that the rate of change on the value of the
0-cochain representing $\star\rho$ at a node is associated to the presence of
a 1-cochain representing $\star J$ along the edges that touch that particular
node, leading to exact charge conservation at the discrete level. To show
this, consider for simplicity the two-dimensional case of a point charge $q$
moving from point $x^{(s)}$ to point $x^{(f)}$ during a time interval $\tau$
inside a triangular cell with nodes $\sigma_{0,0}$, $\sigma_{0,1}$, and
$\sigma_{0,2}$, or simply $0$, $1$, and $2$. At any point $x$ inside this
cell, the 0-form $\star\rho$ can be scattered to these three adjacent nodes
via
$\star\rho=q\sum_{i=1}^{3}\left<x,\omega^{0}_{i}\right>\omega^{0}_{i}$ (60)
where we are again using the short-hand
$\omega^{0}[\sigma_{0,i}]=\omega^{0}_{i}$, and the brackets represent the
pairing expressed by (1). In this case, $p=0$ and the pairing integral in (1)
reduces to a function evaluation at a point. Since Whitney 0-forms are equal
to the barycentric coordinates associated of a given node, that is
$\left<x,\omega^{0}_{i}\right>=\lambda_{i}(x)$, we have the scattered charge
$q\lambda^{s}_{i}\doteq q\lambda_{i}(x^{(s)})$ on node $i$ for a charge $q$ at
$x^{(s)}$, and, similarly, the scattered charge $q\,\lambda^{f}_{i}$ on node
$i$ for a charge $q$ at $x^{(f)}$. The rate of scattered charge variation on a
given node $i$ is therefore equal to
$\dot{q}(\lambda^{f}_{i}-\lambda^{s}_{i})$, where $\dot{q}=q/\tau$.
During $\tau$, the particle travels through a path $\ell$ from $x^{(s)}$ to
$x^{(f)}$, and the corresponding $\star J$ can be expanded as a sum of Whitney
1-forms $\omega^{1}_{\overline{ij}}$ associated to the three adjacent edges
$\overline{ij}=\overline{01},\overline{12},\overline{20}$, that is
$\star
J=\dot{q}\sum_{\overline{ij}}\left<\ell,\omega^{1}_{\overline{ij}}\right>\omega^{1}_{\overline{ij}}$
(61)
The coefficients $\left<\ell,\omega^{1}_{\overline{ij}}\right>$ represent the
(oriented) current flow along the associated oriented edge, that is, the
cochain representation of $\star J$ along edge $\overline{ij}$. Using (16),
the sum of the total current magnitude scattered along edges $\overline{01}$
and $\overline{20}$ that flows into node $0$ is therefore
$\dot{q}\left(-\left<\ell,\omega^{1}_{\overline{01}}\right>+\left<\ell,\omega^{1}_{\overline{20}}\right>\right)=\dot{q}\int_{\ell}\left(-\omega^{1}_{\overline{01}}+\omega^{1}_{\overline{20}}\right)$
(62)
Using
$\omega^{1}_{\overline{ij}}=\lambda_{i}d\lambda_{j}-\lambda_{j}d\lambda_{i}$
and $\lambda_{1}+\lambda_{2}+\lambda_{3}=1$, the above reduces to
$\dot{q}\int_{\ell}d\lambda_{0}=\dot{q}(\lambda^{f}_{0}-\lambda^{s}_{0})$ (63)
which exactly matches the rate of scattered charge variation on node $0$
obtained before. It is clear that similar equalities hold for nodes 1 and 2.
More fundamentally, these equalities are a direct consequence of the
structural property (18).
APPENDIX D: Classification of inconsistencies in naïve discretizations
We provide below a rough classification scheme of inconsistencies arising from
naïve discretizations of the differential calculus on irregular lattices.
(a) Pre-metric inconsistencies of first kind: We call pre-metric
inconsistencies of the first kind those that are related to the primal or dual
lattices taken as separate objects and that occur when the discretization
violates one or more properties of the continuum theory that is invariant
under homeomorphisms—for example, conservations laws that relate a quantity on
a region $S$ with an associated quantity on the boundary of the region,
$\partial S$ (a topological invariant). Perhaps the most illustrative example
is violation of ‘divergence-free’ conditions caused by improper construction
of incidence matrices, whereby the nilpotency of the (adjoint) boundary
operator, $\partial\circ\partial=0$, is not observed. This implies, in a dual
fashion, that the identity $d^{2}=0$ is violated MineP16 . Stated in another
way, the exact sequence property of the underlying de Rham differential
complex is violated Arnold2002 . In practical terms, this leads to the
appearance spurious charges and/or spurious (‘ghost’) modes. As the
classification suggests, these properties are not related to metric aspects of
the lattice, but only to its “topological aspects” that is, on how discrete
calculus operators are defined vis-à-vis the lattice element connectivity. In
more mathematical terms, one can say that the structure of the (co)homology
groups of the continuum manifold is not correctly captured by the cell complex
(lattice). We stress again that, given any dual lattice construction, pre-
metric inconsistencies of the first kind are associated to the primal or dual
lattice taken separately, and not necessarily on how they intertwine.
(b) Pre-metric inconsistencies of second kind: The second type of pre-metric
inconsistency is associated to the breaking of some discrete symmetry of the
Lagrangian. In mathematical terms, this type of inconsistency can occur when
the bijective correspondence between $p$-cells of the primal lattice and
$(n-p)$-cells of the dual lattice (an expression of Poincaré duality at the
level of cellular homology munkres , up to boundary terms) is violated. This
is typified by ‘nonreciprocal’ constructions of derivative operators, where
the boundary operator effecting the spatial derivation on the primal lattice
$K$ is not the dual adjoint (or the incidence matrix transpose) of the
boundary operator on the dual lattice $\mathcal{K}$: for example, the identity
${\tilde{C}}_{ij}^{p}=C_{ji}^{n-1-p}$ (up to boundary terms) used to obtain
eq. (32) is violated. One basic consequence of this violation is that the
resulting discrete equations break time-reversal symmetry. Consequently, the
numerical solutions will violate energy conservation and produce either
artificial dissipation or late-time instabilities MineP16 . Many algorithms
developed over the years for hyperbolic partial differential equations do
indeed violate these properties: they are dissipative and cannot be used for
long integration times ChevalierAP97 ,WhiteMTT01 . It should be noted at this
point that lattice field theories invariably break Lorentz covariance and many
of the continuum Lagrangian symmetries and, as a result, violate conservation
laws (currents) by virtue of Noether’s theorem. For example, angular momentum
conservation does not hold exactly on the lattice because of the lack of
continuous rotational symmetry (note that discrete rotational symmetries can
still be present). However, this latter type of symmetry breaking is of a
fundamentally different nature because it is ‘controllable’, i.e. their effect
on the computed solutions is made arbitrarily small in the continuum limit.
More importantly, discrete transcriptions of the Noether’s theorem can be
constructed for Lagrangian symmetries on a lattice SorkinJMP75 ,christ12 , to
yield exact conservation laws of (properly defined) quantities such as
discrete energy and discrete momentum Chew .
(c) Hodge-star inconsistencies: In the third type of inconsistency, we include
those that arise in connection with metric properties of the lattice. Because
the metric is entirely encoded in the Hodge-star operators MineP16
,BossavitPIER32 ,HiptmairNM01 , such inconsistencies can be simply understood
as inconsistencies on the construction of discrete Hodge-star operators (or
their procedural analogues). For example, it is not uncommon for naïve
discretizations in irregular lattices to yield asymmetric discrete Hodge
operators, as noted in WeilandIJNM96 ,RailtonEL97 . Even if symmetry is
observed, non positive definiteness might ensue that is often associated with
portions of the lattice with highly skewed or obtuse cells Schuhmann98 . Lack
of either of these properties lead to unconditional instabilities that destroy
marching-on-time solutions MineP16 . When very long integration times are
needed, asymmetry in the discrete Hodge matrices can be a problem even if
produced at the level of machine rounding-off errors.
APPENDIX E: Overview of related discretization approaches
We outline below some discretization programs that rely, one way or another,
on tenets exposed above. This delineation is mostly informed mostly by
applications related to electrodynamics and not too sharp as the programs
share much in common. (a) Finite-difference time-domain method: In cubical
lattices, the (lowest-order) Whitney forms can be represented by means of a
product of pulse and ‘rooftop’ functions on the three Cartesian coordinates
chilton2008 . This choice, together with the use of low-order quadrature rules
to compute the Hodge star integrals in (23) and (24), leads to diagonal
matrices $[\star_{\epsilon}]$, $[\star_{\mu^{-1}}]$, and, consequently, also
diagonal $[\star_{\epsilon}]^{-1}$, $[\star_{\mu^{-1}}]^{-1}$ and sparse
$[\Upsilon]$ so that an ultra-local equation results for (33). In this
fashion, one obtains a ‘matrix-free’ algorithm where no linear algebra is
needed during a marching-on-time solution for the fields. This prescription
recovers Yee’s finite-difference time-domain scheme YeeAP69 ,Taflove95
,YeeAP97 . Conventional FDTD adopts the simplest explicit, energy-conserving
(symplectic) time-discretization for eqs. (30) and (33), which can be
constructed by staggering the electric and magnetic fields in time and
replacing time derivatives by central differences. Staggering in both space
and time is consistent with the presence of two staggered hypercubical
spacetime lattices TontiPIER01 ,MattiussiPIER01 . The staggering in time also
provides a $O(\Delta t^{2})$ truncation error.
(b) Finite integration technique: The finite integration technique (FIT)
Weiland84 ,Schuhmann00 ,codecasa04 is closely related to FDTD, the main
distinction being that, assuming piecewise constant fields over each cell, the
latter is equivalent to applying the (discrete version) of the generalized
Stokes’ theorem to the cochains in (30) and (31). Another difference is that
the incidence matrices and material (Hodge star) matrices are treated
separately in FIT, in a manner akin to that exposed in Sections III and IV.
Like FDTD, FIT is based on dual staggered lattices and, for cubical lattices,
it turns out that the lowest-order numerical implementation of FIT is
equivalent to the lowest-order FDTD. The spatial operators in FIT can all be
viewed as discrete incarnations of the exterior derivative for the various
$p$, and as such, the exact sequence property of the underlying de Rham
complex is automatically enforced by construction BossavitIEE88 .
Historically, FIT generalizations to irregular lattices have relied on the use
of either projection operators Schuhmann98 or Whitney forms schuhmann02 to
construct discrete versions of the Hodge star operators (or their procedural
equivalents); however, these generalizations do not necessarily recover the
specific form of the discrete Hodge matrix elements expressed in (23) and
(24).
(c) Cell method: Another related discretization program, based on general
principles originally put forth in TontiPIER01 ,TontiRAL72 ,tonti76 , is the
Cell method bullo04 ,alotto06 ,bullo06 ,alotto08 ,alotto10 ,codecasa . Even
though this program does not rely on Whitney forms for constructing discrete
Hodge star operators (other geometrically-based constructions are used
instead), it is nevertheless still based upon the use of ‘domain-integrated’
discrete variables that conform to the notion of discrete differential forms
or cochains of various degrees and, as such, it is naturally suited for
irregular lattices. The Cell method also employs metric-free discrete
operators that satisfy the exactness property of the de Rham complex and make
explicit use of a dual lattice (but not necessarily barycentric) motivated by
the notion of inner and outer orientations. The relationships between the
various discrete operators and ‘domain-integrated’ field quantities (cochains)
in the Cell method are built into general classification diagrams referred to
as ‘Tonti diagrams’ that reproduce correct commuting diagram properties of the
underlying operators tonti76 ,TontiPIER01 .
(d) Mimetic finite-differences: ‘Mimetic’ finite-difference methods,
originally developed for non-orthogonal hexahedral structured lattices
(‘tensor-product grids’) and later extended for irregular and polyhedral
lattices SteinbergJCP95 ,ShashkovJCP99 ,ShashkovSIAM99 ,ShashkovANM97
,ShashkovPIER01 ,CastilloANM02 ,lipnikov06 ,brezzi10 ,robidoux11 ,lipnikov11
also share many of the properties exposed above. The thrust here is towards
the construction of discrete versions of the differential operators
divergence, gradient, and curl of vector calculus having ‘compatible’ (in the
sense of the exactness property of the underlying de Rham complex) domains and
ranges and such that the resulting discrete equations exactly satisfy discrete
conservation laws. In three dimensions, this naturally leads to the definition
of three ‘natural’ operators and three ‘adjoint’ operators that can be
associated with exterior derivative $d$ and the codifferential $\delta$,
respectively, for $p=1,2,3$ (although the exterior calculus terminology is
often not used explicitly in this context). In mimetic finite-differences, the
discrete analogues of the codifferential operator $\delta$ are full matrices,
and the matrix-free character of FDTD is lacking even on orthogonal lattices.
A very thorough, historical review of mimetic finite-difference is provided in
Lipnikov .
(e) Compatible discretizations and finite element exterior calculus: In recent
years, much attention has been devoted to the development of ‘compatible
discretizations,’ an umbrella term used to denote spatial discretizations of
partial differential equations seeking to provide finite element spaces that
reproduce the exactness of the underlying de Rham complex (or the correct
cohomology in topologically nontrivial domains) arnold02 ,arnold06a ,white06
,bochev06 ,boffi07 ,Bochev12 . In this program, Whitney forms play a role of
providing ‘conforming’ vector-valued functional (finite element) spaces of
Sobolev-type. Specifically, Whitney 1-forms recover the space of ‘Nedelec
edge-elements’ or curl-conforming Sobolev space ${\bf H}(\text{curl},\Omega)$
nedelec80 and Whitney 2-forms recover the space of ‘Raviart-Thomas elements’
or div-conforming Sobolev space ${\bf H}(\text{div},\Omega)$ hiptmairMC99 . In
this regard, a relatively new advance here has been the development of new
finite element spaces, beyond those provided by Whitney forms, based on the
Koszul complex Guil . The latter is key for the stable discretization of
elastodynamics arnold06b . Another recent approach aimed at the stable
discretization of elastodynamics is described in Yavari08 . The link between
stability conditions of some mixed finite element methods nedelec80 and the
complex of Whitney forms has a long history in the context of electrodynamics
BossavitIEE88 ,Bossavitchap ,Bossavit98 ,BossavitEJM91 ,KettunenMAGN98
,KettunenMAGN99 ,MineP27 ,KotiugaJAP93 ,kangas07 ,wong95 ,feliziani98
,castillo04 ,rieben05 .
(f) Discrete exterior calculus: The ‘discrete exterior calculus’ (DEC) is yet
another discretization program aimed at developing ab initio consistent
discrete models to describe field theories squire12 ,Desbrun03 ,Hirani03
,Desbrun05 ,Gillette09 ,perot . This program recognizes the role played by
discrete differential forms to capture and the need for dual lattices to
capture the correct physics. Note that DEC has focused on the use of a
circumcentric dual as opposed to a barycentric dual Hirani03 ,Desbrun05
(despite the fact that the former does not admit a metric-free construction)
and does not emphasize the role of Whitney forms. DEC also recognizes the need
to address group-valued differential forms, as well as the mathematical
objects that exist on the dual-bundle space together with the associated
operators (such as contractions and Lie derivatives), in connection to
discrete problems in mechanics, optimal control, and computer vision/graphics
Desbrun03 . A recent discussion on obstacles associated with some of the DEC
underpinnings is provided in Kotiuga08
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arxiv-papers
| 2013-04-11T21:05:21 |
2024-09-04T02:49:44.232954
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. L. Teixeira",
"submitter": "Fernando Teixeira",
"url": "https://arxiv.org/abs/1304.3485"
}
|
1304.3526
|
# Method of Relative Magnetic Helicity Computation II: Boundary Conditions for
the Vector Potentials
Shangbin Yang 11affiliation: Key Laboratory of Solar Activity, National
Astronomical Observatories, Chinese Academy of Sciences, 100012 Beijing, China
22affiliation: Max-Planck Institute for Solar System Research, 37191
Katlenburg-Lindau, Germany , Jörg Büchner 22affiliation: Max-Planck Institute
for Solar System Research, 37191 Katlenburg-Lindau, Germany , Jean Carlo
Santos 33affiliation: Laboratŕio de Plasmas, Instituto de Física, Universidade
de Brasĺia, Brazil , and Hongqi Zhang 11affiliation: Key Laboratory of Solar
Activity, National Astronomical Observatories, Chinese Academy of Sciences,
100012 Beijing, China
###### Abstract
We have proposed a method to calculate the relative magnetic helicity in a
finite volume as given the magnetic field in the former paper (Yang et al.
Solar Physics, 283, 369, 2013). This method requires that the magnetic flux to
be balanced on all the side boundaries of the considered volume. In this
paper, we propose a scheme to obtain the vector potentials at the boundaries
to remove the above restriction. We also used a theoretical model (Low and
Lou, Astrophys. J. 352, 343, 1990) to test our scheme.
Magnetic helicity
## 1 Introduction
Magnetic helicity is a key geometrical parameter to describe the structure and
evolution of solar coronal magnetic fields ( e.g. Berger, 1999). Magnetic
helicity in a volume $V$ can be determined as
$\centering{H_{\rm M}=\int_{V}\mathbf{A}\cdot\mathbf{B}dV},\@add@centering$
(1)
where A is the vector potential for the magnetic field B in this volume.
Magnetic helicity is conserved in an ideal magneto-plasma (Woltjer, 1958). As
long as the overall magnetic Reynolds number is large, however, it is still
approximately conserved, even in the course of relatively slow magnetic
reconnection (Berger, 1984). The concept of magnetic helicity has successfully
been applied to characterize solar coronal processes, for a recent review
about modeling and observations of photospheric magnetic helicity see, e.g.,
Démoulin and Pariat (2009). Despite of its important role in the dynamical
evolution of solar plasmas, so far only a few attempts have been made to
estimate the helicity of coronal magnetic fields based on observations and
numerical simulations (see, e.g., Thalmann, Inhester, and Wiegelmann, 2011;
Rudenko and Myshyakov, 2011).
Yang et al. (2013) developed a method for an efficient calculation of the
relative magnetic helicity in finite 3D volume which already was applied to a
simulated flaring AR Santos et al. (2011). This method requires the magnetic
flux to be balanced on all the side boundaries of the considered volume. In
this paper, a scheme to remove the restriction has been proposed. In Sec. 2,
we describe the restriction of vector potential in the former paper. In Sec.
3, we present the details of the new scheme to calculate the vector potentials
on the six boundaries. In sec. 4, we use the theoretical model to check our
scheme. The summary and some discussions are given in Sec. 5.
## 2 The former definition of ${\bf A}_{\rm p}$ and A at the boundaries
Let us define a finite three-dimensional (3-D) volume (“box”) in Cartesian
coordinates with a magnetic field ${\bf B}(x,y,z)$ given in this volume. Let
the volume be bounded by $x=[0,l_{x}]$, $y=[0,l_{y}]$, and $z=[0,l_{z}]$.
First one has to provide the values of ${\bf A}_{\rm p}$ and A on all six
boundaries ($x=0,l_{x};y=0,l_{y};z=0,l_{z}$). To take the bottom boundary
($z=0$) for example, we define a new scalar function $\varphi(x,y)$ that
determines the vector potential ${\bf A}_{\rm p}$ of the potential magnetic
field P on this boundary as follows:
${A_{\rm p\it x}=-\frac{{\partial\varphi}}{{\partial y}},\qquad A_{\rm p\it
y}=\frac{{\partial\varphi}}{{\partial x}},\qquad\left.{A_{\rm p\it
z}}\right|_{z=0}=0.}$ (2)
According to the definition of the vector potential, the scalar function
$\varphi(x,y)$ should satisfy the Poisson equation:
$\Delta\varphi(x,y)=B_{z}(x,y,z=0).$ (3)
The value of $\partial\phi/\partial n$ on the four sides of the plane $z=0$ is
set to zero in Equation (3). According to Eq.( 2), $A_{\rm p\it x}$ and
$A_{\rm p\it y}$ will vanish at $y=0,l_{y}$ and at $x=0,l_{x}$, respectively,
on the $z=0$ plane. Thus, the corresponding magnetic flux at the boundary
should also vanish because of Ampère’s law. The values of ${\bf A}_{\rm p}$ on
the other five boundaries could be obtained in a similar way. For the vector
potential A at all boundaries the same values are taken as for ${\bf A}_{\rm
p}$. When the magnetic fluxes at the six boundaries are not zero, we should
calculate the value of vector potentials at the twelve edges of the three-
dimensional (3-D) volume to provide the Neumann boundary for the Poisson
Equation at each side boundary. In next section, we will introduce a scheme to
calculate the vector potentials at the twelve edges.
## 3 new scheme to obtain $\mathbf{A}_{p}$ and $\mathbf{A}$ at the boundaries
For the $\mathbf{A}_{p}$, we define the magnetic flux $\Phi_{i}~{}(i=1,...,6)$
respectively at each side boundary
($z=0;~{}z=l_{z};~{}x=0;~{}x=l_{x};~{}y=0;~{}y=l_{y}$). The integrals of
$\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at the twelve edges are defined as
$a_{i}~{}(i=1,...,12)$. The twelve integrals and the corresponding directions
are represented in Fig. 1.
Figure 1: Magnetic flux $\Phi_{i}~{}(i=1,...,6)$ at the six boundaries and the
integrals $a_{i}~{}(i=1,...,12)$ of $\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at
the twelve edges.
According to the Ampère’s law, the integral value $a_{i}$ satisfy the
following linear equations
${\rm{TX=B},}$ (4)
where
$\textrm{B}=(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5},\Phi_{6})^{T}$,
$\textrm{X}=(a_{1},a_{2},a_{3},...,a_{12})^{T}$ and T is a matrix of
6$\times$12 , which is equal
$\left[\begin{array}[]{cccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&0&0&0&0\\\ 0&0&0&{-1}&0&0&0&{-1}&1&0&0&1\\\
0&{-1}&0&0&0&{-1}&0&0&0&1&1&0\\\ {-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&0&0\\\
0&0&{-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}\\\ \end{array}\right].$ (5)
One can check that the six rows-vector in this matrix is not linear
independent because the magnetic field is divergence free and the sum of
$\Phi_{i}$ at the six boundaries is zero. Moreover, the unknown twelve $a_{i}$
are not unique just under the restriction of above six conditions. Hence, we
need to construct twelve independent conditions to obtain the unique solution
for $a_{i}$. We define the new matrix $\hat{\textrm{T}}$ as follows
$\left[\begin{array}[]{ccccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&0&0&0&0\\\ 0&0&0&{-1}&0&0&0&{-1}&1&0&0&1\\\
0&{-1}&0&0&0&{-1}&0&0&0&1&1&0\\\ {-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&0&0\\\
1&0&{-1}&0&0&0&0&0&0&0&0&0\\\ 0&1&0&{-1}&0&0&0&0&0&0&0&0\\\
0&0&1&0&{-1}&0&0&0&0&0&0&0\\\ 0&0&0&1&0&{-1}&0&0&0&0&0&0\\\
0&0&0&0&1&0&{-1}&0&0&0&1&0\\\ 0&0&0&0&0&1&0&{-1}&0&0&0&0\\\
0&0&0&0&0&0&1&0&{-1}&0&0&0\\\ \end{array}\right].$ (6)
One can check that the determinant of $\hat{\textrm{T}}$ is not zero.
According to Cramer rule, the unique solution is existent for the new linear
equation
${\rm{{\hat{T}}X=\hat{B}},}$ (7)
where $\textrm{X}=(a_{1},a_{2},a_{3},...,a_{12})^{T}$ and
$\hat{\textrm{B}}=(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5},0,0,0,0,0,0,0)^{T}$.
Then we can obtain the integrals of $\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at
the twelve edges. The corresponding vector potential at the twelve edges could
be obtained by using the following equation:
$\begin{split}{\rm{A}}_{{\rm{px}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{x}}}\sin({{\pi x}\mathord{\left/{\vphantom{{\pi
x}{L_{x}}}}\right.\kern-1.2pt}{L_{x}}}),i=1,3,5,7\\\
{\rm{A}}_{{\rm{py}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{y}}}\sin({{\pi y}\mathord{\left/{\vphantom{{\pi
y}{L_{y}}}}\right.\kern-1.2pt}{L_{y}}}),i=2,4,6,8\\\
{\rm{A}}_{{\rm{pz}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{z}}}\sin({{\pi z}\mathord{\left/{\vphantom{{\pi
z}{L_{z}}}}\right.\kern-1.2pt}{L_{z}}}),i=9,10,11,12\end{split}$ (8)
Note that ${\bf A}_{\rm p}$ at the ends of every edge both are zero according
to the above equation. That is the requirement of Eq. (2). Then we resolve the
Poisson equations to obtain ${\bf A}_{\rm p}$ at the six boundaries. For the
vector potential A at all boundaries the same values are taken as for ${\bf
A}_{\rm p}$. Then we can follow the method of Sec. 2.2 and 2.3 of the former
paper Yang et al. (2013) to calculate the relative magnetic helicity in this
volume.
## 4 Testing the scheme
For testing the new scheme to obtain the vector potentials at the boundaries,
we use the axisymmetric nonlinear force-free fields of Low and Lou (1990). We
used the model labeled $P_{1,1}$ with $l=0.3$ and $\Phi=\pi/2$ in the notation
of their paper. We calculated the magnetic field on a uniform grid of
$64\times 64\times 64$. The pixel size in the calculation is assumed to be 1.
We calculate the magnetic fluxes $\Phi_{0}$ at the six boundaries and
substitute it to the Eq. (7) to obtain the integral $a_{i}$ at the twelve
edges of the 3D volume. Then we substitute $a_{i}$ into Eq. (8) respectively
to get the boundary value for resolving the Poisson equation in Eq. (3) at the
six boundaries. After we attain ${\bf A}_{\rm p}$ at the six boundaries, we
could also calculate the magnetic flux $\Phi$ according to the relation
between the vector potential and the magnetic field: ${\bf B}\cdot\hat{\rm
n}=\nabla\times{\bf A}_{\rm p}\cdot\hat{\rm n}$. Table. 1 represents the final
result after we apply the the above scheme. It can be found that the
calculated magnetic fluxes at the six boundaries by using our scheme
respectively coincide well with the original value from the theoretical model.
Note that the total magnetic flux of the theoretical model is not exact zero.
However, it is required that the total magnetic flux is exact zero when
resolving the linear equation Eq. (7), which cause the total magnetic fluxes
of $\oint{\bf A}_{\rm p}\cdot\rm d{\bf l}$ and $\Phi$ are different with that
of $\Phi_{0}$. On the other hand, the numerical errors when resolving the
Poisson equation are also unavoidable, which will also introduce the
difference for the total magnetic flux as well.
Table 1: Testing using the new scheme to a theoretical model. side boundary | $\Phi_{0}^{\tablenotemark{a}}$ | $\oint{\bf A}_{\rm p}\cdot\rm d{\bf l}^{\tablenotemark{b}}$ | $\Phi^{\tablenotemark{c}}$
---|---|---|---
$z=0$ | -3615.81 | -3615.81 | -3487.1068
$z=l_{z}$ | 1461.13 | 1461.13 | 1490.2739
$x=0$ | -1471.95 | -1471.95 | -1563.7657
$x=l_{x}$ | -1471.95 | -1471.96 | -1564.9280
$y=0$ | 4006.42 | 4006.42 | 4087.8426
$y=l_{y}$ | 1068.27 | 1092.17 | 1066.6782
Total flux | -23.89 | 0.00012 | 28.99
## 5 Summary
In this paper, we propose a new scheme to calculate the vector potential at
the boundaries to remove the restrictions in the former paper Yang et al.
(2013). In principle, now we can calculate the relative magnetic helicity of
any magnetic field structure in Cartesian coordinates. In the observations, we
could use force-free extrapolation method to obtain the three-dimensional
magnetic structure to analyze the evolution of relative magnetic helicity. On
the other hand, we can also use a sequence of magnetograms to estimate the
accumulated magnetic helicity in the solar corona (Ref. Démoulin and Pariat,
2009). It will be very interesting to compare the two types of accumulated
magnetic helicity and analyze the correlation between magnetic helicity and
solar eruption (e.g. Jing et al., 2012). In the simulations, we could also
calculate the relative magnetic helicity directly based the known magnetic
field structure to understand how the magnetic helicity plays an important
role in solar reconnection and dynamos.
This study is supported by grants 10733020, 10921303, 41174153,11173033
11178016 and 11103038 of National Natural Science Foundation of China,
2011CB811400 of National Basic Research Program of China, a sandwich-PhD grant
of the Max-Planck Society and the Max-Planck Society Interinstitutional
Research Initiative Turbulent transport and ion heating, reconnection and
electron acceleration in solar and fusion plasmas of Project No. MIF-IF-A-
AERO8047.The authors also like to thank the Supercomputing Center of Chinese
Academy of Sciences (SCCAS) for the allocation of computing time.
## References
* Berger (1984) Berger, M. A., Field, G., B.: 1984, J. Fluid Mech. 147, 133\.
* Berger (1999) Berger, M. A.: 1999, Plasma Phys. Contr. Fusion 41, 167.
* Boulmezaoud (1999) Boulmezaoud, T. Z.: 1999, Étude des champs de Beltrami dans des domaines de R3 bornś et non bornś et applications en astrophysique”, Ph.D. thesis, Univ. Paris VI.
* Démoulin and Pariat (2009) Démoulin, P., Pariat, E.: 2009, Adv. Space Res. 43, 1013\.
* Jing et al. (2012) Jing, J., Park, S., Liu, C., Lee, J., Wiegelmann, T., Xu, Y., Deng, N., & Wang, H. M. : 2012, Astrophys. J. 752, L9
* Low and Lou (1990) Low, B. C., Lou, Y.Q.: 1990, Astrophys. J. 352, 343.
* Rudenko and Myshyakov (2011) Rudenko, G. V., Myshyakov, I. I.: 2011, Solar phys. 270, 165\.
* Santos, Büchner, and Otto (2011) Santos, J. C., Büchner, J., Otto, A.: 2011, Astron. Astrophys. 535, A111.
* Seehafer, Kuzanyan, and Pipin (2003) Seehafer, N., Gellert, M., Kuzanyan, K. M., Pipin, V. V.: 2003, Adv. Space Res. 32, 1819.
* Thalmann, Inhester, Wiegelmann (2011) Thalmann, J. K., Inhester, B., Wiegelmann, T.: 2011, Solar Phys. 272, 243.
* Valori, Démoulin and Pariat (2012) Valori, G., Démoulin, P., Pariat, E.: 2012, Solar Phys. 278, 347.
* Woltjer (1958) Woltjer, L. 1958, Proc. Natl Acad. Sci. USA, 44, 480
* Santos et al. (2011) Santos, J. C., Büchner, J., & Otto, A. 2011, A&A, 535, A111
* Yang et al. (2013) Yang, S., Büchner, J. , Santos, J. C., & Zhang, H.: 2013, Solar Physics,283, 369.
* Yang, Büchner, and Zhang (2009a) Yang, S., Büchner, J., Zhang, H.: 2009, Astrophys. J. Lett. 695, L25.
* Yang, Büchner, and Zhang (2009b) Yang, S., Zhang, H., Büchner, J.: 2009, Astron. Astrophys. 502, 333.
* Zhang (2006) Zhang, H.: 2006, Astrophys. Space Sci. 305, 211.
* Zhang, Flyer, and Low (2006) Zhang, M., Flyer, M., Low, B.: 2006, Astrophys. J. 644, 575\.
|
arxiv-papers
| 2013-04-12T02:41:35 |
2024-09-04T02:49:44.247460
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shangbin Yang, J\\\"org B\\\"uchner, Jean Carlo Santos and Hongqi Zhang",
"submitter": "Shangbin Yang Dr.",
"url": "https://arxiv.org/abs/1304.3526"
}
|
1304.3543
|
# Zeros of Witten zeta functions and absolute limit
N. Kurokawa and H. Ochiai
## 1 Introduction
The Witten zeta function
$\zeta_{G}^{W}(s)=\sum_{\rho\in\hat{G}}\deg(\rho)^{-s}$
was introduced by Witten [W] in 1991, where $G$ is a compact topological group
and $\hat{G}$ denotes the unitary dual, that is, the set of equivalence
classes of irreducible unitary representations. The example
$\zeta^{W}_{SU(2)}(s)=\sum_{m=0}^{\infty}\deg({\operatorname{Sym}}^{m})^{-s}=\sum_{n=1}^{\infty}n^{-s}=\zeta(s),$
where $\zeta(s)$ denotes the Riemann zeta function, suggests fine properties
for general case. In fact, Witten showed arithmetical interpretation for
$\zeta^{W}_{SU(n)}(2m)$ ($m=1,2,3,\dots$) containing Euler’s result ([E1]
1735)
$\zeta^{W}_{SU(2)}(2m)\in\pi^{2m}\mathbf{Q}.$
In this paper we look at the opposite side: special values at negative
integers such as
$\displaystyle\zeta^{W}_{SU(2)}(-1)$ $\displaystyle=\mbox{``\
$\displaystyle\sum_{n=1}^{\infty}n$\ ''}=-\frac{1}{12},$ (1)
$\displaystyle\zeta^{W}_{SU(2)}(-2)$ $\displaystyle=\mbox{``\
$\displaystyle\sum_{n=1}^{\infty}n^{2}$\ ''}=0$ (2)
due to Euler [E2](1749). We notice that the value
$\mbox{``\ $\displaystyle\sum_{n=1}^{\infty}n$\ ''}=-\frac{1}{12}$
appears as the one-dimensional Casimir energy: see Casimir [C] and Hawking
[H]. The equality
$\mbox{``\ $\displaystyle\sum_{n=1}^{\infty}n^{2}$\ ''}=0$
means the vanishing of the two-dimensional Casimir energy.
We notice that
$\zeta^{W}_{G}(-2)=\left|G\right|$
when $G$ is a finite group. We conjecture that
$\zeta^{W}_{G}(-2)=0$ (3)
for infinite groups $G$.
For deeper understanding of the situation, we introduce a new zeta function
(Witten $L$-function)
$\zeta^{W}_{G}(s,g)=\sum_{\rho\in\widehat{G}}\frac{{\operatorname{trace}}(\rho(g))}{\deg(\rho)}\deg(\rho)^{-s}$
(4)
where $G$ is a compact topological group, $g$ is an element of $G$,
$\widehat{G}$ is the set of equivalence classes of irreducible
($\mathbf{C}$-valued) representations of $G$, $\deg(\rho)$ is the degree (the
dimension) of an irreducible representation $\rho\in\widehat{G}$. Note that
${\operatorname{trace}}(\rho(g))$ is the character of the representation
$\rho$. This Witten zeta function $\zeta_{G}^{W}(s,g)$ reduces to the (usual)
Witten zeta function when we specialize $g$ to be the identity element $1\in
G$:
$\zeta^{W}_{G}(s)=\zeta^{W}_{G}(s,1).$
In the case of a finite group $G$ we have
$\zeta^{W}_{G}(-2,g)=\left\\{\begin{array}[]{ll}\left|G\right|&\mbox{ if
}g=1,\\\ 0&\mbox{ otherwise}.\end{array}\right.$
We conjecture that
$\zeta^{W}_{G}(-2,g)=0$ (5)
when $G$ is an infinite group. The following result treats the case $G=SU(2)$.
###### Theorem 1.
Suppose $g\in SU(2)$ is conjugate to
$\left(\begin{array}[]{cc}e^{i\theta}&0\\\ 0&e^{-i\theta}\end{array}\right)$
with $0\leq\theta\leq\pi$.
* (1)
We have an expression
$\zeta^{W}_{SU(2)}(s,g)=\sum_{n=1}^{\infty}\frac{\sin(n\theta)}{n\sin\theta}n^{-s}$
in ${\operatorname{Re}}(s)>1$ The function $\zeta^{W}_{SU(2)}(s,g)$ in $s$ has
a meromorphic continuation to the whole complex plane.
* (2)
For a positive even integer $m$, we have $\zeta_{SU(2)}^{W}(-m,g)=0$ for all
$g\in SU(2)$. Moreover, $s=-2$ is a simple zero of $\zeta_{SU(2)}^{W}(s,g)$,
and the first derivative at $s=-2$ is given as
$\frac{\partial\zeta^{W}_{SU(2)}}{\partial
s}(-2,g)=\left\\{\begin{array}[]{ll}-\frac{\zeta(3)}{4\pi^{2}}&\mbox{ if
}\theta=0,\\\
\frac{1}{4\pi\sin\theta}\left(\zeta(2,\frac{\theta}{2\pi})-\frac{\pi^{2}}{2\sin^{2}\frac{\theta}{2}}\right)>0&\mbox{
if }0<\theta<\pi,\\\ \frac{7\zeta(3)}{4\pi^{2}}&\mbox{ if
}\theta=\pi.\end{array}\right.$
Here $\zeta(s,x)$ denotes the Hurwitz zeta function.
* (3)
The special value at $s=-1$ is given as
$\zeta^{W}_{SU(2)}(-1,g)=\left\\{\begin{array}[]{ll}-\frac{1}{12}&\mbox{ if
}\theta=0,\\\ \frac{1}{4\sin^{2}\frac{\theta}{2}}&\mbox{ if }0<\theta<\pi,\\\
\frac{1}{4}&\mbox{ if }\theta=\pi.\end{array}\right.$
We now introduce a ‘multi’-version of Witten $L$-function. For
$g_{1},\ldots,g_{r}\in G$, we define
$\displaystyle\zeta^{W}_{G}(s;g_{1},\dots,g_{r})$
$\displaystyle:=\sum_{\rho\in\widehat{G}}\frac{{\operatorname{trace}}(\rho(g_{1}))}{\deg(\rho)}\cdots\frac{{\operatorname{trace}}(\rho(g_{r}))}{\deg(\rho)}\times\deg(\rho)^{-s}$
$\displaystyle=\sum_{\rho\in\widehat{G}}\frac{{\operatorname{trace}}(\rho(g_{1}))\cdots{\operatorname{trace}}(\rho(g_{r}))}{\deg(\rho)^{s+r}}.$
It is natural to ask whether the vanishing
$\zeta^{W}_{G}(-2;g_{1},\dots,g_{r})\overset{?}{=}0$ of the special value at
$s=-2$ for this generalization holds. We have a partial answer to this
question.
###### Theorem 2.
We have $\zeta^{W}_{SU(2)}(-m;g_{1},g_{2})=0$ for $g_{1},g_{2}\in SU(2)$, and
a positive even integer $m$.
We also give an example of the non-vanishing for the case $r=3$: for some
$g\in SU(2)$, we prove that $\zeta^{W}_{SU(2)}(-2;g,g,g)\neq 0$. These results
related with the Lie group $SU(2)$ are given in Section 2.
We report further examples of zeros of Witten zeta functions for infinite
groups.
###### Theorem 3.
$\zeta^{W}_{SU(3)}(s)=0$ for $s=-1,-2,\dots.$
The proof of this theorem is given in Section 3.
The next example is not a Lie group, but a totally disconnected group. Let
$\mathbf{Z}_{p}$ be the $p$-adic integer ring for a prime number $p$.
###### Theorem 4.
Suppose $p\neq 2$. Then $\zeta^{W}_{SL_{2}(\mathbf{Z}_{p})}(s)=0$ for
$s=-1,-2$.
Now we consider the congruence subgroups. For a positive integer $m$, we
define a subgroup of $SL_{3}(\mathbf{Z}_{p})$ of finite index by
$SL_{3}(\mathbf{Z}_{p})[p^{m}]=\ker(SL_{3}(\mathbf{Z}_{p})\rightarrow
SL_{3}(\mathbf{Z}_{p}/(p^{m}))).$
###### Theorem 5.
Suppose $p\neq 3$.
* (1)
$\displaystyle\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$
$\displaystyle=p^{8m}\frac{(1-p^{-2-s})(1-p^{-1-s})}{(1-p^{1-2s})(1-p^{2-3s})}$
$\displaystyle\times\left(1+(p^{-1}+p^{-2})p^{-s}+(1+p^{-1})p^{-2s}+p^{-2-3s}\right).$
* (2)
$\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)=0$ for $s=-1,-2$.
* (3)
$\zeta^{W}_{SL_{3}(\mathbf{Z}_{1})[1^{m}]}(s)=\frac{(s+1)(s+2)}{(s-\frac{1}{2})(s-\frac{2}{3})}.$
Here we interpret that if $\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$ has
an expression as an analytic function on $p$, and there is a limit $p\to 1$,
then its limit is denoted by
$\zeta^{W}_{SL_{3}(\mathbf{Z}_{1})[1^{m}]}(s)=\lim_{p\to
1}\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s).$
These results on totally disconnected groups are given in Section 4.
## 2 $SU(2)$
### 2.1 Parametrization of irreducible representations of $SU(2)$
The set of equivalence classes, $\widehat{G}$, of irreducible unitary
representations of $G=SU(2)$ is parametrized by the set of natural numbers.
For a natural number, we denote by $\rho=\rho_{n}\in\widehat{G}$, the
corresponding irreducible representation of $G$. For a
$g=\left(\begin{array}[]{cc}e^{i\theta}&0\\\
0&e^{-i\theta}\end{array}\right)\in G$, we have the character formula
${\operatorname{trace}}(\rho(g))=e^{i(n-1)\theta}+e^{i(n-3)\theta}+\cdots+e^{i(3-n)\theta}+e^{i(1-n)\theta}$
(6)
and the degree
$\deg(\rho)={\operatorname{trace}}(\rho(I_{2}))=n,$ (7)
where $I_{2}=\textstyle\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\in SU(2)$ is the identity matrix. We also see that
${\operatorname{trace}}(\rho(-I_{2}))=(-1)^{n-1}n$.
We start from $g=\pm I_{2}\in SU(2)$. In these cases, $\zeta_{SU(2)}^{W}(s,g)$
is written in terms of Riemann zeta function. We see that
$\zeta_{SU(2)}^{W}(s,I_{2})=\zeta(s)$, and
$\zeta_{SU(2)}^{W}(s,-I_{2})=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}=(1-2^{1-s})\zeta(s).$
(8)
### 2.2 Poly-logarithm function
We recall the poly-logarithm
$\displaystyle Z(s,x)$ $\displaystyle=\sum_{n=1}^{\infty}\frac{x^{n}}{n^{s}},$
which is written also as ${\operatorname{Li}}_{s}(x)$ in literature. This
series converges if $\left|x\right|<1$ and $s\in\mathbf{C}$, or
$\left|x\right|=1$ and ${\operatorname{Re}}(s)>1$. In the following, we
restrict to the case $\left|x\right|=1$.
###### Theorem 6.
Suppose $\left|x\right|=1$ and $x\neq 1$. Then $Z(s,x)$ is analytically
continued to a holomorphic function on $s\in\mathbf{C}$. Moreover, for every
non-negative integer $m$, the function $Z(-m,x)$ can be expressed by a
rational function in $x$. The first several examples are
$Z(0,x)=\frac{x}{1-x},\quad Z(-1,x)=\frac{x}{(1-x)^{2}},\quad
Z(-2,x)=\frac{x(1+x)}{(1-x)^{3}},\ldots.$
###### Proof.
For ${\operatorname{Re}}(s)>1$, we have
$\displaystyle Z(s,x)$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+\sum_{n=3}^{\infty}\frac{x^{n}}{n^{s}}$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+\sum_{n=2}^{\infty}\frac{x^{n+1}}{(n+1)^{s}}$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+\sum_{n=2}^{\infty}x^{n+1}n^{-s}(1+n^{-1})^{-s}$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+\sum_{n=2}^{\infty}x^{n+1}n^{-s}\sum_{k=0}^{\infty}\binom{-s}{k}n^{-k}$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+x\sum_{k=0}^{\infty}\binom{-s}{k}(Z(s+k,x)-x)$
$\displaystyle=x+\frac{x^{2}}{2^{s}}+x(Z(s,x)-x)+x\sum_{k=1}^{\infty}\binom{-s}{k}(Z(s+k,x)-x).$
This shows
$\displaystyle(1-x)Z(s,x)$
$\displaystyle=x+x^{2}(2^{-s}-1)+x\sum_{k=1}^{\infty}\binom{-s}{k}(Z(s+k,x)-x).$
(9)
By the estimates of binomial coefficients, the right-hand side converges
absolutely on the right-half plane ${\operatorname{Re}}(s)>0$. This shows the
analytic continuation of $Z(s,x)$ to ${\operatorname{Re}}(s)>0$. Repeating
this argument, we obtain the analytic continuation to whole $s\in\mathbf{C}$.
To substitute $s=-m$ with $m=0,1,\dots$, we have the recursion equation
$\displaystyle(1-x)Z(-m,x)=x+x^{2}(2^{m}-1)+x\sum_{k=1}^{m}\binom{m}{k}(Z(-(m-k),x)-x).$
(10)
∎
First several examples show
$\displaystyle Z(-3,x)$ $\displaystyle=\frac{x(1+4x+x^{2})}{(1-x)^{4}},\qquad
Z(-4,x)=\frac{x(1+x)(1+10x+x^{2})}{(1-x)^{5}},$ $\displaystyle Z(-5,x)$
$\displaystyle=\frac{x(1+26x+66x^{2}+26x^{3}+x^{4})}{(1-x)^{6}}.$
These examples seem to show
###### Lemma 7.
Suppose $\left|x\right|=1$ with $x\neq 1$. Then
$\displaystyle Z(0,x)+Z(0,x^{-1})=-1,$ (11)
and for every positive integer $m$,
$Z(-m,x)+(-1)^{m}Z(-m,x^{-1})=0.$ (12)
###### Proof.
We start from [Jonquière 1880]
$e^{-\pi is/2}Z(s,e^{i\theta})+e^{\pi
is/2}Z(s,e^{-i\theta})=\frac{(2\pi)^{s}}{\Gamma(s)}\zeta(1-s,\frac{\theta}{2\pi})$
(13)
in Milnor [M]. Putting $s=-m$ with $m=1,2,\dots$, we have
$e^{\pi im/2}Z(-m,e^{i\theta})+e^{-\pi im/2}Z(-m,e^{-i\theta})=0.$
∎
We remark that $Z(0,1)=\zeta(0)=-1/2$. In this sense, the formula (11) is
valid also for $x=1$.
### 2.3 An example
$Z(-1,e^{i\theta})=\frac{1}{(e^{-i\theta/2}-e^{i\theta/2})^{2}}=-\frac{1}{4\sin^{2}(\theta/2)}.$
(14)
and this shows
${\operatorname{Li}}_{-1}(e^{-i\theta})={\operatorname{Li}}_{-1}(e^{i\theta}),$
(15)
an even function in $\theta$.
### 2.4 Proof of Theorem 1(1) and analytic continuation
Now we consider regular elements in $SU(2)$. Suppose $0<\theta<\pi$. Then we
have, for ${\operatorname{Re}}(s)>1$,
$\displaystyle\zeta_{SU(2)}^{W}\left(s,\left(\begin{array}[]{cc}e^{i\theta}&0\\\
0&e^{-i\theta}\end{array}\right)\right)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{e^{in\theta}-e^{-in\theta}}{e^{i\theta}-e^{-i\theta}}\frac{1}{n}n^{-s}$
$\displaystyle=\frac{1}{e^{i\theta}-e^{-i\theta}}\sum_{n=1}^{\infty}\left(\frac{e^{in\theta}}{n^{s+1}}-\frac{e^{-in\theta}}{n^{s+1}}\right)$
$\displaystyle=\frac{1}{e^{i\theta}-e^{-i\theta}}\left\\{Z(s+1,e^{i\theta})-Z(s+1,e^{-i\theta})\right\\}$
$\displaystyle=\frac{1}{2i\sin\theta}\left\\{Z(s+1,e^{i\theta})-Z(s+1,e^{-i\theta})\right\\},$
and the right-hand side has meromorphic continuation to whole
$s\in\mathbf{C}$.
Note that we interpret
$\frac{\sin(n\theta)}{n\sin\theta}=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\theta=0,\\\ (-1)^{n-1}&\mbox{ if }\theta=\pi.\end{array}\right.$ (16)
### 2.5 Proof of Theorem 1(2); vanishing
For $g=\pm I_{2}$ and for positive even integer $m$, we obtain
$\zeta_{SU(2)}^{W}(-m,\pm I_{2})=0$ from $\zeta(-m)=0$.
For $g\neq\pm I_{2}$, suppose $0<\theta<\pi$. Then for a positive integer $m$,
we have
$\zeta_{SU(2)}^{W}(-m,g)=\frac{1}{2i\sin\theta}\left(Z(1-m,e^{i\theta})-Z(1-m,e^{-i\theta})\right).$
(17)
This is zero for even $m$ by the formula (12).
### 2.6 Proof of Theorem 1(2), first derivative
We see that
$\frac{1}{\Gamma(s)}=\frac{s(s+1)}{\Gamma(s+2)}$ (18)
shows that
$\frac{1}{\Gamma(s)}=-(s+1)+O((s+1)^{2}),\quad(s\to-1).$ (19)
We again start from the formula (13)
$e^{-\pi is/2}Z(s,x)+e^{\pi
is/2}Z(s,x^{-1})=\frac{(2\pi)^{s}}{\Gamma(s)}\zeta(1-s,\frac{\theta}{2\pi})$
with $x=e^{i\theta}$. Taking $\left.\frac{\partial}{\partial s}\right|_{s=-1}$
in this formula, we have
$\displaystyle i\frac{\partial Z}{\partial s}(-1,x)+(-i)\frac{\partial
Z}{\partial s}(-1,x^{-1})$ $\displaystyle\quad+(-\pi i/2)(i)Z(-1,x)+(\pi
i/2)(-i)Z(-1,x^{-1})=(2\pi)^{-1}(-1)\zeta(2,\frac{\theta}{2\pi}).$
Then
$\displaystyle i\times
2i\sin\theta\times\frac{\partial\zeta_{SU(2)}^{W}}{\partial s}(-2,g)=-\pi
Z(-1,e^{i\theta})-\frac{1}{2\pi}\zeta(2,\frac{\theta}{2\pi}),$ (20)
and
$\displaystyle 4\pi\sin\theta\times\frac{\partial\zeta_{SU(2)}^{W}}{\partial
s}(-2,g)=2\pi^{2}L(-1,e^{i\theta})+\zeta(2,\frac{\theta}{2\pi})$ (21)
We have
$\zeta(2,t)+\zeta(2,1-t)=\frac{\pi^{2}}{\sin^{2}(\pi t)}$ (22)
since the left-hand side is equal to
$\sum_{n=0}^{\infty}\frac{1}{(n+t)^{2}}+\sum_{n=0}^{\infty}\frac{1}{(n+1-t)^{2}}=\sum_{n=-\infty}^{\infty}\frac{1}{(n+t)^{2}}$
(23)
which is equal to the right-hand side. This shows
$\displaystyle 8\pi\sin\theta\times\frac{\partial\zeta_{SU(2)}^{W}}{\partial
s}(-2,g)=\zeta(2,\frac{\theta}{2\pi})-\zeta(2,1-\frac{\theta}{2\pi})>0$ (24)
since $\frac{\theta}{2\pi}<1-\frac{\theta}{2\pi}$.
### 2.7 Proof of Theorem 1(3)
$\zeta_{SU(2)}^{W}(-1,I_{2})=\zeta(-1)=-\frac{1}{12}$ (25)
and
$\displaystyle\zeta_{SU(2)}^{W}\left(-1,\left(\begin{array}[]{cc}e^{i\theta}&0\\\
0&e^{-i\theta}\end{array}\right)\right)$
$\displaystyle=\frac{Z(0,x)-Z(0,x^{-1})}{x-x^{-1}}$ (28)
$\displaystyle=\frac{-x}{(1-x)^{2}}=\frac{1}{4\sin^{2}(\theta/2)},$ (29)
where $x=e^{i\theta}$ for all $0<\theta\leq\pi$.
### 2.8 An average over the group
Let $G$ be a finite group. The normalized Haar measure $dg$ on $G$ is, by
definition,
$\int_{G}f(g)dg=\frac{1}{\left|G\right|}\sum_{g\in G}f(g).$ (30)
Then we see that, for all $s\in\mathbf{C}$,
$\displaystyle\int_{G}\zeta_{G}^{W}(s,g)dg=1,$ (31)
since the left-hand side is equal to
$\displaystyle=\sum_{\rho\in\widehat{G}}\left(\int_{G}{\operatorname{trace}}(\rho(g))dg\right)\deg(\rho)^{-s-1},$
(32)
where the average is non-zero only for the trivial representation $\rho$.
Now we consider the case where $G$ is a compact group which is not necessarily
a finite group. Again let $dg$ be the normalized Haar measure of $G$ so that
$\int_{G}dg=1$. We ask the value
$\int_{G}\zeta_{G}^{W}(s,g)dg.$ (33)
We can give some example;
$\displaystyle\int_{SU(2)}\zeta_{SU(2)}^{W}(-2,g)dg$ $\displaystyle=0,$ (34)
$\displaystyle\int_{SU(2)}\zeta_{SU(2)}^{W}(-1,g)dg$ $\displaystyle=1.$ (35)
The latter formula is proved by the Weyl integral formula;
$\displaystyle\int_{SU(2)}\zeta_{SU(2)}^{W}(-1,g)dg=\int_{0}^{\pi}\zeta_{SU(2)}^{W}(-1,\left(\begin{array}[]{cc}e^{i\theta}&0\\\
0&e^{-i\theta}\end{array}\right))\frac{2}{\pi}\sin^{2}\theta\ d\theta=1.$ (38)
### 2.9 $r=2$
We now discuss the properties of a generalization of Witten zeta functions
with several characters. We give a proof of Theorem 2.
###### Proof.
${\operatorname{trace}}(\rho(g_{1}))=\frac{x^{n}-x^{-n}}{x-x^{-1}},{\operatorname{trace}}(\rho(g_{2}))=\frac{y^{n}-y^{-n}}{y-y^{-1}}$
with $x=e^{i\theta_{1}}$, $y=e^{i\theta_{2}}$. In the cases $g_{2}=\pm I_{2}$,
we have
$\displaystyle\zeta^{W}_{SU(2)}(s,g_{1},I_{2})$
$\displaystyle=\zeta^{W}_{SU(2)}(s,g_{1}),$ (39)
$\displaystyle\zeta^{W}_{SU(2)}(s,g_{1},-I_{2})$
$\displaystyle=\zeta^{W}_{SU(2)}(s,-g_{1}),$ (40)
Then the problem on the special values is reduced to the case treated in
Theorem 1(2).
Now we may suppose $x,y\neq\pm 1$. Then
$\displaystyle\zeta^{W}_{SU(2)}(s,g_{1},g_{2})$
$\displaystyle=\frac{1}{(x-x^{-1})(y-y^{-1})}\sum_{n=1}^{\infty}\frac{(xy)^{n}+(x^{-1}y^{-1})^{n}-(xy^{-1})^{n}-(x^{-1}y)^{n}}{n^{s+2}}$
(41)
$\displaystyle=\frac{Z(s+2,xy)+Z(s+2,x^{-1}y^{-1})-Z(s+2,xy^{-1})-Z(s+2,x^{-1}y)}{(x-x^{-1})(y-y^{-1})}.$
This shows
$\displaystyle\zeta^{W}_{SU(2)}(-2,g_{1},g_{2})$
$\displaystyle=\frac{(Z(0,xy)+Z(0,x^{-1}y^{-1}))-(Z(0,xy^{-1})+Z(0,x^{-1}y))}{(x-x^{-1})(y-y^{-1})}$
$\displaystyle=0,$ (42)
where we have used the formula (11). ∎
### 2.10 $r=3$
By the similar computation, we obtain
$\displaystyle\zeta^{W}_{SU(2)}(s;g,g,g)$
$\displaystyle=\frac{Z(s+3,x^{3})-3Z(s+3,x)+3Z(s+3,x^{-1})-Z(s+3,x^{-3})}{(x-x^{-1})^{3}}.$
(43)
If $x=i$, then
$\zeta^{W}_{SU(2)}(-2;g,g,g)=\frac{4Z(1,-i)-4Z(1,i)}{(2i)^{3}}=\frac{-2\pi
i}{-8i}=\frac{\pi}{4}\neq 0.$
## 3 $SU(3)$
### 3.1 On analytic continuation
Let $G$ be a compact semisimple Lie group. Then the Witten zeta
$\zeta_{G}^{W}(s)$ has a meromorphic continuation to $\mathbf{C}$. This is a
special case of
$\sum_{m_{1},\dots,m_{r}\geq 1}Q(m_{1},\dots,m_{r})P(m_{1},\dots,m_{r})^{-s}.$
(44)
Analytic continuation of these zeta functions is discussed in [Mellin 1900],
[Mahler 1928].
### 3.2 A special value at a negative integer
Let $n$ be a positive integer. Let $M=2n+2$, and suppose
${\operatorname{Re}}(s)>-n-\frac{1}{2}+\frac{\varepsilon}{2}$, with
$\varepsilon>0$. By [Ma], we have
$\displaystyle\zeta_{SU(3)}^{W}(s)$ $\displaystyle=2^{s}\sum_{m,n\geq
1}\frac{1}{m^{s}n^{s}(m+n)^{s}}$ (45)
$\displaystyle=2^{s}\frac{\Gamma(2s-1)\Gamma(1-s)}{\Gamma(s)}\zeta(3s-1)$
$\displaystyle+2^{s}\sum_{k=0}^{M-1}(-1)^{k}\frac{s(s+1)\cdots(s+k-1)}{k!}\zeta(2s+k)\zeta(s-k)$
$\displaystyle+2^{s}\frac{1}{2\pi\sqrt{-1}}\int_{{\operatorname{Re}}(z)=2n+2-\varepsilon}\frac{\Gamma(s+z)\Gamma(-z)}{\Gamma(s)}\zeta(2s+z)\zeta(s-z)dz.$
Reminding
$\left.\frac{\Gamma(2s-1)}{\Gamma(s)}\right|_{s=-n}=(-1)^{n-1}\frac{n!}{2(2n+1)!},$
(46)
we can put $s=-n$ in this identity and obtain
$\displaystyle\zeta_{SU(3)}^{W}(-n)$
$\displaystyle=2^{-n}(-1)^{n-1}\frac{n!n!}{2(2n+1)!}\zeta(-3n-1)$
$\displaystyle+2^{-n}\sum_{k=0}^{2n}(-1)^{k}\frac{(-n)(1-n)\cdots(k-1-n)}{k!}\zeta(-2n+k)\zeta(-n-k)$
$\displaystyle+2^{-n}(-1)\frac{(-n)(1-n)\cdots(-1)\cdot 1\cdots
n}{(2n+1)!}\frac{1}{2}\zeta(-3n-1).$ (47)
This shows $\zeta_{SU(3)}^{W}(-n)=0$ for a positive odd integer $n$, since
$\zeta(-3n-1)=0$ and $\zeta(-2n+k)\zeta(-n-k)=0$ for $k=0,1,\dots,n$. On the
other hand, for a positive even integer $n$, we have
$\displaystyle\zeta_{SU(3)}^{W}(-n)$
$\displaystyle=-2^{-n}\frac{(n!)^{2}}{(2n+1)!}\zeta(-3n-1)$
$\displaystyle\qquad+2^{-n}\sum_{k=0}^{n}\binom{n}{k}\zeta(-2n+k)\zeta(-n-k)=0,$
(48)
where the last equality follows from the following lemma:
###### Lemma 8.
For a positive even integer $n$, we have
$\sum_{k+l=n,k,l\geq
0}\frac{1}{k!l!}\zeta(-n-k)\zeta(-n-l)=\frac{n!}{(2n+1)!}\zeta(-3n-1).$ (49)
Equivalently,
$\sum_{k+l=n,k,l\geq
0}\frac{1}{k!l!}\frac{B_{n+1+k}}{n+1+k}\frac{B_{n+1+l}}{n+1+l}=-\frac{n!}{(2n+1)!}\frac{B_{3n+2}}{3n+2}.$
(50)
This follows from [CW, Theorem 2] when we substitute $\alpha=\gamma=n-1$ and
$\delta=\varepsilon=1$.∎
This concludes the proof of Theorem 3.
## 4 The groups over $\mathbf{Z}_{p}$
### 4.1 $SL_{2}$
Let $p$ be an odd prime. We denote by $\mathbf{Z}_{p}$ the ring of integers in
the non-archimedean local field $\mathbf{Q}_{p}$. Jaikin-Zapirain [J] obtains
the following explicit formula:
$\zeta_{SL_{2}(\mathbf{Z}_{p})}^{W}(s)=Z_{0}(s)+Z_{\infty}(s),$ (51)
with
$\displaystyle Z_{0}(s)$ $\displaystyle=\zeta_{SL_{2}(\mathbf{F}_{p})}^{W}(s)$
$\displaystyle=1+2\left(\frac{p-1}{2}\right)^{-s}+2\left(\frac{p+1}{2}\right)^{-s}+\frac{p-1}{2}\left(p-1\right)^{-s}$
$\displaystyle\qquad+p^{-s}+\frac{p-3}{2}(p+1)^{-s},$ (52) $\displaystyle
Z_{\infty}(s)$
$\displaystyle=\frac{1}{1-p^{-s+1}}\left(4p\left(\frac{p^{2}-1}{2}\right)^{-s}+\frac{p^{2}-1}{2}(p^{2}-p)^{-s}\right.$
$\displaystyle\qquad\left.+\frac{(p-1)^{2}}{2}(p^{2}+p)^{-s}\right).$ (53)
This deduces
$\displaystyle Z_{0}(-2)$
$\displaystyle=p(p^{2}-1)=\left|SL_{2}(\mathbf{F}_{p})\right|=p(p+1)(p-1),$
(54) $\displaystyle Z_{\infty}(-2)$ $\displaystyle=-p(p^{2}-1),$ (55)
$\displaystyle Z_{0}(-1)$ $\displaystyle=p(p+1),$ (56) $\displaystyle
Z_{\infty}(-1)$ $\displaystyle=-p(p+1),$ (57) $\displaystyle Z_{0}(0)$
$\displaystyle=p+4,$ (58) $\displaystyle Z_{\infty}(0)$
$\displaystyle=-\frac{4}{p-1}-p-4.$ (59)
This shows
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})}^{W}(-2)$ $\displaystyle=0,$ (60)
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})}^{W}(-1)$ $\displaystyle=0,$ (61)
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})}^{W}(0)$
$\displaystyle=-\frac{4}{p-1},$ (62)
which concludes the proof of Theorem 4.
### 4.2 Congruence subgroups of $SL_{2}$
In this subsection, we assume that $p$ is an odd prime. By [AKOV], we obtain
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$
$\displaystyle=p^{3m+2}\frac{1-p^{-2-s}}{1-p^{1-s}}.$ (63)
This shows
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})[p^{m}]}^{W}(-2)$
$\displaystyle=0,$ (64)
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{p})[p^{m}]}^{W}(-1)$
$\displaystyle=-p^{3m+1}/(p+1).$ (65)
By taking an “absolute limit” $p\to 1$, we obtain
$\displaystyle\zeta_{SL_{2}(\mathbf{Z}_{1})[1^{m}]}^{W}(s)$
$\displaystyle=\frac{s+2}{s-1}.$ (66)
### 4.3 Congruence subgroups of $SL_{3}$ and $SU_{3}$
In this subsection, we assume that $p$ is a prime with $p\neq 3$. By [AKOV],
we have
$\displaystyle\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$
$\displaystyle=p^{8m}\frac{1+u(p)p^{-3-2s}+u(p^{-1})p^{-2-3s}+p^{-5-5s}}{(1-p^{1-2s})(1-p^{2-3s})},$
(67)
where $u(X)=X^{3}+X^{2}-X-1-X^{-1}$. We notice that it can be factorized as
$\displaystyle\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$
$\displaystyle=p^{8m}\frac{(1-p^{-2-s})(1-p^{-1-s})}{(1-p^{1-2s})(1-p^{2-3s})}$
$\displaystyle\times\left(1+(p^{-1}+p^{-2})p^{-s}+(1+p^{-1})p^{-2s}+p^{-2-3s}\right).$
(68)
We see that
$\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(-2)=\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(-1)=0.$
The formula (68) shows
$\lim_{p\to
1}\zeta_{SL_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)=\frac{(s+1)(s+2)}{(s-\frac{1}{2})(s-\frac{2}{3})},$
(69)
which is considered to be “an absolute Witten zeta function
$\zeta_{SL_{3}(\mathbf{Z}_{1})[1^{m}]}^{W}(s)$”.
Also by [AKOV],
$\displaystyle\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)$
$\displaystyle=p^{8m}\frac{1+u(p)p^{-3-2s}+u(p^{-1})p^{-2-3s}+p^{-5-5s}}{(1-p^{1-2s})(1-p^{2-3s})}$
(70)
$\displaystyle=p^{8m}\frac{(1-p^{-2-s})(1-p^{-s})(1+p^{-1-s})}{(1-p^{1-2s})(1-p^{2-3s})}$
$\displaystyle\quad\times\left(1+(1-p^{-1}+p^{-2})p^{-s}+p^{-2-2s}\right),$
(71)
where $u(X)=-X^{3}+X^{2}-X+1-X^{-1}$. This shows
$\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(-2)=\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(0)=0,$
while
$\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(-1)=2p^{8m-2}\frac{p-1}{p^{5}-1}=2p^{8m-2}\frac{1}{[5]_{p}}$
(72)
is non-zero where $[n]_{p}=\frac{p^{n}-1}{p-1}$ is a $p$-analogue of an
integer $n$. This shows
$\lim_{p\to 1}\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(-1)=\frac{2}{5}.$ (73)
By the formula (71), we have
$\lim_{p\to
1}\zeta_{SU_{3}(\mathbf{Z}_{p})[p^{m}]}^{W}(s)=\frac{s(s+2)}{(s-\frac{1}{2})(s-\frac{2}{3})}.$
## References
* [AKOV] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, On representation zeta functions of groups and a conjecture of Larsen-Lubotzky, C. R. Acad. Sci. Paris, Ser. I, 348 (2010) 363–367.
* [AKOV2] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of some compact p-adic analytic groups, arXiv:1011.6533.
* [C] H. B. G. Casimir, On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Acad. Wetenschap 51, 793–795 (1948).
* [CW] Venchang Chu and Chenying Wang, Convolution formulae for Bernoulli numbers, Integral Transforms and Special Functions, 21 no. 6, (2010), 437–457.
* [E1] L. Euler, De summis serierum reciprocarum (written in 1735), Commentarii academiae scientiarum Petropolitanae 7 (1740), 123–134; Opera Omnia: Series 1, Volume 14, pp. 73–86.
* [E2] L. Euler, Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques (written in 1749), Memoires de l’academie des sciences de Berlin 17 (1768) 83–106; Opera Omnia: Series 1, Volume 15, pp. 70–90.
* [H] S.W. Hawking, Zeta function regularization of path integrals in curved space time, Comm. Math. Phys. 55 (1977), 133–148.
* [J] A. Jaikin-Zapirain, Zeta function of representations of compact p-adic analytic groups. J. Amer. Math. Soc. 19 (2006), 91–118.
* [Jo] A. Jonquière, Note sur la séries $\sum_{n=1}^{\infty}\frac{x^{n}}{n^{s}}$, Bull. Soc. Math. France 17 (1889), 142–152.
* [L] M. Larsen, Determining a semisimple group from its representation degrees. Internat. Math. Res. Notes 2004 (2004), 1989–2016.
* [LL] M. Larsen and A. Lubotzky, Representation growth of linear groups. J. Eur. Math. Soc. (JEMS) 10 (2008), 351–390.
* [Mah] K. Mahler, Über einen Satz von Mellin, Math. Ann. 100 (1928), 384–398.
* [Ma] K. Matsumoto, On analytic continuation of various multiple zeta-functions, In “Number Theory for the Millennium II, Proc. of the Millennial Conference on Number Theory”, M. A. Bennett et al. (eds.), A K Peters, 2002, 417–440.
* [Me] H. Mellin, Eine Formal für den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Soc. Sci. Fenn., 29 (1900), no. 4.
* [M] J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. (2) 29 (1983), no. 3–4, 281–322.
* [W] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153–209.
* [Z] D. Zagier, Values of zeta functions and their applications. In First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel 1994, 497–512.
Nobushige KUROKAWA
Department of Mathematics, Tokyo Institute of Technology,
Oh-Okayama, Meguro, Tokyo, 152-8551, Japan.
[email protected]
Hiroyuki OCHIAI
Faculty of Mathematics, Kyushu University,
Motooka, Fukuoka, 819-0395, Japan.
[email protected]
|
arxiv-papers
| 2013-04-12T05:59:01 |
2024-09-04T02:49:44.253885
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nobushige Kurokawa and Hiroyuki Ochiai",
"submitter": "Hiroyuki Ochiai",
"url": "https://arxiv.org/abs/1304.3543"
}
|
1304.3558
|
# Warped Alternatives to Froggatt-Nielsen Models
Abhishek M Iyer [email protected] Centre for High Energy Physics,
Indian Institute of Science, Bangalore 560012 Sudhir K Vempati
[email protected] Centre for High Energy Physics, Indian Institute of
Science, Bangalore 560012
###### Abstract
We consider the Randall-Sundrum (RS) set-up to be a theory of flavour, as an
alternative to Froggatt-Nielsen (FN) models instead of as a solution to the
hierarchy problem. The RS framework is modified by taking the low energy brane
to be at the GUT scale. This also alleviates constraints from flavour physics.
Fermion masses and mixing angles are fit at the GUT scale. The ranges of the
bulk mass parameters are determined using a $\chi^{2}$ fit taking in to
consideration the variation in $\mathcal{O}(1)$ parameters. In the hadronic
sector, the heavy top quark requires large bulk mass parameters localising the
right handed top quark close to the IR brane. Two cases of neutrino masses are
considered (a) Planck scale lepton number violation and (b) Dirac neutrino
masses. Contrary to the case of weak scale RS models, both these cases give
reasonable fits to the data, with the Planck scale lepton number violation
fitting slightly better compared to the Dirac case. In the Supersymmetric
version, the fits are not significantly different except for the variation in
$\tan\beta$. If the Higgs superfields and the SUSY breaking spurion are
localized on the same brane then the structure of the sfermion masses are
determined by the profiles of the zero modes of the hypermultiplets in the
bulk. Trilinear terms have the same structure as the Yukawa matrices. The
resultant squark spectrum is around $\sim 2-3~{}\text{TeV}$ required by the
light Higgs mass to be around 125 GeV and to satisfy the flavour violating
constraints.
###### pacs:
73.21.Hb, 73.21.La, 73.50.Bk
## I Introduction
One of the celebrated solutions of the fermion flavour problem is the
Froggatt-Nielsen Mechanism FN . According to this prescription, the symmetry
group of the Standard Model (SM) is augmented by a horizontal $U(1)_{X}$ group
under which all the SM fermions and the Higgs field are charged. The effective
theory includes a flavon field $X$ and the Yukawa couplings are generated from
the higher dimensional operators which are invariant under the $U(1)_{X}$ and
the Standard Model (SM) gauge group. For example, the up-type quark mass
matrix has the form:
$Y^{u}_{ij}(\frac{X}{M_{Pl}})^{c_{Q_{i}}+c_{u_{j}}+c_{H_{u}}}Q_{i}H_{u}U_{j}$,
where $c_{f}$ is $U(1)_{X}$ charge of the $f$ field and $i,j$ are the
generation indices. The flavon field $X$ develops a vacuum expectation value
(vev) such that $0.22\approx\lambda_{c}\approx<X>/M_{Pl}$, $\lambda_{c}$ being
the Cabibbo angle. $Y^{u}_{ij}$ are taken to be $\mathcal{O}(1)$ parameters.
Fermion mass matrices including their mixing patterns can be fit to the data
by choosing appropriate $U(1)_{X}$ charges for various fields. An UV
completion of the model can be constructed by including heavy chiral fermions
in to the theory; integrating these heavy fields would lead to the relevant
non-renormalizable operators (for a review, see Babu ).
The $U(1)_{X}$ symmetry introduces additional anomalies in to the theory and
subsequently, strong constraints on the $U(1)_{X}$ charges for various fields.
In supersymmetric models with a single flavon field, one typically has to
resort to Green-Schwarz (GS) mechanism to cancel the anomalies. The solution
set of $U(1)_{X}$ charges for the fermions and the Higgs which satisfy the
fermion data as well as the anomaly cancellation111 However, with two singlet
flavons there exist a unique solution which is completely non anomalous
Dudas:U(1) requirement have been studied in Dudas:U(1) ; ibarra ; king ;
Ibanezross ; Binetruy1 ; Binetruy2 ; Chun ; Dreiner1 ; Dreiner2 ; King:2004tx
; Ellis ; Joshipura:2000sn and recently updated in lavignac . These models
typically lead to large flavour violations at the weak scale in gravity
mediated supersymmetry breaking models due to contributions from the
$U(1)_{X}$ D-terms. While the constraints from the flavour sector on the
available solutions are very tight, it may still be possible to ease them
without requiring the superpartner masses to be very high Lalak ; Varzielas .
The flavour constraints may also be alleviated to some extent by considering
$U(1)\times U^{\prime}(1)$ class of models Leurer ; flavour . In the present
work, we will study the extra-dimensional alternative ArkaniHamed to
understand the flavour hierarchy in particular concentrating on the
supersymmteric Randall-Sundrum (RS) set up.
The Randall-Sundrum framework RS which elegantly provides a solution to the
hierarchy problem via warping in the extra dimensional space can also thought
to be a theory of flavour. It has been observed sometime ago that the flavour
changing neutral currents (FCNC) can be suppressed due to the so-called RS-GIM
mechanism RSGIM . However, in the absence of additional flavour symmetries the
constraints from FCNC are still very strong(Huber ; Agashe ; Delaunay ;
Petriello ; AgasheSundrum ) (Detailed analysis for the hadronic sector can be
found in Neubert1 ; Neubert2 and references there in. For a recent thorough
analysis in the leptonic sector, please see iyer ). Given these strong
constraints on the RS set up at the weak scale, one can ask the question
whether RS is suitable to be a theory of flavour as well as a solution to the
hierarchy problem simultaneously. It might be that RS as a theory of a flavour
might be better suited at the GUT scale rather than at the weak scale. The
Froggatt-Nielsen models are typically defined at scales closer to the Planck
scale, so perhaps flavour physics might have its origins at the Planck scale.
With this rationale, in the present work we will consider RS to span between
the Planck and the GUT scales. The fermion masses are fit in terms of the bulk
mass parameters of the various fields, which take the role of the $U(1)_{X}$
charges of the FN mechanism. However, these parameters are less constrained
compared to the $U(1)_{X}$ charges, as no additional conditions such as
anomaly cancellations are required on them. While this has been the common
understanding, in Dudas it was pointed out that imposing unification
conditions on gauge couplings in a theory with localization of fermions or
hierarchical wave functions leads to strong constraints which are exactly in
the same way as the Green-Schwarz anomaly cancellation conditions Greenschwarz
222Typically applied in FN models, the Green-Schwarz anomaly cancellation
conditions requires the anomaly factors to be in a particular ratio such that
they are cancelled in String theory. In the present setup we do not impose
these conditions.
Extra dimensions at GUT scale were considered in Hallnomura while the RS
version was considered by the authors in choi1 ; choi2 and later by the
authors in Dudas . Our work, however, is very closely related to the work of
Brummer who have done a thorough analysis of fermion mass spectrum, weak
scale supersymmetric spectrum and flavour phenomenology, assuming a particular
Grand Unified Theory (GUT) model in such a RS setting. However, differences
exist. In the present work we have not assumed any specific GUT model.
Furthermore, we have used a frequentist approach to do the fermion mass
fitting. While this makes it hard to directly compare the results between the
two works, we hope they provide a complementary set of results. We also have
taken in to consideration the constraints from neutrino masses and mixing
angles which can have a significant effect on the lepton flavour violation and
slepton decays.
The equivalent description for the RS set up in four dimensions can be thought
of as a composite Higgs coupled to fermions with couplings which parameterise
the ‘partial compositeness’ of the fermionsRattazzi . In SUSY case, this
partial compositeness can also affect the structure of the soft masses.
In the first part of our work, our aim has been to provide a range of bulk
mass parameters which fit the fermion masses and mixing patterns at the GUT
scale. We believe this can be useful for model builders and other
phenomenologists working in flavour physics and looking for an alternative to
FN models. We have considered both supersymmetric as well as non
supersymmetric versions of the RS framework at the GUT scale while fitting the
data. The supersymmetric case has the added advantage that it could lead to
observable signatures at the weak scale. We consider the case where SUSY
breaking is considered to be on the same brane as where the Higgs is
localized, which is the GUT brane. In this case, the sfermion mass matrices
are determined by the zero mode profiles of the corresponding N=1 superfields
and thus the information of the fermion masses is propagated in to the soft
sector. It is far more striking for the A-terms which follow the same
structure as the Yukawa couplings. The spectrum is highly non-universal at the
high scale, but, its pattern is constrained due to the ranges of bulk mass
parameters which are in turn are fixed by their fits to fermion masses. The
running effects make the diagonal terms large at the weak scale.
The rest of the paper is organized as follows. In section II, we detail the RS
setup we consider and derive the structure of the fermion masses. In section
III we present the fermion mass fits and present the ranges for the bulk mass
parameters for both the non-supersymmetric and the supersymmetric cases. In
section IV we address the issue of supersymmetric breaking and derive
supersymmetric spectrum for a particular supersymmetric breaking case. We end
with summary and outlook in the last section. In Appendices A , Band C, we
have presented plots relevant for fermion mass fits.
## II RS as a theory of flavour
The Randall-Sundrum frame work consists of two branes separated by an single
warped extra dimension RS . The line element for the RS background is given as
$ds^{2}=e^{-2ky}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$ (1)
where $0\leq y\leq\pi R$. Here $y=0$ is identified as the position of the UV
brane and $y=\pi R$ is the position of the IR brane. The scale associated with
physics on the UV brane is $M_{Pl}$ while that on the IR brane is TeV. The
solution to the hierarchy problem is achieved by exponential warping of scales
i.e, $M_{Pl}=e^{kR\pi}~{}M_{\text{weak}}$, where $kR\sim\mathcal{O}(11)$.
In the modified set up we consider here, the scale associated with the IR
brane is $M_{GUT}$. This can be achieved by choosing $kR\sim 1.5$. We define
the hierarchy between the scales, $\epsilon$, for this scenario to be
$\epsilon=\frac{M_{GUT}}{M_{Planck}}\sim 10^{-2}$ (2)
We consider both supersymmetric and non-supersymmetric matter fields to
propagate in the bulk. Localisation of the respective zero modes is dependent
on the corresponding bulk masses. In both the cases, we assume that the Higgs
field (two Higgs fields in the case of supersymmetric models) are localised on
the GUT brane. We now proceed to briefly review the derivation of the mass
matrices and their dependence on the zero mode profiles in both supersymmetric
and non-supersymmetric cases. A couple of points are important to note at this
juncture. Firstly, the typical lowest KK mass for a warped background is given
as $m_{KK}=e^{-kR\pi}k$. In the present set up, the ‘large’ warp factor
ensures the lowest KK modes are very heavy i.e, $m_{KK}=\epsilon k\sim
M_{GUT}$ and thus are decoupled from low energy phenomenology. We do not
consider their effects in this work for low energy phenomenology. Secondly, it
turns out that the dependence of the zero mode mass matrices on the profiles
is very similar in supersymmetric and non-supersymmetric cases. However, the
fermion mass data at the high scale in the supersymmetric case could be
different from the SM one, due to the dependence on tan$\beta$ as well as the
different RGE for the Yukawa coupling as we will discuss in the next section.
### II.1 Standard Model case
The non-supersymmetric case or the Standard Model case has been studied in
many works Neubert1 ; Neubert2 ; Huber ; iyer . The main difference in the
present case is that while those studies have considered a
$kR\sim\mathcal{O}(11)$, while in the present case it is $\mathcal{O}(1)$. We
thus assume a grand desert from the weak scale to the GUT scale, where RS
framework sets in. No attempt is made to solve the hierarchy problem, but the
flavour problem has a solution in terms of the localisation of the fields in
an extra dimension at the GUT scale. We follow the notation of iyer and
present the final formulae for the Yukawa mass matrices. The details of the KK
expansion and the corresponding ortho-normal relations can be found in iyer
and references therein. The five dimensional action has the form:
$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{\text{kin}}+S_{\text{Yuk}}+S_{\nu}+S_{higgs}$ $\displaystyle S_{kin}$
$\displaystyle=$ $\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}~{}\left(~{}\bar{L}(i\not{D}-m_{L})L+\bar{E}(i\not{D}-m_{E})E+\ldots~{}\right)$
$\displaystyle S_{\text{Yuk}}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}\left(~{}Y_{U}\bar{Q}U\tilde{H}+Y_{D}\bar{Q}DH+Y_{E}\bar{L}EH\right)\delta(y-\pi
R)$ $\displaystyle S_{\nu}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int
dy~{}\sqrt{-g}\left(\frac{\mathbf{\kappa}}{\Lambda^{(5)}}LHLH~{}~{}\text{{or}}~{}~{}Y_{N}\bar{L}NH\right)\delta(y-\pi
R)$ (3)
where we used the standard notation with the $Q,U,D$ standing for the quark
doublets, up-type and down type singles respectively, $L$ and $E,N$ stand for
leptonic doublets and charged and neutral singlets respectively. $H$ stands
for the Higgs doublet with $\tilde{H}=i\sigma H^{\star}$. We have suppressed
the Higgs action in the above. Two specific ways for generating non-zero
neutrino masses are considered (a) by a higher dimensional term localised at
the GUT brane and (b) Dirac neutrino mass terms similar to the other fermions.
$\Lambda^{(5)}$ is the five dimensional reduced Planck scale $\sim 2\times
10^{18}$ GeV.
After the Kaluza-Klein (KK) reduction and imposing the orthonormal conditions,
we can derive the 4D mass matrices for the zero-modes of the fermion fields.
They have the form:
$\displaystyle({\mathcal{M}}_{F})_{ij}$ $\displaystyle=$
$\displaystyle\frac{v}{\sqrt{2}}({Y}_{F}^{\prime})_{ij}e^{(1-c_{i}-c^{\prime}_{j})kR\pi}~{}\xi(c_{i})~{}\xi(c^{\prime}_{j})\;\;\;;\;\;$
$\displaystyle\xi(c_{i})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(0.5-c_{i})}{e^{(1-2c_{i})\pi kR}-1}},$ (4)
where $F$ stands for all the Yukawa matrices $F=U,D,E$ and $N$, if the
neutrinos have Dirac masses. $c_{i}$ and $c^{\prime}_{j}$ represent the bulk
masses of the respective matter fields (second line of eq.(II.1)); defined as,
for example, $m_{E_{i}}=c_{E_{i}}k$. $i,j$ denote the generation indices. If
the neutrinos have Dirac masses then their mass matrix is given by Eq.(II.1).
In case the neutrinos attain their masses through higher dimensional operator,
the mass matrix is given by
$({\mathcal{M}}_{\nu})_{ij}=\frac{v^{2}}{2\Lambda^{(5)}}(\kappa^{\prime})_{ij}e^{(2-c_{L_{i}}-c_{L_{j}})kR\pi}\xi(c_{L_{i}})\xi(c_{L_{j}})$
(5)
In Eqs. (II.1, 5), we have defined $Y^{\prime}=kY$ and
$\kappa^{\prime}=2k\kappa$. These are dimensionless $\mathcal{O}(1)$
parameters of anarchical nature333 Note that the Yukawa couplings in Eq.(9)
are dimensionful, with mass dimensions -1.. Eq.(II.1) are used to fit all the
fermion mass data at the GUT scale i.e, up and down type quark masses and the
(Cabibbo-Kobayashi-Masakawa) CKM matrix, charged lepton masses, neutrino mass
differences and the corresponding PMNS mixing matrix. In the case neutrinos
get their masses through higher dimensional operator, Eq. (5) is used instead
to fit their mass differences and mixing angles.
### II.2 Supersymmetric case
In the supersymmetric case, the matter fermions are represented by hyper-
multiplets444 N=1 Supersymmetry in 5D has the particle content of N=2
Supersymmetry in 4D. The hypers can be expressed in N=1, 4D language as two
chiral superfields, where as the Vectors can be expressed as a vector and
chiral superfield Marti ; ArkaniHamed:2001tb . propagating in the bulk. In
terms of the 4D, N=1 SUSY language, they can be expressed as two N=1 chiral
multiplets, $\Phi,\Phi^{c}$. Following Marti ; Gherghetta1 , we write the 5D
action in terms of two chiral fields with a (supersymmetric) bulk mass term to
be
$S_{5}=\int d^{5}x\left[\int d^{4}\theta
e^{-2ky}\left(\Phi^{\dagger}\Phi+\Phi^{c}\Phi^{c\dagger}\right)+\int
d^{2}\theta
e^{-3ky}\Phi^{c}\left(\partial_{y}+M_{\Phi}-\frac{3}{2}k\right)\Phi\right]$
(6)
where $M_{\Phi}=c_{\Phi}k$ is the bulk mass. In writing the above, the radion
field is suppressed by taking its vacuum expectation value, $<Re(T)>=R$. The
super field $\Phi^{c}$ is taken to be odd under $Z_{2}$. Thus, only $\Phi$ has
a zero mode. Since we have a theory at the GUT scale, the KK modes can be
considered to be decoupled from theory. In the effective theory, the profile
of the zero mode of the $\Phi$ is determined byMarti
$\left(\partial_{y}-\left(\frac{3}{2}-c\right)k\right)f^{(0)}=0$ (7)
Thus $f^{(0)}=e^{(\frac{3}{2}-c)ky}$. The superscript (0) stands for the zero
mode, which we will drop subsequently555In the component form, the scalar
component and the fermion components of the chiral super field $\Phi$ have
different bulk masses. However, the solution for the profile for the scalar
and the fermion components turns out to be the same. . In this effective
theory, where the higher KK modes are completely decoupled, we can write the
effective 4-D Kähler terms for the $Z_{2}$ even zero modes as Dudas ; choi1 ;
choi2
$\displaystyle\mathcal{K}^{(4)}$ $\displaystyle=$ $\displaystyle\int
dy\left(e^{(1-2c_{q_{i}})ky}Q^{\dagger}_{i}Q_{i}+e^{(1-2c_{u_{i}})ky}U^{\dagger}_{i}U_{i}+e^{(1-2c_{d_{i}})ky}D^{\dagger}_{i}D_{i}+\ldots\right),$
(8)
where we have substituted for the profile solutions of (7). After integrating
over the extra dimension $y$, the terms in Eq.(8) pick up a factor
$Z_{F}=\frac{1}{(1-2c_{F})k}\left(\epsilon^{2c_{F}-1}-1\right)$ where
$F=Q,U,D,L,E$, as before. We choose to work in a basis in which the Kähler
terms are canonically normalized. We thus re-define the fields as
$\Phi\rightarrow\frac{1}{\sqrt{Z_{F}}}\Phi$.
The effective four dimensional MSSM Yukawa couplings are determined from the
superpotential terms written on the boundary. For Higgs localized on the IR
brane, keeping only the zero modes of the chiral superfields, the effective
four dimensional superpotential is given as Dudas ; Marti
$\displaystyle\mathcal{W}^{(4)}$ $\displaystyle=$ $\displaystyle\int
dye^{-3ky}\left(e^{(\frac{3}{2}-c_{q_{i}})ky}e^{(\frac{3}{2}-c_{u_{j}})ky}Y^{u}_{ij}H_{U}Q_{i}U_{j}+e^{(\frac{3}{2}-c_{q_{i}})ky}e^{(\frac{3}{2}-c_{d_{j}})ky}Y^{d}_{ij}H_{D}Q_{i}D_{j}\right.$
(9) $\displaystyle+$
$\displaystyle\left.e^{(\frac{3}{2}-c_{L_{i}})ky}e^{(\frac{3}{2}-c_{E_{j}})ky}Y^{E}_{ij}H_{D}L_{i}E_{j}+\ldots\right)\delta(y-\pi
R)$
The Higgs fields are canonically normalized as $H_{u,d}\rightarrow
e^{kR\pi}H_{u,d}$. In the canonical basis, after the fields have been
redefined the fermion mass matrices can be derived from Eq. (9) to be
$\displaystyle(\mathcal{M}_{F})_{ij}=\frac{v_{u,d}}{\sqrt{2}}Y^{\prime}_{ij}e^{(1-c_{i}-c^{\prime}_{j})kr\pi}\xi(c_{i})\xi(c^{\prime}_{j})$
(10)
where $c_{i},c^{\prime}_{j}$ denote the bulk mass parameters for various
fields. $\xi(c_{i})$ are defined in Eq.(II.1) The mass matrix in Eq.(10) can
be approximated as
$(\mathcal{M}_{F})_{ij}\sim\frac{v_{u,d}}{\sqrt{2}}\mathcal{O}(1)e^{(1-c_{i}-c^{\prime}_{j})kr\pi}$
where the $\xi(c)$ is absorbed into the $\mathcal{O}$(1) parameters
$Y^{\prime}$ and is now collectively referred to as $\mathcal{O}$(1). This is
true only as long as the $c$ parameter lies between 0 and 1. But as we have
seen earlier, in some realistic cases especially related to neutrino masses
and the top quark, the values of $|c|$ could be large to fit the data.
Redefining the $\mathcal{O}$(1) Yukawa by absorbing the c parameters would
shift the ranges of the $c$ parameters far away from what they are, especially
in the case where $|c|\geq 1$. As in the SM case, we define dimensionless
$\mathcal{O}(1)$ Yukawa couplings as $Y^{\prime}=2kY$ and are of anarchical
nature. While the Dirac masses for the neutrinos have the same structure as
the other fermion mass matrices, the higher dimensional operator has a
different form determined by the super-potential term
$\displaystyle\mathcal{W}^{(4)}=\int dy\delta(y-\pi
R)e^{-3\sigma(y)}\left(e^{(\frac{3}{2}-c_{L_{i}})ky}e^{(\frac{3}{2}-c_{L_{j}})ky}\frac{\kappa_{ij}}{\Lambda^{(5)}}H_{U}H_{U}L_{i}L_{j}\right)$
(11)
The neutrino mass matrix in this case is given as
$(\mathcal{M}_{\nu})_{ij}=\kappa^{\prime}_{ij}\frac{v_{u}^{2}sin^{2}(\beta)}{2\Lambda^{(5)}}e^{(2-c_{L_{i}}-c_{L_{j}})kR\pi}\xi(c_{L_{i}})\xi(c_{L_{j}})$
(12)
where $\kappa^{\prime}=2k\kappa$ is the dimensionless $\mathcal{O}$(1)
parameters. The fermion mass matrices carry the same form as in the SM and the
supersymmetric cases and thus their dependence on $c_{i}$ and $\mathcal{O}(1)$
parameters is the same.
## III Fermion Mass fits
From the previous section, we have seen that in addition to the bulk mass
parameters, the $\mathcal{O}(1)$ Yukawa parameters also play a role in fixing
the fermion masses and mixing angles. We fit the masses and the mixing angles
of the quark sector and the neutrino sector at the GUT scale for both the SM
and the supersymmetric cases. We will use a frequentist approach, i.e, we
minimise the $\chi^{2}$ function, which is defined as follows:
$\chi^{2}=\sum_{j=1}^{N}\left(\frac{y_{j}^{exp}-y_{j}^{theory}}{\sigma_{j}}\right)^{2}$
(13)
where, $y_{j}^{theory}$ is the theory number for the $j^{th}$ observable and
$y_{j}^{exp}$ is its corresponding number quoted by experiments with a
measurement uncertainty of $\sigma_{j}$. In the present case the theory
parameters are just the bulk mass parameters and the $\mathcal{O}(1)$ Yukawa
entries in supersymmetric and non-supersymmetric cases. We define
$0<\chi^{2}<10$ to be a good fit and we try to find regions in the parameters
space of bulk mass parameters and $\mathcal{O}(1)$ Yukawa parameters which
satisfy this condition666We will mention the results with lower $\chi^{2}$ at
relevant places.. The $\mathcal{O}(1)$ Yukawa parameters are varied between -4
and 4, with a lower bound of 0.08 on $|Y|$ to avoid unnaturally small Yukawa
parameters. As far as the bulk mass parameters are concerned, since they are
given as $ck$, we prefer to vary the $c$ parameters between $-1$ to $1$, so as
not to go beyond the 5D cut-off, $k$. This will remove any possible
inconsistencies in the theory due to non-perturbative Yukawa couplings.
However, as we will see it is not always possible to fit the data within this
range of $c$ parameters. We will mention the range chosen specifically for
each case.
The minimization of the $\chi^{2}$ function was performed using MINUIT minuit
. We can minimise the hadronic and the leptonic sectors independently as they
are dependent on different sets of parameters which are uncorrelated. The
methodology is similar to the ones used in fermion mass fitting in GUT models
Joshipura ; Altarelli and also the one used in iyer .
### III.1 Standard Model (SM) Case
In this section, we present the fits in the SM case. For the GUT scale values
of the quark and lepton masses and CKM mixing matrices, we use the results of
xing . In the analysis of xing , two loop RGE have been used to run the Yukawa
couplings of the up-type quarks, down-type quarks and charged leptons from the
weak scale all the way up to the GUT scale. For the neutrino data we used the
publicly available package REAP reap to compute the high scale values. The
masses of the SM fermions at the GUT scale used in our fits are presented in
Table 1. The CKM and PMNS mixing matrices are presented in Table 2.
Table 1: GUT scale masses of fermions for the SM case Mass | Mass | Mass | Mass squared Differences
---|---|---|---
(MeV ) | (GeV) | MeV | $eV^{2}$
$m_{u}=0.48^{+0.20}_{-0.17}$ | $m_{c}=0.235^{+0.035}_{-0.034}$ | $m_{e}=0.4696^{+0.00000004}_{-0.00000004}$ | $\Delta m^{2}_{12}=1.5^{+0.20}_{-0.21}\times 10^{-4}$
$m_{d}=1.14^{+0.51}_{-0.48}$ | $m_{b}=1.0^{+0.04}_{-0.04}$ | $m_{\mu}=99.14^{+0.000008}_{-0.0000089}$ | $\Delta m_{23}^{2}=4.6^{+0.13}_{-0.13}\times 10^{-3}$
$m_{s}=22^{+7}_{-6}$ | $m_{t}=74.0^{+4.0}_{-3.7}$ | $m_{\tau}=1685.58^{+0.19}_{-0.19}$ | -
Table 2: Mixing angles for the hadronic and the leptonic sector for the SM case mixing angles(CKM) | Mixing angles (PMNS)
---|---
$\theta_{12}=0.226^{+0.00087}_{-0.00087}$ | $\theta_{12}=0.59^{+0.02}_{-0.015}$
$\theta_{23}=0.0415^{+0.00019}_{-0.00019}$ | $\theta_{23}=0.79^{+0.12}_{-0.12}$
$\theta_{13}=0.0035^{+0.001}_{-0.001}$ | $\theta_{13}=0.154^{+0.016}_{-0.016}$
#### III.1.1 SM Quark Sector fits
The up and down mass matrices are given in terms of fermion mass matrix of
Eq.(II.1). The theory parameters which are varied simultaneously to minimise
the $\chi^{2}$ in Eq.(13) include: three $c_{Q_{i}}$, each of $c_{u_{i}}$ and
$c_{d_{i}}$ and 18 $\mathcal{O}(1)$ Yukawa parameters. We would expect that
the light quarks would be localised close to the UV brane ( $c>1/2$ ) and the
heavy quarks close to the IR brane ($c<1/2$). However, for this particular
range of $\mathcal{O}(1)$ Yukawa parameters, it is difficult to fit the data
for $|c|$ within unity. We thus enlarged the range for the $c$ parameters.
The range chosen for the scan of the $c$ parameters chosen is:
$-2<c_{Q_{1},Q_{2}}<4$, $-3<c_{Q_{3}}<1$ for the doublets.
$-2<c_{d_{1},d_{2},d_{3}}<3.5$, for the down type singlets and
$-2<c_{u_{1},u_{2}}<4$, $-4<c_{u_{3}}<1$ for the up type singlets. We fit the
quark masses and the CKM mixing angles at the GUT scale. The top quark is
definitely lighter at the GUT scale, but still we see that most of the points
that fit the data lie outside of $|c|~{}\leq~{}1$. This is evident from the
the negative values of the $c_{Q_{3}}$ and $c_{U_{3}}$ that fit the data.
The regions of $c$ parameter space which satisfy the constraint of
$0<\chi^{2}<10$ for the chosen scanning range are shown in Fig.(1) in Appendix
A and the ranges are outlined in Table[3]. We see that the first two
generation bulk mass parameters are concentrated on the positive $c$ values
where as the third generation, the doublet and more so the right handed top is
localised close to the GUT scale brane. Comparing these results with that of
the normal RS, we find that the masses for the light quark fields can be fit
with $c\sim 0.6-0.7$. This can be attributed to the large warping where
$0.5<c<1$ is sufficient to reproduce the masses for light quarks Hubershafi .
Table 3: Allowed range of $c$ parameters in the SM case. These parameters satisfy $0<\chi^{2}<10$ for the SM case. The corresponding figure is 1 in Appendix A. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{Q_{1}}$ | [0,3.0] | $c_{D_{1}}$ | [0.78,4] | $c_{U_{1}}$ | [-0.97,3.98]
$c_{Q_{2}}$ | [-1.95,2.36] | $c_{D_{2}}$ | [0.39,3.02] | $c_{U_{2}}$ | [-1.99,2.43]
$c_{Q_{3}}$ | [-3,1] | $c_{D_{3}}$ | [0.39,2.21] | $c_{U_{3}}$ | [-4,1.0]
#### III.1.2 SM Leptonic Mass fits
Unlike the quark case, the fits in the leptonic sector are far more difficult
and more constraining due to the small mass differences and the large mixing
in the neutrino sector. As mentioned, we will consider two different cases of
neutrino masses while fitting the leptonic data.
(a) LLHH higher dimensional operator
Planck scale lepton number violation is an interesting idea which manifests
itself with higher dimensional operator suppressed by the Planck scale. In
four dimensions such an operator generates too small neutrino mases. It is
typically used as a perturbation over an existing neutrino mass model
umashankar . If not, it needs an enhancement of $\mathcal{O}(10^{3}-10^{4})$
to be consistent with the data. In the standard RS framework close to the weak
scale with bulk fermions, this higher dimensional operator is still
constrained however for different reasons. While the neutrino masses can be
fit by placing the doublet fields $L$ close to the UV brane, the charged
lepton masses become very tiny unless the singlet fields (E) are placed deep
in the IRiyer . This leads to inconsistencies in the theory with large non-
perturbative Yukawa couplings. The question arises whether the situation
repeats itself when we consider the modified RS setup. This can be checked as
follows. The neutrino masses are generated by the higher dimensional operator
as given in Eq.(II.1). The corresponding neutrino mass matrix is given by
Eq.(5) while the mass matrix for the charged leptons is given by Eq.(II.1).
For simplicity assume $c_{L_{i}}=c_{L}\forall$ i. For $c_{L}<0.5$ the mass
matrix in Eq.(5) becomes
$m_{\nu}=\kappa^{\prime}\frac{v^{2}sin^{2}(\beta)}{2\epsilon\Lambda}(1-2c_{L})$
(14)
It is clear that $c_{L}\sim-4$ is required to get neutrino masses
$\mathcal{O}(0.04)$ eV for a warp factor for $\epsilon\sim 10^{-2}$. As
$c_{L}$ increases, beyond 0.5, this formula is no longer valid, the neutrino
masses become smaller and hence do not fit the neutrino mass data with
$\mathcal{O}(1)$ Yukawa couplings. Thus a mildly negative $c_{L}$ should be
able to fit the data without large inconsistencies.
A second enhancement can also come from the $\kappa^{\prime}$, which is the
corresponding $\mathcal{O}(1)$ Yukawa. With this in mind, we enhance the range
of the scanning of the Yukawa couplings from $0.08$ to $4$ to $0.08$ to $10$.
This would help us to accommodate $c_{L}$ values close to $\sim-1$. The final
scanning ranges we have chosen are: the doublets ($c_{L_{i}}$) are varied
between -1.5 and 0.5, while the charged singlets were scanned between 0 and 4.
The region of $c$ values which give a good fit to leptonic masses, i.e,
satisfying the constraint $0<\chi^{2}<10$, is presented in Table[4]. The plots
for these ranges of $c$ values are presented in Figs.(2) in Appendix A.
Table 4: Ranges for scanned regions of the bulk leptonic parameters for the LLHH in the SM case which satisfy $0<\chi^{2}<10$. parameter | range | parameter | range
---|---|---|---
$c_{L_{1}}$ | [ -1.5,-1.15] | $c_{E_{1}}$ | [2.8,4.0]
$c_{L_{2}}$ | [-1.5,-0.97] | $c_{E_{2}}$ | [1.8,2.4]
$c_{L_{3}}$ | [-1.5,-1.22] | $c_{E_{3}}$ | [1.2,1.69]
(b)Dirac type Neutrinos
The case of Dirac neutrinos is interesting possibility though it requires
imposition of a global lepton number conservation777In fact, it is possible to
hide lepton number violation in this case through a careful location of the
right handed fermion fields planckgher . We will not consider this case here..
The running of the neutrino masses from the weak scale to high scale is
different in this case. However with a normal hierarchy of neutrinos and low
tan$\beta$ the differences are insignificantxing2 ; xing3 .
Assuming that there is not much of a difference for normal hierarchy, we
choose the following scanning range for the $c$ parameters. The doublets
($c_{L_{i}}$) and charged lepton singlets ($c_{E_{i}}$) are scanned within the
range -1 to 4.5, while the neutrino singlets were scanned in the range 3.5 to
9. Such a large value of the bulk mass parameters for the singlets is needed
to suppress the corresponding neutrino masses sufficiently. The
$\mathcal{O}$(1) Yukawa parameters were varied between 0.08 and 4. Comparing
the results of Dirac neutrino mass fits with that of the weak scale RS
models,iyer , we find that the $c_{N}$ are roughly a factor $7-8$ larger
compared to the $c_{N}$ at the weak scale. This is purely because of the
weaker warp factor we are considering in the present case. Increasing the
range of the $O(1)$ Yukawa parameters would only make things worse. The ranges
for the $c$ values corresponding to SM fits with Dirac neutrinos case are
presented Table[5]. The plots for the $c$ parameters are presented case in
Fig[3] in Appendix A.
Table 5: Ranges for the scanned regions of the bulk leptonic parameters for the Dirac case which satisfy $0<\chi^{2}<10$ for the SM case. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{L_{1}}$ | [ -1,2.9] | $c_{E_{1}}$ | [0.39,3.62] | $c_{N_{1}}$ | [5.29,8.97]
$c_{L_{2}}$ | [-0.99,2.7] | $c_{E_{2}}$ | [-1.0,2.63] | $c_{N_{2}}$ | [5.31,8.99]
$c_{L_{3}}$ | [-0.99,1.98] | $c_{E_{3}}$ | [-0.99,1.93] | $c_{N_{3}}$ | [5.12,8.97]
### III.2 Supersymmetric Case
The analysis for the case with bulk supersymmetry is similar to the SM case.
The GUT scale values are derived using the supersymmetric RGE at the two loop
instead of the SM ones. For the neutrinos however, one loop RGE were used with
experimental inputs at the weak scale. The running of the masses are not
dependent on the mixing angles for a low tan$\beta$. Supersymmetry threshold
corrections can play an important role while deriving the running masses.
Running masses in the supersymmetric framework were obtained using the
relevant matching conditions. As is well known, these effects are significant
at large tan$\beta$ and the corrections to the neutrino running through
$Y_{D}$ and $Y_{E}$ were considered Antusch .
The GUT scale masses and mixings chosen for the scan corresponded to
tan$\beta=10$ and are given in Table[6] and [7]. The results of the the scan
i.e, the ranges for the $c$ parameters are weakly dependent on tan$\beta$ and
can be applied for studying phenomenology for up to tan$\beta\sim 25$.
Table 6: GUT scale Masses with supersymmetry for tan$\beta=10$ Mass | Mass | Mass | Mass squared Differences
---|---|---|---
(MeV ) | (GeV) | MeV | $eV^{2}$
$m_{u}=0.49^{+0.20}_{-0.17}$ | $m_{c}=0.236^{+0.037}_{-0.036}$ | $m_{e}=0.28^{+0.0000007}_{-0.0000007}$ | $\Delta m^{2}_{12}=1.6^{+0.20}_{-0.21}\times 10^{-4}$
$m_{d}=0.70^{+0.31}_{-0.31}$ | $m_{b}=0.79^{+0.04}_{-0.04}$ | $m_{\mu}=59.9^{+0.000005}_{-0.000005}$ | $\Delta m_{23}^{2}=3.2^{+0.13}_{-0.13}\times 10^{-3}$
$m_{s}=13^{+4}_{-0.4}$ | $m_{t}=92.2^{+9.6}_{-7.8}$ | $m_{\tau}=1021^{+0.1}_{-0.1}$ | -
Table 7: Mixing angles for the quarks and leptons at GUT scale with supersymmetry for tan$\beta=10$ mixing angles(CKM) | Mixing angles (PMNS)
---|---
$\theta_{12}=0.226^{+0.00087}_{-0.00087}$ | $\theta_{12}=0.59^{+0.02}_{-0.015}$
$\theta_{23}=0.0415^{+0.00019}_{-0.00019}$ | $\theta_{23}=0.79^{+0.12}_{-0.12}$
$\theta_{13}=0.0035^{+0.001}_{-0.001}$ | $\theta_{13}=0.154^{+0.016}_{-0.016}$
#### III.2.1 Quark Case
The range chosen for the scan are the same as that for the SM case i.e.
$-2<c_{Q_{1},Q_{2}}<4$, $-3<c_{Q_{3}}<1$ for the doublets.
$-2<c_{d_{1},d_{2},d_{3}}<3.5$, for the down type singlets and
$-2<c_{u_{1},u_{2}}<4$, $-4<c_{u_{3}}<1$ for the up type singlets. The regions
of $c$ parameter space which satisfy the constraint of $0<\chi^{2}<10$ for the
chosen scanning range are shown in Fig.(4) in Appendix B and the ranges are
outlined in Table[8].
Table 8: Ranges for the scanned regions of bulk hadronic parameters which satisfy $0<\chi^{2}<10$ for the supersymmetric case. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{Q_{1}}$ | [-0.16,3.12] | $c_{D_{1}}$ | [-0.5,4] | $c_{U_{1}}$ | [-1.6,4.0]
$c_{Q_{2}}$ | [-1.32,2.34] | $c_{D_{2}}$ | [-1.9,2.5] | $c_{U_{2}}$ | [-2,2.4]
$c_{Q_{3}}$ | [-3,1] | $c_{D_{3}}$ | [-2,1.7] | $c_{U_{3}}$ | [-4,1.0]
#### III.2.2 Leptonic case
Similar to the SM scenario two cases of neutrino mass generation are
considered. The GUT scale input values for the $\chi^{2}$ is given in Table[6]
and [7].
(a)LLHH case
The results of the scan of the LLHH case is very similar for both the SM case
and the supersymmetric case. The expression for the neutrino mass matrix is
given in Eq.(12). For the neutrino sector we allow the $\mathcal{O}$(1) Yukawa
coupling to vary between -10 and 10 with a minimum of 0.08 while that for the
charged leptons are varied between -4 and 4 with a minimum of 0.08. The
doublets were scanned between -1.5 and 0.5 while the charged singlets were
scanned between 0 and 4. The ranges for the $c$ parameters for the LLHH case
for the chosen scanning range satisfying the constraint $0<\chi^{2}<10$, is
presented in Table[9] and the plots for the $c$ values are presented in
Figs.(5) in Appendix B.
Table 9: Ranges for scanned regions of the bulk leptonic parameters for the LLHH scenario in the supersymmetric case which satisfy $0<\chi^{2}<10$. parameter | range | parameter | range
---|---|---|---
$c_{L_{1}}$ | [ -1.5,-0.22] | $c_{E_{1}}$ | [2.6,3.7]
$c_{L_{2}}$ | [-1.5,0.08] | $c_{E_{2}}$ | [2.0,2.57]
$c_{L_{3}}$ | [-1.5,0.04] | $c_{E_{3}}$ | [1.1,1.8]
(b)Dirac Neutrinos
The expression for the mass matrix for the all the leptons is given by Eq.(10)
The scanning range for the $c$ values of all the doublets and charged lepton
singlets was in the range -1 to 4.5, while the neutrino singlets were scanned
in the range 3.5 to 9. The magnitude of $\mathcal{O}$(1) Yukawa parameters
were varied between 0.08 and 4. The regions of the $c$ parameters satisfying
the constraint $0<\chi^{2}<10$ for the scanned ranges are presented in
Table[10]. The ranges are presented in Fig.(6) in Appendix B.
Table 10: Ranges for the scanned regions of the bulk leptonic parameters for the Dirac case with supersymmetry which satisfy $0<\chi^{2}<10$ for the supersymmetric case. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{L_{1}}$ | [ -1,2.6] | $c_{E_{1}}$ | [-0.86,3.46] | $c_{N_{1}}$ | [5.68,8.9]
$c_{L_{2}}$ | [-0.99,2.21] | $c_{E_{2}}$ | [-1,2.24] | $c_{N_{2}}$ | [5.67,8.99]
$c_{L_{3}}$ | [-1,1.54] | $c_{E_{3}}$ | [-1,1.49] | $c_{N_{3}}$ | [5.64,8.99]
To summarize, on comparing the SM and the SUSY fits, we find that within a
given generation, the fields have a tendency to be localized slightly towards
the IR for the SUSY case than for the SM case. This effect is more pronounced
in the down sector and increases with tan$\beta$. A comparison between the
fits for the SM case and the SUSY case for $tan\beta=10$ and $50$ are
presented in Figs.[7] in Appendix C. The underlying features of the fit in
which the first two generations including the neutrinos are elementary from
the ADS/CFT point of view while the third generation fermions $(t_{L},t_{R})$
having a tendency to be partially composite or composite, is maintained for
both the SM and the SUSY case. From the choice of the $c$ parameters, we find
that the LLHH case admits a better fit to the neutrino data than the Dirac
case. This is contrary to the observations made in iyer in normal RS where
the $c$ parameters for all the leptons were close to unity. It thus offered a
more viable alternative than the LLHH case. In order to compensate for the
weak warp factor in the Dirac case the right handed neutrinos had bulk masses
$c_{N}\sim 7$. This weak warping however, works in favour of the LLHH case
where for $c<0.5$ the effective 4D suppression scale is of the
$\mathcal{O}$($M_{GUT}$) resulting in fits with $c$ parameters closer to
unity.
## IV SUSY Spectrum and Flavour Phenomenology
There are several ways to break supersymmetry within this RS set up at the GUT
scale (see for example discussion in choi1 ; choi2 ; Dudas ; Brummer ; nomura
. In the present work, we will consider only one particular set up which
manifestly demonstrates the flavour structure of the fermions within the soft
terms. More detailed analysis of supersymmetric spectrum will be addressed in
iyer2 . We will assume in the following that supersymmetric breaking happens
on the IR brane, or the GUT brane. Unlike the work of Hallnomura and Brummer
we will not arrange the SM fields in any particular GUT representation. As has
been discussed in these works, a GUT structure can be arranged with possible
solutions for proton decay and doublet triplet splitting. Instead we
parameterize SUSY breaking in terms of a single four dimensional spurion
chiral superfield, $X=\theta^{2}F$, which is localized on the GUT brane.
However, for soft masses generated at the Planck brane as in Dudas , one may
then impose the GS anomaly cancellation conditions on bulk masses to ensure
unification of couplings at the Planck scale. We do not impose any such
conditions as the soft masses are generated at the GUT scale.
In the limit, the higher KK modes are decoupled from the GUT scale physics
Marti , the Kähler potential relevant for the scalar mass terms is given by
$\mathcal{K}^{(4)}=\int dy\delta(y-\pi R)e^{-2k\pi
R}k^{-2}X^{\dagger}X\left(\beta_{q,ij}Q^{\dagger}_{i}Q_{j}+\beta_{u,ij}U^{\dagger}_{i}U_{j}+\beta_{d,ij}D^{\dagger}_{i}D_{j}+\gamma_{u,d}H_{u,d}^{\dagger}H_{u,d}+\ldots\right)$
(15)
where $\beta$ have dimensional carrying negative mass dimensions of -1 ( as
the matter fields are five dimensional). $\gamma_{u,d}$ are $\mathcal{O}(1)$
parameters.
The sfermion mass matrix is generated when the $X$ fields get a vacuum
expectation value $m_{\tilde{f}}^{2}\sim k^{-2}<X>^{\dagger}<X>Q^{\dagger}Q$.
The mass matrix will however not be diagonal in flavour space. In the
canonical basis, (15), the mass matrices take the form
$(m_{\tilde{f}}^{2})_{ij}=m_{3/2}^{2}~{}\hat{\beta}_{ij}~{}e^{(1-c_{i}-c_{j})kR\pi}\xi(c_{i})\xi(c_{j})$
(16)
where $\hat{\beta}_{ij}=2k\beta_{ij}$ are dimensionless $\mathcal{O}(1)$
parameters. $\xi(c_{i})$ are defined in Eq.(II.1). And the gravitino mass is
defined as
$m_{3/2}^{2}={<F>^{2}\over k^{2}}={<F>^{2}\over M_{Pl}^{2}}$ (17)
The Higgs fields are localised on the GUT brane, their masses are given by
$m^{2}_{H_{u},H_{d}}=\gamma_{u,d}~{}m_{3/2}^{2}$.
The A-terms are generated from the higher dimensional operators in the super
potential of the type :
$W^{(4)}=\int dy\delta(y-\pi
R)e^{-3ky}k^{-1}X\left(\tilde{A}^{u}_{ij}H_{u}Q_{i}u_{j}+\tilde{A}^{d}_{ij}H_{d}Q_{i}d_{j}+\tilde{A}^{e}_{ij}H_{d}L_{i}E_{j}+\ldots\right)$
(18)
where the $\tilde{A}$ are dimensionful parameters having mass dimension -1.
Substituting for the vev of the $X$, we have for the four dimensional
trilinear couplings at the GUT scale:
$A^{u,d}_{ij}=m_{3/2}A^{\prime}_{ij}e^{(1-c_{i}-c^{\prime}_{j})kR\pi}\xi(c_{i})\xi(c^{\prime}_{j})$
(19)
where we defined the dimensionless $\mathcal{O}(1)$ parameters as
$A^{\prime}=2k\tilde{A}$. The structure of the A terms and the corresponding
fermion mass matrix are similar and they differ only by the choice of the
$\mathcal{O}$(1) parameters. Choosing $A^{\prime}=2kY^{\prime}$, makes the
down sector A terms diagonal in the mass basis of the fermions at the GUT
scale. Henceforth, we shall work in this basis, with the $\mathcal{O}$(1)
parameters of the A terms proportional to the $\mathcal{O}$(1) Yukawa
parameters.
The masses for the gauginos are obtained from the following operator in the
lagrangian
$\mathcal{L}=\int d^{2}\theta
k^{-1}X\mathcal{W}_{A\alpha}\mathcal{W}^{\alpha}_{A}$ (20)
At $M_{GUT}$ their masses will be be $m_{1/2}=fm_{3/2}$ where $f$ is a
$\mathcal{O}(1)$ parameter. $m_{1/2}$ will be treated as an independent
parameter. They are independent of the position of localization of X. as the
profile for the gauginos is flat corresponding to a bulk mass parameter of 0.5
Marti ; Gherghetta1 .
While the above equations set the boundary conditions at the high scale, the
weak scale spectrum is determined by the RGE evolution. In the present case,
the spectrum at the high scale is completely non-universal as determined by
the profiles of the zero modes of the matter chiral superfields. The structure
of soft terms discussed here is similar to the ideas of flavourful
supersymmetry discussed by Nomura1 ; Nomura2 and more recently by Ramond . In
the following we will present two example points one for the LHLH higher
dimensional operator case and another for the Dirac case.
To begin with, in both the examples, we consider that all the $\mathcal{O}(1)$
parameters appearing in the definitions of the soft parameters are
proportional to the unit matrix. We will explicitly mention any deviations as
required by the phenomenology when presenting numerical examples. This would
mean that the matrices, $A^{\prime},\hat{\beta}$ in Eqs. (16, 19) are
proportional to unit matrix and the parameters $\gamma_{u},\gamma_{d}$ in
Eq.(15) are equal to one. However, as we will see below, they play an
important role in low energy phenomenology and one might frequently require to
vary them within the $\mathcal{O}(1)$ range, to satisfy phenomenological
constraints. While studying the flavour phenomenology, we make sure that the
soft terms are present in the super-CKM basis. The low-energy spectrum has
been computed numerically using the spectrum generator SUSEFLAV suseflav .
#### IV.0.1 LHLH operator case
In this case we consider the following point, (21), in the $c$ parameter
space. It has a $\chi^{2}$ of $5.5$ for the hadronic sector and $0.7341$ for
the leptonic sector. As expected it has a mostly composite right handed top
quark. In addition, the leptonic doublets are also significantly composite in
this case.
$\displaystyle c_{Q_{1}}=2.740\hskip 14.22636ptc_{D_{1}}=0.722\hskip
14.22636ptc_{U_{1}}=0.4024\hskip 14.22636ptc_{L_{1}}=-1.497\hskip
14.22636ptc_{E_{1}}=3.634$ $\displaystyle c_{Q_{2}}=1.920\hskip
14.22636ptc_{D_{2}}=0.729\hskip 14.22636ptc_{U_{2}}=0.0652\hskip
14.22636ptc_{L_{2}}=-0.224\hskip 14.22636ptc_{E_{2}}=2.290$ $\displaystyle
c_{Q_{3}}=0.960\hskip 14.22636ptc_{D_{3}}=0.801\hskip
14.22636ptc_{U_{3}}=-3.5615\hskip 14.22636ptc_{L_{3}}=-1.0738\hskip
14.22636ptc_{E_{3}}=1.769$ (21)
The choice of $\mathcal{O}$(1) parameters in the soft sector plays a role in
determining the nature of the low energy spectrum. For a given set of $c$
parameters, a naive choice of one for all the $\mathcal{O}$(1) parameters in
the soft sector may or may not lead to an acceptable spectrum at $M_{susy}$.
For the LLHH case corresponding to the choice in Eq.(21) the $\mathcal{O}$(1)
parameters for all the soft masses are taken to be 1. The $\mathcal{O}$(1)
parameters for the A terms are chosen to be $\hat{A}^{u}=1.02Y^{{}^{\prime}u}$
while $\hat{A}^{u}=Y^{{}^{\prime}d}$ and $\hat{A}^{e}=0.6Y^{{}^{\prime}e}$.
Corresponding to these choices of the $\mathcal{O}$(1) parameters and the $c$
values in Eq.(21), the soft breaking terms at the GUT scale in $GeV$ are given
as:
$m_{Q}=\begin{bmatrix}0.001&-0.03&-0.27\\\ -0.03&0.85&7.6\\\
-0.27&7.6&68.7\end{bmatrix};m_{U}=\begin{bmatrix}11.5&-63.5&156.1\\\
-63.5&349.6&-859.1\\\
156.1&-859.1&2110.8\end{bmatrix};m_{D}=\begin{bmatrix}105.03&-90.4&155.2\\\
-90.4&77.8&-133.6\\\ 155.2&-133.6&229.5\end{bmatrix}$
$A_{U}=\begin{bmatrix}-0.002&-0.09&-0.84\\\ -0.0002&1.09&-6.3\\\
-10^{-6}&0.01&439.4\end{bmatrix};A_{D}=\begin{bmatrix}0.03&0&0\\\ 0&-0.40&0\\\
0&0&-40.6\end{bmatrix};m_{L}=\begin{bmatrix}7.7&8.8&-147.5\\\
8.8&9.9&-167.0\\\ -147.5&-167.0&2798.4\end{bmatrix}$
$m_{E}=\begin{bmatrix}0.001&-0.017&0.05\\\ -0.017&0.26&-0.89\\\
0.05&-0.89&3.04\end{bmatrix};A_{E}=\begin{bmatrix}0.008&0&0\\\ 0&1.81&0\\\
0&0&-31.3\end{bmatrix}$ (22)
A couple of interesting features of the above spectrum are (i) at least one of
the soft masses is tachyonic (ii) significant amount of flavour violation
present at the high scale. However at the weak scale, things are significantly
different. This is because the RG running is quite different for the diagonal
terms compared to the off-diagonal ones. In fact, the off-diagonal entries
barely run, where as the corrections to the diagonal ones are quite
significant. As an illustration, consider the slepton mass matrix at the weak
scale. The analytic output at the weak scale for the diagonal terms can be
approximated as
$\displaystyle\tilde{M}^{2}_{L_{1,2}}\simeq m_{L_{1,2}}^{2}+0.5M_{1/2}^{2}$
(23) $\displaystyle\tilde{M}^{2}_{L_{3}}\simeq m_{L_{3}}^{2}+0.5M_{1/2}^{2}$
$\displaystyle\tilde{M}^{2}_{E_{1,2}}\simeq m_{E_{1,2}}^{2}+0.15M_{1/2}^{2}$
$\displaystyle\tilde{M}^{2}_{E_{3}}\simeq m_{E_{3}}^{2}+0.15M_{1/2}^{2}$
which receive gauge contributions while the off diagonal elements do not. The
A terms are not large enough to make the off diagonal elements of the soft
mass matrices comparable with the diagonal terms. An example for the sleptons
for the case under consideration is given as
$\displaystyle m_{L}^{2}(m_{susy})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}2.2\times 10^{5}&77.0&-2.1\times 10^{4}\\\
77.0&2.2\times 10^{5}&-2.7\times 10^{4}\\\ -2.1\times 10^{4}&-2.7\times
10^{4}&7.7\times 10^{6}\end{bmatrix}\text{GeV}^{2}$ $\displaystyle
m_{E}^{2}(m_{susy})$ $\displaystyle=$ $\displaystyle\begin{bmatrix}1.1\times
10^{6}&3.4\times 10^{-4}&2.2\times 10^{-1}\\\ 3.4\times 10^{-4}&1.1\times
10^{6}&5.9\times 10^{1}\\\ 2.2\times 10^{-1}&5.9\times 10^{1}&5.4\times
10^{5}\end{bmatrix}\text{GeV}^{2}$ (24)
We see that the off-diagonal entry has barely enhanced where as the diagonal
entries have been significantly modified. Constraints from flavour violation
would restrict the mass scales of $m_{3/2}$ and $M_{1/2}$. The most stringent
constraints are from the transitions between the first two generations ie from
$K^{0}\to\bar{K}^{0}$ and $\mu\to e+\gamma$. The expressions for the
$K_{L}-K_{S}$ mass difference and the branching fractions for $\mu\to
e+\gamma$ can be found in Gabbiani ; Gabbiani1 . We impose the flavour
constraints from all the existing data on the $\delta$ parameters.
Bounds from the flavour violating processes are obtained using the mass
insertion approximation Gabbiani and the results of vempati defined in the
Super-CKM basis. The flavour violating indices are defined as
$\delta_{ij}(i\neq j)={(U^{\dagger}M^{2}_{soft}U)_{ij}\over m_{susy}}(i\neq
j)$ are evaluated in the basis in which the down sector is diagonal, with U
being the rotation matrix which rotates the corresponding fermion mass matrix.
The $\delta$ are evaluated at the weak scale and we scale the bounds of
vempati to the present mass scales. In Table[12] we present the low energy
spectrum corresponding to the sample point in Eq.(21). The low energy
$\delta^{\prime}s$ are presented in Table 13.
Table 11: Experimental upper bounds on the $\delta^{down}$ obtained for $\tilde{m}_{q}=2.1$ TeV and $\tilde{m}_{l}=0.7$ TeV (i,j) | $|\delta^{Q}_{LL}|$ | $|\delta^{L}_{LL}|$ | $|\delta^{D}_{LR}|$ | $|\delta^{E}_{LR}|$ | $|\delta^{D}_{RL}|$ | $|\delta^{E}_{RL}|$ | $|\delta^{D}_{RR}|$ | $|\delta^{E}_{RR}|$
---|---|---|---|---|---|---|---|---
12 | 0.053 | $0.0002$ | $0.0003$ | $3.8\times 10^{-6}$ | $0.0003$ | $3.8\times 10^{-6}$ | 0.03 | 0.03
13 | $0.34$ | 0.14 | $0.06$ | $0.03$ | 0.06 | $0.03$ | 0.26 | -
23 | 0.61 | $0.16$ | $0.01$ | 0.04 | $0.02$ | 0.04 | 0.84 | -
Table 12: Soft spectrum for LLHH case: $m_{susy}=1.06$ TeV, $m_{\tilde{g}}=2.64$ TeV, $\mu=3.43$TeV, $tan\beta=25$ Parameter | Mass(TeV) | Parameter | Mass(TeV) | Parameter | Mass(TeV) | Parameter | Mass(Tev) | Parameter | Mass(TeV)
---|---|---|---|---|---|---|---|---|---
$\tilde{t}_{1}$ | 0.47 | $\tilde{b}_{1}$ | 1.01 | $\tilde{\tau}_{1}$ | 0.726 | $\tilde{\nu}_{\tau}$ | 2.78 | $N_{1}$ | 0.465
$\tilde{t}_{2}$ | 1.05 | $\tilde{b}_{2}$ | 2.14 | $\tilde{\tau}_{2}$ | 2.79 | $\tilde{\nu}_{\mu}$ | 0.483 | $N_{2}$ | 0.929
$\tilde{c}_{R}$ | 2.24 | $\tilde{s}_{R}$ | 2.40 | $\tilde{\mu}_{R}$ | 0.478 | $\tilde{\nu}_{e}$ | 0.469 | $N_{3}$ | 3.38
$\tilde{c}_{L}$ | 2.48 | $\tilde{s}_{L}$ | 2.48 | $\tilde{\mu}_{L}$ | 1.05 | - | - | $N_{4}$ | 3.39
$\tilde{u}_{R}$ | 2.24 | $\tilde{d}_{R}$ | 2.40 | $\tilde{e}_{R}$ | 0.476 | - | - | $C_{1}$ | 0.895
$\tilde{u}_{L}$ | 2.48 | $\tilde{d}_{L}$ | 2.48 | $\tilde{e}_{L}$ | 1.05 | - | - | $C_{2}$ | 3.43
$m_{A^{0}}$ | 3.23 | $m_{H}^{\pm}$ | 3.23 | $m_{h}$ | 0.12186 | $m_{H}$ | 3.06 | - | -
Table 13: Low energy $\delta^{\prime}s$ for quarks and leptons corresponding to the points in Eq.(21) for the LLHH case evaluated for $\tilde{m}_{q}=2.1$TeV and $\tilde{m}_{l}=0.7$ TeV (i,j) | $|\delta^{Q}_{LL}|$ | $|\delta^{L}_{LL}|$ | $|\delta^{D}_{LR}|$ | $|\delta^{U}_{LR}|$ | $|\delta^{D}_{RL}|$ | $|\delta^{U}_{RL}|$ | $|\delta^{D}_{RR}|$ | $|\delta^{E}_{RR}|$ | $|\delta^{U}_{RR}|$
---|---|---|---|---|---|---|---|---|---
12 | 0.0003 | $0.0001$ | $10^{-10}$ | $10^{-8}$ | $10^{-8}$ | $10^{-5}$ | $0.001$ | $10^{-10}$ | 0.001
13 | $0.01$ | 0.04 | $10^{-8}$ | $10^{-8}$ | $10^{-6}$ | $0.002$ | $0.005$ | $10^{-7}$ | 0.01
23 | 0.05 | $0.05$ | $10^{-6}$ | $10^{-5}$ | $10^{-5}$ | $0.01$ | $0.003$ | 0.0001 | 0.07
#### IV.0.2 Dirac Case
For the case where neutrinos are of Dirac type the $c$ parameters in Eq.(25)
with $\chi^{2}$ of $0.3211$ for the hadronic sector and $0.1481$ for the
leptonic sector were chosen. The $c$ values for the doublets in this case
indicate they are predominantly elementary from the CFT point of view
especially for the first two generations. The third generation however may be
partially composite as in this case.
$\displaystyle c_{Q_{1}}=1.895\hskip 14.22636ptc_{D_{1}}=1.898\hskip
14.22636ptc_{U_{1}}=1.738\hskip 14.22636ptc_{L_{1}}=1.293\hskip
14.22636ptc_{E_{1}}=2.480\hskip 14.22636ptc_{N_{1}}=6.783$ $\displaystyle
c_{Q_{2}}=1.467\hskip 14.22636ptc_{D_{2}}=1.271\hskip
14.22636ptc_{U_{2}}=1.124\hskip 14.22636ptc_{L_{2}}=1.311\hskip
14.22636ptc_{E_{2}}=1.406\hskip 14.22636ptc_{N_{2}}=7.346$ $\displaystyle
c_{Q_{3}}=-0.137\hskip 14.22636ptc_{D_{3}}=1.394\hskip
14.22636ptc_{U_{3}}=-0.356\hskip 14.22636ptc_{L_{3}}=0.260\hskip
14.22636ptc_{E_{3}}=0.237\hskip 14.22636ptc_{N_{1}}=7.332$ (25)
The generic feature of soft mass matrices, discussed in the LLHH case, of the
spectrum being tachyonic at the high scale and the diagonal terms evolving
more than the off diagonal elements apply to this case as well. Corresponding
to the c values in Eq.(25), the soft masses at the GUT scale have been
evaluated for $m^{GUT}_{3/2}=800$ GeV while choosing $M_{1/2}=1200$ GeV for
the three gauginos. The $\mathcal{O}(1)$ parameters corresponding to
$(m_{Q})_{33}$ and $(m_{U})_{33}$ were chosen to be 4 while for the others
they are set to be 1. $A^{{}^{\prime}u}_{ij}=1.15Y^{{}^{\prime}u}\forall i,j$,
for all the A terms of the up sector while for the down sector and the leptons
they were set equal to the corresponding $\mathcal{O}$(1) Yukawa couplings.
The high scale soft breaking matrices in $GeV$ are given in Eq.(26). In
Table[14] we present the low energy spectrum corresponding to the sample point
in Eq.(25). The low energy $\delta^{\prime}s$ are presented in Table 15.
$m_{Q}=\begin{bmatrix}0.60&1.2&30.0\\\ 1.2&8.7&-19.4\\\
30.0&-19.4&2560.6\end{bmatrix};m_{U}=\begin{bmatrix}0.7&-3.8&17.1\\\
-3.8&32.1&70.3\\\
17.1&70.3&2971.6\end{bmatrix};m_{D}=\begin{bmatrix}0.10&-0.85&-1.7\\\
-0.85&7.1&14.6\\\ -1.7&14.6&29.8\end{bmatrix}$
$A_{U}=\begin{bmatrix}10^{-3}&-0.42&-5.8\\\ 10^{-3}&1.18&0.20\\\
10^{-5}&-0.005&488.8\end{bmatrix};A_{D}=\begin{bmatrix}0.03&0&0\\\ 0&0.57&0\\\
0&0&-40.6\end{bmatrix};m_{L}=\begin{bmatrix}8.7&0.9&62.5\\\ 0.9&0.1&7.0\\\
62.5&7.0&446.3\par\end{bmatrix}$ $m_{E}=\begin{bmatrix}0.2&-0.4&10.5\\\
-0.4&0.7&-17.9\\\
1.0&-17.29&442.7\end{bmatrix};A_{E}=\begin{bmatrix}-0.01&0&0\\\ 0&-3.01&0\\\
0&0&52.1\end{bmatrix}$ (26)
Table 14: Soft spectrum for Dirac case: $m_{susy}=1.05$ TeV, $m_{\tilde{g}}=2.65$ TeV, $\mu=4.32$TeV, $tan\beta=25$ Parameter | Mass(TeV) | Parameter | Mass(TeV) | Parameter | Mass(TeV) | Parameter | Mass(Tev) | Parameter | Mass(TeV)
---|---|---|---|---|---|---|---|---|---
$\tilde{t}_{1}$ | 0.702 | $\tilde{b}_{1}$ | 2.06 | $\tilde{\tau}_{1}$ | 0.480 | $\tilde{\nu}_{\tau}$ | 0.570 | $N_{1}$ | 0.465
$\tilde{t}_{2}$ | 2.31 | $\tilde{b}_{2}$ | 2.32 | $\tilde{\tau}_{2}$ | 0.802 | $\tilde{\nu}_{\mu}$ | 0.624 | $N_{2}$ | 0.928
$\tilde{c}_{R}$ | 2.25 | $\tilde{s}_{R}$ | 2.36 | $\tilde{\mu}_{R}$ | 0.608 | $\tilde{\nu}_{e}$ | 0.625 | $N_{3}$ | 4.26
$\tilde{c}_{L}$ | 2.45 | $\tilde{s}_{L}$ | 2.45 | $\tilde{\mu}_{L}$ | 0.902 | - | - | $N_{4}$ | 4.26
$\tilde{u}_{R}$ | 2.25 | $\tilde{d}_{R}$ | 2.36 | $\tilde{e}_{R}$ | 0.610 | - | - | $C_{1}$ | 0.894
$\tilde{u}_{L}$ | 2.45 | $\tilde{d}_{L}$ | 2.45 | $\tilde{e}_{L}$ | 0.903 | - | - | $C_{2}$ | 4.32
$m_{A^{0}}$ | 4.18 | $m_{H}^{\pm}$ | 4.18 | $m_{h}$ | 0.1235 | $m_{H}$ | 3.96 | - | -
Table 15: Low energy $\delta^{\prime}s$ for the Dirac Case corresponding to the point in Eq.(25) for Dirac case evaluated for $\tilde{m}_{q}=2.1$TeV and $\tilde{m}_{l}=0.7$ TeV (ij) | $|\delta^{Q}_{LL}|$ | $|\delta^{L}_{LL}|$ | $|\delta^{D}_{LR}|$ | $|\delta^{U}_{LR}|$ | $|\delta^{D}_{RL}|$ | $|\delta^{U}_{RL}|$ | $|\delta^{D}_{RR}|$ | $|\delta^{E}_{RR}|$ | $|\delta^{U}_{RR}|$
---|---|---|---|---|---|---|---|---|---
12 | 0.0003 | $10^{-6}$ | $10^{-10}$ | $10^{-8}$ | $10^{-8}$ | $10^{-5}$ | $10^{-7}$ | $10^{-7}$ | 0.00005
13 | $0.01$ | 0.007 | $10^{-8}$ | $10^{-8}$ | $10^{-5}$ | $0.002$ | $10^{-6}$ | $10^{-4}$ | 0.06
23 | 0.06 | $10^{-4}$ | $10^{-6}$ | $10^{-5}$ | $10^{-5}$ | $0.01$ | $10^{-4}$ | 0.0006 | 0.001
## V Outlook
The Randall-Sundrum framework is typically considered to be the geometric
avatar of the Froggatt-Nielsen models. In the present work, we have considered
a warped extra dimension close to the GUT scale. We fit the quark masses and
the CKM mixing angles and determined the range of the $c$ parameters which
give a reasonable $\chi^{2}$ fit. The $\mathcal{O}(1)$ parameters associated
with the Yukawa couplings have also been varied accordingly. Though the top
quark Yukawa is smaller at the high scale compared to the weak scale, it is
still large enough that one requires a large negative bulk mass parameter for
the right handed top quark. For the leptons, we considered two particular
models for neutrino masses (a) with Planck scale lepton number violating
operator and (b) Dirac neutrino masses.
The results show that there is a significant difference in the RS models at
the weak scale and the RS models at the GUT scale especially if one focuses on
the neutrino sector. In the weak scale models, the Planck scale lepton number
violating higher dimensional operator was very hard to accommodate with
perturbative Yukawa couplings and thus was highly disfavoured. The Dirac and
the Majorana cases were favoured though they were strongly constrained by the
data from flavour violating rare decays. The situation is sort of reversed in
the GUT scale RS models, though not exactly. The higher dimensional Planck
scale operator fits the data very well, where as the Dirac case requires
larger $c$ values, some times with bulk mass parameters almost an order of
magnitude larger than the cut-off scale. In the hadronic sector, the situation
is not so dramatic. The top quark Yukawa though reduces at the GUT scale
compared to the weak scale, still requires that the right handed top to be
located close to the IR, making it a composite as in the weak scale models.
The supersymmetric version of the same set up is far more interesting as it
can have possible observable signatures at the weak scale. The main difference
in fitting of the fermion masses is due to the presence of the additional
parameter $\tan\beta$ in the supersymmetric case. The difference in the $c$
values is not very significant for low values of $\tan\beta$. At large
$\tan\beta$, for a given generation, the zero modes are localized more towards
the IR brane as compared to the SM case. This effect is more pronounced in the
down sector as shown in Figures[7]. We parameterise SUSY breaking by a single
spurion field localised on the IR brane. The resultant soft masses depend on
the profiles of the zero modes of the chiral superfields and contain flavour
violation. At the weak scale the constraints from first two generation flavour
transitions rule out light spectrum. Another significant constraint comes the
light Higgs mass at 125 GeV. The trilinear couplings have the same form as the
Yukawa couplings in this model as long as the $\mathcal{O}$(1) parameters
associated with both the parameters are taken to be proportional to each
other. If all the $\mathcal{O}(1)$ parameters at high scale are take to be
exactly unity, the weak scale values of $A_{t}$ are small to generate a 125
GeV light Higgs, for stops of masses $\sim 1.5-2$ TeV. However, with a minor
variation of the $\mathcal{O}(1)$ parameters for $A_{t}$ at the high scale,
125 GeV Higgs is easily possible. More variations of supersymmetric breaking
and the corresponding spectra will be discussed later iyer2 .
Acknowledgments
We thank Emilian Dudas for discussions and collaboration in the initial stages
of this work. We also thank him for a clarification about a point. AI would
like to thank CPhT Ecole Polytechnique for hospitality during his stay and
Gero Von Gersdorff for useful discussions. We also thank Debtosh Chowdhury for
useful inputs. We thank S. Uma Sankar for a reference and communications. SKV
acknowledges support from DST Ramanujam fellowship SR/S2/RJN-25/2008 of Govt.
of India.
## Appendix A Plots for ranges of c parameters for quarks and leptons for SM
fits at the GUT scale
### A.1 Range of c parameters for the quarks
|
---|---
|
|
Figure 1: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10 for the SM fits at the GUT scale. The plot represents the parameter
space for the bulk masses of the quarks.
### A.2 Range of c parameters for the for leptons for the LLHH scenario
|
---|---
|
Figure 2: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10. The plot represents the parameter space for the bulk masses of the
leptons corresponding to the LLHH case.
### A.3 Range of c parameters for the leptons.
|
---|---
|
|
Figure 3: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10 for the SM fits at the GUT scale. The plot represents the parameter
space for the bulk masses of the leptons corresponding to the Dirac type
neutrinos.
## Appendix B Plots for ranges of c parameters for quarks and leptons for the
supersymmetric case
### B.1 Range of c parameters for the quarks
|
---|---
|
|
Figure 4: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10. The plot represents the parameter space for the bulk masses of the
quarks.
### B.2 Range of c parameters for the for leptons for the LLHH scenario
|
---|---
|
Figure 5: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10. The plot represents the parameter space for the bulk masses of the
leptons corresponding to the LLHH case.
### B.3 Range of c parameters for the for leptons for the case of Dirac
neutrinos.
|
---|---
|
|
Figure 6: The points in the above figures correspond to a $\chi^{2}$ between 1
and 10. The plot represents the parameter space for the bulk masses of the
leptons corresponding to the Dirac type neutrinos.
## Appendix C Comparative plot between SM and supersymmetric fits for
fermions
|
---|---
|
|
Figure 7: The plots correspond to comparison between SM and supersymmetric
fits for down sector fermions. Red corresponds to SM, while Blue and Green
corresponds to supersymmetric case for tan$\beta=10$ and tan$\beta=50$
respectively
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* (56) M. J. Perez, P. Ramond, and J. Zhang, “On Mixing Supersymmetry and Family Symmetry Breakings,” Phys.Rev., vol. D87, p. 035021, 2013.
* (57) D. Chowdhury, R. Garani, and S. K. Vempati, “SUSEFLAV: Program for supersymmetric mass spectra with seesaw mechanism and rare lepton flavor violating decays,” Comput.Phys.Commun., vol. 184, pp. 899–918, 2013.
* (58) F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, “A Complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model,” Nucl.Phys., vol. B477, pp. 321–352, 1996.
* (59) F. Gabbiani and A. Masiero, “FCNC in Generalized Supersymmetric Theories,” Nucl.Phys., vol. B322, p. 235, 1989.
* (60) M. Ciuchini, A. Masiero, P. Paradisi, L. Silvestrini, S. Vempati, et al., “Soft SUSY breaking grand unification: Leptons versus quarks on the flavor playground,” Nucl.Phys., vol. B783, pp. 112–142, 2007.
|
arxiv-papers
| 2013-04-12T07:45:21 |
2024-09-04T02:49:44.261390
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhishek M Iyer and Sudhir K Vempati",
"submitter": "Abhishek Iyer M",
"url": "https://arxiv.org/abs/1304.3558"
}
|
1304.3713
|
# Computational Nuclear Quantum Many-Body Problem: The UNEDF Project
S. Bogner A. Bulgac J. Carlson J. Engel G. Fann R.J. Furnstahl S.
Gandolfi G. Hagen M. Horoi C. Johnson M. Kortelainen E. Lusk P. Maris
H. Nam P. Navratil W. Nazarewicz E. Ng G.P.A. Nobre E. Ormand T.
Papenbrock J. Pei S. C. Pieper S. Quaglioni K.J. Roche J. Sarich N.
Schunck M. Sosonkina J. Terasaki I. Thompson J.P. Vary S.M. Wild
Mathematics and Computer Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA Physics Division, Argonne National Laboratory,
Argonne, IL 60439, USA National Nuclear Data Center, Brookhaven National
Laboratory, Upton, NY 11973, USA Central Michigan University, Mount Pleasant,
MI 48859, USA Department of Physics and Astronomy, Iowa State University,
Ames, IA 50011, USA Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA Computational Research Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA Physics Division, Lawrence
Livermore National Laboratory, Livermore, CA 94551, USA National
Superconducting Cyclotron Lab, Michigan State University, East Lansing, MI,
48824, USA Computer Science and Mathematics Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA National Center for Computational
Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Department of Physics, Ohio State University, Columbus, OH 43210, USA
Department of Modeling, Simulation and Visualization Engineering, Old Dominion
University, Norfolk, VA 23529, USA Computational Sciences and Mathematics
Division, Pacific Northwest National Laboratory, Richland, WA 99352, USA
State Key Laboratory of Nuclear Physics and Technology, School of Physics,
Peking University, Beijing 100871, China Department of Physics, San Diego
State University, San Diego, CA 92182, USA TRIUMF, 4004 Westbrook Mall,
Vancouver, BC, V6T 2A3, Canada Department of Physics, P.O. Box 35 (YFL),
FI-40014, University of Jyväskylä, Finland Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996, USA Division of Physics and Center for Computational Sciences,
University of Tsukuba, Tsukuba, 305-8577, Japan Faculty of Physics,
University of Warsaw, 00-681 Warsaw, Poland Department of Physics, University
of Washington, Seattle, WA 98195, USA
###### Abstract
The UNEDF project was a large-scale collaborative effort that applied high-
performance computing to the nuclear quantum many-body problem. UNEDF
demonstrated that close associations among nuclear physicists, mathematicians,
and computer scientists can lead to novel physics outcomes built on
algorithmic innovations and computational developments. This review showcases
a wide range of UNEDF science results to illustrate this interplay.
###### keywords:
Configuration interaction , Coupled-cluster method , Density functional theory
, Effective field theory , High-performance computing , Quantum Monte Carlo
## 1 Introduction to UNEDF
Understanding the properties of atomic nuclei is crucial for a complete
nuclear theory, for element formation, for properties of stars, and for
present and future energy and defense applications. From 2006 to 2012, the
UNEDF (Universal Nuclear Energy Density Functional) collaboration carried out
a comprehensive study of the nuclear many-body problem using advanced
numerical algorithms and extensive computational resources, with a view toward
scaling to petaflop supercomputing platforms and beyond.
The UNEDF project was carried out as part of the SciDAC (Scientific Discovery
through Advanced Computing) program led by Advanced Scientific Computing
Research (ASCR), part of the Office of Science in the U.S. Department of
Energy (DOE). The SciDAC program was started in 2001 as a way to couple the
applied mathematics and computer science research sponsored by ASCR to applied
computational science application projects traditionally supported by other
offices in DOE. UNEDF was funded jointly by ASCR, the Nuclear Physics program
of the Office of Science, and the National Nuclear Security Administration.
Over 50 physicists, applied mathematicians, and computer scientists from 9
universities and 7 national laboratories in the United States, as well as many
international collaborators, participated in UNEDF.
This review describes science outcomes in nuclear many-body physics, with an
emphasis on computational and algorithmic developments, that have resulted
from the successful collaborations within UNEDF among mathematicians and
computer scientists on one side and nuclear physicists on the other. Such
collaborations “across the divide” were newly formed at the early stage of the
project and became its unique feature, with high-performance computing serving
as a catalyst for new interactions. The results described in this paper could
not have been achieved without such couplings.
### 1.1 UNEDF science
The long-term vision initiated with UNEDF is to arrive at a comprehensive,
quantitative, and unified description of nuclei and their reactions that is
grounded in the fundamental interactions between the constituent nucleons [1,
2]. The goal is to replace phenomenological models of nuclear structure and
reactions with a well-founded microscopic theory that delivers maximum
predictive power with well-quantified uncertainties. Specifically, the mission
of UNEDF was threefold:
1. 1.
Find an optimized energy density functional (EDF) using all our knowledge of
the nucleonic Hamiltonian and basic nuclear properties.
2. 2.
Validate the functional using the relevant nuclear data.
3. 3.
Apply the validated theory to properties of interest that cannot be measured.
The main physics areas of UNEDF, defined at the beginning of the project [1],
were ab initio structure, ab initio functionals, density functional theory
(DFT) applications, DFT extensions, and reactions. Few connections between
these areas existed at that time. As UNEDF matured, however, coherence grew
within the effort. Indeed, the project created and facilitated an increasing
interplay among the major areas where none had existed previously. Each of the
main physics areas now includes ongoing collaborations that cross over into
other areas. These interconnections are highlighted in the summary diagram of
the UNEDF strategy shown in Fig. 1. In addition to physics links, numerous
computer science/applied mathematics (CS/AM) interconnections were established
within UNEDF as computational and mathematical tools developed in one area of
UNEDF were used in other parts of the project. These tools, motivated by
nuclear needs, are now available for other areas of science. Access to
leadership-class computing resources and large-scale compute time allocations
were critical for the scientific investigations.
Figure 1: UNEDF project scope. Major science areas are indicated by boxes;
interconnections between areas are marked by arrows. The green boxes indicate
connections to experimental observations.
At the intersection of the ab initio techniques and DFT techniques are
comparisons of observables among the various approaches, particularly through
constraints on density. Such calculations have not been performed before and
require significant computational capability and an increasing sophistication
of data manipulation. Research on the nuclear problem would be incomplete
without a serious effort to understand the nuclear interactions involved and
their connection to DFT. Therefore, the UNEDF project also included elements
that required less computational capability but are integral to the project,
such as the development of nuclear forces using renormalization group
approaches. Another example is research on nuclear reaction properties that
requires both the use and development of algorithms for the largest computers
and more conventional computing needed for algorithmic breakthroughs.
Another new aspect of the nuclear theory effort driven by this project is a
greatly enhanced degree of quality control. Integral to UNEDF was the
verification of methods and codes, the estimation of uncertainties, and other
output assessments. Methods used for verification and validation included the
crosschecking of different theoretical methods and codes, the use of multiple
DFT solvers with benchmarking, and benchmarking of different ab initio methods
using the same Hamiltonian. A new way to estimate theory error bars was to use
multiple Hamiltonians with different energy/momentum cutoffs and then analyze
the cutoff dependence of calculated observables. The UNEDF assessment
component necessitated the development and application of statistical tools to
deliver uncertainty quantification and error analysis for theoretical studies
as well as to assess the significance of new experimental data. Such
technologies are essential as new theories and computational tools are applied
to entirely new nuclear systems and to conditions that are not accessible to
experiment.
### 1.2 Collaborative effort
The successes of the UNEDF project were built upon certain best practices,
some implemented originally and some learned by experience, in organizing and
implementing the scientific effort. In order to foster the close alignment of
the necessary applied mathematics and computer science research with the
necessary physics research, multiple direct partnerships were formed
consisting of computer scientists and applied mathematicians linked with
specific physicists to remove algorithmic and/or computational barriers to
progress. The five-year lifetime of the project provided time for these
collaborations to become deep, and they have continued into follow-up
projects.
All these partnerships have success stories to tell, from greatly improved
load balancing on leadership-class machines, to new DFT solver technologies,
to dramatically improved algorithms for optimization of functionals, to
eigenvalues and eigenfunctions of extremely large matrices, and more.
The SciDAC program aims at transformative science, and this goal has been
fulfilled by the new capabilities stemming from UNEDF. But the outcomes reach
beyond the many compelling nuclear physics calculations. UNEDF has changed for
the better the way that low-energy nuclear theory is carried out, analogous to
the shift in experimental programs, moving from many small groups working
independently to large-scale collaborative efforts.
## 2 Science
The territory of UNEDF science is the chart of the nuclides in the
$(N,Z)$-plane shown in Fig. 2. On this chart, stable nuclei are represented by
black squares, while the yellow squares indicate unstable nuclei that have
been seen in the laboratory. The sizable green area marked “terra incognita”
is populated by unstable isotopes yet to be explored. Above the table of
nuclides are shown three broad classes of theoretical methods, which are also
used in other fields dealing with strongly interacting many-body systems, such
as quantum chemistry and condensed matter physics. Light nuclei and their
reactions can be computed by using ab initio techniques (quantum Monte Carlo,
no-core shell model) described in Sec. 2.1. Medium-mass nuclei can be treated
by configuration interaction (CI) techniques (Sec. 2.2). The bulk of the
nuclides are covered by the nuclear DFT described in Sec. 2.3, which provides
the theoretical underpinning and computational framework for building a
nuclear EDF. Time-dependent phenomena involving complex nuclei, including
nuclear reactions, can be described by means of approaches going beyond static
DFT (Sec. 2.4). By enhancing and exploiting the overlaps with ab initio and CI
approaches, the goal is to construct and validate a nuclear EDF informed by
microscopic interactions as well as experimental data.
Figure 2: Theoretical approaches for solving the nuclear quantum many-body
problem used by UNEDF. The lightest nuclei can be computed by using ab initio
methods based on the bare internucleon interactions (red). Medium-mass nuclei
can be treated by configuration interaction techniques (green). For heavy
nuclei, the density functional theory based on the optimized energy density
functional is the tool of choice. (From [1].)
### 2.1 Ab initio methods and benchmarking
Ab initio methods solve few- and many-body problems by using realistic two-
and three-nucleon interactions and obtain the structure and dynamic properties
of nuclei. The nuclear interaction depends on the spatial, spin, and isospin
coordinates of the nucleons. Consequently, calculations are much more
computationally demanding than typical quantum problems. Items of interest
include nuclear spectra, charge and magnetic ground-state and transition
densities, electron and neutrino scattering, and low-energy reactions. The
main goals are to reproduce known nuclear properties and predict properties
that are difficult or impossible to measure.
Several ab initio methods have been developed for studying light nuclei; all
have analogues in the study of condensed matter and electronic systems.
Quantum Monte Carlo (QMC) methods, including Green’s function Monte Carlo
(GFMC), use Monte Carlo evaluations of path integrals, explicitly summing over
the spin states and isospin states of the system. The most recent GFMC
calculations have concentrated on the 12C nucleus, a fascinating system with a
low-lying excited $0^{+}$ state, the Hoyle state, very near the threshold of
three-alpha particles. QMC methods have also been used to calculate the
properties of neutron matter and neutrons in inhomogeneous potentials.
No-core shell model (NCSM) methods, including the large-scale many-fermion
dynamics nuclear (mfdn) code, expand the interacting states in products of
single-particle states and project the low-lying states through large-scale
matrix operations. mfdn calculations have been used, for example, to explain
the long lifetime of the 14C nucleus used in carbon dating. A combination of
no-core shell model techniques with the resonating group method is currently
used to calculate important low-energy nuclear reactions.
The coupled-cluster method is an ideal microscopic approach to describe nuclei
with closed (sub)shells and their neighbors. It exhibits a low computational
cost (scales polynomially with system size) while capturing the dominant parts
of correlations in the wave function. This method has been employed to
describe and predict the structure and reactions of neutron-rich oxygen and
calcium isotopes.
#### 2.1.1 GFMC
Green’s function Monte Carlo calculations start with an initial trial state
$\Psi_{T}$ and obtain expectation values in the exact eigenfunction $\Psi_{0}$
of the Hamiltonian. These calculations are done by evolution in imaginary time
$\tau$: $\Psi_{0}=\exp[-H\tau]\Psi_{T}$ for sufficiently large $\tau$. The
evolution is done in many small steps of $\tau$, each step being a nested
$3A$-dimensional integral. GFMC was introduced in light nuclei [3, 4] to
include the strong correlations induced by the nuclear interaction. This
method has been used to calculate the spectra of light nuclei up to 12C [4,
5], as well as form factors, electron scattering, and low-energy reactions
[6].
Calculations of 12C require the largest-scale computers available, using a
combination of efficient load-balancing for the Monte Carlo and large-scale
linear algebra for the spin-isospin degrees of freedom. The calculations of
12C required the development of the Asynchronous Dynamic Load Balancing (adlb)
library to efficiently perform the load balancing on more than 100,000 cores
[5].
A program, agfmc, has been developed over the past 15 years to carry out these
calculations [7, 8, 9]. It is a large (80,000 lines) Fortran code that
originally used MPI to manage parallelism. At the beginning of this project,
the agfmc code was scaling well up to around 2,000 processes and performing
satisfactorily on IBM’s Blue Gene/L computer. At that time it was becoming
apparent that if the code were to be able to take advantage of new, petascale
machines expected to come on line during the five-year project to investigate
larger nuclei, a significant increase in the degree of parallelism would need
to be incorporated into its main algorithms. The greater degree of parallelism
(from thousands to tens of thousands of processes) would give rise to load-
balancing problems that would strain the then-used approach.
One of the goals of UNEDF was to construct a software library, intrinsically
general-purpose but with features driven by the requirements of agfmc, to
attack the load-balancing problem. The purposes of the library were to supply
a programming interface that would enable relatively straightforward migration
of the existing agfmc code to the new load-balancing library and to scale the
entire system to much larger degrees of parallelism.
The result is the adlb library [5]. adlb generalizes the classical manager-
worker parallel programming model by allowing application processes (workers)
to _put_ arbitrary independent work units into a shared pool and _get_ them
out to complete them, notifying other processes when they have done so. Work
units are assigned types and priorities by the workers and retrieved according
to these properties, allowing complex algorithms to be implemented, despite
the simple nature of the parallel programming model. Scalability is achieved
by dedicating a small percentage (but still potentially a large number) of the
job’s processes to maintaining this work pool and responding to _put_ and
_get_ requests. These “server” processes execute independently from the
application processes, thus allowing asynchronous load balancing of process
load, memory consumption for the work pool, and message traffic.
Figure 3: Weak scaling of agfmc with adlb in terms of MPI ranks. There are 8
ranks per BG/Q node; each rank is using 6 OpenMP threads. Note the compressed
vertical scale.
This scheme has worked well. Most of the MPI programming in the original agfmc
code has been absorbed into the adlb library, yet the overall code structure
has been maintained. Scalability has been extended to more than 32,000
processes on BG/P and more than 260,000 processes on BG/Q (see Fig. 3),
enabling scientific results unattainable before this project was undertaken.
The 12C nucleus is particularly intriguing because it has a low-lying $0^{+}$
excited state (the “Hoyle” state) very near the energy of the breakup into
three alpha particles. This state is essential for the nucleosynthesis of
carbon in stars through the triple-alpha process. For 12C the $\Psi_{T}$ are
linear combinations of shell-model and alpha-cluster states.
Figure 4: Convergence of the ground state (lower curves) and Hoyle state
(upper curves) for different initial states as a function of imaginary time.
Figure 4 shows the convergence of the calculations of the ground and Hoyle
states in the agfmc calculations. Two different sets of initial states are
propagated to $\tau\approx 1.0\,\mbox{MeV}^{-1}$; they yield consistent
results. The ground-state energy is well reproduced, and the Hoyle state
excitation energy is approximately reproduced (see [10, 11] for complementary
calculations of the Hoyle state). The ground-state form factor of 12C is also
reproduced by these calculations.
Other recent applications of agfmc include pair momentum distributions [12],
electromagnetic transitions [13], and the studies of trapped neutrons
(“drops”) described in Sec. 2.3.4.
#### 2.1.2 NCSM and mfdn
The measured lifetime of 14C, $5730\pm 30$ years, is a valuable chronometer
for many practical applications ranging from archeology to physiology. It is
anomalously long compared with lifetimes of other light nuclei undergoing the
same decay process, allowed Gamow-Teller (GT) beta decay. This lifetime poses
a major challenge to theory because traditional realistic nucleon-nucleon (NN)
interactions alone appear insufficient to produce the effect [14]. Since the
transition operator, in leading approximation, depends on the nucleon spin and
charge but not the spatial coordinates, this decay provides a precision tool
to inspect selected features of the initial and final nuclear states. To
convincingly explain this strongly inhibited transition, we need a microscopic
description that introduces all physically relevant 14-nucleon configurations
in the initial and final states and a realistic Hamiltonian.
Figure 5: Contributions to the 14C beta decay matrix element as a function of
the harmonic oscillator shell when the nuclear structure is described by a
chiral effective field theory interaction (adopted from [15]). The top panel
displays the contributions with (two right bars of each triplet) and without
(leftmost bar of each triplet) the 3NF at $N_{\max}=8$. Contributions are
summed within each shell to yield a total for that shell. The bottom panel
displays the running sum of the GT contributions over the shells. Note the
order-of-magnitude suppression of the $0p$-shell contributions arising from
the 3NFs.
Since the nuclear strong interaction governs the configuration mixing, the
Hamiltonian matrix eigenvalue problem is a very large, sparse matrix in the
configuration space of 14 nucleons. We address this computational challenge
with the mfdn code [16, 17, 18, 19]. Aided by a collaboration with applied
mathematicians on scalable eigensolvers and computational resources on
leadership-class machines, we are able to solve this beta decay problem with
sufficient accuracy to resolve the puzzle: the decay is inhibited by the role
of 3-nucleon forces (3NFs) as shown in Fig. 5 (see [20] for complementary
calculations).
We obtained our results on the Jaguar supercomputer (see Sec. 4) using up to
35,778 hex-core processors (214,668 cores) and up to 6 hours of elapsed time
for each set of low-lying eigenvalues and eigenvectors. The number of
nonvanishing matrix elements exceeded the total memory available and required
matrix element recomputation “on the fly” for the iterative diagonalization
process employing the Lanczos algorithm.
These calculations and many other achievements [21] were made possible by
dramatic improvements to mfdn capabilities during the UNEDF project [22]. The
current scaling performance of mfdn is demonstrated in Fig. 6. Other recent
applications of mfdn include the prediction (before experimental confirmation)
of the spectroscopy of proton-unstable 14F [23] and studies of trapped
neutrons (“drops”) with a variety of interactions and other ab initio
computational methods [24].
Figure 6: Strong scaling for mfdn: speedup for 500 Lanczos iterations (the
most time-consuming phase of the code). Two problems are shown with their
dimension (D) and number of nonzero matrix elements (NNZ) in the legend. The
smaller is 7Li (D=6.2 million, NNZ=118 billion), and the larger is 10B (D=160
million, NNZ=5.2 trillion). The smaller problem needs at least 1 TB in order
to store all nonzero matrix elements in core and needs, therefore, at least
728 cores to fit the problem in core. The larger problem needs at least 42 TB,
and we used between 30,624 and 261,120 cores for that problem.
#### 2.1.3 NCSM and the resonating group method
Weakly bound nuclei, or even unbound exotic nuclei, cannot be understood by
using only bound-state techniques. Our ab initio many-body approach, no-core
shell model with continuum (NCSMC), focuses on a unified description of both
bound and unbound states. With such an approach, we can simultaneously
investigate structure of nuclei and their reactions. The method combines
square-integrable harmonic-oscillator basis (i.e., via the NCSM [21])
accounting for the short- and medium-range many-nucleon correlations with a
continuous basis (i.e., via the NCSM with the resonating group method
(NCSM/RGM) [25, 26]) accounting for long-range correlations between clusters
of nucleons. With this technique, we can predict the ground- and excited-state
energies of light nuclei ($p$-shell, $A{\leq}16$) as well as their
electromagnetic moments and transitions, including weak transitions.
Furthermore, we can investigate properties of resonances and calculate
characteristics of binary nuclear reactions (e.g., cross sections, analyzing
powers).
Recent applications of our ab initio techniques include an investigation of
the unbound 7He [27], calculations of 3H($d$,$n$)4He and 3He($d$,$p$)4He
fusion [28] (see Fig. 7), and calculation of the 7Be($p$,$\gamma$)8B radiative
capture [29], which is important for the standard solar model and neutrino
physics (see Fig. 8). We also developed a three-cluster extension of the
method to describe the Borromean nuclei (e.g., 6He and 11Li).
Figure 7: Experimental results for S-factor of 3He($d$,$p$)4He reaction from
beam-target measurements. The full line represents the ab initio calculation.
No low-energy enhancement is present in the theoretical results, contrary to
the laboratory beam-target data represented by symbols; see [28] for details.
Figure 8: Ab initio calculated 7Be($p$,$\gamma$)8B S-factor (solid line)
compared with experimental data and the calculation used in the latest
evaluation (dashed line); see [29] for details.
#### 2.1.4 Coupled-cluster method
The coupled-cluster method [30, 31, 32, 33] exhibits a favorable scaling that
grows polynomially with the mass number of the nucleus and the size of the
model space. The UNEDF collaboration employed an $m$-scheme-based coupled-
cluster code [34] and an angular-momentum coupled code [35]. The latter
exploits the preservation of angular momentum and pushed ab initio computation
with “bare” interactions from chiral effective field theory [36] to medium-
mass nuclei [37]. Coupled-cluster theory is based on a similarity-transformed
Hamiltonian and employs a nontrivial vacuum such as the Hartree-Fock state. In
practice, one iteratively solves a large set of nonlinear coupled equations.
The exploitation of rotational invariance considerably reduces the number of
degrees of freedom but comes at the cost of working in a much more complicated
scheme (involving angular momentum algebra) that poses challenges for a
scalable and load-balanced implementation.
During UNEDF, several conceptual advances in physics and computing were made
with the coupled-cluster method. On the physics side, these include the
angular-momentum coupled implementation of the coupled-cluster method [37],
the use of a Gamow basis for computation of weakly bound nuclei [38, 39], a
practical solution to the center-of-mass problem in nuclear structure
computations [40], the extension of the method to nuclei with up to two
nucleons outside a closed subshell [41], the approximation of three-nucleon
forces as in-medium correction to nucleon-nucleon forces [42, 43, 44], and the
development of theoretically founded extrapolations in finite oscillator
spaces [45]. On the computational side, scaling was improved by a work-
balancing approach [46, 47] based on MPI and OpenMP such that the model-space
size has increased from ten oscillator shells at the inception of UNEDF [48]
to 20 oscillator shells at UNEDF’s completion [44]. Figure 9 shows how adding
the use of MPI and OpenMP in V2.0 improved the code’s scalability to thousands
of cores, beyond a few hundred cores in V1.0 using MPI only, when calculating
the small system of 40Ca in 12 oscillator shells.
Figure 9: Comparison of runtime for 40Ca in 12 oscillator shells using MPI
only V1.0 and hybrid MPI/OpenMP V2.0. Solid lines show total runtime; dashed
lines show runtime of triples calculation only.
We note that the number of single-particle orbitals grows as the third power
with the number of oscillator shells and that the number of computational
cycles – in the coupled-cluster method with singles and doubles (CCSD)
approximation – grows as $n_{\rm o}^{2}n_{\rm u}^{4}$ (where $n_{\rm o}$ and
$n_{\rm u}$ are the numbers of occupied and unoccupied single-particle states,
respectively). Thus, conceptual and algorithmic improvements during UNEDF
allowed us to solve problems that naïvely required an increase of
computational cycles by about a factor 4,000. The combined efforts culminated
in the computation of neutron-rich isotopes of oxygen [44] and calcium [49].
Doubly magic nuclei are the cornerstones for our understanding of entire
regions of the nuclear chart within the shell model. For this reason, studies
on the evolution of structure in neutron-rich semi-magic isotopes of oxygen,
calcium, nickel, and tin are central to experimental and theoretical efforts.
With 40,48Ca being doubly magic nuclei, many studies were aimed at
understanding the structure of the rare isotopes 52,54Ca and questions
regarding the $N=32,34$ shell closures [50, 51, 52, 53, 54].
A first-principles description of rare calcium isotopes is challenging because
it requires the control and understanding of continuum effects (due to the
weak binding) and 3NFs (as often pivotal contributions arise at next-to-next-
to leading order in chiral effective field theory [55, 56, 57]). Reference
[49] reports coupled-cluster results for neutron-rich isotopes of calcium that
include the effects of the continuum and 3NFs (see [58] for complementary
calculations). It predicts a soft subshell closure in the $N=32$ nucleus 54Ca
and an ordering of single-particle orbitals in neutron-rich calciums that is
at variance with naïve shell-model expectations. Figure 10 shows the computed
energies of the first excited $J^{\pi}=2^{+}$ state in some isotopes of
calcium and compares them with available data. The high excitation energy in
48Ca is due to its double magicity, and the somewhat increased excitation
energies in 52,54Ca suggest that these nuclei exhibit a softer subshell
closure. Where data are available, the theoretical results agree well with
experiment. For 54Ca, theory made a prediction that has recently been verified
experimentally [59].
Figure 10: Excitation energies of $J^{\pi}=2^{+}$ states in Ca isotopes. The
theoretical results (red squares) agree well with data (black circles) and
predict a soft subshell closure in 54Ca.
### 2.2 Configuration interaction
The nuclear shell model has been very effective in describing the physics of
larger nuclei beyond the current reach of pure ab initio methods; indeed,
Eugene Wigner, Maria Goeppert-Mayer, and J. Hans D. Jensen were awarded the
1963 Noble prize for the fundamental symmetries and mean field features that
underlie the successful nuclear shell model. The shell model for larger nuclei
uses the same configuration interaction methods as the NCSM methods described
previously, but with more truncated model spaces where not all nucleons are
“active” and with effective interactions tailored for these spaces.
Since there are numerous challenging physical applications in nuclear physics
that vary across the periodic table, different CI approaches are needed to
efficiently exploit the available computational resources. CI approaches
developed or improved within UNEDF include the following:
* 1.
No-core shell model in the $m$-scheme basis (mfdn [16, 18, 19]; bigstick [60,
61, 62]).
* 2.
No-core shell model in a coupled angular momentum basis (mfdnj [63, 64]).
* 3.
Shell model with a core in a coupled angular momentum basis (nushellx [65,
62]).
UNEDF took advantage of common elements in the various CI approaches to
improve the effectiveness of the nuclear shell model for all nuclei.
These CI codes utilize an input NN interaction file and a Coulomb interaction
between the protons. They all work in the neutron-proton basis (i.e., break
isospin) and allow for charge-dependent NN interactions. In addition, several
of these codes accept 3NFs as input. All these codes evaluate the spectra,
wavefunctions, and a suite of observables for low-lying states of the nucleus.
The implemented algorithms differ considerably among the codes as well as
support systems for processing the output files generated, such as the
wavefunctions and one-body density matrices, both static and transition.
Numerous cross-comparisons between the codes have been accomplished and their
respective accuracies confirmed. Eigenenergies are obtained to the accuracy of
1 keV or better. Other observables are found to differ at the level of a few
percent because of numerical noise in the wavefunctions.
Except for mfdnj (which followed mfdn), the codes evolved along independent
paths, which emphasized various strategic physics and technological goals. For
example, the challenges of addressing heavier nuclei impel working with a
nuclear core; the challenges of working with leadership-class machines versus
local clusters drive some of the algorithmic decisions. The burden of
communications and memory restrictions help resolve the challenge of store-in-
memory versus recompute-on-the-fly strategies that are implemented differently
in these CI codes.
In light of the need to store large amounts of data for retrieval,
postanalyses, and reproducibility, we have developed a prototype database
management system. This prototype records in the database the metadata of
every run. The data referenced in the database may include physical
observables, one-body density matrices, and wavefunctions that result from the
ab initio codes; such data are typically stored on the platforms where runs
are performed. A user can access this database over the web and find out
whether the runs of interest have already been performed and where the results
may be located.
### 2.3 Nuclear density functional theory
Because of the enormous configuration spaces involved, the properties of
complex heavy nuclei are best described by the superfluid nuclear density
functional theory [66] – rooted in the self-consistent Hartree-Fock-Bogoliubov
(HFB), or Bogoliubov-de Gennes, problem. The main ingredient of nuclear DFT is
the effective interaction between nucleons captured by the energy density
functional. Since the nuclear many-body problem involves two kinds of
fermions, protons and neutrons, the EDF depends on two kinds of densities and
currents [67, 68]: isoscalar (neutron-plus-proton) and isovector (neutron-
minus-proton). The coupling constants of the nuclear EDF are usually adjusted
to selected experimental data and pseudodata obtained from ab initio
calculations. The self-consistent HFB equations allow one to compute the
nuclear ground state and a set of quasiparticles that are elementary degrees
of freedom of the system and that can be used to construct better
approximations of the excited states. The HFB equations constitute a system of
coupled integro-differential equations that can be written in a matrix form as
a self-consistent eigenvalue problem, where the dependence of the HFB
Hamiltonian matrix on the eigenvectors (quasiparticle wavefunctions) induces
nonlinearities.
The atomic nucleus is also an open system having unbound states at energies
above the particle emission threshold, and this has implications for the
nuclear DFT. The finiteness of the HFB potential experienced by a nucleon
implies that the energy spectrum of HFB quasiparticles contains discrete bound
states, resonances, and nonresonant continuum states [69]. The size of the
continuum space may become intractable, especially for complex geometries
where self-consistent symmetries are broken. To this end, one has to develop
methods [70] to treat HFB resonances and nonresonant quasiparticle continuum
without resorting to the explicit computation of all states.
The application of high-performance computing, modern optimization techniques,
and statistical methods has revolutionized nuclear DFT during recent years, in
terms of both developing new functionals and carrying out advanced
applications. Optimizing the performance of a single HFB run is crucial for
making the EDF optimization [71, 72] manageable and quickly computing tables
of nuclear observables [73, 74, 75, 76], in order to assess theoretical
uncertainties. These advances are described in the following sections.
#### 2.3.1 DFT solvers
Solutions of HFB equations can be obtained either by direct numerical
integration on a mesh, provided proper boundary conditions are imposed on the
domain, or by expansion on a basis. For the latter case, the harmonic
oscillator (HO) basis proves particularly well-adapted to nuclear structure
problems, as it offers analytical, localized solutions with convenient
symmetry and separability features. Although solving the HFB equations for a
given nuclear configuration is relatively fast on modern computers, accurate
characterization of nuclear properties often requires simultaneous
computations of many different configurations, from a few dozen (e.g., one-
quasiparticle configurations in odd mass nuclei) to a few billion or more in
extreme applications (such as probing multidimensional potential energy
surfaces of heavy nuclei during the fission process).
The two primary DFT solvers based on HO expansion used by the collaboration
are hfbtho [77] and hfodd [78]; see [79] and [80], respectively, for their
latest releases. Both codes solve the HFB equations for generalized Skyrme
functionals in a deformed HO basis and have been carefully benchmarked against
one another up to the 1 eV level. hfbtho assumes axial and time-reversal
symmetry of the solutions, making it a very fast program (execution completes
in typically less than 1 minute on a single node). It is particularly suited
for EDF optimization (see Sec. 2.3.3) or large-scale surveys of nuclear
properties [74, 75]. The solver hfodd is fully symmetry-unrestricted: this
versatility is necessary for science applications such as the computation of
fission pathways [81] or description of high-spin states [82].
The new versions of each solver benefited significantly from recent advances
in high-performance computing and from collaborations with computer scientists
in UNEDF. By expanding the use of tuned blas and lapack libraries, significant
performance gains were reported for both codes and enabled new, large-scale
studies [83]. The speed of hfbtho was further improved by a factor of 2 by
incorporating multithreading; hfodd was turned into a hybrid MPI/OpenMP
program: nuclear configurations are distributed across nodes, while on-node
parallelism is implemented via OpenMP acceleration.
Figure 11: Algorithmic improvements to hfodd. Top: Convergence for a typical
HFB calculation in the ground state of 166Dy with hfodd version 2.49t [80].
Using the Broyden method to iterate the nonlinear HFB equations has provided
significant acceleration compared with traditional linear mixing techniques.
Bottom: Comparison between the augmented Lagrangian method (black squares) and
the standard quadratic penalty method (open squares) for the constrained HFB
calculations of the total energy surface of 252Fm in a two-dimensional plane
of elongation, $Q_{20}$, and reflection-asymmetry, $Q_{30}$. (From [84].)
Figure 11 illustrates two algorithmic improvements to the DFT solver hfodd.
The implementation of the Broyden method for nonlinear iterative problems [85]
has reduced substantially the number of iterations needed to converge the
solution in practical applications. The second example shows the application
of the augmented Lagrangian method (ALM) to fission in 252Fm [84]. This method
is generally used for constrained optimization problems; it allows precise
calculations of multidimensional energy surfaces in the space of collective
coordinates. Indeed, while the standard quadratic penalty method often fails
to produce a solution at the required values of constrained variables on a
rectangular grid, the ALM performs well in all cases. Both improvements
displayed in Fig. 11 are key to producing realistic large-scale surveys of
fission properties in heavy nuclei on leadership-class computers, where
walltime is limited and expensive.
Another HFB solver developed by UNEDF is hfb-ax. It is based on the B-splines
representation of coordinate space and preserves axial symmetry and space
inversion [86]. The solver has been carefully benchmarked with hfbtho and used
in several applications involving complex geometries, such as fission [87] and
competition between normal superfluidity and Larkin-Ovchinnikov (LOFF) phases
of polarized Fermi gases in extremely elongated traps [88]. Hybrid parallel
programming (MPI+OpenMP) has been implemented in hfb-ax to treat large box
sizes that are important for weakly bound heavy nuclei.
New generations of DFT solvers will be taking advantage of emerging
architectures, such as GPUs, and new programming paradigms. In particular, the
cost of performing dense linear algebra in both hfbtho and hfodd can become
prohibitive as the size of the HO basis increases, especially for more
realistic energy functionals involving some form of nonlocality; this
necessitates novel techniques to handle many-body matrix elements [89]. The
massive amount of data generated by large-scale DFT simulations will also
require significant investments in visualization and data-mining techniques.
#### 2.3.2 Multiresolution 3D DFT framework
Figure 12: Quasiparticle wavefunction for a DFT simulation (left, top) and its
six levels of multiresolution structure (left, bottom). The refinement
structure is especially noticeable at levels 5 and 6. Right: The parallel
speedup of one iteration of madness-hfb, for solving the DFT problem for 1,640
3D quasiparticle wavefunctions with over 4.4 billion equations and unknowns;
this simulation was performed within a box with a spatial dimension of 120
fermis, using 8 multiwavelets, up to level 8+ of refinement, and with a
relative precision of $10^{-6}$.
A parallel, adaptive, pseudospectral-based solver, madness-hfb, has been
developed to tackle the fully symmetry-unrestricted HFB problem for both real
and complex wavefunctions in large and asymmetric boxes. The main mathematical
and algorithmic advantage of madness-hfb is its multiscale-multiresolution and
sparse approximation of functions and the application of operators in
coordinate space with guaranteed accuracy but finite precision. madness-hfb
prefers to work with functions and operators with pseudo-spectral
approximations based on a multiwavelet basis (up to order 30). Since the
multiwavelets consist of smooth, singular, and discontinuous functions with
spatial locality (compact support), they are well suited for localized
approximation of weak singularities and discontinuities or regions of high
curvature [90, 91, 92]. Gibbs effects are also reduced. The object-oriented
(OO) nature of the software and template-based programming allow each
wavefunction and each integral or differential operator to have its own
boundary condition and its own sparse pseudospectral expansion. The usual
boundary conditions (e.g., Dirichlet, Neumann, Robin, quasi-periodic, free,
and asymptotic conditions) are supported. Fast applications of Green’s
function for the direct solution of Poisson’s equation and the Yukawa
scattering kernel are available [93, 94, 95]. In the multiwavelet
representation, these approximate Green’s functions and their applications are
again based on sparse data with guaranteed precision, in contrast to dense
tensors based on the use of some other basis sets. Other Green’s functions can
also be constructed.
If desired, the user can specify solvers and routines from other dense and
sparse linear algebra packages such as lapack or scalapack. For example,
parallel and vectorized adaptive quadrature permit the construction of the
Hamiltonian matrix in the usual manner by using the $\ell_{2}$ norm. The
Hamiltonian can be diagonalized by using multithreaded lapack (or a parallel
eigensolver), and the eigenvectors can be converted back to coefficients for
the multiwavelet representation. Other capabilities, such as high-order
approximation of propagators and time-stepping required for the solution of
time-dependent DFT, are also available from applications in time-dependent
molecular DFT, as well as from simulation of attosecond dynamics [96, 97].
Underlying this mathematical capability is a parallel runtime system that
permits the software to scale to hundreds of thousands of processors and runs
on platforms from laptops to leadership-class computers. The ability to use
laptops and workstations is particularly attractive for model and code
development and testing. In addition, the embedding of a parser permits the
OO-based C++ templated codes representing operations on the coefficients of
each wavefunction to be executed as parallel tasks. This parser permits out-
of-order, distributed multithread executions with task- and data-dependency
analysis. This reduces the stalling of execution units due to data
dependencies. A user-configured and executed parallel load-balancing method is
also available, as is a parallel checkpoint and restart method.
The 3D madness-hfb has been benchmarked with the spline-based 2D solver hfb-ax
[86], 3D hfodd [80], and the 1D code hfbrad [98] for a variety of problems.
Because madness-hfb has no limit on the size of the computational domain, we
were able to capture quasiparticle wavefunctions with long tails or
nonsymmetric potentials with steep curvatures and cut-offs to overcome some of
the limitations of the other solvers. The adaptive structure is illustrated in
Fig. 12.
The current madness-hfb approach to the HFB problem is as follows [99]. Let
the coefficients of the wavefunctions in the tensor product multiwavelet
representation be the unknowns. The user provides an initial relative
precision, a set of initial wavefunctions (e.g., in terms of the HO basis,
splines, etc.), and boundary conditions to start the iterative procedure. All
the functions, potentials, operators, and expansion lengths are adaptively
represented as needed by the user-defined precision. A generalized matrix
eigenvalue problem is formed by adaptive quadrature. The solution eigenvectors
are converted to a sparse multiwavelet representation for updating the lengths
of the expansion and the coefficients in the potentials, gradients, and other
terms before the next iteration and diagonalization. The speed and performance
depend on the number of coefficients. Usually, the simulations begin with a
low relative precision, to capture the low-order terms quickly, before
adaptively increasing the order of approximation and the precision for more
accurate results.
#### 2.3.3 EDF optimization
One of the focus areas of UNEDF was the development of an optimization
protocol for determining the coupling constants of nuclear EDFs. In
particular, the collaboration paid special attention to estimating the errors
associated with such a procedure and exploring the correlations among the
coupling constants. The UNEDF optimization protocol was established by
focusing on the Skyrme energy density. We recall that, in this framework, the
energy of an even-even nucleus in its ground state is a functional of the one-
body density matrix and the pairing tensor. The Skyrme energy density reads
$\displaystyle\chi_{t}(\bm{r})$ $\displaystyle=$ $\displaystyle
C_{t}^{\rho\rho}\rho_{t}^{2}+C_{t}^{\rho\tau}\rho_{t}\tau_{t}+C_{t}^{J^{2}}\bm{J}_{t}^{2}$
(1) $\displaystyle+C_{t}^{\rho\Delta\rho}\rho_{t}\Delta\rho_{t}\
+C_{t}^{\rho\nabla J}\rho_{t}\bm{\nabla}\cdot\bm{J}_{t},$
where the isospin index $t$ labels isoscalar ($t$=0) and isovector ($t$=1)
densities, $\rho_{t}$ is the one-body density matrix, and $\tau_{t}$ and
$\bm{J}_{t}$ are derived from $\rho_{t}$ [67]. In the pairing channel, we took
a density-dependent pairing energy density with mixed surface and volume
nature, characterized by the two pairing strengths $V_{0}^{(n)}$ and
$V_{0}^{(p)}$ for neutrons and protons, respectively. The set of coupling
constants $C_{t}^{uu^{\prime}}$, $V_{0}^{(n)}$, and $V_{0}^{(p)}$ are the
parameters $x$ to be determined.
The development of fast DFT solvers (see Sec. 2.3.1), together with the
availability of leadership-class computers, permitted us for the first time to
set up an optimization protocol at a fully deformed HFB level. Our first
parametrization, unedf0, was obtained by considering only three types of
experimental data: nuclear binding energies of both spherical and deformed
nuclei, nuclear charge radii, and odd-even mass differences in selected nuclei
[71]. After recognizing that deformation properties needed to be better
constrained [100], a fourth data type, corresponding to excitation energies of
fission isomers in the actinides, was added. The resulting parametrization,
unedf1, gave a significantly better description of fission properties [72],
see Fig. 13 (bottom). With the oncoming unedf2 parametrization, we will expand
the optimization data set with single-particle level splittings. The new data
are expected to better constrain the tensor coupling constants and improve
single-particle properties.
Figure 13: Top: Performance of the pounders algorithm on the minimization of
the $\chi^{2}$ of Eq. (2) as compared with the standard Nelder-Mead method.
Bottom: Fission pathway for 240Pu along the mass quadrupole moment $Q_{20}$
calculated with SkM∗, unedf0, and unedf1 EDFs. The experimental energy of
fission isomer ($E_{II}$) and the inner ($E_{A}$) and outer ($E_{B}$) barrier
heights are indicated [72].
Formally, we solve the optimization problem
$\min_{x}\left\\{\chi^{2}(x)=\sum\limits_{i=1}^{n_{d}}\left(\frac{s_{i}(x)-d_{i}}{w_{i}}\right)^{2}:x\in\Omega\subseteq\mathbb{R}^{n_{x}}\right\\},\,$
(2)
where $d\in\mathbb{R}^{n_{d}}$ represents the experimental data, $w>0$
represent weights, and the parameters $x$ to be determined are possibly
restricted to lie in a domain $\Omega$. This problem is made difficult because
some of the derivatives with respect to the parameters $x$,
$\nabla_{x}s_{i}(x)$, may be unavailable for some of the theory simulation
observables $s_{i}$.
Traditional approaches for solving (2) in the absence of derivatives typically
either estimate these derivatives by finite differencing or treat $\chi^{2}$
as a black-box function of $x$. The former approach can be sensitive to the
choice of the difference parameter, and care must be taken that the expense of
the differencing does not grow unnecessarily as the number of parameters
$n_{x}$ grows. The latter neglects the structure (in the form of the $n_{d}$
residuals) inherent to (2).
In UNEDF, we instead employed a new optimization solver, pounders, that
exploits the structure in nonlinear least-squares problems and avoids directly
forming computationally expensive derivative approximations. pounders follows
a model-based Newton-like approach, where the first- and second-order
information is inferred by iteratively forming local interpolation models for
each residual. Figure 13 (top) shows the efficiency of the solver: not only
does it converge faster than the standard Nelder-Mead algorithm, but it also
gives a more accurate solution. pounders is available in the open-source
Toolkit for Advanced Optimization (TAO [101]).
#### 2.3.4 Neutron droplets and DFT
The properties of homogeneous and inhomogeneous neutron matter play a key role
in many astrophysical scenarios and in the determination of the symmetry
energy [102, 103, 104]. The equation of state of homogeneous neutron matter
has been studied in many earlier investigations (see, e.g., [105]). Since
neutron matter is not self-bound, inhomogeneous neutron matter has been
theoretically investigated by confining neutrons in external potentials.
Although neutron drops cannot be realized in experimental facilities, they
provide a model to study neutron-rich isotopes [106, 107, 108] and can bridge
ab initio methods and DFT. The external potential confining neutrons has been
chosen to change the geometry and density of the system. A Woods-Saxon form
produces saturation, making neutron drops similar to ordinary nuclei. Instead,
a harmonic potential permits one to better control the calculation of larger
systems and to test the approach to the thermodynamic limit.
Figure 14: Calculated total energies for neutron droplets in
$\hbar\omega=5\,{\rm MeV}$ and $10\,{\rm MeV}$ harmonic potentials as a
function of the neutron number $N$. The figure shows AFDMC, GFMC, SLy4, and
adjusted SLy4 results of [109] together with the unedf0 and unedf1 results.
Nuclear EDFs are commonly optimized to reproduce properties of nuclei close to
stability, with close numbers of protons and neutrons. The use of such
functionals to study neutron-rich nuclei or the neutron star crust requires
large extrapolations in neutron excess. In [109], neutron droplets were
studied by using QMC methods starting from a realistic nuclear Hamiltonian
that includes the Argonne AV8’ two-body interaction supported by the Urbana IX
three-body force. This Hamiltonian fits nucleon-nucleon phase shifts, gives a
satisfactory description of light nuclei, and produces an equation of state of
neutron matter that is compatible with recent neutron star observations [110].
The neutron drop’s energy calculated by using QMC methods was compared with
DFT calculations. The QMC results showed that commonly used Skyrme EDFs
typically overbind neutron drops and that this effect is due mainly to the
neutron density gradient term. The adjustment of the gradient together with
the pairing and spin-orbit terms improves the agreement between ab initio QMC
calculations with Skyrme both for the energy and for neutron densities and
radii [109].
These results can be compared with the predictions of unedf0 and unedf1 EDFs.
Figure 14 shows the calculated total energies for neutron droplets in
$\hbar\omega=5\,{\rm MeV}$ and $10\,{\rm MeV}$ harmonic potentials. The
auxiliary field diffusion Monte Carlo (AFDMC) and GFMC QMC results of [109],
calculated with the AV8’+UIX interactions, agree well with the DFT
calculations [72]. These are encouraging, since neither unedf0 nor unedf1 was
optimized to the pure neutron matter data. Future EDF optimization schemes
will use ab initio results on neutron droplets as pseudodata to improve EDF
properties in very neutron-rich nuclei.
#### 2.3.5 Ab initio functionals
Figure 15: Top: Deformation energy curves for 100Zr calculated using
microscopic EDFs derived from chiral EFT interactions at different orders
[111]. Bottom: Comparison of microscopic EDF calculations of neutron drops at
increasing levels of approximation with full NCFC calculations starting from
the same Hamiltonian [112].
In parallel with efforts to improve the optimization of nuclear EDFs with
conventional Skyrme-type terms, UNEDF members sought to construct ab initio
functionals based on microscopic chiral effective field theory (EFT) [113]. A
pathway to such functionals was opened with the development of new
renormalization group methods, which led to softer nuclear Hamiltonians,
including three-body forces [114]. These soft interactions dramatically
improve convergence properties in many-body calculations [115], extending the
reach of ab initio methods to heavier systems [116, 117, 118]. At the same
time, they make feasible the construction of a microscopically based EDF using
many-body perturbation theory [119] together with improved density matrix
expansion (DME) techniques [120, 121, 122]. Carrying out this long-term
program by individual researchers would be a formidable task, but progress was
made possible within UNEDF by teaming up with two of the physics–CS/AM
partnerships described earlier.
An intermediate step toward a fully ab initio EDF was a new hybrid functional
that incorporated long-range chiral EFT interactions to describe pion-range
physics and a set of Skyrme-like contact interactions with coupling constants
to be fit. The resulting functional has a much richer set of density
dependencies than do conventional Skyrme functionals. These were incorporated
in the DFT solvers, and new preoptimization procedures were developed by the
DFT functional group [111]. A proof-of-principle test in the top panel of Fig.
15 shows deformation energies in 100Zr calculated using the DME functional at
different orders in the chiral expansion (LO, NLO, N2LO). The deviations from
the Skyrme result show nontrivial effects from the finite-range nature of the
underlying NN and 3N interactions [111]. On-going work includes a rigorous
optimization with the procedure outlined in Sec. 2.3.3 and then detailed
evaluations of the predictive power of the DME functional.
In order to directly validate the new DME procedures used in [111], it was
necessary to benchmark against exact results. The first-ever such calculations
were made possible by teaming up with the NCSM–mfdn effort (see Sec. 2.1.2)
using neutron droplets as a controlled theoretical test environment as in Sec.
2.3.4. The DME functional was constructed and evaluated for the _same_ (model)
Hamiltonian used to generate exact results from mfdn [112] for different
numbers of neutrons and varied traps. Figure 15 (bottom) shows the agreement
between no-core full configuration (NCFC) results and microscopic EDF
calculations at different levels of approximation [112], which validates the
optimal strategy used to construct a microscopically based EDF (the points
labeled “fit”), while establishing theoretical error bars. Further important
DME developments made by external collaborators in the FIDIPRO project [123,
124] will be tested in future investigations.
### 2.4 Beyond DFT
Static DFT provides excellent tools for investigating nuclear binding energies
and other ground-state properties. In certain cases, it also can be used to
treat dynamical processes. The path to scission during fission, for example,
sometimes can be predicted accurately by static DFT. A reliable description of
excitation/decay and reactions, however, usually requires methods that go
beyond static DFT. Since an ab initio treatment of the nuclear time-evolution
is difficult, we employ extensions of DFT and related ideas. The simplest
extension, the quasiparticle random phase approximation (QRPA), can be viewed
as an adiabatic approximation to the linear response in time-dependent DFT. It
provides the entire spectrum of excitations with the same EDF used in static
DFT. The adiabatic approximation is, of course, severe (as are the
approximations in the density functional itself) but can be applied in any
nucleus and folded with reaction theory. DFT-based QRPA and its applications
to nuclear excitation and reactions are discussed in Sec. 2.4.1.
DFT-based methods that go beyond the adiabatic approximation are also now in
use. One can exploit the relatively simple dynamics of Fermi gas systems to
construct an approximate time-dependent extension of DFT, the time-dependent
superfluid local density approximation (TDSLDA). The approximation and related
computational techniques can be applied to such classic problems as
photoabsorption but also to other time-dependent processes that go beyond
linear response. The TDSLDA and its applications are discussed in Sec. 2.4.2.
We also need efficient methods to accurately compute average properties of
excited states, such as spin- and parity-dependent level densities, which
suffice to treat reactions that proceed primarily through a compound nucleus.
Obtaining these densities through a direct diagonalization of the nuclear
Hamiltonian and a subsequent level counting is not efficient, but several
techniques based on statistical spectroscopy can be used instead. However,
even statistical spectroscopy poses computational challenges that demand high-
performance computational techniques and resources. Some advances in
computational spectroscopy, leading to the first accurate calculation of
densities of levels with unnatural parity, are described in Sec. 2.4.3.
#### 2.4.1 QRPA and reactions
Members of the UNEDF collaboration developed and exploited both an extremely
accurate spherical Skyrme QRPA code [125] and an equally accurate, though
computationally much more intensive, deformed (axially symmetric) Skyrme QRPA
code [126]. The latter, which can treat both spherical and deformed nuclei, is
at the forefront of the modern QRPA. Other groups have developed their own
versions of the deformed Skyrme, Gogny, or relativistic QRPA [127, 128, 129,
130, 131]; most of these have some disadvantages compared with ours (e.g., a
lack of full self-consistency, oscillator bases that don’t capture continuum
physics, etc.) but also the occasional advantage (e.g., full continuum
wavefunctions rather than the approximate representation of the continuum we
describe below).
Both our spherical and deformed codes diagonalize the traditional QRPA A-B
matrix [132], constructed from single-quasiparticle states in the canonical
basis [132] in a large box (typically 20 fm in each coordinate), so that
continuum states are taken into account in discretized form. Both codes work
with arbitrary Skyrme density functionals plus delta pairing, include all
rearrangement terms, and break neither parity nor time-reversal symmetries.
Both output transition amplitudes to the entire spectrum of excited states.
The two codes have some differences as well. The spherical code gets its
single-quasiparticle wavefunctions, represented on an equidistant mesh, from
an HFB program called hfbmario, which derives from the code hfbrad [69]. The
deformed code takes its wavefunctions from the Vanderbilt HFB program [133],
which uses B-splines to represent wave functions. Each QRPA code represents
those wavefunctions in the same manner as the HFB programs it relies on.
Both QRPA codes have been tested in many ways, including against one another.
With the spherical code, we calculated energy-weighted sums in Ca, Ni, and Sn
isotopes from the proton drip line through the neutron drip line for
$J^{\pi}=0^{+},$ $1^{-}$, and $2^{+}$ multipoles with Skyrme parameter sets
SkM∗ and SLy4 and found excellent agreement with analytical values [125].
Spurious states in the $J^{\pi}=0^{+}$, $1^{+}$, and $1^{-}$ channels are well
separated from physical states in both codes, though the spherical one
performs a bit better because it can include all combinations of HFB two-
quasiparticle states in the QRPA basis without making the calculation
intractable.
The collaboration used the spherical QRPA to study systematics of $2^{+}$
states across the table of isotopes and for microscopic calculations of
reaction rates; they used the deformed version for a more limited study of
$2^{+}$ states and giant resonances in rare-earth nuclei [134].
The collaboration also used transition densities from the spherical QRPA to
calculate nucleon-nucleus scattering. The transition amplitudes produced by
our spherical matrix QRPA, when combined with single-particle wave functions,
yield radial transition densities. These can in turn be folded with the
interaction between the projectile and the nuclear constituents (i.e., the
nucleon-nucleon interaction) to produce transition potentials that excite
target states. References [135, 136, 137] report the development of a code to
fold the densities for all QRPA states below 30 MeV with a Gaussian-shaped
nucleon-nucleon potential. The result is a microscopic coupled-channels
calculation that successfully produces angular distributions and inelastic
cross sections for nucleon-induced reactions—quantities that can be compared
directly with scattering data—at scattering energies between 10 and 70 MeV. To
satisfactorily describe observed absorption, we had to explicitly couple also
to all one-nucleon pickup channels leading to intermediate deuteron formation.
Figure 16 illustrates the effect of such couplings on nucleon-induced
absorption cross sections. The direct connection between the calculated cross
sections and the nuclear structure ingredients makes this kind of reaction
calculation a good test of the structure model.
Figure 16: Reaction cross section as a function of incident energy for $p$ \+
90Zr. The results are shown for couplings to the inelastic states (dash-dotted
line) and to the inelastic and transfer channels with nonorthogonality
corrections (solid line). The Koning-Delaroche [138] optical model
calculations are also shown (dashed line). (Data from [139].)
The collaboration also took significant steps to develop a much more efficient
implementation of the QRPA. The finite amplitude method [140, 141] allows one
to effectively take the derivatives of mean fields that enter the QRPA
equations numerically, through relatively straightforward modifications to the
mean-field codes themselves. A simple iterative procedure then solves the
equations. Our initial application, to monopole resonances in the deformed
nucleus 240Pu [142], consumes a small fraction of the time our matrix QRPA
implementation would use (see [143, 144] for complementary work based on
iterative Arnoldi diagonalization).
#### 2.4.2 Time-dependent DFT for superfluid systems
The application of DFT to nuclear physics requires two nontrivial elements:
the ability to describe both superfluidity and time-dependent phenomena. In
order to avoid the nonlocal character of the DFT extension to superfluid
systems, the superfluid local density approximation (SLDA) and its time-
dependent extension TDSLDA have been developed [145, 146, 147, 148, 149, 150,
151, 152, 153, 154].
SLDA and TDSLDA have been applied to a large number of fermionic systems and
phenomena: vortex structure in neutron matter and cold atomic systems,
generation and dynamics of quantized vortices and their crossing and
reconnection, excitation of the Anderson-Higgs modes, the LOFF phase, quantum
shock waves and excitation of domain walls, one- and two-nucleon separation
energies, giant dipole resonance in superfluid triaxial nuclei, and complex
collisions. In Fig. 17, we illustrate the case of a head-on collision of two
superfluid fermion clouds, which was studied experimentally. Both SLDA and
TDSLDA are derived by using appropriately determined EDFs with QMC input for
homogeneous systems and validating the predictions on independent QMC
calculations of inhomogeneous systems in the well-studied case of a unitary
Fermi gas; see [147, 148, 150] for details. The form of the EDF for a unitary
Fermi gas is largely determined by dimensional arguments; translational,
rotational symmetry, and parity; gauge and Galilean covariance (which
specifies the dependence on current densities); and renormalizability of the
TDSLDA formalism.
For nuclear systems we lack ab initio results of the same quality and rely on
a more phenomenological approach, but with significant microscopic input. The
nuclear EDFs should satisfy the usual symmetries [68] and the consistency with
the best available ab initio results.
The numerical implementation of the SLDA and TDSLDA equations leads to
hundreds of thousands of coupled nonlinear 3D time-dependent PDEs, which are
solved by using the discrete variable representation approach [155, 156] on
desktops [147, 148, 149, 150] and—as a result of UNEDF collaborations with
computer scientists—leadership-class supercomputers [150, 151, 152, 153, 154].
In Fig. 18 we illustrate the first calculation of the photoexcitation of a
triaxial superfluid nucleus performed within TDSLDA (188Os) and two other
axially deformed nuclei, as well as a comparison with the absolute
experimental data (without any fitting parameters). The determination of the
ground-state properties of these nuclei and their subsequent time-evolution
required full diagonalizations of Hermitian matrices of sizes up to $5\cdot
10^{5}\times 5\cdot 10^{5}$ and solutions of $5\cdot 10^{5}$ coupled time-
dependent 3D PDEs. Further studies of excitation of medium- and heavy-mass
nuclei with $\gamma$-rays, neutrons, relativistic heavy ions, and induced
nuclear fission are the next steps.
Figure 17: Three consecutive frames of the head-on collision of two fermion
clouds of $\approx$750 particles in which quantum shock waves and domain
walls/solitons (topological excitations) are formed [152]. The $x$\- and
$y$-directions have an aspect ratio of $\approx$30. Figure 18: Photoabsorption
cross section (solid black line) calculated within TDSLDA using two Skyrme
force parameterizations for three deformed open-shell nuclei and the
experimental $(\gamma,n)$ cross sections (solid purple circles with error
bars); see [153] for details. With dashed (green), dotted (red), and dot-
dashed (blue) lines, we display the contribution to the cross section arising
from exciting the corresponding nucleus along various symmetry axes.
#### 2.4.3 Level densities
The properties of the excited states of nuclei are key to reliably describing
reactions and decays. One important type of reaction mechanism is the compound
nuclear reaction, which can be described with the statistical model of Hauser
and Feshbach [157]. The important ingredient entering the Hauser-Feshbach
theory is the spin- and parity-dependent nuclear level density (NLD).
Experimental information about NLD is limited for stable nuclei and not
available for radioactive nuclides of interest for nuclear astrophysics.
Therefore, a large effort is underway to accurately calculate NLD, and an
interacting shell model approach would be the best model taking into account
the relevant many body correlations beyond DFT. A direct approach by direct CI
diagonalization and level counting is not feasible because of the exponential
increase in CI dimensions. We recently proposed [158] an approach to calculate
shell model spin- and parity-dependent NLD using methods of statistical
spectroscopy. In addition, we showed [159] how one can improve this approach
to calculate the unnatural parity NLD by removing the contribution due to the
spurious center-of-mass excitations. The associated algorithms were
implemented in a high-performance computer code, jmoments, [160, 161, 162],
which runs on massively parallel computers and scales well up to 10,000
processors [160, 162].
Figure 19: Nuclear-level densities for positive parity (red curve) and
negative parity (black curve) of 26Al compared with experimental data; the
solid and dotted staircases represent upper and lower limits, respectively.
Positive parity NLD is larger than negative parity NLD.
Figure 19 shows positive- and negative-parity NLD for 26Al calculated with
jmoments compared with the available experimental data obtained by level
counting. Some known levels have no clear assignment of the parity, which
leads to upper and lower limits. The calculated positive-parity NLD is not
new, an $sd$-shell calculation being available for some time. However, the
negative-parity NLD was calculated only recently by our approach [161].
## 3 Uncertainty Quantification
Figure 20: Top: unedf1 correlation matrix. Presented are the absolute values
of the correlation coefficients between the parameters characterizing the
energy density (1). Bottom: Theoretical extrapolations toward drip lines for
the two-neutron separation energies $S_{2n}$ for the isotopic chain of even-
even Zr isotopes using different EDFs (sly4, sv-min, unedf0, unedf1) [76] and
frdm [163] and hfb-21 [164] mass models. Detailed predictions around
$S_{2n}=0$ are illustrated in the inset. The bars on the sv-min results
indicate statistical errors due to uncertainty in the coupling constants of
the functional.
Uncertainty quantification is a key element for assessing the predictive power
of a model. When working with effective theories with degrees of freedom
relevant to the problem, the parameters of the theoretical model often need to
be adjusted to the empirical input. To quantify the model uncertainties,
sensitivity analysis yields the standard deviations and correlations of the
model parameters, usually encoded as a covariance matrix [165, 166, 167, 168].
The calculation of the covariance matrix requires computing derivatives of the
observables with respect to the model parameters. When a closed-form
expression for the derivatives is not available, we estimate the derivatives
numerically using finite differences. To account for the numerical uncertainty
associated with the underlying DFT-based calculations, we compute the “noise
level” of each observable following the approach in [169]. The difference
parameters used for estimating the Jacobian matrix associated with (2) are
then obtained using these noise levels [170].
Uncertainty quantification was one of the key topics of the EDF optimization
work performed in the UNEDF collaboration [71, 72]. The upper panel of Fig. 20
shows the unedf1 correlation matrix, obtained from the sensitivity analysis.
As can be seen, some of the surface parameters of the unedf1 EDF are strongly
correlated. In [76] we used this information to assess the robustness of the
current EDFs in the predictions of the nuclear landscape limits. This is
illustrated in the lower panel of Fig. 20, which shows calculated and
experimental two-neutron separation energies for the isotopic chain of even-
even zirconium isotopes. The differences between model predictions are small
in the region where data exist and grow steadily when extrapolating toward the
two-neutron drip line ($S_{2n}=0$). Nevertheless, the consistency between the
models was found to be surprisingly good. This study required massive parallel
calculations of the nuclear mass tables [75].
## 4 High-Performance Computing Resources
UNEDF science has benefited from access to some of the largest computers in
the world, provided primarily by DOE’s Innovative and Novel Computational
Impact on Theory and Experiment (INCITE) program [171]. In particular, the
largest computations of UNEDF were carried out on the “Jaguar” machine at Oak
Ridge National Laboratory and the “Intrepid” machine at Argonne National
Laboratory. Jaguar has gone through several processor upgrades during the
project, taking it from 30,976 cores (Cray XT4 in 2008) to 298,592 cores (Cray
XK6 in 2012); Intrepid is an IBM Blue Gene/P with 163,840 processing cores.
Figure 21: UNEDF allocation and utilization (in millions of core-hours) of
leadership-class computing resources from 2008 to 2013.
Figure 21 shows the UNEDF utilization of these computing resources over the
years 2008-2013 provided through INCITE. The figure highlights the increasing
demand for computing time in low-energy nuclear physics research. The combined
2008 INCITE utilization across Jaguar and Intrepid was nearly 20 million core-
hours and by 2012 had increased fourfold. This growth illustrates the
increasing application of high-performance computing in nuclear theory enabled
by the physics/computer science/applied mathematics collaborations fostered by
UNEDF.
For the 2013 calendar year, members of the SciDAC-3 NUCLEI project [172] were
granted the sixth largest allocation of the 61 INCITE projects awarded, with a
total allocation of 155 million core-hours across three leadership-class
computing resources, Titan, Mira, and Intrepid. Titan is a Cray XK7, a hybrid
CPU-GPU system with 299,008 CPU cores and 261,632 GPU streaming
multiprocessors, and Mira is an IBM Blue Gene/Q with 786,432 processing cores.
The substantial changes to computing systems at both Argonne and Oak Ridge,
indicative of future trends in high-performance computing, create new
computational challenges but also new possibilities to achieve larger and more
accurate calculations. Through the close collaborations enabled through UNEDF,
and now NUCLEI, members are working to continuously scale codes to increase
physics capabilities and improve performance for efficient utilization of
these leadership-class resources.
## 5 Conclusions
The examples presented here illustrate the multifaceted outcomes of the UNEDF
project, both in terms of landmark calculations of nuclear structure and
reactions and in terms of how nuclear theory is done. The project was very
productive, as can be assessed by going to the project’s website,
http://unedf.org, which documents the concrete deliverables of UNEDF:
publications, highlights, reports, conference presentations, and computer
codes. UNEDF also placed great importance on recruiting the next generation of
scientists. Annually it provided training to 30 young researchers. The UNEDF
experience has been a springboard for advancement, with many UNEDF postdocs
obtaining permanent positions at U.S. universities, national laboratories, and
overseas institutions.
Figure 22: Physics and computing in NUCLEI. The major areas of research are
marked, together with connections between them and theoretical and
computational tools. For more details, see [172].
By fostering broad new collaborative efforts between physicists,
mathematicians, and computer scientists, the SciDAC-2 UNEDF project showed how
to tackle scientific, algorithmic, and computational challenges in the era of
extreme-scale scientific computing. This effort continues with the SciDAC-3
NUCLEI project [172], which builds on the successful strategies of UNEDF.
Figure 22 shows the key elements of NUCLEI.
## Acknowledgments
Support for the UNEDF and NUCLEI collaborations was provided through the
SciDAC program funded by the U.S. Dept. of Energy (DOE), Office of Science,
Advanced Scientific Computing Research and Nuclear Physics programs. This work
was also supported by DOE Contract Nos. DE-FG02-96ER40963 (Univ. Tenn.), DE-
AC52-07NA27344 (LLNL), DE-AC02-05CH11231 (LBNL), DE-AC05-00OR22725 (ORNL), DE-
AC02-06CH11357 and DE-FC02-07ER41457 (ANL), DE-FC02-09ER41584 (Central
Michigan Univ.), DE-FC02-09ER41582 (Iowa State Univ.), DE-FG02-87ER40371 (Iowa
State Univ.), and DE-FC02-09ER41586 (Ohio State Univ.). This research used the
computational resources of the Oak Ridge Leadership Computing Facility (OLCF)
at ORNL and Argonne Leadership Computing Facility (ALCF) at ANL provided
through the INCITE program. Computational resources were also provided by the
National Institute for Computational Sciences (NICS) at ORNL, the Laboratory
Computing Resource Center (LCRC) at ANL, and the National Energy Research
Scientific Computing Center (NERSC) at LBNL.
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|
arxiv-papers
| 2013-04-12T19:32:07 |
2024-09-04T02:49:44.279147
|
{
"license": "Public Domain",
"authors": "Scott Bogner, Aurel Bulgac, Joseph A. Carlson, Jonathan Engel, George\n Fann, Richard J. Furnstahl, Stefano Gandolfi, Gaute Hagen, Mihai Horoi,\n Calvin W. Johnson, Markus Kortelainen, Ewing Lusk, Pieter Maris, Hai Ah Nam,\n Petr Navratil, Witold Nazarewicz, Esmond G. Ng, Gustavo P.A. Nobre, Erich\n Ormand, Thomas Papenbrock, Junchen Pei, Steven C. Pieper, Sofia Quaglioni,\n Kenneth J. Roche, Jason Sarich, Nicolas Schunck, Masha Sosonkina, Jun\n Terasaki, Ian J. Thompson, James P. Vary, Stefan M. Wild",
"submitter": "Stefan Wild",
"url": "https://arxiv.org/abs/1304.3713"
}
|
1304.3722
|
# Hierarchy of Frustrations as Supplementary Indices in Complex System
Dynamics, Applied to the U.S. Intermarket
Krzysztof Sokalski [email protected] Institute of Computer Science,
Czȩstochowa University of Technology, Al. Armii Krajowej 17, 42-200
Czȩstochowa, Poland
###### Abstract
A definition of frustration is expressed by transitivity of binary
entanglement relation in considered complex system. Extending this definition
into n-ary relation a hierarchy of frustrations’ notions is derived. As a
complex system the U.S. Intermarket is chosen where the correlation
coefficient of Intermarket indices’ sectors play the role of entanglement’s
measure. In each hierarchy level the frustration and the transitivity are
interpreted as values of an order’s measure for corresponding subsystem. The
derived theory is applied to 1983-2012 data of the U.S. Intermarket.
###### pacs:
89.65.Gh, 89.75.-k, 71.45.Gm
## Introduction
Frustration represents situation where several optimization conditions compete
with each other so that a system can not satisfy them simultaneously.
Frustrated systems are characterized by the presence of metastable states
among which the system ”hesitates” to choose. These metastable states often
change their order of stability as a function of external parameters to
exhibit phase transitions from one state to another (bib:KAWAMURA, ). In the
21st century, the research of frustrations has received a revived interest and
new areas. Frustrations appear in different systems and different scales. In
some systems such as modern magnetic materials and superconductors the
frustrations are sources of their expected properties (bib:Cond.Matt, ) The
frustrations are observed also in nature phenomena on the level of molecular
scales (bib:bryng, ). The most recent discovers of frustrations have been done
in Markets. In these systems the frustrations play crucial role in creation of
realistic market’s models (bib:ahlg, ). In this paper we introduce our own
interpretation of the frustration in a plaquette consisting of complex
system’s components. Definition of frustration is expressed by transitivity of
the correlation relation in considered system. Extending this definition into
transitivity of the $m-$ary relation (bib:pickett, ),(bib:usan,
),(bib:cristea, ) we create a hierarchy of frustrations describing the whole
complex system consisting of the $m$ components and its subsystems consisting
of $m-1,m-2,\dots 3,2$ components, respectively. Notion of the transitivity
and frustration are opposite magnitudes of an attribute characterizing $n$
bodies correlation in considered subsystems of the hierarchy, where
$n=m,m-1,m-2,\dots,3,2$. In order to perform investigations of this hierarchy
with respect to the transitivity and frustration we need an appropriate
complex system and empiric data for its members. The paper is organized in the
following way. Section I approaches both system and data. In Section II using
the notion of the transitivity we define frustration on different levels of
hierarchy using notion of the transitivity. So far the transitive and
frustrated subsystems are labelled by the two numbers +1 and -1, respectively.
In order to make them more realistic we introduce in Section III extended
measures of transitivity varying in the continuous domain $[-1,+1]$. Section
IV presents interpretation of transitivity as ordering relation. finally, in
Section V using results of Sections II and III we analyse the 1983-2012 data
of the U.S. Intermarket with respect to transitivity and frustration.
## I 1983-2012 Empiric Data of the Considered U.S. Intermarket
An appropriate complex system should satisfy the following conditions:
1) The system and its subsystems produce data according to probability-based
regime,
2) Produced data should be homogenous, reliable and complete,
3) The data produced by different subsystems should be time synchronous.
We would also like to have data which are easy to get and cheap. Following
Murphy (bib:murp, ) we complete his U. S. Intermarket with Gold. In this way
we choose system which satisfies our requirements. All considerations being
done here base on the data supplied by (bib:NICK, ). On basis of the 1987
Crash’s data of the U. S. markets, Murphy derived the concept of the
Intermarket Technical Analysis involving four sectors. Before the Murphy
invention many people applied very simple technical analysis, such as program
trading and portfolio insurance. Such simple analysis could not predict the
forthcoming stock collapse. The events of 1987 provide a textbook example of
how the intermarket scenario works and make a compelling argument as to why
stock market participants need to monitor the other three market sectors-the
dollar, bonds, commodities and others. The fact that the equity collapse was
global in scope, and not limited to the U.S. markets, many would seem to argue
against such a narrow view and finally supplied enough arguments for the
global analysis of the considered markets. Therefore, Murphy has focused on
the commodity, bond, stock and currency markets, globally. Among the many
conclusions, He presents many arguments that the U.S. dollar contributed to
the weakness in equities. Moreover, he concludes that among the four
considered sectors the role of the U.S. dollar is probably the least precise
and the one most difficult to pin down (bib:murp, ). However, all Murphy’s
considerations are done on the basis of binary relations resulting from the
binary correlations. In this paper we are going to extend his Intermarket
Technical Analysis onto a hierarchy of relations including two, three,
four,…..$m$ -ary relations, where $m$ is number of sectors constituting the
considered complex system. In order to approach this idea we apply concept of
frustrations to the U.S. Intermarket which is constituted by the following
sectors: Stock, Bond, Commodity, Currency and Gold. Therefore, for the further
considerations we select the following list of indexes:
S$\&$P500 - SPX, Treasury Bonds Prices - USB, Commodity Research Bureau
Futures Price Index - CRB, U. S. Dollar Index - USD and Gold Index - XAU.
A distribution of entanglement signs between spins play the crucial role in
typical models of spin glass. In the case of Intermarket the entanglement
between sectors is described by the linear correlation coefficient. Therefore,
the frustration in this system is determined by the distribution of
correlation coefficient’s signs Illustrations of the all concepts derived here
are presented on the example data 1987/07/01-1987/12/31. Whereas, in order to
investigate dynamics of the considered hierarchy of frustrations we analyse
1983 - 2012 statistical data (bib:NICK, ). For a test we applied some data
from (bib:MURsite, ). The correlation coefficients are calculated for each
half of the year. The example correlation coefficients are presented in TABLE
1.
Table 1: Correlation coefficients of the sectors’ indexes for the periods 1987/01/02-1987/06/30 and 1987/07/01-1987/12/31, above and below diagonal, respectively. | CRB | USB | SPX | USD | XAU
---|---|---|---|---|---
CRB | 1 | -0,115 | -0,003 | -0,380 | 0,401
USB | -0,144 | 1 | 0,544 | -0,271 | -0,666
SPX | 0,376 | 0,617 | 1 | -0,124 | -0,182
USD | 0,129 | -0,085 | 0,456 | 1 | -0,195
XAU | 0,750 | -0,081 | 0,235 | -0,351 | 1
## II Frustrations in correlated subsystems
Let us consider the following Intermarket’s sectors represented by their
indexes:
$S=\\{SPX,USB,CRB,USD,XAU\\}$ (1)
and define the following binary relation:
${\bf{{\hat{R}}_{2}}}\subset S\times S\times\\{-,+\\},$ (2)
where $\\{-,+\\}$ is set of two labels corresponding to signs of correlation
coefficients of the pairs belonging to $S\times S$.
Let us determine relation ${\bf{\hat{R}_{2}}}$ on $S^{2}\times\\{-,+\\}$. For
the binary relation we use the following notation:
$(X,Y,\pm)\in{\bf{\hat{R}_{2}}}\equiv X{\bf{\hat{R}_{2}}}Y=\pm.$ (3)
Let $\pm$ are determined by the signs of Pearson’s correlation coefficients
$X{\bf\hat{R}_{2}}Y=sign(\rho(X,Y)),$ (4)
where $X,Y\in S$ and
$\rho(X,Y)=Cov_{XY}/\sqrt{Var_{X}Var_{Y}}.$ (5)
Therefore the considered relation becomes to the following function (for the
period 1987/07/01-1987/12/31):
$\displaystyle CRB{{\bf{\hat{R}_{2}}}}SPX=-,\hskip
5.69054ptCRB{{\bf{\hat{R}_{2}}}}XAU=+,$ $\displaystyle
SPX{{\bf{\hat{R}_{2}}}}USB=-,\hskip 5.69054ptCRB{{\bf{\hat{R}_{2}}}}USD=-,$
$\displaystyle CRB{{\bf{\hat{R}_{2}}}}USB=-,\hskip
5.69054ptUSD{{\bf{\hat{R}_{2}}}}USB=+,$
The remaining mappings of this function are presented in Fig. 1 (for the
period 1987/07/01-1987/12/31).
${CRB}$$USB$${-}$${SPX}$$USD$${+}$${CRB}$$USD$${+}$${CRB}$$SPX$${+}$${SPX}$$XAU$${+}$${CRB}$$XAU$${+}$${XAU}$$USB$${-}$${USB}$$SPX$${+}$${USB}$$USD$${-}$${USD}$$XAU$${-}$
Figure 1: Graphic representation of the relation ${\bf{\hat{R}_{2}}}$ fort
the period 1987/07/01-1987/12/31 reduced to (4)
The values of the correlation coefficients concerning the selected sectors are
presented in TABLE 1. Let us investigate the transitivity of
${\bf{\hat{R}_{2}}}$. Therefore we have to determine superposition of the two
relations like the following example:
$CRB{\bf{\hat{R}_{2}}}SPX{\bf{\hat{R}_{2}}}USB$. Let $X,Y,Z$ are different
members of $S$. According to least squares estimates (bib:Brandt, ) we can
write down the following linear approximations:
$\displaystyle Y=a_{XY}\cdot X+b_{XY},$ (7) $\displaystyle Z=a_{YZ}\cdot
Y+b_{YZ}.$ (8)
Let us create a superposition of (7) and (8):
$Z=a_{XZ}\cdot X+b_{XZ},$ (9)
where
$\displaystyle a_{XZ}=a_{XY}\cdot a_{YZ},$ (10) $\displaystyle
b_{XZ}=b_{XY}\cdot a_{YZ}+b_{YZ}.$ (11)
According to the least squares estimates
$sign(a_{XY})=sign(\rho(X,Y))=\Phi_{R_{2}}(X,Y)$, therefore (4) takes the
following form:
$X{\bf\hat{R}_{2}}Y=\Phi_{R_{2}}(X,Y).$ (12)
Combining (4)-(12) we derive the following superposition’s rule for
${\bf{\hat{R}_{2}}}$:
$X{\bf{\hat{R}_{2}}}Y{\bf{\hat{R}_{2}}}Z=\Phi_{R_{2}}(X,Y)\,\Phi_{R_{2}}(Y,Z).$
(13)
Therefore we formulate and prove the following theorem:
Theorem 1
Let $\forall X,Y,Z\in S_{T}\subset S$
$\displaystyle X{\bf\hat{R}_{2}}Y=\Phi_{R_{2}}(X,Y)\land
Y{\bf\hat{R}_{2}}Z=\Phi_{R_{2}}(Y,Z)\land$ $\displaystyle
X{\bf\hat{R}_{2}}Z=\Phi_{R_{2}}(X,Z),$ (14)
then $\bf{\hat{R}_{2}}$ is transitive in $S_{T}$ if
$\Phi_{R_{2}}(X,Y)\,\Phi_{R_{2}}(Y,Z)\,\Phi_{R_{2}}(X,Z)=+.$ (15)
Proof
By the definition ${\bf{\hat{R}_{2}}}$ is transitive if
$X{\bf\hat{R}_{2}}Y=\Phi_{R_{2}}(X,Y)\land
Y{\bf\hat{R}_{2}}Z=\Phi_{R_{2}}(Y,Z)\Rightarrow
X{\bf\hat{R}_{2}}Z=\Phi_{R_{2}}(X,Z)$. On the basis of this definition and
(13) as well as (14) we derive that ${\bf{\hat{R}_{2}}}$ is transitive if:
$\Phi_{R_{2}}(X,Y)\,\Phi_{R_{2}}(Y,Z)=\Phi_{R_{2}}(X,Z).$ (16)
Multiplying (16) by $\Phi_{R_{2}}(X,Z)$ we derive (15). $\blacksquare$
Taking into account above theorem we obtain:
$\displaystyle
CRB{\bf{\hat{R}_{2}}}USD{\bf{\hat{R}_{2}}}USB=CRB{\bf{\hat{R}_{2}}}USB,$
$\displaystyle
CRB{\bf{\hat{R}_{2}}}USB{\bf{\hat{R}_{2}}}XAU=CRB{\bf{\hat{R}_{2}}}XAU,$ (17)
$\displaystyle
USD{\bf{\hat{R}_{2}}}SPX{\bf{\hat{R}_{2}}}XAU=USD{\bf{\hat{R}_{2}}}XAU.$
$\displaystyle\cdots\hskip 42.67912pt\cdots\hskip 42.67912pt\cdots\hskip
42.67912pt\cdots$
$\displaystyle CRB{\bf{\hat{R}_{2}}}SPX{\bf{\hat{R}_{2}}}USB\neq
CRB{\bf{\hat{R}_{2}}}USB,$ $\displaystyle
CRB{\bf{\hat{R}_{2}}}USD{\bf{\hat{R}_{2}}}XAU\neq CRB{\bf{\hat{R}_{2}}}XAU,$
(18) $\displaystyle USB{\bf{\hat{R}_{2}}}USD{\bf{\hat{R}_{2}}}XAU\neq
USB{\bf{\hat{R}_{2}}}XAU,$ $\displaystyle\cdots\hskip 42.67912pt\cdots\hskip
42.67912pt\cdots\hskip 42.67912pt\cdots$
The remaining tests of transitivity are presented in Fig. 2 and Fig. 3.
Let us call the following subset $\\{X\neq Y\neq Z\neq X\\}\subset S$ a
plaquette, let $V$ be set of the all plaquettes created in $S$. We can see
that the transitivity decomposes $V$ into two components: $V={V_{T}\cup
V_{F}}$, where $\forall\\{X,Y,Z\\}\in V_{T}$ the relation ${\bf{\hat{R}_{2}}}$
is transitive and it is not transitive $\forall\\{X,Y,Z\\}\in V_{F}$, (N, F -
mean no frustration and frustration, respectively).
${CRB}$${USD}$$USB$${-}$${-}$${+}$${CRB}$${XAU}$$USB$${-}$${-}$${+}$${SPX}$${XAU}$$CRB$${+}$${+}$${+}$${USD}$${XAU}$$SPX$${+}$${+}$${+}$${CRB}$${USD}$$SPX$${+}$${+}$${+}$
Figure 2: The subspace $V_{T}$ for the period.1987/07/01-1987/12/31
${CRB}$${SPX}$$USB$${-}$${+}$${+}$${SPX}$${XAU}$$USB$${+}$${-}$${+}$${CRB}$${XAU}$$USD$${-}$${+}$${+}$${USB}$${USD}$$SPX$${+}$${+}$${-}$${USB}$${USD}$$XAU$${-}$${+}$${+}$
Figure 3: The subspace $V_{F}$ for the period 1987/07/01-1987/12/31. (The
signs of the correlations are supplied for convenience in testing of
${\bf{\hat{R}_{2}}}$ with respect to the transitivity)
Definition
Lacking of ${\bf{\hat{R}_{2}}}$’s transitivity in $\\{X,Y,Z\\}\in V_{F}$ we
call frustration of $\\{X,Y,Z\\}$ with respect to ${\bf{\hat{R}_{2}}}$.
### II.1 Hierarchy of frustrations in correlated subsystems
Let us consider the following Intermarket’s sectors represented by their
indexes: $S=\\{SPX,USB,CRB,USD,XAU\\}$ and define the following hierarchy of
relations:
${\bf{\hat{R}_{2}}}\subset S\times S\times\\{-,+\\},$ (19)
where $\\{-,+\\}$ is set of two labels corresponding to signs of correlation
coefficients of the pairs belonging to $S\times S$.
${\bf{\hat{R}_{3}}}\subset S\times S\times S\times\\{F_{R_{2}},T_{R_{2}}\\},$
(20)
where $\\{F_{R_{2}},T_{R_{2}}\\}$ is set of two labels: $F_{R_{2}}$ \-
frustration, $T_{R_{2}}$ \- no frustration with respect to transitivity of
${\bf{\hat{R}_{2}}}$.
${\bf{\hat{R}_{4}}}\subset S\times S\times S\times
S\times\\{F_{R_{3}},T_{R_{3}}\\},$ (21)
where $\\{F_{R_{3}},T_{R_{3}}\\}$ is set of two labels: $F_{R_{3}}$ \-
frustration, $T_{R_{3}}$ \- no frustration with respect to transitivity of
${\bf{\hat{R}_{3}}}$.
${\bf{\hat{R}_{5}}}\subset S\times S\times S\times S\times
S\times\\{F_{R_{4}},T_{R_{4}}\\},$ (22)
where $\\{F_{R_{4}},T_{R_{4}}\\}$ is set of two labels: $F_{R_{4}}$ \-
frustration, $T_{R_{4}}$ \- no frustration with respect to transitivity of
${\bf{\hat{R}_{4}}}$.
#### II.1.1 The ternary relation
Now we are ready to define ${\bf{\hat{R}_{3}}}$.
Definition Let us create all possible points $(X,Y,Z)\in S^{3}$, where
$X,Y,Z\in S$ and $X\neq Y,Y\neq Z,Z\neq X$ and define ${\bf{\hat{R}_{3}}}$ in
the following way: if frustration with respect to transitivity in
$\bf{\hat{R}_{2}}$ occurs in ${X,Y,Z}$ then the point $(X,Y,Z)$ is mapped to
$F_{R_{2}}$ and $(X,Y,Z,F_{R_{2}})\in{\bf{\hat{R}_{3F}}}$ else $(X,Y,Z)$ is
mapped to $T_{R_{2}}$ and $(X,Y,Z,T_{R_{2}})\in{\bf{\hat{R}_{3T}}}$. Finally
${\bf{\hat{R}_{3}}}={\bf{\hat{R}_{3F}}}\cup{\bf{\hat{R}_{3T}}}$. The subsets
corresponding to ${T_{R_{2}}}$ and ${F_{R_{2}}}$:
$\displaystyle{\bf{\hat{R}_{3T}}}\subset S\times S\times
S\times\\{T_{R_{2}}\\},$ (23) $\displaystyle{\bf{\hat{R}_{3F}}}\subset S\times
S\times S\times\\{F_{R_{2}}\\},$ (24)
are presented in Fig. 4 and Fig. 5, respectively. It occurs the following
relation between ${\bf{\hat{R}_{3}}}$ and ${\bf{\hat{R}_{2}}}$:
$XYZ{\bf{\hat{R}_{3}}}=X{\bf{\hat{R}_{2}}}Y\land Y{\bf{\hat{R}_{2}}}Z\land
Z{\bf{\hat{R}_{2}}}X.$ (25)
Therefore, similarly to (12) and according to (25) we determine the values of
${\bf{\hat{R}_{3}}}$ in the following way:
$\displaystyle{\it\Phi_{R_{3}}(X,Y,Z)}=\Phi_{R_{2}}(X,Y)\,\Phi_{R_{2}}({Y,Z})\,\Phi_{R_{2}}({X,Z}),$
$\displaystyle{\it\Phi_{R_{3}}(T,X,Z)}=\Phi_{R_{2}}({T,X})\,\Phi_{R_{2}}({X,Z})\,\Phi_{R_{2}}({T,Z}),$
(26)
$\displaystyle{\it\Phi_{R_{3}}(T,X,Y)}=\Phi_{R_{2}}({T,X})\,\Phi_{R_{2}}(X,Y)\,\Phi_{R_{2}}({T,Y}),$
$\displaystyle{\it\Phi_{R_{3}}(T,Y,Z)}=\Phi_{R_{2}}({T,Y})\,\Phi_{R_{2}}({Y,Z})\,\Phi_{R_{2}}({T,Z}).$
${CRB}$${SPX}$$USB$+${SPX}$${XAU}$$USD$+${CRB}$${XAU}$$USD$+${CRB}$${XAU}$$SPX$+${CRB}$${USD}$$SPX$+${USB}$${XAU}$$USD$+
Figure 4: Subrelation $XYZ{\bf{\hat{R}_{3T}}}={\textbf{+}}$
${CRB}$${USD}$$USB$-${SPX}$${XAU}$$USB$-${CRB}$${XAU}$$USB$-${USB}$${USD}$$SPX$-
Figure 5: Subrelation $XYZ{\bf{\hat{R}_{3F}}}={\textbf{-}}$
In order to extent notion of frustration into ${\bf{\hat{R}_{3}}}$ we have to
define the transitivity with respect to this relation.
Definition
Let
$\\{(X,Y,Z),(T,X,Z),(T,X,Y),(T,Y,Z)\\}\subset S^{3}.$ (27)
If
$\displaystyle XYZ{\bf{\hat{R}_{3}}}=\Phi_{R_{3}}(X,Y,Z)\land$ $\displaystyle
XTZ{\bf{\hat{R}_{3}}}=\Phi_{R_{3}}(X,T,Z)\land$ (28) $\displaystyle
XTY{\bf{\hat{R}_{3}}}=\Phi_{R_{3}}(X,T,Y)\Rightarrow$ $\displaystyle
TYZ{\bf{\hat{R}_{3}}}=\Phi_{R_{3}}(T,Y,Z),$
then $\bf{\hat{R}_{3}}$ is transitive in (27).
Theorem 2
Let $\bf{\hat{R}_{3}}$ be transitive in
$\\{(X,Y,Z),(T,X,Z),(T,X,Y),(T,Y,Z)\\}$:
$\displaystyle XYZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(X,Y,Z)\land$ $\displaystyle
TXZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(T,X,Z)\land$ $\displaystyle
TXY{\bf\hat{R}_{3}}=\Phi_{R_{3}}(X,T,Y)\land$ $\displaystyle
TYZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(T,Y,Z),$ (29)
then
$\displaystyle\Phi_{R_{3}}(X,Y,Z)\,\Phi_{R_{3}}(X,T,Z)$
$\displaystyle\Phi_{R_{3}}(X,T,Y)\,\Phi_{R_{3}}(T,Y,Z)=+.$ (30)
Proof
By the definition ${\bf{\hat{R}_{3}}}$ is transitive if
$XYZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(X,Y,Z)\land
TYZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(T,Y,Z)\land
TXZ{\bf\hat{R}_{3}}=\Phi_{R_{3}}(T,X,Z)\Rightarrow
TXY{\bf\hat{R}_{3}}=\Phi_{R_{3}}(T,X,Y)$. Taking into account (27) and
$\Phi_{R_{2}}^{2}=+1$ as well as (26) we derive that ${\bf{\hat{R}_{3}}}$ is
transitive if:
$\displaystyle{\it\Phi_{R_{3}}(T,X,Z)}=$ (31)
$\displaystyle{\it\Phi_{R_{3}}(T,X,Y)}\,{\it\Phi_{R_{3}}(X,Y,Z)}\,{\it\Phi_{R_{3}}(T,Y,Z)}.$
Multiplying (31) by ${\it\Phi_{R_{3}}(T,X,Z)}$ we get the thesis.
$\blacksquare$
#### II.1.2 The 4-ary and 5-ary relations
Extending results of II.1.1 into the 4-ary and 5-ary relations (21) and (22),
respectively we define the transitivity and frustration for the 4 and 5 point
complexes which are presented in Figure 6 and Figure 7, respectively. For the
considered $S$ system there are five 4-ary relations and one 5-ary relation.
Extending (26) on $\bf{\hat{R}_{4}}$ we derive the following relation:
$\Phi_{R_{4}}(X_{1},X_{2},X_{3},X_{4})=\Phi_{R_{3}}(X_{1},X_{2},X_{3})\prod_{i=1}^{3}\Phi_{R_{2}}(X_{i},X_{4})$
(32)
For presentation of transitivity and frustration in ${\bf{\hat{R_{4}}}}$ we
calculate $\Phi_{R_{4}}$ for the both selected relations (Figure 6):
$\Phi_{R_{4}}(CRB,SPX,USB,USD)=+1,\Phi_{R_{4}}(XAU,SPX,USB,USD)=-1$. Extending
(32) on ${\bf{\hat{R}_{5}}}$ we derive the following value of the transitivity
for the whole considered system: $\Phi_{R_{5}}(S)=+1$. Therefore,
${\bf\hat{R_{4}}}$ is not transitive in $S-CRB$, whereas it is transitive in
$S-XAU$ as well. Since ${\bf{\hat{R}_{5}}}$ is transitive in $S$ whereas it is
not transitive in $S-CRB$ we derive the following conclusion: in the period
1987/07/01-1987/12/31 $CRB$ has been played an ordering role in the considered
Intermarket. Thus, e.g. relating the transitivity’s measures of $\Phi_{R_{4}}$
and $\Phi_{R_{5}}$ we investigate roles of the all Intermarket’s sectors
during 1983-2012 (Section V).
${CRB}$${CRB}$$CRB$${SPX}$${USB}$${USD}$${+}$${+}$${-}$${-}$$\in
V_{T}$${XAU}$${XAU}$$XAU$${SPX}$${USB}$${USD}$${+}$${-}$${-}$${+}$$\in V_{F}$
Figure 6: Subrelation ${\bf{\hat{R}_{4F}}}$.
${SPX}$${XAU}$$USD$$USB$$CRB$$CRB$ Figure 7: Subrelation
${\bf{\hat{R}_{5F}}}$.
#### II.1.3 The n-ary relations
Derivation of ${\bf\hat{R}_{3}},{\bf\hat{R}_{4}},{\bf\hat{R}_{5}},$ and their
properties suggests the following algorithm for creation of the
${\bf\hat{R}_{n}}$ and investigation of the properties:
1\. Write down the relation between ${\bf\hat{R}_{n}}$ and
${\bf\hat{R}_{n-1}}$. Let $(X_{1},X_{2}\dots X_{n})\in S^{n}$, in the
following form:
$X_{1}X_{2}\dots X_{n}{\bf{\hat{R}_{n}}}=\bigwedge_{i=1}^{n}X_{1}X_{2}\dots
X_{i-1}X_{i+1}\dots X_{n}{\bf\hat{R}_{n-1}},$ (33)
where $X_{n+1}=X_{1}$
2\. Express $\Phi_{n}(X_{1},X_{2},\dots X_{n})$ by $\Phi_{2}(X_{i},X_{j})$,
where $i,j=1,2\dots n$:
$\Phi_{n}(X_{1},X_{2},\dots X_{n})=\prod_{i<j\leq
n}\Phi_{R_{2}}(X_{i},X_{j}).$ (34)
3\. Define transitivity of ${\bf{\hat{R}_{n}}}$.
Let $(X_{0},X_{1},X_{2},\dots X_{i-1},X_{i+1},\dots X_{n})\in S^{n}$, where
$i=1,2,3\dots n$ and $X_{n+1}=X_{0}$
If the following relation occurs:
$\bigwedge_{i=1}^{n}\left(X_{0}X_{1}X_{2}\dots X_{i-1}X_{i+1}\dots
X_{n}{\bf\hat{R}_{n}}=\Phi_{R_{n}}(X_{0},X_{1},X_{2},\dots
X_{i-1},X_{i+1},\dots X_{n})\right)\\\ \rightarrow X_{1}X_{2}\dots
X_{n}{\bf\hat{R}_{n}},$ (35)
then ${\bf{\hat{R}_{n}}}$ is transitive, else the subsystem
$(X_{0},X_{1},X_{2},\dots X_{n})\in S^{n+1}$ is frustrated with respect to
${\bf{\hat{R}_{n}}}$.
4\. Derive recurrent formula for $\Phi_{R_{n}}(X_{1},X_{2},\dots X_{n})$:
$\Phi_{R_{n}}(X_{1},X_{2},\dots X_{n})=\Phi_{R_{n-1}}(X_{1},X_{2},\dots
X_{n-1})\prod_{i=1}^{n-1}\Phi_{R_{2}}(X_{i},X_{n}).$ (36)
5\. Derive the superposition rules for $\Phi_{R_{n}}(X_{1},X_{2},\dots
X_{n})$. Writing down the complete system of (36) and performing elimination
of the all $\Phi_{R_{2}}(X_{i},X_{j})$ correlation coefficients we derive:
$\prod_{i=1}^{n}\Phi_{R_{n}}(X_{0},X_{1},X_{2},\dots X_{i-1},X_{i+1},\dots
X_{n})=\Phi_{R_{n}}(X_{1}X_{2}\dots X_{n}).$ (37)
## III Measures of transitivites
For each relation belonging to the hierarchy
$\\{{\bf{{\hat{R}}_{2}}},{\bf{\hat{R}_{3}}},{\bf{\hat{R}_{4}}},{\bf{\hat{R}_{5}}},\dots\\}$
the values of $\Phi_{R_{n}}=\pm 1$ correspond to transitivity or frustration,
respectively. However, they do not describe ”how much” considered system is
transitive or how much frustrated. In order to derive such a measure we come
back to the definition of ${\bf{\hat{R}_{2}}}$. Let us note that
$\rho(X,Y)=\Phi_{R_{2}}(X,Y)\cdot|(\rho(X,Y))|=\rho_{R_{2}}(X,Y)$, where the
first factor informs whether $X$ and $Y$ are correlated or anticorrelated,
whereas the second one describes ”how much”. Therefore, we renamed $\rho(X,Y)$
into $\rho_{R_{2}}(X,Y)$ as an accepted measure of ${\bf{\hat{R}_{2}}}$.
Continuing this way and taking into account (26) we derive the measures of
${\bf{\hat{R}_{3}}},{\bf{\hat{R}_{4}}}$ and ${\bf{\hat{R}_{5}}}$:
$\displaystyle\rho_{R_{3}}(X,Y,Z)=\Phi_{R_{3}}(X,Y,Z)\cdot|\rho(X,Y)|\cdot\
|\rho(Y,Z)|\cdot|\rho(X,Z)|=\rho(X,Y)\cdot\ \rho(Y,Z)\cdot\rho(X,Z).$ (38)
Therefore,
$\displaystyle\rho_{R_{4}}(T,X,Y,Z)=\rho(X,Y)\cdot\rho(Y,Z)\cdot\rho(X,Z)\cdot\rho(T,X)\cdot\rho(T,Y)\cdot\rho(T,Z),$
(39)
$\displaystyle\rho_{R_{5}}(T,X,Y,V,Z)=\rho(X,Y)\rho(Y,Z)\rho(X,Z)\rho(T,X)\rho(T,Y)\rho(T,Z)\rho(X,V)\rho(Y,V)\rho(V,Z)\rho(T,V)$
(40)
In general case $|S|=m$ similarly to (38)-(39) we derive simplified formula
for transitivity measure of the $m$-s member of hierarchy:
$\rho_{R_{m}}(X_{1},X_{2},\dots X_{m})=\prod_{i<j\leq m}\rho(X_{i},X_{j})$
(41)
## IV Transitivity as ordering relation’s property
Proposition
Let us consider two examples of plaquettes, one by one from $V_{T}$ and
$V_{F}$, respectively (Fig. 8).
${CRB}$${SPX}$$USB$${-}$${+}$${-}$$\in
V_{T}$${CRB}$${USD}$$USB$${-}$${-}$${-}$$\in V_{F}$ Figure 8: The three
points complexes: transitive and frustrated. The $\pm$ signs correspond to the
signs of the correlation coefficients.
We argue for the following hypothesis: Transitivity of ${\bf{\hat{R}_{2}}}$ is
responsible for stimulating of sectors to common direction evolution. However,
some sectors interferes with this process leading to the frustration and in
this way they preserve the sectors’ independence.There are two arguments for
such interpretation of the frustration (or its contradiction - transitivity).
The first one is direct. Let us take into account the case $V_{T}$ of Fig. 8.
Let us estimate influence of $CRB$ on $USB$. There are two ways of
entanglement: $CRB\rightarrow USB$ and $CRB\rightarrow SPX\rightarrow USB$.
Both of them push $USB$ into opposite direction with respect to the
evolution’s direction of $CRB$. It is important that both ways push $USB$ into
the same direction (by the direction of X’s evolution we mean its increase or
decrease). Therefore, the resulting effect from the both ways is at least
stronger then the strongest single entanglement in this plaquette. This result
is invariant with respect to a choice of starting point in the considered
plaquette. Now, let us take into account the case $V_{F}$ of Fig. 8, and
estimate influence of $CRB$ on $USD$. There are two ways of entanglement
acting on $USD$: $CRB\rightarrow USD$ and $CRB\rightarrow USB\rightarrow USD$.
However, now they work in opposite directions. Therefore, the resulting effect
is at least weaker then the strongest single entanglement in this plaquette.
Also this result is invariant with respect to a choice of the starting point.
Summarizing, we have shown argument that in a plaquette of the three different
sectors without (with) frustration the influences of sectors between each
other become stronger (weaker). Summarizing, we distinguish an ordering
entanglement in $V_{T}$, whereas in $V_{F}$ such an ordering does not exists.
The second argument is formal and touches the basis of the mathematics. Let
${\bf{\hat{R}_{2T}}},{\bf{\hat{R}_{2F}}}$ be ${\bf{\hat{R}_{2}}}$ constrained
to $V_{T},V_{F}$, respectively. Due to symmetry, reflexibility and
transitivity of ${\bf{\hat{R}_{2T}}}$ this subrelation is an equivalence
relation. Since for the considered Intermarket the sum of the all plaquettes
belonging to $V_{T}$ is equal to $S$:
$\bigcup_{\\{X,Y,Z\\}\in V_{T}}\\{X,Y,Z\\}=S,$ (42)
the structure $(S,{\bf{\hat{R}_{2T}}})$ is a preorder (bib:foldes, ).
Therefore, the transitivity is an inductor of at least a weak kind of order in
the system.
## V Frustrations Hierarchy Analysis of the U.S. Intermarket’s
Applying (41 ) to the Intermarket’s data we have calculated the total
Intermarket’s transitivity measure $\rho_{\bf R_{5}}$ (see Fig.9, Fig.10) and
the five measures $\rho_{\bf R_{4}}$ corresponding to Subintermarkets obtained
by reduction of the considered Intermarket with respect to each its element
$S$\$\\{XYZ\\}$ (see Fig.11\- Fig.15).
### V.1 Discussion of results for $\rho_{\bf R_{5}}$ measure
The values of $\rho_{\bf R_{5}}$ are presented in the two scales. The scales
of Fig.9 and Fig.10 are appropriate for analysis of positive and negative
$\rho_{\bf R_{5}}$, respectively. Combining both figures we see that
$\rho_{\bf R_{5}}$ undergoes variations like the U.S. Business Cycles which
can be described by the Brownian Motion of a Harmonic Oscillator (bib:chen,
)-(bib:Zarn, ). The oscillation’s amplitude for the positive direction of
$\rho_{\bf R_{5}}$ is two orders greater in average then the negative one. The
envelopes of positive and negative values changes according to trends
presented by dashed lines. Let us remind that $\rho_{\bf R_{5}}>0$ corresponds
to transitivity, whereas $\rho_{\bf R_{5}}<0$ corresponds to frustration. From
1983 $(Y=83)$ until 2009 $(Y=109)$ the positive amplitude increases and the
negative one decreases becoming positive. It means that the transitivity
approaches a high value and the frustration disappears. Then from the second
half of 2009 $\rho_{\bf R_{5}}$ exhibits strong fluctuations. One can see from
Fig.10 that the considered system has lost stability when the trend of
frustrations got value equal to zero. On the basis of these observations we
may draw the conclusion that frustration is necessary for the system’s
stability. Due to the strong oscillations which have appeared after the
revealed boom there is a chance to get negative values for the frustration’s
trend and to recover the system’s stability.
Figure 9: The transitivity’s measure ${\bf\rho_{R_{5}}}$ of $S$ v.s. $Y$ for
the period 1983-2012. The vertical scale enables to read the positive values
of ${\bf\rho_{R_{5}}}$, ($Y=Year-1900$). Figure 10: The transitivity’s
measure ${\bf\rho_{R_{5}}}$ of $S$ v.s. $Y$ for the period 1983-2012. The
vertical scale enables to read the negative values of ${\bf\rho_{R_{5}}}$ and
their oscilations around $Y$ axis ($Y=Year-1900$).
The nearest future will show how the frustration analysis applied to an
Intermarket is efficient for the predictions of the economic stability in long
time horizon. Now (January 2013) $\rho_{R_{5}}$ performs large amplitude
oscillations into both directions ($\rho_{R_{5}}>0$ and $\rho_{R_{5}}<0$). The
most probable event is that the envelope of frustrations $\rho_{R_{5}}<0$ will
approach zero by forthcoming decades and different events of U.S. Economy will
generate picks of transitivity. Unlikely but a worse one would be a situation
when $\rho_{R_{5}}$ will oscillate above $Y$ axis approaching zero value
asymptotically for long time.
### V.2 Discussion of results for $\rho_{\bf R_{4}}$ measures
$\rho_{\bf R_{5}}$ is the highest measure of transitivity in the five elements
system. This is a top of the considered hierarchy $\rho_{\bf R_{5}},\rho_{\bf
R_{4}},\rho_{\bf R_{3}},\rho_{\bf R_{2}}$. There are five measures of the
$\rho_{\bf R_{4}}$ describing transitivity in a subset of four Intermarket’s
entities. Let us assume that by removing selected entity from the frustration
analysis we receive an approximation knowledge about the influence of this
Intermarket’s member on the dynamics of the whole system. All the five
$\rho_{\bf R_{4}}$ measures are presented in Fig.11-Fig.15. Comparing
$\rho_{\bf R_{4}}$ of the selected Subintermarket with $\rho_{\bf R_{5}}$ we
will try to answer the question what could be the influence of the removed
entity on the Intermarket’s stability.
Figure 11: The measure ${\bf\rho_{R_{4}}}$ of $S$\$\\{XAU\\}$ v.s. $Y$ for
the period 1983-2012, dots correspond to scaled $\rho_{\bf R_{5}}$, and
$Y=Year-1900$. Figure 12: The measure ${\bf\rho_{R_{4}}}$ of $S$\$\\{USB\\}$
v.s. $Y$ for the period 1983-2012, dots correspond to scaled $\rho_{\bf
R_{5}}$, and $Y=Year-1900$. Figure 13: The measure ${\bf\rho_{R_{4}}}$ of
$S$\$\\{USD\\}$ v.s. $Y$ for the period 1983-2012, dots correspond to scaled
$\rho_{\bf R_{5}}$, and $Y=Year-1900$. Figure 14: The measure
${\bf\rho_{R_{4}}}$ of $S$\$\\{CRB\\}$ v.s. $Y$ for the period 1983-2012, dots
correspond to scaled $\rho_{\bf R_{5}}$, and $Y=Year-1900$. Figure 15: The
measure ${\bf\rho_{R_{4}}}$ of $S$\$\\{SPX\\}$ v.s. $Y$ for the period
1983-2012, dots correspond to scaled $\rho_{\bf R_{5}}$,($Y=Year-1900$).
There are seven possible reactions of picks for removing a sector from the
Intermarket. All of them are listed in TABELE 2 and their interpretations are
indicated. According to this Tabele we present the following discussion.
* •
$S$\$\\{XAU\\}$. In the period 1983-2008 Gold has played crucial role in
Intermarket’s stability. By removing Gold from the Intermarket the high picks
of transitivity have change into deep frustrations. Therefore the Gold was
responsible for blocking of the frustrations. However, in the period 2009-2012
this Sector has loss this influence.
* •
$S$\$\\{USB\\}$. In the period 1983-1994 Treasury Bonds Prices’ influence was
marginal. Whereas, in the period 1995-2012 $USB$ has changed transitivities
into frustrations. Therefore, probably this sector among others was
responsible for the frustrations’ decay.
* •
$S$\$\\{USD\\}$. The analysis of ${\bf\rho_{R_{4}}}$ shows that $USD$ is the
most complicated Intermarket’s sector. The pick of ${\bf\rho_{R_{4}}}$
corresponding to 1985-1986 period has been changed from frustration to
transitivity. Therefore the U. S. Dollar was a creator of frustrations in this
period. The pick corresponding to 1989 was invariant with respect to removing
of USD from the Intermarket. Therefore for this period USD was marginal. The
two next picks of ${\bf\rho_{R_{4}}}$ have appeared at 1995 and 2009. Both of
them were also invariant, however the pick from 1995 had got two sattelite
picks at 1994 and 1996.5. (July of 1996). The pick at 2009 has changed from
frustration into transversity whereas the last one at 2012 reminded to be
invariant. Summarizing, the invariant picks are not correlated with USD,
however those which have been changed from frustration into transfersity
corresponded to frustrations’ creator of ${\bf\rho_{R_{5}}}$.
* •
$S$\$\\{CRB\\}$. Influence of Commodity on the Intermarket was a little
different from the influence of another sectors. In 1985-1986 the pick of
${\bf\rho_{R_{4}}}$ was invariant. The pick at 1989 has changed the sign and
became a measure of frustration. Therefore, in this year Commodity has
protected Intermarket against frustration. A new pick of transitivity has
appeared at 1991.5. In the period 1993.5- 1995.5 a new quality has occur. By
comparing Fig. 14 and Fig. fig:rhoR5g a large increase of the pick at 1995 and
its little satellite at 1996.5 into gigantic measure’s values of transitivity
and frustration have been observed. This can be interpreted as an active role
of Commodity in creation of the frustrations in the considered period. Next,
in 2004-2005 the pair of transitivities’ picks has been collapsed into fuzzy
frustrations.Therefore, the Commodity has stopped the creation of frustrations
. Finally, the events of the period 2008-2011 have shown transition of the
gigantic transitivities’ into picks of transitivity and frustration.
Therefore, from 1993.5 the Commodity has played very important role in the
Intermarket’s stabilization.
* •
$S$\$\\{SPX\\}$. Two picks of transitivity at 1985 and 1989 have been
conserved, whereas the pick of 1995 and its little satellite at 1996 have
changed for frustration, moreover, the intensity of the satellite increased.
Since 1997 it has started the oscillation of $\rho_{R_{4}}$ around zero
characterized by increasing amplitude which was suddenly broken at 2001.5.
There were not any oscillations of the measure ${\bf\rho_{R_{5}}}$
corresponding to that ones. In the four previous cases the events after 2008
were most interesting. In the case of $S$\$\\{SPX\\}$ the gigantic picks of
transversity and frustration has been reversed in time (frustration and
transversity). The five last points in Fig.15 suggest that the black scenario
for developing of ${\bf\rho_{R_{5}}}$ without frustrations would be possible
(black scenario). Note that in the period $1983-2008$ Stock and Intermarket
were strongly anticorrelated, whereas in $2009$ this relation become to be a
strong correlation. Additionaly, this event checks the stabilizing role of
frustrations.
### V.3 The finale remark
The considered Intermarket is a subsystem immersed in the U.S. National
Economy. Therefore, the picks and trends of the transitivity as well as the
frustration measures are related to events of this global system. Analysis of
the results presented here will be related to U.S. Events (bib:USE, ) and
published in forthcoming monograph (bib:SokMor, ).
Table 2: Possible reactions of $\rho_{R_{5}}\rightarrow\rho_{R_{4}}$ for removing a sector from the Intermarket Reaction | Invariant | $F\rightarrow T$ | $T\rightarrow F$ | $F\rightarrow 0$ | $T\rightarrow 0$ | $0\rightarrow F$ | $0\rightarrow T$
---|---|---|---|---|---|---|---
Interpretation for sector’s | No active | Frustration’s | Transisivity’s | Frustartion’s | Transitivity’s | Frustration’s | Transitivity’s
activity | | generator | generator | generator | generator | annihilator | annihilator
## References
* (1) Novel States of Matter Induced by Frustration, Editor: Hikaru Kawamura, J. Phys. Soc. Jpn. (Special Topics), Vol. 79 (2010).
* (2) Proceedings of the International Conference on Highly Frustrated Magnetism, Osaka, Japan,15-19 August 2006, Eds. Hiroi Z. & Tsunetsugu H., Journal of Physics: Cond. Matt.,Vol. 19. No. 14 (2007).
* (3) Bryngleson JD, Wolynes PG, Proc Nat Acad Sci USA Vol. 84 (1987):7524-7528.
* (4) Ahlgren PTH, Jensen MH, Simonsen I, Donangelo R, Sneppen K, Frustration driven stock market dynamics: Leverage effect and assymetry, (2007) Physica A, Vol. 383 :1-4.
* (5) Pickett H.E.,A Note on Generalized Equivalence Relation, Amer. Math. Manthly, Vol. 73, No. 8, (1966) 860-861.
* (6) Us̆an J., S̆es̆elja, Transitive n-Ary Relations and Characterizations of Generalized Equivalences, Rev. of Res. Faculty of Science-Univ. of Novi Sad, Vol. 11 (1981) 231-245.
* (7) Cristea I., Several Aspects on the Hypergroups Associated with n-Ary Relations, An. St. Univ. Ovidius Constanta, Vol. 17(3)(2009) 99-110.
* (8) The Statistical Data purchased from SHARELYNX GOLD, [email protected], January 2010. and January 2012.
* (9) Murphy J. J., Intermarket Technical Analysis: Trading Strategies For The Global Stock, Bond, Commodity And Currency Markets, John Wiley $\&$ Sons, Inc. 1991
* (10) http://stockcharts.com/charts/performance/
perf.html?[IM].
* (11) Brandt S., Data Analysis. Statistical and Computational Methods for Scientists and Engineers, Springer Verlag, New York 1999, Ch. 9.
* (12) Foldes S., Fundamental Structures of Algebra $\&$ Discrete Mathematics, John Wiley $\&$ Sons, Inc. 1994.
* (13) Chen, P. ”Trends, Shocks, Persistent Cycles in Evolving Economy: Business Cycle Measurement in Time-Frequency Representation, in W. A. Barnett, A. P. Kirman, and M. Salmon Eds. Nonlinear Dynamics and Economics, Chapter 13, pp. 307-331, Cambridge University Press (1996a).
* (14) Goodwin, R. M. ”The Nonlinear Accelerator and the Persistence of Business Cycles,” Econometrica, 19, 1-17 (1951).
* (15) Hayek, F. A. Monetary Theory and the Trade Cycle, A.M. Kelley Publishers, New York (1933, 1966).
* (16) Hodrick, R. J., and E. C. Prescott. ”Post-War US. Business Cycles: An Empirical Investigation, ” Discussion Paper No. 451, Carnegie-Mellon University (1981).
* (17) Zarnowitz, V. Business Cycles, Theory, History, Indicators, and Forecasting, pp.196-198, University of Chicago Press, Chicago (1992).
* (18) http://en.wikipedia.org/wiki/1983_in_the_
United_States#Events, and for all the next relevant years.
* (19) Sokalski K, Moroz E., Frustration Analysis of the U.S. Intermarkets, (2014) in preparation.
|
arxiv-papers
| 2013-04-12T19:58:56 |
2024-09-04T02:49:44.294307
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Krzysztof Sokalski",
"submitter": "Krzysztof Sokalski prof",
"url": "https://arxiv.org/abs/1304.3722"
}
|
1304.3888
|
# Minimal Ward-Takahashi Vertices and Pion Light Cone Distribution Amplitudes
from Gauge Invariant, Nonlocal, Dynamical Quark Model
Chuan Li2111Email:[email protected]., Shao-Zhou
Jiang3222Email:[email protected]. and Qing Wang1,2333Email:
[email protected] author
1Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.China
2Department of Physics, Tsinghua University, Beijing 100084,
P.R.China555mailing address
3College of Physics Science and Technology, Guangxi University, Nanning,
Guangxi 530004, P.R.China
###### Abstract
The gauge-invariant, nonlocal, dynamical quark model is proved to generate the
minimal vertices which satisfy the Ward-Takahashi identities. In the chiral
limit, the momentum-dependent quark self-energy results in a flat-like form
with some end point $\delta-$funtions for the light-cone pion distribution
amplitudes, similarly found in the Nambu Jona-Lasino model with constant
constituent mass. The leading order nonzero pion and current quark masses
corrections lead concave type asymptotic-like form modifications to twist-2
pion distribution amplitude with end point pillars and twist-3 tensor pion
distribution amplitude above the flat-like form backgrounds. A by-product of
our investigation shows that the variable $u$ appearing in pion light-cone
distribution amplitudes is just the standard Feynman parameter in the Feynman
parameter integrals; also chiral perturbation works well for these amplitudes.
###### pacs:
11.10.Lm, 12.38.Aw, 12.39.-x, 13.40.Gp
††preprint: TUHEP-TH-13179
The gauge-invariant, nonlocal, dynamical quark (GND) model GND is one of a
phenomenological non-local chiral quark model that can be derived from QCD
first principles by a series approximation WQ2002 . It appears as an
approximate description for the effective interactions among light quarks and
pseudo-scalar mesons induced from the underlying QCD after integrating out
gluon and heavy-quark fields. In the low-energy region, the validity of the
GND model is tested by its resulting low-energy constants GND ; WQ2002 ;
WQ2010 ; WQseries for the well-known Gasser-Leutwyler chiral Lagrangian GS ,
which well match the existing experimental data for pseudo-scalar meson
physics. Considering that in obtaining the GND model the low-energy expansion
is not taken, we expect this model has a larger range of application than the
traditional low-energy region and are interested in its momentum-dependent
behavior beyond the conventional low-energy expansion, which reflects the
interplay between low and high energies. In fact, the typical feature of the
GND model is its quark self-energy (or momentum-dependent quark mass)
$\Sigma(-p^{2})$ which represents the original idea of dynamical perturbation
theory proposed by Pagels and Stokar PS . All dynamics, especially the
momentum-dependent effects of the model, are in the main effectively described
by this quark self-energy.
To investigate the momentum-dependent effects in terms of a nonlocal model in
general, the first obvious problem one faces is its possible violation of the
Ward-Takahashi identities (WTIs), as this happens for most nonlocal chiral
quark models. In the literature, one way to solve the problem is to
artificially revise the vertex Rvertex ; Rvertex1 ; another way is to modify
the model itself. The GND model belongs to this second approach, where the
model is constructed in such a way that it is invariant under local chiral
symmetry transformations for external current sources of light-quark fields.
Except for the GND model, an earlier GNC model GNC took the same tack but
with a complex face factor introduced in the model. Because the local chiral
symmetry is inserted into the model at inception, we expect the WTIs to hold
as a result. This expected validity of the WTIs was claimed but not explicitly
shown in the GNC model, and not even mentioned in the original GND model. It
is the principle aim of this paper to explicitly show that the GND model does
satisfy the WTIs at the chiral limit. A by-product of this demonstration is
that we not only give vector and axial-vector vertices, but also scalar,
pseudo-scalar, and tensor vertices. In fact, due to their non-perturbative
natures, these fundamental vertices have not been very well reported in the
literature. The latest ansatz available for the vector vertex in QED is given
in Refs.VectorVertex which can be traced from the early Refs.Rvertex . For
vector and axial-vector vertices, the WTIs only constrain their longitudinal
parts and cannot fix the transverse parts. For the remaining scalar, pseudo-
scalar and tensor vertices, we even have no corresponding WTIs. Considering
the vertices we obtained are from the unique GND model action and are
constrained by inherent local chiral symmetry of the model, all different
types of resulting vertices are at the same level as the approximate
description of the corresponding QCD ones. We will show that the vector and
axial-vector vertices we obtained in the chiral limit are just the simplest
versions satisfying the WTIs and have exactly the same form of those assumed
by Ref.Holdom ; we call these two vertices and related scalar, pseudo-scalar,
and tensor vertices the minimal Ward-Takahashi vertices.
With proof of validity of the WTIs for the GND model and the resulting minimal
vertices as our starting point to investigate the momentum-dependent effects,
we take the light-cone pion distribution amplitudes (PDAs) as our next target
of study in this paper. The reason these are chosen for discussion is that, at
present, there is still no definite conclusion on whether PDA is in
asymptotic-like form asymp , in Chernyak-Zhitnitsky (CZ)-like form CZ , or in
a flat-like form flat . With progress from experiments, ever more information
and constraints have emerged on the PDAs providing good crosschecks between
theoretical estimations and experimental data WXG .
Theoretically, the model computation of PDAs is mainly through chiral quark
models. Earlier calculations based on the local chiral quark model or local
Nambu Jona-Lasino (NJL)-like models (see Ref.flat and references therein)
yielded the typical result that the lowest twist-2 PDA is in flat-like form in
the chiral limit. In local chiral quark models, quarks have constant
constituent masses and there is ultraviolet divergence in the resultant PDAs.
To avoid the divergence, various regularization schemes are exploited which
often yield confusing results. An improvement is to change to nonlocal chiral
quark models (see Ref.nonlocalPDAs and references therein), where a momentum-
dependent quark mass or quark self-energy is arranged in such a way that it
provides the model with a natural soft ultra-violet cutoff and results in
finite PDAs. Some researchers believe that the momentum dependence of the
quark self-energy will lead to twist-2 PDA deviating from flat-like form and
generate correct end-point behavior nonlocalPDAs1 . As we mentioned
previously, a nonlocal chiral quark model usually violates the WTIs and the
vertices are revised to avoid the defect. Now, our GND model automatically
satisfies the WTIs, so there is no need to make such artificial modification
of the vertices.
Another technical problem met in nonlocal chiral quark model is that momentum
integrations involving quark self-energies are usually difficult to complete,
because usually these have on-shell external momenta which are time-like
whereas conventionally loop momenta is space-like because the Wick rotation of
the integration momenta occurs in Euclidean space. This mixed appearance of
different kinds of momentum variables might cause variable $p^{2}$ in the
quark self-energy $\Sigma(-p^{2})$ in the time-like region or even with
imaginary components. If there exists an analytical expression of
$\Sigma(-p^{2})$, such as for instantons instanton or just some simple ansatz
Holdom , there will be no problem, because the extension to the time-like
region or imaginary region is explicit. If $\Sigma(-p^{2})$ however is defined
as the solution of the Schwinger-Dyson equation (SDE), we meet a difficulty.
The conventional numerical solution of the SDE is only defined in a space-like
region; beyond that, the solution has still not been investigated well
anlycity . In this paper, chiral perturbation is exploited to overcome this
difficulty, with the finding that this perturbation works well.
In the following, we first give a short review of the GND model, next compute
vertices, proving that our vertices satisfy the standard WTIs, and finally
discuss PDAs. We start with the generating functional of QCD,
$\displaystyle
e^{iW[\overline{I},I,J]}=\int\mathcal{D}\overline{\psi}\mathcal{D}\psi\mathcal{D}\overline{\Psi}\mathcal{D}\Psi\mathcal{D}A_{\mu}^{\alpha}~{}e^{i\int
d^{4}x[-\frac{1}{4}G_{\mu\nu}^{\alpha}G^{\alpha\mu\nu}+\overline{\Psi}(i\not{\partial}-g\frac{\lambda^{\alpha}}{2}\not{A}^{\alpha}-M)\Psi+\overline{\psi}(i\not{\partial}-g\frac{\lambda^{\alpha}}{2}\not{A}^{\alpha}+J)\psi+\overline{I}\psi+\overline{I}\psi]}\;,$
(1)
where $\overline{\psi}$ and $\psi$ are light-quark fields (only u and d quark
are taken as light quarks in this paper), $\overline{\Psi}$ and $\Psi$ are
heavy-quark fields with mass $M$, $A_{\mu}^{\alpha}$ is the gluon field and
$G_{\mu\nu}^{\alpha}$ its field strength. $\overline{I}$ and $I$ are external
sources for light-quark fields; $J$ is the external current source for
bilinear local light-quark fields. According to its $\gamma$ matrix structure,
it can be decomposed into vector, axial-vector, scalar, pseudo-scalar, and
tensor parts
$\displaystyle
J(x)=\not{v}(x)+\not{a}(x)\gamma_{5}-s(x)+ip(x)\gamma_{5}+\bar{t}^{\mu\nu}(x)\sigma_{\mu\nu}\;.$
(2)
The current quark mass $m$ for light quarks are absorbed into the scalar
source $s(x)$; i.e. on the vacuum, all external sources vanish, except
$s(x)=m$. After integrating out gluon and heavy-quark fields and integrating
in the pseudo-scalar meson field, according to the discussion of Ref.WQ2002 ,
the generating functional (1) can be approximated by
$\displaystyle
e^{iW[\overline{I},I,J]}=\int\mathcal{D}U\mathcal{D}\overline{\psi}_{\Omega}\mathcal{D}\psi_{\Omega}~{}e^{iS_{\mathrm{GND}}[\overline{\psi}_{\Omega},\psi_{\Omega},J_{\Omega},\overline{I}_{\Omega},I_{\Omega}]}$
(3) $\displaystyle
S_{\mathrm{GND}}[\overline{\psi}_{\Omega},\psi_{\Omega},J_{\Omega},\overline{I}_{\Omega},I_{\Omega}]=\int
d^{4}x\bigg{[}\overline{\psi}_{\Omega}[i\not{\partial}+J_{\Omega}-\Sigma(\overline{\nabla}^{2})]\psi_{\Omega}+\overline{I}_{\Omega}\psi_{\Omega}+\overline{I}_{\Omega}\psi_{\Omega}\bigg{]}$
(4) $\displaystyle\psi_{\Omega}=[\Omega^{{\dagger}}P_{R}+\Omega
P_{L}]\psi\hskip
22.76228pt\overline{\psi}_{\Omega}=\overline{\psi}[\Omega^{{\dagger}}P_{R}+\Omega
P_{L}]\hskip 28.45274ptI_{\Omega}=[\Omega^{{\dagger}}P_{L}+\Omega
P_{R}]I\hskip
22.76228pt\overline{I}_{\Omega}=\overline{I}[\Omega^{{\dagger}}P_{L}+\Omega
P_{R}]$ (5) $\displaystyle J_{\Omega}=[\Omega^{{\dagger}}P_{L}+\Omega
P_{R}][J+i\not{\partial}][\Omega^{{\dagger}}P_{L}+\Omega
P_{R}]=\not{v}_{\Omega}+\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}+\sigma_{\mu\nu}\bar{t}^{\mu\nu}_{\Omega}$
(6) $\displaystyle U=\Omega^{2}\hskip
56.9055pt\overline{\nabla}^{\mu}=\partial^{\mu}-iv_{\Omega}^{\mu}\;,$ (7)
where $U$ and $\Omega$ are unimodular pseudo-scalar meson fields,
$\overline{\psi}_{\Omega}$,$\psi_{\Omega}$ and
$\overline{I}_{\Omega}$,$I_{\Omega}$ are rotated light-quark fields and
corresponding sources, respectively, $J_{\Omega}$ is the rotated source for
the rotated bilinear quark currents. $\Sigma(-p^{2})$ is the self-energy for
the rotated light-quark fields which satisfy the corresponding SDE.
$S_{\mathrm{GND}}[\overline{\psi}_{\Omega},\psi_{\Omega},J_{\Omega},\overline{I}_{\Omega},I_{\Omega}]$
is the action of the GND model; in the original paper GND , $\overline{I}$ and
$I$ were not introduced, but an extra normalization term
$i\mathrm{Trln}[i\not{\partial}+J_{\Omega}]$ appeared. In Ref.WQ2010 , this
extra term was later proven to drop out. Integrating out the rotated light-
quark and pseudo-scalar meson fields of (3), we obtain
$\displaystyle W[\overline{I},I,J]=-i\mathrm{Trln}D^{-1}-\int
d^{4}xd^{4}y\overline{I}_{\Omega}(x)D(x,y)I_{\Omega}(y)+\mbox{pseudo-scalar
meson loop corrections}$ (8) $\displaystyle
D^{-1}(x,y)=[i\not{\partial}_{x}+J_{\Omega}-\Sigma(\overline{\nabla}_{x}^{2})]\delta(x-y)\;,$
(9)
where the pseudo-scalar field $\Omega$ in (8) must satisfy the stationary
equation $\frac{\partial\mathrm{Trln}D^{-1}}{\partial\Omega(x)}=0$ for
$\overline{I}=I=0$. Considering that in the low-energy region, we have shown
in WQ2002 that the Lagrangian of (8) just presumes the standard Gasser-
Leutwyler chiral Lagrangian GS , the stationary equation then can be replaced
with the equation of motion (EOM) derived from the Gasser-Leutwyler chiral
Lagrangian. For convenience in computation, we use below this chiral
Lagrangian-induced EOM for $\Omega$, because it is more common and relatively
simple.
Neglecting pseudo-scalar meson loop corrections, the vertices in the chiral
limit are defined by the following 3-point Green’s function
$\displaystyle\langle
0|\mathbf{T}\psi(x)\overline{\psi}(y)\overline{\psi}(z)\tilde{\gamma}_{i}\tau^{a}\psi(z)|0\rangle_{m=0}=-\frac{\delta^{3}W[\overline{I},I,J]}{\delta\overline{I}(x)\delta
I(y)\delta J_{i}^{a}(z)}\bigg{|}_{J=0,\overline{I}=I=0}\equiv\int
d^{4}x^{\prime}d^{4}y^{\prime}D(x,x^{\prime})\Gamma_{i}^{a}(x^{\prime},y^{\prime},z)D(y^{\prime}y)$
(10) $\displaystyle=-\frac{\delta}{\delta
J_{i}^{a}(z)}\bigg{[}[\Omega^{\dagger}(x)P_{L}+\Omega(x)P_{R}]D(x,y)[\Omega^{\dagger}(y)P_{L}+\Omega(y)P_{R}]\bigg{]}\bigg{|}_{J=0,\overline{I}=I=0}\;,$
which yields
$\displaystyle\Gamma_{i}^{a}(x,y,z)=\bigg{[}\frac{\delta D^{-1}(x,y)}{\delta
J_{i}^{a}(z)}-D^{-1}(x,y)\frac{\delta\Omega(y)}{\delta
J_{i}^{a}(z)}\gamma_{5}-\frac{\delta\Omega(x)}{\delta
J_{i}^{a}(z)}\gamma_{5}D^{-1}(x,y)\bigg{]}\bigg{|}_{J=0,\overline{I}=I=0}\;,$
(11)
where $i=S,P,V,A,T$, $\tau^{a}$ are the Pauli matrices in isospin space with
$\tau^{0}=1$, and
$\displaystyle J_{S}^{a}=-s^{a}\hskip 8.5359ptJ_{P}^{a}=p^{a}\hskip
8.5359ptJ_{V}^{a}=v_{\mu}^{a}\hskip 8.5359ptJ_{A}^{a}=a_{\mu}^{a}\hskip
8.5359ptJ_{T}^{a}=\bar{t}_{\mu\nu}^{a}\hskip
28.45274pt\tilde{\gamma}_{S}=1\hskip
8.5359pt\tilde{\gamma}_{P}=i\gamma_{5}\hskip
8.5359pt\tilde{\gamma}_{V}=\gamma^{\mu}\hskip
8.5359pt\tilde{\gamma}_{A}=\gamma^{\mu}\gamma_{5}\hskip
8.5359pt\tilde{\gamma}_{T}=\sigma^{\mu\nu}$
with the help of the following EOM results in the chiral limit
$\displaystyle\frac{\delta\Omega(x)}{\delta
s^{a}(y)}\bigg{|}_{J=0,\overline{I}=I=0}=\frac{\delta\Omega(x)}{\delta
v_{\mu}^{a}(y)}\bigg{|}_{J=0,\overline{I}=I=0}=\frac{\delta\Omega(x)}{\delta\bar{t}_{\mu\nu}^{a}(y)}\bigg{|}_{J=0,\overline{I}=I=0}=0$
(12) $\displaystyle\frac{\delta\Omega(x)}{\delta
p^{a}(y)}\bigg{|}_{J=0,\overline{I}=I=0}=\tau^{a}(1-\delta_{a0})\frac{iB_{0}}{\partial^{2}_{x}}\delta(x-y)\hskip
28.45274pt\frac{\delta\Omega(x)}{\delta
a_{\mu}^{a}(y)}\bigg{|}_{J=0,\overline{I}=I=0}=\bigg{[}\tau^{a}(1-\delta_{a0})+\delta_{a0}\bigg{]}\frac{i\partial^{\mu}_{x}}{\partial^{2}_{x}}\delta(x-y)\;,$
(13)
where $B_{0}=-\frac{1}{2}\langle\overline{\psi}\psi\rangle/F_{0}^{2}$ is the
$p^{2}$-order low-energy constant of the Gasser-Leutwyler chiral Lagrangian
related to the ratio of the quark condensate
$\langle\overline{\psi}\psi\rangle$ and the square of the pion decay constant,
$F_{0}^{2}$. With the help of (12) and (13), we can compute a series
derivation of rotated sources
$s_{\Omega},p_{\Omega},v_{\Omega},a_{\Omega},\bar{t}_{\Omega}$ to un-rotated
sources $s,p,v,a,\bar{t}$. With these relations, we obtain the chiral-limit
result
$\displaystyle\Gamma_{S}^{a}(x,y,z)=\tau^{a}\delta(x-z)\delta(y-z)\hskip
56.9055pt\Gamma_{T,\mu\nu}^{a}(x,y,z)=\tau^{a}\sigma_{\mu\nu}\delta(x-z)\delta(y-z)$
(14)
$\displaystyle\Gamma_{i}^{a}(x,y,z)=\int\frac{d^{4}pd^{4}q}{(2\pi)^{8}}e^{-iq\cdot
x+ip\cdot y+i(q-p)\cdot z}\tilde{\Gamma}_{i}^{a}(p,q)\hskip
28.45274pti=P,V,A\;.$ (15)
Whereas the resulting scalar and tensor vertices are trivial, the pseudo-
scalar, vector, and axial-vector vertices are non-trivial, their momentum
space expressions being
$\displaystyle\tilde{\Gamma}_{P}^{a}(p,q)=i\tau^{a}\bigg{[}1-B_{0}(1-\delta_{a0})\frac{\Sigma(-q^{2})+\Sigma(-p^{2})}{(q-p)^{2}}\bigg{]}\gamma_{5}$
(16)
$\displaystyle\tilde{\Gamma}_{V,\mu}^{a}(p,q)=\tau^{a}\bigg{[}\gamma_{\mu}-\frac{q_{\mu}+p_{\mu}}{q^{2}-p^{2}}[\Sigma(-q^{2})-\Sigma(-p^{2})]\bigg{]}$
(17)
$\displaystyle\tilde{\Gamma}_{A,\mu}^{a}(p,q)=\tau^{a}\bigg{[}\gamma_{\mu}-\frac{q_{\mu}-p_{\mu}}{(q-p)^{2}}[\Sigma(-q^{2})+\Sigma(-p^{2})]\bigg{]}\gamma_{5}\;.$
(18)
Equations (17) and (18) have exactly the same forms as those assumed by
Ref.Holdom . From (17), the WTI for the vector vertex is
$\displaystyle(q-p)^{\mu}\tilde{\Gamma}_{V,\mu}^{a}=\tau^{a}[S^{-1}(q)-S^{-1}(p)]\hskip
56.9055ptS^{-1}(p)=\not{p}-\Sigma(-p^{2})\;,$ (19)
where $S(p)$ is the light-quark propagator in momentum space with
$D(x,y)\bigg{|}_{J=0}=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip\cdot x}S(p)$. From
(18), the WTI for the axial-vector vertex is
$\displaystyle(q-p)^{\mu}\tilde{\Gamma}_{A,\mu}^{a}=\tau^{a}[S^{-1}(q)\gamma_{5}+\gamma_{5}S^{-1}(p)]\;.$
(20)
The above two WTIs are standard and yield the minimal solutions for (17) and
(18). We call (14), (16), (17) and (18) the minimal WT vertices and the
corresponding GND model is the model which correctly generates these minimal
WT vertices. In other words, if one prefers to trace the minimal WT vertices
back to a source effective action, it is just the GND model action (4).
Considering that the present GND model is only an approximation of the
underlying QCD, we will in the future discuss its corrections in QCD. That
will mean amending the above minimal WT vertices.
We shall now discuss the light-cone PDAs, their definitions being as in
Ref.DAdef
$\displaystyle\langle\vec{p}|\psi(x)\overline{\psi}(0)|0\rangle$
$\displaystyle=$ $\displaystyle-\frac{if}{4}\int_{0}^{1}du~{}e^{i(1-u)p\cdot
x}~{}\Phi(u,p,x)\;,$ (21) $\displaystyle\Phi(u,p,x)$ $\displaystyle=$
$\displaystyle\bigg{[}\not{p}\gamma_{5}\phi(u)-\frac{m_{\pi}^{2}}{2m}\gamma_{5}[\phi_{p}(u)+\sigma_{\mu\nu}p^{\mu}x^{\nu}\frac{\phi_{\sigma}(u)}{6}]+\mbox{high
twist corrections}\bigg{]}_{p^{0}=\sqrt{\vec{p}^{2}+m_{\pi}^{2}}}\;,$ (22)
where $\langle\vec{p}|$ is a pion state with momentum $\vec{p}$,
$m_{\pi},m_{u},m_{d}$ are masses of the pion, u quark, and d quark
respectively. For simplicity, we ignore the mass difference between u and d
quarks setting them to the same current mass $m=m_{u}=m_{d}$. $\phi(u)$ is the
leading twist or twist-2 PDA, $\psi_{p}(u)$ and $\phi_{\sigma}(u)$ are sub-
leading or twist-3 PDAs that correspond to the pseudo-scalar and the pseudo-
tensor structures respectively. In our model, the operator
$\psi(x)\overline{\psi}(0)$ appeared in (21) is related to
$\frac{\delta^{2}W[\overline{I},I,J]}{\delta\overline{I}(x)\delta I(0)}$, and
its contribution to the pion matrix element relies on the part proportional to
the pion field in the result. Parameterizing the pion field $\Pi$ by
$\Omega=e^{i\Pi/2}$, the rotated source becomes
$\displaystyle
J_{\Omega}\bigg{|}_{J=-m,\overline{I}=I=0}=[1+i\frac{\Pi}{2}\gamma_{5}][-m+i\not{\partial}][1+i\frac{\Pi}{2}\gamma_{5}]=-m-(im\Pi+\frac{1}{2}\not{\partial}\Pi)\gamma_{5}+O(\Pi^{2})\;.$
(23)
Hence,
$\displaystyle\langle\vec{p}|\psi(x)\overline{\psi}(0)|0\rangle=\langle\vec{p}|\frac{\delta^{2}W[\overline{I},I,J]}{\delta\overline{I}(x)\delta
I(0)}|0\rangle\bigg{|}_{J=-m,\overline{I}=I=0}$
$\displaystyle=\langle\vec{p}|[1+\frac{i}{2}\Pi(x)\gamma_{5}]\bigg{[}\frac{1}{i\not{\partial}-m-\Sigma(\partial^{2})-(im\Pi+\frac{1}{2}\not{\partial}\Pi)\gamma_{5}}\bigg{]}(x,0)[1+\frac{i}{2}\Pi(x)\gamma_{5}]|0\rangle$
$\displaystyle=i\langle\vec{p}|\Pi(0)|0\rangle\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot
x}\bigg{[}\gamma_{5}e^{ip\cdot
x}\frac{m+\Sigma(-q^{2})}{q^{2}-[m+\Sigma(-q^{2})]^{2}}-\frac{1}{\not{q}\\!+\\!m\\!+\\!\Sigma(-q^{2})}(m+\frac{\not{p}}{2})\gamma_{5}\frac{1}{\not{p}\\!-\\!\not{q}\\!-\\!m\\!-\\!\Sigma[-(p-q)^{2}]}\bigg{]}\;,$
(24)
where the first term, after expanding $e^{iq\cdot x}$ in terms of powers of
$q\cdot x$ and ignoring high twist $O(x^{2})$ terms, contributes to the
twist-3 pseudo-scalar PDA $\phi_{p}(u)$ at the end-point $u=0$, whereas the
second term can be changed to the form of (21) by the standard Feynman
parametrization method,
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot
x}\frac{1}{\not{q}\\!+\\!\tilde{\Sigma}(q)}(m+\frac{\not{p}}{2})\gamma_{5}\frac{1}{\not{p}\\!-\\!\not{q}\\!-\\!\tilde{\Sigma}(p-q)}=\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot
x}\frac{\not{q}\\!-\\!\tilde{\Sigma}(q)}{q^{2}\\!-\\!\tilde{\Sigma}^{2}(q)}(m+\frac{\not{p}}{2})\gamma_{5}\frac{\not{p}\\!-\\!\not{q}\\!+\\!\tilde{\Sigma}(p-q)}{(p-q)^{2}\\!-\\!\tilde{\Sigma}^{2}(p-q)}$
$\displaystyle=\int_{0}^{1}du~{}e^{i(1-u)p{\cdot}x}\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq{\cdot}x}\frac{[\not{q}+(1-u)\not{p}\\!-\\!\tilde{\Sigma}(q+p-up)](m+\frac{\not{p}}{2})[-u\not{p}\\!+\\!\not{q}\\!+\\!\tilde{\Sigma}(up-q)]}{\\{q^{2}+u(1-u)p^{2}-\tilde{\Sigma}^{2}(q+p-up)+(1-u)[\tilde{\Sigma}^{2}(q+p-up)-\tilde{\Sigma}^{2}(up-q)]\\}^{2}}\gamma_{5}\;,$
(25)
where $u$ is the standard Feynman parameter appearing in the well-known
Feynman parameter integration and $\tilde{\Sigma}(q)$ is the abbreviated
expression for $m\\!+\\!\Sigma(-q^{2})$. Conventionally, the Feynman parameter
integration is used to deal with loop-momentum integration with constant mass
in the perturbation theory; here, applied to the momentum integration
involving momentum-dependent quark self-energy, it still works. One can easily
check that if we take the quark self-energy appearing in (25) back to constant
mass as in the traditional NJL model, (25) just becomes the standard Feynman
parameterization formula. Indeed, this identification of the Feynman parameter
with the PDA variable $u$ not only endows the traditional mathematical Feynman
parameter with a physical meaning, but also is valid in any kind of chiral
quark or NJL-like models, as long as we have momentum integration of form (25)
with two quark propagators (the form of vertex between two propagators is not
important for this issue), no matter whether the models are local or non-
local. This result is independent of the GND model investigated in this paper;
the GND model here is taken as an example to exhibit details of the
computation .
With the above Feynman parameterization, and after lengthy computations, we
finally obtain $\Phi(u,p,x)$ introduced in (21) as
$\displaystyle-\frac{if}{4}\Phi(u,p,x)$ $\displaystyle=$ $\displaystyle
i\langle\vec{p}|\Pi(0)|0\rangle\bigg{\\{}\delta(u\\!-\\!0^{+})\int\frac{d^{4}q}{(2\pi)^{4}}\gamma_{5}\frac{\tilde{\Sigma}(q)}{q^{2}-\tilde{\Sigma}^{2}(q)}$
(26)
$\displaystyle-\int\frac{d^{4}q}{(2\pi)^{4}}\frac{I_{3}\\!+\\!\frac{p{\cdot}q}{p^{2}}I_{1}\\!+\\!\left(\frac{q^{2}}{3}\\!-\\!\frac{(p{\cdot}q)^{2}}{3p^{2}}\right)iI_{2}}{\\{q^{2}+u(1-u)p^{2}-\tilde{\Sigma}^{2}(up-q)+u[\tilde{\Sigma}^{2}(up-q)-\tilde{\Sigma}^{2}(q+p-up)]\\}^{2}}\gamma_{5}$
$\displaystyle-\left[-\delta(u\\!-\\!1^{-})\\!+\\!\delta(u\\!-\\!0^{+})\\!+\\!\frac{\partial}{\partial
u}\right]\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\left(\frac{q^{2}}{3}\\!-\\!\frac{(p{\cdot}q)^{2}}{3p^{2}}\right)I_{4}\\!+\\!\frac{p{\cdot}q}{p^{2}}I_{3}+\left(-\frac{q^{2}}{3}\\!+\\!\frac{4(p{\cdot}q)^{2}}{3p^{2}}\right)\frac{I_{1}}{p^{2}}}{\\{q^{2}+u(1-u)p^{2}-\tilde{\Sigma}^{2}(up-q)+u[\tilde{\Sigma}^{2}(up-q)-\tilde{\Sigma}^{2}(q+p-up)]\\}^{2}}\gamma_{5}\bigg{\\}}$
$\displaystyle+\mbox{high twist terms}\;,$
where $p^{2}=m^{2}_{\pi}$ and
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\not{p}[q{\cdot}p+\frac{1}{2}(1-2u)p^{2}+m\tilde{\Sigma}(up-q)-m\tilde{\Sigma}(q+p-up)]+\frac{1}{2}p^{2}[2(1-2u)m+\tilde{\Sigma}(up-q)-\tilde{\Sigma}(q+p-up)]\;,$
(30) $\displaystyle
I_{2}=\frac{1}{2}x_{\mu}p_{\nu}[\gamma^{\mu},\gamma^{\nu}][\frac{1}{2}\tilde{\Sigma}(up-q)-m\\!+\\!\frac{1}{2}\tilde{\Sigma}(q+p-up)]\;,~{}~{}~{}$
$\displaystyle
I_{3}=(m-\not{p})q^{2}+[(1-u)\not{p}\\!-\\!\tilde{\Sigma}(q+p-up)](m+\frac{\not{p}}{2})[-u\not{p}\\!+\\!\tilde{\Sigma}(up-q)]\;,$
$\displaystyle
I_{4}=\frac{1}{2}\tilde{\Sigma}(up-q)+(1-2u)m\\!-\\!\frac{1}{2}\tilde{\Sigma}(q+p-up)\;.$
In (26), we have dropped terms proportional to $\not{x}$, since they belong to
twist-4 terms. The delta function terms are just end-point terms that are from
non-exponential $p{\cdot}x$ terms appearing in the original momentum
integration (25) when we expand $e^{iq\cdot x}$ in terms of powers of $q\cdot
x$ and apply Lorentz invariance in decomposing its Lorentz structure.
Since (26) already has structure of (22), through comparison between the two
equations, we can easily read out general expressions of PDAs $\phi(u)$,
$\psi_{p}(u)$ and $\phi_{\sigma}(u)$ in terms of quark self energy. One can
check that except the pure end point term in the first line of (26), all other
terms satisfying symmetry $u\leftrightarrow 1-u$, since under combined
transformation of $u\leftrightarrow 1-u$ and $q\leftrightarrow-q$, the
integrand of momentum integration is even for the second line and odd for the
third line, respectively. This implies the corresponding symmetry of
$u\leftrightarrow 1-u$ for result PDAs. In the chiral limit, $m=0$ and
$p^{2}=m^{2}_{\pi}=0$, we find (26) simplifies to
$\displaystyle\Phi(u,p,x)\stackrel{{\scriptstyle\mbox{\tiny chiral
limit}}}{{======}}$
$\displaystyle\Phi_{0}(u,p,x)=-\frac{4\langle\vec{p}|\Pi(0)|0\rangle}{f}\bigg{[}\not{p}\gamma_{5}\bigg{(}[\delta(u-1^{-})+\delta(u-0^{+})]\phi_{\delta
0}+\phi_{0}\bigg{)}$ (31) $\displaystyle\hskip
110.96556pt+\delta(u-0^{+})\phi_{p,\delta
0}-\frac{m_{\pi}^{2}}{2m}\gamma_{5}\sigma_{\mu\nu}p^{\mu}x^{\nu}\frac{\phi_{\sigma,0}}{6}\bigg{]}\;,~{}~{}~{}$
where the four coefficients $\phi_{\delta 0}$,$\phi_{0}$,$\phi_{p,\delta 0}$
and $\phi_{\sigma,0}$ are expressed as Euclidean-space momentum integrations
$\displaystyle\phi_{\delta
0}=-\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}\frac{1}{4}X^{2}q_{E}^{2}\Sigma\Sigma^{\prime}\hskip
36.98866pt\phi_{0}=\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}[\frac{1}{4}q_{E}^{2}-\frac{1}{2}\Sigma^{2}+\frac{1}{2}q_{E}^{2}\Sigma\Sigma^{\prime}]$
(32) $\displaystyle\phi_{p,\delta
0}=\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X\Sigma\hskip
71.13188pt\phi_{\sigma,0}=\frac{3m}{m_{\pi}^{2}}\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}q_{E}^{2}\Sigma\;,$
(33) $\displaystyle\Sigma=\Sigma(q_{E}^{2})\hskip
28.45274pt\Sigma^{\prime}=\frac{d\Sigma(q_{E}^{2})}{dq_{E}^{2}}\hskip
28.45274ptX=\frac{1}{q_{E}^{2}+\Sigma^{2}(q_{E}^{2})}\;.$ (34)
Comparing the above result with (22), we find that with the exception of end-
point values for $\phi(u)$ symmetrically at u=0 and u=1 and $\phi_{p}(u)$ non-
symmetrically at u=0, PDAs are all independent of $u$ ($\phi_{p}(u)$ even
vanishes) and therefore take flat-like forms. Graphically, the pictures are
that two symmetric infinitesimal narrow pillars at two end points appear above
the flat-like form backgrounds for $\phi(u)$, one pure non-symmetric pillar at
$u=0$ for $\phi_{p}(u)$ with zero background, and no pillar for
$\phi_{\sigma}(u)$ with just flat-like form background. This result is
completely due to the Feynman parameter description of PDAs, and therefore is
valid for any type of chiral quark or NJL-like models, whether local or non-
local (one can check by replacing quark self energy with constant mass that
the end-point terms from $\delta-$function for $\phi(u)$ and $\phi_{p}(u)$
exist even in local situations). This result is differs from those previously
obtained in the literature, where researchers believed that the momentum
dependence of the quark self-energy would force PDAs to deviate from flat-like
forms in the chiral limit. Considering our result is analytical that does not
rely on technical details of numerical computations and is valid for a large
class of chiral quark models, we believe it is reliable. This result implies
that, at least in the chiral limit, the momentum-dependent behavior of the
quark self-energy, or more fundamentally the related non-locality of its
interaction, causes no difference in the form of PDAs with constant mass or
local interaction such as the NJL model.
For the chiral limit result (31), $\phi(u)$ and $\phi_{\sigma}(u)$ are
ultraviolet divergent. The reason causing this divergence is that the quark
self-energy of the GND model is for the rotated quark field. If the quark
self-energy instead is for the un-rotated quark fields, as the conventional
instanton model does, we need to replace the current quark mass $m$ appearing
in the numerator of the l.h.s. term of (25) with the quark self-energy and
ignore the $\not{p}$ term in the same numerator. This then will suppress the
ultraviolet behavior of the momentum integration decreasing the ultraviolet
divergences of PDAs .
If we want to go beyond the chiral limit, the momentum integrations in the
second and third line of (26) are not easy to achieve. To finish the momentum
integrations, we are used to rotate the momentum integration variable $q$ into
Euclidean space, although the external momentum $p$ must be kept on the pion
mass shell, $p^{2}=m^{2}_{\pi}$. This will create some imaginary components;
for example $\Sigma[-(q-up)^{2}]=\Sigma[-q^{2}-u^{2}p^{2}+2uq{\cdot}p]$ will
become
$\Sigma[q_{E}^{2}-u^{2}p^{2}+2u(iq_{E}^{0}p^{0}-\vec{q}_{E}\cdot\vec{p})]$
after a Wick rotation for integration variable
$q^{\mu}=(q^{0},\vec{q})\rightarrow(iq_{E}^{0},\vec{q}_{E})$. Because the SDE
now cannot provide us with a reliable quark self-energy beyond the space-like
momentum region, we are not able then to directly compute the momentum
integration in (26). One way to avoid this difficulty is to go back to the
original constant quark mass case (or equivalently the NJL model situation),
where all momentum integration can be analytically finished except we need
some momentum cutoff to regularize the integration. An alternative approach is
to set an analytical expression for $\Sigma(-q^{2})$, as for instantons
instanton or introduce some simple ansatz Holdom , then the momentum
integration can still be finished, at least at the level of numerical
computations. Considering that for the former information of the momentum
dependence of the quark self-energy is lost, and the latter is constrained by
specific choices of quark self-energy which also might not precisely describe
its QCD behavior, we propose in this paper to expand the integrand in terms of
powers of $p^{2}=m^{2}_{\pi}$ and $m$. The underlying basis for this expansion
is that when we go to higher-order terms in the expansion, according to (26),
typically we encounter a factor of $[p{\cdot}q/(q^{2}+\Sigma^{2})]^{2}$ or
$p^{2}/(q^{2}+\Sigma^{2})$ or $mp{\cdot}q/(q^{2}+\Sigma^{2})$. In units of the
QCD scale parameter $\Lambda_{\mathrm{QCD}}$, the largest contribution comes
mainly from the platform region of the quark self-energy in which
$\Sigma/\Lambda_{\mathrm{QCD}}\sim 2$ and $q_{E}/\Lambda_{\mathrm{QCD}}\leq 1$
because quark self-energies above $\Lambda_{\mathrm{QCD}}$ decrease with
momentum as $1/q^{2}$ WQ2002 and contribute little. Considering that in our
case $\Lambda_{\mathrm{QCD}}\sim 440$MeV is much larger than the pion mass,
i.e. $p/\Lambda_{\mathrm{QCD}}\sim m_{\pi}/\Lambda_{\mathrm{QCD}}\sim 1/3$,
and $m<10$MeV, then the first factor $[p{\cdot}q/(q^{2}+\Sigma^{2})]^{2}\sim
10^{-2}$, the second factor $p^{2}/(q^{2}+\Sigma^{2})\sim 10^{-1}$, and the
third factor $mp{\cdot}q/(q^{2}+\Sigma^{2})\sim 10^{-3}$; that is, each factor
is at least an order of magnitude small. We expect chiral expansion will works
well. With this analysis, the result of the detail computation gives up to
first order,
$\displaystyle\Phi(u,p,x)$ $\displaystyle=$
$\displaystyle\Phi_{0}(u,p,x)+\Phi_{1}(u,p,x)+O(m^{4}_{\pi},m^{2},mm^{2}_{\pi})\;,$
(35) $\displaystyle\Phi_{1}(u,p,x)$ $\displaystyle=$
$\displaystyle-\frac{4\langle\vec{p}|\Pi(0)|0\rangle}{f}\bigg{[}\not{p}\gamma_{5}\bigg{(}[\delta(u-1^{-})+\delta(u-0^{+})]\phi_{\delta
1}+\phi_{10}+\phi_{11}u(1-u)\bigg{)}$ (36)
$\displaystyle+\delta(u-0^{+})\phi_{p,\delta
10}+[\delta(u-1^{-})+\delta(u-0^{+})]\phi_{p,\delta
11}+\phi_{p1}-\frac{m_{\pi}^{2}}{2m}\gamma_{5}\sigma_{\mu\nu}p^{\mu}x^{\nu}\frac{\phi_{\sigma,10}+\phi_{\sigma,11}u(1-u)}{6}\bigg{]}~{}~{}~{}\;,$
where the eight coefficients $\phi_{\delta
1}$,$\phi_{10}$,$\phi_{11}$,$\phi_{p,\delta 10}$,$\phi_{p,\delta
11}$,$\phi_{p1}$,$\phi_{\sigma,10}$, and $\phi_{\sigma,11}$ are expressed as
Euclidean-space momentum integrations
$\displaystyle\phi_{\delta 1}$ $\displaystyle=$ $\displaystyle
m_{\pi}^{2}\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}\bigg{[}X^{2}(\frac{1}{4}q_{E}^{2}\Sigma\Sigma^{\prime\prime}+\frac{1}{12}q_{E}^{4}\Sigma\Sigma^{\prime\prime}-\frac{1}{6}q_{E}^{4}\frac{m}{m_{\pi}^{2}}\Sigma^{\prime\prime})\bigg{]}$
(37) $\displaystyle\phi_{10}$ $\displaystyle=$ $\displaystyle
m_{\pi}^{2}\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}\bigg{[}X^{2}[(\Sigma-\frac{1}{2}q_{E}^{2}\Sigma^{\prime}+\frac{1}{3}q_{E}^{4}\Sigma^{\prime\prime})\frac{m}{m_{\pi}^{2}}+\frac{1}{2}\Sigma\Sigma^{\prime}-\frac{1}{2}q_{E}^{2}\Sigma\Sigma^{\prime\prime}-\frac{1}{4}q_{E}^{2}{\Sigma^{\prime}}^{2}$
$\displaystyle-\frac{1}{4}q_{E}^{4}\Sigma^{\prime}\Sigma^{\prime\prime}-\frac{1}{4}q_{E}^{4}\Sigma\Sigma^{\prime\prime\prime})+X^{3}(-\frac{1}{2}q_{E}^{2}\Sigma\Sigma^{\prime}-2q_{E}^{2}\Sigma^{2}{\Sigma^{\prime}}^{2}-q_{E}^{2}\Sigma^{3}\Sigma^{\prime\prime}-q_{E}^{4}\Sigma{\Sigma^{\prime}}^{3}-2q_{E}^{4}\Sigma^{2}\Sigma^{\prime}\Sigma^{\prime\prime}-\frac{1}{3}q_{E}^{4}\Sigma^{3}\Sigma^{\prime\prime\prime})\bigg{]}\;,$
$\displaystyle\phi_{11}$ $\displaystyle=$ $\displaystyle
m_{\pi}^{2}\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}\bigg{[}X^{2}(-\frac{1}{2}-\Sigma\Sigma^{\prime}+q_{E}^{2}{\Sigma^{\prime}}^{2}+q_{E}^{2}\Sigma\Sigma^{\prime\prime}+\frac{3}{2}q_{E}^{4}\Sigma^{\prime}\Sigma^{\prime\prime}+\frac{1}{2}q_{E}^{4}\Sigma\Sigma^{\prime\prime\prime})$
$\displaystyle+X^{3}(-\Sigma^{2}-2\Sigma^{3}\Sigma^{\prime}+\frac{1}{2}q_{E}^{2}+4q_{E}^{2}\Sigma\Sigma^{\prime}+11q_{E}^{2}\Sigma^{2}{\Sigma^{\prime}}^{2}+5q_{E}^{2}\Sigma^{3}\Sigma^{\prime\prime}+6q_{E}^{4}\Sigma{\Sigma^{\prime}}^{3}+12q_{E}^{4}\Sigma^{2}\Sigma^{\prime}\Sigma^{\prime\prime}+2q_{E}^{4}\Sigma^{3}\Sigma^{\prime\prime\prime})\bigg{]}\;,$
$\displaystyle\phi_{p,\delta 10}$ $\displaystyle=$
$\displaystyle\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}mX[1-2X\Sigma^{2}]\;,$
(40) $\displaystyle\phi_{p,\delta 11}$ $\displaystyle=$ $\displaystyle
2m\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}\bigg{[}(-\frac{1}{4}q_{E}^{2}-\frac{1}{2}q_{E}^{2}\Sigma\Sigma^{\prime})\frac{m}{m_{\pi}^{2}}-\frac{1}{8}q_{E}^{2}\Sigma^{\prime}-\frac{1}{8}q_{E}^{4}\Sigma^{\prime\prime}\bigg{]}\;,$
(41) $\displaystyle\phi_{p1}$ $\displaystyle=$ $\displaystyle
2m\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}\bigg{[}(\Sigma^{2}+\frac{3}{2}q_{E}^{2}+q_{E}^{2}\Sigma\Sigma^{\prime})\frac{m}{m_{\pi}^{2}}-\frac{1}{2}\Sigma+\frac{1}{4}q_{E}^{4}\Sigma^{\prime\prime}\bigg{]}\;,$
(42) $\displaystyle\phi_{\sigma,10}$ $\displaystyle=$ $\displaystyle
12m\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}\bigg{[}-\frac{1}{4}q_{E}^{2}\frac{m}{m_{\pi}^{2}}-\frac{1}{8}q_{E}^{2}\Sigma^{\prime}-\frac{1}{24}q_{E}^{4}\Sigma^{\prime\prime}\bigg{]}\;,$
(43) $\displaystyle\phi_{\sigma,11}$ $\displaystyle=$ $\displaystyle
12m\int\frac{q_{E}^{2}dq_{E}^{2}}{16\pi^{2}}X^{2}\bigg{[}\frac{1}{4}q_{E}^{2}\Sigma^{\prime}+\frac{1}{12}q_{E}^{4}\Sigma^{\prime\prime}\bigg{]}\;,$
(45)
$\displaystyle\Sigma^{\prime\prime}=\frac{d^{2}\Sigma(q_{E}^{2})}{d(q_{E}^{2})^{2}}\hskip
28.45274pt\Sigma^{\prime\prime\prime}=\frac{d^{3}\Sigma(q_{E}^{2})}{d(q_{E}^{2})^{3}}\;.$
We see that the first-order corrections include three parts
* •
Corrections to chiral limit result which include: heights to two symmetric
infinitesimal narrow pillars at two end points from $\phi_{\delta 1}$ and to
the flat-like form backgrounds from $\phi_{10}$ for $\phi(u)$; heights to non-
symmetric pillar at $u=0$ from $\phi_{p,\delta 10}$ and to zero backgrounds
from $\phi_{p1}$ for $\phi_{p}(u)$; and height for flat-like form background
from $\phi_{\sigma,10}$ for $\phi_{\sigma}(u)$.
* •
Two symmetric infinitesimal narrow negative pillars at two end points from
$\phi_{p,\delta 11}$ for $\phi_{p}(u)$ appear.
* •
Asymptotic-like form concave type corrections asymp proportional to $u(1-u)$
from $\phi_{11}$ and $\phi_{\sigma,11}$ to $\phi(u)$ and $\phi_{\sigma}(u)$
respectively.
To compare with literature’s CZ-like form results, traditional double-hump
structure of PDAs now for $\phi(u)$ is squeezed to two symmetric infinitesimal
narrow pillars at two end points plus concave type asymptotic-like form above
flat-like form background; for $\phi_{p}(u)$ is squeezed to two non-symmetric
infinitesimal narrow pillars at two end points above a flat-like form
background; for $\phi_{\sigma}(u)$ is changed to concave type asymptotic-like
form above flat-like form background. We expect more complex non-asymptotic-
like form corrections will show up in more higher order of our chiral
perturbation expansion.
If we further consider the normalization condition for $\phi(u)$,
$\displaystyle\int_{0}^{1}du~{}\phi(u)=1\;.$ (46)
then our results (31), (35) and (36) imply that
$\displaystyle-4\langle\vec{p}|\Pi(0)|0\rangle(2\phi_{\delta 0}+2\phi_{\delta
1}+\phi_{0}+\phi_{10}+\frac{1}{6}\phi_{11})=1\;.$ (47)
To obtain numerical results, we solve the SDE obtaining a quark self-energy as
in Ref.WQ2002 with model A and $\Lambda_{\mathrm{QCD}}=440$MeV,
$m_{\pi}=139.6$MeV and $m=\frac{m^{2}_{\pi}}{2B_{0}}=12.3$MeV, substitute the
resulting quark self-energy into the above formulae for the coefficients, and
finally perform the numerical computations. Considering that some momentum
integrations are divergent, we take $\Lambda=1$GeV as the cutoff parameter in
the Euclidean space momentum integration. Expressing all results in units of
$m_{\pi}$, values for the coefficients are listed in Table.1
Table 1: The obtained coefficients in unit of $10^{-2}$.
$\displaystyle\begin{array}[]{cccccccccccc}\hline\cr\hline\cr\displaystyle\frac{\phi_{\delta
0}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{\delta
1}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{0}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{10}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{11}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{p,\delta
0}}{m^{3}_{\pi}}&\displaystyle\frac{\phi_{p,\delta
10}}{m^{3}_{\pi}}&\displaystyle\frac{\phi_{p,\delta
11}}{m^{3}_{\pi}}&\displaystyle\frac{\phi_{p1}}{m^{3}_{\pi}}&\displaystyle\frac{\phi_{\sigma,0}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{\sigma,10}}{m^{2}_{\pi}}&\displaystyle\frac{\phi_{\sigma,11}}{m^{2}_{\pi}}\\\
\hline\cr
1.85&-0.15&10.13&0.98&-2.75&82.22&5.49&-0.16&0.86&8.13&-0.72&-0.17\\\
\hline\cr\hline\cr\end{array}$ (50)
We see that the first-order corrections are orders of magnitude smaller than
the leading order result. This verifies our conjecture that chiral
perturbation will work well.
To summarize, we have shown the GND model is a model which can correctly
generate the minimal WT vertices. For PDAs, GND’s result is similar to that of
the NJL-like models, i.e. in the chiral limit except some end point pillars,
PDAs take flat-like forms and non-flat effects of PDAs are due to nonzero pion
and light-quark current mass corrections. We have shown that the variable $u$
in PDAs is just the Feynman parameter of loop calculations of standard
perturbation theory and chiral perturbation works well for PDAs enabling
quantitative estimates of PDA values. These results are valid not only for the
GND model, but also for the larger class of chiral quark and NJL-like models.
Further, the leading order nonzero pion and current quark masses corrections
lead concave type asymptotic-like form modifications for $\phi(u)$ with end-
point pillars and $\phi_{\sigma}(u)$ above the flat-like form backgrounds as a
substitution of original double-hump structure of CZ-like form of PDAs. To
force matching our results with other phenomenological ones especially the
end-point behaviors, one can take present GND model chiral limit result as a
starting point, ignore present corrections from pion and current quark masses,
perform the QCD Gegenbauer evolution for PDAs as done in Ref.flat .
## ACKNOWLEDGMENTS
This work is supported by the National Science Foundation of China (NSFC)
under Grants No.11075085, Specialized Research Fund Grants No.20110002110010
for the Doctoral Program of High Education of China, and Tsinghua University
Initiative Scientific Research Program.
## References
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A.E.Dorokhov, W.Broniowski, and E. Ruiz Arriola, Phys. Rev. D74, 054023(2006).
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|
arxiv-papers
| 2013-04-14T07:05:39 |
2024-09-04T02:49:44.305396
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuan Li, Shao-Zhou Jiang, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/1304.3888"
}
|
1304.3996
|
# Cyber-Physical Security: A Game Theory Model of Humans Interacting over
Control Systems
Scott Backhaus,1 Russell Bent,1 James Bono,2 Ritchie Lee,3 Brendan Tracey,4
David Wolpert,1 Dongping Xie,2 and Yildiray Yildiz3 The authors are with 1Los
Alamos National Laboratory, 2www.bayesoptimal.com, 3NASA Ames, and 4Stanford
University.Manuscript received XXXXX; revised XXXX.
###### Abstract
Recent years have seen increased interest in the design and deployment of
smart grid devices and control algorithms. Each of these smart communicating
devices represents a potential access point for an intruder spurring research
into intruder prevention and detection. However, no security measures are
complete, and intruding attackers will compromise smart grid devices leading
to the attacker and the system operator interacting via the grid and its
control systems. The outcome of these machine-mediated human-human
interactions will depend on the design of the physical and control systems
mediating the interactions. If these outcomes can be predicted via simulation,
they can be used as a tool for designing attack-resilient grids and control
systems. However, accurate predictions require good models of not just the
physical and control systems, but also of the human decision making. In this
manuscript, we present an approach to develop such tools, i.e. models of the
decisions of the cyber-physical intruder who is attacking the systems and the
system operator who is defending it, and demonstrate its usefulness for
design.
## I Introduction
Supervisory Control and Data Acquisition (SCADA) systems form the cyber and
communication components of electrical grids. Human operators use SCADA
systems to receive data from and send control signals to grid devices to cause
physical changes that benefit grid security and operation. If a SCADA system
is compromised by a cyber attack, the attacker may alter these control signals
with the intention of degrading operations or causing widespread damage to the
physical infrastructure.
The increasing connection of SCADA to other cyber systems and the use of off-
the-shelf computer systems for SCADA platforms is creating new
vulnerabilities[1] increasing the likelihood that SCADA systems can and will
be penetrated. However, even when a human attacker has gained some control
over the physical components, the human operators (defenders) retain
significant SCADA observation and control capability. The operators may be
able to anticipate the attacker’s moves and effectively use this remaining
capability to counter the attacker’s moves. The design of the physical and
control system may have a significant impact on the outcome of the SCADA
operator’s defense, however, designing attack resilient systems requires
predictive models of these human-in-the-loop control systems. These machine-
mediated, adversarial interaction between two humans have been described in
previous game-theoretic models of human-in-the-loop collision avoidance
systems for aircraft[2] and our recent extensions of these models to
electrical grid SCADA systems[3]. The current work builds upon and extends
this previous work.
The model of machine-mediated human-human interactions described in [2]
includes two important components. The first is a mathematical framework for
describing the physical state of the system and its evolution as well as the
available information and its flow to both the humans and the automation.
Well-suited to this task is a semi Bayes net[2] which, like a Bayes net,
consists of: a set of nodes representing fixed conditional probability
distributions over the physical state variables and the sets of information
and directed edges describing the flow and transformation of information and
the evolution of the physical state between the nodes. However, a semi Bayes
net also includes “decision” nodes with unspecified conditional probability
distributions that will be used to model the strategic thinking of the humans
in the loop. When these decision nodes incorporate game-theoretic models, the
resulting structure called a semi network-form game (SNFG) of human strategic
behavior.
Game theoretic models of the humans are fundamentally different than models of
the automation and control algorithms. These simpler devices process inputs to
generate outputs without regard for how their outputs affect other components,
nor do they try to infer the outputs of other components before generating
their own output. Strategic humans perform both of these operations. In
adversarial interactions, a strategically thinking human infers the decisions
of his opponent and incorporates this information into his own decision
making. He also incorporates that his opponent is engaged in the same
reasoning. These behaviors distinguish humans from automation making the
principled design of human-in-the-loop control systems challenging. In our
model, we will utilize game theoretic solution concepts to resolve the
circular player-opponent inference problem just described and compute the
conditional probability distributions at the decision nodes in a SNFG
representation of a SCADA system under attack.
A game theoretic model of a decision node includes two important components.
The first is a utility or reward function that captures the goals of the human
represented by the decision node and measures the relative benefit of
different decisions. The second component is a solution concept that
determines how the human goes about making decisions. As a model of human
behavior, the solution concept must be selected to accurately represent the
humans in question. For example, the humans may be modeled as fully rational,
i.e. always selecting the action that maximizes their reward, or as bounded
rational, sometimes taking actions that are less than optimal. Additionally,
if the decisions that the humans are facing are too complex to afford
exhaustive exploration of all options, the mathematical operations we use to
represent the human’s mental approximations are also part of the solution
concept.
The current work builds upon previous game-theoretic models of human-in-the-
loop aircraft collision avoidance systems[2] and our recent extension of these
models to simplified electrical grid SCADA systems[3] where the focus was on
developing the computational model for predicting the outcome of a SCADA
attack where the SCADA operator was certain that an attacker was present. In
the present work, we retain the simplified electrical grid model, but make
several important extensions. First, we remove the SCADA operator’s certainty
that an attacker is present forcing the operator to perform well under both
normal and “attack” conditions. Second, we shift our focus from only
predicting the outcome of an attack to an initial effort at using these
predictions as a tool to design physical and control systems. Third, the
extension to design requires numerical evaluation of many more scenarios, and
we have implemented new computational algorithms that speed our simulations.
To summarize, the designer models and simulates the behavior of the SCADA
operator and the cyber-physical attacker by developing reward functions and
solution concepts that closely represent the decision making processes of
these humans. These game theoretic models are embedded into the decision nodes
of a SNFG that represents the evolution of the physical state and information
available to both human decision nodes and the automation nodes. If the model
is accurate, then the designer can utilize this model to predict the outcomes
of different system designs and, therefore, maximize his own “designer’s
reward function”. This design process closely resembles the economic theory of
mechanism design [4, 5], whereby an external policy-maker seeks to design a
game with specific equilibrium properties. The key difference between our work
and mechanism design is that we do not assume equilibrium behavior, and this
enables us to use the standard control techniques described above[6, 7, 8, 9].
We also make contributions to the growing literature on game theory and
network security [10, 11]. The assumption that human operators infer the
existence of an attacker from the state of the SCADA places this model
alongside work on intrusion detection systems [12, 13, 14, 15, 16]. However,
we also model the human operator’s attempts to mitigate damages when an attack
is detected. So our model contributes to the literature on intrusion response
[17].
The rest of this paper is organized as follows. Section II describes the
simplified electrical distribution circuit and the SCADA used to control it.
Section III reviews the structure of SNFG and points out features and
extensions important for the current work. Section IV describes the solution
concept we apply to our SCADA model. Section V and VI describe the simulation
results and our use of these results to assess design options, respectively.
Section VII gives our conclusions and possible directions for future work.
## II Simplified Electrical Grid Model
To keep the focus of this work on modeling the adversarial interaction between
the defender and attacker, we retain the simplified model of an electrical
grid used in previous work[3]. Specifically, we consider the three-node model
of a radial distribution circuit shown schematically in Fig. 1. The circuit
starts at the under-load tap changer (ULTC) on the low-voltage side of a
substation transformer at node 1, serves an aggregation of loads at node 2,
and connects a relatively large, individually-modeled distributed generator at
node 3. In practice, most systems are considerably more detailed than this
example. This example was chosen to limit the degrees of freedom to allow full
enumeration of the parameter space and improve our understanding of the
model’s salient features. However, it is important to note that the model is
not limited computationally by the size of the power system, rather it is
limited by the number of players and their possible observations and actions.
Extensions to more complex settings is an open challenge for future research.
Figure 1: The simplified distribution feeder line used in this study. Node 1
is at the substation where the SCADA enables control over $V_{1}$ via a tap
changer. Node 2 represents a large aggregate real $p_{2}$ and reactive $q_{2}$
loads that fluctuate within a narrow range. Node three represents a
distributed generator with real and reactive outputs $p_{3}$ and $q_{3}$. The
assume the SCADA system enables control over $q_{3}$ to assist with voltage
regulation along the circuit and that the attacker has taken control over
$q_{3}$. The distribution circuit segments between the nodes have resistance
$r_{i}$ and reactance $x_{i}$. The node injections $p_{i}$ and $q_{i}$
contribute to the circuit segment line flows $Q_{i}$ and $P_{i}$.
In Fig. 1, $V_{i},p_{i},$ and $q_{i}$ are the voltage and real and reactive
power injections at node $i$. $P_{i},Q_{i},r_{i},$ and $x_{i}$ are the real
power flow, reactive power flow, resistance, and reactance of circuit segment
$i$. For this simple setting, we use the LinDistFlow equations [18]
$\displaystyle
P_{2}=-p_{3},\;\;Q_{2}=-q_{3},\;\;P_{1}=P_{2}+p_{2},\;\;Q_{1}=Q_{2}+q_{2}$ (1)
$\displaystyle
V_{2}=V_{1}-(r_{1}P_{1}+x_{1}Q_{1}),\;\;V_{3}=V_{2}-(r_{2}P_{2}+x_{2}Q_{2}).$
(2)
Here, all terms have been normalized by the nominal system voltage $V_{0}$,
and we set $r_{i}=0.03$ and $x_{i}=0.03$.
The attacker-defender game is modeled in discrete time with each simulation
step representing one minute. To emulate the normal fluctuations of consumer
real load, $p_{2}$ at each time step is drawn from a uniform distribution over
the relatively narrow range $[p_{2,min},p_{2,max}]$ with $q_{2}=0.5p_{2}$. The
real power injection $p_{3}$ is of the distributed generator at node 3 is
fixed. Although fixed for any one instance of the game, $p_{2,max}$ and
$p_{3}$ are our design parameters, and we vary these parameters to study how
they affect the outcome of the attacker-defender game. In all scenarios,
$p_{2,min}$ is set 0.05 below $p_{2,max}$
In our simplified game, the SCADA operator (defender) tries to, keep the
voltages $V_{2}$ and $V_{3}$ within appropriate operating bounds (described in
more detail below). Normally, the operator has two controls: the ULTC to
adjust the voltage $V_{1}$ or the reactive power output $q_{3}$ of the
distributed generator. We assume that the system has been compromised, and the
attacker has control of $q_{3}$ while the defender retains control of $V_{1}$.
Changes in $V_{1}$ comprise the defender decision node while control of
$q_{3}$ comprise the attacker decision node.
By controlling $q_{3}$, the attacker can modify the $Q_{i}$ and cause the
voltage $V_{2}$ at the customer node to deviate significantly from $1.0\;p.u.$
– potentially leading to economic losses by damaging customer equipment or by
disrupting computers or computer-based controllers belonging to commercial or
industrial customers[19]. The attacker’s goals are modeled by the reward
function
$R_{A}=\Theta(V_{2}-(1+\epsilon))+\Theta((1-\epsilon)-V_{2}).$ (3)
Here, $\epsilon$ represents the halfwidth of the acceptable range of
normalized voltage. For most distribution systems under consideration,
$\epsilon\sim 0.05$. $\Theta(\cdot)$ is a step function representing the need
for the attacker to cross a voltage deviation threshold to cause damage.
In contrast, the defender attempts to control both $V_{2}$ and $V_{3}$ to near
$1.0\;p.u.$. The defender may also respond to relatively small voltage
deviations that provide no benefit to the attacker. We express these defender
goals through the reward function
$R_{D}=-\left(\frac{V_{2}-1}{\epsilon}\right)^{2}-\left(\frac{V_{3}-1}{\epsilon}\right)^{2}.$
(4)
## III Time-Extended, Iterated Semi Net-Form Game
To predict how system design choices affect the outcome of attacker-defender
interactions, we need a description of when player decisions are made and how
these decisions affect the system state, i.e. a “game” definition.
Sophisticated attacker strategies may be carried out over many time steps
(i.e. many sequential decisions), therefore we need to expand the SNFG
description in the Introduction to allow for this possibility.
Figure 2 shows three individual semi-Bayes networks representing three time
steps of our time-extended attacker-defender interaction. Each semi-Bayes net
has the structure of a distinct SNFG played out at time step $i$. These SNFGs
are “glued” together to form an iterated SNFG by passing the system state
$S^{i}$, the players’ moves/decisions $D_{D}^{i}$ and $D_{A}^{i}$, and the
players’ memories $M_{D}^{i}$ and $M_{A}^{i}$ from the SNFG at time step $i$
to the SNFG at time step $i+1$. Iterated SNFGs are described in more detail in
[3].
In the rest of this section, we describe the nodes in this iterated SNFG and
their relationship to one another.
Figure 2: The iterated semi net-form game (SNFG) used to model attackers and
operators/defenders in a cyber-physical system. The iterated SNFG in the
Figure consists of three individual SNFGs that are “glued” together at a
subset of the nodes in the semi Bayes net that make up each SNFG.
#### III-1 Attacker existence
In contrast to our previous work, we add an ‘A exist” node in Fig. 2–the only
node that is not repeated in each SNFG. This node contains a known probability
distribution that outputs a $1$ (attacker exists) with probability $p$ and a
$0$ (no attacker) with probability $1-p$. When the attacker is not present,
his decision nodes ($D_{A}^{i}$) are disabled and $q_{3}$ is not changed. We
vary $p$ to explore the effect of different attack probabilities.
#### III-2 System state
The nodes $S^{i}$ contain the true physical state of the cyber-physical system
at the beginning of the time step $i$. We note that the defender’s memory
$M_{D}^{i}$ and the attacker’s memory $M_{A}^{i}$ are explicitly held separate
from the $S^{i}$ to indicate that they cannot observed by other player.
#### III-3 Observation Spaces
Extending from $S^{i}$ are two directed edges to defender and attacker
observation nodes $O^{i}_{D}$ and $O^{i}_{A}$. The defender and attacker
observation spaces, $\Omega_{D}$ and $\Omega_{A}$, respectively, are
$\Omega_{D}=[V_{1},V_{2},V_{3},P_{1},Q_{1}],\;\;\Omega_{A}=[V_{2},V_{3},p_{3},q_{3}].$
(5)
These observations are not complete (the players do not get full state
information), they may be binned (indicating only the range of a variable, not
the precise value), and they may be noisy. The content of $\Omega_{D}$ and
$\Omega_{A}$ is an assumption about the capabilities of the players. Here,
$\Omega_{D}$ provides a large amount of system visibility consistent with the
defender being the SCADA operator. However, it does not include $p_{3}$ or
$q_{3}$ as the distributed generator has been taken over by the attacker. In
contrast, $\Omega_{A}$ mostly provides information about node 3 and also
includes $V_{2}$ because a sophisticated attacker would be able to estimate
$V_{2}$ from the other information in $\Omega_{A}$. Although we do not
consider this possibility here, we note that the content of the $\Omega_{D}$
and to some extent the content of $\Omega_{A}$ are potential control system
design variables that would affect the outcome of the attacker-defender
interaction. For example, excluding $V_{3}$ from $\Omega_{D}$ will affect the
decisions made by the defender, and therefore, the outcome of the interaction.
#### III-4 Player Memories
The content and evolution of player memories should be constructed based on
application-specific domain knowledge or guided by human-based experiments. In
this initial work, we assume a defender memory $M_{D}^{i}$ and attacker memory
$M_{A}^{i}$ consisting of a few main components
$M^{i}_{D}=[\Omega^{i}_{D},D^{i-1}_{D},\mathcal{M}^{i}_{D}];\;\;M^{i}_{A}=[\Omega^{i}_{A},D^{i-1}_{A},\mathcal{M}^{i}_{A}].$
(6)
The inclusion of the player’s current observations $\Omega^{i}$ and previous
move $D^{i-1}$ are indicated by directed edges in Fig. 2. The directed edge
from $M^{i-1}$ to $M^{i}$ indicates the carrying forward and updating of a
summary metric $\mathcal{M}^{i}$ that potentially provides a player with
crucial additional, yet imperfect, system information that cannot be directly
observed.
Our defender uses $\mathcal{M}_{D}$ to estimate if an attacker is present. One
mathematical construct that provides this is
$\displaystyle\mathcal{M}^{i}_{D}$ $\displaystyle=$
$\displaystyle(1-1/n)\mathcal{M}^{i-1}_{D}$ (7) $\displaystyle+$
$\displaystyle\textrm{sign}(V^{i}_{1}-V^{i-1}_{1})\;\textrm{sign}(V^{i}_{3}-V^{i-1}_{3})$
The form of statistic in Eq. 7 is similar to the exponentially decaying memory
proposed by Lehrer[20]. For attackers with small $q_{3}$ capability, even full
range changes of $q_{3}$ will not greatly affect $V_{3}$, and the sign of
changes in $V_{3}$ will be the same those of $V_{1}$. The second term on the
RHS of Eq. 7 will always be +1, and $\mathcal{M}_{D}\rightarrow 1$. An
attacker with large $q_{3}$ capability can drive changes in $V_{1}$ and
$V_{3}$ of opposite sign. Several sequential time steps of with opposing
voltage changes will cause $\mathcal{M}_{D}\rightarrow-1$. We note that if the
defender does not change $V_{1}$, the contribution to $\mathcal{M}_{D}$ is
zero, and the defender does not gain any information.
The general form of the attacker’s memory statistic is similar to the
defender’s,
$\displaystyle\mathcal{M}^{i}_{A}$ $\displaystyle=$
$\displaystyle(1-1/n)\mathcal{M}^{i-1}_{A}$ (8) $\displaystyle+$
$\displaystyle\textrm{sign}\left(\textrm{floor}\left(\frac{\Delta
V^{i}_{3}-\Delta q^{i}_{3}x_{2}/V_{0}}{\delta v}\right)\right),$
however the contributions to $\mathcal{M}_{A}$ are designed to track the
defender’s changes to $V_{1}$. If the attacker changes $q_{3}$ by $\Delta
q^{i}_{3}=q^{i}_{3}-q^{i-1}_{3}$, the attacker would expect a proportional
change in $V_{3}$ by $\Delta V^{i}_{3}=V^{i}_{3}-V^{i-1}_{3}\sim\Delta
q^{i}_{3}x_{2}/V_{0}$. If $V_{3}$ changes according to this reasoning, then
the second term on the RHS of Eq. 8 is zero. If instead the defender
simultaneously increases $V_{1}$ by $\delta v$, $\Delta V^{i}_{3}$ will
increase by $\delta v$, and the second term on the RHS of Eq. 8 is then +1. A
similar argument yields -1 if the defender decreases $V_{1}$ by $\delta v$.
Equation 8 then approximately tracks the aggregate changes in $V_{1}$ over the
previous $n$ time steps.
#### III-5 Decision or Move space
Here, we only describe the decision options available to the players. How
decisions are made is discussed in the next Section. Typical hardware-imposed
limits of a ULTC constrain the defender actions at time step $i$ to the
following domain
$D^{i}_{D}=\\{\min(v_{max},V^{i}_{1}+\delta
v),V^{i}_{1},\max(v_{min},V^{i}_{1}-\delta v)\\}$ (9)
where $\delta v$ is the voltage step size for the transformer, and $v_{min}$
and $v_{max}$ represent the absolute min and max voltage the transformer can
produce. In simple terms, the defender may leave $V_{1}$ unchanged or move it
up or down by $\delta v$ as long as $V_{1}$ stays within the range
$[v_{min},v_{max}]$. We take $v_{min}=0.90$, $v_{max}=1.10$, and $\delta
v=0.02$. We allow a single tap change per time step (of one minute) which is a
reasonable approximation tap changer lockout following a tap change.
Hardware limitations on the generator at node 3 constrain the attacker’s range
of control of $q_{3}$. In reality, these limits can be complicated, however,
we simplify the constraints by taking the attacker’s $q_{3}$ control domain to
be
$D^{i}_{A}=\\{-p_{3,max},\ldots,0,\ldots,p_{3,max}\\}.$ (10)
In principle, the attacker could continuously adjust $q_{3}$ within this
range. To reduce the complexity of our computations, we discretize the
attacker’s move space to eleven equally-spaced settings with $-p_{3,max}$ and
$+p_{3,max}$ as the end points.
## IV Solution Concepts
Nodes other than $D^{i}_{D}$ and $D^{i}_{A}$ represent control algorithms,
evolution of a physical system, a mechanistic memory model, or other
conditional probability distributions that can be written down without
reference to any of the other nodes in the semi-Bayes net of Fig. 2.
Specifying nodes $D^{i}_{D}$ and $D^{i}_{A}$ requires a model of human
decision making. In an iterated SNFG with $N$ time steps, our defender would
$3^{N}$ possibilities, and maximizing his average reward
($\sum_{i=1}^{N}R_{D}^{i}/N$) quickly becomes computationally challenging for
reasonably large $N$. However, a human would not consider all $3^{N}$ choices.
Therefore, we seek a different solution concept that better represents human
decision making, which is then necessarily tractable.
### IV-A Policies
We consider a policy-based approach for players’ decisions, i.e. a mapping
from a player’s memory to his action ($M^{i}_{D}\rightarrow D_{D}^{i}$). A
single decision regarding what policy to use for the entire iterated SNFG
greatly reduces the complexity making it independent of $N$. A policy does not
dictate the action at each time step. Rather, the action at time step $i$ is
determined by sampling from the policy based on the actual values of
$M^{i}_{D}$ ($M^{i}_{A}$). We note that the reward garnered by a player’s
single policy decision depends on the policy decisions of other player because
the reward functions of both players depend on variables affect by the other
player’s policy. Policies and the methods for finding optimal policies are
discussed in greater detail in [3].
### IV-B Solution Concept: Level-K Reasoning
The coupling between the players’ policies again increases the complexity of
computing the solution. However, the fully rational procedure of a player
assessing his own reward based on all combinations of the two competing
policies is not a good model of human decision making. We remove this coupling
by invoking level-k reasoning as a solution concept. Starting at the lowest
level-k, a level-1 defender policy is determined by finding the policy that
maximizes the level-1 defender average reward when playing against a level-0
attacker. Similarly, the level-1 attacker policy is determined by optimizing
against a level-0 defender policy. The higher k-level policies are determined
by optimization with regard to the the k-1 policies. We note that the level-0
policies cannot be determined in this manner. They are simply assumptions
about the non-strategic policy behavior of the attacker and defender that are
inputs to this iterative process.
From the perspective of a level-k player, the decision node of his level k-1
opponent is now simply a predetermined conditional probability distribution
making it no different than any other node in the iterated SNFG, i.e. simply
part of his environment. The level-k player only needs to compute his best-
response policy against this fixed level k-1 opponent/environment. The
selection of the levek-k policy is now a single-agent reinforcement learning
problem. Level-k reasoning as a solution concept is discussed in more detail
in [3].
### IV-C Reinforcement Learning
Many standard reinforcement learning techniques can be used to solve the
optimization problem discussed above [21, 22, 23]. In our previous work[3],
the attacker and defender optimization problems were both modeled as Markov
Decision Processes (MDP), even though neither player could observe the entire
state of the grid. The additional uncertainty related to attacker existence
casts doubt on this approach. Instead, we employ a reinforcement learning
algorithm based on [24] which has convergence guarantees for Partially
Observable MDPs (POMDP). This approach has two distinct steps. First is the
policy evaluation step, where the Q-values for the current policy are
estimated using Monte Carlo. Second, the policy is updated by placing greater
weight on actions with higher estimated Q-values. The two steps are iterated
until the policy converges to a fixed point indicating a local maximum has
been found. The details of the algorithm can be found in [24].
## V Simulation Results
Due to space limitations and our desire to explore the design aspects of our
models, we only consider results for a level-1 defender matched against a
level-0 attacker. We retain the level-0 attacker policy that we have used in
our previous work[3]. Although he is only level-0, this level-0 attacker is
modeled as being knowledgeable about power systems and is sophisticated in his
attack policy.
### V-A Level-0 Attacker
The level-0 attacker drifts one step at at time to larger $q_{3}$ if $V_{2}<1$
and smaller $q_{3}$ if $V_{2}>1$. The choice of $V_{2}$ to decide the
direction of the drift is somewhat arbitrary, however, this is simply assumed
level-0 attacker behavior. The drift in $q_{3}$ causes a drift in $Q_{1}$ and,
without any compensating move by the defender, a drift in $V_{2}$. A level-1
defender that is unaware of the attacker’s presence would compensate by
adjusting $V_{1}$ in the opposite sense as $V_{2}$ in order to keep the
average of $V_{2}$ and $V_{3}$ close to 1.0. The level-0 attacker continues
this slow drift forcing the unaware level-1 defender to ratchet $V_{1}$ near
to $v_{min}$ or $v_{max}$. At some point, based on his knowledge of the power
flow equations and the physical circuit, the level-0 attacker determines it is
time to “strike”, i.e. a sudden large change of $q_{3}$ in the opposite
direction to the drift would push $V_{2}$ outside the range
$[1-\varepsilon,1+\epsilon]$. If the deviation of $V_{2}$ is large, it will
take the defender a number of time steps to bring $V_{2}$ back in range, and
the attacker accumulates reward during this recovery time. More formally, this
level-0 attacker policy can be expressed as Level0Attacker$()$ 1
$\ignorespaces V^{*}=\max_{q\in D_{A,t}}|V_{2}-1|;$ 2 if $V^{*}>\theta_{A}$ 3
then $\mbox{\bf return\ }\arg\max_{q\in D_{A,t}}|V_{2}-1|;$ 4 if $V_{2}<1$ 5
then $\mbox{\bf return\ }q_{3,t-1}+1;$ 6 $\ignorespaces\mbox{\bf return\
}q_{3,t-1}-1;$ Here, $\theta_{A}$ is a threshold parameter that triggers the
strike. Throughout this work, we have used $\theta_{A}=0.07>\epsilon$ to
indicate when an attacker strike will accumulate reward.
### V-B Level-1 Defender–Level-0 Attacker Dynamics
We demonstrate our entire modeling and simulation process on two cases. In the
first case, a level-1 defender optimizes his policy against a level-0 attacker
that is present 50% of the time, i.e. $p=0.50$ in the node “A exist” in Fig.
2. In the second case, the level-1 defender optimizes his policy against a
“normal” system, i.e. $p=0.0$ in “A exist”. The behavior of these two level-1
defenders is shown in Fig. 3 where we temporarily depart from the description
of our model. In the first half of these simulations, the level-0 attacker
does not exist, i.e. $p=0.0$, and there are no significant differences between
the two level-1 defenders. At time step 50, a level-0 attacker is introduced
with $p=1.0$. The level-1 defender optimized for $p=0.0$ suffers from the
“drift-and-strike” attacks as described above. In contrast, the level-1
defender with a policy optimized at $p=0.50$ has learned not to follow these
slow drifts and maintains a more or less steady $V_{1}$ even after time step
50. Although $V_{3}$ is out of acceptable bounds for some periods, these are
much shorter than before and $V_{2}$ is never out of bounds.
Figure 3: Typical time evolution of $V_{1}$ (blue), $V_{2}$ (red), and $V_{3}$
(green) for a level-1 defender facing a level-0 attacker. In the left plot,
the level-1 defender’s policy was optimized for $p=0.0$ in “A exist”, i.e. no
level-0 attacker was ever present. In the right plot, the level-1 defender’s
policy was optimized with $p=0.50$. At the start of the simulation, no
attacker is present. The attacker enters the simulation at time step 50. In
these simulations, $p_{2,max}=1.4$, $p_{2,min}=1.35$ and $p_{3,max}=1.0$.
### V-C Policy Dependence on $p$ During Defender Training
Next, we present a few preliminary studies that prepare our model for studying
circuit design tradeoffs. Although policy optimization (i.e. training) and
policy evaluation seem closely related, we carry these out as two distinct
processes. During training, all of the parameters of the system are fixed,
especially the probability of the attacker presence $p$ and the circuit
parameters $p_{2,max}$ and $p_{3,max}$. Many training runs are carried out and
the policy is evolved until the reward per time step generated by the policy
converges to a fixed point. The converged policy can then be evaluated against
the conditions for which is was trained, and in addition, it can be evaluated
for different but related conditions. For example, we can train with one
probability of attacker existence $p$, but evaluate the policy against a
different value of $p^{\prime}$. Next, we carry out just such a study to
determine if a single value of $p$ can used in all of our subsequent level-1
defender training. If using a single training $p$ can be justified, it will
greatly reduce the parameter space to explore during subsequent design
studies.
We consider seven values of $p$ logarithmically spaced from 0.01 to 1.0. A set
of seven level-1 defenders, one for each $p$, is created by optimizing their
individual policies against a level-0 attacker who is present with probability
$p$. Each of these defenders is then simulated seven times, i.e. against the
same level-0 attacker using the same range of $p$ as in the training. In these
simulations, $p_{2,max}=1.4$ and $p_{3,max}=1.0$. During the simulation stage,
the average defender reward per time step is computed and normalized by the
value of $p$ during the simulation stage, i.e. $\sum_{i=1}^{N}R_{D}^{i}/Np$,
creating a measure of level-1 defender performance per time step that the
level-0 attacker is actually present. The results are shown in Fig. 4.
For an achievable number of Monte Carlo samples and for low values of $p$
during policy optimization (i.e. training), there will be many system states
$S$ that are visited infrequently or not at all, particularly those states
where the attacker is present. The reinforcement learning algorithm will
provide poor estimates of the Q values for these states, and the results of
the policy optimization should not be trusted. For these infrequently visited
states, we replace the state-action policy mapping with the mapping given by
the level-0 defender policy used in our previous work[3]. Even with this
replacement, the level-1 defenders trained with $p<0.10$ perform quite poorly.
For $p\geq 0.20$, it appears that enough states are visited frequently enough
such that level-1 defender performance improves. For the remainder of the
studies in this manuscript, we use $p=0.20$ for all of our level-1 defender
training.
Figure 4: Level-1 defender reward per time step of level-0 attacker presence
during simulation ($\sum_{i=1}^{N}R_{D}^{i}/Np$ ) versus the probability of
attacker presence during training. The curves representing different levels of
attacker presence during simulation all show the same general dependence, i.e.
a relatively flat plateau in normalized level-1 defender reward for $p\geq
0.20$ due to the more complete sampling of the states $S$ during training
(i.e. policy optimization). This common feature leads us to select $p=0.20$
for training for all the subsequent work in the manuscript.
## VI Design Procedure and Social Welfare
Significant deviations of $V_{2}$ or $V_{3}$ from 1.0 p.u. can cause economic
loss either from equipment damage or lost productivity due to disturbances to
computers or computer-based industrial controllers[19]. The likelihood of such
voltage deviations is increased because possibility of attacks on the
distributed generator at node 3. However, this generator also provides a
social benefit through the value of the energy it contributes to the grid. The
larger the generator (larger $p_{3,max}$) the more energy it contributes and
the higher this contribution to the social welfare. However, when compromised,
a larger generator increases the likelihood of large voltage deviations and
significant economic loss.
To balance the value of the energy against the lost productivity, we assess
both in terms of dollars. The social welfare of the energy is relatively easy
to estimate because the value of electrical energy, although variable in both
time and grid location, can be assigned a relatively accurate average value.
Here the value of electrical energy is approximated by a flat-rate consumer
price. In heavily regulated markets, the price of electricity can be
distorted, and this approach may be a bad approximation of the true value of
the energy. So while price is a reasonable approximation of value for the
purposes of this model, in practice it may be necessary to adjust for market
distortions on a case-by-case basis. In our work, the generator at node 3 is
installed in a distribution system where we estimate the energy value at
$C_{E}=$$80/MW-hr.
As with estimating the value of energy, estimating the social cost of poor
power quality is also a prerequisite to the power grid design procedure. In
contrast to the value of energy, there is no obvious proxy for this cost
making it difficult to estimate. Studies[19] have concluded that the cost is
typically dominated by a few highly sensitive customers, making this cost also
dependent on grid location and time – the location of the highly sensitive
customers and their periods operation drive this variability. The average cost
of a power quality event has been estimated[19] at roughly
$C_{PQ}=$$300/sensitive customer/per power quality event. Note that social
welfare, including estimates of the value of energy and the social cost of
poor power quality, determines the optimality of the power grid design and
should be carefully chosen for each application.
We now describe a series of numerical simulations and analyses that enable us
to find the social welfare break even conditions for the generator at node 3.
### VI-A Level-1 Defender Performance Versus ($p_{2,max}$,$p_{3,max}$)
Because the output of the node “A exist” in the iterated SNFG in Fig. 2 fixes
the probability of the presence of an attacker for the rest of the $N$ steps
in the simulation, the results from each simulation of the iterated SNFG are
statistically independent. Therefore, if we know the level-1 defender’s
average reward when he is under attack 100% of the time ($p=1$) and 0% of the
time ($p=0$), we can compute his average reward for any intermediate value of
$p$. Taking this into account, we proceed as follows. Using the guidance from
the results in Fig. 4, we train level-1 defenders against level-0 attackers
(using $p=0.2$) for an array of ($p_{2,max}$,$p_{3,max}$) conditions. Next, we
simulate these level-1 defenders with $p=1$ and $p=0$ so that we can compute
their average reward for any $p$. The results for all
($p_{2,max}$,$p_{3,max}$) conditions for $p=0.01$ are shown in Fig. 5. The
results show two important thresholds, i.e. the level-1 defenders’ average
reward falls off quickly when $p_{3,max}>1.5$ or when $p_{2,max}>1.9$. In the
rest of this analysis, we will focus on the region $p_{3,max}<1.5$ before the
large decrease in the defender’s average reward
### VI-B Level-1 Defender $p_{3,max}$ Sensitivity
Using the energy and power quality cost estimates from above, the results in
Fig. 5 could be turned into surface plots of social welfare. However, the
number of design parameters that could be varied would generate a multi-
dimensional set of such surface plots making the results difficult to
interpret. Instead, we seek to reduce this dimensionality and generate results
that provide more design intuition. We first note that the level-1 defenders’
reward falls approximately linearly with $p_{3,max}$ for $p_{3,max}<1.5$. The
slope of these curves is the sensitivity of the level-1 defenders’ average
reward to $p_{3,max}$, and we extract and plot these sensitivities versus
$p_{2,max}$ in Fig. 6.
To further analyze the results in Fig. 6, we must relate the defender’s
average reward to power quality events, which can then be converted into a
social welfare cost using $C_{PQ}$. Equation 4 expresses the defender’s reward
$R_{D}$ as a sum of two smooth functions (one function of $V_{2}$ and another
of $V_{3}$). These individual contributions are equal to $1$ when $V_{2}$ or
$V_{3}$ are equal to either $1+\epsilon$ or $1-\epsilon$. Although these
deviations are not severe, we consider such deviations to constitute a power
quality event, and we estimate its social welfare cost by as $R_{D}C_{PQ}$.
$R_{D}$ increases (decreases) quadratically for larger (smaller) voltage
deviations, and our definition of the social welfare cost captures that these
larger (smaller) deviations result in higher (lower) social welfare costs.
Using $C_{PQ}=$$300/sensitive customer/per power quality event estimated in
[19], our simulation time step of one minute, and assuming there is one
sensitive customer on our circuit, the slopes of $\sim$0.006/(MW of
$p_{3,max}$) in Fig. 6 corresponds to a social welfare cost of $108/(MW of
$p_{3,max})$/hr. At this value of $C_{PG}$, the social value provided by the
energy at $80/MW-hr is outweighed by the social welfare cost caused by the
reduction in power quality.
Figure 5: The level-1 defender’s average reward per simulation time step as a
function of $p_{3,max}$ for a 1% probability of an attack on node 3. Each
curve represents a different value of $p_{2,max}$ in the range $[0.2...2.5]$.
Slightly modifying the analysis just described, we can now find the
energy/power-quality break even points for the social welfare of the generator
at node 3, i.e. the cost of a power quality event that reduces the social
welfare provided by the energy to a net of zero. The break-even power quality
cost is plotted versus $p_{2,max}$ in Fig. 7. Points of $C_{PQ}$ and
$p_{2,max}$ that fall to the lower left of the curve contribute positive
social welfare while those to the upper right contribute negative social
welfare. When applied to more realistic power system models, analysis such as
shown in Fig. 7 can be used to make decisions about whether new distributed
generation should be placed on a particular part of a distribution grid.
Figure 6: The slope of the data in Fig. 5 for $p_{3,max}\leq 1.5$. The slope
measures how quickly the level-1 defender’s reward decreases with $p_{3,max}$
for different values of $p_{2,max}$. Consistent with Fig. 5, the slope is
roughly constant for $p_{2,max}<1.9$ and then rapidly becomes more negative as
$p_{2,max}$ increases beyond 1.9. The rapid decrease demonstrates the level-1
defender is much more susceptable to the level-0 attacker when
$p_{2,max}>1.9$. Figure 7: The cost of a power quality event that yields a
zero social welfare contribution from the distributed generator at node 3
(i.e. the generator’s break even point) versus $p_{2,max}$. To compute the
break even cost of power quality events, we have assumed: the generator is
under attack by a level-0 attacker 1% of the time and the value of the energy
from the generator at node 3 is $80/MW-hr.
## VII Conclusion
We have described a novel time-extended, game theoretic model of humans
interacting with one another via a cyber-physical system, i.e. an interaction
between a cyber intruder and an operator of an electrical grid SCADA system.
The model is used to estimate the outcome of this adversarial interaction, and
subsequent analysis is used to estimate the social welfare of these outcomes.
The modeled interaction has several interesting features. First, the
interaction is asymmetric because the SCADA operator is never completely
certain of the presence of the attacker, but instead uses a simple statistical
representation of memory to attempt to infer the attacker’s existence. Second,
the interaction is mediated by a significant amount of automation, and using
the results of our model or related models, this automation can be
(re)designed to improve the social welfare of these outcomes.
The models in this manuscript can be extended and improved in many ways.
Perhaps the most important of these would be extending the model to
incorporate larger, more realistic grids, such as transmission grids, where
the meshed nature of the physical system would result in more complex impacts
from an attack. In contrast to the setting described here, such complex grids
would have multiple points where a cyber intruder could launch an attack, and
models of the defender, his reward function, and his memory would be equally
more complex.
As discussed earlier, one challenge with our approach is computational. The
size of the physical system itself does not overly increase the computational
requirements (beyond what is normally seen in solving power flow equations in
large-scale systems). However, the number of players and observations does
increase the computational requirements exponentially. This is a major focus
of our current work. In particular, we note that the number of observations
(monitors) that a real human can pay attention to is very limited. One
approach we are investigating for how to overcome the exponential explosion is
to incorporate this aspect of real human limitations into our model The
challenge here will be developing models of how a human chooses which
observations to make to guide their decisions.
## References
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* [21] L. Busoniu, R. Babuska, B. De Schutter, and E. Damien, _Reinforcement Learning and Dynamic Programming Using Function Approximators_ , editor, Ed. CRC Press, 2010.
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|
arxiv-papers
| 2013-04-15T06:27:17 |
2024-09-04T02:49:44.316291
|
{
"license": "Public Domain",
"authors": "Scott Backhaus, Russell Bent, James Bono, Ritchie Lee, Brendan Tracey,\n David Wolpert, Dongping Xie, and Yildiray Yildiz",
"submitter": "Ritchie Lee",
"url": "https://arxiv.org/abs/1304.3996"
}
|
1304.4073
|
# Simultaneous approximation for scheduling problems
Long Wan [email protected]. Department of Mathematics, Zhejiang University,
Hangzhou, 310027, China.
###### Abstract
Motivated by the problem to approximate all feasible schedules by one schedule
in a given scheduling environment, we introduce in this paper the concepts of
strong simultaneous approximation ratio (SAR) and weak simultaneous
approximation ratio (WAR). Then we study the two parameters under various
scheduling environments, such as, non-preemptive, preemptive or fractional
scheduling on identical, related or unrelated machines.
Keywords. scheduling; simultaneous approximation ratio; global fairness
## 1 Introduction
In the scheduling research, people always hope to find a schedule which
achieves the balance of the loads of the machines well. To the end, some
objective functions, such as minimizing makespan and maximizing machine cover,
are designed to find a reasonable schedule. Representative publications can be
found in Graham (1966), Graham (1969), Deuermeyer et al. (1982), and Csirik et
al. (1992) among many others. But these objectives don’t reveal the global
fairness for the loads of all machines. Motivated by the problem to
approximate all feasible schedules by one schedule in a given scheduling
environment and so realizing the global fairness, we present two new
parameters: strong simultaneous approximation ratio (SAR) and weak
simultaneous approximation ratio (WAR).
Our research is also enlightened from the research on global approximation of
vector sets. Related work can be found in Bhargava et al. (2001), Goel et al.
(2001), Goel et al. (2005), Kleinberg et al. (2001) and Kumar and Kleinberg
(2006). Kleinberg et al. (2001) proposed the notion of the coordinate-wise
approximation for the fair vectors of allocations. Based on this notion, Kumar
and Kleinberg (2006) introduced the definitions of the global approximation
ratio and the global approximation ratio under prefix sums.
For a given instance $\mathcal{I}$ of a minimization problem, we use
$V(\mathcal{I})$ to denote the set of vectors induced by all feasible
solutions of $\mathcal{I}$. For a vector $X=(X_{1},X_{2},\cdots,X_{m})\in
V(\mathcal{I})$, we use $\overleftarrow{X}$ to denote the vector in which the
coordinates (components) of $X$ are sorted in non-increasing order, that is,
$\overleftarrow{X}=(X^{\prime}_{1},X^{\prime}_{2},\cdots,X^{\prime}_{m})$ is a
resorting of $(X_{1},X_{2},\cdots,X_{m})$ so that $X^{\prime}_{1}\geq
X^{\prime}_{2}\geq\cdots\geq X^{\prime}_{m}$. For two vectors $X,Y\in
V(\mathcal{I})$, we write $X\preceq_{c}Y$ if $X_{i}\preceq Y_{i}$ for all $i$.
The global approximation ratio of a vector $X\in V(\mathcal{I})$, denoted by
$c(X)$, is defined to be the infimum of $\alpha$ such that
$\overleftarrow{X}\preceq_{c}\alpha\overleftarrow{Y}$ for all $Y\in
V(\mathcal{I})$. Then the best global approximation ratio of instance
$\mathcal{I}$ is defined to be $c^{*}(\mathcal{I})=\inf_{X\in
V(\mathcal{I})}c(X)$. For a vector $X\in V(\mathcal{I})$, we use $\sigma(X)$
to denote the vector in which the $i$-th coordinate is equal to the sum of the
first $i$ coordinates of $X$. We write $X\preceq_{s}Y$ if
$\sigma(\overleftarrow{X})\preceq_{c}\sigma(\overleftarrow{Y})$. The global
approximation ratio under prefix sums of a vector $X\in V(\mathcal{I})$,
denoted by $s(X)$, is defined to be the infimum of $\alpha$ such that
$X\preceq_{s}\alpha Y$ for all $Y\in V(\mathcal{I})$. Then the best global
approximation ratio under prefix sums of instance $\mathcal{I}$ is defined to
be $s^{*}(\mathcal{I})=\inf_{X\in V(\mathcal{I})}s(X)$.
In the terms of scheduling, the above concepts about the global approximation
of vector sets can be naturally formulated as the simultaneous approximation
of scheduling problems. Let $\mathcal{I}$ be an instance of a scheduling
problem ${\cal P}$ on $m$ machines $M_{1},M_{2},\cdots,M_{m}$, and let ${\cal
S}$ be the set of all feasible schedules of $\mathcal{I}$. For a feasible
schedule $S\in{\cal S}$, the _load_ $L^{S}_{i}$ of machine $M_{i}$ is defined
to be the time by which the machine finishes all the process of the jobs and
the parts of the jobs assigned to it. The
$L(S)=(L^{S}_{1},L^{S}_{2},\cdots,L^{S}_{m})$ is called the _load vector_ of
machines under $S$. Then $V(\mathcal{I})$ is defined to be the set of all load
vectors of instance $\mathcal{I}$. We write $c(S)=c(L(S))$ and $s(S)=s(L(S))$
for each $S\in{\cal S}$. Then $c^{*}(\mathcal{I})=\inf_{S\in{\cal S}}c(S)$ and
$s^{*}(\mathcal{I})=\inf_{S\in{\cal S}}s(S)$. The _strong simultaneous
approximation ratio_ of problem ${\cal P}$ is defined to be $SAR({\cal
P})=\sup_{\mathcal{I}}c^{*}(\mathcal{I})$, and the _weak simultaneous
approximation ratio_ of problem ${\cal P}$ is defined to be $WAR({\cal
P})=\sup_{\mathcal{I}}s^{*}(\mathcal{I})$.
A scheduling problem is usually characterized by the machine type and the job
processing mode. In this paper, the machine types under consideration are
identical machines, related machines and unrelated machines, and the job
processing modes under consideration are non-preemptive, preemptive and
fractional. Let $\mathcal{J}=\\{J_{1},J_{2},\cdots,J_{n}\\}$ and
$\mathcal{M}=\\{M_{1},M_{2},\cdots,M_{m}\\}$ be the set of jobs and the set of
machines, respectively. The processing time of $J_{j}$ on $M_{i}$ is $p_{ij}$.
If $p_{ij}=p_{kj}$ for $i\neq k$, the machine type is _identical machines_. In
this case $p_{j}$ is used to denote the processing time of $J_{j}$. If
$p_{ij}=\frac{p_{j}}{s_{i}}$ for all $i$, the machine type is _related
machines_. In this case, $p_{j}$ is called the standard processing time of
$J_{j}$ and $s_{i}$ is called the processing speed of $M_{i}$. If there is no
restriction for $p_{ij}$, the machine type is _unrelated machines_. If each
job must be non-preemptively processed on some machine, the processing mode is
_non-preemptive_. If each job can be processed preemptively and can be
processed on at most one machine at any time, the processing mode is
_preemptive_. If each job can be partitioned into different parts which can be
processed on different machines concurrently, the processing mode is
_fractional_. Each machine can process at most one job at any time under any
processing mode.
Since we cannot avoid the worst schedule in which all jobs are processed on a
common machine, it can be easily verified that, under each processing mode,
$SAR({\cal P})=m$ for identical machines, $SAR({\cal
P})=(s_{1}+s_{2}+\cdots+s_{m})/s_{1}$ for related machines with speeds
$s_{1}\geq s_{2}\geq\cdots\geq s_{m}$, and $SAR({\cal P})=+\infty$ for
unrelated machines.
We then concentrate our research on the weak simultaneous approximation ratio
$WAR({\cal P})$ of the scheduling problems defined above. The main results are
demonstrated in table 1.
| identical machines | related machines | unrelated machines
---|---|---|---
non-preemptive processing | $1<{WAR}\leq\frac{3}{2}$ | $\frac{\sqrt{m}+1}{2}\leq{WAR}\leq\sqrt{m}$ | $\frac{\sqrt{m}+1}{2}\leq{WAR}\leq\sqrt{m}$
preemptive processing | $1$ | $\frac{\sqrt{m}+1}{2}\leq{WAR}\leq\sqrt{m}$ | $\frac{\sqrt{m}+1}{2}\leq{WAR}\leq\sqrt{m}$
fractional processing | $1$ | $\frac{\sqrt{m}+1}{2}$ | $\frac{\sqrt{m}+1}{2}\leq{WAR}\leq\sqrt{m}$
Table 1: The weak simultaneous approximation ratio of various scheduling
problems
For convenience, we use $P$, $Q$ and $R$ to represent identical machines,
related machines and unrelated machines, respectively, and use $NP$, $PP$ and
$FP$ to represent non-preemptive, preemptive and fractional processing,
respectively. Then the notation $Pm(NP)$ represents the scheduling problem on
$m$ identical machines under non-preemptive processing mode. Other notations
for scheduling problems can be similarly understood.
This paper is organizes as follows. In Section 2, we study the weak
simultaneous approximation ratio for scheduling on identical machines. In
Section 3, we study the weak simultaneous approximation ratio for scheduling
on related machines. In Section 4, we study the weak simultaneous
approximation ratio for scheduling on unrelated machines.
## 2 Identical machines
For problem $P2(NP)$, we have $s(S)=1$ for every schedule $S$ which minimizes
the makespan. So $WAR(P2(NP))=1$. For problem $Pm(NP)$ with $m\geq 3$, the
following instance shows that $WAR(Pm(NP))>1$. In the instance, there are $m$
jobs with processing time $m-1$, $(m-1)(m-2)$ jobs with processing time $m$
and a big job with processing time $(m-1)^{2}+r_{m}$, where
$r_{m}=\frac{\sqrt{(m^{3}-m^{2}-m-2)^{2}+4m(m-1)(m-2)}-(m^{3}-m^{2}-m-2)}{2}$.
It can be verified that $0<r_{m}<m-2$. Let $S$ be the schedule in which the
$m$ jobs with processing time $m-1$ are scheduled on one machine, the big job
with with processing time $(m-1)^{2}+r_{m}$ is scheduled on one machine, and
the remaining $(m-1)(m-2)$ jobs with processing time $m$ are scheduled on the
remaining $m-2$ machines averagely. Let $T$ be the schedule in which the big
job is scheduled on one machine together with a job of processing time $m-1$,
and each of the remaining machines has a job of processing time $m-1$ and
$m-2$ jobs of processing time $m$. Then the makespan of schedule $S$ is
$m(m-1)$ and the $(m-1)$-th prefix sum of $\overleftarrow{L(T)}$ is
$m(m-1)^{2}-(m-2-r_{m})$. Now consider an arbitrary schedule $\varrho$. If the
big job is scheduled on one machine solely, then the $(m-1)$-th prefix sum of
$\overleftarrow{L(R)}$ is at least $m(m-1)^{2}$. Thus, by considering the
$(m-1)$-th prefix sums of $\overleftarrow{L(T)}$ and $\overleftarrow{L(R)}$,
we have
$s(R)\geq\frac{m(m-1)^{2}}{m(m-1)^{2}-(m-2-r_{m})}=1+\frac{r_{m}}{m(m-1)}$. If
the big job is scheduled on one machine together with at least one other job,
then the makespan of schedule $R$ is at least $(m-1)+(m-1)^{2}+r_{m}$. Thus,
by considering the makespans of $S$ and $R$, we have $s(R)\geq
1+\frac{r_{m}}{m(m-1)}$. It follows that $WAR(Pm(NP))\geq
1+\frac{r_{m}}{m(m-1)}>1$ for $m\geq 3$.
To establish the upper of $WAR(Pm(NP))$, we first present a simple but useful
lemma.
###### Lemma 1
Let $X,Y$ be two vectors of $n$-dimension and let $X^{\prime},Y^{\prime}$ be
two vectors of two-dimension. If $X\preceq_{s}Y$ and
$X^{\prime}\preceq_{s}Y^{\prime}$, then
$(X,X^{\prime})\preceq_{s}(Y,Y^{\prime})$.
* Proof.
Suppose that $X^{\prime}=(x_{1},x_{2})$ and $Y^{\prime}=(y_{1},y_{2})$.
Without loss of generality, we may further assume that $x_{1}\geq x_{2}$ and
$y_{1}\geq y_{2}$. Then $x_{1}\leq y_{1}$ and $x_{1}+x_{2}\leq y_{1}+y_{2}$.
Let $Z_{x}=(X,X^{\prime})$ and $Z_{y}=(Y,Y^{\prime})$. For
$Z\in\\{Z_{x},Z_{y}\\}$, we use $(\overleftarrow{Z})_{k}$ to denote the $k$-th
coordinate of $\overleftarrow{Z}$, and use $|\overleftarrow{Z}|_{k}$ to denote
the sum of the first $k$ coordinates of $\overleftarrow{Z}$ for $1\leq k\leq
n+2$. Similar notations are also used for $X$ and $Y$. Given an index $k$ with
$1\leq k\leq n+2$, we use $\delta(k,X^{\prime})$ to denote the number of
elements in $\\{x_{1},x_{2}\\}$ included in the first $k$ coordinates of
$\overleftarrow{Z_{x}}$, and $\delta(k,Y^{\prime})$ the number of elements in
$\\{y_{1},y_{2}\\}$ included in the first $k$ coordinates of
$\overleftarrow{Z_{y}}$. Then
$0\leq\delta(k,X^{\prime}),\delta(k,Y^{\prime})\leq 2$.
If $\delta(k,X^{\prime})=\delta(k,Y^{\prime})$, then we clearly have
$|\overleftarrow{Z_{x}}|_{k}\leq|\overleftarrow{Z_{y}}|_{k}$.
If $\delta(k,X^{\prime})=0$, then
$|\overleftarrow{Z_{x}}|_{k}=|\overleftarrow{X}|_{k}\leq|\overleftarrow{Y}|_{k}\leq|\overleftarrow{Z_{y}}|_{k}$.
If $\delta(k,Y^{\prime})=0$ and $\delta(k,X^{\prime})\geq 1$, we suppose that
$x_{1}$ is the $i$-th coordinate of $\overleftarrow{Z_{x}}$. Then, for each
$j$ with $i\leq j\leq k$, $(\overleftarrow{Z_{x}})_{j}\leq x_{1}\leq
y_{1}\leq(\overleftarrow{Z_{y}})_{j}$. Consequently,
$|\overleftarrow{Z_{x}}|_{k}=|\overleftarrow{X}|_{i-1}+\sum_{i\leq j\leq
k}(\overleftarrow{Z_{x}})_{j}\leq|\overleftarrow{Y}|_{i-1}+\sum_{i\leq j\leq
k}(\overleftarrow{Z_{y}})_{j}=|\overleftarrow{Z_{y}}|_{k}$.
If $\delta(k,X^{\prime})=2$ and $\delta(k,Y^{\prime})=1$, then
$(\overleftarrow{Y})_{k-1}\geq y_{2}$. Thus,
$|\overleftarrow{Z_{x}}|_{k}=|\overleftarrow{X}|_{k-2}+x_{1}+x_{2}\leq|\overleftarrow{Y}|_{k-2}+y_{1}+y_{2}\leq|\overleftarrow{Y}|_{k-1}+y_{1}=|\overleftarrow{Z_{y}}|_{k}$.
If $\delta(k,X^{\prime})=1$ and $\delta(k,Y^{\prime})=2$, then
$(\overleftarrow{Y})_{k-1}\leq y_{2}$. Thus,
$|\overleftarrow{Z_{x}}|_{k}=|\overleftarrow{X}|_{k-1}+x_{1}\leq|\overleftarrow{Y}|_{k-1}+y_{1}\leq|\overleftarrow{Y}|_{k-2}+y_{1}+y_{2}=|\overleftarrow{Z_{y}}|_{k}$.
The above discussion covers all possibilities. Then the lemma follows. $\Box$
###### Theorem 2
$WAR(Pm(NP))\leq\frac{3}{2}$ for $m\geq 4$ and
$WAR(P3(NP))\leq\sqrt{5}-1\approx 1.236$.
* Proof.
Consider an instance of $n$ jobs on $m\geq 4$ identical machines with ${\cal
J}=\\{J_{1},J_{2},\cdots,J_{n}\\}$ and ${\cal
M}=\\{M_{1},M_{2},\cdots,M_{m}\\}$. We assume that $p_{1}\geq
p_{2}\geq\cdots\geq p_{n}$. Let $S$ be a schedule produced by LPT algorithm
(which is the LS algorithm with the jobs being given in the LPT order) such
that $L^{S}_{1}\geq L^{S}_{2}\geq\cdots\geq L^{S}_{m}$. Then
$L(S)=\overleftarrow{L(S)}=(L^{S}_{1},L^{S}_{2},\cdots,L^{S}_{m})$. If $n\leq
m$, it is easy to verify that $s(S)=1$. Hence we assume in the following that
$n\geq m+1$. Then some machine has at least two jobs in $S$.
Let $i_{0}$ be the smallest index such that either $M_{i_{0}+1}$ has at least
three jobs in $S$, or $M_{i_{0}+1}$ has exactly two jobs in $S$ and the size
of the shorter job on $M_{i_{0}+1}$ is at most half of the size of the longer
job on $M_{i_{0}+1}$. If there is no such index, we set $i_{0}=m$. Then
$i_{0}\geq 0$, and in the case $i_{0}\geq 1$, each of
$M_{1},M_{2},\cdots,M_{i_{0}}$ has at most two jobs in $S$. Let $J_{k}$ be the
shortest job scheduled on $M_{1},M_{2},\cdots,M_{i_{0}}$ and set ${\cal
J}_{k}=\\{J_{1},J_{2},\cdots,J_{k}\\}$. Then ${\cal J}_{k}$ contains the jobs
scheduled on $M_{1},M_{2},\cdots,M_{i_{0}}$. We use $M_{k^{\prime}}$ to denote
the machine occupied by $J_{k}$ in $S$. Let $T$ be the schedule derived from
$S$ by deleting $J_{k+1},J_{k+2},\cdots,J_{n}$. Then $T$ is an LPT-schedule
for ${\cal J}_{k}$ with $L^{T}_{i}=L^{S}_{i},i=1,2,\cdots,i_{0}$. We claim
that $s(T)=1$. In the case $i_{0}=0$, the claim holds trivially. Hence, we
assume in the following that $i_{0}\geq 1$.
If each of $M_{1},M_{2},\cdots,M_{i_{0}}$ has only one job in $S$, then
$i_{0}=k\leq m$ and it is easy to see that $s(T)=1$.
Suppose in the following that at least one of $M_{1},M_{2},\cdots,M_{i_{0}}$
has exactly two jobs in $S$. Then $m+1\leq k\leq 2m$ and the machine
$M_{k^{\prime}}$ has exactly two jobs, say $J_{t}$ and $J_{k}$, in $S$. Note
that there are at most two jobs on each machine in $T$. (Otherwise, some
machine $M_{i}$ with $i\geq i_{0}+1$ has $r\geq 3$ jobs, say
$J_{h_{1}},J_{h_{2}},\cdots,J_{h_{r}}$, in $T$. By LPT algorithm,
$p_{t}\geq\sum^{r-1}_{j=1}p_{h_{j}}\geq 2p_{k}$, contradicting the choice of
$i_{0}$.) From the LPT algorithm, we have $t={2m+1-k}$. By the choice of
$i_{0}$, we have $p_{k}>\frac{1}{2}p_{2m+1-k}$.
Let $R$ be an arbitrary schedule for ${\cal J}_{k}$. If each machine has at
most two jobs in $R$, we set $R_{1}=R$. If some machine $M_{x}$ has at least
three jobs in $R$, by the pigeonhole principle, a certain machine $M_{y}$ has
either no job or exactly one job in
$\\{J_{2m+1-k},J_{2m+2-k},\cdots,J_{k}\\}$. Let $R^{\prime}$ be the schedule
obtained from $R$ by moving the shortest job, say $J_{x^{\prime}}$, on $M_{x}$
to $M_{y}$. Then $L^{R^{\prime}}_{x}\geq 2p_{k}>p_{2m+1-k}\geq L^{R}_{y}$ and
$L^{R^{\prime}}_{y}=L^{R}_{y}+p_{x^{\prime}}\geq L^{R}_{y}$. Note that
$L^{R}_{x}\geq L^{R^{\prime}}_{x},L^{R^{\prime}}_{y}\geq L^{R}_{y}$ and
$L^{R}_{x}+L^{R}_{y}=L^{R^{\prime}}_{x}+L^{R^{\prime}}_{y}$. Then we have
$L(R^{\prime})\preceq_{s}L(R)$ by lemma 1. This procedure is repeated until we
obtain a schedule $R_{1}$ so that each machine has at most two jobs in
$R_{1}$. Then we have $L(R_{1})\preceq_{s}L(R)$.
If $J_{1},J_{2},\cdots,J_{m}$ are processed on distinct machines,
respectively, in $R_{1}$, we set $R_{2}=R_{1}$. If some machine $M_{x}$ has
two jobs
$J_{x^{\prime}},J_{x^{\prime\prime}}\in\\{J_{1},J_{2},\cdots,J_{m}\\}$ in
$R_{1}$, by the pigeonhole principle, a certain machine $M_{y}$ is occupied by
at most two jobs in $\\{J_{m},J_{m+1},\cdots,J_{k}\\}$. Suppose that
$p_{x^{\prime}}\geq p_{x^{\prime\prime}}$ and $J_{y^{\prime}}$ is the shorter
job on $M_{y}$. Let $R^{\prime}_{1}$ be the schedule obtained from $R_{1}$ by
shifting $J_{x^{\prime\prime}}$ to $M_{y}$ and shifting $J_{y^{\prime}}$ to
$M_{x}$. Then $L^{R_{1}}_{x}\geq
L^{R^{\prime}_{1}}_{x},L^{R^{\prime}_{1}}_{y}\geq L^{R_{1}}_{y}$ and
$L^{R_{1}}_{x}+L^{R_{1}}_{y}=L^{R^{\prime}_{1}}_{x}+L^{R^{\prime}_{1}}_{y}$.
Consequently, by lemma 1, $L(R^{\prime}_{1})\preceq_{s}L(R_{1})$. This
procedure is repeated until we obtain a schedule $R_{2}$ so that
$J_{1},J_{2},\cdots,J_{m}$ are processed on distinct machines, respectively,
in $R_{2}$. Then we have $L(R_{2})\preceq_{s}L(R_{1})$.
Without loss of generality, we assume that $J_{j}$ is processed on $M_{j}$ in
$R_{2}$, $1\leq j\leq m$. Let $t=k-m$. Then the $t$ jobs
$J_{m+1},J_{m+2},\cdots,J_{k}$ are processed on $t$ distinct machines in
$R_{2}$. For convenience, we add another $m-t$ dummy jobs with sizes 0 in
$R_{2}$ so that each machine has exactly two jobs. We define a sequence of $t$
schedules $R_{2}^{(1)},R_{2}^{(2)},\cdots,R_{2}^{(t)}$ for ${\cal J}_{k}$ by
the following way.
Initially we set $R_{2}^{(0)}=R_{2}$. For each $i$ from 1 to $t$, the schedule
$R_{2}^{(i)}$ is obtained from $R_{2}^{(i-1)}$ by exchanging the shorter job
on $M_{m-i+1}$ with job $J_{m+i}$.
We only need to show that $L(R_{2}^{(i)})\preceq_{s}L(R_{2}^{(i-1)})$ for each
$i$ with $1\leq i\leq t$. Note that the jobs
$J_{m+1},J_{m+2},\cdots,J_{m+i-1}$ are processed on machines
$M_{m},M_{m-1},\cdots,M_{m-i+2}$, respectively, in $R_{2}^{(i-1)}$. If
$J_{m+i}$ is processed on $M_{m-i+1}$ in $R_{2}^{(i-1)}$, we have
$R_{2}^{(i)}=R_{2}^{(i-1)}$ and so
$L(R_{2}^{(i)})\preceq_{s}L(R_{2}^{(i-1)})$. Thus we may assume that $J_{m+i}$
is processed on a machine $M_{x}$ with $x\leq{m-i}$ in $R_{2}^{(i-1)}$. Let
$J_{j}$ be the shorter job on $M_{m-i+1}$ in $R_{2}^{(i-1)}$. Then $p_{j}\leq
p_{m+i}$ and $p_{x}\geq p_{m-i+1}$. It is easy to see that
$(L^{R_{2}^{(i)}}_{x},L^{R_{2}^{(i)}}_{m-i+1})=(p_{x}+p_{j},p_{m-i+1}+p_{m+i})\preceq_{s}(p_{x}+p_{m+i},p_{m-i+1}+p_{j})=(L^{R_{2}^{(i-1)}}_{x},L^{R_{2}^{(i-1)}}_{m-i+1})$.
Consequently, by lemma 1, $L(R_{2}^{(i)})\preceq_{s}L(R_{2}^{(i-1)})$.
The above discussion means that
$L(R_{2}^{(t)})\preceq_{s}L(R_{2})\preceq_{s}L(R_{1})\preceq_{s}L(R)$. Since
$R_{2}^{(t)}$ is essentially an LPT-schedule, we have
$\overleftarrow{L(T)}=\overleftarrow{L(R_{2}^{(t)})}$, and so,
$L(T)\preceq_{s}L(R_{2}^{(t)})$. It follows that $L(T)\preceq_{s}L(R)$. The
claim follows.
Now let $\bar{S}$ be an arbitrary schedule for ${\cal J}$, and let $\bar{T}$
be the schedule for ${\cal J}_{k}$ derived from $\bar{S}$ by deleting jobs
$J_{k+1},J_{k+2},\cdots,J_{n}$. Then $L(\bar{T})\preceq_{s}L(\bar{S})$. Assume
without loss of generality that $L^{\bar{S}}_{1}\geq
L^{\bar{S}}_{2}\geq\cdots\geq L^{\bar{S}}_{m}$ and $L^{\bar{T}}_{\pi(1)}\geq
L^{\bar{T}}_{\pi(2)}\geq\cdots\geq L^{\bar{T}}_{\pi(m)}$, where $\pi$ is a
permutation of $\\{1,2,\cdots,m\\}$. For each $i$ with $1\leq i\leq i_{0}$,
the above claim implies that
$\sum^{i}_{j=1}L^{S}_{j}=\sum^{i}_{j=1}L^{T}_{j}\leq\sum^{i}_{j=1}L^{\bar{T}}_{\pi(j)}\leq\sum^{i}_{j=1}L^{\bar{S}}_{j}$.
Write $P=\sum^{n}_{j=1}p_{j}$, $Q=\sum^{i_{0}}_{i=1}L^{S}_{i}$ and
$\bar{Q}=\sum^{i_{0}}_{i=1}L^{\bar{S}}_{i}$. Then $Q\leq\bar{Q}$. Note that,
in the case $i_{0}=0$, we have $Q=\bar{Q}=0$. Let $J_{d}$ be the last job
scheduled on machine $M_{i_{0}+1}$ in $S$. By the choice of $i_{0}$,
$p_{d}\leq\frac{1}{2}(L^{S}_{i_{0}+1}-p_{d})$. From the LPT algorithm, we have
$L^{S}_{i_{0}+1}-p_{d}\leq L^{S}_{j}$, $j=i_{0}+1,i_{0}+2,\cdots,m$. Hence,
$L^{S}_{i_{0}+1}\leq\frac{3}{2}(L^{S}_{i_{0}+1}-p_{d})\leq\frac{3}{2}\cdot\frac{\sum^{m}_{j=i_{0}+1}L^{S}_{j}}{m-i_{0}}=\frac{3}{2}\cdot\frac{1}{m-i_{0}}(P-Q).$
Thus, for each $i$ with $i_{0}+1\leq i\leq m$, we have
$\sum^{i}_{j=1}L^{S}_{j}\leq Q+(i-i_{0})L^{S}_{i_{0}+1}\leq
Q+\frac{3}{2}\cdot\frac{i-i_{0}}{m-i_{0}}(P-Q),$ (1)
and
$\sum^{i}_{j=1}L^{\bar{S}}_{j}\geq\bar{Q}+(i-i_{0})\frac{\sum^{i_{0}+1}_{j=m}L^{\bar{S}}_{j}}{m-i_{0}}=\bar{Q}+\frac{i-i_{0}}{m-i_{0}}(P-\bar{Q})\geq
Q+\frac{i-i_{0}}{m-i_{0}}(P-Q).$ (2)
From (1) and (2), we conclude that
$\sum^{i}_{j=1}L^{S}_{j}\leq\frac{3}{2}\sum^{i}_{j=1}L^{\bar{S}}_{j}$.
Consequently, $s(S)\leq\frac{3}{2}$. It follows that
$WAR(Pm(NP))\leq\frac{3}{2}$ for $m\geq 4$.
Now let us consider problem $P3(NP)$. Let ${\cal I}$ be an instance. Denote by
$S$ the schedule which minimizes the makespan, and by $T$ the schedule which
maximizes the machine cover. Without loss of generality, we may assume that
$L^{S}_{1}\geq L^{S}_{2}\geq L^{S}_{3}$, $L^{T}_{1}\geq L^{T}_{2}\geq
L^{T}_{3}$ and
$L^{S}_{1}+L^{S}_{2}+L^{S}_{3}=L^{T}_{1}+L^{T}_{2}+L^{T}_{3}=1$. Then
$s(S)=\frac{L^{S}_{1}+L^{S}_{2}}{L^{T}_{1}+L^{T}_{2}}$ and
$s(T)=\frac{L^{T}_{1}}{L^{S}_{1}}$. Consequently, $s^{*}({\cal
I})\leq\min\\{\frac{L^{S}_{1}+L^{S}_{2}}{L^{T}_{1}+L^{T}_{2}},\frac{L^{T}_{1}}{L^{S}_{1}}\\}$.
Note that $L^{T}_{1}=1-L^{T}_{2}-L^{T}_{3}\leq 1-2L^{T}_{3}$ and
$L^{S}_{1}\geq\frac{L^{S}_{1}+L^{S}_{2}}{2}=\frac{1-L^{S}_{3}}{2}$. Then
$s^{*}({\cal
I})\leq\min\\{\frac{1-L^{S}_{3}}{1-L^{T}_{3}},\frac{1-2L^{T}_{3}}{\frac{1-L^{S}_{3}}{2}}\\}$.
Set $x=1-2L^{T}_{3}$ and $t=1-L^{S}_{3}$. Then $\frac{2}{3}\leq t\leq 1$ and
$s^{*}({\cal I})\leq\min\\{\frac{2t}{1+x},\frac{2x}{t}\\}$. If
$x\geq\frac{\sqrt{1+4t^{2}}-1}{2}$, then $s^{*}({\cal
I})\leq\frac{2t}{1+x}\leq\frac{2t}{1+\frac{\sqrt{1+4t^{2}}-1}{2}}=\frac{\sqrt{1+4t^{2}}-1}{t}$.
If $x\leq\frac{\sqrt{1+4t^{2}}-1}{2}$, then $s^{*}({\cal
I})\leq\frac{2x}{t}\leq\frac{\sqrt{1+4t^{2}}-1}{t}$. Note that
$\frac{\sqrt{1+4t^{2}}-1}{t}\leq\sqrt{5}-1$ for all $t$ with $\frac{2}{3}\leq
t\leq 1$. It follows that $s^{*}({\cal I})\leq\sqrt{5}-1$. The result follows.
$\Box$
For problem $Pm(PP)$, McNaughton (1959) presented an optimal algorithm to
generate a schedule which minimizes the makespan. A slight modification of the
algorithm can generate a schedule $S$ with $s(S)=1$.
Algorithm $MCR$ (with input $\mathcal{M}$ and $\mathcal{J}$)
* 1.
Finding the longest job $J_{h}$ in $\mathcal{J}$. If
$p_{h}\leq\frac{\sum_{J_{j}\in\mathcal{J}}p_{j}}{|\mathcal{M}|}$, then apply
McNaughton’s algorithm to assign all jobs in $\mathcal{J}$ to the machines in
$\mathcal{M}$ evenly, and stop. Otherwise, assign $J_{h}$ to an arbitrary
machine $M_{i}\in\mathcal{M}$.
* 2.
Reset $\mathcal{M}=\mathcal{M}\setminus\\{M_{i}\\}$ and
$\mathcal{J}=\mathcal{J}\setminus\\{J_{h}\\}$. If $|\mathcal{J}|\neq 0$, then
go back to 1. Otherwise, stop.
###### Lemma 3
Assume $p_{1}\geq p_{2}\geq\cdots\geq p_{n}$ and let $S$ be a preemptive
schedule with $L^{S}_{1}\geq L^{S}_{2}\geq\cdots\geq L^{S}_{m}$. Then
$\sum^{k}_{i=1}p_{i}\leq\sum^{k}_{i=1}L^{S}_{i}$, $k=1,2,\cdots,m$.
* Proof.
Let ${\cal J}_{k}=\\{J_{1},J_{2},\cdots,J_{k}\\}$. Then at most $k$ jobs in
${\cal J}_{k}$ can be processed simultaneously in the time interval
$[0,L^{S}_{k}]$ and at most $k-i$ jobs of ${\cal J}_{k}$ can be processed
simultaneously in the time interval $[L^{S}_{k+1-i},L^{S}_{k-i}]$,
$i=1,2,\cdots,k-1$. Therefore, $\sum^{k}_{i=1}p_{i}\leq
kL^{S}_{k}+\sum^{k-1}_{i=1}(k-i)(L^{S}_{k-i}-L^{S}_{k+1-i})=\sum^{k}_{i=1}L^{S}_{i}$.
The lemma follows. $\Box$
###### Theorem 4
$WAR(Pm(PP))=1$.
* Proof.
Assume that $p_{1}\geq p_{2}\geq\cdots\geq p_{n}$. Let $i_{0}$ be the largest
job index such that $p_{i}>\frac{\sum_{j=i_{0}}^{n}p_{j}}{m-i_{0}+1}$. If
there is no such index, we set $i_{0}=0$. Let $S$ be the preemptive schedule
generated by algorithm $MCR$ with $L^{S}_{1}\geq L^{S}_{2}\geq\cdots\geq
L^{S}_{m}$. Then we have
$L^{S}_{i}=p_{i},\;i=1,2,\cdots,i_{0},$ (3)
and
$L^{S}_{i}=\frac{\sum_{j=i_{0}+1}^{n}p_{j}}{m-i_{0}},\;i=i_{0}+1,i_{0}+2,\cdots,m.$
(4)
Let $T$ be a preemptive schedule with $L^{T}_{1}\geq L^{T}_{2}\geq\cdots\geq
L^{T}_{m}$. If $1\leq k\leq i_{0}$, by lemma 3 and (3),
$\sum^{k}_{i=1}L^{S}_{i}=\sum^{k}_{i=1}p_{i}\leq\sum^{k}_{i=1}L^{T}_{i}$. If
$i_{0}+1\leq k\leq m$, by noting that
$\sum^{i_{0}}_{i=1}L^{S}_{i}\leq\sum^{i_{0}}_{i=1}L^{T}_{i}$, we have
$\sum^{k}_{i=1}L^{S}_{i}=\sum^{i_{0}}_{i=1}L^{S}_{i}+\frac{k-i_{0}}{m-i_{0}}(\sum^{n}_{i=1}p_{i}-\sum^{i_{0}}_{i=1}L^{S}_{i})\leq\sum^{i_{0}}_{i=1}L^{T}_{i}+\frac{k-i_{0}}{m-i_{0}}(\sum^{n}_{i=1}p_{i}-\sum^{i_{0}}_{i=1}L^{T}_{i})\leq\sum^{k}_{i=1}L^{T}_{i}$.
Hence, $WAR(Pm(PP))=1$. The result follows. $\Box$
For problem $Pm(FP)$, the schedule $S$ averagely processing each job on all
machines clearly has $s(S)=1$. Then we have
###### Theorem 5
$WAR(Pm(FP))=1$.
## 3 Related machines
Assume that $s_{1}\geq s_{2}\geq\cdots\geq s_{m}$. We first present the exact
expression of $WAR(Qm(FP))$ on the machine speeds $s_{1},s_{2},\cdots,s_{m}$.
Then we show that it is a lower bound for $WAR(Qm(PP))$ and $WAR(Qm(NP))$.
The fractional processing mode means that all jobs can be merged into a single
job with processing time equal to the sum of processing times of all jobs.
Thus we may assume that ${\cal I}$ is an instance of $Qm(FP)$ with just one
job $J_{\cal I}$. Suppose without loss of generality that $p_{\cal I}=1$. A
schedule $S$ of ${\cal I}$ is called _regular_ if $L^{S}_{1}\geq
L^{S}_{2}\geq\cdots\geq L^{S}_{m}$. Then $\overleftarrow{L(S)}=L(S)$ if $S$ is
regular. The following lemma can be observed from the basic mathematical
knowledge.
###### Lemma 6
Suppose that $x_{1}\geq{x_{2}}\geq\cdots\geq{x_{n}}\geq 0$ and
$y_{1}\geq{y_{2}}\geq\cdots\geq{y_{n}}\geq 0$. Then
$\sum_{i=1}^{n}x_{i}y_{\pi(i)}\leq\sum_{i=1}^{n}x_{i}y_{i}$ for any
permutation $\pi$ of $\\{1,2,\cdots,n\\}$.
###### Lemma 7
For any schedule $T$ of $\mathcal{I}$, there exists a regular schedule $S$
such that $L(S)\preceq_{c}\overleftarrow{L(T)}$.
* Proof.
Let $T$ be a schedule of $\mathcal{I}$ and $\pi$ a permutation of
$\\{1,2,\cdots,m\\}$ such that $L^{T}_{\pi(1)}\geq
L^{T}_{\pi(2)}\geq\cdots\geq L^{T}_{\pi(m)}$. By lemma 6,
$\sum^{m}_{i=1}s_{i}L^{T}_{\pi(i)}\geq\sum^{m}_{i=1}s_{\pi(i)}L^{T}_{\pi(i)}\geq
1$. Let $i_{0}$ be the smallest machine index such that
$\sum^{i_{0}}_{i=1}s_{i}L^{T}_{\pi(i)}\geq 1$. Let $S$ be the schedule in
which a part of processing time $s_{i}L^{T}_{\pi(i)}$ is assigned to $M_{i}$,
$i=1,2,\cdots,i_{0}-1$, and the rest part of processing time
$1-\sum^{i_{0}-1}_{i=1}s_{i}L^{T}_{\pi(i)}$ is assigned to $M_{i_{0}}$. Then
we have $L^{S}_{i}=L^{T}_{\pi(i)}$, for $i=1,2,\cdots,i_{0}-1$,
$L^{S}_{i_{0}}=\frac{1-\sum^{i_{0}-1}_{i=1}s_{i}L^{T}_{\pi(i)}}{s_{i_{0}}}\leq\frac{\sum^{i_{0}}_{i=1}s_{i}L^{T}_{\pi(i)}-\sum^{i_{0}-1}_{i=1}s_{i}L^{T}_{\pi(i)}}{s_{i_{0}}}=L^{T}_{\pi(i_{0})}$,
and $L^{S}_{i}=0\leq L^{T}_{\pi(i)}$ for $i=i_{0}+1,i_{0}+2,\cdots,m$. It can
be observed that $S$ is regular and $L(S)\preceq_{c}\overleftarrow{L(T)}$. The
lemma follows. $\Box$
Let $f(i)$ be the infimum of the sum of the first $i$ coordinates of
$\overleftarrow{L(T)}$ in all feasible schedule $T$ of $\mathcal{I}$,
$i=1,2,\cdots,m$. By lemma 7, we have
$f(i)=\inf\\{\sum_{k=1}^{i}L^{S}_{k}:S\mbox{ is regular}\\},i=1,2,\cdots,m$.
Then, for each schedule $T$ of $\mathcal{I}$ with $L^{T}_{\pi(1)}\geq
L^{T}_{\pi(2)}\geq\cdots\geq L^{T}_{\pi(m)}$ for some permutation $\pi$ of
$\\{1,2,\cdots,m\\}$, we have
$s(T)=\max_{1\leq i\leq
m}\left\\{\frac{\sum_{k=1}^{i}L^{\tau}_{\pi(k)}}{f(i)}\right\\}.$ (5)
The following lemma gives the exact expression for each $f(i)$.
###### Lemma 8
$f(i)=\left\\{\begin{array}[]{cc}\frac{i}{\sum^{m}_{k=1}s_{k}},&i\leq\frac{\sum^{m}_{k=1}s_{k}}{s_{1}};\\\\[5.69046pt]
\frac{1}{s_{1}},&i>\frac{\sum^{m}_{k=1}s_{k}}{s_{1}}.\end{array}\right.$
* Proof.
Fix index $i$ and let $S$ be a regular schedule. Then we have
$L^{S}_{1}\geq L^{S}_{2}\geq\cdots\geq L^{S}_{m}$ (6)
and
$\sum^{m}_{i=1}s_{i}L^{S}_{i}\geq 1.$ (7)
So we only need to find a regular schedule $S$ meeting (6) and (7) such that
$\sum_{k=1}^{i}L^{S}_{k}$ reaches the minimum.
If $i\leq\frac{\sum^{m}_{k=1}s_{k}}{s_{1}}$, by (6) and (7),
$\displaystyle\sum_{t=1}^{i}\left(\frac{\sum^{m}_{k=1}s_{k}}{i}\right)L^{S}_{t}$
$\displaystyle=$
$\displaystyle\sum^{i}_{t=1}s_{t}L^{S}_{t}+\sum_{t=1}^{i}\left(\frac{\sum^{m}_{k=1}s_{k}}{i}-s_{t}\right)L^{S}_{t}$
$\displaystyle\geq$
$\displaystyle\sum^{i}_{t=1}s_{t}L^{S}_{t}+\sum_{t=1}^{i}\left(\frac{\sum^{m}_{k=1}s_{k}}{i}-s_{t}\right)L^{S}_{i+1}$
$\displaystyle=$
$\displaystyle\sum^{i}_{t=1}s_{t}L^{S}_{t}+\left(\sum_{t=i+1}^{m}s_{t}\right)L^{S}_{i+1}$
$\displaystyle\geq$
$\displaystyle\sum^{i}_{t=1}s_{t}L^{S}_{t}+\sum^{m}_{t=i+1}s_{t}L^{S}_{t}=\sum^{m}_{t=1}s_{t}L^{S}_{t}\geq
1.$
The equality holds if and only if
$L^{S}_{1}=L^{S}_{2}=\cdots=L^{S}_{m}=\frac{1}{\sum^{m}_{k=1}s_{k}}$. Then the
regular schedule $S$ can be defined by the way that a part of processing time
$\frac{s_{k}}{\sum^{m}_{k=1}s_{k}}$ is assigned to $M_{k}$, $k=1,2,\cdots,m$.
Thus, $f(i)=\frac{i}{\sum^{m}_{k=1}s_{k}}$.
If $i>\frac{\sum^{m}_{k=1}s_{k}}{s_{1}}$, we can similarly deduce
$\displaystyle\sum_{k=1}^{i}s_{1}L^{S}_{k}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{i}s_{k}L^{S}_{k}+\sum_{k=1}^{i}(s_{1}-s_{k})L^{S}_{k}$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{i}s_{k}L^{S}_{k}+\sum_{k=1}^{i}(s_{1}-s_{k})L^{S}_{i}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{i}s_{k}L^{S}_{k}+\left(is_{1}-\sum^{i}_{k=1}s_{k}\right)L^{S}_{i}$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{i}s_{k}L^{S}_{k}+\left(\sum^{m}_{k=1}s_{k}-\sum^{i}_{k=1}s_{k}\right)L^{S}_{i}$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{i}s_{k}L^{S}_{k}+\sum^{m}_{k=i+1}s_{k}L^{S}_{k}=\sum^{m}_{k=1}s_{k}L^{S}_{k}\geq
1.$
The equality holds if and only if
$L^{S}_{1}=\frac{1}{s_{1}},L^{S}_{2}=\cdots=L^{S}_{m}=0$. Then the regular
schedule $S$ can be defined by the way that $J_{\mathcal{I}}$ is scheduled
totally on $M_{1}$ in $S$. Thus $f(i)=\frac{1}{s_{1}}$. The lemma follows.
$\Box$
By lemma 7, $s^{*}(\mathcal{I})=\inf\\{s(S):S\mbox{ is regular}\\}$. For each
regular schedule $S$, by (5) and lemma 8, we have
$\sum^{i}_{k=1}L^{S}_{k}\leq{s(L(S))}{f(i)}$ for $i=1,2,\cdots,m.$.
Let $s_{m+1}=0$ and $\frac{\sum^{m}_{i=1}s_{i}}{s_{1}}=t+\Delta$, where $t$
with $1\leq t\leq m$ is a positive integer and $0\leq\Delta<1$. By lemma 8, we
have
$i\cdot\frac{s(L(S))}{\sum^{m}_{k=1}s_{k}}\geq\sum^{i}_{k=1}L^{S}_{k},\;i=1,2,\cdots,t.$
(8)
and
$\frac{s(L(S))}{s_{1}}\geq\sum^{i}_{k=1}L^{S}_{k},\;i=t+1,t+2,\cdots,m.$ (9)
From (8) and (9), we have $\sum^{t}_{i=1}(s_{i}-s_{i+1})\cdot
i\cdot\frac{s(L(S))}{\sum^{m}_{i=1}s_{i}}+\sum^{m}_{i=t+1}(s_{i}-s_{i+1})\frac{s(L(S))}{s_{1}}\geq\sum^{t}_{i=1}(s_{i}-s_{i+1})\sum^{i}_{t=1}L^{S}_{t}+\sum^{m}_{i=t+1}(s_{i}-s_{i+1})\sum^{i}_{t=1}L^{S}_{t}=\sum^{m}_{i=1}s_{i}L^{S}_{i}=1$.
Hence,
$s(S)\geq\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\left(\frac{\sum^{m}_{i=1}s_{i}}{s_{1}}-t\right)s_{t+1}}=\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta
s_{t+1}}$. Note that the equality holds if and only if
$L^{S}_{1}=L^{S}_{2}=\cdots=L^{S}_{t}=\frac{1}{\sum^{t}_{i=1}s_{i}+\Delta
s_{t+1}}$, $L^{S}_{t+1}=\frac{\Delta}{\sum^{t}_{i=1}s_{i}+\Delta s_{t+1}}$ and
$L^{S}_{t+2}=L^{S}_{t+3}=\cdots=L^{S}_{m}=0$. Then the corresponding regular
schedule $S$ can be defined by the way that a part of processing time
$\frac{s_{i}}{\sum^{t}_{k=1}s_{k}+\Delta s_{t+1}}$ is assigned to $M_{i}$,
$i=1,2,\cdots,t$, and the rest part of processing time $\frac{\Delta
s_{t+1}}{\sum^{t}_{i=1}s_{i}+\Delta s_{t+1}}$ is assigned to $M_{t+1}$. Hence,
$s^{*}({\mathcal{I}})=\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta
s_{t+1}}$. Consequently,
$WAR(Qm(FP))=\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta s_{t+1}}$
if the machine speeds are fixed.
If the machine speeds are parts of the input, by the fact that $s_{1}\geq
s_{2}\geq\cdots\geq s_{m}$, we have
$\frac{\sum^{t}_{i=2}s_{i}+\Delta
s_{t+1}}{t-1+\Delta}\geq\frac{\sum^{m}_{i=2}s_{i}}{m-1}.$ (10)
Let $\theta=\frac{\sum^{m}_{i=2}s_{i}}{m-1}$ and
$\vartheta=\frac{s_{1}}{\theta}>1$. Then
$t+\Delta=\frac{\sum^{m}_{i=1}s_{i}}{s_{1}}=\frac{s_{1}+(m-1)\theta}{s_{1}}=\frac{\vartheta+m-1}{\vartheta}.$
(11)
Obviously, $\frac{m}{\vartheta-1}+(\vartheta-1)\geq
2\sqrt{\frac{m}{\vartheta-1}(\vartheta-1)}=2\sqrt{m}$. By (10) and (11), we
have $\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta
s_{t+1}}=\frac{s_{1}+(m-1)\frac{\sum^{m}_{i=2}s_{i}}{m-1}}{s_{1}+(t-1+\Delta)\frac{\sum^{t}_{i=2}s_{i}+\Delta
s_{t+1}}{t-1+\Delta}}\leq\frac{s_{1}+(m-1)\frac{\sum^{m}_{i=2}s_{i}}{m-1}}{s_{1}+(t-1+\Delta)\frac{\sum^{m}_{i=2}s_{i}}{m-1}}=1+\frac{m-1}{(\frac{m}{\vartheta-1}+(\vartheta-1))+2}\leq
1+\frac{m-1}{2\sqrt{m}+2}=\frac{\sqrt{m}+1}{2}$. So we have
$s^{*}(\mathcal{I})\leq\frac{\sqrt{m}+1}{2}$ and therefore
$WAR(Qm(FP))\leq\frac{\sqrt{m}+1}{2}$.
To show that $WAR(Qm(FP))=\frac{\sqrt{m}+1}{2}$, we consider the following
instance $\mathcal{I}$ with $p_{\mathcal{I}}=1$, $s_{1}=s=\sqrt{m}+1>1$ and
$s_{2}=s_{3}=\cdots=s_{m}=1$. Let $S$ be a regular schedule and write
$x=sL^{S}_{1}$. Then $\sum^{m}_{t=2}L^{S}_{t}=1-x$. By lemma 8 and (5), we
have
$s(S)\geq\max\left\\{\frac{L^{S}_{1}}{f(1)},\frac{\sum_{i=1}^{m}L^{S}_{i}}{f(m)}\right\\}=\max\left\\{\frac{x(s+m-1)}{s},x+s(1-x)\right\\}\geq\frac{s^{2}+sm-s}{s^{2}+m-1}=\frac{\sqrt{m}+1}{2}$,
where the inequality follows from the fact that $\frac{x(s+m-1)}{s}$ is an
increasing function in $x$ while $x+s(1-x)$ is a decreasing function in $x$
and they meet with $\frac{s^{2}+sm-s}{s^{2}+m-1}$ when
$x=\frac{s^{2}}{s^{2}+m-1}$. Then
$s^{*}(\mathcal{I})\geq\frac{\sqrt{m}+1}{2}$. Consequently,
$WAR(Qm(FP))=\frac{\sqrt{m}+1}{2}$.
The above discussion leads to the following conclusion.
###### Theorem 9
If the machine speeds $s_{1},s_{2},\cdots,s_{m}$ are fixed, then
$WAR(Qm(FP)=\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta s_{t+1}}$,
where $\frac{\sum^{m}_{i=1}s_{i}}{s_{1}}=t+\Delta$, $1\leq t\leq m$ is a
positive integer and $0\leq\Delta<1$. If the machine speeds
$s_{1},s_{2},\cdots,s_{m}$ are parts of the input, then
$WAR(Qm(FP)=\frac{\sqrt{m}+1}{2}$.
###### Lemma 10
If the machine speeds $s_{1},s_{2},\cdots,s_{m}$ are fixed, then
$WAR(Qm(NP))\geq WAR(Qm(FP))$ and $WAR(Qm(PP))\geq WAR(Qm(FP))$.
* Proof.
We only consider the non-preemptive processing mode. For the preemptive
processing mode, the result can be similarly proved. Given a schedule $S$, we
denote by $\pi^{S}$ the permutation of $\\{1,2,\cdots,m\\}$ such that
$L^{S}_{\pi^{S}(1)}\geq L^{S}_{\pi^{S}(2)}\geq\cdots\geq L^{S}_{\pi^{S}(m)}$.
Suppose without loss of generality that $s_{m}=1$. Write $\eta=WAR(Qm(NP))$.
Let $\mathcal{I}$ be an instance of $Q_{m}(FP)$ with only one job
$J_{\mathcal{I}}$ of processing time 1. For each $i$, set $f(i)$ to be the
infimum of $\sum_{k=1}^{i}L^{S}_{\pi^{S}(k)}$ of schedule $S$ over all
fractional schedules of $\mathcal{I}$. We only need to show that
$s^{*}(\mathcal{I})\leq\eta$.
Assume to the contrary that $s^{*}(\mathcal{I})>\eta$. Let $\epsilon>0$ be a
sufficiently small number such that
$\eta(f(i)+i\epsilon)<s^{*}(\mathcal{I})f(i)$, $i=1,2,\cdots,m$. Let
$\mathcal{H}$ be an instance of $Q_{m}(NP)$ such that the total processing
time of jobs is equal to $1$ and the processing time of each job is at most
$\epsilon$. For each $i$, let $g(i)$ be the infimum of
$\sum_{k=1}^{i}L^{S}_{\pi^{S}(k)}$ of schedule $S$ over all feasible schedules
of $\mathcal{H}$. We assert that
$g(i)\leq f(i)+i\epsilon,\;i=1,2,\cdots,m.$ (12)
To the end, let $S_{i}$ be the regular schedule of $\mathcal{I}$ such that
$\sum^{i}_{k=1}L^{S_{i}}_{k}=f(i)$, $i=1,2,\cdots,m$. Fix index $i$, we
construct a non-preemptive schedule $S$ of $\mathcal{H}$ such that
$\sum_{k=1}^{i}L^{S}_{\pi^{S}(k)}\leq f(i)+i\epsilon$. This leads to
$g(i)\leq\sum_{k=1}^{i}L^{S}_{\pi^{S}(k)}\leq f(i)+i\epsilon$, and therefore,
proves the assertion. The construction of $S$ is stated as follows. First, we
assign jobs to $M_{i}$ one by one until $L^{S}_{1}\geq L^{S_{i}}_{1}$. Then we
assign the rest jobs to $M_{2}$ one by one until $L^{S}_{2}\geq
L^{S_{i}}_{2}$. This procedure is repeated until all jobs are assigned.
According to the construction of $S$, we have $L^{S}_{k}\leq
L^{S_{i}}_{k}+\frac{\epsilon}{s_{k}}\leq L^{S_{i}}_{k}+\epsilon$,
$k=1,2,\cdots,m$. Note that $L^{S_{i}}_{1}\geq L^{S_{i}}_{2}\geq\cdots\geq
L^{S_{i}}_{m}$. Then
$\sum_{k=1}^{i}L^{S}_{\pi^{S}(k)}\leq\sum_{k=1}^{i}(L^{S_{i}}_{\pi^{S}(k)}+\epsilon)\leq\sum^{i}_{k=1}L^{S_{i}}_{k}+i\epsilon=f(i)+i\epsilon$.
Let $R$ be the schedule of $\mathcal{H}$ such that $s(R)=s^{*}(\mathcal{H})$.
It can be observed that there exists a schedule $T$ of $\mathcal{I}$ such that
$L(T)\preceq_{c}L(R)$. Hence, for each $i$ with $1\leq i\leq m$, we have
$\sum^{i}_{k=1}L^{T}_{\pi^{T}(k)}\leq\sum^{i}_{k=1}L^{R}_{\pi^{T}(k)}\leq\sum^{i}_{k=1}L^{R}_{\pi^{R}(k)}\leq{s(R)g(i)}\leq{s^{*}(\mathcal{H})(f(i)+i\epsilon)}\leq\eta(f(i)+i\epsilon)<s^{*}(\mathcal{I})f(i)$.
This contradicts the definition of $s^{*}(\mathcal{I})$. So
$s^{*}(\mathcal{I})\leq\eta$. The result follows. $\Box$
By theorem 9 and lemma 10, the following theorem holds.
###### Theorem 11
If the machine speeds $s_{1},s_{2},\cdots,s_{m}$ are fixed, then $WAR({\cal
P})\geq\frac{\sum^{m}_{i=1}s_{i}}{\sum^{t}_{i=1}s_{i}+\Delta s_{t+1}}$ for
${\cal P}\in\\{Qm(NP),Qm(PP)\\}$, where
$\frac{\sum^{m}_{i=1}s_{i}}{s_{1}}=t+\Delta$, $t$ is a positive integer with
$1\leq t\leq m$, and $0\leq\Delta<1$. If the machine speeds
$s_{1},s_{2},\cdots,s_{m}$ are parts of the input, then $WAR({\cal
P})\geq\frac{\sqrt{m}+1}{2}$ for ${\cal P}\in\\{Qm(NP),Qm(PP)\\}$.
## 4 Unrelated machines
Since $Qm$ is a special version of $Rm$, from the results in the previous
section, the weak simultaneous approximation ratio is at least
$\frac{\sqrt{m}+1}{2}$ for each of $Rm(NP)$, $Rm(PP)$ and $Rm(FP)$. The
following lemma establishes an upper bound of the weak simultaneous
approximation ratio for the three problems.
###### Lemma 12
$WAR({\cal P})\leq\sqrt{m}$ for ${\cal P}\in\\{Rm(NP),Rm(PP),Rm(FP)\\}$.
* Proof.
Let ${\cal I}$ be an instance of $R_{m}(NP)$, $R_{m}(PP)$ or $R_{m}(FP)$. Let
$S$ be a schedule which minimizes the makespan with $L^{S}_{1}\geq
L^{S}_{2}\geq\cdots\geq L^{S}_{m}$. Write $p_{[j]}=\min_{1\leq i\leq
m}\\{p_{ij}\\}$.
If $L^{S}_{1}\leq\frac{\sum^{n}_{j=1}p_{[j]}}{\sqrt{m}}$, let $T$ be a
feasible schedule with $L^{T}_{\pi(1)}\geq L^{T}_{\pi(2)}\geq\cdots\geq
L^{T}_{\pi(m)}$ for some permutation $\pi$ of $\\{1,2,\cdots,m\\}$. For each
$i$, we have $\sum^{i}_{k=1}L^{S}_{k}\leq
iL^{S}_{1}\leq\sqrt{m}\cdot\frac{i}{m}\sum^{n}_{j=1}p_{[j]}\leq\sqrt{m}\sum^{i}_{k=1}L^{T}_{\pi(k)}$.
This means that $s^{*}({\cal I})\leq\sqrt{m}$.
If $L^{S}_{1}>\frac{\sum^{n}_{j=1}p_{[j]}}{\sqrt{m}}$, let $R$ be the schedule
in which each job $J_{j}$ is assigned to the machine $M_{i}$ with
$p_{ij}=p_{[j]}$. Let $O$ be an arbitrarily feasible schedule, and let
${\pi}_{1}$ and ${\pi}_{2}$ be two permutations of $\\{1,2,\cdots,m\\}$ such
that $L^{R}_{{\pi}_{1}(1)}\geq L^{R}_{{\pi}_{1}(2)}\geq\cdots\geq
L^{R}_{{\pi}_{1}(m)}$ and $L^{O}_{{\pi}_{2}(1)}\geq
L^{O}_{{\pi}_{2}(2)}\geq\cdots\geq L^{O}_{{\pi}_{2}(m)}$. For each $i$, we
have
$\sum^{i}_{k=1}L^{R}_{{\pi}_{1}(k)}\leq\sum^{m}_{k=1}L^{R}_{{\pi}_{1}(k)}=\sum^{n}_{j=1}p_{[j]}<\sqrt{m}L^{S}_{1}\leq\sqrt{m}L^{O}_{{\pi}_{2}(1)}\leq\sqrt{m}\sum^{i}_{k=1}L^{O}_{{\pi}_{2}(k)}$.
This also means that $s^{*}({\cal I})\leq\sqrt{m}$. The lemma follows. $\Box$
Combining with the results of the previous section, we have the following
theorem.
###### Theorem 13
For each problem ${\cal P}\in\\{Qm(NP),Qm(PP),Qm(FP),Rm(NP),Rm(PP),Rm(FP)\\}$,
we have $\frac{\sqrt{m}+1}{2}\leq WAR({\cal P})\leq\sqrt{m}$.
## Acknowledgments
The authors would like to thank the associate editor and two anonymous
referees for their constructive comments and kind suggestions.
## References
* Bhargava et al. (2001) Bhargava R, Goel A, Meyerson A (2001) Using approximate majorization to characterize protocol fairness. In: Proceedings of the 2001 ACM SIGMETRICS international conference on Measurement and modeling of computer systems (SIGMETRICS’01). ACM, New York, pp 330–331
* Csirik et al. (1992) Csirik J, Kellerer H, Woeginger G (1992) The exact LPT-bound of maximizing the minimum completion time. Operations Research Letters 11(5): 281–287
* Deuermeyer et al. (1982) Deuermeyer BL, Friesen DK, Langston AM (1982) Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM Journal on Discrete Mathematics 3(2): 190–196
* Goel et al. (2001) Goel A, Meyerson A, Plotkin S (2001) Combining fairness with throughput: online routing with multiple objectives. Journal of Computer and System Sciences 63: 62–79
* Goel et al. (2005) Goel A, Meyerson A, Plotkin S (2005) Approximate majorization and fair online load balancing. ACM Transactions on Algorithms 1(2): 338–349
* Graham (1966) Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell System Technical Journal 45(9): 1563–1581
* Graham (1969) Graham RL (1969) Bounds for multiprocessing timing anomalies. SIAM Journal on Applied Mathematics 17(2): 416–429
* Kleinberg et al. (2001) Kleinberg J, Rabani Y, Tardos É (2001) Fairness in Routing and Load Balancing. Journal of Computer and System Sciences 63: 2–20
* Kumar and Kleinberg (2006) Kumar A, Kleinberg J (2006) Fairness measures for resource allocation. SIAM Journal on Computing 36(3): 657–680
* McNaughton (1959) McNaughton R (1959) Scheduling with deadlines and loss functions. Management Science 6(1): 1–12
|
arxiv-papers
| 2013-04-15T12:47:42 |
2024-09-04T02:49:44.336018
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Long Wan",
"submitter": "Long Wan",
"url": "https://arxiv.org/abs/1304.4073"
}
|
1304.4096
|
# The AB equations and the $\bar{\partial}$-dressing method in semi-
characteristic coordinates 00footnotetext:
Junyi Zhu and Xianguo Geng
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan
450001, China Email: [email protected]
###### Abstract
The dressing method based on the $2\times 2$ matrix $\bar{\partial}$-problem
is generalized to study the canonical form of AB equations. The soliton
solutions for the AB equations are given by virtue of the properties of Cauchy
matrix. Asymptotic behaviors of the $N$-soliton solution are discussed.
PACS number: 02.30.IK, 02.30.Jr
## 1 Introduction
The AB equations have important applications in geophysical fluids and in
nonlinear optics [1, 2, 3, 4, 5]. The important features are that the AB
equations are integrable by the inverse scattering transform and can be
reduced to the sine-Gordon equation[6, 7]. The single-phase periodic solution
is studied by the method for improving the effectiveness of one-phase periodic
solutions of integrable equations in [8], the envelope solitary wave and sine
Waves are discussed in [9]. In addition, Guo et al. [10] investigated the
Painlevé property and conservation laws of one type of variable-coefficient AB
equation, and obtained the soliton solutions by Darboux transformation.
The $\bar{\partial}$-dressing method [11, 12, 13, 14, 15] is a powerful tools
to construct and solve integrable nonlinear equations as well as to describe
their transformations and reductions. For a review see [16, 17], and
references therein.
To our knowledge, The $N$-soliton solution of the AB equation has not been
given and $\bar{\partial}$-dressing method for the AB equation is open. In
this paper, we study the AB equations in semi-characteristic coordinates by
extended $\bar{\partial}$-dressing method [18] and give their $N$-soliton
solution.
The present paper is organized as follows. In Sec. 2, the semi-characteristic
coordinates $\xi$ and $\tau$ are introduced in the spectral transform matrix
to derive the Lax pair of these equations, where the $\tau$-dependent linear
spectral problem is obtained by introducing a special singular dispersion
relation. In Sec. 3, suitable symmetry conditions are applied to derive the AB
equations in canonical form. In Sec. 4, the properties of Cauchy matrix are
used to discuss one-soliton, two-soliton, as well as N-soliton solutions of
the equations. In the last section, we study the asymptotic behaviors of the
$N$-soliton solution.
## 2 Spectral transform and Lax pair
In this paper, we consider the $2\times 2$ matrix $\bar{\partial}$-problem in
the complex $k$-plane,
$\bar{\partial}\psi(k,\bar{k})=\psi(k,\bar{k})R(k,\bar{k}),$ (2.1)
where $\bar{\partial}\equiv\partial/\partial\bar{k}$ and $R=R(k,\bar{k})$ is a
spectral transform matrix which will be associated with a nonlinear equation.
It is readily verified that a solution of the $\bar{\partial}$-problem (2.1)
with the canonical normalization can be written as
$\psi(k)=I+\psi RC_{k},$ (2.2)
where $C_{k}$ denotes the Cauchy-Green integral operator acting on the left
$\psi RC_{k}=\frac{1}{2i\pi}\iint\frac{{\rm d}z\wedge{\rm
d}\bar{z}}{z-k}\psi(z)R(z),$
and here we have suppressed the variable $\bar{k}$ dependence in $\psi$ and
$R$. It is readily verified that, for some matrix functions $f(k)$ and $g(k$),
the operator $C_{k}$ satisfies
$\displaystyle
g(k)[f(k)C_{k}]C_{k}+[g(k)C_{k}]f(k)C_{k}=[g(k)C_{k}][f(k)C_{k}],$ (2.3)
The formal solution of $\bar{\partial}$-problem (2.1) in terms of the matrix
$R$ will be given from (2.2) as
$\psi(k)=I\cdot(I-RC_{k})^{-1}.$ (2.4)
For the sake of convenience, we define a pairing
$\langle f,g\rangle=\frac{1}{2i\pi}\iint f(k)g^{\rm T}(k){\rm d}k\wedge{\rm
d}\bar{k},\quad\langle f,g\rangle^{\rm T}=\langle g,f\rangle,$
It is known that the above pairing possesses the following prosperities [13]
$\langle fR,g\rangle=\langle f,gR^{\rm T}\rangle,\quad\langle
fC_{k},g\rangle=-\langle f,gC_{k}\rangle.$ (2.5)
In addition, we can easily prove the following properties
$\displaystyle kf(k)C_{k}=k[f(k)C_{k}]+\langle f(k)\rangle,$ (2.6)
$\displaystyle\frac{1}{\mu-k}f(k)C_{k}=\frac{1}{\mu-k}\\{[f(k)C_{k}]-[f(\mu)C_{\mu}]\\},$
where $\langle f(k)\rangle=\langle f(k),I\rangle$.
The aim of the $\bar{\partial}$ dressing method is to construct the compatible
system of linear equations for $\psi$ and consequently the nonlinear evolution
equations associated the $\bar{\partial}$-problem (2.1). According to the main
idea of the inverse scattering transform method, it is important to introduce
the $\xi,\tau$ dependence in the spectral transform matrix $R(k,\bar{k})$. For
the AB equations, let the $\xi$ and $\tau$-dependence be given by the linear
and solvable equations
$R_{\xi}=ik[\sigma_{3},R],\quad\sigma_{3}={\rm diag}(1,-1),$ (2.7)
and
$R_{\tau}=[\Omega,R],$ (2.8)
where $\Omega(k)$ is a singular dispersion relation, that is
$\Omega(k)=\omega(k)C_{k}\sigma_{3},$ (2.9)
where $\omega(k)$ is some scalar function. Differentiating (2.2) with respect
to $\xi$ and $\tau$, and using (2.7),(2.8), as well as the properties of the
Cauchy-Green operator (2.6), we obtain the Zakharov-Shabat spectral problem
[18]
$\displaystyle\psi_{\xi}-ik[\sigma_{3},\psi]=Q\psi,$ (2.10)
$\displaystyle\quad Q=i[\sigma_{3},\langle\psi R\rangle],$
and the $\tau$-dependent linear equation associated with the singular
dispersion relation
$\psi_{\tau}=\left(\omega\psi\sigma_{3}\psi^{-1}C_{k}\right)\psi-\psi\Omega.$
(2.11)
## 3 The AB equations
In this section, we will derive the AB equations equations associated with
spectral problem (2.10). To the end, differentiating the expression of $Q$ in
(2.10) with respect to $\tau$ yields
$Q_{\tau}=i[\sigma_{3},\langle\psi R\rangle_{\tau}].$ (3.1)
Since $\bar{\partial}(f(k)C_{k})=f(k)$, then
$\displaystyle(\psi R)_{\tau}$
$\displaystyle=\bar{\partial}\psi_{\tau}=\bar{\partial}\left\\{\psi
R_{\tau}C_{k}(I-RC_{k})^{-1}\right\\}$
$\displaystyle=\bar{\partial}\left\\{\psi
R_{\tau}(I-RC_{k})^{-1}C_{k}\right\\}=\psi R_{\tau}(I-RC_{k})^{-1}.$
Hence, in virtue of the properties (2.4), equation (3.1) can be rewritten as
$Q_{\tau}=i[\sigma_{3},\langle\psi
R_{\tau}(I-RC_{k})^{-1},I\rangle]=i[\sigma_{3},\langle\psi
R_{\tau},I\cdot(I+R^{\rm T}C_{k})^{-1}\rangle].$ (3.2)
Based on the identity $\bar{\partial}(\psi^{-1})^{\rm T}=-(\psi^{-1})^{\rm
T}R^{\rm T}$, the same procedure as (2.2) and (2.4) products
$I\cdot(I+R^{\rm T}C_{k})^{-1}=(\psi^{-1})^{\rm T}.$
Therefore, using (2.4) and the definition of pairing $\langle f,g\rangle$,
equation (3.2) takes the form
$\displaystyle Q_{\tau}$
$\displaystyle=i[\sigma_{3},\langle\psi\Omega,(\psi^{-1}R^{\rm
T})\rangle]-i[\sigma_{3},\langle\psi R\Omega,(\psi^{-1})^{\rm T}\rangle]$
$\displaystyle=-i[\sigma_{3},\langle\psi\Omega,\bar{\partial}(\psi^{-1})^{\rm
T}\rangle]-i[\sigma_{3},\langle({\bar{\partial}}\psi)\Omega,(\psi^{-1})^{\rm
T}\rangle]$
$\displaystyle=-i[\sigma_{3},\langle\psi\Omega\bar{\partial}\psi^{-1}\rangle]-i[\sigma_{3},\langle({\bar{\partial}}\psi)\Omega\psi^{-1}\rangle].$
Taking into account the fact that $\Omega\rightarrow 0$ as
$k\rightarrow\infty$, the above equation can be further reduced to
$\displaystyle Q_{\tau}$
$\displaystyle=-i[\sigma_{3},\langle\bar{\partial}(\psi\Omega\psi^{-1})\rangle-\langle\psi(\bar{\partial}\Omega)\psi^{-1}\rangle]$
(3.3)
$\displaystyle=i[\sigma_{3},\langle\omega(k)\psi\sigma_{3}\psi^{-1}\rangle].$
By virtue of the spectral problem (2.10), one can verify that
$U_{\xi}=ik[\sigma_{3},U]+[Q,U],\quad U=\psi\sigma_{3}\psi^{-1},$ (3.4)
and
$Q_{\tau}=i[\sigma_{3},\langle\omega(k)U\rangle].$ (3.5)
In order to derive the $\tau$-dependent linear spectral problem of the AB
equations, we take
$\omega(k)=-i\pi\delta(k),$ (3.6)
then
$V\equiv i\langle\omega U\rangle=-U|_{k=0},$ (3.7)
which implies
$Q_{\tau}=[\sigma_{3},V],\quad V_{\xi}=[Q,V].$ (3.8)
It is noted that the coupled equations (3.8) can also be derived from the
compatibility condition of the linear equations (2.10) and (2.11).
From (3.6), we know that the linear spectral problem (2.11) can be rewritten
as
$\psi_{\tau}+\frac{1}{ik}\psi\sigma_{3}=-\frac{1}{ik}V\psi.$ (3.9)
For the purpose of obtaining the AB equations, we introduce the following
symmetry condition
$Q^{\dagger}=-Q,$ (3.10)
from which we take
$Q=2\left(\begin{matrix}0&-\bar{A}\\\ A&0\\\ \end{matrix}\right).$ (3.11)
Here, the form of the potential function is chosen to ensure that the
normalization condition $|A_{\tau}|^{2}+B^{2}=1$ can be obtained. In addition,
we need another symmetry condition about $\psi(k)$
$\psi^{\dagger}(\bar{k})=\psi^{-1}(k).$ (3.12)
It is noted that this constraint condition can be obtained by using the
symmetry condition (3.10) and the spectral problem (2.10), as well as the
linear equation (2.7).
From (3.8), we know that
$\displaystyle V^{(o)}=\frac{1}{2}\sigma_{3}Q_{\tau},$ $\displaystyle
Q_{\xi\tau}=[\sigma_{3},[Q,V]]=2\sigma_{3}[Q,V^{(d)}],$ $\displaystyle
V^{(d)}_{\xi}=[Q,V^{(o)}]=-\frac{1}{2}\sigma_{3}(Q^{2})_{\tau},$
where $V^{(o)}$ and $V^{(d)}$ denote the off-diagonal and diagonal of the
matrix $V$, respectively. Hence $V=V^{(o)}+V^{(d)}$. According to the above
equations, we take
$V=-\left(\begin{matrix}B&\bar{A}_{\tau}\\\ A_{\tau}&-B\end{matrix}\right),$
(3.13)
then we have the AB equations in canonical form
$A_{\xi\tau}-4AB=0,\quad B_{\xi}+2(|A|^{2})_{\tau}=0.$ (3.14)
It is remarked that the Lax pair of the AB equations is defined by (2.10) and
(3.9), as well as (3.13).
## 4 Soliton solutions
In the section, we will derive the explicit solutions of the AB equations
(3.14) and their soliton solutions. To this end, we introduce the spectral
transform matrix $R$ as
$R(k)=i\pi\sum\limits_{j=1}^{N}\left(\begin{matrix}0&\bar{c}_{j}e^{2ik\xi}\delta(k-\bar{k}_{j})&\\\
c_{j}e^{-2ik\xi}\delta(k-k_{j})&0\\\ \end{matrix}\right),$ (4.1)
where $\\{k_{j}\\}_{1}^{N}$ are complex constants and $c_{j}=c_{j}(\tau)$. The
evolution of these $\tau$-dependent functions can be obtained from (2.8) and
(3.6)
$c_{j,\tau}=-\frac{2}{ik_{j}}c_{j},\quad j=1,2,\cdots,N,$ (4.2)
Substituting (4.1) into (2.10), in view of (3.12), yields
$A=-i\langle\psi R\rangle_{21}=-\hat{\psi}_{22}\cdot g^{T},$ (4.3)
where
$\displaystyle\hat{\psi}_{22}=\left(\psi_{22}(k_{1}),\cdots,\psi_{22}(k_{N})\right),\quad
g=(g_{1},\cdots,g_{N}),$ (4.4) $\displaystyle
g_{j}=c_{j}e^{2k_{j}\xi}=e^{2z_{j}},\quad z_{j}=\theta_{j}-i\varphi_{j},$
$\displaystyle\quad\theta_{j}={\rm Im}k_{j}\xi+\frac{{\rm
Im}k_{j}}{|k_{j}|^{2}}\tau+\kappa_{j},$ $\displaystyle\quad\varphi_{j}={\rm
Re}k_{j}\xi-\frac{{\rm Re}k_{j}}{|k_{j}|^{2}}\tau+\chi_{j},$
where $\\{\kappa_{j},\chi_{j}\\}$ are arbitrary constants. In addition, from
(3.7) and (3.4),(3.13), by virtue of symmetry condition (3.12), we obtain
$B=|\psi_{22}(0)|^{2}-|\psi_{21}(0)|^{2},$ (4.5)
in terms of $\det\psi=1$.
In the following, we will give the expression of $\psi_{ij}$ about the
discrete data. Substitution (4.1) into (2.2) yields
$\displaystyle\psi_{22}(k)$
$\displaystyle=1-i\sum\limits_{j=1}^{N}\frac{\psi_{21}(\bar{k}_{j})\bar{g}_{j}}{\bar{k}_{j}-k},$
(4.6) $\displaystyle\psi_{21}(k)$
$\displaystyle=-i\sum\limits_{j=1}^{N}\frac{\psi_{22}(k_{j})g_{j}}{k_{j}-k},$
which imply that
$\hat{\psi}_{22}=E(I+K\bar{K})^{-1},\quad\tilde{\psi}_{21}=-iEK(I+\bar{K}K)^{-1},$
(4.7)
where the vectors $\hat{\psi}_{22},g$ are defined by (4.4) and
$\displaystyle\tilde{\psi}_{21}=(\psi_{21}(\bar{k}_{1}),\cdots,\psi_{21}(\bar{k}_{N})),\quad
E=(1,\cdots,1),$ (4.8) $\displaystyle K=(K_{nm})_{N\times N},\quad
K_{nm}=\frac{g_{n}}{k_{n}-\bar{k}_{m}}.$
In addition, from (4.6), we have
$\displaystyle\psi_{21}(0)=-i\hat{\psi}_{22}h^{T},\quad\psi_{22}(0)=1-i\tilde{\psi}_{21}\bar{h}^{T},$
(4.9) $\displaystyle\quad h=(h_{1},\cdots,h_{N}),\quad
h_{j}=\frac{g_{j}}{k_{j}}.$
Substituting (4.7) into (4.3) and (4.9) one obtains [19]
$\displaystyle A=$ $\displaystyle-{\rm tr}[(I+M)^{-1}g^{T}E]$ (4.10)
$\displaystyle=$ $\displaystyle-\frac{\det(I+M+g^{T}E)-\det(I+M)}{\det(I+M)},$
$\displaystyle\psi_{21}(0)=$ $\displaystyle-i{\rm tr}[(I+M)^{-1}h^{T}E]$
$\displaystyle=$
$\displaystyle-i\frac{\det(I+M+h^{T}E)-\det(I+M)}{\det(I+M)},$
$\displaystyle\psi_{22}(0)=$ $\displaystyle 1-{\rm
tr}[(I+\tilde{M})\bar{h}^{T}EK]$ $\displaystyle=$ $\displaystyle
1-\frac{\det(I+\tilde{M}+\bar{h}^{T}EK)-\det(I+\tilde{M})}{\det(I+\tilde{M})},$
where
$M=K\bar{K},\quad\tilde{M}=\bar{K}K.$ (4.11)
Indeed, for example, it is easy to see that $A=-E(I+M)^{-1}g^{T}$ in view of
(4.3) and (4.7). Then one may find $A=-{\rm tr}[(I+M)^{-1}g^{T}E]$ by
multiplication of matrices, and $A=-[\det(I+M+g^{T}E)-\det(I+M)]/\det(I+M)$ by
the fact that $\det(g^{T}E)=0$.
In the following, we will give the one-soliton and two-soliton solutions.
Firstly, for $N=1$,
$\displaystyle A=$ $\displaystyle-e^{-2i\varphi}\frac{2{\rm Im}k_{1}}{(2{\rm
Im}k_{1})e^{-2\theta_{1}}+e^{2\theta_{1}}(2{\rm Im}k)^{-1}},$
$\displaystyle\psi_{21}(0)=$
$\displaystyle\frac{e^{-2i\varphi}}{ik_{1}}\frac{2{\rm Im}k_{1}}{(2{\rm
Im}k_{1})e^{-2\theta_{1}}+e^{2\theta_{1}}(2{\rm Im}k)^{-1}},$
$\displaystyle\psi_{22}(0)=$ $\displaystyle
1-\frac{e^{2\theta_{1}}}{i\bar{k}_{1}}\frac{1}{(2{\rm
Im}k_{1})e^{-2\theta_{1}}+e^{2\theta_{1}}(2{\rm Im}k)^{-1}},.$
Let $2{\rm Im}k_{1}=e^{2\beta_{1}}$, then the above expressions give rise to
one-soliton solution in semi-characteristic coordinates
$\displaystyle A=$ $\displaystyle-\frac{1}{2}e^{2(\beta_{1}-i\varphi_{1})}{\rm
sech}2(\theta_{1}-\beta_{1}),$ (4.12) $\displaystyle B=$ $\displaystyle
1+\frac{e^{2(\theta_{1}+\beta_{1})}}{2|k_{1}|^{2}}[{\rm
sech}^{2}2(\theta_{1}-\beta_{1})\sinh 2(\theta_{1}-\beta_{1})-{\rm
sech}2(\theta_{1}-\beta_{1})].$
It is readily verified that $\int_{-\infty}^{\infty}|A|^{2}d\xi={\rm
Im}k_{1}/4$. One can find that the waveform of the envelope solitary wave
travels to the left, and the carrier wave to right, with same velocity
$1/|k_{1}|^{2}$. The graphic of one-soliton solution is shown in Figure 1.
Figure 1: $k_{1}=1.04+0.6i,\kappa_{1}=0,\xi_{1}=0$.
For the case of $N=2$, we have
$\displaystyle\det(I+M)=$ $\displaystyle
1+\frac{|k_{1}-k_{2}|^{4}}{\prod\limits_{j,l=1}^{2}(k_{j}-\bar{k}_{l})^{2}}e^{2(z_{1}+z_{2}+\bar{z}_{1}+\bar{z}_{2})}-\frac{e^{2(z_{1}+\bar{z}_{1})}}{(k_{1}-\bar{k}_{1})^{2}}$
(4.13)
$\displaystyle-\frac{e^{2(z_{2}+\bar{z}_{2})}}{(k_{2}-\bar{k}_{2})^{2}}-\frac{e^{2(z_{1}+\bar{z}_{2})}}{(k_{1}-\bar{k}_{2})^{2}}-\frac{e^{2(\bar{z}_{1}+z_{2})}}{(k_{2}-\bar{k}_{1})^{2}},$
$\displaystyle\det(I+M+$ $\displaystyle g^{T}E)-\det(I+M)$ $\displaystyle=$
$\displaystyle
e^{2z_{1}}\left[1-\frac{(k_{1}-k_{2})^{2}}{(k_{1}-\bar{k}_{2})^{2}(k_{2}-\bar{k}_{2})^{2}}e^{2(z_{2}+\bar{z}_{2})}\right]$
$\displaystyle+e^{2z_{2}}\left[1-\frac{(k_{1}-k_{2})^{2}}{(k_{1}-\bar{k}_{1})^{2}(k_{2}-\bar{k}_{1})^{2}}e^{2(z_{1}+\bar{z}_{1})}\right],$
$\displaystyle\det(I+M+$ $\displaystyle h^{T}E)-\det(I+M)$ $\displaystyle=$
$\displaystyle\frac{e^{2z_{1}}}{k_{1}}\left[1-\frac{\bar{k}_{2}}{k_{2}}\frac{(k_{1}-k_{2})^{2}}{(k_{1}-\bar{k}_{2})^{2}(k_{2}-\bar{k}_{2})^{2}}e^{2(z_{2}+\bar{z}_{2})}\right]$
$\displaystyle+\frac{e^{2z_{2}}}{k_{2}}\left[1-\frac{\bar{k}_{1}}{k_{1}}\frac{(k_{1}-k_{2})^{2}}{(k_{1}-\bar{k}_{1})^{2}(k_{2}-\bar{k}_{1})^{2}}e^{2(z_{1}+\bar{z}_{1})}\right],$
and
$\displaystyle 2\det(I+\tilde{M})$
$\displaystyle-\det(I+\tilde{M}+\bar{h}^{T}EK)$ $\displaystyle=$
$\displaystyle
1-\frac{k_{1}}{\bar{k}_{1}}\frac{e^{2(z_{1}+\bar{z}_{1})}}{(k_{1}-\bar{k}_{1})^{2}}-\frac{k_{2}}{\bar{k}_{2}}\frac{e^{2(z_{2}+\bar{z}_{2})}}{(k_{2}-\bar{k}_{2})^{2}}-\frac{k_{2}}{\bar{k}_{1}}\frac{e^{2(\bar{z}_{1}+z_{2})}}{(k_{2}-\bar{k}_{1})^{2}}$
$\displaystyle-\frac{k_{1}}{\bar{k}_{2}}\frac{e^{2(z_{1}+\bar{z}_{2})}}{(k_{1}-\bar{k}_{2})^{2}}+\frac{k_{1}k_{2}}{\bar{k}_{1}\bar{k}_{2}}\frac{|k_{1}-k_{2}|^{4}}{\prod\limits_{j,l=1}^{2}(k_{j}-\bar{k}_{l})^{2}}e^{2(z_{1}+z_{2}+\bar{z}_{1}+\bar{z}_{2})}.$
To obtain the soliton solutions, we introduce new functions $\omega_{j}$
$e^{2\omega_{j}}=\frac{k_{1}-k_{2}}{(k_{1}-\bar{k}_{j})(k_{2}-\bar{k}_{j})},\quad\omega_{j}=w_{j}+i\phi_{j},\quad
j=1,2.$ (4.14)
Under this definition, equations (4.13) can be rewritten as
$\displaystyle\det(I+M)=$ $\displaystyle
4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\left\\{a\cosh
2\vartheta_{1}\cosh 2\vartheta_{2}+b\sinh 2\vartheta_{1}\sinh
2\vartheta_{2}\right.$ (4.15) $\displaystyle\quad\left.+2{\rm Im}k_{1}{\rm
Im}k_{2}\cos\rho\right\\}$ $\displaystyle\equiv$ $\displaystyle\Delta D_{2},$
$\displaystyle\det(I+M$ $\displaystyle+g^{T}E)-\det(I+M)$ (4.16)
$\displaystyle=$
$\displaystyle-4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\sqrt{a^{2}-b^{2}}\left[{\rm
Im}k_{1}e^{2i(\phi_{2}-\varphi_{1})}\sinh 2(\vartheta_{2}+i\phi_{2})\right.$
$\displaystyle\qquad+\left.{\rm Im}k_{2}e^{2i(\phi_{1}-\varphi_{2})}\sinh
2(\vartheta_{1}+i\phi_{1})\right]$ $\displaystyle\equiv\Delta\Omega_{2},$
$\displaystyle\det(I+M$ $\displaystyle+h^{T}E)-\det(I+M)$ (4.17)
$\displaystyle=$
$\displaystyle-4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\sqrt{a^{2}-b^{2}}\left[\frac{{\rm
Im}k_{1}}{k_{1}}e^{2i(\tilde{\phi}_{2}-\varphi_{1})}\sinh
2(\vartheta_{2}+i\tilde{\phi}_{2})\right.$
$\displaystyle\qquad+\left.\frac{{\rm
Im}k_{2}}{k_{2}}e^{2i(\tilde{\phi_{1}}-\varphi_{2})}\sinh
2(\vartheta_{1}+i\tilde{\phi}_{1})\right]$ $\displaystyle\equiv\Delta\Xi_{2},$
$\displaystyle 2\det(I+\tilde{M})-\det(I+\tilde{M}+\bar{h}^{T}gK)$ (4.18)
$\displaystyle=4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\left\\{e^{i(\arg
k_{1}+\arg k_{2})}[a\cosh(2\vartheta_{1}+i\arg
k_{1})\cosh(2\vartheta_{2}+i\arg k_{2})\right.$
$\displaystyle\qquad\left.b\sinh(2\vartheta_{1}+i\arg
k_{1})\sinh(2\vartheta_{2}+i\arg k_{2})]+2{\rm Im}k_{1}{\rm
Im}k_{2}\cosh(\varepsilon+i\varrho)\right\\}$
$\displaystyle\quad\equiv\Delta\Lambda_{2},$
where
$\displaystyle\vartheta_{j}=\theta_{j}+w_{j},\
\tilde{\phi}_{j}=\phi_{j}+\varpi_{j},\ \varpi_{j}=\arg k_{j},j=1,2$
$\displaystyle|k_{1}-k_{2}|^{2}=a+b,\ |k_{1}-\bar{k}_{2}|^{2}=a-b,$
$\displaystyle\ \rho=2(\varphi_{1}+\phi_{1}-\varphi_{2}-\phi_{2}),$
$\displaystyle\varrho=2(\varphi_{1}-\phi_{1}-\varphi_{2}+\phi_{2})-\varpi_{1}+\varpi_{2},$
$\displaystyle
e^{\varepsilon}=\frac{|k_{1}|}{|k_{2}|},\quad\Delta=4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\cosh
2\vartheta_{1}\cosh 2\vartheta_{2},$
and
$\displaystyle D_{2}=$ $\displaystyle a+b\tanh 2\vartheta_{1}\tanh
2\vartheta_{2}+2{\rm Im}k_{1}{\rm Im}k_{2}\cos\rho{\rm sech}2\vartheta_{1}{\rm
sech}2\vartheta_{2},$ (4.19) $\displaystyle\Omega_{2}=$
$\displaystyle-\sqrt{a^{2}-b^{2}}[{\rm
Im}k_{1}e^{2i(\phi_{2}-\varphi_{1})}{\rm sech}2\vartheta_{1}(\tanh
2\vartheta_{2}\cos 2\phi_{2}+i\sin 2\phi_{2})$ $\displaystyle\quad+{\rm
Im}k_{2}e^{2i(\phi_{1}-\varphi_{2})}{\rm sech}2\vartheta_{2}(\tanh
2\vartheta_{1}\cos 2\phi_{1}+i\sin 2\phi_{1})],$ $\displaystyle\Xi_{2}=$
$\displaystyle-\sqrt{a^{2}-b^{2}}[\frac{{\rm
Im}k_{1}}{k_{1}}e^{2i(\tilde{\phi}_{2}-\varphi_{1})}{\rm
sech}2\vartheta_{1}(\tanh 2\vartheta_{2}\cos 2\tilde{\phi}_{2}+i\sin
2\tilde{\phi}_{2})$ $\displaystyle\quad+\frac{{\rm
Im}k_{2}}{k_{2}}e^{2i(\tilde{\phi}_{1}-\varphi_{2})}{\rm
sech}2\vartheta_{2}(\tanh 2\vartheta_{1}\cos 2\tilde{\phi}_{1}+i\sin
2\tilde{\phi}_{1})],$ $\displaystyle\Lambda_{2}=$ $\displaystyle
e^{i(\varpi_{1}+\varpi_{2})}[a(\cos\varpi_{1}\cos\varpi_{2}-\tanh
2\vartheta_{1}\tanh 2\vartheta_{2}\sin\varpi_{1}\sin\varpi_{2})$
$\displaystyle\quad+b(\tanh 2\vartheta_{1}\tanh
2\vartheta_{2}\cos\varpi_{1}\cos\varpi_{2}-\sin\varpi_{1}\sin\varpi_{2})$
$\displaystyle\quad+ia(\tanh 2\vartheta_{2}\cos\varpi_{1}\sin\varpi_{2}+\tanh
2\vartheta_{1}\sin\varpi_{1}\cos\varpi_{2})$ $\displaystyle\quad+ib(\tanh
2\vartheta_{1}\cos\varpi_{1}\sin\varpi_{2}+\tanh
2\vartheta_{2}\sin\varpi_{1}\cos\varpi_{2})]$ $\displaystyle\qquad+2{\rm
Im}k_{1}{\rm Im}k_{2}\cosh(\varepsilon+i\varrho){\rm sech}2\vartheta_{1}{\rm
sech}2\vartheta_{2}.$
Hence, the two-soliton solution in semi-characteristic coordinates of the AB
equations (3.14) takes the form
$A=\frac{\Omega_{2}}{D_{2}},\quad
B=\frac{|\Lambda_{2}|^{2}-|\Xi_{2}|^{2}}{D_{2}^{2}}.$ (4.20)
The Figure 2 describes two-soliton waves of $|A|$ and $B$ traveling to left,
and Figure 3 (from left to right) shows collision of the two-soliton from
$|A|$ at $\tau_{1}=-3,\tau_{2}=0,\tau_{3}=3$; Figure 4 (from left to right)
shows collision of the two-soliton from $B$ at
$\tau_{1}=-4,\tau_{2}=-1,\tau_{3}=2$. (In the figures, for convenience, we
take variable $\xi$ as $x$, and $\tau$ as $t$.)
Figure 2:
$k_{1}=1.04+0.6i,k_{2}=2+0.4\mathrm{i},\kappa_{j}=0,\xi_{j}=0,j=1,2$.
Figure 3:
$k_{1}=1.04+0.6i,k_{2}=2+0.4\mathrm{i},\kappa_{j}=0,\xi_{j}=0,j=1,2$.
Figure 4:
$k_{1}=1.04+0.6i,k_{2}=2+0.4\mathrm{i},\kappa_{j}=0,\xi_{j}=0,j=1,2$.
Note that the representations (4.15)-(4.18) can be rewritten into another
forms
$\displaystyle\det(I+M)=\tilde{\Delta}\tilde{D_{2}},\quad\det(I+M+g^{T}E)-\det(I+M)=\tilde{\Delta}\tilde{\Omega}_{2}$
(4.21)
$\displaystyle\det(I+M+h^{T}E)-\det(I+M)=\tilde{\Delta}\tilde{\Xi}_{2},$
$\displaystyle
2\det(I+\tilde{M})-\det(I+\tilde{M}+\bar{h}^{T}gK)=\tilde{\Delta}\tilde{\Lambda}_{2},$
where
$\tilde{\Delta}=4\frac{e^{2(\vartheta_{1}+\vartheta_{2})}}{|k_{1}-k_{2}|^{2}}\sinh
2\vartheta_{1}\sinh 2\vartheta_{2},$
and
$\displaystyle\tilde{D_{2}}=$ $\displaystyle b+a\coth 2\vartheta_{1}\coth
2\vartheta_{2}+2{\rm Im}k_{1}{\rm Im}k_{2}\cos\rho{\rm csch}2\vartheta_{1}{\rm
csch}2\vartheta_{2},,$ (4.22) $\displaystyle\tilde{\Omega}_{2}=$
$\displaystyle-\sqrt{a^{2}-b^{2}}[{\rm
Im}k_{1}e^{2i(\phi_{2}-\varphi_{1})}{\rm csch}2\vartheta_{1}(\cos
2\phi_{2}+i\coth 2\vartheta_{2}\sin 2\phi_{2})$ $\displaystyle\quad+{\rm
Im}k_{2}e^{2i(\phi_{1}-\varphi_{2})}{\rm csch}2\vartheta_{2}(\cos
2\phi_{1}+i\coth 2\vartheta_{1}\sin 2\phi_{1})],$
$\displaystyle\tilde{\Xi}_{2}=$ $\displaystyle-\sqrt{a^{2}-b^{2}}[\frac{{\rm
Im}k_{1}}{k_{1}}e^{2i(\tilde{\phi}_{2}-\varphi_{1})}{\rm
csch}2\vartheta_{1}(\cos 2\tilde{\phi}_{2}+i\coth 2\vartheta_{2}\sin
2\tilde{\phi}_{2})$ $\displaystyle\quad+\frac{{\rm
Im}k_{2}}{k_{2}}e^{2i(\tilde{\phi}_{1}-\varphi_{2})}{\rm
csch}2\vartheta_{2}(\cos 2\tilde{\phi}_{1}+i\coth 2\vartheta_{1}\sin
2\tilde{\phi}_{1})],$ $\displaystyle\tilde{\Lambda}_{2}=$ $\displaystyle
e^{i(\varpi_{1}+\varpi_{2})}[a(\coth 2\vartheta_{1}\coth
2\vartheta_{2}\cos\varpi_{1}\cos\varpi_{2}-\sin\varpi_{1}\sin\varpi_{2})$
$\displaystyle\quad+b(\cos\varpi_{1}\cos\varpi_{2}-\coth 2\vartheta_{1}\coth
2\vartheta_{2}\sin\varpi_{1}\sin\varpi_{2})$ $\displaystyle\quad+ia(\coth
2\vartheta_{1}\cos\varpi_{1}\sin\varpi_{2}+\coth
2\vartheta_{2}\sin\varpi_{1}\cos\varpi_{2})$ $\displaystyle\quad+ib(\coth
2\vartheta_{2}\cos\varpi_{1}\sin\varpi_{2}+\coth
2\vartheta_{1}\sin\varpi_{1}\cos\varpi_{2})]$ $\displaystyle\qquad+2{\rm
Im}k_{1}{\rm Im}k_{2}\cosh(\varepsilon+i\varrho){\rm csch}2\vartheta_{1}{\rm
csch}2\vartheta_{2}.$
Hence, we have another form of two-soliton solution of the AB equations (3.14)
$A=\frac{\tilde{\Omega}_{2}}{\tilde{D_{2}}},\quad
B=\frac{|\tilde{\Lambda}_{2}|^{2}-|\tilde{\Xi}_{2}|^{2}}{\tilde{D_{2}}^{2}}.$
(4.23)
It is remarked that the functions $\vartheta_{j}$ can be rewritten as
$\vartheta_{j}={\rm Im}k_{j}(\xi+\frac{\tau}{|k_{j}|^{2}}+\xi_{j}),$ (4.24)
where $\xi_{j}$ is a certain constant. We note that the denominators $D_{2}$
and $\tilde{D}_{2}$ are not zero, because they are derived from the
determinant of an invertible matrix. It is remarked that the graphic of
solution (4.23) has the same form as in Figure 2.
Now, we will derive the $N$-soliton solutions of (3.14). By virtue of the
method of linear algebra, we know that
$\det(I+M)=1+\sum\limits_{\sigma=1}^{N}\sum\limits_{1\leq j_{1}\leq\cdots\leq
j_{\sigma}\leq N}M(j_{1},\cdots,j_{\sigma}),$ (4.25)
where $M(j_{1},\cdots,j_{\sigma})$ denotes the principal minor of $N\times N$
matrix $M$ obtained by taking all the elements of
$(j_{1},\cdots,j_{\sigma})$-th columns and rows. By using the Cauchy-Binet
formula, we can calculate the value of $M(j_{1},\cdots,j_{\sigma})$
$M(j_{1},\cdots,j_{\sigma})=\sum\limits_{1\leq r_{1}\leq\cdots\leq
r_{\sigma}\leq N}K\left(\begin{aligned} j_{1},&j_{2},&\cdots,&j_{\sigma}\\\
r_{1},&r_{2},&\cdots,&r_{\sigma}\end{aligned}\right)\bar{K}\left(\begin{aligned}
r_{1},&r_{2},&\cdots,&r_{\sigma}\\\
j_{1},&j_{2},&\cdots,&j_{\sigma}\end{aligned}\right),$ (4.26)
where $K\left(\begin{aligned} j_{1},&j_{2},&\cdots,&j_{\sigma}\\\
r_{1},&r_{2},&\cdots,&r_{\sigma}\end{aligned}\right)$ denotes the determinant
of the submatrix obtained by preserving the
$(j_{1},j_{2},\cdots,j_{\sigma})$-th rows and
$(r_{1},r_{2},\cdots,r_{\sigma})$-th columns of $K$;
$\bar{K}\left(\begin{aligned} \cdot\\\ \cdot\end{aligned}\right)$ denotes
similarly the determinant of the submatrix for $\bar{K}$. It is noted that $K$
is Cauchy type matrices, then
$M(j_{1},\cdots,j_{\sigma})=\sum\limits_{1\leq r_{1}\leq\cdots\leq
r_{\sigma}\leq
N}(-1)^{\sigma}\prod\limits_{l<l^{\prime},m<m^{\prime}}\frac{|k_{l}-k_{l^{\prime}}|^{2}|\bar{k}_{m^{\prime}}-\bar{k}_{m}|^{2}}{(k_{l}-\bar{k}_{m})^{2}}e^{2(z_{l}+\bar{z}_{m})},$
(4.27)
where
$m\in\\{r_{1},r_{2},\cdots,r_{\sigma}\\};l,l^{\prime}\in\\{j_{1},j_{2},\cdots,j_{\sigma}\\}$
and $\sigma=1,\cdots,N$. Hence, we obtain the explicit representation of
$\det(I+M)$ from (4.25) and (4.27). It is readily verified that
$\det(I+\tilde{M})=\det(I+M)$.
In the following, we will evaluate the numerator of the expressions in (4.10).
To this end, let
$C=M+g^{T}E=GH,$ (4.28)
where $G=(g^{T},K)=(G_{nm})$ and $H=\left(\begin{array}[]{c}E\\\
\bar{K}\end{array}\right)=(H_{mn})$, with
$n\in\\{1,\cdots,N\\},m\in\\{0,1,\cdots,N\\}$. Hence, $\det(I+C)$ takes the
same expansion as (4.25), where
$C(j_{1},\cdots,j_{\sigma})=\sum\limits_{0\leq r_{1}\leq\cdots\leq
r_{\sigma}\leq N}G\left(\begin{aligned} j_{1},&j_{2},&\cdots,&j_{\sigma}\\\
r_{1},&r_{2},&\cdots,&r_{\sigma}\end{aligned}\right)H\left(\begin{aligned}
r_{1},&r_{2},&\cdots,&r_{\sigma}\\\
j_{1},&j_{2},&\cdots,&j_{\sigma}\end{aligned}\right).$ (4.29)
Now, we split the summation on the right hand side of the above equation into
two parts, the first one is $r_{1}=0$, and the second one is $r_{1}\geq 1$. It
is noted that the second one is exactly equal to $M(j_{1},\cdots,j_{\sigma})$.
Thus, the numerator of the expression of $r$ in (4.10) takes the value
$\displaystyle\det(I+M+g^{T}E)-\det(I+M)$
$\displaystyle=\sum\limits_{\sigma=1}^{N}\sum\limits_{(1\leq
j_{1}\leq\cdots\leq j_{\sigma}\leq N)}\sum\limits_{(1\leq r_{2}\leq\cdots\leq
r_{\sigma}\leq N)}G\left(\begin{aligned} j_{1},&j_{2},&\cdots,&j_{\sigma}\\\
0,&r_{2},&\cdots,&r_{\sigma}\end{aligned}\right)H\left(\begin{aligned}
0,&r_{2},&\cdots,&r_{\sigma}\\\
j_{1},&j_{2},&\cdots,&j_{\sigma}\end{aligned}\right)$
$\displaystyle=\sum\limits_{\sigma=1}^{N}\sum\limits_{(1\leq
j_{1}\leq\cdots\leq j_{\sigma}\leq N)}\sum\limits_{(1\leq r_{2}\leq\cdots\leq
r_{\sigma}\leq
N)}(-1)^{\sigma-1}\prod\limits_{l<l^{\prime},m<m^{\prime}}\frac{|k_{l}-k_{l^{\prime}}|^{2}|\bar{k}_{m^{\prime}}-\bar{k}_{m}|^{2}}{(k_{l}-\bar{k}_{m})^{2}}e^{2(z_{l}+\bar{z}_{m})},$
where
$m,m^{\prime}\in\\{r_{2},\cdots,r_{\sigma}\\};l,l^{\prime}\in\\{j_{1},j_{2},\cdots,j_{\sigma}\\}.$
Similarly, for $\psi_{21}(0)$ in (4.10), we have
$\displaystyle\det(I$ $\displaystyle+M+h^{T}E)-\det(I+M)$ (4.30)
$\displaystyle=\sum\limits_{\sigma=1}^{N}\sum\limits_{(1\leq
j_{1}\leq\cdots\leq j_{\sigma}\leq N)}\sum\limits_{(1\leq r_{2}\leq\cdots\leq
r_{\sigma}\leq N)}$
$\displaystyle\quad\times(-1)^{\sigma-1}\prod\limits_{l<l^{\prime},m<m^{\prime}}\frac{\bar{k}_{m}}{k_{l}}\frac{|k_{l}-k_{l^{\prime}}|^{2}|\bar{k}_{m^{\prime}}-\bar{k}_{m}|^{2}}{(k_{l}-\bar{k}_{m})^{2}}e^{2(z_{l}+\bar{z}_{m})},$
where
$(m,m^{\prime}\in\\{r_{2},\cdots,r_{\sigma}\\};l,l^{\prime}\in\\{j_{1},j_{2},\cdots,j_{\sigma}\\}).$
While for $\psi_{22}(0)$, let
$\displaystyle\tilde{M}+\bar{h}^{T}EK=(\bar{h}^{T},\bar{K})\left(\begin{array}[]{c}EK\\\
K\end{array}\right),$
then
$\displaystyle\det(I+\tilde{M}+\bar{h}^{T}gK)-\det(I+\tilde{M})$ (4.31)
$\displaystyle=\sum\limits_{\sigma=1}^{N}\sum\limits_{(1\leq
j_{1}\leq\cdots\leq j_{\sigma}\leq N)}\sum\limits_{(1\leq r_{2}\leq\cdots\leq
r_{\sigma}\leq
N)}(-1)^{\sigma-1}\sum\limits_{r_{0}\in\hat{\sigma}}\prod\limits_{l,m}\frac{k_{r_{0}}-k_{m}}{k_{r_{0}}-\bar{k}_{l}}e^{2z_{r_{0}}}$
$\displaystyle\qquad\times\prod\limits_{l,m}\frac{k_{m}}{\bar{k}_{l}}\frac{e^{2(\bar{z}_{m}+z_{l})}}{(k_{m}-\bar{k}_{l})^{2}}\prod\limits_{l<l^{\prime},m<m^{\prime}}(\bar{k}_{l}-\bar{k}_{l^{\prime}})^{2}(k_{m}-k_{m^{\prime}})^{2}.$
where
$m,m^{\prime}\in\\{r_{2},\cdots,r_{\sigma}\\};l,l^{\prime}\in\\{j_{1},j_{2},\cdots,j_{\sigma}\\}$
and $\hat{\sigma}=\\{1,2,\cdots,N\\}\setminus\\{r_{2},\cdots,r_{\sigma}\\}$
denotes a subset of the set $\\{1,2,\cdots,N\\}$. It is remarked that the
$N$-soliton solution of the AB equations can be obtained from (4.5),(4.10) and
(4.25)-(4.30).
## 5 Asymptotic behaviors of the $N$-soliton solution
In this section, we discuss the asymptotic behaviors of the given $N$-solion
solution. To this end, we assume that
$1<|k_{1}|<|k_{2}|<\cdots<|k_{N}|,\quad{\rm Im}k_{j}>0.$
It is noted that
$g_{j}=e^{2iz_{j}}=e^{2\theta_{j}}e^{-2i\varphi},\quad\theta_{j}={\rm
Im}k_{j}(\xi-v_{j}\tau-\xi_{j}),$ (5.1)
where $v_{j}=-|k_{j}|^{-2}$ and $\xi_{j}$ is a certain real constant. Now the
region of the point $\xi=\xi_{j}+v_{j}\tau$ is denoted by $\Sigma_{j}$. Then,
as $\tau\rightarrow-\infty$, these regions are disjoint and distribute from
left to right as
$\Sigma_{N},~{}\Sigma_{N-1},~{}\cdots,~{}\Sigma_{1},$
in view of $v_{1}<v_{2}<\cdots<v_{N}$. In the region $\Sigma_{j}$, one may
find that
$\displaystyle\xi-\xi_{n}-v_{n}\tau\rightarrow+\infty,$ (5.2)
$\displaystyle|g_{n}|\rightarrow+\infty,\quad n>j;$
and
$\displaystyle\xi-\xi_{m}-v_{m}\tau\rightarrow-\infty,$ (5.3)
$\displaystyle|g_{m}|\rightarrow 0,\quad m<j.$
Thus, in the region $\Sigma_{j}$, as $\tau\rightarrow-\infty$, we find
$\displaystyle\det(I+M)\approx K\left(\begin{aligned} j+1,j+2,\cdots,N\\\
j+1,j+2,\cdots,N\end{aligned}\right)\bar{K}\left(\begin{aligned}
j+1,j+2,\cdots,N\\\ j+1,j+2,\cdots,N\end{aligned}\right)$ (5.4)
$\displaystyle\qquad+K\left(\begin{aligned} j,j+1,\cdots,N\\\
i,j+1,\cdots,N\end{aligned}\right)\bar{K}\left(\begin{aligned}
j,j+1,\cdots,N\\\ j,j+1,\cdots,N\end{aligned}\right)$
$\displaystyle=\left(1+\frac{e^{4\theta_{j}}}{|k_{j}-\bar{k}_{j}|^{2}}\prod\limits_{l=j+1}^{N}\frac{|k_{j}-k_{l}|^{4}}{|k_{j}-\bar{k}_{l}|^{4}}\right)\prod\limits_{j+1\leq
l<l^{\prime}\leq
N}\frac{e^{4\theta_{l}}}{|k_{l}-\bar{k}_{l}|^{2}}\frac{|k_{l}-k_{l^{\prime}}|^{4}}{|k_{l}-\bar{k}_{l^{\prime}}|^{4}},$
and
$\displaystyle\det(I$ $\displaystyle+M+g^{T}E)-\det(I+M)$ (5.5)
$\displaystyle\approx G\left(\begin{aligned} j,j+1,\cdots,N\\\
0,j+1,\cdots,N\end{aligned}\right)H\left(\begin{aligned} 0,j+1,\cdots,N\\\
j,j+1,\cdots,N\end{aligned}\right)$
$\displaystyle=e^{2\theta_{j}}e^{-2i\varphi_{j}}\prod\limits_{l=j+1}^{N}\frac{(k_{j}-k_{l})^{2}}{(k_{j}-\bar{k}_{l})^{2}}\prod\limits_{j+1\leq
l<l^{\prime}\leq
N}\frac{e^{4\theta_{l}}}{|k_{l}-\bar{k}_{l}|^{2}}\frac{|k_{l}-k_{l^{\prime}}|^{4}}{|k_{l}-\bar{k}_{l^{\prime}}|^{4}}.$
Equations (5.4) and (5.5) imply that the solution $A$ in $\Sigma_{j}$ has the
following asymptotic behavior
$A\approx-i{\rm Im}k_{j}e^{-2i(\varphi_{j}+\delta_{j}^{(-)})}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(-)}),\quad\tau\rightarrow-\infty,$ (5.6)
where $\delta_{j}^{(-)}$ and $\gamma_{j}^{(-)}$ are defined by the following
representation
$\frac{1}{\bar{k}_{j}-k_{j}}\prod\limits_{l=j+1}^{N}\frac{(\bar{k}_{j}-\bar{k}_{l})^{2}}{(\bar{k}_{j}-k_{l})^{2}}=e^{2(\gamma_{j}^{(-)}+i\delta_{j}^{(-)})}.$
(5.7)
Similarly, as $\tau\rightarrow+\infty$, the regions are distributed as
$\Sigma_{1},\cdots,\Sigma_{N}$. In this case, the leading term of $\det(I+M)$
in (5.4) will involve $\\{1,\cdots,j-1;1,\cdots,j-1,j\\}$, instead of
$\\{j+1,\cdots,N;j,j+1,\cdots,N\\}$. Also, in the leading term of (5.5) is now
$\\{1,2,\cdots,j-1,j;0,1,\cdots,j-1\\}$. Thus in the region $\Sigma_{j}$, we
find
$A\approx-i{\rm Im}k_{j}e^{-2i(\varphi_{j}+\delta_{j}^{(+)})}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(+)}),\quad\tau\rightarrow+\infty,$ (5.8)
and
$\frac{1}{\bar{k}_{j}-k_{j}}\prod\limits_{l=1}^{j-1}\frac{(\bar{k}_{j}-\bar{k}_{l})^{2}}{(\bar{k}_{j}-k_{l})^{2}}=e^{2(\gamma_{j}^{(+)}+i\delta_{j}^{(+)})}.$
(5.9)
From (5.6) to (5.9), one may find that the $N$-solitons with different
velocity split as $\tau\rightarrow-\infty$, after mutual collisions, split
again as $\tau\rightarrow+\infty$. In this solitary wave collisions, the form
and velocity of each solitary wave do not change, only the center and phase
change from $\delta_{j}^{(-)},\gamma_{j}^{(-)}$ to
$\delta_{j}^{(+)},\gamma_{j}^{(+)}$ for the $j$-th soliton.
Similar considerations apply to $\psi_{21}(0)$ and $\psi_{22}(0)$, we find
$\psi_{21}(0)\approx\left\\{\begin{aligned}
&e^{-i2(\varphi_{j}+\varrho_{j}^{(-)})}\sin\varpi_{j}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(-)}),&\tau\rightarrow-\infty,\\\
&e^{-i2(\varphi_{j}+\varrho_{j}^{(+)})}\sin\varpi_{j}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(+)}),&\tau\rightarrow+\infty,\\\
\end{aligned}\right.$ (5.10)
where
$\varrho_{j}^{(\pm)}=\delta_{j}^{(\pm)}+\rho_{j}^{(\pm)}+\frac{\varpi_{j}}{2},\varpi_{j}=\arg
k_{j},\rho_{j}^{(-)}=\sum\limits_{l=j+1}^{N}\varpi_{l},\rho_{j}^{(+)}=\sum\limits_{l=1}^{j-1}\varpi_{l}$,
and
$\psi_{22}(0)\approx\left\\{\begin{aligned}
&1-e^{-i2(\varphi_{j}+\tilde{\varrho}_{j}^{(-)})}e^{2(\theta_{j}+\mu_{j}^{(-)})}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(-)}),&\tau\rightarrow-\infty,\\\
&1-e^{-i2(\varphi_{j}+\tilde{\varrho}_{j}^{(+)})}e^{2(\theta_{j}+\mu_{j}^{(+)})}{\rm
sech}2(\theta_{j}+\gamma_{j}^{(+)}),&\tau\rightarrow+\infty,\\\
\end{aligned}\right.$ (5.11)
with $\varrho_{j}^{(\pm)}=\delta_{j}^{(\pm)}+\nu_{j}^{(\pm)}$ and
$\mu_{j}^{(\pm)},\nu_{j}^{(\pm)}$ defined by
$\displaystyle\frac{1}{2\bar{k}_{j}}\prod\limits_{l=j+1}^{N}\frac{\bar{k}_{j}-\bar{k}_{l}}{\bar{k}_{j}-k_{l}}\prod\limits_{l=j+1}^{N}\frac{k_{j}(k_{j}-\bar{k}_{l})(\bar{k}_{j}-\bar{k}_{l})}{\bar{k}_{j}(\bar{k}_{j}-k_{l})(k_{j}-k_{l})}=e^{2(\mu_{j}^{(-)}+i\nu_{j}^{(-)})},$
$\displaystyle\frac{1}{2\bar{k}_{j}}\prod\limits_{l=1}^{j-1}\frac{\bar{k}_{j}-\bar{k}_{l}}{\bar{k}_{j}-k_{l}}\prod\limits_{l=1}^{j-1}\frac{k_{j}(k_{j}-\bar{k}_{l})(\bar{k}_{j}-\bar{k}_{l})}{\bar{k}_{j}(\bar{k}_{j}-k_{l})(k_{j}-k_{l})}=e^{2(\mu_{j}^{(+)}+i\nu_{j}^{(+)})}.$
Hence, the asymptotic behavior of the solution $B$ can be characterized by
(4.5) and (5.10), (5.11), and the solitary wave collisions can be discussed
similarly.
## 6 Conclusions and remarks
It is remarked that the $\bar{\partial}$-approach is starting from the
dispersion relations of the AB system, which are introduced in linear
equations of the spectral transform matrix $R$ of the
$\bar{\partial}$-problem. By virtue of the $\bar{\partial}$-dressing method,
we obtain two linear spectral problems, which reduce to the Lax pair of the AB
system by using of the associated symmetry conditions. We note that these
symmetries about potential $Q$ and eigenfunction $\psi$ play a crucial role in
the determination of the form of spectral transform matrix $R$. The solutions
in closed form, including soliton solutions, are obtained by virtue of the
algebraic approach.
From section 5, we find that the envelope solitary wave is
$v_{R}\equiv-1/|k_{j}|^{2}$, and the velocity of the carrier wave is
$v_{I}\equiv 1/|k_{j}|^{2}$. Furthermore, the peculiarity of present solitons
is that the center and the phase difference of solitons are dependent on the
discrete spectrum, which is determined by the symmetry conditions of AB
system. In addition, the present solitons are stable for $|k_{j}|>1$ by the
results in [6, 7, 9], for the reason that the velocity of the envelope
solitary wave $v_{R}$ admits $-1<v_{R}<0$.
Acknowledgments
Project 11001250 and 10871182 were supported by the National Natural Science
Foundation of China.
## References
* [1] Pedlosky J 1970 Finite-amplitude baroclinic waves J. Atmos. Sci. 27 15-30
* [2] Pedlosky J 1972 Finite amplitude baroclinic wave packets J. Atmos. Sci. 29 680-6
* [3] Moroz I M 1981 Slowly modulated baroclinic waves in a three-layer model J. Atmos. Sci. 38 600-8
* [4] Moroz I M and Brindley J 1981 Evolution of baroclinic wave packets in a flow with continuous shear and stratification Proc. Roy. Soc. London A 377 397-404
* [5] Dodd R K, Eilbck J C, Gibbon J D and Morris H C 1982 Solitons and Nonlinear Wave Equations (New York Academic)
* [6] Gibbon J D, James I N and Moroz I 1979 An example of soliton behavior in a rotating baroclinic fluid Proc. Roy. Soc. London A 367 219-37
* [7] Gibbon J D and McGuiness M J 1981 Amplitude equations at the critical points of unstable dispersive physical systems Proc. Roy. Soc. A 337 185-219
* [8] Kamchatnov A M and Pavlov M V 1995 Periodic solutions and Whitham equations for the AB system J. Phys. A: Math, Gen. 28 3279-88
* [9] Tan B and Boyd J P, 2002 Envelope solitary waves and periodic waves in the AB equationss Stud. Appl. Math. 109 67-87
* [10] Guo R and Tian B 2012 Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation Commun. Nonlinear Sci. Numer. Simulat 17 3189-203
* [11] Zakharov V E and Manakov S V 1985 The construction of multidimensional nonlinear integrable systems and their solutions Func. Anal. Appl. 19 89-101
* [12] Bogdanov L V and Manakov S V 1988 The nonlocal $\bar{\partial}$-problem and (2+1)-dimensional soliton equations, J. Phys. A: Math. Gen. 21 L537-44
* [13] Beals R and Coifman R R 1989 Linear spectral problems, non-linear equations and the $\bar{\partial}$-method, Inverse Problems 5 87-130
* [14] Zakharov V E, 1990 On the Dressing Method, in Inverse Problems in Action (ed. Sabatier P S, Springer-Verlag, Berlin) 602-23
* [15] Santini P M 2003 Transformations and reductions of integrable nonlinear equations and the $\bar{\partial}$-problem Geometry And Integrability, Ed.Lionel Mason, Yavuz Nutku, (Cambridge University Press)
* [16] Konopelchenko B G 1993 Solitons in Multidimensions (World Scientific, Singapore)
* [17] Doktorov E V and Lebel S B 2007 A Dressing Method in Mathematical Physics Springer
* [18] Zhu J Y and Geng X G 2012 A hierarchy of coupled evolution equations with self-consistent sources and the dressing method, J. Phys. A: Math. Theor. 46 035204
* [19] Huang N N, 1996 Theory of Solitions and Method of Perturbations, (Shanghai Scientific and Technological Education Publishing House, SHANGHAI) (In Chinese).
|
arxiv-papers
| 2013-04-15T13:45:10 |
2024-09-04T02:49:44.345517
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junyi Zhu, Xianguo Geng",
"submitter": "Junyi Zhu",
"url": "https://arxiv.org/abs/1304.4096"
}
|
1304.4127
|
# Axion mechanism of Sun luminosity,
dark matter and extragalactic background light
V.D. Rusov1111Corresponding author: Vitaliy D. Rusov, E-mail: [email protected],
I.V. Sharf1, V.A. Tarasov1, M.V. Eingorn1,2, V.P. Smolyar1,
D.S. Vlasenko1, T.N. Zelentsova1, E.P. Linnik1, M.E. Beglaryan1
###### Abstract
We show the existence of the strong inverse correlation between the temporal
variations of the toroidal component of the magnetic field in the solar
tachocline (the bottom of the convective zone) and the Earth magnetic field
(the Y-component). The possibility that the hypothetical solar axions, which
can transform into photons in external electric or magnetic fields (the
inverse Primakoff effect), can be the instrument by which the magnetic field
of the Sun convective zone modulates the magnetic field of the Earth is
considered.
We propose the axion mechanism of Sun luminosity and ”solar dynamo –
geodynamo” connection, where the energy of one of the solar axion flux
components emitted in M1 transition in 57Fe nuclei is modulated at first by
the magnetic field of the solar tachocline zone (due to the inverse coherent
Primakoff effect) and after that is resonantly absorbed in the core of the
Earth, thereby playing the role of the energy modulator of the Earth magnetic
field. Within the framework of this mechanism estimations of the strength of
the axion coupling to a photon ($g_{a\gamma}=7.07\cdot 10^{-11}~{}GeV^{-1}$),
the axion-nucleon coupling ($g_{an}=3.20\cdot 10^{-7}$), the axion-electron
coupling ($g_{ae}=5.28\cdot 10^{-11}$) and the axion mass ($m_{a}=17~{}eV$)
have been obtained. It is also shown that the claimed axion parameters do not
contradict any known experimental and theoretical model-independent
limitations.
We consider the effect of dark matter in the form of 17 eV axions on the
extragalactic back-ground light. Our treatment is based on theoretical results
by Overduin and Wesson (Phys. Rep. 402 (2004) 267), who described the axion
halos as a luminous element of a pressureless perfect fluid in the standard
Friedman-Robertson-Walker universe basing on the assumption that axions are
clustered in Galactic halos with nonzero velocity dispersions. We find that
the spectral intensity $I_{\lambda}(\lambda_{0})$ of the extragalactic
background radiation from decaying axions ($m_{a}=17~{}eV$,
$c_{a\gamma\gamma}=0.02$) as a function of the observed wavelength
$\lambda_{0}$ is in good agreement with the known experimental data for the
near ultraviolet, optical and near infrared bands (1500-20000 Å). In the
framework of such approach it is shown that the present density parameter
$\Omega_{a}$ of thermal axions satisfies the inequality
$0.12\leqslant\Omega_{a}\leqslant 0.25$ and is comparable to the density
parameter of dark matter.
1Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, 1 Shevchenko ave., Odessa 65044,
Ukraine
2North Carolina Central University,
1801 Fayetteville st., Durham, North Carolina 27707, USA
###### Contents
1. 1 Introduction
2. 2 Magnetic field of solar tachocline zone and axion mechanism of the solar dynamo – geodynamo connection
1. 2.1 Implication from ”axion helioscope” technique (axion-photon interaction)
2. 2.2 Axion conversion in the Sun magnetic field and the plasma mass of photon
3. 2.3 Channeling of $\gamma$-quanta in periodical structure
4. 2.4 Channeling of $\gamma$-quanta along the magnetic flux tubes (waveguides) in Solar convective zone
5. 2.5 Invisible axions and Solar Equator effect
6. 2.6 Power required to maintain the Earth magnetic field and nuclear georeactor
3. 3 Axion mechanism of Sun luminosity and CUORE experiment
4. 4 Axion mechanism of Sun luminosity and other important experiments
1. 4.1 Axion coupling to a photon
2. 4.2 Axion coupling to an electron
5. 5 Axion dark matter and extragalactic background light
6. 6 Relic axion-like archion and cosmic infrared background
1. 6.1 Decaying axion and relic archion as two components of luminous dark matter
7. 7 Summary and Conclusion
1. 7.1 Axion mechanism of Sun luminosity
2. 7.2 Invisible axions and Solar Equator effect
3. 7.3 Axion mechanism of the solar dynamo – geodynamo connection
4. 7.4 Axion-like particle and extragalactic background light
5. 7.5 Plausible dark matter candidate: hadronic axion or axion-like arhion?
8. A Appendix I. Effect of $\gamma$-quanta channeling in periodic structures
1. A.1 Statement of a problem
2. A.2 Solution of the model problem
3. A.3 Determination of the absorption coefficient angular dependence
4. A.4 Channeling effect onset conditions
5. A.5 On the account of an absorption impact on X-ray intensity when channeling through the solar layered structures
9. B Appendix II. On a possibility of the layered structures formation in the solar convective zone on the basis of the magnetic flux tubes superlattices
1. B.1 Zonal jet streams
2. B.2 Some properties of the magnetic flux tubes in Sun convective zone
3. B.3 The self-confinement of force-free magnetic fields and energy conservation law.
4. B.4 Hydrostatic equilibrium and a sharp tube medium cooling effect
5. B.5 Ideal photon channeling (without absorption) conditions inside the magnetic flux tubes
## 1 Introduction
In the recent paper by Alessandria et al. the results of CUORE experimental
search for axions from the solar core from the 14.4 keV M1 ground-state
nuclear transition in 57Fe were presented [ref001]. The detection technique
employed a search for a peak in the energy spectrum at 14.4 keV when an axion
is absorbed by an electron via the axio-electric effect. The cross-section for
this process is proportional to the photoelectric absorption cross-section for
photons. In this pilot experiment 43.65 $kg\cdot d$ of data were analyzed
resulting in a lower bound on the Peccei-Quinn energy scale of $f_{a}\geqslant
0.76\cdot 10^{6}~{}GeV$ for the value for the flavor-singlet axial vector
matrix element of $S=0.55$; bounds are presented in the graph for values
$0.15\leqslant S\leqslant 0.55$ (Fig. 1). With the numbers quoted in the text,
the limit on $f_{a}$ translates into the axion mass limit $m_{a}<8~{}eV$,
significantly more stringent than in the recent results obtained with 57Fe
detectors [ref002, ref003] and by the Borexino experiment [ref004, ref005].
Figure 1: Expected rate in the axion region as a function of the $f_{a}$ axion
constant for different values of the nuclear $S$ parameter. The horizontal
line indicates the upper limit obtained in CUORE experiment ($f_{a}\sim
0.76\cdot 10^{6}~{}GeV$ for $S$ = 0.55) [ref001].
Despite the fine and elegant experimental implementation of the idea of
detecting the solar axions through the axio-electric effect in TeO2 bolometers
(CUORE detection technique [ref001]), a number of fundamental questions
regarding the appropriateness of some assumptions used in the problem
statement arises immediately.
The first one is rather obvious and lies in the following. Why is the 14.4 keV
M1 ground-state nuclear transition in solar 57Fe chosen as the main mechanism
of solar axions production in CUORE experiment, whereas there are other solar
axion production mechanisms discussed in scientific literature in detail which
also make their respective contributions into the 14.4 keV axions flux, such
as the so called Primakoff effect (e.g. [ref002, ref003]), bremsstrahlung and
the Compton process (e.g. [ref006, ref007]) (see Fig. 2)? From the analysis of
Fig. 2a, where the spectra of the processes under discussion normalized by the
corresponding constants are shown, it follows that this question is absolutely
nontrivial, and the answer depends on the knowledge of the values of all these
constants simultaneously. In fact, as it will be shown below, the real solar
axions spectra may look like the ones depicted on Fig. 2b. Therefore the
question asked above may be reformulated as follows: ”What must be the basic
physical criterion of the accepted problem statement justification, for
example, for the experiment on 14.4 keV solar axions detection?”
Figure 2: Spectra of solar axions at the ground produced by Primakov effect
($g_{a\gamma}$), M1 ground-state nuclear transition in solar 57Fe ($g_{an}$),
bremsstrahlung ($g_{ae}$, green line) and the Compton process ($g_{ae}$, blue
line) correspondingly: (a) $g_{a\gamma}=1$, $g_{an}=1$ ($S=0.5$), $g_{ae}=1$
($S=0.5$); (b) $g_{a\gamma}=7.07\cdot 10^{-11}$ GeV, $g_{an}=3.2\cdot
10^{-7}$, $g_{ae}=5.28\cdot 10^{-11}$. The data by Derbin et al. [ref007] were
used in order to plot the bremsstrahlung and Compton spectra. 14.4 keV axions
are marked with the pink line.
In our opinion, one of the most effective ways of establishing such a
criterion is the search for the models which would describe some
experimentally observed phenomena in the framework of standard or non-standard
solar physics using these properties of axions. If such a model is found, then
the pivotal estimates of e.g. the axion mass or the upper limits on the axion
coupling constants to photons ($g_{a\gamma}$), nucleons ($g_{an}$) and
electrons ($g_{ae}$), obtained in the framework of the given model, may play a
role of the main physical justification criterion for the accepted problem
statement in the 14.4 keV solar axions detection experiment.
In order to justify such a criterion for the future experiments (e.g. CAST,
CUORE, XMASS etc.) we decided to create a modified model of the axion
mechanism of Sun luminosity222It should be noted here that the axion mechanism
of Sun luminosity, which served as a basis for one of the first axion mass
estimates, was described for the first time in 1978 in the paper [ref008]. and
solar dynamo – geodynamo connection, which had been described in our previous
paper [ref009]. The basic idea of such a mechanism, which may be split into
two stages for convenience, is the following. At the first stage the solar
axions flux variations produced by the previously mentioned processes are
modulated by the solar tachocline magnetic field variations through the
inverse Primakoff effect [ref010]. At the second stage the ”modulated” solar
axion flux travels to the Earth, where its ”iron” component containing the
14.4 keV solar axions is resonantly absorbed in the iron-nickel core of the
Earth. If the energy of the axions supplied to the Earth core is enough for
generation and maintaining the geomagnetic field, then this process will
result in a persistent anticorrelation between the variations of the solar
magnetic field and the geomagnetic field (the Y-component)333Note that the
strong (inverse) correlation between the temporal variations of the magnetic
flux in the tachocline zone and the Earth magnetic field (the Y-component) are
observed only for experimental data obtained at that observatories where the
temporal variations of declination ($\delta D/\delta t$) or the closely
associated east component ($\delta Y/\delta t$) are directly proportional to
the westward drift of magnetic features [ref011]. This condition is very
important for understanding of the physical nature of the indicated above
correlation since it is known that it is only the motions of the top layers of
the Earth’s core that are responsible for most magnetic variations and, in
particular, for the westward drift of magnetic features seen on the Earth
surface on the decade time scale. Europe and Australia are geographical
places, where this condition is fulfilled (see Fig. 2 in [ref011]). For more
detailed discussion of this question see below (Section 2). (Fig. 3). This is
extremely important, because such effect of anticorrelation was discovered
recently [ref009], and has a strong experimental basis.
Figure 3: Time evolution of (a) the variations of the magnetic flux at the
bottom (the tachocline zone) of the Sun convective zone (see Fig. 7f in Ref.
[ref012]), (b) the geomagnetic field secular variations (the Y-component, nT /
year) measured at the Eskdalemuir observatory (England) [ref013]. Curves are
smoothed by the sliding intervals in 5 and 11 years. The pink area is a
prediction region.
It should be added that the solar axion flux modulated by the inverse
Primakoff effect in the magnetic field of the solar tachocline must not only
explain the value of solar luminosity, but also describe the solar photon
spectrum from the Active Sun, which in its turn must be equivalent to the data
from accumulated observations [ref014].
Thus, the main purpose of the present report was, on the one hand, to develop
a modified axion model of the Sun luminosity and solar dynamo – geodynamo
connection mechanism; and on the other hand, to obtain the consistent
estimates for the axion mass and the axion coupling constants to photons
($g_{a\gamma}$), nucleons ($g_{an}$) and electrons ($g_{ae}$) through the
comparison and generalization of the model results and the known experiments
including CAST, CUORE and XMASS.
## 2 Magnetic field of solar tachocline zone and axion mechanism of the solar
dynamo – geodynamo connection
It is known that in spite of a long history, the nature of the energy source
maintaining a convection in the liquid core of the Earth, or more exactly the
mechanism of the magnetohydrodynamic dynamo (MHD) generating the magnetic
field of the Earth, still has no clear and unambiguous physical interpretation
[ref015, ref016, ref017, ref018, ref019]. The problem is aggravated by the
fact that none of the candidates for an energy source of the Earth magnetic-
field [ref015] (secular cooling due to the heat transfer from the core to the
mantle, internal heating by radiogenic isotopes, e.g. 40K, latent heat due to
the inner core solidification, compositional buoyancy due to the ejection of
light elements at the inner core surface) can in principle explain one of the
most remarkable phenomena in solar-terrestrial physics, which consists in
strong (inverse) correlation between the temporal variations of the magnetic
flux in the tachocline zone (the bottom of the Sun convective zone) [ref012]
and the Earth magnetic field (the Y-component) [ref013] (Fig. 3).
Figure 4: (a) Geomagnetic filed Y-component variations at different
observatories [ref011]; (b) Secular variation of declination for 2005-2010
[ref020] (the closely associated east Y-component of the geomagnetic field).
(c) Direct impact of the westward drift on the geomagnetic field Y-component
in Europe (e.g. Lerwick) and Australia (e.g. Toolangui). Here $\delta
B_{Y}/\delta t$ is the magnetic variation seen at magnetic observatories; the
$\partial B_{Y}/\partial t$ term accounts for the effects of non-uniform and
north-south motions (as well as the effects of magnetic diffusion),
$U_{\varphi}$ is a magnitude of the westward drift as seen at the Earth
surface and $\partial B_{Y}/\partial\varphi$ is the longitudinal gradient of
the magnetic field as seen at the surface. (d) Time evolution of the
geomagnetic field secular variations (Y-component, $nT/year$), [ref013] and
the variation of the Earth rotation velocity [ref021] (green line). All curves
are smoothed by the sliding intervals in 5 and 11 years. The pink area is a
prediction region.
At the same time, supposing that the transversal (radial) surface area of
tachocline zone, through which the magnetic flux passes, is constant in the
first approximation, we can assume that magnetic flux variations also describe
the temporal variations of the magnetic field in the tachocline zone of the
Sun. In this sense, it is obvious that a future candidate for an energy source
of the Earth magnetic field must not only play the role of a natural trigger
of solar-terrestrial connection, but also directly generate the solar-
terrestrial magnetic correlation by its own participation.
At this point a question about the physical nature of such correlation arises.
Let us turn to the concept of the westward drift of the Earth magnetic field.
The nondipole part of the main field has a characteristic feature – it drifts
westward with time. The phenomenon of the westward drift was noticed as far
back as the XVII century. Each component of the geomagnetic filed has its own
drift speed with the average westward drift speed 0.2∘ per year. It means that
the nondipole field makes one complete revolution around the Earth rotation
axis in 1800 years. A higher rotation velocity of the mantle in comparison
with the outer core is supposed to be the physical mechanism of the westward
drift. Because of electromagnetic forces, the solid mantle of the Earth is
coupled to the core as a whole, and the outer part of the core therefore
travels westward relative to the mantle, carrying the minor features of the
field with it [ref022].
To explain the westward drift of magnetic features we have to distinguish
between the drift effect and other causes of magnetic variations [ref011]. The
magnetic secular variation can be written as:
$\frac{\delta B_{Y}}{\delta t}=\frac{\partial B_{Y}}{\partial
t}+U_{\varphi}\frac{\partial B_{Y}}{\partial\varphi}$ (1)
where $\delta B_{Y}/\delta t$ is the magnetic variation (the Y-component) seen
at magnetic observatories; the $\partial B_{Y}/\partial t$ term accounts for
effects of non-uniform and north-south motions (as well as the less important
effects of magnetic diffusion), $U_{\varphi}$ is the magnitude of the westward
drift as seen at the Earth surface and $\partial B_{Y}/\partial\varphi$ is the
longitudinal gradient of the magnetic field as seen at the surface.
It is known that most of the early magnetic observatories are located in
Europe (Fig. 4a) and fortunately, the $\partial B_{Y}/\partial\varphi$ term
here is large [ref011] and smoothly varying [ref011, ref023] (Fig. 4b).
Moreover, the term $\delta B_{Y}/\delta t$ is dominant and directly
proportional to the magnitude of the westward drift at the geographical places
where the term $\partial B_{Y}/\partial\varphi$ is large and smoothly varying
and the term $\partial B_{Y}/\partial t\to 0$ (for example, Europe and
Australia (see Fig. 4c)):
$\frac{\delta B_{Y}}{\delta t}\sim U_{\varphi}.$ (2)
Consequently, if the westward drift of the magnetic field on the core-mantle
boundary (Fig. 5) is caused by the core-mantle coupling, which induces the
corresponding westward drift of magnetic feature at the Earth surface (Fig.
5), then
$\frac{\delta B_{Y}}{\delta t}\sim U_{\varphi}\sim u_{\varphi}.$ (3)
where $u_{\varphi}$ is the westward drift of the magnetic field on the core-
mantle boundary.
Figure 5: Sketch of Earth magnetic field and westward drift of Earth magnetic
field on the core-mantle boundary ($u_{\varphi}$) and at the Earth surface
($U_{\varphi}$).
On the other hand, as far back as 1953 basing on the Bullard’s model [ref022]
analysis, Vestine [ref024] came to a conclusion that if the core-mantle
coupling mechanism exists, it should also cause a correlation between the
westward drift of the eccentric dipole (the magnetic centre in the Earth core)
and the variations of the Earth rotation velocity $\Omega$. As the further
analysis of the magnetic observations and the Earth rotation variations
[ref011] reveals, such kind of correlation indeed takes place (Fig. 4d), which
is an obvious sign of core-mantle coupling mechanism existence producing the
westward drift of magnetic features at the Earth surface.
It becomes clear therefore that a question about the physical nature of the
strong anticorrelation between the Solar magnetic field and the geomagnetic
field (the Y-component) variations (Fig. 3) comes to a question: ”How can the
Sun know about the processes in the Earth liquid core?”.
The answer is very simple: it governs them! And it governs the magnetic field
in the Earth core by means of some unknown interaction carrier! More precisely
speaking, the Sun governs the processes in the Earth liquid core through some
kind of interaction which must be transmitted by some unknown particles with
their flux controlled (modulated) by the Solar magnetic field.
According to our supposition, these particles may be the axions born primarily
inside the Sun core and may be converted into $\gamma$-quanta in the
tachocline magnetic field. This supposition is the leading idea of the present
paper.
The fact that the solar-terrestrial magnetic correlation has the undoubtedly
fundamental importance for evolution of all the geospheres is confirmed by
existence of stable and strong correlation between temporal variations of the
Earth magnetic field, the Earth angular velocity, the average ocean level and
the number of large earthquakes (with the magnitude M$\geqslant$7), which are
apparently driven by a common physical cause of unknown nature (see e.g.
[ref009]).
In this section we consider the hypothetical particles (solar axions) as the
main carriers of the solar-terrestrial connection, which by virtue of the
inverse coherent Primakoff effect can transform into photons in external
fluctuating electric or magnetic fields [ref010]. At the same time we ground
and develop the axion mechanism of solar dynamo – geodynamo connection, where
the energy of axions, which originate from the Sun core, is modulated at first
by the magnetic field of the solar tachocline zone (due to the inverse
coherent Primakoff effect), and after that is resonantly (57Fe solar axions)
absorbed in the iron core of the Earth, thereby playing the role of an energy
source and a modulator of the Earth magnetic field. Justification of the axion
mechanism of the Sun luminosity and solar dynamo – geodynamo connection is the
goal of the current section.
### 2.1 Implication from ”axion helioscope” technique (axion-photon
interaction)
As it is seen from the Earth, the most important astrophysical source of
axions is the core of the Sun. There, pseudoscalar particles like axions would
be continuously produced in the fluctuating electric and magnetic fields of
the plasma via their coupling to two photons (the Primakoff effect [ref010]).
After production the axions would freely stream out of the Sun without any
further interaction. The resulting differential solar axion flux on the Earth
would be [ref025, ref026]
$\frac{d\Phi_{a}}{dE}=6.02\cdot
10^{10}g_{10}^{2}E^{2.481}\exp\left(-\frac{E}{1.205}\right)~{}~{}cm^{-2}s^{-1}keV^{-1},$
(4)
where $E$ is in keV and $g_{10}=g_{a\gamma}/(10^{-10}~{}GeV^{-1}$).
The spectral energy of the axions (4) follows the thermal energy distribution
between 1 and 100 keV, which peaks at $\approx$3 keV and the average energy
$\langle E_{a}\rangle=4.2$ keV. To be able to compare the expected axion flux
in a specific energy range with available data, by integrating the spectrum
(4) over the energy range of 1 to 100 keV we find the solar axion flux at the
Earth to be
$\Phi_{a}\approx 3.75\cdot 10^{11}g_{10}^{2}~{}~{}cm^{-2}s^{-1}.$ (5)
In the case of the coherent Primakoff effect the number of photons leaving the
magnetic field towards the detector is determined by the probability
$P_{a\to\gamma}$ that an axion converts back to an ”observable” photon inside
the magnetic field [ref027]
$P_{a\rightarrow\gamma}=\left(\frac{Bg_{a\gamma}}{2}\right)^{2}\frac{1}{q^{2}+\Gamma^{2}/4}\left[1+e^{-\Gamma
L}-2e^{-\Gamma L/2}\cos(qL)\right],$ (6)
where $B$ is the strength of the transverse magnetic field along the axion
path, $L$ is the path length traveled by the axion in the magnetic region,
$l=2\pi/q$ is the oscillation length, $\Gamma=\lambda^{-1}$ is the absorption
coefficient for the X-rays in the medium, $\lambda$ is the absorption length
for the X-rays in the medium and the longitudinal momentum difference $q$
between the axion and the X-rays energy $E_{\gamma}=E_{a}$ is
$q=\frac{\left|m_{\gamma}^{2}-m_{a}^{2}\right|}{2E_{a}}$ (7)
with the effective photon mass
$m_{\gamma}\cong\sqrt{\frac{4\pi\alpha
n_{e}}{m_{e}}}=28.9\sqrt{\frac{Z}{A}\rho},$ (8)
where $m_{\gamma}=m_{a}$ is the axion mass, $\alpha$ is the fine-structure
constant, $n_{e}$ is the number of electrons in the medium, $m_{e}$ is the
electron mass, $Z$ is the atomic number of the buffer medium, $A$ is atomic
mass of the medium and its density $\rho$ in $g/cm^{3}$.
On the other hand, it is known that the axion is a neutral pseudoscalar
particle that was introduced in the particle theory to explain the absence of
CP violation in strong interactions [ref028, ref029, ref030]. The most natural
solution to the CP-violation problem was obtained by introducing a new chiral
symmetry, known as Peccei-Quinn (PQ) symmetry [ref001], the spontaneous
breakdown of which at the energy $f_{a}$ fully compensates the CP-nonivariant
term in the QCD Lagrangian and leads to the appearance of the axion [ref029,
ref030]. The axion is not massless because the chiral U(1) PQ-symmetry is
anomalous. As a result, the axion gets a mass of the order [ref031]
$m_{a}\sim\frac{\Lambda_{QCD}^{2}}{f_{a}},$ (9)
where $\Lambda_{QCD}$ is the confining QCD-scale and $f_{a}$ is the energy
scale associated with the breakdown of the U(1) PQ symmetry.
At the same time it is necessary to mention the axion mass estimates obtained
in the framework of the so-called invisible axion models (KSVZ [ref032,
ref032a] and DFSZ [ref033, ref033a]), which restrict the allowed range for
$f_{a}$, or equivalently the range for the axion mass
$m_{a}=6\cdot\frac{10^{6}~{}GeV}{f_{a}}~{}~{}eV,$ (10)
### 2.2 Axion conversion in the Sun magnetic field and the plasma mass of
photon
Let us consider the modulation of the axion flux emerging from the Sun core
and passing through the solar tachocline region (ST) located at the base of
the solar convective zone (Fig. 6c,d). As is known [ref014, ref034], the
equatorial thickness of ST, where the toroidal magnetic field $B\sim 10\div
50$ T dominates [ref034, ref035, ref036], attains $L_{ST}\sim 0.039R_{S}$
(where $R_{S}=6.96\cdot 10^{8}$ m [ref037] is the Sun radius). At the same
time the values of pressure, temperature and density for the ST are
$P_{ST}\sim 6.0\cdot 10^{12}$ Pa, $T_{ST}\sim 2.0\cdot 10^{6}$ K and $\rho\sim
0.2~{}g\cdot cm^{-3}$, respectively.
Figure 6: Examples of simulation of the periodic alternation of layers (the
zonal flow) in the convective structures of the Earth outer core (a, b
[ref038]) and the convective zone of the Sun (c, d). a) View from the north.
Isosurfaces of the axial vorticity, $\omega_{z}$, are shown in red
($\omega_{z}=0.4$) and green ($\omega_{z}=-0.4$) to illustrate the sheet
plumes. Each line forms a closed ring, indicating that the flow is nearly
purely westward; b) Same as a), but viewed from a different angle; c) The
section $AA^{\prime}$ along one of the alternate convective layers of the Sun.
In the tachocline axions are converted into $\gamma$-quanta (see the inset in
d)) channeling in the green area. In the photosphere $\gamma$-quanta are
scattering due to the Compton effect. d) Same as c), but viewed from a
different angle (see the section $AA^{\prime}$ in c)). Alternate layers in the
convective zone, where layers in which the channeling takes place are shown in
green, are also presented. Blue points (in the upper convective zone) and
crosses (in the tachocline) show the direction of output and input of the
magnetic field in the convective zone of the Sun.
To estimate the plasma mass of a photon $m_{\gamma}$ in the hydrogen-helium
medium of ST it is possible, without loss of generality, to use the modified
Eq. (8) in the form [ref025]
$m_{\gamma}(eV)=m_{a}\cong\sqrt{0.02\frac{P_{ST}(mbar)}{T_{ST}(K)}}\cong
25~{}~{}eV,$ (11)
where we use the corresponding parameters $P_{ST}\sim 6.0\cdot 10^{12}$ Pa and
$T_{ST}\sim 2.0\cdot 10^{6}$ K for the hydrogen-helium medium of ST obtained
by Bahcall & Pinsonneault for the standard model of the Sun [ref079].
Thus, the axion mass in the standard model of the Sun is $\sim$25 eV. However,
it will be shown below that in the framework of the axion mechanism of Sun
luminosity the axion mass will be different, since the total energy balance of
the Sun is not violated, but indicates a substantial change in radiation
transport through the radiative zone and the convective zone with respect to
the standard model of the Sun. The energy portion of the axion-independent
radiation transport is rather small here and equals to $\sim
0.015\Lambda_{Sun}$ (see (19)). Since the total energy balance of the Sun is
not violated in the axion model, one may suppose that the basic parameters of
the solar core – the region of the energy generation – remain approximately
the same in both the standard and the ”axion” models. Meanwhile, the
thermodynamic parameters (temperature, pressure, plasma density, electron
density etc.) outside the solar core (between the core and photosphere) are
substantially smaller in the axion model as compared to the corresponding
parameters in the standard model of the Sun.
It should be noted here that the calculation of these parameters for the axion
model is a rather nontrivial task, which, because of its complexity, will be
performed in a separate publication. For this reason from now on let us use
the ”experimental” value of the axion mass found during the extragalactic
background light investigation (see Section 5).
$m_{a}\sim 17~{}~{}eV.$ (12)
Taking into account (11) and (10) it is easy to derive the value of the energy
$f_{a}$:
$f_{a}\cong 0.353\cdot 10^{6}~{}~{}GeV$ (13)
Now we make an important assumption that the axion mass is equal to the plasma
mass of a photon, i.e., $m_{\gamma}=m_{a}\sim 17$ eV. It is obvious, that by
virtue of Eq. (7) $q\to 0$, whence it follows that the oscillation length $l$
becomes an infinite quantity, i.e. $l=2\pi/q\to\infty$. However, taking into
account that in this case the absorption length $\lambda$ is about 0.1 m
[ref014], we have $\Gamma L_{ST}\to\infty$. This means that according to Eq.
(6), the intensity of expected conversion of axions into $\gamma$-quanta is
practically equal to zero in this case.
At the same time, there is a reason to believe (see [ref014] and Refs.
therein) that the conversion of axions into $\gamma$-quanta indeed takes
place, and strangely enough, this process goes on quite effectively. For
example, the reconstructed solar photon spectrum below 10 keV from the Active
Sun (Fig. 7b) is well described by the sum of secondary Compton spectra
obtained e.g. by the simulation of $\gamma$-quanta passage (regenerated from
the solar axion spectrum in the tachocline zone of the Sun (Fig. 7b)) through
the areas of the solar photosphere of different thickness but equal density,
layers with the thickness of $64~{}g/cm^{2}$ and $16~{}g/cm^{2}$.
Figure 7: Reconstructed solar photon spectrum below 10 keV from the Active
(flaring) Sun (the black line) from accumulated observations [ref039] (adapted
from [ref014]). The dashed line is the converted solar axion spectrum. Two
degraded spectra due to multiple Compton scattering are also shown for column
densities above the initial conversion place of $64~{}g/cm^{2}$,
$16~{}g/cm^{2}$. The pink dotted line represents the initial Primakoff axion
spectrum. Note that the Geant4 code photon threshold is at 1 keV and therefore
the turndown around $\sim$1 keV is an artifact.
In other words, despite the fact that the coherent axion-photon conversion by
the Primakoff effect is impossible due to the small absorption length for
$\gamma$-quanta ($l\gg 1$) in the medium (see Eq. (6)
$\Gamma=1/\lambda\to\infty$), there is a good agreement between the relative
theoretical $\gamma$-quantum spectra generated by solar axions and
experimental photon energy spectra detected close to the Sun surface in the
period of its active phase (see Fig. 7). The additional account taken of the
bremsstrahlung in the photosphere will surely enhance the quality of the
theoretical description of the experimenal solar photon spectrum
substantially.
At the same time it is necessary to note that attempts to match the absolute
values of these spectra did not succeed so far [ref014]. It was mainly
associated with the absence of a wish to make efforts, since it was absolutely
unknown how can the $\gamma$-quanta spectra generated by solar axions in the
tachocline be transported in a ”virgin”, i.e. unchanged, form through the
convective zone up to the Solar photosphere.
To overcome the problem of the small absorption length for $\gamma$-quanta and
to reach a resonance in Eq. (6) it is necessary for the refractive gas, in
which the axion-photon oscillation is studied, to have a zero refractive index
[ref040]. It appears that in order to satisfy this condition it is not
necessary to use the so-called metamaterials [ref041] with the negative
permittivity ($\varepsilon$) and magnetic permeability ($\mu$) or results of
the Pendry superlens theory [ref042], which are not practically realized in
nature444Though it should be noted that the metamaterial technology is
frequently used nowadays for laboratory simulations of some celestial
mechanics and cosmology phenomena [ref043, ref044, ref045, ref046]. And the
”…”artificial atoms” used as building blocks in metamaterial design offer much
more freedom in constructing analogues of various exotic spacetime metrics,
such as black holes, wormholes, spinning cosmic strings, and even the metric
of Big Bang itself. Explosive development of this field promises new insights
into the fabric of spacetime, which cannot be gleaned from any other
terrestrial experiments”([ref047] and Refs. therein).. Taking into account the
known difficulties [ref048] induced by the so-called problem of
electromagnetically induced transparency for X-rays and the recent significant
advances in this field [ref049, ref050], let us consider two (possibly
related) alternative ways of solving the problem of ”unperturbed”
$\gamma$-quanta spectra transfer through the Solar convective zone.
### 2.3 Channeling of $\gamma$-quanta in periodical structure
We can use the results from papers [ref051, ref052], where the possibility of
the electromagnetic X-radiation in a microwave range channeling in a multi-
layered metal-dielectric structure is theoretically and experimentally shown.
As it is stated in Appendix A in detail, the essence of the electromagnetic
X-ray channeling in long-period media lies in a fact that the rays are
reflected from the layers of higher electron density when propagating at small
angles to these layers. It leads to a non-uniform intensity distribution over
the cross-sectional plane because of the rays concentration within the
”channels” – the layers with lower electron density. It decreases the
absorption substantially and makes it possible for the rays to penetrate much
deeper into the sample along the layers than in the case of an arbitrary angle
of arrival.
According to [ref051], the intensity $J(x)$ of the photons (see Fig. A.2 and
(A.26)) passed through a sample of a thickness $x$, may be written in the form
$J(x)=J_{0}\exp(-\sigma
x)=J_{0}\exp\left(-\frac{\chi_{0}}{\cos\alpha}x\right)\cdot
Q(\alpha,y_{0},x),$ (14)
where
$Q(\alpha,y_{0},x)=\begin{cases}\exp\left[-\frac{\chi_{0}}{\cos\alpha}\beta^{2}x\left(1-\frac{E(q^{-1})}{K(q^{-1})}\right)\right]&at~{}~{}q>1,\\\
\exp\left[-\frac{\chi_{0}}{q^{2}\cos\alpha}\beta^{2}x\left(1-\frac{E(q)}{K(q)}\right)\right]&at~{}~{}q<1.\end{cases}$
(15)
with the same notation used in expressions (A.26)-(A.27).
Although we give a complete analysis of the Eqs. (14) and (15) in A, let us
make a short remark regarding the physical nature of these equations. Here the
multiplier $J_{0}\exp(-\chi x/\cos\alpha$) in (14) corresponds to the case of
$\gamma$-quanta propagation through a homogeneous medium with the electron
density $N_{e}$ and the absorption coefficient $\chi_{0}$. The additional
multiplier $Q(\alpha,y_{0},x)$ characterizes the influence of the medium
layering.
As is shown in A, the condition $Q(\alpha,y_{0},x)\to 1$ is theoretically
feasible for a majority of the multilayer metal-dielectric structures [ref051,
ref053], which are an effective emulator of a plasma medium (Fig. 6). This
condition is obviously necessary, but not sufficient. The layers with
ultralow, if not with ”quasi-zero” density, are also required for the ideal
photon channeling. Such layers suppress the photon absorption processes almost
completely, i.e. minimize the effect of the multiplier
$J_{0}\exp(-\chi_{0}x/\cos\alpha$) in (14).
Surprisingly enough, it turns out that such long-period (in terms of density)
media with one of the two alternating media having almost zero density can
take place, and not only in plasmas in general, but straight in the convective
zone of the Sun. Here we generally mean the so-called magnetic flux tubes, the
properties of which are examined below (see Appendix B for details).
### 2.4 Channeling of $\gamma$-quanta along the magnetic flux tubes
(waveguides) in Solar convective zone
The idea of the energy flow channeled along a fanning magnetic field has been
suggested for the first time by Hoyle [ref054] as an explanation for darkness
of umbra of sunspots. It was incorporated in a simple sunspot model by Chitre
[ref055]. Zwaan [ref056] extended this suggestion to smaller flux tubes to
explain the dark pores and the bright faculae as well. Summarizing the
research of the convective zone magnetic fields in the form of the isolated
flux tubes, Spruit and Roberts [ref057] suggested a simple mathematical model
for the behavior of thin magnetic flux tubes, dealing with the nature of the
solar cycle, the sunspot structure, the origin of spicules and the source of
mechanical heating in the solar atmosphere. In this model, the so-called thin
tube approximation is used (see [ref057] and Refs. therein), i.e. the field is
conceived to exist in the form of slender bundles of field lines (flux tubes)
embedded in a field-free fluid. Mechanical equilibrium between the tube and
its surrounding is ensured by the reduction of the gas pressure inside the
tube, which compensates the force exerted by the magnetic field. In our
opinion, this is exactly the kind of mechanism Parker [ref058] was thinking
about when he wrote about the problem of flux emergence: ”Once the field has
been amplified by the dynamo, it needs to be released into the convection zone
by some mechanism, where it can be transported to the surface by magnetic
buoyancy” [ref059].
In order to understand magnetic buoyancy, let us consider an isolated
horizontal flux tube in pressure equilibrium with its non-magnetic
surroundings, so that
$p_{int}+\frac{B^{2}}{2\mu_{0}}=p_{ext},$ (16)
where $p_{int}$ and $p_{ext}$ are the internal and external gas pressures
respectively and $\mu_{0}$ is the magnetic permeability of the medium, $B$
denotes the uniform field strength in the flux tube. If the internal and
external temperatures are equal so that $T_{int}=T_{ext}$ (thermal
equilibrium), then since $p_{int}<p_{ext}$, the gas in the tube is less dense
than its surrounding ($\rho_{int}<\rho_{ext}$), implying that the tube will
rise under the influence of gravity.
In spite of the obvious, though turned out to be surmountable, difficulties of
expression (18) application to the real problems, it was shown (see [ref057]
and Refs. therein) that strong buoyancy forces act in magnetic flux tubes of
the required field strength (104-105 G [ref060]). Under their influence tubes
either float to the surface as a whole (e.g. Fig.1 in [ref061]) or they form
loops of which the tops break through the surface (e.g. Fig.1 in [ref056]) and
lower parts descend to the bottom of the convective zone, i.e. to the
overshoot tachocline zone. The convective zone, being unstable, enhances this
process [ref062, ref063]. Small tubes take longer to erupt through the surface
because they feel stronger drag forces. It is interesting to note here that
the phenomenon of the drag force which raises the magnetic flux tubes to the
convective surface with the speeds about 0.3-0.6 km/s was discovered in direct
experiments using the method of time-distance helioseismology [ref064].
Detailed calculations of the process [ref065] show that even a tube with the
size of a very small spot, if located within the convective zone, will erupt
in less than two years. Yet, according to [ref065], the horizontal fields are
needed in the overshoot tachocline zone, which survive for about 11 yr, in
order to produce an activity cycle.
Figure 8: (a) Vertical cut through an active region illustrating the
connection between a sunspot at the surface and its origins in the toroidal
field layer at the base of the convection zone. Horizontal fields are stored
at the base of the convection zone (the overshoot tachocline zone) during the
cycle. Active regions form from sections brought up by buoyancy (one is shown
in the process of rising). After the eruption through the solar surface a
nearly potential field is set up in the atmosphere (broken lines), connecting
to the base of the convective zone via almost vertical flux tube. Hypothetical
small scale structure of a sunspot is shown in the inset (adopted from Spruit
[ref066] and Spruit and Roberts [ref057]).
(b) Detection of emerging sunspot regions in the solar interior [ref064].
Acoustic ray paths with lower turning points between 42 and 75 Mm (1 Mm=1000
km) are crossing the region of the emerging flux. For simplicity, only four
out of a total of 31 ray paths used in this study (the time-distance
helioseismology experiment) are shown here. Adopted from [ref064].
(c) Emerging and anchoring of stable flux tubes in the overshoot tachocline
zone, and its time-evolution in the convective zone. Adopted from [ref067].
(d) Vector magnetogram of the white light image of a sunspot (taken with SOT
on a board of the Hinode satellite – see inset) showing the direction of the
magnetic field and its strength (the length of the bar) in red. The movie
shows the evolution in the photospheric fields that has led to an X class
flare in the lower part of the active region. Adopted from [ref068].
A simplified scenario of magnetic flux tubes (MFT) birth and space-time
evolution (Fig. 8a) may be presented as follows. MFT is born in the overshoot
tachocline zone (Fig. 8d) and rises up to the convective zone surface without
separation from the tachocline (the anchoring effect), where it forms the
sunspot (Fig. 8b) or other kinds of active solar regions when intersecting the
photosphere. There are more fine details of MFT physics expounded in overviews
by Hassan [ref059] and Fisher [ref061], where certain fundamental questions,
which need to be addressed to understand the basic nature of magnetic
activity, are discussed in detail: How is the magnetic field generated,
maintained and dispersed? What are its properties such as structure, strength,
geometry? What are the dynamical processes associated with magnetic fields?
What role do magnetic fields play in energy transport?
Dwelling on the last extremely important question associated with the energy
transport, let us note that it is known that the thin magnetic flux tubes can
support longitudinal (also called sausage), transverse (also called kink),
torsional (also called torsional Alfvén), and fluting modes (e.g. [ref069,
ref070, ref071, ref072, ref073]); for the tube modes supported by wide
magnetic flux tubes see Roberts and Ulmschneider [ref072]. Focusing on the
longitudinal tube waves known to be an important heating agent of solar
magnetic regions, it is necessary to mention the recent papers by Fawzy
[ref075], which showed that the longitudinal flux tube waves are identified as
insufficient to heat the solar transition region and corona in agreement with
previous studies [ref076].
In other words, the problem of generation (the source) and transport of energy
by magnetic flux tubes remains unsolved in spite of its key role in physics of
various types of solar active regions.
Interestingly, this problem may be solved in the natural way in the framework
of the ”axion” model of the Sun. As it is shown in Appendix B, the inner
pressure, temperature and matter density decrease rapidly in a magnetic tube
”growing” between the tachocline and the photosphere. The analysis of these
parameters evolution within the equation of the growing magnetic flux tube
medium state not only gives the ultralow values for them, but also leads to
the so-called hydrostatic condition of an ideal (without absorption)
$\gamma$-quanta channeling inside the thin magnetic flux tubes
$p_{ext}\simeq\frac{|\vec{B}|^{2}}{2\mu_{0}},$ (17)
which is well satisfied for the ”axion” model of the Sun, according to
estimations in Appendix B. It means that such thin magnetic flux tubes are the
ideal $\gamma$-quanta waveguides, which reveal the essence of the unique
energy transport mechanism between the tachocline and the photosphere.
As a matter of fact, the phenomenon of $\gamma$-quanta channeling along the
magnetic flux tubes not only makes it possible to solve a problem of the
energy transport to the photosphere, but may also be a basis for solving other
important and critical problems in solar physics. If we assume that the
vertically oriented thin magnetic flux tubes play the role of waveguides for
$\gamma$-quanta produced in the tachocline via the axion mechanism of Sun
luminosity, virtually all known anomalies of experimental data interpretation
in physics of active solar regions, helioseismology and solar neutrino are
withdrawn. Since this assumption needs to be substantiated, let us describe
our phenomenology, consequences and experimental proofs of this hypothesis
below in short.
First of all, if one takes into account the sufficiently strong magnetic field
in the balance equation of (16) type, it becomes clear that the vertically
oriented thin magnetic flux tubes may serve as the X-ray waveguides for the
radiation originating from the overshoot tachocline zone because of the high
magnetic pressure. Naturally, in this case the X-ray spectrum coincides with
the observed Solar X-ray spectrum. All these facts, i.e. the strong magnetic
field $\sim$200-400T (Fig. 9), X-rays (Fig. 8b) and their spectrum (e.g. Fig.
7 in [ref068]) in active solar regions, are in good agreement with
observational data.
Figure 9: (a) Growth rates for magnetic shear instabilities are plotted as a
function of the initial latitude (vertical axes) and the field strength
(horizontal axes) of a toroidal band. Shaded areas indicate instability in
0.1-100T band (gray) and 200-400T band (green). Contour lines represent $m=1$
and $m=2$ symmetric (S) and antisymmetric (A) modes as indicated. The non-
dimensional model is normalized in such a way that the growth rate of 0.01
corresponds to an e-folding growth time of 1 year. The parameter $s$ is the
fractional angular velocity contrast between equator and pole and the reduced
gravity $G$ (adopted from [ref077]). In addition, a hidden part of the
”latitude – magnetic field in overshoot tachocline zone” dependence, which was
missing on the original plot (Fig.11 in [ref077]), is plotted to the right of
the dashed line.
(b) Solar images at photon energies from 250 eV up to a few keV from the
Japanese X-ray telescope Yohkoh (1991-2001) (adapted from [ref014]). The
following shows solar X-ray activity during the last maximum of the 11-year
solar cycle.
Second, it clears up a way to the solution of the known problem associated
with the over-shoot tachocline anomaly (Fig. 10) which arises when
interpreting the helioseismology and solar abundances data. And here is why.
It is known [ref078] that the problem comes from the attempts to improve
agreement between solar models with low heavy-element abundances and seismic
inference. The low-metallicity models that have the least disagreement with
seismic data require changing all input physics to stellar models beyond their
acceptable ranges. Let us consider the way it happens in the framework of a
solar model built upon the axion mechanism of Sun luminosity.
Figure 10: The relative sound-speed (the panel a) and density differences (the
panel b) between the Sun and the model constructed with the AGS05 abundances
[ref-bahcall2005]. For comparison we also show the results for the model
constructed with the GS98 abundances. MDI 360 day data have been used for the
inversions. The overshoot tachocline anomaly is highlighted with green. Note:
GS98 is a solar model with the solar heavy-element mixture $Z/X=0.0245$; AGS05
is a solar model with low heavy-element mixture $Z/X=0.0122$. Adopted from
[ref078].
Since solar luminosity is determined by the $\gamma$-quanta born in the
tachocline in the framework of the axion mechanism, it is clear that an old
heat flux transport mechanism (from the radiative zone to the overshoot) by
radiation, used in the standard model of the Sun, should be highly depressed
because the major part of the radiation is converted into axions in the core
of the Sun [ref025] and does not get into the radiative interior. It is easy
to see from the traditional statement of the problem involving helioseismology
and solar abundances as described by Basu [ref078] that this is one of the
main and fundamental differences from the standard model of the Sun: ”The most
easily detectable effect of the reduction of heavy-element abundances is a
change in the position of the base of the convection zone. The temperature
gradient in the radiative interior is determined by opacity, and hence, its
structure is affected by the heavy-element abundances. The base of the
convection zone occurs at a point where opacity is just small enough to allow
the entire heat flux to be transported by radiation, and thus the location of
this point depends on the abundance of those heavy elements that are the
predominant sources of opacity in that region. If these abundances are
reduced, opacity reduces, and the depth of the convection zone also reduces.
Since the depth of the convection zone has been measured very accurately, it
is the most sensitive indicator of opacity or heavy-element abundances”.
The axion mechanism of Sun luminosity implies that because of a virtually
complete transparency of the magnetic flux tubes for $\gamma$-radiation there
are no reasons for moving the location of the center of the tachocline ”by
hand”, since the radiative opacity determined by the effect of heavy-element
abundance loses its impact on the location of this point and, consequently,
its significance, because of almost total suppression of the radiative heat
flux transport mechanism itself in this case. In other words, the effect of
absolute magnetic flux tubes transparency for the $\gamma$-radiation is
dominant in the overshoot tachocline zone and levels the influence of
radiative opacity. As a consequence, the value of the temperature gradient in
the radiative interior is almost entirely free from the strict limit
introduced by opacity which is still affected by the heavy element abundances,
but not as dramatically as it is in several other solar models – standard and
nonstandard – that have been published recently [ref078]. The latter opens up
a possibility to build a new standard solar model on the basis of the axion
mechanism of Sun luminosity which may become a key to the solar abundance
problem solution.
Third, let us consider the axion mechanism of Sun luminosity compatibility
with the nuclear energy generation pathways in the solar core and solar
neutrino fluxes generation.
The axion mechanism of Sun luminosity is compatible with the standard nuclear
energy generation pathways scheme and does not disturb the known values for
solar neutrino fluxes, since the ”invisible” axion losses almost do not change
the Sun energy balance in our model (see Section 2.5 below), and therefore do
not introduce any problems related to energy-producing regions (i.e. the solar
core).
It actually means that introducing the axion mechanism of Sun luminosity in
the framework of the standard model of the Sun leads to such value of axion
losses which does not contradict the Gondolo-Raffelt limit on the ”invisible”
axion and Sun luminosities ratio, $\Lambda_{a}^{invis}/\Lambda_{Sun}\leqslant
0.1$ [ref080], for which a good coincidence between the theoretical values and
experimental data of modern helioseismological and solar neutrino experiments
is still observed [ref080, ref081].
And forth, if the vertically oriented thin magnetic flux tubes in the
convective zone play a role of the waveguides for the $\gamma$-quanta born in
the tachocline with total luminosity equal to that of the Sun, what are the
nature and the power of the energy source maintaining the convective processes
on the Sun?
In order to find it out, let us assume that this source is the radiative zone
and perform an estimation of its power basing on the magnetic field $B_{OT}$
in the overshoot tachocline zone dependence on the total ohmic dissipation
$D_{CZ}^{ohmic}$ in the convective zone [ref082].
$D_{CZ}^{ohmic}=\int\frac{\eta}{\mu}\left(\nabla\times\vec{B}_{OT}\right)^{2}dV\propto\frac{2\eta}{H_{p}^{2}}E_{mag},$
(18)
where $E_{mag}$ is the magnetic energy of the field which could be possibly
maintained by the currents that produce the ohmic dissipation of the solar
dynamo [ref083].
It is easy to show [ref082] that the expression (18) may be written down in
the following form:
$D_{CZ}^{ohmic}=\frac{\eta\cdot V_{CZ}}{\mu\cdot H_{p}^{2}}B_{in}^{2}\approx
0.015\Lambda_{Sun},$ (19)
where $D_{CZ}^{ohmic}=0.015\Lambda_{Sun}$ is the heat power of the radiative
zone near the border of the overshoot tachocline zone, equal to the
uncertainty of the known Solar luminosity [ref073, ref084, ref090]; $\eta\sim
10^{4}~{}cm^{2}/s$ is the magnetic diffusivity [ref085, ref086], $V_{CZ}$ is
the volume of the Sun convective zone; $\mu\sim 1$ is the permeability;
$B_{OT}=400~{}T$ is the magnetic field in the overshoot tachocline zone;
$H_{P}=6.5\cdot 10^{3}~{}km$ is the pressure scale height555A larger value of
the pressure scale height is a consequence of the fact that the rigidity
[ref087] of the interior can be provided only by the large-scale magnetic
field (cf. Mestel & Weiss [ref088]; Gough & McIntyre [ref089]) that the
tachocline provides the interface in which radial field lines might connect
the convection zone with the radiative interior only near the latitudes at
which there is essentially no radial shear. [ref056, ref090]
The approximate equality implies that not only the total solar energy balance
is preserved in the framework of the axion mechanism of Sun luminosity, but
also that the temperature transport changes substantially with respect to the
standard model of the Sun. This is because of the fact that the old heat flux
transport mechanism (from the radiative zone to the overshoot) by radiation,
used in the standard model of the Sun, is highly depressed because the
majority of the radiation is converted into axions in the core of the Sun and
therefore does not reach the radiative interior. At the same time, this change
may not seem so dramatic, since almost all known anomalies of the experimental
data interpretation on the active solar regions, helioseismology and solar
neutrino may be leveled as it was noted above.
And, finally, turning back to the possible mechanisms of $\gamma$-quanta
channeling in a periodical structure and along the magnetic flux tubes in the
Sun convective zone, one may suppose with confidence that they are not only
physically compatible, but may turn out to be just two different versions of
the same mechanism, which naturally manifests itself, for example, in the so-
called hexagonal magnetoconvection kinetics (see e.g. [ref091]).
This means, in its turn, that the absorption length $\lambda$ for photons in
such a medium (see (6)) will become considerably greater than the thickness of
the overshoot tachocline zone, i.e., $\lambda\gg L_{OTZ}$. At the same time,
it is obvious that $\Gamma=\lambda^{-1}\to 0$, whence a necessary condition
$\Gamma L_{OTZ}\to 0$ follows. In the particular (non-coherent) case in which
the magnetic field where axions are converted into photons is under vacuum
($\Gamma\to 0$, $m_{\gamma}\to 0$), equation (6) becomes
$P_{a\to\gamma}=\left(\frac{g_{a\gamma}B_{OT}L_{OT}}{2}\right)^{2}\sin^{2}\left(\frac{qL_{OT}}{2}\right)/\left(\frac{qL_{OT}}{2}\right)^{2}$
(20)
where $q=m_{a}^{2}/2E_{a}$ (see (7)).
Obviously, in coherent case $q\to 0$, regardless of the $\gamma$-quanta
channeling mechanism type, the probability (20) for an axion to be converted
back to an ”observable” photon inside the magnetic field may be expressed in
the following simple form
$P_{a\gamma}\simeq\left(\frac{g_{a\gamma}\bar{B}_{OT}\bar{L}_{OT}}{2}\right)^{2},$
(21)
where $\bar{B}_{OT}$ is the mean value of the magnetic field in the overshoot
tachocline zone with the effective thickness $\bar{L}_{OT}$. A value for
$\bar{L}_{OT}$ was chosen basing on the analysis of the following data set.
The most well known results obtained by Charbonneau et al. [ref092] yield a
tachocline thickness of $\Delta_{t}/R_{S}=0.039\pm 0.013$ at the equator and
$\Delta_{t}/R_{S}=0.042\pm 0.013$ at the latitude of 60∘, suggesting that the
tachocline may get somewhat wider at high latitudes but that the result is not
statistically significant. On the other hand, Basu and Antia [ref093] argue
for the statistically significant increase in the tachocline thickness with
the latitude, from $\Delta_{t}/R_{S}\sim 0.016$ at the equator to
$\Delta_{t}/R_{S}\sim 0.038$ at latitudes of 60∘ (when the width is defined as
in [ref093]). Furthermore, they suggest that the variation may not be smooth;
there may be a sharp transition from a narrow tachocline at low latitudes to a
wider tachocline at high latitudes, possibly associated with the sign of the
radial angular velocity gradient which reverses at the latitude of $\sim
35^{\circ}$. Other estimates for the width of the tachocline range from
$0.01R_{S}$ to $0.09R_{S}$ (Kosovichev [ref094], Basu [ref095], Corbard et al.
[ref096], Elliott and Gough [ref097], Basu and Antia [ref098]).
Taking into account that, first, the tachocline is a transition layer between
two distinct rotational regimes (the differrentially rotating solar envelope
and the radiative interior) where the rotation is uniform, second, the maximum
estimate of the tachocline thickness reaches $0.09R_{S}$ and third, the
thickness of the overshoot tachocline zone is somewhat larger than that of the
tachocline, we took the value of $\bar{L}_{OT}$ equal to $0.1R_{S}$.
Then using Eq. (21) and the parameters of the magnetic field, it is possible
to write down the expression for the solar axion flux666Hereinafter we use
rationalized natural units to convert the magnetic field units from Tesla to
$eV^{2}$, where the conversion is 1 T = 195 $eV^{2}$ [ref040]. probability at
the Earth as
$P_{a\rightarrow\gamma}=\frac{1}{4}\left(\frac{g_{a\gamma}}{7.07\cdot
10^{-11}~{}GeV^{-1}}\right)^{2}\left(\frac{\bar{B}_{OT}}{400~{}T}\right)^{2}\left(\frac{\bar{L}_{OT}}{7.25\cdot
10^{7}~{}m}\right)^{2}=1.$ (22)
where the value of the magnetic field $B_{OT}=400~{}T$ (cf. [ref034, ref035,
ref036]) was chosen so that it satisfied the ”experimental” estimates of
200-400 T (see. Fig. 9) and induced via (22) such a value of the axion-photon
coupling constant ($g_{a\gamma}=7.07\cdot 10^{-11}$ GeV-1) that would in its
turn be strictly consistent with the known and very important limits (86)-(87)
taking into account (13). More detailed justification of such self-consistent
choice will be given in Section 5.
It is necessary to make a deviation concerning some important features of the
oscillation length ($l=2\pi/q$) here. It is known that in order to maintain
the maximum conversion probability, i.e. zero momentum transfer ($q\to 0$),
the axion and photon fields, put into some medium ($m_{\gamma}\equiv m_{a}$),
need to remain in phase over the length of the magnetic field. This coherence
condition is met when $qL\leqslant\pi$, and along with (7) lets one obtain the
following remarkable relation [ref014] between the medium density variations
and axion mass variations for the coherent case of $q\to 0$, i.e.
$m_{\gamma}\equiv m_{a}$
$\frac{\Delta\rho}{\rho}=2\frac{\Delta m_{a}}{m_{a}}=\frac{4\pi
E_{a}}{m_{a}^{2}L_{OT}}$ (23)
It is easy to show that for the mean energy $E_{a}=4.2~{}keV$, axion mass
$m_{a}=17~{}eV$ and the thickness of the overshoot tachocline zone
$L_{OT}=7.25\cdot 10^{7}~{}m$ the density variations in (23) are $\sim$10-13.
It means that the inverse coherent Primakoff effect takes place only when the
variations of the medium density inside a cylindric volume of the ”height”
$L_{OT}$ (see Fig. 11b) do not exceed the value of $\sim$10-13. This is a very
strong restriction, since it is hard to imagine any kind of a physical process
in the magnetic flux tube (see Fig. 11b) which would ”freeze” the plasma
(low-Z gas) in this magnetic volume so much so that this restriction on the
density variations is fulfilled.
Figure 11: (a) Magnetic loop tubes formation in the tachocline through the
shear flows instability development; (b) ”Capillary” effect in magnetic tubes
and the sketch of the axions (red arrows) conversion into $\gamma$-quanta
inside the magnetic flux tubes containing the magnetic steps. Here $L_{OT}$ is
the height of the magnetic shear steps. The tubes’ rotation is not shown here
for the sake of simplicity; (c) Emergence of magnetic flux bundle and
coalescence of spots to explain the phenomenology of active region emergence
(adopted from [ref056], [ref066]).
In other words, such mechanism that would validate the possibility of such
locally ”frozen” plasma existence is not known to us. At the same time, there
are some arguments suggesting that such limitation is possible.
First of them is related to the experimentally obtained solar images (Fig. 13,
adapted from [ref014]) from the Japanese X-ray telescope Yohkoh (1991-2001)
which illustrate the solar X-ray activity during the last maximum of the
11-year solar cycle (Fig. 13b). There is currently no model alternative to the
axion mechanism of sun luminosity which would have described the anomalous
distribution of the X-ray radiation over the active Sun surface.
The second argument is related to finding of the axion with mass
$m_{a}=17~{}eV$ during the study of the EBL spectral intensity (see Section
5). On the one hand, this experimentally established fact suggests that the
solar axions with mass $m_{a}=17~{}eV$ are not only a theoretical prediction,
but they really exist; and on the other hand, it is crucial for substantiation
of the axion mechanism of Sun luminosity and the solar dynamo – geodynamo
connection.
Finally, the third argument is related to the lack of understanding the link
between the magnetic tubes formation and lifetime in the tachocline, and the
length of the solar cycle. It means that we do not understand the mechanisms
of solar activity as well as the processes of generation, accumulation and
release of the magnetic energy responsible for the 11-year solar cycle as yet.
This is applies especially to the current level of understanding of the causes
and effects of the differential rotation and meridional circulation in the
tachocline. Let us remind that the tachocline is a thin transitional zone with
the width of only $0.05R_{S}$, where the latitudinal differential rotation of
the convective zone turns into almost solid-body rotation of the radiative
zone (e.g. [ref245]). It is generally believed that the main process of
magnetic field generation – solar dynamo – responsible for the 11-year cycle
takes place in this zone [ref246]. However, there are no evidence of the
11-year variations in the tachocline so far.Instead, mysterious variations of
the rotation velocity with period of 1.3 year are observed here [ref247], and
curiously enough, they coincide with en estimate of the magnetic flux tube
lifetime in the convective zone [ref065].
Although there is no deep and detailed understanding of the magnetic tubes
formation in the tachocline, one may assume the following picture of this
process in the framework of the axion mechanism of Sun luminosity.First of
all, numerous magnetic tubes appear (Fig. 11a) as a consequence of the shear
flows instability development in the tachocline. As is shown in Appendix B,
the pressure inside such tubes is ultralow, which directly leads to formation
and ”floating-up” of the magnetic steps in these tubes (Fig. 11b). In other
words, a kind of ”capillary” effect is observed in this case. The whole
picture of the magnetic tube spatio-temporal evolution in the convective zone
depicted at Fig. 11 leaves the question about the locally ”frozen” plasma
existence open, but at the same time it illustrates the process of axions
conversion into $\gamma$-quanta in the overshoot tachocline and thus the
mechanism of sun luminosity and active solar regions formation in the
photosphere (Fig. 11c).
By normalizing the expression (22) the probability of the total conversion of
axions into photons is assumed to be equal to a unit at given parameters of
the magnetic field, i.e. $P_{a\to\gamma}=1$. The primary criterion for such
choice of the axion-photon coupling strength is the assumption about the
maximum contribution of the luminosity produced by the axion to
$\gamma$-quanta conversion in the tachocline zone (see Fig. 6c,d) into the
total Solar luminosity ($\Lambda_{Sun}$) during the active phase.
$\displaystyle\Delta_{a}\cdot\left[\Phi_{Pr}\cdot\left\langle
E_{a}\right\rangle_{Pr}+\Phi_{Brems}\cdot\left\langle
E_{a}\right\rangle_{Brems}+\Phi_{Compt}\cdot\left\langle
E_{a}\right\rangle_{Compt}+\Phi_{M1}\cdot\left\langle
E_{a}\right\rangle_{M1}\right]\times$ $\displaystyle\times(4\pi
R_{SE}^{2})\times P_{a\rightarrow\gamma}=\Lambda_{Sun},$ (24)
where777In the balance Eq. (24) we considered M1 transition of the 57Fe nuclei
only and ignored the 55Mn and 23Na nuclei, because the favorable Boltzman
factor of 57Fe produces the largest cooling rate near $T_{8}\sim 1$ [ref099].
$\Delta_{a}=0.90$ is a portion of the axion flux that transforms into
$\gamma$-quanta in the tachocline by the inverse Primakoff effect888The choice
of the $\Delta_{a}$ value is dictated by the spatial geometry of the solar
tachocline zone (see Fig. 6c,d) which is described in more detail below (see
Fig. 9). Let us note that the portion of the ”invisible” axions equal to
$(1-\Delta_{a}$) satisfies the neutrino limit on axions, i.e. the Gondolo-
Raffelt criterion [ref080, ref100] at the same time.; $P_{a\to\gamma}=1$;
$\Lambda_{Sun}=3.84\cdot 10^{26}\pm 1.5\%~{}W$ [ref084, ref101];
$R_{SE}=1.496\cdot 10^{13}~{}cm$ is the distance from the Earth to the Sun;
$\langle E_{a}\rangle_{Pr}=4.2~{}keV$, $\langle
E_{a}\rangle_{Brems}=1.6~{}keV$, $\langle E_{a}\rangle_{Compt}=5.1~{}keV$,
$\langle E_{a}\rangle_{M1}=14.4~{}keV$ are the average energies of the solar
axions spectra; $\Phi_{Pr}$, $\Phi_{Brems}$, $\Phi_{Compt}$, $\Phi_{M1}$ are
the integral solar axions fluxes generated by the Primakoff effect, the
bremsstrahlung, the Compton process and the M1 transition in the 57Fe nuclei
on the Sun respectively, which were obtained using the known differential
spectra $d\Phi_{Pr}/dE$ [ref025], $d\Phi_{Brems}/dE$ [ref006],
$d\Phi_{Compt}/dE$ [ref006], $d\Phi_{M1}/dE$ [ref002]:
$\Phi_{Pr}=3.75\cdot 10^{31}g_{a\gamma}^{2}~{}~{}cm^{-2}s^{-1},$ (25)
$\Phi_{Brems}=1.43\cdot 10^{35}g_{ae}^{2}~{}~{}cm^{-2}s^{-1},$ (26)
$\Phi_{Compt}=2.16\cdot 10^{34}g_{ae}^{2}~{}~{}cm^{-2}s^{-1},$ (27)
$\Phi_{M1}=1.66\cdot 10^{23}g_{an}^{2}~{}~{}cm^{-2}s^{-1};$ (28)
where $g_{a\gamma}$, $g_{ae}$, $g_{an}$ are the axion coupling constants to
photons ($g_{a\gamma}$), electrons ($g_{ae}$) and nucleons ($g_{an}$)
respectively999Although Eqs. (25)-(28) were obtained in the framework of the
standard model of the Sun, they may be applied to the axion model as well,
since the basic thermodynamic parameters (temperature, pressure, plasma
density etc.) of the solar core are roughly the same in both models, and
almost all the axions are produced in the solar core, regardless of the exact
way of their birth..
We obtained only the axion-photon coupling constant $g_{a\gamma}$ (see (22))
out of the three axion coupling constants so far. Next it is not too hard to
estimate the axion-nucleon coupling constant basing on the hadronic axion
models [ref102, ref103].
$g_{an}=|g_{0}\beta+g_{3}|,$ (29)
where
$g_{0}=-\frac{m_{N}}{6f_{a}}\left[2S+(3F-D)\frac{1+z-2w}{1+z+w}\right],$ (30)
$g_{3}=-\frac{m_{N}}{2f_{a}}\left[(D+F)\frac{1-z}{1+z+w}\right].$ (31)
Here the value $\beta=-1.19$ for the M1 transition in the 57Fe nucleus was
calculated in [ref102, ref103]; spontaneous breaking of the PQ-symmetry (in
view of (10)-(11)) takes place at the energy $f_{a}\approx 0.119\cdot 10^{6}$
GeV; $m_{N}=939$ MeV is the nucleon mass. The exact values of $D$ and $F$
parameters determined from semileptonic hyperon decays are equal to
$D=0.808\pm 0.006$ and $F=0.462\pm 0.011$ [ref104]. Parameters
$z=m_{u}/m_{d}\approx 0.56$ and $w=m_{u}/m_{s}\approx 0.029$ are quark mass
ratios [ref032, ref033].
The parameter $S$ characterizing the flavor singlet coupling still remains a
poorly constrained one. Its value varies from $S=0.68$ in the naive quark
model down to $S=−0.09$ which is given on the basis of the EMC collaboration
measurements [ref105]. The more stringent boundaries ($0.37\leqslant
S\leqslant 0.53$) and ($0.15\leqslant S\leqslant 0.5$) were found in [ref106]
and [ref107], accordingly. As a result the value of the sum (29) may
significantly decrease and, due to negativity of the parameter $\beta$,
actually vanish. Taking into account that the usually accepted value of $u$\-
and $d$-quark mass ratio $z=0.56$ can vary in $0.35\div 0.6$ range [ref108],
the exact interpretation of experimental results is significantly restricted.
Calculations performed using the expressions (29)-(31) and (13) let one derive
the value of $g_{an}=2.46\cdot 10^{-6}$ for the axion-nucleon coupling
constant at $S=0.55$. However, it should be noted that such a model-dependent
value does not suit our needs, and here is why.
It is easy to show using (28) that the luminosity $\Lambda_{M1}$ of the axions
produced by M1 transition of 57Fe nuclei depends on the solar photon
luminosity $\Lambda_{Sun}$ in the following way101010Here we follow the
calculations by Derbin (e.g. [ref003]), that is why the expression (32)
differs slightly from the analogous expression (2.13) in [ref110]. The
difference mainly comes from the different values of the Doppler spectrum
broadening used during the calculation of the monoenergetic solar axions flux
produced by the ${}^{57}Fe$ nuclear de-excitations (e.g. [ref003, ref110]).:
$\Lambda_{M1}\cong 2.8\cdot 10^{9}g_{an}^{2}\Lambda_{Sun},$ (32)
Substituting the value of $g_{an}=2.46\cdot 10^{-6}$ into (32) we derive that
the relative axion luminosity $\Lambda_{M1}$ is about $\sim 2\%$. This value
is inadmissible for the axion mechanism of Sun luminosity, because otherwise
the resulting solar photon spectrum (Fig. 7) would contain a rather high peak
near $E_{a}=14.4$ keV. As a matter of fact, if it is there, then it is very
weak considering the spectrum uncertainty in this band (see Fig. 9 in
[ref109]).
In this connection let us from now on assume that the relative axion
luminosity $\Lambda_{M1}$ is
$\Lambda_{M1}/\Lambda_{Sun}=0.003.$ (33)
The choice of such relation was made so that the relative axion luminosity
$\Lambda_{M1}$ allowed the 14.4 keV peak existence within the experimental Sun
photon spectrum uncertainty and conformed to the axion-mucleon coupling
constant from the theoretically allowed limitations known as the SN1987A limit
($3\cdot 10^{-7}\leqslant g_{an}\leqslant 10^{-6}$ [ref111, ref112]).
Substituting (33) into (32) we obtain a consistent value of the axion-nucleon
coupling constant111111Hereinafter by $g_{an}$ and $g_{ae}$ we always mean
$|g_{an}|$ and $|g_{ae}|$ respectively.
$g_{an}\cong 3.2\cdot 10^{-7}.$ (34)
Now let us turn again to Eq. (22) which describes the axion mechanism of Sun
luminosity hypothesis. Apparently, the found axion coupling constants to
photons ($g_{a\gamma}=7.07\cdot 10^{-11}~{}GeV^{-1}$) and nucleons
($g_{an}=3.2\cdot 10^{-7}$) let one calculate the axion-electron coupling
constant ($g_{ae}$) by means of Eqs. (22)-(28). It turned out to be
$g_{ae}\cong 5.28\cdot 10^{-11}.$ (35)
It is interesting to note that within such approach, bremsstrahlung ($\sim
2.59\cdot 10^{26}$ W) has the major part in the Solar luminosity, which is
$\sim 67.48\%$, while the Compton process ($\sim 1.24\cdot 10^{26}$ W), the
Primakoff effect ($\sim 3.19\cdot 10^{23}$ W) and the M1 transition of 57Fe
nuclei ($\sim 9.93\cdot 10^{22}$ W) make 32.41%, 0.08% and 0.03% respectively.
Differential solar axions spectra as observed at the Earth originating from
bremsstrahlung, the Compton process, the Primakoff effect and M1 ground-state
nuclear transition in solar 57Fe are shown on Fig. 2b.
Thus, it is obvious that the mechanism of luminosity and the X-ray spectrum
shape for the active and quiet Sun (Fig. 7) can be easily explained by our
model, in which axions are converted into $\gamma$-quanta in the solar
tachocline under certain conditions. First of all, the $\gamma$-quanta energy
spectrum generated by axions in the solar tachocline zone due to the
channeling effect practically does not change to the boundary of the
photosphere of the Sun, where it transforms because of the Compton scattering,
as is shown in Figs. 6c,d and Fig. 7. And, finally, it is obvious that the
integral of this spectrum (see Fig. 7) by virtue of the equality (24)
coincides by the order of magnitude with the estimation of the Sun luminosity.
### 2.5 Invisible axions and Solar Equator effect
As it was already stated before, the solar magnetic field variations (Fig. 3)
must drive (with the help of the solar axions and the inverse Primakoff
effect) the total solar irradiance (TSI) and 14.4 keV axions variations at the
same time (see the inset in Fig. 12). It is important to note here that,
however strange it may seem, TSI variations are not the modulator of the Earth
climatic system (ECS) global temperature, because the strong inverse121212A
lot of climatologists still believe that it is the TSI variations that are
responsible for the Earth global temperature variations obstinately
disregarding the facts of a very small contribution of TSI variations into the
energy balance in the Earth atmosphere and the inverse correlation between TSI
and global temperature variations (see Fig.1 in [ref009], where the ocean
level is a proxy for the global temperature). correlation with the 22-year lag
is observed between them [ref009]. And vice versa, the variations of the solar
axion flux manifest the strong positive correlation with the global
temperature variations with the same time lag. This fact plays a key role in
the new global climate theory [ref113, ref114, ref115], which considers the
variations of the 14.4 keV solar axions (which are resonantly absorbed in the
Earth core) a trigger-like modulator of all the thermal processes in ECS and,
particularly, in the atmosphere. However, this problem requires a special
discussion and therefore will be examined in a separate paper.
Figure 12: Time evolution of $P_{a\to\gamma}$ and TSI during 1975-2010. Inset:
time evolution of (a) the variations of the magnetic flux in the tachocline
zone of the Sun [ref012]), (b) TSI annual variations [ref101]. Curves are
smoothed by the sliding intervals in 5 and 11 years.
On the other hand, as follows from Fig. 12, the TSI variations during the
active phase of the Sun are so small ($\sim 1~{}W/m^{2}$ [ref101]), that the
relative portion of 14.4 keV axions must also be small at the Earth.
$p_{TSI}=\frac{\Delta R_{a}}{R_{a}}=\frac{(TSI)\cdot 4\pi
r_{SE}^{2}}{L_{Sun}}\sim 10^{-3}$ (36)
Therefore their heat power in the Earth core
$\mathcal{R}=p_{TSI}\cdot R_{a}\cdot N_{Fe}^{57}\cdot E_{a}\sim 20~{}~{}W,$
(37)
is not enough not only for the geomagnetic field generation (which requires at
least $\geqslant$0.1 TW [ref018]), but for the geomagnetic field variations
also ($\sim$0.01 TW). Here $R_{a}=5.16\cdot 10^{-3}g_{an}^{4}$,
$N_{Fe}^{57}\sim 3\cdot 10^{47}$ is the number of 57Fe nuclei in the Earth
core, $E_{a}=14.4$ keV is the 57Fe solar axions energy.
At the same time it is not difficult to see that the relative part of the
axions $\Delta_{a}$ that are almost not affected by the Primakoff effect in
the polar ($\Delta_{pol}$) and equatorial ($\Delta_{equ}$) sectors of the
tachocline zone (Fig. 13) is a considerable quantity:
Figure 13: Top: Solar images at photon energies from 250 eV up to a few keV
from the Japanese X-ray telescope Yohkoh (1991-2001) (adapted from [ref014]).
The following is shown: on the left, a composite of 49 of the quietest solar
periods during the solar minimum in 1996. On the right, the solar X-ray
activity during the last maximum of the 11-year solar cycle. According to Fig.
6 (and Fig. 13, bottom) most of the X-ray solar activity (right) occurs at a
wide bandwidth of $\pm 45^{\circ}$ in latitude, being homogeneous in
longitude. Note that $\sim 95\%$ of the solar magnetic activity covers this
bandwidth [ref116] (see also a similar topology for microflares measured with
RHESSI [ref117]).
Bottom: Schematic picture of the solar tachocline zone, the Earth’s liquid
outer (the red region) and inner (the brown region) core. Blue lines on the
Sun designate the magnetic field. In the tachocline axions are converted into
$\gamma$-quanta, which form the experimentally observed solar photon spectrum
(Fig. 7) after passing through the photosphere. Solar axions moving towards
the poles (blue cones) and in the equatorial plane (blue bandwidth) are not
transformed by the Primakoff effect, since the magnetic field vector is almost
collinear to their momentum vector in these regions. Solar axions are then
resonantly absorbed by iron in the Earth core transforming into
$\gamma$-quanta, which are the supplementary energy source in the Earth core
(see the text).
$\Delta_{a}=\Delta_{equ}^{a}+\Delta_{pol}^{a},$ (38)
If we assume that the equatorial surface ($S_{equ}$) formed by two cones going
from the center of the Sun, and a part of the sphere with the radius of the
Sun ($R{Sun}$) has the dihedral angle of $\sim 5^{\circ}$, it is easy to show
that131313This estimate was made on the basis of the numerous computational
experiments on magnetic field evolution in the convective zone of the Sun
(e.g. [ref012] and Refs. therein).
$\Delta_{equ}^{a}=\frac{S_{equ}}{S_{Sun}}\sim 0.05.$ (39)
The expression (39) means that because of the quasi-collinearity of the Earth
and Sun rotation axes, the main axion flux directed towards the Earth
originates from the equatorial sector of the Sun (Fig. 13). In this connection
a short remark should be made regarding the anti-correlation between the solar
magnetic field and geomagnetic field variations. Weak variations of TSI are
obviously produced by the solar magnetic field variations. The same solar
magnetic field variations are the cause of the ”equatorial” axion flux
variations. In other words, the ”equatorial” effect not only generates the
”invisible” axions, but also modulates their intensity inversely proportional
to the solar magnetic field changes, producing the observed inverse
correlation between them (Fig. 11). The latter is supposed to be the main
cause of the anticorrelation between the solar magnetic field variations and
the geomagnetic field variations.
The assumptions used to derive the expressions (16) and (39) are called forth
by the necessity of justification both the axion mechanism of Sun luminosity
and the axion mechanism of solar dynamo – geodynamo connection, and will be
additionally substantiated below.
### 2.6 Power required to maintain the Earth magnetic field and nuclear
georeactor
It is not hard to show that the resonant absorption rate of 14.4 keV solar
axions in the Earth core, which contains the $N_{Fe}^{57}$ nuclei of 57Fe
isotope, is about [ref003]
$R_{a}\cong 5.16\cdot 10^{-3}g_{an}^{4}\cdot
N_{Fe}^{57}\cdot\Delta_{equ}^{a},$ (40)
where $\Delta_{equ}$ is the portion of axions reaching the Earth via the solar
equator effect (see (39)).
It is known, that the number of 57Fe nuclei in the Earth core is
$N_{Fe}^{57}\sim 3\cdot 10^{47}$ and the average energy of 57Fe solar axions
is $E_{a}=14.4$ keV. Then with an allowance for Eq. (40) and the value of the
axion-nucleon coupling constant (35) the maximum energy release rate $\Delta
D_{ohmic}^{a}$ in the Earth core is equal to
$\Delta D_{ohmic}^{a}=R_{a}\cdot E_{a}\simeq 1.2~{}~{}kW.$ (41)
This estimate of the heat power supplied to the Earth core by the absorbed
axions (41) apparently is much less then the value necessary to generate the
magnetic field of the Earth ($\geqslant 0.1$ TW [ref018]). Moreover, it is
small even in comparison with the heat power fluctuations ($\sim$0.01 TW)
responsible for the geomagnetic field variations in the Earth core (see the
inset in Fig. 12).
It is easy to illustrate this using the known dependence of the core magnetic
field $B_{in}$ on ohmic dissipation $D_{ohmic}$ in the Earth core [ref018,
ref082, ref118] in the form:
$D_{ohmic}=\frac{\eta\cdot V}{\mu\cdot l_{B}^{2}}B_{in}^{2}\sim 0.1~{}~{}TW,$
(42)
where $\eta\sim 2~{}m^{2}/s$ is magnetic diffusivity [ref017], $V=(4/3)\pi
R_{C}^{3}$ is the volume of the Earth core, $R_{C}=3480~{}km$ is the radius of
the Earth core, $\mu\sim 1$ is permeability, $l_{B}\sim 0.8\cdot 10^{5}~{}m$
is the characteristic length scale on which the field vector changes [ref017],
$B_{in}\sim 4~{}mT$ is the core magnetic field [ref119].
In order to estimate the heat power fluctuations $\Delta D_{ohmic}$
responsible for the magnetic field variations in the Earth core let us
represent (42) in differential form:
$dD_{ohmic}=\frac{2\eta\cdot V}{\mu\cdot l_{B}^{2}}B_{in}dB_{in},$ (43)
The right-hand side of (43) contains the known estimates except for the
magnetic field fluctuations $\Delta B_{in}\sim dB_{in}$. On the other hand,
there are well known long-term records of magnetic field variations measured
on the Earth surface (e.g. [ref013]). This lets us estimate $dD_{ohmic}$ by
writing down the expression (40) in the following form:
$dD_{ohmic}=\frac{2V}{\mu}B_{in}\frac{dB_{in}}{dt},$ (44)
where the magnetic field variations ($dB_{in}/dt$) in the Earth core appear
after taking into account the magnetic diffusion
$dB_{in}\cong\frac{dB_{in}}{dt}\delta
t=\frac{dB_{in}}{dt}\cdot\frac{l_{B}^{2}}{\eta}.$ (45)
If one also takes into account the known relation between the inner
($dB_{in}/dt$) and outer ($dB_{out}/dt$) geomagnetic field variations
[ref120],
$\frac{dB_{out}}{dt}=\left(\frac{\gamma
R_{C}}{R_{E}}\right)^{2}N\cdot\frac{dB_{in}}{dt},$ (46)
then it is possible to make an estimate of the ohmic dissipation fluctuations
($\Delta D_{ohmic}\sim dD_{ohmic}$) basing on (43)-(45) necessary for inducing
a certain number $N$ of Taylor cells in the core [ref120].
$\Delta D_{ohmic}=\frac{2V}{\mu}B_{in}\left(\frac{R_{E}}{\gamma
R_{C}}\right)^{2}\frac{dB_{out}}{dt}\simeq 0.02~{}~{}TW,$ (47)
Here $R_{E}=6357~{}km$ is the radius of the Earth, $\gamma=0.1$ [ref120],
$(dB_{out}/dt)\approx 20~{}nT/yr$ is the annual variation of the external
magnetic field of the Earth core [ref121, ref020].
A natural question arises from the stated above about the way that 14.4 keV
solar axions may provide an effective mechanism of solar dynamo – geodynamo
connection while supplying a rather low heat power. In other words, how does
this problem reduce to the mechanism of small heat perturbations critical
influence on the convective process in the Earth liquid core. The problem is
stated this way because if there is an effective mechanism of convective
instabilities generation by weak heat perturbations in the Earth liquid core,
then this effect may simultaneously cause substantial weakening of the
convective heat removal from the Earth solid core. The intense weakening of
the heat removal from the Earth solid core surface, in its turn, causes the
corresponding temperature increase in the solid core boundary layer. This is
very important because it promotes the subsequent effective convection
stability recovery in the liquid core.
There are strong grounds to believe that there is a natural nuclear georeactor
operating at the boundary between the Earth solid core and liquid core. The
analysis of the KamLAND experiment neutrino spectra for 2002-2008 shows that
the heat power of this non-stationary traveling wave reactor (TWR) is about 30
TW [ref122, ref123, ref124]. The heat power of such TWR depends on the nuclear
fuel composition and the medium temperature. The latter is because of the fact
that according to [ref123, ref124], 238U and 239Pu capture and fission cross-
sections depend quasi-linearly on the temperature of the neutron-
multiplicating medium in the 3000-5500 K range (Fig. 14).
Figure 14: Dependence of (a) capture cross-sections and (b) fission cross-
sections for 235U (blue), 238U (green), and 239Pu (red) averaged over the
neutron spectrum on the fuel medium temperature for the limiting energy (3kT)
of the Fermi and Maxwell spectra [ref124]. The neutron spectra averaging
procedure was applied for the concentrational fuel composition of the nuclear
georeactor discussed in [ref122].
These peculiarities of TWR are responsible for the positive feedback that
leads to the reactor heat power increase after the corresponding boundary
layer temperature increase (see Fig. 14). The georeactor heat power growth
lasts until the steady heat removal from the TWR is reestablished, which
implies restoration and stabilization of the convection in the Earth liquid
core.
Now let us turn back to the physical essence of the axion mechanism of the
weak convective instability thermal perturbations in the liquid core. In this
connection it should be noted that the convection in the Earth core is
compositional. It means that there are some light elements originating, in
particular, from the 239Pu nuclei fission which take part in the convection
along with the ”iron” component [ref122, ref123, ref124]. It turns out that
the convective instability may appear in such media even under hydrostatically
stable density stratification, i.e. when the density decreases with height
[ref125, ref126, ref127].
It is known that the phenomenon of the convective instability caused by the
double (differential) diffusion was discovered rather long ago and has been
described in numerous overviews and monographs in detail (e.g. [ref125,
ref126, ref127]). The principal role in this case usually belongs to the
difference between the two hydrodynamic components of heat and admixture
[ref125]. The convection caused by double diffusion is generally believed to
appear when the thermal medium stratification is stable, while the weakly
diffusing admixture (e.g. light elements) introduces a destabilizing
contribution into the density stratification. Although this contribution may
be relatively small, it may be enough for destabilization of a stably
stratified (in terms of density) system owing to the mentioned effects
[ref127].
However, we are interested in the conditions of the convective instability
formation in a qualitatively different situation, particularly, when a weakly
diffusing admixture, on the contrary, introduces a stabilizing contribution
into the density stratification. This contribution may even exceed the thermal
instability in absolute value. Such possibility may seem paradoxical at first
glance since due to double diffusion effects the slowly diffusing admixture
usually has the much greater impact on the convective instability than the
quickly transportable heat, all other factors being equal. Let us show that
this is not always the case by analyzing the situation when there is a slow
background motion along the gravity force described in [ref127] in detail.
A problem on convection on the background of the slow (relative to the
characteristic speed of the studied convective motions) vertical motion was
first considered in [ref128]. It is of a considerable interest for us, since
the resonant absorption of 14.4 keV solar axions in the iron nuclei may be
considered as a process that induces a slow descending background motion (Fig.
15a) in the convective medium of the liquid core.
Figure 15: (a) A sketch of the thermal convection with the resonant absortion
of 14.4 keV solar axions by iron nuclei producing the 14.4 keV $\gamma$-quanta
flux (red arrows). Here $\gamma$-quanta flux emulates a slow background
descending motion in the convecting medium. The green arrow denotes the
convection direction. (b) Distortion of the vertical background temperature
distribution by the downward motion: (1) the zero background vertical
velocity, $w=H/h=0$; (2) $w=10$ (adapted from [ref127]).
Following [ref127], let us consider a single-component medium with its density
depending on the temperature $T$ only (neglecting the admixture stratification
effects). In other words, we are considering a modification of the classical
Rayleigh-Bénard problem on the convective stability of a liquid between two
horizontal plates [ref125]. A slow vertical motion along the gravity force is
assumed to be present in the background state. For the sake of simplicity let
us consider a motion with the velocity $W<0$ independent of the vertical
coordinate $z$ counted from the top boundary. Let us also consider the bottom
and top boundaries temperatures $T_{bot}$ and $T_{top}$ fixed and denote the
difference between them by $\Delta T$. Heat transfer in the background flow is
described by the equation
$-W\frac{dT}{dz}=\kappa\frac{d^{2}T}{dz^{2}},$ (48)
where $\kappa$ is the thermal diffusivity. A solution for the two boundary
conditions mentioned above may be written down in the form
$\Theta(z)=\frac{\exp(-\xi)-\exp(-w)}{1-\exp(-w)}.$ (49)
Here $\Theta(z)=[T(z)-T_{top}]/\Delta T$ is the dimensionless temperature
deviation, $\xi=h/\kappa$ is the dimensionless vertical coordinate, and
$h=\kappa/W$ is the reference height associated with the vertical motion
(infinity in the quiescent fluid). The key dimensional parameter is
$w=\frac{H}{h}=W(H/\kappa),$ (50)
where $H$ is the fluid layer thickness. In the absence of the background
vertical motion (in the limit $W\to 0$, $h\to\infty$, and $w\to 0$), the
result is the expected linear profile
$\Theta=1-z/H,$ (51)
i.e. the solution whose stability is analyzed in the classical Rayleigh-Bénard
problem. Fig. 15b shows the vertical profiles of (51) for $w=0$ and $w=10$
[ref127, ref128]. Apparently, the background medium sinking ”pushes” almost
all temperature difference $\Delta T$ to the bottom boundary where it
concentrates within a layer with the thickness $h=\kappa/W$.
Generally speaking, a rigorous stability study of the stationary state with a
curvilinear temperature profile and the background sinking is a rather complex
problem. An estimate of the effective Rayleigh number for the bottom sublayer,
which incorporates virtually all the vertical temperature difference $\Delta
T$ (Fig. 15b), was made on the basis of the simple physical reasoning in the
paper [ref128].
$Ra\sim\frac{\alpha\cdot g\cdot\Delta T\cdot
h^{3}}{\kappa\nu}\sim\frac{\alpha\cdot g\cdot\Delta T\cdot\kappa^{2}}{\nu\cdot
W^{3}},$ (52)
Here $\alpha$ is the thermal expansion coefficient of the fluid, $\nu$ is the
kinematic viscosity, and $g$ is the gravitational acceleration.
Let us denote the value of the effective Rayleigh number, which corresponds to
the loss of stability, by $Ra_{cr}$. The sinking rate sufficient for stability
loss prevention in this case is expressed by the equation
$W_{cr}\sim\left(\frac{\alpha\cdot g\cdot\Delta T\cdot\kappa^{2}}{\nu\cdot
Ra_{cr}}\right)^{1/3}\sim 10^{-4}~{}~{}m\cdot s^{-1},$ (53)
where $Ra_{cr}$ is a complex function of $\varepsilon$ [ref129]
$Ra_{cr}\sim
E^{1.16}\left[0.21\varepsilon^{-2}+22.4(1-\varepsilon^{2})^{1/2}\right]\sim
10^{-6},~{}~{}\varepsilon=R_{in}/R_{out},$ (54)
For example, setting $\kappa\sim 0.1~{}m^{2}\cdot s$, $\nu\sim
10^{3}~{}m^{2}\cdot s$ (effective turbulent transport coefficients
characteristic for the liquid core [ref130]), $\alpha\sim 10^{-5}~{}K^{-1}$
[ref131], $\Delta T\sim 1000~{}K$ [ref131], $g=11~{}m\cdot s^{-2}$ [ref131],
$E\sim 10^{6}$ is the Ekman number for a given $\kappa$ [ref130], one obtains
$W_{cr}\leqslant 10^{-4}~{}m\cdot s^{-1}$.
It is interesting to note that the background sinking leads to an effective
Rayleigh number decrease, i.e. $Ra_{cr}<Ra$, where $Ra>10^{6}$ is Rayleigh
number in Earth core [ref131], and consequently, to a decrease in convective
instability formation probability. This value of downward velocity is also in
good agreement with the value of the characteristic velocity of the Earth core
convection ($\sim 4\cdot 10^{-4}~{}m\cdot s^{-1}$ [ref130]), while the results
obtained in both field experiments and numerical simulations (see Fig. 15b)
demonstrate that downward flow with this velocity suppresses convection
[ref128].
Now let us pass on to a two-component medium which has its unstable thermal
stratification under the absence of the vertical motion overcompensated by a
stable admixture stratification. As is shown in [ref128], if there is a slowly
diffusing admixture stratification along with the temperature stratification
(light elements in our case), admixture is suppressed by the background
movement more effectively than the heat in such two-component medium. In other
words, even the presence of a very slow downward movement may pull the
admixture down as opposed to the heat and thus cancel its stabilizing effect,
and the system becomes unstable. It is also known that the temperature profile
may be deformed as well under more intense background motions. It is important
to keep in mind that the mentioned effects are possible under very small
vertical velocities of the background motions. As the authors of [ref128]
point out, we are dealing with a new kind of instability. Note, however, that
the times required for a system to evolve into the unstable steady states
considered above may be very large [ref128].
It may be concluded that the vertical background motions may prevent the
convective instability formation in the single-component media while leading
to a destabilization of a two-component medium layer. It happens because the
background motions are an immediate cause of the effective vertical drift of a
slowly diffusing admixture. An important fact to remember is that the above-
mentioned effects are possible even under very small vertical background
motion velocities141414It is interesting to note here that a problem on
convection in presence of the slow background motions is extremely urgent for
the known geophysical applications related to e.g. cloud patterns and
atmospheric circulation [ref128, ref132, ref133, ref134]. For example, the
atmospheric and oceanic convection often takes place on the background of the
processes with much larger horizontal scales (cyclones and anticyclones) which
are characterized by the average vertical motions several orders of magnitude
slower than those that appear during a convective instability formation.
According to the natural experiments [ref134], even a slow background sinking
of the medium can effectively suppress the convection in the atmosphere..
The problem on convective instability caused by the background motion is
examined in its simplest form so far. The rotation effects, magnetic field
influence, nonlinear background motion velocity etc. were not taken into
account here, but all of them are actually present in a traditional composite
media magnetohydrodynamics in the Earth core. A detailed consideration of
these effects is beyond the scope of the present paper. The purpose of the
current section is to demonstrate a possibility of a nontrivial impact of
background motions, which may be produced by the resonant absorption of the
”iron” solar axions along with the convection in the Earth liquid core, within
a simple model.
Thus, the essence of the axion mechanism of solar dynamo – geodynamo
connection lies in the following. The resonant absorption of 14.4 keV solar
axions by the iron of the Earth core induces a vertical background motion
along the gravity force (Fig. 15a), which in its turn ”pulls” almost all the
temperature difference $\Delta T$ down to the bottom of the liquid core (Fig.
15b) and concentrates it within a layer of the thickness $h=\kappa/W$. As it
was noted above, this effect takes place both in single-component and in two-
component media.
An important result of these processes is a substantial attenuation of a heat
removal from the Earth solid core surface which leads to a corresponding
temperature increase in the boundary layer between the liquid core and the
solid core where the nuclear georeactor (TWR) resides. As it was shown in
[ref124], one of the peculiarities of such TWR is that its heat power output
depends both on the fuel composition and the medium temperature. It means that
increase of the temperature in the boundary layer at the solid core and liquid
core border leads to a corresponding increase of the nuclear georeactor power
output (see Fig. 14). As a result, the georeactor heat power output grows
until a steady heat removal is re-established, i.e. the convection is re-
established and stabilized in the liquid core (up to the ”next” variation of
the thermal perturbations by axions!).
Therefore if such axion mechanism of solar dynamo – geodynamo connection
exists, then the ohmic dissipation caused by a resonant 14.4 keV solar axions
absorption in the Earth core should be connected with the heat power
perturbations $\Delta D_{ohmic}$, responsible for the magnetic field
variations in the earth core, by the following relations:
$B_{in}=\xi\cdot B_{in}^{a},$ (55)
$dD_{ohmic}=\xi^{2}dD_{ohmic}^{a},$ (56)
where the trigger gain $\xi$ in our case (see (38) and (44)) is equal to
$\xi\sim 3.4\cdot 10^{3}.$ (57)
Here $B_{in}$ and $B_{in}^{a}$ are the magnetic fields in the Earth liquid
core produced by the nuclear georeactor and the solar axions respectively. The
physical sense of the expressions (55)-(56) reveals the reason why all of the
known candidates for an energy source of the Earth magnetic field [ref015]
cannot in principle explain one of the most remarkable phenomena in solar-
terrestrial physics – a strong (inverse) correlation between the temporal
variations of magnetic flux in the overshoot tachocline zone [ref012] and the
Earth magnetic field (Y-component) [ref013] (Fig. 3).
## 3 Axion mechanism of Sun luminosity and CUORE experiment
Recently a CUORE-experimental search was performed for axions from the solar
core from 14.4 keV M1 ground-state nuclear transition in 57Fe [ref001]. The
detection technique employed a search for a peak in the energy spectrum at
14.4 keV when the axion is absorbed by an electron via the axio-electric
effect. The cross section for this process is proportional to the photo-
electric absorption cross section for photons [ref135]:
$\sigma_{ae}=\frac{\sigma_{pe}}{8\pi\alpha_{EM}}\left(\frac{2x_{e}^{\prime}m_{e}c^{2}}{f_{a}}\right)^{2}\left(\frac{\hbar\omega}{m_{e}c^{2}}\right)^{2},$
(58)
where $x_{e}^{\prime}\approx 1$ , $m_{e}c^{2}$ is the electron mass in GeV,
$\alpha_{EM}=1/137$, and $\sigma_{pe}=55.336~{}cm^{2}\cdot gm^{-1}$ is the
photoelectric cross sections for TeO2 (taken from [ref136]).
Substituting the values of these constants one may rewrite Eq. (58) in the
following form:
$\sigma_{ae}=\frac{2.18\cdot
10^{-11}~{}~{}GeV^{2}}{f_{a}^{2}}\cdot\left(\frac{E_{a}}{keV}\right)^{2}\sigma_{pe},$
(59)
where $E_{a}$ is in keV and $f_{a}$ is the Peccei-Quinn scale in GeV.
Hence the estimated absorption rate of 14.4 keV solar axions $N_{CUORE}$
detected by the axio-electric effect (58) in TeO2-detector ($m=3~{}kg$) of the
CUORE experiment is
$\Phi_{a}\cdot\pi\sigma_{ae}\cdot\varepsilon_{D}\leqslant
N_{CUORE}=0.63~{}~{}count\cdot kg^{-1}d^{-1},$ (60)
where $\varepsilon_{D}$ is the detection efficiencies; the flux $\Phi_{a}$ at
the Earth (28) is
$\Phi_{a}=1.66\cdot 10^{23}(g_{an})^{2}~{}~{}cm^{-2}s^{-1},$ (61)
It is the Eq. (60) that made it possible for CUORE collaboration to place a
bound (at $S=0.55$) on the axion coupling constant of $f_{a}\geqslant
0.76\cdot 10^{6}$ GeV at 95% C.L. (Fig. 16b). According to Eq. (10), the limit
on $f_{a}$ translates into a mass limit $m_{a}<8$ eV.
It is necessary to note that if one takes into account the axion mechanism of
Sun luminosity and solar dynamo-geodynamo connection, expression (60) with
respect to solar equator ($\Delta_{equ}^{a}\sim 0.05$), becomes
$\Phi_{a}\cdot\pi\sigma_{ae}\cdot\varepsilon_{D}\leqslant\frac{N_{CUORE}}{\Delta_{equ}^{a}}=\frac{0.63}{\Delta_{equ}^{a}}~{}~{}counts\cdot
kg^{-1}d^{-1},$ (62)
In other words, within the framework of such mechanism, it is necessary to
keep in mind that because of the solar equator effect, only a part of the
total axion flux $\Phi_{a}$ (61) equal to $\Delta_{equ}\Phi_{a}$ arrives to
the Earth. The solution (62) in this case lets us use the expressions151515Let
us point out that that the expressions (63) and (64) used in CUORE experiment
data processing are slightly different from the corresponding expressions
(30)-(31) derived in [ref102, ref103]. [ref001]
$g_{0}=-7.8\cdot 10^{-8}\left(\frac{6.2\cdot
10^{6}~{}GeV}{f_{a}}\right)\left(\frac{3F-D+2S}{3}\right),$ (63)
$g_{3}=-7.8\cdot 10^{-8}\left(\frac{6.2\cdot
10^{6}~{}GeV}{f_{a}}\right)\left((D+F)\frac{1-z}{1+z}\right),$ (64)
to determine (see Fig. 16a) the $f_{a}^{*}$ for a value of $S=0.55$161616It is
appropriate to mention here that E143-experiment provided the value for the
parameter $S$ equal to $0.30\pm 0.06$ [ref137].
$f_{a}^{*}\geqslant 0.353\cdot 10^{6}~{}~{}GeV,$ (65)
calculate (see Fig. 16a) the value of axion-nucleon coupling constant (at
$S=0.55$)
$g_{an}^{*}\leqslant 4.8\cdot 10^{-7},$ (66)
and estimate the axion mass by means of (10) and (65):
$m_{a}^{*}\leqslant 17~{}~{}eV.$ (67)
The estimates (66) and (67) demonstrate and excellent agreement with the
axion-nucleon coupling constant (34) and axion mass (11) obtained in the
framework of axion mechanism of Sun luminosity and solar dynamo – geodynamo
connection.
Figure 16: Expected rate in the axion region as a function of the $f_{a}$
axion constant for different values of the nuclear $S$ parameter. The
horizontal line indicates the upper limit obtained in the present paper
($f_{a}\sim 0.353\cdot 10^{6}$ GeV for $S=0.55$) and in CUORE-experiment
($f_{a}\sim 0.76\cdot 10^{6}$ GeV for $S=0.55$) [ref001].The insets show the
correspondence between the axion-nucleon coupling constant and Peccei-Quinn
energy scale (a) in our model and (b) in CUORE-experiment. Bounds are given
for the interval $0.15\leqslant S\leqslant 0.55$.
These results make it possible to estimate the limit on the axion-electron
coupling constant $g_{ae}$ as well. Derevianko et al. [ref138] and Derbin et
al. [ref007] showed that the cross section $\sigma_{ae}(E_{a})$ for the axio-
electric effect is proportional to the photoelectric cross section
$\sigma_{pe}(E)$, and is given by the formula [ref007]:
$\sigma_{ae}(E)=\sigma_{pe}(E)\frac{g_{ae}^{2}}{\beta}\frac{3}{16\pi\alpha}\left(\frac{E_{a}}{m_{e}c^{2}}\right)^{2}\left(1-\frac{\beta}{3}\right),$
(68)
where $E_{a}$ is the axion total energy, $\beta$ is the axion velocity divided
by the velocity of light, and $g_{ae}$ is the dimensionless axion-electron
coupling constant. At $\beta\to 1$ and $\beta\to 0$, this formula coincides
with the cross sections for relativistic and nonrelativistic axions obtained
in [ref139].
Since the axion mass is small (see (67)), let us consider a relativistic form
of Eq. (68) (i.e. the case of $\beta\to 1$) and compare it to the analogous
expression (58). As a result of such comparison we derive the upper limit on
axion-electron coupling constant with respect to (65):
$g_{ae}^{*}=\frac{2x_{e}^{\prime}m_{e}}{f_{a}^{*}}\leqslant 2.89\cdot
10^{-9}.$ (69)
The limit (69) obviously does not contradict the value of axion-electron
coupling constant (35) obtained in the framework of axion mechanism of Sun
luminosity.
So it may be concluded that the new estimations for the strength of the axion-
photon coupling ($g_{a\gamma}\sim 7.07\cdot 10^{-11}~{}GeV^{-1}$), the axion-
nucleon coupling ($g_{an}\sim 3.2\cdot 10^{-7}$), the axion-electron coupling
($g_{ae}\sim 5.28\cdot 10^{-11}$) and the axion mass ($m_{a}\sim 17$ eV)
obtained in the framework of axion mechanism of Sun luminosity and solar
dynamo – geodynamo connection are in good agreement with the CUORE experiment
data. It brings hope that this hypothesis will be justified by the future
CUORE experiment with the expected exposure of $1.4\cdot 10^{6}~{}kg\cdot day$
which is significantly larger than the current one ($43.65~{}kg\cdot day$)
[ref001].
## 4 Axion mechanism of Sun luminosity and other important experiments
In view of the axion mechanism of Sun luminosity, let us analyze the data of
known experiments on measuring the axion coupling to photon, nucleon and
electron for different axion mass ranges.
### 4.1 Axion coupling to a photon
Fig. 17a shows virtually all the major experiments on estimating the limits of
the axion-photon coupling constant. The statement of the problem in all
presented experiments included the measurement of the axion flux that could be
produced in the Sun by the Primakoff conversion of the thermal photons in the
electric and magnetic fields of the solar plasma. The difference between these
experiments consisted in the axion flux detection technique only: by the
axion-to-photon reconversion (the inverse Primakoff effect) in laboratory
transverse magnetic [ref026, ref110, ref140, ref141, ref142, ref143, ref144,
ref145] and electric (the intense Coulomb field of nuclei in a crystal lattice
of the detector plus the Bragg scattering technique [ref146, ref147, ref096,
ref097, ref098, ref099]) fields.
Figure 17: (a) Exclusion regions in the ”$m_{a}$ – $g_{a\gamma}$”-plane
achieved by CAST in the vacuum [ref140, ref026], 4He and 3He phase [ref141].
We also show constraints from the Tokyo helioscope [ref142, ref143, ref144],
BNL telescope [ref145], SOLAX [ref146], COSME [ref147], CDMS [ref148], DAMA
[ref149] and the hot dark matter limit (HDM) for hadronic axions
$m_{a}<1.05~{}eV/c^{2}$ [ref150] inferred from WMAP observations of the
cosmological large-scale structure. The yellow band represents typical
theoretical models with $|E/N-1.95|=0.07\div 7$. The red solid line
corresponds to $E/N=0$ (the KSVZ model). The field theoretic expectations are
shown together with the string theory $Z_{12-I}$ model of Choi, Kim, and Kim
(the green line) [ref151, ref151a]. (b) Same as (a), but all the experimental
data are corrected with account for the solar equator effect and a
contribution of the Primakoff effect into the total Sun luminosity. A red star
denotes the result obtained in the present paper.
The new constants for the axion mechanism of Sun luminosity, obviously, should
be calculated with due regard for the solar equator effect and Primakoff
effect contribution into the total solar luminosity. It is not hard to show
that in this case they should be
$(g_{a\gamma}^{*})^{2}=\left(\frac{\Phi_{Brems}+\Phi_{Compt}+\Phi_{Pr}+\Phi_{M1}}{\Phi_{Pr}}\cdot\Delta_{equ}^{a}\right)^{-1}\cdot
g_{a\gamma}^{2}$ (70)
Hence, the values of the partial components of Sun luminosity
($\Phi_{Brems}\sim 67.48\%$, $\Phi_{Compt}\sim 32.41\%$, $\Phi_{Pr}\sim
0.08\%$, and $\Phi_{M1}\sim 0.03\%$) and the solar equator effect probability
($\Delta_{equ}\sim 0.05$) lead to a new value of $g_{a\gamma}^{*}$ constant:
$g_{a\gamma}^{*}=1.26\cdot g_{a\gamma}.$ (71)
Fig. 17b shows the new limits obtained for the axion mechanism of Sun
luminosity taking into account (71) for the corresponding experiments. The
change relative to the initial picture (Fig. 17a) is rather small, since the
small contribution of the Primakoff effect into total Sun luminosity
”effectively” compensates the influence of the solar equator effect. At the
same time it is interesting to note that our values for the axion-photon
coupling constant ($g_{a\gamma}^{*}\sim 7.07\cdot 10^{-11}~{}GeV^{-1}$) and
the axion mass ($\sim$17 eV) fit well into the existing limits (a red star in
Fig.17).
### 4.2 Axion coupling to an electron
Let us now consider the paper by Derbin et al. [ref007]. In this paper the
axio-electric effect in silicon atoms is sought for solar axions appearing
owing to bremsstrahlung and the Compton process. Axions are detected using a
Si(Li) detector placed in a low-background setup. As a result, new model-
independent constraints have been obtained for the axion-electron coupling
constant and the mass of the axion. For axions with a mass smaller than 1 keV,
the resulting bound is $g_{ae}\leqslant 2.2\cdot 10^{-10}$ (at 90% C.L.).
Obviously, the new constants for the axion mechanism of Sun luminosity should
be derived with due account taken of the solar equator effect and a
contribution of bremsstrahlung and the Compton process into the total Sun
luminosity. It is not hard to show that in this case they have the following
form:
$(g_{ae})^{2}=\left(\frac{\Phi_{Brems}+\Phi_{Compt}+\Phi_{Pr}+\Phi_{M1}}{\Phi_{Brems}+\Phi_{Compt}}\Delta_{equ}^{a}\right)^{-1}\cdot
g_{ae}^{2}.$ (72)
This time, the values of the partial components of Sun luminosity
($\Phi_{Brems}\sim 67.48\%$, $\Phi_{Compt}\sim 32.41\%$, $\Phi_{Pr}\sim
0.08\%$, and $\Phi_{M1}\sim 0.03\%$) and the solar equator effect probability
($\Delta_{equ}\sim 0.05$) lead to a new estimate of the $g_{ae}^{*}$ constant
limit for the axions with a mass smaller than 1 keV:
$g_{ae}^{*}\leqslant 4.47\cdot g_{ae}\cong 9.8\cdot 10^{-10},$ (73)
where $g_{ae}\leqslant 2.2\cdot 10^{-10}$ [ref007].
Fig. 18b shows the new limitations obtained in the context of the axion
mechanism of Sun luminosity for the corresponding experiments, according to
(72). There is a substantial change in Fig. 18b relative to the initial Fig.
18a, because the relatively high contribution of bremsstrahlung and the
Compton process into the total Sun luminosity cannot compensate the solar
equator effect probability largely. At the same time, the determined values of
the axion-electron coupling constant ($g_{ae}\sim 5.28\cdot 10^{-11}$) and the
axion mass ($\sim$ 17 eV) fit rather well into the known limits (a red star in
Fig. 18)
Figure 18: (a) Summary of limits on axion-electron coupling. The limits shown
include the astrophysical bound from the solar neutrino flux [ref080],
dedicated axion experiments by Derbin using 169Tm [ref006] and Si(Li)
[ref007]; CDMS, CoGeNT, DAMA and XMASS data obtained from [ref148, ref153,
ref154, ref155]; (b) the data from (a) with a due correction associated with
the solar equator effect and a contribution of bremsstrahlung and the Compton
process into the total Sun luminosity, except for the astrophysical bound from
the solar neutrino flux which is corrected for the contribution of the Compton
process into total luminosity only.
## 5 Axion dark matter and extragalactic background light
Naturally, having the complete axion ”portrait”, we are entitled to ask a
question about the conditions of its detectability. In other words, provided
that axions do exist, once they are produced, where are they likely to be
found? Let us turn to the astrophysical observations. Since the current
temperature of the cosmic microwave background radiation is $T\cong 2.35\cdot
10^{-4}~{}eV$ [ref156], axions are nonrelativistic and have been since before
decoupling. Therefore axions should, in accord with the equivalence principle,
fall with baryons and any other particles into the various potential wells
which develop in the Universe [ref157]. The most likely place to find light
relic axions is in clusters of galaxies and the halos of galaxies. It is
always possible, however, that the lines of sight to localized regions are
partially obscured by previously unrecognized amount of absorbing material, so
that the decay photon flux is underestimated [ref158]. To get around this
problem, we shall consider the diffuse extragalactic background (EBL) rather
than the flux from any particular region of the sky.
Before we proceed to the discussion of the axions emission and detectability
questions, let us consider some important theoretical limitations on the axion
parameters. So, within the framework of our mechanism, the new estimations of
the strength of the axion coupling to a photon ($g_{a\gamma}\sim 7.07\cdot
10^{-11}~{}GeV^{-1}$), the axion-nucleon coupling ($g_{an}\sim 3.2\cdot
10^{-7}$) and the axion-electron coupling ($g_{ae}\sim 5.28\cdot 10^{-11}$)
have been obtained. It is necessary to note that obtained estimations cannot
be excluded by the existing experimental data (see Fig. 17 and Fig. 18),
because the discussed above effect of solar axion intensity modulation by
magnetic field variations in the solar tachocline zone was not taken into
account in these observations. The obtained estimates for the strength of the
axion-photon coupling and the axion-nucleon coupling also cannot be ruled out
by the existing theoretical limitations known as the globular cluster star
limit ($g_{a\gamma}<6\cdot 10^{-11}~{}GeV^{-1}$) and the red giant star limit
($g_{ae}<3\cdot 10^{-13}$) [ref100], since these values are highly model-
dependent171717In this context it is interesting to quote a remark from the
paper by Hannestad S., Mirizzi A., Raffelt G.G., and Wong Y.Y.Y. [ref150]:
”…In principle, $f_{a}\leqslant 10^{9}~{}GeV$ is excluded by the supernova SN
1987A neutrino burst duration… However, the sparse data sample, our poor
understanding of the nuclear medium in the supernova interior, and simple
prudence suggest that one should not base far-reaching conclusions about the
existence of axions in this parameter range on a single argument or experiment
alone. Therefore, it remains important to tap other sources of information,
especially if they are easily available”.. That is to say these limitations
have a high level of uncertainty because of the absence of the standard
theoretical model for globular cluster stars and red giant stars (see, for
example the analysis of theoretical models of the hot core in [ref100] the
cross section for axion absorption in [ref112], and a note for Fig. 3 in
[ref100]).
There is one more important fact related to the so-called ”axion trapping”
effect which was not taken into account quantitatively in the paper by Raffelt
[ref100]. It is known [ref159, ref159a, ref160, ref161, ref162, ref163], that
the axion flux from the supernova can be suppressed enough in two parameter
regions [ref162, ref163]. If axion-nucleon-nucleon interaction is weak enough,
the axion cannot be effectively produced in the core of the supernova.
Quantitatively, for $f_{a}\geqslant 10^{9}~{}GeV$, the axion flux can be small
enough not to affect the cooling process [ref159, ref159a, ref160, ref161,
ref162]. On the contrary, if the axion interacts strongly enough, the mean
free path of the axion becomes much shorter than the size of the core, and
hence the axions cannot escape from the supernova. In this case, axion is
trapped inside the so-called ”axion sphere”, and the axion emission is also
suppressed. In this case, axions are emitted only from the surface of the
axion sphere; this type of the axion emission is often called ”axion burst”.
Quantitatively, for $f_{a}\leqslant 2\cdot 10^{6}~{}GeV$ (or equivalently,
$m_{a}\geqslant 3~{}eV$), the axion luminosity from SN1987A is suppressed
enough [ref159, ref159a, ref160, ref161, ref162, ref163].
For $f_{a}\leqslant 2\cdot 10^{6}~{}GeV$ suggested from the cooling of
supernova, following the paper [ref163] we have another constraint from the
detection of axions in water Cherenkov detectors. In this parameter region,
the axion flux from the axion burst is quite sizable for its detection, even
though it does not affect the cooling of SN1987A. If the axion-nucleon-nucleon
coupling is strong enough, axions may excite the oxygen nuclei in the water
Cherenkov detectors (${}^{16}\text{O}+a\to{{}^{16}\text{O}}^{*}$), followed by
radiative decays of the excited state. If this process had happened, the
Kamiokande detector should have observed the photons emitted from the decay of
${}^{16}\text{O}^{*}$. Due to the non-observation of this signal,
$f_{a}\leqslant 3\cdot 10^{5}~{}GeV$ is excluded [ref112].
Alternatively stated, the hadronic axion with the axion decay constant in the
following range is still viable with all the astrophysical
constraints181818Somewhat more conservative estimates performed by Turner
[ref162] and Ressel [ref157] give the following limitations for the hadronic
axion: $2~{}eV\leqslant m_{a}\leqslant 5~{}eV$ and $3~{}eV\leqslant
m_{a}\leqslant 8~{}eV$ respectively. However, we are going to use the
limitation (74) since the uncertainty of above-mentioned estimates is
determined by the factor of $\geqslant 3$ [ref162]. [ref163]
$3\leqslant m_{a}\leqslant 20~{}[eV]$ (74)
The mentioned consequences of the ”axion trapping” effect may be illustrated
as follows (Fig. 19). Given that the value of the axion coupling to a photon
in terms of the axion mechanism of Sun luminosity is $g_{a\gamma}\sim
7.07\cdot 10^{-11}~{}GeV^{-1}$, it is easy to write an equation for a straight
line
$c_{a\gamma\gamma}=\frac{2\pi
f_{a}}{\alpha}g_{a\gamma}=0.06\cdot\frac{f_{a}}{10^{6}~{}GeV},$ (75)
which marks a sort of a ”border” (Fig. 19) between the allowed and forbidden
values of the $c_{a\gamma\gamma}$ constant and the energy scale $f_{a}$
associated with the break-down of the U(1) PQ symmetry.
Figure 19: Astrophysical constraints on the axion mass $m_{a}$ from the
cooling of the supernova, axion burst, cooling of the HB stars, the
extragalactic background light [ref158] (squares), and the emission line in
clusters of galaxies [ref157] (triangles). Shaded region is excluded. A purple
star marks our result ($c_{a\gamma\gamma}\cong 0.02$, $f_{a}=0.353\cdot
10^{6}~{}GeV$). The graph is inspired by [ref163].
Now everything is ready for discussion of the axionic contribution to the EBL.
The axion with a mass $m_{a}=17~{}eV$ has been termed ”invisible” because it
interacts very, very weakly. Its lifetime far exceeds the age of the Universe
$\tau_{a\rightarrow\gamma\gamma}^{*}=\frac{64\pi}{g_{a\gamma}^{2}m_{a}^{3}}\cong
1439\cdot t_{Univ},$ (76)
where the age of the Universe $t_{Univ}\approx 4.34\cdot 10^{17}~{}s$ [ref164,
ref156]. Notice that the lifetime of the axion is longer than the age of the
Universe for $m_{a}=17~{}eV$ and $c_{a\gamma\gamma}\cong 0.02$ and hence
primordial axions are still in the Universe. However, as we will see later,
radiative decay of the axion may affect the background UV photons in spite of
the long lifetime.
The number ($N_{a}$) of axions in a cluster (the mass $M$) is [ref165]
$N_{a}\sim 10^{66}\frac{M}{M_{Sun}}\left(\frac{eV}{m_{a}}\right).$ (77)
It seems to have gone unnoticed that this number may be so large that the
cluster luminosity
$L_{a}=\frac{m_{a}N_{a}}{\tau_{a}^{*}}$ (78)
is easily measured [ref165]. In this connection we explore this possibility of
observable photon luminosity caused by the cluster’s axions decay. Let us
remind that from now on we shall consider the diffuse extragalactic background
(EBL) rather than the galaxy luminosity.
According to the stated above, we shall start with examining the effect of
dark matter in the form the light axions on the extragalactic background
light. In dong so we follow the theoretical results by Overduin and Wesson
[ref166], who assumed that the axions are clustered in Galactic halos with
nonzero velocity dispersions and derived an expression for the intensity of
the axionic contribution which describes the axion halos as a luminous element
of a pressureless perfect fluid in the standard Friedman-Robertson-Walker
universe. To go further and compare our predictions with observational data,
we would like to calculate the intensity of axionic contributions to the EBL
as a function of the wavelength $\lambda_{0}$ after the manner of [ref166]:
$I_{\lambda}(\lambda_{0})=\frac{\Omega_{a}\rho_{crit,0}}{\sqrt{32\pi^{3}}hH_{0}\tau_{a}}\left(\frac{\lambda_{0}}{\sigma_{\lambda}}\right)\cdot\int\limits_{0}^{z_{f}}\frac{\exp\left\\{-\frac{1}{2}\left[\frac{\lambda_{0}/(1+z)-\lambda_{a}}{\sigma_{\lambda}}\right]^{2}\right\\}dz}{(1+z)^{3}\left[\Omega_{m,0}(1+z)^{3}+1-\Omega_{m,0}\right]^{1/2}},$
(79)
where $z_{f}=30$ [ref166];
$\lambda_{a}=24800\text{\AA}\left(\frac{eV}{m_{a}}\right)$ (80)
is a peak wavelength of the decay photons;
$\sigma_{\lambda}=2\frac{v_{c}}{c}\lambda_{a}\approx
220\text{\AA}\left(\frac{eV}{m_{a}}\right)$ (81)
is the standard deviation of the Gaussian spectral energy distribution, for
which the velocity dispersion $\upsilon_{c}$ (for axions bound in galaxy
clusters) rises to as much as 1300 km/s [ref157, ref166];
$\Omega_{a}=5.2\cdot 10^{-3}h_{0}^{-2}\left(\frac{m_{a}}{eV}\right)$ (82)
is the present density parameter of the thermal axions; $h$ is the Planck
constant; $H_{0}=100h_{0}~{}km\cdot s^{-1}\cdot Mpc^{-1}$ is the Hubble
constant, $h_{0}=0.75\pm 0.15$ [ref166] is the usual value of the Hubble
constant expressed in units of $100~{}km\cdot s^{-1}\cdot Mpc^{-1}$;
$\rho_{crit,0}=1.88\cdot 10^{-29}~{}g\cdot cm^{-3}$ is the present critical
density; $\Omega_{m,0}=\Omega_{a}+\Omega_{bar}+\Omega_{\nu}=0.266\pm 0.029$
[ref167] is the present total density parameter of the axions ($\Omega_{a}$),
baryons ($\Omega_{bar}=0.028\pm 0.012$ [ref166]) and neutrinos
($\Omega_{\nu}\leqslant 0.014$ [ref168]); the expression for the decay
lifetime of the axion decay into photon pairs was used in the following form:
$\tau_{a\rightarrow\gamma\gamma}^{*}=\frac{2^{8}\pi^{3}}{c_{a\gamma\gamma}^{2}\alpha_{em}^{2}}\frac{f_{a}^{2}}{m_{a}^{3}}=(1.54\cdot
10^{7}t_{Univ})\zeta^{-2}\left(\frac{m_{a}}{eV}\right)^{-5},~{}~{}\zeta=\frac{c_{a\gamma\gamma}}{0.72};$
(83)
Evaluating Eq. (79) over $1500\text{\AA}\leqslant\lambda_{0}\leqslant
20,000\text{\AA}$ with $\zeta=0.03$ and $z_{f}=30$, we obtain the plots of
$I_{\lambda}(\lambda_{0})$ shown in Fig. 20. Apparently, the theoretical fit
of spectral intensity of the background radiation (83) produced by axion
decays describes the known experimental data in the near ultraviolet and
optical bands very well. They include data from several ground-based telescope
observations (SS78 [ref169], D79 [ref170], BK86 [ref171]), sounding rockets
(H77 [ref172], H78 [ref173]), Apollo-Soyuz mission (P77 [ref174]), the Pioneer
10 spacecraft (T83 [ref175]), the DIRBE instrument aboard the COBE satellite
(H98 [ref176], WR00 [ref177], C01 [ref178]), S2/68 sky-survey telescope aboard
TD-1 satellite (G80 [ref179]).
Figure 20: The spectral intensity $I_{\lambda}(\lambda_{0})$ of the background
radiation from decaying axions as a function of the observed wavelength
$\lambda_{0}$. The curves for the value of $m_{a}=17~{}eV$,
$\zeta=c_{a\gamma\gamma}/0.72=0.03$ correspond to upper, median and lower
limits on $h_{0}$. Also observational upper limits (solid symbols and a heavy
line) and reported detections (empty symbols) over this waveband are shown.
Experimental data depicted in red were not taken into account for the
theoretical fit of the spectral intensity $I_{\lambda}$ (blue line). See
explanations in the text.
It is important to keep in mind that some of the experimental data were not
taken into account at all during the construction of the theoretical fit of
the spectral intensity $I_{\lambda}$. For example, OAO-2 satellite (LW76
[ref180]), sounding rockets (J84 [ref181], T88 [ref182]), the Space Shuttle-
borne Hopkins UVX experiment (M90 [ref183]), and combined Hubble Space
Telescope – Las Campanas Telescope observations (B02 [ref184]), shown in red
in Fig. 20.
One of the reasons for not considering the data from LW76 [ref180], J84
[ref181], T88 [ref182], (M90 [ref183] is that these experiments involved
measurements of certain parts of the sky only. This is a grave methodological
disadvantage which, according to Gondhalekar [ref179], may lead to serious
distortions of the true EBL value, because ”…the individual observation were
taken over different regions of the sky and cover different wavelength ranges.
It should be noted that intensity of the observed inter-stellar radiation
field shows significant variation, not only with galactic latitude but also
with galactic longitude and these variations should be taken into account when
comparing observation taken over different regions of the sky. Strictly, an
accurate determination of the total interstellar radiation density requires
integration of observations made over the whole sky”. Experimental data from
B02 [ref184] were dropped due to the other reason which is related to the fair
criticism by Mattila [ref185], who casts doubt on the data calibration method
and, consequently, on the accuracy of the obtained results.
Let us now make a short comment regarding the quantity $\zeta$. For this
purpose we write the effective Lagrangian density which describes the coupling
$g_{a\gamma}$ of axions to photons:
$L_{a\gamma\gamma}=\frac{g_{a\gamma}}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}a=-g_{a\gamma}\vec{E}\cdot\vec{B}a.$
(84)
where $a$ is the axionic field, $F$ is the electromagnetic field-strength
tensor, $\tilde{F}$ is its dual, and $\vec{E}$ and $\vec{B}$ are the electric
and magnetic fields, respectively. The axion-photon coupling constant is
$g_{a\gamma}=\frac{\alpha}{2\pi f_{a}}c_{a\gamma\gamma}=\frac{\alpha}{2\pi
f_{a}}\left[\frac{E_{PQ}}{N}-\frac{2(4+z)}{3(1+z)}\right],$ (85)
where $E_{PQ}$ and $N$, respectively, are the electromagnetic and color
anomaly of the axial current associated with the axion field. Here
$z=m_{u}/m_{d}$ is the $u$\- and $d$-quarks masses ratio.
Since the theoretical fit (Fig. 20) of the spectral intensity $I_{\lambda}$ of
the background radiation from decaying axions was performed for the value
$\zeta=0.03$, the constant $c_{a\gamma\gamma}$, according to (83), will be
$c_{a\gamma\gamma}=0.72\zeta=\left[\frac{E_{PQ}}{N}-\frac{2(4+z)}{3(1+z)}\right]\simeq
0.02.$ (86)
This value is extremely important, because according to [ref186], it is a
strong indicator of (a) the effects of axion emission on the evolution of
helium burning low-mass stars [ref187], (b) the effect of decaying relic
axions on the diffuse extragalactic background radiation [ref157, ref158].
That is the effects (a) and (b) provide limit to the axion-photon coupling
[ref186]. It can be shown, that a combined action of effects (a) and (b) with
$z=m_{u}/m_{d}=0.56$ and $f_{a}\cong 10^{6}~{}GeV$ leads to the following
important limitation:
$c_{a\gamma\gamma}\leqslant\frac{f_{a}}{10^{7}~{}GeV}=0.02.$ (87)
The exact match of Eqs. (86) and (87) reflects a remarkable fact that the
properties of the axion studied ($m_{a}=17~{}eV$, $f_{a}=0.353\cdot
10^{6}~{}GeV$) conform to conditions (a) and (b) completely. Moreover, they
satisfy the conditions (see [ref186] and Fig. 19) of the axion’s effect on the
neutrino burst from SN 1987A (the so-called ”trapping regime” in which the
axion emission would not have a significant effect on the neutrino burst
[ref159, ref159a, ref160, ref161, ref162]) and the effect of axions emitted
from SN1987A on the Kamiokande II detector, coming from a condition of absence
of a large signal at the Kamiokande detector [ref112]).
Noteworthy, the axion under question ($m_{a}=17~{}eV$, $c_{a\gamma\gamma}\cong
0.02$) may be put into the thermally-produced hadronic axions class (see Fig.
17 and Fig. 18), and it may also be considered as a real candidate for dark
matter at the same time. Here is why.
It is usually assumed that dark matter in standard cosmology models is
produced during the radiation-dominated era. If the axion mass $m_{a}\geqslant
10^{-2}~{}eV$, axions were produced thermally, with cosmological abundance
$\Omega_{a}h_{0}^{2}=\frac{m_{a}}{130~{}eV}\left(\frac{10}{g_{*S,F}}\right),$
(88)
where $g_{*S,F}$ is the effective number of relativistic degrees of freedom
when axions freeze out of equilibrium. Turner showed [ref159, ref159a] that
the vast majority of these would have arisen in the early Universe via thermal
mechanisms such as Primakoff scattering and photo-production. The Boltzmann
equation can be solved to give their present comoving number density as
$n_{a}=(830/g_{*S,F})~{}cm^{-3}$ [ref157], where $g_{*S,F}\approx 15$ counts
the number of relativistic degrees of freedom left in the plasma at the time
when axions ”froze out” of equilibrium. The present density parameter
$\Omega_{a}=n_{a}m_{a}/\rho_{crit,0}$ of thermal axions thus leads to the
expression (82) whence it follows
$0.11\leqslant\Omega_{a}\leqslant 0.25.$ (89)
Here we have taken $0.6\leqslant h_{0}\leqslant 0.9$ as usual. This is
comparable to the density of dark matter.
At the same time, there are axion constraints in the so-called nonstandard
thermal histories. As it was shown in [ref188], the most intriguing one among
the well-known non-standard thermal histories (low-temperature reheating and
kination cosmologies) is the LTR cosmology which allows for the fact that
there is currently no direct evidence for radiation domination prior to big-
bang nucleosynthesis [ref189]. According to this scenario, radiation
domination begins as late as 1 MeV, and is preceded by significant entropy
generation. Thermal axion relic abundances are then suppressed, and
cosmological limits to axions are loosened. However, for reheating
temperatures $T_{rh}\leqslant 35~{}MeV$, the large-scale structure limit to
the axion mass is lifted (see Fig. 21). The remaining constraint from the
total density of matter is significantly relaxed in this case. Constraints are
also relaxed for higher reheating temperatures. It turns out that axions will
be produced thermally, with nonstandard cosmological abundance
$\Omega_{a}^{nth}h_{0}^{2}=\frac{m_{a}}{130~{}eV}\left(\frac{10}{g_{*S,F}}\right)\cdot\gamma(T_{rh}/T_{F}),$
(90)
where $T_{F}$ is the decoupling temperature of axions.
Figure 21: (a) Upper limits to the hadronic axion mass from cosmology,
allowing the possibility of a low-temperature-reheating scenario. The green
region shows the region excluded by the constraint $\Omega_{a}h^{2}<0.135$ as
a function of the reheating temperature $T_{rh}$. The red region shows the
additional part of axion parameter space excluded by WMAP1/SDSS data. At low
reheating temperatures, upper limits to the axion mass are loosened. For
$T_{rh}\geqslant 170~{}MeV$, the usual constraints are recovered. Adapted from
[ref188].
(b) Estimated improvement in the accessible axion parameter space from
including more precise measurements of the matter power spectrum (the region
bounded by the yellow line), corresponding to LSST [ref190, ref191], or from
measurements of clustering on smaller length scales, corresponding to
Lyman-$\alpha$ forest measurements (the region bounded by the blue line)
[ref192]. The red region indicates the parameter space excluded by WMAP1/SDSS
measurements. Adapted from [ref188]. Our result (a red star) has the
coordinates $\left\\{T_{rh}=30~{}MeV;m_{a}=17~{}eV\right\\}$.
Using the abundance $\Omega_{a}^{nth}$ normalized by its standard value
$\Omega_{a}$ as a function of the reheating temperature, obtained by Grin et
al. (Fig. 4 in [ref188]) it is easy to derive for the mass $m_{a}=17~{}eV$ at
$T_{rh}=30-35~{}MeV$ (see Fig. 21)
$\frac{\Omega_{a}^{nth}}{\Omega_{a}}=\gamma(T_{rh}/T_{F})\cong
0.2~{}~{}for~{}~{}T_{rh}=30~{}~{}MeV.$ (91)
Hence,
$0.022\leqslant\Omega_{a}^{nth}\leqslant 0.05.$ (92)
which is rather small and comparable to the baryonic density value
$0.027\leqslant\Omega_{b}\leqslant 0.040$ [ref156] as opposed to the standard
case of (89).
Which one of them is preferable when it comes to description of real physics –
the standard expression (89) for the axion dark matter density or the non-
standard one (92)? Although there is no answer to this question nowadays, one
may still hope that, according to [ref188], future limits to axions in the
standard radiation-dominated and LTR thermal histories may follow from
constraints to their contribution to the energy density of relativistic
particles at $T\sim$1 MeV. This is due to the fact that a comparison between
the abundance of 4He and the predicted abundance from standard big-bang
nucleosynthesis (SBBN) places constraints to the radiative content of the
Universe at $T\sim$1 MeV [ref193]. As the paper [ref188] notes, this can be
stated as a constraint to the effective neutrino number ($N_{\nu}^{eff}$),
because at early times, axions contribute to the total relativistic energy
density (through $N_{\nu}^{eff}$), and thus constraints to 4He abundances can
be turned into constraints on $m_{a}$ and $T_{rh}$.
It is known that in terms of the baryon-number density $n_{b}$, the primordial
4He abundance is characterized by the expression $Y_{P}=4n_{He}/n_{b}$. In
order to translate measurements of primordial 4He abundance $Y_{P}$ to
constraints on $m_{a}$ and $T_{rh}$ we use the scaling relation by Grin et al.
[ref188]
$\Delta
N_{\nu}^{eff}(m_{a},T_{rh})=N_{\nu}^{eff}-3=\frac{43}{7}\left\\{(6.25\cdot\Delta
Y_{p}+1)-1\right\\},$ (93)
which describes the relation between the deviation $\Delta N_{\nu}^{eff}$ for
the given values of $m_{a}$ and $T_{rh}$ and the deviation $\Delta Y_{P}$ of
the primordial 4He abundance. Here the deviation $\Delta Y_{P}$ is thought of
as a deviation from the value $Y_{P}=0.2487\pm 0.0006$ accepted for SBBN-
predicted primordial abundances [ref194].
According to the calculations in [ref188], for the values $m_{a}=17~{}eV$,
$T_{rh}=30~{}MeV$ and the effective neutrino number $N_{\nu}^{eff}\cong 3.44$
(Fig. 7 in [ref188]), the deviation $\Delta N_{\nu}^{eff}$ is 0.44.
Substitution of this value into Eq. (93) gives the following value for the
$\Delta Y_{P}$ of the primordial 4He abundance:
$\Delta Y_{p}=0.0044,$ (94)
Let us present some data for the sake of comparison. One careful study gives
the value $Y_{P}=0.2565\pm 0.0010~{}(stat)\pm 0.0050~{}(syst)$ from 93
H${}_{\text{II}}$ regions [ref195], leading to the sensitivity limit
$N_{\nu}^{eff}\cong 3.61$. Interestingly enough, the deviation $\Delta
N_{\nu}^{eff}=0.61$ may be a sign of the higher mass axions existence, since
according to [ref186], for sufficiently high masses, the axionic contribution
saturates to $\Delta N_{\nu}^{eff}=4/7$ at high reheating temperatures. At the
same time the Planck satellite is expected to reach $\Delta Y_{P}=0.013$
[ref196], yielding sensitivity of $N_{\nu}^{eff}\cong 4.04$, while CMBPol (a
proposed future CMB polarization experiment) is expected to approach $\Delta
Y_{P}=0.0039$, leading to the sensitivity limit $N_{\nu}^{eff}\cong 3.30$
[ref188].
It makes it clear that the answer to a question about the preference of one
expression (standard (89) or non-standard (92)) for the axion dark matter
density over another may be searched for only in direction of a sharp increase
in observations precision. According to [ref197], measuring CMB temperature
and polarization with cosmic variance accuracy would allow to constrain
$Y_{P}$ to within 1.5%, or $\Delta Y_{P}\sim 0.0036$ (assuming flatness). Such
an ideal measurement would be able to discriminate between the BBN-guided,
deuterium based helium value and the current lowest direct helium
observations. In other words, ”…if the CMB-determined helium mass fraction
turns out to be as high as suggested by SBBN calculations combined with the
observed deuterium abundance, this could indicate a systematic error in the
present direct astrophysical helium observations. Alternatively, if the CMB
could independently determine the helium value with sufficient precision to
confirm the present low helium value coming from direct observations, then
this would be a smoking gun for new physics” [ref197]. For example, one could
imagine sterile neutrinos appearing within the nonstandard BBN scenarios,
which would agree with present observations of
$\eta_{10}=10^{10}(n_{b}/n_{\gamma})$, while having a low helium mass
fraction. To put it differently, in our opinion, in spite of the future
possible constraints to axions and low-temperature reheating from the helium
abundance and next-generation large-scale-structure surveys, it is the
appearance of the sterile neutrinos that may effectively solve the problem of
missing dark matter in the framework of LTR cosmology.
And finally, returning to our 17 eV axion ”caught” in the extragalactic
background, we may say that regardless of the scenario which provides a
significant portion of the dark matter, experimentally observed invisible
axions (Fig. 20) must have rest masses in the ”semi-visible” range (1500Å \-
20000Å) where they do contribute significantly to the light of the night sky.
In this sense one may think of our findings as a result of the axion mechanism
of Sun luminosity and the ”…nature’s most versatile dark-matter detector: the
light of the night sky” [ref166].
## 6 Relic axion-like archion and cosmic infrared background
It is well known that a direct measurement of the EBL and, particularly, the
cosmic infrared background (CIB) consists in observing the cumulative emission
from various pregalactic objects, protogalaxies, galaxies, cosmic explosions
and decaying elementary particles (including dark matter particles) throughout
the evolution of the Universe and therefore one can provide important
constraints on the integrated cosmological history of star formation [ref198,
ref199] and control the second most important contribution to the cosmic
electromagnetic background after the Cosmic Microwave Background generated at
the time of recombination at a redshift around 1000 [ref199]. This background
is expected to be composed of three main components [ref199]:
* •
the stellar radiation in galaxies concentrated in the ultraviolet and visible
with a redshifted component in the near InfraRed (IR);
* •
a fraction of the stellar radiation absorbed by dust either in the galaxies or
in the intergalactic medium;
* •
the radiation from active galactic nuclei (a fraction of which is also
absorbed by dust and reradiated in the far-IR).
Its detection is a subject of great scientific interest and the main purpose
of the Diffuse Infrared Background Experiment (DIRBE) on the Cosmic Background
Explorer (COBE) space-craft [ref176, ref177, ref178]. These studies resulted
in upper limits on the EBL in the 1.25-100 $\mu m$ region, and in the
detection of a positive isotropic signal at 140 and 240 $\mu m$.
However, there are serious problems with interpretation of some DIRBE data at
far-IR wavelengths. For example [ref198], the energy sources could either be
yet undetected dust-enshrouded galaxies, or extremely dusty star-forming
regions in observed galaxies, and they may be responsible for the observed
iron enrichment in the intracluster medium. Although there is currently no
compelling need to invoke non-nuclear energy sources to explain the COBE data,
their potential contribution to the observed EBL cannot be ruled out. It leads
to a conclusion that the exact star formation history or scenarios required to
produce the EBL at far-IR wavelengths cannot be unambiguously resolved by the
COBE observations and must await future observations [ref198].
The other type of problems is revealed by the interpretation of the DIRBE data
at near-IR wavelengths. For example, in their studies of EBL at near-IR
wavelengths Cambrésy et al. [ref178] obtain a significantly higher cosmic
background than integrated galaxy counts ($3.6\pm 0.8~{}kJy\cdot sr^{-1}$ and
$5.3\pm 1.2~{}kJy\cdot sr^{-1}$ for 1.25 $\mu m$ and 2.2 $\mu m$,
respectively), suggesting either an increase of the galaxy luminosity function
for magnitudes fainter than 30 or the existence of another contribution to the
cosmic background from primeval stars, black holes, or relic particle decay.
However, models predict other possible contributions to the background at
these wavelengths [ref200] such as a burst of star formation either in
primeval galaxies or in Population III stars ($z\approx 10$), very massive
black holes (accreting from a uniform pre-galactic medium at $z\approx 40$),
massive decaying big bang relic particles ($z\approx 300$). In this regard,
Cambresy et al. [ref178] make a natural conclusion that the new constraints in
the near-infrared should encourage revisiting the importance of those
contributions to the CIB in cosmological models.
Figure 22: The spectral intensity $I_{\lambda}(\lambda_{0})$ of the background
radiation from decaying axions as a function of the observed wavelength
$\lambda_{0}$. The curves for values $m_{a}=17.0~{}eV$,
$\zeta=c_{a\gamma\gamma}/0.72=0.03$ and $m_{a}=2.0~{}eV$, $\zeta=9.0$
correspond to upper, median and lower limits on $h_{0}$. Also observational
upper limits (solid symbols) over these wavebands are shown.
In this connection we consider the effect of dark matter in the form of 2 eV
relic particle on the extragalactic background light at near-IR region (Fig.
22). It was obtained on the basis of the intensity (79) of axionic
contributions to the EBL calculation as a function of the wavelength
$\lambda_{0}$. The theoretical fit (green curves in Fig. 22) of the spectral
intensity of the background radiation (79) produced by axion decays,
apparently, describes the known experimental data for the near-infrared band
very well. They include data from ground-based telescope observations (BK86
[ref171]) and the DIRBE instrument aboard the COBE satellite (H98 [ref176],
WR00 [ref177], C01 [ref178]).
Let us consider some properties of such relic particle with the 2 eV mass. It
may be immediately stated that such particle cannot play the role of a sterile
neutrino with the mass of the order of a few eV [ref201]. And the reason is
the following.
It is known that if neutrinos are massive and if the mass eigenstates are not
degenerate, then it is possible to have a radiative decay of the form
$\nu_{s}\to\nu+\gamma$. According to [ref202], this gives for the decay rate
of Majorana neutrinos
$\Gamma_{\nu_{S}\rightarrow\nu\gamma}^{*}=5.52\cdot
10^{-32}\left(\frac{\sin^{2}\theta}{10^{-10}}\right)\left(\frac{m_{S}}{keV}\right)^{5}~{}~{}s^{-1},$
(95)
where $m_{S}$ is the mass eigenstate most closely associated with the sterile
neutrino, and $\theta$ is the mixing angle between the sterile and active
neutrino. The decay of a nonrelativistic sterile neutrino into two (nearly)
massless particles produces a line at the energy $E{\gamma}=m_{S}/2$.
Obviously, in case of the decay reaction $\nu_{s}\to\nu+\gamma$, Eq. (79) can
be generalized to another relic $\nu_{s}$ by multiplying it by
$I_{\lambda}(\lambda_{0})\cdot\frac{1}{2}\cdot\frac{\Omega_{\nu}}{\Omega_{a}}\cdot\frac{\tau_{a\rightarrow\gamma\gamma}^{*}}{\tau_{\nu_{S}\rightarrow\nu\gamma}^{*}},$
(96)
where the 1/2 is a number of photons produced in each $\nu_{s}$ decay,
$\Omega_{\nu}\leqslant 0.014$ is the present total density parameter of the
neutrinos,
$\tau_{\nu_{\chi}\rightarrow\nu\gamma}^{*}=(\Gamma_{\nu_{S}\rightarrow\nu\gamma}^{*})^{-1}=(1.7\cdot
10^{15}t_{Univ})\zeta^{-2}\left(\frac{m_{S}}{keV}\right)^{-5},$ (97)
where
$\zeta^{-2}=\left(\frac{10^{-10}}{\sin^{2}2\theta}\right).$ (98)
Using the generalization (96), it is rather easy to show that a theoretical
fit of spectral intensity $I_{\lambda}$ of the background radiation is several
orders of magnitude lower than the experimental data in near-infrared band
under any reasonable values of the mixing angle $\sin^{2}2\theta$.
This relic particle with the 2 eV mass may be supposed to belong to a class of
axion-like particles with rather exotic properties. One of them may originate
from the fact that in the distant past their birth in the electromagnetic
field, or in other words interaction with a photon, was not suppressed for
some reason, while at the present time there are simply no conditions suitable
for their birth. To put it differently, the mentioned conditions of such
particles’ birth must be completely suppressed nowadays in order our axion
mechanism of Sun luminosity to be possible.
In our opinion, such axion-like particle, provided that it exists, has the
properties most similar to those of the so-called archion [ref203, ref204,
ref205, ref206, ref207, ref208, ref209]. An archion may be very similar to a
hadronic axion with highly suppressed interaction with leptons under certain
conditions. Let us discuss briefly some of its properties below.
As is generally known, in all the models of an invisible axion this particle
appears as a Goldstone boson connected with the phase of a complex
SU(2)$\times$U(1) singlet Higgs field. The axion coupling to the gauge bosons
appears in these models after the U(1)PQ symmetry violation by means of a
mechanism, specified by the non-vanishing color anomaly U(1)PQ \- SU(3)c \-
SU(3)c [ref210].
In its most generic form the Lagrangian of axion interaction with fermions
(quarks and leptons) and photons is
$L=c_{\alpha\beta}\alpha\cdot\bar{F}_{\alpha}(\sin\theta_{\alpha\beta}+i\gamma_{5}\cos\theta_{\alpha\chi})F_{\beta}+g_{a\gamma}aF_{\mu\nu}\bar{F}^{\mu\nu},$
(99)
where $\alpha,\beta=1,2,3$ indices denote the generation of fermions $F$, and
the constants $\theta_{\alpha\beta}$,
$c_{\alpha\beta}\propto f_{a}^{-1}$ (100)
and
$g_{a\gamma}\propto f_{a}^{-1}$ (101)
depend on the axion model chosen.
A model of an archion [ref203, ref204, ref205, ref206, ref207, ref208, ref209]
arose from a model of horizontal unification, the basis of which is expounded
in a monograph by Khlopov [ref210]. This theory includes the global U(1)H
symmetry, the spontaneous breaking of which leads to prediction of a Goldstone
boson of the invisible axion type. Such boson called ”archion” by authors of
[ref205, ref206, ref207, ref208] has the flavor non-diagonal as well as the
flavor diagonal coupling to fermions.
The global U(1)H symmetry in horizontal unification may be identified with the
Peccei–Quinn symmetry U(1)PQ [ref028, ref029, ref030, ref031, ref032, ref032a,
ref033, ref033a], which is due to the fact of triangle anomaly existence in
axial currents U(1)H interaction with gluons [ref210].
In the simplest variant of the horizontal unification (the gauge symmetry
$S(U)_{c}\otimes SU(2)\otimes U(1)\otimes SU(3)_{H}\otimes U(1)_{H}$ (102)
with a minimal set of heavy fermions) the anomaly is compensated, and the
archion remains almost massless. The interaction of the archion with photons
is absent because of the parallel compensation which corresponds to the
current-photon-photon anomaly.
On the other hand, according to [ref210], within any realistic extension of
the horizontal unification model up to the grand unification symmetry, for
example, within the extension to the SU(5)${}_{H}\otimes$SU(3)H symmetry,
there is no compensation because of the extra heavy fermions, so that the
archion appears to be similar to a hadronic axion with strongly suppressed
interaction with leptons.
In the framework of the archion model, the archion-photon coupling constant
$g_{a\gamma}$ which appears in Eqs. (99) and (101) has the following form
[ref210]:
$g_{a\gamma}=\frac{\alpha}{2\pi f_{a}}c_{a\gamma\gamma}=\frac{\alpha}{4\pi
f_{a}}\frac{A_{c}z^{3/2}}{(1+z)^{2}}\left[\frac{A_{em}}{A_{c}}-\frac{2(4+z)}{3(1+z)}\right]^{-1}.$
(103)
Here as usual $z=m_{u}/m_{d}$ is the $u$\- and $d$-quarks mass ratio, $f_{a}$
is the energy scale associated with the breakdown of the U(1)H (or
equivalently, the U(1)PQ symmetry), $A_{c}$ and $A_{em}$ are the color and
electromagnetic anomalies respectively.
Taking into account the value of the $c_{a\gamma}=0.72\zeta$ constant (see
(83) and Fig. 22), which is $\sim$6.5 for the archion mass of 2 eV, it is easy
to estimate the archion-photon coupling constant from (103) and (10):
$g_{a\gamma}^{*}\simeq 2.5\cdot 10^{-9}~{}~{}GeV^{-1}.$ (104)
Obviously, if the archion indeed exists and has such a high archion-photon
coupling constant, it must be of the relic origin only in a sense that the
conditions of such particles birth must be completely suppressed nowadays.
Otherwise, our axion mechanism of Sun luminosity would not be possible, like
we pointed out earlier191919It is necessary to note that regardless of whether
one assumes the existence of the axion mechanism of Sun luminosity or not,
such high value of the archion-photon coupling constant (104) is forbidden by
the DAMA experiment observations [ref146] presented in Fig. 17a and 17b.. The
value of this constant is also very important because according to [ref186],
it is a strong indicator of (a) the effects of axion emission on the evolution
of helium burning low-mass stars [ref187], (b) the effect of decaying relic
axions on the diffuse extragalactic background radiation [ref157, ref166]. In
other words, an archion with the 2 eV mass, characterized by the archion-
photon coupling constant (104), must be relic in order not to violate the
known limitation (87) which is a consequence of combined action of effects (a)
and (b).
For all invisible axion models, including the archion model, Lagrangian of its
interaction with nucleons has the same form [ref209]. Given (104), it is not
hard to estimate the archion-nucleon coupling constant for the 2 eV mass
archion (Fig. 22) using (29)-(31) and (10):
$g_{an}^{*}\leqslant 5.6\cdot 10^{-8}.$ (105)
If the archion exists with such low value of the archion-nucleon coupling
constant, it obviously must have only relic origin in order not to violate the
SN1987A limit ($3\cdot 10^{-7}\leqslant g_{an}\leqslant 10^{-6}$ [ref111,
ref112]).
This rises the question of why would Nature need a relic axion-like archion in
addition to an ordinary hadronic axion. Curiously enough, the answer suggests
itself and is related to the evolutionary formation of the visible large-scale
cosmological structure against the background of the invisible ”dark”
structure.
Apart from the details of this scenario, our axion-like particles may be among
the primary participants of this creative action, if they do exist. Let us
note however that in spite of the fact that the cosmological structure
formation is provided by the single kind of hidden-mass particles – axions,
the ”axion liquid” turns out to be two-component. The visible structure in the
form of galaxies and superclusters is formed by the shortwave component of the
17 eV axions density perturbations spectrum, while the relic thermal 2 eV
archions background plays an important role in sub-shortwave density
perturbations spectrum evolution. The latter includes, in particular, the
massive halos formation beyond the visible parts of the galaxies. Following
[ref210], it is worth mentioning that the so-called phase-space argument by
Tramaine and Gunn [ref211], which gives rise to a known limit on particles
mass in the halo, may be substantially weakened or even omitted for the Bose
gas [ref212, ref213, ref214, ref215].
If all said above is true, it is more than sufficient for a ”sensible”
existence of axions, and particularly, a relic axion-like archion. On the
other hand, except for the indirect indication of axions existence in the form
of Fig. 22, we have no other – direct – arguments. In order to fill this gap a
little, let us try to substantiate the quantitative relations between the
experimental spectral intensities of the background radiation from decaying
axions and archions shown in Fig. 22 theoretically on the basis of a
simplified cosmological model which takes into account the processes of axion
dark matter radiative decay.
### 6.1 Decaying axion and relic archion as two components of luminous dark
matter
Let us consider the flat (for simplicity) Universe after the recombination
consisting of the following components: relic radiation, usual (light,
visible) matter, dark matter (presented by axions and archions), non-relic
radiation (resulting from their decay) and dark energy (described by the
cosmological constant $\Lambda$), then for the corresponding metrics
$ds^{2}=c^{2}dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (106)
where $a(t)$ is the scale factor, the first Friedmann equation reads
$\frac{3\dot{a}^{2}}{c^{2}a^{2}}=\kappa T_{00}+\Lambda,$ (107)
where a dot denotes the derivative with respect to $t$, $\kappa=8\pi
G_{N}/c^{4}$ ($G_{N}$ is the Newtonian gravitational constant) and the $00$
covariant component of the total energy-momentum tensor reads
$T_{00}=\varepsilon_{rr}+\varepsilon_{\nu
m}+\varepsilon_{a2}+\varepsilon_{a17}+\varepsilon_{r2}+\varepsilon_{r17},$
(108)
where, in their turn, $\varepsilon_{rr,\nu m,a1,a17,r2,r17}$ denote energy
densities of relic radiation (assumed to be independent of all other
components), visible matter (also assumed to be independent), archions (with
the mass $m_{a2}=2~{}eV$ and the disintegration constant
$\lambda_{a2}=3.88\cdot 10^{-22}~{}s$, assumed to be completely
nonrelativistic), axions (with the mass $m_{a17}=17~{}eV$ and the
disintegration constant $\lambda_{a17}=1.91\cdot 10^{-22}s^{-1}$, also assumed
to be completely nonrelativistic), radiation resulting from archion decay and
radiation resulting from axion decay respectively.
Here some additional comments should be made. First, the disintegration
constants are estimated simply as $\lambda=1/\tau^{*}$ on basis of the formula
(83). Second, we assume both dark matter components (archions and axions)
completely nonrelativistic, in other words, we completely neglect their
velocities (or temperatures). This assumption is simultaneously in agreement
and disagreement with [ref166], where the formula (185) (which is equivalent
to (79) in our text) contains simultaneously the scent of the standard
cosmological $\Lambda$CDM-model without matter velocities in the denominator
of the integrand and the scent of the velocity dispersion in its numerator. It
seems that this fact does not mean that there is a self-contradiction in the
formula (185), because the numerator may be much more sensitive to the local
velocity of dark matter than the denominator to the global one. However, in
the subsequent analysis we shall be interested in the evolution of the average
non-relic radiation energy density without taking into account its frequency
distribution and the nonzero dark matter temperature (for simplicity).
Substituting (108) into (107) and introducing the standard portions
$\Omega=\frac{\kappa
c^{2}}{3H_{0}^{2}}\varepsilon_{(0)},~{}~{}\Omega_{\Lambda}=\frac{c^{2}}{3H_{0}^{2}}\Lambda,$
(109)
where the subscript $(0)$ corresponds to the current moment of time $t=0$
(this value may be chosen without loss of generality) and $H=\dot{a}/a$ is the
Hubble parameter, we obtain
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}$
$\displaystyle\left(\Omega_{rr}\frac{\varepsilon_{rr}}{\varepsilon_{rr(0)}}+\Omega_{\nu
m}\frac{\varepsilon_{\nu m}}{\varepsilon_{\nu
m(0)}}+\Omega_{a2}\frac{\varepsilon_{a2}}{\varepsilon_{a2(0)}}+\right.$
$\displaystyle\left.+\Omega_{a17}\frac{\varepsilon_{a17}}{\varepsilon_{a17(0)}}+\Omega_{r2}\frac{\varepsilon_{r2}}{\varepsilon_{r2(0)}}+\Omega_{r17}\frac{\varepsilon_{r17}}{\varepsilon_{r17(0)}}+\Omega_{\Lambda}\right).$
(110)
Here $\varepsilon_{rr}\sim 1/a^{4}$, $\varepsilon_{\nu m}\sim 1/a^{3}$,
consequently, as usual,
$\Omega_{rr}\frac{\varepsilon_{rr}}{\varepsilon_{rr(0)}}+\Omega_{\nu
m}\frac{\varepsilon_{\nu m}}{\varepsilon_{\nu
m(0)}}=\Omega_{rr}\left(\frac{a_{0}}{a}\right)^{4}+\Omega_{\nu
m}\left(\frac{a_{0}}{a}\right)^{3},$ (111)
where $a_{0}$ is the current value of the scale factor: $a(0)=a_{0}$. Further,
$\varepsilon_{a2}:\frac{1}{a^{3}}\exp(-\lambda_{a2}t),~{}~{}\varepsilon_{a17}:\frac{1}{a^{3}}\exp(-\lambda_{a17}t)\Longrightarrow\varepsilon_{a2(0)}:\frac{1}{a_{0}^{3}},~{}~{}\varepsilon_{a17(0)}:\frac{1}{a_{0}^{3}},$
(112)
whence
$\Omega_{a2}\frac{\varepsilon_{a2}}{\varepsilon_{a2(0)}}+\Omega_{a17}\frac{\varepsilon_{a17}}{\varepsilon_{a17(0)}}=\Omega_{a2}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)+\Omega_{a17}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a17}t).$
(113)
Finally, combining first and second Friedmann equations, we come to the
following equations:
$d\left[(\varepsilon_{a2}+\varepsilon_{r2})a^{3}\right]+\frac{1}{3}\varepsilon_{r2}d(a^{3})=0,~{}~{}d\left[(\varepsilon_{a17}+\varepsilon_{r17})a^{3}\right]+\frac{1}{3}\varepsilon_{r17}d(a^{3})=0.$
(114)
From the first one we immediately get
$\frac{d(\varepsilon_{a2}a^{3})}{da}+a^{3}\frac{d\varepsilon_{r2}}{da}+4a^{2}\varepsilon_{r2}=0,~{}~{}\frac{d(\varepsilon_{a2}a^{3})}{da}=\varepsilon_{a2(0)}a_{0}^{3}\exp(-\lambda_{a2}t)\frac{(-\lambda_{a2})}{\dot{a}},$
(115)
$\frac{d\varepsilon_{r2}}{da}+\frac{4\varepsilon_{r2}}{a}=\varepsilon_{a2(0)}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)\frac{\lambda_{a2}}{\dot{a}},$
(116)
whence finally
$\displaystyle\varepsilon_{r2}$
$\displaystyle=\frac{1}{a^{4}}\left(\varepsilon_{a2(0)}a_{0}^{3}\lambda_{a2}\int\limits_{a_{0}}^{a}\exp(-\lambda_{a2}t)\frac{a}{\dot{a}}da+\varepsilon_{r2(0)}a_{0}^{4}\right)=$
$\displaystyle=\frac{1}{a^{4}}\left(\varepsilon_{a2(0)}a_{0}^{3}\lambda_{a2}\int\limits_{0}^{t}a(t)\exp(-\lambda_{a2}t)dt+\varepsilon_{r2(0)}a_{0}^{4}\right).$
(117)
Similarly,
$\varepsilon_{r17}=\frac{1}{a^{4}}\left(\varepsilon_{a17(0)}a_{0}^{3}\lambda_{a17}\int\limits_{0}^{t}a(t)\exp(-\lambda_{a17}t)dt+\varepsilon_{r17(0)}a_{0}^{4}\right).$
(118)
The substitution of (111), (113), (117) and (118) into (110) gives the
equation defining the function $a(t)$ satisfying the condition $a(t)=a_{0}$.
Taking into account that the values of both disintegration constants
$\lambda_{a2}$ and $\lambda_{a17}$ are extremely small and the radiation
energy density decreases faster than the nonrelativistic matter energy density
when $a$ increases, one can neglect all (i.e. both relic and non-relic)
radiation contributions and get
$\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}\left[\Omega_{\nu
m}\left(\frac{a_{0}}{a}\right)^{3}+\Omega_{a2}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a2}t)+\Omega_{a17}\left(\frac{a_{0}}{a}\right)^{3}\exp(-\lambda_{a17}t)+\Omega_{\Lambda}\right].$
(119)
Introducing the dimensionless quantities
$\tilde{t}=H_{0}t,~{}~{}\tilde{a}=\frac{a}{a_{0}},~{}~{}\tilde{\lambda}_{a2}=\frac{\lambda_{a2}}{H_{0}},~{}~{}\tilde{\lambda}_{a17}=\frac{\lambda_{a17}}{H_{0}},$
(120)
from (112) we obtain
$\left(\frac{1}{\tilde{a}}\frac{d\tilde{a}}{d\tilde{t}}\right)^{2}=\frac{\Omega_{\nu
m}}{\tilde{a}^{3}}+\frac{\Omega_{a2}}{\tilde{a}^{3}}\exp(-\tilde{\lambda}_{a2}\tilde{t})+\frac{\Omega_{a17}}{\tilde{a}^{3}}\exp(-\tilde{\lambda}_{a17}\tilde{t})+\Omega_{\Lambda},~{}~{}\tilde{a}(0)=1.$
(121)
According to the recent observations [ref216], we consider the following
values: $\Omega_{\nu m}=0.046$, $\Omega_{a2}+\Omega_{a17}=0.236$,
$\Omega_{\Lambda}=1-\Omega_{\nu m}-(\Omega_{a2}+\Omega_{a17})=0.718$. For
$\Omega_{a2}$ and $\Omega_{a17}$ we use the values $0.016$ and $0.220$
respectively. Again, due to extreme smallness of $\lambda_{a2}$ and
$\lambda_{a17}$ (in the presence of nonzero $\Lambda$) they do not affect
noticeably the dependence $a(t)$. The following graphs confirm this statement:
Figure 23: The numerical solution of Eq. (121) (the blue firm line) is
indistinguishable from its numerical solution for $\lambda_{a2}=0$ and
$\lambda_{a17}=0$ (the yellow dashed line).
From (117) and (118) we obtain respectively
$\tilde{t}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}},$
(122)
$\tilde{t}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}.$
(123)
Let us introduce the convenient functions
$f_{a2}(\tilde{t})=\frac{\varepsilon_{r2}}{\varepsilon_{a2(0)}}\tilde{a}^{4}-\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}}=\tilde{\lambda}_{a2}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t},$
(124)
$f_{a17}(\tilde{t})=\frac{\varepsilon_{r17}}{\varepsilon_{a17(0)}}\tilde{a}^{4}-\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}=\tilde{\lambda}_{a17}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t},$
(125)
For $a=0$ we get $f_{a2}=-9.26\cdot 10^{-5}$ and $f_{a17}=-4.56\cdot 10^{-5}$.
These negative values may be completely compensated by the proper choice of
$\varepsilon_{r2(0)}$ and $\varepsilon_{r17(0)}$ in order to give
$\varepsilon_{r2}=0$ and $\varepsilon_{r17}=0$ at the same moment of time
(when $a=0$). This has clear physical sense: we assume that at the
recombination moment the non-relic radiation is absent (all radiation, present
at that moment of time, may be considered as the relic one). From this
requirement we immediately obtain $\varepsilon_{r2(0)}=9.26\cdot
10^{-5}\varepsilon_{a2(0)}$ and $\varepsilon_{r17(0)}=4.56\cdot
10^{-5}\varepsilon_{a17(0)}$.
Now let us also depict the helpful graphs of the ratios
$\varepsilon_{r2}/\varepsilon_{a2}$ and $\varepsilon_{r17}/\varepsilon_{a17}$
as functions of $\tilde{t}$ (Fig. 24a, red and green lines respectively):
$\frac{\varepsilon_{r2}}{\varepsilon_{a2}}=\frac{\exp(\tilde{\lambda}_{a2}\tilde{t})}{\tilde{a}}\left(\tilde{\lambda}_{a2}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a2}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r2(0)}}{\varepsilon_{a2(0)}}\right),$
(126)
$\frac{\varepsilon_{r17}}{\varepsilon_{a17}}=\frac{\exp(\tilde{\lambda}_{a17}\tilde{t})}{\tilde{a}}\left(\tilde{\lambda}_{a17}\int\limits_{0}^{\tilde{t}}\tilde{a}(\tilde{t})\exp(-\tilde{\lambda}_{a17}\tilde{t})d\tilde{t}+\frac{\varepsilon_{r17(0)}}{\varepsilon_{a17(0)}}\right),$
(127)
In particular, for $t=0$ we get $\varepsilon_{r2}/\varepsilon_{a2}=9.26\cdot
10^{-5}$ and $\varepsilon_{r17}/\varepsilon_{a17}=4.56\cdot 10^{-5}$, as it
should be (Fig. 24a).
Figure 24: Graphs of the ratios $\varepsilon_{r2}/\varepsilon_{a2}$ (red),
$\varepsilon_{r17}/\varepsilon_{a17}$ (green) (a) and
$\varepsilon_{r2}/\varepsilon_{r17}$ (b) as functions of time $\tilde{t}$.
Let us also depict the helpful graph of the ratio
$\varepsilon_{r2}/\varepsilon_{r17}$ as a function of $\tilde{t}$ (Fig. 24b).
The theoretical estimate of the ratio $\varepsilon_{r2}/\varepsilon_{r17}$ is,
apparently, equal to
$\left(\frac{\varepsilon_{r2}}{\varepsilon_{r17}}\right)_{theory}\cong 0.15.$
(128)
On the other hand, it is rather easy to estimate the experimental
$\varepsilon_{r2}/\varepsilon_{r17}$ ratio using the data shown in Fig. 22 and
the expression for the spectral intensity $I_{\lambda}(\lambda_{0})$ of the
background radiation from decaying axions:
$\left(\frac{\varepsilon_{r2}}{\varepsilon_{r17}}\right)_{exper}=\frac{\int\limits_{12400}^{50000}I_{\lambda
2}(\lambda_{0})d\lambda_{0}}{\int\limits_{1500}^{15000}I_{\lambda
17}(\lambda_{0})d\lambda_{0}}\cong 0.23.$ (129)
Curiously enough, the theoretical estimate (128) and the experimental one
(129) agree fairly well, which increases one’s optimism as to the essence of
the results obtained in the current section. At the same time, our estimates
require additional thorough verification, especially when it comes to the
experimental justification of the validity and reliability of the data on the
extragalactic background light in near-ultraviolet, optical and near-infrared
bands (1500-10000Å). Indeed, it is a very important step to perform, because
otherwise our unusual results may seem more like just the luckily guessed
rules of calculation, which do not reflect the actual nature of things.
## 7 Summary and Conclusion
In the present paper we present a self-consistent model of the axion mechanism
of Sun luminosity and solar dynamo – geodynamo connection, in the framework of
which we estimate the values of the axion mass ($m_{a}\sim 17~{}eV$) and the
axion coupling constants to photons ($g_{a\gamma}\sim 7.07\cdot
10^{-11}~{}GeV^{-1}$), nucleons ($g_{an}\sim 3.20\cdot 10^{-7}$) and electrons
($g_{ae}\sim 5.28\cdot 10^{-11}$). Their verification on the basis of the
model results comparison with the known experiments is also provided,
including the CAST-, CUORE- and XMASS-experiments.
In order to explain the solar-terrestrial magnetic connection we propose the
axion mechanism explaining the Sun luminosity physics and ”solar dynamo –
geodynamo” connection, where the total energy of axions, which appear in the
Sun core, is initially modulated by the magnetic field of the solar tachocline
zone due to the inverse coherent Primakoff effect and after that it makes its
way towards the Earth where its ”iron” component containing the 14.4 keV solar
axions is resonantly absorbed inside the nickel-iron core of the Earth. It
results in the fact that the variations of the axion intensity play a role of
an energy source and a modulator of the Earth magnetic field. In other words,
the solar axion mechanism is not only responsible for formation of a thermal
energy source in the liquid core of the Earth necessary for generation and
maintenance of the Earth magnetic field, but unlike other alternative
mechanisms [ref015], naturally explains the cause of the experimentally
observed strong negative correlation of the magnetic field in the tachocline
zone of the Sun and the magnetic field of the Earth.
It is necessary to note that obtained estimations can’t be excluded by the
existing experimental data (see Fig. 17a and Fig. 18a) because the effect of
solar axion intensity modulation by temporal variations of the toroidal
magnetic field of the solar tachocline zone discussed above was not taken into
account in these observations (see Fig. 17b and Fig. 18b). On the other hand,
the obtained estimates for the axion-photon coupling and the axion-nucleon
coupling cannot be ruled out by the existing theoretical limitations known as
the globular cluster star limit ($g_{a\gamma}<6\cdot 10^{-11}~{}GeV^{-1}$) and
the red giant star limit ($g_{ae}<3\cdot 10^{-13}$) [ref100], since these
values are highly model-dependent. It actually means that the axion parameters
obtained in the present paper do not contradict any of the known experimental
and theoretical model-independent limitations.
Let us give some major ideas of the paper which formed a basis for the
statement of the problem justification, and the corresponding experimental
data which comport with these ideas either explicitly or implicitly.
### 7.1 Axion mechanism of Sun luminosity
One of the key ideas behind this mechanism is the effect of $\gamma$-quanta
channeling along the magnetic flux tubes (waveguides) in the Sun convective
zone (Fig. 8), which may be represented by the $\gamma$-quanta channeling in
the periodical structure (Fig. 6) in the particular case. A low refraction
(i.e. a high transparency) of the thin magnetic flux tubes is achieved due to
the ultrahigh magnetic pressure (see (16)), induced by the magnetic field of
about 200-400 T (Fig. 9a, adapted from [ref077]). It is noteworthy that
although such strong magnetic fields have never been used for the explanation
or interpretation of the simulation results such as the stability analysis of
tachocline latitudinal differential rotation and coexisting toroidal band
using an MHD analog of the shallow-water model [ref077], they have always been
present implicitly in the very same simulation results, as our analysis shows.
Fig. 9a provides an illustrative example where a hidden part of the ”latitude
– magnetic field of the overshoot tachocline zone” dependence (which is absent
on the original plot in Fig.11 of [ref077]) is added. A right to exist and
physical validity of this part is confirmed by the bare fact of its direct
correspondence (Fig. 9a,b) with the real X-ray image of the Sun in its active
phase (Fig. 9b) obtained in the experiments performed with the Japanese X-ray
telescope Yohkoh (1991-2001) (adapted from [ref014]).
The direct experiments on monitoring the space-time evolution of the magnetic
flux tubes, rising from the deep layers of the convective zone of the Sun
(Fig. 8b, adapted from [ref064]) proved the existence of the ”hollow” ideal
magnetic waveguides for $\gamma$-quanta. These tubes cross the photosphere and
form the solar active regions such as sunspots which are the sources of X-rays
(Fig. 8d, adapted from [ref068]). There are also some theoretical results
substantiating the effect of anchoring magnetic flux tubes in the tachocline
(Fig. 8c, adapted from [ref067]).
Finally, the most impressive evidence of the axion mechanism of Sun luminosity
are the solar images (Fig. 13, adapted from [ref014]) taken at photon energies
from 250 eV up to a few keV from the Japanese X-ray telescope Yohkoh
(1991-2001), which depict the solar X-ray activity during the last maximum of
the 11-year solar cycle (Fig. 13b). It is hard to imagine another model or
considerations which would explain such anomalous X-ray radiation distribution
over the active Sun surface just as well. One should also keep in mind that
the alternative mechanism of axion transformation into $\gamma$-quanta in the
magnetic field of sunspots or other solar active regions is completely ruled
out, since there are no signs of an X-ray bright spot at the disk center (see
Fig. 13a and b), which should otherwise be observed according to [ref217].
### 7.2 Invisible axions and Solar Equator effect
According to the axion mechanism of Sun luminosity, a part of axions that pass
through the tachocline zone near the equator and poles (Fig. 6 and Fig. 13) is
not converted into $\gamma$-quanta by the Primakoff effect because of the
magnetic field vector collinearity to the axion momentum. It is the equatorial
part (Fig. 13) of invisible axions that reaches the Earth, where its ”iron”
component (i.e. the 14.4 keV solar axions) is resonantly absorbed in the Earth
core. The energy of the solar axions supplied to the Earth core in this way
plays a role of a trigger for the generation and maintenance of the
geomagnetic field, thus giving birth to the effect of the steady
anticorrelation between the variations of the Solar magnetic field and
geomagnetic field (Fig. 3).
In this sense, it is appropriate to make a short remark related to the
anticorrelation between the solar and terrestrial magnetic fields variations.
The small variations of TSI ($\Delta$TSI) are apparently produced by the solar
magnetic field variations. At the same time, these variations of the solar
magnetic field drive the equatorial sector width variations (Fig. 13), and
consequently, the ”equatorial” axion flux (see Fig. 12)
$\Delta_{equ}^{a}=0.05\mp\Delta TSI/\langle TSI\rangle.$ (130)
In other words, the ”equator” effect is not only the source of ”invisible”
axions, it also modulates their intensity inversely proportional to the solar
magnetic field change (see (130)), thus maintaining the inverse correlation
between the solar magnetic field and the ”invisible” axions flux (Fig. 12).
The latter is also the cause of the inverse correlation between the solar and
terrestrial magnetic fields variations (Fig. 3).
### 7.3 Axion mechanism of the solar dynamo – geodynamo connection
This mechanism is responsible for the physical (axionic) nature of the steady
anticorrelation between the variations of the solar magnetic field and the
geomagnetic field (the Y-component) which is directly proportional to the
westward drift of magnetic features (Fig. 4, Fig. 5) at the measurement points
(Western Europe and Australia). Figuratively speaking, in this case the
Y-component of the Earth magnetic field acts as a ”measuring instrument” which
tracks the influence of the ”equatorial” 14.4 keV solar axions on thermal and
magnetic processes in the liquid core of the Earth. The physical scenario for
this may be following.
The ”iron” solar axions that are resonantly absorbed in the Earth core,
activate the vertical background motion along the gravity force. This motion,
in its turn, ”pushes” the temperature gradient to the bottom of the liquid
core more or less heavily (depending on the total energy of axions), thereby
changing the temperature profile in the convective medium of the Earth core.
The change in the temperature profile also changes the thermal conditions near
the nuclear georeactor situated at the boundary of the liquid and solid core
of the Earth. Since the power output of such reactor is proportional to
temperature in the range of 3000-5000C∘ typical for the Earth liquid core, it
means that the variations of the Earth core temperature generated by the
mechanism of solar dynamo-geodynamo connection induce the corresponding
variations of the nuclear georeactor thermal power. If the georeactor
hypothesis is true, the fluctuations of georeactor thermal power, induced by
the variations of the absorbed axions energy in the Earth’s core, can
influence the Earth global climate in the form of anomalous temperature jumps
in the following way. Strong fluctuations of the georeactor thermal power can
lead to partial blocking of the convection in the liquid core [ref113, ref114,
ref115] and the change of the angular velocity of liquid geosphere rotation,
thereby changing the angular velocities of the Earth mantle and crust by
virtue of the total angular momentum conservation law. It means that the heat,
or more precisely the dissipation energy caused by friction of the Earth
surface and lower atmosphere, can make a considerable contribution to the
total energy balance of the atmosphere and thereby influence on the Earth
global climate evolution significantly [ref113, ref114, ref115].
On the other hand, it is clear that understanding of the mechanism of the
solar-terrestrial magnetic correlation can become the clue of the so-called
problem of solar power pacemaker related to possible existence of some hidden
but crucial mechanism of the Sun’s energy influence on the fundamental
geophysical processes. It is interesting that the ”tracks” of this mechanism
have been observed for a long time and manifest themselves in different
problems of solar-terrestrial physics, in particular, in climatology where the
mechanisms, by which small changes in the Sun’s energy output during the solar
cycle can cause change in the weather and climate, have been a puzzle and the
subject of intense research in recent decades.
If we also add the fact that the dissipation energy caused by friction of the
Earth surface and the boundary layer has a considerable value, which is enough
to cover the known deficiency of the total energy balance of the atmosphere,
it becomes clear that in the framework of the axion mechanism of the solar
dynamo – geodynamo connection it is the solar magnetic field that is a host
power pacemaker of the Earth global climate [ref113].
### 7.4 Axion-like particle and extragalactic background light
A complete physical ”portrait” of the axion, obtained from the axion mechanism
of Sun luminosity and the solar dynamo – geodynamo connection, actually
predefined the necessary conditions of its detection in astrophysics. For
example, for the search of the axion with the mass $m_{a}=17~{}eV$ in
radiative decays of the $a\to\gamma\gamma$ type, when the decay photons are
emitted at or near a peak wavelength
$\lambda_{a}=\frac{2hc}{m_{a}c^{2}}\cong 1459\text{\AA},$ (131)
we used the known observational data obtained during the study of the diffuse
extragalactic background which, according to [ref116], are more suitable than
the flux from any particular region of the sky.
Surprisingly, our theoretical fit of the spectral intensity of the background
radiation (Fig. 20) produced by axion decays describes the experimental data
in the near ultraviolet and optical bands including the data from several
ground- and satellite-based telescope observations [ref169, ref170, ref171,
ref172, ref173, ref174, ref175, ref176, ref177, ref178, ref179] with very good
accuracy.
In this sense we may say that our result, obtained within the self-consistent
model of Sun luminosity and solar dynamo – geodynamo connection, has a strong
and evident ”experimental” support represented by the astrophysical solar-like
axions, provided that this is not just a fortuitous coincidence of the data
(Fig. 20), and the axions with the mass $m_{a}=17~{}eV$ really exist.
### 7.5 Plausible dark matter candidate: hadronic axion or axion-like arhion?
Along with the identification of the solar-like axion with the 17 eV mass, we
also found the possible relic axion-like archion with the 2 eV mass in near-
infrared band of the extragalactic background (Fig. 22), which behaves
similarly to the hardronic axion with highly suppressed interaction with
leptons under certain conditions.
Discovering both the hadronic axion and the axion-like archion (with similar
properties, but different physical nature) made us think about what we
actually see and identify. Aren’t these two particles just axion-like archions
in reality? All that we know about the World, tells us that Mother Nature,
although is notable for its Darwinian diversity in the elementary particles
zoo, would not ”multiply entities beyond necessity”. Keeping in mind this
principle of Ockham’s razor and our intuitive notion about the laws of
astroparticle physics, let us give some arguments in favour of the statement
that we are really dealing with the axion-like archions and not with hadronic
axions.
The main reason of our doubts is the following. It is known [ref209] that the
model of an archion automatically avoids the serious problems related to the
necessity of a stable relic supermassive quark existence, predicted by the
hadronic axion model. As a result, the so-called ”wild isotopes” should exist
as a consequence of a corresponding nucleosynthesis process, and they are not
observed today, as is known. On the other hand, including the hadronic axion
model into the Grand Unified model should lead to an inevitable existence of a
supermassive lepton associated with the axion. This is due to the fact that
within such unification, a composite of a supermassive quark with an ordinary
light quark would cause a supermassive quark instability as well as the axion
interaction with leptons, which is incompatible with the hadronic axion
properties [ref210]. In other words, the axion just would not be hadronic in
this case.
Everything is completely different within the archion model. In contrast to
the hadronic axion model, the archion model (i.e. the Berezhiani-Saharov-
Khlopov (BSK) model of horizontal unification [ref209]) naturally gives rise
to a supermassive quark instability and a strong suppression of the axion-like
archion interaction with leptons [ref203, ref204, ref205, ref206, ref207,
ref208, ref209, ref210]. At the same time, because of the suppression of the
archion interaction with the lightest generation of leptons
($u$,$d$,$e$,$\nu_{e}$), the existing limitations on the corresponding scale
(the energy scale $f_{PQ}$ associated with the break-down of the U(1) PQ-
symmetry) of the invisible axion are reduced to
$f_{a}\equiv f_{PQ}\sim 10^{6}~{}~{}GeV,$ (132)
which makes the BSK-model of an archion rather close to the hadronic axion
model [ref032, ref032a, ref033, ref033a, ref102, ref103].
It is interesting that for such values of the energy $f_{a}$ in the BSK-model
of horizontal unification the stable massive neutrinos dominance becomes
possible, or more strictly speaking, the stable sterile neutrinos with the
mass about several keV and small mixing angles with the active neutrinos. If
we also take into account that one or more of these gauge-singlet fermions can
have Majorana masses below the electroweak scale, in which case they appear as
sterile neutrinos in the low-energy theory [ref218], it becomes clear that
these particles have fundamental importance extending the Standard Model by
gauge singlet fermions, which can accommodate the neutrino masses.
Furthermore, if one of the sterile neutrinos has mass in the 1–20 keV range
and has small mixing angles with the active neutrinos, such particle is a
plausible candidate for dark matter [ref219]. The same particle could be
produced in the supernova explosion, and its emission from a cooling neutron
star could explain the pulsar kicks, facilitate core collapse supernova
explosions, and affect the formation of the first stars and black holes (e.g.,
[ref218] and Refs. therein). Therefore, there is a strong motivation to search
for signatures of sterile neutrinos in this mass range, especially as the
first results were obtained recently indicating the detection of the sterile
neutrino mass of 5.0$\pm$0.2 keV and a mixing angle in a narrow range for
which neutrino oscillations can produce all of the dark matter [ref220].
We tried to find a trace of the radiative decay of the sterile neutrino in the
form $\nu_{s}\to\nu+\gamma$ as well. As we noted before, one should not be
embarrassed by the fact that the corresponding life-time of the sterile
neutrino is many orders of magnitude larger than the age of the universe (see
(95) and (97)), because sterile neutrinos are produced in the early universe
by neutrino oscillations [ref219] and, possibly, by other mechanisms as well,
and therefore every dark matter halo should contain some fraction of these
particles. In other words, given a large number of particles in these
astrophysical systems, even a small decay width can make them observable via
the photons produced in the radiative decay.
Figure 25: (a) The cosmic XRB spectrum and the predicted contribution from the
active galactic nuclei (AGNs), that gives origin to the X-ray background. Gray
points: HEAO-1 A2 HED data (Gruber et al. [ref221]). Dark green points: HEAO-1
A4 LED (Gruber et al. [ref221]). Cyan points: Rossi-XTE (Revnivtsev et al.
[ref222]). Red bowtie: HEAO-1 A4 MED (Kinzer et al. [ref223]). Blue bowtie:
ROSAT PSPC data (Georgantopoulos et al. [ref224]). Light green bowtie:
BeppoSAX (Vecchi et al. [ref225]). Purple and yellow bowties: Newton-XMM (Lumb
et al. [ref226]; De Luca & Molendi [ref227]). Solid line: synthesis model
spectrum by Haardt-Madau [ref228], produced by a mixture of absorbed (the
short-dashed black line) and unabsorbed (the long-dashed black line) AGNs.
Adopted from [ref228].
(b) The cosmic XRB spectrum as a function of the wavelength $\lambda_{0}$.
Solid purple line: synthesis model spectrum by Haardt-Madau [ref228], produced
by a mixture of absorbed and unabsorbed AGNs. Black points with bars
(crosses): HEAO A-2 results (Wu et al. [ref229]). Solid green line: our fit of
cosmic XRB spectrum. The orange region is a sum of our fit of spectrum (the
green line) and the spectrum of $\gamma$-quanta (the blue shaded region) born
in the supposed radiative decay of the sterile neutrinos.
We limited our search for possible $\gamma$-spectra of the sterile neutrino
decay to the analysis of the experimental results obtained by different
authors during their study of the diffuse X-ray background (Fig. 25a). The
cosmic XRB spectrum in the 0.1 – 1 keV range was particularly interesting in
this sense, because the X-ray spectrum here is not ”distorted” by the
absorption processes, since unabsorbed AGNs dominate in this energy region
(see predictions by the Haardt-Madau model on Fig. 25a). Allowing for the fact
that in this energy region the problem of Lyman-$\alpha$ forest may be
neglected, it becomes possible to use Eq. (79) for theoretical calculation of
the intensity of sterile neutrino contributions to the diffuse X-ray
background as a function of the wavelength $\lambda_{0}$.
It is necessary to mention some specifics of cosmic XRB spectrum calculation
for this region202020The observations performed using the X-ray spectroscopy
of locations in the Universe are not used here, because we calculate the
spectrum of $\gamma$-quanta born in radiative decays $\nu_{S}\to\nu+\gamma$
for different $z$ (integral method (86) [ref166]) in contrast to the X-ray
spectroscopy of emission lines from the cooling stars (so-called differential
method (e.g. [ref157]). First, the XRB spectrum is highly ”damaged” (see Fig.
25a) by multiplicity of observations with different measuring bases, and
possibly, different methodologies. For this reason we decided to use the
observational data obtained in the single HEAO experiment. It covers the whole
spectrum in 1 – 400 keV range, and also provides the measurements of the
cosmic XRB spectrum in 0.1 – 1 keV region [ref229]. Second, when calculating
the $\gamma$-spectrum formed by the radiative decay of sterile neutrinos, we
used Eqs. (79)-(81) adjusted for expressions (95)-(97), which reflect the
radiative features of sterile neutrinos, and assumed their velocity
distribution (see (81)) to be Gaussian with a variance of $\upsilon_{c}\sim
30~{}km/s$ [ref230, ref230a] and their density to be equal [ref231, ref232]
$\Omega_{S}h_{0}^{2}\approx
0.3\left(\frac{\sin^{2}2\theta}{10^{-10}}\right)\left(\frac{m_{S}}{100~{}keV}\right)^{2}.$
(133)
We also introduced a factor of 3 for the cosmic XRB spectrum calculation (79),
which manually takes into account that the bulk of the EBL contributions in
the decaying-neutrino scenario comes from neutrinos which are distributed on
larger scales. We will refer to these collectively as free-streaming
neutrinos, though some of them may actually be associated with more massive
systems such as clusters of galaxies, and a possible impact of the
astrophysical data non-gaussianity. In support of this assumption let us quote
a remark made in paper [ref236], where a factor of 10 was used: ”…the fact
that a significant part of the dark matter at redshifts $z\leqslant 10$ is
concentrated in galaxies and clusters of galaxies just means that the
strongest signal from the dark matter decay should come from the sum of the
signals from the compact sources at $z\leqslant 10$. Taking into account that
the DM decay signal from $z\leqslant 10$ is some two orders of magnitude
stronger than that from $z\geqslant 10$, while the subtraction of resolved
sources reduces the residual X-ray background maximum by a factor of 10, we
argue that it would be wrong to subtract the contribution from the resolved
sources from the XRB observations when looking for the DM decay signal”.
Our theoretical fit (the yellow curve in Fig. 25b) of spectral intensity of
the background radiation (79) produced by sterile neutrino decays describes
the experimental data from HEAO A-2 in near X-ray band [ref221] adequately.
However, this result requires a serious and thorough verification, since the
data by Wu et al. [ref229] are, in fact, the only cosmic XRB spectrum
measurements of the Large Magellanic Clouds in the 0.16 – 3.5 keV region. From
this point of view, this result is more likely to serve as a demonstration of
the possibilities and peculiarities of the sterile neutrino search in the
diffuse extragalactic212121Wu’s analysis [ref229] of the background data
reveals a limit on the mean absolute intensity for the extragalactic emission
$I_{\lambda}(0.16-3.5~{}keV)\sim 5.6\cdot 10^{-8}~{}ergs\cdot
cm^{-2}s^{-1}sr^{-1}$. X-ray background which sometimes may be more suitable
than the flux (in the form of a narrow band) from any particular region of the
sky.
Turning back to the archion properties, let us point out that such sterile
neutrino ($m_{S}=2.75~{}keV$, $\sin^{2}2\theta=5.1\cdot 10^{-8}$) representing
the dark matter with density $\Omega_{S}\approx 0.20$ (see (133) and Fig. 26)
fits into the archion model very well. This is due to the fact that it is the
sterile neutrino appearance that makes it possible to solve the problem of
missing dark matter (see (92)) which is associated with the archions model (in
a case of $m_{S}>m_{a}$) and emerges in LTR cosmology:
Figure 26: Full parameter space constraints for the sterile neutrino
production model based on the diffuse extragalactic X-ray background, assuming
sterile neutrinos constitute the dark matter. To facilitate comparisons, we
adopt many of the conventions used by Abazajian [ref233]. Favored regions are
in red/magenta colors, disfavored and excluded regions are in blue/turquoise
colors. The favored parameters consistent with pulsar kick generation are in
horizontal hatching (Kusenko et al. [ref234, ref235]). Constraints from X-ray
observations include the diffuse X-ray background (turquoise) (Boyarsky et al.
[ref236]). Also shown is the best current constraint from Chandra, from
observations of contributions of dark matter X-ray decay in the cosmic X-ray
background through the CDFN and CDFS (brown contour, ”Unresolved CXB Milky
Way”). Also shown is an estimate of the sensitivity of a 1 Ms observation of
M31 with IXO (yellow). The region at $m_{S}<1.7~{}keV$ is disfavored by
conservative application of constraints from the Lyman-$\alpha$ forest.
Inset: Supernova bound on sterile neutrino masses $m_{S}$ and mixing angles
$\sin^{2}(2\theta)$, where the purple region is excluded by the energy-loss
argument while the green one by the energy-transfer argument [ref231]. The
excluded region will be extended to the dashed (red) line if the build-up of
degeneracy parameter is ignored, i.e., $\eta(t)=0$. The dot-dashed (green)
line represents the sterile neutrinos as dark matter with the correct relic
abundance $\Omega_{S}h_{0}^{2}=0.1$. The red star ($m_{S}=2.75~{}keV$,
$\sin^{2}2\theta=5.1\cdot 10^{-8}$) marks our result of sterile neutrino
parameters identification.
$\Omega_{DM}=\Omega_{S}+\Omega_{a}^{nth}\sim 0.25,$ (134)
where $\Omega_{a}^{nth}\approx 0.04$ is a non-standard density of the axion
dark matter (92).
Briefly summarizing all said above about the problem of hadronic axion and
axion-like archion, in our opinion, the archion model is preferable as
compared to the hadronic axion model, but as the saying is, time will tell.
And finally let us emphasize two the most painful points of the present paper,
at least from its authors’ point of view.
Regardless of the model type, the simultaneous existence of two axion-like
particles (the hadronic axion with the mass 17 eV and the axion-like arhion
with the mass 2 eV) rises a natural question whether there may be two
different energy scales $f_{a}$ associated with the breakdown of the U(1) PQ-
symmetry, one of them coming from the hadronic axion ($f_{a}=3.5\cdot
10^{5}~{}GeV^{-1}$), and another relic one – from the axion-like archion
($f_{a}=3.0\cdot 10^{6}~{}GeV^{-1}$). Or otherwise stated, can the energy have
two scales, or in general, be hierarchical? If it can, then what kind of
consequences of such fundamental symmetry violations hierarchy may be observed
today? However, if one just winks at the possible existence of the axion-like
particle with the 2 eV mass, the problem just vanishes without any
consequences for the results of the present paper.
The second painful point is related to the key problem of the axion mechanism
of Sun luminosity and is stated rather simply: ”Is the process of axion
conversion into $\gamma$-quanta by the Primakoff effect really possible in the
Solar tachocline magnetic field?” This question is directly connected to the
problem of the hollow magnetic flux tubes existence in the convective zone of
the Sun, which are supposed to connect the tachocline with the photosphere.
So, both the theory and experiment have to answer the question of whether
there are the waveguides in the form of the hollow magnetic flux tubes in the
convective zone of the Sun, which are perfectly transparent for
$\gamma$-quanta, or our model of the axion mechanism of Sun luminosity and
solar dynamo – geodynamo connection is built around simply guessed rules of
calculation which do not reflect any real nature of things.
## References
## Appendix A Appendix I. Effect of $\gamma$-quanta channeling in periodic
structures
It is known [ref-a1.01] that the real part of dielectric susceptibility
$\chi(\omega)$ for the photons with energies exceeding the $K$-electrons
binding energy has the form
$Re\chi(\omega)=-\omega_{e}^{2}/\omega^{2},$ (A.1)
where $\omega_{e}=(4\pi Ne^{2}/m_{e})^{1/2}$ is the electron plasma frequency,
$N$ is the electrons density, $m_{e}$ is the electron mass. Since
$\omega_{e}\sim 10~{}eV$ for the majority of materials, the susceptibility is
small and negative in the X-ray band. It means that X-ray photons may
experience the total reflection on the border of two materials from the one
with the larger value of $|Re~{}\chi(\omega)|$, and the angle of photons
arrival to the border should not exceed $\theta_{c}=|Re~{}\chi|^{1/2}$, where
$\Delta(Re~{}\chi)$ is a dielectric susceptibility step. It is appropriate to
mention that the phenomenon of a small-angle X-ray reflection has been used in
X-ray optic elements for a long time [ref-a1.02], and also for transporting
(e.g. [ref-a1.03]) and turning (e.g. [ref-a1.04]) the X-ray bundles by
cylindrical tubes.
On the other hand, it is also known that according to the optical-mechanical
analogy (Hamilton and Fermat principles identity), in certain cases the
radiation propagation may be described in terms of ray trajectories obeying
the principle of least action. The refractive index, defining a ray
trajectory, plays a role of an external potential in this case. Based on this
analogy, one may conjecture that there exists an effect of high-energy
chargeless particles channeling that leads to the substantial anisotropy after
passing the periodic structures. This assumption is validated using the
example Vinecky and Finegold model problem [ref-a1.51, ref-a1.52] related to
calculation of the ray trajectories and absorption coefficients for the hard
photons in geometrical optics approximation.
### A.1 Statement of a problem
Let us follow [ref-a1.51, ref-a1.52] and consider $\gamma$-radiation with
frequency $\omega=2\pi c/\lambda$ transmission through a medium under the
following condition:
$\lambda_{e}<\lambda\ll a_{0},$ (A.2)
where $\lambda_{e}=\hbar/mc$ is the electron Compton wavelength, $a_{0}$ is a
typical interatomic distance in the medium.
The left-hand side of the inequality (A.2) lets one consider the quanta as
propagating in the continuous medium with a refraction index $n$. As long as
the right-hand side of (A.2) holds, the incident wave frequency is
substantially higher than the lattice vibration eigenfrequencies (at least for
the higher electron shells of the atoms that form the lattice). Therefore the
scattering electrons may be considered free, and a crystal in the large may be
treated as a frozen spatially-inhomogeneous electron plasma (the impact of
nuclei on the scattering is negligible).
The electron plasma dielectric constant222222For a justification of the term
”dielectric constant” applicability for the X-ray and $\gamma$-bands in the
absence of the Lorentz field averaging over the physically small volume
elements see [ref-a1.05]. may be written down in the form
$\varepsilon=\varepsilon_{1}+i\varepsilon_{2}=1-\frac{\omega_{e}^{2}(\vec{r})}{\omega^{2}+i\omega\omega_{\gamma}},$
(A.3)
where $\omega_{\gamma}$ is the damping parameter which defines the wave
absorption in the system.
For the refraction index
$n=\sqrt{\varepsilon}=n_{1}+in_{2}$
from (A.3) taking into account that $\omega\gg\omega_{e}\gg\omega_{\gamma}$ in
the $\gamma$-band, we derive
$n_{1}\approx
1-\frac{1}{2}\left(\frac{\omega_{e}}{\omega}\right)^{2},~{}~{}n_{2}\approx\frac{1}{2}\frac{\omega_{e}^{2}\omega_{\gamma}}{\omega^{3}}.$
(A.4)
In order to perform the quantitative calculations, let us consider a
simplified model of the one-dimensional lattice (a stack of alternating layers
with different, but constant, densities and uniform $N(\vec{r})$ distribution
along the layer (Fig. A.1)).
Figure A.1: Ray trajectory dependence on the angle $\alpha$ in a long-period
structure: the channeling happens when $\alpha<\alpha_{m}$; when
$\alpha>\alpha_{m}$, the ray crosses the surfaces.
Let us choose the distribution in the $y$ direction (perpendicular to the
layers) in the form:
$N(y)=N_{0}\left(1+\beta^{2}\sin^{2}\pi\frac{y}{a}\right).$ (A.5)
Corresponding expressions for $n_{1}$, $n_{2}$ are
$n_{1}=1-\frac{1}{2}\xi^{2}\left(1+\beta^{2}\sin^{2}\frac{1}{2}k_{a}y,~{}~{}n_{2}=\frac{1}{2}\zeta\xi^{2}\left(1+\beta^{2}\sin^{2}\frac{1}{2}k_{a}y\right)\right),$
(A.6)
where
$\xi^{2}\equiv\frac{\omega_{0}^{2}}{\omega^{2}}=\frac{4\pi
N_{0}e^{2}}{m\omega^{2}}\ll
1,~{}~{}\zeta\equiv\frac{\omega_{\gamma}}{\omega}\ll\xi,~{}~{}k_{a}\equiv\frac{2\pi}{a}.$
(A.7)
So the problem reduces to determining the intensity of the $\gamma$-quanta
beam hitting a sample of the thickness $x$ with the angle $\alpha$ to the
$O_{x}$ axis (Fig. A.1).
We shall use the equation for the intensity of a ray that passed a path $l$ in
the absorbing medium:
$J_{1}=J_{0}\exp\left\\{-2k\int n_{2}(l)dl\right\\},~{}~{}k=\frac{2\pi
n_{1}}{\lambda}\approx\frac{2\pi}{\lambda},$ (A.8)
The integral in (A.8) is taken along the ray trajectory, while the trajectory
is determined by means of the Fermat principle [ref-a1.06]
$\delta\int n_{1}dl=0,$
variation of which yields a system of three nonlinear second-order
differential equations describing the ray equation in a parametric form
$x=x(l)$, $y=y(l)$, $z=z(l)$:
$n_{1}\frac{d^{2}x_{i}}{dl^{2}}+\left(\frac{\partial n_{1}}{\partial
x_{j}}\frac{dx_{j}}{dl}\right)\frac{dx_{i}}{dl}=\frac{\partial n_{1}}{\partial
x_{i}},~{}~{}i,j=1,2,3$ (A.9)
($x_{1}$, $x_{2}$, $x_{3}$ correspond to $x$, $y$, $z$). It follows from the
identical relation $dx_{j}dx_{j}=dl^{2}$ that only two equations of (A.9) are
independent.
### A.2 Solution of the model problem
For the model (A.5) the ray trajectory lies in one plane $xy$, and the system
(A.9) reduces to the equation
$n_{1}\frac{d^{2}y}{dl^{2}}+\frac{\partial n_{1}}{\partial
y}\left(\frac{dy}{dl}\right)^{2}=\frac{\partial n_{1}}{\partial y}.$ (A.10)
Let us point out some features of the channeling process that directly follow
from Eq. (A.10). Assuming $dy/dl=0$ everywhere along the ray, we derive the
particular solution (A.10) which describes the propagation in the planes
parallel to the planes of constant density. According to (A.10), such motion
is only possible when $dn_{1}/dy=0$, i.e. within the planes $A_{j}$ and
$B_{j}$, corresponding to the minimum and maximum values of $n_{1}(y)$.
Otherwise, the straight motion of a ray along the planes $n_{1}(y)=const$ is
impossible. Particularly, if the ray is initially parallel to the planes, but
lies neither in $A_{j}$, nor in $B_{j}$, it is bent towards the nearest
$B_{j}$, since $n_{1}$ grows in this direction. Therefore, the motion within
the plane $B_{j}$ is stable, while within $A_{j}$ it is not. In other words,
the rays entering the layers along the $O_{x}$ axis are pushed out into the
”channels” of the low electron density (regions with minimum $N(y)$).
The first integral of Eq. (A.10) is as follows:
$\frac{dy}{dl}=\left[1-\left(\frac{n_{1}(y_{0})}{n_{1}(y)}\cos\alpha\right)^{2}\right]^{1/2},~{}~{}\frac{dy}{dx}=\left[\left(\frac{n_{1}(y)}{n_{1}(y_{0})\cos\alpha}\right)^{2}-1\right]^{1/2}.$
(A.11)
Let us use (A.11) for determining the area of the ray motion depending on
initial values of $\alpha$ and $y_{0}$. By taking $dy/dx=0$ in the ray’s
turning point $y=y_{m}$ in (A.11), we write down
$n_{1}^{2}(y_{m})=n_{1}^{2}(y_{0})\cos^{2}\alpha.$ (A.12)
By substituting (A.6) we derive
$\sin^{2}\left(\frac{1}{2}k_{a}y_{m}\right)=\frac{1-\xi^{2}}{(\beta\xi)^{2}}\sin^{2}\alpha+\cos^{2}\alpha\cdot\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right).$
(A.13)
Eq. (A.13), obviously, has a real root $y_{m}$ only in the case
$\frac{1-\xi^{2}}{(\beta\xi)^{2}}\sin^{2}\alpha+\cos^{2}\alpha\cdot\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right)\leqslant
1,$ (A.14)
whence taking into account (A.7) we obtain
$\alpha\leqslant\alpha_{m}(y_{0}),~{}~{}\alpha_{m}(y_{0})=\arctan\left[\beta\xi\cos\left(\frac{1}{2}k_{a}y_{0}\right)\right]\ll
1.$ (A.15)
The $dy/dx$ function zeros have the form
$y_{m}=\pm k_{a}^{-1}\arccos\left(\cos
k_{a}y_{0}-2(\beta\xi)^{-2}\sin^{2}\alpha\right).$ (A.16)
For each $y_{0}$ a ray with the angle of arrival $\alpha<\alpha_{m}(y_{0})$
”oscillates” between the end points $[y_{m},-y_{m}]$ when passing through a
sample, i.e. it ”channels” between two neighbouring planes $A_{j}$ and
$A_{j+1}$. Hence, $\alpha_{m}(y_{0})$ is the maximum angle of channeling for
the given $y_{0}$. According to (A.15), the angle $\alpha_{m}=0$ corresponds
to $y_{0}-y_{m}=(1/2)a$, which means that the values of $y_{0}$ that make the
channeling possible lie in the region
$\left(-\frac{1}{2}a,\frac{1}{2}a\right),$
thus embracing the whole sample facet exposed to the beam.
If $\alpha>\alpha_{m}(y_{0})$, the $dy/dx$ function has no zeros – a ray that
entered under the angle $\alpha>\alpha_{m}(y_{0})$ does not channel – instead
it crosses the layers one by one and is heavily absorbed by them. Fig. A.1
shows the trajectories of both the ”captured” ray that propagates between two
planes throughout the sample and the non-channeling one.
From this analysis the following picture of the channeling emerges. When a
beam hits the sample in the $\alpha=0$ direction, virtually all the rays pass
through the sample without any substantial absorption – including the ones
that propagate within the strongly absorbing layers as they are pushed out
back into the channels. Under small $\alpha\neq 0$ the thin bands of $\delta
y_{0}$ values appear on both sides of the low density planes, where the
incident rays quit channeling by switching to the crossing trajectories. Since
each crossing is accompanied by a loss of some amount of energy, we obtain two
sets of rays – experiencing the weak and strong absorption respectively. With
$\alpha$ growth (and, consequently, the $\delta y_{0}$ bands growth) first set
of rays shrinks, while the second set grows. When
$\alpha=\alpha_{m}(y_{0}=0)$, the channeling region collapses and all the rays
are strongly absorbed. Therefore the value
$\alpha_{M}\equiv\alpha_{m}(y_{0}=0)=\arctan\beta\xi\approx\beta\xi-\beta\frac{\omega_{e}}{\omega}$
(A.17)
characterizes the maximum possible channeling angle, thus determining the
divergence of a beam for a given sample. As seen from (A.17), the value of
$\alpha_{M}$ decreases with the beam rigidity growth!
Let us estimate the value of $\alpha_{M}$ for the plasma medium in the Solar
tachocline zone. Let $\beta\sim 1$ for simplicity. Taking into account that
the plasma frequency in the tachocline is equal to $\omega_{e}\sim 4.6\cdot
10^{16}~{}s^{-2}$ [see (8) in the main part], and the frequency of
$\gamma$-quanta with the energy $\langle E\rangle=4.2~{}keV$ (or $\lambda\sim
4.6\cdot 10^{-9}~{}cm$) is $\omega=w\pi c/\lambda=2\pi\langle
E\rangle/\hbar\sim 4\cdot 10^{19}~{}s^{-1}$, from (A.17) we obtain
$\alpha_{m}\sim 10^{-3}$. For the sake of channeling effect illustration, Fig.
A.1 shows the model results of the photon ($\lambda\sim 10^{-9}~{}cm$) beam
propagation through the layered media with $\alpha_{M}\sim 10^{-4}$ (see
Section A.4).
Let us now integrate the Eqs. (A.11). For the case of (A.4) these equations
are reducible to the following form (up to the first vanishing terms of the
order $\xi^{2}$ under the radical sign):
$\frac{dy}{dl}=p\sin\alpha\sqrt{1-q^{2}\sin^{2}\left(\frac{1}{2}k_{a}y\right)},~{}~{}\frac{dy}{dx}=p\tan\alpha\sqrt{1-q^{2}\sin^{2}\left(\frac{1}{2}k_{a}y\right)},$
(A.18)
where
$p\equiv\sqrt{1+(\beta\xi)^{2}\cot^{2}\alpha\sin^{2}\left(\frac{1}{2}k_{a}y_{0}\right)},~{}~{}q\equiv\frac{\beta\xi\cot\alpha}{p}=\left(\sin^{2}\frac{\pi
y_{0}}{a}+\frac{\tan^{2}\alpha}{(\beta\xi)^{2}}\right)^{-1/2}.$ (A.19)
By integrating (A.18) we obtain the following trajectory equation:
$x=x_{j}+\frac{2}{k_{a}\beta\xi}\begin{cases}F\left(\arcsin
q\sin\left(\frac{1}{2}k_{a}y\right),q^{-1}\right)&at~{}~{}q\geqslant 1,\\\
qF\left(\frac{1}{2}k_{a}y,q\right)&at~{}~{}q<1\end{cases}$ (A.20)
Here $F(\varphi,q)$ is an elliptic integral of the first kind, and the inverse
function amF (the Jacobian elliptic function) period is equal to $4K(q)$,
where $K(q)\equiv F(\pi/2,q)$ [ref-a1.07]. Correspondingly, the function
$y(x)$ inverse to (A.20a) is periodical with the period
$x_{\tau}=\frac{8}{k_{a}\beta\xi}K(q^{-1})$ (A.21)
and is associated with the channeling trajectories. For the crossing
trajectories (A.29b) which are translation-invariant under the simultaneous
transformations
$x\rightarrow x+jx_{\tau},~{}~{}y\rightarrow y+ja,~{}~{}j=0,1,2,...,$ (A.22)
we have
$x_{\tau}=\frac{8q}{k_{a}\beta\xi}K(q).$ (A.23)
The regions of channeling and crossing trajectories are delimited by the value
of the angle $\alpha=\alpha_{m}(y_{0})$ corresponding to the value $q=1$. When
$\alpha=\alpha_{m}(y_{0})$, the rays asymptotically approach the ”repelling”
planes $A_{j}$ and $x_{\tau}\to\infty$.
A number of trajectory oscillations along the distance $x$ for the channeling
and crossing rays is, respectively
$N_{\tau}=\frac{x}{x_{\tau}}=\frac{1}{8}k_{a}\beta\xi\begin{cases}K^{-1}(q^{-1})&at~{}~{}q>1,\\\
q^{-1}K^{-1}(q)&at~{}~{}q<1.\end{cases}$ (A.24)
### A.3 Determination of the absorption coefficient angular dependence
Let us move on to the absorption coefficient calculation in the exponent
(A.8). By means of (A.18), integration over trajectory reduces to integration
over a variable $u=k_{a}y/2$ within the region
$\begin{cases}0\leqslant u\leqslant u_{m}&at~{}~{}q>1,\\\ 0\leqslant
u\leqslant\pi/2&at~{}~{}q<1.\end{cases}$
Let us remind that $u=k_{a}y_{m}/2=\arcsin(q^{-1})$ by virtue of (A.15) and
(A.19) corresponds to the turning points $y_{m}$ of the channeling
trajectories. Further, substituting (A.8) into the expression for $n_{2}$
(A.6), performing integration and multiplying the obtained integrals, which
correspond to the above mentioned regions, by $N_{\tau}$ (A.24), we obtain
$\sigma=k\zeta\xi^{2}\frac{x}{\cos\alpha}\begin{cases}1+\beta^{2}\left(1-\frac{E(q^{-1})}{K(q^{-1})}\right)&at~{}~{}q>1,\\\
1+\beta^{2}\left(1-\frac{E(q)}{K(q)}\right)&at~{}~{}q<1,\\\ \end{cases}$
(A.25)
where $E(q)$ is the complete elliptic integral of the second kind [ref-a1.08].
According to (A.25), the intensity $J(x)$ (see (A.8)) of a ray that passed
through a sample of the thickness $x$ may be written down as
$J(x)=J_{0}\exp(-\sigma
x)=J_{0}\exp\left(-\frac{\chi_{0}}{\cos\alpha}x\right)\cdot
Q(\alpha,y_{0},x),$ (A.26)
where
$Q(\alpha,y_{0},x)=\begin{cases}\exp\left[-\frac{\chi_{0}}{\cos\alpha}\beta^{2}x\left(1-\frac{E(q^{-1})}{K(q^{-1})}\right)\right]&at~{}~{}q>1,\\\
\exp\left[-\frac{\chi_{0}}{q^{2}\cos\alpha}\beta^{2}x\left(1-\frac{E(q)}{K(q)}\right)\right]&at~{}~{}q<1.\end{cases}$
(A.27)
Here the multiplier $J_{0}\exp(-\chi_{0}x/\cos\alpha)$ corresponds to
$\gamma$-rays propagation through a homogeneous medium with the electron
density $N_{e}$ and the absorption coefficient $\chi_{0}=k\zeta\xi^{2}$. An
additional multiplier $Q(\alpha,y_{0},x)$ characterizes the influence of the
medium layering. Fig. A.2 shows the $Q(\alpha,y_{0},x)$ curves for
$\frac{y_{0}}{a}=0,~{}1/8,~{}1/4,~{}3/8,~{}1/2$ (A.28)
with $\lambda=10^{-9}~{}cm$, $\beta=1$ and $\chi_{0}x=2$, obtained by a
numerical calculation using the elliptic integral value tables [ref-a1.08].
Figure A.2: Transmitted beam relative intensity dependence on the arrival
angle $\alpha$ for different values of $y_{0}$ (blue curves; the symmetric
parts of the curves for $\alpha<0$ are not shown). Red curve represents
$Q(\alpha,y_{0},x)$ averaged over all values of $y_{0}$.
As seen in the figure, the ”total transmission” $(Q=1)$ is observed when
$\alpha=0$, $y_{0}=0$ (the propagation in the $B_{j}$ plane), and the maximum
absorption depending on $\beta$ and $(Q\to\exp(-\chi_{0}\beta x\cos\alpha))$
is observed when $\alpha=0$, $y_{0}=(1/2)a$ (the propagation in the $A_{j}$
plane).
The dips on the decaying parts of the transmission curves when
$y_{0}\neq(1/2)a$ correspond to the critical values of the angle
$\alpha=\alpha_{m}(y_{0})$ (A.15), for which the $\gamma$-quanta beam, that
entered a sample, asymptotically approaches the nearest $A_{j}$ plane and is
absorbed in it. With somewhat bigger values of $\alpha$ the rays pass on to
the neighboring inter-plane space and moving within it leave a sample before
they enter the next $A_{j+1}$ plane. The additional narrow maxima on blue
curves (Fig. A.2) correspond to this type of rays.
Finally, when $\alpha>\alpha_{M}$ and, particularly, when $\alpha\to\pi/2$
(propagation across the layers) from (A.18) we obtain $q\to 0$, which
corresponds to $Q\to\exp(-\chi_{0}\beta^{2}x/2\cos\alpha)$, i.e. there is an
additional absorption due to the averaged additional density of the layers
$N_{1}=N_{0}\beta^{2}$.
Fig. A.2 shows a curve $Q(\alpha,y_{0},x)$ averaged over all values of
$y_{0}$. The side dips and maxima are smoothed out, and there is only a
central maximum. It corresponds to a clearly defined primary $\gamma$-quanta
propagation with their angle of arrival within $\alpha_{M}\sim 2\cdot
10^{-4}$. Therefore, the dense layers ”modulating” a material play a role of
an anisotropic filter which passes the $\gamma$-radiation for a narrow range
of angles $\alpha<\alpha_{M}$ only. The physical mechanism of the arising
transparency anisotropy consists in $\gamma$-quanta channeling between the
layers because of the radiation ”refraction” in a heterogeneous electron
plasma.
### A.4 Channeling effect onset conditions
By definition, channeling occurs for the particles the motion of which in the
transversal phase plane is limited to a region
$\Delta y\Delta p\sim a\alpha\hbar k,$ (A.29)
where $\alpha$ is a channeling angle. According to the uncertainty principle,
the size of this region cannot be less than $\hbar$, i.e.
$ak_{0}\geqslant\alpha^{-1}.$ (A.30)
By letting $\alpha\sim\omega_{e}/\omega$ for the estimation, from (A.30) we
derive
$ak_{0}\geqslant 1,~{}~{}k_{0}\equiv\omega_{0}/c.$ (A.31)
For example, in a monocrystal with $a=a_{0}\sim 3\cdot 10^{-8}~{}cm$,
$\omega_{0}\sim 5\cdot 10^{16}~{}s^{-1}$ we have $a_{0}k_{0}\sim 5\cdot
10^{-2}\ll 1$, i.e. the condition (A.31) does not hold and, consequently, the
channeling is impossible. This prohibition is true for any quanta, since in
the region $\omega\gg\omega_{0}$, where the approximation (A.4) and (A.5) is
applicable, the frequency $\omega$ falls out from the condition (A.31).
However, for a layered system with $a\gg a_{0}$ the condition (A.31) may prove
to be true, and the channeling may be possible. A more strict condition
$ak_{0}\gg 1$ makes the classical description of this effect possible. This is
the case for a long-period structure with $a/a_{0}\gg 10^{2}$ considered
above. It is appropriate to emphasize that we consider here the structures
based on the amorphous matrices which are not monocrystals. Otherwise the
long-period structure would have been overlapped by the short-period
oscillations with the amplitude equal or greater than $\beta N_{0}$, which
would have lead to a noticeable tunnel effect. The question about channeling
possibilities in this case requires a special research.
Thus, the papers [ref-a1.51, ref-a1.52] show that the phenomenon of X-ray and
$\gamma$-radiation channeling exists in layered structures under conditions
when it is possible to use geometric optics. The essence of this phenomenon
lies in the fact that the rays are reflected form the layers with the higher
electron density if they propagate under the small enough angle
($\alpha<\alpha_{M}$) to the layers plane. The initially uniform intensity
distribution in the transversal plane becomes non-uniform, since the rays
concentrate in the ”channels” – the layers with the lower electron density. It
leads to the substantial absorption decrease and deeper radiation penetration
into the sample along the layers than in the case of an arbitrary arrival
angle.
### A.5 On the account of an absorption impact on X-ray intensity when
channeling through the solar layered structures
As it was shown above, the process of $\gamma$-rays channeling through a
homogeneous medium with the electron density $N_{e}$ and the absorption
coefficient $\chi_{0}=k\zeta\xi^{2}$ may be described by the multiplier
$J_{0}\exp(-\chi_{0}x/\cos\alpha)$ in (A.26).
Calculation of this multiplier for an arbitrary point in the solar convective
zone, obviously, requires the estimation of the average Rosseland free path or
so-called Rosseland opacity (the photon absorption coefficient averaged
according to Rosseland) in these points. On the other hand, it is necessary to
know the radial profiles of the temperature and density in the solar
convective zone in order to calculate the Rosseland free path or Rosseland
opacity.
The transmission of photons with the intensity $J_{0}$ normally incident on a
uniform plasma (A.26) is given by
$T(\nu)=J(\nu)/J_{0}(\nu)=\exp(-\chi_{0}x)=\exp\left[-k(\nu)\rho x\right],$
(A.32)
where $h\nu$ is the photon energy and $J(\nu)$ is the attenuated photon
intensity emerging from the plasma, $k(\nu)$ is the opacity per unit mass
typically measured in units of $cm^{2}/g$, $\rho$ is the density, and $x$ is
the optical path length. For plasmas such as the Sun that are much larger than
the photon mean free path, radiation transport is usually described by the
diffusion approximation [ref-a1.09, ref-a1.10, ref-a1.11] using the Rosseland
mean opacity $k_{R}$,
$\frac{1}{k_{R}}=\int d\nu\frac{1}{k(\nu)}\frac{dB}{dT}\Bigg{/}\int
d\nu\frac{dB}{dT},$ (A.33)
where $B$ is the Planck function, $T$ is the plasma temperature, and the
weighting function $dB/dT$ peaks at roughly 3.8 kT. Near the convective zone
(CZ) boundary $T\sim 190~{}eV$ and $dB/dT$ peaks at $h\nu\sim 750~{}eV$ (Fig.
A.3). The frequency dependent opacity near the CZ boundary calculated using
the opacity project model [ref-a1.12, ref-a1.12a, ref-a1.13] is displayed in
Fig. A.3. Comparison with the weighting function for the Rosseland mean shows
that the most important photon energies are approximately
$300<h\nu<1300~{}eV$.
Figure A.3: Frequency dependent opacity [ref-a1.12, ref-a1.12a, ref-a1.13] for
the 17 element solar composition [ref-a1.14] near the base of the solar
convection zone compared to $dB/dT$. The electron temperature and density were
193 eV and $1\cdot 10^{23}~{}cm^{-3}$, respectively. Adopted from [ref-a1.15].
However, it is necessary to point out some essential peculiarities of the
absorption impact on the X-ray intensity when it channels inside the solar
layered structures based on, e.g. magnetic flux tubes superlattices (see
Appendix B).
First, as it was noted in the body text, the total energy balance of the Sun
is not violated in the framework of the axion mechanism of Sun luminosity, but
the radiation transport changes substantially relative to the standard model
of the Sun – the part of radiation transport not related to axions is very
small ($\sim 0.015\Lambda_{Sun}$). It means that the thermodynamic parameters
(temperature, pressure, plasma density, electron density etc.) are
considerably smaller in the axion model of the Sun as compared to the standard
model. On the other hand, the Rosseland free path or Rosseland opacity
calculation inside the thin magnetic flux tubes, which play the role of
$\gamma$-quanta waveguides formed in the tachocline, must take into account
the new values for the mentioned parameters in the framework of the axion
mechanism of Sun luminosity.
Second, we suppose that the decrease of the pressure in the convective zone,
for example, by an order of magnitude may lead to such decrease of the
pressure inside the magnetic flux tubes that it virtually does not influence
the radiation transport in these tubes. In other words, the Rosseland free
paths of $\gamma$-quanta in the thin magnetic flux tubes is so big that the
corresponding Rosseland opacity tends to zero, and so do the absorption
coefficients in (A.32). The low refractivity, or equally the high transparency
of the thin magnetic flux tubes is achieved due to the high magnetic pressure
(see (16) and Fig. 9) able to compensate the outer pressure of the convective
zone completely (see Appendix B for details).
## References
## Appendix B Appendix II. On a possibility of the layered structures
formation in the solar convective zone on the basis of the magnetic flux tubes
superlattices
It is natural to ask a question, whether there is a physical possibility for
the above mentioned long-period structures formation in the
magnetohydrodynamical plasma media typical for the solar dynamo evolution.
Curiously enough, such mechanisms do exist. Let us examine some of them
briefly.
### B.1 Zonal jet streams
The long-period structures in magnetohydrodynamical plasma media may show as
the so-called zonal jet streams, spontaneously generated in turbulent systems.
In fact, zonal jets are very common in nature. Well-known examples are those
in the atmospheres of giant planets and the alternating jet streams found in
the Earth’s world ocean [ref-a2.01]. Zonal flow formation in nuclear fusion
devices are also well studied [ref-a2.02].
As we have already pointed out above, a common feature of these zonal flows is
that they are spontaneously generated in turbulent systems. Because the
Earth’s outer core is believed to be in a turbulent state, it is possible that
there is a zonal flow in the liquid iron of the outer core. It is interesting
that a previously unknown convective regime of the outer core that has a dual
structure comprising inner, sheet-like radial plumes and an outer, westward
cylindrical zonal flow232323Computer simulations have been playing an
important role in the development of our understanding of the geodynamo, but
the direct numerical simulation of the geodynamo with a realistic parameter
regime is still beyond the power of today’s supercomputers. Difficulties in
simulating the geodynamo arise from the extreme conditions of the core, which
are characterized by very large and very small values of the non-dimensional
parameters of the system. Along them, the Ekman number, $E$, has been adopted
as a barometer of the distance of simulations from real core conditions, in
which $E$ is of the order of 10-15. Following the initial computer simulations
of the geodynamo, the Ekman number achieved has been steadily decreasing, with
recent geodynamo simulations performed with $E$ of the order of 10-6
[ref-a2.04]. In work by Miyagoshi et al. [ref-a2.03, ref-a2.04] they present a
geodynamo simulation with the Ekman number of the order of 10-7 – the highest
resolution yet achieved, making use of 4096 processors of the Earth Simulator.
And what is ahead when the magnitude of $E$ becomes closer to its real
value?!, was recently found [ref-a2.03] (Fig. 6 in the body text).
Fig. 6 in the main body of the present paper shows snapshots of the same data
of zonal flow formation in the Earth’s core. The sheet convection structure is
visualized as isosurfaces of the axial vorticity, which are almost straight in
the $z$ direction. The blue curves surrounding the $\omega_{z}$ isosurfaces
are streamlines of the velocity in the outer part of the dual-convection
structure. The ring-like shape of each streamline indicates that the azimuthal
component is dominant.
However, according to (A.26) in A, for the ideal (without absorption) photon
channeling the long-period structures, in which one of the interlaced media
has almost zero density, are necessary. Surprisingly, such long-period (in
terms of density) structures may appear in the plasma media in general, and in
the solar convective zone in particular. We mean here the so-called magnetic
flux tubes, the properties of which are discussed below.
### B.2 Some properties of the magnetic flux tubes in Sun convective zone
Fig. B.1 shows the results of the three-dimensional solar hexagonal
magnetoconvection simulation [ref-a2.05]. In addition to a mere fact of the
long-period layers formation based on the magnetic flux tubes superlattices,
it is necessary to make sure that the matter density inside the tubes is much
smaller than the density of the outer plasma.
Figure B.1: Vertical field component for $R_{m}=400$ when the imposed field is
vertical, displayed by perspective plots. Four times are shown, for the three
levels $z$ = 0, 1/2, and 1. In this and subsequent plots the time unit is
(cell height)/(max. vertical velocity), i.e. $\leqslant$ 1.4 turnover time.
Adopted from [ref-a2.05].
At the same time, as the analysis in Appendix A shows, for the purposes of
photon channeling it is not necessary to guarantee the layers periodicity. In
other words, the channeling requires a large number of necessarily interlaced,
although not necessarily periodic, layers of different density. The total
crosscut area of such interlaced layers must also be comparable to the photon
beam cross-section.
In this connection below we shall examine some properties of the magnetic flux
tubes in the Sun convective zone, which may serve as a basis for estimating
the temperature, pressure and matter density inside the tubes depending on the
similar plasma parameters outside the tube. We shall also derive the zero
refractivity (or absolute transparency) condition for the effective photon
channeling along the magnetic tubes.
### B.3 The self-confinement of force-free magnetic fields and energy
conservation law.
Magnetic field $\vec{B}$ alternating along the vertical axis $z$ induces the
vortex electric field in the magnetic flux tube containing a dense plasma. The
charged particles rotation in plasma with the angular velocity $\omega$ leads
to a centrifugal force per unit volume
$|\vec{F}|\sim\rho|\vec{\omega}|^{2}r.$ (B.1)
If we consider this problem in the rotating noninertial reference frame, such
noninertiality, according to the equivalence principle, is equivalent to
”introducing” a radial non-uniform gravitational field with the ”free fall
acceleration”
$g(r)=|\vec{\omega}|^{2}r.$ (B.2)
In such case the pressure difference inside ($p_{int}$) and outside
($p_{ext}$) the rotating ”liquid” of the tube may be treated by analogy with
the regular hydrostatic pressure (Fig. B.1).
Figure B.2: Representation of the ”hydrostatic equilibrium” in the rotating
”liquid” of a magnetic flux tube.
Let us pick a radial ”liquid column” inside the tube as it is shown in
Fig.B.1. Since the ”gravity” is non-uniform in this column, it is equivalent
to a uniform field with the ”free fall acceleration”
$\left\langle g(r)\right\rangle=\frac{1}{2}|\vec{\omega}|^{2}R.$ (B.3)
where $R$ is the tube radius which plays a role of the ”liquid column height”
in our analogy.
By equating the forces acting on the chosen column similar to the hydrostatic
pressure (Fig. B.1), we derive:
$p_{ext}=p_{int}+\frac{1}{2}\rho|\vec{\omega}|^{2}R^{2}.$ (B.4)
This rises the natural question as to what physics is hidden behind the
”centrifugal” pressure. In this relation, let us consider the magnetic field
energy density
$w_{B}=\frac{|\vec{B}|^{2}}{2\mu_{0}},$ (B.5)
where $\mu_{0}$ is the magnetic permeability of vacuum.
Suppose that the total magnetic field energy of the ”growing” tube grows
linearly between the tachocline and the photosphere. In this case if the
average total energy of the magnetic field in the tube transforms into the
kinetic energy of tube matter rotation completely, it is easy to show that
$\left\langle
E_{B}\right\rangle=\frac{1}{2}w_{B}V=\frac{1}{2}\frac{|\vec{B}|^{2}}{2\mu_{0}}V=\frac{I|\vec{\omega}|^{2}}{2},$
(B.6)
where $I=mR^{2}/2$ is the tube’s moment of inertia about the rotation axis,
$m$ and $V$ are the mass and the volume of the tube medium respectively.
Then from (B.6) it follows that
$\frac{\rho|\vec{\omega}|R^{2}}{2}=\frac{|\vec{B}|^{2}}{2\mu_{0}}.$ (B.7)
Finally, substituting (B.7) into (B.4) we obtain the desired relation
$p_{ext}=p_{int}+\frac{|\vec{B}|^{2}}{2\mu_{0}},$ (B.8)
which is exactly equal to a well-known expression by Parker [ref-a2.06],
describing the so-called self-confinement of force-free magnetic fields.
### B.4 Hydrostatic equilibrium and a sharp tube medium cooling effect
Assume that the tube has the length $l(t)$ by the time $t$. Then its volume is
equal to $S\cdot l(t)$ and the heat capacity is
$c\cdot\rho(t)\cdot Sl(t),$ (B.9)
where $S$ is the tube cross-section, $\rho(t)$ is the density inside the tube,
$c$ is the specific heat capacity.
If the tube becomes longer by $\upsilon(t)dt$ for the time $dt$, then the
magnetic field energy increases by
$\frac{1}{2}\frac{|\vec{B}|}{2\mu_{0}}S\upsilon(t)dt.$ (B.10)
where $\upsilon(t)$ is the tube propagation speed.
The matter inside the tube, obviously, has to cool by the temperature $dT$ so
that the internal energy release maintained the magnetic energy growth.
Therefore, the following equality must hold:
$c\rho(t)l(t)\frac{dT}{dt}S=-\frac{1}{2}\frac{|\vec{B}|^{2}}{2\mu_{0}}\upsilon(t)S.$
(B.11)
Taking into account the fact that the tube grows practically linearly
[ref-a2.07], i.e. $\upsilon t=l$, the Parker relation (B.8) and the tube’s
equation of state
$p_{int}(t)=\frac{\rho}{\mu_{*}}R_{*}T(t)~{}~{}\Longleftrightarrow~{}~{}\rho=\frac{\mu_{*}}{R_{*}}\frac{p_{int}}{T(t)},$
the equality (B.11) may be rewritten (by separation of variables) as follows:
$\frac{dT}{T}=-\frac{R_{*}}{2c\mu_{*}}\left[\frac{p_{ext}}{p_{int}(t)}-1\right]\frac{dt}{t},$
(B.12)
where $\mu_{*}$ is the tube matter molar mass, $R_{*}$ is the universal gas
constant.
After integration of (B.12) we obtain
$\ln\left[\frac{T(t)}{T(0)}\right]=-\frac{R_{*}}{2c\mu_{*}}\int\limits_{0}^{t}\left[\frac{p_{ext}}{p_{int}(\tau)}-1\right]\frac{d\tau}{\tau}.$
(B.13)
It is easy to see that the multiplier ($1/\tau$) in (B.13) assigns the region
near $\tau=0$ in the integral. The integral converges since
$\lim\limits_{\tau\rightarrow
0}\left[\frac{p_{ext}}{p_{int}(\tau)}-1\right]=0.$
Expanding $p_{int}$ into Taylor series and taking into account that
$p_{int}(\tau=0)=p_{ext}$, we obtain
$p_{int}(\tau)=p_{ext}+\frac{dp_{int}(\tau=0)}{d\tau}=p_{ext}(1-\gamma\tau),$
(B.14)
where
$\gamma=-\frac{1}{p_{ext}}\frac{dp_{int}(\tau=0)}{d\tau}=\frac{1}{p_{int}(\tau=0)}\frac{dp_{int}(\tau=0)}{d\tau}.$
(B.15)
From (B.14) it follows that
$\frac{p_{ext}}{p_{int}(\tau)}-1=\frac{1}{1-\gamma\tau}-1\approx\gamma\tau,$
(B.16)
therefore, substituting (B.16) into (B.13), we find its solution in the form
$T(t)=T(0)\exp\left(-\frac{R_{*}}{2c\mu_{*}}\gamma t\right).$ (B.17)
It is extremely important to note here that the solution (B.17) points at the
remarkable fact that at least at the initial stages of the tube formation its
temperature decreases exponentially, i.e. very sharply. The same conclusion
can be made for the pressure and the matter density in the tube. Let us show
it.
Assuming that the relation (B.15) holds not only for $\tau=0$, but also for
small $\tau$ close to zero, we obtain
$p_{int}(t)=p_{int}(0)\exp(-\gamma t),$ (B.18)
i.e. the inner pressure also decreases exponentially, but with the different
exponential factor.
Further, assuming that the heat capacity of a molecule in the tube is
$c=\frac{i}{2}\frac{R_{*}}{\mu_{*}}$ (B.19)
where $i$ is a number of molecule’s degrees of freedom, and substituting the
expressions (B.17) and (B.18) into the tube’s equation of state, we derive the
expression for the matter density in the tube
$\rho(t)=\frac{\mu_{*}}{R_{*}}\frac{p_{int}(0)}{T(0)}\exp\left[-\left(1-\frac{1}{i}\right)\gamma
t\right],$ (B.20)
Taking into account that $i\geqslant 3$, we can see that the density decreases
exponentially just like the temperature (B.17) and pressure (B.18).
### B.5 Ideal photon channeling (without absorption) conditions inside the
magnetic flux tubes
Let us calculate the inner pressure $p_{int}$ of the magnetic flux tube in the
solar tachocline. In the framework of the standard model of the Sun the
pressure in the tachocline zone is about $\sim 6\cdot 10^{12}~{}Pa$, while the
magnetic field strength reaches $400~{}T$, according to our estimates (Fig. 9
in the body text). Then from the hydrostatic condition by Parker (B.8) it
follows that the inner pressure of the magnetic flux tube in the solar
tachocline is equal to
$p_{int}\sim 6\cdot 10^{12}-4\cdot 10^{10}\simeq 6\cdot 10^{12}~{}~{}[Pa],$
(B.21)
which is comparable to the external pressure.
However, the situation changes drastically within the framework of the ”axion”
model of the Sun. We pointed out earlier (A) that the values of thermodynamic
parameters (temperature, pressure and plasma density) in the ”axion” model are
substantially smaller than the corresponding parameters in the standard model
of the Sun. Let us suppose that the pressure in the tachocline zone in the
”axion” model falls by an order of magnitude and is about $\sim 10^{11}~{}Pa$.
In such case it is easy to see that the magnetic field as strong as $\sim
500~{}T$ compensates the outer pressure almost completely, and the inner
pressure
$p_{int}\sim 10^{11}-O(10^{11})\rightarrow 0~{}~{}[Pa]$ (B.22)
becomes ultralow.
The result (B.22) means that the temperature and the matter density decrease
together with the inner pressure, and the decrease is sharply exponential,
because the exponential factor in (B.16) becomes very large
$\gamma\tau=\frac{p_{ext}}{p_{int}(\tau)}-1\rightarrow\infty.$ (B.23)
Because of the fact that the density, pressure and temperature in the tube are
ultralow, they virtually do not influence the radiation transport in these
tubes at all. In other words, the Rosseland free paths for $\gamma$-quanta
inside the tubes are so big that the Rosseland opacity and the absorption
coefficients in (A.32) tend to zero. The low refractivity (or high
transparency) is achieved for the following limiting condition:
$p_{ext}\simeq\frac{|\vec{B}|^{2}}{2\mu_{0}}.$ (B.24)
The obtained results should not be considered as a proof, but rather as some
trial estimates that validate the substantiation of an almost ideal (without
absorption) photon channeling mechanism along the magnetic flux tubes.
A major disadvantage of our reasoning is that, first of all, our estimates
apply to the initial magnetic tube formation stages, and second, that it lacks
the model calculations of the temperature, pressure and density in the
framework of the ”axion” model of the Sun, and consequently, there is no
comparison with the corresponding parameters of the standard model. On the
other hand, we believe that the idea of $\gamma$-quanta channeling along the
magnetic flux tubes is so physically natural and promising, that these
disadvantages will be overcome in the near future in spite of the obvious
serious difficulties.
## References
|
arxiv-papers
| 2013-04-15T15:17:20 |
2024-09-04T02:49:44.358304
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.D. Rusov, I.V. Sharf, V.A. Tarasov, M.V. Eingorn, V.P. Smolyar, D.S.\n Vlasenko, T.N. Zelentsova, E.P. Linnik, M.E. Beglaryan",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/1304.4127"
}
|
1304.4168
|
# Probing Curvature Effects in The Fermi GRB 110920
A. Shenoy11affiliationmark: , E. Sonbas22affiliationmark: 33affiliationmark:
, C. Dermer,44affiliationmark: , L. C. Maximon11affiliationmark: , K. S.
Dhuga11affiliationmark: , P. N. Bhat55affiliationmark: , J.
Hakkila66affiliationmark: , W. C. Parke11affiliationmark: , G. A.
Maclachlan11affiliationmark: , T. N. Ukwatta77affiliationmark: 1Department
of Physics, The George Washington University, Washington, DC 20052, USA
2University of Adiyaman, Department of Physics, 02040 Adiyaman, Turkey 3NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA 4Space Science
Division, Code 7653, Naval Research Laboratory, Washington, D.C. 20375, USA
5CSPAR, University of Alabama in Huntsville, Huntsville, AL 35805, USA
6Department of Physics and Astronomy, College of Charleston, Charleston, S.C.
29424, USA 7Department of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824, USA [email protected]
###### Abstract
Curvature effects in Gamma-ray bursts (GRBs) have long been a source of
considerable interest. In a collimated relativistic GRB jet, photons that are
off-axis relative to the observer arrive at later times than on-axis photons
and are also expected to be spectrally softer. In this work, we invoke a
relatively simple kinematic two-shell collision model for a uniform jet
profile and compare its predictions to GRB prompt-emission data for
observations that have been attributed to curvature effects such as the peak-
flux–peak-frequency relation, i.e., the relation between the $\nu$Fν flux and
the spectral peak, Epk in the decay phase of a GRB pulse, and spectral lags.
In addition, we explore the behavior of pulse widths with energy. We present
the case of the single-pulse Fermi GRB 110920, as a test for the predictions
of the model against observations.
###### Subject headings:
Gamma-ray bursts: general
## 1\. Introduction
Pulses in GRB light curves are thought to be produced by collisions between
relativistic shells ejected from an active central engine (Rees and Meszaros
1994). The interception of a more slowly moving shell by a second shell that
is ejected at a later time, but with greater speed, produces a shock that
dissipates internal energy and accelerates the particles that emit the GRB
radiation. This scenario is widely adopted in order to model pulses in GRB
light curves (e.g., Daigne and Mochkovitch 1998; Zhang et al. 2009). Studies
of pulses are important to determine if GRB sources require engines that are
long-lasting or impulsive, and to determine the likely radiation mechanism(s),
with important implications for the nature of the central engine.
Spectral lags, where low energy photons reach the observer at later times than
high-energy photons, are seen in a significant fraction of GRBs. Cheng et al.
(1995) were the first to analyze the spectral lag of GRBs, which they
determined as the time delay between the peaks in the Burst and Transient
Source Experiment (BATSE) Large Area Detector (LAD) channel 1 (25 - 50 keV)
and channel 3 (100 - 300 keV) light curves. Since then, several authors have
analyzed spectral lags in GRBs, while also extending these observations to the
Swift and Fermi GRB samples (e.g., Norris et al. 1996; Norris, Marani &
Bonnell 2000, Wu & Fenimore 2000; Chen et al. 2005; Ukwatta et al. 2010;
Sonbas et. al. 2012) The leading model to explain the spectral lag is the
curvature effect, i.e. the kinematic effect due to the observer looking at an
increasingly off-axis annulus area relative to the line-of-sight (Fenimore et
al. 1996; Salmonson 2000; Kumar & Panaitescu 2000; Ioka and Nakumura 2001; Qin
2002; Qin et al. 2004; Dermer 2004; Shen et al. 2005; Lu et al. 2006). Softer
low-energy radiation comes from the off-axis annulus area due to smaller
Doppler factors. This radiation is also delayed at the observer end with
respect to on-axis observation due to the geometric curvature of the shell.
The existing models as well as observations suggest that a connection exists
between the observed hard-to-soft spectral evolution of GRB pulses and
spectral lags. It is therefore important to understand the mechanism that
produces this evolution. Tavani (1996) proposed that this hard-to-soft
spectral evolution is caused by the variation of the average Lorentz factor of
pre-accelerated particles and the strength of the local magnetic field at the
GRB site as the synchrotron emission evolves within the burst. Liang (1997)
proposed a physical model of hard-to-soft spectral evolution in which
impulsively accelerated non-thermal leptons cool by saturated Compton up-
scattering of soft photons. Kocevski & Liang (2003) have analyzed a sample of
19 GRBs and found a positive correlation between the decay rate of the peak
energy and the spectral lag. Ukwatta et al. (2010) have analyzed a sample of
31 Swift GRBs with known red-shifts and determined the spectral lags between
fixed source frame bands, 100 – 150 keV and 200 – 250 keV. They also
determined that the source-frame Epk lies beyond the higher-energy band, 100 –
250 keV for a majority of these bursts. Based on this result, they suggest
that spectral evolution may not be the dominant process causing the observed
spectral lag.
Borgonovo and Ryde (2001) studied the spectral evolution in the prompt-
emission in GRBs by performing a time-resolved spectral analysis of BATSE
single pulses and showed that in many cases the $\nu F\nu$ flux at the Epk for
each time segment was $\propto$ E${}_{pk}^{\eta}$ (hereafter referred to as
the peak-flux–peak-frequency relation), with $\eta$ ranging from $\approx 0.6$
to $3$. The exponent $\eta$ was found to stay roughly constant for pulses
within the same GRB. Dermer (2004) modeled and analyzed GRB pulses based on
curvature effects with a Broken-Power-Law (BPL) rest-frame spectrum and showed
that in the curvature limit, $\eta$ was equal to 3 for pulses with a wide
range of temporal properties.
While several studies (Qin 2002; Qin et al. 2004; Shen et al. 2005; Lu et al.
2006) have been performed to determine the role played by curvature effects,
both in the spectral evolution of GRB prompt-emission as well as in producing
spectral lags, a number of questions remain unanswered. Specifically, we note
that the dependence of the lag on the radius at which the emission takes place
is quite unclear at this stage with seemingly contradictory results being
reported in the literature (Shen et al. 2005; Lu et al. 2006). While that is
not the focus of our current study, it is a matter of considerable interest
and will be the central topic of a forthcoming work. In this work we focus on
the role played by the thickness of colliding shells on observables such
spectral lags, the $\nu F\nu$ flux vs. Epk relation, and the behavior of the
corresponding pulse widths as a function of energy. Other studies have also
attempted to determine the role played by the evolution of the rest-frame
spectrum on the observables such as the spectral lags and the evolution of the
pulse-widths with energy within the context of a curvature model(Qin et
al.2005; Lu et al. 2006; Qin et al. 2009; Peng et al. 2011). Again, such
considerations are very important but we do not specifically consider the
effects of the evolution of the rest-frame spectrum in this paper.
The paper is organized as follows: the basic features of the model are
presented in section 2, followed by a description of the sample selection
criteria, analysis methodology, and a case study in section 3. The discussion
of our main results is presented in section 4, followed by a summary of our
conclusions in section 5.
Table 1Selected parameters for the generation of the model light curves unless otherwise stated. $\eta_{r}$ | $\eta_{t}$ | $\eta_{\Delta}$ | $t_{var}(s)$ | $\Gamma$ | z | $\theta_{jet}$ | Epk,0 (keV) | $d_{L}$ (cm) | $u_{0}$ (ergs cm-3)
---|---|---|---|---|---|---|---|---|---
1.0 | 1.0 | 1.0 | 1.0 | 300 | 1.0 | 4.0/$\Gamma$ | 250.0 | $2.2\times 10^{28}$ | 1.0
## 2\. The Model
We have used a particular representation of the internal shock model for our
purposes (Dermer 2004). This model consists of a single two-shell collision
event occurring at a radius $r_{0}$ from the source, generating a uniform
spherical shell, with Lorentz factor $\Gamma$, that radiates for a co-moving
time between $t^{\prime}_{0}$ and $t^{\prime}_{0}+\Delta t^{\prime}$ with
$\Delta t^{\prime}=\eta_{t}\Gamma t_{var}/(1+z)$, where $t_{var}$ is the
observed variability time scale. The rest-frame–emission profile is assumed to
be rectangular with instantaneous rise and decay phases. The opening angle of
the jet is assumed to be $4/\Gamma$. Emission from angles greater than
$4/\Gamma$ are ignored. This is a suitable compromise between considering
emissions from an entire fireball surface ($0<\theta<\pi/2$) and a collimated
jet ($\theta\sim 1/\Gamma$). As noted by Qin et al. 2004, limiting the
radiation to $\theta<1/\Gamma$ leads to a cutoff-tail problem whereas the
contributions from areas at $\theta>1/\Gamma$ fall off very rapidly. We find
such a choice to be suitable for producing pulse profiles that may be directly
compared with observations. The co-moving width of the shell $\Delta
r^{\prime}$ is assumed to remain constant during the period of illumination
and given by $\Delta r^{\prime}=\eta_{\Delta}\Gamma~{}c~{}t_{var}/(1+z)$. It
is in this respect that the chosen model differs from previous studies.
Previous studies (see for instance Qin et al. 2004, Shen et al. 2005) have
studied the effects of curvature from a spherical surface in great detail. As
shown subsequently, the effect of a finite shell where $\Delta r^{\prime}$ is
comparable to $c\Delta t^{\prime}$, has a significant effect on the predicted
observables such as spectral lags and pulse widths as a function of energy
when compared to the models that study curvature effects from a surface. As
$\Delta r^{\prime}<<c\Delta t^{\prime}$, we approach the infinitesimal shell
or emission-surface situation and are able to recover many of the predictions
of the aforementioned models. The emission spectrum in the co-moving frame may
be described by any suitable spectral function such as a Broken Power-Law
(BPL), Band, or Comptonized–Epk, peaking at a co-moving photon energy
E${}^{\prime}_{pk,0}$ = (1 + z) Epk,0/2$\Gamma$ where Epk,0 is the observer
frame Epk at the start of the pulse. The spectral indices at E $<$ Epk,0 and
at E $>$ Epk,0 in counts space are denoted $\alpha$ and $\beta$, respectively.
The curvature constraint requires that $r\lesssim
2\Gamma^{2}c~{}t_{var}/(1+z)$ (Fenimore et al. 1996). The radius is thus
written using the expression $r_{0}=2\eta_{r}\Gamma^{2}c~{}t_{var}/(1+z)$,
with $0\lesssim\eta_{r}\lesssim 1$. The parameters $\eta_{t}$, $\eta_{\Delta}$
and $\eta_{r}$ thus control the blast-wave duration, shell-thickness and
radius of emission respectively.
Figure 1.— Normalized light curves obtained using selected parameters except:
Thin-shell pulse, $\eta_{\Delta}=0.1$; and the Curvature pulse,
$\eta_{\Delta}=\eta_{t}=0.1$. The case of $\eta_{t}=\eta_{\Delta}=\eta_{r}=1$
is referred to as a Causal pulse (see Dermer 2004 for a detailed discussion of
these three generic types of pulses). Figure 2.— Evolution of the spectral
energy distribution due to curvature effects for the case of the causal pulse
in Fig. 1. In the declining phase of the pulse, the value of
$f_{E_{pk}}\propto$ E${}_{pk}^{3}$ is shown by the red line. Figure 3.— Lag
vs. Energy for selected parameters except shell thickness $\eta_{\Delta}$.
Red: $\eta_{\Delta}=1.0$; Green: $\eta_{\Delta}=0.5$; Blue:
$\eta_{\Delta}=0.1$. Figure 4.— Lag vs. Energy due to different rest-frame
spectral functions with selected parameters except Epk,0 = 200 keV. Here, Red:
Broken-power-law with $\alpha=-1/3$, $\beta=-2.5$, Green: Band function with
$\alpha=-0.8$, $\beta=-2.25$ and Blue: Comptonized-Epk with $\alpha=-0.8$.
Figure 5.— Pulse FWHM vs. Energy for different shell-collision parameters
$\eta_{\Delta}$ and $\eta_{t}$ for selected parameters except E${}_{pk}=200$
keV, a Band spectrum with $\alpha=-1.0$ and $\beta=-2.25$ and $t_{var}=1.0$ s,
with: Pink: A pulse with Infinitesimal duration and shell thickness with
$\eta_{\Delta}=0.001$ and $\eta_{t}=0.001$ Red: Curvature pulse with
$\eta_{\Delta}=0.1$ and $\eta_{t}=0.1$; Green: Thin-shell pulse with
$\eta_{\Delta}=0.1$ and $\eta_{t}=1$; and Blue: Causal pulse with
$\eta_{\Delta}=1$ and $\eta_{t}=1$. Also shown is the exponent of the power-
law that best fits the model points for the thin-shell and Causal pulses.
Unless otherwise stated, the selected parameters used for the numerical
calculations of the light curves, spectra and spectral lags are shown in
Table. 1. Fig. 1 shows the normalized, generic pulse shapes obtained using the
selected parameters. As can be seen, the cases of the thin-shell pulse
($\eta_{\Delta}<<\eta_{t}$) and the curvature pulse
($\eta_{\Delta}=\eta_{t}<<\eta_{r}$) produce the sharp featured light curves
noted by Qin et. al. 2004 for emission from a rectangular pulse profile in the
co-moving frame, and which are attributed to the effect of a suddenly-dimming
emission profile. These two cases most closely correspond to a long duration
and a short duration pulse in the co-moving frame respectively, and where the
effects of a finite shell are suppressed. The case of the causal pulse
($\eta_{\Delta}=\eta_{t}=\eta_{r}=1$) however, shows that emission from a
finite shell can produce smooth light curves without the need for a slowly
dimming co-moving emission profile. Fig. 2 shows the evolution of the spectral
energy distribution for the case of the curvature pulse shown in Fig. 1. The
$\nu$Fν peak flux $f_{E_{pk}}$ $\propto$ E${}_{pk}^{3}$ equality line is shown
in the decay portion of the pulse. Dermer (2004) shows that this equality
holds in the declining phase for all pulses (a similar result has been derived
by Qin et al. 2009). We also note that the presence of a finite shell affects
the low energy and high energy fluxes equally and therefore does not effect
the shape of the spectrum as a function of time as first noted by Qin et. al.
2002 in the context of emission from a fireball surface. Fig. 3 shows the
spectral lag as a function of energy for a case with selected parameters but
with varying shell thickness parameter, $\eta_{\Delta}$. Fig. 4 shows the lag
as a function of energy for a case with selected parameters but with different
rest-frame spectral functions (BPL, Band, and Comptonized–Epk). Fig. 5 shows
the pulse Full-width at half-maximum (FWHM) as a function of energy for the
various profiles shown in Fig. 1. For the purpose of comparison we have used a
Band function with $\alpha=-1.0$ and $\beta=-2.25$, identical to Qin et. al.
2005. These authors have shown that the Doppler effect of a relativistically
expanding fireball could lead to a power-law trend for the pulse width as a
function of energy within a certain energy range. By taking a sizable sample
of BATSE GRBs, they demonstrated that the pulse widths exhibit a
plateau/power-law/plateau feature as a function of energy. They also note that
the power-law index depends strongly on Epk and the rest-frame radiation
spectrum. The plateau/power-law/plateau feature reported by Qin et al. 2005 is
well reproduced here. Furthermore, we note that the power-law exponent is
sensitive to the assumed thickness of the shell.
Figure 6.— KRL pulse-fit for the light curve for GRB 110920 with residuals.
Figure 7.— Light curve segments for 110920 with equal fluences. Figure 8.—
Comparable pulses generated using the BPL (E${}_{peak}=300$ keV,
$\alpha=-\frac{1}{3}$, $\beta=-2.5$), Band (E${}_{peak}=334$ keV,
$\alpha=-0.2$, $\beta=-2.65$)and Comptonized–Epk (E${}_{peak}=280.1$ keV,
$\alpha=-0.46$) spectral functions. The light curves have been offset by 20
seconds for better viewing. Figure 9.— $\nu F\nu$ Flux vs. Epk for the data
from GRB 110920. The data were fit with the best-fit Comptonized–Epk function
in the range 100-985 keV. Figure 10.— $\nu F\nu$ Flux vs. Epk from the model
for the three rest-frame spectral functions described in the text with Blue:
BPL; slope = 3.11 +/- 0.04, Pink: Band; slope = 3.39 +/- 0.10 and Black:
Comptonized–Epk; slope = 2.57 +/- 0.01, for time segments identical to those
used for the data. The flux scale has been offset for better viewing. Figure
11.— Lag vs. Energy from the model for the pulses shown in Fig. 7 with Red:
Data, Solid blue squares: Comptonized–Epk, Hollow black squares: Band function
and Pink crosses: BPL. The Band 2 energies are the mid-point of the energy in
the second band. Figure 12.— Pulse FWHM vs. Energy for the data and the model
for GRB 110920 for identical energy bands. Also shown is the exponent of the
power-law that best fits the model points.
## 3\. Sample Selection and Methodology.
As a first step we analyze either single-pulse GRBs, or GRBs with relatively
simple light curves where the individual pulses within a multi-pulse structure
in the light curve can be distinguished. In addition, we require that the GRB
pulses be bright enough and of sufficient duration (the duration of the pulse
is particularly relevant to tests of the peak-flux – peak-frequency relation)
so that we may obtain reliable results from our analyses.
After identifying a potential candidate GRB, we pulse-fit the GRB light curves
using a suitable pulse function (such as the Kocevski-Ryde-Liang (KRL) pulse
function: see Kocevski Ryde & Liang 2003 or the Norris pulse function: see
Norris et al. 2005) in multiple energy bands. In order to support the
supposition that a given strong pulse (obtained from a suitable pulse-fit) is
not made from overlapping multiple pulses (within statistics), we perform a
wavelet based minimum-variability time-scale (MTS) extraction. In essence, the
MTS is a measure of the smallest temporal structure in a lightcurve. The full
details of its extraction, and the technique in general, are given in
MacLachlan et al. (2013). The correlation between MTS and pulse properties
such as rise times and widths is discussed in MacLachlan et al. (2012). The
best-fit pulse profile is then used as a representation of the light curve
from the data.
The time-integrated spectrum of the GRB is fit with a suitable function (Band,
Comptonized–Epk etc.). The best-fit spectral function is used as the rest-
frame emission spectrum in the model. The model parameters are then varied to
generate a light curve that best matches the best-fit pulse profile. The light
curve is subdivided into time segments with equal, and sufficiently high
background-subtracted fluence in order to minimize the effects of varying
signal-to-noise and an Epk is extracted via a spectral fit for each time
segment. The $\nu$Fν flux is extracted at Epk in a range spanned by the Epk-
error. The model light curve is treated in an identical fashion as the data
with regard to segments. Model fluxes and Epk’s are extracted and the peak-
flux – peak-frequency relation is tested. In addition, we extract spectral
lags in suitable, identical energy bands from the data and the model and
compare the predicted and the observed spectral-lag-energy evolution. The
spectral lags are extracted using the cross-correlation-function analysis
method as described in Ukwatta et. al (2010).
### 3.1. GRB 110920 - A Test Case
The Fermi GRB 110920 is a single-pulse burst with a relatively long fast-rise,
exponential-decay structure with a $T_{90}$ of 170 $\pm$ 17 seconds. The best
fit Band parameters (see McGlynn et al. 2012 for a detailed discussion on the
properties of this GRB) for the time interval [$T_{0}+0.003,T_{0}+52.737$]
(where $T_{0}$ is the trigger time) were $\alpha=-0.20\pm 0.02$,
$\beta=-2.65^{+0.07}_{-0.09}$ and $E_{peak}=334\pm 5$ keV with C-stat = 3206.5
(485 d.o.f.). However, when a blackbody component was included in the fit, the
C-stat was reduced to 2848.3 (483 d.o.f.). The peak energy of the Band
component was shifted up to $E_{peak}=978^{+154}_{-121}$ keV and the
temperature of the blackbody was found to be $kT=61.3^{+0.7}_{-0.6}$ keV. The
low energy index $\alpha$ became ($-1.05\pm 0.04$). McGlynn et al. (2012) have
attributed this blackbody component to the photospheric emission (see for
instance: Ryde, 2005; Rees & Meszaros, 2005). Since the redshift (z) is
undetermined for this GRB, they have assumed a value of z = 2 and then used
photospheric emission models to obtain a bulk Lorentz factor, $\Gamma$ of
$\sim$ 440\.
A careful analysis, based partly on the results presented above, allowed us to
separate the underlying pulse structure from the overall structure of the
light curve. Fig. 6 shows the full light curve for the GRB in the energy range
100 – 985 keV along with the best-fit KRL pulse function. The energy range was
so chosen to avoid contamination from the soft component (below 100 keV).
The best-fit KRL function was then used as the representation of the light
curve to be matched by the model. We assumed a Bulk Lorentz factor $\Gamma$ =
440 and z = 2 (as in McGlynn et al. 2012). Fig. 7 shows the segmentation of
the light curve into equal-fluence segments. The time-integrated energy-
spectrum of the pulse was fit with a Band function (as in McGlynn et al 2012,
in the energy range, 8 - 985 keV) and a Comptonized–Epk function (in the range
100-985 keV) and these, along with a theoretical BPL function (originally used
by Dermer 2004) were used as rest-frame spectra for the model. We varied the
shell-collision parameters and extracted best-fit pulses using these three
spectral functions (see Fig. 8 for the pulses as well as details of the
corresponding spectral parameters). The resulting best-fit shell-collision
parameters, together with a best-fit value of $t_{var}=44$ seconds and the
chosen values of $\Gamma$ and z, yielded a radius, $r_{0}$ of 1.5 $\times
10^{17}$ cm, a shell thickness, $\Delta r^{\prime}$ of 8.8 $\times 10^{12}$ cm
and a co-moving frame pulse-duration, $\Delta t^{\prime}$ of 2.0 $\times
10^{3}$ seconds for the Comptonized–Epk spectral function. The corresponding
values for the Band and BPL functions were very similar. Figs. 9 and 10 show
our results for the peak-flux–peak-frequency relation for the data with the
best-fit Comptonized-Epk function and the model using the three different
rest-frame spectra. Fig. 11 shows the spectral lags for the light curves of
Fig. 8.
In order to explore the evolution of the pulse width with energy, we extracted
the FWHM and plotted this as a function of energy. The plot is shown in Fig.
12. The energy bands chosen (in keV) were 8–25, 25–50, 50–100, 100–150,
150–200, 200–250, 250–350, 350–985. These energy bands were so chosen as to
ensure a sufficient number of counts in each energy band for a reliable
measurement of pulse FWHM while also providing a sufficient number and spacing
of bands to show the trend curve. As the KRL function does not fit the pulses
below 150 keV accurately, we employed a Monte-Carlo simulation where 1000
light curves were simulated in each energy band using the square-root of the
counts as their errors (assuming independent Poisson distributions for the
counts), the pulse FWHM was extracted for each light curve, and the mean and
standard deviation of the 1000 pulse FWHMs were used as the pulse FWHM and its
error respectively for each band. We fit a power-law of the form C$E^{a}$ to
the model points and extracted an exponent of -0.31 +/- 0.03.
## 4\. Discussion
Before we turn to the test-case GRB, we note that in the model calculation for
a broken-power-law rest-frame spectrum, the lag shows a relatively well-
defined trend with no lags at energies below $\sim$ 0.3Epk,0 and a constant
lag for all energies above Epk,0 (Fig. 4). Increasing or decreasing the value
of Epk,0, while keeping all other model parameters fixed, only shifts the
entire curve in the direction of increase or decrease. Shen et al. (2005)
studied the lags due to curvature effects using different rest-frame emission
profiles. They found that for an infinitesimal shell, a rectangular profile
produces no lags. We find the situation to be quite different for the case of
a finite shell thickness. In addition, as depicted in Fig 4, a change in the
rest-frame spectrum also has a significant effect in the evolution of the
spectral lag with energy. In the case of a Band or Comptonized–Epk function,
the lags are small for energies small compared to Epk,0, and the lags for a
Comptonized–Epk function do not show a saturation energy. This is primarily
because the Comptonized–Epk function varies monotonically at all energies and
does not have a well-defined break energy. The value of $r_{0}$ obtained for
the test case is consistent with estimates for an internal shock model (see
for e.g. Hascoet et al. 2012). While the value of $\Delta r^{\prime}$ ($\sim
10^{13}$ cm) seems reasonable, it is difficult to infer the significance of
its absolute magnitude in the context of the current analysis. The inclusion
of a finite-shell-thickness component in our curvature model also produces
relatively large lags ($\sim$ a few seconds) without a need for extreme
physical parameters such as $\Gamma<$ 50 (as concluded by Shen et al. 2005),
or a large local pulse width ($\sim 10^{7}$ seconds as concluded by Lu et al.
2006). We find that a rest-frame pulse duration, $\Delta t^{\prime}\sim
10^{3}$ seconds is sufficient to produce such lags.
Our results for the test case show that an internal shock model with a rest-
frame spectrum identical to that used to fit the data (the Comptonized–Epk and
Band functions), reproduces the observed pulse profile as well as the observed
spectral lags. It was difficult to determine if there was a saturation energy
present in the lag-energy-evolution (Fig. 11) as there were insufficient
counts to extract lags in higher energy bands. As noted above, a finite shell
thickness can account for observed lags even for the case of a rectangular
rest-frame emission profile. A second Band function component with a peak
shifted to 1 MeV (when a blackbody component is included in the fit) would
imply a 1 MeV break energy in the lag-energy plot, and would also predict no
lags (or small lags in the case of the Comptonized–Epk) below $\sim$300 keV.
This does not match the observations. It can be seen from Fig. 10 that the
exponent in the peak-flux–peak-frequency relation is close to 3 in all cases
for the model pulses shown in Fig. 8. This confirms the observations of Dermer
(2004) that the exponent at times after the peak of the light curve is close
to 3 even with different choices of rest-frame spectra. However, this does not
match the exponent obtained from the data (Fig. 9). While a connection may
exist between the observed spectral evolution and the spectral lags, it
appears that curvature effects alone cannot describe this connection and
additional emission mechanisms may be needed. We note in passing that Guirec
et al. (2012) have included a blackbody component to describe the temporal and
spectral properties of a number of GRBs.
We have also explored the behavior of the pulse width with energy. As shown in
Fig. 12, the pulse-width decreases with energy approximately as a power law.
Both the data and the model predictions exhibit similar trends although the
low-energy agreement is marginal. The exponent of the power law (-0.31 +/-
0.03) matches well the exponent extracted by Fenimore et. al. (1995), who
analyzed a large sample of bright BATSE bursts and obtained an average power-
law exponent of about -0.4. A similar result was also obtained by Peng et al.
2006 who analyzed a sizeable sample of bright single pulses in BATSE GRBs. In
addition, in a recent work based on an analysis of 51 long-duration FRED-like
single-pulses from the BATSE data, Peng et. al. 2012 showed that the curvature
effect combined with a Band rest-frame spectrum can explain the energy
dependence of the pulse widths. Cohen et al. (1997) have suggested that such
an exponent is consistent with a population of electrons losing energy via
synchrotron radiation, a process for which the exponent is predicted to be
-0.5.
## 5\. Conclusions
We have used a simple two-shell collision model to investigate curvature
effects in the prompt emission of GRBs. We have examined the effects of
emission spectra such as the Band, the Comptonized–Epk, and the BPL functions.
We have focused primarily on the peak-flux – peak-frequency relation and the
evolution of the spectral lags and the pulse widths with energy. We compare
our model results with the results of similar models in the literature and
also present a test case study of GRB 110920.
We summarize our main findings as follows:
* •
We find that introduction of a finite shell thickness in the curvature
formulation can produce smooth light curves, i.e., without a rapid transition
from the rise to the decay portions even for the case of a rectangular rest-
frame emission profile. As we approach the infinitesimal shell (surface)
approximation, we recover the sharp featured light-curve profile of Qin et.
al. 2004 for a rapidly dimming intrinsic emission profile.
* •
While we agree with Shen et al. (2005) that an infinitesimal shell produces no
discernible spectral lag using a rectangular emission-pulse-profile, we find
the situation to be different for a shell of finite thickness i.e., a finite
spectral lag can be produced even with a rectangular pulse profile;
* •
The spectral lag evolution as a function of energy is quite sensitive to the
type of rest-frame spectrum. For example, the Comptonized–Epk model does not
appear to exhibit a saturation energy at which the spectral lags reach a
plateau phase as in the case of the Band and the broken-power-law functions.
We agree with Shen at al (2005) that the spectral lags seem to approach a
maximum when Epk is near the high-energy channel used in extracting the lag.
Most likely this simply reflects the break energy present in the assumed rest-
frame spectrum (i.e., Band and BPL);
* •
All rest-frame spectral models tested exhibit the peak-flux – peak-frequency
relation although with exponents that differ from the predicted exponent of 3.
The significance of this discrepancy is not clear at this stage and warrants
further investigation;
* •
The peak-flux – peak-frequency test for GRB 110920 yields an exponent of 1.64
+/- 0.012 compared to the theoretical one of 2.57 +/- 0.01 (with
Comptonized–Epk as the rest-frame spectrum). We consider this discrepancy to
be significant and the result to be in disagreement with the prediction based
purely on effects of curvature. Similar conclusions were reached by Dermer
(2004), Qin (2009) and Borgonovo and Ryde (2001);
* •
Both the data (test GRB) and the model exhibit a very similar power-law trend
for the pulse width with energy. The plateau/power-law/plateau feature noted
by Qin et al. 2005 is well reproduced with a given choice of key model
parameters. In addition, we note that the power-law exponent is sensitive to
the assumed shell thickness. For the test-case GRB, the power-law exponent
matches well with exponents extracted from a larger sample of GRBs from
earlier studies (Fenimore et al. 1995; Peng et. al. 2006, 2012); and
* •
Relatively good agreement is obtained with all rest-frame spectral models for
the spectral lag versus energy for the test GRB. This is somewhat surprising
given the result of the peak-flux – peak-frequency test noted above. It would
seem that some complex interplay is at work between various model parameters
such as shell thickness, variability time scale, the energy evolution of Epk
and the Lorentz factor. The investigation of the dependencies of these various
parameters is ongoing.
The role of the reported soft component of the light curve for GRB 110920 has
not been fully investigated in this study and is worth pursuing, particularly
with regard to the behavior of the peak-flux–peak-frequency relation. Finally,
we note that these studies are being extended to a larger sample of GRBs.
## Acknowledgements
The authors (AS and KSD) would like to acknowledge A. Eskandarian and O.
Kargaltsev (both from the George Washington University) for their valuable
contributions to the discussions as well as the financial support provided by
them to A. Shenoy at various stages of this work.
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|
arxiv-papers
| 2013-04-15T17:03:51 |
2024-09-04T02:49:44.386776
|
{
"license": "Public Domain",
"authors": "A. Shenoy, E. Sonbas, C. Dermer, L. C. Maximon, K. S. Dhuga, P. N.\n Bhat, J. Hakkila, W. C. Parke, G. A. Maclachlan, T. N. Ukwatta",
"submitter": "Eda Sonbas",
"url": "https://arxiv.org/abs/1304.4168"
}
|
1304.4203
|
Non-dipolar magnetic field at the polar cap of neutron stars and
the physics of pulsar radiation
Niedipolowe pole magnetyczne nad czapą polarną gwiazdy neutronowej
a fizyka promieniowania pulsarów
Andrzej Szary
prof. dr hab. Giorgi Melikidze
Zielona Góra
To Natalia, my daughter, and Beata, my wife, for being there...
CHAPTER: ABSTRACT
Despite the fact that pulsars have been observed for almost half a
century, until now many questions have remained unanswered. One of
the fundamental problems is describing the physics of pulsar radiation.
By trying to find an answer to this fundamental question we use the
analysis of X-ray observations in order to study the polar cap region
of radio pulsars. The size of the hot spots implies that the magnetic
field configuration just above the stellar surface differs significantly
from a purely dipole one. By using the conservation of the magnetic
flux we can estimate the surface magnetic field as of the order of
$10^{14}\,{\rm G}$. On the other hand, the temperature of the hot
spots is about a few million Kelvins. Based on these two facts the
Partially Screened Gap (PSG) model was proposed to describe the Inner
Acceleration Region (IAR). The PSG model assumes that the temperature
of the actual polar cap is equal to the so-called critical value,
i.e. the temperature at which the outflow of thermal ions from the
surface screens the gap completely.
We have found that, depending on the conditions above the polar cap,
the generation of high energetic photons in IAR can be caused either
by Curvature Radiation (CR) or by Inverse Compton Scattering (ICS).
Completely different properties of both processes result in two different
scenarios of breaking the acceleration gap: the so-called PSG-off
mode for the gap dominated by CR and the PSG-on mode for the gap dominated
by ICS. The existence of two different mechanisms of gap breakdown
naturally explains the mode-changing phenomenon. Different characteristics
of plasma generated in the acceleration region for both processes
also explain the pulse nulling phenomenon. Furthermore, the mode changes
of the IAR may explain the anti-correlation of radio and X-ray emission
in very recent observations of PSR B0943+10 [87].
Simultaneous analysis of X-ray and radio properties have allowed to
develop a model which explains the drifting subpulse phenomenon. According
to this model the drift takes place when the charge density in IAR
differs from the Goldreich-Julian co-rotational density. The proposed
model allows to verify both the radio drift parameters and X-ray efficiency
of the observed pulsars.
Pomimo, że pulsary są badane już od prawie pół wieku, do dzisiaj nie
udało się znaleźć odpowiedzi na wiele pytań. Jednym z fundamentalnych
problemów jest opis fizyki promieniowania pulsarów. Próbując znaleźć
odpowiedź na to fundamentalne pytanie, wykorzystujemy analizę obserwacji
rentgenowskich w celu badania obszaru czapy polarnej pulsarów. Rozmiar
obserwowanych gorących plam wskazuje, że konfiguracja pola magnetycznego
na powierzchni gwiazdy różni się znacznie od pola czysto dipolowego.
Wykorzystując prawo zachowania strumienia magnetycznego możemy oszacować
siłę pola magnetycznego w obszarze czapy polarnej, które dla obserwowanych
pulsarów jest rzędu $10^{14}\,{\rm G}$. Z drugiej strony obserwowana
temperatura gorącej plamy jest rzędu kilku milionów kelwinów. Opierając
się na tych dwóch faktach wykorzystujemy model częściowo-ekranowanej
przerwy akceleracyjnej (z ang. Partially Screened Gap - PSG), aby
opisać wewnętrzną przerwę akceleracyjną (z ang. Inner Acceleration
Region - IAR). Model PSG zakłada, że temperatura czapy polarnej jest
bliska do tak zwanej wartości krytycznej tzn. takiej przy, której
termiczny odpływ jonów z powierzchni w pełni ekranuje przerwę akceleracyjną.
W zależności od warunków jakie panują w obszarze czapy polarnej, mechanizmem
odpowiedzialnym za generowanie wysokoenergetycznych fotonów w IAR
może być promieniowanie krzywiznowe (z ang. Curvature Radiation -
CR) lub odwrotne rozpraszanie Comptona (z ang. Inverse Compoton Scattering
- ICS). Całkowicie różne właściwości obu tych procesów prowadzą do
sytuacji, w której możemy wyróżnić dwa scenariusze zamknięcia przerwy
akceleracyjnej: tzw. PSG-off dla przerwy zdominowanej przez promieniowanie
CR, oraz tzw. PSG-on dla przerwy zdominowanej przez ICS. Istnienie
dwóch różnych mechanizmów zamknięcia przerwy w naturalny sposób tłumaczy
zjawisko zmiany trybu promieniowania pulsarów (z ang. mode-changing).
Różna charakterystyka plazmy generowanej w obszarze akceleracyjnym
dla obu tych trybów tłumaczy zjawisko sporadycznego braku pojedynczych
pulsów (z ang. pulse nulling) w obserwacjach radiowych. Co więcej
zmiana trybu w jakim pracuje przerwa akceleracyjna może zostać powiązana
z antykorelacją promieniowania radiowego i rentgenowskiego wykazaną
w ostatnich obserwacjach PSR B0943+10 [87].
Jednoczesna analiza właściwości promieniowania rentgenowskiego i radiowego
pozwoliła na opracowanie modelu dryfujących składowych pulsu pojedynczego
(z ang. subpulses). Model ten zakłada, że dryf jest wynikiem różnicy
gęstości ładunku w IAR w stosunku do gęstości korotacji. Proponowany
model pozwala zarówno na weryfikację wyznaczonych parametrów dryfu
oraz na weryfikację np. efektywności promieniowania rentgenowskiego.
CHAPTER: INTRODUCTION
The history of neutron stars began in the early 1930s when Subrahmanyan
Chandrasekhar calculated the critical mass for a white dwarf. As soon
as the mass of a white dwarf exceeds the critical value (e.g. due
to accretion of matter from a companion star) it collapses and a neutron
star is formed. Chandrasekhar estimated that the critical mass was
approximately $1.4$ solar masses (${\rm M}_{\odot}$). Even before
James Chadwick's discovery of neutrons [31],
Lev Landau anticipated the existence of neutron stars by writing about
stars in which “atomic nuclei come in close contact, forming one
gigantic nucleus”. In 1934 [7] proposed that
the “supernova process represents the transition of an ordinary
star into a neutron star”. Five years later 138,
using the work of 173, computed an upper bound on
the mass of a star composed of neutron-degenerate matter. They assumed
that the neutrons in a neutron star form a cold degenerate Fermi gas
which leads to an upper bound of approximately $0.7\,{\rm M}_{\odot}$.
Modern estimates of the critical mass for neutron stars range from
approximately $1.5\,{\rm M}_{\odot}$ to $3\,{\rm M}_{\odot}$ [24].
This uncertainty reflects the fact that the equation of state for
extremely dense matter is not well known. Let us note that the radius
of a neutron star should be $R\approx10{\rm \, km}$. On the other
hand nobody expected to detect any emission from neutron stars due
to their small size and the lack of theoretical predictions about
any radiation processes, except for thermal radiation. Thus, it took
almost forty years to detect emission from a neutron star.
The breakthrough came on 28 November 1967 with the radio observations
that were performed by Jocelyn Bell-Burnell and Anthony Hewish. They
observed radio pulses separated by $1.33$ seconds. The world “pulsar”
was adopted to reflect the specific property of these celestial objects.
The suggestion that pulsars were rotating neutron stars was put forth
independently by 69 and 139, and
was soon proved beyond a reasonable doubt by the discovery of a pulsar
with a very short ($33$-millisecond) pulse period in the Crab nebula.
It was suggested that this pulsar powers the activity of the nebula
[139]. Nearly 2000 pulsars have been found so far.
Observations of pulsars provide valuable information about neutron
star physics, general relativity, the interstellar medium, celestial
mechanics, planetary physics, the Galactic gravitational potential,
the magnetic field and even cosmology. Studying neutron stars is therefore
a very broad issue and it is beyond the scope of this thesis to describe
the current status of the theory of neutron stars or pulsar population
studies in detail. We rather refer the reader to the literature [129, 128, 68, 179]
and provide only a basic theoretical background that is relevant to
the subject of this thesis.
Following the ideas of 139 and 69
radio pulsars can be interpreted as rapidly spinning, strongly magnetised
neutron stars radiating at the expense of their rotational energy.
Neutron stars consist of compressed matter with density in its core
exceeding nuclear density $\rho_{{\rm nuc}}=2.8\times10^{14}{\rm \, g\, cm^{-3}}$.
Direct and accurate mass measurements come from timing observations
of binary pulsars and are consistent with a typically assumed neutron
star mass $M\approx1.4\,{\rm M}_{\odot}$. Most models predict a radius
of $R\sim10\,{\rm km}$, which is consistent with the theoretical
upper and lower limits. However, the measurements of neutron star
radii are much less reliable than the mass measurements. Therefore,
the moment of inertia for these canonical values ($M=1.4\,{\rm M}_{\odot}$,
$R=10{\rm \, km}$) $I\approx\left(2/5\right)MR^{2}\approx10^{45}{\rm \, g\, cm^{2}}$
may be uncertain by $\sim70\%$. The increase rate of a pulsar period,
$\dot{P}={\rm d}P/{\rm d}t$, is related to the rate of rotational
kinetic energy loss (spin-down luminosity) $\dot{E}=L_{{\rm SD}}=4\pi^{2}I\dot{P}P^{-3}$.
In most cases only a tiny fraction of $\dot{E}$ can be converted
into radio emission. The efficiency, $\chi_{{\rm radio}}=L_{{\rm radio}}/\dot{E}$,
in the radio bands is typically in the range of $\sim10^{-7}-10^{-5}$.
It is assumed that the bulk of the rotational energy is converted
into magnetic dipole radiation. The expected evolution of the angular
velocity ($\Omega=2\pi/P$) of a rotating magnetic dipole can be described
as $\dot{\Omega}\sim\Omega^{n}$, and the breaking index is $n=3$
for the pure dipole radiation. Indeed, the observed values of the
breaking index (e.g. 15) confirm the above statement,
e.g.: for the Crab $n=2.515\pm0.005$, for PSR B1509-58 $n=2.8\text{\ensuremath{\pm}}0.2$,
for PSR B0540-69 $n=2.28\text{\ensuremath{\pm}}0.02$, for PSR J1911-6127
$n=2.91\pm0.05$, for PSR J1846-0258 $n=2.65\text{\ensuremath{\pm}}0.01$,
and for the Vela pulsar $n=1.4\text{\ensuremath{\pm}}0.2$. On the
other hand the observations of pulsar wind nebulae suggest that a
significant fraction of the pulsar rotational energy is carried away
by a pulsar wind. Furthermore, recent observations of high energy
radiation from pulsars show that significantly more energy is radiated
in the form of X-rays and $\gamma$-rays than in the form of radio
emission (e.g. 1). Thus, pure magnetic breaking
does not provide full information about the physical processes that
take place in the pulsar magnetosphere.
Despite the fact that pulsars have been observed for almost half a
century, many questions still remain unanswered. One of the fundamental
problems concerns the physics of pulsar radiation. Radio observations
alone cannot point to the model (e.g. vacuum gap, slot gap, outer
gap, free outflow, etc.) that correctly describes the source of pulsar
activity. Observations carried out by relatively new high-energy instruments,
e.g. Chandra and XMM-Newton, significantly extended
the spectra over which we can study pulsars and their environments.
There is no consensus about the origin of pulsar X-ray emission [129].
We can distinguish two main types of models: the polar gap and the
outer gap. The polar gap models suggest that the emission region is
located in the vicinity of the neutron star polar caps, while the
outer gap models assume that particle acceleration and X-ray emission
take place close to the pulsar light cylinder
[The light cylinder with radius $R_{{\rm LC}}=cP/2\pi$ is defined
as a place where the azimuthal velocity of the co-rotating magnetic
field lines is equal to the speed of light ($c$)
]. In both types of models high energy radiation is generated by relativistic
particles accelerated in charge-depleted regions, while the high energy
photons are emitted by means of Curvature Radiation (CR), Synchrotron
Radiation (SR) and Inverse Compton Scattering (ICS). Both models are
able to interpret existing observational data.
In this thesis we will use the Partially Screened Gap (PSG) model
[63]. The PSG model assumes the existence of the Inner
Acceleration Region (IAR) above the polar cap (a region penetrated
by the open field lines) where the electric field has a component
along the magnetic field. In this region particles (electrons and
positrons) are accelerated in both directions: outward and toward
the stellar surface. Consequently, outflowing particles are responsible
for generation of magnetospheric emission (radio and high-frequency)
while the backflowing particles heat the surface and provide the required
energy for thermal emission. The PSG model is an extension of the
Standard Model developed by 156 and takes into
account the thermionic ion flow from the stellar surface heated up
to a high temperature (a few million Kelvins) by the backstreaming
particles. In such a scenario an analysis of X-ray radiation is an
excellent method of obtaining insight into the most intriguing region
of the neutron star.
CHAPTER: X-RAY EMISSION FROM RADIO PULSARS
§ BRIEF HISTORICAL OVERVIEW
X-ray photons can only be detected by telescopes operating at high
altitudes or above the Earth's atmosphere, thus detectors should be
mounted on high-flying balloons, rockets or satellites. The first
(i.e. carried out from space) X-ray observations were performed by
a team led by Herbert Friedman in 1948. The team estimated the luminosity
of X-ray radiation from the solar corona. They found that X-ray luminosity
is weaker by a factor of $10^{6}$ than luminosity in the optical
wave range. Up until the early 1960s it was widely believed that all
other stars should be so faint in the X-rays that their observations
would be hopeless. The situation changed in 1962 when a team led by
Bruno Rossi and Riccardo Giacconi, when trying to find fluorescent
X-ray photons from the moon, accidentally detected X-rays from Sco
X-1. Subsequent flights launched to confirm these first results detected
Tau X-1, a source in the constellation Taurus which coincided with
the Crab supernova remnant [26]. The search for similar
sources became a source of strong motivation for the further development
of X-ray astronomy.
Before the first direct detection of a neutron star by 90,
it was predicted that neutron stars could be powerful sources of thermal
X-ray emission due to a high surface temperature ($T_{{\rm s}}$).
The expected value of the surface temperature was estimated as $T_{{\rm s}}\sim1\,{\rm MK}$
[41, 174]. The first X-ray observations of isolated
neutron stars
[The term ”isolated” is omitted hereafter in the text however all
X-ray observations presented in this thesis concern isolated neutron
] were initiated by the Einstein Observatory, which was launched
by NASA in 1978. Using a high-resolution imaging camera sensitive
in the $0.2-3.5\,{\rm keV}$ energy range provided unprecedented levels
of sensitivity (hundreds of times better than had previously been
achieved). The Einstein detected X-ray emission from a number
of neutron stars (mainly as compact sources in supernova remnants)
such as the middle-aged radio pulsars B0656+14, B1055-52 and the old
pulsar B0950+08. The Einstein observatory re-entered the
Earth's atmosphere and burned up on 25 March 1982. The next ”decade
of space science” was opened in the 1990s with the launch of the
ROSAT mission that was sensitive in the $0.1-2.4\,{\rm keV}$
energy range. One of the major results achieved with the ROSAT
was the identification of the $\gamma$-ray source Geminga as a pulsar,
hence a neutron star [79].
The current era of X-ray observations of neutron stars was begun with
the launch of two satellites: the XMM-Newton owned by the
European Space Agency and the Chandra owned by the National
Aeronautics and Space Administration. These two grazing-incidence
X-ray telescopes were placed in orbit in 1999. They were equipped
with cameras and high-resolution spectrometers sensitive to low-energy
X-rays: from $0.08$ to $10\,{\rm keV}$ for the Chandra
and from $0.1$ to $15\,{\rm keV}$ for the XMM-Newton. While
the two observatories have similar designs, they are not identical.
The XMM-Newton observatory has three X-ray telescopes that
provide six times the collecting area and a broader spectral range
in images than the Chandra, while the Chandra has
a much finer spatial resolution and a broader spectral range in its
high-resolution spectroscopy than does the XMM-Newton. Both
observatories are in a highly-elliptical orbit that permits continuous
observations of up to 40 hours. The Chandra and XMM-Newton
have greatly increased the quality and availability of observations
of X-ray thermal radiation from neutron star surfaces. The total number
of isolated neutron stars of different types detected in X-rays is
hard to find since not all data have been published. Some authors
estimate that about one hundred rotation-powered pulsars were detected
in the X-rays [185, 15].
§ X-RAY EMISSION FROM ISOLATED NEUTRON STARS
X-ray emission is a common feature of all kinds of neutron stars.
Furthermore, X-ray observations have led to the discovery of other
types of neutron stars that for various reasons were missed in the
standard searches for radio pulsars. These new classes, such as X-ray
Dim Isolated Neutron Stars, Central Compact Objects in supernovae
remnants, Anomalous X-ray Pulsars, and Soft Gamma-ray Repeaters, are
only a small fraction of the whole number of observed pulsars but
provide valuable information on the diversity of the neutron star
X-ray radiation from an isolated neutron star can in general consist
of two distinguishable components: thermal and nonthermal emissions.
The thermal emission can originate either from the entire surface
of a cooling neutron star or from spots around the magnetic poles
on the stellar surface (polar caps and adjacent areas). The temperature
of a neutron star at the moment of its formation is extremely high
- its value is even as high as $10^{10}-10^{11}\,{\rm K}$. Such a
high initial temperature leads to very fast cooling, and after several
minutes the temperature of the star interior falls to $10^{9}-10^{10}{\rm \, K}$.
After $10-100\,{\rm yr}$ the neutron star will cool down to a few
times $10^{6}\,{\rm K}$. At this point, depending on the still poorly
known properties of super-dense matter, the temperature evolution
can follow two different scenarios. The standard cooling scenario
predicts that the temperature decreases gradually, down to $\sim\left(0.3-1\right)\times10^{6}\,{\rm K}$
by the end of the neutrino cooling era and then falls exponentially
to temperatures lower than $\sim10^{5}\,{\rm K}$ in $\sim10^{7}\,{\rm yr}$.
In the accelerated cooling scenario, which implies higher central
densities (up to $10^{15}\,{\rm g\, cm^{-3}}$) and/or exotic interior
composition (e.g. quark plasma), at the age of $\sim10-100\,{\rm yr}$
the temperature decreases rapidly down to $\sim\left(0.3-0.5\right)\times10^{6}\,{\rm K}$
and is followed by a more gradual decrease down to the same $\sim10^{5}\,{\rm K}$
in $\sim10^{7}\,{\rm yr}$ [15]. The thermal evolution
of neutron stars is very sensitive to the composition (and structure)
of their interiors, therefore, measuring surface temperatures is an
important tool in studying super-dense matter. In addition to a thermal
component emitted from the entire surface, other thermal components
can also be seen. One of these additional components could be related
to the reheating of the polar cap region by relativistic backflowing
particles (electron and/or positrons) created and accelerated in the
so-called polar gaps (see Chapter <ref>). The temperature
of these hot spots does not obey the same age dependence as the thermal
evolution of neutron stars. Thus, depending on the pulsar age the
thermal radiation may be dominated by either the entire surface (for
younger neutron stars) or the hot spot components (for older neutron
stars). The nonthermal component is usually attributed to the emission
produced by Synchrotron Radiation (SR) and/or Inverse Compton Scattering
(ICS) of charged relativistic particles accelerated in the pulsar
magnetosphere. As the energy of these particles follows a power-law
distribution, nonthermal emission is also characterised by power-law
The X-ray spectrum of a neutron star (thermal and nonthermal) depends
on many factors, e.g. the age of the star ($\tau$), inclination angle,
strength and geometry of the magnetic field, etc. In most of the very
young pulsars ($\tau\sim1\,{\rm kyr}$) the nonthermal component dominates,
thus making it impossible to accurately measure the thermal flux –
only the upper limits for the surface temperature can be derived.
As a pulsar becomes older, its activity (nonthermal luminosity) decreases
roughly proportionally to its spin-down luminosity $L_{{\rm SD}}$.
A spin-down luminosity generally decreases with the increasing star
age, as $L_{{\rm SD}}\propto\tau^{-m}$, where $m\simeq2-4$ depends
on the pulsar dipole breaking index [186]. With the
increase of the pulsar age the luminosity of the surface thermal radiation
decreases more slowly than the luminosity of the nonthermal one. Thus,
the thermal radiation from an entire stellar surface can dominate
a soft X-ray spectrum of middle-aged ($\tau\sim100\,{\rm kyr}$) and
some younger ($\tau\sim10\,{\rm kyr}$) pulsars. For the old neutron
stars ($\tau>1\,{\rm Myr}$), a surface temperature $T_{{\rm s}}<0.1\,{\rm MK}$
makes it impossible to detect the thermal radiation from the entire
surface by available observatories. However, most of the pulsar models
predict the heating up of polar caps to very high temperatures ($T_{{\rm s}}\apprge1\,{\rm MK}$)
by relativistic particles which are created in the pulsar acceleration
zones. Conventionally, it is assumed that the polar cap radius is
$R_{{\rm dp}}=\sqrt{2\pi R^{3}/cP}$.
Since the spin-down luminosity $L_{{\rm SD}}$ is the source for both
nonthermal (magnetospheric) and thermal (polar cap) components, it
is hard to predict which one would prevail in the X-ray flux of old
neutron stars. Figure <ref>a shows the ratio
of a thermal luminosity to a nonthermal one as a function of the pulsar
age. Since calculating this ratio is possible only for pulsars with
blackbody plus power-law fit, only these pulsars are included in the
Figure. There is also a significant number of pulsars ($16$) with
the spectra dominated by nonthermal components. Let us note that it
is impossible to determine the thermal components for these pulsars.
Most of them are young neutron stars $\sim10^{3}-10^{4}\,{\rm yr}$,
but there are also much older ones ($\sim10^{6}\,{\rm yr}$). In addition,
there is a group of $4$ pulsars with the spectra dominated by thermal
components (without a visible nonthermal component). Their age also
varies in quite a wide range $10^{4}-10^{6}\,{\rm yr}$.
[Nonthermal and thermal components in the X-ray spectra of neutron
stars]Ratio of X-ray luminosities (thermal and nonthermal components) as
a function of $\tau$ (panel a) and $B_{{\rm d}}$ (panel b). The
plots contain only those pulsars for which the BB+PL (Black-Body plus
Power-Law) spectral fit exists. The number labels at the points correspond
to the pulsar numbers in Table <ref>.
As it follows from the left panel of Figure <ref>,
there is no obvious relation between pulsar age and the ratio of luminosities.
The spectra of pulsars with a similar age may be dominated either
by nonthermal (e.g. PSR B1951+32, PSR B1046-58) or thermal (e.g. PSR
B0656+14, PSR J0538+2817) components. It is difficult to provide a
more detailed analysis because, on the one hand, the observational
errors are large and, on the other hand, a separation of thermal and
nonthermal components is often not possible. The ratio of luminosities
also does not show any correlation with the strength of the dipolar
magnetic field (see the right panel of Figure <ref>).
Let us note that the value of the dipolar magnetic field is conventionally
calculated by adopting that the spin-down luminosity is equal to the
power of magneto-dipole radiation (neglecting the influence of a pulsar
wind). Then, assuming a dipolar structure of the neutron star magnetic
field down to the stellar surface, we estimate its strength (measured
in Gauss) at the pole as
\begin{equation}
B_{{\rm d}}=2.02\times10^{12}\left(P\dot{P}_{-15}\right)^{0.5}.
\end{equation}
Here $P$ is a period in seconds and $\dot{P}_{-15}=\dot{P}\times10^{15}$.
The actual strength of the surface magnetic field can greatly exceed
the above value (see Chapter <ref>).
Table <ref> presents the basic parameters of the 48 pulsars
that we use in this thesis, while the results of the X-ray observations
of these pulsars are listed in Tables <ref>
and <ref>.
[Parameters of rotation-powered normal pulsars with detected X-ray
radiation] Parameters of rotation powered normal pulsars with detected X-ray
radiation. The individual columns are as follows: (1) Pulsar name,
(2) Barycentric period $P$ of the pulsar, (3) Time derivative of
barycentric period $\dot{P}$, (4) Canonical value of the dipolar
magnetic field $B_{{\rm d}}$ at the poles, (5) Spin-down energy loss
rate $L_{{\rm SD}}$ (spin-down luminosity) , (6) Dispersion measure
$DM$, (7) Best estimate of pulsar distance $D$ (used in all calculations),
(8) Best estimate of pulsar age or spin-down age $\tau=P/\left(2\dot{P}\right)$,
(9) Pulsar number (used in the Figures). Parameters of the radio pulsar
have been taken from the ATNF catalogue.
Name $P$ $\dot{P}$ $B_{{\rm d}}$ $\log L_{{\rm SD}}$ $DM$ $D$ $\tau$ No. $\left({\rm s}\right)$ $\left(10^{-15}\right)$ $\left(10^{12}\,{\rm G}\right)$ $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm cm^{-3}\, pc}\right)$ $\left({\rm kpc}\right)$ J0108–1431 $0.808$ $0.077$ $0.504$ $30.76$ $2.38$ $0.18$ $166$ Myr 1 J0205+6449 $0.066$ $193.9$ $7.210$ $37.43$ $141$ $3.20$ $5.37$ kyr 2 B0355+54 $0.156$ $4.397$ $1.675$ $34.65$ $57.1$ $1.04$ $564$ kyr 3 B0531+21 $0.033$ $422.8$ $7.555$ $38.66$ $56.8$ $2.00$ $1.24$ kyr 4 J0537–6910 $0.016$ $51.78$ $1.846$ $38.69$ – $47.0$ $4.93$ kyr 5 J0538+2817 $0.143$ $3.669$ $1.464$ $34.69$ $39.6$ $1.20$ $30.0$ kyr 6 B0540–69 $0.050$ $478.9$ $9.934$ $38.18$ $146$ $55.0$ $1.67$ kyr 7 B0628–28 $1.244$ $7.123$ $6.014$ $32.18$ $34.5$ $1.45$ $2.77$ Myr 8 J0633+1746 $0.237$ $10.97$ $3.258$ $34.51$ – $0.16$ $342$ kyr 9 B0656+14 $0.385$ $55.00$ $9.294$ $34.58$ $14.0$ $0.29$ $111$ kyr 10 J0821–4300 $0.113$ $1.200$ $0.743$ $34.52$ – $2.20$ $3.7$ kyr 11 B0823+26 $0.531$ $1.709$ $1.924$ $32.65$ $19.5$ $0.34$ $4.92$ Myr 12 B0833–45 $0.089$ $125.0$ $6.750$ $36.84$ $68.0$ $0.21$ $11.3$ kyr 13 B0834+06 $1.274$ $6.799$ $5.945$ $32.11$ $12.9$ $0.64$ $2.97$ Myr 14 B0943+10 $1.098$ $3.493$ $3.956$ $32.00$ $15.4$ $0.63$ $4.98$ Myr 15 B0950+08 $0.253$ $0.230$ $0.487$ $32.75$ $2.96$ $0.26$ $17.5$ Myr 16 B1046–58 $0.124$ $96.32$ $6.972$ $36.30$ $129$ $2.70$ $20.3$ kyr 17 B1055–52 $0.197$ $5.833$ $2.166$ $34.48$ $30.1$ $0.75$ $535$ kyr 18 J1105–6107 $0.063$ $15.83$ $2.020$ $36.40$ $271$ $7.00$ $63.3$ kyr 19 J1119–6127 $0.408$ $4022$ $81.80$ $36.36$ $707$ $8.40$ $1.61$ kyr 20 9|r|Continued on next page
Table <ref> - continued from previous page
Name $P$ $\dot{P}$ $B_{{\rm d}}$ $\log L_{{\rm SD}}$ $DM$ $D$ $\tau$ No. $\left({\rm s}\right)$ $\left(10^{-15}\right)$ $\left(10^{12}\,{\rm G}\right)$ $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm cm^{-3}\, pc}\right)$ $\left({\rm kpc}\right)$ J1124–5916 $0.135$ $747.1$ $20.31$ $37.08$ $330$ $6.00$ $2.87$ kyr 21 B1133+16 $1.188$ $3.734$ $4.254$ $31.94$ $4.86$ $0.36$ $5.04$ Myr 22 J1210–5226 $0.424$ $0.066$ $0.338$ $31.53$ – $2.45$ $102$ Myr 23 B1259–63 $0.048$ $2.276$ $0.666$ $35.91$ $147$ $2.00$ $332$ kyr 24 J1357–6429 $0.166$ $360.2$ $15.62$ $36.49$ $128$ $2.50$ $7.31$ kyr 25 J1420–6048 $0.068$ $83.17$ $4.810$ $37.00$ $360$ $8.00$ $13.0$ kyr 26 B1451–68 $0.263$ $0.098$ $0.325$ $32.32$ $8.60$ $0.48$ $42.5$ Myr 27 J1509–5850 $0.089$ $9.170$ $1.824$ $35.71$ $138$ $2.56$ $154$ kyr 28 B1509–58 $0.151$ $1537$ $30.73$ $37.26$ $252$ $4.18$ $1.55$ kyr 29 J1617–5055 $0.069$ $135.1$ $6.183$ $37.20$ $467$ $6.50$ $8.13$ kyr 30 B1706–44 $0.102$ $92.98$ $6.235$ $36.53$ $75.7$ $2.50$ $17.5$ kyr 31 B1719–37 $0.236$ $10.85$ $3.234$ $34.52$ $99.5$ $1.84$ $345$ kyr 32 J1747–2958 $0.099$ $61.32$ $4.972$ $36.40$ $102$ $5.00$ $25.5$ kyr 33 B1757–24 $0.125$ $127.9$ $8.075$ $36.41$ $289$ $5.00$ $15.5$ kyr 34 B1800–21 $0.134$ $134.1$ $8.551$ $36.34$ $234$ $4.00$ $15.8$ kyr 35 J1809–1917 $0.083$ $25.54$ $2.936$ $36.26$ $197$ $3.50$ $51.3$ kyr 36 J1811–1925 $0.065$ $44.00$ $3.407$ $36.81$ – $5.00$ $23.3$ kyr 37 B1823–13 $0.101$ $75.06$ $5.575$ $36.45$ $231$ $4.00$ $21.4$ kyr 38 J1846–0258 $0.326$ $7083$ $97.02$ $36.91$ – $6.00$ $0.73$ kyr 39 B1853+01 $0.267$ $208.4$ $15.08$ $35.63$ $96.7$ $2.60$ $20.3$ kyr 40 B1916+14 $1.181$ $212.4$ $31.99$ $33.71$ $27.2$ $2.10$ $88.1$ kyr 41 J1930+1852 $0.137$ $750.6$ $20.47$ $37.08$ $308$ $5.00$ $2.89$ kyr 42 B1929+10 $0.227$ $1.157$ $1.034$ $33.59$ $3.18$ $0.36$ $3.10$ Myr 43 B1951+32 $0.040$ $5.845$ $0.971$ $36.57$ $45.0$ $2.00$ $107$ kyr 44 J2021+3651 $0.104$ $95.60$ $6.361$ $36.53$ $371$ $10.0$ $17.2$ kyr 45 J2043+2740 $0.096$ $1.270$ $0.706$ $34.75$ $21.0$ $1.80$ $1.20$ Myr 46 B2224+65 $0.683$ $9.659$ $5.187$ $33.08$ $36.1$ $2.00$ $1.12$ Myr 47 B2334+61 $0.495$ $191.7$ $19.69$ $34.79$ $58.4$ $3.10$ $40.9$ kyr 48
§ NONTHERMAL X-RAY RADIATION
The nonthermal emission, which is generally observed from radio to
$\gamma$-ray frequencies, should be generated by charged particles
accelerated at the expense of rotational energy in the magnetosphere
of the neutron star. Nonthermal X-ray radiation is characterised by
highly anisotropic emission patterns, which give rise to large pulsed
fractions. The pulse profiles often show narrow (often double) peaks,
however, in many cases nearly sinusoidal profiles are observed. As
the X-ray efficiency is strongly correlated with $L_{{\rm SD}}$,
the most X-ray luminous sources (among rotationally powered pulsars)
are the Crab pulsar and two young pulsars in the Large Magellanic
Cloud, which are the only pulsars with $L_{{\rm SD}}>10^{38}\,{\rm erg\, s^{-1}}$
18 suggested that in the $0.1\lyxmathsym{–}2.4\,{\rm keV}$
band ROSAT sources that are identified as rotation-powered
pulsars exhibit an X-ray efficiency which can be approximated as a
linear function $L_{{\rm X}}=\xi L_{{\rm SD}}$, where the total X-ray
$\xi=\xi_{_{{\rm BB}}}+\xi_{_{{\rm NT}}}\approx10^{-3}$, here $\xi_{_{{\rm BB}}}$
and $\xi_{_{{\rm NT}}}$ are efficiencies of the thermal (without
the cooling component) and nonthermal X-ray emission, respectively.
The higher sensitivity of both the Chandra and XMM-Newton
allows detection of less efficient ($\xi<10^{-3}$) X-ray pulsars
(see Figure <ref>). 15 suggested that
for these faint pulsars the orientation of the magnetic/rotation axes
to the observer's line of sight might not be optimal. We believe that
the efficiency of spin-down energy conversion processes is mostly
affected by the strength and structure of the surface magnetic field.
The variation of $\xi$ is rather due to the nature of physical processes
than the geometrical effects. Let us note that the nonthermal X-ray
luminosities presented in Figure <ref> are calculated
assuming an isotropic radiation pattern. In general, the X-ray emission
pattern differs quite essentially from the isotropic one. Thus, one
should introduce a beaming factor as the ratio of the opening angle
of the radiation cone to the full solid angle $4\pi$. Since a beaming
factor is generally unknown, the actual X-ray efficiency may differ
by up to an order of magnitude (or even more) than we have presented.
http://localhost:9090/pulsars/graphs/ (generate data, ~/Html/pulsar/media/images/xray_sd.dat)
cp ~/Html/pulsar/media/images/xray_sd.dat ~/Programs/studies/phd/xray_sd/.
cp ~/Programs/studies/phd/xray_sd/x_ray.svg ~/Documents/studies/phd/images/x-ray/.
[Nonthermal luminosity within the $0.1-10\,{\rm keV}$ band vs spin-down
luminosity]Nonthermal luminosity within the $0.1-10\,{\rm keV}$ band ($L_{{\rm NT}}$)
vs spin-down luminosity ($L_{{\rm SD}}$). The black solid line corresponds
to the linear fitting for all pulsars, while the blue dotted and red
dashed lines correspond to the linear fit for less luminous ($L_{{\rm SD}}<10^{35}\,{\rm erg\, s^{-1}}$)
and more luminous ($L_{{\rm SD}}>10^{35}\,{\rm erg\, s^{-1}}$) pulsars,
Various fitting parameters and efficiencies of nonthermal X-ray radiation
suggest that the efficiency of processes responsible for the generation
of nonthermal X-ray radiation should highly depend on the pulsar parameters
(see Figure <ref>). The fitting parameters for the data
of all pulsars show a linear trend with $\xi\approx10^{-3}$, however,
if we divide them into two groups of less and more luminous pulsars,
we can see that the fitting parameters for these two groups differ
from one another. The efficiency of less luminous X-ray pulsars depends
on $L_{{\rm SD}}$ to a lesser extent than is the case for more luminous
As we mentioned in the Introduction, there are two main types of models:
the polar cap models and the outer gap models. The outer gap model
was proposed to explain the bright $\gamma$-ray emission from the
Crab and Vela pulsars [36, 37]. Placing
a $\gamma$-ray emission zone at the light cylinder, where the magnetic
field strength is considerably reduced to $B_{{\rm LC}}=B_{{\rm d}}\left(R/R_{{\rm LC}}\right)^{3}$,
provides higher $\gamma$-ray emissivities that are in somewhat better
agreement with the observations. The observational data can be interpreted
with any of the two models, although under completely different assumptions
about pulsar parameters.
§.§ Observations
Generally, the X-ray spectrum of relatively young ($\tau<10\,{\rm kyr}$)
and middle-aged
($\tau<10\,{\rm kyr}$) pulsars is dominated by the nonthermal component.
However, it is not possible to find an exact correlation between $\tau$
and the type of spectra, i.e. which component, thermal or nonthermal,
dominates the spectrum (see the left panel of Figure <ref>).
As we mentioned above, it is quite often impossible to resolve the
components. The Crab pulsar ($\tau=958\,{\rm yrs}$) is the most characteristic
example of a young pulsar. The upper limit for X-ray luminosity of
the Crab pulsar (one of the strongest known X-ray radio pulsars) is
about $L_{{\rm NT}}^{{\rm ^{max}}}=8.9\times10^{35}\,{\rm erg\, s^{-1}}$.
This value is calculated assuming an isotropic radiation pattern,
however, even if we assume an angular anisotropy of the radiation
(beaming factor $\approx1/4\pi$), the lower limit of its luminosity
$L_{{\rm NT}}^{^{{\rm min}}}=7.1\times10^{34}\,{\rm erg\, s^{-1}}$
continues to be very high. The luminosities calculated above correspond
to the following X-ray efficiencies: ${\displaystyle \xi_{_{{\rm NT}}}^{^{{\rm max}}}=10^{-2.71}}$
(isotropic radiation pattern) and $\xi_{_{{\rm NT}}}^{{\rm ^{min}}}=10^{-3.81}$
(anisotropic radiation pattern). Although ${\displaystyle \xi_{_{{\rm NT}}}}$
is quite small, the nonthermal component still obscures all the thermal
ones. To obtain a similar efficiency of the thermal radiation from
the entire stellar surface, its temperature should be $T_{{\rm s}}=5.9\times10^{6}\,{\rm K}$
(assuming $R=10\,{\rm km}$), which vastly exceeds the upper limit
($T_{{\rm s}}<2.3\times10^{6}\,{\rm K}$). Furthermore, the temperature
of the polar caps should be about $2.5\times10^{7}\,{\rm K}$ to obtain
a comparable luminosity.
The Vela-like pulsars compose another characteristic group of pulsars.
This group consists of pulsars with high spin-down luminosities but
considerably low X-ray efficiencies $\xi_{_{{\rm NT}}}\apprle10^{-4}$.
A characteristic age of the Vela is about $1.1\times10^{4}\,{\rm yrs}$
($10$ times older than the Crab), but it can still be classified
as a very young pulsar. The nonthermal luminosity of a Vela pulsar
is $L_{{\rm NT}}^{^{{\rm max}}}=4.2\times10^{32}$ and efficiency
$\xi_{_{{\rm NT}}}^{^{{\rm max}}}=10^{-4.22}$. Some of the Vela-like
pulsars (like the Vela itself) also exhibit a thermal component, which
in some cases can be comparable to the nonthermal component. The thermal
efficiency of the Vela $\xi_{_{{\rm BB}}}=10^{-4.72}$ is quite similar
to $\xi_{_{{\rm NT}}}^{^{{\rm max}}}=10^{-4.22}$, but if we assume
an anisotropic radiation pattern of the nonthermal component than
$\xi_{_{{\rm NT}}}^{^{{\rm min}}}=10^{-5.32}$, thus even less than
$\xi_{_{{\rm BB}}}$.
The third group includes pulsars with low spin-down luminosity $L_{{\rm SD}}\apprle10^{35}$.
In most cases, the X-ray spectra of such pulsars (e.g. PSR 9050+08,
PSR B1929+10) have both thermal and nonthermal components, with similar
efficiencies. Thus, the spectrum fitting procedure is more complicated.
The nonthermal X-ray efficiencies of these pulsars, $\xi_{_{{\rm NT}}}\sim10^{-3}$,
are considerably higher than those of the Vela-like pulsars. Note
that even when the observed spectra are dominated by nonthermal radiation,
we cannot rule out a situation that the thermal component is stronger
than the nonthermal one, but due to unfavourable geometry we cannot
observe it.
Even with the improved quality of X-ray observations performed by
both the Chandra and XMM-Newton, the available data
do not allow us to fully discriminate between the different emission
scenarios. However, these data can be used to verify whether the proposed
model of X-ray emission meets all the requirements. Table <ref>
presents the observed spectral properties of pulsars showing nonthermal
http://127.0.0.1:9090/pulsars/table_pl/ (then rename citet to citetalias
after lyx import of ~/Html/pulsar/download/data/table_pl.tex)
fancy [Observed X-ray spectral properties of rotation-powered pulsars [nonthermal]]Observed spectral properties of rotation-powered pulsars with X-ray
spectrum showing the nonthermal (power-law) component. The individual
columns are as follows: (1) Pulsar name, (2) Additional information,
(3) Spectral components required to fit the observed spectra, PL:
power law, BB: blackbody, (4) Pulse phase average photon index, (5)
Maximum nonthermal luminosity $L_{{\rm NT}}$, (6) Maximum nonthermal
X-ray efficiency $\xi_{_{{\rm NT}}}^{^{{\rm max}}}$, (7) Minimum
nonthermal X-ray efficiency $\xi_{_{{\rm NT}}}^{{\rm ^{min}}}$, (8)
Total thermal luminosity $L_{{\rm BB}}$, (9) Thermal efficiency $\xi_{_{{\rm BB}}}$,
(10) References, (11) Number of the pulsar. Both nonthermal luminosities
and efficiencies were calculated in the $0.1-10\,{\rm keV}$ band.
The maximum value was calculated with the assumption that the X-ray
radiation is isotropic while the minimum value was calculated assuming
strong angular anisotropy of the radiation ($\xi_{_{{\rm NT}}}^{{\rm ^{min}}}\approx1/\left(4\pi\right)\cdot\xi_{_{{\rm NT}}}^{{\rm ^{max}}}$).
Pulsars are sorted by nonthermal X-ray luminosity (5).
Name Comment Spectrum Photon-Index $\log L_{{\rm NT}}$ $\log\xi_{_{{\rm NT}}}^{{\rm ^{max}}}$ $\log\xi_{_{{\rm NT}}}^{^{{\rm min}}}$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ Ref. No. $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ B0540–69 N158A, LMC PL $1.92_{-0.11}^{+0.11}$ $36.90$ $-1.27$ $-2.37$ – – [99], [30] 7 B0531+21 Crab PL $1.63_{-0.07}^{+0.07}$ $35.95$ $-2.71$ $-3.81$ – – [15] 4 J0537–6910 N157B, LMC PL $1.80_{-0.10}^{+0.10}$ $35.95$ $-2.74$ $-3.84$ – – [131] 5 B1509–58 Crab-like pulsar PL $1.19_{-0.04}^{+0.04}$ $35.24$ $-2.00$ $-3.10$ – – [43], [49], [15] 29 J1846–0258 Kes 75 BB + PL $1.90_{-0.10}^{+0.10}$ $35.13$ $-1.78$ $-2.88$ $34.06$ $-2.85$ [136], [85] 39 J1420–6048 PL $1.60_{-0.40}^{+0.40}$ $34.77$ $-2.25$ $-3.35$ – – [152] 26 J2021+3651 PL, BB $1.70_{-0.20}^{+0.30}$ $34.36$ $-2.17$ $-3.27$ $33.78$ $-2.75$ [177],[89] 45 J1617–5055 Crab-like pulsar PL $1.14_{-0.06}^{+0.06}$ $34.25$ $-2.95$ $-4.05$ – – [103], [16] 30 J1747–2958 Mouse PL, BB $1.80_{-0.08}^{+0.08}$ $34.09$ $-2.31$ $-3.41$ – – [57] 33 J1811–1925 G11.2-0.3 PL $0.97_{-0.32}^{+0.39}$ $33.97$ $-2.84$ $-3.94$ – – [153], [151] 37 J1930+1852 Crab-like pulsar PL $1.20_{-0.20}^{+0.20}$ $33.92$ $-3.15$ $-4.25$ – – [114], [29] 42 11|r|Continued on next page
Table <ref> - continued from previous
Name Comment Spectrum Photon-Index $\log L_{{\rm NT}}$ $\log\xi_{_{{\rm NT}}}^{^{max}}$ $\log\xi_{_{{\rm NT}}}^{^{min}}$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ Ref. No. $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ J1105–6107 PL $1.80_{-0.40}^{+0.40}$ $33.91$ $-2.48$ $-3.58$ – – [76] 19 B1757–24 Duck PL $1.60_{-0.50}^{+0.60}$ $33.46$ $-2.95$ $-4.05$ – – [105] 34 B1951+32 CTB 80 BB + PL $1.63_{-0.05}^{+0.03}$ $33.22$ $-3.35$ $-4.45$ $31.95$ $-4.62$ [113] 44 J0205+6449 3C58 BB + PL $1.78_{-0.04}^{+0.02}$ $33.10$ $-4.33$ $-5.43$ $33.60$ $-3.83$ [162] 2 J1119–6127 G292.2-0.5 BB + PL $1.50_{-0.20}^{+0.30}$ $32.95$ $-3.42$ $-4.51$ $33.37$ $-3.00$ [73], [135] 20 J1124–5916 Vela-like pulsar PL $1.60_{-0.10}^{+0.10}$ $32.91$ $-4.17$ $-5.27$ – – [92],[72] 21 B1259–63 Be-star bin PL $1.69_{-0.04}^{+0.04}$ $32.87$ $-3.05$ $-4.15$ – – [39], [40] 24 B0833–45 Vela BB + PL $2.70_{-0.40}^{+0.40}$ $32.62$ $-4.22$ $-5.32$ $32.12$ $-4.72$ [186] 13 B1706–44 G343.1-02.3 BB + PL $2.00_{-0.50}^{+0.50}$ $32.16$ $-4.37$ $-5.47$ $32.78$ $-3.76$ [75] 31 J1357–6429 BB + PL $1.30_{-0.20}^{+0.20}$ $32.15$ $-4.35$ $-5.44$ $32.50$ $-3.99$ [185] 25 B1853+01 W44 PL $1.28_{-0.48}^{+0.48}$ $32.07$ $-3.57$ $-4.66$ – – [146] 40 B1046–58 Vela-like pulsar PL $1.70_{-0.20}^{+0.40}$ $32.04$ $-4.26$ $-5.36$ – – [74] 17 B1916+14 BB, PL $3.50_{-0.70}^{+1.60}$ $32.00$ $-1.71$ $-2.81$ $31.07$ $-2.63$ [194] 41 J1509–5850 MSH 15-52 PL $1.00_{-0.30}^{+0.20}$ $31.80$ $-3.92$ $-5.02$ – – [93] 28 B1823–13 Vela-like BB + PL $1.70_{-0.70}^{+0.70}$ $31.78$ $-4.67$ $-5.77$ $32.19$ $-4.27$ [142] 38 B1800–21 Vela-like pulsar PL + BB $1.40_{-0.60}^{+0.60}$ $31.60$ $-4.74$ $-5.84$ – – [101] 35 11|r|Continued on next page
Table <ref> - continued from previous
Name Comment Spectrum Photon-Index $\log L_{{\rm NT}}$ $\log\xi_{_{{\rm NT}}}^{^{max}}$ $\log\xi_{_{{\rm NT}}}^{^{min}}$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ Ref. No. $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ J1809–1917 BB + PL $1.23_{-0.62}^{+0.62}$ $31.57$ $-4.68$ $-5.78$ $31.69$ $-4.56$ [101] 36 B2334+61 BB + PL $2.20_{-1.40}^{+3.00}$ $31.55$ $-3.24$ $-4.34$ $32.06$ $-2.73$ [120] 48 J2043+2740 BB + PL $2.80_{-0.80}^{+1.00}$ $31.41$ $-3.34$ $-4.44$ $30.77$ $-3.98$ [19] 46 B2224+65 Guitar PL, BB $2.20_{-0.30}^{+0.20}$ $31.21$ $-1.87$ $-2.97$ $30.51$ $-2.57$ [95], [94] 47 B0355+54 BB + PL $1.00_{-0.20}^{+0.20}$ $30.92$ $-3.73$ $-4.83$ $30.40$ $-4.25$ [119],[161] 3 B1055–52 BB+BB+PL $1.70_{-0.10}^{+0.10}$ $30.91$ $-3.57$ $-4.67$ $32.63$ $-1.85$ [48] 18 B0656+14 BB+BB+PL $2.10_{-0.30}^{+0.30}$ $30.26$ $-4.33$ $-5.42$ $32.77$ $-1.81$ [48] 10 J0633+1746 Geminga BB+BB+PL $1.68_{-0.06}^{+0.06}$ $30.24$ $-4.27$ $-5.37$ $31.67$ $-2.84$ [97] 9 B1929+10 BB + PL $1.73_{-0.66}^{+0.46}$ $30.23$ $-3.36$ $-4.46$ $30.06$ $-3.53$ [132] 43 B0628–28 BB + PL $2.98_{-0.65}^{+0.91}$ $30.22$ $-1.94$ $-3.04$ $30.22$ $-1.94$ [169] , [17] 8 B0950+08 BB + PL $1.31_{-0.14}^{+0.14}$ $29.99$ $-2.76$ $-3.86$ $28.92$ $-3.82$ [187] 16 B1451–68 BB + PL $1.40_{-0.50}^{+0.50}$ $29.77$ $-2.56$ $-3.66$ $29.27$ $-3.06$ [147] 27 B1133+16 BB, PL $2.51_{-0.33}^{+0.36}$ $29.52$ $-2.42$ $-3.52$ $28.56$ $-3.38$ [102] 22 B0823+26 PL $1.58_{-0.33}^{+0.43}$ $29.42$ $-3.23$ $-4.33$ – – [19] 12 B0943+10 Chameleon BB, PL $2.60_{-0.50}^{+0.70}$ $29.38$ $-2.64$ $-3.74$ $28.40$ $-3.62$ [191],[102] 15 B0834+06 BB + PL – $28.70$ $-3.41$ $-4.51$ $28.70$ $-3.41$ [61] 14 J0108–1431 BB + PL $3.10_{-0.20}^{+0.50}$ $28.57$ $-2.19$ $-3.29$ $27.94$ $-2.82$ [147], [143] 1
§ THERMAL X-RAY RADIATION
§.§ Modelling of thermal radiation from a neutron star
Thermal X-ray emission seems to be quite a common feature of radio
pulsars. The blackbody fit to the observed thermal spectrum of a neutron
star allows us to obtain the redshifted effective temperature $T^{\infty}$
and redshifted total bolometric flux $F^{\infty}$ (measured by a
distant observer). To estimate the actual (unredshifted) parameters,
one should take into account the gravitational redshift, $g_{{\rm r}}=\sqrt{1-2GM/Rc^{2}}$,
determined by the neutron star mass $M$ and radius $R$, here $G$
is the gravitational constant. Then the actual effective temperature
and actual total bolometric flux can be written as:
\begin{equation}
\begin{split}T & =g_{{\rm r}}^{-1}T^{\infty},\\
F & =g_{{\rm r}}^{-2}F^{\infty}.
\end{split}
\label{eq:x-ray.infty_eff}
\end{equation}
Knowing the distance to the neutron star, $D$, we can use the effective
temperature and total bolometric flux to calculate the size of the
radiating region. If we assume that the radiation is isotropic (same
in all directions, e.g. radiation from the entire stellar surface)
then the radius of the radiating sphere (star) can be calculated as
[185]
\begin{equation}
R_{\perp}^{\infty}=D\sqrt{\frac{F^{\infty}}{\sigma T^{\infty4}}}=g_{{\rm r}}^{-1}R_{\perp},\label{eq:x-ray.r_infty}
\end{equation}
where $\sigma\approx5.6704\times10^{-5}{\rm \, erg\, cm^{-2}\, s^{-1}\, K^{-4}}$
is the Stefan-Boltzmann constant.
Knowing that $L_{{\rm BB}}=4\pi D^{2}F$ and using Equations <ref>
and <ref>, we can write that
\begin{equation}
L_{{\rm BB}}=g_{{\rm r}}^{-2}L_{{\rm BB}}^{\infty}.
\end{equation}
The modelling of thermal radiation is more complicated if we assume
that it comes from the hot spot on the stellar surface. One should
take into account such factors as: time-averaged cosine of the angle
between the magnetic axis and the line of sight $\left<\cos i\right>$,
gravitational bending of light, as well as whether the radiation comes
from two opposite poles of the star or from one hot spot only. In
general, the observed luminosity of the hot spot can be written as:
\begin{equation}
L_{{\rm hs}}^{\infty}=A_{{\rm hs}}^{\infty}\sigma T^{\infty4},\label{x-ray.lbol_spot}
\end{equation}
where $A_{{\rm hs}}^{\infty}=\pi R_{{\rm hs}}^{\infty2}$ is the
observed area of the radiating region.
The observed area of the radiating spot is also influenced by the
geometrical factor $f$. This geometrical factor depends on following
angles: $\zeta$ between the line of sight and the spin axis, and
$\alpha$ between the spin and magnetic axes, as well as on $g_{{\rm r}}$
and whether the radiation comes from the star's two opposite poles
or from a single hot spot only:
\begin{equation}
\begin{split}A_{{\rm hs}}^{\infty} & =g_{{\rm r}}^{-2}fA_{{\rm hs}},\\
R_{{\rm hs}} & =g_{{\rm r}}f^{-1/2}R_{{\rm hs}}^{\infty}.
\end{split}
\label{x-ray.infty_eff2}
\end{equation}
Finally, the hot spot luminosity can be calculated as
\begin{equation}
L_{{\rm hs}}=g_{{\rm r}}^{-2}f^{-1}L_{{\rm hs}}^{\infty}.\label{x-ray.lbol_spot-1}
\end{equation}
The luminosity of a radiating sphere with radius $R_{\perp}$ can
be calculated as
$L_{{\rm sp}}=4A_{\perp}\sigma T^{4}=4\pi R_{\perp}^{2}\sigma T^{4}$.
On the other hand, if we assume that the radiation originates only
from one hot spot we can calculate the luminosity as $L_{{\rm hs}}=A_{{\rm hs}}\sigma T^{4}$.
If the hot spot size is small compared to the star radius ($R_{{\rm hs}}\ll R$)
then the area of the spot can be calculated as $A_{{\rm hs}}\approx\pi R_{\perp}^{2}$.
Thus, we have to remember that the luminosity calculated assuming
a spherical source will be four times higher than the actual luminosity
of a radiating hot spot $L_{{\rm hs}}=1/4\cdot L_{{\rm sp}}$ (see
the next section for details).
§.§ Thermal radiation of hot spots
[Coordinate system co-rotating with a star]Coordinate system co-rotating with a star. The system was chosen
so that the z-axis is along ${\bf \Omega}$ (the angular velocity)
and ${\bf o}$ lies in the x-z plane (fiducial plane, i.e. at longitude
zero). Here, $\boldsymbol{\hat{\mu}}$ is a unit vector in the direction
of the magnetic axis and $\alpha$ is the angle between ${\bf \Omega}$
and $\boldsymbol{\hat{\mu}}$, $\beta$ is the impact parameter.
Let us consider a neutron star with two antipodal hot spots associated
with polar caps of a stellar magnetic field. For simplicity's sake
we assume that the spot size is small compared to the star radius
$R$. If the magnetic axis $\boldsymbol{\hat{\mu}}$ is inclined to
the spin axis by an angle $\alpha\leq90^{\circ}$, the spots periodically
change their position and inclination with respect to a distant observer.
To compute the radiation fluxes from the primary (closer to the observer)
as well as the antipodal spot, we need to know their inclinations:
$\cos i_{1}={\bf n\cdot o}$ and $\cos i_{2}={\bf \bar{n}\cdot o}=-\cos i_{1}$,
where ${\bf n}$ and ${\bf \bar{n}}=-{\bf n}$ are normal vectors
to spots surfaces, and ${\bf o}$ is the unit vector pointing toward
the observer. In the calculations we use a coordinate system co-rotating
with a star. The z-axis is along ${\bf \Omega}$ (the angular velocity)
and ${\bf o}$ lies in the x-z plane (see Figure <ref>).
In the chosen coordinate system we can write that the spherical coordinates
of vectors have the following components:
\begin{equation}
\begin{array}{ccc}
{\bf \Omega} & = & \left(\Omega,\,0,\,0\right);\\
{\bf o} & = & \left(1,\,\alpha+\beta,\,0\right);\\
\boldsymbol{\hat{\mu}} & = & \left(1,\,\alpha,\,\Omega t\right).
\end{array}
\end{equation}
Here the impact parameter $\beta$ represents the closest approach
of the line of sight to the magnetic axis. Note that $\boldsymbol{\hat{\mu}}={\bf n}$
and ${\bf \bar{{\bf n}}}=-\boldsymbol{\hat{\mu}}$; thus, we can write
the following components of Cartesian coordinates:
\begin{equation}
\begin{array}{ccc}
{\bf {\bf o}} & = & \left(\sin\left(\alpha+\beta\right),\,0,\,\cos\left(\alpha+\beta\right)\right);\\
{\bf n} & = & \left(\sin\alpha\cos\Omega t,\,\sin\Omega t\sin\alpha,\,\cos\alpha\right).
\end{array}
\end{equation}
$\left(\sin\left(\pi-\alpha\right)\cos\left(\pi+\Omega t\right),\,\sin\left(\pi+\Omega t\right)\cdot\sin\left(\pi-\alpha\right),\,\cos\left(\pi-\alpha\right)\right)=\left(-\sin\alpha\cos\Omega t,\,-\sin\Omega t\cdot\sin\alpha,\,-\cos\alpha\right)$
Finally, the inclination angle for both primary and antipodal hot
spots can be calculated as
\begin{equation}
\begin{array}{ccc}
\cos i_{1} & = & \sin\alpha\cdot\cos\Omega t\cdot\sin\left(\alpha+\beta\right)+\cos\alpha\cdot\cos\left(\alpha+\beta\right);\\
\cos i_{2} & = & -\cos i_{1}=-\sin\alpha\cdot\cos\Omega t\cdot\sin\left(\alpha+\beta\right)-\cos\alpha\cdot\cos\left(\alpha+\beta\right).
\end{array}
\end{equation}
We can estimate the contributions of the primary and antipodal spots
to the observed X-ray flux by calculating the time-averaged cosine
of the angle between the magnetic axis and the line of sight. Note
that we should take into account only positive values of $\cos i$
since for larger angles ($i>90^{\circ}$) the spot is not visible
(at least in this approximation, see Section <ref>
for more details). Thus, the contribution of the primary spot can
be calculated as follows:
\begin{equation}
\begin{array}{c}
\left\langle \cos i_{1}\right\rangle =\begin{cases}
\begin{split}\int_{0}^{P}\cos\left(i_{1}\right){\rm d}t\end{split}
& {\rm if}\ \frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}<-1\ {\rm or}\ \frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}>1,\\
\begin{split}\int_{0}^{t_{-}}\cos\left(i_{1}\right){\rm d}t\end{split}
+\begin{split}\int_{t_{+}}^{2\pi}\cos\left(i_{1}\right){\rm d}t\end{split}
& {\rm if}\ -1<\frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}<1,
\end{cases}\end{array}
\end{equation}
where integration limits are
\begin{equation}
\end{equation}
On the other hand, the contribution of the antipodal spot can be calculated
\begin{equation}
\begin{array}{c}
\left\langle \cos i_{2}\right\rangle =\begin{cases}
\begin{split}0\end{split}
& {\rm if}\ \frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}<-1\ {\rm or}\ \frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}>1,\\
\begin{split}\int_{t_{-}}^{t_{+}}\cos\left(i_{2}\right){\rm d}t\end{split}
& {\rm if}\ -1<\frac{1}{\tan\alpha\tan\left(\alpha+\beta\right)}<1.
\end{cases}\end{array}
\end{equation}
Depending on the orientation of ${\bf \Omega}$, ${\bf o}$ and $\boldsymbol{\hat{\mu}}$,
the thermal radiation may originate from: (1) both the primary and
antipodal hot spots (see Figure <ref>); (2) mainly
the primary spot but with a small contribution from the antipodal
spot (see Figure <ref>); (3) the primary spot
only (see Figure <ref>).
~/Programs/studies/phd/hot_spots/hot_spots.pu (dim
14x8.7 cm)
[Cosine of the hot spots' inclination angle [PSR B0950+08]]Cosine of the hot spots' inclination angle as a function of the pulsar
phase for PSR B0950+08. The following parameters were used: $\alpha=105.46^{\circ}$,
$\beta=21.1^{\circ}$. For this geometry the thermal radiation of
both primary and antipodal spots has a significant influence on the
observed thermal flux.
[Cosine of the hot spots' inclination angle [PSR B1929+10]]Cosine of the hot spots' inclination angle as a function of the pulsar
phase for PSR B1929+10. The following parameters were used: $\alpha=35.97$,
$\beta=25.55$. For this geometry there is only a small contribution
from the antipodal spot.
[Cosine of the hot spots' inclination angle [PSR B0943+10]]Cosine of the hot spots' inclination angle as a function of the pulsar
phase for PSR B0943+10. The following parameters were used: $\alpha=11.58^{\circ}$,
$\beta=-4.29^{\circ}$. For this geometry only the primary hot spot
is visible.
§.§ Gravitational bending of light near stellar surface
The radius of a neutron star is only a few times larger than the Schwarzschild
radius. The approach presented in the previous section does not include
the gravitational bending effect, which is very strong in neutron
stars. A strong gravitational field just above the stellar surface
causes the bending of light. A photon emitted near a neutron star
surface at an angle $\delta$ with respect to the radial direction
escapes to infinity at a different angle $\delta^{\prime}>\delta$.
As a consequence, even when the spot inclination angle to the line
of sight is $i\gtrsim90^{\circ}$ we can still observe thermal radiation
from this spot. For a Schwarzschild metric we can calculate an observed
flux fraction from the primary $f_{1}=F_{1}/F_{0}$ and antipodal
$f_{2}=F_{2}/F_{0}$ spots. Here $F_{0}$ is the maximum possible
flux that is observed when the primary spot is viewed face-on. The
primary and antipodal fluxes are given by [21]
\begin{equation}
\begin{array}{ccc}
f_{1} & = & \cos\left(i\right)\left(1-\frac{r_{{\rm g}}}{R}\right)+\frac{r_{{\rm g}}}{R},\\
f_{2} & = & -\cos\left(i\right)\left(1-\frac{r_{{\rm g}}}{R}\right)+\frac{r_{{\rm g}}}{R},
\end{array}
\end{equation}
here $r_{{\rm g}}=2GM/c^{2}$ is the Schwarzschild radius. The primary
spot is visible when
$\cos i_{1}>-r_{{\rm g}}/\left(R-r_{{\rm g}}\right)$ and the antipodal
spot when $\cos i_{2}>-r_{{\rm g}}/\left(R-r_{{\rm g}}\right)$. Consequently,
both spots are seen when $-r_{{\rm g}}/\left(R-r_{{\rm g}}\right)<\cos i<r_{{\rm g}}/\left(R-r_{{\rm g}}\right)$,
and then the observed flux fraction is
\begin{equation}
f_{{\rm min}}=f_{1}+f_{2}=\frac{2r_{{\rm g}}}{R}.
\end{equation}
Hence, the blackbody pulse of primary and antipodal spots must display
a plateau whenever both spots are in sight. Depending on the geometry
of a pulsar we can distinguish four classes [21].
Class I: when the antipodal spot is never seen and the primary spot
is visible all the time (see the bottom right panel of Figure <ref>).
For such pulsars the blackbody pulse has a perfect sinusoidal shape.
Class II: when the primary spot is seen all the time and the antipodal
spot is also in the visible zone for some time (see panels a, b and
c of Figure <ref>). For these pulsars the sinusoidal
pulse shape is interrupted by the plateau. Class III: the primary
spot is not visible for a fraction of the period and during this time
only the antipodal spot is seen. The primary sinusoidal profile of
such pulsars is interrupted by the plateau, and the plateau is interrupted
by a weaker sinusoidal subpulse from the antipodal spot. Class IV:
both spots are seen at any time. The observed blackbody flux of such
pulsars is constant.
The gravitational bending of light can significantly increase the
visibility of a pulsar (i.e. the observed flux, compare Figures <ref>
and <ref>). For some specific geometry the gravitational
effects can also drastically change primary to the antipodal flux
ratio (compare Figures <ref> and <ref>).
Our calculations show that for canonical values $M=1.4\,{\rm M}_{\odot}$
and $R=10\,{\rm km}$ the gravitational effect is quite strong and
the observed flux fraction is in the range of $0.85-1$, while the
geometric approach results in the $0.43-1$ range (see Table <ref>).
[Viewing geometry of pulsars]Viewing geometry of pulsars. The individual columns are as follows:
(1) Pulsar name, (2) Inclination angle with respect to the rotation
axis $\alpha$, (3) Opening angle $\rho$, (4) Impact parameter $\beta$,
(5) Total flux correction factor (including gravitational bending
of light) $\left\langle f\right\rangle $ , (6) Flux correction factor
of the primary spot $\left\langle f_{1}\right\rangle $, (7) Flux
correction factor of the antipodal spot $\left\langle f_{2}\right\rangle $,
(8, 9, 10) Time-averaged cosine of the angle between the magnetic
axis and the line of sight: $\left<\cos i\right>$ (the total value),
$\left<\cos i_{1}\right>$(the primary spot), $\left<\cos i_{2}\right>$
(the antipodal spot), (10) Number of the pulsar. The gravitational
bending effect was calculated using $M=1.4\,{\rm M}_{\odot}$ and
$R=10\,{\rm km}$.
Name $\alpha$ $\beta$ $\rho$ $\left\langle f\right\rangle $ $\left\langle f_{1}\right\rangle $ $\left\langle f_{2}\right\rangle $ $\left<\cos i\right>$ $\left<\cos i_{1}\right>$ $\left<\cos i_{2}\right>$ No. $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ B0628–28 $70.0$ $-12.0$ $19.6$ $0.86$ $0.52$ $0.34$ $0.52$ $0.35$ $0.17$ 8 B0834+06 $60.7$ $4.5$ $7.1$ $0.86$ $0.53$ $0.32$ $0.52$ $0.36$ $0.16$ 14 B0943+10 $11.6$ $-4.3$ $4.5$ $0.98$ $0.98$ $0.00$ $0.97$ $0.97$ $0.00$ 15 B0950+08 $105.4$ $22.1$ $25.6$ $0.85$ $0.51$ $0.34$ $0.50$ $0.33$ $0.17$ 16 B1133+16 $52.5$ $4.5$ $8.1$ $0.86$ $0.61$ $0.25$ $0.48$ $0.40$ $0.07$ 22 B1451–68 $37.0$ $-6.0$ – $0.88$ $0.81$ $0.06$ $0.68$ $0.68$ $0.00$ 27 B1929+10 $36.0$ $25.6$ $26.8$ $0.85$ $0.64$ $0.21$ $0.43$ $0.41$ $0.02$ 43
[Comparison of the observed flux fractions for geometric effect only
and for geometric effect with the inclusion of a gravitational bending
of light]Comparison of the observed flux fraction for geometric effect only
(blue dotted line) and for geometric effect with the inclusion of
a gravitational bending of light (red solid line). Individual panels
correspond to the following pulsars: (a) PSR B1133+16, (b) PSR B1929+10,
(c) PSR B0834+06 (d) PSR B0943+10. Parameters used in the calculations
are presented in Table <ref>.
[Observed flux fraction as a function of the rotation phase [PSR
B0950+08]]Observed flux fraction $f$ as a function of the rotation phase for
PSR B0950+08. The following parameters were used: $\alpha=105.46^{\circ}$,
$\beta=21.1^{\circ}$, $M=1.4\,{\rm M}_{\odot}$, $R=10\,{\rm km}$.
The gravitational bending of light increases the flux ratio of the
antipodal to primary spots almost two times ($0.85/0.5=1.7$) and
also increases the antipodal to the primary flux ratio ($\sim1.3$).
[Observed flux fraction as a function of the rotation phase [PSR
B1929+10]]Observed flux fraction $f$ as a function of the rotation phase for
PSR B1929+10. The following parameters were used: $\alpha=35.97$,
$\beta=25.55^{\circ}$, $M=1.4\,{\rm M}_{\odot}$, $R=10\,{\rm km}$.
The gravitational bending of light increases the observed flux fraction
two times ($\left\langle f\right\rangle /\left\langle \cos i\right\rangle =1.98$)
and also increases the flux ratio of the antipodal to primary spots
almost seven times ($\sim6.7$).
§.§ Observations
As we have shown in the previous sections, the blackbody fit to the
X-ray observations allows us to directly obtain the surface temperature
$T_{{\rm s}}$. Using the distance to pulsar $D$ and the luminosity
of thermal emission $L_{{\rm BB}}$ we can estimate the area of spot
$A_{{\rm bb}}$. In most cases, $A_{{\rm bb}}$ differs from the conventional
polar cap area $A_{{\rm dp}}\approx6.2\times10^{4}P^{-1}\,{\rm m^{2}}$.
We use parameter $b=A_{{\rm dp}}/A_{{\rm bb}}$ to describe the difference
between $A_{{\rm dp}}$ and $A_{{\rm bb}}$.
§.§.§ Entire surface radiation and warm spot component (b<1)
In most cases the observed spot area $A_{{\rm bb}}$ is larger than
the conventional polar cap area (see Table <ref>).
We can distinguish two types of pulsars in this group, with $b\ll1$
and $b\lesssim1$.
The first type is associated with observations of a thermal emission
from the entire stellar surface and can be used to test cooling models.
Although the entire surface radiation is strongest for young pulsars
($\tau\lesssim10$ kyr ), observation of this radiation is very difficult
due to the strong nonthermal component. A common practice is to separately
fit the nonthermal (PL) and thermal (BB) components. However, the
temperature obtained in such a BB fit (without the PL component) is
most likely overestimated (e.g. see PSR J2021+3651 in Table <ref>).
The nonthermal luminosity of an aging neutron star decreases proportionally
to its spin-down luminosity $L_{{\rm SD}}$, which is thought to drop
with the star age as $L_{{\rm SD}}\propto\tau^{-m}$, where $m\simeq2-4$
depends on the pulsar dipole breaking index [186].
As a pulsar becomes older, its surface temperature decreases. Depending
on the model, a predicted temperature decrease in the early stages
is gradual (the standard model) or rapid (the accelerated cooling
scenario). For a number of middle-aged ($\tau\sim100\,{\rm kyr}$)
and some younger ($\tau\sim10\,{\rm kyr}$) pulsars the thermal radiation
from the entire stellar surface dominates the radiation at soft X-ray
energies (e.g. PSR J0633+1746, PSR B1055-52, PSR J0821-4300, PSR B0656+14,
PSR J0205+6449, PSR J2021+3651). However, the sample of pulsars is
not sufficient to unambiguously identify the cooling scenario.
The second type is associated with observations of the warm spot area
that is larger than the conventional polar cap area but still significantly
less than the area of the star ($b\lesssim1$). The age of pulsars
in this group varies from very young ($\tau\sim1\,{\rm kyr}$) to
middle-aged ($\tau\sim100\,{\rm kyr}$) neutron stars. There is one
exception, namely PSR J1210-5226, which is very old ($\tau=105\,{\rm Myr}$)
and can still be classified as a pulsar with the large warm spot component.
Note, however, that the age of this pulsar is estimated using a characteristic
value and if the pulsar period at birth is comparable with the current
period then the age is highly overestimated (see, e.g. PSR J0821-4300).
Furthermore, the fit to the X-ray spectrum was performed using only
one thermal component and assuming no nonthermal radiation (PL). We
believe that in many cases the size of the warm spot component and
its temperature are overestimated by neglecting other sources of X-ray
radiation, i.e. the nonthermal component and the hot spot radiation.
The small number of observed X-ray photons in some cases prevents
a full spectrum fit with all thermal and nonthermal components. Therefore,
we need observations with better statistics so that the spectrum fit
can be extended using more spectral components. The non-dipolar structure
of the surface magnetic field may cause significant deviations from
the spherical symmetry of the transport processes in the crust. The
magnetic field slightly enhances heat transport along the magnetic
lines, but strongly suppresses it in the perpendicular direction [78].
Hence, the non-isothermality of the crust strongly depends on the
geometry of the magnetic field [58]. The drastic
difference of the crustal transport process causes significant differences
in the surface temperature distribution [140]. Thus,
the non-dipolar structure of the surface magnetic field can explain
the existence of large warm spot components for young and middle-aged
pulsars. We also suggested a mechanism of heating the surface adjacent
to the polar cap [166]. The model of such heating is
also based on the assumption that the pulsar magnetic field near the
stellar surface differs significantly from the pure dipole one. The
calculations show that it is natural to obtain such a geometry of
the magnetic field lines that allows pair creation in the closed field
line region (see Figure <ref>).
[Cartoon of the magnetic field lines in the polar cap region]Cartoon of the magnetic field lines in the polar cap region. Red
lines are open field lines and green dashed lines correspond to the
dipole field. The blue arrows show the direction of the curvature
photon emission.
The pairs move along the closed magnetic field lines and heat the
surface beyond the polar cap on the opposite side of the star. In
such a scenario the heating energy is generated in IAR, and hence
the luminosity of such a warm spot is limited by the power of the
outflowing particles (for more details see Section <ref>).
In most cases the large size of the emitting area and its high temperature
make it unlikely that the warm spot is related to the particles accelerated
in IAR and is rather connected with the non-isothermality of the crust
(e.g. PSR J1210-5226, PSR J1119-6127).
§.§.§ The hot spot component (b > 1)
In many cases the observed hot spot area $A_{{\rm bb}}$ is less than
the conventional polar cap area ($b>1$). The temperature of the emitting
area of these pulsars is usually higher than the temperature of the
emitting area of pulsars with a warm spot component ($b<1$). The
hot spot component is a natural consequence of the non-dipolar structure
of the surface magnetic field (see Figure <ref>).
In order to define an actual polar cap we need to follow the open
field lines from the light cylinder up to the stellar surface by taking
into account the non-dipolar structure of the surface magnetic field
(see Figure <ref>), which can be estimated
by the magnetic flux conservation law as $b=A_{{\rm dp}}/A_{{\rm bb}}$
= $B_{{\rm s}}/B_{{\rm d}}$. Thus, if $b\gg1$ then $B_{{\rm s}}\gg B_{{\rm d}}$.
In neutron stars with positively charged polar caps (${\bf \Omega}\cdot{\bf B}<0$),
the outflow of iron ions depends on the surface temperature and the
surface binding energy (the so-called cohesive energy) [34, 98, 2, 65].
The cohesive energy of condensed matter increases with magnetic field
strength [122]. If for a given strength of the surface
magnetic field the temperature is below the so-called critical temperature
$T_{{\rm crit}}$ the ions can tightly bind to the condensed surface
and a polar gap can form (see Chapter <ref> for details).
123 calculated the dependence of the critical temperature
(for a vacuum gap formation) on the strength of the surface magnetic
field. In Figure <ref> we present the positions
of pulsars with derived surface temperature $T_{{\rm s}}$ and hot
spot area $A_{{\rm bb}}$ on the $B_{{\rm s}}-T_{{\rm s}}$ diagram,
where $B_{{\rm s}}$ is estimated as $B_{{\rm s}}=bB_{{\rm d}}$.
The red line represents the dependence of the critical temperature
$T_{{\rm crit}}$ on $B_{{\rm s}}$. We can see that in most cases
the pulsars' positions follow the $B_{{\rm s}}-T_{{\rm crit}}$ theoretical
curve. Note that the Figure includes only pulsars with a visible hot
spot component (old pulsars). For younger pulsars (with warm spot
components) it is not possible to estimate the surface magnetic field.
There are a few cases which do not coincide with the theoretical curve.
We believe that they correspond to the observations of warm spot component
but with the area of radiation smaller than the conventional polar
cap area (e.g. due to reheating of the surface beyond the polar cap,
see Section <ref>).
According to our model the actual surface temperature is almost equal
to the critical value $T_{{\rm s}}\approx T_{{\rm crit}}$, which
leads to the formation of the Partially Screened Gap (PSG) above the
polar caps of a neutron star [65]. The hot spot parameters
derived from X-ray observations of isolated neutron stars are presented
in Table <ref>.
http://localhost:9090/pulsars/graphs/ [~/Html/pulsars/data/models.py
cp ~/Html/pulsar/media/images/t6_b14_log_zoom.svg
[Diagram of the surface temperature vs. the surface magnetic field]Diagram of the surface temperature ($T_{6}=T_{{\rm s}}/\left(10^{6}\,{\rm K}\right)$)
vs. the surface magnetic field ($B_{14}=B_{{\rm s}}/\left(10^{14}\,{\rm G}\right)$).
The red line represents the dependence of $T_{{\rm crit}}$ on $B_{14}$
according to 123 and the dashed lines correspond to
uncertainties in the calculations. The diagram includes all pulsars
with $b>1$ with the exception of PSR J2043+2740, for which the blackbody
fit was performed using a fixed radius (estimation of the surface
magnetic field is not possible). Error bars correspond to $1\sigma$.
http://localhost:9090/pulsars/table_bb_age/ (PL instead of age in
~/Html/pulsar/download/data/table_bb_age.tex (import
in lyx and replace citet with citetalias)
fancy [Observed X-ray spectral properties of rotation-powered pulsars [thermal]]Spectral properties of rotation-powered pulsars with detected blackbody
X-ray components. The individual columns are as follows: (1) Pulsar
name, (2) Spectral components required to fit the observed spectra,
PL: power law, BB: blackbody, (3) Radius of the spot obtained from
the blackbody fit $R_{{\rm bb}}$, (4) Surface temperature $T_{{\rm s}}$,
(5) Surface magnetic field strength $B_{{\rm s}}$, (6) $b=A_{{\rm dp}}/A_{{\rm bb}}=B_{{\rm s}}/B_{{\rm d}}$,
$A_{{\rm dp}}$ - conventional polar cap area, $A_{{\rm bb}}$ - actual
polar cap area, (7) Bolometric luminosity of blackbody component $L_{{\rm BB}}$,
(8) Bolometric efficiency $\xi_{_{{\rm BB}}}$, (9) Maximum nonthermal
luminosity $L_{{\rm NT}}^{^{{\rm max}}}$, (10) Maximum nonthermal
X-ray efficiency $\xi_{_{{\rm NT}}}^{^{{\rm max}}}$, (11) Best estimate
of pulsar age or spin down age, (12) References, (13) Number of the
pulsar. Nonthermal luminosity and efficiency were calculated in the
$0.1-10\,{\rm keV}$ band. The maximum value was calculated with the
assumption that the X-ray nonthermal radiation is isotropic. Pulsars
are sorted by $b$ parameter (6).
Name Spectrum $R_{{\rm bb}}$ $T_{{\rm s}}$ $B_{{\rm s}}$ $b$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ $\log L_{{\rm X}}$ $\log\xi_{_{{\rm NT}}}^{^{{\rm max}}}$ $\tau$ Ref. No. $\left(10^{6}{\rm K}\right)$ $\left(10^{14}{\rm G}\right)$ $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ B1451–68 BB + PL $14_{-12.3}^{+24.2}$ m $4.1_{-0.81}^{+1.39}$ $1.36_{-1.18}^{+114}$ $418$ $29.27$ $-3.06$ $29.77$ $-2.56$ $42.5$ Myr [147] 27 B0943+10 BB, PL $12_{-7.7}^{+41.2}$ m $3.1_{-1.07}^{+1.08}$ $4.99_{-4.72}^{+30.45}$ $126$ $28.40$ $-3.62$ $29.38$ $-2.64$ $4.98$ Myr [191],[102] 15 B1929+10 BB + PL $28_{-3.8}^{+4.9}$ m $4.5_{-0.45}^{+0.30}$ $1.26_{-0.35}^{+0.44}$ $122$ $30.06$ $-3.53$ $30.23$ $-3.36$ $3.10$ Myr [132] 43 B1133+16 BB, PL $14_{-9.0}^{+10.5}$ m $3.2_{-0.35}^{+0.46}$ $4.06_{-2.77}^{+31.79}$ $95.5$ $28.56$ $-3.38$ $29.52$ $-2.42$ $5.04$ Myr [102] 22 B0950+08 BB + PL $42_{-26.6}^{+26.6}$ m $2.3_{-0.29}^{+0.29}$ $0.23_{-0.15}^{+1.57}$ $47.9$ $28.92$ $-3.82$ $29.99$ $-2.76$ $17.5$ Myr [187] 16 B2224+65 PL, BB $28_{-18.0}^{+5.6}$ m $5.8_{-1.16}^{+1.16}$ $2.00_{-0.61}^{+13.31}$ $38.6$ $30.51$ $-2.57$ $31.21$ $-1.87$ $1.12$ Myr [95], [94] 47 J0633+1746 BB+BB+PL $62_{-34.0}^{+34.0}$ m $1.7_{-0.23}^{+0.23}$ $0.75_{-0.44}^{+2.92}$ $23.0$ $29.07$ $-5.44$ $30.24$ $-4.27$ $342$ kyr [97] 9 —— $11.17_{-1}^{+1}$ km $0.5_{-0.1}^{+0.1}$ $31.67$ $-2.84$ B0834+06 BB + PL $30_{-15.3}^{+56.4}$ m $2.0_{-0.64}^{+0.75}$ $1.05_{-0.92}^{+3.19}$ $17.7$ $28.70$ $-3.41$ $28.70$ $-3.41$ $2.97$ Myr [61] 14 B0355+54 BB + PL $92_{-53.6}^{+122.5}$ m $3.0_{-1.06}^{+1.51}$ $0.27_{-0.22}^{+1.27}$ $15.9$ $30.40$ $-4.25$ $30.92$ $-3.73$ $564$ kyr [119],[161] 3 J0108–1431 BB + PL $43_{-14.0}^{+24.0}$ m $1.3_{-0.12}^{+0.35}$ $0.07_{-0.04}^{+0.08}$ $14.0$ $27.94$ $-2.82$ $28.57$ $-2.19$ $166$ Myr [147], [143] 1 13|r|Continued on next page
Table <ref> - continued from previous
Name Spectrum $R_{{\rm bb}}$ $T_{{\rm s}}$ $B_{{\rm s}}$ $b$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ $\log L_{{\rm X}}$ $\log\xi_{_{{\rm NT}}}^{^{{\rm max}}}$ $\tau$ Ref. No. $\left(10^{6}{\rm K}\right)$ $\left(10^{14}{\rm G}\right)$ $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ B0628–28 BB + PL $64_{-49.7}^{+70.3}$ m $3.3_{-0.62}^{+1.31}$ $0.25_{-0.19}^{+4.88}$ $4.14$ $30.22$ $-1.94$ $30.22$ $-1.94$ $2.77$ Myr [169], [17] 8 J2043+2740 BB + PL $358_{-153.2}^{+153.2}$ m $1.9_{-0.45}^{+0.45}$ $0.01_{-0.01}^{+0.02}$ $1.70$ $30.77$ $-3.98$ $31.41$ $-3.34$ $1.20$ Myr [19] 46 B1719–37 BB $237_{-122.5}^{+390.6}$ m $3.5_{-0.76}^{+0.91}$ $0.05_{-0.04}^{+0.17}$ $1.57$ $31.19$ $-3.32$ – – $345$ kyr [137] 32 J1846–0258 BB + PL $306_{-153.2}^{+153.2}$ m $13.6_{-3.03}^{+3.03}$ – $0.686$ $34.06$ $-2.85$ $35.13$ $-1.78$ $0.73$ kyr [136], [85] 39 B1055–52 BB+BB+PL $460_{-60.0}^{+60.0}$ m $1.8_{-0.06}^{+0.06}$ – $0.503$ $30.89$ $-3.59$ $30.91$ $-3.57$ $535$ kyr [48] 18 $12.30_{-1}^{+2}$ km $0.8_{-0.03}^{+0.03}$ $32.62$ $-1.86$ J0538+2817 BB $666_{-38.3}^{+38.3}$ m $2.8_{-0.04}^{+0.05}$ – $0.330$ $31.97$ $-2.73$ – – $30.0$ kyr [118] 6 J1809–1917 BB + PL $951_{-693.3}^{+920.2}$ m $2.0_{-0.35}^{+0.35}$ – $0.280$ $31.69$ $-4.56$ $31.57$ $-4.68$ $51.3$ kyr [101] 36 J0821–4300 BB + BB $1.22_{-0.13}^{+0.13}$ km $6.3_{-0.19}^{+0.19}$ – $0.125$ $33.61$ $-0.91$ – – $3.70$ kyr [77] 11 —— $6.02_{-0.4}^{+0.4}$ km $3.2_{-0.10}^{+0.10}$ $33.86$ $-0.66$ B1951+32 BB + PL $2.20_{-0.80}^{+1.40}$ km $1.5_{-0.23}^{+0.23}$ – $0.110$ $31.95$ $-4.62$ $33.22$ $-3.35$ $107$ kyr [113] 44 B0833–45 BB + PL $1.61_{-0.15}^{+0.15}$ km $1.9_{-0.05}^{+0.05}$ – $0.091$ $32.12$ $-4.72$ $32.62$ $-4.22$ $11.3$ kyr [186] 13 J1357–6429 BB + PL $1.91_{-0.38}^{+0.38}$ km $2.2_{-0.26}^{+0.26}$ – $0.034$ $32.50$ $-3.99$ $32.15$ $-4.35$ $7.31$ kyr [185] 25 J1210–5226 BB $1.23$ km $3.8$ – $0.033$ $33.04$ $1.51$ – – $102$ Myr [145] 23 B1823–13 BB + PL $2.52_{-0.00}^{+0.00}$ km $1.6_{-0.07}^{+0.10}$ – $0.032$ $32.19$ $-4.27$ $31.78$ $-4.67$ $21.4$ kyr [142] 38 B1916+14 BB, PL $800_{-100.0}^{+100.0}$ m $1.5_{-0.12}^{+0.12}$ – $0.028$ $31.07$ $-2.63$ $32.00$ $-1.71$ $88.1$ kyr [194] 41 B1706–44 BB + PL $2.76_{-0.69}^{+0.69}$ km $2.2_{-0.20}^{+0.22}$ – $0.027$ $32.78$ $-3.76$ $32.16$ $-4.37$ $17.5$ kyr [75] 31
Table <ref> - continued from previous
Name Spectrum $R_{{\rm bb}}$ $T_{{\rm s}}$ $B_{{\rm s}}$ $b$ $\log L_{{\rm BB}}$ $\log\xi_{_{{\rm BB}}}$ $\log L_{{\rm X}}$ $\log\xi_{_{{\rm NT}}}^{^{{\rm max}}}$ $\tau$ Ref. No. $\left(10^{6}{\rm K}\right)$ $\left(10^{14}{\rm G}\right)$ $\left({\rm erg\, s^{-1}}\right)$ $\left({\rm erg\, s^{-1}}\right)$ B2334+61 BB + PL $1.27_{-0.30}^{+0.45}$ km $2.1_{-0.76}^{+0.46}$ – $0.026$ $32.06$ $-2.73$ $31.55$ $-3.24$ $40.9$ kyr [120] 48 B0656+14 BB+BB+PL $1.80_{-0.15}^{+0.15}$ km $1.2_{-0.03}^{+0.03}$ – $0.017$ $31.45$ $-3.13$ $30.26$ $-4.33$ $111$ kyr [48] 10 —— $20.90_{-4}^{+3}$ km $0.7_{-0.01}^{+0.01}$ $32.74$ $-1.84$ J1119–6127 BB + PL $2.60_{-0.23}^{+1.38}$ km $3.1_{-0.26}^{+0.39}$ – $0.008$ $33.37$ $-3.00$ $32.95$ $-3.42$ $1.61$ kyr [73], [135] 20 J0205+6449 BB + PL $8.1$ km $1.7$ – $0.005$ $33.60$ $-3.83$ $33.10$ $-4.33$ $5.37$ kyr [162] 2 J2021+3651 PL, BB $7.00_{-1.70}^{+4.00}$ km $2.4_{-0.30}^{+0.30}$ – $0.004$ $33.78$ $-2.75$ $34.36$ $-2.17$ $17.2$ kyr [177],[89] 45
CHAPTER: MODEL OF A NON-DIPOLAR SURFACE MAGNETIC FIELD
§ THE MAGNETIC FIELD OF NEUTRON STARS
Generally, the properties of pulsar radio emission support the assumption
that the magnetic field of pulsars is purely dipolar at least in the
radio emission region [148]. However, radio
emission is generated at altitudes $R_{{\rm em}}$ of more than several
stellar radii (e.g. 106, 107, 109
and references therein). Thus, radio observations do not provide information
about the structure of the magnetic field at the surface of the neutron
star. On the other hand, strong non-dipolar surface magnetic fields
have long been thought to be a necessary condition for pulsar activities,
e.g. the vacuum gap model proposed by 156 implicitly
assumes that the radius of curvature of field lines above the polar
cap should be about $10^{6}\,{\rm cm}$ in order to sustain pair production.
This curvature is approximately $100$ times higher than that expected
from a global dipolar magnetic field. Furthermore, to explain radiation
from the Crab Nebula, the Crab pulsar should provide quite a dense
stellar wind, as such a high particle multiplicity is not possible
in a purely dipolar magnetic field.
There are several theoretical studies concerning the formation and
evolution of the non-dipolar magnetic fields of neutron stars (e.g.
23, 108, 155, 5, 33, 59, 133, 140).
According to 182, the magnetic field in neutron
stars results from the fossil field of the progenitor stars which
is amplified during the collapse and remains anchored in the superfluid
core of the neutron star. Several authors also noted that during the
collapse (or shortly after) there is possible magnetic field generation
in the external crust, for instance, by a mechanism like thermomagnetic
instabilities [23]. 175 also showed
that it is possible to form small-scale magnetic field anomalies in
the neutron star crust with a typical size of the order of $100$
The soft X-ray observations of pulsars presented in Chapter <ref>
show non-uniform surface temperatures which can be attributed to small-scale
magnetic anomalies in the crust. Further observational arguments in
favour of the non-dipolar nature of the surface magnetic field can
be found in many articles (e.g. 27, 170, 28, 141, 171, 18, 35, 154, 38, 134, 167, 115).
§ MODELLING OF THE SURFACE MAGNETIC FIELD
In order to model a surface magnetic field we used the scenario proposed
by 67. In this scenario the magnetic field at the neutron
star's surface is non-dipolar in nature, which is due to superposition
of the fossil field in the core and crustal field structures. To calculate
the actual surface magnetic field described by superposition of the
star-centred global dipole $\mathbf{d}$ and the crust-anchored dipole
moment $\mathbf{m}$, let us consider the general situation presented
in Figure <ref>
[Model of a non-dipolar surface magnetic field]Superposition of the star-centred global magnetic dipole $\mathbf{d}$
and crust-anchored local dipole anomaly $\mathbf{m}$ located at $\mathbf{r_{s}}=(r_{s}\sim R,\,\theta=\theta_{r})$
and inclined to the $z$-axis by an angle $\theta_{m}$. The actual
surface magnetic field at radius vector $\mathbf{r}=(r,\,\theta)$
is $\mathbf{B_{s}}=\mathbf{B_{d}}+\mathbf{B_{{\rm m}}}$, where $B_{d}=2d/r^{3}$,
$B_{{\rm m}}=2m/|\mathbf{r}\mathbf{r_{s}}|^{3}$, $r$ is the radius
and $\theta$ - is the polar angle. $R$ is the radius of the neutron
star and L is the external crust thickness. 67
The actual surface magnetic field is a sum of the global magnetic
dipole and crust-anchored local anomalies
\begin{equation}
\mathbf{B_{s}}=\mathbf{B_{d}}+\mathbf{B_{{\rm m}}}+...\label{eq:model.field}
\end{equation}
Using the star-centred spherical coordinates with the $z$-axis directed
along the global magnetic dipole moment we obtain:
\begin{equation}
\mathbf{B_{d}}=\left(\frac{2d\cos\theta}{r^{3}},\,\frac{d\sin\theta}{r^{3}},\,0\right),\label{eq:model.b_d}
\end{equation}
\begin{equation}
\mathbf{B_{{\rm m}}}=\frac{3(\mathbf{r}-\mathbf{r_{s}})(\mathbf{m}\cdot(\mathbf{r}-\mathbf{r_{s}}))-\mathbf{m}|\mathbf{r}-\mathbf{r_{s}}|^{2}}{|\mathbf{r}-\mathbf{r_{s}}|^{5}}.
\end{equation}
Here $\mathbf{r_{s}}=(r_{s},\,\theta_{r},\,\phi_{r})$, $\mathbf{m}=(m,\,\theta_{m},\,\phi_{m})$
and the spherical components of $\mathbf{B_{{\rm m}}}$ are explicitly
given in Equation <ref>.
The global magnetic moment can be written as
\begin{equation}
d=\frac{1}{2}B_{{\rm p}}R^{3},
\end{equation}
where $B_{{\rm p}}=6.4\times10^{19}\left(P\dot{P}\right){}^{1/2}\,{\rm G}$
is the dipole component at the pole derived from pulsar spin-down
energy loss.
The crust-anchored local dipole moment is
\begin{equation}
m=\frac{1}{2}B_{{\rm m}}\Delta R^{3},
\end{equation}
where $\Delta R\sim0.05R<L$ and $L\sim10^{5}\,{\rm cm}$ is the characteristic
crust dimension (for
$R=10^{6}\,{\rm cm}$). For these values a local anomaly can significantly
influence the surface magnetic field ($B_{{\rm m}}>B_{{\rm d}}$)
if $m/d>10^{-4}$.
The system of differential equations for a field line of the vector
field $\mathbf{B=}\left(B_{r},\, B_{\theta},\, B_{\phi}\right)$ in
spherical coordinates can be written as
\begin{equation}
\begin{cases}
\frac{{\rm d}\theta}{{\rm d}r} & =\frac{B_{\theta}}{rB_{r}}\\
\frac{{\rm d}\phi}{{\rm d}r} & =\frac{B_{\phi}}{r\sin(\theta)B_{r}}.
\end{cases}\label{eq:model.diff_eqs}
\end{equation}
The solution of these equations, with the initial conditions $\theta_{0}=\theta(r=R)$
and $\mbox{\ensuremath{\phi_{0}}=\ensuremath{\phi}(r=R)}$ determining
a given field line at the stellar surface, describes the parametric
equation of the magnetic field lines. The spherical components of
$\mathbf{B_{{\rm m}}}$ can be written in the following form
\begin{equation}
\begin{split}B_{r}^{m} & =-\frac{1}{D^{2.5}}\left(3Tr_{r}^{s}-3Tr+Dm_{r}\right),\\
B_{\theta}^{m} & =-\frac{1}{D^{2.5}}\left(3Tr_{\theta}^{s}+Dm_{\theta}\right),\\
B_{\phi}^{m} & =-\frac{1}{D^{2.5}}\left(3Tr_{\phi}^{s}+Dm_{\phi}\right).
\end{split}
\label{eq:model.b_m}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
According to the geometry presented in Figure <ref>,
the components of the radius vector of the origin of the crust-anchored
local dipole anomaly can be written as
\begin{equation}
\begin{split}r_{r}^{s} & =r_{s}\left(\sin\theta_{r}\sin\theta\cos\left(\phi-\phi_{r}\right)+\cos\theta_{r}\cos\theta\right),\\
r_{\theta}^{s} & =r_{s}\left(\sin\theta_{r}\cos\theta\cos\left(\phi-\phi_{r}\right)+\cos\theta_{r}\sin\theta\right),\\
r_{\phi}^{s} & -r_{s}\sin\theta_{r}\sin\left(\phi-\phi_{r}\right).
\end{split}
\end{equation}
The components of the local dipole anomaly are
\begin{equation}
\begin{split}m_{r} & =m\left(\sin\theta_{m}\sin\theta\cos\left(\phi-\phi_{m}\right)+\cos\theta_{m}\cos\theta\right),\\
m_{\theta} & =m\left(\sin\theta_{m}\cos\theta\cos\left(\phi-\phi_{m}\right)+\cos\theta_{m}\sin\theta\right),\\
m_{\phi} & =-m\sin\theta_{m}\sin\left(\phi-\phi_{m}\right).
\end{split}
\end{equation}
Finally, we obtain the system of differential equations from Equation
<ref> by substitutions $B_{r}=B_{r}^{d}+B_{r}^{m}$,
$B_{\theta}=B_{\theta}^{d}+B_{\theta}^{m}$ and $B_{\phi}=B_{\phi}^{d}+B_{\phi}^{m}$
(Equations <ref> and <ref>)
\begin{equation}
\frac{{\rm d}\theta}{{\rm d}r}=\frac{B_{\theta}^{d}+B_{\theta}^{m}}{r\left(B_{r}^{d}+B_{r}^{m}\right)}\equiv\Theta_{1},\label{model.line_diff}
\end{equation}
\begin{equation}
\frac{{\rm d}\phi}{{\rm d}r}=\frac{B_{\phi}^{m}}{r\left(B_{r}^{d}+B-r^{m}\right)\sin\theta}\equiv\Phi_{1}.\label{model.line_diff2}
\end{equation}
§ CURVATURE OF MAGNETIC FIELD LINES
As Curvature Radiation (CR) may play a decisive role in radiation
processes, it is important to calculate the curvature (or curvature
radius) for each field line. The curvature $\rho_{c}=1/\Re$ of field
lines (where $\Re$ is the radius of curvature) is calculated as [67]
\begin{equation}
\rho_{c}=\left(\frac{{\rm d}s}{{\rm d}r}\right)^{-3}\left|\left(\frac{{\rm d}^{2}\mathbf{r}}{{\rm d}r^{2}}\frac{{\rm d}s}{{\rm d}r}-\frac{{\rm d}\mathbf{r}}{{\rm d}r}\frac{{\rm d}^{2}s}{{\rm d}r^{2}}\right)\right|,
\end{equation}
\begin{equation}
\frac{{\rm d}s}{{\rm d}r}=\sqrt{\left[1+r^{2}\Theta_{1}^{2}+r^{2}\Phi_{1}^{2}\sin^{2}(\theta)\right]}.
\end{equation}
Thus, the curvature can be written in the form
\begin{equation}
\rho_{c}=\left(S_{1}\right)^{-3}\left(J_{1}^{2}+J_{2}^{2}+J_{3}^{2}\right)^{1/2},
\end{equation}
\begin{equation}
\begin{split}J_{1}= & X_{2}S_{1}-X_{1}S_{2},\\
J_{2}= & Y_{2}S_{1}-Y_{1}S_{2},\\
J_{3}= & Z_{2}S_{1}-Z_{1}S_{2},\\
X_{1}= & \sin\theta\cos\phi+r\Theta_{1}\cos\theta\cos\phi-r\Phi_{1}\sin\theta\sin\phi,\\
Y_{1}= & \sin\theta\sin\phi+r\Theta_{1}\cos\theta\sin\phi-r\Phi_{1}\sin\theta\cos\phi,\\
Z_{1}= & \cos\theta-r\Theta_{1}\sin\theta,\\
X_{2}= & \left(2\Theta_{1}+r\Theta_{2}\right)\cos\theta\cos\phi-\left(2\Phi_{1}+r\Phi_{2}\right)\sin\theta\sin\phi-\\
& r\left(\Theta_{1}^{2}+\Phi_{1}^{2}\right)\sin\theta\cos\phi+2r\Theta_{1}\Phi_{1}\cos\theta\sin\phi,\\
Y_{2}= & \left(2\Theta_{1}+r\Theta_{2}\right)\cos\theta\sin\phi+\left(2\Phi_{1}+r\Phi_{2}\right)\sin\theta\cos\phi-\\
& r\left(\Theta_{1}^{2}+\Phi_{1}^{2}\right)\sin\theta\sin\phi+2r\Theta_{1}\Phi_{1}\cos\theta\cos\phi,\\
Z_{2}= & -\Theta_{1}\sin\theta-\Theta_{1}\sin\theta-r\Theta_{2}\sin\theta-r\Theta_{1}^{2}\cos\theta,\\
S_{1}= & \sqrt{1+r^{2}\Theta_{1}^{2}+r^{2}\Phi_{1}^{2}\sin^{2}\theta},\\
S_{2}= & S_{1}^{-1}\left(t\Theta_{1}^{2}+r^{2}\Theta_{1}\Theta_{2}+r\Phi_{1}^{2}\sin^{2}\theta+r^{2}\Phi_{1}\Phi_{2}\sin^{2}\theta+r^{2}\Theta_{1}\Phi_{1}^{2}\sin\theta\cos\theta\right),\\
\Theta_{2}\equiv & \frac{{\rm d}\Theta_{1}}{{\rm d}r},\\
\Phi_{2}\equiv & \frac{{\rm d}\Phi_{1}}{{\rm d}r}.
\end{split}
\end{equation}
§.§ Numerical calculation of the curvature
Let us note that when evaluating Equation <ref>
it was assumed that $\sin\theta\neq0$. Thus $\Phi_{1}$ and $\Phi_{2}$
are undefined for $\theta=0$. Figure <ref> presents
the first and second derivative ($\Phi_{1}$, $\Phi_{2}$) of the
$\phi$-coordinate of a magnetic field line with respect to the $r$-coordinate
for a magnetic field structure with $B_{\phi}\neq0$.
The singularity in Equation <ref> may result in
an overestimation of curvature of field lines that cross the $\theta=0$
plane for a complex structure of the surface magnetic field. To solve
this problem, and in addition to the analytical approach, the numerical
calculation of the curvature of magnetic field lines was implemented.
Let us consider three consecutive points of the given magnetic field
line $A$, $B$, $C$ (see Figure <ref>).
~/Programs/studies/phd/curvature/curvature.py (show_angles)
with 99 data set
[First and second derivative of the $\phi$-coordinate of the magnetic
field line]Plot of the first and second derivative of the $\phi$-coordinate
of the magnetic field line with respect to the $r$-coordinate vs.
the distance from the stellar surface. Values were calculated using
the approach described in Section <ref>. Panels
(a) and (b) show the first and second derivative of the $\phi$-coordinate
while panel (c) shows the $\theta$-coordinate of the magnetic field
[Curvature of magnetic field lines (numerical approach)]For any given three points ($A$, $B$, $C$) we can always find
a common plane. We use the following transformations to achieve this:
(I) shift the origin of the system to point $A$ (prime), (II) rotate
the shifted system by an angle $\varsigma_{y}$ around the $y^{\prime}$–axis
and by an angle $\varsigma_{x}$ around the $x^{\prime\prime}$-axis
(double prime). After these transformations the $z^{\prime\prime}$-axis
will be aligned with normal vector $\hat{{\bf N}}$ and all points
will lie in the $x^{\prime\prime}y^{\prime\prime}$-plane of such
a system of coordinates.
To calculate curvature (or radius of curvature) in point $B$ we can
use the following procedure:
* simplify the 3-D problem to 2-D by finding a common plane for all
three points
* move the origin of the coordinate system to point $A$
* rotate the coordinate system to align the z-axis with the normal vector
to the common plane of all three points ($A$, $B$, $C$)
* calculate the radius of the circle passing through all three points
(in a 2-D coordinate system)
§.§.§ 3-D to 2-D transition
To simplify the calculations we shift the origin of the coordinate
system so that point $A$ will be the origin of the new system:
\begin{eqnarray}
A^{\prime} & = & \left(0,\,0,\,0\right);\nonumber \\
B^{\prime} & = & \left(B_{1}-A_{1},\, B_{2}-A_{2},\, B_{3}-A_{3}\right);\\
C^{\prime} & = & \left(C_{1}-A_{1},\, C_{2}-A_{2},\, C_{3}-A_{3}\right).\nonumber
\end{eqnarray}
The unit normal vector to the plane enclosing all three points ($A^{\prime}$,
$B^{\prime}$, $C^{\prime}$) can be calculated as
\begin{equation}
\hat{{\bf N}}=\frac{{\bf b}\times{\bf c}}{\left|{\bf b}\times{\bf c}\right|}=\left(N_{1},\, N_{2},\, N_{3}\right),
\end{equation}
where ${\bf b}=\left(B_{1}^{\prime},\, B_{2}^{\prime},\, B_{3}^{\prime}\right)$
and ${\bf c}=\left(C_{1}^{\prime},\, C_{2}^{\prime},\, C_{3}^{\prime}\right)$.
The next step is to rotate the shifted coordinate system to align
the $z^{\prime}$-axis with normal vector $\hat{{\bf N}}$. In the
new system all three points will lie in the $x^{\prime\prime}y^{\prime\prime}$-plane.
In our calculations we rotate the shifted system by an angle $\varsigma_{y}$
around the $y^{\prime}$-axis, $R_{y}\left(\varsigma_{y}\right)$,
and a rotation by an angle $\varsigma_{x}$ around the $x^{\prime\prime}$-axis,
$R_{x}\left(\varsigma_{x}\right)$. The final rotation matrix can
be written as
\begin{equation}
\cos\varsigma_{y} & \sin\varsigma_{x}\sin\varsigma_{y} & \sin\varsigma_{y}\cos\varsigma_{x}\\
0 & \cos\varsigma_{x} & -\sin\varsigma_{x}\\
-\sin\varsigma_{y} & \cos\varsigma_{y}\sin\varsigma_{x} & \cos\varsigma_{y}\cos\varsigma_{x}
\end{array}\right]
\end{equation}
The Euler angles for these rotations can be calculated as
$\varsigma_{x}={\rm atan2}\left(N_{2},N_{3}\right).$
[where ${\rm atan2}\left(y,\, x\right)$ equals: (1) $\arctan\left(y/x\right)$
if $x>0$; (2) $\arctan\left(y/x\right)+\pi$ if $y\ge0$ and $x<0$;
(3) $\arctan\left(y/x\right)-\pi$ if $y<0$ and $x<0$; (4) $\pi/2$
if $y>0$ and $x=0$; (5) $-\pi/2$ if $y<0$ and $x=0$; (6) is undefined
if $y=0$ and $x=0$. This function is available in many programming
\begin{equation}
\begin{array}{c}
\varsigma_{y}=\begin{cases}
\arctan\left(-\frac{N_{1}}{N_{3}}\cos\varsigma_{x}\right) & {\rm if\ }N_{3}\neq0\\
\arctan\left(-\frac{N_{1}}{N_{2}}\sin\varsigma_{x}\right) & {\rm if\ }N_{2}\neq0\\
\frac{\pi}{2} & {\rm if\ }N_{2}=0\ {\rm and}\ N_{3}=0
\end{cases}\end{array}
\end{equation}
where $\arctan2\left(x,\, y\right)$ is the arc tangent of the two
variables $x$ and $y$. It is similar to calculating the arc tangent
of $x/y$, except that the signs of both arguments are used to determine
the quadrant of the result, which lies in the range $\left[-\pi,\,\pi\right]$.
This function is available in many programming languages and often
is called atan2.
Finally, we can write the components of all three points in our new
(shifted and double-rotated) system of coordinates as follows
\begin{eqnarray}
A^{\prime\prime} & = & R_{yx}A^{\prime}=\left(0,\,0,\,0\right);\nonumber \\
B^{\prime\prime} & = & R_{yx}B^{\prime}=\left(B_{1}^{\prime\prime},\, B_{2}^{\prime\prime},\,0\right);\\
C^{\prime\prime} & = & R_{yx}C^{\prime}=\left(C_{1}^{\prime\prime},\, C_{2}^{\prime\prime},\,0\right).\nonumber
\end{eqnarray}
§.§.§ Circle passing through 3 points [25]
Finding the radius of the circle passing through three consecutive
points of a given magnetic field line ($A=\left(0,\,0\right)$, $B=\left(B_{1},\, B_{2}\right)$,
$C=\left(C_{1},\, C_{2}\right)$) is an exact method for finding the
radius of curvature $\Re$ and hence the curvature $\rho=1/\Re$ of
this line. Note that for simplicity's sake we hereafter describe points
without double prime notation but they refer to coordinates in the
shifted and double-rotated system of coordinates (e.g. $B=\left(B_{1},\, B_{2}\right)=\left(B_{1}^{\prime\prime},\, B_{2}^{\prime\prime}\right)$).
Slope $m_{1}$ of the line joining $A$ to $B$ and slope $m_{2}$
of the line joining $B$ to $C$ (see Figure <ref>)
are given by
\begin{equation}
\begin{split}m_{1}= & \frac{\Delta y}{\Delta x}=\frac{B_{2}}{B_{1}},\\
m_{2}= & \frac{\Delta y}{\Delta x}=\frac{C_{2}-B_{2}}{C_{1}-B_{1}}.
\end{split}
\label{eq:model.slopes}
\end{equation}
In general, the centre of the circle passing through our points is
given by
\begin{equation}
\begin{split}x_{c}= & \frac{m_{1}m_{2}\left(A_{2}-C_{2}\right)+m_{2}\left(A_{1}-B_{1}\right)-m_{1}\left(B_{1}-C_{1}\right)}{2\left(m_{2}-m_{1}\right)},\\
y_{c}= & \frac{1}{m_{1}}\left(x_{c}-\frac{A_{1}+B_{1}}{2}\right)+\frac{A_{2}+B_{2}}{2}.
\end{split}
\end{equation}
cp ~/Programs/magnetic/magnetic/src/model/curvature_circle.pdf
[The radius of curvature of the magnetic field line] The radius of curvature $\Re$ of the magnetic field line at a given
point $B$ can be calculated as the radius of the circle passing through
this and the neighbouring two points ($A$, $C$). The system of coordinates
was moved and double-rotated so that $A$ is in its origin and all
points lie in the $x^{\prime\prime}y^{\prime\prime}$-plane. The slopes
of the lines joining $A$ to $B$ and $B$ to $C$ are described by
Equation <ref>.
Since point $A$ is in the centre of the coordinate system we can
simplify these formulas as follows
\begin{equation}
\begin{split}x_{c}= & \frac{2\, B_{1}^{2}B_{2}-2\, B_{1}B_{2}C_{1}+B_{2}C_{1}^{2}-B_{2}C_{2}^{2}-\left(B_{1}^{2}-B_{2}^{2}\right)C_{2}}{2\,\left(B_{1}C_{2}-B_{2}C_{1}\right)},\\
y_{c}= & \frac{B_{2}^{2}-\left(B_{2}-2\, x_{c}\right)B_{1}}{2\, B_{2}}.
\end{split}
\end{equation}
Finally, we can calculate the radius of curvature simply by finding
the distance between the centre of the circle and any of the points
on the circle (we have chosen point $A$)
\begin{equation}
\Re=\frac{1}{\rho}=\sqrt{x_{c}^{2}+y_{c}^{2}}.
\end{equation}
In this thesis we consider complex structures of the surface magnetic
field, thus the numerical method presented above was used in all the
calculations of curvature. The analytical approach may result in an
overestimation of curvature for points with $\theta\approx0$ (see
Figure <ref>).
~/Programs/studies/phd/curvature/curvature.py (show_curva)
with 99 data set
[Curvature of the magnetic field lines vs. the height above the stellar
surface]Curvature of the magnetic field lines vs. the height above the stellar
surface calculated using the analytical approach described in Section
[<ref>] (green lines) and the numerical approach
presented above (red lines). Panel (a) corresponds to the magnetic
field line which has no $\phi$ component, while panel (b) corresponds
to a more general scenario i.e. the nonzero $\phi$ component of the
magnetic field line. As can be seen, the analytical approach is not
valid for every case. This is caused by the undefined value of the
$\phi$ derivative for $\theta=0$ (see Equation [<ref>]).
Here $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$
and $\Re_{6}=\Re/\left(10^{6}\,{\rm cm}\right)$.
§ SIMULATION RESULTS
In this section we model the surface non-dipolar magnetic field structure
for some pulsars. Note that we can estimate the size of the polar
cap and the strength of the surface magnetic field only for pulsars
with an observed hot spot (see Section <ref>). Here
we present only pulsars listed in Table <ref>.
We use spherical coordinates $\left(r,\,\theta,\,\phi\right)$ to
describe the location and orientation of crust-anchored local anomalies.
The parameters of anomalies are as follows: ${\bf r_{a}}=\left(r_{a},\,\theta_{a},\,\phi_{a}\right)$
is a radius vector which points to the location of the anomaly and
${\bf m_{a}}=\left(m_{a},\,\theta_{a}\,,\phi_{a}\right)$ is its dipole
moment. The value of $m_{a}$ is measured in units of the global dipole
moment $d$, i.e. the moment which corresponds to the pulsar's global
magnetic field. In the Figures showing a possible non-dipolar structure
(e.g. Figure <ref>, <ref>, <ref>)
the dashed lines correspond to the dipolar configuration of the magnetic
field lines, while the solid lines correspond to the actual magnetic
field lines (taking into account the crust-anchored anomalies). Green
and red lines represent the open magnetic field lines for dipolar
and non-dipolar structures, respectively.
§.§ PSR B0628-28
Pulsar B0628-28, a bright radio pulsar, was discovered by 111
during a pulsar search at 408 MHz. The pulsar period $P\approx1.24\,{\rm s}$
and its first derivative $\dot{P}_{-15}\approx7.1$ result in a dipolar
component of magnetic field $B_{{\rm d}}=6\times10^{12}\,{\rm G}$
and a characteristic age $\tau_{c}\approx2.8\,{\rm Myr}$, which makes
it a typical, old pulsar. The large distance to this pulsar $D=1.44\,{\rm kpc}$
(evaluated using the Galactic free electron density model of 42)
makes it impossible to use the parallax method to determine the distance
with better accuracy.
PSR B0628-28 is one of the longest period pulsars among those detected
in X-rays. The pulsar was first detected in the X-ray band by ROSAT
and then later observed with both the Chandra and XMM-Newton.
Observations with the Chandra revealed no pulsations, while
the XMM-Newton observations revealed pulsations with a period
consistent with the period of radio emission [169].
The inconsistency of the observations is a reflection of the fact
that the pulsar is detectable just at the threshold of sensitivity
of both the observatories. The two-component spectral fit (BB+PL)
shows that both the nonthermal and thermal components have a comparable
luminosity (at least if we assume that the nonthermal radiation is
isotropic, see Table <ref>). PSR $\mbox{B0628-28}$
is characterised by one of the largest X-ray efficiencies among the
observed pulsars $\xi_{{\rm BB}}\approx\xi_{{\rm NT}}^{^{{\rm max}}}\approx10^{-2}$.
~/Programs/studies/phd/lines/lines.py (plot_b0628),
400, 401, data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B0628-28]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.95R,\,4^{\circ},\,0^{\circ}\right)$, ${\bf r_{2}}=\left(0.95R,\,5^{\circ},\,180^{\circ}\right)$,
with the dipole moments
${\bf m_{1}}=\left(4.5\times10^{-3}d,\,5^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(4.5\times10^{-3}d,\,170^{\circ},\,180^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim2R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=4.5\times10^{-3}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_b0628_zoom),
400 data set
[Zoom of the polar cap region [PSR B0628-28]]Zoom of the polar cap region of PSR B0628-28. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_b0628),
403 data set
[Curvature of the open magnetic field lines [PSR B0628-28]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B0628-28. The distance is
in units of the stellar radius $z_{6}=z/R$ and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR J0633+1746
Geminga was discovered in 1972 as a $\gamma$-ray source by 56.
The visual magnitude of the pulsar was estimated by 22
to be of the order of $\sim25.5^{^{{\rm mag}}}$. The pulse modulation
was discovered in X-rays [79], in $\gamma$-rays,
and at optical wavelengths [160]. Geminga has been
determined to be a relatively old ($\tau=342\,{\rm kyr}$) radio-quiet
pulsar with a period $P=237\,{\rm ms}$. The distance to the pulsar
$D=0.16\,{\rm kpc}$, evaluated using the parallax method, makes it
the closest pulsar with available X-ray data.
The pulsar exhibits one of the weakest radio luminosities known and
a cutoff at frequencies higher than about $100\,{\rm MHz}$. The model
presented by 66 explains this weak radio emission with
absorption by the magnetised relativistic plasma inside the light
cylinder. As the exact model of radio emission is still unknown (see
Section <ref>), it is difficult to verify
if this weak radio emission is a result of absorption or the absence
of coherent radio emission.
The three-component fit to the X-ray spectrum (PL+BB+BB, see Table
<ref>) reveals the hot spot component with a size
that is considerably smaller than the conventional polar cap size
($b\approx23$). The entire surface temperature $T_{{\rm s}}=0.5\,{\rm MK}$
is consistent with the theoretical value predicted by the cooling
~/Programs/studies/phd/lines/lines.py (plot_j0633),
370, 371data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
J0633+1746]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.95R,\,3^{\circ},\,0^{\circ}\right)$, ${\bf r_{2}}=\left(0.95R,\,6^{\circ},\,180^{\circ}\right)$,
with the dipole moments
${\bf m_{1}}=\left(5.5\times10^{-3}d,\,10^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(5.5\times10^{-3}d,\,160^{\circ},\,0^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim3.1R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=5.5\times10^{-3}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_j0633_zoom),
373 data set
[Zoom of the polar cap region [PSR J0633+1746]]Zoom of the polar cap region of PSR J0633+1746. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_0633),
data set
[Curvature of the open magnetic field lines [PSR J0633+1746]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR J0633+1746. The distance
is in units of the stellar radius $z_{6}=z/R$ and the curvature of
the magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR B0834+06
The bright radio emission of PSR B0834+06 shows frequent nulls (nearly
$9\%$ of the pulses is absent, see 150) . With
a relatively long rotational period $P=1.27\,{\rm s}$ and $\dot{P}_{-15}\approx7.1$
[168], its inferred physical properties, e.g. $B_{{\rm d}}=3\times10^{12}\,{\rm G}$,
are close to the average. The characteristic age $\tau_{c}=2.97\,{\rm Myr}$
implies that the pulsar should be categorised as an old pulsar. The
distance to the pulsar, estimated as $D=0.64\,{\rm kc}$, was derived
from its dispersion measure using the Galactic free-electron density
model of 42. 180 suggest
a drift of subpulses, but the estimated value of a subpulse separation
is larger than the pulse width. Despite the fact that the geometry
based on the carousel model could be fitted to the observations, there
is no clear evidence for a drift of emission between the components
of the pulsar [149].
The pulsar was detected in X-ray by 61 with a total
of $70$ counts from over $50\,{\rm ks}$ exposure time. Because of
the low statistical quality of the X-ray data, it was not possible
to constrain the absorbing column density $N_{H}$. The two-component
spectral fit (BB + PL), as presented in this thesis, was performed
using the assumption that both the thermal and nonthermal fluxes are
of the same order.
~/Programs/studies/phd/lines/lines.py (plot_b0834),
380, 381 data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B0834+06]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.95R,\,2^{\circ},\,0^{\circ}\right)$, ${\bf r_{2}}=\left(0.95R,\,5^{\circ},\,180^{\circ}\right)$,
with the dipole moments
${\bf m_{1}}=\left(3\times10^{-3}d,\,15^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(3\times10^{-3}d,\,150^{\circ},\,0^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim3.2R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=3\times10^{-3}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_b0834_zoom),
380 set
[Zoom of the polar cap region [PSR B0834+06]]Zoom of the polar cap region of PSR B0834+06. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_b0834),
data set
[Curvature of the open magnetic field lines [PSR B0834+06]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B0834+06. The distance is
in units of stellar radius ($z_{6}=z/R$) and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR B0943+10
Pulsar B0943+10 is a relatively old pulsar with a characteristic age
of $\tau_{c}=4.98\,{\rm Myr}$. The pulsar period $P=1.1\,{\rm s}$
and its first derivative $\dot{P}_{-15}\approx3.5$ result in the
dipolar component of a magnetic field $B_{{\rm d}}=4.0\times10^{12}\,{\rm G}$.
Using the Galactic free-electron density model of 42,
we can estimate the distance to the pulsar $D=0.63\,{\rm kpc}$.
PSR B0943+10 is a well-known example of a pulsar exhibiting both the
mode changing and subpulse drifting phenomenon. Strong, regular subpulse
drifting is observed only in radio-bright mode, and only hints of
the modulation feature have been found in the radio-quiescent mode.
Very recent results presented by 87 show synchronous
switching in the radio and X-ray emission properties. When the pulsar
is in a radio-bright mode, the X-rays exhibit only an unpulsed component.
On the other hand, when the pulsar is in a radio-quiet mode, the flux
of X-rays is doubled and a pulsed component is also visible.
~/Programs/studies/phd/lines/lines.py (plot_0943),
910 data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B0943+10]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.96R,\,0^{\circ},\,0^{\circ}\right)$, ${\bf r_{2}}=\left(0.96R,\,15^{\circ},\,0^{\circ}\right)$,
with the dipole moments
${\bf m_{1}}=\left(2.0\times10^{-2}d,\,180^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(6\times10^{-3}d,\,20^{\circ},\,180^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim4.5R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=2\times10^{-2}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_0943),
910 data sets
[Zoom of the polar cap region [PSR B0943+10]]Zoom of the polar cap region of PSR B0943+10. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_0943),
912 data set
[Curvature of the open magnetic field lines [PSR B0943+10]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B0943+10. The distance is
in units of stellar radius ($z_{6}=z/R$) and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR B0950+08
Pulsar B0950+08 is one of the strongest pulsed radio sources in the
metre wavelength range. The pulsar radiation also exhibits an interpulse
located at $152^{\circ}$ from the main pulse [163].
Based on the period $P=1.1\,{\rm s}$ and its first derivative $\dot{P}_{-15}\approx3.5$,
we can estimate the pulsar's characteristic age $\tau_{c}=17.5\,{\rm Myr}$.
PSR B0950+08 has a relatively weak dipolar component of magnetic field
$B_{{\rm d}}=0.5\times10^{12}$. For this pulsar the distance $D=0.26\,{\rm kpc}$
was estimated using the parallax method.
PSR B0950+08 was detected in the ultraviolet-optical range ($2400-4600\,\AA$)
by 144 with the Hubble Space Telescope.
Further observations suggest that the optical radiation of the pulsar
is most likely of a nonthermal origin [130, 193].
X-rays from PSR B0950+08 were first detected with the ROSAT
by 116 ($\sim55$ source counts). Further X-ray
observations revealed pulsations of the X-ray flux at the radio period
of the pulsar [187]. The X-ray spectrum manifests two
components (thermal and nonthermal). Which of the two components dominates
the spectrum depends on the radiation pattern of the nonthermal component
(isotropic or anisotropic). Due to the poor quality of the X-ray data,
the connection of the optical and X-ray spectra remained unclear.
~/Programs/studies/phd/lines/lines.py (plot_b0950),
355, 356 data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B0950+08]]Possible non-dipolar structure of the magnetic field lines of PSR
B0950+08. The structure was obtained using two crust anchored anomalies
located at: ${\bf r_{1}}=\left(0.95R,\,4^{\circ},\,0^{\circ}\right)$,
${\bf r_{2}}=\left(0.95R,\,5^{\circ},\,180^{\circ}\right)$, with
the dipole moments ${\bf m_{1}}=\left(5.9\times10^{-2}d,\,15^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(5.9\times10^{-2}d,\,140^{\circ},\,0^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim5.0R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=5.9\times10^{-2}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_b0950_zoom),
355 data set
[Zoom of the polar cap region [PSR B0950+08]]Zoom of the polar cap region of PSR B0950+08. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_b0959),
357 data set
[Curvature of the open magnetic field lines [PSR B0950+08]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B0950+08. The distance is
in units of stellar radius ($z_{6}=z/R$) and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR B1133+16
Pulsar B1133+16 is one of the brightest pulsating radio sources in
the Northern hemisphere [117]. The relatively long pulse
period $P=1.19\,{\rm s}$ and its first derivative $\dot{P}_{-15}\approx3.5$
result in the following inferred physical properties: $B_{{\rm d}}=4.3\times10^{12}\,{\rm G}$,
$\tau_{c}=5.04\,{\rm Myr}$. The pulsar profile exhibits a classic
double peak along with the usual S-shaped polarisation-angle traverse.
The pulsar also shows the phenomenon of drifting subpulses but only
for some finite time-spans, outside of which the behaviour of individual
pulses is chaotic [91].
PSR B1133+16 is located at a high galactic latitude, thus implying
a low interstellar extinction [159]. 192
suggested a possible optical counterpart with brightness $B=28^{^{{\rm mag}}}$.
X-ray observations performed by 104 with the
Chandra result in a small number of counts ($33$ counts
from over $17\,{\rm ks}$), thus the X-ray spectrum can be described
by various models. The photon statistics are so low that they allowed
only separate fits for the thermal (BB) and nonthermal (PL) components.
~/Programs/studies/phd/lines/lines.py (plot_1133),
340, 341 data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B1133+16]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.95R,\,2^{\circ},\,0^{\circ}\right)$, ${\bf r_{2}}=\left(0.95R,\,5^{\circ},\,180^{\circ}\right)$,
with the dipole moments
${\bf m_{1}}=\left(8\times10^{-3}d,\,20^{\circ},\,0^{\circ}\right)$,
${\bf m_{2}}=\left(8\times10^{-3}d,\,170^{\circ},\,0^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim4.2R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=5\times10^{-2}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_1133_zoom),
340 data set
[Zoom of the polar cap region [PSR B1133+16]]Zoom of the polar cap region of PSR B1133+16. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_1133),
343 data set
[Curvature of the open magnetic field lines [PSR B1133+16]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B1133+16. The distance is
in units of stellar radius ($z_{6}=z/R$) and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
§.§ PSR B1929+10
With a pulse period of $P=0.23\,{\rm s}$ and a period derivative
of $\dot{P}_{-15}\approx1.2$, the pulsar's characteristic age is
determined to be $\tau_{c}=3.1\,{\rm Myr}$. These spin parameters
imply a dipolar component of the magnetic field at the neutron star
magnetic poles $B_{{\rm d}}=1.0\times10^{12}$. The distance to the
pulsar $D=0.36\,{\rm kpc}$ was estimated using the parallax.
144 identified a candidate optical counterpart of
PSR B1929+10 with brightness $U\sim25.7^{^{{\rm mag}}}$, which was
later confirmed by proper motion measurements performed by 130.
The X-ray pulse profile of PSR B1929+10 consists of a single, broad
peak which is in contrast with the sharp radio one of 132.
The two-component spectral fit (BB+PL) suggests that both the thermal
and nonthermal luminosities are of the same order. The derived surface
temperature $T_{{\rm s}}=4.5\,{\rm MK}$ and the surface magnetic
field $B_{{\rm s}}=1.3\times10^{14}\,{\rm G}$ do not coincide with
the theoretical curve $T_{{\rm s}}-B_{{\rm s}}$ of the critical temperature
calculated by 123. We believe that this inconsistency
can be removed by adding an additional blackbody component (the whole
surface or the warm spot radiation).
~/Programs/studies/phd/lines/lines.py (plot_1929),
322 +324 data sets
[Possible non-dipolar structure of the magnetic field lines [PSR
B1929+10]]Possible non-dipolar structure of the magnetic field lines of PSR
The structure was obtained using two crust anchored anomalies located
${\bf r_{1}}=\left(0.95R,\,14^{\circ},\,180^{\circ}\right)$, ${\bf r_{2}}=\left(0.95R,\,0^{\circ},\,0^{\circ}\right)$,
${\bf r_{3}}=\left(0.95R,\,14^{\circ},\,0^{\circ}\right)$, with the
dipole moments
${\bf m_{1}}=\left(1\times10^{-2}d,\,20^{\circ},\,180^{\circ}\right)$,
${\bf m_{2}}=\left(2\times10^{-2}d,\,180^{\circ},\,0^{\circ}\right)$,
${\bf m_{3}}=\left(3\times10^{-2}d,\,10^{\circ},\,0^{\circ}\right)$
respectively (blue arrows). The influence of the anomalies is negligible
at distances $D\gtrsim4.5R$, where $B_{{\rm m}}/B_{{\rm d}}\approx m/d=3\times10^{-2}$
(top panel). For more details on the polar cap region see Figure <ref>.
~/Programs/studies/phd/lines/lines.py (plot_1929_zoom),
315 data sets
[Zoom of the polar cap region [PSR B1929+10]]Zoom of the polar cap region of PSR B1929+10. See Figure <ref>
for a description.
~/Programs/studies/phd/lines/lines.py (curvature_1929),
315 data set
[Dependence of the curvature of open magnetic field lines [PSR B1929+10]]Dependence of a curvature of the open magnetic field lines on the
distance from the stellar surface for PSR B1929+10. The distance is
in units of stellar radius ($z_{6}=z/R$) and the curvature of the
magnetic field lines is $\rho_{-6}=1/\Re_{6}=\rho/\left(10^{-6}\,{\rm cm}^{-1}\right)$.
CHAPTER: PARTIALLY SCREENED GAP
The charge-depleted inner acceleration region above the polar cap
can be formed if a local charge density differs from the co-rotational
charge density [70]. We assume that the crust of
the neutron stars mainly consists of iron $\left({\rm _{26}^{56}Fe}\right)$
formed at the neutron star’s birth (e.g. 110). Depending
on the mutual orientation of ${\bf \Omega}$ and $\boldsymbol{\mu}$,
the stellar surface at the polar caps is either positively (${\bf \Omega}\cdot\boldsymbol{\mu}<0$)
or negatively (${\bf \Omega}\cdot\boldsymbol{\mu}>0$) charged. Therefore,
the charge depletion above the polar cap depends on the binding energy
of either the positive ${\rm _{26}^{56}Fe}$ ions or electrons. In
this thesis we consider the case of positively charged polar caps
(${\bf \Omega}\cdot\boldsymbol{\mu}<0$). We assume that due to the
high cohesive energy of iron ions, the positive charges cannot be
supplied at a rate that would compensate for the inertial outflow
through the light cylinder (see 121, 122, 64).
This is actually possible if the surface temperature $T_{{\rm s}}$
is below the critical value $T_{{\rm crit}}$. Since the number density
of the iron ions in the neutron star crust is many orders of magnitude
larger than the co-rotational charge density (the so-called Goldreich-Julian
density) $\rho_{{\rm GJ}}={\bf \Omega}\cdot{\bf B}/\left(2\pi c\right)$,
then a thermionic emission from the polar cap surface is not simply
described by the usual condition $\epsilon_{{\rm i}}\approx kT_{{\rm s}}$
, where $\epsilon_{i}$ is the cohesive energy and/or work function,
$T_{{\rm s}}$ is the actual surface temperature, and $k$ is the
Boltzman constant. The outflow of iron ions can be described in the
form (65 and references therein)
\begin{equation}
\frac{\rho_{{\rm i}}}{\rho_{{\rm GJ}}}\approx\left(C_{{\rm i}}-\frac{\epsilon_{i}}{kT_{{\rm s}}}\right),
\end{equation}
where $\rho_{{\rm i}}\leq\rho_{{\rm GJ}}$ is the charge density of
the outflowing ions. As soon as the surface temperature $T_{{\rm s}}$
reaches the critical value
\begin{equation}
T_{{\rm crit}}=\frac{\epsilon_{i}}{C_{{\rm i}}k},
\end{equation}
the ion outflow reaches the maximum value $\rho_{{\rm i}}=\rho_{{\rm GJ}}$.
The numerical coefficient
$C_{{\rm i}}=30\pm3$ is determined from the tail of the exponential
function with an accuracy of about 10%. Thus, for a given value of
the cohesive energy, the critical temperature $T_{{\rm crit}}$ is
also estimated within an accuracy of about 10%. The cohesive energy
is mainly defined by the strength of the magnetic field and was calculated
by 121, 122.
§ THE MODEL
As it follows from the X-ray observations (see Section <ref>),
the temperature of the hot spot (which is associated with the actual
polar cap) is more than $10^{6}\,{\rm K}$. As we mentioned above,
in order to sustain such a high temperature bombardment by the backstreaming
particles is required. But particle acceleration (and therefore the
surface heating) is possible only if $T_{{\rm s}}<T_{{\rm crit}}$.
65 introduced the model of the Partially Screened Gap
to describe the polar gap sparking discharge specifically under such
The PSG model assumes the existence of heavy iron ions (${\rm _{26}^{56}Fe}$)
with a density near but still below the co-rotational charge density
($\rho_{{\rm GJ}}$), thus the actual charge density causes partial
screening of the potential drop just above the polar cap. The degree
of screening can be described by screening factor
\begin{equation}
\eta=1-\rho_{{\rm i}}/\rho_{{\rm GJ}}.
\end{equation}
where $\rho_{{\rm i}}$ is the charge density of the heavy ions in
the gap. The thermal ejection of ions from the surface causes partial
screening of the acceleration potential drop
\begin{equation}
\Delta V=\eta\Delta V_{{\rm max}},\label{eq:psg.d_v1}
\end{equation}
where $\Delta V_{{\rm max}}$ is the potential drop in a vacuum gap.
We can express the dependence of the critical temperature on the pulsar
parameters by fitting to the numerical calculations of 122
\begin{equation}
T_{{\rm crit}}=1.6\times10^{4}\left\{ \left[\left(P\dot{P}_{-15}\right)^{0.5}b\right]^{1.1}+17.7\right\} ,\label{eq:psg.t_s}
\end{equation}
or $T_{{\rm crit}}=1.1\times10^{6}\left(B_{14}^{1.1}+0.3\right)$,
where $B_{14}=B_{{\rm s}}/\left(10^{14}\,{\rm G}\right)$ , $B_{{\rm s}}=bB_{{\rm d}}$
is a surface magnetic field (applicable only if hot spot components
are observed, i.e. $b>1$).
The actual potential drop $\Delta V$ should be thermostatically regulated
and a quasi-equilibrium state should be established in which heating
due to the electron/positron bombardment is balanced by cooling due
to thermal radiation (see 65 for more details). The
necessary condition for this quasi-equilibrium state is
\begin{equation}
\sigma T_{{\rm s}}^{4}=\eta e\Delta Vcn_{{\rm GJ}},\label{eq:psg.heating_condition}
\end{equation}
where $\sigma$ is the Stefan-Boltzmann constant, $e$ - the electron
charge, and
$n_{{\rm GJ}}=\rho_{{\rm GJ}}/e=1.4\times10^{11}b\dot{P}_{-15}^{0.5}P^{-0.5}$
is the co-rotational number density. The Goldreich-Julian co-rotational
number density can be expressed in terms of $B_{14}$ as
\begin{equation}
n_{{\rm GJ}}=6.93\times10^{12}B_{14}P^{-1}.\label{eq:psg.n_gj}
\end{equation}
Here we assume that the density of backstreaming relativistic electrons
is $\eta n_{{\rm GJ}}$.
By using Equations <ref>, <ref>
and <ref> we can express the acceleration potential drop
that satisfies the heating condition (Equation <ref>)
as follows
\begin{equation}
\Delta V=7.3\times10^{5}\frac{\left(B_{14}^{1.1}+0.3\right)^{4}P}{\eta B_{14}}.\label{eq:psg.potential_heating}
\end{equation}
The above equation may suggest that the acceleration potential drop
is inversely proportional to the screening factor. In fact, it is
just the opposite (see Equations <ref> and <ref>).
Knowing that $\Delta V=\gamma_{{\rm max}}mc^{2}/e$, where $m$ is
the mass of a particle (electron or positron), we can calculate the
maximum Lorentz factor of the primary particles in PSG as
\begin{equation}
\gamma_{{\rm max}}=450\frac{\left(B_{14}^{1.1}+0.3\right)^{4}P}{\eta B_{14}}.
\end{equation}
§.§ Acceleration potential drop
As the actual polar cap is much smaller than the conventional polar
cap (see section <ref>), we cannot
use the approximation proposed by 156 that the
gap height is of the same order as the gap width ($h\approx h_{\perp}$).
On the contrary, the small polar cap size and subpulse phenomenon
suggest that in the PSG model the spark half-width is considerably
smaller than the gap height ($h_{\perp}<h$). For such a regime we
need to recalculate a formula for the acceleration potential drop
$\Delta V$.
Let us consider a reference frame co-rotating with a star and with
the z-axis aligned with the star's angular velocity ${\bf \Omega}$
(see Figure <ref>).
[Co-rotating frame of reference (acceleration potential drop)]Co-rotating frame of reference with the z-axis aligned with the angular
velocity ${\bf \Omega}$. The magnetic dipole moment $\boldsymbol{\mu}$
is constant in this frame of reference, thus $\partial{\bf B}/\partial t=0$.
Let us underline that we will neglect the effects of non-inertiality
of the co-rotating system. Thus, we assume that in any given moment
we have a system moving with a constant velocity.
In this co-rotating frame of reference we can write the spherical
components of an angular velocity as follows
\begin{equation}
{\bf \Omega}=\left(\Omega\cos\theta,\,-\Omega\sin\theta,\,0\right).
\end{equation}
Gauss's law in the co-rotating frame (after Lorentz transformations)
takes the form
\begin{equation}
\nabla\cdot{\bf E}=4\pi\rho\left({\bf r}\right)-4\pi\left(\frac{{\bf \Omega}\cdot{\bf B}}{2\pi c}\right).\label{eq:psg.divergence}
\end{equation}
While Faraday's law of induction can be written as
\begin{equation}
\nabla\times{\bf E}=0.\label{eq:psg.curl}
\end{equation}
Note that if we consider a drift of plasma in the Inner Acceleration
Region (IAR), we should expect temporal variations of the magnetic
field ($\nabla\times{\bf E}=-\partial{\bf B}/\left(c\partial t\right)$)
[158], but as was shown by 178, even
if we consider fluctuations of the electric current of the order of
the Goldreich-Julian current $\rho_{{\rm GJ}}c$, the resulting variation
of the magnetic field is so small that $\nabla\times{\bf E}=0$ with
a high accuracy, and circulation of the non-co-rotational electric
field along a closed path is zero.
Equation <ref> in the spherical system of coordinates
has the following form
\begin{equation}
\frac{2}{r}E_{r}+\frac{\partial E_{r}}{\partial r}+\frac{\cos\theta}{r\sin\theta}E_{\theta}+\frac{1}{r}\frac{\partial E_{\theta}}{\partial\theta}+\frac{1}{r\sin\theta}\frac{\partial E_{\phi}}{\partial\phi}=4\pi\rho\left(r,\theta,\phi\right)-4\pi\left(\frac{{\bf \Omega}\cdot{\bf B}}{2\pi c}\right).
\end{equation}
The PSG model assumes the existence of ions in the IAR region that
affects the charge density. Using the screening factor, $\eta$, we
can write that
\[
\rho\left(r,\theta,\phi\right)=\left(1-\eta\right)\rho_{{\rm GJ}}\left(r,\theta,\phi\right)=\left(1-\eta\right)\frac{{\bf \Omega}\cdot{\bf B}}{2\pi c}.
\]
In general, $\eta$ depends on the curvature and strength of the magnetic
field, thus it varies across the polar cap, but we can still assume
that $\eta$ is approximately constant at least for a given spark.
\begin{equation}
\frac{2}{r}E_{r}+\frac{\partial E_{r}}{\partial r}+\frac{\cos\theta}{r\sin\theta}E_{\theta}+\frac{1}{r}\frac{\partial E_{\theta}}{\partial\theta}+\frac{1}{r\sin\theta}\frac{\partial E_{\phi}}{\partial\phi}=-4\pi\eta\left(\frac{B_{r}\Omega\cos\theta-B_{\theta}\Omega\sin\theta}{2\pi c}\right).
\end{equation}
Let us change the variables as follows: $r=R+z$ and $\theta=\alpha+\vartheta$.
Here $R$ is the stellar radius and $\alpha$ is the inclination angle
between the rotation and the magnetic axis.
\begin{multline}
\frac{2}{R+z}E_{r}+\frac{\partial E_{r}}{\partial z}+\frac{\cos\left(\alpha+\vartheta\right)}{\left(R+z\right)\sin\left(\alpha+\vartheta\right)}E_{\theta}+\frac{1}{R+z}\frac{\partial E_{\theta}}{\partial\vartheta}+\frac{1}{\left(R+z\right)\sin\left(\alpha+\vartheta\right)}\frac{\partial E_{\phi}}{\partial\phi}=\\
=-4\pi\eta\left(\frac{\left(B_{r}\Omega\cos\theta-B_{\theta}\Omega\sin\theta\right)}{2\pi c}\right).\label{eq:psg.potential_long}
\end{multline}
Assuming that $R\gg z$, which is correct as the gap height is less
than the stellar radius ($h\ll R$), $\alpha\gg\vartheta$, and $B_{r}\gg B_{\theta}$,
which is correct for the polar cap region, we can write Equation <ref>
in the first approximation ($R\rightarrow\infty$) as follows
\begin{equation}
\frac{\partial E_{r}}{\partial z}+\frac{1}{R}\frac{\partial E_{\theta}}{\partial\vartheta}=-4\pi\eta\left(\frac{B_{r}\Omega\cos\theta}{2\pi c}\right).\label{eq:psg.potential_estiamte}
\end{equation}
Note that for spark widths considerably smaller than the stellar radius
$h_{\perp}\ll R$ ($\Delta\vartheta\approx h_{\perp}/R$) we can
write that $\frac{1}{R}\frac{\partial E_{\theta}}{\partial\vartheta}\gg\frac{\cot\left(\alpha+\vartheta\right)}{R}E_{\theta}$.
Let us now consider Faraday's law (Equation <ref>). The
curl of an electric field in spherical coordinates can be written
\begin{equation}
\begin{split}\left({\bf \nabla}\times{\bf E}\right)_{r}= & \frac{1}{r\sin\theta}\left(\frac{\partial}{\partial\theta}\left(E_{\phi}\sin\theta\right)-\frac{\partial E_{\phi}}{\partial\phi}\right)=0,\\
\left({\bf \nabla}\times{\bf E}\right)_{\theta}= & \frac{1}{r}\left(\frac{1}{\sin\theta}\frac{\partial E_{r}}{\partial\phi}-\frac{\partial}{\partial r}\left(rE_{\phi}\right)\right)=0,\\
\left({\bf \nabla}\times{\bf E}\right)_{\phi}= & \frac{1}{r}\left(\frac{\partial}{\partial r}\left(rE_{\theta}\right)-\frac{\partial E{}_{r}}{\partial\theta}\right)=0.
\end{split}
\label{eq:psg.curl_system}
\end{equation}
Using the same change of variables we performed above ($r=R+z$ and
$\theta=\alpha+\vartheta$), the third equation of System <ref>
can be written as
\begin{equation}
R\frac{\partial E_{\theta}}{\partial z}=\frac{\partial E_{r}}{\partial\vartheta}.
\end{equation}
From this equation in the zeroth approximation we can estimate the
variations of the electric field components as
\begin{equation}
R\Delta E_{\theta}\Delta\vartheta\approx\Delta E_{r}\Delta z.
\end{equation}
Since $h_{\perp}\ll R$ we can write that
\begin{equation}
\left\langle h_{\perp}E_{\theta}\right\rangle =\left\langle hE_{r}\right\rangle =\Delta V.\label{eq:psg.potential_est2}
\end{equation}
From Equation <ref> we can also briefly
estimate that
\begin{equation}
\frac{\Delta E_{r}}{h}+\frac{\Delta E_{\theta}}{h_{\perp}}=-4\pi\eta\left(\frac{B_{r}\Omega\cos\theta}{2\pi c}\right).
\end{equation}
Using Equations <ref> and <ref>
we can write that
\begin{equation}
\frac{\left\langle hE_{r}\right\rangle }{h^{2}}+\frac{\left\langle h_{\perp}E_{\theta}\right\rangle }{h_{\perp}^{2}}=\frac{\Delta V}{h^{2}}+\frac{\Delta V}{h_{\perp}^{2}}.
\end{equation}
Finally, we can estimate the potential drop in a spark region
\begin{equation}
\frac{\Delta V}{h^{2}}+\frac{\Delta V}{h_{\perp}^{2}}=\frac{2\eta B_{r}\Omega\cos\left(\alpha+\vartheta\right)}{c}.\label{eq:psg.delta_v_h_hperp}
\end{equation}
If we use the same assumption as 156, i.e.: (1)
the spark half-width is of the same order as the gap height $h_{\perp}=h$,
(2) there is no ion extraction from the stellar surface ($\eta=1$),
and (3) the pulsar magnetic and rotation axes are aligned ($\alpha=0^{\circ}$),
we get:
\begin{equation}
\Delta V_{{\rm RS}}=\frac{B_{r}\Omega}{c}h^{2}.
\end{equation}
Note that the potential drop defined by Equation <ref>
differs from that used in the Standard Model by the screening factor
(as the presence of ions screens the gap) and by the factor of $\cos\left(\alpha+\vartheta\right)$
which also takes into account non-aligned pulsars. In our case the
polar cap size is much smaller than the conventional polar cap size.
It seems reasonable to also consider sparks with widths much smaller
than the gap height ($h_{\perp}\ll h$), in that case the potential
drop can be calculated as
\begin{equation}
\Delta V=\frac{2\eta B_{r}\Omega\cos\left(\alpha+\vartheta\right)}{c}h_{\perp}^{2}.
\end{equation}
Even for a relatively small inclination angle between the rotation
and magnetic axis, we can still write $\vartheta\ll\alpha$, thus
\begin{equation}
\Delta V=\frac{4\pi\eta B_{r}\cos\alpha}{cP}h_{\perp}^{2}.\label{eq:psg.potential_drop}
\end{equation}
§.§ Acceleration path
Since the exact dependence of the electric field on $z$ is unknown
we use the same linear approximation that 156 used.
In the frame of the PSG model as $h_{\perp}<h$ or even $h_{\perp}\ll h$,
we can use Equations <ref> and <ref>
to describe the component of the electric field along the magnetic
field line:
\begin{equation}
E\approx\frac{8\pi\eta B_{{\rm s}}\cos\alpha}{cP}\frac{h_{\perp}^{2}}{h^{2}}\left(h-z\right),\label{eq:psg.acceleration_field}
\end{equation}
which vanishes at the top $z=h$. The Lorentz factor of particles
after passing distance $l_{{\rm acc}}$ can be calculated as follows
\begin{equation}
\gamma_{{\rm acc}}=\frac{e}{mc^{2}}\int_{z_{1}}^{z_{2}}Edz\approx\frac{8\pi\eta B_{{\rm s}}e\cos\alpha}{mc^{3}P}\frac{h_{\perp}^{2}}{h^{2}}\left(z_{2}-z_{1}\right)\left(h-\frac{z_{1}+z_{2}}{2}\right),
\end{equation}
where $m$ is the mass of a particle (electron or positron) and $z_{2}-z_{1}=l_{{\rm acc}}$.
Then we can approximate $z_{1}+z_{2}\approx h$, thus
\begin{equation}
l_{{\rm acc,ap}}=\frac{\gamma_{{\rm acc}}mc^{3}P}{4\pi\eta B_{{\rm s}}e\cos\alpha}\frac{h}{h_{\perp}^{2}}.\label{eq:psg.acceleration}
\end{equation}
Assuming that a non-relativistic particle is accelerated from the
stellar surface ($z_{1}=0$, $\gamma_{0}=1$) we can calculate the
distance $l_{{\rm acc}}$ which it should pass to gain a Lorentz factor
$\gamma_{{\rm acc}}$:
\begin{equation}
l_{{\rm acc}}=h\left(1-\sqrt{1-\frac{2\gamma}{\ell}}\right),\label{eq:psg.acceleration-1}
\end{equation}
where $\ell=8\pi\eta B_{{\rm s}}eh_{\perp}^{2}\cos\left(\alpha\right)/\left(Pc^{3}m\right)$.
Although the approximate formula <ref> is much
more readable, in the calculations we use the exact value (see Equation
<ref>) as for Lorentz factors that are considerably
smaller than the maximum value, the discrepancy is about a factor
of two, $l_{{\rm acc,ap}}\approx2l_{{\rm acc}}$.
§.§ Electron/positron mean free path
The mean free path of a particle (electron and/or positron) $l_{{\rm p}}$
can be defined as the mean length that a particle passes until a $\gamma$-photon
is emitted. In the case of the CR particle, mean free path can be
estimated as a distance that a particle with a Lorentz factor $\gamma$
travels during the time which is necessary to emit a curvature photon
(see 190)
\begin{equation}
l_{{\rm CR}}\sim c\left(\frac{P_{{\rm CR}}}{E_{\gamma,{\rm CR}}}\right)^{-1}=\frac{9}{4}\frac{\hbar\Re c}{\gamma e^{2}},\label{eq:psg.le_cr}
\end{equation}
where $P_{{\rm CR}}=2\gamma^{4}e^{2}c/3\Re^{2}$ is the power of CR,
$E_{\gamma,{\rm CR}}=3\hbar\gamma^{3}c/2\Re$ is the photon characteristic
energy, and $\Re$ is the curvature radius of the magnetic field lines.
For the ICS process calculation of the particle mean free path $l_{{\rm ICS}}$
is not as simple as that of the CR process. Although we can define
$l_{{\rm ICS}}$ in the same way that we defined $l_{{\rm CR}}$,
it is difficult to estimate the characteristic frequency of emitted
photons. We have to take into account photons of various frequencies
with various incident angles. An estimation of the mean free path
of an electron (or positron) to produce a photon is in 184
\begin{equation}
l_{{\rm ICS}}\sim\left[\int_{\mu_{0}}^{\mu_{1}}\int_{0}^{\infty}\sigma^{\prime}\left(\epsilon,\mu\right)\left(1-\beta\mu_{i}\right)n_{{\rm ph}}\left(\epsilon\right)d\epsilon d\mu\right]^{-1}.\label{eq:psg.le_ics}
\end{equation}
Here $\epsilon$ is the incident photon energy in units of $mc^{2}$,
$\mu=\cos\psi$ is the cosine of the photon incident angle, $\beta=v/c$
is the velocity in terms of speed of light, $\sigma^{\prime}$ is
the cross section of ICS in the particle rest frame,
\begin{equation}
n_{{\rm ph}}\left(\epsilon,\, T\right)d\epsilon=\frac{4\pi}{\lambda_{c}^{3}}\frac{\epsilon^{2}}{\exp\left(\epsilon/\mho\right)-1}d\epsilon\label{eq:psg.nph}
\end{equation}
represents the photon number density distribution of semi-isotropic
blackbody radiation, $\mho=kT/mc^{2}$, $k$ is the Boltzmann constant,
and $\lambda_{c}=h/mc=2.424\times10^{-10}$ cm is the electron Compton
wavelength. A detailed description of how to calculate $\sigma^{\prime}$
can be found in Section <ref>.
We should expect two modes of ICS: resonant and thermal-peak (see
Section <ref> for more details). The
Resonant ICS (RICS) takes place if the photon frequency in the particle
rest frame is equal to the electron cyclotron frequency. As shown
in Section <ref>, the particle mean
free path strongly depends on the distance from the polar cap. Both
the photon density and incident angles ($\mu_{0}$ and $\mu_{1}$)
change with increasing altitude. In our calculations we take into
account both of those effects, thus we replace $n_{{\rm ph}}\left(\epsilon,\, T\right)$,
$\mu_{0}$ and $\mu_{1}$ with $n_{{\rm sp}}\left(\epsilon,\, T,\, L\right)$,
$\mu_{{\rm min}}$$\left(L\right)$ and $\mu_{{\rm max}}\left(L\right)$,
respectively (for more details see Section <ref>).
Here, $L$ is the location of the particle, $n_{{\rm sp}}\left(\epsilon,\, T,\, L\right)$
is the photon density at location $L$, and $\mu_{{\rm min}}\left(L\right)$
and $\mu_{{\rm max}}$$\left(L\right)$ correspond to the highest
and lowest angle between the photons and particle at a given location
$L$. Thus, just above the polar cap for RICS the mean free path of
outflowing positrons is:
\begin{equation}
l_{{\rm RICS}}\approx\left[\int_{\mu_{{\rm min}}\left(L\right)}^{\mu_{{\rm max}}\left(L\right)}\int_{\epsilon_{_{{\rm res}}}^{{\rm ^{min}}}}^{\epsilon_{_{{\rm res}}}^{{\rm ^{max}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm sp}}\left(\epsilon,\, T,\, L\right)d\epsilon d\mu\right]^{-1},\label{eq:cascade.ics_free_path-1}
\end{equation}
where the limits of integration over energy, $\epsilon_{{\rm _{res}}}^{{\rm ^{min}}}$
and $\epsilon_{{\rm _{res}}}^{^{{\rm max}}}$, are chosen to cover
the resonant energy (for more details see Section <ref>).
The thermal-peak ICS (TICS) includes all scattering processes of photons
with frequencies around the maximum of the thermal spectrum. As an
example we adopt
$\epsilon_{_{{\rm th}}}^{{\rm ^{min}}}\approx0.05\epsilon_{{\rm _{th}}}$,
and $\epsilon_{_{{\rm th}}}^{{\rm ^{{\rm max}}}}\approx2\epsilon_{_{{\rm th}}}$
where $\epsilon_{_{{\rm th}}}=2.82kT/\left(mc^{2}\right)$ is the
energy, in units of $mc^{2}$, at which blackbody radiation with temperature
$T$ has the largest photon number density. The electron/positron
mean free path for the TICS process is
\begin{equation}
l_{{\rm TICS}}\approx\left[\int_{\mu_{{\rm min}}\left(L\right)}^{\mu_{{\rm max}}\left(L\right)}\int_{\epsilon_{_{{\rm th}}}^{{\rm ^{min}}}}^{\epsilon_{_{{\rm th}}}^{{\rm {\rm ^{max}}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon,\, T,\, L\right)d\epsilon d\mu\right]^{-1}.\label{eq:cascade.t_ics-1}
\end{equation}
§.§ Photon mean free path
The photons with energy $E_{\gamma}>2mc^{2}$ propagating obliquely
to the magnetic field lines can be absorbed by the field, and as a
result, an electron-positron pair is created. To describe the strength
of the magnetic field we use $\beta_{q}=B/B_{q}$, where $B_{q}=m^{2}c^{3}/e\hbar=4.413\times10^{13}\,{\rm G}$
is the critical magnetic field strength.
For strong magnetic fields ($\beta_{q}\gtrsim0.2$, see Section <ref>)
the photon mean free path can be calculated as (see Section <ref>
for more details)
\begin{equation}
l_{{\rm ph}}\approx\Re\frac{2mc^{2}}{E_{\gamma}},\label{eq:psg.l_ph}
\end{equation}
while for weaker magnetic fields ($\beta_{q}\lesssim0.2$) we can
use an asymptotic approximation derived by 55
\begin{equation}
l_{{\rm ph}}=\frac{4.4}{(e^{2}/\hbar c)}\frac{\hbar}{mc}\frac{B_{q}}{B\sin\Psi}\exp\left(\frac{4}{3\chi}\right),\label{eq:psg.l_ph2}
\end{equation}
\begin{equation}
\chi\equiv\frac{E_{\gamma}}{2mc^{2}}\frac{B\sin\Psi}{B_{q}}\hspace{1cm}(\chi\ll1),
\end{equation}
where $\Psi$ is the angle of intersection between the photon and
the local magnetic field.
§ GAP HEIGHT
By knowing the acceleration potential drop in PSG $\Delta V$ we can
evaluate the gap height $h$ and the screening factor $\eta$, which
actually depends on the details of the avalanche pair production in
the gap. First, we need to determine which process, Curvature Radiation
(CR) or Inverse Compton Scattering (ICS), is responsible for the $\gamma$-photon
generation in the gap region. In order to identify the proper process
we need the following parameters: $l_{{\rm acc}}$ - the distance
which a particle should pass to gain the Lorentz factor $\gamma_{{\rm acc}}$,
$l_{{\rm p}}$ - the mean length a particle (electron and/or positron)
travels before a $\gamma$-photon is emitted, and $l_{{\rm ph}}$
- the mean free path of the $\gamma$-photon before being absorbed
by the magnetic field.
As mentioned above, PSG can exist if Equation <ref>
is satisfied. On the other hand, in order to heat the polar cap surface
to high enough temperatures the high enough flux of back-streaming
particles is required. By using Equations <ref>
and <ref> we can find the relationship between
the screening factor, the spark half-width and pulsar parameters
\begin{equation}
\eta h_{\perp}=4.17\frac{\left(B_{14}^{1.1}+0.3\right)^{2}P}{B_{14}\sqrt{\left|\cos\alpha\right|}}.\label{eq:psg.eta_hperp}
\end{equation}
Thus, for specific pulsar parameters we can define a product of the
two main parameters of PSG, namely the screening factor $\eta$ and
the spark half-width $h_{\perp}$.
§.§ Particle mean free paths, CR vs. ICS gap
The Figure <ref> shows the dependence of particle
mean free paths on the Lorentz factor $\gamma$ for some pulsar parameters
(the dependence on pulsar parameters will be discussed in Section
<ref>). Let us note that these free paths
do not depend on the gap height $h$ (see Equations <ref>,
<ref> and <ref>). The results presented
in the Figure do not allow to define the gap height unambiguously.
However, we can find which process is responsible for generation of
the $\gamma$-photon in PSG. For narrow sparks the acceleration potential
drop decreases, and as a result the Lorentz factor of the primary
particles is about $\gamma\sim10^{3}-10^{4}$. In this regime $l_{{\rm ICS}}\ll l_{{\rm CR}}$,
so the gap will be dominated by ICS. Thus, ICS will dominate the gap
if deceleration due to Inverse Compton Scattering prevents further
acceleration by the electric field. Let us remember that ICS is not
efficient for particles with the Lorentz factor $\gamma\gtrsim10^{5}$.
If the sparks are wider or $\eta\approx1$, the acceleration potential
drop increases and the Lorentz factors of primary particles reach
values about $\gamma\sim10^{5}-10^{6}$. In this regime the $\gamma$-photon
emission is dominated by CR. Let us note that the condition $l_{{\rm ICS}}\ll l_{{\rm CR}}$
is satisfied for particles with $\gamma\sim10^{3}-10^{4}$, as one
can see from Figure <ref> (panel a), but this does
not mean that the ICS event happens. Since $l_{{\rm acc}}\ll l_{{\rm ICS}}$,
the particles will be accelerated to higher energies ($\gamma\sim10^{5}-10^{6}$)
before they upscatter the X-ray photons. Thus, the particles start
emission of $\gamma$-photons (via CR) as soon as condition $l_{{\rm acc}}\approx l_{{\rm CR}}$
is met.
~Programs/magnetic/magnetic/src/radiation/gap.py (le_gamma_phd)
[Dependence of the mean free path of the primary particle [CR + ICS]]Dependence of the mean free path of the primary particle on Lorentz
factor $\gamma$ for both the CR and ICS processes. Panel (a) corresponds
to calculations for a relatively higher potential drop (e.g. a wider
spark with $h_{\perp}=3\,{\rm m}$ and $\eta=1$), while panel (b)
corresponds to calculations for a relatively lower potential drop
(e.g. a narrow spark with $h_{\perp}=1\,{\rm m}$ and $\eta=0.1$).
The acceleration paths on both panels were calculated for the same
pulsar parameters ($B_{14}=3.5$, $T_{6}=4.4$, $\Re_{6}=1$, $P=1$,
$\alpha=10^{\circ}$). Note that for the RICS process the particle
mean free paths were calculated for optimal conditions (just above
the polar cap).
§.§ Possible scenarios of the gap breakdown: PSG-on and PSG-off modes
As is seen from Figures <ref> and <ref>,
in the CR-dominated gap the primary particle should travel a distance
comparable with gap height $l_{{\rm acc}}\approx h/2$ in order to
gain an energy corresponding to the characteristic Lorentz factor
$\gamma_{{\rm c}}^{{\rm ^{CR}}}$. On the other hand, the primary
particles in the ICS-dominated gap reach a characteristic value $\gamma_{{\rm c}}^{{\rm ^{ICS}}}$
at altitudes that are considerably smaller than gap height $l_{{\rm acc}}\ll h$
(see Figures <ref> and <ref>).
Thus, $\gamma_{{\rm c}}^{{\rm ^{CR}}}\approx10^{6}\approx\gamma_{{\rm max}}$
is about three orders of magnitude higher than $\gamma_{{\rm c}}^{^{{\rm ICS}}}\approx10^{3}\ll\gamma_{{\rm max}}$,
here $\gamma_{{\rm max}}$ is the value of the Lorentz factor after
the particle travels a distance $h$. Furthermore, the characteristic
energy of CR photons is considerably smaller than the energy of emitting
(primary) particles, e.g. for $\gamma=10^{6}$, $\Re_{6}=1$, $\gamma_{{\rm sec}}\approx10^{2}$.
On the other hand, RICS photons upscattered in an ultrastrong ($B>B_{{\rm crit}}$)
magnetic field gain a significant part of the energy of the scattering
(primary) particle. Therefore, the electron/positron pair created
by the RICS photon has energies comparable with the energy of the
scattering (primary) particle. This will essentially influence the
multiplicity $M_{{\rm pr}}$ in the ICS gap, as all the newly created
particles will participate in further cascade pair-production. Additionally,
RICS in ultrastrong magnetic fields produces approximately the same
amount of photons with $\parallel$ and $\perp$ polarisation (see
Section <ref>), while most of the photons
produced by CR are $\parallel$-polarised (see Section <ref>).
Splitting of the $\perp$-polarised photons will increase the photon
mean free path, but it will also increase the multiplicity in the
ICS gaps.
Figure <ref> presents a sketch of a cascade
formation for CR- and ICS-dominated gaps. The CR photons are emitted
in the upper half of the gap. Most of these photons produce pairs
at about the same height, in the region where the acceleration potential
is almost equal to zero, hereinafter we will call this region the
Zero-Potential Front (ZPF). The newly created particles have much
lower Lorentz factors as compared with the primary particle, thus
they are not able to emit CR photons.
[Sketch of the differences in a cascade formation [CR- and ICS-dominated
gaps]]Sketch of differences in a cascade formation for the CR-dominated
gap (left panel) and the ICS-dominated gap (right panel). In order
to increase readability, only a few points (filled circles) are shown
which correspond to altitudes where $\gamma$-photons are emitted.
The unfilled circles correspond to places where $\gamma$-photons
are also emitted, but those photons (and their evolution) are not
included in the diagram. Note that for the ICS-dominated gap we plot
only the bottom (active) part of the gap ($z\ll h_{{\rm ICS}}$),
furthermore, points of radiation are tracked only for the first population
of newly created particles. The avalanche nature of the ICS-dominated
gap will result in a much higher multiplicity and continuous backflow
of relativistic particles.
Figure <ref> presents the primary particle evolution
and photon mean free paths of
$\gamma$-rays produced in the CR-dominated gap. As can be seen, the
first $\gamma$-photon produces a pair approximately at the same time
(and same place) as the primary particle reaches ZPF. Thus, the multiplicity
in a gap region (the number of particles created by a single primary
particle) in the CR scenario is strictly related to the number of
photons produced by the primary particle $M_{{\rm CR}}\approx2\times N_{{\rm ph}}^{{\rm ^{CR}}}$.
(find_solution_cr_psgoff_plot ,plot_solution_cr, plot_solution_acr,
$n_{end}=50$ $B_{14}=2.3$, $B_{{\rm d}}=0.02978$, $T_{6}=3.0$,
$P=1.273768291578$, $\Re_{6}=0.5$, $\alpha=60.7^{\circ}$
[Cascade formation for a CR-dominated gap]Cascade formation for a CR-dominated gap. Blue lines represent the
mean free path of $\gamma$-photons. The filled circles correspond
to places of $\gamma$-photon emission. Panel (a) includes the free
paths of $\gamma$-photons which produce pairs below ZPF (red circles)
while panel (b) includes the free paths of $\gamma$-photons which
produce pairs above the acceleration gap (blue circles). The results
were obtained using the following parameters: $N_{{\rm ph}}^{^{{\rm CR}}}=50$,
$B_{{\rm s}}=2.3\times10^{14}\,{\rm G}$, $B_{{\rm d}}=2.9\times10^{14}\,{\rm G}$,
$T_{}=3\,{\rm MK}$, $P=1.3\,{\rm s}$, $\Re_{6}=0.5$, and $\alpha=60.7^{\circ}$.
The energy of $\gamma$-photons produced by ICS depends on the Lorentz
factor of the primary particles and on the strength of the magnetic
field. In ultrastrong magnetic fields the energy of newly created
particles is comparable with the energy of the scattering particle
$\gamma_{{\rm new}}\approx\gamma_{{\rm c}}/2$. Figure <ref>
shows schematically the locations at which $\gamma$-photons are emitted
by ICS. The first $\gamma$-photon is produced already at altitudes
of about a few metres and then converted to an electron-positron pair
well below ZPF. Note that already at relatively low altitudes ($z\gtrsim100\,{\rm m}$)
the photon density decreases rapidly (see Section <ref>),
furthermore, the small size of the polar cap entails a rapid change
of the particle-photon incident angles (see Section <ref>).
Those two effects make the ICS process significant only in the lower
parts of the gap ($z\lesssim20\,{\rm m}$). On the other hand, the
multiplicity in the ICS-dominated gap is enhanced by all newly created
particles which are created in the lower part of the gap. Furthermore,
the ICS is more effective for backstreaming particles (see Figure
<ref>), thus most of the $\gamma$-photons in the
gap region will be created by scatterings on electrons. For the ICS
scenario it is not possible to evaluate a simple expression for the
multiplicity produced by a single primary particle in a gap region.
Furthermore, it is not possible to determine the actual value of $N_{{\rm ph}}$
required to break the gap (both for CR and ICS) without a full cascade
(read_data(910), find_solution_hperp(200.), plot_solution()
[Cascade formation for an ICS-dominated gap]Cascade formation for an ICS-dominated gap. Blue lines represent
the mean free path of $\gamma$-photons. The filled circles correspond
to places of $\gamma$-photon emission. The results were obtained
using the following parameters: $N_{{\rm ph}}^{^{{\rm ICS}}}=15$,
$B_{{\rm s}}=2.3\times10^{14}\,{\rm G}$, $B_{{\rm d}}=2.9\times10^{14}\,{\rm G}$,
$T_{{\rm s}}=3\,{\rm MK}$, $P=1.3\,{\rm s}$, $\Re_{6}=0.5$, and
The differences between the CR and ICS gaps that we mention above
have drastic consequences on the cascade formation process. Since
the cooling time of the hot spot is very short ($\tau_{{\rm cool}}\lesssim10^{-8}\,{\rm s}$
, see 65), to sustain the hot spot temperature just
below the critical temperature a continuous backflow of relativistic
particles is required. An energetic enough flux of backstreaming particles
can be produced only in ICS-dominated gaps. The heating of the surface
will sustain the outflow of iron ions from the crust, maintaining
$\eta<1$, hence we call this mode the PSG-on mode. As the temperature
of the polar cap is in quasi equilibrium with the backstreaming particles
(temperature is close to the critical value) the gap can break only
due to production of a dense enough plasma $n_{p}\gg\eta n_{{\rm GJ}}$
in the gap region. The multiplicity in the PSG-on mode is much higher
than the multiplicity of CR-dominated gaps. Moreover, in the gap dominated
by CR the particles are created in a cloud-like fashion (see Figure
<ref>). The successive clouds heat up the surface
once per $\tau_{{\rm 0}}\approx2h/c$, which for a typical gap height
$h\approx100\,{\rm m}$ is much longer than the time needed for the
surface to cool down $\tau_{{\rm 0}}\approx6\times10^{-7}\gg\tau_{{\rm cool}}$.
Therefore, in the CR-dominated gaps the backstreaming particles cannot
sustain the temperature that is close to the critical value during
$\tau_{{\rm 1}}\gg\tau_{0}\gg\tau_{{\rm cool}}$, thus for most of
the time the screening factor is $\eta\approx1$ and we call this
mode the PSG-off mode. The low multiplicity of a cascade in the PSG-off
mode can cause that the gap to breakdown only due to overheating of
the surface, but not due to production of a dense enough plasma. The
growth of particle density will continue to the point when the backstreaming
particles heat up the surface to a temperature equal to or higher
than the critical temperature, $\tau_{{\rm heat}}\gg\tau_{0}$. Let
us note that the primary particles in the PSG-off mode are very energetic
$\gamma\approx10^{6}$, and hence the density of particles required
to close gap $\rho_{c}$ is much lower than the Goldreich-Julian density.
To describe this difference we use the overheating parameter $\kappa=\rho_{c}/\rho_{{\rm GJ}}$.
Knowing that in the PSG-off mode $\eta\approx1$, we use Equation
<ref> and the relation $\Delta V=\gamma_{{\rm acc}}mc^{2}/e$
to calculate the overheating parameter:
\begin{equation}
\kappa=\frac{\sigma\, T^{4}}{n_{{\rm GJ}}\,\gamma_{{\rm max}}\, mc^{3}}.\label{eq:psg.overheating_parameter}
\end{equation}
§.§ PSG-off mode
Curvature emission by a primary particle is effective for Lorentz
factors $\gamma\gtrsim10^{5}$ (when $l_{{\rm CR}}\leq l_{{\rm acc}}$).
An equilibrium between acceleration and deceleration (by reaction
force) would be established if the CR power were equal to the ”electric
power”. In our case ($\Re_{6}\approx1$, $\gamma_{{\rm c}}\approx10^{6}$),
the reaction force is not high enough to stop acceleration by the
electric field. In the PSG-off mode the spark region is free from
ions ($\eta\approx1$), thus the heating condition (Equations <ref>
and <ref>) is no longer satisfied. Taking into account
the curvature of magnetic field lines just above the stellar surface,
we can estimate the dependence of the minimum spark half-width on
the gap height (see Figure <ref>):
\begin{equation}
h_{\perp}^{{\rm ^{min}}}=\Re-\sqrt{\Re^{2}-h^{2}}.\label{eq:psg.hperp_min}
\end{equation}
Figure <ref> presents the minimum spark half-width
for three different radii of curvature: $\Re_{6}=0.1$, $\Re_{6}=0.5$,
$\Re_{6}=1$. Note that as long as the gap height does not exceed
some specific value ($h\approx40\,{\rm m}$, $h\approx100\,{\rm m}$,
$h\approx140\,{\rm m}$, respectively for the given curvature radii)
the minimum spark half-width is well below $1\,{\rm m}$.
[Diagram of the minimum spark half-width]Diagram of the minimum spark half-width $h_{\perp}^{{\rm ^{min}}}$
for a given gap height $h$ and a radius of curvature $\Re$.
[Minimum spark half-width vs. the gap height]Minimum spark half-width vs. gap height calculated for three different
radii of curvature: $\Re_{6}=0.1$ - red solid line, $\Re_{6}=0.5$
- green dashed line, and $\Re_{6}=1$ - blue dotted line.
On the other hand, we can estimate the acceleration potential $\Delta V$
(and thus the spark half-width $h_{\perp}^{^{{\rm N_{{\rm ph}}}}}$)
required to produce a specified number of photons $N_{{\rm ph}}^{^{{\rm CR}}}$
within a gap. Figure <ref> presents the dependence
of both $h_{\perp}^{^{{\rm min}}}$ and $h_{\perp}^{{\rm ^{N_{{\rm ph}}}}}$
on the gap height. As results from the Figure, the gap height in PSG-off
does not change drastically with $N_{{\rm ph}}^{^{{\rm CR}}}$, and
for these specific parameters of a pulsar it is $h\approx240\,{\rm m}$.
For historical reasons, hereafter unless stated otherwise, we will
use $N_{{\rm ph}}^{^{{\rm CR}}}=50$ to calculate the gap parameters
of the PSG-off mode. Note that in order to find the gap height, we
assume $h_{\perp}^{^{{\rm min}}}=h_{\perp}^{^{N_{{\rm ph}}}}$, which
results in a gap that allows both overheating of the entire spark
surface by backstreaming particles and the creation of the required
number of photons $N_{{\rm ph}}^{^{{\rm CR}}}$.
(show_solution_cr_psgoff, plot_psgoff_cr (or just this to use
files) $B_{14}=2.3$, $B_{{\rm d}}=0.02978$, $T_{6}=3.0$, $P=1.291578$,
$\Re_{6}=1.0$, $\alpha=60.7^{\circ}$)
[Dependence of a spark half-width on the gap height [PSG-off mode]]Dependence of a spark half-width on the gap height for the PSG-off
mode. The results were obtained using the following pulsar parameters:
$B_{14}=2.3$, $T_{6}=3.0$, $P=1.3\,{\rm s}$, $\Re_{6}=1.0$, $\alpha=60.7^{\circ}$.
In our calculations we use the algorithm presented in Figure <ref>
to find the gap height in the PSG-off mode for given pulsar parameters:
a pulsar period $P$, a pulsar inclination angle $\alpha$, a surface
magnetic field strength $B_{{\rm s}}$, and a curvature radius of
field lines $\Re$.
[Flowchart of the algorithm used to estimate the gap height [PSG-off
mode]]Flowchart of the algorithm used to estimate the gap height in the
PSG-off mode. The initial gap height from which we begin our calculations
is an arbitrary set to $h_{{\rm init.}}=10\,{\rm m}$, while the step
$\Delta h$ depends on the required accuracy. The number of $\gamma$-ray
photons created in a spark by a single primary particle is set to
$N_{{\rm ph}}^{{\rm ^{CR}}}=50$ (see text for more details).
Figure <ref> presents the result of finding
the gap height in the PSG-off mode for PSR B0943+10. The presented
solution corresponds to the magnetic field structure presented in
Section <ref>. The average radius of curvature in
the gap region is relatively high, $\Re_{6}=0.7$, hence the inclination
of the gap region. The polar gap conditions, the strength of magnetic
field $B_{14}=2.4$ ($R_{{\rm bb}}=17\,{\rm m}$) and the polar cap
temperature $T_{6}=3.0$ were restrained to follow the observed values
(see Table <ref>). The presented solution corresponds
to the following PSG parameters: gap height $h=166\,{\rm m}$, spark
half-width $h_{\perp}=1.9\,{\rm m}$, $\eta=1$ (fixed), $\kappa=7\times10^{-3}$,
$\gamma_{{\rm c}}=1.4\times10^{6}$. Note that the primary particles
will gain $\gamma_{{\rm max}}=1.9\times10^{6}$ as the CR efficiency
is not high enough to stop the acceleration.
(plot_cr), set_=318, ds=1e2
[Gap structure in the PSG-off mode [PSR B0943+10]]Gap structure in the PSG-off mode for PSR B0943+10. Filled columns
represent the locations and sizes of the active regions of sparks.
Here we assumed that the active region of a spark (the place where
acceleration is high enough to produce a cascade) has a size comparable
with the spark half-width. The iron ions extracted from the surface
(due to a high surface temperature) are represented by circle-plus
§.§ PSG-on mode
In the PSG-on mode, radiation of the surface just below the spark
is in quasi-equilibrium with the flux of backstreaming particles.
When the surface temperature rises, the density of iron ions increases,
thus resulting in a decrease in the potential drop, which in turn,
reduces the flux of backstreaming particles. On the other hand, when
the surface temperature decreases it entails the drop of iron ion
density and, consequently, an increase in the flux of backstreaming
particles. Thus the polar cap temperature is maintained slightly below
the critical value. This quasi-equilibrium state prevents the gap
breakdown due to surface overheating. However, a high multiplicity
in the PSG-on mode leads to a production of dense plasma. When the
density of the plasma $n_{p}\gg\eta n_{{\rm GJ}}$, the acceleration
potential drop will be completely screened due to charge separation.
Alongside the pulsar parameters the gap height in the PSG-on mode
also depends on the spark half-width $h_{\perp}$ and on the number
of scatterings by the first population of newly created particles
$N_{{\rm ph}}^{{\rm ^{ICS}}}$. For a sample of pulsars we can use
drift information to put constraints on the spark half-width (see
Section <ref>). Figure <ref>
presents the procedure of finding the gap height in the PSG-on mode
for the following pulsar parameters: a pulsar period $P$, a pulsar
inclination angle $\alpha$, a surface magnetic field strength $B_{{\rm s}}$,
a surface temperature $T_{{\rm s}}$, a curvature radius of magnetic
field lines $\Re$, and a spark half-width $h_{\perp}$. First we
use Equation <ref> to estimate the screening
factor $\eta$ which defines the electric field, and thus the particle
acceleration. Then we estimate the number of scatterings for a single
outflowing particle $N_{{\rm ph}}^{{\rm ^{pr}}}$ for the initial
gap height. The initial gap height from which we begin our calculations
is an arbitrary set to $h_{{\rm init.}}=10\,{\rm m}$. We track the
propagation of $\gamma$-photons produced by ICS on a primary particle
to find the location $L_{{\rm new}}$ where pairs are created. Then
we calculate their propagation through the acceleration region and
we estimate the number of scatterings by every newly created particle
of the first population $N_{{\rm ph}}^{{\rm ^{new}}}$. If the total
number of scatterings by the first population (including the primary
particle) is $N_{{\rm ph}}<N_{{\rm ph}}^{{\rm ^{ICS}}}$ , we resume
our calculations assuming a higher gap until the $N_{{\rm ph}}\geq N_{{\rm ph}}^{{\rm ^{ICS}}}$
is met.
[Flowchart of the algorithm used to estimate the gap height [PSG-on
mode]]Flowchart of algorithm used to estimate the gap height in PSG-on
mode for a given spark half-width (see text for more details).
As a result we obtain the gap parameters: the gap height $h$, the
screening factor $\eta$, the characteristic Lorentz factor of a particle
at the moment of ICS photon emission $\gamma_{{\rm c}}$, the maximum
value of the Lorentz factor $\gamma_{{\rm max}}$, and the characteristic
Lorentz factor of iron ions $\gamma_{{\rm i}}$. In our calculations,
if not stated otherwise, we use $N_{{\rm ph}}^{^{{\rm ICS}}}=25$
to calculate the gap parameters of the PSG-on mode. Note that in this
approximation we take into account only the first population of newly
created particles. In fact, the avalanche nature of the ICS-dominated
gap will result in a much higher multiplicity than in the PSG-off
mode $M_{{\rm ICS}}\gg M_{{\rm CR}}$. For details of particle/photon
propagation, see Chapter <ref>. Subpulse drift observations
are available only for a few X-ray pulsars with the hot spot component.
Thus, to find the approximate gap parameters for pulsars without the
predicted spark half-width we use $h_{\perp}=2\,{\rm {\rm m}}$.
Figure <ref> presents the result of finding
the gap height in the PSG-on mode for PSR B0943+10. In this model
the gap parameters, such as the magnetic field strength $B_{{\rm s}}$
and the surface temperature $T_{{\rm s}}$, were restrained to follow
the observed values (see Table <ref>). The result
was obtained for the non-dipolar structure of a surface magnetic field
presented in Section <ref> and for the predicted
value of a spark half-width $h_{\perp}\approx2\,{\rm m}$ (see Table
<ref>). The height required to produce $N_{{\rm ph}}^{{\rm ^{ICS}}}=25$
photons by the first population of particles was estimated as $h\approx92\,{\rm m}$.
Other gap parameters for this solution can be found in Table <ref>.
(plot_ics, 2, old in 0)
[Gap structure in the PSG-on mode [PSR B0943+10]]Gap structure in the PSG-on mode for PSR B0943+10. Filled columns
represent the locations and sizes of the active regions of sparks.
Here we assumed that the active region of a spark (the place where
acceleration is high enough to produce a cascade) has a size comparable
with the spark half-width. The iron ions extracted from the surface
(due to a high surface temperature) are represented by circle-plus
symbols. Note that iron ions are present in both the non-active and
active regions. The density of ions in the non-active regions is so
high that it prevents cascade formation of pairs.
§.§ Results
In Table <ref> we present the results of finding
the gap height for the sample of pulsars. For the PSG-on mode we show
the estimated PSG parameters found using the predicted spark half-width
and the spark half-width $h_{\perp}=2\,{\rm m}$. The only exception
is Geminga (PSR J0633+1746), for which drift information is not available
and we can only present calculations for $h_{\perp}=2\,{\rm m}$.
For PSR B0628-28 the predicted spark half-width is large ($h_{\perp}=3.9\,{\rm m}$),
which entails a high acceleration potential. For such wide sparks
it is not possible to find the PSG-on solution with the required number
of scatterings $N_{{\rm ph}}^{^{{\rm ICS}}}$. We believe that for
this specific pulsar the predicted spark half-width is overestimated.
Actually, if a spark is narrower ($h_{\perp}=2\,{\rm m}$), it can
operate in the PSG-on mode (see Table <ref>). This
result may suggest that for this specific pulsar the parameters of
the subpulse phenomenon could be overestimated (e.g. due to aliasing).
On the other hand, X-ray observations of Geminga suggest a relatively
low temperature of the hot spot
($T_{{\rm s}}\approx1.9\,{\rm MK}$). The low density of the background
photons requires the formation of narrow sparks ($h_{{\rm \perp}}=1\,{\rm m}$)
to allow the gap to operate in the PSG-on mode. We believe that the
relatively large hot spot ($R_{{\rm pc}}=44.5\,{\rm m}$) of Geminga
causes the width of the sparks to grow so fast that it can operate
only in the PSG-off mode. We believe that this can explain the very
weak radio luminosity of the Geminga pulsar.
Results from results.log. files in: 403 (PSR B0628-28), 373
(PSR J0633+1746), 383 (PSR B0834+06), 315 (PSR
B0943+10), 350 (PSR B0950+08), 341 (PSR B1133+16),
322 (PSR B1929+10)
[Estimated parameters of PSG for the sample of pulsars]Estimated parameters of PSG for the sample of pulsars. The conditions
in the polar cap region: surface temperature, magnetic field strength,
polar cap radius, and curvature radius of the field lines are given
the in headers next to the pulsar name. The individual columns are
as follows: (1) PSG mode (see Section <ref>),
(2) Gap height, (3) Spark half-width, (4) Screening factor, (5) Overheating
parameter, (6) Characteristic Lorentz factor of scattering particles
, (7) Maximum Lorentz factor of primary particles, (8) Lorentz factor
of iron ions (if they are relativistic), (9) Particle mean free path,
and (10) Photon mean free path. The results are presented for two
different gap breakdown scenarios: the PSG-off and PSG-on modes (see
Section <ref> for more details).
$^{a}$ The modes correspond to calculations using the predicted
spark half-width (see Table <ref>)
$^{b}$ The modes correspond to calculations with a spark half-width
$h_{\perp}=2\,{\rm m}$
mode $h$ $h_{\perp}$ $\eta$ $\kappa$ $\gamma_{{\rm c}}$ $\gamma_{{\rm max}}$ $\gamma_{{\rm i}}$ $l_{{\rm p}}$ $l_{{\rm ph}}$$\left(N_{{\rm ph}}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$10|c|10|c|PSR B0628-28$T_{6}=2.8$ $B_{14}=2.2$
$R_{{\rm pc}}=21.3\,{\rm m}$ $\Re_{6}=0.6$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $198.3$ $3.2$ – $0.007$ $1.3\times10^{6}$ $1.6\times10^{6}$ – $1.4$ $58.9$on$^{a}$ – $3.6$ – – – – – – –on$^{b}$ $78.6$ $2.0$ $0.15$ – $6.1\times10^{3}$ $8.9\times10^{4}$ $23$ $1.9$ $1.3$10|r|Continued on next page
Table <ref> - continued from previous page
mode $h$ $h_{\perp}$ $\eta$ $\kappa$ $\gamma_{{\rm c}}$ $\gamma_{{\rm max}}$ $\gamma_{{\rm i}}$ $l_{{\rm p}}$ $l_{{\rm ph}}$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$10|c|10|c|PSR B0628-28$T_{6}=2.8$ $B_{14}=2.2$
$R_{{\rm pc}}=21.3\,{\rm m}$ $\Re_{6}=0.6$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $198.3$ $3.2$ – $0.007$ $1.3\times10^{6}$ $1.6\times10^{6}$ – $1.4$ $58.9$on$^{a}$ – $3.6$ – – – – – – –on$^{b}$ $78.6$ $2.0$ $0.15$ – $6.1\times10^{3}$ $8.9\times10^{4}$ $23$ $1.9$ $1.3$ 1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR J0633+1746$T_{6}=1.9$ $B_{14}=1.5$
$R_{{\rm pc}}=44.5\,{\rm m}$ $\Re_{6}=2.1$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $252.1$ $1.5$ – $0.0002$ $2.9\times10^{6}$ $3.5\times10^{6}$ – $2.2$ $66.6$on$^{b}$ – $2.0$ – – – – – – – 1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR B0834+06$T_{6}=2.4$ $B_{14}=1.9$
$R_{{\rm pc}}=22.7\,{\rm m}$ $\Re_{6}=0.6$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $172.5$ $3.2$ – $0.0027$ $1.2\times10^{6}$ $1.4\times10^{6}$ – $1.3$ $52.1$on$^{a}$ $82.0$ $1.8$ $0.12$ – $5.3\times10^{3}$ $7.0\times10^{4}$ $18$ $2.3$ $1.5$on$^{b}$ $102.9$ $2.0$ $0.11$ – $4.9\times10^{3}$ $7.8\times10^{4}$ $20$ $2.4$ $1.8$ 1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR B0943+10$T_{6}=3.1$ $B_{14}=2.4$
$R_{{\rm pc}}=17.6\,{\rm m}$ $\Re_{6}=0.7$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $168.3$ $1.9$ – $0.0068$ $1.6\times10^{6}$ $2.0\times10^{6}$ – $1.4$ $46.5$on$^{a,b}$ $71.6$ $2.0$ $0.1$ – $8.8\times10^{3}$ $2.5\times10^{5}$ $63$ $1.1$ $1.1$ 10|r|Continued on next page
Table <ref> - continued from previous page
mode $h$ $h_{\perp}$ $\eta$ $\kappa$ $\gamma_{{\rm c}}$ $\gamma_{{\rm max}}$ $\gamma_{{\rm i}}$ $l_{{\rm p}}$ $l_{{\rm ph}}$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR B0950+08 $T_{6}=2.6$ $B_{14}=2.0$
$R_{{\rm pc}}=14.0\,{\rm m}$ $\Re_{6}=0.8$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $172.8$ $1.9$ – $0.0009$ $1.7\times10^{6}$ $2.0\times10^{6}$ – $1.6$ $47.9$on$^{a}$ $16.9$ $0.7$ $0.09$ – $3.9\times10^{3}$ $2.3\times10^{4}$ $6$ $1.4$ $0.6$on$^{b}$ $61.7$ $2.0$ $0.03$ – $5.1\times10^{3}$ $6.6\times10^{4}$ $17$ $1.8$ $1.6$ 1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR B1133+16$T_{6}=2.9$ $B_{14}=2.3$
$R_{{\rm pc}}=17.9\,{\rm m}$ $\Re_{6}=0.6$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $167.4$ $2.4$ – $0.0076$ $1.4\times10^{6}$ $1.7\times10^{6}$ – $1.3$ $48.4$on$^{a}$ $95.9$ $2.9$ $0.08$ – $7.0\times10^{3}$ $1.9\times10^{5}$ $47$ $1.2$ $1.3$on$^{b}$ $54.3$ $2.0$ $0.11$ – $7.9\times10^{3}$ $1.3\times10^{5}$ $33$ $1.4$ $1.1$ 1|c 1c 1c 1c 1c 1c 1c 1c 1c 10|c|PSR B1929+10$T_{6}=3.0$ $B_{14}=2.4$
$R_{{\rm pc}}=20\,{\rm m}$ $\Re_{6}=0.6$1|c 1c 1c 1c 1c 1c 1c 1c 1c off $112.7$ $1.0$ – $0.0012$ $1.8\times10^{6}$ $2.1\times10^{6}$ – $1.1$ $28.0$on$^{a}$ $50.5$ $1.6$ $0.02$ – $9.8\times10^{3}$ $1.2\times10^{5}$ $31$ $1.6$ $1.0$on$^{b}$ $75.1$ $2.0$ $0.02$ – $8.4\times10^{3}$ $1.5\times10^{5}$ $39$ $1.5$ $1.5$
§ PSG MODEL PARAMETERS
We can distinguish two types of PSG parameters: observed and derived.
As we have mentioned above, in some cases when X-ray observations
are available we can directly estimate the surface magnetic field
$B_{{\rm s}}$. On the one hand, $B_{{\rm s}}$ can be calculated
using the size of the hot spot $A_{{\rm bb}}$, and on the other hand
we can find $B_{{\rm s}}$ by using the estimation of the critical
temperature and the assumption that $T_{{\rm s}}=T_{{\rm crit}}$.
One of the most important requirements for the PSG model is that these
two estimations should coincide with each other. As is clear from
Figure <ref>, in most cases when the hot spot
parameters are available this requirement is fulfilled. Thus, we can
assume that the characteristic values of $B_{{\rm s}}$ vary in the
range of $(1-4)\times10^{14}\,{\rm G}$, which corresponds to the
critical surface temperature in the range of $(1.3-5)\times10^{6}\,{\rm K}$
(see Table <ref>). By using these values we can
estimate the derived parameters of PSG, such as the gap height $h$,
the screening factor $\eta$ (or the overheating parameter $\kappa$
in the PSG-off mode) and the characteristic Lorentz factor of primary
particles $\gamma_{{\rm c}}$. Let us note that these parameters also
depend on the curvature radius of the magnetic field lines $\Re$.
The curvature can be neither observed nor derived, but modelling of
the surface magnetic field (see Chapter <ref>) indicates
that the curvature radius varies in the range of $(0.1-10)\times10^{6}$
cm. Below we will discuss the influence of pulsar parameters, such
as the magnetic field, the curvature of field lines and the period
on derived PSG parameters.
§.§ Influence of the magnetic field
The conditions in PSG are mainly defined by the surface magnetic field.
In Figure <ref>, panel (a) we present the dependence
of the gap height on the surface magnetic field calculated according
to the approach described in Section <ref>.
It is clear that in the PSG-off mode the gap height decreases as the
surface magnetic field increases. In the PSG-on mode, on the other
hand, the gap height shows a minimum at a specific value of the magnetic
field strength (for a given pulsar's parameters it is $B_{14}\approx3$).
This behaviour is the result of an increasing potential acceleration
drop with an increasing surface magnetic field. When the magnetic
field strength exceeds the optimal value, which corresponds to acceleration
when ICS is more effective, the increase in the acceleration potential
results in less effective scattering. Panel (b) shows the dependence
of the screening factor (or the overheating parameter $\kappa$ in
the PSG-off mode) on the surface magnetic field. We can see that for
stronger magnetic fields both $\eta$ and $\kappa$ increase, which
means that: (1) the density of heavy ions above the polar cap in the
PSG-on mode decreases, (2) the density of particles required to overheat
(and thus to close) the polar cap increases. Let us note that the
surface temperature $T_{{\rm s}}$ stays very near to the critical
temperature $T_{{\rm crit}}$, which is shown on the top axis of the
Figures. In panel (c) the red, solid and dotted lines correspond to
characteristic and maximum Lorentz factors ($\gamma_{{\rm c}}$, $\gamma_{{\rm max}}$)
in the PSG-on mode, while the blue, dashed and dashed-dotted lines
correspond to $\gamma_{{\rm c}}$ and $\gamma_{{\rm max}}$ in the
PSG-off mode. We see that especially for the PSG-off mode $\gamma_{{\rm c}}$
does not depend on the magnetic field strength. Note also that in
the PSG-off mode (CR-dominated gap), the characteristic Lorentz factor
(the Lorentz factor for which most of the gamma photons are produced)
slightly differs from the maximum value, $\gamma_{{\rm c}}\approx\gamma_{{\rm max}}$.
On the other hand, in the PSG-on mode $\gamma_{{\rm c}}\ll\gamma_{{\rm max}}$,
which reflects the fact that most of the scatterings take place in
the bottom part of the gap.
(show_b14, show_b14_cr)
[Dependence of the PSG model parameters on the surface magnetic field.]Dependence of the gap height (panel a), the screening factor or the
overheating parameter (panel b), and the particle Lorentz factor (panel
c) on the surface magnetic field. Solid red lines correspond to the
PSG-on mode (ICS-dominated gaps) while dashed blue lines correspond
to the PSG-off mode (CR-dominated gaps). Calculations were performed
using the following parameters: $P=0.23$, $\Re_{6}=0.6$, $B_{{\rm d}}=1.2\times10^{12}\,{\rm G}$,
and $\alpha=36^{\circ}$. The actual polar cap radius was calculated
separately for a given surface magnetic field as $R_{{\rm pc}}=R_{{\rm dp}}\sqrt{B_{{\rm d}}/B_{{\rm s}}}$.
In panel (c) the red solid and dotted lines correspond to characteristic
and maximum Lorentz factors ($\gamma_{{\rm c}}$, $\gamma_{{\rm max}}$)
in the PSG-on mode while blue dashed and dashed-dotted lines correspond
to $\gamma_{{\rm c}}$ and $\gamma_{{\rm max}}$ in the PSG-off mode.
Corresponding critical temperature is shown on top axis of the figures.
§.§ Influence of the curvature radius
The curvature of the magnetic field lines significantly affects the
gap height in the PSG-off mode (see Figure <ref>, panel
a). In the case of the CR-dominated gap, the curvature of the magnetic
field lines affects not only the photons' mean free path (for higher
curvature the magnetic field will absorb photons faster), but also
the particle mean free path and, more importantly, the energy of photons
generated in the gap region. The higher energy of photons further
reduces the photon mean free path, thus resulting in lower heights
of the PSG. In contrast, the gap height in the PSG-on mode is only
slightly affected by changes in the curvature of the magnetic field
lines. In this case the most important parameter which determines
the cascade properties is the primary particle mean free path which
does not depend on the curvature of the magnetic field lines.
The overheating parameter in the PSG-off mode inversely depends on
the radius of curvature of the magnetic field lines (see Figure <ref>,
panel b). The higher the curvature, the higher the overheating parameter,
which means that the sparks are narrower. This is consistent with
the expectation that for a higher curvature of the magnetic field
lines, the gap breakdown is easier to develop and takes place before
the sparks manage to grow in width. On the other hand, the screening
factor in the PSG-off mode does not depend on the curvature of the
magnetic field lines.
With an increasing radius of curvature the Lorentz factor of primary
particles (both $\gamma_{{\rm c}}$ and $\gamma_{{\rm max}}$) required
to close the gap in the PSG-off mode also increases. This reflects
the fact that in order to produce a sufficient number of photons in
the gap region, the primary particles should be accelerated to higher
energies (if the curvature is lower). Higher energies of the primary
particles will increase the emitted $\gamma$-photon energy, thereby
they will partly inhibit the growth of the photon mean free path due
to the lower curvature. As mentioned above, the gap height in the
PSG-on mode very weakly depends on the photon mean free path, thus
both $\gamma_{{\rm c}}$ and $\gamma_{{\rm max}}$ are not affected
by the increase in the radius of curvature.
[Dependence of the PSG model parameters on the curvature radius of
magnetic field lines.]Dependence of the gap height (panel a), the screening factor or the
overheating parameter (panel b), and the particle Lorentz factor (panel
c) on the curvature radius of magnetic field lines. Calculations were
performed using the following parameters: $P=0.23$, $B_{{\rm d}}=1.2\times10^{12}\,{\rm G}$,
$B_{{\rm s}}=2.4\times10^{14}\,{\rm G}$, $\alpha=36^{\circ}$. For
a more detailed description see Figure <ref>.
§.§ Influence of the pulsar period
As we can see from Figure <ref>, panel (a) and panel
(c), in the PSG-on mode the gap height and the Lorentz factor of primary
particles do not depend on the pulsar period. The increase in the
screening factor (see Figure <ref>b) compensates the
increase in the acceleration potential drop (see Equation <ref>).
Thus the particles in the gap region are accelerated in the same way
independently of the pulsar period. On the other hand, the gap height
in the PSG-off mode increases with the increasing pulsar period. This
reflects the fact that in the PSG-off mode the acceleration potential,
and hence $\gamma_{{\rm c}}$ and $\gamma_{{\rm max}}$, decreases
with longer periods (see Equation <ref>). Longer
pulsar periods entail an increase in the screening factor (in the
PSG-on mode, see Equation <ref>) and in the overheating
parameter (in the PSG-off mode). Note that for periods longer than
some specific value (for a given pulsar's parameters it is $P_{{\rm max}}\approx9\,{\rm s}$),
the screening factor in the PSG-on mode would exceed unity. This means
that the PSG-on mode cannot be responsible for the gap breakdown for
pulsars with such long periods.
[Dependence of the PSG model parameters on the pulsar period.]Dependence of the gap height (panel a), the screening factor or a
overheating parameter (panel b), and the particle Lorentz factor (panel
c) on the pulsar period. Calculations were performed using the following
parameters: $P=0.23$, $B_{{\rm d}}=1.2\times10^{12}\,{\rm G}$, $B_{{\rm s}}=2.4\times10^{14}\,{\rm G}$,
$\alpha=36^{\circ}$, $\Re_{6}=0.6$. The actual polar cap radius
was calculated separately for a given pulsar period as $R_{{\rm pc}}=R_{{\rm dp}}\sqrt{B_{{\rm d}}/B_{{\rm s}}}$,
where $R_{{\rm dp}}=\sqrt{2\pi R^{3}/\left(cP\right)}$. For a more
detailed description see Figure <ref>.
§ DRIFT MODEL
The existence of IAR in general causes a rotation of the plasma relative
to the NS, as the charge density differs from the Goldreich-Julian
co-rotational density. The power spectrum of radio emission must have
a feature due to this plasma rotation. This feature is indeed observed
and is called the drifting subpulse phenomenon.
§.§ Aligned pulsars
An explanation for drifting subpulses was offered by 156
as being due to a rotating carousel of sub-beams within a hollow emission
cone. According to this model a pair cascades may not occur simultaneously
across the whole polar cap but is localised in the form of discharges
of small regions in the polar gap. Such sparks may produce plasma
columns that stream into the magnetosphere to produce the observed
radio emission. The location of the discharges on the polar cap determines
the geometrical pattern of instantaneous subpulses within a pulsar's
integrated pulse profile.
In the PSG model the stable pattern of subpulses is due to heating
of the inactive part of the spark (the place where no cascade forms
due to a low acceleration potential) by all the neighbouring discharges.
The lifetime of a single spark is very short. On the other hand, an
inactive region is continuously heated by all the neighbouring sparks.
Even when one of them dies, the temperature is still high enough (high
ion density) to prevent spark formation in this region. As the discharges
do not exchange information (they are not synchronised) and their
lifetime is very small, the geometrical pattern of sparks on the polar
cap should be stable.
For pulsars with an aligned magnetic and rotation axis the sparks
circulate around the rotation axis. Note that this circulation is
not related with the magnetic axis but with the direction of the co-rotational
velocity. Namely, the drift velocity is opposed to the co-rotational
We can calculate the drift velocity of an aligned pulsar using the
following approximation (see Figure <ref>)
\begin{equation}
v_{{\rm dr}}\approx\frac{2\pi R_{{\rm pc}}}{PP_{3}}\frac{\beta}{\rho}\frac{P_{2}^{\circ}}{360^{\circ}},
\end{equation}
where $R_{{\rm pc}}$ is the actual polar cap size, $P_{2}^{\circ}$
is the characteristic spacing between subpulses in the pulse longitude,
$P_{3}$ is the period at which a pattern of subpulses crosses the
pulse window (in units of the pulsar period), $\beta$ is the impact
angle, and $\rho$ is the opening angle.
[Top view of a polar cap region of an aligned pulsar]Top view of a polar cap region of an aligned pulsar. Small circles
represent sparks, while the red line corresponds to the line of sight.
If we neglect the transition from a non-dipolar structure of the magnetic
field on the stellar surface to a dipolar structure in the region
where radio emission is produced, we can assume that the observed
subpulse separation $P_{2}^{\circ}$ also describes spark separation
$\varrho_{s}$ (angular separation between the adjacent sparks on
the polar cap).
In such an approximation the assumption that only half of the spark
is active can be written as
\begin{equation}
\frac{P_{2}^{\circ}}{360^{\circ}}\approx\frac{2h_{\perp}}{2\pi R_{{\rm pc}}\frac{\beta}{\rho}}.
\end{equation}
Finally, we can define the observed drift velocity of aligned pulsars
\begin{equation}
\end{equation}
§.§ Non-aligned pulsars
Most observed pulsars are non-aligned rotators. It is very common
to apply the carousel model to interpret observations of the drifting
subpulses of non-aligned pulsars [156]. Despite the
fact that the carousel model can explain some properties of subpulses
(for example the change in intensity), we believe that this model
is not suitable for describing the spark's behaviour on the polar
cap. There is no physical reason for a spark to circulate around the
magnetic axis. The circulation in aligned pulsars is caused by a lack
of coronation with respect to the rotation axis. For non-aligned pulsars,
the co-rotation velocity in the polar cap region has more or less
the same direction: what is more, if we assume circulation around
the magnetic axis we will get plasma with a velocity that is higher
than the co-rotational velocity, which is difficult to explain in
a region where the charge density is lower than the co-rotational
As in our model, the drift is caused by a lack of charge in IAR, thus
the plasma should drift in approximately the same direction, i.e.
in the direction opposite to the co-rotation velocity. We believe
that the change in subpulse intensity is caused by the observation
of a different part of a spark and/or different conditions across
the polar cap at which the spark is formed (e.g. magnetic field strength,
curvature of the magnetic field lines, background photon flux).
For pulsars with a relatively high inclination angle $\alpha$ we
can calculate the drift velocity using the following approximation
\begin{equation}
v_{dr}\approx\frac{2R_{{\rm pc}}\frac{W}{W_{\beta0}}}{PP_{3}}\frac{P_{2}^{\circ}}{W}\approx\frac{2R_{{\rm pc}}}{PP_{3}}\frac{P_{2}^{\circ}}{W_{\beta0}},\label{eq:psg.vdr_nona_approx}
\end{equation}
where $W$ is the profile width and $W_{\beta0}\approx W/\sqrt{1-\left(\frac{\beta}{\rho}\right)^{2}}$
is the profile width calculated assuming $\beta=0$ (see Figure <ref>).
Using the assumption that only half of the spark is active, we can
write that
\begin{equation}
\frac{P_{2}^{\circ}}{W}\approx\frac{2h_{\perp}}{2R_{{\rm pc}}\frac{W}{W_{\beta0}}}\longrightarrow\frac{P_{2}^{\circ}}{W_{\beta0}}\approx\frac{h_{\perp}}{R_{{\rm pc}}},\label{eq:psg.hperp_wb0}
\end{equation}
and the drift velocity
\begin{equation}
\end{equation}
The spark half-width can be calculated using Equation <ref>
as follows
\begin{equation}
h_{\perp}=R_{{\rm pc}}\frac{P_{2}^{\circ}}{W_{\beta0}}.\label{eq:psg.hperp_p2}
\end{equation}
[Top view of the polar cap region in the case of a non-aligned pulsar]Top view of the polar cap region in the case of a non-aligned pulsar.
Small circles represent sparks, the red line corresponds to the line
of sight. In general, the observed subpulse separation $P_{2}^{\circ}$
does not describe the actual spark separation $\varrho_{s}$ (the
angular separation between the adjacent sparks on the polar cap).
In order to calculate the distance between the sparks we use an approximation
from Equation <ref>.
§.§ Screening factor
In our model the drift is caused by a lack of charge in IAR, thus
we can write the equation for the drift velocity as follows
\begin{equation}
{\bf v_{\perp}={\bf v}}_{{\rm dr}}=\frac{c{\bf \Delta E}\times{\bf B}}{B^{2}},
\end{equation}
where $\Delta{\bf E}$ is the electric field caused by the difference
of an actual charge density from the Goldreich-Julian co-rotational
density. We can a use calculation of the circulation of an electric
field, Equations <ref> and <ref>,
to find the dependence of the drift velocity on the screening factor:
\begin{equation}
v_{{\rm dr}}=c\frac{E_{\theta}B_{r}}{B_{r}^{2}}=\frac{4\pi\eta h_{\perp}\cos\alpha}{P}.\label{eq:psg.vdr_shielding}
\end{equation}
Finally, by using Equations <ref> and <ref>
we can find the dependence of the screening factor on the observed
drift parameters
\begin{equation}
\eta=\frac{1}{2\pi P_{3}\cos\alpha}.\label{eq:psg.eta_p3}
\end{equation}
§.§ Profile width and subpulse separation
The key parameters in the above calculations are the pulse width $W$
(or $W_{\beta0}$), the characteristic spacing between subpulses $P_{2}^{\circ}$,
and the period at which a pattern of subpulses crosses the pulse window
$P_{3}$. Of these three only $P_{3}$ is easy to apply, both $W$
and $P_{2}^{\circ}$ need serious study before they can be used.
In general, the profile width depends on the frequency at which we
observe the pulsar, and most normal pulsars show a systematic increase
in pulse width and the separation of profile components when observed
at lower frequencies. The model known as radius-to-frequency mapping
explains this effect as a direct consequence of the emission at higher
frequencies being produced closer to the neutron star surface than
at lower frequencies. For this reason both the pulse width and the
spacing between subpulses should be measured at the same frequency.
Note that $P_{3}$ is not affected by this effect since its determination
involves analyses of many pulses and does not depend on the pulse
width. The observed pulse width $W$, measured in longitude of rotation,
can be calculated by applying simple spherical geometry [60]:
\begin{equation}
\sin^{2}\frac{W}{4}=\frac{\sin^{2}\left(\rho/2\right)-\sin^{2}\left(\beta/2\right)}{\sin\alpha\sin\left(\alpha+\beta\right)}.
\end{equation}
In the above calculations we are using the $W_{\beta0}\approx W/\sqrt{1-\left(\frac{\beta}{\rho}\right)^{2}}$approximation,
where $W_{\beta0}$ is the pulse width calculated assuming $\beta=0$.
In the first approximation we can assume that $W_{\beta0}$ corresponds
to the distance $2R_{{\rm pc}}$ at the polar cap which, is valid
for non-aligned pulsars with a relatively high inclination angle.
A more accurate value can be calculated using formulas presented in
60. The running polar coordinates along the line of
sight trajectory can be expressed in the form
\begin{equation}
\rho\left(\varphi\right)=2\arcsin\left(\sqrt{\sin^{2}\frac{\varphi}{2}\sin\alpha\sin\left(\alpha+\beta\right)+\sin^{2}\frac{\beta}{2}}\right),
\end{equation}
\begin{equation}
\sigma\left(\varphi\right)=\arctan\left(\frac{\sin\varphi\sin\alpha\sin\left(\alpha+\beta\right)}{\cos\left(\alpha+\beta\right)-\cos\alpha\cos\rho\left(\varphi\right)}\right).
\end{equation}
In numerical calculations of $\sigma\left(\varphi\right)$ it is convenient
to use the “${\rm atan2}$” function which takes into account
the signs of both components and places the angle in the correct quadrant
(see the footnote on page fn:model.atan2). Figure <ref>
presents the geometry of the emission region for pulsars with available
radio observations of the subpulse drift and X-ray observations of
the hot spot. By knowing the actual polar cap radius $R_{{\rm pc}}$
we can determine the transverse size of the region responsible for
the generation of plasma clouds in IAR (the spark half-width).
[Top view of the polar cap region of pulsars with radio and X-ray observations]Top view of the polar cap region of pulsars with radio drift observations
and X-ray hot spot radiation. Red lines correspond to the line of
sight while green dashed lines correspond to the theoretical lines
of sight calculated with an assumption that $\beta=0^{\circ}$. The
geometry of pulsars can be found in Table <ref>.
In our model the motion of sparks and the progressively different
positions of the associated plasma columns are responsible for the
observed drift of subpulses. For some pulsars it is possible to measure
directly the subpulse separation using a single pulse. In most calculations
it is assumed that the observed subpulses correspond to the adjacent
sparks. In general, this is not necessarily true. The distribution
of sparks on the polar cap is unknown and it is very likely that the
line of sight does not cross the adjacent sparks but it omits some
sparks in between. Therefore, the observed value of $P_{2}^{\circ}$
should be considered rather as an upper limit for spark separation.
Furthermore, for many pulsars the observed value $P_{2}^{\circ}>W$,
which means that it is not related to any structure at the polar cap
but that it corresponds to some other periodicity. We can use Equations
<ref> and <ref> to calculate the
spark half-width as follows
\begin{equation}
\end{equation}
Finally, using Equation <ref> we can determine the
predicted value of the subpulse separation
\begin{equation}
\tilde{P}_{2}^{\circ}\approx\frac{26.2\left(B_{14}^{1.1}+0.3\right)^{2}PP_{3}\sqrt{\left|\cos\alpha\right|}}{B_{14}R_{{\rm pc}}}W_{\beta0}.
\end{equation}
§.§ Heating efficiency
The spin-down energy loss is
\begin{equation}
L_{{\rm SD}}=3.9\times10^{31}\frac{\dot{P}_{-15}}{P^{3}}.
\end{equation}
We can use Equations <ref>, <ref>
and <ref> to calculate the dependence of the acceleration
potential drop on the parameters of drifting subpulses:
\begin{equation}
\Delta V\approx2.824\times10^{10}\left(\frac{\dot{P}_{-15}}{P^{3}}\right)^{0.5}\frac{1}{P_{3}}\left(\frac{P_{2}^{\circ}}{W_{\beta0}}\right)^{2}.\label{eq:psg.potential_drop_radio}
\end{equation}
The power of heating by backstreaming particles can be calculated
as follows
\begin{equation}
L_{{\rm heat}}=\eta n_{{\rm GJ}}c\left(\Delta Ve\right)\pi R_{{\rm pc}}^{2}.\label{eq:psg.l_heat}
\end{equation}
The number density of the Goldreich-Julian co-rotational charge can
be calculated using
\begin{equation}
n_{{\rm GJ}}=\frac{{\bf \Omega}\cdot{\bf B_{{\rm s}}}}{2\pi ce},
\end{equation}
where $\Omega=2\pi/P$ is an angular velocity, $B_{{\rm s}}=bB_{{\rm d}}$
is the surface magnetic field,
$b=R_{dp}^{2}/R_{pc}^{2}$, $B_{{\rm d}}=2.02\times10^{12}\sqrt{P\dot{P}_{-15}}\,{\rm G}$,
and $R_{{\rm dp}}=\sqrt{2\pi R^{3}/\left(cP\right)}\approx1.45\times10^{4}P^{-0.5}$.
The Goldreich-Julian density in terms of observed parameters can be
written as
\begin{equation}
n_{{\rm GJ}}=2.9\times10^{19}\left(P^{-3}\dot{P}_{-15}\right)^{1/2}\frac{\cos\alpha}{R_{{\rm pc}}^{2}}.\label{eq:psg.ngj_radio}
\end{equation}
Finally, using Equations <ref>, <ref>
and <ref> we can estimate the dependence of the
heating power and thus the X-ray luminosity of the hot spot radiation
on the parameters of radio observations as follows
\begin{equation}
L_{{\rm heat}}=L_{{\rm X}}=6\times10^{30}\left(\frac{\dot{P}_{-15}}{P^{3}}\right)\left(\frac{1}{P_{3}}\frac{P_{2}^{\circ}}{W_{\beta0}}\right)^{2}.
\end{equation}
The heating efficiency by the backstreaming particles can be calculated
\begin{equation}
\xi_{_{{\rm heat}}}=\frac{L_{{\rm heat}}}{L_{{\rm SD}}}=0.15\left(\frac{1}{P_{3}}\frac{P_{2}^{\circ}}{W_{\beta0}}\right)^{2}.
\end{equation}
§.§ Ion luminosity
In the PSG-on mode the bulk of energy is transferred to the iron ions
which shield the acceleration potential drop. Similar as for the backstreaming
particles, we can estimate the power of ion acceleration as
\begin{equation}
L_{{\rm ion}}=\left(1-\eta\right)n_{{\rm GJ}}c\left(\Delta Vq_{{\rm ion}}\right)\pi R_{{\rm pc}}^{2},\label{eq:psg.l_ion}
\end{equation}
where $q_{{\rm ion}}=26e=1.25\times10^{-8}\,{\rm erg^{0.5}cm^{0.5}}$
is the ion charge. Using the same approach as for electron, we can
calculate the dependence of energy transformed to the ions per second
on the parameters of the radio observations as follows
\begin{equation}
L_{{\rm ion}}=9.75\times10^{32}\left(1-\eta\right)\frac{\dot{P}_{-15}}{P^{3}P_{3}}\left(\frac{P_{2}^{\circ}}{W_{\beta0}}\right)^{2}\cos\alpha.
\end{equation}
It is clearly visible that if the screening factor is low $\eta\ll1$,
most of the energy in IAR is transferred to the iron ions. Using Equations
<ref> and <ref> we can show that
\begin{equation}
\frac{L_{{\rm ion}}}{L_{{\rm heat}}}=\frac{26\left(1-\eta\right)}{\eta}\approx\frac{26}{\eta}.
\end{equation}
Finally, the ion acceleration efficiency can be calculated as
\begin{equation}
\xi_{{\rm _{ion}}}=\frac{L_{{\rm ion}}}{L_{{\rm SD}}}\approx25\frac{1}{P_{3}}\left(\frac{P_{2}^{\circ}}{W_{\beta0}}\right)^{2}\cos\alpha.
\end{equation}
It may seem that ion luminosity exceeds the spin-down luminosity,
but note that both $P_{3}>1$ and $P_{2}^{\circ}<W_{\beta0}$. The
predicted values of heating efficiency $\xi_{{\rm heat}}$ and ion
acceleration efficiency $\xi_{{\rm ion}}$ are presented in the next
§.§ Observations
In this section we confront the values of the subpulse drift and X-ray
radiation as estimated by other authors with predicted values estimated
using the approach presented above. In Table <ref>,
alongside our predicted value of $\tilde{P}_{2}^{\circ}$ we present
two other estimates: (1) the subpulse separation estimated using the
carousel model developed by 156, $P_{2,{\rm RS}}^{\circ}$;
(2) the subpulse separation found using the analysis of the Longitude-Resolved
Fluctuation Spectrum [8] and the integrated Two-Dimensional
Fluctuation Spectrum [54], performed by 180,
$P_{2,{\rm W}}^{\circ}$. We have found that the subpulse separation
estimated using the fluctuations spectra is overestimated (in most
cases $P_{2,{\rm W}}^{\circ}>W$). By definition $P_{2}^{\circ}$
should correspond to the structure within a single pulse, thus if
the geometry is not extreme ($\varrho>1^{\circ}$) it should comply
with $P_{2}^{\circ}\leq W$. For this specific sample of pulsars $P_{2,{\rm W}}^{\circ}$
should not be interpreted as the actual subpulse separation. On the
other hand, $\tilde{P}_{2}^{\circ}$ is in good agreement with $P_{2,{\rm RS}}^{\circ}$.
The predicted values $\tilde{P}_{2}^{\circ}$ for B0834+06 and B0943+10
suggest that $P_{2,{\rm RS}}^{\circ}$ for those pulsars could be
overestimated due to the aliasing phenomenon ($P_{2,{\rm RS}}^{\circ}\approx2\tilde{P}_{2}^{\circ}$).
For B1929+10 we do not list $P_{2,{\rm RS}}^{\circ}$ as its value
presented in 61 does not comply with the $P_{2,{\rm RS}}^{\circ}\leq W$
condition. We believe that the overestimated value of $P_{2,{\rm RS}}^{\circ}$
for B1929+10 is a result of using the fluctuations spectra presented
in 180 to calculate the number of sparks in the
carousel model. Note that the coincidence of $\tilde{P}_{2}^{\circ}$
and $P_{2,{\rm RS}}^{\circ}$ is yet to be clarified, as in our model
there is no physical reason for sparks to circulate around the magnetic
axis. In fact, the PSG model assumes the non-dipolar structure of
the magnetic field lines in the gap region and the actual position
of the polar cap is not necessarily coincident with the global dipole
(e.g. see Figures <ref>, <ref>,
[Details of a subpulse drift for pulsars with X-ray hot spot radiation]Details of a subpulse drift for pulsars with X-ray hot spot radiation.
The individual columns are as follows: (1) Pulsar name, (2) Predicted
characteristic spacing between subpulses in the pulse longitude, $\tilde{P}_{2}^{\circ}$;
(3) Spacing between subpulses, found in the literature, estimated
using the carousel model, $P_{2,{\rm RS}}^{\circ}$; (4) Spacing between
subpulses estimated using fluctuations spectra, $P_{2,{\rm W}}^{\circ}$;
(5) Period at which a pattern of subpulses crosses the pulse window
(in units of the pulsar period), $P_{3}$; (6) Number of sparks estimated
using the carousel model, $N$; (7) Profile width at 10%, $W$; (8)
Profile width calculated assuming $\beta=0$, $W_{\beta0}$; (9) Angular
width of the observed region on the polar cap $\varrho$ (see Figure
<ref>); (10) References; (11) Number of the pulsar.
Name $\tilde{P}_{2}^{\circ}$ $P_{2,{\rm RS}}^{\circ}$ $P_{2,{\rm W}}^{\circ}$ $P_{3}$ $N$ $W$ $W_{\beta0}$ $\varrho$ Ref. No. $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ $\left(P\right)$ $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ $\left({\rm deg}\right)$ B0628–28 $8.4$ $6$ $30$ $7.0$ $24$ $38.1$ $48.2$ $98.0$ [64], [180] $8$B0834+06 $1.4$ $3$ $20$ $2.2$ $14$ $12.2$ $15.8$ $105.5$ [149], [180] $14$B0943+10 $9.8$ $17$ – $1.8$ $20$ $25.6$ $84.7$ $27.5$ [52], [53], [6] $15$B0950+08 $4.0$ – – $6.5$ – $32.2$ $63.7$ $55.1$ [181] $16$B1133+16 $3.2$ $4$ $130$ $3.0$ $11$ $14.4$ $18.1$ $107.9$ [62], [86], [180] $22$B1929+10 $5.8$ $?$ $90$ $9.8$ $?$ $24.5$ $80.8$ $43.6$ [61], [180] $42$
Table <ref> presents the observed and derived parameters
of PSG for pulsars with available radio and X-ray observations. In
the calculations we used the predicted value of subpulse separation
$\tilde{P_{2}^{\circ}}$. Note that we consider only pulsars with
a visible hot spot component since only for these pulsars we can estimate
the size of the polar cap. The low value of the estimated screening
factor ($\eta\ll1$) suggests that when the drift is visible, the
pulsar operates in the PSG-on mode. If the pulsar operates in the
PSG-off mode, $\eta\approx1$, the drift velocity is much higher (see
Equation <ref>) and the drift phenomenon should
be more chaotic and much difficult to identify. Observations of PSR
0943+10 show a strong, regular subpulse drifting in the radio-bright
mode, with only a hint of modulation in the radio-quiescent mode.
Based on this fact we believe that the two different scenarios of
the gap breakdown (PSG-on and PSG-off modes) can explain the mode
[Derived parameters of PSG for pulsars with available radio observations
of the subpulse drift and X-ray hot spot radiation]Derived parameters of PSG for pulsars with available radio observations
of the subpulse drift and X-ray hot spot radiation. The individual
columns are as follows: (1) Pulsar name, (2) Screening factor, $\eta$;
(3) Predicted heating efficiency, $\xi_{{\rm heat}}$; (4) Observed
bolometric efficiency, $\xi_{_{{\rm BB}}}$; (5) Predicted ion acceleration
efficiency, $\xi_{{\rm _{ion}}}$; (6) Surface temperature, $T_{{\rm s}}$;
(7) Strength of the surface magnetic field, $B_{{\rm s}}$; (8) Observed
polar cap radius, $R_{{\rm pc}}$; (9) Estimated spark half-width,
$h_{\perp}$; (10) Number of the pulsar. $T_{{\rm s}}$, $R_{{\rm pc}}$,
$b$ were chosen to fit $1\sigma$ uncertainty. Note that in the calculations
$\tilde{P}_{2}^{\circ}$ was used.
Name $\eta$ $\log\xi_{_{{\rm heat}}}$ $\log\xi_{_{{\rm BB}}}$ $\log\xi_{_{{\rm ion}}}$ $T_{{\rm s}}$ $B_{{\rm s}}$ $R_{{\rm pc}}$ $h_{\perp}$ No. $\left({\rm radio}\right)$ $\left({\rm x-ray}\right)$ $\left({\rm ions}\right)$ $\left(10^{6}{\rm K}\right)$ $\left(10^{14}{\rm G}\right)$ $\left({\rm m}\right)$ $\left({\rm m}\right)$ B0628–28 $0.07$ $-4.03$ $-3.61$ $-1.43$ $2.5$ $2.0$ $23$ $3.9$ $8$B0834+06 $0.15$ $-3.60$ $-3.34$ $-1.35$ $3.0$ $2.4$ $20$ $1.8$ $14$B0943+10 $0.09$ $-3.24$ $-3.27$ $-0.75$ $3.3$ $2.5$ $17$ $2.0$ $15$B0950+08 $0.09$ $-5.08$ $-4.54$ $-2.62$ $2.6$ $2.1$ $14$ $0.7$ $16$B1133+16 $0.09$ $-3.29$ $-3.13$ $-0.81$ $3.4$ $2.7$ $17$ $2.9$ $22$B1929+10 $0.02$ $-5.10$ $-4.17$ $-1.98$ $4.2$ $2.0$ $22$ $1.6$ $42$
CHAPTER: CASCADE SIMULATION
In this chapter we present the approach of calculating the
pair cascades developed by 124 which has been applied
to cases with non-dipolar structure of magnetic field. The original
approach was adapted to perform full three-dimensional calculations
and extended with effects that may have a greater importance for non-dipolar
configuration of surface magnetic fields (e.g. aberration). Additionally,
to perform a thorough analysis of the Inverse Compton Scattering we
present the detailed description of calculating the ICS cross section
originally developed by 71.
Following the approach presented by 124 we
can divide the cascade simulation into three parts:
* propagation of the primary particle (including photon emission),
* photon propagation in strong magnetic field (pair production, photon
* propagation and photon emission of the secondary
[In this thesis the term ”secondary” refers to any newly created
particle except for the primary particles accelerated in IAR, e.g.
the third generation of electrons and positrons are all considered
as ”secondary” particles.
We use the ”co-rotating” frame of reference (the frame which rotates
with the star) to track both photons and particles. In calculations
we consider regions far inside the light cylinder. Thus, following
124, we ignore any bending of the photon path due
to rotation of the star. Furthermore, we also ignore effects of general
relativity on trajectories of photons and particles.
Figure <ref> presents a summary flowchart
of the algorithm used to calculate the properties of secondary plasma
and the spectrum of radiation for a given structure of a neutron's
star magnetic field and gap parameters.
[Flowchart of the algorithm used to calculate a cascade simulation]Flowchart of algorithm used to calculate a cascade simulation.
§ CURVATURE RADIATION
As we have shown in Chapter <ref>, an ultrastrong surface
magnetic field ($B_{{\rm s}}>10^{14}\,{\rm G}$) is accompanied by
high curvature (curvature radius $\Re_{6}\approx0.1-10$). This suggests
that one of the important processes of radiation which should be considered
is Curvature Radiation (CR).
CR is quite similar to ordinary synchrotron radiation (radiation of
ultrarelativistic particles in the magnetic field), the only difference
being that the radius of circular motion (the gyroradius) is in fact
the curvature radius of magnetic field lines. Due to beaming effects
the radiation appears to be concentrated in a narrow set of directions
about the velocity of the particle. The angular width of the cone
of emission is of the order $\sim1/\gamma$, where $\gamma$ is a
Lorentz factor of an emitting particle (for more details see 157).
We track the primary particle above the acceleration zone (the gap
region) as it moves along the magnetic field line. The length of the
step $\Delta s$ is chosen so as to achieve sufficient accuracy even
for large curvature of the magnetic field line, $\Delta s\approx0.01\Re_{{\rm min}}$,
where $\Re_{{\rm min}}$ is the minimum radius of curvature. The distribution
of CR photon energy can be written as (see Equation 14.93 in 96)
\begin{equation}
\frac{{\rm d}N}{{\rm d}\epsilon}=\frac{E}{\epsilon_{{\rm _{CR}}}}\frac{9\sqrt{3}}{8\pi}\int_{\epsilon/\epsilon_{_{{\rm CR}}}}^{\infty}K_{5/3}(t){\rm d}t,\label{eq:cascade.dn_deps}
\end{equation}
where $E=4\pi e^{2}\gamma^{4}/3\Re$ is the total energy radiated
per revolution, $\epsilon_{_{{\rm CR}}}=3\gamma^{3}\hbar c/(2\Re)$
is the characteristic energy of curvature photons, and $K_{5/3}$
is the $n=5/3$ Bessel function of the second kind. The total energy
radiated by a particle after it passes the length $\Delta s$, $E_{_{\Delta s}}$,
can be written as
\begin{equation}
E_{_{\Delta s}}=E\frac{\Delta s}{2\pi\Re}.\label{eq:cascade.i_ds}
\end{equation}
Thus, by using Equations <ref> and <ref>
we can write the formula for the distribution on CR photon energy
after a particle passes length $\Delta s$
\begin{equation}
\frac{{\rm d}N}{{\rm d}\epsilon}=\frac{\Delta s}{2\pi\Re}\frac{\sqrt{3}e^{2}\gamma}{\hbar c}\int_{\epsilon/\epsilon_{{\rm _{CR}}}}^{\infty}K_{5/3}(t){\rm d}t.
\end{equation}
It is convenient to divide the spectrum of photon energy into discrete
bins. Then, the number of photons in each energy bin can be calculated
\begin{equation}
N_{\epsilon}=\int_{\epsilon_{_{i}}}^{\epsilon_{_{i}}+\Delta\epsilon}\frac{dN}{d\epsilon}{\rm d}\epsilon,
\end{equation}
where $\epsilon_{_{i}}$ is the lowest energy for the $i$-th bin
and $\Delta\epsilon$ is the energy bin width. Our simulation uses
$50$ bins with an energy range of $\epsilon_{_{0}}=4\times10^{-2}\,{\rm keV}$
(soft X-ray) to
$\epsilon_{_{49}}=4\times10^{5}\,{\rm MeV}$ (hard $\gamma$-rays).
Depending on the photon frequency the polarisation fraction of CR
photons is between $50\%$ and $100\%$ polarised parallel to the
magnetic field (see 96, 157).
Therefore, using similar approach as 124 we randomly
assign the polarisation in the ratio of one photon $\perp$-polarised
to every seven $\parallel$-polarised photons, which corresponds to
$75\%$ parallel polarisation.
§ PHOTON PROPAGATION
To explain some of the properties of pulsars and their surroundings
(e.g. nebulae radiation), large magnetospheric plasma densities exceeding
the Goldreich-Julian density (see Equation <ref>) by
many orders of magnitude are required. In order to simulate the process
of generation of such a dense plasma it is necessary to check the
conditions of photon decay into electron-positron pairs.
A photon with energy $E_{\gamma}>2mc^{2}$ and propagating with a
nonzero angle $\Psi$ with respect to an external magnetic field can
be absorbed by the field and, as a result an electron-positron pair
is created. The concurrent process is photon splitting $\gamma\rightarrow\gamma\gamma$,
which may occur even if the photon energy is below the pair creation
threshold ($E_{\gamma}<2mc^{2}$).
In the cascade simulation the photon is emitted (or scattered in the
case of ICS) from point $P_{{\rm ph}}$ in a direction tangent to
the magnetic field line $\Delta\mathbf{s_{\parallel}}$. The direction
vector is calculated as the value of the magnetic field at the point
of photon creation (see Equations <ref>, <ref>
and <ref>) normalised so that its length is equal to
the desired step $\Delta\mathbf{s_{\parallel}}=\mathbf{B}\Delta s/B$.
However, the direction of the magnetic field at the point of photon
emission does not take into account the randomness of the emission
direction due to the relativistic beaming effect. In Section <ref>
we describe a procedure to include the beaming effect in the emission
process which alters $\Delta\mathbf{s_{\parallel}}\rightarrow{\bf \Delta s_{ph}}$.
Finally, we can write that at the point of curvature emission photons
are created with energy $\epsilon_{{\rm ph}}$, polarisation $\parallel$
or $\perp$, weighting factor $N_{\epsilon}$ (number of photons),
and with both optical depths (for pair production $\tau$ and for
photon splitting $\tau_{{\rm sp}}$) set to zero. Since we neglect
any banding of the photon path we assume that from the point of emission
it travels in a straight line. In each following step the photon travels
a distance ${\bf \Delta s_{ph}}$. In the co-rotating frame of reference
in every step we need to take into the account aberration due to pulsar
rotation. In order to do so, in every step we alter the photon position
according to the procedure described in Section <ref>.
As stated by 124 we can calculate the change in the
pair production optical depth , $\Delta\tau$, and in the photon splitting
optical depth $\Delta\tau_{{\rm sp}}$, at the new position as:
\begin{equation}
\Delta\tau\simeq\Delta s_{{\rm ph}}R_{\|,\perp},
\end{equation}
\begin{equation}
\Delta\tau_{{\rm sp}}\simeq\Delta s_{{\rm ph}}R_{\|,\perp}^{{\rm sp}},
\end{equation}
where $R_{\|,\perp}$ and $R_{\|,\perp}^{{\rm sp}}$ are the attenuation
coefficients for $\|$ or $\perp$ polarised photons for pair production
and photon splitting, receptively.
§.§ Relativistic beaming (emission direction)
Due to relativistic beaming the emission direction should be modified
by an additional emission angle of order $\sim1/\gamma$. We use the
following steps to include the beaming effect in our simulation (see
Figure <ref>).
(I) The first step is rotation of the $xyz$ frame of reference in
order to align the $z$-axis with $\mathbf{\Delta s_{\parallel}}$.
In our calculations we used rotation by angle $\varsigma_{y}$ around
the $y$-axis, $R_{y}\left(\varsigma_{y}\right)$, and rotation by
angle $\varsigma_{x}$ around the $x$-axis, $R_{x}\left(\varsigma_{x}\right)$.
The final rotation matrix can be written as
\begin{equation}
\cos\varsigma_{y} & \sin\varsigma_{x}\sin\varsigma_{y} & \sin\varsigma_{y}\cos\varsigma_{x}\\
0 & \cos\alpha & -\sin\alpha\\
-\sin\varsigma_{y} & \cos\varsigma_{y}\sin\varsigma_{x} & \cos\varsigma_{y}\cos\varsigma_{x}
\end{array}\right].
\end{equation}
[Relativistic beaming effect of photon emission]Relativistic beaming effect of photon emission (for both CR and ICS).
In the simulation we include the beaming effect by performing three
steps: (I) rotation of the $xyz$ frame of reference in order to align
the $z$-axis with $\mathbf{\Delta s_{\parallel}}$, (II) transformation
of the step vector from a Cartesian to a spherical system of coordinates
and alteration of the $\theta$ and $\phi$ components with random
values $1/\gamma\cos\Lambda$ and $\Pi$, respectively, (III) transformation
of the step vector from a spherical to a Cartesian system of coordinates
and rotation back to the original system of reference. Note that after
these steps we get a new vector ${\bf \Delta s_{ph}}$ inclined to
the primary one, ${\bf \Delta s_{ph}}$, at an angle ranging from
$0$ to $1/\gamma$.
The Euler angles for rotations can be calculated as
\begin{equation}
\begin{array}{c}
\varsigma_{x}={\rm atan2}\left(s_{y},s_{z}\right),\\
\varsigma_{y}=\begin{cases}
\arctan\left(-\frac{s_{x}}{s_{z}}\cos\varsigma_{x}\right) & {\rm if\ }s_{z}\neq0\\
\arctan\left(-\frac{s_{x}}{s_{y}}\sin\varsigma_{x}\right) & {\rm if\ }s_{y}\neq0\\
\frac{\pi}{2} & {\rm if\ }s_{x}=0\ {\rm and}\ s_{y}=0.
\end{cases}
\end{array}
\end{equation}
Note that in order to increase readability, the $\Delta$ symbol and
${\rm \parallel}$ index were discarded (e.g. $s_{x}=\Delta s_{{\rm \parallel},x}$).
(II) The second step is the transformation of the step vector's coordinates
in the double rotated frame of reference ${\bf \Delta s_{{\rm ph}}^{\prime\prime}=}\left(s_{x}^{\prime\prime},\ s_{y}^{\prime\prime},\ s_{z}^{\prime\prime}\right)$
to spherical system of coordinates and alteration of the $\theta$
and $\phi$ components as follows
\begin{eqnarray}
s_{r}^{\prime\prime} & = & \sqrt{s_{x}^{\prime\prime2}+s_{y}^{\prime\prime2}+s_{z}^{\prime\prime2}},\nonumber \\
s_{\theta}^{\prime\prime} & = & \arccos\left(\frac{s_{z}^{\prime\prime}}{\sqrt{s_{x}^{\prime\prime2}+s_{y}^{\prime\prime2}+s_{z}^{\prime\prime2}}}\right)+\frac{1}{\gamma}\cos\Lambda,\nonumber \\
s_{\phi}^{\prime\prime} & = & \arctan\left(\frac{s_{y}^{\prime\prime}}{s_{x}^{\prime\prime}}\right)+\Pi,
\end{eqnarray}
where $\Lambda$ and $\Pi$ are random angles between $0$ and $2\pi$.
The inverse tangent denoted in the $\phi$-coordinate must be suitably
defined by taking into account the correct quadrant (see the “${\rm atan2}$”
description in the footnote on page fn:model.atan2).
(III) The last step is the transformation of vector components to
the Cartesian system of coordinates, ${\bf s_{ph}^{\prime\prime}}=\left[s_{r}^{\prime\prime}\sin\left(s_{\theta}^{\prime\prime}\right)\cos\left(s_{\phi}^{\prime\prime}\right),\, s_{r}^{\prime\prime}\sin\left(s_{\theta}^{\prime\prime}\right)\sin\left(s_{\phi}^{\prime\prime}\right),\, s_{r}^{\prime\prime}\cos\left(s_{\theta}^{\prime\prime}\right)\right]$
and rotation back to the original coordinate system ${\bf \Delta s_{ph}}=\left(R_{yx}\right)^{-1}{\bf {\bf s_{ph}^{\prime\prime}}}$.
The rotation matrix of this transformation can be written as
\begin{equation}
\left(R_{yx}\right)^{-1}=\left(R_{yx}\right)^{T}=\left[\begin{array}{ccc}
\cos\varsigma_{y} & 0 & -\sin\varsigma_{y}\\
\sin\varsigma_{x}\sin\varsigma_{y} & \cos\varsigma_{x} & \sin\varsigma_{x}\cos\varsigma_{y}\\
\sin\varsigma_{y}\cos\varsigma_{x} & -\sin\varsigma_{x} & \cos\varsigma_{x}\cos\varsigma_{y}
\end{array}\right].
\end{equation}
§.§ Aberration due to pulsar rotation
Note that in our frame of reference (co-rotating with a star) the
path of the photon should be curved (see 84).
In the dipolar case the angular deviation increases approximately
as $s_{{\rm ph}}\Omega/c=s_{{\rm ph}}/R_{LC}$. When the configuration
of magnetic field in non-dipolar inclusion of an aberration is even
more important. Therefore, the location of photon decay should be
modified to include the growth of the photon-magnetic field intersection
In our simulation we include the aberration effect by alteration of
photon position $P_{{\rm ph}}$ in every step ${\bf \Delta s_{ph}}$
(see Figure <ref>).
[Aberration due to pulsar rotation] Aberration due to pulsar rotation. We use the following procedure
to include the aberration effect: (I) rotation around the $y$-axis
to align $\Omega$ with $\mu$, (II) rotation by angle
$\omega=2\pi\Delta s_{{\rm bm}}/\left(cP\right)$ around the $z$-axis
(which reflects the pulsar rotation), (III) rotation back to the original
frame of reference (in which $\mu$ is aligned with the $z$-axis).
We use the three-step procedure to alter the photon position.
(I) Rotation of the $xyz$ frame of reference around the $y$-axis
by angle $\alpha$,
${\bf P_{ph}^{\prime}}=R_{y}\left(\alpha\right){\bf P_{ph}}$. Note
that here $\alpha$ refers to the inclination of the magnetic axis
with respect to the rotation axis and we assume that the pulsar's
angular velocity vector ${\bf \Omega}$ lies in the $xz$-plane. The
rotation matrix of this transformation can be written as
\begin{equation}
\cos\alpha & 0 & \sin\alpha\\
0 & 1 & 0\\
-\sin\alpha & 0 & \cos\alpha
\end{array}\right].
\end{equation}
(II) After step (I) the $z$-axis is aligned to $\Omega$, and in
order to include the rotation of the pulsar we need to again rotate
the frame of reverence by angle $\omega=2\pi\Delta s_{{\rm bm}}/\left(cP\right)$
around the $z$-axis, ${\bf P_{ph}^{\prime\prime}}=R_{z}\left(\omega\right){\bf P_{ph}^{\prime}}$.
We use the following rotation matrix
\begin{equation}
\cos\omega & -\sin\omega & 0\\
\sin\omega & \cos\omega & 0\\
0 & 0 & 1
\end{array}\right].
\end{equation}
(III) The final step is a rotation back to the original frame of reference,
${\bf P_{ph}^{\prime\prime\prime}=}\left(R_{y}\left(\alpha\right)\right)^{-1}{\bf P_{ph}^{\prime\prime}}$,
using the following rotation matrix
\begin{equation}
\left(R_{y}\left(\alpha\right)\right)^{-1}=\left(R_{y}\left(\alpha\right)\right)^{T}=\left[\begin{array}{ccc}
\cos\alpha & 0 & -\sin\alpha\\
0 & 1 & 0\\
\sin\alpha & 0 & \cos\alpha
\end{array}\right].
\end{equation}
§.§ Pair production attenuation coefficient
The pair production attenuation coefficient can be written as [124]
\begin{equation}
\end{equation}
where $R^{\prime}$ is the attenuation coefficient in the frame where
the photon propagates perpendicular to the local magnetic field (the
so-called ”perpendicular” frame), $\Psi$ is the intersection
angle between the propagation direction of the photon and the local
magnetic field. To increase readability we suppress the subscripts
$\parallel$ and $\perp$, but english$R^{\prime}$
has to be calculated for both polarisations separately.
As stated by 124 the total attenuation coefficient
for pair production can be calculated as $R^{\prime}=\sum_{jk}R^{\prime}{}_{j,k}$,
where $R{}_{j,k}^{\prime}$ is the attenuation coefficient for the
process producing an electron in Landau level $j$ and a positron
in Landau level $k$. For the electron-positron pair the sum is taken
over all possible states ($j$ and $k$). Note that production of
electron-positron pairs is symmetric $R{}_{jk}^{\prime}=R{}_{kj}^{\prime}$.
Thus, to represent the pair creation probability in either the $\left(jk\right)$
or $\left(kj\right)$ state we will use $R{}_{jk}^{\prime}$. For
a given Landau levels $j$ and $k$, the pair production threshold
condition is [124]
\begin{equation}
\end{equation}
where $E_{\gamma}^{\prime}=E_{\gamma}\sin\Psi$ is the photon energy
in the perpendicular frame and
$E_{n}^{\prime}=mc^{2}\sqrt{1+2\epsilon_{_{B}}n}$ is the minimum
energy of a particle (electron or positron) in Landau Level $n$.
This condition can be written in a dimensionless form as
\begin{equation}
\end{equation}
where $\epsilon_{{\rm _{B}}}=\hbar eB/\left(mc\right)$ is the cyclotron
energy of a particle (electron or positron) in magnetic field $B$
in units of $mc^{2}$.
The first nonzero pair production attenuation coefficients for both
polarisations ($\perp$ and $\parallel$) are [45, 124]
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
where $a_{0}$ is the Bohr radius (let us note that $R'_{\perp,00}=0$).
In the above equations for all channels except $00$ the pair production
attenuation coefficients are multiplied by a factor of two (see the
text above Equation <ref>).
The pair production optical depth is defined as [124]:
\begin{equation}
\tau=\int_{0}^{s_{{\rm ph}}}R(s){\rm d}s=\int_{0}^{s_{{\rm ph}}}R^{\prime}(s)\sin\Psi{\rm d}s.\label{eq:cascade.tau_in}
\end{equation}
We can assume $\Psi\ll1$, because all high-energy photons ($x>1$)
will produce pairs much earlier than $\Psi$ reaches a value near
unity. In this limit $\sin\Psi\simeq s_{{\rm ph}}/\Re$, so the relation
between $x$ and $s_{{\rm ph}}$ can be expressed by
\begin{equation}
x\simeq\frac{s_{{\rm ph}}}{\Re}\frac{E_{\gamma}}{2mc^{2}}.
\end{equation}
Equation <ref> can be rewritten as
\[
\tau=\tau_{1}+\tau_{\|,2}+\tau_{\perp,2}+...;
\]
\begin{equation}
\tau_{1}=\int_{s_{0}}^{s_{1}}R_{\|,00}{\rm d}s,\hspace{0.5cm}\tau_{\|,2}=\int_{s_{1}}^{s_{2}}\left(R_{\|,00}+R_{\|,01}\right){\rm d}s,\hspace{0.5cm}\tau_{\perp,2}=\int_{s_{1}}^{s_{2}}R_{\perp,01}{\rm d}s,
\end{equation}
where $s_{0}$ and $s_{1}$ are distances which the photon should
pass in order to have energy $x_{0}$ and $x_{1}$, respectively (in
the perpendicular frame of reference). Let us note that $s_{0}$,
$s_{1}$ and $s_{2}$ are of the same order, and if $s<s_{0}$ the
attenuation coefficient is zero.
The pair production optical depth to reach the second threshold is
\begin{equation}
\int_{s_{0}}^{s_{1}}{\rm d}sR_{\|,00}(s)=\frac{\epsilon_{{\rm _{B}}}}{2a_{0}}\left(\frac{2mc^{2}}{E_{\gamma}}\right)^{2}\Re\int_{x_{1}}^{x_{2}}\frac{{\rm d}x}{x\sqrt{x^{2}-1}}e^{-2x^{2}/\epsilon_{{\rm _{B}}}},
\end{equation}
where $s_{0}$ is the distance travelled by the photon to reach the
threshold $x_{0}\equiv1$, and $s_{1}$ is the distance travelled
by the photon to reach the second threshold $x_{1}\equiv\left(1+\sqrt{1+2\epsilon_{{\rm _{B}}}}\right)/2$.
(a - show_tau, 500 MeV, Re=1e6, size=1e2, b - show_tau_psi, 500
MeV, b=B_crit, Re=1e6, size=1e4, )
[Pair production optical depth ]Panel (a) presents the dependence of the pair production optical
depth on the magnetic field strength ($\beta_{q}=B/B_{q}$). Panel
(b) presents the dependence of the optical depth on photon energy
in the perpendicular frame of reference ($x=\epsilon\sin\Psi/\left(2mc^{2}\right)$).
On both panels the photon energy is $\epsilon=500\,{\rm MeV}$, while
panel (b) was obtained for magnetic field strength $\beta_{q}=1$.
§.§ Photon mean free path
As was shown in the previous section (see Figure <ref>)
for strong magnetic fields (e.g. $\beta_{q}\gtrsim0.2$), $\tau_{1}$,
$\tau_{\|,2}$, and $\tau_{\perp,2}$ are much larger than one. Therefore,
the pair production process takes place according to two scenarios
(see also 124). If $\beta_{q}\gtrsim0.2$ pairs
are produced by photons almost immediately upon reaching the first
threshold, the created pairs will be in the low Landau levels ($n\lesssim2$).
If $\beta_{q}\lesssim0.2$, the photons will travel longer distances
to be absorbed and the created pairs will be in the higher Landau
Thus, for strong magnetic fields ($\beta_{q}\gtrsim0.2$) the photon
mean free path can be approximated as
\begin{equation}
l_{{\rm ph}}\approx s_{0}=\Re\frac{2mc^{2}}{E_{\gamma}},\label{eq:cascade.l_ph}
\end{equation}
while for relatively weak magnetic fields ($\beta_{q}\lesssim0.2$)
we can use the asymptotic approximation derived by 55:
\begin{equation}
l_{{\rm ph}}\approx\frac{4.4}{(e^{2}/\hbar c)}\frac{\hbar}{mc}\frac{B_{q}}{B\sin\Psi}\exp\left(\frac{4}{3\chi}\right),\label{eq:cascade.l_ph2}
\end{equation}
\begin{equation}
\chi\equiv\frac{E_{\gamma}}{2mc^{2}}\frac{B\sin\Psi}{B_{q}}\hspace{1cm}(\chi\ll1).
\end{equation}
§.§ Photon-splitting attenuation coefficient
In our calculations we include photon splitting by following the approach
presented by 124. Since only the $\perp\rightarrow\parallel\parallel$
process is allowed, for $\parallel$-polarised photons the photon
splitting attenuation coefficient is zero $R_{\parallel}^{sp}=0$
(3, 176, 11).
To calculate the splitting attenuation coefficient in the perpendicular
frame for $\perp$-polarised photons we use the formula adopted from
the numerical calculation of 10 :
\begin{equation}
R{}_{\perp}^{\prime{\rm sp}}\simeq\frac{\frac{\alpha_{f}^{2}}{60\pi^{2}}\left(\frac{26}{315}\right)^{2}\left(2x\right)^{5}\epsilon_{{\rm _{B}}}^{6}}{\left[\epsilon_{_{B}}^{3}\exp\left(-0.6x^{3}\right)+0.05\right]\left[0.25\epsilon_{{\rm _{B}}}^{3}\exp\left(-0.6x^{3}\right)+20\right]}.
\end{equation}
For photon energies $x\leq1$ this expression underestimates the results
of 10 at $\beta_{q}=1$ by less than $30\%$, while
at both $\beta_{q}\le0.5$ and $\beta_{q}\gg1$ the discrepancy is
less than $10\%$. As can be seen from Figure <ref>,
the attenuation coefficient $R{}_{\perp}^{\prime{\rm sp}}$ drops
rapidly with the magnetic field strength for $\beta_{q}<1$, thus
photon splitting is unimportant for $\beta_{q}\lesssim0.5$ (e.g.
11, 124).
[Photon-splitting attenuation coefficient]Dependence of the photon-splitting attenuation coefficient on the
energy of the photon in the perpendicular frame ($x=\epsilon\sin\Psi/\left(2mc^{2}\right)$,
vertical axis) and on the strength of the magnetic field ($\beta_{q}=B/B_{q}$,
horizontal axis).
§.§ Pair creation vs photon splitting
As noted by 124, even though the photon splitting
attenuation coefficient above the first threshold ($x>x_{0}$) is
much smaller than for pair production (see Figure <ref>),
in ultrastrong magnetic fields ($\beta_{q}\gtrsim0.5$) the $\perp$-polarised
photons split before reaching the first threshold (see Figure <ref>).
On the other hand, the $\parallel$-polarised photons produce
pairs in the zeroth Landau level.
[Attenuation coefficients of pair production and photon splitting]Attenuation coefficients of pair production and photon splitting
in the perpendicular frame of reference. Panel (a) was obtained using
photon energy $E_{\gamma}=10^{3}\,{\rm MeV}$ and magnetic field strength
$B=B_{q}=4.414\times10^{13}\,{\rm G}$ ($\beta_{q}=1$). Panel (b)
presents calculations for photon energy $E_{\gamma}=10^{3}\,{\rm MeV}$
and magnetic field strength $B=2.5\times10^{14}\,{\rm G}$ ($\beta_{q}=5.7$).
[Optical depth for pair production and photon splitting]Optical depth for pair production and photon splitting for $\perp$-polarised
photons. Panel (a) presents results for $E_{\gamma}=10^{3}\,{\rm MeV}$
and $B=B_{q}=4.4\times10^{13}\,{\rm G}$ ($\beta_{q}=1$), while panel
(b) was obtained using the same photon energy but a stronger magnetic
field $B=2.5\times10^{14}\,{\rm G}$ ($\beta_{q}=5.7$). If $\beta_{q}=1$
the photon creates an electron-positron pair, while in an ultrastrong
magnetic field ($\beta_{q}=5.7$) the photon splits before it reaches
the first threshold, $x=x_{0}$.
§.§ Secondary plasma
Following the approach presented by 124 whenever $\tau\geq1$
and the threshold for pair production is reached ($x=x_{0}$ for $\parallel$-polarised
photons and $x=x_{1}$ for $\perp$-polarised photons), the photon
is turned into an electron-positron pair. Whereas if $\tau_{{\rm sp}}\geq1$
the photon is turned into two photons. Following the results of 10
we assume that the energy of parent photon is equally distributed
between both newly created photons. A new $\parallel$-polarised photon
is created with an energy $0.5\epsilon_{{\rm ph}}$ and weighting
factor $2\Delta N_{\epsilon}$. We assume that the newly created photon
travels in the same direction as the parent photon, ${\bf \Delta s_{ph}}$.
Note that the photon should split with probability $1-e^{-\tau}$,
but as shown by 124 for cascade results this effect
is negligible.
For $\beta_{q}\lesssim0.1$, the particles are produced in high Landau
levels with energy equal to half of the photon energy each (see 45).
In our calculations we assume that the newly created particles (electron-positron
pairs) travel in the same direction as the photon. When $\beta_{q}\gtrsim0.1$,
on the other hand, we choose the maximum allowed values of $j$ and
$k$ for the newly created electron and positron. Note that for $\beta_{q}\gtrsim0.1$
the particles are created in low Landau levels.
Figure <ref> presents the spectrum of Curvature
Radiation for a dipolar and non-dipolar structure of magnetic field
lines. Note the characteristic three peaks in the CR distribution
for the non-dipolar structure.
(379 - gamma=3.5e6, h=252e2 - manual photon_evolution=0, 373 photon_evolution=0)
(plot_spectrum2 or t2)
[Distribution of CR-photons produced by a single primary particle] Distribution of CR photons produced by a single primary particle
for a dipolar (blue line) and non-dipolar (red line) structure of
the magnetic field. The minimum radius of curvature in thee dipolar
case is $\Re_{6}^{^{{\rm min}}}\approx50$, while in the non-dipolar
case $\Re_{6}^{^{{\rm min}}}\approx2$. In both cases the radiation
was calculated up to a distance of $D=100R$, and with an initial
Lorentz factor of the primary particle $\gamma_{{\rm c}}=3.5\times10^{6}$.
Formation of the peaks is caused by the fact that the particle passes
regions with three different values of curvature: (I) just above the
stellar surface, $z\approx1\,{\rm km}$, where curvature is the highest;
(II) at altitudes where the influence of anomalies is comparable with
the global dipole, $z\approx2.5\,{\rm km}$, also with strong curvature;
(III) and at altitudes where the influence of anomalies is negligible,
$z\gtrsim3.1R$, with approximately dipolar curvature (see Figure
<ref>). Hence, the spectrum is a sum of radiation
generated in a highly non-dipolar magnetic field (high energetic and
soft $\gamma$-rays) and with radiation at higher altitudes where
the magnetic field is dipolar (X-rays). The primary particle loses
about $63\%$ and $1\%$ of its initial energy in the non-dipolar
and dipolar case, respectively. As can be seen from the Figure, to
get high emission of CR photons and, thus, a significant density of
secondary plasma, a non-dipolar structure of the magnetic field is
The high energetic photons produced in a strongly non-dipolar magnetic
field will either split or create electron-positron pairs. Figure
<ref> presents the distribution of particle
energy created by CR photons. Note that for $\beta_{q}\lesssim0.1$
the pairs are created in high Landau levels and in order to get the
final distribution of secondary plasma energy we should consider the
loss of particle energy due to Synchrotron Radiation (see Section
(plot_pairs, data/373_cr_1e04_100m_CR_noSR_PH_e5_leftline/
[Distribution of particle energy created by CR photons ] Distribution of particle energy created by CR photons calculated
for a non-dipolar structure of the magnetic field. For this specific
magnetic field configuration and initial parameters (see the caption
of Figure <ref>) the secondary plasma multiplicity
is $M_{{\rm sec}}\approx6\times10^{3}$. Note that this result does
not include Synchrotron Radiation and the actual energies of the created
pairs are lower as they lose their transverse momenta (see Section
§ SYNCHROTRON RADIATION
When pairs (electrons and positrons) are created in high Landau Levels
they radiate away their transverse momentum through Synchrotron Radiation
(SR). The secondary positron (or electron) is created with energy
$\gamma mc^{2}$ and pitch angle $\Psi$, which corresponds to a specific
value of Landau Level $n$. Following 124 we choose
the frame in which the particle has no momentum along the direction
of external magnetic field. In such a frame of reference the particle
propagates in a circular motion transverse to the magnetic field (the
so-called ”circular” frame). The relation of the energy of the
newly created particle in the circular frame of reference ($E_{\perp}=\gamma_{\perp}mc^{2}$)
with the particle energy in the co-rotating frame can be written as
[124]
\begin{equation}
\gamma_{\perp}=\sqrt{\gamma^{2}\sin^{2}\Psi+\cos^{2}\Psi}=\sqrt{1+2\epsilon_{_{B}}n}.\label{eq:cascade.gamma_perp}
\end{equation}
The power of synchrotron emission, $P_{{\rm SR}}$, can calculated
as follows
\begin{equation}
P_{{\rm SR}}=\frac{2e^{2}}{3c^{3}}\left(\gamma_{\perp}^{2}-1\right)c^{2}\epsilon_{_{B}}^{2},
\end{equation}
In the circular frame $E_{\perp}$, is radiated away through synchrotron
emission after a particle travels a distance
\begin{equation}
l_{{\rm p}}^{{\rm SR}}\approx\left|\frac{E_{\perp}}{P_{{\rm SR}}}c\right|=\frac{\gamma_{\perp}mc^{3}}{\frac{2e^{2}}{3c^{3}}\left(\gamma_{\perp}^{2}-1\right)c^{2}\epsilon_{{\rm _{B}}}^{2}}.
\end{equation}
The particle (electron or positron) mean free path for SR is much
shorter than for other relevant cascade processes (see Section <ref>
for Curvature Radiation, and Section <ref> for ICS).
In fact, it is so short that in our calculations we assume that before
moving from its initial position the particle loses all of its perpendicular
momentum $p_{\perp}$ due to SR (see 44, 124).
Once the particle reaches the ground Landau level ($n=0$, $p_{\perp}=0$)
its final energy can be calculate as
\begin{equation}
\gamma_{\parallel}=\left(1-\beta^{2}\cos^{2}\Psi\right)^{-1/2}=\gamma/\gamma_{\perp},\label{eq:cascade.gamma_par}
\end{equation}
here $\beta=v/c=\sqrt{1-1/\gamma^{2}}$ is the particle velocity in
units of speed of light.
Following the approach presented by 124, to simplify
the simulation we assume that in the circular frame synchrotron photons
are emitted isotropically in the plane of motion such that there is
no perpendicular velocity change of the particle (the Lorentz factors
$\gamma$ and $\gamma_{\perp}$ decrease but $\gamma_{\parallel}$
is constant). Thus, the Equation <ref> remains
valid until the particle reaches the ground state. In order to simulate
the full SR process the following procedure was adopted: the particle
Lorentz factor in the circular frame $\gamma_{\perp}$ drops from
its initial value to $\gamma_{\perp}=1$ (i.e., $n=0$) in a series
of steps. Each step entails emission of one synchrotron photon, with
energy $\epsilon_{_{\perp}}$ depending on the current value of $\gamma_{\perp}$.
After the photon emission the energy of the particle is reduced by
$\epsilon_{_{\perp}}$, $\Delta\gamma_{\perp}=\epsilon_{_{\perp}}/mc^{2}$.
Subsequently, the particle with reduced energy emits a photon with
a new value of $\epsilon_{_{\perp}}$. This process continues until
the particle is at $n=0$ Landau level. Depending on the particle's
Landau level $n$, the SR photon energy $\epsilon_{_{\perp}}$ is
chosen in one of three ways.
(I) When the particle is created in a high Landau Level ($n\geq3$),
we choose the energy of the photon randomly but according to a probability
based on the asymptotic synchrotron spectrum (e.g. 164,
\begin{equation}
\frac{{\rm d}^{2}N}{{\rm d}t{\rm d}\epsilon_{\perp}}=\frac{\sqrt{3}}{2\pi}\frac{\alpha_{f}\epsilon_{{\rm _{B}}}}{\epsilon_{\perp}}\times\left[f\cdot F\left(\frac{\epsilon_{\perp}}{f\epsilon_{{\rm _{SR}}}}\right)+\left(\frac{\epsilon_{\perp}}{\gamma_{\perp}mc^{2}}\right)^{2}G\left(\frac{\epsilon_{_{\perp}}}{f\epsilon_{{\rm _{SR}}}}\right)\right],\label{eq:cascade.synchrotron_spectrum}
\end{equation}
\begin{equation}
\epsilon_{_{{\rm SR}}}=\frac{3}{2}\gamma_{\perp}^{2}\hbar\epsilon_{_{B}}
\end{equation}
is the characteristic energy of the synchrotron photons, $f=1-\epsilon_{_{\perp}}/\left(\gamma_{\perp}mc^{2}\right)$
is the fraction of the electron's energy after photon emission, $F\left(x\right)=x\int_{x}^{\infty}K_{5/3}\left(t\right){\rm d}t$,
and $G\left(x\right)=xK_{2/3}\left(x\right)$. The functions $K_{5/3}$
and $K_{2/3}$ correspond to modified Bessel functions of the second
kind. The expression in Equation <ref>
differs from the classical synchrotron spectrum (e.g. 157)
by a factor of $f=1-\epsilon_{_{\perp}}/\left(\gamma_{\perp}mc^{2}\right)$
which appears in several places in Equation <ref>
and by a term with the function $G\left(x\right)$. Note that in the
classical expressions for the total radiation spectra these terms
cancel out. However, as noted by 124 when the quantum
effects are considered there is asymmetry between the perpendicular
and parallel polarisations such that term $G\left(x\right)$ remain.
(II) If $n=2$, the photon's energy is either that required to lower
the particle energy to its first excited state ($n=1$) or to the
ground state ($n=0$). The probability of each process depends on
the local magnetic field strength. We use the simplified prescription
based on the results of 88 to calculate the transition
rates (see also 83). If $\beta_{q}<1$ the energy
of the photon is set to lower the particle energy to the first excited
state, $\epsilon_{_{\perp}}=mc^{2}\left(\sqrt{1+4\beta_{q}}-\sqrt{1+2\beta_{q}}\right)$.
If $\beta_{q}\gtrsim1$ the photon's energy is randomly chosen to
be that which is required to lower the particle energy to either the
first excited state, or the ground state ($\epsilon_{\perp}=mc^{2}\left(\sqrt{1+4\beta_{q}}-1\right)$),
with probability $50\%$ each.
(III) When $n=1$, the photon's energy is chosen to lower the particle's
energy to its ground state, $\epsilon_{_{\perp}}=mc^{2}\left(\sqrt{1+2\beta_{q}}-1\right)$.
If after emission of SR photon the particle is not in the ground state,
$\gamma_{\perp}$ is recalculated and a new energy of photon is chosen.
The photon energy in the co-rotating frame can be calculated as
\begin{equation}
\epsilon=\gamma_{\parallel}\epsilon_{_{\perp}}.
\end{equation}
The weighting factor of the emitted photon is the same as the secondary
particle that emitted it ($\Delta N_{\epsilon}$ ). In the circular
frame the photon is emitted in a random direction perpendicular to
the magnetic field. Hence, in the co-rotating frame the emission angle
can be calculated using Equations <ref> and
<ref> as follows
\begin{equation}
\Psi=\arcsin\sqrt{\frac{\gamma_{\perp}^{2}-1}{\gamma_{\perp}^{2}\gamma_{\parallel}^{2}-1}}\cos\Pi,
\end{equation}
where $\Pi$ is a random number from $0$ to $2\pi$. In our simulation
we include this emission angle by using the same approach as presented
in Section <ref>, but as the maximum value we
use $\Psi$ instead of $1/\gamma$.
The polarisation fraction of SR photons is the exact opposite of the
CR case and it ranges from $50\%$ to $100\%$ polarised perpendicular
to the magnetic field. Following the approach presented by 124
in our calculations the photon polarisation is randomly assign in
the ratio of one $\parallel$ to every seven $\perp$ photons, which
corresponds to a $75\%$ perpendicular polarisation.
Figure <ref> presents the distribution of SR
produced by a single secondary particle. To show the nature of the
distribution, a relatively high pitch angle was used. Note that when
a particle is created at a distance where the magnetic field is relatively
weak (e.g. $\beta_{q}=10^{-5}$ for $\gamma=10^{2}$) then most of
the energy is radiated in the range of $1-10\,{\rm keV}$. Thus, we
believe that if a strong enough instability forms (that increases
the particle's pitch angle), the SR process could be responsible for
the production of a non-thermal component of the X-ray spectrum.
~/Programs/magnetic/magnetic/src/cascade/sr.py (show_spectrum_new)
[Distribution of SR produced by a single secondary particle] Distribution of SR produced by a single secondary particle with
Lorentz factor $\gamma=10^{2}$. We have assumed that the particle
was created in a region where the magnetic field strength was $B=4.14\times10^{8}$
($\beta_{q}=10^{-5}$) and with a pitch angle $\Psi=7^{\circ}$. For
such a relatively high pitch angle the particle loses most of its
energy ending with Lorentz factor $\gamma_{{\rm end}}\approx6$.
Figure <ref> presents the final spectrum produced
by a single primary particle with an initial Lorentz factor of $\gamma_{{\rm c}}=3.5\times10^{6}$
for a non-dipolar configuration of the surface magnetic field of PSR
J0633+1746 (see Section <ref>). Due to CR the particle
loses about $68\%$ of its initial energy ($\Delta\epsilon=2.2\times10^{6}mc^{2}$),
which is radiated mainly in close vicinity of a neutron star, where
curvature of the magnetic field is the highest. As the $\gamma$-photons
propagate they will split (only if the magnetic field is strong enough)
and eventually most photons will be absorbed by the magnetic field
- as a result electron-positron pairs emerge. These pairs radiate
away their transverse momenta through SR, producing mainly X-ray photons
(at larger distances) and only a few $\gamma$-photons (in a strong
magnetic field just above the stellar surface). Note that at the end
(after pair production) only $14\%$ of the primary particle's energy
($\Delta\epsilon_{{\rm ph}}=4\times10^{5}mc^{2}$) is converted into
photons and the bulk of its energy, $54\%$ ($\Delta\epsilon_{{\rm pairs}}=1.8\times10^{6}mc^{2}$),
is allocated into secondary plasma. The multiplicity for this specific
simulation is of the order $M_{{\rm sec}}=10^{4}$. Note that we use
$M_{{\rm sec}}$ to describe the multiplicity of secondary plasma
in contrast to $M_{{\rm pr}}$ which describes particle multiplicity
in the IAR.
[Final photon distribution produced by a single primary particle [CR]]Final photon distribution produced by a single primary particle.
The blue line corresponds to the initial CR photons distribution for
a non-dipolar structure of the magnetic field, while the red line
presents the final distribution with the inclusion of photon splitting,
pair production and SR.
Figure <ref> presents the distribution of
particle energy created by CR photons but with the inclusion of SR
emission (red line). Note that synchrotron emission both lowers the
particle energy (after SR maximum at $\gamma\approx5-8$, while without
SR at $\gamma\approx15-20$) and increases the multiplicity of secondary
plasma $M_{{\rm sec}}\approx10^{4}$.
(plot_pairs2;373_cr_1e05_10m_CR_noSR_PH_e4_leftline, 373_cr_1e05_10m_CR_SR_e4_leftline)
[Distribution of particle energy created by CR photons] Distribution of particle energy created by CR photons calculated
for a non-dipolar structure of the magnetic field. For this specific
magnetic field configuration and initial parameters (see the caption
of Figure <ref>) the secondary plasma multiplicity
is $M_{{\rm sec}}\approx10^{4}$. Note that this result does not include
Synchrotron Radiation and the actual energies of the created pairs
are lower as they lose their transverse momenta (see Section <ref>).
§ INVERSE COMPTON SCATTERING
The Inverse Compton Scattering (hereafter ICS) process in the neutron
star vicinity has been studied extensively by 183, 100, 184, 46, 50, 51, 20, 32, 165, 189, 190, 188, 82,
etc. According to these studies, the ICS process may play a significant
role in the physics of a neutron star's magnetosphere. Relativistic
particles (positrons and electrons) can Compton-scatter thermal radiation
from the neutron star surface. As a particle with a certain relativistic
velocity scatters the thermal photons with a blackbody distribution,
it will produce radiation in quite a wide energy range. However, we
can distinguish two characteristic frequencies of upscattered photons:
one is the frequency due to resonant scattering, another is the range
of frequencies contributed by the scattering of photons with frequencies
around the ”thermal-peak”. The Resonant Inverse Compton Scattering
(RICS) corresponds to a scenario when the scattering cross section
is largest. On the other hand, Thermal-peak Inverse Compton Scattering
(TICS) corresponds to interactions with photons with the maximum number
density. These two modes are very different when it comes to the nature
of the process. The photons' energy in RICS depends on the strength
of the magnetic field, thus at low altitudes (where the field is very
strong), it can power pair cascades, while TICS can be responsible
for magnetospheric radiation at much higher altitudes. Note that for
some specific combinations of magnetic strength and distribution of
background photons, RICS and TICS are indistinguishable as the resonance
frequency falls into the thermal peak range.
§.§ The cross section of ICS
Due to the rapid time scale for synchrotron emission (see section
<ref>), a particle in an excited Landau level
almost instantaneously de-excites to the ground level. The particle
motion is therefore strongly confined to the magnetic field direction.
In our calculations we consider the geometry illustrated in Figure
<ref>. In the observer's frame of reference (OF),
a particle with Lorentz factor $\gamma$ travelling along the magnetic
field line scatters a photon. Let $\psi=\arccos\mu$ be the angle
between the magnetic field line (particle propagation) and the direction
of photon propagation in OF and $\psi^{\prime}=\arccos\mu^{\prime}$
in the particle rest frame (PRF). The energy of the photon in PRF
is given by
\begin{equation}
\epsilon^{\prime}=\gamma\epsilon\left(1-\beta\mu\right).\label{cascade.erf_phot_eng}
\end{equation}
After scattering, the photon energy is denoted by $\epsilon{}_{s}^{\prime}$
in PRF and $\epsilon_{s}$ in OF. The angle between the direction
of propagation of the scattered photon and ${\bf B}$ (which describes
the direction of particle propagation) is denoted by $\psi_{s}=\arccos\mu_{s}$
in OF and $\psi{}_{s}^{\prime}=\arccos\mu{}_{s}^{\prime}$, where
in PRF [51].
[Geometry of Inverse Compton Scattering] Reproduction of the Figure from 51. Geometry of
the ICS event in the observer's frame (left) and the particle rest
frame (right). A particle with Lorentz factor $\gamma$, beamed along
the direction of the magnetic field, scatters a photon with energy
$\epsilon$ directed at angle $\psi$ with respect to the magnetic
field line. After scattering, the energy and angle of the photon are
denoted by $\epsilon_{s}$ and $\psi_{s}$, respectively. Quantities
in the particle rest frame are denoted by a prime.
§.§.§ ICS cross section in the Thomson regime
Restriction to the Thomson regime requires that $\gamma\epsilon\left(1-\mu\right)\ll1$.
In the particle rest frame, the angle $\psi^{\prime}=\arcsin\left\{ \gamma^{-1}\left[\sin\psi/\left(1-\beta\cos\psi\right)\right]\right\} $,
and when $\gamma\gg1$, $\left|\mu^{\prime}\right|\to1$. In the Thomson
regime the only important Compton scattering process involves transitions
between ground-state Landau levels. 46 and 50
calculated the differential cross section (after summing over polarisation
modes and integrating over azimuth) for a photon scattered from $\psi^{\prime}=0^{\circ}$
into angle $\psi{}_{s}^{\prime}=\arccos\mu{}_{s}^{\prime}$ as follows
\begin{equation}
\frac{{\rm d}\sigma^{\prime}}{{\rm d}\mu{}_{s}^{\prime}}=\frac{3\sigma_{_{{\rm T}}}}{16}\left(1+\mu{}_{s}^{\prime2}\right)\left[\frac{\epsilon{}^{\prime2}}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right){}^{2}}+\frac{\epsilon{}^{\prime2}}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right){}^{2}+\left(\Gamma/2\right){}^{2}}\right],\label{cascade.ics_simple_cross}
\end{equation}
where $\Gamma=4\alpha_{f}\epsilon_{_{B}}^{2}/3$ is the resonant width
[47, 184], $\sigma_{_{{\rm T}}}$ is the Thomson
cross section, and $\alpha_{f}=e^{2}/\hbar c$ is the fine-structure
constant. In the Thomson limit $\epsilon^{\prime}\ll1$, and thus
the scattered photon energy in PRF can be approximated as
\begin{equation}
\epsilon{}_{s}^{\prime}\simeq\epsilon^{\prime}+\epsilon{}^{\prime2}(\mu^{\prime}-\mu{}_{s}^{\prime})^{2}/2\approx\epsilon^{\prime}.\label{cascade.ics_simple_scateng}
\end{equation}
Equations <ref> and <ref>
show that a differential magnetic Compton scattering cross section
when $\gamma\gg1$ is similar in form to a nonmagnetic Thomson cross
section. The important difference is that the magnitude of the cross
section is enhanced when $\epsilon^{\prime}$ approaches $\epsilon_{_{B}}$
and is depressed at energies $\epsilon^{\prime}<\epsilon_{_{B}}$.
The total cross section for magnetic Compton scattering, obtained
by integrating Equation <ref> over $\mu{}_{s}^{\prime}$,
was calculated by 50, 190 and is given by
\begin{equation}
\sigma^{\prime}=\frac{\sigma_{_{{\rm IC}}}}{2}\left[\frac{u^{2}}{\left(u+1\right){}^{2}}+\frac{u^{2}}{\left(u-1\right){}^{2}+a^{2}}\right],\label{eq:cascade.ics_cross}
\end{equation}
where $\sigma_{_{{\rm IC}}}=\sigma_{{\rm _{T}}}$, $\sigma_{{\rm _{T}}}$
is the Thomson cross section, $u=\epsilon^{\prime}/\epsilon_{_{B}}$,
§.§.§ ICS cross section in the Klein-Nishina regime
The Klein-Nishina regime includes quantum effects due to the relativistic
nature of scattering, and it requires that $\gamma\epsilon\left(1-\mu\right)\gtrsim1$.
The principal effect is to reduce the cross section from its classical
value as the photon energy in PRF becomes large. In the Klein-Nishina
regime instead of $\sigma_{{\rm _{IC}}}=\sigma_{_{{\rm T}}}$ we can
use the following relationship
\begin{equation}
\sigma_{_{{\rm IC}}}=\sigma_{_{{\rm KN}}}=\frac{3}{4}\sigma_{_{{\rm T}}}\left\{ \frac{1+\epsilon^{\prime}}{\epsilon^{\prime3}}\left[\frac{2\epsilon^{\prime}\left(1+\epsilon^{\prime}\right)}{1+2\epsilon^{\prime}}-\ln\left(1+2\epsilon^{\prime}\right)\right]+\frac{1}{2\epsilon^{\prime}}\ln\left(1+2\epsilon^{\prime}\right)-\frac{1+3\epsilon^{\prime}}{\left(1+2\epsilon^{\prime}\right)^{2}}\right\} .
\end{equation}
In an extreme relativistic regime $\epsilon^{\prime}\gg1$ the Klein-Nishina
formula can be simplified to
\begin{equation}
\sigma_{{\rm _{KN}}}\approx\frac{3}{8}\sigma_{_{{\rm T}}}\epsilon^{\prime-1}\left[\ln\left(2\epsilon^{\prime}\right)+\frac{1}{2}\right].\label{eq:cascade.kn_cross}
\end{equation}
The above formula clearly shows that Inverse Compton Scattering is
less efficient for photons with energy in PRF significantly exceeding
particle rest energy.
§.§.§ QED Compton Scattering cross section
Previous studies on upscattering and energy loss by relativistic particles
have used the non-relativistic, magnetic Thomson cross section for
resonant scattering or the Klein-Nishina cross section for thermal-peak
scattering. As noted by 71, this approach does
not account for the relativistic quantum effects of strong magnetic
fields ($B>10^{12}\,{\rm G}$). When the photon energy exceeds $mc^{2}$
in the particle rest frame, the strong magnetic field significantly
lowers the Compton scattering cross section below and at the resonance.
71 developed expressions for the scattering of
ultrarelativistic electrons with $\gamma\gg1$ moving parallel to
the magnetic field. Because of the large Lorentz Factor of particle
$\gamma$, the photon incident angle $\psi$ gets Lorentz concentrated
to $\psi^{\prime}\approx\psi/2\gamma\approx0^{\circ}$ in the PRF.
The differential cross section in the rest frame of the particle can
be written as
\begin{equation}
\frac{{\rm d}\sigma{}_{\|,\perp}^{\prime}}{{\rm d}\cos\psi{}_{s}^{\prime}}=\frac{3\sigma_{_{{\rm T}}}}{16\pi}\frac{\epsilon{}_{s}^{\prime2}e^{-\epsilon{}_{s}^{\prime2}\sin^{2}\left(\psi_{s}^{\prime}/2\epsilon_{_{B}}\right)}}{\epsilon^{\prime}\left(2+\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\right)\left[\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right]}\frac{1}{l!}\left(\frac{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon_{_{B}}}\right)G_{\|,\perp},\label{eq:cascade.cross_gonthier2}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\begin{split}\hat{G}_{\mathrm{no-flip}}^{\|}= & \int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\|,\|}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\perp,\|}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left\{ \left[\left(B_{1}+B_{3}+B_{7}\right)\cos\psi{}_{s}^{\prime}-\left(B_{2}+B_{6}\right)\sin\psi{}_{s}^{\prime}\right]^{2}+\left(B_{4}\cos\psi{}_{s}^{\prime}-B_{5}\sin\psi{}_{s}^{\prime}\right)^{2}\right\} ,\\
\hat{G}_{\mathrm{no-flip}}^{\perp}= & \int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\|,\perp}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{no-flip}}^{\perp,\perp}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left[\left(B_{1}-B_{3}-B_{7}\right)^{2}+B_{4}^{2}\right],\\
\hat{G}_{\mathrm{flip}}^{\|}= & \int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\|,\|}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\perp,\|}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left\{ \left[\left(C_{1}+C_{3}+C_{7}\right)\cos\psi{}_{s}^{\prime}-\left(C_{2}+C_{6}\right)\sin\psi{}_{s}^{\prime}\right]^{2}+\left(C_{4}\cos\psi{}_{s}^{\prime}-C_{5}\sin\psi{}_{s}^{\prime}\right)^{2}\right\} ,\\
\hat{G}_{\mathrm{flip}}^{\perp}= & \int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\|,\perp}\right|^{2}{\rm d}\phi^{\prime}=\int_{0}^{2\pi}\left|G_{\mathrm{flip}}^{\perp,\perp}\right|^{2}{\rm d}\phi^{\prime}=\\
= & 2\pi\left[\left(C_{1}-C_{3}-C_{7}\right)^{2}+C_{4}^{2}\right].
\end{split}
\end{equation}
The imaginary terms and the $\phi^{\prime}$ dependence are isolated
in the polarisation components and in the phase exponentials, leading
to elementary integrations over the azimuthal angle, $\phi^{\prime}$
The differential cross section depends on the final Landau state $l$,
thus a sum must be calculated over all the contributing Landau states.
The energy of the scattered photon is given by [71]
\begin{equation}
\epsilon{}_{s}^{\prime}=\frac{2\left(\epsilon^{\prime}-l\epsilon_{_{B}}\right)}{1+\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)+\left\{ \left[1+\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)\right]^{2}-2\left(\epsilon^{\prime}-l\epsilon_{_{B}}\right)\sin^{2}\psi{}_{s}^{\prime}\right\} ^{\frac{1}{2}}},
\end{equation}
where $l$ is the final Landau level of the scattered particle. Each
final state has an energy threshold of $l\epsilon_{_{B}}$, thus the
maximum contributing Landau state $l_{{\rm max}}$ can be expressed
as: $\epsilon^{\prime}/\epsilon_{_{B}}-1<l_{{\rm max}}<\epsilon^{\prime}/\epsilon_{_{B}}$.
To obtain the energy-dependent cross section, the Romberg's method
can be used to numerically integrate the differential cross section
over $\psi{}_{s}^{\prime}$. For this particular case (scattering
of relativistic particles) there is only one resonance appearing at
the fundamental cyclotron frequency $\epsilon_{_{B}}=\beta_{q}=eB/\left(mc\right)$.
The values of $B$ and $C$ can be expressed as:
\begin{equation}
\begin{split}B_{1}= & \frac{2\epsilon^{\prime}-\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)}{2\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
B_{2}= & -\frac{\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\left(2l\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)+2l\epsilon_{_{B}}\epsilon^{\prime}}{2\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
B_{3}= & \frac{l\epsilon_{_{B}}\left(2l\epsilon_{_{B}}-2\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)}{\epsilon{}_{s}^{\prime2}\sin^{2}\left[\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)\right]},\\
B_{4}= & -\frac{2\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
B_{5}= & -\frac{\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}}{2\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
B_{6}= & \frac{l\epsilon_{_{B}}\cos\psi{}_{s}^{\prime}}{\sin\psi{}_{s}^{\prime}\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
B_{7}= & \frac{2l\left(l-1\right)\epsilon_{_{B}}^{2}}{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left[\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
C_{1}= & \sqrt{2l\epsilon_{_{B}}}\frac{\epsilon^{\prime}}{2\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{2}= & -\sqrt{2l\epsilon_{_{B}}}\frac{2\epsilon^{\prime}+2\epsilon^{\prime}{}^{2}-\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-2l\epsilon_{_{B}}+\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon^{\prime}{}_{s}\sin\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{3}= & \sqrt{2l\epsilon_{_{B}}}\frac{\left(\epsilon^{\prime}-\epsilon_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)\left(2l\epsilon_{_{B}}-2\epsilon_{_{B}}-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\right)}{2\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)},\\
C_{4}= & -\sqrt{2l\epsilon_{_{B}}}\frac{\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}}{2\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
C_{5}= & \sqrt{2l\epsilon_{_{B}}}\frac{\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}}{2\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}-\epsilon_{_{B}}\right]},\\
C_{6}= & -\sqrt{2l\epsilon_{_{B}}}\frac{2\epsilon{}_{s}^{\prime}+\epsilon^{\prime}\epsilon{}_{s}^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}}{2\epsilon{}_{s}^{\prime}\sin\psi{}_{s}^{\prime}\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]},\\
C_{7}= & \sqrt{2l\epsilon_{_{B}}}\frac{\left(l-1\right)\epsilon_{_{B}}\left(\epsilon^{\prime}-\epsilon{}_{s}^{\prime}\cos\psi{}_{s}^{\prime}\right)}{\epsilon{}_{s}^{\prime2}\sin^{2}\psi{}_{s}^{\prime}\left[\epsilon{}_{s}^{\prime}\epsilon^{\prime}\left(1-\cos\psi{}_{s}^{\prime}\right)-\epsilon^{\prime}+\epsilon_{_{B}}\right]}.
\end{split}
\end{equation}
Although the expressions presented above describe the exact cross
section for ICS in strong magnetic fields, due to their complexity
their usage in cascade simulation is limited.
§.§.§ Approximate cross section (final states l=0)
An approximation to the exact $l=0$ differential cross section can
be given by assuming that the scattering is significantly below the
resonance, where $\epsilon^{\prime}<\epsilon_{_{B}}$ and also $\epsilon^{\prime}<1$.
71 showed that by keeping only terms to first
order in $\epsilon^{\prime}$ and $\epsilon{}_{s}^{\prime}$ in the
region of validity, it agrees very well with the exact $l=0$ cross
section. The approximation overestimates the exact $l=0$ cross section
above the region of validity $\epsilon^{\prime}>\epsilon_{_{B}}$.
However, the approximation is close to the total cross section for
both energy regions ($\epsilon^{\prime}<\epsilon_{_{B}}$ and $\epsilon^{\prime}>\epsilon_{_{B}}$
), even for high magnetic field strengths (see Figure <ref>).
According to 71, the polarisation-dependent and
averaged, approximate cross section can be calculated as:
\begin{equation}
\sigma{}^{\prime\|\rightarrow\|}=\sigma{}^{\prime\perp\rightarrow\|}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)-h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],
\end{equation}
\begin{equation}
\sigma{}^{\prime\|\rightarrow\perp}=\sigma{}^{\prime\perp\rightarrow\perp}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[f\left(\epsilon^{\prime}\right)-2\epsilon^{\prime}h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],
\end{equation}
\begin{equation}
\sigma{}_{{\rm avg}}^{\prime}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)+f\left(\epsilon^{\prime}\right)-\left(1+2\epsilon^{\prime}\right)h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right],\label{cascade.cross_gonthier}
\end{equation}
\begin{equation}
\begin{array}{c}
\end{array}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
h\left(\epsilon^{\prime}\right)=\left\{ \begin{array}{ll}
\frac{\epsilon{}^{\prime2}}{\sqrt{\epsilon^{\prime}\left(2-\epsilon^{\prime}\right)}}\arctan\left[\frac{\sqrt{\epsilon^{\prime}\left(2-\epsilon^{\prime}\right)}}{1+\epsilon^{\prime}}\right] & \mbox{ for \ensuremath{\epsilon^{\prime}<2}},\\
\frac{\epsilon^{\prime}{}^{2}}{2\sqrt{\epsilon^{\prime}\left(\epsilon^{\prime}-2\right)}}\ln\left[\frac{\left(1+\epsilon^{\prime}+\sqrt{\epsilon^{\prime}(\epsilon^{\prime}-2)}\right)^{2}}{1+4\epsilon^{\prime}}\right] & \mbox{ for \ensuremath{\epsilon^{\prime}>2}}.
\end{array}\right.
\end{equation}
Figure <ref> presents the total approximate
cross section of Compton scattering, the exact QED cross section (summed
over all contributing final electron/positron Landau states) and the
exact cross section for final Landau state $l=0$ as a function of
energy of the incident photon in PRF (in units of cyclotron energy,
$\epsilon^{\prime}/\epsilon_{_{B}}$). As mentioned above, the approximation
is valid in the region below the resonance, $\epsilon^{\prime}<\epsilon_{_{B}}$.
Although the approximation overestimates the cross section for $l=0$
final Landau state in the regime of high energetic photons ($\epsilon^{\prime}>\epsilon_{_{B}}$),
it can be used in this regime as the approximation of the total cross
section. In our simulation we use this approach to calculate the total
ICS cross section in both regimes, $\epsilon^{\prime}<\epsilon_{_{B}}$
and $\epsilon^{\prime}>\epsilon_{_{B}}$. Calculation of the cross
section for the resonance frequency ($\epsilon^{\prime}=\epsilon_{{\rm _{B}}}$)
is presented in the next section.
[Total cross section of ICS as a function of an incident photon energy] Total cross section of Compton scattering (in Thomson units) as
a function of an incident photon energy in PRF (in units of the cyclotron
energy) calculated for a magnetic field strength $B_{14}=3.5$. The
exact QED scattering cross section, summed over all contributing final
electron/positron Landau states, is indicated as the red dotted curve.
The cross section for final Landau states $l=0$ is plotted as a blue
dashed line.
§.§ Resonant Compton Scattering
This section describes an approach used to calculate the RICS cross
section for ultrastrong magnetic fields ($B>10^{12}\,{\rm G}$). For
weaker fields the calculations are much simpler and resonance is already
included in Equation <ref>.
The trend as $\beta_{q}$ increases is for the magnitude of the cross
section to drop at all energies. For weaker magnetic fields ($\beta_{q}<1$)
the width of the resonance increases with increasing $\beta_{q}$,
but for $\beta_{q}\ge1$ this width actually declines. Since the resonance
is formally divergent, the common practice (see 184, 112, 46, 51, 80, 9, 81, 12, 13)
is to truncate it at $\epsilon^{\prime}=\epsilon_{_{B}}$ by introducing
a finite width $\Gamma$. The procedure is to replace the resonant
$\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}$ denominator
(see Equations <ref> and <ref>)
by $\left[\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}+\Gamma^{2}/4\right]$.
In the $\beta_{q}\ll1$ regime, the cyclotron decay width assumes
the well-known result $\Gamma\approx4\alpha_{f}\epsilon_{_{B}}^{2}/3$
in dimensionless units. When $\beta_{q}\gg1$, quantum and recoil
effects generate $\Gamma\approx\alpha_{f}\epsilon_{_{B}}\left(1-1/\tilde{e}\right)$
where $\tilde{e}$ is Euler's number (e.g. see 14).
These widths lead to areas under the resonance being independent of
$\epsilon_{_{B}}$ in the magnetic Thomson regime of $\beta_{q}\ll1$
and scaling as $\epsilon_{_{B}}^{1/2}$ when $\beta_{q}\gg1$. These
results can be deduced using the $l=0$ approximation derived in Equation
<ref>. By using this approach the averaged,
approximate cross section can be written as
\begin{equation}
\sigma{}_{{\rm avg}}^{\prime}=\frac{3\sigma_{_{{\rm T}}}}{16}\left[g\left(\epsilon^{\prime}\right)+f\left(\epsilon^{\prime}\right)-\left(1+2\epsilon^{\prime}\right)h\left(\epsilon^{\prime}\right)\right]\left[\frac{1}{\left(\epsilon^{\prime}-\epsilon_{_{B}}\right)^{2}+\Gamma{}^{2}/4}+\frac{1}{\left(\epsilon^{\prime}+\epsilon_{_{B}}\right)^{2}}\right].\label{eq:cascade.sigma_gont}
\end{equation}
The common practice to calculate a resonant cross section in an ultrastrong
magnetic fields is to use the Dirac delta function as follows (e.g.
\begin{equation}
\sigma_{{\rm res}}^{\prime}\simeq2\pi^{2}\frac{e^{2}\hbar}{mc}\delta\left(\epsilon{}_{s}^{\prime}-\epsilon_{_{B}}\right)\label{eq:cascade.res_cross_a}
\end{equation}
This simplified approach, however, does not include scatterings of
photons whose energy in a particle rest frame is not equal but very
close to the resonance frequency. The relativistic quantum effects
of strong magnetic fields that are included in the approximate solution
increase the cross section, and thus the efficiency of the ICS process
in previous estimates could be underestimated.
According to 124 in ultrastrong magnetic fields the
ICS polarisation fraction is about $50\%$ (approximately $50\%$
of the photons are slightly above resonance and $50\%$ are slightly
below). Therefore, the polarisation of ICS photons is randomly assigned
in the ratio of one $\perp$ (perpendicular to the field) to every
$\parallel$ photon.
§.§ Particle mean free path
For the ICS process the calculation of the particle mean free path
$l_{{\rm ICS}}$ is not as simple as that of the CR process. Although
we can define $l_{{\rm ICS}}$ in the same way as we defined $l_{{\rm CR}}$,
it is difficult to estimate a characteristic frequency of emitted
photons. We have to take into account photons of various frequencies
with various incident angles. An estimation of the mean free path
of a positron (or electron) to produce a photon is [184]
\begin{equation}
l_{{\rm ICS}}\approx\left[\int_{\mu_{0}}^{\mu_{1}}\int_{0}^{\infty}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1},
\end{equation}
where (as before) $\beta=v/c$ is the velocity in terms of speed of
light, $n_{{\rm ph}}$ represents the photon number density distribution
of semi-isotropic blackbody radiation (see Equation <ref>).
Here $\sigma^{\prime}$ is the average cross section of scattering
in the particle rest frame (see Equation <ref>).
We should expect two modes of the ICS process, i.e. Resonant ICS and
Thermal-peak ICS.
§.§.§ Resonant ICS
The RICS takes place if the photon frequency in the particle rest
frame is equal to the cyclotron electron frequency. Using Equation
<ref> we can write that the incident photon
energy is $\epsilon=\epsilon_{_{B}}/\left[\gamma\left(1-\beta\mu\right)\right]$.
For altitudes of the same order as the polar cap size we use $\mu_{0}=1$,
$\mu_{1}=0$ as incident angle limits for outflowing particles, and
$\mu_{0}=0$, $\mu_{1}=-1$ as incident angle limits for backflowing
particles. Thus, for outflowing particles the electron/positron mean
free path above a polar cap for the RICS process is
\begin{equation}
l_{{\rm RICS}}\approx\left[\int_{0}^{1}\int_{\epsilon_{{\rm _{res}}}^{^{{\rm min}}}}^{\epsilon_{{\rm _{res}}}^{{\rm ^{max}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1},\label{eq:cascade.ics_free_path}
\end{equation}
where limits of integration, $\epsilon_{{\rm _{res}}}^{^{{\rm min}}}$
and $\epsilon_{_{{\rm res}}}^{{\rm ^{max}}}$, are chosen to cover
the resonance. In our simulation we use such limits to include the
region where the integrated function decreases up to about two orders
of magnitude from its maximum:
\begin{equation}
\epsilon_{_{{\rm res}}}^{^{{\rm min/max}}}=\frac{\epsilon_{_{B}}\pm\frac{3}{2}\sqrt{11}\Gamma}{\gamma\left(1-\beta\mu\right)}.
\end{equation}
Here $\Gamma$ is the finite width introduced in Section <ref>
to describe the decay of an excited intermediate particle state.
Figure <ref> presents the dependence of
the integrand from Equation <ref> on the
incident photon energy for a given incident angle. The maximum of
the integrand shows a significant decline for stronger magnetic fields.
This is due to both the drop of the cross section at all energies
with an increasing magnetic field (see Section <ref>)
and due to the fact that for this specific incident angle resonance
is in a different range of photon energy. In stronger magnetic fields
resonance occurs not only for higher energetic photons but also the
width of the resonance is wider (see the right panel of Figure <ref>).
[Dependence of the integrand from Equation <ref>
on the energy of the incident photon]Dependence of the integrand from Equation <ref>
on the energy of the incident photon. Both panels were calculated
for surface temperature $T=3\times10^{6}\,{\rm K}$, cosine of the
incident angle $\mu=0.1$ and Lorentz factor of particle $\gamma=10^{3}$.
The left panel corresponds to resonance in magnetic field $B=10^{14}\,{\rm G}$,
while the right panel was obtained using $B=3\times10^{14}\,{\rm G}$.
Note that both plots do not include the dependence of the photon density
on distance from the stellar surface. Depending on whether the radiation
originates from the whole stellar surface or from the polar cap only,
the dependence of the photon number density on the height above the
surface can differ significantly (see Section <ref>).
§.§.§ Thermal-peak ICS
TICS includes all scattering processes of photons with frequencies
around the maximum of the thermal spectrum. In our simulation we adopt
$\epsilon_{{\rm _{th}}}^{{\rm ^{min}}}\approx0.05\epsilon_{{\rm _{th}}}$,
and $\epsilon_{{\rm _{th}}}^{^{{\rm max}}}\approx2\epsilon_{{\rm _{th}}}$
where $\epsilon_{{\rm _{th}}}=2.82kT/\left(mc^{2}\right)$ is the
energy, in units of $mc^{2}$, at which blackbody radiation with temperature
$T$ has the largest photon number density. The electron/positron
mean free path for the TICS process can be calculated as
\begin{equation}
l_{{\rm TICS}}\approx\left[\int_{\mu_{0}}^{\mu_{1}}\int_{\epsilon_{_{{\rm th}}}^{{\rm ^{min}}}}^{\epsilon_{{\rm _{th}}}^{{\rm ^{max}}}}\left(1-\beta\mu\right)\sigma^{\prime}\left(\epsilon,\mu\right)n_{{\rm ph}}\left(\epsilon\right){\rm d}\epsilon{\rm d}\mu\right]^{-1}.\label{eq:cascade.t_ics}
\end{equation}
Figure <ref> presents the dependence of the
integrand from Equation <ref> on photon energy for
two different incident angles of background photons. As the number
density depends exponentially on the photon energy, TICS is important
only for small incident angles ($\mu\approx1$). Note that for some
specific combination of magnetic field strength, the Lorentz factor
of the primary particle and the incident angle of background photons
the resonance is in region of thermal peak. In such a case the resonant
component is dominating (a much higher cross section) and the particle
mean free path should be calculated using the approach described in
the previous section.
[The integrand from Equation <ref> vs. photon number
density]Comparison of the integrand from Equation <ref>
with photon number density. The bottom panels present the dependence
of the photon number density on photon energy in OF. The red dashed
lines correspond to limits used to calculate the particle mean free
path for TICS. The top panels present the dependence of the integrand
on photon energy in PRF. Both panels were obtained using surface temperature
$T=3\times10^{6}\,{\rm K}$, Lorentz factor of the particle $\gamma=10^{2}$
and magnetic field strength $B=10^{12}\,{\rm G}$. The cosine of the
incident angle, $\mu=0.975$ and $\mu=0.96$, was used for the left
and right panel, respectively.
§.§.§ Calculation results
For ultrastrong magnetic fields quite a wide range of the particle
Lorentz factor falls into the peak of background photons (see Figure
<ref>). In such a case RICS is enhanced by the
fact that it involves photons with very high density. Furthermore,
the RICS process for such particles is indistinguishable from the
TICS (see Figure <ref>). For particles with
Lorentz Factor $\gamma\gtrsim10^{5}$, the dominant process of radiation
is CR. The exact value of this limit depends on conditions such as:
density of background photons, incident angles between particles and
photons, and curvature of magnetic field lines ($1/\Re$).
(show_le_gamma - Graphs, $B_{14}=2.$)
[Dependence of a particle mean free path on its Lorentz factor]Dependence of a particle mean free path on its Lorentz factor for
three different processes: CR, RICS and TICS. The calculations were
performed for magnetic field strength $B_{14}=2$, radius of curvature
of magnetic field lines $\Re_{6}=1$ (for the CR process) and hot
spot temperature $T_{6}=3$ (for RICS and TICS). Both RICS and TICS
were calculated for a full range of incident angles ($\mu_{0}=0$,
$\mu_{1}=1$). Note that for a Lorentz factor in the range of $\gamma\approx2\times10^{3}-10^{5}$
the particle mean free paths of RICS and TICS are equal as the resonance
falls into the peak of the background photons.
Figure <ref> presents the dependence of
a particle mean free path on the magnetic field strength and the particle
Lorentz factor for RICS. The minimum of the mean free path for relatively
weak magnetic fields ($B_{14}=0.5$) is for particles with Lorentz
factor $\gamma\approx2\times10^{3}$, while for relatively stronger
magnetic fields ($B_{14}=3.5$) the RICS is most efficient for particles
with energy an order of magnitude larger ($\gamma\approx2\times10^{4}$).
This is a natural consequence of the fact that resonance takes place
when the photon energy in PRF is equal to the electron cyclotron energy,
which in stronger fields is higher. As can be seen from the Figure,
the particle mean free paths for RICS in stronger magnetic fields
increase. This is due to the decreasing resonant cross section with
increasing magnetic field strength (see Figure <ref>).
Note, however, that this behaviour does not include the fact that
photon density in regions with weaker magnetic fields is considerably
smaller. In fact, the results of the cascade simulation presented
in Chapter <ref> show that RICS is efficient only in
the immediate vicinity of a neutron star since photon density at higher
altitudes drops rapidly.
(show_le3d_gonthier, $T_{6}=2$)
[Dependence of a particle mean free path on magnetic field strength
and the Lorentz factor of a particle [RICS] ]Dependence of a particle mean free path on magnetic field strength
($B_{14}$) and the Lorentz factor of a particle ($\gamma$) for the
RICS process. The particle mean free path was calculated for semi-isotropic
blackbody radiation ($\mu_{0}=0$, $\mu_{1}=1$) with temperature
§.§ Background photons
§.§.§ Photon density
One of the main parameters affecting ICS above the stellar surface
is photon density. The initial photon density (at altitude $z=0$)
highly depends on the temperature of the radiating surface. As shown
in Chapter <ref> (e.g. see Table <ref>),
the entire surface has the lowest temperature ($T_{6}\lesssim0.8$),
thus the initial photon density is up to about two orders of magnitude
lower than warm spot radiation ($T_{6}\lesssim3$) and up to about
three orders magnitude lower than hot spot radiation ($T_{6}\lesssim5$).
However, the density of the photons strongly depends on the distance
from the source of radiation (especially for the hot spot). Therefore,
we used the simplified method presented in Figure <ref>
to calculate photon density at a given point $L=\left(r,\,\theta,\,\phi\right)$.
Then the relative density of photons originating from the entire surface
can be calculated as
\begin{equation}
\frac{n_{{\rm st}}\left(\epsilon,\, T_{{\rm st}},\, L\right)}{n_{0}\left(\epsilon,\, T_{{\rm st}}\right)}=\sin^{2}\left(\frac{\Delta\theta_{{\rm st}}}{2}\right)=\left(\frac{R}{r}\right)^{2},\label{eq:cascade.n_ph_dist}
\end{equation}
where $n_{{\rm st},0}\left(\epsilon,\, T_{{\rm st}}\right)$ is the
density of photons with energy $\epsilon$ at the stellar surface
with temperature $T_{{\rm st}}$, and $\Delta\theta_{{\rm st}}$ is
the angular diameter of the star at a distance from the star centre
Likewise, we can write a formula for the relative density of photons
originating from a spot (warm or hot) as
\begin{equation}
\frac{n_{{\rm sp}}\left(\epsilon,\, T_{{\rm sp}},\, L\right)}{n_{{\rm sp},0}\left(\epsilon,\, T_{{\rm sp}}\right)}=\sin^{2}\left(\frac{\Delta\theta}{2}\right),
\end{equation}
where $n_{{\rm sp},0}\left(\epsilon,\, T_{{\rm sp}}\right)$ is the
density of photons with energy $\epsilon$ at the spot surface (either
hot or warm) with temperature $T_{{\rm sp}}$. The angular diameter
of the spot can be calculated as
\begin{equation}
\Delta\theta=\arccos\left(\frac{r_{1}^{2}+r_{2}^{2}-4R_{{\rm sp}}}{2r_{1}r_{2}}\right),
\end{equation}
here $R_{{\rm sp}}$ is the spot radius and $r_{1}$, $r_{2}$ are
the distances to the outer edges of the spot (see Figure <ref>).
[Simplified method used for calculating the background photon density]Simplified method used for calculation of a photon density originating
from an entire stellar surface (blue lines) and from a hot/warm spot
(red lines). Here $R_{{\rm sp}}$ is a spot radius (either hot or
warm). Let us note that the simplified method is valid for the entire
surface component regardless of the $\phi$ component of location
$L$, while for the spot component it can be used only for small values
of $\phi$. In a more general case the spot should be projected on
the surface perpendicular to the radius vector ${\bf r}$ and passing
through point $L$.
Figure <ref> presents the dependence
of the relative photon density ($n\left(z\right)/n_{0}$) on the distance
from the stellar surface. Due to the small size of a polar cap (hot
$R_{{\rm hs}}=50\,{\rm m}$) the density of the photons drops rapidly
and already at a distance of about $z=150\,{\rm m}$ it is one order
of magnitude lower than at the polar cap surface. On the other hand,
for a larger size of the warm spot ($R_{{\rm hs}}=1\,{\rm km}$) the
photon density is reduced by an order of magnitude at a distance of
about $z=3\,{\rm {\rm km}}$. From Equation <ref>
it can easily be seen that the photon density of radiation from the
entire stellar surface decreases by an order of magnitude at a distance
of about $z\approx3R\approx30\,{\rm km}$.
[Dependence of the relative photon density on the distance from the
stellar surface]Dependence of the relative photon density on the distance from the
stellar surface for three different thermal components (the entire
stellar surface, the warm spot and the hot spot). The following parameters
were used for the calculations: star radius $R=10\,{\rm km}$, warm
spot radius $R_{{\rm ws}}=1\,{\rm km}$ and hot spot radius $R_{{\rm hs}}=50\,{\rm m}$.
The very small size of the polar cap also has an additional implication
to the background photons' density. Namely, the density of the background
photons just above the polar cap highly depends not only on the distance
from the surface, but also on the position relative to the cap centre.
Figure <ref> presents the dependence
of the relative photon density originating from a polar cap (the hot
spot) on the distance from the stellar surface for three different
starting points on the polar cap. The distance was calculated for
points which follow the magnetic field structure of PSR B0656+14.
Note that for the extreme magnetic line (which starts at the cap edge)
already at a distance of about $z_{2}\approx5\,{\rm m}$ the photon
density decreases twice, while for the central ($\theta_{0}$) and
middle line ($\theta_{1}$) the distances are respectively $z_{0}\approx45\,{\rm m}$
and $z_{1}\approx30\,{\rm m}$. This result is important as the background
photon density directly translates to the particle mean free path
in ICS (see Section <ref>). This means
that for ICS-dominated gaps the sparks' height will vary depending
on their location. The breakdown of the gap (spark) in the central
region of a polar cap is easier to develop as the particle mean free
path is lower, and eventually it will result in lower heights of the
central sparks. This will influence the properties of plasma produced
in the central region of open magnetic field lines, and depending
on the conditions may result in the formation of plasma either suitable
to produce radio emission (core emission) or unsuitable to produce
radio emission (conal emission but with the line of sight crossing
the centre of the beam).
To find the dominant component of thermal radiation at a given altitude
we need to take into account the initial flux of radiation and how
it changes with the distance. Below we present the calculations of
a radiation flux (Figure <ref>)
for PSR B0656+14. The parameters of an entire surface and warm spot
components are in agreement with the observations (see Table <ref>),
while the hot spot component was calculated using parameters derived
from the modelling of a non-dipolar structure of the magnetic field
(see Chapter <ref>).
(show_hotspot, 430)
[Dependence of the relative photon density on the distance from the
stellar surface for a hot spot component [PSR B0656+14]]Dependence of the relative photon density on the distance from the
stellar surface for a hot spot component of PSR B0656+14. The relative
photon density was calculated for three different starting positions:
$\theta_{0}$ (central), $\theta_{1}$ (at the half distance to the
edge), and $\theta_{2}$ (the cap edge). The altitude ($z$) was calculated
for points which follow the magnetic field structure of
PSR B0656+14.
show(430 r_surf=2e6, r_ws=1.8e5, r_hs=5e3, t_surf=0.7e6,t_ws=1.2e6,
t_hs=2.9e6), plot_b0656
[Dependence of the radiation flux on the distance from the stellar
surface [PSR B0656+14]]Dependence of the radiation flux for three different components (the
entire stellar surface, the warm spot and the hot spot) on the distance
from the stellar surface for PSR B0656+14. The following parameters
were used for the calculations: entire stellar surface radiation,
$T_{{\rm st}}=0.7\,{\rm MK}$, $R_{{\rm st}}=20\,{\rm km}$; warm
spot, $T_{{\rm ws}}=1.2\,{\rm MK}$, $R_{{\rm ws}}=1.8\,{\rm km}$;
and hot spot, $T_{{\rm hs}}=2.9\,{\rm MK}$, $R_{{\rm hs}}=50\,{\rm m}$.
Already at a distance of $240\,{\rm m}$ the flux of the warm spot
radiation becomes higher than the flux of the hot spot radiation.
Furthermore, already at a height of $750\,{\rm m}$ flux the radiation
originating from the polar cap (hot spot) becomes lower than the flux
of radiation from the entire stellar surface. With an increasing distance
the flux of the warm spot decreases faster than the flux of the entire
surface radiation and at a distance of $6.3\,{\rm km}$ the thermal
radiation from the entire stellar surface becomes the dominant component
of the background photons.
The results may suggest that up to a height of about $240\,{\rm m}$
(for PSR B0656+14) the hot spot radiation should be the main source
of the background photons involved in ICS. However, the actual height
is smaller as the results do not include the efficiency of ICS, which
also depends on the incident angle between the photons and the particles
(see the next Section).
§.§.§ Photon incident angles
Another parameter that significantly affects the ICS is the incident
angle between the background photons and the relativistic particles.
Especially for Resonant Inverse Compton Scattering is the incident
angle of great importance. Figure <ref> presents
the dependence of a particle mean free path for ICS on a maximum value
of the incident angle $\psi_{{\rm crit}}$. If incident angles are
low, the resonance is outside of the photon spectrum and results in
very high values of particle mean free paths. The lower the energy
of the particle (lower Lorentz factor), the incident angles should
be larger to ensure that the resonance falls into an energy range
with high photon density.
[Dependence of the particle mean free path on the maximum value of
the incident angle] Dependence of the particle mean free path on the maximum value of
the incident angle $\psi_{{\rm crit}}$. The particle mean free path
$l_{{\rm p}}$ was calculated for magnetic field strength $B=10^{14}\,{\rm G}$
assuming background blackbody radiation with a temperature $T=3\,{\rm MK}$.
Two different particle Lorentz factors were used for the calculations:
$\gamma=10^{3}$ (dashed lines) and $\gamma=10^{4}$ (solid lines).
The red lines correspond to Resonant Inverse Compton Scattering, while
the blue lines correspond to Thermal-peak Inverse Compton Scattering.
TICS for a given magnetic field strength and the Lorentz factor of
particles is not significant (high particle mean free paths) unless
the angles of the incident photons are high enough. Note the characteristic
drop of the particle mean free path for TICS at $\psi_{{\rm crit}}\approx20^{\circ}$
(for $\gamma=10^{4}$) and $\psi_{{\rm crit}}\approx75^{\circ}$ (for
$\gamma=10^{3}$). For such high incident angles the resonance takes
place at the thermal peak of the background photons. Therefore, TICS
and RICS are indistinguishable, which results in an almost equal particle
mean free path (see the text above Figure <ref>
for more details).
Due to the very small size of the polar cap the influence of the hot
spot component will by lower not only because of the change of photon
density, but also because of the rapid change of the incident angle
between the photons and particles. Figure <ref>
presents the dependence of the maximum incident angle on the altitude
above the stellar surface for three thermal components (the entire
surface, the warm spot and the hot spot). As follows from the Figure,
already at an altitude of $z\approx90\,{\rm m}$ does the maximum
value of the incident angle between the photons from the hot spot
and the particles drop to $\psi_{{\rm crit}}=30^{\circ}$, which significantly
lowers the efficiency of ICS for this source of background photons
(see Figure <ref>). Since the size of the
warm spot component is larger, the warm spot radiation will be significant
for up to higher altitudes, but already at a distance of $z\approx1.5\,{\rm {\rm km}}$
the maximum value of the incident angle also drops to $\psi_{{\rm crit}}=30^{\circ}$.
show_three, plot_three_psi
[Dependence of the maximum incident angle on the distance from the
stellar surface] Dependence of the maximum incident angle on the altitude above the
stellar surface for three thermal components (the entire surface,
the warm spot and the hot spot radiation).
Note that in the Figure we have calculated the maximum value of the
intersection angle at altitudes which correspond to radial progression
from the stellar surface. In fact, the actual maximum value of the
incident angle also depends on the structure of the magnetic field.
Figure <ref> presents the actual
maximum value of the incident angle of photons originating from the
hot spot for three different magnetic field lines calculated for PSR
B0656+14. The actual values of the maximum incident angle just above
the surface exceed $90^{\circ}$, but its rapid decline (especially
for extreme lines) causes the radiation of the hot spot component
to become insignificant for ICS at relatively low altitudes $z\approx20\,{\rm m}$.
show_hotspot, plot_hotspot_psi
[Dependence of the maximum incident angle on the distance from the
stellar surface [PSR B0656+14]]Dependence of the maximum incident angle on the altitude above the
stellar surface for the hot spot component of PSR B0656+14. The maximum
incident angle was calculated for three different starting positions:
$\theta_{0}$ (central), $\theta_{1}$ (at the half distance to the
edge), and $\theta_{2}$ (the cap edge).
Both the decrease of photon density and the decrease of the maximum
inclination angle cause the parameters of plasma produced by RICS
to highly depend on the properties (size and temperature) of the background
photons source. The hot spot component will be the dominant source
of background photons for ICS in the gap region ($z\lesssim20\,{\rm m}$),
while the radiation of the warm spot and the entire surface will be
the main source of the background photons for ICS at higher altitudes.
CHAPTER: PHYSICS OF PULSAR RADIATION
§ INNER ACCELERATION REGION
§.§ Gamma-ray emission
In our model most of the $\gamma$-photons are produced in the Inner
Acceleration Region or in close vicinity of a neutron star. Due to
an ultrastrong surface magnetic field, the most energetic $\gamma$-photons
are produced by Inverse Compton Scattering in the PSG-on mode. If
a pulsar is in the PSG-off mode, Curvature Radiation produces fewer
energetic photons than ICS in the PSG-on mode. Photons produced in
IAR (both the ICS and CR) are absorbed by strong magnetic fields creating
positron-electron plasma in the gap region, thereby enhancing a cascade,
or just above the gap enhancing a secondary plasma population. The
absorption of $\gamma$-photons in close vicinity of NS makes it impossible
to directly observe the radiation produced in IAR. However, a characteristic
of this emission defines the parameters of the gap (e.g. multiplicity
in the gap region, gap height, etc.), and thus the parameters of secondary
§.§.§ PSG-off mode
In general, the existence of high potential in IAR (e.g. wide sparks
or $\eta\approx1$) results in solutions for which CR is responsible
for the emission of $\gamma$-photons. The energy of such radiation
depends on the Lorentz factor of primary particles and curvature of
the magnetic field lines. Figures <ref>
and <ref> present the histogram of
photons produced in IAR by CR for PSR B0628-28 and Geminga, respectively.
The curvature in IAR of Geminga is lower ($\Re_{6}\approx2.1$, see
Section <ref>), thus the primary particle should be
accelerated to higher energies in order to produce the required number
of photons in the gap region. Eventually the higher Lorentz factor
of primary particles will result in the emission of $\gamma$-photons
with energy up to $10\,{\rm GeV}$ for Geminga. On the other hand,
the curvature magnetic lines for PSR B0628-28 ($\Re_{6}=0.6$, see
Section <ref>) is higher, which reduces the photon
mean free path and it is possible to produce the required number of
photons in the gap region $N_{{\rm ph}}^{{\rm CR}}$ for lower the
Lorentz factor of primary particles.
In CR-dominated gaps we can distinguish three types of photons: (I)
radiation with energy below $1\,{\rm MeV}$ which is unaffected by
the magnetic field (except the splitting) and can be detected by a
distant observer, (II) soft $\gamma$-ray photons which create pairs
above ZPF, (III) and high energetic $\gamma$-photons responsible
for pair production below ZPF. In an ultrastrong magnetic field the
photons from the third group will produce particles just after reaching
the first threshold. Due to the fact that most CR photons are $\parallel$-polarised,
photon splitting is insignificant in cascade pair production in the
PSG-off mode.
t3 for 404 and 373
[Distribution of photons produced in IAR in the PSG-off mode [PSR
B0628-28]]Distribution of photons produced in IAR by a single particle for
PSR B0628-28. In the calculations we used parameters of the gap in
the PSG-off mode as presented in Table <ref>. We
also assumed a linear change in the acceleration electric field (see
Equation <ref>).
[Distribution of photons produced in IAR in the PSG-off mode [PSR
J0633-1746]]Distribution of photons produced in IAR by a single particle for
PSR J0633-1746. In the calculations we used parameters of the gap
in the PSG-off mode as presented in Table <ref>.
§.§.§ PSG-on mode
When the acceleration potential is low enough (narrow sparks with
$\eta<1$) to satisfy the condition for effective ICS ($l_{{\rm ICS}}\lesssim l_{{\rm acc}}$),
the gap will operate in the PSG-on mode. The energy of ICS radiation
in the gap region (RICS) depends on the Lorentz factor of primary
particles and the strength of magnetic field. In an ultrastrong magnetic
field of IAR implied by the PSG model, the primary particle loses
most of its energy during the scattering of background photons. Such
extremely energetic photons produce pairs on the zero-th Landau level
($\parallel$-polarised photons) or split to less energetic photons
before reaching the first threshold (see Section <ref>).
After the photons split the resulting photons are still very energetic
and create an electron-positron pair enhancing the avalanche production
of particles. In contrast to the PSG-off, most of the electron-positron
pairs in the PSG-on mode are created well below ZPF. Furthermore,
there is no additional radiation at lower energies ($\epsilon<1\,{\rm MeV}$)
which could be detected by a distant observer. Figures <ref>
and <ref> present the distribution
of photons produced by the first population of newly created particles
for PSR B0950+08 and PSR B1929+10, respectively. In both cases the
energy of the $\gamma$-photons ranges from $1\,{\rm GeV}$ to $\approx20\,{\rm GeV}$.
The narrow predicted spark half-width of PSR B0950+08 results in a
lower potential in IAR, thus increasing the efficiency of ICS (more
photons produced by the first population of particles). The particle
mean free path for ICS is smaller for backstreaming particles (see
Section <ref> for more details), thus most
photons in the PSG-on mode are produced in the direction towards the
stellar surface. Note that not all photons will produce electron-positron
pairs since some
$\gamma$-photons are produced so close to the stellar surface that
they reach its surface before they manage to reach the first threshold
for pair production.
t + plot_ics for 322 and 355
[Distribution of photons produced in IAR in the PSG-on mode [PSR
B0950+08]]Distribution of photons produced in IAR by the first population of
newly created particles for PSR B0950+08. In the calculations we used
the parameters of the gap in the PSG-on mode as presented in Table
[Distribution of photons produced in IAR in the PSG-on mode [PSR
B1929+10]]Distribution of photons produced in IAR by the first population of
newly created particles for PSR B1929+10. In the calculations we used
the parameters of the gap in the PSG-on mode as presented in Table
§.§ X-ray and less energetic emission
An negligible fraction of energy radiated by a primary particle in
the PSG-off mode falls in the X-ray band. What is more, in the PSG-on
mode all photons produced by ICS have energy which exceeds an electron's
rest energy by many orders of magnitude. Thus, IAR may be responsible
only for generating the thermal component of the X-ray spectrum in
the process of heating the stellar surface.
§.§.§ Thermal emission
As shown in Section <ref>, thermal emission is
a common feature of neutron stars. Due to the large uncertainties
in X-ray observations, it is not possible to distinguish all three
thermal components (entire surface radiation, warm spot component
and hot spot radiation) for one specific pulsar. Furthermore, only
for a few pulsars (e.g. Geminga, PSR B0656+14) was it possible to
distinguish two thermal components alongside the nonthermal one. In
this thesis we focus on an analysis of pulsars with a visible hot
spot component ($b>1$), since only for these pulsars is it possible
to estimate the size of the actual polar cap. Most of these pulsars
are old neutron stars and only for one of them (Geminga) was the whole
surface radiation found in the X-ray spectrum. Figure <ref>
presents the observed X-ray components of the Geminga pulsar: the
whole surface radiation, the polar cap (hot spot) and the nonthermal
component. The maximum of energy for the whole surface radiation is
in extreme ultraviolet and in soft X-rays for the hot spot component.
Taking into account the very small area of the polar cap, radiation
from the hot spot is unlikely to be observed in wavelengths off the
~/Programs/studies/phd/spectrum/spectrum.py (uncomment
[Observed flux of radiation [PSR J0633+1746]]Observed flux of radiation for PSR J0633+1746. In the figure we present
three components of radiation: the nonthermal one (green line), the
entire surface radiation (blue line), and the hot spot component (red
line). The dashed lines correspond to uncertainties in observations
(see Table <ref>).
Figure <ref> presents the X-ray spectrum
of PSR B1133+16. The small number of counts detected resulted in the
fact that only separate fits for the BB and PL components were performed.
Both the BB and PL fits describe the observed spectrum with similar
accuracy. In the Figure we present additional thermal components (the
entire surface radiation and the warm spot) which have not been determined
by the observations. The Figure shows that the overlapping thermal
components can mimic the power-law dependence of the spectrum at frequencies
below $2\,{\rm keV}$.
~/Programs/studies/phd/spectrum/spectrum.py (uncomment
[X-ray spectrum [PSR B1133+16]]X-ray spectrum of PSR B1133+16. In addition to the observed thermal
radiation (red solid line), two other thermal components are presented:
the warm spot radiation (green dashed line) and the entire surface
radiation (blue dotted line).
Although this specific combination of thermal components for PSR B1133+16
would result in a photon index greater than the observed one $\Gamma=2.51$,
the spectral fits for all pulsars should be extended to include more
BB components in order to examine the effect of thermal components
overlapping at lower frequencies. The results of our calculations
suggest that the nonthermal X-ray radiation should dominate the spectrum
at higher frequencies $\approx3-10\,{\rm keV}$, but the power-law-like
behaviour at lower frequencies could be the result of the overlapping
of thermal components anticipated in the PSG scenario (see Section
§.§.§ Nonthermal emission
The polarisation of ICS radiation in an ultrastrong magnetic field
is $50\%$ (one $\parallel$ to every $\perp$-polarised photon).
Synchrotron Radiation of secondary particles created by $\perp$-polarised
photons would generate hard X-ray photons, however, as was mentioned
in Section <ref>, these photons will split
before they reach the first threshold to produce pairs. Therefore,
regardless of whether the gap is dominated by CR or by ICS, Synchrotron
Radiation in IAR is not significant.
§.§.§ Warm spot component
Apart from the obvious X-ray component corresponding to the whole
surface radiation, the PSG model can explain both the hot and warm
spot radiation. The hot spot radiation is a natural consequence of
heating the actual polar cap region by the backstreaming particles
(see Section <ref>). As was mentioned in Section <ref>,
the warm spot component can have two different sources: (I) the drastic
difference of the crustal transport process due to the non-dipolar
structure of the surface magnetic field (for young and middle-aged
pulsars), (II) and a mechanism of heating the surface adjacent to
the polar cap. In this section we present the second mechanism, i.e.
heating of the surface adjacent to the polar cap, which can be applied
to both young and old pulsars.
Figure <ref> presents the mechanism
of heating the area adjacent to the polar cap for PSR B0950+08. When
the gap operates in the PSG-off mode the primary plasma (see Section
<ref>) will lose a significant part
of its energy via CR as the particles propagate through the region
of high curvature. For this particular magnetic line's configuration
the region of high CR extends up to an altitude about $4\,{\rm km}$
above the stellar surface. The most energetic CR photons emitted in
this region have a relatively short mean free path and they produce
electron-positron pairs in the region of open magnetic field lines.
However, both the less energetic CR photons and $\gamma$-photons
produced by SR have a large enough photon mean free path to produce
pairs in the region of the closed magnetic field lines. All newly
created pairs move along the closed magnetic field lines and heat
the surface beyond the polar cap on the opposite side of the star.
[The warm spot component [PSR B0950+08]]Global structure of magnetic field lines for PSR B0950+08. The structure
was obtained using two crust-anchored anomalies (see Section <ref>).
Green lines correspond to the outer open magnetic field lines, while
the red lines correspond to the closed magnetic field lines at which
secondary pairs are produced. Blue, yellow and red dots represent
the locations of secondary pair production for the outer left, the
middle and the outer right open field lines, respectively.
The fraction of energy transferred to the region of the closed field
lines highly depends on the region of open magnetic field lines considered
in CR/SR emission. In the Figure we use three different colours (blue,
yellow and red) to show the positions of pair creation for three characteristic
open magnetic field lines (the outer left, the middle and the outer
right). The simulation results in the following fractions of energy
transferred to the region of the closed field lines are: $0.02\%$,
$0.1\%$, $6\%$ for all three lines, respectively. For this specific
magnetic field configuration the transferred energy fraction increases
as we move towards the region with the highest curvature. We can roughly
estimate that for the proposed magnetic field configuration of PSR
B0950+08, about $1\%$ of the outflowing energy is responsible for
heating of the surface beyond the polar cap on the opposite side of
the star. Note that due to strong anisotropy of the outflowing and
backflowing stream of particles (see Section <ref>),
this fraction could be enough to obtain the warm spot component with
a luminosity equal or in some cases even higher than the luminosity
of the hot spot component.
[The warm spot component [PSR B0943+10]]Global structure of magnetic field lines for PSR B0943+10. The structure
was obtained using two crust-anchored anomalies (see Section <ref>).
Green lines correspond to the outer open magnetic field lines, while
the red lines correspond to the closed magnetic field lines at which
secondary pairs are produced. Blue, yellow and red dots represent
the locations of secondary pair creation for the outer left, the middle
and the outer right open field lines, respectively.
The fraction of energy transferred to the region of closed field lines
highly depends on the magnetic field configuration. A more complicated
structure of the magnetic field lines proposed for PSR B0943+10 (see
Figure <ref>) results in a much wider
area of closed field lines at which pairs are created, and hence higher
fractions of energy transferred to the region of closed field lines.
For this specific structure of the magnetic field these fractions
are: $7\%$, $1\%$, $2\%$ for three characteristic lines, respectively.
We can roughly estimate that about $3-5\%$ of the outflowing energy
is responsible for the heating. Note that the magnetic field structure
of PSR B0950+08 results in the heating of only one side beyond the
polar cap, while in the case of PSR B0943+10 the whole surface around
the polar cap is heated. The actual size of the warm spot also depends
on the magnetic field configuration in the heating zone, and can either
be decreased or increased.
§.§ Primary plasma
As we mentioned in Section <ref>, PSG-off and
PSG-on modes differ essentially by the Lorentz factor of primary particles
produced in the gap region. Furthermore, different scenarios of the
gap breakdown (due to surface overheating or due to production of
dense enough plasma) cause the evolution of primary particles in the
two modes to completely different.
We assume that in the PSG-off mode the gap breakdown is due to surface
overheating; hence the plasma cloud moving away from the stellar surface
is a mixture of ions and electron-positron plasma. In this scenario
the ions are the main source of charge density required to screen
the gap (see Equation <ref>). As the
plasma cloud moves away from the stellar surface both the spark height
and the spark width increase, which results in an increase of the
acceleration potential drop. When the particles gain the Lorentz factors
$\gamma\gtrsim10^{5}$, CR begins to produce $\gamma$-photons. In
the PSG-off mode most of the
$\gamma$-photons are created near ZPF (see Figure <ref>).
All particles created by $\gamma$-photons above the ZPF do not contribute
to the heating of the surface. Furthermore, the acceleration in the
upper parts of the gap is relatively weak, and electrons produced
in this region will also escape from the gap, thus not contributing
to the surface heating. Depending on the details of the cascade formation,
the process described above may result in the creation of strong streaming
anisotropies, where the flux of the backstreaming particles is considerably
smaller than the flux of the outstreaming particles. Note that the
density of the backstreaming particles required to overheat the surface
is significantly lower than the co-rotational density $n_{{\rm CR}}\ll n_{{\rm GJ}}$
(see Table <ref>).
In the PSG-on mode the quasi-equilibrium of the flux of backstreaming
particles and the flux of the polar cap radiation can cause the gap
to break only due to the production of dense enough plasma. Thus,
the surplus of positrons is the main source of the charge in the plasma
cloud moving away from the stellar surface. The ICS process responsible
for the cascade production of particles is effective only in the bottom
part of the gap. Hence, the backstreaming electrons will hit the surface
with a Lorentz factor $\gamma_{{\rm c}}$ well below the $\gamma_{{\rm max}}$.
As there is no strong pair production near (or above) ZPF, the backstreaming/outstreaming
anisotropy arises only due to the difference of the Lorentz factor
of electrons hitting the stellar surface and the Lorentz factor of
positrons accelerated in the gap $\gamma_{{\rm max}}/\gamma_{{\rm c}}\approx10$.
The actual density of newly created plasma to completely screen the
gap can be calculated only in the full cascade simulation. However,
as shown by 172, this density should significantly
exceed the co-rotational Goldreich-Julian density $n_{{\rm ICS}}\gg n_{{\rm GJ}}$.
We describe the difference between the co-rotational density and the
actual density of primary plasma required to completely screen the
ICS-dominated gap by factor $N_{{\rm ICS}}=n_{{\rm ICS}}/n_{{\rm GJ}}\gg1$.
§ INNER MAGNETOSPHERE OF A PULSAR
§.§ Gamma-ray emission
In general there are three processes which can produce $\gamma$-ray
emission in the inner magnetosphere of a pulsar ($R_{{\rm pc}}\ll z\ll R_{{\rm LC}}$):
CR, ICS and SR. Which of them produces the majority of $\gamma$-photons
depends on the parameters of the primary particles, and thus mainly
depends on the mode in which the gap operates. Additionally, the efficiency
of the ICS process strongly depends on the source of the background
§.§.§ Curvature Radiation of primary particles
When the gap operates in the PSG-off mode, high-energetic particles
are produced
$\gamma_{{\rm c}}\gtrsim10^{6}$. As they pass the region with high
curvature ($\Re_{6}\approx1$) they radiate a significant part of
their energy through CR (see Section <ref>).
Figure <ref> presents the distribution of CR
photons produced by a single primary particle moving along the open
magnetic field line of PSR B1133+16 (see Section <ref>
for the details of the magnetic field configuration). The initial
Lorentz factor of the particle $\gamma_{{\rm max}}=1.7\times10^{6}$
was set according to the value presented in Table <ref>.
As the particle advanced through the region with high curvature, it
lost about $46\%$ of its initial energy, which was mainly converted
to high-energetic $\gamma$-photons with an energy up to about $2\,{\rm GeV}$.
The $\gamma$-photons are produced in a region of a strong magnetic
field, thus after passing a relatively short distance the most energetic
photons are absorbed by the magnetic field and electron-positron pairs
emerge. The red colour in the Figure corresponds to the final spectrum
(after photon splitting, pair production and SR) produced by a single
primary particle in the PSG-off mode. Most of the energy radiated
by the primary particle was converted into the secondary plasma (see
Section <ref>) and only about $5\%$
of the particle's initial energy ended in the form of radiation with
a cut-off at about $30\,{\rm MeV}$.
(read_data_final, plot_spectrum_final, 341_cr_1e04_10m_new)
[Final photon distribution produced by a single primary particle [PSR
B1133+16]]Final photon distribution produced by a single primary particle for
PSR B1133+16. The blue line corresponds to the initial CR distribution,
while the red line presents the final distribution with the inclusion
of photon splitting, pair production and SR.
To increase the amount of photons reaching the observer, the emission
zone, i.e. the region with the highest curvature, should by located
in the area with a weaker magnetic field. Such a configuration allows
a photon to travel a longer distance before it is absorbed by the
magnetic field. As a result the electron-positron pairs are created
at higher Landau levels, which enhances SR. Figure <ref>
presents the distribution of CR photons for PSR B0950+08. The calculations
were performed for the initial Lorentz factor of the particle $\gamma_{{\rm max}}=2.0\times10^{6}$
(see Table <ref>).
(read_data_final, plot_spectrum_final, 355_cr_1e04_10m_new)
[Final photon distribution produced by a single primary particle [PSR
B0950+08]]Final photon distribution produced by a single primary particle for
PSR B0950+08. The blue line corresponds to the initial CR distribution,
while the red line presents the final distribution with the inclusion
of photon splitting, pair production and SR.
Due to CR the primary particle lost about $40\%$ of its initial energy.
In this case about a half of the energy radiated by the primary particle
was converted into the secondary plasma and the same amount of energy
(about $20\%$ of the particle's initial energy) ended in the form
of radiation. For both PSR B1133+16 and PSR B0950+08, the maximum
of the curvature is of the same order. However, the maximum of curvature
PSR B1133+16 is located at an altitude of about $800\,{\rm m}$, while
for PSR B0950+08 it is located at an altitude of about $1.75\,{\rm km}$
(compare Figures <ref> and <ref>).
§.§.§ Inverse Compton Scattering of primary particles
In the PSG-on mode the maximum Lorentz factor of primary particles
is in the range of $10^{4}-10^{5}$ (see Table <ref>).
As it follows from Figures <ref> and <ref>,
the ICS process is most effective for particles with a Lorentz factor
in the range of $10^{3}-10^{4}$. Particles with high energies ($\gamma\gtrsim10^{5}$)
will upscatter thermal photons only just above the stellar surface,
where the density of the background photons is very high (see Section
<ref>). Thus, if there is no additional source
of background photons, the most energetic particles ($\gamma\gtrsim10^{5}$)
will escape from the inner magnetosphere without losing their energy
by ICS. However, the plasma cloud produced by the ICS-dominated gap
has a density exceeding the co-rotational Goldreich-Julian density
even by a few orders of magnitude (see Section <ref>).
Such a high charge density reduces the acceleration [172]
and, consequently, the bulk of particles will escape from the IAR
with lower Lorentz factors. It is not possible to estimate the actual
Lorentz factor of particles in the plasma cloud at the moment of gap
breakdown without performing a full cascade simulation. Thus, in this
thesis we assume that at the moment of gap breakdown most of the particles
will have an energy that is about the characteristic value at which
the acceleration is stopped by ICS in the bottom parts of the IAR
$\gamma_{{\rm c}}$. To increase readability for cascade simulations
with very low surface temperature, in all the Figures of the ICS distribution
we present $\gamma$-photons produced by $50$ primary particles with
Lorentz factors in the range of $0.5\gamma_{{\rm c}}-2\gamma_{{\rm c}}$.
In Figure <ref> we present the distribution
of ICS photons produced by the upscattering of surface thermal radiation
with temperature $T_{{\rm s}}=0.3\,{\rm MK}$ for PSR B0834+06. Even
for such a low surface temperature the whole surface radiation is
the dominant source of background photons for ICS up to altitudes
of about one stellar radii. During the scattering the primary particles
lose about $30\%$ of their initial energy while producing $\gamma$-photons
with energy up to $1\,{\rm GeV}$. Since the $\gamma$-photons are
very energetic and are produced in a region with a strong magnetic
field, they will be absorbed by the magnetic field, thus giving rise
to the secondary plasma population (see Section <ref>).
All pairs in the inner magnetosphere of a pulsar are created in the
nonzero Landau level, thus the pair production process is also accompanied
by strong SR (see the next section).
(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics2_t03_fifty)
[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=0.3\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of surface
($T_{{\rm s}}=0.3\,{\rm MK}$) for PSR B0834+06. The plot includes
all $\gamma$-photons upscatterd by $50$ primary particles with Lorentz
factors in the range of $2.5\times10^{3}-10^{4}$.
A natural way of increasing the number of $\gamma$-photons produced
by ICS in the inner magnetosphere is to increase the number of background
photons. Figure <ref> presents the distribution
of ICS photons produced by an upscattering of the surface thermal
radiation with temperature $T_{{\rm s}}=0.4\,{\rm MK}$ for PSR B0834+06.
During the ICS process the primary particles lose about $65\%$ of
their initial energy. For higher surface temperatures the ICS produces
$\gamma$-photons up to higher altitudes (about two stellar radii),
thus photons with lower energy emerge $\epsilon_{{\rm min}}\approx3\,{\rm MeV}$.
These less energetic photons will reach the observer, but their total
energy is significantly lower than the total energy of the secondary
plasma created by more energetic $\gamma$-photons.
(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics2_t04_fifty)
[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=0.4\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of the surface
radiation ($T_{{\rm s}}=0.4\,{\rm MK}$) for PSR B0834+06. The plot
includes all $\gamma$-photons upscatterd by $50$ primary particles
with Lorentz factors in the range of $2.5\times10^{3}-10^{4}$.
Note that although for PSR B0834+06 the X-ray spectral fit was performed
with only one BB component, the surface temperatures used in the calculations
($0.3\,{\rm MK}$ and $0.4\,{\rm MK}$) are in good agreement with
the predicted surface temperature of an old neutron star.
Another source of background photons which could be relevant for ICS
in the inner magnetosphere is the warm spot component. As shown in
Section <ref>, if the antipodal spot
operates in the PSG-off mode and if the magnetic field structure is
suitable then the warm spot is formed in the region adjacent to the
polar cap. With a temperature lower than the hot spot but a much larger
area, the warm spot is the main source of the background photons at
altitudes up to about half a stellar radius.
In Figure <ref> we present the distribution
of ICS photons produced by an upscattering of warm spot radiation
with temperature $T_{{\rm s}}=1.0\,{\rm MK}$ and radius $R_{{\rm ws}}=1\,{\rm km}$
for PSR B0834+06. When the warm spot is the main source of background
photons, the ICS process starts at lower altitudes. As a consequence,
the scattering produces photons with higher energy and the primary
particles lose up to $90\%$ of their initial energy. All these high
energetic $\gamma$-photons are absorbed by the magnetic field producing
electron-positron pairs. Note that for this specific pulsar the existence
of such a strong warm spot component is unlikely, but as mentioned
in Section <ref> the X-ray spectral
fits should be extended to include more thermal components to put
better constraints on the X-ray emission of pulsars.
(read_data_ics, plot_hist_ics, 384_ics_1e03_10m_ics1_t10_fifty_warm)
[Distribution of ICS photons for PSR B0834+06 [$T_{{\rm s}}=1.0\,{\rm MK}$,
$R_{{\rm ws}}=1\,{\rm km}$]]Distribution of ICS photons produced by an upscattering of warm spot
radiation ($T_{{\rm s}}=1.0\,{\rm MK}$, $R_{{\rm ws}}=1\,{\rm km}$)
for PSR B0834+06. The plot includes all $\gamma$-photons upscatterd
by $50$ primary particles with Lorentz factors in the range of $2.5\times10^{3}-10^{4}$.
§.§.§ Synchrotron Radiation
In both PSG-off and PSG-on modes SR plays a significant role in the
generation of soft $\gamma$-ray photons. Figure <ref>
presents the places at which SR-photons are generated (left panel)
and the SR spectrum (right panel) in the PSG-off mode for Geminga.
The most energetic photons are generated close to the stellar surface
($z\approx500\,{\rm m}$), while the less energetic ones are produced
at altitudes $z>2\,{\rm km}$, where the magnetic field is weaker.
SR in the PSG-off mode produces photons with energy in the range of
from $30\,{\rm keV}$ to $1\,{\rm GeV}$. Again, the high energetic
$\gamma$-photons produce electron-positron pairs in a strong magnetic
field, thus its observation is not possible.
~/Programs/studies/phd/lines/lines.py t0633_cr
[Synchrotron Radiation in the PSG-off mode [PSR J0633+1746]]Synchrotron Radiation in the PSG-off mode for PSR J0633+1746. The
left panel presents the places at which SR-photons are generated,
while the right panel presents the SR-photons distribution. Plots
were obtained in a cascade simulation calculated for a single primary
particle moving along the extreme left open magnetic field line.
In Figure <ref> we present the places
of SR-photon generation (left panel) and the energy distribution of
photons (right panel) in the PSG-on mode for Geminga.
~/Programs/studies/phd/lines/lines.py t0633_ics
[Synchrotron Radiation in the PSG-on mode [PSR J0633+1746]]Synchrotron Radiation in the PSG-on mode for PSR J0633+1746. The
left panel presents places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line. The ICS process was
calculated using the whole surface radiation with temperature $T_{{\rm s}}=0.5\,{\rm MK}$
(see Table <ref>).
The production of SR-photons in the PSG-off mode starts at altitudes
about $z\approx1\,{\rm km}$ and ends at altitudes $z\approx4.5\,{\rm km}$.
Thus, the energy range of SR-photons is narrower, with the minimum
and maximum photon energy $\epsilon_{{\rm min}}\approx40\,{\rm keV}$
and $\epsilon_{{\rm max}}\approx50\,{\rm MeV}$, respectively. Note
the significant difference in the number of photons produced by SR
in the PSG-off and PSG-on modes. The difference is a direct consequence
of low secondary plasma multiplicity in the PSG-on mode (see Section
§.§ X-ray emission
The main source of X-ray photons produced in the inner magnetosphere
is SR. As mentioned in Section <ref>, to
increase the amount of photons reaching the observer the emission
zone should by located in the area with a weaker magnetic field. In
this section we focus on the results of PSR B0943+10 and PSR 1929+10
for which the proposed configuration of a magnetic field satisfies
this requirement (see Sections <ref> and <ref>).
In the PSG-off mode most of the X-ray photons are produced by the
SR of newly created electron-positron pairs. Figure <ref>
presents the final photon distribution produced by a single primary
particle of PSR B1929+10 in the PSG-off mode. For a single primary
particle we can estimate that only about $0.7\%$ of the total photon
energy is in the range of $1-10\,{\rm keV}$. The bulk of the energy
is carried away by newly created particles ($73\%$) and high energetic
photons ($27\%$).
t1929_cr_photons (read_data_final, plot_spectrum_final, 322_cr_1e04_10m_new)
[Final photon distribution produced by a single primary particle [PSR
1929+10]]Final photon distribution produced by a single primary particle for
PSR 1929+10 in the PSG-off mode. The blue line corresponds to the
initial CR photons distribution, while the red line presents the final
distribution with the inclusion of photon splitting, pair production
and SR.
In Figure <ref> we present the locations
and the photon distribution of SR for PSR 1929+10 in the PSG-off mode.
All SR-photons produced closer to the stellar surface will contribute
to $\gamma$-ray emission, while the SR-photons produced at higher
altitudes will produce photons in the X-ray band. As it results from
Figure <ref>, the curvature at an altitude
of $z\approx3.5\,{\rm km}$ is only $50\%$ higher than at $z\approx2\,{\rm km}$.
Furthermore, before the particle reaches the region with a relatively
low magnetic field ($z\approx3.5\,{\rm km}$), it radiates a significant
part of its energy at lower heights.
~/Programs/studies/phd/lines/lines.py t1929
[Synchrotron Radiation in the PSG-off mode [PSR B1929+10]]Synchrotron Radiation in the PSG-off mode for PSR B1929+10. The left
panel presents the places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line.
To increase radiation in the $1-10\,{\rm keV}$ energy band we should
apply the magnetic field structure with considerably higher curvature
at altitudes where X-ray photons are generated. Although the curvature
will not directly affect the SR, it will enhance CR, and thus it will
increase the number of pairs produced in the region of a relatively
weak magnetic field. In Figure <ref> we present
the final photon distribution produced by a single primary particle
for PSR B0943+10 calculated using the magnetic field configuration
as presented in Section <ref>.
t0943_cr_photons (read_data_final, plot_spectrum_final, 911_cr_1e04_10m_leftline)
[Final photon distribution produced by a single primary particle [PSR
B0943+10]]Final photon distribution produced by a single primary particle for
PSR B0943+10 in the PSG-off mode. The blue line corresponds to the
initial CR photons distribution, while the red line presents the final
distribution with the inclusion of photon splitting, pair production
and SR.
For this magnetic field structure about $3\%$ of the total photon
energy is in the range of $1-10\,{\rm keV}$. The newly created particles
carry away about $63\%$ of the energy radiated by the primary particle,
while about $37\%$ of the energy remains in the form of photons.
The structure of the magnetic field of PSR B0943+10 allows enhanced
pair production in a region of a weaker magnetic field (see Figure
<ref>). The SR that accompanies pair
production at higher altitudes ($z>3\,{\rm km}$) essentially increases
the amount of energy radiated in the $1-10\,{\rm keV}$ energy band.
Note, however, that the fraction of energy radiated in this band is
still relatively low ($3\%$), and in order to be a substantial part
of the observed X-ray spectrum the strong anisotropy of backstreaming
and outstreaming plasma is required (see Section <ref>).
~/Programs/studies/phd/lines/lines.py t0943_cr
[Synchrotron Radiation in the PSG-off mode [PSR B0943+10]]Synchrotron Radiation in the PSG-off mode for PSR B0943+10. The left
panel presents the places at which SR-photons are generated, while
the right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for a single primary particle moving
along the extreme left open magnetic field line.
In the PSG-on mode even for a complicated structure of the magnetic
field most of the outflowing energy is converted to secondary plasma.
Figure <ref> presents the ICS-photons distribution
produced in the PSG-on mode for PSR B0628-28. The bulk of energy is
radiated in the form of high energetic $\gamma$-photons which are
responsible for pair production, and thus the formation of secondary
plasma. Taking into account not so high backstreaming/outstreaming
anisotropy in the PSG-on mode, the ICS process is not relevant for
the production of X-ray photons.
[Distribution of ICS photons for PSR B0628-28 [$T_{{\rm s}}=0.5\,{\rm MK}$]]Distribution of ICS photons produced by an upscattering of the whole
surface radiation ($T_{{\rm s}}=0.5\,{\rm MK}$) for PSR B0628-28.
The plot includes all photons upscatterd by $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$.
The SR which accompanies the pair creation process in the PSG-on mode
mostly produces soft $\gamma$-photons (see the right panel of Figure
<ref>). Although the secondary pairs
are produced at similar altitudes in both modes, (compare the left
panels of Figures <ref> and <ref>),
the higher Lorentz factor of secondary plasma produced in the PSG-on
mode results in higher energy of the SR-photons. The results suggest
that when the gap operates in the PSG-on mode we should expect lower
efficiencies of nonthermal X-ray emission than in the PSG-off mode.
Note, however, that the final efficiency of X-ray radiation in the
PSG-off mode highly depends on the backstreaming/outstreaming anisotropy
and the structure of magnetic field lines.
~/Programs/studies/phd/lines/lines.py t0628_ics_xrays
[Synchrotron Radiation in the PSG-on mode [PSR B0628-28]]Synchrotron Radiation in the PSG-on mode for PSR B0628-28. The left
panel presents places at which SR-photons are generated, while the
right panel presents the SR-photons distribution. Plots were obtained
in a cascade simulation calculated for $50$ primary particles with
Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$ moving
along the extreme left open magnetic field line.
§.§ Secondary plasma
The multiplicity of secondary particles in the PSG-off mode is much
higher than in the PSG-on mode. However, the primary plasma produced
in the IAR of CR-dominated gaps has a density considerably lower than
the Goldreich-Julian co-rotational density (see Equation <ref>).
Figure <ref> presents the energy histogram
of secondary plasma for Geminga (left panel) and PSR B1133+16 (right
panel). Despite major differences in the magnetic field structure
and conditions in the IAR for both pulsars, the secondary plasma distribution
shows many similarities. The only significant difference is the maximum
Lorentz factor of secondary plasma, which for Geminga is about $\gamma_{{\rm sec}}^{{\rm max}}\approx10^{4}$,
while for PSR B1133+16 is is a few times smaller $\gamma_{{\rm sec}}^{{\rm max}}\approx3\times10^{3}$.
By using the overheating parameters presented in Table <ref>
we can roughly estimate that the final multiplicity of particles in
the plasma cloud in the PSG-off mode ranges from $M=\kappa\cdot M_{{\rm sec}}\approx2$
(for Geminga) to $M=\kappa\cdot M_{{\rm sec}}\approx100$ (for PSR
B1133+16). Note, however, that these values do not take into account
the anticipated anisotropy of backstreaming and outstreaming particles.
The existence of such an anisotropy could further increase the final
multiplicity of particles in the plasma cloud leaving the inner magnetosphere.
Despite the fact that without a full cascade simulation in the IAR
we cannot unambiguously determine the final multiplicity in the plasma
cloud, it can be clearly seen that depending on the details of the
gap operating in the PSG-off mode, the produced plasma may be suitable
(e.g. PSR B1133+16) or unsuitable (e.g. Geminga) to generate radio
emission (see Section <ref>). The main
factor determining the parameters of the CR-dominated gap, and thus
determining whether it is possible to effectively produce radio emission,
is the radius of curvature of the magnetic field lines (see Section
t0633_cr_pairs, t1133_cr_pairs (change to right)
[Secondary plasma in the PSG-off mode [PSR J0633+1746, PSR B1133+16]]Energy histogram of secondary plasma in the PSG-off mode. The left
panel was obtained in a cascade simulation calculated for a single
primary particle moving along the extreme left open magnetic field
line of PSR J0633+1746, while the right panel corresponds to a cascade
simulation for PSR B1133+16.
In ICS-dominated gaps, on the other hand, the density of primary plasma
produced in IAR exceeds the co-rotational density. Thus, the development
of dense enough plasma for radio emission is much easier in the PSG-on
mode. In Figure <ref> we present
the locations of pair production and energy distribution of secondary
plasma in the PSG-on mode for PSR B0628-28. The final multiplicity
of particles in the plasma cloud in the PSG-on mode can be calculated
as $M=N_{{\rm ICS}}\times M_{{\rm sec}}$. As mentioned in Section
<ref>, the exact value of $N_{{\rm ICS}}$
can be found only by performing the full cascade simulation in IAR
but, as shown by 172, we should expect a full screening
of the acceleration region when $N_{{\rm ICS}}$ reaches a value as
high as $20-100$. Thus we can roughly estimate that for the whole
surface radiation with temperature $T_{{\rm s}}=0.3\,{\rm MK}$, the
final multiplicity of secondary plasma in the PSG-on mode for PSR
B0628-28 is of the order of $M\approx100$.
~/Programs/studies/phd/lines/lines.py t0628_ics_a50
[Secondary plasma in the PSG-on mode [PSR B0628-28, $T_{{\rm s}}=0.3\,{\rm MK}$]]Secondary plasma produced in the PSG-on mode for PSR B0628-28. The
left panel presents places at which pairs are produced, while the
right panel presents the histogram of particle energy. Plots were
obtained in a cascade simulation calculated for $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$
moving along the extreme left open magnetic field line. The ICS process
was calculated using the whole surface radiation with temperature
$T_{{\rm s}}=0.3\,{\rm MK}$.
In the PSG-on mode the main factor which determines the final multiplicity
of secondary plasma is the source of the background photons. As shown
in Section <ref>, the polar cap radiation
(the hot spot component) has a negligible impact on the ICS process
above the IAR. Figure <ref> presents
the location of pair production and energy distribution of secondary
plasma for PSR B0628-28 calculated assuming the whole surface radiation
with temperature $T_{{\rm s}}=0.5\,{\rm MK}$. The increase in the
number of background photons results in enhancement of the ICS process,
and thus an increase of the secondary multiplicity $M_{{\rm sec}}\approx60$.
For such conditions the final multiplicity of secondary plasma in
the PSG-on mode is of the order of $M\approx10^{3}-10^{5}$.
~/Programs/studies/phd/lines/lines.py t0628_ics_b
[Secondary plasma in the PSG-on mode [PSR B0628-28, $T_{{\rm s}}=0.5\,{\rm MK}$]]Secondary plasma produced in the PSG-on mode for PSR B0628-28. The
left panel presents the places at which pairs are produced, while
the right panel presents the histogram of particle energy. Plots were
obtained in a cascade simulation calculated for $50$ primary particles
with Lorentz factors in the range of $3\times10^{3}-1.2\times10^{4}$
moving along the extreme left open magnetic field line. The ICS process
was calculated using the whole surface radiation with temperature
$T_{{\rm s}}=0.5\,{\rm MK}$.
CHAPTER: CONCLUSIONS
The hot spot component identified in X-ray observations implies the
non-dipolar structure of surface magnetic field. We used the Partially
Screened Gap model to explain both the
X-ray radiation of radio pulsars and production of secondary plasma
suitable for generation of radio emission.
§ A SPECIAL CASE (PSR B0943+10)
Our model predicts two additional sources of X-ray emission: (I) the
warm spot component and (II) enhanced SR radiation in the PSG-off
mode. The warm spot component is associated with particles originating
from the antipodal polar cap, while the high luminosity of X-ray photons
produced in the PSG-off mode is a result of strong anisotropy of backstreaming
and outstreaming particles.
Very recent results presented by 87 show the anti-correlation
of radio and X-ray emission of PSR B0943+10. The authors suggest an
unpulsed, non-thermal component in radio-bright mode and a $100\%$-pulsed
thermal component along with a nonthermal component in a radio-quiet
mode. In our model it is not possible to produce an unpulsed, nonthermal
X-ray component without the accompanying blackbody radiation of the
polar cap. Although it is possible to produce nonthermal X-ray radiation
which obscures the thermal component (strong SR in the PSG-off mode
with a high predominance of outstreaming particles), the resulting
radiation should be pulsed. We believe that the X-ray radiation of
PSR B0943+10 in the radio bright mode was misinterpreted as the nonthermal
one. As shown in Figure <ref> (panel d), for
a derived geometry of PSR B0943+10 the polar cap produces unpulsed,
thermal radiation. Furthermore, as reported by the authors, in the
radio-bright mode both the absorbed blackbody (BB) and the absorbed
power-law (PL) models fit the spectrum equally well (see Table S4
in 87). We believe that the observed radiation
modes of PSR B0943+10 correspond to a mode switch between the PSG-on
(radio-bright) and the PSG-off mode (radio-quiet). When pulsar is
in the PSG-on we observe both the radio emission and thermal radiation
which originates from the polar cap. In the PSG-off mode the secondary
plasma is not suitable to produce so strong radio emission as in the
PSG-on mode, but the polar cap radiation is accompanied by pulsed,
nonthermal emission produced by SR (see Sec. <ref>).
§ GAMMA-RAY PULSARS
As was shown in Sections <ref> and <ref>,
$\gamma$-rays produced in IAR and the inner magnetosphere cannot
reach the observer due to efficient pair production in those regions.
Current models of $\gamma$-ray emission propose that the emission
comes from outer magnetospheric gaps. The non-dipolar structure of
a magnetic field has two key implications on $\gamma$-ray emission
models: (I) the formation of slot gaps is not possible as pairs are
produced along all open magnetic field lines, (II) the high density
of electron-positron plasma ($n_{p}\gg n_{{\rm GJ}}$) produced in
the inner magnetosphere prevents the outer gap formation. The high-density
plasma which crosses the null line will screen the outer magnetospheric
region due to plasma separation (acceleration of electrons and deceleration
of positrons). Thus, the formation of outer gaps is possible only
in special cases when the pulsar operates in the PSG-off mode and
produces secondary plasma with low density $n_{p}\approx n_{{\rm GJ}}$.
As recently reported by 4: “It is possible for
relativistic populations of electrons and positrons in the current
sheet of a pulsar’s wind right outside the light cylinder to emit
synchrotron radiation that peaks in the ${\rm sub-GeV}$ to ${\rm GeV}$
regime, with $\gamma$-ray efficiencies similar to those observed
for the Fermi/LAT pulsars.” We believe that the observed high-energetic
$\gamma$-rays are produced in the not yet well explored region right
outside the light cylinder.
§ RADIO EMISSION
Pulsed radio emission remains one of the most intriguing puzzles of
astrophysics. It is remarkable that despite the large ranges in $P$,
$B_{{\rm d}}$, the variations in the pulse profile between different
classes of neutron stars (young, old, millisecond, magnetars) are
similar to those within classes [125]. The radio emission
of most pulsars can be characterised by: a relatively narrow frequency
range, $\sim100\,{\rm MHz}$ to $\sim10\,{\rm GHz}$, and a high degree
of polarisation with a characteristic sweep of the position angle.
The extremely high brightness temperature of pulsar radio emission
(typically $T_{b}>10^{25}\,{\rm K}$) implies that a coherent emission
mechanism is involved. Many radio emission mechanisms have been proposed,
but no consensus on a specific emission mechanism has emerged. The
radio observations alone cannot identify the emission mechanism and,
hence, a model of the magnetosphere is needed to put constraints on
the radio emission model. An acceptable emission mechanism must involve
some form of instability to produce coherent radiation. The main difficulty
in finding a specific emission mechanism is that many of the predicted
features are common all proposed models. Furthermore, the polarisation
can also be regarded as generic rather than associated with a specific
emission mechanism [126].
The X-ray observations have allowed us to put constraints on the polar
cap region of pulsars. The non-dipolar structure of the surface magnetic
field causes plasma to form under similar conditions regardless of
the global configuration of the magnetic field. We have showed that
depending on the details of IAR, the resulting plasma either meets
the requirements for efficient radio emission (suitable multiplicity
and energy distribution of secondary plasma) or is not suitable to
produce efficient radio emission (e.g. Geminga). Furthermore, the
proposed drift model allows to find a connection between radio and
X-ray emission processes (see Section <ref>).
§ THE MIXED MODE
Although in the thesis we consider the PSG-on and PSG-off mode separately,
in a real case both of these modes can coexist either on two separate
polar caps or on the same polar cap occupying its different parts.
In the latter case the change of modes is associated with varying
degrees of intensity of the two modes. Furthermore, if specific conditions
are met, the ICS process can be a main source of $\gamma$-photons
in the lower parts of the gap, while the CR process can produce $\gamma$-photons
in the upper parts of the acceleration region. In such a case distinguishing
between the two modes is even more difficult.
§ SUMMARY
The main propositions associated with this thesis are as follows:
* The size of the hot spots implies that the magnetic field configuration
just above the stellar surface differs significantly from a purely
dipole one.
* The analysis of X-ray observations shows that the temperature of the
actual polar cap is equal to the so-called critical value, i.e. the
temperature at which the outflow of thermal ions from the surface
screens the gap completely.
* The non-dipolar structure of a surface magnetic field and the high
multiplicity of particles produced in IAR prevents the formation of
slot and outer gaps.
* The PSG model predicts the existence of two scenarios of gap breakdown:
the PSG-off mode for CR-dominated gaps and the PSG-on mode for ICS-dominated
* The two different scenarios of gap breakdown can in a natural way
explain the mode-changing phenomenon when both modes produce plasma
suitable to generate radio emission, and pulse nulling when the radio
emission is not generated in one of the modes.
* The mode changes of the IAR may explain the anti-correlation of radio
and X-ray emission in very recent observations of PSR B0943+10 [87].
* The regular drift of subpulses can be expected only when the gap operates
in the PSG-on mode. The proposed model of drift allows to connect
the drift information obtained by radio observations with the X-ray
data of rotation-powered pulsars.
CHAPTER: ACKNOWLEDGEMENTS
I would like to express my deep gratitude to Professor Giorgi Melikidze,
my research supervisor, for his patient guidance, enthusiastic encouragement
and useful critique of this research work. I would also like to thank
Professor Janusz Gil for his advice and support which allowed me to
complete this thesis. This research project would not have been possible
without the support of many people. I would like to thank all my colleagues
at the Institute of Astronomy who taught me a lot and never refused
to help: Professor Ulrich Geppert, Professor Dorota Gondek-Rosińska,
Professor Jarosław Kijak, Dr. Krzysztof Krzeszowki, Dr. Wojciech Lewandowski,
Professor Andrzej Maciejewski, Dr. Krzysztof Maciesiak, Dr. Olaf Maron,
Dr. Roberto Mignani, Dr. Marek Sendyk, and Dr. Agnieszka Słowikowska.
And a special thanks to Mrs Emilia Gil for her assistance in all the
administrative issues.
I would also like to extend my thanks to friends and family for their
support, sacrifice, patience and wisdom. My special thanks are extended
to my parents for their support and encouragement throughout my studies.
Thank you.
Image by Karolina Rożko
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|
arxiv-papers
| 2013-04-15T19:07:00 |
2024-09-04T02:49:44.402471
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrzej Szary",
"submitter": "Andrzej Szary M.Sc.",
"url": "https://arxiv.org/abs/1304.4203"
}
|
1304.4339
|
# The unified method for the three-wave equation on the half-line
Jian Xu School of Mathematical Sciences
Fudan University
Shanghai 200433
People’s Republic of China [email protected] and Engui Fan School of
Mathematical Sciences, Institute of Mathematics and Key Laboratory of
Mathematics for Nonlinear Science
Fudan University
Shanghai 200433
People’s Republic of China correspondence author:[email protected]
###### Abstract.
We present a Riemann-Hilbert problem formalism for the initial-boundary value
problem of the three-wave equation:
$p_{ij,t}-\frac{b_{i}-b_{j}}{a_{i}-a_{j}}p_{ij,x}+\sum_{k}(\frac{b_{k}-b_{j}}{a_{k}-a_{j}}-\frac{b_{i}-b_{k}}{a_{i}-a_{k}})p_{ik}p_{kj}=0,\quad
i,j,k=1,2,3,$
on the half-line.
###### Key words and phrases:
Riemann-Hilbert problem, Three-wave equation, Initial-boundary value problem
## 1\. Introduction
The 3-wave resonant interaction model described by the equations
$\begin{array}[]{ll}p_{ij,t}-\frac{b_{i}-b_{j}}{a_{i}-a_{j}}p_{ij,x}+\sum_{k}(\frac{b_{k}-b_{j}}{a_{k}-a_{j}}-\frac{b_{i}-b_{k}}{a_{i}-a_{k}})p_{ik}p_{kj}=0,\\\\[4.0pt]
i,j,k=1,2,3;\ a_{i}\neq a_{j},b_{i}\neq b_{j},\ {\rm for}\ i\neq
j,\end{array}$ (1.1)
is one of the important nonlinear models with numerous applications to physics
[14]. The $3$\- and $N$-wave interaction models describe a special class of
wave-wave interactions that are not sensitive on the physical nature of the
waves and bear an universal character. This explains why they find numerous
applications in physics and attract the attention of the scientific community
over the last few decades [23, 24, 25, 20, 19, 22, 21, 9] and the references
therin.
The 3-wave equations can be solved through the inverse scattering method due
to the fact that equation (1.1) admits a Lax representation [24, 25]. But
until the 1990s, the inverse scattering method was pursued almost entirely for
pure initial value problems. In 1997, Fokas announced a new unified approach
for the analysis of inital-boundary value problems for linear and nonlinear
integrable PDEs [1, 2, 3]. The Fokas method provides a generalization of the
inverse scattering formalism from initial value to IBV problems, and over the
last fifteen years, this method has been used to analyze boundary value
problems for several of the most important integrable equations with $2\times
2$ Lax pairs, such as KdV, Schrödinger, sine-Gordon, and stationary
axisymmetric Einstein equations, see e.g. [4, 6]. Just like the IST on the
line, the unified method yields an expression for the solution of an initial-
boundary value problem with that of a Riemann-Hilbert problem. In particular,
the asymptotic behavior of the solution can be analyzed in an effective way by
employing the Riemann-Hilbert problem and the steepest descent method
introduced by Deift and Zhou [10]. Recently, Lenells develop a methodology for
analyzing initial-boundary value problems for integrable evolution equations
with Lax pairs involving $3\times 3$ matrices [7]. He also used this method to
analyze the Degasperis-Procesi equation in [8].
Pelloni and Pinotsis also studied the boundary value problem of the $N-$wave
equation by using the unified method [11]. Recently, Gerdjikov and Grahovski
considered Cauchy problem of the 3-wave equation with with non-vanishing
initial values [12]. In this paper we analyze the initial-boundary value
problem of the three-wave equation (1.1) on the half-line. Compared with these
two papers, there are two differences in our paper. The first difference is
that we get the residue conditions of matrix function $M$ in the Riemann-
Hilbert problem (see (2.27) in the next section 2). The second difference is
that we the jump matrix $J$ is explicitly constructed ( see the equations
(2.14) and (2.22) in the next section 2). Of course, the initial-boundary
value problem for the $3-$wave equation does not need to analysis the global
relation, because the initial data and the boundary data are all known.
The organization of the paper is as follows. In the following section 2, we
perform the spectral analysis of the associated Lax pair. And we formulate the
main Riemann-Hilbert problem in section 3.
## 2\. Spectral Analysis
Our goal in this section is to define analytic eigenfunctions of the Lax pair
(2.1) which are suitable foe the formulation of a Riemann-Hilbert problem.
### 2.1. Lax pair
We first consider the three-wave equations (1.1), with $(x,t)\in\Omega$, and
$\Omega$ denoting the half-line domain
$\Omega=\\{0<x<+\infty,0<t<T\\}$
and $T>0$ being a fixed final time. We denote the initial and boundary values
by $p_{ij,0}(x)$ and $q_{ij,0}(t)$, respectively
$p_{ij,0}(x)=p_{ij}(x,0),\quad q_{ij,0}(t)=p_{ij}(0,t)$
with $p_{ij,0}(x)$ and $q_{ij,0}(t)$ are rapidly decaying. Equation (1.1)
admits the following Lax representation [14]
$\left\\{\begin{array}[]{l}\phi_{x}=M\phi,\\\
\phi_{t}=N\phi,\end{array}\right.$ (2.1)
where $M=i\lambda A+P$ and $N=i\lambda B+Q$, with
$\begin{array}[]{ll}A=\left(\begin{array}[]{ccc}a_{1}&0&0\\\ 0&a_{2}&0\\\
0&0&a_{3}\end{array}\right)&P=\left(\begin{array}[]{ccc}0&p_{12}&p_{13}\\\
p_{21}&0&p_{23}\\\ p_{31}&p_{32}&0\end{array}\right)\\\
B=\left(\begin{array}[]{ccc}b_{1}&0&0\\\ 0&b_{2}&0\\\
0&0&b_{3}\end{array}\right)&Q=\left(\begin{array}[]{ccc}0&n_{12}p_{12}&n_{13}p_{13}\\\
n_{21}p_{21}&0&n_{23}p_{23}\\\
n_{31}p_{31}&n_{32}p_{32}&0\end{array}\right).\end{array}$ (2.2)
Obviously, $trace(A)=trace(B)=0$. We also assume that $a_{1}>a_{2}>a_{3}$ and
$b_{1}<b_{2}<b_{3}$. By using transformation
$\mu=\phi e^{-i\lambda Ax-i\lambda Bt},$
we change Lax pair (2.1) in the form
$\left\\{\begin{array}[]{l}\mu_{x}-i\lambda[A,\mu]=P\mu,\\\
\mu_{t}-i\lambda[B,\mu]=Q\mu,\end{array}\right.$ (2.3)
which can be further written in differential form as
$d(e^{-i\lambda\hat{A}x-i\lambda\hat{B}t}\mu)=W(x,t,\lambda),$ (2.4)
where
$W(x,t,\lambda)=e^{-i\lambda\hat{A}x-i\lambda\hat{B}t}(Pdx+Qdt)\mu$ (2.5)
and $e^{\hat{A}}X=e^{A}Xe^{-A}$.
### 2.2. Spectral functions
We define three eigenfunctions $\\{\mu_{j}\\}_{1}^{3}$ of (2.3) by the
Volterra integral equations
$\mu_{j}(x,t,\lambda)=\mathbb{I}+\int_{\gamma_{j}}e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}W_{j}(x^{\prime},t^{\prime},\lambda).\qquad
j=1,2,3.$ (2.6)
where $W_{j}$ is given by (2.5) with $\mu$ replaced with $\mu_{j}$, and the
contours $\\{\gamma_{j}\\}_{1}^{3}$ are showed in Figure 1.
Figure 1. The three contours $\gamma_{1},\gamma_{2}$ and $\gamma_{3}$ in the
$(x,t)-$domain.
And we have the following inequalities on the contours:
$\begin{array}[]{ll}\gamma_{1}:&x-x^{\prime}\geq 0,t-t^{\prime}\leq 0,\\\
\gamma_{2}:&x-x^{\prime}\geq 0,t-t^{\prime}\geq 0,\\\
\gamma_{3}:&x-x^{\prime}\leq 0.\end{array}$ (2.7)
So, these inequalities imply that the functions $\\{\mu_{j}\\}_{1}^{3}$ are
bounded and analytic for $\lambda\in{\mathbb{C}}$ such that $\lambda$ belongs
to
$\begin{array}[]{ll}\mu_{1}:&(D_{2},\emptyset,D_{1}),\\\
\mu_{2}:&\emptyset,\\\ \mu_{3}:&(D_{1},\emptyset,D_{2}),\end{array}$ (2.8)
where $\\{D_{n}\\}_{1}^{2}$ denote two open, pairwisely disjoint subsets of
the complex $\lambda$ plane showed in Figure 2.
Figure 2. The sets $D_{n}$, $n=1,2$, which decompose the complex $k-$plane.
And the sets $\\{D_{n}\\}_{1}^{2}$ has the following properties:
$\begin{array}[]{l}D_{1}=\\{\lambda\in{\mathbb{C}}|\mathrm{Re}{l_{1}}<\mathrm{Re}{l_{2}}<\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}>\mathrm{Re}{z_{2}}>\mathrm{Re}{z_{3}}\\},\\\
D_{2}=\\{\lambda\in{\mathbb{C}}|\mathrm{Re}{l_{1}}>\mathrm{Re}{l_{2}}>\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}<\mathrm{Re}{z_{2}}<\mathrm{Re}{z_{3}}\\},\\\
\end{array}$
where $l_{i}(\lambda)$ and $z_{i}(\lambda)$ are the diagonal entries of
matrices $i\lambda A$ and $i\lambda B$, respectively.
### 2.3. Matrix valued FUNCTIONS $M_{n}$’s
For each $n=1,2$, define a solution $M_{n}(x,t,\lambda)$ of (2.3) by the
following system of integral equations:
$(M_{n})_{ij}(x,t,\lambda)=\delta_{ij}+\int_{\gamma_{ij}^{n}}(e^{(-i\lambda\hat{A}x-i\lambda\hat{B}t)}W_{n}(x^{\prime},t^{\prime},\lambda))_{ij},\quad\lambda\in
D_{n},\quad i,j=1,2,3.$ (2.9)
where $W_{n}$ an $M_{n}$ are given by (2.5), and the contours
$\gamma_{ij}^{n}$, $n=1,2$, $i,j=1,2,3$ are defined by
$\gamma_{ij}^{n}=\left\\{\begin{array}[]{lclcl}\gamma_{1}&if&\mathrm{Re}l_{i}(\lambda)<\mathrm{Re}l_{j}(\lambda)&and&\mathrm{Re}z_{i}(\lambda)\geq\mathrm{Re}z_{j}(\lambda),\\\
\gamma_{2}&if&\mathrm{Re}l_{i}(\lambda)<\mathrm{Re}l_{j}(\lambda)&and&\mathrm{Re}z_{i}(\lambda)<\mathrm{Re}z_{j}(\lambda),\\\
\gamma_{3}&if&\mathrm{Re}l_{i}(\lambda)\geq\mathrm{Re}l_{j}(\lambda)&&.\\\
\end{array}\right.\quad\mbox{for }\quad\lambda\in D_{n}.$ (2.10)
The following proposition ascertains that the $M_{n}$’s defined in this way
have the properties required for the formulation of a Riemann-Hilbert problem.
###### Proposition 2.1.
For each $n=1,2$, the function $M_{n}(x,t,\lambda)$ is well-defined by
equation (2.9) for $\lambda\in\bar{D}_{n}$ and $(x,t)\in\Omega$. For any fixed
point $(x,t)$, $M_{n}$ is bounded and analytic as a function of $\lambda\in
D_{n}$ away from a possible discrete set of singularities $\\{\lambda_{j}\\}$
at which the Fredholm determinant vanishes. Moreover, $M_{n}$ admits a bounded
and continuous extension to $\bar{D}_{n}$ and
$M_{n}(x,t,\lambda)=\mathbb{I}+O(\frac{1}{\lambda}),\qquad\lambda\rightarrow\infty,\quad\lambda\in
D_{n}.$ (2.11)
###### Proof.
The bounedness and analyticity properties are established in appendix B in
[7]. And substituting the expansion
$M=M_{0}+\frac{M^{(1)}}{\lambda}+\frac{M^{(2)}}{\lambda^{2}}+\cdots,\qquad\lambda\rightarrow\infty.$
into the Lax pair (2.3) and comparing the terms of the same order of $\lambda$
yield the equation (2.11). ∎
### 2.4. The jump matrices
We define spectral functions $S_{n}(\lambda)$, $n=1,2$, and
$S_{n}(\lambda)=M_{n}(0,0,\lambda),\qquad\lambda\in D_{n},\quad n=1,2.$ (2.12)
Let $M$ denote the sectionally analytic function on the complex
$\lambda-$plane which equals $M_{n}$ for $\lambda\in D_{n}$. Then $M_{n}$
satisfies the jump conditions
$M_{1}=M_{2}J,\qquad\lambda\in{\mathbb{R}},$ (2.13)
where the jump matrices $J(x,t,\lambda)$ are defined by
$J=e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}(S_{2}^{-1}S_{1}).$ (2.14)
According to the definition of the $\gamma^{n}$, we find that
$\begin{array}[]{ll}\gamma^{1}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{1}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\end{array}\right)&\gamma^{2}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{1}&\gamma_{1}&\gamma_{3}\end{array}\right)\\\ \end{array}$ (2.15)
### 2.5. The adjugated eigenfunctions
We will also need the analyticity and boundedness properties of the minors of
the matrices $\\{\mu_{j}(x,t,\lambda)\\}_{1}^{3}$. We recall that the adjugate
matrix $X^{A}$ of a $3\times 3$ matrix $X$ is defined by
$X^{A}=\left(\begin{array}[]{ccc}m_{11}(X)&-m_{12}(X)&m_{13}(X)\\\
-m_{21}(X)&m_{22}(X)&-m_{23}(X)\\\
m_{31}(X)&-m_{32}(X)&m_{33}(X)\end{array}\right),$
where $m_{ij}(X)$ denote the $(ij)$th minor of $X$.
It follows from (2.3) that the adjugated eigenfunction $\mu^{A}$ satisfies the
Lax pair
$\left\\{\begin{array}[]{l}\mu_{x}^{A}+[i\lambda A,\mu^{A}]=-P^{T}\mu^{A},\\\
\mu_{t}^{A}+[i\lambda B,\mu^{A}]=-Q^{T}\mu^{A}.\end{array}\right.$ (2.16)
where $V^{T}$ denote the transform of a matrix $V$. Thus, the eigenfunctions
$\\{\mu_{j}^{A}\\}_{1}^{3}$ are solutions of the integral equations
$\mu_{j}^{A}(x,t,\lambda)=\mathbb{I}-\int_{\gamma_{j}}e^{-i\lambda\hat{A}(x-x^{\prime})-i\lambda
B(t-t^{\prime})}(P^{T}dx+Q^{T})\mu^{A},\quad j=1,2,3.$ (2.17)
Then we can get the following analyticity and boundedness properties:
$\begin{array}[]{ll}\mu_{1}^{A}:&(D_{1},\emptyset,D_{2}),\\\
\mu_{2}^{A}:&\emptyset,\\\ \mu_{3}^{A}:&(D_{2},\emptyset,D_{1}).\end{array}$
(2.18)
### 2.6. The computation of jump matrices
Let us define the $3\times 3-$matrix value spectral functions $s(\lambda)$ and
$S(\lambda)$ by
$\mu_{3}(x,t,\lambda)=\mu_{2}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}s(\lambda),$
(2.19a)
$\mu_{1}(x,t,\lambda)=\mu_{2}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}S(\lambda).$
(2.19b)
Thus,
$s(\lambda)=\mu_{3}(0,0,\lambda),\qquad S(\lambda)=\mu_{1}(0,0,\lambda).$
(2.20)
And we deduce from the properties of $\mu_{j}$ and $\mu_{j}^{A}$ that
$s(\lambda)$ and $S(\lambda)$ have the following boundedness properties:
$\begin{array}[]{ll}s(\lambda):&(D_{1},\emptyset,D_{2}),\\\
S(\lambda):&(D_{2},\emptyset,D_{1}),\\\
s^{A}(\lambda):&(D_{2},\emptyset,D_{1}),\\\
S^{A}(\lambda):&(D_{1},\emptyset,D_{2}).\end{array}$
Moreover,
$M_{n}(x,t,\lambda)=\mu_{2}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}S_{n}(\lambda),\quad\lambda\in
D_{n}.$ (2.21)
###### Proposition 2.2.
The $S_{n}$ can be expressed in terms of the entries of $s(\lambda)$ and
$S(\lambda)$ as follows:
$\begin{array}[]{l}S_{1}=\left(\begin{array}[]{ccc}s_{11}&\frac{m_{33}(s)M_{21}(S)-m_{23}(s)M_{31}(S)}{(s^{T}S^{A})_{11}}&\frac{S_{13}}{(S^{T}s^{A})_{33}}\\\
s_{21}&\frac{m_{33}(s)M_{11}(S)-m_{13}(s)M_{31}(S)}{(s^{T}S^{A})_{11}}&\frac{S_{23}}{(S^{T}s^{A})_{33}}\\\
s_{31}&\frac{m_{23}(s)M_{11}(S)-m_{13}(s)M_{21}(S)}{(s^{T}S^{A})_{11}}&\frac{S_{33}}{(S^{T}s^{A})_{33}}\end{array}\right),\\\
S_{2}=\left(\begin{array}[]{ccc}\frac{S_{11}}{(S^{T}s^{A})_{11}}&\frac{m_{21}(s)M_{33}(S)-m_{31}(s)M_{23}(S)}{(s^{T}S^{A})_{33}}&s_{13}\\\
\frac{S_{21}}{(S^{T}s^{A})_{11}}&\frac{m_{11}(s)M_{33}(S)-m_{31}(s)M_{13}(S)}{(s^{T}S^{A})_{33}}&s_{23}\\\
\frac{S_{31}}{(S^{T}s^{A})_{11}}&\frac{m_{11}(s)M_{23}(S)-m_{21}(s)M_{13}(S)}{(s^{T}S^{A})_{33}}&s_{33}\end{array}\right),\end{array}$
(2.22)
where $m_{ij}$ and $M_{ij}$ denote that the $(i,j)-$th minor of $s$ and $S$,
respectively.
###### Proof.
Let $\gamma_{3}^{X_{0}}$ denote the contour $(X_{0},0)\rightarrow(x,t)$ in the
$(x,t)-$plane, here $X_{0}>0$ is a constant. We introduce
$\mu_{3}(x,t,k;X_{0})$ as the solution of (2.6) with $j=3$ and with the
contour $\gamma_{3}$ replaced by $\gamma_{3}^{X_{0}}$. Similarly, we define
$M_{n}(x,t,\lambda;X_{0})$ as the solution of (2.9) with $\gamma_{3}$ replaced
by $\gamma_{3}^{X_{0}}$. We will first derive expression for
$S_{n}(\lambda;X_{0})=M_{n}(0,0,\lambda;X_{0})$ in terms of $S(\lambda)$ and
$s(\lambda;X_{0})=\mu_{3}(0,0,\lambda;X_{0})$. Then (2.22) will follow by
taking the limit $X_{0}\rightarrow\infty$.
First, We have the following relations:
$\left\\{\begin{array}[]{l}M_{n}(x,t,\lambda;X_{0})=\mu_{1}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}R_{n}(\lambda;X_{0}),\\\
M_{n}(x,t,\lambda;X_{0})=\mu_{2}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}S_{n}(\lambda;X_{0}),\\\
M_{n}(x,t,\lambda;X_{0})=\mu_{3}(x,t,\lambda)e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}T_{n}(\lambda;X_{0}).\end{array}\right.$
(2.23)
Then we get $R_{n}(\lambda;X_{0})$ and $T_{n}(\lambda;X_{0})$ are defined as
follows:
$R_{n}(\lambda;X_{0})=e^{-i\lambda\hat{B}T}M_{n}(0,T,\lambda;X_{0}),$ (2.24a)
$T_{n}(\lambda;X_{0})=e^{-i\lambda\hat{A}X_{0}}M_{n}(X_{0},0,\lambda;X_{0}).$
(2.24b)
The relations (2.23) imply that
$s(\lambda;X_{0})=S_{n}(\lambda;X_{0})T^{-1}_{n}(\lambda;X_{0}),\qquad
S(\lambda)=S_{n}(\lambda;X_{0})R^{-1}_{n}(\lambda;X_{0}).$ (2.25)
These equations constitute a matrix factorization problem which, given
$\\{s(\lambda),S(\lambda)\\}$ can be solved for the $\\{R_{n},S_{n},T_{n}\\}$.
Indeed, the integral equations (2.9) together with the definitions of
$\\{R_{n},S_{n},T_{n}\\}$ imply that
$\left\\{\begin{array}[]{lll}(R_{n}(\lambda;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{1},\\\
(S_{n}(\lambda;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{2},\\\
(T_{n}(\lambda;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{3}.\end{array}\right.$
(2.26)
It follows that (2.25) are 18 scalar equations for 18 unknowns. By computing
the explicit solution of this algebraic system, we find that
$\\{S_{n}(\lambda;X_{0})\\}_{1}^{2}$ are given by the equation obtained from
(2.22) by replacing $\\{S_{n}(\lambda),s(\lambda)\\}$ with
$\\{S_{n}(\lambda;X_{0}),s(\lambda;X_{0})\\}$. taking $X_{0}\rightarrow\infty$
in this equation, we arrive at (2.22). ∎
### 2.7. The residue conditions
Since $\mu_{2}$ is an entire function, it follows from (2.21) that M can only
have singularities at the points where the $S_{n}^{\prime}s$ have
singularities. We infer from the explicit formulas (2.22) that the possible
singularities of $M$ are as follows:
* •
$[M]_{1}$ could have poles in $D_{2}$ at the zeros of
$(S^{T}s^{A})_{11}(\lambda)$;
* •
$[M]_{2}$ could have poles in $D_{1}$ at the zeros of
$(s^{T}S^{A})_{11}(\lambda)$;
* •
$[M]_{2}$ could have poles in $D_{2}$ at the zeros of
$(s^{T}S^{A})_{33}(\lambda)$;
* •
$[M]_{3}$ could have poles in $D_{1}$ at the zeros of
$(S^{T}s^{A})_{33}(\lambda)$.
We denote the above possible zeros by $\\{\lambda_{j}\\}_{1}^{N}$ and assume
they satisfy the following assumption.
###### Assumption 2.3.
We assume that
* •
$(S^{T}s^{A})_{11}(\lambda)$ has $n_{0}$ possible simple zeros in $D_{2}$
denoted by $\\{\lambda_{j}\\}_{1}^{n_{0}}$;
* •
$(s^{T}S^{A})_{11}(\lambda)$ has $n_{1}-n_{0}$ possible simple zeros in
$D_{1}$ denoted by $\\{\lambda_{j}\\}_{n_{0}+1}^{n_{1}}$;
* •
$(s^{T}S^{A})_{33}(\lambda)$ has $n_{2}-n_{1}$ possible simple zeros in
$D_{2}$ denoted by $\\{\lambda_{j}\\}_{n_{1}+1}^{n_{2}}$;
* •
$(S^{T}s^{A})_{33}(\lambda)$ has $n_{3}-n_{2}$ possible simple zeros in
$D_{1}$ denoted by $\\{\lambda_{j}\\}_{n_{2}+1}^{N}$;
and that none of these zeros coincide. Moreover, we assume that none of these
functions have zeros on the boundaries of the $D_{n}$’s.
We determine the residue conditions at these zeros in the following:
###### Proposition 2.4.
Let $\\{M_{n}\\}_{1}^{2}$ be the eigenfunctions defined by (2.9) and assume
that the set $\\{\lambda_{j}\\}_{1}^{N}$ of singularities are as the above
assumption. Then the following residue conditions hold:
$\displaystyle\begin{array}[]{l}{Res}_{\lambda=\lambda_{j}}[M]_{1}=\frac{1}{\dot{(S^{T}s^{A})_{11}(\lambda_{j})}}\frac{(S_{11}s_{23}-S_{21}s_{13})(\lambda_{j})}{m_{31}(\lambda_{j})}e^{\theta_{21}(\lambda_{j})}[M(\lambda_{j})]_{2},\\\
\quad 1\leq j\leq n_{0},\lambda_{j}\in D_{2}\end{array},$ (2.27c)
$\displaystyle\begin{array}[]{l}{Res}_{\lambda=\lambda_{j}}[M]_{2}=-\frac{1}{\dot{(s^{T}S^{A})_{11}(\lambda_{j})}}\frac{M_{21}(S^{T}s^{A})_{33}(\lambda_{j})}{(S_{13}(\lambda_{j})s_{31}(\lambda_{j})-S_{33}(\lambda_{j})s_{11}(\lambda_{j}))}e^{\theta_{12}(\lambda_{j})}[M(\lambda_{j})]_{1},\\\
\quad n_{0}<j\leq n_{1},\lambda_{j}\in D_{1}\end{array},$ (2.27f)
$\displaystyle\begin{array}[]{l}{Res}_{\lambda=\lambda_{j}}[M]_{2}=-\frac{1}{\dot{(s^{T}S^{A})_{33}(\lambda_{j})}}\frac{M_{23}(S^{T}s^{A})_{11}(\lambda_{j})}{(s_{13}(\lambda_{j})S_{31}(\lambda_{j})-s_{33}(\lambda_{j})S_{11}(\lambda_{j}))}e^{\theta_{32}(\lambda_{j})}[M(\lambda_{j})]_{3},\\\
\quad n_{1}<j\leq n_{2},\lambda_{j}\in D_{2}\end{array},$ (2.27i)
$\displaystyle\begin{array}[]{l}{Res}_{\lambda=\lambda_{j}}[M]_{3}=\frac{1}{\dot{(S^{T}s^{A})_{33}(\lambda_{j})}}\frac{(S_{13}s_{21}-S_{23}s_{11})(\lambda_{j})}{m_{33}(\lambda_{j})}e^{\theta_{23}(\lambda_{j})}[M(\lambda_{j})]_{2},\\\
\quad n_{2}<j\leq N,\lambda_{j}\in D_{1}\end{array},$ (2.27l)
where $\dot{f}=\frac{df}{d\lambda}$, and $\theta_{ij}$ is defined by
$\theta_{ij}(x,t,\lambda)=(l_{i}-l_{j})x+(z_{i}-z_{j})t,\quad i,j=1,2,3.$
(2.28)
###### Proof.
We will prove (2.27c), (2.27f), the other conditions follow by similar
arguments. Equation (2.21) implies the relation
$M_{1}=\mu_{2}e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}S_{1},$ (2.29a)
$M_{2}=\mu_{2}e^{(i\lambda\hat{A}x+i\lambda\hat{B}t)}S_{2}.$ (2.29b)
In view of the expressions for $S_{1}$ and $S_{2}$ given in (2.22), the three
columns of (2.29a) read:
$\displaystyle[M_{1}]_{1}=[\mu_{2}]_{1}s_{11}(\lambda)+[\mu_{2}]_{2}e^{\theta_{21}}s_{21}(\lambda)+[\mu_{2}]_{3}e^{\theta_{31}}s_{31}(\lambda),$
(2.30a)
$\displaystyle\begin{array}[]{ll}[M_{1}]_{2}=&[\mu_{2}]_{1}e^{\theta_{12}}\frac{m_{33}M_{21}-m_{23}M_{31}}{(s^{T}S^{A})_{11}}(\lambda)+[\mu_{2}]_{2}\frac{m_{33}M_{11}-m_{13}M_{31}}{(s^{T}S^{A})_{11}}(\lambda)\\\
&+[\mu_{2}]_{3}e^{\theta_{32}}\frac{m_{23}M_{11}-m_{13}M_{21}}{(s^{T}S^{A})_{11}}(\lambda)\end{array},$
(2.30d)
$\displaystyle[M_{1}]_{3}=[\mu_{2}]_{1}e^{\theta_{13}}\frac{S_{13}}{(S^{T}s^{A})_{33}}(\lambda)+[\mu_{2}]_{2}e^{\theta_{23}}\frac{S_{23}}{(S^{T}s^{A})_{33}}(\lambda)+[\mu_{2}]_{3}\frac{S_{33}}{(S^{T}s^{A})_{33}}(\lambda).$
(2.30e)
while the three columns of (2.29b) read:
$\displaystyle[M_{2}]_{1}=[\mu_{2}]_{1}\frac{S_{11}}{(S^{T}s^{A})_{11}}(\lambda)+[\mu_{2}]_{2}e^{\theta_{21}}\frac{S_{21}}{(S^{T}s^{A})_{11}}(\lambda)+[\mu_{2}]_{3}e^{\theta_{31}}\frac{S_{31}}{(S^{T}s^{A})_{11}}(\lambda),$
(2.31a)
$\displaystyle\begin{array}[]{ll}[M_{2}]_{2}=&[\mu_{2}]_{1}e^{\theta_{12}}\frac{m_{21}M_{33}-m_{31}M_{23}}{(s^{T}S^{A})_{33}}(\lambda)+[\mu_{2}]_{2}\frac{m_{11}M_{33}-m_{31}M_{13}}{(s^{T}S^{A})_{33}}(\lambda)\\\
&+[\mu_{2}]_{3}e^{\theta_{32}}\frac{m_{11}M_{23}-m_{21}M_{13}}{(s^{T}S^{A})_{33}}(\lambda)\end{array},$
(2.31d)
$\displaystyle[M_{2}]_{3}=[\mu_{2}]_{1}s_{13}e^{\theta_{13}}+[\mu_{2}]_{2}s_{23}e^{\theta_{23}}+[\mu_{2}]_{3}s_{33}.$
(2.31e)
We first suppose that $\lambda_{j}\in D_{2}$ is a simple zero of
$(S^{T}s^{A})_{11}(\lambda)$. Solving (2.31d) and (2.31e) for $[\mu_{2}]_{1}$
and $[\mu_{2}]_{2}$ and substituting the result in to (2.31a), we find
$\begin{array}[]{rl}[M_{1}]_{1}=&\frac{S_{11}s_{23}-S_{21}s_{13}}{(S^{T}s^{A})_{11}m_{31}}e^{\theta_{21}}[M_{2}]_{2}+\frac{M_{33}}{m_{31}}e^{\theta_{31}}[M_{2}]_{3}\\\
&+\frac{1}{m_{31}}e^{\theta_{31}}[\mu_{2}]_{3}\end{array}.$
Taking the residue of this equation at $\lambda_{j}$, we find the condition
(2.27c) in the case when $\lambda_{j}\in D_{2}$. Similarly, we can get the
equation (2.27l).
Then let us consider that $\lambda_{j}\in D_{1}$ is a simple zero of
$(s^{T}S^{A})_{11}(\lambda)$. Solving (2.30a) and (2.30e) for $[\mu_{2}]_{1}$
and $[\mu_{2}]_{3}$ and substituting the result in to (2.30d), we find
$\begin{array}[]{rl}[M_{1}]_{2}=&-\frac{M_{21}(S^{T}s^{A})_{33}}{(s^{T}S^{A})_{11}(S_{13}s_{31}-S_{33}s_{11})}e^{\theta_{12}}[M_{1}]_{1}-\frac{(S^{T}s^{A})_{33}}{S_{13}s_{31}-S_{33}s_{11}}[\mu_{2}]_{2}\\\
&-\frac{m_{23}(S^{T}s^{A})_{33}}{S_{13}s_{31}-S_{33}s_{11}}e^{\theta_{32}}[M_{1}]_{3}\end{array}.$
Taking the residue of this equation at $\lambda_{j}$, we find the condition
(2.27f) in the case when $\lambda_{j}\in D_{1}$. Similarly, we can get the
equation (2.27i). ∎
## 3\. The Riemann-Hilbert problem
The sectionally analytic function $M(x,t,\lambda)$ defined in section 2
satisfies a Riemann-Hilbert problem which can be formulated in terms of the
initial and boundary values of $p_{ij}(x,t)$. By solving this Riemann-Hilbert
problem, the solution of (1.1) can be recovered for all values of $x,t$.
###### Theorem 3.1.
Suppose that $p_{ij}(x,t)$ are a solution of (1.1) in the half-line domain
$\Omega$ with sufficient smoothness and decays as $x\rightarrow\infty$. Then
$p_{ij}(x,t)$ can be reconstructed from the initial value
$\\{p_{ij,0}(x)\\}_{i,j=1}^{3}$ and boundary values
$\\{q_{ij,0}(t)\\}_{i,j=1}^{3}$ defined as follows,
$p_{ij,0}(x)=p_{ij}(x,0),\quad q_{ij,0}(t)=p_{ij}(0,t).$ (3.1)
Use the initial and boundary data to define the jump matrices $J(x,t,\lambda)$
as well as the spectral $s(\lambda)$ and $S(\lambda)$ by equation (2.19).
Assume that the possible zeros $\\{\lambda_{j}\\}_{1}^{N}$ of the functions
$(S^{T}s^{A})_{33}(\lambda),(s^{T}S^{A})_{11}(\lambda)$,
$(s^{T}S^{A})_{33}(\lambda),(S^{T}s^{A})_{33}(\lambda)$ are as in assumption
2.3.
Then the solution $\\{p_{ij}(x,t)\\}_{i,j=1}^{3}$ is given by
$p_{ij}(x,t)=-i(a_{i}-a_{j})\lim_{\lambda\rightarrow\infty}(\lambda
M(x,t,\lambda))_{ij}.$ (3.2)
where $M(x,t,\lambda)$ satisfies the following $3\times 3$ matrix Riemann-
Hilbert problem:
* •
$M$ is sectionally meromorphic on the complex $\lambda-$plane with jumps
across the contour ${\mathbb{R}}$, see Figure 2.
* •
Across the contour ${\mathbb{R}}$, $M$ satisfies the jump condition
$M_{1}(x,t,\lambda)=M_{2}(x,t,\lambda)J(x,t,\lambda),\quad\lambda\in{\mathbb{R}}.$
(3.3)
where the jump $J$ is defined by the equation (2.14).
* •
$M(x,t,\lambda)=\mathbb{I}+O(\frac{1}{\lambda}),\qquad\lambda\rightarrow\infty$.
* •
The residue condition of $M$ is showed in Proposition 2.4.
###### Proof.
It only remains to prove (3.2) and this equation follows from the large
$\lambda$ asymptotics of the eigenfunctions.
We write the large $\lambda$ asymptotics of $M$ as follows:
$M(x,t,\lambda)=M_{0}(x,t)+\frac{M_{1}(x,t)}{\lambda}+\cdots.\qquad\lambda\rightarrow+\infty.$
(3.4)
And insert this equation into the equation (2.3) and compare the coeffients of
the same order $\lambda$, for the $O(\lambda)$, we have $M_{0}$ is a diagonal
matrix ; for the $O(1)$, we get $M_{0}=\mathbb{I}$ by comparing the diagonal
elements, and we can have the following equation by comparing the other
elements
$-i[A,M_{1}]=P,$ (3.5)
this equation reads the required result of $p_{ij}(x,t)$
$p_{ij}(x,t)=-i(a_{i}-a_{j})M_{1,ij}(x,t).$ (3.6)
∎
Acknowledgements The work of Xu was partially supported by Excellent Doctor
Research Funding Project of Fudan University. The work described in this paper
was supported by grants from the National Science Foundation of China (Project
No.10971031;11271079), Doctoral Programs Foundation of the Ministry of
Education of China, and the Shanghai Shuguang Tracking Project (project
08GG01).
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|
arxiv-papers
| 2013-04-16T06:36:14 |
2024-09-04T02:49:44.449784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Xu, Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1304.4339"
}
|
1304.4387
|
# Lagrangian transport in a microtidal coastal area: the Bay of Palma, island
of Mallorca, Spain
Ismael Hernández-Carrasco Cristóbal López Emilio Hernández-García IFISC,
Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), 07122
Palma de Mallorca, Spain Alejandro Orfila IMEDEA, Instituto Mediterráneo de
Estudios Avanzados (CSIC-UIB), 07190 Esporles, Spain
###### Abstract
Coastal transport in the Bay of Palma, a small region in the island of
Mallorca, Spain, is characterized in terms of Lagrangian descriptors. The data
sets used for this study are the output for two months (one in autumn and one
in summer) of a high resolution numerical model, ROMS, forced atmospherically
and with a spatial resolution of 300 m. The two months were selected because
its different wind regime, which is the main driver of the sea dynamics in
this area. Finite-size Lyapunov Exponents (FSLEs) were used to locate semi-
persistent Lagrangian coherent structures (LCS) and to understand the
different flow regimes in the Bay. The different wind directions and
regularity in the two months have a clear impact on the surface Bay dynamics,
whereas only topographic features appear clearly in the bottom structures. The
fluid interchange between the Bay and the open ocean was studied by computing
particle trajectories and Residence Times (RT) maps. The escape rate of
particles out of the Bay is qualitatively different, with a 32$\%$ more of
escape rate of particles to the ocean in October than in July, owing to the
different geometric characteristics of the flow. We show that LCSs separate
regions with different transport properties by displaying spatial
distributions of residence times on synoptic Lagrangian maps together with the
location of the LCSs. Correlations between the time-dependent behavior of FSLE
and RT are also investigated, showing a negative dependence when the stirring
characterized by FSLE values moves particles in the direction of escape.
## I Introduction
The study of transport and mixing in coastal flows is of major interest
because of their economic and ecological importance. Due to the
particularities that they present, like influence of complex topography,
coastline shape and the direct driving at the surface by highly variable wind
forcing, coastal flow dynamics remains still poorly understood.
Recently, coastal observations and modeling efforts in different regions have
been addressed from the Lagrangian point of view: Lekien et al. (2005) showed
that Lagrangian Coherent Structures (LCSs) computed from velocity fields
obtained from HF Radar measurements can be used to predict pollutant
dispersion in the coast of Florida; Gildor et al. (2009) and Shadden et al.
(2009) detected LCSs with HF Radar data in the Gulf of Eliat, Israel, and in
Monterey Bay, respectively. Haza et al. (2010) studied small-scale properties
of dispersion measurements obtained from HF Radar data in the Gulf of La
Spezia, Italy. Also Nencioli et al. (2011) have detected LSCs in a coastal
region with a Lyapunov method based on in situ observations. Besides Radar
measurements, LCSs obtained from velocity data of high resolution numerical
models have been used to analyze the effect of the waves on LCS in the Bay of
Palma, Spain (Galan et al., 2012), to study the transport in the tidal flow of
Ria de Vigo, Spain (Huhn et al., 2012), or to study the water quality of a
very small coastal region, the Hobie Beach, USA (Fiorentino et al., 2012)
Also, data from drifters released in the Santa Barbara Channel were used by
Ohlmann et al. (2012) to characterize relative dispersion, very useful to
improve Lagrangian stochastic models. The application of Lagrangian techniques
to study the dynamics in a shallow lake (small closed basin) has been
performed in Pattantyús-Ábrahám et al. (2008).
Palma is the largest city in the Balearic Islands. Human activities, in
particular recreational ones, give to water quality in the Bay of Palma a
large economic value. A proper analysis of transport can be useful to
understand the fluid dynamics in the Bay and therefore help protect the
coastal water. Previous studies performed in the Bay of Palma used Eulerian
techniques to understand the coastal dynamics (Jordi et al., 2009, 2011). In
this work we study some transport properties in the Bay of Palma using
Lagrangian techniques developed from dynamical systems theory. Computing both
LCSs and residence times the Bay of Palma can be sorted in regions of
different properties, for example having more or less connectivity with the
open ocean. This kind of studies have demonstrated to be useful to identify
pollution pathways or conditions for red tides (Lekien et al., 2005;
Fiorentino et al., 2012). The Bay is a semi-enclosed basin located in the
southwest of the island of Mallorca (western Mediterranean sea), whose coastal
flow is mainly induced by the wind (Jordi et al., 2009, 2011). Forcing by
tides is almost negligible with a tidal amplitude of less than 0.25$m$. This
makes the dynamics here different form other locations (e.g. Shadden et al.
(2009); Huhn et al. (2012)) where tides are dominant, and then provides the
opportunity to test the performance of dynamical systems tools in this
situation in which forcing only acts directly on the sea surface, and in which
there are rather different forcing regimes depending on the season. The
Lagrangian diagnosis will be obtained from velocity data of a realistic
numerical model at high resolution, which resolves spatial scales of a few
hundred of meters. We investigate the surface horizontal transport during two
months corresponding to different seasons (autumn and summer), and therefore
to different wind conditions, in order to highlight the effect of the wind on
transport. In the case of July we also study the deepest bottom layer. We
compute the barriers and avenues to transport (LCS) from lines of high values
of Finite-Size Lyapunov Exponents (FSLE). We also present calculations of
residence times and show synoptic Lagrangian maps (SLM) of these times
(Lipphardt et al., 2006), which will allow us a detailed visualization of the
interchange of fluid particles between the Bay and the open sea. The
relationship between LCSs and areas of different residence times will be
analyzed.
The organization of this paper is as follows. The data set used in the
computations and the area of study is described in Section II. Section III
presents a brief overview of the Lagrangian tools that are used. Before
presenting the Lagrangian results, we show in Section IV a short summary of
Eulerian results by studying the velocties in the Bay. We present in Section V
a characterization of stirring in the Bay of Palma in terms of FSLE and
residence times. Using the definition of LCS given in Section III, Lagrangian
barriers are identified in the domain of interest. We compute escape rates and
residence times of fluid particles to describe the transport relation between
the Bay and the open ocean. We provide possible mechanisms to explain
differences in the residences times and FSLE between different seasonal
months. Finally we summarize the main results in Section VI.
Figure 1: Bathymetry contours (in meters) of the model domain. The black box
indicates the Palma Bay and the inset graphics give the geographical location
of Mallorca Island in the western Mediterranean Sea.
## II Data and characteristics of the study region
### II.1 Area of study
The island of Mallorca (Fig. 1) is part of the Balearic Islands Archipelago
and is located in the center of the western Mediterranean (between 39∘ and
40∘N and 2.50∘ and 3.50∘ E). The Bay of Palma is a nearly semi-circular and
semi-enclosed basin located in the southwest coast of Mallorca and it can
reach depths of more than 60 $m$. The Bay of Palma is defined as the water
mass inside the square in Fig.1, consisting of a northern limit at 39∘34′N, a
southern limit at 39∘24′N, and 2∘30′E and 2∘45′E as the western and eastern
limits, respectively. The open boundary to the sea is in the southern part and
it is 20 km wide.
The size of the Bay is smaller than the Rossby radius of deformation at these
latitudes, and the main circulation is determined by the bathymetry at the
bottom layer and by local and remote winds at the surface layer. In particular
the studies by Jordi et al. (2009, 2011) have shown that the major forcing
mechanisms come from wind-induced island trapped waves (ITW) propagating at an
island scale and by locally wind-induced mass balance. The intense ITW can
produce new instabilities which can generate coastal gyres at submesoscale
(see Jordi et al. (2011)). During summer there are persistent sea breeze
conditions. In July and August, the weather is often almost identical from one
day to the next. In the vicinity of the Bay and along the southern coast of
Mallorca the breeze blows from the south-west. Several studies (Ramis and
Alonso, 1988; Ramis and Romero, 1995), have pointed out that the
meteorological conditions of Mallorca (intense solar radiation, clear skies,
soil water deficit, dryness, weak surface pressure gradients, etc.) favors the
development of sea breeze, often from April to October, and almost every day
during July and August. Winds in autumn, and particularly in late September
and October are more irregular, with episodes of strong storm activity (Tudurí
and Ramis, 1997).
### II.2 Data
The velocity data sets were obtained from the numerical model ROMS (Regional
Ocean Model System). ROMS is a free surface, hydrostatic, primitive equation
ocean model. The model uses a stretched, generalized nonlinear coordinate
system to follow bottom topography in the vertical, and orthogonal curvilinear
coordinates in the horizontal (Song and Haidvogel, 1994; Haidvogel et al.,
2000). At each grid point, horizontal resolution $\Delta_{0}$ is the same in
both the longitudinal, $\phi$, and latitudinal, $\theta$, directions.
We run the simulation with a resolution of $\Delta_{0}=0.0027^{\circ}$
($\sim$300$m$, ROMS300), which is itself nested into a larger and coarser grid
with $\Delta_{0}$ =1/74∘ ($\sim$1500$m$). Boundary conditions for the coarser
domain were taken from daily outputs of the Mediterranean Forecasting System
(Dobricic et al., 2007; Oddo et al., 2009). The ROMS300 domain covers
$39^{\circ}$12′N - $39^{\circ}$36′N (latitude), and $2^{\circ}$24′E -
$3^{\circ}$6′E (longitude). The total number of grid nodes is 260 $\times$
148\. Vertical resolution is variable with $10$ layers in total. All domains
were forced using realistic winds provided by the PSU/NCAR mesoscale model
MM5. The initial vertical structure of temperature and salinity was obtained
from the Levitus database (Locarnini et al., 2006; Antonov et al., 2006).
We will manage velocity data from the surface layer and the bottom layer for
the grid of $\Delta_{0}$ $\approx$ 300$m$. This domain allows us to analyze
the fluid interchange between the Bay and the open ocean, using a high
resolution velocity field. Only horizontal velocities are considered, so that
vertical displacements are neglected in the surface layer, and particles in
the bottom remain in the bottom layer. This is justified by the small
integration times we will use. Nevertheless, close to the coast they can have
an impact that will be the subject of future work. The output of the model was
compared with data from drifters (see Galan et al. (2012)) and a reasonable
agreement was found, although it improved when adding the influence of wave
intensity. Thus the present study should be considered as a simplified
baseline case against which to compare the future consideration of the full 3d
dynamics, or the influence of small scale process such as waves (Galan et al.,
2012). We will study two different intervals of time corresponding to two
different wind regimes: one starting on October $5$th, $2008$ and finishing on
October $29$th, $2008$; and the other extending from July $1$st, $2009$ until
July $26$th, $2009$. Temporal resolution is 15 minutes and 10 minutes for
October and July, respectively, resulting in a total of $2375$ snapshots of
the velocity field for October, and $3744$ for July.
## III Methodology
### III.1 LCSs and particle dispersion from FSLE
Our methodology is based on the Lagrangian analysis of marine flows. In the
Lagrangian view, particles are advected by the flow and their horizontal
motion (neglecting motions between model layers) is governed by the
differential equations
$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle{v_{x}(x,y,t)},$
(1) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$
$\displaystyle{v_{y}(x,y,t)},$ (2)
where ($x(t),y(t)$) are the west-east and the south-north coordinates of the
trajectories and ($v_{x},v_{y}$) are the eastwards and northwards components
of the velocity. Because of the small sizes involved, we will use a Cartesian
coordinate system.
LCSs (Haller and Yuan, 2000; d’Ovidio et al., 2004; Shadden et al., 2005), are
roughly defined as the material lines organizing the transport in the flow.
They are the analogs, for time-dependent flows, of the unstable and stable
manifolds of hyperbolic fixed points. Among other approaches (Mancho et al.,
2006; Mendoza and Mancho, 2010; Mezić et al., 2010; Rypina et al., 2011;
Haller and Beron-Vera, 2012), ridges of the local Lyapunov Exponents provide a
convenient tool to locate them. In our case, we use the so-called Finite-Size
Lyapunov Exponents (FSLEs) which are the adaptation of the asymptotic
classical Lyapunov Exponent to finite spatial scales (Aurell et al., 1997;
Boffetta et al., 2001). FSLEs are a local measure of particle dispersion and
thus of stirring and mixing, as a function of the spatial resolution, serving
to isolate the different regimes corresponding to different length scales of
the oceanic flows, very useful in coastal systems (Cencini et al., 2010). In
fact the first applications of the FSLE technique in oceanography were for
closed or semi-closed basins (Buffoni et al., 1996, 1997).
For two particles of fluid, one of them located at x, the FSLE at time $t_{0}$
and at the spatial point x is given by the formula:
$\lambda(\textbf{x},t_{0},\delta_{0},\delta_{f})=\frac{1}{|\tau|}\ln{\frac{\delta_{f}}{\delta_{0}}},$
(3)
where $\delta_{0}$ is the initial distance of the two given particles, and
$\delta_{f}$ is their final distance. Thus, to compute the FSLEs we need to
calculate the minimal time, $\tau$, needed for the two particles initially
separated $\delta_{0}$, to get a final distance $\delta_{f}$ (in this way the
FSLE represents the inverse time scale for mixing up fluid parcels between
length scales $\delta_{0}$ and $\delta_{f}$). To obtain this time we need to
know the trajectories of the particles (from Eqs. (1) and (2)) which gives the
Lagrangian character to this quantity. The FSLEs are computed for the points x
of a square lattice with lattice spacing coincident with the initial
separation of fluid particles $\delta_{0}$. We can obtain a good estimation of
the minimal $\tau$ at each site by selecting the trajectory which diverges the
first among the four trajectories starting at the neighbors of the given site
in the grid of initial conditions. Numerically we integrate the equations of
motion using a standard, fourth-order Runge-Kutta scheme, with an integration
time step corresponding to the time resolution of the velocity data: $dt=15$
minutes in October and $dt=10$ minutes in July. We have checked in selected
trajectories that using in July the same time step $dt=15$ as in October does
not alter the trajectories. Since velocity information is provided just in a
discrete space-time grid, spatiotemporal interpolation of the velocity data is
achieved by bilinear interpolation. For the spatial scales that define FSLEs,
we take $\delta_{f}=0.1^{\circ}$, i.e., final separations of about $10\ km$,
because of the size of the Bay. On the other side, we take $\delta_{0}$ equal
to $75\ m$, four times smaller than the resolution of the velocity field,
$\Delta_{0}=300\ m$. Since we are interested only in fast time scales, our
integrations are restricted to 5 days. Locations for which the final
separation at the end of this period has not reached the prescribed
$\delta_{f}=10\ km$ (or for which particles have been trapped by land) are
assigned a value $\lambda=0$.
FSLEs can be computed from trajectory integration backwards and forward in
time. Their highest values as a function of the initial location, x, organize
in filamental structures approximating relevant manifolds: ridges in the
spatial distribution of backward (forward) FSLEs identify regions of locally
maximum compression (separation), approximating attracting (repelling)
material lines or unstable (stable) manifolds of hyperbolic trajectories,
which can be identified with the LCSs (Haller and Yuan, 2000; d’Ovidio et al.,
2004; Shadden et al., 2005; Tew Kai et al., 2009; Hernández-Carrasco et al.,
2011), and characterize the flow from the Lagrangian point of view (Joseph and
Legras, 2002; Koh and Legras, 2002). Attracting LCSs associated to backward
integration (the unstable manifolds) have a direct physical interpretation
(Joseph and Legras, 2002; d’Ovidio et al., 2004, 2009). Tracers (chlorophyll,
temperature, …) spread along these attracting LCSs, thus creating their
typical filamental structure (Tél and Gruiz, 2006; Lehan et al., 2007; Tew Kai
et al., 2009; Calil and Richards, 2010). When not stated explicitly, by FSLE
we will mean the backwards FSLE values. In addition to locate spatial
structures, time-averages of FSLE give an indication of the intensity of
stirring in given areas, which we analyze in Sect. V.1.
We close this section by noting that the relationship between LCSs and
Lyapunov exponents is based on heuristic arguments which may not be correct in
some cases (see for example Haller (2011)). We identify as possible LCSs only
the locations having the largest values of FSLE, which align in linear
structures. In this way we effectively select only the highest FSLE ridges
which are more likely to organize the flow. Even in this case, it is possible
that the FSLE technique identifies regions of high shear which are not
hyperbolic and then may lack some of the properties of bona fide LCSs. Thus,
direct inspection of particle trajectories and comparison with complementary
techniques would be needed to confirm the validity of the FSLE approach in
this situation. One of such complementary techniques is residence time maps
that we present in the following section.
### III.2 Escape and residence times
Another characteristic time-scale for transport processes in open flows is the
so-called escape rate (Lai and Tel, 2011). This quantity measures how quickly
particles trajectories escape from a domain. If we initiate $N(0)$ particles
in a flow, we can measure how the trajectories escape the preselected region.
In the case in which the decay in the number of particles remaining in the
region up to time $t$, $N(t)$, decays exponentially with time, $N(t)/N(0)\sim
e^{-\kappa t}$, there is a well-defined escape time defined as the inverse of
the escape rate $\kappa$: $\tau_{e}=1/\kappa$. For the range of times explored
in our work, we will see that the particle escape is close to exponential and
then we can estimate the value of $\tau_{e}$.
$\tau_{e}$ is a global quantity associated to the whole basin. A more detailed
description of the transport processes can be obtained by other suitable
Lagrangian quantities such as residence times (Buffoni et al., 1996, 1997;
Falco et al., 2000; Orfila et al., 2005). The particle residence time (RT) is
defined as the interval of time that a fluid particle remains in a region
before crossing a particular boundary. For each fluid particle inside the Bay
at an initial time, we need to compute two times: the forward exit time,
$t_{f}$, computed as the time needed for a particle to cross the line
delimiting the Bay, taking the forward-in-time dynamics; and the backward exit
time, $t_{b}$, the same but in the backward-in-time dynamics. The residence
time is defined as $RT=t_{f}+t_{b}$. RTs can be displayed in plots named
Lagrangian Synoptic Maps (Lipphardt et al., 2006), in which the residence time
of each fluid particle is referenced to its initial position on the grid.
## IV Preliminary Eulerian description
A first approach to the transport process in the Bay can be a description from
the Eulerian point of view, by studying averages of the velocity field. To do
this we consider separately the meridional $v_{y}$ and zonal $v_{x}$
components of the surface flow, and we analyze the time evolution of their
spatial averages.
Figure 2: a) Complete time series throughout October of the zonal (top panels)
and meridional (bottom panels) of the spatial average of the surface velocity
field (black line). The red line is a running daily average. b) the same as a)
but for July. Figure 3: Spectra for the zonal (left panels) and meridional
component (right panels) of the surface velocity field ($m^{2}/s$)in October
(top) and July (bottom)
Figures 2 a) and b) show the time series of data taken every 15 min in October
and 10 min in July (black lines), and daily average time series (red lines) of
$v_{x}$ and $v_{y}$ for October and July, respectively. The impact of the more
variable and stormy weather in October is clear in the high frequency
variability of the time-series. During the two months both components of the
flow present daily variability related to the presence of land and sea
breezes. In July the zonal fluctuations are much more noticeable and regular
than the meridional ones, being $\langle v_{y}\rangle$ very small. We have
computed the power spectra for both months (see Fig. 3). In October, in
addition to higher power at high frequencies, there are also stronger low-
frequency fluctuations. From such features in their spectra of ADCP-derived
velocities Jordi et al. (2011) identified wind-induced island trapped waves as
the main source of variability in the Bay dynamics, in addition to the local
wind (essentially sea breeze). In contrast, the dominant role of sea breeze in
July is seen as the very strong dominance of the daily frequency peak at the
July zonal spectrum.
Comparing the velocity components of both months we observe quantitative
differences. The values of $v_{y}$ in the case of October range from -1.0 to
1.5 $m/s$ (bottom panel of Fig. 2 a), while in the case of July, $v_{y}$ is
two orders of magnitude smaller, ranging from -0.1 to 0.02 $m/s$ (bottom panel
of Fig. 2 b). On other hand, $v_{x}$ are similar during October and July. In
October, $v_{x}$ ranges from -1.5 to 0.5 $m/s$ (top panel of Fig. 2 a), the
same order of magnitude than the meridional velocity, resulting in circular
motions (clockwise along the Bay). In July the situation is significantly
different. The zonal velocity ranges from -1.5 to 0.7 $m/s$ (top panel of Fig.
2 b), much larger than the meridional velocity, resulting in a flow consisting
on oscillations along the zonal direction. In October the mean values (and the
standard deviations given in parenthesis) of the time series are
$<v_{x}>=-0.0704~{}(0.1897)m/s$ and $<v_{y}>=0.0440~{}(0.1982)m/s$. In July we
have $<v_{x}>=0.0013~{}(0.3052)m/s$ and $<v_{y}>=-0.0140~{}(0.0134)m/s$. The
large standard deviation in the zonal velocity in July is an indicator of the
large (breeze induced) daily fluctuations in this month, but restricted to a
single direction of motion.
## V Lagrangian Results
### V.1 Average characterization of stirring
We now describe our Lagrangian results. First we compute the temporal average
(over the months of October and July) of the FSLEs for the surface layer, and
for July in the bottom layer. This calculation helps us to unveil areas of
different stirring and the differences between layers and months.
Figure 4: Spatial distribution of the time average of 6-hourly FSLEs maps over
different months and at different layers: a) October at surface layer, b) July
at surface layer, c) July at bottom layer.
The surface computations for the different seasonal months, October and July
(Fig. 4 a, b) show different values and spatial distributions of stirring. We
use the same colorbar to compare the stirring in both months. The Bay of Palma
appears to be an area with important activity. Average FSLE field looks more
homogeneous in July than in October. During October filamental structures of
high values of FSLE are accumulated over the northeast side of the Bay,
forming a linear structure running from north to south-east which comes from
similar structures in the instantaneous (non-averaged) fields that can act as
barriers, therefore dividing the Bay in two flow regions of qualitatively
different dynamics. The difference in wind regularity and intensity between
these months, and the fact that local and remote winds are the main drivers of
the bay dynamics, explains the difference in mean stirring distribution among
the two months. The importance of wind will be replaced by bottom topography
when going to the deep layers. The effect of the terrain topography on
stirring is clear in Fig. 4 c), where FSLEs are computed at the deepest layer
for July. The high values of time-averaged FSLEs are located close to a region
of high bathymetry gradient, which seems to act as a barrier along which the
flow is stretched.
Figure 5: Evolution of the locations of two sets of particles in the Palma Bay
during a night of October, superimposed on the spatial distributions of high
values of backward FSLEs. The colorbars specify FSLE values in days-1. Zero
FSLE values, displayed as white, are assigned to locations for which the
particles do not attain the prescribed $\delta_{f}=10km$ separation after
5-days integration. Note the highest values of FSLEs (green lines) act as a
barrier practically dividing the Bay in two parts. The two sets of particles
are deployed from both sides of the barrier. (a) Initial conditions of the
particles on October 8 at 20:00 GMT, 2008.; (b) October 9 at 00:00 GMT, 2008 ;
(c) October 9 at 04:00 GMT, 2008 ; (d) October 9 at 08:00 GMT, 2008. Particles
marked by black dots were released in the right side (northeast) of the
barrier while the particles marked with red were released on the left side of
the barrier.
### V.2 Coastal LCSs
The temporal averages computed in Sec. V.1 give us a rough idea of stirring in
the Bay. More detailed information is obtained by looking at non-averaged
quantities, that may reveal the existence of barriers to transport. Figure 5
shows the location of the high backward FSLE values (LCSs), appearing as a
network of lines, computed at successive instants of time in October. These
temporary structures can remain for one or more days, as happens in October,
or they can appear in the same location periodically (not shown). We stress
here the appearance of a clear barrier, from north to south-east, that divides
the Bay in two areas that correlates with the temporal average in Fig. 4 a).
This barrier appears in almost the same position in different days, remaining
without displacing too much. To effectively see that it acts as a barrier we
have considered the evolution of virtual particles released at both sides of
the barrier. Red and black particles do not mix and they tend to spread along
the barrier (confirming that, as expected, it is an attracting line).
In July the situation is rather different. Lines of high Lyapunov exponents
(forward and backwards) are mainly oriented zonally in the bay (except close
to the opening to the sea), which is also the dominant direction of motion.
Thus, it does not seem that they represent hyperbolic LCS, but rather lines of
intense shear between zonally moving strips.
Figure 6: Average of 15 subsequent (started at $t_{0}$ values separated 18
hours) estimations of $N(t)$, the number of particles remaining in Palma Bay
at least for a lapse of time $t$ after release at $t_{0}$. Black and red lines
are for surface layer in October and July respectively. Dashed lines are the
measured averages, and the solid lines are the indicated exponential fits.
### V.3 Transport between the Bay of Palma and the open sea
In this section we study the surface transport of particles in and out of the
Bay. To have an idea of the time scales involved in this interchange we
proceed by computing the number of particles remaining in the Bay, $N(t)$,
averaged over different starting times (separated by $18$hours in order to
collect the information of diurnal and nocturnal signal; this gives us 15
different simulations to be averaged for each month) as a function of the
integration time $t$. A particle is considered to leave the Bay when crossing
the red open-sea boundary in Fig. 1, so that particles landing on the coast
are considered not escaped. Fig. 6 shows the different average decays for
October and July. In both cases $N(t)$ is reasonably fitted by an exponential
in the considered time-range, thus identifying the escape rates $\kappa$ =
0.62 and 0.47 $days^{-1}$, respectively. The corresponding escape times, given
by the inverse of the escape rate, are, respectively, $\tau_{e}$ = 1.61 and
2.12 $days$. The relative difference of the escape rates of July with respect
to October, $(\kappa_{October}-\kappa_{July})/\kappa_{July}$, is 0.32. Thus
the exchange of fluid particles between the Bay and the open ocean is a 32$\%$
more active in autumn than in summer.
Next we compute synoptic maps of the residence times. As was indicated in Sec.
III.2 the residence time of the particles throughout the study area is
considered as the sum of the entry time ($t_{b}$) and the escape time
($t_{f}$). To compute $t_{f}$ and $t_{b}$ particles are initialized every 6
hours in a regular grid of 75$m$ spacing and they are integrated forward and
backward in time during 5 days. We consider that 5 days is a proper
integration time according with the time scales associated with the coastal
processes of this small Bay, and also owing to the short period of the
available data. In these computations we assign the maximum possible value of
$t_{f}$ and $t_{b}$ (5 days) to the fluid particles that remain in the pre-
selected area after the 5 days of integration.
Figure 7: Lines are the locations of top values of FSLE (greater than
$0.5days^{-1}$ in October and greater than $1.5days^{-1}$ in July). Backward
FSLE lines are colored in black and forward FSLEs in white. They are
superimposed on spatial distributions of residence times in Palma Bay for
different dates. The colorbars give the residence times in days. a) and b)
correspond to two different days in October, and c) and d) in July.
In Fig. 7 we color the initial positions of particles in the Bay attending to
the time they transit through the Bay ($RT=t_{f}+t_{b}$) for different days.
Initial positions of particles with short residence times are indicated in
blue in Fig. 7. Regions from where particles have longer residence time (i.e.
take more time between entry and escape) are marked in red/brown.
These maps show that the spatial distribution of residence times of particles
can be complex and time depending, presenting different patterns at different
times. A number of small structures can be observed, including thin filaments
or small lobes. Comparing both months, one can see differences in the RT
distributions. The most noticeable is the approximate east-west alignment of
the zones of similar RT in July, which is not seen in October. Also, in
October the values of residence times of the particles are smaller, in
agreement with the global rate estimations showed before. A common feature is
the southwest region with low values of residence times, because in this
region there are not coastal boundaries and it is totally open to the ocean.
Figure 8: Spatial distributions of time averages of 60 snapshots of 6-hourly
RT values collected over 15 days in Palma Bay for a) October and b) July. The
colorbar units are $days$.
In order to reveal regions with different persistent transport properties we
compute time averages of the spatial distributions of residence times. We
average 6-hourly snapshots of RT during 15 days (i.e. 60 snapshots) for each
month. The results, plotted in Fig. 8 a) and b), show the common features
that, in general, the low values of RT are for particles initiated close to
the open ocean, specially in the southwest part, and high values are for
particles started near the coast, as expected. However, on average, the
residence time is larger in July than in October (3.25 days in July and 1.51
days in October) consistent with the behavior of the corresponding values of
$\tau_{e}$. Also, in July there is a clear boundary between the interior of
the Bay to the north, with large average residence times and the open sea to
the south, whereas the boundary between high and low residence times in
October is well inside the Bay, aligned with the Lagrangian structure
identified from the FSLE analysis, as will be discussed in the next section.
Another feature observed in movies of particle trajectories (not shown) is
that in October fluid particles tend to circulate mostly clockwise, while in
July they are oscillating along the zonal direction (see Section IV). This
difference, arising as discussed in Sect. IV from the different regimes of
wind forcing, is likely to be responsible for most of the different behavior
between both months.
### V.4 Relation between LCSs and RTs
Figure 9: Snapshots of top values of FSLE (greater than $0.5days^{-1}$)
plotted over residence times. a) is for backward integration in time, and b)
is for forward integration. Both plots correspond to October 19th at 00:00h
(GMT) The colorbar units are $days$.
We now examine the connection between regions of different residence times
with LCSs. To compare RT and FSLE we have superimposed in Fig. 7 the filaments
of high values of forward (white) and backward FSLE (black) values on the
spatial distribution of residence times. The Figure shows a good
correspondence between many structures of RT and FSLE. Note that RT is the sum
of a forward and a backward exit time, so that some of the strong gradients of
RT will correspond to forward and some others to backwards LCSs. Fig 9 shows
clearer the correspondence between FSLE backwards in time with entry times,
and FSLE forward with escape times. At least in October, the LCSs given by
high values of FSLE clearly separate regions with different values of
residence times, confirming the value of the FSLE technique to identify
boundaries between different flow regions and barriers to transport. There are
however some lines of FSLE that are not associated to gradients of RT, and
viceversa. For the first this happens mainly because we only integrate 5 days
backward and 5 days forward in time, and the time assigned to the particles
that not cross the open-ocean boundary in the 5 days of integration
corresponds to the maximum time integration (5 days). This makes the spatial
distribution of the residence time more homogeneous. We need to integrate the
trajectories longer to unveil more areas with different RT. In the same way
not all abrupt changes in RT are captured by FSLE lines since we only plot the
highest ridges. This illustrates that both techniques have limitations and
that the complementary use of both could give a rather complete overview of
the geometric structure of flow in marine areas. Pattantyús-Ábrahám et al.
(2008) studied the relation between residence time and FSLE for a wind-forced
hydrodynamical model of a shallow lake. They found that areas with long
residence time visualize the stable manifolds of the so-called chaotic saddle,
a structure controlling the escape properties at long times. In our case, our
integrations are restricted to times too short to characterize long-time
chaotic behavior, but still there is a good correspondence between the FSLE
structures characterizing attracting or repelling trajectories, and escape or
residence times.
Fig. 8 a) shows a time average over October of the spatial distribution of RT,
to be compared with the corresponding average figure (Fig. 4 a) for FSLE. It
is evident that the region in the north and east side of the Bay with high
values of RT is separated from the rest by a region of high values of FSLE.
This can be explained by the presence of persistent barriers which do not
allow particles to escape from the northeast side of the Bay, and thus
separating the Bay in regions with different residence times. In July the
situation is different, because the spatial distribution of FSLE (Fig.4 b) and
RT (Fig. 8 b) is almost homogeneous, with higher values over the whole area of
the Bay, and lower values in the small region bordering the open ocean. This
indicates that the instantaneous configurations of high FSLE lines (Fig. 7)
are not persistent, and that only there is a persistent large difference
between the interior of the Bay and its opening to the ocean. The
predominantly zonal direction of particle motion in the Bay is consistent with
the orientation of the boundaries between areas of different RTs in July.
### V.5 Variability of RT and FSLE
The differences of residence times and FSLEs in the two considered months
indicate that the dynamics of the flow is qualitatively different, as
anticipated by the different wind regimes that are the main drivers of the
Bay. Now we analyze the time evolution of their spatial averages.
Figure 10: a) Time series throughout October of the spatial average of
residence times (top) and spatial average of FSLE (bottom) for the surface
layer of Palma Bay. b) the same that a) but for July.
Figures 10 a) and b) show the time series of the spatial mean of residence
times (top panel) and backward FSLE (bottom panel) for October and July,
respectively.
Comparison between time evolution of spatial average of the RT for the
different months confirms, again, that particles tend to stay longer times in
the Bay during July than in October. The values of RT vary approximately from
0.25 days to 3 days in October (Fig. 10 a, top), and from 2 days to 6 days in
July (Fig. 10 b, top). The same happens with FSLE, higher values correspond to
July and lower ones to October. Diurnal fluctuations, likely related to the
effect of the sea breeze, are evident in RT and FSLE for both months. In
October there are some large fluctuations of low frequency in RT, probably
induced by the variability of remote forcing winds. On the other hand, during
October, minima of RT correspond to maxima of FSLE, and maxima of RT
correspond to minima of FSLE. In July, the relationship between FSLE and RT is
looser and only observed in the high-frequency fluctuations. To be more
quantitative, Pearson correlation coefficient between RT and FSLE time series
is -0.526 in October, but just -0.223 in July. This difference in behavior
probably arises from the fact that high values of FSLE determine clear and
well-defined barriers to transport only in the case of October (see Sect. V.3
and V.4). In July lines of high FSLE remain nearly zonal and parallel to
dominant particle direction of motion. Thus, as commented in Sect. V.2, they
probably represent regions of high zonal shear everywhere in the Bay. Their
high or low values indicate large or small differences in East-West velocities
but by themselves they do not imply stronger or weaker escape towards the
South.
Figure 11: a) Snapshot of spatial distributions of residence times at the
bottom layer in Palma Bay corresponding to July 17, 2009 at 18:00h (GMT).
Lines are the locations of top values of FSLE (greater than $0.3days^{-1}$).
Backward FSLE lines are colored in black and forward FSLEs in white. b)
spatial distribution of time average of 60 snapshots of 6-hourly maps of RT
collected over 15 days in July at the bottom layer.
### V.6 Transport at the bottom layer
In this subsection we compare the main Lagrangian characteristics at the
bottom layer, not driven directly by wind, with those at the surface. In Fig.
11 a) we show an instantaneous map of the residence times in the bottom layer
one day of July, overlayed with lines of high FSLE values. Again, the spatial
distribution is inhomogeneous, and we find high values of RT over all the Bay
except very close to the ocean. The correlation of RT values with FSLE lines
is weaker than in the upper layer, but still we see that the relatively lower
values of RT in the western part of the Bay at that particular day appear
bounded by backwards FSLE lines, indicating a temporal escape route of
particles in that region towards the southwest. The spatial distribution of
the time average of RT plotted in Fig. 11 b) shows that highest average values
of RT are concentrated in the northwestern region of the Bay. Fig. 4 c)
displays high values of FSLE located precisely in the same region where the RT
qualitatively change to high values. This suggests the presence of persistent
barriers that separate this southeastern region from the rest in this bottom
layer. The formation of these persistent LCSs is associated to the gradient of
the bathymetry (see Fig. 1).
The time evolution of the spatial average of RT and FSLE are plotted in top
and bottom panel of Fig. 12 respectively. Contrarily to the surface, in the
bottom layer the diurnal fluctuations in the time series of RT disappear,
showing that the flow at this depth is not directly influenced by breeze. The
RT values are larger than in the surface, and therefore the interchange
between the ocean and the Bay is less intense at bottom layers. This is a
consequence of the slowness of the flow produced by the absence of direct wind
forcing at deepest levels. There is a strong negative correlation ($r=-0.803$)
between stirring and residence times: when the flow is more dispersive the
particles transit during less time over the Bay, so that maxima of RT
correspond to minima of FSLE and vice-versa.
Figure 12: Time series throughout July of (top) spatial average of residence
times and (bottom) spatial average of FSLE computed for the bottom layer.
## VI Conclusions
Properties of coastal transport in the Bay of Palma, which is a small semi-
enclosed region of the Island of Mallorca, were studied in a Lagrangian
framework, by using model velocity data at high resolution. We have applied
two complementary Lagrangian methods (FSLEs and RT) to analyze the small
scales of these coastal currents. LCSs have been detected as high ridges of
FSLE, and virtual experiments with particle trajectories have shown that these
structures really act as barriers in most cases, organizing the coastal flow.
Global and average aspects of the transport in different seasonal months show
that, in the period studied, in autumn there is more exchange between the Bay
and the open ocean than in summer. This arises from the different wind regimes
in both months, that during July induce a flow that restricts motion of the
coastal marine surface to the zonal direction, preventing the flow to enter or
escape toward the open ocean. The transport of particles at the deepest layer
is less active than at the surface and not directly driven by wind, but
influenced by the bottom topography. Regions with different values of RT are
generally separated by ridges of FSLE, proving the fact that FSLE separate
regions of qualitatively different dynamics also in small coastal regions.
Thus, we think that these Lagrangian quantities can be used as key variables
able to determine the dynamics and health of other bays or estuaries,
particularly in relation with human activities. Future improvements include
the adaptation of these methods to three-dimensional spaces and capture three-
dimensional effects, such as upwelling and downwelling in coastal areas, and
analyzing longer periods of time.
## Acknowledgments
I.H-C, C.L and E.H-G acknowledge support from MICINN and FEDER through
projects FISICOS (FIS2007-60327), INTENSE@COSYP (FIS2012-30634), and ESCOLA
(CTM2012-39025-C02-01). AO acknowledges financial support from MICINN and EU-
Med Programme through projects BUS2 (CGL2011-22964) and TOSCA (2GMED-09-425).
We acknowledge the ICMAT Severo Ochoa project (SEV-2011-0087) for the funding
of the publication charges of this article.
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|
arxiv-papers
| 2013-04-16T10:17:47 |
2024-09-04T02:49:44.455943
|
{
"license": "Public Domain",
"authors": "Ismael Hern\\'andez-Carrasco and Crist\\'obal L\\'opez and Alejandro\n Orfila and Emilio Hern\\'andez-Garc\\'ia",
"submitter": "Ismael Hernandez-Carrasco",
"url": "https://arxiv.org/abs/1304.4387"
}
|
1304.4455
|
# The gravitational wave signal from isolated objects
Jinzhong Liu 1 Yu Zhang 1 1National Astronomical Observatory/Xinjiang
Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi,
Xinjiang 830011, China
email: [email protected]
(2012)
###### Abstract
According to the theoretical study, a deformation object (e.g., a spinning
non-axisymmetric pulsar star) will radiate a gravitational wave (GW) signal
during an accelaration motion process by LIGO science project. These types of
disturbance sources with a large bump or dimple on the equator would survive
and be identifiable as GW sources. In this work, we aim to provide a method
for exploring GW radiation from isolated neutron stars (NSs) with deformation
state using some observational results, which can be confirmed by the next
LIGO project. Combination with the properties in observation results (e.g.,
PSR J1748-2446, PSR 1828-11 and Cygnus X-1), based on a binary population
synthesis (BPS) approach we give a numerical GW radiation under the assumption
that NS should have non-axisymmetric and give the results of energy spectrum.
We find that the GW luminosity of $L_{GW}$ can be changed from about
$10^{40}\rm erg/s$– $10^{55}\rm erg/s$.
###### keywords:
gravitational waves, neutron star, evolution.
††volume: 290††journal: Feeding compact objects: Accretion on all
scales††editors: C.M. Zhang, T. Belloni, M. Méndez & S.N. Zhang, eds.
## 1 Introduction
In Einstein s theory of general relativity, gravitational wave (GW) is
considered as a phenomenon resulting from a space perturbation of the metric
traveling at the speed of light, and the observation of the binary pulsar PSR
1913+16 has given an indirect evidence of GW radiation (e.g. Hulse &Taylor
1975). Nowadays, these ripples in space-time due to GW have still not been
directly observed on the ground detectors. The various frequency ranges of the
GW detectors can respectively fix different GW sources (Jaffe & Backer 2003;
Belczynski, Kalogera & Bulik 2002; Liu 2009; Liu et al. 2010A; Liu et al.
2010B; Liu et al. 2012). Here, we focus on the other, much less studied groups
of isolated neutron stars due to asymmetric mechanism, which can be divided
into two groups according to the difference of GW radiation: I) the intrinsic
asymmetry of NSs, II) the relative motion of the asymmetric part of NSs (e.g.,
Papaloizou & Pringle 1978 ). In this work, we aim to explore the GW radiation
from group I. The three formation mechanisms in group I can be summarized as
follows: i) a rotating NS with asymmetrical ellipsoid. (e.g., Hessels et al.
2006); ii) a oblique-dipole-rotator model (e.g., PRS1828-11); iii) the mass
deformations due to an eccentricity in the equatorial plane of NS (the typical
example is the low-mass X ray binaries: Cygnus X-1). The purpose of this
poster is to study the GW radiation from an isolated object with asymmetric
structure.
## 2 Computations
The GW luminosity $L_{GW}$ and dimensionless strain h of a rigidity object
with rotating process are predicted by Press & Thore (1972). The description
of our physics parameters and assumptions are as follows: (I) In the single-
star evolution code, we trace back to the formation of a NS from the zero-age
main sequence to remnant stages. (II) We give the fitting curves of physical
properties according to the NS dynamic structure model and equation of state
in left panel of Fig. 1.(III) For the eccentricity e, we obtain it from
uniform distribution in the range $10^{-3}$ to $10^{-11}$.(IV)We present a
Gaussian distribution of D from the GW sources to the earth in the Galaxy. In
order to compare with observation, we download 109 NSs with rotation period
less than 0.05s from the website
(http://www.atnf.csiro.au/research/pulsar/psrcat/). In middle panel of Fig. 1,
we give the distribution of the rotation period between our model results and
observations.
Figure 1: Left:The fitting physical property curves of NS; Middle:The
distribution of rotation period; Right:The spectral energy distribution of
rapid rotating NS.
## 3 Results and Discussion
In general, the total energy ($Mc^{2}$) of NS is about $10^{54}$erg, the
rotation energy is $~{}10^{53}$ erg. Therefore, for the e$>10^{-5}$, the most
energy of NS can be radiation as GW signal during several years. All these
calculations of isolated objects can examine that these sources are expected
to be a type of GW sources. In our model, the influence parameter $\xi$ of
eccentricity can be changed to the influence of mass, which is $~{}2.8\times
10^{-4}<\xi<8.9\times 10^{-9}$, corresponding to the value of strain amplitude
$0.8\times 10^{-24}<h<10^{-32}$. Our prediction agrees with that of Crab
nebula and Virgo cluster ($10^{-24}-10^{-25}$). Meanwhile, the right panel of
Fig. 1 gives the spectral energy distribution of rapid rotating NS and implies
that the GW luminosity of $L_{GW}$ can be changed from about $10^{40}\rm
erg/s$– $10^{55}\rm erg/s$.
###### Acknowledgements.
This work is supported by the program of the light in China’s Western Region
(LCWR) (No. XBBS201022), Natural Science Foundation (No. 11103054) and
Xinjiang Natural Science Foundation (No. 2011211A104). This
project/publication was made through the support of a grant from the John
Templeton Foundation and National Astronomical Observatories, Chinese Academy
of Sciences (No. 100020101).
## References
* [] Belczynski K., Kalogera V., Bulik T., 2002, ApJ, 572, 407
* [] Hessels, J. W. T., et al. 2006, Science, 311, 1901
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|
arxiv-papers
| 2013-04-16T14:12:42 |
2024-09-04T02:49:44.463588
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinzhong Liu and Yu Zhang",
"submitter": "Zhang Yu",
"url": "https://arxiv.org/abs/1304.4455"
}
|
1304.4500
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-063 LHCb-PAPER-2013-015 April 16, 2013
Measurement of the effective $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ lifetime
The LHCb collaboration†††Authors are listed on the following pages.
This paper reports the first measurement of the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime and an updated measurement
of its time-integrated branching fraction. Both measurements are performed
with a data sample, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$ of $pp$ collisions, recorded by the LHCb experiment in 2011
at a centre-of-mass energy of $7\>\mathrm{\,Te\kern-1.00006ptV}$. The results
are: $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}=1.75\pm
0.12\>(\text{stat})\pm 0.07\>(\text{syst})\>\text{ps}$ and ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})=(1.97\pm 0.23)\times 10^{-5}$. For
the latter measurement, the uncertainty includes both statistical and
systematic sources.
Published in Nucl. Phys. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A.
Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez58, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain57, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal57, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
In the Standard Model (SM), $C\\!P$ violation arises through a single phase in
the CKM quark mixing matrix [1, *Cabibbo:1963yz]. In decays of neutral $B$
mesons ($B$ stands for a $B^{0}$ or $B^{0}_{s}$ meson) to a final state
accessible to both $B$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$, the
interference between the amplitude for the direct decay and the amplitude for
decay via oscillation leads to time-dependent $C\\!P$ violation. A measurement
of the time-dependent $C\\!P$ asymmetry in the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode allows for a determination of the $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing phase $\phi_{d}$. In the SM
it is equal to $2\beta$ [3], where $\beta$ is one of the angles of the
unitarity triangle in the quark mixing matrix. This phase has already been
well measured by the $B$ factories [4, 5], but further improvements are still
necessary to conclusively resolve possible small tensions with the other
measurements constraining the unitarity triangle [6, *Charles:2004jd]. The
latest average composed by the Heavy Flavour Averaging Group (HFAG) is
$\sin\phi_{d}=0.682\pm 0.019$ [8]. To achieve precision below the percent
level, knowledge of the doubly Cabibbo-suppressed higher order perturbative
corrections, originating from penguin topologies, becomes mandatory. These
contributions are difficult to calculate reliably and therefore need to be
determined directly from experimentally accessible observables.
From a theoretical perspective, the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode is the most promising candidate for this task. It is related to the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode through the interchange of all
$d$ and $s$ quarks ($U$-spin symmetry, a subgroup of $SU(3)$) [9], leading to
a one-to-one correspondence between all decay topologies of these two modes,
as illustrated in Fig. 1. Moreover, the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ penguin topologies are not CKM suppressed relative to the tree diagram, as
is the case for their $B^{0}$ counterparts. A further discussion regarding the
theory of this decay and its potential use in LHCb is given in Ref. [10,
*DeBruyn:2010ge].
To determine the parameters related to the penguin contributions in these
decays, a time-dependent $C\\!P$ violation study of the $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode is required. The determination
of its branching fraction, previously measured by CDF [12] and LHCb [13], was
an important first step, allowing a test of the $U$-spin symmetry assumption
that lies at the basis of the proposed approach. The second step towards the
time-dependent $C\\!P$ violation study is the measurement of the effective
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime, formally defined as [14]
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}\equiv\frac{\int_{0}^{\infty}t\>\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S})\rangle\>\mathrm{d}t}{\int_{0}^{\infty}\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})\rangle\>\mathrm{d}t}\>,$ (1)
where
$\displaystyle\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})\rangle$ $\displaystyle=$
$\displaystyle\Gamma(B^{0}_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})+\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})$ (2) $\displaystyle=$ $\displaystyle
R_{\mathrm{H}}e^{-\Gamma_{\mathrm{H}}t}+R_{\mathrm{L}}e^{-\Gamma_{\mathrm{L}}t}$
(3)
is the untagged decay time distribution, under the assumption that $C\\!P$
violation in $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing can be neglected [8].
Due to the non-zero decay width difference
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{H}}-\Gamma_{\mathrm{L}}=0.106\pm
0.013\>\text{ps}^{-1}$ [15] between the heavy and light $B^{0}_{s}$ mass
eigenstates, the effective lifetime does not coincide with the $B^{0}_{s}$
lifetime $\tau_{B^{0}_{s}}\equiv 1/\Gamma_{s}=1.513\pm 0.011\>\text{ps}$ [15],
where $\Gamma_{s}=(\Gamma_{\mathrm{H}}+\Gamma_{\mathrm{L}})/2$ is the average
$B^{0}_{s}$ decay width. Instead, it depends on the decay mode specific
relative contributions $R_{\mathrm{H}}$ and $R_{\mathrm{L}}$. These two
parameters also define the $C\\!P$ observable
$\mathcal{A}_{\Delta\Gamma_{s}}\equiv\frac{R_{\mathrm{H}}-R_{\mathrm{L}}}{R_{\mathrm{H}}+R_{\mathrm{L}}}\>,$
(4)
which allows the effective lifetime to be expressed as [14]
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}=\frac{\tau_{B^{0}_{s}}}{1-y_{s}^{2}}\frac{1+2\>\mathcal{A}_{\Delta\Gamma_{s}}y_{s}+y_{s}^{2}}{1+\mathcal{A}_{\Delta\Gamma_{s}}y_{s}}\>,$
(5)
where $y_{s}\equiv\Delta\Gamma_{s}/2\Gamma_{s}$ is the normalised decay width
difference. For the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode, the value of $\mathcal{A}_{\Delta\Gamma_{s}}$ depends on the penguin
contributions, and in particular on their relative weak phase $\phi_{s}$ [9].
Using the latest estimates on the size of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ penguin contributions [16] gives $\mathcal{A}_{\Delta\Gamma_{s}}=0.944\pm
0.066$ and the SM prediction
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}\Big{|}_{\text{SM}}=1.639\pm 0.022\>\text{ps}\>.$ (6)
Effective lifetime measurements have been performed for the $B^{0}_{s}$
$\rightarrow$ $K^{+}$$K^{-}$ [17] and $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$f_{0}(980)$ [18] decay modes.
Figure 1: Decay topologies contributing to the $B_{d(s)}$$\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ channel: (left) tree diagram and (right) penguin diagram.
This paper presents the first measurement of the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime, as well as an update of the
time-integrated branching fraction measurement in Ref. [13], performed with a
data sample, corresponding to an integrated luminosity of
$1.0\>\mbox{\,fb}^{-1}$ of $pp$ collisions, recorded at a centre-of-mass
energy of $7\>\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment in 2011.
The LHCb detector [19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$) with
respect to the beam direction. Charged hadrons are identified using two ring-
imaging Cherenkov detectors [20]. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers.
Events are selected by a trigger system [21] consisting of a hardware trigger,
which requires muon or hadron candidates with high $p_{\rm T}$, followed by a
two-stage software trigger. In the first stage a partial event reconstruction
is performed. For this analysis, events are required to have either two
oppositely charged muons with combined mass above
$2.7\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or at least one muon or one
high-$p_{\rm T}$ track ($\mbox{$p_{\rm
T}$}>1.8\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) with a large impact parameter
with respect to all $pp$ interaction vertices (PVs). In the second stage a
full event reconstruction is performed and only events containing
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$
$\mu^{+}$$\mu^{-}$ candidates are retained.
The signal simulation samples used for this analysis are generated using
Pythia 6.4 [22] with a specific LHCb configuration [23]. Decays of hadronic
particles are described by EvtGen [24] in which final state radiation is
generated using Photos [25]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [26,
*Agostinelli:2002hh] as described in Ref. [28].
## 2 Data samples and initial selection
Candidate $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ decays are reconstructed in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$ $\mu^{+}\mu^{-}$
and $K^{0}_{\rm\scriptscriptstyle S}$ $\rightarrow$ $\pi^{+}\pi^{-}$ final
state. Candidate ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$
$\mu^{+}\mu^{-}$ decays are required to form a good quality vertex and have a
mass in the range $[3030,3150]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
This interval corresponds to about eight times the $\mu^{+}\mu^{-}$ mass
resolution at the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass and
covers part of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ radiative
tail. The selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate is
required to satisfy the trigger decision at both software trigger stages. The
$K^{0}_{\rm\scriptscriptstyle S}$ selection requires two oppositely charged
particles reconstructed in the tracking stations placed on either side of the
magnet, both with hits in the vertex detector (‘long
$K^{0}_{\rm\scriptscriptstyle S}$’ candidate) or without (‘downstream
$K^{0}_{\rm\scriptscriptstyle S}$’ candidate). The long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ $\rightarrow$ $\pi^{+}\pi^{-}$ candidates
are required to form a good quality vertex and have a mass within
$35\>(64)\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass [29]. Moreover, to remove contamination
from $\mathchar 28931\relax$ $\rightarrow$ $p$$\pi^{-}$ decays, the
reconstructed $p$$\pi^{-}$ mass of the long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ candidates is required to be more than
$6\>(10)\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ away from the known
$\mathchar 28931\relax$ mass [29]. Furthermore, the
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have a flight
distance that is at least five times larger than its uncertainty.
Candidate $B$ mesons are selected from combinations of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidates with mass
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ in the range $[5180,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
reconstructed mass and decay time are obtained from a kinematic fit [30] that
constrains the masses of the $\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$ pairs to
the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ masses [29], respectively, and constrains
the $B$ candidate to originate from the PV. In case the event has multiple
PVs, all combinations are considered. The $\chi^{2}$ of the fit, which has
eight degrees of freedom, is required to be less than $72$ and the estimated
uncertainty on the $B$ mass must not exceed
$30\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Candidates are required to
have a decay time larger than $0.2\>{\rm\,ps}$. To remove misreconstructed
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{*0}$
background that survives the requirement on the $K^{0}_{\rm\scriptscriptstyle
S}$ flight distance, the mass of the long $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ candidates computed under the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{\pm}$$\pi^{\mp}$ mass hypotheses must not be within
$20\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}$ mass [29].
## 3 Multivariate selection
The loose selection described above does not suppress the combinatorial
background sufficiently to isolate the small $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal. The initial selection is therefore followed by a multivariate
analysis, based on a neural network (NN) [31]. The NN classifier’s output is
used as the final selection variable.
The NN is trained entirely on data, using the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal as a proxy for the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ decay. The training sample is taken from the mass windows
$[5180,5340]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5390,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, thus avoiding the
$B^{0}_{s}$ signal region. A normalisation sample consisting of one quarter of
the candidates, selected at random, is left out of the NN training to allow an
unbiased measurement of the $B^{0}$ yield. The signal and background weights
are determined using the _sPlot_ technique [32] and obtained by performing an
unbinned maximum likelihood fit to the mass distribution of the candidates
surviving the loose selection criteria. The fitted probability density
function (PDF) is defined as the sum of a $B^{0}$ signal component and a
combinatorial background. The parametrisation of the individual components is
described in more detail in the next section.
Due to the differences in the distributions of the input variables of the NN,
as well as the different initial signal to background ratio, the multivariate
selection is performed separately for the $B$ candidate samples containing
long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidates. In the
remainder of this paper, these two datasets will be referred to as the long
and downstream $K^{0}_{\rm\scriptscriptstyle S}$ sample, respectively. The NN
classifiers use information about the candidate kinematics, vertex and track
quality, impact parameter, particle identification information from the RICH
and muon detectors, as well as global event properties like track and
interaction vertex multiplicities. The variables that are used in the NN are
chosen to avoid correlations with the reconstructed $B$ mass.
Final selection requirements on the NN classifier outputs are chosen to
optimise the expected sensitivity to the $B^{0}_{s}$ signal observation. The
expected signal and background yields entering the calculation of the figure
of merit [33] are obtained from the normalisation sample by scaling the number
of fitted $B^{0}$ candidates, and by counting the number of events in the mass
ranges $[5180,5240]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5400,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. After
applying the final requirement on the NN classifier output associated with the
long (downstream) $K^{0}_{\rm\scriptscriptstyle S}$ sample, the multivariate
selection rejects, relative to the initial selection, 98.7% (99.6%) of the
background while keeping 71.5% (50.2%) of the $B^{0}$ signal. Due to the worse
initial signal to background ratio, the final requirement on the NN classifier
output is much tighter in the downstream $K^{0}_{\rm\scriptscriptstyle S}$
sample than in the long $K^{0}_{\rm\scriptscriptstyle S}$ sample.
After applying the full selection, the $B$ candidate can still be associated
with more than one PV in about 1% of the events. Likewise, about $0.1\%$ of
the selected events have several candidates sharing one or more tracks. In
these cases, respectively one of the surviving PVs and one of the candidates
is used at random.
## 4 Event yields
Long $K^{0}_{\rm\scriptscriptstyle S}$
Downstream $K^{0}_{\rm\scriptscriptstyle S}$
Figure 2: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate mass distributions and
their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples, after applying
the final requirement on the NN classifier outputs.
For the candidates passing the NN requirements, the ratio of $B^{0}_{s}$ and
$B^{0}$ yields is determined from an unbinned maximum likelihood fit to the
mass distribution of the reconstructed $B$ candidates. The fitted PDF is
defined as the sum of a $B^{0}$ signal component, a $B^{0}_{s}$ signal
component and a combinatorial background. The $B^{0}_{s}$ component is
constrained to have the same shape as the $B^{0}$ PDF, shifted by the known
$B^{0}_{s}$–$B^{0}$ mass difference [34]. The mass lineshapes of the $B$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ modes in both data and simulation
exhibit non-Gaussian tails on both sides of their signal peaks due to final
state radiation, the detector resolution and its dependence on the decay
angles. Each individual signal shape is parametrised by a double-sided Crystal
Ball (CB) function [35]. The parameters describing the CB tails are taken from
simulation; all other parameters are allowed to vary in the fit. The
background contribution is described by an exponential function.
The results of the fits are shown in Fig. 2, and the fitted yields are listed
in Table 1. The $B^{0}$ yield is determined in the normalisation sample and
scaled to the full sample, whereas the $B^{0}_{s}$ yield is obtained directly
from the full sample. The scaled $B^{0}$ yield, obtained from the unbiased
sample, differs from the corresponding fit result in the full sample by
$-211\pm 211$ events for the long $K^{0}_{\rm\scriptscriptstyle S}$ sample and
by $213\pm 273$ events for the downstream $K^{0}_{\rm\scriptscriptstyle S}$
sample. Both results are in good agreement, showing that the NN is not
overtrained. The yield ratios obtained from the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ samples are compatible with each other and
are combined using a weighted average.
Table 1: Signal yields from the unbinned maximum likelihood fits to the $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate mass distributions. The uncertainties are statistical only. The yield ratio is calculated from the quantities highlighted in boldface, where the fitted $B^{0}$ yield is first multiplied by a factor of four. Sample | Yield | Long $K^{0}_{\rm\scriptscriptstyle S}$ | Downstream $K^{0}_{\rm\scriptscriptstyle S}$
---|---|---|---
Normalisation | $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $\textbf{2205}\ \pm\ \textbf{47}$ | $\textbf{3651}\ \pm\ \textbf{61}$
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $21\ \pm\ 5$ | $49\ \pm\ 8$
Background | $56\ \pm\ 11$ | $110\ \pm\ 16$
Full | $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $9031\ \pm\ 96$ | $\text{14,391}\ \pm\ 122$
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $\textbf{115}\ \pm\ \textbf{12}$ | $\textbf{158}\ \pm\ \textbf{15}$
Background | $287\ \pm\ 23$ | $490\ \pm\ 32$
Yield ratio $R\equiv N_{B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{Full}}/4N_{B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{Norm}}$ | $0.0131\ \pm\ 0.0014$ | $0.0108\ \pm\ 0.0010$
Average yield ratio $R$ | $0.0116\pm 0.0008$
## 5 Decay time distribution
Following the procedure explained in Ref. [36], the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime is determined by fitting a
single exponential function $g(t)\propto\exp(-t/\tau_{\text{single}})$ to the
decay time distribution of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal candidates. In this analysis, the exponential shape parameter
$\tau_{\text{single}}$ is determined from a two-dimensional unbinned maximum
likelihood fit to the mass and decay time distribution of the reconstructed
$B$ candidates. The fitted PDF is again defined as the sum of a $B^{0}$ signal
component, a $B^{0}_{s}$ signal component and a combinatorial background. The
freely varying parameters in the fit are the signal and background yields, and
the parameters describing the acceptance, mass and background decay time
distributions.
The decay time distribution of each of the two signal components needs to be
corrected with a decay time resolution and acceptance model to account for
detector effects. The shape of the acceptance function affecting the
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode is, like the lineshape of its
mass distribution, assumed to be identical to that of the $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ component. The acceptance function is
obtained directly from the data using the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode. Contrary to the $B^{0}_{s}$ system, the $B^{0}$ system has a
negligible decay width difference $\Delta\Gamma_{d}$ [29]. The decay time
distribution of the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ channel is therefore fully described by a single exponential function with
known lifetime $\tau_{B^{0}}=1.519\>\text{ps}$ [8]. Hence, fixing the $B^{0}$
lifetime to its known value allows the acceptance parameters to be determined
from the fit.
From simulation studies it is found that the decay time acceptance of both
signal components is well modelled by the function
$f_{\text{Acc}}\>(t)=\frac{1+\beta\>t}{1+(\lambda\>t)^{-\kappa}}\>.$ (7)
The parameter $\beta$ describes the fall in the acceptance at large decay
times [15]. The parameters $\kappa$ and $\lambda$ model the turn-on curve,
caused by the use of decay time biasing triggers, the initial selection
requirements and, most importantly, the NN classifier outputs.
The decay time resolution for the signal and background components is
determined from candidates that have an unphysical, negative decay time. Due
to the requirement of $0.2\>\text{${\rm\,ps}$}$ on the decay time of the $B$
candidates applied in the initial selection, such events are not present in
the analysed data sample. Instead, a second sample, that is prescaled and does
not have the decay time requirement, is used. This sample consists primarily
of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons produced at the PV
which are combined with a random $K^{0}_{\rm\scriptscriptstyle S}$ candidate.
The decay time distribution for these events is a good measure of the decay
time resolution and is modelled by the sum of three Gaussian functions sharing
a common mean. Two of the Gaussian functions parametrise the inner core of the
resolution function, while the third describes the small fraction of outliers.
Long $K^{0}_{\rm\scriptscriptstyle S}$
Downstream $K^{0}_{\rm\scriptscriptstyle S}$
Figure 3: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate decay time distributions
and their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples, after applying
the final requirement on the NN classifier outputs.
Long $K^{0}_{\rm\scriptscriptstyle S}$
Downstream $K^{0}_{\rm\scriptscriptstyle S}$
Figure 4: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate decay time distributions
and their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidates in the
$B^{0}_{s}$ signal region $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5340,5390]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, after applying
the final requirement on the NN classifier outputs.
The background decay time distributions are studied directly using the data.
Their shape is obtained from background candidates that are isolated using the
background weights determined by the _sPlot_ technique, and cross-checked
using the high mass sideband. The exact values of the shape parameters are
determined in the nominal fit. Because of the differences induced by the
multivariate selection, the background decay time distribution of the long and
downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples cannot be parametrised
using the same background model. For the long $K^{0}_{\rm\scriptscriptstyle
S}$ sample, the background is modelled by two exponential functions,
describing a short-lived and a long-lived component, respectively. In the
downstream $K^{0}_{\rm\scriptscriptstyle S}$ sample such a short-lived
component is not present due to the tighter requirement on the NN classifier
output. Its decay time distribution is better described by a single
exponential shape corrected by the acceptance function in Eq. (7) with
independent parameters $(\kappa^{\prime},\alpha^{\prime},\beta^{\prime})$. The
parameter $\beta^{\prime}$ is set to zero because we also fit the lifetime of
the single exponential function itself, and the combination of both parameters
would result in ambiguous solutions.
The decay time distributions resulting from the two-dimensional fits are shown
in Figs. 3 and 4 for candidates in the full mass range
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5180,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and in the
$B^{0}_{s}$ signal region $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5340,5390]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively.
The fitted values are $\tau_{\text{single}}=1.54\pm 0.17\>\text{ps}$ and
$\tau_{\text{single}}=1.96\pm 0.17\>\text{ps}$ for the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ sample, respectively. The $1.7\sigma$
difference between both results is understood as a statistical fluctuation.
The two main fit results are therefore combined using a weighted average,
leading to
$\tau_{\text{single}}=1.75\pm 0.12\>\text{ps}\>,$
where the uncertainty is statistical only. The event yields obtained from the
two-dimensional fits are compatible with the results quoted in Table 1.
## 6 Corrections and systematic uncertainties
A number of systematic uncertainties affecting the relative branching fraction
$\cal B$($B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$)/$\cal B$($B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$) and the effective lifetime are considered. The sources affecting the
ratio of branching fractions are discussed first, followed by those
contributing to the effective lifetime measurement.
The largest systematic uncertainty on the yield ratio comes from the mass
shape model, and in particular from the uncertainty on the fraction of the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ component’s high mass tail extending
below the $B^{0}_{s}$ signal. The magnitude of this effect is studied by
allowing both tails of the CB shapes to vary in the fit. The largest observed
deviation in the yield ratios is 3.4%, which is taken as a systematic
uncertainty. The mass resolution, and hence the widths of the CB shapes, is
assumed to be identical for the $B^{0}$ and $B^{0}_{s}$ signal modes, but
could in principle depend on the mass of the reconstructed $B$ candidate. This
effect is studied by multiplying the CB widths of the $B^{0}_{s}$ signal PDF
by different scale factors, obtained by comparing $B^{0}$ and $B^{0}_{s}$
signal shapes in simulation. The largest observed difference in the yield
ratios is 1.4%, which is taken as a systematic uncertainty. Varying the
$B^{0}_{s}$–$B^{0}$ mass difference within its uncertainty has negligible
effect on the yield ratios.
The selection procedure is designed to be independent of the reconstructed $B$
mass. Simulated data is used to check this assumption, and to evaluate the
difference in selection efficiency arising from the different shapes of the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ decay time distributions. The ratio of total selection efficiencies is
equal to $0.968\pm 0.007$, and is used to correct the yield ratio.
The stability of the multivariate selection is verified by comparing different
training schemes and optimisation procedures, as well as by calculating the
yield ratios for different subsets of the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ sample. All of these tests give results that
are compatible with the measured ratio.
The corrections and systematic uncertainties affecting the branching fraction
ratio are listed in Table 2. The total systematic uncertainty is obtained by
adding all the uncertainties in quadrature.
Table 2: Corrections and systematic uncertainties on the yield ratio. Source | Value
---|---
Fit model | $1.000\ \pm\ 0.034$
$B^{0}_{s}$ mass resolution | $1.000\ \pm\ 0.014$
Selection efficiency | $0.968\ \pm\ 0.007$
Total correction $f_{\text{corr}}^{{\cal B}}$ | $0.968\ \pm\ 0.034$
The main systematic uncertainties affecting the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime arise from modelling the
different components of the decay time distribution. Their amplitudes are
evaluated by comparing the results from the nominal fit to similar fits using
alternative parametrisations. All tested fit models give compatible results.
The largest observed deviations in $\tau_{\text{single}}$ are 3.9% due to
modelling of the background decay time distribution, 0.47% due to the
acceptance function and 0.39% due to the reconstructed $B$ mass description,
all of which are assigned as systematic uncertainties. Variations in the decay
time resolution model are found to have negligible impact on
$\tau_{\text{single}}$.
The assumed value of the $B^{0}$ lifetime has a significant impact on the
shape of the acceptance function, and the $\beta$ parameter in particular,
which in turn affects the fitted value of $\tau_{\text{single}}$. This effect
is studied by varying the $B^{0}$ lifetime within its uncertainty [29]. The
largest observed deviation in $\tau_{\text{single}}$ is 0.52%, which is taken
as a systematic uncertainty.
The fit method is tested on simulated data using large sets of pseudo-
experiments, which have the same mass and decay time distributions as the
data. Different datasets are generated using the fitted two-dimensional signal
and background distributions, and $\tau_{\text{single}}$ is then again fitted
to these pseudo-experiments. The fit result is compared with the input value
to search for possible biases. From the spread in the fitted values and the
accompanying residual distributions, a small bias is found. This bias is
attributed to the limited size of the background sample, and the resulting
difficulty to constrain the background decay time parameters. A correction
factor of $1.002\pm 0.002$ is assigned to account for this potential bias.
Due to the presence of a non-trivial acceptance function, the result of
fitting a single exponential to the untagged $B^{0}_{s}$ decay time
distribution does mathematically not agree with the formal definition of the
effective lifetime in Eq. (1), as explained in Ref. [36]. The size and sign of
the difference between $\tau_{\text{single}}$ and
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ depend on the values of
$\tau_{B^{0}_{s}}$, $y_{s}$, $\mathcal{A}_{\Delta\Gamma_{s}}$, and the shape
of the acceptance function. The difference is calculated with pseudo-
experiments that sample the acceptance parameters, $\tau_{B^{0}_{s}}$ and
$y_{s}$ from Gaussian distributions related to their respective fitted and
known values. Since $\mathcal{A}_{\Delta\Gamma_{s}}$ is currently not
constrained by experiment, it is sampled uniformly from the interval
$[{-1},1]$. The average difference between
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ and
$\tau_{\text{single}}$, obtained using the acceptance function affecting the
long (downstream) $K^{0}_{\rm\scriptscriptstyle S}$ sample, is found to be
$-0.001\>\text{ps}$ $(-0.003\>\text{ps})$. A correction factor of $0.999\pm
0.001$ is assigned to account for this bias.
The presence of a production asymmetry between the $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons could potentially alter
the measured value of the effective lifetime, but even for large estimates of
the size of this asymmetry, the effect is found to be negligible.
Finally, the systematic uncertainties in the momentum and the decay length
scale propagate to the effective lifetime. The size of the former contribution
is evaluated by recomputing the decay time while varying the momenta of the
final state particles within their uncertainty. The systematic uncertainty due
to the decay length scale mainly comes from the track-based alignment. Both
effects are found to be negligible.
The stability of the fit is verified by comparing the nominal results with
those obtained using different fit ranges, or using only subsets of the long
and downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples. All these tests give
compatible results.
The corrections and systematic uncertainties affecting the effective
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime are listed in Table 3. The
total systematic uncertainty is obtained by adding all the uncertainties in
quadrature.
Table 3: Corrections and systematic uncertainties on the effective $B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime. Source | Value
---|---
Background model | $1.000\ \pm\ 0.039$
Acceptance model | $1.000\ \pm\ 0.005$
Mass model | $1.000\ \pm\ 0.004$
$B^{0}$ lifetime | $1.000\ \pm\ 0.005$
Fit method | $1.002\ \pm\ 0.002$
Effective lifetime definition | $0.999\ \pm\ 0.001$
Total correction $f_{\text{corr}}^{\text{eff}}$ | $1.001\ \pm\ 0.040$
## 7 Results and conclusion
Using the measured ratio $R=0.0116\pm 0.0008$ of $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ and $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ yields, the correction factor
$f_{\text{corr}}^{{\cal B}}=0.968\pm 0.034$, and the ratio of hadronisation
fractions $f_{s}/f_{d}=0.256\pm 0.020$ [37], the ratio of branching fractions
is computed to be
$\displaystyle\frac{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})}$ $\displaystyle=$ $\displaystyle
R\times f_{\text{corr}}^{{\cal B}}\times\frac{f_{d}}{f_{s}}$ $\displaystyle=$
$\displaystyle 0.0439\pm 0.0032\>\text{(stat)}\pm 0.0015\>\text{(syst)}\pm
0.0034\>(f_{s}/f_{d})\>,$
where the quoted uncertainties are statistical, systematic, and due to the
uncertainty in $f_{s}/f_{d}$, respectively.
Using the known $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}$ branching fraction [29], the ratio of branching fractions can
be converted into a measurement of the time-integrated $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ branching fraction. Taking into
account the different rates of $B^{+}$$B^{-}$ and $B^{0}$$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pair production at the $\mathchar
28935\relax{(4S)}$ resonance $\Gamma(B^{+}{}B^{-})/\Gamma(B^{0}{}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})=1.055\pm 0.025$ [29], the above
result is multiplied by the corrected value ${\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0})=(8.98\pm 0.35)\times 10^{-4}$ and gives
$\displaystyle{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})=$ $\displaystyle\left[1.97\pm
0.14\>\text{(stat)}\pm 0.07\>\text{(syst)}\pm 0.15\>(f_{s}/f_{d})\pm
0.08\>({\cal B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}))\right]\times 10^{-5}\>,$
where the last uncertainty comes from the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$ branching fraction. This
result is compatible with, and more precise than, previous measurements [12,
13], and supersedes the previous LHCb measurement. The branching fraction is
consistent with expectations from $U$-spin symmetry [13].
Using $\tau_{\text{single}}=1.75\pm 0.12\>\text{ps}$ and the correction factor
$f_{\text{corr}}^{\text{eff}}=1.001\pm 0.040$, the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime is given by
$\displaystyle\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ $\displaystyle=$
$\displaystyle f_{\text{corr}}^{\text{eff}}\times\tau_{\text{single}}$
$\displaystyle=$ $\displaystyle 1.75\pm 0.12\>(\text{stat})\pm
0.07\>(\text{syst})\>\text{ps}\>.$
This is the first measurement of this quantity. The result is compatible with
the SM prediction given in Eq. (6).
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-16T15:53:24 |
2024-09-04T02:49:44.468967
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van\n den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M. Dogaru, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D. Johnson, C.R. Jones, C.\n Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R.\n Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G.\n Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero\n Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Kristof De Bruyn",
"url": "https://arxiv.org/abs/1304.4500"
}
|
1304.4518
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-062 LHCb-PAPER-2013-014
Searches for violation of lepton flavour and baryon number in tau lepton
decays at LHCb
The LHCb collaboration†††Authors are listed on the following pages.
Searches for the lepton flavour violating decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and the lepton flavour and baryon
number violating decays $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ have been carried out using proton-
proton collision data, corresponding to an integrated luminosity of $1.0$
$\mbox{\,fb}^{-1}$, taken by the LHCb experiment at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. No evidence has been found for any
signal, and limits have been set at $90\%$ confidence level on the branching
fractions: ${\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})<8.0\times
10^{-8}$, ${\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})<3.3\times
10^{-7}$ and ${\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})<4.4\times
10^{-7}$. The results for the $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ decay modes represent the first direct
experimental limits on these channels.
Submitted to Physics Letters B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A.
Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez58, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain57, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal57, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The observation of neutrino oscillations was the first evidence for lepton
flavour violation (LFV). As a consequence, the introduction of mass terms for
neutrinos in the Standard Model (SM) implies that LFV exists also in the
charged sector, but with branching fractions smaller than $\sim 10^{-40}$ [1,
2]. Physics beyond the Standard Model (BSM) could significantly enhance these
branching fractions. Many BSM theories predict enhanced LFV in $\tau^{-}$
decays with respect to $\mu^{-}$ decays111The inclusion of charge conjugate
processes is implied throughout this Letter., with branching fractions within
experimental reach [3]. To date, no charged LFV decays such as
$\mu^{-}\rightarrow e^{-}\gamma$, $\mu^{-}\rightarrow e^{-}e^{+}e^{-}$,
$\tau^{-}\rightarrow\ell^{-}\gamma$ and
$\tau^{-}\rightarrow\ell^{-}\ell^{+}\ell^{-}$ (with $\ell^{-}=e^{-},\mu^{-}$)
have been observed [4]. Baryon number violation (BNV) is believed to have
occurred in the early universe, although the mechanism is unknown. BNV in
charged lepton decays automatically implies lepton number and lepton flavour
violation, with angular momentum conservation requiring the change
$|\Delta(B-L)|=0$ or $2$, where $B$ and $L$ are the net baryon and lepton
numbers. The SM and most of its extensions [1] require $|\Delta(B-L)|=0$. Any
observation of BNV or charged LFV would be a clear sign for BSM physics, while
a lowering of the experimental upper limits on branching fractions would
further constrain the parameter spaces of BSM models.
In this Letter we report on searches for the LFV decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and the LFV and BNV decay modes
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$ at LHCb [5]. The inclusive $\tau^{-}$ production cross-
section at the LHC is relatively large, at about $80\,\upmu$b (approximately
$80\%$ of which comes from $D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau}$),
estimated using the $b\bar{b}$ and $c\bar{c}$ cross-sections measured by LHCb
[6, 7] and the inclusive $b\rightarrow\tau$ and $c\rightarrow\tau$ branching
fractions [8]. The $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and
$\tau\rightarrow p\mu\mu$ decay modes222In the following $\tau\rightarrow
p\mu\mu$ refers to both the $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ channels. are of particular interest at
LHCb, since muons provide clean signatures in the detector and the ring-
imaging Cherenkov (RICH) detectors give excellent identification of protons.
This Letter presents the first results on the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ decay mode from a hadron collider
and demonstrates an experimental sensitivity at LHCb, with data corresponding
to an integrated luminosity of $1.0$$\mbox{\,fb}^{-1}$, that approaches the
current best experimental upper limit, from Belle, ${\cal
B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})<2.1\times 10^{-8}$ at 90%
confidence level (CL) [9]. BaBar and Belle have searched for BNV $\tau$ decays
with $|\Delta(B-L)|=0$ and $|\Delta(B-L)|=2$ using the modes
$\tau^{-}\rightarrow\mathchar 28931\relax h^{-}$ and $\bar{\mathchar
28931\relax}h^{-}$ (with $h^{-}=\pi^{-},K^{-}$), and upper limits on branching
fractions of order $10^{-7}$ were obtained [4]. BaBar has also searched for
the $B$ meson decays $B^{0}\rightarrow\mathchar 28931\relax_{c}^{+}l^{-}$,
$B^{-}\rightarrow\mathchar 28931\relax l^{-}$ (both having $|\Delta(B-L)|=0$)
and $B^{-}\rightarrow\bar{\mathchar 28931\relax}l^{-}$ ($|\Delta(B-L)|=2$),
obtaining upper limits at 90% CL on branching fractions in the range
$(3.2-520)\times 10^{-8}$ [10]. The two BNV $\tau$ decays presented here,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, have $|\Delta(B-L)|=0$ but they could have rather different
BSM interpretations; they have not been studied by any previous experiment.
In this analysis the LHCb data sample from 2011, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, is used. Selection criteria are
implemented for the three signal modes,
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, and for the calibration and normalisation channel, which is
$D_{s}^{-}\rightarrow\phi\pi^{-}$ followed by $\phi\rightarrow\mu^{+}\mu^{-}$,
referred to in the following as
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$. These initial, cut-based
selections are designed to keep good efficiency for signal whilst reducing the
dataset to a manageable level. To avoid potential bias,
$\mu^{-}\mu^{+}\mu^{-}$ and $p\mu\mu$ candidates with mass within $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}~{}(\approx 3\sigma_{m})$ of the
$\tau$ mass are initially blinded from the analysis, where $\sigma_{m}$
denotes the expected mass resolution. For the $3\mu$ channel, discrimination
between potential signal and background is performed using a three-dimensional
binned distribution in two likelihood variables and the mass of the $\tau$
candidate. One likelihood variable is based on the three-body decay topology
and the other on muon identification. For the $\tau\rightarrow p\mu\mu$
channels, the use of the second likelihood function is replaced by cuts on the
proton and muon particle identification (PID) variables. The analysis strategy
and limit-setting procedure are similar to those used for the LHCb analyses of
the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$
channels [11, 12, *Junk_99].
## 2 Detector and triggers
The LHCb detector [5] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two RICH detectors. Photon, electron and
hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.
The trigger [14] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction. The hardware trigger selects muons with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The software trigger requires
a two-, three- or four-track secondary vertex with a high sum of the $p_{\rm
T}$ of the tracks and a significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter chi-squared
(IP $\chi^{2}$), with respect to the $pp$ collision vertex, greater than 16.
The IP $\chi^{2}$ is defined as the difference between the $\chi^{2}$ of the
PV reconstructed with and without the track under consideration. A
multivariate algorithm is used for the identification of secondary vertices.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [15] with a
specific LHCb configuration [16]. Particle decays are described by EvtGen [17]
in which final-state radiation is generated using Photos [18]. For the three
signal $\tau$ decay channels, the final-state particles are distributed
according to three-body phase space. The interaction of the generated
particles with the detector, and its response, are implemented using the
Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].
## 3 Signal candidate selection
The signal and normalisation channels have the same topology, the signature of
which is a vertex displaced from the PV, having three tracks that are
reconstructed to give a mass close to that of the $\tau$ lepton (or $D_{s}$
meson for the normalisation channel). In order to discriminate against
background, well-reconstructed and well-identified muon, pion and proton
tracks are required, with selections on track quality criteria and a
requirement of $p_{\rm T}$ $>300$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Furthermore, for the $\tau\rightarrow p\mu\mu$ signal and normalisation
channels the muon and proton candidates must pass loose PID requirements and
the combined $p_{\rm T}$ of the three-track system is required to be greater
than $4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. All selected tracks are required
to have IP $\chi^{2}>9$. The fitted three-track vertex has to be of good
quality, with a fit $\chi^{2}<15$, and the measured decay time, $t$, of the
candidate forming the vertex has to be compatible with that of a heavy meson
or tau lepton ($ct>100\,\upmu\rm m$). Since the $Q$-values in decays of charm
mesons to $\tau$ are relatively small, poorly reconstructed candidates are
removed by a cut on the pointing angle between the momentum vector of the
three-track system and the line joining the primary and secondary vertices. In
the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ channel, signal candidates with
a $\mu^{+}\mu^{-}$ mass within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\phi$ meson mass are
removed, and to eliminate irreducible background near the signal region
arising from the decay
$D_{s}^{-}\rightarrow\eta(\mu^{+}\mu^{-}\gamma)\mu^{-}{\bar{\nu}_{\mu}}$,
candidates with a $\mu^{+}\mu^{-}$ mass combination below
$450{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are also rejected (see Section
6). Finally, to remove potential contamination from pairs of reconstructed
tracks that arise from the same particle, same-sign muon pairs with mass lower
than 250${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are removed in both the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$ channels. The signal regions are defined by $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\ (\approx 2\sigma_{m})$ windows
around the nominal $\tau$ mass, but candidates within wide mass windows, of
$\pm$400${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ decays and
$\pm$250${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\tau\rightarrow
p\mu\mu$ decays, are kept to allow evaluation of the background contributions
in the signal regions. A mass window of $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is also used to define the signal
region for the $D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ channel, with
the $\mu^{+}\mu^{-}$ mass required to be within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\phi$ meson mass.
## 4 Signal and background discrimination
After the selection each $\tau$ candidate is given a probability to be signal
or background according to the values of several likelihoods. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ three likelihoods are used: a
three-body likelihood, ${\rm\mathcal{M}_{3body}}$, a PID likelihood,
${\rm\mathcal{M}_{PID}}$, and an invariant mass likelihood. The likelihood
${\rm\mathcal{M}_{3body}}$ uses the properties of the reconstructed $\tau$
decay to distinguish displaced three-body decays from $N$-body decays (with
$N>3$) and combinations of tracks from different vertices. Variables used
include the vertex quality and its displacement from the PV, and the IP and
fit $\chi^{2}$ values of the tracks. The likelihood ${\rm\mathcal{M}_{PID}}$
quantifies the compatibility of each of the three particles with the muon
hypothesis using information from the RICH detectors, the calorimeters and the
muon stations; the value of ${\rm\mathcal{M}_{PID}}$ is taken as the smallest
one of the three muon candidates. For $\tau\rightarrow p\mu\mu$, the use of
${\rm\mathcal{M}_{PID}}$ is replaced by cuts on PID quantities. The invariant
mass likelihood uses the reconstructed mass of the $\tau$ candidate to help
discriminate between signal and background.
For the ${\rm\mathcal{M}_{3body}}$ likelihood a boosted decision tree [22] is
used, with the AdaBoost algorithm [23], and is implemented via the TMVA [24]
toolkit. It is trained using signal and background samples, both from
simulation, where the composition of the background is a mixture of
$b\bar{b}\rightarrow\mu\mu X$ and $c\bar{c}\rightarrow\mu\mu X$ according to
their relative abundance as measured in data. The ${\rm\mathcal{M}_{PID}}$
likelihood uses a neural network, which is also trained on simulated events.
The probability density function shapes are calibrated using the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ control channel and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ data for the ${\rm\mathcal{M}_{3body}}$ and
${\rm\mathcal{M}_{PID}}$ likelihoods, respectively. The shape of the signal
mass spectrum is modelled using
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ data. The
${\rm\mathcal{M}_{3body}}$ response as determined using the training from the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ samples is used also for the
$\tau\rightarrow p\mu\mu$ analyses.
For the ${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ likelihoods
the binning is chosen such that the separation power between the background-
only and signal-plus-background hypotheses is maximised, whilst minimising the
number of bins. For the ${\rm\mathcal{M}_{3body}}$ likelihood the optimum
number of bins is found to be six for the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis and five for
$\tau\rightarrow p\mu\mu$, while for the ${\rm\mathcal{M}_{PID}}$ likelihood
the optimum number of bins is found to be five. The lowest bins in
${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ do not contribute to
the sensitivity and are later excluded from the analyses. The distributions of
the two likelihoods, along with their binning schemes, are shown in Fig. 1 for
the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis.
\begin{overpic}[width=432.48048pt]{figs/supp/4a_new.pdf}
\put(40.0,120.0){\small{(a)}} \end{overpic}
\begin{overpic}[width=432.48048pt]{figs/supp/4b_new.pdf}
\put(40.0,120.0){\small{(b)}} \end{overpic}
Figure 1: Distribution of (a) ${\rm\mathcal{M}_{3body}}$ and (b)
${\rm\mathcal{M}_{PID}}$ for $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ where
the binning corresponds to that used in the limit calculation. The short
dashed (red) lines show the response of the data sidebands, whilst the long
dashed (blue) and solid (black) lines show the response of simulated signal
events before and after calibration. Note that in both cases the lowest
likelihood bin is later excluded from the analysis.
For the $\tau\rightarrow p\mu\mu$ analysis, further cuts on the muon and
proton PID hypotheses are used instead of ${\rm\mathcal{M}_{PID}}$ and are
optimised, for a $2\sigma$ significance, on simulated signal events and data
sidebands using the figure of merit from Ref. [25], with the distributions of
the PID variables corrected according to those observed in data. The expected
shapes of the invariant mass spectra for the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and $\tau\rightarrow p\mu\mu$
signals, with the appropriate selections applied, are taken from fits to the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ control channel in data as
shown in Fig. 2. The signal distributions are modelled with the sum of two
Gaussian functions with a common mean, where the narrower Gaussian contributes
70% of the total signal yield, while the combinatorial backgrounds are
modelled with linear functions. The expected widths of the $\tau$ signals in
data are taken from simulation, scaled by the ratio of the widths of the
$D_{s}^{-}$ peaks in data and simulation. The data are divided into eight
equally spaced bins in the $\pm 20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
mass window around the nominal $\tau$ mass.
\begin{overpic}[width=432.48048pt]{figs/1a.pdf} \put(40.0,120.0){\small{(a)}}
\end{overpic}
\begin{overpic}[width=432.48048pt]{figs/1b.pdf} \put(40.0,120.0){\small{(b)}}
\end{overpic}
Figure 2: Invariant mass distribution of $\phi(\mu^{+}\mu^{-})\pi^{-}$ after
(a) the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ selection and (b) the
$\tau\rightarrow p\mu\mu$ selection and PID cuts. The solid (blue) lines show
the overall fits, the long dashed (green) and short dashed (red) lines show
the two Gaussian components of the signal and the dot dashed (black) lines
show the backgrounds.
## 5 Normalisation
To measure the signal branching fraction for the decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ (and similarly for $\tau\rightarrow
p\mu\mu$) we normalise to the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ calibration channel using
$\displaystyle{{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})}$
$\displaystyle\quad={{\cal
B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})}\times\frac{f^{D_{s}}_{\tau}}{{\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})}\times\frac{\rm{\epsilon\mathstrut_{cal}^{REC\&SEL}}}{\rm\epsilon\mathstrut_{sig}^{REC\&SEL}}\times\frac{\rm{\epsilon\mathstrut_{cal}^{TRIG}}}{\rm\epsilon\mathstrut_{sig}^{TRIG}}\times\frac{N_{\rm
sig}}{N_{\rm cal}}$ $\displaystyle\quad=\alpha\times N_{\rm sig}\,,$ (1)
where $\alpha$ is the overall normalisation factor and $N_{\rm sig}$ is the
number of observed signal events. The branching fraction ${\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})$ is taken from Ref. [26]. The
quantity $f^{D_{s}}_{\tau}$ is the fraction of $\tau$ leptons that originate
from $D_{s}^{-}$ decays, calculated using the $b\bar{b}$ and $c\bar{c}$ cross-
sections as measured by LHCb [6, 7] and the inclusive $b\rightarrow\tau$,
$c\rightarrow\tau$, $b\rightarrow D_{s}$ and $c\rightarrow D_{s}$ branching
fractions [8]. The corresponding expression for the $\tau\rightarrow p\mu\mu$
decay is identical except for the inclusion of a further term,
${\rm\epsilon\mathstrut_{cal}^{PID}}/{\rm\epsilon\mathstrut_{sig}^{PID}}$, to
account for the effect of the PID cuts.
The reconstruction and selection efficiencies, $\rm\epsilon^{REC\&SEL}$, are
products of the detector acceptances for the particular decays, the muon
identification efficiencies and the selection efficiencies. The combined muon
identification and selection efficiency is determined from the yield of
simulated events after the full selections have been applied. In the sample of
simulated events, the track IPs are smeared to describe the secondary-vertex
resolution of the data. Furthermore, the events are given weights to adjust
the prompt and non-prompt $b$ and $c$ particle production fractions to the
latest measurements [8]. The difference in the result if the weights are
varied within their uncertainties is assigned as a systematic uncertainty. The
ratio of efficiencies is corrected to account for the differences between data
and simulation in efficiencies of track reconstruction, muon identification,
the $\phi(1020)$ mass window cut in the normalisation channel and the $\tau$
mass window cut, with all associated systematic uncertainties included. The
removal of candidates in the least sensitive bins in the
${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ classifiers is also
taken into account.
The trigger efficiency for selected candidates, $\rm\epsilon^{TRIG}$, is
evaluated from simulation while its systematic uncertainty is determined from
the difference between trigger efficiencies of $B^{-}\rightarrow J/\psi K^{-}$
decays measured in data and in simulation.
For the $\tau\rightarrow p\mu\mu$ channels the PID efficiency for selected and
triggered candidates, $\rm\epsilon^{PID}$, is calculated using data
calibration samples of $J/\psi\rightarrow\mu^{+}\mu^{-}$ and $\mathchar
28931\relax\rightarrow p\pi^{-}$ decays, with the tracks weighted to match the
kinematics of the signal and calibration channels. A systematic uncertainty of
1% per corrected final-state track is assigned [7], as well as a further 1%
uncertainty to account for differences in the kinematic binning of the
calibration samples between the analyses.
The branching fraction of the calibration channel is determined from a
combination of known branching fractions using
${\cal B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})=\frac{{\cal
B}(D_{s}^{-}\rightarrow\phi(K^{+}K^{-})\pi^{-})}{{\cal B}(\phi\rightarrow
K^{+}K^{-})}{\cal B}(\phi\rightarrow\mu^{+}\mu^{-})=(1.33\pm 0.12)\times
10^{-5}\,,$ (2)
where ${\cal B}(\phi\rightarrow K^{+}K^{-})$ and ${\cal
B}(\phi\rightarrow\mu^{+}\mu^{-})$ are taken from [8] and ${\cal
B}(D_{s}^{-}\rightarrow\phi(K^{+}K^{-})\pi^{-})$ is taken from the BaBar
amplitude analysis [27], which considers only the $\phi\rightarrow K^{+}K^{-}$
resonant part of the $D_{s}^{-}$ decay. This is motivated by the negligible
contribution of non-resonant $D_{s}^{-}\rightarrow\mu^{+}\mu^{-}\pi^{-}$
events seen in our data. The yields of
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ candidates in data, $N_{\rm
cal}$, are determined from the fits to reconstructed
$\phi(\mu^{+}\mu^{-})\pi^{-}$ mass distributions, shown in Fig. 2. The
variations in the yields if the relative contributions of the two Gaussian
components are varied in the fits are considered as systematic uncertainties.
Table 1 gives a summary of all contributions to $\alpha$; the uncertainties
are taken to be uncorrelated.
Table 1: Terms entering in the normalisation factor $\alpha$ for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, and their combined statistical and systematic uncertainties.
$\begin{array}[]{c|r@{\hspace{1mm}\pm\hspace{1mm}}l|r@{\hspace{1mm}\pm\hspace{1mm}}l|r@{\hspace{1mm}\pm\hspace{1mm}}l}&\lx@intercol\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\tau^{-}\rightarrow
p\mu^{-}\mu^{-}\hfil\lx@intercol\\\ \hline\cr{\cal
B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})&\lx@intercol\hfil(1.33\hskip
2.84526pt\pm\hskip 2.84526pt&\lx@intercol 0.12)\times
10^{-5}\hfil\lx@intercol\\\ \hline\cr f^{D_{s}}_{\tau}&\lx@intercol\hfil
0.78\hskip 2.84526pt\pm\hskip 2.84526pt&\lx@intercol 0.05\hfil\lx@intercol\\\
\hline\cr{\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})&\lx@intercol\hfil
0.0561\hskip 2.84526pt\pm\hskip 2.84526pt&\lx@intercol
0.0024\hfil\lx@intercol\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{REC\&SEL}/\rm{\epsilon\mathstrut_{sig}}^{REC\&SEL}&1.49\hskip
2.84526pt\pm\hskip 2.84526pt&0.12&1.35\hskip 2.84526pt\pm\hskip
2.84526pt&0.12&1.36\hskip 2.84526pt\pm\hskip 2.84526pt&0.12\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{TRIG}/\rm{\epsilon\mathstrut_{sig}}^{TRIG}&0.753\hskip
2.84526pt\pm\hskip 2.84526pt&0.037&1.68\hskip 2.84526pt\pm\hskip
2.84526pt&0.10&2.03\hskip 2.84526pt\pm\hskip 2.84526pt&0.13\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{PID}/\rm{\epsilon\mathstrut_{sig}}^{PID}&\lx@intercol\hskip
28.45274pt\rm{n/a}\hfil\lx@intercol\vrule\lx@intercol&1.43\hskip
2.84526pt\pm\hskip 2.84526pt&0.07&1.42\hskip 2.84526pt\pm\hskip
2.84526pt&0.08\\\ \hline\cr N_{\rm cal}&48\,076\hskip 2.84526pt\pm\hskip
2.84526pt&840&\lx@intercol\hfil 8\,145\pm 180\hfil\lx@intercol\\\
\hline\cr\alpha&(4.34\hskip 2.84526pt\pm\hskip 2.84526pt&0.65)\times
10^{-9}&(7.4\hskip 2.84526pt\pm\hskip 2.84526pt&1.2)\times 10^{-8}&(9.0\hskip
2.84526pt\pm\hskip 2.84526pt&1.5)\times 10^{-8}\\\ \end{array}$
## 6 Background studies
The background processes for the decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ consist mainly of decay chains of
heavy mesons with three real muons in the final state or with one or two real
muons in combination with two or one misidentified particles. These
backgrounds vary smoothly in the mass spectra in the region of the signal
channel. The most important peaking background channel is found to be
$D_{s}^{-}\rightarrow\eta(\mu^{+}\mu^{-}\gamma)\mu^{-}{\bar{\nu}_{\mu}}$,
about $80\%$ of which is removed (see Section 3) by a cut on the dimuon mass.
The small remaining background from this process is consistent with the smooth
variation in the mass spectra of the other backgrounds in the mass range
considered in the fit. Based on simulations, no peaking backgrounds are
expected in the $\tau\rightarrow p\mu\mu$ analyses.
The expected numbers of background events within the signal region, for each
bin in ${\rm\mathcal{M}_{3body}}$, ${\rm\mathcal{M}_{PID}}$ (for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$) and mass, are evaluated by fitting
the candidate mass spectra outside of the signal windows to an exponential
function using an extended, unbinned maximum likelihood fit. The small
differences obtained if the exponential curves are replaced by straight lines
are included as systematic uncertainties. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ the data are fitted over the mass
range $1600-1950$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, while for
$\tau\rightarrow p\mu\mu$ the fitted mass range is
$1650-1900$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, excluding windows
around the expected signal mass of $\pm
30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\mu^{-}\mu^{+}\mu^{-}$ and
$\pm 20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $p\mu\mu$. The
resulting fits to the data sidebands for a selection of bins for the three
channels are shown in Fig. 3.
\begin{overpic}[width=455.24408pt]{figs/2a.pdf} \put(40.0,120.0){\small{(a)}}
\put(80.0,125.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.65,1.0]$}}
\put(80.0,115.0){\tiny{${\rm\mathcal{M}_{PID}}$ $\in[0.725,1.0]$}}
\end{overpic}
\begin{overpic}[width=455.24408pt]{figs/2b.pdf} \put(40.0,120.0){\small{(b)}}
\put(60.0,120.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.40,1.0]$}}
\end{overpic}
\begin{overpic}[width=455.24408pt]{figs/2c.pdf} \put(40.0,120.0){\small{(c)}}
\put(35.0,105.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.40,1.0]$}}
\end{overpic}
Figure 3: Invariant mass distributions and fits to the mass sidebands in data
for (a) $\mu^{+}\mu^{-}\mu^{-}$ candidates in the four merged bins that
contain the highest signal probabilities, (b) ${\bar{p}}\mu^{+}\mu^{-}$
candidates in the two merged bins with the highest signal probabilities, and
(c) $p\mu^{-}\mu^{-}$ candidates in the two merged bins with the highest
signal probabilities.
## 7 Results
Tables 2 and 3 give the expected and observed numbers of candidates for all
three channels investigated, in each bin of the likelihood variables, where
the uncertainties on the background likelihoods are used to compute the
uncertainties on the expected numbers of events. No significant evidence for
an excess of events is observed. Using the $\textrm{CL}_{\textrm{s}}$ method
as a statistical framework, the distributions of observed and expected
$\textrm{CL}_{\textrm{s}}$ values are calculated as functions of the assumed
branching fractions. The aforementioned uncertainties and the uncertainties on
the signal likelihoods and normalisation factors are included using the
techniques described in Ref. [12, *Junk_99]. The resulting distributions of
$\textrm{CL}_{\textrm{s}}$ values are shown in Fig. 4.
The expected limits at $90\%~{}(95\%)$ CL for the branching fractions are
$\displaystyle{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 8.3~{}(10.2)\times 10^{-8},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 4.6~{}(5.9)\times 10^{-7},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})$ $\displaystyle<$
$\displaystyle 5.4~{}(6.9)\times 10^{-7},$
while the observed limits at $90\%~{}(95\%)$ CL are
$\displaystyle{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 8.0~{}(9.8)\times 10^{-8},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 3.3~{}(4.3)\times 10^{-7},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})$ $\displaystyle<$
$\displaystyle 4.4~{}(5.7)\times 10^{-7}.$
All limits are given for the phase-space model of $\tau$ decays. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$, the efficiency is found to vary by
no more than $20\%$ over the $\mu^{-}\mu^{-}$ mass range and by $10\%$ over
the $\mu^{+}\mu^{-}$ mass range. For $\tau\rightarrow p\mu\mu$, the efficiency
varies by less than $20\%$ over the dimuon mass range and less than $10\%$
with $p\mu$ mass.
In summary, a first limit on the lepton flavour violating decay mode
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ has been obtained at a hadron
collider. The result is compatible with previous limits and indicates that
with the additional luminosity expected from the LHC over the coming years,
the sensitivity of LHCb will become comparable with, or exceed, those of BaBar
and Belle. First direct upper limits have been placed on the branching
fractions for two $\tau$ decay modes that violate both baryon number and
lepton flavour, $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$.
Table 2: Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ signal window in the different likelihood bins for the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis. The likelihood values for ${\rm\mathcal{M}_{PID}}$ range from $0$ (most background-like) to $+1$ (most signal-like), while those for ${\rm\mathcal{M}_{3body}}$ range from $-1$ (most background-like) to $+1$ (most signal-like). The lowest likelihood bins have been excluded from the analysis. ${\rm\mathcal{M}_{PID}}$ | ${\rm\mathcal{M}_{3body}}$ | Expected | Observed
---|---|---|---
| $-$0.48 – | 0.05 | 345.0 $\pm$ | 6.7 | 409
| 0.05 – | 0.35 | 83.8 $\pm$ | 3.3 | 68
0.43 – 0.6 | 0.35 – | 0.65 | 30.2 $\pm$ | 2.0 | 35
| 0.65 – | 0.74 | 4.3 $\pm$ | 0.8 | 2
| 0.74 – | 1.0 | 1.4 $\pm$ | 0.4 | 1
| $-$0.48 – | 0.05 | 73.1 $\pm$ | 3.1 | 64
| 0.05 – | 0.35 | 18.3 $\pm$ | 1.5 | 15
0.6 – 0.65 | 0.35 – | 0.65 | 8.6 $\pm$ | 1.1 | 7
| 0.65 – | 0.74 | 0.4 $\pm$ | 0.1 | 0
| 0.74 – | 1.0 | 0.6 $\pm$ | 0.2 | 2
| $-$0.48 – | 0.05 | 45.4 $\pm$ | 2.4 | 51
| 0.05 – | 0.35 | 11.7 $\pm$ | 1.2 | 6
0.65 – 0.725 | 0.35 – | 0.65 | 5.3 $\pm$ | 0.8 | 3
| 0.65 – | 0.74 | 0.8 $\pm$ | 0.2 | 1
| 0.74 – | 1.0 | 0.4 $\pm$ | 0.1 | 0
| $-$0.48 – | 0.05 | 44.5 $\pm$ | 2.4 | 62
| 0.05 – | 0.35 | 10.6 $\pm$ | 1.2 | 13
0.725 – 0.86 | 0.35 – | 0.65 | 7.3 $\pm$ | 1.0 | 7
| 0.65 – | 0.74 | 1.0 $\pm$ | 0.2 | 2
| 0.74 – | 1.0 | 0.4 $\pm$ | 0.1 | 0
| $-$0.48 – | 0.05 | 5.9 $\pm$ | 0.9 | 7
| 0.05 – | 0.35 | 0.7 $\pm$ | 0.2 | 1
0.86 – 1.0 | 0.35 – | 0.65 | 1.0 $\pm$ | 0.2 | 1
| 0.65 – | 0.74 | 0.5 $\pm$ | 0.0 | 0
| 0.74 – | 1.0 | 0.4 $\pm$ | 0.1 | 0
Table 3: Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ mass window in the different likelihood bins for the $\tau\rightarrow p\mu\mu$ analysis. The likelihood values for ${\rm\mathcal{M}_{3body}}$ range from $-1$ (most background-like) to $+1$ (most signal-like). The lowest likelihood bin has been excluded from the analysis. | $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ | $\tau^{-}\rightarrow p\mu^{-}\mu^{-}$
---|---|---
${\rm\mathcal{M}_{3body}}$ | Expected | Observed | Expected | Observed
$-$0.05 – | 0.20 | 37.9 $\pm$ | 0.8 | 43 | 41.0 $\pm$ | 0.9 | 41
0.20 – | 0.40 | 12.6 $\pm$ | 0.5 | 8 | 11.0 $\pm$ | 0.5 | 13
0.40 – | 0.70 | 6.76 $\pm$ | 0.37 | 6 | 7.64 $\pm$ | 0.39 | 10
0.70 – | 1.00 | 0.96 $\pm$ | 0.14 | 0 | 0.49 $\pm$ | 0.12 | 0
\begin{overpic}[width=455.24408pt]{figs/3a.pdf} \put(40.0,120.0){\small{(a)}}
\end{overpic}
\begin{overpic}[width=455.24408pt]{figs/3b.pdf} \put(40.0,120.0){\small{(b)}}
\end{overpic}
\begin{overpic}[width=455.24408pt]{figs/3c.pdf} \put(40.0,120.0){\small{(c)}}
\end{overpic}
Figure 4: Distribution of $\textrm{CL}_{\textrm{s}}$ values as functions of
the assumed branching fractions, under the hypothesis to observe background
events only, for (a) $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$, (b)
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and (c) $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$. The dashed lines indicate the expected curves and the solid
lines the observed ones. The light (yellow) and dark (green) bands cover the
regions of 68% and 95% confidence for the expected limits.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-16T17:01:10 |
2024-09-04T02:49:44.476694
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van\n den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M. Dogaru, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D. Johnson, C.R. Jones, C.\n Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R.\n Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G.\n Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero\n Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Jonathan Harrison",
"url": "https://arxiv.org/abs/1304.4518"
}
|
1304.4524
|
# Investigating Randomly Generated Adjacency Matrices For Their Use In
Modeling Wireless Topologies
Gautam Bhanage and Sanjit Kaul
{gautamb, sanjit}@winlab.rutgers.edu
WINLAB, Rutgers University, North Brunswick, NJ 08902, USA
###### Abstract
Generation of realistic topologies plays an important role in determining the
accuracy and validity of simulation studies. This study presents a discussion
to justify why, and how often randomly generated adjacency matrices may not
not conform to wireless topologies in the physical world. Specifically, it
shows through analysis and random trials that, _more than $90\%$ of times, a
randomly generated adjacency matrix will not conform to a valid wireless
topology, when it has more than $3$ nodes_. By showing that node triplets in
the adjacency graph need to adhere to rules of a geometric vector space, the
study shows that the number of randomly chosen node triplets failing
consistency checks grow at the order of $O(base^{3})$, where $base$ is the
granularity of the distance metric. Further, the study models and presents a
probability estimate with which any randomly generated adjacency matrix would
fail realization. This information could be used to design simpler algorithms
for generating _k-connected_ wireless topologies.
## I Introduction
Simulation studies can be easily setup for wired networks by generating a
random adjacency matrix for modeling a random topology. As long as finite non-
negative entries are chosen for the adjacency matrix, it could be used to
represent a valid wired topology. However, in this paper we discuss how, and
why this may not hold true in the case of wireless topologies.
Specifically, this study addresses the following questions:
1. 1.
_Correctness:_ Are randomly generated topologies always valid, if not under
what conditions.
2. 2.
_Frequency of Failure:_ What percentage of randomly generated matrices are
invalid?
3. 3.
_Dominant Failure Factor:_ What feature of the matrix decides the probability
of the topology being invalid?
4. 4.
_Implication:_ Using this understanding, we propose designing algorithms with
a relatively direct approach for generating k-connected graphs.
Rest of the paper is organized as follows. Section II shows an example where a
randomly generated adjacency matrix does not represent a wireless topology.
Section III describes the problem statement, and present our approach for
determination of valid matrices. Section IV presents a comparison of results
from random trials with an approximation generated by our probability
function. Finally, we present a brief conclusion.
## II Discussion
Figure 1: Mapping problem definition as seen in $\Re^{1}$ space. If we have
node B connected to A and C, we cannot have another node D with connectivity
to A and C but not connected to B on the number line in $\Re^{1}$ space.
### II-A Example Of An Invalid Wireless Topology
We first address the question of whether a randomly generated adjacency matrix
can result in a non-realizable wireless topology. Figure 1 shows the positions
of three previously mapped points A, B, and C in a one dimensional metric
space ($\Re^{1}$). For this problem, we consider that all nodes have similar
radio capabilities and can communicate with each other only if they are within
_unit_ distance of each other. As per this condition, we have node B connected
to nodes A and C.
Now consider a case where the adjacency matrix generating the topology in
Figure 1 has an additional entry for a fourth node D, which has links to A and
C but is not within coverage of node B. Such a wireless topology is physically
not possible in the one dimensional metric space ($\Re^{1}$). Note that this
problem cannot be solved by using a different channel, since all the nodes
will need to be on the same frequency to be connected111We refer to a
_connection_ between any nodes 1 and 2, as a general term to signify that 1
and 2 have a significant SNR to communicate with each other. This
_connectivity_ is at the layer-1 and is independent of any access control
mechanisms used at a higher layer in the network stack.. It is also important
to observe that this failure occurs even when we do not have any planarity
constraints like requiring non-intersecting graph edges. Non-uniform radio
coverage for nodes D and B also fails to solve the problem. This is because
both nodes D and B need to be on the line, and a non-uniform radio coverage in
either directions (left or right) will result in disconnection from nodes A
and/or C.
This problem can be extended to all higher dimensions in metric spaces
$\Re^{n}$, $n>0$, which could result in invalid physical topologies. The only
factor that varies across these dimensions is the nature of the wireless
coverage. In $\Re^{1}$, we consider a line of unit (manhattan) distance on
each side of the node, in $\Re^{2}$, the coverage can be assumed in the form
of a unit circle in the plane (euclidean distance), similarly, a unit sphere
in $\Re^{3}$ and so on.
### II-B Problem Statement
Now that we have shown an example of an invalid wireless topology generation,
we will explicitly define the problem. Consider a network graph G which is
generated by a random adjacency matrix $A_{adj}[\texttt{ }]_{n\times n}$,
where $n$ denotes the number of nodes in the graph G. The individual entries
in $A_{adj}$ will denote the link conditions between corresponding wireless
nodes. In this study, given a specific $A_{adj}[\texttt{ }]$, we will define a
function _F_() to tell us whether the given adjacency matrix is capable of
realizing a valid wireless topology or not:
$F:G(A_{adj}[\texttt{ }]_{n\times n})\mapsto\\{Valid,Invalid\\}$ (1)
Once determined, _F_() can be used as a test for incrementally adding neighbor
nodes to an adjacency matrix for generating _k-connected_ graphs. We also
calculate the probability $(P_{F})$ with which _F_() will fail, which could be
used as a metric for determining the average number of trials that would be
required for valid wireless topology generation.
## III Modeling
### III-A Wireless Topologies $\&$ Vector Spaces
To determine _F_() defined above, we briefly discuss why the random adjacency
matrix used for Figure 1 fails. If we consider, the first three nodes A, B, C,
we observe that they satisfy the triangle inequality requirements in the
$\Re^{1}$ metric space. Let $\parallel.\parallel$ represent an arbitrary
distance norm. Triangle inequality requirement states that the sum of the
lengths of any two sides (say $\parallel x\parallel+\parallel y\parallel$) has
to be greater than the third side ($\parallel x+y\parallel$). While mapping
the fourth node D, with the requirement $A_{adj}(A,D)=1$, $A_{adj}(D,C)=1$,
and $A_{adj}(B,D)=0$, we observe that the triangle inequality fails for the
node sets $\\{B,D,A\\}$ and $\\{B,D,C\\}$. Thus using simple triangle
inequality as a test for the function $F$, i.e by determining if the generated
wireless topology fits in a geometric vector space, we can classify random
matrices as representing valid or invalid wireless topologies.
### III-B Estimating Adjacency Matrix Failure Probability $(P_{F})$
A randomly generated adjacency matrix $A_{adj}$ for a wireless topology with
$n$ nodes will fail when any one combination of three links fails the triangle
inequality check. Hence, the probability of at least one failure is:
$P_{F}=1-P_{NF},$ where the $P_{NF}$ is the probability that no combination in
the adjacency graphs fails the triangle inequality check. Thus, $P_{F}$ can be
calculated as:
$P_{F}=1-(1-P_{\triangle})^{N_{pairs}},$ (2)
where $N_{pairs}$ denotes the number of combinations of nodes checked in a
randomly generated matrix, and $P_{\triangle}$ denotes the probability of
failure of the triangle inequality on any random adjacency triplet. We define
an _adjacency triplet_ as any single combination of three values
$A_{adj}(P,Q)$, $A_{adj}(Q,R)$ and $A_{adj}(P,R)$ that describe link
conditions between any three nodes P, Q and R. The $N_{pairs}$ are determined
by the number of non-diagonal entries in the adjacency matrix. Since the
adjacency matrix is representing a wireless topology, it should be symmetric
akin to a metric space distance matrix. Hence,
$N_{pairs}=(\frac{n^{2}-n}{2})\times(\frac{n^{2}-n}{2}-1)$.
Figure 2: Probability of failure of triangle inequality tests for unique
combinations and permutations of side triplets.
### III-C Determining $P_{\triangle}$
To estimate $P_{\triangle}$, we use the complete set of adjacency triplets
($S_{3}$) described as:
$\texttt{ }S_{3}=\\{A_{adj}(P,Q),\texttt{ }A_{adj}(Q,R),\texttt{
}A_{adj}(P,R)\\},$ (3)
defined $\forall P,Q,R\in A_{adj}$. To determine $P_{\triangle}$ we can either
use combinations or permutations on $S_{3}$ to determine failure probability
of combinations. For all such possible permutations and combinations over
$S_{3}$, we determine $P_{\triangle}$ by calculating the fraction of adjacency
triplets that fail strict ($\leq$) and non-strict ($\leq$) triangle inequality
checks. While evaluating, we vary the _base_ , which denotes the maximum
number of discretized values that can be used to represent the link between
two points. E.g. when we chose the base as 1, the link represented in the
adjacency matrix can either take the values as 0 (off) or 1(on). If the base
is 2, the possible values are 0,1,2 and so on. Results for this model are as
described in Figure 2.
We observe that the fraction of permutations or combinations resulting in the
triangle inequality failure remain fairly constant, irrespective of the
increasing number base. Also, we note that the number of combinations being
evaluated are growing as $O(base^{3})$. Hence, we conclude that, the number of
unique combinations failing are also increasing at $O(base^{3})$ to keep the
ratio constant.
## IV Monte Carlo Tests
In this section we estimate and compare the probability with which a randomly
generated adjacency matrix will fail when mapped as a wireless topology. We
compare our failure probability estimate ($P_{F}$) with the failure
probability obtained through randomized trials. In these comparisons, we use
the discreteness of link qualities (or the discreteness of distance) and the
size of wireless topologies as the two varying parameters.
### IV-A Discreteness Of Link Connectivity Representation
Figure 3: Probability of failure of a randomly generated adjacency matrix in
representing a wireless topology as a function of the discreteness of distance
or connectivity.
In this test, as with the estimation of $P_{\triangle}$, we vary the maximum
number of discretized values (_base_) that can be used to represent the link
between two points. Results from our estimates and those from monte-carlo
tests are correspondingly marked as _Estimate:*_ and _MTC:*_ in the Figure 3.
The results show that our estimate of $P_{F}$ is able to closely match the
failure probability obtained from trials of $1000$ randomly generated
adjacency matrices for every distance _base_. For all topology sizes: $2,3,4$
(nodes each), we observe that the probability of the matrix failing to conform
to a wireless topology ($P_{F}$) can be high when the link connectivity is
coarsely described (E.g. on or off). This result is a direct consequence of:
$P_{F}\propto P_{\triangle}$. Hence, we observe that as $P_{\triangle}$
stabilizes for higher values of the distance _base_ , $P_{F}$ stabilizes too.
Figure 4: Comparison of estimated and observed failure probability as a
function of varying topology size.
### IV-B Impact Of Varying Topology Size
The size of a topology can be changed by varying the number of nodes. Edges
are not explicitly used as a factor for changing topology size since the
number and type of edges are randomly decided. In this experiment, we vary the
size of the wireless topology from 1 to 10 nodes. For every topology, an edge
can have a value uniformly distributed among the number of discretized
distance values given by the _base_. We generate $1000$ random matrices for
each topology size.
As shown in the results in Figure 4 the estimated failure probability $P_{F}$
(denoted by _Estimate:*_) closely matches that obtained from random trials
(_MTC:*_). Further, we observe that failure probability quickly approaches 1.
This matches with our estimate since, $P_{F}\propto N_{pairs}$, and
$N_{pairs}$ increase at least as $O(n^{2})$. An important implication of this
result is that as the size of the wireless topology goes beyond $3$ nodes, it
is almost certain that a randomly generated adjacency matrix will not conform
to a wireless topology.
## V Related Work
A class of studies has focussed on enumerating the characteristics of wired
networks [1, 2] that need to be taken into consideration while generating
topologies from random graphs. Consequently, a parallel area of research is
focussed on efficient generation [3] and improvement of the features of random
graphs to model real wired networks [4].
With concerns to wireless networks, [5] investigates the impact of spatial
distribution of nodes on the minimum node degree, and the k-connectivity in
random network graphs. We take a completely opposite view of the problem in
determining if a randomly generated adjacency could be used for faithfully
representing a realistic wireless topology.
## VI Conclusions
This study describes an approach for determining if randomly generated
adjacency matrices can conform to wireless topologies in the physical world.
It is shown that these random matrices are prone to failure, specially for
topologies with more than 3 nodes. Using this information, an alternative
approach can be taken for random wireless topology creation. Instead of
designing simulation studies based on random placement of nodes and then
generating k-connected graphs, algorithms could be designed that would
iteratively add nodes to the adjacency graph based on k-connectivity
requirements as long as they do not violate constraints of the geometric
space.
## References
* [1] M. B. Doar and A. Nexion, “A better model for generating test networks,” in _IEEE Global Telecommunications Conf. (Globecomm)_ , 1996, pp. 86–93.
* [2] E. Zegura, K. Calvert, and S. Bhattacharjee, “How to model an internetwork,” in _In Proceedings of IEEE INFOCOM_ , 1996, pp. 594–602.
* [3] J. Leskovec, D. Chakrabarti, J. Kleinberg, and C. Faloutsos, “Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication,” _Lecture Notes in Computer Science: Knowledge Discovery In Databases_ , vol. 3721, pp. 133–145, 2005.
* [4] A. Rodionov and H. Choo, “On generating random network structures: Connected graphs,” _Lecture Notes in Computer Science: Information Networking_ , vol. 3090, pp. 483–491, 2004.
* [5] C. Bettstetter, “On the minimum node degree and connectivity of a wireless multihop network,” in _MobiHoc ’02: Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking & computing_. New York, NY, USA: ACM, 2002, pp. 80–91.
|
arxiv-papers
| 2013-04-16T17:23:20 |
2024-09-04T02:49:44.483427
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gautam Bhanage, Sanjit Kaul",
"submitter": "Gautam Bhanage",
"url": "https://arxiv.org/abs/1304.4524"
}
|
1304.4530
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-051 LHCb-PAPER-2013-010
Observation of
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The decays
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ are observed for the first time
using a dataset, corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in proton-proton
collisions at centre-of-mass energies of $\sqrt{s}$ = 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$. The statistical significance for both
signals is in excess of 9 standard deviations. The following ratios of
branching fractions are measured to be
$\displaystyle\dfrac{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}\right)}$ $\displaystyle=$ $\displaystyle 2.90\pm 0.57\pm
0.24,$ $\displaystyle\dfrac{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}$ $\displaystyle=$ $\displaystyle
2.37\pm 0.56\pm 0.10,$
where the first uncertainties are statistical and the second systematic.
The mass of the $\mathrm{B}_{\mathrm{c}}^{+}$ meson is measured to be
$m_{\mathrm{B}_{\mathrm{c}}^{+}}=6276.28\pm 1.44\,\mathrm{(stat)}\pm
0.36\,\mathrm{(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$
using the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ decay mode.
Published in Physical Review D.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk57, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
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Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
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Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D.
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Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van
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Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
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C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2,
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P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
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E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
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T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
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T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
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J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
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McNab53, R. McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40,
D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53,
P. Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
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Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
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Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
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Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
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Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
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V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
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Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The $\mathrm{B}_{\mathrm{c}}^{+}$ meson, the ground state of the
$\bar{\mathrm{b}}{}\mathrm{c}$ system, is unique, being the only weakly
decaying heavy quarkonium system. Its lifetime [1, 2] is almost three times
smaller than that of other beauty mesons, pointing to the important role of
the charm quark in weak $\mathrm{B}_{\mathrm{c}}^{+}$ decays. The
$\mathrm{B}_{\mathrm{c}}^{+}$ meson was first observed through its
semileptonic decay
$\mathrm{B}_{\mathrm{c}}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\ell^{+}\upnu_{\ell}\mathrm{X}$ [3]. Only three hadronic modes have been
observed so far:
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ [4],
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}{}\uppi^{+}{}\uppi^{-}$ [5] and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}\uppsi{(2\mathrm{S})}{}\uppi^{+}$
[6].
The first observations of the decays
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ are reported in this paper. The
leading Feynman diagrams of these decays are shown in Fig. 1. The decay
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ is expected to proceed mainly through
spectator and colour-suppressed spectator diagrams. In contrast to decays of
other beauty hadrons, the weak annihilation topology is not suppressed and can
contribute significantly to the decay amplitude.
Figure 1: Feynman diagrams for
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ decays: (a) spectator, (b) colour-
suppressed spectator and (c) annihilation topology.
Assuming that the spectator diagram dominates and that factorization holds,
the following approximations can be established
$\displaystyle\mathcal{R}_{\mathrm{D}^{+}_{\mathrm{s}}\mskip-6.0mu/\uppi^{+}}$
$\displaystyle\equiv$
$\displaystyle\dfrac{\Gamma\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{\Gamma\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}\right)}\approx\dfrac{\Gamma\left(\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{\Gamma\left(\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\uppi^{+}\right)},$
(1a)
$\displaystyle\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}$
$\displaystyle\equiv$
$\displaystyle\dfrac{\Gamma\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}{\Gamma\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}\approx\dfrac{\Gamma\left(\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}{\Gamma\left(\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\mathrm{D}^{+}_{\mathrm{s}}\right)},$
(1b)
where $\mathrm{B}$ stands for $\mathrm{B}^{+}$ or $\mathrm{B}^{0}$ and
$\overline{\mathrm{D}}{}^{\ast}$ denotes $\overline{\mathrm{D}}^{\ast 0}$ or
$\mathrm{D}^{\ast-}$. Phase space corrections amount to ${\cal O}(0.5\%)$ for
Eq. (1a) and can be as large as 28% for Eq. (1b), depending on the relative
orbital momentum. The relative branching ratios estimated in this way,
together with more detailed theoretical calculations, are listed in Table 1,
where the branching fractions for the
$\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\mathrm{D}^{+}_{\mathrm{s}}$
and $\mathrm{B}{}\rightarrow{}\overline{\mathrm{D}}{}^{\ast}{}\uppi^{+}$
decays are taken from Ref. [1].
Table 1: Predictions for the ratios of $\mathrm{B}_{\mathrm{c}}^{+}$ meson branching fractions. In the case of $\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}$ the second uncertainty is related to the unknown relative orbital momentum. $\mathcal{R}_{\mathrm{D}^{+}_{\mathrm{s}}\mskip-6.0mu/\uppi^{+}}$ | $\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}$ |
---|---|---
$2.90\pm 0.42$ | $2.20\pm 0.35\pm 0.62$ | Eqs. (1) with $\mathrm{B}^{0}$
$1.58\pm 0.34$ | $2.07\pm 0.52\pm 0.52$ | Eqs. (1) with $\mathrm{B}^{+}$
1.3 | 3.9 | Ref. [7]
2.6 | 1.7 | Ref. [8]
2.0 | 2.9 | Ref. [9]
2.2 | — | Ref. [10]
1.2 | — | Ref. [11]
The analysis presented here is based on a data sample, corresponding to an
integrated luminosity of 3$\mbox{\,fb}^{-1}$, collected with the LHCb detector
during 2011 and 2012 in $\mathrm{pp}$ collisions at centre-of-mass energies of
7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, respectively. The decay
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ is used as a normalization channel for the measurement of
the branching fraction $\cal
B$($\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$). In addition, the low energy release
($Q$-value) in the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ mode allows a determination of the
$\mathrm{B}_{\mathrm{c}}^{+}$ mass with small systematic uncertainty.
## 2 LHCb detector
The LHCb detector [12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $\mathrm{pp}$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system has momentum resolution
$\Delta p/p$ that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger [13] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction.
This analysis uses events collected by triggers that select the decay products
of the dimuon decay of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ meson with high efficiency. At the hardware stage either one or two
identified muon candidates are required. In the case of single muon triggers
the transverse momentum, $p_{\rm T}$, of the candidate is required to be
larger than 1.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For dimuon candidates a
requirement on the product of the $p_{\rm T}$ of the muon candidates is
applied, $\sqrt{\mbox{$p_{\rm T}$}_{1}\mbox{$p_{\rm
T}$}_{2}}>1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. At the subsequent software
trigger stage, two muons with invariant mass in the interval
$2.97<m_{\upmu^{+}\upmu^{-}}<3.21{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
consistent with originating from a common vertex are required.
The detector acceptance and response are estimated with simulated data.
Proton-proton collisions are generated using Pythia 6.4 [14] with the
configuration described in Ref. [15]. Particle decays are then simulated by
EvtGen [16] in which final state radiation is generated using Photos [17]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [18, *Agostinelli:2002hh] as described in
Ref. [20].
## 3 Event selection
Track quality of charged particles is ensured by requiring that the $\chi^{2}$
per degree of freedom, $\chi^{2}_{\rm{tr}}/\mathrm{ndf}$, is less than $4$.
Further suppression of fake tracks created by the reconstruction is achieved
by a neural network trained to discriminate between these and real particles
based on information from track fit and hit pattern in the tracking detectors.
A requirement on the output of this neural network,
$\mathcal{P}_{\mathrm{fake}}<0.5$ allows to reject half of the fake tracks.
Duplicate particles created by the reconstruction are suppressed by requiring
the symmetrized Kullback-Leibler divergence [21, *Kullback1, *Kullback3],
$\Delta^{\rm min}_{\rm KL}$, calculated with respect to all particles in the
event, to be in excess of 5000. In addition, the transverse momentum is
required to be greater than 550 (250)${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$
for each muon (hadron) candidate.
Well identified muons are selected by requiring that the difference in
logarithms of the likelihood of the muon hypothesis, as provided by the muon
system, with respect to the pion hypothesis,
$\Delta^{\upmu/\uppi}\ln\mathcal{L}$ [24], is greater than zero. Good quality
particle identification by the ring-imaging Cherenkov detectors is ensured by
requiring the momentum of the hadron candidates, $p$, to be between
$3.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the pseudorapidity to be in the
range $2<\eta<5$. To select well-identified kaons (pions) the corresponding
difference in logarithms of the likelihood of the kaon and pion hypotheses
[25] is required to be $\Delta^{\mathrm{K}/\uppi}\ln\mathcal{L}>2(<0)$. These
criteria are chosen to be tight enough to reduce significantly the background
due to misidentification, whilst ensuring good agreement between data and
simulation.
To ensure that the hadrons used in the analysis are inconsistent with being
directly produced in a pp interaction vertex, the impact parameter $\chi^{2}$,
defined as the difference between the $\chi^{2}$ of the reconstructed pp
collision vertex formed with and without the considered track, is required to
be $\chi^{2}_{\mathrm{IP}}>9$. When more than one vertex is reconstructed,
that with the smallest value of $\chi^{2}_{\mathrm{IP}}$ is chosen.
As in Refs. [26, 27, 28] the selection of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\rightarrow{}\upmu^{+}\upmu^{-}$ candidates proceeds from pairs of
oppositely-charged muons forming a common vertex. The quality of the vertex is
ensured by requiring the $\chi^{2}$ of the vertex fit,
$\chi^{2}_{\mathrm{vx}}$, to be less than 30. The vertex is forced to be well
separated from the reconstructed pp interaction vertex by requiring the decay
length significance, $\mathcal{S}_{\mathrm{flight}}$, defined as the ratio of
the projected distance from pp interaction vertex to $\upmu^{+}\upmu^{-}$
vertex on direction of $\upmu^{+}\upmu^{-}$ pair momentum and its uncertainty,
to be greater than 3. Finally, the mass of the dimuon combination is required
to be within $\pm 45{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass [1], which
corresponds to a $\pm 3.5\sigma$ window, where $\sigma$ is the measured
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass resolution.
Candidate $\mathrm{D}^{+}_{\mathrm{s}}$ mesons are reconstructed in the
$\mathrm{D}^{+}_{\mathrm{s}}{}\rightarrow\left(\mathrm{K}^{+}{}\mathrm{K}^{-}{}\right)_{\upphi}\uppi^{+}$
mode using criteria similar to those in Ref. [29]. A good vertex quality is
ensured by requiring $\chi^{2}_{\mathrm{vx}}<25$. The mass of the kaon pair is
required to be consistent with the decay
$\upphi{}\rightarrow\mathrm{K}^{+}{}\mathrm{K}^{-}$,
$\left|m_{\mathrm{K}^{+}{}\mathrm{K}^{-}}-m_{\upphi}\right|<20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Finally, the mass of the candidate is required to be within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$\mathrm{D}^{+}_{\mathrm{s}}$ mass [1], which corresponds to a $\pm 3.5\sigma$
window, where $\sigma$ is the measured $\mathrm{D}^{+}_{\mathrm{s}}$ mass
resolution, and its transverse momentum to be
$>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Candidate $\mathrm{B}_{\mathrm{c}}^{+}$ mesons are formed from
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ pairs with transverse momentum in excess
of 1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The candidates should be
consistent with being produced in a $\mathrm{pp}$ interaction vertex by
requiring $\chi^{2}_{\mathrm{IP}}<9$ with respect to reconstructed pp
collision vertices. A kinematic fit is applied to the
$\mathrm{B}_{\mathrm{c}}^{+}$ candidates [30]. To improve the mass and
lifetime resolution, in this fit, a constraint on the pointing of the
candidate to the primary vertex is applied together with mass constraints on
the intermediate ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\mathrm{D}^{+}_{\mathrm{s}}$ states. The value of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass is taken from
Ref. [1]. For the $\mathrm{D}^{+}_{\mathrm{s}}$ meson the value of
$m_{\mathrm{D}^{+}_{\mathrm{s}}}=1968.31\pm
0.20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is used, that is the average of
the values given in Ref. [31] and Ref. [1]. The $\chi^{2}$ per degree of
freedom of this fit, $\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$, is required to be
less than $5$. The decay time of the $\mathrm{D}^{+}_{\mathrm{s}}$ candidate,
$c\tau\left(\mathrm{D}^{+}_{\mathrm{s}}\right)$, determined by this fit, is
required to satisfy $c\tau>75\,\upmu\rm m$. The corresponding signed
significance, $\mathcal{S}_{c\tau}$, defined as the ratio of the measured
decay time and its uncertainty, is required to be in excess of $3$. Finally,
the decay time of the $\mathrm{B}_{\mathrm{c}}^{+}$ candidate,
$c\tau\left(\mathrm{B}_{\mathrm{c}}^{+}\right)$, is required to be between
$75\,\upmu\rm m$ and 1$\rm\,mm$. The upper edge, in excess of 7 lifetimes of
$\mathrm{B}_{\mathrm{c}}^{+}$ meson, is introduced to remove badly
recontructed candidates.
## 4 Observation of
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$
The mass distribution of the selected
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ candidates is shown in Fig. 2. The peak
close to the known mass of the $\mathrm{B}_{\mathrm{c}}^{+}$ meson [1, 32]
with a width compatible with the expected mass resolution is interpreted as
being due to the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ decay. The wide structure between 5.9 and
$6.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is attributed to the decay
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{\ast+}_{\mathrm{s}}$, followed by
$\mathrm{D}^{\ast+}_{\mathrm{s}}\rightarrow{}\mathrm{D}^{+}_{\mathrm{s}}{}\upgamma$
or
$\mathrm{D}^{\ast+}_{\mathrm{s}}\rightarrow{}\mathrm{D}^{+}_{\mathrm{s}}{}\uppi^{0}$
decays, where the neutral particles are not detected. The process
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{\ast+}_{\mathrm{s}}$ being the decay of a pseudoscalar
particle into two vector particles is described by three helicity amplitudes:
$\mathcal{A}_{++}$, $\mathcal{A}_{00}$ and $\mathcal{A}_{--}$, where indices
correspond to the helicities of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\mathrm{D}^{\ast+}_{\mathrm{s}}$ mesons. Simulation studies show that the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$$\mathrm{D}^{+}_{\mathrm{s}}$ mass distributions are the same for the
$\mathcal{A}_{++}$ and $\mathcal{A}_{--}$ amplitudes. Thus, the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$$\mathrm{D}^{+}_{\mathrm{s}}$ mass spectrum is described by a model
consisting of the following components: an exponential shape to describe the
combinatorial background, a Gaussian shape to describe the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ signal and two helicity components to
describe the
$\mathrm{B}_{\mathrm{c}}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{D}^{\ast+}_{\mathrm{s}}$ contributions corresponding to the
$\mathcal{A}_{\pm\pm}$ and $\mathcal{A}_{00}$ amplitudes. The shape of these
components is determined using the simulation where the branching fractions
for
$\mathrm{D}^{\ast+}_{\mathrm{s}}\rightarrow\mathrm{D}^{+}_{\mathrm{s}}\upgamma$
and
$\mathrm{D}^{\ast+}_{\mathrm{s}}\rightarrow\mathrm{D}^{+}_{\mathrm{s}}\uppi^{0}$
decays are taken from Ref. [1].
To estimate the signal yields, an extended unbinned maximum likelihood fit to
the mass distribution is performed. The correctness of the fit procedure
together with the reliability of the estimated uncertainties has been
extensively checked using simulation. The fit has seven free parameters: the
mass of the $\mathrm{B}_{\mathrm{c}}^{+}$ meson,
$m_{\mathrm{B}_{\mathrm{c}}^{+}}$, the signal resolution,
$\sigma_{\mathrm{B}_{\mathrm{c}}^{+}}$, the relative amount of the
$\mathcal{A}_{\pm\pm}$ helicity amplitudes of total
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ decay rate, $\mathrm{f}_{\pm\pm}$,
the slope parameter of the exponential background and the yields of the two
signal components,
$N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}}$ and
$N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}}$, and of the background. The values
of the signal parameters obtained from the fit are summarized in Table 2. The
fit result is also shown in Fig. 2.
Figure 2: Mass distributions for selected ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$$\mathrm{D}^{+}_{\mathrm{s}}$ pairs. The solid curve represents the result of a fit to the model described in the text. The contribution from the $\mathrm{B}_{\mathrm{c}}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{D}^{\ast+}_{\mathrm{s}}$ decay is shown with thin green dotted and thin yellow dash-dotted lines for the $\mathcal{A}_{\pm\pm}$ and $\mathcal{A}_{00}$ amplitudes, respectively. The insert shows a zoom of the $\mathrm{B}_{\mathrm{c}}^{+}$ mass region. Table 2: Signal parameters of the unbinned extended maximum likelihood fit to the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$$\mathrm{D}^{+}_{\mathrm{s}}$ mass distribution. Parameter | Value
---|---
$m_{\mathrm{B}_{\mathrm{c}}^{+}}$ | $\left[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $6276.28\pm 1.44\phantom{000}$
$\sigma_{\mathrm{B}_{\mathrm{c}}^{+}}$ | $\left[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $\phantom{0}7.0\pm 1.1\phantom{0}$
$N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}}$ | | $28.9\pm 5.6\phantom{0}$
$\dfrac{N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}}}{N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}}}$ | | $2.37\pm 0.56$
$\mathrm{f}_{\pm\pm}$ | $\left[\%\right]$ | $52\pm 20$
To check the result, the fit has been performed with different models for the
signal: a double-sided Crystal Ball function [33, 34], and a modified
Novosibirsk function [35]. For these tests the tail and asymmetry parameters
are fixed using the simulation values, while the parameters representing the
peak position and resolution are left free to vary. As alternative models for
the background, the product of an exponential function and a fourth-order
polynomial function are used. The fit parameters obtained are stable with
respect to the choice of the fit model and the fit range interval.
The statistical significance for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ signal is estimated from the change in
the likelihood function
$\mathcal{S}_{\sigma}=\sqrt{2\ln\tfrac{\mathcal{L}_{\mathcal{B}+\mathcal{S}}}{\mathcal{L}_{\mathcal{B}}}}$,
where $\mathcal{L}_{\mathcal{B}}$ is the likelihood of a background-only
hypothesis and $\mathcal{L}_{\mathcal{B}+\mathcal{S}}$ is the likelihood of a
background-plus-signal hypothesis. The significance has been estimated
separately for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ signals. To exclude the look-
elsewhere effect [36, *Gross:2010], the mass and resolution of the peak are
fixed to the values obtained with the simulation. The minimal significance
found varying the fit model as described above is taken as the signal
significance. The statistical significance for both the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ signals estimated in this way is in
excess of 9 standard deviations.
The low $Q$-value for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ decay mode allows the
$\mathrm{B}_{\mathrm{c}}^{+}$ mass to be precisely measured. This makes use of
the $\mathrm{D}^{+}_{\mathrm{s}}$ mass value, evaluated in Sect. 3, taking
correctly into account the correlations between the measurements. The
calibration of the momentum scale for the dataset used here is detailed in
Refs. [38, 31]. It is based upon large calibration samples of
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{K}^{+}$ and ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ decays and leads to an accuracy in the
momentum scale of $3\times 10^{-4}$. This translates into an uncertainty of
0.30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ on the
$\mathrm{B}_{\mathrm{c}}^{+}$ meson mass. A further uncertainty of
0.11${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ arises from the knowledge of
the detector material distribution [38, 31, 32, 39] and the signal modelling.
The uncertainty on the $\mathrm{D}^{+}_{\mathrm{s}}$ mass results in a
0.16${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ uncertainty on the
$\mathrm{B}_{\mathrm{c}}^{+}$ meson mass. Adding these in quadrature gives
$m_{\mathrm{B}_{\mathrm{c}}^{+}}=6276.28\pm 1.44\,\mathrm{(stat)}\pm
0.36\,\mathrm{(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$
The uncertainty on the $\mathrm{D}^{+}_{\mathrm{s}}$ meson mass and on the
momentum scale largely cancels in the mass difference
$m_{\mathrm{B}_{\mathrm{c}}^{+}}-m_{\mathrm{D}^{+}_{\mathrm{s}}}=4307.97\pm
1.44\,\mathrm{(stat)}\pm
0.20\,\mathrm{(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$
## 5 Normalization to the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ decay mode
A large sample of
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ decays serves as a normalization channel to measure the
ratio of branching fractions for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ modes. Selection of
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ events is performed in a manner similar to that described
in Sect. 3 for the signal channel. To further reduce the combinatorial
background, the transverse momentum of the pion for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ mode is required to be in excess of
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The mass distribution of the selected
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ candidates is shown in Fig. 3.
Figure 3: Mass distribution for selected
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ candidates. The results of a fit to the model described in
the text are superimposed (solid line) together with the background component
(dotted line).
To determine the yield, an extended unbinned maximum likelihood fit to the
mass distribution is performed. The signal is modelled by a double-sided
Crystal Ball function and the background with an exponential function. The fit
gives a yield of $3009\pm 79$ events. As cross-checks, a modified Novosibirsk
function and a Gaussian function for the signal component and a product of
exponential and polynomial functions for the background are used. The
difference is treated as systematic uncertainty.
The ratio of the total efficiencies (including acceptance, reconstruction,
selection and trigger) for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ modes is determined with simulated data to be $0.148\pm
0.001$, where the uncertainty is statistical only. As only events explicitly
selected by the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
triggers are used, the ratio of the trigger efficiencies for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ modes is close to unity.
## 6 Systematic uncertainties
Uncertainties on the ratio
$\mathcal{R}_{\mathrm{D}^{+}_{\mathrm{s}}\mskip-6.0mu/\uppi^{+}}$ related to
differences between the data and simulation efficiency for the selection
requirements are studied using the abundant
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ channel. As an example, Fig. 4 compares the distributions
of $\chi^{2}_{\mathrm{fit}}(\mathrm{B}_{\mathrm{c}}^{+})$ and
$\chi^{2}_{\mathrm{IP}}(\mathrm{B}_{\mathrm{c}}^{+})$ for data and simulated
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ events. For background subtraction the sPlot techinque [40]
has been used. It can be seen that the agreement between data and simulation
is good. In addition, a large sample of selected
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\left(\mathrm{K}^{+}{}\mathrm{K}^{-}\right)_{\upphi}\mathrm{K}^{+}$
events has been used to quantify differences between data and simulation.
Based on the deviation, a systematic uncertainty of 1% is assigned.
The agreement of the absolute trigger efficiency between data and simulation
has been validated to a precision of 4% using the technique described in Refs.
[41, 34, 13] with a large sample of
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\left(\mathrm{K}^{+}{}\mathrm{K}^{-}\right)_{\upphi}\mathrm{K}^{+}$
events. A further cancellation of uncertainties occurs in the ratio of
branching fractions resulting in a systematic uncertainty of $1.1\%$.
Figure 4: Distributions of (a)
$\chi^{2}_{\mathrm{fit}}(\mathrm{B}_{\mathrm{c}}^{+})$ and (b)
$\chi^{2}_{\mathrm{IP}}(\mathrm{B}_{\mathrm{c}}^{+})$ for
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ events: background subtracted data (red points with error
bars), and simulation (blue histogram).
The systematic uncertainties related to the fit model, in particular to the
signal shape, mass and resolution for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ mode and the fit interval have been
discussed in Sects. 4 and 5. The main part comes from the normalization
channel
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$.
Other systematic uncertainties arise from differences in the efficiency of
charged particle reconstruction between data and simulation. The largest of
these arises from the knowledge of the hadronic interaction probability in the
detector, which has an uncertainty of $2\%$ per track [41]. A further
uncertainty related to the reconstruction of two additional kaons in the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ mode with respect to the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ mode is estimated to be $2\times 0.6\%$ [42]. Further
uncertainties are related to the track quality selection requirements
$\chi^{2}_{\mathrm{tr}}<4$ and $\mathcal{P}_{\mathrm{fake}}<0.5$. These are
estimated from a comparison of data and simulation in the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ decay mode to be $0.4\%$ per final state track.
The uncertainty associated with the kaon identification criteria is studied
using the combined
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ signals. The efficiency to identify
a kaon pair with a selection on $\Delta^{\mathrm{K}/\uppi}\ln\mathcal{L}$ has
been compared for data and simulation for various selection requirements. The
comparison shows a $(-1.8\pm 2.9)\%$ difference between data and simulation in
the efficiency to identify a kaon pair with
$2\leq\min\Delta^{\mathrm{K}/\uppi}\log\mathcal{L}$. This estimate has been
confirmed using a kinematically similar sample of reconstructed
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\left(\mathrm{K}^{+}{}\mathrm{K}^{-}\right)_{\upphi}\mathrm{K}^{+}$
events. An uncertainty of 3% is assigned.
The limited knowledge of the $\mathrm{B}_{\mathrm{c}}^{+}$ lifetime leads to
an additional systematic uncertainty due to the different decay time
acceptance between the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ decay modes. To estimate this effect, the decay time
distributions for simulated events are reweighted to change the
$\mathrm{B}_{\mathrm{c}}^{+}$ lifetime by one standard deviation from the
known value [1], as well as the value recently measured by the CDF
collaboration [2], and the efficiencies are recomputed. An uncertainty of
$1\%$ is assigned.
Possible uncertainties related to the stability of the data taking conditions
are tested by studying the ratio of the yields of
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{K}^{+}{}\uppi^{+}\uppi^{-}$ and
$\mathrm{B}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{K}^{+}$ decays for different data taking periods and dipole
magnet polarities. This results in a further $2.5\%$ uncertainty.
The largest systematic uncertainty is due to the knowledge of the branching
fraction of the
$\mathrm{D}^{+}_{\mathrm{s}}{}\rightarrow{}\left(\mathrm{K}^{-}{}\mathrm{K}^{+}{}\right)_{\upphi}{}\uppi^{+}$
decay, with a kaon pair mass within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\upphi$ meson mass.
The value of $(2.24\pm 0.11\pm 0.06)\%$ from Ref.[43] is used in the analysis.
The systematic uncertainties on
$\mathcal{R}_{\mathrm{D}^{+}_{\mathrm{s}}\mskip-6.0mu/\uppi^{+}}$ are
summarized in Table 3.
Table 3: Relative systematic uncertainties for the ratio of branching fractions of $\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and $\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\uppi^{+}$. Source | Uncertainty $\left[\%\right]$
---|---
Simulated efficiencies | 1.0
Trigger | 1.1
Fit model | 1.8
Track reconstruction | $2\times 0.6$
Hadron interactions | $2\times 2.0$
Track quality selection | $2\times 0.4$
Kaon identification | 3.0
$\mathrm{B}_{\mathrm{c}}^{+}$ lifetime | 1.0
Stability for various data taking conditions | 2.5
${\cal B}\left(\mathrm{D}^{+}_{\mathrm{s}}\rightarrow\left(\mathrm{K}^{-}\mathrm{K}^{+}\right)_{\upphi}\uppi^{+}\right)$ | 5.6
Total | 8.4
The ratio
$\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}$
is estimated as
$\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}=\frac{N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}}}{N_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}}},$ (2)
where the ratio of yields is given in Table 2. The uncertainty associated with
the assumption that the efficiencies for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ modes are equal, is evaluated by
studying the dependence of the relative yields for these modes for loose (or
no) requirements on the $\chi^{2}_{\mathrm{IP}}(\mathrm{B}_{\mathrm{c}}^{+})$,
$\chi^{2}_{\mathrm{fit}}(\mathrm{B}_{\mathrm{c}}^{+})$ and
$c\tau({}\mathrm{B}_{\mathrm{c}}^{+})$ variables. For this selection the
measured ratio of
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ to
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ events changes to $2.27\pm 0.59$. An
uncertainty of 4% is assigned to the
$\mathcal{R}_{\mathrm{D}{}^{\ast+}_{\mathrm{s}}\mskip-6.0mu/\mathrm{D}^{+}_{\mathrm{s}}}$
ratio.
The uncertainty on the fraction of the $\mathcal{A}_{\pm\pm}$ amplitude,
$\mathrm{f}_{\pm\pm}$, has been studied with different fit models for the
parameterization of the combinatorial background, as well as different mass
resolution models. This is negligible in comparison to the statistical
uncertainty.
## 7 Results and summary
The decays
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ have been observed for the first
time with statistical significances in excess of 9 standard deviations. The
ratio of branching fractions for
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ is calculated as
$\dfrac{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}\right)}=\dfrac{1}{{\cal
B}_{\mathrm{D}^{+}_{\mathrm{s}}}}\times\dfrac{\varepsilon^{\mathrm{tot}}_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}}}{\varepsilon^{\mathrm{tot}}_{\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}}}\times\dfrac{N\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{N\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}\right)},$ (3)
where the value of ${\cal B}_{\mathrm{D}^{+}_{\mathrm{s}}}={\cal
B}\left(\mathrm{D}^{+}_{\mathrm{s}}{}\rightarrow{}\left(\mathrm{K}^{-}{}\mathrm{K}^{+}{}\right)_{\upphi}{}\uppi^{+}\right)$
[43] with the mass of the kaon pair within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known value of the $\upphi$
mass is used, together with the ratio of efficiencies, and the signal yields
given in Sects. 4 and 5. This results in
$\dfrac{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}\right)}=2.90\pm 0.57\,\mathrm{(stat)}\pm
0.24\,\mathrm{(syst)}.$
The value obtained is in agreement with the naïve expectations given in Eq.
(1a) from $\mathrm{B}^{0}$ decays, and the values from Refs. [8, 10, 9] but
larger than predictions from Refs. [7, 11] and factorization expectations from
$\mathrm{B}^{+}$ decays.
The ratio of branching fractions for the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ and
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}$ decays is measured to be
$\dfrac{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}{{\cal
B}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}^{+}_{\mathrm{s}}\right)}=2.37\pm 0.56\,\mathrm{(stat)}\pm
0.10\,\mathrm{(syst)}.$
This result is in agreement with the naïve factorization hypothesis (Eq. (1b))
and with the predictions of Refs. [9, 8].
The fraction of the $\mathcal{A}_{\pm\pm}$ amplitude in the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}$ decay is measured to be
$\dfrac{\Gamma_{\pm\pm}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}{\Gamma_{\mathrm{tot}}\left(\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\mathrm{D}{}^{\ast+}_{\mathrm{s}}\right)}=(52\pm 20)\%,$
in agreement with a simple estimate of $\tfrac{2}{3}$, the measurements [44,
45] and factorization predictions [46] for
$\mathrm{B}^{0}\rightarrow\mathrm{D}^{*-}\mathrm{D}_{\mathrm{s}}^{\ast+}$
decays, and expectations for
$\mathrm{B}_{\mathrm{c}}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\ell^{+}\upnu_{\ell}$ decays from Refs. [47, 48].
The mass of the $\mathrm{B}_{\mathrm{c}}^{+}$ meson and the mass difference
between the $\mathrm{B}_{\mathrm{c}}^{+}$ and $\mathrm{D}^{+}_{\mathrm{s}}$
mesons are measured to be
$\displaystyle m_{\mathrm{B}_{\mathrm{c}}^{+}}$ $\displaystyle=$
$\displaystyle 6276.28\pm 1.44\,\mathrm{(stat)}\pm
0.36\,\mathrm{(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle
m_{\mathrm{B}_{\mathrm{c}}^{+}}-m_{\mathrm{D}^{+}_{\mathrm{s}}}$
$\displaystyle=$ $\displaystyle 4307.97\pm 1.44\,\mathrm{(stat)}\pm
0.20\,\mathrm{(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$
The $\mathrm{B}_{\mathrm{c}}^{+}$ mass measurement is in good agreement with
the previous result obtained by LHCb in the
$\mathrm{B}_{\mathrm{c}}^{+}{}\rightarrow{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{}\uppi^{+}$ mode [32] and has smaller systematic uncertainty.
## Acknowledgements
We thank A. Luchinsky and A.K. Likhoded for advice on aspects of
$\mathrm{B}_{\mathrm{c}}^{+}$ physics. We express our gratitude to our
colleagues in the CERN accelerator departments for the excellent performance
of the LHC. We thank the technical and administrative staff at the LHCb
institutes. We acknowledge support from CERN and from the national agencies:
CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region
Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN
(Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS/IFA (Romania);
MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal
and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC
(United Kingdom); NSF (USA). We also acknowledge the support received from the
ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France),
KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC
(Spain), GridPP (United Kingdom). We are thankful for the computing resources
put at our disposal by Yandex LLC (Russia), as well as to the communities
behind the multiple open source software packages that we depend on.
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|
arxiv-papers
| 2013-04-16T17:34:58 |
2024-09-04T02:49:44.488784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto,\n H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D.\n Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi,\n J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck,\n T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen,\n E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P.\n Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik,\n P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina,\n I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki,\n J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev,\n Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R.\n McNulty, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello,\n R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B.\n Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K.\n Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak,\n S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas,\n J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch,\n A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1304.4530"
}
|
1304.4542
|
# Spintronics in ${\rm MoS}_{2}$ monolayer quantum wires
Jelena Klinovaja Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland Daniel Loss Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland
###### Abstract
We study analytically and numerically spin effects in ${\rm MoS}_{2}$
monolayer armchair quantum wires and quantum dots. The interplay between
intrinsic and Rashba spin orbit interactions induced by an electric field
leads to helical modes, giving rise to spin filtering in time-reversal
invariant systems. The Rashba spin orbit interaction can also be generated by
spatially varying magnetic fields. In this case, the system can be in a
helical regime with nearly perfect spin polarization. If such a quantum wire
is brought into proximity to an $s$-wave superconductor, the system can be
tuned into a topological phase, resulting in midgap Majorana fermions
localized at the wire ends.
###### pacs:
71.70.Ej, 85.75.-d, 73.63.Kv, 78.67.-n
Introduction. Atomic monolayers such as graphene sheets Novoselov_2009 have
attracted much attention over the years. However, the small spin orbit
interaction (SOI) in graphene makes spin effects negligibly small. kane_mele ;
cnt_ext_kuemmeth ; klinovaja_cnt ; cnt_helical_2011 ; kane_mele ; izumida ;
fabian In contrast, transition-metal dichalcogenide semiconductors,
frindt_mos ; Morrison_mos ; mos_nanotubes ; mos_ribbons ; mos_Fuhrer ; mos_gap
; mos_ribbons_defects ; mos_transistor ; mos_etching ; mos_steele ; mos_sc ;
MOS_review in particular ${\rm MoS}_{2}$, possess giant values of SOI.
MOS_review Combined with a direct band gap this SOI makes these materials
attractive for optical effects. monolayer_optics_exp_2010 ;
Zeng_optics_exp_2012 ; optics_Nature ; Yao_2012 ; Niu_valley_Hall_2007 ;
Niu_optics_rules_2008 However, previous work emphasized valleytronics in
${\rm MoS}_{2}$, Niu_valley_Hall_2007 ; Niu_optics_rules_2008 while the spin
degrees of freedom have received much less attention, despite the fact that
these materials can be expected to display interesting spintronics effects,
such as helical states in quantum wires, Majorana fermions, spin qubits in
quantum dots, electrical control of spin, etc. This gap of understanding has
motivated the present work where we will propose and analyze spin effects
specifically for quantum confined structures in ${\rm MoS}_{2}$ monolayers.
One of our main findings is that the intrinsic SOI needs to be complemented by
Rashba-like SOI to obtain interesting spin effects in the conduction band. In
particular, we will focus on suitably defined quantum wires of armchair type
and show that they allow for helical modes, with and without time-reversal
symmetry. The Rashba-like interaction can be generated by breaking structure
inversion symmetry with gates or adatoms or, alternatively, by nanomagnets
with alternating magnetization direction. Helical modes serve as basis for
spin filters streda but also as platform for exotic quantum states such as
Majorana fermions alicea_review_2012 or fractionally charged fermions.
Two_field_Klinovaja We finally discuss quantum dots with well-defined Kramers
doublets that can serve as spin qubits. kloeffel_prospects_2013
Figure 1: (a) The ${\rm MoS}_{2}$ monolayer lattice consists of ${\rm Mo}$
(large green dots) each connected to six ${\rm S}$ (small blue dots). The
armchair quantum wire in the monolayer can be formed by metallic gates (yellow
area) that fix the propagation direction (defined as the $y$ axis) to be
perpendicular to one of the lattice translation vectors, say $\bf a_{1}$ (red
arrow). (b) Brillouin zone where the valleys $K$ and $-K$ lie on the $k_{x}$
axis which is perpendicular to the direction of propagation given by the
$k_{y}$ axis. Note that the boundaries of a ${\rm MoS}_{2}$ flake are not
important for this setup. Figure 2: (a) The energy spectrum of $H_{0}$
[$\epsilon(\tilde{k}_{y})\equiv E(\sqrt{3}k_{y}a)-\Delta$] for a quantum wire
of width $W=50a$ as obtained by numerical diagonalization. Parameters are
chosen as $t=1.27\ {\rm eV}$, $\Delta=0.83\ {\rm eV}$, and $a=0.32\ {\rm nm}$.
All levels are degenerate only in spin. This spectrum is in good agreement
with our analytical predictions, see Eq. (3). (b) The Rashba SOI term
$H_{Rx}$, $\alpha_{R}=10\ {\rm meV}$, lifts the spin-degeneracy and results in
the spin-dependent shift of the wavevector $\tilde{k}_{y}$. (c) The remaining
SOI terms: intrinsic SOI $H_{so}$, $\alpha=38\ {\rm meV}$, and Rashba SOI
$H_{Ry}$, $\alpha_{R}=10\ {\rm meV}$, lead to the anticrossings in the
spectrum (red dashed circles).
Bandstructure. A molybdenum disulphide (${\rm MoS}_{2}$) monolayer consists of
two layers of ${\rm S}$ atoms stacked over each other forming an effective
trigonal lattice and of $\rm Mo$ atoms located in the center of the sulphur
lattice, see Fig. 1a. The Brillouin zone consists of a hexagon with two
nonequivalent corners (valleys) at $\mathbf{K}$ $(\tau_{z}=1)$ and
$-\mathbf{K}$ $(\tau_{z}=-1)$ (see Fig. 1b) that determine the low energy
spectrum of the monolayer. Eriksson_2009_cones ; Yao_2012 ; dft_mos_2012 ;
Ataca_chemistry ; kuc_2011 This part of the spectrum is dominated by three
$d$ orbitals of ${\rm Mo}$: the conduction band by
$\left|\psi_{c}\right\rangle=\left|d_{z^{2}}\right\rangle$ and the valence
band by
$\left|\psi_{v}^{\tau_{z}}\right\rangle=(\left|d_{x^{2}-y^{2}}\right\rangle+i\tau_{z}\left|d_{xy}\right\rangle)/\sqrt{2}$.
The effective Hamiltonian is given by
$\displaystyle
H_{0}=\hbar\upsilon_{F}(k_{x}\tau_{z}\sigma_{1}+k_{y}\sigma_{2})+\Delta\sigma_{3},$
(1)
where the Pauli matrices $\sigma_{i}$ act on the $d$ orbital space. The
momenta $k_{x}$ and $k_{y}$ are calculated from the corresponding valley
characterized by $\tau_{z}$. Here, $\upsilon_{F}\approx 0.53\times 10^{6}\
{\rm m/s}$ is the Fermi velocity and the mass term, $\Delta\approx 830\ {\rm
meV}$, arising from broken inversion symmetry, has been extracted from DFT
calculations. Yao_2012 The spectrum of $H_{0}$ is given by
$E_{c,v}=\pm\sqrt{(\hbar\upsilon_{F})^{2}(k_{x}^{2}+k_{y}^{2})+\Delta^{2}}$.
The large gap $2\Delta$ between the valence and conduction bands makes the
monolayer attractive for optical effects. monolayer_optics_exp_2010 ;
Niu_optics_rules_2008 ; Zeng_optics_exp_2012 ; optics_Nature ; Yao_2012 The
wavefunctions at fixed energy $E$ and momentum $k_{y}$ are written in the
basis $(\psi_{c},\psi_{v}^{\tau_{z}})$ as
$\psi_{\tau_{z},p}(x)=e^{i(\tau_{z}K+pk_{x})x+ik_{y}y}\begin{pmatrix}b_{\tau_{z},p}\\\
1\end{pmatrix}_{{\tau_{z}}},$ (2)
where $b_{\tau_{z},p}=\hbar\upsilon_{F}(\tau_{z}pk_{x}-ik_{y})/({E-\Delta})$,
$k_{x}>0$, and the index $p$ labels right- and left- movers in $x$ direction.
We focus now on the quasi-one-dimensional limit where the system forms a
quantum wire. Similarly to carbon-based materials, nanoribbon_KL ;
bilayer_MF_2012 this quantum regime can be achieved in two ways: either by
growing a nanoribbon of $\rm MoS_{2}$ with particular boundaries or by
electrostatically confining the electrons into a quantum wire by placing
metallic gates on a $\rm MoS_{2}$ monolayer flake (with unspecified
boundaries), see Fig. 1. In both cases we consider the armchair regime where
the direction of propagation is perpendicular to a lattice translation vector.
The details of the boundaries are not essential provided that they do not
suppress the transport. In addition, we assume for both cases hard-wall type
boundary conditions. A most characteristic feature of such armchair quantum
wires is that the two valleys [$\pm\mathbf{K}=(\pm 4\pi/3a,0)$], projected
onto the propagation direction along the $y$ axis, coincide at $k=0$. The
valleys easily hybridize (lifting their degeneracy), for instance, by
impurities, irregular boundaries or, in particular, by our hard-wall
boundaries. brey_2006 ; nanoribbon_KL
We determine now the spectrum for a quantum wire of width $W=Na$, where $a$ is
the lattice constant and $N$ the number of unit cells in the $x$ direction. To
find the quantization conditions on $k_{x}$, we virtually extend our quantum
wire by two sides, $W^{\prime}=(N+2)a$, and impose the boundary conditions at
these virtual sites on the total wavefunction
$\psi(x)=\sum_{\tau_{z},p}a_{\tau_{z},p}\psi_{\tau_{z},p}(x)\equiv\sum_{j}\psi_{j}(x)$,
where $\psi_{j}(x)$ is the probability amplitude to find the electron in one
of the orthogonal states $j=d_{z^{2}},d_{x^{2}-y^{2}},d_{xy}$. This gives us
$six$ conditions (three at each edge) for four fundamental solutions [see Eq.
(2)]. To solve this overconstrained boundary value problem we use mixed
boundary conditions. On the orbitals $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ we
impose Dirichlet boundary conditions,
$\psi_{d_{z^{2}},d_{x^{2}-y^{2}}}(x=0,W^{\prime})=0$, while on the orbital
$d_{xy}$ we impose von Neumann boundary conditions,
$\partial_{x}\psi_{d_{xy}}(x=0,W^{\prime})=0$.
These boundary conditions can be fulfilled only for those values of
$k_{x}=|\kappa_{m}|$ that satisfy $(K+\kappa_{m})W^{\prime}=\pi m$, where $m$
is an integer. If $W=(3M+1)a$, where $M$ is a positive integer, this leads to
$\kappa_{m}=\pi m/W^{\prime}$. We note that in this case all energy levels
except the lowest one in the conduction band and the highest one in the
valence band are two-fold degenerate. If $W=(3M+2)a$ [$W=3Ma$], this leads to
$\kappa_{m}=(\pi m+2\pi/3)/W^{\prime}$ [$\kappa_{m}=(\pi
m-2\pi/3)/W^{\prime}$]. In this case, the energy levels are non-degenerate.
The conduction band spectrum for small momenta is quadratic,
$E_{c,w}=\sqrt{(\hbar\upsilon_{F}\kappa_{m})^{2}+(\hbar\upsilon_{F}k_{y})^{2}+\Delta^{2}}\approx\Delta_{m}+\frac{(\hbar\upsilon_{F}k_{y})^{2}}{2\Delta_{m}},$
(3)
where we define the minimum energy for the $m$th subband as
$\Delta_{m}=\sqrt{(\hbar\upsilon_{F}\kappa_{m})^{2}+\Delta^{2}}$, see Fig. 2a.
We note that the slope of the spectrum branches at small momenta is decreasing
with increasing $m$, resulting in crossings of different subbands, see Fig.
2c. The corresponding wavefunction is given by
$\displaystyle\psi_{m}(x)=\psi_{1,p}(x)-\psi_{-1,-p}(x),$ (4)
where $p={\rm sgn}\kappa_{m}$. Again, we note that for $W=(3M+1)a$ the
subbands $m$ and $-m$ have the same energy.
To confirm our analytical results numerically, we develop a tight-binding
model for a honeycomb lattice composed of two kinds of atoms, $A$
(representing $d_{z^{2}}$ orbitals) and $B$ (representing
$d_{x^{2}-y^{2}}+i\tau_{z}d_{xy}$ orbitals). The effective Hamiltonian
$\bar{H}_{0}$ consists of an on-site energy term and a term describing hopping
between nearest neighbours, respectively,
$\bar{H}_{0}=\sum_{i}\varepsilon_{i}c_{i\mu}^{\dagger}c_{i\mu}+\sum_{<ij>}t_{ij}c_{i\mu}^{\dagger}c_{j\mu},$
(5)
where $c_{i\mu}^{\dagger}$ creates an electron with spin $\mu$ at site $i$.
The on-site energy $\varepsilon_{i}$ is equal to $\Delta$ ($-\Delta$) on $A$
($B$) sublattice. The hopping matrix element $t_{ij}$ is assumed to be
uniform, $t_{ij}\equiv t=2\hbar\upsilon_{F}/\sqrt{3}a$. The Hamiltonians
$H_{0}$ and $\bar{H}_{0}$ are equivalent for momenta close to $\pm\mathbf{K}$
and result in the same low energy spectrum, see Fig. 2a.
Intrinsic spin orbit interaction. The intrinsic spin orbit interaction in $\rm
MoS_{2}$ monolayer is much larger than in other monolayers, for example, in
graphene, arising from $d$ orbitals of the heavier atom. Symmetry arguments
confirmed by DFT calculations lead to the intrinsic SOI Hamiltonian of the
form Yao_2012
$H_{so}=\alpha\tau_{z}s_{z}(1-\sigma_{3}),$ (6)
where Pauli matrices $s_{i}$ act on spin space, and $\alpha=38\ {\rm meV}$ is
the SOI strength.
Rashba spin orbit interaction. Breaking of structure inversion symmetry by an
electric field $\mathbf{E}$ along the $z$ axis perpendicular to the monolayer
leads to a Rashba term of the form kane_mele $H_{R}=H_{Rx}+H_{Ry}$,
$H_{Rx}=-\alpha_{R}s_{x}\sigma_{2},\ \
H_{Ry}=\alpha_{R}\tau_{z}s_{y}\sigma_{1},$ (7)
where the Rashba SOI strength, in general, is proportional to the electric
field strength. Such an electric field could be produced by gates kane_mele
or by doping with adatoms. Franz_2012 ; Rashba_2012 For both cases,
$\alpha_{R}$ is best determined by ab initio calculations or experimentally.
An alternative way to generate Rashba SOI is to apply a spatially varying
magnetic field, Braunecker_Jap_Klin_2009 produced, for instance, by
nanomagnets. exp_field ; Flensberg_Rot_Field For example, a magnetic field
${\bf B}_{n}$ rotating in the plane of a quantum wire produces the Zeeman term
$H_{Z}^{\parallel}=\Delta_{Z}[s_{x}\cos(k_{n}y)+s_{y}\sin(k_{n}y)],$ (8)
where $\Delta_{Z}=g\mu_{B}B_{n}/2$, and the period of the rotating field is
$\lambda_{n}=2\pi/k_{n}$. The unitary spin-dependent transformation
$U_{n}=\exp(-ik_{n}ys_{z}/2)$ allows us to gauge away the coordinate dependent
term $H_{Z}^{\parallel}$ in the Hamiltonian
$H=H_{0}+H_{so}+H_{Z}^{\parallel}$. This results in
$H_{n}=U_{n}^{\dagger}HU_{n}$,
$\displaystyle H_{n}=H_{0}+H_{so}-\alpha_{Rn}s_{z}\sigma_{2}+\Delta_{Z}s_{x},$
(9)
where $\alpha_{Rn}=\hbar\upsilon_{F}k_{n}/2$, so the strength of the induced
Rashba SOI depends only on the rotation period $\lambda_{n}$ but not on the
magnetic field strength $B_{n}$. We note that the induced Rashba SOI described
by $H_{Rn}=-\alpha_{Rn}s_{z}\sigma_{2}$ reaches $\alpha_{Rn}\approx 11\ {\rm
meV}$ for the nanomagnets placed with a period $\lambda_{n}=100\ {\rm nm}$.
Here we note that for a quantum wire created by gates we can estimate that the
misalignment angle should be less than $a/\lambda_{n}$ (i.e. $\lesssim
1^{\circ}$). bilayer_MF_2012 If one works with a nanoribbon, then the
propagation in the armchair or zigzag direction should be favoured by the
growth process. nanoribbon_production ; nanoribbon_production_CNT We note
that also a Zeeman term $H_{Z}=\Delta_{Z}s_{x}$, which breaks the time-
reversal invariance of the system, inevitably arises.
To account for the Rashba SOI of Eq. (7) in the tight-binding model, we allow
for spin-flip hoppings, kane_mele ; nanoribbon_KL
$\bar{H}_{R}=\frac{3i\alpha_{R}}{4}\sum_{<ij>,\mu,\mu^{\prime}}c_{i\mu}^{\dagger}({\boldsymbol{e}}_{ij}\times{\boldsymbol{e}}_{z})\cdot{\mathbf{s}}_{\mu\mu^{\prime}}c_{j\mu^{\prime}},$
(10)
where ${\mathbf{s}}=(s_{x},s_{y},s_{z})$, and where the unit vectors
${\boldsymbol{e}}_{z}$ points along $z$ and ${\boldsymbol{e}}_{ij}$ along the
bond connecting sites $i$ and $j$. The intrinsics SOI, see Eq. (6), can be
modeled by
$\bar{H}_{so}=\frac{2i\alpha}{3\sqrt{3}}\sum_{\ll
ij\gg,\mu,\mu^{\prime}}\nu_{ij}c_{i\mu}^{\dagger}s_{z,\mu\mu^{\prime}}c_{j\mu^{\prime}},$
(11)
where the sum runs over the next-nearest neighbour sites belonging to the $B$
sublattice. The spin dependent amplitude $\nu_{ij}=-\nu_{ji}=\pm 1$ depends on
whether the electron takes a right or left turn by hopping from $i$ to $j$.
kane_mele These two terms $\bar{H}_{so}$ and $\bar{H}_{R}$ are constructed in
such a way that they are equivalent to $H_{so}$ and ${H}_{R}$ in the low-
energy sector. We note that by taking only part of $\bar{H}_{R}$ and changing
$s_{x}$ to $s_{z}$, we can model $H_{Rn}$. The Zeeman term $H_{z}$ is given by
$\bar{H}_{Z}=\Delta_{Z}\sum_{i,\mu,\mu^{\prime}}c_{i\mu}^{\dagger}{s}_{x,\mu\mu^{\prime}}c_{i\mu^{\prime}}.$
(12)
Spectrum with SOI. A part of the Rashba SOI, $H_{Rx}$ ($H_{Rn}$), can be
easily included in $H_{0}$. The spin $s_{x}$ ($s_{z}$) is a good quantum
number for the Hamiltonian $H_{0}+H_{Rx}$ ($H_{0}+H_{Rn}$). In this case, the
SOI only results in the spin-dependent shift of the momentum, $k_{y}\to
k_{y}-s_{x}\alpha_{R}/\hbar\upsilon_{F}$ ($k_{y}\to
k_{y}-s_{z}\alpha_{Rn}/\hbar\upsilon_{F}$), see Fig. 2b. Next, we treat the
remaining SOI terms, $H^{\prime}=H_{so}+H_{Ry}$ ($H_{so}$), as a perturbation.
We note that $H^{\prime}$ ($H_{so}$) is proportional to $\tau_{z}$. At the
same time, the wavefunctions $\psi_{m}(x)$ [see Eq. (4)] are eigenstates of
the Pauli matrix $\tau_{1}$, so the intrasubband matrix elements vanish,
footnote_1 which is consistent with Kramers degeneracy at $k_{y}=0$ for a
time-reversal invariant Hamiltonian. The intersubband matrix elements
$t_{mm^{\prime}}$, however, are non-zero, but they contain a strong
suppression factor arising from the sublattice degree of freedom as follows.
At small momenta, the mass term $\Delta\sigma_{3}$ dominates in the
Hamiltonian, so the wavefunctions $\psi_{m}(x)$ are close to the eigenstate of
$\sigma_{3}$. As a result, the sublattice terms in $H^{\prime}$ ($H_{so}$),
$1-\sigma_{3}$ and $\sigma_{1}$, lead to a suppression of SOI effects, where
the intrinsic SOI is suppressed by a factor $(E-\Delta)/\Delta\ll 1$ and the
Rashba SOI by a factor $\sqrt{(E-\Delta)/\Delta}$. Thus, the corrections to
the spectrum are small in the parameter
$t_{mm^{\prime}}/\omega_{mm^{\prime}}$, where $\omega_{mm^{\prime}}$ denotes
the subband splitting at given $k_{y}$. However, these terms lead to an
anticrossing between two different subbands with opposite spin (with the same
spin) along $x$ (along $z$), see Fig. 2c. We note here that in spite of having
strong SOI, the spin degeneracy is not lifted in case of quantum wires.
Helical modes via electric field. The Rashba SOI induced by an electric field
offers the possibility to generate helical modes in a time-reversal invariant
system. As shown above, $H^{\prime}$ results in subband anticrossings, see
Fig. 2c. Sufficiently far away from them,
$t_{mm^{\prime}}\ll\omega_{mm^{\prime}}$, the subbands are spin-polarized by
the Rashba SOI $H_{Rx}$ in the $x$ direction, see Fig. 3. However, passing
through the anticrossing the spin polarization goes through zero and changes
sign. All this suggests that if the Fermi level is tuned close to the
anticrossing (see Fig. 2c) in such a way that there are four propagating modes
(two left and two right), the system is in a quasi-helical regime. The lowest
subband $n=1$ is almost fully spin-polarized and transports opposite spins
into opposite directions, whereas the next subband $n=2$ is only partially
polarized. This means that scattering due to impurities between subbands is
allowed and helical modes are not protected from backscattering.
Figure 3: The spin polarization $\left<s_{x}\right>$ along $x$ direction as
function of momentum $\tilde{k}_{y}$ for the $n$th level defined in Fig. 2c.
The spin projections onto the $y$ and $z$ directions vanish. Away from the
anticrossings the spin is almost perfectly aligned along $x$. If the chemical
potential $\mu$, defining the Fermi momentum $\tilde{k}_{F}^{(n)}$ for the
$n$th subband, is tuned close to the anticrossing (see Fig. 2c), the total
polarization $\left<s_{x}\right>$ of a left (right) propagating electron is
non-zero.
Helical modes via magnetic field. If a Rashba SOI (along $x$) is generated by
a spatially varying magnetic field, the time-reversal invariance of the system
is broken, giving rise to a Zeeman term $H_{Z}$. The corresponding magnetic
field, pointing along $z$, is perpendicular to the spin quantization axis
determined by the Rashba SOI. Thus, the spin degeneracy at $k_{y}=0$ gets
lifted, and a gap of size $2\Delta_{Z}$ is opened, see Fig. 4. The spin
polarization along $z$ is given by
$\left<s_{z}\right>=\frac{\omega_{\downarrow\uparrow}}{\sqrt{\omega_{\downarrow\uparrow}+4\Delta_{Z}^{2}}},$
(13)
where $\omega_{\uparrow\downarrow}$ is the energy difference between spin up
and spin down states at given momentum $k_{y}$ for the unperturbed problem
$H_{0}+H_{Rn}+H_{so}$. footnote2 If the chemical potential $\mu$ is tuned
inside the gap, there is one mode propagating to the left and one to the
right. Moreover, these two modes carry opposite spins with almost perfect
polarization, $\left|\left<s_{z}\right>\right|\approx 1$, provided that
$\Delta_{Z}\ll 16\alpha_{Rn}^{2}/\Delta$. Thus, the system is in a helical
regime.
Figure 4: (a) The two lowest energy levels of $H_{0}+H_{Rn}+H_{so}+H_{Z}$,
cf. Fig. 2. The Zeeman term lifts the Kramers degeneracy at $\tilde{k}_{y}=0$
and opens a gap $2\Delta_{Z}$. If $\mu$ is tuned inside the gap, the system is
in a helical regime with a left (right) propagating mode with spin down (up).
(b) The spin polarization $\left<s_{z}\right>$ as function of the momentum
$\tilde{k}_{y}$ for the lowest level. The parameters are chosen as $W=50a$,
$\alpha_{Rn}=10\ {\rm meV}$, and $\Delta_{Z}=0.05\ {\rm meV}$.
Majorana fermions. Helical modes as in Fig. 4 have attracted considerable
attention in various candidate systems not only as a platform for spin-filters
streda but also as a platform for generating MFs. alicea_review_2012 MFs are
particles that are their own antiparticles. When the quantum wire is brought
into tunnel contact with an $s$-wave superconductor inducing a proximity gap
$\Delta_{sc}$, states with opposite spins and momenta are coupled giving rise
to an effective $p$-wave pairing. In the topological phase, MFs emerge as
midgap boundstates, one localized at each end of the quantum wire. This phase
emerges if $\Delta_{Z}^{2}>\Delta_{sc}^{2}+{\mu}^{2}$ is satisfied, where
$\mu$ is now counted from the middle of the gap. Since the derivation is
similar to previously studied cases, MF_wavefunction_klinovaja_2012 ;
MF_CNT_2012 ; nanoribbon_KL ; Two_field_Klinovaja we defer the details to
App. A. The $\rm MoS_{2}$ monolayer quantum wires offer the unique possibility
to probe MFs not only by transport but also by optical spectroscopy.
Quantum dots. In contrast to gapless graphene, Guido_Nature ; Guido_dots
quantum dots Spin_qubits ; kloeffel_prospects_2013 in $\rm MoS_{2}$ can be
created by gates. Lieven_exp ; Amir_exp We note that the confining potential
should be sharp enough to lift the valley degeneracy, as shown above for the
quantum wires, and, in addition, to insure a non-equidistant spectrum, which
is more suitable for optical experiments. If vanishing boundary conditions are
imposed also along $y$, the momentum $k_{y}$ is quantized, $\kappa_{yn}=\pi
n/(W_{y}+2a/\sqrt{3})$, where $W_{y}$ is width in the $y$ direction, and $n$
is a positive integer. The dot spectrum then becomes
$E_{n,m}=\sqrt{(\hbar\upsilon_{F})^{2}(\kappa_{m}^{2}+\kappa_{yn}^{2})+\Delta^{2}}$.
Each level is spin degenerate and the intrinsic SOI can neither lift this
Kramers degeneracy, nor, due to its symmetry, change substantially splittings
between levels [in the small parameter $\alpha(E-\Delta)/\Delta\ll\alpha$, see
above]. The spin degeneracy can be lifted with a magnetic field, say, along
$z$. Similar to nanotubes, klinovaja_cnt EDSR can then be achieved by
applying an oscillatory electric field $E$ also along $z$, causing
$\alpha_{R}$ to oscillate and thereby inducing spin rotations at a Rabi
frequency $\sim|\alpha_{R}|$. Thus, we conclude that quantum dots in $\rm
MoS_{2}$ host well-defined Kramers doublets that can serve as platform for
spin qubits. Spin_qubits ; kloeffel_prospects_2013
We acknowledge stimulating discussions with Parisa Fallahi, Richard Warburton,
and Dominik Zumbuhl. This work is supported by the Swiss NSF, NCCR
Nanoscience, and NCCR QSIT.
## Appendix A Majorana Fermions
We give here more details of the derivation of the Majorana fermions (MFs)
introduced in the main text. Thereby we closely follow the derivation given in
Ref. MF_wavefunction_klinovaja_2012, which requires a few minor modifications
for the present case. If the chemical potential $\mu$ is tuned inside the gap
$2\Delta_{Z}$ opened by the magnetic field ${\bf B}_{n}$ at $k_{y}=0$, the two
propagating modes are helical, see Fig. 4. The same helical states can be
obtained by a Rashba SOI induced by an electric field in the presence of a
uniform magnetic field giving rise to a Zeeman splitting $2\Delta_{Z}$. If
such a quantum wire is brought into tunnel contact with an $s$-wave
superconductor, a superconducting proximity gap $\Delta_{sc}$ is induced in
the wire. Through the pairing mechanism coupling Kramers partners, the helical
states get paired into a $p$-wave-like superconducting state. Sato ;
lutchyn_majorana_wire_2010 ; oreg_majorana_wire_2010 ; alicea_review_2012
There are no propagating modes inside the gap but there could exist
boundstates localized at the ends of the wire. If a certain topological
criterion is satisfied, these states are MFs, particles that are their own
antiparticles. To find this criterion, we describe the system by an effective
linearized model for the exterior ($\chi=e$, states with momenta close to the
Fermi momentum, $k_{e}=k_{F}$) and the interior branches ($\chi=i$, states
with momenta close to $k_{i}=0$). MF_wavefunction_klinovaja_2012 The electron
operator is represented as $\Psi(y)=\sum_{\rho={\pm 1},\chi=e,i}e^{i\rho
k_{\chi}y}\Psi_{\rho\chi}$, where the sum runs over the right ($R$, $\rho=1$)
and left ($L$, $\rho~{}=~{}-1$) movers and $\Psi_{\rho\chi}$ is an
annihilation operator for the $(\rho,\chi)$ branch of the spectrum. The
effective Hamiltonian becomes
$\displaystyle
H=-i\hbar\upsilon\rho_{3}\chi_{3}\partial_{y}+\Delta_{Z}\eta_{3}\rho_{1}(1+\chi_{3})/2$
$\displaystyle\hskip
20.0pt+{\Delta}_{sc}\eta_{2}\rho_{2}(1+\chi_{3})/2+{\bar{\Delta}}_{sc}\eta_{2}\rho_{2}(1-\chi_{3})/2.$
(14)
in the basis
$\widetilde{\Psi}=(\Psi_{Re},\Psi_{Le},\Psi_{Re}^{\dagger},\Psi_{Le}^{\dagger},\Psi_{Li},\Psi_{Ri},\Psi^{\dagger}_{Li},\Psi^{\dagger}_{Ri}),$
where the Pauli matrices $\chi_{i}$ ($\eta_{i}$) act in the interior-exterior
branch (electron-hole) space.
Here, $\Delta_{Z}=g\mu_{B}B_{n}/2$ is the Zeeman energy, and
$\upsilon=(\partial E/\partial\hbar k_{y})|_{k_{y}=k_{F}}$ is the velocity at
the Fermi level. In the limit of strong Rashba SOI
($\alpha_{Rn}\gg\Delta_{Z},\Delta_{sc}$), the strength of the effective
proximity induced superconductivity acting on the exterior branches
$\bar{\Delta}_{sc}$ due to the nearly perfect spin polarization at the Fermi
wavevector $k_{F}$ is equal to $\Delta_{sc}$. In the opposite limit of weak
Rashba SOI ($\alpha_{Rn}\ll\Delta_{Z}$), $\bar{\Delta}_{sc}$ is getting
suppressed by the magnetic field, $\bar{\Delta}_{sc}=\Delta_{sc}k_{n}/k_{F}$,
where $k_{n}=2\pi/\lambda_{n}$ ($k_{n}=2\alpha_{R}/\hbar\upsilon_{F}$) for
Rashba SOI induced by rotating magnetic fields (by electric fields). Note that
the Fermi wavevector $k_{F}$ grows with magnetic field as
$k_{F}\propto\sqrt{\Delta_{Z}}$.
All this together leads us to the criterion for the topological phase given by
$\displaystyle\Delta_{Z}>\sqrt{\Delta_{sc}^{2}+\mu^{2}},$ (15)
where the chemical potential $\mu$ is now calculated from the middle of gap
$2\Delta_{Z}$.
Similarly, following Refs. MF_wavefunction_klinovaja_2012, ;
Two_field_Klinovaja, , we can also obtain the localization length of the MFs.
For example, in the strong SOI regime and for $\mu=0$, the wavefunction of the
left localized MF is written in the basis
${\bar{\Psi}}=({\Psi}_{\uparrow},{\Psi}_{\downarrow},{\Psi}_{\uparrow}^{\dagger},{\Psi}_{\downarrow}^{\dagger})$
as
$\displaystyle\varPhi_{M}(y)=\begin{pmatrix}i\\\ 1\\\ -i\\\
1\end{pmatrix}e^{-k_{-}^{(i)}y}-\begin{pmatrix}i\ e^{ik_{F}y}\\\
e^{-ik_{F}y}\\\ -i\ e^{-ik_{F}y}\\\ e^{ik_{F}y}\end{pmatrix}e^{-k^{(e)}y},$
(16)
where $k_{-}^{(i)}=(\Delta_{Z}-\Delta_{sc})/\hbar\upsilon$ and
$k^{(e)}=\Delta_{sc}/\hbar\upsilon$ MF_wavefunction_klinovaja_2012 . The
localization length is determined by the smallest gap in the system $\xi={\rm
max}\\{1/k_{-}^{(i)},1/k^{(e)}\\}$. Here,
${\Psi}_{\uparrow(\downarrow)}^{\dagger}$ is a creation operator of the
electron with spin up (down), where the spin quantization axis is determined
by the Rashba SOI. In the weak SOI regime deeply in the topological phase, the
left localized MF wavefunction is written as
$\varPhi_{M}(y)=\begin{pmatrix}e^{-i\pi/4}\\\ ie^{i\pi/4}\\\ e^{i\pi/4}\\\
-ie^{-i\pi/4}\end{pmatrix}\sin(k_{F}y)e^{-\bar{k}^{(e)}y},$ (17)
where $\bar{k}^{(e)}=\bar{\Delta}_{sc}/\hbar\upsilon$
MF_wavefunction_klinovaja_2012 . Note that both solutions explicitly satisfy
the MF condition of being self-conjugate,
$\varPhi_{M}\cdot{\bar{\Psi}}=[\varPhi_{M}\cdot{\bar{\Psi}}]^{\dagger}$.
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|
arxiv-papers
| 2013-04-16T18:17:50 |
2024-09-04T02:49:44.496235
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jelena Klinovaja and Daniel Loss",
"submitter": "Jelena Klinovaja",
"url": "https://arxiv.org/abs/1304.4542"
}
|
1304.4586
|
# The Fokas method to the Sasa-Satsuma equation on the half-line
Jian Xu School of Mathematical Sciences
Fudan University
Shanghai 200433
People’s Republic of China [email protected] and Engui Fan School of
Mathematical Sciences, Institute of Mathematics and Key Laboratory of
Mathematics for Nonlinear Science
Fudan University
Shanghai 200433
People’s Republic of China correspondence author:[email protected]
###### Abstract.
We present a Riemann-Hilbert problem formalism for the initial-boundary value
problem for the Sasa-Satsuma(SS) equation: on the half-line. And we also
analysis the global relation in this paper.
###### Key words and phrases:
Riemann-Hilbert problem, Sasa-Satsuma equation, Initial-boundary value problem
## 1\. Introduction
Several of the most important PDEs in mathematics and physics are integrable.
Integrable PDEs can be analyzed by means of the Inverse Scattering Transform
(IST) formalism. Until the 1990s the IST methodology was pursued almost
entirely for pure initial value problems. However, in many laboratory and
field situations, the wave motion is initiated by what corresponds to the
imposition of boundary conditions rather than initial conditions. This
naturally leads to the formulation of an initial-boundary value (IBV) problem
instead of a pure initial value problem.
In 1997, Fokas announced a new unified approach for the analysis of IBV
problems for linear and nonlinear integrable PDEs [1, 2](see also [3]). The
Fokas method provides a generalization of the IST formalism from initial value
to IBV problems, and over the last fifteen years, this method has been used to
analyze boundary value problems for several of the most important integrable
equations with $2\times 2$ Lax pairs, such as the Korteweg de Vries, the
nonlinear Schrö dinger, the sine-Gordon, and the stationary axisymmetric
Einstein equations, see e.g. [4, 9]. Just like the IST on the line, the
unified method yields an expression for the solution of an IBV problem in
terms of the solution of a Riemann-Hilbert problem. In particular, the
asymptotic behavior of the solution can be analyzed in an effective way by
using this Riemann-Hilbert problem and by employing the nonlinear version of
the steepest descent method introduced by Deift and Zhou [15].
It is well known that the nonlinear Schrödinger(NLS) equation
$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q=0$ (1.1)
describes slowly varying wave envelopes in dispersive media and arises in
various physical systems such as water waves, plasma physics, solid-state
physics and nonlinear optics. One of the most successful among them is the
description of optical solitons in fibers. But, by the advancement of
experomenal accuracy, several phenomena which can not be explained by equation
(1.1) have been observed. In order to understand such phenomena, Kodama and
Hasegawa proposed a higer-order nonlinear Schrödinger equation
$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q+i\varepsilon\\{\beta_{1}q_{xxx}+\beta_{2}|q|^{2}q_{X}+\beta_{3}q(|q|^{2})_{X}\\}=0.$
(1.2)
In general, equation (1.2) may not be completely integrable. However, if some
restrictions are imposed on the real parameters $\beta_{1},\beta_{2}$ and
$\beta_{3}$, then we can apply the IST to solve its initial value problems.
Until now, the following four cases besides the NLS equation itself are konwn
to be solvable:
* •
the derivative NLS equation-type i@($\beta_{1}:\beta_{2}:\beta_{3}$=0:1:1),
* •
the derivative NLS equation-type ii@($\beta_{1}:\beta_{2}:\beta_{3}$=0:1:0),
* •
the Hirota equation($\beta_{1}:\beta_{2}:\beta_{3}$=1:6:0),
* •
the Sasa-Satsuma equation($\beta_{1}:\beta_{2}:\beta_{3}$=1:6:3).
$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q+i\varepsilon(q_{XXX}+6|q|^{2}q_{X}+3q(|q|^{2})_{X})=0$
(1.3)
Recently, Lenells develop a methodology for analyzing IBV problems for
integrable evolution equations with Lax pairs involving $3\times 3$ matrices
[12]. He also used this method to analyze the Degasperis-Procesi equation in
[13]. In this paper we analyze the initial-boundary value problem of the Sasa-
Satsuma equation on the half-line by using this method. The IST formalism for
the initial value problem of the Sasa-Satsuma equation has been obtained in
[10].
According to [10] we introduce variable transformations,
$u(x,t)=q(X,T)\exp\\{\frac{-i}{6\varepsilon}(X-\frac{T}{18\varepsilon})\\},$
(1.4a) $t=T,$ (1.4b) $x=X-\frac{T}{12\varepsilon}.$ (1.4c)
Then equation (1.2) is reduce to a complex modified KdV-type equation
$u_{t}+\varepsilon\\{u_{xxx}+6|u|^{2}u_{x}+3u(|u|^{2})_{x}\\}=0.$ (1.5)
Organization of the paper.In section 2 we perform the spectral analysis of the
associated Lax pair. And we formulate the main Riemann-Hilbert problem in
section 3. We also analysis the global relation in section 4.
## 2\. Spectral analysis
The Lax pair of equation (1.5) is [10],
$\Psi_{x}=U\Psi,\quad\Psi=\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\\\
\Psi_{3}\end{array}\right).$ (2.1a) $\Psi_{t}=V\Psi.$ (2.1b)
where
$U=-ik\Lambda+V_{1}.$ (2.2)
and
$V=-4i\varepsilon k^{3}\Lambda+V_{2}$ (2.3)
here
$\Lambda=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right),V_{1}=\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\
-\bar{u}&-u&0\end{array}\right),V_{2}=k^{2}V_{2}^{(2)}+kV_{2}^{(1)}+V_{2}^{(0)}.$
(2.4)
where
$\begin{array}[]{l}V_{2}^{(2)}=4\varepsilon\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\ -\bar{u}&-u&0\end{array}\right),\\\
V_{2}^{(1)}=2i\varepsilon\left(\begin{array}[]{ccc}|u|^{2}&u^{2}&u_{x}\\\
\bar{u}^{2}&|u|^{2}&\bar{u}_{x}\\\
\bar{u}_{x}&u_{x}&-2|u|^{2}\end{array}\right),\\\
V_{2}^{(0)}=-4|u|^{2}\varepsilon\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\
-\bar{u}&-u&0\end{array}\right)-\varepsilon\left(\begin{array}[]{ccc}0&0&u_{xx}\\\
0&0&\bar{u}_{xx}\\\
-\bar{u}_{xx}&-u_{xx}&0\end{array}\right)+\varepsilon(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right)\end{array}$ (2.5)
In the following, we let $\varepsilon=1$ for the convenient of the analysis.
### 2.1. The closed one-form
Suppose that $u(x,t)$ is sufficiently smooth function of $(x,t)$ in the half-
line domain $\Omega=\\{0<x<\infty,0<t<T\\}$ which decay as
$x\rightarrow\infty$. Introducing a new eigenfunction $\mu(x,t,k)$ by
$\Psi=\mu e^{-i\Lambda kx-4i\Lambda k^{3}t}$ (2.6)
then we find the Lax pair equations
$\left\\{\begin{array}[]{l}\mu_{x}+[ik\Lambda,\mu]=V_{1}\mu,\\\
\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.\end{array}\right.$ (2.7)
the equations in (A.2) can be written in differential form as
$d(e^{(ikx+4ik^{3}t)\hat{\Lambda}}\mu)=W,$ (2.8)
where $W(x,t,k)$ is the closed one-form defined by
$W=e^{(ikx+4ik^{3}t)\hat{\Lambda}}(V_{1}dx+V_{2}dt)\mu.$ (2.9)
### 2.2. The $\mu_{j}$’s
We define three eigenfunctions $\\{\mu_{j}\\}_{1}^{3}$ of (A.2) by the
Volterra integral equations
$\mu_{j}(x,t,k)=\mathbb{I}+\int_{\gamma_{j}}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}W_{j}(x^{\prime},t^{\prime},k).\qquad
j=1,2,3.$ (2.10)
where $W_{j}$ is given by (2.9) with $\mu$ replaced with $\mu_{j}$, and the
contours $\\{\gamma_{j}\\}_{1}^{3}$ are showed in Figure 1.
Figure 1. The three contours $\gamma_{1},\gamma_{2}$ and $\gamma_{3}$ in the
$(x,t)-$domain.
The first, second and third column of the matrix equation (2.10) involves the
exponentials
$\begin{array}[]{ll}\mbox{$[\mu_{j}]_{1}$:}&e^{2ik(x-x^{\prime})+8ik^{3}(t-t^{\prime})},\\\
\mbox{$[\mu_{j}]_{2}$:}&e^{2ik(x-x^{\prime})+8ik^{3}(t-t^{\prime})},\\\
\mbox{$[\mu_{j}]_{3}$:}&e^{-2ik(x-x^{\prime})-8ik^{3}(t-t^{\prime})},e^{-2ik(x-x^{\prime})-8ik^{3}(t-t^{\prime})}.\end{array}$
(2.11)
And we have the following inequalities on the contours:
$\begin{array}[]{ll}\gamma_{1}:&x-x^{\prime}\geq 0,t-t^{\prime}\leq 0,\\\
\gamma_{2}:&x-x^{\prime}\geq 0,t-t^{\prime}\geq 0,\\\
\gamma_{3}:&x-x^{\prime}\leq 0.\end{array}$ (2.12)
So, these inequalities imply that the functions $\\{\mu_{j}\\}_{1}^{3}$ are
bounded and analytic for $k\in{\mathbb{C}}$ such that $k$ belongs to
$\begin{array}[]{ll}\mu_{1}:&(D_{2},D_{2},D_{3}),\\\
\mu_{2}:&(D_{1},D_{1},D_{4}),\\\ \mu_{3}:&(D_{3}\cup D_{4},D_{3}\cup
D_{4},D_{1}\cup D_{2}).\end{array}$ (2.13)
where $\\{D_{n}\\}_{1}^{4}$ denote four open, pairwisely disjoint subsets of
the Riemann $k-$sphere showed in Figure 2.
Figure 2. The sets $D_{n}$, $n=1,\ldots,4$, which decompose the complex
$k-$plane.
And the sets $\\{D_{n}\\}_{1}^{4}$ has the following properties:
$\begin{array}[]{l}D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}>\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}>\mathrm{Re}{z_{3}}\\},\\\
D_{2}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}>\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}<\mathrm{Re}{z_{3}}\\},\\\
D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}<\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}>\mathrm{Re}{z_{3}}\\},\\\
D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}<\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}<\mathrm{Re}{z_{3}}\\},\\\
\end{array}$
where $l_{i}(k)$ and $z_{i}(k)$ are the diagonal entries of matrices
$-ik\Lambda$ and $-4ik^{3}\Lambda$, respectively.
In fact, for $x=0$, $\mu_{1}(0,t,k)$ has enlarged domain of boundedness:
$(D_{2}\cup D_{4},D_{2}\cup D_{4},D_{1}\cup D_{3})$, and $\mu_{2}(0,t,k)$ has
enlarged domain of boundedness: $(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup
D_{4})$.
### 2.3. The $M_{n}$’s
For each $n=1,\ldots,4$, define a solution $M_{n}(x,t,k)$ of (A.2) by the
following system of integral equations:
$(M_{n})_{ij}(x,t,k)=\delta_{ij}+\int_{\gamma_{ij}^{n}}(e^{(-ikx-4ik^{3}t)\hat{\Lambda}}W_{n}(x^{\prime},t^{\prime},k))_{ij},\quad
k\in D_{n},\quad i,j=1,2,3.$ (2.14)
where $W_{n}$ is given by (2.9) with $\mu$ replaced with $M_{n}$, and the
contours $\gamma_{ij}^{n}$, $n=1,\ldots,4$, $i,j=1,2,3$ are defined by
$\gamma_{ij}^{n}=\left\\{\begin{array}[]{lclcl}\gamma_{1}&if&\mathrm{Re}l_{i}(k)<\mathrm{Re}l_{j}(k)&and&\mathrm{Re}z_{i}(k)\geq\mathrm{Re}z_{j}(k),\\\
\gamma_{2}&if&\mathrm{Re}l_{i}(k)<\mathrm{Re}l_{j}(k)&and&\mathrm{Re}z_{i}(k)<\mathrm{Re}z_{j}(k),\\\
\gamma_{3}&if&\mathrm{Re}l_{i}(k)\geq\mathrm{Re}l_{j}(k)&&.\\\
\end{array}\right.\quad\mbox{for }\quad k\in D_{n}.$ (2.15)
The following proposition ascertains that the $M_{n}$’s defined in this way
have the properties required for the formulation of a Riemann-Hilbert problem.
###### Proposition 2.1.
For each $n=1,\ldots,4$, the function $M_{n}(x,t,k)$ is well-defined by
equation (2.14) for $k\in\bar{D}_{n}$ and $(x,t)\in\Omega$. For any fixed
point $(x,t)$, $M_{n}$ is bounded and analytic as a function of $k\in D_{n}$
away from a possible discrete set of singularities $\\{k_{j}\\}$ at which the
Fredholm determinant vanishes. Moreover, $M_{n}$ admits a bounded and contious
extension to $\bar{D}_{n}$ and
$M_{n}(x,t,k)=\mathbb{I}+O(\frac{1}{k}),\qquad k\rightarrow\infty,\quad k\in
D_{n}.$ (2.16)
###### Proof.
The bounedness and analyticity properties are established in appendix B in
[12]. And substituting the expansion
$M=M_{0}+\frac{M^{(1)}}{k}+\frac{M^{(2)}}{k^{2}}+\cdots,\qquad
k\rightarrow\infty.$
into the Lax pair (A.2) and comparing the terms of the same order of $k$ yield
the equation (2.16). ∎
### 2.4. The jump matrices
We define spectral functions $S_{n}(k)$, $n=1,\ldots,4$, and
$S_{n}(k)=M_{n}(0,0,k),\qquad k\in D_{n},\quad n=1,\ldots,4.$ (2.17)
Let $M$ denote the sectionally analytic function on the Riemann $k-$sphere
which equals $M_{n}$ for $k\in D_{n}$. Then $M$ satisfies the jump conditions
$M_{n}=M_{m}J_{m,n},\qquad k\in\bar{D}_{n}\cap\bar{D}_{m},\qquad
n,m=1,\ldots,4,\quad n\neq m,$ (2.18)
where the jump matrices $J_{m,n}(x,t,k)$ are defined by
$J_{m,n}=e^{(-ikx-4ik^{3}t)\hat{\Lambda}}(S_{m}^{-1}S_{n}).$ (2.19)
According to the definition of the $\gamma^{n}$, we find that
$\begin{array}[]{ll}\gamma^{1}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{2}&\gamma_{2}&\gamma_{3}\end{array}\right)&\gamma^{2}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{1}&\gamma_{1}&\gamma_{3}\end{array}\right)\\\
\gamma^{3}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\end{array}\right)&\gamma^{4}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{2}\\\
\gamma_{3}&\gamma_{3}&\gamma_{2}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\end{array}\right).\end{array}$ (2.20)
### 2.5. The adjugated eigenfunctions
We will also need the analyticity and boundedness properties of the minors of
the matrices $\\{\mu_{j}(x,t,k)\\}_{1}^{3}$. We recall that the adjugate
matrix $X^{A}$ of a $3\times 3$ matrix $X$ is defined by
$X^{A}=\left(\begin{array}[]{ccc}m_{11}(X)&-m_{12}(X)&m_{13}(X)\\\
-m_{21}(X)&m_{22}(X)&-m_{23}(X)\\\
m_{31}(X)&-m_{32}(X)&m_{33}(X)\end{array}\right),$
where $m_{ij}(X)$ denote the $(ij)$th minor of $X$.
It follows from (A.2) that the adjugated eigenfunction $\mu^{A}$ satisfies the
Lax pair
$\left\\{\begin{array}[]{l}\mu_{x}^{A}-[ik\Lambda,\mu^{A}]=-V_{1}^{T}\mu^{A},\\\
\mu_{t}^{A}-[4ik^{3}\Lambda,\mu^{A}]=-V_{2}^{T}\mu^{A}.\end{array}\right.$
(2.21)
where $V^{T}$ denote the transform of a matrix $V$. Thus, the eigenfunctions
$\\{\mu_{j}^{A}\\}_{1}^{3}$ are solutions of the integral equations
$\mu_{j}^{A}(x,t,k)=\mathbb{I}-\int_{\gamma_{j}}e^{ik(x-x^{\prime})+4ik^{3}(t-t^{\prime})\hat{\Lambda}}(V_{1}^{T}dx+V_{2}^{T})\mu^{A},\quad
j=1,2,3.$ (2.22)
Then we can get the following analyticity and boundedness properties:
$\begin{array}[]{ll}\mu_{1}^{A}:&(D_{3},D_{3},D_{2}),\\\
\mu_{2}^{A}:&(D_{4},D_{4},D_{1}),\\\ \mu_{3}^{A}:&(D_{1}\cup D_{2},D_{1}\cup
D_{2},D_{3}\cup D_{4}).\end{array}$ (2.23)
In fact, for $x=0$, $\mu_{1}^{A}(0,t,k)$ has enlarged domain of boundedness:
$(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup D_{4})$, and $\mu_{2}^{A}(0,t,k)$
has enlarged domain of boundedness: $(D_{2}\cup D_{4},D_{2}\cup
D_{4},D_{1}\cup D_{3})$.
### 2.6. The $J_{m,n}$’s computation
Let us define the $3\times 3-$matrix value spectral functions $s(k)$ and
$S(k)$ by
$\mu_{3}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}s(k),$ (2.24a)
$\mu_{1}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S(k),$ (2.24b)
Thus,
$s(k)=\mu_{3}(0,0,k),\qquad S(k)=\mu_{1}(0,0,k).$ (2.25)
And we deduce from the properties of $\mu_{j}$ and $\mu_{j}^{A}$ that $s(k)$
and $S(k)$ have the following boundedness properties:
$\begin{array}[]{ll}s(k):&(D_{3}\cup D_{4},D_{3}\cup D_{4},D_{1}\cup
D_{2}),\\\ S(k):&(D_{2}\cup D_{4},D_{2}\cup D_{4},D_{1}\cup D_{3}),\\\
s^{A}(k):&(D_{1}\cup D_{2},D_{1}\cup D_{2},D_{3}\cup D_{4}),\\\
S^{A}(k):&(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup D_{4}).\end{array}$
Moreover,
$M_{n}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{n}(k),\quad
k\in D_{n}.$ (2.26)
###### Proposition 2.2.
The $S_{n}$ can be expressed in terms of the entries of $s(k)$ and $S(k)$ as
follows:
$\begin{array}[]{l}S_{1}=\left(\begin{array}[]{ccc}\frac{m_{22}(s)}{s_{33}}&\frac{m_{21}(s)}{s_{33}}&s_{13}\\\
\frac{m_{12}(s)}{s_{33}}&\frac{m_{11}(s)}{s_{33}}&s_{23}\\\
0&0&s_{33}\end{array}\right),\\\
S_{2}=\left(\begin{array}[]{ccc}\frac{m_{22}(s)m_{33}(S)-m_{32}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{21}(s)m_{33}(S)-m_{31}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}&s_{13}\\\
\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{11}(s)m_{33}(S)-m_{31}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&s_{23}\\\
\frac{m_{12}(s)m_{23}(S)-m_{22}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{11}(s)m_{23}(S)-m_{21}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&s_{33}\end{array}\right),\\\
\end{array}$ (2.27a)
$\begin{array}[]{ll}S_{3}=\left(\begin{array}[]{ccc}s_{11}&s_{12}&\frac{S_{13}}{(S^{T}s^{A})_{33}}\\\
s_{21}&s_{22}&\frac{S_{23}}{(S^{T}s^{A})_{33}}\\\
s_{31}&s_{32}&\frac{S_{33}}{(S^{T}s^{A})_{33}}\end{array}\right),&S_{4}=\left(\begin{array}[]{ccc}s_{11}&s_{12}&0\\\
s_{21}&s_{22}&0\\\
s_{31}&s_{32}&\frac{1}{m_{33}(s)}\end{array}\right).\end{array}$ (2.27b)
###### Proof.
Let $\gamma_{3}^{X_{0}}$ denote the contour $(X_{0},0)\rightarrow(x,t)$ in the
$(x,t)-$plane, here $X_{0}>0$ is a constant. We introduce
$\mu_{3}(x,t,k;X_{0})$ as the solution of (2.10) with $j=3$ and with the
contour $\gamma_{3}$ replaced by $\gamma_{3}^{X_{0}}$. Similarly, we define
$M_{n}(x,t,k;X_{0})$ as the solution of (2.14) with $\gamma_{3}$ replaced by
$\gamma_{3}^{X_{0}}$. We will first derive expression for
$S_{n}(k;X_{0})=M_{n}(0,0,k;X_{0})$ in terms of $S(k)$ and
$s(k;X_{0})=\mu_{3}(0,0,k;X_{0})$. Then (2.27) will follow by taking the limit
$X_{0}\rightarrow\infty$.
First, We have the following relations:
$\left\\{\begin{array}[]{l}M_{n}(x,t,k;X_{0})=\mu_{1}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}R_{n}(k;X_{0}),\\\
M_{n}(x,t,k;X_{0})=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{n}(k;X_{0}),\\\
M_{n}(x,t,k;X_{0})=\mu_{3}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}T_{n}(k;X_{0}).\end{array}\right.$
(2.28)
Then we get $R_{n}(k;X_{0})$ and $T_{n}(k;X_{0})$ are fedined as follows:
$R_{n}(k;X_{0})=e^{4ik^{3}T\hat{\Lambda}}M_{n}(0,T,k;X_{0}),$ (2.29a)
$T_{n}(k;X_{0})=e^{ikx\hat{\Lambda}}M_{n}(X_{0},0,k;X_{0}).$ (2.29b)
The relations (2.28) imply that
$s(k;X_{0})=S_{n}(k;X_{0})T^{-1}_{n}(k;X_{0}),\qquad
S(k)=S_{n}(k;X_{0})R^{-1}_{n}(k;X_{0}).$ (2.30)
These equations constitute a matrix factorization problem which, given
$\\{s,S\\}$ can be solved for the $\\{R_{n},S_{n},T_{n}\\}$. Indeed, the
integral equations (2.14) together with the definitions of
$\\{R_{n},S_{n},T_{n}\\}$ imply that
$\left\\{\begin{array}[]{lll}(R_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{1},\\\
(S_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{2},\\\
(T_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{3}.\end{array}\right.$
(2.31)
It follows that (2.30) are 18 scalar equations for 18 unknowns. By computing
the explicit solution of this algebraic system, we find that
$\\{S_{n}(k;X_{0})\\}_{1}^{4}$ are given by the equation obtained from (2.27)
by replacing $\\{S_{n}(k),s(k)\\}$ with $\\{S_{n}(k;X_{0}),s(k;X_{0})\\}$.
taking $X_{0}\rightarrow\infty$ in this equation, we arrive at (2.27). ∎
### 2.7. The global relation
The spectral functions $S(k)$ and $s(k)$ are not independent but satisfy an
important relation. Indeed, it follows from (2.24) that
$\mu_{1}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S^{-1}(k)s(k)=\mu_{3}(x,t,k),\quad
k\in(D_{3}\cup D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (2.32)
Since $\mu_{1}(0,T,k)=\mathbb{I}$, evaluation at $(0,T)$ yields the following
global relation:
$S^{-1}(k)s(k)=e^{4ik^{3}T\hat{\Lambda}}c(T,k),\quad k\in(D_{3}\cup
D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (2.33)
where $c(T,k)=\mu_{3}(0,T,k)$.
### 2.8. The residue conditions
Since $\mu_{2}$ is an entire function, it follows from (2.26) that M can only
have sigularities at the points where the $S_{n}^{\prime}s$ have
singularities. We infer from the explicit formulas (2.27) that the possible
singularities of $M$ are as follows:
* •
$[M]_{1}$ could have poles in $D_{1}\cup D_{2}$ at the zeros of $s_{33}(k)$;
* •
$[M]_{1}$ could have poles in $D_{2}$ at the zeros of $(s^{T}S^{A})_{33}(k)$;
* •
$[M]_{2}$ could have poles in $D_{1}\cup D_{2}$ at the zeros of $s_{33}(k)$;
* •
$[M]_{2}$ could have poles in $D_{2}$ at the zeros of $(s^{T}S^{A})_{33}(k)$;
* •
$[M]_{3}$ could have poles in $D_{3}$ at the zeros of $(S^{T}s^{A})_{33}(k)$;
* •
$[M]_{3}$ could have poles in $D_{3}\cup D_{4}$ at the zeros of
$m_{33}(s)(k)$;
We denote the above possible zeros by $\\{k_{j}\\}_{1}^{N}$ and assume they
satisfy the following assumption.
###### Assumption 2.3.
We assume that
* •
$s_{33}(k)$ has $n_{0}$ possible simple zeros in $D_{1}$ denoted by
$\\{k_{j}\\}_{1}^{n_{0}}$;
* •
$s_{33}(k)$ has $n_{1}-n_{0}$ possible simple zeros in $D_{2}$ denoted by
$\\{k_{j}\\}_{n_{0}+1}^{n_{1}}$;
* •
$(s^{T}S^{A})_{33}(k)$ has $n_{2}-n_{1}$ possible simple zeros in $D_{2}$
denoted by $\\{k_{j}\\}_{n_{1}+1}^{n_{2}}$;
* •
$(S^{T}s^{A})_{33}(k)$ has $n_{3}-n_{2}$ possible simple zeros in $D_{3}$
denoted by $\\{k_{j}\\}_{n_{2}+1}^{n_{3}}$;
* •
$m_{33}(s)(k)$ has $n_{4}-n_{3}$ possible simple zeros in $D_{3}$ denoted by
$\\{k_{j}\\}_{n_{3}+1}^{n_{4}}$;
* •
$m_{33}(s)(k)$ has $n_{5}-n_{4}$ possible simple zeros in $D_{3}$ denoted by
$\\{k_{j}\\}_{n_{4}+1}^{n_{5}}$;
* •
$m_{33}(s)(k)$ has $N-n_{5}$ possible simple zeros in $D_{4}$ denoted by
$\\{k_{j}\\}_{n_{5}+1}^{N}$;
and that none of these zeros coincide. Moreover, we assume that none of these
functions have zeros on the boundaries of the $D_{n}$’s.
We determine the residue conditions at these zeros in the following:
###### Proposition 2.4.
Let $\\{M_{n}\\}_{1}^{4}$ be the eigenfunctions defined by (2.14) and assume
that the set $\\{k_{j}\\}_{1}^{N}$ of singularitues are as the above
assumption. Then the following residue conditions hold:
${Res}_{k=k_{j}}[M]_{1}=\frac{m_{12}(s)(k_{j})}{\dot{s}_{33}(k_{j})s_{23}(k_{j})}e^{\theta_{31}(k_{j})}[M(k_{j})]_{3},\quad
1\leq j\leq n_{0},k_{j}\in D_{1}$ (2.34a)
${Res}_{k=k_{j}}[M]_{2}=\frac{m_{12}(s)(k_{j})}{\dot{s}_{33}(k_{j})s_{13}(k_{j})}e^{\theta_{32}(k_{j})}[M(k_{j})]_{3},\quad
1\leq j\leq n_{0},k_{j}\in D_{1}$ (2.34b)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{1}=\frac{m_{12}(s)(k_{j})m_{33}(S)(k_{j})-m_{32}(s)(k_{j})m_{13}(S)(k_{j})}{\dot{(s^{T}S^{A})_{33}(k_{j})}s_{23}(k_{j})}e^{\theta_{31}(k_{j})}[M(k_{j})]_{3}\\\
\quad n_{1}+1\leq j\leq n_{2},k_{j}\in D_{2},\end{array}$ (2.34c)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{2}=\frac{m_{21}(s)(k_{j})m_{33}(S)(k_{j})-m_{31}(s)(k_{j})m_{23}(S)(k_{j})}{\dot{(s^{T}S^{A})_{33}(k_{j})}s_{13}(k_{j})}e^{\theta_{32}(k_{j})}[M(k_{j})]_{3}\\\
\quad n_{1}+1\leq j\leq n_{2},k_{j}\in D_{2},\end{array}$ (2.34d)
$\begin{array}[]{rl}Res_{k=k_{j}}[M]_{3}=&\frac{S_{13}(k_{j})s_{32}(k_{j})-S_{33}(k_{j})s_{12}(k_{j})}{\dot{(S^{T}s^{A})_{33}(k_{j})}m_{23}(s)(k_{j})}e^{\theta_{13}(k_{j})}[M(k_{j})]_{1}\\\
&+\frac{S_{33}(k_{j})s_{11}(k_{j})-S_{13}(k_{j})s_{31}(k_{j})}{\dot{(S^{T}s^{A})_{33}(k_{j})}m_{23}(s)(k_{j})}e^{\theta_{23}(k_{j})}[M(k_{j})]_{2},n_{2}+1\leq
j\leq n_{3},k_{j}\in D_{3},\end{array}$ (2.34e)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{3}=\frac{s_{12}(k_{j})}{\dot{m}_{33}(s)(k_{j})m_{23}(s)(k_{j})}e^{\theta_{13}(k_{j})}[M(k_{j})]_{1}-\frac{s_{11}(k_{j})}{\dot{m}_{33}(s)(k_{j})m_{23}(s)(k_{j})}e^{\theta_{23}(k_{j})}[M(k_{j})]_{2}\\\
\quad n_{4}+1\leq j\leq N,k_{j}\in D_{4}.\end{array}$ (2.34f)
where $\dot{f}=\frac{df}{dk}$, and $\theta_{ij}$ is defined by
$\theta_{ij}(x,t,k)=(l_{i}-l_{j})x+(z_{i}-z_{j})t,\quad i,j=1,2,3.$ (2.35)
that implies that
$\theta_{ij}=0,i,j=1,2;\quad\theta_{13}=\theta_{23}=-\theta_{32}=-\theta_{31}=-2ikx-8ik^{3}t.$
###### Proof.
We will prove (2.34a), (2.34c), (2.34e), (2.34f), the other conditions follow
by similar arguments. Equation (2.26) implies the relation
$M_{1}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{1},$ (2.36a)
$M_{2}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{2}.$ (2.36b)
$M_{3}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{3},$ (2.36c)
$M_{4}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{4},$ (2.36d)
In view of the expressions for $S_{1}$ and $S_{2}$ given in (2.27), the three
columns of (2.36a) read:
$[M_{1}]_{1}=[\mu_{2}]_{1}\frac{m_{22}(s)}{s_{33}}+[\mu_{2}]_{2}e^{\theta_{21}}\frac{m_{12}(s)}{s_{33}},$
(2.37a)
$[M_{1}]_{2}=[\mu_{2}]_{1}e^{\theta_{12}}\frac{m_{21}(s)}{s_{33}}+[\mu_{2}]_{2}\frac{m_{11}(s)}{s_{33}},$
(2.37b)
$[M_{1}]_{3}=[\mu_{2}]_{1}e^{\theta_{13}}s_{13}+[\mu_{2}]_{2}e^{\theta_{23}}s_{23}+[\mu_{2}]_{3}s_{33}.$
(2.37c)
while the three columns of (2.36b) read:
$\begin{array}[]{rl}[M_{2}]_{1}&=[\mu_{2}]_{1}\frac{m_{22}(s)m_{33}(S)-m_{32}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}\\\
&+[\mu_{2}]_{2}\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{21}}\\\
&+[\mu_{2}]_{3}\frac{m_{12}(s)m_{23}(S)-m_{22}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{31}}\end{array}$
(2.38a)
$\begin{array}[]{rl}[M_{2}]_{2}&=[\mu_{2}]_{1}\frac{m_{21}(s)m_{33}(S)-m_{31}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{12}}\\\
&+[\mu_{2}]_{2}\frac{m_{11}(s)m_{33}(S)-m_{31}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}\\\
&+[\mu_{2}]_{3}\frac{m_{11}(s)m_{23}(S)-m_{21}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{32}}\end{array}$
(2.38b)
$[M_{2}]_{3}=[\mu_{2}]_{1}s_{13}e^{\theta_{13}}+[\mu_{2}]_{2}s_{23}e^{\theta_{23}}+[\mu_{2}]_{3}s_{33}.$
(2.38c)
and the three columns of (2.36c) read:
$[M_{3}]_{1}=[\mu_{2}]_{1}s_{11}+[\mu_{2}]_{2}s_{21}e^{\theta_{21}}+[\mu_{2}]_{3}s_{31}e^{\theta_{31}},$
(2.39a)
$[M_{3}]_{2}=[\mu_{2}]_{1}s_{12}e^{\theta_{12}}+[\mu_{2}]_{2}s_{22}+[\mu_{2}]_{3}s_{32}e^{\theta_{32}},$
(2.39b)
$[M_{3}]_{3}=[\mu_{2}]_{1}\frac{S_{13}}{(S^{T}s^{A})_{33}}e^{\theta_{13}}+[\mu_{2}]_{2}\frac{S_{23}}{(S^{T}s^{A})_{33}}e^{\theta_{23}}+[\mu_{2}]_{3}\frac{S_{33}}{(S^{T}s^{A})_{33}}.$
(2.39c)
the three columns of (2.36d) read:
$[M_{4}]_{1}=[\mu_{2}]_{1}s_{11}+[\mu_{2}]_{2}s_{21}e^{\theta_{21}}+[\mu_{2}]_{3}s_{31}e^{\theta_{31}},$
(2.40a)
$[M_{4}]_{2}=[\mu_{2}]_{1}s_{12}e^{\theta_{12}}+[\mu_{2}]_{2}s_{22}+[\mu_{2}]_{3}s_{32}e^{\theta_{32}},$
(2.40b) $[M_{4}]_{3}=[\mu_{2}]_{3}\frac{1}{m_{33}(s)}.$ (2.40c)
We first suppose that $k_{j}\in D_{1}$ is a simple zero of $s_{33}(k)$.
Solving (2.37c) for $[\mu_{2}]_{2}$ and substituting the result in to (2.37a),
we find
$[M_{1}]_{1}=\frac{m_{12}(s)}{s_{33}s_{23}}e^{\theta_{31}}[M_{1}]_{3}+\frac{m_{32}(s)}{s_{23}}[\mu_{2}]_{2}-\frac{m_{12}(s)}{s_{23}}e^{\theta_{31}}[\mu_{2}]_{3}.$
Taking the residue of this equation at $k_{j}$, we find the condition (2.34a)
in the case when $k_{j}\in D_{1}$. Similarly, Solving (2.38c) for
$[\mu_{2}]_{2}$ and substituting the result in to (2.38a), we find
$[M_{2}]_{1}=\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}s_{23}}e^{\theta_{31}}[M_{1}]_{3}-\frac{m_{32}(s)}{s_{23}}[\mu_{2}]_{1}-\frac{m_{12}(s)}{s_{23}}e^{\theta_{31}}[\mu_{2}]_{3}.$
Taking the residue of this equation at $k_{j}$, we find the condition (2.34c)
in the case when $k_{j}\in D_{2}$.
In order to prove (2.34e), we solve (2.39a) and (2.39b) for $[\mu_{2}]_{1}$
and $[\mu_{2}]_{3}$, then substituting the result into (2.39c), we find
$[M_{3}]_{3}=\frac{S_{13}s_{32}-S_{33}s_{12}}{(S^{T}s^{A})_{33}m_{23}(s)}e^{\theta_{13}}[M_{3}]_{1}+\frac{S_{33}s_{11}-S_{13}s_{31}}{(S^{T}s^{A})_{33}(k_{j})m_{23}(s)}e^{\theta_{23}}[M_{3}]_{2}+\frac{1}{m_{23}(s)}[\mu_{2}]_{3}.$
Taking the residue of this equation at $k_{j}$, we find the condition (2.34e)
in the case when $k_{j}\in D_{3}$. Similarly, solving (2.40a) and (2.40b) for
$[\mu_{2}]_{1}$ and $[\mu_{2}]_{3}$, then substituting the result into
(2.40c), we find
$[M_{4}]_{3}=\frac{s_{12}}{m_{33}(s)m_{23}(s)}e^{\theta_{13}}[M_{4}]_{1}-\frac{s_{11}}{m_{33}(s)m_{23}(s)}e^{\theta_{13}}[M_{4}]_{2}-\frac{1}{m_{23}(s)}e^{\theta_{23}}[\mu_{2}]_{2}.$
Taking the residue of this equation at $k_{j}$, we find the condition (2.34f)
in the case when $k_{j}\in D_{4}$. ∎
## 3\. The Riemann-Hilbert problem
The sectionally analytic function $M(x,t,k)$ defined in section 2 satisfies a
Riemann-Hilbert problem which can be formulated in terms of the initial and
boundary values of $u(x,t)$. By solving this Riemann-Hilbert problem, the
solution of (1.5)(then (1.3)) can be recovered for all values of $x,t$.
###### Theorem 3.1.
Suppose that $u(x,t)$ is a solution of (1.5) in the half-line domain $\Omega$
with sufficient smoothness and decays as $x\rightarrow\infty$. Then $u(x,t)$
can be reconstructed from the initial value $\\{u_{0}(x)\\}$ and boundary
values $\\{g_{0}(t),g_{1}(t),g_{2}(t)\\}$ defined as follows,
$u_{0}(x)=u(x,0),\quad g_{0}(t)=u(0,t),\quad g_{1}(t)=u_{x}(0,t),\quad
g_{2}(t)=u_{xx}(0,t).$ (3.1)
Use the initial and boundary data to define the jump matrices $J_{m,n}(x,t,k)$
as well as the spectral $s(k)$ and $S(k)$ by equation (2.24). Assume that the
possible zeros $\\{k_{j}\\}_{1}^{N}$ of the functions
$s_{33}(k),(s^{T}S^{A})_{33}(k),(S^{T}s^{A})_{33}(k)$ and $m_{33}(s)(k)$ are
as in assumption 2.3.
Then the solution $\\{u(x,t)\\}$ is given by
$u(x,t)=2i\lim_{k\rightarrow\infty}(kM(x,t,k))_{13}.$ (3.2)
where $M(x,t,k)$ satisfies the following $3\times 3$ matrix Riemann-Hilbert
problem:
* •
$M$ is sectionally meromorphic on the Riemann $k-$sphere with jumps across the
contours $\bar{D}_{n}\cap\bar{D}_{m},n,m=1,\cdots,4$, see Figure 2.
* •
Across the contours $\bar{D}_{n}\cap\bar{D}_{m}$, $M$ satisfies the jump
condition
$M_{n}(x,t,k)=M_{m}(x,t,k)J_{m,n}(x,t,k),\quad
k\in\bar{D}_{n}\cap\bar{D}_{m},n,m=1,2,3,4.$ (3.3)
* •
$M(x,t,k)=\mathbb{I}+O(\frac{1}{k}),\qquad k\rightarrow\infty$.
* •
The residue condition of $M$ is showed in Proposition 2.4.
###### Proof.
It only remains to prove (3.2) and this equation follows from the large $k$
asymptotics of the eigenfunctions, see the appendix A. ∎
## 4\. Non-linearizable Boundary Conditions
A major difficulty of initial-boundary value problems is that some of the
boundary values are unkown for a well-posed problem. All boundary values are
needed for the definition of $S(k)$, and hence for the formulation of the
Riemann-Hilbert problem. Our main result expresses the spectral function
$S(k)$ in terms of the prescribed boundary data and the initial data via the
solution of a system of nonlinear integral equations.
### 4.1. Asymptotics
An analysis of (A.2) shows that the eigenfunctions $\\{\mu_{j}\\}_{1}^{3}$
have the following asymptotics as $k\rightarrow\infty$ (see the appendix A):
$\begin{array}[]{l}\mu_{j}(x,t,k)=\mathbb{I}+\frac{1}{k}\left(\begin{array}[]{lll}\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta&\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}&\frac{1}{2i}u\\\
\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}&\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta&\frac{1}{2i}\bar{u}\\\
\frac{1}{2i}\bar{u}&\frac{1}{2i}u&-i\int_{(x_{j},t_{j})}^{(x,t)}\Delta\end{array}\right)\\\
+\frac{1}{k^{2}}\left(\begin{array}[]{lll}-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}(\eta+\nu_{1})&-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}\eta_{1}&\mu^{(2)}_{13}\\\
-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}\eta_{2}&-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}(\eta+\nu_{2})&\mu^{(2)}_{23}\\\
\mu^{(2)}_{31}&\mu^{(2)}_{32}&\int_{(x_{j},t_{j})}^{(x,t)}\eta_{3}\\\
\end{array}\right)\\\
+\frac{1}{k^{3}}\left(\begin{array}[]{lll}\mu^{(3)}_{11}&\mu^{(3)}_{12}&\mu^{(3)}_{13}\\\
\mu^{(3)}_{21}&\mu^{(3)}_{22}&\mu^{(3)}_{23}\\\
\mu^{(3)}_{31}&\mu^{(3)}_{32}&\mu^{(3)}_{33}\end{array}\right)+O(\frac{1}{k^{4}})\end{array}$
(4.1a)
where
$\begin{array}[]{l}\Delta=-|u|^{2}dx+(u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4})dt\\\
\Delta_{1}=-u^{2}dx+(uu_{xx}+u_{xx}u-(u_{x})^{2}+6|u|^{2}u^{2})dt\\\
\Delta_{2}=-\bar{u}^{2}dx+(\bar{u}\bar{u}_{xx}+\bar{u}_{xx}\bar{u}-(\bar{u}_{x})^{2}+6|u|^{2}\bar{u}^{2})dt\end{array}$
(4.2a)
$\begin{array}[]{l}\mu^{(2)}_{13}=-\frac{1}{2}u\int_{(x_{j},t_{j})}^{(x,t)}\Delta+\frac{1}{4}u_{x}\\\
\mu^{(2)}_{23}=-\frac{1}{2}\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta+\frac{1}{4}\bar{u}_{x}\\\
\mu^{(2)}_{31}=\frac{1}{4}(\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta+u\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2})-\frac{1}{4}\bar{u}_{x}\\\
\mu^{(2)}_{32}=\frac{1}{4}(\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+u\int_{(x_{j},t_{j})}^{(x,t)}\Delta)-\frac{1}{4}u_{x}.\end{array}$
(4.2b)
$\begin{array}[]{l}\eta=d[\frac{1}{2}(\int_{(x_{j},t_{j})}^{(x,t)}\Delta)^{2}]\\\
\nu_{1}=\Delta_{1}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}+(u\bar{u}_{x})dx+(u\bar{u}_{t}-2|u|^{2}(u\bar{u}_{x}-u_{x}\bar{u})-(u_{xx}\bar{u}_{x}-u_{x}\bar{u}_{xx}))dt\\\
\eta_{1}=\int_{(x_{j},t_{j})}^{(x,t)}\Delta\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+u^{2}\\\
\eta_{2}=\int_{(x_{j},t_{j})}^{(x,t)}\Delta\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}+\bar{u}^{2}\\\
\nu_{2}=\Delta_{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+(\bar{u}u_{x})dx+(\bar{u}u_{t}-2|u|^{2}(\bar{u}u_{x}-\bar{u}_{x}u)-(\bar{u}_{xx}u_{x}-\bar{u}_{x}u_{xx}))dt,\\\
\eta_{3}=d[-\frac{1}{2}(\int_{(x_{j},t_{j})}^{(x,t)}\Delta)^{2}-\frac{1}{4}|u|^{2}].\end{array}$
(4.2c) and in the following we just use
$\mu^{(3)}_{13},\mu^{(3)}_{23},\mu^{(3)}_{31}$ and $\mu^{(3)}_{32}$, so we
only compute these functions
$\begin{array}[]{l}\mu^{(3)}_{13}=\frac{1}{2i}u\mu^{(2)}_{33}+\frac{1}{4}u_{x}\mu^{(1)}_{33}+\frac{i}{4}|u|^{2}u+\frac{i}{8}u_{xx}\\\
\mu^{(3)}_{23}=\frac{1}{2i}\bar{u}\mu^{(2)}_{33}+\frac{1}{4}\bar{u}_{x}\mu^{(1)}_{33}+\frac{i}{4}|u|^{2}\bar{u}+\frac{i}{8}\bar{u}_{xx}\\\
\mu^{(3)}_{31}=\frac{1}{2i}(\bar{u}\mu^{(2)_{11}}+u\mu^{(2)}_{21})-\frac{1}{4}(\bar{u}_{x}\mu^{(1)_{11}}+u_{x}\mu^{(1)}_{21})+\frac{i}{4}|u|^{2}\bar{u}+\frac{i}{8}\bar{u}_{xx}\\\
\mu^{(3)}_{32}=\frac{1}{2i}(\bar{u}\mu^{(2)_{12}}+u\mu^{(2)}_{22})-\frac{1}{4}(\bar{u}_{x}\mu^{(1)_{12}}+u_{x}\mu^{(1)}_{22})+\frac{i}{4}|u|^{2}u+\frac{i}{8}u_{xx}\end{array}$
(4.2d)
From the global relation (2.33)and replacing $T$ by $t$, we find
$\mu_{2}(0,t,k)e^{-4ik^{3}t\hat{\Lambda}}s(k)=c(t,k),\quad k\in(D_{3}\cup
D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (4.3)
We define functions $\\{\Phi_{13}(t,k),\Phi_{23}(t,k),\Phi_{33}(t,k)\\}$ and
$\\{c_{j}(t,k)\\}_{1}^{3}$ by:
$\mu_{2}(0,t,k)=\left(\begin{array}[]{lll}\Phi_{11}(t,k)&\Phi_{12}(t,k)&\Phi_{13}(t,k)\\\
\Phi_{21}(t,k)&\Phi_{22}(t,k)&\Phi_{23}(t,k)\\\
\Phi_{31}(t,k)&\Phi_{32}(t,k)&\Phi_{33}(t,k)\end{array}\right),\quad\frac{[c(t,k)]_{3}}{s_{33}(k)}=\left(\begin{array}[]{l}c_{1}(t,k)\\\
c_{2}(t,k)\\\ c_{3}(t,k)\end{array}\right).$ (4.4)
we can write the $(13)$ and $(23)$ entries of the global relation as
$\Phi_{11}(t,k)e^{-8ik^{3}t}\frac{s_{13}}{s_{33}}+\Phi_{12}(t,k)e^{-8ik^{3}t}\frac{s_{23}}{s_{33}}+\Phi_{13}(t,k)=c_{1}(t,k),\quad
k\in D_{1}\cup D_{2},$ (4.5a)
$\Phi_{21}(t,k)e^{-8ik^{3}t}\frac{s_{13}}{s_{33}}+\Phi_{22}(t,k)e^{-8ik^{3}t}\frac{s_{23}}{s_{33}}+\Phi_{23}(t,k)=c_{2}(t,k),\quad
k\in D_{1}\cup D_{2},$ (4.5b)
The functions $\\{c_{j}(t,k)\\}_{1}^{3}$ are analytic and bounded in
$D_{1}\cup D_{2}$ away from the possible zeros of $s_{33}(k)$ and of order
$O(\frac{1}{k})$ as $k\rightarrow\infty$.
From the asymptotic of $\mu_{j}(x,t,k)$ in (4.1a) we have
$\left(\begin{array}[]{l}s_{13}(k)\\\ s_{23}(k)\\\
s_{33}(k)\end{array}\right)=\left(\begin{array}[]{l}0\\\ 0\\\
1\end{array}\right)+\frac{1}{2ik}\left(\begin{array}[]{l}u(0,0)\\\
\bar{u}(0,0)\\\
2\int_{(\infty,0)}^{(0,0)}\Delta\end{array}\right)+O(\frac{1}{k^{2}}).$ (4.6)
and
$\Phi_{j3}(t,k)=\frac{\Phi_{j3}^{(1)}(t)}{k}+\frac{\Phi_{j3}^{(2)}(t)}{k^{2}}+\frac{\Phi_{j3}^{(3)}(t)}{k^{3}}+O(\frac{1}{k^{4}}),$
(4.7a)
$\Phi_{33}(t,k)=1+\frac{\Phi_{33}^{(1)}(t)}{k}+\frac{\Phi_{33}^{(2)}(t)}{k^{2}}+O(\frac{1}{k^{3}}),\quad
k\rightarrow\infty,k\in D_{1}\cup D_{2}.$ (4.7b) where
$\begin{array}[]{ll}\Phi_{j3}^{(1)}(t)=\frac{1}{2i}g_{0}(t)^{T},&\Phi_{j3}^{(2)}(t)=\frac{1}{4}g_{1}(t)^{T}-\frac{1}{2}g_{0}^{T}\int_{(0,0)}^{(x,t)}\Delta\\\
\Phi_{j3}^{(3)}(t)=\frac{1}{2i}g_{0}^{T}\Phi^{(2)}_{33}+\frac{1}{4}g_{1}^{T}\Phi^{(1)}_{33}+\frac{i}{4}|u|^{2}g_{0}^{T}+\frac{i}{8}g_{2}^{T},&\\\
\Phi_{33}^{(1)}(t)=-i\int_{(0,0)}^{(x,t)}\Delta,&\Phi_{33}^{(2)}(t)=\int_{(x_{j},t_{j})}^{(x,t)}\eta_{3}.\end{array}$
Here the definition of $\Phi_{j3}(t,k)$ can be found in the appendix A.
In particular, we find the following expressions for the boudary values:
$g_{0}^{T}=2i\Phi_{j3}^{(1)}(t),$ (4.8a)
$g_{1}^{T}=2ig_{0}^{T}\Phi_{33}^{(1)}(t)+4\Phi_{j3}^{(2)}(t),$ (4.8b)
$g_{2}^{T}=-2|g_{0}|^{2}g_{0}^{T}+2ig_{1}^{T}\Phi_{33}^{(1)}(t)+4g_{0}^{T}\Phi_{33}^{(2)}(t)-8i\Phi_{j3}^{(3)}(t).$
(4.8c)
We will also need the asymptotic of $c_{j}(t,k)$,
###### Lemma 4.1.
The global relation (4.5) implies that the large $k$ behavior of $c_{j}(t,k)$
satisfies
$c_{j}(t,k)=\frac{\Phi_{j3}^{(1)}(t)}{k}+\frac{\Phi_{j3}^{(2)}(t)}{k}+\frac{\Phi_{j3}^{(3)}(t)}{k}+O(\frac{1}{k^{4}}),\quad
k\rightarrow\infty,k\in D_{1}.$ (4.9)
###### Proof.
See the appendix B. ∎
### 4.2. The Dirichlet and Neumann problems
We can now derive effective characterizations of spectral function $S(k)$ for
the Dirichlet ($g_{0}$ prescribed), the first Neumann ($g_{1}$ prescribed),
and the second Neumann ($g_{2}$ prescribed) problems.
Define $\alpha$ by $\alpha=e^{\frac{2\pi i}{3}}$ and let
$\\{\Pi_{j}(t,k),\hat{\Pi}_{j}(t,k),\tilde{\Pi}_{j}(t,k)\\}_{1}^{3}$ denote
the following combinations formed from $\\{\Phi_{j3}(t,k)\\}_{1}^{3}$:
$\begin{array}[]{l}\Pi_{j}(t,k)=\Phi_{j3}(t,k)+\alpha\Phi_{j3}(t,\alpha
k)+\alpha^{2}\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3,\\\
\hat{\Pi}_{j}(t,k)=\Phi_{j3}(t,k)+\alpha^{2}\Phi_{j3}(t,\alpha
k)+\alpha\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3,\\\
\tilde{\Pi}_{j}(t,k)=\Phi_{j3}(t,k)+\Phi_{j3}(t,\alpha
k)+\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3.\end{array}$ (4.10)
And let $R(k)=\Phi_{11}\frac{s_{13}}{s_{33}}+\Phi_{12}\frac{s_{23}}{s_{33}}$.
Let $D_{1}=D_{1}^{{}^{\prime}}\cup D_{1}^{{}^{\prime\prime}}$ where
$D_{1}^{{}^{\prime}}=D_{1}\cap\\{\mathrm{Re}k>0\\}$ and
$D_{1}^{{}^{\prime\prime}}=D_{1}\cap\\{\mathrm{Re}k<0\\}$. Similarly, let
$D_{4}=D_{4}^{{}^{\prime}}\cup D_{4}^{{}^{\prime\prime}}$ where
$D_{4}^{{}^{\prime}}=D_{4}\cap\\{\mathrm{Re}k>0\\}$ and
$D_{4}^{{}^{\prime\prime}}=D_{1}\cap\\{\mathrm{Re}k<0\\}$.
###### Theorem 4.2.
Let $T<\infty$. Let $u_{0}(x),u\geq 0$, be a function of Schwartz class.
For the Dirichlet problem it is assumed that the function $g_{0}(t),0\leq
t<T$, has sufficient smoothness and is compatible with $u_{0}(x)$ at $x=t=0$.
For the first Neumann problem it is assumed that the function $g_{1}(t),0\leq
t<T$, has sufficient smoothness and is compatible with $u_{0}(x)$ at $x=t=0$.
Similarly, for the second Neumann problem it is assumed that the function
$g_{2}(t),0\leq t<T$, has sufficient smoothness and is compatible with
$u_{0}(x)$ at $x=t=0$.
Suppose that $s_{33}(k)$ has a finite number of simple zeros in $D_{1}$.
Then the spectral function $S(k)$ is given by
$S(k)=\left(\begin{array}[]{ccc}A(k)&B(k)&e^{8ik^{3}T}C(k)\\\
D(k)&E(k)&e^{8ik^{3}T}F(k)\\\
e^{-8ik^{3}T}G(k)&e^{-8ik^{3}T}H(k)&I(k)\end{array}\right)$ (4.11)
where
$\begin{array}[]{ll}A(k)=\Phi_{22}(k)\Phi_{33}(k)-\Phi_{23}(k)\Phi_{32}(k)&B(k)=\Phi_{13}(k)\Phi_{22}(k)-\Phi_{12}(k)\Phi_{33}(k)\\\
C(k)=\Phi_{12}(k)\Phi_{23}(k)-\Phi_{13}(k)\Phi_{22}(k)&D(k)=\Phi_{23}(k)\Phi_{31}(k)-\Phi_{21}(k)\Phi_{33}(k)\\\
E(k)=\Phi_{11}(k)\Phi_{33}(k)-\Phi_{13}(k)\Phi_{31}(k)&F(k)=\Phi_{21}(k)\Phi_{13}(k)-\Phi_{11}(k)\Phi_{23}(k)\\\
G(k)=\Phi_{21}(k)\Phi_{32}(k)-\Phi_{22}(k)\Phi_{31}(k)&H(k)=\Phi_{12}(k)\Phi_{31}(k)-\Phi_{11}(k)\Phi_{32}(k)\\\
I(k)=\Phi_{11}(k)\Phi_{22}(k)-\Phi_{12}(k)\Phi_{21}(k)&\end{array}$
and the complex-value functions $\\{\Phi_{l3}(t,k)\\}_{l=1}^{3}$ satisfy the
following system of integral equations:
$\begin{array}[]{rl}\Phi_{13}(t,k)&=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{13}\right.\\\
&\left.+g_{0}^{2}\Phi_{23}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12a)
$\begin{array}[]{rl}\Phi_{23}(t,k)&=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{13}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{23}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12b)
$\begin{array}[]{rl}\Phi_{33}(t,k)&=1+\int_{0}^{t}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{13}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{23}+-4ik|g_{0}|^{2}\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12c)
and $\\{\Phi_{l1}(t,k)\\}_{l=1}^{3},\\{\Phi_{l2}(t,k)\\}_{l=1}^{3}$ satisfy
the following system of integral equations:
$\begin{array}[]{rl}\Phi_{11}(t,k)&=1+\int_{0}^{t}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{11}\right.\\\
&\left.+g_{0}^{2}\Phi_{21}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13a)
$\begin{array}[]{rl}\Phi_{21}(t,k)&=\int_{0}^{t}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{11}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{21}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13b)
$\begin{array}[]{rl}\Phi_{33}(t,k)&=\int_{0}^{t}e^{8ik^{3}(t-t^{\prime})}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{11}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{21}+-4ik|g_{0}|^{2}\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13c)
$\begin{array}[]{rl}\Phi_{12}(t,k)&=\int_{0}^{t}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{12}\right.\\\
&\left.+g_{0}^{2}\Phi_{22}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14a)
$\begin{array}[]{rl}\Phi_{22}(t,k)&=1+\int_{0}^{t}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{12}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{22}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14b)
$\begin{array}[]{rl}\Phi_{32}(t,k)&=\int_{0}^{t}e^{8ik^{3}(t-t^{\prime})}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{12}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{22}+-4ik|g_{0}|^{2}\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14c)
1. (i)
For the Dirichlet problem, the unknown Neumann boundary values $g_{1}(t)$ and
$g_{2}(t)$ are given by
$\begin{array}[]{rl}g_{1}(t)=&\frac{2g_{0}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk+\frac{2}{\pi i}\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk\\\ &-\frac{2}{\pi
i}\int_{\partial D_{3}}ke^{-8ik^{3}t}[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)]dk\\\
&+4\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.15a) and
$\begin{array}[]{rl}g_{2}(t)=&g_{0}(t)^{3}-\frac{4}{\pi}\int_{\partial
D_{3}}\left[k^{2}\Pi_{1}(t,k)-\frac{3kg_{0}(t)}{2i}\right]dk\\\
&+\frac{4}{\pi}\int_{\partial D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)R(\alpha
k)+(1-\alpha^{2})R(\alpha^{2}k)\right]dk\\\
&-8i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k)\\\
&+\frac{4g_{0}(t)}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{3}(t,k)dk+\frac{2g_{1}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.15b)
2. (ii)
For the first Neumann problem, the unknown boundary values $g_{0}(t)$ and
$g_{2}(t)$ are given by
$\begin{array}[]{rl}g_{0}(t)=&\frac{1}{\pi}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk-\frac{1}{\pi}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
&+2i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k),\end{array}$
(4.16a) and
$\begin{array}[]{rl}g_{2}(t)=&g_{0}^{3}(t)-\frac{4}{\pi}\int_{\partial
D_{3}}\left(k^{2}\hat{\Pi}_{1}(t,k)-\frac{3}{\pi i}\int_{\partial
D_{3}}l\hat{\Pi}_{1}(t,l)dl\right)dk\\\ &+\frac{4}{\pi}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha^{2})R(\alpha
k)+(1-\alpha)R(\alpha^{2}k)\right]dk\\\ &-8i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k)\\\
&+\frac{4g_{0}(t)}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{3}(t,k)dk+\frac{2g_{1}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.16b)
3. (iii)
For the second Neumann problem, the unknown boundary values $g_{0}(t)$ and
$g_{1}(t)$ are given by
$\begin{array}[]{rl}g_{0}(t)=&\frac{1}{\pi}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk-\frac{1}{\pi}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
&+2i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k),\end{array}$
(4.17a) and $\begin{array}[]{rl}g_{1}(t)=&\frac{2g_{0}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk+\frac{2}{\pi i}\int_{\partial
D_{3}}k\tilde{\Pi}_{1}(t,k)dk\\\ &-\frac{2}{\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-1)R(\alpha
k)+(\alpha-1)R(\alpha^{2}k)\right]dk\\\ &+4\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.17b)
###### Proof.
The representations (4.11) follow from the relation
$S(k)=e^{8ik^{3}T}\mu_{2}^{A}(0,T,k)^{T}$. And the system (4.12) is the direct
result of the Volteral integral equations of $\mu_{2}(0,t,k)$.
1. (i)
In order to derive (4.15a) we note that equation (4.8b) expresses $g_{1}$ in
terms of $\Phi_{33}^{(1)}$ and $\Phi_{13}^{(2)}$. Furthermore, equation (4.7)
and Cauchy theorem imply
$-\frac{2\pi i}{3}\Phi_{33}^{(1)}(t)=2\int_{\partial
D_{2}}[\Phi_{33}(t,k)-1]dk=\int_{\partial D_{4}}[\Phi_{33}(t,k)-1]dk$
and
$-\frac{2\pi i}{3}\Phi_{13}^{(2)}(t)=2\int_{\partial
D_{2}}\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk=\int_{\partial
D_{4}}\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk.$
Thus,
$\begin{array}[]{l}i\pi\Phi_{33}^{(1)}(t)=-\left(\int_{\partial
D_{2}}+\int_{\partial D_{4}}\right)[\Phi_{33}(t,k)-1]dk=\left(\int_{\partial
D_{1}}+\int_{\partial D_{3}}\right)[\Phi_{33}(t,k)-1]dk\\\ =\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk+\alpha\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk+\alpha^{2}\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk\\\ =\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.18)
Similarly,
$\begin{array}[]{l}i\pi\Phi_{13}^{(2)}(t)=\left(\int_{\partial
D_{3}}+\int_{\partial
D_{1}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk\\\
=\left(\int_{\partial D_{3}}+\alpha^{2}\int_{\partial
D_{1}^{{}^{\prime}}}+\alpha\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk+I(t)\\\
=\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk+I(t).\end{array}$
(4.19)
where $I(t)$ is defined by
$I(t)=\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk$
The last step involves using the global relation to compute $I(t)$
$\begin{array}[]{r}I(t)=\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[kc_{1}(t,k)-\frac{g_{0}(t)}{2i}\right]dk\\\
-\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)ke^{-8ik^{3}t}R(k)dk\end{array}$ (4.20)
Using the asymptotic (4.9) and Cauchy theorem to compute the first term on the
right-hand side of equation (4.20) and using the transformation
$k\rightarrow\alpha k$ and $k\rightarrow\alpha^{2}k$ in the second term on the
right-hand side of (4.20), we find
$\begin{array}[]{r}I(t)=-i\pi\Phi_{13}^{(2)}(t)-\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)\right]dk\\\ +2\pi
i\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}ke^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.21)
Equations (4.19) and (4.21) imply
$\begin{array}[]{l}\Phi_{13}^{(2)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk\\\ -\frac{1}{2\pi
i}\int_{\partial D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
This equation together with (4.8b) and (4.18) yields (4.15a).
In order to derive (4.15b), we note that (4.8c) expresses $g_{2}$ in terms of
$\Phi_{13}^{(3)}$, $\Phi_{33}^{(2)}$ and $\Phi_{33}^{(1)}$. Equation (4.15b)
follows from the expression (4.18) for $\Phi_{33}^{(1)}$ and the following
formulas:
$\Phi_{33}^{(2)}(t)=\frac{1}{\pi i}\int_{\partial D_{3}}k\hat{\Pi}_{3}dk,$
(4.22a) $\begin{array}[]{l}\Phi_{13}^{(3)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k^{2}\Pi_{1}(t,k)-\frac{3kg_{0}(t)}{2i}\right]dk\\\ -\frac{1}{2\pi
i}\int_{\partial D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)R(\alpha
k)+(1-\alpha^{2})R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.22b)
2. (ii)
In order to derive the representations (4.16) relevant for the first Neumann
problem, we use (4.8) together with (4.18), (4.22a) and the following
formulas:
$\begin{array}[]{l}\Phi_{13}^{(1)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-\alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.23a) $\Phi_{13}^{(2)}(t)=\frac{1}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{1}dk,$ (4.23b)
$\begin{array}[]{l}\Phi_{13}^{(3)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k^{2}\hat{\Pi}_{1}(t,k)-3\Phi_{13}^{(2)}\right]dk\\\
-\frac{1}{2\pi i}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha^{2})R(\alpha
k)+(1-\alpha)R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.23c)
3. (iii)
In order to derive the representations (4.17) relevant for the second Neumann
problem, we use (4.8) together with (4.18) and the following formulas:
$\begin{array}[]{l}\Phi_{13}^{(1)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\tilde{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-1)R(\alpha
k)+(\alpha^{2}-1)R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.24a) $\begin{array}[]{l}\Phi_{13}^{(2)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}k\tilde{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-1)R(\alpha
k)+(\alpha-1)R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.24b)
∎
### 4.3. Effective characterizations
Substituting into the system (4.12) the expressions
$\Phi_{ij}=\Phi^{(0)}_{ij}+\varepsilon\Phi^{(1)}_{ij}+\varepsilon^{2}\Phi^{(2)}_{ij}+\cdots,\quad
i,j=1,2,3.$ (4.25a) $g_{0}=\varepsilon g_{01}+\varepsilon^{2}g_{02}+\cdots,$
(4.25b) $g_{1}=\varepsilon g_{11}+\varepsilon^{2}g_{12}+\cdots,$ (4.25c)
$g_{2}=\varepsilon g_{21}+\varepsilon^{2}g_{22}+\cdots,$ (4.25d)
where $\varepsilon>0$ is a small parameter, we find that the terms of $O(1)$
give $\Phi^{(0)}_{13}=\Phi^{(0)}_{23}=0$ and $\Phi^{(0)}_{33}=1$. Moreover,
the terms of $O(\varepsilon)$ give $\Phi^{(1)}_{33}=0$ and
$O(\varepsilon):\quad\Phi^{(1)}_{13}(t,k)=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}(4k^{2}g_{01}+2ikg_{11}-g_{21})(t^{\prime},k)dt^{\prime},$
(4.26)
From the above equation (4.26) we can get
$\Pi^{(1)}_{1}(t,k)=12k^{2}\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{01}(t^{\prime})dt^{\prime},$
(4.27a)
$\hat{\Pi}^{(1)}_{1}(t,k)=6ik\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{11}(t^{\prime})dt^{\prime},$
(4.27b)
$\tilde{\Pi}^{(1)}_{1}(t,k)=-3\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{11}(t^{\prime})dt^{\prime},$
(4.27c)
The Dirichlet problem can now be solved perturbatively as follows: assuming
for simplicity that $s_{33}(k)$ has no zeros and expanding (4.15a) and
(4.15b), we find
$\begin{array}[]{rl}g_{11}=&\frac{2}{\pi i}\int_{\partial
D_{3}}\left[k\Pi^{(1)}_{1}(t,k)-\frac{3g_{01}(t)}{2i}\right]dk\\\
&-\frac{2}{\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}[(\alpha^{2}-\alpha)s_{131}(\alpha
k)+(\alpha-\alpha^{2})s_{131}(\alpha^{2}k)]dk\end{array}$ (4.28a)
$\begin{array}[]{rl}g_{21}=&-\frac{4}{\pi}\int_{\partial
D_{3}}\left[k^{2}\Pi^{(1)}_{1}(t,k)-\frac{3kg_{01}(t)}{2i}\right]dk\\\
&+\frac{4}{\pi}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)s_{131}(\alpha
k)+(1-\alpha^{2})s_{131}(\alpha^{2}k)\right]dk\end{array}$ (4.28b)
Using equation (4.27a) to determine $\Pi^{(1)}_{1}$, we can determine
$g_{11},g_{21}$ from (4.28), then $\Phi^{(1)}_{13}$ can be found from (4.26),
And these arguments can be extended to higher orders and also can be extended
to the systems (4.13a) and (4.14a), thus yields a constructive scheme for
computing $S(k)$ to all orders.
Similarly, these arguments also can be used to the first Neumann problem and
the second Neumann problem. That is to say, in all cases, the system can be
solved perturbatively to all orders.
## Appendix A The asymptotic behavior of the functions
$\\{\mu_{j}(x,t,k)\\}_{1}^{3}$
We denote some symbols as follows:
$\Lambda=\left(\begin{array}[]{ll}\mathbb{I}_{2\times 2}&0\\\
0&-1\end{array}\right),$ (A.1a)
$\begin{array}[]{l}V_{1}=\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right),\\\
V_{2}^{(2)}=4\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right),\\\
V_{2}^{(1)}=2i\left(\begin{array}[]{ll}U^{T}\bar{U}&U_{x}^{T}\\\
\bar{U}_{x}&-2|u|^{2}\end{array}\right),\\\
V_{2}^{(0)}=-4|u|^{2}\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right)-\left(\begin{array}[]{ll}0&U_{xx}^{T}\\\
-\bar{U}_{xx}&0\end{array}\right)+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}&0\\\
0&0\end{array}\right).\end{array}$ (A.1b) where $\mathbb{I}_{2\times
2}=\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$ and $U=(u,\bar{u})$.
From the Lax pair of $\mu$
$\left\\{\begin{array}[]{l}\mu_{x}+[ik\Lambda,\mu]=V_{1}\mu,\\\
\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.\end{array}\right.$ (A.2)
Suppose that
$\mu(x,t,k)=D_{0}+\frac{D_{1}}{k}+\frac{D_{2}}{k^{2}}+\frac{D_{3}}{k^{3}}+\cdots.$
(A.3)
We substitute the equation (A.3) into the Lax pair (A.2), and compare the
order of $k$, we find that:
$\begin{array}[]{ll}O(k):&[i\Lambda,D_{0}]=0,\\\
O(1):&D_{0x}+[i\Lambda,D_{1}]=V_{1}D_{0},\\\
O(k^{-1}):&D_{1x}+[i\Lambda,D_{2}]=V_{1}D_{1},\\\
O(k^{-2}):&D_{2x}+[i\Lambda,D_{3}]=V_{1}D_{2},\\\ \end{array}$ (A.4a)
$\begin{array}[]{ll}O(k^{3}):&[4i\Lambda,D_{0}]=0,\\\
O(k^{2}):&[4i\Lambda,D_{1}]=V_{2}^{(2)}D_{0},\\\
O(k^{1}):&[4i\Lambda,D_{2}]=V_{2}^{(2)}D_{1}+V_{2}^{(1)}D_{0},\\\
O(1):&D_{0t}+[4i\Lambda,D_{3}]=V_{2}^{(2)}D_{2}+V_{2}^{(1)}D_{1}+V_{2}^{(0)}D_{0},\\\
O(k^{-1}):&D_{1t}+[4i\Lambda,D_{4}]=V_{2}^{(2)}D_{3}+V_{2}^{(1)}D_{4}+V_{2}^{(0)}D_{1},\\\
O(k^{-2}):&D_{2t}+[4i\Lambda,D_{5}]=V_{2}^{(2)}D_{4}+V_{2}^{(1)}D_{3}+V_{2}^{(0)}D_{2},\\\
\end{array}$ (A.4b)
And we denote the $D_{l}$ by $D_{l}=\left(\begin{array}[]{ll}D_{2\times
2}^{(l)}&D_{j3}^{(l)}\\\ D_{3j}^{(l)}&D_{33}^{(l)}\end{array}\right),\quad
j=1,2$.
Then, from $O(k^{3})$,we have
$D_{j3}^{(0)}=0,\quad D_{3j}^{(0)}=0.$ (A.5)
$O(k^{2})$, we get
$4i\left(\begin{array}[]{ll}0&2D_{j3}^{(1)}\\\
-2D_{3j}^{(1)}&0\end{array}\right)=4\left(\begin{array}[]{ll}0&U^{T}D_{33}^{(0)}\\\
-\bar{U}D_{2\times 2}^{(0)}&0\end{array}\right),$ (A.6a) this implies that
$\left\\{\begin{array}[]{l}D_{j3}^{(1)}=-\frac{i}{2}U^{T}D_{33}^{(0)}\\\
D_{3j}^{(1)}=-\frac{i}{2}\bar{U}D_{2\times 2}^{(0)}.\end{array}\right.$ (A.6b)
$O(k)$, we find
$\begin{array}[]{l}4i\left(\begin{array}[]{ll}0&2D_{j3}^{(2)}\\\
-2D_{3j}^{(2)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(1)}&U^{T}D_{33}^{(1)}\\\
-\bar{U}D_{2\times
2}^{(1)}&-\bar{U}D_{j3}^{(1)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(0)}&U_{x}^{T}D_{33}^{(0)}\\\ -\bar{U}_{x}D_{2\times
2}^{(0)}&-2|u|^{2}D_{33}^{(0)}\end{array}\right),\end{array}$ (A.7a) this
implies that
$\left\\{\begin{array}[]{l}D_{j3}^{(2)}=-\frac{i}{2}U^{T}D_{33}^{(1)}+\frac{1}{4}U_{x}^{T}D_{33}^{0}\\\
D_{3j}^{(2)}=-\frac{i}{2}\bar{U}D_{2\times
2}^{(1)}-\frac{1}{4}\bar{U}_{x}D_{2\times 2}^{(0)}.\end{array}\right.$ (A.7b)
$O(1)$, we have
$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times 2t}^{(0)}&0\\\
0&D_{33t}^{(0)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(3)}\\\
-2D_{3j}^{(3)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(2)}&U^{T}D_{33}^{(2)}\\\
-\bar{U}D_{2\times
2}^{(2)}&-\bar{U}D_{j3}^{(2)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(1)}+U_{x}^{T}D_{3j}^{(1)}&U^{T}\bar{U}D_{j3}^{(1)}+U_{x}^{T}D_{33}^{(1)}\\\
-\bar{U}_{x}D_{2\times
2}^{(1)}-2|u|^{2}D_{3j}^{(1)}&\bar{U}_{x}D_{j3}^{(1)}-2|u|^{2}D_{33}^{(1)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}0&U^{T}D_{33}^{(0)}\\\ -\bar{U}D_{2\times
2}^{(0)}&0\end{array}\right)-\left(\begin{array}[]{ll}0&U_{xx}^{T}D_{33}^{(0)}\\\
-\bar{U}_{xx}D_{2\times 2}^{(0)}&0\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(0)}&0\\\ 0&0\end{array}\right).\end{array}$ (A.8a) this implies that
$\begin{array}[]{l}D_{2\times 2t}^{(0)}=0\quad D_{33t}^{(0)}=0\\\
\left\\{\begin{array}[]{l}D_{j3}^{(3)}=-\frac{i}{2}U^{T}D_{33}^{(2)}+\frac{1}{4}U_{x}^{T}D_{33}^{(1)}+\frac{i}{4}|u|^{2}U^{T}D_{33}^{(0)}+\frac{i}{8}U_{xx}^{T}D_{33}^{(0)}\\\
D_{3j}^{(3)}=-\frac{i}{2}\bar{U}D_{2\times
2}^{(2)}-\frac{1}{4}\bar{U}_{x}D_{2\times
2}^{(1)}+\frac{i}{4}|u|^{2}\bar{U}D_{2\times
2}^{(0)}+\frac{i}{8}\bar{U}_{xx}D_{2\times
2}^{(0)}.\end{array}\right.\end{array}$ (A.8b)
$O(k^{-1})$, we get
$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times
2t}^{(1)}&D_{j3t}^{(1)}\\\
D_{3jt}^{(1)}&D_{33t}^{(1)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(4)}\\\
-2D_{3j}^{(4)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(3)}&U^{T}D_{33}^{(3)}\\\
-\bar{U}D_{2\times
2}^{(3)}&-\bar{U}D_{j3}^{(3)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(2)}+U_{x}^{T}D_{3j}^{(2)}&U^{T}\bar{U}D_{j3}^{(2)}+U_{x}^{T}D_{33}^{(2)}\\\
-\bar{U}_{x}D_{2\times
2}^{(2)}-2|u|^{2}D_{3j}^{(2)}&\bar{U}_{x}D_{j3}^{(2)}-2|u|^{2}D_{33}^{(2)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}U^{T}D_{3j}^{(1)}&U^{T}D_{33}^{(1)}\\\
-\bar{U}D_{2\times
2}^{(1)}&-\bar{U}D_{j3}^{(1)}\end{array}\right)-\left(\begin{array}[]{ll}U_{xx}^{T}D_{3j}^{(1)}&U_{xx}^{T}D_{33}^{(1)}\\\
-\bar{U}_{xx}D_{2\times
2}^{(1)}&-\bar{U}_{xx}D_{j3}^{(1)}\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(1)}&\sigma_{3}D_{j3}^{(1)}\\\ 0&0\end{array}\right).\end{array}$ (A.9a)
this implies that $\begin{array}[]{l}\left\\{\begin{array}[]{l}D_{2\times
2t}^{(1)}=\frac{i}{2}\\{U^{T}\bar{U}_{xx}+U_{xx}\bar{U}-U_{x}^{T}\bar{U}_{x}+6|u|^{2}U^{T}\bar{U}\\}D_{2\times
2}^{(0)}\\\
D_{33t}^{(1)}=-i\\{u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4}\\}D_{33}^{(0)}.\end{array}\right.\\\
\left\\{\begin{array}[]{l}D_{j3}^{(4)}=\frac{1}{16}U_{t}^{T}D_{33}^{(0)}-\frac{i}{2}U^{T}D_{33}^{(3)}+\frac{1}{4}U_{x}^{T}D_{33}^{(2)}+\frac{i}{4}|u|^{2}U^{T}D_{33}^{(1)}+\frac{i}{8}U_{xx}^{T}D_{33}^{(1)}+\frac{1}{8}|u|^{2}U_{x}^{T}D_{33}^{(0)}\\\
D_{3j}^{(3)}=-\frac{1}{16}\bar{U}_{t}D_{2\times
2}^{(0)}-\frac{i}{2}\bar{U}D_{2\times
2}^{(3)}-\frac{1}{4}\bar{U}_{x}D_{2\times
2}^{(2)}+\frac{i}{4}|u|^{2}\bar{U}D_{2\times
2}^{(1)}+\frac{i}{8}\bar{U}_{xx}D_{2\times
2}^{(1)}-\frac{1}{8}|u|^{2}\bar{U}_{x}D_{2\times
2}^{(0)}.\end{array}\right.\end{array}$ (A.9b)
$O(k^{-2})$, we get
$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times
2t}^{(2)}&D_{j3t}^{(2)}\\\
D_{3jt}^{(2)}&D_{33t}^{(2)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(5)}\\\
-2D_{3j}^{(5)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(4)}&U^{T}D_{33}^{(4)}\\\
-\bar{U}D_{2\times
2}^{(4)}&-\bar{U}D_{j3}^{(4)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(3)}+U_{x}^{T}D_{3j}^{(3)}&U^{T}\bar{U}D_{j3}^{(3)}+U_{x}^{T}D_{33}^{(3)}\\\
-\bar{U}_{x}D_{2\times
2}^{(3)}-2|u|^{2}D_{3j}^{(3)}&\bar{U}_{x}D_{j3}^{(3)}-2|u|^{2}D_{33}^{(3)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}U^{T}D_{3j}^{(2)}&U^{T}D_{33}^{(2)}\\\
-\bar{U}D_{2\times
2}^{(2)}&-\bar{U}D_{j3}^{(2)}\end{array}\right)-\left(\begin{array}[]{ll}U_{xx}^{T}D_{3j}^{(2)}&U_{xx}^{T}D_{33}^{(2)}\\\
-\bar{U}_{xx}D_{2\times
2}^{(2)}&-\bar{U}_{xx}D_{j3}^{(2)}\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(2)}&\sigma_{3}D_{j3}^{(2)}\\\ 0&0\end{array}\right).\end{array}$ (A.10a)
this implies that $\left\\{\begin{array}[]{l}\begin{array}[]{rl}D_{2\times
2t}^{(2)}=&\frac{i}{2}\\{U^{T}\bar{U}_{xx}+U_{xx}\bar{U}-U_{x}^{T}\bar{U}_{x}+6|u|^{2}U^{T}\bar{U}\\}D_{2\times
2}^{(1)}\\\
&+\\{-\frac{1}{4}U^{T}\bar{U}_{t}+\frac{1}{2}|u|^{2}(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}+\frac{1}{4}(u_{xx}\bar{u}_{x}-u_{x}\bar{u}_{xx})\sigma_{3}\\}\end{array}\\\
D_{33t}^{(2)}=-i\\{u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4}\\}D_{33}^{(1)}-\frac{1}{4}(|u|^{2})_{t}D_{33}^{(0)}.\end{array}\right.$
(A.10b)
Also, from the $x-$part of the Lax pair, we have the following equations
$D_{2\times 2x}^{(0)}=0,\quad D_{33x}^{(0)}=0.$ (A.11a)
$\left\\{\begin{array}[]{l}D_{2\times
2x}^{(1)}=-\frac{i}{2}U^{T}\bar{U}D_{2\times 2}^{(0)}\\\
D_{33x}^{(1)}=i|u|^{2}D_{33}^{(0)}.\end{array}\right.$ (A.11b)
$\left\\{\begin{array}[]{l}D_{2\times
2x}^{(2)}=-\frac{i}{2}U^{T}\bar{U}D_{2\times
2}^{(1)}-\frac{1}{4}U^{T}\bar{U}_{x}D_{2\times 2}^{(0)}\\\
D_{33x}^{(2)}=i|u|^{2}D_{33}^{(1)}-\frac{1}{4}(|u|^{2})_{x}D_{33}^{(0)}.\end{array}\right.$
(A.11c)
Then from the integral contours $\gamma_{j}$, we can get
$D_{2\times 2}^{(0)}=\mathbb{I}_{2\times 2},\quad D_{33}^{(0)}=1.$ (A.12)
## Appendix B The asymptotic behavior of $c_{j}(t,k)$
Let
$\mu_{2}(0,t,k)=\left(\begin{array}[]{ll}\Phi_{2\times 2}&\Phi_{j3}\\\
\Phi_{3j}&\Phi_{33}\end{array}\right).$
The global relation shows that
$\Phi_{2\times 2}\frac{s_{j3}}{s_{33}}e^{-8ik^{3}t}+\Phi_{j3}=c_{j}.$ (B.1)
And from equation
$\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.$
we get
$\begin{array}[]{l}\left(\begin{array}[]{ll}\Phi_{2\times 2}&\Phi_{j3}\\\
\Phi_{3j}&\Phi_{33}\end{array}\right)_{t}+4ik^{3}\left(\begin{array}[]{ll}0&2\Phi_{j3}\\\
-2\Phi_{3j}&0\end{array}\right)=4k^{2}\left(\begin{array}[]{ll}U^{T}\Phi_{3j}&U^{T}\Phi_{33}\\\
-\bar{U}\Phi_{2\times 2}&-\bar{U}\Phi_{j3}\end{array}\right)\\\
+2ik\left(\begin{array}[]{ll}U^{T}\bar{U}\Phi_{2\times
2}+U_{x}^{T}\Phi_{3j}&U^{T}\bar{U}\Phi_{j3}+U_{x}^{T}\Phi_{33}\\\
\bar{U}_{x}\Phi_{2\times
2}-2|u|^{2}\Phi_{3j}&-\bar{U}_{x}\Phi_{j3}-2|u|^{2}\Phi_{33}\end{array}\right)-4|u|^{2}\left(\begin{array}[]{ll}U^{T}\Phi_{3j}&U^{T}\Phi_{33}\\\
-\bar{U}\Phi_{2\times 2}&-\bar{U}\Phi_{j3}\end{array}\right)\\\
-\left(\begin{array}[]{ll}U_{xx}^{T}\Phi_{3j}&U_{xx}^{T}\Phi_{33}\\\
-\bar{U}_{xx}\Phi_{2\times
2}&-\bar{U}_{xx}\Phi_{j3}\end{array}\right)+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}\Phi_{2\times
2}&\sigma_{3}\Phi_{j3}\\\ 0&0\end{array}\right).\end{array}$ (B.2)
From the second column of the equation (B.2) we get
$\left\\{\begin{array}[]{l}\begin{array}[]{rl}\Phi_{j3t}+8ik^{3}\Phi_{j3}=&4k^{2}U^{T}\Phi_{33}+2ik(U^{T}\bar{U}\Phi_{j3}+U^{T}_{x}\Phi_{33})\\\
&-4|u|^{2}U^{T}\Phi_{33}-U_{xx}^{T}\Phi_{33}+(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}\Phi_{j3}\end{array}\\\
\Phi_{33t}=-4k^{2}\bar{U}\Phi_{j3}+2ik(\bar{U}_{x}\Phi_{j3}-2|u|^{2}\Phi_{33})+4|u|^{2}\bar{U}\Phi_{j3}+\bar{U}_{xx}\Phi_{j3}.\end{array}\right.$
(B.3)
Suppose
$\left(\begin{array}[]{l}\Phi_{j3}\\\
\Phi_{33}\end{array}\right)=(\alpha_{0}(t)+\frac{\alpha_{1}(t)}{k}+\frac{\alpha_{2}(t)}{k^{2}}+\cdots)+(\beta_{0}(t)+\frac{\beta_{1}(t)}{k}+\frac{\beta_{2}(t)}{k^{2}}+\cdots)e^{-8ik^{3}t}$
(B.4)
where the coefficients $\alpha_{l}(t)$ and $\beta_{l}(t)$, $l\geq 0$, are
independent of $k$. To determine these coefficients,we substitute the above
equation into equation (B.3) and use the initial conditions
$\alpha_{0}(0)+\beta_{0}(0)=(0_{1\times
2},1)^{T},\quad\alpha_{1}(0)+\beta_{1}(0)=(0_{1\times 2},0)^{T}.$
Then we get
$\begin{array}[]{rl}\left(\begin{array}[]{l}\Phi_{j3}\\\
\Phi_{33}\end{array}\right)=&\left(\begin{array}[]{l}0_{1\times 2}\\\
1\end{array}\right)+\frac{1}{k}\left(\begin{array}[]{l}\Phi_{j3}^{(1)}\\\
\Phi_{33}^{(1)}\end{array}\right)+\frac{1}{k^{2}}\left(\begin{array}[]{l}\Phi_{j3}^{(2)}\\\
\Phi_{33}^{(2)}\end{array}\right)+\cdots\\\
&+\left[-\frac{1}{k}\left(\begin{array}[]{l}\Phi_{j3}^{(1)}(0)\\\
0\end{array}\right)+\cdots\right]e^{-8ik^{3}t}\end{array}$ (B.5)
From the first column of the equation (B.2) we get
$\left\\{\begin{array}[]{l}\begin{array}[]{rl}\Phi_{2\times
2t}=&4k^{2}U^{T}\Phi_{3j}+2ik(U^{T}\bar{U}\Phi_{2\times
2}+U_{x}^{T}\Phi_{3j})\\\
&-4|u|^{2}U^{T}\Phi_{3j}-U_{xx}^{T}\Phi_{3j}+(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}\Phi_{2\times
2}\end{array}\\\ \Phi_{3jt}-8ik^{3}\Phi_{3j}=-4k^{2}\bar{U}\Phi_{2\times
2}+2ik(\bar{U}_{x}\Phi_{2\times
2}-2|u|^{2}\Phi_{3j})+4|u|^{2}\bar{U}\Phi_{2\times
2}+\bar{U}_{xx}\Phi_{2\times 2}.\end{array}\right.$ (B.6)
Suppose
$\left(\begin{array}[]{l}\Phi_{2\times 2}\\\
\Phi_{3j}\end{array}\right)=(\xi_{0}(t)+\frac{\xi_{1}(t)}{k}+\frac{\xi_{2}(t)}{k^{2}}+\cdots)+(\nu_{0}(t)+\frac{\nu_{1}(t)}{k}+\frac{\nu_{2}(t)}{k^{2}}+\cdots)e^{8ik^{3}t}$
(B.7)
where the coefficients $\xi_{l}(t)$ and $\nu_{l}(t)$, $l\geq 0$, are
independent of $k$. To determine these coefficients,we substitute the above
equation into equation (B.6) and use the initial conditions
$\xi_{0}(0)+\nu_{0}(0)=(\mathbb{I}_{2\times 2},0_{2\times 1})^{T},$
Then we get
$\begin{array}[]{rl}\left(\begin{array}[]{l}\Phi_{2\times 2}\\\
\Phi_{3j}\end{array}\right)=&\left(\begin{array}[]{l}\mathbb{I}_{2\times 2}\\\
0_{2\times
1}\end{array}\right)+\frac{1}{k}\left(\begin{array}[]{l}\Phi_{2\times
2}^{(1)}\\\ \Phi_{3j}^{(1)}\end{array}\right)+\cdots\\\
&+\left[\frac{1}{k^{2}}\left(\begin{array}[]{l}0\\\
\nu_{2}^{(2)}\end{array}\right)+\cdots\right]e^{8ik^{3}t}\end{array}$ (B.8)
So, from the equation (B.1) and the asymptotic of $s_{j3}(k)$ and $s_{33}(k)$,
we get the asymptotic behavior of $c_{j}(t,k)$ as $k\rightarrow\infty$,
$c_{j}(t,k)=\frac{\Phi_{j3}^{(1)}}{k}+\frac{\Phi_{j3}^{(2)}}{k^{2}}+\frac{\Phi_{j3}^{(3)}}{k^{3}}+\cdots.$
(B.9)
## References
* [1] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A 453(1997), 1411-1443.
* [2] A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys. 230(2002), 1-39.
* [3] A.S. Fokas, A Unified Approach to Boundary Value Problems, in: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.
* [4] A. Boutet De Monvel, A.S. Fokas, D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys. 263 (2006) 133 C172.
* [5] A. Boutet de Monvel,A.S.Fokas,D.Shepelsky, The mKDV equation on the half-line, J. Inst. Math. Jussieu.3(2004), 139-164.
* [6] A. S. Fokas, A. R. Its and L. Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity. 18(2005), 1771-1822.
* [7] S. Kamvissis, Semiclassical nonlinear Schr dinger on the half line, J. Math. Phys. 44 (2003) 5849 5868.
* [8] J. Lenells, Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole, Comm. Math. Phys. 304 (2011) 585-635.
* [9] J. Lenells, A.S. Fokas, Boundary-value problems for the stationary axisymmetric Einstein equations: a rotating disc, Nonlinearity 24 (2011) 177-206.
* [10] N. Sasa, J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schr dinger equation, J. Phys. Soc. Japan 60 (1991) 409 417.
* [11] D.J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class $\psi_{xxx}+6Q\psi_{x}+6R\psi=\lambda\psi$, Stud. Appl. Math. 62 (1980) 189-216.
* [12] J. Lenells, Initial-boundary value problems for integrable evolution equations with $3\times 3$ Lax pairs, Physica D 241(2012) 857-875.
* [13] J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Analysis 76(2013) 122-139.
* [14] R. Beals and R. Coifman, Scattering and inverse scattering for first order systems, Comm. in Pure and Applied Math. 37(1984), 39–90.
* [15] P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Ann. of Math. (2) 137(1993), 295-368.
* [16] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure. Appl. Math.21(1968), 467-490.
* [17] A. S. Forkas and J. Lenells, The unified method: [email protected] problem on the half-line, J. Phys. A: Math. Theor. 45(2012) 195201;
* [18] J. Lenells and A. S. Forkas, The unified method: ii@. NLS on the half-line t-periodic boundary conditions, J. Phys. A: Math. Theor. 45(2012) 195202;
* [19] J. Lenells and A. S. Forkas, The unified method: iii@. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor. 45(2012) 195203;
|
arxiv-papers
| 2013-04-16T06:43:56 |
2024-09-04T02:49:44.502505
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Xu, Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1304.4586"
}
|
1304.4632
|
Lifting Automorphisms of Quotients by Central Subgroups
Ben Kane, Andrew Shallue
Department of Mathematics
University of Wisconsin-Madison
480 Lincoln Dr
Madison, WI 53706, USA
[email protected] [email protected], [email protected]
[email protected]
###### Abstract.
Given a finitely presented group $G$, we wish to explore the conditions under
which automorphisms of quotients $G/N$ can be lifted to automorphisms of $G$.
We discover that in the case where $N$ is a central subgroup of $G$, the
question of lifting can be reduced to solving a certain matrix equation. We
then use the techniques developed to show that $Inn(G)$ is not characteristic
in $Aut(G)$, where $G$ is a metacyclic group of order $p^{n}$, $p\neq 2$.
###### Key words and phrases:
finitely presented group, automorphism group, lift
###### 2000 Mathematics Subject Classification:
20F28
## 1\. Introduction
Let $F$ be the free group on $n$ letters, and let $G$ be a quotient of that
group. We will be working with a given presentation of $G$, namely
$G:=\left<x_{1},x_{2},\dots
x_{n}|r_{1}(\text{\boldmath{$x$}}),r_{2}(\text{\boldmath{$x$}}),\dots
r_{m}(\text{\boldmath{$x$}})\right>,$
where $x$ represents the $n$-tuple $(x_{1},x_{2},\dots,x_{n})$. This vector
notation continues throughout the paper. For later ease of exposition, we will
think of the relations $r_{k}$ as noncommutative monomials on $n$ variables
$(x_{1},\dots,x_{n})$ defined by
$r_{k}(\text{\boldmath{$x$}})=\overset{s_{k}}{\underset{l=1}{\prod}}{x_{j_{k,l}}^{e_{k,l}}}$
with $e_{k,l}\in\\{\pm 1\\}$.
It is a well known fact that if $N$ is a characteristic subgroup of $G$, then
automorphisms of $G$ induce automorphisms of $G/N$ canonically by acting on
the coset representatives. However, much less is known about the conditions
under which a lift of an element of $Aut(G/N)$ exists.
Study so far has focused on the case where $G$ is the free group $F$. Many
techniques have been developed to show whether automorphisms of various
quotients of $F$ are tame (i.e. for which lifts to $F$ exist). See for
instance [1, 2, 5, 6].
We show in the case where $N$ is a central subgroup of an arbitrary group $G$
that automorphisms of $G/N$ are in one-to-one correspondence with solutions to
a certain set of matrix equations that depend only on the relations.
## Acknowledgements
The authors are grateful to N. Boston for giving motivation to the problem and
for valuable conversation.
## 2\. Homomorphic Lifts
First, assume that $G$ is a finitely presentable group, and $N$ is a cyclic,
central subgroup of $G$, generated by an element $z$. We will later generalize
to the case where $N$ is not cyclic. We wish to study when automorphisms of
$G/N$ can be lifted to automorphisms of $G$. We first give a condition for
when such automorphisms lift to homomorphisms of $G$, so in this section lift
means homomorphic lift.
We will also consider $G$ as the image of $F$ under the canonical quotient map
$\pi:F\to F/R$, where $R$ is the normal closure of the set
$\\{r_{1}(\text{\boldmath{$x$}})\dots r_{m}(\text{\boldmath{$x$}})\\}$.
However, this will be for convenience only. In practice, working with the
relations will suffice, as evidenced by the following lemma, the proof of
which is immediate.
###### Lemma 1.
Let $\theta\in Hom(F,H)$ be given. Then
$\theta(r_{k}(\text{\boldmath{$x$}}))=1$ for all $k$ if and only if
$\theta(R)=1$.
The following definition will help to clarify our direction in this paper:
###### Definition 1.
We say that an $n$-tuple $\text{\boldmath{$g$}}=(g_{1},g_{2},\dots,g_{n})\in
G^{n}$ _extends_ to a homomorphism if the homomorphism $\theta:F\mapsto G$,
defined by $\theta(x_{i})=g_{i}$, factors through $R$. In this case
$\theta\circ\pi^{-1}$ is well defined and defines an element $\psi\in
Hom(G,G)$.
###### Lemma 2.
An $n$-tuple $\text{\boldmath{$g$}}\in G^{n}$ extends to a homomorphism if and
only if $r_{k}(\text{\boldmath{$g$}})=1$ for every $k=1,\dots,m$.
###### Proof.
Let $\text{\boldmath{$g$}}\in G^{n}$ be given. Consider $\theta$ defined as
above. Note that $g$ extends to a homomorphism if and only if $\theta(R)=1$.
By lemma 1, $\theta(R)=1$ if and only if
$\theta(r_{k}(\text{\boldmath{$x$}}))=1$ for all $k$. So it remains to show
that $\theta(r_{k}(\text{\boldmath{$x$}}))=1$ if and only if
$r_{k}(\text{\boldmath{$g$}})=1$. However, since $\theta$ is a homomorphism,
$\theta(r_{k}(\text{\boldmath{$x$}}))=r_{k}(\theta(\text{\boldmath{$x$}}))=r_{k}(\text{\boldmath{$g$}}).$
∎
The following definition is vital, since homomorphic lifts are the focus of
this paper.
###### Definition 2.
A _lift_ of $\varphi\in End(G/N)$ is $\psi\in End(G)$ such that
$\psi(g)N=\varphi(gN)\qquad\text{for every }g\in G.$
###### Theorem 1.
If $\varphi\in End(G/N)$, then there is a one-to-one correspondence between
lifts of $\varphi$ to $End(G)$ and $n$-tuples $\text{\boldmath{$g$}}\in G^{n}$
such that $g_{i}N=\varphi(x_{i}N)$ and $r_{k}(\text{\boldmath{$g$}})=1$ for
every $k\in 1,\dots,m$.
###### Proof.
First suppose $\psi$ is a lift of $\varphi$. Then by definition
$\varphi(x_{i}N)=\psi(x_{i})N$, so we set $g_{i}:=\psi(x_{i})$. Also,
$1=\psi(1)=\psi(r_{k}(\text{\boldmath{$x$}}))=r_{k}(\psi(\text{\boldmath{$x$}}))=r_{k}(\text{\boldmath{$g$}})$
since $\psi$ is a homomorphism.
Conversely, suppose we have an $n$-tuple $g$ such that
$g_{i}N=\varphi(x_{i}N)$ and $r_{k}(\text{\boldmath{$g$}})=1$ for every
$k=1\dots m$. Then by Lemma 2, $g$ extends to $\psi\in End(G)$, where
$\psi(x_{i})=g_{i}$. Thus
$\psi(x_{i})N=g_{i}N=\varphi(x_{i}N)$
and this is exactly the definition of $\psi$ being a lift of $\varphi$. ∎
We now give a nice characterization of the lifts of $\varphi$. For this we
define a certain matrix and a vector.
Define $m_{ij}$ to be the degree of $x_{j}$ in the commutative image of the
word $r_{i}(\text{\boldmath{$x$}})$. Note that since
$r_{i}(\text{\boldmath{$x$}})=\overset{s_{i}}{\underset{l=1}{\prod}}x_{j_{i,l}}^{e_{i,l}}$,
$m_{ij}=\overset{}{\underset{\overset{}{\underset{j_{i,l}=j}{l\in
1..s_{i}}}}{\sum}}e_{i,l}.$
We consider the matrix $M:=\left(m_{ij}\right)$. To make the construction of
$M$ clear, we give an example.
###### Example 1.
For
$r_{1}(\text{\boldmath{$x$}})=x_{1}^{2}\cdot x_{2}^{-1}\cdot x_{1}^{-5}\cdot
x_{2}^{-1},\qquad r_{2}(\text{\boldmath{$x$}})=x_{1}\cdot x_{2}^{-3}\cdot
x_{1}^{7},$
we have
$M=\left(\begin{smallmatrix}-3&-2\\\ 8&-3\end{smallmatrix}\right)$
For $\varphi\in Aut(G/N)$, fix a set of coset representatives
$\overline{x_{i}}\in\varphi(x_{i}N)$. Since
$N=\varphi(r_{k}(\text{\boldmath{$x$}})N)=r_{k}(\varphi(\text{\boldmath{$x$}}N))=r_{k}(\overline{\text{\boldmath{$x$}}})N,$
it is clear that $r_{k}(\overline{\text{\boldmath{$x$}}})\in N$. Since $N$ is
generated by $z$, we can choose $w_{i}$ such that
$r_{i}(\overline{\text{\boldmath{$x$}}})=z^{-w_{i}}$. Note that
$\text{\boldmath{$w$}}:=(w_{1},\dots,w_{m})$ is only defined up to the order
of $N$.
###### Theorem 2.
The lifts of $\varphi$ are in one-to-one correspondence with solutions
$\text{\boldmath{$v$}}=(v_{1},\dots v_{n})$ to the matrix equation
$M\text{\boldmath{$v$}}=\text{\boldmath{$w$}}\pmod{\\#N}$
where, if $N$ is infinite, we simply mean the matrix equation on the integers.
###### Proof.
We know from above that lifts are in one-to-one correspondence with
$\text{\boldmath{$g$}}\in G^{n}$ such that $r_{k}(\text{\boldmath{$g$}})=1$
and $g_{i}N=\varphi(x_{i}N)$.
However, if $g_{i}N=\varphi(x_{i}N)=\overline{x_{i}}N$, then
$g_{i}\in\overline{x_{i}}N$. But then $g_{i}=\overline{x_{i}}z^{v_{i}}$ for
some $i$. So $g_{i}N=\varphi(x_{i}N)$ if and only if
$g_{i}=\overline{x_{i}}z^{v_{i}}$ for some $v_{i}$. So lifts are in one-to-one
correspondence with $\text{\boldmath{$g$}}\in G^{n}$ such that
$r_{k}(\text{\boldmath{$g$}})=1$ and $g_{i}=\overline{x_{i}}z^{v_{i}}$.
Since $z$ is central,
$\begin{array}[]{lcl}r_{i}(\overline{x_{1}}z^{v_{1}},\dots,\overline{x_{n}}z^{v_{n}})&=&r_{i}(\overline{\text{\boldmath{$x$}}})r_{i}(z^{v_{1}},z^{v_{2}},\dots,z^{v_{n}})\\\
&=&z^{-w_{i}}z^{\overset{n}{\underset{j=1}{\sum}}m_{ij}v_{j}}\\\
&=&z^{-w_{i}+\overset{n}{\underset{j=1}{\sum}}m_{ij}v_{j}}\end{array}$
But this is equal to $1$ if and only if
$-w_{i}+\overset{n}{\underset{j=1}{\sum}}m_{ij}v_{j}=0\pmod{\\#N}$. This
corresponds exactly to a solution of the above matrix equation.
∎
Having shown the result for $N$ cyclic, it is easy to generalize to the case
when $N$ is not cyclic.
If $N$ is a central subgroup of $G$, generated by $z_{1},\dots,z_{t}$, then we
have
$r_{k}(\overline{\text{\boldmath{$x$}}})=z_{1}^{-w_{1,k}}z_{2}^{-w_{2,k}}\cdots
z_{t}^{-w_{t,k}}$
###### Corollary 1.
The lifts of $\varphi$ are in one-to-one correspondence with solutions of the
matrix equation
$\left(\begin{smallmatrix}M&&\\\ &\ddots&\\\ &&M\\\
\end{smallmatrix}\right)\text{\boldmath{$v$}}=\left(\begin{smallmatrix}\text{\boldmath{$w$}}_{1}\\\
\vdots\\\ \text{\boldmath{$w$}}_{t}\end{smallmatrix}\right)\pmod{\\#N},$
where, if $N$ is infinite, we simply mean the matrix equation on the integers.
###### Proof.
The proof follows from the proof of the previous theorem, noting that each
generator of $N$ commutes. ∎
## 3\. Automorphic Lifts
In this section we investigate when such homomorphic lifts are automorphic. As
before, we assume that $G$ is finitely presented and $N$ is a central subgroup
of $G$.
###### Lemma 3.
If $N$ is abelian, finitely generated, and $\psi\in End(N)$, then $\psi$
surjective implies $\psi$ injective.
This lemma follows from the fundamental theorem of abelian groups and the fact
that the rank of the image plus the rank of the kernel equals the rank of $N$.
###### Lemma 4.
A lift $\psi\in End(G)$ of $\varphi\in Aut(G/N)$ is an automorphism if and
only if $\psi(N)=N$.
###### Proof.
Consider $K:=Ker(\psi)$ and $H:=Im(\psi)$.
Since $\psi$ is a lift of $\varphi$, we have the identity
$N=\psi(K)N=\varphi(KN).$
As $\varphi$ is injective, it follows that $KN=N$, and hence $K\subseteq N$.
So $K=Ker(\psi|_{N})$. Therefore $\psi$ is injective on $G$ if and only if it
is injective when resticted to $N$. Because $N$ is abelian and finitely
generated by assumption, it will suffice to show $\psi|_{N}$ is surjective,
even if $N$ is infinite.
Moreover, since $\psi(G)N=\varphi(GN)=G$, it follows that $HN=G$. If
$\psi(N)=N$, then $N\subseteq H$, so that $G=HN=H$. Conversely, $N$ is in the
image of $\psi$, and since for $g\in G$, $\psi(g)N=\varphi(gN)$, and
$\varphi(gN)=N$ if and only if $g\in N$, we know that the preimage of $N$ is
$N$. So we have $\psi$ surjective if and only if $\psi(N)=N$.
∎
In the case where $\\#N$ is finite and squarefree, the following result will
show that the previous work for finding homomorphic lifts will suffice for
showing that there is an automorphic lift. The proof relies heavily on finite
group theory, for which a good reference is [3].
###### Theorem 3.
The automorphism $\varphi$ lifts to $\psi\in End(G)$ if and only if $\varphi$
lifts to some $\psi^{\prime}\in Aut(G)$.
###### Proof.
Let $\psi\in End(G)$ a lift of $\varphi$ be given. Let $K=Ker(\psi)$ and
$H=Im(\psi)$. We will show that $G=K\times H$, from which the theorem follows
directly, since $\psi|_{H}$ is an isomorphism and $(Id,\psi|_{H})$ will be a
lift as desired, where $Id$ stands for the identity on $K$
From the proof of Lemma 4, $K\subseteq N$. Since $\\#N$ is squarefree and $N$
is abelian, it splits completely. In particular, by the first isomorphism
theorem, $N=K\times H_{N}$, where $H_{N}=Im(\psi|_{N})$. We know from the
above proof that $HN=G$. Then
$G=H(K\times H_{N})=(HK)(HH_{N})=HKH=HK.$
Let $h\in H\cap K$ be given. Notice first that $Im(\psi)\cap N=Im(\psi|_{N})$
since the preimage of $N$ is contained in $N$. Since $h\in K\subseteq N$, we
have $h\in H_{N}$. But $H_{N}\cap K=1$, so $h=1$. Hence $G=H\ltimes K$.
Therefore, since $K$ is central, $G=H\times K$. ∎
We are now going to construct a set of matrix equations whose solutions
correspond to automorphic lifts of $\varphi$. Let us first assume $N$ is
cyclic and generated by $z$. The key idea is that matching exponents of $z$
reduces to solving linear equations. We first fix a noncommutative monomial
$f(\text{\boldmath{$x$}}):=\overset{s}{\underset{l=1}{\prod}}x_{j_{l}}^{e_{l}}$
such that $f(\text{\boldmath{$x$}})=z$. Define
$M_{m+1,j}:=\overset{}{\underset{\overset{}{\underset{j_{l}=j}{l\in
1..s}}}{\sum}}e_{l},$
which is the exponent of $x_{j}$ in the commutative image of
$f(\text{\boldmath{$x$}})$. Since $\varphi(N)=N$, we can choose $w_{m+1}$ such
that $z^{-w_{m+1}}=f(\overline{\text{\boldmath{$x$}}})$, where
$\overline{x_{i}}$ is as above.
For an automorphism, the image of $z$ must be a generator of $N$. We will
define a matrix equation for each of the possible generators of $N$. For $N$
infinite, define $w_{m+1}^{(1)}:=w_{m+1}+1$ and $w_{m+1}^{(2)}:=w_{m+1}-1$ and
set
$M^{\prime}:=\left(\begin{array}[]{c}M\\\ M_{m+1}\end{array}\right).$
For $N$ finite, with $p$ the smallest prime dividing $\\#N$, define
$w_{m+1}^{(k)}:=w_{m+1}+k\frac{\\#N}{p}$ for $k=1,\dots,p-1$ and set
$M^{\prime}:=\left(\begin{array}[]{c}M\\\
\frac{\\#N}{p}M_{m+1}\end{array}\right).$
In either case, set
$\text{\boldmath{$w$}}^{(k)}:=\left(\begin{array}[]{c}\text{\boldmath{$w$}}\\\
w_{m+1}^{(k)}\end{array}\right)$ for every $k$.
###### Theorem 4.
Automorphic lifts of $\varphi\in Aut(G/N)$ are in one-to-one correspondence
with solutions $\text{\boldmath{$v$}}=(v_{1},\dots,v_{n})$ to the matrix
equations
$M^{\prime}\text{\boldmath{$v$}}=\text{\boldmath{$w$}}^{(k)}\pmod{\\#N}$
for $k=1,2$ if $N$ is infinite, and $k=1,\dots,p-1$ for $N$ finite and $p$ the
smallest prime dividing $\\#N$.
###### Proof.
For a solution $v$ to the equation
$M\text{\boldmath{$v$}}=\text{\boldmath{$w$}}$, we have a lift $\psi\in
End(G)$ by Theorem 2. By Lemma 4, $\psi\in Aut(G)$ if and only if $\psi(z)$
generates $N$. But then
$\displaystyle\psi(z)$ $\displaystyle=$ $\displaystyle
f(\psi(\text{\boldmath{$x$}}))=f(\overline{x_{1}}z^{v_{1}},\dots,\overline{x_{n}}z^{v_{n}})$
$\displaystyle=$ $\displaystyle
f(\overline{\text{\boldmath{$x$}}})f(z^{\text{\boldmath{$v$}}})=z^{-w_{m+1}+\overset{n}{\underset{j=1}{\sum}}m_{m+1,j}v_{j}}$
If $N$ is infinite we need $\psi(z)=z^{\pm 1}$ to generate $N$. This is
equivalent to $-w_{m+1}+\overset{n}{\underset{j=1}{\sum}}m_{m+1,j}v_{j}=\pm
1$. These are exactly the matrix equations listed above.
If $N$ is finite, then we need $o(\psi(z))=\\#N$, so we simply need
$\psi(z)^{\frac{\\#N}{p}}\neq 1$. But this is equivalent to
$-\frac{\\#N}{p}w_{m+1}+\frac{\\#N}{p}\overset{n}{\underset{j=1}{\sum}}m_{m+1,j}v_{j}\neq
0\pmod{\\#N}.$
But, if this sum is not 0, it must be $k\frac{\\#N}{p}$ for some
$k=1,\dots,p-1$. These correspond to the above vectors.
∎
Similarly to above, we can generalize this to the case when $N$ is not cyclic.
If $N$ is generated by $z_{1},\dots z_{t}$, then, following the above process,
we can choose a presentation of each element $z_{i}$ and add another row to
$M$ with this presentation, generating a matrix $M^{\prime\prime}$. Next, we
choose for each $z_{i}$ an element of $z_{i}^{\prime}$ of $N$ which we would
like to map $z_{i}$ to, such that the $z_{i}^{\prime}$ also generate $N$. Note
that $z_{i}^{\prime}$ must have the same order as $z_{i}$, since we have a
homomorphism.
We write for each $k$ from 1 to the number of possible generator sets of $N$
$z_{i}^{\prime}=\overset{t}{\underset{j=1}{\prod}}z_{j}^{{w_{i,m+j}}^{(k)}}.$
###### Corollary 2.
Automorphic lifts of $\varphi\in Aut(G/N)$ are in one-to-one correspondence
with solutions to the matrix equations
$\left(\begin{smallmatrix}M^{\prime\prime}&&\\\ &\ddots&\\\
&&M^{\prime\prime}\\\
\end{smallmatrix}\right)\text{\boldmath{$v$}}=\left(\begin{smallmatrix}\text{\boldmath{$w$}}_{1}^{(k)}\\\
\vdots\\\ \text{\boldmath{$w$}}_{t}^{(k)}\end{smallmatrix}\right)\pmod{\\#N},$
###### Remark.
Note that if we have 2 elements of infinite order as generators of $N$, then
there are infinitely many choices for our $\text{\boldmath{$w$}}^{(k)}$, but
otherwise we have a finite number of choices as above.
## 4\. An Application of the Above Techniques
In this section, we give an application of the techniques developed above.
Given a metacyclic group of order $p^{n}$ ($p$ an odd prime) represented by
$G:=\left<x,y|\ x^{p^{n-1}},y^{p},x^{y}=x^{1+p^{n-2}}\right>,$
we will show that $Inn(G)$ is not characteristic in $Aut(G)$.
To do this, we need information about the structure of $A:=Aut(G)$. Schulte in
[4] found the presentation for $A$
$A=\left<x_{1},x_{2},x_{3}\left|\begin{smallmatrix}x_{1}^{p},\ x_{2}^{p},\
x_{3}^{(p-1)p^{n-2}},\ x_{1}^{-a}\cdot x_{3}^{-1}\cdot x_{1}\cdot x_{3}\cdot
x_{3}^{j(p-1)p^{n-3}},\\\ x_{2}^{-a^{-1}}\cdot x_{3}^{-1}\cdot x_{2}\cdot
x_{3}\cdot x_{3}^{k(p-1)p^{n-3}},\ x_{1}^{-1}\cdot x_{2}^{-1}\cdot x_{1}\cdot
x_{2}\cdot x_{3}^{-(p-1)p^{n-3}}\end{smallmatrix}\right.\right>,$
where $a$ is a generator for the multiplicative group
$(\mathbb{Z}/p^{n-2}\mathbb{Z})^{\times}$, $a^{-1}$ is the multiplicative
inverse $\pmod{p}$, and $j$ and $k$ are integers which can be determined
explicitly.
We shall henceforth refer to the center of $A$ as $Z:=Z(A)$. By the commutator
relations above $\left<x_{3}^{p-1}\right>\subseteq Z$. Note that
$A/\left<x_{3}^{p-1}\right>$ equals $(C_{p}\times C_{p})\rtimes C_{p-1}$.
Since that group has trivial center, by the correspondence theorem,
$\left<x_{3}^{p-1}\right>=Z$. For ease of exposition, we will denote
$x_{3}^{p-1}$ by $z$.
Another important fact is that
$I:=Inn(G)=\left<x_{1},x_{3}^{(p-1)p^{n-3}}\right>$. This follows from [4] by
showing that conjugation by $x$ and $y$ correspond to the elements $x_{1}$ and
$x_{3}^{(p-1)p^{n-3}}$, respectively.
From Section 1, we have a technique for determining when homomorphisms can be
lifted from quotient groups. Since $Z$ is large, cyclic, central, and
characteristic in $A$, $A/Z$ is a natural candidate for this process.
We will show that all elements of $Aut(A/Z)$ lift to $Aut(A)$, and then show
that some $\varphi\in Aut(A/Z)$ does not fix $IZ/Z$. We then conclude that
some $\psi\in Aut(A)$ does not fix $IZ$ and so $IZ$ is not characteristic in
$A$. Since $Z$ is characteristic, it follows that $I$ is not characteristic.
###### Theorem 5.
The canonical mapping $\pi:Aut(A)\mapsto Aut(A/Z)$ is surjective.
###### Proof.
Let $\varphi\in Aut(A/Z)$ be given and set $K:=\left<x_{1},x_{2}\right>$.
Studying $K$ in some detail will simplify the ensuing calculations.
Since $KZ$ is the unique Sylow $p$-subgroup of $A/Z$, it is characteristic,
and hence $\varphi(KZ)=KZ$. Therefore, without loss of generality, we can
choose representatives $(\overline{x_{1}},\overline{x_{2}},\overline{x_{3}})$
such that $\varphi(x_{i}Z)=\overline{x_{i}}Z$ and
$\overline{x_{1}},\overline{x_{2}}\in K$.
Since $A/K$ is cyclic and hence abelian, it follows that $A^{\prime}\subseteq
K$. Notice that $K\cap Z=\left<[x_{1},x_{2}]\right>=\left<z^{p^{n-3}}\right>$
and $K^{p}=1$. Also, the exponent of $A$ is $(p-1)p^{n-2}$, so
$A^{(p-1)p^{n-3}}$ contains only elements of order $p$, and hence is contained
in $K$, since every element of order $p$ is in $K$.
Since $Z$ is cyclic and central, Theorem 2 tells us that homomorphisms of $A$
that are lifts of $\varphi$ are in one-to-one correspondence with solutions to
$\left(\begin{smallmatrix}p&0&0\\\ 0&p&0\\\ 0&0&(p-1)p^{n-2}\\\
1-a&0&j(p-1)p^{n-3}\\\ 0&1-a^{-1}&k(p-1)p^{n-3}\\\
0&0&-(p-1)p^{n-3}\end{smallmatrix}\right)\left(\begin{smallmatrix}\\\ \\\
\text{\boldmath{$v$}}\\\ \\\ \\\
\end{smallmatrix}\right)=\left(\begin{smallmatrix}\\\ \\\
\text{\boldmath{$w$}}\\\ \\\ \\\ \end{smallmatrix}\right)\pmod{\\#Z}.$
So $\varphi$ lifts to a homomorphism if and only if this matrix is not
degenerate. We shall show that this matrix has $p^{n-3}$ solutions.
Instead of calculating $w$, we only need to show that $w_{1},w_{2},w_{3}=0$,
and $p^{n-3}\mid w_{4},w_{5},w_{6}$. Notice that the latter statement is
equivalent to $z^{w_{i}}\in K$.
For $i=1,2$, we note that $K^{p}=1$ and for $i=3$, we have
$r_{3}(\overline{\text{\boldmath{$x$}}})=1$ because the exponent of $A$ is
$(p-1)p^{n-2}$, so any choice for $\overline{x_{3}}$ will give $w_{3}=0$.
For $i=4,5,6$, notice that since $\overline{x_{1}},\overline{x_{2}}\in K$,
$r_{i}(\overline{\text{\boldmath{$x$}}})\in KA^{\prime}A^{(p-1)p^{n-3}}.$
From our comments above, we know that $A^{\prime},A^{(p-1)p^{n-3}}\subseteq
K$. Hence $r_{i}(\overline{\text{\boldmath{$x$}}})\in K$, and we have
$p^{n-3}\mid w_{i}$.
Since the matrix is taken mod $\\#Z=p^{n-2}$, the third row of the matrix is
all zeroes and $w_{3}=0$, so we can remove this redundancy. Rows 1 and 2
correspond to $pv_{1}=0$ and $pv_{2}=0$, respectively. This is equivalent to
$p^{n-3}\mid v_{1},v_{2}$.
Therefore, if $v$ is a solution, it must be the case that $p^{n-3}\mid
v_{1},v_{2}$. Hence, we can remove the first three rows and write the matrix
in the form:
$p^{n-3}\left(\begin{smallmatrix}(1-a)&0&j(p-1)\\\ 0&(1-a^{-1})&k(p-1)\\\
0&0&-(p-1)\end{smallmatrix}\right)\left(\begin{smallmatrix}\\\
\text{\boldmath{$v$}}\\\ \\\
\end{smallmatrix}\right)=p^{n-3}\left(\begin{smallmatrix}\\\
\frac{\text{\boldmath{$w$}}}{p^{n-3}}\\\ \\\
\end{smallmatrix}\right)\pmod{\\#Z}$
With $v_{1},v_{2}\in\mathbb{Z}/p\mathbb{Z}$, and
$v_{3}\in\mathbb{Z}/p^{n-2}\mathbb{Z}$.
Solutions to this matrix will correspond to solutions of
$\left(\begin{smallmatrix}1-a&0&-j\\\ 0&1-a^{-1}&-k\\\
0&0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}\\\
\text{\boldmath{$v$}}\\\ \\\
\end{smallmatrix}\right)=\left(\begin{smallmatrix}\\\
\frac{\text{\boldmath{$w$}}}{p^{n-3}}\\\ \\\ \end{smallmatrix}\right)\pmod{p}$
The determinant of this matrix is $D:=(1-a)(1-a^{-1})$, but $a$ is a generator
for the multiplicative group
$\left(\mathbb{Z}/(p^{n-1}\mathbb{Z})\right)^{\times}$, so $a\neq 1\pmod{p}$
and $a^{-1}\neq 1\pmod{p}$. Thus, $D$ is invertible, so the matrix is
solvable. Moreover, since $v_{3}\in\mathbb{Z}/p^{n-2}\mathbb{Z}$, and this
solution only fixes $v_{3}\pmod{p}$, we have $p^{n-3}$ choices for $v_{3}$,
and hence $\varphi$ lifts to $p^{n-3}$ homomorphisms of $A$.
We would now like to show that each of these homomorphisms is moreover an
automorphism. By Lemma 4, we know that $\psi$ is an automorphic lift exactly
when $\psi(z)^{p^{n-3}}=[\psi(x_{1}),\psi(x_{2})]\neq 1$.
###### Lemma 5.
We have the equality
$Z(K)=Z\cap K.$
###### Proof.
Let $h=x_{1}^{a}x_{2}^{b}\left(z^{p^{n-3}}\right)^{c}\in Z(K)$ be given. Then,
since $[x_{1},x_{2}]=z^{p^{n-3}}$,
$hx_{1}=x_{1}h\left(z^{p^{n-3}}\right)^{-b}\text{ and
}hx_{2}=x_{2}h\left(z^{p^{n-3}}\right)^{a}.$
Thus, $b=0=a$. ∎
We see that for $h\in K\backslash Z(K)$, $C_{K}(h)=\left<h\right>Z(K)$, since
$p^{3}=\\#K>\\#C_{K}(h)\geq p^{2}.$
But $\psi(x_{1})\notin\left<\psi(x_{2})\right>Z(K)$, as
$\varphi(x_{1}Z)\notin\left<\varphi(x_{2}Z)\right>$. Therefore, since
$\psi(x_{1})$ and $\psi(x_{2})$ do not commute, $[\psi(x_{1}),\psi(x_{2})]\neq
1$. So the canonical mapping is surjective. ∎
We will now proceed to show that $IZ/Z$ is not characteristic in $A/Z$. Note
that
$A/Z=(IZ\times HZ)\rtimes\left<x_{3}\right>Z\cong\left(C_{p}\times
C_{p}\right)\rtimes C_{p-1},$
with the action of $x_{3}$ on $IZ$ being $x\mapsto x^{a}$ and the action of
$x_{3}$ on $HZ$ being $x\mapsto x^{a^{-1}}$.
We would like to show that
$(\overline{\text{\boldmath{$x$}}})=\left(x_{2}^{a^{-1}},x_{1}^{a},x_{3}^{-1}\right)$
extends to a homomorphism. The presentation for $A/Z$ is the same as the
presentation for $A$, adding the relation that $z=x_{3}^{p-1}=1$. It is
straightforward to calculate that $r_{k}(\overline{\text{\boldmath{$x$}}})=1$
for every $k$. Moreover,
$\left<x_{2}^{a^{-1}},x_{1}^{a},x_{3}^{-1}\right>=A/Z$, so this homomorphism
is an automorphism. This automorphism is the desired element of $Aut(A/Z)$.
## References
* [1] S. Andreadakis, On the Automorphisms of Free Groups and Free Nilpotent Groups., Proc. London Math. Soc. (3) 15 (1965) 239-268.
* [2] S. Bachmuth, Automorphisms of Free Metabelian Groups, Trans. Amer. Math. Soc. 118 (1965) 93-104.
* [3] S. Lang, Algebra, Addison-Wesley, Reading, 1993.
* [4] M. Schulte, Automorphisms of Metacyclic $p$-groups With Cyclic Maximal Subgroups, Rose-Hulman Undergraduate Mathematics Journal 2 (2001) (available at http://www.rose-hulman.edu/mathjournal/2001/vol2-n2/paper4/v2n2-4pd.pdf).
* [5] V. Shpilrain, Non-Commutative determinants and automorphisms of groups, Comm. Algebra 25 (1997) 559-574.
* [6] E. Turner, D. Voce, Tame Automorphisms of Finitely Generated Abelian Groups, Proc. Edinburgh Math. Soc. 41 (1998) 277-287.
|
arxiv-papers
| 2013-04-16T22:05:52 |
2024-09-04T02:49:44.514035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ben Kane, Andrew Shallue",
"submitter": "Andrew Shallue",
"url": "https://arxiv.org/abs/1304.4632"
}
|
1304.4653
|
∎
11institutetext: Yang Liu 22institutetext: Department of Mathematics, Zhejiang
Normal University, Jinhua 321004, China
Fax: +86-57982298897
22email: [email protected] 33institutetext: Zhihua Chen 44institutetext:
Department of Mathematics, Tongji University, Shanghai 200092, China
55institutetext: Yifei Pan 66institutetext: Department of Mathematical
Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, Indiana
46805, USA
# A variant of Hörmander’s $L^{2}$ theorem for Dirac operator in Clifford
analysis
Yang Liu Zhihua Chen Yifei Pan
(Received: date / Accepted: date)
###### Abstract
In this paper, we give the Hörmander’s $L^{2}$ theorem for Dirac operator over
an open subset $\Omega\in\mathbb{R}^{n+1}$ with Clifford algebra. Some
sufficient condition on the existence of the weak solutions for Dirac operator
has been found in the sense of Clifford analysis. In particular, if $\Omega$
is bounded, then we prove that for any $f$ in $L^{2}$ space with value in
Clifford algebra, there exists a weak solution of Dirac operator such that
$\overline{D}u=f$
with $u$ in the $L^{2}$ space as well. The method is based on Hörmander’s
$L^{2}$ existence theorem in complex analysis and the $L^{2}$ weighted space
is utilised.
###### Keywords:
Hörmander’s $L^{2}$ theoremClifford analysis weak solutionDirac operator
###### MSC:
32W50 15A66
## 1 Introduction
The development of function theories on Clifford algebras has proved a useful
setting for generalizing many aspects of one variable complex function theory
to higher dimensions. The study of these function theories is referred to as
Clifford analysis Brackx et al (1982); Huang et al (2006); Gong et al (2009);
Ryan (2000), which is closely related to a number of studies made in
mathematical physics, and many applications in this area have been found in
recent years. In Ryan (1995), Ryan considered solutions of the polynomial
Dirac operator, which afforded an integral representation. Furthermore, the
author gave a Pompeiu representation for $C^{1}$-functions in a Lipschitz
bounded domain. In Ryan (1990), the author presented a classification of
linear, conformally invariant, Clifford-algebra-valued differential operators
over $\mathbb{C}^{n}$, which comprised the Dirac operator and its iterates. In
Qian and Ryan (1996), Qian and Ryan used Vahlen matrices to study the
conformal covariance of various types of Hardy spaces over hypersurfaces in
$\mathbb{R}^{n}$. In De Ridder et al (2012), the discrete Fueter polynomials
was introduced, which formed a basis of the space of discrete spherical
monogenics. Moreover, the explicit construction for this discrete Fueter
basis, in arbitrary dimension $m$ and for arbitrary homogeneity degree $k$ was
presented as well.
In Hörmander (1965), the famous Hörmander’s $L^{2}$ existence and
approximation theorems was given for the $\bar{\partial}$ operator in pseudo-
convex domains in $\mathbb{C}^{n}$. When $n=1$, the existence theorem of
complex variable can be deduced. The aim of this paper is to establish a
Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{n+1}$ with Clifford analysis, and
present sufficient condition on the existence of the weak solutions for Dirac
operator in the sense of Clifford algebra.
Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ and the corresponding norm on $\mathcal{A}$ is
given by $|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$ (see subsection 2.1).
Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$,
$L^{2}(\Omega,\mathcal{A},\varphi)$ be a right Hilbert $\mathcal{A}$-module
for a given function $\varphi\in C^{2}(\Omega,\mathbb{R})$ with the norm given
by Definition 2.9. (see subsection 2.3). $\overline{D}$ denotes the Dirac
differential operator and the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ is given by (4). For
$x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$,
$\Delta=\sum_{i=0}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$. Then we can
obtain our main results as follows.
###### Theorem 1.1
Given $f\in L^{2}(\Omega,\mathcal{A},\varphi)$, there exists $u\in
L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\begin{split}\overline{D}u=f\end{split}$ (1)
with
$\begin{split}\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}c\end{split}$ (2)
if
$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}\leq
c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=c\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).\end{split}$ (3)
Conversely, if there exists $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
(1) is satisfied with
$\begin{split}\|u\|^{2}\leq c\end{split}$
Then we can get the inequality (3) for norm estimation.
The factor $2^{2n}$ in (2) comes from the definition of the norm in Clifford
analysis. If $n=1$, then the factor would disappear which gives a necessary
and sufficient condition in the theorem. From the above theorem, we give the
following sufficient condition on the existence of weak solutions for Dirac
operator.
###### Theorem 1.2
Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ and $n>1$; $\Delta\varphi\geq 0$,
and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$. Then for all $f\in L^{2}(\Omega,\mathcal{A},\varphi)$
with $\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
###### Remark 1.3
Assuming $x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$, it is easy to see
that $\varphi(x)=x_{0}^{2}$ satisfies the conditions in Theorem 1.2. Another
simple example would be
$\varphi(x)=(n+1)x_{0}^{2}-\sum_{i=1}^{n}x_{i}^{2}.$
It is obvious that $\Delta\varphi(x)=2$, $\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}=-2$, and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}=0,~{}i\neq j,~{}1\leq i,j\leq n$.
###### Corollary 1.4
Given $\varphi\in C^{2}(\Omega,\mathbb{R}),$ and $\varphi(x)=\varphi(x_{0})$
with $\varphi^{\prime\prime}(x_{0})\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx.$
It is noticed that there is nothing to do with the boundary conditions of
$\Omega$ in the above results. This phenomenon is totally different with the
famous Hörmander’s $L^{2}$ existence theorems of several complex variables in
Hörmander (1965). Then we can also have the following theorem on global
solutions.
###### Theorem 1.5
Given $\varphi\in C^{2}(\mathbb{R}^{n+1},\mathbb{R})$ with all derivative
conditions in Theorem 1.1 satisfied. Then for all $f\in
L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ with
$\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ satisfying
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\mathbb{R}^{n+1}}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
On the other hand, if the boundary of $\Omega$ is concerned, we consider a
special kind of domain ${\Omega}_{0}=\\{x\in\mathbb{R}^{n+1}:a\leq x_{0}\leq
b\\}$ for any $a,~{}b\in\mathbb{R}$ with $a<b$, then we can get the following
theorem within $L^{2}$ space instead of $L^{2}$ weighted space.
###### Theorem 1.6
Let $f\in L^{2}({\Omega_{0}},\mathcal{A})$. Then there exists a $u\in
L^{2}({\Omega_{0}},\mathcal{A})$ such that
$\overline{D}u=f$
with
$\int_{\Omega_{0}}|u|^{2}_{0}dx\leq
2^{2n}c(a,b)\int_{\Omega_{0}}{|f|^{2}_{0}}dx$
and $c(a,b)$ is a factor depending on $a,~{}b$.
###### Proof
Let $\varphi(x)=x_{0}^{2}$. It can be obtained that
$L^{2}({\Omega_{0}},\mathcal{A})=L^{2}({\Omega_{0}},\mathcal{A},\varphi)$ for
the boundary of $x_{0}$. Then the theorem is proved with Theorem 1.2.
###### Remark 1.7
In particular, any bounded domain $\Omega$ in $\mathbb{R}^{n+1}$ can be
regarded as one type of $\Omega_{0}$. Therefore, it comes from Theorem 1.6
that for any $f\in L^{2}(\Omega,\mathcal{A})$, we can find a weak solution of
Dirac operator $\overline{D}u=f$ with $u\in L^{2}(\Omega,\mathcal{A})$.
## 2 Preliminaries
To make the paper self-contained, some basic notations and results used in
this paper are included.
### 2.1 The Clifford algebra $\mathcal{A}$
Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ with orthogonal basis
$e:=\\{e_{0},e_{1},...,e_{n}\\}$, where $e_{0}=1$ is a unit element in
$\mathbb{R}^{n+1}$. Furthermore,
$\left\\{\begin{aligned} e_{i}e_{j}+e_{j}e_{i}&=0,~{}i\neq j\\\
e_{i}^{2}&=-1,~{}i=1,...,n.\end{aligned}\right.$
Then $\mathcal{A}$ has its basis
$\\{e_{A}=e_{h_{1}\cdots h_{r}}=e_{h_{1}}\cdots e_{h_{r}}:1\leq
h_{1}<...<h_{r}\leq n,1\leq r\leq n\\}.$
If $i\in\\{h_{1},...,h_{r}\\}$, we denote $i\in A$ and if
$i\not\in\\{h_{1},...,h_{r}\\}$, we denote $i\not\in A$. $A-{i}$ means
$\\{h_{1},...,h_{r}\\}\setminus\\{i\\}$ and $A+{i}$ means
$\\{h_{1},...,h_{r}\\}\cup\\{i\\}$. So the real Clifford algebra is composed
of elements having the type $a=\sum\limits_{A}x_{A}e_{A}$, in which
$x_{A}\in\mathbb{R}$ are real numbers. For $a\in\mathcal{A}$, we give the
inversion in the Clifford algebra as follows:
$a^{*}=\sum\limits_{A}x_{A}e_{A}^{*}$ where $e_{A}^{*}=(-1)^{|A|}e_{A}$ and
$|A|=n(A)$ is the $r\in\mathbb{Z}^{+}$ as $e_{A}=e_{h_{1}\cdots h_{r}}$. When
$A=\emptyset$, $e_{A}=e_{0}$, $|A|=0$. Next, we define the reversion in the
Clifford algebra, which is given by
$a^{\dagger}=\sum\limits_{A}x_{A}e_{A}^{\dagger}$ where
$e_{A}^{\dagger}=(-1)^{(|A|-1)|A|/2}e_{A}.$ Now we present the involution
which is a combination of the inversion and the reversion introduced above.
$\bar{a}=\sum\limits_{A}x_{A}\bar{e}_{A}$
where $\bar{e}_{A}=e_{A}^{*{\dagger}}=(-1)^{(|A|+1)|A|/2}e_{A}.$ From the
definition, one can easily deduce that $e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1.$
Furthermore, we have
$\overline{\lambda\mu}=\bar{\mu}\bar{\lambda},~{}~{}\forall\lambda,\mu\in\mathcal{A}.$
Let $a=\sum\limits_{A}x_{A}e_{A}$ be a Clifford number. The coefficient
$x_{A}$ of the $e_{A}$-component will also be denoted by $[a]_{A}$. In
particular the coefficient $x_{0}$ of the $e_{0}$-component will be denoted by
$[a]_{0}$, which is called the scalar part of the Clifford number $a$. An
inner product on $\mathcal{A}$ is defined by putting for any
$\lambda,\mu\in\mathcal{A}$,
$(\lambda,\mu)_{0}:=2^{n}[\lambda\bar{\mu}]_{0}=2^{n}\sum\limits_{A}\lambda_{A}\mu_{A}$.
The corresponding norm on $\mathcal{A}$ reads
$|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$.
We define a real functional on $\mathcal{A}$ that
$\tau_{e_{A}}:\mathcal{A}\rightarrow\mathbb{R}$
$\langle\tau_{e_{A}},\mu\rangle=2^{n}(-1)^{(|A|+1)|A|/2}\mu_{A}.$
In the special case where $A=\emptyset$ we have
$\langle\tau_{e_{0}},\mu\rangle=2^{n}\mu_{0}.$
Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$. Then functions $f$
defined in $\Omega$ and with values in $\mathcal{A}$ are considered. They are
of the form
$f(x)=\sum_{A}f_{A}(x)e_{A}$
where $f_{A}(x)$ are functions with real value. Let $\overline{D}$ denotes the
Dirac differential operator
$\overline{D}=\sum_{i=0}^{n}e_{i}\partial_{x_{i}},$
its action on functions from the left and from the right being governed by the
rules
$\overline{D}f=\sum_{i,A}e_{i}e_{A}\partial_{x_{i}}f_{A}~{}\mbox{and}~{}f\overline{D}=\sum_{i,A}e_{A}e_{i}\partial_{x_{i}}f_{A}.$
$f$ is called left-monogenic if $\overline{D}f=0$ and it is called right-
monogenic if $f\overline{D}=0$. The conjugate operator is given by
$D=\sum_{i=0}^{n}\bar{e}_{i}\partial_{x_{i}}.$
It can be found that
$\overline{D}D=D\overline{D}=\Delta$
where $\Delta$ denotes the classical Laplacian in $\mathbb{R}^{n+1}$. When
$n=1$, one can think of $x_{0}$ as the real part and of $x_{1}$ as the
imaginary part of the variable and to identify $e_{1}$ with $i$. the operator
$\overline{D}$ then take the form
$\overline{D}=\partial_{x_{0}}+i\partial_{x_{1}}$, which is similar with the
operator $\bar{\partial}$ in complex analysis.
### 2.2 Modules over Clifford algebras
This subsection is to give some general information concerning a class of
topological modules over Clifford algebras. In the sequel definitions and
properties will be stated for left $\mathcal{A}$-module and their duals, the
passage to the case of right $\mathcal{A}$-module being straight-forward.
###### Definition 2.1
(unitary left $\mathcal{A}$-module) Let $X$ be a unitary left
$\mathcal{A}$-module, i.e. $X$ is abelian group and a law
$(\lambda,f)\rightarrow\lambda f:\mathcal{A}\times X\rightarrow X$ is defined
such that $\forall\lambda,\mu\in\mathcal{A}$, and $f,~{}g\in X$
1. (1)
$(\lambda+\mu)f=\lambda f+\mu f$,
2. (2)
$\lambda\mu f=\lambda(\mu f)$,
3. (3)
$\lambda(f+g)=\lambda f+\lambda g$,
4. (4)
$e_{0}f=f$.
Moreover, when speaking of a submodule $E$ of the unitary left
$\mathcal{A}$-module $X$, we mean that $E$ is a non empty subset of $X$ which
becomes a unitary left $\mathcal{A}$-module too when restricting the module
operations of $X$ to $E$.
###### Definition 2.2
(left $\mathcal{A}$-linear operator) If $X,Y$ are unitary left
$\mathcal{A}$-modules, then $T:X\rightarrow Y$ is said to be a left
$\mathcal{A}$-linear operator, if $\forall~{}f,~{}g\in X$ and
$\lambda\in\mathcal{A}$
$T(\lambda f+g)=\lambda T(f)+T(g).$
The set of all $``T"$ is denoted by $L(X,Y)$. If
$Y=\mathcal{A},~{}L(X,\mathcal{A})$ is called the algebraic dual of $X$ and
denoted by $X^{*alg}$. Its elements are called left $\mathcal{A}$-linear
functionals on $X$ and for any $T\in X^{*alg}$ and $f\in X$, we denote by
$\langle T,f\rangle$ the value of $T$ at $f$.
###### Definition 2.3
(bounded functional) An element $T\in X^{*alg}$ is called bounded, if there
exist a semi-norm $p$ on $X$ and $c>0$ such that for all $f\in X$
$|\langle T,f\rangle|_{0}\leq c\cdot p(f).$
###### Theorem 2.4
(Hahn-Banach type theorem)Brackx et al (1982) Let $X$ be a unitary left
$\mathcal{A}$-module with semi-norm $p$, $Y$ be a submodule of $X$, and $T$ be
a left $\mathcal{A}$-linear functional on $Y$ such that for some $c>0,$
$|\langle T,g\rangle|_{0}\leq c\cdot p(g),~{}~{}\forall g\in Y$
Then there exists a left $\mathcal{A}$-linear functional $\widetilde{T}$ on
$X$ such that
1. (1)
$\widetilde{T}\mid_{Y}=T$,
2. (2)
for some $c^{*}>0$, $|\langle\widetilde{T},f\rangle|_{0}\leq c^{*}\cdot p(f)$,
$\forall f\in X$.
###### Definition 2.5
(inner product on a unitary right $\mathcal{A}$-module) Let $H$ be a unitary
right $\mathcal{A}$-module, then a function $(~{},~{}):~{}H\times
H\rightarrow\mathcal{A}$ is said to be a inner product on $H$ if for all
$f,g,h\in H$ and $\lambda\in\mathcal{A}$,
1. (1)
$(f,g+h)=(f,g)+(f,h)$,
2. (2)
$(f,g\lambda)=(f,g)\lambda$,
3. (3)
$(f,g)=\overline{(g,f)}$,
4. (4)
$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0~{}\mbox{if and only if}~{}f=0$,
5. (5)
$\langle\tau_{e_{0}},(f\lambda,f\lambda)\rangle\leq|\lambda|^{2}_{0}\langle\tau_{e_{0}},(f,f)\rangle$.
From the definition on inner product, putting for each $f\in H$
$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle,$
then it can be obtained that for any $f,g\in H,$
$\begin{split}|\langle\tau_{e_{0}},~{}(f,g)\rangle|\leq\|f\|\|g\|,\|f+g\|\leq\|f\|+\|g\|.\end{split}$
Hence, $\|\cdot\|$ is a proper norm on $H$ turning it into a normed right
$A$-module. Moreover, we have the following Cauchy-Schwarz inequality.
###### Proposition 2.6
Brackx et al (1982) For all $f,g\in H,$ $|(f,g)|_{0}\leq\|f\|\|g\|.$
###### Definition 2.7
(right Hilbert $\mathcal{A}$-module) Let $H$ be a unitary right
$\mathcal{A}$-module provided with an inner product $(~{},~{})$. Then is it
called a right Hilbert $\mathcal{A}$-module if it is complete for the norm
topology derived from the inner product.
###### Theorem 2.8
(Riesz representation theorem)Brackx et al (1982) Let $H$ be a right Hilbert
$\mathcal{A}$-modules and $T\in H^{*alg}$. Then $T$ is bounded if and only if
there exists a (unique) element $g\in H$ such that for all $f\in H$,
$T(f):=\langle T,f\rangle=(g,f).$
### 2.3 Hilbert space of square integrable functions
Now we extend the standard Hilbert space of square integrable functions to
Clifford algebra. First, we denote $L^{1}(\Omega,\mu)$ and $L^{2}(\Omega,\mu)$
be the sets of all integrable or square integrable functions defined on the
domain $\Omega\in\mathbb{R}^{n+1}$ with respect to the measure $\mu$. Then
$L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
defined as the sets of functions $f:\Omega\rightarrow\mathcal{A}$ which are
integrable or square integrable with respect to $\mu$, i.e., if
$f=\sum\limits_{A}f_{A}e_{A}$, then for each $A$, $f_{A}\in L^{1}(\Omega,\mu)$
and $f^{2}_{A}\in L^{1}(\Omega,\mu)$, respectively. Then one may easily check
that $L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
unitary bi-$\mathcal{A}$-module, i.e., unitary left-$\mathcal{A}$-module and
unitary right-$\mathcal{A}$-module. Furthermore, for any $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$, $\bar{f}\in L^{2}(\Omega,\mathcal{A},\mu)$
while $\bar{f}g\in L^{1}(\Omega,\mathcal{A},\mu)$, where
$\bar{f}(x)=\overline{f(x)}$ and $(\bar{f}g)(x)=\bar{f}(x)g(x),~{}x\in\Omega$.
Consider as a right $\mathcal{A}$-module, define for $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$ that
$(f,g)=\int_{\Omega}\bar{f}(x)g(x)d\mu.$
Furthermore for any real linear functional $T$ on $\mathcal{A}$
$\langle T,(f,g)\rangle=\langle
T,\int_{\Omega}\bar{f}(x)g(x)d\mu\rangle=\int_{\Omega}\langle
T,\bar{f}(x)g(x)\rangle d\mu.$
Consequently, taking $T=\tau_{e_{0}}$ we find that
$\begin{split}\langle\tau_{e_{0}},(f,f)\rangle&=\langle\tau_{e_{0}},\int_{\Omega}\bar{f}(x)f(x)d\mu\rangle=\int_{\Omega}\langle\tau_{e_{0}},\bar{f}(x)f(x)\rangle
d\mu\\\ &=\int_{\Omega}|f(x)|^{2}_{0}d\mu.\end{split}$
Hence, for all $f\in L^{2}(\Omega,\mathcal{A},\mu)$,
$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0$ if and only if $f=0$ a.e. in $\Omega$.
Then it is easy to see that under the inner product defined, all conditions
for $L^{2}(\Omega,\mathcal{A},\mu)$ to be a unitary right inner product
$\mathcal{A}$-module are satisfied. Since
$L^{2}(\Omega,\mathcal{A},\mu)=\prod_{A}L^{2}(\Omega,\mu)$, we have that
$L^{2}(\Omega,\mathcal{A},\mu)$ is complete; in other words
$L^{2}(\Omega,\mathcal{A},\mu)$ is a right Hilbert $\mathcal{A}$-module, with
the norm
$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle=\int_{\Omega}|f(x)|^{2}_{0}d\mu$
for $f\in L^{2}(\Omega,\mathcal{A},\mu)$.
###### Definition 2.9
(weighted $L^{2}$ space) Similar with $L^{2}(\Omega,\mathcal{A},\mu)$, we can
define the weighted $L^{2}(H,\mathcal{A},\varphi)$ for a given function
$\varphi\in C^{2}(\Omega,\mathbb{R})$. First, let
$L^{2}(\Omega,\varphi)=\big{\\{}f|f:\Omega\rightarrow\mathbb{R},~{}\int_{\Omega}|f(x)|^{2}e^{-\varphi}~{}dx<+\infty\big{\\}}.$
Then we denote
$L^{2}(H,\mathcal{A},\varphi)=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L^{2}(\Omega,\varphi)\\}.$
Moreover, for all $f,g\in L^{2}(H,\mathcal{A},\varphi)$, we define
$(f,g)_{\varphi}=\int_{\Omega}\bar{f}(x)g(x)e^{-\varphi}dx.$
Then it is also easy to see $L^{2}(\Omega,\mathcal{A},\varphi)$ is a right
Hilbert $\mathcal{A}$-module, with the norm
$\begin{split}\|f\|^{2}=\langle\tau_{e_{0}},(f,f)_{\varphi}\rangle=\int_{\Omega}|f(x)|^{2}_{0}e^{-\varphi}dx\end{split}$
for $f\in L^{2}(\Omega,\mathcal{A},\varphi)$.
### 2.4 Cauchy’s integral formula
Let $M$ be an (n+1)-dimensional differentiable and oriented manifold contained
in some open subset $\Sigma$ of $\mathbb{R}^{n+1}$. By means of the n-forms
$d\hat{x}_{i}=dx_{0}\wedge\cdots\wedge dx_{i-1}\wedge
dx_{x_{i+1}}\wedge\cdots\wedge dx_{n},~{}i=0,1,...,n,$
an $\mathcal{A}$-valued n-form is introduced by putting
$d\sigma=\sum_{i=0}^{n}(-1)^{i}e_{i}d\hat{x}_{i},$
similarly, denote
$d\bar{\sigma}=\sum_{i=0}^{n}(-1)^{i}\bar{e}_{i}d\hat{x}_{i}.$
Furthermore the volume-element
$dx=dx_{0}\wedge\cdots\wedge dx_{n}$
is used.
###### Proposition 2.10
(Stokes-Green Theorem)Brackx et al (1982) If $f,g\in
C^{1}(\Sigma,\mathcal{A})$ then for any (n+1)-chain $\Omega$ on
$M\subset\Sigma$,
$\int_{\partial\Omega}fd\sigma
g=\int_{\Omega}(f\overline{D})gdx+\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\partial\Omega}fd\bar{\sigma}g=\int_{\Omega}(fD)gdx+\int_{\Omega}f(Dg)dx.$
###### Remark 2.11
Denote $C^{\infty}_{0}(\Omega,\mathbb{R})$ as the set of all smooth real-
valued functions with compact support in $\Omega$ and
$C^{\infty}_{0}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
C^{\infty}_{0}(\Omega,\mathbb{R})\\}.$ If $f$ or $g\in
C^{\infty}_{0}(\Omega,\mathcal{A})$, then we have from the Stokes-Green
theorem that
$\int_{\Omega}(f\overline{D})gdx=-\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\Omega}(fD)gdx=-\int_{\Omega}f(Dg)dx.$
###### Lemma 2.12
If $u(x)\in C^{1}(\Omega,\mathcal{A})$, then
$\overline{\overline{D}u}=\bar{u}D$.
###### Proof
Let $u(x)=\sum_{A}e_{A}u_{A}$. Then
$\begin{split}\overline{\overline{D}u}=\sum_{i,A}\overline{e_{i}e_{A}}\partial_{x_{i}}u_{A}=\sum_{i,A}\bar{e}_{A}\bar{e}_{i}\partial_{x_{i}}u_{A}=\bar{u}D.\end{split}$
###### Lemma 2.13
Huang et al (2006) If $u(x)=\sum_{A}e_{A}u_{A}$,
$v(x)=\sum_{i=0}^{n}e_{i}v_{i}$, then
$\overline{D}(uv)=(\overline{D}u)v+u(\overline{D}v)+\sum\limits^{n}_{j=1}(e_{j}u-ue_{j})\partial_{x_{j}}v.$
### 2.5 Weak solutions
Let
$L_{loc}^{1}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L_{loc}^{1}(\Omega,\mathbb{R})\\}$. Then we define the weak solution in the
sense of Clifford algebra as follows.
###### Definition 2.14
($\overline{D}$ solution in weak sense) If $f\in
L_{loc}^{1}(\Omega,\mathcal{A})$, $u:\Omega\rightarrow\mathcal{A}$ is a weak
solution of
$\overline{D}u=f~{}(\mbox{or}~{}{D}u=f)$
if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha\overline{D})udx~{}(\mbox{or}~{}\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha{D})udx).$
It should be noticed that if $u$ is a weak solution of Dirac equation
$\overline{D}u=0$, in addition, if $u$ is smooth in $\Omega$, then it is left-
monogenic. Now it is natural to give the definition of $\Delta$ solution in
the weak sense.
###### Definition 2.15
($\Delta$ solution in weak sense) If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$,
$u:\Omega\rightarrow\mathcal{A}$ is a weak solution of
$\Delta u=f$
if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\int_{\Omega}\alpha fdx=\int_{\Omega}({\Delta}\alpha)udx.$
###### Theorem 2.16
If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$, and $\overline{D}f=0$ in weak
sense, then $f$ is left-monogenic at any point of $\Omega$.
###### Proof
: Since $\overline{D}f=0$ in weak sense, then $\Delta f=0$ in weak sense. By
Weyl’s lemma, $f$ is smooth in $\Omega$ and has $\Delta f=0$ in classical
sense, then of course $f$ is left-monogenic at any point of $\Omega$.
###### Remark 2.17
This is useful to deal with uniqueness of weak solutions. for example, if
$u,~{}v\in L_{loc}^{1}(\Omega,\mathcal{A})$ are two weak solutions of
$\overline{D}u=f$, then $u=v+w$ with any $w$ left-monogenic.
###### Remark 2.18
An important example of a left monogenic function is the generalized Cauchy
kernel
$G(x)=\frac{1}{\omega_{n+1}}\frac{\overline{x}}{|x|^{n+1}},$
where $\omega_{n+1}$ denotes the surface area of the unit ball in
$\mathbb{R}^{n+1}$. This function obviously belongs to
$L_{loc}^{1}(\Omega,\mathcal{A})$ and is a fundamental solution of the Dirac
equation in the classical sense at any point of $\mathbb{R}^{n+1}$ except 0.
However, it is not a weak solution of the Dirac operator. In fact, if it
satisfies $\overline{D}f=0$ in the weak sense, then from Theorem 2.16, it must
be left-monogenic in the any point of $\Omega$ which could include $0$.
Therefore, we get a contradiction.
For $f\in L^{2}(\Omega,\mathcal{A},\varphi)$,
$u:\Omega\rightarrow\mathcal{A}$. If $\overline{D}u=f$, based on the Stokes-
Green theorem, we can define the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ under the inner product of $L^{2}(\Omega,\mathcal{A},\varphi)$.
For any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\begin{split}(\alpha,f)_{\varphi}=&~{}\int_{\Omega}\bar{\alpha}fe^{-\varphi}dx=\int_{\Omega}\bar{\alpha}e^{-\varphi}fdx\\\
=&~{}\int_{\Omega}(\bar{\alpha}e^{-\varphi})(\overline{D}u)dx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}udx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}e^{\varphi}ue^{-\varphi}dx\\\
=&~{}\int_{\Omega}\overline{-e^{\varphi}D(\alpha
e^{-\varphi})}ue^{-\varphi}dx\\\ =&~{}(-e^{-\varphi}D(\alpha
e^{-\varphi}),u)_{\varphi}\triangleq(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi},\end{split}$
(4)
where $\overline{D}^{*}_{\varphi}\alpha=-e^{\varphi}D(\alpha
e^{-\varphi})=\alpha(D\varphi)-D\alpha$, i.e.
$(\alpha,\overline{D}u)_{\varphi}=(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi}.$
In the same way, we also have
$(\overline{D}u,\alpha)_{\varphi}=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}.$
## 3 The proof of Theorem 1.1
Now we are in the position of proving Theorem 1.1.
###### Proof
($Sufficiency$) From the definition of dual operator and Cauchy-Schwarz
inequality in Proposition 2.6, we have
$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}=&|(\overline{D}u,\alpha)_{\varphi}|^{2}_{0}=|(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}|^{2}_{0}\\\
\leq&~{}\|u\|^{2}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\\\
\leq&~{}c\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$
($necessity$) We aim to prove the necessity with Riesz representation theorem.
First, we denote the submodule
$E=\\{\overline{D}^{*}_{\varphi}\alpha,~{}\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}),~{}\varphi\in
C^{2}(\Omega,\mathbb{R})\\}\subset L^{2}(\Omega,\mathcal{A},\varphi).$
Then we define a linear functional $L_{f}$ on $E$, i.e., $L_{f}\in E^{*alg}$
for a fixed $f\in L^{2}(\Omega,\mathcal{A},\varphi)$ as follows,
$\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(f,\alpha)_{\varphi}=\int_{\Omega}\bar{f}\cdot\alpha\cdot
e^{-\varphi}dx\in\mathcal{A}.$
From (3), we have
$|\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle|_{0}=|(f,\alpha)_{\varphi}|_{0}\leq\sqrt{c}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|,$
which meas that $L_{f}$ is a bounded functional from Definition 2.3. By the
Hahn-Banach type theorem in Theorem 2.4, $L_{f}$ can be extended to a linear
functional $\widetilde{L}_{f}$ on $L^{2}(\Omega,\mathcal{A},\varphi)$, and
with
$\begin{split}|\langle\widetilde{L}_{f},g\rangle|_{0}\leq\sqrt{c^{*}}\|g\|,~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi),\end{split}$ (5)
where $\sqrt{c^{*}}=\sqrt{c}\cdot|e_{0}|_{0}$, since $|e_{A}|_{0}=2^{n/2}$,
then $c^{*}=2^{n}c$ from Brackx et al (1982). Now we are in the position to
use the Riesz representation theorem for the operator $\widetilde{L}_{f}$.
From Theorem 2.8, there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such
that
$\begin{split}\langle\widetilde{L}_{f},g\rangle=(u,g)_{\varphi},~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi).\end{split}$ (6)
For $\forall\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$, let
$g=\overline{D}^{*}_{\varphi}\alpha$. Then
$\begin{split}(f,\alpha)_{\varphi}=&\langle\widetilde{L}_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}=(\overline{D}u,\alpha)_{\varphi},\end{split}$
which deduces that
$\int_{\Omega}\bar{f}\alpha
e^{-\varphi}dx=\int_{\Omega}\overline{(\overline{D}u)}{\alpha}e^{-\varphi}dx.$
Conjugating both sides of above equation leads to
$\int_{\Omega}\bar{\alpha}f\cdot
e^{-\varphi}dx=\int_{\Omega}\bar{\alpha}(\overline{D})ue^{-\varphi}dx.$
Let $\alpha=\bar{\alpha}e^{\varphi}$, it can be obtained that
$\int_{\Omega}\alpha
fdx=\int_{\Omega}\alpha(\overline{D}u)dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$
Therefore,
$\overline{D}u=f$
is proved from the definition of weak solutions.
Next, we give the bound for the norm of $u$. Let $g=u=\sum_{A}e_{A}u_{A}\in
L^{2}(\Omega,\mathcal{A},\varphi)$, from (5) and (6), we get that
$\begin{split}|(u,u)_{\varphi}|_{0}\leq\sqrt{c^{*}}\|u\|.\end{split}$ (7)
On the other hand,
$\begin{split}|(u,u)_{\varphi}|_{0}^{2}=&\big{|}\int_{\Omega}\bar{u}ue^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}2^{n}\cdot\big{[}\int_{\Omega}\bar{u}ue^{-\varphi}dx\cdot\overline{\int_{\Omega}\bar{u}ue^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx\cdot\overline{\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}(\int_{\Omega}\sum\limits_{A}u^{2}_{A}e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}\big{]},\end{split}$
and
$\begin{split}\|u\|^{2}=&~{}\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}[\bar{u}u]_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx\end{split}$
So we have $\|u\|^{4}=2^{2n}\cdot(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}$. Hence,
$|(u,u)_{\varphi}|_{0}^{2}=2^{n}[(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}]\geq 2^{-n}\|u\|^{4}.$
Combining with (7), it is obtained that
$\|u\|^{2}\leq 2^{n/2}|(u,u)_{\varphi}|_{0}\leq 2^{n/2}\sqrt{c^{*}}\|u\|,$
and
$\|u\|^{2}\leq 2^{2n}{c}.$
The proof is completed.
## 4 The proof of Theorem 1.2
It should be noticed that inequality (3) in Theorem 1.1 is related with
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$. In the following, we will give
another sufficient condition that has nothing to do with the space
$C^{\infty}_{0}(\Omega,\mathcal{A})$. First, we need to compute the norm of
$\|\overline{D}^{*}_{\varphi}\alpha\|$ for any $\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$
$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=&\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha\rangle
e^{-\varphi}dx\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha
e^{-\varphi}dx\rangle\\\
=&\langle\tau_{e_{0}},(\overline{D}^{*}_{\varphi}\alpha,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}(\alpha(D\varphi)-D\alpha))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\alpha(D\varphi)+\alpha\Delta\varphi-\Delta\alpha+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha)+\alpha\Delta\varphi+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}+(\alpha,\alpha\Delta\varphi)_{\varphi}+(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&I_{1}+I_{2}+I_{3},\end{split}$
where
$\begin{split}I_{1}=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle=\langle\tau_{e_{0}},(\overline{D}\alpha,\overline{D}\alpha)_{\varphi}\rangle=\|\overline{D}\alpha\|^{2},\\\
I_{2}=&\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle=\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx,\end{split}$
and
$\begin{split}I_{3}=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial
x_{j}}(\sum_{i=0}^{n}\bar{e}_{i}\frac{\partial\varphi}{\partial
x_{i}}))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}})_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}e^{-\varphi}dx\rangle\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle e^{-\varphi}dx.\end{split}$
It should be noticed that if $n=1$, i.e., the space $\mathbb{R}^{2}$ is
considered, then $I_{3}=0.$
Since for $1\leq i,j\leq n$ and $i\neq j$,
$e_{j}\bar{e}_{i}=-e_{j}{e}_{i}=e_{i}{e}_{j}=-{e}_{i}\bar{e}_{j}$. For
simplicity, let
$\begin{split}I_{4}=&\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum\limits_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}\sum\limits_{i=1}^{n}(\bar{\alpha}e_{j}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{i}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\ =&I_{5}+I_{6}+I_{7}.\end{split}$
Assume
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}\in\mathcal{A},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$,
then for any $1\leq i\leq n,$
$\begin{split}\bar{\alpha}e_{i}\alpha\bar{e}_{i}=&~{}\sum\limits_{A}\alpha_{A}\bar{e}_{A}e_{i}\cdot\sum\limits_{A}\alpha_{A}e_{A}\bar{e}_{i}\\\
=&~{}\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i}\end{split}$
Therefore
$\begin{split}I_{5}=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha_{A}^{2}e_{A}e_{i}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha^{2}_{A}\cdot\overline{e_{A}e_{i}}\cdot
e_{A}e_{i}\cdot(-1)^{\frac{(|A|+1)(|A|+2)}{2}}\\\ &+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\cdot\alpha^{2}_{A}\cdot\overline{e_{A-{i}}}\cdot
e_{A-{i}}\cdot(-1)^{\frac{(|A|-1)(|A|)}{2}})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|+1)(|A|+2)}{2}}\cdot\alpha^{2}_{A}\\\
&+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|-1)(|A|)}{2}}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{|A|^{2}}\cdot\alpha^{2}_{A}+\sum\limits_{i\in
A}(-1)^{|A|^{2}+1}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}(-2)\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is
even}}(-2)\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\\\
=&-2^{n+1}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^{2}~{}\mbox{is
odd}}\alpha^{2}_{A}+\sum\limits_{i\in A,|A|^{2}~{}\mbox{is
even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}.\end{split}$ (8)
To consider $I_{7}$, we first study $\bar{\alpha}e_{j}\alpha$ for any $1\leq
j\leq n$. Without loss of generality, let
$e_{j}=e_{1},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A},~{}\alpha=\sum\limits_{A}\alpha_{A}e_{A}$.
Then
$\bar{\alpha}e_{1}\alpha=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}(\sum\limits_{A}\alpha_{A}e_{A})$.
When $e_{A}=e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}}\cdot e_{1}e_{h_{2}}e_{h_{3}}\cdots
e_{h_{r}}\cdot e_{1}\\\ =&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}=\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (9)
When $e_{A}=e_{1}$,
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1}\\\
\alpha_{A}e_{A}e_{1}=&-\alpha_{1}.\end{split}$ (10)
When $e_{A}=e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}}\cdot e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\cdot
e_{1}\\\ =&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1}=\alpha_{h_{2}\cdots h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (11)
When $e_{A}=e_{0}$,
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{0}e_{1}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{0}e_{1}.\end{split}$ (12)
To compute $I_{7}$, one needs to know the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha-\bar{\alpha}\alpha e_{1}$. It means that we should
find out the corresponding terms of $e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$
and $e_{h_{2}}\cdots e_{h_{r}}$ in $\bar{\alpha}e_{1}$ and $\alpha$, in
$\bar{\alpha}$ and $\alpha e_{1}$.
Case a1. For $\bar{\alpha}e_{1}\alpha$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in
$\bar{\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}$ and
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and
$\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to
$\begin{split}(-1)&{}^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r-1)(r)}{2}+r-1}(-1)^{\frac{(r)(r+1)}{2}}\cdot\overline{e_{1}\cdots
e_{h_{r}}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r-1+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (13)
On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in $\bar{\alpha}e_{1}$ and
$\alpha$ are $\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ and
$\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to
$\begin{split}(-1)&{}^{\frac{(r)(r+1)}{2}+r}e_{h_{2}\cdots h_{r}}\cdot
e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r}(-1)^{\frac{(r-1)(r)}{2}}\cdot\overline{e_{h_{2}\cdots
h_{r}}}\cdot e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (14)
From (13) and (14), these two terms vanish.
Case a2. For $\bar{\alpha}e_{1}\alpha$, from (12), the corresponding terms of
$e_{1}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}e_{1}$, respectively. Multiplying these terms leads to
$\begin{split}\alpha_{0}e_{1}\alpha_{1}e_{1}=-\alpha_{0}\alpha_{1}.\end{split}$
(15)
On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (10), the corresponding
terms of $e_{0}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{1}$ and
$\alpha_{0}$, respectively. Multiplying these terms leads to
$\alpha_{0}\alpha_{1}$. Combining with (15), these two terms also vanish.
From Cases a1 and a2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,
$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{0}}\rangle=0.\end{split}$ (16)
Case b1. For $\bar{\alpha}\alpha e_{1}$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in ${\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}{e}_{A})e_{1}$ and
$\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and $\alpha_{1h_{2}\cdots
h_{r}}\overline{e_{1}e_{h_{2}}\cdots e_{h_{r}}}$, respectively. Multiplying
these terms leads to
$\begin{split}(\alpha_{1h_{2}\cdots h_{r}}&\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{1h_{2}\cdots h_{r}}\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}})\\\
=&~{}(-1)^{r-1}\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}.\end{split}$ (17)
On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in ${\alpha}e_{1}$ and
$\bar{\alpha}$ are $\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}$ and $\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}}$, respectively. Multiplying these terms leads to
$\begin{split}(\alpha_{h_{2}\cdots h_{r}}&\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{1h_{2}\cdots h_{r}}e_{1}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}})\\\ =&~{}(-1)^{r}\alpha_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}.\end{split}$ (18)
From (17) and (18), these two terms vanish.
Case b2. For $\bar{\alpha}\alpha e_{1}$, from (12), the corresponding terms of
$e_{1}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}\bar{e}_{1}$, respectively. Multiplying these terms leads to
$\begin{split}\alpha_{0}e_{1}\alpha_{1}\bar{e}_{1}=\alpha_{0}\alpha_{1}.\end{split}$
(19)
On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (10), the
corresponding terms of $e_{0}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are
$-\alpha_{1}$ and $\alpha_{0}$, respectively. Multiplying these terms leads to
$-\alpha_{0}\alpha_{1}$. Combining with (19), these two terms also cancel.
From Cases b1 and b2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,
$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle=0.\end{split}$ (20)
Thus, $I_{7}=0$ from (16) and (20).
To compute $I_{6}$, i.e., to get $[\bar{\alpha}e_{i}\alpha\bar{e}_{j}]_{0}$
for $i\neq j$, similar with the analysis of $I_{7}$, we should divide the
vectors in $\bar{\alpha}e_{i}$ and $\alpha\bar{e}_{j}$ into four cases.
Case c1. $i\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B-{j}$.
For this case, firstly, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$
and $h_{p(i)}=i$, $e_{B}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and
$h_{p(j)}=j$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)}e_{h_{1}}\cdots e_{i}^{2}\cdots
e_{h_{r}},\\\ =&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r-p(j)}e_{h_{1}}\cdots
e_{j}\bar{e}_{j}\cdots e_{h_{r}},\\\
=&\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}}\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1+r-p(j)+\frac{r(r-1)}{2}}\overline{e_{A-{i}}}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}.\end{split}$ (21)
Case c2. $i\not\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B+{j}$.
We assume $e_{A}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and $h_{p(j)}=j$,
$e_{B}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$ and $h_{p(i)}=i$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot{e}_{j}\\\ =&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot e_{i}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+\frac{(r+1)(r+2)}{2}}\overline{e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}}e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot
e_{i}\\\ =&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}.\end{split}$
Case c3. $i\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B+{j}$.
For this case, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{p(j)}\cdots
h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$. Without loss of generality, we
assume $i<j$. Furthermore, let $e_{B}=e_{h_{1}\cdots h_{r}}$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{j}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+2-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot e^{2}_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}(-1)\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j}}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$
Case c4. $i\not\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B-{j}$.
For this case, we assume $e_{A}=e_{h_{1}\cdots h_{r}}$, $e_{B}=e_{h_{1}\cdots
h_{p(i)}\cdots h_{p(j)}\cdots h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$ and
$i<j$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r+2}}\cdot e_{j}\bar{e}_{j}\\\
=&\alpha_{B}(-1)^{r+2-h(j)+r+2-h(i)-1}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ =&\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ \end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)}\cdot e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$
Combining cases c1-c4, we have
$\begin{split}I_{6}=&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{j}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{i}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\rangle\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&2^{n}\sum\limits^{n}_{j\neq i}\Big{(}\sum_{i\in
A,~{}j\not\in A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\
&+\sum_{i\not\in A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\
&+\sum_{i\in A,~{}j\in
A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\ &+\sum_{i\not\in
A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}.\end{split}$
In all,
$\begin{split}I_{3}=&\int_{\Omega}I_{4}e^{-\varphi}dx\\\
=&\int_{\Omega}(I_{5}+I_{6}+I_{7})e^{-\varphi}dx\\\
=&-2^{n+1}\int_{\Omega}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}e^{-\varphi}dx\\\ &+2^{n}\int_{\Omega}\sum\limits^{n}_{j\neq
i}\Big{(}\sum_{i\in A,~{}j\not\in
A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\ &+\sum_{i\not\in
A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\ &+\sum_{i\in
A,~{}j\in A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\
&+\sum_{i\not\in A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}e^{-\varphi}dx.\end{split}$
Then
$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=\|\overline{D}\alpha\|^{2}+\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx+I_{3}.\end{split}$ (22)
If $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$, we have $I_{3}\geq 0$, and
$\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\geq\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx.$
With the above analysis, we can prove Theorem 1.2 easily.
###### Proof
It is sufficient to prove the theorem if condition (3) in Theorem 1.1 is
presented. By Cauchy-Schwarz inequality in Proposition 2.6, we have for any
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$ that
$\begin{split}|({f},\alpha)_{\varphi}|^{2}_{0}=&\big{|}\int_{\Omega}\bar{f}\cdot\alpha
e^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}\big{|}\int_{\Omega}\bar{f}\cdot\frac{1}{\sqrt{\Delta\varphi}}\cdot\alpha\cdot\sqrt{\Delta\varphi}\cdot
e^{-\varphi}dx\big{|}^{2}_{0}\\\
\leq&~{}\big{\|}\bar{f}\frac{1}{\sqrt{\Delta\varphi}}\big{\|}^{2}\cdot\big{\|}\alpha\cdot\sqrt{\Delta\varphi}\big{\|}^{2}\\\
=&~{}\int_{\Omega}\big{|}\frac{\bar{f}}{\sqrt{\Delta\varphi}}\big{|}^{2}_{0}e^{-\varphi}dx\cdot\int_{\Omega}\big{|}\alpha\cdot\sqrt{\Delta\varphi}\big{|}^{2}_{0}e^{-\varphi}dx\\\
\leq&c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$
The proof is completed with Theorem 1.1.
It should be noticed that when $n=1$, $I_{3}=0$. Then it comes from equation
(22) that the Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{2}$ could be
described which equals the classical Hörmander’s $L^{2}$ theorem in
$\mathbb{C}$.
###### Corollary 4.1
Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ with $\Omega$ being an open subset
of $\mathbb{R}^{2}$; $\Delta\varphi\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there
exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}\leq\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
## 5 Conclusion
In this paper, based on the Hörmander’s $L^{2}$ theorem in complex analysis,
the Hörmander’s $L^{2}$ theorem for Dirac operator in $\mathbb{R}^{n+1}$ has
been obtained by Clifford algebra. When $n=1$, the result is equivalent to the
classical Hörmander’s $L^{2}$ theorem in complex variable. Moreover, for any
$f$ in $L^{2}$ space over a bounded domain with value in Clifford algebra,
there is a weak solution of Dirac operator with the solution in the $L^{2}$
space as well. The potential applications of the results will be studied in
our future work.
###### Acknowledgements.
This work was supported by the National Natural Science Foundations of China
(No. 11171255, 11101373) and Doctoral Program Foundation of the Ministry of
Education of China (No. 20090072110053).
## References
* Brackx et al (1982) Brackx F, Delanghe R, Sommen F (1982) Clifford Analysis, Research Notes in Mathematics. London, Pitman
* De Ridder et al (2012) De Ridder H, De Schepper H, Sommen F (2012) Fueter polynomials in discrete Clifford analysis. Mathematische Zeitschrift 272 (2012) :253–268.
* Gong et al (2009) Gong Y, Leong IT, Qian T (2009) Two integral operators in Clifford analysis. Journal of Mathematical Analysis and Applications 354(2):435–444
* Hörmander (1965) Hörmander L (1965) $l^{2}$ estimates and existence theorems for the operator. Acta Mathematica 113(1):89–152
* Huang et al (2006) Huang S, Qiao YY, Wen GC (2006) Real and Complex Clifford Analysis, Advances in Complex Analysis and Its Applications. New York, Springer
* Qian and Ryan (1996) Qian T, Ryan J (1996) Conformal transformations and Hardy spaces arising in Clifford analysis. Journal of Operator Theory 35(2):349–372
* Ryan (1990) Ryan J (1990) Iterated Dirac operators in $c^{n}$. Zeitschrift für Analysis und ihre Anwendungen 9:385–401
* Ryan (1995) Ryan J (1995) Cauchy-Green type formulae in Clifford analysis. Transactions of the American Mathematical Society 347(4):1331–1342
* Ryan (2000) Ryan J (2000) Basic Clifford analysis. Cubo Matemática Educacional 2:226–256
|
arxiv-papers
| 2013-04-17T00:39:58 |
2024-09-04T02:49:44.520096
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Liu, Zhihua Chen and Yifei Pan",
"submitter": "Yang Liu",
"url": "https://arxiv.org/abs/1304.4653"
}
|
1304.4681
|
# Long-time asymptotic for the derivative nonlinear Schrödinger equation with
step-like initial value
Jian Xu School of Mathematical Sciences
Fudan University
Shanghai 200433
People’s Republic of China [email protected] , Engui Fan School of
Mathematical Sciences, Institute of Mathematics and Key Laboratory of
Mathematics for Nonlinear Science
Fudan University
Shanghai 200433
People’s Republic of China correspondence author: [email protected] and
Yong Chen Shanghai Key Laboratory of Trustworthy Computing
East China Normal University,
Shanghai 200062, People s Republic of China. [email protected]
###### Abstract.
We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the
derivative nonlinear Schrödinger (DNLS) equation:
$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0.$
with steplike initial data: $q(x,0)=0$ for $x\leq 0$ and $q(x,0)=Ae^{-2iBx}$
for $x>0$,where $A>0$ and $B\in{\mathbb{R}}$ are constants.The paper aims at
studying the long-time asymptotics of the solution to this problem.We show
that there are four regions in the half-plane $-\infty<x<\infty,t>0$,where the
asymptotics has qualitatively different forms:a slowly decaying self-similar
wave of Zakharov-Manakov type for $x>-4tB$, a plane wave
region:$x<-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$, an elliptic
region:$-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})<x<-4tB$. The main tool is the
asymptotic analysis of an associated matrix Riemann-Hilbert problem.
###### Key words and phrases:
Riemann-Hilbert problem, GI-DNLS equation, Long-time asymptotic, steplike
initial value problem
## 1\. Introduction
The classical, mathematical model for non-linear pulse propagation in the
picosecond time scale in the anomalous dispersion regime in an isotropic,
homogeneous, lossless, non-amplifying, polarization-preserving single-mode
optical fibre is the non-linear Schrödinger(NLS) equation [2]. However, in the
subpicosecond-femtosecond time scale, experiments and theories on the
propagation of high-power ultrashort pulses in long monomode optical fibres
have shown that the NLS equation is no longer valid and that additional non-
linear terms (dispersive and dissipative) and higher-order linear dispersion
should be taken into account, you can see [36] and the references therein. In
this case, subpicosecond-femtosecond pulse propagation is described (in
dimensionless and normalized form) by the following non-linear evolution
equation (NLEE)
$iu_{\xi}+\frac{1}{2}u_{\tau\tau}+|u|^{2}u+is(|u|^{2}u)_{\tau}=-i\tilde{\Gamma}u+i\tilde{\delta}u_{\tau\tau\tau}+\frac{\tau_{n}}{\tau_{0}}u(|u|^{2})_{\tau},$
(1.1)
where $u$ is the slowly varying amplitude of the complex field envelope, $\xi$
is the propagation distance along the fibre length, $\tau$ is the time
measured in a frame of reference moving with the pulse at the group velocity
(the retarded frame), $s(>0)$ governs the effects due to the intensity
dependence of the group velocity (self-steepening), $\tilde{\Gamma}$ is the
intrinsic fibre loss, $\tilde{\delta}$ governs the effects of the third-order
linear dispersion, and $\frac{\tau_{n}}{\tau_{0}}$, where $\tau_{0}$ is the
normalized input pulsewidth and $\tau_{n}$ is related to the slope of the
Raman gain curve (assumed to vary linearly in the vicinity of the mean carrier
frequency, $\omega_{0}$), governs the soliton self-frequency shift (SSFS)
effect, [36] and the references therein.
We set the right-hand side of (1.1) equal to zero, we obtain the following
equation,
$iu_{\xi}+\frac{1}{2}u_{\tau\tau}+|u|^{2}u+is(|u|^{2}u)_{\tau}=0,$ (1.2)
This equation is related to the Kaup-Newell type of derivative nonlinear
Schrödinger equation,
$iq_{t}(x,t)=-q_{xx}(x,t)+(\bar{q}q^{2})_{x}$ (1.3)
by change of variables
$u(\xi,\tau)=q(x,t)e^{i(\frac{t}{4s^{4}}-\frac{x}{2s^{2}})},\quad\xi=\frac{t}{2s^{2}},\quad\tau=-\frac{x}{2s}+\frac{t}{2s^{3}}.$
And we note that if we replace $x$ by $-x$,equation (1.3) changes into
$iq_{t}(x,t)=-q_{xx}(x,t)-(\bar{q}q^{2})_{x}.$ (1.4)
But, we also know if we formulate a Riemann-Hilbert problem for the solution
of the inverse spectral problem of the equation (1.4), we find we cannot find
solutions of its spectral problem which approach the $2\times 2$ identity
matrix $\mathbb{I}$ as $k\rightarrow\infty$.It is well-known that there are
three kinds of celebrated DNLS equations, including Kaup-Newell equation ( i.e
Eq.(1.4)), Chen-Lee-Liu equation [37]
$iq_{t}+q_{xx}+i|q|^{2}q_{x}=0,$
and Gerdjikov-Ivanov(GI) equation [38, 40]
$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0$ (1.5)
It has been found that they may be transformed into each other by gauge
transformations [38, 39]. And in [40], the GI-type has the required property
of the solutions of its spectral problem which approach the $2\times 2$
identity matrix $\mathbb{I}$ as $k\rightarrow\infty$. So,we focus on the GI-
type of derivative nonlinear Schrödinger equation. In the following of this
paper we also name the GI-type DNLS equation as DNLS equation.
Initial value problems for nonlinear evolution equations with step-like
initial data have attracted much attention since the early 1970s [16, 17, 18,
19], but only a few rigorous results concerning the long-time behavior of
solutions of such problems were available.In 1980s-1990s, a considerable
progress was achieved following the development of the theory of Whitham
deformations [20] and the analysis of matrix Riemann-Hilbert problem
representations of solutions of initial value problems, see [21, 22, 23] and
references therein.Most complete results,obtained by using this approach,were
related to integrable equations,for which linear operators from the associated
Lax pair were self-adjoint and thus their spectrum was real.In [22],Bikbaev
considered the case of the focusing nonlinear Schrödinger equation,which
required the development of a much more complicated complex form of the theory
of Whitham deformations.
A completely rigorous approach for studying asymptotics of solutions of
integrable nonlinear equations was introduced by Deift and Zhou [9](this
approach was inspired by earlier works of Manakov [24] and Its [25];see [10]
for a detailed historical review) and further extended by Deift,Venakides,and
Zhou [26, 27]. This approach is based on the development of the nonlinear
steepest descent method for Riemann-Hilbert problems associated with
integrable nonlinear equations. Being originally introduced for studying
initial value problems with decaying initial data, this approach was recently
adapted by Buckingham and Venakides [28] to problems with shock-type
oscillating initial data for focusing nonlinear Schrödinger equation. A
central role in this development is played by the so-called $g-$function
mechanism allowing to deform the original Riemann-Hilbert problem to a form
that can be asymptotically treated with the help of associated singular
integral equations.
The Riemann-Hilbert problem approach to initial value problems with
nondecaying step-like initial data shares many issues with the adaptation of
this approach for studying initial-boundary value problems with non-decaying
boundary data [29, 30, 31].However,there is an important difference: in the
latter case,the construction of the associated Riemann-Hilbert problem
normally requires the knowledge of spectral functions associated with
overspecified initial and boundary data,which leads to the fact that
results(in particular,the asymptotic results,see [29]) have,in a certain
sense,a conditional character.As for the initial value problems of the type
considered in this paper,the Riemann-Hilbert construction requires only
initial data,and thus,the issue of overdetermination does not arise.
In this paper,we consider a pure step-like initial value problem for the DNLS
equation:
$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0,\qquad
x\in{\mathbb{R}},t>0,$ (1.6a)
$q(x,0)=q_{0}(x)=\left\\{\begin{array}[]{lr}0&\mbox{if }x\geq 0,\\\
Ae^{-2iBx}&\mbox{if }x<0,\end{array}\right.$ (1.6b)
where $A>0$ and $B\in{\mathbb{R}}$ are some constants. Kitaev and Vartanian
got the leading order long-time asymptotic for the KN-type of DNLS equation
with the decaying initial value,in [34], and the higher order long-time
asymptotic in [36].
Since the DNLS equation (1.6a) has a plane wave solution
$q^{p}(x,t)=Ae^{-2iBx+2i\omega t},$ (1.7)
with
$\omega:=A^{2}B-2B^{2}+\frac{A^{4}}{4},$ (1.8)
which is consistent with (1.6b) for $x<0$,that is,$q^{p}(x,0)=q_{0}(x)$,we
assume that the solution $q(x,t)$ of the initial value problem (1.6a)
evaluated at any $t>0$ has the following behavior as $x\rightarrow\pm\infty$:
$q(x,t)=o(1),\qquad x\rightarrow+\infty,$ (1.9) $q(x,t)=q^{p}(x,t)+o(1),\qquad
x\rightarrow-\infty,$ (1.10)
where $o(1)$ means sufficiently fast decay to $0$.This assumption can be
justified a posteriori,by evaluating the large-$x$ behavior of the solution of
the Riemann-Hilbert problem formulated in Section 3.
Recently, in [32],A.Boutet de Monvel,V.P.Kotlyarov, and D.Shepelsky considered
the long-time dynamics of the initial value problem for the focusing nonlinear
Schrödinger equation with step-like data.The strategy of the Riemann-Hilbert
problem deformations that we adopt in this paper is similar,though not
identical,to that in [28].In particular,the realization of the $g-$function
mechanism is different as well as the resulting asymptotic picture.
As we have already mentioned, the main tool available now for studying
rigorously the long-time asymptoitcs of solutions of initial and initial
boundary value problems for integrable nonlinear equations is the asymptotic
analysis of associated Riemann-Hilbert problems,whose construction involves
dedicated solutions of the system of two linear equations,the Lax pair
associated with the integrable nonlinear equation.
For the DNLS equation (1.6a), a Lax pair is as follows [34]:
$\begin{split}&\Psi_{x}(x,t;k)=M(x,t;k)\Psi(x,t;k),\\\
&\Psi_{t}(x,t;k)=N(x,t;k)\Psi(x,t;k),\end{split}$ (1.11)
where
$\begin{split}&M(x,t;k)=-ik^{2}\sigma_{3}+kQ+\frac{i}{2}|q|^{2}\sigma_{3},\\\
&N(x,t;k)=-2ik^{4}\sigma_{3}+2k^{3}Q+ik^{2}|q|^{2}\sigma_{3}-ikQ_{x}\sigma_{3}+\frac{i}{4}|q|^{4}\sigma_{3}+\frac{1}{2}(q{\bar{q}}_{x}-\bar{q}q_{x})\sigma_{3},\end{split}$
(1.12)
with $\sigma_{3}=\left(\begin{array}[]{lc}1&0\\\ 0&-1\end{array}\right)$, and
$\Psi(x,t;k)$ is a $2\times 2$ matrix-value function,$k\in{\mathbb{C}}$ is a
spectral parameter, and the matrix coefficient $Q$ is expressed in terms of a
scalar function $q$:
$Q=\left(\begin{array}[]{lc}0&q\\\ -\bar{q}&0\end{array}\right),$ (1.13)
It is well-known [34] that this over-determined system of equations (1.11) is
compatible if and only if $q(x,t)$ solves the DNLS equation (1.6a).
In Section 2 we present these dedicated solutions(eigenfunctions) and
associated spectral functions.All these functions are then used in Section 3
for constructing a basic Riemann-Hilbert problem,whose solution gives the
solution of the initial value problem (1.6a),(1.6b).Section 4 develops the
asymptotic analysis of this Riemann-Hilbert problem leading to asymptotic
formulas for the solution of the original Cauchy problem (1.6).
## 2\. Eigenfunctions
Let $Q^{p}$ be defined by (1.13) with $q^{p}$ instead of $q$. A particular
solution of the system (1.11),with $Q^{p}$ instead of $Q$,is given by
$\Psi^{p}(x,t;k)=e^{i(\omega
t-Bx)\sigma_{3}}E(k)e^{-i(xX(k)+t\Omega(k))\sigma_{3}},$ (2.1)
where
$X(k)=\sqrt{(k^{2}-B-\frac{A^{2}}{2})^{2}+k^{2}A^{2}},$ (2.2)
$\Omega(k)=2(k^{2}+B)X(k).$ (2.3)
$E(k)=\frac{1}{2}\left(\begin{array}[]{lc}\varphi(k)+\frac{1}{\varphi(k)}&\varphi(k)-\frac{1}{\varphi(k)}\\\
\varphi(k)-\frac{1}{\varphi(k)}&\varphi(k)+\frac{1}{\varphi(k)}\end{array}\right)$
(2.4)
with
$\varphi(k)=(\frac{k^{2}-B-\frac{A^{2}}{2}-ikA}{k^{2}-B-\frac{A^{2}}{2}+ikA})^{\frac{1}{4}},$
(2.5)
The branch cut for $X$ and $\varphi$ is taken along the segment
$\gamma\cup\bar{\gamma}:=\\{k\in{\mathbb{C}}|k_{1}^{2}-k_{2}^{2}=B,k_{1}^{2}\leq
C^{2}\\},$ (2.6)
where $\gamma=\\{k\in{\mathbb{C}}|k_{1}^{2}-k_{2}^{2}=B,k_{1}^{2}\leq
C^{2},\mathrm{Im}k^{2}>0\\}$, $C^{2}=B+\frac{A^{2}}{4}$,
$k_{1}=\mathrm{Re}{k}$ and $k_{2}=\mathrm{Im}{k}$. And the branches are fixed
by the asymptotics:
$X(k)=k^{2}-B+O(\frac{1}{k^{2}}),\qquad\mbox{as }k\rightarrow\infty,$
$\varphi(k)=1+O(\frac{1}{k}),\qquad\mbox{as }k\rightarrow\infty.$
We find that $\Omega(k)=2k^{4}+\omega+O(\frac{1}{k}),\mbox{as
}k\rightarrow\infty$. We also find that $\mathrm{Im}{X(k)}=0$ is
$k_{1}k_{2}(k_{1}^{2}-k_{2}^{2}-B)=0,$ (2.7)
which is on
$\Sigma:={\mathbb{R}}\cup i{\mathbb{R}}\cup\gamma\cup\bar{\gamma}.$ (2.8)
Thus, for any $t\geq 0$, $\Psi^{p}(x,t;k)$ is bounded in $x$ if and only if
$k\in\Sigma$.
Let $q(x,t)$ be a solution of the Cauchy problem (1.6a),(1.6b) satisfying the
asymptotic conditions (1.9),(1.10), and let $Q(x,t)$ and $Q^{p}(x,t)$ be
defined by (1.13), in terms of $q$ and $q^{p}$, respectively.Define the
$2\times 2$ matrix-value functions $\mu_{j}(x,t;k)$, $j=1,2$,
$-\infty<x<\infty,0\leq t<\infty$, as the solutions of the Volterra integral
equations:
$\mu_{1}(x,t;k)=\mathbb{I}+\int_{+\infty}^{x}e^{ik^{2}(y-x)\sigma_{3}}(kQ\mu_{1})(y,t;k)e^{-ik^{2}(y-x)\sigma_{3}},\qquad
k^{2}\in{\mathbb{R}},$ (2.9) $\displaystyle\mu_{2}(x,t;k)$ $\displaystyle=$
$\displaystyle e^{i(\omega t-Bx)\sigma_{3}}E(k)$
$\displaystyle+\int_{-\infty}^{x}\Gamma^{p}(x,y,t,k)k[Q-Q^{p}](y,t)\mu_{2}(y,t,k)e^{-ik^{2}(y-x)\sigma_{3}},k\in\Sigma,$
where
$\Gamma^{p}(x,y,t,k):=\Psi^{p}(x,t,k)[\Psi^{p}(y,t,k)]^{-1}.$
Note that $\Gamma^{p}$ can be written in the form
$\Gamma^{p}(x,y,t,k)=e^{i(\omega t-Bx)\sigma_{3}}G^{p}(x,y,k)e^{-i(\omega
t-By)\sigma_{3}},$
where
$G^{p}(x,y,k)=\left(\begin{array}[]{cc}\alpha+i(k^{2}-B-\frac{A^{2}}{2})\beta&-kA\beta\\\
kA\beta&\alpha-i(k^{2}-B-\frac{A^{2}}{2})\beta\end{array}\right),$
with
$\alpha=\cos[(y-x)X(k)],\qquad\beta=\frac{\sin[(y-x)X(k)]}{X(k)}.$
For any $(x,y)\in{\mathbb{R}}^{2}$,$G^{p}(x,y,k)$ is an entire function of $k$
with asymptotic behavior
$G^{p}(x,y,k)=e^{i(y-x)(k^{2}-B-\frac{A^{2}}{2})\sigma_{3}}[\mathbb{I}+O(\frac{1}{k})],\qquad\mbox{as
}k\rightarrow\infty,\quad\mathrm{Im}{k^{2}}=0.$
The analytic properties of the $2\times 2$ matrices $\mu_{j}(x,t;k)$, $j=1,2$,
that follow from (2.9) and (2) are collected in the following proposition.We
denote by $\mu_{j}^{(1)}(x,t,k)$ and $\mu_{j}^{(2)}(x,t,k)$ the columns of
$\mu_{j}(x,t;k)$.
###### Proposition 2.1.
The matrices $\mu_{1}(x,t;k)$ and $\mu_{2}(x,t;k)$ have the following
properties:
1. (i)
$det\mu_{1}(x,t,k)=\mu_{2}(x,t;k)=1$.
2. (ii)
The functions $\Phi(x,t,k)$ and $\Psi(x,t,k)$ defined by
$\Psi(x,t,k):=\mu_{1}(x,t,k)e^{-ik^{2}x\sigma_{3}-2ik^{4}t\sigma_{3}},$
$\Phi(x,t,k):=\mu_{2}(x,t;k)e^{-ixX(k)\sigma_{3}-it\Omega(k)\sigma_{3}}.$
satisfy the Lax pair equations (1.11).
3. (iii)
$\mu_{1}^{(1)}(x,t,k)$ is analytic in $\mathrm{Im}k^{2}<0$ and
$\mu_{1}^{(1)}(x,t,k)=\left(\begin{array}[]{c}1\\\
0\end{array}\right)+O(\frac{1}{k}),\mbox{as
}k\rightarrow\infty,\quad\mathrm{Im}k^{2}\leq 0.$
4. (iv)
$\mu_{1}^{(2)}(x,t,k)$ is analytic in $\mathrm{Im}k^{2}>0$ and
$\mu_{1}^{(2)}(x,t,k)=\left(\begin{array}[]{c}0\\\
1\end{array}\right)+O(\frac{1}{k}),\mbox{as
}k\rightarrow\infty,\quad\mathrm{Im}k^{2}\geq 0.$
5. (v)
$\mu_{2}^{(1)}(x,t,k)$ is analytic in $\mathrm{Im}k^{2}>0\backslash\gamma$,has
a jump across $\gamma$, and
$\mu_{2}^{(1)}(x,t,k)=\left(\begin{array}[]{c}1\\\
0\end{array}\right)+O(\frac{1}{k}),\mbox{as
}k\rightarrow\infty,\quad\mathrm{Im}k^{2}\geq 0.$
6. (vi)
$\mu_{2}^{(2)}(x,t,k)$ is analytic in
$\mathrm{Im}k^{2}<0\backslash\bar{\gamma}$,has a jump across $\bar{\gamma}$,
and
$\mu_{2}^{(2)}(x,t,k)=\left(\begin{array}[]{c}0\\\
1\end{array}\right)+O(\frac{1}{k}),\mbox{as
}k\rightarrow\infty,\quad\mathrm{Im}k^{2}\leq 0.$
7. (vii)
Moreover,
$\mu_{j}^{(1)}(x,t,k)=\mathbb{I}+\frac{\tilde{\mu}(x,t)}{ik}+o(\frac{1}{k})$
as $k\rightarrow\infty$ along curves transversal to the real and image axis,
where
$[\sigma_{3},\tilde{\mu}(x,t)]=\left(\begin{array}[]{cc}0&q(x,t)\\\
-\bar{q}(x,t)&0\end{array}\right)$
8. (viii)
$\mu_{2}^{(2)}(x,t,k)(k-E)^{\frac{1}{4}}$ is boundary near $k=E$ and
$\mu_{2}^{(2)}(x,t,k)(k-\bar{E})^{\frac{1}{4}}$ is boundary near $k=\bar{E}$.
Since the eigenfunctions $\Psi(x,t,k)$ and $\Phi(x,t,k)$ satisfy both
equations of the Lax pair, we have
$\Phi(x,t,k)=\Psi(x,t,k)S(k),\qquad k^{2}\in{\mathbb{R}},$ (2.11)
where $S(k)$ is independent of $(x,t)$. Since (see (2.9) and (2) for $t=0$)
$\Psi(x,0,k)=e^{-ik^{2}x\sigma_{3}},\qquad\mbox{for }x\geq 0,$
$\Phi(x,0,k)=e^{-iBx\sigma_{3}}E(k)e^{-ixX(k)\sigma_{3}},\qquad\mbox{for
}x\leq 0,$
we conclude that
$S(k)=\Psi^{-1}(0,0,k)\Phi(0,0,k)=\Phi(0,0,k)=E(k).$ (2.12)
Thus, we have
$S(k)=\left(\begin{array}[]{cc}\bar{a}(\bar{k})&b(k)\\\
-\bar{b}(\bar{k})&a(k)\end{array}\right)=\left(\begin{array}[]{cc}a(k)&b(k)\\\
b(k)&a(k)\end{array}\right),$ (2.13)
where
$\begin{split}&a(k)=\bar{a}(\bar{k})=\frac{1}{2}[\varphi(k)+\frac{1}{\varphi(k)}],\\\
&b(k)=-\bar{b}(\bar{k})=\frac{1}{2}[\varphi(k)-\frac{1}{\varphi(k)}].\end{split}$
(2.14)
## 3\. The basic Riemann-Hilbert problem
The scattering relation (2.11) involving the eigenfunctions $\Psi(x,t,k)$ and
$\Phi(x,t,k)$ can be rewritten in the form of conjugation of boundary values
of a piecewise analytic matrix-value function on a contour in the complex
$k-$plane,namely:
$M_{+}(x,t,k)=M_{-}(x,t,k)J(x,t,k),\qquad k\in\Sigma,$ (3.1)
where $M_{\pm}(x,t,k)$ denote the boundary vales of $M(x,t,k)$ according to a
chosen orientation of $\Sigma$, and $\Sigma={\mathbb{R}}\cup
i{\mathbb{R}}\cup\gamma\cup\bar{\gamma}$.
Indeed,let us write (2.11) in the vector form:
$\begin{split}&\frac{\Phi^{(1)}(x,t,k)}{a(k)}=\Psi^{(1)}(x,t,k)+r(k)\Psi^{(2)}(x,t,k),\\\
&\frac{\Phi^{(2)}(x,t,k)}{a(k)}=r(k)\Psi^{(1)}(x,t,k)+\Psi^{(2)}(x,t,k),\end{split}$
(3.2)
where
$r(k):=\frac{b(k)}{a(k)}=\frac{i}{kA}[k^{2}-B-\frac{A^{2}}{2}-X(k)],$ (3.3)
and define the matrix $M(x,t,k)$ as follows:
$M(x,t,k)=\left\\{\begin{array}[]{cc}(\begin{array}[]{cc}\frac{\Phi^{(1)}(x,t,k)}{a(k)}e^{it\theta(k)}&\Psi^{(2)}(x,t,k)e^{-it\theta(k)}\end{array}),&k\in\\{k\in{\mathbb{C}}|\mathrm{Im}k^{2}>0\backslash\gamma\\},\\\
(\begin{array}[]{cc}\Psi^{(1)}(x,t,k)e^{it\theta(k)}&\frac{\Phi^{(2)}(x,t,k)}{a(k)}e^{-it\theta(k)}\end{array}),&k\in\\{k\in{\mathbb{C}}|\mathrm{Im}k^{2}<0\backslash\bar{\gamma}\\},\end{array}\right.$
(3.4)
where
$\theta(k):=2k^{4}+\frac{x}{t}k^{2},$ (3.5)
Then the boundary values $M_{+}(x,t,k)$ and $M_{-}(x,t,k)$ relative to
$\Sigma$ are related by (3.1),where
$J(x,t,k)=\left\\{\begin{array}[]{lc}\left(\begin{array}[]{cc}1-r^{2}(k)&-r(k)e^{-2it\theta(k)}\\\
r(k)e^{2it\theta(k)}&1\end{array}\right),&k^{2}\in{\mathbb{R}},\\\
\left(\begin{array}[]{cc}1&0\\\
f(k)e^{2it\theta(k)}&1\end{array}\right),&k^{2}\in\gamma,\\\
\left(\begin{array}[]{cc}1&f(k)e^{-2it\theta(k)}\\\
0&1\end{array}\right),&k^{2}\in\bar{\gamma},\end{array}\right.$ (3.6)
with
$f(k):=r_{+}(k)-r_{-}(k).$ (3.7)
The jump relation (3.1) considered together with the properties of the
eigenfunctions listed in Proposition 1 suggests a way of representing the
solution to the Cauchy problem (1.6a) and (1.6b) in terms of the solution of
the Riemann-Hilbert problem, which is specified by the initial conditions
(1.6b) via the associated spectral function $r(k)$.
The solution $q(x,t)$ of the initial value problem (1.6a) and (1.6b) can be
expressed in terms of the solution of the basic Riemann-Hilbert problem as
follows:
$q(x,t)=2i\lim_{k\rightarrow\infty}(kM(x,t,k))_{12}.$ (3.8)
where $M$ is the solution of the following Riemann-Hilbert problem:
Basic Riemann-Hilbert problem i@.
Given $r(k),k^{2}\in{\mathbb{R}}$ and
$f(k)=r_{+}(k)-r_{-}(k),k^{2}\in\gamma\cup\bar{\gamma}$, and
$\Sigma={\mathbb{R}}\cup i{\mathbb{R}}\cup\gamma\cup\bar{\gamma}$, find a
$2\times 2$ matrix-value function $M(x,t,k)$ such that
1. (i)
$M(x,t,k)$ is analytic in $k\in{\mathbb{C}}\backslash\Sigma$.
2. (ii)
$M(x,t,k)$ is bounded at the end points $E$ and $\bar{E}$.
3. (iii)
The boundary value $M_{\pm}(x,t,k)$ at $\Sigma$ satisfy the jump condition
$M_{+}(x,t,k)=M_{-}(x,t,k)J(x,t,k),\quad k\in\Sigma$
where the jump matrix $J(x,t,k)$ is defined in terms of $r(k)$ and $f(k)$ by
(3.6).
4. (iv)
Behavior at $\infty$
$M(x,t,k)=\mathbb{I}+O(\frac{1}{k}),\qquad\mbox{as }k\rightarrow\infty.$
If we try to analysis the long-time asymptotic behavior of the GI-type of DNLS
equation (1.6a) and (1.6b) with step-like initial value problem, this type of
Riemann-Hilbert problem has a contradiction in the plane wave region. So we
try to derive a new Riemann-Hilbert problem, which is similar to the type of
nonlinear Schrödinger equation, to overcome this contradiction. That means we
arrive at the following Riemann-Hilbert problem.
We define
$N(x,t,k)=k^{-\frac{\hat{\sigma}_{3}}{2}}M(x,t,k),$ (3.9)
then the jump condition for $N$ is
$N_{+}(x,t,k)=N_{-}(x,t,k)e^{-i(k^{2}x+2k^{4}t)\hat{\sigma}_{3}}J_{N}(x,t,k).$
(3.10)
introducing $\lambda=k^{2}$ and control the branch of $k$ as
$Sign\mathrm{Im}k=Sign\mathrm{Im}\lambda$, and define the modified scattering
data $\rho(\lambda)=\frac{r(k)}{k}$, [13].
Then
$X(\lambda)=\sqrt{(\lambda-B-\frac{A^{2}}{2})^{2}+\lambda
A^{2}}=\sqrt{(\lambda-B)^{2}+\frac{A^{4}}{4}+A^{2}B},$ (3.11)
$\Omega(\lambda)=2(\lambda+B)X(\lambda).$ (3.12)
and the segment
$\gamma\cup\bar{\gamma}:=\\{\lambda\in{\mathbb{C}}|\lambda_{1}=B,\lambda_{2}^{2}\leq
D^{2}\\},$ (3.13)
where $\gamma=\\{k\in{\mathbb{C}}|\lambda_{1}=B,\lambda_{2}^{2}\leq
D^{2},\mathrm{Im}\lambda_{2}>0\\}$, $D^{2}=A^{2}B+\frac{A^{4}}{4}$,
$\lambda_{1}=\mathrm{Re}{\lambda}$ and $\lambda_{2}=\mathrm{Im}{\lambda}$. Let
$E=B+iD$, then $\gamma=[E,B]$ and $\bar{\gamma}=[B,\bar{E}]$. And the jump
condition for $N$ is
$N_{+}(x,t,\lambda)=N_{-}(x,t,\lambda)e^{-i(\lambda
x+2\lambda^{2}t)\hat{\sigma}_{3}}J_{N}(x,t,\lambda).$ (3.14)
where
$J_{N}(x,t,\lambda)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1-\lambda\rho(\lambda)^{2}&-\rho(\lambda)e^{-2it\theta(\lambda)}\\\
\lambda\rho(\lambda)e^{2it\theta(\lambda)}&1\end{array}\right),&\lambda\in{\mathbb{R}},\\\
\left(\begin{array}[]{cc}1&0\\\ \lambda
f(\lambda)e^{2it\theta(\lambda)}&1\end{array}\right),&\lambda\in\gamma,\\\
\left(\begin{array}[]{cc}1&f(\lambda)e^{-2it\theta(\lambda)}\\\
0&1\end{array}\right),&\lambda\in\bar{\gamma},\end{array}\right.$ (3.15)
where
$f(\lambda)=\rho(\lambda)_{+}-\rho(\lambda)_{-}.$ (3.16)
Figure 1. The oriented contour
$\Sigma={\mathbb{R}}\cup\gamma\cup\bar{\gamma}$.
In other word,we have the following basic Riemann-Hilbert problem
Basic Riemann-Hilbert problem ii@.
Given $\rho(\lambda),\lambda\in{\mathbb{R}}$ and
$f(\lambda)=\rho(\lambda)_{+}-\rho(\lambda)_{-},\lambda\in\gamma\cup\bar{\gamma}$,
and $\Sigma={\mathbb{R}}\cup\gamma\cup\bar{\gamma}$, find a $2\times 2$
matrix-value function $N(x,t,\lambda)$ such that
1. (i)
$N(x,t,\lambda)$ is analytic in $\lambda\in{\mathbb{C}}\backslash\Sigma$.
2. (ii)
$N(x,t,\lambda)$ is bounded at the end points $E$ and $\bar{E}$.
3. (iii)
The boundary value $N_{\pm}(x,t,\lambda)$ at $\Sigma$ satisfy the jump
condition
$N_{+}(x,t,\lambda)=N_{-}(x,t,\lambda)J_{N}(x,t,\lambda),\quad\lambda\in\Sigma\backslash\\{E,\bar{E},B\\},$
where the jump matrix $J_{N}(x,t,k)$ is defined in terms of $\rho(\lambda)$
and $f(\lambda)$ by (3.15).
4. (iv)
Behavior at $\infty$
$N(x,t,\lambda)=\mathbb{I}+O(\frac{1}{\lambda}),\qquad\mbox{as
}\lambda\rightarrow\infty.$
## 4\. Long-time Asymptotics
The representation of the solution $q(x,t)$ of the initial value problem (1.6)
in terms of the solution of an associated basic Riemann-Hilbert problem allows
using the ideas of the asymptotic analysis of oscillating Riemann-Hilbert
problems [9, 28, 10, 11, 32] for studying the long-time asymptotics of
$q(x,t)$. The key fact leading to different asymptotics in different regions
of the $(x,t)$ half-plane is that the behavior of the jump matrix of the basic
Riemann-Hilbert problem as a function of the large parameter $t$ is different
in these regions. Indeed, as seen on (3.15), this behavior is governed by the
sign of $\mathrm{Im}\theta(\lambda)$, which itself depends on
$\xi=\frac{x}{4t}$. As we have already written, three regions are to be
distinguished:
1. (i)
A Zakharov-Manakov region:$\xi>-B$.
2. (ii)
A plane wave region:$\xi<-\sqrt{2}D-B$.
3. (iii)
An elliptic wave region:$-\sqrt{2}D-B<\xi<-B$.
Figure 2. The different regions of the $(x,t)-$plane.
### 4.1. The Zakharov-Manakov region:$\xi>-B$
In this region $\xi>-B$, we have $\mathrm{Im}\theta(\lambda)>0$ for all
$\lambda\in\gamma$ and $\mathrm{Im}\theta(\lambda)<0$ for all
$\lambda\in\bar{\gamma}$. Therefore, the exponentials in the jump matrix
$J_{N}$, see (3.15), are decaying as $t\rightarrow+\infty$ for
$\lambda\in\Sigma\backslash{\mathbb{R}}$.
This implies that one can follow the technique of asymptotic analysis proposed
for the first time in [9]. The basic step of the procedure is a deformation of
the original Riemann-Hilbert problem, with the help of the solution of an
appropriate scalar Riemann-Hilbert problem, in order to obtain an equivalent
Riemann-Hilbert problem whose jump matrix decays, in $t$, to a constant (in
$\lambda$) matrix. This leads to model Riemann-Hilbert problems whose
solutions can be given explicitly.
A particular feature of the Riemann-Hilbert problem under consideration is
that the contour of the modified Riemann-Hilbert problem contains neither the
real axis, where the jump matrix for the original Riemann-Hilbert problem
oscillates with $t$, see (3.15), nor the finite parts $\gamma$ and
$\bar{\gamma}$. This happens due to the pure step-like initial conditions,
which in turn implies that the associated spectral functions $\rho(\lambda)$
and $\lambda\rho(\lambda)$ can be analytically extended from the contour to
the whole $\lambda$-plane.
#### 4.1.1. First transformation
The first transform is as usual:
$N^{(1)}(x,t,\lambda)=N(x,t,\lambda)\delta^{-\sigma_{3}}(\lambda),$ (4.1)
where ([41])
$\delta(\lambda)=\exp{\frac{1}{2\pi
i}}\int_{-\infty}^{\lambda_{0}}\frac{\log{(1-\lambda^{\prime}\rho(\lambda^{\prime})^{2})}}{\lambda^{\prime}-\lambda}d\lambda^{\prime},$
(4.2)
is the solution of the following scalar Riemann-Hilbert problem:
* •
$\delta(\lambda)$ is analytic in
${\mathbb{C}}\backslash(-\infty,\lambda_{0}]$,
* •
$\delta(\lambda)\rightarrow 1$ as $\lambda\rightarrow\infty$,
* •
$\delta(\lambda)$ satisfies the jump relation
$\delta_{+}(\lambda)=\delta_{-}(\lambda)(1-\lambda\rho^{2}(\lambda)),\qquad\lambda\in(-\infty,\lambda_{0}).$
(4.3)
Here, $\lambda_{0}$ is the stationary point of the phase function
$\theta(\lambda)=2\lambda^{2}+4\xi\lambda$, that is,
$\theta^{\prime}(\lambda_{0})=0$:
$\lambda_{0}=-\xi=\frac{-x}{4t}.$
Then $N^{(1)}(x,t,\lambda)$ satisfies the jump condition
$\begin{split}&N_{+}^{(1)}(x,t,\lambda)=N_{-}^{(1)}(x,t,N)J_{N}^{(1)}(x,t,\lambda),\\\
&\lambda\in\Sigma^{(1)}=\Sigma,\end{split}$ (4.4)
where
$J_{N}^{(1)}(x,t,\lambda)=\delta_{-}^{\sigma_{3}}J_{N}\delta_{+}^{-\sigma_{3}},$
that is
$J_{N}^{(1)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}e^{-it\theta\hat{\sigma}_{3}}\left(\begin{array}[]{cc}\frac{\delta_{-}}{\delta_{+}}(1-\lambda\rho(\lambda)^{2})&-\rho\delta_{+}\delta_{-}\\\
\frac{\lambda\rho}{\delta_{+}\delta_{-}}&\frac{\delta_{+}}{\delta_{-}}\end{array}\right),&\qquad\lambda\in{\mathbb{R}},\\\
\left(\begin{array}[]{cc}\frac{\delta_{-}}{\delta_{+}}&0\\\ \frac{\lambda
f}{\delta_{+}\delta_{-}}e^{2it\theta\sigma_{3}}&\frac{\delta_{+}}{\delta_{-}}\end{array}\right),&\qquad\lambda\in\gamma,\\\
\left(\begin{array}[]{cc}\frac{\delta_{-}}{\delta_{+}}&f\delta_{+}\delta_{-}e^{-2it\theta\sigma_{3}}\\\
0&\frac{\delta_{-}}{\delta_{+}}\end{array}\right),&\qquad\lambda\in\bar{\gamma}.\end{array}\right.$
(4.5)
From the Riemann-Hilbert problem of the $\delta$, we can find
$J_{N}^{(1)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}e^{-it\theta\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1-\lambda\rho^{2}&-\rho\delta^{2}\\\
\frac{\lambda\rho}{\delta^{2}}&1\end{array}\right),&\qquad\lambda>\lambda_{0},\\\
e^{-it\theta\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1&\frac{-\rho}{1-\lambda\rho^{2}}\delta_{-}^{2}\\\
\frac{\lambda\rho}{1-\lambda\rho^{2}}\frac{1}{\delta_{+}^{2}}&1-\lambda\rho^{2}\end{array}\right),&\qquad\lambda<\lambda_{0},\\\
\left(\begin{array}[]{cc}1&0\\\ \frac{\lambda
f}{\delta^{2}}e^{2it\theta}&1\end{array}\right),&\qquad\lambda\in\gamma,\\\
\left(\begin{array}[]{cc}1&f\delta^{2}e^{-2it\theta}\\\
0&1\end{array}\right),&\qquad\lambda\in\bar{\gamma}.\end{array}\right.$ (4.6)
#### 4.1.2. Second transformation
The next transformation is:
$N^{(2)}(x,t,\lambda)=N^{(1)}(x,t,\lambda)G(\lambda),$ (4.7)
where
$G(\lambda)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1&\frac{\rho}{1-\lambda\rho^{2}}\delta_{-}^{2}e^{-2it\theta}\\\
0&1\end{array}\right),&\qquad\lambda\in D_{1},\\\
\left(\begin{array}[]{cc}1&0\\\
\frac{\lambda\rho}{1-\lambda\rho^{2}}\frac{1}{\delta_{+}^{2}}e^{2it\theta}&1\end{array}\right),&\qquad\lambda\in
D_{2},\\\ \left(\begin{array}[]{cc}1&-\rho\delta^{2}e^{-2it\theta}\\\
0&1\end{array}\right),&\qquad\lambda\in D_{3},\\\
\left(\begin{array}[]{cc}1&0\\\
\frac{-\lambda\rho}{\delta^{2}}e^{2it\theta}&1\end{array}\right),&\qquad\lambda\in
D_{4},\\\ \mathbb{I},&\qquad\lambda\in D_{5}\cup D_{6}.\end{array}\right.$
(4.8)
The domains $D_{1},\ldots D_{6}$ are shown on the following Figure.
Figure 3. The oriented contour $\Sigma^{(2)}=L_{1}\cup L_{2}\cup L_{3}\cup
L_{4}$.
This new function $N^{(2)}$ solves the equivalent Riemann-Hilbert problem:
$\begin{split}&N_{+}^{(2)}(x,t,\lambda)=N_{-}^{(2)}(x,t,\lambda)J_{N}^{(2)}(x,t,\lambda),\\\
&\lambda\in\Sigma^{(2)},\end{split}$
where
$\small
J_{N}^{(2)}(x,t,\lambda)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1&\frac{-\rho}{1-\lambda\rho^{2}}\delta_{-}^{2}e^{-2it\theta}\\\
0&1\end{array}\right),&\qquad\lambda\in L_{1},\\\
\left(\begin{array}[]{cc}1&0\\\
\frac{\lambda\rho}{1-\lambda\rho^{2}}\frac{1}{\delta_{+}^{2}}e^{2it\theta}&1\end{array}\right),&\qquad\lambda\in
L_{2},\\\ \left(\begin{array}[]{cc}1&-\rho\delta^{2}e^{-2it\theta}\\\
0&1\end{array}\right),&\qquad\lambda\in L_{3},\\\
\left(\begin{array}[]{cc}1&0\\\
\frac{\lambda\rho}{\delta^{2}}e^{2it\theta}&1\end{array}\right),&\qquad\lambda\in
L_{4}.\end{array}\right.$ (4.9)
#### 4.1.3. The last transformation
Now $J_{N}^{(2)}(x,t,\lambda)$ decays exponentially fast to the identity
matrix, as $t\rightarrow+\infty$, and uniformly outside any neighborhood of
$\lambda=\lambda_{0}$. Thus, we are in a situation where the asymptotic
analysis of [41] works. Particularly,
$N^{(2)}(x,t,\lambda)=Z(x,t,\lambda)N^{as}(x,t,\lambda),$
where $N^{as}(x,t,\lambda)$ is a solution of the model problem explicitly
given in terms of parabolic cylinder functions whereas $Z(x,t,\lambda)$ can be
estimated:
$Z(x,t,\lambda)=\mathbb{I}+O(\frac{logt}{t^{\frac{1}{2}}}).$
Therefore, the final asymptotic result is as in [41] giving the main term of
the asymptotic in terms of the modified reflection coefficient
$\rho(\lambda)$:
###### Theorem 4.1.
(The Zakharov-Manakov region) In the region $x>-4tB$, the asymptotics, as
$t\rightarrow+\infty$, of the solution $q(x,t)$ of the initial value problem
(1.6) is described by the Zakharov-Manakov type formula
$q(x,t)=q_{as}(x,t)+O(\frac{\log t}{t})$ (4.10)
where
$\begin{array}[]{l}q_{as}=\frac{1}{\sqrt{t}}\alpha(\lambda_{0})e^{\frac{ix^{2}}{4t}-i\nu(\lambda_{0})\log
t},\\\
|\alpha(\lambda_{0})|^{2}=\frac{\nu(\lambda_{0})}{2}=-\frac{1}{4\pi}\log(1-\lambda_{0}|\rho(\lambda_{0})|^{2}),\\\
\arg\alpha(\lambda_{0})=-3\nu\log 2-\frac{\pi}{4}+\arg\Gamma(i\nu)-\arg
r(\lambda_{0})+\frac{1}{\pi}\int_{-\infty}^{\lambda_{0}}\log|\lambda-\lambda_{0}|d\log(1-\lambda|\rho(\lambda)|^{2}),\\\
\lambda_{0}=-\frac{x}{4t}.\end{array}$ (4.11)
### 4.2. The plane wave region: $\xi<-\sqrt{2}D-B$
For $x<-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$, that means,
$\mathrm{Im}\theta(\lambda)$ is negative on $\gamma$ and positive on
$\bar{\gamma}$, which implies that the exponentials in (3.15) increase with
$t$. Thus, the jump matrix $J_{N}$ for the Riemann-Hilbert problem does not
converge to a reasonable limit as $t\rightarrow\infty$.
To bypass this difficulty, one deforms the Riemann-Hilbert problem in such a
way that the phase $\mathrm{Im}\theta(\lambda)$ is replaced by another
function, $g(\lambda)$, providing suitable behavior of the modified jump
matrix. The extension of the nonlinear steepest descent method for Riemann-
Hilbert problems, involving the $g$-function mechanism was first proposed by
Deift, Venakides, and Zhou, see [26, 27].
#### 4.2.1. The $g$ function
A natural choice for a $g$-function appropriate for the region adjacent to the
half-axis $x<0$, $t=0$, is the phase appearing in the explicit expression for
the eigenfunction $\Psi^{p}$, see (2.1), associated with the potential
$q^{p}$. Setting
$g(x,t,\lambda)=xX(\lambda)+t\Omega(\lambda),$ (4.12)
where $X(\lambda)$ and $\Omega(\lambda)$ are defined in (3.11) and (3.12),we
have
$\Psi^{p}(x,t,k)=e^{i(\omega
t-Bx)\sigma_{3}}E(\lambda)e^{-ig(x,t,\lambda)\sigma_{3}}$ (4.13)
The signature table for $\mathrm{Im}g(\lambda;\xi)$ is the partition of the
$\lambda$-plane into maximal domains where the sign of
$\mathrm{Im}g(\lambda;\xi)$ is constant. Its form can be controlled by the
zeros of the differential $dg(\lambda)$. Indeed,
$dg(\lambda)=4\frac{(\lambda-\mu_{+})(\lambda-\mu_{-})}{X(\lambda)}d\lambda,$
(4.14)
where
$\mu_{\pm}=\frac{B-\xi}{2}\pm\sqrt{\frac{(B+\xi)^{2}}{4}-\frac{\frac{A^{4}}{4}+A^{2}B}{2}},$
(4.15)
Thus, for $\xi<-(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$, $\mu_{\pm}$ are both
real. Moreover,
$B<\mu_{-}<\mu_{+}<-\xi.$
In what follows the signature table of the function $\mathrm{Im}g(\lambda)$
for different values of $\xi$ plays a very important role. The lines of
separation between the different domains are the real axile
$\lambda_{2}=0,$
and the algebraic curve
$\lambda_{2}^{2}(\lambda_{1}+\xi)=(\lambda_{1}+B+2\xi)[(\lambda_{1}-B)(\lambda_{1}+\xi)+\frac{\frac{A^{4}}{4}+A^{2}B}{2}],$
(4.16)
They are indeed given by $\mathrm{Im}g(\lambda)=0$. Because of
$\mathrm{Im}g(\lambda)=4\lambda_{2}\\{(\lambda_{1}+B+2\xi)[(\lambda_{1}-B)(\lambda_{1}+\xi)+\frac{\frac{A^{4}}{4}+A^{2}B}{2}]-\lambda_{2}^{2}(\lambda_{1}+\xi)\\}$
The equation (4.16) can be written:
$\lambda_{2}^{2}(\lambda_{1}+\xi)=(\lambda_{1}+B+2\xi)[(\lambda_{1}-\mu_{+})(\lambda_{1}-\mu_{-})].$
And the signature table of the function $\mathrm{Im}g(\lambda)$ is shown in
the following Figure 4.
Figure 4. The curves of $\mathrm{Im}g(\lambda)=0$ for
$x<-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$.
The advantage of the signature table shown in Figure 4 is that there is a
finite arc connecting the branch points $E$ and $\bar{E}$ such that
$\mathrm{Im}g(\lambda)=0$ for all $\lambda$ along this arc. Since the jump
matrix depends on $t$ via exponentials of type $e^{\pm ig(\lambda)}$, it is
oscillatory along an arc where $\mathrm{Im}g(\lambda)=0$.
This suggests to deform the original contour $\gamma\cup\bar{\gamma}$ of the
basic Riemann-Hilbert problem to a new contour
$\gamma_{g}\cup\bar{\gamma}_{g}$ which depends on $\xi$ and where
$\mathrm{Im}g(\lambda)=0$, and to view $X(\lambda)$, thus also $g(\lambda)$ as
functions with branch cut $\gamma_{g}\cup\bar{\gamma}_{g}$.
Another important feature of $g(\lambda;\xi)$ is that it has, up to a
constant, the same large $\lambda$ asymptotic behavior as the phase function
$\theta(\lambda)$:
$g(\lambda;\xi)=t(2\lambda^{2}+4\xi\lambda+g(\infty;\xi))+O(\frac{1}{\lambda}),\qquad\lambda\rightarrow\infty,$
(4.17)
where
$g(\infty;\xi)=(\omega-4B\xi).$ (4.18)
#### 4.2.2. The first transformation
We put
$N^{(1)}(x,t,\lambda)=e^{-itg(\infty,\xi)\sigma_{3}}N(x,t,\lambda)e^{-i(\lambda
x+2\lambda^{2}t-g(\lambda))\sigma_{3}},$
Then the matrix-value function $N^{(1)}(x,t,\lambda)$ satisfies the following
Riemann-Hilbert problem:
$N_{+}^{(1)}(x,t,\lambda)=N_{-}^{(1)}(x,t,\lambda)J_{N}^{(1)}(x,t,\lambda),\qquad\lambda\in\Sigma^{(1)}={\mathbb{R}}\cup\gamma_{g}\cup\bar{\gamma}_{g},$
with the jump matrix
$J_{N}^{(1)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}\left(\begin{array}[]{cc}1-\lambda\rho^{2}(\lambda)&-\rho(\lambda)e^{-2ig(\lambda)}\\\
\lambda\rho(\lambda)e^{2ig(\lambda)}&1\end{array}\right),&\qquad\lambda\in{\mathbb{R}},\\\
\left(\begin{array}[]{cc}e^{-2ig_{-}(\lambda)}&0\\\ \lambda
f(\lambda)&e^{2ig_{-}(\lambda)}\end{array}\right),&\qquad\lambda\in\gamma_{g},\\\
\left(\begin{array}[]{cc}e^{-2ig_{-}(\lambda)}&f(\lambda)\\\
0&e^{2ig_{-}(\lambda)}\end{array}\right),&\qquad\lambda\in\bar{\gamma}_{g}.\end{array}\right.$
(4.19)
Here $g_{\pm}(\lambda)$ are boundary values of $g$ on
$\gamma_{g}\cup\bar{\gamma}_{g}$, and they are real. We also use the equation
$g_{+}(\lambda)=-g_{-}(\lambda)$.
#### 4.2.3. The second transformation
The next transformation is similar to the first transformation applied in the
Zakharov Manakov region, see Section 4.1.1. It involves the solution
$\delta(\lambda)$ of the scalar Riemann-Hilbert problem 4.3) but with
$\mu_{+}$ instead of $\lambda_{0}$,where $\mu_{+}$ is the stationary point of
the new phase function $g(\lambda)$. With this new scalar function
$\delta(\lambda)$, we set
$N^{(2)}(x,t,\lambda)=N^{(1)}(x,t,\lambda)\delta^{-\sigma_{3}}(\lambda),$
Then the matrix-value function $N^{(2)}(x,t,\lambda)$ satisfies the following
Riemann-Hilbert problem
$N^{(2)}_{+}(x,t,\lambda)=N^{(2)}_{-}(x,t,\lambda)J_{N}^{(2)}(x,t,\lambda),\qquad\lambda\in\Sigma^{(2)}=\Sigma^{(1)},$
(4.20)
where $J_{N}^{(2)}(x,t,\lambda)$ is defined as follows:
$J_{N}^{(2)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}e^{-ig\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1-\lambda\rho^{2}&-\rho\delta^{2}\\\
\frac{\lambda\rho}{\delta^{2}}&1\end{array}\right),&\qquad\lambda>\mu_{+},\\\
e^{-ig\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1&\frac{-\rho}{1-\lambda\rho^{2}}\delta_{-}^{2}\\\
\frac{\lambda\rho}{1-\lambda\rho^{2}}\frac{1}{\delta_{+}^{2}}&1-\lambda\rho^{2}\end{array}\right),&\qquad\lambda<\mu_{+},\\\
\left(\begin{array}[]{cc}e^{-2ig_{-}(\lambda)}&0\\\ \frac{\lambda
f}{\delta^{2}}&e^{2ig_{-}(\lambda)}\end{array}\right),&\qquad\lambda\in\gamma_{g},\\\
\left(\begin{array}[]{cc}e^{-2ig_{-}(\lambda)}&f\delta^{2}\\\
0&e^{2ig_{-}(\lambda)}\end{array}\right),&\qquad\lambda\in\bar{\gamma}_{g}.\end{array}\right.$
(4.21)
#### 4.2.4. The third transformation
The subsequent transformation
$N^{(3)}(x,t,\lambda)=N^{(2)}(x,t,\lambda)G(\lambda),$
involves $G(\lambda)$ defined similarly to (4.8), with $t\theta$ replaced by
$g$ and $\lambda_{0}$ replaced by $\mu_{+}$. Then $N^{(3)}(x,t,\lambda)$
satisfies the jump relation
$N_{+}^{(3)}(x,t,\lambda)=N_{-}^{(3)}(x,t,\lambda)J_{N}^{(3)}(x,t,\lambda),$
across to the contour
$\Sigma^{(3)}=L_{1}\cup L_{2}\cup L_{3}\cup
L_{4}\cup\gamma_{g}\cup\bar{\gamma}_{g},$
shown in Figure 5.
Figure 5. The contour $\Sigma^{(3)}=L_{1}\cup L_{2}\cup L_{3}\cup
L_{4}\cup\gamma_{g}\cup\bar{\gamma}_{g}$ of the Riemann-Hilbert problem for
$N^{(3)}$ for $x<-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$.
And we notice that
1.For $\lambda\in L_{1}\cup L_{2}\cup L_{3}\cup L_{4}$ the jump matrix
$J_{N}^{(3)}(x,t,\lambda)$ decays to the identity matrix, as
$t\rightarrow\infty$, exponentially fast and uniformly outside any
neighborhood of $\lambda=\mu_{+}$.
2.For $\lambda\in\gamma_{g}$, the jump matrix $J_{N}^{(3)}(x,t,\lambda)$
factorizes as
$\small\left(\begin{array}[]{cc}1&(\frac{-\rho}{1-\lambda\rho^{2}})_{-}\delta^{2}e^{-2ig_{-}(\lambda)}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}e^{-2ig_{(}\lambda)}&0\\\
\lambda
f(\lambda)\delta^{-2}(\lambda)&e^{2ig_{-}(\lambda)}\end{array}\right)\left(\begin{array}[]{cc}1&(\frac{\rho}{1-\lambda\rho^{2}})_{+}\delta^{2}e^{2ig_{-}(\lambda)}\\\
0&1\end{array}\right)$ (4.22)
3.For $\lambda\in\bar{\gamma}_{g}$, the jump matrix $J_{N}^{(3)}(x,t,\lambda)$
factorizes as
$\small\left(\begin{array}[]{cc}1&0\\\
(\frac{-\lambda\rho}{1-\lambda\rho^{2}})_{-}\delta^{-2}e^{2ig_{-}(\lambda)}&1\end{array}\right)\left(\begin{array}[]{cc}e^{-2ig_{-}(\lambda)}&f(\lambda)\delta^{2}(\lambda)\\\
0&e^{2ig_{-}(\lambda)}\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
(\frac{\lambda\rho}{1-\lambda\rho^{2}})_{+}\delta^{-2}e^{2ig_{-}(\lambda)}&1\end{array}\right)$
(4.23)
4.Using the identities
$1+\lambda f(\frac{-\rho}{1-\lambda\rho^{2}})_{-}=0,$
$1+f(\frac{\lambda\rho}{1-\lambda\rho^{2}})_{+}=0,$
we find
$J_{N}^{(3)}(x,t,k)=\left\\{\begin{array}[]{lc}\left(\begin{array}[]{cc}0&-(\lambda
f)^{-1}(\lambda)\delta^{2}(\lambda)\\\ \lambda
f(\lambda)\delta^{-2}(\lambda)&0\end{array}\right),&\qquad\lambda\in\gamma_{g},\\\
\left(\begin{array}[]{cc}0&f(\lambda)\delta^{2}(\lambda)\\\
-f^{-1}(\lambda)\delta^{-2}(\lambda)&0\end{array}\right),&\qquad\lambda\in\bar{\gamma}_{g},\end{array}\right.$
(4.24)
In order to arrive at a Riemann-Hilbert problem whose jump matrix does not
depend on $\lambda$, we introduce a factorization involving a scalar function
$F(\lambda)$ to be defined;
$J_{N}^{(3)}(x,t,\lambda)=\left(\begin{array}[]{cc}F_{+}^{-1}(\lambda)&0\\\
0&F_{+}(\lambda)\end{array}\right)\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)\left(\begin{array}[]{cc}F_{-}(\lambda)&0\\\
0&F_{-}^{-1}(\lambda)\end{array}\right),$ (4.25)
in such a way that the boundary values $F_{\pm}(\lambda)$ of $F(\lambda)$
along the two sides of $\gamma_{g}\cup\bar{\gamma}_{g}$ satisfy
$F_{-}(\lambda)F_{+}(\lambda)=\left\\{\begin{array}[]{lc}-i\lambda
f(\lambda)\delta^{-2}(\lambda)&\qquad\lambda\in\gamma_{g},\\\
if^{-1}(\lambda)\delta^{-2}(\lambda)&\qquad\lambda\in\bar{\gamma}_{g}.\end{array}\right.$
(4.26)
Indeed, once (4.25) is satisfied, one can absorb the diagonal factors into a
new piecewise analytic function whose jump across
$\gamma_{g}\cup\bar{\gamma}_{g}$ is only the constant middle factor in (4.25).
Thus, we arrive at the following scalar Riemann-Hilbert problem:
Scalar Riemann-Hilbert problem.
Find a scalar function $F(\lambda)$ such that
* •
$F(\lambda)$ and $F^{-1}(\lambda)$ are analytic in
${\mathbb{C}}\backslash\\{\gamma_{g}\cup\bar{\gamma}_{g}\\}$.
* •
$F(\lambda)$ satisfies the jump relation:
$F_{+}(\lambda)F_{-}(\lambda)=\left\\{\begin{array}[]{lc}-i\lambda
f(\lambda)\delta^{-2}(\lambda)=a_{+}^{-1}(\lambda)a_{-}^{-1}(\lambda)\sqrt{\lambda}\delta^{-2}(\lambda),&\qquad\lambda\in\gamma_{g},\\\
if^{-1}(\lambda)\delta^{-2}(\lambda)=a_{+}(\lambda)a_{-}(\lambda)\sqrt{\lambda}\delta^{-2}(\lambda),&\qquad\lambda\in\bar{\gamma}_{g}.\end{array}\right.$
(4.27)
where the contour $\gamma_{g}\cup\bar{\gamma}_{g}$ is oriented from $E$ to
$\bar{E}$, and
* •
$F(\lambda)$ is bounded at $\lambda=\infty$.
Introducing
$H(\lambda)=\left\\{\begin{array}[]{lc}F(\lambda)a(\lambda),&\qquad\lambda\in{\mathbb{C}}_{+}\backslash\gamma_{g},\\\
\frac{F(\lambda)}{a(\lambda)},&\qquad\lambda\in{\mathbb{C}}_{-}\backslash\bar{\gamma}_{g}.\end{array}\right.$
(4.28)
then the jump relation (4.27) transforms to
$[\frac{\log{H(\lambda)}}{X(\lambda)}]_{+}-[\frac{\log{H(\lambda)}}{X(\lambda)}]_{-}=\left\\{\begin{array}[]{ll}\frac{\log{\sqrt{\lambda}\delta^{-2}(\lambda)}}{X(\lambda)_{+}},&\qquad\lambda\in\gamma_{g}\cup\bar{\gamma}_{g},\\\
\frac{\log{a^{2}(\lambda)}}{X(\lambda)},&\qquad\lambda\in{\mathbb{R}}.\end{array}\right.$
(4.29)
The Sokhotski-Plemelj formula shows that this last jump relation is satisfied
by
$H(k)=\exp\\{\frac{X(\lambda)}{2\pi
i}[\int_{\gamma_{g}\cup\bar{\gamma}_{g}}\frac{\log{\sqrt{s}}+\log{\delta^{-2}(s,\xi)}}{s-\lambda}\frac{ds}{X_{+}(s)}+\int_{{\mathbb{R}}}\frac{\log
ab(s)}{s-\lambda}\frac{ds}{X(s)}]\\}$ (4.30)
Then $F(\lambda)$ is defined in terms of $H(\lambda)$ by (4.28). At
$\lambda=\infty$ we find
$F(\infty)=H(\infty)=e^{i\phi(\xi)},$
where
$\phi(\xi)=\frac{1}{2\pi}[\int_{\gamma_{g}\cup\bar{\gamma}_{g}}\frac{\log{\sqrt{s}\delta^{-2}(s,\xi)}}{X_{+}(s)}ds+\int_{{\mathbb{R}}}\frac{\log
a^{2}(s)}{X(s)}ds]$ (4.31)
with
$\delta(\lambda,\xi)=\exp{\frac{1}{2\pi
i}}\int_{-\infty}^{\mu_{+}}\frac{\log{(1-\lambda^{\prime}\rho(\lambda^{\prime})^{2})}}{\lambda^{\prime}-\lambda}d\lambda^{\prime},$
(4.32)
Using the relation $1-\lambda\rho^{2}(\lambda)=a^{-2}(\lambda)$, we find a
simpler expression for $\phi(\xi)$:
$\phi(\xi)=\frac{1}{2\pi}[\int_{\mu_{+}}^{+\infty}\log{a^{2}(\lambda)}\frac{d\lambda}{X(\lambda)}+\int_{\gamma_{g}\cup\bar{\gamma}_{g}}\frac{\log{\sqrt{\lambda}}}{X_{+}(\lambda)}d\lambda]$
#### 4.2.5. The fourth transformation
The factorization (4.25) suggests a fourth transformation
$N^{(4)}(x,t,\lambda)=F^{\sigma_{3}}(\infty,\xi)N^{(3)}(x,t,\lambda)F^{-\sigma_{3}}(\lambda,\xi),$
Then we have
$N^{(4)}_{+}(x,t,\lambda)=N^{(4)}_{-}(x,t,\lambda)J_{N}^{(4)}(x,t,\lambda)$
For $\lambda\in\gamma_{g}\cup\bar{\gamma}_{g}$ the jump matrix
$J_{N}^{(4)}(x,t,\lambda)$ is constant
$J_{N}^{(4)}(x,t,\lambda)=J_{N}^{mod}=\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right).$
1.For $\lambda\in\gamma_{g}\cup\bar{\gamma}_{g}$ the jump matrix
$J_{N}^{(4)}(x,t,\lambda)$ is constant:
$J_{N}^{(4)}(x,t,\lambda)=J_{N}^{mod}=\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right).$
2.For $\lambda\in L\cup\bar{L}$, the jump matrix $J_{N}^{(4)}(x,t,\lambda)$
decays to the identity
$J_{N}^{(4)}(x,t,\lambda)=\mathbb{I}+O(\frac{1}{e^{\varepsilon t}}).$
#### 4.2.6. The final transformation
Finally, we can express $N^{(4)}$ in the form
$N^{(4)}(x,t,\lambda)=N^{err}(x,t,\lambda)N^{mod}(x,t,\lambda),$
where $N^{mod}(x,t,\lambda)$ solves the model problem:
$N_{-}^{mod}(x,t,\lambda)=N_{+}^{(mod)}(x,t,\lambda)J_{N}^{mod},\qquad\lambda\in\gamma_{g}\cup\bar{\gamma}_{g},$
(4.33)
with constant jump matrix
$J_{N}^{mod}=\left(\begin{array}[]{cc}0&i\\\ i&0\end{array}\right),$
and $N^{err}(x,t,\lambda)=\mathbb{I}+O(t^{-\frac{1}{2}})$.
As for the model problem, since $\varphi(\lambda)_{-}=i\varphi(\lambda)_{+}$
on $\gamma_{g}\cup\bar{\gamma}_{g}$, its solution can be given explicitly in
terms of $\varphi(\lambda)$:
$N^{mod}(x,t,\lambda)=\frac{1}{2}\left(\begin{array}[]{cc}\varphi(\lambda)+\frac{1}{\varphi(\lambda)}&\varphi(\lambda)-\frac{1}{\varphi(\lambda)}\\\
\varphi(\lambda)-\frac{1}{\varphi(\lambda)}&\varphi(\lambda)+\frac{1}{\varphi(\lambda)}\end{array}\right).$
#### 4.2.7. Back to the original problem
Let $N^{*}(x,t,\lambda)$, $*=$ (1),(2),(3),(4),mod, denote the solution of the
Riemann-Hilbert problem $RH^{*}$, and let
$m_{12}^{*}(x,t)=\lim_{\lambda\rightarrow\infty}(\lambda
M^{*}(x,t,\lambda))_{12},$
Then, going back to the determination of $q(x,t)$ in terms of the solution of
the basic Riemann-Hilbert problem, we have
$\begin{split}q(x,t)=&2im(x,t)_{12}=2ie^{2ig(\infty,\xi)}m^{(1)}(x,t)_{12}\\\
=&2ie^{2ig(\infty,\xi)}m^{(2)}(x,t)_{12}+O(t^{-\frac{1}{2}})\\\
=&2ie^{2ig(\infty,\xi)}m^{(3)}(x,t)_{12}+O(t^{-\frac{1}{2}})\\\
=&2ie^{2ig(\infty,\xi)}m^{(4)}(x,t)_{12}F^{-2}(\infty,\xi)+O(t^{-\frac{1}{2}})\\\
=&2ie^{2ig(\infty,\xi)}m^{mod}(x,t)_{12}F^{-2}(\infty,\xi)+O(t^{-\frac{1}{2}}).\end{split}$
(4.34)
Taking into account that $g(\infty,\xi)=\omega t-4Bx$, $2im^{mod}(x,t)_{12}=A$
and $F^{-2}(\infty,\xi)=e^{-2i\phi(\xi)}$we arrive at the following theorem:
###### Theorem 4.2.
(Plane wave region) In the region
$x<-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$,the asymptotics, as
$t\rightarrow+\infty$, of the solution $q(x,t)$ of the initial value problem
(1.6) takes the form of a plane wave:
$q(x,t)=Ae^{2i(\omega t-Bx-\phi(\xi))}+O(t^{-\frac{1}{2}}),\qquad
t\rightarrow+\infty.$ (4.35)
###### Remark 4.3.
If we let $\xi\rightarrow+\infty$, then $\mu_{+}\rightarrow+\infty$, then
$\phi(\xi)\rightarrow\phi$, with
$\phi=\frac{1}{2\pi}\int_{\gamma_{g}\cup\bar{\gamma}_{g}}\frac{\log{\sqrt{\lambda}}}{X_{+}(\lambda)}d\lambda$,
and then the above equation (4.61) reduce to $q(x,t)=Ae^{2i(\omega
t-Bx-\phi)}$, this is correspondence to our initial condition up to a phase
shift.
### 4.3. The elliptic region:$-4t(B+\sqrt{2}D)<x<-4tB$
For the limit case $\xi_{0}=-(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$, we have
$\mu_{+}(\xi_{0})=\mu_{-}(\xi_{0})$, see Figure 7, whereas for
$\xi>-(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$, $\mu_{+}$ and $\mu_{-}$ become
non-real, complex conjugated numbers. As a result, the $g$-function mechanism
with $g(\lambda;\xi)$ as in the plane wave region fails. This shows that there
is a break in the qualitative picture of the asymptotic behavior at
$\xi=\xi_{0}$.
#### 4.3.1. The new $g$-function
A suitable $g$-function for $\xi>-(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})$ can be
obtained as follows. First, we need to introduce a new real stationary point
$\mu(\xi)$ which must be a zero of the new differential $d\hat{g}$. On the
other hand we have to preserve the asymptotic behavior of the $g$-function for
large $\lambda$. To do so we must change the denominator of the differential
$d\hat{g}$. Thus the new differential takes the form:
$d\hat{g}(\lambda,\xi)=4\frac{(\lambda-\mu(\xi))(\lambda-\mu_{-}(\xi))(\lambda-\mu_{+}(\xi))}{\sqrt{(\lambda-E)(\lambda-\bar{E})(\lambda-d(\xi))(\lambda-\bar{d}(\xi))}}d\lambda,$
(4.36)
where $\mu(\xi),\mu_{\pm}(\xi)$, and $d(\xi),\bar{d}(\xi)$ are to be
determined.
If $\mu=d=\bar{d}$, then the new differential coincides with the previous one,
that is $dg=d\hat{g}$, which is expected to hold for the value $\xi_{0}$ of
$\xi$ limiting the two adjacent asymptotic regions.
Now we consider $d\hat{g}$ as an Abelian differential of the second kind with
poles at $\infty_{\pm}$ on the Riemann-Hilbert surface of
$\omega(\lambda)=\sqrt{(\lambda-E)(\lambda-\bar{E})(\lambda-d(\xi))(\lambda-\bar{d}(\xi))},$
with
$E=B+iD,\quad d(\xi)=d_{1}(\xi)+id_{2}(\xi)$
The branch of the square root is fixed by the asymptotics on the upper sheet:
$\omega(\lambda)=\lambda^{2}+O(\lambda),\qquad\lambda\rightarrow\infty_{+}.$
We choose on this Riemann surface a basis $\\{a,b\\}$ of cycles as follows.
The $b$-cycle is a closed clock-wise oriented simple loop around the arc
$\gamma_{E,d}$ joining $E$ and $d$. The $a$-cycle starts on the upper sheet
from the left side of the cut $\gamma_{E,d}$, goes to the left side of the cut
$\gamma_{\bar{d},\bar{E}}$, proceeds to the lower sheet, and then returns to
the starting point.
We can also write the Abelian differential $d\hat{g}(\lambda)$ in the form:
$d\hat{g}(\lambda)=4\frac{\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda+c_{0}}{\omega(\lambda)}d\lambda,$
(4.37)
and normalize it so that its $a-$period vanishes. This determines $c_{0}$:
$c_{0}=-\frac{\int_{\bar{d}}^{d}(\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda)\frac{d\lambda}{\omega(\lambda)}}{\int_{\bar{d}}^{d}\frac{d\lambda}{\omega(\lambda)}}\in{\mathbb{R}}.$
We also require that $\hat{g}(\lambda)$ has the same large-$\lambda$ behavior
as the original phase function $\theta(\lambda)$:
$\hat{g}(\lambda)=2\lambda^{2}t+4\lambda
x+O(1),\qquad\lambda\rightarrow\infty_{+}.$
This condition implies
$c_{1}=(B-\xi)d_{1}-B\xi+\frac{1}{2}(d_{2}^{2}+D^{2}),$ $c_{2}=\xi-B-d_{1},$
Define $\hat{g}(\lambda)$ as the sum of two Abelian integrals:
$\hat{g}(\lambda,\xi)=2(\int_{E}^{\lambda}+\int_{\bar{E}}^{\lambda})\frac{\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda+c_{0}}{\omega(\lambda)}d\lambda.$
(4.38)
Then it evidently has real $b-$period
$B_{\hat{g}}=2(\int_{E}^{d}+\int_{\bar{E}}^{\bar{d}})\frac{\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda+c_{0}}{\omega(\lambda)}d\lambda.$
(4.39)
Now notice that $\hat{g}(\lambda)$ can be written as a single Abelian integral
$\hat{g}(\lambda)=4\int_{E}^{k}\frac{\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda+c_{0}}{\omega(\lambda)}d\lambda$
and indeed
$B_{\hat{g}}=\int_{b}d\hat{g}.$
The large-$\lambda$ asymptotics of $\hat{g}(\lambda,\xi)$ can now be specified
as
$\hat{g}(\lambda,\xi)=2\lambda^{2}t+4\xi\lambda
t+\hat{g}(\infty,\xi)+O(\lambda^{-1}).$
where
$\hat{g}(\infty,\xi)=t(2(\int_{E}^{\infty}+\int_{\bar{E}}^{\infty})[\frac{\lambda^{3}+c_{2}\lambda^{2}+c_{1}\lambda+c_{0}}{\omega(\lambda)}-(\lambda+\xi)]d\lambda+2D^{2}-2B^{2}-4B\xi)$
(4.40)
is a real function of $\xi$.
###### Remark 4.4.
For $\xi=-B$, if we set $\mu(-B)=d_{1}(-B)=B$ and $d_{2}(-B)=D$, that is,
$d(-B)=E$ and $\bar{d}(-B)=\bar{E}$, then $\hat{g}(\lambda,-B)$ coincide(up to
a constant) with $\theta(\lambda,-B)$:
$\hat{g}(\lambda,-B)=\theta(\lambda,-B)+2|E|^{2}.$
which provides matching at the interface with the Zakharov-Manakov region.
In order to define $\mu,\mu_{\pm}$ and $d$ as functions of $\xi$, let us
compare the forms (4.36) and (4.37) of the differential $d\hat{g}$. This gives
$(\mu_{\pm}=\mu_{1}\pm i\mu_{2}):$
$\begin{array}[]{l}\mu+2\mu_{1}-d_{1}=B-\xi,\\\
2\mu\mu_{1}+\mu_{1}^{2}+\mu_{2}^{2}+(\xi-B)d_{1}-\frac{1}{2}d_{2}^{2}=\frac{1}{2}D^{2}-B\xi,\\\
\mu(\mu_{1}^{2}+\mu_{2}^{2})=-c_{0}(\xi,d_{1},d_{2}).\end{array}$
The local expansion of $\hat{g}(\lambda)$ at $\lambda=d$ is of the form
$\hat{g}(\lambda)=B_{\hat{g}}+g_{1}(\lambda-d)^{1/2}+g_{2}(\lambda-d)^{3/2}+\cdots,$
where $B_{\hat{g}}$ is real. The signature table for
$\mathrm{Im}\hat{g}(\lambda)$ must have three branches of the curve
$\mathrm{Im}\hat{g}(\lambda)=0$ going out from the point $d$, see Figure 6.
Indeed:
* •
Since $\hat{g}(E)=0$, one branch should connect $d$ with $E$.
* •
There should exist a branch separating the basins of $+$ and $-$ near the real
axis.
* •
Since $\hat{g}(\lambda)$ behaves like $\theta(\lambda)$ for large $\lambda$,
there should be an infinite branch going to infinity along the asymptotic line
$\mathrm{Re}\lambda=-\xi$.
Figure 6. The curves of $\mathrm{Im}\hat{g}(\lambda)=0$ for
$-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})<x<-4tB$.
Therefore, we arrive at the requirement $g_{1}=0$, that is
$(\lambda-d)^{1/2}\hat{g}^{\prime}(\lambda)|_{\lambda=d}=4\frac{(d-\mu(\xi))(d-\mu_{-}(\xi))(d-\mu_{+}(\xi))}{\sqrt{(\lambda-E)(\lambda-\bar{E})(d-\bar{d})}}=0$
The fact that $\mu$ is real implies that $\mu_{+}=d$ and $\mu_{-}=\bar{d}$,
which finally leads to the following ansatz for $d\hat{g}(\lambda)$:
$d\hat{g}(\lambda)=4(\lambda-\mu(\xi))\sqrt{\frac{(\lambda-d(\xi))(\lambda-\bar{d}(\xi))}{(\lambda-E)(\lambda-\bar{E})}}d\lambda,$
where $\mu(\xi),d_{1}(\xi)$ and $d_{2}(\xi)$ ($d=d_{1}+id_{2},d_{2}\geq 0$)
satisfy the equations:
$\mu=B-\xi-d_{1},$ (4.41a) $d_{2}^{2}=D^{2}-2(B-\mu)(B-d_{1}),$ (4.41b)
$\int_{B-iD}^{B+iD}\sqrt{\frac{(\lambda-
d_{1})^{2}+d_{2}^{2}}{(\lambda-B)^{2}+D^{2}}}(\lambda-\mu)d\lambda=0.$ (4.41c)
Recall that (4.41a) and (4.41b) follow from the requirement that
$d\hat{g}(\lambda)=(4\lambda+4\xi+O(\lambda^{-2}))d\lambda,\qquad\mbox{as
}\lambda\rightarrow\infty.$
while (4.41c) is the normalization condition
$\int_{\bar{E}}^{E}d\hat{g}(\lambda)=0$.
Substituting (4.41a) and (4.41b) into (4.41c) yields an equation relating
implicitly $d_{1}$ and $\xi$. In terms of the variables $u$ and $v$, where
$u=\frac{B-d_{1}}{D},\qquad v=\frac{\xi+B}{2D}.$
this equation reads
$\mathcal{F}(u,v)=\int_{-1}^{1}\sqrt{\frac{(i\tau+1)^{2}+1-4uv+2u^{2}}{1-\tau^{2}}}(i\tau+2v-u)d\tau=0.$
(4.42)
which is considered for $0\leq v\leq\frac{\sqrt{2}}{2}$ and $u\geq 0$. It is
easy to check that $\mathcal{F}(0,v)=4v$(and thus $\mathcal{F}(0,v)>0$ for
$v>0$),
$\mathcal{F}(+\infty,v)<0,\mathcal{F}(0,0)=\mathcal{F}(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})=0$
and $\mathcal{F}_{u}(u,v)<0$ for
$(u,v)\neq(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Therefore, (4.42)
determines a unique function $u=u(v),v\in[0,\frac{\sqrt{2}}{2}]$ such that
$u(0)=0$ and $u(\frac{\sqrt{2}}{2})=\frac{\sqrt{2}}{2}$. Consequently, we have
that the system (4.41) determines uniquely $d_{1}(\xi),d_{2}(\xi)$ and
$\mu(\xi)$, such that $d_{1}(-B-\sqrt{2}D)=B+\sqrt{2}D$ and $d_{1}(-B)=B$.
We have now specified a $g-$function $\hat{g}(\lambda)$ whose signature table
is as in Figure 8. Hence, we can begin deforming the basic Riemann-Hilbert
problem.
#### 4.3.2. The first deformation
We deform the part $\gamma\cup\bar{\gamma}$ of the contour of the basic
Riemann-Hilbert problem into a contour $\gamma_{E,\bar{E}}$ connecting $E$ and
$\bar{E}$ in such a way that it contains:
1. (i)
Two arcs $\gamma_{d}$ and $\bar{\gamma}_{d}$ connecting, respectively, $E$
with $d$ and $\bar{d}$ and $\bar{E}$, and where
$\mathrm{Im}\hat{g}(\lambda)=0$;
2. (ii)
An arc $\gamma_{\mu}$ connecting $d$ and $\bar{d}$, passing through $\mu$, and
along which $\mathrm{Im}\hat{g}(\lambda)<0$ for $\mathrm{Im}\lambda<0$ and
$\mathrm{Im}\hat{g}(\lambda)>0$ for $\mathrm{Im}\lambda>0$.
Figure 7. The contour $\Sigma^{(3)}=L_{1}\cup L_{2}\cup L_{3}\cup
L_{4}\cup\gamma_{d}\cup\bar{\gamma}_{d}\cup\gamma_{\mu}\cup\bar{\gamma}_{\mu}$
for $-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})<x<-4tB$.
Supplying $\gamma_{E,\bar{E}}=\gamma_{\mu}\cup\gamma_{d}\cup\bar{\gamma}_{d}$
with the orientation as going from $E$ to $\bar{E}$, we fix the branch of
$\hat{g}(\lambda)$ as having a jump across $\gamma_{E,\bar{E}}$:
$\begin{array}[]{ll}\hat{g}(\lambda)_{+}+\hat{g}(\lambda)_{-}=0,&\qquad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d};\\\
\hat{g}(\lambda)_{+}-\hat{g}(\lambda)_{-}=B_{\hat{g}},&\qquad\lambda\in\gamma_{\mu},\\\
\mbox{with $\mathrm{Im}B_{\hat{g}}=0$}&\end{array}$
#### 4.3.3. The second transformation
The further series of transformations
$N(x,t,\lambda)\leadsto N^{(1)}(x,t,\lambda)\leadsto
N^{(2)}(x,t,\lambda)\leadsto N^{(3)}(x,t,\lambda)$
is similar to that for the plane wave region but
1. (i)
with $g(\lambda)$ replaced by $\hat{g}(\lambda)$,
2. (ii)
with $\mu$, which is the real stationary point of $\hat{g}(\lambda)$ instead
of $\mu_{+}$,
3. (iii)
with the partition into domains with boundaries $L$ as shown in Figure 7.
The jump matrix $J_{N}^{(3)}(x,t,\lambda)$ is as follows:
* •
For $\lambda\in L_{j}$ at a fixed positive distance from the stationary point
$\lambda=\mu(\xi)$,
$J_{N}^{(3)}(x,t,\lambda)=\mathbb{I}+O(e^{-\varepsilon t})\mbox{ as
}t\rightarrow+\infty.$
* •
For $\lambda\in\gamma_{\mu}$ we have
$J_{N}^{(3)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}\left(\begin{array}[]{cc}e^{-itB_{\hat{g}}}&0\\\
\lambda
f(\lambda)\delta^{-2}(\lambda)e^{it(\hat{g}_{+}(\lambda)+\hat{g}_{-}(\lambda))}&e^{itB_{\hat{g}}}\end{array}\right),&\quad\mathrm{Im}\lambda>0,\\\
\left(\begin{array}[]{cc}e^{-itB_{\hat{g}}}&f(\lambda)\delta^{2}(\lambda)e^{-it(\hat{g}_{+}(\lambda)+\hat{g}_{-}(\lambda))}\\\
0&e^{itB_{\hat{g}}}\end{array}\right),&\quad\mathrm{Im}\lambda<0,\end{array}\right.$
(4.43)
Thus, away from $d$,$\mu$ and $\bar{d}$ and as $t\rightarrow+\infty$,
$J_{N}^{(3)}(x,t,\lambda)$ is close to a diagonal matrix:
$J_{N}^{(3)}(x,t,\lambda)=\left(\begin{array}[]{cc}e^{-itB_{\hat{g}}}&0\\\
0&e^{itB_{\hat{g}}}\end{array}\right)+O(e^{-\varepsilon t}),\qquad
t\rightarrow+\infty.$ (4.44)
* •
For $\lambda\in\gamma_{d}\cup\bar{\gamma}_{d}$, similarly to the plane wave
region, $J_{N}^{(3)}(x,t,\lambda)$ reduces to
$J_{N}^{(3)}(x,t,\lambda)=\left\\{\begin{array}[]{lc}\left(\begin{array}[]{cc}0&-f^{-1}(\lambda)\delta^{2}(\lambda)\\\
\lambda
f(\lambda)\delta^{-2}(\lambda)&0\end{array}\right),&\qquad\lambda\in\gamma_{d},\\\
\left(\begin{array}[]{cc}0&f(\lambda)\delta^{2}(\lambda)\\\ -\lambda
f^{-1}(\lambda)\delta^{-2}(\lambda)&0\end{array}\right),&\qquad\lambda\in\bar{\gamma}_{d},\end{array}\right.$
(4.45)
In order to arrive at a Riemann-Hilbert problem with a jump matrix independent
of $\lambda$, we proceed as in the plane wave region.
Scalar Riemann-Hilbert problem. We are looking for a scalar function
$F(\lambda)$ analytic in
${\mathbb{C}}\backslash\gamma_{d}\cup\bar{\gamma}_{d}$ such that
$F_{-}(\lambda)F_{+}(\lambda)=h(\lambda)\sqrt{\lambda}\delta^{-2}(\lambda),\qquad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d},$
(4.46)
where
$h(\lambda)=\left\\{\begin{array}[]{lc}-i\sqrt{\lambda}f(\lambda),&\qquad\lambda\in\gamma_{g},\\\
i\sqrt{\lambda}^{-1}f^{-1}(k),&\qquad\lambda\in\bar{\gamma}_{g}.\end{array}\right.$
(4.47)
After solving this scalar problem, $J_{N}^{(3)}(x,t,\lambda)$ can be
factorized as in (4.25). This factorization allows absorbing the diagonal
factors into a new Riemann-Hilbert problem with constant jump matrix on
$\gamma_{d}\cup\bar{\gamma}_{d}$.
However, an important difference with the plane wave region is that now the
jump conditions (4.46) for $F(\lambda)$ are specified on two disjoint arcs.
This implies that in order to arrive at a jump condition in additive form, we
are led to use
$\omega(\lambda)=\sqrt{(\lambda-E)(\lambda-\bar{E})(\lambda-d(\xi))(\lambda-\bar{d}(\xi))}$
Indeed, (4.46) can be rewritten as
$[\frac{\log{F(\lambda)}}{\omega(\lambda)}]_{+}-[\frac{\log{F(\lambda)}}{\omega(\lambda)}]_{-}=\frac{\log{h(\lambda)}}{\omega_{+}(\lambda)},\quad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d},$
(4.48)
and thus for $F(\lambda)$, we have
$F(\lambda)=\exp\\{\frac{\omega(\lambda)}{2\pi
i}\int_{\gamma_{d}\cup\bar{\gamma}_{d}}\frac{\log{h(s)}}{\omega_{+}(s)}\frac{ds}{s-\lambda}\\}$
(4.49)
But now $F(\lambda)$ has an essential singularity at infinity:
$F(\lambda)=F_{\infty}e^{i\Delta\lambda}(1+O(\lambda^{-1})),\qquad\lambda\rightarrow\infty.$
where
$\Delta=\Delta(\xi)=\frac{1}{2\pi}\int_{\gamma_{d}\cup\bar{\gamma}_{d}}\frac{\log{h(\lambda)}}{\omega_{+}(\lambda)}d\lambda.$
(4.50)
and
$F_{\infty}(\xi)=\exp\\{\frac{i}{2\pi}\int_{\gamma_{d}\cup\bar{\gamma}_{d}}(s-e_{1})\frac{\log{h(s)}}{\omega_{+}(s)}ds\\}$
with
$e_{1}=\frac{E+\bar{E}+d+\bar{d}}{2}.$ (4.51)
To account for this singularity, let us introduce the normalized, that is, its
$a-$period vanishes, Abelian integral $w(\lambda)$ of the second kind with
simple poles at $\infty_{\pm}$:
$w(\lambda)=\int_{E}^{\lambda}\frac{z^{2}-e_{1}z+e_{0}}{\omega_{(}z)}dz,$
where $e_{1}$ is the same as in (4.51) and $e_{0}$ is determined by the
condition $\int_{a}dw(\lambda)=0$:
$e_{0}=-\frac{\int_{d}^{\bar{d}}(z^{2}-e_{1}z+e_{0})\frac{dz}{\omega_{(}z)}}{\int_{d}^{\bar{d}}\frac{dz}{\omega_{(}z)}}.$
The large-$\lambda$ expansion of $w(\lambda)$ is of the form
$w(\lambda)=\lambda+w_{\infty}(\xi)+O(\lambda^{-1}),\qquad\lambda\rightarrow\infty,$
where
$\begin{split}w_{\infty}&=\int_{E}^{\infty}[\frac{z^{2}-e_{1}z+e_{0}}{\omega_{(}z)}-1]dz-E\\\
&=\frac{1}{2}(\int_{E}^{\infty}+\int_{\bar{E}}^{\infty})[\frac{z^{2}-e_{1}z+e_{0}}{\omega_{(}z)}-1]dz-B\end{split}$
(4.52)
The jump conditions for $w(\lambda)$ are as follows:
$\begin{split}w_{+}(\lambda)+w_{-}(\lambda)=0,&\qquad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d},\\\
w_{+}(\lambda)-w_{-}(\lambda)=B_{w},&\qquad\lambda\in\gamma_{\mu}.\end{split}$
Here $B_{w}$ is the $b-$period of $w(\lambda)$:
$B_{w}=\int_{b}dw=2\int_{E}^{d}\frac{z^{2}-e_{1}z+e_{0}}{\omega_{(}z)}dz=(\int_{E}^{d}+\int_{\bar{E}}^{\bar{d}})\frac{z^{2}-e_{1}z+e_{0}}{\omega_{(}z)}dz\in{\mathbb{R}}.$
(4.53)
Now introduce
$\hat{F}(\lambda)=F(\lambda)e^{-i\Delta w(\lambda)},$ (4.54)
This new function is clearly bounded at $\lambda=\infty$:
$\hat{F}(\infty,\xi)=e^{i\hat{\phi}(\xi)}.$ (4.55)
with
$\hat{\phi}(\xi)=\frac{1}{2\pi}\int_{\gamma_{d}\cup\bar{\gamma}_{d}}(s-e_{1})\log{[h(s)\delta^{-2}(s,\xi)]}\frac{ds}{\omega_{+}(s)}-\Delta(\xi)w_{\infty}(\xi).$
Also, $\hat{F}(\lambda)$ has the same jumps as $F(\lambda)$ across
$\gamma_{d}$ and $\bar{\gamma}_{d}$. On the other hand, the price for
introducing the exponential factor in (4.54) is that $\hat{F}(\lambda)$ has a
jump across $\gamma_{\mu}$:
$\frac{\hat{F}_{+}(\lambda)}{\hat{F}_{-}(\lambda)}=e^{-i\Delta
B_{w}},\qquad\lambda\in\gamma_{\mu}.$
Now we can absorb $\hat{F}(\lambda)$ into the Riemann-Hilbert problem for
$N^{(4)}(x,t,\lambda)$:
$N^{(4)}(x,t,\lambda)=\hat{F}^{\sigma_{3}}(\infty)N^{(3)}(x,t,\lambda)\hat{F}^{-\sigma_{3}}(\lambda),$
which leads to the jump conditions
$N^{(4)}_{+}(x,t,\lambda)=N^{(4)}_{-}(x,t,\lambda)J_{N}^{(4)}(x,t,\lambda),$
where
$J_{N}^{(4)}(x,t,\lambda)=\left\\{\begin{array}[]{ll}J_{N}^{mod}+O(e^{-\varepsilon
t}),&\quad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d}\cup\gamma_{\mu},\\\
\mathbb{I}+O(e^{-\varepsilon t}),&\quad\lambda\in
L\cup\bar{L}.\end{array}\right.$
with
$J_{N}^{(mod)}=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),&\quad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d},\\\
\left(\begin{array}[]{cc}e^{-itB_{\hat{g}}-i\Delta B_{w}}&0\\\
0&e^{itB_{\hat{g}}+i\Delta
B_{w}}\end{array}\right),&\quad\lambda\in\gamma_{\mu},\end{array}\right.$
(4.56)
#### 4.3.4. The model problem
Thus, we arrive at the model Riemann-Hilbert problem:
$N^{mod}_{+}(x,t,\lambda)=N^{mod}_{-}(x,t,\lambda)J_{N}^{mod}(x,t,\lambda),\qquad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d}\cup\gamma_{\mu},$
(4.57a)
$N^{mod}(x,t,\lambda)=\mathbb{I}+O(\frac{1}{\lambda}),\qquad\lambda\rightarrow\infty.$
(4.57b)
The solution of this model Riemann-Hilbert problem approximates
$N^{(4)}(x,t,\lambda)$:
$N^{(4)}(x,t,\lambda)=(\mathbb{I}+O(t^{-\frac{1}{2}}))N^{mod}(x,t,\lambda),$
(4.58)
The model problem (4.57) can be solved in terms of elliptic theta functions.
Let
$U(\lambda)=\frac{1}{c}\int_{E}^{\lambda}\frac{dz}{\omega(z)}$
be the normalized Abelian integral, that is
$c=2\int_{\bar{d}}^{d}\frac{dz}{\omega(z)}$
Then, define
$\tau=\tau(\xi)=\frac{2}{c}\int_{E}^{d}\frac{dz}{\omega(z)}$ (4.59)
with $\mathrm{Im}\tau>0$. Furthermore, the following relations are valid:
$\begin{split}\begin{array}[]{ll}U_{+}(\lambda)+U_{-}(\lambda)=0,&\quad\lambda\in\gamma_{d},\\\
U_{+}(\lambda)+U_{-}(\lambda)=-1,&\quad\lambda\in\bar{\gamma}_{d},\\\
U_{+}(\lambda)-U_{-}(\lambda)=\tau,&\quad\lambda\in\gamma_{\mu},\end{array}\end{split}$
(4.60)
Next, define
$\nu(\lambda)=(\frac{(\lambda-E)(\lambda-d)}{(\lambda-\bar{E})(\lambda-\bar{d})})^{\frac{1}{4}},$
where the branch is fixed by specifying the branch cut $\gamma_{E,\bar{E}}$
and the behavior as $\lambda\rightarrow\infty$;
$\nu(\lambda)=1+\frac{D+d_{2}}{2i\lambda}+O(\lambda^{-2}),\qquad\lambda\rightarrow\infty.$
Along the cut,we have
$\nu_{+}(\lambda)=\left\\{\begin{array}[]{ll}-i\nu_{-}(\lambda),&\quad\lambda\in\gamma_{d}\cup\bar{\gamma}_{d},\\\
-\nu_{-}(\lambda),&\quad\lambda\in\gamma_{\mu}.\end{array}\right.$
Finally, introduce the theta function
$\theta_{3}(z)=\sum_{m\in{\mathbb{Z}}}e^{\pi i\tau m^{2}+2\pi imz},$
and define the $2\times 2$ matrix-value function
$\Theta(\lambda)=\Theta(t,\xi,\lambda)$ with entries:
$\begin{array}[]{c}\Theta_{11}(\lambda)=\frac{1}{2}[\nu(\lambda)+\frac{1}{\nu(\lambda)}]\frac{\theta_{3}[U(\lambda)-U_{0}-\frac{1}{2}-\frac{B_{\hat{g}}t}{2\pi}-\frac{B_{w}\Delta}{2\pi}]}{\theta_{3}[U(\lambda)-U_{0}]},\\\
\Theta_{12}(\lambda)=\frac{1}{2}[\nu(\lambda)-\frac{1}{\nu(\lambda)}]\frac{\theta_{3}[U(\lambda)+U_{0}+\frac{1}{2}+\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}]}{\theta_{3}[U(\lambda)+U_{0}]},\\\
\Theta_{21}(\lambda)=\frac{1}{2}[\nu(\lambda)-\frac{1}{\nu(\lambda)}]\frac{\theta_{3}[U(\lambda)+U_{0}-\frac{1}{2}-\frac{B_{\hat{g}}t}{2\pi}-\frac{B_{w}\Delta}{2\pi}]}{\theta_{3}[U(\lambda)+U_{0}]},\\\
\Theta_{22}(\lambda)=\frac{1}{2}[\nu(\lambda)+\frac{1}{\nu(\lambda)}]\frac{\theta_{3}[U(\lambda)-U_{0}+\frac{1}{2}+\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}]}{\theta_{3}[U(\lambda)-U_{0}]},\end{array}$
where $U_{0}$ is to be chosen so that the unique zero of
$\theta_{3}(U(\lambda)-U_{0})$, as a function on the Riemann surface, lying on
the first sheet is compensated by the zero of
$\nu(\lambda)+\frac{1}{\nu(\lambda)}$ where $\theta_{3}(U(\lambda)+U_{0})$ has
no zero on this sheet. Setting
$U_{0}=U(E_{0})+\frac{1}{2}+\frac{\tau}{2},$
where
$E_{0}=\frac{Ed-\bar{E}\bar{d}}{E-\bar{E}+d-\bar{d}}$
satisfies this requirement, and thus $\Theta(\lambda)$ can be viewed as a
function analytic in ${\mathbb{C}}\backslash\gamma_{E,\bar{E}}$. On the other
hand, due to the properties of theta function:
$\theta_{3}(-z)=\theta_{3}(z),\quad\theta_{3}(z+1)=\theta_{3}(z),\quad\theta_{3}(z\pm\tau)=e^{-\pi
i\tau\mp 2\pi iz}\theta_{3}(z)$
$\Theta(\lambda)$ satisfies the jump conditions (4.57a)-(4.56) of the model
Riemann-Hilbert problem. Taking into account the normalization condition
(4.57b), the solution of the model Riemann-Hilbert problem is given by
$N^{mod}(x,t,\lambda)=\Theta^{-1}(t,\xi,\infty)\Theta(t,\xi,\lambda).$
#### 4.3.5. Back to the original problem
Now, following the sequence of equations of type (4.34) (with $g$ and $F$
replaced, respectively, by $\hat{g}$ and $\hat{F}$) and taking into account
the equations $\hat{g}$ and $\hat{F}$, and the explicit formula for
$n^{mod}_{12}(x,t,\lambda)$
$2in^{mod}_{12}(x,t,\lambda)=[D+d_{2}]\frac{\theta_{3}[\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}+U_{0}+\frac{1}{2}+U(\infty)]}{\theta_{3}[\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}+U_{0}+\frac{1}{2}-U(\infty)]}\frac{\theta_{3}[U_{0}-U(\infty)]}{\theta_{3}[U_{0}+U(\infty)]}$
and $\hat{F}^{-2}(\infty)=e^{-2i\hat{\phi}(\xi)}$, we obtain the asymptotics
in the region $-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})<x<-4tB$.
###### Theorem 4.5.
(Elliptic wave region) In the region
$-4t(B+\sqrt{2A^{2}(B+\frac{A^{2}}{4})})<x<-4tB$,the asymptotics, as
$t\rightarrow+\infty$, of the solution $q(x,t)$ of the initial value problem
(1.6) takes the form of a modulated elliptic wave:
$q(x,t)=[D+\mathrm{Im}d(\xi)]\frac{\theta_{3}[\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}+V_{+}(\xi)]}{\theta_{3}[\frac{B_{\hat{g}}t}{2\pi}+\frac{B_{w}\Delta}{2\pi}+V_{-}(\xi)]}\frac{\theta_{3}[V_{-}(\xi)-\frac{1}{2}]}{\theta_{3}[V_{+}(\xi)-\frac{1}{2}]}+O(t^{-\frac{1}{2}}),t\rightarrow+\infty.$
(4.61)
Here $B_{\hat{g}},B_{w}$ and $\Delta$ are functions of the variable
$\xi=\frac{x}{4t}$ defined, respectively, by (4.39), (4.53) and (4.50), and
$V_{\pm}(\xi)=U_{0}+\frac{1}{2}\pm U(\infty)$. Furthermore,
$\theta_{3}(z)=\sum_{z\in{\mathbb{Z}}}e^{\pi i\tau m^{2}+2\pi imz}$
is the theta function of invariant $\tau=\tau(\xi)$ defined in (4.59),
$\hat{g}(\infty,\xi)=t(2(\int_{E}^{\infty}+\int_{\bar{E}}^{\infty})[(z-\mu(\xi))\sqrt{\frac{(z-d(\xi))(z-\bar{d}(\xi))}{(z-E)(z-\bar{E})}}-(z+\xi)]dz+2D^{2}-2B^{2}-4B\xi)$
and the phase shift $\phi(\xi)$ is given by
$\phi(\xi)=\frac{1}{2\pi}\int_{\gamma_{d}\cup\gamma_{\bar{d}}}\frac{[s-e_{1}(\xi)-\omega_{\infty}(\xi)]\log[h(s)\sqrt{s}\delta^{-2}(s,\xi)]}{[(s-E)(s-\bar{E})(s-d(\xi))(s-\bar{d}(\xi))]^{1/2}}ds$
where
$\begin{array}[]{l}h(\lambda)=\left\\{\begin{array}[]{ll}a_{+}^{-1}(\lambda)a_{-}^{-1}(\lambda),&\lambda\in\gamma_{d}\\\
a_{+}(\lambda)a_{-}(\lambda),&\lambda\in\gamma_{\bar{d}}\end{array}\right.\\\
\delta(\lambda,\xi)=\exp\\{\frac{1}{2\pi
i}\int_{-\infty}^{\mu(\xi)}\frac{\log(1+\lambda\rho^{2}(\lambda))}{s-\lambda}ds\\}.\end{array}$
and $e_{1}(\xi),\omega_{\infty}$ and $\mu(\xi)$ are defined, respectively, by
(4.51), (4.52) and (4.41).
Acknowledgments. J.X want to thank Prof. V.P.Kotlyarov for the helpful
suggestions.
## References
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* [33] V.Kotlyarov and A.Minakov, Riemann-Hilbert problem to the modified Korteveg-de Vries equation: Long-time dynamics of the steplike initial data., Journal of Mathematical Physics 51(2010),093506.
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* [35] A.V.Kitaev and A.H.Vartanian, Asymptotics of solutions to the modified nonlinear Schrödinger equation: Solution on a nonvanishing continuous background., SIAM Journal of Mathematical Analysis 30,no.4(1999),787-832.
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|
arxiv-papers
| 2013-04-17T04:20:32 |
2024-09-04T02:49:44.529484
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Xu, Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1304.4681"
}
|
1304.4733
|
# Formation of relativistic non-viscous fluid in central collisions of protons
with energy 0.8 TeV with photoemulsion nuclei
U. U. Abdurakhmanov V. V. Lugovoi [email protected] Physical-Technical
Institute of Uzbek Academy of Science, Tashkent, Uzbekistan
###### Abstract
By the methods of mathematical statistics we test a qualitative prediction of
the old theory of relativistic hydrodynamics non-viscous liquid which can be
used as a part of the process of hadronization within the modern
hydrodynamical approach for the description of the quark-gluon plasma.
Experimental data on the interaction of protons with the energies of 0.8 TeV
with emulsion nuclei are used. Results do not contradict the formation of
relativistic ideal non-viscous liquid in rare central collisions.
## I Introduction
In the ion-ion collisions at CERN and RHIC it was discovered a collective
behaviour of the quark-gluon medium, which manifests itself in the possibility
of the quarks and gluons to free themselves off nucleons, interact strongly
with each other and quite a long time to move as a unit. This movement is well
described within the hydrodynamic theory of liquids of low viscosity, which is
formed in the central collisions of ions (see reviews Dremin-UFN-2010 ;
Shuryak-2009 ). Such a hot quark-gluon plasma (QGP) expands and cools to a
temperature $T\approx\mu c^{2}$ ($\mu$ is a pion mass). This results in the
more $"$cold$"$ hadrons. Hadronization of quarks and gluons is a serious yet
insuperable theoretical problem. Therefore, the hydrodynamic approach in the
stage of hadronization uses the fitting parameters (see Dremin-UFN-2010 .)
Therefore, it might be useful to use old result related to the hydrodynamic
theory, which seems logically to fit into the model of the modern approach to
the hadronization of QGP. Namely, for a description of the second stage of
hadronization, when the temperature is close to $T\approx\mu c^{2}$, but QGP
has already started to form hadrons, among which there is still a colored
interaction, that is, these hadrons still yet form a substance with the
properties of the relativistic non-viscous liquid. This state of substance is
considered as the starting point in the Landau approach Landau53 ; Landau52 ,
which showed that, in this case, after further expansion of matter according
to the laws of relativistic hydrodynamics of non-viscous liquid, the not
interacting each other hadrons are born according to the Gaussian distribution
on the quasirapidity $\eta=-ln\;tg\frac{\theta}{2}$, where $\theta$ is the
polar angle of the particle. In the modern approach, like it was in Landau53 ;
Landau52 , the central collisions with the large multiplicity of secondary
hadrons are taken into account. However, the presence of fast valence quarks
creates the fluctuations NuclPhys of the average values of quasirapidities of
the hadrons which are formed from the QGP, where the parton density increases
with energy, in particular, due to the production of vacuum pairs of leading
to the birth of the relatively soft hadrons. These quasirapidity fluctuations
lead to the total non-Gaussian inclusive distribution of particles in all
events. A variation of mean quasirapidity does not change the form of the
quasirapidity distribution in each event. Therefore, it would be interesting
to determine the form of experimental particle distribution on the
quasirapidity in the every individual central nucleon-nucleus and nucleus-
nucleus collision. It is not possible to verify visually. However, in the
mathematical statistics, there are well-developed methods using which we can
verify that the given distribution has a Gauss type. In this paper we will use
these techniques.
Our experimental data (Baton Rouge-Krakow-Moscow-Tashkent Collaboration Collab
) are (central) collisions of protons with energies of 0.8 TeV with emulsion
nuclei. This energy is less than the energy at which ATLAS Dremin-UFN-2010 ;
Shuryak-2009 collaboration is working, and so the cross section of the hard
jet production is small. Thus, the hard jets can not distort [1] the form of
quasirapidy distribution. Therefore, at 0.8 TeV energy, this distortion
practically will not be. However, in the papers Landau53 ; Landau52 it was
predicted that non-viscous liquid can be formed at the incident proton
energies above $1TeV$. However, this value is close to the energy at which our
experimental data was obtained. Therefore, to test the theoretical predictions
for the properties of the relativistic ideal non-viscous liquid, our
experimental data can be used.
## II Parametrically invariant variables
The theoretical Gaussian distribution
$f(\eta)\propto(\sigma\sqrt{2\pi})^{-1}\cdot
exp(-\frac{(\nu-\eta)^{2}}{2\sigma^{2}})\;$ has two parameters (a mathematical
expectation $\nu$ and variance $\sigma$), which depend on the physical
conditions that arise in each collision (see NuclPhys ). Therefore, for
example, the total inclusive theoretical and experimental Collab
distributions differ from the Gaussian distribution. The theory of
mathematical statistics Kramer offers asymmetry $g_{1}$ and excess $g_{2}$,
which do not depend on these parameters $\nu$ and $\sigma$ but they are
sensitive to the shape of the distribution :
$\displaystyle
g_{1}=m_{3}m_{2}^{-3/2}\;,\;\;\;\;g_{2}=m_{4}m_{2}^{-2}-3\;,\;\;\;\;m_{k}=\frac{1}{n}\sum_{i=1}^{n}\left(\eta_{i}-\bar{\eta}\;\right)^{k}\;\;,\;\;\;\;\;\;\bar{\eta}=\frac{1}{n}\sum_{i=1}^{n}\eta_{i}\;\;.$
(1)
Here $n$ is the number of particles in the event (interaction).
In order to use an approach proposed by the mathematical statistics, we divide
an ensemble Kramer of the theoretical central collisions into subensembles so
that the number of particles $n$ and the value of $\nu$ (the average
quasirapidity of particles in the event) were constant in the events of each
subensemble. In this case, if the values of $\eta_{1}$, $\eta_{2}$, … ,
$\eta_{n}$ are mutually independent in the events of subensemble and
distributed according to the Gaussian law with parameters $\nu$ and $\sigma$,
then the distribution of $g_{1}$ and $g_{2}$ is independent on the parameters
$\nu$ and $\sigma$ and uniquely determined by the number of particles $n$ in
the event of subensemble. The mathematical expectation and variance of $g_{1}$
and $g_{2}$ values are Kramer
$\displaystyle\nu_{g_{1}}(n)=0\;\;,\;\;\;\sigma_{g_{1}}^{2}(n)=6(n-2)(n+1)^{-1}(n+3)^{-1}\;\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\nu_{g_{2}}(n)=-6(n+1)^{-1}\;\;,\;\;\sigma_{g_{2}}^{2}(n)=24n(n-2)(n-3)(n+1)^{-2}(n+3)^{-1}(n+5)^{-1}\;\;,$
(2)
and the values of Kramer
$\displaystyle
d_{1}=\left[g_{1}-\nu_{g_{1}}(n)\right]\;\sigma_{g_{1}}^{-1}(n)\;\;\;,\;\;\;\;d_{2}=\left[g_{2}-\nu_{g_{2}}(n)\right]\;\sigma_{g_{2}}^{-1}(n)\;\;,$
(3)
according to the represented form, have the mathematical expectations equal to
$0$ and variances equal to $1$ in each subensemble, and so in an ensemble of
all the events.
In accordance with the logic of mathematical statistics, we can group the
events with different n into so-called complex tests (groups), containing $N$
of events. Now we use the central limit theorem of the probability theory,
namely, when a large $N$ each of the quantities
$\displaystyle\bar{d_{1}}\;\sqrt{N}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}d_{1i}\;\;,\;\;\;\bar{d_{2}}\;\sqrt{N}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}d_{2i}\;\;$
(4)
has approximately a normal distribution with parameters $0$ and $1$, and its
absolute value must be less than two111Given the asymptotic normality of the
variables $d_{1}$ and $d_{2}$ for large $n$ Kramer , this conclusion can be
considered as valid for a small number of $N$, but a large number of $n$. with
probability $\approx 95\%$. In the next section we use this theoretical
result.
## III RESULTS
We use experimental data222 The details of the experiment were described in
Collab . Collab , which contain 1685 collisions of protons with an energy of
0.8 TeV with emulsion nuclei. For secondary charged particles, the azimuthal
angles $\varphi$ and their emission angles $\theta$ with respect to the
direction of the projectile were measured. The quasirapidity $\eta$ of
secondary particle is determined by the formula
$\eta=-\;ln\;tg\frac{\theta}{2}$. he average multiplicities of weakly ionizing
particles and all charged particles are, respectively, 20 and 25. Particles
for which $I<1.4I_{0}$, where $I_{0}$ is the ionization along the tracks of
singly charged relativistic particles, were taken to be weakly ionizing
particles.
If in the event a large number of gray particles are produced, it is likely
the result of the intranuclear cascade, rather than a central collision.
However, the relativistic particles of ideal inviscid fluid can be produced in
central collisions from the narrow relativistic disks Dremin-UFN-2010 ;
Shuryak-2009 . So in this case we can expect the formation of the largest
possible number of weakly ionizing particles and the minimum number of gray
particles. This is the first qualitative criterion of the selection of events.
The second quantitative selection criterion of events means that each of two
values $\mid\bar{d_{1}}\;\sqrt{N}\mid$ and $\mid\bar{d_{2}}\;\sqrt{N}\mid$
should be less than two (see section 2).
These criteria are completely fulfilled, that is,
$\mid\bar{d_{1}}\;\sqrt{N}\mid=2.0$ and $\mid\bar{d_{2}}\;\sqrt{N}\mid=0.4$,
in eight stars, where the multiplicity of relativistic singly charged
particles is $n\geq$ 55 and there is complete absence of gray particle.
Thus, only small fraction of the events meets the criteria formation of the
relativistic ideal inviscid fluid. This may be connected with the fact that
the energy $E_{lab}=0.8$ TeV is equal to the minimum energy at which the
theoretical prediction was done Landau53 ; Landau52 for, in fact, very rare
absolutely central collisions. Moreover, for example, an excess is a moment of
high order and so it is very sensitive to the form of (quasirapidity)
distribution at the tails of the distribution. Therefore we use a very strict
statistical selection criterion. Thus, we can conclude that our result does
not contradict the formation of the relativistic ideal non-viscous liquid, and
in the same time, shows that it would be interesting to carry out similar
calculations for higher energy. If the result of the comparison will be
positive, the theoretical prediction of Landau53 ; Landau52 could be
considered as a part of the process of hadronization in the modern
hydrodynamic theory of QGP.
## Acknowledgements
The authors are grateful to V.Sh. Novotny and V.M. Chudakov for helpful
discussions, and the participants of cooperation (Baton Rouge-Krakow-Moscow-
Tashkent Collaboration) for the provided experimental data.
## References
* (1)
## References
* (2) I.M. Dremin and A.V. Leonidov, $"$The quark-gluon medium$"$, Uspekhi Fizicheskikh Nauk, Vol. 180, No. 11, 2010, pp. 1167 - 1196.
* (3) E. Shuryak,$"$Physics of Strongly coupled Quark-Gluon Plasma$"$, Prog. Part. Nucl. Phys., Vol. 62, 2009, pp.48-124; arXiv:0807.3033v2 [ hep-ph].
* (4) L.D. Landau, $"$About multiple production of particles in collisions of fast particles$"$, Izvestija AN SSSR, Vol.17 , 1953, pp.51-64.
* (5) S.Z. Belenkiy and L.D. Landau, $"$The hydrodynamic theory of multiparticle production$"$, Uspekhi Fizicheskikh Nauk, Vol.56, No.3-4, 1955, pp.309-348.
* (6) A. Abduzhamilov, L.M. Barbier, L.P. Chernova et all., $"$Charged-particle multiplicity and angular distributions in proton-emulsion interactions at 800 GeV$"$, Phys. Rev. D Part Fields, Vol. 35, No.1, 1987, pp.3537-3540.
* (7) K.G. Gulamov, S.I. Zhokhova, V.V. Lugovoi, et all., $"$Pseudorapidity Configurations in Collisions between Gold Nuclei and Track-Emulsion Nuclei.$"$, Phys.Atom.Nucl. Vol.73, 2010, pp. 1185-1190 ; Yad.Fiz. Vol.73, 2010, pp. 1225-1230.
* (8) G. Kramer, $"$The mathematical methods of statistics$"$, IL, Moscow, 1948, pp.1-648.
|
arxiv-papers
| 2013-04-17T08:49:40 |
2024-09-04T02:49:44.537765
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "U. U. Abdurakhmanov and V. V. Lugovoi",
"submitter": "Vladimir Lugovoi",
"url": "https://arxiv.org/abs/1304.4733"
}
|
1304.4741
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-054 LHCb-PAPER-2013-006 16 April 2013
Precision measurement of the $B^{0}_{s}$-$\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}$ oscillation frequency with the
decay $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
The LHCb collaboration†††Authors are listed on the following pages.
A key ingredient to searches for physics beyond the Standard Model in
$B^{0}_{s}$ mixing phenomena is the measurement of the $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation frequency, which
is equivalent to the mass difference $\Delta m_{s}$ of the $B^{0}_{s}$ mass
eigenstates. Using the world’s largest $B^{0}_{s}$ meson sample accumulated in
a dataset, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by the LHCb experiment at the CERN LHC in 2011,
a measurement of $\Delta m_{s}$ is presented. A total of about 34,000
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ signal decays are reconstructed,
with an average decay time resolution of 44 fs. The oscillation frequency is
measured to be $\Delta m_{s}$ = 17.768 $\pm$ 0.023 (stat) $\pm$ 0.006 (syst)
ps-1, which is the most precise measurement to date.
Submitted to New Journal of Physics
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk57, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, I. De Bonis4, K. De
Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De Paula2, W. De
Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, D.
Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M.
Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A.
Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C.
Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F.
Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C.
Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M.
Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, B. Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37,
T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, L. Martin54, A. Martín Sánchez7, M. Martinelli40,
D. Martinez Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A.
McNab53, R. McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40,
D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53,
P. Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren 35,o, B.K. Pal57,
A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, V.
Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K.
Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The Standard Model (SM) of particle physics, despite its great success in
describing experimental data, is considered an effective theory only valid at
low energies, below the $\mathrm{\,Te\kern-1.00006ptV}$ scale. At higher
energies, new physics phenomena are predicted to emerge. For analyses looking
for physics beyond the SM (BSM) there are two conceptually different
approaches: direct and indirect searches. Direct searches are performed at the
highest available energies and aim at producing and detecting new heavy
particles. Indirect searches focus on precision measurements of quantum-loop
induced processes. Accurate theoretical predictions are available for the
heavy quark sector in the SM. It is therefore an excellent place to search for
new phenomena [1, 2], since any deviation from these predictions can be
attributed to contributions from BSM.
In the SM, transitions between quark families (flavours) are possible via the
charged current weak interaction. Flavour changing neutral currents (FCNC) are
forbidden at lowest order, but are allowed in higher order processes. Since
new particles can contribute to these loop diagrams, such processes are highly
sensitive to contributions from BSM. An example FCNC transition is neutral
meson mixing, where neutral mesons can transform into their antiparticles.
Particle-antiparticle oscillations have been observed in the $K^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ system [3], the $B^{0}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ [4] system, the $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system [5, 6] and the
$D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ system [7, 8, 9,
10]. The frequency of $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillations is the highest.
On average, a $B^{0}_{s}$ meson changes its flavour nine times between
production and decay. This poses a challenge to the detector for the
measurement of the decay time. Another key ingredient of this measurement is
the determination of the flavour of the $B^{0}_{s}$ meson at production, which
relies heavily on good particle identification and the separation of tracks
from the primary interaction point.
The observed particle and antiparticle states $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ are linear combinations of the
mass eigenstates $B_{\rm H}$ and $B_{\rm L}$ with masses $m_{\rm H}$ and
$m_{\rm L}$ and decay widths $\Gamma_{\rm H}$ and $\Gamma_{\rm L}$,
respectively [11]. The $B^{0}_{s}$ oscillation frequency is equivalent to the
mass difference $\Delta m_{s}=m_{\rm H}-m_{\rm L}$. The parameter $\Delta
m_{s}$ is an essential ingredient for all studies of time-dependent
matter–antimatter asymmetries involving $B^{0}_{s}$ mesons, such as the
$B^{0}_{s}$ mixing phase $\phi_{s}$ in the decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
[12]. It was first observed by the CDF experiment [6]. The LHCb experiment
published a measurement of this frequency using a dataset, corresponding to an
integrated luminosity of 37$\mbox{\,pb}^{-1}$, taken in 2010 [13]. This
analysis complements the previous result and is obtained in a similar way,
using a data sample, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by LHCb in 2011.
## 2 The LHCb experiment
The LHCb experiment is designed for precision measurements in the beauty and
charm hadron systems. At a center-of-mass energy of $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$, about $3\cdot 10^{11}$ $b\overline{}b$ pairs
were produced in 2011. The LHCb detector [14] is a single-arm forward
spectrometer covering the pseudorapidity range from two to five. The excellent
decay time resolution necessary to resolve the fast $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation is provided by a
silicon-strip vertex detector surrounding the $pp$ interaction region. At
nominal position the sensitive region of the vertex detector is only 8 mm away
from the beam. Impact parameter (IP) resolution of 20$\,\upmu\rm m$ for tracks
with high transverse momentum ($p_{\rm T}$) is achieved.
Charged particle momenta are measured with the LHCb tracking system consisting
of the aforementioned vertex dector, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system has momentum resolution
$\Delta p/p$ that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Since this analysis is performed with decays involving only hadrons in the
final state, excellent particle identification is crucial to suppress
background. Charged hadrons are identified using two ring-imaging Cherenkov
detectors[15]. Photon, electron, and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter, and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.
The first stage of the trigger [16] is implemented in hardware, based on
information from the calorimeter and muon systems, and selects events that
contain candidates with large transverse energy and transverse momentum. This
is followed by a software stage which applies a full event reconstruction. The
software trigger used in this analysis requires a two-, three- or four-track
secondary vertex with a significant displacement from the primary interaction,
a large sum of $p_{\rm T}$ of the tracks, and at least one track with
$\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In addition an
IP $\chi^{2}$ with respect to the primary interaction greater than 16 and a
track fit $\chi^{2}$ per degree of freedom $<2$ is required. The IP $\chi^{2}$
is defined as the difference between the $\chi^{2}$ of the primary vertex
reconstructed with and without the considered track. A multivariate algorithm
is used for the identification of the secondary vertices.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [17] with a
specific LHCb configuration [18]. Decays of hadronic particles are described
by EvtGen [19], in which final state radiation is generated using Photos [20].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [21, *Agostinelli:2002hh], as
described in Ref. [23].
## 3 Signal selection and analysis strategy
The analysis uses $B^{0}_{s}$ candidates reconstructed in the flavour-specific
decay mode111Unless explicitly stated, inclusion of charge-conjugated modes is
implied. $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ in five $D^{-}_{s}$ decay
modes, namely $D^{-}_{s}\\!\rightarrow\phi(K^{+}K^{-})\pi^{-}$,
$D^{-}_{s}\\!\rightarrow K^{*0}(K^{+}\pi^{-})K^{-}$, $D^{-}_{s}\\!\rightarrow
K^{+}K^{-}\pi^{-}$ nonresonant, $D^{-}_{s}\\!\rightarrow K^{-}\pi^{+}\pi^{-}$,
and $D^{-}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{-}$. To avoid double counting,
events that contain a candidate passing the selection criteria of one mode,
are not considered for the subsequent modes, using the order listed above. All
reconstructed decays are flavour-specific final states, thus the flavour of
the $B^{0}_{s}$ candidate at the time of its decay is given by the charges of
the final state particles. A combination of tagging algorithms is used to
identify the $B^{0}_{s}$ flavour at production. The algorithms provide for
each candidate a tagging decision as well as an estimate of the probability
that this decision is wrong (mistag probability). These algorithms have been
optimized using large event samples of flavour-specific decays [24, 25].
To be able to study the effect of selection criteria that influence the decay
time spectrum, we restrict the analysis to those events in which the signal
candidate passed the requirements of the software trigger algorithm used in
this analysis. Specific features, such as the masses of the intermediate
$\phi$ and $K^{*0}$ resonances or the Dalitz structure of the
$D^{-}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{-}$ decay mode, are exploited for
the five decay modes. The most powerful quantity to separate signal from
background common to all decay modes is the output of a boosted decision tree
(BDT) [26]. The BDT exploits the long $B^{0}_{s}$ lifetime by using as input
the IP $\chi^{2}$ of the daughter tracks, the angle of the reconstructed
$B^{0}_{s}$ momentum relative to the line between the reconstructed primary
vertex, and the $B^{0}_{s}$ vertex and the radial flight distance in the
transverse plane of both the $B^{0}_{s}$ and the $D^{-}_{s}$ meson. Additional
requirements are applied on the sum of the $p_{\rm T}$ of the $B^{0}_{s}$
candidate’s decay products as well as on particle identification variables,
and on track and vertex quality. The reconstructed $D^{-}_{s}$ mass is
required to be consistent with the known value [27]. After this selection, a
total of about 47,800 candidates remain in the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ invariant mass window of 5.32 – 5.98
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
An unbinned likelihood method is employed to simultaneously fit the
$B^{0}_{s}$ invariant mass and decay time distributions of the five decay
modes. The probability density functions (PDFs) for signal and background in
each of the five modes can be written as
$\mathcal{P}=\mathcal{P}_{m}(m)\,\mathcal{P}_{t}(t,q|\sigma_{t},\eta)\,\mathcal{P}_{\sigma_{t}}(\sigma_{t})\,\mathcal{P}_{\eta}(\eta),$
(1)
where $m$ is the reconstructed invariant mass of the $B^{0}_{s}$ candidate,
$t$ is its reconstructed decay time and $\sigma_{t}$ is an event-by-event
estimate of the decay time resolution. The tagging decision $q$ can be 0 if no
tag is found, $-1$ for events with different flavour at production and decay
(mixed) or $+1$ for events with the same flavour at production and decay
(unmixed). The predicted event-by-event mistag probability $\eta$ can take
values between 0 and 0.5. The functions $\mathcal{P}_{m}$ and
$\mathcal{P}_{t}$ describe the invariant mass and the decay time probability
distributions, respectively. $\mathcal{P}_{t}$ is a conditional probability
depending on $\sigma_{t}$ and $\eta$. The functions $\mathcal{P}_{\sigma_{t}}$
and $\mathcal{P}_{\eta}$ are required to ensure the proper relative
normalization of $\mathcal{P}_{t}$ for signal and background [28]. The
functions $\mathcal{P}_{\sigma_{t}}$ and $\mathcal{P}_{\eta}$ are determined
from data, using the measured distribution in the upper $B^{0}_{s}$ invariant
mass sideband for the background PDF and the sideband subtracted distribution
in the invariant mass signal region for the signal PDF.
This measurement has been performed “blinded”, meaning that during the
analysis process the fitted value of $\Delta m_{s}$ was shifted by an unknown
value, which was removed after the analysis procedure had been finalized.
## 4 Invariant mass description
Figure 1: Invariant mass distributions for $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ candidates with the $D^{-}_{s}$ meson decaying as a)
$D^{-}_{s}\\!\rightarrow\phi(K^{+}K^{-})\pi^{-}$, b) $D^{-}_{s}\\!\rightarrow
K^{*0}(K^{+}\pi^{-})K^{-}$, c) $D^{-}_{s}\\!\rightarrow K^{+}K^{-}\pi^{-}$
nonresonant, d) $D^{-}_{s}\\!\rightarrow K^{-}\pi^{+}\pi^{-}$, and e)
$D^{-}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{-}$. The fits and the various
background components are described in the text. Misidentified backgrounds
refer to background from $B^{0}$ and $\mathchar 28931\relax^{0}_{b}$ decays
with one misidentified daughter particle.
The invariant mass of each $B^{0}_{s}$ candidate is determined in a vertex fit
constraining the $D^{-}_{s}$ invariant mass to its known value [27]. The
invariant mass spectra for the five decay modes after all the selection
criteria are applied are shown in Fig. 1. The fit to the five distributions
takes into account contributions from signal, combinatorial background and
$b$-hadron decay backgrounds. The signal components are described by the sum
of two Crystal Ball (CB) functions [29], which are constrained to have the
same peak parameter. The parameters of the CB function describing the tails
are fixed to values obtained from simulation, whereas the mean and the two
widths are allowed to vary. These are constrained to be the same for all five
decay modes. It has been checked on data that the mass resolution is
compatible among all modes.
The $b$-hadron decay background includes $B^{0}$ and $\mathchar
28931\relax^{0}_{b}$ decays with one misidentified daughter particle. Their
mass shapes are derived from simulated samples. The yields for the different
$b$-hadron decay backgrounds are allowed to vary individually for each of the
five decay modes. Another component originates from $B^{0}_{s}\\!\rightarrow
D^{\mp}_{s}K^{\pm}$ decays, in which the kaon is misidentified as a pion. This
contribution is treated as signal in the decay time analysis.
Table 1: Number of candidates and $B^{0}_{s}$ signal fractions in the mass range 5.32 – 5.98 ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$. Decay mode | ($D^{-}_{s}$ $\pi^{+}$) candidates | $f_{\mbox{$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$}}$ | $f_{\mbox{$B^{0}_{s}\\!\rightarrow D^{\mp}_{s}K^{\pm}$}}$
---|---|---|---
$D^{-}_{s}\\!\rightarrow\phi(K^{+}K^{-})\pi^{-}$ | 14691 | 0.834 | $\pm$ 0.008 | |
$D^{-}_{s}\\!\rightarrow K^{*0}(K^{+}\pi^{-})K^{-}$ | 10866 | 0.857 | $\pm$ 0.009 | |
$D^{-}_{s}\\!\rightarrow K^{+}K^{-}\pi^{-}$ nonresonant | 11262 | 0.595 | $\pm$ 0.009 | |
$D^{-}_{s}\\!\rightarrow K^{-}\pi^{+}\pi^{-}$ | 4288 | 0.437 | $\pm$ 0.014 | |
$D^{-}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{-}$ | 6674 | 0.599 | $\pm$ 0.008 | 0.019 | $\pm$0.010
Total | 47781 | 0.714 | $\pm$ 0.004 | 0.019 | $\pm$0.010
The requirement that the invariant mass be larger than 5.32
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ rejects background candidates from
$B^{0}_{s}$ decays with additional particles in the decay not reconstructed,
such as $B^{0}_{s}\\!\rightarrow D^{*-}_{s}\pi^{+}$ ($D^{*-}_{s}\\!\rightarrow
D^{-}_{s}\pi^{0}$ or $D^{-}_{s}$ $\gamma$). The fitted number of signal
candidates does not change with respect to a fit in a larger mass window. The
high mass sideband region 5.55 – 5.98
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ provides a sample of mainly
combinatorial background candidates. The mass distribution is described by an
exponential function, whose parameters are allowed to vary individually for
the five decay modes. By including this region in the fit, we are able to
determine the decay time distribution as well as the tagging behaviour of the
combinatorial background.
The number of used candidates, along with the signal fractions extracted from
the two dimensional fit in mass and decay time, are reported in Table 1. One
complication arises from the fact that the shape of the invariant mass
distribution of the $B^{0}_{s}\\!\rightarrow D^{\mp}_{s}K^{\pm}$ events is
very similar to that of the $B^{0}$ background. Therefore the fraction of
$B^{0}_{s}\\!\rightarrow D^{\mp}_{s}K^{\pm}$ candidates has been determined in
a fit to the $D^{-}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{-}$ mode only, in
which no $B^{0}$ background is present. Subsequently this value is used for
all other modes.
## 5 Decay time description
The decay time of a particle is measured as
$t=\frac{Lm}{p},$ (2)
where $L$ is the distance between the production vertex and the decay vertex
of the particle, $m$ its reconstructed invariant mass, and $p$ its
reconstructed momentum. We use the decay time calculated without the
$D^{-}_{s}$ mass constraint to avoid a systematic dependence of the
$B^{0}_{s}$ decay time on the reconstructed invariant mass. The theoretical
distribution of the decay time, $t$, ignoring the oscillation and any detector
resolution, is
$\mathcal{P}_{t}\propto\Gamma_{s}\,e^{-\Gamma_{s}t}\,\cosh\left(\frac{\Delta\Gamma_{s}}{2}t\right)\,\theta(t),$
(3)
where $\Gamma_{s}$ is the $B^{0}_{s}$ decay width and $\Delta\Gamma_{s}$ the
decay width difference between the light and heavy mass
eigenstate.222$\Delta\Gamma_{s}$ and $\Delta m_{s}$ are measured in units with
$\hbar$ = 1 throughout this paper. The value for $\Delta\Gamma_{s}$ is fixed
to the latest value measured by LHCb [12] $\Delta\Gamma_{s}$ = 0.106 $\pm$
0.011 $\pm$ 0.007${\rm\,ps^{-1}}$. It is varied within its uncertainties to
assess the systematic effect on the measurement of $\Delta m_{s}$. The
Heaviside step function $\theta(t)$ restricts the PDF to positive decay times.
To account for detector resolution effects, the decay time PDF is convolved
with a Gaussian distribution. The width $\sigma_{t}$ is taken from an event-
by-event estimate returned by the fitting algorithm that reconstructs the
$B^{0}_{s}$ decay vertex. Due to tracking detector resolution effects
$\sigma_{t}$ needs to be calibrated. A data-driven method, combining prompt
$D^{-}_{s}$ mesons from the primary interaction with random $\pi^{+}$ mesons,
forms fake $B^{0}_{s}$ candidates. The decay time distribution of these
candidates, each divided by its event-by-event $\sigma_{t}$, is fitted with a
Gaussian function. The width provides a scale factor $S_{\sigma_{t}}$ = 1.37,
by which each $\sigma_{t}$ is multiplied, such that it represents the correct
resolution. By inspecting different regions of phase space of the fake
$B^{0}_{s}$ candidates, the uncertainty range on this number is found to be
$1.25<S_{\sigma_{t}}<1.45$. The variation is taken into account as part of the
$\Delta m_{s}$ systematic studies. The resulting average decay time resolution
is $S_{\sigma_{t}}\times\langle\sigma_{t}\rangle=44$ fs.
Some of the selection criteria influence the shape of the decay time
distribution, e.g. the requirement of a large IP for $B^{0}_{s}$ daughter
tracks. Thus a decay time acceptance function $\mathcal{E}_{t}(t)$ has to be
taken into account. Its parametrization is determined from simulated data and
the parameter describing its shape is allowed to vary in the fit to the data,
while $\Gamma_{s}$ is fixed to the nominal value [27]. Taking into account
resolution and decay time acceptance, the PDF given in Eq.(3) is modified to
$\mathcal{P}_{t}(t|\sigma_{t})\propto\left[\Gamma_{s}e^{-\Gamma_{s}\,t}\,\cosh\left(\frac{\Delta\Gamma_{s}}{2}t\right)\,\theta(t)\right]\otimes
G(t;0,S_{\sigma_{t}}\sigma_{t})\,\,\mathcal{E}_{t}(t),$ (4)
with $G(t;0,S_{\sigma_{t}}\sigma_{t})$ being the resolution function
determined by the method mentioned above. The decay time PDFs for the $B^{0}$
and $\mathchar 28931\relax^{0}_{b}$ backgrounds are identical to the signal
PDF, except for $\Delta\Gamma$ being zero, and $\Gamma_{s}$ being replaced by
their respective decay widths [27]. The shape of the decay time distribution
of the combinatorial background is determined with high mass sideband data. It
is parametrized by the sum of two exponential functions multiplied by a second
order polynomial distribution. The exponential and polynomial parameters are
allowed to vary in the fit and are constrained to be the same for the five
decay modes.
## 6 Flavour tagging
To determine the flavour of the $B^{0}_{s}$ meson at production, both
opposite-side (OST) and same-side (SST) tagging algorithms are used. The OST
exploits the fact that $b$ quarks at the LHC are predominantly produced in
quark–antiquark pairs. By partially reconstructing the second $b$ hadron in
the event, conclusions on the flavour at production of the signal $B^{0}_{s}$
candidate can be drawn. The OST have been optimized on large samples of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B\rightarrow\mu^{+}D^{*-}X$, and $B^{0}\\!\rightarrow D^{-}\pi^{+}$ decays
[24].
The SST takes advantage of the fact that the net strangeness of the $pp$
collision is zero. Therefore, the $s$ quark needed for the hadronization of
the $B^{0}_{s}$ meson must have been produced in association with an
$\overline{}s$ quark, which in about 50% of the cases hadronizes to form a
charged kaon. By identifying this kaon, the flavour at production of the
signal $B^{0}_{s}$ candidate is determined. The optimization of the SST was
performed on a data sample of $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
decays, which has a large overlap with the sample used in this analysis [25].
However, since the oscillation frequency is not correlated with the parameters
describing tagging performance, this does not bias the $\Delta m_{s}$
measurement.
The decisions given by both tagging algorithms have a probability $\omega$ to
be incorrect. Each tagging algorithm provides an estimate for the mistag
probability $\eta$ which is the output of a neural network combining various
event properties. The true mistag probability $\omega$ can be parametrized as
a linear function of the estimate $\eta$ [24, 25]
$\omega=p_{0}+p_{1}\times\left(\eta-\langle\eta\rangle\right),$ (5)
with $\langle\eta\rangle$ being the mean of the distribution of $\eta$. This
parametrization is chosen to minimize the correlations between $p_{0}$ and
$p_{1}$. The calibration is performed separately for the OST and SST.
The sets of calibration parameters $(p_{0},p_{1})_{\rm OST}$ and
$(p_{0},p_{1})_{\rm SST}$ are allowed to vary in the fit. The figure of merit
of these tagging algorithms is called the effective tagging efficiency
$\varepsilon_{\rm eff}$. It gives the factor by which the statistical power of
the sample is reduced due to imperfect tagging decisions. In this analysis,
$\varepsilon_{\rm eff}$ is found to be $(2.6\pm 0.4)\%$ for the OST and
$(1.2\pm 0.3)\%$ for the SST. Uncertainties are statistical only.
## 7 Measurement of $\Delta m_{s}$
Adding the information of the flavour tagging algorithms, the decay time PDF
for tagged signal candidates is modified to
$\displaystyle\mathcal{P}_{t}(t|\sigma_{t})$ $\displaystyle\propto$
$\displaystyle\left\\{\Gamma_{s}e^{-\Gamma_{s}\,t}\,\frac{1}{2}\left[\cosh\left(\frac{\Delta\Gamma_{s}}{2}t\right)\,+q\left[1-2\omega(\eta_{\rm
OST},\eta_{\rm SST})\right]\cos(\Delta m_{s}t)\right]\,\theta(t)\right\\}$ (6)
$\displaystyle\otimes~{}G(t,S_{\sigma_{t}}\sigma_{t})\,\,\mathcal{E}_{t}(t)\,\epsilon,$
where $\epsilon$ gives the fraction of candidates with a tagging decision.
Signal candidates without a tagging decision are still described by Eq.(4)
multiplied by an additional factor $(1-\epsilon)$ to ensure the relative
normalization.
The information provided by the opposite-side and same-side taggers for the
signal is combined to a single tagging decision $q$ and a single mistag
probability $\omega(\eta_{\rm OST},\eta_{\rm SST})$ using their respective
calibration parameters $p_{0_{\rm OST/SST}}$ and $p_{1_{\rm OST/SST}}$. The
individual background components show different tagging characteristics for
candidates tagged by the OST or SST. The $b$ hadron backgrounds show the same
opposite-side tagging behaviour ($q$ and $\omega$) as the signal, while the
combinatorial background shows random tagging behaviour. For same-side tagged
events, we assume random tagging behaviour for all background components. We
introduce tagging asymmetry parameters to allow for different numbers of
candidates being tagged as mixed or unmixed, and other parameters to describe
the tagging efficiencies for these backgrounds. As expected, the fitted values
of these asymmetry parameters are consistent with zero within uncertainties.
All tagging parameters, as well as the value for $\Delta m_{s}$, are
constrained to be the same for the five decay modes. The result is $\Delta
m_{s}$ = 17.768 $\pm$ 0.023 ${\rm\,ps^{-1}}$ (statistical uncertainty only).
The likelihood profile was examined and found to have a Gaussian shape up to
nine standard deviations. The decay time distributions for candidates tagged
as mixed or unmixed are shown in Fig. 2, together with the decay time
projections of the PDF distributions resulting from the fit.
Figure 2: Decay time distribution for the sum of the five decay modes for
candidates tagged as mixed (different flavour at decay and production; red,
continuous line) or unmixed (same flavour at decay and production; blue,
dotted line). The data and the fit projections are plotted in a signal window
around the reconstructed $B^{0}_{s}$ mass of 5.32 – 5.55
${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$.
## 8 Systematic uncertainties
With respect to the first measurement of $\Delta m_{s}$ at LHCb [13] all
sources of systematic uncertainties have been reevaluated.
The dominant source is related to the knowledge of the absolute value of the
decay time. This has two main contributions. First, the imperfect knowledge of
the longitudinal ($z$) scale of the detector contributes to the systematic
uncertainty. It is obtained by comparing the track-based alignment and survey
data and evaluating the track distribution in the vertex detector. This
results in 0.02% uncertainty on the decay time scale and thus an absolute
uncertainty of $\pm 0.004{\rm\,ps^{-1}}$ on $\Delta m_{s}$.
The second contribution to the uncertainty of the decay time scale comes from
the knowledge of the overall momentum scale. This has been evaluated by an
independent study using mass measurements of well-known resonances. Deviations
from the reference values [27] are measured to be within 0.15%. However, since
both the measured invariant mass and momentum enter the calculation of the
decay time, this effect cancels to some extent. The resulting systematic on
the decay time scale is evaluated from simulation to be 0.02%. This again
translates to an absolute uncertainty of $\pm 0.004{\rm\,ps^{-1}}$ on $\Delta
m_{s}$.
The next largest systematic uncertainty is due to a possible bias of the
measured decay time given by the track reconstruction and the selection
procedure. This is estimated from simulated data to be less than about 0.2 fs,
and results in $\pm 0.001{\rm\,ps^{-1}}$ systematic uncertainty on $\Delta
m_{s}$.
Various other sources contributing to the systematic uncertainty have been
studied such as the decay time acceptance, decay time resolution, variations
of the value of $\Delta\Gamma_{s}$, different signal models for the invariant
mass and the decay time resolution, variations of the signal fraction and the
fraction of $B^{0}_{s}\\!\rightarrow D^{\mp}_{s}K^{\pm}$ candidates. They are
all found to be negligible. The sources of systematic uncertainty on the
measurement of $\Delta m_{s}$ are summarized in Table 2.
Table 2: Systematic uncertainties on the $\Delta m_{s}$ measurement. The total systematic uncertainty is calculated as the quadratic sum of the individual contributions. Source | Uncertainty [ps-1]
---|---
$z$-scale | 0.004
Momentum scale | 0.004
Decay time bias | 0.001
Total systematic uncertainty | 0.006
## 9 Conclusion
A measurement of the $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation frequency $\Delta
m_{s}$ is performed using $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ decays in
five different $D^{-}_{s}$ decay channels. Using a data sample corresponding
to an integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$ collected by LHCb in
2011, the oscillation frequency is found to be
$\Delta m_{s}=17.768\pm 0.023\mathrm{~{}(stat)}\pm
0.006\mathrm{~{}(syst)}~{}\mathrm{ps}^{-1},$
in good agreement with the first result reported by the LHCb experiment [13]
and the current world average, $17.69\pm 0.08~{}\mathrm{ps}^{-1}$ [27]. This
is the most precise measurement of $\Delta m_{s}$ to date, and will be a
crucial ingredient in future searches for BSM physics in $B^{0}_{s}$
oscillations.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-17T09:08:06 |
2024-09-04T02:49:44.542570
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M.\n De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff,\n L. Del Buono, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen,\n E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P.\n Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik,\n P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina,\n I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki,\n J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev,\n Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R.\n McNulty, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello,\n R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B.\n Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K.\n Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak,\n S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas,\n J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch,\n A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Sebastian Wandernoth",
"url": "https://arxiv.org/abs/1304.4741"
}
|
1304.4839
|
# Non-commuting graphs of nilpotent groups
Alireza Abdollahi Department of Mathematics, University of Isfahan, Isfahan
81746-73441, Iran;
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5746 , Tehran, Iran. [email protected] and Hamid
Shahverdi School of Mathematics, Institute for research in fundamental
science (IPM), P.O. Box 19395-5746, Tehran, Iran. Iran.
[email protected]
###### Abstract.
Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-
commuting graph $\Gamma_{G}$ associated to $G$ is the graph whose vertex set
is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only
if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups
with irregular isomorphic non-commuting graphs, then $|G|=|H|$.
###### Key words and phrases:
Non-commuting graph; nilpotent groups; graph isomorphism; groups with abelian
centralizers
###### 2000 Mathematics Subject Classification:
20D15; 20D60
## 1\. Introduction and results
Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting
graph $\Gamma_{G}$ of $G$ is a graph whose vertex set is $G\setminus Z(G)$ and
two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. The non-
commuting graph of a group was first considered by Paul Erdős in 1975 [7].
Many people have studied the non-commuting graph (e.g., [1, 2, 3, 9, 10]). In
[2] the following conjecture was put forward:
###### Conjecture 1.1 (Conjecture 1.1 of [2]).
Let $G$ and $H$ be two finite non-abelian groups such that
$\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.
Conjecture 1.1 was refuted by an example due to Isaacs in [6], however it is
valid whenever one of $G$ or $H$ is a non-abelian finite simple group [3] or
whenever one of $G$ or $H$ has prime power order [1]. The counterexample given
in [6] is a pair $(G,H)$ of nilpotent non-abelian groups with regular non-
commuting graph; recall that a graph is called regular if the degree of all
vertices are the same, otherwise the graph is called irregular. It follows
from a result of Ito [5] that a finite group with a regular non-commuting
graph is a direct product of a non-abelian $p$-group for some prime $p$ and an
abelian group.
Here we study pairs $(G,H)$ of non-abelian finite groups which provide a
counterexample to Conjecture 1.1. It follows from the main result of [1], that
if a pair $(G,H)$ provides a counterexample then none of $G$ and $H$ are of
prime power order. Here we prove that if a pair of non-abelian finite
nilpotent groups provides a counterexample for Conjecture 1.1 then their non-
commuting graphs must be regular.
###### Theorem 1.2.
Let $G$ and $H$ be two finite non-abelian nilpotent groups with irregular non-
commuting graphs such that $\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.
We conjecture that the word “nilpotent” in Theorem 1.2 is sufficient for one
of the groups $G$ and $H$.
## 2\. Non-commuting graphs of nilpotent groups
A non-abelian group is called an $AC$-group if the centralizer of every non-
central element is abelian. For a group $G$ and an element $g\in G$, $g^{G}$
denotes the conjugacy class of $g$ in $G$.
###### Lemma 2.1.
Let $G$ and $H$ be two finite non-abelian groups. If
$\phi:\Gamma_{G}\rightarrow\Gamma_{H}$ is a graph isomorphism and $g$ is a
non-central element of $G$, then the following hold:
1. (1)
$|G|-|Z(G)|=|H|-|Z(H)|$.
2. (2)
$|G|-|C_{G}(g)|=|H|-|C_{H}(\phi(g))|$.
3. (3)
$|C_{G}(g)|-|Z(C_{G}(g))|=|H|-|Z(C_{H}(\phi(g)))|$, where $C_{G}(g)$ is not
abelian.
4. (4)
If $C_{G}(g)$ is not abelian, then
$\Gamma_{C_{G}(g)}\cong\Gamma_{C_{H}(\phi(g))}$.
5. (5)
Suppose that $C_{1}=C_{G}(g_{1})$ and $C_{i}=C_{C_{i-1}}(g_{i})$ for $i\geq
2$, where $g_{1}\in G\setminus Z(G)$ and $g_{i}\in C_{i-1}\setminus
Z(C_{i-1})$. Then there exists $k\in\mathbb{N}$ such that $C_{k}$ is an
$AC$-group.
6. (6)
$|G|=|H|$ if and only if $|C_{G}(g)|=|C_{H}(\phi(g))|$ if and only if
$|Z(G)|=|Z(H)|$.
###### Proof.
It is straightforward. To prove (5), note that if the centralizer $C_{i}$ is
not an $AC$-group, then some proper centralizer in $C_{i}$ is not abelian
guaranteeing the existence of an element $g_{i+1}$. On the other hand, $G$ is
finite so the series $C_{1}>C_{2}>\cdots>C_{i}>\cdots$ will eventually
terminate in an $AC$-group. ∎
###### Lemma 2.2 (see e.g. Theorem 2.1 of [1]).
Let $G$ be a finite non-abelian group and $H$ be a group such that
$\Gamma_{G}\cong\Gamma_{H}$. Then the following hold:
1. (1)
$|C_{H}(h)|$ divides $(|g^{G}|-1)(|Z(G)|-|Z(H)|)$ for any $g\in G\setminus
Z(G)$ and $h=\phi(g)$, where $\phi$ is any graph isomorphism from $\Gamma_{G}$
to $\Gamma_{H}$.
2. (2)
If $|Z(G)|\geq|Z(H)|$ and $G$ contains a non-central element $g$ such that
${|C_{G}(g)|}^{2}\geq|G|\cdot|Z(G)|$, then $|G|=|H|$.
We need the following result concerning a number theoretic conjecture due to
Goormaghtigh.
###### Theorem 2.3 (see e.g. Theorem 1.3 of [4]).
Let $x,y,m,n$ be integers such that $y>x>1$ and $m,n>1$. Then the following
equation has at most one pair $(m,n)$ of solution for every fixed pair
$(x,y)$:
$\frac{y^{n}-1}{y-1}=\frac{x^{m}-1}{x-1}.$
###### Theorem 2.4.
Let $G$ be a nilpotent group with at least two distinct non-abelian Sylow
subgroups. Suppose also that $H$ is any non-abelian group such that
$|Z(G)|\geq|Z(H)|$ and $\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.
###### Proof.
Suppose $G=P\times Q\times S$, where $P$ and $Q$ are non-abelian Sylow $p$,
$q$-subgroups of $G$ such that $p\neq q$ and $S$ is a subgroup of $G$. If
$x\in P\setminus Z(P)$ and $y\in Q\setminus Z(Q)$, then
$|Z(G)|<|C_{P}(x)||C_{Q}(y)||S|$. Therefore
$|G||Z(G)|<|C_{P}(x)||C_{Q}(y)||P||Q||S|^{2}=|C_{G}(x)||C_{G}(y)|.$
It follows that
$|G||Z(G)|<\max\\{|C_{G}(x)|^{2},|C_{G}(y|^{2}\\}.$
Now, Lemma 2.2(2) completes the proof. ∎
###### Corollary 2.5.
Let $G$ and $H$ be two nilpotent groups each of which have at least two non-
abelian Sylow subgroups. If $\Gamma_{G}\cong\Gamma_{H}$, then $|G|=|H|$.
Both groups $G$ and $H$ in the counterexample of Conjecture 1.1 due to Isaacs
in [6] have the same shape, that is, they are direct products of a non-abelian
group of prime power order $P$ and a non-trivial abelian group $A$ such that
$\gcd(|P|,|A|)=1$ and all non-trivial conjugacy class sizes of $G$ or $H$ have
equal order. The latter property was first studied by Ito [5] and we want to
prove Theorem 1.2 for all nilpotent groups except those satisfying the latter
shape.
## 3\. Proof of Theorem 1.2
Now, we prove Theorem 1.2 in four cases. In this section $G$ and $H$ are
finite non-abelian nilpotent groups with irregular non-commuting graphs and
$\phi:\Gamma_{G}\rightarrow\Gamma_{H}$ is a graph isomorphism. By Corollary
2.5, we may assume that $G$ has exactly one non-abelian Sylow subgroup. If $G$
is of prime power order, the main result of [1] implies that $|G|=|H|$. Thus
we may assume $G=P\times A$, where $P$ is a non-abelian Sylow $p$-subgroup of
$G$ and $A$ is a non-trivial abelian subgroup whose order is prime to $p$.
Also, set $|P|=p^{n}$ and $|Z(P)|=p^{r}$.
Case (a): $H=P_{1}\times B$ for some non-abelian Sylow $p$-subgroup $P_{1}$ of
$H$ and for some non-trivial abelian subgroup $B$ of $H$.
We use the following notation: $|P_{1}|=p^{m}$, $|Z(P_{1})|=p^{s}$ and
$\phi(g_{i})=h_{i}$, where $g_{1},\dots,g_{k}$ are non-central elements of $G$
chosen from conjugacy classes of $G$ with pairwise distinct sizes such that
$|{g_{i}}^{G}|=p^{a_{i}}$ and $|{h_{i}}^{H}|=p^{b_{i}}$ and
$a_{1}<\dots<a_{k}$ and $b_{1}<\dots<b_{k}$. Notice that $k\geq 2$, since
$\Gamma_{G}$ and $\Gamma_{H}$ are irregular.
Since $\Gamma_{G}\cong\Gamma_{H}$,
(1) $|A|p^{r}(p^{n-r}-1)=|B|p^{s}(p^{m-s}-1)$ (2)
$|A|p^{n-a_{i}}(p^{a_{i}}-1)=|B|p^{m-b_{i}}(p^{b_{i}}-1)$
for every $1\leq i\leq k$. Equation (1) implies that $r=s$ and equation (2)
implies that $n-a_{i}=m-b_{i}$. Since $\Gamma_{G}$ is not regular, graph
isomorphism implies that
(3) $|A|(p^{n-a_{1}}-p^{n-a_{2}})=|B|(p^{m-b_{1}}-p^{m-b_{2}}).$
Therefore $|A|=|B|$. Now, equation (2) implies that $a_{1}=b_{1}$. Hence
$|P|=|P_{1}|$.
Case (b): $H=P_{1}\times X$, where $P_{1}$ is a non-abelian Sylow $p$-subgroup
of $H$ and $X$ is an arbitrary group such that $\gcd(p,|X|)=1$.
Suppose $H$ is a minimal counterexample. Also suppose by way of contradiction
that $X$ is a non-abelian group. Then $P_{1}$ and $X$ are $AC$-group. Let
$x\in X\setminus Z(X)$. Then $C_{H}(x)=P_{1}\times B$, where $B\subseteq X$ is
an abelian subgroup of $X$. Therefore Case (a) implies that
$|C_{H}(x)|=|C_{G}(\phi^{-1}(x))|$. Since $\Gamma_{G}\cong\Gamma_{H}$, we have
$|G|=|H|$. Now, $|G|=|H|$ implies that $|P|=|P_{1}|$. Set
$|C_{G}(\phi^{-1}(x))|=p^{n-\alpha}|A|$ for some integer $1<\alpha<n$. By
graph isomorphism, we have
$(p^{n}-p^{n-\alpha})|A|=p^{n}(|X|-|C_{X}(x)|).$
The largest $p$-power dividing the right-hand side of the equation is $\geq
p^{n}$ and the left is $p^{n-\alpha}$. This is a contradiction. Hence $X$ is
abelian and Case (a) completes the proof.
Case (c): $H=Q_{1}\times X$, where $Q_{1}$ is a Sylow $q_{1}$-subgroup of $H$
and $X$ is a non-abelian nilpotent group.
If $p=q_{1}$, then Case (b) completes the proof. We claim that $p\neq q_{1}$
is not possible. Let $H$ be a minimal counterexample. Therefore $Q_{1}$ and
$X$ are $AC$-groups. By the characterization of $AC$-groups [8], a nilpotent
$AC$-group is a direct product of a non-abelian group of prime power order and
an abelian group. Therefore $X=Q_{2}\times B$, where $Q_{2}$ is a non-abelian
$q_{2}$-group for some prime $q_{2}$, $B$ is an abelian group and
$\gcd(|Q_{2}|,|B|)=1$. Let $h_{i}\in Q_{i}\setminus Z(Q_{i})$ for
$i\in\\{1,2\\}$. Also, set $\phi^{-1}(h_{i})=g_{i}$ for $i\in\\{1,2\\}$ and
$|C_{G}(g_{i})|=|A|p^{n-{a_{i}}}$, where $1<a_{i}<n$. If $q_{2}=p$, then again
Case (b) implies that $Q_{1}\times B$ is an abelian group. This is a
contradiction. Therefore $p\neq q_{1},q_{2}$.
We have $C_{H}(h_{1})=C_{Q_{1}}(h_{1})\times Q_{2}\times B$ and
$C_{H}(h_{2})=Q_{1}\times C_{Q_{2}}(h_{2})\times B$ and
$Z(C_{H}(h_{1}))=C_{Q_{1}}(h)\times Z(Q_{2})\times B$ and
$Z(C_{H}(h_{2}))=Z(Q_{1})\times C_{Q_{2}}(h_{1})\times B$. So $Z(H)\subsetneqq
Z(C_{H}(h_{i}))$. Therefore graph isomorphism implies that $Z(G)\subsetneqq
Z(C_{G}(g_{i}))$. Set $|Z(C_{G}(g_{i}))|=|A|p^{d_{i}}$ for $i\in\\{1,2\\}$ and
$|Z(G)|=|A|p^{r}$. It is clear that $d_{i}>r$. Now,
$\Gamma_{G}\cong\Gamma_{H}$ and
$\Gamma_{C_{G}(g_{i})}\cong\Gamma_{C_{H}(h_{i})}$ for $i\in\\{1,2\\}$ imply
that
(4)
$|C_{H}(h_{2})|-|Z(C_{H}(h_{2}))|=(|Q_{1}|-|Z(Q_{1})|)|C_{Q_{2}}(h_{2})||B|=|A|(p^{n-a_{2}}-p^{d_{2}})$
(5)
$|Z(C_{H}(h_{1}))|-|Z(H)|=(|C_{Q_{1}}(h_{1})|-|Z(Q_{1})|)|Z(Q_{2})||B|=|A|(p^{d_{1}}-p^{r})$
(6)
$|H|-|C_{H}(h_{1})|=(|Q_{1}|-|C_{Q_{1}}(h_{1})|)|Q_{2}||B|=|A|(p^{n}-p^{n-a_{1}}).$
Since
$|B|(|Q_{1}|-|Z(Q_{1})|)=|B|(|Q_{1}|-|C_{Q_{1}}(h_{1})|)+|B|(|C_{Q_{1}}(h_{1})|-|Z(Q_{1})|)$,
by equation (5) and (6) the largest $p$-power dividing the right-hand side of
the latter equation is $p^{r}$ and by equation (4) the largest $p$-power
dividing the left hand side is $p^{d_{2}}$. This is a contradiction.
Case (d): $H=Q\times B$, where $Q$ is a non-abelian Sylow $q$-subgroup for
some prime $q\neq p$ and $B$ is a non-trivial abelian subgroup.
Suppose by way of contradiction that $|G|\neq|H|$. Since $\Gamma_{G}$ is not
regular, there exist $g_{1},g_{2}\in G\setminus Z(G)$ such that
$|g_{1}^{G}|=p^{a_{1}}\neq p^{a_{2}}=|g_{2}^{G}|$. Set $|Q|=q^{m}$,
$|Z(Q)|=q^{s}$, $\phi(g_{i})=h_{i}$ for $i\in\\{1,2\\}$ and
$|h_{i}^{H}|=q^{b_{i}}$. Since $\Gamma_{G}\cong\Gamma_{H}$,
(7) $|A|(p^{n}-p^{r})=|B|(q^{m}-q^{s})$ (8)
$|A|(p^{n-a_{i}}-p^{r})=|B|(q^{m-b_{i}}-q^{s}).$
If $u=\gcd(a_{1},a_{2},n-r)$ and $v=\gcd(b_{1},b_{2},m-s)$, by considering
equations (7) and (8) and taking greatest common divisors, we have
(9) $|A|p^{r}(p^{u}-1)=|B|q^{s}(q^{v}-1).$
Now, by dividing equations (7) and (9), we have
(10) $\frac{p^{n-r}-1}{p^{u}-1}=\frac{q^{m-s}-1}{q^{v}-1}$
and by dividing equations (8) and (9), we have
(11) $\frac{p^{n-a_{i}-r}-1}{p^{u}-1}=\frac{q^{m-b_{i}-s}-1}{q^{v}-1}.$
Note that it is not possible that $n-a_{1}-r=u=n-a_{2}-r$, since $a_{1}\neq
a_{2}$. Now, Theorem 2.3 and equation (10) and (11) yield a contradiction.
$\hfill\Box$
Acknowledgments
The authors are grateful to the referee for his/her invaluable comments. The
first author was financially supported by the Center of Excellence for
Mathematics, University of Isfahan. This research was in part supported by
grants IPM (No. 91050219) and IPM (No. 91200045).
## References
* [1] A. Abdollahi, S. Akbari, H. Dorbidi and H. Shahverdi, _Commutativity pattern of non-abelian $p$-groups determine their orders_, Comm. Algebra, 41 (2013) 451-461.
* [2] A. Abdollahi, S. Akbari and H. R. Maimani, _Non-commuting graph of a group_ , J. Algebra, 298 (2006) 468-492.
* [3] M.R. Darafsheh, _Groups with the same non-commuting graph_ , Discrete Appl. Math., 157 no. 4 (2009) 833-837.
* [4] B. He and A. Togbe, _On the number of solutions of Goormaghtigh equation for given $x$ and $y$_, Indag. Mathern. N.S., 19 no. 1 (2008) 65-72.
* [5] N. Ito, _On finite groups with given conjugate types_ , Nagoya Math. J., 6 (1953) 17-28.
* [6] A. R. Moghaddamfar, _About noncommuting graphs_ , Siberian Math. J., 47 no. 5 (2005) 1112-1116.
* [7] B.H. Neumann, _A problem of Paul Erdős on groups_ , J. Aust. Math. Soc. Ser. A, 21 (1976) 467-472.
* [8] R. Schmidt, _Zenralisatorverbande endlicher gruppen_ , Rend. Sem. Mat. Univ. Padova, 44 (1975) 55-75.
* [9] R. Solomon and A. Woldar, _Simple non-abelian groups are characterized by their non-commuting graph_ , preprint 2012.
* [10] L. Wang and W. Shi. _A new characterization of $A_{10}$ by its non-commuting graph_, Comm. Algebra, 36 (2008) 533-540.
|
arxiv-papers
| 2013-04-17T14:31:39 |
2024-09-04T02:49:44.549917
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alireza Abdollahi and Hamid Shahverdi",
"submitter": "Alireza Abdollahi",
"url": "https://arxiv.org/abs/1304.4839"
}
|
1304.4924
|
# On the push-out spaces
M. Fathy University of Applied Science
and Technology of East Azarbayjan Cooperation,
Tabriz, Iran [email protected] and M. Faghfouri University of Tabriz,
Tabriz, Iran [email protected]
###### Abstract.
Let $f:M^{m}\longrightarrow\mathbb{R}^{m+k}$ be an immersion where $M$ is a
smooth connected $m$-dimensional manifold without boundary. Then we construct
a subspace $\Omega(f)$ of $\mathbb{R}^{k}$, namely push-out space. which
corresponds to a set of embedded manifolds which are either parallel to $f$,
tubes around $f$ or, ingeneral, partial tubes around $f$. This space is
invariant under the action of the normal holonomy group, $\mathcal{H}ol(f)$.
Moreover, we construct geometrically some examples for normal holonomy group
and push-out space in ${\mathbb{R}}^{3}$. These examples will show that
properties of push-out space that are proved in the case $\mathcal{H}ol(f)$ is
trivial, is not true in general.
###### Key words and phrases:
Singular points, Normal holonomy group, push-out space.
###### 1991 Mathematics Subject Classification:
53C40, 53C42.
## 1\. Introduction
In this paper we introduce push-out space for an immersion
$f:M^{m}\longrightarrow\mathbb{R}^{m+k}$, where $M$ is a smooth connected
$m$-dimensional manifold without boundary. To do this, we give some examples
in 3-dimensional Euclidean space, $\mathbb{R}^{3}$, infact, in these examples
we calculate normal holonomy group and push-out space geometrically. We
consider the case when $\mathcal{H}ol(f)$ is non-trivial. This extends the
work of Carter and Senturk [2], who obtained results about the case when
$\mathcal{H}ol(f)$ is trivial. In these examples we show that some of the
properties of push-out space which they obtained is not true for the case when
$\mathcal{H}ol(f)$ is non-trivial.
## 2\. Basic definitions
###### Definition 2.1 ([2]).
Let $f:M^{m}\longrightarrow\mathbb{R}^{m+k}$ be a smooth immersion where $M$
is a smooth connected $m$-dimensional manifold without boundary. The total
space of the normal bundle of $f$ is defined by
$N(f)=\\{(p,x)\in M\times\mathbb{R}^{m+k}:<x,v>=0\quad\forall v\in
f_{*}T_{p}(M)\\}$
The endpoint map $\eta:N(f)\longrightarrow\mathbb{R}^{m+k}$ is defined by
$\eta(p,x)=f(p)+x$ and, the set of singular points of $\eta$ is subset
$\Sigma(f)\subset N(f)$ called the set of critical normals of $f$ and the set
of focal points of $\eta$ is a subset
$\eta({\Sigma(f)})\subset\mathbb{R}^{m+k}$.
For $p\in M$, we put $N_{p}(f)=\\{x:(p,x)\in N(f)\\}$ and
${\Sigma}_{p}(f)=\\{x:(p,x)\in\Sigma(f)\\}$ respectively, normal space at $p$
and the set can be thought of as focal points with base $p$.
###### Definition 2.2 ([1]).
For $p_{0}\in M$ and $p\in M$ and path $\gamma:[0,1]\longrightarrow M$ from
$p_{0}$ to $p$ define $\varphi_{p,\gamma}:N_{p_{0}}(f)\longrightarrow
N_{p}(f)$ by parallel transport along $\gamma$. The $\varphi_{p,\gamma}$’s are
isometries. The normal holonomy group on $N_{p_{0}}(f)$, is
$\mathcal{H}ol(f)=\\{\varphi_{p_{0},\gamma}:\gamma:[0,1]\longrightarrow
M,\quad\gamma(0)=\gamma(1)=p_{0}\\}$
If the closed path $\gamma$ at $p_{0}$ is homotopically trivial then
$\varphi_{p_{0},\gamma}$ is an element of the restricted normal holonomy group
${\mathcal{H}ol}_{0}(f)$.
###### Definition 2.3 ([3]).
For a fix $p_{0}\in M$ the push-out space for an immersion
$f:M^{m}\longrightarrow\mathbb{R}^{m+k}$ is defined by
$\Omega(f)=\\{x\in N_{p_{0}}(f):\forall p\in M,\forall\gamma\mbox{ s.t.
}\gamma(0)=p_{0},\gamma(1)=p\mbox{ then
}\varphi_{p,\gamma}x\notin\Sigma_{p}(f)\\}$
(i.e. $\forall p\in M$, $f(p)+\varphi_{p,\gamma}(x)$ is not a focal point with
base $p$ when $x$ belongs to $\Omega(f)$). Therefore $\Omega(f)$ is the set of
normals at $p_{0}$, where transported parallely along all curves, do not meet
focal points. So $\Omega(f)$ is invariant under the action of
$\mathcal{H}ol(f)$.
###### Definition 2.4 ([4]).
Let $B\subset N(f)$ be a smooth subbundle with type fiber S where
1) S is a smooth submanifold of $\mathbb{R}^{k}$
2) $B\cap\Sigma(f)=\emptyset$
3) B is invariant under parallel transport (along any curve in M). Then B is a
smooth manifold and $g\equiv\eta|_{B}:B\longrightarrow\mathbb{R}^{m+k}$ is a
smooth immersion called a partial tube about f.
###### Theorem 2.5 ([2]).
Let $\mathcal{H}ol(f)$ is trivial and $M$ be a compact manifold. Then each
path-connected component of $\Omega(f)$ is open in $\mathbb{R}^{k}$.
###### Theorem 2.6 ([2]).
Let $\mathcal{H}ol(f)$ is trivial then Each path-connected component of
$\Omega(f)$ is convex.
###### Remark 2.7.
In Example 3.2, if $\frac{\alpha}{\pi}$ is irrational then $\Omega(\bar{f})$
is not open in $\mathbb{R}^{2}$ but $M=\mathbb{S}^{1}$ is compact. Also, in
Example 3.5, $\Omega({f})=\\{O\\}$ hence $\Omega({f})$ is closed in
$\mathbb{R}^{2}$ but $\mathcal{H}ol(f)$ is trivial. This shows that Theorem
2.5 is false when M is not compact or $\mathcal{H}ol(f)$ is non-trivial.
###### Remark 2.8.
In Example 3.6 one of path-connected components of $\Omega(f)$, which is the
complement space of cone and two other components in $\mathbb{R}^{3}$, is not
convex. This shows that Theorem 2.6 is false when $\mathcal{H}ol(f)$ is non-
trivial.
we conclude that the properties of push-out space that are proved in the case
$\mathcal{H}ol(f)$ is trivial, is not true in general.
## 3\. Examples of normal holonomy groups and push-out spaces
###### Example 3.1.
We start with a curve as below
suppose this curve is given by $s\mapsto(\xi(s),\eta(s))$ where $s\in[0,1]$
and at $(1,0,0)$:$s=0$ ,${\frac{\partial\xi}{\partial
s}}=1,({\frac{\partial}{\partial s}})^{r}\eta=0$ for all $r\geq 0$ and at
$(0,1,0)$:$s=1$,${\frac{\partial\eta}{\partial
s}}=1,({\frac{\partial}{\partial s}})^{r}\xi=0$ for all $r\geq 0$. Now,we take
this curve in $\mathbb{R}^{3}$ and consider the same curves in $yz$-plane and
$xz$-plane and fit together to make a smooth closed curve in
$\mathbb{R}^{3}$.Now by identifying $\mathbb{S}^{1}$ with
$\frac{\mathbb{R}}{3\mathbb{Z}}$, the curve in $\mathbb{R}^{3}$ can be
redefined as $f:\mathbb{S}^{1}\longrightarrow\mathbb{R}^{3}$ where:
$f(s)=\left\\{\begin{array}[]{cc}(\xi(s),\eta(s),0)\qquad\quad\quad 0\leq
s\leq 1\\\ (0,\xi(s-1),\eta(s-1))\quad 1\leq s\leq 2\\\
(\eta(s-2),0,\xi(s-2))\quad 2\leq s\leq 3\\\ \end{array}\right.$
To find the normal holonomy group of the above curve, we will consider normal
vector to the curve under parallel transport. As each part of the curve lies
in a 2-plane, the normal plane at a point of the curve is spanned by the
perpendicular direction to the 2-planes.
Step1.Start with the normal vector at (1,0,0), in the diagram, it stays in the
xy-plane under parallel transport.
The normal vector (0,1,0) at (1,0,0) goes to normal vector (-1,0,0)at (0,1,0).
Step2.At (0,1,0)the the normal vector (-1,0,0) is perpendicular to the yz-
plane, it stays perpendicular to the yz-plane under parallel transport form
(0,1,0) to (0,0,1).
The normal vector (-1,0,0)at(0,1,0) goes to normal vector(-1,0,0) at (0,0,1).
Step3.The the normal vector (-1,0,0) is in the xz-plane at (0,0,1) and stays
in the xz-plane from (0,0,1) to (1,0,0).
The normal vector (-1,0,0,) at (1,0,0) by going once around the curve the
normal vector will turn about $\frac{\pi}{2}$.
Going around of curve again, the normal vector moves through another
$\frac{\pi}{2}$ and after four times around the curve back to its original
position. this shows that $\mathcal{H}ol(f)$ is generated by a rotation
through $\frac{\pi}{2}$.
Now we find the push-out space of $f$. Except at end-points of three areas,
locally the curve lies in a 2-plane so the focal points with base
$s$,$f(s)+\Sigma_{s}(f)$, consists of a straight line through the center of
curvature, $c(s)$, of the curve at $s$, perpendicular to the line joining
$c(s)$ and $f(s)$.
At end-points of three areas, and possibly some other points, the focal set is
empty as the center of curvature ”at infinity”.
so $\Sigma_{s}(f)$ is a line in $N_{s}(f)$. The image of $\Sigma_{s}(f)$ under
normal holonomy group is obtained by rotating it through $\frac{\pi}{2}$ until
it returns to the original position.
Now, fix the normal plane $N_{s_{0}}(f)$ at $f(S_{0})=(1,0,0)$ where $s_{0}=0$
and use parallel transport to identify all the normal planes with the normal
plane $N_{s_{0}}(f)$. The push-out space is complement of all the
$\Sigma_{s}(f)$ and their images under normal holonomy group.
Therefore the push-out space of
$f:\mathbb{S}^{1}\longrightarrow\mathbb{R}^{3}$ is an open square $Q$ with
sides of length $2\rho$ where $\rho$ is the minimum absolute value of the
radius of curvature of the original curve in the xy-plane. (i.e.
$\Omega(f)$)is the interior of the smallest square on $N_{s_{0}}(f)$.)
###### Example 3.2.
We consider the immersion $\bar{f}$ as in Example 3.1 except that the xz-plane
is tilted through an angle $\alpha$.
In other words, $\bar{f}=Lof$ where f is the immersion in example 3.1 and $L$
is the linear transformation given by
$L=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&\tan\alpha\\\ 0&0&1\\\
\end{array}\right)$ where $0<\alpha<\frac{\pi}{2}$. So, we have
$\bar{f}(s)=\begin{cases}(\xi(s),\eta(s),0)&0\leq s\leq 1\\\
(0,\xi(s-1)+\eta(s-1)\tan{\alpha},\eta(s-1))&1\leq s\leq 2\\\
(\eta(s-2),\xi(s-2)\tan{\alpha},\xi(s-2))&2\leq s\leq 3\end{cases}$
The end-points of three areas of this immersion are (1,0,0),(0,1,0) and
(0,$\tan\alpha$,1). Note that at these points the tangent to the curve is the
radial line form (0,0,0) so unit tangent at (1,0,0) is (1,0,0)and unit tangent
at (0,$\tan\alpha$,1) is $\frac{(0,\tan\alpha,1)}{\sqrt{1+\tan^{2}\alpha}}$
etc.
As in Example 3.1, under parallel transport, the normal vector (0,1,0) at
(1,0,0) goes to the normal vector (-1,0,0) at (0,1,0), which goes to the
normal vector (-1,0,0) at (0,$\tan\alpha$,1), which goes to the normal
vector$\frac{(0,\tan\alpha,1)}{\sqrt{1+\tan^{2}\alpha}}$ at (1,0,0). So going
once around the curve the normal vector has moved through
$\frac{\pi}{2}-\alpha$.
This shows that $\mathcal{H}ol(\bar{f})$ is generated by a rotation through
$\frac{\pi}{2}-\alpha$ As in Example 3.1, the image of $\Sigma_{s}(\bar{f})$
under normal holonomy group is obtained by rotating the line
$\Sigma_{s}(\bar{f})$ through $\frac{\pi}{2}-\alpha$. It depends on $\alpha$
and is obtained as:
If $R$ is the rotating through an angle $\frac{\pi}{2}-\alpha$ then
$\Omega({\bar{f}})=\bigcap\\{(R)^{n}Q:n\in\mathbb{Z}\\}$ where Q is a square
as in Example 3.1, thus if $\frac{\alpha}{\pi}$ is rational then
$(R)^{n}(\Sigma_{s}(\bar{f}))=\Sigma_{s}(\bar{f})$ for some $n\in\mathbb{Z}$
and so,$\Omega({\bar{f}})$ is the interior of the smallest polygon. If
$\frac{\alpha}{\pi}$ is irrational then
$(R)^{n}(\Sigma_{s}(\bar{f}))\neq\Sigma_{s}(\bar{f})$ for any $n\in\mathbb{Z}$
and so,$\Omega({\bar{f}})$ is an open disk of radius $\rho$ together with a
dense set of points on the boundary circle where $\rho$ is minimum absolute
value of the radius of curvature of the immersed curve by $\bar{f}$.
###### Example 3.3.
In Example 3.2, we replace the immersion $\bar{f}$ with the immersion
$\bar{f}oh$ where
$h:\mathbb{R}\longrightarrow\mathbb{S}^{1}\equiv\frac{\mathbb{R}}{3\mathbb{Z}}$
is covering projection. Since ${\mathbb{R}}$ is simply connected, for any
arbitrary point $s\in{\mathbb{R}}$, any closed path at $s$ is nulhomotopic
with constant path at $s$, hence definition 2.2 shows that, the normal
holonomy group of $\bar{f}Oh$ is trivial
(i.e.$\mathcal{H}ol(\bar{f}oh)=\mathcal{H}ol_{0}(\bar{f}oh)$ ). To calculate
$\Omega(\bar{f}oh)$, we prove the theorem 3.4, in general. It will show that
$\Omega(\bar{f}oh)=\Omega(\bar{f})$.
###### Theorem 3.4.
Let $f:M^{m}\longrightarrow\mathbb{R}^{m+k}$ be an immersion and $\hat{M}$ be
any covering space with covering projection $h:\hat{M}\longrightarrow M$. If
$\hat{f}=foh$, then $\Omega(\hat{f})=\Omega({f})$.
###### Proof.
Let $x\in\Omega({f})$ and fix $p_{0}\in M$. Then definition 2.3 implies that,
$\forall p\in M,\forall\gamma$ s.t. $\gamma(0)=p_{0},\gamma(1)=p$;
$\varphi_{p,\gamma}x\neq\Sigma_{p}(f).$
we define the total space of the normal bundle of $\hat{f}$ by
$N(\hat{f})=\\{(\hat{p},x)\in\hat{M}\times\mathbb{R}^{m+k}:<x,v>=0\quad\forall
v\in\hat{f}_{*}T_{\hat{p}}(\hat{M})\\}$
Also, for any $\hat{p}\in h^{-1}(p)$ we have
$\displaystyle\hat{f}_{*}T_{\hat{p}}(\hat{M})$
$\displaystyle=(foh)_{*}T_{\hat{p}}(\hat{M})$
$\displaystyle=(f_{*}oh_{*})T_{\hat{p}}(\hat{M})$
$\displaystyle=f_{*}T_{p}(M)$
this shows that, for any $\hat{p}\in h^{-1}(p)$ we have
$N_{\hat{p}}(\hat{f})=N_{p}(f)$ and
so$\Sigma_{\hat{p}}(\hat{f})=\Sigma_{p}(f)$. Further, we fix$\hat{p_{0}}\in
h^{-1}(p_{0})$ then $\hat{\varphi}_{\hat{p},\hat{\gamma}}=\varphi_{p,\gamma}$
where $\hat{\gamma}:[0,1]\longrightarrow\hat{M}$ s.t.
$\hat{\gamma(0)}=\hat{p_{0}},\hat{\gamma}(1)=\hat{p}.$
Therefore,$\forall\hat{p}\in\hat{M},\forall\hat{\gamma}$;
$\hat{\varphi}_{\hat{p},\hat{\gamma}}x\neq\Sigma_{\hat{p}}(\hat{f}).$ Now
using definition 2.3 again, follows that, $x\in\Omega(\hat{f})$. By the same
way proves that $\Omega(\hat{f})\subseteq\Omega(f)$. ∎
###### Example 3.5.
If $\frac{\pi}{2}-\alpha=\frac{2\pi}{n}$, then Example 3.3 can be modified by
replacing h by the n-fold covering
$\bar{h}:\mathbb{S}^{1}\longrightarrow\mathbb{S}^{1}$.
Going once around the first $\mathbb{S}^{1}$ in
$\bar{h}:\mathbb{S}^{1}\longrightarrow\mathbb{S}^{1}$ corresponds to moving n
times around the second $\mathbb{S}^{1}$ so parallely transporting a normal n
times around the second $\mathbb{S}^{1}$ which gives a rotation of
$\displaystyle n(\frac{\pi}{2}-\alpha)$ $\displaystyle=n(\frac{2\pi}{n})$
$\displaystyle=2\pi$
i.e. the identity, so
$\mathcal{H}ol(\bar{f}o\bar{h})=\mathcal{H}ol_{0}(\bar{f}o\bar{h})$. Since,
the immersed curve by $\bar{f}o\bar{h}$ and the immersed curve by $\bar{f}$
have same figure in $\mathbb{R}^{3}$ and $\mathcal{H}ol(\bar{f}o\bar{h})$ is
trivial so the singular sets of them also the same (i.e.
$\Sigma(\bar{f}o\bar{h})=\Sigma(\bar{f})$). This implies that
$\Omega(\bar{f}o\bar{h})=\Omega(\bar{f})$.
###### Example 3.6.
We consider a sequence of curves $f_{n}$ in $\mathbb{R}^{3}$ defined as in
Example 3.1 except that $||f_{n}(s)||$ and the curvature tends to infinity
with n when $s=\frac{1}{2},\frac{3}{2}$ or $\frac{5}{2}$ but is bounded
otherwise.
Now, we define the immersion $f:\mathbb{R}\longrightarrow\mathbb{R}^{3}$ by
$f(s\pm 3n)=f_{n}(s)$. When n tends to infinity, the immersion
$f:\mathbb{R}\longrightarrow\mathbb{R}^{3}$ has a sequence of points where the
curvature tends to infinity and the radius of curvature at these points can be
arbitrary small; in other words, $\exists s$ where $\Sigma_{s}(f)$ is
arbitrary close to ”O” in $N_{s}(f)$. So $\\{O\\}$ is the only point not in
the image of $\Sigma_{s}(f)$ under normal holonomy group for all
$s\in\mathbb{R}.$ Then $\Omega(f)=\\{O\\}$. In this case because $\mathbb{R}$
is simply connected then $\mathcal{H}ol(f)$ is trivial.
The following results have been proved in [2], when $\mathcal{H}ol(f)$ is
trivial.
###### Theorem 3.7.
Let M be a compact manifold, then each path-connected component of $\Omega(f)$
is open in $\mathbb{R}^{k}$.
###### Theorem 3.8.
Each path-connected component of $\Omega(f)$ is convex.
###### Remark 3.9.
In Example 3.2, if $\frac{\alpha}{\pi}$ is irrational then $\Omega(\bar{f})$
is not open in $\mathbb{R}^{2}$ but $M=\mathbb{S}^{1}$ is compact. Also, in
Example 3.5 $\Omega({f})=\\{O\\}$ so $\Omega({f})$ is closed in
$\mathbb{R}^{2}$ but $\mathcal{H}ol(f)$ is trivial. This shows that Theorem
3.7 is false when M is not compact or $\mathcal{H}ol(f)$ is non-trivial.
###### Remark 3.10.
In Example 3.6 one of the path-connected components of $\Omega(f)$, which is
the complement space of cone and two other components in $\mathbb{R}^{3}$, is
not convex. This shows that Theorem 3.8 is false when $\mathcal{H}ol(f)$ is
non-trivial.
Thus the properties of push-out space that are proved in the case
$\mathcal{H}ol(f)$ is trivial, is not true in general.
## References
* [1] J. Berndt, S. Console and C. Olmos, Submanifolds and Holonomy,Research Notes in Mathematics 434, CHAPMAN & HALL/CRC, 2003.
* [2] S. Carter and Z. Senturk, The space of immersions parallel to a given immersion,J. London Math. Soc. (2) 50 (1994), 404-416.
* [3] S. Carter, Z. Senturk and A. West, The push-out space of a submanifold, Geometry and Topology of submanifolds VI, (1994),50-57.
* [4] S. Carter and A. West, partial tubes about immersed manifolds, Geom. Dedicata 54 (1995), 145-169.
|
arxiv-papers
| 2013-04-17T19:23:32 |
2024-09-04T02:49:44.556986
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Morteza Fathy and Morteza Faghfouri",
"submitter": "Morteza Faghfouri",
"url": "https://arxiv.org/abs/1304.4924"
}
|
1304.5016
|
# Note on Bessel functions of type $A_{N-1}$.
Béchir Amri
(University of Tunis, Preparatory Institut of Engineer Studies of Tunis,
Department of Mathematics, 1089 Montfleury Tunis, Tunisia
[email protected] )
###### Abstract
Through the theory of Jack polynomials we give an iterative method for
integral formula of Bessel function of type $A_{N-1}$ and a partial product
formula for it. 111Key words and phrases: Dunkl operators, Heckman-Opdam
polynomials, Jack polynomials. 2010 Mathematics Subject Classification:
33C52,33C67, 05E05. Author partially supported by DGRST project 04/UR/15-02
and CMCU program 10G 1503.
## 1 Introduction and backgrounds
Dunkl operators which were first introduced by C. F. Dunkl [6] in the late
80ies are commuting differential-difference operators, associated to a finite
reflection groups on a Euclidean space. Their eigenfunctions are called Dunkl
kernels and appear as a generalization of the exponential functions. Although
attempts were made to study them and except the reflection group
$\mathbb{Z}_{2}^{N}$ the explicit forms or behaviors of these kernels are
remain unknown. In the present work we will be concerned with generalized
Bessel functions $J_{k}$ defined through symmetrization of Dunkl kernels in
the case of the symmetric group $S_{N}$. We will obtain the following
$\displaystyle
J_{k}(\mu,\lambda)=\int_{\mathbb{R}^{N-1}}e^{\langle\mu,x\rangle}\delta_{k}(\lambda,x)dx.$
(1.1)
where the function $\delta_{k}$ can be explicitly computed using a recursive
formula on the dimension $N$. The key ingredient is the integral formula of A.
Okounkov and G. Olshanski [10] for Jack polynomials. As the last are connected
with Heckman-opdam-Jacobi polynomials [2] the formula (1.1) follow by limit
transition. We should note here that when $N=3$, the formula (1.1) is
comparable to that obtained by C. F. Dunkl [5] for intertwining operator.
Let us start with some well-known facts about Heckman Opdam Jacobi
polynomials, Jack polynomials and Dunkl kernels associated with a root system
$R$. The standard references are [2, 4, 8, 11, 16, 15]. Here $\mathbb{R}^{N}$
is equipped with the usual inner product $\langle,\;.\;,\rangle$ and the
canonical orthonormal basis $(e_{1},e_{2},...,e_{N})$. Further, we shall
assume that $R$ is reduced and crystallographic, that is a finite subset of
$\mathbb{R}^{N}\backslash\\{0\\}$ which satisfies:
(i) $R$ spanned $\mathbb{R}^{N}$.
(ii) $R$ is invariant under $r_{\alpha}$ the reflection in the hyperplane
orthogonal to any $\alpha\in R$.
(iii) $\alpha.\mathbb{R}\cap R=\\{\pm\alpha\\}$ for all $\alpha\in R$
(iii) for all $\alpha,\;\beta\in R$;
$\langle\alpha,\breve{\beta}\rangle\in\mathbb{Z}$,
$\breve{\beta}=\frac{2\beta}{\|\beta\|^{2}}$
We assume that the reader is familiar with the basics of root systems and
their Weyl groups, see for examples Humphreys [9].
### 1.a Heckman Opdam Jacobi polynomials.
Let $R$ be a reduced root system with $\\{\alpha_{1},...,\alpha_{N}\\}$ be a
basis of simple roots and $R_{+}$ be the set of positive roots determined by
this basis. The fundamental weights $\\{\beta_{1},...,\beta_{N}\\}$ are given
by $\langle\;\beta_{j},\;\check{\alpha}_{i}\;\rangle=\delta_{i,j}$,
$\displaystyle{\check{\alpha_{i}}=\frac{2\alpha_{i}}{\|\alpha_{i}\|^{2}}}$.
Let $\displaystyle Q=\bigoplus_{i=1}^{N}\mathbb{Z}\alpha_{i}$, $\displaystyle
P=\bigoplus_{i=1}^{N}\mathbb{Z}\beta_{i}$, $\displaystyle
Q^{+}=\bigoplus_{i=1}^{N}\mathbb{N}\alpha_{i}$ and $\displaystyle
P^{+}=\bigoplus_{i=1}^{N}\mathbb{N}\beta_{i}$. We define a partial ordering on
$P$ by $\lambda\preceq\mu$ if $\mu-\lambda\in Q^{+}$
The group algebra $\mathbb{C}[P]$ of the free Abelian group $P$ is the algebra
generated by the formal exponentials $e^{\lambda}$, $\lambda\in P$ subject to
the multiplication relation $e^{\lambda}e^{\mu}=e^{\lambda+\mu}$. The Weyl
group W acts on $\mathbb{C}[P]$ by $we^{\lambda}=e^{w\lambda}$. The orbit-sums
$\displaystyle m_{\lambda}=\sum_{\mu\in W.\lambda}e^{\mu}$, $\lambda\in P^{+}$
form a basis of $\mathbb{C}[P]^{W}$, the subalgebra of $W$-invariant elements
of $\mathbb{C}[P]$. Here $W.\lambda$ denotes the W-orbit of $\lambda$.
Let $\mathbb{T}=\mathbb{R}^{d}/2\pi\check{Q}$ where
$\displaystyle\check{Q}=\bigoplus_{i=1}^{d}\mathbb{Z}\check{\alpha_{i}}$. The
algebra $\mathbb{C}[P]$ can be realized explicitly as the algebra of
polynomials on the torus $\mathbb{T}$ through the identification
$e^{\lambda}(\dot{x})=e^{i\langle\lambda,x\rangle}$ where
$\dot{x}\in\mathbb{T}$ is the image of $x\in\mathbb{R}^{d}$. Let
$k:R\rightarrow[0,+\infty[$ be a $W$-invariant function, called multiplicity
function. We equip $C[P]^{W}$ with the inner product
$(f,g)_{k}=\int_{\mathbb{T}}f(x)\overline{g(x)}\delta_{k}(x)dx$
where
$\delta_{k}=\prod_{\alpha\in
R^{+}}\left|e^{\frac{\alpha}{2}}-e^{-\frac{\alpha}{2}}\right|^{2k_{\alpha}}$
and $dx$ is the Haar measure on $\mathbb{T}$.
The Heckman Opdam Jacobi polynomials are introduced by Heckman and Opdam [8]
as the unique family of elements $P_{\lambda}\in\mathbb{C}[P]^{W}$,
$\lambda\in P^{+}$ satisfying the following conditions:
* (i)
$P_{\lambda}=m_{\lambda}+\sum_{\mu\prec\lambda}a_{\lambda\mu}m_{\mu}$
* (ii)
$\langle P_{\lambda},m_{\mu}\rangle=0$ if $\mu\in P_{+}$, $\lambda\prec\mu$.
( Note that in [8], these polynomials are indexed by $-P_{+}$ instead of
$P_{+}$ ). They form an orthogonal basis of $\mathbb{C}[P]^{W}$ and satisfy
the second differential equation
$\Big{(}\Delta+\sum_{\alpha\in R_{+}}k_{\alpha}\coth(\frac{1}{2}\langle
x,\alpha\rangle)\partial_{\alpha}\Big{)}P_{\lambda}(x)=\langle\lambda,\lambda+\sum_{\alpha\in
R_{+}}k_{\alpha}\alpha\rangle P_{\lambda}(x).$
where $\Delta$ is the Laplace operator on $\mathbb{R}^{N}$.
The Cherednik operator $T_{\xi}$, $\xi\in\mathbb{R}^{N}$, associated with the
root system $R$ and the multiplicity $k$ is defined by
$T_{\xi}^{k}=\partial_{\xi}+\sum_{\alpha\in
R_{+}}k_{\alpha}\langle\alpha,\;\xi\rangle\;\frac{1-r_{\alpha}}{1-e^{{}^{\alpha}}}-\langle\rho_{k},\;\xi\rangle,$
where $\displaystyle{\rho_{k}=\frac{1}{2}\sum_{\alpha\in
R_{+}}k_{\alpha}\alpha}$. The hypergeometric function $F_{k}$ is defined as
the unique holomorphic W-invariant function on
$\mathbb{C}^{N}\times(\mathbb{R}^{N}+iU)$ ( U is a W-invariant neighborhood of
$0$ ) which satisfies the system of differential equations:
$p(T_{e_{1}},...T_{e_{N}})F_{k}(\lambda,.)=p(\lambda)F_{k}(\lambda,.);\qquad
F(\lambda,0)=1$
for all $\lambda\in\mathbb{C}^{N}$ and all $W$-invariant polynomial $p$ on
$\mathbb{R}^{N}$. The Heckman opdam Jacobi polynomials are related to the
hypergeometric function $F_{k}$ by ( see [7] )
$\displaystyle
F_{k}(\lambda+\rho_{k},x)=c(\lambda+\rho_{k})P_{\lambda}(x);\qquad\lambda\in
P^{+},\;x\in\mathbb{R}^{N},$ (1.2)
where the function $c$ is given on $\mathbb{R}^{N}$ by
$\displaystyle c(\lambda)=\prod_{\alpha\in
R^{+}}\frac{\Gamma(\langle\lambda,\check{\alpha}\rangle)\Gamma(\langle\rho,\check{\alpha}\rangle+k_{\alpha})}{\Gamma(\langle\lambda,\check{\alpha}\rangle+k_{\alpha})\Gamma(\langle\rho,\check{\alpha}\rangle)}.$
(1.3)
### 1.b Jack polynomials
Let $k>0$, the symmetric group $S_{N}$ acts on the ring of polynomials
$\mathbb{Q}(k)[x_{1},...,x_{N}]$ by
$\tau p(x_{1},...,x_{N})=p(x_{\tau(1)},...,x_{\tau(N)})$
Let $\Lambda_{N}$ the subspace of symmetric polynomials,
$\Lambda_{N}=\\{p\in\mathbb{C}[x_{1},...,x_{N}],\;\tau p=p,\;\forall\tau\in
S_{N}\\}.$
We call partition all $\lambda=(\lambda_{1},...\lambda_{N})\in\mathbb{N}^{N}$
such that $\lambda_{1}\geq...\geq\lambda_{N}$. The weight of a partition
$\lambda$ is the sum $|\lambda|=\lambda_{1}+...+\lambda_{N}$ and its length
$\ell(\lambda)=\max\left\\{j;\;\lambda_{j}\neq 0\right\\}.$ The set of all
partitions are partially ordered by the dominance order:
$\lambda\leq\mu\Leftrightarrow|\lambda|=|\mu|\quad\text{
and}\quad\lambda_{1}+\lambda_{2}+...+\lambda_{i}\leq\mu_{1}+\mu_{2}+...+\mu_{i}$
for all $i=1,2,...,N$. The simplest basis of $\Lambda_{N}$ is given by the
monomial symmetric polynomials,
$m_{\lambda}(x)=\sum_{\mu\in S_{N}\lambda}x_{1}^{\mu_{1}}...x_{N}^{\mu_{N}}.$
We define an inner product on $\Lambda_{N}$ by
$\langle
f,g\rangle_{k}=\int_{T}f(z)\overline{g(z)}\prod_{i<j}|z_{i}-z_{j}|^{2k}dz$
where $T=\\{(z_{1},...,z_{N})\in\mathbb{C}^{N};|z_{j}|=1,\;\forall\;1\leq
j\leq N\\}$ is the $N$-dimensional torus and $dz$ is the haar measure on $T$.
Jack symmetric polynomials $j_{\lambda}$ indexed by a partitions $\lambda$ can
be defined as the unique polynomials such that
* (i)
$j_{\lambda}=m_{\lambda}+\sum_{\mu\prec\lambda}m_{\mu}$,
* (ii)
$\langle j_{\lambda},m_{\mu}\rangle_{k}=0$ if $\lambda\leq\mu$.
By a result of I. G. Macdonald ([13], p: 383 ) they form a family of
orthogonal polynomials. Jack polynomials can be defined as eigenfunctions of
certain Laplac-Beltrami type operator ( coming in the theory of Calogero
integrable systems and in random matrix theory ),
$L_{k}=\sum_{i=1}^{d}x_{i}^{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+2k\sum_{i\neq
j}\frac{x_{i}^{2}}{x_{i}-x_{j}}\frac{\partial}{\partial x_{i}}.$
Jack polynomials $j_{\lambda}$ are homogeneous of degree $|\lambda|$ and
satisfy the compatibility relation
$\displaystyle
j_{(\lambda_{1},...,\lambda_{N-1},0)}(x_{1},...,x_{N-1},0)=j_{(\lambda_{1},...,\lambda_{N-1})}(x_{1},...,x_{N-1}).$
(1.4)
The relationship between Heckman Opdam Jacobi polynomials and Jack polynomials
can be illustrated as follows ( see [2] ): Let $\mathbb{V}$ be the hyperplane
orthogonal to the vector $e=e_{1}+...+e_{N}$. In $V$ we consider the root
system of type $A_{N-1}$,
$R_{A}=\\{\pm(e_{i}-e_{j}),\;1\leq i<j\leq N\\}.$
The fundamental weights are given by $\pi_{N}(\omega_{i})$,
$\omega_{i}=e_{1}+...+e_{i}$, where $\pi_{N}$ denote the orthogonal projection
along $e$ onto V,
$\pi_{N}(x)=x-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right)e=\left(x_{1}-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right),...,x_{N}-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right)\right)$
and then $P_{A}^{+}=\\{\pi_{N}(\lambda),\lambda\;\text{partition}\\}$. The
result is that:
$\displaystyle j_{\lambda}(e^{x})=P_{\pi_{N}(\lambda)}(x),$ (1.5)
For all partition $\lambda$ and all $x\in\mathbb{V}$ with
$e^{x}=(e^{x_{1}},...,e^{x_{N}})$.
### 1.c Dunkl kernels and Dunkl-Bessel functions
The Dunkl operator $D_{\xi}$, $\xi\in\mathbb{R}^{N}$ associated with a root
system $R$ and a multiplicity function $k$ is defined by
$D_{\xi}=\partial_{\xi}+\sum_{\alpha\in
R^{+}}k(\alpha)\langle\alpha,\xi\rangle\frac{1-r_{\alpha}}{\langle\alpha,.\rangle}.$
The Dunkl intertwining operator $V_{k}$ is the unique isomorphism on the
polynomials space $\mathbb{C}[\mathbb{R}^{N}]$ such that
$V_{k}(1)=1,\quad V_{k}(\mathcal{P}_{n})=\mathcal{P}_{n}\quad\text{and}\quad
D_{\xi}V_{k}=V_{k}\partial_{\xi}$
where $\mathcal{P}_{n}$ is the subspace of homogeneous polynomials of degree
$n\in\mathbb{N}$. For $r>0$ , $V_{k}$ extends to a continuous linear operator
on the Banach space
$A_{r}=\\{f=\sum_{n=0}^{\infty}f_{n},\;f_{n}\in\mathcal{P}_{n},\;\|f\|_{A_{r}}=\sum_{n=0}^{\infty}\sup_{|x|\leq
r}|f_{n}(x)|<\infty\\}$
by
$V_{k}(f)=\sum_{n=0}^{\infty}V_{k}(f_{n}).$
A remarkable result due to M. Rösler [14] says that for each
$x\in\mathbb{R}^{N}$,
$V_{k}(f)(x)=\int_{\mathbb{R}^{d}}f(\xi)d\mu_{x}(\xi)$
where $\mu_{x}$ is a probability measure supported in $co(x)$ the convex hull
of the orbit W.x.
The Dunkl kernel $E_{k}$ is given by
$\displaystyle
E_{k}(x,y)=V_{k}(e^{\langle\;.\;,\;y\;\rangle})(x)=\int_{\mathbb{R}^{d}}e^{\langle\xi,y\rangle}d\mu_{x}(\xi),\quad
x\in\mathbb{R}^{N},\;y\in\mathbb{C}^{N}$
and having the following properties:
* (i)
For each $y\in\mathbb{C}^{N}$ the function $E_{k}(.,y)$ is the unique solution
of eigenvalue problem:
$D_{\xi}f(x)=\langle\xi,y\rangle
f(x)\;\forall\;\quad\xi\in\mathbb{R}^{N}\;\text{and}\;f(0)=1.$
* (ii)
$E_{k}$ extends to a holomorphic function on
$\mathbb{C}^{N}\times\mathbb{C}^{N}$ and for all
$(x,y)\in\mathbb{C}^{N}\times\mathbb{C}^{N}$, $w\in W$ and $t\in\mathbb{C}$:
$E_{k}(x,y)=E_{k}(y,x),\quad E_{k}(wx,wy)=E_{k}(x,y)\quad\text{and}\quad
E_{k}(x,ty)=E_{k}(tx,y)$
We define the Bessel function associated with $R$ and $k$ by,
$\displaystyle J_{k}(x,y)=\frac{1}{|W|}\sum_{w\in W}E_{k}(x,wy).$
The limit transition between hypergeometric functions $F_{k}$ and Dunkl Bessel
function is expressed by ( see (2.21) of [14] )
$\displaystyle
J_{k}(x,y)=\lim_{n\rightarrow+\infty}F_{k}(nx+\rho_{k},\frac{y}{n})\;.$ (1.6)
According to these preliminaries we can now formulate the main result of this
note.
## 2 Integral formula for $J_{k}$
The starting point is the following remarkable integral identity obtained by
[10] which connecting jack polynomials of $N$ variables to Jack polynomials of
$N-1$ variables. For $\lambda=(\lambda_{1},...,\lambda_{N})\in\mathbb{R}^{N}$
we use the notation $|\lambda|=\lambda_{1}+...+\lambda_{N}$.
###### Proposition 1 ([10]).
Suppose that the partition $\mu$ has less than $N$ parts and
$\lambda\in\mathbb{R}^{N}$ such that $\lambda_{1}\geq...\geq\lambda_{N}$. Then
$\displaystyle
j_{\mu}(\lambda)=\frac{1}{U(\mu)V(\lambda)^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}j_{\mu}(\nu)V(\nu)\Pi(\lambda,\nu)d\nu$
(2.1)
where
$U(\mu)=\prod_{j=1}^{N-1}\beta(\mu_{j}+(N-j)k,k),\quad V(\lambda)=\prod_{1\leq
i<j\leq N}(\lambda_{i}-\lambda_{j})$
and
$\Pi(\lambda,\nu)=\prod_{i\leq
j}(\lambda_{i}-\nu_{j})^{k-1}\prod_{i>j}(\nu_{j}-\lambda_{i})^{k-1}.$
We follow three simple steps that lead to our formula. All functions of $N$
variables will be indexed by $N$ and by $N-1$ if it considered as $N-1$
variables.
Step 1: For any partition $\mu=(\mu_{1},...,\mu_{N})$ we set
$\widetilde{\mu}=(\mu_{1}-\mu_{N},...,\mu_{N-1}-\mu_{N},0)\quad\text{and}\quad\overline{\mu}=(\mu_{1}-\mu_{N},...,\mu_{N-1}-\mu_{N})\in\mathbb{R}^{N-1}.$
By Homogeneity of Jack polynomials we have that
$j_{\mu,N}(\lambda)=\left(\prod_{j=1}^{N}\lambda_{j}\right)^{\mu_{N}}j_{\widetilde{\mu},N}(\lambda)$
and from (2.1) and (1.4) we may write
$j_{\mu,N}(\lambda)=\frac{\left(\prod_{j=1}^{N}\lambda_{j}\right)^{\mu_{N}}}{U_{N}(\widetilde{\mu})V_{N}(\lambda)^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}j_{\overline{\mu},N-1}(\nu)V(\nu)\Pi(\lambda,\nu)d\nu.$
Taking $\lambda$ in $\mathbb{V}$ and making use a change of variables we get
that
$\displaystyle j_{\mu,N}(e^{\lambda})$ $\displaystyle=$
$\displaystyle\frac{1}{U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|}j_{\overline{\mu},N-1}(e^{\nu})V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu$
$\displaystyle=$
$\displaystyle\frac{1}{U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|(1+\frac{|\overline{\mu}|}{N-1})}j_{\overline{\mu},N-1}(e^{\pi_{N-1}(\nu)})V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu.$
In order by (1.2), (1.5) and (1.3) we have
$F_{N}(\pi_{N}(\mu)+\rho_{k,N},\lambda)=c_{N}(\pi_{N}(\mu)+\rho_{k,N})j_{\mu,N}(e^{\lambda})=c_{N}(\mu+\rho_{k,N})j_{\mu,N}(e^{\lambda}),$
where here
$\rho_{k,N}=\frac{k}{2}\sum_{i=1}^{N}(N-2i+1)e_{i}=\left(\frac{k(N-1)}{2},...,\frac{k(N-2i+1)}{2},...,\frac{-k(N-1)}{2}\right)\in\mathbb{R}^{N}$
and
$c_{N}(\mu+\rho_{k,N})=\prod_{1\leq i<j\leq
N}\frac{\Gamma(\mu_{i}-\mu_{j})\Gamma(k(j-i+1))}{\Gamma(\mu_{i}-\mu_{j}+k)\Gamma(k(j-i))}.$
Therefore,
$\displaystyle
F_{N}(\pi_{N}(\mu)+\rho_{k,N},\lambda)=\frac{c_{N}(\mu+\rho_{k,N})}{c_{N-1}(\overline{\mu}+\rho_{k,N-1})U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|(1+\frac{|\overline{\mu}|}{N-1})}F_{N-1}(\pi_{N-1}(\overline{\mu})+\rho_{k,N-1},\pi_{N-1}(\nu))V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu.$
Step 2: Now we apply (1.6), by using the following when $n\rightarrow+\infty$
$\displaystyle U_{N}(n\widetilde{\mu})\sim
n^{-k(N-1)}\Gamma(k)^{N-1}\prod_{j=1}^{N-1}(\mu_{j}-\mu_{N})^{-k},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle V_{N}(e^{\frac{\lambda}{n}})\sim
n^{-\frac{N(N-1)}{2}}V_{N}(\lambda),$ $\displaystyle
c_{N}(n\mu+\rho_{k,N})\sim n^{\frac{-kN(N-1)}{2}}V_{N}(\mu)^{-k}\prod_{1\leq
i<j\leq N}\frac{\Gamma(k(j-i+1))}{\Gamma(k(j-i))},$ $\displaystyle
c_{N-1}(n\overline{\mu}+\rho_{k,N-1})\sim
n^{\frac{-k(N-1)(N-2)}{2}}V_{N-1}(\overline{\mu})^{-k}\prod_{1\leq i<j\leq
N-1}\frac{\Gamma(k(j-i+1))}{\Gamma(k(j-i))},$
$\displaystyle\frac{c_{N}(n\mu+\rho_{k,N})}{c_{N-1}(n\overline{\mu}+\rho_{k,N-1})}\sim
n^{-k(N-1)}\frac{\Gamma(Nk)}{\Gamma(k)}\prod_{j=1}^{N-1}(\mu_{j}-\mu_{N})^{-k},$
$\displaystyle\Pi_{N}(e^{\frac{\lambda}{n}},e^{\frac{\nu}{n}})\sim
n^{-N(N-1)(k-1)}\Pi(\lambda,\nu).$
Thus
$\displaystyle J_{k,N}(\pi_{N}(\mu),\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}$
(2.2) $\displaystyle\qquad\qquad
e^{|\overline{\mu}|\frac{|\nu|}{N-1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$
Step 3:
The formula (2.2) is valid only for a partition $\mu$, to keep it for any
$\mu\in\mathbb{R}^{N}$ we proceed as follows. Let $r\in(0,+\infty)$ and $\mu$
be a partition. We obtain after a change of variables
$\displaystyle J_{k,N}(\pi_{N}(r\mu),\lambda)=J_{k,N}(\pi_{N}(\mu),r\lambda)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\overline{r\mu}|\frac{|\nu|}{N-1}}$
$\displaystyle\qquad\qquad
J_{k,N-1}(\pi_{N-1}(\overline{r\mu}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$
Since the set $\\{r\mu;\quad r\in(0,+\infty),\quad\mu\;\text{partitions}\;\\}$
is dense in the set
$H=\\{\mu\in\mathbb{R}^{N},\quad 0\leq\mu_{N}\leq...\leq\mu_{1}\\}$
and $J_{k,N}$ is $S_{N}$-invariant continuous function then (2.2) can be
extended to all $\mu\in H$. Now for $\mu\in\mathbb{R}^{N}$ we denote by
$\mu^{+}$ the unique element of $S_{N}.\mu$ so that
$\mu^{+}_{N}\leq...\leq\mu^{+}_{1}$. So we have
$J_{k,N}(\pi_{N}(\mu),\lambda)=J_{k,N}(\pi_{N}(\mu^{+}),\lambda)=J_{k,N}(\pi_{N}(\widetilde{\mu^{+}}),\lambda)$
and since $\widetilde{\mu^{+}}\in H$ then
$\displaystyle J_{k,N}(\pi_{N}(\mu),\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}$
$\displaystyle\qquad\qquad
e^{|\overline{\mu^{+}}|\frac{|\nu|}{N-1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu^{+}}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$
Now when restricted to the space $\mathbb{V}$ we state the following.
###### Theorem 1.
For all $\mu,\lambda\in\mathbb{V}$ we have
$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}^{+}}^{\lambda_{1}^{+}}...\int_{\lambda_{N}^{+}}^{\lambda_{N-1}^{+}}e^{|\overline{\mu^{+}}|\frac{|\nu|}{N-1}}$
(2.3) $\displaystyle\qquad\qquad
J_{k,N-1}(\pi_{N-1}(\overline{\mu^{+}}),\pi_{N-1}(\nu))V_{N-1}(\nu)\Pi_{N}(\lambda^{+},\nu)d\nu.$
In what follows, we shall restrict ourselves to the case $N=2,3,4$ where we
give representations of $J_{k,N}$ as Laplace-type integrals,
$\displaystyle
J_{k,N}(\mu,\lambda)=\int_{\mathbb{R}^{N}}e^{\langle\mu,x\rangle}d\nu_{\lambda}(x).$
where $\nu_{\lambda}$ is a probability measure supported in the convex hall of
the orbit $S_{N}.\lambda$.
### 2.a Bessel function of type $A_{1}$
When $N=2$ we have that $\mathbb{V}=\mathbb{R}(e_{1}-e_{2})$,
$\mu^{+}=(|\mu_{1}|,-|\mu_{1}|)$, $\overline{\mu^{+}}=2|\mu_{1}|$ and
$\lambda^{+}=(|\lambda_{1}|,-|\lambda_{1}|)$. It is obvious that $J_{k,1}=1$,
so we get from (2.3)
$\displaystyle J_{k,2}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{(\Gamma(k))^{2}(2|\lambda_{1}|)^{2k-1}}\int_{-|\lambda_{1}|}^{|\lambda_{1}|}e^{2|\mu_{1}|\nu}(\lambda_{1}^{2}-\nu^{2})^{k-1}d\nu$
$\displaystyle=$
$\displaystyle\frac{\Gamma(k+\frac{1}{2})}{\sqrt{\pi}\Gamma(k)}\int_{-1}^{1}e^{(2|\mu_{1}||\lambda_{1}|\nu}(1-\nu^{2})^{k-1}d\nu$
$\displaystyle=$
$\displaystyle\mathcal{J}_{k-\frac{1}{2}}(2\mu_{1}\lambda_{1})$
where $\mathcal{J}_{k-\frac{1}{2}}$ is the modified Bessel function given by
$\mathcal{J}_{k-\frac{1}{2}}(z)=\Gamma(k+\frac{1}{2})\sum_{n=0}^{\infty}\frac{1}{n!\Gamma(n+k+\frac{1}{2})}(\frac{z}{2})^{2n}.$
However, it is usual to identify $\mathbb{V}=\mathbb{R}\varepsilon$,
$\varepsilon=\frac{e_{1}-e_{2}}{\sqrt{2}}$ with $\mathbb{R}$ and write
$\displaystyle
J_{k,2}(\mu,\lambda)=\mathcal{J}_{k-\frac{1}{2}}(\mu\lambda),\qquad\mu,\lambda\in\mathbb{R}.$
### 2.b Bessel function of type $A_{2}$
Let $\mu=(\mu_{1},\mu_{2},\mu_{3})$ and
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})$ in the fundamental Weyl
chamber
$C=\\{(u_{1},u_{2},u_{3});\quad u_{1}\geq u_{2}\geq u_{3},\quad
u_{1}+u_{2}+u_{3}=0\\}.$
With $\overline{\mu}=(\mu_{1}-\mu_{3},\mu_{2}-\mu_{3},)$ and
$\pi_{2}(\overline{\mu})=(\frac{\mu_{1}-\mu_{2}}{2},\frac{\mu_{2}-\mu_{1}}{2})$
the formula (2.3) gives
$\displaystyle J_{k,3}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(3k)}{V(\lambda)^{2k-1}\Gamma(k)^{3}}\int_{\lambda_{2}}^{\lambda_{1}}\int_{\lambda_{3}}^{\lambda_{2}}e^{\frac{(\mu_{1}+\mu_{2}-2\mu_{3})(\nu_{1}+\nu_{2})}{2}}\mathcal{J}_{k-\frac{1}{2}}(\frac{(\mu_{1}-\mu_{2})(\nu_{1}-\nu_{2})}{2})(\nu_{1}-\nu_{2})$
$\displaystyle\qquad\Big{(}(\lambda_{1}-\nu_{1})(\lambda_{1}-\nu_{2})(\lambda_{2}-\nu_{2})(\nu_{1}-\lambda_{2})(\nu_{1}-\lambda_{3})(\nu_{2}-\lambda_{3})\Big{)}^{k-1}d\nu_{1}d\nu_{2}.$
Using the change of variables: $x=\frac{\nu_{1}+\nu_{2}}{2}$,
$z=\frac{\nu_{1}-\nu_{2}}{2}$ we have
$\displaystyle J_{k,3}(\mu,\lambda)=$
$\displaystyle\frac{4\Gamma(3k)}{V(\lambda)^{2k-1}\Gamma(k)^{3}}\int_{\mathbb{R}}\int_{\mathbb{R}}ze^{(\mu_{1}+\mu_{2}-2\mu_{3})x}\mathcal{J}_{k-\frac{1}{2}}(\mu_{1}-\mu_{2})z)\;\chi_{[\lambda_{2},\lambda_{1}]}(x+z)\;\chi_{[\lambda_{3},\lambda_{2}]}(x-z)$
(2.4)
$\displaystyle\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}dxdz.$
Now recall that
$\displaystyle\mathcal{J}_{k-\frac{1}{2}}((\mu_{1}-\mu_{2})z)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{2^{2k-1}\Gamma(k)^{2}}\int_{\mathbb{R}}e^{(\mu_{1}-\mu_{2})zt}(1-t^{2})^{k-1}\chi_{[-1,1]}(t)dt.$
(2.5) $\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{2^{2k-1}\Gamma(k)^{2}}\int_{\mathbb{R}}e^{(\mu_{1}-\mu_{2})y}(1-\frac{y^{2}}{z^{2}})^{k-1}\chi_{[-1,1]}(\frac{y}{z})z^{-1}dy$
then inserting (2.5) in (2.4) with the use of Fubini’s Theorem we can write
$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{(\mu_{1}+\mu_{2}-2\mu_{3})x+(\mu_{1}-\mu_{2})y}\Delta_{k}(\lambda,x,y)dxdy$
where
$\displaystyle\Delta_{k}(\lambda,x,y)=$
$\displaystyle\frac{4\Gamma(2k)\Gamma(3k)}{2^{2k-3}\Gamma(k)^{5}V(\lambda)^{2k-1}}\int_{\mathbb{R}}\left(\frac{z^{2}-y^{2}}{z^{2}}\right)^{k-1}\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\chi_{[-1,1]}(\frac{y}{z})\chi_{[\lambda_{1},\lambda_{2}]}(x+z)\chi_{[\lambda_{3},\lambda_{2}]}(x-z)dz$
We note here that
$\displaystyle\chi_{[-1,1]}(\frac{y}{z})\chi_{[\lambda_{1},\lambda_{2}]}(x+z)\chi_{[\lambda_{3},\lambda_{2}]}(x-z)=\chi_{\max(|y|,|x-\lambda_{2}|)\leq
z\leq\min(x-\lambda_{3},\lambda_{1}-x)}.$
Thus we have
$\displaystyle\Delta_{k}(\lambda,x,y)=$
$\displaystyle\frac{4\Gamma(2k)\Gamma(3k)}{2^{2k-3}\Gamma(k)^{5}V(\lambda)^{2k-1}}\int_{\max(|y|,|x-\lambda_{2}|)}^{\min(x-\lambda_{3},\lambda_{1}-x)}\left(\frac{z^{2}-y^{2}}{z^{2}}\right)^{k-1}\qquad\qquad$
$\displaystyle\qquad\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}dz$
if
$\displaystyle\max(|y|,|x-\lambda_{2}|)\leq\min(x-\lambda_{3},\lambda_{1}-x)$
and $\Delta_{k}(\lambda,x,y)=0$, otherwise. Making the change of variables
$\nu_{1}=x+y,\qquad\nu_{2}=x-y$
and put $\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ with
$\nu_{3}=-(\nu_{1}+\nu_{2})$ we obtain
$\displaystyle
J_{3}^{k}(\mu,\lambda)=\frac{1}{2}\int_{\mathbb{R}^{2}}e^{\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}}\Delta_{k,2}\left(\lambda,\frac{\nu_{1}+\nu_{2}}{2},\frac{\nu_{1}-\nu_{2}}{2}\right)d\nu_{1}d\nu_{2}$
But we can identify $\mathbb{R}^{2}$ with the space $\mathbb{V}$ via the basis
$(e_{1}-e_{2},e_{2}-e_{3})$, since for
$\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ we have
$\nu=\nu_{1}(e_{1}-e_{2})+\nu_{2}(e_{1}-e_{3})$. Then we get
$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}^{2}}e^{\langle\mu,\nu\rangle}\delta_{k,2}\left(\lambda,\nu\right)d\nu_{1}d\nu_{2}.$
(2.6)
with
$\displaystyle\delta_{k,2}(\lambda,\nu)=\frac{1}{2}\Delta_{k,2}\left(\lambda,\frac{\nu_{1}+\nu_{2}}{2},\frac{\nu_{1}-\nu_{2}}{2}\right).$
Now considering the orthonormal basis $(\varepsilon_{1},\varepsilon_{2})$ of
$\mathbb{V}$,
$\varepsilon_{1}=\frac{1}{\sqrt{6}}(e_{1}+e_{2}-2e_{3}),\quad\varepsilon_{2}=\frac{1}{\sqrt{2}}(e_{1}-e_{2})$
we can write
$\mu=\frac{(\mu_{1}+\mu_{2}-2\mu_{3})}{\sqrt{6}}\;\varepsilon_{1}+\frac{\mu_{1}-\mu_{2}}{\sqrt{2}}\;\varepsilon_{2}$
and for $x=x_{1}\varepsilon_{1}+x_{2}\varepsilon_{2}$
$\langle\mu,x\rangle=\frac{(\mu_{1}+\mu_{2}-2\mu_{3})}{\sqrt{6}}x_{1}+\frac{\mu_{1}-\mu_{2}}{\sqrt{2}}x_{2}$
Then using change of variables $x_{1}=\sqrt{6}\;x$ and $x_{2}=\sqrt{2}\;y$ in
the formula (LABEL:k3) we obtain
$\displaystyle
J_{3}^{k}(\mu,\lambda)=\frac{1}{\sqrt{12}}\int_{\mathbb{R}^{2}}e^{\langle\mu,x\rangle}\Delta_{k}\left(\lambda,\frac{x_{1}}{\sqrt{6}},\frac{x_{2}}{\sqrt{2}}\right)dx_{1}dx_{2}.$
###### Proposition 2.
For all $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in C$ the function
$\delta_{k,2}\left(\lambda,.\right)$ is supported in the closed convex hull
$co(\lambda)$ of the $S_{3}$-orbit of $\lambda$, described by:
$\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ such that
$\displaystyle\lambda_{3}\leq\min(\nu_{1},\nu_{2},\nu_{3})\leq\max(\nu_{1},\nu_{2},\nu_{3})\leq\lambda_{1}.$
###### Proof.
In view of (2.b) and (2.b) the support of $\delta_{k,2}\left(\lambda,.\right)$
is contain is the set
$\left\\{\nu\in\mathbb{V};\quad\max\left(\frac{|\nu_{1}-\nu_{2}|}{2},\left|\frac{\nu_{1}+\nu_{2}}{2}-\lambda_{2}\right|\right)\leq\min\Big{(}\frac{\nu_{1}+\nu_{2}}{2}-\lambda_{3},\lambda_{1}-\frac{\nu_{1}+\nu_{2}}{2}\Big{)}\right\\}$
which by straightforward calculus reduced to the set
$\\{\nu\in\mathbb{V};\quad\lambda_{3}\leq\min(\nu_{1},\nu_{2},\nu_{3})\leq\max(\nu_{1},\nu_{2},\nu_{3})\leq\lambda_{1}\\}.$
However, we known that
$\nu\in
co(\lambda)\quad\Leftrightarrow\quad\lambda^{+}-\nu^{+}\in\bigoplus_{i=1}^{N}\mathbb{R}_{+}\alpha_{i}$
and here
$\lambda^{+}-\nu^{+}=\lambda-\nu^{+}=(\lambda_{1}-\nu^{+}_{1})(e_{1}-e_{2})+(\nu^{+}_{3}-\lambda_{3})(e_{2}-e_{3})$
Then
$\nu\in
co(\lambda)\quad\Leftrightarrow\quad\nu^{+}_{1}\leq\lambda_{1}\quad\text{and}\quad\nu^{+}_{3}\geq\lambda_{3},$
which proves the proposition, since
$\nu^{+}_{1}=\max(\nu_{1},\nu_{2},\nu_{3})$ and
$\nu^{+}_{3}=\min(\nu_{1},\nu_{2},\nu_{3})$. ∎
### 2.c Bessel function of type $A_{3}$
Let $\mu,\lambda\in C$, the Weyl chamber. We have
$\displaystyle|\overline{\mu}|$ $\displaystyle=$
$\displaystyle\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4},$
$\displaystyle\pi_{4}(\overline{\mu})$ $\displaystyle=$
$\displaystyle(\mu_{1}+\frac{\mu_{4}}{3},\mu_{2}+\frac{\mu_{4}}{3},\mu_{3}+\frac{\mu_{4}}{3}),$
$\displaystyle\pi_{4}(\nu)$ $\displaystyle=$
$\displaystyle(\frac{2\nu_{1}-\nu_{2}-\nu_{3}}{3},\frac{2\nu_{2}-\nu_{1}-\nu_{3}}{3},\frac{2\nu_{3}-\nu_{1}-\nu_{2}}{3}).$
Taking (2.3) with the change of variables
$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\frac{\nu_{1}+\nu_{2}+\nu_{3}}{3},$ $\displaystyle x_{1}$
$\displaystyle=$ $\displaystyle\frac{2\nu_{1}-\nu_{2}-\nu_{3}}{3},$
$\displaystyle x_{2}$ $\displaystyle=$
$\displaystyle\frac{2\nu_{2}-\nu_{1}-\nu_{3}}{3}$
and put $x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, $x_{1}+x_{2}+x_{3}=0$, we
have
$\displaystyle
J_{k,4}(\mu,\lambda)=\int_{\mathbb{R}^{3}}e^{(\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}}J_{k,3}(\pi_{3}(\overline{\mu}),x)V_{3}(x)$
$\displaystyle\Pi_{4}(x_{1}+z_{1},x_{2}+z_{1},x_{3}+z_{1},\lambda)\chi_{[\lambda_{2},\lambda_{1}]}(x_{1}+z_{1})\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+z_{1})\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+z_{1})dz_{1}dx_{1}dx_{2}.$
By inserting (2.6)
$\displaystyle J_{k,4}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{5}}e^{\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}+(\mu_{1}-\mu_{3})z_{2}+(\mu_{2}-\mu_{3})z_{3}}\delta_{k,2}((z_{2},z_{3},-(z_{2}+z_{3})),x)V_{3}(x)$
$\displaystyle\Pi_{4}(x_{1}+z_{1},x_{2}+z_{1},x_{3}+z_{1},\lambda)\chi_{[\lambda_{2},\lambda_{2}]}(x_{1}+z_{1})\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+z_{1})\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+z_{1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
dz_{1}dz_{2}dz_{3}dx_{1}dx_{2}.$
Now with the change of variables
$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle z_{1}+z_{2},$
$\displaystyle Z_{2}$ $\displaystyle=$ $\displaystyle z_{1}+z_{3},$
$\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle z_{1}-(z_{2}+z_{3})$
and with $Z=(Z_{1},Z_{2},Z_{3},Z_{4})\in\mathbb{R}^{4}$, such that
$Z_{1}+Z_{2}+Z_{3}+Z_{4}=0$ we have that
$(\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}+(\mu_{1}-\mu_{3})z_{2}+(\mu_{2}-\mu_{3})z_{3}=\mu_{1}Z_{1}+\mu_{2}Z_{2}+\mu_{3}Z_{3}+\mu_{4}Z_{4}=\langle\mu,Z\rangle.$
Therefore we can write
$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}^{3}}e^{\langle\mu,Z\rangle}\delta_{k,3}(Z,\lambda)dZ_{1}dZ_{2}dZ_{3},$
where
$\displaystyle\delta_{k,3}(Z,\lambda)=$
$\displaystyle\int_{\mathbb{R}^{2}}\Pi_{4}(x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),\lambda)$
$\displaystyle\delta_{k,2}(\frac{1}{3}(2Z_{1}-Z_{2}-Z_{3}),\frac{1}{3}(2Z_{2}-Z_{1}-Z_{3}),\frac{1}{3}(2Z_{3}-Z_{1}-Z_{2}),x)\chi_{[\lambda_{2},\lambda_{1}]}(x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))$
$\displaystyle\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))dx_{1}dx_{2}.$
(2.7)
Let us now describe the support of $\delta_{k,3}$. In fact,
$\delta_{k,3}(Z,\lambda)\neq 0$ if the variables $x$ and $Z$ of the integrant
(2.c) satisfy:
$\displaystyle(1)$ $\displaystyle\quad\lambda_{2}\leq
x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{1},$ $\displaystyle(2)$
$\displaystyle\quad\lambda_{3}\leq
x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{2},$ $\displaystyle(3)$
$\displaystyle\quad\lambda_{4}\leq
x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{3},$ $\displaystyle(4)$
$\displaystyle\quad x_{3}\leq\frac{1}{3}(2Z_{1}-Z_{2}-Z_{3})\leq x_{1},$
$\displaystyle(5)$ $\displaystyle\quad
x_{3}\leq\frac{1}{3}(2Z_{2}-Z_{1}-Z_{3})\leq x_{1},$ $\displaystyle(6)$
$\displaystyle\quad x_{3}\leq\frac{1}{3}(2Z_{3}-Z_{1}-Z_{2})\leq x_{1}.$
It Follows that
$\displaystyle(1)+(4)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}\leq\lambda_{1},$ $\displaystyle(1)+(5)\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad Z_{2}\leq\lambda_{1},$ $\displaystyle(1)+(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad Z_{3}\leq\lambda_{1},$
$\displaystyle(1)+(2)+(3)\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad Z_{4}\leq\lambda_{1},$ $\displaystyle(1)+(2)-(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)-(5)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{3}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)-(4)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{3}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(2)+(3)-(4)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{4}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(2)+(3)-(5)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{4}\leq\lambda_{1}+\lambda_{2}$ $\displaystyle(2)+(3)-(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)+(3)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}+Z_{3}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(4)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(5)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(6)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3}.$
These inequalities can be expressed in terms of
$Z^{+}=(Z^{+}_{1},Z^{+}_{2},Z^{+}_{3},Z^{+}_{4})$ as
$\displaystyle Z^{+}_{1}=\max(Z_{1},Z_{2},Z_{3},Z_{4})\leq\lambda_{1}$
$\displaystyle
Z^{+}_{1}+Z_{2}^{+}=\max(Z_{1}+Z_{2},Z_{1}+Z_{3},Z_{1}+Z_{4},Z_{2}+Z_{3},Z_{2}+Z_{4},Z_{3}+Z_{4})\leq\lambda_{1}+\lambda_{2}$
$\displaystyle
Z^{+}_{1}+Z_{2}^{+}+Z^{+}_{3}=\max(Z_{1}+Z_{2}+Z_{3},Z_{2}+Z_{3}+Z_{4},Z_{1}+Z_{2}+Z_{4},Z_{1}+Z_{3}+Z_{4})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\lambda_{1}+\lambda_{2}+\lambda_{3}$
which imply that $Z^{+}\preceq\lambda$ and therefore $Z\in co(\lambda)$.
### 2.d Case for arbitrary $N$
After having idea about the case $N=2,3$ it is not hard to see that the
formula (1.1) can be found using recurrence. In fact, let $\mu,\lambda\in C$
the Weyl chamber. Put for $\nu\in\mathbb{R}^{N-1}$
$\Omega(\lambda,\nu)=\prod_{i=1}^{N-1}\chi_{[\lambda_{i+1},\lambda_{i}]}(\nu).$
With the change of variables
$\displaystyle z_{1}=\frac{|\nu|}{N-1}=\frac{\nu_{1}+...+\nu_{N-1}}{N-1}$
$\displaystyle x_{i}=\nu_{i}-\frac{|\nu|}{N-1};\qquad 1\leq i\leq N-2$
the formula (2.3) becomes,
$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}e^{|\overline{\mu}|z_{1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu}),x)V_{N-1}(x)$
$\displaystyle\Pi_{N}(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))\Omega(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))dz_{1}dx_{1}...dx_{N-2}$
where we put $x=(x_{1},...,x_{N-1})$ with $x_{N-1}=-(x_{1}+...+x_{N-2})$. The
recurrence hypothesis says that
$\displaystyle J_{k,N-1}(\pi_{N-1}(\overline{\mu}),x)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}_{N-1}}e^{\langle\pi_{N-1}(\overline{\mu}),z\rangle}\delta_{k,N-1}(x,z)dz.$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N-2}}e^{\sum_{i=1}^{N-1}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}}\delta_{k,N-1}(x,z)dz_{2}...dz_{N-2}.$
where $z=(z_{2},...,z_{N})$ with $z_{N}=-(z_{2}+...+z_{N-1})$ and
$\delta_{k,N-1}(.,x)$ is supported in the convex hull of $S_{N-1}.x$ in
$\mathbb{R}^{N-1}$. Hence we get
$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}\int_{\mathbb{R}^{N-2}}e^{|\overline{\mu}|z_{1}+\sum_{i=1}^{N-1}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}}\delta_{k,N-1}(z,x)$
$\displaystyle
V_{N-1}(x)\Pi_{N}(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))\Omega(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))$
$\displaystyle\qquad\qquad\qquad\qquad
dz_{1}dz_{2}...dz_{N-1}dx_{1}...dx_{N-2}.$
Now observing that
$\displaystyle|\overline{\mu}|z_{1}+\sum_{i=1}^{{}^{N-1}}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}$
$\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{N-1}\mu_{i}-(N-1)\mu_{N}\right)z_{1}+\sum_{i=1}^{N-1}\mu_{i}z_{i+1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N-1}\mu_{i}(z_{1}+z_{i+1})-(N-1)\mu_{N-1}z_{1}.$
Then making the change of variables
$\displaystyle Z_{i}=z_{1}+z_{i+1},\qquad 1\leq i\leq N-1$
and put $Z=(Z_{1},...,Z_{N})$ with $Z_{N}=-(Z_{1}+...+Z_{N-1}$, we have
$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}\int_{\mathbb{R}^{N-2}}e^{\sum_{i=1}^{N}\mu_{i}Z_{i}}\delta_{k,N-1}(\phi(Z),x)V_{N-1}(x)\Pi_{N}(\lambda,\theta(Z,x))$
$\displaystyle\qquad\qquad\qquad\Omega_{N}(\lambda,\theta(Z,x))dZ_{1}dZ_{2}...dZ_{N-1}dx_{1}...dx_{N-2}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N-1}}e^{\sum_{i=1}^{N}\mu_{i}Z_{i}}\delta_{k,N}(\lambda,Z)dZ_{1}...dZ_{N-1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{V}_{N}}e^{\langle\mu,Z\rangle}\delta_{k,N}(\lambda,Z)dZ,$
with
$\displaystyle\phi(Z)$ $\displaystyle=$
$\displaystyle\left(Z_{1}-\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1},...,Z_{N_{-1}}-\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\right)$
$\displaystyle\theta(Z,x)$ $\displaystyle=$
$\displaystyle\left(x_{1}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1},...,x_{N-1}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\right)$
$\displaystyle\delta_{k,N}(\lambda,Z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-2}}\delta_{k,N-1}(\phi(Z),x)$
(2.8) $\displaystyle\qquad\qquad\qquad
V_{N-1}(x)\Pi_{N}(\lambda,\theta(Z,x))\Omega_{N}(\lambda,\theta(Z,x))dx_{1}...dx_{N-2}.$
Now we write sufficient conditions for which the integrant (2.8) does not
vanish
$\displaystyle(\Lambda_{i})$ $\displaystyle\qquad\lambda_{i+1}\leq
x_{i}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\leq\lambda_{i}$
$\displaystyle(\Lambda_{I})$ $\displaystyle\qquad\sum_{i\in
I}Z_{i}-\frac{|I|}{N-1}\sum_{i=1}^{N-1}Z_{i}\leq\sum_{i=1}^{|I|}x_{i}$
for all $I\subset\\{1,2,...,N-1\\}$ of cardinally $|I|$. It follows that
$\sum_{i=1}^{|I|}\Lambda_{i}+\Lambda_{I}\quad\Rightarrow\quad\sum_{i\in
I}Z_{i}\leq\sum_{i=1}^{|I|}\lambda_{i}$
which proves that $Z^{+}\leq\lambda$ and then $Z\in co(\lambda)$.
## 3 Partially product formula for $J_{k}$
We will first establish a product formula for $J_{k}$ provided that a
conjecture of Stanley on the multiplication of Jack polynomials is true. The
conjecture says that for all partitions $\mu$ and $\lambda$
$\displaystyle
j_{\mu}j_{\lambda}=\sum_{\nu\leq\mu+\lambda}g_{\mu,\lambda}^{\nu}j_{\nu}$
where $g_{\mu,\lambda}^{\nu}$ ( the Littlewood-Richardson coefficients ) is a
polynomial in $k$ with nonnegative integer coefficients. In particular,
$g_{\mu,\lambda}^{\nu}\geq 0$, what is the interesting facts in our setting.
Hence we have for all $\mu,\lambda$ partitions,
$\displaystyle
F(\pi(\mu)+\rho_{k},.)F(\pi(\lambda)+\rho_{k},.)=\sum_{\nu\leq\mu+\lambda}f_{\mu,\lambda}^{\nu}F(\pi(\nu)+\rho_{k},.)$
with $f_{\mu,\lambda}^{\nu}\geq 0$ and
$\sum_{\nu}f_{\mu,\lambda}^{\nu}=1$
But if $\nu\leq\mu+\lambda$ as partitions then we also have
$\pi(\nu)\preceq\pi(\mu)+\pi(\lambda)$ in the dominance ordering ([2], Lemma
3.1 ). This allows us to write for all $\mu,\lambda\in P^{+}$
$\displaystyle F(\mu+\rho_{k},.)F(\lambda+\rho_{k},.)=\sum_{\nu\in
P^{+};\;\nu\preceq\mu+\lambda}f_{\mu,\lambda}^{\nu}F(\nu+\rho_{k},.).$
To arrive at product formula for $J_{k}$ we follow the technic used by M.
Rösler in [14]. We first write
$F(n\mu+\rho_{k},\frac{z}{n})F(n\lambda+\rho_{k},\frac{z}{n})=\int_{\mathbb{R}^{N}}F(nx+\rho_{k},\frac{z}{n})d\gamma_{\mu,\lambda}^{n}(x),\qquad
z\in\mathbb{V}.$
where
$d\gamma_{\mu,\lambda}^{n}=\sum_{\nu\in
P^{+};\;\nu\preceq\mu+\lambda}f_{\mu,\lambda}^{\nu}\;\delta_{\frac{\nu}{n}}.$
According to ([14], Lemma 3.2) the probability measure
$\gamma_{\mu,\lambda}^{n}$ is supported in the convex hull $co(\mu+\lambda)$.
So, from Prohorov’s theorem (see [3] ) there exists a probability measure
$\gamma_{\mu,\lambda}$ supported in $co(\mu+\lambda)$ and a subsequence
$(\gamma_{\mu,\lambda}^{n_{j}})_{j}$ which converges weakly to
$\gamma_{\mu,\lambda}$. Then by using (1.6) it follows that
$J_{k}(\mu,z)J_{k}(\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,\lambda}(\xi)$
for all $z\in\mathbb{V}$ and $\mu,\lambda\in P^{+}$.
Now let $r,s\in\mathbb{Q}^{+}$ with $r=\frac{a}{b}$ and $s=\frac{c}{b}$,
$a,b,c\in\mathbb{N}$, $b\neq 0$. We write
$\displaystyle J_{k}(r\mu,z)J_{k}(s\lambda,z)$ $\displaystyle=$ $\displaystyle
J_{k}(a\mu,\frac{z}{b})J_{k}(c\lambda,\frac{z}{b})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}}J_{k}(\xi,\frac{z}{b})d\gamma_{a\mu,c\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}}J_{k}(\frac{\xi}{b},z)d\gamma_{a\mu,c\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$
Defining $\gamma_{r\mu,s\lambda}$ as the image measure of
$\gamma_{a\mu,c\lambda}$ under the dilation $\xi\rightarrow\frac{\xi}{b}$. We
get
$J_{k}(r\mu,z)J_{k}(s\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{r\mu,s\lambda}(\xi).$
Now we apply the density argument, since
$\mathbb{Q}^{+}.P^{+}\times\mathbb{Q}^{+}.P^{+}$ is dense in $C\times C$,
where $C$ is the Weyl chamber. Then Prohorov’s theorem yields
$\displaystyle
J_{k}(\mu,z)J_{k}(\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$
for all $\mu,\lambda\in C$ with $supp(\gamma_{\mu,\lambda})\subset
co(\mu+\lambda)$. This finish our approach for the product formula.
An important special case of the Stanley conjecture called Peiri formula is
where the partition $\lambda=(n)$, $n\in\mathbb{N}$. Since this formula has
already been proved (see [16]) then we can state the following partial result
###### Theorem 2.
For all $\mu\in C$ and all $t\geq 0$ there exists a probability measure
$\gamma_{\mu,t}$ such that
$\displaystyle
J_{k}(\mu,z)J_{k}(t\beta_{1},z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,t}(\xi);\quad
z\in\mathbb{R}^{d}$
where $\beta_{1}=\pi(e_{1})$. The measure $\gamma_{\mu,t}$ is supported in
$co(\mu+t\beta_{1})$.
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* [9] J.H. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
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* [11] E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Comp. Math. 85 (1993) 333–373.
* [12] E. M. Opdam, Harmonic analysis for certain representations of the graded Hecke algebra, Acta Math. 175 (1995), 75–121.
* [13] I. G. Macdonald, Symmetric functions and Hall polynomials 2nd ed. Oxford: Clarendon Press 1995.
* [14] M. Rösler, M. Voit, Positivity of Dunkl’s intertwining operator via the trigonometric setting. Int. Math. Res. Not. 63 (2004), 3379–3389.
* [15] M. Rösler, Dunkl operators: theory and applications. Lecture Notes in Math., 1817, Orthogonal polynomials and special functions, Leuven, 2002, (Springer, Berlin, 2003) 93–135.
* [16] R. Stanley, Some combinatorial properties of Jack symmetric functions. Advances Math. 77 (1989), 76–115.
|
arxiv-papers
| 2013-04-18T04:38:59 |
2024-09-04T02:49:44.565287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B\\'echir Amri",
"submitter": "Amri Bechir B. Amri",
"url": "https://arxiv.org/abs/1304.5016"
}
|
1304.5062
|
# Exact general relativistic lensing versus thin lens approximation: the
crucial role of the void
M. Parsi Mood Department of Physics, Sharif University of Technology, Tehran,
Iran [email protected] Javad T. Firouzjaee School of Astronomy
and Physics, Institute for Research in Fundamental Sciences (IPM), Tehran,
Iran [email protected] Reza Mansouri Department of Physics, Sharif
University of Technology, Tehran, Iran and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran [email protected]
###### Abstract
We have used an exact general relativistic model structure within a FRW
cosmological background based on a LTB metric to study the gravitational
lensing of a cosmological structure. The integration of the geodesic equations
turned out to be a delicate task. We realized that the use of the rank 8(7)
and 10(11) Runge-Kutta numerical method leads to a numerical effect and is
therefore unreliable. The so-called semi-implicit Rosenbrock method, however,
turned out to be a viable integration method for our problem. The deviation
angle calculated by the integration of the geodesic equations for different
density profiles of the model structure was then compared to those of the
corresponding thin lens approximation. Using the familiar NFW density profile,
it is shown that independent of the truncation details the thin lens
approximation differ substantially from the exact relativistic calculation.
The difference in the deflection angle for different impact parameters may be
up to about 30 percent. However, using the modified NFW density profile with a
void before going over to the FRW background, as required by an exact general
relativistic model, the thin lens approximation coincides almost exactly with
the general relativistic calculation.
###### pacs:
98.80.Jk, 98.62.Js, 98.62.Ck , 95.35.+d
The thin lens (Th-L) approximation in the gravitational lensing is the
prevailing method to estimate cosmological parameters and the mass of large
scale structures leading to dark matter and dark energy contents of the
universe GLenses , hoekstra . The current view is that this Th-L approximation
is accurate enough at the cosmological scales where we are faced with very
weak gravitational fields and potentials. There has already been attempts to
compare the Th-L approximation with the integration of null geodesics in a
perturbed cosmological background (Sas93 ; Fut95 ; FriKling11 , see also
FriKling11 and the references there). However, a full general relativistic
calculation based on an exact model is still missing. There are two sources of
misinterpretation of astrophysical phenomenon in a weak gravity environment,
depending on the local or quasi-local phenomena under consideration. In the
case of local phenomena the familiar perturbation theories maybe valid to some
extend. There are already detailed studies on this subject (see Wald1 , Wald2
, Wald3 ). However, if quasi-local phenomena or structures come into play we
may encounter counter-intuitive effects not detected in the perturbational
approach to the weak field limits. The definition of quasi-local mass in
general relativity is one of these issues which has been extensively studied
in general relativity Szabados . We have already shown numerically how
different various quasi-local mass definitions of a general relativistic
structure may be taghizadeh . Another quasi-local effect relevant to the
gravitational lensing is how a spherically symmetric structure is matched to a
FRW background. Such a general relativistic matching is only possible through
an underdensity region or a void khakshournia ; a fact not realized in the
post-Newtonian approaches or cosmological perturbations relevant to lensing,
and missed in all studies comparing the Th-L approaches to a more exact
general relativistic lensing calculation.
We are interested in the exact general relativistic lensing by an exact
solution of Einstein Equations representing a cosmological structure defined
by a spherically symmetric overdensity structure within a FRW universe. There
is already an exact general relativistic model structure within an FRW
universe based on a Lemaître, Tolman and Bondi (LTB) metric Lem97 ; Tol34 ;
Bon47 representing an inhomogeneous cosmological model with a structure at
its centertaghizadeh . Choosing such a model for an extended spherical lens,
we study the gravitational lensing in a dynamical cosmological background in
the framework of general relativity by integrating numerically the null
geodesic equations to obtain the deflection angle. The result is then compared
with the corresponding Th-L approximation to understand the accuracy of this
technology and its possible flaws in interpreting the structure and the mass
of cluster of galaxies. The effect of the cosmological constant in the lensing
is negligible in small scales we are considering ishak and only effect the
cosmological distances which we will take into account. That is why we have
neglected the cosmological constant in our exact model to avoid unnecessary
complexities
Take a spherically symmetric cosmological structure in a FRW matter dominated
universe with the density $\rho(r,t)$. This is modeled by a LTB solution of
the Einstein equations which is written in the comoving coordinates as
($G=1,c=1$)
$ds^{2}=-dt^{2}+X^{2}(r,t)dr^{2}+R^{2}(t,r)d\Omega^{2}.$ (1)
satisfying
$\displaystyle\rho(r,t)$ $\displaystyle=$
$\displaystyle\frac{M^{\prime}(r)}{4\pi R^{2}R^{\prime}},$ (2) $\displaystyle
X$ $\displaystyle=$ $\displaystyle\frac{R^{\prime}}{\sqrt{1+E(r)}},$ (3)
$\displaystyle\dot{R}^{2}$ $\displaystyle=$ $\displaystyle
E(r)+\frac{2M(r)}{R}.$ (4)
Here $M$ and $E$ are integrating functions, where dot and prime denote partial
derivatives with respect to the coordinates $t$ and $r$ respectively. Equation
(4) has three different analytic solution, depending on the value of $E$. The
solution for negative $E$ we are interested in is given by
$\displaystyle R$ $\displaystyle=$ $\displaystyle-\frac{M}{E}(1-\cos\eta),$
$\displaystyle\eta-\sin\eta$ $\displaystyle=$
$\displaystyle\frac{(-E)^{3/2}}{M}(t-t_{b}(r)).$ (5)
The solution has three free functions: $t_{b}(r)$, $E(r)$, and $M(r)$. Given
that the metric is covariant under the rescaling $r\rightarrow\tilde{r}(r)$
one of these functions may be fixed.
The geodesic equations may be written in the arbitrary plane of
$\theta=\frac{\pi}{2}$ due to the spherical symmetry:
$\displaystyle
t:\frac{d^{2}t}{d\lambda^{2}}+X\dot{X}\left(\frac{dr}{d\lambda}\right)^{2}+R\dot{R}\left(\frac{d\phi}{d\lambda}\right)^{2}=0,$
(6) $\displaystyle
r:\frac{d^{2}r}{d\lambda^{2}}+2\frac{\dot{X}}{X}\frac{dr}{d\lambda}\frac{dt}{d\lambda}+\frac{X^{\prime}}{X}\left(\frac{dr}{d\lambda}\right)^{2}-\frac{RR^{\prime}}{X^{2}}\left(\frac{d\phi}{d\lambda}\right)^{2}=0,$
(7)
$\displaystyle\phi:\frac{d^{2}\phi}{d\lambda^{2}}+2\frac{\dot{R}}{R}\frac{dt}{d\lambda}\frac{d\phi}{d\lambda}+2\frac{R^{\prime}}{R}\frac{dr}{d\lambda}\frac{d\phi}{d\lambda}=0,$
(8)
where $\lambda$ is an affine parameter. Equation (8) expresses the
conservation of the angular momentum:
$L=R^{2}\frac{d\phi}{d\lambda}=Const.$ (9)
We are interested in the light-like geodesics. From the metric we obtain the
light-like condition in the form
$\left(\frac{dt}{d\lambda}\right)^{2}=X^{2}\left(\frac{dr}{d\lambda}\right)^{2}+R^{2}\left(\frac{d\phi}{d\lambda}\right)^{2}$
(10)
These partial non-linear differential equations can not be solved
analytically. To integrate them numerically one has to specify the three
functions $M(r),t_{b}(r)$, and $E(r)$ and all derivatives of the metric
functions, using a procedure proposed in KH01 ; BKCH . We start with a generic
density profile and specify it at two different times $t_{1},t_{2}$ as a
function of the coordinate $r$. Now, the numerical procedure is based on the
choice of $r$-coordinate such that $M(r)=r$. This is due to the fact that
$M(r)$ is an increasing function of $r$. Therefore, $E$ and $t_{b}$ become
functions of $M$. For the initial time we choose the time of the last
scattering surface: $t_{1}\simeq 3.77\times 10^{5}yr$. The initial density
profile should show a small over-density near the center imitating otherwise a
FRW universe. Therefore, we add a Gaussian peak to the FRW background density
$\rho_{b}$. We know already that having an over-density in an otherwise
homogeneous universe needs a void to compensate for the extra mass within the
over-density region. Therefore, to model this void we subtract a wider
gaussian peak:
$\rho(R,t_{1})=\rho_{b}(t_{1})\left[\left(\delta_{1}e^{-\left(\frac{R}{R_{0}}\right)^{2}}-b_{1}\right)e^{-\left(\frac{R}{R_{1}}\right)^{2}}+1\right],$
(11)
where $\delta_{1}$ is the density contrast of the Gaussian peak, $R_{0}$ is
the width of the Gaussian peak, and $R_{1}$ is the width of the negative
Gaussian profile. The mass compensation condition leads to an equation for
$b_{1}$. For the final time we choose the time when our null geodesy has the
nearest distance to the center of our model structure. For instance if we set
our lens at the redshift $z\simeq 0.2$ then $t_{2}\simeq 6.98Gyr$.
The density profile we choose for the final time is the universal halo density
profile (NFW) NFW95 convolved with a negative Gaussian profile to compensate
the mass plus the background density at that time:
$\rho(R,t_{2})=\left(\rho_{NFW}-b_{2}\rho_{b}(t_{2})\right)e^{-\left(\frac{R}{R_{2}}\right)^{2}}+\rho_{b}(t_{2}),$
(12)
where
$\rho_{NFW}=\rho_{b}(t_{2})\frac{\delta_{c}}{\left(\frac{R}{R_{s}}\right)\left(1+\frac{R}{R_{s}}\right)^{2}}$
(13)
and
$\delta_{c}=\frac{200}{3}\frac{c^{3}}{\ln(1+c)-\frac{c}{1+c}}.$ (14)
In our numerical calculation we will use typical NFW values $R_{s}=0.5Mpc$ and
$c=5$ for a galaxy cluster. Note that at the time $t_{2}$ a black hole
singularity covered by an apparent horizon has already been evolved.
Therefore, the NFW profile has to be modified and a black hole mass greater
than a minimum value has to be added to it at the center. This physical fact
is reflected in a shell crossing singularity if we take the familiar NFW
profile similar to that assumed for the time $t_{1}$. The mass we have assumed
for this black hole singularity is about one thousandth of the mass up to the
$R_{s}$ and equal to $5.66\times 10^{11}M_{\odot}$. Figs. 1 and 2 shows the
LTB functions $E$ and $t_{b}$ as a result of these boundary assumptions. Using
these LTB functions, the density profile of our model structure is obtained
and depicted in Fig. 3.
Figure 1: $E$ as a function of $M$ for a cluster with NFW density profile. $M$
is given in the unit of the Sun mass. Figure 2: $t_{b}$ as a function of $M$
for a cluster with NFW density profile. $t_{b}$ is given in the unit of
$3.263Gyr$. Figure 3: Density profile for a cluster. The dot line corresponds
to the familiar NFW profile and the solid line corresponds to the modified NFW
with a void.
To solve these equations we have to specify four initial conditions taking
into account the light-like condition (10). The freedom of choosing the affine
parameter reduces the initial conditions to three. Now, the integration of the
geodesics happens by a backshooting procedure. Our initial conditions are
taken to be the time of observation, distance of the observer to the lens
expressed in terms of the redshift of the lens at the time of the observation,
and angle between the line of sight to the image of source and the line of
sight to the lens ($\theta$ in Fig. 4):
$\left.\tan\theta\right|_{O}=\left.\frac{R\frac{d\phi}{d\lambda}}{R^{\prime}\frac{dr}{d\lambda}}\right|_{\text{Null}}.$
(15)
The integration is done from the observer to the source at a specific
redshift. Assuming there is no lens, the model reduces to a homogenous flat
FRW universe and the geodesics are straight lines (in comoving coordinates)
allowing us to determine the angle between the source and the lens ($\beta$ in
Fig. 4):
$\tan\beta=\frac{\sin\phi_{e}}{\frac{r_{o}}{r_{e}}-\cos\phi_{e}},\\\ $ (16)
where $\phi_{e}$ is the $\widehat{OLS}$ angle, $r_{o}$ is the comoving
distance of the observer, and $r_{e}$ is the comoving distance of the source
from the center of coordinate system in the absence of lens at the time
$t_{e}$. From the geodesic equations the $t_{e}$ is given by
$\left(t_{o}^{\frac{1}{3}}-t_{e}^{\frac{1}{3}}\right)^{2}=\frac{1}{9}\left[\frac{R_{o}^{2}}{t_{o}^{\frac{4}{3}}}+\frac{R_{e}^{2}}{t_{e}^{\frac{4}{3}}}-\frac{2R_{o}R_{e}}{t_{o}^{\frac{2}{3}}t_{e}^{\frac{2}{3}}}\cos\phi_{e}\right].$
(17)
Figure 4: GL diagram: O is observer, S is source, S’ is image in source plane,
L is lens and $\gamma$ is deflection angle.
We then write the lens equation and determine the deflection angle $\gamma$:
$\gamma=(\theta-\beta)\frac{D_{OS}}{D_{LS}},$ (18)
where we have assumed that the presence of the lens has not a significant
effect on the distances and we may use the corresponding FRW ones.
The validity of the numerical method chosen to integrate such complex system
of partial differential equations is a delicate issue. We first started with
the familiar Runge-Kutta adaptive step size algorithm with proportional and
integral feedback (PI control) NR07 in which the step size is adjusted to
keep local error under a suitable threshold. We started with the so-called
embedded Runge-Kutta of the rank 5(4). It turned out, however, that its
accuracy is too low. Therefore, we tried the rank 8(7) and then the rank
11(10) algorithm. The difference between these two last ranks, however, turned
out to be marginal and below one percent. Given the time-consuming rank 11(10)
algorithm, we preferred to use the rank 8(7) one. Now, as a fist test for the
accuracy of this numerical method we tried the trivial example of the LTB
model, namely the FRW case, expecting a null result. The result was a non-
negligible deflection angle of the order of few milliarcseconds. Suspecting to
face a numerical effect, and trying to understand the numerical algorithm and
the source of this numerical effect, we continued to calculate a more concrete
and non-trivial LTB case. The result for the rank 8(7) Runge-Kutta numerical
method applied to a structure with a compact density profile did agree with
the thin lens approximation. However, in the case of a more diffuse density
profile the result showed a deflection angle up to an order of magnitude
higher than the thin lens approximation. We did interpret this result as a
sign not to trust the Runge-Kutta method and turned to an alternative
numerical method!
The root of this numerical deficiency could be due to the term
$\frac{d\phi}{d\lambda}$ in our equations, which is almost zero in the most
part of the path of the light ray and changes suddenly to $\pi$ in the
vicinity of the lens. This is a well-known phenomenon in the numerical method
of integrating differential equations called as ”stiff” DV84 . The
characteristic property of such stiff equations is the presence of two quite
different scales. In our case we have on one side the cosmological distance
scale of the source relative to the lens and the observer, and on the other
side the scale of the structure or the nearest distance of the ray to the
lens. Realizing this stiffness property, we turned to the so-called semi-
implicit Rosenbrock method of the numerical integration of partial
differential equations DV84 ; NR07 . As a first test we calculated again the
trivial case of a FRW model which gave an acceptable null result. We,
therefore, decided to integrate our geodesic equations using the semi-implicit
Rosenbrock method instead of the Runge-Kutta one.
The null geodesics equations of our exact general relativistic structure model
is now integrated using the modified NFW density profile with a void before
matching to the background FRW universe to obtain the deflection angle. Note
that the density in the NFW density profile is taken to be the oversdensity in
an otherwise FRW model, namely $\rho-\rho_{b}$. However, for the Th-L
approximation we have used two different density profiles namely the familiar
one and the modified one with a void before matching to the background
density. In the case of familiar NFW density profile without a void, the
corresponding equations can be integrated analytically to give the deviation
angle bart96 ; keet02 :
$\displaystyle\gamma(x)$ $\displaystyle=$
$\displaystyle{\frac{4M_{sing}}{xR_{s}}}+{16\pi\rho_{b}\delta_{c}\frac{R_{s}^{2}}{x}}{\left(\log{\frac{x}{2}}+F(x)\right)}$
(19) $\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}\frac{\textrm{arctanh}({\sqrt{1-x^{2}}})}{\sqrt{1-x^{2}}}&x<1\\\
1&x=1\\\
\frac{\arctan({\sqrt{x^{2}-1}})}{\sqrt{x^{2}-1}}&x>1\end{array}\right.$ (23)
Assuming the same modified NFW profile as in general relativistic case for the
Th-L approximation we have also calculated the deflection angle applying the
lens equation GLenses
$\theta-\beta=\frac{D_{LS}}{D_{OL}D_{OS}}\frac{d\Psi(\theta)}{d\theta},$ (24)
where $\Psi$ is the lens potential.
Figure 5: Deviation angle for three cases: the general relativistic result is
indicated by plus points; the thin lens approximation using our modified NFW
is shown by the continuous line; and the dashed line is for the familiar NFW
profile without the void (formula (19)).
The result for the three cases, the exact general relativistic model with our
modified NFW profile, thin lens approximation using the modified NFW with
void, and the thin lens approximation using the familiar NFW without a void is
depicted in Fig. 5. Obviously the two cases of the thin lens approximation
with the modified NFW density profile including the void and the LTB exact
method almost coincide.
Figure 6: Deviation angle for NFW density profiles with different parameters.
Horizontal axis is normalized to $R_{s}$ and vertical axis is normalized to
the maximum of the deflection angle in each case. Dash line is for NFW model
without void (formula (19)).
The thin lens approximation with the familiar density profile without a void,
however, differ from the exact LTB model. The difference in the deviation
angle can be more than 30 percent depending on the impact parameter. The
difference between the exact general relativistic LTB model and the thin lens
approximation is due to the absence of the void in the familiar NFW profile
used in the literature. To see the implications of the NFW parameters in this
difference we have also calculated the deviation angle for different NFW
profiles, with and without void. The result is depicted in the Fig. 19. We see
again that the Th-L approximation using different modified NFW profiles
including a void almost coincide with the exact LTB model. Models with the NFW
profiles without void, however, differ substantially from the exact model. The
difference is higher the bigger the $c_{s}$ parameter is, i.e. the less the
concentration of the density of structure is.
We, therefore, conclude that by interpreting astrophysical data of
gravitational lensing by clusters using a familiar NFW density profile without
a void we are deviating from the exact result and the Th-L approximation is no
longer valid. The Th-L approximation may, however, be considered as precise
enough if one modify the density profile and add the corresponding void to it,
as require by general relativity for a quasi-local structure. The detail of
the void, such as its density contrast,its depth and length, depends on the
detail of the model and the deviation from the familiar NFW may even be much
higher for other choices. Also note that the effect of the void is higher for
larger impact parameter. In the case of strongly lensed objects in
astrophysical applications we are usually faced with small impact parameter
where this effect is negligible. For example in the case of Abell 2261 cluster
($z=0.225$) with many strong lensing arcs, D. Coe et al. coe have assigned
$c_{s}=6.2\pm 0.3$ and $M_{vir}=2.2\pm 0.2\times 10^{15}M_{\odot}$. The exact
general relativistic results according to our model would lead to $c_{s}=6.23$
and $M_{vir}=2.23\times 10^{15}M_{\odot}$. In the case of weak lensing,
however, we expect this effect to have significant impact on the cosmological
parameters. Work in this direction is in progress.
## References
* (1) P. Schneider, J. Ehlers, E.E. Falco, _Gravitational Lenses_ , Springer-Verlag (1992).
* (2) H. Hoekstra, M. Bartelmann, H. Dahle, H. Israel, M. Limousin, M. Meneghetti, [arXiv:1303.3274].
* (3) M. Sasaki, Prog. Theor. Phys., 90, No. 4 (1993).
* (4) T. Futamase, Prog. Theor. Phys., 93, No. 3 (1995).
* (5) S. Frittelli, T. P. Kling, Mon. Not. R. Astron. Soc., 415, 3599-3608 (2011).
* (6) S. R. Green, R. M. Wald, Phys. Rev. D, 83, 084020 (2011).
* (7) S. R. Green, R. M. Wald, Phys. Rev. D, 85, 063512 (2012).
* (8) S. R. Green, R. M. Wald, [arXiv:1304.2318].
* (9) L. B. Szabados, Living Rev. Relativity, 4, (2004).
* (10) J. T. Firouzjaee, M. Parsi Mood, R. Mansouri, Gen. Rel. Grav., 44, 639 (2012).
* (11) S. Khakshournia, R. Mansouri, Phys. Rev. D, 65, 027302, (2001).
* (12) W. Rindler,M. Ishak, Phys. Rev. D,76, 043006, (2007).
* (13) G. A. Lemaître, Gen. Rel. Grav., 29, 5 (1997)(reprint).
* (14) R. C. Tolman, Proc. Nat. Acad. Sci., 20, 169 (1934).
* (15) H. Bondi, Mon. Not. R. Astron. Soc., 107, 410 (1947).
* (16) A. Krasiński, C. Hellaby, Phys. Rev. D, 65,023501 (2001).
* (17) K. Bolejko, A. Krasiński, M. Célérier, C. Hellaby, _Structures in the Universe by Exact Methods: Formation, Evolution, Interactions_ , Cambridge University Press (2010).
* (18) J. F. Navarro, C. S. Frenk, S. D. M. White, Mon. Not. R. Astron. Soc., 275, 720 (1995).
* (19) W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, _Numerical Recipes: The Art of Scientific Computing_ , 3rd Edition, Cambridge University Press (2007).
* (20) K. Dekker, J. G. Verwer, _Stability of Runge-Kutta methods for stiff nonlinear differential equations_ , North-Holland (1984).
* (21) S. Weinberg, _Cosmology_ , Oxford University Press, (2008).
* (22) K. Van Acoleyen, J. Cosmol. Astropart. Phys., 10, 028, (2008).
* (23) A. Paranjape, T. P. Singh, J. Cosmol. Astropart. Phys., 03, 023, (2008).
* (24) M. Bartelmann, Astron. Astrophys., 313,697 (1996).
* (25) C. R. Keeton, [arXiv:astro-ph/0102341]
* (26) C. Giocoli, M. Meneghetti,S. Ettori,L. Moscardini, Mon. Not. R. Astron. Soc., 426, 1558, (2011).
* (27) D. Coe, et al., Astrophy. J., 757, 22C, (2012).
|
arxiv-papers
| 2013-04-18T09:37:27 |
2024-09-04T02:49:44.572287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Parsi Mood, Javad T. Firouzjaee and Reza Mansouri",
"submitter": "Mojahed Parsi Mood",
"url": "https://arxiv.org/abs/1304.5062"
}
|
1304.5063
|
# Combinaison d’information visuelle, conceptuelle, et contextuelle pour la
construction automatique de hi rarchies s mantiques adapt es l’annotation
d’images
Hichem Bannour Céline Hudelot
Laboratoire de Math matiques Appliqu es aux Syst mes (MAS)
cole Centrale Paris
Grande Voie des Vignes
92295 Ch tenay-Malabry, France
{Hichem.bannour, Celine.hudelot}@ecp.fr
### Résumé
Ce papier propose une nouvelle m thode pour la construction automatique de hi
rarchies s mantiques adapt es la classification et l’annotation d’images. La
construction de la hi rarchie est bas e sur une nouvelle mesure de similarit s
mantique qui int gre plusieurs sources d’informations: visuelle, conceptuelle
et contextuelle que nous d finissons dans ce papier. L’objectif est de fournir
une mesure qui est plus proche de la s mantique des images. Nous proposons
ensuite des r gles, bas es sur cette mesure, pour la construction de la hi
rarchie finale qui encode explicitement les relations hi rarchiques entre les
diff rents concepts. La hi rarchie construite est ensuite utilis e dans un
cadre de classification s mantique hi rarchique d’images en concepts visuels.
Nos exp riences et r sultats montrent que la hi rarchie construite permet d’am
liorer les r sultats de la classification.
### Mots Clef
Construction de hi rarchies s mantiques, s mantique d’images, annotation
d’images, mesures de similarit s mantiques, classification hi rarchique
d’images.
### Abstract
This paper proposes a new methodology to automatically build semantic
hierarchies suitable for image annotation and classification. The building of
the hierarchy is based on a new measure of semantic similarity. The proposed
measure incorporates several sources of information: visual, conceptual and
contextual as we defined in this paper. The aim is to provide a measure that
best represents image semantics. We then propose rules based on this measure,
for the building of the final hierarchy, and which explicitly encode
hierarchical relationships between different concepts. Therefore, the built
hierarchy is used in a semantic hierarchical classification framework for
image annotation. Our experiments and results show that the hierarchy built
improves classification results.
### Keywords
Semantic hierarchies building, image semantics, image annotation, semantic
relatedness measure, hierarchical image classification.
## 1 Introduction
Avec l’explosion des donn es images, il devient essentiel de fournir une
annotation s mantique de haut niveau ces images pour satisfaire les attentes
des utilisateurs dans un contexte de recherche d’information. Des outils
efficaces doivent donc tre mis en place pour permettre une description s
mantique pr cise des images. Depuis les dix derni res ann es, plusieurs
approches d’annotation automatique d’images ont donc t propos es [5, 19, 14,
2, 27] pour essayer de r duire le probl me bien connu du _foss s mantique_
[29]. Cependant, dans la plupart de ces approches, la s mantique est souvent
limit e sa manifestation perceptuelle, i.e. au travers de l’apprentissage
d’une fonction de correspondance associant les caract ristiques de bas niveau
des concepts visuels de plus haut niveau s mantique [5, 19]. Cependant, malgr
une efficacit relative concernant la description du contenu visuel d’une
image, ces approches sont incapables de d crire la s mantique d’une image
comme le ferait un annotateur humain. Elles sont galement confront es au probl
me du passage l’ chelle [21]. En effet, les performances de ces approches
varient consid rablement en fonction du nombre de concepts et de la nature des
donn es cibl es [18]. Cette variabilit peut tre expliqu e d’une part par la
large variabilit visuelle intra-concept, et d’autre part par une grande
similarit visuelle inter-concept, qui conduisent souvent des annotations
imparfaites.
R cemment, plusieurs travaux se sont int ress s l’utilisation de hi rarchies s
mantiques pour surmonter ces probl mes [30, 3, 4]. En effet, l’utilisation de
connaissances explicites, telles que les hi rarchies s mantiques, peut am
liorer l’annotation en fournissant un cadre formel qui permet d’argumenter sur
la coh rence des informations extraites des images. En particulier, les hi
rarchies s mantiques se sont av r es tre tr s utiles pour r duire le foss s
mantique [11]. Trois types de hi rarchies pour l’annotation et la
classification d’images ont t r cemment explor es : 1) les hi rarchies bas es
sur des connaissances textuelles (nous ferons r f rence ce type de
connaissances par information conceptuelle dans le reste du papier) 111Exemple
d’information textuelle utilis e pour la construction des hi rarchies: les
tags, contexte environnant, WordNet, Wikipedia, etc. [23, 31, 12], 2) les hi
rarchies bas es sur des informations visuelles (ou perceptuelles), i.e. caract
ristiques de bas niveau de l’image [28, 6, 33], 3) les hi rarchies que nous
nommerons s mantiques bas es la fois sur des informations textuelles et
visuelles [20, 13, 32]. Les deux premi res cat gories d’approches ont montr un
succ s limit dans leur usage. En effet, d’un c t l’information conceptuelle
seule n’est pas toujours en phase avec la s mantique de l’image, et est alors
insuffisante pour construire une hi rarchie ad quate pour l’annotation
d’images [32]. De l’autre cot , l’information perceptuelle ne suffit pas non
plus elle seule pour la construction d’une hi rarchie s mantique ad quate
(voir le travail de [28]). En effet, il est difficile d’interpr ter ces hi
rarchies dans des niveaux d’abstraction plus lev s. Ainsi, la combinaison de
ces deux sources d’information semble donc obligatoire pour construire des hi
rarchies s mantiques adapt es l’annotation d’images.
La suite de ce papier est organis e comme suit: dans la section 2 nous pr
sentons les travaux connexes. La section 3 pr sente la mesure s mantique
propos e dans un premier temps, puis les r gles utilis es pour la construction
de la hi rarchie s mantique. Les r sultats exp rimentaux sont pr sent s dans
la section 4. La section 5 pr sente nos conclusions et perspectives.
## 2 tat de l’art
Plusieurs m thodes [20, 13, 23, 31, 28, 6] ont t propos es pour la
construction de hi rarchies de concepts d di es l’annotation d’images. Dans
cette section nous pr senterons ces diff rentes m thodes en suivant l’ordre
propos dans l’introduction.
Marszalek & al. [23] ont propos de construire une hi rarchie par l’extraction
du graphe pertinent dans WordNet reliant l’ensemble des concepts entre eux. La
structure de cette hi rarchie est ensuite utilis e pour construire un ensemble
de classifieurs hi rarchiques. Deng & al. [12] ont propos _ImageNet_ , une
ontologie grande chelle pour les images qui repose sur la structure de
WordNet, et qui vise peupler les 80 000 synsets de WordNet avec une moyenne de
500 1000 images s lectionn es manuellement. L’ontologie LSCOM [24] vise
concevoir une taxonomie avec une couverture de pr s de 1 000 concepts pour la
recherche de vid o dans les bases de journaux t l vis s. Une m thode pour la
construction d’un espace s mantique enrichi par les ontologies est propos e
dans [31]. Bien que ces hi rarchies soient utiles pour fournir une
structuration compr hensible des concepts, elles ignorent l’information
visuelle qui est une partie importante du contenu des images.
D’autres travaux se sont donc bas s sur l’information visuelle [28, 6, 33].
Une plateforme (I2T) d di e la g n ration automatique de descriptions
textuelles pour les images et les vid os est propos e dans [33]. I2T est bas e
principalement sur un graphe AND-OR pour la repr sentation des connaissances
visuelles. Sivic & al. [28] ont propos de regrouper les objets dans une hi
rarchie visuelle en fonction de leurs similarit s visuelles. Le regroupement
est obtenu en adaptant, pour le domaine de l’image, le mod le d’Allocation
Dirichlet Latente hi rarchique (hLDA) [7]. Bart & al. [6] ont propos une m
thode bay sienne pour organiser une collection d’images dans une arborescence
en forme d’arbre hi rarchique. Dans [17], une m thode pour construire
automatiquement une taxonomie pour la classification d’images est propos e.
Les auteurs sugg rent d’utiliser cette taxonomie afin d’augmenter la rapidit
de la classification au lieu d’utiliser un classifieur multi-classe sur toutes
les cat gories. Une des principales limitations de ces hi rarchies visuelles
est qu’elles sont difficiles interpr ter. Ainsi, une hi rarchie s mantique
compr hensible et adequate pour l’annotation d’images devrait tenir compte la
fois de l’information conceptuelle et de l’information visuelle lors du
processus du construction.
Parmi les approches pour la construction de hi rarchies s mantiques, Li & al.
[20] ont pr sent une m thode bas e la fois sur des informations visuelles et
textuelles (les tiquettes associ es aux images) pour construire
automatiquement une hi rarchie, appel e "semantivisual", selon le mod le hLDA.
Une troisi me source d’information que nous nommerons information contextuelle
est aussi utilis e pour la construction de telles hierarchies. Nous discutons
plus pr cis ment de cette information dans le paragraphe suivant. Fan & al.
[15] ont propos un algorithme qui int gre la similarit visuelle et la
similarit contextuelle entre les concepts. Ces similarit s sont utilis es pour
la construction d’un r seau de concepts utilis pour la d sambigu sation des
mots. Une m thode pour la construction de hi rarchies bas es sur la similarit
contextuelle et visuelle est propos e dans [13]. La "distance de Flickr" est
propos e dans [32]. Elle repr sente une nouvelle mesure de similarit entre les
concepts dans le domaine visuel. Un r seau de concepts visuels (VCNet) bas sur
cette distance est galement propos dans [32]. Ces hi rarchies s mantiques ont
un potentiel int ressant pour am liorer l’annotation d’images.
Discussion
Comme nous venons de le voir, plusieurs approches de construction de
hierarchies se basent sur WordNet [23, 12]. Toutefois, WordNet n’est pas tr s
appropri la mod lisation de la s mantique des images. En effet, l’organisation
des concepts dans WordNet suit une structure psycholinguistique, qui peut tre
utile pour raisonner sur les concepts et comprendre leur signification, mais
elle est limit e et inefficace pour raisonner sur le contexte de l’image ou
sur son contenu. En effet, les distances entre les concepts similaires dans
WordNet ne refl tent pas n cessairement la proximit des concepts dans un cadre
d’annotation d’images. Par exemple, selon la distance du plus court chemin
dans WordNet, la distance entre les concepts "Requin" et "Baleine" est de 11
(nœuds), et entre "Humain" et "Baleine" est de 7. Cela signifie que le concept
"Baleine" est plus proche (similaire) de "Humain" que de "Requin". Ceci est
tout fait coh rent d’un point de vue biologique, parce que "Baleine" et
"Humain" sont des mammif res tandis que "Requin" ne l’est pas. Cependant, dans
le domaine de l’image il est plus int ressant d’avoir une similarit plus lev e
entre "Requin" et "Baleine", puisqu’ils vivent dans le m me environnement,
partagent de nombreuses caract ristiques visuelles, et il est donc plus fr
quent qu’on les retrouve conjointement dans une m me image ou un m me type
d’images (ils partagent un m me contexte). Donc, une hi rarchie s mantique
appropri e devrait repr senter cette information ou permettre de la d duire,
pour aider comprendre la s mantique de l’image.
## 3 M thode Propos e
En se basant sur la discussion pr c dente, nous d finissons les hypoth ses
suivantes sur lesquelles repose notre approche:
_Une hi rarchie s mantique appropri e pour l’annotation d’images doit: 1) mod
liser le contexte des images (comme d fini dans la section pr c dente), 2)
permettre de regrouper des concepts selon leurs caract ristiques visuelles et
textuelles, 3) et refl ter la s mantique des images, i.e. l’organisation des
concepts dans la hi rarchie et leurs relations s mantiques est fid le la s
mantique d’images._
Figure 1: Illustration de la mesure propos e bas e sur les similarit s
normalis es: visuelle $\overline{\varphi}$, conceptuelle $\overline{\pi}$ et
contextuelle $\overline{\gamma}$ entre concepts.
Nous proposons dans ce papier une nouvelle m thode pour la construction de hi
rarchies s mantiques appropri es l’annotation d’images. Notre m thode se base
sur une nouvelle mesure pour estimer les relations s mantiques entre concepts.
Cette mesure int gre les trois sources d’information que nous avons d crites
pr c demment. Elle est donc bas e sur 1) une similarit visuelle qui repr sente
la correspondance visuelle entre les concepts, 2) une similarit conceptuelle
qui d finit un degr de similarit entre les concepts cibles, bas e sur leur d
finition dans WordNet, et 3) une similarit contextuelle qui mesure la d
pendance statistique entre chaque paire de concepts dans un corpus donn (cf.
Figure 1). Ensuite cette mesure est utilis e dans des r gles qui permettent de
statuer sur la vraisemblance des relations de parent entre les concepts, et
permettent de construire une hi rarchie.
tant donn un ensemble de couples image/annotation, o chaque annotation d crit
un ensemble de concepts associ s l’image, notre approche permet de cr er
automatiquement une hi rarchie s mantique adapt e l’annotation d’images. Plus
formellement, nous consid rons $I=<i_{1},i_{2},\cdots,i_{\mathcal{L}}>$
l’ensemble des images de la base consid r e, et
$C=<c_{1},c_{2},\cdots,c_{\mathcal{N}}>$ le vocabulaire d’annotation de ces
images, i.e. l’ensemble de concepts associ s ces images. L’approche que nous
proposons consiste alors identifier $\mathcal{M}$ nouveaux concepts qui
permettent de relier tous les concepts de $C$ dans une structure hi rarchique
repr sentant au mieux la s mantique d’images.
### 3.1 Similarit Visuelle
Soit $x_{i}^{v}$ une repr sentation visuelle quelconque de l’image $i$
(vecteur de caract ristiques visuelles), on apprend pour chaque concept
$c_{j}$ un classifieur qui permet d’associer ce concept ses caract ristiques
visuelles. Pour cela, nous utilisons $\mathcal{N}$ machines vecteurs de
support (SVM) [10] binaires (un-contre-tous) avec une fonction de d cision
$\mathcal{G}(x^{v})$:
$\mathcal{G}(x^{v})=\sum_{k}\alpha_{k}y_{k}\mathbf{K}(x_{k}^{v},x^{v})+b$ (1)
o : $\mathbf{K}(x_{i}^{v},x^{v})$ est la valeur d’une fonction noyau pour l’
chantillon d’apprentissage $x_{i}^{v}$ et l’ chantillon de test $x^{v}$,
$y_{i}\in\\{1,-1\\}$ est l’ tiquette de la classe de $x_{i}^{v}$, $\alpha_{i}$
est le poids appris de l’ chantillon d’apprentissage $x_{i}^{v}$, et $b$ est
un param tre seuil appris. Il est noter que les chantillons d’apprentissage
$x_{i}^{v}$ avec leurs poids $\alpha_{i}>0$ forment _les vecteurs de support_.
Apr s avoir test diff rentes fonction noyau sur notre ensemble
d’apprentissage, nous avons d cid d’utiliser une fonction noyau base radiale:
$\mathbf{K}(x,y)=exp\Big{(}\frac{\|x-y\|^{2}}{\sigma^{2}}\Big{)}$ (2)
Maintenant, compte tenu de ces $\mathcal{N}$ SVM appris o les repr sentations
visuelles des images sont les entr es et les concepts (classes d’images) sont
les sorties, nous voulons d finir pour chaque classe de concept un centro de
$\vartheta(c_{i})$ qui soit repr sentatif du concept $c_{i}$. Les centro des d
finis doivent alors minimiser la somme des carr s l’int rieur de chaque
ensemble $S_{i}$:
$\underset{S}{\operatorname{argmin}}\sum_{i=1}^{\mathcal{N}}\sum_{x_{j}^{v}\in
S_{i}}\|x_{j}^{v}-\mu_{i}\|^{2}$ (3)
o $S_{i}$ est l’ensemble de _vecteurs de support_ de la classe $c_{i}$,
$S=\\{S_{1},S_{2},\cdots,S_{\mathcal{N}}\\}$, et $\mu_{i}$ est la moyenne des
points dans $S_{i}$.
L’objectif tant d’estimer une distance entre ces classes afin d’ valuer leurs
similarit s visuelles, nous calculons le centro de $\vartheta(c_{i})$ de
chaque concept visuel $c_{i}$ en utilisant:
$\vartheta(c_{i})=\frac{1}{|S_{i}|}\sum_{x_{j}\in S_{i}}x_{j}^{v}$ (4)
La similarit visuelle entre deux concepts $c_{i}$ et $c_{j}$, est alors
inversement proportionnelle la distance entre leurs centro des respectifs
$\vartheta(c_{i})$ et $\vartheta(c_{j})$:
$\varphi(c_{i},c_{j})=\frac{1}{1+d(\vartheta(c_{i}),\vartheta(c_{j}))}$ (5)
o $d(\vartheta(c_{i}),\vartheta(c_{j}))$ est la distance euclidienne entre les
deux vecteurs $\vartheta(c_{i})$ et $\vartheta(c_{j})$ d finie dans l’espace
des caract ristiques visuelles.
### 3.2 Similarit Conceptuelle
La similarit conceptuelle refl te la relation s mantique entre deux concepts
d’un point de vue linguistique et taxonomique. Plusieurs mesures de similarit
ont t propos es dans la litt rature [8, 26, 1]. La plupart sont bas s sur une
ressource lexicale, comme WordNet [16]. Une premi re famille d’approches se
base sur la structure de cette ressource externe (souvent un r seau s mantique
ou un graphe orient ) et la similarit est alors calcul e en fonction des
distances des chemins reliant les concepts dans cette structure [8].
Cependant, comme nous l’avons d j dit pr c demment, la structure de ces
ressources ne refl te pas forcement la s mantique des images, et ce type de
mesures ne semble donc pas adapt notre probl matique. Une approche alternative
pour mesurer le degr de similarit s mantique entre deux concepts est
d’utiliser la d finition textuelle associ e ces concepts. Dans le cas de
WordNet, ces d finitions sont connues sous le nom de glosses. Par exemple,
Banerjee et Pedersen [1] ont propos une mesure de proximit s mantique entre
deux concepts qui est bas e sur le nombre de mots communs (chevauchements)
dans leurs d finitions (glosses).
Dans notre approche, nous avons utilis la mesure de similarit propos e par
[25], qui se base sur WordNet et l’exploitation des vecteurs de co-occurrences
du second ordre entre les glosses. Plus pr cis ment, dans une premi re tape un
espace de mots de taille $\mathcal{P}$ est construit en prenant l’ensemble des
mots significatifs utilis s pour d finir l’ensemble des synsets222Synonym set:
composante atomique sur laquelle repose WordNet, compos e d’un groupe de mots
interchangeables d notant un sens ou un usage particulier. A un concept
correspond un ou plusieurs synsets. de WordNet. Ensuite, chaque concept
$c_{i}$ est repr sent par un vecteur $\overrightarrow{w}_{c_{i}}$ de taille
$\mathcal{P}$, o chaque _i me_ l ment de ce vecteur repr sente le nombre
d’occurrences du _i me_ mot de l’espace des mots dans la d finition de
$c_{i}$. La similarit s mantique entre deux concepts $c_{i}$ et $c_{j}$ est
alors mesur e en utilisant la similarit cosinus entre
$\overrightarrow{w}_{c_{i}}$ et $\overrightarrow{w}_{c_{j}}$:
$\eta(c_{i},c_{j})=\frac{\overrightarrow{w}_{c_{i}}\cdot\overrightarrow{w}_{c_{j}}}{|\overrightarrow{w}_{c_{i}}||\overrightarrow{w}_{c_{j}}|}$
(6)
Certaines d finitions de concepts dans WordNet sont tr s concises et rendent
donc cette mesure peu fiable. En cons quence, les auteurs de [25] ont propos
d’ tendre les glosses des concepts avec les glosses des concepts situ s dans
leur voisinage d’ordre 1. Ainsi, pour chaque concept $c_{i}$ l’ensemble
$\Psi_{c_{i}}$ est d fini comme l’ensemble des glosses adjacents connect s au
concept $c_{i}$ ($\Psi_{c_{i}}$={gloss($c_{i}$), gloss(hyponyms($c_{i}$)),
gloss(meronyms($c_{i}$)), etc.}). Ensuite pour chaque l ment $x$ (gloss) de
$\Psi_{c_{i}}$ , sa repr sentation $\overrightarrow{w}_{x}$ est construite
comme expliqu ci-dessus. La mesure de similarit entre deux concepts $c_{i}$ et
$c_{j}$ est alors d finie comme la somme des cosinus individuels des vecteurs
correspondants:
$\theta(c_{i},c_{j})=\frac{1}{|\Psi_{c_{i}}|}\sum_{x\in\Psi_{c_{i}},y\in\Psi_{c_{j}}}\frac{\overrightarrow{w}_{x}\cdot\overrightarrow{w}_{y}}{|\overrightarrow{w}_{x}||\overrightarrow{w}_{y}|}$
(7)
o $|\Psi|=|\Psi_{i}|=|\Psi_{j}|$.
Enfin, chaque concept dans WordNet peut correspondre plusieurs sens (synsets)
qui diff rent les uns des autres dans leur position dans la hi rarchie et leur
d finition. Une tape de d sambigu sation est donc n cessaire pour
l’identification du bon synset. Par exemple, la similarit entre "Souris"
(animal) et "Clavier" (p riph rique) diff re largement de celle entre "Souris"
(p riph rique) et "Clavier" (p riph rique). Ainsi, nous calculons d’abord la
similarit conceptuelle entre les diff rents sens (synset) de $c_{i}$ et
$c_{j}$. La valeur maximale de similarit est ensuite utilis e pour identifier
le sens le plus probable de ces deux concepts, i.e. d sambig iser $c_{i}$ et
$c_{j}$. La similarit conceptuelle est alors calcul e par la formule suivante:
$\pi(c_{i},c_{j})=\underset{\delta_{i}\in s(c_{i}),\delta_{j}\in
s(c_{j})}{\operatorname{argmax}}\theta(\delta_{i},\delta_{j})$ (8)
o $s(c_{x})$ est l’ensemble des synsets qu’il est possible d’associer aux diff
rents sens du concept $c_{x}$.
### 3.3 Similarit Contextuelle
Comme cela a t expliqu dans la section 2, l’information li e au contexte
d’apparition des concepts est tr s importante dans un cadre d’annotation
d’images. En effet, cette information, dite contextuelle, permet de relier des
concepts qui apparaissent souvent ensemble dans des images ou des m mes types
d’images, bien que s mantiquement loign s du point de vue taxonomique. De
plus, cette information contextuelle peut aussi permettre d’inf rer des
connaissances de plus haut niveau sur l’image. Par exemple, si une photo
contient "Mer" et "Sable", il est probable que la sc ne repr sent e sur cette
photo est celle de la plage. Il semble donc important de pouvoir mesurer la
similarit contextuelle entre deux concepts. Contrairement aux deux mesures de
similarit pr c dentes, cette mesure de similarit contextuelle d pend du
corpus, ou plus pr cis ment d pend de la r partition des concepts dans le
corpus.
Dans notre approche, nous mod lisons la similarit contextuelle entre deux
concepts $c_{i}$ et $c_{j}$ par l’information mutuelle PMI [9] (Pointwise
mutual information) $\rho(c_{i},c_{j})$:
$\rho(c_{i},c_{j})=\log\frac{P(c_{i},c_{j})}{P(c_{i})P(c_{j})}$ (9)
o , $P(c_{i})$ est la probabilit d’apparition de $c_{i}$, et $P(c_{i},c_{j})$
est la probabilit jointe de $c_{i}$ et de $c_{j}$. Ces probabilit s sont estim
es en calculant les fr quences d’occurrence et de cooccurrence des concepts
$c_{i}$ et $c_{j}$ dans la base d’images.
tant donn $\mathcal{N}$ le nombre total de concepts dans notre base d’images,
$\mathcal{L}$ le nombre total d’images, $n_{i}$ le nombre d’images annot es
par $c_{i}$ (fr quence d’occurrence de $c_{i}$) et $n_{ij}$ le nombre d’images
co-annot es par $c_{i}$ et $c_{j}$, les probabilit s pr c dentes peuvent tre
estim es par:
$\begin{array}[]{cc}\widehat{P(c_{i})}=\frac{n_{i}}{\mathcal{L}},&\widehat{P(c_{i},c_{j})}=\frac{n_{ij}}{\mathcal{L}}\\\
\end{array}$ (10)
Ainsi:
$\rho(c_{i},c_{j})=\log\frac{{\mathcal{L}*n_{ij}}}{n_{i}*n_{j}}$ (11)
$\rho(c_{i},c_{j})$ quantifie la quantit d’information partag e entre les deux
concepts $c_{i}$ et $c_{j}$. Ainsi, si $c_{i}$ et $c_{j}$ sont des concepts
ind pendants, alors $P(c_{i},c_{j})=P(c_{i})\cdot P(c_{j})$ et donc
$\rho(c_{i},c_{j})=log\leavevmode\nobreak\ 1=0$. $\rho(c_{i},c_{j})$ peut tre
n gative si $c_{i}$ et $c_{j}$ sont corr l s n gativement. Sinon,
$\rho(c_{i},c_{j})>0$ et quantifie le degr de d pendance entre ces deux
concepts. Dans ce travail, nous cherchons uniquement mesurer la d pendance
positive entre les concepts et donc nous ramenons les valeurs n gatives de
$\rho(c_{i},c_{j})$ 0.
Enfin, afin de la normaliser dans l’intervalle [0,1], nous calculons la
similarit contextuelle entre deux concepts $c_{i}$ et $c_{j}$ dans notre
approche par:
$\gamma(c_{i},c_{j})=\frac{\rho(c_{i},c_{j})}{-\log[\max(P(c_{i}),P(c_{j}))]}$
(12)
Il est noter que la mesure PMI d pend de la distribution des concepts dans la
base. Plus un concept est rare plus sa PMI est grande. Donc si la distribution
des concepts dans la base n’est pas uniforme, il est pr f rable de calculer
$\rho$ par:
$\rho(c_{i},c_{j})=P(c_{i},c_{j})\log\frac{P(c_{i},c_{j})}{P(c_{i})P(c_{j})}$
(13)
### 3.4 Mesure de Similarit Propos e
Pour deux concepts donn s, les mesures de similarit visuelle, conceptuelle et
contextuelle sont d’abord normalis es dans le m me intervalle. La
normalisation est faite par la normalisation Min-Max. Puis en combinant les
mesures pr c dentes, nous obtenons la mesure de similarit s mantique adapt e
l’annotation suivante:
$\phi(c_{i},c_{j})=\omega_{1}\cdot\overline{\varphi}(c_{i},c_{j})+\omega_{2}\cdot\overline{\pi}(c_{i},c_{j})+\omega_{3}\cdot\overline{\gamma}(c_{i},c_{j})$
(14)
o : $\sum_{i=1}^{3}\omega_{i}=1$; $\overline{\varphi}(c_{i},c_{j})$,
$\overline{\pi}(c_{i},c_{j})$ et $\overline{\gamma}(c_{i},c_{j})$ sont
respectivement la similarit visuelle, la similarit conceptuelle et la
similarit contextuelle normalis es.
Le choix des pond rations $\omega_{i}$ est tr s important. En effet, selon
l’application cibl e, certains pr f reront construire une hi rarchie sp
cifique un domaine (qui repr sente le mieux une particularit d’un domaine ou
d’un corpus), et pourront donc attribuer un plus fort poids la similarit
contextuelle ($\omega_{3}\nearrow$). D’autres pourront vouloir cr er une hi
rarchie g n rique, et devront donc donner plus de poids la similarit
conceptuelle ($\omega_{2}\nearrow$). Toutefois, si le but de la hi rarchie est
plut t de construire une plateforme pour la classification de concepts
visuels, il est peut tre avantageux de donner plus de poids la similarit
visuelle ($\omega_{1}\nearrow$).
### 3.5 R gles pour la cr ation de la hi rarchie
La mesure propos e pr c demment ne permet que de donner une information sur la
similarit entre les concepts deux deux. Notre objectif est de regrouper ces
diff rents concepts dans une structure hi rarchique. Pour cela, nous d
finissons un ensemble de r gles qui permettent d’inf rer les relations
d’hypernymie entre les concepts.
Nous d finissons d’abord les fonctions suivantes sur lesquelles se basent nos
r gles de raisonnement:
* •
$Closest(c_{i})$ qui retourne le concept le plus proche de $c_{i}$ selon notre
mesure:
$\begin{split}Closest(c_{i})=\underset{c_{k}\in\mathcal{C}\backslash\\{c_{i}\\}}{\operatorname{argmax}}\phi(c_{i},c_{k})\end{split}$
(15)
* •
$LCS(c_{i},c_{j})$ permet de trouver l’anc tre commun le plus proche (_Least
Common Subsumer_) de $c_{i}$ et $c_{j}$ dans WordNet:
$\begin{split}LCS(c_{i},c_{j})=\underset{c_{l}\in\\{H(c_{i})\cap
H(c_{j})\\}}{\operatorname{argmin}}len(c_{l},root)\end{split}$ (16)
o $H(c_{i})$ permet de trouver l’ensemble des hypernymes de $c_{i}$ dans la
ressource WordNet, $root$ repr sente la racine de la hi rarchie WordNet et
$len(c_{x},root)$ renvoie la longueur du plus court chemin entre $c_{x}$ et
$root$ dans WordNet.
* •
$Hits_{3}(c_{i})$ renvoie les 3 concepts les plus proche de $c_{i}$ au sens de
la fonction $Closest(c_{i})$.
(a) $1^{ere}$ R gle.
(b) $2^{ieme}$ R gle.
(c) $3^{ieme}$ R gle.
Figure 2: R gles pour inf rer les liens de parent entre les diff rents
concepts. En rouge les pr conditions devant tre satisfaites, en noir les
actions de cr ation de nœuds dans la hi rarchie.
Nous d finissons ensuite trois r gles qui permettent d’inf rer les liens de
parent entre les diff rents concepts. Ces diff rentes r gles sont repr sent es
graphiquement sur la figure 2. Ces r gles sont ex cut es selon l’ordre d crit
dans la figure 2. La premi re r gle v rifie si un concept $c_{i}$ est class
comme le plus proche par rapport plusieurs concepts
($(Closest(c_{j})=c_{i}),\forall j\in\\{1,2,\cdots\\}$). Si oui et si ces
concepts $\\{c_{j}\\},\forall j\in\\{1,2,\cdots\\}$, sont r ciproquement dans
$Hits_{3}(c_{i})$, alors en fonction de leur LCS ils seront soit reli s
directement leur LCS ou dans une structure 2 niveaux, comme illustr dans
Figure 2(a). Dans la seconde, si $(Closest(c_{i})=c_{j})$ et
$(Closest(c_{j})=c_{i})$ (peut aussi tre crite
$Closest(Closest(c_{i}))=c_{i}$) alors $c_{i}$ et $c_{j}$ sont fortement
apparent s et seront reli s leur LCS. La troisi me r gle concerne le cas o
$(Closest(c_{i})=c_{j})$ et $(Closest(c_{j})=c_{k})$ \- voir Figure 2(c).
La construction de la hi rarchie suit une approche ascendante (i.e. commence
partir des concepts feuilles) et utilise un algorithme it ratif jusqu’
atteindre le nœud racine. tant donn un ensemble de concepts associ s aux
images dans un ensemble d’apprentissage, notre m thode calcule la similarit
$\phi(c_{i},c_{j})$ entre toutes les paires de concepts, puis relie les
concepts les plus apparent s tout en respectant les r gles d finies pr c
demment. La construction de la hi rarchie se fait donc pas- -pas en ajoutant
un ensemble de concepts inf r s des concepts du niveau inf rieur. On it re le
processus jusqu’ ce que tous les concepts soient li s un nœud racine.
## 4 R sultats Exp rimentaux
Pour valider notre approche, nous comparons la performance d’une
classification plate d’images avec une classification hi rarchique exploitant
la hi rarchie construite avec notre approche sur les donn es de Pascal
VOC’2010 (11 321 images, 20 concepts).
### 4.1 Repr sentation Visuelle
Pour calculer la similarit visuelle des concepts, nous avons utilis dans notre
approche le mod le de sac-de-mots visuels (Bag of Features) (BoF). Le mod le
utilis BoF est construit comme suit: d tection de caract ristiques visuelles
l’aide des d tecteurs DoG de Lowe [22], description de ces caract ristiques
visuelles en utilisant le descripteur SIFT [22], puis g n ration du
dictionnaire eu utilisant un K-Means. Le dictionnaire g n r est un ensemble de
caract ristiques suppos es tre repr sentatives de toutes les caract ristiques
visuelles de la base. tant donn e la collection de patches (point d’int r t) d
tect s dans les images de l’ensemble d’apprentissage, nous g n rons un
dictionnaire de taille $D=1000$ en utilisant l’algorithme k-Means. Ensuite,
chaque patch dans une image est associ au mot visuel le plus similaire dans le
dictionnaire en utilisant un arbre KD. Chaque image est alors repr sent e par
un histogramme de $1000$ mots visuels (1000 tant la taille du codebook), o
chaque bin dans l’histogramme correspond au nombre d’occurrences d’un mot
visuel dans cette image.
### 4.2 Pond ration
Comme ce travail vise construire une hi rarchie adapt e l’annotation et la
classification d’images, nous avons fix les facteurs de pond ration de mani re
exp rimentale comme suit : $\omega_{1}=0.4$, $\omega_{2}=0.3$, et
$\omega_{3}=0.3$. Nos exp rimentations sur l’impact des poids ($\omega_{i}$)
ont galement montr que la similarit visuelle est plus repr sentative de la
similarit s mantique des concepts, comme cela est illustr sur la figure 3 avec
la hi rarchie produite. Cette hi rarchie est construite sur les donn es de
Pascal VOC’2010.
Figure 3: La hi rarchie s mantique construite sur les donn es de Pascal VOC en
utilisant la mesure propos e et les r gles de construction. Les nœuds en
double octogone sont les concepts de d part, le nœud en diamant est la racine
de la hi rarchie construite et les autres sont les nœuds inf r s.
$\phi(c_{i},c_{j})=0.4\cdot\overline{\varphi}(c_{i},c_{j})+0.3\cdot\overline{\pi}(c_{i},c_{j})+0.3\cdot\overline{\gamma}(c_{i},c_{j})$
### 4.3 Evaluation
Figure 4: Comparaison de la Pr cision Moyenne (AP) entre la classification
plate et hi rarchique sur les donn es de Pascal VOC’2010.
(a) Concept Person.
(b) Concept Tv_monitor.
Figure 5: Courbes Rappel/Pr cision pour la classification hi rarchique (en +)
et plate (en trait) pour les concepts "Personne" et "TV_Monitor".
Pour valuer notre approche, nous avons utilis 50% des images du challenge
Pascal VOC’2010 pour l’apprentissage des classifieurs et les autres pour les
tests. Chaque image peut appartenir une ou plusieurs des 20 classes (concepts)
existantes. La classification plate est faite par l’apprentissage de
$\mathcal{N}$ SVM binaires un-contre-tous, o les entr es sont les repr
sentations en BoF des images de la base et les sorties sont les r ponses du
SVM pour chaque image (1 ou -1) - pour plus de d tails voir la section 3.1. Un
probl me important dans les donn es de Pascal VOC est que les donn es ne sont
pas quilibr es, i.e. plusieurs classes ne contiennent qu’une centaine d’images
positives parmi les 11321 images de la base. Pour rem dier ce probl me, nous
avons utilis la validation crois e d’ordre 5 en prenant chaque fois autant
d’images positives que n gatives.
La classification hi rarchique est faite par l’apprentissage d’un ensemble de
($\mathcal{N}$+$\mathcal{M}$) classifieurs hi rarchiques conformes la
structure de la hi rarchie d crite dans la figure 3. $\mathcal{M}$ est le
nombre de nouveaux concepts cr s lors de la construction de la hi rarchie.
Pour l’apprentissage de chacun des concepts de la hi rarchie, nous avons pris
toutes les images des nœuds fils (d’un concept donn ) comme positives et
toutes les images des nœuds fils de son anc tre imm diat comme n gatives. Par
exemple, pour apprendre un classifieur pour le concept "Carnivore", les images
de "Dog" et "Cat" sont prises comme positives et les images de "Bird",
"Sheep", "Horse" et "Cow" comme n gatives. Ainsi chaque classifieur apprend
diff rencier une classe parmi d’autres dans la m me cat gorie. Durant la phase
de test de la classification hi rarchique et pour une image donn e, on
commence partir du nœud racine et on avance par niveau dans la hi rarchie en
fonction des r ponses des classifieurs des nœuds interm diaires, jusqu’
atteindre un nœud feuille. Notons qu’une image peut prendre plusieurs chemins
dans la hi rarchie. Les r sultats sont valu s avec les courbes rappel/pr
cision et le score de pr cision moyenne.
La Figure 4 compare les performances de nos classifieurs hi rarchiques avec
les performances de la classification plate. L’utilisation de la hi rarchie
propos e comme un cadre de classification hi rarchique assure des meilleures
performances qu’une classification plate, avec une am lioration moyenne de
+8.4%. Notons que ces r sultats sont obtenus en n’utilisant que la moiti des
images du jeu d’apprentissage de Pascal VOC. En effet, en l’absence des images
de test utilis es dans le challenge, nous avons utilis le reste de l’ensemble
d’apprentissage pour faire les tests. Nous avons aussi inclus les images marqu
es comme difficiles dans les valuations de notre m thode. La pr cision moyenne
de notre classification hi rarchique est de 28,2%, alors que la classification
plate reste 19,8%. On peut donc remarquer une nette am lioration des
performances avec l’utilisation de la hi rarchie propos e. La Figure 5 montre
les courbes de rappel/pr cision des concepts "Personne" et "TV_Monitor" en
utilisant la classification hi rarchique et plate. Une simple comparaison
entre ces courbes montre que la classification hi rarchique permet d’avoir un
meilleur rendement tous les niveaux de rappel. Cependant, il serait int
ressant de tester notre approche sur une plus grande base, avec plus de
concepts, pour voir si la hi rarchie construite pour la classification des
images passe l’ chelle.
## 5 Conclusion
Ce papier pr sente une nouvelle approche pour construire automatiquement des
hi rarchies adapt es l’annotation s mantique d’images. Notre approche est bas
e sur une nouvelle mesure de similarit s mantique qui prend en compte la
similarit visuelle, conceptuelle et contextuelle. Cette mesure permet
d’estimer une similarit s mantique entre concepts adapt e la probl matique de
l’annotation. Un ensemble de r gles est propos pour ensuite effectivement
relier les concepts entre eux selon la pr c dente mesure et leur anc tre
commun le plus proche dans WordNet. Ces concepts sont ensuite structur s en hi
rarchie. Nos exp riences ont montr que notre m thode fournit une bonne mesure
pour estimer la similarit des concepts, qui peut aussi tre utilis e pour la
classification d’images et/ou pour raisonner sur le contenu d’images. Nos
recherches futures porteront sur l’ valuation de notre approche sur des plus
grandes bases d’images (MirFlicker et ImageNet) et sa comparaison avec l’ tat
de l’art.
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|
arxiv-papers
| 2013-04-18T09:40:12 |
2024-09-04T02:49:44.577457
|
{
"license": "Public Domain",
"authors": "Hichem Bannour and C\\'eline Hudelot",
"submitter": "Hichem Bannour",
"url": "https://arxiv.org/abs/1304.5063"
}
|
1304.5193
|
# Acoustic emission from magnetic flux tubes in the solar network
G Vigeesh1 and S S Hasan2 1 Department of Astronomy, New Mexico State
University, Las Cruces, NM, U.S.A. 2 Indian Institute of Astrophysics,
Koramangala, Bangalore, India [email protected], [email protected]
###### Abstract
We present the results of three-dimensional numerical simulations to
investigate the excitation of waves in the magnetic network of the Sun due to
footpoint motions of a magnetic flux tube. We consider motions that typically
mimic granular buffeting and vortex flows and implement them as driving
motions at the base of the flux tube. The driving motions generates various
MHD modes within the flux tube and acoustic waves in the ambient medium. The
response of the upper atmosphere to the underlying photospheric motion and the
role of the flux tube in channeling the waves is investigated. We compute the
acoustic energy flux in the various wave modes across different boundary
layers defined by the plasma and magnetic field parameters and examine the
observational implications for chromospheric and coronal heating.
## 1 Introduction
Observations of the solar surface in various filtergrams show a distribution
of bright points, which are generally associated with an underlying
concentrated magnetic field located in intergranular lanes [1]. Magnetic flux
accumulates here after being swept in by granular flow forming strong vertical
flux tubes at the junction of granules. The continually changing solar
photosphere perturbs these magnetic structures resulting in the generation of
magnetohydrodynamic (MHD) waves that propagate within the flux tube and also
into the ambient plasma. The waves are excited by impulses imparted by
granules. Eventually, the flux tubes are advected towards regions of stronger
downdrafts present at the intersections of two or more granules where
convectively driven vortex flows occur. The footpoints of the magnetic field
structures are dragged in the vortex resulting in, presumably, a different
source of wave excitation in the flux tube [2]. The conditions that prevail in
the lower atmosphere impacts the upper layers, primarily through magnetic
fields. Estimating the energy supplied by various photospheric disturbance
will help us to determine the fraction of the overall heating of the upper
atmosphere by wave sources. In order to examine energy transport to the outer
layers of the quiet solar atmosphere mediated by magnetic fields [3, 4, 5, 6],
we need to clearly understand how the magnetic flux tube reacts to different
types of perturbations. Despite being idealized representations of the actual
processes, studies have shown that various wave-driven energy transport
mechanisms are possible in a magnetized atmosphere [7, 8, 9, 10, 11, 12].
Further investigations could reveal how these processes contribute to the
overall energy budget and shed light on the significance of wave related
sources. The purpose of this paper is to study two such scenarios with the aim
of evaluating them in terms of their contribution to the heating of
chromosphere and corona.
In this paper, we investigate the excitation of waves in magnetic flux tubes
extending vertically through the solar atmosphere. This work is an extension
of our previous study [12] that dealt with 3D modeling of MHD wave propagation
in a magnetic flux tube embedded in a stratified atmosphere. We showed that
there are possibly more than one mechanism of wave production effective which
can lead to temporal variations of the emission that occur at chromospheric
heights above these elements. In this paper our focus is to investigate the
generations of acoustic waves excited by these perturbations and estimate the
acoustic energy flux at different levels in the flux-tube. We calculate and
compare the acoustic emission from various boundaries in the flux-tube and
also from different $\beta$ (the ratio of gas to magnetic pressure) surfaces
as a result of the excitation.
## 2 Equilibrium Model and Boundary Conditions
Details of the equilibrium model that we consider in this study are given in
[12]. Briefly, we consider an intense (kG field strength at the base) axially
symmetric magnetic flux tube in a stratified solar model atmosphere that is in
magnetostatic equilibrium. The temperature in the model increases with height
to include the chromospheric temperature rise. The footpoint of the flux tube
is located in the solar photosphere. The mathematical construction of the flux
tube is described in the Appendix of [12] and the equilibrium properties of
the flux tube are given in Table 1.
Table 1: The equilibrium model parameters on the axis of the flux tube and ambient medium (values shown within brackets). Height | T | $\rho$ | P | cS | vA | B | $\beta$
---|---|---|---|---|---|---|---
| (K) | (kg m-3) | (N m-2) | (km s-1) | (km s-1) | (G) |
$z$=1 Mm | 7263 | 3.4 $\times$ 10-8 | 1.6 | 8.8 | 77.7 | 161 | 64
(7195) | (1.0 $\times$ 10-7) | (4.8) | (8.7) | (39.2) | (141) | (16)
$z$=0 Mm | 4768 | 1.3 $\times$ 10-4 | 4.2 $\times$ 103 | 7.1 | 10.9 | 1435 | 1.9
(4766) | (4.0 $\times$ 10-4) | (1.2 $\times$ 104) | (7.1) | (0.003) | (0.77) | (1.9 $\times$ 10-7)
| | | | | | |
The plasma-$\beta=1$ surface essentially outlines the flux tube boundary in
the lower part of the tube. In these layers, $\beta<1$ inside the flux tube as
shown in Fig 1.
Figure 1: Two-dimensional representation of the flux-tube model. The solid
lines mark the magnetic field lines on a $x$-$z$ plane at $y=0$ Mm. The
$\beta=1$ () and $\beta=0.1$ () contours are also shown.
Wave excitation is carried out by implementing velocity drivers at the bottom
boundary. We use a horizontal driver to mimic the granular buffeting motion
and a torsional driver to mimic the effect of vortex-like motion. The driving
motions at the bottom boundary are specified using the following velocity
drivers.
$\displaystyle V_{x}(x,y,0,t)=\left\\{\begin{array}[]{l l}\displaystyle
V_{0}\sin(2\pi t/P)&\mbox{for}\quad 0\leq t\leq P/2\,,\\\\[4.30554pt]
\displaystyle 0&\mbox{for}\quad 0>t>P/2\,.\end{array}\right.{\rm(Horizontal)}$
(3) $\displaystyle V_{\phi}(x,y,0,t)=\left\\{\begin{array}[]{l
l}\displaystyle-V_{0}\tanh\left(\frac{2\pi r}{\delta
r}\right)\sin\left(\frac{2\pi t}{P}\right)&\quad 0\leq t\leq
P/2,\\\\[4.30554pt] \displaystyle
0&\quad\phantom{0>}t>P/2.\end{array}\right.{\rm(Torsional)}$ (6)
These driving motions generate slow (predominantly acoustic) and fast
(predominantly magnetic) waves in the model along with the intermediate Alfvén
wave which we do not consider in this study.
## 3 Numerical Simulation
The three-dimensional numerical simulations were carried out using the
Sheffield Advanced Code [13]. The code uses a modified version of the set of
MHD equations to deal with a strongly stratified magnetized medium. In this
study, we solve the full set of ideal magnetohydrodynamic equations in three
dimensions to study the propagation of waves in the computational domain. The
domain is a 1 Mm $\times$ 1 Mm $\times$ 1 Mm cube discretized on 100 $\times$
100 $\times$ 100 grid points. We use transmitting boundary conditions at the
top and side boundaries to allow the waves to propagate out of the simulation
box.
The simulation starts with a localized, time dependent perturbation at the
bottom boundary resulting in the excitation of various kinds of MHD modes with
different strength. The gas pressure perturbations drive slow magneto-acoustic
wave (SMAW) within the $\beta<$1 region and fast magneto-acoustic wave (FMAW)
in the ambient medium where $\beta>$1\. The magneto-acoustic wave propagation
can be seen in the velocity and temperature perturbation in the medium as
shown in Fig 2. We notice that the horizontal excitation generates strong SMAW
within the flux tube as can be seen in the temperature fluctuations. Since the
SMAW propagates along field lines, we also see strong velocity parallel to the
field lines. The torsional excitation on the other hand generates SMAW that
are weaker than those generated by the horizontal driver.
Figure 2: Top: Temperature perturbations at t=20 s,60 s, and 100 s at
different heights as a result of a transversal uni-directional excitation.
Bottom: Temperature fluctuations as a result of torsional excitation. The
projected velocity vectors on the $x$-$y$ plane are shown at each height. The
thick blue curve depicts the $\beta$=1 region and the blue dashed curve shows
the $\beta$=0.1 region.
In the previous study [12], we estimated the energy transport by MHD waves for
the two driving cases. The energy fluxes were calculated on a representative
field line and we showed that there is a strong acoustic flux associated with
SMAW in the case of the horizontal excitation, which is two orders of
magnitude more than that for the case of torsional excitation. However, these
studies were restricted to a single field line and hence it was not possible
to assess the response of the whole flux tube to the different perturbations
that were considered. In this paper, we extend the previous study by looking
at the flux tube as a whole and estimating the total acoustic emission from
various physical surfaces relevant to the model. Due to the nature of the
drivers, the analysis on a single field line depends wholly on the location of
the footpoint of the chosen field line with respect to the driving motion.
Especially, in the case of a horizontal motion, where the strongest
perturbations is localized to a small region on either side of the $y=0$
plane. Also, since the velocity driver acts in region where $\beta<1$ (close
to the axis) as well as $\beta>1$ (outer regions), a field line located in
either of these region sees a different mode of magneto-acoustic wave.
Considering an ensemble of field lines will give a better idea about the
reaction of the flux tube as a whole to the excitation at the base.
## 4 Results
### 4.1 Longitudinal & Lateral emission
Apart from acting as a conduit for MHD waves to the overlying layers, the
perturbed flux tube also transfers energy to the ambient medium in the form of
acoustic waves. It is interesting to look at acoustic energy flux leaking out
of the flux tube boundary and how it depends on the nature of the excitation
that the flux tube undergoes. Since we consider here a thick flux tube, a
strict definition of the flux tube boundary is ambiguous. However, for the
purpose of this study, we choose the surfaces of equal magnetic potential or
the magnetic isosurface as representative levels in the flux tube for purposes
of calculating the fluxes. The cross-section of the magnetic isosurface at any
given height is a circle centered at the axis, due to the cylindrical symmetry
of the initial model. This allows us to define a circle at a radial distance
$r$ of the flux tube on any horizontal plane. We calculate the acoustic flux
on three magnetic isosurfaces crossing radial distance of $r=0.2~{}$Mm,
$0.3~{}$Mm and $0.4~{}$Mm from the axis of the flux tube at $z=1~{}$Mm. The
magnetic field lines that define this isosurface can be identified from the
equilibrium model and they remain the same throughout the simulation, allowing
us to calculate physical quantities on this surface as it evolves. The
velocity vector at any point on this surface can be decomposed into three
orthogonal components, viz. parallel (${v_{s}}$), normal (${v_{n}}$) and
azimuthal (${v_{\phi}}$) component. The calculation of these component for a
single field line is described in [12]. Using these velocity components for a
collection of field lines defining various isosurfaces, we calculate the
parallel and normal acoustic fluxes according to,
$F_{\rm s,n}=\Delta p{v_{\rm s,n}}$ (7)
where $\Delta p$ is the gas pressure perturbation from the equilibrium.
Acoustic fluxes are calculated on field lines corresponding to three magnetic
isosurfaces. In Figure 3, the left panel shows the time averaged parallel
acoustic fluxes ($F_{\rm s}$) on field lines associated with different
isosurfaces as a function of height. The horizontal excitation results in
relatively stronger acoustic emission compared to torsional excitation in the
lower part of the flux tube. But, the averaged fluxes tend to be similar as we
go higher up in the atmosphere, since in the horizontal excitation case, only
a small fraction of field lines on either side of the tube partake in
transporting the longitudinal fluxes. The right panel of Fig. 3 shows the time
averaged acoustic flux directed normal to the flux-tube boundaries, which is
significantly lower than the fluxes directed along the field lines in the
tube.
Figure 3: Left: Time-averaged longitudinal acoustic flux on the three
isosurfaces for the horizontal (Solid lines) and torsional(dashed lines)
excitation cases. Right Normal acoustic flux on the three isosurfaces.
### 4.2 $\beta$-surface emission
The surface of equipartition ($\beta$=1 layer) between thermal and magnetic
energy density influences the MHD modes by acting as region of wave
transmission and conversion. As a preliminary step towards understanding this
region more closely, we look at the acoustic energy flux crossing normal to
the surfaces of constant plasma $\beta$ surface ($\beta$ isosurfaces). In this
study, we mainly focus on the $\beta=1$ isosurface, since this is where the
different MHD modes undergo strong coupling which eventually leads to the
generation of Alfvén waves in the upper chromosphere. The whole process
proceeds in two stages. Initially, the magneto-acoustic modes encounter the
equipartition zone as they propagate to the upper layers of the atmosphere.
The energy in the modes get redistributed partially by transmission and
partially by conversion to other modes. The transmission and conversion
depends on the properties of the wave and the background magnetic field across
this zone of influence [14]. In the second stage, the mode converted magnetic
wave (FMAW) propagates up to a certain height after which it gets partially
reflected down and partially gets converted to Alfvén waves due to steep
gradients in the Alfvén speed [15]. A better understanding of the energy
distribution during the initial stage where mode coupling occurs across the
$\beta=1$ surface is nessasary to further evaluate the energy available for
Alfvén wave. To this end, we look at the acoustic flux directed normal to the
surface of $\beta=1$ and compare it with that from a $\beta=0.1$ isosorface
which lies well above in the atmosphere. It should be noted that in the lower
part of the flux tube, the $\beta=1$ surface normal points towards the axis of
the flux-tube.
Figure 4 shows the spatially averaged acoustic flux directed normal to the
surfaces of constant $\beta$ for $\beta$=1 and 0.1 as a function of time. We
clearly see that the acoustic emission from the $\beta$=1 surface is stronger
compared to the emission on the $\beta$=0.1 surface for both horizontal and
torsional excitations. The $\beta$=1 surface responds more efficiently to a
horizontal uni-directional motion at the foot-point by emitting more acoustic
flux normal to the surface. Our analysis is limited by the fact that we
consider a strong flux tube where the $\beta=1$ surface dips below the bottom
boundary near the axis of the flux tube. To have a better understanding about
this layer, we need to look at flux tubes models with $\beta=1$ surfaces at
different levels in the atmosphere.
Figure 4: Time evolution of the normal acoustic flux on $\beta$=1 (solid line)
and $\beta$=0.1 (dotted) surface for horizontal (red line) and torsional
(blue) excitation of a magnetic flux tube
## 5 Summary & Conclusions
Three dimensional numerical simulation of wave propagation in a magnetic flux
tube embedded in a solar atmosphere were carried out. We investigated two
types of excitation mechanism, viz. transversal and torsional, that are
characteristic processes by which waves can be generated in the real solar
atmosphere. We calculated the acoustic energy components at various levels in
the flux tube as well as on the surfaces defined by constant plasma $\beta$ of
the flux tube due to the driving footpoint motions. We observe that the
acoustic power is predominantly directed vertically upward along the flux tube
in both cases. The lateral acoustic emission from the boundary of the flux
tube in both cases of horizontal and torsional excitations is much lower. The
magnetic flux tube acts as an efficient conduit for acoustic waves, with
negligible acoustic leakage from the tube boundary. Most of the acoustic
energy produced due to photospheric disturbances are efficiently transported
and are made available to higher layers regardless of the source of the
disturbance. As far as the comparison between the two excitation scenarios in
terms of acoustic energy transport to upper layers is concerned, we have not
been able to conclusively point out which of the two mechanisms is more
efficient. This is partly due to the axisymmetric equilibrium model that we
have used and the idealistic driving mechanisms that we consider in the study.
Nevertheless, our analysis supports that the magnetic field mediates the
coupling between various photospheric disturbances and the upper layers.
However meagre their contribution to the overall energy output is, these
scenarios must be considered when evaluating the available energy sources for
the heating of chromosphere and corona. The surface of equipartition between
magnetic and thermal energy density ($\beta=1$) is a strong source of acoustic
emission in both excitation scenarios and would be the focus of future
investigations.
## References
* [1] de Wijn A G, Stenflo J O, Solanki S K and Tsuneta S 2009 Space Sci. Rev. 144 275–315
* [2] Jess D B, Mathioudakis M, Erdélyi R, Crockett P J, Keenan F P and Christian D J 2009 Science 323 1582–1585
* [3] Hasan S S and van Ballegooijen A A 2008 Astrophys. J. 680 1542–1552
* [4] van Ballegooijen A A, Asgari-Targhi M, Cranmer S R and DeLuca E E 2011 Astrophys. J. 736
* [5] Kato Y, Steiner O, Steffen M and Suematsu Y 2011 Astrophys. J. Lett. 730 L24
* [6] Wedemeyer-Böhm S, Scullion E, Steiner O, Rouppe van der Voort L, de La Cruz Rodriguez J, Fedun V and Erdélyi R 2012 Nature 486 505–508
* [7] Hasan S S, van Ballegooijen A A, Kalkofen W and Steiner O 2005 Astrophys. J. 631 1270–1280
* [8] Khomenko E, Collados M and Felipe T 2008 Sol. Phys. 251 589–611
* [9] Vigeesh G, Hasan S S and Steiner O 2009 Astron. Astrophys. 508 951–962
* [10] Kitiashvili I N, Kosovichev A G, Mansour N N and Wray A A 2011 Astrophys. J. Lett. 727
* [11] Fedun V, Verth G, Jess D B and Erdélyi R 2011 Astrophys. J. Lett. 740 L46
* [12] Vigeesh G, Fedun V, Hasan S S and Erdélyi R 2012 Astrophys. J. 755 18
* [13] Shelyag S, Fedun V and Erdélyi R 2008 Astron. Astrophys. 486 655–662
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|
arxiv-papers
| 2013-04-18T17:16:58 |
2024-09-04T02:49:44.585059
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Vigeesh and S. S. Hasan",
"submitter": "G Vigeesh",
"url": "https://arxiv.org/abs/1304.5193"
}
|
1304.5260
|
Currently at ]Institute of Evolutionary Sciences, University of Montpellier
II, Montpellier 34095, France Also at ]UC Berkeley and San Francisco Art
Institute, San Francisco, California, USA
# Effects of mixing in threshold models of social behavior
Andrei R. Akhmetzhanov [email protected] [ Lee Worden [email protected] [
Jonathan Dushoff [email protected] Theoretical Biology Laboratory,
Department of Biology, McMaster University, Hamilton, Ontario L8S4K1, Canada
###### Abstract
We consider the dynamics of an extension of the influential Granovetter model
of social behavior, where individuals are affected by their personal
preferences and observation of the neighbors’ behavior. Individuals are
arranged in a network (usually, the square lattice) and each has a state and a
fixed threshold for behavior changes. We simulate the system asynchronously
either by picking a random individual and either update its state or exchange
it with another randomly chosen individual (mixing). We describe the dynamics
analytically in the fast-mixing limit by using the mean-field approximation
and investigate it mainly numerically in case of a finite mixing. We show that
the dynamics converge to a manifold in state space, which determines the
possible equilibria, and show how to estimate the projection of manifold by
using simulated trajectories, emitted from different initial points.
We show that the effects of considering the network can be decomposed into
finite-neighborhood effects, and finite-mixing-rate effects, which have
qualitatively similar effects. Both of these effects increase the tendency of
the system to move from a less-desired equilibrium to the “ground state”. Our
findings can be used to probe shifts in behavioral norms and have implications
for the role of information flow in determining when social norms that have
become unpopular (such as foot binding or female genital cutting) persist or
vanish.
Social norms, Threshold models, Random-field Ising models, Mixing
###### pacs:
89.65.-s,05.40.-a,89.75.-k
## I Introduction
In this paper, we investigate a simple model of behavior, the threshold model
(TM) Schelling1971 ; Granovetter1978 . It consists of $N$ individuals arranged
in a network. Each individual, described by a state variable $s_{i}$
($i=1,\ldots,N$), has either adopted or rejected the behavior in question and
has a tendency to switch to adopting (rejecting) if the proportion of
individuals in its neighborhood adopting the behavior is greater (less) than
its (constant) threshold $T_{i}$. Individuals are chosen at random to be
“updated” – i.e., to consider, and possibly change (“flip”) their state. We
make an analogy with physics by thinking of the individual’s state as a ‘spin’
with value $+1$ ($-1$) for those who adopt (reject) the behavior.
Threshold models are relevant to questions of how patterns of behavior
persist, even when attitudes change, and how these patterns can sometimes
change rapidly. A currently relevant example is the practice of _female
genital cutting_ (FGC), which goes back at least to ancient Egypt Kennedy2009
. Despite a public health consensus that the practice is harmful WHO2008 ,
traditional practice remains widespread in various societies TagEldin2008 ;
UNICEF2010 . A similar example is the Chinese practice of footbinding, which
was widely practiced for hundreds of years, before disappearing rapidly
Mackie1996 . These practices can be considered in the context of the theory of
“social norms”, behaviors which individuals prefer to follow, _given that they
think that others will conform, and that others expect them to conform_
Bicchieri2006 .
Many similar individual-based models are also based on individuals making
binary choices SanMiguel2005 ; Castellano2009 . Usually, the voter model
Liggett1999 is associated with imitation process, since a randomly chosen
individual adopts the behavior of one of its neighbors. In this sense, the TM
puts the social pressure in the framework: the adoption or rejection of the
behavior by an individual depends on the current level of adoption in its
neighborhood Vilone2012 . The majority rule model (MR, see Krapivsky2003 ) is
a special case of the TM, since all thresholds are one half (the randomly
chosen individual tends to flip if the _majority_ of its neighbors have
opposite spin).
It has been shown that the majority rule model can be described by the
classical Ising model with zero external magnetic field Krapivsky2003 and
that the general TM can be described as a random-field Ising model (RFIM)
Barra2012 . The study of RFIMs in physics often focuses on critical
temperature phenomena Dorogovtsev2008 or metastable states and hysteresis
loop phenomena at zero temperature Sethna1993 ; Rosinberg2009b . Instead of
using the notion of the thermodynamic temperature, where individuals
probabilistically flip in a non-preferred direction, see, for example,
Brock2001 ; Malarz2011 , we chose to set the thermodynamic temperature to zero
and study the effects of mixing on the dynamics.
We simulate our model on a two-dimensional lattice, with global mixing. We
implement mixing by allowing individuals to exchange places within the network
at rate $\mu$ (relative to the update rate). The importance of mixing in
sociological and ecological studies has been demonstrated in other contexts
Levin1974 ; Blasius1999 ; Agliari2006 ; Reichenbach2007 . Introducing global
mixing on a two-dimensional lattice is similar in concept to using a “small-
world” network Watts1998 . Both cases have regular connections, and random
global connections – the difference is that we implement random global
connections by switching individuals.
We simulate behavior change by either choosing an individual at random to
update or mixing two individuals in each step of the simulation. Mixing
consists of exchanging two randomly chosen individuals rather than updating in
a given simulation step, with probability $\mu/(1+\mu)$, so that we have an
average of $\mu$ switches per update. We increment the clock by $1/N$ per
update event. This gives us update events at rate 1 per individual and mixing
events at rate $\mu$ per individual. When we mix individuals, we exchange
their states and thresholds, leaving the network otherwise unchanged. A
synchronous process or a pure Poisson process would be expected to give
qualitatively similar results, but this asynchronous process is simpler to
simulate, and can be analyzed using an ordinary differential equation (ODE)
framework derived from master equations.
Most analytical results in the field of TMs/RFIMs are obtained using mean-
field approximations Dominicis2006 ; Krapivsky2010 . This can be achieved
either by considering a complete network (where every node is a neighbor of
every other node), or by setting the mixing rate $\mu\gg 1$.
The intermediate mixing case $\mu\sim 1$ is not so easily treated. If we write
equations for the moments of different order for the distribution of states
among individuals, we get a hierarchical system of coupled equations
Krapivsky2010 . There are then various methods to “close” the system by
approximating higher moments in terms of lower moments Bolker1999 ;
Murrell2004 ; Murrell2009 .
In Toral2007 , the authors considered a MR model, and concluded that the
behavior of their system resembles the movement of a Brownian particle in a
potential field that is unknown _a priori_. We describe such an “effective
potential” function for our threshold model and calculate the analytical
potential form for the mean-field version of the model.
We can use the effective potential to provide an additional perspective on the
dynamical properties of the system. The bifurcation where the system changes
from having one stable equilibrium to two, for example, corresponds to a
change from a single-welled to a double-welled effective potential function.
In terms of an Ising-like model, this would correspond to a phase transition
of the first order. This bifurcation is relevant from a sociological point of
view, since a transformation from a potential consisting of two wells to a
potential consisting of one well, due to a change of mixing rate, could give
rise to sudden abandonment or adoption of a social norm. In contrast, if such
transformation does not occur, even when one well is much deeper than the
other, this might help to explain why a human society sometimes continues to
support a fairly unpopular social norm for many years Bicchieri2006 .
In this paper we explore the dynamics of this system using a Gaussian
threshold distribution. It is not necessary to truncate the distribution,
since we can simply assume that if the threshold is ${<}0$ (or ${>}1$), an
individual will always be updated to its preferred state independent of its
neighborhood configuration. We expect other flexible distributions to give
similar qualitative results. For example, even uniform distributions show the
same basic bifurcations that we explore here Granovetter1978 . We have also
tried simulations with bimodal superpositions of Gaussians, and again seen
qualitatively similar results.
Here is how we organize the rest of the paper. First we consider the mean-
field dynamics which are given by the fast-mixing and large-scale ($N\gg 1$)
limits. Then we consider the intermediate mixing rates and state the main
results of our paper: we discuss the bifurcation phenomena found in the TM and
we demonstrate the appearance of a manifold in the dynamics that is approached
by any trajectory of the TM. Finally, we interpret our main results from a
sociological point of view, and draw conclusions.
## II Mean-field approximation
First, consider the case in which the neighborhood size is so large that each
spin is connected with all other spins in the network. In this case, we
recover Granovetter’s threshold model for collective behavior Granovetter1978
, with dynamics in which the probability of a spin being updated from minus to
plus is given by $\mathbb{P}(\uparrow|y)=(1-y)F(y)$, where $y$ denotes the
proportion of plus spins and $F(\cdot)$ is the cumulative distribution
function of the thresholds’ PDF $f(x)$:
$F(x)=\int_{-\infty}^{x}f(\xi)\mathrm{d}\xi\;.$ (1)
The probability of a spin being flipped in opposite direction from plus to
minus is $\mathbb{P}(\downarrow|y)=y(1-F(y))$. In this case, master equations
can be written in terms of the probability function $p(y_{k},t)$
($y_{k}=k/N$), which provides the probability to find the TM in a state with
$k$ spins in a plus state and $N-k$ spins in a minus state at a given moment
of time $t$:
$\frac{\mathrm{d}p(y_{k},t)}{\mathrm{d}t}=\mathbb{P}(\uparrow|y_{k-1})p(y_{k-1},t)\\\
{}+\mathbb{P}(\downarrow|y_{k+1})p(y_{k+1},t)-\mathbb{P}(\updownarrow|y_{k})p(y_{k},t)\,,\\\
k=0,\ldots,N\,,$ (2)
where
$\mathbb{P}(\updownarrow|y_{k})=\mathbb{P}(\uparrow|y_{k})+\mathbb{P}(\downarrow|y_{k})$.
Letting $N\rightarrow\infty$ and scaling the time as $t\rightarrow Nt$, we can
treat the discrete variable $y_{k}$ as continuously changing $y\in[0,1]$ and
transform (2) to the Hamilton-Jacobi equation, which is the first order
partial differential equation:
$\frac{\partial p(y,t)}{\partial t}=-\frac{\partial}{\partial
y}[(F(y)-y)p(y,t)]\;.$ (3)
During such transformation, the diffusive terms, consisting of second order
partial derivatives, vanish due to the large-scale limit $N\gg 1$.
If the initial configuration is strictly defined such that
$p(y,0)=\delta(y-y_{0})$, where $\delta(\cdot)$ is the Dirac delta, the
solution of (3) is given by the following ODE, see p. 53–54 Gardiner2004 :
$\dot{y}\equiv\frac{\mathrm{d}y}{\mathrm{d}t}=F(y)-y,\quad y(0)=y_{0}\;.$ (4)
The equilibria of this system are all values $y_{*}$ for which
$F(y_{*})=y_{*}$.
Notice that (4) can also be written in terms of the potential function:
$V(x)=-\int^{x}(F(\xi)-\xi)\mathrm{d}\xi$, such that
$\frac{\mathrm{d}y}{\mathrm{d}t}=-V^{\prime}(y)$. Thus, the equilibrium points
can be also defined by the extrema of $V(y)$.
Figure 1: Simulations of the threshold model (TM) on a two-dimensional lattice
of size $100^{2}$ with no mixing among individuals, and different
neighbourhood sizes. The thresholds are normally distributed with the mean
$0.45$ and standard deviation $0.3$. The initial pattern of thresholds and
initial states is the same for all simulations shown. Activated individuals
are shown in orange (black). The $\infty$-symbol denotes an equilibrium, which
is reached in the TM.
We now consider a case where each individual’s updates depend on the states in
a _finite_ neighborhood. First simulations of the TM on a two-dimensional
lattice with 8 nearest neighbors for each individual and with no mixing among
them reveal complex patterns, see Fig. 1. This figure presents initial,
intermediate and final states of the lattice for four different neighborhood
sizes, but with identical initial distributions of states and thresholds. We
see that increasing neighborhood size can shift the outcome of the system’s
dynamics from a low level of conformity to a very high level. Moreover, the
equilibrium distribution preserves some noticeable clustering for small
neighborhood sizes.
However, in the large-scale ($N\rightarrow\infty$), fast-mixing
($\mu\rightarrow\infty$) limit, the behavior of the TM can still be described
analytically by a mean-field model, in which a spin and all its neighbors are
chosen _de novo_ at each update event.
The probability that a randomly selected individual will choose to adopt is
equal to the probability that the activation level of its randomly selected
neighborhood exceeds its threshold. In a regular network where each individual
has $n$ neighbors, this is given by:
$F_{n}(y)=\sum_{k=0}^{n}F\\!\left(\frac{k}{n}\right)C_{n}^{k}y^{k}(1-y)^{n-k}\,,$
(5)
where $C_{n}^{k}=\frac{n!}{k!(n-k)!}$ is a binomial coefficient (see Barra2012
). Then (2)-(4) remain valid for this system once we substitute $F_{n}(y)$ for
$F(y)$ in (5), and they give us the dynamics of the TM with finite
neighborhood size in the mean-field approximation. (This approach also works
for networks of variable degree; if neighborhood sizes are distributed with
probability density $\mathcal{P}(n)$, we average over the distribution to get
$F_{\mathcal{P}}=\sum_{n=0}^{\infty}\mathcal{P}(n)F_{n}(y)$.)
Figure 2: (Color online) The mean-field dynamics of the TM in the phase plane
$(y,\mathrm{d}y/\mathrm{d}t)$ for different neighborhood sizes: $n=4$ (magenta
(light)), 12 (red (medium)) and 24 (green (dark)). The dashed curve
corresponds to the case of infinitely large neighborhood size. The
corresponding potential functions $V_{n}(y)$ are shown in the inset. The
thresholds’ PDF is Gaussian with the mean $0.6$ and standard deviation
$0.225$. The crosses show the results of numerical simulations of the TM on a
two-dimensional lattice of the size $100^{2}$ at mixing rate $\mu=4$ and
$y(0)=1$
Fig. 2 illustrates the difference between the functions (5) and (1) as well as
the difference in the mean-field dynamics of the TM for different
neighborhoods. Parameters of the thresholds’ PDF $f$ for Fig. 2 were chosen in
such a way that there are two stable equilibria for large neighborhoods, and
only one stable equilibrium for small neighborhoods. Different neighborhood
sizes can lead to very different outcomes, even when distributions and initial
conditions are the same. For example, in simulations starting with everybody
adopting the behavior ($y(0)=1$), the TM reaches a high equilibrium (few
individuals change), when neighborhood size is large, but a low equilibrium
(almost everybody rejects the behavior) when neighborhood size is small. Note
that simulations done on a two-dimensional lattice of size $100^{2}$ at mixing
rate $\mu=4$ give a good approximation to the large-scale, fast-mixing limit
in this case; later we will show that this is not true for smaller mixing
rates, though.
In case of a Gaussian distribution for the thresholds, the curve
$y^{\prime}=F_{n}(y)$ has up to three crossings with the diagonal
$y^{\prime}=y$. If there is only one crossing with the diagonal, there exists
a globally stable equilibrium $y_{-}\in[0,1]$. If there are three crossings,
we have three equilibrium points, which we denote as $y_{-}$, $y_{*}$ and
$y_{+}$, such that $y_{-}<y_{*}<y_{+}$. Two of them, $y_{\pm}$, are stable
equilibria and one of them, $y_{*}$, is an unstable equilibrium.
We can define a potential, analogous to the mean-field case:
$V_{n}(y)=\int^{y}(F_{n}(\xi)-\xi)\>\mathrm{d}\xi$. Then $y_{\pm}$ are the
minima and $y_{*}$ is the maximum of $V_{n}(y)$, see Fig. 2 (inset). The case
of two crossings represents the bifurcation point between these two generic
cases.
If the mean of the Gaussian distribution is not exactly $0.5$, the potential
function $V_{n}(y)$ is asymmetric, with one well deeper than the other.
Without loss of generality, we assume that the norm is intrinsically unpopular
(i.e., the mean of the threshold distribution $>\\!0.5$), so that the “lower”
equilibrium $y_{-}$ corresponds to the deeper well, and the “upper”
equilibrium and $y_{+}$ to the shallower well, when it exists. These values
refer to the case where $\mu\to\infty$. For clarity, we will sometimes add
$\infty$ as a superscript.
## III Intermediate mixing
Simulations show that reducing the mixing rate away from the fast-mixing limit
has a similar qualitative effect to reducing neighborhood size (as seen in
Fig. 2). In the case where the mean-field system has one stable equilibrium,
reducing mixing rates does not lead to a qualitative change in the dynamics.
In the case where the mean-field system has two stable equilibria, as the
mixing rate gets smaller we often find a bifurcation to a single equilibrium;
i.e., the equilibrium with the shallower potential well disappears.
We consider a two-dimensional lattice, with initial activation level $y(0)$.
If we simulate, starting from a value between the two stable equilibria, the
system will move to the upper equilibrium with probability $p_{+}$; otherwise
it moves to the lower equilibrium. The result depends on the random selection
of thresholds, initial states and the order in which sites are updated.
Figure 3: (Color online) The probability $p_{+}$ that the TM will approach the
upper equilibrium as a function of $\mu$ or scaled $\mu/L$ (the inset). The TM
is on a 2d-lattice of size $N=L^{2}$ with the neighborhood $8$. The
thresholds’ PDF is Gaussian with the mean $0.55$ and standard deviation
$0.225$. Different colors (shades) stand for different initial values $y(0)$,
while the symbols stand for different values of $L$ (see the legend in the
bottom right corner). In the main figure the points with $y(0)=0.71$ are shown
in cyan (light), with $y(0)=0.69$ in blue (dark). In the inset overlapping
points in the middle correspond to $y(0)=y_{*}^{\infty}$, the ones above them
to $y(0)=0.69$, and the ones below them to $y(0)=0.66$. The bifurcation value
$\tilde{\mu}(y(0)=0.69)\approx 1.556$ is indicated. To estimate the
probability, we performed $10^{5}$ simulations with different random initial
individual states and thresholds, and update order
Fig. 3 shows how the probability $p_{+}$ depends on the mixing rate $\mu$,
using two different scaling approaches. In either case, as we move away from
the fast-mixing case (from right to left on the figure), the system becomes
increasingly certain to end in the deeper well, and eventually the shallower
well disappears altogether.
The main picture of Fig. 3 shows the probability $p_{+}$ vs. $\mu$ for values
of $y(0)>y_{*}^{\infty}$. In this case the system stops in the shallow well
for large values of $\mu$, and moves to the deeper well for smaller values.
This transition becomes steeper as the size of the network increases. The
curves for a given starting point intersect where $p_{+}=1/2$. That is to say,
for a given value of $\mu$, the value of $y(0)$ that falls “in the middle” of
the two wells – so that the system is equally likely to go to either one –
does not change with lattice size. We call this value $y_{\times}^{\mu}$,
because it is related to the unstable equilibrium $y_{*}^{\mu}$, but not
equivalent (as we will see below). The dependence of $y_{\times}^{\mu}$ on
$\mu$ has a hyperbolic shape and its minimal value $\bar{\mu}$ is reached at
$y_{\times}^{\mu}=1$, which is shown in Fig. 4.
Figure 4: The dependence of $y_{\times}^{\mu}$ on $\mu$. The TM is posed on a
2-d lattice of size $200^{2}$ with neighborhood $8$ and Gaussian distribution
of thresholds with the mean $0.55$ and standard deviation $0.225$. The
extrapolation curve, shown by the dashed line, gives
$\bar{\mu}=\mu(y_{\times}^{\mu}=1.0)\approx 0.168$.
The inset in Fig. 3 shows the same data, with $p_{+}$ plotted against a
_scaled_ version of the mixing rate $\mu/L$ (where $L=N^{1/2}$ is the length
of the two-dimensional lattice). A surprising pattern emerges. For
$y(0)=y_{*}^{\infty}$, all of the curves approximately align onto a single
curve, with $p_{+}\rightarrow 1/2$ for $\mu\rightarrow\infty$, as we expect,
since we are approaching the well-mixed case, where $y(0)$ is the unstable
equilibrium. For other values of $y(0)$, the curves do not intersect in this
scaling: instead, as $N$ gets larger, the system becomes less likely to
“switch” to the equilibrium on the other side of $y_{*}^{\infty}$.
If we visualize the trajectories in the phase subspace
$(y,\mathrm{d}y/\mathrm{d}t)$, we find that all of them approach the same
curve $F_{n}^{\mu}$, shown in Fig. 5, presumably because they are collapsing
onto a lower-dimensional slow manifold. Thus, the behavior of the TM can be
well-approximated by the ODE: ${\mathrm{d}y}/{\mathrm{d}t}=F_{n}^{\mu}(y)-y$,
on some time interval $t\in[t_{1},\infty)$, which is similar to (4), where
$F_{n}^{\infty}\equiv F_{n}$. The equilibrium points $y_{*}^{\mu}$ can be
determined as $F_{n}^{\mu}(y_{*}^{\mu})=y_{*}^{\mu}$ and the effective
potential can be introduced by
$V_{n}^{\mu}(y)=-\int^{y}(F_{n}^{\mu}(\xi)-\xi)\mathrm{d}\xi$. Thus, we can
describe the behavior of the TM qualitatively, by studying the properties of
the manifold-projection curve $F_{n}^{\mu}$.
Figure 5: (Color online) The projection curve $F_{n}^{\mu}$ on a two-
dimensional lattice of size $800^{2}$ with neighborhood $8$ and the Gaussian
distribution of thresholds (the mean 0.55 and standard deviation $0.225$ and
mixing rate equals $0.278$. The trajectories, shown in red (thin) lines, start
from $y(0)=i/10$ ($i=1,\ldots,9$), while the green (solid) curve consists of
the trajectories initiated at $y(0)=0$, $y(0)=1$ and
$y(0)=y_{\times}^{\mu}=0.76$. Initially, the states and thresholds are
distributed randomly among individuals, hence all initial points fall along
the curve $F_{n}(y)\equiv F_{n}^{\infty}(y)$ (dashed curve)
When $y$ approaches 0 or 1, mixing does not affect the dynamics. Therefore, we
can construct at least part of the curve $F_{n}^{\mu}$ (for any value of
$\mu$) by simulating trajectories starting from $y(0)=1$ and $y(0)=0$. When
there is only one equilibrium in $[0,1]$, this method generates the whole
projection curve. When there are two stable equilibria, this method generates
only the part “outside” them. Completing the curve requires that we start
simulations from one or more intermediate initial points $y(0)\in(0,1)$. In
fact, we need only one additional starting point, which is precisely
$y_{\times}^{\mu}$, since any trajectory initiated at that point goes through
the point $y_{*}^{\mu}$ on the curve $F_{n}^{\mu}$ to the upper or lower
equilibrium with equal probability one half.
Note that if we take the minimal value $\bar{\mu}$ of the mixing rate, which
can defined using Fig. 4, $F_{n}^{\mu}$ will have two fixed points, one of
them will be a double root of $F_{n}^{\mu}(y_{*}^{\mu})=y_{*}^{\mu}$, which
corresponds to the bifurcation, described above.
### Transition times
Figure 6: (Color online) The mean transition time $\langle T_{R}\rangle$,
necessary for the TM to evolve from complete activation $y(0)=1.0$ to a given
intermediate value $y(T_{R})=y_{*}^{\infty}\approx 0.6752$. The TM is posed on
a two-dimensional lattice of size $N=L^{2}$ with neighborhood $8$ and Gaussian
distribution of the thresholds with the mean 0.55 and standard deviation
0.225. The inset illustrates the distribution of $T_{R}$ at $\mu=0.1$ for
$L=100$ (green (light)), $200$ (magenta (medium)) and $400$ (blue (dark))
For mixing rates $\mu<\bar{\mu}$, the system will always move towards the
lower equilibrium $y_{-}^{\mu}$. We therefore explore the “transition time”
$T_{R}$ – how long it takes to move from the fully activated state $y(0)=1.0$
to some intermediate activation level, chosen to be near, but to the right of,
the lower equilibrium.
From simulations, we see that the distribution of transition times becomes
narrower for larger $N$, see the inset Fig. 6. Hence, it is important to look
how the mean value $\langle T_{R}\rangle$ changes for $N\gg 1$. It turns out
that the dependence $\langle T_{R}\rangle$ vs. $\mu$ is concave and has a
minimum at some intermediate value $\hat{\mu}$. We see that it becomes
arbitrarily large for small mixing rates and exponentially decreases as $\mu$
approaches $\hat{\mu}$. From the other side the value $\langle T_{R}\rangle$
increases as $\mu$ becomes closer to $\bar{\mu}$, see Fig. 6.
The minimum in the transition-time curve can be explained in terms of two
countervailing effects of increased mixing. When the mixing rate is very slow,
any changes in behavior take a long time to spread through the lattice. When
the mixing rate is high, for our parameters, exchange between neighborhoods
tends to preserve the “upper equilibrium”, leading to an exponentially slow
transition on the finite lattice (and no transition at all for the infinite
system). Thus, the most rapid transition from the activated state to the lower
equilibrium occurs at an intermediate value.
## IV Conclusions
Offering an explanation why collective behavior can shift abruptly from
avoidance to adoption of an alternative, or _vice versa_ , Granovetter1978
provides an example where a slight change in distribution of individual
thresholds leads to completely different outcomes: a system initially at one
stable equilibrium switches to another one due to a change of the threshold of
one individual. This change can be visualized as a change in the shape of the
$F_{n}$-curve, which we will call the “activation curve”. We investigate other
factors that can change this curve and lead to similar phenomena, including
abrupt changes in outcome when an equilibrium disappears.
We model a population on a lattice, with a finite interaction neighborhood,
and “mixing” – implemented by exchanging random individuals. To disentangle
the effects of neighborhood size and “locality”, we first considered a lattice
with finite neighborhoods in the infinite-mixing limit. We showed that the
effect of finite neighborhoods is to “flatten” the activation curve, often
leading eventually to the elimination of the “weaker” equilibrium, as
neighborhood size gets smaller.
We then consider the effect of “locality”, by reducing the mixing rate to be
of the same order as the update rate. This system is harder to analyze, but we
show that it tends to converge towards a manifold, whose projection can be
interpreted in a way similar to the activation curve. This interpretation
allows us to define an effective potential in the finite mixing case, which
can aid in qualitative analysis. We find that the effect of locality on the
projection curve is similar to the effect of finite neighborhood size: it
flattens the curve and eventually leads to the disappearance of the weaker
equilibrium.
The flattening due to finite neighborhood size Fig. 2 can be understood in
terms of averaging. If each individual evaluates a random, finite subset of
the population when updating, the realized activation curve is a weighted
average of the original curve. This averaging tends to flatten out curvature:
in the limit of considering a single neighbor, the activation curve becomes a
straight line. Finite mixing has a similar effect Fig. 5. Individuals’ states
will be correlated with those of their neighbors, since they are responding to
each other. This increases the variance in neighborhood activation perceived
by individuals, for a given value of the mean activation, accentuating the
effect of averaging and further smoothing the activation curve.
Here we, as others in the past Krapivsky2010 ; Barra2012 , use the mapping
between the threshold model and the random field Ising model so that it is
possible also to apply tools from statistical physics to the question. Another
possibly useful analogy can be made between the TM and a spin gas. We neglect
the structure of the network and consider particles stochastically moving in
uniform medium, and affected primarily by nearby particles. In this case, the
mixing rate can be associated directly with thermodynamic temperature. There
is then an analogy between the tendency of all spins to be at the lower
equilibrium for small mixing rate in the original system, and low-temperature
Bose-Einstein condensation in the spin gas Leggett2001 . This mapping may be
worth future study.
From sociological point of view, mixing is associated with the rate of
information flow in a given society or people mobility. We might imagine
“activists” who have high mixing rates, and who are eager to change the
prevalent behavior. We have seen that large mixing rates can actually prevent
the system from switching to a desirable equilibrium, so that an unpopular
social norm persists, while low mixing rates facilitate the abandonment of the
social norm. However, very low mixing rates make the transition very slow, so
that in many cases the transition will happen fastest at intermediate mixing
rates.
## Acknowledgments
Authors were supported by J.S. McDonnell Foundation. This work was made
particularly by using the facilities of the Shared Hierarchical Academic
Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul
Canada. Authors are grateful to anonymous reviewers for their remarks, A.R.A.
is thankful to Gustavo Düring (NYU, USA) and Yevgeni Sh. Mamasakhlisov
(Yerevan State University, Armenia) for helpful discussions.
## Appendix A: Ising model framework
In our model, each node $i\in\\{1\ldots N\\}$ of the network has a state (or
spin) $s_{i}$ and a (constant) threshold $T_{i}$. The classical Ising model
translates to the a majority rule (MR) model, where all thresholds are exactly
$0.5$: a spin tends to flip to the state where it will be aligned with more
than one half its neighboring spins, see Ch. 8 Krapivsky2010 . This system is
Hamiltonian with the energy function:
$\mathcal{H}=-\frac{1}{2}\sum\nolimits_{i;j\in\langle i\rangle}s_{i}s_{j}$,
where $i\in 1\ldots N$, and $\langle i\rangle$ refers to the network neighbors
of node $i$.
When the thresholds are randomly distributed with a given probability
distribution function (PDF), the model becomes equivalent to the spin system
under a magnetic field which describes effects of locality between spins, and
such that the strength of nearest interactions depends on the connectivity of
the network. In this case, the system also obeys Hamiltonian dynamics and its
energy function has the form
$\mathcal{H}=-\sum\limits_{i;j\in\langle
i\rangle}\frac{s_{i}s_{j}}{2n_{i}}+\sum_{i}(2T_{i}-1)s_{i}\;.$
where $n_{i}$ is the number of connections for a spin $s_{i}$ and $T_{i}$ is a
given threshold of it. Thus, the induced magnetic field is $h_{i}=2T_{i}-1$.
To simulate the TM dynamics, the following underlying update rule is posed for
each update event
$s_{i}\mapsto\mathop{\rm
sign}\nolimits\left(-\frac{\partial\mathcal{H}}{\partial
s_{i}}\right)=\mathop{\rm
sign}\nolimits\\!\left(\frac{1}{n_{i}}\sum\limits_{j\in\langle
i\rangle}s_{j}-h_{i}\right)\,,$ (6)
while the thermodynamic temperature, determining the rate of random flips of
spins, is set to zero. Hence, we use only the first part of the Metropolis
algorithm Glauber1963 that consists only of (6) in order to simulate the
dynamics of the TM. The second part when the spin might be flipped even if it
was not updated due to (6) is omitted.
Note that (6) indeed allows a sociological interpretation of the TM: if the
proportion of plus spins (individuals adopting the social norm) in the
neighborhood of a spin $s_{i}$ is written as
$y_{i}^{\circ}=\frac{1}{n_{i}}\sum\nolimits_{j\in\langle
i\rangle}\frac{1+s_{j}}{2}$, such that $n_{i}$ is the connectivity of a spin
$s_{i}$, then (6) transforms to the following form: $s_{i}\mapsto\mathop{\rm
sign}\nolimits(y_{i}^{\circ}-T_{i})$. In a particular case of the MR, it
translates to the simplest form: $s_{i}\mapsto\mathop{\rm
sign}\nolimits\sum\nolimits_{j\in\langle i\rangle}s_{j}=\mathop{\rm
sign}\nolimits(y_{i}^{\circ}-1/2)$.
## Appendix B: Technical details of simulations
In our simulations, individuals’ initial states and thresholds are
independently identically distributed with a given initial activation level
$y(0)$ and PDF of thresholds. After that, we initialize the simulation
process, using the Metropolis algorithm. At each event, we either update with
probability $(1+\mu)^{-1}$ a state of a randomly chosen individual or exchange
with complementary probability the locations of two randomly chosen
individuals. We associate the time only with update events, by defining the
time quanta $1/N$. We say that the TM is located near the equilibrium point if
it fluctuates near it over a sufficiently long period of time, compared with
the time of observation.
To construct Fig. 3 and Fig. 4, we considered the TM with
$y_{-}^{\infty}<y(0)<y_{+}^{\infty}$. In this case, we expect the system to
move toward either the lower or upper equilibrium. We run the simulations
until they traverse 85% of the distance from the starting point to one of the
mean-field equilibria; $p_{+}$ is the probability that it has moved toward the
upper equilibrium.
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|
arxiv-papers
| 2013-04-18T21:40:41 |
2024-09-04T02:49:44.591256
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrei R. Akhmetzhanov, Lee Worden, Jonathan Dushoff",
"submitter": "Andrei R. Akhmetzhanov",
"url": "https://arxiv.org/abs/1304.5260"
}
|
1304.5287
|
∎
11institutetext: Yang Liu 22institutetext: Department of Mathematics, Zhejiang
Normal University, Jinhua 321004, China
Fax: +86-57982298897
22email: [email protected]; [email protected] 33institutetext: Zhihua
Chen 44institutetext: Department of Mathematics, Tongji University, Shanghai
200092, China
44email: [email protected] 55institutetext: Yifei Pan 66institutetext:
Department of Mathematical Sciences, Indiana University-Purdue University Fort
Wayne, Fort Wayne, Indiana 46805, USA
66email: [email protected]
# A variant of Hörmander’s $L^{2}$ existence theorem for Dirac operator in
Clifford analysis
Yang Liu Zhihua Chen Yifei Pan
(Received: date / Accepted: date)
###### Abstract
In this paper, we give the Hörmander’s $L^{2}$ theorem for Dirac operator over
an open subset $\Omega\in\mathbb{R}^{n+1}$ with Clifford algebra. Some
sufficient condition on the existence of the weak solutions for Dirac operator
has been found in the sense of Clifford analysis. In particular, if $\Omega$
is bounded, then we prove that for any $f$ in $L^{2}$ space with value in
Clifford algebra, there exists a weak solution of Dirac operator such that
$\overline{D}u=f$
with $u$ in the $L^{2}$ space as well. The method is based on Hörmander’s
$L^{2}$ existence theorem in complex analysis and the $L^{2}$ weighted space
is utilised.
###### Keywords:
Hörmander’s $L^{2}$ theoremClifford analysis weak solutionDirac operator
###### MSC:
32W50 15A66
## 1 Introduction
The development of function theories on Clifford algebras has proved a useful
setting for generalizing many aspects of one variable complex function theory
to higher dimensions. The study of these function theories is referred to as
Clifford analysis Brackx et al (1982); Huang et al (2006); Gong et al (2009);
Ryan (2000), which is closely related to a number of studies made in
mathematical physics, and many applications in this area have been found in
recent years. In Ryan (1995), Ryan considered solutions of the polynomial
Dirac operator, which afforded an integral representation. Furthermore, the
author gave a Pompeiu representation for $C^{1}$-functions in a Lipschitz
bounded domain. In Ryan (1990), the author presented a classification of
linear, conformally invariant, Clifford-algebra-valued differential operators
over $\mathbb{C}^{n}$, which comprised the Dirac operator and its iterates. In
Qian and Ryan (1996), Qian and Ryan used Vahlen matrices to study the
conformal covariance of various types of Hardy spaces over hypersurfaces in
$\mathbb{R}^{n}$. In De Ridder et al (2012), the discrete Fueter polynomials
was introduced, which formed a basis of the space of discrete spherical
monogenics. Moreover, the explicit construction for this discrete Fueter
basis, in arbitrary dimension $m$ and for arbitrary homogeneity degree $k$ was
presented as well.
In Hörmander (1965), the famous Hörmander’s $L^{2}$ existence and
approximation theorems was given for the $\bar{\partial}$ operator in pseudo-
convex domains in $\mathbb{C}^{n}$. When $n=1$, the existence theorem of
complex variable can be deduced. The aim of this paper is to establish a
Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{n+1}$ with Clifford analysis, and
present sufficient condition on the existence of the weak solutions for Dirac
operator in the sense of Clifford algebra.
Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ and the corresponding norm on $\mathcal{A}$ is
given by $|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$ (see subsection 2.1).
Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$,
$L^{2}(\Omega,\mathcal{A},\varphi)$ be a right Hilbert $\mathcal{A}$-module
for a given function $\varphi\in C^{2}(\Omega,\mathbb{R})$ with the norm given
by Definition 2.9. (see subsection 2.3). $\overline{D}$ denotes the Dirac
differential operator and the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ is given by (4). For
$x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$,
$\Delta=\sum_{i=0}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$. Then we can
obtain our main results as follows.
###### Theorem 1.1
Given $f\in L^{2}(\Omega,\mathcal{A},\varphi)$, there exists $u\in
L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\begin{split}\overline{D}u=f\end{split}$ (1)
with
$\begin{split}\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}c\end{split}$ (2)
if
$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}\leq
c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=c\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).\end{split}$ (3)
Conversely, if there exists $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
(1) is satisfied with
$\begin{split}\|u\|^{2}\leq c\end{split}$
Then we can get the inequality (3) for norm estimation.
The factor $2^{2n}$ in (2) comes from the definition of the norm in Clifford
analysis. If $n=1$, then the factor would disappear which gives a necessary
and sufficient condition in the theorem. From the above theorem, we give the
following sufficient condition on the existence of weak solutions for Dirac
operator.
###### Theorem 1.2
Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ and $n>1$; $\Delta\varphi\geq 0$,
and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$. Then for all $f\in L^{2}(\Omega,\mathcal{A},\varphi)$
with $\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
###### Remark 1.3
Assuming $x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$, it is easy to see
that $\varphi(x)=x_{0}^{2}$ satisfies the conditions in Theorem 1.2. Another
simple example would be
$\varphi(x)=(n+1)x_{0}^{2}-\sum_{i=1}^{n}x_{i}^{2}.$
It is obvious that $\Delta\varphi(x)=2$, $\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}=-2$, and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}=0,~{}i\neq j,~{}1\leq i,j\leq n$.
###### Corollary 1.4
Given $\varphi\in C^{2}(\Omega,\mathbb{R}),$ and $\varphi(x)=\varphi(x_{0})$
with $\varphi^{\prime\prime}(x_{0})\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx.$
It is noticed that there is nothing to do with the boundary conditions of
$\Omega$ in the above results. This phenomenon is totally different with the
famous Hörmander’s $L^{2}$ existence theorems of several complex variables in
Hörmander (1965). Then we can also have the following theorem on global
solutions.
###### Theorem 1.5
Given $\varphi\in C^{2}(\mathbb{R}^{n+1},\mathbb{R})$ with all derivative
conditions in Theorem 1.1 satisfied. Then for all $f\in
L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ with
$\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ satisfying
$\overline{D}u=f$
with
$\|u\|^{2}=\int_{\mathbb{R}^{n+1}}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
On the other hand, if the boundary of $\Omega$ is concerned, we consider a
special kind of domain ${\Omega}_{0}=\\{x\in\mathbb{R}^{n+1}:a\leq x_{0}\leq
b\\}$ for any $a,~{}b\in\mathbb{R}$ with $a<b$, then we can get the following
theorem within $L^{2}$ space instead of $L^{2}$ weighted space.
###### Theorem 1.6
Let $f\in L^{2}({\Omega_{0}},\mathcal{A})$. Then there exists a $u\in
L^{2}({\Omega_{0}},\mathcal{A})$ such that
$\overline{D}u=f$
with
$\int_{\Omega_{0}}|u|^{2}_{0}dx\leq
2^{2n}c(a,b)\int_{\Omega_{0}}{|f|^{2}_{0}}dx$
and $c(a,b)$ is a factor depending on $a,~{}b$.
###### Proof
Let $\varphi(x)=x_{0}^{2}$. It can be obtained that
$L^{2}({\Omega_{0}},\mathcal{A})=L^{2}({\Omega_{0}},\mathcal{A},\varphi)$ for
the boundary of $x_{0}$. Then the theorem is proved with Theorem 1.2.
###### Remark 1.7
In particular, any bounded domain $\Omega$ in $\mathbb{R}^{n+1}$ can be
regarded as one type of $\Omega_{0}$. Therefore, it comes from Theorem 1.6
that for any $f\in L^{2}(\Omega,\mathcal{A})$, we can find a weak solution of
Dirac operator $\overline{D}u=f$ with $u\in L^{2}(\Omega,\mathcal{A})$.
## 2 Preliminaries
To make the paper self-contained, some basic notations and results used in
this paper are included.
### 2.1 The Clifford algebra $\mathcal{A}$
Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ with orthogonal basis
$e:=\\{e_{0},e_{1},...,e_{n}\\}$, where $e_{0}=1$ is a unit element in
$\mathbb{R}^{n+1}$. Furthermore,
$\left\\{\begin{aligned} e_{i}e_{j}+e_{j}e_{i}&=0,~{}i\neq j\\\
e_{i}^{2}&=-1,~{}i=1,...,n.\end{aligned}\right.$
Then $\mathcal{A}$ has its basis
$\\{e_{A}=e_{h_{1}\cdots h_{r}}=e_{h_{1}}\cdots e_{h_{r}}:1\leq
h_{1}<...<h_{r}\leq n,1\leq r\leq n\\}.$
If $i\in\\{h_{1},...,h_{r}\\}$, we denote $i\in A$ and if
$i\not\in\\{h_{1},...,h_{r}\\}$, we denote $i\not\in A$. $A-{i}$ means
$\\{h_{1},...,h_{r}\\}\setminus\\{i\\}$ and $A+{i}$ means
$\\{h_{1},...,h_{r}\\}\cup\\{i\\}$. So the real Clifford algebra is composed
of elements having the type $a=\sum\limits_{A}x_{A}e_{A}$, in which
$x_{A}\in\mathbb{R}$ are real numbers. For $a\in\mathcal{A}$, we give the
inversion in the Clifford algebra as follows:
$a^{*}=\sum\limits_{A}x_{A}e_{A}^{*}$ where $e_{A}^{*}=(-1)^{|A|}e_{A}$ and
$|A|=n(A)$ is the $r\in\mathbb{Z}^{+}$ as $e_{A}=e_{h_{1}\cdots h_{r}}$. When
$A=\emptyset$, $e_{A}=e_{0}$, $|A|=0$. Next, we define the reversion in the
Clifford algebra, which is given by
$a^{\dagger}=\sum\limits_{A}x_{A}e_{A}^{\dagger}$ where
$e_{A}^{\dagger}=(-1)^{(|A|-1)|A|/2}e_{A}.$ Now we present the involution
which is a combination of the inversion and the reversion introduced above.
$\bar{a}=\sum\limits_{A}x_{A}\bar{e}_{A}$
where $\bar{e}_{A}=e_{A}^{*{\dagger}}=(-1)^{(|A|+1)|A|/2}e_{A}.$ From the
definition, one can easily deduce that $e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1.$
Furthermore, we have
$\overline{\lambda\mu}=\bar{\mu}\bar{\lambda},~{}~{}\forall\lambda,\mu\in\mathcal{A}.$
Let $a=\sum\limits_{A}x_{A}e_{A}$ be a Clifford number. The coefficient
$x_{A}$ of the $e_{A}$-component will also be denoted by $[a]_{A}$. In
particular the coefficient $x_{0}$ of the $e_{0}$-component will be denoted by
$[a]_{0}$, which is called the scalar part of the Clifford number $a$. An
inner product on $\mathcal{A}$ is defined by putting for any
$\lambda,\mu\in\mathcal{A}$,
$(\lambda,\mu)_{0}:=2^{n}[\lambda\bar{\mu}]_{0}=2^{n}\sum\limits_{A}\lambda_{A}\mu_{A}$.
The corresponding norm on $\mathcal{A}$ reads
$|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$.
We define a real functional on $\mathcal{A}$ that
$\tau_{e_{A}}:\mathcal{A}\rightarrow\mathbb{R}$
$\langle\tau_{e_{A}},\mu\rangle=2^{n}(-1)^{(|A|+1)|A|/2}\mu_{A}.$
In the special case where $A=\emptyset$ we have
$\langle\tau_{e_{0}},\mu\rangle=2^{n}\mu_{0}.$
Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$. Then functions $f$
defined in $\Omega$ and with values in $\mathcal{A}$ are considered. They are
of the form
$f(x)=\sum_{A}f_{A}(x)e_{A}$
where $f_{A}(x)$ are functions with real value. Let $\overline{D}$ denotes the
Dirac differential operator
$\overline{D}=\sum_{i=0}^{n}e_{i}\partial_{x_{i}},$
its action on functions from the left and from the right being governed by the
rules
$\overline{D}f=\sum_{i,A}e_{i}e_{A}\partial_{x_{i}}f_{A}~{}\mbox{and}~{}f\overline{D}=\sum_{i,A}e_{A}e_{i}\partial_{x_{i}}f_{A}.$
$f$ is called left-monogenic if $\overline{D}f=0$ and it is called right-
monogenic if $f\overline{D}=0$. The conjugate operator is given by
$D=\sum_{i=0}^{n}\bar{e}_{i}\partial_{x_{i}}.$
It can be found that
$\overline{D}D=D\overline{D}=\Delta$
where $\Delta$ denotes the classical Laplacian in $\mathbb{R}^{n+1}$. When
$n=1$, one can think of $x_{0}$ as the real part and of $x_{1}$ as the
imaginary part of the variable and to identify $e_{1}$ with $i$. the operator
$\overline{D}$ then take the form
$\overline{D}=\partial_{x_{0}}+i\partial_{x_{1}}$, which is similar with the
operator $\bar{\partial}$ in complex analysis.
### 2.2 Modules over Clifford algebras
This subsection is to give some general information concerning a class of
topological modules over Clifford algebras. In the sequel definitions and
properties will be stated for left $\mathcal{A}$-module and their duals, the
passage to the case of right $\mathcal{A}$-module being straight-forward.
###### Definition 2.1
(unitary left $\mathcal{A}$-module) Let $X$ be a unitary left
$\mathcal{A}$-module, i.e. $X$ is abelian group and a law
$(\lambda,f)\rightarrow\lambda f:\mathcal{A}\times X\rightarrow X$ is defined
such that $\forall\lambda,\mu\in\mathcal{A}$, and $f,~{}g\in X$
1. (1)
$(\lambda+\mu)f=\lambda f+\mu f$,
2. (2)
$\lambda\mu f=\lambda(\mu f)$,
3. (3)
$\lambda(f+g)=\lambda f+\lambda g$,
4. (4)
$e_{0}f=f$.
Moreover, when speaking of a submodule $E$ of the unitary left
$\mathcal{A}$-module $X$, we mean that $E$ is a non empty subset of $X$ which
becomes a unitary left $\mathcal{A}$-module too when restricting the module
operations of $X$ to $E$.
###### Definition 2.2
(left $\mathcal{A}$-linear operator) If $X,Y$ are unitary left
$\mathcal{A}$-modules, then $T:X\rightarrow Y$ is said to be a left
$\mathcal{A}$-linear operator, if $\forall~{}f,~{}g\in X$ and
$\lambda\in\mathcal{A}$
$T(\lambda f+g)=\lambda T(f)+T(g).$
The set of all $``T"$ is denoted by $L(X,Y)$. If
$Y=\mathcal{A},~{}L(X,\mathcal{A})$ is called the algebraic dual of $X$ and
denoted by $X^{*alg}$. Its elements are called left $\mathcal{A}$-linear
functionals on $X$ and for any $T\in X^{*alg}$ and $f\in X$, we denote by
$\langle T,f\rangle$ the value of $T$ at $f$.
###### Definition 2.3
(bounded functional) An element $T\in X^{*alg}$ is called bounded, if there
exist a semi-norm $p$ on $X$ and $c>0$ such that for all $f\in X$
$|\langle T,f\rangle|_{0}\leq c\cdot p(f).$
###### Theorem 2.4
(Hahn-Banach type theorem)Brackx et al (1982) Let $X$ be a unitary left
$\mathcal{A}$-module with semi-norm $p$, $Y$ be a submodule of $X$, and $T$ be
a left $\mathcal{A}$-linear functional on $Y$ such that for some $c>0,$
$|\langle T,g\rangle|_{0}\leq c\cdot p(g),~{}~{}\forall g\in Y$
Then there exists a left $\mathcal{A}$-linear functional $\widetilde{T}$ on
$X$ such that
1. (1)
$\widetilde{T}\mid_{Y}=T$,
2. (2)
for some $c^{*}>0$, $|\langle\widetilde{T},f\rangle|_{0}\leq c^{*}\cdot p(f)$,
$\forall f\in X$.
###### Definition 2.5
(inner product on a unitary right $\mathcal{A}$-module) Let $H$ be a unitary
right $\mathcal{A}$-module, then a function $(~{},~{}):~{}H\times
H\rightarrow\mathcal{A}$ is said to be a inner product on $H$ if for all
$f,g,h\in H$ and $\lambda\in\mathcal{A}$,
1. (1)
$(f,g+h)=(f,g)+(f,h)$,
2. (2)
$(f,g\lambda)=(f,g)\lambda$,
3. (3)
$(f,g)=\overline{(g,f)}$,
4. (4)
$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0~{}\mbox{if and only if}~{}f=0$,
5. (5)
$\langle\tau_{e_{0}},(f\lambda,f\lambda)\rangle\leq|\lambda|^{2}_{0}\langle\tau_{e_{0}},(f,f)\rangle$.
From the definition on inner product, putting for each $f\in H$
$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle,$
then it can be obtained that for any $f,g\in H,$
$\begin{split}|\langle\tau_{e_{0}},~{}(f,g)\rangle|\leq\|f\|\|g\|,\|f+g\|\leq\|f\|+\|g\|.\end{split}$
Hence, $\|\cdot\|$ is a proper norm on $H$ turning it into a normed right
$A$-module. Moreover, we have the following Cauchy-Schwarz inequality.
###### Proposition 2.6
Brackx et al (1982) For all $f,g\in H,$ $|(f,g)|_{0}\leq\|f\|\|g\|.$
###### Definition 2.7
(right Hilbert $\mathcal{A}$-module) Let $H$ be a unitary right
$\mathcal{A}$-module provided with an inner product $(~{},~{})$. Then is it
called a right Hilbert $\mathcal{A}$-module if it is complete for the norm
topology derived from the inner product.
###### Theorem 2.8
(Riesz representation theorem)Brackx et al (1982) Let $H$ be a right Hilbert
$\mathcal{A}$-modules and $T\in H^{*alg}$. Then $T$ is bounded if and only if
there exists a (unique) element $g\in H$ such that for all $f\in H$,
$T(f):=\langle T,f\rangle=(g,f).$
### 2.3 Hilbert space of square integrable functions
Now we extend the standard Hilbert space of square integrable functions to
Clifford algebra. First, we denote $L^{1}(\Omega,\mu)$ and $L^{2}(\Omega,\mu)$
be the sets of all integrable or square integrable functions defined on the
domain $\Omega\in\mathbb{R}^{n+1}$ with respect to the measure $\mu$. Then
$L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
defined as the sets of functions $f:\Omega\rightarrow\mathcal{A}$ which are
integrable or square integrable with respect to $\mu$, i.e., if
$f=\sum\limits_{A}f_{A}e_{A}$, then for each $A$, $f_{A}\in L^{1}(\Omega,\mu)$
and $f^{2}_{A}\in L^{1}(\Omega,\mu)$, respectively. Then one may easily check
that $L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
unitary bi-$\mathcal{A}$-module, i.e., unitary left-$\mathcal{A}$-module and
unitary right-$\mathcal{A}$-module. Furthermore, for any $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$, $\bar{f}\in L^{2}(\Omega,\mathcal{A},\mu)$
while $\bar{f}g\in L^{1}(\Omega,\mathcal{A},\mu)$, where
$\bar{f}(x)=\overline{f(x)}$ and $(\bar{f}g)(x)=\bar{f}(x)g(x),~{}x\in\Omega$.
Consider as a right $\mathcal{A}$-module, define for $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$ that
$(f,g)=\int_{\Omega}\bar{f}(x)g(x)d\mu.$
Furthermore for any real linear functional $T$ on $\mathcal{A}$
$\langle T,(f,g)\rangle=\langle
T,\int_{\Omega}\bar{f}(x)g(x)d\mu\rangle=\int_{\Omega}\langle
T,\bar{f}(x)g(x)\rangle d\mu.$
Consequently, taking $T=\tau_{e_{0}}$ we find that
$\begin{split}\langle\tau_{e_{0}},(f,f)\rangle&=\langle\tau_{e_{0}},\int_{\Omega}\bar{f}(x)f(x)d\mu\rangle=\int_{\Omega}\langle\tau_{e_{0}},\bar{f}(x)f(x)\rangle
d\mu\\\ &=\int_{\Omega}|f(x)|^{2}_{0}d\mu.\end{split}$
Hence, for all $f\in L^{2}(\Omega,\mathcal{A},\mu)$,
$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0$ if and only if $f=0$ a.e. in $\Omega$.
Then it is easy to see that under the inner product defined, all conditions
for $L^{2}(\Omega,\mathcal{A},\mu)$ to be a unitary right inner product
$\mathcal{A}$-module are satisfied. Since
$L^{2}(\Omega,\mathcal{A},\mu)=\prod_{A}L^{2}(\Omega,\mu)$, we have that
$L^{2}(\Omega,\mathcal{A},\mu)$ is complete; in other words
$L^{2}(\Omega,\mathcal{A},\mu)$ is a right Hilbert $\mathcal{A}$-module, with
the norm
$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle=\int_{\Omega}|f(x)|^{2}_{0}d\mu$
for $f\in L^{2}(\Omega,\mathcal{A},\mu)$.
###### Definition 2.9
(weighted $L^{2}$ space) Similar with $L^{2}(\Omega,\mathcal{A},\mu)$, we can
define the weighted $L^{2}(H,\mathcal{A},\varphi)$ for a given function
$\varphi\in C^{2}(\Omega,\mathbb{R})$. First, let
$L^{2}(\Omega,\varphi)=\big{\\{}f|f:\Omega\rightarrow\mathbb{R},~{}\int_{\Omega}|f(x)|^{2}e^{-\varphi}~{}dx<+\infty\big{\\}}.$
Then we denote
$L^{2}(H,\mathcal{A},\varphi)=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L^{2}(\Omega,\varphi)\\}.$
Moreover, for all $f,g\in L^{2}(H,\mathcal{A},\varphi)$, we define
$(f,g)_{\varphi}=\int_{\Omega}\bar{f}(x)g(x)e^{-\varphi}dx.$
Then it is also easy to see $L^{2}(\Omega,\mathcal{A},\varphi)$ is a right
Hilbert $\mathcal{A}$-module, with the norm
$\begin{split}\|f\|^{2}=\langle\tau_{e_{0}},(f,f)_{\varphi}\rangle=\int_{\Omega}|f(x)|^{2}_{0}e^{-\varphi}dx\end{split}$
for $f\in L^{2}(\Omega,\mathcal{A},\varphi)$.
### 2.4 Cauchy’s integral formula
Let $M$ be an (n+1)-dimensional differentiable and oriented manifold contained
in some open subset $\Sigma$ of $\mathbb{R}^{n+1}$. By means of the n-forms
$d\hat{x}_{i}=dx_{0}\wedge\cdots\wedge dx_{i-1}\wedge
dx_{x_{i+1}}\wedge\cdots\wedge dx_{n},~{}i=0,1,...,n,$
an $\mathcal{A}$-valued n-form is introduced by putting
$d\sigma=\sum_{i=0}^{n}(-1)^{i}e_{i}d\hat{x}_{i},$
similarly, denote
$d\bar{\sigma}=\sum_{i=0}^{n}(-1)^{i}\bar{e}_{i}d\hat{x}_{i}.$
Furthermore the volume-element
$dx=dx_{0}\wedge\cdots\wedge dx_{n}$
is used.
###### Proposition 2.10
(Stokes-Green Theorem)Brackx et al (1982) If $f,g\in
C^{1}(\Sigma,\mathcal{A})$ then for any (n+1)-chain $\Omega$ on
$M\subset\Sigma$,
$\int_{\partial\Omega}fd\sigma
g=\int_{\Omega}(f\overline{D})gdx+\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\partial\Omega}fd\bar{\sigma}g=\int_{\Omega}(fD)gdx+\int_{\Omega}f(Dg)dx.$
###### Remark 2.11
Denote $C^{\infty}_{0}(\Omega,\mathbb{R})$ as the set of all smooth real-
valued functions with compact support in $\Omega$ and
$C^{\infty}_{0}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
C^{\infty}_{0}(\Omega,\mathbb{R})\\}.$ If $f$ or $g\in
C^{\infty}_{0}(\Omega,\mathcal{A})$, then we have from the Stokes-Green
theorem that
$\int_{\Omega}(f\overline{D})gdx=-\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\Omega}(fD)gdx=-\int_{\Omega}f(Dg)dx.$
###### Lemma 2.12
If $u(x)\in C^{1}(\Omega,\mathcal{A})$, then
$\overline{\overline{D}u}=\bar{u}D$.
###### Proof
Let $u(x)=\sum_{A}e_{A}u_{A}$. Then
$\begin{split}\overline{\overline{D}u}=\sum_{i,A}\overline{e_{i}e_{A}}\partial_{x_{i}}u_{A}=\sum_{i,A}\bar{e}_{A}\bar{e}_{i}\partial_{x_{i}}u_{A}=\bar{u}D.\end{split}$
###### Lemma 2.13
Huang et al (2006) If $u(x)=\sum_{A}e_{A}u_{A}$,
$v(x)=\sum_{i=0}^{n}e_{i}v_{i}$, then
$\overline{D}(uv)=(\overline{D}u)v+u(\overline{D}v)+\sum\limits^{n}_{j=1}(e_{j}u-ue_{j})\partial_{x_{j}}v.$
### 2.5 Weak solutions
Let
$L_{loc}^{1}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L_{loc}^{1}(\Omega,\mathbb{R})\\}$. Then we define the weak solution in the
sense of Clifford algebra as follows.
###### Definition 2.14
($\overline{D}$ solution in weak sense) If $f\in
L_{loc}^{1}(\Omega,\mathcal{A})$, $u:\Omega\rightarrow\mathcal{A}$ is a weak
solution of
$\overline{D}u=f~{}(\mbox{or}~{}{D}u=f)$
if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha\overline{D})udx~{}(\mbox{or}~{}\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha{D})udx).$
It should be noticed that if $u$ is a weak solution of Dirac equation
$\overline{D}u=0$, in addition, if $u$ is smooth in $\Omega$, then it is left-
monogenic. Now it is natural to give the definition of $\Delta$ solution in
the weak sense.
###### Definition 2.15
($\Delta$ solution in weak sense) If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$,
$u:\Omega\rightarrow\mathcal{A}$ is a weak solution of
$\Delta u=f$
if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\int_{\Omega}\alpha fdx=\int_{\Omega}({\Delta}\alpha)udx.$
###### Theorem 2.16
If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$, and $\overline{D}f=0$ in weak
sense, then $f$ is left-monogenic at any point of $\Omega$.
###### Proof
: Since $\overline{D}f=0$ in weak sense, then $\Delta f=0$ in weak sense. By
Weyl’s lemma, $f$ is smooth in $\Omega$ and has $\Delta f=0$ in classical
sense, then of course $f$ is left-monogenic at any point of $\Omega$.
###### Remark 2.17
This is useful to deal with uniqueness of weak solutions. for example, if
$u,~{}v\in L_{loc}^{1}(\Omega,\mathcal{A})$ are two weak solutions of
$\overline{D}u=f$, then $u=v+w$ with any $w$ left-monogenic.
###### Remark 2.18
An important example of a left monogenic function is the generalized Cauchy
kernel
$G(x)=\frac{1}{\omega_{n+1}}\frac{\overline{x}}{|x|^{n+1}},$
where $\omega_{n+1}$ denotes the surface area of the unit ball in
$\mathbb{R}^{n+1}$. This function obviously belongs to
$L_{loc}^{1}(\Omega,\mathcal{A})$ and is a fundamental solution of the Dirac
equation in the classical sense at any point of $\mathbb{R}^{n+1}$ except 0.
However, it is not a weak solution of the Dirac operator. In fact, if it
satisfies $\overline{D}f=0$ in the weak sense, then from Theorem 2.16, it must
be left-monogenic in the any point of $\Omega$ which could include $0$.
Therefore, we get a contradiction.
For $f\in L^{2}(\Omega,\mathcal{A},\varphi)$,
$u:\Omega\rightarrow\mathcal{A}$. If $\overline{D}u=f$, based on the Stokes-
Green theorem, we can define the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ under the inner product of $L^{2}(\Omega,\mathcal{A},\varphi)$.
For any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,
$\begin{split}(\alpha,f)_{\varphi}=&~{}\int_{\Omega}\bar{\alpha}fe^{-\varphi}dx=\int_{\Omega}\bar{\alpha}e^{-\varphi}fdx\\\
=&~{}\int_{\Omega}(\bar{\alpha}e^{-\varphi})(\overline{D}u)dx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}udx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}e^{\varphi}ue^{-\varphi}dx\\\
=&~{}\int_{\Omega}\overline{-e^{\varphi}D(\alpha
e^{-\varphi})}ue^{-\varphi}dx\\\ =&~{}(-e^{-\varphi}D(\alpha
e^{-\varphi}),u)_{\varphi}\triangleq(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi},\end{split}$
(4)
where $\overline{D}^{*}_{\varphi}\alpha=-e^{\varphi}D(\alpha
e^{-\varphi})=\alpha(D\varphi)-D\alpha$, i.e.
$(\alpha,\overline{D}u)_{\varphi}=(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi}.$
In the same way, we also have
$(\overline{D}u,\alpha)_{\varphi}=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}.$
## 3 The proof of Theorem 1.1
Now we are in the position of proving Theorem 1.1.
###### Proof
($Sufficiency$) From the definition of dual operator and Cauchy-Schwarz
inequality in Proposition 2.6, we have
$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}=&|(\overline{D}u,\alpha)_{\varphi}|^{2}_{0}=|(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}|^{2}_{0}\\\
\leq&~{}\|u\|^{2}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\\\
\leq&~{}c\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$
($necessity$) We aim to prove the necessity with Riesz representation theorem.
First, we denote the submodule
$E=\\{\overline{D}^{*}_{\varphi}\alpha,~{}\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}),~{}\varphi\in
C^{2}(\Omega,\mathbb{R})\\}\subset L^{2}(\Omega,\mathcal{A},\varphi).$
Then we define a linear functional $L_{f}$ on $E$, i.e., $L_{f}\in E^{*alg}$
for a fixed $f\in L^{2}(\Omega,\mathcal{A},\varphi)$ as follows,
$\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(f,\alpha)_{\varphi}=\int_{\Omega}\bar{f}\cdot\alpha\cdot
e^{-\varphi}dx\in\mathcal{A}.$
From (3), we have
$|\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle|_{0}=|(f,\alpha)_{\varphi}|_{0}\leq\sqrt{c}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|,$
which meas that $L_{f}$ is a bounded functional from Definition 2.3. By the
Hahn-Banach type theorem in Theorem 2.4, $L_{f}$ can be extended to a linear
functional $\widetilde{L}_{f}$ on $L^{2}(\Omega,\mathcal{A},\varphi)$, and
with
$\begin{split}|\langle\widetilde{L}_{f},g\rangle|_{0}\leq\sqrt{c^{*}}\|g\|,~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi),\end{split}$ (5)
where $\sqrt{c^{*}}=\sqrt{c}\cdot|e_{0}|_{0}$, since $|e_{A}|_{0}=2^{n/2}$,
then $c^{*}=2^{n}c$ from Brackx et al (1982). Now we are in the position to
use the Riesz representation theorem for the operator $\widetilde{L}_{f}$.
From Theorem 2.8, there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such
that
$\begin{split}\langle\widetilde{L}_{f},g\rangle=(u,g)_{\varphi},~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi).\end{split}$ (6)
For $\forall\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$, let
$g=\overline{D}^{*}_{\varphi}\alpha$. Then
$\begin{split}(f,\alpha)_{\varphi}=&\langle\widetilde{L}_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}=(\overline{D}u,\alpha)_{\varphi},\end{split}$
which deduces that
$\int_{\Omega}\bar{f}\alpha
e^{-\varphi}dx=\int_{\Omega}\overline{(\overline{D}u)}{\alpha}e^{-\varphi}dx.$
Conjugating both sides of above equation leads to
$\int_{\Omega}\bar{\alpha}f\cdot
e^{-\varphi}dx=\int_{\Omega}\bar{\alpha}(\overline{D})ue^{-\varphi}dx.$
Let $\alpha=\bar{\alpha}e^{\varphi}$, it can be obtained that
$\int_{\Omega}\alpha
fdx=\int_{\Omega}\alpha(\overline{D}u)dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$
Therefore,
$\overline{D}u=f$
is proved from the definition of weak solutions.
Next, we give the bound for the norm of $u$. Let $g=u=\sum_{A}e_{A}u_{A}\in
L^{2}(\Omega,\mathcal{A},\varphi)$, from (5) and (6), we get that
$\begin{split}|(u,u)_{\varphi}|_{0}\leq\sqrt{c^{*}}\|u\|.\end{split}$ (7)
On the other hand,
$\begin{split}|(u,u)_{\varphi}|_{0}^{2}=&\big{|}\int_{\Omega}\bar{u}ue^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}2^{n}\cdot\big{[}\int_{\Omega}\bar{u}ue^{-\varphi}dx\cdot\overline{\int_{\Omega}\bar{u}ue^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx\cdot\overline{\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}(\int_{\Omega}\sum\limits_{A}u^{2}_{A}e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}\big{]},\end{split}$
and
$\begin{split}\|u\|^{2}=&~{}\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}[\bar{u}u]_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx\end{split}$
So we have $\|u\|^{4}=2^{2n}\cdot(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}$. Hence,
$|(u,u)_{\varphi}|_{0}^{2}=2^{n}[(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}]\geq 2^{-n}\|u\|^{4}.$
Combining with (7), it is obtained that
$\|u\|^{2}\leq 2^{n/2}|(u,u)_{\varphi}|_{0}\leq 2^{n/2}\sqrt{c^{*}}\|u\|,$
and
$\|u\|^{2}\leq 2^{2n}{c}.$
The proof is completed.
## 4 The proof of Theorem 1.2
It should be noticed that inequality (3) in Theorem 1.1 is related with
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$. In the following, we will give
another sufficient condition that has nothing to do with the space
$C^{\infty}_{0}(\Omega,\mathcal{A})$. First, we need to compute the norm of
$\|\overline{D}^{*}_{\varphi}\alpha\|$ for any $\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$
$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=&\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha\rangle
e^{-\varphi}dx\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha
e^{-\varphi}dx\rangle\\\
=&\langle\tau_{e_{0}},(\overline{D}^{*}_{\varphi}\alpha,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}(\alpha(D\varphi)-D\alpha))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\alpha(D\varphi)+\alpha\Delta\varphi-\Delta\alpha+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha)+\alpha\Delta\varphi+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}+(\alpha,\alpha\Delta\varphi)_{\varphi}+(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&I_{1}+I_{2}+I_{3},\end{split}$
where
$\begin{split}I_{1}=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle=\langle\tau_{e_{0}},(\overline{D}\alpha,\overline{D}\alpha)_{\varphi}\rangle=\|\overline{D}\alpha\|^{2},\\\
I_{2}=&\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle=\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx,\end{split}$
and
$\begin{split}I_{3}=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial
x_{j}}(\sum_{i=0}^{n}\bar{e}_{i}\frac{\partial\varphi}{\partial
x_{i}}))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}})_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}e^{-\varphi}dx\rangle\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle e^{-\varphi}dx.\end{split}$
It should be noticed that if $n=1$, i.e., the space $\mathbb{R}^{2}$ is
considered, then $I_{3}=0.$
Since for $1\leq i,j\leq n$ and $i\neq j$,
$e_{j}\bar{e}_{i}=-e_{j}{e}_{i}=e_{i}{e}_{j}=-{e}_{i}\bar{e}_{j}$. For
simplicity, let
$\begin{split}I_{4}=&\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum\limits_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}\sum\limits_{i=1}^{n}(\bar{\alpha}e_{j}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{i}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\ =&I_{5}+I_{6}+I_{7}.\end{split}$
Assume
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}\in\mathcal{A},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$,
then for any $1\leq i\leq n,$
$\begin{split}\bar{\alpha}e_{i}\alpha\bar{e}_{i}=&~{}\sum\limits_{A}\alpha_{A}\bar{e}_{A}e_{i}\cdot\sum\limits_{A}\alpha_{A}e_{A}\bar{e}_{i}\\\
=&~{}\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i}\end{split}$
Therefore
$\begin{split}I_{5}=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha_{A}^{2}e_{A}e_{i}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha^{2}_{A}\cdot\overline{e_{A}e_{i}}\cdot
e_{A}e_{i}\cdot(-1)^{\frac{(|A|+1)(|A|+2)}{2}}\\\ &+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\cdot\alpha^{2}_{A}\cdot\overline{e_{A-{i}}}\cdot
e_{A-{i}}\cdot(-1)^{\frac{(|A|-1)(|A|)}{2}})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|+1)(|A|+2)}{2}}\cdot\alpha^{2}_{A}\\\
&+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|-1)(|A|)}{2}}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{|A|^{2}}\cdot\alpha^{2}_{A}+\sum\limits_{i\in
A}(-1)^{|A|^{2}+1}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}(-2)\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is
even}}(-2)\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\\\
=&-2^{n+1}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^{2}~{}\mbox{is
odd}}\alpha^{2}_{A}+\sum\limits_{i\in A,|A|^{2}~{}\mbox{is
even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}.\end{split}$ (8)
To consider $I_{7}$, we first study $\bar{\alpha}e_{j}\alpha$ for any $1\leq
j\leq n$. Without loss of generality, let
$e_{j}=e_{1},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A},~{}\alpha=\sum\limits_{A}\alpha_{A}e_{A}$.
Then
$\bar{\alpha}e_{1}\alpha=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}(\sum\limits_{A}\alpha_{A}e_{A})$.
When $e_{A}=e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}}\cdot e_{1}e_{h_{2}}e_{h_{3}}\cdots
e_{h_{r}}\cdot e_{1}\\\ =&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}=\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (9)
When $e_{A}=e_{1}$,
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1}\\\
\alpha_{A}e_{A}e_{1}=&-\alpha_{1}.\end{split}$ (10)
When $e_{A}=e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}}\cdot e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\cdot
e_{1}\\\ =&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1}=\alpha_{h_{2}\cdots h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (11)
When $e_{A}=e_{0}$,
$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{0}e_{1}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{0}e_{1}.\end{split}$ (12)
To compute $I_{7}$, one needs to know the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha-\bar{\alpha}\alpha e_{1}$. It means that we should
find out the corresponding terms of $e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$
and $e_{h_{2}}\cdots e_{h_{r}}$ in $\bar{\alpha}e_{1}$ and $\alpha$, in
$\bar{\alpha}$ and $\alpha e_{1}$.
Case a1. For $\bar{\alpha}e_{1}\alpha$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in
$\bar{\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}$ and
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and
$\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to
$\begin{split}(-1)&{}^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r-1)(r)}{2}+r-1}(-1)^{\frac{(r)(r+1)}{2}}\cdot\overline{e_{1}\cdots
e_{h_{r}}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r-1+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (13)
On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in $\bar{\alpha}e_{1}$ and
$\alpha$ are $\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ and
$\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to
$\begin{split}(-1)&{}^{\frac{(r)(r+1)}{2}+r}e_{h_{2}\cdots h_{r}}\cdot
e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r}(-1)^{\frac{(r-1)(r)}{2}}\cdot\overline{e_{h_{2}\cdots
h_{r}}}\cdot e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (14)
From (13) and (14), these two terms vanish.
Case a2. For $\bar{\alpha}e_{1}\alpha$, from (12), the corresponding terms of
$e_{1}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}e_{1}$, respectively. Multiplying these terms leads to
$\begin{split}\alpha_{0}e_{1}\alpha_{1}e_{1}=-\alpha_{0}\alpha_{1}.\end{split}$
(15)
On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (10), the corresponding
terms of $e_{0}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{1}$ and
$\alpha_{0}$, respectively. Multiplying these terms leads to
$\alpha_{0}\alpha_{1}$. Combining with (15), these two terms also vanish.
From Cases a1 and a2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,
$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{0}}\rangle=0.\end{split}$ (16)
Case b1. For $\bar{\alpha}\alpha e_{1}$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in ${\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}{e}_{A})e_{1}$ and
$\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and $\alpha_{1h_{2}\cdots
h_{r}}\overline{e_{1}e_{h_{2}}\cdots e_{h_{r}}}$, respectively. Multiplying
these terms leads to
$\begin{split}(\alpha_{1h_{2}\cdots h_{r}}&\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{1h_{2}\cdots h_{r}}\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}})\\\
=&~{}(-1)^{r-1}\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}.\end{split}$ (17)
On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in ${\alpha}e_{1}$ and
$\bar{\alpha}$ are $\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}$ and $\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}}$, respectively. Multiplying these terms leads to
$\begin{split}(\alpha_{h_{2}\cdots h_{r}}&\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{1h_{2}\cdots h_{r}}e_{1}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}})\\\ =&~{}(-1)^{r}\alpha_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}.\end{split}$ (18)
From (17) and (18), these two terms vanish.
Case b2. For $\bar{\alpha}\alpha e_{1}$, from (12), the corresponding terms of
$e_{1}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}\bar{e}_{1}$, respectively. Multiplying these terms leads to
$\begin{split}\alpha_{0}e_{1}\alpha_{1}\bar{e}_{1}=\alpha_{0}\alpha_{1}.\end{split}$
(19)
On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (10), the
corresponding terms of $e_{0}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are
$-\alpha_{1}$ and $\alpha_{0}$, respectively. Multiplying these terms leads to
$-\alpha_{0}\alpha_{1}$. Combining with (19), these two terms also cancel.
From Cases b1 and b2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,
$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle=0.\end{split}$ (20)
Thus, $I_{7}=0$ from (16) and (20).
To compute $I_{6}$, i.e., to get $[\bar{\alpha}e_{i}\alpha\bar{e}_{j}]_{0}$
for $i\neq j$, similar with the analysis of $I_{7}$, we should divide the
vectors in $\bar{\alpha}e_{i}$ and $\alpha\bar{e}_{j}$ into four cases.
Case c1. $i\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B-{j}$.
For this case, firstly, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$
and $h_{p(i)}=i$, $e_{B}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and
$h_{p(j)}=j$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)}e_{h_{1}}\cdots e_{i}^{2}\cdots
e_{h_{r}},\\\ =&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r-p(j)}e_{h_{1}}\cdots
e_{j}\bar{e}_{j}\cdots e_{h_{r}},\\\
=&\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}}\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1+r-p(j)+\frac{r(r-1)}{2}}\overline{e_{A-{i}}}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}.\end{split}$ (21)
Case c2. $i\not\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B+{j}$.
We assume $e_{A}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and $h_{p(j)}=j$,
$e_{B}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$ and $h_{p(i)}=i$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot{e}_{j}\\\ =&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot e_{i}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+\frac{(r+1)(r+2)}{2}}\overline{e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}}e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot
e_{i}\\\ =&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}.\end{split}$
Case c3. $i\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B+{j}$.
For this case, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{p(j)}\cdots
h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$. Without loss of generality, we
assume $i<j$. Furthermore, let $e_{B}=e_{h_{1}\cdots h_{r}}$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{j}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+2-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot e^{2}_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}.\end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}(-1)\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j}}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$
Case c4. $i\not\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B-{j}$.
For this case, we assume $e_{A}=e_{h_{1}\cdots h_{r}}$, $e_{B}=e_{h_{1}\cdots
h_{p(i)}\cdots h_{p(j)}\cdots h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$ and
$i<j$. We have
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r+2}}\cdot e_{j}\bar{e}_{j}\\\
=&\alpha_{B}(-1)^{r+2-h(j)+r+2-h(i)-1}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ =&\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ \end{split}$
Then
$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)}\cdot e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$
Combining cases c1-c4, we have
$\begin{split}I_{6}=&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{j}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{i}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\rangle\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&2^{n}\sum\limits^{n}_{j\neq i}\Big{(}\sum_{i\in
A,~{}j\not\in A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\
&+\sum_{i\not\in A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\
&+\sum_{i\in A,~{}j\in
A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\ &+\sum_{i\not\in
A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}.\end{split}$
In all,
$\begin{split}I_{3}=&\int_{\Omega}I_{4}e^{-\varphi}dx\\\
=&\int_{\Omega}(I_{5}+I_{6}+I_{7})e^{-\varphi}dx\\\
=&-2^{n+1}\int_{\Omega}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}e^{-\varphi}dx\\\ &+2^{n}\int_{\Omega}\sum\limits^{n}_{j\neq
i}\Big{(}\sum_{i\in A,~{}j\not\in
A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\ &+\sum_{i\not\in
A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\ &+\sum_{i\in
A,~{}j\in A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\
&+\sum_{i\not\in A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}e^{-\varphi}dx.\end{split}$
Then
$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=\|\overline{D}\alpha\|^{2}+\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx+I_{3}.\end{split}$ (22)
If $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$, we have $I_{3}\geq 0$, and
$\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\geq\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx.$
With the above analysis, we can prove Theorem 1.2 easily.
###### Proof
It is sufficient to prove the theorem if condition (3) in Theorem 1.1 is
presented. By Cauchy-Schwarz inequality in Proposition 2.6, we have for any
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$ that
$\begin{split}|({f},\alpha)_{\varphi}|^{2}_{0}=&\big{|}\int_{\Omega}\bar{f}\cdot\alpha
e^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}\big{|}\int_{\Omega}\bar{f}\cdot\frac{1}{\sqrt{\Delta\varphi}}\cdot\alpha\cdot\sqrt{\Delta\varphi}\cdot
e^{-\varphi}dx\big{|}^{2}_{0}\\\
\leq&~{}\big{\|}\bar{f}\frac{1}{\sqrt{\Delta\varphi}}\big{\|}^{2}\cdot\big{\|}\alpha\cdot\sqrt{\Delta\varphi}\big{\|}^{2}\\\
=&~{}\int_{\Omega}\big{|}\frac{\bar{f}}{\sqrt{\Delta\varphi}}\big{|}^{2}_{0}e^{-\varphi}dx\cdot\int_{\Omega}\big{|}\alpha\cdot\sqrt{\Delta\varphi}\big{|}^{2}_{0}e^{-\varphi}dx\\\
\leq&c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$
The proof is completed with Theorem 1.1.
It should be noticed that when $n=1$, $I_{3}=0$. Then it comes from equation
(22) that the Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{2}$ could be
described which equals the classical Hörmander’s $L^{2}$ theorem in
$\mathbb{C}$.
###### Corollary 4.1
Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ with $\Omega$ being an open subset
of $\mathbb{R}^{2}$; $\Delta\varphi\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there
exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
$\overline{D}u=f$
with
$\|u\|^{2}\leq\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$
## 5 Conclusion
In this paper, based on the Hörmander’s $L^{2}$ theorem in complex analysis,
the Hörmander’s $L^{2}$ theorem for Dirac operator in $\mathbb{R}^{n+1}$ has
been obtained by Clifford algebra. When $n=1$, the result is equivalent to the
classical Hörmander’s $L^{2}$ theorem in complex variable. Moreover, for any
$f$ in $L^{2}$ space over a bounded domain with value in Clifford algebra,
there is a weak solution of Dirac operator with the solution in the $L^{2}$
space as well. The potential applications of the results will be studied in
our future work.
###### Acknowledgements.
This work was supported by the National Natural Science Foundations of China
(No. 11171255, 11101373) and Doctoral Program Foundation of the Ministry of
Education of China (No. 20090072110053).
## References
* Brackx et al (1982) Brackx F, Delanghe R, Sommen F (1982) Clifford Analysis, Research Notes in Mathematics. London, Pitman
* De Ridder et al (2012) De Ridder H, De Schepper H, Sommen F (2012) Fueter polynomials in discrete Clifford analysis. Mathematische Zeitschrift 272 (2012) :253–268.
* Gong et al (2009) Gong Y, Leong IT, Qian T (2009) Two integral operators in Clifford analysis. Journal of Mathematical Analysis and Applications 354(2):435–444
* Hörmander (1965) Hörmander L (1965) $l^{2}$ estimates and existence theorems for the operator. Acta Mathematica 113(1):89–152
* Huang et al (2006) Huang S, Qiao YY, Wen GC (2006) Real and Complex Clifford Analysis, Advances in Complex Analysis and Its Applications. New York, Springer
* Qian and Ryan (1996) Qian T, Ryan J (1996) Conformal transformations and Hardy spaces arising in Clifford analysis. Journal of Operator Theory 35(2):349–372
* Ryan (1990) Ryan J (1990) Iterated Dirac operators in $C^{n}$. Zeitschrift für Analysis und ihre Anwendungen 9:385–401
* Ryan (1995) Ryan J (1995) Cauchy-Green type formulae in Clifford analysis. Transactions of the American Mathematical Society 347(4):1331–1342
* Ryan (2000) Ryan J (2000) Basic Clifford analysis. Cubo Matemática Educacional 2:226–256
|
arxiv-papers
| 2013-04-19T00:32:06 |
2024-09-04T02:49:44.597959
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liu Yang, Chen Zhihua and Pan Yifei",
"submitter": "Yang Liu",
"url": "https://arxiv.org/abs/1304.5287"
}
|
1304.5315
|
# Quality-Aware Coding and Relaying for 60 GHz Real-Time Wireless Video
Broadcasting
Joongheon Kim†, Member, IEEE, Yafei Tian♮, Member, IEEE, Stefan Mangold§,
Member, IEEE, and
Andreas F. Molisch‡, Fellow, IEEE †‡Communication Sciences Institute,
University of Southern California, Los Angeles, CA 90089, USA
♮School of Electronics and Information Engineering, Beihang University,
Beijing 100191, China
§Disney Research, 8092 Zurich, Switzerland
Emails: †[email protected], ♮[email protected],
§[email protected], ‡[email protected]
###### Abstract
Wireless streaming of high-definition video is a promising application for 60
GHz links, since multi-Gigabit/s data rates are possible. In particular we
consider a sports stadium broadcasting system where video signals from
multiple cameras are transmitted to a central location. Due to the high
pathloss of 60 GHz radiation over the large distances encountered in this
setting, the use of relays is required. This paper designs a quality-aware
coding and relaying algorithm for maximization of the overall video quality.
We consider the setting that the source can split its data stream into
parallel streams, which can be transmitted via different relays to the
destination. For this, we derive the related formulation and re-formulate it
as convex programming, which can guarantee optimal solutions.
## I Introduction
Wireless video streaming in the millimeter-wave range has received a lot of
attention in both the academic and industrial communities. In particular the
60 GHz frequency range is of great interest: around 7 GHz bandwidth (58-65
GHz) has been made available, which enables multi-Gbit/s high-definition video
streaming in an uncompressed, or less compressed, manner. Therefore, two
industry consortia, i.e., WirelessHD and Wireless Gigabit Alliance (WiGig),
have developed related specifications; there are also two activities within
the IEEE, namely IEEE 802.15.3c [1] and IEEE 802.11ad [2].
In this paper, we design and analyze such a 60 GHz video transmission system
for outdoor applications, in particular in a sports stadium. In this system,
there are multiple wireless video cameras in a stadium for high-quality real-
time broadcasting, all send their signals to a broadcasting center. To
transmit uncompressed HD video streams in real-time, a data rate of around
$1.5$ Gbit/s is required [3]. Since the distance between wireless cameras and
a broadcasting center is on the order of several hundred meters, the high
pathloss at 60 GHz is one of key challenges that limits communication ranges.
One promising approach to deal with this problem is using relays for extending
the coverage [3]. Additionally, we take the complexity of the antennas into
account. In order to compensate for the high pathloss, as well as to reduce
interference, high-gain antennas need to be employed. We also consider the
situation where the antenna at the camera (video source) can form multiple
beams, so that it can split its data stream into multiple streams and send
them to the destination via parallel links. By introducing multiple beams in
each relay, our framework operates even though the number of relays is smaller
than the number of sources. In this case, appropriate compression and routing
of multiple streams via the same relay can be used. Relaying for sum rate
maximization has been analyzed in many papers. However, for video streaming,
we are more interested in video quality. For this, the proposed quality-aware
formulation selects the relays and decides the coding rates for every single
video stream. With this formulation, optimal solutions are obtained by convex
optimization techniques.
Thus, the contribution of the proposed scheme is achieving joint rate and
relay selection with video quality consideration and an interference-free
operation. This combination of special features makes it different from other
schemes.
(a) Wireless Video Camera (Source)
(b) Relay
(c) Broadcasting Center
Figure 1: System Components (Camera (a), Relay (b), Broadcasting Center (c))
Figure 2: Overall Architecture
## II Related Work
There are two salient factors in our broadcasting setup: (i) stream splitting
via the multiple-beam antennas, and (ii) rate control for video quality
maximization. In the following, we outline why these aspects make the setup
different from other scenarios that have been treated in the literature. There
is, of course, a huge number of papers (too numerous to reference here)
dealing with the topic of routing (relay selection) and sum rate maximization
in multi-node networks with multi-beam relays [5] and with multi-beam sources
and relays [6]. However these papers do not consider the control of the video
coding rate (compression), and are thus not directly applicable to our
scenario. For video networks, example publications include [7, 8, 9]. The
scheme in [7] is for video streaming over IEEE 802.11 networks. The proposed
scheme is efficient for the multi-hop networks it investigates; however, it
does not consider the video stream splitting via the multi-beam antennas and
route selection. Ref. [8] considers video streaming in multi-hop networks. It
considers networks similar to ours (when specialized to the two-hop case), but
again does not investigate multi-beam antennas and the splitting of the data
streams. The formulation in [9] considers path selection for video streaming
in MANET. It concentrates on the consideration of interference, a factor that
does not play a role in our 60 GHz channel, where the high directionality of
the links prevents inter-stream interference. None of these papers consider
the control of the coding rate (compression). In previous research on video
streaming, schemes usually considered multipath transmission to combat the
limited bandwidth [8][9]. Also, some of the research considered retransmission
of frames and tried to reduce transmission time [7]. However, thanks to the
extremely large bandwidth at 60 GHz, these factors are not longer critical in
our system. The representative work which considers both rate control and
routing appeared in [10]: However, the relays cannot aggregate streams, which
is required when the number of relays is smaller than the number of flows. In
addition, the proposed framework does not consider the properties of video. In
our previous work [3], we considered the properties of 60 GHz channel, rate
control, and video quality, but we restricted ourselves to the cases that the
number of relays exceeds the number of sources (i.e., no consideration for
multi-beam antennas) and the numbers of sources and destinations are
identical. We finally note that wireless video for sports stadiums [11];
however the fundamental setup differs from ours in that [11] considers content
distribution to wireless devices of the audience in the stadium, while our
system is for real-time streaming to a broadcasting center in the stadium.
## III A Reference System Architecture
### III-A Link Budget Analysis
Shannon’s equation for the capacity is used for the data rate:
$C=B\cdot\log_{2}\left(1+\text{SNR}\right)$ (1)
where SNR is equal to $P_{\text{signal}}/P_{\text{noise}}$ on a linear scale,
$P_{\text{signal}}$ and $P_{\text{noise}}$ stand for the signal power and
noise power, and $B$ stands for bandwidth ($2.16$ GHz in WiGig [2]) [12]. The
signal power expressed in dB, $P_{\text{signal, dB}}$, is obtained as:
$P_{\text{signal,dB}}=E+G_{r}-W-O(d)+F(d)$ (2)
where $E$ denotes the EIRP (equivalent isotropically radiated power), which is
limited to $40$ dBm in the USA and $57$ dBm in Europe. $G_{r}$ means the
receiver antenna gain and is set to $40$ dB, which corresponds to high-gain 60
GHz scalar horn antennas [13], which we propose to achieve long range.
Shadowing can be either temporally variant (due to people walking close to
LOS), or time-invariant (due to objects that are (partly) shadowing off the
LOS. While the shadowing variances envisioned for our deployment scenarios are
on the order of a few dB, we use a $10$ dB shadowing margin to provide high
link reliability. $F(d)$ is the mean pathloss, which depends on the distance
$d$ between transmitter and receiver
$F(d)=10\log_{10}\left\\{\lambda/(4\pi d)\right\\}^{n}$ (3)
where the pathloss coefficient $n$ is set to $2.5$ [14] and the wavelength
($\lambda$) is $5$ millimeter at $60$ GHz. $O(d)$ denotes the oxygen
attenuation, which can be computed as $O(d)=\frac{15}{1000}d$ when $d>200$ m.
Otherwise, it is ignored [14]. The noise power in dB, $P_{\text{noise,dB}}$ is
computed as:
$P_{\text{noise,dB}}=10\log_{10}\left(k_{B}T_{e}\cdot B\right)+F_{N}$ (4)
where $k_{B}T_{e}$ stands for the noise power spectral density ($-174$ dBm/Hz)
and $F_{N}$ is the noise figure of the receiver ($6$ dB). By combining the
above equations, approximately $200-300$ m is the maximum distance for
obtaining $1.5$ GBit/s data rate, i.e., successful uncompressed video
transmission.
### III-B 60 GHz Outdoor Broadcasting Systems
As shown in Sec. III-A, the assistance of relays is required if the distance
between wireless cameras and a broadcasting center is more than $200-300$ m.
Furthermore, the size of a sports stadium (from wireless cameras to a
broadcasting center) is generally not more than $500$ m. Thus, we restrict the
number of relays to one. In our 60 GHz broadcasting system, three components
are existing, i.e., wireless video cameras, relays, and a broadcasting center.
As presented in Fig. 1(a), the proposed wireless video cameras have scalable
video coding (SVC) functionalities that reproduce the recorded video signals
as layered SVC-coded bitstreams. If the achievable rate of a 60 GHz link is
sufficient for uncompressed video streaming (i.e., more than $1.5$ Gbit/s),
all layers can be transmitted. Otherwise, the optimal coding level decision
module has to determine the number of layers. Each wireless video camera has
multiple-beam antennas. Therefore, each antenna can form $N$ independent
beams, so that the multiple streams created by SVC-encoding are divided into
$N$ parts and each part is assigned to a beam to be concurrently transmitted.
We furthermore assume that the relays have multiple-beam antennas for
reception, see Fig. 1(b). The relays aggregate the received signals and
transmit them towards a broadcasting center. As presented in Fig. 1(c), the
proposed broadcasting center has multiple antennas which are facing the
relays. We emphasize that due to the narrow beamwidth
($1.5^{\circ}$-$10^{\circ}$ [13]) of the antennas, multiple streams arriving
at the broadcasting center or relays do not interfere with each other.
### III-C Objective
For this given system, our objective is the maximization of the delivered
total video quality. As shown in [4], the quality of video is related to the
data rate in a nonlinear fashion as a sublinearly, but monotonically,
increasing form. Following is one example of a quality function:
$f_{q}(a)=\frac{1}{\log_{\beta}(a_{\text{max}}+1)}\log_{\beta}(a+1)$ (5)
$\beta$ is a base ($1<\beta$), $a_{\text{max}}$ is a desired data rate for
uncompressed video streaming, and $a$ is a given data rate.
## IV Mathematical Optimization Formulation
Fig. 2 shows the reference model with a set of sources $\mathcal{S}$, a set of
relays $\mathcal{R}$, and a single destination $D$. In the relay-destination
region (RDR) of Fig. 2, all relays are connected to $D$. Then the maximum
achievable rates of all relay and destination pairs can be computed (i.e.,
$a_{r_{1}\rightarrow D}^{\text{RDR}},\cdots,a_{r_{|\mathcal{R}|}\rightarrow
D}^{\text{RDR}}$). Our assumption is that $D$ can form a sufficient number of
independent beams so that it has no limitations concerning the number of
relays. Thus, we wish to find optimal combinations between sources and relays
in source-relay region (SRR) for the settings that both sources and relays can
form multiple beams. Then, our formulation for the maximization of delivered
total video quality is as follows:
$\max\sum_{j=1}^{|\mathcal{R}|}\sum_{i=1}^{|\mathcal{S}|}f_{q}\left(\frac{1}{2}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\right)x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ (6)
subject to
$\displaystyle\sum_{i=1}^{|\mathcal{S}|}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$
$\displaystyle\leq$ $\displaystyle\mathcal{A}_{r_{j}\rightarrow
D}^{\text{RDR}},\forall j,$ (7)
$\displaystyle\sum_{i=1}^{|\mathcal{S}|}x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}$ $\displaystyle\leq$ $\displaystyle B_{r_{j}},\forall j,$
(8) $\displaystyle\sum_{j=1}^{|\mathcal{R}|}x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}$ $\displaystyle\leq$ $\displaystyle B_{s_{i}},\forall i,$
(9) $\displaystyle\underline{a}_{s_{i}}$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{|\mathcal{R}|}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}x_{s_{i}\rightarrow r_{j}}^{\text{SRR}},\forall i,$ (10)
$\displaystyle a_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ $\displaystyle\leq$
$\displaystyle\mathcal{A}_{s_{i}\rightarrow r_{j}}^{\text{SRR}},\forall
i,\forall j,$ (11) $\displaystyle x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$
$\displaystyle\in$ $\displaystyle\\{0,1\\},\forall i,\forall j,$ (12)
where $a_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ and $x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}$ stand for the Data rate between $s_{i}$ and $r_{j}$ and
Boolean connectivity index between $s_{i}$ and $r_{j}$, respectively. Note
that $s_{i}$ and $r_{j}$ stand for the source $i$, $\forall
i\in\\{1,\cdots,|\mathcal{S}|\\}$ and the relay $j$, $\forall
j\in\\{1,\cdots,|\mathcal{R}|\\}$, respectively. If $s_{i}$ and $r_{j}$ are
connected, $x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ is $1$ by (12).
Otherwise, $x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ is $0$ by (12). The
relays are all connected to $D$, thus, $x_{r_{j}\rightarrow
D}^{\text{RDR}}=1$. The $\mathcal{A}_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$
and $\mathcal{A}_{r_{j}\rightarrow D}^{\text{RDR}}$ are maximum achievable
rates computed by (1). In addition, the desired data rates between $s_{i}$ and
$r_{j}$ are less than $\mathcal{A}_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ as
shown in (11) where $f_{q}\left(\cdot\right)$ is a function for the
relationship between video quality and data rate (logarithmically and
monotonically increasing form). As shown in (10), the desired data rates
between $s_{i}$ and $r_{j}$ should exceed the defined minimum rates
($\underline{a}_{s_{i}}$, $\forall s_{i}$), which are minimum data rates for
guaranteeing the required minimum video qualities for each flow. Here,
$\mathcal{A}_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ from $s_{i}$ to $r_{j}$
and $\mathcal{A}_{r_{j}\rightarrow D}^{\text{RDR}}$ from $r_{j}$ to $D$ are
fixed. For each individual source, there are multiple outgoing flows (multiple
beams) toward relays, as formulated in (9) where $B_{s_{i}}$ stands for the
number of antenna-beams at source $i,\forall
i\in\\{1,\cdots,|\mathcal{S}|\\}$. Similarly, each relay can form multiple
beams in receiving mode, thus the number of incoming flows from sources can be
$B_{r_{j}}$ as formulated in (8) where it means the number of antenna-beams at
relay $j,\forall j\in\\{1,\cdots,|\mathcal{R}|\\}$. In (7), for each relay,
the summation of incoming rates from sources cannot exceed the data rate
between the relay and $D$. Finally, (6) describes the objective of finding the
pairs between sources and relays as well as finding the corresponding data
rates for maximizing the total video quality and the data rate value becomes
$1/2$ due to the half-duplex constraint.
###### Theorem 1.
The formulation in Section IV is non-convex.
###### Proof.
This proof considers the setting of one-source and one-relay. Then the
objective function becomes
$\displaystyle f\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}},x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}\right)$
$\displaystyle\triangleq$ $\displaystyle f_{q}\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\right)x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ (13)
$\displaystyle=$
$\displaystyle\mathcal{K}\log_{\beta}\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1\right)x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ (14)
where $\mathcal{K}=\frac{1}{\log_{\beta}(a_{\text{max}}+1)}$ is constant and
$x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}$ is relaxed, i.e., $0\leq
x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}\leq 1$. To show that this is non-
convex, the second-order Hessian of this should not be positive definite [15].
The Hessian $\nabla^{2}f\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}},x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}\right)$ is:
$\begin{bmatrix}0&\frac{\mathcal{K}/\ln\beta}{a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1}\\\ \frac{\mathcal{K}/\ln\beta}{a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1}&-x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\cdot\frac{\mathcal{K}/\ln\beta}{\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1\right)^{2}}\end{bmatrix}$ (15)
and then the corresponding two eigenvalues are
$\frac{\mathcal{I}}{2}\pm\frac{1}{2}\sqrt{\mathcal{I}^{2}+\left(\frac{2\mathcal{K}/\ln\beta}{a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1}\right)^{2}}$ (16)
where $\mathcal{I}=\frac{-\frac{\mathcal{K}}{\ln\beta}\cdot
x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}}{\left(a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}+1\right)^{2}}$, $0\leq a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\leq 1.5$, $0\leq x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\leq 1$.
These are not all positive, thus Hessian is not positive definite, which
proves the formulation is non-convex. ∎
For non-convex MINLP, heuristic searches can find approximate solutions but
cannot guarantee optimality [15]. With the following Theorem, our non-convex
MINLP can be re-formulated as a convex program form.
###### Theorem 2.
For the given formulation, (6)-(12), introducing
$a_{s_{i}\rightarrow r_{j}}^{\text{SRR}}\leq\mathcal{A}_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\cdot x_{s_{i}\rightarrow r_{j}}^{\text{SRR}},\forall
i,\forall j$ (17)
instead of (11) makes the formulation convex.
###### Proof.
For the non-convex MINLP formulation in Section IV, $x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}=0$ means the link is disconnected. Thus the corresponding
rate becomes $0$ and (17) leads to the same result when $x_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}=0$, i.e.,
$a_{s_{i}\rightarrow r_{j}}^{\text{SRR}}\leq\mathcal{A}_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\cdot 0=0,\forall i,\forall j.$ (18)
Otherwise, if $x_{s_{i}\rightarrow r_{j}}^{\text{SRR}}=1$, then this term is
equivalent to (11). Therefore, in turn, (6) is also updated as
$\max\sum_{j=1}^{|\mathcal{R}|}\sum_{i=1}^{|\mathcal{S}|}f_{q}\left(\frac{1}{2}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}\right)$ (19)
and (7) and (10) are also updated as following (20) and (21):
$\displaystyle\sum_{i=1}^{|\mathcal{S}|}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}}$ $\displaystyle\leq$
$\displaystyle\mathcal{A}_{r_{j}\rightarrow D}^{\text{RDR}},\forall j.$ (20)
$\displaystyle\underline{a}_{s_{i}}$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{|\mathcal{R}|}a_{s_{i}\rightarrow
r_{j}}^{\text{SRR}},\forall i,$ (21)
Then now there are no non-convex terms in the program. ∎
Finally, our convex MINLP, which can guarantee optimal solutions, is as
follows: (19) subject to (20), (8), (9), (17), (21), (12) where $\forall
i\in\\{1,\cdots,|\mathcal{S}|\\},\forall j\in\\{1,\cdots,|\mathcal{R}|\\}$.
## V Performance Evaluation
To verify the performance of our scheme, i.e., video quality maximization
(named as VQM), we compare it with the following schemes:
* •
The joint video coding and relaying under the consideration of sum rate
maximization (named as SRM). In this case, the proposed objective function
(19) is:
$\max\sum_{j=1}^{|\mathcal{R}|}\sum_{i=1}^{|\mathcal{S}|}\frac{1}{2}a_{s_{i}\rightarrow
r_{j}}^{\text{SRC}}$ (22)
due to the fact that the quality is no longer considered.
* •
The scheme in [10], which is an efficient algorithm that considers joint rate
selection and routing (named as JRSR) in terms of sum-rate maximization with
cooperative communication mode selection and no multiple-beams. For fair
comparisons, we adapt the scheme to our outdoor-stadium architecture (one-tier
relay) and allow only decode-and-forward relaying.
TABLE I: Expectation of Achieved Normalized Aggregated Video Quality | Multiple-Beams at $s_{i}$ and $r_{j}$
---|---
$|\mathcal{S}|$ | $|\mathcal{R}|$ | Setting | VQM | SRM | JRSR
5 | 10 | I | 4.166 | 3.873 | 3.331
5 | 10 | II | 4.934 | 4.647 | 4.165
5 | 10 | III | 4.681 | 4.397 | 3.632
10 | 5 | I | 4.164 | 3.871 | 3.352
10 | 5 | II | 4.954 | 4.620 | 4.182
10 | 5 | III | 4.664 | 4.371 | 3.650
10 | 10 | I | 8.813 | 8.451 | 5.483
10 | 10 | II | 9.817 | 9.452 | 6.336
10 | 10 | III | 9.312 | 8.958 | 5.795
10 | 15 | I | 8.883 | 8.574 | 5.633
10 | 15 | II | 9.872 | 9.563 | 6.456
10 | 15 | III | 9.383 | 9.074 | 5.927
15 | 10 | I | 13.420 | 11.765 | 6.483
15 | 10 | II | 14.902 | 13.255 | 7.376
15 | 10 | III | 14.403 | 12.755 | 6.839
For the setting, the cameras are uniformly distributed on top of the stadium.
Between stadium and broadcasting center, multiple relays are uniformly
deployed along a line. To vary the settings, we consider this line to be near
the cameras (Setting I), in the middle between cameras and broadcasting center
(Setting II), and near the center (Setting III). As our performance measure,
we consider the cumulative probability distribution (cdf) of the aggregate
video quality. The cdf is obtained as follows: we consider multiple
realizations of the deployment of sources and relays, i.e., the random
deployment of relays with Setting I, Setting II, and Setting III. For each
such realization, we optimize coding rates and relay selection; thus each run
gives us one realization of the aggregate video quality. We finally plot the
cdf of this quality. For the simulation of VQM, the lower bounds
($\underline{a}_{s_{i}}$, $\forall i\in\\{1,\cdots,|\mathcal{S}|\\}$) of each
source are set as $0.75$ Gbit/s ($50\%$ of $1.5$ Gbit/s). More detailed
scenarios and simulation results can be found in [4].
### V-A CDF of Aggregate Video Quality
Figure 3: Impact of Various Lower Bound Setting:
$|\mathcal{S}|=10,|\mathcal{R}|=15$
(a) Setting I
(b) Setting II
(c) Setting III
Figure 4: Simulation Results: Number of Sources ($|\mathcal{S}|=5,10,15$) and
Fixed Number of Relays ($|\mathcal{R}|=10$)
Fig. 4 plots the cases that the number of sources is smaller, equal, or larger
than the number of relays (i.e., $|\mathcal{S}|=5,10,15$, and
$|\mathcal{R}|=10$). The mean achieved normalized aggregated video qualities
are in Table I. In Table I, the video quality values from each source are
normalized as $1$ for performance evaluation. Thus, if we have $N$ cameras in
the system, the maximum achievable aggregated video quality is $N$. As shown
in this result, the performance of JRSR is worse than that of both SRM and
VQM. The latter (i.e., JRSR), by design, does not allow the exploitation of
the multiple-beam antennas at relays, and thus shows worse performance. The
performance gains of SRM are more pronounced than JRSR in Settings I and III,
i.e., when the relays are close to either the sources or the destination. More
importantly, we find that the relative performance advantage drastically
increases as the number of sources increases relative to the number of relays.
This is not surprising, as for these situations the ability of the sources to
split their streams and flexibly route them via the relays becomes more
important. We also see that SRM shows lower performance than VQM due to the
fact that SRM aims to the maximization of sum data rates, while VQM aims to
maximize the overall delivered video quality. The relative advantage of VQM
also increases as the number of sources increases. Again, this is not
surprising, as the bandwidth limitations become more stringent as the number
of sources increases.
### V-B Impact of Lower Bound Setting
In previous simulation, the lower bounds for the data rate per data stream are
set as $0.75$ Gbit/s. Here, we vary this value from $0$ Gbit/s (no lower
bound) to $1.5$ Gbit/s (allowing only uncompressed video) in steps of $0.1$
Gbit/s. As a performance quality measure, we define “stream outage” (i.e., the
probability that at least one stream does not have the minimum required
quality). As shown in Fig. 3, Setting III suffers significantly from the
higher required per-stream quality. With Setting III, the data rates between
sources and relays are lower than the others. Thus, when we set the lower
bound quite high, all flows are disconnected. Thus, it achieves the lowest
performance. On the other hand, in Setting I, all flows between sources and
relays have enough capacity to support uncompressed video transmission, thus,
a higher setting for minimum quality does not have a strong impact. Fig. 3
also shows that VQM has better performance than SRM for all settings.
## VI Conclusion
This paper suggests and discusses quality-aware coding and routing for 60 GHz
multi-Gbit/s real-time video streaming in an outdoor broadcasting system. In
the system, there are multiple wireless video cameras distributed throughout
the stadium. We presented an optimization framework for finding the
combination of wireless link pairs between wireless cameras and relays that
can maximize the overall or per-flow qualities of delivered video to a
broadcasting center. An initial non-convex MINLP is re-formulated as a convex
program, which allows optimum solutions. Simulations show that this
methodology outperforms other methods that do not take the peculiarities of
millimeter-wave video links into account.
## References
* [1] IEEE 802.15.3c Millimeter-wave-based Alternative Physical Layer Extension, October 2009.
* [2] IEEE 802.11ad VHT Specification Version 1.0, December 2012.
* [3] J. Kim, Y. Tian, A.F. Molisch, and S. Mangold, “Joint Optimization of HD Video Coding Rates and Unicast Flow Control for IEEE 802.11ad Relaying,” in Proc. IEEE PIMRC, 2011.
* [4] J. Kim, Y. Tian, S. Mangold, and A.F. Molisch, “Joint Scalable Coding and Routing for 60 GHz Real-Time Live HD Video Streaming Applications,” Submitted to IEEE Trans. on Broadcasting., Available on Request.
* [5] E. Yılmaz, R. Zakhour, D. Gesbert, and R. Knopp, “Multi-pair Two-way Relay Channel with Multiple Antenna Relay Station,” in Proc. IEEE ICC, 2010.
* [6] J. Liu, N.B. Shroff, and H.D. Sherali, “Optimal Power Allocation in Multi-Relay MIMO Cooperative Networks: Theory and Algorithms,” JSAC, 30(2):331-340, Feb. 2012.
* [7] M.-H. Lu, P. Steenkiste, and T. Chen, “Time-Aware Opportunistic Relay for Video Streaming over WLANs,” in Proc. IEEE ICME, 2007.
* [8] S. Mao, X. Cheng, Y.T. Hou, H.D. Sherali, and J. Reed, “On Joint Routing and Server Selection for MD Video Streaming in Ad Hoc Networks,” IEEE Trans. Wireless Comm., 6(1):338-347, Jan. 2007.
* [9] W. Wei and A. Zakhor, “Interference Aware Multipath Selection for Video Streaming in Wireless Ad Hoc Networks,” IEEE Trans. CSVT, 19(2):165-178, Feb. 2009.
* [10] S. Sharma, et. al., “Joint Flow Routing and Relay Node Assignment in Cooperative Multi-Hop Networks,” JSAC, 30(2):254-262, Feb. 2012.
* [11] D. Giustiniano, V. Vukadinovic, and S. Mangold, “Wireless Networking for Automated Live Video Broadcasting: System Architecture and Research Challenges,” in Proc. IEEE WoWMoM, 2011.
* [12] A.F. Molisch, Wireless Communications, 2nd Ed., IEEE, Feb. 2011.
* [13] Comotech Corporation; http://www.comotech.com/en/index.html
* [14] P. Smulders, “Exploiting the 60 GHz Band for Local Wireless Multimedia Access: Prospects and Future Directions,” IEEE Communications Magazine, 40(1):140-147, Jan. 2002.
* [15] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
|
arxiv-papers
| 2013-04-19T05:49:49 |
2024-09-04T02:49:44.604667
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joongheon Kim, Yafei Tian, Stefan Mangold, Andreas F. Molisch",
"submitter": "Joongheon Kim",
"url": "https://arxiv.org/abs/1304.5315"
}
|
1304.5480
|
December 20, 2013
LA-UR-13-22745
arXiv:1304.5480
# A Mesonic Analog of the Deuteron
T. Goldman [email protected] Theoretical Division, MS-B283, Los Alamos
National Laboratory, Los Alamos, NM 87545
and
Dept. of Physics and Astronomy, University of New Mexico, Albuquerque, NM
87501 Richard R. Silbar [email protected] Theoretical Division, MS-B283,
Los Alamos National Laboratory, Los Alamos, NM 87545
###### Abstract
Using the LAMP model for nuclear quark structure, we calculate the binding
energy and quark structure of a $B$ meson merging with a $D$ meson. The
larger-than-nucleon masses of the two heavy quarks allow for a more reliable
application of the Born-Oppenheimer-like approximation of the LAMP. With the
absence of quark-level Pauli Exclusion Principle repulsive effects, the
appearance of a bound state is unsurprising. Our variational calculation shows
that the molecular, deuteron-like state structure changes rather abruptly, as
the separation between the two mesons decreases, at a separation of about 0.45
fm, into a four-quark bound state, although one maintaining an internal
structure rather than that of a four-quark bag. Unlike the deuteron, pion
exchange does not provide any contribution to the $\approx 150$ MeV binding.
Keywords: heavy meson, four-quark, relativistic, variational, pion-less
## I Introduction
What would nuclear physics look like without pion exchange? The long range of
the nuclear force due to pion exchange between nucleons, along with the
empirical short distance repulsion between nucleons, supports the established
view of nuclear physics as due to the interaction of effective degrees of
freedom that bear a very close resemblance to free space nucleons.
Calculations of nuclear structure for small nuclei, using potential
interactions fit to scattering data, succeed quite accurately.Carlson
Effective field theory expansions, with or without pions, claim successes Bira
as well. For large nuclei, elaborations of the shell model can also reproduce
experimentally known results.
However, all of these approaches ignore the internal structure of the three-
quark states that are on-shell nucleons in free space but not so well defined
off-shell degrees of freedom in the nucleus. In particular, the basis for off-
shell nucleon form factors resembling those of on-shell nucleons is weak, and
conflicts with the experimental results of deep inelastic scattering (DIS) on
nuclei. Those results are not well represented by multiplying the results of
DIS on free space nucleons by the number of nucleons in the target nucleus.
This is known as the “EMC effect”.EMC
The relativistic Los Alamos Model Potential GMSS ; GBS (LAMP) has been used
to describe the binding and structure of 3He and 4He, including a good
description BG of the deep inelastic structure function of 3He. It was
explicitly constructed to access the internal quark structure of the baryonic
components of the nucleus without the presumption of a free space nucleon
approximation. As such, except for the difficulties of carrying out
calculations, it provides a less biased view (although not a systematic
expansion) of the hadronic structure of nuclei than do the conventional models
referred to above.
The LAMP does not describe the deuteron at all due to the very large
separation of the nucleons and the dominance of single-pion exchange
contributions there.Friar The LAMP, lacking quark-exchange correlations, best
encompasses medium and short-range meson exchanges (two-pion, $\rho$, etc.).
It must therefore be supplemented with long-range single-pion-exchange
contributions piqk for a better description of nuclear binding energies.
However, in this model, we can ask: What would nuclear physics, and in
particular, the deuteron, look like in the absence of long-range pion exchange
interactions? If bound states exist, the constituents would be much closer
together than in actual nuclei and disruption of the internal structure could
be much more significant than suggested by the LAMP as applied to nucleons or
the results of conventional nuclear physics. Could one still identify
nucleonic effective degrees of freedom even when the multi-quark hadronic
objects are in such close proximity that their average separation is less than
their internal structure? This is to be contrasted with real nuclei where the
mean separation between nucleons is quite close to twice their root-mean-
square radii.
In this paper, we make an initial address to this question by considering a
simpler problem, the binding of two heavy mesons. Large mass quarks are used
to mimic the large mass of the nucleon, but one light antiquark in each stands
in for the diquark in the nucleons and so simplifies the calculations. Since
no quark-exchange correlations are included, no ($t$-channel) quark-antiquark
combinations with pion quantum numbers contribute any more significantly than
higher mass mesons. However, the extension/size of the mesonic states is
comparable to that of nucleons due to the spread of the light quark
wavefunction.
In fact, for this case, all “light” meson exchanges are prevented, and the
interactions have solely to do with the structure of the light antiquark wave
functions under the influence of the color confining force, represented here
by a collective potential. This is somewhat analogous, in principle, to the
nuclear shell model potential although significantly different in form to be
consistent with known models of confinement.
In particular, we examine here the structure of a four-quark system derived
from $B^{-}=b\bar{u}$ and $D^{+}=c\bar{d}$ mesons for a bound state, or their
neutral equivalents when the light antiquarks are exchanged between them.
Because these mesons are considerably more massive than nucleons, localization
energy is much reduced. This brings them into closer proximity than the
nucleons in a deuteron, or indeed, even in a large nucleus. The larger-than-
nucleon masses of the two heavy quarks also allow for a more reliable
application of the Born-Oppenheimer-like approximation of the LAMP.
Furthermore, the quark content chosen here does not involve any pairs of
quarks with the same (internal) quantum numbers, so there are no (quark) Pauli
exclusion effects such as those that contribute to the short-range repulsion
between nucleons. Thus, this is a system in which one can expect greater
accuracy of the LAMP and a significantly more deeply bound state than the
deuteron.
When this $B$-$D$ bound state is observed, the deviation from our predictions
here will provide a very good measure of the center of mass motion and
breathing mode collective excitations. These are difficult to remove in the
LAMP due to its relativistic nature. Since the non-relativistic model
analogous to the LAMP, the Quark Delocalization and Color Screening Model of
Wang et al. fanwang , gives very similar results to the LAMP after removing
such effects, we expect the corrections due to these effects to be small. Thus
our predictions here should be reasonably accurate.
There have been many different approaches, going back to the Cornell potential
eichten , along lines comparable to the LAMP, to modelling quark-antiquark
states using potentials. We note here a few recent references others . There
have also been many papers devoted to the study of four-quark systems, with a
view to identifying exotic states constructed of more than three quarks or one
quark and one antiquark. See, for example, the references in the recent review
of Brambilla et al. nora and some very early papers heller as well.
Generally, however, these papers have focused on states more likely to appear
in hadronic collisions, such as those with the quark content of $B$ and
$\bar{B}$ or $D$ and $\bar{D}$ mesons and their excited state partners (for a
recent example, see PB ), since strong production of heavy quarks proceeds in
a pairwise fashion. (Some, such as Ref.(heller ), have also included
consideration of the case studied here, albeit without the intricacies
available in the LAMP). In general, the mixing of these states with the
charmonium and bottomonium spectra, however, make for difficulties in
extracting them unambiguously from experimental observations and may require
the determination of exotic quantum numbers. No such problems occur in the
case considered here, although the reduced probability of production must
certainly be recognized. In any event, our interest is not in the prediction
of exotic states, but in the elucidation of the origins of the nature of
nuclear structure and thus a deeper understanding of it.
### I.1 Initial Concepts
The LAMP treats the confining potential for quarks (and antiquarks) as a fixed
scalar interaction in a Born-Oppenheimer-like picture, with the location of
the potential minimum defining the system location. Quarks bound in a baryon
or meson are treated as being bound within this potential rather than directly
to each other. As such, there are immediate concerns about removing center-of-
mass and breathing mode contributions to the evaluated state energy. This
concern is ameliorated by comparing the energy of the interacting system of
the two heavy mesons with the value at large (essentially infinite)
separation.
In this paper, in addition to the confining Lorentz scalar potential of the
LAMP, we have included a Lorentz vector potential, as is required from the
observed small spin-orbit interaction in the non-relativistic quark model.PGG
In fact, the vector potential is also taken as linear, attractive, but without
a Coulomb-like contribution, as discussed in Ref.(Convolve ).
In the LAMP, the confining potentials for each hadron are distributed in an
array and are truncated on the mid-planes between them. While complex in
general, for the case of interest here – two heavy mesons – the structure is
very similar to that of the hydrogen molecule in the Born-Oppenheimer
approximation, except for the linear vs. inverse distance form of the
potential. In this case, the large masses of the $c$ and $b$ quarks further
enhance the credibility of the approximation – they may be taken in the
conventional heavy quark limit HQET as the fixed origins of the confining
potentials for the light anti-quarks that complete each meson.
At large separation between the heavy quarks, confinement guarantees the
isolation of the light quark wave functions from each other. However, as the
two mesons approach within a distance less than a few times their root-mean-
square radii, the truncation of the confining potential allows for tunneling
of each light anti-quark wave function into the confinement region of the
other heavy quark rather than the one to which the light anti-quark is
initially bound. The concept behind this is that a quark can only be confined
to nearest center of color attraction, as in string-flip models stringflip ,
for example. This spreading out, or delocalization, of the wave functions
naturally reduces the localization energy and provides an initial source of
binding between the two hadrons.
### I.2 Color magnetic and quantum number issues
In nuclei and other systems, this basic consideration is complicated by
additional elements: there are color 6 combinations of quarks and color-
magnetic spin interactions of significance on the scale of the binding energy.
Here again, the concerns raised by these considerations are considerably
reduced – the color magnetic interactions between the heavy quarks are reduced
by their large masses. The light quark color magnetic interactions with the
heavy quarks are also reduced. Only the light-quark to light-quark color
magnetic interaction remains comparable to that inferred in simple quark
models of light-quark states. This energy is at most $\approx 50$ MeV as seen
PDG in individual hadrons (nucleons, $\Delta$’s, light spin-0, and spin-1
mesons) where it depends on the color and spin-strong-isospin combinations
determined by the constraints of statistics. Furthermore, here the presence of
both color 6 and color 3 combinations, as well as spin-1 and spin-0 elements,
make it clear that strong cancellations of these color magnetic effects to low
levels are to be expected. Therefore, in this paper we will largely ignore
these contributions, since our emphasis here is to determine whether the $B$
and $D$ form a four-quark bound state or a more molecular-like combination of
two identifiable mesons. We also will neglect the very small electro-magnetic
contributions.
Because of these simplifications, in this paper we can also ignore the fact
that there are two neutral states ($B^{-}D^{+}$ and $\bar{B^{0}}D^{0}$) that
should exist and mix, splitting to form states of definite strong isospin (0
and 1) although both have $I_{3}=0$. They also allow us to ignore the detailed
spin structures, ranging from $J=0$ to $J=2$, the last with all of the quark
spins aligned. Also unlike the individual nucleon case, the $c$ and $b$ quarks
may combine anti-symmetrically to form a color ${\bf{\bar{3}}}$ state or
symmetrically to form a color 6. In the first case, the light anti-quarks must
form a color 3 antisymmetrically, thus requiring the spin-isospin combination
to be symmetric ($I=0$, $J=0$, or $I=1$, $J=1$) and in the latter, the
opposite is true – a color ${\bf{\bar{6}}}$ and ($I=1$, $J=0$, or $I=0$,
$J=1$).
Again, these allowed spin-isospin combinations for the light quarks would only
produce significant energy differences if the color magnetic interaction were
larger than the overall binding due to delocalization. The color 6 combination
of the heavy quarks would not be expected to produce any attraction, as indeed
no such components appear in baryons, but the color ${\bf{\bar{3}}}$
combination would. Neither of these effects is included here as the channel to
color neutralization by decomposition into two color-singlet mesons ($B$ and
$D$) is almost open, so overall color confinement issues should not be
significant. In any event, symmetrization and antisymmetrization between the
$c$ and $b$ quarks is moot as they are distinguishable.
We turn now to the detailed calculations of the light (anti)quark wave
functions in the double well defined by the Born-Oppenheimer-fixed heavy
quarks.
## II The Two-Well Wave Function
Figure 1: Two-well linear potential. In this and all the following figures,
distances, energies, and wave functions are dimensionless.
For two wells separated by $2\delta$ at dimensionless positions ${\bf
w}_{\pm}=\\{0,\;0,\;\pm\,\delta\\}$ along the $z$-axis (see Fig. 1), we define
the wave function
$\Psi_{L}({\bf r})=\psi({\bf r}_{-})+\epsilon\;\psi({\bf r}_{+}),\quad\text{
where }\quad{\bf r}_{\pm}={\bf r}+{\bf w}_{\pm}=\\{x,y,z\pm\delta\\}\ .$ (1)
This represents, for example, a light $\bar{u}$-quark (which we assume to be
massless) mostly moving and confined in the well at $r_{-}$ (the “right”)
provided by the heavy $b$-quark. There may be some “leakage,” represented by
$\epsilon$, into the “left” well at $r_{+}$, provided by the heavy $c$-quark.
As mentioned above, we assume that the $b$ and $c$ quark masses are large
enough to justify a Born-Oppenheimer approximation of this sort. There is a
similar wave function $\Psi_{R}$ with $r_{-}$ and $r_{+}$ interchanged in Eq.
(1) for a light $\bar{d}$-quark mostly confined to the well at $r_{+}$ with
$\epsilon$-leakage into the well at $r_{-}$.
We will determine variationally what the best values of the parameters
$\delta$ and $\epsilon$ are that provide a four-quark or molecular-like
binding that form a $b\,\bar{u}\;c\,\bar{d}$ system. The $b$ and $c$ are well
separated compared with their Compton sizes. Since they have little, if any,
wave function overlap and have distinct quantum numbers, anti-symmetrization
issues are irrelevant. For the rest of the paper we will drop the subscripts
$L$ and $R$ on $\Psi$, but it should be borne in mind when we finally compose
the $b\,\bar{u}\;c\,\bar{d}$ four-quark state.
In this paper we work as much as possible with dimensionless quantities (with
$\hbar=c=1$). That is, $\delta$, ${\bf r}$, etc., are all dimensionless
distances. The dimensionless potentials $V(r)$ and $S(r)$ given below in Eq.
(4) are related to dimension-full potentials $\cal V$ and $\cal S$ by a factor
of $\kappa^{2}$, which has dimensions of GeV/fm. For example, $\cal S$ would
be defined as ${\cal S}({\sf r})=\kappa^{2}\;{\sf r}$, where ${\sf
r}=r/\kappa$ has dimensions in fm. In GMSS GMSS , to cite one reference,
$\kappa^{2}$ was chosen to be 0.9 GeV/fm, corresponding to $\kappa=2.21$ fm-1.
In this paper we have used a larger value, $\kappa^{2}$ = 1.253 GeV/fm, or
$\kappa=2.520$ fm-1, as found in our fitting of charmonia masses.Convolve
We take the $\psi$’s in Eq. (1) to be dimensionless four-component Dirac wave
functions for light massless $u$\- and $d$-quarks. They are solutions of
$H_{D}\;\psi=[-i\mbox{\boldmath$\alpha$}\cdot{\bf\nabla}+V({\bf r})+\beta
S({\bf r})]\;\psi=E\;\psi\ .$ (2)
Here $V({\bf r})$ is the time component of a Lorentz four-vector and $S({\bf
r})$ is a Lorentz scalar potential (both to be specified below). With the
Pauli spinor $\chi$ assumed to be quantized along the $z$-direction with spin-
projection $m_{s}$, the normalized four-component $s$-wave Dirac wave function
$\psi({\bf r})$ is
$\psi_{m_{s}}({\bf
r})=\frac{1}{\sqrt{4\pi}}\left(\begin{array}[]{c}\psi_{a}(r)\;\chi_{m_{s}}\\\
i\mbox{\boldmath$\sigma$}\cdot{\bf
r}\;\psi_{b}(r)\;\chi_{m_{s}}\end{array}\right)\ .$ (3)
The upper and lower radial wave functions $\psi_{a}(r)$ and $\psi_{b}(r)$ can
be chosen real. We have calculated them by solving the coupled radial Dirac
equations EJP for (dimensionless) linear Lorentz vector and scalar potentials
of the form
$V(r)=r-R\quad\text{ and }\quad S(r)=r\ .$ (4)
Here $-R\,$ is a negative displacement pushing the vector potential $V(r)$
down below the scalar potential $S(r)$, so that confinement trumps Klein-
Gordon pair creation.PGG
Figure 2: Normalized massless quark $1s$ wave functions $\psi_{a}(r)$ (above
the axis) and $r\psi_{b}(r)$ (below).
The curves in Fig. 2 show the calculated (dimensionless) $1S$ wave functions
$\psi_{a}(r)$ and $r\psi_{b}(r)$ when the potentials have $R=1.92$,
$\kappa^{2}=1.253$ GeV/fm. Physical dimensions can be obtained by dividing the
dimensionless $r$, $R$, etc., by $\kappa=2.52$ fm-1. The ground state
eigenenergy resulting from this calculation is 0.375 GeV. These potentials
provide a reasonable fit to the $c\,\bar{c}$ spectrum.Convolve
## III Expanding $\left<H^{\ 2}_{D}\right>$
The idea is that we will want to minimize the expectation value
$\left<\,H^{2}_{D}\,\right>^{1/2}$ with respect to the parameters $\epsilon$
and $\delta$ to bound (approximately) the energy for the four-quark system
consisting of $b$, $c$, $\bar{u}$, and $\bar{d}$. The square $\left<\,H^{\
2}_{D}\,\right>$ is required for a variational bound as, due to negative
energy states, $\left<\,H_{D}\,\right>$ itself is unbounded below. The Dirac
Hamiltonian $H_{D}$ is displayed in Eq. (2) but now, for the two-well case
(Fig. 1), the potentials are
$V({\bf r})=\left\\{\begin{array}[]{ll}r_{-}-R,&\mbox{ if $z>0$}\\\
r_{+}-R,&\mbox{ if $z<0$}\end{array}\right.\qquad\mbox{ and }\qquad S({\bf
r})=\left\\{\begin{array}[]{ll}r_{-},&\mbox{ if $z>0$}\\\ r_{+},&\mbox{ if
$z<0$}\end{array}\right.\ .$ (5)
As already mentioned, $-R\,$ is a negative offset so the vector potential lies
below the scalar.
The exact two-well energy $E$ is in principle found by solving for the
eigenvalue of
$H_{D}\;\Psi({\bf r})=E\;\Psi({\bf r})\ ,$ (6)
with $\Psi$ given in Eq. (1). This being difficult, we instead chose to find
an approximate value of the four-quark energy $E$ by the above-mentioned
minimization of $\left<H^{2}_{D}\right>^{1/2}$.
After some algebra one finds
$\displaystyle H^{2}_{D}$ $\displaystyle=$
$\displaystyle-\nabla^{2}+V^{2}({\bf r})+S^{2}({\bf r})+2\beta\,V({\bf
r})\,S({\bf r})$ (7)
$\displaystyle\quad-i\mbox{\boldmath$\alpha$}\cdot\left[\left({\bf\nabla}V({\bf
r})\right)+\beta\left({\bf\nabla}S({\bf r})\right)\right]-2i\,V({\bf
r})\;\mbox{\boldmath$\alpha$}\cdot{\bf\nabla}\ .$
The lack of a term like $-2i\,S({\bf
r})\;\mbox{\boldmath$\alpha$}\cdot{\bf\nabla}$ is because of a cancellation
(the Dirac operators $\alpha$ and $\beta$ anti-commute). The first four terms
of $H^{2}_{D}$ are “diagonal” (generically, ${\cal O}_{D}$) in that they
connect $\psi_{a}$ to $\psi_{a}$ and $\psi_{b}$ to $\psi_{b}$, while the last
two terms are “off-diagonal” (${\cal O}_{OD}$) connecting $\psi_{a}$ to
$\psi_{b}$.
An Appendix describes, in detail, how we calculate the expectation values of
the terms in Eq. (7). For brevity, we now present the numerical results of
these calculations for $H_{D}^{\ \ 2}$ and its components.
Figure 3: Plot of all the diagonal contributions to $<H_{D}^{\ \
2}(\epsilon,\delta)>$.
## IV Plotting $<H_{D}^{\ \ 2}>$ to find a minimum energy
We combine all the expectation integrals discussed in the Appendix together to
get an analytic expression for $<H_{D}^{\ \ 2}>$, which we can plot to look
for a minimum squared energy.
First, we define the (unnormalized) contribution, as a function of $\epsilon$
and $\delta$, from the diagonal pieces,
$\displaystyle<H_{D,\;{\rm diag}}^{\ \ 2}(\epsilon,\delta)>$ $\displaystyle=$
$\displaystyle\sum_{i,j}a_{i}\,a_{j}\,\left[\,(1+\epsilon^{2})\,\left(I_{<\nabla^{2}>}^{(0)}+4\,I_{ij,<r_{\pm}^{2}>}^{(0)}-4\,R\,I_{ij,<r_{\pm}>}^{(0)}+R^{2}\,I_{ij,<1>}^{(0)}\right)\right.$
(8)
$\displaystyle\left.\qquad\qquad\qquad+\,\epsilon\,\left(I_{ij,<\nabla^{2}>}^{(1)}+4\,I_{ij,<r_{\pm}^{2}>}^{(1)}-4\,R\,I_{ij,<r_{\pm}>}^{(1)}+R^{2}\,I_{ij,<1>}^{(1)}\right)\,\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum_{i,j}b_{i}\,b_{j}\,\left[\,(1+\epsilon^{2})\,\left(J_{ij,<\nabla^{2}>}^{(0)}+R^{2}\,J_{ij,<1>}^{(0)}\right)+\,\epsilon\,\left(J_{ij,<\nabla^{2}>}^{(1)}+R^{2}\,J_{ij,<1>}^{(1)}\right)\,\right]\
,$
using the expressions for the integrals $I$ and $J$ given in the Appendix.
Figure 3 displays a three-dimensional plot of the normalized $<H_{D,\;{\rm
diag}}^{\ \ 2}(\epsilon,\delta)>/N^{2}(\epsilon,\delta)$, where
$N^{2}(\epsilon,\delta)$ is also discussed and displayed in the Appendix. It
shows a relatively shallow minimum at $\epsilon=1$ and $\delta\approx 0.8$.
Note the large value, a dimensionless squared-energy of $\approx 4$, which
must be largely cancelled by the off-diagonal contributions to achieve a
squared-energy similar to that for the one-well case, $E^{2}=0.5685$.
The off-diagonal (unnormalized) contributions are
$\displaystyle<H_{D,\;{\rm off-diag}}^{\ \ 2}(\epsilon,\delta)>$
$\displaystyle=$
$\displaystyle\sum_{i,j}a_{i}\,b_{j}\,\left[\,(1+\epsilon^{2})\,\left(K_{ij,<\nabla
VS>}^{(0)}+K_{ij,<V\nabla>}^{(0)}\right)\right.$ (9)
$\displaystyle\left.\qquad\qquad\qquad+\;\epsilon\,\left(K_{ij,<\nabla
VS>}^{(1)}+K_{ij,<V\nabla>}^{(1)}\right)\,\right]\ ,$
with integrals $K$ also from the Appendix.
Figure 4: Plot of all the off-diagonal contributions to $<H_{D}^{\ \
2}(\epsilon,\delta)>$.
Figure 4 gives the three-dimensional plot of $<H_{D,\;{\rm off-diag}}^{\ \
2}(\epsilon,\delta)>/N^{2}(\epsilon,\delta)$. In contrast with $<H_{D,\;{\rm
diag}}^{\ \ 2}>/N^{2}$, it has a repulsive hump around $\delta\approx 1$ as
well as a shallow valley running from $\epsilon=0$ to 1 for $\delta\approx
0.2$. In the final sum of diagonal and off-diagonal contributions that hump
will fill in the minimum seen in Fig. 3.
Thus we finally combine the two contributions, defining a normalized
$<H_{D}^{\ \ 2}(\epsilon,\delta)>=\left[<H_{D,\;{\rm off-diag}}^{\ \
2}(\epsilon,\delta)>+<H_{D,\;{\rm off-diag}}^{\ \
2}(\epsilon,\delta)>\right]/N^{2}(\epsilon,\delta)\ .$ (10)
Figure 5: Plot of the final $<H_{D}^{\ \ 2}(\epsilon,\delta)>$.
Figure 5 plots how $H_{D}^{\ \ 2}$, as a function of $\epsilon$ and $\delta$,
develops a long, flat valley for all values of $\epsilon$ at a separation of
$\delta\approx 0.2$ (i.e., recalling the value of $\kappa$, a separation of
$\approx 0.45$ fm). Also important is the hump (reminiscent of a fission
barrier) around $\delta\approx 0.9$ that will help to confine this four-quark
system at $\delta\approx 0.2$. This hump corresponds to a repulsion between
two $Q-\bar{q}$ asymptotic meson states preventing the light quarks from
delocalizing. There is very little, if any, barrier to coalescence at
$\epsilon$ = 0.
Figure 6: $H_{D}^{\ \ 2}(\epsilon=1,\delta)$, with a valley at $\delta=0.18$
and a “fission barrier” at $\delta\approx 0.9$.
It is easier to see this behavior with a two-dimensional plot, Fig. 6, showing
$H_{D}^{\ \ 2}$ as a function of $\delta$ at $\epsilon=1$, where the valley is
deepest and the hump is highest.
Figure 7: Plot of how the nearly flat valley at $\delta=0.18$ decreases from
$\epsilon=0$ to $\epsilon=1$ .
The dimensionless squared-energy valley-depth at $\epsilon=1.0$ and
$\delta=0.18$, $\Delta H_{D}^{\ \ 2}=0.097$, corresponds to a binding energy
of 155 MeV for this $b\,c\,\bar{u}\,\bar{d}$ four-quark mesonic state. The
valley is surprisingly flat, as shown in Fig. 7, dropping only 0.0023 squared
dimensionless energy units from $\epsilon=0$ to $\epsilon=1$. This corresponds
to an energy drop of about 24 MeV, a rather small energy difference. This
suggests that Zitterbewegung may play an important role in the nature of this
meson.
## V Discussion
Figure 8 is a contour plot of the binding energy of the state in the
$\epsilon$-$\delta$ plane. It displays two remarkable features: The first is
that, at very small $\epsilon$, appropriate to the approach towards each other
of the two asymptotic ($B$ and $D$) mesons, there is no evidence of a
repulsive barrier to the fusion of those mesons. The second is that the valley
of attraction at small meson separation is very flat between small $\epsilon$
($\sim 0.2$) and $\epsilon=1$. This indicates that there is little energy
associated with fluctuations in the $\epsilon$ collective variable of the
light quarks in the state. There may be a more significant amount associated
with the $\delta$ collective variable, but this effect is suppressed by the
large masses associated with the Born-Oppenheimer centers defined by the heavy
quarks, at least when viewed non-relativistically as seems appropriate for
them, due to their relatively large masses. We therefore expect little
correction to our estimates of the mass of the four-quark state due to
collective variable effects.
Figure 8: Contour plot of $H_{D}^{\ \ 2}(\epsilon,\delta)$. The dashed curve
illustrates how two well-separated $Q-\bar{q}$ mesons at $\epsilon=0$ and
large $\delta$ come together and slide down the valley at $\delta\approx 0.2$
to form a four-quark state at $\epsilon=1$.
The dashed curve in Figure 8 illustrates how two well-separated $B$ and $D$
mesons at $\epsilon=0$ and large $\delta$ would come together to
$\delta\approx 0.2$ and $\epsilon\approx 0.2$, corresponding to a heavy quark
separation of about $0.45$ fm. As we have emphasized above, this small
separation makes it clear that long-range pion-exchange effects do not
contribute significantly. From $\epsilon\approx 0.2$, the four-quark state
then slides gently down the nearly flat valley to $\epsilon=1$ where it is
most bound. Such a state is prevented from falling apart because of the
“fission barrier” around $\delta\approx 0.9$.
We have ignored the possible color magnetic contributions from the interaction
of the two light antiquarks, but this must be less than 50 MeV and we expect
it to be even less than half this value. These corrections, which we will deal
with in a future publication, are not large compared to the extracted
variational upper bound on the binding energy of order 150 MeV found in our
calculations. Thus, by comparing our binding energy with the threshold for $B$
and $D$ mesons, we predict a set of states in the region of 7 GeV/$c^{2}$.
Finally, we comment on the surprisingly small difference in binding energy
between the “molecular” form of the bound state, ($\epsilon\approx 0.2$, as in
nuclei GMSS )) and the four-quark limit ($\epsilon=1$). If this feature is
widespread in such heavy quark systems, it could go far towards explaining why
it has been so difficult to identify unambiguous four-quark states.
In any event, as our interest here is in nuclear physics, we note that the
small separation compared to root-mean-square size of the meson states argues
against the identification of the system as that of two slightly off-shell
free space mesons, at least, at $\epsilon\sim 1$. However, the small
difference in energy between that region and $\epsilon\sim 0.2$ suggests to
the contrary, that since the binding energy is not large, at least some of the
time, the system would appear to be one described as two slightly off-shell
free space mesons, with substantial fluctuations between the two pictures.
Difficult as it was historically, we conclude that nuclear physics would have
been even more difficult to understand if it had similar properties.
## VI Acknowledgments
This work was carried out in part under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy at Los Alamos
National Laboratory under Contract No. DE-AC52-06NA25396.
## References
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* (2) See, e.g., U. van Kolck, Prog. Part. Nucl. Phys. 43 (1999) 337.
* (3) European Muon Collaboration (J. Ashman et al.), Z.Phys.C57 (1993) 211 and earlier papers cited there.
* (4) T. Goldman, K. R. Maltman, G. J. Stephenson, Jr., and K. E. Schmidt, Nucl. Phys. A481 (1988) 621.
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* (8) E.g., T. Goldman and R. R. Silbar, Phys. Rev. C77 (2008) 065203.
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* (10) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. -M. Yan, Phys. Rev. D17 (1978) 3090.
* (11) Some recent examples include: Takayuki Matsuki and Koichi Seo, Phys. Rev. D85 (2012) 014036; Stanislaw D. Glazek, “Hypothesis of quark binding by condensation of gluons in hadrons”, presented at LIGHTCONE 2011, 23 - 27 May, 2011, Dallas, TX, arXiv:1110.1430; Jin-Hee Yoon, Byeong-Noh Kim, Horace W. Crater and Cheuk-Yin Wong, “On the Mass Difference between pion and rho meson using a Relativistic Two-Body Model”, Talk presented at the Fifth Asia-Pacific Conference on Few-Body Problems in Physics, August 22-26, 2011, Seoul, Republic of Korea, to be published in Few-Body Systems, arXiv:1110.1598; M. Blank and A. Krassnigg, Phys. Rev. D84 (2011) 096014.
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* (14) Carlos Peña and David Blaschke, Acta Phys. Pol. B, Proc. Suppl. 5 (2012 963).
* (15) P. R. Page, T. Goldman, and J. N. Ginocchio, Phys. Rev. Lett. 86 (2001) 204.
* (16) T. Goldman and R. R. Silbar, Phys. Rev. C85 (2012) 015203.
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* (18) See, e.g., M. Oka, Phys. Rev. D31 (1985) 2274, Phys. Rev. D31 (1985) 2773, and subsequent related articles.
* (19) J. Beringer et al. (Particle Data Group), Phys. Rev. D86 (2012) 010001. A convenient access to this data is to go on-line to http://pdglive.lbl.gov/.
* (20) R. R. Silbar and T. Goldman, Eur. J. Phys. 32 (2011) 217.
## Appendix A Calculational Details
### A.1 Approximating $\psi_{a}$ and $\psi_{b}$ as a sum of Gaussians
For the calculations presented below, the $\psi_{a,b}$ have both been fitted
to sums of Gaussians,
$\psi_{a}(r)=\sum_{i}a_{i}\;e^{-\mu_{i}r^{2}/2},\quad\psi_{b}(r)=\sum_{i}b_{i}\;e^{-\mu_{i}r^{2}/2}\
,$ (11)
where the $a_{i}$, $b_{i}$, and $\mu_{i}$ are dimensionless numbers. We found
it necessary to go to six terms, so that evaluating the upper and lower
components of the left-hand-side of the Dirac equation [Eq. (2) in the main
text] gives reasonable agreement with the right-hand-side. The fitted
parameters are
$\displaystyle\mu_{i}$ $\displaystyle=$ $\displaystyle\
{\;1.0,1.3,1.6,2.0,4.0,8.0}\;\\}$ $\displaystyle a_{i}$ $\displaystyle=$
$\displaystyle\\{\;0.492649,-0.687482,1.84609,-0.00246039,0.258295,0.0956581\;\\}$
(12) $\displaystyle b_{i}$ $\displaystyle=$
$\displaystyle\\{\;-0.0571296,1.03367,-1.18398,1.33989,0.162575,0.299479\;\\}\
.$
The fitted $\psi_{a}(r)$ and $r\psi_{b}(r)$ are shown as the dashed curves in
Fig. 2, largely overlying the solid curves from the solution of the Dirac
equation. To check the quality of the fits we have evaluated the single-quark
expectation value of the Hamiltonian, $<\,H_{D}\,>=0.7545$, which is slightly
larger than the (dimensionless) energy eigenvalue $E=0.7540$ (which, for a
variational trial function, is as it should be). As a second check on our
Gaussian fits of $\psi_{a}$ and $\psi_{b}$, Eqs. (11) and (4), we also
evaluated the single-well expectation $<\,H_{D}^{2}\,>$ to be 0.5691, again
slightly larger than $E^{2}=0.5685$, as it should be.
### A.2 General Remarks on calculating the expectations
The reason for approximating our numerical radial wave functions $\psi_{a}$
and $\psi_{b}$ as sums of Gaussians is that it allows us to calculate the
expectation values of each of the terms of $H_{D}^{2}$ analytically. Given an
analytic expression for $H_{D}^{2}$ allows us to plot it quickly and precisely
as a function of the variational parameters $\delta$ and $\epsilon$. To do
these integrations, we have relied heavily on programs such as Mathematica and
Maple. As will be seen, the final results can sometimes be messy and often
involve error functions111See, e.g., M. Abramovitz and I. A. Stegun, Handbook
of Mathematical Functions, (Dover, New York, 1965), Chap. 7 because of the
Gaussians being integrated.
For the diagonal operators of $H^{2}_{D}$ we will calculate the upper and
lower contributions separately,
$\left<\Psi|{\cal O}_{D}|{\Psi}\right>=\left<\Psi|{\cal
O}_{D}|\Psi\right>_{A}+\left<\Psi|{\cal O}_{D}|\Psi\right>_{B}\ .$ (13)
The $B$-expectations are more complicated than those for $A$ because of the
factors of $-i\mbox{\boldmath$\sigma$}\cdot{\bf{r_{\pm}}}$ multiplying the
radial $\psi_{b}$’s. However, for some diagonal operators, as will be seen
below, the $B$-expectations are not always needed. In any case, from Eq. (11)
we expand these diagonal operator expectations as
$\left<\Psi|{\cal O}_{D}|\Psi\right>_{A}=\sum_{i,j}a_{i}\,a_{j}\;I_{ij}\
,\qquad\left<\Psi|{\cal O}_{D}|\Psi\right>_{B}=\sum_{i,j}b_{i}\,b_{j}\;J_{ij}\
,$ (14)
where the $I_{ij}$ and $J_{ij}$ are integrals over Gaussians.
First, we separate out the quadratic dependence on $\epsilon$ as
$I_{ij}=I_{ij}^{(0)}+\epsilon\;I_{ij}^{(1)}+\epsilon^{2}\;I_{ij}^{(2)}=(1+\epsilon^{2})\;I_{ij}^{(0)}+\epsilon\;I_{ij}^{(1)}\
,$ (15)
and likewise for the lower-component $B$-integrals $J_{ij}$. The second
equality here comes about because parity symmetry ensures that the
$I_{ij}^{(2)}=I_{ij}^{(0)}$, etc. We will refer to the $I_{ij}^{(0)}$ as
“direct terms,” in that they connect Gaussians with $\mu_{j}\,r_{-}^{2}/2$ to
those with $\mu_{i}\,r_{-}^{2}/2$ (and similarly for $I_{ij}^{(2)}$ with
$r_{+}$). Recalling the $1/4\pi$ from the normalization of the $\psi$’s, we
ensure the symmetry under the interchange of indices $i$ and $j$ by writing
$I_{ij}^{(0)}=\frac{1}{8\pi}\;\int
d^{3}r\;\left\\{\;e^{-\mu_{i}\,r^{2}_{-}/2}\;{\cal
O}_{D}\;e^{-\mu_{j}\,r^{2}_{-}/2}+e^{-\mu_{j}\,r^{2}_{-}/2}\;{\cal
O}_{D}\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\right\\}\ .$ (16)
The direct integrals $J_{ij}^{(0)}$ have a similar form but with
$\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-}\;{\cal
O}_{D}\;\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-}$ in place of the ${\cal
O}_{D}$.
The “cross terms” $I_{ij}^{(1)}$ are more complicated integrals than the
$I_{ij}^{(0)}$, and likewise for $J_{ij}^{(1)}$. They connect Gaussians with
$\mu_{j}\,r_{-}^{2}/2$ to $\mu_{j}\,r_{+}^{2}/2$ and vice versa. Thus, on
symmetrizing in $i$ and $j$,
$\displaystyle I_{ij}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\;\int
d^{3}r\;\left\\{\left[e^{-\mu_{i}\,r^{2}_{-}/2}\;{\cal
O}_{D}\;e^{-\mu_{j}\,r^{2}_{+}/2}+e^{-\mu_{i}\,r^{2}_{+}/2}\;{\cal
O}_{D}\;e^{-\mu_{j}\,r^{2}_{-}/2}+\right]\right.$ (17)
$\displaystyle\left.\qquad\qquad+\;\left[e^{-\mu_{j}\,r^{2}_{-}/2}\;{\cal
O}_{D}\;e^{-\mu_{i}\,r^{2}_{+}/2}+e^{-\mu_{j}\,r^{2}_{+}/2}\;{\cal
O}_{D}\;e^{-\mu_{i}\,r^{2}_{-}/2}\right]\right\\}$
The $J_{ij}^{(1)}$ have a similar form but with ${\cal O}_{D}$ replaced by
$\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-}\;{\cal
O}_{D}\;\mbox{\boldmath$\sigma$}\cdot{\bf r}_{+}\ $ or $\
\mbox{\boldmath$\sigma$}\cdot{\bf r}_{+}\;{\cal
O}_{D}\;\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-}$, as appropriate.
Each of the off-diagonal operators in Eq. (9) of the main text has the general
form
${\cal O}_{OD}=-i\mbox{\boldmath$\alpha$}\cdot{\bf
X}=\left[\begin{array}[]{cc}0&-i\;\mbox{\boldmath$\sigma$}\cdot{\bf X}_{12}\\\
-i\;\mbox{\boldmath$\sigma$}\cdot{\bf X}_{21}&\quad 0\end{array}\ \right]\ $
(18)
where the ${\bf X}_{12}$ and ${\bf X}_{21}$ are vector-operators that may not
be equal because of the possible presence of the diagonal $\beta$ matrix in
${\cal O}_{OD}$.
The direct terms of the off-diagonal expectation $<{\cal O}_{OD}>$ involve
several terms because the upper component of $\Psi^{\dagger}({\bf r}_{-})$
connects through $-i\;\mbox{\boldmath$\sigma$}\cdot{\bf X}_{12}$ to the lower
component of $\Psi({\bf r}_{-})$ at the same time that the lower component of
$\Psi^{\dagger}({\bf r}_{-})$ connects through
$-i\;\mbox{\boldmath$\sigma$}\cdot{\bf X}_{21}$ to the upper component of
$\Psi({\bf r}_{-})$. We therefore have to keep the sums over the $a$’s and
$b$’s in Eq. (11) as parts of the integrand. Again symmetrizing in $i$ and
$j$,
$\displaystyle<{\cal O}_{OD}^{(0)}>$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\;\sum_{i,j}\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[-a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-})\right.\right.$ (19)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\;a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[-a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\left.+\;a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right\\}\ .$
The cross terms of $<{\cal O}_{OD}>$ have even more terms because the
$\Psi^{\dagger}({\bf r}_{+})$ connects to $\Psi({\bf r}_{-})$ at the same time
that $\Psi^{\dagger}({\bf r}_{-})$ connects to $\Psi({\bf r}_{+})$. It becomes
$\displaystyle<{\cal O}_{OD}^{(1)}>$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\;\sum_{i,j}\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[-a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-})\right.\right.$ (20)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\;a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[-a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{+})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\;a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\left[-a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\;a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[-a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{12})(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{+})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\left.+\;a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
X}_{21})\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}\right\\}\ .$
The integrations for the $I$’s, $J$’s, and in Eqs. (19) and (20) can best be
done using (dimensionless) cylindrical coordinates,
$\rho=\left({x^{2}+y^{2}}\right)^{1/2}$, $\theta$, and $z$. The $\theta$
integrations are trivial, providing a factor of $2\pi$, which will cancel with
the $1/4\pi$ coming from the normalizations of the $\psi$’s in Eq. (3) to give
an overall factor of $1/2$ before each double integral over $\rho$ and $z$. It
usually is easier to do the $\rho$-integration (from 0 to $+\infty$) first.
Because $V({\bf r})$ and $S({\bf r})$ depend on $r_{-}$ when $z>0$ and on
$r_{+}$ when $z<0$, we need to do the $z$-integration separately for those
regions, i.e., for $z$ from $-\infty$ to 0 and then for $z$ from 0 to
$+\infty$. The separate results are then added and simplified to give the
final integral.
We will distinguish the results for the expectations of the different
operators in Eq. (7) by an appropriate subscript. For example, for
$O_{D}=\nabla^{2}$, we will write $I_{ij}^{(0,1)}$ as
$I_{ij,\;<\nabla^{2}>}^{(0,1)}$, and similarly for the $J_{ij}$ integrals.
### A.3 Normalizing $\Psi$
While the Dirac $\psi$’s are themselves properly normalized, the two-well
$\Psi$ is not. For this we need to calculate the expectation values of ${\cal
O}_{D}=1$ to find
$N^{2}(\delta,\epsilon)=\int
d^{3}r\;\Psi^{\dagger}\Psi=\left<\Psi|1|\Psi\right>=\left<\Psi|1|\Psi\right>_{A}+\left<\Psi|1|\Psi\right>_{B}\
.$ (21)
We make the expansion in $\epsilon$ as in Eq. (15) above. The direct-term
integrals for the expectation $\left<\,1\,\right>$ are, noting that for the
$J_{ij,\;<1>}^{(0)}$ we also have a factor of
$(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})=r_{-}^{2}$ in the integrand,
$\displaystyle I_{ij,\;<1>}^{(0)}$ $\displaystyle=$
$\displaystyle\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{3}}\right]^{1/2}$ (22)
$\displaystyle J_{ij,\;<1>}^{(0)}$ $\displaystyle=$ $\displaystyle
3\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\ ,$ (23)
both independent of $\delta$.
The cross-term integrals do depend on $\delta$. For the $J_{ij,\;<1>}^{(1)}$
we need the factor
$(\mbox{\boldmath$\sigma$}\cdot{\bf r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})={\bf r}_{+}\cdot{\bf r}_{-}=\rho^{2}+z^{2}-\delta^{2}\ $ (24)
in the integrand. Proceeding as in Sec. A.2, we find
$\displaystyle I_{ij,\;<1>}^{(1)}$ $\displaystyle=$
$\displaystyle\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{3}}\right]^{1/2}\,e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\
,$ (25) $\displaystyle J_{ij,\;<1>}^{(1)}$ $\displaystyle=$
$\displaystyle\left[\;3(\mu_{i}+\mu_{j})-4\;\mu_{i}\mu_{j}\;\delta^{2}\;\right]\;\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{7}}\right]^{1/2}\,e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\
.$ (26)
Note that, when $\delta=0$, $I_{ij,\;<1>}^{(1)}=2\;I_{ij,\;<1>}^{(0)}$, and
$J_{ij,\;<1>}^{(1)}=2\;J_{ij,\;<1>}^{(0)}$. This is a common feature for all
the expectations here and below. This is necessary so that, for example, when
$\delta=0$ and $\epsilon=1$, one recovers a result that is four times that
when $\delta=0$ and $\epsilon=0$.
Figure 9: Typical plots of $I$’s, $J$’s, and $K$’s as functions of $\delta$.
The $y$-axes are in arbitrary units.
We see from Eq. (25) that $I_{ij,\;<1>}^{(1)}$,as a function of $\delta$, is a
decaying Gaussian (as in Fig. 9, plot A). On the other hand,
$J_{ij,\;<1>}^{(1)}$ falls off from its peak at $\delta=0$, goes through zero,
and has a mild minimum before decaying to zero at large $\delta^{2}$ (as in
Fig. 9, plot B).
Combining all terms,
$\displaystyle N^{2}(\epsilon,\delta)$ $\displaystyle=$
$\displaystyle\sum_{i,j}a_{i}\,a_{j}\,\left[\,(1+\epsilon^{2})\,I_{<1>}^{(0)}+\epsilon\,I_{<1>}^{(1)}\,\right]+\sum_{i,j}b_{i}\,b_{j}\,\left[\,(1+\epsilon^{2})\,J_{<1>}^{(0)}+\epsilon\,J_{<1>}^{(1)}\,\right]$
(27)
and the normalized $\Psi$ is obtained by multiplying Eq. (1) by
${1/N(\epsilon,\delta)}$.
Figure 10: Three-dimensional plot of $N^{2}(\epsilon,\delta)$.
Figure 10 shows a plot of $N^{2}(\epsilon,\delta)$ for the values of the
$a$’s, $b$’s, and $\mu$’s that were fitted to the normalized $\psi_{a}$ and
$\psi_{b}$, Eq. (A.1). We have checked that, for these values,
$N^{2}(0,0)=0.9858\approx 1$ and $N^{2}(1,0)=3.9430\approx 4$, as they should
but with some deviation ($\approx 2$%) coming from the inexactness of the
fitting. The ratio of the two values is 4 to high accuracy.
Figure 11: Plot of a normalized $\Psi_{a}(\rho,z)$ for $\epsilon=0.5$ and
$\delta=1.0$.
To illustrate what ”leakage” from one well to the other might look like, Fig.
11 shows a plot of the upper component of the normalized $\Psi$ as a function
of $\rho$ (running from 0 to 2) and $z$ (running from -3.5 to +3.5) for
$\epsilon=0.5$ and $\delta=1.1$.
### A.4 Evaluating the diagonal expectation $\left<\,-\nabla^{2}\,\right>$
First, note that, for ${\bf r}_{\pm}=\\{x,\,y,\,z\pm\delta\\}$, the $i$th
component of the gradient
$\nabla_{i}=\frac{\partial}{\partial
x_{i}}=\nabla_{i}^{\prime}=\frac{\partial}{\partial
x_{i}^{\prime}}\quad\mbox{for}\quad{\bf
r}^{\prime}=\\{x^{\prime}=x,\,y^{\prime}=y,\,z^{\prime}=z\pm\delta\\}={\bf
r}_{\pm}\,$ (28)
since each $\partial x_{i}^{\prime}/\partial x_{i}=1$. Thus we can replace the
result of the Laplacian with respect to $r$ acting on a function such as
$\psi_{a}(r_{-})$ with that for a Laplacian with respect to $r_{-}$ acting on
that function. For spherical coordinates, $-\nabla^{2}$ on the angle-
independent $e^{-\mu_{j}\,r_{-}^{2}/2}$ then becomes
$-\nabla^{2}\;e^{-\mu_{j}r_{-}^{2}/2}=-\nabla^{\prime\,2}\;e^{-\mu_{j}\,r_{-}^{2}/2}=-\frac{1}{r_{-}}\frac{d^{2}}{d\,r_{-}^{2}}\left(\;r_{-}e^{-\mu_{j}\,r_{-}^{2}/2}\;\right)=-\mu_{j}(\mu_{j}\,r_{-}^{2}-3)\;e^{-\mu_{j}\,r_{-}^{2}/2}\
,$ (29)
whence the three-dimensional integral reduces, after symmetrizing and
cancelling factors of $4\pi$, to
$I_{ij,\;<-\nabla^{2}>}^{(0)}=3\;\mu_{i}\mu_{j}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2},$
(30)
independent of $\delta$.
For the $B$-integrals things are more complicated because of the
$\mbox{\boldmath$\sigma$}\cdot{\bf r}_{-}$ factor to the right of the
Laplacian. After some algebra,
$-({\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-}}\nabla^{2}{\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-}})\,e^{-\mu_{j}r_{-}^{2}/2}=-r_{-}^{2}\;\mu_{j}\,(\mu_{j}r_{-}^{2}-5)\,e^{-\mu_{j}r_{-}^{2}/2}$
(31)
whence
$J_{ij,\;<-\nabla^{2}>}^{(0)}=\;15\;\mu_{i}\mu_{j}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{7}}\right]^{1/2}\
,$ (32)
also independent of $\delta$.
The cross terms, again, do depend on $\delta$.
$I_{ij,\;<-\nabla^{2}>}^{(1)}=\mu_{i}\mu_{j}\,[\,3\,(\mu_{i}+\mu_{j})-4\mu_{i}\mu_{j}\,\delta^{2}\,]\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{7}}\right]^{1/2}\;e^{-2\mu_{i}\mu_{j}\,\delta^{2}/(\mu_{i}+\mu_{j})}\
.$ (33)
This integral as a function of $\delta$ looks like Fig. 9B.
For the corresponding $B$-cross term, one proceeds in the same manner but,
instead of Eq. (31), we need222 Because we have separated the two wells along
the $z$-direction, the cross product ${\bf r}_{+}\times{\bf r_{-}}$ only has
$x$ and $y$ components. Since we have assumed the Pauli spinor $\chi_{m_{s}}$
to be polarized along the $z$-axis, the term from the product of two Pauli
$\sigma$ matrices that gives a $i\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+}\times{\bf r_{-}}$ contribution vanishes.
$-({\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+}}\nabla^{2}{\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-}})\,e^{-\mu_{j}r_{-}^{2}/2}=-(\rho^{2}+z^{2}-\delta^{2})\;\mu_{j}\,(\mu_{j}r_{-}^{2}-5)\,e^{-\mu_{j}r_{-}^{2}/2}\
.$ (34)
We find
$\displaystyle J_{ij,\;<-\nabla^{2}>}^{(1)}$ $\displaystyle=$
$\displaystyle\mu_{i}\mu_{j}\,[\,15\,(\mu_{i}+\mu_{j})^{2}-40\,\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})\,\delta^{2}+16\,\mu_{i}^{2}\mu_{j}^{2}\;\delta^{4}\,]\times$
(35)
$\displaystyle\qquad\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{11}}\right]^{1/2}e^{-2\mu_{i}\mu_{j}\,\delta^{2}/(\mu_{i}+\mu_{j})}\
.$
This integral as a function of $\delta$ also looks like Fig. 9B, but because
it is quartic, it is slightly positive beyond $\delta=1.7$.
### A.5 Evaluating the expectation of $V^{2}+S^{2}+2\beta\,VS$
This is also a diagonal operator. The linear vector potential $V({\bf r})$
differs from the linear scalar potential $S({\bf r})$ by a negative offset
$-\;R$. In $\left<V^{2}({\bf r})\right>$ the integrals of
$\left<r_{\pm}^{2}\right>$ are the same as those for $\left<S^{2}({\bf
r})\right>$. Here, $\left<r_{\pm}^{2}\right>$ means the integration of
$r_{-}^{2}$ when $z>0$ and of $r_{+}^{2}$ when $z<0$. Thus we (schematically)
expand the diagonal $V^{2}+S^{2}+2\beta\,VS$ as
$\left<V^{2}+S^{2}+2\beta\,VS\right>=2\left<\;r_{\pm}^{2}\;\right>(1+\beta)-2\;R\left<\;r_{\pm}\;\right>(1+\beta)+\;R^{2}\left<\;1\;\right>\
.$ (36)
The factor of $(1+\beta)$ ensures that only the upper components of $\Psi$
contribute to the first two expectation values. That is, we only need to
calculate the $A$-integrals (the $I$’s) for those terms. The expectation value
$\left<\,1\,\right>$ multiplying $R^{2}$ does have contributions from the
lower components and their integrals $I_{<1>}^{(0)}$, $I_{<1>}^{(1)}$,
$J_{<1>}^{(0)}$, and $J_{<1>}^{(1)}$ are given in Sec. A.3.. The integrals for
the operators $\left<r_{\pm}^{2}\right>$ and $\left<r_{\pm}\right>$ are rather
more complicated and their analytic forms are presented next.
#### A.5.1 Expectation of ${\cal O}_{D}=r_{\pm}^{2}$
The direct integral for this operator is
$\displaystyle I_{ij,\;<r_{\pm}^{2}>}^{(0)}$ $\displaystyle=$
$\displaystyle-\;\frac{2\delta}{(\mu_{i}+\mu_{j})^{2}}e^{-(\mu_{i}+\mu_{j})\;\delta^{2}/2}$
(37)
$\displaystyle+\;\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\\!\left[3+2(\mu_{i}+\mu_{j})\;\delta^{2}\;\text{Erfc}\left(\sqrt{\frac{(\mu_{i}+\mu_{j})}{2}}\;\delta\right)\right]\
.$
Note the linear dependence on $\delta$, which gives rise to a shallow minimum
near the origin before the function returns to its initial value, as in Fig.
9C.
The cross-term integral for $<r_{\pm}^{2}>$ is
$\displaystyle I_{ij,\;<r_{\pm}^{2}>}^{(1)}$ $\displaystyle=$
$\displaystyle\;-\;\frac{4\delta}{(\mu_{i}+\mu_{j})^{2}}e^{-(\mu_{i}+\mu_{j})\;\delta^{2}/2}$
$\displaystyle\;+\;\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{7}}\right]^{1/2}e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\times$
$\displaystyle\quad\quad\left\\{3(\mu_{i}+\mu_{j})+2\;(\mu_{i}^{2}+\mu_{j}^{2})\;\delta^{2}-2\;(\mu_{i}^{2}-\mu_{j}^{2})\;\delta^{2}\;\text{Erf}\left(\frac{(\mu_{i}-\mu_{j})}{\sqrt{2(\mu_{i}+\mu_{j})}}\;\delta\right)\right\\},$
which also has odd terms in $\delta$. In this case, as a function of $\delta$,
$I_{ij,\;<r_{\pm}^{2}>}^{(1)}$ falls off smoothly to zero from its peak value
at $\delta=0$, as in Fig. 9D. $I_{ij,\;<r_{\pm}^{2}>}^{(1)}$ is symmetric in
$i$ and $j$ because $\text{Erf}(-x)=-\text{Erf}(x)$,
$I_{ij,\;<r_{\pm}^{2}>}^{(1)}=I_{ji,\;<r_{\pm}^{2}>}^{(1)}$. Also, as
expected,
$I_{<r_{\pm}^{2}>}^{(1)}=\;2\;I_{<r_{\pm}^{2}>}^{(0)}=3\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\
$ (39)
when $\delta=0$.
#### A.5.2 Expectation of ${\cal O}_{D}=r_{\pm}$
The direct term for this operator is
$\displaystyle I_{ij,\;<r_{\pm}>}^{(0)}$ $\displaystyle=$
$\displaystyle\;\frac{1}{2(\mu_{i}+\mu_{j})^{2}}\;[4+2\;e^{-2(\mu_{i}+\mu_{j})\;\delta^{2}}-3\;e^{-(\mu_{i}+\mu_{j})\;\delta^{2}/2}]$
$\displaystyle-\
\frac{1}{2\delta}\;\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\times$
$\displaystyle\qquad\qquad\left\\{\;(\mu_{i}+\mu_{j})\,\delta^{2}-\
\left(1+4(\mu_{i}+\mu_{j})\;\delta^{2}\right)\;\text{Erf}\left(\sqrt{2\,(\mu_{i}+\mu_{j})}\
\delta\right)\right.$ $\displaystyle\left.\qquad\qquad\qquad\ +\
\left(1+3(\mu_{i}+\mu_{j})\;\delta^{2}\right)\;\text{Erf}\left(\sqrt{(\mu_{i}+\mu_{j})/2}\
\delta\,\right)\right\\}\ .$
This integral also has an odd term in $\delta$, like
$I_{ij,\;<r_{\pm}^{2}>}^{(0)}$. As a function of $\delta$ it resembles that
shown in Fig. 9C. That is, despite the $1/\delta$ factor in the last term,
$I_{ij,\;<r_{\pm}>}^{(0)}$ is not singular at $\delta=0$ (i.e., when there is
no separation between the two wells): $I_{ij,\;<r_{\pm}>}^{(0)}\rightarrow
2/(\mu_{i}+\mu_{j})^{2}$ as $\delta\rightarrow 0\ $.
The cross term for $\left<r_{\pm}\right>$ is
$\displaystyle I_{ij,\;<r_{\pm}>}^{(1)}$ $\displaystyle=$
$\displaystyle\;\frac{2}{(\mu_{i}+\mu_{j})^{2}}\
\left(e^{-2\mu_{i}\;\delta^{2}}+e^{-2\mu_{j}\;\delta^{2}}-e^{-\frac{1}{2}(\mu_{i}+\mu_{j})\;\delta^{2}}\right)$
(41) $\displaystyle+\
\frac{1}{2\,\delta\,\mu_{i}\mu_{j}}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\left\\{\left(\mu_{i}+\mu_{j}\right)^{2}\;\text{Erfc}\left(\sqrt{\frac{\mu_{i}+\mu_{j}}{2}}\,\delta\right)\right.$
$\displaystyle\quad\quad\left.-\;2\;\mu_{j}\left(\mu_{i}+\mu_{j}+4\,\mu_{i}^{2}\,\delta^{2}\right)\;e^{-\frac{2\delta^{2}\mu_{i}\mu_{j}}{\mu_{i}+\mu_{j}}}\;\text{Erfc}\left(\sqrt{\frac{2}{\mu_{i}+\mu_{j}}}\;\mu_{i}\,\delta\right)\right.$
$\displaystyle\quad\quad\left.+\;2\text{
}\mu_{i}\left(\mu_{i}+\mu_{j}+4\,\mu_{j}^{2}\,\delta^{2}\right)\;e^{-\frac{2\delta^{2}\mu_{i}\mu_{j}}{\mu_{i}+\mu_{j}}}\;\text{Erf}\left(\sqrt{\frac{2}{\mu_{i}+\mu_{j}}}\;\mu_{j}\,\delta\right)\right.$
$\displaystyle\quad\quad\left.-\;\left(\mu_{i}-\mu_{j}\right)\left(\mu_{i}+\mu_{j}-4\;\mu_{i}\mu_{j}\,\delta^{2}\right)e^{-\frac{2\delta^{2}\mu_{i}\mu_{j}}{\mu_{i}+\mu_{j}}}\;\text{Erfc}\left(\frac{\left(\mu_{i}-\mu_{j}\right)\,\delta}{\sqrt{2\,(\mu_{i}+\mu_{j})}}\right)\right\\}$
Note that $I_{ij,\;<r_{\pm}>}^{(1)}$ is also symmetric under the interchange
of $i$ and $j$ and, again, at $\delta=0$, we have
$I_{ij,\;<r_{\pm}>}^{(1)}=4/(\mu_{i}+\mu_{j})^{2}=2\;I_{ij,\;<r_{\pm}>}^{(0)}$.
Its behavior as a function of $\delta$ is similar to that shown in Fig. 9D,
again partly due to the presence of odd terms in $\delta$.
### A.6 The off-diagonal expectation of
$-i\mbox{\boldmath$\alpha$}\cdot\left[\left({\bf\nabla}V({\bf
r})\right)+\beta\left({\bf\nabla}S({\bf r})\right)\right]$
For the linear potentials of Eq. (5)
${\bf\nabla}V({\bf r})\;=\;{\bf\nabla}S({\bf
r})=\left\\{\begin{array}[]{ll}{\bf\hat{r}_{-}}&\ \mbox{if $z>0$}\\\
{\bf\hat{r}_{+}}&\ \mbox{if $z<0$}\end{array}\right.\ $ (42)
and we again have a simplification from the $(1+\beta)$, namely,
$-i\mbox{\boldmath$\alpha$}\cdot\left[\left({\bf\nabla}V({\bf
r})\right)+\beta\left({\bf\nabla}S({\bf
r})\right)\right]=-i\mbox{\boldmath$\alpha$}\cdot{\bf\hat{r}_{\pm}}(1+\beta)=\left[\begin{array}[]{cc}0&\quad
0\\\ -2i\;\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}_{\pm}}&\quad 0\end{array}\
\right]\ ,$ (43)
i.e., the operator ${\bf X}_{12}$ in Eq. (18) vanishes and ${\bf X}_{21}$ is
doubled. The latter operator connects the upper component of $\Psi^{\dagger}$
to the lower component of $\Psi$.
For the direct terms, Eq. (19) reduces to two terms
$\displaystyle<[\nabla VS]^{(0)}>$ $\displaystyle=$
$\displaystyle-2\;\frac{1}{4\pi}\;\sum_{i,j}\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[a_{j}b_{i}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
(44)
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[a_{i}b_{j}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right\\}$
$\displaystyle=$ $\displaystyle\sum_{i,j}\left[a_{j}b_{i}\,K_{ij,\,<\nabla
VS>}^{(0)}+a_{i}b_{j}\,K_{ji,\,<\nabla VS>}^{(0)}\right]\ ,$
where
$K_{ij,<\nabla VS>}^{(0)}=-2\;\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\
.$ (45)
The Pauli matrices here reduce to
$(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\;(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}_{\pm}})=r_{-}\;({\bf\hat{r}_{-}}\cdot{\bf\hat{r}_{\pm}})\
.$ (46)
For the integration over $z>0$ the integrand becomes simply $r_{-}$, which is
the same as that already needed for getting to the final result for
$I_{<r_{\pm}>}^{(0)}$ in subsection A.5.2 above. For the integration over
negative $z$, however, Eq. (46) becomes
$r_{-}\;({\bf\hat{r}_{-}}\cdot{\bf\hat{r}_{+}})={\bf r}_{-}\cdot{\bf
r}_{+}/r_{+}=(\rho^{2}+z^{2}-\delta^{2})/\sqrt{\rho^{2}+(z+\delta)^{2}}\ ,$
(47)
which involves a new integrand, but which nonetheless can still be done
analytically. (Here it is much easier to do the $\rho$-integration first.) We
find
$\displaystyle\\!\\!\\!\\!\\!\\!K_{ij,<\nabla VS>}^{(0)}$ $\displaystyle=$
$\displaystyle-\
\frac{2}{(\mu_{i}+\mu_{j})^{2}}\left[2-e^{-(\mu_{i}+\mu_{j})\;\delta^{2}/2}\right]$
(48) $\displaystyle-\
\frac{1}{\delta}\,\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\left[\text{Erf}\left(\sqrt{2(\mu_{i}+\mu_{j})}\;\delta\,\right)-\text{Erf}\left(\sqrt{(\mu_{i}+\mu_{j})/2}\;\delta\,\right)\right]\
.$
This result is, again, symmetric and non-singular with $K_{ij,<\nabla
VS>}^{(0)}=-4/(\mu_{i}+\mu_{j})^{2}$ at $\delta=0$. In this case there are no
odd terms (!) in $\delta$. Versus $\delta$ it is similar to that shown in Fig.
9C, but with the initial slope at the origin being zero.
Because $K_{ij,<\nabla VS>}^{(0)}=K_{ji,<\nabla VS>}^{(0)}$, we can finally
write the direct term contributions for this expectation as
$<[\nabla VS]^{(0)}>\
=\sum_{i,j}\left(a_{j}b_{i}+a_{i}b_{j}\right)\;\,K_{ij,\,<\nabla VS>}^{(0)}\
,$ (49)
regaining explicit symmetry.
The cross term integral $K_{<\nabla VS>}^{(1)}$ is more complicated but is
done similarly. As ${\bf X}_{12}=0$, there are now four terms remaining from
Eq. (20),
$\displaystyle<[\nabla VS]^{(1)}>$ $\displaystyle=$
$\displaystyle-2\;\frac{1}{4\pi}\;\sum_{i,j}\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[a_{j}b_{i}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
(50)
$\displaystyle\qquad\qquad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[a_{j}b_{i}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}\right.$
$\displaystyle\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\left[a_{i}b_{j}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right.$
$\displaystyle\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[a_{i}b_{j}\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{i}\,r^{2}_{+}/2}\right\\}$
$\displaystyle=$ $\displaystyle\sum_{i,j}\int
d^{3}r\;\left[a_{j}b_{i}\;K_{ij,\,<\nabla
VS>}^{(1)}+a_{i}b_{j}\;K_{ji,\,<\nabla VS>}^{(1)}\right]\ ,$
where
$\displaystyle K_{ij,\,<\nabla VS>}^{(1)}$ $\displaystyle=$
$\displaystyle-2\;\frac{1}{4\pi}\;\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
(51)
$\displaystyle\qquad\qquad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[\,(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}}_{\pm})\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}\,\right\\}$
In addition to Eq. (46) we also need
$(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})\;(\mbox{\boldmath$\sigma$}\cdot{\bf\hat{r}_{\pm}})=r_{+}\;({\bf\hat{r}_{+}}\cdot{\bf\hat{r}_{\pm}})\
,$ (52)
which becomes $r_{+}$ for the $z<0$ integration and
$(\rho^{2}+z^{2}-\delta^{2})/\sqrt{\rho^{2}+(z-\delta)^{2}}$ for the $z>0$
integration.
The integrations over $z$ go much easier if one re-defines the integrations
over $z$ in terms of $\mu=\mu_{i}+\mu_{j}$ and $\nu=\mu_{i}-\mu_{j}$. The
resulting integrals in $\mu$ and $\nu$ can then be converted back to $\mu_{i}$
and $\mu_{j}$. We find
$\displaystyle K_{ij,\,<\nabla VS>}^{(1)}$ $\displaystyle=$ $\displaystyle\
\frac{1}{\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})^{2}}\;\left[\;2\,\mu_{j}\,(\mu_{j}-\mu_{i})\;e^{-2\mu_{i}\;\delta^{2}}+2\,\mu_{i}\,(\mu_{i}-\mu_{j})\;e^{-2\mu_{j}\;\delta^{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad-\;(\mu_{i}-\mu_{j})^{2}\;e^{-(\mu_{i}+\mu_{j})\;\delta^{2}/2}\;\right]$
$\displaystyle-\
\frac{1}{2\,\delta\;\mu_{i}^{2}\mu_{j}^{2}}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{5}}\right]^{1/2}\times$
$\displaystyle\qquad\qquad\left\\{(\mu_{i}+\mu_{j})^{3}\,\left(\,\mu_{i}+\mu_{j}-2\mu_{i}\mu_{j}\;\delta^{2}\,\right)\right.\;\text{Erfc}\left(\sqrt{(\mu_{i}+\mu_{j})/2}\;\delta\,\right)$
$\displaystyle\qquad\qquad\quad+\
2\,\mu_{i}^{2}\;\left[\;(\mu_{i}^{2}+4\mu_{i}\mu_{j}+3\mu_{j}^{2})-4\mu_{j}^{2}(\mu_{i}-\mu_{j})\;\delta^{2}\;\right]\
$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erf}\left(\sqrt{2/(\mu_{i}+\mu_{j})}\;\mu_{j}\;\delta\,\right)$
$\displaystyle\qquad\qquad\quad-\
2\,\mu_{j}^{2}\;\left[\;3\mu_{i}^{2}+4\mu_{i}\mu_{j}+\mu_{j}^{2}-4\mu_{i}^{2}(\mu_{j}-\mu_{i})\;\delta^{2}\;\right]\
$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erfc}\left(\sqrt{2/(\mu_{i}+\mu_{j})}\;\mu_{i}\;\delta\,\right)$
$\displaystyle\qquad\qquad\quad-\
\left[\;(\mu_{i}^{3}+5\,\mu_{i}^{2}\mu_{j}+5\,\mu_{i}\mu_{j}^{2}+\mu_{j}^{3})-8\,\mu_{i}^{2}\mu_{j}^{2}\;\delta^{2}\;\right]\
$
$\displaystyle\qquad\qquad\qquad\quad\left.\times\,(\mu_{i}-\mu_{j})\,e^{-2\mu_{i}\mu_{j})\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erfc}\left(\frac{(\mu_{i}-\mu_{j})\;\delta}{\sqrt{2(\mu_{i}+\mu_{j})}}\,\right)\right\\}\
,$
which also is symmetric and goes to
$-8/(\mu_{i}+\mu_{j})^{2}=2\,K_{ij,\,<\nabla VS>}^{(0)}$ at $\delta=0$. This
integral does have some odd terms in $\delta$. As a function of $\delta$ it
resembles a Gaussian, i.e., looks like that shown in Fig. 9A.
Because $K_{ij,\,<\nabla VS>}^{(1)}=K_{ji,\,<\nabla VS>}^{(1)}$ we can again
finally write
$<[\nabla VS]^{(1)}>\
=\sum_{i,j}\left(a_{j}b_{i}+a_{i}b_{j}\right)\;\,K_{ij,\,<\nabla VS>}^{(1)}\
,$ (54)
mirroring the form of Eq. (49).
### A.7 The off-diagonal expectation $-2i\;V({\bf
r})\;\mbox{\boldmath$\alpha$}\cdot\nabla$
For this off-diagonal operator ${\bf X}_{12}={\bf X}_{21}=-2\,V({\bf
r})\,\nabla$ in Eq. (18) and the direct term expectation, Eq. (19), has all
four terms
$\displaystyle\\!\\!\\!\\!\\!\\!<[2\,V\nabla]^{(0)}>$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\;\sum_{i,j}\int d^{3}r\,V({\bf
r})\;\left\\{e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[\;2\,a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\right.\right.$ (55)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.-\;2\,b_{i}a_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}$
$\displaystyle\qquad\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[\;2\,a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\;2\,b_{j}a_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right\\}\,.$
With
$\nabla_{k}\;e^{-\mu_{i}\,r_{-}^{2}/2}=-\mu_{i}({\bf
r}_{-})_{k}\;e^{-\mu_{i}\,r_{-}^{2}/2},\quad\nabla_{k}({\bf
r}_{-})_{l}=\delta_{kl},\quad\text{ and
}\quad(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})=3\,$ (56)
we have, for the first terms in the square brackets of Eq. (55),
$\displaystyle(\mbox{\boldmath$\sigma$}\cdot\nabla)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\;e^{-\mu_{i}\,r_{-}^{2}/2}=\;e^{-\mu_{i}\,r_{-}^{2}/2}\;(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})+\mbox{\boldmath$\sigma$}\cdot\;[(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\nabla\;e^{-\mu_{i}\,r_{-}^{2}/2}]$ (57) $\displaystyle=$
$\displaystyle[3-\mu_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})]\;e^{-\mu_{i}\,r_{-}^{2}/2}=\,(3-\mu_{i}\,r_{-}^{2})\
e^{-\mu_{i}\,r_{-}^{2}/2}\ $
and similarly when acting on $e^{-\mu_{j}\,r_{-}^{2}/2}$.
For the second terms in the square brackets of Eq. (55),
$\displaystyle(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;e^{-\mu_{i}\,r_{-}^{2}/2}$
$\displaystyle=$ $\displaystyle-\mu_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})\;e^{-\mu_{i}\,r_{-}^{2}/2}=-\mu_{i}r_{-}^{2}\;e^{-\mu_{i}\,r_{-}^{2}/2}\
$ (58)
and, again, similarly when acting on $e^{-\mu_{j}\,r_{-}^{2}/2}$.
With Eqs. (57) and (58), Eq. (55) reduces to
$\displaystyle\\!\\!\\!\\!\\!\\!\\!<[2\,V\nabla]^{(0)}>$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\sum_{i,j}\int
d^{3}r\;\left\\{e^{-\mu_{i}\,r^{2}_{-}/2}\;V(r_{\pm})\left[a_{i}b_{j}\;(3-\mu_{j}r_{-}^{2})\;+\;b_{i}a_{j}\;\mu_{j}r_{-}^{2}\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
(59)
$\displaystyle\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;V(r_{\pm})\left[a_{j}b_{i}\;(3-\mu_{i}r_{-}^{2})\;+\;b_{j}a_{i}\;\mu_{i}r_{-}^{2}\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{i,j}\left\\{a_{i}b_{j}\;K_{ij,\,<2V\nabla>}^{(0)}+a_{j}b_{i}\;K_{ji,\,<2V\nabla>}^{(0)}\right\\}$
where, with $V(r_{\pm})=r_{\pm}-R,$
$\displaystyle K_{ij,\,<2V\nabla>}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;(r_{\pm}-R)\;[(\mu_{i}-\mu_{j})\,r_{-}^{2}-3]\;e^{-\mu_{j}\,r_{-}^{2}/2}\
.$ (60) $\displaystyle=$
$\displaystyle(\mu_{i}-\mu_{j})\;K_{ij,\,a}^{(0)}-(\mu_{i}-\mu_{j})\;R\;K_{ij,\,b}^{(0)}+3\;K_{ij,\,c}^{(0)}-3\;R\;K_{ij,\,d}^{(0)}$
where these four integrals are
$\displaystyle K_{ij,\,a}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;r_{\pm}\,r_{-}^{2}\;e^{-\mu_{j}\,r_{-}^{2}/2}$
(61) $\displaystyle=$ $\displaystyle\
\frac{1}{2\,(\mu_{j}+\mu_{i})^{3}}\left[\;16+6\;e^{-2(\mu_{j}+\mu_{i})\;\delta^{2}}-11\;e^{-(\mu_{j}+\mu_{i})\;\delta^{2}/2}\;\right]$
$\displaystyle+\frac{1}{2\delta}\left[\frac{\pi}{2(\mu_{j}+\mu_{i})^{7}}\right]^{1/2}\left\\{\;[5+9(\mu_{j}+\mu_{i})\delta^{2}]\;\text{Erfc}\left(\sqrt{(\mu_{j}+\mu_{i})/2}\;\delta\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\;[5+12(\mu_{j}+\mu_{i})\delta^{2}]\;\text{Erfc}\left(\sqrt{2(\mu_{j}+\mu_{i})}\;\delta\right)\right\\}\
,$ $\displaystyle K_{ij,\,b}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;r_{-}^{2}\;e^{-\mu_{j}\,r_{-}^{2}/2}=3\;\left[\frac{\pi}{2\,(\mu_{j}+\mu_{i})^{5}}\right]^{1/2}\
,$ (62) $\displaystyle K_{ij,\,c}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;r_{\pm}\;e^{-\mu_{j}\,r_{-}^{2}/2}\;=\;I_{ij,\;<r_{\pm}>}^{(0)}\
,$ (63) $\displaystyle K_{ij,\,d}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;e^{-\mu_{i}\,r^{2}_{-}/2}\;e^{-\mu_{j}\,r_{-}^{2}/2}\;=\;I_{ij,\;<1>}^{(0)}\
.$ (64)
$K_{ij,\,a}^{(0)}$ has an odd term in $\delta$ and its plot resembles that
shown in Fig. 9D. All four of the above integrals are symmetric in $i$ and
$j$, so we can finally write
$<[2\,V\nabla]^{(0)}>\;=\;\sum_{i,j}\left(a_{j}b_{i}+a_{i}b_{j}\right)\;\,K_{ij,\,<2\,V\nabla>}^{(0)}=2\;\sum_{i,j}a_{j}b_{i}\;K_{ij,\,<2\,V\nabla>}^{(0)}\
.$ (65)
For the cross term, from Eqs. (57) and (58) and the like, Eq. (20) becomes
$\displaystyle<[2\,V\nabla]^{(1)}>\ =\ \frac{1}{8\pi}\;\sum_{i,j}\int
d^{3}r\;V(r_{\pm})\ \times$ $\displaystyle\qquad\qquad\
\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[\;2\,a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})-\;2\,b_{i}a_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[\;2\,a_{i}b_{j}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})-\;2\,b_{i}a_{j}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}\right.$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\left[\;2\,a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})-\;2\,b_{j}a_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right.$
$\displaystyle\qquad\qquad\quad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[\;2\,a_{j}b_{i}(\mbox{\boldmath$\sigma$}\cdot\nabla)(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{+})-\;2\,b_{j}a_{i}(\mbox{\boldmath$\sigma$}\cdot{\bf
r}_{-})(\mbox{\boldmath$\sigma$}\cdot\nabla)\;\right]\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\right\\}$
$\displaystyle\qquad=\ \frac{1}{4\pi}\;\sum_{i,j}\int
d^{3}r\;\;V(r_{\pm})\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[a_{i}b_{j}(3-\mu_{j}r_{-}^{2})+\;b_{i}a_{j}\mu_{j}({\bf
r}_{+}\cdot{\bf r}_{-})\right]\;e^{-\mu_{j}\,r^{2}_{-}/2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[a_{i}b_{j}(3-\mu_{j}r_{+}^{2})+\;b_{i}a_{j}\mu_{j}({\bf
r}_{+}\cdot{\bf r}_{-})\right]\;e^{-\mu_{j}\,r^{2}_{+}/2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\left[a_{j}b_{i}(3-\mu_{i}r_{-}^{2})+\;b_{j}a_{i}\mu_{i}({\bf
r}_{+}\cdot{\bf r}_{-})\right]\;e^{-\mu_{i}\,r^{2}_{-}/2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\;e^{-\mu_{j}\,r^{2}_{-}/2}\;\left[a_{j}b_{i}(3-\mu_{i}r_{+}^{2})+\;b_{j}a_{i}\mu_{i}({\bf
r}_{+}\cdot{\bf r}_{-})\right]\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\right\\}$
$\displaystyle\qquad=\
\sum_{i,j}\left\\{\;a_{i}b_{j}\;K_{ij,\,<2V\nabla>}^{(1)}+a_{j}b_{i}\;K_{ji,\,<2V\nabla>}^{(1)}\;\right\\}\
,$ (66)
where
$\displaystyle K_{ij,\,<2\,V\nabla>}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\;\int
d^{3}r\;(r_{\pm}-R)\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\left[(3-\mu_{j}r_{-}^{2})+\mu_{i}({\bf
r}_{+}\cdot{\bf r}_{-})\right]e^{-\mu_{j}\,r^{2}_{-}/2}\;\right.$ (67)
$\displaystyle\qquad\qquad\qquad\quad\left.+\;e^{-\mu_{i}\,r^{2}_{-}/2}\;\left[(3-\mu_{j}r_{+}^{2})+\mu_{i}({\bf
r}_{+}\cdot{\bf r}_{-})\right]e^{-\mu_{j}\,r^{2}_{+}/2}\;\right\\}$
$\displaystyle=$
$\displaystyle-\mu_{j}\,K_{ij,\,a}^{(1)}+\mu_{j}\,R\,K_{ij,\,b}^{(1)}+\mu_{i}\,K_{ij,\,c}^{(1)}-\mu_{i}\,R\,K_{ij,\,d}^{(1)}+3\,K_{ij,\,e}^{(1)}-3\,R\,K_{ij,\,f}^{(1)}\
.$
The first integral,
$K_{ij,\,a}^{(1)}=\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;r_{\pm}\,r_{-}^{2}\;e^{-\mu_{j}\,r_{-}^{2}/2}+e^{-\mu_{i}\,r^{2}_{-}/2}\;r_{\pm}\,r_{+}^{2}\;e^{-\mu_{j}\,r_{+}^{2}/2}\right\\}\
,$ (68)
can be done using $\mu=\mu_{i}+\mu_{j}$ and $\nu=\mu_{i}-\mu_{j}$, noting that
$\mu>|\nu|$. Writing
$\displaystyle
e^{-\mu_{i}\,r^{2}_{+}/2}\,r_{-}^{2}\;e^{-\mu_{j}\,r_{-}^{2}/2}+e^{-\mu_{i}\,r^{2}_{-}/2}\,r_{+}^{2}\;e^{-\mu_{j}\,r_{+}^{2}/2}$
$\displaystyle\qquad\qquad=2\;e^{-\mu\,(\rho^{2}+z^{2}+\delta^{2})/2}\left\\{\;(\rho^{2}+z^{2}+\delta^{2})\cosh(\nu\delta
z)+(2z\delta)\sinh(\nu\delta z)\right\\}$ (69)
displays the $i,\;j$ symmetric and anti-symmetric parts explicitly. After
converting back to $\mu_{i}$ and $\mu_{j}$,
$\displaystyle K_{ij,\,a}^{(1)}$ $\displaystyle\;=\
\frac{2}{\mu_{j}(\mu_{i}+\mu_{j})^{4}}\left\\{\;[\;5\,\mu_{j}\,(\mu_{i}+\mu_{j})+4\,\mu_{i}^{2}\mu_{j}\;\delta^{2}\;]\;e^{-2\mu_{i}\;\delta^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\quad+\
[\;(\mu_{i}+\mu_{j})(-2\mu_{i}+3\mu_{j})+4\,\mu_{i}^{2}\mu_{j}\;\delta^{2}\;]\;e^{-2\mu_{j}\;\delta^{2}}$
$\displaystyle\qquad\qquad\qquad\quad\left.+\
[\,(\mu_{i}+\mu_{j})(\mu_{i}-4\,\mu_{j})-4\,\mu_{i}^{2}\mu_{j}\;\delta^{2}\,]\;e^{-\frac{1}{2}(\mu_{i}+\mu_{j})\;\delta^{2}}\;\right\\}$
$\displaystyle+\
\frac{1}{2\,\delta\,\mu_{i}\,\mu_{j}^{2}}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{9}}\right]^{1/2}\times$
$\displaystyle\left\\{\;2\,[\;\mu_{i}\,(\mu_{i}+\mu_{j})^{2}(2\,\mu_{i}+5\,\mu_{j})+\;4\,\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})(\mu_{i}^{2}-2\,\mu_{i}\mu_{j}+3\,\mu_{j}^{2})\;\delta^{2}+16\,\mu_{i}^{3}\mu_{j}^{3}\;\delta^{4}\;]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erf}\left(\sqrt{2/(\mu_{i}+\mu_{j})}\;\mu_{j}\;\delta\;\right)$
$\displaystyle\quad-\
2\,\mu_{j}^{2}[\;3\,(\mu_{i}+\mu_{j})^{2}+24\,\mu_{i}^{2}(\mu_{i}+\mu_{j})\;\delta^{2}+16\,\mu_{i}^{4}\;\delta^{4}\;]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erfc}\left(\sqrt{2/(\mu_{i}+\mu_{j})}\;\mu_{i}\;\delta\;\right)$
$\displaystyle\quad-\
[\;(\mu_{i}+\mu_{j})^{2}(2\,\mu_{i}^{2}+5\,\mu_{i}\mu_{j}-3\,\mu_{j}^{2})+\;4\,\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})(\mu_{i}^{2}-8\,\mu_{i}\mu_{j}+3\,\mu_{j}^{2})\;\delta^{2}$
$\displaystyle\qquad\qquad\qquad-\;16\;\mu_{i}^{3}\mu_{j}^{2}\,(\mu_{i}-\mu_{j})\;\delta^{4}]\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\;\text{Erfc}\left(\frac{(\mu_{i}-\mu_{j})}{\sqrt{2(\mu_{i}+\mu_{j})}}\;\delta\;\right)$
$\displaystyle\quad+\;\left.(\mu_{i}+\mu_{j})^{3}\,(2\,\mu_{i}+3\,\mu_{j})\;\text{Erfc}\left(\sqrt{(\mu_{i}+\mu_{j})/2}\;\delta\;\right)\right\\}\
,$
which is, as expected, not symmetric in $i$ and $j$. It is, however, non-
singular: $K_{ij,\,a}^{(1)}=16/(\mu_{i}+\mu_{j})^{3}$ at $\delta=0$. Its plot
resembles that in Fig. 9D.
The second integral is much simpler,
$\displaystyle K_{ij,\,b}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\,r_{-}^{2}\;e^{-\mu_{j}\,r_{-}^{2}/2}+e^{-\mu_{i}\,r^{2}_{-}/2}\;\,r_{+}^{2}\;e^{-\mu_{j}\,r_{+}^{2}/2}\right\\}$
(71) $\displaystyle=$ $\displaystyle\
\left[\frac{2\pi}{(\mu_{i}+\mu_{j})^{7}}\right]^{1/2}\left[\;3\,(\mu_{i}+\mu_{j})+4\,\mu_{j}^{2}\;\delta^{2}\;\right]e^{-2\,\mu_{i}\mu_{j}\,\delta^{2}/(\mu_{i}+\mu_{j})}\
,$
which is also non-symmetric, but only because of the term proportional to
$\delta^{2}$. As a function of $\delta$ it looks like Fig. 9E.
Almost as complicated as $K_{ij,\,a}^{(1)}$, the third integral is
$\displaystyle K_{ij,\,c}^{(1)}=\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\,r_{\pm}\,({\bf r}_{+}\cdot{\bf
r}_{-})\;e^{-\mu_{i}\,r_{-}^{2}/2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\
\left.e^{-\mu_{j}\,r^{2}_{-}/2}\;\,r_{\pm}\,({\bf r}_{+}\cdot{\bf
r}_{-})\;e^{-\mu_{i}\,r_{+}^{2}/2}\right\\}$ $\displaystyle\qquad=\
\frac{1}{\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})^{4}}\left\\{\;2\mu_{j}\;[\;(\mu_{i}+\mu_{j})(4\mu_{i}-\mu_{j})-4\mu_{i}^{2}\mu_{j}\;\delta^{2}\;]\;e^{-2\mu_{i}\;\delta^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\
2\mu_{i}\;[\;(\mu_{i}+\mu_{j})(4\mu_{j}-\mu_{i})-4\mu_{i}\mu_{j}^{2}\;\delta^{2}\;]\;e^{-2\mu_{j}\;\delta^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.+\
[\;(\mu_{i}+\mu_{j})(\mu_{i}^{2}-8\,\mu_{i}\mu_{j}+\mu_{j}^{2})+8\,\mu_{i}^{2}\mu_{j}^{2}\;\delta^{2}\;]\;e^{-\frac{1}{2}(\mu_{i}+\mu_{j})\;\delta^{2}}\;\right\\}$
$\displaystyle\qquad\qquad+\
\frac{1}{2\,\delta\,\mu_{i}^{2}\,\mu_{j}^{2}}\left[\frac{\pi}{2(\mu_{i}+\mu_{j})^{9}}\right]^{1/2}\times$
(72)
$\displaystyle\qquad\qquad\qquad\left\\{\;2\,\mu_{j}^{2}[\;(\mu_{i}+\mu_{j})^{2}(4\mu_{i}+\mu_{j})+8\,\mu_{i}^{2}\,(\mu_{i}+\mu_{j})(2\mu_{i}-\mu_{j})\;\delta^{2}-16\,\mu_{i}^{4}\mu_{j}\;\delta^{4}\;]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erf}\left(\sqrt{\frac{2}{\mu_{i}+\mu_{j}}}\;\mu_{i}\;\delta\;\right)$
$\displaystyle\qquad\qquad\qquad-\
2\,\mu_{i}^{2}[\;(\mu_{i}+\mu_{j})^{2}(4\mu_{j}+\mu_{i})+8\,\mu_{j}^{2}\,(\mu_{i}+\mu_{j})(2\mu_{j}-\mu_{i})\;\delta^{2}-16\,\mu_{i}\mu_{j}^{4}\;\delta^{4}\;]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\text{Erfc}\left(\sqrt{\frac{2}{\mu_{i}+\mu_{j}}}\;\mu_{j}\;\delta\;\right)$
$\displaystyle\qquad\qquad\qquad+\
[\;(\mu_{i}+\mu_{j})^{2}(\mu_{i}^{2}+5\,\mu_{i}\mu_{j}+\mu_{j}^{2})\;-24\,\mu_{i}^{2}\mu_{j}^{2}(\mu_{i}+\mu_{j})\;\delta^{2}+16\,\mu_{i}^{3}\mu_{j}^{3}\;\delta^{4}\;]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times(\mu_{i}-\mu_{j})\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\;\;\left[1+\text{Erf}\left(\frac{(\mu_{i}-\mu_{j})}{\sqrt{2(\mu_{i}+\mu_{j})}}\;\delta\;\right)\right]$
$\displaystyle\qquad\qquad\qquad+\
\left.(\mu_{i}+\mu_{j})^{3}\;[\,(\mu_{i}^{2}+3\,\mu_{i}\mu_{j}+\mu_{j}^{2})-2\,\mu_{i}\mu_{j}(\mu_{i}+\mu_{j})\;\delta^{2}\,]\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left.\;\text{Erfc}\left(\sqrt{(\mu_{i}+\mu_{j})/2}\;\delta\;\right)\right\\}\
,$
which is surprisingly both symmetric, $K_{ji,\,c}^{(1)}=K_{ij,\,c}^{(1)}$, and
non-singular: $K_{ij,\,c}^{(1)}=16/(\mu_{i}+\mu_{j})^{3}$ at $\delta=0$. This
integral as a function of $\delta$ looks like Fig. 9B.
The fourth integral is also simple,
$\displaystyle K_{ij,\,d}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{j}\,r^{2}_{+}/2}\;\,({\bf r}_{+}\cdot{\bf
r}_{-})\;e^{-\mu_{i}\,r_{-}^{2}/2}+e^{-\mu_{j}\,r^{2}_{-}/2}\;\,({\bf
r}_{+}\cdot{\bf r}_{-})\;e^{-\mu_{i}\,r_{+}^{2}/2}\right\\}$ (73)
$\displaystyle=$
$\displaystyle\left[\frac{2\pi}{(\mu_{j}+\mu_{i})^{7}}\right]^{1/2}\;[\;3(\mu_{j}+\mu_{i})-4\mu_{i}\mu_{j}\;\delta^{2}\,]\;e^{-2\mu_{i}\mu_{j}\;\delta^{2}/(\mu_{i}+\mu_{j})}\
.$
Its $\delta$ dependence, Fig. 9F, shows a relatively deeper minimum than that
depicted in Fig. 9B. The fifth and sixth integrals are already familiar,
$\displaystyle K_{ij,\,e}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\,r_{\pm}\;e^{-\mu_{j}\,r_{-}^{2}/2}+e^{-\mu_{i}\,r^{2}_{-}/2}\;\,r_{\pm}\;e^{-\mu_{j}\,r_{+}^{2}/2}\;\right\\}\;=\;I_{ij,\,<r_{\pm}>}^{(1)}$
(74) $\displaystyle K_{ij,\,f}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d^{3}r\;\left\\{\;e^{-\mu_{i}\,r^{2}_{+}/2}\;\;e^{-\mu_{j}\,r_{-}^{2}/2}+e^{-\mu_{i}\,r^{2}_{-}/2}\;\;e^{-\mu_{j}\,r_{+}^{2}/2}\;\right\\}\;=\;I_{ij,\,<1>}^{(1)}\
.$ (75)
These last three integrals, $K_{ij,\,d}^{(1)}$ through $K_{ij,\,f}^{(1)}$, are
all symmetric in $i$ and $j$.
|
arxiv-papers
| 2013-04-19T17:00:56 |
2024-09-04T02:49:44.612778
|
{
"license": "Public Domain",
"authors": "T. Goldman and Richard R. Silbar",
"submitter": "Richard R. Silbar",
"url": "https://arxiv.org/abs/1304.5480"
}
|
1304.5486
|
Inferring evolutionary histories of pathway regulation from transcriptional
profiling data
Joshua G. Schraiber1, Yulia Mostovoy2, Tiffany Y. Hsu2,3 Rachel B. Brem2,∗
1 Department of Integrative Biology, University of California, Berkeley, CA,
USA
2 Department of Molecular and Cellular Biology, University of California,
Berkeley, CA, USA
3 Present Address: Graduate Program in Biological and Biomedical Sciences,
Harvard Medical School, Boston, MA, USA
$\ast$ E-mail: Corresponding [email protected]
## Abstract
One of the outstanding challenges in comparative genomics is to interpret the
evolutionary importance of regulatory variation between species. Rigorous
molecular evolution-based methods to infer evidence for natural selection from
expression data are at a premium in the field, and to date, phylogenetic
approaches have not been well-suited to address the question in the small sets
of taxa profiled in standard surveys of gene expression. We have developed a
strategy to infer evolutionary histories from expression profiles by analyzing
suites of genes of common function. In a manner conceptually similar to
molecular evolution models in which the evolutionary rates of DNA sequence at
multiple loci follow a gamma distribution, we modeled expression of the genes
of an _a priori_ -defined pathway with rates drawn from an inverse gamma
distribution. We then developed a fitting strategy to infer the parameters of
this distribution from expression measurements, and to identify gene groups
whose expression patterns were consistent with evolutionary constraint or
rapid evolution in particular species. Simulations confirmed the power and
accuracy of our inference method. As an experimental testbed for our approach,
we generated and analyzed transcriptional profiles of four _Saccharomyces_
yeasts. The results revealed pathways with signatures of constrained and
accelerated regulatory evolution in individual yeasts and across the
phylogeny, highlighting the prevalence of pathway-level expression change
during the divergence of yeast species. We anticipate that our pathway-based
phylogenetic approach will be of broad utility in the search to understand the
evolutionary relevance of regulatory change.
## Author Summary
Comparative transcriptomic studies routinely identify thousands of genes
differentially expressed between species. The central question in the field is
whether and how such regulatory changes have been the product of natural
selection. Can the signal of evolutionarily relevant expression divergence be
detected amid the noise of changes resulting from genetic drift? Our work
develops a theory of gene expression variation among a suite of genes that
function together. We derive a formalism that relates empirical observations
of expression of pathway genes in divergent species to the underlying strength
of natural selection on expression output. We show that fitting this type of
model to simulated data accurately recapitulates the parameters used to
generate the simulation. We then make experimental measurements of gene
expression in a panel of single-celled eukaryotic yeast species. To these data
we apply our inference method, and identify pathways with striking evidence
for accelerated or constrained regulatory evolution, in particular species and
across the phylogeny. Our method provides a key advance over previous
approaches in that it maximizes the power of rigorous molecular-evolution
analysis of regulatory variation even when data are relatively sparse. As
such, the theory and tools we have developed will likely find broad
application in the field of comparative genomics.
## Introduction
Comparative studies of gene expression across species routinely detect
regulatory variation at thousands of loci [1]. Whether and how these
expression changes are of evolutionary relevance has become a central question
in the field. In landmark cases, experimental dissection of model phenotypes
has revealed evidence for adaptive regulatory change at individual genes [2,
3, 4, 5]. These findings have motivated hypothesis-generating, genome-scale
searches for signatures of natural selection on gene regulation. In addition
to molecular-evolution analyses of regulatory sequence [6, 7, 8, 9],
phylogenetic methods have been developed to infer evidence for non-neutral
evolutionary change from measurements of gene expression [10, 11, 12]. Two
classic models of continuous character evolution have been used for the latter
purpose: Brownian motion models, which can specify lineage-specific rates of
evolution on a phylogenetic tree [13, 14, 15, 16] and have been used to model
the neutral evolution of gene expression [17, 11], and the Ornstein-Uhlenbeck
model, which by describing lineage-specific forces of drift and stabilizing
selection [13, 18, 19] can be used to test for evolutionary constraint on gene
expression [11, 12]. To date, phylogenetic approaches have had relatively
modest power to infer lineage-specific rates or selective optima of gene
expression levels. This limitation is due in part to the sparse species
coverage typical of transcriptomic surveys, in contrast to studies of
organismal traits where observations in hundreds of species can be made to
maximize the power of phylogenetic inference [20, 21, 22].
As a complement to model-based phylogenetic methods, more empirical approaches
have also been proposed that detect expression patterns suggestive of non-
neutral evolution [23, 24, 25]. We previously developed a paradigm to detect
species changes in selective pressure on the regulation of a pathway, or suite
of genes of common function, in the case where multiple independent variants
drive expression of pathway genes in the same direction [24, 26]. Broadly,
pathway-level analyses have the potential to uncover evidence for changes in
selective pressure on a gene group in the aggregate, when the signal at any
one gene may be too weak to emerge from genome-scale scans. However, the
currently available tests for directional regulatory evolution are not well
suited to cases in which some components of a pathway are activated, and
others are down-regulated, in response to selection.
In this work, we set out to combine the rigor of phylogenetic methods to
reconstruct histories of continuous-character evolution with the power of
pathway-level analyses of regulatory change. We reasoned that an integration
of these two families of methods could be used to detect cases of pathway
regulatory evolution from gene expression data, without assuming a directional
model. To this end, we aimed to develop a phylogenetic model of pathway
regulatory change that accounted for differences in evolutionary rate between
the individual genes of a pathway. We sought to use this model to uncover gene
groups whose regulation has undergone accelerated evolution or been subject to
evolutionary constraint, over and above the degree expected by drift during
species divergence as estimated from genome sequence. As an experimental
testbed for our inference strategy, we used the _Saccharomyces_ yeasts. These
microbial eukaryotes span an estimated 20 million years of divergence and have
available well-established orthologous gene calls [27], and yeast pathways are
well-annotated based on decades of characterization of the model organism _S.
cerevisiae_. We generated a comparative transcriptomic data set across
Saccharomycetes by RNA-seq, and we used the data to search for cases of
pathway regulatory change.
## Results
### Modeling the rates of regulatory evolution across the genes of a pathway
The Brownian-motion model of expression of a gene predicts a multivariate
normal distribution of observed expression levels in the species at the tips
of a phylogenetic tree. The variance-covariance matrix of this multivariate
normal distribution reflects both the relatedness of the species and the rate
of regulatory evolution along each branch of the tree. We sought to apply this
model to interpret expression changes in a pre-defined set of genes of common
function, which we term a pathway. Our goal was to test for accelerated or
constrained regulatory variation in a pathway relative to the expectation from
DNA sequence divergence, as specified by a genome tree. To avoid the potential
for over-parameterization if the rate of each gene in a pathway were fit
separately, we instead developed a formalism, detailed in Methods, to model
regulatory evolution in the pathway using a parametric distribution of
evolutionary rates across the genes. This strategy parallels well-established
models of the rate of DNA sequence evolution across different sites in a locus
or genome [28]. Briefly, we assumed that each gene in the pathway draws its
rate of evolution from an inverse gamma distribution, and we derived the
relationship between the parameters of this distribution and the likelihood of
expression observations at the tips of the tree. For each gene, we modeled the
contrasts of the expression level in each species relative to an arbitrary
species used as a reference, to eliminate the need to estimate the ancestral
expression level. A further normalization step, recentering the distribution
of expression across pathway genes in each species to a mean of $0$, corrected
for the effects of coherent regulatory divergence due to drift. This formalism
enabled a maximum-likelihood fit of the parameters describing the pathway
expression distribution, given empirical expression data, and could
accommodate models of lineage-specific regulatory evolution, in which a
particular subtree was described by distinct evolutionary rate parameters
relative to the rest of the phylogeny. As a point of comparison, we
additionally made use of an Ornstein-Uhlenbeck (OU) model [19]: here the rate
of regulatory evolution of each gene in a pathway, across the entire
phylogeny, was drawn from an inverse-gamma distribution, and all genes of the
pathway were subject to the same degree of stabilizing selection, again across
the entire tree.
Our ultimate application of the method given a set of expression data was to
enumerate all possible Brownian motion models in which pathway expression
evolved at a distinct rate along the lineages of a subtree relative to the
rest of the phylogeny, and for each such model, apply our fitting strategy and
tabulate the likelihood of the data under the best-fit parameter set. To
compare these likelihoods and the analogous likelihood from the best-fit OU
model of universal constraint, we applied a standard Akaike information
criterion (AIC) [21, 29, 30] to identify strongly supported models.
### Simulation testing of inference of pathway regulatory evolution
As an initial test of our approach, we sought to assess the performance of our
phylogenetic inference scheme in the ideal case in which rates of regulatory
evolution of the genes of a pathway were simulated from, and thus conformed
to, the models of our theoretical treatment. In keeping with our experimental
application below which used a comparison of _Saccharomyces_ yeast species as
a testbed, we developed a simulation scheme using a molecular clock-calibrated
_Saccharomyces_ phylogeny [27] (see Figure 1a inset). We simulated the
expression of a multi-gene pathway in which rates of evolution of the member
genes were drawn from an inverse gamma distribution. With the simulated
expression data in hand from a given generating model, we fit an OU model, an
equal-rates model, and models of evolutionary rate shifts in each subtree in
turn.
Figure 1 shows the results of inferring the mode and rate of evolution from
data simulated under a model of accelerated regulatory change on the branch
leading to _S. paradoxus_ , and similar results can be seen in Figures S1
through S5 for other rate shift models. As expected, for very small gene
groups, inference efforts did not achieve high power or recapitulate model
parameters (Figure 1a, leftmost data point; Figure 1b, leftmost point in each
cluster), reflecting the challenges of the phylogenetic approach when applied
on a gene-by-gene basis to relatively sparse trees like the _Saccharomyces_
species set. By contrast, for pathways of ten genes or more, we observed
strong AIC support for the true generating model in cases of lineage-specific
regulatory evolution, approaching AIC weights of 100% for the correct model if
a pathway contained more than 50 genes (Figure 1a, Figure S1 and panel a of
Figures S2-S5). In these simulations our method also inferred the correct
magnitudes of lineage-specific shifts with high confidence, for all but the
smallest pathways (Figure 1b and panel b of Figures S2-S5). Likewise, when
applied to simulated expression data generated under models of phylogeny-wide
constraint, our method successfully identified OU as the correct model (Figure
2a), though with biased estimates of the magnitude of the constraint parameter
when the latter was large (Figure 2b), likely due to a lack of identifiability
with the inverse-gamma rate parameter (Figure S6).
We also sought to evaluate the robustness of our method to violations of the
underlying model. To explore the effect of our assumption of independence
between genes, we simulated a pathway in which expression of the individual
genes was coupled to one another and evolving under an equal-rates Brownian
motion model, and we inferred evolutionary histories either including or
eliminating the mean-centering normalization step of our analysis pipeline.
With the latter step in place, our method correctly yielded little support for
shifts in evolutionary rates in the simulated data except in the case of
extremely tight correlation between genes, a regime unlikely to be
biologically relevant (Figure S7). Additionally, to test the impact of our
assumption that the genes of a pathway were all subject to similar
evolutionary pressures, we simulated a heterogeneous pathway in which
expression of only a fraction of the gene members was subject to a lineage-
specific shift in evolutionary rate. Inferring parameters from these data
revealed accurate detection of rate shifts even when a large proportion of the
genes in the pathway deviated from the rate shift model (Figure S8). Taken
together, our results make clear that the pathway-based phylogenetic approach
is highly powered to infer evolutionary histories of gene expression change,
particularly lineage-specific evolutionary rate shifts. As a contrast to the
poor performance of phylogenetic inference when applied to one or a few genes,
our findings underscore the utility of the multi-gene paradigm in identifying
candidate cases of evolutionarily relevant expression divergence.
### Phylogenetic inference of regulatory evolution from experimental
measurements of _Saccharomyces_ expression
We next set out to apply our method for evolutionary reconstruction of
regulatory change to experimental measurements of gene expression. The total
difference in gene expression between any two species is a consequence of
heritable differences that act in _cis_ on the DNA strand of a gene whose
expression is measured, and of variants that act in _trans_ , through a
soluble factor, to impact gene expression of distal targets. Effects of _cis_
-acting variation can be surveyed on a genomic scale using our previously
reported strategy of mapping of RNA-seq reads to the individual alleles of a
given gene in a diploid inter-specific hybrid [24], whereas the joint effects
of _cis_ and _trans_ -acting factors can be assessed with standard
transcriptional profiling approaches in cultures of purebred species. To apply
these experimental paradigms we chose a system of _Saccharomyces sensu
stricto_ yeasts. We cultured two biological replicates for each of a series of
hybrids formed by the mating of _S. cerevisiae_ to _S. paradoxus_ , _S.
mikatae_ , and _S. bayanus_ in turn, as well as homozygotes of each species.
We measured total expression in the species homozygotes, and allele-specific
expression in the hybrids, of each gene by RNA-seq, using established mapping
and normalization procedures (see Methods). In each set of expression data, we
made use of _S. cerevisiae_ as a reference: we normalized expression in the
homozygote of a given species, and expression of the allele of a given species
in a diploid hybrid, relative to the analogous measurement from _S.
cerevisiae_.
To search for evidence of evolutionary constraint and lineage-specific shifts
in evolutionary rate in our yeast expression data, we considered as pathways
the pre-defined sets of genes of common function from the Gene Ontology (GO)
process categories. For the genes of each GO term, we used normalized
expression measurements in yeast species and, separately, measurements of
_cis_ -regulatory variation from interspecific hybrids, as input into our
phylogenetic analysis pipeline. Thus, for each of the two classes of
expression measurements, for a given GO term we fit models of a lineage-
specific rate shift in regulatory evolution incorporating inverse-gamma-
distributed rates across genes; an analogous model with no lineage-specific
rate shift; and an OU model of universal constraint. The results revealed a
range of inferred evolutionary models and AIC support across GO terms (Figure
3, Tables 1 and 2, and Tables S3 and S4), and this complete data set served as
the basis for manual inspection of biologically interesting features.
Among the inferences of pathway regulatory evolution from our method, we
observed many cases of evolutionary interest whose best-fitting model had
strong AIC support (Figure 3). For each of 15 GO terms, _cis_ -regulatory
expression variation measurements yielded inference of an evolutionary model
with $>$80% AIC weight (Figure 3a and Table 1). Many such GO terms represented
candidate cases of polygenic regulatory evolution, in which multiple
independent variants, at the unlinked genes that make up a pathway, have been
maintained in some yeast species in response to a lineage-specific shift in
selective pressure on expression of the pathway components. For example, in
replicative cell aging genes (GO term 0001302), _cis_ -regulatory variation
measured in interspecific hybrids supported a model of polygenic, accelerated
evolution in _S. paradoxus_ (Figure 4a), with some pathway components
upregulated and some downregulated in the latter species relative to other
yeasts. The total expression levels of cell aging genes in species homozygotes
were also consistent with rapid evolution in _S. paradoxus_ (Figure 4a),
arguing against a model of compensation between _cis_ \- and _trans_ -acting
regulatory variation, and highlighting this pathway as a particularly
compelling potential case of a lineage-specific change in selective pressure.
In other instances, expression measurements in species homozygotes alone
supported models of lineage-specific evolution, with each such pathway
representing a candidate case of accelerated or constrained evolution at
_trans_ -acting regulatory factors. For a total of 41 GO terms, our method
inferred models with $>$80% AIC weight from homozygote species expression data
(Figure 3b and Table 2). These top-scoring pathways included a set of
components of the transcription machinery (GO term 0006351), whose expression
levels in _S. bayanus_ were less volatile than those of other yeasts and thus
supported a model of lineage-specific constraint (Figure 4b). Additionally,
expression of a number of pathways in species homozygotes conformed to the OU
model of universal constraint, such as a set of genes annotated in transport
(GO term 0006281), whose expression varied less across all species than would
be expected from the genome tree (Figure 4c). Taken together, our findings
indicate that evolutionary histories can be inferred with high confidence from
experimental measurements of pathway gene expression. In our yeast data, many
pathways exhibit expression signatures consistent with non-neutral regulatory
evolution, in particular lineages and across the phylogeny.
Another emergent trend was the prevalence, across many GO terms, of models of
distinct regulatory evolution in the lineage to _S. paradoxus_ as the best fit
to expression measurements in species homozygotes (Figure 3b). We noted no
such recurrent model in analyses of _cis_ -regulatory variation (Figure 3a),
implicating _trans_ -acting variants as the likely source of the regulatory
divergence in _S. paradoxus_. To validate these patterns, we applied our
phylogenetic inference method to expression measurements from all genes in the
genome analyzed as a single group, rather than to each GO term in turn. When
we used expression data from species homozygotes as input for this genome-
scale analysis, our method assigned complete AIC support to a model in which
the rate of evolution was $2.5$ times faster on the branch leading to _S.
paradoxus_ (AIC weight $=1$), consistent with results from individual GO terms
(Figure 3b). An analogous inference calculation using measurements of _cis_
-regulatory variation, for all genes in the genome, yielded essentially
complete support for an OU model of universal constraint (AIC weight $=.99$).
We conclude that constraint on the _cis_ -acting determinants of gene
expression, of roughly the same degree in all yeasts, is the general rule from
which changes in selective pressure on particular functions may drive
deviations in individual pathways. However, for many genes, expression in the
_S. paradoxus_ homozygote is distinct from that of other yeasts out of
proportion to its sequence divergence, suggestive of derived, _trans_ -acting
regulatory variants with pleotropic effects.
## Discussion
The effort to infer evolutionary histories of gene expression change has been
a central focus of modern comparative genomics. Against a backdrop of a few
landmark successes [11, 12], progress in the field has been limited by the
relatively weak power of phylogenetic methods when applied, on a gene-by-gene
basis, to measurements from small sets of species. In this work, we have met
this challenge with a method to infer evolutionary rates of any suite of
independently measured continuous characters that can be analyzed together
across species. We have derived the mathematical formalism for this model, and
we have illustrated the power and accuracy of our approach in simulations. We
have generated yeast transcriptional profiles that complement available data
sets [31, 32] by measuring _cis_ -regulatory contributions to species
expression differences as well as the total variation between species. With
these data, we have demonstrated that our phylogenetic inference method yields
robust, interpretable candidate cases of pathway regulatory evolution from
experimental measurements.
The defining feature of our phylogenetic inference method is that it gains
power by jointly leveraging expression measurements of a group of genes, while
avoiding a high-dimensional evolutionary model. Rather than requiring an
estimate of the evolutionary rate at each gene, our strategy estimates the
parameters of a distribution of evolutionary rates across genes. We thus apply
the assumption of [10] and model expression of the individual genes of a
pathway as independent draws from the same distribution, mirroring the
standard assumption of independence across sites in phylogenetic analyses of
DNA sequence [33]. Any observation of lineage-specific _cis_ -acting
regulatory variation from our approach is of immediate evolutionary interest:
a species-specific excess of variants at unlinked loci of common function
would be unlikely under neutrality, and would represent a potential signature
of positive selection if fixed across individuals of the species. In the study
of _trans_ -acting regulatory variation, _a priori_ a case of apparent
accelerated evolution of a pathway could be driven by a single mutation of
large effect maintained by drift in a species, as in any phenomenological
analysis of trait evolution [13, 34]. Our results indicate that for correlated
gene groups, the latter issue can be largely resolved by a simple
transformation in which expression of each gene is normalized against the mean
of all genes in the pathway. Additional corrections could be required under
more complex models of correlation among pathway genes, potentially to be
incorporated with matrix-regularization techniques that highlight patterns of
correlation in transcriptome data [35]. Similarly, although the assumption of
independence across genes could upwardly bias the likelihoods of best-fit
models in our inferences, model choice and parameter estimates will still be
correct on average even with the scheme implemented here [36].
Our strategy also assumes that the genes of a pre-defined pathway are subject
to similar evolutionary pressures. Simulation results indicate that this
assumption does not compromise the performance of our method, as we observed
robust inference to be the rule rather than the exception even in a quite
heterogeneous pathway, if a proportion of the genes evolved under a rate shift
model. Although we have used pathways defined by Gene Ontology in this study,
our method can easily be applied to gene modules defined on the basis of
protein or genetic interactions or coexpression. Any such module is likely to
contain both activators and repressors, or other classes of gene function
whose expression may be quantitatively tuned in response to selection by
alleles with effects of opposite sign [37, 38]. The phylogenetic approach we
have developed here is well-suited to detect these non-directional regulatory
patterns, rather than relying on the coherence of up- or down-regulation of
pathway genes [24, 25, 39, 40, 26, 41]. Ultimately, a given case of strong
signal in our pathway evolution paradigm, when the best-fit model is one of
lineage-specific accelerated regulatory evolution, can be explained either as
a product of relaxed purifying selection or positive selection on pathway
output. Our approach thus serves as a powerful strategy to identify candidates
for population-genetic [26] and empirical [42, 40] tests of the adaptive
importance of pathway regulatory change. We have developed an R package,
PIGShift (Polygenic Inverse Gamma rateShift), to facilitate the usage of our
method. The pathway-level approach is not contingent on the Gaussian models of
regulatory evolution we have used here, and future work will evaluate the
advantages of compound Poisson process [43, 10] or more general Lévy process
[44] models of gene expression.
The advent of RNA-seq has enabled expression surveys across non-model species
in many taxa. Maximizing the biological value of these data requires methods
that evaluate expression variation in the context of sequence divergence
between species. As rigorous phylogenetic interpretation of expression data
becomes possible, these measurements will take their place beside genome
sequences as a rich source of hypotheses, in the search for the molecular
basis of evolutionary novelty.
## Methods
### Basic model
Our basic assumption, following [10], is that the average expression levels of
genes in a pathway evolve as independent replicates of the same Brownian
motion or Ornstein-Uhlenbeck process. However, instead of assuming that each
gene in the pathway has the same rate of evolution, we allow the different
genes in a pathway to draw their rate of evolution from a parametric
distribution.
As a point of departure, we begin by considering the likelihood of a group of
genes whose expression evolves independently, each with its own rate of
evolution. Throughout, we use uppercase letters to represent random variables
and matrices and lowercase letters to represent nonrandom variables. Assume
that we have measured expression of the genes of a pathway in $n$ species, and
that we have a fixed, time-calibrated phylogeny from genome sequence data
describing the relationships between those species. We let
$\mathbf{X}_{i}=(X_{i,1},X_{i,2},\ldots,X_{i,n})$ be the observations of the
expression level of the $i$th gene of the pathway, in each of $n$ species.
Both the Brownian- motion and Ornstein-Uhlenbeck (OU) models predict that the
vector $\mathbf{X}_{i}$ is a draw from a multivariate normal distribution with
variance-covariance matrix $\sigma_{i}^{2}\mathbf{V}$ (where $\sigma_{i}^{2}$
is a scalar—the rate of evolution—and the elements of $\mathbf{V}$ depend on
whether evolution follows the Brownian or Ornstein-Uhlenbeck model; see
below). Hence, the likelihood of the data is
$g(\mathbf{X})=\prod_{i}\frac{1}{\sqrt{(2\pi\sigma_{i}^{2})^{n}\det(\mathbf{V})}}e^{-\frac{1}{2\sigma_{i}^{2}}(\mathbf{x}_{i}-\mathbf{\mu}_{i})^{\prime}\mathbf{V}^{-1}(\mathbf{x}_{i}-\mathbf{\mu}_{i})}$
(1)
where $\mathbf{\mu}_{i}$ is a vector representing the mean expression value at
the tips of the phylogenetic tree for gene $i$. Note that
$\sigma_{i}^{2}V_{j,k}=\text{Cov}(X_{i,j},X_{i,k})$ where $V_{i,j}$ is the
$i,j$th element of $\mathbf{V}$.
If we assume that there is no branch-specific directionality to evolution, we
can avoid the need to estimate $\mu$ in either the Brownian motion model or
the OU model by a renormalization of the data. We first arbitrarily choose the
gene expression measurements in a single species (say species 1), and define
the new random vector $\mathbf{Z}_{i}=(Z_{i,2},Z_{i,3},\ldots,Z_{i,n})$ by
$Z_{i,j}=X_{i,j}-X_{i,1}.$
By our assumption that there is no branch-specific directionality,
$\mathbb{E}(X_{i,j})=\mathbb{E}(X_{i,1})$ so $\mathbb{E}(Z_{i,j})=0$ for all
$i$ and $j$. Because each $\mathbf{X}_{i}$ is multivariate normally
distributed with dimension $n$, each $\mathbf{Z}_{i}$ will also be
multivariate normally distributed with dimension $n-1$ and a slightly
different covariance structure. Letting $\mathbf{W}$ be the covariance matrix
corresponding to the $\mathbf{Z}_{i}$, elementary calculations taking into
account variances and covariances of sums of random variables reveal that
$W_{i-1,j-1}=\begin{cases}V_{i,i}+V_{1,1}-2V_{i,1}&\text{if }i=j\\\
V_{i,j}+V_{1,1}-V_{i,1}-V_{j,1}&\text{if }i\neq j.\end{cases}$
Next, we wish to incorporate into the Brownian motion and OU models a scheme
in which the rates of evolution of the genes of a pathway are not specified
independently but instead are drawn from an inverse-gamma distribution. In
this context, the genes in a pathway share $\mathbf{W}$, the variance-
covariance structure due to the tree, but the rate of evolution
$\sigma_{i}^{2}$ for each gene is an independent draw from an inverse-gamma
distribution. The inverse-gamma distribution has density
$h(y)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}y^{-(\alpha+1)}e^{-\frac{\beta}{y}},$
(2)
where $\Gamma(\cdot)$ is the gamma function and $\alpha$ and $\beta$ are shape
and scale parameters. The moments of this distribution are
$\mathbb{E}(Y)=\frac{\beta}{\alpha-1}$
and
$\text{Var}(Y)=\frac{\beta^{2}}{(\alpha-1)^{2}(\alpha-2)},$
from which it follows that the inverse-gamma distribution has no mean if
$\alpha<1$ and no variance if $\alpha<2$. These properties allow for the
distribution of rates of gene expression evolution in a pathway to be
relatively broad; in addition, the inverse gamma density has no mass at $0$,
which prevents any gene in a pathway from not evolving at all. Also, as
$\alpha\rightarrow\infty$ and $\beta\rightarrow\infty$ as
$\frac{\beta}{\alpha-1}=\mu$ stays fixed, the distribution converges to a
point mass at $\mu$. Thus, a model where there is one rate for every gene is
nested within the inverse-gamma distributed rates model.
Computation of the the likelihood of the data under this model is simplified
by the fact that the inverse-gamma distribution is the conjugate prior to the
variance of a normal distribution. Hence, we see that the likelihood of the
observed expression data $\mathbf{Z}$ is
$\displaystyle L(\mathbf{Z})$ $\displaystyle=$
$\displaystyle\idotsint_{0}^{\infty}g(\mathbf{Z})h(\sigma_{1}^{2})h(\sigma_{2}^{2})\cdots
h(\sigma_{n}^{2})d(\sigma_{1}^{2})d(\sigma_{2}^{2})\cdots d(\sigma_{n}^{2})$
(3) $\displaystyle=$
$\displaystyle\prod_{i}\int_{0}^{\infty}\frac{1}{\sqrt{(2\pi\sigma^{2})^{n-1}\det(\mathbf{W})}}e^{-\frac{1}{2\sigma^{2}}\mathbf{z_{i}}^{\prime}\mathbf{W}^{-1}\mathbf{z_{i}}}\frac{\beta^{\alpha}}{\Gamma(\alpha)}(\sigma^{2})^{-(\alpha+1)}e^{-\frac{\beta}{\sigma^{2}}}d(\sigma^{2})$
$\displaystyle=$
$\displaystyle\prod_{i}\frac{1}{\sqrt{(2\pi)^{n-1}\det(\mathbf{W})}}\frac{\beta^{\alpha}}{(\frac{1}{2}\mathbf{z}_{i}^{\prime}\mathbf{W}^{-1}\mathbf{z}_{i}+\beta)^{\alpha+(n-1)/2}}\frac{\Gamma(\alpha+(n-1)/2)}{\Gamma(\alpha)}.$
The second line follows recognizing that each integral is independent. Thus,
the likelihood of the observations of transcriptome-wide gene expression
across the pathway in $n$ taxa, normalized by the expression level in taxon
$1$, is given by (3).
For the application to simulated and experimental data as described below,
given observations of gene expression of the species at the tips of the tree,
and a model that specifies the covariance matrix $\mathbf{V}$ detailed in the
next section, we optimized the log likelihood function using the L-BFGS-B
optimization routine in R [45].
### Covariance matrix
In the previous section, we left the unnormalized covariance matrix
$\mathbf{V}$ unspecified. Here we briefly recall the forms of $\mathbf{V}$
under Brownian motion and the Ornstein-Uhlenbeck process. Define the height of
the evolutionary tree to be $T$ and and the height of the node containing the
common ancestor of taxa $i$ and $j$ by $t_{ij}$. Then the covariance matrix
for Brownian motion is
$V_{i,j}=\begin{cases}t_{ij}&\text{if }i\neq j\\\ T&\text{if }i=j\end{cases}$
and the covariance matrix for the Ornstein-Uhlenbeck process is
$V_{i,j}=\begin{cases}\frac{1}{2\theta}e^{-2\theta(T-t_{ij})}(1-e^{2\theta
t_{ij}})&\text{if }i\neq j\\\ \frac{1}{2\theta}(1-e^{2\theta T})&\text{if
}i=j\end{cases}$
where $\theta$ quantifies the strength of stabilizing selection, with large
$\theta$ corresponding to stronger selection.
To model lineage-specific shifts in the evolutionary rate of gene expression
in the context of the Brownian motion model, we adopt a framework similar to
that of O’Meara _et al._[15]. We assume that in a specified subtree of the
total phylogeny, the rate of evolution of every gene is multiplied by a
constant, compared to the rest of the tree. Under the Brownian motion model,
this is equivalent to multiplying the branch lengths in that part of the tree
by that same constant; hence, shifts in evolutionary rate are incorporated by
multiplying the branch lengths of affected branches by the value of the rate
shift.
### Comparing likelihoods among fitted models
To evaluate the support for the distinct models we fit to expression data for
a given pathway, we require a strategy that will be broadly applicable in
cases where no _a priori_ expectation of the correct model is available, such
that nested hypothesis testing schemes [15] are not applicable. Instead, given
likelihoods $L$ from fitting of each model in turn to expression data from the
genes of a pathway, we use the Akaike Information Criterion, $2k-2ln(L)$ [46],
to report the strength of the support for each, where $k$ is the number of
parameters in the model ($k=2$ for the Brownian motion model in which the rate
of evolution is the same along all lineages in the phylogeny, and $k=3$ for
all other models).
### Simulations
For all simulations, we used a phylogenetic tree adapted from [27] by removing
the branch leading to _Saccharomyces kudriavzevii_ (see inset of Figure 1a and
Figures S1-S5). To simulate under models in which each gene in a pathway
evolves independently, we generated expression data for one gene at a time as
follows. We first drew the rate of evolution from the appropriately
parameterized inverse-gamma distribution. Then, without loss of generality, we
specified that the expression level at the root of the phylogeny was equal to
$0$, and we simulated evolution along the branches of the yeast phylogeny
according to either a Brownian motion or an Ornstein-Uhlenbeck process (with
optimal expression level equal to $0$), using the terminal expression level on
a branch as the initial expression level of its daughter branches. To account
for lineage-specific shifts in evolutionary rate in a simulated pathway, we
multiplied the rate of evolution of each gene by the rate shift parameter for
evolution along the branches affected by the rate shift. For each Brownian
motion-based rate shift model applicable to the tree, we simulated 100
replicate datasets for each of a range of gene group sizes, in each case
setting $\alpha=3$, $\beta=2$, and the rate shift parameter as specified in
Figure 1 and Figures S1-S5. For the Ornstein-Uhlenbeck model, we simulated 100
replicate datasets for each of a range of pathway sizes with $\alpha=3$,
$\beta=2$, and $\theta$ as specified in Figure 2.
To simulate under models in which expression of genes in a pathway was
correlated with coefficient $\rho$, we first drew $(\sigma^{2}_{i},1\leq i\leq
n)$, the rate of evolution for each gene, from an inverse-gamma distribution
with $\alpha=3$, $\beta=2$. We then parameterized the instantaneous variance-
covariance matrix of the $n$-dimensional Brownian motion by
$\Sigma_{i,j}=\begin{cases}\sigma^{2}_{i}&\text{if }i=j\\\
\rho\sigma_{i}\sigma_{j}&\text{if }i\neq j\end{cases}$
so that the distribution of trait change along a lineage was multivariate
normal with mean 0 and variance covariance matrix $\Sigma$. Separate simulated
expression data sets were generated with $\rho$ varying from 0 (complete
independence) to 1 (complete dependence) using 100 replicate simulations for
each value.
### Yeast strains, growth conditions, and RNA-seq
Strains used in this study are listed in Table S1. For pairwise comparisons of
_S. cerevisiae_ and each of _S. paradoxus_ , _S. mikatae_ , and _S. bayanus_ ,
two biological replicates of each diploid parent species and each
interspecific hybrid were grown at 25∘C in YPD medium [47] to log phase
(between 0.65-0.75 OD at 600 nm). Total RNA was isolated by the hot acid
phenol method [47] and treated with Turbo DNA-free (Ambion) according to the
manufacturer’s instructions. Libraries for a strand-specific RNA-seq protocol
on the Illumina sequencing platform, which delineates transcript boundaries by
sequencing poly-adenylated transcript ends, were generated as in [48] with the
following modifications: 1) AmpureXP beads (Beckman) were used to clean up
enzymatic reactions; 2) the gel purification and size-selection step was
eliminated; 3) the oligo-dT primer used for cDNA synthesis was
phosphorothioated at position ten (TTTTTTTTTT*TTTTTTTTTTVN, V=A,C,G,
N=A,C,G,T, *=phosphorothioate linkage, Integrated DNA Technologies); and 4) 12
PCR cycles were performed. Libraries were sequenced using 36 bp paired-end
modules on an Illumina IIx Genome Analyzer (Elim Biopharmaceuticals).
### RNA-seq mapping and normalization
Bioinformatic analyses were conducted in Python and R. RNA-seq reads were
stripped of their putative poly-A tails by removing stretches of consecutive
Ts flanking the sequenced fragment; reads without at least two such Ts were
discarded, as were reads with Ts at both ends. To ensure that expression data
from hybrid diploids and purebred species could be compared, for each class of
expression measurement for a given pair of species we mapped reads to both
species genomes from http://www.saccharomycessensustricto.org [27] using
Bowtie [49] with default settings and flags -m1 -X1000. These settings allowed
us to retain only those reads that were unambiguously assigned to one of the
two species in each pairwise comparison. A mapped read was inferred to have
originated from the plus strand of the genome if its poly-A tail corresponded
to a stretch of As at the 3′ end of the fragment, and a read was assigned to
the minus strand if its poly-A tail corresponded to a stretch of Ts at the 5′
end of the fragment relative to the reference genome. To filter out cases in
which inferred poly-A tails originated from stretches of As or Ts encoded
endogenously in the genome, we eliminated from analysis all reads whose
stretch of As or Ts contained more than 50% matches to the reference genome.
In order to filter out cases of potential oligo-dT mispriming during cDNA
synthesis, we also eliminated from analysis all reads that contained 10 or
more As in the 20 nucleotides upstream of their transcription termination
site. Read mapping statistics can be found in Table S2.
We controlled for read abundance biases due to differing GC content as
follows. For each lane of sequencing, we grouped sets of overlapping reads and
normalized abundance according to GC content of the overlapping region using
full-quantile normalization as implemented in the package EDASeq [50].
Normalized abundance was divided by raw abundance to generate a weight that
was assigned to every read in the group. These weights were used in place of
raw read counts in all downstream analyses. All expression data are available
through the Gene Expression Omnibus under identification number GSE38875.
### Transcript annotation
Coordinates of orthologous open reading frames (ORFs) in each genome were
taken from
http://www.saccharomycessensustricto.org. These ORF boundaries in _S.
cerevisiae_ differed, in some cases, from ORF definitions in the
_Saccharomyces_ Genome Database [51, SGD, using the definitions from December
22, 2007]; genes for which the two sets of definitions did not overlap were
discarded. For cases where the definitions overlapped but differed by more
than ten base pairs at either end, we used the boundaries defined by SGD and
adjusted ortholog boundaries in other species accordingly after performing
local multiple alignment [52] of the orthologous regions and flanking
sequences as defined by [27].
For most genomic loci, each sense transcript feature was defined as the region
from 50 bp upstream to 500 bp downstream of its respective ORF. If sequence
within this window for a given target ORF overlapped with the boundaries of an
adjacent gene or known non-coding RNA on the same strand, the sense feature
boundaries of the target were trimmed to eliminate the overlap. For tandem
gene pairs, the 3′ boundary of the upstream gene sense feature was set to 500
bp past the coding stop or the coding start of the downstream gene sense
feature, whichever was closer; the 5′ boundary of the downstream gene sense
feature was set to 50 bp upstream of its coding start or the 3′ end of the
upstream gene sense feature, whichever was closer.
We tabulated the GC-normalized expression counts (see above) that mapped to
each transcript feature for each RNA-seq sample. Given the full set of such
counts across all features and all samples, we then applied the upper-quartile
between-lane normalization method implemented in EDASeq [50]. The normalized
counts from this latter step for a given species were averaged across all
biological replicates to yield a final expression level for the feature, which
we then $\log_{2}$ transformed and used in all analysis in this work.
### Yeast pathways
We downloaded the list of genes associated with each Gene Ontology process
term from the _Saccharomyces_ Genome Database and filtered for terms
containing at least 10 genes. The resulting set comprised 333 terms.
### Visualizing distributions of interspecific expression variation
For visual inspection of expression differences between species in Figure 4,
we normalized experimentally measured data by branch lengths ascertained from
genome sequence as follows. If expression evolution follows the same Gaussian-
based model on all lineages of the yeast phylogeny, when the expression level
of gene $j$ in taxon $i$ is compared to that in taxon $1$ used as a reference,
the marginal distribution $Z_{i,j}$ (the difference in expression between
taxon $i$ and taxon $1$ at gene $j$) is distributed according to a univariate
analog of equation (3). In this case, dividing $Z_{i,j}$ by the absolute
branch length according to DNA sequence between taxon $i$ and taxon $1$
eliminates the dependence of the distribution on the total divergence time
between taxa, and the density of this normalized quantity will be the same for
all species comparisons. In the case of lineage-specific shifts in
evolutionary rate or universal selective constraint, one or more taxa will
exhibit distinct densities of the normalized expression divergence measure.
Thus, we generated each distribution in Figure 4 by tabulating the log fold-
change in expression between the indicated species and _S. cerevisiae_ , and
then dividing this quantity by the divergence time between the indicated
species and _S. cerevisiae_ according to the genome tree. After this
normalization, if a pathway has been subject to accelerated regulatory
evolution in one lineage, the distribution of expression log fold-changes
corresponding to the species at the tip of that lineage will be wider than
expected based on the length of the branch from DNA sequence, and hence it
will stand out against the other distributions when plotted as in Figure 4;
likewise, constraint on expression evolution of a pathway in a particular
species will manifest as a narrower distribution for that species. In the case
of a pathway subject to the same degree of regulatory constraint on all
branches of the yeast phylogeny, branch lengths ascertained from genome
sequence will be large relative to the modest expression divergence, with the
most dramatic disparity manifesting when divergent species are compared,
yielding the narrowest distribution of normalized expression levels. When
visualized as in Figure 4, the width of the distribution of log fold-changes
across genes of the pathway in a given species will thus be inversely
proportional to the species distance from _S. cerevisiae_ , with the narrowest
distribution for _S. bayanus_ and the widest for _S. paradoxus_.
## Acknowledgments
The authors thank Davide Risso and Oh Kyu Yoon for generously providing
software before publication; Daniela Delnieri, Chris Todd Hittinger and Oliver
Zill for providing _Saccharomyces_ strains; and John Huelsenbeck, Mason Liang,
Nicholas Matzke, Rasmus Nielsen, Benjamin Peter, Jeremy Roop, and Montgomery
Slatkin for helpful discussions.
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## Figure Legends
Figure 1: Phylogenetic inference of the evolutionary history of yeast pathway
regulation from data simulated under a model of a lineage-specific,
accelerated evolutionary rate. Each panel reports results of the inference of
evolutionary history from expression of the genes of a pathway in yeast
species, simulated under a model of a shift in evolutionary rate on the branch
leading to _S. paradoxus_ (dark line in inset phylogeny in (a)). (a), Each
trace reports the strength of support for one evolutionary model in inferences
from simulated expression in pathways of varying size. The $x$ axis reports
the number of genes in the pathway and the $y$ axis reports the Akaike weight
of the indicated model. Data were simulated under a Brownian motion model in
which the rate of regulatory evolution for each gene was drawn from an
inverse-gamma distribution with $\alpha=3$, $\beta=2$ and, for the branch
leading to _S. paradoxus_ , increased by a factor of $5$. In the legend, ER
denotes an equal-rates Brownian motion model in which rates of evolution were
the same on each branch of the phylogeny; OU denotes an Ornstein-Uhlenbeck
model of evolution; and species name abbreviations denote Brownian motion
models of accelerated evolutionary rate on the subtrees leading to the
respective taxa. (b), Each set of symbols reports results from expression data
simulated under a Brownian motion model in which the rate of regulatory
evolution for each gene was drawn from an inverse-gamma distribution with
$\alpha=3$, $\beta=2$ and, for the branch leading to _S. paradoxus_ ,
increased by the factor indicated on the $x$ axis. In a given set of symbols,
filled circles report the mean, and vertical bars report the standard
deviation of the sampling distribution, of the inferred rate shift parameter
in simulations of pathways containing, from left to right, 2, 10, 50, and 100
genes. Results from simulations of expression under models of evolutionary
rate shifts on other branches of the yeast phylogeny, and simulations of
expression in the absence of a lineage-specific evolutionary rate shift, are
reported in Supplmentary Figures 1-5.
Figure 2: Phylogenetic inference of the evolutionary history of yeast pathway
regulation from data simulated under an Ornstein-Uhlenbeck (OU) model. (a),
Data are as in Figure 1a except that expression measurements were simulated
under an OU model in which the phylogeny-wide rate of regulatory evolution for
each gene was drawn from an inverse-gamma distribution with $\alpha=3$,
$\beta=2$ and the phylogeny-wide constraint parameter had a value of 10. (b),
Data are as in Figure 1b except that expression measurements were simulated
under an OU model in which the phylogeny-wide rate of regulatory evolution for
each gene was drawn from an inverse-gamma distribution with $\alpha=3$,
$\beta=2$ and the phylogeny-wide constraint parameter had the value indicated
on the $x$ axis.
Figure 3: Inference of regulatory evolution in yeast pathways from
experimental expression measurements. Each panel reports results of
phylogenetic inference of evolutionary histories of gene expression change
from one set of experimental transcriptional profiling data. In a given panel,
each vertical bar reports results of maximum-likelihood fits of Brownian-
motion and Ornstein-Uhlenbeck models to expression of the genes of one Gene
Ontology process term; the total proportion of a bar corresponding to a
particular color indicates the Akaike weight of the corresponding model
(legend at right, with labels as in Figure 1). Bars are sorted by the model
with maximum Akaike weight. (a), Inference of _cis_ -regulatory variation from
interspecies hybrids; numerical indices correspond to rows in Table S3. (b),
Inference from measurements of total expression in species homozygotes;
numerical indices correspond to rows in Table S4.
Figure 4: Lineage-specific regulatory evolution and constraint in yeast
pathways, inferred from experimental expression measurements. Each panel shows
kernel density estimates of the distributions of experimental gene expression
measurements among the genes of one yeast Gene Ontology process term, whose
evolutionary history was inferred with strong support. In a given panel, each
trace reports the expression levels of the genes of the indicated pathway,
from the allele of the indicated yeast species in a hybrid or in the purebred
homozygote of a species, normalized with respect to the analogous measurement
in _S. cerevisiae_ and with respect to branch length. Inset cartoons represent
the model inferred with AIC weight $>$80% for the indicated pathway (see
Tables 1 and 2). (a) Allele-specific expression from measurements in diploid
hybrids (left) and total expression measurements in species homozygotes
(right) for the 38 genes of GO:0001302, replicative cell aging, supporting a
model of accelerated evolution in _S. paradoxus_ ; in the inset, the number
above the bolded branch reports the inferred shift in the rate of regulatory
evolution along that lineage. (b) Allele-specific expression from measurements
in diploid hybrids for the 462 genes of GO:0006351, transport, supporting a
model of constraint in _S. bayanus_ ; in the inset, the number above the
bolded branch reports the inferred shift in the rate of regulatory evolution
along that lineage. (c) Total expression measured in species homozygotes for
the 175 genes of GO:0006281, transcription, supporting an Ornstein-Uhlenbeck
model of universal constraint; in the inset, the number above the tree reports
the inferred value of the constraint parameter. Note that in (c), the width of
the distribution of expression differences between a given species and _S.
cerevisiae_ correlates inversely with the sequence divergence of that species,
as expected if selective constraint on expression renders the estimate of
evolutionary distance from genome sequence an increasing over-estimate of
expression change.
## Tables
Table 1: Top-scoring fitted models of _cis_ -regulatory evolution in yeast
pathways from experimental expression measurements.
GO term | $N$ | Model | wAIC | Constraint or shift parameter
---|---|---|---|---
34599 | 57 | Ornstein-Uhlenbeck | 0.899405768 | 49.97745883
6355 | 433 | _S. bayanus_ shift | 0.837382338 | 0.230918849
6351 | 462 | _S. bayanus_ shift | 0.849912647 | 0.258701476
1302 | 38 | _S. paradoxus_ shift | 0.859866949 | 3.197059161
6897 | 73 | _S. paradoxus_ shift | 0.965743399 | 4.292287639
6338 | 45 | _S. cerevisiae_ shift | 0.840339574 | 0.037806902
42254 | 136 | Ornstein-Uhlenbeck | 0.924785133 | 3.733770466
6364 | 177 | Ornstein-Uhlenbeck | 0.902358815 | 3.079387696
44255 | 13 | _S. paradoxus_ shift | 0.945799302 | 11.43989834
54 | 11 | _S. paradoxus_ shift | 0.91523272 | 9.314688245
16310 | 188 | _S. bayanus_ shift | 0.902247359 | 0.188381056
8152 | 243 | _S. bayanus_ shift | 0.844716856 | 0.043114988
6629 | 136 | _S. bayanus_ shift | 0.91650274 | 0.005082617
122 | 71 | _S. bayanus_ shift | 0.819216472 | 0.040060263
30437 | 45 | _S. paradoxus_ shift | 0.931136455 | 4.060128813
Each row reports the results of phylogenetic inference of the evolutionary
history of gene regulation for one yeast Gene Ontology process term, from
experimental measurements of _cis_ -regulatory variation in interspecific
yeast hybrids. $N$, number of genes in the indicated GO term for which
expression measurements were available in all species. Model, best-fit model
from among the five possible Brownian motion models of evolutionary rate shift
in lineages of the _Saccharomyces_ phylogeny (see Figure 1a), the Ornstein-
Uhlenbeck (OU) model of universal constraint, and the equal-rates model
involving no lineage-specific differences in evolutionary rate. wAIC, Akaike
Information Criterion weight of the indicated model. Constraint or shift
parameter, fitted value of the strength of purifying selection or the shift in
the rate of regulatory evolution on the indicated lineage, when the best-fit
model was the OU model of constraint or a Brownian motion lineage-specific
evolutionary rate model, respectively.
Table 2: Top-scoring fitted models of species regulatory evolution in yeast
pathways from experimental expression measurements.
GO term | $N$ | Model | wAIC | Constraint or shift parameter
---|---|---|---|---
6397 | 151 | _S. paradoxus_ shift | 0.965171603 | 3.028130303
8033 | 69 | _S. paradoxus_ shift | 0.969683391 | 3.714749932
71038 | 15 | _S. paradoxus_ shift | 0.89725301 | 6.751073973
480 | 29 | _S. paradoxus_ shift | 0.928296518 | 4.460579672
42274 | 25 | _S. paradoxus_ shift | 0.958076119 | 8.083546161
472 | 31 | _S. paradoxus_ shift | 0.953733629 | 4.686648741
15031 | 362 | _S. bayanus_ shift | 0.872939854 | 0.183834463
1302 | 38 | _S. paradoxus_ shift | 0.999927135 | 6.671016575
6006 | 22 | _S. paradoxus_ shift | 0.816341854 | 4.6555377
6260 | 72 | _S. paradoxus_ shift | 0.831407464 | 3.043207869
30163 | 15 | _S. paradoxus_ shift | 0.82364567 | 7.009201233
6897 | 73 | _S. paradoxus_ shift | 0.970677101 | 4.408614609
6412 | 228 | _S. paradoxus_ shift | 0.981277345 | 2.770778823
7121 | 16 | _S. paradoxus_ shift | 0.998579562 | 16.81960721
6914 | 49 | Ornstein-Uhlenbeck | 0.810293525 | 41.38598192
30488 | 18 | _S. paradoxus_ shift | 0.893282646 | 7.945094861
42254 | 163 | _S. paradoxus_ shift | 0.99999983 | 6.856141937
6200 | 34 | _S. paradoxus_ shift | 0.81144199 | 5.590943868
6468 | 120 | _S. paradoxus_ shift | 0.990399439 | 2.655209273
16567 | 71 | _S. paradoxus_ shift | 0.959694914 | 3.313920599
6364 | 177 | _S. paradoxus_ shift | 0.999995709 | 5.841035759
6754 | 18 | _S. paradoxus_ shift | 0.816303046 | 4.668929462
422 | 27 | Ornstein-Uhlenbeck | 0.877576591 | 57.08946364
463 | 20 | _S. paradoxus_ and _S. cerevisiae_ shift | 0.958484282 | 10.39289039
6414 | 23 | _S. paradoxus_ and _S. cerevisiae_ shift | 0.906687775 | 8.121469425
19236 | 29 | _S. paradoxus_ shift | 0.989881765 | 6.821984459
31505 | 72 | _S. paradoxus_ shift | 0.955855579 | 3.032267535
32259 | 65 | _S. paradoxus_ shift | 0.998665437 | 4.546902844
6506 | 29 | _S. paradoxus_ shift | 0.982054204 | 5.468542886
16310 | 188 | _S. paradoxus_ shift | 0.99652632 | 2.487101867
447 | 39 | _S. paradoxus_ shift | 0.994506418 | 5.252074336
6281 | 175 | Ornstein-Uhlenbeck | 0.882367142 | 3.410968446
71042 | 13 | _S. paradoxus_ shift | 0.804318406 | 6.030946867
6378 | 18 | _S. cerevisiae_ shift | 0.845112064 | 1.00E-04
7165 | 63 | _S. paradoxus_ shift | 0.811091269 | 4.465389345
6810 | 681 | Ornstein-Uhlenbeck | 0.859937275 | 2.618523967
6812 | 28 | _S. paradoxus_ shift | 0.898839416 | 4.312524185
8150 | 723 | _S. paradoxus_ shift | 0.999962114 | 2.871955612
6417 | 45 | _S. paradoxus_ shift | 0.925463092 | 5.339113187
6407 | 18 | _S. paradoxus_ shift | 0.988260506 | 8.792447836
462 | 55 | _S. paradoxus_ shift | 0.817627126 | 7.291083934
Data are as in Table 1 except that inferences were made from experimental
measurements of expression in purebred yeast homozygotes.
## Supplementary Figure Legends
Figure S1. Phylogenetic inference of the evolutionary history of yeast pathway
regulation under a Brownian motion model with equal rates on each branch of
the tree. Data are as in Figure 1a of the main text except that expression
data were simulated under a model in which no yeast lineage was subject to a
change in evolutionary rate.
Figure S2. Phylogenetic inference of the evolutionary history of yeast pathway
regulation under a model with a rate shift on the subtree leading to _S.
paradoxus_ and _S. cerevisiae_. (a), Data are as in Figure 1a of the main
text, except that expression measurements were simulated under a Brownian
motion model in which the rate of regulatory evolution for each gene was drawn
from an inverse-gamma distribution with $\alpha$ = 3, $\beta$ = 2 and, for the
subtree leading to _S. paradoxus_ and _S. cerevisiae_ , increased by a factor
of 5. (b), Data are as in Figure 1b of the main text, except that expression
measurements were simulated under a Brownian motion model in which the rate of
regulatory evolution for each gene was drawn from an inverse-gamma
distribution with $\alpha$ = 3, $\beta$ = 2 and, for the subtree leading to
_S. paradoxus_ and _S. cerevisiae_ , increased by the factor indicated on the
$x$ axis.
Figure S3. Phylogenetic inference of the evolutionary history of yeast pathway
regulation under a model with a rate shift on the branch leading to _S.
cerevisiae_. (a), Data are as in Figure 1a of the main text, except that
expression measurements were simulated under a Brownian motion model in which
the rate of regulatory evolution for each gene was drawn from an inverse-gamma
distribution with $\alpha$ = 3, $\beta$ = 2 and, for the branch leading to _S.
cerevisiae_ , increased by a factor of 5. (b), Data are as in Figure 1b of the
main text, except that expression measurements were simulated under a Brownian
motion model in which the rate of regulatory evolution for each gene was drawn
from an inverse-gamma distribution with $\alpha$ = 3, $\beta$ = 2 and, for the
branch leading to _S. cerevisiae_ , increased by the factor indicated on the
$x$ axis.
Figure S4. Phylogenetic inference of the evolutionary history of yeast pathway
regulation under a model with a rate shift on the branch leading to _S.
mikatae_. (a), Data are as in Figure 1a of the main text, except that
expression measurements were simulated under a Brownian motion model in which
the rate of regulatory evolution for each gene was drawn from an inverse-gamma
distribution with $\alpha$ = 3, $\beta$ = 2 and, for the branch leading to _S.
mikatae_ , increased by a factor of 5. (b), Data are as in Figure 1b of the
main text, except that expression measurements were simulated under a Brownian
motion model in which the rate of regulatory evolution for each gene was drawn
from an inverse-gamma distribution with $\alpha$ = 3, $\beta$ = 2 and, for the
branch leading to _S. mikatae_ , increased by the factor indicated on the $x$
axis.
Figure S5. Phylogenetic inference of the evolutionary history of yeast pathway
regulation under a model with a rate shift on the branch leading to _S.
bayanus_. (a), Data are as in Figure 1a of the main text, except that
expression measurements were simulated under a Brownian motion model in which
the rate of regulatory evolution for each gene was drawn from an inverse-gamma
distribution with $\alpha$ = 3, $\beta$ = 2 and, for the branch leading to _S.
bayanus_ , increased by a factor of 5. (b), Data are as in Figure 1b of the
main text, except that expression measurements were simulated under a Brownian
motion model in which the rate of regulatory evolution for each gene was drawn
from an inverse-gamma distribution with $\alpha$ = 3, $\beta$ = 2 and, for the
branch leading to _S. bayanus_ , increased by the factor indicated on the $x$
axis.
Figure S6. Relationship between inferred values of parameters in phylogenetic
reconstruction of the evolutionary history of yeast pathway regulation, under
an Ornstein-Uhlenbeck model. In the main plot, each data point reports the
results of inference of the evolutionary history of regulation of a yeast
pathway of size 100: expression data were simulated under an Ornstein-
Uhlenbeck (OU) model in which the rates of regulatory evolution of pathway
genes were drawn from an inverse-gamma distribution with $\alpha=3$ and
$\beta=2$ and the OU constraint parameter $\theta$ was set to 10, after which
parameter values for an OU model were optimized against the simulated
expression data. For histograms at top and left, the independent variable is
shared with the axis of the main plot and reports the indicated parameter
value, and the dependent variable reports the proportion of simulated data
sets in which the corresponding value was inferred. Note that inferences from
most simulated data sets accurately estimate $\beta$ and $\theta$, but for a
few data sets, large parameter values are inferred.
Figure S7. Mean-centering pathway expression levels in each species corrects
for spurious inference of non-neutral regulatory evolution arising from gene
co-regulation. Each trace reports the results of inference of the evolutionary
history of regulation of a yeast pathway of size 100, from expression data
simulated under a Brownian motion model in which evolutionary rates were the
same on all branches of the yeast phylogeny, and pathway genes were correlated
with one another with respect to expression throughout the phylogeny. Each
line style reports one scheme for normalization of simulated expression data
before evolutionary inference: expression measurements were analyzed as is
(Uncentered), or the distribution of expression across pathway genes for each
species in turn was normalized to have a mean of $0$ (Centered). The $x$ axis
reports the value of the correlation coefficient between genes in the group,
and the $y$ axis reports the fraction of 500 simulations that resulted in a
model other than the Brownian motion equal-rates model having an Akaike weight
greater than 0.8.
Figure S8. Heterogeneity in the mode of regulatory evolution across the genes
of a pathway has little impact on inference of evolutionary histories from
expression data. Each trace reports the results of inference of the
evolutionary history of regulation of a yeast pathway of size 100, from
expression data simulated under a Brownian motion model in which the rate of
regulatory evolution for each gene was drawn from an inverse-gamma
distribution with $\alpha=3$, $\beta=2$ and, for the branch leading to _S.
paradoxus_ , increased by a factor of $5$ for a subset of pathway genes. The
$x$ axis reports the fraction of genes in the group without a rate shift, and
the $y$ axis reports the average Akaike weight assigned to each model. Line
styles are as in Figure 1a of the main text.
## Supplementary Table Legends
Table S1. Strains used in this work. Table S2. Read mapping statistics from
yeast RNA-seq. Each set of rows reports the mapping statistics for reads from
RNA-seq libraries used for a comparison of two yeast species. For a given set,
in row headings, numerals indicate biological replicates, single species names
indicate homozygotes, and species name pairs separated by a slash indicate
diploid interspecies hybrids. Each row reports results from one library. Total
reads, the full set of reads sequenced. Have polyT, the number of reads
containing at least two consecutive Ts at only one end. Uniquely mapped, the
number of reads mapping uniquely, with no mismatches, to the concatenated
genomes of the two species of the set. Passed through filters, the number of
reads whose poly-A tails were unlikely to have originated from oligo-dT
mispriming to A-rich regions of the genome; see Methods. Table S3. Fitted
models of $cis$-regulatory evolution in yeast pathways. Data are as in Table 1
of the main text except that results for all pathways are shown. Table S4.
Fitted models of species regulatory evolution in yeast pathways. Data are as
in Table 2 of the main text except that results for all pathways are shown.
|
arxiv-papers
| 2013-04-19T17:35:23 |
2024-09-04T02:49:44.621831
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joshua G. Schraiber, Yulia Mostovoy, Tiffany Y. Hsu, Rachel B. Brem",
"submitter": "Joshua Schraiber",
"url": "https://arxiv.org/abs/1304.5486"
}
|
1304.5618
|
# Maximal Subalgebras for Lie Superalgebras of Cartan Type
Wei Bai111Supported by NSF of the Education Department of HLJP (12521158)
Department of Mathematics, Harbin Institute of Technology,
Harbin 150006, P. R. China School of Mathematical Sciences, Harbin Normal
University,
Harbin 150025, P. R. China Wende Liu222Supported by NSF of China (11171055)
and NSF of HLJP (JC201004, A2010-03) Department of Mathematics, Harbin
Institute of Technology,
Harbin 150006, P. R. China Xuan Liu Applied Mathematics Department, The
University of Western Ontario,
London, N6A 5B7, Canada Hayk Melikyan333Supported by part NSF Grant # 0833184
Department of Mathematics and Computer Science, North Carolina Central
University
Durham, NC 27713, USA
###### Abstract
The maximal graded subalgebras for four families of Lie superalgebras of
Cartan type over a field of prime characteristic are studied. All maximal
reducible graded subalgebras are described completely and their isomorphism
classes, dimension formulas are found. The classification of maximal
irreducible graded subalgebras is reduced to the classification of the maximal
irreducible subalgebras for the classical Lie superalgebras
$\mathfrak{gl}(m,n)$, $\mathfrak{sl}(m,n)$ and $\mathfrak{osp}(m,n)$.
###### keywords:
Lie superalgebras; maximal graded subalgebras Mathematics Subject
Classification 2010: 17B50, 17B05.
††journal: Journal of Algebra and its Applications.
3.6cm3.6cm2.4cm2cm
## 0 Introduction
Since V. G. Kac [1] classified the finite dimensional simple Lie superalgebras
over algebraically closed fields of characteristic zero, the theory of Lie
superalgebras has undergone a significant development (for example [2, 3]).
Over a field of finite characteristic, however, the classification problem is
still open for the finite dimensional simple Lie superalgebras [4, 5]. Even
recently, new simple Lie superalgebras over a field of characteristic $p=3$
were constructed [5, 6].
In general, study of the maximal subsystems of an algebraic system, such as
finite groups, Lie groups, Lie (super)algebras, is an essential part of
structural characterization of the system. In classical Lie theory, the
classification of maximal subalgebras of simple Lie algebras over the field of
complex numbers is one of the beautiful results of that theory which was due
to E. Dynkin [7, 8]. In classical modular Lie theory there is a series of
papers by G. Seitz and his students devoted to the study of the maximal
subgroups of simple algebraic groups over fields of positive characteristic.
These investigations were summarized by G. Seitz in his two publications [9,
10] which generalize E. Dynkin’s classification of the maximal subgroups of
simple Lie groups over the field of complex numbers [8] to simple algebraic
groups over fields of characteristic $p>7$. The study of maximal subalgebras
of different classes of (super)algebras has been the focus of several
researchers. The maximal subalgebras of Jordan (super)algebras were studied by
M. Racine [11, 12], A. Elduque, J. Laliena and S. Sacristan [13, 14]. The
maximal graded subalgebras of affine Kac-Moody algebras were classified in
[15]. The fourth author of the present paper summarized his investigations on
maximal subalgebras in Cartan type simple Lie algebras over the field of
characteristic $p>3$ in his paper [16].
Let $L$ be a finite dimensional simple Lie superalgebras of Cartan type $W$,
$S$, $H$ or $K$ with a $\mathbb{Z}$-grading $L=\oplus_{i\geq-2}L_{i}$. The
present paper is devoted to characterizing the maximal graded subalgebras of
$L$. To this end, we construct a series of graded subalgebras of $L$ and state
the necessary and sufficient conditions for their maximality. Moreover, the
number of isomorphism classes and the dimension formulas of all maximal graded
subalgebras are completely determined except for maximal irreducible graded
subalgebras. Note that the null of $L$ is isomorphic to a classical Lie
superalgebra (see Lemma 2.1(3)). Thus the classification of the maximal
irreducible graded subalgebras of $L$ is reduced to that of the maximal
irreducible subalgebras of a classical Lie superalgebra. Moreover, we give
necessary and sufficient conditions for the existence of maximal irreducible
graded subalgebras of $L$. We should mention that the present work which
partially generalizes the results of [16] is motivated by a paper by A. I.
Kostrikin and I. R. Shafarevich [17] on the structure theory of modular Lie
algebras.
We close this introduction by establishing the following conventions: The
underlying field $\mathbb{F}$ is an algebraically closed field of
characteristic $p>3$. In addition to the standard notation $\mathbb{Z},$ we
write $\mathbb{N}$ for the set of nonnegative integers. The field of two
elements is denoted by $\mathbb{Z}_{2}=\\{\bar{0},\bar{1}\\}.$ For a
proposition $P$, put $\delta_{P}=1$ if $P$ is true and $\delta_{P}=0$
otherwise. All subspaces, subalgebras and submodules are assumed to be
$\mathbb{Z}_{2}$-graded and all the homomorphisms of $\mathbb{Z}$-graded
superalgebras are both $\mathbb{Z}_{2}$-homogeneous and
$\mathbb{Z}$-homogeneous.
## 1 Basics
Fix two positive integers $m,n\in\mathbb{N}\backslash\\{1\\}.$ Put
$\mathbf{I}_{0}=\overline{1,m},\ \mathbf{I}_{1}=\overline{m+1,m+n},\
\mathbf{I}=\mathbf{I}_{0}\cup\mathbf{I}_{1},$
where $\overline{k,s}=\\{k,k+1,\ldots,s\\}$ with the convention
$\overline{k,s}=\emptyset$ whenever $k>s.$ Write
$\mathbf{A}(m)=\\{\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}\mid
0\leq\alpha_{i}\leq p-1,i\in\mathbf{I}_{0}\\}.$
Let $\mathcal{O}(m)$ be the divided power algebra with $\mathbb{F}$-basis
$\\{x^{(\alpha)}\mid\alpha\in\mathbf{A}(m)\\}$ and $\Lambda(n)$ be the
exterior superalgebra of $n$ variables $x_{m+1},x_{m+2},\ldots,x_{m+n}$. The
tensor product
$\mathcal{O}(m,n)=\mathcal{O}(m)\otimes\Lambda(n)$
is an associative superalgebra with respect to the usual
$\mathbb{Z}_{2}$-grading. Let
$\mathbf{B}(n)=\\{\langle i_{1},i_{2},\ldots,i_{k}\rangle\mid 0\leq k\leq
n;m+1\leq i_{1}<i_{2}<\cdots<i_{k}\leq m+n\\}$
be the set of $k$-tuples of strictly increasing integers in $\mathbf{I}_{1}$,
$0\leq k\leq n.$ For $u=\langle
i_{1},i_{2},\ldots,i_{k}\rangle\in\mathbf{B}(n)$, write
$x^{u}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}$ ($x^{\emptyset}=1$). If
$g\in\mathcal{O}(m)$ and $f\in\Lambda(n)$, we write $gf$ instead of $g\otimes
f$. Then $\mathcal{O}(m,n)$ has a $\mathbb{Z}_{2}$-homogeneous
$\mathbb{F}$-basis
$\\{x^{(\alpha)}x^{u}\mid\alpha\in\mathbf{A}(m),u\in\mathbf{B}(n)\\}.$
For $i\in\textbf{I}_{0}$ and
$\varepsilon_{i}=(\delta_{i1},\delta_{i2},\ldots,\delta_{im})$, write $x_{i}$
for $x^{(\varepsilon_{i})}.$ Let $\partial_{1},\ldots,\partial_{m+n}$ be the
superderivations of the superalgebra $\mathcal{O}(m,n)$ such that
$\partial_{i}(x_{j})=\delta_{i=j}.$ The parity of $\partial_{i}$ is
$|\partial_{i}|=\bar{0}$ if $i\in\mathbf{I}_{0}$ and $\bar{1}$ if
$i\in\mathbf{I}_{1}$. Hereafter the symbol $|x|$ implies that $x$ is a
$\mathbb{Z}_{2}$-homogeneous element. Put
$W(m,n)=\mathrm{span}_{\mathbb{F}}\left\\{a\partial_{i}\mid
a\in\mathcal{O}(m,n),i\in\mathbf{I}\right\\},$
which is a finite dimensional simple Lie superalgebra, called Witt
superalgebra. Consider the linear mapping called divergence:
$\mathrm{div}:W(m,n)\longrightarrow\mathcal{O}(m,n),\quad\mathrm{div}(f\partial_{i})=(-1)^{|f||\partial_{i}|}\partial_{i}(f).$
Set $S(m,n)=[\overline{S}(m,n),\overline{S}(m,n)]$, where
$\overline{S}(m,n)=\ker(\mathrm{div})$. Then we have
$S(m,n)=\mathrm{span}_{\mathbb{F}}\left\\{D_{ij}(a)\mid
a\in\mathcal{O}(m,n),i,j\in\mathbf{I}\right\\},$
where
$D_{ij}(a)=(-1)^{|\partial_{i}||\partial_{j}|}\partial_{i}(a)\partial_{j}-(-1)^{(|\partial_{i}|+|\partial_{j}|)|a|}\partial_{j}(a)\partial_{i}\quad\mbox{for
}a\in\mathcal{O}(m,n).$
$S(m,n)$ is a simple Lie superalgebra, called special superalgebra.
For $j\in\\{1,\ldots,2\lfloor\frac{m}{2}\rfloor,m+1,\ldots,m+n\\}$, we put
$\sigma(j)=\left\\{\begin{array}[]{ll}-1,&j\in\overline{\lfloor\frac{m}{2}\rfloor+1,2\lfloor\frac{m}{2}\rfloor};\\\
1,&\mbox{ otherwise }\end{array}\right.\mbox{ and }\
j^{\prime}=\left\\{\begin{array}[]{lll}j+\lfloor\frac{m}{2}\rfloor,&j\in\overline{1,\lfloor\frac{m}{2}\rfloor};\\\
j-\lfloor\frac{m}{2}\rfloor,&j\in\overline{\lfloor\frac{m}{2}\rfloor+1,2\lfloor\frac{m}{2}\rfloor};\\\
j,&\mbox{ otherwise}.\end{array}\right.$
Suppose $m=2r$ is even. Define an even linear mapping
$D_{H}:\mathcal{O}(m,n)\longrightarrow W(m,n)$ by
$D_{H}(a)=\sum_{i\in\mathbf{I}}\sigma(i)(-1)^{|\partial_{i}||a|}\partial_{i}(a)\partial_{i^{\prime}}.$
Put
$\overline{H}(m,n)=\mathrm{span}_{\mathbb{F}}\\{D_{H}(a)\mid
a\in\mathcal{O}(m,n)\\}.$
Write $\bar{\mathcal{O}}(m,n)$ for the quotient superspace
$\mathcal{O}(m,n)/\mathbb{F}\cdot 1$. We can view $D_{H}$ as a linear operator
of $\bar{\mathcal{O}}(m,n)$ since the kernel of $D_{H}$ is $\mathbb{F}\cdot
1$. Thus we have $\overline{H}(m,n)\cong(\bar{\mathcal{O}}(m,n),[\;,\;])$,
where the bracket is:
$[a,b]=D_{H}\left(a\right)\left(b\right)\ \mbox{ for
}a,b\in\bar{\mathcal{O}}(m,n).$
Its derived algebra ${H}(m,n)$ is simple, called Hamiltonian Lie superalgebra.
Suppose $m=2r+1$ is odd. Define an even linear mapping
$D_{K}:\mathcal{O}(m,n)\longrightarrow W(m,n)$ by
$D_{K}(a)=D_{H}(a)+\partial_{m}(a)\mathfrak{D}+\Delta(a)\partial_{m},$ where
$\mathfrak{D}=\sum_{i\in\mathbf{I}\backslash\\{m\\}}x_{i}\partial_{i}$ and
$\Delta(a)=2a-\mathfrak{D}(a)$. Put
$\overline{K}(m,n)=\mathrm{span}_{\mathbb{F}}\\{D_{K}(a)\mid
a\in\mathcal{O}(m,n)\\}.$
Since $D_{K}$ is injective, we have
$\overline{K}(m,n)\cong(\mathcal{O}(m,n),[\;,\;]),$ where the bracket is:
$\begin{split}[a,b]&=D_{H}(a)b+\Delta(a)\partial_{m}(b)-\partial_{m}(a)\Delta(b)\
\mbox{ for }a,b\in{\mathcal{O}}(m,n).\end{split}$
Its derived algebra $K(m,n)$ is simple, called contact Lie superalgebra.
For simplicity, hereafter, we write $X$ for $X(m,n)$, where $X=\mathcal{O}$,
$\bar{\mathcal{O}}$, $W$, $\overline{S}$, $S$, $\overline{H}$, $H$,
$\overline{K}$ or $K$. Let us consider the standard $\mathbb{Z}$-grading of
$L$, where $L=\mathcal{O}$, $W$, $S$, $\overline{H}$, $H$ or $K$. Define the
$\mathbb{Z}$-degrees of $x_{i}$ and $\partial_{i}$ to be
$\mathrm{zd}(x_{i})=-\mathrm{zd}(\partial_{i})=1+\delta_{L=K}\delta_{i=m}$,
$i\in\mathbf{I}$. Hereafter, the symbol $\mathrm{zd}(x)$ always implies that
$x$ is a $\mathbb{Z}$-homogeneous element. Put $\xi=(m+\delta_{L=K})(p-1)+n$.
Then we have:
$\displaystyle\mathcal{O}=\oplus_{i=-1}^{\xi}\mathcal{O}_{i},\ \
\mathcal{O}_{i}=\mathrm{span}_{\mathbb{F}}\\{f\in\mathcal{O}\mid\mathrm{zd}(f)=i\\};$
$\displaystyle W=\oplus_{i=-1}^{\xi-1}W_{i},\ \
W_{i}=\mathrm{span}_{\mathbb{F}}\\{f\partial_{j}\mid f\in\mathcal{O}_{i+1},\
j\in\mathbf{I}\\};$ $\displaystyle S=\oplus_{i=-1}^{\xi-2}S_{i},\ \
S_{i}=\mathrm{span}_{\mathbb{F}}\\{D_{jk}(f)\in W\mid f\in\mathcal{O}_{i+2},\
j,k\in\mathbf{I}\\};$ $\displaystyle\overline{H}=\oplus_{i=-1}^{\xi-2}H_{i},\
\ H_{i}=\mathrm{span}_{\mathbb{F}}\\{f\mid f\in\mathcal{O}_{i+2}\\};\
H=\oplus_{i=-1}^{\xi-3}H_{i};$
$\displaystyle{K}=\oplus_{i=-2}^{\xi-2-\delta_{n-m-3\equiv
0\;(\mathrm{mod}\;p)}}K_{i},\ \ K_{i}=\mathrm{span}_{\mathbb{F}}\\{f\mid
f\in\mathcal{O}_{i+2}\\}.$
We adopt the following conventions:
* (1)
$L=H$ implies that $m=2r$ is even; $L=K$ implies that $m=2r+1$ is odd.
* (2)
$K$ can be viewed as a $\mathbb{Z}$-graded subalgebra of $W$ when
$\mathrm{zd}(x_{m})=-\mathrm{zd}(\partial_{m})=2$ for $W$. Thus, $L$ is a
$\mathbb{Z}$-graded subalgebra of $W$, where $L=S$, $H$ or $K$.
* (3)
For $L=K$, we write $z$ for $x_{m}.$
* (4)
Write $\mathrm{alg}(S)$ for the subalgebra of $L$ generated by a subset $S$.
A proper subalgebra $M$ of a $\mathbb{Z}$-graded Lie superalgebra $L$ is
called a maximal graded subalgebra (MGS) provided that $M$ is
$\mathbb{Z}$-graded and no nontrivial $\mathbb{Z}$-graded subalgebras of $L$
strictly contains $M$. Since $L_{-1}$ is an irreducible $L_{0}$-module, it is
clear that $\oplus_{i\geq 0}L_{i}$ is an MGS of $L$. Any other MGS, $M$, must
satisfy exactly one of the following conditions:
* $\mathrm{(I)}$
$M_{-1}=L_{-1}$ and $M_{0}=L_{0};$
* $\mathrm{(II)}$
$M_{-1}$ is a nontrivial proper subspace of $L_{-1};$
* $\mathrm{(III)}$
$M_{-1}=L_{-1}$ and $M_{0}\neq L_{0}.$
Let $\mathfrak{G}_{0}$ be a subalgebra of $L_{0}$. $\mathfrak{G}_{0}$ is
called reducible (resp. irreducible) if the $\mathfrak{G}_{0}$-module $L_{-1}$
is reducible (resp. irreducible). An MGS
$\mathfrak{G}=\sum_{i\geq-2}\mathfrak{G}_{i}$ of $L$ is called maximal
reducible graded (resp. maximal irreducible graded) if $\mathfrak{G}_{0}$ is
reducible (resp. irreducible).
## 2 Preliminary Results
In order to simplify our considerations, in this section, we establish some
technical lemmas. For $L=H$ or $K$, we redescribe $L$ in an appropriate form
and establish a suitable automorphism of $L$ by virtue of a nondegenerate skew
supersymmetric bilinear form on $L_{-1}$.
As in the case of Lie superalgebras of characteristic 0 [1] or modular Lie
algebras [17, 18, 19], it is easy to show the following:
###### Lemma 2.1.
Let $L=W,S,H$ or $K$.
* $(1)$
$L$ is transitive.
* $(2)$
$L$ is generated by its local part, $L=\mathrm{alg}(L_{-1}+L_{0}+L_{1})$.
* $(3)$
For the null of $L$, the following conclusions hold:
$\displaystyle W(m,n)_{0}\cong\mathfrak{gl}(m,n);\
S(m,n)_{0}\cong\mathfrak{sl}(m,n);$ $\displaystyle
H(2r,n)_{0}\cong\mathfrak{osp}(2r,n);\
K(2r+1,n)_{0}\cong\mathfrak{osp}(2r,n)\oplus\mathbb{F}I_{2r+n}.$
When $L=W$ or $S$, we know that $L_{-1}$ is spanned by the standard ordered
$\mathbb{F}$-basis
$\displaystyle\\{\partial_{i}\mid i\in\mathbf{I}\\}.$ (2.1)
For a $\mathbb{Z}_{2}$-graded subspace $V=V_{\bar{0}}\oplus V_{\bar{1}}$ of
$L_{-1}$, the super-dimension is denoted by
$\mathrm{superdim}V=(\mathrm{dim}V_{\bar{0}},\mathrm{dim}V_{\bar{1}}).$
When $L=H$ or $K$, we redescribe $L$ in a desired form. For
$i\in\mathbb{N}\backslash{\\{0\\}}$, write $A_{i}$ for an $i\times i$ matrix,
and particularly, let $I_{i}$ be the $i\times i$ unit matrix. Denote by
$\sqrt{a}$ a fixed solution of the equation $x^{2}=a$ in $\mathbb{F}$, where
$a=-1,2$. Put
$y_{i}=\left\\{\begin{array}[]{lll}{x_{i}},&i\in\mathbf{I}_{0}\cup\overline{m+2q+1,m+n};\\\
\frac{x_{i}+\sqrt{-1}x_{i+q}}{\sqrt{2}},&i\in\overline{m+1,m+q};\\\
\frac{x_{i-q}-\sqrt{-1}x_{i}}{\sqrt{2}},&i\in\overline{m+q+1,m+2q},\end{array}\right.$
where $0\leq d\leq n$, $q=\lfloor\frac{n-d}{2}\rfloor$. Then there exists an
invertible matrix $A_{m+n}$ such that
$(y_{1},\ldots,y_{m+n})A=(x_{1},\ldots,x_{m+n})$. Obviously, $|y_{i}|=|x_{i}|$
and $\mathrm{zd}(y_{i})=\mathrm{zd}(x_{i}),$ $i\in\mathbf{I}$. By [20, Lemma
2.5], we have:
$\\{y^{(\alpha)}y^{u}\mid\alpha\in\mathbf{A}(m),\ u\in\mathbf{B}(n)\\}$
is an $\mathbb{F}$-basis of $\mathcal{O}$, where $y^{(\alpha)}=x^{(\alpha)}$
and $y^{u}=y_{i_{1}}y_{i_{2}}\cdots y_{i_{k}}$ when $u=\langle
i_{1},i_{2},\ldots,i_{k}\rangle$. The basis-element $y^{(\alpha)}y^{u}$ is
called a monomial.
Write $(D_{1},\ldots,D_{m+n})=(\partial_{1},\ldots,\partial_{m+n})A^{t}$. Then
we have
$D_{i}=\left\\{\begin{array}[]{lll}\partial_{i},&i\in\mathbf{I}_{0}\cup\overline{m+2q+1,m+n};\\\
\frac{\partial_{i}-\sqrt{-1}\partial_{i+q}}{\sqrt{2}},&i\in\overline{m+1,m+q};\\\
\frac{\partial_{i-q}+\sqrt{-1}\partial_{i}}{\sqrt{2}},&i\in\overline{m+q+1,m+2q}.\end{array}\right.$
By a direct computation, we have:
$D_{i}(y_{j})=\delta_{i=j},\ \
{\sum}_{i\in\mathbf{I}\backslash\\{2r+1\\}}y_{i}D_{i}=\mathfrak{D}.$
When $m=2r$, define an even linear mapping $E_{H}:\mathcal{O}\longrightarrow
W$ by
$E_{H}(a)={\sum}_{i\in\mathbf{I}}\sigma(i)(-1)^{|D_{i}||a|}D_{i}(a)D_{\widetilde{i}},$
where
$\widetilde{i}=\left\\{\begin{array}[]{lll}i^{\prime},&i\in\mathbf{I}_{0}\cup\overline{m+2q+1,m+n};\\\
i+q,&i\in\overline{m+1,m+q};\\\
i-q,&i\in\overline{m+q+1,m+2q}.\end{array}\right.$
When $m=2r+1$, define an even linear mapping $E_{K}:\mathcal{O}\longrightarrow
W$ by
$\begin{split}E_{K}(a)&=E_{H}(a)+D_{m}(a)\mathfrak{D}+\Delta(a)D_{m}.\end{split}$
A direct computation shows that $D_{L}=E_{L}$. Note that $L_{-1}$ is spanned
by the standard ordered $\mathbb{F}$-basis
$\displaystyle\\{y_{i}\mid i\in\overline{1,2r}\cup\overline{m+1,m+n}\\}.$
(2.2)
Define an even bilinear form $\beta:L_{-1}\times
L_{-1}\longrightarrow\mathbb{F}$ satisfying
$\beta(u,v)={\sum}_{i\in\mathbf{I}}\sigma(i)(-1)^{|D_{i}||u|}D_{i}(u)D_{\widetilde{i}}(v)\
\mbox{ for }u,v\in L_{-1}.$
Then the matrix of $\beta$ in the ordered basis (2.2) is
$J=\left(\begin{tabular}[]{c|c}\begin{tabular}[]{c|c}0&$I_{r}$\\\
\hline\cr$-I_{r}$&0\\\ \end{tabular}&0\\\ \hline\cr
0&\begin{tabular}[]{c|c|c}0&$-I_{q}$&0\\\ \hline\cr$-I_{q}$&0&0\\\ \hline\cr
0&0&$-I_{n-2q}$\end{tabular}\end{tabular}\right).$ (2.3)
Clearly, $\beta$ is a nondegenerate skew supersymmetric bilinear form on
$L_{-1}$.
An $\mathbb{F}$-basis of $L_{-1}$ in which the matrix of $\beta$ is $J$ is
called generalized orthosymplectic. Let $V=V_{\bar{0}}\oplus V_{\bar{1}}$ be a
subspace of $L_{-1}$. Suppose $2a$ (resp. $d$) is the rank of $\beta$
restricted to $V_{\bar{0}}$ (resp. $V_{\bar{1}}$). A
$\mathbb{Z}_{2}$-homogeneous basis of $V$
$\displaystyle\\{e_{1},\ldots,e_{a},e_{r+1},\ldots,e_{r+a};e_{a+1},\ldots,e_{b}\mid
e_{m+1},\ldots,e_{m+c};e_{m+n-d+1},\ldots,e_{m+n}\\}$ (2.4)
is called a $\beta$-basis of $V$, if
$\\{e_{1},\ldots,e_{a},e_{r+1},\ldots,e_{r+a};e_{a+1},\ldots,e_{b}\\},\ 0\leq
a\leq b\leq r$
is an $\mathbb{F}$-basis of $V_{\bar{0}}$ satisfying
$\beta(e_{i},e_{j})=-\beta(e_{j},e_{i})=\left\\{\begin{array}[]{ll}1,&1\leq
i\leq a,j=\widetilde{i};\\\ 0,&\mbox{otherwise}\end{array}\right.$
and
$\\{e_{m+1},\ldots,e_{m+c};e_{m+n-d+1},\ldots,e_{m+n}\\},\ 0\leq d\leq n,\
0\leq c\leq\lfloor\frac{n-d}{2}\rfloor$
is an $\mathbb{F}$-basis of $V_{\bar{1}}$ satisfying
$\beta(e_{i},e_{j})=\beta(e_{j},e_{i})=\left\\{\begin{array}[]{ll}-1,&m+n-d+1\leq
i=j\leq m+n;\\\ 0,&\mbox{otherwise}.\end{array}\right.$
The 4-tuple $(a,b,c,d)$ is called the $\beta$-dimension of $V$, denoted by
$\beta$-$\dim V=(a,b,c,d)$. $V$ is nondegenerate (with respect to $\beta$) if
$a=b$ and $c=0$. $V$ is isotropic if $a=0$ and $d=0$. Clearly, for any
$\mathbb{Z}_{2}$-graded subspace of $L_{-1}$, there exists a $\beta$-basis of
it, which can extend to a generalized orthosymplectic basis of $L_{-1}$.
Now, suppose $L=W$, $S$, $H$ or $K$. Put
$\mathfrak{V}^{L}=\\{V\mid V\mbox{ is a nontrivial subspace of }L_{-1}\\}.$
$V\in\mathfrak{V}^{L}$ is called a standard element if $V$ is spanned by
$\\{\partial_{1},\ldots,\partial_{k}\mid\partial_{m+1},\ldots,\partial_{m+l}\\},$
when $L=W$ or $S$, $0\leq k\leq m$, $0\leq l\leq n$; if $V$ is spanned by
$\\{y_{1},\ldots,y_{a},y_{r+1},\ldots,y_{r+a};y_{a+1},\ldots,y_{b}\mid
y_{m+1},\ldots,y_{m+c};y_{m+n-d+1},\ldots,y_{m+n}\\},$
when $L=H$ or $K$, $0\leq a\leq b\leq r$, $0\leq d\leq n$, $0\leq
c\leq\lfloor\frac{n-d}{2}\rfloor$. Hereafter, for
$V,V^{\prime}\in\mathfrak{V}^{L}$, the symbol $V\cong V^{\prime}$ always means
$\mathrm{superdim}V=\mathrm{superdim}V^{\prime}$ when $L=W$ or $S$ and means
$\beta$-$\dim V=\beta$-$\dim V^{\prime}$ when $L=H$ or $K$.
###### Lemma 2.2.
Let $L=W,S,H$ or $K$. Suppose $V$, $V^{\prime}\in\mathfrak{V}^{L}$ satisfying
$V\cong V^{\prime}$. Then there exists a $\mathbb{Z}$-homogeneous automorphism
${\Phi}_{L}$ of $L$ such that ${\Phi}_{L}(V)=V^{\prime}$.
###### Proof.
Without loss of generality, we may assume that $V$ is a standard element in
$\mathfrak{V}^{L}$. When $L=W$ or $S$, suppose
$\mathrm{superdim}V=\mathrm{superdim}V^{\prime}=(k,l).$ Let
$(E_{1},\ldots,E_{k}\mid E_{m+1},\ldots,E_{m+l})$ be a
$\mathbb{Z}_{2}$-homogeneous basis of $V^{\prime}$. It extends to a
$\mathbb{Z}_{2}$-homogeneous basis of $W_{-1}$:
$(E_{1},\ldots,E_{m}\mid E_{m+1},\ldots,E_{m+n}),$
where $|E_{i}|=|\partial_{i}|$, $i\in\mathbf{I}$. There exists an even
invertible matrix $A_{m+n}$ such that
$\displaystyle(E_{1},\ldots,E_{m+n})=(\partial_{1},\ldots,\partial_{m+n})A^{t}.$
(2.5)
Let $(\xi_{1},\ldots,\xi_{m+n})=(x_{1},\ldots,x_{m+n})A^{-1}$. Consider the
mapping $\phi$ such that
$\phi(x_{i})=\xi_{i}\quad\mbox{for all }i\in\mathbf{I}.$
Notice that $|x_{i}|=|\xi_{i}|$, since $A$ is even. By [20, Lemma 2.5], $\phi$
can extend to an endomorphism of $\mathcal{O}$, which is still written as
$\phi$. Then we have:
$\displaystyle(\phi^{-1}(x_{1}),\ldots,\phi^{-1}(x_{m+n}))=(x_{1},\ldots,x_{m+n})A$
(2.6)
We denote by ${\Phi}$ the automorphism of $W$ which is induced by $\phi$
according to the formula
${\Phi}(D)=\phi D\phi^{-1}\ \mbox{ for }\ D\in W.$
Clearly, ${\Phi}$ is $\mathbb{Z}$-homogeneous. By (2.5) and (2.6), we have:
$\displaystyle\Phi(\partial_{i})=\phi\partial_{i}\phi^{-1}=E_{i}\quad\mbox{for
all }i\in\mathbf{I}.$ (2.7)
Furthemore, for $D=\sum_{i\in\mathbf{I}}f_{i}\partial_{i}\in W$, one can
verify that
${\Phi}(D)=\phi
D\phi^{-1}={\sum}_{i,j\in\mathbf{I}}\partial_{i}(\phi^{-1}(x_{j}))\phi(f_{i})\partial_{j}.$
By virtue of (2.5) and (2.7), we have:
$\mathrm{div}({\Phi}(D))=\phi(\mathrm{div}D).$
This shows that ${\Phi}({S})={S}$ since $S=[\overline{S},\overline{S}]$. Then
$\Phi_{L}=\Phi\big{|}_{L}$ is desired.
When $L=H$ or $K$, suppose $\beta$-$\dim V=\beta$-$\dim V^{\prime}=(a,b,c,d)$.
Let $\\{e_{i}\mid i\in\overline{1,2r}\cup\overline{m+1,m+n}\\}$ be an
extension of $\beta$-basis (2.4) of $V^{\prime}$ to a generalized
orthosymplectic basis of $L_{-1}$. Then, there exist two even invertible
matrices
$\displaystyle A=\left(\begin{array}[]{c|c}A_{2r}&0\\\ \hline\cr
0&A_{n}\end{array}\right)\mbox{ and
}A^{\prime}=\left(\begin{array}[]{c|c}\begin{array}[]{c|c}A_{2r}&0\\\
\hline\cr 0&I_{1}\end{array}&0\\\ \hline\cr 0&A_{n}\end{array}\right)$
satisfying
$\displaystyle(e_{1},\ldots,e_{2r}\mid
e_{m+1},\ldots,e_{m+n})A=(y_{1},\ldots,y_{2r}\mid y_{m+1},\ldots,y_{m+n}),$
$\displaystyle(e_{1},\ldots,e_{2r},e_{2r+1}\mid
e_{m+1},\ldots,e_{m+n})A^{\prime}=(y_{1},\ldots,y_{2r},y_{2r+1}\mid
y_{m+1},\ldots,y_{m+n}).$
Thus, we obtain that
$\displaystyle A^{-1}J(A^{t})^{-1}=J.$ (2.10)
By virtue of [20, Lemma 2.5], there exists a unique automorphism of
$\mathcal{O}$ denoted by $\phi_{L}$ satisfying $\phi_{L}(y_{i})=e_{i}$,
$i\in\mathbf{I}.$ As in the case $L=W$, we denote by $\overline{{\Phi}}_{L}$
the $\mathbb{Z}$-homogeneous automorphism of $W$ which is induced by
$\phi_{L}$. From (2.10), we have:
$\displaystyle(\overline{{\Phi}}_{H}(D_{1}),\ldots,\overline{{\Phi}}_{H}(D_{m+n}))=(D_{1},\ldots,D_{m+n})A^{t}.$
(2.11)
$\displaystyle(\overline{{\Phi}}_{K}(D_{1}),\ldots,\overline{{\Phi}}_{K}(D_{m+n}))=(D_{1},\ldots,D_{m+n})A^{\prime
t}.$ (2.12)
For any $D=\sum_{i\in\mathbf{I}\backslash\\{2r+1\\}}f_{i}D_{i}\in W$ and
$fD_{2r+1}\in W$, from (2.10)-(2.12) we have:
$\displaystyle\overline{{\Phi}}_{L}(D)={\sum}_{i\in\mathbf{I}\backslash\\{2r+1\\}}\phi_{L}(f_{i})\overline{{\Phi}}_{L}(D_{i}),$
(2.13) $\displaystyle\overline{{\Phi}}_{K}(fD_{2r+1})=\phi_{K}(f)D_{2r+1}.$
(2.14)
For any $f\in\mathcal{O}$, we have:
$\displaystyle\overline{{\Phi}}_{K}(D_{2r+1}(f)\mathfrak{D})=D_{2r+1}(\phi_{K}(f))\mathfrak{D}.$
(2.15)
$\displaystyle\overline{{\Phi}}_{K}((2-\mathfrak{D})(f)D_{2r+1})=(2-\mathfrak{D})\phi_{K}(f)D_{2r+1}.$
(2.16)
By virtue of (2.10)-(2.16), we have:
$\overline{{\Phi}}_{L}(E_{L}(f))=E_{L}(\phi_{L}(f))\ \mbox{ for any
}f\in\mathcal{O}.$
It follows that $\overline{\Phi}_{L}({L})={L}$ since
$L=[\overline{L},\overline{L}]$. Then
$\Phi_{L}=\overline{\Phi}_{L}\big{|}_{L}$ is desired. ∎
For convenience, we introduce the following notations. Let $L=W$ or $S$. For
any $V\in\mathfrak{V}^{L}$ with $\mathrm{superdim}V=(k,l)$, put
$\displaystyle\mathbf{I}(k,l)=\overline{1,k}\cup\overline{m+1,m+l},\qquad\overline{\mathbf{I}}(k,l)=\mathbf{I}\backslash\mathbf{I}(k,l),$
If $V$ is standard, we have:
$V=\mathrm{span}_{\mathbb{F}}\\{\partial_{i}\mid i\in\mathbf{I}(k,l)\\}.$
(2.17)
Let $L=H$ or $K$. For any $V\in\mathfrak{V}^{L}$ with $\beta$-$\dim
V=(a,b,c,d)$, put
$\displaystyle I_{01}=\overline{1,a};\ \bar{I}_{01}=\overline{r+1,r+a};\
I_{02}=\overline{a+1,b};\ \bar{I}_{02}=\overline{r+a+1,r+b};$
$\displaystyle{I}_{11}=\overline{m+n-d+1,m+n};\ {I}_{12}=\overline{m+1,m+c};\
\bar{I}_{12}=\overline{m+q+1,m+q+c};$
$\displaystyle{I}_{03}=\overline{b+1,r}\cup\overline{r+b+1,m};\
{I}_{13}=\overline{m+c+1,m+q}\cup\overline{m+q+c+1,m+n-d}.$ (2.18)
Obviously, $\mathbf{I}=J_{1}\cup J_{2}\cup\bar{J}_{2}\cup J_{3},$ where
$\displaystyle J_{1}={I}_{01}\cup\bar{I}_{01}\cup{I}_{11},\ \
J_{2}={I}_{02}\cup{I}_{12},\ \ \bar{J}_{2}=\bar{I}_{02}\cup\bar{I}_{12}\
\mbox{ and }\ J_{3}={I}_{03}\cup{I}_{13}.$ (2.19)
We call $J_{i}$ to be single (resp. twinned) if $\mathbf{I}_{0}\cap
J_{i}=\emptyset$ and there exists only one element in $\mathbf{I}_{1}\cap
J_{i}$ (resp. there exist two elements in $\mathbf{I}_{1}\cap J_{i})$,
$i=1,3$. If $V$ is standard, we have:
$V=\mathrm{span}_{\mathbb{F}}\\{y_{i}\mid i\in J_{1}\cup J_{2}\\}.$ (2.20)
For any $i\in\mathbf{I}$, let us assign to each $y_{i}$ a value as follows:
$\nu(y_{i})=\left\\{\begin{array}[]{llll}1&i\in J_{1};\\\ 0&i\in J_{2};\\\
\frac{1}{3}&i\in\bar{J}_{2};\\\ 2&i\in J_{3}.\end{array}\right.$ (2.21)
If $a=y_{1}^{\alpha_{1}}y_{2}^{\alpha_{2}}\cdots y_{m}^{\alpha_{m}}y^{u}$,
define $\nu(a)=\prod_{i\in\mathbf{I}_{0}}\nu(y_{i})^{\alpha_{i}}\prod_{i\in
u}\nu(y_{i}).$
###### Remark 2.3.
Let $T$ be a torus of $L$, $L=W,S,H$ or $K$. Consider the weight space
decompositions with respect to $T$:
$\displaystyle L=L^{\theta}\oplus\oplus_{\gamma\in\Delta}L^{\gamma},\ \
L_{i}=L_{i}^{\theta}\oplus\oplus_{\gamma\in\Delta_{i}}L^{\gamma}_{i},$
where $\Delta_{i}\subset\Delta\subset T^{*}$ and $\theta$ is the zero weight.
Notice the standard facts below.
* $(1)$
For $t\in T$, suppose $x=x_{1}+x_{2}+\cdots+x_{n}\in L$ is a sum of
eigenvectors of $\mathrm{ad}t$ associated with mutually distinct eigenvalues.
Then all $x_{i}$’s lie in $\mathrm{alg}(\\{t,x\\})$.
* $(2)$
$T=\mathrm{span}_{\mathbb{F}}\\{y_{i}y_{\widetilde{i}}\mid
i\in\overline{1,r}\cup\overline{m+1,m+q}\\}$ is a torus of $L$, $L=H$ or $K$,
where $0\leq d\leq n$, $q=\lfloor\frac{n-d}{2}\rfloor$. Define $\epsilon_{j}$
to be the linear function on $T$ by
$\epsilon_{j}(y_{i}y_{\widetilde{i}})=\delta_{j{\widetilde{i}}}-\delta_{ji}.$
For $i,j,k\in\mathbf{I}\backslash\\{\\{2r+1\\}\cup\overline{m+2q+1,m+n}\\}$,
if $\epsilon_{i}+\epsilon_{j}\in\Delta_{0}$, we have:
$\dim L^{\epsilon_{i}}_{-1}=1;\ \ \dim L^{\epsilon_{i}+\epsilon_{j}}_{0}=1;\ \
L^{\epsilon_{k}}_{0}={\sum}_{l=m+2q+1}^{m+n}\mathbb{F}y_{k}y_{l}.$
## 3 MGS of Type (I)
To formulate the MGS of type $(\textrm{I})$, we introduce the following
notations.
For $i\geq 1$, write
$L_{i}^{\prime}=\overline{S}_{i}=\\{D\in
L_{i}\mid\mathrm{div}D=0\\}\quad\mbox{and}\quad
L_{i}^{\prime\prime}=\\{f\mathfrak{D}\mid f\in\mathcal{O}_{i}\\},$ (3.22)
where $L=W$ or $S$, $\mathfrak{D}$ is the degree derivation of $\mathcal{O}$;
that is, $\mathfrak{D}=\sum_{k\in\mathbf{I}}x_{k}\partial_{k}$. Clearly, both
$W_{i}^{\prime}$ and $W_{i}^{\prime\prime}$ are nontrivial subspaces of
$W_{i}$.
For $i\geq 0$, write
$K_{ij}=\big{\\{}u\in K_{i}\mid
u=fz^{j},f\in\mathcal{O}_{i+2-2j},[1,f]=0\big{\\}}.$
Clearly, $K_{ij}$ is a nontrivial subspace of $K_{i}$.
###### Theorem 3.1.
All MGS of type $\mathrm{(I)}$ are characterized as follows:
* $\mathrm{(1)}$
If $m-n+1\equiv 0\pmod{p}$ then $W$ has exactly one MGS of type
$\mathrm{(I)}:$
$W_{-1}+W_{0}+W_{1}^{\prime}+W_{2}^{\prime}+\cdots+W_{\xi-2}^{\prime}$
with dimension $(m+n-1)2^{n}p^{m}+2$;
If $m-n+1\not\equiv 0\pmod{p}$ then $W$ has exactly two MGS of type
$\mathrm{(I)}:$
$W_{-1}+W_{0}+W_{1}^{\prime\prime}\quad\mbox{and}\quad
W_{-1}+W_{0}+W_{1}^{\prime}+W_{2}^{\prime}+\cdots+W_{\xi-2}^{\prime}$
with dimensions $(m+n)(m+n+2)$ and $(m+n-1)2^{n}p^{m}+2$, respectively.
* $\mathrm{(2)}$
If $m-n+1\equiv 0\pmod{p}$ then $S$ has exactly one MGS of type
$\mathrm{(I)}:$
$S_{-1}+S_{0}+S_{1}^{\prime\prime}$
with dimension $(m+n)^{2}+2(m+n)-1$;
If $m-n+1\not\equiv 0\pmod{p}$ then $S$ has exactly one MGS of type
$\mathrm{(I)}:$
$S_{-1}+S_{0}$
with dimension $(m+n)^{2}+(m+n)-1$.
* $\mathrm{(3)}$
$H$ has exactly one MGS of type $\mathrm{(I)}:$
$H_{-1}+H_{0}$
with dimension $(m+n)^{2}+m$.
* $\mathrm{(4)}$
$K$ has exactly two MGS of type $\mathrm{(I)}:$
$K_{-2}+K_{-1}+K_{0}+{\sum}_{i=1}^{2r(p-1)+n}K_{i0}\ \mbox{ and }\
K_{-2}+K_{-1}+K_{0}+K_{11}+K_{22}$
with dimensions $2^{n}p^{2r}+1$ and $(2r+n)^{2}+4r+n+3$, respectively.
We note that many preliminary results in this section are analogous to the
ones of Lie algebras (see [16, 17, 18]). We will need the following formulas
which are easy to verify by direct calculations.
###### Lemma 3.2.
For $f\in\mathcal{O}_{s}$ and $g\in\mathcal{O}_{t},$
$\displaystyle\mathrm{div}(f\mathfrak{D})=(m-n+s)f\quad\mbox{for}~{}f\in\mathcal{O}_{s},$
$\displaystyle[f\mathfrak{D},g\mathfrak{D}]=(t-s)fg\mathfrak{D}.$ (3.23)
###### Lemma 3.3.
The following statements hold.
* $(1)$
$W_{s}^{\prime}$ and $W^{\prime\prime}_{s}$ are $W_{0}$-submodules of $W_{s}$.
Moreover, $W_{s}^{\prime\prime}$ is irreducible.
* $(2)$
If $m-n+s\not\equiv 0\pmod{p}$ then $W_{s}=W_{s}^{\prime}\oplus
W_{s}^{\prime\prime}$;
* $(3)$
If $m-n+s\equiv 0\pmod{p}$ then $W_{s}^{\prime\prime}\subset W_{s}^{\prime}$.
###### Proof.
Note that $\mathrm{div}$ is a derivation from $W$ to $\mathcal{O}$ as
$W$-module. Thus, (1), (2) and (3) hold by virtue of Lemma 3.2. ∎
Below, the $1$-component $W_{1}$ will be a focus of our attention. For
convenience, we introduce two concepts, by which our arguments are largely
simplified: An element $\mathscr{L}$ in $W_{1}$ is called a leader if it is of
the form
$\displaystyle\mathscr{L}=x_{1}^{2}\partial_{1}+{\sum}_{i=2}^{m+n}f_{i}\partial_{i}\quad\mbox{where}\;f_{i}\in\mathcal{O}_{2};$
An element in $W_{1}$ is called $1$-defective if it is of the form
${\sum}_{i=2}^{m+n}f_{i}\partial_{i}\ \mbox{ where }f_{i}\in\mathcal{O}_{2}.$
###### Lemma 3.4.
Let $D\in W_{1}$.
* $(1)$
$[x_{1}\partial_{j},D]=0$ for all $j\geq 2$ if and only if $D=\lambda
x_{1}\mathfrak{D}+x_{1}^{2}\sum_{j\geq 2}k_{j}\partial_{j}$ for some
$\lambda,k_{j}\in\mathbb{F}.$
* $(2)$
$[x_{1}\partial_{j},D]\in W_{1}^{\prime\prime}$ for all $j\geq 2$ if and only
if $D=f\mathfrak{D}+x_{1}^{2}\sum_{j\geq 2}k_{j}\partial_{j}$ for some
$f\in\mathcal{O}_{1}$ and $k_{j}\in\mathbb{F}.$
###### Proof.
(1) Suppose $[x_{1}\partial_{j},D]=0$ for all $j\geq 2$ and write
$D=\sum_{i}a_{i}\partial_{i}$. Then
$x_{1}\partial_{j}(a_{i})-\delta_{i=j}(-1)^{|x_{1}\partial_{j}||a_{1}\partial_{1}|}a_{1}=0\quad\mbox{for
all}\;j\geq 2.$ (3.24)
Then $a_{1}=kx_{1}^{2}$ for some $k\in\mathbb{F}.$ If $k=0$, it follows from
(3.24) that $\partial_{j}(a_{i})=0$ for all $j,i\geq 2$. That is,
$a_{j}=k_{j}x_{1}^{2}$ for all $j\geq 2.$ Hence $D=x_{1}^{2}\sum_{j\geq
2}k_{j}\partial_{j}.$ If $k\neq 0$ then write $a_{1}=x_{1}^{2}.$ From (3.24)
one deduces
$\partial_{j}(a_{i})=\delta_{i=j}x_{1}\quad\mbox{for all}\;i,j\geq 2.$
It follows that $a_{j}=k_{j}x_{1}^{2}+x_{1}x_{j}$ for $j\geq 2$ and one
direction holds. The other one is clear.
(2) Write $[x_{1}\partial_{j},D]=f_{j}\mathfrak{D}$,
$f_{j}\in\mathcal{O}_{1},$ $j\geq 2.$ By acting on $x_{i}$ with $i\neq 1,j,$
we have $x_{1}[\partial_{j},D](x_{i})=f_{j}x_{i}$ and then $f_{j}=k_{j}x_{1}$
for some $k_{j}\in\mathbb{F}.$ Thus
$[x_{1}\partial_{j},D]=k_{j}x_{1}\mathfrak{D}\quad\mbox{for all}\;j\geq 2.$
(3.25)
Since $[x_{1}\partial_{j},x_{i}\mathfrak{D}]=\delta_{i=j}x_{1}\mathfrak{D},$
from (3.25) we have $\left[x_{1}\partial_{j},D-\left(\sum_{i\geq
2}k_{i}x_{i}\right)\mathfrak{D}\right]=0$ for all $j\geq 2.$ Now the
conclusion follows from (1). ∎
###### Lemma 3.5.
Let $M$ be a nonzero $W_{0}$-submodule of $W_{1}.$
* $\mathrm{(1)}$
$M$ contains a leader.
* $\mathrm{(2)}$
If $M$ contains a leader which does not lie in $W_{1}^{\prime\prime}$ then $M$
contains a nonzero 1-defective element.
* $\mathrm{(3)}$
If $M$ contains a nonzero $1$-defective element then $M\supset
W_{1}^{\prime}$. In particular, as $W_{0}$-module, $W_{1}^{\prime}$ is
generated by $x_{1}^{2}\partial_{j}$ for any fixed $j\geq 2$.
* $\mathrm{(4)}$
As $W_{0}$-module, $W_{1}$ is generated by $x_{1}^{2}\partial_{1}.$
* $\mathrm{(5)}$
Any nonzero $W_{0}$-submodule of $W_{1}$ different from $W_{1}^{\prime\prime}$
must contain $W_{1}^{\prime}.$
###### Proof.
(1), (3) and (4) need only a straightforward verification.
(2) Let $D=x_{1}^{2}\partial_{1}+\cdots$ be a leader in $M\setminus
W_{1}^{\prime\prime}$. Then $[x_{1}\partial_{j},D]\in M$ are 1-defective for
all $j\geq 2.$ If they are not all zero, we are done. Otherwise, by Lemma
3.4(1),
$D=x_{1}\mathfrak{D}+x_{1}^{2}\sum_{j\geq 2}k_{j}\partial_{j}$ for some
$k_{j}\in\mathbb{F}.$
Clearly, $\sum_{j\geq 2}k_{j}\partial_{j}\neq 0,$ say, $k_{2}\neq 0.$
Consequently, $x_{1}^{2}\partial_{2}=k_{2}^{-1}[x_{2}\partial_{2},D]\in M$.
(5) Let $M$ be a nonzero $W_{0}$-submodule and $M\neq W_{1}^{\prime}$. Let us
show that $M\supset W_{1}^{\prime}.$ By (1), (2) and (3) we may assume that
all the leaders of $M$ lie in $W^{\prime\prime}$. Then
$W^{\prime\prime}\subset M,$ since $W^{\prime\prime}$ as $W_{0}$-module is
irreducible by Lemma 3.3(1). For $D\in M\setminus W^{\prime\prime}$, if there
is some $i\geq 2$ such that $E=[x_{1}\partial_{i},D]\not\in W^{\prime\prime},$
then $[x_{1}\partial_{j},E]$ is a leader or 1-defective for any $j\geq 2$. By
(3), one may assume that there is $D\in M\setminus W^{\prime\prime}$ which is
pulled into $W^{\prime\prime}$ by any $x_{1}\partial_{j}$ with $j\geq 2.$ Then
by Lemma 3.4(2), $M$ contains a nonzero 1-defective element and then $M\supset
W^{\prime}.$ ∎
###### Lemma 3.6.
The following statements hold.
* $\mathrm{(1)}$
$W_{1}^{\prime}$ is a maximal $W_{0}$-submodule of $W_{1}$.
* $\mathrm{(2)}$
If $m-n+1\not\equiv 0\pmod{p}$, $W_{0}$-module $W_{1}^{\prime}$ is
irreducible. In particular, $W_{1}$ has a decomposition of irreducible
$W_{0}$-submodules:
$W_{1}=W_{1}^{\prime}\oplus W_{1}^{\prime\prime}.$
* $\mathrm{(3)}$
If $m-n+1\equiv 0\pmod{p}$, $W_{1}$ has exactly a composition series of
$W_{0}$-submodules:
$0\subset W_{1}^{\prime\prime}\subset W_{1}^{\prime}\subset W_{1}.$
###### Proof.
(1) Let $M$ be a submodule of $W_{1}$ containing strictly $W_{1}^{\prime}$.
Note that
$\mathrm{div}:\mathrm{span}_{\mathbb{F}}\\{x_{1}x_{1}\partial_{1},x_{2}x_{1}\partial_{1},\ldots,x_{m+n}x_{1}\partial_{1}\\}\longmapsto\mathcal{O}_{1}$
is surjective. Pick any $D\in M\setminus W_{1}^{\prime}$. Then there exists
$E=fx_{1}\partial_{1},$ $f\in\mathcal{O}_{1},$ such that
$\mathrm{div}E=\mathrm{div}D$. That is, $E-D\in W_{1}^{\prime}\subset M$ and
then $0\neq E\in M.$ If $\partial_{j}(f)=0$ for all $j\geq 2$ then
$E=\partial_{1}(f)x_{1}^{2}\partial_{1}$ and hence $M=W_{1}$ by Lemma 3.5(4).
Suppose $\partial_{j}(f)\neq 0$ for some $j\neq 1.$ Then
$x_{j}x_{1}\partial_{1}={\partial_{j}(f)^{-1}}[x_{j}\partial_{j},E]\in M$. It
follows that
$x_{1}^{2}\partial_{1}-(-1)^{|\partial_{j}|}x_{j}x_{1}\partial_{j}=[x_{1}\partial_{j},x_{j}x_{1}\partial_{1}]\in
M.$
Note that
$\frac{1}{2}x_{1}^{2}\partial_{1}-(-1)^{|\partial_{j}|}x_{j}x_{1}\partial_{j}$
is in $W_{1}^{\prime}\subset W.$ It follows that $x_{1}^{2}\partial_{1}\in M$
and $M=W_{1}$ by Lemma 3.5(4), showing that $W_{1}^{\prime}$ is maximal.
(2) and (3) are immediate consequences of Lemmas 3.3 and 3.5(5). ∎
###### Corollary 3.7.
The following statements hold.
* $\mathrm{(1)}$
If $m-n+1\not\equiv 0\pmod{p}$ then $S_{1}$ is an irreducible $S_{0}$-module.
* $\mathrm{(2)}$
If $m-n+1\equiv 0\pmod{p}$ then $S_{1}^{\prime\prime}$ is the unique
nontrivial $S_{0}$-submodule of $S_{1}$.
###### Proof.
Note that $W_{0}=S_{0}+\mathbb{F}\mathfrak{D}$ and that
$W^{\prime}_{1}=S_{1}$. If $m-n+1\equiv 0\pmod{p}$ then
$S_{1}^{\prime\prime}=W_{1}^{\prime\prime}$. The lemma follows directly from
Lemma 3.6. ∎
###### Lemma 3.8.
$\\{D\in W_{2}\mid[W_{-1},D]\subset W_{1}^{\prime\prime}\\}=0.$
###### Proof.
Write $D=\sum_{i\in\mathbf{I}}a_{i}\partial_{i}\in W_{2}$ and suppose $D$ is
pulled into $W_{1}^{\prime\prime}$ by $W_{-1}$. Then, each $a_{i}$ must be a
multiple of $x_{i}^{2}$ and in particular, $a_{j}=0$ for all $j>m.$ Write
$D=\sum_{i\leq m}f_{i}x_{i}^{2}\partial_{i}$, where $f_{i}\in\mathcal{O}_{1}.$
Since $[\partial_{j},D]\in W_{1}^{\prime\prime}$, one deduces that
$\partial_{j}(f_{i})=0$ for $j>m$ and $i\leq m.$ Now it is clear that $D$ is
not in $W_{1}^{\prime\prime}$ unless it is zero. ∎
Let
$\displaystyle
M^{\prime}=W_{-1}+W_{0}+W_{1}^{\prime}+W_{2}^{\prime}+\cdots+W_{\xi-2}^{\prime},$
$\displaystyle M^{\prime\prime}=W_{-1}+W_{0}+W_{1}^{\prime\prime}.$
Using (3.23) and keeping in mind that $\mathrm{div}$ is a derivation from $W$
to $\mathcal{O}$, one may verify that $M^{\prime}$ and $M^{\prime\prime}$ are
subalgebras of $W$.
###### Lemma 3.9.
Suppose $M$ is a proper subalgebra containing $W_{-1}\oplus W_{0}\oplus
W_{1}^{\prime}.$ Then $M\subset M^{\prime}$.
###### Proof.
Assume conversely that $M\not\subset M^{\prime}$. Then there exists $D\in
M\cap\sum_{i\geq 1}W_{i}$ satisfying $\mathrm{div}D\not=0$. Using the formula
$\mathrm{div}[\partial_{j},D]=\partial_{j}(\mathrm{div}D)$ for all
$j\in\mathbf{I}$, one sees that $M\supset W_{1}$ by Lemma 3.6(1). By Lemma
2.1(2), $M=W$, a contradiction. ∎
Proof of (1) and (2) in Theorem 3.1 (1) Claim A: $M^{\prime}$ is maximal. This
follows immediately from Lemma 3.9.
Claim B: $M^{\prime\prime}$ is maximal if $m-n+1\not\equiv 0\pmod{p}.$ Let $M$
be a subalgebra strictly containing $M^{\prime\prime}$. By transitivity and
Lemma 3.8, $M\cap W_{1}$ must strictly contain $W_{1}^{\prime\prime}$. Lemma
3.6(2) forces $M\supset W_{1}$ and therefore, $M=W$ by Lemma 2.1(2).
Claim C: $M^{\prime}$ and $M^{\prime\prime}$ exhaust all the maximal
subalgebras of type (I). Let $M$ be a maximal subalgebra of type (I). By
transitivity, $M$ must contain a nonzero element of $W_{1}$ and therefore,
$M\cap W_{1}\neq 0$ is a nonzero $W_{0}$-submodule of $W_{1}$. By Lemma
3.5(5), we have $M\cap W_{1}=W_{1}^{\prime\prime}$ or $M\cap W_{1}\supset
W_{1}^{\prime}$.
Case 1. Suppose $m-n+1\not\equiv 0\pmod{p}.$ If $M\cap
W_{1}=W_{1}^{\prime\prime}$ then Claim B forces $M=M^{\prime\prime}.$ Suppose
$M\cap W_{1}\supset W_{1}^{\prime}$. By Lemma 3.9, we have $M\subset
M^{\prime}$ and then $M=M^{\prime}$ by the maximality of $M$.
Case 2. Suppose $m-n+1\equiv 0\pmod{p}.$ We have $M\supset W^{\prime\prime}$.
Since $W^{\prime\prime}\subsetneq W^{\prime}$ in this situation, one sees
$M\supsetneq W^{\prime\prime}$. By transitivity and Lemma 3.8, $M\cap
W_{1}\supsetneq W_{1}^{\prime\prime}$ and hence $M\cap W_{1}\supset
W_{1}^{\prime}$ by Lemma 3.5(5). It follows from Lemma 3.9 that
$M=M^{\prime}.$ This completes the proof of (1).
(2) First of all, $S_{-1}+S_{0}$ and $S_{-1}+S_{0}+S_{1}^{\prime\prime}$
($m+n-1\equiv 0\pmod{p}$) are subalgebras of $S$. Let $M$ be a maximal
subalgebra of $S$ containing $S_{-1}+S_{0}$. Note that
$W_{1}^{\prime\prime}=S_{1}^{\prime\prime}$ when $m-n+1\equiv 0\pmod{p}$. By
the transitivity of $S$, Lemmas 2.1(2), 3.8 and Corollary 3.7, we obtain that
$M=S_{-1}+S_{0}+S_{1}^{\prime\prime}$ when $m+n-1\equiv 0\pmod{p}$;
$M=S_{-1}+S_{0}$ when $m+n-1\not\equiv 0\pmod{p}$. The process shows also that
these two subalgebras are indeed maximal. This completes the proof of (2). ∎
###### Remark 3.10.
For $W$ and $S$, the arguments for MGS of the other types will be reduced to
the case of type $\mathrm{(I)}$ MGS by the method of minimal counterexample.
###### Lemma 3.11.
The following statements hold.
* $\mathrm{(1)}$
$H_{1}$ is an irreducible $H_{0}$-module.
* $\mathrm{(2)}$
For $i\geq 0$, $K_{i}$ is a direct sum of $K_{0}$-submodules $K_{ij}$.
Moreover, $K_{10}$ and $K_{11}$ are irreducible $K_{0}$-modules.
###### Proof.
Using the results in the case of modular Lie algebras [17] and by a direct
computation, it is easy to show that (1) holds. Since $K_{10}\cong H_{1}$ and
$K_{11}\cong H_{-1}$ as $H_{0}$-modules, by irreducibilities of $H_{-1}$ and
$H_{1}$, (2) holds. ∎
Let
$\displaystyle M^{\prime}=K_{-2}+K_{-1}+K_{0}+{\sum}_{i=1}^{2r(p-1)+n}K_{i0},$
$\displaystyle M^{\prime\prime}=K_{-2}+K_{-1}+K_{0}+K_{11}+K_{22}.$
By a standard and direct computation, one may verify that $M^{\prime}$ and
$M^{\prime\prime}$ are subalgebras of $K$.
Proof of (3) and (4) in Theorem 3.1 (3) This statement follows immediately
from Lemmas 2.1(2) and 3.11(1).
(4) Claim A: $M^{\prime}$ is maximal. For any $0\not=u\in K$, $u\not\in
M^{\prime}$, put $\overline{M}=\mathrm{alg}(M^{\prime}+\mathbb{F}{u})$. Note
that there exist $k\in\mathbb{N}$ and $v_{1},\ldots,v_{s}\in K_{-1}$ such that
$0\not=u_{1}=fz+\alpha
z^{2}=[v_{1},[\cdots[v_{s},(\mathrm{ad}1)^{k}u]\cdots]]\in\overline{M}\backslash
M^{\prime},$
where $f\in\mathcal{O}$ satisfying $[1,f]=0$ and $\alpha\in\mathbb{F}$. Then
there exists $i\in\mathbf{I}$ such that
$[y_{\widetilde{i}},u_{1}]-y_{\widetilde{i}}f\not=0$. It follows that
$0\not=(\sigma(\widetilde{i})(-1)^{i}D_{i}(f)+\alpha
y_{\widetilde{i}})z\in\overline{M}.$
From Lemma 2.1(1), there exists a nonzero element in $\overline{M}\cap
K_{11}.$ By Lemmas 2.1(2) and 3.11(2), we have $\overline{M}=K$. Thus
$M^{\prime}$ is maximal.
Claim B: $M^{\prime\prime}$ is maximal. For any $0\not=u\in K$, $u\not\in
M^{\prime\prime}$, put
$\overline{M}=\mathrm{alg}(M^{\prime\prime}+\mathbb{F}{u})$. It is sufficient
to show that there exists a nonzero element in $K_{10}\cap\overline{M}$. When
$\mathrm{zd}(u)>2$, by transitivity, there exist $v_{1},\ldots,v_{s}\in
K_{-1}$ such that
$0\not=u_{3}=u_{30}+u_{31}+u_{32}=[v_{1},[\cdots[v_{s},u]\cdots]]\in\overline{M},$
where $u_{3i}\in K_{3i}$, $i=0,1,2$. Note that $[1,u_{32}]\in K_{11}$. If
$u_{31}\not=0$, then
$0\not=[1,u_{31}]=[1,u_{3}-u_{30}-u_{32}]\in\overline{M}\cap K_{10}.$
If $u_{31}=0$, there exists $j\in\mathbf{I}$ such that
$0\not=u_{2}=\sigma(\widetilde{j})(-1)^{j}D_{j}(u_{30})+y_{\widetilde{j}}D_{2r+1}(u_{32})\in\overline{M}\backslash{M^{\prime\prime}},$
Note that $\mathrm{zd}(u_{2})=2$. Thus, it remains to consider the case
$\mathrm{zd}(u)=2$. Assume that $u=u_{20}+u_{21}$, where $u_{2i}\in K_{2i}$,
$i=0,1$. If $u_{21}=0$ the conclusion follows. Notice that ${H_{0}}\cong
K_{21}$ as $H_{0}$-module. If $u_{21}\not=0$, by Remark 2.3 and a direct
computation, we obtain that there exists $i\in\mathbf{I}_{0}$ such that
$u_{2}=u^{\prime}_{20}+y^{(2\varepsilon_{i})}z\in\overline{M}$, where
$u^{\prime}_{20}\in K_{20}.$ Since $[K_{-1},u_{2}]\subset\overline{M}$, there
exists $j\in\mathbf{I}$ such that
$0\not=\sigma(\widetilde{j})(-1)^{j}D_{j}(u^{\prime}_{20})+y_{\widetilde{j}}y^{(2\varepsilon_{i})}\in\overline{M}\cap
K_{10}.$
Thus the conclusion holds.
Claim C: $M^{\prime}$ and $M^{\prime\prime}$ exhaust all the maximal graded
subalgebras of type $\mathrm{(I)}$. Suppose $N$ is a maximal graded subalgebra
of $K$ containing $K_{-1}+K_{0}$. By transitivity, there exists
$0\not=D=D_{10}+D_{11}\in N$, where $D_{1i}\in K_{1i}$, $i=0,1$. Since
$N\subsetneq K$, we claim that $D_{11}=0$ or $D_{10}=0$. Indeed, if
$D_{11}\not=0$ and $D_{10}\not=0$, by the irreducibility of $K_{10}$, we have
$w=y^{(2\varepsilon_{1})}y_{\widetilde{1}}+{\sum}_{i=1}^{2r+n}\alpha_{i}y_{i}z\in
N,\ \alpha_{i}\in\mathbb{F}.$
We consider the following cases.
Case 1. For all $i$, $\alpha_{i}=0$. Obviously, $N=K$ by the irreducibility of
$K_{1i}$ and $D_{1i}\not=0$, $i=0,1.$
Case 2. There exists $k$ such that $\alpha_{k}\not=0.$ If
$k\not=1,\widetilde{1}$, for $\widetilde{k}\not=j\in\mathbf{I}_{1}$, we have:
$0\not=[y_{j}y_{\widetilde{k}},w]\in N\cap K_{11}.$
Similar to Case 1, we have $N=K$.
If $k=1$ or $\widetilde{1}$, then
$w=y^{(2\varepsilon_{1})}y_{\widetilde{1}}+\alpha_{1}y_{1}z+\alpha_{\widetilde{1}}y_{\widetilde{1}}z$.
For $j\in\mathbf{I}_{1}$, we have:
$y_{j}y_{1}y_{\widetilde{1}}=[[y_{j}y_{1},w],y^{(2\varepsilon_{\widetilde{1}})}]\in
N\cap K_{10}.$
Similar to Case 1, we have $N=K$.
Consequently, $N=M^{\prime}$ when $D_{11}=0$ and $N=M^{\prime\prime}$ when
$D_{10}=0$. ∎
## 4 MGS of Type (II)
Let $L=W,S,H$ or $K$. Recall
$\mathfrak{V}^{L}=\\{V\mid V\mbox{ is a nontrivial subspace of }L_{-1}\\}.$
To describe the MGS of type $(\textrm{II})$ of $L$, for any
$V\in\mathfrak{V}^{L}$, we define
$\mathcal{M}(V)=\oplus_{i\geq-2}\mathcal{M}_{i}(V),$
where
$\displaystyle\mathcal{M}_{-1}(V)=V;\ \
\mathcal{M}_{-2}(V)=[\mathcal{M}_{-1}(V),\mathcal{M}_{-1}(V)];$
$\displaystyle\mathcal{M}_{i}(V)=\\{u\in
L_{i}\mid[V,u]\subset\mathcal{M}_{i-1}(V)\\}\qquad\mbox{for}\;i\geq 0.$ (4.26)
###### Theorem 4.1.
Suppose $L=W$ or $S$. All MGS of type $\mathrm{(II)}$ of $L$ are characterized
as follows:
* $\mathrm{(1)}$
All MGS of type $\mathrm{(II)}$ of $L$ are precisely:
$\\{\mathcal{M}(V)\mid V\in\mathfrak{V}^{L}\\}.$
* $\mathrm{(2)}$
For any $V$ and $V^{\prime}$ in $\mathfrak{V}^{L}$,
$\mathcal{M}(V)\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(V^{\prime})\Longleftrightarrow
V{\cong}V^{\prime}.$
* $\mathrm{(3)}$
$L$ has exactly $(m+1)(n+1)-2$ isomorphism classes of MGS of type
$\mathrm{(II)}$.
* $\mathrm{(4)}$
If $\mathrm{superdim}V=(k,l),$ then
$\displaystyle\mathrm{dim}\mathcal{M}(V)=\left\\{\begin{array}[]{ll}2^{n}p^{m}(m+n)-2^{l}p^{k}(m+n-k-l),&L=W;\\\
2^{n}p^{m}(m+n-1)+1-2^{l}p^{k}(m+n-k-l),&L=S.\end{array}\right.$
When $L=H$ or $K$, recall definitions (2.18)-(2.20) mentioned in Section 2.
Put
$\displaystyle\mathcal{V}^{L}=\left\\{V\in\mathfrak{V}^{L},\mbox{ satisfying
}J_{3}\mbox{ is neither single nor twinned}\right\\};$
$\displaystyle\mathcal{W}^{K}=\left\\{V\in\mathfrak{V}^{K},\mbox{ satisfying
}J_{3}\mbox{ is not twinned}\right\\}.$
Suppose $V\in\mathfrak{V}^{K}$ is isotropic. Put
$\mathcal{M}^{K}(1,V)=\oplus_{i\geq-2}\mathcal{M}_{i}^{K}(1,V),$
where
$\displaystyle\mathcal{M}^{K}_{-2}(1,V)=\mathbb{F};\ \
\mathcal{M}^{K}_{-1}(1,V)=V;$
$\displaystyle\mathcal{M}^{K}_{i}(1,V)=\\{u\in{K}_{i}\mid[V,u]\subset\mathcal{M}^{K}_{i-1}(1,V),\
[1,u]\in\mathcal{M}^{K}_{i-2}(1,V)\\},\ i\geq 0.$
###### Theorem 4.2.
All MGS of type $\mathrm{(II)}$ of $H$ and $K$ are characterized as follows:
For ${H}$,
* $\mathrm{(1)}$
All MGS of type $\mathrm{(II)}$ are precisely:
$\left\\{\mathcal{M}(V)\mid V\in\mathcal{V}^{H}\right\\}.$
* $\mathrm{(2)}$
For any $V$ and $V^{\prime}$ in $\mathcal{V}^{H}$,
$\mathcal{M}(V)\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(V^{\prime})\Longleftrightarrow
V{\cong}V^{\prime}.$
* $\mathrm{(3)}$
$H$ has exactly $\phi(r,n)$ isomorphism classes of MGS of type
$\mathrm{(II)}$, where
$\phi(r,n)=\left\\{\begin{array}[]{ll}8^{-1}(r+1)(r(n+2)^{2}+2n^{2}+6-r)-2,&n\
\mbox{ is odd};\\\ 8^{-1}(r+1)(r(n+2)^{2}+2n^{2}+8)-2,&n\ \mbox{ is
even}.\end{array}\right.$
* $\mathrm{(4)}$
For $V\in\mathcal{V}^{H}$, if $\beta$-$\mathrm{dim}V=(a,b,c,d),$ then
$\mathrm{dim}\mathcal{M}(V)=p^{m}2^{n}+p^{2a}2^{d}-p^{a+b}2^{c+d}(m-2b+n-d-2c+1)-2.$
For ${K}$,
* $(1^{\prime})$
All MGS of type $\mathrm{(II)}$ are precisely:
$\displaystyle\left\\{\mathcal{M}(V)\mid V\in\mathcal{V}^{K}\mbox{ is neither
nondegenerate nor isotropic}\right\\}$ $\displaystyle\cup$
$\displaystyle\left\\{\mathcal{M}(V)\mid V\in\mathcal{W}^{K}\mbox{ is
nondegenerate or isotropic}\right\\}$ $\displaystyle\cup$
$\displaystyle\left\\{\mathcal{M}^{K}(1,V)\mid V\in\mathcal{V}^{K}\mbox{ is
isotropic}\right\\}.$
* $(2^{\prime})$
For all MGS of type $\mathrm{(II)}$,
$\displaystyle\mathcal{M}(V)\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(V^{\prime})\Longleftrightarrow
V{\cong}V^{\prime},$
$\displaystyle\mathcal{M}^{K}(1,V)\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}^{K}(1,V^{\prime})\Longleftrightarrow
V{\cong}V^{\prime}.$
* $(3^{\prime})$
$K$ has exactly $\phi{(r,n)}$ isomorphism classes of MGS of type
$\mathrm{(II)}$, where
$\phi(r,n)=\left\\{\begin{array}[]{ll}8^{-1}(r+1)(r(n+2)^{2}+2n^{2}+4n+2-r)+r-1,&n\
\mbox{ is odd};\\\ 8^{-1}(r+1)(r(n+2)^{2}+2n^{2}+4n+8)+r-2,&n\ \mbox{ is
even}.\end{array}\right.$
* $(4^{\prime})$
Let $\delta=1$ when $n-m-3=0\pmod{p}$ and $\delta=0$, otherwise. Suppose
$\beta$-$\mathrm{dim}V=(a,b,c,d).$
* (a)
If $V$ is isotropic, then
$\displaystyle\mathrm{dim}\mathcal{M}(V)=p^{m}2^{n}-p^{b}2^{c}(m-2b+n-2c)-\delta,\mbox{
when }\ V\in\mathcal{W}^{K};$
$\displaystyle\mathrm{dim}\mathcal{M}^{K}(1,V)=p^{m}2^{n}-p^{b+1}2^{c}(m-2b+n-2c)+p-\delta,\mbox{
when }\ V\in\mathcal{V}^{K}.$
* (b)
If $V$ is not isotropic, then
$\mathrm{dim}\mathcal{M}(V)=p^{m}2^{n}-(2r-2a+n-d)p^{2a+1}2^{d}-p,$
when $V\in\mathcal{W}^{K}$ is nondegenerate satisfying $J_{3}$ is single;
$\mathrm{dim}\mathcal{M}(V)=p^{m}2^{n}+p^{2a+1}2^{d}-p^{a+b+1}2^{c+d}(m-2b+n-d-2c)-\delta,$
when $V\in\mathcal{W}^{K}$ is nondegenerate satisfying $J_{3}$ is not single
or $V\in\mathcal{V}^{K}$ is not nondegenerate.
###### Lemma 4.3.
Suppose $L=W,S,H$ or $K$.
* $\mathrm{(1)}$
$\mathcal{M}(V)$ is a $\mathbb{Z}$-graded subalgebra of $L$.
$\mathcal{M}^{K}(1,V)$ is a $\mathbb{Z}$-graded subalgebra of $K$.
* $\mathrm{(2)}$
Suppose $\Phi$ is a $\mathbb{Z}$-homogeneous automorphism of $L$. Then
$\Phi(\mathcal{M}_{i}(V))=\mathcal{M}_{i}(\Phi(V))\ \mbox{ for all }\
i\geq-2.$
Moreover,
$\Phi(\mathcal{M}(V))=\mathcal{M}(\Phi(V)).$
For $K$,
$\Phi(\mathcal{M}^{K}_{i}(1,V))=\mathcal{M}^{K}_{i}(1,\Phi(V))\ \mbox{ for all
}\ i\geq-2.$
Moreover,
$\Phi(\mathcal{M}^{K}(1,V))=\mathcal{M}^{K}(1,\Phi(V)).$
* $\mathrm{(3)}$
If $M$ is an MGS of type $\mathrm{(II)}$ of $L$, then $M=\mathcal{M}(M_{-1})$
unless $L=K$, $M_{-1}$ is isotropic and $M_{-2}\not=0$. However, in the latter
case, $M=\mathcal{M}^{K}(1,M_{-1})$.
###### Proof.
The approach is analogous to that used in the case of modular Lie algebras
[16]. ∎
###### Remark 4.4.
Suppose $L=W,S,H$ or $K$. In view of Lemmas 2.2 and 4.3, for any
$V\in\mathfrak{V}^{L}$, we may assume that $V$ is standard [see (2.17),
(2.20)].
Now, we consider the case $L=W$ or $S$. Suppose $V\in\mathfrak{V}^{L}$ with
$\mathrm{superdim}V=(k,l)$. For $L=W$, it is easy to verify that
$\mathcal{M}_{0}(V)$ has a standard $\mathbb{F}$-basis
$\mathcal{A}_{1}\cup\mathcal{A}_{2},$ where
$\displaystyle\mathcal{A}_{1}=\\{x_{i}\partial_{j}\mid
i,j\in\mathbf{I}(k,l)\\},$
$\displaystyle\mathcal{A}_{2}=\\{x_{i}\partial_{j}\mid
i\in\overline{\mathbf{I}}(k,l),j\in\mathbf{I}\\}.$
Similarly, for $L=S$, $\mathcal{M}_{0}(V)$ has a standard $\mathbb{F}$-basis
$\mathcal{C}_{1}\cup\mathcal{C}_{2}\cup\mathcal{C}_{3}$, where
$\displaystyle\mathcal{C}_{1}=\\{x_{i}\partial_{j}\mid
i,j\in\mathbf{I}(k,l),i\neq j\\},$
$\displaystyle\mathcal{C}_{2}=\\{x_{i}\partial_{j}\mid
i\in\overline{\mathbf{I}}(k,l),j\in\mathbf{I},i\neq j\\},$
$\displaystyle\mathcal{C}_{3}=\\{x_{1}\partial_{1}-(-1)^{|\partial_{i}|}x_{i}\partial_{i}\mid
i\in\mathbf{I}\backslash\\{1\\}\\}.$
Moreover, in any case of $L=W$ or $S$, $\mathcal{M}_{0}(V)$ has a standard co-
basis in $W_{0}$:
$\mathcal{A}_{3}=\\{x_{i}\partial_{j}\mid
i\in\mathbf{I}(k,l),j\in\overline{\mathbf{I}}(k,l)\\}.$ (4.27)
###### Lemma 4.5.
Suppose $U$, $V\in\mathfrak{V}^{L}$, $L=W$ or $S$.
* $\mathrm{(1)}$
$\mathcal{M}_{0}(V)$ is a maximal subalgebra of $L_{0}$.
* $\mathrm{(2)}$
$\mathcal{M}_{0}(U)=\mathcal{M}_{0}(V)$ if and only if $U=V.$
###### Proof.
(1) Let $\mathfrak{G}_{0}$ be a subalgebra of $L_{0}$ which strictly contains
$\mathcal{M}_{0}(V)$. It is clear that $\mathfrak{G}_{0}$ contains a nonzero
element of form $B=\sum_{h,t\geq 1}\alpha_{ht}x_{i_{h}}\partial_{j_{t}},$
where
$0\neq\alpha_{ht}\in\mathbb{F},i_{h}\in\mathbf{I}(k,l),j_{t}\in\overline{\mathbf{I}}(k,l)$.
When $L=W$, for any $i\in\mathbf{I}(k,l)$ and
$j\in\overline{\mathbf{I}}(k,l),$ one has
$x_{i}\partial_{i_{1}}\in\mathcal{A}_{1}$ and
$x_{j_{1}}\partial_{j}\in\mathcal{A}_{2}$. Then
$x_{i}\partial_{j}=\alpha_{11}^{-1}[x_{i}\partial_{i_{1}},[B,x_{j_{1}}\partial_{j}]]\in\mathfrak{G}_{0},$
showing that the co-basis $\mathcal{A}_{3}\subset\mathfrak{G}_{0}$. Hence
$\mathfrak{G}_{0}=W_{0}$.
When $L=S$, suppose $|\mathbf{I}(k,l)|>1$ and
$|\overline{\mathbf{I}}(k,l)|>1$. Choosing $x_{j_{1}}\partial_{j}$ in
$\mathcal{C}_{2}$ with $j\in\overline{\mathbf{I}}(k,l)\backslash\\{j_{1}\\}$
and $x_{i}\partial_{i_{1}}$ in $\mathcal{C}_{1}$ with
$i\in\mathbf{I}(k,l)\backslash\\{i_{1}\\}$, we have
$x_{i}\partial_{j}=[x_{i}\partial_{i_{1}},[B,x_{j_{1}}\partial_{j}]]\in\mathfrak{G}_{0},$
showing that the co-basis $\mathcal{A}_{3}\subset\mathfrak{G}_{0}$ and then
$\mathfrak{G}_{0}=S_{0}$. For the remaining case $|\mathbf{I}(k,l)|=1$ or
$|\overline{\mathbf{I}}(k,l)|=1$, the argument is similar and much easier.
(2) One direction is obvious. Note that one may choose bases of $U$ and $V$ as
follows:
$\overbrace{E_{1},\ldots,E_{r}}^{\textrm{cobasis
in}\;U},\overbrace{F_{1},\ldots,F_{s}}^{\textrm{basis of}\;U\cap
V},\overbrace{G_{1},\ldots,G_{t}}^{\textrm{cobasis in}\;V}$
where $(E_{1},\ldots,E_{r},F_{1},\ldots,F_{s},G_{1},\ldots,G_{t})$ is a
permutation of $\partial_{i}$’s. Keeping in mind the standard co-basis (4.27),
we are done by a similar argument as in (1). ∎
###### Proposition 4.6.
$\mathcal{M}(V)$ is maximal in $L$ for any $V\in\mathfrak{V}^{L}$, $L=W$ or
$S$.
###### Proof.
Let $M$ be an MGS containing $\mathcal{M}(V)$. Then $\mathcal{M}_{i}(V)\subset
M_{i}$ for all $i\geq-1.$ In particular, because of the maximality of
$\mathcal{M}_{0}(V),$ it must be $M_{0}=\mathcal{M}_{0}(V)$ or $M_{0}=L_{0}.$
Case 1. Suppose $M_{0}=\mathcal{M}_{0}(V)$. By induction, it is routine to
verify that $M_{i}=\mathcal{M}_{i}(V)$ for all $i\geq 0.$ Assume on the
contrary that $M$ strictly contains $\mathcal{M}(V)$. Then
$M_{-1}\supsetneq\mathcal{M}_{-1}(V)=V.$ Note that
$\mathcal{M}_{0}(V)=M_{0}=\mathcal{M}_{0}(M_{-1})$ from Lemma 4.3(3). Thus,
Lemma 4.5(2) forces $M_{-1}=L_{-1}$. Pick any $i\in\mathbf{I}(k,l)$,
$j\in\overline{\mathbf{I}}(k,l)$ and any $h\neq i,j.$ We are able to check
that
$A=(-1)^{|x_{h}|}x_{i}x_{j}\partial_{j}-(-1)^{|x_{j}|}x_{i}x_{h}\partial_{h}\in
S_{1}\subset W_{1}.$
Moreover, $A\in\mathcal{M}_{1}(V)=M_{1}.$ Since $M_{-1}=W_{-1}=S_{-1}$, we
have
$x_{i}\partial_{j}=(-1)^{(|x_{h}|+|x_{i}||x_{j}|)}[\partial_{j},A]\in
M_{0}=\mathcal{M}_{0}(V).$
This contradicts the fact that $x_{i}\partial_{j}\in\mathcal{A}_{3}$ [see
(4.27)]. Therefore, $M=\mathcal{M}(V).$
Case 2. Suppose $M_{0}=L_{0}$. In this case, since $L_{-1}$ is irreducible as
$L_{0}$-module, we have $M_{-1}=L_{-1}.$ Hence $M$ is an MGS of type (I). By
Theorem 3.1(1) and (2), $M_{1}=W_{1}^{\prime}$, $W_{1}^{\prime\prime},$
$S_{1}^{\prime\prime}$, or $\\{0\\}.$ In Case 1, we have shown that
$A\in\mathcal{M}_{1}(V)$. However, it is clear that $A$ does not belong to
$W_{1}^{\prime}$, $W_{1}^{\prime\prime},$ $S^{{}^{\prime\prime}}$, or
$\\{0\\}.$ Hence $\mathcal{M}_{1}(V)\not\subset M_{1}.$ This contradicts the
assumption that $M$ is a graded subalgebra containing $\mathcal{M}(V).$ ∎
Proof of Theorem 4.1 (1), (2) and (3) are immediate consequences of Lemmas
2.2, 4.3(3) and Proposition 4.6. It remains to show the dimension formulas.
For $W$, $\mathcal{M}(V)$ has a standard $\mathbb{F}$-basis which is a
disjoint union:
$\displaystyle\\{x^{(\alpha)}x^{u}\partial_{i}\mid\alpha\in\mathbf{A}(m),u\in\mathbf{B}(n);\;i\in\mathbf{I}(k,l)\\}$
$\displaystyle\cup$
$\displaystyle\\{x^{(\alpha)}x^{u}\partial_{i}\mid\;i\in\overline{\mathbf{I}}(k,l)\;\mbox{and}\;\exists
j\in\overline{\mathbf{I}}(k,l)\;\mbox{such
that}\;\partial_{j}(x^{(\alpha)}x^{u})\neq 0\\}.$
A standard and direct computation shows that:
$\mathrm{dim}\mathcal{M}(V)=2^{n}p^{m}(m+n)-2^{l}p^{k}(m+n-k-l).$
Similarly, for $S$, we have:
$\mathrm{dim}\mathcal{M}(V)=2^{n}p^{m}(m+n-1)+1-2^{l}p^{k}(m+n-k-l).$
∎
Next, we consider the case $L=H$ or $K$. In this case, we shall frequently use
the standard facts mentioned in Remark 2.3 without notice. Suppose
$V\in\mathfrak{V}^{L}$ with $\beta$-$\dim V=(a,b,c,d)$. In order to prove
Theorem 4.2 we list the following assertions. For simplicity, we write
$\lambda_{i,j}$ for a nonzero element in $\mathbb{F}$, where
$i,j\in\mathbf{I}$. Recall definitions (2.18)-(2.21). Put
$\displaystyle\mathcal{V}_{\mathfrak{i}}^{L}=\left\\{V\in\mathfrak{V}^{L}\mid
V\mbox{ is isotropic and }J_{3}\mbox{ is not twinned}\right\\};$
$\displaystyle\mathcal{V}_{\mathfrak{n}}^{L}=\left\\{V\in\mathfrak{V}^{L}\mid
V\mbox{ is nondegenerate and }J_{i}\mbox{ is not twinned, }i=1,3\right\\};$
$\displaystyle\mathcal{V}_{\mathfrak{d}}^{L}=\left\\{V\in\mathfrak{V}^{L}\mid
V\mbox{ is degenerate},J_{3}\mbox{ is empty and }J_{1}\mbox{ is not
twinned}\right\\}.$ (4.28)
###### Lemma 4.7.
For $H$, put
$A_{i}=\mathrm{span}_{\mathbb{F}}\\{u\in H_{i}\mid\nu(u)=0,1\mbox{ or
}(\frac{1}{3})^{k}2^{l},\ k,l\in\mathbb{N},\ l>1\\}.$
Then
* $(1)$
$A_{i}=\mathcal{M}_{i}(V)$, $i\geq-1$.
* $(2)$
The subalgebra $A_{0}=\mathcal{M}_{0}(V)$ is maximal in $H_{0}$ if and only if
$V\in\mathcal{V}_{\mathfrak{i}}^{H}\cup\mathcal{V}_{\mathfrak{n}}^{H}\cup\mathcal{V}_{\mathfrak{d}}^{H}.$
###### Proof.
(1) It follows by using induction on $i$, $i\geq-1$.
(2) Obviously, the torus $T$ mentioned in Remark 2.3(2) is contained in
$\mathcal{M}_{0}(V)$. For any $h\in H_{0}$ and $h\not\in\mathcal{M}_{0}(V)$,
put $\overline{M}=\mathrm{alg}(\mathcal{M}_{0}(V)+\mathbb{F}h)$. Firstly, we
show the maximality of $\mathcal{M}_{0}(V).$ It suffices to prove
$H_{0}=\overline{M}$.
Case 1. $V\in\mathcal{V}_{\mathfrak{i}}^{H}$. Notice that
$\nu(y_{i})=\left\\{\begin{array}[]{llll}0&i\in J_{2};\\\
\frac{1}{3}&i\in\bar{J}_{2};\\\ 2&i\in J_{3}\end{array}\right.\ \ \mbox{ and
}\ \ \mathcal{M}_{0}(V)=\mathrm{span}_{\mathbb{F}}\\{y_{i}y_{j}\mid(i,j)\in
J_{2}\times\mathbf{I}\cup J_{3}\times J_{3}\\}.$
We may assume that $h$ is a monomial with $\nu(h)=\frac{1}{9}$ or
$\frac{2}{3}$. When $h=y_{i}y_{j}$, $(i,j)\in\bar{J}_{2}\times\bar{J}_{2}$, we
have:
$\displaystyle
y_{k}y_{l}=\lambda_{k,l}[[y_{i}y_{j},y_{\widetilde{j}}y_{k}],y_{\widetilde{i}}y_{l}]\in\overline{M}\
\mbox{ for all }\ (k,l)\in\bar{J}_{2}\times\bar{J}_{2},$ $\displaystyle
y_{k}y_{s}=\lambda_{k,s}[y_{k}y_{l},y_{\widetilde{l}}y_{s}]\in\overline{M}\
\mbox{ for all }\ s\in{J}_{3}.$
Thus, $H_{0}=\overline{M}$. When $h=y_{i}y_{j}$,
$(i,j)\in\bar{J}_{2}\times{J}_{3}$, if $j\in I_{03}$ or $I_{03}$ is not empty,
we get $H_{0}=\overline{M}$ in an analogous way as above. Otherwise, we may
assume that $I_{03}$ is empty. If $J_{3}$ is single, we have:
$\displaystyle
y_{k}y_{m+n-d}=\lambda_{k,m+n-d}[y_{i}y_{m+n-d},y_{\widetilde{i}}y_{k}]\in\overline{M}\
\mbox{ for all }\ k\in\bar{J}_{2},$ $\displaystyle
y_{k}y_{l}=\lambda_{k,l}[y_{k}y_{m+n-d},y_{m+n-d}y_{l}]\in\overline{M}\ \mbox{
for all }\ l\in\bar{J}_{2}.$
It follows that $H_{0}=\overline{M}$. If $J_{3}$ is neither single nor
twinned, for $s\in J_{13}$, $s\not=\widetilde{j}$, we have:
$\displaystyle
y_{k}y_{s}=\lambda_{k,s}[[y_{i}y_{j},y_{\widetilde{i}}y_{k}],y_{\widetilde{j}}y_{s}]\in\overline{M}\
\mbox{ for all }\ k\in\bar{J}_{2},$ $\displaystyle
y_{k}y_{\widetilde{j}}=\lambda_{k,\widetilde{j}}[y_{k}y_{s},y_{\widetilde{s}}y_{\widetilde{j}}]\in\overline{M}\
\ s\not=j\mbox{ and }\widetilde{j},$ $\displaystyle
y_{k}y_{l}=\lambda_{k,l}[y_{j}y_{k},y_{\widetilde{j}}y_{l}]\in\overline{M}\
\mbox{ for all }l\in\bar{J}_{2}.$
Thus, $H_{0}=\overline{M}$.
Case 2. $V\in\mathcal{V}_{\mathfrak{n}}^{H}$. Notice that
$\nu(y_{i})=\left\\{\begin{array}[]{llll}1&i\in J_{1};\\\ 2&i\in
J_{3}\end{array}\right.\ \ \mbox{ and }\ \
\mathcal{M}_{0}(V)=\mathrm{span}_{\mathbb{F}}\\{y_{i}y_{j}\mid(i,j)\in
J_{1}\times J_{1}\cup J_{3}\times J_{3}\\}.$
We may assume that $h$ is a linear combination of monomials with value 2. When
$h=y_{i}y_{j}$, $(i,j)\in(I_{01}\cup\bar{I}_{01})\times{J}_{3}$, using the
same method as in Case 1, we get $H_{0}=\overline{M}$. When $h=\sum_{i\in
I_{11}}a_{i}y_{k}y_{i}$, where $k\in J_{3}$, $a_{i}\in\mathbb{F}$,
$a_{j}\not=0$, we get $H_{0}=\overline{M}$ if $I_{01}$ is not empty or $J_{1}$
is single by a similar argument as in Case 1. Thus, it suffices to consider
the condition that $I_{01}$ is empty and $J_{1}$ is neither single nor
twinned. For distinct $l,s,j\in I_{11}$, we have
$\displaystyle
y_{k}y_{s}=(a_{j})^{-1}\lambda_{k,s}[y_{l}y_{s},[y_{j}y_{l},h]]\in\overline{M},$
$\displaystyle y_{e}y_{k}=\lambda_{e,k}[y_{k}y_{s},y_{s}y_{e}]\in\overline{M}\
\mbox{ for any }\ s\not=e\in I_{11}.$
For any $i\in I_{11}$, $f\in I_{03}$ and $t\in I_{13}$, we have
$y_{f}y_{i}=\lambda_{f,i}[y_{k}y_{i},y_{\widetilde{k}}y_{f}]\in\overline{M}\
\mbox{ and }\
y_{t}y_{i}=\lambda_{t,i}[y_{f}y_{i},y_{\widetilde{f}}y_{t}]\in\overline{M}.$
Thus, $H_{0}=\overline{M}.$
Case 3. $V\in\mathcal{V}_{\mathfrak{d}}^{H}$. Notice that
$\nu(y_{i})=\left\\{\begin{array}[]{llll}1&i\in J_{1};\\\ 0&i\in J_{2};\\\
\frac{1}{3}&i\in\bar{J}_{2}\end{array}\right.\ \ \mbox{ and }\ \
\mathcal{M}_{0}(V)=\mathrm{span}_{\mathbb{F}}\\{y_{i}y_{j}\mid(i,j)\in{J}_{2}\times\mathbf{I}\cup
J_{1}\times{J}_{1}\\}.$
We have $H_{0}=\overline{M}$ by the same method as in Cases 1 and 2.
Conversely, we consider the co-basis of $\mathcal{M}_{0}(V)$ in $H_{0}$:
$\\{y_{i}y_{j}\mid(i,j)\in{J}_{1}\times\bar{J}_{2}\cup
J_{1}\times{J}_{3}\cup\bar{J}_{2}\times\bar{J}_{2}\cup\bar{J}_{2}\times{J}_{3}\\}.$
(4.29)
Notice that if
$V\not\in\mathcal{V}_{\mathfrak{i}}^{H}\cup\mathcal{V}_{\mathfrak{n}}^{H}\cup\mathcal{V}_{\mathfrak{d}}^{H}$
, then $V\in\mathfrak{V}^{H}$ must satisfy one of the following conditions:
(i) None of $J_{1},J_{2},J_{3}$ is empty. In this case, we choose a monomial
$h$ of $H_{0}$ with $\nu(h)=2$. Then there do not exist monomials with value
$\frac{1}{3}$ in $\overline{M}$.
(ii) $J_{3}$ is twinned, i.e., $J_{3}=\\{j,{\widetilde{j}}\\}$, where
$j\not=\widetilde{j}\in\mathbf{I}_{1}$. In this case, let $h=y_{i}y_{j}$,
where
$i\in\left\\{\begin{array}[]{ll}\bar{J}_{2},&J_{1}\ \mbox{ is empty};\\\
{J}_{1},&\ \mbox{ otherwise}.\end{array}\right.$
Then $y_{i}y_{\widetilde{j}}\not\in\overline{M}$.
(iii) $J_{1}$ is twinned, i.e., $J_{1}=\\{m+n-1,m+n\\}$. In this case, let
$h=y_{k}(y_{m+n-1}+\sqrt{-1}y_{m+n})$, where
$k\in\left\\{\begin{array}[]{ll}\bar{J}_{2},&J_{3}\ \mbox{ is empty};\\\
{J}_{3},&\ \mbox{ otherwise}.\end{array}\right.$
Then $y_{k}y_{m+n}\not\in\overline{M}$.
Therefore, $\overline{M}$ is a nontrivial subalgebra of $H_{0}$ strictly
containing $\mathcal{M}_{0}(V)$ when (i), (ii) or (iii) holds, which implies
that $\mathcal{M}_{0}(V)$ is not a maximal subalgebra of $H_{0}$. ∎
###### Proposition 4.8.
The subalgebra $\mathcal{M}(V)$ is maximal in $H$ if and only if
$V\in\mathcal{V}^{H}$.
###### Proof.
If $J_{3}=\\{m+n-d\\}$, from Lemma 4.7(1), we know that
$\mathcal{M}_{i}(V)=\mathrm{span}_{\mathbb{F}}\\{u\in H_{i}\mid\nu(u)=1,0\\},\
i\geq-1,$
which implies that
$\mathrm{alg}(\mathcal{M}(V)+\mathbb{F}y_{m+n-d})\subset\mathrm{span}_{\mathbb{F}}\\{u\in
H\mid\nu(u)=1,0\\}+\mathbb{F}y_{m+n-d}.$
Thus,
$y_{i},y_{j}y_{m+n-d}\not\in\mathrm{alg}(\mathcal{M}(V)+\mathbb{F}y_{m+n-d})$
if $\nu(y_{i})={\frac{1}{3}}$ or $\nu(y_{j})=1,$ which contradicts the
maximality of $\mathcal{M}(V)$.
If $J_{3}=\\{j,{\widetilde{j}}\\}$, where
$j\not=\widetilde{j}\in\mathbf{I}_{1}$, from Lemma 4.7(1), we know that
$\mathcal{M}_{i}(V)=\mathrm{span}_{\mathbb{F}}\\{u\in
H_{i}\mid\nu(u)=1,0,(\frac{1}{3})^{k}4\\},\ i\geq 0.$
Then for any monomial $u\in\mathcal{M}_{i}(V)$, we have:
$[y_{j},u]=0\ \ \mbox{ or }\ \ [y_{j},u]=wy_{j},$
where $0\not=w\in H_{i-1}$ with $D_{\widetilde{j}}(w)=0$, which implies that
$y_{\widetilde{j}}\not\in\mathrm{alg}(\mathcal{M}(V)+\mathbb{F}y_{j}).$
This contradicts the maximality of $\mathcal{M}(V)$.
Conversely, let us prove the maximality of $\mathcal{M}(V)$. By definition
(4.26), it is sufficient to show that
$\overline{M}=\mathrm{alg}(\mathcal{M}(V)+\mathbb{F}h)=H$, where $h=y_{i}$,
$i\in\bar{J}_{2}\cup J_{3}$. Note that $\mathcal{M}_{1}(V)\not=0$ for
$|\mathbf{I}_{0}|\geq 2$. From Lemmas 2.1(2) and 3.11(1), it suffices to prove
$H_{-1},H_{0}\subset\overline{M}$. For $V\in\mathcal{V}^{H}$, we discuss the
following cases:
Case 1. $J_{2}$ is not empty. When $i\in\bar{J}_{2}$, since
$y_{\widetilde{i}}\in V\ \mbox{ and }\
y_{j}=\lambda_{i,j}[y_{i},y_{\widetilde{i}}y_{j}]\in\overline{M}\ \mbox{ for
}\ \widetilde{i}\not=j\in\mathbf{I},$
we have $H_{-1}\subset\overline{M}$. When $i\in{J}_{3}$, for all $j\in J_{3}$
with $j\not=i,\widetilde{i}$, we have:
$y_{j}=\lambda_{i,j}[y_{i},y_{\widetilde{i}}y_{j}]\ \ \mbox{ and }\ \
y_{\widetilde{i}}=\lambda_{\widetilde{i},j}[y_{j},y_{\widetilde{i}}y_{\widetilde{j}}].$
Note that
$y_{l}=\lambda_{l,i}[y_{i},[y_{\widetilde{i}},y_{i}y_{\widetilde{i}}y_{l}]]\in\overline{M}\
\mbox{ for all }\ l\in\bar{J}_{2}.$
Thus we have $H_{-1}\subset\overline{M}$. Note that for an arbitrary monomial
$u\in H_{0}$, there exists $k\in\mathbf{I}$ such that $uy_{k}\not=0$ and
$\nu(uy_{k})=0$. Then we have
$u=\lambda_{\widetilde{k},k}[y_{\widetilde{{k}}},uy_{k}]\in\overline{M},$
which implies that $H_{0}\subset\overline{M}$. Thus, we have $\overline{M}=H$.
Case 2. $J_{2}$ is empty. Obviously, $J_{1}$ and $J_{3}$ are not empty. Then
we have $H_{-1},H_{0}\subset\overline{M}$ by the same method as in Case 1. ∎
To avoid confusion, we rewrite $\mathcal{M}^{L}_{i}(V)$ for
$\mathcal{M}_{i}(V)$, $\mathcal{M}^{L}(V)$ for $\mathcal{M}(V)$, $L=H$ or $K$.
###### Lemma 4.9.
Let $\gamma=\lfloor\frac{i+2}{2}\rfloor$ for $i>0$. Put
$\widetilde{\mathcal{M}}_{j}(V)=\left\\{\begin{array}[]{ll}0,&j>\eta-2;\\\
\mathcal{M}^{H}_{j}(V),&j<\eta-2;\end{array}\right.\
\widetilde{\mathcal{M}}_{\eta-2}(V)=\left\\{\begin{array}[]{ll}0,&V\mbox{ is
nondegenerate}\\\ &\mbox{and}\ J_{3}\ \mbox{is single};\\\
\mathbb{F}y^{(\pi)}y^{\omega},&\mbox{otherwise, }\end{array}\right.$
where $\pi=(p-1,\ldots,p-1)\in\mathbb{N}^{2r}$, $\eta=2r(p-1)+n$ and
$\omega=\langle m+1,\ldots,m+n\rangle$. Then
* $\mathrm{(1)}$
$\mathcal{M}^{K}_{0}(V)=\mathcal{M}^{H}_{0}(V)\oplus\mathbb{F}z$.
* $\mathrm{(2)}$
If $V$ is not isotropic, for $i>0$,
$\mathcal{M}^{K}_{i}(V)=\widetilde{\mathcal{M}}_{i}(V)\oplus\widetilde{\mathcal{M}}_{i-2}(V)z\oplus\cdots\oplus\widetilde{\mathcal{M}}_{i-2\gamma}(V)z^{\gamma}.$
* $\mathrm{(3)}$
If $V$ is isotropic, for $i>0$,
$\mathcal{M}^{K}_{i}(V)=\widetilde{\mathcal{M}}_{i}(V)\oplus\overline{H}_{i-2}z\oplus\cdots\oplus\overline{H}_{i-2\gamma}z^{\gamma}.$
###### Proof.
(1) It is obvious.
(2) Use induction on $i$. Clearly,
$\widetilde{\mathcal{M}}_{i}(V)\subset\mathcal{M}^{K}_{i}(V)$.
$``\supset"$: For $gz^{k}\in\widetilde{\mathcal{M}}_{i-2k}(V)z^{k}$,
$0<k\leq\gamma$, we know that
$[y_{l},gz^{k}]=[y_{l},g]z^{k}+y_{l}gz^{k-1}.$
Note that
$y_{l}g\in\widetilde{\mathcal{M}}_{i-2(k-1)-1}(V)\ \mbox{ for
}\nu(y_{l})=1,0.$
By induction on $\mathrm{zd}(g)$, we have:
$y_{l}gz^{k-1},\ [y_{l},g]z^{k}\in\mathcal{M}^{K}_{i-1}(V).$
Thus, $gz^{k}\in\mathcal{M}^{K}_{i}(V)$.
$``\subset"$: For any $u\in\mathcal{M}^{K}_{i}(V)$, by Lemma 3.11(2), we may
assume that
$u=u_{i}+u_{i-2}z+\cdots+u_{i-2\gamma}z^{\gamma},$
where $u_{j}\in\overline{H}_{j}$ for $i-2\gamma\leq j\leq i$. Note that
$\mathcal{M}^{K}_{-2}(V)\not=0$, since $V$ is not isotropic. Then we have:
$u_{i-2}+u_{i-4}z+\cdots+u_{i-2\gamma}z^{\gamma-1}=2^{-1}[1,u]\in\mathcal{M}^{K}_{i-2}(V).$
By induction, we have $u_{j}\in\widetilde{\mathcal{M}}_{j}(V)$ for
$i-2\gamma\leq j\leq i-2$. Moreover,
$u_{i-2}z+u_{i-4}z^{2}+\cdots+u_{i-2\gamma}z^{\gamma}\in\mathcal{M}^{K}_{i}(V).$
Consequently, $u_{i}\in\widetilde{\mathcal{M}}_{i}(V)$.
(3) When $V$ is isotropic, note that $\nu(y_{k})=0$ for all $y_{k}\in V$. The
remaining discussion is analogous to that of the condition (2). ∎
###### Proposition 4.10.
The subalgebra $\mathcal{M}^{K}(V)$ is maximal in $K$ if and only if
$V\in\mathcal{V}^{K}$ when $V$ is neither nondegenerate nor isotropic;
$V\in\mathcal{W}^{K}$, otherwise.
###### Proof.
The proof of the necessity is similar to the one of Proposition 4.8. We only
consider the sufficiency. For any $u\in K$, $u\not\in\mathcal{M}^{K}(V)$, put
$\overline{M}=\mathrm{alg}(\mathcal{M}^{K}(V)+\mathbb{F}u)$. Then there exist
$v_{1},\ldots,v_{i}\in V$ such that
$0\not=h=[v_{1},[\cdots,[v_{i},u]\cdots]]\in\overline{M}\cap K_{-1}\ \mbox{
and }\ h\not\in V.$
When $J_{3}$ is neither single nor twinned, by Proposition 4.8, we have
$H\subset\overline{M}.$ When $J_{3}$ is single and $V$ is isotropic, we may
assume that $h=y_{i}$, $i\in\bar{J}_{2}\cup\\{m+n-d\\}$. If $i\in\bar{J}_{2}$,
for any $j,k\in\bar{J}_{2}$, we obtain that
$y_{m+n-d}=\lambda_{i,\widetilde{i}}[y_{\widetilde{i}},[y_{i},y_{m+n-d}z]],\ \
y_{m+n-d}y_{j}y_{k}=\lambda_{j,k}[y_{m+n-d},y_{j}y_{k}z]$
are in $\overline{M}$ from Lemma 4.9(3). Moreover,
$y_{j}y_{k}=-[y_{m+n-d},y_{m+n-d}y_{j}y_{k}],\ \
y_{m+n-d}y_{k}=\lambda_{m+n-d,k}[y_{\widetilde{j}},y_{m+n-d}y_{j}y_{k}]$
are in $\overline{M}$. Keeping in mind the co-basis (4.29), we have
$H_{0}\subset\overline{M}$, which also holds when $J_{3}$ is single and $V$ is
nondegenerate. From the irreducibility of $K_{-1}$, $K_{10}$ and $K_{11}$, as
well as Lemma 4.9, we obtain that $\mathcal{M}^{K}(V)$ is maximal in $K$. ∎
In the same way as in Proposition 4.10, one may check the following
proposition.
###### Proposition 4.11.
If $V$ is isotropic, the subalgebra $\mathcal{M}^{K}(1,V)$ is maximal in $K$
if and only if $V\in\mathcal{V}^{K}$.
###### Convention 4.12.
For simplicity, put
$\mathcal{O}_{X}=\mathrm{span}_{\mathbb{F}}\\{y_{i_{1}}\cdots y_{i_{s}}\mid
i_{1},\ldots,i_{s}\in X,s\geq 1\\}$,
$\mathcal{Q}_{X}=\mathrm{span}_{\mathbb{F}}\\{y_{i}\mid i\in X\\}$ and
$\mathcal{Y}^{+}_{X}=\mathcal{Y}_{X}\oplus\mathbb{F}\cdot 1$, where $X$ is a
subset of $\mathbf{I}$ and $\mathcal{Y}=\mathcal{O}$ or $\mathcal{Q}$.
Proof of Theorem 4.2 For (1) and $(1^{\prime})$, the proofs follow from Lemma
4.3(3), Propositions 4.8, 4.10 and 4.11.
For $H$, from Lemma 4.7(1) and (1), we obtain that
$\dim{\mathcal{M}^{H}(V)}=\dim
H-\dim(\mathcal{O}^{+}_{J_{1}}\mathcal{O}_{\bar{J}_{2}}\oplus\mathcal{O}^{+}_{J_{1}\cup\bar{J}_{2}}\mathcal{Q}_{{J}_{3}}).$
For $K$, from Lemma 4.9 and $(1^{\prime})$, we obtain that
$\displaystyle\dim{\mathcal{M}^{K}(1,V)}=p(\dim\mathcal{M}^{H}(V)+2);$
$\displaystyle\dim{\mathcal{M}^{K}(V)}=\left\\{\begin{array}[]{ll}\dim\mathcal{M}^{H}(V)+1+(p-1)(\dim
H+2),&V\in\mathcal{W}^{K}\mbox{ is isotropic};\\\
p(\dim\mathcal{M}^{H}(V)+2),&\mbox{ otherwise}.\end{array}\right.$
By a standard and direct computation we get the formulas (4) and
$(4^{\prime})$. Noting that
$\dim\mathcal{M}_{0}(V)=\dim\mathcal{M}_{0}(V^{\prime})$ if
$\mathcal{M}(V)\cong$$\mathcal{M}(V^{\prime})$ and using the same method as in
Theorem 4.1(2), (2) and $(2^{\prime})$ hold. From (1), $(1^{\prime})$ and (2),
$(2^{\prime})$, we obtain that (3) and $(3^{\prime})$ hold.∎
## 5 MGS of Type (III)
Suppose $L=W,S,H$ or $K$. Recall that an MGS of type (III) of $L$, $M$,
satisfies the condition
$M_{-1}=L_{-1}\quad\mbox{and}\quad M_{0}\neq L_{0}.$
Let $\mathfrak{G}_{0}$ be a nontrivial subalgebra of $L_{0}$. Define a graded
subspace of $L$ as follows:
$\mathcal{M}(L_{-1},\mathfrak{G}_{0})=\oplus_{i\geq-2}\mathcal{M}_{i}(L_{-1},\mathfrak{G}_{0}),$
where
$\displaystyle\mathcal{M}_{-i}(L_{-1},\mathfrak{G}_{0})=L_{-i},\
i<0;\quad\mathcal{M}_{0}(L_{-1},\mathfrak{G}_{0})=\mathfrak{G}_{0};$
$\displaystyle\mathcal{M}_{i}(L_{-1},\mathfrak{G}_{0})=\\{u\in
L_{i}\mid[L_{-1},u]\subset\mathcal{M}_{i-1}(L_{-1},\mathfrak{G}_{0})\\}\
\mbox{ for }i>0.$ (5.30)
It is easy to see that $\mathcal{M}(L_{-1},\mathfrak{G}_{0})$ is a graded
subalgebra satisfying the condition (III). We call $\mathfrak{G}$ a maximal
R-subalgebra (resp. maximal S-subalgebra) of $L$ if $\mathfrak{G}$ is maximal
reducible (resp. irreducible) graded and satisfies the condition (III). All
the MGS of type (III) can be split into the disjoint union of maximal
R-subalgebras and maximal S-subalgebras.
###### Theorem 5.1.
Suppose $L=W$ or $S$.
* $\mathrm{(1)}$
All maximal R-subalgebras of $L$ are precisely:
$\\{\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))\mid V\in\mathfrak{V}^{L}\\}.$
* $\mathrm{(2)}$
For any $V,V^{\prime}\in\mathfrak{V}^{L}$,
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(L_{-1},\mathcal{M}_{0}(V^{\prime}))\Longleftrightarrow
V{\cong}V^{\prime}.$
* $\mathrm{(3)}$
$L$ has exactly $(m+1)(n+1)-2$ isomorphism classes of maximal R-subalgebras.
* $\mathrm{(4)}$
Suppose $V\in\mathfrak{V}^{L}$ with $\mathrm{superdim}V=(k,l),$ then
$\displaystyle\mathrm{dim}\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2^{n-l}p^{m-k}(m+n-k-l)+2^{n}p^{m}(k+l),&L=W\\\
2^{n-l}p^{m-k}(m+n)+2^{n}p^{m}(k+l-1)-k-1,&L=S.\end{array}\right.$
Let $L=H$ or $K$ and put $V^{\bot}=\\{u\in L_{-1}\mid\beta(u,V)=0\\}$ for
$V\in\mathfrak{V}^{L}$. Recall definitions (2.18)–(2.20) and (4.28).
###### Theorem 5.2.
All maximal R-subalgebras of $H$ and $K$ are characterized as follows:
For $H$,
* $(1)$
All maximal R-subalgebras of $H$ are precisely:
$\left\\{\mathcal{M}(H_{-1},\mathcal{M}_{0}(V))\mid
V\in\mathcal{V}_{\mathfrak{n}}^{H}\cup\mathcal{V}_{\mathfrak{i}}^{H}\right\\}.$
* $(2)$
Suppose
$V,V^{\prime}\in\mathcal{V}_{\mathfrak{n}}^{H}\cup\mathcal{V}_{\mathfrak{i}}^{H}$.
Then
$\mathcal{M}(H_{-1},\mathcal{M}_{0}(V))\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(H_{-1},\mathcal{M}_{0}(V^{\prime}))$
if and only if one of the following conditions holds.
* (i)
$V{\cong}V^{\prime}.$
* (ii)
$V^{\bot}{\cong}V^{\prime}$ when $V$ and $V^{\prime}$ are both nondegenerate.
* $(3)$
$H$ has exactly $\phi(r,n)$ isomorphism classes of maximal R-subalgebras,
where
$\phi(r,n)=\left\\{\begin{array}[]{ll}2^{-1}(nr+3n+2r-2)+\lfloor\frac{r}{2}\rfloor(n+1),&n\
\mbox{ is even};\\\ 2^{-1}(nr+3n+r-1)+\lfloor\frac{r}{2}\rfloor(n+1),&n\
\mbox{ is odd}.\end{array}\right.$
* $(4)$
Suppose $V\in\mathcal{V}_{\mathfrak{n}}^{H}\cup\mathcal{V}_{\mathfrak{i}}^{H}$
with $\beta$-$\mathrm{dim}V=(a,b,c,d),$ then
$\dim\mathcal{M}(H_{-1},\mathcal{M}(V))=\left\\{\begin{array}[]{ll}p^{2a}2^{d}+p^{2(r-a)}2^{(n-d)}-2,&V\in\mathcal{V}_{\mathfrak{n}}^{H};\\\
p^{m-b}2^{n-c}+(b+c)p^{b}2^{c}-1,&V\in\mathcal{V}_{\mathfrak{i}}^{H}.\end{array}\right.$
For $K$,
* $(1^{\prime})$
All maxima R-subalgebras of $K$ are precisely:
$\left\\{\mathcal{M}(K_{-1},\mathcal{M}_{0}(V))\mid
V\in\mathcal{V}_{\mathfrak{i}}^{K}\right\\}.$
* $(2^{\prime})$
Suppose $V,V^{\prime}\in\mathcal{V}_{\mathfrak{i}}^{K}$. Then
$\mathcal{M}(K_{-1},\mathcal{M}_{0}(V))\stackrel{{\scriptstyle\mathrm{algebra}}}{{\cong}}\mathcal{M}(K_{-1},\mathcal{M}_{0}(V^{\prime}))\Longleftrightarrow
V{\cong}V^{\prime}.$
* $(3^{\prime})$
$K$ has exactly $\phi(r,n)$ isomorphism classes of maximal R-subalgebras,
where
$\phi(r,n)=\left\\{\begin{array}[]{ll}2^{-1}(rn+n+2r-2),&n\ \mbox{ is
even};\\\ 2^{-1}(rn+n+r-1),&n\ \mbox{ is odd}.\end{array}\right.$
* $(4^{\prime})$
Suppose $V\in\mathcal{V}_{\mathfrak{i}}^{K}$ with
$\beta$-$\mathrm{dim}V=(0,b,c,0),$ then
$\displaystyle\dim\mathcal{M}(K_{-1},\mathcal{M}_{0}(V))=\left\\{\begin{array}[]{ll}p^{b+1}2^{c}(b+c+1),&J_{3}\
\mbox{ is empty };\\\ p^{b}2^{c}(p^{m-2b}2^{n-2c}+b+c),&\mbox{
otherwise}.\end{array}\right.$
Unfortunately, for maximal S-subalgebras, we have not obtained a similar
description as for the maximal graded subalgebras of type $(\mathrm{I})$ or
$(\mathrm{II})$ as well as for the maximal R-subalgebras. However, the
classification of maximal S-subalgebras of $L$ can be reduced to that of the
maximal irreducible subalgebras of the classical Lie superalgebras (see Lemma
2.1(3)).
###### Theorem 5.3.
Suppose $L=W,S,H$ or $K$. All maximal S-subalgebra of $L$ are characterized as
follows:
Every maximal S-subalgebra of $L$ is of the form
$\mathcal{M}(L_{-1},\mathfrak{G}_{0}),$ where $\mathfrak{G}_{0}$ is a maximal
irreducible subalgebra of $L_{0}$.
Suppose $\mathfrak{G}_{0}$ is a maximal irreducible subalgebra of $L_{0}$.
* $\mathrm{(a)}$
For $L=W$, $\mathcal{M}(W_{-1},\mathfrak{G}_{0})$ is maximal in $W$ if and
only if
$\mathrm{div}(\mathcal{M}_{1}(W_{-1},\mathfrak{G}_{0}))\neq 0.$
* $\mathrm{(b)}$
For $L=S$ or $H$, $\mathcal{M}(L_{-1},\mathfrak{G}_{0})$ is maximal in $L$ if
and only if
$\mathcal{M}_{1}(L_{-1},\mathfrak{G}_{0})\neq 0.$
* $\mathrm{(c)}$
For $L=K$, $\mathcal{M}(K_{-1},\mathfrak{G}_{0})$ is a maximal in $K$ if and
only if there exists $u\in\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})$ satisfying
$[1,u]\not=0.$
Let $L=W,S,H$ or $K$. As in the case of modular Lie algebras [16], it is easy
to show the following lemmas.
###### Lemma 5.4.
Let $\mathfrak{G}_{0}$ be a nontrivial subalgebra of $L_{0}$. If $\Phi$ is a
$\mathbb{Z}$-homogeneous automorphism of $L$. Then
$\Phi(\mathcal{M}_{i}(L_{-1},\mathfrak{G}_{0}))=\mathcal{M}_{i}(L_{-1},\Phi(\mathfrak{G}_{0}))\
\mbox{ for all }i\geq-2.$
Moreover,
$\Phi(\mathcal{M}(L_{-1},\mathfrak{G}_{0}))=\mathcal{M}(L_{-1},\Phi(\mathfrak{G}_{0})).$
###### Lemma 5.5.
Let $M=L_{-1}+M_{0}+M_{1}+M_{2}+\cdots$ be any MGS of $L$. Then $M_{0}$ is
maximal in $L_{0}$ unless $M_{0}=L_{0}$.
###### Lemma 5.6.
If $M$ is an MGS of type $\mathrm{(III)}$ of $L$ then $M_{0}$ is maximal in
$L_{0}$ and $M=\mathcal{M}(L_{-1},M_{0})$.
###### Lemma 5.7.
If $\mathfrak{G}_{0}$ is a maximal reducible subalgebra of $L_{0}$ then there
exists a $V\in\mathfrak{V}^{L}$ such that
$\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$ and
$\mathcal{M}_{i}(L_{-1},\mathfrak{G}_{0})\subset\mathcal{M}_{i}(V)$ for $i\geq
0$. Conversely, $\mathcal{M}_{0}(V)$ is a reducible maximal subalgebra of
$L_{0}$ if $V\in\mathfrak{V}^{L}$ when $L=W$ or $S$; if
$V\in\mathcal{V}_{\mathfrak{n}}^{L}\cup\mathcal{V}_{\mathfrak{i}}^{L}\cup\mathcal{V}_{\mathfrak{d}}^{L}$
when $L=H$ or $K$.
###### Proof.
Since $\mathfrak{G}_{0}$ is reducible, $L_{-1}$ has a nontrivial
$\mathfrak{G}_{0}$-submodule $V$. From definition (4.26) and the maximality of
$\mathfrak{G}_{0}$, we have $\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$. From
definitions (4.26) and (5.30), we have
$\mathcal{M}_{i}(L_{-1},\mathfrak{G}_{0})\subset\mathcal{M}_{i}(V)$ for $i\geq
0$. The second statement follows immediately from Lemmas 4.5(1) and 4.7(2). ∎
###### Remark 5.8.
In view of Lemmas 2.2, 5.4 and 5.7, if $\mathfrak{G}_{0}$ is a maximal
reducible subalgebra of $L_{0}$, we may assume that $V$ is a standard element
in $\mathfrak{V}^{L}$ [see (2.17), 2.20)] such that
$\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$.
###### Proposition 5.9.
Suppose $L=W$ or $S$. $\mathcal{M}(L_{-1},\mathfrak{G}_{0})$ is a maximal
R-subalgebra, if $\mathfrak{G}_{0}$ is a maximal reducible subalgebra of
$L_{0}$.
###### Proof.
Let us show that $M=\mathcal{M}(L_{-1},\mathfrak{G}_{0})$ is maximal. Assume
that $\overline{M}$ is a maximal graded subalgebra containing $M$. Clearly,
$\overline{M}_{-1}=L_{-1}$. Since $\mathfrak{G}_{0}$ is a maximal subalgebra
of $L_{0}$, we have $\overline{M}_{0}=\mathfrak{G}_{0}$ or $L_{0}$. If
$\overline{M}_{0}=\mathfrak{G}_{0}$ then $\overline{M}$ is an MGS of type
(III). By Lemma 5.6,
$\overline{M}=\mathcal{M}(L_{-1},\mathfrak{G}_{0})=M$
and we are done. Let us consider the remaining case; $\overline{M}_{0}=L_{0}$.
Clearly, $\overline{M}$ is an MGS of type (I) and by Theorem 3.1,
$M_{1}\subset\overline{M}_{1}=W_{1}^{\prime},W_{1}^{\prime\prime},S_{1}^{\prime\prime},\;\mbox{or}\;\\{0\\}.$
(5.31)
On the other hand, by Lemma 5.7, there exists a $V\in\mathfrak{V}^{L}$ such
that $\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$. Assume that $V$ has a standard
basis:
$(\partial_{1},\ldots,\partial_{k}\mid\partial_{m+1},\ldots,\partial_{m+l}).$
Hence $\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$ has a standard co-basis (4.27) in
$W_{0}$:
$\mathcal{A}_{3}=\\{x_{i}\partial_{j}\mid
i\in\mathbf{I}(k,l),j\in\overline{\mathbf{I}}(k,l)\\}.$
To reach a contradiction, in view of (5.31), it is sufficient to find an
element belonging to $M_{1}$ but not $W^{\prime},W^{\prime\prime}$ for $W$,
but not $S^{\prime\prime}$ or $\\{0\\}$ for $S$. For $L=W$,
$x_{j}x_{i}\partial_{i}$ with $i\in\mathbf{I}(k,l)$ and an arbitrarily chosen
$j$ is a desired element. Here we have used the fact that both
$|\mathbf{I}(k,l)|\geq 1$ and $|\overline{\mathbf{I}}(k,l)|\geq 1$, since
$V\in\mathfrak{V}^{W}$. For $L=S$, pick distinct $i,j,r$ with
$i\in\mathbf{I}(k,l)$ and with $j,r$ chosen arbitrarily. Here note that the
general assumption ensures $|\mathbf{I}|\geq 4.$ Then
$x_{j}x_{r}\partial_{i}\in S_{1}$ is a desired candidate for $S$. The proof is
complete. ∎
Proof of Theorem 5.1 (1) This follows from Lemmas 5.5, 5.6, 5.7 and
Proposition 5.9.
(2) One implication is obvious. Suppose $\Phi$ is an isomorphism of
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))$ onto
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V^{\prime}))$. Consequently,
$\Phi(L_{-1})=L_{-1}$ and
$\Phi(\mathcal{M}_{0}(V))=\mathcal{M}_{0}(V^{\prime}).$ A standard
verification shows that $\Phi(\mathcal{M}_{0}(V))=\mathcal{M}_{0}(\Phi(V))$.
By Lemma 4.5(2), we have $\Phi(V)=V^{\prime}$.
(3) This is a direct consequence of (2).
(4) Suppose $V$ is a standard element in $\mathfrak{V}^{L}$. Then
$\mathcal{M}(W_{-1},\mathcal{M}_{0}(V))$ has a standard $\mathbb{F}$-basis
$\displaystyle\\{x^{(\alpha)}x^{u}\partial_{i}\mid\alpha\in\mathbf{A}(m),u\in\mathbf{B}(n);\;i\in\mathbf{I}(k,l)\\}$
$\displaystyle\cup$
$\displaystyle\\{x^{(\alpha)}x^{u}\partial_{i}\mid\alpha_{1}=\cdots=\alpha_{k}=0,u\subset\overline{m+l+1,m+n};\;i\in\overline{\mathbf{I}}(k,l)\\}.$
Thus, we have:
$\mathrm{dim}\mathcal{M}(W_{-1},\mathcal{M}_{0}(V))=2^{n-l}p^{m-k}(m+n-k-l)+2^{n}p^{m}(k+l).$
Note that
$\overline{S}=S\oplus\sum_{i\in\mathbf{I}_{0}}x^{(\pi-(p-1)\varepsilon_{i})}x^{\omega}\partial_{i}$,
where $\pi=(p-1,\ldots,p-1)\in\mathbb{N}^{m}$ and $\omega=\langle
m+1,\ldots,m+n\rangle$. Then we have:
$\mathrm{dim}\mathcal{M}(S_{-1},\mathcal{M}_{0}(V))=2^{n-l}p^{m-k}(m+n)+2^{n}p^{m}(k+l-1)-k-1.$
∎
We call $\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$ is degenerate if
$V\in\mathcal{V}_{\mathfrak{i}}^{L}\cup\mathcal{V}_{\mathfrak{d}}^{L}$.
###### Proposition 5.10.
Let $\mathfrak{G}_{0}$ be a maximal reducible subalgebra of $H_{0}$ or
$K_{0}$.
* $(1)$
$\mathcal{M}(H_{-1},\mathfrak{G}_{0})$ is maximal in $H$.
* $(2)$
$\mathcal{M}(K_{-1},\mathfrak{G}_{0})$ is maximal in $K$ if and only if
${\mathfrak{G}_{0}}$ is degenerate.
###### Proof.
For any $0\neq h\in L$, $h\not\in\mathcal{M}(L_{-1},\mathfrak{G}_{0})$, put
$\overline{M}=\mathrm{alg}(\mathcal{M}(L_{-1},\mathfrak{G}_{0})+\mathbb{F}h)$.
By the maximality of $\mathfrak{G}_{0}$, we have $L_{0}\subset\overline{M}$.
For $H$, choose $k\in I_{0i}$ if $I_{0i}$ is not empty where $i=1,2$ or 3. It
follows that $y^{3}_{k}\in\mathcal{M}_{1}(H_{-1},\mathfrak{G}_{0})$. For $K$,
suppose ${\mathfrak{G}_{0}}$ is degenerate. Using the same method as for $H$,
we can find $0\not=v_{i}\in\overline{M}\cap K_{1i}$, where $i=0,1$. From
Lemmas 2.1(2) and 3.11, we have $\overline{M}=L$.
It remains to show that ${\mathfrak{G}_{0}}$ is degenerate if
$\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})$ is maximal. Assume on the contrary
that $V_{\mathfrak{G}_{0}}\in\mathcal{V}_{\mathfrak{n}}^{K}$ is a
nondegenerate irreducible $\mathfrak{G}_{0}$-module. For any
$u\in\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})$, by Lemmas 4.9(2) and 5.7, we
may assume that
$u=f_{-1}z+f_{1},\mbox{ where }\ f_{-1}\in V_{\mathfrak{G}_{0}}\ \mbox{ and }\
f_{1}\in\widetilde{\mathcal{M}}_{1}(V_{\mathfrak{G}_{0}}).$
Note that ${f_{-1}}$ is a linear combination of monomials with value 1. Let
$f_{1}=f^{1}+f^{4}+f^{8}$ where $f^{i}$ is a linear combination of monomials
with value $i$, $i=1,4$ or $8$. We claim that $f_{-1}=0$. Indeed, for any
$y_{i}\in K_{-1}$ with value 2, we have
$\sigma(i)(-1)^{i}(D_{\widetilde{i}}(f_{1})+D_{\widetilde{i}}(f_{-1})z)+y_{i}f_{-1}=[y_{i},u]\in\mathcal{M}_{0}(V_{\mathfrak{G}_{0}})=\mathfrak{G}_{0},$
which implies that $f_{-1}=0$ when $J_{3}$ is single. Otherwise, the following
equation holds:
$\displaystyle\sigma(i)(-1)^{i}D_{\widetilde{i}}(f^{4})=-y_{i}f_{-1}.$ (5.32)
Then there exists $g_{1}\in K_{1}$ with $D_{\widetilde{i}}(g_{1})=0$
satisfying
$\displaystyle\sigma(i)(-1)^{i}f^{4}=-y_{\widetilde{i}}y_{i}f_{-1}+g_{1}.$
(5.33)
By equations (5.32) and (5.33), we have
$D_{i}(g_{1})=(-1)^{i}2y_{\widetilde{i}}f_{-1}$ which contradicts
$D_{\widetilde{i}}(g_{1})=0$ if $f_{-1}\not=0$. Consequently,
$\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})\subset K_{10}$. Using induction on
$k$ and the transitivity of $K$, we have
$\mathcal{M}_{k}(K_{-1},\mathfrak{G}_{0})\subset K_{k0}$ for $k>0.$ It follows
that $\mathcal{M}(K_{-1},\mathfrak{G}_{0})$ is strictly contained in
$K_{-2}+K_{-1}+K_{0}+\sum_{i=1}^{2r(p-1)+n}K_{i0}$. The latter is a maximal
graded subalgebra (see Theorem 3.1(4)). This contradicts the maximality of
$\mathcal{M}(K_{-1},\mathfrak{G}_{0})$. The proof is complete. ∎
###### Lemma 5.11.
Suppose $L=H$ or $K$.
* $(1)$
Suppose $V\in\mathcal{V}_{\mathfrak{d}}^{L}$. Then $V$ contains a subspace
$V^{\prime}\in\mathcal{V}_{\mathfrak{i}}^{L}$ such that
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))=\mathcal{M}(L_{-1},\mathcal{M}_{0}(V^{\prime})).$
* $(1)$
If
$V,V^{\prime}\in\mathcal{V}_{\mathfrak{n}}^{L}\cup\mathcal{V}_{\mathfrak{i}}^{L}$,
then
$\mathcal{M}_{0}(V)=\mathcal{M}_{0}(V^{\prime})$
if and only if one of the following conditions holds.
* $(i)$
$V=V^{\prime}.$
* $(ii)$
$V^{\bot}=V^{\prime}$ when $V$ and $V^{\prime}$ are nondegenerate.
###### Proof.
For (1), we may assume that $V=\mathrm{span}_{\mathbb{F}}\\{y_{i}\mid i\in
J_{1}\cup J_{2}\\}$. Then $V^{\prime}=\mathrm{span}_{\mathbb{F}}\\{y_{i}\mid
i\in J_{2}\\}$ is desired. For (2), by a similar argument as in Lemma 4.5(2),
we get the desired conclusion. ∎
###### Lemma 5.12.
The following statements hold.
* $(1)$
If $V\in\mathcal{V}_{\mathfrak{n}}^{H}$, then
$\mathcal{M}(H_{-1},\mathcal{M}_{0}(V))=\mathcal{O}_{J_{1}}\oplus\mathcal{O}_{{J}_{3}}.$
* $(2)$
If $V\in\mathcal{V}_{\mathfrak{i}}^{H}$, then
$\mathcal{M}(H_{-1},\mathcal{M}_{0}(V))=\mathcal{O}_{J_{2}\cup
J_{3}}\oplus\mathcal{O}^{+}_{J_{2}}\mathcal{Q}_{\bar{J}_{2}}.$
###### Proof.
(1) For $V\in\mathcal{V}_{\mathfrak{n}}^{H}$, a direct computation shows that
$\mathcal{M}_{i}(H_{-1},\mathcal{M}_{0}(V))=\mathrm{span}_{\mathbb{F}}\\{u\in
H_{i}\mid u\ \mbox{ is a monomial with }\ \nu(u)=1,2^{i+2}\\}.$
(2) For $V\in\mathcal{V}_{\mathfrak{i}}^{H}$, using induction on $i$, we
obtain that $\mathcal{M}_{i}(H_{-1},\mathcal{M}_{0}(V))$ is spanned by
monomials in $H$ as follows:
* $(a)$
$u_{1}u_{2}\in H_{i}$, where $u_{1}$ is a monomial with the variables of value
0 and $u_{2}$ is a monomial with the variables of value $2$.
* $(b)$
$y_{j}u_{3}\in H_{i}$, where $j\in\bar{J}_{2}$ and $u_{3}$ is a monomial with
the variables of value $0$.
Then, the conclusions hold. ∎
For $u\in K$, put $Z(u)=i$ if $(ad1)^{i+1}u=0$ and $(ad1)^{i}u\not=0$.
###### Lemma 5.13.
Suppose $V\in\mathcal{V}_{\mathfrak{i}}^{K}$. For any element $u\in K$ with
$[1,u]\not=0$, $u\in\mathcal{M}(K_{-1},\mathcal{M}_{0}(V))$ if and only if $u$
is a linear combination of elements of the form $f(z+x)^{j}+g$, where
$g\in\mathcal{O}^{+}_{J_{2}\cup
J_{3}}\oplus\mathcal{O}^{+}_{J_{2}}\mathcal{Q}_{\bar{J}_{2}}$,
$f\in\left\\{\begin{array}[]{ll}\mathcal{O}^{+}_{J_{2}}\mathcal{Q}^{+}_{\bar{J}_{2}},&V\in\mathcal{V}_{\mathfrak{i}}^{K}\mbox{
satisfying }J_{3}\ \mbox{ is empty};\\\ \mathcal{O}^{+}_{J_{2}\cup
J_{3}},&V\in\mathcal{V}_{\mathfrak{i}}^{K}\mbox{ satisfying }J_{3}\ \mbox{ is
not empty},\end{array}\right.$
$x=\sum_{i\in J_{2}}y_{i}y_{\widetilde{i}}$ and $0<j<p$.
###### Proof.
Let $\mathfrak{G}_{0}=\mathcal{M}_{0}(V)$. Notice that $g$, $x$ and
$z+x\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})$. Firstly, for any
$i\in\mathbf{I}$, one computes
$\displaystyle[y_{i},z+x]\in\mathcal{O}^{+}_{J_{2}\cup J_{3}},$ $\displaystyle
f[y_{i},z+x]\in\mathcal{O}_{J_{2}\cup
J_{3}}+\mathcal{O}_{J_{2}}\mathcal{Q}_{\bar{J}_{2}},$
$\displaystyle[y_{i},f]\in\left\\{\begin{array}[]{ll}\mathcal{O}^{+}_{J_{2}}\mathcal{Q}^{+}_{\bar{J}_{2}},&V\in\mathcal{V}_{\mathfrak{i}}^{K}\mbox{
satisfying }J_{3}\ \mbox{ is empty};\\\ \mathcal{O}^{+}_{J_{2}\cup
J_{3}},&V\in\mathcal{V}_{\mathfrak{i}}^{K}\mbox{ satisfying }J_{3}\ \mbox{ is
not empty}.\end{array}\right.$
Using induction on $\mathrm{zd}(f)$ and $j$, respectively, we have
$f(z+x),\ (z+x)^{j}\in\mathcal{M}(K_{-1},\mathfrak{G}_{0}).$
Furthermore, $f(z+x)^{j}\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})$.
Conversely, let us use induction on $Z(u)$. When $Z(u)=1$, we consider the
following cases.
Case 1. $u\in\mathcal{M}_{0}(K_{-1},\mathfrak{G}_{0}).$ By Lemmas 4.9(1) and
5.7, we may assume that $u=z+u_{0}$, where $u_{0}\in\mathfrak{G}_{0}\cap
H_{0}$, which means that $u_{0}\in\mathcal{O}_{J_{2}\cup
J_{3}}+\mathcal{O}_{J_{2}}\mathcal{Q}_{\bar{J}_{2}}$. Thus, $u=z+x+(u_{0}-x)$
is desired.
Case 2. $u\in\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0}).$ From Remark 2.3, we
may assume that $u=y_{t}z+u_{1}$, where $u_{1}\in H_{1}$ and $t\in\mathbf{I}$.
Notice that, when $y_{t}(z+x)\in\mathcal{M}(K_{-1},\mathfrak{G}_{0}),$
$u_{1}-y_{t}x=u-y_{t}(z+x)\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})\cap H,$
which follows that $u=y_{t}(z+x)+(y_{t}x-u_{1})$ is desired. Thus, by the
necessity of this lemma, it is sufficient to consider the case of
$t\in\bar{J}_{2}$ when $J_{3}$ is not empty. From Lemmas 4.9(3) and 5.7, we
may assume that
$u_{1}=h^{(\frac{1}{3},2,2)}+h^{(0,\frac{1}{3},\frac{1}{3})}+h^{(0,\frac{1}{3},2)}+h,$
where
$h^{(\alpha,\beta,\gamma)}=\mathrm{span}_{\mathbb{F}}\\{y_{i}y_{j}y_{k}\mid\nu({y_{i}})=\alpha,\nu({y_{j}})=\beta,\nu({y_{k}})=\gamma\\},$
$h\in\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})\cap H$. For any $y_{l}\in
K_{-1}$, we have:
$\sigma(l)(-1)^{l}(D_{\widetilde{l}}(y_{t})z+D_{\widetilde{l}}u_{1})+y_{l}y_{t}=[y_{l},u]\in\mathfrak{G}_{0},$
which means that
$\sigma(l)(-1)^{l}D_{\widetilde{l}}u_{1}+y_{l}y_{t}\in\mathfrak{G}_{0}.$
(5.34)
When $\nu(y_{l})=2$, from equation (5.34) we have:
$\sigma(l)(-1)^{l}D_{\widetilde{l}}h^{(\frac{1}{3},2,2)}+y_{l}y_{t}\in\mathfrak{G}_{0},$
which is a linear combination of elements with value $\frac{2}{3}$. It follows
that
$\sigma(l)(-1)^{l}D_{\widetilde{l}}h^{(\frac{1}{3},2,2)}+y_{l}y_{t}=0.$
When $J_{3}$ is single, we have $y_{l}y_{t}=0$, a contradiction. When $J_{3}$
is not single, there exist distinct $k,\widetilde{k}\in J_{3}$ such that
$h^{(\frac{1}{3},2,2)}=-\sigma(k)(-1)^{k}y_{\widetilde{k}}y_{k}y_{t}+h^{\prime},$
(5.35)
where $D_{\widetilde{k}}h^{\prime}=0$ and
$\sigma(\widetilde{k})(-1)^{k}D_{k}(h^{(\frac{1}{3},2,2)})+y_{\widetilde{k}}y_{t}=0.$
From equation (5.35), we have
$D_{k}(h^{\prime})=D_{k}(h^{(\frac{1}{3},2,2)})+\sigma(k)y_{\widetilde{k}}y_{t}=2\sigma(k)y_{\widetilde{k}}y_{t},$
which contradicts $D_{\widetilde{k}}h^{\prime}=0$. Thus, an element of the
form $y_{t}z+u_{1}$, $t\in\bar{J}_{2}$ is not in
$\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})$ when $J_{3}$ is not empty.
Case 3. $u\in\mathcal{M}_{i}(K_{-1},\mathfrak{G}_{0})$ for $i>1$. We may
assume that
$u=g_{i-2}z+g_{i},\ g_{j}\in\overline{H}_{j},\ j=i-2,i.$
Note that the elements of the form $h_{2}z+h$ are not in
$\mathcal{M}(K_{-1},\mathfrak{G}_{0})$, where $h_{2}$ is a linear combination
of monomials with value $\frac{1}{9}$. By induction on $i$, we obtain that
$g_{i-2}$ is in $\mathcal{O}_{J_{2}}\mathcal{Q}_{\bar{J}_{2}}$ if $J_{3}$ is
empty; in $\mathcal{O}_{J_{2}\cup J_{3}}$, otherwise. Thus,
$g_{i-2}(z+x)\in\mathcal{M}_{i}(K_{-1},\mathfrak{G}_{0})$. Moreover,
$g_{i}-g_{i-2}x\in\mathcal{M}_{i}(K_{-1},\mathfrak{G}_{0})\cap\overline{H}$.
Then $u=g_{i-2}(z+x)+(g_{i}-g_{i-2}x)$ is desired.
When $Z(u)=k>1$, suppose
$u=u_{k}z^{k}+u_{k-1}z^{k-1}+\cdots+u_{1}z+u_{0},\ u_{j}\in\overline{H},\
j=0,\ldots,k.$
Obviously,
$u_{k}z+u_{k-1}=2^{(1-k)}(\mathrm{ad}1)^{k-1}(u)\in\mathcal{M}(K_{-1},\mathfrak{G}_{0}).$
Thus, $u_{k}$ is in $\mathcal{O}^{+}_{J_{2}}\mathcal{Q}^{+}_{\bar{J}_{2}}$
when $J_{3}$ is empty; in $\mathcal{O}^{+}_{J_{2}\cup J_{3}}$, otherwise.
Consequently, $u_{k}(z+x)^{k}\in\mathcal{M}(K_{-1},\mathfrak{G}_{0}).$ Thus,
$v=u-u_{k}(z+x)^{k}\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})$
and $Z(v)<k$. By the inductive hypothesis, $v$ is a linear combination of the
desired form. So is $u$. The proof is complete. ∎
Proof of Theorem 5.2. For (2) and ($2^{\prime}$), sufficiency is obvious. For
necessity, suppose $\Phi$ is an isomorphism of
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V))$ onto
$\mathcal{M}(L_{-1},\mathcal{M}_{0}(V^{\prime}))$. Then, $\Phi(L_{-1})=L_{-1}$
and $\Phi(\mathcal{M}_{0}(V))=\mathcal{M}_{0}(V^{\prime})$, which implies that
$\dim\mathcal{M}_{0}(V)=\dim\mathcal{M}_{0}(V^{\prime})$. It follows that $V$
and $V^{\prime}$ are both nondegenerate or are both isotropic. Notice that
$\Phi(\mathcal{M}_{0}(V))\subset\mathcal{M}_{0}(\Phi(V))$. For the maximality
of $\mathcal{M}_{0}(V^{\prime})$, we have
$\mathcal{M}_{0}(V^{\prime})=\mathcal{M}_{0}(\Phi(V)).$ By virtue of Lemma
5.11(2), we have $V^{\prime}=\Phi(V)$ or $V^{\prime}=\Phi(V)^{\bot}$ when
$V^{\prime}$ and $\Phi(V)$ are both nondegenerate. Thus, we have $\dim V=\dim
V^{\prime}$ or $\dim V=m+n-\dim V^{\prime}$. We can obtain the desired
conclusions by a direct computation. (3) and $(3^{\prime})$ are direct
consequences of (2) and $(2^{\prime})$.
The remaining statements hold from Lemmas 5.5, 5.6, 5.11, 5.12, 5.13 and
Proposition 5.10.∎
Finally, we consider the maximal S-subalgebras of $L$, where $L=W,S,H$ or $K$.
As in the case of modular Lie algebras [16], it easy to show the following:
###### Lemma 5.14.
Suppose $\mathfrak{G}_{0}$ is a maximal irreducible subalgebra of $L_{0}$. The
subalgebra $\mathcal{M}(L_{-1},\mathfrak{G}_{0})$ is not maximal in $L$ if
$\mathcal{M}_{1}(L_{-1},\mathfrak{G}_{0})=0$.
Proof of Theorem 5.3. (1) This is nothing but Lemma 5.6.
(2) Let $\mathfrak{G}_{0}$ be a maximal irreducible subalgebra of $L_{0}.$
(a) Suppose $L=W$ and $\mathcal{M}(W_{-1},\mathfrak{G}_{0})$ is maximal in
$W$. Assume on the contrary that
$\mathrm{div}(\mathcal{M}_{1}(W_{-1},\mathfrak{G}_{0}))=0$. By induction on
$i$, one has $\mathcal{M}_{i}(W_{-1},\mathfrak{G}_{0})\subset W^{\prime}_{i}$
for all $i\geq 1.$ Since $\mathfrak{G}_{0}$ is a nontrivial subalgebra of
$W_{0}$, we have
$\mathcal{M}(W_{-1},\mathfrak{G}_{0})\subsetneq
W_{-1}+W_{0}+W^{\prime}_{1}+W^{\prime}_{2}+\cdots$
By Theorem 3.1, the latter is an MGS of $W$. This contradicts the maximality
of $\mathcal{M}(W_{-1},\mathfrak{G}_{0})$.
Conversely, to show the maximality of $\mathcal{M}(W_{-1},\mathfrak{G}_{0})$,
assume that $M$ is an MGS strictly containing
$\mathcal{M}(W_{-1},\mathfrak{G}_{0})$. By definition (5.30), it must be that
$M_{0}\supsetneq\mathfrak{G}_{0}$ and therefore, $M_{0}=W_{0}$ by the
maximality of $\mathfrak{G}_{0}$. Thus $M$ is an MGS of type (I) and thereby
$M_{1}=W_{1}^{\prime}\;\mbox{or}\;W_{1}^{\prime\prime}.$ (5.36)
Note that $W^{\prime\prime}_{1}$ is an irreducible $\mathfrak{G}_{0}$-module,
which follows from the irreducibility of $\mathfrak{G}_{0}$ and a simple fact
that, as $W_{0}$-modules,
$W^{\prime\prime}_{1}\cong(W_{-1})^{*}.$
By our assumption, there is a
$D\in\mathcal{M}_{1}(W_{-1},\mathfrak{G}_{0})\subset M_{1}$ with
$\mathrm{div}D$ $\neq 0$. Assert that $D\notin W^{\prime\prime}_{1}$. Assuming
on the contrary, by the irreducibility of $W^{\prime\prime}_{1}$, we have
$W^{\prime\prime}_{1}\subset\mathcal{M}_{1}(W_{-1},\mathfrak{G}_{0})$ and
thereby
$W_{0}=\mathrm{alg}([W_{-1},W^{\prime\prime}_{1}])\subset\mathrm{alg}([W_{-1},\mathcal{M}_{1}(W_{-1},\mathfrak{G}_{0})])\subset\mathfrak{G}_{0}.$
This contradicts the assumption that $\mathfrak{G}_{0}$ is a nontrivial
subalgebra of $W_{0}$ and hence the assertion holds. This proves that $D$
belongs to neither $W_{1}^{\prime}$ nor $W_{1}^{\prime\prime}$, contradicting
(5.36).
(b) For $S$, from Lemma 5.14, one implication is obvious. As in (a), we have
$\mathcal{M}(S_{-1},\mathfrak{G}_{0})$ is maximal when
$\mathcal{M}_{1}(S_{-1},\mathfrak{G}_{0})\neq 0$. For $H$, the conclusion
follows from Lemmas 3.11(1) and 5.14.
(c) Suppose $L=K$. Assume on the contrary that $[1,u]=0$ for every
$u\in\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})$. Then,
$\mathcal{M}_{1}(K_{-1},\mathfrak{G}_{0})\subset K_{10}.$
As in the proof of Proposition 5.10(2), we have
$\mathcal{M}(K_{-1},\mathfrak{G}_{0})$ is not maximal.
Conversely, suppose $u=u_{0}+u_{1}$ where $u_{i}\in K_{1i}$, $i=0,1$ and
$u_{1}\not=0$. We claim that $u_{0}\not=0$. Indeed, by a direct computation,
$[K_{-1},K_{11}]=K_{0}$ holds. Assuming on the contrary that $u_{0}=0$, we
have $u_{1}\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})$. $K_{-1}$ is an
irreducible $\mathfrak{G}_{0}$-module, and so is $K_{11}$. Moreover,
$K_{11}\subset\mathcal{M}(K_{-1},\mathfrak{G}_{0})$. Thus,
$[K_{-1},K_{11}]\subset[K_{-1},\mathcal{M}(K_{-1},\mathfrak{G}_{0})]\subset\mathfrak{G}_{0}\subsetneq
K_{0},$
which contradicts $[K_{-1},K_{11}]=K_{0}$.
Put $h\in K$, $h\not\in\mathcal{M}(K_{-1},\mathfrak{G}_{0})$ and
$\overline{M}=\mathrm{alg}{(\mathcal{M}(K_{-1},\mathfrak{G}_{0})+\mathbb{F}h)}$.
By definition (5.30) and the maximality of $\mathfrak{G}_{0}$, we have
$K_{0}\subset\overline{M}$. Since the torus $T$ is in $\overline{M}$, from
Remark 2.3, there exists
$v=v_{0}+y_{i}z\in\mathrm{alg}{(\mathbb{F}u+T)}\subset\overline{M},\
i\in\mathbf{I},\ v_{0}\in K_{10}.$
For $K_{0}\subset\overline{M}$, without loss of generality, we may assume that
$i\in\mathbf{I}_{0}.$ If $v_{0}=0$, we have $y_{i}z\in\overline{M}$. For
$u_{0}\not=0$, the conclusion holds. Otherwise, we claim that there exists a
nonzero element in $\overline{M}\cap K_{10}$. Indeed, it is sufficient to
consider the following cases.
Case 1. $D_{\widetilde{i}}(v_{0})\not=0$. Note that
$0\not=[y^{(2\varepsilon_{i})},v]\in\overline{M}\cap K_{10}.$
Case 2. $D_{\widetilde{i}}(v_{0})=0$ and there exists $t\in\mathbf{I}$,
$t\not=i$, $\widetilde{i}$, such that $D_{\widetilde{t}}(v_{0})\not=0$. Note
that $0\not=[y_{t}y_{i},v]\in\overline{M}\cap K_{10}.$
Case 3. $D_{{t}}(v_{0})=0$, for all $t\in\mathbf{I}\backslash\\{i\\}$. Then
$v_{0}=y^{(3\varepsilon_{i})}.$ Note that
$y^{(3\varepsilon_{i})}=(2\sigma(\widetilde{i}))^{-1}([y_{i}y_{\widetilde{i}},v]-\sigma(\widetilde{i})v)\in\overline{M}\cap
K_{10}.$
By Lemmas 2.1(2), 3.11(2) and $u_{i}\not=0$ for $i=0,1$, the conclusion
follows.∎
## References
* [1] V. G. Kac. Lie superalgebras. Adv. Math. 26 (1977): 8–96.
* [2] M. Scheunert. Theory of Lie Superalgebras. Lecture Notes in Math. vol. 716, Springer-Verlag, Berlin, 1979.
* [3] V. G. Kac. Classification of infinite dimensional simple linearly compact Lie superalgebras. Adv. Math. 139 (1998): 1–55.
* [4] Y.-Z. Zhang. Finite-dimensional Lie superalgebras of Cartan-type over fields of prime characteristic. Chinese. Sci. Bull. 42(9) (1997): 720–724.
* [5] S. Bouarroudj and D. Leites. Simple Lie superalgebras and nonintegrable distributions in characteristic $p$. J. Math. Sci. 141(4) (2007): 1390–1398.
* [6] A. Elduque. New simple Lie superalgebras in characteristic 3. J. Algebra 296(1) (2006): 196–233.
* [7] E. B. Dynkin. Semisimple subalgebras of semisimple Lie algebras. Mat. Sb. (N. S.) 30(72) (1952): 349–462; transl. AMS Transl. 6(2) (1957): 111–244.
* [8] E. B. Dynkin. Maximal subgroups of the classical groups. Trudy Moskow. Mat. Obsh. 1 (1952): 39–166. transl. AMS Transl. 6(2) (1957): 245–378.
* [9] G. M. Seitz. The maximal subgroups of classical algebraic groups. Memories of the AMS 67 (1987).
* [10] G. M. Seitz. Maximal subgroups of exceptional algebraic groups. Memories of the AMS 90 (1991).
* [11] M. Racine. On maximal subalgebras. J. Algebra 30(1) (1974): 155–180.
* [12] M. Racine. Maximal subalgebras of exceptional Jordan algebras. J. Algebra 46 (1977): 12–21.
* [13] A. Elduque, J. Laliena, and S. Sacristan. Maximal subalgebras of associative superalgebras. J. Algebra 275(1) (2004): 40–58.
* [14] A. Elduque J. Laliena, and S. Sacristan. The Kac Jordan superalgebra: Automorphisms amd maximal subalgebras. Proc. AMS 135(12) (2007): 3805–3813.
* [15] Y. Barnea, A. Shalev and E. I. Zelmanov. Graded subalgebras of affine Kac-Moody algebras. Israel J. Math. 104 (1998): 321–334.
* [16] H. Melikyan. Maximal subalgebras of simple modular Lie algebras. J. Algebra 284 (2005): 824–856.
* [17] A. I. Kostrikin, I. R. Shafarevich. Graded Lie algebras of finite characteristic. Izv. Akad. Nauk. SSSR Ser. Mat. 33 (1969): 251–322 (in Russian); transl. Math. USSR Izv. 3 (1969): 237–304.
* [18] H. Strade. Simple Lie Algebras over Fields of Positive Characteristic I. Structure Theory. de Gruyter Exp. Math., vol. 38, Walter de Gruyter, Berlin, 2004.
* [19] H. Strade and R. Farnsteiner. Modular Lie Algebras and Their Representations. Monographys and Textbooks in Pure Appl. Math. vol 116, Marcel Dekker, New York, 1988.
* [20] W.-D. Liu and Y.-Z. Zhang. Automorphism groups of restricted Cartan-type Lie superalgebras. Comm. Algebra 34(10) (2006): 1–18.
|
arxiv-papers
| 2013-04-20T09:39:34 |
2024-09-04T02:49:44.633759
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Bai, Wende Liu, Xuan Liu, Hayk Melikyan",
"submitter": "Wende Liu Professor",
"url": "https://arxiv.org/abs/1304.5618"
}
|
1304.5637
|
# Tucker Tensor Regression
and Neuroimaging Analysis
Xiaoshan Li, Hua Zhou and Lexin Li
North Carolina State University
###### Abstract
Large-scale neuroimaging studies have been collecting brain images of study
individuals, which take the form of two-dimensional, three-dimensional, or
higher dimensional arrays, also known as tensors. Addressing scientific
questions arising from such data demands new regression models that take
multidimensional arrays as covariates. Simply turning an image array into a
long vector causes extremely high dimensionality that compromises classical
regression methods, and, more seriously, destroys the inherent spatial
structure of array data that possesses wealth of information. In this article,
we propose a family of generalized linear tensor regression models based upon
the Tucker decomposition of regression coefficient arrays. Effectively
exploiting the low rank structure of tensor covariates brings the ultrahigh
dimensionality to a manageable level that leads to efficient estimation. We
demonstrate, both numerically that the new model could provide a sound
recovery of even high rank signals, and asymptotically that the model is
consistently estimating the best Tucker structure approximation to the full
array model in the sense of Kullback-Liebler distance. The new model is also
compared to a recently proposed tensor regression model that relies upon an
alternative CANDECOMP/PARAFAC (CP) decomposition.
11footnotetext: Address for correspondence: Lexin Li, Department of
Statistics, North Carolina State University, Box 8203, Raleigh, NC 27695-8203.
Email: [email protected].
Key Words: CP decomposition; magnetic resonance image; tensor; Tucker
decomposition.
## 1 Introduction
Advancing technologies are constantly producing large scale scientific data
with complex structures. An important class arises from medical imaging, where
the data takes the form of multidimensional array, also known as _tensor_.
Notable examples include electroencephalography (EEG, 2D matrix), anatomical
magnetic resonance images (MRI, 3D array), functional magnetic resonance
images (fMRI, 4D array), among other image modalities. In medical imaging data
analysis, a primary goal is to better understand associations between brains
and clinical outcomes. Applications include using brain images to diagnose
neurodegenerative disorders, to predict onset of neuropsychiatric diseases,
and to identify disease relevant brain regions or activity patterns. This
family of problems can collectively be formulated as a regression with
clinical outcome as response, and image, or tensor, as predictor. However, the
sheer size and complex structure of image covariate pose unusual challenges,
which motivate us to develop a new class of regression models with image
covariate.
Most classical regression models take vector as covariate. Naively turning an
image array into a vector is evidently unsatisfactory. For instance, a typical
MRI image of size 128-by-128-by-128 implicitly requires $128^{3}=2,097,152$
regression parameters. Both computability and theoretical guarantee of the
classical regression models are severely compromised by this ultra-high
dimensionality. More seriously, vectorizing an array destroys the inherent
spatial structure of the image array that usually possesses abundant
information. A typical solution in the literature first employs the subject
knowledge to extract a vector of features from images, and then feeds the
feature vector into a classical regression model (Mckeown et al.,, 1998;
Blankertz et al.,, 2001; Haxby et al.,, 2001; Kontos et al.,, 2003; Mitchell
et al.,, 2004; LaConte et al.,, 2005; Shinkareva et al.,, 2006). Alternatively
one first applies unsupervised dimension reduction, often some variant of
principal components analysis, to the image array, and then fits a regression
model in the reduced dimensional vector space (Caffo et al.,, 2010). Both
solutions are intuitive and popular, and have enjoyed varying degrees of
success. At heart, both transform the problem to a classical vector covariate
regression. However, there is no consensus on what choice best summarizes a
brain image even for a single modality, whereas unsupervised dimension
reduction like principal components could result in information loss in a
regression setup. In contrast to constructing an image feature vector, the
functional approach views image as a function and then employs functional
regression models (Ramsay and Silverman,, 2005). Reiss and Ogden, (2010)
notably applied this idea to regression with 2D image predictor. Extending
their method to 3D and higher dimensional images, however, is far from trivial
and requires substantial research, due to the large number of parameters and
multi-collinearity among imaging measures.
In a recent work, Zhou et al., (2013) proposed a class of generalized linear
_tensor_ regression models. Specifically, for a response variable $Y$, a
vector predictor ${\bm{Z}}\in\mathrm{I\\!R}\mathit{{}^{p_{0}}}$ and a
$D$-dimensional tensor predictor
${\bm{X}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\ldots\times p_{D}}}$, the
response is assumed to belong to an exponential family where the linear
systematic part is of the form,
$\displaystyle g(\mu)=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle{\bm{B}},{\bm{X}}\rangle.$ (1)
Here $g(\cdot)$ is a strictly increasing link function,
$\mu=E(Y|{\bm{X}},{\bm{Z}})$,
$\mbox{\boldmath$\gamma$}\in\mathrm{I\\!R}\mathit{{}^{p_{0}}}$ is the regular
regression coefficient vector,
${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times p_{D}}}$ is the
coefficient array that captures the effects of tensor covariate ${\bm{X}}$,
and the inner product between two arrays is defined as
$\langle{\bm{B}},{\bm{X}}\rangle=\langle\mathrm{vec}{\bm{B}},\mathrm{vec}{\bm{X}}\rangle=\sum_{i_{1},\ldots,i_{D}}\beta_{i_{1}\ldots
i_{D}}x_{i_{1}\ldots i_{D}}$. This model, if with no further simplification,
is prohibitive given its gigantic dimensionality:
$p_{0}+\prod_{d=1}^{D}p_{d}$. Motivated by a commonly used tensor
decomposition, Zhou et al., (2013) introduced a low rank structure on the
coefficient array ${\bm{B}}$. That is, ${\bm{B}}$ is assumed to follow a
rank-$R$ CANDECOMP/PARAFAC (CP) decomposition (Kolda and Bader,, 2009),
$\displaystyle{\bm{B}}=\sum_{r=1}^{R}\mbox{\boldmath$\beta$}_{1}^{(r)}\circ\cdots\circ\mbox{\boldmath$\beta$}_{D}^{(r)},$
(2)
where $\mbox{\boldmath$\beta$}_{d}^{(r)}\in\mathrm{I\\!R}\mathit{{}^{p_{d}}}$
are all column vectors, $d=1,\ldots,D,r=1,\ldots,R$, and $\circ$ denotes an
outer product among vectors. Here the outer product
${\bm{b}}_{1}\circ{\bm{b}}_{2}\circ\cdots\circ{\bm{b}}_{D}$ of $D$ vectors
${\bm{b}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}}}$, $d=1,\ldots,D$, is defined
as the $p_{1}\times\cdots\times p_{D}$ array with entries
$({\bm{b}}_{1}\circ{\bm{b}}_{2}\circ\cdots\circ{\bm{b}}_{D})_{i_{1}\cdots
i_{D}}=\prod_{d=1}^{D}b_{di_{d}}$. For convenience, this CP decomposition is
often represented by a shorthand
${\bm{B}}=\llbracket{\bm{B}}_{1},\ldots,{\bm{B}}_{D}\rrbracket$, where
${\bm{B}}_{d}=[\mbox{\boldmath$\beta$}_{d}^{(1)},\ldots,\mbox{\boldmath$\beta$}_{d}^{(R)}]\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times
R}}$, $d=1,\ldots,D$. Combining (1) and (2) yields generalized linear tensor
regression models of Zhou et al., (2013), where the dimensionality decreases
to the scale of $p_{0}+R\times\sum_{d=1}^{D}p_{d}$. Under this setup,
ultrahigh dimensionality of (1) is reduced to a manageable level, which in
turn results in efficient estimation and prediction. For instance, for a
regression with 128-by-128-by-128 MRI image and 5 usual covariates, the
dimensionality is reduced from the order of $2,097,157=5+128^{3}$ to
$389=5+128\times 3$ for a rank-1 model, and to $1,157=5+3\times 128\times 3$
for a rank-3 model. Zhou et al., (2013) showed that this low rank tensor model
could provide a sound recovery of even high rank signals.
In the tensor literature, there has been an important development parallel to
CP decomposition, which is called Tucker decomposition, or higher-order
singular value decomposition (HOSVD) (Kolda and Bader,, 2009). In this
article, we propose a class of _Tucker tensor regression models_. To
differentiate, we call the models of Zhou et al., (2013) _CP tensor regression
models_. Specifically, we continue to adopt the model (1), but assume that the
coefficient array ${\bm{B}}$ follows a Tucker decomposition,
$\displaystyle{\bm{B}}=\sum_{r_{1}=1}^{R_{1}}\cdots\sum_{r_{D}=1}^{R_{D}}g_{r_{1},\ldots,r_{D}}\mbox{\boldmath$\beta$}_{1}^{(r_{1})}\circ\cdots\circ\mbox{\boldmath$\beta$}_{D}^{(r_{D})},$
(3)
where
$\mbox{\boldmath$\beta$}_{d}^{(r_{d})}\in\mathrm{I\\!R}\mathit{{}^{p_{d}}}$
are all column vectors, $d=1,\ldots,D,r_{d}=1,\ldots,R_{d}$, and
$g_{r_{1},\ldots,r_{D}}$ are constants. It is often abbreviated as
${\bm{B}}=\llbracket{\bm{G}};{\bm{B}}_{1},\ldots,{\bm{B}}_{D}\rrbracket$,
where ${\bm{G}}\in\mathrm{I\\!R}\mathit{{}^{R_{1}\times\cdots\times R_{D}}}$
is a $D$-dimensional _core tensor_ with entries $({\bm{G}})_{r_{1}\ldots
r_{D}}=g_{r_{1},\ldots,r_{D}}$, and
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R_{d}}}$ are the factor
matrices. ${\bm{B}}_{d}$’s are usually orthogonal and can be thought of as the
_principal components_ in each dimension (and thus the name, HOSVD). The
number of parameters of a Tucker tensor model is in the order of
$p_{0}+\sum_{d=1}^{D}R_{d}\times p_{d}$. Comparing the two decompositions (2)
and (3), the key difference is that CP fixes the number of basis vectors $R$
along each dimension of ${\bm{B}}$ so that all ${\bm{B}}_{d}$’s have the
_same_ number of columns (ranks). In contrast, Tucker allows the number
$R_{d}$ to differ along different dimensions and ${\bm{B}}_{d}$’s could have
_different_ ranks.
This difference between the two decompositions seems minor; however, in the
context of tensor regression modeling and neuroimging analysis, it has
profound implications, and such implications motivate this article. On one
hand, the Tucker tensor regression model shares the advantages of the CP
tensor regression model, in that it effectively exploits the special structure
of the tensor data, it substantially reduces the dimensionality to enable
efficient model estimation, and it provides a sound low rank approximation to
a potentially high rank signal. On the other hand, Tucker tensor regression
offers a much more _flexible_ modeling framework than CP regression, as it
allows distinct order along each dimension. When the orders are all identical,
it includes the CP model as a special case. This flexibility leads to several
improvements that are particularly useful for neuroimaging analysis. First, a
Tucker model could be more parsimonious than a CP model thanks to the
flexibility of different orders. For instance, suppose a 3D signal
${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{16\times 16\times 16}}$ admits a Tucker
decomposition (3) with $R_{1}=R_{2}=2$ and $R_{3}=5$. It can only be recovered
by a CP decomposition with $R=5$, costing 230 parameters. In contrast, the
Tucker model is more parsimonious with only 131 parameters. This reduction of
free parameters is valuable for medical imaging studies, as the number of
subjects is often limited. Second, the freedom in the choice of different
orders is useful when the tensor data is skewed in dimensions, which is common
in neuroimaging data. For instance, in EEG, the two dimensions consist of
electrodes (channels) and time, and the number of sampling time points usually
far exceeds the number of channels. Third, even when all tensor modes have
comparable sizes, the Tucker formulation explicitly models the interactions
between factor matrices ${\bm{B}}_{d}$’s, and as such allows a finer grid
search within a larger model space, which in turn may explain more trait
variance. Finally, as we will show in Section 2.3, there exists a duality
regarding the Tucker tensor model. Thanks to this duality, a Tucker tensor
decomposition naturally lends itself to a principled way of imaging data
downsizing, which, given the often limited sample size, again plays a
practically very useful role in neuroimaging analysis.
For these reasons, we feel it important to develop a complete methodology of
Tucker tensor regression and its associated theory. The resulting Tucker
tensor model carries a number of useful features. It performs dimension
reduction through low rank tensor decomposition but in a supervised fashion,
and as such avoids potential information loss in regression. It works for
general array-valued image modalities and/or any combination of them, and for
various types of responses, including continuous, binary, and count data.
Besides, an efficient and highly scalable algorithm has been developed for the
associated maximum likelihood estimation. This scalability is important
considering the massive scale of imaging data. In addition, regularization has
been studied in conjunction with the proposed model, yielding a collection of
regularized Tucker tensor models, and particularly one that encourages
sparsity of the core tensor to facilitate model selection among the defined
Tucker model space.
Recently there have been some increasing interests in matrix/tensor
decomposition and their applications in brain imaging studies (Crainiceanu et
al.,, 2011; Allen et al.,, 2011; Hoff,, 2011; Aston and Kirch,, 2012).
Nevertheless, this article is distinct in that we concentrate on a regression
framework with scalar response and tensor valued covariates. In contrast,
Crainiceanu et al., (2011) and Allen et al., (2011) studied unsupervised
decomposition, Hoff, (2011) considered model-based decomposition, whereas
Aston and Kirch, (2012) focused on change point distribution estimation. The
most closely related work to this article is Zhou et al., (2013); however, we
feel our work is _not_ a simple extension of theirs. First of all, considering
the complex nature of tensor, the development of the Tucker model estimation
as well as its asymptotics is far from a trivial extension of the CP model of
Zhou et al., (2013). Moreover, we offer a detailed comparison, both
analytically (in Section 2.4) and numerically (in Sections 6.3 and 6.4), of
the CP and Tucker decompositions in the context of regression with
imaging/tensor covariates. We believe this comparison is crucial for an
adequate comprehension of tensor regression models and supervised tensor
decomposition in general.
The rest of the article is organized as follows. Section 2 begins with a brief
review of some preliminaries on tensor, and then presents the Tucker tensor
regression model. Section 3 develops an efficient algorithm for maximum
likelihood estimation. Section 4 derives inferential tools such as score,
Fisher information, identifiability, consistency, and asymptotic normality.
Section 5 investigates regularization method for the Tucker regression.
Section 6 presents extensive numerical results. Section 7 concludes with some
discussions and points to future extensions. All technical proofs are
delegated to the Appendix.
## 2 Model
### 2.1 Preliminaries
We start with a brief review of some matrix/array operations and results.
Extensive references can be found in the survey paper (Kolda and Bader,,
2009).
A _tensor_ is a multidimensional array. _Fibers_ of a tensor are the higher
order analogue of matrix rows and columns. A fiber is defined by fixing every
index but one. A matrix column is a mode-1 fiber and a matrix row is a mode-2
fiber. Third-order tensors have column, row, and tube fibers, respectively. We
next review some important operators that transform a tensor into a
vector/matrix. The _vec operator_ stacks the entries of a $D$-dimensional
tensor ${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times p_{D}}}$
into a column vector. Specifically, an entry $b_{i_{1}\ldots i_{D}}$ maps to
the $j$-th entry of $\mathrm{vec}\,{\bm{B}}$ where
$j=1+\sum_{d=1}^{D}(i_{d}-1)\prod_{d^{\prime}=1}^{d-1}p_{d^{\prime}}$. For
instance, when $D=2$, the matrix entry at cell $(i_{1},i_{2})$ maps to
position $j=1+i_{1}-1+(i_{2}-1)p_{1}=i_{1}+(i_{2}-1)p_{1}$, which is
consistent with the more familiar $\mathrm{vec}$ operator on a matrix. The
_mode- $d$ matricization_, ${\bm{B}}_{(d)}$, maps a tensor ${\bm{B}}$ into a
$p_{d}\times\prod_{d^{\prime}\neq d}p_{d^{\prime}}$ matrix such that the
$(i_{1},\ldots,i_{D})$ element of the array ${\bm{B}}$ maps to the $(i_{d},j)$
element of the matrix ${\bm{B}}_{(d)}$, where $j=1+\sum_{d^{\prime}\neq
d}(i_{d^{\prime}}-1)\prod_{d^{\prime\prime}<d^{\prime},d^{\prime\prime}\neq
d}p_{d^{\prime\prime}}$. When $D=1$, we observe that $\mathrm{vec}\,{\bm{B}}$
is the same as vectorizing the mode-1 matricization ${\bm{B}}_{(1)}$. The
_mode-( $d,d^{\prime}$) matricization_
${\bm{B}}_{(dd^{\prime})}\in\mathrm{I\\!R}\mathit{{}^{p_{d}p_{d^{\prime}}\times\prod_{d^{\prime\prime}\neq
d,d^{\prime}}p_{d^{\prime\prime}}}}$ is defined in a similar fashion. We then
define the _mode- $d$ multiplication_ of the tensor ${\bm{B}}$ with a matrix
${\bm{U}}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times q}}$, denoted by
${\bm{B}}\times_{d}{\bm{U}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times
q\times\cdots\times p_{D}}}$, as the multiplication of the mode-$d$ fibers of
${\bm{B}}$ by ${\bm{U}}$. In other words, the mode-$d$ matricization of
${\bm{B}}\times_{d}{\bm{U}}$ is ${\bm{U}}{\bm{B}}_{(d)}$.
We also review two properties of a tensor ${\bm{B}}$ that admits a Tucker
decomposition (3). The mode-$d$ matricization of ${\bm{B}}$ can be expresses
as
$\displaystyle{\bm{B}}_{(d)}={\bm{B}}_{d}{\bm{G}}_{(d)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1})^{\mbox{\tiny{\sf
T}}},$
where $\otimes$ denotes the Kronecker product of matrices. If applying the
$\mathrm{vec}$ operator to ${\bm{B}}$, then
$\displaystyle\mathrm{vec}{\bm{B}}=\mathrm{vec}{\bm{B}}_{(1)}=\mathrm{vec}({\bm{B}}_{1}{\bm{G}}_{(1)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{2})^{\mbox{\tiny{\sf
T}}})=({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1})\mathrm{vec}{\bm{G}}.$
These two properties are useful for our subsequent Tucker regression
development.
### 2.2 Tucker Regression Model
We elaborate on the Tucker tensor regression model introduced in Section 1. We
assume that $Y$ belongs to an exponential family with probability mass
function or density (McCullagh and Nelder,, 1983),
$\displaystyle
p(y_{i}|\theta_{i},\phi)=\exp\left\\{\frac{y_{i}\theta_{i}-b(\theta_{i})}{a(\phi)}+c(y_{i},\phi)\right\\}$
with the first two moments $E(Y_{i})=\mu_{i}=b^{\prime}(\theta_{i})$ and
$\mathrm{Var}(Y_{i})=\sigma_{i}^{2}=b^{\prime\prime}(\theta_{i})a_{i}(\phi)$.
$\theta$ and $\phi>0$ are, respectively, called the natural and dispersion
parameters. We assume the systematic part of GLM is of the form
$\displaystyle g(\mu)=\eta=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle\sum_{r_{1}=1}^{R_{1}}\cdots\sum_{r_{D}=1}^{R_{D}}g_{r_{1},\ldots,r_{D}}\mbox{\boldmath$\beta$}_{1}^{(r_{1})}\circ\cdots\circ\mbox{\boldmath$\beta$}_{D}^{(r_{D})},{\bm{X}}\rangle.$
(4)
That is, we impose a Tucker structure on the array coefficient ${\bm{B}}$. We
make a few remarks. First, in this article, we consider the problem of
estimating the core tensor ${\bm{G}}$ and factor matrices ${\bm{B}}_{d}$
simultaneously given the response $Y$ and covariates ${\bm{X}}$ and
${\bm{Z}}$. This can be viewed as a _supervised_ version of the classical
unsupervised Tucker decomposition. It is also a supervised version of
principal components analysis for higher-order multidimensional array. Unlike
a two-stage solution that first performs principal components analysis and
then fits a regression model, the basis (principal components) ${\bm{B}}_{d}$
in our models are estimated under the guidance (supervision) of the response
variable. Second, the CP model of Zhou et al., (2013) corresponds to a special
case of the Tucker model (4) with
$g_{r_{1},\ldots,r_{D}}=1_{\\{r_{1}=\cdots=r_{D}\\}}$ and
$R_{1}=\ldots=R_{D}=R$. In other words, the CP model is a specific Tucker
model with a super-diagonal core tensor ${\bm{G}}$. The CP model has a rank at
most $R$ while the general Tucker model can have a rank as high as $R^{D}$. We
will further compare the two model sizes in Section 2.4.
### 2.3 Duality and Tensor Basis Pursuit
Next we investigate a duality regarding the inner product between a general
tensor and a tensor that admits a Tucker decomposition.
###### Lemma 1 (Duality).
Suppose a tensor ${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times
p_{D}}}$ admits Tucker decomposition
${\bm{B}}=\llbracket{\bm{G}};{\bm{B}}_{1},\ldots,{\bm{B}}_{D}\rrbracket$.
Then, for any tensor
${\bm{X}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times p_{D}}}$,
$\langle{\bm{B}},{\bm{X}}\rangle=\langle{\bm{G}},\tilde{\bm{X}}\rangle$, where
$\tilde{\bm{X}}$ admits a Tucker decomposition
$\tilde{\bm{X}}=\llbracket{\bm{X}};{\bm{B}}_{1}^{\mbox{\tiny{\sf
T}}},\ldots,{\bm{B}}_{D}^{\mbox{\tiny{\sf T}}}\rrbracket$.
This duality gives some important insights to the Tucker tensor regression
model. First, if we consider
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R_{d}}}$ as fixed and
known basis matrices, then Lemma 1 says fitting the Tucker tensor regression
model (4) is equivalent to fitting a tensor regression model in ${\bm{G}}$
with the _transformed_ data
$\tilde{\bm{X}}=\llbracket{\bm{X}};{\bm{B}}_{1}^{\mbox{\tiny{\sf
T}}},\ldots,{\bm{B}}_{D}^{\mbox{\tiny{\sf
T}}}\rrbracket\in\mathrm{I\\!R}\mathit{{}^{R_{1}\times\cdots\times R_{D}}}$.
When $R_{d}\ll p_{d}$, the transformed data $\tilde{\bm{X}}$ effectively
_downsize_ the original data. We will further illustrate this downsizing
feature in the real data analysis in Section 6.4. Second, in applications
where the numbers of basis vectors $R_{d}$ are unknown, we can utilize
possibly over-complete basis matrices ${\bm{B}}_{d}$ such that $R_{d}\geq
p_{d}$, and then estimate ${\bm{G}}$ with sparsity regularizations. This leads
to a tensor version of the classical basis pursuit problem (Chen et al.,,
2001). Take fMRI data as an example. We can adopt the wavelet basis for the
three image dimensions and the Fourier basis for the time dimension.
Regularization on ${\bm{G}}$ can be achieved by either imposing a low rank
decomposition (CP or Tucker) on ${\bm{G}}$ (hard thresholding) or penalized
regression (soft thresholding). We will investigate Tucker regression
regularization in details in Section 5.
### 2.4 Model Size: Tucker vs CP
In this section we investigate the size of the Tucker tensor model. Comparison
with the size of the CP tensor model helps gain better understanding of both
models. In addition, it provides a base for data adaptive selection of
appropriate orders in a Tucker model.
First we quickly review the number of free parameters $p_{\text{C}}$ for a CP
model ${\bm{B}}=\llbracket{\bm{B}}_{1},\ldots,{\bm{B}}_{d}\rrbracket$, with
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R}}$. For $D=2$,
$p_{\text{C}}=R(p_{1}+p_{2})-R^{2}$, and for $D>2$,
$p_{\text{C}}=R(\sum_{d=1}^{D}p_{d}-D+1)$. For $D=2$, the term $-R^{2}$
adjusts for the nonsingular transformation indeterminacy for model
identifiability; for $D>2$, the term $R(-D+1)$ adjusts for the scaling
indeterminacy in the CP decomposition. See Zhou et al., (2013) for more
details. Following similar arguments, we obtain that the number of free
parameters $p_{\text{T}}$ in a Tucker model
${\bm{B}}=\llbracket{\bm{G}};{\bm{B}}_{1},\ldots,{\bm{B}}_{d}\rrbracket$, with
${\bm{G}}\in\mathrm{I\\!R}\mathit{{}^{R_{1}\times\cdots\times R_{d}}}$ and
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R_{d}}}$, is
$\displaystyle
p_{\text{T}}=\sum_{d=1}^{D}p_{d}R_{d}+\prod_{d=1}^{D}R_{d}-\sum_{d=1}^{D}R_{d}^{2},$
for any $D$. Here the term -$\sum_{d=1}^{D}R_{d}^{2}$ adjusts for the non-
singular transformation indeterminancy in the Tucker decomposition. We
summarize these results in Table 1.
Next we compare the two model sizes (degrees of freedom) under an additional
assumption that $R_{1}=\cdots=R_{d}=R$. The difference becomes:
$\displaystyle p_{\text{T}}-p_{\text{C}}=\begin{cases}0&\textrm{ when }D=2,\\\
R(R-1)(R-2)&\textrm{ when }D=3,\\\ R(R^{3}-4R+3)&\textrm{ when }D=4,\\\
R(R^{D-1}-DR+D-1)&\textrm{ when }D>4.\end{cases}$
Based on this formula, when $D=2$, the Tucker model is essentially the same as
the CP model. When $D=3$, Tucker has the same number of parameters as CP for
$R=1$ or $R=2$, but costs $R(R-1)(R-2)$ more parameters for $R>2$. When $D>3$,
Tucker and CP are the same for $R=1$, but Tucker costs substantially more
parameters than CP for $R>2$. For instance, when $D=4$ and $R=3$, Tucker model
takes 54 more parameters than the CP model. However, one should bear in mind
that the above discussion assumes $R_{1}=\cdots=R_{d}=R$. In reality, Tucker
could require _less_ free parameters than CP, as shown in the illustrative
example given in Section 1, since Tucker is more flexible and allows different
order $R_{d}$ along each dimension.
Table 1: Number of free parameters in Tucker and CP models. | CP | Tucker
---|---|---
$D=2$ | $R(p_{1}+p_{2})-R^{2}$ | $p_{1}R_{1}+p_{2}R_{2}+R_{1}R_{2}-R_{1}^{2}-R_{2}^{2}$
$D>2$ | $R(\sum_{d}p_{d}-D+1)$ | $\sum_{d}p_{d}R_{d}+\prod_{d}R_{d}-\sum_{d}R_{d}^{2}$
Figure 1 shows an example with $D=3$ dimensional array covariates. Half of the
true signal (brain activity map) ${\bm{B}}$ is displayed in the left panel,
which is by no means a low rank signal. Suppose 3D images ${\bm{X}}_{i}$ are
taken on $n=1,000$ subjects. We simulate image traits ${\bm{X}}_{i}$ from
independent standard normals and quantitative traits $Y_{i}$ from independent
normals with mean $\langle{\bm{X}}_{i},{\bm{B}}\rangle$ and unit variance.
Given the limited sample size, the hope is to infer a reasonable low rank
approximation to the activity map from the 3D image covariates. The right
panel displays the model deviance versus the degrees of freedom of a series of
CP and Tucker model estimates. The CP model is estimated at ranks
$R=1,\ldots,5$. The Tucker model is fitted at orders
$(R_{1},R_{2},R_{3})=(1,1,1)$, $(2,2,2)$, $(3,3,3)$, $(4,4,3)$, $(4,4,4)$,
$(5,4,4)$, $(5,5,4)$, and $(5,5,5)$. We see from the plot that, under the same
number of free parameters, the Tucker model could generally achieve a better
model fit with a smaller deviance. (Note that the deviance is in the log
scale, so a small discrepancy between the two lines translates to a large
value of difference in deviance.)
$\begin{array}[]{cc}\includegraphics[width=166.2212pt]{fig_skull_half}&\includegraphics[width=166.2212pt]{fig_skull_dev_vs_dof}\end{array}$
Figure 1: Left: half of the true signal array ${\bm{B}}$. Right: Deviances of
CP regression estimates at $R=1,\ldots,5$, and Tucker regression estimates at
orders $(R_{1},R_{2},R_{3})=(1,1,1)$, $(2,2,2)$, $(3,3,3)$, $(4,4,3)$,
$(4,4,4)$, $(5,4,4)$, $(5,5,4)$, and $(5,5,5)$. The sample size is $n=1000$.
The explicit model size formula of the Tucker model is also useful for
choosing appropriate orders $R_{d}$’s along each direction given data. This
can be treated as a model selection problem, and we can employ a typical model
selection criterion, e.g., Bayesian information criterion (BIC). It is of the
form: $-2\log\ell+\log(n)p_{e}$, where $\ell$ is the log-likelihood, and
$p_{e}=p_{\text{T}}$ is the effective number of parameters of the Tucker model
as given in Table 1. We will illustrate this BIC criterion in the numerical
Section 6.1, and will discuss some heuristic guidelines of selecting orders in
Section 6.4.
## 3 Estimation
We pursue the maximum likelihood estimation (MLE) for the Tucker tensor
regression model and develop a scalable estimation algorithm in this section.
The key observation is that, although the systematic part (4) is not linear in
${\bm{G}}$ and ${\bm{B}}_{d}$ _jointly_ , it is linear in them _separately_.
This naturally suggests a block relaxation algorithm, which updates each
factor matrix ${\bm{B}}_{d}$ and the core tensor ${\bm{G}}$ _alternately_.
The algorithm consists of two core steps. First, when updating
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times R_{d}}}$ with the rest
${\bm{B}}_{d^{\prime}}$’s and ${\bm{G}}$ fixed , we rewrite the array inner
product in (4) as
$\displaystyle\langle{\bm{B}},{\bm{X}}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bm{B}}_{(d)},{\bm{X}}_{(d)}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bm{B}}_{d}{\bm{G}}_{(d)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1})^{\mbox{\tiny{\sf
T}}},{\bm{X}}_{(d)}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bm{B}}_{d},{\bm{X}}_{(d)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1}){\bm{G}}_{(d)}^{\mbox{\tiny{\sf
T}}}\rangle.$
Then the problem turns into a GLM regression with ${\bm{B}}_{d}$ as the
“parameter” and the term
${\bm{X}}_{(d)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1}){\bm{G}}_{(d)}^{\mbox{\tiny{\sf
T}}}$ as the “predictor”. It is a low dimensional GLM with only $p_{d}R_{d}$
parameters and thus is easy to solve. Second, when updating
${\bm{G}}\in\mathrm{I\\!R}\mathit{{}^{R_{1}\times\cdots\times R_{D}}}$ with
all ${\bm{B}}_{d}$’s fixed,
$\displaystyle\langle{\bm{B}},{\bm{X}}\rangle$ $\displaystyle=$
$\displaystyle\langle\mathrm{vec}{\bm{B}},\mathrm{vec}{\bm{X}}\rangle$
$\displaystyle=$
$\displaystyle\langle({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1})\mathrm{vec}{\bm{G}},\mathrm{vec}{\bm{X}}\rangle$
$\displaystyle=$
$\displaystyle\langle\mathrm{vec}{\bm{G}},({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1})^{\mbox{\tiny{\sf
T}}}\mathrm{vec}{\bm{X}}\rangle.$
This implies a GLM regression with $\mathrm{vec}{\bm{G}}$ as the “parameter”
and the term $({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1})^{\mbox{\tiny{\sf
T}}}\mathrm{vec}{\bm{X}}$ as the ”predictor”. Again this is a low dimensional
regression problem with $\prod_{d}R_{d}$ parameters. For completeness, we
summarize the above alternating estimation procedure in Algorithm 1. The
orthogonality between the columns of factor matrices ${\bm{B}}_{d}$ is not
enforced as in unsupervised HOSVD, because our primary goal is approximating
tensor signal instead of finding the principal components along each mode.
Initialize:
$\mbox{\boldmath$\gamma$}^{(0)}=\mbox{argmax}_{\mbox{\boldmath$\gamma$}}\,\ell(\mbox{\boldmath$\gamma$},{\bf
0},\ldots,{\bf 0})$, ${\bm{B}}_{d}^{(0)}\in$
$\mathrm{I\\!R}\mathit{{}^{p_{d}\times R_{d}}}$ a random matrix for
$d=1,\ldots,D$, and
${\bm{G}}^{(0)}\in\mathrm{I\\!R}\mathit{{}^{R_{1}\times\cdots\times R_{D}}}$ a
random matrix.
repeat
for $d=1,\ldots,D$ do
${\bm{B}}_{d}^{(t+1)}=\mbox{argmax}_{{\bm{B}}_{d}}\,\ell(\mbox{\boldmath$\gamma$}^{(t)},{\bm{B}}_{1}^{(t+1)},\ldots,{\bm{B}}_{d-1}^{(t+1)},{\bm{B}}_{d},{\bm{B}}_{d+1}^{(t)},\ldots,{\bm{B}}_{D}^{(t)},{\bm{G}}^{(t)})$
end for
${\bm{G}}^{(t+1)}=\mbox{argmax}_{{\bm{G}}}\,\ell(\mbox{\boldmath$\gamma$}^{(t)},{\bm{B}}_{1}^{(t+1)},\ldots,{\bm{B}}_{D}^{(t+1)},{\bm{G}})$
$\mbox{\boldmath$\gamma$}^{(t+1)}=\mbox{argmax}_{\mbox{\boldmath$\gamma$}}\,\ell(\mbox{\boldmath$\gamma$},{\bm{B}}_{1}^{(t+1)},\ldots,{\bm{B}}_{D}^{(t+1)},{\bm{G}}^{(t+1)})$
until
$\ell(\mbox{\boldmath$\theta$}^{(t+1)})-\ell(\mbox{\boldmath$\theta$}^{(t)})<\epsilon$
Algorithm 1 Block relaxation algorithm for fitting the Tucker tensor
regression.
Next we study the convergence properties of the proposed algorithm. As the
block relaxation algorithm monotonically increases the objective value, the
stopping criterion is well-defined and the convergence properties of iterates
follow from the standard theory for monotone algorithms (de Leeuw,, 1994;
Lange,, 2010). The proof of next result is given in the Appendix.
###### Proposition 1.
Assume (i) the log-likelihood function $\ell$ is continuous, coercive, i.e.,
the set
$\\{\mbox{\boldmath$\theta$}:\ell(\mbox{\boldmath$\theta$})\geq\ell(\mbox{\boldmath$\theta$}^{(0)})\\}$
is compact, and bounded above, (ii) the objective function in each block
update of Algorithm 1 is strictly concave, and (iii) the set of stationary
points (modulo nonsingular transformation indeterminacy) of
$\ell(\mbox{\boldmath$\gamma$},{\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ are
isolated. We have the following results.
1. 1.
(Global Convergence) The sequence
$\mbox{\boldmath$\theta$}^{(t)}=(\mbox{\boldmath$\gamma$}^{(t)},{\bm{G}}^{(t)},{\bm{B}}_{1}^{(t)},\ldots,{\bm{B}}_{D}^{(t)})$
generated by Algorithm 1 converges to a stationary point of
$\ell(\mbox{\boldmath$\gamma$},{\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$.
2. 2.
(Local Convergence) Let
$\mbox{\boldmath$\theta$}^{(\infty)}=(\mbox{\boldmath$\gamma$}^{(\infty)},{\bm{G}}^{(\infty)},{\bm{B}}_{1}^{(\infty)},\ldots,{\bm{B}}_{D}^{(\infty)})$
be a strict local maximum of $\ell$. The iterates generated by Algorithm 1 are
locally attracted to $\mbox{\boldmath$\theta$}^{(\infty)}$ for
$\mbox{\boldmath$\theta$}^{(0)}$ sufficiently close to
$\mbox{\boldmath$\theta$}^{(\infty)}$.
## 4 Statistical Theory
In this section we study the usual large $n$ asymptotics of the proposed
Tucker tensor regression. Regularization is treated in the next section for
the small or moderate $n$ cases. For simplicity, we drop the classical
covariate ${\bm{Z}}$ in this section, but all the results can be
straightforwardly extended to include ${\bm{Z}}$. We also remark that,
although the usually limited sample size of neuroimging studies makes the
large $n$ asymptotics seem irrelevant, we still believe such an asymptotic
investigation important, for several reasons. First, when the sample size $n$
is considerably larger than the effective number of parameters $p_{\text{T}}$,
the asymptotic study tells us that the model is consistently estimating the
best Tucker structure approximation to the full array model in the sense of
Kullback-Liebler distance. Second, the explicit formula for score and
information are not only useful for asymptotic theory but also for
computation, while the identifiability issue has to be properly dealt with for
the given model. Finally, the regular asymptotics can be of practical
relevance, for instance, can be useful in a likelihood ratio type test in a
replication study.
### 4.1 Score and Information
We first derive the score and information for the tensor regression model,
which are essential for statistical estimation and inference. The following
standard calculus notations are used. For a scalar function $f$, $\nabla f$ is
the (column) gradient vector, $df=[\nabla f]^{\mbox{\tiny{\sf T}}}$ is the
differential, and $d^{2}f$ is the Hessian matrix. For a multivariate function
$g:\mathrm{I\\!R}\mathit{{}^{p}}\mapsto\mathrm{I\\!R}\mathit{{}^{q}}$,
$Dg\in\mathrm{I\\!R}\mathit{{}^{p\times q}}$ denotes the Jacobian matrix
holding partial derivatives $\frac{\partial g_{j}}{\partial x_{i}}$. We start
from the Jacobian and Hessian of the systematic part $\eta\equiv g(\mu)$ in
(4).
###### Lemma 2.
1. 1.
The gradient
$\nabla\eta({\bm{B}}_{1},\ldots,{\bm{B}}_{D})\in\mathrm{I\\!R}\mathit{{}^{\prod_{d}R_{d}+\sum_{d=1}^{D}p_{d}R_{d}}}$
is
$\displaystyle\nabla\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})=[{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\,\,{\bm{J}}_{2}\,\,\cdots\,\,{\bm{J}}_{D}]^{\mbox{\tiny{\sf
T}}}(\mathrm{vec}{\bm{X}}),$
where ${\bm{J}}_{d}\in\mathrm{I\\!R}\mathit{{}^{\prod_{d=1}^{D}p_{d}\times
p_{d}R_{d}}}$ is the Jacobian
$\displaystyle{\bm{J}}_{d}=D{\bm{B}}({\bm{B}}_{d})=\mbox{\boldmath$\Pi$}_{d}\\{[({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1}){\bm{G}}_{(d)}^{\mbox{\tiny{\sf
T}}}]\otimes{\bm{I}}_{p_{d}}\\}$ (5)
and $\mbox{\boldmath$\Pi$}_{d}$ is the
$(\prod_{d=1}^{D}p_{d})$-by-$(\prod_{d=1}^{D}p_{d})$ permutation matrix that
reorders $\mathrm{vec}{\bm{B}}_{(d)}$ to obtain $\mathrm{vec}{\bm{B}}$, i.e.,
$\mathrm{vec}{\bm{B}}=\mbox{\boldmath$\Pi$}_{d}\,\mathrm{vec}{\bm{B}}_{(d)}.$
2. 2.
Let the Hessian
$d^{2}\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})\in\mathrm{I\\!R}\mathit{{}^{(\prod_{d}R_{d}+\sum_{d}p_{d}R_{d})\times(\prod_{d}R_{d}+\sum_{d}p_{d}R_{d})}}$
be partitioned into four blocks
${\bm{H}}_{{\bm{G}},{\bm{G}}}\in\mathrm{I\\!R}\mathit{{}^{\prod_{d}R_{d}\times\prod_{d}R_{d}}}$,
${\bm{H}}_{{\bm{G}},{\bm{B}}}={\bm{H}}_{{\bm{B}},{\bm{G}}}^{\mbox{\tiny{\sf
T}}}\in\mathrm{I\\!R}\mathit{{}^{\prod_{d}R_{d}\times\sum_{d}p_{d}R_{d}}}$ and
${\bm{H}}_{{\bm{B}},{\bm{B}}}\in\mathrm{I\\!R}\mathit{{}^{\sum_{d}p_{d}R_{d}\times\sum_{d}p_{d}R_{d}}}$.
Then ${\bm{H}}_{{\bm{G}},{\bm{G}}}={\bf 0}$, ${\bm{H}}_{{\bm{G}},{\bm{B}}}$
has entries
$\displaystyle h_{(r_{1},\ldots,r_{D}),(i_{d},s_{d})}$ $\displaystyle=$
$\displaystyle
1_{\\{r_{d}=s_{d}\\}}\sum_{j_{d}=i_{d}}x_{j_{1},\ldots,j_{D}}\prod_{d^{\prime}\neq
d}\beta_{j_{d^{\prime}}}^{(r_{d^{\prime}})},$
and ${\bm{H}}_{{\bm{B}},{\bm{B}}}$ has entries
$\displaystyle h_{(i_{d},r_{d}),(i_{d^{\prime}},r_{d^{\prime}})}=1_{\\{d\neq
d^{\prime}\\}}\sum_{j_{d}=i_{d},j_{d^{\prime}}=i_{d^{\prime}}}x_{j_{1},\ldots,j_{D}}\sum_{s_{d}=r_{d},s_{d^{\prime}}=r_{d^{\prime}}}g_{s_{1},\ldots,s_{D}}\prod_{d^{\prime\prime}\neq
d,d^{\prime}}\beta_{j_{d^{\prime\prime}}}^{(s_{d^{\prime\prime}})}.$
Furthermore, ${\bm{H}}_{{\bm{B}},{\bm{B}}}$ can be partitioned in $D^{2}$ sub-
blocks as
$\displaystyle\left(\begin{array}[]{cccc}{\bf 0}&*&*&*\\\ {\bm{H}}_{21}&{\bf
0}&*&*\\\ \vdots&\vdots&\ddots&*\\\ {\bm{H}}_{D1}&{\bm{H}}_{D2}&\cdots&{\bf
0}\end{array}\right).$
The elements of sub-block
${\bm{H}}_{dd^{\prime}}\in\mathrm{I\\!R}\mathit{{}^{p_{d}R_{d}\times
p_{d^{\prime}}R_{d^{\prime}}}}$ can be retrieved from the matrix
${\bm{X}}_{(dd^{\prime})}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{d^{\prime}+1}\otimes{\bm{B}}_{d^{\prime}-1}\otimes\cdots\otimes{\bm{B}}_{1}){\bm{G}}_{(dd^{\prime})}^{\mbox{\tiny{\sf
T}}}.$
${\bm{H}}_{{\bm{G}},{\bm{B}}}$ can be partitioned into $D$ sub-blocks as
$({\bm{H}}_{1},\ldots,{\bm{H}}_{D})$. The sub-block
${\bm{H}}_{d}\in\mathrm{I\\!R}\mathit{{}^{\prod_{d}R_{d}\times p_{d}R_{d}}}$
has at most $p_{d}\prod_{d}R_{d}$ nonzero entries which can be retrieved from
the matrix
$\displaystyle{\bm{X}}_{(d)}({\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{d+1}\otimes{\bm{B}}_{d-1}\otimes\cdots\otimes{\bm{B}}_{1}).$
Let $\ell({\bm{B}}_{1},\ldots,{\bm{B}}_{D}|y,{\bm{x}})=\ln
p(y|{\bm{x}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ be the log-density of GLM.
Next result derives the score function, Hessian, and Fisher information of the
Tucker tensor regression model.
###### Proposition 2.
Consider the tensor regression model defined by (2.2) and (4).
1. 1.
The score function (or score vector) is
$\displaystyle\nabla\ell({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})=\frac{(y-\mu)\mu^{\prime}(\eta)}{\sigma^{2}}\nabla\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$
(7)
with $\nabla\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ given in Lemma 2.
2. 2.
The Hessian of the log-density $\ell$ is
$\displaystyle H({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ (8)
$\displaystyle=$
$\displaystyle-\left[\frac{[\mu^{\prime}(\eta)]^{2}}{\sigma^{2}}-\frac{(y-\mu)\theta^{\prime\prime}(\eta)}{\sigma^{2}}\right]\nabla\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})d\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$
$\displaystyle+\frac{(y-\mu)\theta^{\prime}(\eta)}{\sigma^{2}}d^{2}\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D}),$
with $d^{2}\eta$ defined in Lemma 2.
3. 3.
The Fisher information matrix is
$\displaystyle{\bm{I}}({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ (9)
$\displaystyle=$ $\displaystyle
E[-H({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})]$ $\displaystyle=$
$\displaystyle\mathrm{Var}[\nabla\ell({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})d\ell({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})]$
$\displaystyle=$
$\displaystyle\frac{[\mu^{\prime}(\eta)]^{2}}{\sigma^{2}}[{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\ldots{\bm{J}}_{D}]^{\mbox{\tiny{\sf
T}}}(\mathrm{vec}{\bm{X}})(\mathrm{vec}{\bm{X}})^{\mbox{\tiny{\sf
T}}}[{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\ldots{\bm{J}}_{D}].$
Remark 2.1: For canonical link, $\theta=\eta$, $\theta^{\prime}(\eta)=1$,
$\theta^{\prime\prime}(\eta)=0$, and the second term of Hessian vanishes. For
the classical GLM with linear systematic part ($D=1$),
$d^{2}\eta({\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})$ is zero and thus the
third term of Hessian vanishes. For the classical GLM ($D=1$) with canonical
link, both second and third terms of the Hessian vanish and thus the Hessian
is non-stochastic, coinciding with the information matrix.
### 4.2 Identifiability
The Tucker decomposition (3) is unidentifiable due to the nonsingular
transformation indeterminacy. That is
$\displaystyle\llbracket{\bm{G}};{\bm{B}}_{1},\ldots,{\bm{B}}_{D}\rrbracket=\llbracket{\bm{G}}\times_{1}{\bm{O}}_{1}^{-1}\times\cdots\times_{D}{\bm{O}}_{D}^{-1};{\bm{B}}_{1}{\bm{O}}_{1},\ldots,{\bm{B}}_{D}{\bm{O}}_{D}\rrbracket$
for any nonsingular matrices
${\bm{O}}_{d}\in\mathrm{I\\!R}\mathit{{}^{R_{d}\times R_{d}}}$. This implies
that the number of free parameters for a Tucker model is
$\sum_{d}p_{d}R_{d}+\prod_{d}R_{d}-\sum_{d}R_{d}^{2}$, with the last term
adjusting for nonsingular indeterminacy. Therefore the Tucker model is
identifiable only in terms of the equivalency classes.
For asymptotic consistency and normality, it is necessary to adopt a specific
constrained parameterization. It is common to impose the orthonormality
constraint on the factor matrices ${\bm{B}}_{d}^{\mbox{\tiny{\sf
T}}}{\bm{B}}_{d}={\bm{I}}_{R_{d}}$, $d=1,\ldots,D$. However the resulting
parameter space is a manifold and much harder to deal with. We adopt an
alternative parameterization that fixes the entries of the first $R_{d}$ rows
of ${\bm{B}}_{d}$ to be ones
$\displaystyle{\cal{\bm{B}}}=\\{\llbracket{\bm{G}};{\bm{B}}_{1},\ldots,{\bm{B}}_{D}\rrbracket:\beta_{i_{d}}^{(r)}=1,i_{d}=1,\ldots,R_{d},d=1,\ldots,D\\}.$
The formulae for score, Hessian and information in Proposition 2 require
changes accordingly. The entries in the first $R_{d}$ rows of ${\bm{B}}_{d}$
are fixed at ones and their corresponding entries, rows and columns in score,
Hessian and information need to be deleted. Choice of the restricted space
$\mathcal{{\bm{B}}}$ is obviously arbitrary, and excludes arrays with any
entries in the first rows of ${\bm{B}}_{d}$ equal to zeros. However the set of
such exceptional arrays has Lebesgue measure zero. In specific applications,
subject knowledge may suggest alternative restrictions on the parameters.
Given a finite sample size, conditions for global identifiability of
parameters are in general hard to obtain except in the linear case ($D=1$).
Local identifiability essentially requires linear independence between the
“collapsed” vectors
$[{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\ldots{\bm{J}}_{D}]^{\mbox{\tiny{\sf
T}}}\mathrm{vec}{\bm{x}}_{i}\in\mathrm{I\\!R}\mathit{{}^{\sum_{d}p_{d}R_{d}+\prod_{d}R_{d}-\sum_{d}R_{d}^{2}}}$.
###### Proposition 3 (Identifiability).
Given iid data points $\\{(y_{i},{\bm{x}}_{i}),i=1,\ldots,n\\}$ from the
Tucker tensor regression model. Let ${\bm{B}}_{0}\in\mathcal{{\bm{B}}}$ be a
parameter point and assume there exists an open neighborhood of ${\bm{B}}_{0}$
in which the information matrix has a constant rank. Then ${\bm{B}}_{0}$ is
locally identifiable if and only if
$\displaystyle
I({\bm{B}}_{0})=[{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\ldots{\bm{J}}_{D}]^{\mbox{\tiny{\sf
T}}}\left[\sum_{i=1}^{n}\frac{\mu^{\prime}(\eta_{i})^{2}}{\sigma_{i}^{2}}(\mathrm{vec}\,{\bm{x}}_{i})(\mathrm{vec}\,{\bm{x}}_{i})^{\mbox{\tiny{\sf
T}}}\right][{\bm{B}}_{D}\otimes\cdots\otimes{\bm{B}}_{1}\,\,{\bm{J}}_{1}\ldots{\bm{J}}_{D}]$
is nonsingular.
### 4.3 Asymptotics
The asymptotics for tensor regression follow from those for MLE or
M-estimation. The key observation is that the nonlinear part of tensor model
(4) is a degree-$D$ polynomial of parameters and the collection of polynomials
$\\{\langle{\bm{B}},{\bm{X}}\rangle,{\bm{B}}\in\mathcal{{\bm{B}}}\\}$ form a
Vapnik-C̆ervonenkis (VC) class. Then the classical uniform convergence theory
applies (van der Vaart,, 1998). For asymptotic normality, we need to establish
that the log-likelihood function of tensor regression model is quadratic mean
differentiable (Lehmann and Romano,, 2005). A sketch of the proof is given in
the Appendix.
###### Theorem 1.
Assume ${\bm{B}}_{0}\in\mathcal{{\bm{B}}}$ is (globally) identifiable up to
permutation and the array covariates ${\bm{X}}_{i}$ are iid from a bounded
underlying distribution.
1. 1.
(Consistency) The MLE is consistent, i.e., $\hat{\bm{B}}_{n}$ converges to
${\bm{B}}_{0}$ in probability, in following models. (1) Normal tensor
regression with a compact parameter space
$\mathcal{{\bm{B}}}_{0}\subset\mathcal{{\bm{B}}}$. (2) Binary tensor
regression. (3) Poisson tensor regression with a compact parameter space
$\mathcal{{\bm{B}}}_{0}\subset\mathcal{{\bm{B}}}$.
2. 2.
(Asymptotic Normality) For an interior point
${\bm{B}}_{0}\in\mathcal{{\bm{B}}}$ with nonsingular information matrix
${\bm{I}}({\bm{B}}_{0})$ (9) and $\hat{\bm{B}}_{n}$ is consistent,
$\sqrt{n}(\mathrm{vec}\hat{\bm{B}}_{n}-\mathrm{vec}{\bm{B}}_{0})$ converges in
distribution to a normal with mean zero and covariance matrix
${\bm{I}}^{-1}({\bm{B}}_{0})$.
In practice it is rare that the true regression coefficient
${\bm{B}}_{\text{true}}\in\mathrm{I\\!R}\mathit{{}^{p_{1}\times\cdots\times
p_{D}}}$ is exactly a low rank tensor. However the MLE of the rank-$R$ tensor
model converges to the maximizer of function
$M({\bm{B}})=\mathbb{P}_{{\bm{B}}_{\text{true}}}\ln p_{{\bm{B}}}$ or
equivalently
$\mathbb{P}_{{\bm{B}}_{\text{true}}}\ln(p_{{\bm{B}}}/p_{{\bm{B}}_{\text{true}}})$.
In other words, the MLE consistently estimates the best approximation (among
models in ${\cal{\bm{B}}}$) of ${\bm{B}}_{\text{true}}$ in the sense of
Kullback-Leibler distance.
## 5 Regularized Estimation
Regularization plays a crucial role in neuroimaging analysis for several
reasons. First, even after substantial dimension reduction by imposing a
Tucker structure, the number of parameters $p_{\text{T}}$ can still exceed the
number of observations $n$. Second, even when $n>p_{\text{T}}$, regularization
could potentially be useful for stabilizing the estimates and improving the
risk property. Finally, regularization is an effective way to incorporate
prior scientific knowledge about brain structures. For instance, it may
sometimes be reasonable to impose symmetry on the parameters along the coronal
plane for MRI images.
In our context of Tucker regularized regression, there are two possible types
of regularizations, one on the core tensor ${\bm{G}}$ _only_ , and the other
on both ${\bm{G}}$ and ${\bm{B}}_{d}$ _simultaneously_. Which regularization
to use depends on the practical purpose of a scientific study. In this
section, we illustrate the regularization on the core tensor, which
simultaneously achieves sparsity in the number of outer products in Tucker
decomposition (3) and shrinkage. Toward that purpose, we propose to maximize
the regularized log-likelihood
$\displaystyle\ell(\mbox{\boldmath$\gamma$},{\bm{G}},{\bm{B}}_{1},\ldots,{\bm{B}}_{D})-\sum_{r_{1},\ldots,r_{D}}P_{\eta}(|g_{r_{1},\ldots,r_{D}}|,\lambda),$
where $P_{\eta}(|x|,\lambda)$ is a scalar penalty function, $\lambda$ is the
penalty tuning parameter, and $\eta$ is an index for the penalty family. Note
that the penalty term above only involves elements of the core tensor, and
thus regularization on ${\bm{G}}$ only. This formulation includes a large
class of penalty functions, including power family (Frank and Friedman,,
1993), where $P_{\eta}(|x|,\lambda)=\lambda|x|^{\eta}$, $\eta\in(0,2]$, and in
particular lasso (Tibshirani,, 1996) ($\eta=1$) and ridge ($\eta=2$); elastic
net (Zou and Hastie,, 2005), where
$P_{\eta}(|x|,\lambda)=\lambda[(\eta-1)x^{2}/2+(2-\eta)|x|]$, $\eta\in[1,2]$;
SCAD (Fan and Li,, 2001), where
$\partial/\partial|x|P_{\eta}(|x|,\lambda)=\lambda\left\\{1_{\\{|x|\leq\lambda\\}}+(\eta\lambda-|x|)_{+}/(\eta-1)\lambda
1_{\\{|x|>\lambda\\}}\right\\}$, $\eta>2$; and MC+ penalty (Zhang,, 2010),
where
$P_{\eta}(|x|,\lambda)=\left\\{\lambda|x|-x^{2}/(2\eta)\right\\}1_{\\{|x|<\eta\lambda\\}}+0.5\lambda^{2}\eta
1_{\\{|x|\geq\eta\lambda\\}}$, among many others.
Two aspects of the proposed regularized Tucker regression, parameter
estimation and tuning, deserve some discussion. For regularized estimation, it
incurs only slight changes in Algorithm 1. That is, when updating ${\bm{G}}$,
we simply fit a penalized GLM regression problem,
$\displaystyle{\bm{G}}^{(t+1)}=\mbox{argmax}_{{\bm{G}}}\,\ell(\mbox{\boldmath$\gamma$}^{(t)},{\bm{B}}_{1}^{(t+1)},\ldots,{\bm{B}}_{D}^{(t+1)},{\bm{G}})-\sum_{r_{1},\ldots,r_{D}}P_{\eta}(|g_{r_{1},\ldots,r_{D}}|,\lambda),$
for which many software packages exist. Our implementation utilizes an
efficient Matlab toolbox for sparse regression (Zhou et al.,, 2011). Other
steps of Algorithm 1 remain unchanged. For the regularization to remain
legitimate, we constrain the column norms of ${\bm{B}}_{d}$ to be one when
updating factor matrices ${\bm{B}}_{d}$. For parameter tuning, one can either
use the general cross validation approach, or employ Bayesian information
criterion to tune the penalty parameter $\lambda$.
## 6 Numerical Study
We have carried out intensive numerical experiments to study the finite sample
performance of the Tucker regression. Our simulations focus on three aspects:
first, we demonstrate the capacity of the Tucker regression in identifying
various shapes of signals; second, we study the consistency property of the
method by gradually increasing the sample size; third, we compare the
performance of the Tucker regression with the CP regression of Zhou et al.,
(2013). We also examine a real MRI imaging data to illustrate the Tucker
downsizing and to further compare the two tensor models.
### 6.1 Identification of Various Shapes of Signals
In our first example, we demonstrate that the proposed Tucker regression
model, though with substantial reduction in dimension, can manage to identify
a range of two dimensional signal shapes with varying ranks. In Figure 2, we
list the 2D signals ${\bm{B}}\in\mathrm{I\\!R}\mathit{{}^{64\times 64}}$ in
the first row, along with the estimates by Tucker tensor models in the second
to fourth rows with orders $(1,1),(2,2)$ and $(3,3)$, respectively. Note that,
since the orders along both dimensions are made equal, the Tucker model is to
perform essentially the same as a CP model in this example, and the results
are presented here for completeness. We will examine differences of the two
models in later examples. The regular covariate vector
${\bm{Z}}\in\mathrm{I\\!R}\mathit{{}^{5}}$ and image covariate
${\bm{X}}\in\mathrm{I\\!R}\mathit{{}^{64\times 64}}$ are randomly generated
with all elements being independent standard normals. The response $Y$ is
generated from a normal model with mean
$\mu=\mbox{\boldmath$\gamma$}^{\mbox{\tiny{\sf
T}}}{\bm{Z}}+\langle{\bm{B}},{\bm{X}}\rangle$ and variance
$\textrm{var}(\mu)/10$. The vector coefficient $\mbox{\boldmath$\gamma$}={\bf
1}_{5}$, and the coefficient array ${\bm{B}}$ is binary, with the signal
region equal to one and the rest zero. Note that this problem differs from the
usual edge detection or object recognition in imaging processing (Qiu,, 2005,
2007). In our setup, all elements of the image ${\bm{X}}$ follow the same
distribution. The signal region is defined through the coefficient matrix
${\bm{B}}$ and needs to be inferred from the relation between $Y$ and
${\bm{X}}$ after adjusting for ${\bm{Z}}$. It is clearly see in Figure 2 that,
the Tucker model yields a sound recovery of the true signals, even for those
of high rank or natural shape, e.g., “disk” and “butterfly”. We also
illustrate in the plot the BIC criterion in Section 2.4.
---
Figure 2: True and recovered image signals by Tucker regression. The matrix
variate has size 64 by 64 with entries generated as independent standard
normals. The regression coefficient for each entry is either 0 (white) or 1
(black). The sample size is 1000. TR$(r)$ means estimate from the Tucker
regression with an $r$-by-$r$ core tensor.
### 6.2 Performance with Increasing Sample Size
In our second example, we continue to employ a similar model as in Figure 2
but with a three dimensional image covariate. The dimension of ${\bm{X}}$ is
set as $p_{1}\times p_{2}\times p_{3}$, with $p_{1}=p_{2}=p_{3}=16$ and $32$,
respectively. The signal array ${\bm{B}}$ is generated from a Tucker
structure, with the elements of core tensor ${\bm{G}}$ and the factor matrices
${\bm{B}}$’s all coming from independent standard normals. The dimension of
the core tensor ${\bm{G}}$ is set as $R_{1}\times R_{2}\times R_{3}$, with
$R_{1}=R_{2}=R_{3}=2,5$, and $8$, respectively. We gradually increase the
sample size, starting with an $n$ that is in hundred and no smaller than the
degrees of freedom of the generating model. We aim to achieve two purposes
with this example: first, we verify the consistency property of the proposed
estimator, and second, we gain some practical knowledge about the estimation
accuracy with different values of the sample size. Figure 3 summarizes the
results. It is clearly seen that the estimation improves with the increasing
sample size. Meanwhile, we observe that, unless the core tensor dimension is
small, one would require a relatively large sample size to achieve a good
estimation accuracy. This is not surprising though, considering the number of
parameters of the model and that regularization is not employed here. The
proposed tensor regression approach has been primarily designed for imaging
studies with a reasonably large number of subjects. Recently, a number of such
large-scale brain imaging studies are emerging. For instance, the Attention
Deficit Hyperactivity Disorder Sample Initiative (ADHD,, 2013) consists of
over 900 participants from eight imaging centers with both MRI and fMRI
images, as well as their clinical information. Another example is the
Alzheimer’s Disease Neuroimaging Initiative (ADNI,, 2013) database, which
accumulates over 3,000 participants with MRI, fMRI and genomics data. In
addition, regularization discussed in Section 5 and the Tucker downsizing in
Section 2.3 can both help improve estimation given a limited sample size.
$p_{1}=p_{2}=p_{3}=16$ | $p_{1}=p_{2}=p_{3}=32$
---|---
|
|
|
Figure 3: Root mean squared error (RMSE) of the tensor parameter estimate
versus the sample size. Reported are the average and standard deviation of
RMSE based on 100 data replications. Top: $R_{1}=R_{2}=R_{3}=2$; Middle:
$R_{1}=R_{2}=R_{3}=5$; Bottom: $R_{1}=R_{2}=R_{3}=8$.
### 6.3 Comparison of the Tucker and CP Models
In our third example, we focus on comparison between the Tucker tensor model
with the CP tensor model of Zhou et al., (2013). We generate a normal
response, and the 3D signal array ${\bm{B}}$ with dimensions
$p_{1},p_{2},p_{3}$ and the $d$-ranks $r_{1},r_{2},r_{3}$. Here, the $d$-rank
is defined as the column rank of the mode-$d$ matricization ${\bm{B}}_{(d)}$
of ${\bm{B}}$. We set $p_{1}=p_{2}=p_{3}=16$ and $32$, and
$(r_{1},r_{2},r_{3})=(5,3,3),(8,4,4)$ and $(10,5,5)$, respectively. The sample
size is 2000. We fit a Tucker model with $R_{d}=r_{d}$, and a CP model with
$R=\max r_{d}$, $d=1,2,3$. We report in Table 2 the degrees of freedom of the
two models under different setup, as well as the root mean squared error
(RMSE) out of 100 data replications. It is seen that the Tucker model requires
a smaller number of free parameters, while it achieves a more accurate
estimation compared to the CP model. Such advantages come from the flexibility
of the Tucker decomposition that permits different orders $R_{d}$ along
directions.
Table 2: Comparison of the Tucker and CP models. Reported are the average and standard deviation (in the parenthesis) of the root mean squared error, all based on 100 data replications. Dimension | Criterion | Model | $(5,3,3)$ | $(8,4,4)$ | $(10,5,5)$
---|---|---|---|---|---
$16\times 16\times 16$ | Df | Tucker | 178 | 288 | 420
| | CP | 230 | 368 | 460
| RMSE | Tucker | 0.202 (0.013) | 0.379 (0.017) | 0.728 (0.030)
| | CP | 0.287 (0.033) | 1.030 (0.081) | 2.858 (0.133)
$32\times 32\times 32$ | Df | Tucker | 354 | 544 | 740
| | CP | 470 | 752 | 940
| RMSE | Tucker | 0.288 (0.013) | 0.570 (0.023) | 1.236 (0.045)
| | CP | 0.392 (0.046) | 1.927 (0.172) | 16.238 (3.867)
### 6.4 Attention Deficit Hyperactivity Disorder Data Analysis
We analyze the attention deficit hyperactivity disorder (ADHD) data from the
ADHD-200 Sample Initiative (ADHD,, 2013) to illustrate our proposed method as
well as the Tucker downsizing. ADHD is a common childhood disorder and can
continue through adolescence and adulthood. Symptoms include difficulty in
staying focused and paying attention, difficulty in controlling behavior, and
over-activity. The data set that we analyzed is part of the ADHD-200 Global
Competition data sets. It was pre-partitioned into a training data of 770
subjects and a testing data of 197 subjects. We removed those subjects with
missing observations or poor image quality, resulting in 762 training subjects
and 169 testing subjects. In the training set, there were 280 combined ADHD
subjects, 482 normal controls, and the case-control ratio is about 3:5. In the
testing set, there were 76 combined ADHD subjects, 93 normal controls, and the
case-control ratio is about 4:5. T1-weighted images were acquired for each
subject, and were preprocessed by standard steps. The data we used is obtained
from the Neuro Bureau after preprocessing (the Burner data,
http://neurobureau.projects.nitrc.org/ADHD200/Data.html). In addition to the
MRI image predictor, we also include the subjects’ age and handiness as
regular covariates. The response is the binary diagnosis status.
The original image size was $p_{1}\times p_{2}\times p_{3}=121\times 145\times
121$. We employ the Tucker downsizing in Section 2.3. More specifically, we
first choose a wavelet basis for
${\bm{B}}_{d}\in\mathrm{I\\!R}\mathit{{}^{p_{d}\times\tilde{p}_{d}}}$, then
transform the image predictor from ${\bm{X}}$ to
$\tilde{\bm{X}}=\llbracket{\bm{X}};{\bm{B}}_{1}^{\mbox{\tiny{\sf
T}}},\ldots,{\bm{B}}_{D}^{\mbox{\tiny{\sf T}}}\rrbracket$. We pre-specify the
values of $\tilde{p}_{d}$’s that are about tenth of the original dimensions
$p_{d}$, and equivalently, we fit a Tucker tensor regression with the image
predictor dimension downsized to
$\tilde{p}_{1}\times\tilde{p}_{2}\times\tilde{p}_{3}$. In our example, we have
experimented with a set of values of $\tilde{p}_{d}$’s, and the results are
qualitatively similar. We report two sets, $\tilde{p}_{1}=12$,
$\tilde{p}_{2}=14$, $\tilde{p}_{3}=12$, and $\tilde{p}_{1}=10$,
$\tilde{p}_{2}=12$, $\tilde{p}_{3}=10$. We have also experimented with the
Haar wavelet basis (Daubechies D2) and the Daubechies D4 wavelet basis, which
again show similar qualitative patterns.
For $\tilde{p}_{1}=12,\tilde{p}_{2}=14,\tilde{p}_{3}=12$, we fit a Tucker
tensor model with $R_{1}=R_{2}=R_{3}=3$, resulting in 114 free parameters, and
fit a CP tensor model with $R=4$, resulting in 144 free parameters. For
$\tilde{p}_{1}=10,\tilde{p}_{2}=12,\tilde{p}_{3}=10$, we fit a Tucker tensor
model with $R_{1}=R_{2}=2$ and $R_{3}=3$, resulting in 71 free parameters, and
fit a CP tensor model with $R=4$, resulting in 120 free parameters. We have
chosen those orders based on the following considerations. First, the number
of free parameters of the Tucker and CP models are comparable. Second, at each
step of GLM model fit, we ensure that the ratio between the sample size $n$
and the number of parameters under estimation in that step
$\tilde{p}_{d}\times R_{d}$ satisfies a heuristic rule of greater than two in
normal models and greater than five in logistic models. In the Tucker model,
we also ensure the ratio between $n$ and the number of parameters in the core
tensor estimation $\prod_{d}R_{d}$ satisfies this rule. We note that this
selection of Tucker orders is heuristic; however, it seems to be a useful
guideline especially when the data is noisy. We also fit a regularized Tucker
model and a regularized CP model with the same orders, while the penalty
parameter is tuned based on 5-fold cross validation of the training data.
We evaluate each model by comparing the misclassification error rate on the
independent testing set. The results are shown in Table 3. We see from the
table that, the regularized Tucker model performs the best, which echoes the
findings in our simulations above. We also remark that, considering the fact
that the ratio of case-control is about 4:5 in the testing data, the
misclassification rate from 0.32 to 0.36 achieved by the regularized Tucker
model indicates a fairly sound classification accuracy. On the other hand, we
note that, a key advantage of our proposed approach is its capability of
suggesting a useful model rather than the classification accuracy per se. This
is different from black-box type machine learning based imaging classifiers.
Table 3: ADHD testing data misclassification error. Basis | Reduced dimension | Reg-Tucker | Reg-CP | Tucker | CP
---|---|---|---|---|---
Haar (D2) | $12\times 14\times 12$ | 0.361 | 0.367 | 0.379 | 0.438
| $10\times 12\times 10$ | 0.343 | 0.390 | 0.379 | 0.408
Daubechies (D4) | $12\times 14\times 12$ | 0.337 | 0.385 | 0.385 | 0.414
| $10\times 12\times 10$ | 0.320 | 0.396 | 0.367 | 0.373
It is also of interest to compare the run times of the two tensor model
fittings. We record the run times of fitting the Tucker and CP models with the
ADHD training data in Table 4. They are comparable.
Table 4: ADHD model fitting run time (in seconds). Basis | Reduced dimension | Reg-Tucker | Reg-CP | Tucker | CP
---|---|---|---|---|---
Haar (D2) | $12\times 14\times 12$ | 3.68 | 4.39 | 31.25 | 22.43
| $10\times 12\times 10$ | 1.36 | 2.79 | 9.08 | 25.10
Daubechies (D4) | $12\times 14\times 12$ | 3.30 | 2.18 | 16.87 | 26.34
| $10\times 12\times 10$ | 1.92 | 1.90 | 9.96 | 17.10
## 7 Discussion
We have proposed a tensor regression model based on the Tucker decomposition.
Including the CP tensor regression (Zhou et al.,, 2013) as a special case,
Tucker model provides a more flexible framework for regression with imaging
covariates. We develop a fast estimation algorithm, a general regularization
procedure, and the associated asymptotic properties. In addition, we provide a
detailed comparison, both analytically and numerically, of the Tucker and CP
tensor models.
In real imaging analysis, the signal hardly has an exact low rank. On the
other hand, given the limited sample size, a low rank estimate often provides
a reasonable approximation to the true signal. This is why the low rank models
such as the Tucker and CP could offer a sound recovery of even a complex
signal.
The tensor regression framework established in this article is general enough
to encompass a large number of potential extensions, including but not limited
to imaging multi-modality analysis, imaging classification, and longitudinal
imaging analysis. These extensions consist of our future research.
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|
arxiv-papers
| 2013-04-20T15:04:08 |
2024-09-04T02:49:44.644032
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoshan Li and Hua Zhou and Lexin Li",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1304.5637"
}
|
1304.5743
|
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# Genericity versus expressivity - an exercise in semantic interoperable
research information systems for Web Science
Christophe Guéret
Tamy Chambers
Linda Reijnhoudt
Frank van der Most
Andrea Scharnhorst
Data Archiving and Networked Services (DANS) Den Haag, The Netherlands
[email protected] School of Library & Information Sciences
Indiana University, USA [email protected] Data Archiving and Networked
Services (DANS) Den Haag, The Netherlands [email protected] Data
Archiving and Networked Services (DANS) Den Haag, The Netherlands
[email protected] Data Archiving and Networked Services (DANS)
Den Haag, The Netherlands [email protected]
###### Abstract
The web does not only enable new forms of science, it also creates new
possibilities to study science and new digital scholarship. This paper brings
together multiple perspectives: from individual researchers seeking the best
options to display their activities and market their skills on the academic
job market; to academic institutions, national funding agencies, and countries
needing to monitor the science system and account for public money spending.
We also address the research interests aimed at better understanding the self-
organising and complex nature of the science system through researcher
tracing, the identification of the emergence of new fields, and knowledge
discovery using large-data mining and non-linear dynamics. In particular this
paper draws attention to the need for standardisation and data
interoperability in the area of research information as an indispensable pre-
condition for any science modelling. We discuss which levels of complexity are
needed to provide a globally, interoperable, and expressive data
infrastructure for research information. With possible dynamic science model
applications in mind, we introduce the need for a “middle-range” level of
complexity for data representation and propose a conceptual model for research
data based on a core international ontology with national and local
extensions.
###### keywords:
Ontology; VIVO; NARCIS; CERIF; CRIS; Linked Data
###### category:
I.2.4 Knowledge Representation Formalisms and Methods
## 1 Introduction
The science of the 21st century, to a large extent is team science [10],
operating globally, often cross disciplinary, and fully entangled with the
web. The study of science as a specific, complex, and social system has been
addressed by many research disciplines for quite some time. The availability
of digital traces of scholarly activities at unknown scale and variety,
together with the urgent need to monitor and control this growing system, is
at heart of knowledge economies and has brought the question how best to
measure, model, and forecast science back on to the research agenda [32].
When reviewing the current models of science, it is clear there is no
consistent framework of science models yet [7]. Existing models are often
driven by the available data. For example, interdisciplinary bibliographic
databases (such as the Web of Science or SCOPUS) use the principle of citation
indexing [17] from the field of scientometrics to analyse the science system
based on formal scholarly communication. Typical output indicators are counts
of publications, citations, and patents. They form the heart of the current
“measurement of science” and have been taken up as data by network science [5]
and Web Science [23].
This specific kind of output is, however, only a tiny fraction of information
on science dynamics. Traditionally, the measurement of science encompasses
input indicators (human capital, expenditure), output indicators, and. where
possible, process information [18]. Research Information Systems, around since
WWII in Europe, are marking the shift to “big science” [29]. However, the
input side to science dynamics, in particular researchers, has been
underrepresented in quantitative science studies for quite some time. This is
partly due to the lack of databases and the problem of author ambiguity in the
existing database [33, 30]. Information on researchers has been mainly
collected, documented, and curated locally at individual scientific
institutions - and in nation-wide research information systems, at least in
European countries.
The emergence of the web has transformed this situation completely. The web
has become an important, if not the most important, information source for
researchers and a platform for collaboration [6]. The extent and diversity of
the traces scholars leave on the web has called for alt metrics [39]. It has
also triggered the development of standards and ontologies capable of
automatically harvesting this wealth of information, beyond existing
traditional bibliographic reference.
The wealth of information provided on the web about researcher activities and
their relations carries the potential for new insights into the global
research landscape. But we are not yet at the point where this data can be
both expressive enough to be useful and easy enough to consume.
To illustrate the current situation we display the conceptual space of
communities dealing with research information in form of four mind maps (c.f.
Figure 1). In the upper left corner we brought together concepts, which are
relevant from the perspective of scientific career research and often
conducted qualitatively, with rich factual evidence, which is hardly
interoperable or scalable. For this mind node we drew on current discussions
and first results [37] in a FP7 framework programme ACUMEN, Academic Careers
understood by Measurements and Norms (see http://research-acumen.eu/), where
sociologists and scientometricians work together. In the right lower corner we
display the main classes of an ontology for research information
(VIVO111http://www.vivoweg.org) developed in the US. In the upper right
corner, the main tables of a Dutch Research Information Database (NOD-NARCIS)
are displayed, and in the lower left corner is a selection of information and
concepts which can be retrieved using different fields in one of the leading
cross-disciplinary bibliographic databases - the Web of Knowledge. Although,
the mind map sketches are different in nature, from formal schemes to
collection of aspects, this illustration shows their difference in size,
granularity, scope, and expression or semantics.
Figure 1: Conceptual space of four different communities dealing with research
information. The variation among these mind maps illustrates the difference in
size, granularity, scope, and expression of the different information systems
with which they are associated.
In this work we argue for the need of a scalable, interoperable, and multi-
layered data representation model for research information system (RIS).
Science of science and modeling of science dynamics raise and fall with a
consistent measurement system for the sciences. The contributions of this
paper are as follows:
* •
A highlight of information loss happening when expressing data with generic
ontologies;
* •
The introductions of the notion of levels of semantic agreement for expressing
research data;
* •
A multi-layered ontology based on the above definition.
The remainder of the paper describes the landscape of research data
publication before diving into the details of a specific Dutch case. We
thereafter introduce our proposed multi-layer conceptual model for a research
ontology and conclude in its potential for documenting research.
## 2 Current landscape of RIS
### 2.1 Publishing research data
In order to publish re-usable research data, one has to think in terms of
standards and publication media. While the web imposes itself as the
publication platform, the question of standards remains open and has been long
investigated.
First efforts in standardisation have been undertaken from the traditional
research information communities. One example is the “CERIF” standard
developed by EuroCRIS222http://www.eurocris.org. This standard defines a set
of generic classes and properties used to describe research data. The
serialisation format used for the data is XML, although an RDF version is
being considered333http://spi-fm.uca.es/neologism/cerif. The content
management system (CMS) “METIS”, popular in the Netherlands, uses this
standard to store and expose research data. This standard has also been used
for the Dutch portal “NARCIS”444http://www.narcis.nl/.
The Web of Linked Data is a way of combining the publication platform and the
standards. More recent efforts have been made in this direction via a number
of ontologies and publication platforms. The initiative
LinkedUniversities555http://linkeduniversities.org/lu/ provides a reference
towards these systems and highlights their practical use. VIVO a United States
based open source semantic web application is another such a system. The
application both describes and publishes data, using RDF to encode the data
and OWL for the logical structure.In addition to its own classes and
properties, the VIVO ontology incorpates other standard ontolgies thus
increasing its interoperability [8]. However, the ontology relies heavly on
the US academic model which limits its ability to accurately represent
researchers in other systems.
VIVO and CERIF based CMS have been successfully put in use at many
institutions. Still, the landscape of research information is very scattered
and far from being connected. One of the reasons for this is a lack of
agreement upon semantics for the data. Efforts have been made to align VIVO
and CERIF [25] but the main problem remains that data publishers essentially
have to choose between using a globally agreed upon representation, which is
less expressive as a result of covering a vast amount of heterogeneous
information (CERIF), or a very expressive and specialised ontology (VIVO),
which is difficult to map to other ontologies of similar complexity.
### 2.2 The Dutch case
In the Netherlands, we find the following situation. All 13 universities (14
with the Open University) use a system called METIS to register and document
their research information [14]. In practice, information is usually entered
in METIS centrally by a person in the administration although, sometimes
individual accounts to METIS are created. Aside from those unconnected local
implementations of one system, higher education in the Netherlands embraced
the Open Access Movement with a project called DARE. This lead to an open
repository for scientific publications. Moreover, a web portal to Dutch
research information exists - NARCIS - which harvests publications from open
repositories, but also entails a very well curated (and still manually edited)
research information database (NOD) with information about the scientific
staff of about 400 university and outside university research institutions
[13, 31].
As Oskam and other Dutch researchers already pointed out in 2006, “the
researcher is key” [27]. Outside of institutional RIS this idea is prolific in
Web 2.0. platforms such as Mendeley and Academia.edu. They have been designed
around the needs of scholars. General social network sites such as LinkedIn -
which is very popular for professionals in the Netherlands - and Facebook also
profile themselves as outlets for individual researchers. This leads to a
situation where user-content driven systems compete for the limited time and
resources of an individual researcher and where, as a result, snippets of the
oeuvre and academic journey of a researcher can be found at different places,
recorded in different standards, and with different accuracy. The question
raised in the 2006 paper: “How can we make the CRIS666CRIS stands for “Current
Research Information System” a valuable and attractive (career) tool for the
researcher?” [27, p. 168] is still waiting to be answered in a standardized
way.
The purpose of documentation of science (and of careers of researchers) has
grown far beyond the effective information exchange. Research evaluation
relies heavily on indicators computed (semi) automatically from databases and
the web. Currently, individual careers of researchers are very much influenced
by indicators which are built on activities for which large amounts of
standardised data are available. Prominent examples are journal impact factor
or the H index. But, a researcher is not just a “paper publication machine”.
Grant acquisition is another important “currency” in the academic market - for
individuals on the job market, as well as, for institutions competing for
funding. Teaching is an area which is monitored locally and institutionally,
but for which no cross-institutional databases exist. Moreover, researchers
are no longer loyal to one institution, one country, or one discipline for
their whole life. There is an increasing need for cross-discipline and cross-
institutional mapping of whole careers.
### 2.3 Tracing scientific careers
Projects such as ACUMEN look into current practices of evaluation and peer
review to empower the individual researcher and develop guidelines for how
best to present your academic profile to the outside world. “ACUMEN” is the
acronym for Academic Careers Understood through MEasurements and Norms. In
this project, we analyse the use of a wide range of indicators - ranging from
traditional bibliometrics to alt-metrics and metrics based on Web 2.0 - for
the evaluation of the work of individual academics. One of the author of the
present work, Frank van der Most, also conducted interviews to investigate the
impact or influence of evaluations on individual careers. For his work the
following events are of interesting in tracking an academic career:
* •
Birth of the academic;
* •
Acquisition of diploma’s and titles, in particular MA diplomas (and
equivalents), PhD/Dr. diplomas, habilitiation, professorships of sorts and
levels;
* •
Jobs, in universities and academic research institutes, but also in non-
academic organisations. The latter is interesting because people move in, out,
and sometimes back into academia;
* •
Particular functions within or as part of the job(s): director of studies
(teaching), research-coordinator, head of department, dean, vice dean (for
research, education, or other), vice-chancellor/rector, board member of
faculty/school/university/institute;
* •
Launch of start-ups/spin-outs or people’s own companies. It could simply be a
form of employment, but start-ups or own companies may indicate economic or
other societal value of academic work;
* •
Prizes;
* •
Retirement and decease.
For the study of the impact, or influence, of evaluations an overview of
someone’s career is necessary to “locate” influential evaluations. This
“location” has multiple dimensions. One is the calendar time, i.e. on which
date or in which year did an influential evaluation take place. Based on time,
geographic, and institutional location the context of a particular evaluation
event can be reconstructed. Scientific careers follow patterns which are
influenced by current regimes of science dynamics (including evaluations).
Another important dimension concerns the location of an evaluation (or any
event) within someone’s career. If two academics apply for the same job, the
location in time and place is the same, but if one is an early-career
researcher and the other is halfway through his/her career, this clearly makes
a large difference to how their applications are being evaluated and how the
evaluation results are likely to impact their respective careers. A rejection
may have a bigger impact on the early-career researcher than on the mid-career
researcher.
Another ACUMEN sub-project investigates gender effects of evaluations and
includes an analysis of performance indicators on research careers. This is
planned to be a statistical analysis which would require some form of career
descriptions.
One of ACUMEN’s central aims is to identify and investigate bibliometric
indicators that can be used in the evaluation of the work of individual
researchers. A major point discussed in the ACUMEN workshops is the
realisation that researchers have a career or a life-cycle which
contextualises the values of bibliometric indicators.
Although the events listed above are interesting for ACUMEN, these events, or
a sub-set or extension thereof, is likely to be interesting to many career
studies. For example, productivity-studies would relate academic production of
texts [11, 15, 24], courses taught, and other outputs to someone’s career
stage or career paths. An academic’s epistemic development (their research
agenda) could be studied in relation to career stages [22] or mobility.
To be able to trace the co-evolution of individual career paths and the social
process of science for larger part of science, one would need a different kind
of information depending on the study being undertaken.
## 3 Towards a core research vocabulary
The challenge when designing a standard for sharing data is to make it generic
enough so that aggregation makes sense, while being specific enough so
institutions can express the data they need.
As it is highlighted by the two most popular search tools, consuming data
exposed via VIVO from a number of external sources777See
http://nrn.cns.iu.edu/ and http://beta.vivosearch.org/ at the international
level, only the most general concepts such as “People” make sense. On the
opposite, the search features offered by a national portal such as NARCIS
proposes a number of refined search criteria. These two extremes of the data
mash-up scale show that depending on the study being done, different levels of
semantics agreement are likely to be put into use.
In contrary to XML schemas, Semantic Web technologies make it possible to
express data using an highly specified model while also making it available
using a more general model. The technology of particular importance here is
“reasoning”, that is the entailment of other factual valid information from
the facts already contained in the knowledge base. For instance, if an RDF
knowledge base contains a fact assessing that “A is a researcher” and another
stating that “Every researcher is a person”, the system will infer that “A is
a person”.
Leveraging this, it is possible to extend ontologies by refining the
definition of classes and properties. The most refined versions of the
concepts will inherit from their parents. We argue that for research
information systems, three levels are necessary (see Figure 2). First, an
international level containing a set of core concepts that can be used to
build data mash-up on an international scale. Then, a national level extending
the previous core level with concepts commonly agreed upon nation wide (e.g.
positions). Last, an institutional level where every institution is free to
further refine the previous level with its own concepts and properties that
matter to its network.
Figure 2: The proposed model of multi-layer ontology and its trade-off between
scope and expressivity. At the lowest level, institutionaly defined semantics
have the highest expressivity but the lowest scope.
As a feasibility assessment and to propose a first model, we hereafter
introduce a core ontology and two national extensions. This proposal is based
on related work, existing ontologies, and our personal experience but stands
more as a first iteration of work in progress rather than a definitive model.
### 3.1 Conceptual models
Conceptual models allows for the representation of classes and properties of a
knowledge base, along with their relations, in an abstracted way. The proposed
conceptual models that we hereafter introduce are not dependent on the
technical solution implementing them. There is however, as highlighted
previously, an advantage in using Semantic Web technologies for this. This
point is discussed in details in the following, after the introduction and the
description of the three proposed conceptual models.
#### 3.1.1 Core model
The model depicted in Figure 3 is a proposal for a core research ontology
based on the work being done on CERIF, the VIVO ontology, the Core
vocabularies [4], and the data needs of ACUMEN. As part of its goal to study
the scientific career through the research data made available, ACUMEN needs a
number of information related to individuals, such as but not limited to:
* •
Grants/project applications - both applied and granted. This in relation to
persons (applicants of various sorts) and organisations (applying/receiving
institutes, main and sub-contractors, funding institutes);
* •
Skills. For instance, “Leadership” or “Artificial Intelligence”. There is no
limit to the definition and several thesaurus could be implied;
* •
Networks or network relations. Relation between persons and organisations, but
also between persons and results are of particular importance;
* •
Memberships of scientific associations or academies;
* •
Conferences visited or organised.
The model contains classes to define individuals, projects, scientific output,
positions and tasks. A generic “Relation” can be established between authors
and papers, or teachers and courses taught. The exact meaning of the relation
is to be defined either by sub-classes of it or by using the property “role”.
Figure 3: Conceptual model of the core ontology. This model describes the
minimum set of classes, relationships, and properties needed to describe a
natural person and trace his scientific career. These classes can be further
extended by national and local ontologies to account for specificity. As an
example, the coloured classes are extended in two national ontologies in
Figure 4
#### 3.1.2 National extensions
The second level of semantic agreement is that of national extensions. Based
on the core concepts, these extensions allows for the modeling of concepts
actually used in the country - using the language and terminology of that
country. When building such an extension, the main assumption made is that
there is a level of agreement that can be reached on a national basis.
An example of national extension is given in Figure 4. This extension extends
the core “Position” and “Organization” classes to define the type of positions
and organisation commonly found in the Netherlands (Figure 4(a)) and the US
(Figure 4(b)). The classes depicted in the Dutch extension are those found in
NARCIS, and as such represent the union set of all the specific classes used
within the research institutions in the Netherlands888We must note here that
this classes are not defined by an authority but are rather crowd-sourced. A
more accurate, authoritative, list would have to be defined by an national
entity..
(a) Conceptual model of the extension for the Netherlands
(b) Conceptual model of the extension for the US
Figure 4: Example of two national extensions of the core model. These
extensions allow for expressing the particularities found in the national
system while grounding their semantic on the more generic concepts.
It can be observed that the Dutch extensions shows a high level of variety,
with some classes that could be replaced with other model mechanisms, such as
the “part time Hoogleraar” class which is actually a “Hoogleraar” contracted
with less hours.
We also note from Figure 4(b) that the national level has to be kept generic
in the US because of the variation observed locally. In the US, many titles
and/or positions are essentially at the discretion of the individual
institutions (with some direction from the American Association of University
Professors (AAUP)), thus a very detailed national ontology is not appropriate.
However, for countries with a more centralised model and using title and
positions officially described, more detail can be added at this level thus
increasing semantic understanding. The national level allows for this grey
area adaption instead of the current two level “very general” to “very
specific” model.
#### 3.1.3 Local extensions
Local extensions are the most specific level of specification we propose for
this approach. These can be used to specify concepts and relations that are
understood within a given sub community inside a country. For instance, in the
Netherlands, the research institution KNAW defines an additional position
“AkademieHoogleraar” for “Hoogleraar” which are appointed to universities but
directly affiliated to KNAW. This additional position is only used by some
institutions and for this academy - here, the “Akademie” in
“AkademieHoogleraar” implicitly refers to KNAW.
### 3.2 Implementation
Prior to its concrete use, the proposed conceptual models have to be turned
into an RDF based vocabulary. This vocabulary also has to be hosted under a
domain name.
#### 3.2.1 Vocabulary terms
There are a large number of vocabularies published on the Web. The proposed
models can effectively leverage most of their properties and classes from one
of these existing sources of terms, having fewer new terms to introduce. In
particular, the following vocabularies are to be considered:
* •
FOAF999http://xmlns.com/foaf/spec/, for the description of the persons;
* •
BIBO101010http://bibliontology.com/, for the publications;
* •
LODE111111http://linkedevents.org/ontology/, for the description of events;
* •
SKOS121212http://www.w3.org/2004/02/skos/, for the description of thesaurus
terms such as those used to describe researchers’ skills;
* •
PROV-O131313http://www.w3.org/TR/prov-o/, to add additional provenance
information to the data being served.
We also note that, by design, there is a significant overlap between the
conceptual model of Figure 3 and that defined in the Core Vocabularies for
Person, Location and registered Organisations in [4, page 10]. This allows for
the proposed core vocabulary for research to be defined based on these other
core vocabularies defined by JoinUp and formalised by the W3C in the context
of the Working Group on Governmental Linked Data (GLD)
141414http://www.w3.org/2011/gld/wiki/Main_Page.
#### 3.2.2 Ontology hosting
The domain name at which an ontology is being served is, as for the data
itself, often seen as indication of the person, or entity, in charge of
supporting the ontology. To account for this, we envision the hosting of the
core ontology and its extensions done at institutions matching the scope of
the level of agreement. That is, an international organisation for the
international layer, a national organisation for the national layer, and the
institutions themselves for the local extensions. More concretely, such an
hosting plan could be materialised as having: the core ontology being served
by the W3C, the Dutch national ontology by the VSNU151515the association in
charge of the collective labour agreement for Dutch universities and other
cross-institution regulations on salaries and positions, and the local
extension from the KNAW by the KNAW.
## 4 Conclusion
This paper operates at different levels. At the core it proposes a model to
semantically describe data in Research Information Systems in a way which
allows to aggregate but also to deconstruct if needed. It does so based on
experiences with standards and data representation in the past and looking
into very concrete practices - taking a VIVO implementation exercise in the
Netherlands as point of reference and departure. A next shell of
considerations around those specific mappings is added when we incorporate
research outside of the traditional area of scientific information and
documentation. Science and technology studies, science of science, and
scientometrics have produced over decades of insights in the structure and
dynamics of the science system. A wealth of information is available in this
area, most of it case-based evidence. We include the aims and achievements of
an on-going EU FP7 funded project (ACUMEN) which, in itself tries to combine
bibliometric and indicator-based research with interviews, survey, and
literature studies. The target subject of this project is the researcher. It
is also the researcher which is targeted by Research Information Systems, and
it is the researcher which is the innovative driver for science dynamics.
Bibliometric indicators are heavily based on standards, part of them shared
with RIS. What makes the ACUMEN project and the perspective of scientific
career research so interesting for the design of future research information
systems is the identification of factors relevant for career development which
are not yet covered by current standards, databases, or ontologies. The last
and most visionary shell in this paper is to design research information
systems which can be used for science modeling. In the general framework
developed by Borner et al. science models can be developed at different scales
of the science system, from the individual research up to the global science
system; they can differ in geographic coverage, as well as, in scales of time.
In any case, the ideal would be having one data representation which can be
scaled up and down along those different dimensions, and not singular data
samples in incomparable measurement units not relatable for particular areas
of the dynamics of science. Our main argument is to provide a data
representation which is retraceable - if needed - towards its specific roots
and at the same time can be aggregated. In such a “measurement system” we
would find a middle layer of data granularity on which basis complex, non-
linear models can be validated and implemented, to better monitor and
understand the science system.
## 5 Acknowledgments
This work has been supported by the ACUMEN project FP7 framework. We would
like to think our colleagues Ying Ding, Katy Borner, and Chris Baars for their
comments and support during this work.
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* [35] Sheppard, N. Learning how to play nicely: Repositories and cris, 2010. http://www.ariadne.ac.uk/issue64/wrn-repos-2010-05-rpt/ ,Accessed January 25, 2013.
* [36] Shvaiko, P., and Euzenat, J. Ontology matching: State of the art and future challenges. IEEE Trans. Knowl. Data Eng. 25, 1 (2013), 158–176.
* [37] Van der Most, F. The role of evaluations in the development of researchers’ careers. a conceptual frame and research strategy for a comparative study. Poster presented at the conference ‘How to track researchers’ careers.’, Luxembourg, 9-10 February 2012. (unpublished, contact author), 2012\.
* [38] Wouters, P. Academic careers understood through measurements and norms. http://research-acumen.eu/, Accessed January 26, 2013.
* [39] Wouters, P., and Costas, R. Users, narcissism and control - tracking the impact of scholarly publications in the 21 st century. http://www.surffoundation.nl/nl/publicaties/Documents/Users%20narcissism%20and%20control.pdf, Accessed January 25, 2013.
|
arxiv-papers
| 2013-04-21T14:41:23 |
2024-09-04T02:49:44.652869
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christophe Gu\\'eret, Tamy Chambers, Linda Reijnhoudt, Frank van der\n Most, Andrea Scharnhorst",
"submitter": "Andrea Scharnhorst",
"url": "https://arxiv.org/abs/1304.5743"
}
|
1304.5772
|
# Singular 2-webs and Polar Curves
Fernando Etayo111Departamento de Matemáticas, Estadística y Computación.
Facultad de Ciencias. Universidad de Cantabria. Avda. de los Castros, s/n,
39071 Santander, SPAIN. e-mail: [email protected]
###### Abstract
A 2-web in the plane is given by two everywhere transverse 1-foliations. In
this paper we introduce the study of singular 2-webs, given by any two
foliations, which may be tangent in some points. We show that such two
foliations are tangent along a curve, which will be called the polar curve of
the 2-web, and we study the relationship between the contact order of leaves
of both foliations and the singularities of the polar curve.
AMS Classification: 53A60, 53C15.
Keywords: Singular 2-web, Foliations, Pfaffian forms, Paracomplex structure.
## 1 Introduction
A 2-web in the real plane is given by two everywhere transverse 1-dimensional
foliations, i.e., two families of curves such that in any point the curves
passing through it have different tangent lines. These 2-webs are always
locally diffeomorphic to that of vertical and horizontal lines, and then they
have no local invariants. The case of 3-webs is completely different because
of the curvature of the Blaschke-Chern connection, which measures how far the
web is to be hexagonal [2, 6].
A 2-web can be defined by a (1,1) tensor field $F$ of maximum rank such that
$F^{2}=I$, where $I$ denotes the identity tensor field. The eigenspaces
associated to the eigenvalues $\pm 1$ define two distributions, which are
involutive in the case of the real plane. $F$ is said to be a paracomplex
structure.
One could think that 2-webs have no interest. Nevertheless, we want to go into
an unknown landscape, which has not been explored yet, as far as the author
knows. Let us consider two different 1-dimensional foliations in the plane.
What can you say about the set of points in which both foliations are tangent?
If this set is non-empty, we shall say that both foliations define a _singular
2-web_ and the set will be called the _polar curve_ of the singular 2-web. We
use this terminology following a similar idea [3] developed in the context of
a holomorphic foliation in $\mathbb{C}^{2}$, where the polar curve is defined
from the foliation and a direction in the plane, i.e., it is defined from
2-web defined by the holomorphic foliation and the foliation of lines parallel
to the direction.
Polar curves should not be confused with polar foliations. A polar foliation
[1] is a singular foliation in a complete Riemannian manifold such that for
each regular point $p$, there is an immersed submanifold $\Sigma_{p}$, called
section, that passes through $p$ and that meets all the leaves and always
perpendicularly.
In the previous paper [4] we have shown some explicit examples. In the present
one we state some results about the polar curve. The following example is
introduced as a motivating case.
###### Example 1
Let us consider the punctured plane $\mathbb{R}^{2}-\\{(0,0)\\}$ and the
foliations given by
* •
${\cal F}_{1}$ is the set of circles with center $(0,0)$.
* •
${\cal F}_{2}$ is the set of vertical lines.
Obviously, this is a singular 2-web having the horizontal axis as polar curve.
We can prove it by using different techniques: analyzing the associated
distributions, the paracomplex structure and the Pfaffian forms. As is well
known 1-dimensional distributions are always integrable, and then we can work
interchangeably with distributions and foliations. We shall show carefully
these three techniques, because we should choose the best one in order to
ahead more complex situations.
* •
Associated distributions: The tangent vector of the foliation ${\cal F}_{1}$
at the point $(x,y)$ is $X=-y\frac{\partial}{\partial
x}+x\frac{\partial}{\partial y}$. The tangent vector to ${\cal F}_{2}$ is
$Y=\frac{\partial}{\partial y}$. These vectors are colinear in the polar curve
$\\{y=0\\}-\\{(0,0)\\}$.
* •
The paracomplex structure [5] is the $(1,1)$-tensor field $F$ such that their
eigenspaces are the distributions tangent to the foliations. A straightforward
calculation shows that
$F=\left(\begin{array}[]{cc}1&0\\\ -\frac{2x}{y}&-1\end{array}\right)$
One can esaily check that $F^{2}=id;F(X)=X;F(Y)=-Y$. The paracomplex structure
$F$ is well defined in all the punctured plane unless the polar curve
$\\{y=0\\}-\\{(0,0)\\}$. Then the polar curve appears as the set of points
where the structure cannot be defined.
* •
The Pfaffian forms defining ${\cal F}_{1}$ and ${\cal F}_{2}$ are
$\omega=x\,dx+y\,dy$ and $\eta=dx$. The points where these 1-forms are
dependent are those where $0=\omega\wedge\eta=(x\,dx+y\,dy)\wedge
dx=-ydx\wedge dy$, which are those of the polar curve.
The best option is to work with Pfaffian forms, because one has only to check
a product of 1-forms.
## 2 Algebraic foliations and paracomplex structure
As is well known, a foliation is said to be algebraic if it is given by a
1-form $\omega=\omega_{1}(x,y)dx+\omega_{2}(x,y)dy$, where $\omega_{i}(x,y)$
are polynomials. This doesn’t mean that the algebraic curves of the foliation
have to be algebraic curves. For example, the 1-form $\omega=ydx-dy$ is
algebraic and the curves of this foliation are the exponential
$C_{k}=\\{y=ke^{x}\\},k\in\mathbb{R}$.
In the same way, we can say that a (1,1)-tensor field
$F=\frac{\partial}{\partial x^{i}}\otimes F^{i}_{j}dx^{j}$ is algebraic (resp.
rational) if the functions $F^{i}_{j}$ are polynomial (resp. rational).
Then we have,
###### Theorem 2
Let $\omega=\omega_{1}(x,y)dx+\omega_{2}(x,y)dy$ and
$\eta=\eta_{1}(x,y)dx+\eta_{2}(x,y)dy$ be two 1-forms in an open subset of the
real plane. Then,
(1) The polar curve of the singular 2-web defined by $\omega$ and $\eta$ is
the curve $\\{\omega_{1}\eta_{2}-\omega_{2}\eta_{1}=0\\}$. Besides, if the
coefficient functions $\omega_{1},\omega_{2},\eta_{1},\eta_{2}$ are polynomial
of degrees $p,q,r,s$ then the polar curve is an algebraic curve of degree less
or equal to ${\rm max}(p+s,q+r)$.
(2) The paracomplex structure $F$ defined by $\omega$ and $\eta$ is
$F=\frac{1}{-\omega_{2}\eta_{1}+\omega_{1}\eta_{2}}\left(\begin{array}[]{cc}-\eta_{1}\omega_{2}-\omega_{1}\eta_{2}&-2\omega_{2}\eta_{2}\\\
2\omega_{1}\eta_{1}&\eta_{2}\omega_{1}+\omega_{2}\eta_{1}\end{array}\right)$
(3) If $\omega$ and $\eta$ are algebraic, then $F$ is rational.
_Proof_.
(1) Observe that
$\omega\wedge\eta=(\omega_{1}\eta_{2}-\omega_{2}\eta_{1})dx\wedge dy$, thus
showing that both 1-forms are dependent on the curve
$C=\\{\omega_{1}\eta_{2}-\omega_{2}\eta_{1}=0\\}$.
(2) In the points $p\in\mathbb{R}^{2}-\\{C\\}$, where $c$ denotes the polar
curve, one has $T_{p}\mathbb{R}^{2}=(ker\omega)_{p}\oplus(ker\eta)_{p}$. A
basis of $ker\omega$ is $-\omega_{2}\frac{\partial}{\partial
x^{1}}+\omega_{1}\frac{\partial}{\partial x^{2}}=(-\omega_{2},\omega_{1})$. A
basis of $ker\eta$ is $(-\eta_{2},\eta_{1})$. Then, a vector field
$v=v_{1}\frac{\partial}{\partial x^{1}}+v_{2}\frac{\partial}{\partial x^{2}}$
can be decomposed as
$v=\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right)=\alpha\left(\begin{array}[]{c}-\omega_{2}\\\
\omega_{1}\end{array}\right)+\beta\left(\begin{array}[]{c}-\eta_{2}\\\
\eta_{1}\end{array}\right)$
which produces, by using the Cramer rule,
$\alpha=\frac{\mid\begin{array}[]{cc}v_{1}&-\eta_{2}\\\
v_{2}&\eta_{1}\end{array}\mid}{\mid\begin{array}[]{cc}-\omega_{2}&-\eta_{2}\\\
\omega_{1}&\eta_{1}\end{array}\mid}\hskip 14.22636pt;\hskip
14.22636pt\beta=\frac{\mid\begin{array}[]{cc}-\omega_{2}&v_{1}\\\
\omega_{1}&v_{2}\end{array}\mid}{\mid\begin{array}[]{cc}-\omega_{2}&-\eta_{2}\\\
\omega_{1}&\eta_{1}\end{array}\mid}$
Then, the paracomplex structure $F$ defined as $F|_{ker\omega}=I$,
$F|_{ker\eta}=-I$ is that given by
$F(v)=\alpha\left(\begin{array}[]{c}-\omega_{2}\\\
\omega_{1}\end{array}\right)-\beta\left(\begin{array}[]{c}-\eta_{2}\\\
\eta_{1}\end{array}\right)=\frac{1}{-\omega_{2}\eta_{1}+\omega_{1}\eta_{2}}\left(\begin{array}[]{cc}-\eta_{1}\omega_{2}-\omega_{1}\eta_{2}&-2\omega_{2}\eta_{2}\\\
2\omega_{1}\eta_{1}&\eta_{2}\omega_{1}+\omega_{2}\eta_{1}\end{array}\right)\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right)$
(3) It is a direct consequence of (2).
###### Remark 3
Observe that the polar curve corresponds to the locus where the paracomplex
structure $F$ cannot be defined. Observe that formula (2) does not depend on
the algebricity of foliations.
###### Remark 4
If one changes the 1-forms $\omega$ and $\eta$ by proportional 1-forms
$f\omega$ and $g\eta$, f,g beings functions, then the 2-web is the same, and
the paracomplex structure $F$ remains invariable in the above theorem. This is
important: the 1-forms are not uniquely defined, but the paracomplex structure
is uniquely defined, up to sign.
## 3 Singular points of the polar curve
Let us assume that $\omega$ and $\eta$ are two algebraic foliations with polar
curve $C$. As $C$ is an algebraic curve, $C$ may have singular points. The
following examples suggest that singular points of $C$ correspond to the
points where the curves of each foliations have contact of order greater than
two. Remember the classical definitions:
* •
Two curves $\alpha$ and $\beta$ are said to have _contact of order_ $k$ at a
point $p$ if their derivatives of order $0,1,\ldots,k$ coincide at the point
and the derivatives of order $k+1$ are different. We shall denote it as
$ord_{\alpha\beta}(p)=k$.
* •
The multiplicity $mult_{C}(p)$ of a curve C at p is the order of the first
non-vanishing term in the Taylor expansion of f at p, where
$C=\\{f(x,y)=0\\}$. The point is said to be a _regular_ point if
$mult_{C}(p)=1$, and _singular_ if $mult_{C}(p)\geq 2$.
* •
A 1-form $\omega=\omega_{1}(x,y)dx+\omega_{2}(x,y)dy$ is said to define an
_exact differential equation_ if $\frac{\partial\omega_{1}(x,y)}{\partial
y}=\frac{\partial\omega_{2}(x,y)}{\partial x}$. In this case, a curve
$f(x,y)=0$ is an integral curve iff $\frac{\partial f}{\partial
x}=\omega_{1}(x,y)$ and $\frac{\partial f}{\partial y}=\omega_{2}(x,y)$. As it
is well known, multiplying by an integrating factor $\mu(x,y)$ any 1-form can
be transformed into an exact differential equation, although in many cases
obtaining the integrating factor is not easy.
###### Example 5
Let us consider the foliations ${\cal F}_{\omega}$ and ${\cal F}_{\eta}$ given
by the 1-forms and their dual vector fields:
$\omega=y^{2}dx-dy,\;X_{\omega}=\frac{\partial}{\partial
x}+y^{2}\frac{\partial}{\partial y};$
$\eta=-x^{3}dx+dy,\;X_{\eta}=-\frac{\partial}{\partial
x}-x^{3}\frac{\partial}{\partial y}$.
The first one is the set of hyperbolas $\\{y=\frac{1}{k-x},k\in\mathbb{R}\\}$
and the horizontal axis, and the second one that of quartics
$\\{y=k+\frac{x^{4}}{4},k\in\mathbb{R}\\}$. The polar curve is the cubic
$C=\\{y^{2}=x^{3}\\}$, which has a singular point of multiplicity two (a cusp
point) in the origin. The curves of ${\cal F}_{\omega}$ and ${\cal F}_{\eta}$
through the origin are $\\{y=0\\}$ and $\\{y=\frac{x^{4}}{4}\\}$. The last one
has an inflection point in the origin, thus proving that both curves have
contact of order three.
###### Example 6
Let us consider the foliation ${\cal F}_{\alpha}$ given by the 1-form
$\alpha=dy$, and the foliation ${\cal F}_{\eta}$ given by the same 1-form
$\eta$ of the above example. Then de polar curve is $C=\\{x^{3}=0\\}$, which
is the vertical axis, being all of its points singular of multiplicity three,
equal to the contact order of tangent curves of each foliation.
We can state:
###### Theorem 7
Let ${\cal F}_{\omega}$ and ${\cal F}_{\eta}$ be two algebraic foliations
given by two exact differential equations $\omega$ and $\eta$ and let $C$ be
their polar curve. If the curves $\gamma_{\omega}$ and $\gamma_{\eta}$ of
${\cal F}_{\omega}$ and ${\cal F}_{\eta}$ through $p$ have contact of order
$\geq 2$ then $p\in c$ is a singular point of the polar curve. In this case,
$ord_{\gamma_{\omega}\gamma_{\eta}}(p)\geq mult_{C}(p)$.
_Proof_
First we introduce some notation. Let
$\omega=\omega_{1}(x,y)dx+\omega_{2}(x,y)dy$ and
$\eta=\eta_{1}(x,y)dx+\eta_{2}(x,y)dy$ the 1-forms corresponding to ${\cal
F}_{\omega}$ and ${\cal F}_{\eta}$ and let
$C=\\{\omega_{1}\eta_{2}-\omega_{2}\eta_{1}=0\\}=\\{f(x,y)=0\\}$
be their polar curve. Then a point $p\in C$ is singular iff
$\frac{\partial f}{\partial x}(p)=0=\frac{\partial f}{\partial y}(p)$
Thus, $p$ is a singular point iff the following equations hold at $p$:
(1) $\left\\{\begin{array}[]{cc}\frac{\partial\omega_{1}}{\partial
x}\eta_{2}+\omega_{1}\frac{\partial\eta_{2}}{\partial
x}-\frac{\partial\omega_{2}}{\partial
x}\eta_{1}-\omega_{2}\frac{\partial\eta_{1}}{\partial x}&=0\\\ &\\\
\frac{\partial\omega_{1}}{\partial
y}\eta_{2}+\omega_{1}\frac{\partial\eta_{2}}{\partial
y}-\frac{\partial\omega_{2}}{\partial
y}\eta_{1}-\omega_{2}\frac{\partial\eta_{1}}{\partial
y}&=0\end{array}\right\\}$
Let $\gamma_{\omega}$ and $\gamma_{\eta}$ the integral curves of ${\cal
F}_{\omega}$ and ${\cal F}_{\eta}$ through the point $p$. As the 1-forms
$\omega$ and $\eta$ are exact, we have:
(2) $\left\\{\begin{array}[]{cc}\frac{\partial\gamma_{\omega}}{\partial
x}&=\omega_{1}\\\ &\\\ \frac{\partial\gamma_{\omega}}{\partial
y}&=\omega_{2}\end{array}\right\\}\hskip 28.45274pt{\rm and}\hskip
28.45274pt\left\\{\begin{array}[]{cc}\frac{\partial\gamma_{\eta}}{\partial
x}&=\eta_{1}\\\ &\\\ \frac{\partial\gamma_{\eta}}{\partial
y}&=\eta_{2}\end{array}\right\\}$
Then, $ord_{\gamma_{\omega}\gamma_{\eta}}\geq 2$ iff the following equations
hold at $p$:
(3) $\left\\{\begin{array}[]{ccc}\frac{\partial\gamma_{\omega}}{\partial
x}=\frac{\partial\gamma_{\eta}}{\partial
x};&\frac{\partial\gamma_{\omega}}{\partial
y}=\frac{\partial\gamma_{\eta}}{\partial y}&\\\ &&\\\
\frac{\partial^{2}\gamma_{\omega}}{\partial
x^{2}}=\frac{\partial^{2}\gamma_{\eta}}{\partial
x^{2}};&\frac{\partial^{2}\gamma_{\omega}}{\partial x\partial
y}=\frac{\partial^{2}\gamma_{\eta}}{\partial x\partial
y};&\frac{\partial^{2}\gamma_{\omega}}{\partial
x^{y}}=\frac{\partial^{2}\gamma_{\eta}}{\partial x^{y}}\end{array}\right\\}$
Then, taking into account equation (2) one easily check that
$(3)\Rightarrow(1)$, thus proving the first statement of the theorem. The
second one follows from a similar reasoning.
The condition in the above theorem of being $\omega$ and $\eta$ exact
differential equations is necessary, as the following example in
$\mathbb{R}^{2}-\\{(\pm 1,0)\\}$ shows:
###### Example 8
Let ${\cal F}$ be the vertical axis and the set of circles $C_{a}$, with
center in the horizontal axis passing through the points $(a,0)$ and
$(1/a,0)$, where $a\neq\pm 1$.
Points $(a,0),(1/a,0),(1,0),(-1,0)$ define a harmonic quadruple of points,
i.e., their cross ratio is -1. The center of $C_{a}$ is the point
$\frac{a^{2}+1}{2a}$ and the radius is $\frac{a^{2}-1}{2a}$ thus obtaining the
equation
$C_{a}=\left\\{\left(x-\frac{a^{2}+1}{2a}\right)^{2}\,+y^{2}\,=\,\left(\frac{a^{2}-1}{2a}\right)^{2}\right\\}=\left\\{x^{2}-\frac{a^{2}+1}{a}\,x+y^{2}+1=0\right\\}$
Let us denote by $f(x,y)=0$ is the equation of the curve $C_{a}$. The tangent
line in $p=(x,y)\in C_{a}$ is $f_{x}(x-p_{1})+f_{y}(y-p_{2})$, whose direction
is generated by the vector $(-f_{y}(p),f_{x}(p))$, and the corresponding dual
form will be $f_{x}(p)dx+f_{y}(p)dy$. In our case,
$\omega=\left(2x-\frac{a^{2}+1}{a}\right)\,dx+\,2y\,dy$
Then, for the equation of $C_{a}$, we can deduce that
$\frac{a^{2}+1}{a}=\frac{x^{2}+y^{2}+1}{x}$
thus showing
$\omega=\frac{x^{2}-y^{2}-1}{x}\,dx\,+\,2y\,dy\>;\>{\rm if}\,x\neq 0$
We can replace the Pffaf form by another one obtained multiplying by the
function $\mu(x,y)=x$, and then we would have coefficients of degree two.
Then, we can take
$\omega=(x^{2}-y^{2}-1)\,dx\,+\,2xy\,dy$
which is not an exact differential equation.
Let us consider the singular 2-web given by the foliations:
* •
${\cal F}_{\omega}$ is the vertical axis and the set of circles $C_{a}$, with
center in the horizontal axis passing through the points $(a,0)$ and
$(1/a,0)$.
* •
${\cal F}_{\eta}$ is the set vertical lines, whose Pfaffian form is $\eta=dx$.
Then the polar curve is the reducible curve $\\{xy=0\\}$ given by both axis
and has a unique singular point. Nevertheless, both foliations have in common
the leaf $\\{x=0\\}$, and then in all of its points have contact of order
infinite, thus showing that there are regular points of the polar curve
corresponding to higher order contact of both foliations.
On the other hand, by using Theorem 2 one can obtain the paracomplex structure
associated to this 2-web:
$F=\left(\begin{array}[]{cc}1&0\\\
\frac{-(x^{2}-y^{2}-1)}{xy}&-1\end{array}\right)$
thus showing again that the polar curve is $\\{xy=0\\}$. But one cannot obtain
information about the contact order of leaves of both foliations.
In the present case, as one can easily check, $\mu(x,y)=\frac{1}{x^{2}}$ is an
integrating factor of $\omega$, and then we can re-write $\omega$ as
$\omega=\frac{x^{2}-y^{2}-1}{x^{2}}\,dx\,+\,\frac{2y}{x}\,dy$
Then, the equation of the polar curve is $C=\\{-\frac{2y}{x}=0\\}$, showing
the special property of the vertical axis $\\{x=0\\}$. The points where the
polar curve cannot be defined when it is obtained from exact differential
equations corresponds with those of a common leaf of both foliations. But this
example also shows that multiplying the 1-form by an integrating factor can
add points where the 1-form is not defined: in the example those of the
vertical axis $\\{x=0\\}$.
## 4 Conclusions
We write down some global conclusions.
* •
The following mathematical objects are equivalent in the real plane: a 2-web,
two differential equations, two vector fields, two 1-forms, two distributions.
Assume that each of them has no singularities nor zeros (restricting to an
open subset of the plane, if necessary). Then, we have two families of curves,
and we ask where they are tangent. This is a natural question and, as far as
the author knows, there was not yet any answer about it.
* •
The points where the foliations are tangent define a curve, called the polar
curve of the 2-web. If one defines the 2-web by means of two 1-forms it is
very easy to find an equation for the polar curve. If the 1-forms are
algebraic, the polar curve is an algebraic curve.
* •
The 1-forms associated to a 2-web are not unique. If one takes exact
differential equations for them, then we have proved that higher order contact
points of the foliations are singular points of the polar curve.
* •
There exist integrating factors which allow to obtain an exact differential
equation for any 1-form. In general, integrating factors are difficult to be
calculated. Besides they can exclude points of the plane.
* •
The paracomplex structure is unique up to a sign and the points where it is
not defined define the polar curve. One can derive the expression of the
paracomplex structure from those of the 1-forms, but there is no a general way
for the reverse.
## References
* [1] Alexandrino, M. M.: On polar foliations and fundamental group. _Results in Mathematics_ , 60, Issue 1 (2011), 213–223
* [2] Blaschke, W.: _Geometrie der Gewebe_. Springer, 1938.
* [3] Corral, N.: Infinitesimal adjunction and polar curves. _Bull. Braz. Math. Soc._ (N.S.) 40 (2009), no. 2, 181 -224.
* [4] Etayo, F: Singular 2-webs. An Introduction. _Contribuciones matemáticas en homenaje a Juan Tarrés_. Ed. Complutense, 2012, 141-147. ISBN: 978-84-695-4421-1.
* [5] Etayo, F. ; Santamaría, R; Trías, U. R.: The geometry of a bi-Lagrangian manifold. _Differential Geom. Appl_. 24 (2006), 33–59.
* [6] Grifone, J.; Salem, E.: _Web theory and related topics_. World Scientific Publishing Co. Pte. Ltd. (2001)
|
arxiv-papers
| 2013-04-21T18:06:47 |
2024-09-04T02:49:44.663574
|
{
"license": "Public Domain",
"authors": "Fernando Etayo",
"submitter": "Fernando Etayo",
"url": "https://arxiv.org/abs/1304.5772"
}
|
1304.6036
|
# Sub-barrier capture reactions with 16,18O and 40,48Ca beams
V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
Various sub-barrier capture reactions with beams 16,18O and 40,48Ca are
treated within the quantum diffusion approach. The role of neutron transfer in
these capture reactions is discussed. The quasielastic and capture barrier
distributions are analyzed and compared with the recent experimental data.
###### pacs:
25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, neutron transfer, quantum diffusion approach
## I Introduction
From the present experimental data the role of the neutron transfer channel in
the capture (fusion) process cannot be unambiguously inferred Bertulani ; Jia
; Kol ; EPJSub ; EPJSub1 . The fusion excitation functions have been recently
measured for the reactions 16O+76Ge and 18O+74Ge at energies near and below
the Coulomb barrier and the fusion barrier distributions have been extracted
from the corresponding excitation functions Jia . The fusion enhancement due
to the positive $Q_{2n}$-value two neutron ($2n$) transfer channel for
18O+74Ge has not been revealed as compared with the reference system 16O+76Ge
Jia . This is very different from the situation for the reactions
40Ca+124,132Sn Kol and other systems in literature, which show considerable
sub-barrier enhancements. The enhancement appears to be related to the
existence of positive $Q$ values for neutron transfer.
The purpose of this paper is the theoretical explanation of these experimental
observations. Within the quantum diffusion approach EPJSub ; EPJSub1 we try
to answer the question how strong the influence of neutron transfer in sub-
barrier capture (fusion) reactions
18O+74Ge,52,50Cr,94,92Mo,112,114,118,120,124,126Sn and 40,48Ca+124,132Sn. This
study seems to be important for future experiments indicated in Ref. Jia . In
addition, the new structures of the quasielastic and capture barrier
distributions at deep sub-barrier energies will be discussed.
## II Model
In the quantum diffusion approach EPJSub ; EPJSub1 the collisions of nuclei
are described with a single relevant collective variable: the relative
distance between the colliding nuclei. This approach takes into consideration
the fluctuation and dissipation effects in collisions of heavy ions which
model the coupling with various channels (for example, coupling of the
relative motion with low-lying collective modes such as dynamical quadrupole
and octupole modes of the target and projectile Ayik333 ). We have to mention
that many quantum-mechanical and non-Markovian effects accompanying the
passage through the potential barrier are taken into consideration in our
formalism EPJSub1 . The nuclear deformation effects are taken into account
through the dependence of the nucleus-nucleus potential on the deformations
and mutual orientations of the colliding nuclei. To calculate the nucleus-
nucleus interaction potential $V(R)$, we use the procedure presented in Refs.
EPJSub1 . For the nuclear part of the nucleus-nucleus potential, the double-
folding formalism with the Skyrme-type density-dependent effective nucleon-
nucleon interaction is used. With this approach many heavy-ion capture
reactions at energies above and well below the Coulomb barrier have been
successfully described EPJSub1 . Note that the diffusion models, which include
the quantum statistical effects, were also treated in Refs. Hofman .
Following the hypothesis of Ref. Broglia , we assume that the sub-barrier
capture in the reactions under consideration mainly depends on the possible
two-neutron transfer with the positive $Q_{2n}$-value. Our assumption is that,
just before the projectile is captured by the target-nucleus (just before the
crossing of the Coulomb barrier) which is a slow process, the $2n$-transfer
($Q_{2n}>0$) occurs that can lead to the population of the excited collective
states in the recipient nucleus SSzilner . So, the motion to the $N/Z$
equilibrium starts in the system before the capture because it is
energetically favorable in the dinuclear system in the vicinity of the Coulomb
barrier. For the reactions considered, the average change of mass asymmetry is
related to the two-neutron transfer. In these reactions the $2n$-transfer
channel is more favorable than $1n$-transfer channel ($Q_{2n}>Q_{1n}$). Since
after the $2n$-transfer the mass numbers, the deformation parameters of the
interacting nuclei, and, correspondingly, the height $V_{b}=V(R_{b})$ of the
Coulomb barrier are changed, one can expect an enhancement or suppression of
the capture. This scenario was verified in the description of many reactions
EPJSub1 .
## III Results of calculations
All calculated results are obtained with the same set of parameters as in Ref.
EPJSub1 and are rather insensitive to the reasonable variation of them
EPJSub1 . Realistic friction coefficient in the momentum $\hbar\lambda$=2 MeV
is used which is close to those calculated within the mean field approaches
obzor . The parameters of the nucleus-nucleus interaction potential $V(R)$ are
adjusted to describe the experimental data at energies above the Coulomb
barrier corresponding to spherical nuclei. The absolute values of the
quadrupole deformation parameters $\beta_{2}$ of even-even deformed nuclei are
taken from Ref. Ram . In Ref. Ram , the quadrupole deformation parameters
$\beta_{2}$ are given for the first excited 2+ states of nuclei. For the
nuclei deformed in the ground state, the $\beta_{2}$ in 2+ state is similar to
the $\beta_{2}$ in the ground state and we use $\beta_{2}$ from Ref. Ram in
the calculations. For the double magic nucleus 16O, in the ground state we
take $\beta_{2}=0$. Since there are uncertainties in the definition of the
values of $\beta_{2}$ in light- and medium-mass nuclei, one can extract the
quadrupole deformation parameters of these nuclei from a comparison of the
calculated capture cross sections with the existing experimental data. By
describing the reactions 18O+208Pb, where there are no neutron transfer
channels with positive $Q$-values, we extract $\beta_{2}=0.1$ for the ground-
state of 18O EPJSub1 . This extracted value is used in our calculations.
### III.1 Effect of neutron transfer in reactions with beams 40,48Ca
To eliminate the influence of the nucleus-nucleus potential on the capture
(fusion) cross section and to make conclusions about the role of deformation
of colliding nuclei and the nucleon transfer between interacting nuclei in the
capture (fusion) cross section, a reduction procedure is useful Gomes . It
consists of the following transformations:
$E_{\rm c.m.}\rightarrow x=\dfrac{E_{\rm
c.m.}-V_{b}}{\hbar\omega_{b}},\qquad\sigma_{cap}\rightarrow\sigma_{cap}^{red}=\dfrac{2E_{\rm
c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma_{cap},$
where $\sigma_{cap}=\sigma_{cap}(E_{\rm c.m.})$ is the capture cross section
at bombarding energy $E_{\rm c.m.}$. The frequency
$\omega_{b}=\sqrt{V^{{}^{\prime\prime}}(R_{b})/\mu}$ is related with the
second derivative $V^{{}^{\prime\prime}}(R_{b})$ of the total nucleus-nucleus
potential $V(R)$ (the Coulomb + nuclear parts) at the barrier radius $R_{b}$
and the reduced mass parameter $\mu$. With these replacements we compared the
reduced calculated capture (fusion) cross sections $\sigma_{cap}^{red}$ for
the reactions 40,48Ca+124,132Sn (Fig. 1). The choice of the projectile-target
combination is crucial, and for the systems studied one can make unambiguous
statements regarding the neutron transfer process with a positive $Q$-value
when the interacting nuclei are double magic or semi-magic spherical nuclei.
In this case one can disregard the strong direct nuclear deformation effects.
In Fig. 1, one can see that the reduced capture cross sections in the
reactions 40Ca+124,132Sn with the positive $Q_{2n}$-values strongly deviate
from those in the reactions 48Ca+124,132Sn, where the neutron transfers are
suppressed because of the negative $Q$-values.
Figure 1: (Colour online) The calculated reduced capture cross sections versus
$(E_{\rm c.m.}-V_{b})/(\hbar\omega_{b})$ in the reactions 40Ca+124Sn (solid
line), 48Ca+124Sn (dashed line), 48Ca+124Sn (dotted line), and 48Ca+132Sn
(dash-dotted line).
After two-neutron transfer in the reactions
40Ca($\beta_{2}=0$)+124Sn($\beta_{2}=0.1$)$\to^{42}$Ca($\beta_{2}=0.25$)+122Sn($\beta_{2}=0.1$)
($Q_{2n}$=5.4 MeV) and
40Ca($\beta_{2}=0$)+132Sn($\beta_{2}=0$)$\to^{42}$Ca($\beta_{2}=0.25$)+130Sn($\beta_{2}=0$)
($Q_{2n}$=7.3 MeV) the deformation of the light nucleus increases and the mass
asymmetry of the system decreases and, thus, the value of the Coulomb barrier
decreases and the capture cross section becomes larger (Fig. 1). So, because
of the transfer effect the systems 40Ca+124,132Sn show large sub-barrier
enhancements with respect to the systems 48Ca+124,132Sn. We observe that the
$\sigma_{cap}^{red}$ in the 40Ca+124Sn (48Ca+124Sn) reaction are larger than
those in the 40Ca+132Sn (48Ca+132Sn) reaction. The reason of that is the
nonzero quadrupole deformation of the heavy nucleus 124Sn. It should be
stressed that there are almost no difference between $\sigma_{cap}^{red}$ in
the reactions 40,48Ca+124,132Sn at energies above the Coulomb barrier.
Figure 2: The calculated capture cross sections versus $E_{\rm c.m.}$ for the
reactions 40Ca+124Sn (solid line) and 48Ca+124Sn (dashed line). The
experimental data for the reactions 40Ca+124Sn (solid squares) and 48Ca+124Sn
(open squares) are from Ref. Kol . In the calculations the barriers were
adjusted to the experimental values. Figure 3: The calculated capture cross
sections versus $E_{\rm c.m.}$ for the reactions 40Ca+132Sn (solid line) and
48Ca+132Sn (dashed line). The experimental data for the reactions 40Ca+132Sn
(solid squares) and 48Ca+132Sn (open squares) are from Ref. Kol . In the
calculations the barriers were adjusted to the experimental values.
In Figs. 2 and 3 one can see a good agreement between the calculated results
and the experimental data in the reactions 40,48Ca+124,132Sn. This means that
the observed capture enhancements in the reactions 40Ca+124,132Sn at sub-
barrier energies are related to the two-neutron transfer effect. Note that the
slope of the excitation function strongly depends on the deformations of the
interacting nuclei and, respectively, on the neutron transfer effect.
To describe the reactions 40,48Ca+132Sn (Fig. 2) and 48Ca+124,132Sn (Fig. 3),
we extracted the values of the corresponding Coulomb barrier $V_{b}$ for the
spherical nuclei. There are differences between the calculated and extracted
$V_{b}$. From the direct calculations of the nucleus-nucleus potentials (with
the same set of parameters), we obtained
$V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=2.3 MeV,
$V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=2.2 MeV,
$V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=1.3 MeV, and
$V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=1.2 MeV. From the extractions, we got
$V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=1.1 MeV
$V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=1.0 MeV,
$V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=-0.3 MeV, and
$V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=-0.4 MeV, which seem to be
unrealistically small. However, these differences of $V_{b}$ do not influence
the slopes of the excitation functions but only lead to the shifting of the
energy scale. With realistic isospin trend of $V_{b}$
$\sigma_{cap}$(40Ca+124Sn)$<\sigma_{cap}$(48Ca+124Sn) and
$\sigma_{cap}$(40Ca+132Sn)$<\sigma_{cap}$(48Ca+132Sn) at energies above the
corresponding Coulomb barriers.
### III.2 Effect of neutron transfer in reactions with beams 16,18O
Figures 4-7 show the capture excitation function for the reactions
16,18O+76,74Ge, 16,18O+94,92Mo, 16,18O+114,112,120,118,126,124Sn, and
16,18O+52,50Cr as a function of bombarding energy. One can see a rather good
agreement between the calculated results and the experimental data Jia ;
16OAGe ; AO92Mo ; AOASn for the reactions 16O+76Ge, 16,18O+92Mo, and
18O+112,118,124Sn.
Figure 4: (Colour online) The calculated (solid line) capture cross sections
versus $E_{\rm c.m.}$ for the reactions 16O+76Ge and 18O+74Ge (the curves
coincide). For the 18O+74Ge reaction, the calculated capture cross sections
without neutron transfer are shown by dotted line. The experimental data for
the reactions 16O+76Ge (open circles) and 18O+74Ge (open squares) are from
Ref. Jia . The experimental data for the 16O+76Ge reaction (solid circles) are
from Ref. 16OAGe . Figure 5: (Colour online) The calculated capture cross
sections versus $E_{\rm c.m.}$ for the reactions 16O+92Mo (dashed line) and
18O+92Mo (solid line). For the 18O+92Mo reaction, the calculated capture cross
sections without the neutron transfer are shown by dotted line. The
experimental data for the reactions 16O+92Mo (solid stars) and 18O+92Mo (solid
squares) are from Ref. AO92Mo . Figure 6: The calculated capture cross
sections versus $E_{\rm c.m.}$ for the reactions 16O+114Sn and 18O+112Sn
(solid line), 16O+120Sn and 18O+118Sn (dashed line), 16O+126Sn and 18O+124Sn
(dotted line). The calculated results for the reactions 16O+114,120,126Sn and
18O+112,118,124Sn coincide, respectively. The experimental data for the
reactions 18O+112Sn (solid squares), 18O+118Sn (open squares), and 18O+124Sn
(open stars) are from Ref. AOASn . Figure 7: (Coulor online) The calculated
capture cross sections versus $E_{\rm c.m.}$ for the reactions 16O+52Cr
(dashed line) and 18O+50Cr (solid line).
The $Q_{2n}$-values for the $2n$-transfer processes are positive (negative)
for all reactions with 18O (16O). Thus, the neutron transfer can be important
for the reactions with the 16O beam. However, our results show that cross
sections for reactions 16O+76Ge (16O+114,120,126Sn,52Cr) and 18O+74Ge
(18O+114,118,124Sn,50Cr) are very similar. The reason of such behavior is that
after the $2n$-transfer in the system 18O+A-2X$\to^{16}$O+AX the deformations
remain to be similar. As a result, the corresponding Coulomb barriers of the
systems 18O+A-2X and 16O+AX are almost the same and, correspondingly, their
capture cross sections coincide. Just the same behavior was observed in the
recent experiments 16,18O+76,74Ge Jia .
One can see in Figs. 4-7 that at energies above and near the Coulomb barrier
the cross sections with and without two-neutron transfer are quite similar.
After the $2n$-transfer (before the capture) in the reactions
18O($\beta_{2}=0.1$) + 92Mo($\beta_{2}=0.05$)$\to^{16}$O($\beta_{2}=0$) +
94Mo($\beta_{2}=0.151$), 18O($\beta_{2}=0.1$) +
74Ge($\beta_{2}=0.283$)$\to^{16}$O($\beta_{2}=0$) + 76Ge($\beta_{2}=0.262$),
18O($\beta_{2}=0.1$)+112Sn($\beta_{2}=0.123$)$\to^{16}$O($\beta_{2}=0$)+114Sn($\beta_{2}=0.121$),
18O($\beta_{2}=0.1$)+118Sn($\beta_{2}=0.111$)$\to^{16}$O($\beta_{2}=0$)+120Sn($\beta_{2}=0.104$),
and
18O($\beta_{2}=0.1$)+124Sn($\beta_{2}=0.095$)$\to^{16}$O($\beta_{2}=0$)+126Sn($\beta_{2}=0.09$)
the deformations of the nuclei decrease and the values of the corresponding
Coulomb barriers increase. As a result, the transfer suppresses the capture
process at the sub-barrier energies. The suppression becomes stronger with
decreasing energy. As examples, in Fig. 4 and 5 we show this effect for the
reactions 18O+74Ge,92Mo.
### III.3 Capture and quasielastic barrier distributions
In Figs. 8 and 9, the calculated capture barrier distributions
$D=d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm c.m.}^{2}$
for the reactions 16O+76Ge,144,154Sm have only one pronounced maximum around
$E_{\rm c.m.}=V_{b}$ as in the experiments Jia ; Timmers . The calculated
barrier distributions in Figs. 8 and 9 are slightly wider and fit the
experimental data better than those obtained with the couple-channels approach
in Fig. 5 of Ref. Jia . The capture (fusion) cross sections for the reactions
16O+76Ge,144,154Sm were well described with the quantum diffusion model in
Ref. EPJSub1 . With almost spherical (deformed) target-nucleus we obtain a
more narrow (wide) barrier distribution for the 16O+144Sm (16O+154Sm)
reaction.
We compared the capture and the quasielastic barrier distributions for these
reactions (Figs. 8 and 9).
Figure 8: (Colour online) (a) The calculated values of the quasielastic $\pi
R_{b}^{2}D_{qe}$ (solid line) and capture $D$ (dotted line) barrier
distributions for the reactions 16O + 76Ge and 18O + 74Ge. The curves coincide
for these reactions. The calculated $D$ for the spherical interacting nuclei
is shown by dashed line. The experimental data for the reactions 16O + 76Ge
(solid circles) and 18O + 74Ge (open circles) are from Ref. Jia . (b) The
calculated values of $\pi R_{b}^{2}D_{qe}$ (solid line) and $D$ (dotted line)
are shown in the logarithmic scale. Figure 9: (Colour online) The calculated
values of quasielastic $D_{qe}$ (solid line) and capture $D/(\pi R_{b}^{2})$
(dotted line) barrier distributions for the reactions 16O + 144Sm (a) and 16O
+ 154Sm (b). The experimental $D_{qe}$ (open squares) and $D/(\pi R_{b}^{2})$
(solid circles) are from Ref. Timmers . The calculated $D$ for the spherical
interacting nuclei is shown by dashed line (b).
There is a direct relationship between the capture and the quasielastic
scattering processes because any loss from the quasielastic channel
contributes directly to the capture (the conservation of the reaction flux):
$P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)=1$
and
$dP_{cap}/dE_{\rm c.m.}=-dP_{qe}/dE_{\rm c.m.},$
where $P_{qe}$ is the reflection probability and $P_{cap}$ is the capture
(transmission) probability ($J$ is the partial wave). The quasielastic barrier
distribution is extracted by taking the first derivative of the $P_{qe}(E_{\rm
c.m.},J=0)$ or $P_{cap}(E_{\rm c.m.},J=0)$ with respect to $E_{\rm c.m.}$,
that is,
$D_{qe}(E_{\rm c.m.})=-dP_{qe}(E_{\rm c.m.},J=0)/dE_{\rm c.m.}=dP_{cap}(E_{\rm
c.m.},J=0)/dE_{\rm c.m.}.$
So, by employing the quantum diffusion approach and calculating
$dP_{cap}(E_{\rm c.m.},J=0)/dE_{\rm c.m.}$, one can obtain $D_{qe}(E_{\rm
c.m.})$. One can see in Figs. 8 and 9 that the shapes of the quasielastic and
capture barrier distributions are similar. The same conclusion was
experimentally obtained for the 20Ne+208Pb reaction in Ref. Piasecki . As in
the case of capture barrier distribution, one can show that the width of the
quasielastic barrier distribution increases with the deformation of the
target-nucleus. In addition to the mean peak position of the $D_{qe}$ around
the barrier height, we observe the sharp change of the slope of $D_{qe}$ or
$D$ below the Coulomb barrier energy because of a change of the regime of
interaction (the external turning point leaves the region of the nuclear
forces and friction EPJSub ; EPJSub1 ) in the deep sub-barrier capture process
(Fig. 8(b)).
## IV Summary
As shown with the quantum diffusion approach, the capture cross sections for
the reactions 16O+52Cr,76Ge,94Mo,114,120,126Sn and
18O+50Cr,74Ge,92Mo,112,118,124Sn, respectively, almost match. The fusion
enhancement due to the positive $Q_{2n}$-value $2n$-transfer for 18O+74Ge has
not been observed Jia because the deformations of nuclei slightly decrease
after the neutron transfer. This is different from the situation for the
reactions 40Ca+124,132Sn Kol with large positive $Q_{2n}$-values. The strong
enhancements have been observed Kol in these reactions at sub-barrier
energies because the deformation of light nucleus strongly increases (the
heavy nucleus is spherical before and after transfer) after the two-neutron
transfer.
We found that the shapes of the quasielastic and capture barrier distributions
are similar. The sharp change of the slope of the quasielastic or capture
barrier distribution is predicted at deep sub-barrier energy. This anomalous
behavior of the barrier distribution is expected to be the experimental
indication of a change of the regime of interaction in the sub-barrier
capture. One concludes that the quasielastic technique could be an important
tool in capture (fusion) research.
This work was supported by DFG, NSFC, and RFBR. The IN2P3(France) -
JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully
acknowledged. H.Q. Zhang is grateful to Chinese NSFC for the partial support.
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|
arxiv-papers
| 2013-04-22T18:03:36 |
2024-09-04T02:49:44.671280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1304.6036"
}
|
1304.6173
|
The LHCb collaboration
# First observation of $C\\!P$ violation in the decays of $B^{0}_{s}$ mesons
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
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Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal58, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
###### Abstract
Using $pp$ collision data, corresponding to an integrated luminosity of
$1.0~{}\mathrm{fb}^{-1}$, collected by LHCb in 2011 at a center-of-mass energy
of $7\\!$ $\mathrm{\,Te\kern-1.00006ptV}$, we report the measurement of direct
$C\\!P$ violation in $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays,
$A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})=0.27\pm 0.04\,\mathrm{(stat)}\pm
0.01\,\mathrm{(syst)}$, with significance exceeding five standard deviations.
This is the first observation of $C\\!P$ violation in the decays of
$B^{0}_{s}$ mesons. Furthermore, we provide an improved determination of
direct $C\\!P$ violation in $B^{0}\rightarrow K^{+}\pi^{-}$ decays,
$A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})=-0.080\pm 0.007\,\mathrm{(stat)}\pm
0.003\,\mathrm{(syst)}$, which is the most precise measurement of this
quantity to date.
###### pacs:
Valid PACS appear here
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
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| | CERN-PH-EP-2013-068
| | LHCb-PAPER-2013-018
| | June 7, 2013
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
Submitted to Phys. Rev. Lett.
The non-invariance of fundamental interactions under the combined action of
the charge conjugation ($C$) and parity ($P$) transformations is
experimentally well established in the $K^{0}$ and $B^{0}$ meson systems
Christenson:1964fg ; Aubert:2001nu ; Abe:2001xe ; PDG2012 . The Standard Model
(SM) description of $C\\!P$ violation, as given by the Cabibbo-Kobayashi-
Maskawa (CKM) theory of quark-flavor mixing Cabibbo:1963yz ; Kobayashi:1973fv
, has been very successful in describing existing data. However, the source of
$C\\!P$ violation in the SM is known to be too small to account for the
matter-dominated universe Cohen:1993nk ; Riotto:1999yt ; Hou:2008xd .
The study of $C\\!P$ violation in charmless charged two-body decays of neutral
$B$ mesons provides stringent tests of the CKM picture in the SM, and is a
sensitive probe to search for the presence of non-SM physics Deshpande:1994ii
; He:1998rq ; Fleischer:1999pa ; Gronau:2000md ; Lipkin:2005pb ;
Fleischer:2007hj ; Fleischer:2010ib . However, quantitative SM predictions for
$C\\!P$ violation in these decays are challenging because of the presence of
hadronic factors in the decay amplitudes, which cannot be accurately
calculated from quantum chromodynamics (QCD) at present. It is crucial to
combine several measurements from such two-body decays, exploiting approximate
flavor symmetries in order to cancel the unknown parameters. An experimental
program for measuring the properties of these decays has been carried out
during the last decade at the $B$ factories Lees:2013bb ; PhysRevD.87.031103
and at the Tevatron Aaltonen:2011qt , and is now continued by LHCb with
increased sensitivity. The discovery of direct $C\\!P$ violation in the
$B^{0}\rightarrow K^{+}\pi^{-}$ decay dates back to 2004 Aubert:2004qm ;
Chao:2004jy . This observation raised the question of whether the effect could
be accommodated by the SM or was due to non-SM physics. A simple but powerful
model-independent test was proposed in Refs. He:1998rq ; Lipkin:2005pb , which
required the measurement of direct $C\\!P$ violation in the
$B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decay. However, $C\\!P$ violation has
never been observed with significance exceeding five Gaussian standard
deviations ($\sigma$) in any $B^{0}_{s}$ meson decay so far.
In this Letter we report measurements of direct $C\\!P$-violating asymmetries
in $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays using $pp$ collision data, corresponding to an integrated luminosity of
$1.0~{}\mathrm{fb}^{-1}$, collected with the LHCb detector in 2011 at a
center-of-mass energy of $7\\!$ $\mathrm{\,Te\kern-1.00006ptV}$. The present
results supersede those given in Ref. LHCb-PAPER-2011-029 . The inclusion of
charge-conjugate decay modes is implied except in the asymmetry definitions.
The direct $C\\!P$ asymmetry in the $B^{0}_{(s)}$ decay rate to the final
state $f_{(s)}$, with $f=K^{+}\pi^{-}$ and $f_{s}=K^{-}\pi^{+}$, is defined as
$A_{C\\!P}\\!\\!\left(\\!B^{0}_{(s)}\\!\\!\rightarrow\\!\\!f_{(s)}\\!\right)\\!\\!=\\!\Phi\\!\left[\Gamma\\!\left(\\!\overline{B}^{0}_{(s)}\\!\\!\rightarrow\\!\\!\bar{f}_{(s)}\\!\right)\\!\\!,\,\Gamma\\!\left(\\!B^{0}_{(s)}\\!\\!\rightarrow\\!\\!f_{(s)}\\!\right)\right]\\!\\!,$
(1)
where $\Phi[X,\,Y]=(X-Y)/(X+Y)$ and $\bar{f}_{(s)}$ denotes the charge-
conjugate of $f_{(s)}$.
The LHCb detector Alves:2008zz is a single-arm forward spectrometer covering
the pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The trigger LHCb-DP-2012-004 consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage that applies a full event reconstruction. The
hadronic hardware trigger selects large transverse energy clusters in the
hadronic calorimeter. The software trigger requires a two-, three-, or four-
track secondary vertex with a large sum of the transverse momenta
($p_{\mathrm{T}}$) of the tracks and a significant displacement from the
primary $pp$ interaction vertices (PVs). At least one track should have
$p_{\mathrm{T}}$ and impact parameter (IP) $\chi^{2}$ with respect to all PVs
exceeding given thresholds. The IP is defined as the distance between the
reconstructed trajectory of a particle and a given $pp$ collision vertex, and
the IP $\chi^{2}$ is the difference between the $\chi^{2}$ of the PV
reconstructed with and without the considered track. A multivariate algorithm
is used for the identification of secondary vertices consistent with the decay
of a $b$ hadron. In order to improve the trigger efficiency on hadronic two-
body decays, a dedicated two-body software trigger is also used. This trigger
imposes requirements on the following quantities: the quality of the online-
reconstructed tracks, their $p_{\mathrm{T}}$ and IP; the distance of closest
approach of the decay products of the $B$ meson candidate, its
$p_{\mathrm{T}}$, IP and the decay time in its rest frame.
Figure 1: Invariant mass spectra obtained using the event selection adopted
for the best sensitivity on (a, b) $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$
and (c, d) $A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$. Panels (a) and (c)
represent the $K^{+}\pi^{-}$ invariant mass, whereas panels (b) and (d)
represent the $K^{-}\pi^{+}$ invariant mass. The results of the unbinned
maximum likelihood fits are overlaid. The main components contributing to the
fit model are also shown.
More selective requirements are applied offline. Two sets of criteria have
been optimized with the aim of minimizing the expected statistical uncertainty
either on $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ or on
$A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$. In addition to the
requirements on the kinematic variables already used in the trigger,
requirements on the largest $p_{\mathrm{T}}$ and IP of the $B$ daughter
particles are applied. In the case of $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays, a tighter selection is needed to achieve stronger rejection of
combinatorial background. For example, the decay time is required to exceed
$1.5$ ps, whereas in the $B^{0}\rightarrow K^{+}\pi^{-}$ selection a lower
threshold of $0.9$ ps is applied. This is because the probability for a $b$
quark to form a $B^{0}_{s}$ meson, which subsequently decays to the
$K^{-}\pi^{+}$ final state, is one order of magnitude smaller than that to
form a $B^{0}$ meson decaying to $K^{+}\pi^{-}$ LHCb-PAPER-2012-002 . The two
samples are then subdivided according to the various final states using the
particle identification (PID) provided by the two ring-imaging Cherenkov
(RICH) detectors LHCb-DP-2012-003 . Two sets of PID selection criteria are
applied: a loose set optimized for the measurement of
$A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ and a tight set for that of
$A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$. More details on the event
selection can be found in Ref. LHCb-PAPER-2011-029 .
To determine the amount of background events from other two-body $b$-hadron
decays with a misidentified pion or kaon (cross-feed background), the relative
efficiencies of the RICH PID selection criteria must be determined. This is
achieved by means of a data-driven method that uses $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $\mathchar 28931\relax\rightarrow p\pi^{-}$
decays as control samples. The production and decay kinematic properties of
the $D^{0}\rightarrow K^{-}\pi^{+}$ and $\mathchar 28931\relax\rightarrow
p\pi^{-}$ channels differ from those of the $b$-hadron decays under study.
Since the RICH PID information is momentum dependent, a calibration procedure
is performed by reweighting the distributions of the PID variables obtained
from the calibration samples, in order to match the momentum distributions of
signal final-state particles observed in data.
Unbinned maximum likelihood fits to the mass spectra of the selected events
are performed. The $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
K^{-}\pi^{+}$ signal components are described by double Gaussian functions
convolved with a function that describes the effect of final-state radiation
Baracchini:2005wp . The background due to partially reconstructed three-body
$B$ decays is parameterized by means of two ARGUS functions Albrecht:1989ga
convolved with a Gaussian resolution function. The combinatorial background is
modeled by an exponential function and the shapes of the cross-feed
backgrounds, mainly due to $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays with one misidentified particle in
the final state, are obtained from simulation. The cross-feed background
yields are determined from the $\pi^{+}\pi^{-}$, $K^{+}K^{-}$, $p\pi^{-}$ and
$pK^{-}$ mass spectra, using events passing the same selection as the signal
and taking into account the appropriate PID efficiency factors. The
$K^{+}\pi^{-}$ and $K^{-}\pi^{+}$ mass spectra for the events passing the two
selections are shown in Fig. 1. The average invariant mass resolution is about
$22~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
From the two mass fits we determine the signal yields $N(B^{0}\rightarrow
K^{+}\pi^{-})=41\hskip 1.42262pt420\pm 300$ and $N(B^{0}_{s}\rightarrow
K^{-}\pi^{+})=1065\pm 55$, as well as the raw asymmetries
$A_{\mathrm{raw}}(B^{0}\rightarrow K^{+}\pi^{-})=-0.091\pm 0.006$ and
$A_{\mathrm{raw}}(B^{0}_{s}\rightarrow K^{-}\pi^{+})=0.28\pm 0.04$, where the
uncertainties are statistical only. In order to derive the $C\\!P$ asymmetries
from the observed raw asymmetries, effects induced by the detector acceptance
and event reconstruction, as well as due to interactions of final-state
particles with the detector material, must be accounted for. Furthermore, the
possible presence of a $B^{0}_{(s)}-\overline{B}^{0}_{(s)}$ production
asymmetry must also be considered.
The $C\\!P$ asymmetry is related to the raw asymmetry by
$A_{C\\!P}=A_{\mathrm{raw}}-A_{\Delta}$, where the correction $A_{\Delta}$ is
defined as
$A_{\Delta}(B^{0}_{(s)}\rightarrow
K\pi)=\zeta_{d(s)}A_{\mathrm{D}}(K\pi)+\kappa_{d(s)}A_{\mathrm{P}}(B^{0}_{(s)}),$
(2)
with $\zeta_{d}=1$ and $\zeta_{s}=-1$. The instrumental asymmetry
$A_{\mathrm{D}}(K\pi)$ is given in terms of the detection efficiencies
$\varepsilon_{\mathrm{D}}$ of the charge-conjugate final states by
$A_{\mathrm{D}}(K\pi)=\Phi[\varepsilon_{\mathrm{D}}(K^{-}\pi^{+}),\,\varepsilon_{\mathrm{D}}(K^{+}\pi^{-})]$,
and the production asymmetry $A_{\mathrm{P}}(B^{0}_{(s)})$ is defined in terms
of the $\overline{B}^{0}_{(s)}$ and $B^{0}_{(s)}$ production rates,
$R(\overline{B}^{0}_{(s)})$ and $R(B^{0}_{(s)})$, as
$A_{\mathrm{P}}(B^{0}_{(s)})=\Phi[R(\overline{B}^{0}_{(s)}),\,R(B^{0}_{(s)})]$.
The factors $\kappa_{d}$ and $\kappa_{s}$ take into account dilutions due to
$B^{0}$ and $B^{0}_{s}$ meson mixing, respectively. Their values also depend
on event reconstruction and selection, and are $\kappa_{d}=0.303\pm 0.005$ and
$\kappa_{s}=-0.033\pm 0.003$ LHCb-PAPER-2011-029 . The factor $\kappa_{s}$ is
ten times smaller than $\kappa_{d}$, owing to the large $B^{0}_{s}$
oscillation frequency.
The instrumental charge asymmetry $A_{\mathrm{D}}(K\pi)$ is measured from data
using $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ and $D^{*+}\rightarrow
D^{0}(K^{-}K^{+})\pi^{+}$ decays. The combination of the time-integrated raw
asymmetries of these two decay modes is used to disentangle the various
contributions to each raw asymmetry. The presence of open charm production
asymmetries arising from the primary $pp$ interaction constitutes an
additional complication. We write the following equations relating the
observed raw asymmetries to the physical $C\\!P$ asymmetries
$\displaystyle A^{*}_{\mathrm{raw}}(K\pi)$ $\displaystyle=$ $\displaystyle
A^{*}_{\mathrm{D}}(\pi_{s})+A^{*}_{\mathrm{D}}(K\pi)+A_{\mathrm{P}}(D^{*}),$
(3) $\displaystyle A^{*}_{\mathrm{raw}}(KK)$ $\displaystyle=$ $\displaystyle
A_{C\\!P}(KK)+A^{*}_{\mathrm{D}}(\pi_{s})+A_{\mathrm{P}}(D^{*}),$ (4)
where $A^{*}_{\mathrm{raw}}(K\pi)$ and $A^{*}_{\mathrm{raw}}(KK)$ are the
time-integrated raw asymmetries in $D^{*}$-tagged $D^{0}\rightarrow
K^{-}\pi^{+}$ and $D^{0}\rightarrow K^{-}K^{+}$ decays, respectively;
$A_{C\\!P}(KK)$ is the $D^{0}\rightarrow K^{-}K^{+}$ $C\\!P$ asymmetry;
$A^{*}_{\mathrm{D}}(K\pi)$ is the detection asymmetry in reconstructing
$D^{0}\rightarrow K^{-}\pi^{+}$ and $\overline{D}^{0}\rightarrow K^{+}\pi^{-}$
decays; $A^{*}_{\mathrm{D}}(\pi_{s})$ is the detection asymmetry in
reconstructing positively- and negatively-charged pions originated from
$D^{*}$ decays; and $A_{\mathrm{P}}(D^{*})$ is the production asymmetry for
prompt charged $D^{*}$ mesons. In Eq. (3) any possible $C\\!P$ asymmetry in
the Cabibbo-favored $D^{0}\rightarrow K^{-}\pi^{+}$ decay is neglected
Bianco:2003vb . By subtracting Eqs. (3) and (4), one obtains
$A^{*}_{\mathrm{raw}}(K\pi)-A^{*}_{\mathrm{raw}}(KK)=A^{*}_{\mathrm{D}}(K\pi)-A_{C\\!P}(KK).$
(5)
Once the raw asymmetries are measured, this equation determines unambiguously
the detection asymmetry $A^{*}_{\mathrm{D}}(K\pi)$, using the world average
for the $C\\!P$ asymmetry of the $D^{0}\rightarrow K^{-}K^{+}$ decay. Since
the measured value of the time-integrated asymmetry depends on the decay-time
acceptance, the existing measurements of $A_{C\\!P}(KK)$ Staric:2008rx ;
Aubert:2007if ; Aaltonen:2011se are corrected for the difference in
acceptance with respect to LHCb LHCb-PAPER-2011-023 . This leads to the value
$A_{C\\!P}(KK)=(-0.24\pm 0.18)\%$. Furthermore, $B$ meson production and decay
kinematic properties differ from those of the $D$ decays being considered, and
different trigger and selection algorithms are applied. In order to correct
the raw asymmetries of $B$ decays, using the detection asymmetry
$A^{*}_{\mathrm{D}}(K\pi)$ derived from $D$ decays, a reweighting procedure is
needed. We reweight the $D^{0}$ momentum, transverse momentum and azimuthal
angle in $D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{0}\rightarrow K^{-}K^{+}$
decays, to match the respective $B^{0}_{(s)}$ distributions in
$B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays. The raw asymmetries are determined by means of $\chi^{2}$ fits to the
reweighted $\delta m=M_{D^{*}}-M_{D^{0}}$ distributions, where $M_{D^{*}}$ and
$M_{D^{0}}$ are the reconstructed $D^{*}$ and $D^{0}$ candidate invariant
masses, respectively.
From the raw asymmetries, values for the quantity $\Delta
A=A_{\mathrm{D}}(K\pi)-A_{C\\!P}(KK)$ are determined. We obtain the values
$\Delta A=(-0.91\pm 0.15)\%$ and $\Delta A=(-0.98\pm 0.11)\%$, using as target
kinematic distributions those of $B$ candidates passing the event selection
optimized for $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ and for
$A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$, respectively. Using these two
values of $\Delta A$ and the value of $A_{C\\!P}(KK)$, we obtain the
instrumental asymmetries $A_{\mathrm{D}}(K\pi)=(-1.15\pm 0.23)\%$ for the
$B^{0}\rightarrow K^{+}\pi^{-}$ decay and $A_{\mathrm{D}}(K\pi)=(-1.22\pm
0.21)\%$ for the $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decay.
Assuming negligible $C\\!P$ violation in the mixing, as expected in the SM and
confirmed by current experimental determinations bib:hfagbase , the decay rate
of a $B^{0}_{(s)}$ meson with production asymmetry $A_{\mathrm{P}}$, decaying
into a flavor-specific final state $f_{(s)}$ with $C\\!P$ asymmetry
$A_{C\\!P}$ and detection asymmetry $A_{\mathrm{D}}$, can be written as
$\mathcal{R}(t;\,p)\propto\left(1\\!-\\!pA_{C\\!P}\right)\left(1\\!-\\!pA_{\mathrm{D}}\right)\left[H_{+}\\!\left(t\right)\\!-\\!pA_{\mathrm{P}}H_{-}\\!\left(t\right)\right]\\!,$
(6)
where $t$ is the reconstructed decay time of the $B$ meson and $p$ assumes the
values $p=+1$ for the final state $f_{(s)}$ and $p=-1$ for the final state
$\bar{f}_{(s)}$. The functions $H_{+}\left(t\right)$ and $H_{-}\left(t\right)$
are defined as
$\displaystyle H_{+}\\!\left(t\right)\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left[e^{-\Gamma_{d(s)}t^{\prime}}\\!\\!\\!\cosh\\!{\left(\\!\\!\frac{\Delta\Gamma_{d(s)}}{2}t^{\prime}\\!\\!\right)}\\!\\!\otimes\\!R\\!\left(t,\,t^{\prime}\right)\right]\\!\varepsilon_{d(s)}\\!\left(t\right)\\!,$
(7) $\displaystyle H_{-}\\!\left(t\right)\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left[e^{-\Gamma_{d(s)}t^{\prime}}\\!\\!\\!\cos\\!{\left(\Delta
m_{d(s)}t^{\prime}\right)}\\!\otimes\\!R\\!\left(t,\,t^{\prime}\right)\right]\\!\varepsilon_{d(s)}\\!\left(t\right)\\!,$
(8)
where $\Gamma_{d(s)}$ is the average decay width of the $B^{0}_{(s)}$ meson,
$\Delta\Gamma_{d(s)}$ and $\Delta m_{d(s)}$ are the decay width and mass
differences between the two $B^{0}_{(s)}$ mass eigenstates respectively,
$R\left(t,\,t^{\prime}\right)$ is the decay time resolution ($\sigma\simeq
50~{}\mathrm{fs}$ in our case) and the symbol $\otimes$ stands for
convolution. Finally $\varepsilon_{d(s)}\left(t\right)$ is the acceptance as a
function of the $B^{0}_{(s)}$ decay time. Using Eq. (6) we obtain the
following expression for the time-dependent asymmetry
$\displaystyle\mathcal{A}\left(t\right)$ $\displaystyle=$
$\displaystyle\Phi\\!\left[\mathcal{R}\left(t;\,-1\right)\\!,\,\mathcal{R}\left(t;\,+1\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\left(A_{C\\!P}\\!+\\!A_{\mathrm{D}}\right)\\!H_{+}\\!\left(t\right)\\!+\\!A_{\mathrm{P}}\left(1\\!+\\!A_{C\\!P}A_{\mathrm{D}}\right)\\!H_{-}\\!\left(t\right)}{\left(1\\!+\\!A_{C\\!P}A_{\mathrm{D}}\right)\\!H_{+}\\!\left(t\right)\\!+\\!A_{\mathrm{P}}\left(A_{C\\!P}\\!+\\!A_{\mathrm{D}}\right)\\!H_{-}\\!\left(t\right)}.$
For illustrative purposes only, we consider the case of perfect decay time
resolution and negligible $\Delta\Gamma$, retaining only first-order terms in
$A_{C\\!P}$, $A_{\mathrm{P}}$ and $A_{\mathrm{D}}$. In this case, Eq. (First
observation of $C\\!P$ violation in the decays of $B^{0}_{s}$ mesons) reduces
to the expression
$\mathcal{A}\left(t\right)\approx
A_{C\\!P}+A_{\mathrm{D}}+A_{\mathrm{P}}\cos{\left(\Delta m_{d(s)}t\right)},$
(10)
i.e., the time-dependent asymmetry has an oscillatory term with amplitude
equal to the production asymmetry $A_{\mathrm{P}}$. By studying the full time-
dependent decay rate it is then possible to determine $A_{\mathrm{P}}$
unambiguously.
In order to measure the production asymmetry $A_{\mathrm{P}}$ for $B^{0}$ and
$B^{0}_{s}$ mesons, we perform fits to the decay time spectra of the $B$
candidates, separately for the events passing the two selections. The $B^{0}$
production asymmetry is determined from the sample obtained applying the
selection optimized for the measurement of $A_{C\\!P}(B^{0}\rightarrow
K^{+}\pi^{-})$, whereas the $B^{0}_{s}$ production asymmetry is determined
from the sample obtained applying the selection optimized for the measurement
of $A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$. We obtain
$A_{\mathrm{P}}(B^{0})=(0.1\pm 1.0)\%$ and $A_{\mathrm{P}}(B^{0}_{s})=(4\pm
8)\%$. Figure 2 shows the raw asymmetries as a function of the decay time,
obtained by performing fits to the invariant mass distributions of events
restricted to independent intervals of the $B$ candidate decay times.
Figure 2: Raw asymmetries as a function of the decay time for (a)
$B^{0}\rightarrow K^{+}\pi^{-}$ and (b) $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays. In (b), the offset $t_{0}=1.5$${\rm\,ps}$ corresponds to the minimum
value of the decay time required by the $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
event selection. The curves represent the asymmetry projections of fits to the
decay time spectra.
By using the values of the detection and production asymmetries, the
correction factors to the raw asymmetries $A_{\Delta}(B^{0}\rightarrow
K^{+}\pi^{-})=\left(-1.12\pm 0.23\pm 0.30\right)\\!\%$ and
$A_{\Delta}(B^{0}_{s}\rightarrow K^{-}\pi^{+})=\left(1.09\pm 0.21\pm
0.26\right)\\!\%$ are obtained, where the first uncertainties are due to the
detection asymmetry and the second to the production asymmetry.
Systematic uncertainties on the asymmetries are related to PID calibration,
modeling of the signal and background components in the maximum likelihood
fits and instrumental charge asymmetries. In order to estimate the impact of
imperfect PID calibration, we perform mass fits to determine raw asymmetries
using altered numbers of cross-feed background events, according to the
systematic uncertainties affecting the PID efficiencies. An estimate of the
uncertainty due to possible mismodeling of the final-state radiation is
determined by varying the amount of emitted radiation Baracchini:2005wp in
the signal shape parameterization, according to studies performed on fully
simulated events, in which final state radiation is generated using Photos
Golonka:2005pn . The possibility of an incorrect description of the signal
mass model is investigated by replacing the double Gaussian function with the
sum of three Gaussian functions, where the third component has fixed fraction
($5\%$) and width ($50\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$), and is
aimed at describing long tails, as observed in simulation. To assess a
systematic uncertainty on the shape of the partially reconstructed
backgrounds, we remove the second ARGUS function. For the modeling of the
combinatorial background component, the fit is repeated using a straight line.
Finally, for the case of the cross-feed backgrounds, two distinct systematic
uncertainties are estimated: one due to a relative bias in the mass scale of
the simulated distributions with respect to the signal distributions in data,
and another accounting for the difference in mass resolution between
simulation and data. All shifts from the relevant baseline values are
accounted for as systematic uncertainties. Systematic uncertainties related to
the determination of detection asymmetries are calculated by summing in
quadrature the respective uncertainties on $A_{\Delta}(B^{0}\rightarrow
K^{+}\pi^{-})$ and $A_{\Delta}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$ with an
additional uncertainty of $0.10\%$, accounting for residual differences in the
trigger composition between signal and calibration samples.
The systematic uncertainties for $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$
and $A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$ are summarized in Table 1.
Since the production asymmetries are obtained from the fitted decay time
spectra of $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
K^{-}\pi^{+}$ decays, their uncertainties are statistical in nature and are
then propagated to the statistical uncertainties on
$A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ and $A_{C\\!P}(B^{0}_{s}\rightarrow
K^{-}\pi^{+})$.
Table 1: Systematic uncertainties on $A_{C\\!P}(B^{0}\rightarrow
K^{+}\pi^{-})$ and $A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$. The total
systematic uncertainties are obtained by summing the individual contributions
in quadrature.
Systematic uncertainty | $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ | $A_{C\\!P}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$
---|---|---
PID calibration | 0.0006 | 0.0012
Final state radiation | 0.0008 | 0.0020
Signal model | 0.0001 | 0.0064
Combinatorial background | 0.0004 | 0.0042
Three-body background | 0.0005 | 0.0027
Cross-feed background | 0.0010 | 0.0033
Detection asymmetry | 0.0025 | 0.0023
Total | 0.0029 | 0.0094
In conclusion, the parameters of $C\\!P$ violation in $B^{0}\rightarrow
K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays have been
measured to be
$\displaystyle A_{C\\!P}(B^{0}\\!\rightarrow\\!K^{+}\pi^{-})$ $\displaystyle=$
$\displaystyle-0.080\pm 0.007\,\mathrm{(stat)}\pm 0.003\,\mathrm{(syst)},$
$\displaystyle A_{C\\!P}(B^{0}_{s}\\!\rightarrow\\!K^{-}\pi^{+})$
$\displaystyle=$ $\displaystyle 0.27\pm 0.04\,\mathrm{(stat)}\pm
0.01\,\mathrm{(syst)}.$
Dividing the central values by the sum in quadrature of statistical and
systematic uncertainties, the significances of the measured deviations from
zero are $10.5\sigma$ and $6.5\sigma$, respectively. The former is the most
precise measurement of $A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ to date,
whereas the latter represents the first observation of $C\\!P$ violation in
decays of $B^{0}_{s}$ mesons with significance exceeding $5\sigma$. Both
measurements are in good agreement with world averages bib:hfagbase and
previous LHCb results LHCb-PAPER-2011-029 .
These results allow a stringent test of the validity of the relation between
$A_{C\\!P}(B^{0}\rightarrow K^{+}\pi^{-})$ and $A_{C\\!P}(B^{0}_{s}\rightarrow
K^{-}\pi^{+})$ in the SM given in Ref. Lipkin:2005pb as
$\Delta=\frac{A_{C\\!P}(B^{0}\\!\rightarrow\\!K^{+}\pi^{-})}{A_{C\\!P}(B^{0}_{s}\\!\rightarrow\\!K^{-}\pi^{+})}+\frac{\mathcal{B}(B^{0}_{s}\\!\rightarrow\\!K^{-}\pi^{+})}{\mathcal{B}(B^{0}\\!\rightarrow\\!K^{+}\pi^{-})}\frac{\tau_{d}}{\tau_{s}}=0,$
(11)
where $\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})$ and
$\mathcal{B}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$ are $C\\!P$-averaged
branching fractions, and $\tau_{d}$ and $\tau_{s}$ are the $B^{0}$ and
$B^{0}_{s}$ mean lifetimes, respectively. Using additional results for
$\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})$ and
$\mathcal{B}(B^{0}_{s}\rightarrow K^{-}\pi^{+})$ LHCb-PAPER-2012-002 and the
world averages for $\tau_{d}$ and $\tau_{s}$ bib:hfagbase , we obtain
$\Delta=-0.02\pm 0.05\pm 0.04$, where the first uncertainty is from the
measurements of the $C\\!P$ asymmetries and the second is from the input
values of the branching fractions and the lifetimes. No evidence for a
deviation from zero of $\Delta$ is observed with the present experimental
precision.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-23T05:54:19 |
2024-09-04T02:49:44.680612
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van\n den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M. Dogaru, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach,\n I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G.\n Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero\n Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Vincenzo Maria Vagnoni",
"url": "https://arxiv.org/abs/1304.6173"
}
|
1304.6190
|
# $X(1870)$ and $\eta_{2}(1870)$: Which can be assigned as a hybrid state?
Bing Chen111Corresponding author: [email protected] School of Physics and
Electrical Engineering, Anyang Normal University, Anyang 455000, China Ke-Wei
Wei School of Physics and Electrical Engineering, Anyang Normal University,
Anyang 455000, China Ailin Zhang Department of Physics, Shanghai University,
Shanghai 200444, China
###### Abstract
The mass spectrum and strong decays of the $X(1870)$ and $\eta_{2}(1870)$ are
analyzed. Our results indicate that $X(1870)$ and $\eta_{2}(1870)$ are the two
different resonances. The narrower $X(1870)$ seems likely a good hybrid
candidate. We support the $\eta_{2}(1870)$ as the $\eta_{2}(2^{1}D_{2})$
quarkonium. We suggest to search the isospin partner of $X(1870)$ in the
channels of $J/\psi\rightarrow\rho f_{0}(980)\pi$ and $J/\psi\rightarrow\rho
b_{1}(1235)\pi$ in the future. The latter channel is very important for
testing the hybrid scenario.
###### pacs:
12.38.Lg, 13.25.Jx
††preprint: AHEP(Hadron)/AYNU[2013]
## I INTRODUCTION
A isoscalar resonant structure of $X(1870)$ was observed by the BESIII
Collaboration with a statistical significance of 7.2$\sigma$ in the processes
$J/\psi\rightarrow\omega X(1870)\rightarrow\omega\eta\pi^{+}\pi^{-}$ recently
Bes6 . Its mass and width were given as
$M=1877.3\pm 6.3^{+3.4}_{-7.4}$MeV, $\Gamma=57\pm 12^{+19}_{-4}$MeV.
Here the first errors are statistical and the second ones are systematic. The
product branching fraction of $\mathcal{B}(J/\psi\rightarrow\omega
X(1870))\cdot\mathcal{B}(X(1870)\rightarrow
a^{\pm}_{0}(980)\pi^{\mp})\cdot\mathcal{B}(a^{\pm}_{0}(980)\rightarrow\eta\pi^{\pm})=[1.5\pm
0.26(stat)^{+0.72}_{-0.36}(syst)]\times 10^{-4}$ was also presented Bes6 . But
the quantum numbers of $X(1870)$ are still unknown, then the partial wave
analysis is required in future.
The mass of $X(1870)$ is consistent with the $\eta_{2}(1870)$, but the width
is much narrower than the $\eta_{2}(1870)$. In the tables of the Particle Data
Group (PDG) PDG , the available mass and width of $\eta_{2}(1870)$ are
_M_ = 1842 $\pm$ 8MeV, $\Gamma$ = 225 $\pm$ 14MeV.
The $\eta_{2}(1870)$ has been observed in $\gamma\gamma$ reactions Crystall ;
DESY , $p\bar{p}$ annihilation pp1 ; pp2 ; WA1021 ; WA1022 and radiative
$J/\psi$ decays Bes7 . It should be stressed that radiative $J/\psi$ decay
channels (Fig.1[A]) and $p\bar{p}$ annihilation prosesses are the ideal
glueball hunting grounds. But the glueball production is suppressed in
$\gamma\gamma$ reaction. By contrast, the hadronic $J/\psi$ decay are
considered ``hybrid rich'' (Fig.1[B]).
Figure 1: [A]. A prior production of glueball in the $J/\psi\rightarrow\gamma
X_{G}$; [B]. A prior production of hybrid in the $J/\psi\rightarrow\omega
X_{H}$.
Furthermore, the branching ratio
$\mathcal{R}_{1}=\frac{\Gamma(\eta_{2}(1870)\rightarrow
a_{2}(1320)\pi)}{\Gamma(\eta_{2}(1870)\rightarrow a_{0}(980)\pi)}=32.6\pm
12.6$ reported by the WA102 Collaboration indicates that the decay channel of
$a_{0}(980)\pi$ is tiny for $\eta_{2}(1870)$ WA1021 . This has been confirmed
by an extensive re-analysis of the Crystal Barrel data Bugg1 . Differently,
the analysis of BESIII Collaboration indicates that the $X(1870)$ primarily
decay via the $a_{0}(980)\pi$ channel Bes6 . Then the present measurements of
the decay widths, productions, and decay properties suggest that
$\eta_{2}(1870)$ and $X(1870)$ are two different isoscalar mesons.
If the production process $J/\psi\rightarrow\omega X(1870)$ is mainly
hadronic, the quantum numbers of $X(1870)$ should be $0^{+}0^{-+}$,
$0^{+}1^{++}$ or $0^{+}2^{-+}$. One notices that the predicted masses for the
light $0^{+}0^{-+}$, $0^{+}1^{++}$ and $0^{+}2^{-+}$ hybrids overlap 1.8GeV in
the Bag model Bag1 ; Bag2 , the flux-tube model tube1 ; tube2 and the
constituent gluon model gluon . In addition, the decay width of isoscalar
$2^{-+}$ hybrid is expected to be narrow Swanson . Therefore, $X(1870)$
becomes a possible $2^{-+}$ hybrid candidate.
In addition, the predicted masses of $0^{-+}$ and $2^{-+}$ glueball are much
higher than 1.8 GeV by lattice gauge theory latt1 ; latt2 ; latt3 . Therefore,
$X(1870)$ is not likely to be a glueball state. Moreover, the molecule and
fourquark states are not expected in this region Swanson . Then the unclear
structure $X(1870)$ looks more like a good hybrid candidate. But the actual
situation is much complicated because the nature of $\eta_{2}(1870)$ is still
ambiguous:
1. (i)
Since no evidences have been found in the decay mode of $K\bar{K}\pi$, the
$\eta_{2}(1870)$ disfavors the $1^{1}D_{2}$ $s\bar{s}$ quarkonium assignment.
The mass of $\eta_{2}(1870)$ seems much smaller for the $2^{1}D_{2}$
$n\bar{n}$ ($n\bar{n}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$) state in the
Godfrey-Isgur (GI) quark model Isgur . Therefore the $\eta_{2}(1870)$ has been
assigned as the $2^{-+}$ hybrid state Bugg1 ; Bugg2 ; Klmept1 ; Amsler1 .
2. (ii)
However, Li and Wang pointed out that the mass, production, total decay width,
and decay pattern of the $\eta_{2}(1870)$ do not appear to contradict with the
picture of it as being the conventional $2^{1}D_{2}$ $n\bar{n}$ state Li3 .
Therefore, systematical study of the mass spectrum and strong decay properties
is urgently required for $X(1870)$ and $\eta_{2}(1870)$. Some valuable
suggestions for the experiments in future are also needed.
The paper is organized as follows. In Sec.II, the masses of $X(1870)$ and
$\eta_{2}(1870)$ will be explored in the GI relativized quark model and the
Regge trajectories (RTs) framework. In Sec.III, the decay processes that a
isoscalar meson decays into light scalar (below 1 GeV) and pseudoscalar mesons
will discussed. The two-body strong decays $X(1870)$ and $\eta_{2}(1870)$ will
be calculated within the ${}^{3}P_{0}$ model and the flux-tube model. Finally,
our discussions and conclusions will be presented in Sec.IV.
## II Mass spectrum
In the Godfrey-Isgur relativized potential model Isgur , the Hamiltonian
consists of the central potential and a kinetic term in a ``relativized'' form
$H=\sqrt{\vec{p}_{q}^{2}+m^{2}_{q}}+\sqrt{\vec{p}_{\bar{q}}^{2}+m^{2}_{\bar{q}}}+V_{q\bar{q}}(r).$
(1)
The funnel-shaped potentials which include a color coulomb term at short
distances and a linear scalar confining term at large distances are usually
incorporated as the zeroth-order potential. The typical funnel-shaped
potential was proposed by the Cornell group (Cornell potential) with the form
Eichten
$V_{q\bar{q}}(r)=-\frac{4}{3}\frac{\alpha_{s}}{r}+\sigma r+C.$ (2)
The strong coupling constant $\alpha_{s}$, the string tension $\sigma$ and the
constant _C_ are the model parameters which can be fixed by the well
established experimental states. The remaining spin-dependent terms for mass
shifts are usually treated as the leading-order perturbations which include
the spin-spin contact hyperfine interaction, spin-orbit and tensor
interactions and a longer-ranged inverted spin-orbit term. They arise from one
gluon exchange (OGE) forces and the assumed Lorentz scalar confinement. The
expressions for these terms may be found in Ref. Isgur .
It should be pointed out that the nonperturbative contribution may dominate
for the hyperfine splitting of light mesons, which is not like the heavy
quarkonium Badalian . For example, the hyperfine shift of the $h_{c}(1P)$
meson with respect to the center gravity of the $\chi_{c}(1P)$ mesons is much
small: $M_{cog}(\chi_{c})-M(h_{c})=-0.02\pm 0.19\pm 0.13$MeV CLEO . However,
for the light isovector mesons $a_{0}(1450)$, $a_{1}(1260)$, $a_{2}(1320)$,
and $b_{1}(1235)$, the hyperfine shift is $76.7\pm 44.4$ MeV. Here the masses
of $a_{0}$, $a_{1}$, $a_{2}$, and $b_{1}$ are taken from PDG PDG . For the
complexities of nonperturbative interactions, then we are not going to
calculate the hyperfine splitting.
Now, the spin-averaged mass, $\bar{M}_{nl}$, of $nL$ multiplet can be obtained
by solving the spinless Salpeter equation
$[\sqrt{\vec{p}_{q}^{2}+m^{2}_{q}}+\sqrt{\vec{p}_{\bar{q}}^{2}+m^{2}_{\bar{q}}}+V_{q\bar{q}}(r)]\psi(r)=E\psi(r).$
(3)
Here we employ a variational approach described in Ref. var to solve the
Eq.(3). This variational approach has been applied well in solving the
Salpeter equation for $c\bar{s}$ cs , $c\bar{c}$ and $b\bar{b}$ cc mass
spectrum.
In the calculations, the basic simple harmonic oscillator (SHO) functions are
taken as the trial wave functions. It is given by
$\psi_{nl}(r,\beta)=\beta^{3/2}\sqrt{\frac{2(2n-1)!}{\Gamma(n+l+\frac{1}{2})}}(\beta
r)^{l}e^{-\frac{\beta^{2}r^{2}}{2}}L^{l+1/2}_{n-1}(\beta^{2}r^{2})$
in the position space. Here the SHO function scale $\beta$ is the variational
parameter.
By the Fourier transform, the SHO radial wave function in the momentum is
$\displaystyle\psi_{nl}(p,\beta)=\frac{(-1)^{n}}{\beta^{3/2}}\sqrt{\frac{2(2n-1)!}{\Gamma(n+l+\frac{1}{2})}}(\frac{p}{\beta})^{l}e^{-\frac{p^{2}}{2\beta^{2}}}L^{l+1/2}_{n-1}(\frac{p^{2}}{\beta^{2}}).$
The wave functions of $\psi_{nl}(r,\beta)$ and $\psi_{nl}(p,\beta)$ meet the
normalization conditions:
$\int^{\infty}_{0}\psi^{2}_{nl}(r,\beta)r^{2}dr=1;~{}~{}~{}~{}\int^{\infty}_{0}\psi^{2}_{nl}(p,\beta)p^{2}dp=1.$
In the variational approach, the corresponding $\bar{M}_{nl}$ are given by
minimizing the expectation value of $H$
$\frac{d}{d\beta}E_{nl}(\beta)=0.$ (4)
where
$E_{nl}(\beta)\equiv\langle
H\rangle_{nl}=\langle\psi_{nl}|H|\psi_{nl}\rangle.$ (5)
When all the parameters of the potential model are known, the values of the
harmonic oscillator parameter $\bar{\beta}$ can be fixed directly. With the
values of $\bar{\beta}$, all the spin-averaged mass $\bar{M}_{nl}$ will be
obtained easily. $\bar{M}_{nl}$ obtained in this way trend to be better for
the higher-excited states Roberts .
It is unreasonable to treat the spin-spin contact hyperfine interaction as a
perturbation for the ground states, because the mass splitting between
pseudoscalar mesons and vector mesons are much large. Then we consider the
contributions of $V_{\vec{s}\cdot\vec{s}}(r)$ for the $1S$ mesons. The
following Gaussian-smeared contact hyperfine interaction ss is taken for
convenience,
$V_{q\bar{q}}^{\vec{s}\cdot\vec{s}}(r)=\frac{32\pi\alpha_{s}}{9m^{2}_{q}}(\frac{\kappa}{\sqrt{\pi}})^{3}e^{-\kappa^{2}r^{2}}\vec{S}_{q}\cdot\vec{S}_{\bar{q}}.$
(6)
In this work, we choose the model parameters as follows: $m_{u}$ = $m_{d}$ =
0.220 GeV, $m_{s}=$ 0.428 GeV, $\alpha_{s}=$ 0.6, $\sigma=$ 0.143 GeV2,
$\kappa=$ $0.37$ GeV, and $C=$ $-0.37$ GeV. We take the smaller value of
$\sigma$ here rather than the value in Ref. Isgur . The smaller $\sigma$ was
obtained by the relation between the slope of the Regge trajectory for the
Salpeter equation $\alpha^{\prime}$ and the slope $\alpha^{\prime}_{st}$ in
the string picture Badalian . The Gaussian smearing parameter $\kappa$ seems a
little smaller than that in Ref. Isgur . However, the $\kappa$ is usually
fitted by the hyperfine splitting of low-excited $nS$ states in the
literatures with a certain arbitrariness.
The values of $\bar{M}_{nL}$ and $\bar{\beta}$ for the states $2S$, $3S$,
$4S$, $1P$, $2P$, $3P$, $1D$, $2D$, $3D$, $1F$, $2F$, $1G$ and $1H$ are listed
in Table 1. The experimental masses for the relative mesons are taken from PDG
PDG .
States | $\bar{M}_{nl}(n\bar{n})$ | $\bar{\beta}$ | Expt. PDG | $\bar{M}_{nl}(s\bar{s})$ | $\bar{\beta}$ | Expt. PDG
---|---|---|---|---|---|---
1S | - | 0.44 0.34 | - | - | 0.42 0.39 | -
2S | 1.399 | 0.310 | 1.389 | 1.631 | 0.330 | 1.629
3S | 1.859 | 0.295 | | $\underline{2.069}$ | 0.310 |
4S | 2.240 | 0.290 | | 2.436 | 0.300 |
1P | 1.252 | 0.310 | 1.257 | 1.460 | 0.340 | 1.478
2P | 1.711 | 0.294 | | $\underline{1.926}$ | 0.315 |
3P | 2.110 | 0.290 | | 2.308 | 0.300 |
1D | 1.661 | 0.280 | 1.672 | $\underline{1.883}$ | 0.300 |
2D | $\underline{2.067}$ | 0.276 | | 2.272 | 0.292 |
3D | 2.417 | 0.275 | | 2.609 | 0.288 |
1F | 1.924 | 0.277 | | 2.128 | 0.295 |
2F | 2.287 | 0.275 | | 2.478 | 0.290 |
1G | 2.161 | 0.275 | | 2.350 | 0.292 |
1H | 2.377 | 0.273 | | 2.554 | 0.287 |
Table 1: The spin-averaged mass (unit: GeV) and the harmonic oscillator
parameter $\bar{\beta}$ (unit: GeV-1) of the states $2S$, $3S$, $4S$, $1P$,
$2P$, $3P$, $1D$, $2D$, $3D$, $1F$, $2F$, $1G$, and $1H$.
Obviously, the spin-averaged masses of the $2S$, $1P$, $1D$ $n\bar{n}$ and
$1P$, $2S$ $s\bar{s}$ mesons are consistent with the experimental data.
Indeed, the predicted masses of higher excited states here are also
reasonable, $e.g.$, $a_{4}(2040)$ and $f_{4}(2050)$ are very possible the
$F-$wave $n\bar{n}$ isovector and isoscalar mesons with the masses of
$1996^{+10}_{-9}$MeV and $2018\pm{11}$MeV, respectively PDG . The predicted
spin-averaged mass of $1F$ is not incompatible with experiments. Our results
are also overall in good agreement with the expectations from Ref. long1 . The
trend that a higher excited state corresponds to a smaller $\bar{\beta}$
coincides with Ref. Godfrey ; Close ; Li . For considering the spin-spin
contact hyperfine interaction, there are two $\bar{\beta}$s for the $1S$
mesons. The larger one corresponds to the $1^{1}S_{0}$ state, the smaller one
the $1^{3}S_{1}$ state.
As shown in Ref. long1 ; long2 , the confinement potential $V_{conf}(r)$ is
determinant for the properties of higher excited states. In Ref. long1 , the
masses for higher excited states with $\sigma=0.143$GeV2 and $\alpha_{s}=0$
are closer to experimental data than the results given in Ref. Isgur . Then we
ignored the Coulomb interaction for $1D$, $2D$, $1F$, $1G$ and $1H$ states. In
this way, $\bar{M}_{nl}$ for these states increase about 100MeV.
The masses of $\eta^{\prime}(3^{1}S_{0})$, $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ and $\eta_{2}(2^{1}D_{2})$ are usually within
$1.8\sim 2.1$GeV in various quark potential models Isgur ; Ebert ; Sorace ;
Vijande (see in Table 2). The predicted spin-averaged masses of
$3S(s\bar{s})$, $2P(s\bar{s})$, $1D(s\bar{s})$ and $2D(n\bar{n})$ are also
within this mass regions (see in Table 1). Due to the uncertainty of the
potential models, absolute deviation from experimental data are usually about
100$\sim$150 MeV for the higher excited states. Comparing with these predicted
masses, $X(1870)$ disfavors the $\eta^{\prime}(3^{1}S_{0})$ assignment for its
low mass. But the possibilities of $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ and $\eta_{2}(2^{1}D_{2})$ still exist. Here
we don't consider the possibility of $X(1870)$ as the $\eta(3^{1}S_{0})$ state
because $\eta(1760)$ looks more like a good $\eta(3^{1}S_{0})$ candidate Li5 ;
Liu ; Yu .
States | $\eta^{\prime}(3^{1}S_{0})$ | $f^{\prime}_{1}(2^{3}P_{1})$ | $\eta^{\prime}_{2}(1^{1}D_{2})$ | $\eta_{2}(2^{1}D_{2})$
---|---|---|---|---
Ref. Isgur | $-$ | 2030 | 1890 | 2130†
Ref. Ebert | 2085 | 2016 | 1909 | 1960
Ref. Sorace | 2099 | 1988 | 1851 | $-$
Ref. Vijande | $-$ | $-$ | 1853 | 1863
Table 2: The masses predicted for $3^{1}S_{0}$($\eta^{\prime}$),
$2^{3}P_{1}$($\eta^{\prime}$), $1^{1}D_{2}$($\eta^{\prime}$) and
$2^{1}D_{2}$($\eta$) in Refs. Isgur ; Ebert ; Sorace ; Vijande .
Regge trajectories (RTs) is another useful tool for studying the mass spectrum
of the light flavor mesons. In Ref. Regge1 , the authors fitted the RTs for
all light-quark meson states listed in the PDG tables. A global description
was constructed as
$M^{2}=1.38(4)n+1.12(4)J-1.25(4).$ (7)
Here, _n_ and _J_ mean the the radial and angular-momentum quantum number.
Recently, the authors of Ref. Regge1 repeated their fits with the subset
mesons of the paper Regge2 . They found a little smaller averaged slopes of
$\mu^{2}=1.28(5)$GeV2 and $\beta^{2}=1.09(6)$GeV2, to be compared with
$\mu^{2}=1.38(4)$GeV2 and $\beta^{2}=1.12(4)$GeV2 in the Eq.(7). Here the
$\mu^{2}$ and $\beta^{2}$ are the weighted averaged slope for radial and
angular-momentum RTs Regge1 ; Regge3 .
Now $h_{1}(1380)$, $f_{1}(1420)$ and $\eta^{\prime}(1475)$ have been
established as the $1^{1}P_{1}$, $1^{3}P_{1}$ and $2^{1}S_{0}$ $s\bar{s}$
states in PDG PDG . With the differences between the mass squared of $X(1870)$
and these states (Table 3), $X(1870)$ could be assigned for the
$\eta^{\prime}(3^{1}S_{0})$ and $f^{\prime}_{1}(2^{1}P_{1})$. The mass of
$X(1870)$ is too large for the $\eta^{\prime}_{2}(1^{1}D_{2})$ state in the
RTs. $\eta_{2}(1645)$ has been assigned as the $1^{1}D_{2}$ $n\bar{n}$ meson
PDG . Since $M^{2}(X(1870))-M^{2}(\eta_{2}(1640))=0.91^{+0.04}_{-0.03}$GeV2
which is much smaller than $1.38(4)$GeV2, $X(1870)$ looks unlike the
$2^{1}D_{2}$ $n\bar{n}$ state for its low mass. However, the difference of
$M^{2}(X(1870))-M^{2}(h_{1}(1170))=2.16^{+0.06}_{-0.05}$GeV2 matches the
slopes $2.37(11)$GeV2 well. Then the RTs can't exclude the possibility of
$X(1870)$ as the $2^{1}D_{2}$ $n\bar{n}$ state.
Four possible states for _X_(1870)
---
$\eta^{\prime}(3^{1}S_{0})$ | $f_{1}^{\prime}(2^{3}P_{1})$ | $\eta^{\prime}_{2}(1^{1}D_{2})$ | $\eta_{2}(2^{1}D_{2})$
$\eta^{\prime}(1475)$ | $f_{1}(1420)$ | $h_{1}(1380)$ | $\eta_{2}(1645)$
$\mu^{2}=$1.34${}^{+0.04}_{-0.03}$ | $\mu^{2}=$1.38${}^{+0.04}_{-0.03}$ | $\beta^{2}=$1.60${}^{+0.06}_{-0.06}$ | $\mu^{2}=$0.91${}^{+0.04}_{-0.03}$
Table 3: $X(1870)$ calculated in RTs for different states are shown. The
masses of $\eta^{\prime}(1475)$, $f_{1}(1420)$, $h_{1}(1380)$ and
$\eta_{2}(1645)$ are taken from PDG PDG .
As mentioned in the Introduction, $X(1870)$ is also a good hybrid candidate
since its mass overlaps the predictions given by different models. The
predicted masses for $0^{+}0^{-+}$, $0^{+}1^{++}$ and $0^{+}2^{-+}$
$n\bar{n}g$ states by these models are collected in Table 4.
States | $\eta_{H}(0^{+}0^{-+})$ | $f_{H}(0^{+}1^{++})$ | $\eta_{H}(0^{+}2^{-+})$
---|---|---|---
Bag Bag1 ; Bag2 | 1.3 | heavier | 1.9
Flux tube tube1 ; tube2 | 1.7$\sim$1.9 | 1.7$\sim$1.9 | 1.7$\sim$1.9
Constituent gluon gluon | 1.8$\sim$2.2 | 1.3$\sim$1.8 | 1.8$\sim$2.2
Table 4: The masses predicted for $\eta_{H}(0^{+}0^{-+})$ $f_{H}(0^{+}1^{++})$
and $\eta_{H}(0^{+}2^{-+})$ hybrid states in Refs. Bag1 ; Bag2 ; tube1 ; tube2
; gluon .
In this section, the mass of $X(1870)$ has been studied in the GI quark
potential model and the RTs framework. In the GI quark potential model,
$X(1870)$ can be interpreted as the $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ or $\eta_{2}(2^{1}D_{2})$ state with a
reasonable uncertainty. In the RTs, $X(1870)$ favors the
$\eta^{\prime}(3^{1}S_{0})$ and $f^{\prime}_{1}(2^{3}P_{1})$ assignments. But
the $\eta_{2}(2^{1}D_{2})$ assignment can't be excluded thoroughly. $X(1870)$
is also a good hybrid state candidate. Since the masses of $X(1870)$ and
$\eta_{2}(1870)$ are nearly equal, the possible assignments of $X(1870)$ also
suit $\eta_{2}(1870)$. The investigations of the strong decay properties will
be more helpful to distinguish the $\eta_{2}(1870)$ and $X(1870)$.
## III The strong decay
### III.1 The final mesons include the scalar mesons below 1 GeV
Despite many theoretical efforts, the scalar nonet of $q\bar{q}$ mesons has
never well-established. The lowest-lying scalar mesons including $\sigma(500)$
(or $f_{0}(600)$), $\kappa(800)$, $a_{0}(980)$ and $f_{0}(980)$ are difficult
to be described as $q\bar{q}$ states, _e.g_., $a_{0}(980)$ is associated with
nonstrange quarks in the $q\bar{q}$ scheme. If this is true, its high mass and
decay properties are difficult to be understood simultaneously. So
interpretations as exotic states were triggered, _i.e_., as two clusters of
two quarks and two antiquarks Maiani , particular quasimolecular states
molecule , and uncorrelated four quark states $qq\bar{q}\bar{q}$ tetraquark1 ;
tetraquark2 ; tetraquark3 have been proposed.
Though the structures of these scalar mesons below 1 GeV are still in dispute,
the viewpoint that these scalar mesons can constitute a complete nonet states
has been reached in the most literatures (as illustrated in Fig.2). In the
following, we will denote this nonet as `` $\mathcal{S}$ '' multiplet for
convenience.
Figure 2: The `` $\mathcal{S}$ '' nonet below 1 GeV shown in $Y-I_{3}$ plane.
Due to the unclear nature of the $\mathcal{S}$ mesons, it seems much difficult
to study the decay processes when the final mesons includes a $\mathcal{S}$
member. As an approximation, $a_{0}(980)$, $\sigma(500)$ and $f_{0}(980)$ were
treated as $1^{3}P_{0}$ $q\bar{q}$ mesons in Refs. Yu ; Liu1 . In Refs. Li3 ;
Liu , this kind of decay channel was ignored. However, this kind of decay mode
maybe predominant for some mesons. For example, the observations indicate that
$f_{1}(1285)$, $\eta(1405)$ and $X(1870)$ primarily decay via the
$a_{0}(980)\pi$ channel Bes6 .
In what follows, we will extract some useful information about this kind of
decay mode by the SU(3) flavor symmetry. We will show that $a_{0}(980)\pi$,
$\sigma_{0}\eta$ and $f_{0}\eta$ are the main decay channels for the isoscalar
$n\bar{n}$ and the $n\bar{n}g$ mesons when they decay primarily through ``
$\mathcal{S}$ $+$ P'' mesons, where the sign ``P'' denotes a light
pseudoscalar meson. This will explain why $X(1870)$ has been first observed in
the $\eta\pi^{+}\pi^{-}$ channel.
We noticed that the $\mathcal{S}$ nonet could be interpreted like the
$q\bar{q}$ nonet in the diquark-antidiquark scenario. In Wilczek and Jaffe's
terminology Jaffe ; Wilczek , the $\mathcal{S}$ mesons consist of a ``good''
diquark and a ``good'' antidiquark. When $u$, $d$ quarks forms a ``good''
diquark, it means that the two light quarks, _u_ and _d_ , could be treated as
a quasiparticle in color $\bar{3}$, flavor $\bar{3}$ and the spin singlet. The
``good'' $u$, $d$ diquark is usually denoted as $[ud]$.
In the diquark-antidiquark limit, the parity of a tetraquark is determined by
$P=(-1)^{L_{12-34}}$ Santopinto where the $L_{12-34}$ refer to the relative
angular momentum between two clusters. Thus the $\mathcal{S}$ mesons are the
lightest tetraquark states in the diquark-antidiquark model with
$L_{12-34}=0$. The $\mathcal{S}$ nonet in the full set of flavor
representations is
$\displaystyle(3\otimes 3)_{\bar{3}}\otimes(\bar{3}\otimes\bar{3})_{3}=8\oplus
1$
Because the SU(3) flavor symmetry is not exact, the two physical isoscalar
mesons, $\sigma_{0}$ and $f_{0}$, are usually the mixing states of the
$|8\rangle_{I=0}$ and $|1\rangle_{I=0}$ states Maiani ,
$\displaystyle\begin{aligned} \left(\begin{array}[]{c}f_{0}\\\ \sigma_{0}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\vartheta&\sin\vartheta\\\
-\sin\vartheta&\cos\vartheta\\\
\end{array}\right)\left(\begin{array}[]{c}|8\rangle_{I=0}\\\
|1\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (8)
When the mixing angle $\vartheta$ equals the so-called ideal mixing angle,
$i.e.$, $\vartheta$ = 54.74∘, the composition of the $\sigma(500)$ and
$f_{0}(980)$ are
$\displaystyle\begin{aligned} \left(\begin{array}[]{c}f_{0}\\\ \sigma_{0}\\\
\end{array}\right)=\left(\begin{array}[]{c}|\frac{1}{\sqrt{2}}([su][\bar{s}\bar{u}]+[sd][\bar{s}\bar{d}])\rangle\\\
|[ud][\bar{u}\bar{d}]\rangle\\\ \end{array}\right).\end{aligned}$
It seems that the deviation from the ideal mixing angle of the $\sigma(500)$
and $f_{0}(980)$ is small Maiani . In the following calculations, we will
treat them in the ideal mixing scheme.
Under the SU(3) flavor assumption, all the members of the octet have the same
basic coupling constant in one type of reaction, while the singlet member have
a different coupling constant. Particularly, when a quarkonium decays into
$\mathcal{S}$ and $q\bar{q}$ mesons, there are five independent coupling
constants, $i.e.$, $g_{A88}$, $g_{A81}$, $g_{A18}$, $g_{B88}$ and $g_{B11}$,
corresponding to five different channels
$\begin{cases}\mid 8\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A88}\\\ \mid 8\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 1\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A81}\\\ \mid 8\rangle_{q\bar{q}}\rightarrow\mid
1\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A18}\\\ \mid 1\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{B88}\\\ \mid 1\rangle_{q\bar{q}}\rightarrow\mid
1\rangle_{\mathcal{S}}\otimes\mid 1\rangle_{q\bar{q}}:\hskip
28.45274ptg_{B11}\\\ \end{cases}$
In order to determine the relations between these coupling constants, we shall
assume the process that the $q\bar{q}$ or $q\bar{q}g$ meson decays into a
$\mathcal{S}$ and another $q\bar{q}$ mesons obeys the OZI (Okubo-Zweig-Iizuka)
rule, $i.e.$, the two quarks in the mother meson go into two daughter mesons,
respectively. Therefore, there are four forbidden processes:
$X(\frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d}))\nrightarrow a_{0}+s\bar{s}$,
$X(\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d}))\nrightarrow\sigma_{0}+s\bar{s}$,
$X(s\bar{s})\nrightarrow f_{0}+\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$ and
$X(s\bar{s})\nrightarrow\sigma_{0}+s\bar{s}$. With the help of the SU(3)
Clebsch$-$Gordon coefficients Coefficients , the ratios between the five
coupling constants are extracted as
$\displaystyle g_{A81}:g_{A18}:g_{B88}:g_{B11}:g_{A88}$
$\displaystyle=\sqrt{2}:-\sqrt{\frac{2}{5}}(\sqrt{5}+1):-\frac{2}{\sqrt{5}}(\sqrt{5}+1):-\sqrt{\frac{2}{5}}(\sqrt{5}+1):1$
(9) $\displaystyle\approx 1.41:-2.05:-2.89:-2.05:1.00$
Figure 3: The coefficients $\zeta^{2}$ of the isoscalar meson $\xi$ versus the
mixing angle $\theta$. Figure 4: The coefficients $\zeta^{2}$ of the isoscalar
meson $\xi^{\prime}$ versus the mixing angle $\theta$.
It is well known that the physical states, $\eta(548)$ and
$\eta^{\prime}(958)$ are the mixture of the SU(3) flavor octet and singlet.
They can be written in terms of a mixing angle, $\theta_{p}$, as follows
$\displaystyle\begin{aligned} \left(\begin{array}[]{c}\eta(548)\\\
\eta^{\prime}(958)\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{p}&-\sin\theta_{p}\\\
\sin\theta_{p}&\cos\theta_{p}\\\
\end{array}\right)\left(\begin{array}[]{c}|8\rangle_{I=0}\\\
|1\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (10)
The mixing angle $\theta_{p}$ has been measured by various means. However,
there is still uncertainty for $\theta_{p}$. An excellent fit to the tensor
meson decay widths was performed under the SU(3) symmetry, and
$\theta_{p}\simeq-17^{o}$ was obtained Amsler1 . In our calculation,
$\theta_{p}$ is taken as $-17^{o}$. The excited mixtures of $n\bar{n}$ and
$s\bar{s}$ are denoted as
$\displaystyle\begin{aligned} \left(\begin{array}[]{c}\xi\\\ \xi^{\prime}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{c}|1\rangle_{I=0}\\\
|8\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (11)
In this scheme, the ideal mixing occurs with the choice of $\theta=35.3^{o}$.
When $\xi$ and $\xi^{\prime}$ decay into a $\mathcal{S}$ and pseudoscalar
mesons, the relations of decay amplitudes are governed by the coefficients
$\zeta^{2}$ which are model-independent in the limitation of SU(3)f symmetry.
With the coupling constants in hand, the coefficients $\zeta^{2}$ of $\xi$ and
$\xi^{\prime}$ versus the mixing angle $\theta$ are shown in the Fig.3 and
Fig.4. When $\xi$ and $\xi^{\prime}$ occurs in the ideal mixing, the values of
$\zeta^{2}$ are presented in Table 5. In the factorization framework, the
decay difference of a hybrid and excited $q\bar{q}$ mesons comes from the
spatial contraction Burns . Then the coefficients $\zeta^{2}$ for hybrid
states are same as these of $q\bar{q}$ quarkoniums.
Decay | $a_{0}\pi$ | $\sigma\eta$ | $\kappa K$ | $f_{0}\eta$ | $\sigma\eta^{\prime}$
---|---|---|---|---|---
channels
$\zeta^{2}[n\bar{n}(g)]$ | 2.17 | 3.56 | 0.47 | 1.07 | 0.44
$\zeta^{2}[s\bar{s}(g)]$ | 0.00 | 0.00 | 0.72 | 1.08 | 0.00
Table 5: .The coefficients $\zeta^{2}$ of $\xi$ and $\xi^{\prime}$ in the
ideal mixing.
Here the mixing of $\eta(548)$ and $\eta^{\prime}(958)$ has been considered.
It is sure that the $\zeta^{2}$ are zero for the processes,
$\xi^{\prime}\rightarrow a_{0}\pi$, $\xi^{\prime}\rightarrow\sigma\eta$ and
$\xi^{\prime}\rightarrow\sigma\eta^{\prime}$, since they are OZI-forbidden.
$\zeta^{2}$ of $\xi^{\prime}\rightarrow f_{0}\eta^{\prime}$ hasn't been
considered in Table 5 since $X(1870)$ lies below the threshold of
$f_{0}\eta^{\prime}$.
As illustrated in the Fig.3 and Fig.4, the primary decay channels of a
$s\bar{s}$ or $s\bar{s}g$ predominant excitation are $f_{0}\eta$ and $\kappa
K$. If the deviation of $\theta$ from the ideal mixing angle is not large,
$X(1870)$ should be a $n\bar{n}$ or $n\bar{n}g$ predominant state since
$X(1870)$ primarily decay via the $a_{0}(980)\pi$ channel. At present, only
the ground $0^{-+}$ and the $0^{++}$ isoscalar mesons deviate from the ideal
mixing distinctly. In addition, if the $X(1870)$ is produced via a diagram of
Fig.1 [B], its should also be $n\bar{n}$ or $n\bar{n}g$ predominant state.
Of course, the SU(3)f symmetry breaking will effect the ratios of these
channels listed in Table 5, because the three-momentum of the these products
are different. However, the coefficients $\zeta^{2}$ have presented the
valuable information for these specific decay channels. When $\eta_{2}(1870)$
occupies the $2^{1}D_{2}$ $n\bar{n}$ state, $X(1870)$ becomes a good
$n\bar{n}g$ candidate. In the following subsection, we will explore the two-
body strong decays of $X(1870)$ within the ${}^{3}P_{0}$ model and the flux-
tube model. Of course, the analysis of $X(1870)$ also suit $\eta_{2}(1870)$
for their nearly equal masses.
### III.2 The strong decays of $\eta_{2}(1870)$ and $X(1870)$
In Ref. Li3 , the ${}^{3}P_{0}$ model Micu ; Oliver1 ; Oliver2 and the flux-
tube model Kokoski were employed to study the two-body strong decays of
$\eta_{2}(1870)$. There, the pair production (creation) strength $\gamma$ and
the simple harmonic oscillator (SHO) wave function scale parameter, $\beta$s,
were taken as constants.
However, a series of studies indicate that the strength $\gamma$ may depend on
both the flavor and the relative momentum of the produced quarks Ackleh ;
Bonnaz . $\gamma$ may also depend on the reduced mass of quark-antiquark pair
of the decaying meson Segovia . Firstly, the relations of the ${}^{3}P_{0}$
model to ``microscopic'' QCD decay mechanisms have been studied in Ref. Ackleh
. There, the authors found that the constant $\gamma$ corresponds
approximately to the dimensionless combination, $\sigma/m_{q}\beta$, where
$m_{q}$ is the mass of produced quark, $\beta$ means the meson wave function
scale and $\sigma$ is the string tension. Secondly, the momentum dependent
manner of $\gamma$ has been studied in Ref. Bonnaz . It was found that
$\gamma$ is dependent on the relative momentum of the created $q\bar{q}$ pair,
and the form of $\gamma(k)=A+B\exp(-Ck^{2})$ with
$k=|\vec{k}_{3}-\vec{k}_{4}|$ was suggested. Thirdly, J.Segovia, _et al_.,
proposed that $\gamma$ is a function of the reduced mass of quark-antiquark
pair of the decaying meson Segovia . Based on the first and third points
above, $\gamma$ will depend on the flavors of both the decaying meson and
produced pairs. In our calculations, we will treat the $\gamma$ as a free
parameter and fix it by the well-measured partial decay widths.
In addition, the amplitudes given by the ${}^{3}P_{0}$ model and the flux-tube
model often contain the nodal-type Gaussian form factors which can lead to a
dynamic suppression for some channels. Then the values of $\beta$ are
important to exact the decay width for the higher excited mesons in these two
strong decay models.
In the following, the two-body strong decay of $X(1870)$ will be investigated
in the ${}^{3}P_{0}$ model where the strength $\gamma$ will be extracted by
fitting the experimental data. The SHO wave function scale parameter,
$\beta$s, will be borrowed from the Table 1 which are extracted by the GI
relativized potential model. We will also check the possibility of $X(1870)$
as a possible hybrid state by the flux-tube model.
In the non relativistic limit, the transition operator $\mathcal{\hat{T}}$ of
the ${}^{3}P_{0}$ model is depicted as
$\displaystyle\mathcal{\hat{T}}$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{\text{\emph{m}}}\langle 1,m;1,-m|0,0\rangle\iint
d^{3}\vec{k}_{3}d^{3}\vec{k}_{4}\delta^{3}(\vec{k}_{3}+\vec{k}_{4})\mathcal{Y}_{1}^{m}(\frac{\vec{k}_{3}-\vec{k}_{4}}{2})\omega_{0}^{(3,4)}\varphi^{(3,4)}_{0}\chi^{(3,4)}_{1,-m}d^{\dagger}_{3i}(\vec{k}_{3})d^{\dagger}_{4j}(\vec{k}_{4})$
(12)
Where the $\omega_{0}^{(3,4)}$ and $\varphi^{(3,4)}_{0}$ are the color and
flavor wave functions of the $q_{3}\bar{q}_{4}$ pair created from vacuum.
Thus, $\omega_{0}^{(3,4)}=(R\bar{R}+G\bar{G}+B\bar{B})/\sqrt{3}$,
$\varphi^{(3,4)}_{0}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ are color and
flavor singlets. The pair is also assumed to carry the quantum numbers of
$0^{++}$, suggesting that they are in a ${}^{3}P_{0}$ state. Then
$\chi^{(3,4)}_{1,-m}$ represents the pair production in a spin triplet state.
The solid harmonic polynomial
$\mathcal{Y}_{1}^{m}(\vec{k})\equiv|\vec{k}|\mathcal{Y}_{1}^{m}(\theta_{k},\phi_{k})$
reflects the momentum-space distribution of the $q_{3}\bar{q}_{4}$.
The helicity amplitude $\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p)$ of
$A\rightarrow B+C$ is given by
$\displaystyle\langle
BC|\mathcal{\hat{T}}|A\rangle=\delta^{3}(\vec{P}_{A}-\vec{P}_{B}-\vec{P}_{C})\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p),$
(13)
where _p_ represents the momentum of the outgoing meson in the rest frame of
the meson _A_. When the mock state Hayne is adopted to describe the spatial
wave function of a meson, the helicity amplitude
$\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p)$ can be constructed in the
$L-S$ basis easily Oliver1 ; Oliver2 . The mock state for _A_ meson is
$\displaystyle|A({n_{A}}$
$\displaystyle{}^{2S_{A}+1}L_{A}^{J_{A}M_{J_{A}}}(\vec{P}_{A})\rangle$ (14)
$\displaystyle\equiv$
$\displaystyle\sqrt{2E_{A}}\sum_{{M_{L_{A}}}{M_{S_{A}}}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle\omega_{A}^{12}\phi_{A}^{12}\chi_{S_{A}M_{S_{A}}}^{12}$
$\displaystyle\times\int
d\vec{P}_{A}\psi_{n_{A}}^{L_{A}M_{L_{A}}}(\vec{k}_{1},\vec{k}_{2})|q_{1}(\vec{k}_{1})q_{2}(\vec{k}_{2})\rangle.$
To obtain the analytical amplitudes, the SHO wave functions are usually
employed for $\psi_{n_{A}}^{L_{A}M_{L_{A}}}(\vec{k}_{1},\vec{k}_{2})$. For
comparison with experiments, one obtains the partial decay width
$\mathcal{M}^{JL}(p)$ via the Jacob-Wick formula Jacob
$\displaystyle\mathcal{M}_{LS}(p)=$
$\displaystyle\frac{\sqrt{2L+1}}{2J_{A}+1}\sum_{\text{$M_{J_{B}}$,$M_{J_{C}}$}}\langle
L0JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$ (15) $\displaystyle\times\langle
J_{B},M_{J_{B}}J_{C},M_{J_{C}}|JM_{J_{A}}\rangle\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p).$
Finally, the decay width $\Gamma(A\rightarrow BC)$ is derived analytically in
terms of the partial wave amplitudes
$\displaystyle\Gamma(A\rightarrow
BC)=2\pi\frac{E_{B}E_{C}}{M_{A}}p\sum_{LS}|\mathcal{M}_{LS}(p)|^{2}.$ (16)
More technical details of the ${}^{3}P_{0}$ model can be found in Ref. Oliver2
. The inherent uncertainties of the ${}^{3}P_{0}$ decay model itself have been
discussed in the Refs. Bonnaz ; model1 ; model2 .
The dimensionless parameter $\gamma$ will be fixed by the 8 well-measured
partial decay widths which are listed in Table6. The $\mathcal{M}_{LS}$
amplitudes of these decay channels are presented explicitly in the Appendix A.
Decay channels | p (GeV) | $\gamma(10^{3})$ | $\gamma$Bonnaz | Decay channels | p (GeV) | $\gamma(10^{3})$ | $\gamma$Bonnaz
---|---|---|---|---|---|---|---
$\rho\rightarrow\pi\pi$ | 0.362 | 17.8 | 9.18 | $f^{\prime}_{1}\rightarrow K^{*}\bar{K}$ | 0.158 | 4.9 | -
$a_{2}\rightarrow\eta\pi$ | 0.535 | 11.5 | - | $f_{2}\rightarrow K\bar{K}$ | 0.401 | 2.9 | 6.11
$f_{2}\rightarrow\pi\pi$ | 0.622 | 7.8 | 7.13 | $a_{2}\rightarrow K\bar{K}$ | 0.434 | 2.3 | 3.91
$\rho_{3}\rightarrow\pi\pi$ | 0.833 | 4.2 | - | $f^{\prime}_{2}\rightarrow K\bar{K}$ | 0.579 | 2.0 | 5.66
Table 6: . Values of $\gamma$ in different channels and comparison with the
results given in Ref.Bonnaz . Here, $\rho(770)$, $a_{2}(1320)$,
$f^{\prime}_{1}(1420)$, $f_{2}(1270)$, $f^{\prime}_{2}(1525)$ and
$\rho_{3}(1690)$ have been studied.
As mentioned before, $\gamma$ may depend on the flavors of both the decaying
meson and produced pairs. Then we divide the 8 decay channels into two groups:
one is $n\bar{n}\rightarrow n\bar{n}+n\bar{n}$, the other includes
$s\bar{s}\rightarrow n\bar{s}+s\bar{n}$ and $n\bar{n}\rightarrow
n\bar{s}+s\bar{n}$. The values of $\gamma$ here are a little different from
these given in Ref.Bonnaz where an _AL_ 1 potential (for details of _AL_ 1
potential, see Ref.Roberts ) was selected to determine the meson wave
functions. Of course, the meson wave function given by different potentials
will influence the values of $\gamma$.
It is clear in Table 6 that $\gamma$ decrease with _p_ increase. In addition,
our calculation indicates that $\gamma$ depend on flavors of both the decaying
meson and the produced quark pairs. For example, values of $\gamma$ fixed by
$a_{2}\rightarrow K\bar{K}$ and $f^{\prime}_{2}\rightarrow K\bar{K}$ are
roughly equal.
In the following calculations, we assume that the values of $\gamma$
corresponding to the processes of $s\bar{s}\rightarrow n\bar{s}+s\bar{n}$ and
$n\bar{n}\rightarrow n\bar{s}+s\bar{n}$ are determined by one function.
Similarly, we take the function, $\gamma(p)=A+B\exp(-Cp^{2})$, for the
creation vertex. _The function of the creation vertex here is different with
the one used in the RefBonnaz _. With the four decay channels listed in fifth
column of Table 6, we fix the function as $\gamma(p)=1.8+4\exp(-10p^{2})$. For
the processes of $n\bar{n}\rightarrow n\bar{n}+n\bar{n}$ (the first column of
Table 6), we fix the creation vertex function as
$\gamma(p)=3.0+25\exp(-4p^{2})$. The dependence of $\gamma$ on the momentum
_p_ are plotted in the Fig. 5. Obviously the functions can describe the
dependence of $\gamma$ and _p_ well. The functions of creation vertex given
here need further test.
Figure 5: The functions of $\gamma(p)=A+B\exp(-Cp^{2})$ in different decay
processes. The symbols of red ``⚫'' and black ``◼'' denote $\gamma$ values
determined by the experimental data.
Since we neglected the mass splitting within the isospin multiplet, the
partial width into the specific charge channel should be multiplied by the
flavor multiplicity factor $\mathcal{F}$ (Table 7). This $\mathcal{F}$ factor
also incorporates the statistical factor 1/2 if the final state mesons _B_ and
_C_ are identical (as illustrated in Fig.6). More details of $\mathcal{F}$ can
be found in the Appendix A of Ref.Barnes .
Figure 6: Two topological diagrams for a $q\bar{q}$ meson decay in the ${}^{3}P_{0}$ decay model. We refer to the left one as _d_ 1 where the produced quark goes into meson _C_ , and _d_ 2 where it goes into _B_. Decay | $\mathcal{I}_{flavor}(d1)$ | $\mathcal{I}_{flavor}(d1)$ | $\mathcal{F}$
---|---|---|---
channels
$\rho\rightarrow\pi\pi$ | $+1/\sqrt{2}$ | $-1/\sqrt{2}$ | 1
$f_{2}\rightarrow\pi\pi$ | $-1/\sqrt{2}$ | $-1/\sqrt{2}$ | 3/2
$f_{2}\rightarrow KK$ | 0 | $-1/\sqrt{2}$ | 2
$f^{\prime}_{1}\rightarrow K^{*}K$ | +1 | 0 | 4
$f^{\prime}_{2}\rightarrow KK$ | +1 | 0 | 2
$a_{2}\rightarrow KK$ | 0 | -1 | 1
$a_{2}\rightarrow\eta\pi$ | $+1/2$ | $+1/2$ | 1
$\eta_{2}\rightarrow\omega\omega$ | $-1/\sqrt{2}$ | $-1/\sqrt{2}$ | 1/2
$\eta_{2}\rightarrow a_{i}\pi$ | $-1/\sqrt{2}$ | $-1/\sqrt{2}$ | 3
$\eta_{2}\rightarrow f_{i}\eta$ | $+1/2$ | $+1/2$ | 1
Table 7: The second and third columns for the flavor weight factors corresponding to two topological diagrams shown in Fig.6. The last column for the the flavor multiplicity factor $\mathcal{F}$. Here, $|\eta\rangle=(|n\bar{n}\rangle-|s\bar{s}\rangle)/\sqrt{2}$ and $|\eta^{\prime}\rangle=(|n\bar{n}\rangle+|s\bar{s}\rangle)/\sqrt{2}$ have been taken for simplicity. Decay | $\eta_{2}(2^{1}D_{2})$ | | | $\eta_{H}(0^{+}0^{-+})$ | $f_{H}(0^{+}1^{++})$ | $\eta_{H}(0^{+}2^{-+})$
---|---|---|---|---|---|---
channels | Our | Ref. Li3 | Our | Ref. Swanson | Our | Ref. Swanson | Our | Ref. Swanson
$K^{*}K$ | 0.5 | 17.7 | 19.3 | 12.6 | 10 5 | 4.9 | 24.1 18.0 | 3.2 | 2.0 1.0
$\rho\rho$ | 12.9 | 52.2 | 56.8 | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$
$\omega\omega$ | 4.2 | 16.9 | 18.4 | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$
$K^{*}K^{*}$ | 0.2 | 2.1 | 2.3 | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$ | $\times$ | $\times$ $\times$
$a_{0}(1450)\pi$ | 16.0 | 2.4 | 2.6 | 56.3 | 70 175 | 0.5 | $\times$ 6 | 0.5 | 0.0 0.6
$a_{1}(1260)\pi$ | 0.0 | 15.2 | 16.6 | $\times$ | $\times$ $\times$ | 57.3 | 14 232 | $\times$ | 0.3 $\times$
$f_{1}(1280)\eta$ | 0.0 | 0.0 | 0.0 | $\times$ | $\times$ $\times$ | 2.5 | $-$ $-$ | $\times$ | 0.0 $\times$
$a_{2}(1320)\pi$ | 54.2 | 102.5 | 111.6 | 8.8 | 1 16 | 35.1 | 5.0 179.4 | 26.7 | 25.1 67
$f_{2}(1275)\eta$ | 15.1 | 17.5 | 19.0 | 0.0 | $\times$ $\times$ | 1.0 | $-$ $-$ | 4.6 | 0.0 0.0
$\sum\Gamma_{i}$ | 103.3 | 226.5 | 246.7 | 77.7 | 81 196 | 101.3 | 43.1 435.4 | 35.0 | 27.4 68.6
Expt (MeV) | 225$\pm$14 PDG | | | | | 57$\pm$12${}^{+19}_{-4}$ Bes6
Table 8: The partial widths of $X(1870)$ and compared with results from Refs.
Li3 ; Swanson . The symbol ``$\times$'' indicates that the decay modes are
forbidden and ``$-$'' denotes that the decay channels can be ignored. Here, we
collected the results given by the ${}^{3}P_{0}$ model from Ref. Li3 in the
left column, the right column by the flux-tube model. In Ref. Swanson , the
masses are taken as 1.8 GeV for the $0^{-+}$, $1^{++}$ and $2^{-+}$ for the
hybrid states.
The partial decay widths of $X(1870)$ are shown in Table 8 except the channels
of $\mathcal{S}$ $+$ $P$ mesons. $a_{2}(1320)\pi$ and $f_{2}(1275)\eta$ are
large channels for the $\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state in our work and
the Ref. Li3 , which are consistent with the experimental observations of the
$\eta_{2}(1870)$. The partial widths of $K^{*}K$, $\rho\rho$ and
$\omega\omega$ are narrower in our work than the expectations from Ref. Li3 .
$\eta_{2}(1870)$ has been observed by the BES Collaboration in the radiative
decay channel of $J/\psi\rightarrow\gamma\eta\pi\pi$ Li3 . However, no
apparent $\eta_{2}(1870)$ signals were detected in the channels of
$J/\psi\rightarrow\gamma\rho\rho$ Mark1
and$J/\psi\rightarrow\gamma\omega\omega$ Mark2 ; Bes4 . Therefore, improved
experimental measurements of the radiative $J/\psi$ decay channels are needed
for the $\eta_{2}(1870)$ in future.
Figure 7: The diagram for the `` $\mathcal{S}$ $+$ P'' channels through a
virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson.
Nextly, we shall evaluate the partial widths of ``$\mathcal{S}+P$'' channels
which have not been listed in the table8. _The scheme is proposed as
following._ As illustrated in the Fig.7, we assume $X(1870)$ decay into
$a_{0}(980)\pi$ via a virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson . We
notice the $\eta(1295)$ also dominantly decay into the $\eta\pi\pi$ Bes6 . Its
three-body decay can occur via three intermediate processes:
$\eta(1295)\rightarrow\eta\sigma/a_{0}(980)\pi/\eta(\pi\pi)_{S-wave}\rightarrow\eta\pi\pi$PDG
. With the ratio $\Gamma(a_{0}(980)\pi)/\Gamma(\eta\pi\pi)=0.65\pm 0.10$ and
$\Gamma(\eta(1295))=55\pm 5$MeV, the partial width of $\eta(1295)$ decaying
into $a_{0}(980)\pi$ is estimated no more than 45MeV. By the ${}^{3}P_{0}$
model, the ratio of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(1^{3}P_{0})\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(1^{3}P_{0})\pi)}$
can be reached easily. If the uncertainty of the coupling vertex of
$\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow a_{0}(980))$ (see in Fig.7) is
assumed to be canceled in the ratio of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(1^{3}P_{0})\pi)\cdot\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980))}{\Gamma(\eta(1295)\rightarrow
a_{0}(1^{3}P_{0})\pi)\cdot\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980))}$, the value of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(980)\pi)}$ can be extracted
roughly. Although the assumption above seems a little rough, we just need to
evaluate magnitudes of these decay channels.
$\eta(1295)$ is proposed to be the first radial excited state of $\eta(550)$.
Then the total decay widths of $\Gamma(X(1870)\rightarrow\mathcal{S}+P)$ is
evaluated no more than 12.6MeV and $\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)\leq 3.8$MeV. The BESIII Collaboration claimed that $X(1870)$
primarily decay via $a_{0}(980)\pi$ Bes6 . The small partial width of
$\Gamma(X(1870)\rightarrow a_{0}(980)\pi)$ also indicates that the $X(1870)$
can't be interpreted as the $2^{1}D_{2}$ $q\bar{q}$ state.
In addition, our results do not support $X(1870)$ as the
$\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state since its observed decay width is much
smaller than the theoretical estimate. The $a_{2}(1320)\pi$ is the largest
decay channel in our numerical results and in Ref. Li3 for the
$\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state (Table 8). If the partial width of
$a_{0}(980)\pi$ channel is as large as $a_{2}(1320)\pi$, the predicted width
of $X(1870)$ will be much larger than the observed value.
We adopt the flux tube model to check the possibility of $X(1870)$ as a hybrid
meson. The partial widths are also listed in Table 8 for the comparison.
Details of the flux model are collected in the Appendix B.
Two groups of the partial widths predicted in the Ref. Swanson are quoted in
the Table 8. The left column was given by the flux tube decay model of Isgur,
Kokoski, and Paton (IKP) with the ``standard parameters'' IKP . The right
column was by the developed flux tube decay model of Swanson-Szczepaniak (SS).
In Ref. Swanson , the masses are taken as 1.8 GeV for the $0^{-+}$, $1^{++}$
and $2^{-+}$ for the hybrid states.
For a hybrid meson, $X(1870)$ seems most possible to be the
$\eta_{H}(0^{+}2^{-+})$ state because the total widths exclude the channels of
$\mathcal{S}+P$ are much narrow in our work and in Ref.Swanson . It is
consistent with the narrow width of $X(1870)$.
As shown in Table 8, $X(1870)$ is impossible to be the $\eta_{H}(0^{+}0^{-+})$
hybrid state. The predicted width in both our work and in Ref.Swanson are
broader. In addition, $\eta\pi$ is a visible channel for both $a_{0}(1450)$. A
week signal was found in the region of 1200$\sim$1400MeV in the analysis of
$\eta\pi^{\pm}$ (Fig.2(b) of Ref.Bes6 ), which contradicts the large
$a_{0}(1450)\pi$ channel of the $\eta_{H}(0^{+}0^{-+})$ state. We can exclude
the possibility of $X(1870)$ as the $\eta_{H}(0^{+}0^{-+})$ hybrid state
preliminarily.
The assignment for $X(1870)$ as the $f_{H}(0^{+}1^{++})$ hybrid seems
impossible since the theoretical width of $a_{1}(1260)\pi$ is rather broad in
our results and in the IKP model. If the partial width of $a_{0}(980)\pi$
channel is as large as $a_{1}(1260)\pi$, the total widths of $X(1870)$ will be
much broader than the experimental value. But the width given by the SS flux
tube decay model for the $f_{H}(0^{+}1^{++})$ hybrid is much small. So the
possibility of $X(1870)$ as a $f_{H}(0^{+}1^{++})$ hybrid can not be excluded.
We suggest to detect the decay channel of $a_{1}(1260)\pi$ because this
channel is forbidden for the $\eta_{H}(0^{+}2^{-+})$ state in the IKP flux
tube decay model and very small in the SS flux tube decay model (see Table8).
Then the channel of $a_{1}(1260)\pi$ can discriminate the state
$f_{H}(0^{+}1^{++})$ and $\eta_{H}(0^{+}2^{-+})$ for $X(1870)$.
Finally, if $\eta_{2}(1870)$ is the $\eta_{2}(2^{1}D_{2})$ state, its decay
width is predicted about 100MeV which is much smaller than the experiments.
However, the difference can be explained by the remedy of mixing effect. If
$X(1870)$ and $\eta_{2}(1870)$ have the same quantum numbers, $0^{+}2^{-+}$,
they should mix with each other with a visible mixing angle. Then the
interference enhancement will enlarge the width of $\eta_{2}(1870)$. The broad
decay width of $\eta_{2}(1870)$ could be explained naturally. On the other
hand, $\eta_{2}(1870)$ has been observed in the channel of $a_{0}(980)\pi$.
However, this channel seems much small if $\eta_{2}(1870)$ is a pure
$2^{1}D_{2}$ $n\bar{n}$ meson. The mixing effect will also enlarge this
partial width. Here, we don't plan to discuss the mixing of $X(1870)$ and
$\eta_{2}(1870)$ further for the complex mechanism.
## IV DISCUSSIONS AND CONCLUSIONS
A isoscalar resonant structure of $X(1870)$ was observed by BESIII in the
channels $J/\psi\rightarrow\omega X(1870)\rightarrow\omega\eta\pi^{+}\pi^{-}$
recently. Although the mass of $X(1870)$ is consistent with the
$\eta_{2}(1870)$, the production, decay width and decay properties are much
different. In this paper, the mass spectrum and strong decays of the $X(1870)$
and $\eta_{2}(1870)$ are analyzed.
Firstly, the mass spectrum are studied in the GI potential model and the RTs
framework. In the GI potential model, both $X(1870)$ and $\eta_{2}(1870)$
could be the $\eta^{\prime}_{2}(1^{1}D_{2})$, $f^{\prime}_{1}(2^{3}P_{1})$ and
$\eta_{2}(2^{1}D_{2})$ states. In RTs, the possible assignments are the
$\eta(3^{1}S_{0})$, $f^{\prime}_{1}(2^{3}P_{1})$ and $\eta_{2}(2^{1}D_{2})$
states. For the mass spectrum, they are also good hybrid candidates since the
masses overlap the predictions given by different models (see Table4).
Secondly, the processes of a $n\bar{n}$ quarkonium or a $n\bar{n}g$ hybrid
meson decaying into the ``$\mathcal{S}+P$'' mesons are studied under the
SU(3)f symmetry and the diquark-antidiquark description of the $\mathcal{S}$
mesons. We assumed the processes obey the OZI rule. We find that the channels
of $a_{0}\pi$, $\sigma\eta$ and $f_{0}\eta$ are the dominant when a $n\bar{n}$
quarkonium or a $n\bar{n}g$ hybrid meson decays primarily through this kind of
processes. This result can explain why $X(1870)$ has been first observed in
the $\eta\pi\pi$ channel.
Thirdly, the two-body strong decay of $X(1870)$ is computed in the
${}^{3}P_{0}$ model. As the $\eta_{2}(2^{1}D_{2})$ quarkonium, the predicted
width of $X(1870)$ looks much larger than the observations. The broad
resonance, $\eta_{2}(1870)$, can be a natural candidate for the $2^{1}D_{2}$
$n\bar{n}$ meson. There, we fix the creation strength, $\gamma$, in two kinds
of processes: ①.$n\bar{n}\rightarrow n\bar{n}+n\bar{n}$; ②.
$n\bar{n}\rightarrow n\bar{s}+s\bar{n}$ and $s\bar{s}\rightarrow
n\bar{s}+s\bar{n}$. The functions of creation vertex are determined as
$\gamma(p)=3.0+25\exp(-4p^{2})$ and $\gamma(p)=1.8+4\exp(-10p^{2})$
respectively. Meanwhile, the SHO wave function scale, $\beta$s, are obtained
by the GI potential model.
We have evaluated the magnitude of the partial widths of ``$\mathcal{S}+P$''
channels by the ratio, $\frac{\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(980)\pi)}$, under a rather
crude assumption that $\eta(1295)/X(1870)\rightarrow a_{0}(980)\pi$ through a
virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson (see Fig.7). Then the
uncertainties of the coupling vertex for $1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980)$ are assumed to be canceled in the ratio. The total widths of
``$\mathcal{S}+P$'' are evaluated no more than 12.6MeV and
$\Gamma(X(1870)\rightarrow a_{0}(980)\pi)\leq 3.8$MeV. Since $X(1870)$
primarily decay via $a_{0}(980)\pi$, it also indicated that the $X(1870)$
can't be interpreted as the $2^{1}D_{2}$ $n\bar{n}$ state.
We also study the $X(1870)$ as a hybrid state in the flux tube model. Our
results agree well with most of predictions given by Ref. Swanson . $X(1870)$
looks most like the $\eta_{H}(0^{+}2^{-+})$ state for the narrow predicted
width, which is consistent with the experiments. But we can't exclude the
possibility of $0^{+}1^{++}$. A precise measurement of $a_{1}(1260)\pi$ is
suggested to pin down this uncertainty.
Finally, some important arguments and useful suggestions are given as follows.
➀.If $\eta_{2}(1870)$ is the $\eta_{2}(2^{1}D_{2})$ state, the broad
$\pi_{2}(1880)$ should be isovector partner of $\eta_{2}(1870)$.
$\pi_{2}(1880)$ has been interpreted as the conventional $2^{1}D_{2}$
$q\bar{q}$ meson in Ref. Li4 . In deed, the decay channel of $\omega\rho$ is
large enough for $\pi_{2}(1880)$ E852 . This observation disfavors the
$\pi_{2}(1880)$ as a $2^{-+}$ hybrid candidate for the selection rule that a
hybrid meson decaying into two S-wave mesons is strongly suppressed page .
➁.If $X(1870)$ is a hybrid meson, we suggest to search its isospin partner in
the decay channels of $J/\psi\rightarrow\rho f_{0}(980)\pi$ and
$J/\psi\rightarrow\rho b_{1}(1235)\pi$, which are accessible at BESIII, Belle
and BABAR Collaborations. The decay channel of $b_{1}(1235)\pi$ is forbidden
for the $\pi_{2}(2^{1}D_{2})$ quarkonium due to the ``spin selection rule''
Burns ; flux1 . We also suggest to search the $\eta_{2}(1870)$ in the decay
channels of $J/\psi\rightarrow\gamma\rho\rho$ and
$J/\psi\rightarrow\gamma\omega\omega$ since these channels are forbidden for
the hybrid production.
###### Acknowledgements.
Bing Chen thanks Jun-Long Tian and D.V. Bugg for very helpful discussions.
This work is supported by the Key Program of the He'nan Educational Committee
of China (No.13A140014), the National Natural Science Foundation of China
under grant No. 11305003, No. 11075102, No. 11005003, and U1204115, the
Innovation Program of Shanghai Municipal Education Commission under grant No.
13ZZ066, and the Program of He'nan Technology Department (No. 11147201).
## Appendix A The expressions of amplitudes
We have omitted a exponential factor in following decay amplitudes
$\mathcal{M}_{LS}$ for compactness,
$\exp(-\frac{2\lambda\mu-\nu^{2}}{4\mu}p^{2}).$ (17)
where we defined
$\displaystyle\mu=\frac{1}{2}(\frac{1}{\beta_{A}^{2}}+\frac{1}{\beta_{B}^{2}}+\frac{1}{\beta_{C}^{2}});\hskip
3.41418pt\nu=\frac{m_{1}}{(m+m_{1})\beta_{B}^{2}}+\frac{m_{2}}{(m+m_{2})\beta_{C}^{2}}.$
and
$\displaystyle\lambda=\frac{m_{1}^{2}}{(m+m_{1})^{2}\beta_{B}^{2}}+\frac{m_{2}^{2}}{(m+m_{2})^{2}\beta_{C}^{2}};\hskip
10.243pt\eta=\frac{m_{1}}{m+m_{1}}.$
For $1^{3}S_{1}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,
$\mathcal{M}_{10}=\frac{2\mu-\nu}{8\sqrt{3}\pi^{5/4}\mu^{5/2}(\beta_{A}\beta_{B}\beta_{C})^{3/2}}p$
(18)
For $1^{3}P_{2}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,
$\mathcal{M}_{20}=\frac{2\mu\beta_{B}^{3/2}-(p^{2}\nu^{2}+2\mu(1-p^{2}\nu))\beta_{C}^{3/2}}{8\sqrt{15}\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}$
(19)
For $1^{3}P_{2}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,
$\mathcal{M}_{21}=-\sqrt{3/2}\mathcal{M}_{20}$.
For $1^{3}P_{1}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,
$\mathcal{M}_{01}=\frac{4\mu\beta_{B}^{3/2}+(p^{2}\nu^{2}+2\mu(1-p^{2}\nu))\beta_{C}^{3/2}}{24\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}$
(20)
$\mathcal{M}_{21}=\frac{(2\mu-\nu)\nu}{24\sqrt{2}\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}p^{2}$
(21)
For $1^{3}D_{3}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,
$\mathcal{M}_{30}=-\frac{2\mu-\nu}{16\sqrt{35}\pi^{5/4}\mu^{9/2}\beta_{A}^{7/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu^{2}p^{3}$
(22)
For $2^{1}S_{0}\rightarrow 1^{3}P_{0}+1^{1}S_{0}$,
$\displaystyle\mathcal{M}_{11}=$
$\displaystyle\frac{1}{96\pi^{5/4}\mu^{11/2}\beta_{A}^{7/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(23)
$\displaystyle\times(-24p^{2}\eta\mu^{3}-p^{4}\nu^{4}+2p^{2}\mu\nu^{2}(-10+p^{2}(1+\eta)\nu)-4\mu^{2}(15-5p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})$
$\displaystyle+6\mu^{2}(4p^{2}\eta\mu^{2}+p^{2}\nu^{2}-2\mu(-3+p^{2}(1+\eta)\nu))\beta_{A}^{2})$
For $2^{1}D_{2}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,
$\mathcal{M}_{11}=\frac{-p^{4}\nu^{4}+2p^{2}\nu^{2}\mu(\nu
p^{2}-14)+28\mu^{2}(\nu p^{2}-5)+14\mu^{2}(p^{2}\nu^{2}-2(\nu
p^{2}-5)\mu\beta_{A}^{2})}{160\sqrt{21}\pi^{5/4}\mu^{13/2}\beta_{A}^{11/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu
p$ (24) $\mathcal{M}_{31}=\frac{-28\mu^{2}+\nu^{3}p^{2}-2\mu\nu(\nu
p^{2}-9)+14(2\mu-\nu)\mu^{2}\beta_{A}^{2}}{160\sqrt{14}\pi^{5/4}\mu^{13/2}\beta_{A}^{11/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu^{2}p^{3}$
(25)
For $2^{1}D_{2}\rightarrow 1^{3}S_{1}+1^{3}S_{1}$,
$\mathcal{M}^{\prime}_{11}=\sqrt{2}\mathcal{M}_{11}$ and
$\mathcal{M}^{\prime}_{31}=\sqrt{2}\mathcal{M}_{31}$.
For $2^{1}D_{2}\rightarrow 1^{3}P_{0}+1^{1}S_{0}$
$\displaystyle\mathcal{M}_{20}=$
$\displaystyle-\frac{1}{192\sqrt{35}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(26)
$\displaystyle\times(p^{4}\nu^{5}-2p^{2}\mu\nu^{3}(-18+p^{2}(1+\eta)\nu)+4\mu^{2}\nu(63-11p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})+56\mu^{3}(-2+\eta(-2+p^{2}\nu))$
$\displaystyle-14\mu^{2}(p^{2}\nu^{3}-2\mu\nu(-7+p^{2}(1+\eta)\nu)+4\mu^{2}(-2+\eta(-2+p^{2}\nu)))\beta_{A}^{2})\nu
p^{2}.$
For $2^{1}D_{2}\rightarrow 1^{3}P_{1}+1^{1}S_{0}$
$\displaystyle\mathcal{M}_{21}=-\frac{p^{2}\nu^{2}-14\mu^{2}\beta^{2}_{A}+14\mu}{16\sqrt{35}\pi^{5/4}\mu^{11/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}(\eta-1)\nu
p^{2}.$ (27)
For $2^{1}D_{2}\rightarrow 1^{3}P_{2}+1^{1}S_{0}$
$\displaystyle\mathcal{M}_{02}=$
$\displaystyle\frac{1}{480\sqrt{14}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(28)
$\displaystyle\times(p^{6}\mu^{6}-2p^{4}\mu\nu^{4}(p^{2}(1+\eta)\nu-21)+4p^{2}\mu^{2}\nu^{2}(105-14p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})+56\mu^{3}(15-5p^{2}(1+\eta)\nu$
$\displaystyle+p^{4}\eta\nu^{2})-14\mu^{2}(p^{4}\nu^{4}-2p^{2}\mu\nu^{2}(-10+p^{2}(1+\eta)\nu)+4\mu^{2}(15-5p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2}))\beta_{A}^{2}).$
$\displaystyle\mathcal{M}_{22}=$
$\displaystyle-\frac{1}{672\sqrt{5}\pi^{5/4}\mu^{15/2}\beta_{A}^{15/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(29)
$\displaystyle\times(p^{4}\nu^{5}-2p^{2}\mu\nu^{3}(-18+p^{2}(1+\eta)\nu)+2\mu^{2}\nu(126-25p^{2}(1+\eta)\nu+2p^{4}\eta\nu^{2})+28\mu^{3}(-7+\eta(-7+2p^{2}\nu))$
$\displaystyle-14(\mu^{2})(p^{2}\nu^{3}-2\mu\nu(-7+p^{2}(1+\eta)\nu)+2\mu^{2}(-7+\eta(-7+2p^{2}\nu)))\beta_{A}^{2})\nu
p^{2}.$
$\displaystyle\mathcal{M}_{42}=$
$\displaystyle-\frac{1}{1120\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(30)
$\displaystyle\times(\nu(p^{2}\nu^{3}-36\mu^{2}+2\mu\nu(11-p^{2}\nu))+2\eta\mu(28\mu^{2}-p^{2}\nu^{3}+2\mu\nu(p^{2}\nu-9))-14\mu^{2}(2\mu-\nu)(2\eta\mu-\nu)\beta_{A}^{2})\nu^{2}p^{4}.$
For $2^{1}D_{2}\rightarrow 2^{3}P_{1}+1^{1}S_{0}$
$\displaystyle\mathcal{M}_{21}=$
$\displaystyle\frac{1}{160\sqrt{14}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{9/2}\beta_{C}^{3/2}}$
(31)
$\displaystyle\times(-112\eta\mu^{3}+252\mu^{2}\nu+56p^{2}\eta^{2}\mu^{3}\nu-80p^{2}\eta\mu^{2}\nu^{2}+36p^{2}\mu\nu^{3}+4p^{4}\eta^{2}\mu^{2}\nu^{3}-4p^{4}\eta\mu\nu^{4}+p^{4}\nu^{5}$
$\displaystyle-10\mu^{2}\nu(14\mu+p^{2}\nu^{2})\beta_{B}^{2}-14\mu^{2}\beta_{A}^{2}(14\mu\nu+4p^{2}\eta^{2}\mu^{2}\nu+p^{2}\nu^{3}-4\eta\mu(2\mu+p^{2}\nu^{2})-10\mu^{2}\nu\beta_{B}^{2}))(\eta-1)p^{2}.$
$m_{1}$ and $m_{2}$ are the masses of quarks in the decaying meson _A_. _m_ is
the mass of the created quark from the vacuum. For calculating the decay
widths, the masses of quarks are taken as: $m_{u}$ = $m_{d}$ = 0.220 GeV,
$m_{s}=$ 0.428 GeV, which are as same as these in the Section II. The above
amplitudes, $\mathcal{M}_{LS}$, can be reduced further in the approximation of
$m_{1}=m_{2}=m$ and $\beta_{A}=\beta_{B}=\beta_{C}=\beta$. The reduced
$\mathcal{M}_{LS}$ are consistent with these given by Ref. Barnes except for
an unimportant factor, $-2^{9/2}$, since this factor can be absorbed into the
coefficient $\gamma$.
## Appendix B Hybrid decay in the flux tube model
The flux tube model was motivated by the strong coupling expansion of the
lattice QCD. In this model, decay occurs when the flux-tube breaks at any
point along its length, with a $q\bar{q}$ pair production in a relative
$J^{PC}=0^{++}$ state. It is similar to the ${}^{3}P_{0}$ decay model but with
an essential difference. The flux tube model extend the nonrelativistic
constituent quark model to include gluonic degrees of freedom in a very simple
and intuitive way, where the gluonic field is regarded as tubes of color flux.
Then it can be extended to the hybrid research. When the hybrid mesons are
assumed to be narrow, and the threshold effects aren't taken into account, the
partial decay width $\Gamma_{LS}(H\rightarrow BC)$ is given by the flux model
as flux1
$\Gamma_{LS}(H\rightarrow
BC)=\frac{p}{(2J_{A}+1)\pi}\frac{\tilde{M}_{B}\tilde{M}_{C}}{\tilde{M}_{A}}|\mathcal{M}_{LS}(H\rightarrow
BC)|^{2}$ (32)
where $\tilde{M}_{A}$, $\tilde{M}_{B}$, $\tilde{M}_{C}$ are the ``mock-meson''
masses of A, B, C Kokoski . When a hybrid meson decay into _P_ -wave and
pseudoscalar mesons, the partial wave amplitude $\mathcal{M}_{L}(H\rightarrow
BC)$ (with $S=S_{B}$) is given as the following form
$\mathcal{M}_{L}(H\rightarrow
BC)=\langle\phi_{B}\phi_{C}|\phi_{A}\phi_{0}\rangle(\frac{a\tilde{c}}{9\sqrt{3}}\frac{1}{2}A^{0}_{00}\sqrt{\frac{fb}{\pi}})\frac{\kappa\sqrt{b}}{(1+fb/(2\tilde{\beta}^{2}))^{2}}\sqrt{\frac{2\pi}{3\Gamma(3/2+\delta)}}\frac{\beta_{A}^{3/2+\delta}}{\tilde{\beta}}\tilde{\mathcal{M}}_{L}(H\rightarrow
BC)$ (33)
The flavor matrix element $\langle\phi_{B}\phi_{C}|\phi_{A}\phi_{0}\rangle$
have been discussed before. $\tilde{\mathcal{M}}_{L}(H\rightarrow BC)$ are
listed in Table 9 for the states of $\eta_{H}(0^{+}0^{-+})$,
$f_{H}(0^{+}1^{++})$ and $\eta_{H}(0^{+}2^{-+})$.
_B_ | _H_($0^{+}0^{-+}$) | _H_($0^{+}1^{++}$) | _H_($0^{+}2^{-+}$)
---|---|---|---
$0^{++}$ | $+\sqrt{2}\mathcal{M}_{S}/3$ | $-\sqrt{2}\mathcal{M}_{P_{2}}/\sqrt{3}$ | $+\mathcal{M}_{D}/3$
$1^{++}$ | $-$ | $-\mathcal{M}_{P_{1}}/\sqrt{2}$ | -
$2^{++}$ | $+\mathcal{M}_{D}/3$ | $-\mathcal{M}_{P_{4}}/\sqrt{30}$ | $-\sqrt{5}\mathcal{M}_{S}/\sqrt{18}$
| | $+\mathcal{M}_{F}/\sqrt{5}$ | $-\sqrt{7}\mathcal{M}_{D}/3$
Table 9: Partial wave amplitudes $\tilde{\mathcal{M}}_{L}(H\rightarrow BC)$
for an initial hybrid _H_ decaying into a _P_ -wave and pseudoscalar mesons.
Here the $\mathcal{M}_{S}$, $\mathcal{M}_{D}$, $\mathcal{M}_{P_{i}}$ and
$\mathcal{M}_{F}$ are defined as
$\mathcal{M}_{S}=-(3\tilde{h}_{0}-\tilde{g}_{1}+4\tilde{h}_{2})$,
$\mathcal{M}_{D}=(\tilde{g}_{1}+5\tilde{h}_{2})$,
$\mathcal{M}_{P_{1}}=-i(2\tilde{g}_{0}+3\tilde{h}_{1}-\tilde{g}_{2})$,
$\mathcal{M}_{P_{2}}=-i(\tilde{g}_{0}+\tilde{g}_{2})$,
$\mathcal{M}_{P_{4}}=-i(10\tilde{g}_{0}+9\tilde{h}_{1}+\tilde{g}_{2})$ and
$\mathcal{M}_{F}=-3i(\tilde{g}_{2}+\tilde{h}_{3})$. The analytical expressions
of $\tilde{g}_{i}$ and $\tilde{h}_{i}$ are given as
$\tilde{g}_{n}=2^{3+\delta}\frac{M^{n}m}{(M+m)^{n+1}}(2\beta_{A}^{2}+\tilde{\beta}^{2})^{-\frac{n+\delta+3}{2}}\Gamma(\frac{n+\delta+3}{2})_{1}F_{1}[\frac{n+\delta+3}{2},n+1,-(\frac{M}{M+m})^{2}\frac{p^{2}}{2\beta_{A}^{2}+\tilde{\beta}^{2}}]p^{n+1}$
(34)
$\tilde{h}_{n}=2^{3+\delta}\tilde{\beta}^{2}(\frac{M}{M+m})^{n}(2\beta_{A}^{2}+\tilde{\beta}^{2})^{-\frac{n+\delta+4}{2}}\Gamma(\frac{n+\delta+4}{2})_{1}F_{1}[\frac{n+\delta+4}{2},n+1,-(\frac{M}{M+m})^{2}\frac{p^{2}}{2\beta_{A}^{2}+\tilde{\beta}^{2}}]p^{n}$
(35)
where ${}_{1}F_{1}[\cdots]$ are the confluent hypergeometric functions. Here
we don't take account of the decay channels of $H\rightarrow 2S+1S$ because
they are forbidden by the conservation laws, or the ``spin selection rule'',
or the phase space, _e.g._ , the decay channel of $\pi(1300)+\pi$ is forbidden
for the $f_{H}(0^{+}1^{++})$ state by the ``spin selection rule''. In this
work, we choose to follow the Refs. flux1 and take the combination
$(a\tilde{c}/9\sqrt{3})\frac{1}{2}A^{0}_{00}\sqrt{\frac{fb}{\pi}}$ as 0.64
which was fixed by the conventional mesons Kokoski , $M=m=m_{u,d}=330$MeV,
$\tilde{M}^{I=0}_{B}$ $=\tilde{M}^{I=1}_{B}=1250$MeV,
$\tilde{M}^{I=0}_{C}=720$MeV, $\tilde{M}^{I=1}_{C}=850$MeV,
$\beta_{A}=$0.27GeV, $\delta=$0.62, $b=$0.18GeV2 and $\kappa=$0.9. Final
states containing $\pi$ have $\tilde{\beta}=$0.36GeV, otherwise
$\tilde{\beta}=$0.40GeV.
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|
arxiv-papers
| 2013-04-23T07:41:27 |
2024-09-04T02:49:44.688419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Bing, Ke-Wei Wei, Ailin Zhang",
"submitter": "Chen Bing",
"url": "https://arxiv.org/abs/1304.6190"
}
|
1304.6317
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-070 LHCb-PAPER-2013-022 June 4, 2013
Measurement of the branching fractions of the decays
$B^{0}_{s}\rightarrow\kern
4.14793pt\overline{\kern-4.14793ptD}{}^{0}K^{-}\pi^{+}$ and
$B^{0}\rightarrow\kern 4.14793pt\overline{\kern-4.14793ptD}{}^{0}K^{+}\pi^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The first observation of the decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ is reported. The
analysis is based on a data sample, corresponding to an integrated luminosity
of $1.0\mbox{\,fb}^{-1}$ of $pp$ collisions, collected with the LHCb detector.
The branching fraction relative to that of the topologically similar decay
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ is measured to be
$\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm
0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$
In addition, the relative branching fraction of the decay
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
is measured to be
$\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm
0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,.$
Submitted to Phys. Rev. D.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann60, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal58, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, R. Wallace12,
S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The precise measurement of the angle $\gamma$ of the CKM Unitarity Triangle
[1, 2] is one of the primary objectives in contemporary flavour physics.
Measurements from the experiments BaBar, Belle and LHCb are based mainly on
studies of $B^{+}\rightarrow DK^{+}$ decays, where the notation $D$ implies
that the neutral $D$ meson is an admixture of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ states. Each experiment currently
gives constraints on $\gamma$ with a precision of $\sim 15^{\circ}$ [3, 4, 5].
Significant reduction of this uncertainty is well motivated and the use of
additional channels to further improve the precision is of great interest.
The decay $B^{0}\rightarrow DK^{+}\pi^{-}$, including the resonant
contribution from $B^{0}\rightarrow DK^{*0}$, is one of the modes with the
potential to make significant impact on the overall determination of $\gamma$
[6]. A first measurement of $C\\!P$ observables in $B^{0}\rightarrow DK^{*0}$
decays has been reported by LHCb [7]. This decay is particularly sensitive to
$\gamma$ owing to the interference of $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ amplitudes, which for this decay are of similar
magnitude. It has been noted that an amplitude analysis of $B^{0}\rightarrow
DK^{+}\pi^{-}$ decays can further improve the sensitivity and also resolve the
ambiguities in the result [8, 9].
The decays $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ can be mediated by the
decay diagrams shown in Fig. 1. Both $B^{0}$ and $B^{0}_{s}$ decays are
flavour-specific, with the charge of the kaon identifying the flavour of the
decaying $B$ meson, though the charges are opposite in the two cases. In
addition to these colour-allowed tree-level diagrams, colour-suppressed tree-
level diagrams contribute to $B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$ decays ($K\pi$ denotes the sum
over both charge combinations). Both colour-allowed and colour-suppressed
diagrams contribute to the CKM-suppressed $B^{0}_{(s)}\rightarrow D^{0}K\pi$
modes.
Figure 1: Decay diagrams for (a) favoured $B^{0}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{+}\pi^{-}$ decays and (b)
favoured $B^{0}_{s}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{-}\pi^{+}$ decays.
A first study of the decay $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ has been performed by
BaBar [10], giving a branching fraction measurement ${\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)=(88\pm 15\pm
9)\times 10^{-6}$, where the contribution from the $B^{0}\rightarrow
D^{*-}K^{+}$ decay is excluded. There is no previous branching fraction
measurement for the inclusive three-body process $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$, although that of the
resonant contribution $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ has been measured by LHCb [11].
Since the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ and the related
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ decays form
potentially serious backgrounds to the $B^{0}$ $\rightarrow$ $D$ $K^{+}$
$\pi^{-}$ channel, measurements of their properties will be necessary to
reduce systematic uncertainties in the determination of $\gamma$.
In this paper the results of a study of neutral $B$ meson decays to $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$, including inspections of
their Dalitz plot distributions, are presented. The $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ final states are
combined, and the inclusion of charge conjugate processes is implied
throughout the paper. In order to reduce systematic uncertainties in the
measurements, the topologically similar decay $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$, which has been
studied in detail previously [12, 13], is used as a normalisation channel. In
this paper, $D\pi\pi$ denotes the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ final state and
$DK\pi$ denotes the sum over the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ final states. The
neutral $D$ meson is reconstructed using the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ final
state; therefore the signal yields measured include small contributions from
$D^{0}\rightarrow K^{+}\pi^{-}$ decays, but such contributions are expected to
be small and are neglected hereafter. The analysis uses a data sample,
corresponding to an integrated luminosity of $1.0\mbox{\,fb}^{-1}$ of $pp$
collisions at a centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$,
collected with the LHCb detector during 2011.
## 2 Detector, trigger and selection
The LHCb detector [14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4 % at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6 % at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors [15]. Photon, electron and hadron candidates are identified
by a calorimeter system consisting of scintillating-pad and preshower
detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons
are identified by a system composed of alternating layers of iron and
multiwire proportional chambers.
The LHCb trigger [16] consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage that applies a
full event reconstruction. In this analysis, signal candidates are accepted if
one of the final state particles created a cluster in the hadronic calorimeter
with sufficient transverse energy to fire the hardware trigger. Events that
are triggered at the hardware level by another particle in the event are also
retained.
The software trigger requires a two-, three- or four-track secondary vertex
with a high sum of the transverse momentum, $p_{\rm T}$, of the tracks and a
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter $\chi^{2}$,
$\chi^{2}_{\rm IP}$, with respect to the primary interaction greater than 16.
The $\chi^{2}_{\rm IP}$ is the difference between the $\chi^{2}$ of the PV
reconstruction with and without the considered track. A multivariate algorithm
[17] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron.
Candidates that satisfy the software trigger selection and are consistent with
the decay chain $B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{\pm}\pi^{\mp}$, $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ are
selected, with requirements similar to those in the LHCb study of the decay
$B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$ [18]. The $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate invariant mass is
required to satisfy
$1844<m_{K\pi}<1884{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Tracks are
required to be consistent with either the kaon or pion hypothesis, as
appropriate, based on particle identification (PID) information primarily from
the RICH detectors [15]. All other selection criteria were tuned on the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ channel. The large
yield available for the $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ normalisation sample
allows the selection to be based on data, though the efficiencies are
determined using simulated events. In the simulation, $pp$ collisions are
generated using Pythia 6.4 [19] with a specific LHCb configuration [20].
Decays of hadronic particles are described by EvtGen [21] in which final state
radiation is generated using Photos [22]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [23, *Agostinelli:2002hh] as described in Ref. [25].
Loose selection requirements are applied to obtain a visible signal peak in
the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$
normalisation channel. The selection includes criteria on the quality of the
tracks forming the signal candidate, their $p$, $p_{\rm T}$ and $\chi^{2}_{\rm
IP}$. Requirements are also placed on the corresponding variables for
candidate composite particles ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, $B^{0}_{(s)}$) together with
restrictions on the consistency of the decay fit ($\chi^{2}_{\rm vertex}$),
the flight distance significance ($\chi^{2}_{\rm flight}$), and the cosine of
the angle between the momentum vector and the line joining the PV under
consideration to the $B^{0}_{(s)}$ vertex ($\cos\theta_{\rm dir}$) [11].
A boosted decision tree (BDT) [26] that identifies $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ candidates
is used to suppress backgrounds from $b$-hadron decays to final states that do
not contain charmed particles and backgrounds where the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ does not decay to the
$K^{+}\pi^{-}$ final state. This “$D^{0}$ BDT” [27, 28] is trained using a
large high-purity sample obtained from $B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ decays. The BDT takes
advantage of the kinematic similarity of all $b$-hadron decays and avoids
using any topological information from the $B^{0}_{(s)}$ decay. Properties of
the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate and its
daughter tracks, containing kinematic, track quality, vertex and PID
information, are used to train the BDT.
Further discrimination between signal and background categories is achieved by
calculating weights, using the sPlot technique [29], for the remaining $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ candidates. The
weights are based on a simplified fit to the $B$ candidate invariant mass
distribution from the $D\pi\pi$ data sample. The weights are used to train a
neural network [30] to maximise the separation between the categories. A total
of 10 variables are used in the network. They include the $p_{\rm T}$,
$\chi^{2}_{\rm IP}$, $\chi^{2}_{\rm vertex}$, $\chi^{2}_{\rm flight}$ and
$\cos\theta_{\rm dir}$ of the $B^{0}_{(s)}$ candidate, the output of the
$D^{0}$ BDT and the $\chi^{2}_{\rm IP}$ of the two pion tracks that originate
from the $B^{0}_{(s)}$ vertex. The $p_{\rm T}$ asymmetry and track
multiplicity in a cone with half-angle of 1.5 units in the plane of
pseudorapidity and azimuthal angle (measured in radians) [31] around the
$B^{0}_{(s)}$ candidate flight direction are also used. The input quantities
to the neural network only depend weakly on the kinematics of the
$B^{0}_{(s)}$ decay. A requirement on the network output is imposed that
reduces the combinatorial background by an order of magnitude while retaining
about 70 % of the signal.
To improve the $B^{0}_{(s)}$ candidate invariant mass resolution, the four-
momenta of the tracks from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate are adjusted [32] so
that their combined invariant mass matches the world average value [33]. An
additional $B^{0}_{(s)}$ mass constraint is applied in the calculation of the
Dalitz plot coordinates, $m^{2}(DK)$ and $m^{2}(D\pi)$, which are used in the
determination of event-by-event efficiencies. The coordinates are calculated
twice: once each with a $B^{0}$ and a $B^{0}_{s}$ mass constraint. A small
fraction ($\sim 1\,\%$ within the fitted mass range) of candidates with
invariant masses far from the $B^{0}_{(s)}$ peak fail one or both of these
mass-constrained fits, and are removed from the analysis.
To remove the large background from $B^{0}\rightarrow D^{*-}\pi^{+}$ decays,
candidates in both samples are rejected if the mass difference
$m_{D\pi}$–$m_{D}$ (for either pion charge in the combinations $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K\pi$) lies within $\pm
2.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $D^{*-}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mass difference [33]. Candidates
in the $DK\pi$ sample are also rejected if the mass difference
$m_{DK}$–$m_{D}$ calculated under the pion mass hypothesis satisfies the same
criterion. A potential background contribution from $B^{0}_{s}\rightarrow
D^{\mp}K^{\pm}$ decays is removed by requiring that the pion from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate together with the kaon
and the pion do not form an invariant mass in the range
$1850$–$1885{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Further $DK\pi$
candidates are rejected by requiring that the kaon from the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidate together with the kaon
and the pion do not form an invariant mass in the range
$1955$–$1975{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which removes potential
background from $B^{0}_{s}\rightarrow D^{\mp}_{s}\pi^{\pm}$ decays. A muon
veto is applied to all four final state tracks to remove potential background
from $B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ decays and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
candidates are required to travel at least $1\rm\,mm$ from the $B^{0}_{(s)}$
decay vertex to remove charmless backgrounds that survive the $D^{0}$ BDT
requirement.
Candidates are retained for further analysis if they have an invariant mass in
the range $5150$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $D\pi\pi$
or $5200$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $DK\pi$. After
all selection requirements are applied, fewer than 1 % of events with at least
one candidate also contain a second candidate. Such multiple candidates are
retained and treated in the same manner as other candidates; the associated
systematic uncertainty is negligible.
## 3 Determination of signal yields
The signal yields are obtained from unbinned maximum likelihood fits to the
invariant mass distributions. In addition to signal contributions and
combinatorial background, candidates may be formed from misidentified or
partially reconstructed $b$-hadron decays. Contributions from partially
reconstructed decays are reduced by the lower bounds on the invariant mass
regions used in the fits. Sources of misidentified backgrounds are
investigated using simulation. Most potential sources are found to have broad
invariant mass distributions, and are absorbed in the combinatorial background
shapes used in the fits described below. Backgrounds from $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ [34] and
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays may, however,
give contributions with distinctive shapes in the mass distributions of
$D\pi\pi$ and $DK\pi$ candidates, respectively, and are therefore explicitly
modelled in the fits.
The $D\pi\pi$ fit includes a double Gaussian shape to describe the signal,
where the two Gaussian functions share a common mean, together with an
exponential component for partially reconstructed background, and a
probability density function (PDF) for $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays. This
PDF is modelled using a smoothed non-parametric function obtained from
simulated data, reweighted so that the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$ invariant mass distribution
matches that observed in data. The shape of the combinatorial background is
essentially linear, but is multiplied by a function that accounts for the fact
that candidates with high invariant masses are more likely to fail the
$B^{0}_{(s)}$ mass constrained fit. There are ten free parameters in the
$D\pi\pi$ fit: the double Gaussian peak position, the widths of the two
Gaussian shapes and the relative normalisation of the two Gaussian functions,
the linear slope of the combinatorial background, the exponential shape
parameter of the partially reconstructed background, and the yields of the
four categories. The result of the fit to the $D\pi\pi$ candidates is shown in
Fig. 2(a) and yields $8558\pm 134$ $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays.
Figure 2: Fits to the $B^{0}_{(s)}$ candidate invariant mass distributions
for the (a) $D\pi\pi$ and (b) $DK\pi$ samples. Data points are shown in black,
the full fitted PDFs as solid blue lines and the components as detailed in the
legends.
The $DK\pi$ fit includes a second double Gaussian component to account for the
presence of both $B^{0}$ and $B^{0}_{s}$ decays. The peaking background PDF
for $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decays is modelled
using a smoothed non-parametric function derived from simulation, reweighted
in the same way as described for $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays above.
The dominant partially reconstructed backgrounds in the $DK\pi$ fit are from
$B^{0}_{s}$ decays and these extend into the $B^{0}$ signal region. Instead of
an exponential component, a background PDF for $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ decays is included,
modelled using a smoothed non-parametric function obtained from simulation.
Studies using simulated data show that this function can account for all
resonant contributions to the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}K^{-}\pi^{+}$ final state. The
function describing the combinatorial background has the same form as for the
$D\pi\pi$ fit. The $DK\pi$ fit has eight free parameters; the parameters of
the double Gaussian functions are constrained to be identical for the $B^{0}$
and $B^{0}_{s}$ signals, with an offset in their mean values fixed to the
known $B^{0}$–$B^{0}_{s}$ mass difference [33]. The relative width of the
broader to the narrower Gaussian component and the relative normalisation of
the two Gaussian functions are constrained within their uncertainties to the
values obtained in simulation. The result of the fit is shown in Fig. 2(b) and
yields $815\pm 55$ $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ and $2391\pm 81$
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ decays. All background
yields in both fits are consistent with their expectations within
uncertainties, based on measured or predicted production rates and branching
fractions and background rejection factors determined from simulations.
## 4 Calculation of branching fraction ratios
The ratios of branching fractions are obtained after applying event-by-event
efficiencies as a function of the Dalitz plot position. The branching fraction
for the $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ decay is determined as
$R_{B^{0}}\equiv\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=\frac{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-})}{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-})}\,,$ (1)
and the branching fraction of the $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ mode is determined as
$R_{B^{0}_{s}}\equiv\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=\left(\frac{f_{s}}{f_{d}}\right)^{-1}\frac{N^{\rm
corr}(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+})}{N^{\rm
corr}(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-})}\,,$ (2)
where the efficiency corrected yield is $N^{\rm
corr}=\sum_{i}W_{i}/\epsilon^{\rm tot}_{i}$. Here the index $i$ runs over all
candidates in the fit range, $W_{i}$ is the signal weight for candidate $i$,
determined using the procedure described in Ref. [29], from the fits shown in
Fig. 2 and $\epsilon^{\rm tot}_{i}$ is the efficiency for candidate $i$ as a
function of its Dalitz plot position. The ratio of fragmentation fractions is
$f_{s}/f_{d}=0.256\pm 0.020$ [35]. The statistical uncertainty on the
branching fraction ratio incorporates the effects of the shape parameters that
are allowed to vary in the fit and the dilution due to event weighting. Most
potential systematic effects cancel in the ratio.
The PID efficiency is measured using a control sample of
$D^{*-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-},\,\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decays to
obtain background-subtracted efficiency tables for kaons and pions as a
function of their $p$ and $p_{\rm T}$ [36, 15]. The kinematic properties of
the particles in signal decays are obtained from simulation in which events
are uniformly distributed across the phase space, allowing the PID efficiency
for each event to be obtained from the tables, while taking into account the
correlation between the $p$ and $p_{\rm T}$ values of the two tracks. The
other contributions to the efficiency (detector acceptance, selection criteria
and trigger effects) are determined from phase space simulation, and validated
using data. All are found to be approximately constant across the Dalitz
plane, apart from some modulations seen near the kinematic boundaries and, for
the $DK\pi$ channels, a variation caused by different PID requirements on the
pion and the kaon. The efficiency for each mode, averaged across the Dalitz
plot, is given in Table 1 together with the contributions from geometrical
acceptance, trigger and selection requirements and particle identification.
Table 1: Summary of the efficiencies for $D\pi\pi$ and $DK\pi$ in phase space simulation. Contributions from geometrical acceptance ($\epsilon^{\rm geom}$), trigger and selection requirements ($\epsilon^{\rm trig\&sel}$) and particle identification ($\epsilon^{\rm PID}$) are shown. The geometrical acceptance is evaluated for $B$ mesons produced within the detector acceptance. Values given are in percent. | $B^{0}\rightarrow D\pi\pi$ | $B^{0}\rightarrow DK\pi$ | $B^{0}_{s}\rightarrow DK\pi$
---|---|---|---
$\epsilon^{\rm geom}$ | 44.70 | 46.60 | 46.50
$\epsilon^{\rm trig\&sel}$ | 01.32 | 01.25 | 01.25
$\epsilon^{\rm PID}$ | 89.30 | 74.80 | 75.00
$\epsilon^{\rm tot}$ | 00.53 | 00.44 | 00.44
The Dalitz plots obtained from the signal weights are shown in Fig. 3. The
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ plot, Fig. 3(a),
shows contributions from the $\rho^{0}(770)$ and $f_{2}(1270)$ resonances
(upper diagonal edge of the Dalitz plot) and from the $D_{2}^{*-}(2460)$ state
(horizontal band), as expected from previous studies of this decay [12, 13].
The $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$ plot, Fig. 3(b), shows
contributions from the $K^{*0}(892)$ (upper diagonal edge) and from the
$D_{2}^{*-}(2460)$ (vertical band) resonances, also as expected [10]. The
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ plot, Fig. 3(c), shows
contributions from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ (upper diagonal edge) and
from the $D_{s2}^{*-}(2573)$ (horizontal band) states. The former contribution
is as expected [11]. The decay $B^{0}_{s}\rightarrow D_{s2}^{*-}(2573)\pi^{+}$
has not been observed previously but is expected to exist given the
observation of the $B^{0}_{s}\rightarrow D_{s2}^{*-}(2573)\mu^{+}\nu X$ decay
[37].
Figure 3: Efficiency corrected Dalitz plot distributions for (a)
$B^{0}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\pi^{+}\pi^{-}$, (b)
$B^{0}\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{+}\pi^{-}$
and (c) $B^{0}_{s}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{-}\pi^{+}$ candidates obtained
from the signal weights.
## 5 Systematic uncertainties and cross-checks
Systematic uncertainties are assigned to both branching fraction ratios due to
the following sources (summarised in Table 2). Note that all uncertainties are
relative. The variation of efficiency across the Dalitz plot may not be
correctly modelled in simulation. A two-dimensional polynomial is used to fit
the variation across the Dalitz region of each of the four contributions to
the efficiency (detector acceptance, selection criteria, PID and trigger
effects). These polynomials are used to generate 1000 simulated pseudo-
experiments, varying the fit parameters within their uncertainties. Each set
of simulations is used to calculate the efficiency corrected yield. The
standard deviation from a Gaussian fit to these yields is used to provide a
systematic uncertainty for each decay mode. This leads to a systematic
uncertainty of 3.4 % (3.1 %) for $R_{B^{0}}$ ($R_{B^{0}_{s}}$). The $DK\pi$
fit model is varied by scaling the signal PDF width ratio to account for the
different masses of the $B^{0}$ and $B^{0}_{s}$ mesons, replacing the PDFs of
the background components with unsmoothed versions, adding components for
potential background from $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ decays, and
replacing the double Gaussian signal components with double Crystal Ball [38]
functions. The $D\pi\pi$ fit model is varied by replacing the PDF of the
$\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\overline{}p\pi^{+}$ component with
an unsmoothed version, varying the slope of the combinatorial background and
replacing the exponential partially reconstructed background component with a
PDF for $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\pi^{+}\pi^{-}$ decays. Combined in
quadrature, these contribute 6.3 % (4.3 %) to $R_{B^{0}}$ ($R_{B^{0}_{s}}$).
Variations in the $D^{*\pm}$, $D^{\pm}$ and $D^{\pm}_{s}$ vetoes contribute to
$R_{B^{0}}$ ($R_{B^{0}_{s}}$), at the level of $<$0.1 %, 2.0 % and 0.2 % (1.0
%, 0.5 % and 0.2 %), respectively. In addition, the possible differences in
the data to simulation ratios of trigger and PID efficiencies between the two
channels (both 1.0 %) and the limited statistics of the simulated data samples
used to calculate efficiencies (2.0 %) affect both $R_{B^{0}}$ and
$R_{B^{0}_{s}}$. The uncertainty on the quantity $f_{s}/f_{d}$ (7.8 %) affects
only $R_{B^{0}_{s}}$. The total systematic uncertainties are obtained as the
quadratic sums of all contributions.
Table 2: Systematic uncertainties on $R_{B^{0}}$ and $R_{B^{0}_{s}}$. The total is obtained from the sum in quadrature of all contributions. Note that all uncertainties are relative. | Uncertainty (%)
---|---
Source | $<$ $B^{0}$ | $B^{0}_{s}$
Modelling of efficiency | $<$ 3.4 | 3.1
Fit model | $<$ 6.3 | 4.3
$D^{*\pm}$ veto | $<$ 0.1 | 1.0
$D^{\pm}$ veto | $<$ 2.0 | 0.2
$D^{\pm}_{s}$ veto | $<$ 0.2 | 0.5
Trigger | $<$ 1.0 | 1.0
Particle identification | $<$ 1.0 | 1.0
Simulation statistics | $<$ 2.0 | 2.0
$f_{s}/f_{d}$ | $<$ – | 7.8
Total | $<$ 7.8 | 9.8
A number of cross-checks are performed to test the stability of the results.
Based upon the hardware trigger decision, candidates are separated into three
groups: events in which a particle from the signal decay created a cluster
with enough energy in the calorimeter to fire the trigger, events that were
triggered independently of the signal decay and those events that were
triggered by both the signal decay and the rest of the event. The data sample
is divided by dipole magnet polarity. The neural network and PID requirements
are both tightened and loosened. The PID efficiency is evaluated using the
kinematic properties from $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ data instead of from
simulation. The requirement for the $B^{0}_{(s)}$ mass constrained fits to
converge is removed. All cross-checks give consistent results.
## 6 Results and conclusions
In summary, the decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ has been observed for
the first time, and its branching fraction relative to that of the
$B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}$ decay is measured to
be
$\frac{{\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm
0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$
The current world average value of ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)=(8.4\pm 0.4\pm
0.8)\times 10^{-4}$ [12] assumes equal production of $B^{+}B^{-}$ and
$B^{0}\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ at the $\mathchar
28935\relax{(4S)}$ resonance and uses the $D^{0}$ branching fraction ${\cal
B}\left(D^{0}\rightarrow K^{-}\pi^{+}\right)=(3.80\pm 0.07)\,\%$. Using the
current world average values of $\Gamma(\mathchar 28935\relax{(4S)}\rightarrow
B^{+}B^{-})/\Gamma(\mathchar 28935\relax{(4S)}\rightarrow B^{0}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})=1.055\pm 0.025$ [33] and ${\cal
B}\left(D^{0}\rightarrow K^{-}\pi^{+}\right)=(3.88\pm 0.05)\,\%$ [33], the
branching fraction of the normalisation channel becomes ${\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)=(8.5\pm 0.4\pm
0.8)\times 10^{-4}$. This corrected value gives
${\cal B}\left(B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}\right)=(1.00\pm
0.04\,\text{(stat.)}\pm 0.10\,\text{(syst.)}\pm 0.10\,\text{(}{\cal
B}\text{)})\times 10^{-3}\,,$
where the third uncertainty arises from ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)$. The
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
decay has also been measured, with relative branching fraction
$\frac{{\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm
0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,.$
Using the corrected value of ${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}\pi^{-}\right)$ gives
${\cal B}\left(B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}\right)=(9.0\pm
0.6\,\text{(stat.)}\pm 0.7\,\text{(syst.)}\pm 0.9\,\text{(}{\cal
B}\text{)})\times 10^{-5}\,,$
which is the most precise measurement of this quantity to date. Future studies
of the Dalitz plot distributions of these decays will provide insight into the
dynamics of hadronic $B$ decays. In addition, the $B^{0}\rightarrow
DK^{+}\pi^{-}$ decay may be used to measure the $C\\!P$ violating phase
$\gamma$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-04-23T15:20:29 |
2024-09-04T02:49:44.700616
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van\n den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M. Dogaru, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach,\n I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G.\n Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero\n Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Daniel Craik",
"url": "https://arxiv.org/abs/1304.6317"
}
|
1304.6325
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-074 LHCb-PAPER-2013-019 8 July 2013
Differential branching fraction
and angular analysis of
the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The angular distribution and differential branching fraction of the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ are studied using a data sample,
collected by the LHCb experiment in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $1.0\mbox{\,fb}^{-1}$. Several angular observables are measured
in bins of the dimuon invariant mass squared, $q^{2}$. A first measurement of
the zero-crossing point of the forward-backward asymmetry of the dimuon system
is also presented. The zero-crossing point is measured to be $q_{0}^{2}=4.9\pm
0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, where the uncertainty is the sum
of statistical and systematic uncertainties. The results are consistent with
the Standard Model predictions.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
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H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
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K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
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Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
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Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
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J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
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G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
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P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
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van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
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Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
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A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
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N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
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Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4,
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Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
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V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki58, N.A.
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B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
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S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
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Xie49,37, F. Xing54, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay,111Charge conjugation is
implied throughout this paper unless stated otherwise. where
$K^{*0}\\!\rightarrow K^{+}\pi^{-}$, is a $b\rightarrow s$ flavour changing
neutral current process that is mediated by electroweak box and penguin type
diagrams in the Standard Model (SM). The angular distribution of the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ system offers particular sensitivity to
contributions from new particles in extensions to the SM. The differential
branching fraction of the decay also provides information on the contribution
from those new particles but typically suffers from larger theoretical
uncertainties due to hadronic form factors.
The angular distribution of the decay can be described by three angles
($\theta_{\ell},\theta_{K}$ and $\phi$) and by the invariant mass squared of
the dimuon system ($q^{2}$). The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
decay is self-tagging through the charge of the kaon and so there is some
freedom in the choice of the angular basis that is used to describe the decay.
In this paper, the angle $\theta_{\ell}$ is defined as the angle between the
direction of the $\mu^{+}$ ($\mu^{-}$) and the direction opposite that of the
$B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) in the dimuon
rest frame. The angle $\theta_{K}$ is defined as the angle between the
direction of the kaon and the direction of opposite that of the $B^{0}$
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) in in the $K^{*0}$
($\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) rest frame. The angle
$\phi$ is the angle between the plane containing the $\mu^{+}$ and $\mu^{-}$
and the plane containing the kaon and pion from the $K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) in the $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The basis is designed
such that the angular definition for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay is a $C\\!P$ transformation
of that for the $B^{0}$ decay. This basis differs from some that appear in the
literature. A graphical representation, and a more detailed description, of
the angular basis is given in Appendix A.
Using the notation of Ref. [1], the decay distribution of the $B^{0}$
corresponds to
$\begin{split}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\phi}=\frac{9}{32\pi}&\left[\frac{}{}{I_{1}^{s}}\sin^{2}\theta_{K}+{I_{1}^{c}}\cos^{2}\theta_{K}~{}+\right.\\\
&\left.~{}\frac{}{}{I_{2}^{s}}\sin^{2}\theta_{K}\cos
2\theta_{\ell}+{I_{2}^{c}}\cos^{2}\theta_{K}\cos 2\theta_{\ell}~{}+\right.\\\
&\left.~{}\frac{}{}{I_{3}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+{{I_{4}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi}}~{}+\right.\\\
&~{}\frac{}{}\left.{{{I_{5}}\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi}}+I_{6}\sin^{2}\theta_{K}\cos\theta_{\ell}~{}+\right.\\\
&~{}\frac{}{}\left.{{{I_{7}}\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi}}+{{{I_{8}}\sin 2\theta_{K}\sin
2\theta_{\ell}\sin\phi}}~{}+\right.\\\
&~{}\frac{}{}\left.I_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi\frac{}{}~{}\right]~{},\end{split}$ (1)
where the 11 coefficients, $I_{j}$, are bilinear combinations of $K^{*0}$
decay amplitudes, ${\cal A}_{m}$, and vary with $q^{2}$. The superscripts $s$
and $c$ in the first two terms arise in Ref. [1] and indicate either a
$\sin^{2}\theta_{K}$ or $\cos^{2}\theta_{K}$ dependence of the corresponding
angular term. In the SM, there are seven complex decay amplitudes,
corresponding to different polarisation states of the $K^{*0}$ and chiralities
of the dimuon system. In the angular coefficients, the decay amplitudes appear
in the combinations $|{\cal A}_{m}|^{2}$, ${\rm Re}({\cal A}_{m}{\cal
A}_{n}^{*})$ and ${\rm Im}({\cal A}_{m}{\cal A}_{n}^{*})$. Combining $B^{0}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and assuming
there are equal numbers of each, it is possible to build angular observables
that depend on the average of, or difference between, the distributions for
the $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay,
$S_{j}=\left.\left(I_{j}+\bar{I}_{j}\right)\middle/\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\right.~{}\text{or}~{}\left.A_{j}=\left(I_{j}-\bar{I}_{j}\right)\middle/\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\right.~{}.$
(2)
These observables are referred to below as $C\\!P$ averages or $C\\!P$
asymmetries and are normalised with respect to the combined differential decay
rate, $\mathrm{d}\Gamma/\mathrm{d}q^{2}$, of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. The observables $S_{7}$,
$S_{8}$ and $S_{9}$ depend on combinations ${\rm Im}({\cal A}_{m}{\cal
A}_{n}^{*})$ and are suppressed by the small size of the strong phase
difference between the decay amplitudes. They are consequently expected to be
close to zero across the full $q^{2}$ range not only in the SM but also in
most extensions. However, the corresponding $C\\!P$ asymmetries, $A_{7}$,
$A_{8}$ and $A_{9}$, are not suppressed by the strong phases involved [2] and
remain sensitive to the effects of new particles.
If the $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays
are combined using the angular basis in Appendix A, the resulting angular
distribution is sensitive to only the $C\\!P$ averages of each of the angular
terms. Sensitivity to $A_{7}$, $A_{8}$ and $A_{9}$ is achieved by flipping the
sign of $\phi$ ($\phi\rightarrow-\phi$) for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. This procedure results in a
combined $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
angular distribution that is sensitive to the $C\\!P$ averages $S_{1}-S_{6}$
and the $C\\!P$ asymmetries of $A_{7}$, $A_{8}$ and $A_{9}$.
In the limit that the dimuon mass is large compared to the mass of the muons,
$q^{2}\gg 4m_{\mu}^{2}$, the $C\\!P$ average of $I_{1}^{c}$, $I_{1}^{s}$,
$I_{2}^{c}$ and $I_{2}^{s}$ ($S_{1}^{c}$, $S_{1}^{s}$, $S_{2}^{c}$ and
$S_{2}^{s}$) are related to the fraction of longitudinal polarisation of the
$K^{*0}$ meson, $F_{\rm L}$ ($S_{1}^{c}=-S_{2}^{c}=F_{\rm L}$ and
$\frac{4}{3}S_{1}^{s}=4S_{2}^{s}=1-F_{\rm L}$). The angular term, $I_{6}$ in
Eq. 1, which has a $\sin^{2}\theta_{K}\cos\theta_{\ell}$ dependence, generates
a forward-backward asymmetry of the dimuon system, $A_{\rm FB}$ [3] ($A_{\rm
FB}=\frac{3}{4}S_{6}$). The term $S_{3}$ is related to the asymmetry between
the two sets of transverse $K^{*0}$ amplitudes, referred to in literature as
$A_{\rm T}^{2}$ [4], where $S_{3}=\frac{1}{2}\left(1-F_{\rm L}\right)A_{\rm
T}^{2}$.
In the SM, $A_{\rm FB}$ varies as a function of $q^{2}$ and is known to change
sign. The $q^{2}$ dependence arises from the interplay between the different
penguin and box diagrams that contribute to the decay. The position of the
zero-crossing point of $A_{\rm FB}$ is a precision test of the SM since, in
the limit of large $K^{*0}$ energy, its prediction is free from form-factor
uncertainties [3]. At large recoil, low values of $q^{2}$, penguin diagrams
involving a virtual photon dominate. In this $q^{2}$ region, $A_{\rm T}^{2}$
is sensitive to the polarisation of the virtual photon which, in the SM, is
predominately left-handed, due to the nature of the charged-current
interaction. In many possible extensions of the SM however, the photon can be
both left- or right-hand polarised, leading to large enhancements of $A_{\rm
T}^{2}$ [4].
The one-dimensional $\cos\theta_{\ell}$ and $\cos\theta_{K}$ distributions
have previously been studied by the LHCb [5], BaBar [6], Belle [7] and CDF [8]
experiments with much smaller data samples. The CDF experiment has also
previously studied the $\phi$ angle. Even with the larger dataset available in
this analysis, it is not yet possible to fit the data for all 11 angular
terms. Instead, rather than examining the one dimensional projections as has
been done in previous analyses, the angle $\phi$ is transformed such that
$\hat{\phi}=\begin{cases}\phi+\pi&\text{~{}if~{}}\phi<0\\\
\phi&\text{~{}otherwise}\end{cases}$ (3)
to cancel terms in Eq. 1 that have either a $\sin\phi$ or a $\cos\phi$
dependence. This provides a simplified angular expression, which contains only
$F_{\rm L}$, $A_{\rm FB}$, $S_{3}$ and $A_{9}$,
$\begin{split}\frac{1}{\mathrm{d}\Gamma/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}=\frac{9}{16\pi}&\left[\frac{}{}F_{\rm
L}\cos^{2}\theta_{K}+\frac{3}{4}(1-F_{\rm
L})(1-\cos^{2}\theta_{K})~{}~{}-\right.\\\ &\left.~{}\frac{}{}\,F_{\rm
L}\cos^{2}\theta_{K}(2\cos^{2}\theta_{\ell}-1)~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}\frac{1}{4}(1-F_{\rm
L})(1-\cos^{2}\theta_{K})(2\cos^{2}\theta_{\ell}-1)~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}S_{3}(1-\cos^{2}\theta_{K})(1-\cos^{2}\theta_{\ell})\cos
2\hat{\phi}~{}~{}+\right.\\\ &\left.~{}\frac{}{}~{}\frac{4}{3}A_{\rm
FB}(1-\cos^{2}\theta_{K})\cos\theta_{\ell}~{}~{}+\right.\\\
&\left.~{}\frac{}{}~{}A_{9}(1-\cos^{2}\theta_{K})(1-\cos^{2}\theta_{\ell})\sin
2\hat{\phi}\frac{}{}~{}\right]~{}.\end{split}$ (4)
This expression involves the same set of observables that can be extracted
from fits to the one-dimensional angular projections.
At large recoil it is also advantageous to reformulate Eq. 4 in terms of the
observables $A_{\rm T}^{2}$ and $A_{\rm T}^{\rm Re}$, where $A_{\rm
FB}=\frac{3}{4}\left(1-F_{\rm L}\right)A_{\rm T}^{\rm Re}$. These so called
“transverse” observables only depend on a subset of the decay amplitudes (with
transverse polarisation of the $K^{*0}$) and are expected to come with reduced
form-factor uncertainties [4, 9]. A first measurement of $A_{\rm T}^{2}$ was
performed by the CDF experiment [8].
This paper presents a measurement of the differential branching fraction
($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$), $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and
$A_{9}$ of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in six bins of
$q^{2}$. Measurements of the transverse observables $A_{\rm T}^{2}$ and
$A_{\rm T}^{\rm Re}$ are also presented. The analysis is based on a dataset,
corresponding to 1.0$\mbox{\,fb}^{-1}$ of integrated luminosity, collected by
the LHCb detector in $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions
in 2011. Section 2 describes the experimental setup used in the analyses.
Section 3 describes the event selection. Section 4 discusses potential sources
of peaking background. Section 5 describes the treatment of the detector
acceptance in the analysis. Section 6 discusses the measurement of
$\mathrm{d}{\cal B}/\mathrm{d}q^{2}$. The angular analysis of the decay, in
terms of $\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\hat{\phi}$, is described
in Sec. 7. Finally, a first measurement of the zero-crossing point of $A_{\rm
FB}$ is presented in Sec. 8.
## 2 The LHCb detector
The LHCb detector [10] is a single-arm forward spectrometer, covering the
pseudorapidity range $2<\eta<5$, that is designed to study $b$ and $c$ hadron
decays. A dipole magnet with a bending power of 4 Tm and a large area tracking
detector provide momentum resolution ranging from 0.4% for tracks with a
momentum of 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% for a momentum of
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. A silicon microstrip detector,
located around the $pp$ interaction region, provides excellent separation of
$B$ meson decay vertices from the primary $pp$ interaction and impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with high transverse
momentum ($p_{\rm T}$). Two ring-imaging Cherenkov (RICH) detectors [11]
provide kaon-pion separation in the momentum range
$2-100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons are identified based on
hits created in a system of multiwire proportional chambers interleaved with
layers of iron. The LHCb trigger [12] comprises a hardware trigger and a two-
stage software trigger that performs a full event reconstruction.
Samples of simulated events are used to estimate the contribution from
specific sources of exclusive backgrounds and the efficiency to trigger,
reconstruct and select the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ signal.
The simulated $pp$ interactions are generated using Pythia 6.4 [13] with a
specific LHCb configuration [14]. Decays of hadronic particles are then
described by EvtGen [15] in which final state radiation is generated using
Photos [16]. Finally, the Geant4 toolkit [17, *Agostinelli:2002hh] is used to
simulate the detector response to the particles produced by Pythia/EvtGen, as
described in Ref. [19]. The simulated samples are corrected for known
differences between data and simulation in the $B^{0}$ momentum spectrum, the
detector impact parameter resolution, particle identification [11] and
tracking system performance using control samples from the data.
## 3 Selection of signal candidates
The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates are selected from
events that have been triggered by a muon with $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, in the hardware trigger. In the
first stage of the software trigger, candidates are selected if there is a
reconstructed track in the event with high impact parameter ($>125\,\upmu\rm
m$) with respect to one of the primary $pp$ interactions and $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the second stage of the
software trigger, candidates are triggered on the kinematic properties of the
partially or fully reconstructed $B^{0}$ candidate [12].
Signal candidates are then required to pass a set of loose (pre-)selection
requirements. Candidates are selected for further analysis if: the $B^{0}$
decay vertex is separated from the primary $pp$ interaction; the $B^{0}$
candidate impact parameter is small, and the impact parameters of the charged
kaon, pion and muons are large, with respect to the primary $pp$ interaction;
and the angle between the $B^{0}$ momentum vector and the vector between the
primary $pp$ interaction and the $B^{0}$ decay vertex is small. Candidates are
retained if their $K^{+}\pi^{-}$ invariant mass is in the range
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
A multivariate selection, using a boosted decision tree (BDT) [20] with the
AdaBoost algorithm[21], is applied to further reduce the level of
combinatorial background. The BDT is identical to that described in Ref. [5].
It has been trained on a data sample, corresponding to 36$\mbox{\,pb}^{-1}$ of
integrated luminosity, collected by the LHCb experiment in 2010. A sample of
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$)
candidates is used to represent the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
signal in the BDT training. The decay $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is used throughout this
analysis as a control channel. Candidates from the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ upper mass sideband
($5350<m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
are used as a background sample. Candidates with invariant masses below the
nominal $B^{0}$ mass contain a significant contribution from partially
reconstructed $B$ decays and are not used in the BDT training or in the
subsequent analysis. They are removed by requiring that candidates have
$m({K^{+}\pi^{-}\mu^{+}\mu^{-}})>5150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The BDT uses predominantly geometric variables, including the variables used
in the above pre-selection. It also includes information on the quality of the
$B^{0}$ vertex and the fit $\chi^{2}$ of the four tracks. Finally the BDT
includes information from the RICH and muon systems on the likelihood that the
kaon, pion and muons are correctly identified. Care has been taken to ensure
that the BDT does not preferentially select regions of $q^{2}$,
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass or of the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ angular distribution. The multivariate selection
retains 78% of the signal and 12% of the background that remains after the
pre-selection.
Figure 1: Distribution of $\mu^{+}\mu^{-}$ versus $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass of selected $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
candidates. The vertical lines indicate a $\pm
50{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ signal mass window around the
nominal $B^{0}$ mass. The horizontal lines indicate the two veto regions that
are used to remove ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays. The $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal is clearly visible outside of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ windows.
Figure 1 shows the $\mu^{+}\mu^{-}$ versus $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass of the selected candidates. The $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal, which peaks in $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass, and populates the full range of the dimuon invariant mass
range, is clearly visible.
## 4 Exclusive and partially reconstructed backgrounds
Several sources of peaking background have been studied using samples of
simulated events, corrected to reflect the difference in particle
identification (and misidentification) performance between the data and
simulation. Sources of background that are not reduced to a negligible level
by the pre- and multivariate-selections are described below.
The decays $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $B^{0}\\!\rightarrow K^{*0}\psi{(2S)}$, where
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$, are removed by rejecting candidates
with $2946<m({\mu^{+}\mu^{-}})<3176{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $3586<m({\mu^{+}\mu^{-}})<3766{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
These vetoes are extended downwards by
150${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in $m({\mu^{+}\mu^{-}})$ for
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates with masses
$5150<m({K^{+}\pi^{-}\mu^{+}\mu^{-}})<5230{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
to account for the radiative tails of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons. They
are also extended upwards by 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
candidates with masses above the $B^{0}$ mass to account for the small
percentage of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$
decays that are misreconstructed at higher masses. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ vetoes are
shown in Fig. 1.
The decay $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ can also form a source of peaking background if the kaon or pion is
misidentified as a muon and swapped with one of the muons from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay. This background is
removed by rejecting candidates that have a $K^{+}\mu^{-}$ or $\pi^{-}\mu^{+}$
invariant mass (where the kaon or pion is assigned the muon mass) in the range
$3036<m({\mu^{+}\mu^{-}})<3156{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ if the
kaon or pion can also be matched to hits in the muon stations. A similar veto
is applied for the decay $B^{0}\\!\rightarrow K^{*0}\psi{(2S)}$.
The decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$, where
$\phi\\!\rightarrow K^{+}K^{-}$, is removed by rejecting candidates if the
$K^{+}\pi^{-}$ mass is consistent with originating from a $\phi\\!\rightarrow
K^{+}K^{-}$ decay and the pion is kaon-like according to the RICH detectors. A
similar veto is applied to remove $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax^{*}(1520)\mu^{+}\mu^{-}$ ($\mathchar
28931\relax^{*}(1520)\\!\rightarrow pK^{-}$) decays.
There is also a source of background from the decay $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ that appears in the upper mass sideband and has a peaking
structure in $\cos\theta_{K}$. This background arises when a $K^{*0}$
candidate is formed using a pion from the other $B$ decay in the event, and is
removed by vetoing events that have a $K^{+}\mu^{+}\mu^{-}$ invariant mass in
the range
$5230<m({K^{+}\mu^{+}\mu^{-}})<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The fraction of combinatorial background candidates removed by this veto is
small.
After these selection requirements the dominant sources of peaking background
are expected to be from the decays $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ (where the kaon or pion is
misidentified as a muon and a muon as a pion or kaon),
$B^{0}_{s}\\!\rightarrow\phi$$\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ at the levels of $(0.3\pm 0.1)\%$, $(1.2\pm 0.5)\%$ and
$(1.0\pm 1.0)\%$, respectively. The rate of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ is estimated using the fragmentation fraction
$f_{s}/f_{d}$ [22] and assuming the branching fraction of this decay is
suppressed by the ratio of CKM elements $|V_{td}/V_{ts}|^{2}$ with respect to
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$. To estimate the systematic
uncertainty arising from the assumed $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal, the expectation is varied by 100%. Finally, the
probability for a decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ to be
misidentified as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ is estimated to be
$(0.85\pm 0.02)\%$ using simulated events.
## 5 Detector acceptance and selection biases
The geometrical acceptance of the detector, the trigger, the event
reconstruction and selection can all bias the angular distribution of the
selected candidates. At low $q^{2}$ there are large distortions of the angular
distribution at extreme values of $\cos\theta_{\ell}$
($|\cos\theta_{\ell}|\sim 1$). These arise from the requirement that muons
have momentum $p{~{}\raise 1.49994pt\hbox{$>$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to traverse
the LHCb muon system. Distortions are also visible in the $\cos\theta_{K}$
angular distribution. They arise from the momentum needed for a track to reach
the tracking system downstream of the dipole magnet, and from the impact
parameter requirements in the pre-selection. The acceptance in
$\cos\theta_{K}$ is asymmetric due to the momentum imbalance between the pion
and kaon from the $K^{*0}$ decay in the laboratory frame (due to the boost).
Acceptance effects are accounted for, in a model-independent way by weighting
candidates by the inverse of their efficiency determined from simulation. The
event weighting takes into account the variation of the acceptance in $q^{2}$
to give an unbiased estimate of the observables over the $q^{2}$ bin. The
candidate weights are normalised such that they have mean 1.0. The resulting
distribution of weights in each $q^{2}$ bin has a root-mean-square in the
range $0.2-0.4$. Less than 2% of the candidates have weights larger than 2.0.
The weights are determined using a large sample of simulated three-body
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ phase-space decays. They are
determined separately in fine bins of $q^{2}$ with widths:
$0.1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ for
$q^{2}<1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$;
$0.2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ in the range
$1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$; and
$0.5\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ for
$q^{2}>6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$. The width of the $q^{2}$
bins is motivated by the size of the simulated sample and by the rate of
variation of the acceptance in $q^{2}$. Inside the $q^{2}$ bins, the angular
acceptance is assumed to factorise such that
$\varepsilon(\cos\theta_{\ell},\cos\theta_{K},\phi)=\varepsilon(\cos\theta_{\ell})\varepsilon(\cos\theta_{K})\varepsilon(\phi)$.
This factorisation is validated at the level of 5% in the phase-space sample.
The treatment of the event weights is discussed in more detail in Sec. 7.1,
when determining the statistical uncertainty on the angular observables.
Event weights are also used to account for the fraction of background
candidates that were removed in the lower mass
($m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5230{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
and upper mass
($m(K^{+}\pi^{-}\mu^{+}\mu^{-})>5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
sidebands by the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ vetoes described in Sec. 4 (and shown in Fig. 1). In each $q^{2}$
bin, a linear extrapolation in $q^{2}$ is used to estimate this fraction and
the resulting event weights.
## 6 Differential branching fraction
The angular and differential branching fraction analyses are performed in six
bins of $q^{2}$, which are the same as those used in Ref. [7]. The
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution of candidates in
these $q^{2}$ bins is shown in Fig. 2.
The number of signal candidates in each of the $q^{2}$ bins is estimated by
performing an extended unbinned maximum likelihood fit to the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution. The signal shape is
taken from a fit to the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control sample and is
parameterised by the sum of two Crystal Ball [23] functions that differ only
by the width of the Gaussian component. The combinatorial background is
described by an exponential distribution. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, which forms a peaking background, is assumed to have a
shape identical to that of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
signal, but shifted in mass by the $B^{0}_{s}-B^{0}$ mass difference [24].
Contributions from the decays $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
(where the $\mu^{-}$ is swapped with the $\pi^{-}$) are also included. The
shapes of these backgrounds are taken from samples of simulated events. The
sizes of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
backgrounds are fixed with respect to the fitted $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal yield according to the ratios described in Sec.
4. These backgrounds are varied to evaluate the corresponding systematic
uncertainty. The resulting signal yields are given in Table 1. In the full
$0.1<q^{2}<19.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ range, the fit yields
$883\pm 34$ signal decays.
Figure 2: Invariant mass distributions of $K^{+}\pi^{-}\mu^{+}\mu^{-}$
candidates in the six $q^{2}$ bins used in the analysis. The candidates have
been weighted to account for the detector acceptance (see text). Contributions
from exclusive (peaking) backgrounds are negligible after applying the vetoes
described in Sec. 4.
The differential branching fraction of the decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$, in each $q^{2}$ bin, is estimated by normalising the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ yield, $N_{\text{sig}}$, to the
total event yield of the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control sample,
$N_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and correcting for
the relative efficiency between the two decays,
$\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\varepsilon_{K^{*0}\mu^{+}\mu^{-}}$,
$\frac{\mathrm{d}{\cal
B}}{\mathrm{d}q^{2}}=\frac{1}{q^{2}_{\text{max}}-q^{2}_{\text{min}}}\frac{N_{\text{sig}}}{N_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\frac{\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\varepsilon_{K^{*0}\mu^{+}\mu^{-}}}\times{\cal B}(B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})\times{\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})~{}.$ (5)
The branching fractions ${\cal B}(B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ and ${\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-})$
are $(1.31\pm 0.03\pm 0.08)\times 10^{-3}$ [25] and $(5.93\pm 0.06)\times
10^{-2}$ [24], respectively.
The efficiency ratio,
$\varepsilon_{K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\varepsilon_{K^{*0}\mu^{+}\mu^{-}}$, depends on the unknown angular
distribution of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay. To avoid
making any assumption on the angular distribution, the event-by-event weights
described in Sec. 5 are used to estimate the average efficiency of the
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates and the signal candidates in each $q^{2}$ bin.
### 6.1 Comparison with theory
The resulting differential branching fraction of the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ is shown in Fig. 3 and in Table 1.
The bands shown in Fig. 3 indicate the theoretical prediction for the
differential branching fraction. The calculation of the bands is described in
Ref. [26].222A consistent set of SM predictions, averaged over each $q^{2}$
bin, have recently also been provided by the authors of Ref. [27]. In the low
$q^{2}$ region, the calculations are based on QCD factorisation and soft
collinear effective theory (SCET) [28], which profit from having a heavy
$B^{0}$ meson and an energetic $K^{*0}$ meson. In the soft-recoil, high
$q^{2}$ region, an operator product expansion in inverse $b$-quark mass
($1/m_{b}$) and $1/\sqrt{q^{2}}$ is used to estimate the long-distance
contributions from quark loops [29, 30]. No theory prediction is included in
the region close to the narrow $c\overline{}c$ resonances (the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$) where the
assumptions from QCD factorisation, SCET and the operator product expansion
break down. The treatment of this region is discussed in Ref. [31]. The form-
factor calculations are taken from Ref. [32]. A dimensional estimate is made
of the uncertainty on the decay amplitudes from QCD factorisation and SCET of
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ [33]. Contributions from light-quark
resonances at large recoil (low $q^{2}$) have been neglected. A discussion of
these contributions can be found in Ref. [34]. The same techniques are
employed in calculations of the angular observables described in Sec. 7.
Table 1: Signal yield ($N_{\text{sig}}$) and differential branching fraction ($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$) of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in the six $q^{2}$ bins used in this analysis. Results are also presented in the $1<q^{2}<6\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range where theoretical uncertainties are best controlled. The first and second uncertainties are statistical and systematic. The third uncertainty comes from the uncertainty on the $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ branching fractions. The final uncertainty on $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ comes from an estimate of the pollution from non-$K^{*0}$ $B^{0}\\!\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the $792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ mass window (see Sec. 7.3.2). $q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $N_{\text{sig}}$ | $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ $(10^{-7}\mathrm{\,Ge\kern-1.00006ptV}^{-2}c^{4})$
---|---|---
$\phantom{0}0.10-\phantom{0}2.00$ | $140\pm 13$ | $0.60\pm 0.06\pm 0.05\pm 0.04\,^{+0.00}_{-0.05}$
$\phantom{0}2.00-\phantom{0}4.30$ | $\phantom{0}73\pm 11$ | $0.30\pm 0.03\pm 0.03\pm 0.02\,^{+0.00}_{-0.02}$
$\phantom{0}4.30-\phantom{0}8.68$ | $271\pm 19$ | $0.49\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.04}$
$10.09-12.86$ | $168\pm 15$ | $0.43\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.03}$
$14.18-16.00$ | $115\pm 12$ | $0.56\pm 0.06\pm 0.04\pm 0.04\,^{+0.00}_{-0.05}$
$16.00-19.00$ | $116\pm 13$ | $0.41\pm 0.04\pm 0.04\pm 0.03\,^{+0.00}_{-0.03}$
$\phantom{0}1.00-\phantom{0}6.00$ | $197\pm 17$ | $0.34\pm 0.03\pm 0.04\pm 0.02\,^{+0.00}_{-0.03}$
Figure 3: Differential branching fraction of the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decay as a function of the dimuon invariant mass
squared. The data are overlaid with a SM prediction (see text) for the decay
(light-blue band). A rate average of the SM prediction across each $q^{2}$ bin
is indicated by the dark (purple) rectangular regions. No SM prediction is
included in the region close to the narrow $c\overline{}c$ resonances.
### 6.2 Systematic uncertainty
The largest sources of systematic uncertainty on the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ differential branching fraction come from the $\sim 6\%$
uncertainty on the combined $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
branching fractions and from the uncertainty on the pollution of non-$K^{*0}$
decays in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass
window. The latter pollution arises from decays where the $K^{+}\pi^{-}$
system is in an S- rather than P-wave configuration. For the decay
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, the
S-wave pollution is known to be at the level of a few percent [35]. The effect
of S-wave pollution on the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ is
considered in Sec. 7.3.2. No S-wave correction needs to be applied to the
yield of $B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ decays in the present analysis, since the branching fraction used in
the normalisation (from Ref. [25]) corresponds to a measurement of the decay
$B^{0}\\!\rightarrow K^{+}\pi^{-}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
over the same $m(K^{+}\pi^{-})$ window used in this analysis.
The uncertainty associated with the data-derived corrections to the
simulation, which were described in Sec. 2, is estimated to be $1-2\%$.
Varying the level of the peaking backgrounds within their uncertainties
changes the differential branching fraction by 1% and this variation is taken
as a systematic uncertainty. In the simulation a small variation in the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass resolution is seen between
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays at low and high $q^{2}$, due
to differences in the decay kinematics. The maximum size of this variation in
the simulation is 5%. A conservative systematic uncertainty is assigned by
varying the mass resolution of the signal decay by this amount in every
$q^{2}$ bin and taking the deviation from the nominal fit as the uncertainty.
## 7 Angular analysis
This section describes the analysis of the $\cos\theta_{\ell}$,
$\cos\theta_{K}$ and $\hat{\phi}$ distribution after applying the
transformations that were described earlier. These transformations reduce the
full angular distribution from 11 angular terms to one that only depends on
four observables: $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and $A_{9}$. The
resulting angular distribution is given in Eq. 4 in Sec. 1.
In order for Eq. 4 to remain positive in all regions of the allowed phase
space, the observables $A_{\rm FB}$, $F_{\rm L}$, $S_{3}$ and $A_{9}$ must
satisfy the constraints
$|A_{\rm FB}|\leq\frac{3}{4}(1-F_{\rm
L})~{},~{}|A_{9}|\leq\frac{1}{2}(1-F_{\rm
L})~{}~{}\text{and}~{}~{}|S_{3}|\leq\frac{1}{2}(1-F_{\rm L})~{}.$
These requirements are automatically taken into account if $A_{\rm FB}$ and
$S_{3}$ are replaced by the theoretically cleaner transverse observables,
$A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$,
$A_{\rm FB}=\frac{3}{4}(1-F_{\rm L})A_{\rm T}^{\rm
Re}~{}~{}\text{and}~{}~{}S_{3}=\frac{1}{2}(1-F_{\rm L})A_{\rm T}^{2}~{},$
which are defined in the range $[-1,1]$.
In each of the $q^{2}$ bins, $A_{\rm FB}$ ($A_{\rm T}^{\rm Re}$), $F_{\rm L}$,
$S_{3}$ ($A_{\rm T}^{2}$) and $A_{9}$ are estimated by performing an unbinned
maximum likelihood fit to the $\cos\theta_{\ell}$, $\cos\theta_{K}$ and
$\hat{\phi}$ distributions of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
candidates. The $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass of the candidates
is also included in the fit to separate between signal- and background-like
candidates. The background angular distribution is described using the product
of three second-order Chebychev polynomials under the assumption that the
background can be factorised into three single angle distributions. This
assumption has been validated on the data sidebands
($5350<m({K^{+}\pi^{-}\mu^{+}\mu^{-}})<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$).
A dilution factor ($\mathcal{D}=1-2\omega$) is included in the likelihood fit
for $A_{\rm FB}$ and $A_{9}$, to account at first order for the small
probability ($\omega$) for a decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ to be misidentified
as $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$. The value of $\omega$ is fixed
to $0.85\%$ in the fit (see Sec. 4).
Two fits to the dataset are performed: one, with the signal angular
distribution described by Eq. 4, to measure $F_{\rm L}$, $A_{\rm FB}$, $S_{3}$
and $A_{9}$ and a second replacing $A_{\rm FB}$ and $S_{3}$ with the
observables $A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$. The angular observables
vary with $q^{2}$ within the $q^{2}$ bins used in the analysis. The measured
quantities therefore correspond to averages over these $q^{2}$ bins. For the
transverse observables, where the observable appears alongside $1-F_{\rm L}$
in the angular distribution, the averaging is complicated by the $q^{2}$
dependence of both the observable and $F_{\rm L}$. In this case, the measured
quantity corresponds to a weighted average of the transverse observable over
$q^{2}$, with a weight $(1-F_{\rm L})\mathrm{d}\Gamma/\mathrm{d}q^{2}$.
### 7.1 Statistical uncertainty on the angular observables
The results of the angular fits are presented in Table 2 and in Figs. 4 and 5.
The 68% confidence intervals are estimated using pseudo-experiments and the
Feldman-Cousins technique [36].333Nuisance parameters are treated according to
the “plug-in” method (see, for example, Ref. [37]). This avoids any potential
bias on the parameter uncertainty that could have otherwise come from using
event weights in the likelihood fit or from boundary issues arising in the
fitting. The observables are each treated separately in this procedure. For
example, when determining the interval on $A_{\rm FB}$, the observables
$F_{\rm L}$, $S_{3}$ and $A_{9}$ are treated as if they were nuisance
parameters. At each value of the angular observable being considered, the
maximum likelihood estimate of the nuisance parameters (which also include the
background parameters) is used when generating the pseudo-experiments. The
resulting confidence intervals do not express correlations between the
different observables. The treatment of systematic uncertainties on the
angular observables is described in Sec. 7.3.
The final column of Table 2 contains the p-value of the SM point in each
$q^{2}$ bin, which is defined as the probability to observe a difference
between the log-likelihood of the SM point compared to the best fit point
larger than that seen in the data. They are estimated in a similar way to the
Feldman-Cousins intervals by: generating a large ensemble of pseudo-
experiments, with all of the angular observables fixed to the central value of
the SM prediction; and performing two fits to the pseudo-experiments, one with
all of the angular observables fixed to their SM values and one varying them
freely. The data are then fitted in a similar manner and the p-value estimated
by comparing the ratio of likelihoods obtained for the data to those of the
pseudo-experiments. The p-values lie in the range $0.18-0.72$ and indicate
good agreement with the SM hypothesis.
As a cross-check, a third fit is also performed in which the sign of the angle
$\phi$ for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays is
flipped to measure $S_{9}$ in place of $A_{9}$ in the angular distribution.
The term $S_{9}$ is expected to be suppressed by the size of the strong phases
and be close to zero in every $q^{2}$ bin. $A_{\rm FB}$ has also been cross-
checked by performing a counting experiment in bins of $q^{2}$. A consistent
result is obtained in every bin.
Table 2: Fraction of longitudinal polarisation of the $K^{*0}$, $F_{\rm L}$, dimuon system forward-backward asymmetry, $A_{\rm FB}$ and the angular observables $S_{3}$, $S_{9}$ and $A_{9}$ from the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in the six bins of dimuon invariant mass squared, $q^{2}$, used in the analysis. The lower table includes the transverse observables $A_{\rm T}^{\rm Re}$ and $A_{\rm T}^{2}$, which have reduced form-factor uncertainties. Results are also presented in the $1<q^{2}<6\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range where theoretical uncertainties are best controlled. In the large-recoil bin, $0.1<q^{2}<2.0\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$, two results are given to highlight the size of the correction needed to account for changes in the angular distribution that occur when $q^{2}{~{}\raise 1.34995pt\hbox{$<$}\kern-7.65005pt\lower 3.15005pt\hbox{$\sim$}~{}}1\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ (see Sec. 7.2). The value of $F_{\rm L}$ is independent of this correction. The final column contains the p-value for the SM point (see text). No SM prediction, and consequently no p-value, is available for the $10.09<q^{2}<12.86\mathrm{\,Ge\kern-0.90005ptV}^{2}/c^{4}$ range. $q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $F_{\rm L}$ | $A_{\rm FB}$ | $S_{3}$ | $S_{9}$
---|---|---|---|---
$0.10-2.00$ | $0.37\,^{+0.10}_{-0.09}\,{}^{+0.04}_{-0.03}$ | $-0.02\,^{+0.12}_{-0.12}\,{}^{+0.01}_{-0.01}$ | $-0.04\,^{+0.10}_{-0.10}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.05\,^{+0.10}_{-0.09}\,{}^{+0.01}_{-0.01}$
(uncorrected) | | | |
$0.10-2.00$ | $0.37\,^{+0.10}_{-0.09}\,{}^{+0.04}_{-0.03}$ | $-0.02\,_{-0.13}^{+0.13}\,{}_{-0.01}^{+0.01}$ | $-0.05\,_{-0.12}^{+0.12}\,{}_{-0.01}^{+0.01}$ | $\phantom{-}0.06\,_{-0.12}^{+0.12}\,{}_{-0.01}^{+0.01}$
(corrected) | | | |
$2.00-4.30$ | $0.74\,^{+0.10}_{-0.09}\,{}^{+0.02}_{-0.03}$ | $-0.20\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.01}$ | $-0.04\,^{+0.10}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $-0.03\,^{+0.11}_{-0.04}\,{}^{+0.01}_{-0.01}$
$4.30-8.68$ | $0.57\,^{+0.07}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.16\,^{+0.06}_{-0.05}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.08\,^{+0.07}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.01\,^{+0.07}_{-0.08}\,{}^{+0.01}_{-0.01}$
$10.09-12.86$ | $0.48\,^{+0.08}_{-0.09}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.28\,^{+0.07}_{-0.06}\,{}^{+0.02}_{-0.02}$ | $-0.16\,^{+0.11}_{-0.07}\,{}^{+0.01}_{-0.01}$ | $-0.01\,^{+0.10}_{-0.11}\,{}^{+0.01}_{-0.01}$
$14.18-16.00$ | $0.33\,^{+0.08}_{-0.07}\,{}^{+0.02}_{-0.03}$ | $\phantom{-}0.51\,^{+0.07}_{-0.05}\,{}^{+0.02}_{-0.02}$ | $\phantom{-}0.03\,^{+0.09}_{-0.10}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.00\,^{+0.09}_{-0.08}\,{}^{+0.01}_{-0.01}$
$16.00-19.00$ | $0.38\,^{+0.09}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.30\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.02}$ | $-0.22\,^{+0.10}_{-0.09}\,{}^{+0.02}_{-0.01}$ | $\phantom{-}0.06\,^{+0.11}_{-0.10}\,{}^{+0.01}_{-0.01}$
$1.00-6.00$ | $0.65\,^{+0.08}_{-0.07}\,{}^{+0.03}_{-0.03}$ | $-0.17\,^{+0.06}_{-0.06}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.03\,^{+0.07}_{-0.07}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.07\,^{+0.09}_{-0.08}\,{}^{+0.01}_{-0.01}$
$q^{2}$ $(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | $A_{9}$ | $A_{\rm T}^{2}$ | $A_{\rm T}^{\rm Re}$ | p-value
$0.10-2.00$ | $\phantom{-}0.12\,_{-0.09}^{+0.09}\,{}^{+0.01}_{-0.01}$ | $-0.14\,_{-0.30}^{+0.34}\,{}^{+0.02}_{-0.02}$ | $-0.04\,_{-0.24}^{+0.26}\,{}^{+0.02}_{-0.01}$ | 0.18
(uncorrected) | | | |
$0.10-2.00$ | $\phantom{-}0.14\,_{-0.11}^{+0.11}\,{}_{-0.01}^{+0.01}$ | $-0.19\,_{-0.35}^{+0.40}\,{}_{-0.02}^{+0.02}$ | $-0.06\,_{-0.27}^{+0.29}\,{}_{-0.01}^{+0.02}$ | –
(corrected) | | | |
$2.00-4.30$ | $\phantom{-}0.06\,_{-0.08}^{+0.12}\,{}^{+0.01}_{-0.01}$ | $-0.29\,_{-0.46}^{+0.65}\,{}^{+0.02}_{-0.03}$ | $-1.00\,_{-0.00}^{+0.13}\,{}^{+0.04}_{-0.00}$ | 0.57
$4.30-8.68$ | $-0.13\,_{-0.07}^{+0.07}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.36\,_{-0.31}^{+0.30}\,{}^{+0.03}_{-0.03}$ | $\phantom{-}0.50\,_{-0.14}^{+0.16}\,{}^{+0.01}_{-0.03}$ | 0.71
$10.09-12.86$ | $\phantom{-}0.00\,_{-0.11}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $-0.60\,_{-0.27}^{+0.42}\,{}^{+0.05}_{-0.02}$ | $\phantom{-}0.71\,_{-0.15}^{+0.15}\,{}^{+0.01}_{-0.03}$ | –
$14.18-16.00$ | $-0.06\,_{-0.08}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.07\,_{-0.28}^{+0.26}\,{}^{+0.02}_{-0.02}$ | $\phantom{-}1.00\,_{-0.05}^{+0.00}\,{}^{+0.00}_{-0.02}$ | 0.38
$16.00-19.00$ | $\phantom{-}0.00\,_{-0.10}^{+0.11}\,{}^{+0.01}_{-0.01}$ | $-0.71\,_{-0.26}^{+0.35}\,{}^{+0.06}_{-0.04}$ | $\phantom{-}0.64\,_{-0.15}^{+0.15}\,{}^{+0.01}_{-0.02}$ | 0.28
$1.00-6.00$ | $\phantom{-}0.03\,^{+0.08}_{-0.08}\,{}^{+0.01}_{-0.01}$ | $\phantom{-}0.15\,_{-0.41}^{+0.39}\,{}^{+0.03}_{-0.03}$ | $-0.66\,_{-0.22}^{+0.24}\,{}^{+0.04}_{-0.01}$ | 0.72
Figure 4: Fraction of longitudinal polarisation of the $K^{*0}$, $F_{\rm L}$,
dimuon system forward-backward asymmetry, $A_{\rm FB}$ and the angular
observables $S_{3}$ and $A_{9}$ from the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decay as a function of the dimuon invariant mass
squared, $q^{2}$. The lowest $q^{2}$ bin has been corrected for the threshold
behaviour described in Sec. 7.2. The experimental data points overlay the SM
prediction described in the text. A rate average of the SM prediction across
each $q^{2}$ bin is indicated by the dark (purple) rectangular regions. No
theory prediction is included for $A_{9}$, which is vanishingly small in the
SM.
Figure 5: Transverse asymmetries $A_{\rm T}^{2}$ and $A_{\rm T}^{\rm Re}$ as a
function of the dimuon invariant mass squared, $q^{2}$, in the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay. The lowest $q^{2}$ bin has
been corrected for the threshold behaviour described in Sec. 7.2. The
experimental data points overlay the SM prediction that is described in the
text. A rate average of the SM prediction across each $q^{2}$ bin is indicated
by the dark (purple) rectangular regions.
### 7.2 Angular distribution at large recoil
In the previous section, when fitting the angular distribution, it was assumed
that the muon mass was small compared to that of the dimuon system. Whilst
this assumption is valid for $q^{2}>2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
it breaks down in the $0.1<q^{2}<2.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$
bin. In this bin, the angular terms receive an additional $q^{2}$ dependence,
proportional to
$\frac{1-{4m_{\mu}^{2}}/{q^{2}}}{1+2m_{\mu}^{2}/q^{2}}~{}~{}\text{or}~{}~{}\frac{(1-4m_{\mu}^{2}/q^{2})^{1/2}}{1+2m_{\mu}^{2}/q^{2}}~{},$
(6)
depending on the angular term $I_{j}$ [1].
As $q^{2}$ tends to zero, these threshold terms become small and reduce the
sensitivity to the angular observables. Neglecting these terms leads to a bias
in the measurement of the angular observables. Previous analyses by LHCb,
BaBar, Belle and CDF have not considered this effect.
The fraction of longitudinal polarisation of the $K^{*0}$ meson, $F_{\rm L}$,
is the only observable that is unaffected by the additional terms; sensitivity
to $F_{\rm L}$ arises mainly through the shape of the $\cos\theta_{K}$
distribution and this shape remains the same whether the threshold terms are
included or not.
In order to estimate the size of the bias, it is assumed that $A_{9}$ and
$A_{T}^{2}$ are constant over the
$0.1<q^{2}<2\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region and $A_{\rm
T}^{\rm Re}$ rises linearly (with the constraint that $A_{\rm T}^{\rm Re}=0$
at $q^{2}=0$). Even though $F_{\rm L}$ is in itself unbiased, an assumption
needs to be made about the $q^{2}$ dependence of $F_{\rm L}$ when determining
the bias introduced on the other observables. An empirical model,
$F_{\rm L}(q^{2})=\frac{aq^{2}}{1+aq^{2}}~{}~{},$ (7)
is used. This functional form displays the correct behaviour since it tends to
zero as $q^{2}$ tends to zero and rises slowly over the $q^{2}$ bin,
reflecting the dominance of the photon penguin at low $q^{2}$ and the
transverse polarisation of the photon.
The coefficient $a=0.67\,^{+0.54}_{-0.30}$ is estimated by assigning each
(background subtracted) signal candidate a value of $F_{\rm L}$ according to
Eq. 7, averaging $F_{\rm L}$ over the candidates in the $q^{2}$ bin and
comparing this to the value that is obtained from the fit to the
$0.1<q^{2}<2.0\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region (in Table 2).
Different values of the coefficient $a$ are tried until the two estimates
agree.
To remain model independent, the bias on the angular observables is similarly
estimated by summing over the observed candidates. A concrete example of how
this is done is given in Appendix B for the observable $A_{\rm T}^{2}$. The
typical size of the correction is $10-20\%$. The values of the angular
observables, after correcting for the bias, are included in Table 2. A similar
factor is also applied to the statistical uncertainty on the fit parameters to
scale them accordingly. No systematic uncertainty is assigned to this
correction.
The procedure to calculate the size of the bias that is introduced by
neglecting the threshold terms has been validated using large samples of
simulated events, generated according to the SM prediction and several other
scenarios in which large deviations from the SM expectation of the angular
observables are possible. In all cases an unbiased estimate of the angular
observables is obtained after applying the correction procedure. Different
hypotheses for the $q^{2}$ dependence of $F_{\rm L}$, $A_{\rm FB}$ and $A_{\rm
T}^{\rm Re}$ do not give large variations in the size of the correction
factors.
### 7.3 Systematic uncertainties in the angular analysis
Sources of systematic uncertainty are considered if they introduce either an
angular or $q^{2}$ dependent bias to the acceptance correction. Moreover,
three assumptions have been made that may affect the interpretation of the
result of the fit to the $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass or
angular distribution: that $q^{2}\gg 4m_{\mu}^{2}$; that there are equal
numbers of $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
decays; and that there is no contribution from non-$K^{*0}$
$B^{0}\\!\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass
window. The first assumption was addressed in Sec. 7.2 and no systematic
uncertainty is assigned to this correction. The number of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates in the data set is very
similar [38]. The resulting systematic uncertainty is addressed in Sec. 7.3.1.
The final assumption is discussed in Sec. 7.3.2 below.
The full fitting procedure has been tested on $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays. In this larger
data sample, $A_{\rm FB}$ is found to be consistent with zero (as expected)
and the other observables are in agreement with the results of Ref. [39].
There is however a small discrepancy between the expected parabolic shape of
the $\cos\theta_{K}$ distribution and the distribution of the
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates after weighting the candidates to correct for the detector
acceptance. This percent-level discrepancy could point to a bias in the
acceptance model. To account for this discrepancy, and any breakdown in the
assumption that the efficiencies in $\cos\theta_{\ell}$, $\cos\theta_{K}$ and
$\phi$ are independent, systematic variations of the weights are tried in
which they are conservatively rescaled by 10% at the edges of
$\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\phi$ with respect to the centre.
Several possible variations are explored, including variations that are non-
factorisable. The variation which has the largest effect on each of the
angular observables is assigned as a systematic uncertainty. The resulting
systematic uncertainties are at the level of $0.01-0.03$ and are largest for
the transverse observables.
The uncertainties on the signal mass model have little effect on the angular
observables. Of more importance are potential sources of uncertainty on the
background shape. In the angular fit the background is modelled as the product
of three second-order polynomials, the parameters of which are allowed to vary
freely in the likelihood fit. This model describes the data well in the
sidebands. As a cross-check, alternative fits are performed both using higher
order polynomials and by fixing the shape of the background to be flat in
$\cos\theta_{\ell}$, $\cos\theta_{K}$ and $\hat{\phi}$. The largest shifts in
the angular observables occur for the flat background model and are at the
level of $0.01-0.06$ and $0.02-0.25$ for the transverse observables (they are
at most 65% of the statistical uncertainty). These variations are extreme
modifications of the background model and are not considered further as
sources of systematic uncertainty.
The angular distributions of the decays
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ are both poorly known. The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ is yet to be observed. A first measurement of
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ has been made in Ref. [40]. In the
likelihood fit to the angular distribution these backgrounds are neglected. A
conservative systematic uncertainty on the angular observables is assigned at
the level of ${~{}\raise 1.49994pt\hbox{$<$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}0.01$ by assuming that the peaking backgrounds have
an identical shape to the signal, but have an angular distribution in which
each of the observables is either maximal or minimal.
Systematic variations are also considered for the data-derived corrections to
the simulated events. For example, the muon identification efficiency, which
is derived from data using a tag-and-probe approach with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays, is varied within its
uncertainty in opposite direction for high
($p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and low
($p<10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) momentum muons. Similar
variations are applied to the other data-derived corrections, yielding a
combined systematic uncertainty at the level of $0.01-0.02$ on the angular
observables. The correction needed to account for differences between data and
simulation in the $B^{0}$ momentum spectrum is small. If this correction is
neglected, the angular observables vary by at most 0.01. This variation is
associated as a systematic uncertainty.
The systematic uncertainties arising from the variations of the angular
acceptance are assessed using pseudo-experiments that are generated with one
acceptance model and fitted according to a different model. Consistent results
are achieved by varying the event weights applied to the data and repeating
the likelihood fit.
A summary of the different contributions to the total systematic uncertainty
can be found in Table 3. The systematic uncertainty on the angular observables
in Table 2 is the result of adding these contributions in quadrature.
Table 3: Systematic contributions to the angular observables. The values given are the magnitude of the maximum contribution from each source of systematic uncertainty, taken across the six principal $q^{2}$ bins used in the analysis. Source | $A_{\rm FB}$ | $F_{\rm L}$ | $S_{3}$ | $S_{9}$ | $A_{9}$ | $A_{\rm T}^{2}$ | $A_{\rm T}^{\rm Re}$
---|---|---|---|---|---|---|---
Acceptance model | $\phantom{<}0.02$ | $\phantom{<}0.03$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.02$ | $\phantom{<}0.01$
Mass model | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$
$B^{0}\rightarrow\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mis-id | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$
Data-simulation diff. | $\phantom{<}0.01$ | $\phantom{<}0.03$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.03$ | $\phantom{<}0.01$
Kinematic reweighting | $<0.01$ | $\phantom{<}0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $\phantom{<}0.01$ | $<0.01$
Peaking backgrounds | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.01$
S-wave | $\phantom{<}0.01$ | $\phantom{<}0.01$ | $\phantom{<}0.02$ | $\phantom{<}0.01$ | $<0.01$ | $\phantom{<}0.05$ | $\phantom{<}0.04$
$B^{0}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ asymmetries | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$ | $<0.01$
#### 7.3.1 Production, detection and direct ${\boldmath C\\!P}$ asymmetries
If the number of $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays are not equal in the
likelihood fit then the terms in the angular distribution no longer correspond
to pure $C\\!P$ averages or asymmetries. They instead correspond to admixtures
of the two, e.g.
$S_{3}^{\text{obs}}\approx S_{3}-A_{3}\left(\mathcal{A}_{\rm
CP}+\kappa\mathcal{A}_{\rm P}+\mathcal{A}_{\rm D}\right)~{},$ (8)
where $\mathcal{A}_{\rm CP}$ is the direct $C\\!P$ asymmetry between
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays;
$\mathcal{A}_{\rm P}$ is the production asymmetry between $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons, which is diluted by a
factor $\kappa$ due to $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing; and $\mathcal{A}_{\rm D}$
is the detection asymmetry between the $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays (which might be non-zero
due to differences in the interaction cross-section with matter between
$K^{+}$ and $K^{-}$ mesons). In practice, the production and detection
asymmetries are small in LHCb and $\mathcal{A}_{\rm CP}$ is measured to be
$\mathcal{A}_{\rm CP}=-0.072\pm 0.040\pm 0.005$ [38], which is consistent with
zero. Combined with the expected small size of the $C\\!P$ asymmetry or
$C\\!P$-averaged counterparts of the angular observables measured in this
analysis, this reduces any systematic bias to $<0.01$.
#### 7.3.2 Influence of S-wave interference on the angular distribution
The presence of a non-$K^{*0}$ $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ component, where the $K^{+}\pi^{-}$ system is in
an S-wave configuration, modifies Eq. 4 to
$\begin{split}\frac{1}{\mathrm{d}\Gamma^{\prime}/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma^{\prime}}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}=&~{}(1-F_{\rm
S})\left[\frac{1}{\mathrm{d}\Gamma/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\,\mathrm{d}\cos\theta_{\ell}\,\mathrm{d}\cos\theta_{K}\,\mathrm{d}\hat{\phi}}\right]\\\
&~{}~{}\,+\frac{9}{16\pi}\left[\frac{2}{3}F_{\rm
S}(1-\cos^{2}\theta_{\ell})+\frac{4}{3}A_{\rm
S}\cos\theta_{K}(1-\cos^{2}\theta_{\ell})\right]~{},\\\ \end{split}$ (9)
where $F_{\rm S}$ is the fraction of $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ S-wave in the
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window.
The partial width, $\Gamma^{\prime}$, is the sum of the partial widths for the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay and the $B^{0}\\!\rightarrow
K^{+}\pi^{-}\mu^{+}\mu^{-}$ S-wave. A forward-backward asymmetry in
$\cos\theta_{K}$, $A_{\rm S}$, arises due to the interference between the
longitudinal amplitude of the $K^{*0}$ and the S-wave amplitude [41, 42, 43,
44].
The S-wave is neglected in the results given in Table 2. To estimate the size
of the S-wave component, and the impact it might have on the
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ angular analysis, the phase shift
of the $K^{*0}$ Breit-Wigner function around the $K^{*0}$ pole mass is
exploited. Instead of measuring $F_{\rm S}$ directly, the average value of
$A_{\rm S}$ is measured in two bins of $K^{+}\pi^{-}$ invariant mass, one
below and one above the $K^{*0}$ pole mass. If the magnitude and phase of the
S-wave amplitude are assumed to be independent of the $K^{+}\pi^{-}$ invariant
mass in the range
$792<m({K^{+}\pi^{-}})<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the
P-wave amplitude is modelled by a Breit-Wigner function, the two $A_{\rm S}$
values can then be used to determine the real and imaginary components of the
S-wave amplitude (and $F_{\rm S}$).444In the decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ there are actually two pairs of amplitudes involved,
left- and right-handed longitudinal amplitudes and left- and right-handed
S-wave amplitudes (where the handedness refers to the chirality of the dimuon
system). In order to exploit the interference and determine $F_{\rm S}$ it is
assumed that the phase difference between the two left-handed amplitudes is
the same as the difference between the two right-handed amplitudes, as
expected from the expression for the amplitudes in Refs. [41, 42].
For a small S-wave amplitude, the pure S-wave contribution, $F_{\rm S}$, to
Eq. 9 has only a small effect on the angular distribution. The magnitude of
$A_{\rm S}$ arising from the interference between the S- and P-wave can
however still be sizable and this information is exploited by this phase-shift
method. The method, described above, is statistically more precise than
fitting Eq. 9 directly for $A_{\rm S}$ and $F_{\rm S}$ as uncorrelated
variables. For the $B^{0}\\!\rightarrow
K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ control mode, the gain in
statistical precision is approximately a factor of three.
Due to the limited number of signal candidates that are available in each of
the $q^{2}$ bins, the bins are merged in order to estimate the S-wave
fraction. In the range $0.1<q^{2}<19\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
$F_{\rm S}=0.03\pm 0.03$, which corresponds to an upper limit of $F_{\rm
S}<0.04$ at $68\%$ confidence level (CL). The procedure has also been
performed in the region $1<q^{2}<6\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
where both $F_{\rm L}$ and $F_{\rm S}$ are expected to be enhanced. This gives
$F_{\rm S}=0.04\pm 0.04$ and an upper limit of $F_{\rm S}<0.07$ at $68\%$ CL.
In order to be conservative, $F_{\rm S}=0.07$ is used to estimate a systematic
uncertainty on the differential branching fraction and angular analyses. The
$B^{0}\\!\rightarrow K^{*0}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ data
has been used to validate the method.
For the differential branching fraction analysis, $F_{\rm S}$ scales the
observed branching fraction by up to 7%. For the angular analysis, $F_{\rm S}$
dilutes $A_{\rm FB}$, $S_{3}$ and $A_{9}$. The impact on $F_{\rm L}$ however,
is less easy to disentangle. To assess the possible size of a systematic bias,
pseudo-experiments have been carried out generating with, and fitting without,
the S-wave contribution in the likelihood fit. The typical bias on the angular
observables due to the S-wave is $0.01-0.03$.
## 8 Forward-backward asymmetry zero-crossing point
In the SM, $A_{\rm FB}$ changes sign at a well defined value of $q^{2}$,
$q^{2}_{0}$, whose prediction is largely free from form-factor uncertainties
[3]. It is non-trivial to estimate $q^{2}_{0}$ from the angular fits to the
data in the different $q^{2}$ bins, due to the large size of the bins
involved. Instead, $A_{\rm FB}$ can be estimated by counting the number of
forward-going ($\cos\theta_{\ell}>0$) and backward-going
($\cos\theta_{\ell}<0$) candidates and $q^{2}_{0}$ determined from the
resulting distribution of $A_{\rm FB}(q^{2})$.
The $q^{2}$ distribution of the forward- and backward-going candidates, in the
range $1.0<q^{2}<7.8\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, is shown in Fig.
6. To make a precise measurement of the zero-crossing point a polynomial fit,
$P(q^{2})$, is made to the $q^{2}$ distributions of these candidates. The
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass is included in the fit to separate
signal from background. If $P_{\rm F}(q^{2})$ describes the $q^{2}$ dependence
of the forward-going, and $P_{\rm B}(q^{2})$ the backward-going signal decays,
then
$A_{\rm FB}(q^{2})=\frac{P_{\rm F}(q^{2})-P_{\rm B}(q^{2})}{P_{\rm
F}(q^{2})+P_{\rm B}(q^{2})}~{}~{}.$ (10)
The zero-crossing point of $A_{\rm FB}$ is found by solving for the value of
$q^{2}$ at which $A_{\rm FB}(q^{2})$ is zero.
Using third-order polynomials to describe both the $q^{2}$ dependence of the
signal and the background, the zero-crossing point is found to be
$q_{0}^{2}=4.9\pm 0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}~{}~{}.$
The uncertainty on $q^{2}_{0}$ is determined using a bootstrapping technique
[45]. The zero-crossing point is largely independent of the polynomial order
and the $q^{2}$ range that is used. This value is consistent with SM
predictions, which are typically in the range
$3.9-4.4\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ [46, 47, 48] and have
relative uncertainties below the 10% level, for example,
$q_{0}^{2}=4.36\,^{+0.33}_{-0.31}\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$
[47].
The systematic uncertainty on the zero-crossing point of the forward-backward
asymmetry is negligible compared to the statistical uncertainty. To generate a
large systematic bias, it would be necessary to create an asymmetric
acceptance effect in $\cos\theta_{\ell}$ that is not canceled when combining
$B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays. The
combined systematic uncertainty is at the level of $\pm
0.05\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.
Figure 6: Dimuon invariant mass squared, $q^{2}$, distribution of forward-
going (left) and backward-going (right) candidates in the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass window
$5230<m(K^{+}\pi^{-}\mu^{+}\mu^{-})<5330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The polynomial fit to the signal and background distributions in $q^{2}$ is
overlaid.
## 9 Conclusions
In summary, using a data sample corresponding to 1.0$\mbox{\,fb}^{-1}$ of
integrated luminosity, collected by the LHCb experiment in 2011, the
differential branching fraction, $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$, of the
decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ has been measured in bins of
$q^{2}$. Measurements of the angular observables, $A_{\rm FB}$ ($A_{\rm
T}^{\rm Re}$), $F_{L}$, $S_{3}$ ($A_{\rm T}^{2}$) and $A_{9}$ have also been
performed in the same $q^{2}$ bins.
The complete set of results obtained in this paper are provided in Tables 1
and 2. These are the most precise measurements of $\mathrm{d}$$\cal
B$/$\mathrm{d}$$q^{2}$ and the angular observables to date. All of the
observables are consistent with SM expectations and together put stringent
constraints on the contributions from new particles to $b\rightarrow s$
flavour changing neutral current processes. A bin-by-bin comparison of the
reduced angular distribution with the SM hypothesis indicates an excellent
agreement with p-values between 18 and 72%.
Finally, a first measurement of the zero-crossing point of the forward-
backward asymmetry has also been performed, yielding $q_{0}^{2}=4.9\pm
0.9\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$. This measurement is again
consistent with SM expectations.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
Appendix
## Appendix A Angular basis
Figure 7: Graphical representation of the angular basis used for
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\\!\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays in this
paper. The notation $\hat{n}_{ab}$ is used to represent the normal to the
plane containing particles $a$ and $b$ in the $B^{0}$ (or $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}$) rest frame. An explicit
description of the angular basis is given in the text.
The angular basis used in this paper is illustrated in Fig. 7. The angle
$\theta_{\ell}$ is defined as the angle between the direction of the $\mu^{+}$
($\mu^{-}$) in the dimuon rest frame and the direction of the dimuon in the
$B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The
angle $\theta_{K}$ is defined as the angle between the direction of the kaon
in the $K^{*0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) rest
frame and the direction of the $K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) in the $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. The angle $\phi$ is
the angle between the plane containing the $\mu^{+}$ and $\mu^{-}$ and the
plane containing the kaon and pion from the $K^{*0}$. Explicitly,
$\cos\theta_{\ell}$ and $\cos\theta_{K}$ are defined as
$\cos\theta_{\ell}=\left(\hat{p}_{\mu^{+}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(\hat{p}_{\mu^{+}\mu^{-}}^{(B^{0})}\right)=\left(\hat{p}_{\mu^{+}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(-\hat{p}_{B^{0}}^{(\mu^{+}\mu^{-})}\right)~{},$
(11)
$\cos\theta_{K}=\left(\hat{p}_{K^{+}}^{(K^{*0})}\right)\cdot\left(\hat{p}_{K^{*0}}^{(B^{0})}\right)=\left(\hat{p}_{K^{+}}^{(K^{*0})}\right)\cdot\left(-\hat{p}_{B^{0}}^{(K^{*0})}\right)$
(12)
for the $B^{0}$ and
$\cos\theta_{\ell}=\left(\hat{p}_{\mu^{-}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(\hat{p}_{\mu^{+}\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)=\left(\hat{p}_{\mu^{-}}^{(\mu^{+}\mu^{-})}\right)\cdot\left(-\hat{p}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}^{(\mu^{+}\mu^{-})}\right)~{},$
(13)
$\cos\theta_{K}=\left(\hat{p}_{K^{-}}^{(K^{*0})}\right)\cdot\left(\hat{p}_{K^{*0}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)=\left(\hat{p}_{K^{-}}^{(K^{*0})}\right)\cdot\left(-\hat{p}_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}^{(K^{*0})}\right)$ (14)
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. The
definition of the angle $\phi$ is given by
$\cos\phi=\left(\hat{p}_{\mu^{+}}^{(B^{0})}\times\hat{p}_{\mu^{-}}^{(B^{0})}\right)\cdot\left(\hat{p}_{K^{+}}^{(B^{0})}\times\hat{p}_{\pi^{-}}^{(B^{0})}\right)~{},$
(15)
$\sin\phi=\left[\left(\hat{p}_{\mu^{+}}^{(B^{0})}\times\hat{p}_{\mu^{-}}^{(B^{0})}\right)\times\left(\hat{p}_{K^{+}}^{(B^{0})}\times\hat{p}_{\pi^{-}}^{(B^{0})}\right)\right]\cdot\hat{p}_{K^{*0}}^{(B^{0})}$
(16)
for the $B^{0}$ and
$\cos\phi=\left(\hat{p}_{\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\mu^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\cdot\left(\hat{p}_{K^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\pi^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)~{},$ (17)
$\sin\phi=-\left[\left(\hat{p}_{\mu^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\mu^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\times\left(\hat{p}_{K^{-}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\times\hat{p}_{\pi^{+}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}\right)\right]\cdot\hat{p}_{\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}$ (18)
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decay. The
$\hat{p}_{X}^{(Y)}$ are unit vectors describing the direction of a particle
$X$ in the rest frame of the system $Y$. In every case the particle momenta
are first boosted to the $B^{0}$ (or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) rest frame. In this basis, the
angular definition for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
decay is a $C\\!P$ transformation of that for the $B^{0}$ decay.
## Appendix B Angular distribution at large recoil
An explicit example of the bias on the angular observables that comes from the
threshold terms is provided below for $A_{\rm T}^{2}$. Sensitivity to $A_{\rm
T}^{2}$ comes through the term in Eq. 1 with
$\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ angular dependence. In the
limit $q^{2}\gg m_{\mu}^{2}$, this term is simply
$\frac{1}{2}\left(1-F_{\rm L}(q^{2})\right)A_{\rm
T}^{2}(q^{2})\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ (19)
and the differential decay width is
$\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}=|A_{0,{\rm
L}}|^{2}+|A_{\parallel,{\rm L}}|^{2}+|A_{\perp,{\rm L}}|^{2}+|A_{0,{\rm
R}}|^{2}+|A_{\parallel,{\rm R}}|^{2}+|A_{\perp,{\rm R}}|^{2}~{},$ (20)
where $A_{0}$, $A_{\parallel}$ and $A_{\perp}$ are the $K^{*0}$ spin-
amplitudes and the L/R index refers to the chirality of the lepton current
(see for example Ref. [1]). If $q^{2}{~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}1\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, these
expressions are modified to
$\frac{1}{2}\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]\left(1-F_{\rm
L}(q^{2})\right)A_{\rm
T}^{2}(q^{2})\sin^{2}\theta_{\ell}\sin^{2}\theta_{K}\cos 2\phi$ (21)
and
$\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}=\left[1+2m_{\mu}^{2}/q^{2}\right]\left(|A_{0,{\rm
L}}|^{2}+|A_{\parallel,{\rm L}}|^{2}+|A_{\perp,{\rm L}}|^{2}+|A_{0,{\rm
R}}|^{2}+|A_{\parallel,{\rm R}}|^{2}+|A_{\perp,{\rm R}}|^{2}\right).$ (22)
In an infinitesimal window of $q^{2}$, the difference between an experimental
measurement of $A_{\rm T}^{2}$, $A_{\rm T}^{2~{}\text{exp}}$, in which the
threshold terms are neglected and the value of $A_{\rm T}^{2}$ defined in
literature is
$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]~{}~{}.$
(23)
Unfortunately, in a wider $q^{2}$ window, the $q^{2}$ dependence of $F_{\rm
L}$, $A_{\rm T}^{2}$ and the threshold terms needs to be considered and it
becomes less straightforward to estimate the bias due to the threshold terms.
If $A_{\rm T}^{2}$ is constant over the $q^{2}$ window,
$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\frac{\displaystyle\int_{q^{2}_{\text{min}}}^{q^{2}_{\text{max}}}\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\left[\frac{1-4m_{\mu}^{2}/q^{2}}{1+2m_{\mu}^{2}/q^{2}}\right]\left[1-F_{\rm
L}(q^{2})\right]\mathrm{d}q^{2}}{\displaystyle\int_{q^{2}_{\text{min}}}^{q^{2}_{\text{max}}}\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}\left[1-F_{\rm
L}(q^{2})\right]\mathrm{d}q^{2}}~{}~{}.$ (24)
In practice the integration in Eq. 24 can be replaced by a sum over the signal
events in the $q^{2}$ window
$\frac{A_{\rm T}^{2~{}\text{exp}}}{A_{\rm
T}^{2}}=\frac{\sum\limits_{i=0}^{N}\left[\frac{1-4m_{\mu}^{2}/q^{2}_{i}}{1+2m_{\mu}^{2}/q^{2}_{i}}\right](1-F_{\rm
L}(q_{i}^{2}))\omega_{i}}{\sum\limits_{i=0}^{N}(1-F_{\rm
L}(q_{i}^{2}))\omega_{i}}~{},$ (25)
where $\omega_{i}$ is a weight applied to the $i^{\text{th}}$ candidate to
account for the detector and selection acceptance and the background in the
$q^{2}$ window.
Correction factors for the other observables can be similarly defined if it is
assumed that they are constant over the $q^{2}$ window. In the case of $A_{\rm
FB}$ (and $A_{\rm T}^{\rm Re}$) that are expected to exhibit a strong $q^{2}$
dependence, the $q^{2}$ dependence of the observable needs to be considered.
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|
arxiv-papers
| 2013-04-23T15:33:18 |
2024-09-04T02:49:44.709219
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van\n den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K.\n Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti, B.\n Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P.\n David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De\n Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M. Dogaru, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach,\n I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, G.\n Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero\n Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang,\n D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Thomas Blake",
"url": "https://arxiv.org/abs/1304.6325"
}
|
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