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+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\nThe quark model of Gell-Mann~\\cite{GellMann:1964nj} and Zweig~\\cite{Zweig:1964} classifies mesons \n({\\ensuremath{\\quark\\quarkbar}}\\xspace) and baryons ({\\ensuremath{\\Pq}}\\xspace\\quark{\\ensuremath{\\Pq}}\\xspace) into multiplets, and also allows for hadrons \nwith more than the minimal quark contents.\nIn 2015, LHCb observed two pentaquark states in the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace}$\\cite{LHCb-PAPER-2015-029}. In the decay  \nchannel  \n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$,\\footnote{Unless explicitly noted, charge conjugate decays are implied.}  \ncharmed dibaryon resonant states \ncould be present. As discussed \nin Ref.~\\cite{MAIANI201537}, such states could manifest via the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace + [cd][ud][ud] = {\\ensuremath{\\overline \\proton}}\\xspace + \\mathscr{D}_c^+}$, where\n${\\mathscr{D}_c^+}$ is the dibaryon state with a mass below 4682\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.\nThe subsequent decay of the ${\\mathscr{D}_c^+}$ dibaryon could proceed either via\nquark rearrangement to the final state ${{\\ensuremath{\\Pp}}\\xspace\\PSigma_c^0}$, with\n${\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, or via string breaking to the final state $\\mathscr{P}_c^0(\\bar{u}[cd][ud])$, \nwhich could involve\na lighter, yet undiscovered $\\mathscr{P}_c^0$  pentaquark state,  \n$\\mathscr{D}_c^+\\ensuremath{\\rightarrow}\\xspace \\mathscr{P}_c^0(\\bar{u}[cd][ud]) {\\ensuremath{\\Pp}}\\xspace$,  \nwith ${\\mathscr{P}_c^0 \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$~\\cite{MAIANI201537}.\nThe discovery of any of these decay modes would test the predictions \nof quantum chromodynamics and the fundamental workings of the Standard Model.\n\nIn this Letter, the first observation of the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, \nreferred to as the signal channel, is reported. \nA measurement is made of its  \nbranching fraction relative to the normalisation channel ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.  \nResonance structures within the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ system are also investigated. \nWhile no evidence for dibaryon resonances is found, \nsignificant contributions from the $\\PSigma_c(2455)^0$ and $\\PSigma_c^{*}(2520)^0$ resonances \nare found in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ invariant mass spectrum. \nThe ratios of branching fractions between decays via these resonances, \nhereinafter denoted as $\\PSigma_c^0$ and $\\PSigma_c^{*0}$, and the \n${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ inclusive decay are also reported.\nThe measurements in this Letter  are based on a data sample of $pp$ collisions collected  \nwith the LHCb detector at centre-of-mass energies of $\\sqrt{s}=$\n7\\ifthenelse{\\boolean{inbibliography}}{\\ensuremath{~T\\kern -0.05em eV}}{\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}}\\xspace in 2011 and $\\sqrt{s}=$ 8\\ifthenelse{\\boolean{inbibliography}}{\\ensuremath{~T\\kern -0.05em eV}}{\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}}\\xspace in 2012, corresponding to an integrated luminosity of\n3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace.\n\n\n\n\\section{Detector and simulation}\n\\label{sec:Detector}\n\nThe \\mbox{LHCb}\\xspace detector~\\cite{Alves:2008zz,LHCb-DP-2014-002} is a single-arm forward\nspectrometer covering the \\mbox{pseudorapidity} range ${2<\\eta <5}$,\ndesigned for the study of particles containing {\\ensuremath{\\Pb}}\\xspace or {\\ensuremath{\\Pc}}\\xspace\nquarks. The detector includes a high-precision tracking system\nconsisting of a silicon-strip vertex detector surrounding the $pp$\ninteraction region, a large-area silicon-strip detector located\nupstream of a dipole magnet with a bending power of about\n$4{\\mathrm{\\,Tm}}$, and three stations of silicon-strip detectors and straw\ndrift tubes placed downstream of the magnet.\nDifferent types of charged hadrons are distinguished using information\nfrom two ring-imaging Cherenkov (RICH) detectors.\nPhotons, electrons and hadrons are identified by a calorimeter system consisting of\nscintillating-pad and preshower detectors, an electromagnetic\ncalorimeter and a hadronic calorimeter. Muons are identified by a\nsystem composed of alternating layers of iron and multiwire\nproportional chambers.\nThe online event selection is performed by a trigger~\\cite{LHCb-DP-2012-004},\nwhich consists of a hardware stage, based on information from the calorimeter and muon\nsystems, followed by a software stage, in which all charged particles\nwith ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>500\\,(300)\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$ are reconstructed for 2011\\,(2012) data, where \\mbox{$p_{\\mathrm{ T}}$}\\xspace is the transverse momentum~\\cite{LHCb-DP-2012-004}.  \nAt the hardware trigger stage, events are required to contain a \nmuon or dimuon pair with high \\mbox{$p_{\\mathrm{ T}}$}\\xspace, or a hadron, photon or electron \nwith high transverse energy deposited in the calorimeters.\nThe software trigger requires a two-, three- or four-track\nsecondary vertex with a significant displacement from \nany primary proton-proton interaction vertices (PVs).  \nAt least one charged particle\nmust have a ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace > 1.7~(1.6)\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ for 2011 (2012) data, and be\ninconsistent with originating from a PV.\nA multivariate algorithm~\\cite{BBDT} is used for\nthe identification of secondary vertices consistent with the decay\nof a {\\ensuremath{\\Pb}}\\xspace hadron.\n\n\nSimulated samples of the signal, the normalisation channels and \nbackgrounds produced in $pp$ collisions  are generated using\n\\mbox{\\textsc{Pythia}}\\xspace~\\cite{Sjostrand:2007gs,*Sjostrand:2006za}\nwith a specific \\mbox{LHCb}\\xspace\nconfiguration~\\cite{LHCb-PROC-2010-056}. Decays of hadronic particles\nare described by \\mbox{\\textsc{EvtGen}}\\xspace~\\cite{Lange:2001uf}, in which final-state\nradiation is generated using \\mbox{\\textsc{Photos}}\\xspace~\\cite{Golonka:2005pn}. The\ninteraction of the generated particles with the detector, and its response,\nare implemented using the \\mbox{\\textsc{Geant4}}\\xspace\ntoolkit~\\cite{Allison:2006ve, *Agostinelli:2002hh} as described in\nRef.~\\cite{LHCb-PROC-2011-006}.\n\n\n\\section{Candidate selection}\nThe ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ candidates \nare reconstructed using the decay ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$. \nAn offline selection is applied,  \nbased on a loose preselection, followed by a multivariate analysis. \nTo minimize the systematic uncertainty on the ratio of efficiencies  \nbetween the signal and the normalisation channels, the selection criteria on the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates   \nare similar  between the two channels. \n\n\nReconstructed final-state particles in ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and \n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ candidate decays are required to have \na momentum ${p>1\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ and ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>100\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$.  \nProtons and antiprotons are required to \nhave ${p>10\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$ to improve particle identification.  \nAll final-state particles are also required to be inconsistent with originating from any PV, \nby rejecting the tracks with a small \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace, where \n\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace is the difference in the vertex-fit \\ensuremath{\\chi^2}\\xspace of a given PV \nwith or without the track considered, requiring ${\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace > 4}$. \nCandidate {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays are required to have at least one decay product with ${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>500\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c}}\\xspace}$ and \n${p>5\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, a good vertex-fit quality, and an invariant mass \nwithin $\\pm$15\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace of the known {\\ensuremath{\\Lz^+_\\cquark}}\\xspace mass~\\cite{PDG2016}. The scalar sum of the transverse momenta of \nthe {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay products is required to be greater than 1.8\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace. \n\nThe {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace candidate is reconstructed by combining a {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nwith a pion, and the signal candidate is reconstructed by combining a {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nwith a pion, a proton and an antiproton.  \nThese combinations must form a {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate with a good-quality vertex and  \nbe consistent with \noriginating from the associated PV,  \ndefined as that for which the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate has the least \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace. \nFurthermore, the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate is required to decay downstream of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace \ndecay vertex. The {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay time, calculated as ${t = m_{\\ensuremath{\\Lz^0_\\bquark}}\\xspace L\/p}$, is required to be \ngreater than 0.2\\ensuremath{{\\mathrm{ \\,ps}}}\\xspace, where $m_{\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ is the mass, $L$ is the decay length and $p$ is \nthe momentum of the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ candidate.\nThe {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is also required to have at least one final-state particle in the decay chain with \n${\\mbox{$p_{\\mathrm{ T}}$}\\xspace>1.7\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, ${p>10\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c}}\\xspace}$, and have at least one track significantly \ninconsistent with originating from the associated PV by requiring the track to have  ${\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace > 16}$. \nFinal-state tracks of signal and normalisation channel candidates  \nmust pass strict particle-identification requirements based on the RICH\ndetectors, calorimeters and muon stations. \nA constrained fit~\\cite{Hulsbergen:2005pu} is applied to the candidate decay chain for \nboth the signal and the normalisation channels,  \nrequiring the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate to come from the associated PV and constraining the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace particle to its known \nmass~\\cite{PDG2016}. In the case of the search of the resonant  contributions, the mass of the \n{\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is also constrained to the known mass~\\cite{PDG2016}. \n\nTrigger signals are associated with reconstructed\nparticles from the decays of the signal channel or of the normalisation channel. \nSelection requirements can therefore be made on the trigger\nselection itself and on whether the decision was due to the reconstructed candidate decay, \nother particles produced in the $pp$ collision, or a combination of the two.\nThis association makes it possible to use a data-driven method for the correction\nand systematic uncertainty estimation on the trigger efficiencies~\\cite{LHCb-DP-2012-004}.\nTo take advantage of the similarity between the signal and the normalisation channels,  \nwhich helps to  minimize the systematic uncertainty on the ratio of their efficiencies,\ncandidates are classified in one of the following two hardware trigger categories.\nIn the first category, called Triggered On Signal ({TOS}), the candidate must include a hadron consistent with originating from\nthe decay of a ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace$ candidate and which deposited enough transverse energy  in the calorimeter\nto satisfy the hardware trigger requirements. The typical value of the transverse energy\nthreshold is around 3.5\\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V\\!\/}c^2}}\\xspace.\nAs the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace baryon is a {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay product for both the signal  and the \nnormalisation channels, this choice minimizes the difference between the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay modes. \nThe second category,\ncalled Triggered Independent of Signal ({TIS}), comprises events\nwhich satisfied the hardware trigger through signatures unassociated with the complete {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay chains, \neither due to a muon with high \\mbox{$p_{\\mathrm{ T}}$}\\xspace,\nor a hadron, photon, or electron with high transverse\nenergy deposited in the calorimeters.\nThe efficiencies of the TIS and TOS requirements are different, so\nthe data are divided into two\nstatistically independent samples, one TIS, and the other TOS and not TIS,\nwhich will be referred to as TOS for the rest of this Letter.\n\nThe so-called cross-feed backgrounds, contributing under the peak of the \ninvariant mass of the normalisation channel or of the signal channel  \nfrom the  ${{\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace)\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) {\\ensuremath{\\pion^-}}\\xspace}$ and  \n${{\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace) \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace}$ decays, respectively, with ${{\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace) \\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$ \nor ${{\\ensuremath{\\D^+}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip}$,  \nwhere either the kaon or pion is misidentified as a proton,  \nare explicitly vetoed when both of the following two conditions are satisfied. First, the mass hypothesis \nof the proton from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate is replaced with either the kaon or pion hypothesis,  \nand the resulting invariant mass of the combination is consistent  \nwith the known ${\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace)$ mass~\\cite{PDG2016} within $\\pm$15\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.\nSecond, the invariant mass of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate \nis set to the known ${\\ensuremath{\\D^+}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace)$ mass~\\cite{PDG2016}, and the resulting invariant mass of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is consistent with  \nthe known ${\\ensuremath{\\Bbar{}^0}}\\xspace({\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace)$ mass~\\cite{PDG2016} within $\\pm$25\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, and within  \n$\\pm$45\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays.\n\nFurther background reduction is achieved using a multivariate\nanalysis based on a gradient boosted decision tree (BDTG)\\cite{Breiman}. The BDTG\nis trained using twelve variables: the vertex-fit quality of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \nand {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates, the decay-vertex displacement along the beamline between the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace and {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates, \nthe displacement between the decay vertex of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate and the associated PV, the \n\\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate, the angle between the reconstructed {\\ensuremath{\\Lz^0_\\bquark}}\\xspace momentum and \nthe direction of flight from the associated PV to the decay vertex, \nthe smallest \\mbox{$p_{\\mathrm{ T}}$}\\xspace  \nand smallest \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace among the three {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay products, the \\mbox{$p_{\\mathrm{ T}}$}\\xspace and \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the pion originating directly  \nfrom the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay, and the smallest \\mbox{$p_{\\mathrm{ T}}$}\\xspace and smallest \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace between  \nthe {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace originating directly from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay.   \nThe BDTG training is performed using simulated samples for the signal,  \nand data distributions for the background, \nwith reconstructed invariant mass \nwell above the known {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass~\\cite{PDG2016}. Cross-feed backgrounds from the\ndecays ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$, ${{\\ensuremath{\\Bbar{}^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$\nare explicitly vetoed during the BDTG-training process by requiring the difference between the    \nreconstructed {\\ensuremath{\\Pb}}\\xspace-hadron mass and its known mass to be  \ngreater than $\\pm$30\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace. The BDTG selection is optimized for the figure of merit  \n$S\/\\sqrt{S+B}$, where $S$ and $B$ are the expected signal and background yields  \nwithin $\\pm$30\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace of the known {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass~\\cite{PDG2016}. The initial value of $S$ and $B$  \nwithout BDTG selection is obtained from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass spectrum in data.   \nNo improvement in the normalisation channel\nis found using a similar procedure, therefore no BDTG selection is applied. \nA systematic uncertainty is assessed for this choice in Section~\\ref{sec:syst}.   \n\nDue to the large number of final-state particles in the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decays, particles   \nwith the same charge may share track segments, representing a possible background. \nThese  tracks are referred to as clones, \nand are suppressed by requiring    \nthat the opening angle between any same-charged tracks in the candidate is larger than 0.5 mrad. \nThis selection\nremoves 2\\% of candidates in the signal sample and 0.1\\% in the normalisation sample. \nIf multiple {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates are reconstructed in one single event, \none candidate is chosen at random in the following two cases. \nFirst, if the proton from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays is exchanged with that directly from \nthe {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decays, forming two candidates with nearly the same {\\ensuremath{\\Lz^0_\\bquark}}\\xspace mass. \nSecond, if a track from one candidate shares a segment with a track from another  \ncandidate. \nWith these criteria, 2.5\\% of candidates \nin the signal sample and 0.1\\% in the normalisation sample are vetoed. \nAfter these selections, 0.8\\% of events in the signal sample  \nand 0.2\\% in the normalisation sample contain multiple {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidates. \nThese remaining multiple candidates mainly originate from the random \ncombinations of the final-state tracks, and have a negligible influence on the \nestimation of the signal yields.  \nNo further vetoes on these candidates are applied. \n\n\n \n \n\n\n\\section{Efficiencies}\n\\label{sec:eff}\nThe total efficiencies of the signal and the normalisation decays are given by\n\\begin{equation}\n\\epsilon_{\\rm total} = \\epsilon_{\\rm a}\\cdot\\epsilon_{\\rm rec\\&sel|a}\\cdot\\epsilon_{\\rm trig|sel}\\cdot\\epsilon_{\\rm PID},\n\\end{equation}\nwhere $\\epsilon_{\\rm a}$ represents the geometrical acceptance of the \\mbox{LHCb}\\xspace detector,\n$\\epsilon_{\\rm rec\\&sel|a}$ is the efficiency of reconstruction and selection  \ncalculated on candidates in the acceptance, $\\epsilon_{\\rm trig|sel}$ is the trigger efficiency  \nof the selected candidates, and $\\epsilon_{\\rm PID}$ is the particle-identification efficiency.\nAll efficiencies except $\\epsilon_{\\rm PID}$ and $\\epsilon_{\\rm trig|sel}$\nare determined from simulation. The particle-identification efficiency is determined  \nfrom calibration data specific to each data-taking year, binned in momentum and pseudorapidity\nof the track in question, as well as in the multiplicity of the event~\\cite{LHCb-PROC-2011-008}.\nThe trigger efficiency is determined from a combination of\nsimulation and data-driven techniques where the agreement between data and simulation\nis explicitly verified using the normalisation sample satisfying the TIS requirement. \nAll efficiencies are calculated separately for the TIS and TOS\ntrigger samples, and for data-taking year, due to the difference in\ncentre-of-mass energies. Agreement between data and simulation is improved by applying\na per-candidate weight to the \\mbox{$p_{\\mathrm{ T}}$}\\xspace and rapidity, $y$, of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon in simulated events to match\nthe normalisation sample in the TIS category, which is largely independent of trigger\nconditions. \nThe \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ distributions of {\\ensuremath{\\Lz^0_\\bquark}}\\xspace produced in $pp$ collision \nare identical for the signal and the normalisation channels, so the same per-candidate weights \nare applied to the signal sample. \nThe simulated \\ensuremath{\\chi^2_{\\text{IP}}}\\xspace of the final-state particles and the vertex-fit \\ensuremath{\\chi^2}\\xspace of {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidates  \nare weighted to reproduce the data distributions. \nThe ratio between the efficiencies of the signal and the normalisation channels,  $\\epsilon_r$, \nis $(10.00\\pm0.12)\\%$ for the TIS sample and $(11.39\\pm0.22)\\%$ for the\nTOS sample, including uncertainties due to the limited size of the simulated sample.\n\n\n\\section{Fit model and the ratio of branching fractions}\n\\label{sec:eff}\nThe yields in both the signal and  \nthe normalisation channels are determined  \nfrom an unbinned extended maximum-likelihood fit to the corresponding invariant-mass spectra \nwith both the TIS and TOS samples combined.  \nThe signal  \nis modelled by a sum of two Crystal Ball functions\\cite{Skwarnicki:1986xj} with\na common mean of the Gaussian core, and with the tail parameters fixed from simulation. For both the\nsignal and the normalisation channels, the background from random combinations of final-state  \nparticles is described by an exponential function, whose parameters are left free in the fits \nand are independent between the signal and the normalisation channels.  \nFor the\nnormalisation channel, background from the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace$ decays, with\n${\\ensuremath{\\rhomeson^-}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ is modelled by the convolution of an empirical threshold function  \nwith a Gaussian resolution. The contribution due to misidentification of the kaon to pion from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$\nis modelled by a sum of two Crystal Ball functions. The parameters of these two background sources are\ntaken from simulation. The fits to the invariant-mass distributions for the signal \nand the normalisation channels are shown in Figure~\\ref{fig:fits}. In this figure, the TIS and TOS \nsamples are combined. From these fits, $926\\pm43$\n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and $(167.00\\pm0.50)\\times10^3$   \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays are observed.\n\n\\begin{figure}[!tbp]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Fig1a.pdf}\n\\includegraphics[width=0.48\\textwidth]{Fig1b.pdf}\n\\caption{Invariant mass distributions of the (a) ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ \nand (b) ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ candidates. Fit results are overlaid as a solid blue line.\nFor (a), the red dotted line represents the signal component and the green dotted\nline the background due to random combinations. For (b), the red dotted line is the\nsignal component, the green dotted line is the random combination background,\nthe purple dashed line is the contribution from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace$ and the brown\ndashed-dotted line represents the contribution from ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$.}\n\\label{fig:fits}\n\\end{center}\n\\end{figure}\n\n\nTo determine the ratio of  branching fractions  \n${\\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)} }$, \nindicated \u00a0in the  following by $\\mathcal{B}_r$,  \na simultaneous fit is performed to the signal and the normalisation channels, each divided into the two  \nindependent trigger categories. The yield of the normalisation sample, $N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$, is \na free parameter in the fits, whereas the yield of the signal sample is calculated as\n${N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) = \\mathcal{B}_r\\times\\epsilon_r\\times N({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)}$, where\n$\\epsilon_r$ is the ratio between the total efficiency of the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays.  \nThe ratio of branching fractions $\\mathcal{B}_r$  \nis the same for the TIS and TOS subsamples and is measured to be  \n${\\mathcal{B}_r = 0.0542 \\pm 0.0023}$. The corresponding\nsignal yields are 677 $\\pm$ 29 for the TIS subsample and 259 $\\pm$ 11 for the TOS subsample; the yields in the normalisation\nsample are ${(124.9 \\pm 0.4)\\times 10^3}$ for the TIS subsample and ${(41.9 \\pm 0.2) \\times10^3}$ for the TOS subsample.\n\n\n\\section{Systematic uncertainties}\n\\label{sec:syst}\nThe systematic uncertainties on the measurement of the ratio of branching \nfractions are listed in Table~\\ref{tab:systemerr}. The total systematic uncertainty  \nis determined from the sum in quadrature of all terms. \n\nFirst, the uncertainty\nrelated to the background modelling is considered.\nIn the signal sample, \nthe exponential function is replaced with a second-order polynomial for the background component. \nFor the normalisation\nchannel, the model is varied by using the sum of\ntwo exponential functions. The resulting uncertainty on the ratio of branching  \nfractions is 0.6\\%. The uncertainties due to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace}$ shape parameters are assessed\nby increasing the width of the Crystal Ball functions  \nby 10\\%, corresponding to two standard deviations, resulting in a change of 0.1\\%.\nThe uncertainty due to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\rhomeson^-}}\\xspace}$ contribution is estimated by varying\nthe shape parameters by one standard deviation, resulting in an uncertainty of 0.4\\%.\nThe total uncertainty on the ratio of the branching fractions due to the background modelling \nis 0.7\\%.  \n\nThe signal-model parameterization is changed to a single Hypatia function\\cite{Santos:2013gra},\nwhere the mean and width are allowed to float and all other parameters are taken  \nfrom simulation, resulting in an uncertainty of 0.1\\%. \n\n\n\\begin{table}[!tbp]\n\\centering\n\\caption{Summary of systematic uncertainties and correction factors\nto the ratio of branching fractions measurement. \nAll uncertainties \nare given as a percentage of the ratio  \nof branching fractions.}\n\\label{tab:systemerr}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lSS}\\hline\\hline\n\\multicolumn{1}{l}{Source} & \\multicolumn{1}{c}{Uncertainty (\\%)}  & \\multicolumn{1}{c}{Correction factor} \\\\\\hline\nBackground fit model & 0.7 & $\\rm{-}$ \\\\\nSignal fit model & 0.1 & $\\rm{-}$ \\\\\nPID efficiency &0.3 & $\\rm{-}$ \\\\\nTracking efficiency calibration &0.8 & 0.985\\\\\nKinematic range of final-state tracks&0.7 & $\\rm{-}$ \\\\\nHadron interaction &4.4 & $\\rm{-}$ \\\\\n$\\mbox{$p_{\\mathrm{ T}}$}\\xspace, ~y$ weighting  & 1.0 & $\\rm{-}$ \\\\\nTrigger efficiency  &2.9 & $\\rm{-}$\\\\\nSimulated sample size  &1.3 & $\\rm{-}$\\\\\nCandidates with clone tracks and multiple candidates & 0.2&$\\rm{-}$ \\\\\nVeto of the reflection background & 0.4 & $\\rm{-}$ \\\\\n{\\ensuremath{\\Lz^+_\\cquark}}\\xspace Dalitz weighting &0.2  &0.984\\\\\n{\\ensuremath{\\Lz^+_\\cquark}}\\xspace polarization & 0.3  & 0.987\\\\\nResonant structures & 1.8& 1.041\\\\\n\\hline\nTotal&6.0 &0.996 \\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\nThe uncertainty on the relative efficiency of the particle identification is assessed by\ngenerating pseudoexperiments. For each pseudoexperiment, efficiencies in different \nmomentum, pseudorapidity and multiplicity bins are  \ndetermined from independent Gaussian distributions with mean values equal to the nominal efficiencies and widths \ncorresponding to their uncertainties. \nThis procedure is repeated 1000 times, and\nthe width of the resulting efficiency is taken as the\nsystematic uncertainty. This procedure, performed separately for the TIS and TOS\nsamples, results in a 0.13\\% uncertainty for both samples. Binning effects on \nthe efficiency are estimated by halving the bin size of the momentum distributions, resulting\nin a relative change of 0.2\\% for the TIS sample and 0.1\\% for the TOS sample.\nThe total uncertainty on the relative efficiency for the TIS and TOS samples is   \n0.24\\% and 0.16\\%, respectively, \ncorresponding to an uncertainty of 0.3\\% on the ratio of the branching fractions. \n\nTracking efficiencies are determined with simulated\nevents weighted to match the kinematic properties of dedicated calibration samples.\nThe weights are determined  \nas a function of the kinematic variables,  \nseparately for each data-taking year~\\cite{LHCb-DP-2013-002}. \nThe kinematic properties of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace\ndecay products are similar for\nthe signal and the normalisation samples and therefore provide minor    \ncontributions to the total tracking efficiency ratio. The dominant contribution to the systematic uncertainty  \ncomes from the knowledge of the {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace tracking efficiencies, \nwhose systematic uncertainties are fully correlated. \nThe efficiency correction procedure gives a change in efficiency\nof 2.0\\% for the TIS sample and 1.4\\% for the TOS sample, yielding a total\ncorrection factor of 0.985 for the ratio of branching fractions, and \na systematic uncertainty of 0.4\\% for each of the {\\ensuremath{\\Pp}}\\xspace and {\\ensuremath{\\overline \\proton}}\\xspace \n mainly stemming from the finite size of the calibration sample~\\cite{LHCb-DP-2013-002}. \n\nDue to distinct trigger requirements, the kinematic acceptance of the\ncalibration samples differs slightly from the signal and the normalisation channels.\nA nonnegligible  fraction of candidates have final-state  \nparticles in a kinematic range  outside of the regions covered \nby the calibration samples.  \nAbout 20\\% of the candidates from both channels fall in this category \ndue to the low-momentum pion from the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decay. In addition, 10\\% \nof the candidates from the signal channel are also affected, mainly due to the pion originating from the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay. \nFor all of these outside-range candidates, the efficiency correction in the nearest available \nbin is used. As the effects for {\\ensuremath{\\Lz^+_\\cquark}}\\xspace decays cancel in the relative efficiency, only the additional \n10\\% candidates in the signal channel contribute a 0.7\\% uncertainty on the relative efficiency.   \n\nHadronic interactions with the \\mbox{LHCb}\\xspace\ndetector contribute an additional uncertainty of 2.2\\% on the ratio \nof the branching fractions for each {\\ensuremath{\\Pp}}\\xspace or\n{\\ensuremath{\\overline \\proton}}\\xspace (4.4\\% in total), which is obtained from simulation, accounting for the imperfect knowledge of  \nmaterial budget of the \\mbox{LHCb}\\xspace detector\\cite{LHCbVELOGroup:2014uea}.\n\nPer-candidate weights depending on \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon are applied in \nsimulated events to improve the agreements between data and simulation.  \nSystematic uncertainties for the weighting due to the finite\nsize of the normalisation sample are assessed with pseudoexperiments.\nIn each pseudoexperiment, the weights are varied within their uncertainties, \nand the results are propagated  to the ratio of branching fractions. The\nstandard deviation of the obtained distributions is taken as a systematic uncertainty,\nresulting in 0.65\\% for the TIS sample and 0.65\\% for the TOS sample. \nThe systematic uncertainties due to the binning scheme of the weighting in \\mbox{$p_{\\mathrm{ T}}$}\\xspace and $y$ are   \nestimated by halving the bin size, or using the gradient boosting~\\cite{friedman2001greedy}\\cite{Belov:2010xm}, which is an unbinned method of weighting, \nto check the changes on the relative efficiencies.  \nThe resulting systematic uncertainties\nare 0.43\\% for the TIS sample and 1.5\\% for the TOS sample.  \nAfter propagation through the entire fit procedure, this results\nin an uncertainty of 1.0\\% on the ratio of the branching fractions.\n \nTrigger efficiencies for the TOS samples are also assessed using pseudoexperiments which are propagated\nto the final measurement, resulting in a final uncertainty of 0.1\\%.\nThe trigger efficiency of the TIS sample is taken from simulation. \nIts systematic uncertainty is computed from the difference between \nthe \nTIS efficiency taken from data and simulation for events which are \ntriggered both on the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace candidate  and also on other tracks unassociated to the signal decay.  \nAs a result, a systematic uncertainty of 3.9\\% is assigned for the relative trigger efficiency of the TIS sample, \ncorresponding to an uncertainty of 2.9\\% on the ratio of the branching fractions.\n\n\n\n\nThe effect of the finite size of the simulated samples is assessed by considering\nthe possible variation of the efficiency with weighted samples in a bin of \\mbox{$p_{\\mathrm{ T}}$}\\xspace and\nrapidity of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate, and the corresponding systematic uncertainty  \non the efficiency of the signal or normalisation channel, TIS or TOS sample, is given by\n\\begin{equation}\n\\sigma_\\epsilon = \\sqrt{\\sum_{i}\\epsilon_{i}(1-\\epsilon_{i})N_{i}w_{i}}\/\\sum_{i}N_{i}w_{i},\n\\end{equation}\nwhere\nfor each bin $i$, $N_{i}$ is the number of candidates, $w_{i}$ is the single event weight,\nand $\\epsilon_{i}$ is the single event efficiency. \nThe total uncertainty on the relative efficiency for the TIS and TOS samples is  \n1.2\\% and 1.9\\%, respectively,\ncorresponding to an uncertainty of 1.3\\% on the ratio of the branching fractions.\n\n\n\nThe uncertainty due to the removal of candidates reconstructed \nwith clone tracks and multiple candidates is assessed by applying\nthe same procedure to simulation, resulting in a difference of 0.2\\%.\n\nVetoes on the invariant mass of possible cross-feed backgrounds may bias the signal mass distributions.\nAn uncertainty of 0.4\\% is determined \nby changing the fit range of the normalisation\nsample to begin at 5450\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace, instead of 5350\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace.  \n\nThe agreement between data and simulation in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}$ decay\nis also tested by comparing the Dalitz plot distributions. The normalisation sample  is\nweighted in the $m^2({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace)$ versus $m^2({\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)$ plane. Due to the smaller\nsample size of the signal channel, weights obtained from the normalisation channel are\napplied to the signal. The resulting procedure renders all distributions consistent\nwithin one statistical standard deviation. The difference in the ratio of  branching\nfractions is 1.3\\% smaller than the nominal result, providing a correction\nfactor of 0.984. \nAn uncertainty of 0.2\\% is determined by using an alternative binning scheme \n and varying the Dalitz-plot weights by their statistical uncertainties. \n\nThe polarization of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace particles has been measured to be consistent with zero\\cite{LHCb-PAPER-2012-057}, but   \nthe weak decay of the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon may induce a polarization in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace system.  \nIn the simulation, it is assumed that the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace particle is unpolarized, leading  \nto a difference in angular distributions between simulation and data. A possible effect \ndue to the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace polarization is assessed by applying\na weighting procedure to the distribution of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace helicity angle,\nwhich is defined as the angle between the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace flight direction in the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace rest frame and the\ndirection of the ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ pair in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace rest frame. \nThis weight is obtained through a comparison between the angular distributions in simulation \nand data for the signal and the normalisation channels individually. \nApplying this weight\nto both the signal and the normalisation channels does not change the efficiency with\nrespect to any of the other possible angles, and leads to a change of 1.1\\% in the\nrelative efficiency for the TOS sample and 1.4\\% for the TIS sample. Propagation of these\nuncertainties leads to \na correction factor of 0.987 on the ratio of the branching fractions. \nAn uncertainty of 0.3\\% is determined by using an alternative binning scheme\n and varying the single-candidate weights by their statistical uncertainties.\n\nSimulated data are generated using a phase-space model for the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace decay,\nwhich does not take into\naccount possible resonances in the ${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ system. Upon\ninspection, clear signals from the $\\PSigma_c^0$ and $\\PSigma_c^{*0}$\nresonances are found, as described in Section~\\ref{sec:sigc}.  \nTo assess the\neffect of these resonances, the simulation is weighted to reproduce the data.\nWeights are applied in two invariant mass dimensions, namely the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass\nand another invariant mass of any two or three body combination. \nAmong these weighting strategies, applying weights in $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ and $m({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ (option 1) leads \nto the smallest $\\mathcal{B}_r$, while weights in $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ and $m({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ (option 2) leads to the \nlargest $\\mathcal{B}_r$. \nA correction factor is computed as the average of the central values of the ratio of  branching fractions \nfor the two options divided by the nominal branching fraction, with an \nuncertainty determined by half the difference between the two ratios of branching fractions.\nThis leads to a correction factor of 1.041 and a resulting systematic uncertainty of\n1.8\\%.\n\n\nUncertainties due to the use of the BDTG are tested by repeating the BDTG  training  \nand selection procedure to the normalisation channel \nwithout variables related to the {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace pair; \nthe ratio of branching fractions is found to be consistent.\n\n\n\\section{\\boldmath Resonance structures in the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass spectrum}\n\\label{sec:sigc}\n\nAs the resonant structure of ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays is  \nunexplored, the \nresonances in the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ system are studied.\nAn unbinned maximum-likelihood fit of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass is performed \nfor those candidates which pass all the selection criteria for the signal\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, to determine if there are \nresonant contributions. In this case the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace candidate is constrained to its known\nmass~\\cite{PDG2016} when obtaining the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass spectrum.  \n\n\nThe signal shapes of the $\\PSigma_c^0$ \nand $\\PSigma_c^{*0}$ resonances are given as the modulus squared of the relativistic \nBreit-Wigner function\\cite{PDG2016}, \n\\begin{equation}\n\\left|{\\rm BW}(m|M_{0},\\Gamma_{0})\\right|^2 = \\left|1\/(M_{0}^2-m^2 -iM_{0}\\Gamma(m))\\right|^2, \n\\end{equation}\nmultiplied by $m \\Gamma(m)$, and convolved with a Gaussian resolution determined from simulation.\nHere, $M_0$ is the known value of the $\\PSigma_c^0$ or $\\PSigma_c^{*0}$ mass~\\cite{PDG2016}, \n$m$ is the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant mass, \nand $\\Gamma_0$ is the mass-independent width of the resonance, namely 1.83\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for the \n$\\PSigma_c^0$ and 15.3\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace for the $\\PSigma_c^{*0}$ resonance. The mass-dependent width\nis given by \n\\begin{equation} \n\\Gamma(m) = \\Gamma_0 \\times \\left(\\frac{q}{q_0}\\right)^{2L+1}\\frac{M_0}{m}B_L(q,q_0,d)^2, \n\\end{equation}\nwhere $L$ is the angular momentum in the resonance decay, $q$ is the momentum of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace baryon\nin the $\\PSigma_c^{(*)0}$ rest frame, ${q_0 \\equiv q(m=M_0)}$\nand $d$ stands for the size of the $\\PSigma_c^{(*)0}$ particles. From parity\nand angular momentum conservation, it follows that $L=1$. The width also\ndepends on the Blatt-Weisskopf factor $B_L(q,q_0,d)$\\cite{VonHippel:1972fg}, where the value of $d$\nis set to be 1\\ensuremath{\\mathrm{ \\,fm}}\\xspace (5 GeV$^{-1}$ in natural units).  \nThe ratio of widths of the Gaussian resolution functions for the $\\PSigma_c^0$ and\n$\\PSigma_c^{*0}$ resonances is fixed from simulation to be 1.96.\nThe background is described with an empirical\nthreshold function. The fit shown in Figure~\\ref{fig:sigcres}\nyields  $59\\pm10$ ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^0{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$ decays and $104\\pm17$\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^{*0}{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$ decays.\n\n\\begin{figure}[!tbp]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig2.pdf}\n\\caption{Invariant mass of the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace system from the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.\nThe $\\PSigma_c^0$ and $\\PSigma_c^{*0}$ resonances are indicated. The fit to the\ndata is shown as a blue continuous line, with the background component shown as a green dotted line, \nthe $\\PSigma_c^{0}$ shape shown as a dashed red line, and \n the $\\PSigma_c^{*0}$ shape shown as a dash-dotted magenta line.}\n\\label{fig:sigcres}\n\\end{center}\n\\end{figure}\n\nThe relative efficiencies for the decays  ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^0{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$, with ${\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ \nand ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\PSigma_c^{*0}{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$, with ${\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$  \nwith respect to ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays are determined\nwith an analogous  procedure as that for the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays relative\nto the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ decays, but with the trigger samples combined due to\nlimited sample size. The efficiencies are $0.685\\pm0.021$\nfor the $\\PSigma_c^0$ mode and $0.904\\pm0.021$ for the $\\PSigma_c^{*0}$ mode, relative to\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$.\n\nMany of the systematic uncertainties cancel out in the measurement\nof the ratio of  branching fractions, with the remaining systematic\nuncertainties stemming from the yield determination. The value of $d$  \n in the Blatt-Weisskopf factor is varied between 1.5 and 0.5\\ensuremath{\\mathrm{ \\,fm}}\\xspace, with\nthe largest variation for each resonance taken as the systematic uncertainty,\nresulting in 3.4\\% for the $\\PSigma_c^0$ resonance and 1.9\\% for the $\\PSigma_c^{*0}$ resonance. The\nbackground shape is changed to a third-order polynomial, with a relative difference\nof 1.7\\% for the $\\PSigma_c^0$ resonance and 10.6\\% for the $\\PSigma_c^{*0}$ resonance taken as the systematic\nuncertainty. The masses and widths of the $\\PSigma_c^{(*)0}$ resonances are allowed\nto float within one standard deviation of their known values~\\cite{PDG2016}, resulting in a 3.8\\%\ndifference of the raw yield for the $\\PSigma_c^0$ resonance and 2.2\\% difference for the\n$\\PSigma_c^{*0}$ resonance. All uncertainties in the relative efficiency cancel, except for\nthose related to the weighting due to resonant structures in the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace system. The scaling factor of\n1.041, with an uncertainty of 1.8\\% on the relative efficiency, which is shown in Table~\\ref{tab:systemerr}, \nis therefore used here as well. The resulting ratios of branching fractions are\n\n\\begin{align*}\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^0 p\\overline{p})\\times\\mathcal{B}(\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)} = 0.089\\pm0.015\\pm0.006,\\\\\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^{*0} p\\overline{p})\\times\\mathcal{B}(\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace \\pi^-)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  p \\overline{p}\\pi^-)} = 0.119\\pm0.020\\pm0.014,\n\\end{align*}\nwhere the first uncertainty is statistical and the second is systematic.\n\n\n\\section{Search for dibaryon resonances}\nThe existence of dibaryon resonances, \n${\\mathscr{D}_c^+\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace\\PSigma_c^0}$, is investigated in the \n${{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\Pp}}\\xspace}$ mass spectrum of background-subtracted data.  \nThe full $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ spectrum is considered, while the signal regions of \n$\\PSigma_c^0$ and $\\PSigma_c^{*0}$ resonances are defined  by the ranges \n${ 2450 < m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) <  2458\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace }$ and  ${2488 < m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) <  2549\\ensuremath{{\\mathrm{\\,Me\\kern -0.1em V\\!\/}c^2}}\\xspace}$, respectively.   \nThe background is subtracted with the \\mbox{\\em sPlot}\\xspace technique\\cite{Pivk:2004ty}.\nNo peaking structures are observed in the \ndistributions shown in Figure~\\ref{fig:dibaryon}. \nThe two-dimensional distribution of $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ versus $m({\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$ \nhas been checked and does not exhibit  \nany clear structure. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.32\\textwidth]{Fig3a.pdf}\n\\includegraphics[width=0.32\\textwidth]{Fig3b.pdf}\n\\includegraphics[width=0.32\\textwidth]{Fig3c.pdf}\n\\caption{Background-subtracted mass spectrum of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\Pp}}\\xspace$ system from the decay\n${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ in (a) the full ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass spectrum,\n(b) the  signal region of the $\\PSigma_c^0$ resonance, and (c) \n  the signal region of the $\\PSigma_c^{*0}$ resonance. In all figures, the\n  black points are data and the red points  \n  are simulated events where the {\\ensuremath{\\Lz^0_\\bquark}}\\xspace baryon decays to the {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace final state \n  (a) based on a uniform-phase-space model, (b) through the $\\PSigma_c^0$ resonance and (c) \n  through the $\\PSigma_c^{*0}$ resonance.  \n  No evident peaking shapes are visible.}\n\\label{fig:dibaryon}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion}\nThe first observation of the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ is presented. The\nratio of the branching fractions using the decay ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace}$ \nas the  normalisation  channel  \nis measured to be\n\\begin{equation*}\n    \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)} = 0.0540 \\pm 0.0023 \\pm 0.0032,\n\\end{equation*}\nusing data corresponding to an integrated luminosity of 3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace collected during 2011 and 2012 with the \\mbox{LHCb}\\xspace detector.\nContributions from the $\\PSigma_c(2455)^0$ and $\\PSigma_c^{*}(2520)^0$ resonances are observed,\nand the ratios of their branching fractions with respect to the ${{\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace  {\\ensuremath{\\pion^-}}\\xspace}$ decays  are measured to be\n\\begin{align*}\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^0 {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace )\\times\\mathcal{B}(\\PSigma_c^0\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace  {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)} = 0.089\\pm0.015\\pm0.006,\\\\\n  \\frac{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PSigma_c^{*0} {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace )\\times\\mathcal{B}(\\PSigma_c^{*0}\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{\\mathcal{B}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\overline \\proton}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)} = 0.119\\pm0.020\\pm0.014.\n\\end{align*}\nIn all of the above results, the first uncertainty is statistical and the second is systematic.\n\nThe mass spectra of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ final state are also inspected for possible dibaryon\nresonances, but no evidence of peaking structures is observed.\n\n\n\\section*{Acknowledgements}\n\\noindent We express our gratitude to our colleagues in the CERN\naccelerator departments for the excellent performance of the LHC. We\nthank the technical and administrative staff at the LHCb\ninstitutes. We acknowledge support from CERN and from the national\nagencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC\n(China); CNRS\/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN\n(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN\/IFA\n(Romania); MinES and FASO (Russia); MinECo (Spain); SNSF and SER\n(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA).  We\nacknowledge the computing resources that are provided by CERN, IN2P3\n(France), KIT and DESY (Germany), INFN (Italy), SURF (The\nNetherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex\nLLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil),\nPL-GRID (Poland) and OSC (USA). We are indebted to the communities\nbehind the multiple open-source software packages on which we depend.\nIndividual groups or members have received support from AvH Foundation\n(Germany), EPLANET, Marie Sk\\l{}odowska-Curie Actions and ERC\n(European Union), ANR, Labex P2IO and OCEVU, and R\\'{e}gion\nAuvergne-Rh\\^{o}ne-Alpes (France), Key Research Program of Frontier\nSciences of CAS, CAS PIFI, and the Thousand Talents Program (China),\nRFBR, RSF and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain),\nHerchel Smith Fund, the Royal Society, the English-Speaking Union and\nthe Leverhulme Trust (United Kingdom).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe growing importance of simulation tools in analysis and design has directed\nthe interest of the research community towards building accurate,  yet \ninexpensive,  mathematical models.\nThe need of  accuracy  leads  invariably to the incorporation of a large number of state variables. This, in turn, leads to large scale models which in general may be difficult to simulate owing to time or storage constraints. Model reduction methods alleviate this issue by constructing simplified models which retain prescribed features of the original system~\\citep{antoulas2005approximation}.\n\nModel reduction methods for linear systems belong broadly to two main classes~\\citep{antoulas2005approximation}. The first class makes use of the  singular value decomposition~\\citep{moore1981principal,glover1984all},  while the second class is based on the concept of moment matching or on Krylov projectors~\\citep{georgiou1983partial,antoulas1990solution,georgiou1999interpolation}. While moment matching methods are generally more efficient and reliable from a numerical point of view, methods based on the singular value decomposition not only preserve important features of the original system, but also offer error bounds~\\citep{antoulas2005approximation}.   \n\nOver the past decades, several nonlinear counterparts of linear model reduction methods have been developed~\\citep{berkooz1993proper,scherpen1993balancing,hahn2002improved,astolfi2010model}. \nThis line of research has mostly focused on the construction of reduced order models for \\textit{local} behaviors around equilibria. However, the problem of approximating the \\textit{global} behavior of a nonlinear system is still open. Besselink and co-authors have addressed this problem in~\\citep{besselink2009error,besselink2013model} and shown that (incremental) stability properties can be preserved for systems that can be decomposed as the feedback interconnection of a linear system and a nonlinear system. The  main idea is to reduce the linear dynamics using standard model reduction methods and to impose (incremental) small gain conditions to guarantee existence, uniqueness, and incremental stability   of a  steady-state equilibrium solution.\n\n\nThe present work extends the approach of~\\citep{besselink2009error} to multistable and oscillatory Lure systems \nusing dominance theory~\\citep{forni2018differential,felix2018analysis,padoan2019feedback,padoan2019norm}.\nThe goal is to develop a model reduction theory for non-equilibrium behaviors.\n The proposed framework   is based on two key ingredients: \n(i) the dynamics is split into dominant and non-dominant components, and standard model reduction methods are used to reduce the non-dominant one; \n(ii) the asymptotic behavior of the closed-loop system is characterized using small gain conditions from dominance theory. These conditions guarantee that the behavior of the original system is well captured by that of the reduced order model when the approximation error is small. \nThe framework is illustrated by developing a balanced truncation method for dominant Lure systems inspired by the method in~\\citep{besselink2009error}.  \n\n\nThe remainder of the paper is organized as follows. \nSection~\\ref{sec:problem-formulation} provides the problem formulation.\nSection~\\ref{sec:preliminaries} recalls  some   preliminary results  from   dominance theory. \nSection~\\ref{sec:main-results} contains the main results of the paper, where  a general model reduction framework for dominant Lure systems is presented. \nSection~\\ref{sec:mr}  discusses a balanced truncation method for dominant Lure systems inspired by the method presented in~\\citep{besselink2009error}.   \nSection~\\ref{sec:example} provides an illustrative example,   in which\nthe oscillatory behavior of a discretized heat flow control system\nneeds to be approximated.   \nSection~\\ref {sec:conclusion} summarizes our results  and outlines future research directions.   \n\n\n\n\\section{Problem formulation} \\label{sec:problem-formulation}\n\nConsider a continuous-time, single-input, single-output, time-invariant  \\textit{Lure system} described by the equations \n\\begin{equation} \\label{eq:system-Lure-mr}\n\\dot{x} = Ax +B_u u +B_w w, \\, y = C_y x,  \\, z = C_z x,  \\, w = - \\varphi(z),  \\! \\!\n\\end{equation}\nin which ${x\\in\\R^n}$, ${u\\in\\R}$, ${y\\in\\R}$,  ${w\\in\\R}$, ${z\\in\\R}$, ${A\\in\\R^{n \\times n}}$, ${B_u\\in\\R^{n \\times 1}}$, ${B_w\\in\\R^{n \\times 1}}$, ${C_y\\in\\R^{1\\times n}}$, and ${C_z\\in\\R^{1\\times n}}$ are constant matrices, and ${\\varphi:\\R \\to \\R}$ is a continuously differentiable function\\footnote{Local Lipschitz continuity would be sufficient. Continuous differentiability is assumed to simplify the exposition.} such that ${\\varphi(0) =0}$. Let \n\\begin{equation} \\nn\nG(s) \n=\nC (sI-A)^{-1} B\n= \n\\bma\n\\begin{array}{cc}\nG_{yu}(s) & G_{yw}(s)\\\\\nG_{zu}(s) & G_{zw}(s)\n\\end{array}\n\\ema\n,\n\\end{equation}\nin which ${B = [\\, B_{u} \\ B_{w} \\,]}$ and  ${C = [\\, C_{y}^{\\transpose} \\ C_{z}^{\\transpose} \\,]^{\\transpose}}$, respectively.\n\nSuppose we wish to construct a reduced order model   of   order ${\\nu < n}$ of system~\\eqref{eq:system-Lure-mr} described by the equations\n\\begin{equation} \\label{eq:system-ROM}\n\\dot{\\hat{x}} = \\hat{A}\\hat{x} +\\hat{B}_u  \\hat{u}   +\\hat{B}_w \\hat{w}, \\, \\hat{y} = \\hat{C}_y \\hat{x},  \\, \\hat{z} = \\hat{C}_z \\hat{x},  \\, \\hat{w} = - \\varphi(\\hat{z}),  \\! \\!\n\\end{equation}\nwith ${\\hat{x}\\in\\R^\\nu}$, ${ \\hat{u} \\in\\R}$, ${\\hat{y}\\in\\R}$,  ${\\hat{w}\\in\\R}$, ${\\hat{z}\\in\\R}$, ${\\hat{A}\\in\\R^{\\nu \\times \\nu}}$, ${\\hat{B}_u\\in\\R^{\\nu \\times 1}}$, ${\\hat{B}_w\\in\\R^{\\nu \\times 1}}$, ${\\hat{C}_y\\in\\R^{1\\times \\nu}}$, and ${\\hat{C}_z\\in\\R^{1\\times \\nu}}$. Let \n\\begin{equation} \\nn\n\\hat{G}(s) \n=\n\\hat{C} (sI-\\hat{A})^{-1} \\hat{B}\n= \n\\bma\n\\begin{array}{cc}\n\\hat{G}_{yu}(s) & \\hat{G}_{yw}(s)\\\\\n\\hat{G}_{zu}(s) & \\hat{G}_{zw}(s)\n\\end{array}\n\\ema\n,\n\\end{equation}\nin which ${\\hat{B} = [\\, \\hat{B}_{u} \\ \\hat{B}_{w} \\,]}$ and  ${\\hat{C} = [\\, \\hat{C}_{y}^{\\transpose} \\ \\hat{C}_{z}^{\\transpose} \\,]^{\\transpose}}$, respectively.  \nThe transfer function of the error system is defined as\n\\begin{equation} \\nn\n\\tilde{G}(s) = G(s) - \\hat{G}(s). \n\\end{equation}\nThe reduced order model~\\eqref{eq:system-ROM} is constructed by replacing the linear dynamics of the original system\n\\begin{equation} \\label{eq:system-Lure-mr-linear}\n\\dot{x} = Ax +B_u u +B_w w, \\ y = C_y x,  \\ z = C_z x, \n\\end{equation}\nwith a linear system described by the equations\n\\begin{equation} \\label{eq:system-ROM-linear}\n\\dot{\\hat{x}} = \\hat{A}\\hat{x} +\\hat{B}_u  \\hat{u}   +\\hat{B}_w \\hat{w}, \\ \\hat{y} = \\hat{C}_y \\hat{x},  \\ \\hat{z} = \\hat{C}_z \\hat{x}, \n\\end{equation}\nand by leaving the static nonlinearity ${\\varphi}$ unchanged, as illustrated in Fig.~\\ref{fig:model_reduction}. \n\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=.75, black,every node\/.style={transform shape}]\n \n\\draw[line width = .5 pt]  (10.3125,3.5) rectangle (11.3125,2.5);\n\\node at (10.8125,3) {$G$};\n\\draw[rounded corners,line width = .5 pt]  (10.3125,2.3125) rectangle (11.3125,1.3125);\n\\node at (10.8125,1.8125) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (11.3125,2.75) -- (12.3125,2.75) -- (12.3125,1.8125) -- (11.3125,1.8125);\n\\draw[-latex, line width = .5 pt] (10.3125,1.8125) -- (9.3125,1.8125) -- (9.3125,2.75) -- (10.3125,2.75) ;\n\\draw [-latex, line width = .5 pt](9.3125,3.25) -- (10.3125,3.25);\n\\node at (9.8125,3.5) {$u$};\n\\node at (11.8125,3.5) {$y$};\n\\node at (9.8125,2.5) {$w$};\n\\node at (11.8125,2.5) {$z$}; \n\\draw [-latex,line width = .5 pt](11.3125,3.25) -- (12.3125,3.25);\n\n  \n\\draw[line width = .5 pt]  (16.6875,3.5) rectangle (17.6875,2.5);\n\\node at (17.1875,3) {$\\hat{G}$};\n\\draw[rounded corners,line width = .5 pt]  (16.6875,2.3125) rectangle (17.6875,1.3125);\n\\node at (17.1875,1.8125) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (17.6875,2.75) -- (18.6875,2.75) -- (18.6875,1.8125) -- (17.6875,1.8125);\n\\draw[-latex, line width = .5 pt] (16.6875,1.8125) -- (15.6875,1.8125) -- (15.6875,2.75) -- (16.6875,2.75) ;\n\\draw [-latex, line width = .5 pt](15.6875,3.25) -- (16.6875,3.25);\n\\node at (16.1875,3.5) {$\\hat{u}$};\n\\node at (18.1875,3.5) {$\\hat{y}$};\n\\node at (16.1875,2.5) {$\\hat{w}$};\n\\node at (18.1875,2.5) {$\\hat{z}$}; \n\\draw [-latex,line width = .5 pt](17.6875,3.25) -- (18.6875,3.25);\n\n\\draw [thick, -latex](12.8125,2.5) -- (15.1875,2.5);\n\\node at (14,3) { Model reduction};\n \n\\end{tikzpicture} \n\\centering\n\\caption{Diagrammatic illustration of the original system~\\eqref{eq:system-Lure-mr} (left) and of the reduced order model~\\eqref{eq:system-ROM} (right).\n}\n\\label{fig:model_reduction}\n\\end{figure\n\n\nThis work addresses basic, yet open, questions about the \\textit{global} behavior of the reduced order model~\\eqref{eq:system-ROM}. For example, \nsuppose that the original system~\\eqref{eq:system-Lure-mr} is multistable or oscillatory. Further, suppose an error bound is available for the approximation of the linear dynamics. One can expect that~\\eqref{eq:system-ROM} is indeed a good reduced order model if the error is small, but how does one \\textit{quantify} this? Can one construct  better  reduced order models in a systematic and computationally appealing fashion? Is it sensible to use the reduced order model for analysis and design?\n\nThe paper~\\citep{besselink2009error} has shown that these questions can be addressed for (incrementally) \\textit{stable} Lure systems using sector conditions and small gain conditions.\nThe next sections show that similar conclusions can be drawn for \\textit{dominant} Lure systems by combining existing linear model reductions methods with dominance theory.\n\n\n\n\n\\section{Dominance theory} \\label{sec:preliminaries}\n\n\nConsider a continuous-time, nonlinear, time-invariant system and its linearization described by the equations\n\\begin{align}\n\\,\\Sigma : \\quad \\ \\dot{x} &= f(x) +Bu, \\quad \\quad \\quad \\ \\ \\ y = \\ Cx , \\label{eq:system-nonlinear} \\\\\n\\delta\\Sigma :\\quad \\dot{\\delta x} &= \\partial f(x) \\delta x + B \\delta u, \\quad ~\\, \\delta y =~ C \\delta x,   \\label{eq:system-nonlinear-linearization}\n\\end{align}\nin which ${x\\in\\R^n}$, ${u\\in\\R^m}$,  ${y\\in\\R^l}$,  \n${f:\\R^{n} \\to \\R^{n}}$ is a continuously differentiable vector field,   ${B\\in\\R^{n\\times m}}$,   and ${C\\in\\R^{m\\times n}}$ are constant matrices, ${\\delta x\\in\\R^n}$, ${\\delta u\\in\\R^m}$,   ${\\delta y\\in\\R^l}$   (identified with the respective tangent spaces), and ${\\partial f}$ is the Jacobian of the vector field $f$.\n\n\n\n\n\\begin{definition} \\label{def:dominance}\nThe system~\\eqref{eq:system-nonlinear} is $p$-dominant with rate ${\\lambda \\in \\Rge}$ (for ${u=0}$) if there exist ${\\varepsilon \\in \\Rge}$ and a symmetric matrix ${P\\in\\R^{n\\times n}}$, with inertia\\footnote{The inertia of the matrix ${A \\in \\R^{n \\times n}}$ is defined as $\\operatorname{In}(A)=(n_{-},n_{0},n_{+})$, where $n_{-}$ is the number of eigenvalues of $A$ in the open left half-plane, $n_{0}$ is the number of eigenvalues of $A$ on the imaginary axis, and $n_{+}$ is the number of eigenvalues of $A$ in the open right half-plane, respectively.  \n} $\\operatorname{In}(P) = ( p, 0,n-p)$, such that the prolonged system~\\eqref{eq:system-nonlinear}-\\eqref{eq:system-nonlinear-linearization}\nsatisfies\n\\begin{equation}  \\label{eq:dominance-LMI}\n\\bma\n\\begin{array}{c}\n \\dot{\\delta x}  \\\\\n\\delta x\n\\end{array}\n\\ema^{\\transpose}\n\\bma\n\\begin{array}{cc}\n0 & P \\\\\nP &  2\\lambda P + \\varepsilon I\n\\end{array}\n\\ema\n\\bma\n\\begin{array}{c}\n\\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema \\le 0 \n\\end{equation} \nfor every $(x,\\delta x) \\in \\R^n \\times \\R^n$. The property is strict if ${\\varepsilon >}0$.\n\\end{definition}\n\n\nThe property of $p$-dominance ensures the existence of a splitting between $p$ dominant modes and $n-p$ transient modes.  In particular, the matrix ${\\partial f}(x) +\\lambda I$ has $p$ unstable eigenvalues and $n-p$ stable eigenvalues for every ${x\\in\\R^n}$. For linear, time-invariant systems, such eigenvalues are referred to as \\textit{dominant and non-dominant}, respectively.\n\n\n\\begin{definition} \\label{def:pgain}\nThe system~\\eqref{eq:system-nonlinear} is said to have \\textit{finite (differential) ${\\L_{2,p}}$-gain (from $u$ to $y$) less than ${\\gamma \\in \\Rge}$ with rate ${\\lambda \\in \\Rge}$} if there exist ${\\varepsilon \\in \\Rge}$ and a symmetric matrix ${P\\in\\R^{n\\times n}}$, with inertia   $(p, 0, n-p)$,   such that the conic constraint\n\\begin{equation} \\label{eq:system-linear-MIMO-conic-IO}\n\\begingroup \n\\setlength\\arraycolsep{1.3pt}\n\\bma\n\\begin{array}{c}\n\\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema^{\\transpose} \\!\n\\bma\n\\begin{array}{cc}\n0 & P \\\\\nP &  2\\lambda P + \\varepsilon I\n\\end{array}\n\\ema \\!\n\\bma\n\\begin{array}{c}\n  \\dot{\\delta x} \\\\\n\\delta x\n\\end{array}\n\\ema \n\\le\n\\bma\n\\begin{array}{c}\n\\delta y \\\\\n\\delta u\n\\end{array}\n\\ema^{\\transpose} \\!\n\\bma\n\t\\begin{array}{cc}\n\t-I & 0 \\\\\n\t 0 & \\gamma^2 I\n\t\\end{array}\n\t\\ema \\!\n\\bma\n\\begin{array}{c}\n\\delta y \\\\\n\\delta u\n\\end{array}\n\\ema  \\! \\! \n\\endgroup\n\\end{equation}\nholds along the solutions of the prolonged system~\\eqref{eq:system-nonlinear}-\\eqref{eq:system-nonlinear-linearization}.\nThe \\textit{(differential) $\\L_{2,p}$-gain of system~\\eqref{eq:system-nonlinear}  (from $u$ to $y$) with rate $\\lambda$} is defined as ${\\gamma_{\\lambda} = \\inf \\left\\{ \\gamma \\in \\Rge  :  \\text{\\eqref{eq:system-linear-MIMO-conic-IO} holds} \\right\\}}.$ The properties are strict if $\\varepsilon >0$. \n\\end{definition}\n\nThe asymptotic behavior of a $p$-dominant system is $p$-dimensional and, hence, its attractors\nare severely constrained for small values of $p$~\\citep{forni2018differential}.\n\n\\begin{theorem}  \\label{thm:asymptotic}\nAssume system~\\eqref{eq:system-nonlinear} is strictly $p$-dominant with rate ${\\lambda \\in \\Rge}$. Then every bounded solution of~\\eqref{eq:system-nonlinear} converges asymptotically to\n\\begin{description}\n\\item[{\\small $\\bullet$}] the unique equilibrium  point if ${p=0}$; \n\\item[{\\small $\\bullet$}] a (possibly non-unique) equilibrium  point if ${p=1}$;\n\\item[{\\small $\\bullet$}] a simple attractor if ${p=2}$, \\textit{i.e.} an equilibrium  point, a set of equilibrium  points and their connected arcs or a limit cycle.\n\\end{description} \n\\end{theorem} \n\nThe dominance properties of an interconnected system can be studied\nusing small ${\\L_{2,p}}$-gain conditions~\\citep{forni2018differential,padoan2019norm}.\n\n\n\\begin{theorem}[Small-gain theorem for $p$-dominance] \\label{thm:small-gain}\nLet $\\Sigma_i$ be a system with input ${u_i\\in\\R^{m_i}}$, output ${y_i\\in\\R^{l_i}}$, and (strict) $\\L_{2,p_i}$-gain less than ${\\gamma_i\\in\\Rge}$ with rate ${\\lambda \\in \\Rge}$, with ${i \\in \\{1,2\\}}$. Then the closed-loop system defined by the feedback interconnection equations  ${u_1 = -y_2}$ and ${u_2 = y_1}$  is strictly ${(p_1+p_2)}$-dominant with rate $\\lambda$ if ${\\gamma_1 \\gamma_2 <1}$. \n\\end{theorem}\n\n\\subsection{Dominant systems in the frequency domain}\n\nConsider a continuous-time, linear, time-invariant system described by the equations \n\\begin{equation} \\label{eq:system-linear-MIMO}\n\\quad \\dot{x} = Ax+Bu, \\quad y=Cx,\n\\end{equation}\nin which ${x \\in\\R^n}$, ${u\\in\\R^m}$, ${y\\in\\R^l}$, and ${A\\in\\R^{n \\times n}}$, ${B\\in\\R^{n \\times m}}$, and ${C\\in\\R^{l\\times n}}$, are constant matrices, respectively. Let $G(s) = C(sI-A)^{-1} B$ be its transfer function and, given ${\\lambda \\in \\Rge}$, let ${G_\\lambda(s) = G (s-\\lambda)}$ be its shifted transfer function, provided $G$ is well-defined on the axis ${\\Re(s) = - \\lambda}$.\n\nThe ${\\L_{2,p}}$ gain is intimately connected to the \\textit{${\\H_{\\infty,p}}$ norm} of the transfer function of system~\\eqref{eq:system-linear-MIMO}, defined as\n\\begin{equation} \\label{eq:Hq-norm-lambda-strip} \n\\norm{G}_{\\infty,p} = \\displaystyle \\sup_{\\omega \\in  \\R } {\\sigma_{\\max}} (G_{\\lambda}(i\\omega)) , \n\\end{equation} \nwhere ${\\sigma_{\\max}}(M)$ is the largest singular value of the matrix ${M\\in\\Co^{l\\times m}}$ and ${G}$ has $p$ poles in the open half-plane ${\\Re(s)>-\\lambda}$, respectively. The ${\\H_{\\infty,p}}$ norm is not uniquely defined by the integer $p$, as it depends on the parameter $\\lambda$.\n\n\n\\begin{theorem} \\citep{padoan2019norm} \\label{thm:pgain}\nAssume system~\\eqref{eq:system-linear-MIMO} has (strict) $\\L_{2,p}$-gain  ${\\gamma_{\\lambda} }$ with rate ${\\lambda \\in \\Rge}$. Then ${\\gamma_{\\lambda}  =  \\norm{G}_{\\infty,p}.}$\n\\end{theorem}\n\n\n \n\n\n\nThe property of $p$-dominance can be also characterized \\textit{graphically} in the frequency domain~\\citep{felix2018analysis}.  Consider a single-input, single-output, Lure system described by the equations \n\\begin{equation} \\label{eq:system-Lure}\n\\dot{x} = Ax +Bu, \\quad y = Cx,  \\quad u =  -   \\varphi(y),\n\\end{equation}\nin which ${x\\in\\R^n}$, ${u \\in\\R}$, ${y \\in\\R}$, ${A\\in\\R^{n \\times n}}$, ${B\\in\\R^{n \\times 1}}$, and ${C\\in\\R^{1\\times n}}$  are constant matrices, and ${\\varphi:\\R \\to \\R}$ is a continuously differentiable function which satisfies \nthe \\textit{differential sector condition} ${\\partial \\varphi \\in  [\\alpha, \\beta]}$, defined as\n\\begin{equation} \\label{eq:sector-condition} \n(\\partial \\varphi (y) \\delta y - \\alpha \\delta y) (\\partial \\varphi (y) \\delta y - \\beta \\delta y) \\leq 0, \\  \\forall \\, y \\in \\R,\n\\end{equation}\nwith ${-\\infty \\le \\alpha < \\beta \\le \\infty}$. Let \n${G(s) = C(sI-A)^{-1}B}$.\n\n\\begin{theorem}[Circle criterion for $p$-dominance] \\label{thm:circle_criterion}\nConsider \\linebreak system~\\eqref{eq:system-Lure} and let $\\lambda \\in \\Rge$. Assume \n$\\partial \\varphi \\in [\\alpha, \\beta]$, $G_{\\lambda}$ has $q$ unstable poles and no poles along the imaginary axis,\nthe Nyquist diagram of $G_{\\lambda}$ encircles ${(p-q)}$ times the point $-\\tfrac{1}{\\alpha}$  in the clockwise direction, and one of the following  mutually exclusive   conditions hold\\footnote{$D(\\alpha, \\beta)$  denotes the closed disk in the complex plane associated with the sector $[\\alpha, \\beta]$, with ${\\alpha < \\beta}$. The notation is standard and is defined, \\textit{e.g.}, in~\\citep[p.82]{felix2018analysis}.}\n\\begin{itemize}\n\\item[(a)] ${\\alpha \\beta \\ge 0}$ and the Nyquist diagram of $G_{\\lambda}$ lies \\textit{outside} the disk $D(\\alpha, \\beta)$.\n\\item[(b)] ${\\alpha \\beta <0}$ and the Nyquist diagram of $G_{\\lambda}$ lies \\textit{inside} the disk $D(\\alpha, \\beta)$.\n\\end{itemize}\nThen system~\\eqref{eq:system-Lure} is strictly $p$-dominant with rate $\\lambda$.\n\\end{theorem}\n\n\n\\section{Main results} \\label{sec:main-results}\n\nThe first step of our model reduction approach for dominant Lure systems is to split the linear dynamics into dominant and non-dominant components. Standard model reduction methods are then applied to the non-dominant component, as illustrated in Fig.~\\ref{fig:model_reduction_detail_new}. The main steps are summarized in Algorithm~\\ref{alg:mr-Lure}.  This approach preserves by construction the dominance properties of the linear dynamics of the original system. Furthermore, taking advantage of small-gain conditions for $p$-dominance, it guarantees that the behavior of the original original system is well captured by that of the reduced order model, when the approximation error is small. This is formalized in Theorem~\\ref{thm:1}.\n \n\\begin{algorithm}[h!] \n\\algnewcommand\\algorithmicto{\\textbf{to}}\n\\algnewcommand\\algorithmicbreak{\\textbf{break}}\n\\algnewcommand\\algorithmicstop{\\textbf{stop}}\n\\algnewcommand\\algorithmicass{\\textbf{Assumption}}\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output}}\n\\caption{}\n \\label{alg:mr-Lure}  \n\\algorithmicrequire{: The system~\\eqref{eq:system-Lure-mr}}\\\\\n\\algorithmicensure{: The reduced order model~\\eqref{eq:system-ROM}}\\\\\n\\algorithmicass{: {Strict\\,$p$-dominance\\,with\\,rate\\,$\\lambda$\\,of\\,system\\,\\eqref{eq:system-Lure-mr}}} \n\\begin{algorithmic}[1] \n\\State Consider the linear dynamics~\\eqref{eq:system-Lure-mr-linear} and, upon a possible change of coordinates, let\n\\begin{equation}  \\nn\nA =\n\\bma\n\\begin{array}{cc}\nA^+ & 0 \\\\\n0 & A^-\n\\end{array}\n\\ema, \\\nB =\n\\bma\n\\begin{array}{c}\nB^+ \\\\\nB^-\n\\end{array}\n\\ema, \\\nC =\n\\bma\n\\begin{array}{cc}\nC^+  &\nC^- \n\\end{array}\n\\ema, \n\\end{equation}\nwhere ${\\spectrum{A^+}}$ and ${\\spectrum{A^-}}$ contain $p$ dominant eigenvalues and $n-p$ non-dominant eigenvalues,  respectively.  \n\\State Construct a reduced order model \n\\begin{equation} \\nn\n\\dot{\\hat{x}}^- = \\hat{A}^-\\hat{x}^- +\\hat{B}^-  \\hat{v}^-, \\quad \\hat{y}^- = \\hat{C}^- \\hat{x}^-, \n\\end{equation}\nof the non-dominant linear dynamics \n\\begin{equation} \\nn\n\\dot{x}^- = {A}^- {x}^- + {B}^-{v}^-, \\quad {y}^- = {C}^- {x}^-, \n\\end{equation}\nusing a stability-preserving model reduction method. \n\\State Define the reduced order model~\\eqref{eq:system-ROM} as\n\\begin{equation}  \\nn\n\\hat{A} =\n\\bma\n\\begin{array}{cc}\nA^+ & 0 \\\\\n0 & \\hat{A}^-\n\\end{array}\n\\ema, \\\nB =\n\\bma\n\\begin{array}{c}\nB^+ \\\\\n\\hat{B}^-\n\\end{array}\n\\ema, \\\nC =\n\\bma\n\\begin{array}{cc}\nC^+  &\n\\hat{C}^- \n\\end{array}\n\\ema .\n\\end{equation} \n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=.75, black,every node\/.style={transform shape}]\n      \n\\draw[line width = .5 pt]  (10.6875,-0.375) rectangle (11.6875,-1.375);\n\\node at (11.1875,-0.875) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (10.6875,-2.6964) rectangle (11.6875,-3.6964);\n\\node at (11.1875,-3.1964) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (12.1875,-2.0625)   -- (12.1875,-3.1964) -- (11.6875,-3.1964);\n\\draw[-latex, line width = .5 pt] (10.6875,-3.1964) -- (10.1875,-3.1964) -- (10.1875,-2) -- (10.6875,-2) ;\n\\draw [-latex, line width = .5 pt] (10.1875,-0.875) --(10.1875,0.4466) -- (10.6875,0.4466);\n\\draw [-latex,line width = .5 pt](11.6875,0.4466) -- (12.1875,0.4466)  -- (12.1875,-0.8194);\n\\draw[fill = white] (12.1875,-0.875) circle [radius=0.05];\n\\draw[semithick, dashed] (10,1.0716) rectangle (12.6875,-0.1159);\n\\draw[semithick, dashed] (9.9992,-0.3) rectangle (12.6883,-2.6076);\n\\draw  (10.6875,0.9466) rectangle (11.6875,-0.0534);\n\\node at (11.1875,0.4466) {$\\tilde{G}_{zw}^{-}$};\n\\draw[-latex, line width = .5 pt] (11.6875,-0.875) -- (12.1319,-0.875); \n\\draw[-latex, line width = .5 pt]  (10.6875,-1.5) rectangle (11.6875,-2.5);\n\\node at (11.1875,-2) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](10.1875,-2) -- (10.1875,-0.875) -- (10.6875,-0.875);\n\\draw[fill = white] (12.1875,-2) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](12.1875,-0.9375) -- (12.1875,-1.9375);\n\\draw [-latex, line width = .5 pt](11.6875,-2) -- (12.125,-2);\n\\node at (12.4375,0.8216) {$\\tilde{G}$};\n\\node at (12.4375,-2.2951) {$\\hat{G}$};\n\n \n\\draw [thick, -latex](12.875,-1.25) -- (15.25,-1.25);\n\\node at (14.0625,-0.75) { Model reduction};\n  \n\\draw[line width = .5 pt]  (16.1875,-0.375) rectangle (17.1875,-1.375);\n\\node at (16.6875,-0.875) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (16.1875,-2.6964) rectangle (17.1875,-3.6964);\n\\node at (16.6875,-3.1964) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (17.6875,-2.0625)   -- (17.6875,-3.1964) -- (17.1875,-3.1964);\n\\draw[-latex, line width = .5 pt] (16.1875,-3.1964) -- (15.6875,-3.1964) -- (15.6875,-2) -- (16.1875,-2) ;\n\n\n\\draw[semithick, dashed] (15.4992,-0.3) rectangle (18.1883,-2.6076);\n  \n\\draw[-latex, line width = .5 pt]  (16.1875,-1.5) rectangle (17.1875,-2.5);\n\\node at (16.6875,-2) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](15.6875,-2) -- (15.6875,-0.875) -- (16.1875,-0.875);\n\\draw[fill = white] (17.6875,-2) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](17.1875,-0.875) -- (17.6875,-0.8661) -- (17.6875,-1.9375);\n\\draw [-latex, line width = .5 pt](17.1875,-2) -- (17.625,-2);\n \n\\node at (17.9375,-2.2951) {$\\hat{G}$};\n\n\\end{tikzpicture} \n\\centering\n\\caption{Diagrammatic illustration of the model reduction method described in Algorithm~\\ref{alg:mr-Lure}.\n}\n\\label{fig:model_reduction_detail_new}\n\\end{figure\n\n\n\\begin{theorem} \\label{thm:1}\nConsider system~\\eqref{eq:system-Lure-mr} and a reduced order model~\\eqref{eq:system-ROM} constructed as  in Algorithm~\\ref{alg:mr-Lure}.  Assume\n\\begin{description}\n\\item[(A1)]  \\eqref{eq:system-Lure-mr-linear} is strictly   $p$-dominant with rate ${\\lambda\\in\\Rge}$,\n\\item[(A2)] ${\\partial \\varphi \\in  [-\\mu, \\mu]}$, with  ${\\mu\\in \\Rge}$,\n\\item[(A3)] ${\\norm{\\tilde{G}}_{\\infty,0} \\le \\epsilon}$, with ${0<\\epsilon<\\mu^{-1}}$,\n\\item[(A4)] ${\\norm{\\hat{G}_{zw}}_{\\infty,p} < \\mu^{-1} - \\epsilon}$.\n\\end{description}\nThen system~\\eqref{eq:system-Lure-mr} is strictly $p$-dominant with rate ${\\lambda}$.\n\\end{theorem}\n\n\n\n\n\n\n\\begin{pf}\nBy the circle criterion for $p$-dominance, system~\\eqref{eq:system-Lure-mr} is $p$-dominant with rate ${\\lambda}$ if \n\\begin{equation} \\label{eq:thm1-proof-1}\n\\norm{G_{zw}}_{\\infty,p} < \\mu^{-1},\n\\end{equation}\nsince, by assumptions (A1) and (A2), \\eqref{eq:system-Lure-mr} is $p$-dominant with rate ${\\lambda}$ (and, thus, the transfer function ${G(s-\\lambda)}$ has $p$ unstable poles and no poles on the imaginary axis) and ${\\partial \\varphi \\in  [-\\mu, \\mu]}$. However, \\eqref{eq:thm1-proof-1} follows directly from assumptions (A3) and (A4), as\n\\begin{align*}\n\\norm{G_{zw}}_{\\infty,p}\n\\stackrel{\\text{def}}{=} \\norm{\\hat{G}_{zw}  + \\tilde{G}_{zw}}_{\\infty,p} \n\\stackrel{(\\text{A3})}{\\le} \\norm{\\hat{G}_{zw}}_{\\infty,p} + \\epsilon  \n\\stackrel{(\\text{A4})}{<} \\mu^{-1}.\n\\end{align*}\n\\end{pf} \n\n \n\n\nTheorem~\\ref{thm:1} establishes that the dominance properties of the \noriginal closed-loop system can be inferred from those of the reduced order model if the approximation error between the original linear component $G$ and the reduced linear component $\\hat{G}$ is sufficiently small.  \nFrom Algorithm~\\ref{alg:mr-Lure}, by construction, Assumption (A1) guarantees that the reduced linear dynamics $\\hat{G}$ is $p$-dominant with rate $\\lambda$. \nAssumption (A2) and the circle criterion for $p$-dominance \nensure that the closed-loop reduced order model is also $p$-dominant \nwith the same rate  if ${\\norm{\\hat{G}_{zw}}_{\\infty,p} < \\mu^{-1} }$ (which, in turn, is implied by assumption (A4)). The approximation error bound of assumption (A3) expresses the fact that the  linear   dynamics of the reduced order model is ``close'' to that of the original system in the ${\\H_{\\infty,0}}$ norm.\nFinally, the small ${\\LTp}$ gain condition of assumption (A4) ensures that the dominance properties of the original system are captured by those of the the reduced order model when the approximation error is regarded as a perturbation.\n\n\n\\begin{remark}\nTheorem~\\ref{thm:1} may be conservative as it requires only minimal information about the static nonlinearity $\\varphi$; it applies to \\textit{any} static nonlinearity $\\varphi$ such that ${\\varphi(0) =0}$ and ${\\partial \\varphi \\in  [-\\mu, \\mu]}$. The model reduction framework is therefore robust to nonlinear uncertainty.\n\\hspace*{\\fill} $\\blacktriangle$\n\\end{remark}\n\n\n\n\nTheorem~\\ref{thm:1} can be seen an application the small ${\\LTp}$ gain theorem.  Assumptions (A3) and (A4) imply the small ${\\LTp}$ condition ${\\gamma_1 \\gamma_2 < 1}$, in which ${\\tilde{\\gamma} = \\norm{\\tilde{G} }_{\\infty,0}}$ and ${\\hat{\\gamma} = (\\mu^{-1} - \\norm{\\hat{G}}_{\\infty,p} )^{-1}}.$ This guarantees that the closed-loop system in Fig.~\\ref{fig:theorem_new} (left) is $p$-dominant, as the product of the differential  gain  ${\\tilde{\\gamma}}$ (from $w$ to $\\tilde{z}$) and the differential gain ${\\hat{\\gamma}}$ (from $\\tilde{z}$ and $w$) is strictly less than one. \\hspace*{\\fill} \n\n  \nThe small ${\\LTp}$ condition (A4)    admits a nice graphical interpretation. From a geometrical viewpoint, it implies that the distance between the critical circle and the Nyquist diagram of the shifted transfer function ${\\hat{G}_{\\lambda}}$ must be at least ${\\epsilon}$, as illustrated in Fig.~\\ref{fig:theorem_new} (right). This implies that $D(-\\hat{\\mu}, \\hat{\\mu})$, with ${\\hat{\\mu} = (\\mu^{-1} - \\epsilon)^{-1}}$, is a $p$-disk margin for the reduced order model~\\citep{padoan2019feedback}. This suggests that claim (i) can be established by a homotopy argument  (given, \\textit{e.g.}, in   \\cite[Theorem 3]{glover1983robust}).\n\n \n\n\n\n\\begin{figure}[h!]\n\\centering\n\\usetikzlibrary{arrows.meta}\n\\tikzset{>={Latex[width=3pt,length=3pt]}}\n\\usetikzlibrary{decorations.pathreplacing}\n\\begin{tikzpicture}[black, scale=.7, every node\/.style={transform shape}]\n \n \n\\draw[line width = .5 pt]  (11.8125,3.3125) rectangle (12.8125,2.3125);\n\\node at (12.3125,2.8125) {$\\hat{G}_{zw}^{-}$};\n\\draw[rounded corners,line width = .5 pt]  (11.8125,0.9375) rectangle (12.8125,-0.0625);\n\\node at (12.3125,0.4375) {$-\\varphi(\\cdot)$};\n\\draw[-latex, line width = .5 pt] (13.3125,1.625)   -- (13.3125,0.4375) -- (12.8125,0.4375);\n\\draw[-latex, line width = .5 pt] (11.8125,0.4375) -- (11.3125,0.4375) -- (11.3125,1.6875) -- (11.8125,1.6875) ;\n\\draw [-latex, line width = .5 pt] (11.3125,2.8125) --(11.3125,4.5625) -- (11.8125,4.5625);\n\\node at (11.125,3.6875) {$w$};\n\\node at (13.5625,3.6875) {$\\tilde{z}$};\n\\draw [-latex,line width = .5 pt](12.8125,4.5625) -- (13.3125,4.5625)  -- (13.3125,2.8681);\n\\draw[fill = white] (13.3125,2.8125) circle [radius=0.05];\n\\draw[semithick, dashed] (11.125,5.1875) rectangle (13.8125,4);\n\\node at (15,4.5625) {{ Gain $\\tilde{\\gamma}$}};\n\\draw[semithick, dashed] (11.1242,3.3875) rectangle (13.8133,1.0625);\n\\node at (15,1.75) {{ Gain ${\\hat{\\gamma}}$}};\n\\draw  (11.8125,5.0625) rectangle (12.8125,4.0625);\n\\node at (12.3125,4.5625) {$\\tilde{G}_{zw}^{-}$};\n\\draw[-latex, line width = .5 pt] (12.8125,2.8125) -- (13.2569,2.8125); \n\\draw[-latex, line width = .5 pt]  (11.8125,2.1875) rectangle (12.8125,1.1875);\n\\node at (12.3125,1.6875) {${G_{zw}^+}$};\n\\draw [-latex, line width = .5 pt](11.3125,1.6875) -- (11.3125,2.8125) -- (11.8125,2.8125);\n\\draw[fill = white] (13.3125,1.6875) circle [radius=0.05];\n  \\draw [-latex, line width = .5 pt](13.3125,2.75) -- (13.3125,1.75);\n\\draw [-latex, line width = .5 pt](12.8125,1.6875) -- (13.25,1.6875);\n\\node at (13.5625,4.9375) {$\\tilde{G}$};\n\\node at (13.5625,1.375) {$\\hat{G}$};\n\n\\draw [-latex](18.0625,0.125) -- (18.0625,5);\n\\draw [-latex](16.0625,2.5) -- (21,2.5);\n\\draw[thick] plot[smooth cycle, tension=.7] coordinates {(18.6733,2.5) (18.5069,2.8888) (18.0625,2.9444) (17.8405,2.5832) (18.0625,2.4168) (18.2289,2.4168)(18.2289,2.5832) (18.0625,2.5832) (17.8405,2.4168) (18.0625,2.0556) (18.5069,2.1112)} ;\n\\node at (20.5625,2.125) {{ $\\mathbf{Re} \\,\\hat{G}_{\\lambda}(i\\omega)$}};\n\\node at (17.0625,4.75) {{ $\\mathbf{Im} \\,\\hat{G}_{\\lambda}(i\\omega)$}};\n\\node at (16.1875,2.125) {{\\large $-\\tfrac{1}{\\mu}$}}; \n\\node at (19.25,3.75) {{\\large${\\mathbf{{\\epsilon}}} $}}; \n\n\\begin{scope}[even odd rule, shift={(2.0625,0)}] \n\\clip (16,2.5) ellipse (1.5 and 1.5) ellipse (1.0 and 1.0);\n\\draw[thick, pattern=north west lines,  pattern color=gray!50]  (16,2.5) ellipse (1.9 and 1.9);\n\\end{scope} \n\\draw[semithick,color=gray!80 ]  (18.0625,2.5) ellipse (1.0 and 1.0);\n\\draw[semithick,color=gray!80 ]  (18.0625,2.5) ellipse (1.5 and 1.5);\n\\draw[<->,thick] (18.7301,3.2232) -- (19.0901,3.5832);\n\n\\draw [thick, decorate, decoration={brace, amplitude=5pt}](14.0625,5.1875) -- (14.0625,4);\n\\draw [thick, decorate, decoration={brace, amplitude=5pt}](14.0625,3.5) -- (14.0625,0);\n\\end{tikzpicture} \n\\centering\n\\caption{Theorem~\\ref{thm:1}: a small ${\\LTp}$ gain intepretation (left) and a graphical interpretation (right), respectively.\n}\n\\label{fig:theorem_new}\n\\end{figure\n\n\nA reduced order model can be very useful for analysis. As such, it should be able to reproduce faithfully the behavior of the original system. This point  is  addressed by the next result, which provides conditions under which the dominance properties of the original system are preserved by the reduced order model. The proof is omitted as it closely parallels that of Theorem~\\ref{thm:1}, with  ${G}$ and ${-\\tilde{G}}$ playing the roles of ${\\hat{G}}$ and ${\\tilde{G}}$, respectively. \n\n\\begin{corollary} \\label{corollary:1}\nConsider system~\\eqref{eq:system-Lure-mr} and  a  reduced order model~\\eqref{eq:system-ROM}\nconstructed as in Algorithm~\\ref{alg:mr-Lure}. Assume\n\\begin{description}\n\\item[(A1)$^{\\star}$]\\eqref{eq:system-ROM-linear} is  strictly   $p$-dominant with rate ${\\lambda\\in\\Rge}$,\n\\item[(A2)$^{\\star}$] ${\\partial \\varphi \\in  [-\\mu, \\mu]}$, with  ${\\mu\\in \\Rge}$,\n\\item[(A3)$^{\\star}$] ${\\norm{\\tilde{G}}_{\\infty,0} \\le \\epsilon}$, with ${0<\\epsilon<\\mu^{-1}}$;\n\\item[(A4)$^{\\star}$] ${\\norm{G_{zw}}_{\\infty,p} < \\mu^{-1} - \\epsilon}$.\n\\end{description}\nThen the reduced order model~\\eqref{eq:system-ROM} is strictly $p$-dominant with rate ${\\lambda}$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Balanced truncation of Lure systems} \\label{sec:mr}\n \nThe results developed in the previous section require a stable  reduced order model of the non-dominant linear dynamics   together with an \\textit{a priori} error bound. In general, any model reduction method that produces a reduced order model with these properties can be used to reduce the original Lure system. \n\nModel reduction by balanced truncation~\\citep[Section 7]{antoulas2005approximation} is an obvious candidate, as it preserves stability and provides an error bound that is easy to compute. We briefly recall this method for completeness.\nGiven a linear, time-invariant, minimal, stable system~\\eqref{eq:system-linear-MIMO},  balancing consists in finding a coordinates transformation ${\\bar{x} = T^{-1} x }$  such that the reachability gramian $P\\in\\R^{n \\times n}$ and the observability gramian $Q\\in\\R^{n \\times n}$ of the system, defined implicitly by the Lyapunov equations\n\\begin{align}\nA P +PA^{\\transpose} +BB^{\\transpose} = 0, \\\\\nA^{\\transpose} Q +QA  +C^{\\transpose}C = 0, \n\\end{align} \nare both diagonal and equal. A \\textit{balancing transformation} $T$ acts on the reachability and observability gramians as \n\\begin{equation}\nT^{-1} P T^{-\\transpose}  = T^{\\transpose} Q T =   \\operatorname{diag}(\\sigma_1, \\ldots, \\sigma_n), \n\\end{equation}\nin which case the corresponding realization is said to be \\textit{(principal-axis) balanced}. The elements ${\\sigma_1, \\ldots, \\sigma_n}$ are the \\textit{Hankel singular values} of the system. These are system invariants  which  measure the  influence  of each state  on  the overall input-output behavior of the system.\nModel reduction by balanced truncation consists in ordering the Hankel singular values and in eliminating ${n-\\nu}$ state variables by truncation. The resulting reduced order model  is of order $\\nu$, stable,   and satisfies the error bound\n\\begin{equation}\n\\norm{G - \\hat{G}}_{\\infty} \\le 2 \n\\sum_{j=\\nu+1}^{n} \\sigma_{j} =\\epsilon ,\n\\end{equation}\nin which $G$ and $\\hat{G}$ are the transfer functions of the original system and  of  the reduced order model, respectively.\n\n\nAs anticipated above, balanced truncation can be applied to  a  dominant Lure system as follows. First, one decomposes the linear dynamics of the original system~\\eqref{eq:system-Lure} into the parallel interconnection of two subsystems described by the transfer functions $G^+$ and  $G^-$, which are strictly $p$-dominant and $0$-dominant with rate $\\lambda$, respectively. Then a reduced order model of order $\\nu^{-}$  of system ${G}^-$ is obtained by balanced truncation using the Lyapunov equations\n\\begin{align}\n(A^{-}+\\lambda I) P +P(A^{-}+\\lambda I)^{\\transpose} +B^{-}(B^{-})^{\\transpose} = 0, \\label{eq:Lyapunov-reachability}\\\\\n(A^{-}+\\lambda I)^{\\transpose} Q +Q(A^{-}+\\lambda I) +(C^{-})^{\\transpose}C^{-} = 0, \\label{eq:Lyapunov-observability}\n\\end{align} \nwith $(A^{-}, B^{-}, C^{-})$ a minimal realization of order $n^{-}$ of ${G}^-$. The resulting reduced order model $\\hat{G}^-$  is strictly $0$-dominant with rate $\\lambda$ and satisfies the error bound\n\\begin{equation} \\label{eq:error-dominant}\n\\norm{G^{-} - \\hat{G}^{-}}_{\\infty,0} \\le 2 \n\\sum_{j=\\nu^{-}+1}^{n^{-}} \\sigma_{j} =\\epsilon .\n\\end{equation}\nFinally, a reduced order model is constructed by considering first the parallel interconnection of two subsystems $G^+$ and  $\\hat{G}^-$ and then the feedback interconnection of the resulting system with the static nonlinearity $\\varphi$, as illustrated in Fig.~\\ref{fig:model_reduction_detail_new}. Balanced truncation allows for direct control over the error bound $\\epsilon$ according to~\\eqref{eq:error-dominant} by appropriately selecting the order $\\nu^{-}$\nof the reduced model $\\hat{G}^-$.  In the context of dominant systems, the error bound $\\epsilon$   is related to the rate $\\lambda$ selected to compute the  $H_{\\infty,0}$ norm. In principle, a careful selection of this additional parameter could lead to tighter reduction errors. This requires additional investigation.  \n\n\n\n\n\n\n \n \n \n\\section{An illustrative example}  \\label{sec:example}\n\nThe approximation of large-scale systems arising from the spatial discretization of the heat equation is a paradigmatic model reduction problem~\\citep{chahlaoui2002collection,antoulas2005approximation}. The goal is often to capture the behavior of these systems around equilibrium. This example, instead, focuses on oscillatory regimes, motivated by the fact that the heat equation describes key physical phenomena involving transport, which play a fundamental role in fluid dynamics and neuroscience~\\citep{keener1998mathematical}.\n\nConsider the problem of regulating heat flow oscillations in a homogeneous rod of unitary length via saturated\nproportional control, as illustrated in Fig.~\\ref{fig:setup}.\n\\begin{figure}[h!]\n\\centering\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{patterns}\n\\usetikzlibrary{arrows, arrows.meta}\n\\tikzset{>={Latex[width=3pt,length=3pt]}}\n\\begin{tikzpicture}[black, scale=.8, every node\/.style={transform shape}]\n\\draw  (1.5,3) rectangle (2.5,2);\n\\node at (2,2.5) {$-k_P$};\n\\draw[rounded corners]  (3.5,3) rectangle (4.5,2);\n\n\\draw[ line width = 1 pt] (3.7,2.35) -- (3.85,2.35) -- (4.15,2.65) -- (4.35,2.65);\n\\draw[-latex, line width = 0.5 pt] (5.5556,2.5) -- (6.35,2.5);\n\\draw [-latex, line width = 0.5 pt](2.5,2.5) -- (3.5,2.5);\n\\draw [-latex, line width = 0.5 pt](9.5,2.5) -- (10.5,2.5) -- (10.5,1.5) -- (0.5,1.5) -- (0.5,2.5) -- (1.5,2.5);\n\\draw[fill = white] (5.5,2.5) circle [radius=0.05];\n\\draw [-latex, line width = 0.5 pt](5.5,3.5) -- (5.5,2.5556);\n\\node at (10.25,2.75) {$y =z$};\n\\node at (5.7776,3.1112) {$u$};\n   \n\\draw[top color=white,bottom color=black!70] (6.5,28mm) arc (90:270:0.15 and 0.3)--++(3cm,0) arc (-90:-270:0.15 and 0.3)-- cycle;\n\\draw (9.5cm,28mm) arc (90:-90:0.15 and 0.3);\n\\draw (7cm,28mm) arc (90:270:0.15 and 0.3);\n\\draw (7.5cm,28mm) arc (90:270:0.15 and 0.3);\n\\draw (9cm,28mm) arc (90:270:0.15 and 0.3);\n\\node at (8.2,2.5) {$\\cdots$};\n \\node at (7.05,3) { $x_1$};\n\\node at (7.55,3) { $x_2$};\n\\node at (9.2,3) { $x_n$};\n\\node at (6.75,1.85) {\\footnotesize $h$};\n\\draw [<->, thick](6.5,2) -- (7,2);\n\\draw [-latex, line width = 0.5 pt](4.5,2.5) -- (5.4444,2.5);\n\\node at (5,2.7776) {$w$};\n\\end{tikzpicture}\n\\centering\n\\caption{Heat flow control system for a homogeneous rod of unitary length via saturated proportional control.\n}\n\\label{fig:setup}\n\\end{figure\n\nThe rod is insulated along its length and at the end. All temperature changes result from heat transfer at one end of the rod and by heat conduction along the rod.  By Fourier's law of heat conduction, the propagation of heat along the rod is described by the equation\n\\begin{equation}\n\\frac{\\partial T}{\\partial t} = \\kappa \\frac{\\partial^2 T}{\\partial \\xi^2},\n\\end{equation}\nin which ${T \\in \\Rge}$ is the temperature field on the rod, ${t \\in \\Rge}$ is the time variable, ${\\xi \\in (0,1)}$ is the spatial variable, and ${\\kappa > 0}$ is the thermal diffusivity, respectively. The temperature is measured at the end of the rod, \\textit{i.e.} ${y(t) = z(t) = T(1,t)}$. The temperature is initially zero and the heat flow source is placed at the beginning of the rod. The initial condition is\n\\begin{equation}\n T(\\xi,0) = 0, \\quad \\xi \\in (0,1)\n\\end{equation}\nand the (Neumann) boundary conditions are\n\\begin{equation}\n\\frac{\\partial T}{\\partial \\xi}(0,t) = u(t)+w(t) , \\quad  \\frac{\\partial T}{\\partial \\xi}(1,t) = 0,\n\\end{equation} \nin which ${u \\in \\R}$ is the exogenous input and ${w \\in \\R}$ is defined as $w = -\\sat(k_P z )$, where ${k_P>0}$ is the proportional gain and ${{\\sat(z) = \\tanh(z)}}$ is the saturation function.  The spatial domain is discretized into segments of length ${h = \\tfrac{1}{n+1}}$ using the second order central difference scheme\n\\begin{equation}\n\\frac{\\partial^2 T}{\\partial z^2}(kh,t) \\approx \\frac{1}{h^2}(T_{k+1}(t) - 2T_{k}(t) + T_{k+1}(t)),\n\\end{equation} \nwhere ${T_{k}(t) = T(kh,t)}$. This yields a system described by the equations~\\eqref{eq:system-Lure}, with \n${x = [\\, T_{1} \\ \\cdots \\ T_{n} \\,]^{\\transpose}}$, ${x(0) = 0}$, and \n\\begin{equation}  \\!\n\\scalebox{0.925}{$\n\\setlength{\\arraycolsep}{1.25 pt}\n\\renewcommand{\\arraystretch}{0.5}\nA =\n\\tfrac{\\kappa}{h^{2}}\n\\bma\n\\begin{array}{rrcrr} \n-1 &1 & & & \\\\\n1 &-2 &\\ddots & &\\  \\\\\n &\\ddots &\\ddots & \\ddots & \\\\ \n & &\\ddots &-2  &1 \\\\\n &  \t\t & &1  &-1 \n\\end{array}\n\\ema  \\! , \\, \nB=\n\\tfrac{\\kappa}{h}\n\\bma\n\\begin{array}{cc} \n1 & 1\\\\\n0 & 0 \\\\\n\\vdots & \\vdots\\\\\n0 & 0\n\\end{array}\n\\ema \\! , \\,\nC=\n\\bma\n\\begin{array}{cc} \n0 & 0\\\\\n0 & 0 \\\\\n\\vdots & \\vdots\\\\\n1 & 1\n\\end{array}\n\\ema^{\\transpose} \\!\\!  . $} \\!\\!\\! \n\\end{equation}\nThe transfer function of the system is\n\\begin{equation} \\label{eq:example-transfer-function}\nG(s) \n= \n\\bma\n\\begin{array}{cc}\nG_{yu}(s) & G_{yw}(s)\\\\\nG_{zu}(s) & G_{zw}(s)\n\\end{array}\n\\ema\n=\n\\bgroup\n\\setlength{\\arraycolsep}{3 pt}\n\\renewcommand{\\arraystretch}{1}\n\\tfrac{\\gamma_0}{\\prod_{j=1}^n \\left(s+\\lambda_k\\right)} \n\\bma\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & 1 \n\\end{array}\n\\ema\n\\egroup\n,\n\\end{equation}\nwith ${\\gamma_0 = \\frac{ \\kappa^n}{h^{2n-1}}}$ and ${\\lambda_k = 2\\tfrac{\\kappa}{h^2} -2\\tfrac{\\kappa}{h^2}\\cos\\tfrac{(k-1)\\pi}{n}}$ for every integer ${ k \\in [1,n]}$. The static nonlinearity is ${\\varphi(z) = \\text{sat}(k_Pz)}$, which satisfies the differential sector condition ${\\partial {\\varphi} \\in [0,k_P]}$.\n\nThe system captures the main features of the Goodwin model~\\citep{murray2002mathematical},  a classical model in cellular physiology described by a cascade of reactions modelled by first-order lags and a negative feedback modelled by a Hill function. Thus, while approximating the heat equation away from equilibrium might not have significant engineering relevance, this could be crucial to understand, simulate and control biological rhythms.\n\n\nTo illustrate our model reduction framework and to ensure that assumptions (A1) and (A2) of Theorem~\\ref{thm:1} hold, we consider the loop transformation defined as \n\\begin{equation} \\label{eq:loop-transformation}\n\\bar{\\varphi}(z) = \\varphi(z) - \\tfrac{k_P}{2} z .\n\\end{equation}\nThis converts the differential sector condition ${\\partial {\\varphi} \\in [0,k_P]}$ into ${\\partial \\bar{\\varphi} \\in [- \\mu, \\mu]}$, with  ${\\mu = \\tfrac{k_P}{2}}$, and the transfer function $G_{zw}$ into\n\\begin{equation} \\label{eq:example-transfer-function-loop}\nH_{zw}(s) = \\frac{G_{zw}(s)}{1+ \\tfrac{k_P}{2} G_{zw}(s)}.\n\\end{equation} \nThe root locus of the transfer function $H_{zw}$ reveals that for any choice of the step size $h$ it is possible to establish strict $2$-dominance for ${k_P\\in[\\underline{k}_P,\\overbar{k}_P]}$, with \n${0<\\underline{k}_P <\\overbar{k}_P}$. For example, let ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$  and ${\\lambda = 12}$. Then $H_{zw}$ has exactly two unstable poles for ${k_P = 20}$.  Direct inspection of the Nyquist diagram and the circle criterion for $p$-dominance imply that the system is strictly $2$-dominant with the same rate, as illustrated in Fig.~\\ref{fig:nyquist-P}. The system has therefore a unique limit cycle, since all solutions are bounded and the equilibrium at the origin is unstable.\n\n\\begin{figure}[h!]\n\\centering \n\\begin{tikzpicture}[black, every tick label\/.append style={font=\\scriptsize}]\n\\matrix{\n\\begin{axis}[\nheight = 0.23\\textwidth,\nwidth = 0.27\\textwidth,\nxmin=-.1,xmax=.1,\nymin=-.1,ymax=.1,\nx tick label style={\n        \/pgf\/number format\/.cd,\n            fixed,\n            fixed zerofill,\n            precision=2,\n        \/tikz\/.cd\n    },\ny tick label style={\n        \/pgf\/number format\/.cd,\n            fixed,\n            fixed zerofill,\n            precision=2,\n        \/tikz\/.cd\n    }    ,\ngrid\n]\n\\draw[name path =A] (axis cs:-0.1,-0.1) -- (axis cs:-0.1,0.1) -- (axis cs:0.1,0.1) -- (axis cs:0.1,-0.1) -- cycle;\n\\addplot[name path =B,no markers, black] table [x index = {0}, y index = {1},  col sep=comma]{circle.csv};\n\\tikzfillbetween[of=A and B]{pattern=north west lines};  \n\\addplot [black, very thick,  %\npostaction={decorate}, %\ndecoration={\n  markings,\n    mark=at position 0.5*\\pgfdecoratedpathlength-7.5pt\n  with {\\arrow[xshift=2.5\\pgflinewidth,>=stealth]{>}},\n  mark=at position 0.5*\\pgfdecoratedpathlength+7.5pt\n  with {\\arrow[xshift=2.5\\pgflinewidth,>=stealth]{>}}\n} %\n]  table [x index = {0}, y index = {1},  col sep=comma]{nyquist_system.csv};\n\\end{axis}\n~&~\n\\begin{axis}[\nheight = 0.23\\textwidth,\nwidth = 0.27\\textwidth,\nxmin=-10,xmax=10,\nymin=-20,ymax=20,\ngrid\n]\n\\fill [color=gray!50,fill=gray!50,fill opacity=0.5] (axis cs:-10,-10) rectangle (axis cs:10,10);\n\\addplot[black, very thick]  table [x index = {0}, y index = {1},  col sep=comma]{sector_condition.csv};\n\\end{axis}\n\\\\[1 pt]\n};\n\\end{tikzpicture}\n\\vspace{-.5cm}\n\\centering\n\\caption{{Left}: Nyquist diagram of the shifted transfer function ${H_{zw}}$ defined by~\\eqref{eq:example-transfer-function-loop} for ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$ and ${\\lambda = 12}$. The disk $D(-\\mu,\\mu)$, with  ${\\mu =  \\tfrac{k_P}{2} =  10}$, is represented by diagonal lines. {Right}: Derivative of the static nonlinearity ${\\bar{\\varphi}}$. The differential sector condition ${ \\partial \\bar{\\varphi} \\in [-\\mu,\\mu]}$  is represented by the shaded area.\n}\n\\label{fig:nyquist-P}\n\\end{figure\n \n\nSimulations have been run to assess the properties of different reduced order models. Algorithm~\\ref{alg:mr-Lure} and classical balanced truncation have been used to obtain reduced order models of the original system of order ${\\nu = 3, 4, 5}$.  Assumption (A3) of Theorem~\\ref{thm:1} holds, as the linear dynamics of the resulting reduced models are such that ${\\norm{\\tilde{H}_{zw}}_{\\infty,0} \\le \\epsilon_{\\nu}}$, with ${\\epsilon_3 = 3.27\\cdot 10^{-3}}$,    ${\\epsilon_4 = 2.55\\cdot10^{-4}}$, ${\\epsilon_5 = 2.28\\cdot10^{-5}}$, indicating that the approximation error ${\\epsilon_{\\nu}}$ is a strictly decreasing function of $\\nu$.  Figure~\\ref{fig:bode_bt} shows the magnitude of the shifted frequency response of the transfer function ${H_{zw}}$ defined by~\\eqref{eq:example-transfer-function-loop}, with ${\\kappa = 1}$, ${n=29}$, and  ${\\lambda = 12}$, (solid) and the shifted frequency response of the transfer function ${\\hat{H}_{zw}}$ of the reduced order model obtained via balanced truncation for ${\\nu = 3}$ (dashed), ${\\nu = 4}$ (dashdotted), ${\\nu = 5}$ (dotted), respectively. Note that assumption (A4) is satisfied by all reduced order models, as the magnitude of the shifted frequency response of the transfer function ${\\hat{H}_{zw}}$ is below the lines defined  by ${\\mu^{-1} - \\epsilon_{\\nu}}$. Figure~\\ref{fig:bode_bt} only shows the (solid) line defined by ${\\mu^{-1} = 0.1}$ as this is almost overlapped with the lines defined  by ${\\mu^{-1} - \\epsilon_{\\nu}}$. Theorem~\\ref{thm:1} guarantees that the original system is also strictly $2$-dominant, since balanced truncation preserves $2$-dominance of the linear dynamics and since ${\\norm{H_{zw}}_{\\infty,p}<\\mu^{-1} - \\epsilon_\\nu }$ for ${\\nu = 3, 4, 5}$. This is confirmed by the fact that the magnitude of the shifted frequency response of the transfer function ${{H}_{zw}}$ is below the (solid) line defined by ${\\mu^{-1} = 0.1}$. \n \n\n\n\\begin{figure}[t!]\n\\centering\n\\usetikzlibrary{spy}\n\\tikzset{new spy style\/.style={spy scope={%\n magnification=5,\n size=1.25cm, \n connect spies,\n every spy on node\/.style={\n   circle,\n   draw,\n   },\n every spy in node\/.style={\n   draw,\n   circle,\n   fill=gray!10,\n   }\n  }\n }\n} \n\\begin{tikzpicture}[new spy style,every tick label\/.append style={font=\\scriptsize}]\n\\begin{loglogaxis}[\nheight = 0.225\\textwidth,\nwidth = \\columnwidth,\nxmin=1e-0, xmax=1e03,\nymin=1e-6, ymax=1e0,\nxlabel={{$\\omega$} \\textrm{[rad\/s]}},          \nxlabel near ticks,\ngrid\n]\n\\node[left]  \t\t\tat (axis cs: 9*1e2,3*1e-2) {{\\footnotesize ${\\tfrac{1}{\\mu}=0.1}$}};\n\\node[left, rotate=-10]  at (axis cs: 9*1e2,6*1e-4) {{\\footnotesize ${\\nu=3}$}};\n\\node[left, rotate=-11]  at (axis cs: 9*1e2,7*1e-5) {{\\footnotesize ${\\nu=4}$}};\n\\node[left, rotate=-12]  at (axis cs: 9*1e2,8*1e-6) {{\\footnotesize ${\\nu=5}$}};\n\n\\addplot [black, thick]  table [x index = {0}, y index = {1},  col sep=comma]{bode_system.csv};\n\\addplot [black, thick,  dashed]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom1.csv};\n\\addplot [black, thick, dashdotted]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom2.csv};\n\\addplot [black, thick, densely dotted]  table [x index = {0}, y index = {1},  col sep=comma]{bode_rom3.csv};\n\\addplot [blue, thick ]  table [x index = {0}, y index = {1},  col sep=comma]{bode_mu_bound.csv};\n\\end{loglogaxis}\n\\spy[size=1.5cm,magnification=7] on (0.125,2.1) in node at (1.5,0.9);\n\\end{tikzpicture}\n\\vspace{-.25cm}\n\\centering\n\\caption{Magnitude  of the frequency response of the transfer function ${H_{zw}(s -\\lambda)}$  defined by~\\eqref{eq:example-transfer-function-loop},  with ${\\kappa = 1}$, ${n=29}$, ${k_P=20}$, ${\\lambda = 12}$, and   the frequency response of the transfer function ${\\hat{H}_{zw}(s -\\lambda)}$  of the reduced order model obtained via balanced truncation. \n}\n\\label{fig:bode_bt}\n\\end{figure}%\n\n\n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\n\n\nA model reduction framework geared towards the analysis and design of systems that switch and oscillate has been described. Classical balanced truncation for linear, time-invariant systems has been revisited to develop a model reduction method for Lure systems which preserves dominance properties. The theory has been illustrated by the approximation of an oscillatory Lure system arising from the discretization of a heat flow control problem. \nA future research direction is to extend the proposed framework to systems that can be decomposed as the feedback interconnection of a linear system and a dominant nonlinear system using the small $\\LTp$ gain theorem.  \n\n\n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\n\nLow mass X-ray binaries (LMXBs) are interacting binaries where a low mass \ndonor transfers matter onto a neutron star or black hole. \n4U 1636-536 (=V801 Ara) and 4U 1735-444 (=V926 Sco) are among the optically \nbrighter members in the class of {\\it persistent} LMXBs, characterized by  \n$L_{\\rm x} \\simeq 10^{37-38}$ erg s$^{-1}$ and blue spectra with weak\nhigh-excitation emission lines (mainly HeII $\\lambda$4686 and the \nCIII\/NIII Bowen blend at $\\lambda$4640). They also share similar \nproperties: they are both atoll sources (as based on the pattern traced \nin X-ray color-color diagrams, see e.g.\\citealt{hasin89}) with frequent \nburst activity and short orbital periods (3.80 and 4.65 hrs respectively) \nrevealed through optical photometry (\\citealt{corbet86}, \\citealt{pedersen81}). \nTheir lightcurves display shallow sinusoidal modulations which have been \ninterpreted as due to the geometrically varying visibility of the irradiated \ndonor star (e.g. \\citealt{van88}). Therefore, the photometric maxima \nsupposedly define orbital phase 0.5 i.e. inferior conjunction of the \ncompact object. Note, however, that this assumption requires confirmation \nbecause photometric maxima can sometimes be associated with asymmetries in the \ndisc structure such as the visibility of the irradiated inner disc bulge \nat phase $\\sim 0.3$ (e.g. 4U 1822-371 \\citealt{hell89}) or superhump\nactivity (see \\citealt{haswell01}). \n\nOnly a few \nspectroscopic studies have been presented on these two binaries up to now. \nFor example, \\cite{smale91} report H$_{\\alpha}$ spectroscopy of V926 Sco \nshowing that  the line core is dominated by emission from the disc bulge or\nsplash region where the gas stream interacts with the outer disc rim. \nOn the other hand, \\cite{aug98} (A98 hereafter) present radial velocity curves \nof HeII $\\lambda$4686 and the Bowen blend at $\\lambda$4640 for both V926 Sco \nand V801 Ara. They conclude that these high excitation lines are also tracing \nthe motion of the disc bulge.  \n\nV801 Ara is particularly remarkable since it is one of only 14 bursters where \n``burst oscillations\" (i.e. nearly coherent high-frequency pulsations) \nhave been detected during several thermonuclear X-ray bursts \n(\\citealt{giles02}, G02 hereafter). \nFurthermore, a train of burst oscillations was also discovered in a \n13 min  interval during a ``superburst\", showing a frequency drift which has \nbeen interpreted as the orbital motion of the neutron star (\\citealt{stro02},\nSM02 hereafter). \nBy fitting the frequency evolution with a circular orbit \nSM02 constrain the radial velocity amplitude of the neutron star to \nthe range $90<K_1<175$ km s$^{-1}$. These constraints on $K_1$ \nwill be readdressed and improved in this paper. \nFurther constraints on the system parameters and the component masses, \nhowever, require dynamical information on the donor\nstar which, unfortunately, is normally overwhelmed by the optical emission \nfrom the X-ray irradiated disc.    \n                                                                                \nA new indirect method to extract dynamical information from donor stars in \npersistent LMXBs was proposed by \\cite{stee02} (SC02 hereafter). They detected \nnarrow high excitation emission lines from the irradiated surface of the donor \nstar in Sco X-1 which led to the first determination of its radial velocity \ncurve and mass function. These lines are strongest in the Bowen blend, a \ncombination of C\\,III and N\\,III lines which are powered by photoionization and \nfluorescence (respectively) due to UV photons arising from the hot inner \ndisk (\\citealt{mc75}). This technique has been amply confirmed by subsequent \nstudies of the eclipsing ADC (Accretion Disc Corona) and X-ray pulsar \n4U 1822-371 (\\citealt{casa03}), the black hole soft X-ray transient \nGX 339-4 during its 2002 outburst (\\citealt{hynes03}) and   \nAql X-1 during its 2004 outburst (\\citealt{cornel06}).\n \nIn this paper we apply this method to the X-ray bursters \n4U 1636-536 (=V801 Ara) and 4U 1735-444 (=V926 Sco) and present the first \ndetection of the donor stars in these two LMXBs. The paper is organized as\nfollows: Section 2 summarizes the observation details and data reduction. \nThe average spectra and main emission line parameters are presented in Sect. \n3, with multigaussian deconvolution of the Bowen blend. In Section 4 we analyse \nthe radial velocities and orbital variability of the strong Bowen blend and \nHeII $\\lambda$4686 emission lines. Estimates of the systemic velocities are \nobtained through the Double Gaussian technique applied to the wings of the \nHeII line. Using these systemic \nvelocities we compute Doppler tomograms of HeII $\\lambda$4686 and the Bowen \nfluorescence NIII $\\lambda$4640 which are presented in Section 5. The NIII maps \ndisplay evidence of irradiated emission from the donor star, which is used to\nrefine the absolute phasing and systemic velocities. Finally, in Section \n6 we provide an improved determination of $K_1$  for V801 Ara and present our \nconstraints on the masses in the two binaries.\n\n\\section{Observations and Data Reduction} \\label{observations}\n\nV801 Ara and V926 Sco were observed on the nights of 23 and 25 June 2003 using \nthe FORS2 Spectrograph attached to the 8.2m Yepun Telescope (UT4) at the \nObservatorio Monte Paranal (ESO). A total of 42 spectra of 600s of V801 \nAra and 102 exposures of 200s of V926 Sco were \nobtained with the R1400V holographic grating, covering a complete \norbital cycle per night for each target. We used a 0.7 arcsec slit width which \nrendered a wavelength coverage of $\\lambda\\lambda$4514-5815 at \n70 km s$^{-1}$ (FWHM) resolution, as measured from Gaussian fits to the arc \nlines. The seeing was variable between 0.6\" -- 1.2\" during \nour run. The flux standard Feige 110 was also observed with the \nsame instrumental configuration to correct for the instrumental \nresponse of the detector.\n\nThe images were de-biased and flat-fielded, and the spectra subsequently \nextracted using conventional optimal extraction techniques in order to  \noptimize the signal-to-noise ratio of the output spectra (Horne 1986). A \nHe+Ne+Hg+Cd \ncomparison lamp image was obtained in daytime to provide the wavelength \ncalibration scale. This was obtained by a 4th-order polynomial fit to \n19 lines, resulting in a dispersion of 0.64 \\AA~pix$^{-1}$ and rms scatter \n$<$ 0.05 \\AA. Instrumental flexure was monitored through cross-correlation \nbetween the sky spectra and was found to be very small, always within 14 km\ns$^{-1}$ (0.4 pix.) on each night. These velocity drifts were removed from each \nindividual spectrum, and the zero point of the final wavelength scale \nwas established from the position of the strong OI $\\lambda$5577.338 sky line.  \nAll the spectra were calibrated in flux using observations of the flux \nstandard Feige 110. However, due to light loss caused by our narrow slit and \nvariable seeing conditions, our flux calibration is only accurate to $\\sim$ 50\\%. \n\n\\section{Average Spectra and Emission Line Parameters}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig1.eps}\n\\caption{Summed spectra of V801 Ara and V926 Sco with the principal emission \nlines indicated.}\n\\label{fig1}\n\\end{figure}\n\n\nFigure~\\ref{fig1} presents the average spectra of V801 Ara and V926 Sco in \n$f_{\\lambda}$ flux units. They show a blue continuum with broad high excitation \nemission lines of HeII $\\lambda$4686, $\\lambda$5411, the Bowen blend at \n$\\lambda\\lambda$4630-50 and H$\\beta$, which are typical of X-ray active \nLMXBs. Possible HeI lines at $\\lambda$4922 and $\\lambda$5015 \nare also identified but these are significantly weaker. Table 1 summarizes the \nFWHM, EW and centroid $\\lambda_{\\rm c}$ of the main emission lines obtained \nthrough simple Gaussian fits. \nWe note that the average spectra (including the line strengths) look very \nsimilar to those presented by A98, which seems to imply that there has been no \nlarge, long-term variations between the two data epochs. \nIncidentally, the EWs and FWHMs of all lines do not show any significant \nnigh-to-night variability nor modulation with the orbital period.\nAside from the intrinsically broad Bowen blend, which is \na blend of at least three CIII\/NIII transitions, we note that emission lines \nare a factor $\\sim$2 narrower in V926 Sco than in V801 Ara. Given  the \nsimilarities in their orbital periods, this suggests a projection effect with a \nlower inclination angle for V926 Sco. In addition, the emission line centroids \nin V926 Sco are significantly blueshifted, probably due to a larger \n(approaching) systemic velocity. \nThis also applies to the Bowen complex since the difference in\n  wavelength between V801 Ara and V926 Sco (-110 km\/s) is consistent with \n  the difference in velocities found for the other lines. \nThe blue continuum is also steeper for \nV926 Sco, as expected because of its lower reddening (see A98).  \n\n\\begin{table}\n\\centering\n\\caption[]{Emission line Parameters} \n\\begin{tabular}{lcccc}\n\\hline \n{\\em Line} & {\\em FWHM} & {\\em EW} & {\\em Centroid} &  {\\em $\\Delta V$} \\\\\n &  {\\em (km s$^{-1}$)} & {\\em (\\AA)} & {\\em (\\AA)} & {\\em (km s$^{-1}$)} \\\\\n\\hline\n{\\bf V801 Ara} & & & \\\\\nBowen               & 1848 $\\pm$ 65 & 4.25 $\\pm$ 0.03 & 4643.2 $\\pm$ 0.4 & -- \\\\ \nHeII $\\lambda$4686  & 1216 $\\pm$ 45 & 3.31 $\\pm$ 0.03 & 4685.6 $\\pm$ 0.3 & -9  $\\pm$ 11 \\\\\n$H\\beta$            &  963 $\\pm$ 56 & 1.67 $\\pm$ 0.02 & 4860.9 $\\pm$ 0.3 & -25 $\\pm$ 21  \\\\\n\\hline\n\\hline\n{\\bf V926 Sco} & & & \\\\\nBowen               & 1662 $\\pm$ 65 & 3.99 $\\pm$ 0.02 & 4641.5 $\\pm$ 0.4 & -- \\\\ \nHeII $\\lambda$4686  &  657 $\\pm$ 26 & 2.18 $\\pm$ 0.02 & 4683.9 $\\pm$ 0.2 & -118 $\\pm$ 12 \\\\\n$H\\beta$            &  507 $\\pm$ 39 & 0.43 $\\pm$ 0.02 & 4859.2 $\\pm$ 0.4 & -134 $\\pm$ 25  \\\\\n\\hline\n\\label{lines}\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig2.eps}\n\\caption{Combined fit to HeII $\\lambda$4686 and the Bowen blend using three \nGaussians at $\\simeq$4634\\AA~ (NIII), $\\simeq$4641\\AA~(NIII) and \n$\\simeq$4651\\AA~ (CIII). See text for details.}\n\\label{fig2}\n\\end{figure}\n \nThe Bowen blend mainly consists of emissions corresponding to the NIII \ntransitions at $\\lambda\\lambda$4634,4641,4642 and CIII at  \n$\\lambda\\lambda$4647,4651,4652 (\\citealt{scha89}). In an attempt to estimate \nthe relative contribution of the different transitions we have performed a \ncombined multi-Gaussian fit to \nthe average emission profiles of HeII $\\lambda$4686 and the Bowen blend for \nthe two LMXBs. The fit consists of four Gaussians, one for the HeII line, two \nfor the NIII emissions at $\\lambda$4634 and $\\lambda$4641-2 and another for \nthe CIII emissions at $\\lambda$4647-52. Free parameters are the \nline widths, which are set to be equal for all the lines, individual line \ncentroids and intensities. The line widths are mainly driven by the fit to the\nunblended HeII line and hence the latter is effectively used as a template to \nconstrain the line profiles within the Bowen blend. Figure~\\ref{fig2} presents \nthe results of the fit. Because widths are set to be the same for all lines, \nflux ratios are given by simple peak ratios. Our best fit yields NIII ratio \n$I(\\lambda4634)\/I(\\lambda\\lambda4641-2)= 0.31 \\pm 0.10$ and $0.59 \\pm 0.02$ \nfor V801 Ara and V926 Sco, respectively. This is a factor $\\leq$ 2 lower than \nthe theoretical NIII ratio of 0.71, computed by \\cite{nuss71} for Bowen \nfluorescence, but consistent with the results of \\cite{scha89} for \nSco X-1. Similarly, Hynes (private communication) finds a NIII ratio in the \nrange 0.36-0.56 (depending on time) for GX 339-4 during its 2002 outburst. \nWe can also calculate the CIII\/NIII ratio \n$I(\\lambda\\lambda4647-52)\/I(\\lambda4634 + \\lambda\\lambda4641-2)$ \nand find $0.38 \\pm 0.06$ and $0.35 \\pm 0.01$ for V801 Ara and V926 Sco \nrespectively. For comparison, we have \nperformed the same analysis on average spectra of Sco X-1 and 4U 1822-371, \nusing data presented in SC02 and \\cite{casa03}, and find \n$0.44 \\pm 0.03$ and $0.49 \\pm 0.01$ respectively. The lower CIII\/NIII ratio in \nV801 Ara and V926 Sco may indicate possible evidence of CNO processed material \nbut we have to be cautious here because of the oversimplification in our \nfitting model. In particular, we cannot rule \nout other possible transitions (most likely OII lines at \n$\\lambda$4641.8 and $\\lambda$4649.1; see \\citealt{mc75}) \ncontributing to our components at $\\sim$4641\\AA~and $\\sim$4651\\AA, \nwhich so far we have assumed to be dominated by NIII and CIII transitions. \nFurthermore, since the CIII and \nOII lines are not powered by Bowen fluorescence but photoionization, \ndifferent line ratios may stem from differences in efficiency of these two \nexcitation mechanisms rather than true CNO abundance variations. \n\n\\section{Radial Velocity Study and Orbital Variability}\n\nFigure~\\ref{fig3} displays the radial velocities of the Bowen blend and HeII \n$\\lambda$4686 line for V801 Ara and V926 Sco, obtained by cross-correlating \nthe individual spectra with Gaussians of fixed FWHM as given in Table 1. \nWe have assumed $\\lambda_c = 4643.0$ for the central \nwavelength of the Bowen blend. \nThe velocities are folded in orbital phase, using the\nephemerides of G02 and A98 for V801 Ara \nand V926 Sco but shifted in phase by +0.5 to make phase zero \ncoincide with the inferior conjunction of the donor star. These will be  \ncalled the {\\it photometric ephemerides} and will be used throughout this \nSection. The accumulated phase uncertainty at the time of our observations is \n$\\pm 0.06$ for both systems. Note that this phase convention assumes that \nlightcurve maxima are driven by the visibility of the irradiated donor star.  \n\nThe radial velocity curves for V801 Ara show maxima at phase $\\sim$ 1. \nThis phasing suggests that the velocity variations contain a significant \ncomponent arising in the disc bulge (which has its maximum visibility at \nphase $\\sim$ 0.75), caused by the \ninteraction of the gas stream with the outer edge of the disc, as \nis typically seen in persistent LMXBs (e.g. 4U 1822-371; \\citealt{cowley82}). \nOn the other hand, the radial velocity curves for V926 Sco \nshow maxima at phase $\\sim$ 0.7. \nBy comparing the HeII curves in the two binaries we clearly see  \nevidence for a large systemic velocity of $\\sim$ -150 km s$^{-1}$ in \nV926 Sco, in good agreement with \\cite{smale91}. We also note that the \nCIII\/NIII blend in V801 Ara is modulated with \na much larger amplitude than in V926 Sco i.e. 280 km s$^{-1}$ versus \n70 km s$^{-1}$, respectively. \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig3.eps}\n\\caption{Radial velocity curves of the Bowen blend (top) and HeII \n$\\lambda$4686 in V801 Ara and V926 Sco.}\n\\label{fig3}\n\\end{figure}\n\nFigure~\\ref{fig4} presents the trailed spectra of the Bowen blend and \nHeII$\\lambda$4686 for V801 Ara and V926 Sco, after co-adding our individual \nspectra in 15 phase bins to improve statistics. The core of the emission lines \nin V801 Ara show clear S-wave components. The S-wave in the Bowen blend is \nquite narrow and it may arise from the heated face of the donor star, as \nobserved in Sco X-1 (SC02). \nThe S-wave in HeII is rather extended and is likely produced in a \nregion with a large velocity dispersion such as the disc bulge. \nOn the other hand, the Bowen blend in V926 Sco is rather noisy and does not \nexhibit any clear components that are visible by eye. The HeII trailed \nspectra do show a clear orbital modulation with complex structure. \n \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig4.eps}\n\\caption{Trailed spectra showing the orbital evolution of the Bowen blend \nand HeII $\\lambda$4686 in 15 phase bins using the {\\it photometric ephemerides}}\n\\label{fig4}\n\\end{figure}\n    \nFollowing our work on Sco X-1 (SC02) and 4U 1822-371\n(\\citealt{casa03}), we have applied the \ndouble-Gaussian technique (\\citealt{sch80}) to the wings of the HeII \n$\\lambda$4686 line in an attempt to estimate the radial velocity curve of \nthe compact object. The trailed spectra presented in Fig.~\\ref{fig4} \ndemonstrate that the line cores are dominated by \nstrong, complex, low-velocity components associated with asymmetric emission \nfrom the outer disc and\/or the donor star, which we want to avoid. \nTherefore, by convolving the emission line with a double-Gaussian filter of \nsufficiently large Gaussian \n separation we can extract radial velocity curves from the wings of \nthe profile, which are expected to follow the motion of the compact star. We \nhave used a double-Gaussian bandpass with $FWHM=100$ km s$^{-1}$ and \nrelative Gaussian separations in the range  \n$a= 400-1400$ km s$^{-1}$ in steps of 100 km s$^{-1}$. In order to \nimprove statistics, we have co-added our spectra in 15 phase bins using the \n{\\it photometric ephemerides}. The radial velocity curves\nobtained for different Gaussian separations are subsequently fitted with \nsine-waves of the form $V(\\phi)=\\gamma + K \\sin 2\\pi (\\phi-\\phi_{0})$, fixing \nthe period to the orbital value. The best fitting parameters \nare displayed as a function of the Gaussian separation $a$ in a Diagnostic \nDiagram (see Fig.~\\ref{fig5}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=84mm]{fig5.eps}\n\\caption{Diagnostic Diagram for HeII $\\lambda$4686 in V801 Ara (solid circles) \nand V926 Sco (open circles). The dotted horizontal line marks $\\phi_{0}=0.5$ \nor the expected inferior conjunction of the compact object, according to the\n{\\it photometric ephemerides}.}\n\\label{fig5}\n\\end{figure}\n\nIn both cases we see how the line cores are dominated by high-amplitude \n($K\\sim 200$ km s$^{-1}$) S-waves which fade as we move to the line wings,\nwhere lower $K-$amplitudes of a few tens of km s$^{-1}$ are found. On the other\nhand, the $\\gamma$ velocities are very steady throughout the profiles, with \naverage values of~$\\sim -30$ (V801 Ara) and~$\\sim -130$ km s$^{-1}$ (V926 Sco). \nThe blue-to-red crossing phase for V926 Sco \ndisplays a smooth decreasing trend from the line core to the wings, \nwhereas it is rather constant for V801 Ara, with an average value of \n$\\phi_{0} \\sim 0.7$. \nWe estimate that the velocity points start to be\ncorrupted by continuum noise for Gaussian separations larger than \n$a\\sim1000$ as indicated by the diagnostic parameter $\\sigma(K)\/K$ \n(see \\citealt{shafter86}). Therefore, we decided to adopt the average values \nfor the parameters obtained from the last two separations before $\\sigma(K)\/K$ \nstarts to rise i.e. $a=900-1000$ km s$^{-1}$ for V801 Ara and 1000-1100 km \ns$^{-1}$ for V926 Sco. \nThis yields $K_1 = 93 \\pm 25$ km s$^{-1}$, $\\gamma = -42 \\pm 4$ km \ns$^{-1}$,  $\\phi_{0}=0.68 \\pm 0.03$ for V801 Ara and $K_1 = 96 \\pm 9$ \nkm s$^{-1}$, $\\gamma = -121 \\pm 7$ km s$^{-1}$, $\\phi_{0}=0.20 \n\\pm 0.01$ for V926 Sco. The errorbars have been adjusted to incorporate the\nscatter between different values within our preferred range. Note that \n$\\phi_{0}$ of V801 Ara is delayed by \n$\\sim0.18$ with respect to the inferior conjunction of the compact \nobject (dotted line in Fig.~\\ref{fig5}), as predicted by the {\\it photometric \nephemerides}. \nThis is classically observed in interacting binaries and interpreted as \ncontamination of the line wings by residual emission from the \ndisc-bulge\/hot-spot (\\citealt{marsh98}).   \nOn the other hand, $\\phi_{0}$ of V926 Sco leads the inferior \nconjunction of the compact star by $\\sim0.30$, according to the \n{\\it photometric ephemerides}. Such a large \nshift is unexpected and a full discussion is diverted to Sect. 6.\nFurthermore, we also note the very asymmetric distributions obtained \nin the HeII Doppler maps (see next Sect.) which may invalidate the \ndouble-Gaussian technique. Therefore, the constraints on zero-phase, \n$\\gamma$-velocity and $K_1$ (for V801 Ara) will be superseded in the \nfollowing sections. \n\n\\section{Bowen Fluorescence from the Irradiated Companion}   \n\nDoppler tomography enables us to map the brightness \ndistribution of a binary system in velocity space through combining \norbital phase spectra (see details in \\citealt{marsh01}). This is \nparticularly effective when dealing with weak emission\nfeatures which are barely detected or embedded by noise in individual \nspectra, such as our narrow Bowen emission lines. In order to compute Doppler \nimages of the NIII $\\lambda$4640 fluorescence line in V801 Ara and V926 Sco \nwe have rectified the individual spectra by subtracting a low order \nspline fit to the continuum regions and rebinned them into a uniform \nvelocity scale of 37 km s$^{-1}$ per pixel. Doppler images were subsequently \ncomputed using the {\\it photometric ephemerides} and $\\gamma=-42$ and $-121$ \nkm s$^{-1}$ for V801 Ara and V926 Sco, respectively. The maps show compact \nspots shifted in phase by $+0.03$ and $-0.18$ with respect to the expected \nlocation of the donor star in  the velocity maps (i.e. along the vertical\n$V_{\\rm y}$ axis)\\footnote{Note that the greyscale in the NIII maps have \nbeen set to enhance the contrast of the sharp components. This is the reason \nwhy these maps do not show any trace of emission from the underlying broad \ncomponent.}. Assuming that these spots \nare produced on the irradiated hemisphere of the donor star, as has been\nshown to be the case in Sco X-1 (SC02) and 4U 1822-371 \n(\\citealt{casa03}), we correct the previous {\\it photometric ephemerides} and \nderive the following {\\it spectroscopic ephemerides}, which will be used in the \nremainder of the paper:   \n\n\\begin{equation} \nT_{0}(HJD) = 245 2813.531(2) + 0.15804693(16) E    \n\\end{equation}\n\n\\begin{equation} \nT_{0}(HJD) = 245 2813.495(3) + 0.19383351(32) E      \n\\end{equation}\n\n\\noindent\nwhere the zero-phase error comes from the uncertainty in the centroid\nposition of the spots. Equation (1) corresponds to the ephemerides of  \nV801 Ara and equation (2) to V926 Sco.\nWith these ephemerides, the HeII $\\lambda$4686 maps of the two binaries show \na crescent shape brightness distribution pointing towards emission from an \nextended disc bulge.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig6.eps}\n\\caption{Doppler maps of HeII $\\lambda$4686 (left panels) and NIII \n$\\lambda$4640 (right panels) for V801 Ara and V926 Sco, computed using our \n{\\it spectroscopic ephemerides}. Systemic velocities \n$\\gamma = -34$ km s$^{-1}$ and $\\gamma = -140$ km s$^{-1}$ were adopted for \nV801 Ara and V926 Sco, respectively. For comparison we also plot the gas\nstream trajectory (in units of 0.1 R$_L1$) and Roche lobe of the donor star \nfor the case of $K_2=360$ km s$^{-1}$ (V801 Ara) and $K_2=298$ km \ns$^{-1}$  (V926 Sco). A mass ratio $q=0.27$ has been assumed for the two\nbinaries. \n }\n\\label{fig6}\n\\end{figure}\n\nFollowing SC02 we have used the HeII Doppler maps to refine the \nsystemic velocities derived in the previous section. The \n$\\chi^2$ value of the map was calculated for a range of $\\gamma$'s, and the \nbest fit in terms of mimimal $\\chi^2$ was achieved for $\\gamma\\simeq -39 $  \nkm s$^{-1}$ (V801 Ara) and $\\gamma \\simeq -132 $  km s$^{-1}$ (V926 Sco). \nThe use of an incorrect systemic velocity has the effect  \nof blurring bright spots into , as opposed to V801 Ara,elongated ``defocused\" \nfeatures in Doppler \nimages. Therefore, as a further test, we have also computed NIII  \n$\\lambda$4640 maps for a set of $\\gamma$-velocities in the range \n-200 -- +200 km s$^{-1}$ and we looked for the best focused NIII spots by \ncomputing the\nskewness using box sizes of 5, 10 and 20 pixels. We find that the most \nsymmetric and compact spots are found for $\\gamma$ values in the\nrange -29 -- -39 km s$^{-1}$ (V801 Ara) and 137-143  km s$^{-1}$ (V926 Sco). \nTherefore, we decided to adopt $\\gamma=-34 \\pm 5$ km s$^{-1}$ (V801Ara) \nand $\\gamma=-140 \\pm 3$ km s$^{-1}$ (V926 Sco) and these are the \nvalues that we have used in the final Doppler maps, which are presented in\nFig.~\\ref{fig6}. The NIII maps reveal compact bright spots \nwhich we interpret as irradiated emission from the inner hemisphere of\nthe donor star. These spots are obviously located along the vertical axis\nbecause of our ephemerides definition. We have calculated the position of \nthese spots by computing their centroids using the centering algorithm \nCENTROID in IRAF\\footnote{IRAF is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association of Universities \nof Research in Astronomy, Inc., under cooperative agreement with the\nNacional Science Foundation.} \nand find $K_{\\rm em}= 277 \\pm 22$ km \ns$^{-1}$ (V801 Ara) and $226 \\pm 22$ km s$^{-1}$ (V926 Sco). We note that \n$K_{\\rm em}$ is very weakly dependent on $\\gamma$, with only a 10 km s$^{-1}$ \ndrift when $\\gamma$ varies by $\\pm 20$ km s$^{-1}$ around our central value. \nTherefore, our $K_{\\rm em}$ determinations seem very robust and they represent \nstrict lower limits to the true radial velocity semiamplitude $K_2$ of the \ndonor stars because they must arise from the irradiated hemisphere facing the \nneutron star. Further confirmation of our system parameters is provided by \nthe appearance of the sharp CIII\/NIII transitions after coadding all the \nspectra in the rest frame of the NIII $\\lambda$4640 emission line region \n(see Fig.~\\ref{fig7}). \nNote that we have not attempted to compute Doppler images of the H$\\beta$ \nlines because these are contaminated by phase-variable absorption components, \nas was also the case in 4U 1822-371 (e.g. \\citealt{casa03}). Absorption \nviolates the principles of Doppler Tomography and makes it very difficult to\ninterpret the corresponding Doppler images.     \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig7.eps}\n\\caption{Average spectra of V801 Ara (left panel) and V926 Sco (right panel).\nTop spectra have been Doppler corrected in the rest frame of the donor using \nthe spectroscopic ephemerides and the $K$-velocity amplitudes derived from the \nbright spots in the NIII $\\lambda4640$ Doppler images. The narrow NIII and \nCIII lines are clearly seen in the Bowen blend. These are smeared out and cannot\nbe detected in the straight averages shown at bottom. Noise has been filtered \nusing a 2 pixel boxcar smoothing.}\n\\label{fig7}\n\\end{figure}\n\n\\section{DISCUSSION}\n\nThe detection of NIII $\\lambda$4640 fluorescence emission from the irradiated \ndonor in V801 Ara and V926 Sco provides new absolute (spectroscopic) \nephemerides and opens the possibility to derive the first constraints on the \ndynamical masses of these two LMXBs. In the light of the new ephemerides, the\nphotometric lightcurve maxima of V801 Ara take place at phase $0.47 \n\\pm 0.06$ (with the uncertainty dominated by error propagation of the G02 \nephemerides) and, therefore, they are consistent with X-ray irradiation of the \ndonor star. On the other hand, lightcurve maxima in V926 Sco are located at\norbital phase $0.82 \\pm 0.06$. This is in between the maximum visibility of \nthe irradiated donor and the disc-bulge and hence it would imply that \nlightcurve maxima arise by a combination of these two emitting regions. \nHowever, we note that the uncertainty in the photometric ephemerides of V926 \nSco is likely to be higher than stated in A98 due to the limited number of\narrival times fitted and their large scatter (Augusteijn, private\ncommunication). Therefore, it is possible that lightcurve maxima in V926 Sco are\nalso consistent with superior conjunction of the irradiated donor star. This is\nthe most likely scenario because it would be hard to understand an emission\nsource peaking at phase $\\sim 0.8$ in an irradiation dominated environment\nwithout invoking some strange (and stable) disc configuration.  \n\nWe have also determined the systemic velocities for the two binaries i.e. \n$-34\\pm5$ km s$^{-1}$ (V801 Ara) and $-140\\pm3$ km s$^{-1}$ (V926 Sco). \nThese can be compared with $V_{\\rm r}$, the radial velocity,  \nrelative to the Local Standard of Rest, due to Galactic \nrotation at the location of the binaries. For the case of \nV801 Ara, we take $l=332.9^{\\circ}$, $b=-4.8^{\\circ}$ \nand $d=6-7$ kpc, based on peak fluxes of radius-expansion \nX-ray bursts for $M_1=1.4-2$ M$_{\\odot}$ (\\citealt{galloway05}). Assuming \nthe Galactic rotation curve of \\cite{naka03} we find \n$V_{\\rm r}$ in the range -103 to -111 km s$^{-1}$. This is much larger \nthan our systemic velocity which could be explained through the recoil \nvelocity gained by the binary during the supernova explosion which formed \nthe neutron star. Thus, we can set a lower limit to the kick velocity in \nV801 Ara of $\\simeq70$ km s$^{-1}$. Regarding V926 Sco, we take \n$l=346.1^{\\circ}$, $b=-7.0^{\\circ}$ and a distance $d=9.1$ kpc \n(for a canonical 1.4 M$_{\\odot}$ neutron \nstar, see A98). The Galactic rotation curve at the position of \nV926 Sco yields $V_{\\rm r}= -111$ km s$^{-1}$, close but \nsignificantly lower than our systemic velocity. The difference might also be \nascribed to a kick velocity received by the neutron star at birth or,\nalternatively, a neutron star mass of $\\simeq$1.7 M$_{\\odot}$. Note that the \nlatter implies a larger distance of $d=10$ kpc and hence $V_{\\rm r}=-140$ km \ns$^{-1}$.  \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=84mm]{fig8.eps} \n\\caption{Top panel: Dynamic Fourier power contours showing the excursion \nin oscillation frequency during the 2001 February 22 superburst from 4U \n1636-53 (V801 Ara) with the best fitting orbit model overlayed. Time zero for \nthe x-axi is MJD 51962.7102530038 (TDB) \nBottom panel: Pulse phase residuals for the best fitting model.} \n\\label{fig8}\n\\end{figure}\n\n\\subsection{System Parameters for V801 Ara}   \n\nThe NIII $\\lambda$4640 spot in the Doppler map yields $K_{\\rm em}= 277 \\pm \n22$ km s$^{-1}$. It must arise on the inner hemisphere of the donor\nand hence it sets a lower limit to the velocity semi-amplitude of the \ncompanion's center of mass $K_2$. In order to find the real $K_2$ we need to\ncalculate the $K-correction$ or $K_{\\rm em}\/K_2$ for the case of emission \nlines in illuminated atmospheres, which depends mainly on the binary mass \nratio $q=M_2\/M_1$ (with $M_2$, $M_1$ the masses of the donor and compact \nstar respectively) and the disc flaring angle $\\alpha$. The $K-correction$ has \nbeen calculated by \\cite*{munoz05} using an irradiation binary code which \nincludes shadowing by an axisymmetric flared disc. The results are \ntabulated as a function of $q$, $\\alpha$ and the inclination angle $i$ in \nTable 1 of their paper, although the dependence on $i$ is very weak. The \n$K-correction$ is constrained between $\\alpha=0$ (i.e. we neglect disc \nshadowing) and the geometric limit set by emission from the irradiated limb of \nthe donor, i.e. $K_{\\rm em}\/K_2 < 1-0.213 q^{2\/3} (1+q)^{1\/3}$ (see \n\\citealt{munoz05}). \n\nAn upper limit on $q$ is established by taking $M_1\\ge1.4 $M$_{\\odot}$ and \n$M_2\\le 0.48$ M$_{\\odot}$, the largest possible zero-age main sequence (ZAMS) \nstar fitting in a 3.79 hr period Roche lobe (\\citealt{tout96}), which leads to \n$q\\le 0.34$. As a first approach, one can assume the empirical Mass-Radius \nrelation for low-mass stars in cataclysmic variables and LMXBs (see e.g. \n\\citealt{smith98}, \\citealt{warner95}) which yields $M_2 \\simeq 0.32$. This, \ncombined with a canonical neutron star of $M_1\\simeq1.4$ M$_{\\odot}$, \nwould lead to \na plausible $q\\sim 0.23$. The $K-correction$ for $K_{\\rm em}=277\\pm22$ km \ns$^{-1}$ and $q\\sim0.23$ would yield $303<K_2<404$ km s$^{-1}$, where we have \nadopted the coefficients for the case $i=40^{\\circ}$ in Table 1 of \n\\cite{munoz05} because V801 Ara is not eclipsing.\n  \nA more refined $K-correction$ requires a knowledge of $q$. This can be\nconstrained by the rotational broadening $V_{\\rm rot} \\sin i$ of the companion\nstar which, for the case of synchronous rotation, is related to $q$ through \n\n\\begin{equation}\nV_{\\rm rot} \\sin i \\simeq 0.462 K_2 q^{1\/3} (1+q)^{2\/3}\n\\end{equation}\n\n\\noindent\n(\\citealt{wadehorne88}). \nA lower limit to $V_{\\rm rot} \\sin i$ can be estimated from the width \nof the sharp NIII fluorescence emission in the Doppler corrected average \nspectrum. This is because emission lines arise not from the entire\nRoche lobe but from the irradiated part only. \nA multigaussian fit to the Bowen profile presented in Fig.~\\ref{fig7} gives \n$FWHM (NIII \\lambda4640) = 140 \\pm 21$ km s$^{-1}$,  which includes the effect \nof our intrinsic instrumental resolution (70 km s$^{-1}$). This has been \naccounted for by broadening a Gaussian template of $FWHM=70$ km s$^{-1}$ \nbetween 10 and 200 km s$^{-1}$, in steps of 10 km s$^{-1}$, using a Gray \nrotational profile (\\citealt{gray92}) without limb-darkening, because \nfluorescence lines arise in optically thin conditions. Our simulation \nindicates that $V_{\\rm rot} \\sin i=92\\pm16$ km s$^{-1}$ \nis equivalent to $FWHM = 140 \\pm 21$ km s$^{-1}$. On the other hand, by \nsubstituting $V_{\\rm rot} \\sin i \\ge 76$ km s$^{-1}$ and the $K-correction$ for \n$\\alpha>0^{\\circ}$ into equation (3), one finds a secure lower limit $q\\ge \n0.08$. \n\nAn upper limit to the rotational broadening of the donor star is set by \n$V_{\\rm rot} \\sin i=2\\pi R_2 \\sin i \/P$. By assuming that the donor \nmust be more evolved than a ZAMS star within a 3.79 hr Roche lobe, then \n$R_2\\le 0.44 R_{\\odot}$ (\\citealt{tout96}). This, together with  \n$i\\le 78^{\\circ}$ (lack of X-ray eclipses for $q\\ge0.08$), yields \n$V_{\\rm rot} \\sin i \\le 138$ km s$^{-1}$. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig9.eps} \n\\caption{Constraints on $K_2$ and $q$ for V801 Ara. The disc flaring angle \nmust be between $\\alpha=0^{\\circ}$ and $\\alpha=max$, whereas the rotational \nbroadening is constrained between the observed $V_{\\rm rot} \\sin i \\simeq 76$ \nkm s$^{-1}$ and 138 km s$^{-1}$, the maximum allowed for a ZAMS star in a 3.8hr \nperiod and for $i\\le 78^{\\circ}$. \nThe shaded area represents the $K_1$ solution obtained from a reanalysis of \nthe superburst pulsations. These constrain the binary mass ratio to be \n$q=0.21-0.34$ and $K_2=286-433$ km s$^{-1}$. For comparison we also mark  \n(dashed line) the solution for a hypothetic disc flaring angle of \n$\\alpha=12^{\\circ}$.}\n\\label{fig9}\n\\end{figure}\n\nFurther constraints are provided by the study of burst oscillations which \ncan set limits to the neutron star's projected velocity $K_1 (= q K_2)$.  \nSM02 derived $90\\le K_1\\le 175$ km s$^{-1}$ by fitting the frequency\ndrift of highly coherent X-ray pulsations observed in an 800s interval\nduring a superburst. They used a circular orbit model and fixed the zero phase \nto the ephemeris of G02. \nWe have used our new spectroscopic ephemeris to reanalyse the superburst\npulsation data in order to better constrain $K_1$.  Based on our spectroscopic\nephemeris, the pulsation interval during the superburst from V801 Ara \ncomes slightly earlier in orbital phase by 0.032 cycles as compared to the\nG02 ephemeris. We fit the pulsation data (see SM02 for details on the phase \ntiming analysis) to a circular orbit model with the\nreference epoch fixed to our new $T_0$, and we also fixed the orbital period\n(using the G02 value). This\nleaves two free parameters, the projected neutron\nstar velocity, $K_1$, and the rest-frame spin frequency, $\\nu_0$.  We find\nacceptable fits  with a reference epoch within the $\\pm 1 \\sigma$ range for\nour spectroscopic $T_0$.  The inferred $K_1$ velocity ranges from 90 - 113 km \ns$^{-1}$ as $T_0$ ranges over $\\pm 1\\sigma$.  \nThe best fit $\\chi^2$ values range from 14.6 to 10.6 (with 7 dof) over this \nsame range, that is, higher velocities are modestly favored, but given the \nshortness of the pulse\ntrain compared to the orbital period, we do not consider these differences \nas significant. With the reference epoch fixed, the statistical error on the\nvelocity is much smaller than the range given above, so 90 - 113 km s$^{-1}$ is \na robust range for $K_1$ at the $\\pm 1\\sigma$ limits of $T_0$.  The best fit \n$\\nu_0$ ranges from 582.04143 to 582.09548 Hz, and which \nmay help constrain future searches for a persistent millisecond pulsar in V801  \nAra. Figure ~\\ref{fig8} summarizes the results of the new timing analysis. \n\nFig.~\\ref{fig9} summarizes all our restrictions in the $K_2-q$ parameter \nspace i.e. $0^{\\circ}\\le \\alpha\\le max$, $V_{\\rm rot} \\sin i =76-138$ km \ns$^{-1}$ and $K_1=90-113$ km s$^{-1}$. The shaded area \nindicates the region allowed by our constraints, which yields $q=0.21-0.34$ \nand $K_2=286-433$ km s$^{-1}$. These imply a mass function $f(M)=M_1 \n\\sin^3 i \/(1+q)^2=0.76\\pm0.47$ M$_{\\odot}$ and hence $M_1 \n\\sin^3 i=1.24\\pm0.77$ M$_{\\odot}$. \nClearly the large uncertainties are dominated by the error in \n$K_{\\rm em}$ and the uncertainty in the disc flaring angle.\n\n\n\\subsection{System Parameters for V926 Sco}   \n\nThe same reasoning can be applied to V926 Sco. The spot in the NIII \n$\\lambda$4640 map yields $K_{\\rm em}=226 \\pm 22$ km s$^{-1}$ whereas \nthe ZAMS Mass-Radius relation for a 4.65 hr Roche lobe leads to $M_2 \n\\le 0.58 M_{\\odot}$ (\\citealt{tout96}) and hence $q\\le 0.41$.\nOn the other hand, we measure $FWHM (NIII \\lambda4640) = 115 \\pm 23$ km \ns$^{-1}$ which, after deconvolution of the instrumental resolution using Gray\nrotation profiles as above, yields $V_{\\rm rot} \\sin i \\ge 71 \\pm 21$ km \ns$^{-1}$. This lower limit, combined with the $K-correction$ for \n$\\alpha>0^{\\circ}$ and equation (3), yields $q\\ge 0.05$ and, hence \n$i\\le 80^{\\circ}$. And, under the assumption that the donor star is more \nevolved than a ZAMS star, the Roche lobe geometry implies $R_2 \\le 0.54 \nR_{\\odot}$ which, for $i\\le 80^{\\circ}$, leads to $V_{\\rm rot} \\sin i \n\\le 137$ km s$^{-1}$. \n\nAll these restrictions translate into constraints on the $K_2-q$ plane \nwhich are presented in Fig.~\\ref{fig10}. The allowed region results in \n$q=0.05-0.41$ and $K_2=215-381$ km s$^{-1}$, depending on the value of \n$\\alpha$. These numbers imply  $f(M)=0.53\\pm0.44$ \nM$_{\\odot}$ and, thus, $M_1 \\sin^3 i= 0.80 \\pm 0.71$ M$_{\\odot}$.  \nUnfortunately we do not have any contraints on $K_1$ from  \nburst oscillations and hence our mass restrictions are not as \nwell constrained as for V801 Ara.  \nOur best estimates of the system parameters for both objects \nare presented in Table 2. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig10.eps} \n\\caption{Same as Fig.~\\ref{fig9} but for V926 Sco, but with no pulsation \nconstraints yet.}\n\\label{fig10}\n\\end{figure}\n\n \n\\begin{table}\n\\centering\n\\caption[]{System Parameters.~$T_0$ indicates zero phase or inferior conjunction\nof the donor star. Orbital periods $P_{\\rm orb}$ are from G02 and A98.} \n\\begin{tabular}{lcc}\n\\hline \nParameter  & {\\em 4U 1636-536} & {\\em 4U 1735-444} \\\\\n           & {\\em V801 Ara}  & {\\em V926 Sco}  \\\\\n\\hline\n$P_{\\rm orb}$ (days)       & 0.15804693(16) & 0.19383351(32)         \\\\\n$T_0$ (HJD$-$2\\,452\\,000)  & 813.531 $\\pm$ 0.002    & 813.495 $\\pm$ 0.003 \\\\\n$\\gamma$ (km~s$^{-1}$)     & -34     $\\pm$ 5        & -140    $\\pm$ 3     \\\\\n$K_{\\rm em}$ (km~s$^{-1}$) & 277     $\\pm$ 22       & 226     $\\pm$ 22    \\\\\n$q$ ($M_2\/M_1$)            &      0.21-0.34         &       0.05-0.41     \\\\\n$K_1$ (km~s$^{-1}$)        & 90-113         &     ---    \\\\\n$K_2$ (km~s$^{-1}$)        & 360     $\\pm$ 74       & 298     $\\pm$ 83    \\\\\n$f(M)$ (M$_\\odot$)         & 0.76    $\\pm$ 0.47     & 0.53    $\\pm$ 0.44  \\\\\n$i$ (deg)                  & 36-60                  & 27-60               \\\\\n\\hline\n\\label{param}\n\\end{tabular}\n\\end{table}\n \n\\subsection{Constraints on the inclination}\n\nFurther constraints on the stellar masses requires a knowledge of the \nbinary inclination. While strict upper limits are set by the absence \nof X-ray eclipses, lower limits can be established by combining \n$M_1 \\le 3.1$ M$_{\\odot}$ (the maximum mass allowed for a stable \nneutron star, \\citealt{rho74}) with our mass function and $q$ restrictions. \nThis leads to $i=36^{\\circ}-74^{\\circ}$ and $i=27^{\\circ}-80^{\\circ}$ for \nV801 Ara and V926 Sco, respectively. In addition, the physical model of \n\\cite{frank87} indicates $i\\le60^{\\circ}$ due to the lack of X-ray dips. \n\nOn the other hand, we mentioned in Section 3 that the factor $\\sim$2 narrowness \nof the HeII $\\lambda$4686 profile in V926 Sco with respect to V801 Ara \nindicates a lower inclination. To test this hypothesis\nwe have produced Doppler corrected averages for V801 \nAra and V926 Sco by coadding all the spectra in two bins centered at orbital \nphases 0.0 and 0.5, when the visibility of the irradiated face of the donor\nis minimum and maximum, respectively. These spectra are presented in \nFig.~\\ref{fig11} and show a marked difference for the two binaries: the narrow \nCIII\/NIII lines become significantly enhanced around phase 0.5 (top spectra) \nfor V801 Ara, but not much difference is seen in V926 Sco between phase 0.5 and\nphase 0. This clearly supports a higher inclination angle in V801 Ara. \nFurthermore, the relative contibution of the NIII $\\lambda$4640 line with \nrespect to the broad base, as estimated through a multigaussian fit, is a \nfactor $\\sim 2$ larger for V801 Ara than for V926 Sco. This can be taken \nas the relative contributions of the heated donor and disc, and hence, \nstrongly suggests that V801 Ara is seen at a higher inclination angle than \nV926 Sco. This seems at odds with the fact that the optical lightcurves \nin V801 Ara and V926 Sco display similar amplitudes, $A\\simeq$ 0.2 mags  \n(\\citealt{vanamer87}, \\citealt{van90}). However, \\cite{dejong96} have shown \nthat lightcurve amplitudes are more sensitive to $\\alpha$ than $i$, which  \nmight simply imply a thicker disc in V926 Sco  and \nhence lightcurve amplitudes cannot be taken as a simple indication of the \ninclination angle. \n\n\\subsection{Mass Estimate and Evolutionary Scenarios} \n\n\\cite{meyer82} analyzed the\neffects of X-ray heating in the vertical structure of discs and demonstrated \nthat these are strongly correlated. Even in the absence of irradiation, they\nfind a disc opening angle of $\\sim6^{\\circ}$. De Jong et al. (1996) \nmodelled the effects of irradiation in optical lightcurves and, by\ncomparing with observations of LMXBS, they infer a mean disc opening angle of \n12$^{\\circ}$. Most of their systems are short period LMXBs, just like  \nV801 Ara and V926 Sco, and hence it seems justified to speculate what our  \nsystem parameters would be in this particular case. \n\nRegarding V801 Ara, $\\alpha=12^{\\circ}$ leads to  \n$q\\simeq0.25-0.30$, $K_2\\simeq356-375$ km s$^{-1}$ (see dashed line \nin Fig.~\\ref{fig9}) and hence $f(M)\\simeq 0.81 \\pm 0.07$ M$_{\\odot}$ and \n$M_1 \\sin^3 i \\simeq 1.32 \\pm 0.13$ M$_{\\odot}$. Asssuming that the donor star \nis a 0.48 M$_{\\odot}$ \nZAMS, then $M_1\\simeq1.6-1.9$ M$_{\\odot}$ and $i\\simeq 60-71^{\\circ}$. On the\nother hand, a canonical 1.4 M$_{\\odot}$ would imply an evolved \ndonor with $M_2\\simeq0.35-0.42$ M$_{\\odot}$ and $i\\ge71^{\\circ}$, which \nis difficult to reconcile with the absence of X-ray  eclipses, dips and the\nlow amplitude of the optical modulation (see Sect. 6.3). Therefore, a flaring \nangle of $\\alpha\\simeq12^{\\circ}$ seems to support scenarios with ZAMS donors \nand massive neutron stars. These masses can be reproduced by recent \nevolutionary scenarios where some LMXBs are suposed to descend from binaries\nwith high mass donors (\\citealt{schenker02}). These are significantly evolved \nat the begining of the mass transfer phase and mass is transferred in a thermal \ntimescale until mass ratio reverses, which can result in the accretion of \nseveral tenths solar masses by the neutron star. This would make V801 Ara \nsimilar to other LMXBs such as 4U 1822-371 (\\citealt{munoz05}). \n\nHowever, it is also possible to accommodate a \ncanonical neutron star with a MS donor and a plausible inclination angle by \npushing the disc flaring angle to higher values.   \nFor instance, $\\alpha=16^{\\circ}$ leads to $q\\simeq0.30-0.34$,   \n$f(M)\\simeq 0.51$ M$_{\\odot}$ and $M_1 \\sin^3 i \\simeq \n0.88$ M$_{\\odot}$. Then, for $M_1=1.4$ M$_{\\odot}$, \n$M_2\\simeq0.45$ M$_{\\odot}$ and $i\\simeq57^{\\circ}$. These masses would be \nconsistent with standard evolutionary pictures where donors in LMXBs descend\nfrom slightly evolved low-mass stars through dynamically stable mass transfer\nand result in slightly undermassive stars.  \n        \nNote also that $\\alpha<12^{\\circ}$ are virtually ruled out \nbecause this would imply higher $K_2$ and $f(M)$ values which result in massive\nneutron stars and very high inclinations. For instance, $\\alpha=8^{\\circ}$ \nleads to $f(M)\\simeq 0.96$ M$_{\\odot}$ and $M_1 \\sin^3 i \n\\simeq 1.5$ M$_{\\odot}$. Then, for $i\\leq76^{\\circ}$ (lack of X-ray \neclipses), $M_1\\ge1.7$ M$_{\\odot}$ \nwhereas a plaussible $i\\simeq57^{\\circ}$ yields $M_1\\simeq2.6$ M$_{\\odot}$.   \n\nThe case of V926 Sco is unfortunately unconstrained because of the lack \nof an X-ray mass function. For example, assuming $\\alpha=12^{\\circ}$ leads to \n$q\\simeq0.09--0.38$, $f(M)\\simeq 0.22--0.68$ M$_{\\odot}$ and $M_1 \\sin^3 i \n\\simeq 0.29--1.07$ M$_{\\odot}$. The allowed range in $M_1$ and $M_2$ is too\nwide as to impose any useful restriction on possible evoultionary scenarios.    \nMore, higher resolution data are required to measure the \nNIII $\\lambda$4640 flux and $V_{\\rm rot} \\sin i$ as a function of orbital \nphase, from which tighter constraints on the inclination and disc flaring\nangle can be set. This, \ntogether with the determination of the X-ray mass function for V926 Sco \nthrough pulse delays of burst oscillations and smaller errors in the \n$K_{\\rm em}$ determinations, is expected to provide stronger  \nlimits on the stellar masses and the evolutionary models for these two LMXBs \n(see e.g. \\citealt{munoz05}).\n   \n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=84mm]{fig11.eps}\n\\caption{Doppler corrected averages of V801 Ara (left panel) and V926 Sco \n(right panel) in the rest frame of the donor star. Top spectra present\naverages between orbital phases 0.25-0.75 and bottom spectra between \n0.75-0.25. The sharp CIII\/NIII components in V801 Ara are clearly more  \nintense around phase 0.5 than 0, suggesting a higher inclination\nbinary. Spectra have been smoothed by a 2 pixel boxcar.}\n\\label{fig11}\n\\end{figure}\n\n\\section{Conclusions} \n\nThe main results of the paper are summarized as follows:\n\n\\begin{itemize}\n\n\\item We have presented the first detection of the donor stars in the bursters \nLMXBs V801 Ara (=4U 1636-536) and V926 Sco (=4U 1735-444) through NIII\n$\\lambda$4640 fluorescent emission caused by irradiation. \n\n\\item The narrow NIII $\\lambda$4640 spots in the Doppler maps define \n$K_{\\rm em}=277\\pm22$ km s$^{-1}$ (V801 Ara) and $K_{\\rm em}=226\\pm22$ km\ns$^{-1}$ (V926 Sco), and a new set of {\\it spectroscopic} ephemerides which \nlend support to the assumption that photometric modulation is driven by the\nvisibility of the irradiated donor stars. On the other hand, the phasing of the\nradial velocity curves of HeII $\\lambda$4684 and the Bowen blend suggest\nthat these lines are mainly associated with the disc bulge. Our spectroscopic \nephemerides, combined with burst oscillations data for V801 Ara, enables us \nto refine the neutron star's projected velocity to $K_1=90-113$ km s$^{-1}$ \nfor this particular system.\n\n\\item Following \\cite{munoz05}, we have computed the $K-corrections$ for \nthe two LMXBs and obtain $K_2=360\\pm74$ km s$^{-1}$, $q=0.21-0.34$, \n$f(M)=0.76\\pm0.47$ M$_{\\odot}$ (V801 Ara) and $K_2=298\\pm83$ km s$^{-1}$, \n$q=0.05-0.41$, $f(M)=0.53\\pm0.44$ M$_{\\odot}$ (V926 Sco). \nBoth systems are seen at intermediate inclination angles in the range \n$i\\simeq30-60^{\\circ}$, with V801 Ara having the higher inclination of the two.\n\n\\item Regarding V801 Ara, disc flaring angles $\\alpha \\le 8^{\\circ}$ seem to be\nruled out because of the high inclinations implied. Opening angles \n$\\alpha\\simeq 12^{\\circ}$ support massive neutron stars and main-sequence \ndonors which may descend from intermediate-mass X-ray binaries as predicted by \nsome evolutionary models (\\citealt{schenker02}). Alternatively, higher opening \nangles $\\alpha\\simeq16^{\\circ}$ are consistent with canonical neutron stars and   \nmain-sequence (or slightly evolved) donors which have evolved from standard\nLMXBs through a dynamically stable mass tranfer phase. \n\n\\item The lack of an X-ray mass function in V926 Sco prevents to set tighter \nconstraints to the $K-correction$, mass estimates and evolutionary history in\nthis LMXB. \n\n\\end{itemize}\n\n \n  \n\n\n\\section*{Acknowledgments}\n\nWe thank the referee Thomas Augusteijn for helful comments to the manuscript. \nMOLLY and DOPPLER software developed by T.R. Marsh is gratefully \nacknowledged. JC acknowledges support from the Spanish Ministry of Science and\nTechnology through the project AYA2002-03570. DS acknowledges a Smithsonian \nAstrophysical Observatory Clay Fellowship as well as support through NASA GO \ngrants NNG04GG96G and NNG04G014G. Based on data collected \nat the European Southern Observatory, Monte Paranal, Chile.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTaking the flat-space limit (zero cosmological constant limit)\nof asymptotically AdS spacetimes results in asymptotically flat geometries.\nThis procedure can be done by taking\nthe $\\ell\\to\\infty$ limit where $\\ell$ is the radius of AdS spacetime.\nFrom the field theory perspective,\none could expect that the $\\ell\\to\\infty$ limit in the bulk theory\nhas a holographic description at the boundary.\nRecently, it has been argued that the flat-space limit of AdS gravity\nis dual to the $\\dot {\\rm I}$n$\\ddot {\\rm o}$n$\\ddot {\\rm u}$-Wigner\ncontraction of the boundary conformal field theory (CFT) \\cite{Bagchi:2010zz,Bagchi:2012cy}.\\footnote{It is worth noticing\nthat another interesting approach to understanding\nflat space quantum gravity is given in \\cite{Krishnan:2013wta}.\nTherein, the authors showed that interpreting the inverse AdS$_{3}$ radius $1\/\\ell$\nas a Grassmann variable leads to a map from gravity in AdS$_{3}$ to gravity in flat space.}\n\nThe so-called flat\/contracted conformal field theory (CCFT) correspondence has received a great deal of attention recently.\nFor example, in \\cite{Bagchi:2012xr} a Cardy-like formula has been obtained for the\ntwo-dimensional CCFT which yields the correct entropy of the three-dimensional cosmological solution.\nThese asymptotically flat spacetimes can be obtained by taking the flat-space limit,\nas in \\cite{Cornalba:2002fi}, of nonextremal Ba\\~{n}ados-Teitelboim-Zanellii (BTZ) black holes.\nAfter taking the flat-space limit, the outer horizon of BTZ is mapped to infinity;\nhowever, the value of the inner horizon remains finite and defines the cosmological horizon. \nThe entropy of the cosmological solution has been identified with the area of the cosmological horizon.\nIn the literature (see for example \\cite{Detournay:2012ug}), a modified Cardy formula has been introduced\nwhich reproduces the entropy of the inner horizon of BTZ black holes.\nThe CFT origin of this formula has not been well understood yet, but the observation of \\cite{Fareghbal:2014qga,Riegler:2014bia}\nis that if we accept the modified Cardy formula related to the inner horizon of the BTZ\nand contract it by using appropriate parameters of CCFT, the final result\nis exactly the CCFT Cardy-like formula which yields the entropy of the cosmological horizon.\n\nFurthermore, in \\cite{Fareghbal:2013ifa}, the authors found the correlation functions\nof CCFT energy-momentum tensor by using the contraction of CFT correlation functions\nfor finding the quasilocal stress tensor of the asymptotically\nflat spacetimes which gives the correct conserved charges of these geometries.\n\nThe Flat\/CCFT correspondence can also propose a dual field theory\nwhich lives at the horizon of nonextreme black holes.\nThe idea begins from the appearance of Rindler spacetime in the near horizon limit of\nnonextreme black holes.\nIf one starts with the Rindler-AdS\/CFT correspondence \\cite{Czech:2012be,Parikh:2012kg} and takes\nthe flat-space limit in the bulk, which results in the Rindler spacetime,\nthe boundary field theory is given by the contraction of the parent CFT.\nThis proposal has been used in \\cite{Fareghbal:2014oba}.\nTherein, the authors found the Bekenstein-Hawking entropy of the nonextreme BTZ black hole\nby counting the CCFT microsates.\n\nMoreover,   Bagchi {\\it et al.}\\,\ncalculated the entanglement entropy of a two-dimensional field theory with\nGalilean conformal symmetry\\footnote{The group of symmetries of CCFT$_{2}$ is isomorphic to Galilean Conformal Algebra\nbut in higher dimensions these are not the same.} recently \\cite{Bagchi:2014iea}.\nThe authors used the Wilson lines approach and found the holographic entanglement entropy\nthey computed is in precise agreement with the ones obtained in the field theory side. \nFor an almost complete list of papers related to the Flat\/CCFT correspondence, see the references of \\cite{Bagchi:2014iea}.\n\nIn (2+1)-dimensional Einstein gravity, black holes can exist only in the\npresence of a negative cosmological constant \\cite{Ida:2000jh}.\nIn order to find asymptotically flat black holes in three dimensions one has  to consider higher-derivative gravity theories.\nThe entropy of these black holes can be obtained by Wald\\rq{}s formula \\cite{Wald:1993nt}.\nA successful theory of quantum gravity should be able to give a microscopic description of this semiclassical entropy.\nAn alternative approach for this study is using holography.\nThe Falt\/CCFT correspondence as a duality between quantum gravity in the asymptotically flat backgrounds\nand a field theory with contracted conformal symmetry can be an appropriate context to study this problem.\nThe current paper is focused in this direction.\n\nWe consider a theory of gravity which is given by taking the flat-space limit of  new massive gravity (NMG) \\cite{Bergshoeff:2009hq}.\nThis theory possesses remarkable properties.\nIn \\cite{Deser:2009hb}, it was shown that it is a ghost-free and power-counting UV finite, three-dimensional gravity.\nWe use the dictionary of the Flat\/CCFT correspondence for finding quasi local stress tensor\nof the new type of asymptotically flat black hole \\cite{Oliva:2009ip}.\nUsing the holographic stress tensor along with the Brown and York\\rq{}s method \\cite{Brown:1992br},\nwe compute the conserved charges of this black hole.\nWe also take the limit from the Cardy formula and find a Cardy-like formula for the\nCCFT and show that this gives agreement with the Wald\\rq{}s semiclassical approach.\n\nThe next sections are devoted to two main parts.\nIn Sec.(\\ref{Bulk Solutions}) we introduce the bulk solutions.\nWe start from NMG and review its  asymptotically AdS rotating hairy black hole.\nThen we take the flat-space limit from the\naction and its black hole solution and introduce the asymptotically\nflat rotating hairy black hole with some novel properties.\nWe calculate its entropy using Wald\\rq{}s formula and verify for it the first law\nof black hole thermodynamics.\nIn Sec.(\\ref{Dual Boundary Theory}) we argue about the dual boundary theory of the bulk solution.\nWe shortly review the known results about the dual CFT of NMG\nand then try to contract these results and find a CCFT which is dual to the\nhigher-derivative gravity theory of \\cite{Deser:2009hb}.\nThis work is another check for the correctness of the Flat\/CCFT correspondence.\n\n\\section{Bulk Solutions}\n\\label{Bulk Solutions}\n\n\\subsection{Rotating hairy black hole of NMG}\nWe consider the three-dimensional higher-derivative gravity theory known as NMG.\nThis theory is described by the parity-invariant action \\cite{Bergshoeff:2009hq}\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{BHTaction}\nS=\\frac{1}{16\\pi G}\\int d^{3}x \\sqrt{-g}\\left[R-2\\Lambda+\\frac{1}{m^{2}}K\\right],\n\\ee\nwhere\n\\begin{equation}}   \\def\\ee{\\end{equation}\nK=R_{\\mu\\nu}R^{\\mu\\nu}-\\frac{3}{8}R^{2}.\n\\ee\n\nThe above theory (\\ref{BHTaction}) for the special case $m^{2}\\ell^{2}=1\/2$\nhas the following rotating black hole solution \\cite{Oliva:2009ip}\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{BHTbh}\nds^{2}=-NFdt^{2}+\\frac{dr^{2}}{F}+r^{2}\\left(d\\phi+N^{\\phi}dt\\right)^{2},\n\\ee\nwhere $N$, $N^{\\phi}$, and $F$ are  given by\n\\begin{align}\nN  &  =\\left[  1+\\frac{b\\ell^{2}}{4H}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  \\right]\n^{2}\\ ,\\nonumber\\\\\nN^{\\phi}  &  =-\\frac{a}{2r^{2}}\\left(  4GM-bH\\right)  \\ ,\\label{Ns&F}\\\\\nF  &  =\\frac{H^{2}}{r^{2}}\\left[  \\frac{H^{2}}{\\ell^{2}}+\\frac{b}{2}\\left(\n1+\\Xi^{\\frac{1}{2}}\\right)  H+\\frac{b^{2}\\ell^{2}}{16}\\left(  1-\\Xi^{\\frac{1}{2}\n}\\right)  ^{2}-4GM\\ \\Xi^{\\frac{1}{2}}\\right]  \\ ,\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\nonumber H&=\\left[  r^{2}-2GM\\ell^{2}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  -\\frac{b^{2}\\ell^{4}\n}{16}\\left(  1-\\Xi^{\\frac{1}{2}}\\right)  ^{2}\\right]  ^{\\frac{1}{2}},\n\\label{H}\\\\\n\\Xi&=1-a^{2}\/\\ell^{2}.\n\\end{align}\nIt is labeled by three parameters:\nthe mass $M$, the angular momentum $J=M a$, and an additional \\lq\\lq{}gravitational hair\\rq\\rq{} parameter $b$.\nThe rotation parameter $a$ is bounded according to $-\\ell\\leq a\\leq \\ell$.\n\nThe angular velocity of the horizon is\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Omega_{+}=\\frac{1}{a}\\left(  \\Xi^{\\frac{1}{2}}-1\\right)  \\ . \\label{Omega+}\n\\ee\nWe can associate a Hawking temperature and entropy to it\n\\begin{align}\nT&=\\frac{\\Xi^{\\frac{1}{2}} }{\\pi \\ell}\\sqrt{2G\\Delta M \\left( 1+\\Xi^{\\frac{1}{2}} \\right) ^{-1}},  \\label{Temperature}\\\\\nS&=\\pi \\ell\\sqrt{\\frac{2}{G}\\Delta M\n\\left( 1+\\Xi^{\\frac{1}{2}} \\right) },  \\label{Entropy}\n\\end{align}\nwhere\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta M=M+\\frac{b^{2}\\ell^{2}}{16G}\\ .\n\\ee\nThese quantities fulfill the relation\n\\begin{equation}}   \\def\\ee{\\end{equation}\nTdS=\\Xi^{\\frac{1}{2}} \\text{ }dM+\\frac{b\\ell^{2}}{8G}\\Xi^{\\frac{1}{2}} \n\\text{ }db-\\frac{1}{a}\\left( 1-\\Xi^{\\frac{1}{2}} \\right) \\Delta M \\text{ }da\\ .  \\label{calentate}\n\\ee\nNot much more work would bring this equation to the familiar form of the first law of black hole thermodynamics\n\\begin{equation}}   \\def\\ee{\\end{equation}\nd(\\Delta M)=TdS-\\Omega_{+}d(\\Delta J)\\, ,\n\\ee\nwhere we defined\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta J=a \\Delta M\\, .\n\\ee\n\n\\subsection{The flat-space limit of NMG and its black hole solution}\nIn order to have a well-defined flat-space limit ($\\Lambda\\to 0$ or $\\ell\\to\\infty$)\nfor \\eqref{BHTaction} in the special point $m^2\\ell^2=1\/2$, we also need to scale Newton\\rq{}s constant to infinity while keeping fixed $\\kappa=\\ell^{2}\/G$ .\nThus, the flat-space limit of NMG action \\eqref{BHTaction} becomes\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Flataction}\nS=\\frac{\\kappa}{8\\pi}\\int d^{3}x \\sqrt{-g} K.\n\\ee\nMoreover, a well-defined flat-space limit for the black hole solution \\eqref{BHTbh}\nneeds a scaling of the mass parameter $M$ such that $\\mu=M G$ remains fixed.\nThe final line element for the asymptotically flat rotating hairy black hole is given by \n \\begin{equation}}   \\def\\ee{\\end{equation}\\label{FlatBH}\nds^{2}=-{\\cal F}dt^{2}+\\frac{r^{2}}{{\\cal F}\\Delta}dr^{2}+a{\\cal F}dtd\\phi+r^{2}d\\phi^{2},\n\\ee\nwhere $\\Delta(r)$ and ${\\cal F}(r)$  are functions of the radial coordinate $r$, given by\n\\begin{align}\n\\Delta &  = r^{2}- \\mu a^{2}-\\left(\\frac{a^{2}b}{8}\\right)^{2} \\ ,\\nonumber\\\\\n{\\cal F} &  = b\\sqrt{\\Delta}-4 \\mu\\,.\n\\end{align}\nThe Ricci scalar of this asymptotically flat black hole can be written as\n\\begin{equation}}   \\def\\ee{\\end{equation}\nR=-\\frac{16\\, b}{a^2 b+8\\sqrt{\\Delta}}\\, .\n\\ee\nOne can verify that \\eqref{FlatBH} satisfies the\nequations of motion resulting from the action \\eqref{Flataction}.\nIt is worth noting that in \\cite{Deser:2009hb}, it was argued that the three-dimensional gravity\ntheory described by \\eqref{Flataction} is  ghost free and finite.\n\nHorizons of \\eqref{FlatBH} are at\n\\begin{equation}}   \\def\\ee{\\end{equation}\nr_{+}= \\frac{a^2 b}{8}+\\frac{4\\mu}{b}\\, ,\n\\qquad r_{-}=\\sqrt{\\left(\\frac{a^{2} b}{8}\\right)^{2}+\\mu a^{2}}\\, ,\n\\ee\nand one can calculate the entropy of the outer horizon using the Wald\\rq{}s formula.\nThis formula gives the black hole entropy in an arbitrary diffeomorphism invariant theory and is given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Wald}\nS=-\\frac{2\\pi}{16\\pi G}\\int_{\\Sigma_{h}}\\,\\frac{\\delta L}{\\delta R_{\\alpha\\beta\\gamma\\delta}}\\,\n\\epsilon_{\\alpha\\beta}\\,\\epsilon_{\\gamma\\delta}\\,\\bar{\\epsilon}\\ ,\n\\ee\nwhere $L$ is the Lagrangian, and $\\bar{\\epsilon}$, $\\epsilon_{\\mu\\nu}$,\ndenote the volume form and the binormal vector to the spacelike\nbifurcation surface $\\Sigma_{h}$, respectively.\n$\\epsilon_{\\mu\\nu}$ is normalized as $\\epsilon^{\\mu\\nu}\\epsilon_{\\mu\\nu}=-2$.\nFor the action \\eqref{Flataction} and the asymptotically flat rotating hairy black hole solution \\eqref{FlatBH} we obtain \n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial L}{\\partial R_{\\alpha\\beta\\gamma\\delta}}\n&=\\frac{3}{8}R\n\\left(g^{\\alpha\\delta}g^{\\beta\\gamma}-g^{\\alpha\\gamma}g^{\\beta\\delta}\\right)\n+\\frac{1}{2}\\left(g^{\\alpha\\gamma}R^{\\beta\\delta}\n-g^{\\alpha\\delta}R^{\\beta\\gamma}\n-g^{\\beta\\gamma}R^{\\alpha\\delta}\n+g^{\\beta\\delta}R^{\\alpha\\gamma}\\right),\\\\\n\\epsilon_{\\alpha\\beta}&=-\n\\left(\\frac{a^{2}{\\cal F}+4r^{2}}{\\Delta}\\right)^{\\frac{1}{2}}\n\\delta^{t}_{[\\alpha}\\delta^{r}_{\\beta]}\\, .\n\\end{align}\n\\end{subequations}\nTherefore, the Wald\\rq{}s entropy for the new type of asymptotically flat black hole becomes\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{flat entropy}\nS_{{\\rm flat}}=\\frac{\\pi\\kappa b}{2}\\, .\n\\ee\n\nIt is instructive to derive the above entropy by taking the flat-space limit of the entropy \\eqref{Entropy}.\nThe entropy can, therefore, be computed as follows:\n\\begin{align}\\label{FEntropy}\n\\lim_{\\ell\\to\\infty} S&=\\lim_{\\ell\\to\\infty}\\pi \\ell\\sqrt{\\frac{2}{G}\\Delta M\\left(  1+\\Xi^{\\frac{1}{2}}\\right)  }\\nonumber\\\\\n&=\\lim_{\\ell\\to\\infty}\\frac{\\pi\\kappa b}{2}\\sqrt{1+\\frac{16\\mu}{b^{2}\\ell^{2}}}\\nonumber\\\\\n&=\\frac{\\pi\\kappa b}{2}=S_{{\\rm flat}}\\, .\n\\end{align}\nNow, consider the Hawking temperature.\nFrom (\\ref{Temperature}) it follows that\n\\begin{align}\\label{FTemperature}\n\\lim_{\\ell\\to\\infty}T  &  =\\lim_{\\ell\\to\\infty}\\frac{1}{\\pi\\ell}\\Xi^{\\frac{1}{2}}\\sqrt{2G\\Delta M\\left(  1+\\Xi^{\\frac\n{1}{2}}\\right)  ^{-1}}\\ ,\\nonumber\\\\\n&=\\lim_{\\ell\\to\\infty}\\frac{b}{4\\pi}\\sqrt{1+\\frac{16\\mu}{b^{2}\\ell^{2}}}\\nonumber\\\\\n&=\\frac{b}{4\\pi}=T_{{\\rm flat}}\\, .\n\\end{align}\nThese quantities fulfill the relation\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Fcalentate}\nT_{\\text{flat}}\\,dS_{\\text{flat}}=\\frac{b \\kappa}{8}\\,db\\, .\n\\ee\nThis agrees precisely with the $\\ell\\to\\infty$ limit of (\\ref{calentate}).\nA direct calculation by using \\eqref{FlatBH} or taking the flat-space limit of \\eqref{Omega+}\nshows that the angular velocity of the black hole \\eqref{FlatBH} at the outer horizon\nis zero ($\\Omega_{{+}_{\\rm flat}}=0$) though it has a nonvanishing angular momentum.\n\nFrom \\eqref{flat entropy} and \\eqref{FTemperature}, it is clear that the hair parameter $b$\ndetermines the entropy and the temperature of the outer horizon.\nIn the $b\\to 0$ limit the hairy black hole \\eqref{FlatBH} is reduced to the cosmological solution of \\cite{Cornalba:2002fi}.\nIn this limit, $r_{+}$ is mapped to infinity; however, $r_{-}$ remains finite and defines the cosmological horizon. \n\n\\section{Dual Boundary Theory}\n\\label{Dual Boundary Theory}\n\n\\subsection{CFT dual to NMG}\nIn \\cite{Bergshoeff:2009aq, Giribet:2009qz}, it was proposed that NMG has a dual description in terms of a CFT.\\footnote{In 2013, de Buyl {\\it et al.}\nconsidered the asymptotically dS case, $\\Lambda=+1\/\\ell^{2}>0$,\nwithin the context of dS\/CFT correspondence \\cite{deBuyl:2013ega}.\nWe would like to thank the referee for bringing this paper to our attention.}\nThe charges associated to the asymptotic symmetries enhance the isometry of asymptotically $AdS_3$ spacetimes\nto two copies of the Virasoro algebra.\nThe central charges are given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\nc_{\\pm}=c=\\frac{3\\ell}{2G}\\left(1+\\frac{1}{2m^{2}\\ell^{2}}\\right)\\ .\\label{central charge}\n\\ee\nAt the spacial point  $m^{2}\\ell^{2}=1\/2$, the central charges are twice\nthe values proposed by Brown and  Henneaux for the Einstein gravity\nwith negative cosmological constant \\cite{Brown:1986nw}, i.e.,\n\\begin{equation}}   \\def\\ee{\\end{equation}\nc=\\frac{3\\ell}{G}\\ .\\label{central charge}\n\\ee\nThe entropy of the black hole \\eqref{BHTbh} can be given by the Cardy formula\n\\begin{equation}}   \\def\\ee{\\end{equation}\nS=2\\pi\\sqrt{\\frac{c_{+}\\Delta_{+}}{6}}+2\\pi\\sqrt{\\frac{c_{-}\\Delta_{-}}{6}\n}\\ ,\\label{Cardy formula}\n\\ee\nwhere $\\Delta_\\pm$ are the eigenvalues of the left and right Virasoro generators $L_{0}^{\\pm}$ and are given by\n\\begin{equation}}   \\def\\ee{\\end{equation}\n\\Delta_{\\pm}=\\frac{1}\n{2}\\Delta M\\left(  \\ell\\pm a\\right)  \\ .\\label{Delta+-}\n\\ee\nUsing (\\ref{central charge}) and (\\ref{Delta+-}), this is\n\\begin{equation}}   \\def\\ee{\\end{equation}\nS =\\pi \\ell\\sqrt{\\frac{2}{G}\\left(  1+\\Xi^{\\frac{1}{2}}\\right)  \\Delta M}\\ ,\n\\ee\nin precise agreement with  (\\ref{Entropy}).\n\n\\subsection{CCFT dual to the flat-space limit of NMG}\nIn this section we want to propose a dual description for the theory of gravity given by \\eqref{Flataction}.\nTo do so, we will use the idea which was first proposed in papers \\cite{Bagchi:2010zz,Bagchi:2012cy}.\nThat is, if we start from the AdS\/CFT correspondence,\nthe large AdS radius limit in the bulk is equivalent to a contraction\nof spacetime coordinates in the boundary CFT.\\footnote{We refer to \\cite{Bagchi:2012cy} for\na full-scale investigation into the CCFT representation.}\n\nWe shall first show how one obtains the appropriate coordinate which must be contracted in the parent CFT.\nLet us look at the conformal boundary of the black hole \\eqref{BHTbh} for an arbitrary large $\\ell$.\nIt could be written as follows:\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{CB}\nds_{{\\rm C.B.}}^{2}=\\frac{r^{2}}{\\kappa^{2}}\\left(-\\frac{\\kappa^{2}}{\\ell^{2}}dt^{2}+\\kappa^{2}d\\phi^{2}\\right)\\, .\n\\ee\nWe have  used $\\kappa$ in the conformal factor to make it dimensionless.\nMoreover, the fact that $\\kappa$ is fixed in our flat-space limit makes\nthe conformal factor well defined for all large values of $\\ell$.\nNow, $\\ell$ can be absorbed by defining new time coordinate as $\\tau=\\kappa t\/\\ell$.\nThe dual CFT lives on a cylinder with coordinates $(\\tau, \\phi)$ and radius $\\kappa$.\nTaking the $\\ell\\to\\infty$ limit (or $\\kappa\/\\ell\\to 0$ limit),\nit is obvious that the flat-space limit in the bulk induces a contraction in\n$t$ of the boundary CFT reducing it to the two-dimensional CCFT.\n\n\\subsubsection{Symmetries of CCFT}\nAccording to the proposal of \\cite{Bagchi:2010zz,Bagchi:2012cy}, the symmetries of CCFT\nrealize the group of  asymptotic symmetries of the asymptotically flat  spacetimes at null infinity,\nnamely the Bondi-Metzner-Sachs (BMS) group \\cite{BMS,Barnich aspects}.\nThere is a very precise procedure, called the $\\dot {\\rm I}$n$\\ddot {\\rm o}$n$\\ddot {\\rm u}$-Wigner contraction, by which one\ncan obtain the CCFT algebra from the relativistic conformal algebra of the parent CFT.\nLet us consider two copies of the Virasoro algebra\\footnote{The relativistic conformal algebra\nconsists of two copies of the Virasoro algebra.}\n\\begin{align}\\label{Virasoro}\n  \\nonumber [L^+_m,L^+_n]&=(m-n)L^+_{n+m}+{c^+\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\\n  \\nonumber [L^-_m,L^-_n]&=(m-n)L^-_{n+m}+{c^-\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\ \n            [L^+_m,L^-_n]&=0\\, .\n  \\end{align}  \nFor a small parameter $\\epsilon$, at the level of the algebra, if we define\\footnote{It is clear from (3.8)\nthat other positive powers of $\\epsilon$ are meaningless\nsince one can redefine $\\epsilon$ and rewrite them finally as in (3.8).\nThis definition is consistent with the observation of \\cite{bar-comp}.\nThat is, one can obtain the BMS$_{3}$ algebra by taking the flat-space limit\nfrom the asymptotic symmetry algebra of three-dimensional asymptotically AdS spacetimes.}\n\\begin{align}\\label{def of BMS generators}\n\\nonumber L_n &=L^+_n-L^-_{(-n)}\\, ,\\\\\n          M_n &=\\epsilon\\left(L^+_n+L^-_{(-n)}\\right), \n\\end{align}\nwe can see that the CCFT algebra is generated from the Virasoro algebras, on taking the $\\epsilon\\to0$ limit, i.e.,\n\\begin{align}\\label{BMS}\n  \\nonumber [L_m,L_n]&=(m-n)L_{n+m}+{c_L\\over 12}m(m^2-1)\\delta_{m+n,0}\\,,\\\\\n  \\nonumber [L_m,M_n]&=(m-n)M_{n+m}+{c_M\\over 12}m(m^2-1)\\delta_{m+n,0}\\, ,\\\\ \n            [M_m,M_n]&=0\\, ,    \n  \\end{align}\nwhere the central charges $c_L$ and $c_M$ are given by the linear combination of the parent relativistic central charges\n\\begin{equation}\\label{def of cC of CCFT}\nc_L=\\lim_{\\epsilon\\to 0 }(c^+-c^-)\\, ,\\qquad c_M=\\lim_{\\epsilon\\to 0 }\\epsilon(c^++c^-)\\, .\n\\end{equation}\nThe algebra \\eqref{BMS} which is given by the contraction of the Virasoro algebra in the boundary theory is exactly\nthe (centrally extended) BMS$_{3}$ algebra \\cite{bar-comp}.\n\nWe would expect the same symmetry group for the CCFT dual to\nthe theory described by the action \\eqref{Flataction} at the special point $m^2\\ell^2=1\/2$.\nAs we stated earlier, the $\\epsilon\\to 0$ limit in the boundary corresponds to the flat-space limit\nor, more precisely, the $\\kappa\/\\ell\\to 0$ limit in the bulk side.\nUsing \\eqref{central charge} and \\eqref{def of cC of CCFT}, for the problem in hand we find\n\\begin{equation}}   \\def\\ee{\\end{equation}\n c_L=c_M=0\\, .\n \\ee\n\nWe will show that although the CCFT algebra has vanishing central charges,\nit is possible to find a Cardy-like formula for the asymptotic growth of the number of states\nwhich reproduces the entropy of the black hole \\eqref{FlatBH}.\nTo add strength to this claim, let us find more evidence about\nthe correctness of our proposal using the CCFT energy-momentum tensor.\n\n\\subsubsection{Quasi local stress tensor}\nThe one-point function of the CCFT energy-momentum operator\ncorresponds to the quasilocal stress tensor of the bulk theory.\nIt was argued in \\cite{Fareghbal:2013ifa} that the definition \\eqref{def of BMS generators} provides a recipe\nto calculate the components of the stress tensor in the asymptotically flat spacetimes.\nTherefore, we can write\n\\begin{eqnarray}\\label{def of energy-momentum }\n \\nonumber \\tilde T_{++}+\\tilde T_{--}&=&\\lim_{\\epsilon\\to 0}\\epsilon\\left(T_{++}+T_{--}\\right)\\, ,\\\\\n  \\nonumber\\tilde T_{++}-\\tilde T_{--}&=&\\lim_{\\epsilon\\to 0}\\left(T_{++}-T_{--}\\right)\\, ,\\\\\n  \\tilde T_{+-}&=&\\lim_{\\epsilon\\to 0} T_{+-}\\, ,\n  \\end{eqnarray}\nwhere $T_{ij}$ and $\\tilde T_{ij}$ are, respectively, the stress tensor of the asymptotically AdS and flat spacetimes\nand $x^{\\pm}$ are the light-cone coordinates constructed from the nonradial coordinates of the metrics.\nIn the above definition it was assumed that both the asymptotically AdS\nand flat spacetimes are given in the BMS gauge \\cite{Fareghbal:2013ifa}. \n\nThe nonzero components of the stress tensor at the boundary\nof the asymptotically AdS black hole \\eqref{BHTbh} are given by \\cite{Kwon:2011jz} \n\\begin{align}\n\\nonumber T_{tt}&=\\frac{1}{8\\pi G \\ell }\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right), \\\\\n\\nonumber T_{t\\phi}&=-\\frac{a}{8\\pi G \\ell}\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right), \\\\\n          T_{\\phi\\phi}&=\\frac{\\ell}{8\\pi G }\\left(\\frac{b^{2}\\ell^{2}}{4}+4 M G\\right).\n\\end{align}\nThe formula \\eqref{def of energy-momentum } results in a stress tensor $\\tilde T_{ij}$\nfor the asymptotically flat black hole \\eqref{FlatBH} as follows:\n\\begin{equation}\\label{flat stress tensor}\n\\tilde T_{tt}={b^2\\over 32\\pi}\\, ,\\qquad \\tilde T_{\\phi\\phi}={\\kappa^2 b^2\\over 32 \\pi}\\, , \\qquad \\tilde T_{t\\phi}=-{a b^2\\over 32\\pi}\\, .\n\\end{equation}\nUsing $\\tilde T_{ij}$ we can calculate the conserved charges of the black hole \\eqref{FlatBH}.\n\n Let us denote the hypersurface of the spacetime where CCFT lives with $\\partial {\\cal M}$.\nIts line element is given by taking the $\\ell\\to \\infty$ limit of the conformal boundary \\eqref{CB},\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{bundary of flat}\nds_{\\partial {\\cal M}}^{2}=\\frac{r^{2}}{\\kappa^{2}}\\left(-dt^{2}+\\kappa^{2}d\\phi^{2}\\right).\n\\ee\nFollowing Brown and York\\rq{}s method \\cite{Brown:1992br},\nthe charges associated to a boundary Killing vector $\\xi^\\mu$ are given by\n\\begin{equation}\\label{BY formula}\nQ_{\\xi}=\\int_{\\Sigma} d\\phi \\sqrt{\\sigma}\\xi^\\mu n^\\nu \\tilde T_{\\mu\\nu}\\, ,\n\\end{equation}\nwhere $\\Sigma$ is the spacelike surface embedded in $\\partial {\\cal M}$ with induced metric $\\sigma_{\\mu\\nu}$.\nMoreover, $n^{\\mu}$ is the timelike unit normal to $\\Sigma$.\nUsing \\eqref{bundary of flat} and \\eqref{BY formula},\nthe mass and the angular momentum of the asymptotically flat black hole \\eqref{FlatBH} are \n\\begin{equation}\\label{charges}\n{\\cal M}=Q_{\\partial_t}={\\kappa b^2\\over 16}\\, ,\\qquad {\\cal J}=Q_{\\partial_\\phi}=-{\\kappa a b^2\\over 16}\\, .\n\\end{equation}\nIt is clear that $|{\\cal J}|\/{\\cal M}=a$ as expected.\n\nGiven the expressions above, together with \\eqref{Fcalentate},\nit is straightforward to check that the first law of black hole thermodynamics is satisfied, i.e.\n\\begin{equation}}   \\def\\ee{\\end{equation}\nd{\\cal M}=T_{\\rm flat}\\,dS_{\\rm flat}-\\Omega_{{+}_{\\rm flat}}\\,d{\\cal J}\\, .\n\\ee\n\n\n\\subsubsection{Cardy-like formula}\\label{entropy}\nIf the gravity theory \\eqref{Flataction} has a dual description in terms of a CCFT,\nthen the entropy of the black hole \\eqref{FlatBH} must be given by\nthe asymptotic growth of the number of states in the boundary theory.\nIn \\cite{Bagchi:2012xr}, the authors found a Cardy-like formula\nby computing the CCFT partition function using the saddle-point approximation.\nHowever, in the recent papers \\cite{Fareghbal:2014qga,Riegler:2014bia}\nit was shown that the Cardy-like formula of \\cite{Bagchi:2012xr} can be obtained\nif one writes the Cardy formula in terms of CCFT parameters and then takes the $\\epsilon\\to 0$ limit.\nIn the current work we will use the same approach and take the $\\epsilon\\to 0$ limit from the Cardy formula in the parent CFT.\n  \nThe CCFT algebra is given by \\eqref{BMS}.\nWe denote the eigenvalues of $L_{0}$ and $M_{0}$ by $\\Delta_L$ and $\\Delta_M$, respectively.\nFor the current problem $c_{L}=c_{M}=0$, however, the eigenvalues of $L_{0}$ and $M_{0}$ are nonzero.\nFrom the viewpoint of the limit \\eqref{def of BMS generators} we see that the two labels $\\Delta_L$ and $\\Delta_M$\nare related to the conformal weights in the two-dimensional CFT as\n\\footnote{\nThe Hilbert space construction of a CCFT is analogous to that of the relativistic 2D CFT.\nNow, the states are labeled by the eigenvalues under $L_{0}$ and $M_{0}$.\nWe shall use the cylinder representation of the CCFT algebra \\cite{Bagchi:2012cy}.\nThe Hilbert space of the 2D CCFT are constructed by considering the states having definite scaling dimensions.\nWe define primary states by demanding that the states in the theory be annihilated by all generators with $n>0$.\nOne can build up a tower of operators by acting on a primary state with the creation operators $L_{-n}$ and $M_{-n}$ ($n>0$).\n}\n\\begin{align}\\label{def of D_LM}\n\\Delta_{L}&=\\lim_{\\epsilon\\to 0}\\left(\\Delta_{+}-\\Delta_{-}\\right)\\, ,\n\\qquad \\Delta_{M}=\\lim_{\\epsilon\\to 0}\\epsilon\\left(\\Delta_{+}+\\Delta_{-}\\right)\\, .\n\\end{align}\nWe would like to remind the reader that the $\\epsilon\\to 0$ limit in the boundary field theory\ncorresponds to the $\\kappa\/\\ell\\to 0$ limit in the bulk.\nTherefore, using \\eqref{Delta+-} and \\eqref{def of D_LM} one can easily find\n\\begin{equation}\\label{CCFT quantities}\n\\Delta_{L}=\\frac{a\\kappa b^{2}}{16}\\, ,\\qquad \\Delta_{M}=\\frac{\\kappa^2 b^{2}}{16}\\, .\n\\end{equation}\nLet us consider the Cardy formula \\eqref{Cardy formula} and try to take its $\\epsilon\\to 0$  limit.\nFor our current problem the relativistic central charges are $c_+=c_-=3\\epsilon$.\nUsing \\eqref{CCFT quantities}, we obtain\n\\begin{align}\\label{CCFT cardy}\n\\nonumber \\lim_{\\epsilon\\to 0} {S_{\\text{CFT}}}&= \\lim_{\\epsilon\\to 0} 2\\pi\\left(\\sqrt{c_+\\Delta_+\\over 6}+\\sqrt{c_-\\Delta_-\\over 6}\\right)\\\\\n\\nonumber &=\\lim_{\\epsilon\\to 0} \\pi\\left[\\sqrt{{\\epsilon}\\left({\\Delta_M\\over\\epsilon}+\\Delta_L\\right)}+\\sqrt{{\\epsilon}\\left({\\Delta_M\\over\\epsilon}-\\Delta_L\\right)}\\right]\\\\\n&=2\\pi\\sqrt{\\Delta_M}=S_{\\text{CCFT}}\\, .\n\\end{align}\nThis is  the Cardy-like formula for the CCFT dual to the flat-space limit of NMG.\nInserting (\\ref{CCFT quantities}) into (\\ref{CCFT cardy}), we finally recover the entropy (\\ref{flat entropy}),\n\\begin{equation}}   \\def\\ee{\\end{equation}\\label{Flat\/CCFT}\nS_{\\text{CCFT}}=S_{\\text{flat}}\\, ,\n\\ee\nas we wanted to show.\nIt is a quite nontrivial result since the theory has vanishing central charges.\n\n\\section{Conclusions}\n\nIn this paper, we have proposed a flat space generalization of the AdS$_3$\/CFT$_{2}$ holographic correspondence.\\footnote{We note that in \\cite{Hasanpour:2011ji},\nthe authors considered an asymptotically flat geometry which is a solution to\nthree-dimensional Einstein gravity conformally coupled to a scalar field\nand discussed gravity\/CFT correspondence for this background.}\nWe have provided the first example of a holographic dual of an asymptotically flat black hole solution.\nDue to the absence of black hole solutions in three-dimensional Einstein gravity with vanishing cosmological constant,\nwe have considered higher-derivative gravity theories which admit asymptotically flat black hole solutions.\nThe theory we have investigated is given by taking the flat-space limit ($\\Lambda\\to 0)$ of NMG.\nWe argued that the dual field theory of the black hole solution of this theory is a CCFT.\nFor this purpose, we have constructed a stress tensor for the\nasymptotically flat black hole solution and computed the conserved charges.\nWe then verified it using the first principle of black hole thermodynamics.\nFurthermore, we have used the Flat\/CCFT correspondence to find the black hole entropy\nin terms of the asymptotic growth of the number of CCFT states.\n\nIt is interesting to note that the symmetry algebra of the corresponding CCFT had vanishing central charges though\nthe asymptotic growth of  states were nonzero.\nThis remarkable point can be used for finding holographic duals of four-dimensional asymptotically flat spacetimes.\nAccording to the proposal of Flat\/CCFT correspondence, the dual of four-dimensional asymptotically flat black holes\nare field theories with BMS$_{4}$ symmetry  \\cite{Barnich:2009se, Barnich aspects}.\nIn \\cite{Barnich:2011mi}, the authors constructed the field-dependent central extension of BMS$_{4}$ algebra\nand found that for the Kerr black hole some of the charges involved divergent integrals on the 2-sphere\nif they used extended BMS algebra with both supertranslations and superrotations.\nThus, at first sight, it seems that counting CCFT$_{3}$ states would be a problematic issue, but our current work\nshows that counting the asymptotic growth of CCFT states can be done whatever the central charges are.\n\nAlthough our current study gives a holographic description of asymptotically flat black holes in three-dimensional higher-derivative gravity,\nwe believe that the Flat\/CCFT correspondence can be extended to find a holographic description of\nblack holes in higher dimensions and, specifically, the four-dimensional Kerr black hole.\nWe hope to explore other intriguing aspects of the relation between asymptotically flat spacetimes and CCFTs in our future works.\n\n\\section*{Acknowledgments}\nWe would like to thank Ali Naseh for useful discussions.\nWe are grateful to \\'{A}lvaro V\\'{e}liz-Osorio and Arjun Bagchi for their useful\ncomments on the revised version of the manuscript.\nS.M.H. is supported in part by INFN.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThis paper is the first part of our project to integrate\nrepresentations up to homotopy of Lie algebras (algebroids). Our\noriginal motivation is to integrate the standard Courant algebroid\n$TM\\oplus T^*M$ since it is this Courant algebroid that is much used\nin Hitchin and Gualtieri's program of generalized complex geometry.\nCourant algebroids are Lie 2-algebroids in the sense of Roytenberg\nand \\v{S}evera \\cite{royt, s:funny}. The general procedure to\nintegrate Lie $n$-algebras (algebroids) is already described in\n\\cite{getzler, henriques, s:funny}. We want to pursue some explicit formulas\nfor the special case of the  standard Courant algebroid. It turns\nout that the sections of the Courant algebroid $TM \\oplus T^*M$ form\na semidirect product of a Lie algebra with a representation up to\nhomotopy.   Abad and Crainic \\cite{abad-crainic:rep-homotopy}\nrecently studied the representations up to homotopy of Lie algebras,\nLie groups, and even Lie algebroids, Lie groupoids, in general. Just\nas one can form the semidirect product of a Lie algebra with a\nrepresentation, one can form the semidirect product with\nrepresentations up to homotopy too. In our case, the semidirect\nproduct coming from the standard Courant algebra is a Lie 2-algebra.\nBut using the fact that it is also a semidirect product, the\nintegration becomes easier. The integration result is related to the\nsemidirect product of Lie groups with its representation up to\nhomotopy, which will be discussed in Section \\ref{sec:gp}. However\nit turns out that the concept of representation up to homotopy of\nLie groups of Abad and Crainic will not be general enough to cover\nall the integration results. This we will continue in a forthcoming\npaper \\cite{sheng-zhu:II}.\n\nIn this paper we focus on exhibiting more examples of representation\nup to homotopy and  their semidirect products to demonstrate the\nimportance of our integration procedure. The examples are all sorts\nof variations of Courant algebroids. One is Chen and Liu's omni-Lie\nalgebroids, which generalizes Weinstein's omni-Lie algebras. Hence\nwe expect to give an integration to Weinstein's omni-Lie algebras\nvia Lie 2-algebras in the next paper \\cite{sheng-zhu:II}.\n\n\nAnother example comes from the so called string Lie 2-algebra. It is\nessentially  a Courant algebroid over a point (see Section\n\\ref{sec:string}),  namely  a Lie algebra with an adjoint-invariant\ninner product. This sort of Lie algebra is usually called a {\\em\nquadratic Lie algebra}. This concept also appears in the context of\nManin triples and double Lie algebras. The example  $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g\n\\oplus \\g^*$ that we consider in this paper is an analogue of the\nstandard Courant algebroid, and is basically a special case taken\nfrom \\cite{lu-weinstein} \\footnote{private conversation\n  with Jiang-Hua Lu}. We give an integration of\nthe string Lie 2-algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g \\oplus \\g^*$  at the end.\n\nUsually people require the base Lie algebra of a string Lie algebra\nto be semisimple and of compact type (see Remark\n\\ref{rk:semisimple}). For such usual sort of string Lie 2-algebras,\nBaez et al \\cite{baez:2gp} have proved a no-go theorem, namely such\nstring Lie 2-algebras can not be integrated to finite dimensional\nsemi-strict Lie 2-groups. Here a {\\em semi-strict Lie 2-group} is a\ngroup object in $\\rm DiffCat$, where  $\\rm DiffCat$ is the\n2-category consisting of categories, functors, and natural\ntransformations in the category of differential manifolds, or\nequivalently $\\rm DiffCat$ is a 2-category with Lie groupoids as\nobjects, strict morphisms of Lie groupoids as morphisms, and\n2-morphisms of Lie groupoids as 2-morphisms.  Our semi-strict Lie\n2-group is actually called a Lie 2-group by the authors in\n\\cite{baez:2gp}. However, we call it a semi-strict Lie 2-group\nbecause  compared to the Lie 2-group in the sense of Henriques\n\\cite{henriques}, or equivalently the stacky group in the sense of\nBlohmann \\cite{blohmann},  it is stricter. Basically,  their Lie\n2-group is a group object in the 2-category with objects as Lie\ngroupoids, morphisms as Hilsum-Skandalas bimodules (or generalized\nmorphisms), 2-morphisms as 2-morphisms of Lie groupoids.\nSchommer-Pries realizes the string 2-group as such a Lie 2-group\nwith a finite dimensional model \\cite{schommer:string-finite-dim}\nand the integration of a string Lie 2-algebra to such a model is a\nwork in progress \\cite{ccc:integration-string}.\n\nIt is not needed in the\ndefinition of the string Lie 2-algebra for the base Lie algebra to be\nsemisimple of compact type. One only needs a quadratic Lie algebra.\nAs soon as we relax this condition on compactness, we find out that\none can integrate  $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z \\to \\g \\oplus \\g^*$ to a finite dimensional\nsemi-strict Lie\n2-group in the sense of Baez et al. The integrating object is actually\na special Lie 2-group (very close to a strict Lie 2-group) in the\nsense of Baez et al.\n\nThen of course, as we relax the\ncondition, we are in danger that the class corresponding to this\nLie 2-algebra in $H^3(\\g\\oplus \\g^*, \\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$ might be trivial, and\nconsequently our Lie 2-algebra might be strict. Then what\nwe have done\nwould not have been a big surprise  because  a strict\nLie 2-algebra corresponds to a crossed module of Lie algebras, and it\neasily integrates to a strict Lie 2-group by integrating the crossed module.  However,\nwe verified that when $\\g$ itself (not $\\g \\oplus \\g^*$) is semisimple, this Lie 2-algebra is not strict.\n\n\n\n{\\bf Acknowledgement:} We give warmest thanks to  Zhang-Ju Liu,\nJiang-Hua Lu, Giorgio Trentinagia and Marco Zambon for\nuseful comments and discussion. Y. Sheng gives his warmest thanks to Courant Research Centre ``Higher Order Structures'', G\\\"{o}ttingen University, where this work was\ndone when he visited there.\n\n\\section{Representations up to homotopy of Lie algebras}\nIn this section, we first consider the 2-term representation up to\nhomotopy of Lie algebras. We give explicit formulas of the\ncorresponding 2-term $L_\\infty$-algebra, which is their semidirect\nproduct. Then we give several interesting examples including Courant\nalgebroids and omni-Lie algebroids.\n\n\\subsection{Representation up to homotopy of Lie algebras and their semidirect products}\n\n$L_\\infty$-algebras,  sometimes  called strongly homotopy Lie\nalgebras,  were introduced by Drinfeld and Stasheff\n\\cite{stasheff:shla} as a model for ``Lie algebras that satisfy\nJacobi up to all higher homotopies''. The following convention of\n$L_\\infty$-algebras has the same grading as in \\cite{henriques} and\n\\cite{rw}.\n\n\\begin{defi}\nAn $L_\\infty$-algebra is a graded  vector space $L=L_0\\oplus\nL_1\\oplus\\cdots$ equipped with a system $\\{l_k|~1\\leq k<\\infty\\}$ of\nlinear maps $l_k:\\wedge^kL\\longrightarrow L$ with degree\n$\\deg(l_k)=k-2$, where the exterior powers are interpreted in the\ngraded sense and the following relation with Koszul sign ``Ksgn'' is\nsatisfied for all $n\\geq0$:\n\\begin{equation}\n\\sum_{i+j=n+1}(-1)^{i(j-1)}\\sum_{\\sigma}\\mathrm{sgn}(\\sigma)\\mathrm{Ksgn}(\\sigma)l_j(l_i(x_{\\sigma(1)},\\cdots,x_{\\sigma(i)}),x_{\\sigma(i+1)},\\cdots,x_{\\sigma(n)})=0,\n\\end{equation}\nwhere the summation is taken over all $(i,n-i)$-unshuffles with\n$i\\geq1$.\n\\end{defi}\n\n\nLet $n=1$, we have\n$$\nl_1^2=0,\\quad l_1:L_{i+1}\\longrightarrow L_i,\n$$\nwhich means that $L$ is a complex and we usually write $\\mathrm{d}=l_1$.\nLet $n=2$, we have\n$$\n\\mathrm{d} l_2(x,y)=l_2(\\mathrm{d} x,y)+(-1)^pl_2(x,\\mathrm{d} y),\\quad \\forall~x\\in L_p,\ny\\in L_q,\n$$\nwhich means that $\\mathrm{d} $ is a derivation with respect to $l_2$. We\nusually view $l_2$ as a bracket $[\\cdot,\\cdot]$. However, it is not\na Lie bracket, the obstruction of Jacobi identity is controlled by\n$l_3$: \\begin{eqnarray} \\nonumber\n&&l_2(l_2(x,y),z)+(-1)^{(p+q)r}l_2(l_2(y,z),x)+(-1)^{qr+1}l_2(l_2(x,z),y)\\\\\n&&=-\\mathrm{d} l_3(x,y,z)-l_3(\\mathrm{d} x,y,z)+(-1)^{pq}l_3(\\mathrm{d}\ny,x,z)-(-1)^{(p+q)r}l_3(\\mathrm{d} z, x,y),\n\\end{eqnarray}\nwhere $x\\in L_p,~ y\\in L_q,~z\\in L_q$ and $l_3$ also satisfies\nhigher coherence laws.\n\nIn particular, if the $k$-ary brackets are zero for all $k>2$, we\nrecover the usual notion of {\\bf differential graded Lie algebras}\n(DGLA). If $L$ is concentrated in degrees $<n$, $L$ is called an\n {\\bf $n$-term $L_\\infty$-algebra}.\n\nIn this paper, we mainly consider\n 2-term $L_\\infty$-algebras, which are equivalent  to Lie 2-algebras\n given in \\cite{baez:2algebras} by John Baez and Alissa Crans.   In\nthis special case, $l_4$ is always zero. Thus restricting the coherence\nlaw satisfied by $l_3$ on degree-0, we obtain\n\\begin{equation}\nl_3(l_2(x,y),z,w)+c.p.-\\big(l_2(l_3(x,y,z),w)+c.p.\\big)=0,\\quad\\forall~x,y,z,w\\in\nL_0.\n\\end{equation} Lie 2-algebras are categorified\n version of Lie algebras. In a Lie 2-algebra, the Jacobi identity\n is replaced by an isomorphism which is called the {\\bf\n Jacobiator}. The Jacobiator  satisfies a certain law of its own.  Given a 2-term\n $L_\\infty$-algebra $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$, the\n underlying 2-vector space of the corresponding Lie 2-algebra is\nmade up by $L_0$  as the vector space of objects and $L_0\\oplus\n L_1$ as the vector space of morphism. Please see \\cite[Theorem\n 4.3.6]{baez:2algebras} for more details.\n\n\nRecall from \\cite{baez:2algebras} that a Lie 2-algebra is skeletal\nif isomorphic objects are equal. Viewing a Lie 2-algebra as a 2-term\n$L_\\infty$-algebra, explicitly we have\n \\begin{defi}\\label{defi:skeletal L 2}\nA 2-term $L_\\infty$-algebra $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$\nis called {\\bf skeletal} if $\\mathrm{d}=0$.\n \\end{defi}\n\nThey also prove the following theorem\n\\begin{thm}{\\rm\\cite{baez:2algebras}}\\label{thm:skeletal Lie 2 a}  There is a\n one-to-one correspondence between 2-term skeletal $L_\\infty$-algebra\n $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$ and quadruples\n $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$ where $\\mathfrak k_1$ is a Lie algebra, $\\mathfrak k_2$ is a\n vector space, $\\phi$ is  a representation of $\\mathfrak k_1$ on $\\mathfrak k_2$ and\n $\\theta$ is a 3-cocycle on $\\mathfrak k_1$ with values in $\\mathfrak k_2$.\\end{thm}\n\n\n\nWe briefly recall that given a 2-term skeletal $L_\\infty$-algebra\n $L_1\\stackrel{\\mathrm{d}}{\\longrightarrow}L_0$,\n $\\mathfrak k_1$ is $L_0$, $\\mathfrak k_2$ is $L_1$, the representation $\\phi$ comes from $l_2$\n and the 3-cocycle $\\theta$ is obtained from $l_3$.\n\n Please see \\cite{abad-crainic:rep-homotopy, abad} for\nthe general theory of  representation up to homotopy of Lie\nalgebroids. In this paper we only consider the 2-term representation\nup to homotopy of Lie algebras.\n\\begin{defi}{\\rm\\cite{abad-crainic:rep-homotopy}}\nA 2-term representation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$\nconsists of\n\\begin{itemize}\n\\item[\\rm 1.] A 2-term complex of vector spaces\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$.\n\n\\item[\\rm 2.] Two linear maps $\\mu_i: \\g \\to End(V_i)$, which are compatible with $\\mathrm{d}$, i.e. for\nany $X\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g,~\\xi\\in V_1$, we have\n\\begin{equation}\\label{eqn:d mu}\n\\mathrm{d}\\circ\\mu_1(X)(\\xi)=\\mu_0(X)\\circ\\mathrm{d}(\\xi).\n\\end{equation}\n\n\\item[\\rm 3.] A linear map $\\nu:\\wedge^2\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\to\\mathrm{Hom}(V_0,V_1)$ such that\n\\begin{eqnarray}\n\\label{eqn:mu0}\\mu_0[X_1,X_2]-[\\mu_0(X_1),\\mu_0(X_2)]&=&\\mathrm{d}\\circ\n\\nu(X_1,X_2),\\\\\n\\label{eqn:mu1}\\mu_1[X_1,X_2]-[\\mu_1(X_1),\\mu_1(X_2)]&=&\n\\nu(X_1,X_2)\\circ\\mathrm{d},\n\\end{eqnarray}\nas well as\n\\begin{equation}\\label{eqn:d k}\n[\\mu_0(X_1) + \\mu_1(X_1),\\nu(X_2,X_3)]+c.p.= \\nu([X_1,X_2],X_3)+c.p.,\n\\end{equation}\nwhere $c.p.$ stands for  cyclic permutation.\n\\end{itemize}\nWe usually write $\\mu=\\mu_0+\\mu_1$ and denote a 2-term\nrepresentation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ by\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$.\n\\end{defi}\n\n\nIn \\cite[Example 3.25]{abad-crainic:rep-homotopy}, the authors\nproved that associated to any representation up to homotopy\n$V_\\bullet$ of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, one can form a new $L_\\infty\n$-algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet$, which is their semidirect\nproduct. Here we make this construction explicit in the 2-term case.\n\nLet $(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$ be a 2-term\nrepresentation up to homotopy of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, then we can form a new\n2-term complex\n$$\n(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet,\\mathrm{d}):V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0).\n$$\nDefine  $l_2:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet)\\longrightarrow\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet $ by setting\n\\begin{equation}\\label{2bracket}\n\\left\\{\\begin{array}{l}l_2(X+\\xi,Y+\\eta)=[X,Y]+\\mu_0(X)(\\eta)-\\mu_0(Y)(\\xi),\\\\\nl_2(X+\\xi,f)=\\mu_1(X)(f),\\\\\nl_2(f,g)=0,\\end{array}\\right.\n\\end{equation}\nfor any $X+\\xi,Y+\\eta\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0$ and $f,~g\\in V_1$.  One\nshould note that $l_2$ is not a Lie bracket, but we have\n$$\nl_2(l_2(X+\\xi,Y+\\eta),Z+\\gamma)+c.p.=\\mathrm{d}(\\nu(X,Y)(\\gamma))+c.p..\n$$\nDefine $l_3:\\wedge^3(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes\nV_\\bullet)\\longrightarrow\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\ltimes V_\\bullet $ by setting:\n\\begin{equation}\\label{3bracket}\nl_3(X+\\xi,Y+\\eta,Z+\\gamma)=-\\nu(X,Y)(\\gamma)+c.p.,\n\\end{equation} then we have\n\\begin{pro}\\label{pro:Lie 2}\nWith the above notation, if\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,\\mu,\\nu)$ is a 2-term\nrepresentation up to homotopy of a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$, then\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus V_0),l_2,l_3)$ is a\n2-term $L_\\infty$-algebra.\n\\end{pro}\n\n\n\n\\subsection{Example I: Courant algebroids $TM\\oplus T^*M$}\n\nCourant algebroids were first introduced in \\cite{lwx} to study the\ndouble of Lie bialgebroids. It is a vector bundle $E\\longrightarrow\nM$ equipped with a nondegenerate symmetric bilinear form\n$\\pairing{\\cdot,\\cdot}$ on the bundle, an antisymmetric bracket\n$\\Courant{\\cdot,\\cdot}$ on the section space $\\Gamma(E)$ and a\nbundle map $\\rho:E\\longrightarrow TM$ such that a set of axioms are\nsatisfied. It can be viewed \\cite{royt} as a Lie 2-algebroid with a\n``degree 2 symplectic form''. The first example  is the standard\nCourant algebroid $(\\mathcal{T}=TM\\oplus\nT^*M,\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$ associated\nto a manifold $M$, where $\\rho:\\mathcal{T}\\longrightarrow TM$ is the\nprojection, the canonical pairing $\\pairing{\\cdot,\\cdot}$ is given\nby\n\\begin{equation}\\label{eqn:pair}\n\\pairing{X+\\xi,Y+\\eta}=\\frac{1}{2}\\big(\\xi(Y)+\\eta(X)\\big),\\quad\\forall\n~X,Y\\in\\mathfrak X(M),~\\xi,\\eta\\in\\Omega^1(M),\n\\end{equation}\nthe antisymmetric bracket $\\Courant{\\cdot,\\cdot}$ is given by\n\\begin{equation}\\label{eqn:bracket}\n\\Courant{X+\\xi,Y+\\eta}\\triangleq[X,Y]+L_X\\eta-L_Y\\xi+\\frac{1}{2}d(\\xi(Y)-\\eta(X)),\\quad\\forall~X+\\xi,~Y+\\eta\\in\\Gamma(\\mathcal{T}).\n\\end{equation}\nIt is not a Lie bracket, but we have\n\\begin{equation}\\label{eqn:jacobi}\n\\Courant{\\Courant{e_1,e_2},e_3}+c.p.=d T(e_1,e_2,e_3),\\quad\n\\forall~e_1,e_2,e_3\\in\\Gamma(\\mathcal{T}),\n\\end{equation}\nwhere $T(e_1,e_2,e_3) $ is given by\n\\begin{equation}\\label{T}\nT(e_1,e_2,e_3)=\\frac{1}{3}(\\pairing{\\Courant{e_1,e_2},e_3}+c.p.).\n\\end{equation}\n\n\nNow we realize the section space of $\\mathcal{T}$ as the semidirect\nproduct of the Lie algebra $\\mathfrak X(M)$ of vector fields with the\nnatural 2-term deRham complex\n\\begin{equation}\\label{eq:natural-cx}\nC^\\infty(M)\\stackrel{d}{\\longrightarrow}\\Omega^1(M).\n\\end{equation}\nFor this we need to define a representation up to homotopy of $\\mathfrak X(M)$\non this complex. For any $X\\in\\mathfrak X(M)$, define linear actions $\\mu_0$ and $\\mu_1$ by\n\\begin{eqnarray}\n\\label{C 1}\\mu_0(X)(\\xi)&\\triangleq &\\Courant{X,\\xi}=L_X\\xi-\\frac{1}{2} d(\\xi(X)),\\quad\\forall ~\\xi\\in\\Omega^1(M)\\\\\n\\label{C 2}\\mu_1(X)(f)&\\triangleq &\\pairing{X,d\nf}=\\frac{1}{2}X(f),\\quad\\forall~f\\in C^\\infty(M).\n\\end{eqnarray}\nDefine $\\nu:\\wedge^2\\mathfrak X(M)\\to\\mathrm{Hom}(\\Omega^1(M),C^\\infty(M))$ by\n\\begin{equation}\\label{C 3}\n\\nu(X,Y)(\\xi)=T(X,Y,\\xi),\\quad\\forall\n~X,Y\\in\\mathfrak X(M),~\\xi\\in\\Omega^1(M). \\end{equation}\n\\begin{pro}\\label{pro:rep up to homotopy}\nWith the above notations,\n$(C^\\infty(M)\\stackrel{d}{\\longrightarrow}\\Omega^1(M),\\mu=\\mu_0+\\mu_1,\\nu)$\n is a representation up to homotopy of the Lie algebra $\\mathfrak X(M)$.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  For any $f\\in C^\\infty(M)$, we have\n$$\n\\mu_0(X)(d f)=L_X df-\\frac{1}{2}d X( f)=\\frac{1}{2}d X( f),\n$$\nwhich implies $\\mu_0\\circ d=d\\circ\\mu_1$, i.e. $\\mu_i$'s are compatible\nwith the differential $d$. By straightforward computations, we have\n\\begin{eqnarray*}\n\\mu_0[X,Y](\\xi)-[\\mu_0(X),\\mu_0(Y)](\\xi)&=&\\Courant{\\Courant{X,Y},\\xi}+c.p.\\\\\n&=&d T(X,Y,\\xi) =d \\big(\\nu(X,Y)(\\xi)\\big),\\\\\n\\mu_1[X,Y](f)-[\\mu_1(X),\\mu_1(Y)](f)&=&\\frac{1}{2}[X,Y](f)-\\frac{1}{4}\\big(X(Y(f))-Y(X(f))\\big)\\\\\n&=&\\frac{1}{4}[X,Y](f),\\\\\n\\nu(X,Y)(d f)&=&T(X,Y,d\nf)=\\frac{1}{3}\\big(\\frac{1}{2}[X,Y](f)+\\frac{1}{4}\\big(X(Y(f))-Y(X(f))\\big)\\\\\n&=&\\frac{1}{4}[X,Y](f),\n\\end{eqnarray*}\nwhich implies (\\ref{eqn:mu0}) and (\\ref{eqn:mu1}). At last we need\nto prove (\\ref{eqn:d k}), which is obviously equivalent to\n\\begin{equation}\\label{c temp}\n\\mu_1(X)T(Y,Z,\\xi)-T(Y,Z,\\mu_0(X)(\\xi))+c.p.(X,Y,Z)=T([X,Y],Z,\\xi)+c.p.(X,Y,Z).\n\\end{equation}\nObserve that since $\\mu_0(X)(\\xi)=\\Courant{X,\\xi}$, we have\n$$T(Y,Z,\\mu_0(X)(\\xi))+c.p.(X,Y,Z)+T([X,Y],Z,\\xi)+c.p.(X,Y,Z)=T([X,Y],Z,\\xi)+c.p.(X,Y,Z,\\xi).$$\nFurthermore, since for any $f\\in C^{\\infty}(M)$, $\\mu_1(X)(f)=\\pairing{X,df}$\nand the cotangent bundle $T^*M$ is isotropic under the pairing\n$(\\ref{eqn:pair})$, we have\n$$\n\\mu_1(X)T(Y,Z,\\xi)+c.p.(X,Y,Z)=\\pairing{X,dT(Y,Z,\\xi)}+c.p.(X,Y,Z,\\xi).\n$$\n\nThus, (\\ref{c temp}) is equivalent to\n$$\n\\pairing{X,d T(Y,Z,\\xi)}+c.p.(X, Y, Z, \\xi)=T([X,Y],Z,\\xi)+c.p.(X, Y,\nZ, \\xi),\n$$\nwhich holds by Lemma 4.5 in \\cite{rw}. \\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\nBy Proposition \\ref{pro:Lie 2}, we have\n\n\\begin{cor}\\label{cor:courant}\n$\\big(C^\\infty(M) \\xrightarrow{\\mathrm{d}=0\\oplus d }(\\mathfrak X(M)\\oplus\\Omega^1(M)),l_2,l_3\\big)$\nis a 2-term $L_\\infty$-algebra, where $l_2$ and $l_3$ are given by\n(\\ref{2bracket}) and (\\ref{3bracket}), in which $\\mu_0$, $\\mu_1$ and\n$\\nu$ are given by (\\ref{C 1}),~(\\ref{C 2}) and (\\ref{C 3})\nrespectively.\n\\end{cor}\n\\begin{rmk}\\label{rmk:L infty}\n\n\nIn \\cite{rw}  the authors proved  that  the sections of  a Courant\nalgebroid $({\\mathcal{C}}}%{\\mathcal{C},\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$\nform an $L_\\infty$-algebra. In the case when ${\\mathcal{C}}}%{\\mathcal{C}=\\mathcal{T}$ the\nstandard Courant algebroid, the 2-term $L_\\infty$-algebra is given\nby\n$$\nC^{\\infty}(M) \\xrightarrow{\\mathrm{d}=0\\oplus d } \\mathfrak X(M) \\oplus\n\\Omega^1(M)=\\Gamma(\\mathcal{T}),\n$$\nwith brackets  given by\n$$\nl_2(e_1,e_2)=\\Courant{e_1,e_2},\\quad l_2(e_1,f)=\\pairing{e_1,\\mathrm{d}\nf},\\quad l_3(e_1,e_2,e_3)=-T(e_1,e_2,e_3), \\quad l_{i\\ge 4} =0,\n$$\nfor any $e_1,e_2,e_3\\in\\Gamma(\\mathcal{T}),~f\\in C^{\\infty}(M)$. Here $T$ is\ndefined by (\\ref{T}). It is easy to verify that this is the same as\nour 2-term $L_\\infty$-algebra in Corollary \\ref{cor:courant}.\n\\end{rmk}\n\n\n\n\nWe can also modify our complex \\eqref{eq:natural-cx} to\n$$\n\\Omega^1(M)\\stackrel{\\rm{Id}}{\\longrightarrow}\\Omega^1(M).\n$$\nFollowing the same procedure, we obtain another representation up to\nhomotopy of the Lie algebra $\\mathfrak X(M)$ and therefore obtain another\n2-term $L_\\infty$-algebra which is also totally determined by the\nCourant algebroid\n$(\\mathcal{T},\\pairing{\\cdot,\\cdot},\\Courant{\\cdot,\\cdot},\\rho)$. More\nprecisely, for any $X\\in \\mathfrak X(M),~\\xi\\in\\Omega^1(M)$, $\\mu_0=\\mu_1$\nis given by\n$$\n\\mu_0(X)(\\xi)\\triangleq\\Courant{X,\\xi},\n$$\nand\n$\\nu:\\mathfrak X(M):\\longrightarrow\\Omega^2(\\mathfrak X^2(M),\\mathrm{End}(\\Omega^1(M),\\Omega^1(M)))$\nis given by\n$$\n\\nu(X,Y)(\\xi)\\triangleq d T(X,Y,\\xi).\n$$\n\\begin{pro}\nWith the above notations,\n$(\\Omega^1(M)\\stackrel{\\rm{Id}}{\\longrightarrow}\\Omega^1(M),\\mu=\\mu_0=\\mu_1,\\nu)$\nis a representation up to homotopy of the Lie algebra $\\mathfrak X(M)$.\n\\end{pro}\n\n\\subsection{Example \\textup{II}: Courant algebroids over a point and\n  string Lie 2-algebras} \\label{sec:string}\n\nA Courant algebroid over a point is literally  a quadratic Lie\nalgebra, namely a Lie algebra $\\mathfrak k$ together with nondegenerate\ninner product $\\pairing{\\cdot,\\cdot}$ which is invariant under the\nadjoint action.  People often think that a Courant algebroid over a\npoint  is a string Lie 2-algebra \\footnote{private conversation with\n  John Baez and Urs Schreiber}. Here we justify this thinking.\n\\begin{defi}\\label{defi:string-alg}\nThe  string Lie 2-algebra associated to a quadratic Lie algebra\n$(\\mathfrak k, \\pairing{\\cdot,\\cdot})$, is a  2-term $L_\\infty$-algebra\n$\\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak k$, whose degree-$0$\npart is $\\mathfrak k$, degree-$1$ part is $\\mathbb R$,  and $l_2,~l_3$ are\ngiven by\n\\begin{eqnarray*}\n~l_2(e,c)&=&0,\\\\\n~l_2((e_1,c_1),(e_2,c_2))&=&([e_1,e_2],0),\\\\\n~l_3((e_1,c_1),(e_2,c_2),(e_3,c_3))&=&(0,\\pairing{[e_1,e_2],e_3}),\n\\end{eqnarray*}\nwhere $e,~e_1,~e_2,~e_3\\in\\mathfrak k,~c,~c_1,~c_2,~c_3\\in\\mathbb R$.\n\\end{defi}\n\nThe representation $\\phi $ of $\\mathfrak k$ on $\\mathbb R$ and the\n3-cocycle $\\theta:\\wedge^3\\mathfrak k\\longrightarrow \\mathbb R$ in the\ncorresponding quadruple in Theorem \\ref{thm:skeletal Lie 2 a} is\ngiven by\n\\begin{eqnarray}\n\\nonumber\\rho(e)(c)&=&l_2(e,c)\\\\\n\\label{3cocycle}\\theta(e_1,e_2,e_3)&=&\\pairing{[e_1,e_2],e_3}.\n\\end{eqnarray}\n\n\\begin{rmk}\\label{rk:semisimple}\nIn the definition of string Lie 2-algebras \\cite{baez:string,\n  henriques} the base Lie algebra $\\mathfrak k$ is usually required\nto be semisimple and of compact type, such that the Jacobiator gives\nrise to the generator of $H^3(\\mathfrak k, \\mathbb Z)=\\mathbb Z$. This is\nbecause  Witten's original motivation is to obtain a $3$-connected\ncover of $Spin(n)$, and $\\mathfrak{so}(n)$ is simple and of compact\ntype. However, to write down the structure of the string Lie\n2-algebra, we only need a quadratic Lie algebra. Then $H^3(\\mathfrak k,\n\\mathbb Z)$ is not necessarily $\\mathbb Z$ for a general quadratic\nLie algebra $\\mathfrak k$. For example, for the abelian Lie algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z$,\nany inner product is adjoint-invariant, and $H^3(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z, \\Z)=0$.  We\nthus face the danger that sometimes, the string Lie 2-algebra is\ntrivial, that is the Jacobiator corresponds to the trivial element\nin $H^3(\\mathfrak k, \\mathbb Z)$. Then what we have is a strict Lie\n2-algebra, which is a crossed module of Lie algebras. Then the\nintegration of a crossed module of Lie algebras is simply a crossed\nmodule of Lie groups.  However, we will verify that the example we\nconsider is not such a case.\n\\end{rmk}\n\n\n\nThe standard Courant algebroid\nmotivates us to consider the case of  the direct sum\n$\\mathfrak k=\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ of  a Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$\nand its dual with  the\nsemidirect product  Lie algebra structure:\n$$\n[X+\\xi,Y+\\eta]=[X,Y]_\\g+\\mathrm{ad}_X^*\\eta-\\mathrm{ad}_Y^*\\xi,\n$$here $[\\cdot,\\cdot]_\\g$ is the Lie bracket of $\\g$. The nondegenerate invariant pairing\n$\\pairing{\\cdot,\\cdot}$ on $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$  is given by\n$$\n\\pairing{X+\\xi,Y+\\eta}=\\frac{1}{2}(\\eta(X)+\\xi(Y)),\\quad\\forall~X+\\xi,Y+\\eta\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*.\n$$ With these definitions,\n$(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,[\\cdot,\\cdot],\\pairing{\\cdot,\\cdot})$ is a\nquadratic Lie algebra. In fact, we have\n\\begin{eqnarray*}\n\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}&=&\\pairing{[X_1,X_2]_\\g+\\mathrm{ad}_X^*\\xi_2-\\mathrm{ad}_Y^*\\xi_1,X_3+\\xi_3}\\\\\n&=&\\pairing{[X_1,X_2]_\\g,\\xi_3}+c.p..\n\\end{eqnarray*}\nSimilarly, we have\n$$\n\\pairing{X_2+\\xi_2,[X_1+\\xi_1,X_3+\\xi_3]}=\\pairing{[X_1,X_3]_\\g,\\xi_2}+c.p.,\n$$\nwhich implies that\n$\\pairing{X_2+\\xi_2,[X_1+\\xi_1,X_3+\\xi_3]}+\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}=0$,\ni.e. the nondegenerate inner product $\\pairing{\\cdot,\\cdot}$ is\ninvariant under the adjoint action. This example is a special case\nof \\cite[Theorem 1.12]{lu-weinstein} with $\\g^*$ equipped with $0$\nLie bracket. Thus $(\\g, \\g^*)$ forms a Lie bialgebra or equivalently\n$(\\g\\oplus \\g^*, \\g, \\g^*)$ is a Manin triple. However, honestly we\nhave not found other Lie bialgebras (Manin triples) giving rise to\nLie 2-algebras of the form of semidirect products.\n\nWe  denote the corresponding string Lie 2-algebra of $(\\g \\oplus \\g^*,\n[\\cdot, \\cdot], \\pairing{\\cdot,\\cdot})$ by\n\\begin{equation}\\label{Lie 2 g g dual}\n\\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*.\n\\end{equation}\nThe corresponding 3-cocycle (see \\eqref{3cocycle}), which we denote\nby $\\widetilde{\\nu}:\\wedge^3(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*)\\longrightarrow\n\\mathbb R$ is\n\\begin{eqnarray}\\label{eqn:nu3}\n\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)&=&\\pairing{[X_1+\\xi_1,X_2+\\xi_2],X_3+\\xi_3}\\\\\n\\nonumber&=&\\pairing{[X_1,X_2]_\\g,\\xi_3}+c.p\n\\end{eqnarray}\n\n\\begin{pro}  $(\\mathbb\nR\\stackrel{0}{\\longrightarrow} \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mu_1=0,\n\\mu_0=\\mathrm{ad}^*,\\nu=[\\cdot,\\cdot]_\\g)$ is a 2-term representation up to\nhomotopy of the Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$. Moreover the string Lie\n2-algebra $ \\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^* $ is the semidirect product of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g $ and the complex\n$\\mathbb R\\stackrel{0}{\\longrightarrow} \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$.\n\\end{pro}\n\\begin{proof}\nSince $\\mathrm{d}=0$, we only need to verify that $\\mu_0$ and $\\mu_1$ are\nLie algebra morphisms and \\eqref{eqn:d k}. $\\mathrm{ad}^*: \\g \\to\n\\mathrm{End}(\\g^*)$ and $0$ are both Lie algebra morphisms. \\eqref{eqn:d k}\nfollows from Jacobi identity of $[\\cdot,\\cdot]_\\g$.\n\nThen  it is not hard to see that the string Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ with formulas in\nDefinition \\ref{defi:string-alg} is exactly the semidirect product\nof $\\g$ with the complex  $(\\mathbb R\\stackrel{0}{\\longrightarrow}\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mu_1=0, \\mu_0=\\mathrm{ad}^*,\\nu=[\\cdot,\\cdot]_\\g)$ with the\nformulas \\eqref{2bracket}, \\eqref{3bracket}.\n\\end{proof}\n\n\n\\begin{pro}\\label{pro:nondegenerate}\nIf the Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Lie algebra 3-cocycle\n$\\widetilde{\\nu}$ which is given by (\\ref{eqn:nu3}) is not exact.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  Let $\\pairing{\\cdot,\\cdot}_k$ be the Killing form on $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$.\nThe proof follows from the fact that the Cartan 3-form\n$\\pairing{[\\cdot,\\cdot],\\cdot}_k$   on a semisimple Lie algebra is\nnot exact.\nSince $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Killing form\n$\\pairing{\\cdot,\\cdot}_k$ is nondegenerate. Identify $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ and\n$\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ by using the Killing form $\\pairing{\\cdot,\\cdot}_k$ and let\n$\\mathcal{K}$ be the corresponding isomorphism,\n$$\n\\pairing{\\mathcal{K}(\\xi),X}_k=\\pairing{\\xi,X}.\n$$\nAssume that $\\widetilde{\\nu}=d\\phi$ for some\n$\\phi:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*)\\longrightarrow \\mathbb R$, define\n$\\varphi:\\wedge^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g)\\longrightarrow \\mathbb R$ by\n$$\n\\phi(X+\\xi,Y+\\eta)=\\varphi(X+\\mathcal{K}(\\xi),Y+\\mathcal{K}(\\eta)),\n$$then we have\n\\begin{eqnarray*}\n\\widetilde{\\nu}(X,Y,\\xi)&=&d\\phi(X,Y,\\xi)\\\\\n&=&-\\phi([X,Y],\\xi)+\\phi(\\mathrm{ad}^*_X\\xi, Y)-\\phi(\\mathrm{ad}^*_Y\\xi, X)\\\\\n&=&-\\varphi([X,Y],\\mathcal{K}(\\xi))+\\varphi([X,\\mathcal{K}(\\xi)],\nY)-\\varphi([Y,\\mathcal{K}(\\xi)], X)\\\\\n&=&d\\varphi(X,Y,\\mathcal{K}(\\xi)).\n\\end{eqnarray*}\nOn the other hand, we have\n$$\n\\widetilde{\\nu}(X,Y,\\xi)=\\pairing{[X,Y],\\xi}=\\pairing{[X,Y],\\mathcal{K}(\\xi)}_k,\n$$\nwhich implies that the Cartan 3-from\n$$\n\\pairing{[X,Y],\\mathcal{K}(\\xi)}_k=d\\varphi(X,Y,\\mathcal{K}(\\xi)),\n$$ is exact.\nThis is a contradiction. \\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\n\nIn the last section, we will give  the integration of the string Lie\n2-algebra $ \\mathbb R\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^* $ by using the semidirect product of a Lie group with its\n2-term representation up to homotopy. It turns out that this string\nLie 2-algebra can be integrated to a special Lie 2-group with a\nfinite dimensional model.\n\n\n\n\\subsection{Example III: Omni-Lie algebroids $\\mathfrak{D} E\\oplus \\mathfrak{J} E$}\n\nThe notion of  omni-Lie algebroids, which was introduced by Chen and\nLiu in \\cite{clomni}, is a generalization of Weinstein's omni-Lie\nalgebras. Just as Dirac structures of an omni-Lie algebra\ncharacterize Lie algebra structures on a vector space, Dirac\nstructures of an omni-Lie algebroid  characterize Lie algebroid\nstructures on a vector bundle. See \\cite{clsdirac} for more details.\nIn fact, the role of omni-Lie algebroids $\\mathfrak{D} E\\oplus \\mathfrak{J} E$ in\n$E$-Courant algebroids, which was introduced in \\cite{clsecourant},\nis the same as the role of standard Courant algebroids $TM\\oplus\nT^*M$ in Courant algebroids.\n\n\nNow we briefly recall the notion of omni-Lie algebroids and we will\nsee that it gives rise to a 2-term $L_\\infty$-algebra which is a\nsemidirect product. In this subsection $E$ is a vector bundle over a\nsmooth manifold $M$, $\\Gamma(E)$ is the section space of $E$.\n\nFor a vector bundle $E$, let $\\mathfrak{D} E$ be its covariant differential\noperator bundle. The associated Atiyah sequence is given by\n\\begin{equation}\\label{Seq:DE}\n\\xymatrix@C=0.5cm{0 \\ar[r] & \\mathfrak {gl}(E)  \\ar[rr]^{\\mathbbm{i}} &&\n                \\mathfrak{D}{E}  \\ar[rr]^{a} && TM \\ar[r]  & 0.\n                }\n\\end{equation}\nThe associated $1$-jet vector bundle ${\\mathfrak{J}} E$ is defined as\nfollows. For any $m\\in M$, $({\\mathfrak{J}}{E})_m$ is defined as a quotient\nof local sections of $E$. Two local sections $u_1$ and $u_2$ are\nequivalent and we denote this by $u_1\\sim u_2$ if\n$$ u_1(m)=u_2(m) ~~\\mbox{ and }~~ d\\pairing{u_{1\n},\\xi}_m=d\\pairing{u_{2 },\\xi}_m, \\quad\\forall~ \\xi\\in\\Gamma(E^*).$$\nSo any $\\mu\\in ({\\mathfrak{J}}{E})_m$ has a representative $u\\in\\Gamma(E)$\nsuch that\n  $\\mu=[u]_m$.\nLet $\\mathbbm{p}$ be the projection which sends $[u]_m$ to $u(m)$.  Then\n$\\mathrm{Ker}\\mathbbm{p} \\cong  \\mathrm{Hom}(TM,E)$ and there is a short exact sequence,\ncalled the jet sequence of $E$,\n\\begin{equation}\\label{Seq:JetE}\n\\xymatrix@C=0.5cm{0 \\ar[r] & \\mathrm{Hom}(TM,E)  \\ar[rr]^{~\\qquad \\mathbbm{e}~} &&\n                {\\mathfrak{J}}{E} \\ar[rr]^{\\mathbbm{p}} && E \\ar[r]  & 0,\n                }\n\\end{equation}\nfrom which it is straightforward to see that $\\mathfrak{J} E$ is a finite\ndimensional vector bundle. Moreover, $\\Gamma(\\mathfrak{J}{E})$ is isomorphic\nto $\\Gamma(E) \\oplus \\Gamma (T^*M \\otimes E)$ as an $\\mathbb R$-vector\nspace,\n   and any\n$u\\in \\Gamma (E)$ has a lift $\\mathbbm{d} u\\in\\Gamma ({\\mathfrak{J}} E)$ by taking\nits equivalence class, such that\n\\begin{equation}\\label{Eqt:CWMmoduleGammaE}\n\\mathbbm{d}(fu)=f \\mathbbm{d} u+ d f\\otimes u, \\quad \\forall~ f\\in C^{\\infty}(M).\n\\end{equation}\nIn \\cite{clomni} the authors proved that\n\\begin{eqnarray*}  {\\mathfrak{J} E} \\cong\n\\set{\\nu\\in \\mathrm{Hom}(\\mathfrak{D}{E},E)\\,|\\,\n\\nu(\\Phi)=\\Phi\\circ\\nu(\\rm{Id}_{E}),\\quad\\forall~ ~ ~\\Phi\\in \\mathfrak {gl}(E)}.\n\\end{eqnarray*}\n\nTherefore, there is   an $E$-pairing between $\\mathfrak{J}{E}$ and $\\mathfrak{D}{E}$\nby setting:\n\\begin{equation}\n\\label{Eqt:conpairingE} \\conpairing{\\mu,\\mathfrak d}_E~ \\triangleq\n\\mathfrak d(u),\\quad\\forall~ ~~ \\mu\\in (\\mathfrak{J}{E})_m,~\\mathfrak d\\in(\\mathfrak{D}{E})_m,\n\\end{equation}\nwhere $u\\in \\Gamma(E)$ satisfies $\\mu=[u]_m$. Particularly, one has\n\\begin{eqnarray}\\label{conpairing1}\n\\conpairing{\\mu,\\Phi}_E &=& \\Phi\\circ \\mathbbm{p}(\\mu),\\quad\\forall~ ~~ \\Phi\\in \\mathfrak {gl}(E),~\\mu\\in\\mathfrak{J}{E};\\\\\n\\label{conpairing2} \\conpairing{{\\mathfrak y},\\mathfrak d}_{E} &=& {\\mathfrak y}\\circ\na(\\mathfrak d),\\quad\\forall~ ~~ \\mathfrak y\\in \\mathrm{Hom}(TM,E),~\\mathfrak d\\in\\mathfrak{D}{E}.\n\\end{eqnarray}\n Furthermore, we claim that $\\Gamma (\\mathfrak{J} E)$ is an invariant\nsubspace of the Lie derivative $\\frkL_{\\mathfrak d}$ for any\n $\\mathfrak d \\in\\Gamma(\\mathfrak{D}{E})$, which is defined by the\n Leibniz rule as follows:\n\\begin{eqnarray}\\nonumbe\n\\conpairing{\\frkL_{\\mathfrak d}\\mu,\\mathfrak d^\\prime}_{E}&\\triangleq&\n\\mathfrak d\\conpairing{\\mu,\\mathfrak d^\\prime}_{E}-\\conpairing{\\mu,[\\mathfrak d,\\mathfrak d^\\prime]_{\\mathfrak{D}}}_{E},\n\\quad\\forall~ \\mu \\in \\Gamma(\\mathfrak{J}{E}), ~\n~\\mathfrak d^\\prime\\in\\Gamma(\\mathfrak{D}{E}).\n\\end{eqnarray}\n\nIntroduce a nondegenerate symmetric $E$-valued $2$-form\n$\\ppairingE{\\cdot,\\cdot}$ on $ \\mathcal{E}\\triangleq\\mathfrak{D}{E}\\oplus \\mathfrak{J}{E}$\nby:\n$$\n\\ppairingE{\\mathfrak d+\\mu,\\mathfrak r+\\nu}\\triangleq \\frac{1}{2}(\\conpairing{\\mathfrak d,\\nu}_E\n+\\conpairing{\\mathfrak r,\\mu}_E),\\quad\\forall~ ~~\n\\mathfrak d,\\mathfrak r\\in\\mathfrak{D}{E},~\\mu,\\nu\\in\\mathfrak{J}{E}.\n$$\nFurthermore, we define an antisymmetric bracket\n$\\Courant{\\cdot,\\cdot}$ on $\\Gamma(\\mathcal{E})$ by:\n\\begin{eqnarray*}\n\\Courant{\\mathfrak d+\\mu,\\mathfrak r+\\nu}&\\triangleq&\n[\\mathfrak d,\\mathfrak r]_{\\mathfrak{D}}+\\frkL_{\\mathfrak d}\\nu-\\frkL_{\\mathfrak r}\\mu +\n\\frac{1}{2}\\big(\\mathbbm{d}\\conpairing{\\mu,\\mathfrak r}_E-\\mathbbm{d}\\conpairing{\\nu,\\mathfrak d}_E\\big).\n\\end{eqnarray*}\n\nIn \\cite{clomni} the authors  call  the quadruple\n$(\\mathcal{E},\\Courant{\\cdot,\\cdot},\\ppairingE{\\cdot,\\cdot},\\rho_{\\varepsilon})$ the\n{\\bf omni-Lie algebroid} \\footnote{It is slightly different from the\nnotion given in \\cite{clomni}, where the bracket is not\nskewsymmetric.} associated to the vector bundle $E$,  where $\\rho$\nis the projection of $\\mathcal{E}$ onto $\\mathfrak{D}{E}$. Even though\n$\\Courant{\\cdot,\\cdot}$ is antisymmetric,  it is not a Lie bracket.\nMore precisely, for any $X,Y,Z\\in \\Gamma(\\mathcal{E})$, we have\n$$\\Courant{\\Courant{X,Y},Z}+c.p.=\\mathbbm{d} T(X,Y,Z),$$\nwhere $T:\\Gamma(\\wedge^3\\mathcal{E})\\longrightarrow \\Gamma(E)$ is defined\nby \\be\\label{T}\nT(X,Y,Z)=\\frac{1}{3}\\big(\\ppairingE{\\Courant{X,Y},Z}+c.p.\\big). \\ee\n\n\nNow let us construct a 2-term $L_\\infty$-algebra from the omni-Lie\nalgebroid $\\mathcal{E}$. Obviously, $\\Gamma(\\mathfrak{D} E)$ is a Lie algebra and\nthere is a natural 2-term complex\n$$\n\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J} E).\n$$\n\n\nFor any $\\mathfrak d\\in\\Gamma(\\mathfrak{D} E)$, define linear actions $\\mu_0$ and\n$\\mu_1$ by\n\\begin{equation}\\label{O 1}\\left\\{\\begin{array}{l}\n\\mu_0(\\mathfrak d)(\\mu)\\triangleq \\Courant{\\mathfrak d,\\mu}=\\frkL_\\mathfrak d\\mu-\\mathbbm{d}\\ppairingE{\\mu,\\mathfrak d},\\quad\\forall~\\mu\\in \\Gamma(\\mathfrak{J} E),\\\\\n\\mu_1(\\mathfrak d)(u)\\triangleq \\ppairingE{\\mathfrak d,\\mathbbm{d}\nu}=\\frac{1}{2}\\mathfrak d(u),\\quad\\forall~u\\in\\Gamma(E).\\end{array}\\right.\n\\end{equation}\nDefine $\\nu:\\wedge^2\\Gamma(\\mathfrak{D} E)\\longrightarrow\\mathrm{Hom}(\\Gamma(\\mathfrak{J}\nE),\\Gamma( E))$ by\n\\begin{equation}\\label{O 2}\n\\nu(\\mathfrak d,\\mathfrak t)(\\xi)=T(\\mathfrak d,\\mathfrak t,\\mu),\\quad\\forall\n~\\mathfrak d,\\mathfrak t\\in\\Gamma(\\mathfrak{D} E),~\\mu\\in\\Gamma(\\mathfrak{J} E).\n\\end{equation}\nSimilar to Proposition \\ref{pro:rep up to homotopy}, we prove\n\\begin{pro}\nWith the above notations,\n$(\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J}\nE),\\mu=\\mu_0+\\mu_1,\\nu)$\n is a representation up to homotopy of the Lie algebra $\\Gamma(\\mathfrak{D}\nE)$.\n\\end{pro}\n\\begin{cor}\n$\\big(\\Gamma(E)\\stackrel{\\mathrm{d}=0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{D}\nE)\\oplus\\Gamma(\\mathfrak{J} E),l_2,l_3\\big)$ is a 2-term $L_\\infty$-algebra,\nwhere $l_2$ and $l_3$ are given by (\\ref{2bracket}) and\n(\\ref{3bracket}), in which $\\mu$ and $\\nu$ are given by (\\ref{O 1})\nand (\\ref{O 2}) respectively.\n\\end{cor}\n\n\n\\begin{rmk}\nIf the base manifold $M$ is a point, i.e. $E$ is a vector space, for\nwhich  we use a new notation $V$, then $\\mathfrak{D} V=\\mathfrak {gl}(V),~\\mathfrak{J} V=V$,\nand we recover the notion of omni-Lie algebras. The complex\n$\\Gamma(V)\\stackrel{0\\oplus\\mathbbm{d}}{\\longrightarrow}\\Gamma(\\mathfrak{J} V)$\nreduces to $V\\stackrel{\\rm{Id}}{\\longrightarrow}V$, which is a\nrepresentation up to homotopy of $\\mathfrak {gl}(V)$ with $(\\mu_0=\\mu_1, \\nu)$\ngiven by\n\\begin{equation}\\label{omni mu nu}\n\\mu_0(A)(u)=\\frac{1}{2} Au,\\quad\n\\nu(A,B)=\\frac{1}{4}[A,B],\\quad\\forall~A,B\\in\\mathfrak {gl}(V),~u\\in V.\n\\end{equation}\nHence even though an omni-Lie algebra   $\\mathfrak {gl}(V)\\oplus V$ is not a\nLie algebra,  we can extend it to a  2-term $L_\\infty$-algebra, of\nwhich $L_0=\\mathfrak {gl}(V)\\oplus V,~L_1=V$, $l_2$ and $l_3$ are given by\n(\\ref{2bracket}) and (\\ref{3bracket}), in which $\\mu$ and $\\nu$ are\ngiven by (\\ref{omni mu nu}). This 2-term $L_\\infty$-algebra is a\nsemidirect product of $\\mathfrak {gl}(V)$ with $V\\xrightarrow{\\rm{Id}} V$.\n\nWe will study the global object of the 2-term $L_\\infty$-algebra\nassociated to an omni-Lie algebra in the forthcoming paper \\cite{sheng-zhu:II}.\n\\end{rmk}\n\n\\section{Representation up to homotopy of Lie groups and semidirect\n  product} \\label{sec:gp}\n\nThe representation up to homotopy of a Lie group was introduced in\n\\cite{abad}. In this section we define the semidirect product of a\nLie group with a 2-term representation up to homotopy and prove that\nthe semidirect product is a Lie 2-group. Thus we first recall some\nbackground on Lie 2-groups.\n\nA group is a monoid where every element has an inverse. A 2-group is\na monoidal category where every object has a weak inverse and every\nmorphism has an inverse. Denote the category of smooth manifolds and\nsmooth maps by $\\rm Diff$, a semistrict\\footnote{Please see the\n  introduction for the reason why we call it a semistrict Lie 2-group.} Lie 2-group is  a 2-group\nin $\\rm DiffCat$, where  $\\rm DiffCat$ is the 2-category consisting\nof categories, functors, and natural transformations in $\\rm Diff$.\nFor more details, see \\cite{baez:2gp} and here  we only recall the\nexpanded definition:\n\\begin{defi}{\\rm\\cite{baez:2gp}}\nA semistrict Lie 2-group consists of an object $C$ in $\\rm\nDiffCat$, i.e. \\[ \\xymatrix{ C_1 \\ar@<1ex>[r]^{s} \\ar[r]_{t} & C_0},\n\\]\nwhere $C_1,~C_0$ are objects in $\\rm Diff$, $s,~t$ are the source\nand target maps, and there is a vertical multiplication\n$\\cdot_\\mathrm{v}:C\\times C\\longrightarrow C$, together with\n\\begin{itemize}\n\\item[$\\bullet$] a  functor (horizontal\nmultiplication) $\\cdot_\\mathrm{h}:C\\times C\\longrightarrow C$,\n\\item[$\\bullet$] an identity object $1$,\n\\item[$\\bullet$]a contravariant  functor  $\\mathrm{inv}:C\\longrightarrow C$\n\\end{itemize}\nand the following natural isomorphisms:\n\\begin{itemize}\n\\item[$\\bullet$] the {\\bf associator}\n$$\na_{x,y,z}:(x\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h} z\\longrightarrow x\\cdot_\\mathrm{h}\n(y\\cdot_\\mathrm{h} z),\n$$\n\\item[$\\bullet$]the {\\bf left} and {\\bf right unit}\n$$\nl_x:1\\cdot_\\mathrm{h} x\\longrightarrow x,\\quad r_x:x\\cdot_\\mathrm{h}\n1\\longrightarrow x,\n$$\n\\item[$\\bullet$]the  {\\bf  unit} and {\\bf  counit}\n$$\ni_x:1\\longrightarrow x\\cdot_\\mathrm{h} \\mathrm{inv}(x),\\quad e_x:\\mathrm{inv}(x)\\cdot_\\mathrm{h}\nx\\longrightarrow 1,\n$$\n\\end{itemize}\nsuch that the following diagrams commute:\n\\begin{itemize}\n\\item[$\\bullet$] the {\\bf pentagon identity } for the associator\n\\[\n \\xymatrix{ & (w\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h}(y\\cdot_\\mathrm{h} z)\\ar[dr]^{a_{w,x, y\\cdot_\\mathrm{h} z}}&  \\\\\n((w\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h} z\\ar[ur]^{a_{(w\\cdot_\\mathrm{h}\nx),y,z}}\\ar[dr]_{a_{w,x,y}\\cdot_\\mathrm{h} 1_z}&&w\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h}\n(y\\cdot_\\mathrm{h} z))\\\\\n&(w\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h} y))\\cdot_\\mathrm{h} z\\stackrel{a_{w,x\\cdot_\\mathrm{h}\ny,z}}{\\longrightarrow} w\\cdot_\\mathrm{h} ((x\\cdot_\\mathrm{h} y)\\cdot_\\mathrm{h}\nz)\\ar[ur]^{1_w\\cdot_\\mathrm{h} a_{x,y,z}}&}\n\\]\n\\item[$\\bullet$] the {\\bf triangle identity} for the left and right\nunit lows:\n\\[\n \\xymatrix{\n( x\\cdot_\\mathrm{h} 1)\\cdot_\\mathrm{h} y\\ar[rr]^{a_{x,1,y}}\\ar[dr]^{r_x\\cdot_\\mathrm{h} 1_y}&&x\\cdot_\\mathrm{h}(1\\cdot_\\mathrm{h} y)\\ar[dl]^{1_x\\cdot_\\mathrm{h} l_y}\\\\\n&x\\cdot_\\mathrm{h} y& }\n\\]\n\\item[$\\bullet$]the {\\bf first zig-zag identity}:\n\\[\n \\xymatrix{\n&(x\\cdot_\\mathrm{h} \\mathrm{inv}(x))\\cdot_\\mathrm{h} x\\stackrel{a_{x,\\mathrm{inv}(x),x}}{\\longrightarrow}x\\cdot_\\mathrm{h}(\\mathrm{inv}(x)\\cdot_\\mathrm{h} x)\\ar[dr]^{1_x\\cdot_\\mathrm{h} e_x}&\\\\\n1\\cdot_\\mathrm{h} x\\ar[dr]^{l_x}\\ar[ur]^{i_x\\cdot_\\mathrm{h} 1_x}&&x\\cdot_\\mathrm{h} 1\\\\\n&x\\ar[ur]^{r_x^{-1}}&}\n\\]\n\\item[$\\bullet$]the {\\bf second zig-zag identity}:\n\\[\n \\xymatrix{\n&\\mathrm{inv}(x)\\cdot_\\mathrm{h} (x\\cdot_\\mathrm{h} \\mathrm{inv}(x))\\stackrel{a_{\\mathrm{inv}(x),x,\\mathrm{inv}(x)}}{\\longrightarrow}(\\mathrm{inv}(x)\\cdot_\\mathrm{h} x)\\cdot_\\mathrm{h}\\mathrm{inv}(x)\\ar[dr]^{e_x\\cdot_\\mathrm{h} 1_{\\mathrm{inv}(x)}}&\\\\\n\\mathrm{inv}(x)\\cdot_\\mathrm{h} 1\\ar[dr]^{r_{\\mathrm{inv}(x)}}\\ar[ur]^{1_{\\mathrm{inv}(x)}\\cdot_\\mathrm{h} i_x}&&1\\cdot_\\mathrm{h} \\mathrm{inv}(x).\\\\\n&\\mathrm{inv}(x)\\ar[ur]^{l_{\\mathrm{inv}(x)}^{-1}}&}\n\\]\n\\end{itemize}\n\\end{defi}\n\nIn the special case where $a_{x,y,z},~l_x,~r_x,~i_x,~e_x$ are all\nidentity isomorphisms, we call such a Lie 2-group a {\\em strict Lie\n  2-group}\\footnote{The notion of strict Lie 2-groups is the same as \\cite{baez:2gp}.}.\n\nIn the same reference, they also defined the notion of special\n2-groups, which we recall here,\n\\begin{defi}\\label{defi:special 2 G}\nA {\\bf special Lie 2-group}  is a Lie 2-group of which the source\nand target coincide and  the left unit law $l$, the right unit law\n$r$, the unit $i$ and the counit $e$ are identity isomorphisms.\n\\end{defi}\n\nFor classification of special Lie 2-groups,  we need the group cohomology with smooth cocycles, that is we\nconsider the cochain complex with smooth morphisms $G^{\\times\nn}\\to M$ with $G$ a Lie group, $M$ its module. And the differential is defined as usual for\ngroup cohomology. We denote this cohomology by $H_{sm}^\\bullet(G, M)$.\n\n\\begin{thm}{\\rm \\cite[Theorem 8.3.7]{baez:2gp}}\\label{thm:special 2 G}  There is a\none-to-one correspondence between special Lie 2-groups and\nquadruples $(K_1,K_2,\\Phi,\\Theta)$ consisting a Lie group $K_1$, an\nabelian group $K_2$, an action $\\Phi$ of $K_1$ as automorphisms of\n$K_2$ and a normalized smooth 3-cocycle $\\Theta:K_1^3\\longrightarrow\nK_2$. Two special Lie 2-groups are isomorphic if and only if they\ncorresponds to the same\\footnote{up to isomorphisms of groups of\n  course} $(K_1, K_2, \\Phi)$ and the\ncorresponding 3-cocycles represent the same element in $H^3_{sm}(K_1,\nK_2)$.\n\\end{thm}\n\\begin{rmk}\nWe briefly recall that given a quadruple $(K_1,K_2,\\Phi,\\Theta)$ the\ncorresponding semistrict Lie 2-group has the Lie group $K_1$ as the space\nof objects and the semidirect product Lie group $K_1 \\ltimes_{\\Phi}\nK_2$ as the space of morphisms. The associator is given by $\\Theta$.\n\\end{rmk}\n\\begin{defi}\nA unital 2-term representation up to homotopy of a Lie group $G$\nconsists of\n\\begin{itemize}\n\\item[\\rm 1.] A 2-term complex of vector spaces\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$.\n\n\\item[\\rm 2.] A nonassociative action $F_1$ on $V_0$ and $V_1$\n  satisfying $$ \\mathrm{d} F_1 = F_1 \\mathrm{d}, \\quad F_1(1_G)=\\rm{Id}.$$\n\n\\item[\\rm 3.] A smooth map $F_2:G\\times G\\longrightarrow\\mathrm{End}(V_0,V_1)$ such that\n\\begin{equation}\\label{eqn:F fail}\nF_1(g_1)\\cdot F_1(g_2)-F_1(g_1\\cdot g_2)=[\\mathrm{d},F_2(g_1,g_2)],\n\\end{equation}\nas well as\n\\begin{equation}\\label{eqn:F closed}\nF_1(g_1)\\circ F_2(g_2,g_3)-F_2(g_1\\cdot g_2,g_3)+F_2(g_1, g_2\\cdot\ng_3)-F_2(g_1, g_2)\\circ F_1(g_3)=0.\n\\end{equation}\n\\end{itemize}\n\\end{defi}\nWe denote this 2-term representation up to homotopy of the Lie group\n$G$ by $(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,F_1,F_2)$. One should\nbe careful that even if $F_1$ is a usual associative action,\n\\eqref{eqn:F closed} is not equivalent to $F_2$ being a 2-cocycle. This is strangely different from the Lie algebra case (see Section\n\\ref{sec:int}). Define $\\widetilde{F_2}:(G\\ltimes\nV_0)^3\\longrightarrow V_1$ by\n\\begin{equation}\\label{F2tuta}\n\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))=F_2(g_1,\ng_2)(\\xi_3).\n\\end{equation}\nIf  $F_1$ is a usual associative action, we form  $G\\ltimes V_0$\nthe semidirect product. Then\n$V_1$ is a  $G\\ltimes V_0$-module with an associated action\n$\\widetilde{F_1}$ of $G\\ltimes V_0$ on $V_1$\n$$\n\\widetilde{F_1}(g,\\xi)(m)=F_1(g)(m),\\quad\\forall~m\\in V_1.\n$$\n\n\n\\begin{pro}\\label{pro:3cocycle} If  $F_1$ is the usual associative action of the Lie group $G$  on the complex\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$, then $\\widetilde{F_2}$\ndefined by (\\ref{F2tuta}) is a group 3-cocycle representing an\nelement in $H^3_{sm}(G\\ltimes V_0,V_1)$.\n\\end{pro}\n\\noindent{\\bf Proof.}\\  By direct computations, we have\n\\begin{eqnarray*}\n&&d\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))\\\\&=&\\widetilde{F_1}(g_1,\\xi_1)\\widetilde{F_2}((g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))\\\\\n&&-\\widetilde{F_2}((g_1,\\xi_1)\\cdot(g_2,\\xi_2),(g_3,\\xi_3),(g_4,\\xi_4))+\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2)\\cdot(g_3,\\xi_3),(g_4,\\xi_4))\\\\\n&&-\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3)\\cdot(g_4,\\xi_4))+\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))\\\\\n&=&F_1(g_1)F_2(g_2,g_3)(\\xi_4)-F_2(g_1\\cdot g_2,g_3)(\\xi_4)+F_2(g_1,\ng_2\\cdot g_3)(\\xi_4)\\\\\n&&-F_2(g_1,g_2)(\\xi_3+F_1(g_3)(\\xi_4))+F_2(g_1,g_2)(\\xi_3)\\\\\n&=&\\big(F_1(g_1)\\circ F_2(g_2,g_3)-F_2(g_1\\cdot g_2,g_3)+F_2(g_1,\ng_2\\cdot g_3)-F_2(g_1, g_2)\\circ F_1(g_3)\\big)(\\xi_4).\n\\end{eqnarray*}\nBy (\\ref{eqn:F closed}), $\\widetilde{F_2}$ is a Lie group 3-cocycle.\n\\hfill ~\\vrule height6pt width6pt depth0pt\\vspace{3mm}\n\n\n\nSimilar to the fact that associated to any representation of a Lie\ngroup, we can form a new Lie group which is their semidirect\nproduct, for the 2-term representation up to homotopy of a Lie\ngroup, we can form a Lie 2-group.\n\n\\begin{thm}\\label{thm:main 1}\nGiven a 2-term representation up to homotopy\n$(V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0,F_1,F_2)$ of a Lie group\n$G$, its semidirect product with $G$ is defined to be\n\\begin{equation}\\begin{array}{c}\nG\\times V_0\\times V_1\\\\\n\\vcenter{\\rlap{s }}~\\Big\\downarrow\\Big\\downarrow\\vcenter{\\rlap{t }}\\\\\nG\\times V_0.\n \\end{array}\\end{equation}\nThen it is a Lie 2-group with the following structure maps:\n\n The source and target are given by\n\\be\\label{s t} s(g,\\xi,m)=(g,\\xi),\\quad t(g,\\xi,m)=(g,\\xi+\\mathrm{d} m).\n\\ee The vertical multiplication $\\cdot_\\mathrm{v}$ is given by\n$$\n(h,\\eta,n)\\cdot_\\mathrm{v}(g,\\xi,m) =(g,\\xi,m+n),\\quad \\mbox{where}~\nh=g,\\eta=\\xi+\\mathrm{d} m.\n$$\nThe horizontal multiplication $\\cdot_\\mathrm{h}$ of objects is given by\n\\be\\label{m o} (g_1,\\xi)\\cdot_\\mathrm{h} (g_2,\\eta)=(g_1\\cdot\ng_2,\\xi+F_1(g_1)(\\eta)), \\ee the horizontal multiplication\n$\\cdot_\\mathrm{h}$ of morphisms is given by \\be\\label{m m}\n(g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)=(g_1\\cdot\ng_2,\\xi+F_1(g_1)(\\eta),m+F_1(g_1)(n)). \\ee The inverse map $\\mathrm{inv}$ is\ngiven by \\be \\mathrm{inv}(g,\\xi)=(g^{-1},-F_1(g^{-1})(\\xi)). \\ee The\nidentity object is $(1_G,0)$.\\\\\nThe  associator\n$$\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}:\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\longrightarrow\n(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\n$$\nis given by \\be\\label{associator}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}=(g_1\\cdot g_2\\cdot\ng_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),F_2(g_1,g_2)(\\gamma)). \\ee The  unit\n$i_{(g,\\xi)}:(1_G,0)\\longrightarrow (g,\\xi)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi)$ is\ngiven by \\be\\label{unit} i_{(g,\\xi)}=(1_G,0,-F_2(g,g^{-1})(\\xi)).\n\\ee All the other natural isomorphisms are identity isomorphisms.\n\\end{thm}\n\\noindent{\\bf Proof.}\\  By (\\ref{s t}), (\\ref{m o}) and (\\ref{m m}), it is\nstraightforward to see that\n\\begin{eqnarray*}\ns\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)&=&s(g_1,\\xi,m)\\cdot_\\mathrm{h} s(g_2,\\eta,n),\\\\\nt\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)&=&t(g_1,\\xi,m)\\cdot_\\mathrm{h}\nt(g_2,\\eta,n).\n\\end{eqnarray*}\nThus the multiplication $\\cdot_\\mathrm{h}$ respects  the source and target\nmap. Furthermore, it is not hard  to check that the horizontal and\nvertical multiplications commute, i.e.\n$$\n\\big((g,\\xi+\\mathrm{d} m,n)\\cdot_\\mathrm{h}(g^\\prime,\\eta+\\mathrm{d}\np,q)\\big)\\cdot_\\mathrm{v}\\big((g,\\xi,m)\\cdot_\\mathrm{h}(g^\\prime,\\eta,p)\\big)=\\big((g,\\xi+\\mathrm{d}\nm,n)\\cdot_\\mathrm{v}(g,\\xi,m)\\big)\\cdot_\\mathrm{h}\\big((g^\\prime,\\eta+\\mathrm{d}\np,q)\\cdot_\\mathrm{v}(g^\\prime,\\eta,p)\\big)\n$$\n\n\n\n\n\\be\\label{m commute with v} \\xymatrix@C+2em{\n  \\bullet &\n  \\ar@\/_2pc\/[l]_{(g,\\xi)}_{}=\"0\"\n  \\ar[l]|{(g,\\xi+\\mathrm{d} m)}^{}=\"1\"_{}=\"1b\"\n  \\ar@\/^2pc\/[l]^{(g,\\xi+\\mathrm{d} (m+n))}^{}=\"2\"\n  \\ar@{=>} \"0\";\"1\"^{m}\n  \\ar@{=>} \"1b\";\"2\"^{n}\n  \\bullet &\n  \\ar@\/_2pc\/[l]_{(g^\\prime,\\eta)}_{}=\"3\"\n  \\ar[l]|{(g^\\prime,\\eta+\\mathrm{d} p)}^{}=\"4\"_{}=\"4b\"\n  \\ar@\/^2pc\/[l]^{(g^\\prime,\\eta+\\mathrm{d} (p+q))}^{}=\"5\"\n  \\ar@{=>} \"3\";\"4\"^{p}\n  \\ar@{=>} \"4b\";\"5\"^{q}\n  \\bullet.\n} \\ee\n\nIt follows from (\\ref{eqn:F fail}) that the associator\n$a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}$ defined by\n(\\ref{associator}) is indeed a morphism from\n$\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)$ to\n$(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)$. To see that\nit is natural, we need to show that \\be\\label{left} a_{(g_1,\\xi+\\mathrm{d}\nm),(g_2,\\eta+\\mathrm{d} n),(g_3,\\gamma+\\mathrm{d} k)}\\cdot_\\mathrm{h}\n\\Big(\\big((g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)\\big)\\cdot_\\mathrm{h}\n(g_3,\\gamma,k)\\Big) \\ee is equal to \\be\\label{right}\n\\Big((g_1,\\xi,m)\\cdot_\\mathrm{h} \\big((g_2,\\eta,n)\\cdot_\\mathrm{h}\n(g_3,\\gamma,k)\\big)\\Big)\\cdot_\\mathrm{h}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}, \\ee i.e. the following\n diagram commutates:\n\\[\n\\xymatrix{\n\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\ar[d]\\ar[r]^{a}&(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\\ar[d]\\\\\n\\big((g_1,\\xi+\\mathrm{d} m)\\cdot_\\mathrm{h}(g_2,\\eta+\\mathrm{d}\nn)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma+\\mathrm{d} k)\\ar[r]^{a}&(g_1,\\xi+\\mathrm{d}\nm)\\cdot_\\mathrm{h}\\big((g_2,\\eta+\\mathrm{d} n)\\cdot_\\mathrm{h}(g_3,\\gamma+\\mathrm{d} k)\\big). }\n\\]\nBy straightforward computations, we  obtain that (\\ref{left}) is\nequal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),m+F_1(g_1)(n)+F_1(g_1\\cdot\ng_2)(k)+F_2(g_1,g_2)(\\gamma+\\mathrm{d} k)\\big),\n$$\nand (\\ref{right}) is equal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma),m+F_1(g_1)(n)+F_1(g_1)\\cdot\nF_1(g_2)(k)+F_2(g_1,g_2)(\\gamma)\\big).\n$$\n Hence (\\ref{left}) is equal to (\\ref{right}) by (\\ref{eqn:F fail}).\nThis implies that $a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}$ defined by\n(\\ref{associator}) is a natural isomorphism.\n\nBy (\\ref{eqn:F fail}) and the fact that $F_1(1_G)=\\rm{Id}$, the unit\ngiven by (\\ref{unit}) is indeed  a morphism from $(1_G,0)$ to\n$(g,\\xi)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi)$. To see that it is natural, we need to\nprove\n$$\n\\big((g,\\xi,m)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi,m)\\big)\\cdot_\\mathrm{h}\ni_{(g,\\xi)}=i_{(g,\\xi+\\mathrm{d} m)},\n$$\ni.e. the following commutative diagram:\n\n\n \n \n$$\n\\xymatrix{\n  &  (1_G,0)\\ar[dr]^{i_{(g,\\xi)}} \\ar_{i_{(g,\\xi+\\mathrm{d} m)}}[dl]\\\\\n(g,\\xi+\\mathrm{d} m)\\cdot_\\mathrm{h} \\mathrm{inv}(g,\\xi+\\mathrm{d} m)\n&&\\ar[ll]_{\\quad\\qquad\\qquad~(g,\\xi,m)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi,m)}(g,\\xi)\\cdot_\\mathrm{h}\\mathrm{inv}(g,\\xi)\n}\n$$\n This follows from\n $$\nF_2(g,g^{-1})(\\mathrm{d} m)=F_1(g)\\cdot F_1(g^{-1})(m)-F_1(g\\cdot\ng^{-1})(m)=F_1(g)\\cdot F_1(g^{-1})(m)-m,\n $$ which is a special case of (\\ref{eqn:F fail}).\n\nSince $F(1_G)=\\rm{Id}$, we have $$(1_G,0)\\cdot_\\mathrm{h} (g,\\xi)=(g,\\xi),\\quad\n(g,\\xi)\\cdot_\\mathrm{h}(1_G,0)= (g,\\xi).$$ Hence the left unit and the right\nunit can also be taken as the identity isomorphism.\n\n The counit\n$e_{(g,\\xi)}:\\mathrm{inv}(g,\\xi)\\cdot_\\mathrm{h} (g,\\xi)\\longrightarrow (1_G,0)$ can\nbe taken as the identity morphism since we have\n$$\n\\mathrm{inv}(g,\\xi)\\cdot_\\mathrm{h} (g,\\xi)=(g^{-1},-F_1(g^{-1})(\\xi))\\cdot_\\mathrm{h}\n(g,\\xi)=(1_G,0).\n$$\nAt last, we need to show\n\\begin{itemize}\n\\item[$\\bullet$] the  pentagon identity  for the associator,\n\\item[$\\bullet$] the  triangle identity for the left and right\nunit laws,\n\\item[$\\bullet$]the  first zig-zag identity,\n\\item[$\\bullet$]the  second zig-zag identity.\n\\end{itemize}\nWe only give the proof of the  pentagon identity, the others can be\nproved similarly and we leave them to the readers. In fact, the\npentagon identity is equivalent to\n\\begin{eqnarray*}\n&&a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)\\cdot_\\mathrm{h}(g_4,\\theta)}\\cdot_\\mathrm{h}\na_{(g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta),(g_3,\\gamma),(g_4,\\theta)}=\\\\\n&&\\big((g_1,\\xi)\\cdot_\\mathrm{h}\na_{(g_2,\\eta),(g_3,\\gamma),(g_4,\\theta)}\\big)\\cdot_\\mathrm{h}\na_{(g_1,\\xi),(g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma),(g_4,\\theta)}\\cdot_\\mathrm{h}\n\\big(a_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}\\cdot_\\mathrm{h}(g_4,\\theta)\\big)\n\\end{eqnarray*}\nBy straightforward computations, the left hand side is equal to\n$$\n\\big(g_1\\cdot g_2\\cdot g_3\\cdot g_4,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma)+F_1(g_1\\cdot g_2\\cdot g_3)(\\theta),F_2(g_1\\cdot g_2,\ng_3)(\\theta)+F_2(g_1, g_2)(\\gamma+F_1(g_3)(\\theta))\\big),\n$$\nand the right hand side is equal to\n\\begin{eqnarray*}\n&\\big(g_1\\cdot g_2\\cdot g_3\\cdot g_4,\\xi+F_1(g_1)(\\eta)+F_1(g_1\\cdot\ng_2)(\\gamma)+F_1(g_1\\cdot g_2\\cdot g_3)(\\theta),\\\\&F_2(g_1,\ng_2)(\\gamma)+F_2(g_1, g_2\\cdot g_3)(\\theta)+F_1(g_1)\\circ F_2(g_2,\ng_3)(\\theta)\\big).\n\\end{eqnarray*}\nBy (\\ref{eqn:F closed}), they are equal. \\hfill ~\\vrule height6pt width6pt depth0pt\n\n\\section{Integrating string Lie 2-algebra $\\mathbb R\\longrightarrow \\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$}\\label{sec:int}\n\nAs an application of Theorem \\ref{thm:main 1}, we consider the\nintegration of the string Lie 2-algebra $\\mathbb R\\longrightarrow\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ given by (\\ref{Lie 2 g g dual}). Now we\nrestrict to the case that $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is finite dimensional. Obviously,\ngiven a quadruple $(K_1,K_2,\\Phi,\\Theta)$ which represents a special\nLie 2-group (see Theorem \\ref{thm:special 2 G}), by differentiation,\nwe obtain a quadruple $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$, which\nrepresents a 2-term skeletal $L_\\infty$-algebra.\n\\begin{defi}\nA special Lie 2-group which is represented by\n$(K_1,K_2,\\Phi,\\Theta)$ is an integration of a 2-term skeletal\n$L_\\infty$-algebra which is represented by\n$(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$ if the differentiation of\n$(K_1,K_2,\\Phi,\\Theta)$ is $(\\mathfrak k_1,\\mathfrak k_2,\\phi,\\theta)$.\n\\end{defi}\n\nIf the differential $\\mathrm{d}$ in a 2-term complex\n$V_1\\stackrel{\\mathrm{d}}{\\longrightarrow}V_0$ is 0, a representation up to\nhomotopy of Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on\n$V_1\\stackrel{0}{\\longrightarrow}V_0$ consists of two strict\nrepresentations $\\mu_1$ and $\\mu_0$, and  a liner map\n$\\nu:\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\wedge\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\longrightarrow\\mathrm{Hom}(V_0,V_1)$ satisfying\nequation (\\ref{eqn:d k}).  This equation implies that $\\nu$ is a Lie\nalgebra 2-cocycle representing an element in\n$H^2(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g,\\mathrm{Hom}(V_0,V_1))$, with the representation\n$[\\mu(\\cdot),\\cdot]$ of  $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on $\\mathrm{Hom}(V_0,V_1)$  defined by\n$$\n[\\mu(\\cdot),\\cdot](X)(A)\\triangleq[\\mu(X), A]=\\mu_1(X)\\circ A-A\\circ\n\\mu_0(X),\\quad \\forall ~X\\in \\mathfrak g}\\newcommand{\\g}{\\mathfrak g,~A\\in\\mathrm{Hom}(V_0,V_1).\n$$\n\n\\begin{lem}\nDefine $\\widetilde{\\nu}:\\wedge^3(\\g\\oplus V_0)\\longrightarrow V_1$\nby\n$$\n\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)=\\nu(X_1,X_2)(\\xi_3)+c.p.\n$$\nthen $\\nu$ is a 2-cocycle if and only if $\\widetilde{\\nu}$ is a\n3-cocycle where the representation $\\widetilde{\\mu}$ of $\\g\\oplus\nV_0$ on $V_1$ is given by\n$$\n\\widetilde{\\mu}(X+\\xi)(m)=\\mu(X)(m).\n$$\n\\end{lem}\n\\noindent{\\bf Proof.}\\  By direct computations, for any $X_i+\\xi_i\\in\\g\\oplus\nV_0,~i=1,2,3,4,$ we have\n$$d\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3,X_4+\\xi_4)=d\\nu(X_1,X_2,X_3)(\\xi_4)+c.p.,$$\nwhich implies the conclusion.\\hfill ~\\vrule height6pt width6pt depth0pt\n\n\n The  Lie algebra homomorphism $\\mu$ from $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ to\n$\\mathrm{End}(V_0)\\oplus\\mathrm{End}(V_1)$ integrates to a Lie group homomorphism\n$F_1$ from the simply connected Lie group $G$ of $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ to\n$GL(V_0)\\oplus GL(V_1)$ with\n$$\n\\mu(X)=\\frac{d}{dt}\\Big|_{t=0}F_1(\\exp tX),\\quad\\forall~X\\in\\mathfrak g}\\newcommand{\\g}{\\mathfrak g.\n$$\n Consequently, $\\mathrm{Hom}(V_0,V_1)$ is a $G$-module with $G$ action\n$$\ng\\cdot A=F_1(g)\\circ A \\circ F_1(g)^{-1},\\quad\\forall ~g\\in G,~A\\in\n\\mathrm{Hom}(V_0,V_1).\n$$ The Lie algebra 2-cocycle\n$\\nu:\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\wedge\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\longrightarrow\\mathrm{Hom}(V_0,V_1)$ can integrate to\na smooth Lie group 2-cocycle $\\overline{F_2}:G\\times G\\longrightarrow\n\\mathrm{Hom}(V_0,V_1)$, satisfying\n\\begin{equation}\\label{eqn:F2 closed}\nF_1(g_1)\\circ(\\overline{F_2})(g_2,g_3)\\circ\nF_1(g_1)^{-1}-(\\overline{F_2})(g_1\\cdot\ng_2,g_3)+(\\overline{F_2})(g_1, g_2\\cdot\ng_3)-(\\overline{F_2})(g_1,g_2)=0,\n\\end{equation}\nand $\\overline{F_2}(1_G,1_G)=0.$ Let us explain how.\n\nThe classical theory of cohomology of discrete groups tells us that\nthe equivalence classes of extensions of $G$ by a $G$ module $M$,\none to one correspond to the elements in $H^2(G, M)$. In our case,\nthe same theory tells us that  $H^2_{sm}(G, \\mathrm{Hom}(V_0, V_0))$\nclassifies  the equivalence classes of splitting extensions of $G$\nby the $G$-module $\\mathrm{Hom}(V_0, V_1)$, which is a splitting short exact\nsequence of Lie groups, with $\\mathrm{Hom}(V_0, V_1)$ endowed with an\nabelian group structure,\n\\begin{equation}\\label{eq:extension} \\mathrm{Hom}(V_0, V_1) \\to \\hat{G} \\to G. \\end{equation}\nIn a general extension $\\hat{G}$ is a principal bundle over $G$,\nthus it usually does not permit a smooth lift $G\n\\xrightarrow{\\sigma} \\hat{G}$. It permits such a lift if and only if\nthe sequence splits. However in our case, since the abelian group\n$\\mathrm{Hom}(V_0, V_1) $ is a vector space, we have  $H^1(X,\\mathrm{Hom}(V_0, V_1)\n)=0$ for any manifold $X$. The proof makes use of a partition of\nunity and similar to the proof showing that $H^1(X,\n\\underline{\\mathbb R}\\newcommand{\\Z}{\\mathbb Z})=0$ for the sheaf cohomology. Hence all $ \\mathrm{Hom}(V_0,\nV_1)$ principal bundles are trivial. Therefore \\eqref{eq:extension}\nalways splits. On the other hand\n it is well-known that when $G$ is simply connected, there is a\n one-to-one correspondence between extensions of $G$\n \\cite[Theorem 4.15]{brahic} and extensions of its Lie algebra\n $\\g$, which in turn are classified by the Lie algebra cohomology\n $H^2(\\g, \\mathrm{Hom}(V_0, V_1))$. Hence in our case the differentiation\n map\n\\[ H^2_{sm} (G, \\mathrm{Hom}(V_0, V_1)) \\to H^2(\\g, \\mathrm{Hom}(V_0, V_1)) \\]\nis an isomorphism. Hence $\\nu$  always integrates to a smooth Lie\ngroup 2-cocycle unique up to exact 2-cocycles. Then\n$\\overline{F_2}(1_G,1_G)=0$ can be arranged too, because we can\nalways modify the section $\\sigma: G \\to \\hat{G}$ to satisfy\n$\\sigma(1_G)=1_{\\hat{G}}$ and the modification of sections results\nin an exact term. Then combined with (\\ref{eqn:F2 closed}), it is\nnot hard to see that\n\\begin{equation}\\label{temp1}\n\\overline{F_2}(1_G,g)=\\overline{F_2}(g,1_G)=0,\\quad\\forall~g\\in\nG.\\end{equation} Thus $\\overline{F_2}$ is a normalized 2-cocycle.\n\n\n\n\\begin{pro}\\label{pro:int mu nu }\nFor any 2-term representation up to homotopy $(\\mu,\\nu)$ of a Lie\nalgebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ on the complex\n$V_1\\stackrel{0}{\\longrightarrow}V_0$, there is an associated\nrepresentation up to homotopy $(F_1,F_2)$ of the Lie group $G$ on\nthe complex $V_1\\stackrel{0}{\\longrightarrow}V_0$, where $F_1$ is\nthe integration of $\\mu$ and $F_2:G\\times G\\longrightarrow\n\\mathrm{End}(V_0,V_1)$ is defined by\n\\begin{equation}\\label{eqn:F2}\nF_2(g_1,g_2)=\\overline{F_2}(g_1,g_2)\\circ F_1(g_1\\cdot g_2).\n\\end{equation}\n\\end{pro}\n\\noindent{\\bf Proof.}\\  Obviously, (\\ref{eqn:F fail}) is satisfied. To see (\\ref{eqn:F\nclosed}) is also satisfied, combine  (\\ref{eqn:F2}) with\n(\\ref{eqn:F2 closed}). By the fact that $F_1$ is a homomorphism, we\nobtain\n\\begin{eqnarray*}\n&F_1(g_1)\\circ F_2(g_2,g_3)\\circ F_1(g_2\\cdot g_3)^{-1}\\circ\nF_1(g_1)^{-1}-F_2(g_1\\cdot g_2,g_3)\\circ F_1(g_1\\cdot g_2\\cdot\ng_3)^{-1}\\\\&+F_2(g_1, g_2\\cdot g_3)\\circ F_1(g_1\\cdot g_2\\cdot\ng_3)^{-1}-F_2(g_1,g_2)\\circ F_1(g_1\\cdot g_2)^{-1}=0.\n\\end{eqnarray*}\nComposed with $F_1(g_1\\cdot g_2\\cdot g_3)$ on the right hand side,\nwe obtain (\\ref{eqn:F closed}). \\hfill ~\\vrule height6pt width6pt depth0pt\n\nBy Proposition \\ref{pro:int mu nu } and Theorem \\ref{thm:main 1}, we\nhave\n\\begin{thm}\\label{thm:main 2}\nLet $G$ be the simply connected Lie group integrating $\\g$, then the\nstring Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ given by (\\ref{Lie\n2 g g dual}) integrates to the following Lie 2-group,\n\\begin{equation}\\label{2 group}\\begin{array}{c}\nG\\times \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*\\times \\mathbb R\\\\\n\\vcenter{\\rlap{s }}~\\Big\\downarrow\\Big\\downarrow\\vcenter{\\rlap{t }}\\\\\nG\\times \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\n \\end{array}\\end{equation}\nin which the source and target are given by \\be\ns(g,\\xi,m)=t(g,\\xi,m)=(g,\\xi)\\ee the vertical multiplication\n$\\cdot_\\mathrm{v}$ is given by\n$$\n(h,\\eta,n)\\cdot_\\mathrm{v}(g,\\xi,m) =(g,\\xi,m+n),\\quad \\mbox{where}~\nh=g,\\eta=\\xi,\n$$\nthe horizontal multiplication $\\cdot_\\mathrm{h}$ of objects is given by\n$$\n (g_1,\\xi)\\cdot_\\mathrm{h} (g_2,\\eta)=(g_1\\cdot\ng_2,\\xi+\\mathrm{Ad}^*_{g_1}\\eta),\n$$\nthe horizontal multiplication $\\cdot_\\mathrm{h}$ of morphisms is given by\n$$ (g_1,\\xi,m)\\cdot_\\mathrm{h} (g_2,\\eta,n)=(g_1\\cdot\ng_2,\\xi+\\mathrm{Ad}^*_{g_1}\\eta,m+n),\n$$\nthe inverse map $\\mathrm{inv}$ is given by $$\n\\mathrm{inv}(g,\\xi)=(g^{-1},-\\mathrm{Ad}^*_{g^{-1}}\\xi), $$ The\nidentity object is $(1_G,0)$,\\\\\nthe  associator\n$$\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}:\\big((g_1,\\xi)\\cdot_\\mathrm{h}(g_2,\\eta)\\big)\\cdot_\\mathrm{h}(g_3,\\gamma)\\longrightarrow\n(g_1,\\xi)\\cdot_\\mathrm{h}\\big((g_2,\\eta)\\cdot_\\mathrm{h}(g_3,\\gamma)\\big)\n$$\nis given by \\be\\label{associator}\na_{(g_1,\\xi),(g_2,\\eta),(g_3,\\gamma)}=(g_1\\cdot g_2\\cdot\ng_3,\\xi+\\mathrm{Ad}^*_{g_1}\\eta+\\mathrm{Ad}^*_{g_1\\cdot\ng_2}\\gamma,F_2(g_1,g_2)(\\gamma)), \\ee\nAll the other structures are identity isomorphisms.\n\\end{thm}\n\\noindent{\\bf Proof.}\\  Since $F_1$ is a usual associative action, we may modify the\nunit (\\ref{unit}) given in Theorem \\ref{thm:main 1} to be the\nidentity natural transformation. It turns out that (\\ref{2 group})\nis a special Lie 2-group and is represented by $(G\\ltimes\n\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R,\\rm{Id},\\widetilde{F_2})$, where $G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ is\nthe semidirect product with the coadjoint action of $G$ on $\\g^*$,\n$\\rm{Id}$ is the constant map $G\\ltimes \\g^* \\to \\mathrm{Aut}(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$ by mapping\neverything to $\\rm{Id} \\in \\mathrm{Aut}(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)$, and  $\\widetilde{F_2}$ is given by\n\\begin{equation}\\label{eq:tf2}\n\\widetilde{F_2}((g_1,\\xi_1),(g_2,\\xi_2),(g_3,\\xi_3))=F_2(g_1,\ng_2)(\\xi_3)=\\overline{F_2}(g_1, g_2)\\circ F_1(g_1\\cdot g_2)(\\xi_3).\n\\end{equation} Since $\\overline{F_2}$ is normalized, $\\widetilde{F_2}$ is also\nnormalized.  The string Lie 2-algebra $\\mathbb\nR\\stackrel{0}{\\longrightarrow}\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*$ is  skeletal and\nis represented by $(\\mathfrak g}\\newcommand{\\g}{\\mathfrak g\\oplus\\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb\nR,0,\\widetilde{\\nu})$, where $\\widetilde{\\nu}$ is given by\n(\\ref{eqn:nu3}). Thus to show that our Lie 2-group is an integration\nof the string Lie algebra, we only need to show that the\ndifferential of the Lie group 3-cocycle $\\widetilde{F_2}$ is the Lie\nalgebra 3-cocycle $\\widetilde{\\nu}$.  By direct computations\n\\cite[Lemma 7.3.9]{brylinski}, we have\n\\begin{eqnarray*}\n&&\\frac{\\partial^3}{\\partial_{t_1}\\partial_{t_2}\\partial_{t_3}}\\Big|_{t_i=0}\\sum_{\\sigma\\in\nS_3}\\epsilon(\\sigma)\\widetilde{F_2}\\big((e^\n{t_{\\sigma(1)}X_{\\sigma(1)}},t_{\\sigma(1)}\\xi_{\\sigma(1)}),(e^\n{t_{\\sigma(2)}X_{\\sigma(2)}},t_{\\sigma(2)}\\xi_{\\sigma(2)}),(e^\n{t_{\\sigma(3)}X_{\\sigma(3)}},t_{\\sigma(3)}\\xi_{\\sigma(3)})\\big)\\\\\n&=&\\frac{\\partial^3}{\\partial_{t_1}\\partial_{t_2}\\partial_{t_3}}\\Big|_{t_i=0}\\Big(\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{ t_1X_1}\\cdot e^\n{t_2X_2})(t_3\\xi_3)\\Big)+c.p.\\\\\n&=&\\frac{\\partial^2}{\\partial_{t_1}\\partial_{t_2}}\\Big|_{t_i=0}\\Big(\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{ t_1X_1}\\cdot e^ {t_2X_2})(\\xi_3)\\Big)+c.p.\\\\\n&=&\\frac{\\partial}{\\partial_{t_1}}\\Big|_{t_1=0}\\Big(\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}\\overline{F_2}(e^{\nt_1X_1},e^ {t_2X_2})\\circ F_1(e^{\nt_1X_1})(\\xi_3)+\\overline{F_2}(e^{t_1X_1},1_G)\\circ\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}F_1(e^{\nt_1X_1}\\cdot e^{ t_2X_2})\\Big)\\\\&&+c.p.\\\\\n&=&\\frac{\\partial}{\\partial_{t_1}}\\frac{\\partial}{\\partial_{t_2}}\\Big|_{t_i=0}\\overline{F_2}(e^{\nt_1X_1},e^\n{t_2X_2})(\\xi_3)+\\frac{\\partial}{\\partial_{t_2}}\\big|_{t_2=0}\\overline{F_2}(1_G,e^\n{t_2X_2})\\circ \\frac{\\partial}{\\partial_{t_1}}\\Big|_{t_1=0}F_1(e^{\nt_1X_1})(\\xi_3)\\\\\n&&+c.p.\\quad\\mbox{by (\\ref{temp1})}\\\\\n&=&\\nu(X_1,X_2)(\\xi_3)+c.p.\\\\\n&=&\\widetilde{\\nu}(X_1+\\xi_1,X_2+\\xi_2,X_3+\\xi_3)  \\quad\\mbox{by\n(\\ref{eqn:nu3})},\n\\end{eqnarray*}\nwhich completes the proof. \\hfill ~\\vrule height6pt width6pt depth0pt\n\\begin{cor}\nIf Lie algebra $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is semisimple, the Lie group 3-cocycle\n$\\widetilde{F_2}$ is not exact, i.e. $[\\widetilde{F_2}]\\neq0$\nin $H^3_{sm}(G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)$.\n\\end{cor}\n\\noindent{\\bf Proof.}\\  By Theorem \\ref{thm:main 2}, the differentiation of the Lie\ngroup 3-cocycle $\\widetilde{F_2}$ is the Lie algebra 3-cocycle\n$\\widetilde{\\nu}$. We only need to show that when $\\mathfrak g}\\newcommand{\\g}{\\mathfrak g$ is\nsemisimple, the Lie algebra 3-cocycle $\\widetilde{\\nu}$ is not\nexact. This fact is proved in Proposition \\ref{pro:nondegenerate}.\n\\hfill ~\\vrule height6pt width6pt depth0pt\n\n\\begin{remark}Since $G\\ltimes \\g^*$ is a fibration over $G$, the spectral sequence with $E_2^{p,q}=H^p_{sm}(G, H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z))$ calculates the group cohomology $H^3_{sm}(G\\ltimes\n  \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)$. Since $\\g^*$ is an abelian group, $H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\wedge^q\\g^*$. Thus when $G$ is compact, $ H^p_{sm}(G, H^q_{sm}( \\g^*,\n  \\mathbb R}\\newcommand{\\Z}{\\mathbb Z))  = (\\wedge^q\\g^*)^G$ if $p=0$ and 0 otherwise, where\n  $(\\wedge^q\\g^*)^G$ denotes the set of invariant elements of\n  $\\wedge^q \\g^*$  under the coadjoint action of $G$. Thus when $G$ is compact, $H^3_{sm}(G\\ltimes \\mathfrak g}\\newcommand{\\g}{\\mathfrak g^*,\\mathbb R)=(\\wedge^3\ng^*)^G \\neq 0$ because the Cartan 3-form is an element of\n$(\\wedge^3 g^*)^G$.\n\\end{remark}\n\nNotice that our 2-cocycle $\\overline{F_2}$ is unique only up to\nexact terms. Hence by Theorem \\ref{thm:special 2 G},  to verify that our\nconstruction is unique up to isomorphism, we need the following\nlemma,\n\n\\begin{lemma}\nIf $\\overline{F_2}= \\mathrm{d} \\alpha$ is exact, then $\\widetilde{F_2}=\\mathrm{d} \\beta$ is also exact\nwith\n\\[\\beta((g_1, \\xi_1), (g_2, \\xi_2))=\\alpha(g_1)F_1(g_1)(\\xi_2). \\]\n\\end{lemma}\n\\begin{proof}\nIt is a direct calculation. Since $\\overline{F_2}= \\mathrm{d} \\alpha$, we have\n\\[\\overline{F_2}(g_1, g_2) = F_1 (g_1) \\alpha(g_2) F_1(g_1)^{-1} - \\alpha(g_1\ng_2) + \\alpha(g_1). \\]\nFrom the definition of $F_2$, we know that\n\\[ F_2(g_1, g_2)= F_1(g_1) \\alpha(g_2) F_1(g_1)^{-1} F_1(g_1 g_2) -\n\\alpha(g_1 g_2) F_1(g_1 g_2) + \\alpha(g_1) F_1(g_1g_2). \\]\nBy \\eqref{eq:tf2}, we have\n\\[\n\\begin{split}\n\\widetilde{F_2}((g_1, \\xi_1), (g_2, \\xi_2),(g_3, \\xi_3))= &F_1(g_1) \\alpha(g_2)\nF_1(g_1)^{-1} F_1(g_1g_2) (\\xi_3) - \\alpha(g_1g_2) F_1(g_1g_2) (\\xi_3)\n+ \\alpha(g_1) F_1(g_1g_2) (\\xi_3) \\\\\n= &\\mathrm{d} \\beta((g_1, \\xi_1), (g_2, \\xi_2),(g_3, \\xi_3)),\n\\end{split}\n \\] since $F_1$ is a group homomorphism.\n\\end{proof}\n\n\\begin{remark}\nOur Lie 2-group as a stacky group has the underlying differential\nstack $G \\times \\g^* \\times B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z$. Thus it is  0,1,2-connected (i.e.\nit has $\\pi_0=\\pi_1=\\pi_2=0$) since $\\pi_2(B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\pi_1(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=0$ and\n$\\pi_1(B\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=\\pi_0(\\mathbb R}\\newcommand{\\Z}{\\mathbb Z)=0$. Thus it is the unique 0,1,2-connected\nstacky Lie group integrating the string Lie 2-algebra $\\mathbb R}\\newcommand{\\Z}{\\mathbb Z\n\\xrightarrow{0} \\g\\oplus \\g^*$ in the sense of \\cite{z:lie2}.\n\\end{remark}\n\\bibliographystyle{habbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}