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+{"text":"\\section{Introduction}\n\n\\vspace{1mm}\n\\noindent\n\nThe spin properties of hadrons inclusively produced in high energy \ninteractions are related to the fundamental properties of quarks and \ngluons and to their elementary interactions in a much more subtle way \nthan unpolarized quantities. They test unusual basic dynamical \nproperties and reveal how the usual  models for quark distribution and\nhadronization -- successful in predicting unpolarized cross-sections -- may \nnot be adequate to describe spin effects. We consider here such two cases,\nsingle spin asymmetries in Deep Inelastic Scattering and the polarization\nof mesons produced in the fragmentation of polarized quarks at LEP.\n\n\\section{Single spin asymmetries in DIS}\n\n\\vspace{1mm}\n\\noindent\n\nWe discuss first single spin asymmetries in DIS processes. \nWe start by reminding that single spin asymmetries in large $p_{_T}$ inclusive \nhadronic reactions are forbidden in leading-twist perturbative QCD, \nreflecting the fact that single spin asymmetries are zero at the partonic \nlevel and that collinear parton configurations inside hadrons do not allow \nsingle spin dependences. However, experiments tell us in several cases, \n\\cite{ada1,ada2} that single spin asymmetries are large and indeed non \nnegligible.\n\nThe usual arguments to explain this apparent disagreement between pQCD \nand experiment invoke the moderate $p_{_T}$ values of the data -- a few \nGeV, not quite yet in the true perturbative regime -- and the importance \nof higher-twist effects. Several phenomenological models have recently\nattempted to explain the large single spin asymmetries observed in\n$p^\\uparrow p \\to \\pi X$ as twist-3 effects which might be due to intrinsic \npartonic $\\mbox{\\boldmath $k$}_\\perp$ in the fragmentation \\cite{col1} and\/or distribution \nfunctions \\cite{siv1}-\\cite{noi}. \n\nLet us consider a process in which one has convincing evidence \nthat partons and perturbative QCD work well and successfully describe the \nunpolarized and leading-twist spin data, namely Deep Inelastic Scattering \n(DIS). In particular we shall discuss single spin asymmetries in the inclusive,\n$\\ell N^\\uparrow \\to \\ell + jets$ and $\\ell N^\\uparrow \\to hX$, \nreactions looking at possible origins of such asymmetries and \ndevising strategies to isolate and discriminate among them \\cite{alm}.\n\nAccording to the QCD hard scattering picture and the factorization theorem \n\\cite{col1,col2,col3} the cross-section for the $\\ell N^\\uparrow \\to hX$ \nreaction is given by\n\\begin{eqnarray}\n& & \\frac{E_h \\, d^3\\sigma^{\\ell + N,S \\to h + X}} {d^{3} \\mbox{\\boldmath $p$}_h} = \n\\sum_{q; \\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}, \\lambda^{\\,}_h} \n\\int \\frac {dx \\, d^2\\mbox{\\boldmath $k$}_\\perp d^2\\mbox{\\boldmath $k$}_\\perp^\\prime} {\\pi z} \\label{gen} \\\\  \n& & \\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\n\\frac{d\\hat\\sigma^{q,P_q}}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\>\n\\rho^{q^{\\prime}}_{\\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}}\n(x, \\mbox{\\boldmath $k$}_\\perp, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\> \\widetilde \nD_{\\lambda^{\\,}_h, \\lambda^{\\,}_h}^{\\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}}\n(z, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\,. \\nonumber\n\\end{eqnarray}\n\nLet us briefly discuss the different quantities appearing in the above \nequation; more details can be found in Ref. \\cite{alm}.\n$\\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp)$ is the quark distribution function,\nthat is the total number density of quarks $q$  with momentum fraction \n$x$ and intrinsic transverse momentum $\\mbox{\\boldmath $k$}_\\perp$ inside a polarized nucleon\n$N$ with spin four-vector $S$. \n\n$d\\hat\\sigma^{q,P_q} \/ d\\hat t$ is the cross-section for the $\\ell \nq^\\uparrow \\to \\ell q$ process, with an unpolarized lepton and an initial \nquark with polarization $P_q$, while the final quark and lepton polarization\nare summed over. Notice that for helicity conserving elementary \ninteractions $d\\hat\\sigma^{q,P_q} \/ d\\hat t$ equals the unpolarized \ncross-section $d\\hat\\sigma^q \/ d\\hat t$.\n$\\rho^{q^\\prime}$ is the helicity density matrix of the final quark \nproduced in the $\\ell q^\\uparrow$ interaction and \n\\begin{equation}\n\\widetilde D^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) = \\sum_{\\lambda^{\\,}_q, \\lambda^{\\prime}_q} \n\\rho^q_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} \\> \\widetilde D_{\\lambda^{\\,}_h, \n\\lambda^{\\,}_h}^{\\lambda^{\\,}_q, \\lambda^{\\prime}_q}(z, \\mbox{\\boldmath $k$}_\\perp)\n\\end{equation}\ndescribes the fragmentation process of a polarized quark $q$ \nwith spin $s_q$ into a hadron $h$ with helicity $\\lambda^{\\,}_h$, \nmomentum fraction $z$ and intrinsic transverse momentum $\\mbox{\\boldmath $k$}_\\perp$ with \nrespect to the jet axis. It is simply the inclusive cross-section for\nthe $q^\\uparrow \\to hX$ fragmentation process. As it will be shown in the \nsecond part we can safely neglect the coherent interactions of the \nfragmenting quark, as we are not looking at the spin state of the final \nhadron. The usual unpolarized fragmentation function is given by\n\\begin{equation}\nD^h_q(z) = {1\\over 2} \\sum_{\\lambda^{\\,}_q, \\lambda^{\\,}_h}\n\\int d^2\\mbox{\\boldmath $k$}_\\perp \\>\n\\widetilde D^{\\lambda^{\\,}_q,\\lambda^{\\,}_q}_{\\lambda^{\\,}_h,\\lambda^{\\,}_h}\n(z, \\mbox{\\boldmath $k$}_\\perp) \\,.\n\\label{fr}\n\\end{equation}\n\nSimilar formulae hold also when the elementary interaction\nis $\\ell q \\to \\ell q g$ rather than $\\ell q \\to \\ell q$: in the latter \ncase two jets are observed in the final state -- the target jet and the \ncurrent quark jet -- and in the former case three -- the target jet and \n$q$ + $g$ current jets.\n\nIn Eq. (\\ref{gen}) we have taken into account intrinsic transverse momenta \nboth in the distribution and the fragmentation process;\nthe $\\mbox{\\boldmath $k$}_\\perp$ dependences are expected to have negligible effects on \nunpolarized variables for which they are indeed usually neglected, but \nthey can be of crucial importance for some single spin observables, as \ndiscussed in Refs. \\cite{col1}-\\cite{noi}.\n\nWe discuss now possible sources of single spin effects in Eq. (\\ref{gen}).\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $k_\\perp$}} {\\bf effects in fragmentation process}\n\\cite{col1}\n\\vskip 6pt\n\\nobreak\n \nThe fragmentation process of a transversely polarized quark into a hadron \n$h$ (whose polarization in not observed) with fixed $z$ and $\\mbox{\\boldmath $k$}_\\perp$ \nmay depend on the quark spin orientation, provided the quark spin \n$s_q$ has a component perpendicular to the hadron-quark plane (otherwise \nany spin dependence would be forbidden by parity conservation). That is, \nthere might be a non zero {\\it quark analysing power} \\cite{col1}:\n\\begin{equation}\nA_q^h \\equiv {\\widetilde D^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) \n- \\widetilde D^h_{q,-s_q}(z, \\mbox{\\boldmath $k$}_\\perp) \\over \\widetilde \nD^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) + \\widetilde D^h_{q,-s_q}(z,\\mbox{\\boldmath $k$}_\\perp)} \\,\\cdot\n\\label{qap}\n\\end{equation}\nBy rotational invariance $\\widetilde D^h_{q,-s_q}(z,\\mbox{\\boldmath $k$}_\\perp) =\n\\widetilde D^h_{q, s_q}(z,-\\mbox{\\boldmath $k$}_\\perp)$, which shows immediately how the \nquark analysing power vanishes for $\\mbox{\\boldmath $k$}_\\perp = 0$. \n \n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $k_\\perp$}} {\\bf effects in distribution functions}\n\\cite{siv1}-\\cite{noi}\n\\vskip 6pt\n\\nobreak\n\nA similar idea had been previously proposed in Refs. \\cite{siv1, siv2}\nand later rediscovered in Ref. \\cite{noi}, concerning the distribution\nfunctions: that is, the number of quarks (whose spin is not observed)\nwith fixed $x$ and $\\mbox{\\boldmath $k$}_\\perp$ inside a transversely polarized nucleon\nmay depend on the nucleon spin orientation and the function\n\\begin{equation}\n\\Delta\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) \\equiv\n\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N^\\downarrow}(x, \\mbox{\\boldmath $k$}_\\perp)\n= \\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N,^\\uparrow}(x,-\\mbox{\\boldmath $k$}_\\perp)\n\\label{nap}\n\\end{equation}\nmay be different from zero. $\\uparrow$ and $\\downarrow$ refer to the \nnucleon spin up or down with respect to the quark-nucleon plane. \n\nIn terms of the usual light-cone operator definition of structure \nfunctions one has \\cite{col1,dra}\n\\begin{eqnarray}\n\\Delta\\tilde f_{q\/N}^{N^\\uparrow}\n&=& 2 \\, {\\rm Im} \\int {dy^- d\\mbox{\\boldmath $y$}_\\perp \\over(2 \\pi)^3}\ne^{-i x p^+ y^- + i \\mbox{\\boldmath $k$}_\\perp\\cdot\\mbox{\\boldmath $y$}_\\perp} \\nonumber\\\\\n&&\\quad\\langle p, -|\\bar\\psi_a(0,y^-,y_\\perp)\n{\\gamma^+\\over 2}\\psi_a(0)|p, + \\rangle \\,. \n\\label{del+-}\n\\end{eqnarray}\n\nIn Ref. \\cite{col1} it is argued that such off-diagonal (in the helicity\nbasis) matrix elements are zero due to the time-reversal invariance of QCD, \nand indeed this is proven by exploiting the time-reversal and parity \ntransformation properties of free Dirac spinors. However, in Ref. \\cite{dra} \nit has been shown that this need not be so in chiral models with quark moving \nin a background of chiral fields.\n\nBoth $\\widetilde D^h_{q^\\uparrow}(z, \\mbox{\\boldmath $k$}_\\perp) \n- \\widetilde D^h_{q^\\downarrow}(z, \\mbox{\\boldmath $k$}_\\perp)$ and \n$\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N^\\downarrow}(x, \\mbox{\\boldmath $k$}_\\perp)$ can be considered as \nnew fundamental spin and $\\mbox{\\boldmath $k$}_\\perp$ dependent non perturbative \nfunctions describing respectively quark fragmentation and distribution\nproperties. In the sequel we shall devise strategies to test their\nrelevance. \n\n\\vskip 12pt\n\\noindent\n{\\bf Single spin effects in the elementary interactions} \n\\vskip 6pt\n\nAs we already mentioned both perturbative QED and QCD at high energy \ndo not allow single helicity flips in the $\\ell q$ interactions, so that \nthere cannot be any dependence on the quark polarization in \n$d\\hat\\sigma^{q,P_q}\/d\\hat t$. Similarly, the perturbative QCD evolution\nof the distribution and fragmentation functions is not expected to\nintroduce any single spin dependence. We must conclude that the hard \nelementary interactions are unlikely to introduce any single\nspin effect: however, this basic QED and QCD property should also\nbe tested. \n\n\\vskip 6pt\nLet us now describe a set of possible measurements which could single out\nsome of the above mechanisms and test them. \n\n\\goodbreak\n\\vskip 6pt\n\\noindent\n$a) \\> \\ell N^\\uparrow \\to \\ell + 2\\,jets$\n\\vskip 4pt\n\\nobreak\n\nHere one avoids any fragmentation effect by looking at the fully \ninclusive cross-section for the process $\\ell N^\\uparrow \\to \\ell + 2\\, jets$,\nthe 2 jets being the target and current ones; this is the usual DIS, \nand Eq. (\\ref{gen}) becomes  \n\\begin{equation}\n\\frac{d^2\\sigma^{\\ell + N,S \\to \\ell + X}} {dx \\, dQ^2} = \\sum_q\n\\int d^2\\mbox{\\boldmath $k$}_\\perp \\> \\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\n\\frac{d\\hat\\sigma^{q,P_q}}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp) \\,. \n\\label{gena}\n\\end{equation}\n\nIn this case the elementary interaction is a pure QED, \nhelicity conserving one, $\\ell q \\to \\ell q$, and $d\\hat\\sigma^{q,P_q}\/d\\hat t$\ncannot depend on the quark polarization. Some spin dependence might only\nremain in the distribution function, due to intrinsic $\\mbox{\\boldmath $k$}_\\perp$ effects\n\\cite{siv1}-\\cite{noi}, \\cite{dra} and we have\n\\begin{eqnarray}\n\\frac{d^2\\sigma^{\\ell N^\\uparrow \\to \\ell + X}} {dx \\, dQ^2} \n&-& \\frac{d^2\\sigma^{\\ell N^\\downarrow \\to \\ell + X}} {dx \\, dQ^2} \n= \\sum_q \\int d^2\\mbox{\\boldmath $k$}_\\perp \\nonumber \\\\\n&\\times& \\Delta\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\\>\n\\frac{d\\hat\\sigma^q}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp) \\,. \n\\label{asymg}\n\\end{eqnarray}\nDespite the fact that $\\Delta\\tilde f_{q\/N}^{N^\\uparrow}$ is an odd\nfunction of $\\mbox{\\boldmath $k$}_\\perp$ a non zero value of the above difference \n-- of ${\\cal O}(k_\\perp\/\\sqrt {Q^2})$, twist 3 -- \nmight remain even after integration on $d^2\\mbox{\\boldmath $k$}_\\perp$ because of\nthe $\\mbox{\\boldmath $k$}_\\perp$ dependence of $d\\hat\\sigma^q\/d\\hat t$, similarly \nto what happens in $pp^\\uparrow \\to \\pi X$ \\cite{noi}. The observation\nof a non vanishing value of the single spin effect of Eq. (\\ref{asymg})\nwould be a decisive test in favour of the mechanism suggested in \nRefs. \\cite{siv1}-\\cite{noi} and would allow an estimate of the\nnew function (\\ref{nap}).\n\n\\vskip 6pt\n\\noindent\n$b) \\> \\ell N^\\uparrow \\to h + X \\> (2\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp \\not= 0)$\n\\vskip 4pt\n\nOne looks for a hadron $h$, with transverse momentum $\\mbox{\\boldmath $k$}_\\perp$, inside \nthe quark current jet; the final lepton may or may not be observed.\nThe elementary subprocess is $\\ell q \\to \\ell q$ and Eq. (\\ref{gen}) yields\n\\begin{eqnarray}\n& & \\frac{E_h \\, d^5\\sigma^{\\ell + N^\\uparrow \\to h + X}} \n{d^{3} \\mbox{\\boldmath $p$}_h d^2 \\mbox{\\boldmath $k$}_\\perp} \n- \\frac{E_h \\, d^5\\sigma^{\\ell + N^\\downarrow \\to h + X}} \n{d^{3} \\mbox{\\boldmath $p$}_h d^2 \\mbox{\\boldmath $k$}_\\perp} \\label{coll} \\\\\n&=& \\sum_q \\int \\frac {dx} {\\pi z} \\>    \n\\Delta_{_T} q(x) \\> \\Delta_{_N} \\hat\\sigma^q (x, \\mbox{\\boldmath $k$}_\\perp) \\,\n\\left[ \\tilde D^h_{q^\\uparrow}(z, \\mbox{\\boldmath $k$}_\\perp)\n- \\tilde D^h_{q^\\uparrow}(z, - \\mbox{\\boldmath $k$}_\\perp) \\right]\n\\nonumber\n\\end{eqnarray}\nwhere $\\Delta_{_T}q$ is the polarized number density for transversely spinning \nquarks $q$ and $\\Delta_{_N} \\hat\\sigma^q$ is the elementary cross-section \ndouble spin asymmetry\n\\begin{equation}\n\\Delta_{_N} \\hat\\sigma^q = {d\\hat \\sigma^{\\ell q^\\uparrow \\to \n\\ell q^\\uparrow} \\over d\\hat t} - {d\\hat \\sigma^{\\ell q^\\uparrow \\to \n\\ell q^\\downarrow} \\over d\\hat t} \\,\\cdot\n\\label{del}\n\\end{equation}\n\nIn Eq. (\\ref{coll}) we have neglected the $\\mbox{\\boldmath $k$}_\\perp$ effect in the \ndistribution function, which can be done once the asymmetry discussed\nin $a)$ turns out to be negligible. We are then testing directly the \nmechanism suggested in Ref. \\cite{col1} and a non zero value of\nthe l.h.s. of Eq. (\\ref{coll}) would be a decisive test in its favour \nand would allow an estimate of the new function appearing in the\nnumerator of Eq. (\\ref{qap}). Notice again that even upon integration over\n$d^2\\mbox{\\boldmath $k$}_\\perp$ the spin asymmetry of Eq. (\\ref{coll}) might survive,\nat higher twist order $k_\\perp\/p_{_T}$, due to some $\\mbox{\\boldmath $k$}_\\perp$ dependence \nin $\\Delta_{_N} \\hat\\sigma^q$.\n\n\\goodbreak\n\\vskip 6pt\n\\noindent\n$c) \\> \\ell N^\\uparrow \\to h + X \\> (2\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp = 0)$\n\\vskip 4pt\n\\nobreak\n\nBy selecting events with the final hadron collinear to the jet axis\n($\\mbox{\\boldmath $k$}_\\perp = 0$) one forbids any single spin effect in the fragmentation\nprocess. As in the fully inclusive case $a)$ the observation of a\nsingle spin asymmetry in such a case would imply single, $\\mbox{\\boldmath $k$}_\\perp$\ndependent, spin effects in the distribution functions. \n\n\\vskip 6pt\n\\noindent\n$d) \\> \\ell N^\\uparrow \\to h + X \\> (3\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp \\not= 0)$\n\\vskip 4pt\n \nThe elementary process is now $\\ell q \\to \\ell qg$ and one looks at \nhadrons with $\\mbox{\\boldmath $k$}_\\perp \\not= 0$ inside the $q$ current jet. Single\nspin asymmetries can originate either from single spin effects in the \nfragmentation process or distribution functions. One should not, in \nprinciple, forget possible spin effects in the elementary QCD interaction.\n\n\\vskip 6pt\n\\noindent\n$e) \\> \\ell N^\\uparrow \\to \\ell + 3\\,jets$ or \n$\\ell N^\\uparrow \\to h + X \\> (3\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp = 0)$\n\\vskip 4pt\n\nThese cases are analogous to $a)$ and $c)$ respectively: the measurement \neliminates spin effects arising from the fragmentation functions. \nThe only possible origin of a single spin asymmetry would reside \nin the distribution function. However, if no effect is observed in cases\n$a)$ and $c)$, but some effect is observed here, then one has to \nconclude that there must be some single spin effect in the elementary \nQCD interaction. Utterly unexpectedly, this would question the\nvalidity of quark helicity conservation, a fundamental property of\npQCD which has never been directly tested. \n\n\\vskip 6pt\nIn summary, a study of single transverse spin asymmetries in DIS could\nprovide a series of profound tests of our understanding of large $p_{_T}$\nQCD-controlled reactions.\n\n\\section{{\\mbox{\\boldmath $\\rho^{\\,}_{1,-1}(V)$}} in the process \n{\\mbox{\\boldmath $e^- e^+ \\to q\\bar q \\to V + X$}}}\n\n\\vspace{1mm}\n\\noindent\n\nWe consider now the spin properties of hadrons produced at LEP. It was pointed \nout in Refs. \\cite{akp} and \\cite{aamr} that final state interactions \nbetween the $q$ and $\\bar q$ created in $e^+ e^-$ annihilations \n-- usually neglected, but indeed necessary -- might give origin to non \nzero spin observables which would otherwise be forced to vanish: \nthe off-diagonal element $\\rho^{\\,}_{1,-1}$ of the helicity density matrix of \nvector mesons may be sizeably different from zero \\cite{akp} due to a \ncoherent fragmentation process which takes into account $q \\bar q$ \ninteractions. The incoherent fragmentation of a single independent quark \nleads instead to zero values for such off-diagonal elements. \n\nWe present predictions \\cite{abmq} for $\\rho^{\\,}_{1,-1}$ of several vector \nmesons $V$ provided they are produced in two jet events, carry\na large momentum or energy fraction $z=2E_{_V}\/\\sqrt s$, and have a small\ntransverse momentum $p_{_T}$ inside the jet. Our estimates are in agreement\nwith the existing data and are crucially related both to the presence \nof final state interactions and to the Standard Model couplings of the \nelementary $e^- e^+ \\to q \\bar q$ interaction. \n\nThe helicity density matrix of a hadron $h$ inclusively produced in the \ntwo jet event $e^- e^+ \\to q\\bar q \\to h + X$ can be written as \n\\cite{akp, aamr}\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\prime}_h}(h) \n= {1\\over N_h} \\sum_{q,X,\\lambda^{\\,}_X,\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q},\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q) \\>\\> \nD^*_{\\lambda^{\\prime}_h \\lambda^{\\,}_X; \\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \\,,\n\\label{rhoh}\n\\end{equation}\nwhere $\\rho(q\\bar q)$ is the helicity density matrix of the $q\\bar q$ state \ncreated in the annihilation of the unpolarized $e^+$ and $e^-$,\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q)\n= {1\\over 4N_{q\\bar q}} \\sum_{\\lambda^{\\,}_{-}, \\lambda^{\\,}_{+}}\nM^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_{\\bar q};\\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\>\nM^*_{\\lambda^{\\prime}_q \\lambda^{\\prime}_{\\bar q}; \\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\,.\n\\label{rhoqq}\n\\end{equation}\nThe $M$'s are the helicity amplitudes for the $e^-e^+ \\to q\\bar q$ process and\nthe $D$'s are the fragmentation amplitudes, {\\it i.e.} the helicity\namplitudes for the process $q\\bar q \\to h+X$; the $\\sum_{X,\\lambda_X}$ stands \nfor the phase space integration and the sum over spins of all the unobserved \nparticles, grouped into a state $X$. The normalization factors $N_h$ and\n$N_{q\\bar q}$ are given by:\n\\begin{equation}\nN_h = \\sum_{q,X; \\lambda^{\\,}_h, \\lambda^{\\,}_X, \\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q},\n\\lambda^{\\prime}_q, \\lambda^{\\prime}_{\\bar q}} \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q) \\>\\> \nD^*_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \\,\n= \\sum_q D^h_q \\,,\n\\label{nh}\n\\end{equation}\nwhere $D^h_q$ is the usual fragmentation function of quark $q$ into \nhadron $h$, and \n\\begin{equation}\nN_{q\\bar q} = {1\\over 4} \n\\sum_{\\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q}; \\lambda^{\\,}_{-}, \\lambda^{\\,}_{+}} \\vert \nM^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_{\\bar q}; \\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\vert^2 \\,.\n\\label{nqq}\n\\end{equation}\n\nThe helicity density matrix for the $q\\bar q$ state can be computed \nin the Standard Model and its non zero elements are given by\n\\begin{eqnarray}\n\\rho^{\\,}_{+-;+-}(q\\bar q) &=& 1 - \\rho^{\\,}_{-+;-+}(q\\bar q) \\> \\simeq \\> \n{1\\over 2}\\,{(g_{_V} - g_{_A})^2_q \\over (g^2_{_V} + g^2_{_A})_q}\n\\label{rhoqqd} \\\\\n\\rho^{\\,}_{+-;-+}(q\\bar q) &=& \\rho^*_{+-;-+}(q\\bar q) \\> \\simeq \\> \n{1\\over 2}\\,{(g^2_{_V} - g^2_{_A})_q \\over (g^2_{_V} + g^2_{_A})_q} \n\\, {\\sin^2\\theta \\over 1+ \\cos^2\\theta} \\, \\cdot\n\\label{rhoqqap}\n\\end{eqnarray}\nThese expressions are simple but approximate and hold at the $Z_0$ pole, \nneglecting electromagnetic contributions, masses and terms proportional \nto $g_{_V}^l$; the full correct expressions can be found in Ref. \\cite{abmq}.\n\nNotice that, inserting the values of the coupling constants\n\\begin{equation}\ng_{_V}^{u,c,t} =  \\>\\> {1\\over 2} - {4\\over 3}\\sin^2\\theta_{_W} \\quad\\quad\ng_{_V}^{d,s,b} = -{1\\over 2} + {2\\over 3}\\sin^2\\theta_{_W} \\quad\\quad\ng_{_A}^{u,c,t} = - g_{_A}^{d,s,b} = {1\\over 2} \\label{cc}\n\\nonumber\n\\end{equation} \none has\n\\begin{equation}\n\\rho^{\\,}_{+-;-+}(u\\bar u, c\\bar c, t\\bar t) \n\\simeq -0.36 {\\sin^2\\theta \\over 1 + \\cos^2\\theta} \n\\quad\\quad \n\\rho^{\\,}_{+-;-+}(d\\bar d, s\\bar s, b\\bar b) \n\\simeq -0.17 {\\sin^2\\theta \\over 1 + \\cos^2\\theta} \n\\, \\cdot \\label{rho+-ap}\n\\end{equation}\n\nEq. (\\ref{rho+-ap}) clearly shows the $\\theta$ dependence of \n$\\rho^{\\,}_{+-;-+}(q\\bar q)$. In case of pure electromagnetic interactions\n($\\sqrt s \\ll M_{_Z}$) one has exactly:\n\\begin{equation}\n\\rho^{\\gamma}_{+-;-+}(q\\bar q) = {1\\over 2}\n\\,{\\sin^2\\theta \\over 1+ \\cos^2\\theta} \\, \\cdot\n\\label{rhoqqelm}\n\\end{equation}\nNotice that Eqs. (\\ref{rho+-ap}) and (\\ref{rhoqqelm}) have the same\nangular dependence, but a different sign for the coefficient in front,\nwhich is negative for the $Z$ contribution.\n\nBy using the above equations for $\\rho(q\\bar q)$ into Eq. (\\ref{rhoh}) one \nobtains the most general expression of $\\rho(h)$ in terms of the $q\\bar q$ spin \nstate and the unknown fragmentation amplitudes.\n\nDespite the ignorance of the fragmentation process some predictions\ncan be made \\cite{abmq} by considering the production of hadrons almost \ncollinear with the parent jet: the $q \\bar q \\to h + X$ fragmentation is \nthen essentially a c.m. forward process and the unknown $D$ amplitudes must \nsatisfy the angular momentum conservation relation \\cite{bls}\n\\begin{equation}\nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \n\\sim \\left( \\sin{\\theta_h\\over 2} \\right)^{\n\\vert \\lambda^{\\,}_h - \\lambda^{\\,}_X - \\lambda^{\\,}_q + \\lambda^{\\,}_{\\bar q} \\vert} \n\\simeq \\left( {p_{_T}\\over z\\sqrt s} \\right)^{\n\\vert \\lambda^{\\,}_h - \\lambda^{\\,}_X - \\lambda^{\\,}_q + \\lambda^{\\,}_{\\bar q} \\vert} \\,,\n\\label{frd}\n\\end{equation}\nwith $\\theta_h$ the angle between the hadron momentum, \n$\\mbox{\\boldmath $h$} = z \\mbox{\\boldmath $q$} + \\mbox{\\boldmath $p$}_{_T}$, and the quark momentum $\\mbox{\\boldmath $q$}$.\n\nThe bilinear combinations of fragmentation amplitudes contributing to\n$\\rho(h)$ are then not suppressed by powers of $(p_{_T}\/z\\sqrt s)$\nonly if the exponent in Eq. (\\ref{frd}) is zero, which greatly reduces the \nnumber of relevant helicity configurations.\n\nThe fragmentation process is a parity conserving one and the fragmentation\namplitudes must then also satisfy the forward parity relationship\n\\begin{equation}\nD^{\\,}_{-\\lambda^{\\,}_h -\\lambda^{\\,}_X; -+} = (-1)^{S^{\\,}_h + S^{\\,}_X + \n\\lambda^{\\,}_h - \\lambda^{\\,}_X} \\> \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; +-} \\,.\n\\label{par}\n\\end{equation}\n\nBefore presenting analytical and numerical results for the coherent quark \nfragmentation let us remember that in case of incoherent single quark \nfragmentation Eq. (\\ref{rhoh}) becomes\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\prime}_h}(h) \n= {1\\over N_h} \\sum_{q,X,\\lambda^{\\,}_X,\\lambda^{\\,}_q,\\lambda^{\\prime}_q}\nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} \\>\\>\nD^*_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q} \\,,\n\\label{rhohp1}\n\\end{equation}\nwhere $\\rho(q)$ is the quark $q$ helicity density matrix related to\n$\\rho(q\\bar q)$ by\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} = \\sum_{\\lambda^{\\,}_{\\bar q}}\n\\rho^{\\,}_{\\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q}; \\lambda^{\\prime}_q,\n\\lambda^{\\,}_{\\bar q}} (q\\bar q) \\,.\n\\end{equation}\n\nIn such a case angular momentum conservation for the collinear quark \nfragmentation requires $\\lambda^{\\,}_q = \\lambda^{\\,}_h + \\lambda^{\\,}_X$;\nthe Standard Model computation of $\\rho(q)$ gives only diagonal\nterms [$\\rho^{\\,}_{++}(q) = \\rho^{\\,}_{+-;+-}(q\\bar q)$, \n$\\rho^{\\,}_{--}(q) = \\rho^{\\,}_{-+;-+}(q\\bar q)$], and one ends up\nwith the usual probabilistic expression\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_h}(h) \n= {1\\over N_h} \\sum_{q, \\lambda^{\\,}_q}\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_q} \\>\\>\nD_{q,\\lambda^{\\,}_q}^{h, \\lambda^{\\,}_h} \\,,\n\\label{rhohp2}\n\\end{equation}\nwhere $D_{q,\\lambda^{\\,}_q}^{h, \\lambda^{\\,}_h}$ is the polarized fragmentation \nfunction of a $q$ with helicity $\\lambda^{\\,}_q$ into a hadron $h$ with \nhelicity $\\lambda^{\\,}_h$. Off-diagonal elements of $\\rho(h)$ are all zero.\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $e^- e^+ \\to BX, \\> (S_{_B} = 1\/2, \\> p_{_T}\/\\sqrt s \\to 0)$}}\n\\vskip 6pt\n\\nobreak\nLet us consider first the case in which $h$ is a spin 1\/2 baryon. It was\nshown in Ref. \\cite{aamr} that in such a case the coherent quark \nfragmentation only induces small corrections to the usual incoherent \ndescription \n\\begin{eqnarray}\n\\rho^{\\,}_{++}(B) &=& {1 \\over N_{_B}} \\sum_q \n\\left[ \\rho^{\\,}_{+-;+-}(q\\bar q) \\> D_{q,+}^{B,+} + \\rho^{\\,}_{-+;-+}(q\\bar q) \\> \nD_{q,-}^{B,+} \\right] \\\\\n\\rho^{\\,}_{+-}(B) &=& {\\cal O} \\left[ \\left(\n{p_{_T} \\over z \\sqrt s} \\right) \\right] \\label{rho+-b}\\,.\n\\end{eqnarray}\n\nThat is, the diagonal elements of $\\rho(B)$ are the same as those given by \nthe usual probabilistic formula (\\ref{rhohp2}), with small corrections\nof the order of $(p_{_T}\/z\\sqrt s)^2$, while off-diagonal elements are \nof the order $(p_{_T}\/z\\sqrt s)$ and vanish in the $p_{_T}\/\\sqrt s \\to 0$\nlimit.\n\nThe matrix elements of $\\rho(B)$ are related to the longitudinal ($P_z$)\nand transverse ($P_y$) polarization of the baryon:\n\\begin{equation}\nP_z = 2\\rho_{++} - 1, \\quad\\quad\\quad\\quad P_y = -2\\,{\\rm Im} \\rho_{+-} \\,.\n\\end{equation}\nSome data are available on $\\Lambda$ polarization, both longitudinal and\ntransverse, from ALEPH Collaboration \\cite{aleph} and they do agree with the\nabove equations. In particular the transverse polarization, at \n$\\sqrt s = M_{_Z}$, $p_{_T} \\simeq 0.5$ GeV\/$c$ and $z \\simeq 0.5$ is \nindeed of the order 1\\%, as expected from Eq. (\\ref{rho+-b}).\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $e^- e^+ \\to VX, \\> (S_{_V} = 1, \\> p_{_T}\/\\sqrt s \\to 0)$}}\n\\vskip 6pt\n\\nobreak\nIn case of final spin 1 vector mesons one has, always in the limit of small\n$p_{_T}$ \\cite{akp}, \\cite{abmq} \n\\begin{eqnarray}\n\\rho^{\\,}_{00}(V) &=& {1 \\over N_{_V}} \\sum_q D_{q,+}^{V,0} \\\\\n\\rho^{\\,}_{11}(V) &=& {1 \\over N_{_V}} \\sum_q\n\\left[ \\rho_{+-;+-}(q\\bar q) D_{q,+}^{V,1} + \\rho_{-+;-+}(q\\bar q) D_{q,-}^{V,1}\n\\right] \\\\\n\\rho^{\\,}_{1,-1}(V) &=& {1 \\over N_{_V}} \\sum_{q,X}\nD^{\\,}_{10;+-} \\> D^*_{-10;-+} \\> \\rho^{\\,}_{+-;+-}(q\\bar q) \\,.\n\\end{eqnarray}\n\nAgain, the diagonal elements have the usual probabilistic expression; \nhowever, there is now an off-diagonal element, $\\rho^{\\,}_{1,-1}$, \nwhich may survive even in the $p_{_T}\/\\sqrt s \\to 0$ limit. In the sequel\nwe shall concentrate on it. Let us first notice that, in the collinear limit,\none has\n\\begin{eqnarray}\nD^{V,0}_{q,+} &=& \\sum_X \\vert D^{\\,}_{0-1;+-} \\vert^2 = D^{V,0}_{q,-} \\\\\nD^{V,1}_{q,+} &=& \\sum_X \\vert D^{\\,}_{10;+-} \\vert^2 = D^{V,-1}_{q,-} \\\\\nD^{V,1}_{q,-} &=& \\sum_X \\vert D^{\\,}_{12;+-} \\vert^2 = D^{V,-1}_{q,+} \\,,\n\\end{eqnarray}\nwith $ D_q^V = D^{V,0}_{q,+} + D^{V,1}_{q,+} + D^{V,-1}_{q,+}$ and \n$N_{_V} = \\sum_q D_q^V$. We also notice that the two fragmentation \namplitudes appearing in Eq. (30) are related by parity and their product\nis always real. $\\rho^{\\,}_{00}$ and $\\rho^{\\,}_{1,-1}$ can be measured\nthrough the angular distribution of two body decays of $V$. \n\nIn order to give numerical estimates of $\\rho^{\\,}_{1,-1}$ we make some\nplausible assumptions\n\\begin{equation}\nD^{h,1}_{q,-} = D^{h,-1}_{q,+} = 0 \\quad\\quad\\quad\nD^{h,0}_{q,+} = \\alpha^V_q \\> D^{h,1}_{q,+} \\label{ass} \\,.\n\\end{equation}\nThe first of these assumptions simply means that quarks with helicity\n1\/2 ($-1\/2$) cannot fragment into vector mesons with helicity $-1$ ($+1$).\nThis is true for valence quarks assuming vector meson wave functions \nwith no orbital angular momentum, like in $SU(6)$. The second assumption \nis also true in $SU(6)$ with $\\alpha^V_q = 1\/2$ for \nany valence $q$ and $V$. Rather than taking  \n$\\alpha^V_q = 1\/2$ we prefer to relate the value of $\\alpha^V_q$ to the \nvalue of $\\rho^{\\,}_{00}(V)$ which can be or has been measured.\nIn fact, always in the $p_{_T} \\to 0$ limit, one has \\cite{abmq}\n\\begin{equation}\n\\rho^{\\,}_{00}(V) = {\\sum_q \\alpha^V_q \\, D^{h,1}_{q,+}\n\\over \\sum_q \\> (1+\\alpha^V_q) \\, D^{h,1}_{q,+}} \\,\\cdot\n\\label{rho00}\n\\end{equation}\nIf $\\alpha^V_q$ is the same for all valence quarks in $V$ \n($\\alpha^V_q = \\alpha^V$) \none has, for the valence quark contribution:\n\\begin{equation}\n\\alpha^V = {\\rho^{\\,}_{00}(V) \\over 1 - \\rho^{\\,}_{00}(V)} \\,\\cdot\n\\label{alrho}\n\\end{equation}\n\nFinally, one obtains \\cite{abmq}\n\\begin{equation}\n\\rho^{\\,}_{1,-1}(V) \\simeq [1 - \\rho^{\\,}_{0,0}(V)] \\,\n{\\sum_q \\, D^{V,1}_{q,+} \\> \\rho_{+-;-+}(q\\bar q) \n\\over \\sum_q \\, D^{V,1}_{q,+}} \\,\\cdot\n\\label{rho1-1tss}\n\\end{equation}\n\nWe shall now consider some specific cases in which we expect \nEq. (\\ref{rho1-1tss}) to hold; let us remind once more that our \nconclusions apply to spin 1 vector mesons produced in \n$e^- e^+ \\to q \\bar q \\to V+X$ processes in the limit of small $p_{_T}$\nand large $z$, {\\it i.e.}, to vector mesons produced in two jet events\n($e^- e^+ \\to q\\bar q$) and collinear with one of them ($p_{_T} = 0$), \nwhich is the jet generated by a quark which is a valence quark for the\nobserved vector meson (large $z$). These conditions should be met \nmore easily in the production of heavy vector mesons. \n\nAmong other results one obtains \\cite{abmq}:\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^{*}) &\\simeq& [1 - \\rho^{\\,}_{0,0}(D^{*})] \\>\n\\rho_{+-;-+}(c \\bar c) \\label{rhoda} \\\\\n\\rho^{\\,}_{1,-1}(\\phi)  &\\simeq& [1 - \\rho^{\\,}_{0,0}(\\phi)] \\>\n\\rho_{+-;-+}(s\\bar s) \\label{rhopa} \\\\\n\\rho^{\\,}_{1,-1}(K^{*0}) &\\simeq& {1\\over 2} \\> [1 - \\rho^{\\,}_{0,0}(K^{*0})] \n\\> [\\rho_{+-;-+}(d\\bar d) + \\rho_{+-;-+}(s\\bar s)] \\,. \\label{rhok0a} \n\\end{eqnarray}\n\nEqs. (\\ref{rhoda})-(\\ref{rhok0a}) show how the value of $\\rho^{\\,}_{1,-1}(V)$\nare simply related to the off-diagonal helicity density matrix element \n$\\rho_{+-;-+}(q\\bar q)$ of the $q\\bar q$ pair created in the elementary \n$e^- e^+ \\to q\\bar q$ process; such off-diagonal elements would not appear \nin the incoherent independent fragmentation of a single quark, yielding \n$\\rho^{\\,}_{1,-1}(V)=0$.\n\nBy inserting into the above equations the value of $\\rho^{\\,}_{00}$ when\navailable \\cite{opal} and the expressions of $\\rho^{\\,}_{+-;-+}$,\nEq. (18), one has:\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^{*}) &\\simeq& -(0.216 \\pm 0.007) \\ \n{\\sin^2\\theta \\over 1 + \\cos^2\\theta} \\\\\n\\rho^{\\,}_{1,-1}(\\phi) &\\simeq& -(0.078 \\pm 0.014) \\ \n{\\sin^2\\theta \\over 1 + \\cos^2\\theta} \\\\\n\\rho^{\\,}_{1,-1}(K^{*0}) &\\simeq& -0.170 \\ \n[1- \\rho^{\\,}_{0,0}(K^{*0})] \\ {\\sin^2\\theta \\over 1 + \\cos^2\\theta}  \\,\\cdot\n\\end{eqnarray}\nFinally, in case one collects all meson produced at different angles in\nthe full available $\\theta$ range (say $\\alpha < \\theta < \\pi -\\alpha, \n\\> |\\cos\\theta| < \\cos\\alpha$) an average should be taken in $\\theta$, \nweighting the different values of $\\rho^{\\,}_{1,-1}(\\theta)$ with the \ncross-section for the $e^-e^+ \\to V+X$ process; this gives \\cite{abmq}:\n\\begin{eqnarray}\n\\langle \\rho^{\\,}_{1,-1}(D^{*}) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -(0.216 \\pm 0.007) \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\label{rhodn} \\\\\n\\langle \\rho^{\\,}_{1,-1}(\\phi) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -(0.078 \\pm 0.014) \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\label{rhopn} \\\\\n\\langle \\rho^{\\,}_{1,-1}(K^{*0}) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -0.170 \\ [1- \\rho^{\\,}_{0,0}(K^*)] \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\,\\cdot \\label{rhok0n} \n\\end{eqnarray}\n\nThese results have to be compared with data \\cite{opal}\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^*) &=& -0.039 \\pm 0.016 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.5 \\quad\\quad \\cos\\alpha = 0.9 \\\\ \n\\rho^{\\,}_{1,-1}(\\phi) &=& -0.110 \\pm 0.070 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.7 \\quad\\quad \\cos\\alpha = 0.9 \\\\ \n\\rho^{\\,}_{1,-1}(K^{*0}) &=& -0.090 \\pm 0.030 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.3 \\quad\\quad \\cos\\alpha = 0.9  \n\\end{eqnarray}\nwhich shows a good qualitative agreement with the theoretical\npredictions. We notice that while the mere fact that $\\rho_{1,-1}$ differs \nfrom zero is due to a coherent fragmentation of the $q\\bar q$ pair, the actual \nnumerical values depend on the Standard Model coupling constants; for example,\n$\\rho_{1,-1}$ would be positive at smaller energies, at which the one gamma\nexchange dominates, while it is negative at LEP energy where the one $Z$\nexchange dominates. $\\rho_{1,-1}$ has also a peculiar dependence on \nthe meson production angle, being small at small and large angles\nand maximum at $\\theta = \\pi\/2$. Such angular dependence has been tested\nin case of $K^{*0}$ production, assuming no dependence of $\\rho_{00}$\non $\\theta$, and indeed one has \\cite{opal}, in agreement \nwith Eqs. (\\ref{rhok0a}) and (\\ref{rho+-ap})\n\\begin{equation}\n\\left[ {\\rho^{\\,}_{1,-1} \\over 1- \\rho^{\\,}_{00}} \\right]_{|\\cos\\theta|<0.5} \n\\cdot \n\\left[ {\\rho^{\\,}_{1,-1} \\over 1- \\rho^{\\,}_{00}} \\right]^{-1}\n_{|\\cos\\theta|>0.5} = 1.5 \\pm 0.7\n\\end{equation}\n\nThese results are encouraging; it would be interesting to have more and \nmore detailed data, possibly with a selection of final hadrons with the \nrequired features for our results to hold. It would also be interesting to \ntest the coherent fragmentation of quarks in other processes, like\n$\\gamma\\gamma \\to VX$, $pp \\to D^*X$ and $\\ell p \\to VX$. The first two\nprocesses are similar to $e^-e^+ \\to VX$ in that a $q\\bar q$ pair is created \nwhich then fragments coherently into the observed vector meson; one assumes\nthat the dominating elementary process in $pp \\to D^*X$ is $gg \\to c \\bar c$.\nIn both these cases one has for $\\rho^{\\,}_{+-;-+}(q\\bar q)$ the same value\nas in Eq. (\\ref{rhoqqelm}), so that one expects a {\\it positive} value\nof $\\rho^{\\,}_{1,-1}(V)$. \n\nIn the last process, the production of vector mesons in DIS, the quark \nfragmentation is in general a more complicated interaction of the quark \nwith the remnants of the proton and it might be more difficult to obtain \nnumerical predictions. However, if one observes $D^*$ mesons one can \nassume or select kinematical regions for which the underlying elementary \ninteraction is $\\gamma^* g \\to c\\bar c$: again, one would have the same \n$\\rho^{\\,}_{+-;-+}(c\\bar c)$ as in Eq. (\\ref{rhoqqelm}), and one would \nexpect a positive value of $\\rho^{\\,}_{1,-1}(D^*)$. It would indeed be\ninteresting to perform these simple tests of coherent fragmentation \neffects.\n\n\\vskip 18pt\n\\noindent\n{\\bf Acknowledgements}\n\\vskip 6pt\n\\noindent\nI would like to thank the organizers of the Workshop and DESY for financial\nsupport\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip\n\nGiven an open bounded connected domain $\\Omega \\subset \\mathbb{R}^{N}$ with\na sufficiently regular (locally Lipschitz) boundary, $\\partial \\Omega $, let\nus consider the integra\n\\begin{equation}\nI\\left( u\\right) :=\\int_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right) \\,dx\n\\label{Functional}\n\\end{equation\nto be minimized on a class of Sobolev functions $u:\\Omega \\rightarrow \n\\mathbb{R}^{d}$ with a kind of boundary conditions to be described later.\nAll over the paper we assume the integrand $W:\\mathbb{R}^{d\\times\nN}\\rightarrow \\mathbb{R\\cup }\\left\\{ +\\infty \\right\\} $ to be \\emph\npolyconvex}. This means that the representatio\n\\begin{equation*}\nW\\left( \\xi \\right) =g\\left( \\mathbb{T}\\left( \\xi \\right) \\right) \n,\\;\\; \\xi \\in \\mathbb{R}^{d\\times N},\n\\end{equation*}\nholds for some convex function $g:\\mathbb{R}^{\\tau \\left( d,N\\right)\n}\\rightarrow \\mathbb{R\\cup }\\left\\{ +\\infty \\right\\} $\n\\begin{equation*}\n\\tau( d,N) :=\\underset{s=1}{\\overset{d\\wedge N}{\\dsum }}\n\\varkappa(s) ,\\;\\; \\varkappa(s) :=\\dbinom{\n}{d}\\dbinom{s}{N}=\\frac{d!N!}{(s!) ^{2}( d-s)!(\nN-s) !},\n\\end{equation*}\nwher\n\\begin{equation*}\n\\mathbb{T}\\left(\\xi \\right) :=\\left( \\mathrm{Adj}_{1}\\xi ,\\mathrm{Adj\n_{2}\\xi ,\\mathrm{Adj}_{3}\\xi ,\\dots ,\\mathrm{Adj}_{d\\wedge N}\\xi \\right) \n,\\;\\; \\xi \\in \\mathbb{R}^{d\\times N},\n\\end{equation*\nand $\\mathrm{Adj}_{k}\\xi $ is the vector of all \\emph{minors} of the matrix \n\\xi $ of order $k=1,2,\\dots ,d\\wedge N$, respectively. In particular, \n\\mathrm{Adj}_{1}\\xi =\\xi $ and $\\mathrm{Adj}_{d}\\xi =\\det \\,\\xi $ whenever \nd=N$.\n\nIt is known that, under strong coercivity assumptions on $W$ to assure weak convergence of the minors of gradients for the minimizing sequence, the functional $I$ attains its minimum on $\\bar{u}\\left( \\cdot \\right) +\\mathbf{\n}_{0}^{1,p}\\left( \\Omega ;\\mathbb{R}^{d}\\right) $, $p\\geq 1$. We refer to the fundamental work by J. Ball \\cite{B} motivated by problems coming from nonlinear elasticity and to \\cite{ADM, MQY, My} for further improvements.\n\n\nThe lower semicontinuity for general polyconvex integrands with respect to the weak convergence in $W^{1,p}(\\Omega;\\mathbb{R}^{d}),$       $\\Omega\\subset \\mathbb{R}^{N},$ has been the subject of many investigations.\nNamely, Marcellini showed in \\cite{M} that this property holds whenever $p<N$. Later, this result was improved by Dacorogna and Marcellini in \\cite{DM} who proved the lower semicontinuity for $p>N-1$ while Mal\\'{y} in \\cite{My} exhibited a counterexample for $p<N-1$. The limit case $p=N-1$ was adressed in \\cite{ADM, DMS, CDM, FH}. Very recently (see \\cite{FFLMMV} and \\cite{DPMF})  the limit case was studied for polyconvex integrands depending on $x$ and\/or on $u$.\n\nBesides the Dirichlet boundary condition $u\n\\bar{u}$ for the displacement one considered, for instance, boundary condition on traction, which somehow depends on the normal\nderivatives of $u\\left( \\cdot \\right) $ (generalizing the Neumann boundary\ndata). Observe that these conditions (displacement, traction or a mixed one)\ncan be applied either to the whole boundary $\\partial \\Omega $, or to some\nof its subsets of positive Hausdorff measure leaving the rest free.\nMoreover, some restrictions on the \\emph{jacobian} may be relevant and\npractically justified. For example, the constraint $\\det \\,\\nabla u\\left(\nx\\right) >0$ means that the minimum is searched among the deformations\npreserving orientation, while $\\det \\,\\nabla u\\left( x\\right) =1$ refers to\nthe case of incompressible elastic body.\n\nOne of the possible applications of the above variational problem is\nregarded to plastic surgery, namely, in the woman breast reduction, where we\ndeal with a sort of very elastic and soft tissue. Some recent publications\n(see, e.g., \\cite{Ay, D, S1, PY, S2}) were devoted to mathematical setting\nof the related problems and to their numerical simulations. Medical\nexaminations allow to consider the involved tissue as a neo-Hookean\ncompressible material (see \\cite{Z}). We have a more precise model when the strain energy is  defined\nby the integral (\\ref{Functional}) with the density $W:\\mathbb{R}^{3\\times\n3}\\rightarrow \\mathbb{R}$\n\\begin{eqnarray}\n&&W\\left( \\xi \\right) :=\\mu \\left( \\limfunc{tr}\\left( \\xi \\cdot \\xi\n^{T}\\right) -3-2\\ln \\left( \\det \\xi \\right) \\right)   \\notag \\\\\n&&+\\lambda \\left( \\det \\xi -1\\right) ^{2}+\\beta \\limfunc{tr}\\mathrm{\\,\n\\left( \\mathrm{Adj\\,}\\xi \\cdot \\mathrm{Adj\\,}\\xi ^{T}\\right) \\,,\n\\label{Int_ex}\n\\end{eqnarray\nwhere \"$\\limfunc{tr}$\" means the trace of a matrix, $\\mathrm{Adj\\,}\\xi :\n\\mathrm{Adj}_{2}\\xi $, and the symbol \"$T$\" stands for the matrix\ntransposition.  One of the steps of the\n(breast reduction) surgery is the suturing, which mathematically can\nbe seen as an identification of points of some surface piece $\\Gamma\n^{+}\\subset \\partial \\Omega $ with points of another one $\\Gamma ^{-}\\subset\n\\partial \\Omega $. Denoting the respective correspondence between the points\nof $\\Gamma ^{+}$ and $\\Gamma ^{-}$ by $\\sigma $, we are led to a new type of constrain\n\\begin{equation}\nu(x) =(u\\circ \\sigma)(x), \\;x\\in\n\\Gamma ^{+},  \\label{sutur_cond}\n\\end{equation\ncalled the \\emph{knitting boundary condition}. Let us note that the one-to-one\nmapping $\\sigma $ is not \\emph{a priori} given and should be chosen to\nguarantee the minimum value to the functional (\\ref{Functional}). In other\nwords, a minimizer of (\\ref{Functional}) (if any) should be a pair $\\left(\nu,\\sigma \\right) $ where $u\\in \\mathbf{W}^{1,p}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $, $p\\geq 1$, and $\\sigma :\\Gamma ^{+}\\rightarrow \\Gamma ^{-}$\nis sufficiently regular. We set the natural hypothesis that $\\sigma $ and\nits inverse $\\sigma ^{-1}$ are Lipschitz transformations (with the same\nLipschitz constant $L>0$). Practically this means that the sutured tissue\ncan not be extended nor compressed too much.\n\n\n\nMotivated by the problem coming from the plastic surgery we will consider\njust the case $p=2$ and $d=N=3$, although the results remain\ntrue in the case $p>2$ and arbitrary $d=N\\geq 2$ as well.\n\nThe paper is organized as follows. In the next section we give the exact\nsetting of the variational problem together with the main hypotheses on the\nintegrand $W$. For simplicity of references we put here also some important\nfacts regarded with the Sobolev functions. In Section 3 we justify first the\nwell-posedness of the problem by showing that the composed function from the\nknitting condition (\\ref{sutur_cond}) belongs to the respective Lebesgue\nclass. Afterwards, we prove existence of a minimizer as an accumulation\npoint of an arbitrary minimizing sequence (the so called \\emph{direct method\n, see \\cite{Dac}). The paper is concluded with a necessary optimality\ncondition for the given problem (see Section 4) allowing to construct\neffective numerical algorithms, which can be successively applied in the\nmedical practice.\n\n\\bigskip\n\n\\section{\\protect\\bigskip Main hypotheses and auxiliary results}\n\nIn what follows we fix a nonempty open bounded and connected set $\\Omega\n\\subset \\mathbb{R}^{3}$ whose boundary $\\partial \\Omega $ is assumed to be\nlocally Lipschitz (see, e.g., \\cite[p. 354]{L}). By the symbol $\\mathcal{L\n^{m}$ ($dx$) we denote the Lebesgue measure in the space $\\mathbb{R}^{m}$, \nm=2,3$, while $\\mathcal{H}^{2}$ means the two-dimensional \\emph{Hausdorff\nmeasure} (see, e.g., \\cite{F}).\n\nLet us divide the surface $\\partial \\Omega $ into several parts $\\Gamma _{i}\n, $i=1,2,3,4$, in such a way that $\\mathcal{H}^{2}\\left( \\Gamma _{i}\\cap\n\\Gamma _{j}\\right) =0$ for $i\\neq j$. Moreover, we set $\\Gamma _{4}:=\\Gamma\n^{+}\\cup \\Gamma ^{-}$ where $\\Gamma ^{\\pm }\\subset \\Gamma $ with $\\mathcal{H\n^{2}\\left( \\Gamma ^{\\pm }\\right) >0$ and $\\mathcal{H}^{2}\\left( \\Gamma\n^{+}\\cap \\Gamma ^{-}\\right) =0$ are also given.\n\nSuppose that $W:\\mathbb{R}^{3\\times 3}\\rightarrow \\mathbb{R}$ is a \\emph\npolyconvex function} satisfying the \\emph{growth\\ assumption}\n\\begin{equation}\nW(\\xi) \\geq c_{0}+c_{1}\\left\\vert \\xi \\right\\vert\n^{2}+c_{2}\\left\\vert \\mathrm{Adj\\,}\\xi \\right\\vert ^{2}+c_{3}\\left(\\det \\xi\n\\right)^{2}, \\;\\; \\xi \\in \\mathbb{R}^{3\\times 3},\n\\label{growth_cond}\n\\end{equation\nwhere $c_{0}\\in \\mathbb{R}$ and $c_{i}>0$, $i=1,2,3$, are some given\nconstants. Here and in what follows by $\\left\\vert \\cdot \\right\\vert $ we\ndenote the norm of both a vector in $\\mathbb{R}^{n}$ and a $3\\times 3\n-matrix.\n\nTaking into account that $\\limfunc{tr}\\left( \\xi \\cdot \\xi ^{T}\\right)\n=\\left\\vert \\xi \\right\\vert ^{2}$ for each matrix $\\xi \\in \\mathbb{R\n^{3\\times 3}$, we see that the integrand (\\ref{Int_ex}) satisfies the above\nproperties. Indeed, it is convex as a function of $\\mathbb{T}\\left( \\xi\n\\right) $ being represented as a sum of three terms, which are convex w.r.t. \n$\\xi $, $\\det \\xi $ and $\\mathrm{Adj\\,}\\xi $, respectively. Furthermore\n\\begin{equation*}\nW(\\xi) =-3\\mu +\\lambda +\\mu \\left\\vert \\xi \\right\\vert\n^{2}+\\beta \\left\\vert \\mathrm{Adj\\,}\\xi \\right\\vert ^{2}+f( \\det \\xi), \\;\\xi \\in \\mathbb{R}^{3\\times 3},\n\\end{equation*\nwhere the functio\n\\begin{equation*}\nf\\left( t\\right) :=\\frac{\\lambda }{2}t^{2}-2\\lambda t-2\\mu \\ln t, \\;t>\n,\n\\end{equation*\nis lower bounded by some (negative) constant.\n\nSince on various pieces of the surface $\\partial \\Omega $ the boundary conditions\nare structurally different (some part of $\\partial \\Omega $ can be left even\nfree), to set the problem we use the notion of the \\emph{trace operator},\nwhich associates to each $u\\in $ $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ a function $\\limfunc{Tr}u$ defined on the boundary, $\\partial\n\\Omega,$ which can be interpreted as the \"\\emph{boundary values}\" of $u$. We refer to \\cite[pp. 465-474]{L}, where\nthe existence and uniqueness of the trace operator were proved for scalar Sobolev functions $u \\in \\mathbf{W}^{1,p}(\\Omega),   p>1.$ For vector-valued functions $u: \\Omega\\rightarrow \\mathbb{R}^{3}$ instead, we can argue componentwise. So, applying \\cite[Theorem 15.23]{L}, we define the trace as the linear and bounded operator\n$\\limfunc{Tr}:\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) \\rightarrow \\mathbf{L}^{2}(\\partial \\Omega ;\\mathbb{R\n^{3})$ satisfying the following properties:\n\n\\begin{enumerate}\n\\item $\\limfunc{Tr}u\\left( x\\right) =u\\left( x\\right) $, $x\\in \\partial\n\\Omega $, whenever $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) \\cap \\mathbf{C}\\left( \\overline{\\Omega };\\mathbb{R}^{3}\\right) $;\n\n\\item for each $u\\in \\mathbf{W}^{1,2}(\\Omega;\\mathbb{R}^3)$\nand any test function $\\varphi \\in \\mathbf{C}^{1}(\\overline{\\Omega };\\mathbb{R}^{3})$ the equalitie\n\\begin{equation*}\n\\int_{\\Omega}u_{j}\\frac{\\partial \\varphi_{j}}{\\partial x_{i}}dx=-\\int_{\\Omega}\\varphi_{j}\\frac{\\partial u_{j}}{\\partial x_{i}}dx+\\int_{\\partial \\Omega }\\varphi_{j}\n\\limfunc{Tr}(u_j)\\nu_i d\\mathcal{H}^{2}\n\\end{equation*\nhold for each $ i, j=1,2,3$ where $\\nu:=(\\nu_1,\\nu_2,\\nu_3)^{T}$ means the unit outward normal to \n\\partial \\Omega $.\n\\end{enumerate}\n \nIn addition to the properties above, observe that the trace operator gives a compact embedding into the space $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R}^{3}\\right)$ that will be crucial to obtain the main result in Section 3. Namely, the following proposition takes place.\n\n\\begin{proposition}\n\\label{traceconv}Let $\\Omega \\subset \\mathbb{R}^{3}$ be an open bounded set\nwith locally Lipschitz boundary. Then for each $\\left\\{ u_{n}\\right\\}\n\\subset \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ converging to \n$u$ weakly in $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ the\nsequence of traces $\\left\\{ \\limfunc{Tr}u_{n}\\right\\} $ converges to \n\\limfunc{Tr}u$ strongly in $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R\n^{3}\\right)$.\n\\end{proposition}\n\nThe proof is based essentially on the following lemma giving a nice estimate\nfor the surface integral of the trace operator.\n\n\\begin{lemma}\n\\label{Lemmatrace}Let $\\Omega \\subset \\mathbb{R}^{3}$ be as in Proposition \\ref{traceconv}. Then there exists a constant\n$C>0$ such that \n\\begin{equation}\n\\int_{\\partial \\Omega }\\left\\vert \\limfunc{Tr}u\\right\\vert ^{2}d\\mathcal{H\n^{2}\\leq C\\left( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\left\\vert u\\right\\vert\n^{2}dx+\\varepsilon \\int_{\\Omega }\\left\\vert \\nabla u\\right\\vert\n^{2}dx\\right)   \\label{Estimate_tr}\n\\end{equation\nfor any $\\varepsilon >0$ and any $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$\n\\end{lemma}\n\n\\begin{proof}\nGiven $x\\in \\partial \\Omega ,$ due to the Lipschitz hypothesis there exists\na neighborhood $U_{x}$ of $x$ such that $U_{x}\\cap \\partial \\Omega $ can be\nrepresented as the graph of a Lipschitz function w.r.t. some (local)\ncoordinates. Without loss of generality, we may assume that \n\\begin{equation*}\nU_{x}\\cap \\Omega =\\left\\{ \\left( y^{\\prime },y_{3}\\right) :f_{x}\\left(\ny^{\\prime }\\right) <y_{3}\\leq f_{x}\\left( y^{\\prime }\\right) +\\delta\n_{x},~y^{\\prime }\\in G_{x}\\right\\} \n\\end{equation*\nan\n\\begin{equation*}\nU_{x}\\cap \\partial \\Omega =\\left\\{ \\left( y^{\\prime },f_{x}\\left( y^{\\prime\n}\\right) \\right) :y^{\\prime }\\in G_{x}\\right\\} ,\n\\end{equation*\nwhere $\\delta _{x}>0,$ $G_{x}\\subset \\mathbb{R}^{2}$ is an open set in the\nspace of the first two coordinates and $f_{x}:G_{x}\\rightarrow \\mathbb{R}$\nis a Lipschitz function. By compactness there exists a finite number of\npoints $x^{i}\\in \\partial \\Omega \\,,~i=1,\\dots ,q,$ such that \n\\begin{equation*}\n\\partial \\Omega =\\tbigcup_{i=1}^{q}\\left( U_{x^{i}}\\cap \\partial \\Omega\n\\right) .\n\\end{equation*\nSet $\\delta _{i}:=\\delta _{x^{i}},~U_{i}:=U_{x^{i}},~G_{i}:=G_{x^{i}}$ and~ \nf_{i}:=f_{x^{i}}$, $i=1,\\dots ,q$. Denote by $L>0$ the biggest Lipschitz\nconstant of the functions $f_{i}.$\n\nLet us choose $\\varepsilon <\\min \\left\\{ \\delta _{i}:i=1,\\dots ,q\\right\\} $\nand consider first the function $u\\in \\mathbf{C}^{1}\\left( \\overline{\\Omega \n\\right) \\cap \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$ Given \ni\\in \\left\\{ 1,\\dots ,q\\right\\} ,$ by the Newton-Leibniz formula, for each \nx^{\\prime }\\in G_{i}$ and $\\ x_{3}\\in \\mathbb{R}$ with $f_{i}\\left(\nx^{\\prime }\\right) \\leq x_{3}<f_{i}\\left( x^{\\prime }\\right) +\\varepsilon $\nwe have \n\\begin{equation*}\nu\\left( x^{\\prime },f_{i}\\left( x^{\\prime }\\right) \\right) =u\\left(\nx^{\\prime },x_{3}\\right) -\\int_{f_{i}\\left( x^{\\prime }\\right) }^{x_{3}\n\\frac{\\partial u}{\\partial x_{3}}\\left( x^{\\prime },s\\right) ~ds.\n\\end{equation*\nIn turn, by Cauchy-Schwartz inequality\n\\begin{eqnarray*}\n\\left\\vert u\\left( x^{\\prime },f_{i}\\left( x^{\\prime }\\right) \\right)\n\\right\\vert ^{2} &\\leq &2\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right)\n\\right\\vert ^{2}+\\left( \\int_{f_{i}\\left( x^{\\prime }\\right) }^{f_{i}\\left(\nx^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}{\\partial x_{3}\n\\left( x^{\\prime },s\\right) \\right\\vert ds\\right) ^{2}\\right)  \\\\\n&\\leq &2\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right) \\right\\vert\n^{2}+\\varepsilon \\int_{f_{i}\\left( x^{\\prime }\\right) }^{f_{i}\\left(\nx^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}{\\partial x_{3}\n\\left( x^{\\prime },s\\right) \\right\\vert ^{2}ds\\right) .\n\\end{eqnarray*\nHence,\n\n\\begin{eqnarray*}\n&&\\left\\vert u\\left( x^{\\prime },f_{i}(x^{\\prime })\\right) \\right\\vert ^{2}\\sqrt\n1+\\left\\vert \\nabla f_{i}\\left( x^{\\prime }\\right) \\right\\vert ^{2}} \\\\\n&\\leq &2\\sqrt{1+L^{2}}\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right)\n\\right\\vert ^{2}+\\varepsilon \\int_{f_{i}\\left( x^{\\prime }\\right)\n}^{f_{i}\\left( x^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}\n\\partial x_{3}}\\left( x^{\\prime },s\\right) \\right\\vert ^{2}ds\\right) .\n\\end{eqnarray*\nIntegrating both parts of the previous inequality in the cylinder \n\\begin{equation*}\n\\Omega _{i}:=\\left\\{ \\left( x^{\\prime },x_{3}\\right) :f\\left( x^{\\prime\n}\\right) \\leq x_{3}\\leq f\\left( x^{\\prime }\\right) +\\varepsilon ,~x^{\\prime\n}\\in G_{i}\\right\\} \\subset U_{i}\\cap \\Omega ,\n\\end{equation*\ngive\n\\begin{eqnarray*}\n\\varepsilon \\int_{U_{i}\\cap \\partial \\Omega }\\left\\vert u\\left( x\\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  &\\leq &2\\sqrt{1+L^{2}\n\\left( \\int_{\\Omega _{i}}\\left\\vert u\\left( x\\right) \\right\\vert\n^{2}dx+\\varepsilon ^{2}\\int_{\\Omega _{i}}\\left\\vert \\frac{\\partial u}\n\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right)  \\\\\n&\\leq &2\\sqrt{1+L^{2}}\\left( \\int_{\\Omega }\\left\\vert u\\left( x\\right)\n\\right\\vert ^{2}dx+\\varepsilon ^{2}\\int_{\\Omega }\\left\\vert \\frac{\\partial \n}{\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right) .\n\\end{eqnarray*}\n\nSumming in $i=1,\\dots ,q$ we hav\n\\begin{eqnarray}\n&&\\int_{\\partial \\Omega }\\left\\vert u\\left( x\\right) \\right\\vert ^{2}\n\\mathcal{H}^{2}\\left( x\\right)   \\label{traceu} \\\\\n&\\leq &2q\\sqrt{1+L^{2}}\\left( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\left\\vert\nu\\left( x\\right) \\right\\vert ^{2}dx+\\varepsilon \\int_{\\Omega }\\left\\vert \n\\frac{\\partial u}{\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right) .\n\\notag\n\\end{eqnarray\nThe inequality (\\ref{Estimate_tr}) for each $u \\in \\mathbf{W}^{1,2}(\\Omega ;\\mathbb{R}^{3})$ follows now from (\\ref{traceu}) by the\nproperties of the trace operator and by the density of smooth functions.\n\\end{proof}\n\n\\medskip\n\n\\begin{proof}[Proof of Proposition 1]\nGiven a sequence $\\left\\{ u_{n}\\right\\} \\subset \\mathbf{W}^{1,2}\\left(\n\\Omega ;\\mathbb{R}^{3}\\right) $ converging weakly to $u$ in $\\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $, there exists $M>0$ with \n\\begin{equation*}\n\\int_{\\Omega }\\left\\vert \\nabla u_{n}\\left( x\\right) \\right\\vert ^{2}dx\\leq M\n\\end{equation*\nfor all $n\\in \\mathbb{N}.$ Then, by the \\emph{Rellich-Kondrachov theorem}\n(see \\cite[Theorem 11.21, p. 326]{L}), $u_{n}\\rightarrow u$ $\\ $strongly$\\ \nin$~\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$ Applying Lemma \\re\n{Lemmatrace} to $\\limfunc{Tr}\\left( u_{n}-u\\right) =\\limfunc{Tr}u_{n}\n\\limfunc{Tr}u,$ we hav\n\\begin{align*}\n\\int_{\\partial \\Omega }\\vert \\limfunc{Tr}u_{n}-\\limfunc{Tr}\nu\\vert ^{2}d\\mathcal{H}^{2}&\\leq C(\\frac{1}{\\varepsilon }\\int_{\\Omega }\\vert\nu_{n}-u\\vert ^{2}dx+\\varepsilon \\int_{\\Omega }\\vert \\nabla\nu_{n}-\\nabla u\\vert ^{2}dx)\\\\\n&\\leq C( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\vert\nu_{n}-u\\vert ^{2}dx+4\\varepsilon M) .\n\\end{align*}\nHenc\n\\begin{equation*}\n\\underset{n\\rightarrow \\infty }{\\lim \\sup }\\int_{\\partial \\Omega }\\left\\vert \n\\limfunc{Tr}u_{n}-\\limfunc{Tr}u\\right\\vert ^{2}d\\mathcal{H}^{2}\\leq\n4\\varepsilon CM.\n\\end{equation*\nLetting $\\varepsilon \\rightarrow 0^{+}$ concludes the proof.\\medskip \n\\end{proof}\n\nWe will use also the so called \\emph{generalized\nPoincar\\'{e} inequality} (see \\cite[Theorem 6.1-8, p. 281]{C}).\n\n\\begin{proposition}\n\\label{poincare}Given an open bounded domain $\\Omega \\subset \\mathbb{R}^{3}$\nwith locally Lipschitz boundary and a measurable subset $\\Gamma \\subset\n\\partial \\Omega $ with $\\mathcal{H}^{2}\\left( \\Gamma \\right) >0$ there\nexists a constant $C>0$ such that \n\\begin{equation}\n\\underset{\\Omega }{\\dint }\\left\\vert u\\left( x\\right) \\right\\vert ^{2}dx\\leq\nC\\left[ \\underset{\\Omega }{\\dint }\\left\\vert \\nabla u\\left( x\\right)\n\\right\\vert ^{2}\\,dx+\\left\\vert \\underset{\\Gamma }{\\dint }\\limfunc{Tr\nu\\left( x\\right) \\,d\\mathcal{H}^{2}\\left( x\\right) \\right\\vert ^{2}\\right] \n\\label{poincare_ineq}\n\\end{equation\nfor each $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $.\n\\end{proposition}\n\n\\medskip\nLet us formulate now the boundary conditions in terms of the trace operator.\nConsider first a surface $\\mathcal{S}\\subset \\mathbb{R}^{3}$ defined by some\ncontinuous function $h:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}$\n\\begin{equation*}\n\\mathcal{S}:=\\left\\{ u\\in \\mathbb{R}^{3}:h\\left( u\\right) =0\\right\\} .\n\\end{equation*}\nThen, given $L\\geq 1$ we denote by $\\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma\n^{-}\\right) $ the set of all functions $\\sigma :\\Gamma ^{+}\\rightarrow\n\\Gamma ^{-}$ satisfying the inequalitie\n\\begin{equation}\n\\frac{1}{L}\\left\\vert x-y\\right\\vert \\leq \\left\\vert \\sigma \\left( x\\right)\n-\\sigma \\left( y\\right) \\right\\vert \\leq L\\left\\vert x-y\\right\\vert \n\\label{Lipschitz}\n\\end{equation\nfor all $x,y\\in \\Gamma ^{+}$, and introduce the set $\\mathcal{W}_{L}\\subset \n\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) \\times \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ of all pairs $\\left( u,\\sigma\n\\right) $ satisfying the (boundary) conditions:\n\n\\begin{enumerate}\n\\item[(C$_{1}$)] $\\limfunc{Tr}u\\left( x\\right) =x$ for $\\mathcal{H}^{2}\n-a.e. $x\\in \\Gamma _{1}$;\n\n\\item[(C$_{2}$)] $h\\left( \\limfunc{Tr}u\\left( x\\right) \\right) =0$ for \n\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma _{2}$;\n\n\\item[(C$_{3}$)] $\\limfunc{Tr}u\\left( x\\right) =\\limfunc{Tr}u\\left( \\sigma\n\\left( x\\right) \\right) $ for $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$.\n\\end{enumerate}\nThus, we can write the \\emph{knitting variational problem} in the for\n\\begin{equation}\n\\min \\left\\{ \\int_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right)dx:\\left( u,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\}.  \\label{varproblem}\n\\end{equation\n\n\\begin{figure}[htb]\t\n\\begin{center}\n\\begin{tikzpicture}\n\\node[label=below:~]  (x1) at (2,2)  {$\\bullet$};\n\\node[label=below:~ ]  (x2) at (2,0)  {$\\bullet$};\n\\node[label=below:~ ]  (x3) at (0,0)  {$\\bullet$};\n\\node[label=below:~ ]  (x4) at (1,1)  {$\\bullet$};\n  \\node[label=below:~ ]  (x5) at (0,1)  {$\\bullet$};\n\n  \\draw (x1) .. controls (3,1)  .. (x2);\n \\draw (x2) .. controls (1,-0.3)  .. (x3);\n \\draw (x3) -- (x4);\n \\draw (x4) -- (x5);\n  \\draw (x5) .. controls (1,3)  .. (x1);\n  \n  \n\\node[label=below:~]  (z1) at (8,2)  {$\\bullet$};\n\\node[label=below:~ ]  (z2) at (8,0)  {$\\bullet$};\n\\node[label=below:~ ]  (z3) at (6,0)  {$\\bullet$};\n\\node[label=below:~ ]  (z4) at (7,1)  {$\\bullet$};\n\n\n  \\draw (z1) .. controls (9,1)  .. (z2);\n \\draw (z2) -- (z3);\n \\draw[very thick] (z3) -- (z4);\n\n  \\draw (z3) .. controls (6,2)  .. (z1);  \n  \n\\draw[->] (3.5,1) -- (5.3,1);  \n  \n\\node [above] at (4.4,1.2) {$u$};\n\n\\node [above] at (3,1.2) {$\\Gamma_1$};\n\\node [above] at (10,1.2) {$\\Gamma_1=u(\\Gamma_1)$};\n\n\\node [below] at (1,-0.3) {$\\Gamma_2$};\n\\node [below] at (7.5,-0.3) {$u(\\Gamma_2),\\;\\; h(x)=0$};\n\n\\node [above] at (0.2,1.8) {$\\Gamma_3$};\n\\node [above] at (6.2,1.9) {$u(\\Gamma_3)$};\n\n\\node [above] at (0.8,1.1) {$\\Gamma_+$};\n\n\n\\node [below] at (0.8,0.6) {$\\Gamma_-$};\n\\node [below] at (7,0.65) {$u(\\Gamma_{\\pm})$};\n\n\\draw[->] (0.2,0.9) -- (0.3,0.4);  \n\\node [below] at (0,0.8) {$\\sigma$};\n\n\\node [below] at (2,1.3) {$\\Omega$};\n\\node [below] at (8,1.3) {$u ({\\Omega})$};\n  \n\\end{tikzpicture}\n\\end{center}\n\\caption{General scheme of plastic surgery.}\n\\end{figure}\n\nLet us emphasize the difference between the boundary conditions (C$_{1}$)$-\n(C$_{3}$) above (see Fig. 1). The first condition (C$_{1}$) means that the surface \n\\Gamma _{1}$ is a part of the elastic body (breast) that remains fixed. The condition (C$_{2}$),\ninstead, can be interpreted as the knitting of a part of the incised  breast \n(surface $\\Gamma _{2}$) to the fixed surface $\\mathcal{S}$ of the woman's\nchest, while (C$_{3}$) means the knitting of the cut breast surface $\\Gamma\n_{4}:=\\Gamma ^{+}\\cup \\Gamma ^{-},$ of $\\Gamma ^{+}$ into $\\Gamma ^{-}$.\nFinally, the piece $\\Gamma _{3}$ of the boundary $\\partial \\Omega $ remains\nfree and admits an arbitrary configuration depending on the knitting process.\n\n\\bigskip\n\n\\section{\\protect\\bigskip Existence of minimizers}\n\nBefore proving the main existence theorem let us justify that for each \n\\sigma \\in \\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ the boundary\ncondition (C$_{3}$) makes sense.\n\n\\begin{lemma}\n\\label{measurability_comp}Let $\\Omega \\subset \\mathbb{R}^{3}$ be an open\nbounded connected domain with locally Lipschitz boundary and $\\Gamma ^{\\pm\n}\\subset \\partial \\Omega $ be $\\mathcal{H}^{2}$-measurable sets with \n\\mathcal{H}^{2}\\left( \\Gamma ^{\\pm }\\right) >0$ and $\\mathcal{H}^{2}\\left(\n\\Gamma ^{+}\\cap \\Gamma ^{-}\\right) =0$. Then for each $u\\in \\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ and $\\sigma \\in \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $, $L\\geq 1$, the composed\nfunction $\\limfunc{Tr}u\\circ \\sigma :\\Gamma ^{+}\\rightarrow \\mathbb{R}^{3}$\nis measurable w.r.t. the measure $\\mathcal{H}^{2}$ on $\\Gamma ^{+}$.\n\\end{lemma}\n\n\\begin{proof}\nGiven $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ by the\ndensity argument there exists a sequence of continuous functions $v_{n}\n\\overline{\\Omega }\\rightarrow \\mathbb{R}^{3}$ converging to $u$ in the space \n$\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $. Consequently (see the properties $1$ and $2$ of traces), \nv_{n}=\\limfunc{Tr}v_{n}\\rightarrow \\limfunc{Tr}u$, as $\\ n\\rightarrow \\infty $,\nin $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R}^{3}\\right) $. Then, up\nto a subsequence, we hav\n\\begin{equation}\nv_{n}\\left( y\\right) \\rightarrow \\limfunc{Tr}u\\left( y\\right) \\;\\;\n\\forall y\\in \\Gamma ^{-}\\setminus E_{0}^{-}  \\label{appr_cont_a.e.}\n\\end{equation\nwhere $E_{0}^{-}\\subset \\Gamma ^{-}$ is some set with null Hausdorff\nmeasure. So, it remains to prove tha\n\\begin{equation}\n\\mathcal{H}^{2}\\left( \\sigma ^{-1}\\left( E_{0}^{-}\\right) \\right) =0\n\\label{measure_0}\n\\end{equation\n$\\,$because in such case we deduce from (\\ref{appr_cont_a.e.}) that the\nsequence of (continuous) functions $\\left\\{ v_{n}\\left( \\sigma \\left(\nx\\right) \\right) \\right\\} $ converges to $\\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) $ for all $x\\in \\Gamma ^{+}$ up to the negligible set of\npoints $E_{0}^{+}:=\\sigma ^{-1}\\left( E_{0}^{-}\\right) $.\n\nOn the other hand, (\\ref{measure_0}) follows easily from the left inequality\nin (\\ref{Lipschitz}) and from the definition of \\emph{Hausdorff measure}\n(see \\cite[p. 171]{F})\n\\begin{equation*}\n\\mathcal{H}^{2}\\left( E\\right) :=\\frac{\\pi }{4}\\lim_{\\varepsilon \\rightarrow\n0}\\,\\inf \\,\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}A_{i}\\right) ^{2}\n\\end{equation*\nwhere the infimum is taken over all coverings $\\left\\{ A_{i}\\right\\}\n_{i=1}^{\\infty }$ of $E$ with the diameters $\\limfunc{diam}A_{i}\\leq\n\\varepsilon $. In fact, given $\\eta >0$ we find $\\varepsilon >0$ and a\nfamily $\\left\\{ A_{i}\\right\\} _{i=1}^{\\infty }$ with $\\tbigcup_{i=1}^{\\infty\n}A_{i}\\supset E_{0}^{-}$ , $\\limfunc{diam}A_{i}\\leq \\varepsilon \/L$ an\n\\begin{equation*}\n\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}A_{i}\\right) ^{2}\\leq \\frac{\\eta }\nL^{2}}.\n\\end{equation*\nSince due to (\\ref{Lipschitz}) obviously $\\limfunc{diam}\\,\\left( \\sigma\n^{-1}\\left( A_{i}\\right) \\right) \\leq L\\limfunc{diam}\\,A_{i}\\leq \\varepsilon \n$, $i=1,2,\\dots $; \n\\\\the family $\\left\\{ \\sigma ^{-1}\\left( A_{i}\\right) \\right\\}\n_{i=1}^{\\infty }$ is a covering of $E_{0}^{+}$ an\n\\begin{equation*}\n\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}\\,\\left( \\sigma ^{-1}\\left(\nA_{i}\\right) \\right) \\right) ^{2}\\leq L^{2}\\sum_{i=1}^{\\infty }\\left( \n\\limfunc{diam}A_{i}\\right) ^{2}\\leq \\eta ,\n\\end{equation*\nwe conclude that $\\mathcal{H}^{2}\\left( E_{0}^{+}\\right) =0$, and the \n\\mathcal{H}^{2}$-measurability of $x\\mapsto \\limfunc{Tr}u\\left( \\sigma\n\\left( x\\right) \\right) $ on $\\Gamma ^{+}$ follows.\n\\end{proof}\n\nLet us give now an \\emph{a priori} estimate for the \"weighted\" integra\n\\begin{equation*}\n\\dint\\limits_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) ,\n\\end{equation*\nimplying, in particular, that the composed function $\\limfunc{Tr}u\\circ\n\\sigma $ belongs to the class $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb\nR}^{3}\\right) $.\n\n\\begin{lemma}\n\\label{estimate_w}Let $\\Omega \\subset \\mathbb{R}^{3}$ and $\\Gamma ^{\\pm\n}\\subset \\partial \\Omega $ be such as in Lemma \\ref{measurability_comp}. Then\ngiven $L\\geq 1$ there exists a constant $\\mathfrak{L}_{L}>0$ such that the inequalit\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathfrak{L}_{L\n\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr}u\\left( y\\right) \\right\\vert ^{2}\n\\mathcal{H}^{2}\\left( y\\right)   \\label{estimate_traces}\n\\end{equation\nholds whenever $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $\nand $\\sigma \\in \\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $.\n\\end{lemma}\n\n\\begin{proof}\nTo prove the estimate (\\ref{estimate_traces}) we employ the local\nlipschitzianity of the surfaces $\\Gamma ^{\\pm }$. Namely, given $y\\in \\Gamma\n^{-}$ we choose an (open) ball $B\\left( y,\\varepsilon _{y}\\right) $, \n\\varepsilon _{y}>0$, such that $\\Gamma ^{-}\\cap B\\left( y,\\varepsilon\n_{y}\\right) $ can be represented as the graph of a Lipschitz continuous\nfunction with respect to some (local) coordinates. Without loss of\ngenerality we can suppose that this function (say $f_{y}^{-}$) is defined on\nan open set $D_{y}^{-}$ from the cartesian product of the first two\ncomponents $x^{\\prime }:=\\left( x_{1},x_{2}\\right) $ and admits as values\nthe component $x_{3}$, i.e., \n\\begin{equation}\n\\label{Gamma_-cap}\n\\Gamma ^{-}\\cap B\\left( y,\\varepsilon _{y}\\right) =\\left\\{ \\left( z^{\\prime\n},f_{y}^{-}\\left( z^{\\prime }\\right) \\right) :z^{\\prime }\\in\nD_{y}^{-}\\right\\}.\n\\end{equation\nBy the compactness of $\\Gamma^{-}$ one can find a finite number of points \ny^{1},\\dots ,y^{q}\\in \\Gamma^{-}$ with \n\\begin{equation}\n\\Gamma ^{-}=\\Gamma ^{-}\\cap \\tbigcup\\limits_{j=1}^{q}B\\left( y^{j},\\frac\n\\varepsilon_{_{y^{j}}}}{2}\\right) .  \\label{repres.Gamma_-}\n\\end{equation\nSet $\\varepsilon _{j}:=\\varepsilon _{y^{j}}$, $D_{j}^{-}:=D_{y^{j}}^{-}$ and \n$f_{j}^{-}\\left( z^{\\prime }\\right) :=f_{y^{j}}^{-}\\left( z^{\\prime }\\right) \n$, $z^{\\prime }\\in D_{j}^{-}$, $j=1,\\dots ,q$.\n\nSimilarly, for any $x\\in \\Gamma ^{+}$ there exist $\\delta _{x}>0$, an open\ndomain $D_{x}^{+}\\subset \\mathbb{R}^{2}$ and a Lipschitz function \nf_{x}^{+}:D_{x}^{+}\\rightarrow \\mathbb{R}$ such that \n\\begin{equation*}\n\\Gamma ^{+}\\cap B\\left( x,\\delta _{x}\\right) =\\left\\{ \\left( z^{\\prime\n},f_{x}^{+}\\left( z^{\\prime }\\right) \\right) :z^{\\prime }\\in\nD_{x}^{+}\\right\\} .\n\\end{equation*\nWe do not loose generality assuming that the value of $f_{x}^{+}$ is the\nlast component of the vector $z\\in \\Gamma ^{+}$ (as in (\\ref{Gamma_-cap\n)). Again due to the compactness argument there exists a finite number of\npoints $x^{1},\\dots ,x^{r}\\in \\Gamma ^{+}$ such tha\n\\begin{equation*}\n\\Gamma ^{+}=\\Gamma ^{+}\\cap \\tbigcup\\limits_{i=1}^{r}B\\left( x^{i},\\delta\n_{i}\\right) \n\\end{equation*\nwher\n\\begin{equation}\n\\delta _{i}:=\\underset{1\\leq j\\leq q}{\\min }\\left( \\delta _{x^{i}},\\tfrac\n\\varepsilon _{j}}{2L}\\right) ,\\;\\; i=1,\\dots ,r.\n\\label{def_delta}\n\\end{equation\nSet also $D_{i}^{+}:=D_{x^{i}}^{+}$ and $f_{i}^{+}\\left( z^{\\prime }\\right)\n:=f_{x^{i}}^{+}\\left( z^{\\prime }\\right) $, $z^{\\prime }\\in D_{i}^{+}$, \ni=1,\\dots ,r$, and denote by $L_{\\Gamma }$ the maximal Lipschitz constant of\nthe functions $f_{j}^{-}:D_{j}^{-}\\rightarrow \\mathbb{R}$, $j=1,\\dots ,q$,\nand $f_{i}^{+}:D_{i}^{+}\\rightarrow \\mathbb{R}$, $i=1,\\dots ,r$.\n\nWe claim that given $i=1,\\dots ,r$ and $\\sigma \\in \\Sigma _{L}$, the image \n\\sigma \\left( \\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) \\right) $ is\ncontained in $\\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon _{j}\\right) $ for\nsome $j=1,\\dots ,q$. Indeed, let us choose $j$ such that $\\sigma \\left(\nx^{i}\\right) \\in B\\left( y^{j},\\varepsilon _{j}\/2\\right) $ (see (\\re\n{repres.Gamma_-})). Taking an arbitrary $z\\in \\Gamma ^{+}\\cap B\\left(\nx^{i},\\delta _{i}\\right) $ we, in particular, have $\\left\\vert\nz-x^{i}\\right\\vert <\\frac{\\varepsilon _{j}}{2L}$ (see (\\ref{def_delta})) and\nby (\\ref{Lipschitz}) $\\left\\vert \\sigma \\left( z\\right) -\\sigma \\left(\nx^{i}\\right) \\right\\vert <\\varepsilon _{j}\/2$. However, assuming that \n\\sigma \\left( z\\right) \\notin \\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon\n_{j}\\right) $ we have $\\left\\vert \\sigma \\left( z\\right) -y^{j}\\right\\vert\n\\geq \\varepsilon _{j}$ and henc\n\\begin{equation*}\n\\left\\vert \\sigma \\left( z\\right) -\\sigma \\left( x^{i}\\right) \\right\\vert\n\\geq \\left\\vert \\sigma \\left( z\\right) -y^{j}\\right\\vert -\\left\\vert \\sigma\n\\left( x^{i}\\right) -y^{j}\\right\\vert \\geq \\varepsilon _{j}-\\frac\n\\varepsilon _{_{j}}}{2}=\\frac{\\varepsilon _{_{j}}}{2},\n\\end{equation*\nwhich is a contradiction. In what follows we associate to each $\\sigma \\in \\Sigma _{L}$ and to each $i=1,\\dots ,r$ an index \nj=j\\left( \\sigma ,i\\right) \\in \\left\\{ 1,\\dots ,q\\right\\} $ such tha\n\\begin{equation*}\n\\sigma \\left( \\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) \\right)\n\\subset \\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon _{j}\\right) .\n\\end{equation*}\n\nThe latter inclusion allows us to define correctly the (injective) mapping \n\\psi _{\\sigma }^{i}:D_{i}^{+}\\rightarrow D_{j\\left( \\sigma ,i\\right) }^{-}$\nsuch tha\n\\begin{equation}\n\\sigma \\left( x\\right) =\\sigma \\left( x^{\\prime },f_{i}^{+}\\left( x^{\\prime\n}\\right) \\right) =\\left( \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right)\n,f_{j}^{-}\\left( \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) \\right)\n\\right) ,x^{\\prime }\\in D_{i}^{+}.  \\label{def_psi}\n\\end{equation\n>From (\\ref{Lipschitz}) it follows immediately that $\\psi _{\\sigma }^{i}$ is\nLipschitz\n\\begin{eqnarray}\n\\left\\vert \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) -\\psi _{\\sigma\n}^{i}\\left( z^{\\prime }\\right) \\right\\vert  &\\leq &\\left\\vert \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) -\\sigma \\left(\nz^{\\prime },f_{i}^{+}\\left( z^{\\prime }\\right) \\right) \\right\\vert   \\notag\n\\\\\n&\\leq &L\\left( \\left\\vert x^{\\prime }-z^{\\prime }\\right\\vert ^{2}+\\left(\nf_{i}^{+}\\left( x^{\\prime }\\right) -f_{i}^{+}\\left( z^{\\prime }\\right)\n\\right) ^{2}\\right) ^{1\/2}  \\label{Lip_psi} \\\\\n&\\leq &L\\sqrt{1+L_{\\Gamma }^{2}}\\left\\vert x^{\\prime }-z^{\\prime\n}\\right\\vert ,\\;\\; x^{\\prime },z^{\\prime }\\in D_{i}^{+}. \n\\notag\n\\end{eqnarray\nSo, by \\emph{Rademacher's Theorem,} $\\psi _{\\sigma }^{i}$ is $\\mathcal{L}^{2}\n-$a.e. differentiable on $D_{i}^{+}$ with $\\mathcal{L}^{2}$-measurable\ngradient, and the inequalit\n\\begin{equation*}\n\\left\\vert \\nabla \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) \\right\\vert\n\\leq M:=L\\sqrt{1+L_{\\Gamma }^{2}}  \\label{bound_grad_psi}\n\\end{equation*\nholds for a.e. $x^{\\prime }\\in D_{i}^{+}$. Notice that the inverse mapping \n\\left( \\psi _{\\sigma }^{i}\\right) ^{-1}$ is well defined on $G_{\\sigma\n}^{i}:=\\psi _{\\sigma }^{i}\\left( D_{i}^{+}\\right) \\subset D_{j\\left( \\sigma\n,i\\right) }^{-}$ by the formula similar to (\\ref{def_psi}), namely\n\\begin{equation*}\n\\sigma ^{-1}\\left( y^{\\prime },f_{j}^{-}\\left( y^{\\prime }\\right) \\right)\n=\\left( \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime }\\right)\n,f_{i}^{+}\\left( \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime\n}\\right) \\right) \\right) ,y^{\\prime }\\in G_{\\sigma }^{i}.\n\\end{equation*\nIn the same way as (\\ref{Lip_psi}) we deduce, from (\\ref{Lipschitz}), that \n\\left( \\psi _{\\sigma }^{i}\\right) ^{-1}$ is Lipschitz on the (open) set \nG_{\\sigma }^{i}$ and, the estimat\n\\begin{equation}\n\\left\\vert \\nabla \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime\n}\\right) \\right\\vert \\leq M  \\label{bound_grad_psi_inv}\n\\end{equation\nholds for $\\mathcal{L}^{2}$-a.e. $y^{\\prime }\\in G_{\\sigma }^{i}$.\n\nIntegrating the function $\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) \\right\\vert ^{2}$ on the surface piece $\\Gamma ^{+}\\cap\nB\\left( x^{i},\\delta _{i}\\right) $ we pass first to the double integra\n\\begin{eqnarray}\n&&\\underset{\\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) }{\\int \n\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right)   \\notag \\\\\n&=&\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) \\right) \\right\\vert\n^{2}\\sqrt{1+\\left\\vert \\nabla f_{i}^{+}\\left( x^{\\prime }\\right) \\right\\vert\n^{2}}dx^{\\prime }  \\label{int_change_var} \\\\\n&\\leq &\\sqrt{1+L_{\\Gamma }^{2}}\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc\nTr}u\\left( \\sigma \\left( x^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right)\n\\right) \\right) \\right\\vert ^{2}\\,dx^{\\prime }.  \\notag\n\\end{eqnarray\nDue to the representation (\\ref{def_psi}) we can make the change of\nvariables $y^{\\prime }=\\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) $, \nx^{\\prime }\\in D_{i}^{+}$, in the integral (\\ref{int_change_var}), and\nreturning then to the surface integral on a piece of $\\Gamma ^{-}$, we hav\n\\begin{eqnarray}\n&&\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) \\right) \\right\\vert\n^{2}\\,dx^{\\prime }  \\notag \\\\\n&=&\\diint\\limits_{G_{\\sigma }^{i}}\\left\\vert \\limfunc{Tr}u\\left( y^{\\prime\n},f_{j}^{-}\\left( y^{\\prime }\\right) \\right) \\right\\vert ^{2}\\left\\vert \\det\n\\nabla \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime }\\right)\n\\right\\vert \\,dy^{\\prime }  \\notag \\\\\n&\\leq &6M^{3}\\diint\\limits_{G_{\\sigma }^{i}}\\left\\vert \\limfunc{Tr}u\\left(\ny^{\\prime },f_{j}^{-}\\left( y^{\\prime }\\right) \\right) \\right\\vert ^{2}\\sqrt\n1+\\left\\vert \\nabla f_{j}^{-}\\left( y^{\\prime }\\right) \\right\\vert ^{2}\ndy^{\\prime }  \\label{change_var1} \\\\\n&\\leq &6M^{3}\\underset{\\Gamma ^{-}}{\\int }\\left\\vert \\limfunc{Tr}u\\left(\ny\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) .  \\notag\n\\end{eqnarray\nHere we used the estimate (\\ref{bound_grad_psi_inv}) and the obvious\ninequality $\\left\\vert \\det A\\right\\vert \\leq 6\\left\\Vert A\\right\\Vert ^{3}$\n($A$ is an arbitrary $3\\times 3$-matrix). Since the sets $\\Gamma ^{+}\\cap\nB\\left( x^{i},\\delta _{i}\\right) $, $i=1,\\dots ,r$, cover the surface \n\\Gamma ^{+}$, taking into account (\\ref{int_change_var}) and (\\re\n{change_var1}) we conclude tha\n\\begin{eqnarray*}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  &\\leq &\\underset{i=\n}{\\overset{r}{\\dsum }}\\underset{\\Gamma ^{+}\\cap B\\left( x^{i},\\delta\n_{i}\\right) }{\\int }\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  \\\\\n&\\leq &\\mathfrak{L}\\underset{\\Gamma ^{-}}{\\int }\\left\\vert \\limfunc{Tr\nu\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) \n\\end{eqnarray*\nwhere $\\mathfrak{L}:=6rM^{3}\\sqrt{1+L_{\\Gamma }^{2}}>0$ depends just on the\nLipschitz constant $L\\geq 1$ and on the properties of the domain $\\Omega $\n(namely, of its boundary).\\bigskip \n\\end{proof}\n\n\\medskip\n\nProving the existence theorem we pay the main attention to the validity of\nthe boundary condition (C$_{3}$) where Lemma \\ref{estimate_w} is crucial.\n\n\\begin{theorem}\n\\label{Th_exist} Let $W:\\mathbb{R}^{3\\times 3}\\rightarrow \\mathbb{R\\cup \n\\left\\{ +\\infty \\right\\} $ be a polyconvex function satisfying the growth\nassumption (\\ref{growth_cond}). Then problem (\\ref{varproblem}) admits a\nminimizer whenever there exists at least one pair $\\omega :=\\left( u,\\sigma\n\\right) \\in \\mathcal{W}_{L}$ wit\n\\begin{equation*}\nI\\left( \\omega \\right) :=\\dint_{\\Omega }W\\left( \\nabla u\\left( x\\right)\n\\right) \\,\\,dx<+\\infty .\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nLet us consider a minimizing sequence $\\left\\{ \\left( u_{n},\\sigma\n_{n}\\right) \\right\\} \\subset \\mathcal{W}_{L}$ of the functional (\\re\n{Functional}), e.g., such a\n\\begin{equation}\n\\dint_{\\Omega }W\\left( \\nabla u_{n}\\left( x\\right) \\right) \\,dx\\leq \\inf\n\\left\\{ \\dint_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right) \\,dx:\\left(\nu,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\} +\\frac{1}{n}<+\\infty .\n\\label{min_seq}\n\\end{equation\nTaking into account the estimate (\\ref{growth_cond}) we deduce from (\\re\n{min_seq}) that the sequences $\\left\\{ \\nabla u_{n}\\right\\} $, $\\left\\{ \n\\mathrm{Adj\\,}\\nabla u_{n}\\right\\} $ and $\\left\\{ \\det \\,\\nabla\nu_{n}\\right\\} $ are bounded in $\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R\n^{3\\times 3}\\right) $ and in $\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R}\\right) \n$, respectively. Applying Proposition \\ref{poincare} and the boundary\ncondition (C$_{1}$) we find a constant $C>0$ such that the inequalit\n\\begin{equation*}\n\\underset{\\Omega }{\\dint }\\left\\vert u_{n}\\left( x\\right) \\right\\vert\n^{2}dx\\leq C\\left[ \\underset{\\Omega }{\\dint }\\left\\vert \\nabla u_{n}\\left(\nx\\right) \\right\\vert ^{2}\\,dx+\\left\\vert \\dint\\limits_{\\Gamma _{1}}x\\,\n\\mathcal{H}^{2}\\left( x\\right) \\right\\vert \\right] \n\\end{equation*\nholds for each  $n\\geq 1$. So, the sequence $\\left\\{ u_{n}\\right\\}$ is bounded in $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ and by the \\emph{Banach-Alaoglu theorem}, up to a subsequence,\nconverges weakly to some function $\\bar{u}\\in \\mathbf{W}^{1,2}\\left( \\Omega \n\\mathbb{R}^{3}\\right) $. Without loss of generality, we can also assume that \n$\\left\\{ \\mathrm{Adj\\,}\\nabla u_{n}\\right\\} $ and $\\left\\{ \\mathrm{\\det \\,\n\\nabla u_{n}\\right\\} $ converge weakly to some functions $\\xi \\in \\mathbf{L\n^{2}\\left( \\Omega ;\\mathbb{R}^{3\\times 3}\\right) $ and $\\eta \\in \\mathbf{L\n^{2}\\left( \\Omega ;\\mathbb{R}\\right) $, respectively. Now, by Theorem 8.20 \n\\cite[pp. 395-396]{Dac} due to the uniqueness of the limit we deduce that \n\\xi \\left( x\\right) =\\mathrm{Adj\\,}\\nabla u\\left( x\\right) $ and $\\eta\n\\left( x\\right) =\\mathrm{\\det \\,}\\nabla u\\left( x\\right) $ for almost all \nx\\in \\Omega $. Thus we have the weak convergence\\ of the sequence $\\left\\{ \n\\mathbb{T}\\left( \\nabla u_{n}\\right) \\right\\} $ to the vector-function \n\\mathbb{T}\\left( \\nabla u\\right) $ in the space $\\mathbf{L}^{2}\\left( \\Omega\n;\\mathbb{R}^{\\tau \\left( 3,3\\right) }\\right) $.\n\nOn the other hand, recalling that $\\left\\{ \\sigma _{n}\\right\\} \\subset\n\\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ (see (\\ref{Lipschitz}))\nby \\emph{Ascoli's theorem} up to a subsequence, not relabeled, $\\left\\{\n\\sigma _{n}\\right\\} $ converges uniformly to $\\overline{\\sigma }\\in \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $.\n\nSince the integrand $W$ is polyconvex, it can be represented as $W\\left( \\xi\n\\right) =g\\left( \\mathbb{T}\\left( \\xi \\right) \\right) $, $\\xi \\in \\mathbb{R\n^{3\\times 3}$, with some convex function $g:\\mathbb{R}^{\\tau \\left(\n3,3\\right) }\\rightarrow \\mathbb{R}$, and, therefore\n\\begin{eqnarray*}\n\\underset{\\Omega }{\\dint }W\\left( \\nabla \\overline{u}\\left( x\\right) \\right)\n\\,\\,dx &=&\\underset{\\Omega }{\\dint }g\\left( \\mathbb{T}\\left( \\nabla \n\\overline{u}\\left( x\\right) \\right) \\right) \\,\\,dx\\leq \\underset\nn\\rightarrow \\infty }{\\lim \\inf }\\,\\underset{\\Omega }{\\dint }g\\left( \\mathbb\nT}\\left( \\nabla u_{n}\\left( x\\right) \\right) \\right) \\,\\,dx \\\\\n&\\leq &\\inf \\left\\{ \\underset{\\Omega }{\\dint }W\\left( \\nabla u\\left(\nx\\right) \\right) \\,\\,dx:\\left( u,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\} \n.\n\\end{eqnarray*\nThus, it remains just to prove that $\\bar{\\omega}:=\\left( \\overline{u}\n\\overline{\\sigma }\\right) \\in \\mathcal{W}_{L}$ (i.e., that the Sobolev\nfunction $\\overline{u}$ satisfies the boundary conditions (C$_{1}$)$-$(C$_{3}\n$) above with the transformation $\\overline{\\sigma }$). The validity of (C\n_{1}$) and (C$_{2}$) follows immediately from Proposition \\ref{traceconv}.\nIn fact, the weak convergence of $\\left\\{ u_{n}\\right\\} $ in $\\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ implies the strong convergence\nof traces $\\left\\{ \\limfunc{Tr}u_{n}\\right\\} $ in $\\mathbf{L}^{2}\\left(\n\\partial \\Omega ;\\mathbb{R}^{3}\\right) $. So, up to a subsequence, $\\limfunc\nTr}u_{n}\\left( x\\right) \\rightarrow \\limfunc{Tr}\\bar{u}\\left( x\\right) $ for \n$\\mathcal{H}^{2}$-a.e. $x\\in \\partial \\Omega $. In particular, $\\limfunc{Tr\n\\bar{u}\\left( x\\right) =x$ and $h\\left( \\limfunc{Tr}\\bar{u}\\left( x\\right)\n\\right) =0$ almost everywhere on $\\Gamma _{1}$ and on $\\Gamma _{2}$,\nrespectively (w.r.t. the Hausdorff measure).\n\nIn order to verify the condition (C$_{3}$) we observe first tha\n\\begin{equation}\n\\limfunc{Tr}{u}_{n}(x) =\\limfunc{Tr}{u}_{n}({\n\\sigma}_{n}(x)) ,n=1,2,\\dots ,\n\\label{cond_C3_n}\n\\end{equation\nfor $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$ and consider the surface\nintegra\n\\begin{equation*}\n\\mathcal{J}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( x\\right)\n-\\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right) .\n\\end{equation*\nBy the Minkowski's inequality we hav\n\\begin{equation}\n\\mathcal{J}^{1\/2}\\leq \\left( \\mathcal{J}_{1}^{n}\\right) ^{1\/2}+\\left( \n\\mathcal{J}_{2}^{n}\\right) ^{1\/2}+\\left( \\mathcal{J}_{3}^{n}\\right) ^{1\/2}\n\\label{estimate_int}\n\\end{equation\nwher\n\\begin{eqnarray*}\n&&\\mathcal{J}_{1}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u\n\\left( x\\right) -\\limfunc{Tr}u_{n}\\left( \\sigma _{n}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  {;} \\\\\n&&\\mathcal{J}_{2}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u_{n}\\left(\n\\sigma _{n}\\left( x\\right) \\right) -\\limfunc{Tr}\\bar{u}\\left( \\sigma\n_{n}\\left( x\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \n {;} \\\\\n&&\\mathcal{J}_{3}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u\n\\left( \\sigma _{n}\\left( x\\right) \\right) -\\limfunc{Tr}\\bar{u}\\left( \\bar\n\\sigma}\\left( x\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left(\nx\\right) .\n\\end{eqnarray*}\n\nTaking into account the equalities (\\ref{cond_C3_n}) by using Proposition\n\\ref{traceconv} we immediately obtain that $\\mathcal{J}_{1}^{n}\\rightarrow 0$\nas $n\\rightarrow \\infty $.\n\nDue to the linearity of the trace operator, applying Lemma \\ref{estimate_w}\nand again Proposition \\ref{traceconv} we arrive a\n\\begin{equation*}\n\\mathcal{J}_{2}^{n}\\leq \\mathfrak{L}_L\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr\n\\left( u_{n}-\\bar{u}\\right) \\left( x\\right) \\right\\vert ^{2}d\\mathcal{H\n^{2}\\left( x\\right) \\rightarrow 0 { \\ \\ as \\ \\ }n\\rightarrow \\infty \n.\n\\end{equation*}\n\nLet us approximate now $\\bar{u}\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ by a sequence of continuous functions $v_{k}:\\overline{\\Omega \n\\rightarrow \\mathbb{R}^{3}$, $k=1,2,\\dots $ (with respect to the norm of \n\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $). Then (see\nProposition \\ref{traceconv}) $v_{k}=\\limfunc{Tr}v_{k}\\rightarrow \\limfunc{Tr\n\\bar{u}$ as $k\\rightarrow \\infty$ in $\\mathbf{L}^{2}\\left( \\partial \\Omega\n;\\mathbb{R}^{3}\\right) $. In particular, given $\\varepsilon >0$ there exists\nan index $k^{\\ast }\\geq 1$ such tha\n\\begin{equation*}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( y\\right) -v_{k^{\\ast\n}}\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) \\leq\n\\varepsilon .\n\\end{equation*\nBy using Lemma \\ref{estimate_w} similarly as was done to estimate the\nintegral $\\mathcal{J}_{2}^{n}$ we hav\n\\begin{eqnarray}\n&&\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( \\sigma _{n}\\left(\nx\\right) \\right) -v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)   \\notag \\\\\n&\\leq &\\mathfrak{L}_{L}\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left(\ny\\right) -v_{k^{\\ast }}\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H\n^{2}\\left( y\\right) \\leq \\mathfrak{L}_{L}\\varepsilon ,\\;\\; n=1,2,\\dots \n,  \\label{Int_3_1}\n\\end{eqnarray\nand similarly\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left(\nx\\right) \\right) -v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathfrak{L}_{L}\\varepsilon \n.  \\label{Int_3_2}\n\\end{equation\nOn the other hand, by the uniform continuity of $v_{k^{\\ast }}$ and the uniform\nconvergence $\\sigma _{n}\\rightarrow \\overline{\\sigma }$ as $n\\rightarrow\n\\infty $, we find a number $n^{\\ast }\\geq 1$ such tha\n\\begin{equation*}\n\\left\\vert v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right) \\right)\n-v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert \\leq\n\\varepsilon \n\\end{equation*\nfor all $n\\geq n^{\\ast }$ and all $x\\in \\Gamma ^{+}$, and, consequently\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right)\n\\right) -v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathcal{H}^{2}\\left( \\Gamma\n^{+}\\right) \\varepsilon ^{2},n\\geq n^{\\ast }.  \\label{Int_3_3}\n\\end{equation\nJoining together the inequalities (\\ref{Int_3_1}), (\\ref{Int_3_2}) and (\\re\n{Int_3_3}) we obtain tha\n\\begin{equation*}\n\\left( \\mathcal{J}_{3}^{n}\\right) ^{1\/2}\\leq \\left( \\mathfrak{L}_L\\varepsilon\n\\right) ^{1\/2}+\\left( \\mathfrak{L}_L\\varepsilon \\right) ^{1\/2}+\\left( \\mathcal\nH}^{2}\\left( \\Gamma ^{+}\\right) \\varepsilon ^{2}\\right) ^{1\/2},n\\geq\nn^{\\ast }.\n\\end{equation*\nSince $\\varepsilon >0$ is arbitrary and the constant $\\mathfrak{L}_L$ does not\ndepend on $n=1,2,\\dots $, we conclude that all the three integrals in the\nright-hand side of (\\ref{estimate_int}) tend to zero as $n\\rightarrow \\infty \n$. Thus $\\mathcal{J}=0$, or, in other words, $\\limfunc{Tr}\\bar{u}\\left(\nx\\right) -\\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left( x\\right) \\right) =0$\nfor $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$, and the theorem is proved.\n\\end{proof}\n\n\n\\section{Necessary conditions of optimality}\n\nIn this section, under some additional hypotheses, we deduce necessary conditions of optimality for problem (\\ref{varproblem}).\n\nTo simplify, assume that the function $W$ is twice continuously differentiable  and $h$ is continuously differentiable.\nMoreover, suppose that the surfaces $\\Gamma_1, \\Gamma_2, \\Gamma^+, \\Gamma^-, \\Gamma_4$ are sufficiently smooth. \n\nGiven $\\Gamma \\subset \\partial\\Omega,$ with $\\mathcal{H}^{2}(\\Gamma)>0,$ in what follows we denote by $\\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3})$ the family of restrictions to $\\Gamma$ of all functions $u:\\Omega\\rightarrow \\mathbb{R}^{3},$ whose gradients are continuous up to the boundary. Let us supply $\\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})$ with the natural sup-norm. \n\nWe consider the problem (\\ref{varproblem}) defined in the space $\\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3})$.\n\n\\begin{theorem}\nLet $(\\bar{u},\\bar{\\sigma})\\in \\mathbf{C}^{1}({\\Omega};\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})$ be a minimizer of problem (\\ref{varproblem}). Assume that $\\nabla h(\\bar{u}(x))\\neq 0$, $x\\in \\Gamma_2$ and ${\\rm det}\\nabla\\bar{u}(\\bar{\\sigma}(x))\\neq 0$, $x\\in\\Omega$. Then the following conditions are satisfied:\n\\begin{eqnarray}\n&& {\\rm Div}(\\nabla W)(\\nabla\\bar{u}(x))=0,\\;\\; x\\in\\Omega; \\label{f1}\\\\\n\\smallskip\n&& \\nabla W(\\nabla\\bar{u}(x))\\nu(x)=0,\\;\\; x\\in \\Gamma_3; \\label{f2}\\\\\n\\smallskip\n&& \\nabla W(\\nabla\\bar{u}(x))\\nu(x)\\times\\nabla h(\\bar{u}(x))=0,\\;\\; x\\in \\Gamma_2; \\label{f3}\\\\\n\\smallskip\n&&\\nabla W(\\nabla\\bar{u}(x))\\nu(x)=0,\\;\\; x\\in \\Gamma^{\\pm}. \\label{f4}\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof} Let us write the constraints in the minimization problem (\\ref{varproblem}) as $F(u,\\sigma)=0$ where the map  \n$\nF:\\mathbf{C}^{1}({\\Omega};\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})\\rightarrow \\mathbf{C}^{1}({\\Gamma}_1;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}_2;\\mathbb{R})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})$ is given by\n$$\nF(u,\\sigma ):=(u(x)-x,h(u(x)),u(x)-u(\\sigma(x))).\n$$\nUnder our assumptions the map $F$ and the functional $I$ are both Fr\\'{e}chet differentiable. In particular, for the (Fr\\'{e}chet) derivative of $F$ at the point $(\\bar{u},\\bar{\\sigma})$ we have\n$$\nDF(\\bar{u},\\bar{\\sigma})(\\tilde{u},\\tilde{\\sigma})(x)=\n\\left(\n\\begin{array}{c}\n\\tilde{u}(x)\\\\\n\\langle\\nabla h(\\bar{u}(x)),\\tilde{u}(x)\\rangle\\\\\n\\tilde{u}(x)-\\tilde{u}(\\bar{\\sigma}(x))-\\nabla \\bar{u}(\\bar{\\sigma}(x))\\tilde{\\sigma}(x)\n\\end{array}\n\\right).\n$$\nHere, and in what follows, $\\langle\\cdot,\\cdot\\rangle$ denotes the inner product in $\\mathbb{R}^{3}.$ Taking into account that $\\nabla h(\\bar{u}(x))\\neq0$ on $\\Gamma_2$ and that the jacobian matrix $\\nabla\\bar{u}(\\bar{\\sigma}(x))$ is not degenerated, we have that the linear operator $DF(\\bar{u},\\bar{\\sigma})$ is onto the space $\\mathbf{C}^{1}({\\Gamma}_1;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})\\times \\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3}).$\nBy the Lagrange multipliers rule (see, e.g., \\cite{IT}) there exist linear continuous functionals $\\lambda_1,\\;\\lambda_2,\\;\\lambda^+$ on $\\mathbf{C}^{1}(\\Gamma_1;\\mathbb{R}^{3})$,\\; $\\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})$ and $\\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3}),$ respectively, such that\n\\begin{equation*}\n\\int_{\\Omega}\\sum_{i,j=1}^{3}\\frac{\\partial W(\\nabla\\bar{u}(x))} {\\partial\\xi_{ij}}\\frac{\\partial\\tilde{u}_i(x)}{\\partial x_j}dx\n+\\lambda_1(\\tilde{u})+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)\n+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma})=0.\n\\end{equation*}\n\n\\noindent Applying the Divergence theorem we get\n\\begin{eqnarray}\n&&-\\int_{\\Omega}\\left\\langle {\\rm Div}{(\\nabla W(\\nabla\\bar{u}(x)))},\\tilde{u}(x)\\right\\rangle dx\n+\\int_{\\partial\\Omega}\\left\\langle{\\nabla W(\\nabla\\bar{u}(x))\\nu(x),\\tilde{u}(x)}\\right\\rangle d\\mathcal{H}^{2}(x) \\nonumber \\\\\n&&+\\lambda_1(\\tilde{u})+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma}))=0. \\label{f5} \n\\end{eqnarray}\nHere $\\nu(x)$ is the unit outer normal to the boundary. Varying $\\tilde{u}$ in (\\ref{f5}) such that $\\tilde{u}(x)=0$ on $\\partial{\\Omega}$, we obtain  (\\ref{f1}). Taking then $\\tilde{u} \\in \\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})$ with $\\tilde{u}(x)=0$ on $\\partial\\Omega\\setminus \\Gamma_3$ we arrive at (\\ref{f2}).\nFurthermore, choosing appropriate functions $\\tilde{u}$ in (\\ref{f5}) we obtain\n\\begin{equation}\n\\label{f6}\n\\int_{\\Gamma_2}\\left\\langle\\nabla W(\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^{2}(x)\n+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)=0 \n\\end{equation}\nwhenever $\\tilde{u}(x)=0,\\; x\\in\\partial\\Omega\\setminus \\Gamma_2$, and\n\\begin{equation}\n\\label{f7}\n\\int_{\\Gamma^+\\cup\\Gamma^-}\\left\\langle{\\nabla W(\\nabla\\bar{u}(x)) }\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^{2}(x)\n+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma})=0\\;\\; \n\\end{equation}\nwhenever $\\tilde{u}(x)=0,\\;\\; x\\in\\partial\\Omega\\setminus (\\Gamma^+\\cup\\Gamma^-)$.\n\n\nDenote by $\\Gamma_2^0$ the part of $\\Gamma_2$ where the vectors $a(x):=\\nabla W(\\nabla\\bar{u}(x)) \\nu (x)$ and\n$b(x):=\\nabla h(\\bar{u})(x)$ are co-linear. Taking an arbitrary $c \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})$ such that $c(x)=0$ and $\\nabla c(x)=0$ on $\\Gamma_{2}^0$, let us define\n$$\n\\hat{u}(x):=\n\\left\\{\n\\begin{array}{cl}\n\\frac{a(x)\\langle a(x),b(x)\\rangle-b(x)|a(x)|^2}{\\langle a(x),b(x)\\rangle^2-|a(x)|^2|b(x)|^2}c(x),& x\\in\\Gamma_2\\setminus\\Gamma_2^0,\\\\\n\\smallskip\n0, & x\\in\\Gamma_2^0. \n\\end{array}\n\\right.\n$$\nObviously, $\\hat{u} \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R}^3)$, $\\langle\\hat{u}(x), a(x)\\rangle=0$ and $\\langle\\hat{u}(x),b(x)\\rangle=c(x)$ for $x\\in \\Gamma_2$. From (\\ref{f6}) we\nget $\\lambda_2(c)=\\lambda_2(\\langle b,\\hat{u}\\rangle)=0$. Hence, varying $\\tilde{u} \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R}^3)$ in (\\ref{f6}) in a suitable way (in particular, setting $\\hat u(x)=0$ on $\\Gamma_2^0$) we get $a(x)=0$ in $\\Gamma_2\\setminus\\Gamma_2^0.$ Thus the equality (\\ref{f3}) follows.\n\n\nFinally, taking $\\tilde{u}=0$, from (\\ref{f7}) we get $\\lambda^+=0$, and, as a consequence, \n$$\n\\int_{\\Gamma^-}\\left\\langle\\nabla W (\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^2(x)\n+\\int_{\\Gamma^+}\\left\\langle\\nabla W (\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^2(x)=0,\n$$\nwhich implies (\\ref{f4}). \n\\end{proof}\n\n\\bigskip\n\n\\bigskip\n\n\\noindent  {\\bf Acknowledgements}\n\n\\bigskip \n\nThe authors are grateful to Hor\\'{a}cio Costa and Augusta Cardoso for fruitful discussion \nof medical aspects of the problem and also to Giovanni Leoni, who kindly communicate the proof of Lemma \\ref\n{Lemmatrace}.\n\nThis research was supported by Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e\nTecnologia (FCT), Portuguese Operational Programme for Competitiveness\nFactors (COMPETE), Portuguese National Strategic Reference Framework (QREN)\nand European Regional Development Fund (FEDER) through Project VAPS\n(EXPL\/MAT-NAN\/0606\/2013).\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAlthough we have known since 2012 that there exists a neutral spin-0 resonance with properties (mass and couplings) that are compatible within the experimental error, with those of the scalar SM-Higgs boson~ \\cite{Aad:2012tfa, Chatrchyan:2012xdj}, these data do not exclude the existence of more fields of this sort.\nIn fact, almost all the extensions of the standard model (SM) include extra scalar \nmultiplets: complex~\\cite{Chikashige:1980ui} or real singlets~\\cite{Hill:1987ea,Davoudiasl:2004be,vanderBij:2006ne}, two~\\cite{Branco:2011iw} or more doublets~\\cite{Machado:2012ed}, and Hermitian~\\cite{Brdar:2013iea} and\/or non-Hermitian triplets~\\cite{Konetschny:1977bn,Magg:1980ut,Cheng:1980qt,Escobar:1982dp}. Moreover, extra scalar multiplets usually are introduced in a given model just in order to give masses to the neutrino and\/or charged leptons. Hence, they may have small vacuum expectation values (VEV). \nHowever, this usually implies that there might be light neutral scalars which can be easily ruled out by phenomenology. In two-Higgs doublets this is not the case when there is a positive quadratic term $\\mu^2>0$, which behaves like a positive mass square term in the scalar potential. In this case such parameters may dominate the contributions to the masses of the multiplets' members, which are almost mass degenerated, i.e. in the context of models with one or more inert doublets, see~\\cite{Machado:2012ed} and \\cite{Ma:2006km}. \n\nThe 3-3-1 models are intrinsically multi-Higgs models. For instance,\nin the minimal 3-3-1 model (here denoted by m331 for short), the charged leptons gain mass from a triplet and a sextet and the neutrino gain Majorana masses only through  the sextet ~\\cite{Pisano:1991ee,Foot:1992rh,Frampton:1992wt}. \nOn the other hand, in the model with heavy charged~\\cite{Pleitez:1992xh} or neutral leptons~\\cite{Montero:1992jk}, only the triplets are needed, if right-handed neutrinos are introduced and the type-I seesaw mechanism is implemented. \n\nBecause of the sextet, the scalar potential in the m331 becomes more complicated, for this reason it was pointed out in Ref.~\\cite{Montero:2001tq}  that the sextet can be omitted if a dimension-five effective operator, involving only triplets, is in charge of the mass generation of the charged leptons and neutrinos.  The m331 model without the sextet was called the \"reduced\" m331 model in Ref.~\\cite{Ferreira:2011hm} because only the triplets $\\rho$ and $\\chi$ are introduced. However, there are important differences between our model and those in Refs.~\\cite{Montero:2001tq,Ferreira:2011hm} as we will discuss in Sec.~\\ref{sec:con}. Moreover, as in the case of the SM, the question now is, how does this effective operator arise at tree and\/or loop level~\\cite{Ma:1998dn,Bonnet:2012kz,Sierra:2014rxa}? In the context of the m331 model, mechanisms for generating effective dimension-five operators for the case of the neutrino masses were given in Ref.~\\cite{Montero:2001ts}. \nHowever, in those works thePontecorvo-Maki-Nakawaga-Sakata (PMNS) matrix was not considered. It is far from obvious that the same parameters that allow to obtain the correct lepton masses also accommodate a realistic PMNS matrix. We show that this is possible in the m331 model with a heavy sextet which implements a sort of type-II seesawlike mechanism in the charged lepton sector and, by introducing right-handed neutrinos, neutrino masses arise from a type-I seesaw mechanism. \n\nIn fact, we show that the sextet is just a way to generate, at the tree level, the effective five-dimensional operator proposed in Ref.~\\cite{Montero:2001tq} in order to give the charged leptons their correct masses. This happens if all fields in this multiplet are heavy and its neutral components gain a small ($s^0_2$) or a zero ($s^0_1$) VEV. We also study in this model the conditions upon the dimensionless coupling constants that imply a scalar potential bounded from below, with a global minimum as well. We obtain a realistic PMNS mixing matrix as well. \n\nThe outline of this paper is the following. In Sec.~\\ref{sec:scalars} we give the scalar representation content of the m331 model and the scalar potential of the model. In Sec.~\\ref{sec:masses} we obtain the scalar mass spectra of the model under the conditions of $Z_7\\otimes Z_3$ discrete symmetries. In Sec.~\\ref{sec:pmns} we obtain the charged lepton and neutrino masses and the corresponding PMNS matrix.\nOur conclusions appear in Sec.~\\ref{sec:con}. The conditions for a stable minimum, at tree level, of the scalar potential are obtained in Appendix~\\ref{sec:appendixa}. In Appendix B we consider the Goldstone bosons in the model with exact mass matrices. \n\n\\section{The scalar sector in the m331 model}\n\\label{sec:scalars}\n\nThe scalar potential in several 3-3-1 models was considered in Refs.~\\cite{Diaz:2003dk,Nguyen:1998ui,Giraldo:2011gd,Hernandez:2014vta}. Here, however, we\nwill study only the  m331 model in the situation in which the sextet gain a small VEV, the extra scalars in the sextet are heavy,  and there is no explicit total lepton number violation in the scalar potential, which is avoided by an appropriate discrete symmetry.\n\nIn the m331 model the scalar sector is composed of a sextet $S\\sim(\\textbf{6},0)$ \nand three  triplets: $\\eta=(\\eta^0\\,\\;-\\!\\eta^{-}_1\\,\\eta^+_2)^T\\sim({\\bf3},0)$,\n$\\rho=(\\rho^+\\,\\rho^0\\,\\rho^{++})^T\\sim({\\bf3},1)$, and\n$\\chi=(\\chi^-\\,\\chi^{--}\\,\\chi^0)^T\\sim({\\bf3},-1)$, where $(x,y)$ refer to the $(SU(3)_L,U(1)_X)$ transformations. Only the triplet $\\eta$ and the sextet~$S$,\n\\begin{equation}\nS=\\left(\n\\begin{array}{ccc}\ns^0_1& \\frac{s^-_1}{\\sqrt2} & \\frac{s^+_2}{\\sqrt2}\\\\\n\\frac{s^-_1}{\\sqrt2}& S^{--}_1&\\frac{s^0_2}{\\sqrt2}\\\\\n\\frac{s^+_2}{\\sqrt2}&\\frac{s^0_2}{\\sqrt2}&S^{++}_2\n\\end{array}\n\\right),\n\\label{sextet1}\n\\end{equation}\ncouple to the leptons through the Yukawa interactions $\\overline{(\\Psi_L)^c}\\Psi_LS^*$ and $\\overline{(\\Psi_L)^c}\\Psi_L\\eta$. \n\nWe can write the $SU(3)$ multiplets above in terms of the $SU(2)$ ones. For the triplets we write\n\\begin{equation}\n\\eta=\n\\left(\n\\begin{array}{c}\n\\Phi_\\eta \\\\ \\eta^+_2\\end{array}\\right)\\sim({\\bf3},0),\\quad\n\\rho=\\left(\\begin{array}{c}\n\\Phi_\\rho \\\\ \\rho^{++}\\end{array} \\right)\\sim({\\bf3},1),\\quad\n\\chi=\\left(\n\\begin{array}{c}\n\\Phi_\\chi\\\\ \\chi^0\\end{array} \\right)\\sim({\\bf3},-1).\n\\label{tripletos}\n\\end{equation}\nThe sextet in Eq.~(\\ref{sextet1}) can be written as\n\\begin{equation}\nS=\\left(\n\\begin{array}{cc}\nT & \\frac{\\Phi_s}{\\sqrt2} \\\\\n\\frac{\\Phi^t_s}{\\sqrt2}  & S^{--}_2\n\\end{array}\n\\right),\\quad S^*=\\left(\n\\begin{array}{cc}\nT^* & \\frac{\\Phi^*_s}{\\sqrt2} \\\\\n\\frac{\\Phi^\\dagger_s}{\\sqrt2}  & S^{++}_2\n\\end{array}\n\\right),\n\\label{sextet2}\n\\end{equation}\nwhere $\\Phi^t_s$ means the transpose of the doublet $\\Phi_s$. Under the $SU(2)\\otimes U(1)_Y$ group the multiplets $\\Phi_{\\eta,\\rho,\\chi,s}$  in Eqs.~(\\ref{tripletos}) and (\\ref{sextet2}) transform as\n\\begin{equation}\n\\Phi_\\eta=\\left(\n\\begin{array}{c}\n\\eta^0\\\\ -\\eta^-_1\n\\end{array}\n\\right),\\;  \\Phi_\\rho=\\left(\n\\begin{array}{c}\n\\rho^+\\\\ \\rho^0\n\\end{array}\n\\right),\\; \\Phi_\\chi=\\left(\n\\begin{array}{c}\n\\chi^-\\\\ \\chi^{--}\n\\end{array}\n\\right), \\; \\Phi_s=\\left(\n\\begin{array}{c}\ns^+_2\\\\ s^0_2\n\\end{array}\n\\right),\n\\label{dubletos}\n\\end{equation}\nwhere these are doublets with weak hypercharge $Y=-1,+1,-3,+1$,  and $T$ in Eq.~(\\ref{sextet2})\n\\begin{equation}\nT=\\left( \\begin{array}{cc}\ns^0_1 &\\frac{s^+_1}{\\sqrt2} \\\\\n\\frac{s^+_1}{\\sqrt2} & S^{--}_1\n\\end{array}\n\\right),\n\\label{triplet}\n\\end{equation}\nis a triplet with $Y=2$. The $SU(2)$ singlets $\\eta^+_2,\\rho^{++},\\chi^0,S^{--}_2$ have $Y=+2,+4,0,+4$, respectively. \n\nThe total lepton number assignment in the scalar sector is~\\cite{Liu:1993gy}\n\\begin{equation} \nL(T^*,\\eta^-_2,\\Phi_\\chi,\\rho^{--},S^{--}_2)=+2,\\quad L(\\Phi_{\\eta,\\rho,s},\\chi^0)=0.\n\\label{ln}\n\\end{equation}\nNotice that the  only scalar doublet carrying lepton number is $\\Phi_\\chi$, and both members of the doublet have electric charge; for this reason, $\\langle \\Phi_\\chi\\rangle=0$ always. The existence of scalars carrying lepton number implies the possibility of explicit breaking of this quantum number in the scalar potential. It is possible to avoid such terms by imposing an appropriate discrete symmetry. We show one possibility in Table~\\ref{z7}.\nIn the table, $Q_{1,2}$ denote the quark triplets and $Q_3$, the quark antitriplet, $j_{mR}$ and $J$ are exotic quarks carrying electric charge of -4\/3 and 5\/3, in units of the positron charge $e$. For more details, see Ref.~\\cite{Machado:2013jca}.  \n\nSince the complex triplet  $T$ and the singlet (under $SU(2)$) $S^{++}_2$   carry lepton number they do not mix with  $\\Phi_{\\eta,\\rho,\\chi}$ if there are no lepton number violating terms in the scalar potential. As we will show below, there is some range of the parameter space that allows $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle\/v_W\\ll1$, where $v_W=246$ GeV is the electroweak energy scale.  In this situation the neutral scalar $s^0_1$ does not participate in the spontaneous symmetry breaking and $s^0_2$ has a small effect on the vector and charged lepton masses. At this stage, active neutrinos are massless and the charged leptons gain a rather small mass. However, these scalar fields are heavy and the charged leptons gain the appropriate mass through the interaction with the triplet $\\eta$ and an effective interaction involving the triplets $\\rho$ and $\\chi$. Similar to the standard model a non-Hermitian scalar triplet generates, at tree level, the neutrino masses by the interaction $(1\/\\Lambda)\\phi^0\\phi^0\\nu\\nu$ through the exchange of a complex triplet~\\cite{Ma:1998dn} (see  Fig.~\\ref{fig1}).\n\n\\begin{table}\n\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n\t\t& $Q_{(1,2)L}$ & $Q_{3L}$ & $U_{aR}$ & $D_{aR}$ & $\\Psi_{aL}$ & $\\nu_{aR}$ & $\\eta$ & $\\chi$ & $\\rho$ & $S$ & $j_{mR}$ & $J_{R}$ \\\\ \\hline\n\t\t$Z_{7}$ & 1 & $\\omega^6$ & $\\omega^5$ & $\\omega^3$ & $\\omega^2$ & $\\omega^6$ & $\\omega^3$ & $\\omega^6$\n\t\t& $\\omega^5$ & $\\omega^4$ & $\\omega^6$ & $\\omega^2$ \\\\ \\hline\n\t\t$Z_{3}$ & 1 & w$^2$ & w  & w & w  & w  & w  & 1 \n\t\t& w & w  & 1 & w \\\\ \\hline \\hline\n\t\\end{tabular}\n\t\\caption{Transformation properties of the fermion and scalar fields under $Z_{7} \\otimes Z_{3} $. Here $\\omega=e^{i2\\pi\/7}$ and w$=e^{i2\\pi\/3}$. } \n\t\\label{z7}\n\n\\end{table}\n\nThe most general scalar potential involving the three triplets and the sextet is~\\cite{Liu:1993gy}\n\\begin{equation}\nV(\\eta,\\rho,\\chi,S)=V^{(2)}+V^{(3)}+V^{(4a)}+\\cdots +V^{(4e)},\n\\label{potential}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nV^{(2)}&=&\\sum_{X=\\eta,\\rho,\\chi,S}\\mu^2_X\\textrm{Tr}(X^\\dagger X),\\nonumber \\\\\nV^{(3)}&=& \\frac{1}{3!} \\,f_1\\epsilon_{ijk} \\eta_i\\rho_j\\chi_k+f_2 (\\chi^T S^* \\rho+\\rho^T S^*\\chi) +f_3 \\eta^T S^* \\eta+\n\\frac{f_4}{3!}\\,\\epsilon_{ijk}\\epsilon_{mnl}\\;S^*_{im}S^*_{jn}S^*_{kl} ,\n\\nonumber \\\\\nV^{(4a)}&=&a_1(\\eta^\\dagger \\eta)^2+a_2(\\rho^\\dagger\\rho)^2+ a_3(\\chi^\\dagger \\chi)^2+\n\\chi^\\dagger\\chi(a_4\\eta^\\dagger\\eta+a_5\\rho^\\dagger\\rho)+a_6(\\eta^\\dagger\\eta)(\\rho^\\dagger \\rho)\\nonumber\\\\&& \na_7(\\chi^\\dagger \\eta)(\\eta^\\dagger\\chi)+a_8(\\chi^\\dagger\\rho)(\\rho^\\dagger\\chi)+\na_9(\\eta^\\dagger\\rho)(\\rho^\\dagger\\eta)+[a_{10}(\\chi^\\dagger\\eta)(\\rho^\\dagger\\eta)+H.c.],\\nonumber \\\\\nV^{(4b)}&=& b_1\\chi^\\dagger S\\hat{\\chi}\\eta+b_2\\rho^\\dagger S\\hat{\\rho}\\eta+b_3\\eta^\\dagger S[\\hat{\\chi}\\rho-\\hat{\\rho}\\chi]+H.c.,\\nonumber \\\\\nV^{(4c)}&=&   c_1\\textrm{Tr}[\\hat{\\eta}S\\hat{\\eta}S]+c_2\\textrm{Tr}[\\hat{\\rho}S\\hat{\\chi}S]+H.c.,\\nonumber \\\\\nV^{(4d)}&=&d_1(\\chi^\\dagger \\chi)\\textrm{Tr}S S^*+d_2 [(\\chi^\\dagger S)(S^* \\chi)]+d_3(\\eta^\\dagger\\eta)\\textrm{Tr}(S S^*)+d_4\\textrm{Tr}[(S^*\\eta)(\\eta^\\dagger S)]\\nonumber \\\\\n&&+d_5(\\rho^\\dagger\\rho)\\textrm{Tr}S S^*+d_6 \\textrm{Tr}[(S^* \\rho)(\\rho^\\dagger S)],\\nonumber \\\\\nV^{(4e)}&=& e_1(\\textrm{Tr}S S^*)^2+e_2\\textrm{Tr}(S S^*S S^*),\n\\label{potential2}\n\\end{eqnarray} \nand we have defined in the $V^{(b)}$ and $V^{(c)}$ terms $\\hat{x}_{ij}= \\epsilon_{ijk}x_k$, with $x=\\eta,\\rho,\\chi$. Notice also that $S^\\dagger=S^*$ since $S$ is a symmetric matrix. The conditions for having a potential bounded from below in Eq.~(\\ref{potential2}), under the conditions in Table~\\ref{z7}, are given in Appendix~\\ref{sec:appendixa}.\n\nConcerning the vacuum alignment and the conservation of the lepton number $L$, five possibilities can be considered (see also Ref.~\\cite{Liu:1993gy}) \n\\begin{enumerate}\n\\item[a)] Explicit $L$ violation and $\\langle s^0_{1,2}\\rangle\\not=0$ and arbitrary. This is the most general case and it has not been consider in the literature.  \n\\item[b)] Explicit lepton number violation in the scalar potential and  $\\langle s^0_1\\rangle = 0$ and  $\\langle s^0_2\\rangle\\not=0$ at tree level, but $\\langle s^0_1\\rangle\\not=0$  by loop corrections  \\cite{Pleitez:1992xh,Frampton:1993wu}.\n\\item[c)] No explicit $L$ violation and $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle=0$. Notwithstanding, the latter condition is not stable under radiative corrections unless a fine-tuning\nis imposed. \n \n\\item[d)] No explicit lepton number violation but $\\langle s^0_1\\rangle\\not=0$ and $\\langle s^0_2\\rangle\\not=0$. In this case there is a triplet Majoron that has been ruled out by the $Z$ invisible width~\\cite{GonzalezGarcia:1989zh}.\n\n\\item[e)] No explicit $L$ violation and $\\langle s^ 0_1\\rangle=0$ but $\\langle  s^0_2\\rangle\\not=0$. Although $L$ is conserved, there is violation of the family numbers $L_{e,\\mu,\\tau}$. In this case $\\langle s^0_1\\rangle=0$ is stable at tree and higher-order level~\\cite{Foot:1992rh,Pleitez:1992xh}. \n\\end{enumerate}\n\nHere we will consider the last case, (e), with $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle\/v_W \\ll 1$.  \nThis case occurs if the constraint $v^2_W=\\sum v^2_i=(246\\,\\textrm{GeV})^2$ (note that $i = \\rho, \\eta, s_1, s_2$) is saturated with the $v^2_\\eta$ and $v^2_\\rho$ as in Ref.~\\cite{Machado:2013jca}. \nMoreover, as we said before, in order to simplify the scalar potential, we impose a $Z_7$ discrete symmetry which forbids the $L$ violating terms, $f_3,f_4,a_{10},b_3,c_2=0$, but also the terms $c_1$ and $b_{1,2}$ are forbidden if we impose an additional discrete $Z_3$ symmetry. See Table~\\ref{z7}.\n\n\\section{Scalar mass spectra in the model}\n\\label{sec:masses}\n \nLet us consider the scalar potential in Eq.~(\\ref{potential2}) with the $Z_{7}\\otimes Z_3$ symmetries given in Table~\\ref{z7}. We make as usual $y^0=(1\/\\sqrt{2})(v_y+X^0_y+iI^0_y)$, where $y=\\eta,\\rho,\\chi$ and $s_2$.\nThe constraint equations, obtained by imposing that $\\partial V\/\\partial v_y=0$, being $V$ the potential in (\\ref{potential2}), under the conditions of the item e) above and considering all VEVs real, are given by\n\\begin{eqnarray}\n&& v_\\eta\\left[\\mu^2_\\eta+a_1v^2_\\eta+\\frac{a_6}{2}v^2_\\rho+\\frac{a_4}{2}v^2_\\chi+ d_3v^2_{s_2}+\\frac{f_1v_\\rho v_\\chi}{2\\sqrt{2}v_\\eta}\\right]=0,\\nonumber \\\\&&\nv_\\rho\\left[\\mu^2_\\rho+\\frac{a_6}{2}v^2_\\eta+ a_2v^2_\\rho+\\frac{a_5}{2}v^2_\\chi+ \\frac{d_{56}}{2} v^2_{s_2}+\\frac{(f_1v_\\eta+f_2v_{s_2})v_\\chi} {2\\sqrt{2}v_\\rho}\\right]=0,\\nonumber \\\\&&\nv_\\chi\\left[\\mu^2_\\chi+\\frac{a_4}{2}v^2_\\eta+\\frac{a_5}{2}v^2_\\rho+a_3v^2_\\chi+ d_{12}v^2_{s_2}+\\frac{(f_1v_\\eta v_\\rho+f_2v_{s_2})v_\\rho}{2\\sqrt{2}v_\\chi}\\right]=0,\\nonumber \\\\ && \nv_{s_2}\\left[2\\mu^2_S+d_3 v^2_\\eta+\\frac{d_{56}}{2} v^2_\\rho+\\frac{d_{12}}{2}v^2_\\chi+2e_{12}v^2_{s_2}+\\frac{f_2v_\\rho v_\\chi}{2\\sqrt{2}v_{s_2}}\\right]=0,\n\\label{ce}\n\\end{eqnarray} \nwhere we have defined $d_{56}=2d_5+d_6$, $d_{12}=2d_1+d_2$, and $e_{12}=2e_1+e_2$.\nNotice that no VEV can be zero unless $f_1=f_2=0$. However, if this were the case, the scalar potential has a non-Abelian symmetry larger than the rest of the Lagrangian. Hence, $f_1,f_2\\not=0$ in order that the gauge symmetry of the scalar potential is the same as the other terms of the Lagrangian. \nIn the case of the sextet, even if we had begun with $\\mu^2_S>0$ and $v_{s_2}=0$, the term $f_2\\not=0$ induces a tadpole which implies a counterterm leaving this VEV arbitrary. Hence, we assume $\\mu_S^2<0$ and $v_{s_2}\\not=0$ but is small in the sense that $v_{s_2}\/v_W\\ll1$. In Appendix~\\ref{sec:appendixb} we show explicitly the Goldstone bosons. \n\nHere, we will consider the mass matrices of the $C\\!P$-even scalars and other mass matrices in Appendix~\\ref{sec:appendixb}, assuming that $v_{s_2}\/v_\\chi<<1$. We also disregard some off-diagonal terms besides the ones from the assumption just mentioned, assuming that the respective diagonal elements are much bigger, to further simplify the matrices. In this approximation it is possible to obtain exact eigenvectors, but the case of the $CP$-even neutral scalars is more complicated and we will not consider here in detail. We show only that at least the two neutral scalars in the sextet are heavy. Analytical expressions, within the approximation above, of the masses and the eigenstates of the neutral $C\\!P$-odd and the charged sectors are given.\n \n\\subsection{Neutral $C\\!P$-even scalars}\n\\label{subsec:cpeven2}\n\nIn this sector the mass matrix $m^2$ is $5\\times5$ decompose, in the approximation used, into $4\\times4$ + $1\\times1$,  where \nthe $4\\times4$ matrix, in the basis $(X_\\eta^0,X_\\rho^0, X_\\chi^0, X_{s2}^0 )$, is given by\n\\begin{equation}\n\\left(\n\\begin{array}{cccc}\n2 a_1 v_\\eta^2-\\frac{f_1 v_\\rho v_\\chi}{\\sqrt{2} v_\\eta} & \t a_6v_\\eta v_\\rho+\\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\rho}{\\sqrt{2}}+a_4 v_\\eta v_\\chi & d_3 v_\\eta v_{s_2} \\\\\n& \\frac{4 a_2 v_\\rho^3-(\\sqrt{2} f_1 v_\\eta -f_2 v_{s_2}) v_\\chi}{2 v_\\rho} & \\frac{f_1 v_\\eta}{\\sqrt{2}}+\\frac{f_2 v_{s_2}}{2}+a_5 v_\\rho v_\\chi & \\frac{1}{2} [(2 d_5+d_6) v_\\rho v_{s_2}+f_2 v_\\chi] \\\\\n& & \\frac{4 a_3 v_\\chi^3-\\sqrt{2} f_1 v_\\eta v_\\rho-f_2 v_\\rho v_{s_2}}{2 v_\\chi} & \\frac{1}{2} [f_2 v_\\rho+(2 d_1+d_2) v_{s_2} v_\\chi] \\\\\n&  &  & (2 e_1+e_2) v_{s_2}^2-\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\label{cpeven1}\n\\end{equation}\nand the $1\\times1$ part implies the eigenvalue is $m^2_5=\\frac{(d_2  v_\\chi^2+2 d_4 v_\\eta^2 -d_6 v_\\rho^2 -2 e_2 v_{s_2}^2)v_{s_2}-2 f_2 v_\\rho v_\\chi}{4 v_{s_2}}>0$, which implies\n$d_2v^2_\\chi\/4-2 f_2 v_\\rho v_\\chi\/v_{s_2}>(2d_4v^2_\\eta- d_6 v_\\rho^2 -2 e_2 v_{s_2}^2)\/4$. Recall that $f_2<0$.\nHere the mass eigenstates are denoted by $m^2_i,\\;i=1,\\cdots,5$.\nIn fact, we can see that $m^2_5\\approx (1\/4)d_2v^2_\\chi-f_2v_\\rho v_\\chi\/v_{s_2}$; hence, this is a large mass.  One of them must be mainly that of the 125 GeV discovery at the LHC. Although we have not given details, according to the results in Ref.~\\cite{Machado:2013jca}, if $Re\\,\\rho^0=0.42 h+\\cdots$ this scalar has the same coupling with the top quarks that is numerically equal to that from the Higgs boson in the SM.  \n\n\\subsection{Neutral CP-odd scalars}\n\\label{subsec:cpodd2}\n\nThe mass matrix in Eq.~(\\ref{i2}), in the approximation of subsection ~\\ref{subsec:cpeven2},\nalso decomposes in  $3\\times3$ and  diagonal $2\\times2$ matrices. The $3\\times3$ matrix,\nin the basis $(I_\\eta^0,I_\\rho^0, I_\\chi^0) $, is\n\\begin{equation}\nM^2\\approx \\frac{1}{2} \\left(\n\\begin{array}{ccc}\n\\frac{f_1 v_\\rho v_\\chi}{\\sqrt{2} v_\\eta} & \\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\rho}{\\sqrt{2}} \\\\\n\\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\eta v_\\chi}{\\sqrt{2} v_\\rho} & \\frac{f_1 v_\\eta}{\\sqrt{2}} \\\\\n\\frac{f_1 v_\\rho}{\\sqrt{2}} & \\frac{f_1 v_\\eta}{\\sqrt{2}} & \\frac{f_1 v_\\eta v_\\rho}{\\sqrt{2} v_\\chi} \\\\\n\\end{array}\n\\right).\n\\label{i1}\n\\end{equation}\n\nThis matrix has two zero eigenvalues and another nonzero one: \n\\begin{equation}\n\\begin{array}{cl}\nM^2_1&=M^2_2=0 \\\\\nM^2_3&= -\\frac{f_1 \\left(v_\\eta^2 \\left(v_\\rho^2+v_\\chi^2\\right)+v_\\rho^2 v_\\chi^2\\right)}{\\sqrt{2} v_\\eta v_\\rho v_\\chi}\\approx -\\frac{f_1v_\\eta v_\\chi}{\\sqrt{2}v_\\rho}>0,\\;\\;f_1<0,\n\\label{a1}\n\\end{array}\n\\end{equation}\nwith respective eigenvectors given by the columns of the matrix:\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n-\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} & -\\frac{v_\\eta v_\\chi^2}{\\sqrt{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2\\right)}} & \\frac{v_\\chi}{v_\\eta \\sqrt{\\left(\\frac{1}{v_\\rho^2}+\\frac{1}{v_\\eta^2}\\right) v_\\chi^2+1}} \\\\\n0 & v_\\rho \\sqrt{\\frac{v_\\eta^2+v_\\chi^2}{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} & \\frac{v_\\eta v_\\chi}{\\sqrt{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} \\\\\n\\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} & -\\frac{v_\\eta^2 v_\\chi}{\\sqrt{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2\\right)}} & \\frac{v_\\eta v_\\rho}{\\sqrt{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} \\\\\n\\end{array}\n\\right).\n\\label{e1}\n\\end{equation}\n\nThe eigenvalues for the $2\\times2$ part are $M^2_4$ and $M^2_5$:\n\\begin{equation}\n\\begin{array}{cl}\nM^2_4&=-\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}}>0,\\;\\;f_2<0 \\\\\nM^2_5&=-\\frac{d_2 v_{s_2} v_\\chi^2+2 f_2 v_\\rho v_\\chi}{4 v_{s_2}}>0,\\;\\; 2\\vert f_2\\vert v_\\rho>d_2v_{s_2}.\n\\label{a2} \n\\end{array}\n\\end{equation}\nWe see from Eqs.~(\\ref{a1}) and (\\ref{a2}), that the three physical pseudoscalar fields are heavy and also induce the type-II seesawlike mechanism in the charged lepton sector.\n\n\\subsection{Singly charged scalars 1}\n\\label{subsec:charged12}\n\nIf the lepton number is conserved in the scalar potential (\\ref{potential}) under the conditions in Table~\\ref{z7}, as we are assuming, the charged scalar mass matrix splits in two sectors $(\\rho^+,\\eta_1^+,h^+)$ and $(\\chi^+, \\eta_2^+, h_2^+)$. \n\nIn the basis $(\\rho^+,\\eta_1^+,h^+)$, the mass matrix is\n\\begin{equation}\nm^2_+\\approx \\frac{1}{2}\\left(\n\\begin{array}{ccc}\n-\\frac{2 \\sqrt{2} f_1 v_\\eta v_\\chi-2 a_9 v_\\eta^2 v_\\rho}{4 v_\\rho} & \\frac{f_1 v_\\chi}{\\sqrt{2}}-\\frac{a_9 v_\\eta v_\\rho}{2} & 0 \\\\\n\\frac{f_1 v_\\chi}{\\sqrt{2}}-\\frac{a_9 v_\\eta v_\\rho}{2} & \\frac{1}{4} \\left(2 a_9 v_\\rho^2-\\frac{2 \\sqrt{2} f_1 v_\\rho v_\\chi}{v_\\eta}\\right) & 0 \\\\\n0 & 0 & -\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nwith the following eigenvalues,\n\\begin{equation}\n\\begin{array}{cl}\nm^2_{+1}&=0 \\\\\nm^2_{+2}&= -\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}}>0,\\;\\;f_2<0 \\\\\nm^2_{+3}&= \\frac{\\left(v_\\eta^2+v_\\rho^2\\right) \\left(a_9 v_\\eta v_\\rho-\\sqrt{2} f_1 v_\\chi\\right)}{2 v_\\eta v_\\rho}\\approx -\\frac{1}{\\sqrt2}\\frac{f_1 v_\\eta v_\\chi}{v_\\rho}>0,\n\\end{array}\n\\label{a3}\n\\end{equation}\nand the eigenvectors are given by the column of the matrix\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n\\frac{v_\\rho}{\\sqrt{v_\\eta^2+v_\\rho^2}} & 0 & -\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\rho^2}} \\\\\n\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\rho^2}} & 0 & \\frac{v_\\rho}{\\sqrt{v_\\eta^2+v_\\rho^2}} \\\\\n0 & 1 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nAccording to (\\ref{a3}), both physical charged scalar in this sector are very heavy. \n\n\\subsection{Singly charged scalars 2}\n\\label{subsec:charged22}\n\nIn the other singly charged sector the mass matrix in the basis\n$(\\chi^+, \\eta_2^+, h_2^+)$ is \n\\begin{equation}\nM_+^2 \\approx \\frac{1}{2} \\left(\n\\begin{array}{ccc}\n-\\frac{2 \\sqrt{2} f_1 v_\\eta v_\\rho-2 a_7 v_\\eta^2 v_\\chi}{4 v_\\chi} & \\frac{1}{2} \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right) & 0 \\\\\n\\frac{1}{2} \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right) & \\frac{1}{2} v_\\chi \\left(a_7 v_\\chi-\\frac{\\sqrt{2} f_1 v_\\rho}{v_\\eta}\\right) & 0 \\\\\n0 & 0 & -\\frac{v_\\chi (2 f_2 v_\\rho+d_2 v_{s_2} v_\\chi)}{4 v_{s_2}} \\\\\n\\end{array}\n\\right)\n\\end{equation}\n\nthe mass eigenvalues are,\n\n\\begin{equation}\n\\begin{array}{cl}\nM^2_{+1}&=0 \\\\\nM^2_{+2}&= -\\frac{v_\\chi (d_2 v_{s_2} v_\\chi+2 f_2 v_\\rho)}{4 v_{s_2}}>0, \\;\\;2\\vert f_2\\vert v_\\rho>d_2v_{s_2}v_\\chi\\\\\nM^2_{+3}&= \\frac{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right)}{2 v_\\eta v_\\chi}>0,\n\\label{a4}\n\\end{array}\n\\end{equation}\nwith the following eigenvectors:\n\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n-\\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} & 0 & \\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} \\\\\n\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} & 0 & \\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} \\\\\n0 & 1 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nAgain, the charged scalar masses in this sector might be heavy [see Eq.~(\\ref{a4})].\n\n\\subsection{Double charge scalars}\n\\label{subsec:2charges}\n\nThe mass matrix in the basis $(\\chi^{++},\\rho^{++},S_1^{++},S_2^{++})$ is \n\\begin{equation}\nM_{++}^2 \\approx \\frac{1}{2}\\left(\n\\begin{array}{cccc}\n\\frac{v_\\rho \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\chi} & \\frac{1}{2} \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right) & 0 & 0 \\\\\n\\frac{1}{2} \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right) & \\frac{v_\\chi \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\rho} & 0 & 0 \\\\\n0 & 0 & -\\frac{v_\\chi (2 f_2 v_\\rho+d_2 v_{s_2} v_\\chi)}{4 v_{s_2}} & 0 \\\\\n0 & 0 & 0 & \\frac{v_\\chi (d_2 v_{s_2} v_\\chi-2 f_2 v_\\rho)}{4 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nand the masses squared of the fields are,\n\n\\begin{equation}\n\\begin{array}{cl}\nM^2_{++1}&=0 \\\\\nM^2_{++2}&= \\frac{v_\\chi (d_2 v_{s_2} v_\\chi-2 f_2 v_\\rho)}{4 v_{s_2}}>0,\\;\\; \\\\\nM^2_{++3} &= -\\frac{v_\\chi (d_2 v_{s_2} v_\\chi+2 f_2 v_\\rho)}{4 v_{s_2}}>0 \\\\\nM^2_{++4}&= \\frac{\\left(v_\\rho^2+v_\\chi^2\\right) \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\rho v_\\chi}>0,\n\\end{array}\n\\end{equation}\nwith the respective eigenvectors:\n\\begin{equation}\n\\left(\n\\begin{array}{cccc}\n-\\frac{v_\\chi}{\\sqrt{v_\\rho^2+v_\\chi^2}} & 0 & 0 & \\frac{v_\\rho}{\\sqrt{v_\\rho^2+v_\\chi^2}} \\\\\n\\frac{v_\\rho}{\\sqrt{v_\\rho^2+v_\\chi^2}} & 0 & 0 & \\frac{v_\\chi}{\\sqrt{v_\\rho^2+v_\\chi^2}} \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nIn this doubly charged sector, all physical scalars might be heavy. \n\n\\section{The lepton masses and the PMNS matrix}\n\\label{sec:pmns}\n\nThe problem of the leptonic mixing matrix has already been considered in the context of some 331 models without quarks with exotic charge, right-handed neutrinos  transforming nontrivially under $SU(3)_L$, three right-handed Majorana leptons, and with flavor  symmetries like $T_7$ \\cite{Hernandez:2015cra}, $A_4$~\\cite{Hernandez:2015tna} and extra scalars transforming as singlets under $SU(3)$. In these sort of models an $S_3$~\\cite{Hernandez:2014lpa} symmetry is also used. Here, however, we will consider the m331 using only the gauge symmetries and also right-handed neutrinos as the extra degrees of freedom. \n\nAs said before, the possibility that the lepton masses are generated by an effective dimensional operator was presented in Ref.~\\cite{Montero:2001tq}. However, here we will consider the case when the sextet is introduced but its degrees of freedom are heavy enough to generate an effective nonrenormalizable interaction. This implements a type-II seesaw mechanism in the charged lepton sector. The latter situation arises when the members of the sextet are very heavy and one of the neutral scalars gains a zero VEV and the other one a small VEV. \n\nThe Yukawa interactions are given by\n\\begin{eqnarray}\n-\\mathcal{L}^{lep}_1&=&-\\frac{1}{2}\\epsilon_{ijk}\\,\\overline{(\\Psi_{ia})^c}G^\\eta_{ab} \\Psi_{jb}\\eta_k+\\frac{1}{2\\Lambda_l}\\,\n\\overline{(\\Psi_{a})^c} \\tilde{G}^s_{ab} (\\chi^*\\rho^\\dagger+ \\rho^*\\chi^\\dagger)\\Psi_{b} \\nonumber \\\\ &+&\n\\overline{(\\Psi_{aL})}(G^\\nu)_{ab}\\nu_{aR}\\eta+ \\overline{(\\nu_{aR})^c}(M_R)_{ab}\\nu_{bR}+\\overline{(\\Psi_{ia})^c}G^s_{ab} \\Psi_{jb}S_{ij}+H.c.,\n\\label{effective1}\n\\end{eqnarray}\nwhere $\\Lambda_l$ is a mass scale related to the origin of the dimension five interaction.\nThe second term in the first line of Eq.~(\\ref{effective1}), the dimension-five operator, is generated by the loop in Fig.~\\ref{fig1}. Notice that in Eq.~(\\ref{effective1}) the interactions with the sextet appear and, although they do not contribute significantly to the charged lepton masses, the degrees of freedom in this multiplet might be exited at high energies.\n\nWe will assume, for the sake of simplicity, that $M_R$ is diagonal and that $m_{3R}\\equiv M>m_{R1},m_{R2}$, and $M^{-1}_R=(1\/M)\\bar{M}_R$, where \n$\\bar{M}=\\textrm{diag}(r_{1},r_{2},1))$ and $r_1\\equiv M\/m_{R1},r_2\\equiv M\/m_{R2}$. In this case we have the mass matrices in the lepton sector\n\\begin{equation}\nM^\\nu\\approx -\\frac{v^2_\\eta}{2}G^\\nu\\frac{\\bar{M}_R}{M}G^{\\nu T},\\quad\nM^l_{ab}=G^\\eta_{ab}\\frac{v_\\eta}{\\sqrt2}+\\frac{1}{\\Lambda_l}\\tilde{G}^s_{ab}v_\\rho v_\\chi.\n\\label{mmassa3}\n\\end{equation}\nIf it is the sextet, through the interaction $\\overline{(\\Psi_{ai})^c} G^s_{ab} \\Psi_{jb}S_{ij}$ and the $f_2$ trilinear term in the scalar potential involving $\\rho,\\chi$ and $S$, then  we have $1\/\\Lambda_l=f_2\/m^2_{s_2} $ and\n$\\tilde{G}^s=G^s$. \n\nThese mass matrices are diagonalized as follows: \n\\begin{equation} \n\\hat{M}^\\nu=V^{\\nu T}_LM^\\nu V^\\nu_L,\\;\\; \\hat{M}^l=V^{l\\dagger}_L M^l V^l_R,\n\\label{def}\n\\end{equation} \nwhere $\\hat{M}^\\nu = diag (m_1, m_2, m_3),\\;\\hat{M}^l = diag (m_e, m_\\mu, m_\\tau)$. The relation between symmetry eigenstates (primed) and mass eigenstates (unprimed) are $l^\\prime_{L,R}=V^l_{L,R}l_{L,R}$\nand $\\nu^\\prime_L=V^\\nu_L \\nu_L$, where\n$l^\\prime_{L,R}=(e^\\prime,\\mu^\\prime,\\tau^\\prime)^T_{L,R}$, $l_{L,R}=(e,\\mu,\\tau)^T_{L,R}$  and\n$\\nu ^\\prime_L=(\\nu_e\\,\\nu_\\mu\\,\\nu_\\tau)^T_L$\nand $\\nu_L=(\\nu_1\\,\\nu_2\\,\\nu_3)_L$. \n\nIn the following we assume  $v_\\chi\\approx \\Lambda_l$ and, as in Ref.~\\cite{Machado:2013jca}, $v_\\rho\\sim 54,v_\\eta\\sim 240$ GeV. \nThe neutrino mass matrix is as in Eq.~(\\ref{mmassa3}). Solving simultaneously the following equations:\n\\begin{equation}\n\\hat{M}^\\nu_{ab}=V^{\\nu T}_L M^\\nu_{ab}V^\\nu_L,\\quad \nV^{l\\dagger}_L M^l M^{l\\dagger}V^l_L=V^{l\\dagger}_R M^{l\\dagger}  M^lV^l_R=(\\hat{M}^l)^2,\\quad V_{PMNS}=V^{l\\dagger}_LV^\\nu_L,\n\\end{equation}\nwhere $M^\\nu$ and $M^l$ are defined in Eq.~(\\ref{mmassa3}), and $V_{PMNS}$ is the mixing matrix in the lepton sector (PMNS), the values for the charged lepton masses obtained are (in MeV) $(m_e,m_\\mu,m_\\tau)=(0.509648,105.541,1775.87)$\nand for the neutrinos masses (in eV) $(m_1,m_2,m_3)=(0.051,-0.0194,0.0174)$ which are consistent with \n$\\Delta m^2_{23}=2.219\\times10^{-3}\\,(\\textrm{eV})^2$ and $\\Delta m^2_{21}=7.5\\times 10^{-5}\\,(\\textrm{eV})^2$. \nThese values for the masses arise from the following values for the Yukawa matrices: $v^2_\\eta\/M=0.33$ eV, \n$G^\\nu_{11}=0.109$, $G^\\nu_{12}=0.097$, \n$G^\\nu_{13}=0.101$, $G^\\nu_{22}=0.09$, $G^\\nu_{23}=-0.02$, $G^\\nu_{33}=0.0106$ in the neutrino sector; and \n$G^s_{11}=-0.0453,G^s_{12}=-0.0076,G^s_{13}=-0.0008,G^s_{22}=0.0015,\nG^s_{23}=0.0001,G^s_{33}=1.84\\times10^{-5}$,\n$G^\\eta_{12}=,G^\\eta_{13}=G^\\eta_{13}=-0.00001$ in the charged lepton sector. The only way to avoid the latter fine-tunning is \nto consider $v_\\eta$ smaller, but in the context of the Ref.~\\cite{Machado:2013jca} this VEV is already fixed.\n\nWe obtain for the diagonalization matrices\n\\begin{equation}\nV^\\nu_L\\approx\\left(\\begin{array}{ccc}\n-0.24825& -0.57732& 0.77786\\\\\n0.73980 &-0.40539  &0.53698 \\\\\n-0.62535 &-0.70877 &0.32647 \\\\\n\\end{array}\\right)\n\\label{lep1}\n\\end{equation}\nand\n\\begin{equation}\nV^l_L\\approx\\left(\\begin{array}{ccc}\n-0.00985 &0.01457 &-0.99984 \\\\\n-0.31848& -0.94787 &-0.01067 \\\\\n0.94788&-0.31833 & -0.01398\\\\\n\\end{array}\\right),\\quad \nV^l_R\\approx\\left(\\begin{array}{ccc}\n0.00501&0.00716 & 0.99996\\\\\n0.00261&0.9910  & -0.00717\\\\\n0.99998 &-0.00265 & -0.00499\\\\\n\\end{array}\\right)\n\\label{lep2}\n\\end{equation}\n\nNotice that we have defined the lepton mixing matrix as $V_{PMNS}=V^{l\\dagger}_LV^\\nu_L$, which means that this matrix appears in the charged currents coupled to $W^-$.  We obtain from Eqs.~(\\ref{lep1}) and (\\ref{lep2}) the following values for the PMNS matrix:\n\\begin{equation}\n\\vert V_{PMNS}\\vert \\approx\\left(\\begin{array}{ccc}\n0.826& 0.548 & 0.130\\\\\n0.506&0.618  &0.602 \\\\\n0.249 &0.563 &0.788 \\\\\n\\end{array}\\right),\n\\label{pmns}\n\\end{equation}\nwhich is in agreement, within 3$\\sigma$, with the experimental data given in Ref.~\\cite{GonzalezGarcia:2012sz},\n\\begin{equation}\n\\vert V_{PMNS}\\vert \\approx\\left(\\begin{array}{ccc}\n0.795-0.846& 0.513-0.585 & 0.126-0.178\\\\\n0.205-0.543&0.416-0.730  &0.579 - 0.808 \\\\\n0.215 - 0.548 &0.409 - 0.725 &0.567 -0.800 \\\\\n\\end{array}\\right),\n\\label{pmnsexp}\n\\end{equation}\nand we see that it is possible to accommodate all lepton masses and the PMNS matrix. We do not consider $CP$ violation here.\n\n\\section{Conclusions}\n\\label{sec:con}\n\nWe have shown that even if we introduce the sextet in such a way that it practically does not contribute to the lepton masses because of the small VEVs, and since its components are very heavy, it might generate the dimension-five operator involving only the triplets $\\rho$ and $\\chi$ in Eq.~(\\ref{effective1}) trough a process like the one shown in Fig.~\\ref{fig1}. A similar operator can be obtained for the neutrinos as in Ref.~\\cite{Montero:2001tq}, but here we prefer to introduce right-handed neutrinos in order to implement a type-I seesaw mechanism. Moreover, in the charged lepton sector the mass generation is similar to the type-II seesaw mechanism inducing small masses for neutrinos through the exchange of a heavy non-Hermitian triplet~\\cite{Cheng:1980qt,Konetschny:1977bn,Magg:1980ut}.\nThe existence of several mechanisms to generate this interaction in the context of the standard model have been shown in the literature~\\cite{Ma:1998dn,Bonnet:2012kz,Sierra:2014rxa}.  Notwithstanding, the effective operator in Eq.~(\\ref{effective1}) can be originated by the effects of higher-dimension operators. \n\nIn our case, the sextet is introduced in the model as in the m331 model, and through its interactions with the other scalars in the scalar potential and with leptons in the Yukawa interactions, their degrees of freedom can be exited at high energies mainly in lepton colliders.\nNotice, that all extra scalars in the model are heavy except for two neutral scalars that correspond to the fields in a two-Higgs-doublet extension of the SM.\nWe also show the conditions under which we have a weak copositivity of the scalar potential and generate the vacuum stability at tree level and a global minimum as well. It is interesting to study the same problem at the one-loop level. For instance, in a model with two doublets (one of them inert) taking into account a neutral scalar with a mass of 125 GeV, the stability of the vacuum was shown in Ref.~\\cite{Goudelis:2013uca}; however, such analysis is beyond the scope of our paper.\n\nIn order to obtain the correct mass for the charged leptons, besides the effective interaction, it is necessary to consider the interactions with the triplet $\\eta$. For neutrinos, as we are considering that $v_{s_1}=0$, we have introduced right-handed components to generate the type-I seesaw mechanism. With the unitary (orthogonal if we neglect phases) matrices that diagonalize the mass matrices in the lepton sector, it is possible to accommodate a realistic Pontecorvo-Maki-Nakawaga-Sakata matrix. The constraints on the masses of the extra particles in the m331 model coming from lepton violation processes will be considered elsewhere.\nIn the present context we recall that it is the neutral scalar $\\rho^0$ which has the larger projection on the neutral scalar with a mass of near 125 GeV. For details see Ref.~\\cite{Machado:2013jca}.\n\nFinally, we would like to discuss the differences between our present model and those in Refs.~\\cite{Montero:2001tq,Ferreira:2011hm}. Our model is the usual minimal 3-3-1 (in the sense that the lepton sector consists only of the known leptons) and the four scalar multiplets, three triplets, and one sextet \\textit{plus} right-handed neutrinos. Although the sextet has interactions mediated by scalars [see Eq.~(\\ref{effective1})], its neutral components do not contribute significantly to the lepton masses.\nThe degrees of freedom in the sextet decouple at low energies; however, its interactions have to be taken into account for some phenomenology at sufficiently high energy. For instance, if this mechanism is implemented at the 100 GeV--1TeV scale, the interactions of the charged leptons with the left-handed neutrinos could have some signature at the LHC or other colliders. (See \\cite{Ng:2015hba} and references therein.)\n\nIn  Ref.~\\cite{Montero:2001tq}, only three triplets $\\eta,\\rho$ and $\\chi$ were considered, being the sextet avoided at all, with the charged leptons and neutrinos gain mass only through nonrenormalizable interactions. The model of Ref.~\\cite{Ferreira:2011hm} is  considered the so called \"reduced\" 3-3-1 model with only the triplets $\\rho$ and $\\chi$ (with no $\\eta$ and $S$ at all). Therefore, for generating all the fermion masses they need, besides the usual Yukawa interactions with these triplets, nonrenormalizable interactions involving the same triplets. Although this is an interesting situation, the model with only the triplets $\\rho$ and $\\chi$ has experimental troubles if we accept the existence of the Landau-like pole. See Ref.~\\cite{Dong:2014bha} for a discussion of the 3-3-1 models with only two triplets, in particular the troubles with the ``reduced\" minimal 3-3-1 model~\\cite{Ferreira:2011hm}.   \n\nThus, in the limit of a heavy sextet, our model is a 3-3-1 model with three triplets and the sextet, but the latter one does not contribute to the lepton masses and almost not at all to the spontaneous symmetry breaking since its VEVs are zero or very small in the sense of $v_{s_2}\/v_W\\ll1$. The lepton interactions with the sextet and the interactions among all the scalars \nbecome important only at high energies. Our case is more similar to the type-II seesaw mechanism in which a complex heavy triplet is in charge of generating the neutrino masses~\\cite{Ma:1998dn}.\n\n\n\\acknowledgments\n\nThe authors would like to thank for full support to CNPq  (GDC), CAPES (ACBM) and for partial\nsupport to CNPq (VP).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider the problem of online learning in Markov decision processes (MDPs) where a learner sequentially interacts with an environment \nby repeatedly taking actions that influence the future states of the environment while incurring some immediate costs. The goal of the \nlearner is to choose its actions in a way that the accumulated costs are as small as possible. Several variants of this problem have been \nwell-studied in the literature, primarily in the case where the costs are assumed to be independent and identically distributed \n\\citep{sutton,Puterman1994,ndp,Sze10}. In the current paper, we consider the case where the costs are generated by an arbitrary external \nprocess and the learner aims to minimize its total loss during the learning procedure---conforming to the learning paradigm known as \n\\emph{online learning} \\citep{CBLu06:Book,SS12}. In the online-learning framework, the performance of the learner is measured in terms of \nthe \\emph{regret} defined as the gap between the total costs incurred by the learner and the total costs of the best comparator chosen\nfrom a pre-specified class of strategies. In the case of online learning in MDPs, a natural class of strategies is the set of all \nstate-feedback policies: several works studied minimizing regret against this class both in the \nstationary-cost~\\citep{bartlett09regal,jaksch10ucrl,AYSze11} and the non-stochastic \nsetting~\\citep{even-dar09OnlineMDP,yu09ArbitraryRewards,neu10o-ssp,neu12ssp-trans,ZiNe13,DGS14,neu14o-mdp-full,AYBK14}. In the \nnon-stochastic setting, most works \nconsider MDPs with unstructured, finite state spaces and guarantee that the regret increases no faster than~$O(\\sqrt{T})$ as the number of \ninteraction rounds $T$ grows large. A notable exception is the work of \\citet{AYBK14}, who consider the special \ncase of (continuous-state) linear-quadratic control with arbitrarily changing target states, and propose an algorithm that guarantees a \nregret bound of $O(\\log^2 T)$.\n\nIn the present paper, we study another special class of MDPs that turns out to allow fast rates. Specifically, we consider the class of \nso-called \\emph{linearly solvable MDPs} (in short, LMDPs), first proposed and named so by \\citet{Tod06}.\nThis class takes its name \nafter the special property that the Bellman optimality equations characterizing\nthe optimal behavior policy take the form of a system of linear equations,\nwhich makes optimization remarkably straightforward in such problems. The\ncontinuous formulation (in both space and time) was discovered independently\nby~\\citet{Kap05} and is known as \\emph{path integral control}.\nLMDPs have many interesting properties. For example, optimal control laws for\nLMDPs can be linearly combined to derive composite optimal control laws\nefficiently~\\citep{Tod09}. Also, the inverse optimal control problem in LMDPs can be expressed as a convex optimization problem~\\citep{dvijotham2010inverse}.\nLMDPs generalize an existing duality between optimal control computation and Bayesian inference~\\citep{todorov2008general}.\nIndeed, the popular belief propagation algorithm used in dynamic probabilistic graphical models is\nequivalent the the power iteration method used to solve LMDPs~\\citep{KGO12}.\n\nThe LMDP framework has found applications in robotics~\\citep{latentkl,ariki}, crowdsourcing~\\citep{Abbasi}, and controlling the growth \ndynamics of complex networks~\\citep{klnetwork}. The related path integral control framework of \\citet{Kap05} has been applied in several \nreal-world tasks, including robot navigation~\\citep{KinjoFN2013}, motor skill \nreinforcement learning~\\citep{TheodorouJMLR10a,rombokas2013reinforcement,pireps}, aggressive car maneuvering~\\citep{aggressive} or \nautonomous flight of teams of quadrotors~\\citep{pi_uavs}.\n\n\nIn the present paper, we show that besides the aforementioned properties, the structure of LMDPs also enables constructing efficient online \nlearning procedures with very low regret. In particular, we show that, under some mild assumptions on the structure of the LMDP, the \n(conceptually) simplest online learning strategy of \\emph{following the leader} guarantees a regret of order $\\log^2 T$, vastly improving \nover the best known previous result by \\citet*{GRW12}, who prove a regret bound of order $T^{3\/4+\\epsilon}$ for arbitrarily small \n$\\epsilon>0$ under the same assumptions. Our approach is based on the observation that the optimal control law arising from the LMDP \nstructure is a smooth function of the underlying cost function, enabling rapid learning without any regularization whatsoever.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:bck} introduces the formalism of LMDPs and summarizes some basic facts that \nour technical content is going to rely on. Section~\\ref{sec:online} describes our online learning model. Our learning algorithm is \ndescribed in Section~\\ref{sec:ftl} and analyzed in Section~\\ref{sec:analysis}. Finally, we draw conclusions in Section~\\ref{sec:discussion}.\n\n\\paragraph{Notation.} We will consider several real-valued functions over a finite state-space $\\mathcal{X}$, and we will often treat these functions \nas finite-dimensional (column) vectors endowed with the usual definitions of the $\\ell_p$ norms. The set of probability distributions over \n$\\mathcal{X}$ will be denoted as $\\Delta(\\mathcal{X})$. Indefinite sums with running variables $x,y$ or $s$ are understood to run through all $\\mathcal{X}$.\n\n\\section{Background on linearly solvable MDPs}\\label{sec:bck}\nThis section serves as a quick introduction into the formalism of linearly solvable MDPs (LMDPs, \\cite{Tod06}). These decision processes \nare defined by the tuple $\\ev{\\mathcal{X},P,c}$, where $\\mathcal{X}$ is a finite set of \\emph{states}, $P:\\mathcal{X} \\rightarrow \\Delta(\\mathcal{X})$ is a transition kernel called \nthe \\emph{passive dynamics} (with $P(x'|x)$ being the probability of the process moving to state $x'$ given the previous state $x$) and \n$c:\\mathcal{X}\\rightarrow[0,1]$ is the \\emph{state-cost function}. Our Markov decision process is a sequential decision-making problem where the initial \nstate $X_0$ is drawn from some distribution $\\mu_0$, and the following steps are repeated for an indefinite number of rounds $t=1,2,\\dots$:\n\\begin{enumerate}[leftmargin=.7cm]\n \\item The learner chooses a transition kernel $Q_t:\\mathcal{X}\\rightarrow \\Delta(\\mathcal{X})$ satisfying $\\mathop{supp}Q_t(\\cdot|x) \\subseteq \n\\mathop{supp}P(\\cdot|x)$ \nfor all $x\\in\\mathcal{X}$.\n \\item The learner observes $X_t\\in\\mathcal{X}$ and draws the next state $X_{t+1}\\sim Q(\\cdot|X_t)$.\n \\item The learner incurs the cost\n \\[\n  \\ell(X_t,Q_t) = c(X_t) + \\kl{Q_t(\\cdot|X_t)}{P(\\cdot|X_t)},\n \\]\n where $\\kl{q}{p}$ is the relative entropy (or Kullback-Leibler divergence) between the probability distributions $p$ \nand $q$ defined as $\\kl{q}{p}=\\sum_x q(x)\\log \\frac{q(x)}{p(x)}$.\n\\end{enumerate}\n\nThe state-cost function $c$ should be thought of as specifying the objective for the learner in the MDP, while the relative-entropy term \ngoverns the costs associated with significant deviations from the passive dynamics. Accordingly, we refer to this component as the \n\\emph{control \ncost}. A central question in the theory of Markov decision problems is finding a behavior policy that minimizes (some notion of) the \nlong-term total costs. In this paper, we consider the problem of \\emph{minimizing the long-term average cost-per-stage} \n$\\lim\\sup_{T\\rightarrow\\infty} \\frac 1T \\sum_{t=1}^T \\ell(X_t,Q_t)$. Assuming that the passive dynamics $P$ is aperiodic and irreducible, this \nlimit is minimized by a \\emph{stationary} policy $Q$ (see, e.g., \\citet[Sec.~8.4.4]{Puterman1994}). Below, we provide two distinct \nderivations for the optimal stationary policy that minimizes the average costs under this assumption.\n\n\\subsection{The Bellman equations}\\label{sec:bellman}\nWe first take an approach rooted in dynamic programming \\citep{Ber07:DPbookVol2}, following \\citet{Tod06}. Under our assumptions, the \noptimal stationary policy minimizing the average cost is given by finding the solution to the Bellman optimality equation\n\\begin{equation}\\label{eq:bellman_V}\n v(x) = c(x) - \\lambda + \\min_{q\\in\\Delta(\\mathcal{X})} \\ev{\\kl{q}{P(\\cdot|x)} + \\sum_{x'} q(x') v(x')}\n\\end{equation}\nfor all $x\\in\\mathcal{X}$, where $v$ is called the \\emph{optimal value function} and $\\lambda\\in\\mathbb{R}$ is the average cost associated with the \noptimal policy\\footnote{This solution is guaranteed to be unique up to a constant shift of the values: if $v$ is a solution, then so is \n$v + a$ for any $a\\in \\mathbb{R}$. Unless stated otherwise, we will assume that $v$ is such that $v(x_0) = 0$ holds for a fixed state \n$x_0\\in\\mathcal{X}$.}.  Linearly solvable MDPs get their name from the fact that the Bellman optimality \nequation can be rewritten in a simple \nlinear form. To see this, observe that by elementary calculations involving Lagrange multipliers, we have\n\\begin{align*}\n \\min_{q\\in\\Delta(\\mathcal{X})} \\ev{\\kl{q}{P(\\cdot|x)} + \\sum_{x'} q(x') v(x')} =& -\\log \\sum_{x'} P(x'|x) e^{-v(x')},\n\\end{align*}\nso, after defining the exponentiated value function $z(x) = e^{-v(x)}$ for all $x$, plugging into Equation~\\eqref{eq:bellman_V} and \nexponentiating both sides gives\n\\begin{equation}\\label{eq:bellman_Z}\n z(x) = e^{\\lambda - c(x)} \\sum_{x'} P(x'|x) z(x').\n\\end{equation}\nRewriting the above set of equations in matrix form, we obtain the linear equations\n\\[\n e^{-\\lambda} z = GPz,\n\\]\nwhere $G$ is a diagonal matrix with $G_{ii} = e^{-c(i)}$. By the \\emph{Perron-Frobenius theorem} (see, e.g., Chapter~8 of \\cite{Mey00}) \nconcerning positive matrices, the above system of linear equations has a unique\\footnote{As in the case of the Bellman equations, this \nsolution is unique up to a \\emph{scaling} of $z$.} solution satisfying $z(x)\\ge 0$ for all $x$, and this \neigenvector corresponds to the largest eigenvalue $e^{-\\lambda}$ of $GP$. Since the solution of the Bellman optimality equation \n\\eqref{eq:bellman_V} is unique (up to a constant shift corresponding to a constant scaling of $z$), we obtain that $\\lambda$ is the average \ncost of the optimal policy. In summary, the Bellman optimality \nequation takes the form of a \\emph{Perron--Frobenius eigenvalue problem}, which can be efficiently solved by iterative methods such as the \nwell-known power method for finding top eigenvectors. Finally, getting back to the basic form~\\eqref{eq:bellman_V} of the Bellman equations, \nwe can conclude after simple calculations that the optimal policy can be computed  for all $x,x'$ as\n\\[\n Q(x'|x) = \\frac{P(x'|x) z(x')}{\\sum_y P(y|x) z(y)}.\n\\]\n\n\\subsection{The convex optimization view}\\label{sec:convex}\nWe also provide an alternative (and, to our knowledge, yet unpublished) view of the optimal control problem in LMDPs, based on convex \noptimization. For the purposes of this paper, we find this form to be more insightful, as it enables us to study our \nlearning problem in the framework of online convex optimization \\citep{Haz11,Haz16,SS12}. To derive this form, observe that under \nour assumptions, every feasible policy $Q$ induces a stationary distribution $\\mu_Q$ over the state space $\\mathcal{X}$ satisfying $\\mu_Q^\\mathsf{\\scriptscriptstyle T} \n= \\mu_Q^\\mathsf{\\scriptscriptstyle T} Q$. This stationary distribution and the policy together induce a distribution $\\pi_Q$ over $\\mathcal{X}^2$ defined for all $x,x'$ \nas $\\pi_Q(x,x') = \\mu_Q(x) Q(x'|x)$. We will call $\\pi_Q$ as the \\emph{stationary transition measure} induced by $Q$, which is motivated by \nthe observation that $\\pi_Q(x,x')$ corresponds to the probability of observing the transition $x\\rightarrow x'$ in the equilibrium state: \n$\\pi_Q(x,x') = \\lim_{T\\rightarrow \\infty} \\frac 1T \\sum_{t=1}^T \\PP{X_t = x, X_{t+1}=x'}$. Notice that, with this notation, the average \ncost-per-stage of policy $Q$ can be rewritten in the form\n\\[\n\\begin{split}\n &\\lim_{T\\rightarrow\\infty} \\frac 1T \\sum_{t=1}^T \\EE{\\ell(X_t,Q)} = \\sum_{x} \\mu_Q(x) \\Bpa{c(x) + \\kl{Q(\\cdot|x)}{P(\\cdot|x)}}\n \\\\\n &\\qquad\\qquad= \\sum_{x,x'} \\pi_Q(x,x') \\pa{c(x) + \\log\\frac{\\pi_Q(x,x')}{P(x'|x)\\sum_y \\pi_Q(x,y)}}\n \\\\\n &\\qquad\\qquad= \\sum_{x,x'} \\pi_Q(x,x') \\log\\frac{\\pi_Q(x,x')}{\\sum_y \\pi_Q(x,y)} + \\sum_{x,x'} \\pi_Q(x,x') \\pa{c(x) - \\log\\pa{P(x'|x)}}.\n\\end{split}\n\\]\nThe first term in the final expression above is the \\emph{negative conditional entropy} of $X'$ relative to $X$, where $(X,X')$ is a pair \nof random states drawn from $\\pi_Q$. Since the negative conditional entropy is convex in $\\pi_Q$ (for a proof, see \nAppendix~\\ref{app:convexity}) and the second term in the expression is linear in $\\pi_Q$, we can see that $\\lambda$ is a convex function of \n$\\pi_Q$. This suggests that we can \nview the optimal control problem as having to find a feasible stationary transition measure $\\pi$ that minimizes the expected costs. \nIn short, defining \n\\begin{equation}\\label{eq:statcost}\n f(\\pi;c) = \\sum_{x,x'} \\pi(x,x') \\pa{c(x) + \\log\\frac{\\pi(x,x')}{P(x'|x)\\sum_y \\pi(x,y)}}\n\\end{equation}\nand $\\Delta(M)$ as the (convex) set of feasible stationary transition measures $\\pi$ satisfying\n\\begin{equation}\\label{eq:statdist}\n\\begin{split}\n \\sum_{x'} \\pi(x,x') &= \\sum_{x''} \\pi(x'',x) \\;\\;\\;\\;\\;\\,\\qquad (\\forall x),\n \\\\\n \\sum_{x,x'} \\pi(x,x')&=1,\n \\\\\n \\pi(x,x') &\\ge 0 \\qquad\\qquad\\qquad\\qquad (\\forall x,x'),\n \\\\\n \\pi(x,x') &= 0 \\qquad\\qquad\\qquad\\qquad (\\forall x,x': P(x'|x)=0),\n\\end{split}\n\\end{equation}\nthe optimization problem can be succinctly written as $\\min_{\\pi\\in\\Delta(M)} f(\\pi;c)$. In Appendix~\\ref{app:dual}, we provide a \nderivation of the optimal control given by Equation~\\eqref{eq:bellman_Z} starting from the formulation given above. We also remark that our \nanalysis will heavily rely on the fact that $f(\\pi;c)$ is affine in $c$. \n\n\n\n\n\n\n\\section{Online learning in linearly solvable MDPs}\\label{sec:online}\nWe now present the precise learning setting that we consider in the present paper.\nWe will study an online learning scheme where for each round $t=1,2,\\dots,T$, the following steps are repeated:\n\\begin{enumerate}[leftmargin=.7cm]\n \\item The learner chooses a transition kernel $Q_t:\\mathcal{X}\\rightarrow \\Delta(\\mathcal{X})$ satisfying $\\mathop{supp}Q_t(\\cdot|x) \\subseteq \n\\mathop{supp}P(\\cdot|x)$ for all $x\\in\\mathcal{X}$.\n \\item The learner observes $X_t\\in\\mathcal{X}$ and draws the next state $X_{t+1}\\sim Q_t(\\cdot|X_t)$.\n \\item Obliviously to the learner's choice, the environment chooses state-cost function $c_t:\\mathcal{X}\\rightarrow[0,1]$.\n \\item The learner incurs the cost\n \\[\n  \\ell_t(X_t,Q_t) = c_t(X_t) + \\kl{Q_t(\\cdot|X_t)}{P(\\cdot|X_t)}.\n \\]\n \\item The environment reveals the state-cost function $c_t$.\n\\end{enumerate}\nThe key change from the stationary setting described in the previous section is that the state-cost function now may \\emph{change \narbitrarily} between each round, and the learner is only allowed to observe the costs \\emph{after it has made its decision}. We stress \nthat we assume that the learner \\emph{fully knows} the passive dynamics, so the only difficulty comes from having to deal with the \nchanging costs. As usual in the online-learning literature, our goal is to do nearly as well as the best \\emph{stationary} policy \nchosen in hindsight after observing the entire sequence of cost functions. To define our precise performance measure, we first define the \naverage reward of a policy $Q$ as\n\\[\n \\mathcal{L}_T(Q) = \\EE{\\sum_{t=1}^T \\ell_t(X_t',Q)},\n\\]\nwhere the state trajectory $X_t'$ is generated sequentially as $X_t'\\sim Q(\\cdot|X_{t-1}')$ and the expectation integrates over the \nrandomness of the \ntransitions. Having this definition in place, we can specify the best stationary policy\\footnote{The existence of the minimum is warranted \nby the fact that $\\mathcal{L}_T$ is a continuous function bounded from below on its compact domain.} $Q^*_T = \\mathop{\\mbox{ \\rm arg\\,min}}_{Q} \\mathcal{L}_T(Q)$ and define \nour \nperformance measure as the (total expected) \\emph{regret} against $Q^*_T$:\n\\[\n R_T = \\EE{\\sum_{t=1}^T \\ell_t(X_t,Q_t)} - \\mathcal{L}_T(Q^*_T),\n\\]\nwhere the expectation integrates over both the randomness of the state transitions and the potential randomization used by the \nlearning algorithm. Having access to this definition, we can now formally define the goal of the learner as having to come up \nwith a sequence of policies $Q_1,Q_2,...$ that guarantee that the total regret grows sublinearly, that is, that the average per-round \nregret asymptotically converges to zero.\n\nFor our analysis, it will be useful to define an idealized version of the above online optimization problem, where the learner is \nallowed to \\emph{immediately switch} between the stationary distributions of the chosen policies. By making use of the convex-optimization \nview given in Section~\\ref{sec:convex}, we define an auxiliary online convex optimization (or, in short, OCO, see, e.g., \n\\citealp{Haz11,SS12}) problem called the \n\\emph{idealized OCO problem} where in each round $t$, the following steps are repeated:\n\\begin{enumerate}\n \\item The learner chooses the stationary transition measure $\\pi_t\\in\\Delta(M)$.\n \\item Obliviously to the learner's choice, the environment chooses the loss function $\\widetilde{\\ell}_t = f(\\cdot;c_t)$.\n \\item The learner incurs a loss of $\\widetilde{\\ell}_t(\\pi_t)$. \n \\item The environment reveals the loss function $\\widetilde{\\ell}_t$.\n\\end{enumerate}\nThe performance of the learner \nin this setting is measured by the \\emph{idealized regret}\n\\[\n \\overline{R}_T = \\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi_t) - \\min_{\\pi\\in\\Delta(M)}\\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi).\n\\]\n\nThroughout the paper, we will consider \\emph{oblivious environments} that choose the sequence of state-cost functions without taking into \naccount the states visited by the learner. This assumption will enable us to simultaneously reason about the expected costs under any \nsequence of state distributions, and thus to make a connection between the idealized regret $\\overline{R}_T$ and the true regret $R_T$. \nThis technique was first used by \\citet{even-dar09OnlineMDP} and was shown to be essentially inevitable by \\citet{yu09ArbitraryRewards}: As \ndiscussed in their Section~3.1, no learning algorithm can avoid linear regret if the environment is not oblivious.\n\n\n\n\\section{Algorithm and main result}\\label{sec:ftl}\nIn this section, we propose a simple algorithm for online learning in LMDPs based on the ``follow-the-leader'' (FTL) strategy. On a high \nlevel, the idea of this algorithm is greedily betting on the policy that seems to have been optimal for the total costs observed so \nfar. While this strategy is known to fail catastrophically in several simple learning problems (see, e.g., \\citealt{CBLu06:Book}), it is \nknown to perform well in several important scenarios such as sequential prediction under the logarithmic loss \\citep{MF92} or prediction \nwith expert advice under bounded losses, given that losses are stationary \\citep{Kot16} and often serves as a strong benchmark \nstrategy \\citep{REGK14,SNL14}. In our learning problem, following the leader is a very natural choice of algorithm, as the convex \nformulation of Section~\\ref{sec:convex} suggests that we can effectively build on the analysis of Follow-the-Regularized-Leader-type \nalgorithms without having to explicitly regularize the objective.\n\nIn precise terms, our algorithm computes the sequence of policies $Q_1,Q_2,\\dots,Q_T$ by running FTL \\emph{in the idealized setting}: in \nround $t$, the algorithm chooses the stationary transition measure \n\\[\n\\begin{split}\n \\pi_t &= \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \\sum_{s=1}^{t-1} \\widetilde{\\ell}_s(\\pi) = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \\sum_{s=1}^{t-1} f(\\pi;c_s)\n \\\\\n &= \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} (t-1)\\cdot f\\pa{\\pi;\\frac{1}{t-1}\\sum_{s=1}^{t-1}c_s} = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \nf\\pa{\\pi;\\overline{c}_t},\n\\end{split}\n\\]\nwhere the third equality uses the fact that $f$ is affine in its second argument and the last step introduces the average state-cost \nfunction $\\overline{c}_t = \\frac {1}{t-1} \\sum_{s=1}^{t-1} c_s$. This form implies that $\\pi_t$ can be computed as the optimal control for the \nstate-cost function $\\overline{c}_t$, which can be done by following the procedure described in Section~\\ref{sec:bellman}. Precisely, we define the  \ndiagonal matrix $G_t$ with its $i$\\th~diagonal element $e^{-\\overline{c}_t(i)}$, let $\\gamma_t$ be the largest eigenvalue of $G_tP$ and $z_t$ be the \ncorresponding (unit-norm) right eigenvector. Also, let $v_t = -\\log z_t$ and $\\lambda_t = -\\log\\gamma_t$, and note that $\\lambda_t = \nf(\\pi_t;\\overline{c}_t)$ is the optimal average-cost-per-stage of $\\pi_t$ given the cost function $\\overline{c}_t$.\nFinally, we define the policy used in round $t$ as\n\\begin{equation}\\label{eq:optQ}\n Q_t(x'|x) = \\frac{P(x'|x) z_t(x')}{\\sum_y P(y|x) z_t(y)}\n\\end{equation}\nfor all $x'$ and $x$. We denote the induced stationary distribution by $\\mu_t$. The algorithm is presented as Algorithm~\\ref{alg:ftl}.\n\n\\begin{algorithm}\n \\textbf{Input:} Passive dynamics $P$.\n \\\\\\textbf{Initialization:} $\\overline{c}_1(x) = 0$ for all $x\\in\\mathcal{X}$.\n \\\\\\textbf{For $t=1,2,\\dots,T$, repeat}\n \\begin{enumerate}[leftmargin=.7cm]\n  \\item Construct $G_t = \\left[\\mbox{diag}(e^{-\\overline{c}_t})\\right]$.\n  \\item Find the right eigenvector $z_t$ of $G_t P$ corresponding to the largest eigenvalue.\n  \\item Compute the policy\n  \\[\n Q_t(x'|x) = \\frac{P(x'|x) z_t(x')}{\\sum_y P(y|x) z_t(y)}.\n\\]\n  \\item Observe state $X_{t}$ and draw $X_{t+1}\\sim Q_t(\\cdot|X_{t})$.\n  \\item Observe state-cost function $c_t$ and update $\\overline{c}_{t+1} = \\frac{\\pa{t-1}\\overline{c}_t + c_t}{t}$.\n \\end{enumerate}\n  \\caption{Follow The Leader in LMDPs} \\label{alg:ftl}\n\\end{algorithm}\n\n\nNow we present our main result. First, we state two key assumptions about the underlying passive dynamics; both of these assumptions are  \nalso made by \\citet{GRW12}.\n\\begin{assumption}\\label{ass:irred}\n The passive dynamics $P$ is irreducible and aperiodic. In particular, there exists a natural number $H>0$ such that $\\pa{P^n}(y|x)>0$ for \nall $n\\ge H$ and all $x,y\\in\\mathcal{X}$. We will refer to $H$ as the (worst-case) \\emph{hitting time}.\n\\end{assumption}\n\\begin{assumption}\\label{ass:ergod}\n The passive dynamics $P$ is ergodic in the sense that its Markov--Dobrushin ergodicity coefficient is strictly less than $1$:\n \\[\n  \\alpha(P) = \\max_{x,y\\in\\mathcal{X}} \\onenorm{P(\\cdot|x)-P(\\cdot|y)} < 1.\n \\]\n\\end{assumption}\nA standard consequence (see, e.g., \\citealt{Sen2006}) of Assumption~\\ref{ass:ergod} is that the passive dynamics mixes quickly: for any \ndistributions $\\mu,\\mu'\\in\\Delta(\\mathcal{X})$, we have\n\\[\n \\onenorm{\\pa{\\mu-\\mu'}^\\mathsf{\\scriptscriptstyle T} P}\\le \\alpha(P)\\onenorm{\\mu-\\mu'}.\n\\]\nWe will sometimes refer to $\\tau(P) = \\pa{\\log\\pa{1\/\\alpha(P)}}^{-1}$ as the \\emph{mixing time} associated with $P$. \nNow we are ready to state our main result:\n\\begin{theorem}\\label{thm:main}\n Suppose that the passive dynamics satisfies Assumptions~\\ref{ass:irred} and~\\ref{ass:ergod}. Then, the regret of Algorithm~\\ref{alg:ftl} \nsatisfies $R_T = O(\\log^2 T)$.\n\\end{theorem}\nThe asymptotic notation used in the theorem hides a number of factors that depend only on the \npassive dynamics $P$. In particular, the bound scales polynomially with the worst-case mixing time $\\tau$ of \nany optimal policy, and shows no \\emph{explicit} dependence on the number of states.\\footnote{Of course, the mixing time time does depend \non the size of the state space in general.} We explicitly state the bound at the end of the proof \npresented in the next section as Equation~\\eqref{eq:fullbound}, when all terms are formally defined.\n\n\\section{Analysis}\\label{sec:analysis}\nIn this section, we provide a series of lemmas paving the way towards proving Theorem~\\ref{thm:main}. The attentive reader may find some of \nthese lemmas familiar from related work: indeed, we build on several technical results from \\citet{even-dar09OnlineMDP,neu14o-mdp-full} and \n\\citet{GRW12}. Our main technical contribution is an efficient combination of these tools that enables us to go way beyond the \nbest known bounds for our problem, proved by \\citet{GRW12}. Throughout the section, we will assume that the conditions of \nTheorem~\\ref{thm:main} are satisfied.\n \nBefore diving into the analysis, we state some technical results that we will use several times. We defer all proofs to \nAppendix~\\ref{sec:app}. First, we present some important facts regarding LMDPs with bounded state-costs. In particular, we define $Q^*(c)$ \nas the optimal policy with respect to an arbitrary state-cost function $c$ and let $\\mathcal{C}$ be the set of all state-costs bounded \nin $[0,1]$. We define $\\mathcal{Q}^*$ as the set of optimal policies induced by state-cost functions in $\\mathcal{C}$: $\\mathcal{Q}^* = \nQ^*(\\mathcal{C})$. Observe that $Q_t\\in\\mathcal{Q}^*$ for all $t$, as $Q_t = Q^*(\\overline{c}_t)$ and $\\overline{c}_t \\in \\mathcal{C}$ for all $t$. Below, we \ngive several useful results concerning policies in $\\mathcal{Q}^*$.\nFor stating these results, let $c\\in\\mathcal{C}$ and $Q = Q^*(c)$. We first note that the average cost $\\lambda$ of $Q$ is bounded in \n$[0,1]$: By the Perron-Frobenius theorem (see, e.g., \\citealp[Chapter~8]{Mey00}), we have that the largest eigenvalue of $GP$ is \nbounded by the maximal and minimal row sums of $GP$: $e^{-\\lambda}\\in[e^{-\\max_x c(x)},e^{-c(x)}]$, which translates to \nhaving $\\lambda\\in[0,1]$ under our assumptions. The next key result bounds the value functions and the control costs in terms of the \nhitting time:\n\\begin{lemma}\\label{lem:vbound}\n For all $x,y$ and $t$, the value functions satisfy $v_t(x)-v_t(y) \\le H$. Furthermore, all policies $Q\\in\\mathcal{Q}^*$ satisfy\n\\[\n \\max_x \\kl{Q(\\cdot|x)}{P(\\cdot|x)} \\le H+1.\n\\]\n\\end{lemma}\nThe proof is loosely based on ideas from \\citet{bartlett09regal}.\nThe second statement guarantees that the mixing time $\\tau(Q) = \\pa{\\log(1\/\\alpha(Q))}^{-1}$ is finite for all policies in $\\mathcal{Q}^*$:\n\\begin{lemma}\\label{lem:mixing}\n The Markov--Dobrushin coefficient $\\alpha(Q)$ of any policy $Q\\in\\mathcal{Q}^*$ is bounded as\n \\[\n  \\alpha(Q) \\le \\alpha(P) + \\pa{1-\\alpha(P)} \\pa{1-e^{-H-2}}<1.\n \\]\n\\end{lemma}\nThe proof builds on the previous lemma and uses standard ideas from Markov-chain theory.\nIn what follows, we will use $\\tau = \\max_{Q\\in\\mathcal{Q}^*} \\tau(Q)$ and $\\alpha = \\max_{Q\\in\\mathcal{Q}^*} \\alpha(Q)$ to denote the worst-case \nmixing time and ergodicity coefficient, respectively. With this notation, we can state the following lemma that establishes that the value \nfunctions are $2\\tau$-Lipschitz with respect to the state-cost function. For pronouncing and proving the statement, it is useful to define \nthe \\emph{span seminorm} $\\spannorm{c} = \\max_x c(x) - \\min_y c(y)$. Note \nthat it is easy to show that $\\spannorm{\\cdot}$ is indeed a seminorm as it satisfies all the requirements to be a norm except that it maps \nall constant vectors (and not just zero) to zero. \n\\begin{lemma}\\label{lem:c_to_v}\n Let $f$ and $g$ be two state-cost functions taking values in the interval $[0,1]$ and let $v_f$ and $v_g$ be the corresponding optimal \nvalue \nfunctions. Then,\n\\[\n \\spannorm{v_f - v_g} \\le 2\\tau \\infnorm{f-g}.\n\\]\n\\end{lemma}\n The proof roughly follows the proof of Proposition~3 of \\citet{GRW12}, with the slight difference that we make the constant factor in the \nbound explicit. A consequence of this result is our final key lemma in this section that actually makes our fast rates possible: a bound on \nthe change-rate of the policies chosen by the algorithm.\n\\begin{lemma}\\label{lem:change} \n$\\max_x \\onenorm{Q_t(\\cdot|x) - Q_{t+1}(\\cdot|x)} \\le \\frac{\\tau}{t}$.\n\\end{lemma}\nThe proof is based on ideas by \\citet{GRW12}. As for the proof of Theorem~\\ref{thm:main}, we follow the path of \n\\citet{even-dar09OnlineMDP,neu14o-mdp-full,GRW12}, and first analyze the idealized setting where the learner is allowed to directly pick \nstationary distributions instead of policies. Then, we show how to relate the idealized regret of FTL to its true regret in the original \nproblem.\n\n\\subsection{Regret in the idealized OCO problem}\nLet us now consider the idealized online convex optimization problem described at the end of Section~\\ref{sec:online}. \nIn this setting, our algorithm can be formally stated as choosing the stationary transition measure $\\pi_t = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \nf(\\pi;\\overline{c}_t)$. This view enables us to follow a standard proof technique for analyzing online convex optimization algorithms, going back to \nat least \\citet{MF92}.\nThe first ingredient of our proof is the so-called ``follow-the-leader\/be-the-leader'' lemma \\citet[Lemma~3.1]{CBLu06:Book}:\n\\begin{lemma} \\label{lem:btl}\n$\\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi_{t+1}) \\le \\min_\\pi \\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi)$.\n\\end{lemma}\nThe second step exploits the bound on the change rate of the policies to show that looking one step into the future does not buy much \nadvantage. Note however that controlling the change rate is not sufficient by itself, as our loss functions are effectively unbounded.\n\\begin{lemma}\\label{lem:price}\n$\\sum_{t=1}^T \\pa{\\widetilde{\\ell}_t(\\pi_{t}) - \\widetilde{\\ell}_t(\\pi_{t+1})} \\le 2 \\pa{\\tau^2+1} \\pa{1+\\log T}$.\n\\end{lemma}\nIn the interest of space, we only provide a proof sketch here and defer the full proof to Appendix~\\ref{app:price}.\n\\begin{proofsketch}\n Let us define $\\Delta_t = \\overline{c}_{t+1} - \\overline{c}_{t}$. By exploiting the affinity of $f$ in its second argument, we can start by proving\n$\\lambda_t - \\lambda_{t+1} \\le \\infnorm{\\Delta_t}$. Furthermore, by using the form of the optimal policy $Q_t$ given in Eq.~\\eqref{eq:optQ} \nand the form of $f$ given in Eq.~\\eqref{eq:statcost}, we can obtain\n\\begin{align*}\n \\widetilde{\\ell}_t(\\pi_{t}) - \\widetilde{\\ell}_t(\\pi_{t+1}) &= \\pa{\\mu_{t} - \\mu_{t+1}}^\\mathsf{\\scriptscriptstyle T} \\pa{c_t + \\overline{c}_{t}} + \\mu_{t+1}^\\mathsf{\\scriptscriptstyle T} \\pa{\\overline{c}_{t} - \n\\overline{c}_{t+1}}\n + \\lambda_{t} - \\lambda_{t+1}\n \\\\\n &\\le 2 \\onenorm{\\mu_{t+1} - \\mu_t} + 2\\infnorm{\\Delta_t}.\n\\end{align*}\nThe first term can be bounded by a simple argument (see, e.g., Lemma~4 of \\citealt{neu14o-mdp-full}) that leads to\n\\[\n\\onenorm{\\mu_{t+1} - \\mu_t} \\le \\max\\ev{\\tau(Q_t),\\tau(Q_{t+1})} \\max_x \\onenorm{Q_{t+1}(\\cdot|x)-Q_t(\\cdot|x)}.\n\\]\nNow, the first factor can be bounded by $\\tau$ and the second by appealing to Lemma~\\ref{lem:change}. The proof is \nconcluded by plugging the above bounds into Equation~\\eqref{eq:lpidiff}, using $\\infnorm{\\Delta_t} \\le 1\/t$, summing up both sides, and \nnoting that $\\sum_{t=1}^T 1\/t \\le 1 + \\log T$.\n\\end{proofsketch}\nPutting Lemmas~\\ref{lem:btl} and~\\ref{lem:price} together, we obtain the following bound on the idealized regret of FTL:\n\\begin{lemma}\\label{lem:ideal}\n$\\overline{R}_T \\le 2\\pa{\\tau^2+1}\\pa{1+\\log T}$.\n\\end{lemma}\n\n\n\\subsection{Regret in the reactive setting}\nWe first show that the advantage of the true best policy $Q^*_T$ over our final policy $Q_{T+1}$ is bounded. \n\\begin{lemma} Let $p^* = \\min_{x,x':P(x'|x)>0} P(x'|x)$ be the smallest non-zero transition probability under the passive dynamics and $B = \n-\\log p^*$. Then,\n$ \\sum_{t=1}^T \\overline{\\loss}_t(\\pi_{T+1}) - \\mathcal{L}_T(Q_T^*) \\le \\pa{2\\tau + 2}\\pa{B+1}$.\n\\end{lemma}\nThe proof follows from applying Lemma~1 from \\citet{neu14o-mdp-full} and observing that $\\ell_t(X_t,Q_T^*) \\le B+1$ holds for all $t$.\nIt remains to relate the total cost of FTL to the total idealized cost of the algorithm. This is done in the following lemma:\n\\begin{lemma} \\label{lem:trueloss}\n$ \\sum_{t=1}^T \\pa{\\EE{\\ell_t(Q_t,X_t)} - \\overline{\\loss}_t(\\pi_{t})} \\le \\pa{\\tau+1}^3 \\pa{1+\\log T}^2 + 2\\pa{\\tau + 1}\\pa{3+\\log T}$.\n\\end{lemma}\n\\begin{proof}\n  Let $p_t(x) = \\PP{X_t = x}$. Similarly to the proof of Lemma~\\ref{lem:price}, we rewrite $\\overline{\\loss}_t(\\pi_t)$ using \nEquation~\\eqref{eq:lpiform} to obtain\n \\[\n \\begin{split}\n  \\EE{\\ell_t(Q_t,X_t) - \\overline{\\loss}_t(\\pi_{t})}\n  &=\n  \\sum_{x} \\pa{p_t(x) - \\mu_t(x)} \\pa{c_t(x) + v_t(x) + \\lambda_{t} - \\overline{c}_{t}(x) - \\sum_{x'} Q_t(x'|x) v_t(x')}\n  \\\\\n  &\\le  \\sum_{x} p_t(x) \\pa{v_t(x) - \\sum_{x'} Q_t(x'|x) v_{t}(x')} + \\onenorm{p_t- \\mu_t},\n \\end{split}\n \\]\n where the last step uses $\\sum_{x} \\mu_t(x) Q_t(x'|x) = \\mu_t(x')$ and $\\infnorm{c_t - \\overline{c}_t}\\le 1$.\n Now, noticing that $\\sum_{x} p_t(x) Q_t(x'|x) = p_{t+1}(x')$, we obtain\n \\[\n \\begin{split}\n  &\\sum_{t=1}^T \\EE{\\ell_t(Q_t,X_t) - \\overline{\\loss}_t(\\pi_{t})}\n  \\le \\sum_{t=1}^T \\pa{p_t-p_{t+1}}^\\mathsf{\\scriptscriptstyle T} v_t + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t}\n  \\\\\n  &\\quad= \\sum_{t=1}^{T} p_t^\\mathsf{\\scriptscriptstyle T} \\pa{v_t - v_{t-1}} + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t} - p_{T+1}^\\mathsf{\\scriptscriptstyle T} v_T\n  \\le \\sum_{t=1}^T \\frac{2\\tau}{t} + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t} - p_{T+1}^\\mathsf{\\scriptscriptstyle T} v_T,\n \\end{split}\n \\]\n where the last inequality uses Lemma~\\ref{lem:c_to_v} and $\\infnorm{\\overline{c}_{t} - \\overline{c}_{t-1}}\\le 1\/t$ to bound the first term. By \nLemma~\\ref{lem:c_to_v}, this last term can be bounded by $\\norm{v_T}_s = \\norm{v_T - v_0}_s\\le 2\\tau\\infnorm{\\overline{c}_T} \\le 2\\tau$, where $v_0$ \nis the value function corresponding to the all-zero state-cost function.\n \n In the rest of the proof, we are going to prove the inequality\n \\begin{equation}\\label{eq:pmudiff}\n \\onenorm{\\mu_t - p_t} \\le 2 e^{-\\pa{t-1}\/\\tau} +  \\frac{2(\\tau+1)^3\\pa{1+ \\log t}}{t}.\n \\end{equation}\n It is easy to see that this trivially holds for $\\pa{2 \\tau \\log t}\/t \\ge \n1$, so we will assume that the contrary holds in the following derivations.\nTo prove Equation~\\eqref{eq:pmudiff} for larger values of $t$, we can follow the proofs \nof Lemma~5 of \\citet{neu14o-mdp-full} or Lemma~5.2 of \\citet{even-dar09OnlineMDP} to obtain\n\\begin{equation}\\label{eq:mu_to_p}\n \\onenorm{\\mu_t - p_t} \\le 2 e^{-\\pa{t-1}\/\\tau} +  \\tau \\pa{\\tau+1} \\sum_{n=1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n}.\n\\end{equation}\nFor completeness, we include a proof in Appendix~\\ref{app:mu_to_p}.\nFor bounding the last term, we split the sum at $B=\\left\\lfloor t - \\tau\\log t\\right\\rfloor$:\n\\[\n \\begin{split}\n  \\sum_{n=1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n} &= \n  \\sum_{n=1}^{B} \\frac{e^{-(t-n)\/\\tau}}{n} + \\sum_{n=B+1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n}\n \\\\\n &= e^{-(t-B)\/\\tau} \\sum_{n=1}^{B} \\frac{e^{-(B-n)\/\\tau}}{n} + \\sum_{n=B+1}^{t} \\frac{e^{-(t-n)\/\\tau}}{n}\n \\\\\n &\\le \\frac{1}{t} \\cdot \\frac{1}{1-e^{-1\/\\tau}} + \\frac{\\tau\\log t}{t-\\tau\\log t} \\le \n \\frac{\\tau }{t} + \\frac{\\tau\\log t}{t} \\cdot \\frac{1}{1-\\pa{\\tau\\log t}\/ t}\n \\\\\n &\\le \\frac{\\tau}{t} + \\frac{2\\tau \\log t}{t} \\le \\frac{2\\tau\\pa{1 + \\log t}}{t},\n \\end{split}\n\\]\nwhere the first inequality follows from bounding the $1\/n$ factors by $1$ and $1\/B$, respectively, and bounding the sums by \nthe full geometric sums. The second-to-last inequality follows from our assumption that $(2\\tau\\log t)\/t \\le 1$.  That is, we have \nsuccessfully proved Equation~\\eqref{eq:pmudiff}. Now the statement of the lemma follows from summing up for all $t$ and noting that \n$\\sum_{t=1}^T \\frac{1}{t} \\le 1 + \\log T$ and $\\sum_{t=1}^T e^{-\\pa{t-1}\/\\tau} \\le \\tau + 1$.\n\\end{proof}\nNow the proof of Theorem~\\ref{thm:main} follows easily from combining the bounds of Lemmas~\\ref{lem:ideal}--\\ref{lem:trueloss}. The result \nis\n\\begin{equation}\\label{eq:fullbound}\n R_T \\le \n 2\\pa{\\tau+1}^3 \\pa{1+\\log T}^2 + 2\\pa{\\tau^2 + \\tau + 2}  (3+\\log T) + \\pa{2\\tau+2}\\pa{B+2}.\n\\end{equation}\nThus, we can see that the bound indeed demonstrates a polynomial dependence on the mixing time $\\tau$, \nand depends logarithmically on the smallest non-zero transition probability $p^*$ via $B = -\\log p^*$.\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this paper, we have shown that, besides the well-established computational advantages, linearly solvable MDPs also admit a remarkable\ninformation-theoretic advantage: fast learnability in the online setting. In particular, we show that achieving a regret of $O(\\log^2 T)$ \nis achievable by the simple algorithm of following the leader, thus greatly improving on the best previously known regret bounds of \n$O(T^{3\/4})$. At first sight, our improvement may appear dramatic: in their paper, \\citet{GRW12} pose the possibility of improving their \nbounds to $O(\\sqrt{T})$ as an important open question (Sec.~VII.). In light of our results, these conjectured improvements are also grossly \nsuboptimal. On the other hand, our new results can be also seen to complement well-known results on fast rates in online learning (see, \ne.g., \\citealt{EGMRW15} for an excellent summary). Indeed, our learning setting can be seen as a generalized variant of sequential \nprediction under the  relative-entropy loss (see, e.g., \\citealp[Sec.~3.6]{CBLu06:Book}), which is known to be \\emph{exp-concave}. Such \nexp-concave losses are well-studied in the online learning literature, and are known to allow logarithmic regret bounds \\citep{KW99,HAK07}.\n\nInspired by these related results, we ask the question: Is the loss function $f$ defined in Section~\\ref{sec:convex} exp-concave? While our \nderivations Appendix~\\ref{app:convexity} indicate that $f$ has curvature in certain directions, we were not able to prove its \nexp-concavity. Similarly to the approach of \\citet{MF92}, our analysis in the current paper merely exploits the Lipschitzness of the optimal \npolicies with respect to the cost functions, but otherwise does not explicitly make use of the curvature of $f$. We hope that our work \npresented in this paper will inspire future studies that will clarify the exact role of the LMDP structure in efficient online learnability, \npotentially also leading to a better understanding of policy gradient algorithms for LMDPs \\citep{Tod10}.\n\nFinally, let us comment on the tightness of our bounds. Regardless of whether the loss function $f$ is exp-concave or not, \nwe are almost certain that our rates can be improved to at least $O(\\log T)$ by using a more sophisticated algorithm. While our focus in \nthis paper was on improving the asymptotic regret guarantees, we also slightly improve on the results \\citet{GRW12} in that we make the \nleading constants more explicit. However, we expect that the dependence on these constants may also be improved in future work. \nNote however that the potential \nlooseness of our bounds does not impact the performance of the algorithm itself, as it never makes use of any problem-dependent constants.\n\n\\paragraph{Acknowledgements}\nThis work was supported by the UPFellows Fellowship (Marie Curie COFUND program n${^\\circ}$ \n600387) and the Ramon y Cajal program RYC-2015-18878 (AEI\/MINEICO\/FSE, UE). \nThe authors wish to thank the three anonymous reviewers for their valuable comments that helped to improve the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDeriving the light hadron spectrum from the first principles of\nQCD has been a major subject of lattice QCD\nsimulations\\cite{ref:reviews}.\nA precise determination of the known hadron spectrum\nwould lead us to a fundamental verification of QCD.\nWe should also clarify the nature of observed hadrons,\nprovide predictions for hadrons not in the quark model,\nand give informations for quantities of phenomenological\nimportance.\n\nIn order to achieve these goals, understanding and control\nof various systematic errors are required.\nOne of major sources of systematic errors is that of\na finite lattice spacing.\nRecent progress in reducing this systematic error\nhas been made in two ways.\nFor the quenched QCD spectrum,\ndevelopment of computer power has enabled to push simulations \ntoward smaller lattice spacings on physically larger lattices with \nhigher statistics than the previous attempts.\nAs a result we are now in a status to discuss the problem of how well\nquenched QCD describes the experimental spectrum.\nAnother progress in reducing scaling violation is brought \nwith the use of improved quark actions.\nTests of improvement, previously made mainly in quenched QCD, \nhave been extended this year to full QCD.\n\nFinite size effects and chiral extrapolations have been\nstudied extensively in the past.\nSeveral studies to investigate these systematic errors were \nalso reported at the Symposium.\n\nIn this review we attempt to describe the present status of\nspectroscopic studies.\nProgress in quenched QCD spectrum is summarized in\nsec.~\\ref{sec:progress}, emphasizing results in the continuum limit.\nDiscussions on several issues in spectroscopic studies follow\nin sec.~\\ref{sec:issues},\nwhich include study of finite size effects, chiral extrapolations,\nand quenching error in meson decay constants.\nAfter discussions on improvement of quark actions\nin sec.~\\ref{sec:improve},\nattempts toward a realistic calculation in full QCD are presented\nin sec.~\\ref{sec:fullQCD}.\nSec.~\\ref{sec:other} is devoted to results for masses of glueballs\nand exotics.  Our conclusions are given in sec.~\\ref{sec:conclusions}.\n\n\n\\section{Progress in Quenched QCD Spectrum}\\label{sec:progress}\n\\subsection{major simulations}\n\nRecent quenched simulations made with the plaquette gauge action\nare compiled in Table \\ref{tab:table-q}. \nSee sec.~\\ref{sec:improve} for those with improved \ngauge actions. \n\nDeriving precise quenched results in the continuum limit \nis a first step toward understanding the light hadron spectrum. \nThe GF11 collaboration\\cite{ref:GF11mass} carried out\nthe first systematic effort to achieve this goal \nwith the Wilson quark action \nusing three lattices with $a^{-1} = 1.4-2.8$ GeV  and \nthe spatial size $La \\approx 2.3$ fm.\n \nThis year the CP-PACS collaboration reported further effort \nin this direction\\cite{ref:CPPACS}.\nThey made high statistics simulations\non four lattices with $a^{-1} = 2.0 - 4.2$ GeV and $La \\approx 3$ fm.\nHadron masses are calculated for five quark masses\ncorresponding to $m_\\pi\/m_\\rho$ = 0.75, 0.7, 0.6, 0.5\nand 0.4, the last point being closer to the chiral limit\nthan ever attempted for the Wilson action. \nThey reported continuum values of hadron masses with a statistical \nerror of 0.5 \\% for mesons and 1--3 \\% for baryons.\n\nAnother trend in this year's simulations is a pursuit \nof reduction of scaling violation with the use of \nthe Sheikholeslami-Wohlert\\cite{ref:SW} or clover action.\nEfforts in this direction were made by \nthe UKQCD\\cite{ref:UKQCDlat96,ref:UKQCDb57,ref:UKQCDlat97} \nand JLQCD \\cite{ref:JLQCD-fB} collaborations\nfor the tadpole-improved\\cite{ref:LM} clover action\nand by the UKQCD, QCDSF\\cite{ref:QCDSFlatest} and APETOV\\cite{ref:APETOV}\ncollaborations for the non-perturbatively $O(a)$-improved\\cite{ref:NPI} \nclover action (see also Ref.\\cite{ref:Wittig} on this subject).\nThese studies have not yet reached the level of simulations with the Wilson \naction, being restricted to the parameter range \n$m_\\pi\/m_\\rho \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.5$, $a^{-1} \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 3$ GeV, and\n$La \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 2.0$ fm.\n\nFor the Kogut-Susskind (KS) quark action, \nthe MILC collaboration\\cite{ref:MILClat96} last year \nreported a result of nucleon mass in the continuum limit based on\nsimulations on four lattices with $a^{-1}=0.6 - 2.4$ GeV and \n$La \\approx 2.7$ fm.\nNot much progress has been made this \nyear\\cite{ref:MILC-Tsukuba-Gottlieb,ref:KimOhta}.\n\n\\begin{table*}[t]\n\\caption{Recent spectrum runs in quenched QCD\nwith the standard gauge action. \nNew results since Lattice 96 are marked by double asterisks and\nthose with increased statistics by asterisks.\nQuark actions are denoted in parentheses by\nW: Wilson, C: clover, and KS: Kogut-Susskind. \nClover coefficients are denoted by 1: tree level, TP: tadpole improved,\nTP1: one-loop tadpole improved, and NP: non-perturbatively improved.}\n\\label{tab:table-q}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n        &  $\\beta$   & size &  (fm) & \\#conf. & \\ \\ \\ $m_\\pi\/m_\\rho$   & \\#m & ref.\\\\\n\\hline\n\\hline\nMILC (W)** & 5.70 & $(12-24)^3\\times48$ & 1.7--3.4 & 404-170 & 0.90-0.50 & 6  & \n\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat97} \\\\\n\\hline\nCP-PACS (W)**  & 5.90 & $32^3\\times56$ & 3.21 & 800 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.10 & $40^3\\times70$ & 3.04 & 600 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.25 & $48^3\\times84$ & 3.03 & 420 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.47 & $64^3\\times112$& 3.03 &  91 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\n\\hline\n\\hline\nUKQCD (C=TP)  & 5.70 & $(12,16)^3\\times24$ & (2.1,2.8) & (482,145) & 0.78,0.65 & 2 & \n\\cite{ref:UKQCDlat96,ref:UKQCDb57} \\\\\nUKQCD (C=TP)  & 6.00 & $16^3\\times48$ & 1.6  & 499 & 0.76-0.62 & 3 & \n\\cite{ref:UKQCDlat96,ref:UKQCDlat97}\\\\\nUKQCD (C=TP)* & 6.20 & $24^3\\times48$ & 1.8  & 218 & 0.75-0.49 & 3 & \n\\cite{ref:UKQCDlat96,ref:UKQCDlat97} \\\\\nUKQCD (C=NP)** & 6.00 & $(16,32)^3\\times48$ & (1.7,3.3)  & (497,70) & 0.77-0.50 & 3 & \n\\cite{ref:UKQCDlat97}\\\\\nUKQCD (C=NP)** & 6.20 & $24^3\\times48$ & 1.7  & 251 & 0.71-0.54 & 3 & \n\\cite{ref:UKQCDlat97} \\\\\n\\hline\nQCDSF (C=1)** & 5.70& $16^3\\times32$ & 2.4 & & 0.66-0.44 & 3 & \n\\cite{ref:QCDSFlatest} \\\\\nQCDSF (C=NP)** & 5.70& $16^3\\times32$ & 3.3& & 0.77-0.56 & 6 & \n\\cite{ref:QCDSFlatest} \\\\\nQCDSF (W)*     & 6.00& $(16,24)^3\\times32$ & (1.4,2.0) & O(5000,100)& 0.93-0.50 \n& (4,3) & \\cite{ref:QCDSFlatest,ref:QCDSFb60}\\\\\nQCDSF (C=NP)* & 6.00& $(16,24)^3\\times32$ & (1.7,2.6) & O(1000,200) & 0.90-0.41 & \n(6,3) & \\cite{ref:QCDSFlatest,ref:QCDSFb60} \\\\\nQCDSF (W)**    & 6.20& $24^3\\times48$ & 1.6 & O(100) & 0.94-0.61 & 5 & \n\\cite{ref:QCDSFlatest}\\\\\nQCDSF (C=NP)** & 6.20& $24^3\\times48$ & 1.8 & O(300) & 0.90-0.59  & 5 & \n\\cite{ref:QCDSFlatest}\\\\\nQCDSF (C=NP)** & 6.20& $32^3\\times64$ & 2.4 & O(40) & 0.55-0.39  & 3 &\n\\cite{ref:QCDSFlat97-2} \\\\\n\\hline\nAPETOV (W)**& 6.20 & $24^3\\times48$ & 1.7 & 50 & & 7 & \n\\cite{ref:APETOV} \\\\\nAPETOV (C=NP)**&6.20 & $24^3\\times48$ & 1.9 & 50 & 0.98-0.56 & 7 & \n\\cite{ref:APETOV} \\\\\n\\hline\nJLQCD (C=TP1)** & 5.90& $16^3\\times40$ & 2.0 & 400 & 0.76-0.56 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\nJLQCD (C=TP1)** & 6.10& $24^3\\times64$ & 2.1 & 200 & 0.77-0.50 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\nJLQCD (C=TP1)** & 6.30& $32^3\\times80$ & 2.2 & 100 & 0.81-0.52 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\n\\hline\n\\hline\nKim-Ohta (KS)* & 6.50 & $48^3\\times64$ & 2.6 & 350 & 0.65-0.28& 4 & \n\\cite{ref:KimOhta}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table*}\n\n\n\\subsection{quenched spectrum in the continuum limit}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{spectrum.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Quenched light hadron spectrum in the continuum limit \nreported by GF11\\protect\\cite{ref:GF11mass} and\nCP-PACS\\protect\\cite{ref:CPPACS} as compared to experiment.}\n\\label{fig:spectrum}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:spectrum} we plot the result for the quenched \nlight hadron spectrum reported by the CP-PACS collaboration \nas compared to the GF11 result and experiment.  \nThe quenched spectrum depends on the choice of hadron masses to set \nthe lattice scale and light quark masses.  Results for two choices are shown \nin Fig.~\\ref{fig:spectrum}, one employing $m_\\pi, m_\\rho$ and $m_K$ and \nthe other replacing $m_K$ with $m_\\phi$.\nThe disagreement of about 5--10\\%\nobserved for strange hadrons between the two choices represent a \nmanifestation of quenching error.\n\nThe GF11 result, albeit not covering the entire spectrum, \nshowed agreement with experiment within the quoted error \nof 2\\% for mesons and 4--8\\% for baryons.\nComparing their result with the CP-PACS result obtained with \nthe same input (filled circles), \none finds a sizable difference for $K^*, \\phi, \\Xi^*$ and $\\Omega$. \nIn fact the CP-PACS result with significantly reduced errors \nexhibits a clear systematic deviation from experiment both for \nmesons and baryons.  \n\n\\subsection{meson spectrum}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{Kstar-phi.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_{K^{*}}$ and $m_\\phi$ with $m_K$ as input\nfor the Wilson (filled symbols) and clover (open symbols) \nactions. In parentheses of legends are physical lattice sizes \nin fm. Lines are continuum extrapolation adopted by GF11 and CP-PACS. \nLeft-most triangles are GF11 estimates for infinite volume.\n}\n\\label{fig:Kstar-phi}\n\\vspace{-7mm}\n\\end{figure}\n\nThe CP-PACS result in the continuum shows that the value of $m_{K^*}$ is \n3\\%(6$\\sigma$) smaller than experiment and $m_\\phi$ by \n5\\% (7$\\sigma$) if $m_K$ is used as input.  Alternatively, with \n$m_\\phi$ as input, they find that $m_{K^{*}}$ agrees with\nexperiment to 0.6\\%, but $m_K$ is larger by 9\\%(7$\\sigma$).\nThis means that a small value of hyperfine splitting, previously\nobserved at finite lattice spacings\\cite{ref:GF11mass,ref:LANLmass},\nremains in the continuum limit, which is different from the conclusion \nof the GF11 collaboration after the continuum extrapolation. \n\nThe origin of the discrepancy is clearly seen in Fig.\\ref{fig:Kstar-phi}\nwhere the continuum extrapolations of $m_{K^{*}}$ and $m_\\phi$ are plotted.\nThe CP-PACS data (filled circles) show very small scaling violation, \nin contrast to an increase exhibited by the GF11 results. \nThe continuum extrapolation of GF11 strongly\ndepends on the small values of results at $\\beta=5.7$ obtained on a \nlattice of size $La \\approx 2.3$~fm ($L=16$).  Their additional \nresults for a larger lattice with $La \\approx 3.4$~fm ($L=24$), also \nshown in Fig.~\\ref{fig:Kstar-phi},  are higher \nby 2--3\\%, and are more compatible with the CP-PACS results.  \nWhether one can attribute the difference of the GF11 results \nbetween $L=16$ and 24 to finite-size effects \nis not clear since values of the two groups \nfor smaller lattice spacings are consistent.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{hyperfine.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Meson hyperfine splitting obtained with $m_K$ as input. }\n\\label{fig:hyperfine}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:hyperfine} we plot the meson hyperfine splitting as \na function of the pseudo-scalar meson mass squared where $m_K$ is used \nas input.  \nThe CP-PACS data at four values of $\\beta$ (filled symbols) \nscale well and do not reproduce the experimental \nvalue of $K$--$K^{*}$ mass splitting.\n\nIn Figs.~\\ref{fig:Kstar-phi} and \\ref{fig:hyperfine}, the clover results \nhave also been plotted with open symbols. \nWe observe that they lie slightly above the Wilson results.\nThis agrees with the expectation that the clover term should increase \nthe hyperfine splitting compared to that of the Wilson action. \nHowever, there is a problematical feature that the difference \nof results for the two actions increases toward the continuum \nlimit rather than decreasing as $O(a)$. \nIn fact the UKQCD collaboration\\cite{ref:UKQCDlat97}\nconcluded this year \nthat $m_{K^{*}}$ linearly extrapolated to the continuum limit is\nconsistent with experiment using either $m_K$ or $m_\\phi$ as input. \n\nWe should emphasize that the difference of meson masses for the two actions\nis tiny(1--2\\%) and no more than a 3$\\sigma$ effect at finite $\\beta$.\nLattice sizes of $La\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 2$~fm employed in the clover studies may be \ntoo small to avoid finite-size errors at this level of precision.\nStatistical errors of the clover results,  \nwhich are larger by a factor 2--3 compared to those of the Wilson action, \nalso need to be reduced to resolve the discrepancy. \n\nWe compile results for the $J$ parameter\\cite{ref:J} in Fig.~\\ref{fig:J}.  \nAs has been known, results for the Wilson action and its improved ones\nconsistently lie below the experimental value for a wide range\nof lattice spacing. Results for the KS action\\cite{ref:JLQCD-BKKS}\nalso converge to a similar value from above.\n \nA small value of $J$ is equivalent to a small hyperfine splitting if the \nlatter is a linear function of quark mass.\nThis correspondence is satisfied for the Wilson results, \nwhile it is apparently not for the clover case.  This represents \nanother problem which needs to be understood in the quenched meson \nspectrum.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{J.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Results for $J$ parameter. \nData are taken from CP-PACS\\protect\\cite{ref:CPPACS},\nJLQCD\\protect\\cite{ref:JLQCD1000,ref:JLQCD-BKKS}\nfor the Wilson and KS actions, respectively,\nLANL\\protect\\cite{ref:LANLmass},\nSCRI\\protect\\cite{ref:SCRIlat96},\nAlford {\\it et al.}\\protect\\cite{ref:Alfordlat95,ref:Alfordlat96}\nfor the D234 and D234(2\/3) actions, respectively.\nLines are fits to the CP-PACS results and the KS results.}\n\\label{fig:J}\n\\vspace{-7mm}\n\\end{figure}\n\n\\subsection{baryon spectrum}\n\nIn Fig.~\\ref{fig:ndE} we plot the continuum extrapolation of\nrepresentative baryon masses reported by the GF11 and CP-PACS \ncollaborations.  \nThe quenched value of nucleon mass has been a long debated issue.  \nPrevious high statistics results\\cite{ref:APE-OLD,ref:QCDPAX,ref:LANLmass}\n(see also Ref.\\cite{ref:UKQCDlat97}) at $\\beta\\approx 5.7-6.2$ \nobtained by a chiral extrapolation \nfrom $m_\\pi\/m_\\rho\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.5$ yielded a value higher than \nexperiment. The GF11 results also shared this feature, and agreement \nwith experiment in the continuum limit was obtained only after \na finite-size correction.\n\nThe CP-PACS data down to $m_\\pi\/m_\\rho\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.4$ show \nthat the nucleon and $\\Lambda$ masses have a negative\ncurvature in terms of $1\/K$ toward the chiral limit.  \nThe bending significantly lowers the nucleon mass even at finite $\\beta$\nas shown in Fig.~\\ref{fig:ndE}, and a linear continuum extrapolation \nleads to a value 2.3\\% lower than experiment, albeit consistent \nwithin a 3\\% statistical error.\nThe nucleon mass for the KS action from the MILC \ncollaboration\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat96}\nis also consistent with experiment.\nSee Sec.~\\ref{sec:chiral-N} for further discussion on the chiral \nextrapolation.\n\nFor $\\Delta$ and $\\Omega$ masses, the GF11 and CP-PACS results \nare reasonably consistent at similar lattice spacings. \nThe continuum extrapolation is different, especially for $\\Omega$, \nwith the GF11 case strongly \naffected by the results at $\\beta=5.7$ on an $L=16$ lattice.  \n\nIn the continuum limit, the CP-PACS results \nshow a systematic deviation from experiment.  \nFor the octet, the non-strange nucleon mass is consistent with \nexperiment, while strange baryon masses are lower \nby 5--8\\% (3--5\\%) with $m_K$ ($m_\\phi$) as input.\nHowever, the Gell-Mann-Okubo (GMO) relation is well\nsatisfied at a 1\\% level.\n\nThe GMO relation is also well satisfied for the decuplet, where \nit takes the form of an equal spacing rule, \nwith at most 10\\% deviations.\nHowever, the average spacing is too small by 30\\% (20\\%) with\n$m_K$ ($m_\\phi$) as input.\n\nBaryon mass splittings were extensively studied at $\\beta=6.0$ on \na $32^3\\times 64$ lattice in Ref.~\\cite{ref:LANLmass}, which \nreported the validity of the GMO relations and \nthe smallness of the decuplet mass splitting. \nThe CP-PACS data confirm these results and extend them as the \nproperty of the quenched baryon spectrum in the continuum.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\vspace*{-8mm}\n\\epsfxsize=7.8cm \\epsfbox{ndE.ps}\n\\end{center}\n\\vspace{-15mm}\n\\caption{Continuum extrapolations of baryon masses\nfrom CP-PACS (filled symbols) and GF11 (open symbols). } \n\\label{fig:ndE}\n\\vspace{-6mm}\n\\end{figure}\n\n\\subsection{quark mass for the Wilson action}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{mstrange.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Comparison of strange quark masses obtained \nfrom the Ward identity and perturbation.\nMasses are in $\\overline {\\rm MS}$ scheme at $\\mu=2$ GeV.}\n\\label{fig:strange-mass}\n\\vspace{-8mm}\n\\end{figure}\n\nThe Wilson action explicitly breaks chiral symmetry at finite lattice spacing.\nOne of its manifestations is that quark mass $m_q^{WI}$ defined \nby the Ward identity \\cite{ref:Bo,ref:Itoh86,ref:MM} \ndoes not agree with quark mass $m_q^P$ defined perturbatively\nat finite lattice spacings\\cite{ref:Itoh86,ref:LANLmass}.\n\nThis problem was examined by four groups this year. \nThe CP-PACS collaboration compared the two definitions\nfor the Wilson action, and reported that they linearly extrapolate to \na consistent value in the continuum limit\\cite{ref:CPPACS}.  \nThe JLQCD collaboration employed an extended current and found indications\nthat scaling violation for $m_q^{WI}$ becomes smaller than that for the local \ncurrent\\cite{ref:JLQCD-Kura}.  \nThe QCDSF collaboration\\cite{ref:QCDSFlatest} reported that\nthe two definitions give consistent results in the continuum limit\nalso for the non-perturbatively $O(a)$ improved clover action.\nThe Ape collaboration\\cite{ref:Ape-QM} reported that $m_q^{WI}$\nare compatible with $m_q^{P}$ at each $\\beta$ when\nrenormalization factors determined non-perturbatively are used.\n\nWe summarize results for the strange quark mass in \nFig.~\\ref{fig:strange-mass}.  \nThe agreement of $m_q^{WI}$ with $m_q^{P}$ \nin the continuum limit supports our expectation that \nchiral symmetry of the Wilson and clover actions is recovered\nin the continuum limit.\nThe disagreement of the values \n$m_s\\approx 135$ MeV obtained with $m_\\phi$ as input and \n$m_s\\approx 110$ MeV found with $m_K$ as input originates \nfrom the small meson hyperfine splitting, and hence represents a \nquenching uncertainty.\nFurther results on quark masses are reviewed in Ref.~\\cite{ref:Gupta}.\n\n\\section{Issues in Spectroscopic Studies}\\label{sec:issues}\n\n\n\\subsection{finite size effects in quenched QCD}\\label{sec:FS}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=6.5cm \\epsfbox{finite.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Nucleon mass for various source\/sink and lattice sizes \nat $\\beta=5.7$ and $m_\\pi\/m_\\rho \\approx 0.5$.\nData are slightly shifted in horizontal axis for clarity.\n}\n\\label{fig:finite}\n\\vspace{-7mm}\n\\end{figure}\n\nIn quenched QCD finite-size effects of hadron masses are expected to \nbe smaller than in full QCD due to Z(3) symmetry. \nFor the nucleon mass with the KS action, the magnitude has been \nestimated to be less than 2\\% at $m_\\pi\/m_\\rho \\approx 0.5$\nfor $La\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 2$ fm\\cite{ref:Aoki-FS,ref:MILC-FS-Q}.\nOn the other hand, the GF11 result\\cite{ref:GF11mass} for the Wilson \naction at $\\beta=5.7$ showed \na larger effect of 5\\% between the sizes $L=16$ (2.3~fm) to 24 (3.4~fm).\n \nThe MILC collaboration carried out extensive runs \nat $\\beta=5.7$ with the Wilson action for the sizes $L=12-24$, \nand we reproduce their results for \nthe nucleon mass\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat97} \ntogether with those of GF11 in Fig.~\\ref{fig:finite}.\n\nThe GF11 result for $L=16$ significantly depends on the source\/sink size, \nwith the value for the size 4 consistent with those for $L=24$. \nThe MILC results for $L=16$ \ndo not show a source size dependence.  Their values for the sizes \n$L=12-24$ mutually agree within the statistical error of about 2\\%, \nand are also consistent with the GF11 results for $L=24$.\n\nThese comparisons strongly suggest that finite-size effect at \n$La\\approx 2$~fm is already 2\\% or less also for the Wilson action, \nrather than 5\\% estimated by GF11.  This implies that finite-size \neffects are negligible for $La \\approx 3$ fm as employed by \nthe CP-PACS collaboration.\n\n\n\\subsection{chiral extrapolation of nucleon mass}\\label{sec:chiral-N}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{chiral.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{(a) Chiral extrapolations of the CP-PACS nucleon masses \nat $\\beta=5.9$. (b) Continuum extrapolations of the nucleon masses\nobtained by various chiral extrapolations.\n}\n\\label{fig:chiral}\n\\vspace{-7mm}\n\\end{figure}\n\nLast year the MILC \ncollaboration\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat96} \nemphasized the difficulties in reliable chiral extrapolation for the \nnucleon mass using their high precision data with the \nKS action.  The results obtained for light quarks down to \n$m_\\pi\/m_\\rho \\approx 0.3-0.4$ exhibit a negative curvature, \nand the mass in the chiral limit is sensitive to the choice of \nfitting functions.\n \nThe CP-PACS data for the nucleon mass for the Wilson action measured \ndown to $m_\\pi\/m_\\rho \\approx 0.4$ also show a negative curvature.\nThey tried to fit their data using four fitting\nfunctions; a cubic function in quark mass,\na form predicted by chiral perturbation theory ($\\chi$PT)\nin full QCD\\cite{ref:xPT} given by  \n$m_N = c_0 + c_1 m_\\pi^2 + c_2 m_\\pi^3$, and two forms \nin quenched QCD (Q$\\chi$PT)\\cite{ref:QxPT-S,ref:QxPT-BG,ref:LSc1} given by \n$m_N = c_0 + c_1 m_\\pi + c_2 m_\\pi^2$ and \n$m_N = c_0 - 0.53 m_\\pi + c_1 m_\\pi^2 + c_2 m_\\pi^3$\nwhere in the latter the coefficient of the linear term  \nis fixed to a value estimated from experiment. \nAs shown in Fig.~\\ref{fig:chiral}(a),\nthe four fitting functions describe data equally well, but deviate \nsignificantly toward the chiral limit. \n\n\\begin{figure}[t]\n\\vspace*{-3mm}\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{decay-ratio.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$f_K\/f_\\pi-1$. \nData are from CP-PACS\\protect\\cite{ref:CPPACS},\nGF11\\protect\\cite{ref:GF11decay},\nLANL\\protect\\cite{ref:LANLdecay},\nUKQCD\\protect\\cite{ref:UKQCDlat97}, \nQCDSF\\protect\\cite{ref:QCDSFlatest},\nJLQCD\\protect\\cite{ref:JLQCD-fB}, and\nOSU\\protect\\cite{ref:OSU}.}\n\\label{fig:decay-ratio}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:chiral}(b) we show how the choice of chiral extrapolations \naffects the nucleon mass in the continuum limit.  \nHaving precision results \ndown to $m_\\pi\/m_\\rho=0.4$ at each $\\beta$ helped to constrain the \nuncertainty in the continuum limit almost within the statistical error \nof 3\\%.  \n\nA major difficulty in exploring the chiral limit in quenched QCD simulations \nis the presence of exceptional configurations.\nA method has recently been proposed to avoid this \ndifficulty\\cite{ref:Eichten}.  It would be very interesting to see \nif the method allows to obtain reliable results near the chiral limit as \nclose as $m_\\pi\/m_\\rho \\approx 0.2$, \nwhich would be needed to control the chiral extrapolation at a few \\% \nprecision level.\n\n\nThe APETOV collaboration\\cite{ref:APETOV} studied quark mass\ndependence of octet baryon masses for the non-perturbatively $O(a)$ \nimproved action\nfor the range of $m_\\pi\/m_\\rho = 0.98-0.56$.\nThey found that linearity is better \nif one includes the $O(m_qa)$ improvement term in the \ndefinition of quark mass.\n\n\n\n\\subsection{decay constants and quenching error}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{decay.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Results for $f_\\pi$.\nSee the caption of Fig.~\\protect\\ref{fig:decay-ratio} for\nreferences except JLQCD\\protect\\cite{ref:JLQCD-Kura}\nfor the Wilson action.}\n\\label{fig:decay}\n\\vspace{-7mm}\n\\end{figure}\n\nIt has been observed for the Wilson action that\n$f_K\/f_\\pi-1$ in quenched QCD is much smaller than experiment,\nwhich is considered to be a quenching error \n(see Ref.~\\cite{ref:Sharpelat96} for a recent review).\nIn Fig.~\\ref{fig:decay-ratio} we compile recent results for the \nratio.  Small values in the range 0.1--0.15 are also obtained for the clover \nand KS actions.  A discrepancy of 30--40\\% with experiment \nroughly agrees with estimates based on quenched chiral perturbation \ntheory\\cite{ref:QxPT-BG}.\n\nIn Fig.~\\ref{fig:decay} we summarize the status with the determination \nof the pion decay constant.  Continuum values for the Wilson action \nreported by various groups are consistent with each other, and \nare slightly smaller than experiment, while the situation with the clover \nresults is very unsatisfactory, suffering from a large discrepancy among \ngroups. \n\n\\section{Improvement of Quark Actions}\\label{sec:improve}\n\\begin{table*}[t]\n\\caption{Tests of improved quark actions with improved gauge actions.\nAbbreviations for gauge actions in brackets are \nTILW: tadpole-improved \nL\\\"uscher-Weisz\\protect\\cite{ref:TILW-LW,ref:LM,ref:TILW-Al},\nTISY: tadpole-improved Symanzik\\protect\\cite{ref:Symanzik,ref:LM},\nSY: Symanzik\\protect\\cite{ref:Symanzik}.}\n\\label{tab:table-I}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n    &  $\\beta_{pl}$   & size &  (fm) & \\#conf. & \\ \\ \\ $m_\\pi\/m_\\rho$    & \\#m \\ \n& ref. \\\\\n\\hline\n\\hline\nSCRI (C=NP)[TILW]   & 7.75-12 &  $8^3\\times15$ & & O(1000) & & 1 &  \\cite{ref:SCRIlat97} \\\\\n\\hline\nAlford {\\it et al.} (D234c,C)[TISY] & 1.157 & $5^3\\times18$ & 2.0 & & 0.76,0.70 & 2 & \n\\cite{ref:Alfordlat97}\\\\\nAlford {\\it et al.} (D234c,C)[TISY] & 1.719 & $8^3\\times20$ & 2.0 & & 0.76,0.70 & 2 & \n\\cite{ref:Alfordlat97}\\\\\n\\hline\nDeGrand          & \\multicolumn{6}{c}{Fixed point actions} & \\cite{ref:DeGrand} \\\\\n\\hline\n\\hline\nMILC  (KS,Naik)[TILW] & 7.60 & $16^3\\times32$ & & 100 &0.82-0.3 & 5 & \n\\cite{ref:MILClat97}\\\\\nMILC  (KS,Naik)[TILW] & 7.75 & $16^3\\times32$ & & 200 &0.76-0.33 & 5 & \n\\cite{ref:MILClat97}\\\\\nMILC  (KS,Naik)[TILW] & 7.90 & $16^3\\times32$ & & 200 &0.80-0.27 & 6 & \n\\cite{ref:MILClat97}\\\\\n\\hline\nBielefeld  (fat)[SY] & 4.1 & $16^3\\times30$ & & 57 & $\\approx 0.65$ &  & \n\\cite{ref:Bielefeld}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-7mm}\n\\end{table*}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{rho07-Q.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Comparison of $m_N\/m_\\rho$ at $m_\\pi\/m_\\rho=0.7$\nfor various quark actions.\nC-ML and D234c-ML employ mean link for the tadpole factor. \nGauge actions are denoted in brackets.\nLattice spacings are set with the string tension \n($\\protect\\sqrt\\sigma=427$ MeV)\nexcept for results with TISY gauge action which use \nthe charmonium spectrum.} \n\\label{fig:Rho07-Q}\n\\vspace{-7mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\vspace*{-3mm}\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{mv07.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_V$ at $m_\\pi\/m_\\rho=0.7$. Symbols are the \nsame as in Fig.~\\protect\\ref{fig:Rho07-Q}.\nSold lines are extrapolation of SCRI \ndata\\protect\\cite{ref:SCRIlat96}.\nLattice spacings for the C-ML and D234c-ML actions are \nrecalibrated by us to those given by $\\protect\\sqrt\\sigma$.\n}\n\\label{fig:mv07}\n\\vspace{-7mm}\n\\end{figure}\n\nSeveral groups have been testing improved quark actions with improved \ngauge actions.  In this section we discuss quenched results in this category.  \nNew simulations since Lattice 96 are listed in Table \\ref{tab:table-I}.\n\n\\subsection{improvement of the Wilson action}\n\nImprovement of the Wilson quark action by adding the clover term \nhas been extensively investigated both\nwith the standard gauge \naction\\cite{ref:UKQCDlat96,ref:UKQCDb57,ref:UKQCDlat97,ref:QCDSFlatest,ref:QCDSFb60,ref:QCDSFlat97-2,ref:APETOV,ref:JLQCD-fB}\nand with improved gauge \nactions\\cite{ref:Bock,ref:SCRIlat96,ref:Alfordlat96,ref:Alfordlat97}.\nWe plot in Fig.~\\ref{fig:Rho07-Q} the mass ratio $m_N\/m_\\rho$ at \n$m_\\pi\/m_\\rho =0.7$.  We clearly observe that the clover term \nsignificantly reduces scaling violation so that the ratio agrees \nwith the phenomenological value\\cite{ref:Ono} within 5\\% \nalready at $a\\approx 0.4$~fm.\n\nThe D234 action\\cite{ref:D234} is designed to achieve improvement  \nbeyond the clover action.  \nResults\\cite{ref:Alfordlat95,ref:Alfordlat96,ref:Alfordlat97,ref:Bock}\nfor a class of D234 actions, however, do not show clear improvement\nfor the mass ratio compared with those for the clover action.\n\n\n\nScaling test of hadron masses themselves at a fixed $m_\\pi\/m_\\rho$\nis useful to examine the functional dependence of scaling violation \non the lattice spacing.\nUsing the tadpole-improved L\\\"uscher-Weisz (TILW) gauge action \nfor which we expect only small scaling violation, the \nSCRI group\\cite{ref:SCRIlat96} showed last year that mass results for\nthe tadpole-improved clover action are consistent with an \n$O(a^2)$ scaling behavior,  \nwhile Wilson data need both $O(a)$ and $O(a^2)$ terms.\n \nIn Fig.\\ref{fig:mv07} we reproduce their figure for the vector \nmeson mass at $m_\\pi\/m_\\rho=0.7$,\nadding new results for the Wilson\\cite{ref:CPPACS}(open circles) \nand clover\\cite{ref:UKQCDlat97} (filled circles) actions on\nthe standard plaquette gauge action.\nThe results for the two actions lie on the respective \nextrapolation curves of the SCRI results, showing a reduction of \nscaling violation with the clover action also for the plaquette \ngauge action. \n\nThe Cornell group\\cite{ref:Alfordlat97}\ntested improvement using mean value of link in the Landau \ngauge rather than plaquette for the tadpole factor(right triangles).  \nThey reported that the mean link is superior in reducing scaling\nviolation effects over plaquette.\n\nLet us also mention that\nnon-perturbative determinations of\nthe clover coefficient with improved gauge actions \nhave been attempted\\cite{ref:SCRIlat97,ref:Klassen}.\nSpectrum calculations are in progress.\n\n\n\n\n\n\\subsection{improvement of the KS action}\n\nThe MILC\ncollaboration\\cite{ref:MILClat96}\nstudied the KS and Naik\\cite{ref:Naik} three-link actions\nusing the TILW gauge action, and compared them with those for\nthe KS action on the standard gauge acton.\nThey found that $m_N\/m_\\rho$ is improved by the use of the\nimproved gauge action, but the Naik improvement has a relatively\nsmall effect on the mass ratio. \nPushing the calculation toward higher $\\beta$\\cite{ref:MILClat97}, \nthey found little difference between the Naik and KS actions.\n\nAnother direction of improvement tested by the MILC \ncollaboration\\cite{ref:MILCfat} \nis the use of fat link, \nin which one replaces a link variable with a weighted sum of \nthe link and staples.  \nThis is expected to improve flavor symmetry,\nand indeed they found a substantial reduction \nin the mass difference between the Goldstone and non-Goldstone pions. \n\nThe Bielefeld group\\cite{ref:Bielefeld} studied the fat link \nimprovement with the Symanzik gauge action.\nThey also observed improvement of flavor symmetry for this quark \naction, while $O(p^2)$ and $O(p^4)$ improved actions \nwhich include many link paths do not show any significant \nimprovement of flavor symmetry. \n\n\n\\section{Toward Full QCD Spectrum}\\label{sec:fullQCD}\n\\begin{table*}[t]\n\\caption{Recent spectrum runs in full QCD for $N_f$=2.\nNew results since Lattice 96 are marked by double asterisks and\nthose with increased statistics by asterisks.}\n\\label{tab:table-F}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n   &       $\\beta$    &  size   &  (fm) & traj.    &  $m_\\pi\/m_\\rho$ & \\#m  & ref.\\\\\n\\hline\n\\hline\nSESAM (W)*    &  5.6   &  $16^3\\times32$ &  1.4  & $200\\times 25$ &  0.84-0.7 & 3 &\n\\cite{ref:SESAMlat96,ref:Hoeber}\\\\\nT$\\chi$L (W)* &  5.6   &  $24^3\\times40$ &  2.0  & O(3000)     &  0.7,0.55 & 2 &\n\\cite{ref:TxLlat96,ref:Hoeber} \\\\\n\\hline\nUKQCD (C=1.76)**  &  5.2  & $12^3\\times24$ &     & 50 conf. & 0.85-0.75 & 4 &\n\\cite{ref:Talevi}\\\\\n\\hline\nCP-PACS (W,C=1,TP)** &  & $(12,16)^3\\times32$ & \\multicolumn{4}{c}\n{study of action improvement} & \\cite{ref:CPPACS-F} \\\\\n\\hline\n\\hline\nMILC (KS)  & 5.30 & $12^3\\times32$ & 3.7 &1000-5000 & 0.8-0.3   & 8 &\n\\cite{ref:MILClat96} \\\\\nMILC (KS)* & 5.415& $16^3\\times32$ & 3.2 &1000-2000  & 0.77-0.44 & 6 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\nMILC (KS)* & 5.415& $12^3\\times24$ & 2.4 &2000      & 0.46       & 1 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\nMILC (KS)  & 5.50 & $24^3\\times64$ & 3.6 &1000-2000 & 0.69-0.63 & 2 &\n\\cite{ref:MILClat96}\\\\\nMILC (KS)  & 5.50 & $20^3\\times48$ & 3.0 &2000      & 0.56-0.48 & 2 &\n\\cite{ref:MILClat96}\\\\\nMILC (KS)* & 5.60 & $24^3\\times64$ & 2.6 &1500-2000 & 0.75-0.53 & 4 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\n\\hline\nColumbia(KS,$N_f$=2)*& 5.70 & $16^3\\times32(40)$& 1.5 & 1400-4900 &0.70-0.57& 4 & \n\\cite{ref:Columbialat96,ref:Columbialat97} \\\\\nColumbia(KS,$N_f$=4)*& 5.40 & $16^3\\times32$& 1.5 & 2700-4500&0.72-0.67& 2 &\n \\cite{ref:Columbialat96,ref:Columbialat97} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{-3mm}\n\\end{table*}\n\nWith progress of our understanding of the quenched spectrum, \nincreasingly larger efforts are beginning to be spent in simulations of \nfull QCD.  Here we summarize recent work listed in \nTable \\ref{tab:table-F}.\n\n\\subsection{progress with the KS action}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{edplot.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Edinburgh plot for $N_f$=2 KS quarks reported by \nMILC\\protect\\cite{ref:MILClat97-F} together with those \nfrom HEMCGC\\protect\\cite{ref:HEMCGC},\nColumbia\\protect\\cite{ref:Columbialat97,ref:Columbia32},\nand Kyoto-Tsukuba\\protect\\cite{ref:K-T}.\nIn parentheses are lattice sizes.}\n\\label{fig:Ed-KS-Nf2}\n\\vspace{-7mm}\n\\end{figure}\n\nThe MILC collaboration\\cite{ref:MILClat96} continued their study \nof the $N_f=2$ KS spectrum for $\\beta=5.3-5.6$ employing large lattices \nof a size $La \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 2.6$~fm. In Fig.~\\ref{fig:Ed-KS-Nf2} we show their  \nresults in the Edinburgh plot together with those of previous \nstudies\\cite{ref:Columbialat97,ref:HEMCGC,ref:Columbia32,ref:K-T}. \n\nThe ratio $m_N\/m_\\rho$ decreases toward weak coupling. \nTaking advantage of improved precision of their results \nas is clear from Fig.~\\ref{fig:Ed-KS-Nf2}, \nthe MILC collaboration attempted a continuum extrapolation of \n$m_N\/m_\\rho$ for a fixed value of $m_\\pi\/m_\\rho$.  They find \n$m_N\/m_\\rho=1.252(37)$ at the physical point in the continuum limit. \n\nAlso of interest is the problem of\nhow the KS spectrum depends on the number of dynamical \nquark flavors.\nColumbia group\\cite{ref:Columbialat97} showed that\nthe four flavor hadron spectrum is nearly parity doubled on \na $16^3\\times 32$ lattice at $\\beta=5.4$.\nChiral symmetry breaking effects are smaller for four\nflavors than for two or zero flavors.\n\n\\subsection{progress with the Wilson action}\nSimulations of full QCD with the Wilson quark action for $N_f$=2\nhave been pushed forward by the SESAM\\cite{ref:SESAMlat96,ref:Hoeber}\nand T$\\chi$L\\cite{ref:TxLlat96,ref:Hoeber} collaborations.  \nSimulations were initially made at $\\beta=5.6$ on a $16^3$ \nspatial lattice ($La \\approx 1.4$ fm) for $m_\\pi\/m_\\rho=0.85-0.7$ (SESAM), \nwhich have been extended to those on a larger lattice $24^3$ \n($La \\approx 2.0$ fm) and closer to the chiral limit with \n$m_\\pi\/m_\\rho =$0.7 and 0.55 (T$\\chi$L).\n\nAn important aspect of their study is a careful examination of \nvarious algorithmic issues of full QCD simulation, including\ndevelopment and tuning of efficient Wilson marix \ninverter\\cite{ref:Frommer}\nand a detailed autocorrelation study. \n\nFor the spectrum, they observed 3\\% (5\\%) finite-size effects\nfor $\\rho$-meson (nucleon) at $m_\\pi\/m_\\rho \\approx 0.7$.\nThe magnitude is comparable to that for the KS \naction\\cite{ref:K-T,ref:MILC-FS-F}.\nThey estimated strange hadron masses,\ntreating the strange quark as a valence quark\nin the presence of two light dynamical quarks.\nThe $K-K^{*}$ mass splitting is smaller than experiment by \n15\\%, contrary to the expectation that dynamical sea quark \neffects alleviate the small hyperfine splitting of quenched \nQCD.  It is possible that dynamical quarks employed is still \ntoo heavy to improve the splitting significantly.\n\nSESAM and T$\\chi$L also studied the static potential and several \nhadron matrix elements to explore effects of sea quarks. See \nRef.~\\cite{ref:Gusken} for a review.\n\n\\subsection{full QCD with improved actions}\n\nTill last year there were only sporadic attempts toward full \nQCD simulations of the light hadron spectrum with improved \nactions\\cite{ref:SCRIlat96-F}.\nThis year the CP-PACS collaboration\\cite{ref:CPPACS-F} \nand the UKQCD collaboration\\cite{ref:Talevi} presented \npreliminary results of a systematic attempt in this direction. \n\nThe CP-PACS collaboration made a comparative study of improvement \nat a coarse lattice $a^{-1} \\approx 0.9-1.5$ GeV employing \nthe plaquette and an RG-improved action\\cite{ref:Iwasaki} for gluons and\nthe Wilson and tadpole-improved clover action for quarks.\nFor one action combination, they also explored the chiral limit \ndown to $m_\\pi\/m_\\rho\\approx 0.4$ with simulations on a $16^3\\times 32$ \nlattice.\nThe UKQCD collaboration employed the plaquette action at $\\beta=5.2$ and \nthe clover action with a clover coefficient of 1.76.\nSimulations were made for four values of sea quark masses\nand the spectrum is calculated for four values of valence quark \nmasses on each dynamical quark ensemble.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{rho07.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_N\/m_\\rho$ in full QCD with $N_f=2$ as a function of \n$m_\\rho a$ both calculated at $m_\\pi\/m_\\rho =0.7$. \nAbbreviations for gauge actions are\nP: plaquette and  R: RG-improved\\protect\\cite{ref:Iwasaki}, and \nfor quark actions W: Wilson and C: clover.\nData are taken from CP-PACS\\protect\\cite{ref:CPPACS-F,ref:CPPACS},\nSCRI\\protect\\cite{ref:SCRI-W2}, and\nSESAM\\protect\\cite{ref:Hoeber}.\n}\n\\label{fig:rho07}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.\\ref{fig:rho07} we compile full QCD results for $m_N\/m_\\rho$ \nas a function of $m_\\rho a$, both calculated at $m_\\pi\/m_\\rho =$ 0.7.\nResults for the Wilson quark action have large scaling violation \nand approximately lie on a single \ncurve, irrespective of the choice of gauge actions. \nIn contrast the lattice spacing dependence is much \nweaker for the clover actions, again irrespective of the gauge action, \nand  the value of the ratio is close to a phenomenological estimate\neven on a very coase lattice of $a^{-1} \\approx 1.0$ GeV. \nThese results show that a significant improvement of $m_N\/m_\\rho$ \ndue to the clover term observed for the quenched case also \nholds in full QCD. \n\nAnother interesting question in full QCD is to what extent \nthe lattice scale obtained from the hadron spectrum agrees with \nthat from the static potential.\nThe clover term is important also in this regard. \nA mismatch of the scale determined from $m_\\rho$ in the chiral limit \nand that with the string tension observed for the Wilson action\nat $a^{-1} \\approx 1.0$ GeV is much reduced by the use of the clover \naction\\cite{ref:CPPACS-F}.\nThe UKQCD collaboration reported that the scale determined from $m_{K^{*}}$ \napproximately agrees with that from $r_0$ for each value of \ndynamical quark.\n\nFor an effect of improvement of gauge actions, \nrotational symmetry of the potential is improved to a great extent\nalso in full QCD\\cite{ref:CPPACS-F}.\n\nThe effects of improvement summarized here are parallel to those \nobserved in quenched QCD, and come mainly from valence quarks rather than \ndynamical sea quarks.  \nNovertheless, they are important since they show that \nrealistic full QCD simulations are possible without having \nto reduce the lattice spacing below $a^{-1}\\approx 2$ GeV which is  \nneeded with the standard action.\n\n\\section{Other Topics}\\label{sec:other}\nCalculation of glueball masses in quenched QCD\nhas reached a stage to pinpoint the mass\nranges at least for the scalar glueball. \nThe GF11 collaboration\\cite{ref:GF11lat97} \nreported $m_{0^{++}}=1710(63)$ MeV as the infinite volume value \nin the continuum from a reanalysis of their data\\cite{ref:GF11gb}.\nThis value is consistent or slightly higher than the previous\nresults by other groups\\cite{ref:UKQCDgb,ref:MPn,ref:Luo}.\n\nThe central effort of the GF11 collaboration has been a \ncalculation of the mass of the $s\\bar s$ scalar \nmeson\\cite{ref:GF11lat97,ref:GF11lat96}, for which they \nfound values below $m_{s\\bar s} < 1500$ MeV.\nThey conclude that the observed meson $f_J(1710)$ is mainly a\nscalar glueball, while $f_0(1500)$ is mainly an $s\\bar s$\nquarkonium.  \n\nThe SESAM collaboration\\cite{ref:SESAMgb}\nmade a glueball mass measurement with their \nfull QCD runs.\nNo clear dynamical quark effects are seen in the glueball\nmasses. Instead, they observed strong finite size effects\nin the scalar glueball mass, which may be an indication of\nthe presence of mixing between the glueball and \nthe $s\\bar s$ scalar meson. \n\nTwo groups have contributions for spin exotic meson masses. \nThe UKQCD collaboration\\cite{ref:UKQCDlat97-ex} increased\nstatistics since last year. \nCalculating masses at one combination of $\\beta$\nand the quark mass and employing a model to estimate masses at\nthe strange quark,  \nthey obtained $m_{1^{-+}}(s\\bar s) =2000(200)$ MeV.\nThe MILC collaboration\\cite{ref:MILClat97-ex}\nmade simulations at $\\beta$=5.85 and 6.15. \nExtrapolation to the strange quark mass was made to \nobtain $m_{1^{-+}}(s\\bar s)=2170(80)$ MeV.\nThe two results are consistent within 10\\%.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nA number of interesting studies have been made this year, making a step\nforward toward a precise determination of the light hadron spectrum.\n\nFor the quenched spectrum, a systematic deviation from experiment\nhas been uncovered both in the meson and baryon sectors.\nQuantitative results have been accumulated with improved actions \nboth for quenched and full QCD, clarifiing to what extent improving \nactions reduce scaling violations\nin the light hadron spectrum.\nQuenched clover simulations are moving toward high precision determination \nof physical quantities exploiting the improved scaling behavior,  and  \nsimilar effort should be pursued with other improved actions.\n\nAnd finally, attempts toward a realistic simulation in full QCD have begun.\nIn my opinion, there is real hope that such a calculation could be achieved \nwith the current generation of computers through application\nof improved actions.\n\n\\ \\\\\nI am deeply indebted to all the colleagues who made their results available\nto me before the conference.\nI also would like to thank Y.~Iwasaki and A.~Ukawa for \ncritical comments and suggestions on the manuscript.\nThis work is in part supported by the Grant-in-Aid of Ministry of\nEducation, Science and Culture (Nos. 08NP0101 and 09304029). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}