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+{"text":"\n\\section{Summary}\n\nThe CMS Collaboration has searched for narrow resonances in the\ninvariant mass spectrum of  dimuon and dielectron final\nstates in event samples corresponding to an integrated luminosity of\n$40$\\pbinv and $35$\\pbinv, respectively. The spectra are consistent\nwith standard model expectations and upper limits on the\ncross section times branching fraction for $\\cPZpr$ into\nlepton pairs relative to standard model $\\cPZ$ boson\nproduction have been set.\nMass limits have been set on neutral gauge bosons $\\cPZpr$\nand RS Kaluza--Klein gravitons $\\GKK$.  A $\\cPZpr$ with\nstandard-model-like couplings can be excluded below 1140\\GeV, the\nsuperstring-inspired $\\ZPPSI$ below 887\\GeV, and RS\nKaluza--Klein gravitons below 855 (1079)\\GeV for couplings of\n0.05 (0.10), all at 95\\%~C.L.\nThe higher centre of mass energy used in this search, compared to that\nof previous experiments, has resulted in limits that are comparable to or\nexceed those previously published, despite the much lower\nintegrated luminosity accumulated at the LHC thus far.\n\n\\section*{Acknowledgments}\n\nWe wish to congratulate our colleagues in the CERN accelerator\ndepartments for the excellent performance of the LHC machine. We thank\nthe technical and administrative staff at CERN and other CMS\ninstitutes, and acknowledge support from: FMSR (Austria); FNRS and FWO\n(Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria);\nCERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES\n(Croatia); RPF (Cyprus); Academy of Sciences and NICPB (Estonia);\nAcademy of Finland, ME, and HIP (Finland); CEA and CNRS\/IN2P3\n(France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH\n(Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN\n(Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT,\nSEP, and UASLP-FAI (Mexico); PAEC (Pakistan); SCSR (Poland); FCT\n(Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MST\nand MAE (Russia); MSTD (Serbia); MICINN and CPAN (Spain); Swiss\nFunding Agencies (Switzerland); NSC (Taipei); TUBITAK and TAEK\n(Turkey); STFC (United Kingdom); DOE and NSF (USA).\n\n\n\n\\subsection{\\texorpdfstring{\\cPZ\/$\\gamma^*$}{Z\/gamma*} Backgrounds}\n\nThe shape of the dilepton invariant mass spectrum is obtained from\nDrell--Yan production using a MC simulation based on the\n\\PYTHIA event generator.   The simulated\nspectrum at the invariant mass peak of the $\\cPZ$ boson is normalized to\nthe data.  The dimuon analysis uses the data events in the $\\cPZ$\nmass interval of 60--120\\GeV; the dielectron analysis\nuses data events in the narrower interval of 80--100\\GeV in order to\nobtain a comparably small background contamination.\n\nA contribution to the uncertainty attributed to the extrapolation of the\nevent yield and the shape of the Drell--Yan background to high\ninvariant masses arises from higher order QCD corrections.\nThe next-to-next-to-leading order (NNLO) $k$-factor is computed using\n{\\sc FEWZz v1.X}~\\cite{Melnikov:2006kv}, with \\PYTHIA~{\\sc v6.409} and\n{\\sc CTEQ6.1} PDF~\\cite{Stump:2003yu}\nas a baseline. It is found that the variation of the\n$k$-factor with mass does not exceed 4\\% where\nthe main difference arises from the comparison of \\PYTHIA and {\\sc FEWZz} calculations.\nA further source of uncertainty arises from the PDFs. The  {\\sc lhaglue}~\\cite{Bourilkov:2003kk}\ninterface to the {\\sc lhapdf-5.3.1}~\\cite{Whalley:2005nh} library is used to evaluate these uncertainties,\nusing the error PDFs from the {\\sc CTEQ6.1} and the MRST2006nnlo~\\cite{Martin:2007bv}\nuncertainty eigenvector sets.\nThe uncertainty on the ratio of the background in\nthe high-mass region to that in the region of the $\\cPZ$ peak is below\n4\\% for both PDF sets and masses below 1 TeV. Combining the higher order QCD and PDF uncertainties in\nquadrature, the resulting uncertainty in the number of events normalized to those expected at the\n$\\cPZ$ peak is about 5.7\\% for masses between 200\\GeV and 1\\TeV.\n\n\\subsection{Other Backgrounds with Prompt Lepton Pairs\n\\label{sec:e-mu}  }\n\nThe dominant non-Drell--Yan electroweak contribution to high\n$m_{\\ell\\ell}$ masses is {$\\ttbar$}; in addition there are contributions\nfrom $\\tq\\PW$ and diboson production. In the $\\cPZ$ peak\nregion, $\\cPZ \\rightarrow \\Pgt\\Pgt$ decays also contribute.\nAll these processes are flavour symmetric and produce twice as\nmany $\\Pe\\Pgm$ pairs as $\\Pe\\Pe$ or $\\Pgm\\Pgm$ pairs.\nThe invariant mass spectrum from\n$\\Pe^\\pm\\Pgm^\\mp$ events is expected to have the same shape as that of same\nflavour $\\ell^+\\ell^-$ events but without significant contamination\nfrom Drell--Yan production.\n\nFigure~\\ref{fig:muonselectrons} shows the\nobserved $\\Pe^\\pm\\Pgm^\\mp$ dilepton invariant mass spectrum\nfrom a dataset corresponding to 35\\pbinv, overlaid on\nthe prediction from simulated background processes. This spectrum was\nobtained using the same single-muon trigger as in the dimuon analysis\nand by requiring oppositely charged leptons of different flavour. Using an electron\ntrigger, a very similar spectrum is produced.\nDifferences in the geometric acceptances and efficiencies result in the predicted ratios\nof $\\Pgmp\\Pgmm$ and $\\Pe\\Pe$ to $\\Pe^\\pm\\Pgm^\\mp$ being approximately 0.64\nand 0.50, respectively. In the data, shown in\nFig.~\\ref{fig:muonselectrons}, there are 32 (7) $\\Pe^\\pm\\Pgm^\\mp$ events\nwith invariant mass above 120 (200)\\GeV. This yields an expectation of\nabout 20 (4) dimuon events and 16 (4) dielectron events.  A direct\nestimate from MC simulations of the processes involved\npredicts $20.1\\pm3.6$ $(5.3\\pm 1.0)$ dimuon events and\n$13.2\\pm2.4$ $(3.5\\pm0.6)$ dielectron events. The uncertainty\nincludes  both statistical and systematic contributions, and is dominated\nby the theoretical uncertainty  of 15\\% on the $\\ttbar$ production cross\nsection~\\cite{Campbell:2010ff,Kleiss:1988xr}.\nThe agreement\nbetween the observed and predicted distributions provides a validation\nof the estimated contributions from the backgrounds from prompt leptons\nobtained using MC simulations.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsElectronsOppSign.pdf}\n\\end{center}\n\\caption{\\label{fig:muonselectrons}\nThe observed opposite-sign $\\Pe^\\pm\\Pgm^\\mp$ dilepton invariant mass spectrum\n(data points). The uncertainties on the data points (statistical only)  \nrepresent 68\\% confidence intervals for the Poisson means.\nFilled histograms show contributions to the\nspectrum from  $\\ttbar$, other sources of prompt leptons\n ($\\tq\\PW$, diboson production, $\\cPZ\\to\\Pgt\\Pgt$),  and\nthe multi-jet background (from Monte Carlo simulation).\n}\n\\end{figure}\n\n\\subsection{Events with Misidentified and Non-Prompt Leptons}\n\nA further source of background arises when objects are falsely\nidentified as prompt leptons.  The misidentification of jets as\nleptons, the principle source of such backgrounds, is more likely to\noccur for electrons than for muons.\n\nBackgrounds arising from jets that are misidentified as electrons\ninclude $\\PW\\to \\Pe\\cPgn$ + jet events with\none jet misidentified as a prompt electron, and also multi-jet events with\ntwo jets misidentified as prompt electrons.\nA prescaled single EM cluster trigger is used for collecting a sample of\nevents to determine the rate of jets misreconstructed as electrons and\nto estimate the backgrounds from misidentified electrons.\nThe events in this sample are\nrequired to have no more than one\nreconstructed electron, and missing transverse energy of less than 20\\GeV,\nto suppress the contribution from $\\cPZ$\nand $\\PW$ events respectively.\nThe probability for an EM cluster with $H\/E<5\\%$ to be\nreconstructed as an electron is determined in bins of $\\ET$ and $\\eta$ from\na data sample dominated by multi-jet events and is used to weight\nappropriately  events which have two such clusters passing\nthe double EM trigger.\nThe estimated background\ncontribution to the dielectron mass spectrum due to misidentified jets is\n 8.6$\\pm$3.4 (2.1$\\pm$0.8)  for $m_{\\Pe\\Pe} > 120$ (200)\\GeV.\n\nIn order to estimate the residual contribution from background events\nwith at least one non-prompt or misidentified muon, events are\nselected from the data sample with single muons that pass all\nselection cuts except the isolation requirement.\nA map is created, showing the isolation probability for these muons as\na function of $\\pt$ and $\\eta$.\nThis probability map is corrected for the expected\ncontribution from events with single prompt muons from $\\ttbar$ and\n$\\PW$ decays and for the observed correlation between the\nprobabilities for two muons in the same event. The probability map is\nused to predict the number of background events with two isolated\nmuons based on the sample of events that have two non-isolated muons.\nThis procedure, which has been validated using simulated events,\npredicts a mean background to $m_{\\Pgm\\Pgm} > 120$ (200)\\GeV of\n$0.8\\pm0.2\\ (0.2\\pm0.1)$.\n\nAs the signal sample includes the requirement that the muons in the\npair have opposite electric charge, a further cross-check of the\nestimate is performed using events with two isolated muons of the same\ncharge.  There are\nno events with same-charge muon pairs and $m_{\\Pgm\\Pgm} > 120$\\GeV,\na result which is statistically compatible with both the figure of\n $1.6\\pm0.3$ events predicted from SM process using MC simulation and\nthe figure of $0.4\\pm0.1$ events obtained using methods based on data.\n\n\n\\subsection{Cosmic Ray Muon Backgrounds}\n\nThe $\\Pgmp\\Pgmm$ data sample is susceptible to contamination from\ntraversing cosmic ray muons, which may  be misreconstructed as a pair\nof oppositely charged, high-momentum muons.\nCosmic ray events can be removed from the data sample\nbecause of their distinct topology (collinearity of two tracks\nassociated with  the\nsame muon),\nand their uniform distribution of impact parameters with respect to the collision vertex.\nThe residual mean expected background from cosmic ray muons is\nmeasured using sidebands to be less than 0.1 events with\n$m_{\\Pgm\\Pgm} > 120$\\GeV.\n\n\n\\section{Dilepton Invariant Mass Spectra}\n\nThe measured dimuon and dielectron invariant mass spectra are displayed in\nFigs.~\\ref{fig:spectra}(left) and (right) respectively,\nalong with the expected signal from $\\ZPSSM$ with a mass of 750\\GeV.\nIn the dimuon sample, the highest invariant mass\nevent has $m_{\\Pgm\\Pgm}=463$\\GeV, with the $\\pt$ of the two\nmuons measured to be 258 and 185\\GeV.\nThe highest invariant mass event in the dielectron sample\nhas $m_{\\Pe\\Pe}=419$\\GeV, with the electron candidates having $\\ET$ of 125 and 84\\GeV.\n\nThe expectations from the various background sources,\n$\\cPZ{\/}\\gamma^*$, $\\ttbar$, other sources of  prompt leptons ($\\tq\\PW$, diboson\nproduction, $\\cPZ\\to\\Pgt\\Pgt$) and multi-jet events\nare also overlaid in Fig.~\\ref{fig:spectra}.  For the dielectron\nsample, the multi-jet background estimate was obtained directly from\nthe data. The prediction for Drell--Yan production of $\\cPZ{\/}\\gamma^*$\nis normalized to the observed $\\cPZ\\to\\ell\\ell$ signal.  All other MC\npredictions are normalized to the expected cross sections.\nFigures~\\ref{fig:cum_spectra}(left) and (right) show the corresponding\ncumulative distributions of the spectra for the dimuon and\ndielectron samples.  Good agreement is observed between data and the\nexpectation from SM processes over the mass region above the Z peak.\n\nSearches for narrow resonances at the Tevatron~\\cite{D0_Zp,CDF_Zp}\nhave placed lower limits in the mass range $600$\\GeV to\n$1000$\\GeV. The region with dilepton masses $120\\GeV < m_{\\ell\\ell} <\n200\\GeV$ is part of the region for which resonances have been excluded\nby previous experiments, and thus should be dominated by SM processes.  The\nobserved good agreement between the data and the prediction in this\ncontrol region gives confidence that the SM expectations and the detector\nperformance are well understood.\n\nIn the $\\cPZ$~peak mass region defined as $60 < m_{\\ell\\ell} <\n120$\\GeV, the number of dimuon and dielectron candidates are 16\\,515\nand 8\\,768 respectively, with very small backgrounds.  The\ndifference in the electron and muon numbers is due to the higher $E_T$\ncut in the electron analysis and lower electron identification\nefficiencies at these energies.  The expected yields in the control\nregion (120--200\\GeV) and high invariant mass regions ($>$ 200\\GeV) are listed in\nTable~\\ref{tab:event_yield}.  The agreement between the observed data\nand expectations, while not used in the shape-based analysis,\nis good.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsPlusMuonsMinus_log.pdf}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/massHist35ForPaper.pdf}\n\\end{center}\n\\caption{\\label{fig:spectra}\nInvariant mass spectrum of $\\Pgmp\\Pgmm$ (left) and $\\Pe\\Pe$ (right) events. The points with\nerror bars represent the data. The uncertainties on the data points (statistical only) \nrepresent 68\\%  confidence intervals for the Poisson means. The\nfilled histograms represent the expectations from\nSM processes: $\\cPZ{\/}\\gamma^*$, $\\ttbar$, other sources of \nprompt leptons ($\\tq\\PW$, diboson production, $\\cPZ\\to\\Pgt\\Pgt$), and\nthe multi-jet backgrounds.\nThe open histogram  shows the  signal expected for a $\\ZPSSM$ with\na mass of 750\\GeV.\n}\n\\end{figure}\n\n\\begin{figure*}[htbp!]\n\\begin{center}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/MuonsPlusMuonsMinus_cumulative_log.pdf}\n\\includegraphics[angle=90,width=0.49\\textwidth]{figures\/cMassHist35ForPaper.pdf}\n\\end{center}\n\\caption{\\label{fig:cum_spectra} Cumulative distribution of invariant\nmass spectrum of $\\Pgmp\\Pgmm$ (left) and $\\Pe\\Pe$ (right) events.  The\npoints with error bars represent the data, and the filled histogram\nrepresents the expectations from SM processes.  }\n\\end{figure*}\n\n\n\n\\begin{table*}[htb!]\n\\centering\n\\caption{Number of dilepton events with invariant mass in the control\nregion $120~<~m_{\\ell\\ell}~<~200$\\GeV and the search region $m_{\\ell\\ell} > 200$\\GeV.\nThe expected number of $\\cPZpr$ events is given within ranges of 328\\GeV and \n120\\GeV for the dimuon sample and the dielectron sample respectively, centred \non 750\\GeV. The total background is the sum of the SM processes listed.\nThe MC yields are normalized to the expected cross sections. \nUncertainties include both statistical and systematic components\nadded in quadrature.}\n\\label{tab:event_yield}\n\\begin{tabular}{l|c|c|c|c}\n\\hline\\hline\nSource & \\multicolumn{4}{c}{Number of events} \\\\\n       & \\multicolumn{2}{c|}{Dimuon sample }\n       & \\multicolumn{2}{c}{Dielectron sample} \\\\\n &  ($120-200$)\\GeV  & $ >$200\\GeV\n &  ($120-200$)\\GeV  & $ >$200\\GeV  \\\\ \\hline\n CMS data                   &  227             &  35              & 109             & 26            \\\\\n$\\ZPSSM$ (750\\GeV)         & ---              & $15.0 \\pm 1.9$   & ---             & $8.7\\pm1.1$   \\\\\nTotal background           & $204 \\pm 23$     & $36.3 \\pm 4.3$   & $120 \\pm 14$    & $24.4\\pm 3.0$ \\\\ \\hline\n$\\cPZ{\/}\\gamma^*$          & $187 \\pm 23$     & $30.2 \\pm 3.6$   & $104 \\pm 14$    & $18.8\\pm 2.3$   \\\\\n$\\ttbar$                   & $12.3 \\pm 2.3$   & $4.2 \\pm 0.8$    & $7.6 \\pm 1.4$   & $2.7 \\pm 0.5$ \\\\\nOther prompt leptons      & $4.4 \\pm 0.5$    & $1.7 \\pm 0.2$    & $2.1 \\pm 0.2$   & $0.8 \\pm 0.1$  \\\\\nMulti-jet events           & $0.6 \\pm 0.2$    & $0.2 \\pm 0.1$    & $6.5 \\pm 2.6$   & $2.1 \\pm 0.8$  \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\n\n\n\n\n\\section{Event Samples and Selection}\n\nSimulated event samples for the signal and associated backgrounds\nwere generated with the \\PYTHIA~{\\sc v6.422}~\\cite{Sjostrand:2006za}\nMC event generator, and with\n\\MADGRAPH~\\cite{MADGRAPH} and\n{\\sc powheg v1.1}~\\cite{Alioli:2008gx, Nason:2004rx, Frixione:2007vw}\ninterfaced with the \\PYTHIA parton-shower generator \nusing the {\\sc CTEQ6L1}~\\cite{Pumplin:2002vw} PDF set.\nThe response of the detector was simulated in detail using\n\\GEANTfour~\\cite{GEANT4}. These samples were further processed\nthrough the trigger emulation and event reconstruction chain of the CMS\nexperiment.\n\nFor both dimuon and dielectron final states, \ntwo isolated same flavour leptons that pass\nthe lepton identification criteria described in Section~\\ref{sec:leptonID} are required.\nThe two charges are\nrequired to have opposite sign in the case of dimuons (for which a charge\nmisassignment implies a large momentum measurement error), but not in the\ncase of dielectrons (for which charge assignment is decoupled from\nthe ECAL-based energy measurement). \nAn opposite-charge requirement for\ndielectrons would lead to a loss of signal efficiency of a few percent.\n\n\nOf the two muons selected, one is required to satisfy the ``tight''\ncriteria. \nThe electron sample requires at least one\nelectron candidate in the barrel because events with both electrons in the endcaps\nwill have a lower signal-to-background ratio.  \nFor both channels, each event is required to have a\nreconstructed vertex with at least four associated tracks, located\nless than 2~cm from the centre of the detector in the direction\ntransverse to the beam and less than 24\\cm in the direction along the\nbeam. This requirement provides protection against cosmic rays.\nAdditional suppression of cosmic ray muons\nis obtained by requiring the\nthree-dimensional opening angle between the two muons to be smaller than $\\pi - 0.02$ radians.\n\n\n\\section{Introduction}\n\nMany models of new physics predict the existence of narrow\nresonances, possibly at the TeV mass scale, that decay to a pair of\ncharged leptons. This Letter describes a search for resonant signals\nthat can be detected by the Compact Muon Solenoid (CMS) detector at\nthe Large Hadron Collider (LHC)~\\cite{lhc} at CERN. \nThe Sequential Standard Model $\\ZPSSM$ with standard-model-like couplings, the\n$\\ZPPSI$ predicted by grand unified theories~\\cite{Leike:1998wr},\nand Kaluza--Klein graviton excitations arising in \nthe Randall-Sundrum (RS) model of extra\ndimensions~\\cite{Randall:1999vf, Randall:1999ee} were used as benchmarks. \nThe RS model has two free parameters: the mass of the first graviton excitation and \nthe coupling $k\/\\overline{M}_{\\rm Pl}$, \nwhere $k$ is the curvature of the extra dimension and \n$\\overline{M}_{\\rm Pl}$ is the reduced effective Planck scale. \nTwo values of the coupling parameter were considered: $k\/\\overline{M}_{\\rm Pl}$~=~0.05 and 0.1.\nFor a resonance mass of 1\\TeV, the widths are 30, 6\\ and 3.5\\ (14)\\GeV \nfor a $\\ZPSSM$, $\\ZPPSI$, and $\\GKK$ with $k\/\\overline{M}_{\\rm Pl}$~=~0.05 (0.1), respectively.\n\nThe results of searches for narrow $\\cPZpr \\rightarrow \\ell^+ \\ell^-$ and\n$\\GKK \\rightarrow \\ell^+ \\ell^-$ resonances in $\\Pp\\Pap$ collisions at the\nTevatron with over\n5\\fbinv of integrated luminosity at centre-of-mass energy\nof 1.96\\TeV have previously\nbeen reported~\\cite{D0_RS,D0_Zp,CDF_RS,CDF_Zp}. \nIndirect constraints have been placed on the mass of the virtual \n$\\cPZpr$ bosons by LEP-II experiments~\\cite{delphi,aleph,opal,l3}\nby examining the cross sections and angular distribution\nof dileptons and hadronic final states in $\\Pep\\Pem$ collisions.\n\nThe results presented in this Letter were obtained from an analysis of\ndata recorded in 2010, corresponding to an integrated luminosity of $40 \\pm\n4$\\pbinv in the dimuon channel, and $35 \\pm 4$\\pbinv in the dielectron\nchannel, obtained from $\\Pp\\Pp$ collisions at a centre-of-mass energy\nof 7\\TeV.  The total integrated luminosity used for the electron\nanalysis is smaller than that for the muon analysis because of the\ntighter quality requirements imposed on the data.\nThe search for resonances is based on a shape analysis of dilepton mass spectra,\nin order to be robust against uncertainties in the absolute background\nlevel.  By examining the dilepton-mass spectrum from below the $\\cPZ$\nresonance to the highest mass events recorded, we obtain\nlimits on the ratio of the production cross section times branching\nfraction for high-mass resonances to that of the $\\cPZ$.  Using\nfurther input describing the dilepton mass dependence on effects of\nparton distribution functions (PDFs) and $k$-factors,\nmass bounds are calculated for specific models. \nIn addition, model-independent limit contours are determined in the\ntwo-parameter $(c_d,c_u)$ plane~\\cite{Carena:2004xs}. Selected\nbenchmark models for $\\cPZpr$ production are illustrated in this plane,\nwhere where $c_u$ and $c_d$ are model-dependent couplings of the\n$\\cPZpr$ to up- and down-type quarks, respectively allowing lower bounds \nto be determined. \n\n\n\\section{The CMS Detector}\n\nThe central feature of the CMS~\\cite{JINST} apparatus is a\nsuperconducting solenoid, of 6~m internal diameter, providing an axial field\nof 3.8~T. Within the field volume are the silicon pixel and strip\ntrackers, the crystal electromagnetic calorimeter (ECAL) and the\nbrass\/scintillator hadron calorimeter (HCAL). The endcap hadronic\ncalorimeters are segmented in the z-direction.  Muons are measured in\ngas-ionization detectors embedded in the steel return yoke.  In\naddition to the barrel and endcap detectors, CMS has extensive forward\ncalorimetry.\n\n\nCMS uses a right-handed coordinate system, with the origin at the\nnominal interaction point, the $x$-axis pointing to the centre of the\nLHC, the $y$-axis pointing up (perpendicular to the LHC plane), and\nthe $z$-axis along the anticlockwise-beam direction. The polar angle,\n$\\theta$, is measured from the positive $z$-axis and the azimuthal\nangle, $\\phi$, is measured in the $x$-$y$ plane.\n\nMuons are measured in the pseudorapidity range $|\\eta|< 2.4$, with\ndetection planes based on one of three technologies: drift tubes in the barrel\nregion, cathode strip chambers in the endcaps, and resistive plate chambers\nin the barrel and part of the endcaps.\nThe inner tracker (silicon pixels and strips) detects charged particles within the pseudorapidity range\n$|\\eta| < 2.5$.\n\nThe electromagnetic calorimeter consists of nearly 76\\,000 lead\ntungstate crystals which provide coverage in pseudorapidity $|\\eta| < 1.479$\nin the barrel region (EB, with crystal size\n$\\Delta\\eta=0.0174$ and $\\Delta\\phi = 0.0174$) and $1.479 < |\\eta| <\n3.0$ in the two endcap regions (EE, with somewhat larger crystals.).\nA preshower detector\nconsisting of two planes of silicon sensors interleaved with a total of\n3$\\,X_0$ of lead is located in front of the EE.\n\nThe first level (L1) of the CMS trigger system, composed of custom\nhardware processors, selects the most interesting events\nusing information from the calorimeters and muon detectors.\nThe High Level Trigger (HLT) processor farm further decreases the\nevent rate employing the full event information, including the\ninner tracker.  The muon selection algorithms in the HLT use information from the\nmuon detectors and the silicon pixel and strip trackers.  The electromagnetic (EM)\nselection algorithms use the energy deposits in the ECAL and HCAL; the\nelectron selection in addition requires tracks matched to clusters. Events\nwith muons or electromagnetic clusters with $\\pt$ above L1 and HLT\nthresholds are recorded.\n\n\n\\section{Electron and Muon Selection \\label{sec:lepton}}\n\n\n\\subsection{Triggers \\label{sec:triggers}}\n\nThe events used in the dimuon channel analysis were collected using a\nsingle-muon trigger. \nThe algorithm requires a muon candidate to be found in the muon detectors by the L1 trigger.\nThe candidate track is then matched to a silicon tracker track, forming an HLT muon.\nThe HLT muon is required to have $\\pt >9$ to 15\\GeV, depending on the running period.  \n\nA double EM cluster trigger was used to select\nthe events for the dielectron channel.  ECAL clusters are formed by\nsumming energy deposits in crystals surrounding a ``seed'' that is\nlocally the highest-energy crystal. \nThe clustering algorithm takes into account the emission of bremsstrahlung.\nThis trigger requires two clusters with the ECAL transverse energy $\\ET$\nabove a threshold of 17 to 22\\GeV, depending on the running period. \nFor each of these clusters, the ratio $H\/E$, where $E$ is the energy of the ECAL cluster and\n $H$ is the energy in the HCAL cells situated behind it, is required to be less than 15\\%.\nAt least one of these clusters must\nhave been associated with an energy deposit identified by the L1 trigger.\n\n\\subsection{Lepton Reconstruction\\label{sec:leptonID}}\n\nThe reconstruction, identification, and calibration of muons\nand electrons\nfollow standard CMS methods~\\cite{EWK-10-002-PAS}.\nCombinations of test beam, cosmic ray muons, and \ndata from proton collisions \nhave been used to calibrate the relevant detector systems\nfor both muons and electrons.\n\n\nMuons are reconstructed independently as tracks in both the muon\ndetectors and the silicon tracker~\\cite{MUONPAS}. \nThe two tracks can be matched and fitted simultaneously to\nform a ``global muon''. Both muons in the event must be identified as\nglobal muons, with at least 10 hits in the silicon tracker and with\n$\\pt > 20$\\GeV.  All muon candidates that satisfy these criteria\nare classified as ``loose'' muons.  At least one of the two muons in\neach event must be further classified as a ``tight'' muon by passing the\nfollowing additional requirements: a transverse impact parameter with\nrespect to the collision point less than 0.2~cm; a $\\chi^2$ per degree\nof freedom less than 10 for the global track fit; at least one hit in\nthe pixel detector; hits from the muon tracking system in at least two\nmuon stations on the track; and correspondence with the single-muon trigger.\n\nElectrons are reconstructed by associating a cluster in the ECAL with\na track in the tracker~\\cite{EGMPAS}.  Track reconstruction, which \nis specific to\nelectrons to account for bremsstrahlung emission,  is seeded from the\nclusters in the ECAL, first using the cluster position and energy to\nsearch for compatible hits in the pixel detector, and then using these\nhits as seeds to reconstruct a track in the silicon tracker. A minimum\nof five hits is required on each track.  Electron candidates are required to be\nwithin the barrel or endcap acceptance regions, with pseudorapidities\nof $|\\eta|<1.442$ and $1.560<|\\eta|<2.5$, respectively. A candidate\nelectron is required to deposit most of its energy in the ECAL and\nrelatively little in the HCAL ($H\/E<5\\%$). The transverse shape of the\nenergy deposit is required to be consistent with that expected for an\nelectron, and the associated track must be well-matched in $\\eta$ and\n$\\phi$. Electron candidates must have $\\ET > 25$~GeV.\n\nIn order to suppress misidentified leptons from jets and non-prompt muons from\nhadron decays, both lepton selections impose isolation requirements.\nCandidate leptons are required to be isolated within a narrow cone of\nradius $\\Delta R = \\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2} = 0.3$,\ncentred on the lepton. Muon isolation requires that the sum of the\n$\\pt$ of all tracks within the cone, excluding the muon,\nis less than 10\\% of the $\\pt$ of the muon.\nFor electrons, the sum of the $\\pt$ of the tracks, excluding\nthe tracks within an inner cone of $\\Delta R = 0.04$, is required to be less than 7$\\GeV$\nfor candidates reconstructed within the barrel acceptance and 15$\\GeV$\nwithin the endcap acceptance. The calorimeter isolation\nrequirement for electron candidates within the barrel acceptance is that,\nexcluding the $\\ET$ of the candidate, the sum of the $\\ET$ resulting from\ndeposits in the ECAL and the HCAL within a cone of $\\Delta R=0.3$ be less than\n0.03$\\ET$ + 2\\GeV. For candidates within the endcap acceptance, the\nsegmentation of the HCAL in the $z$-direction is exploited. For candidates with $\\ET$\nbelow 50\\GeV (above 50\\GeV), the isolation energy is required to be\nless than 2.5\\GeV ($0.03(\\ET-50) + 2.5\\GeV)$, where $\\ET$ is\ndetermined using the ECAL and the first layer of the segmented HCAL. The\n$\\ET$ in the second layer of the HCAL is required to be less than\n0.5\\GeV.  These requirements ensure that the candidate electrons are\nwell-measured and have minimal contamination from jets.\n\nThe performance of the detector systems for the data sample presented in this paper is \nestablished using measurements of standard model (SM) $\\PW$ and $\\cPZ$ processes \nwith leptonic final states~\\cite{EWK-10-002-PAS} \nand using traversing cosmic ray muons~\\cite{CMS_CFT_09_014}.\n\nMuon momentum resolution varies from 1\\% at momenta\nof a few tens of \\GeV to 10\\% at momenta of several hundred \\GeV,\nas verified with measurements made with cosmic rays.\nThe alignment of the muon and inner tracking systems is important\nfor obtaining the best momentum resolution, and hence mass resolution,\nparticularly at the high masses relevant to the $\\cPZpr$ search.\nAn additional contribution to the momentum\nresolution arises from the presence of distortion modes in the tracker\ngeometry that are not completely constrained by the alignment procedures.\nThe dimuon mass resolution is estimated to have an rms of\n5.8\\% at 500\\GeV and 9.6\\% at 1\\TeV.\n\nThe ECAL has an ultimate energy resolution of better than $0.5\\%$ for\nunconverted photons with transverse energies above $100\\GeV$.\nThe ECAL energy resolution obtained thus far is\non average 1.0\\% for the barrel and 4.0\\% for the endcaps.\nThe mass resolution is estimated to be\n1.3\\% at 500\\GeV and 1.1\\% at 1\\TeV.\nElectrons from $\\PW$ and $\\cPZ$ bosons were used to calibrate ECAL\nenergy measurements.  \nFor both muons and electrons, the energy scale\nis set using the $\\cPZ$ mass peak, except for electrons in the barrel\nsection of the ECAL, where the energy scale is set using neutral pions,\nand then checked using the $\\cPZ$ mass peak.\nThe ECAL energy scale uncertainty is 1\\% in the barrel and 3\\% in the\nendcaps.\n\n\n\n\\subsection{Efficiency Estimation \\label{sec:eff}}\n\nThe efficiency for identifying and reconstructing lepton candidates is\nmeasured with the tag-and-probe method~\\cite{EWK-10-002-PAS}.\nA tag lepton is established by applying tight cuts\nto one lepton candidate; the other candidate is used as a probe. A\nlarge sample of high-purity probes is obtained by requiring that the\ntag-and-probe pair have an invariant mass consistent with the $\\cPZ$\nboson mass ($80 < m_{\\ell\\ell} < 100\\GeV)$.  \nThe factors contributing to the overall efficiency are measured in the data. They are:\nthe trigger efficiency, the reconstruction efficiency in the silicon tracker,\nthe electron clustering efficiency, and the lepton reconstruction and\nidentification efficiency. All efficiencies and scale factors\nquoted below are computed\nusing events in the $\\cPZ$ mass region.\n\nThe trigger efficiencies are defined relative to the full offline\nlepton requirements.  For the dimuon events, the efficiency of the\nsingle muon trigger with respect to loose muons is measured to be\n$89\\% \\pm 2\\%$~\\cite{EWK-10-002-PAS}.  The overall efficiency, defined\nwith respect to particles within the physical acceptance of the\ndetector, for loose (tight) muons is measured to be $94.1\\%\\pm1.0\\%$\n($81.2\\%\\pm1.0\\%$).  Within the statistical precision allowed by the\ncurrent data sample, the dimuon efficiency is constant as a function of\n$\\pt$ above 20\\GeV, as is the ratio of the efficiency in the data to that in the Monte Carlo (MC)\nof 0.977 $\\pm$ 0.004. For dielectron events, the double EM cluster trigger is\n100\\% efficient (99\\% during the early running period).  The total\nelectron identification efficiency is 90.1\\% $\\pm$ 0.5\\%\\ (barrel) and 87.2\\% $\\pm$ 0.9\\%\\ (endcap). \nThe ratio of the electron efficiency measured from the data to that\ndetermined from MC simulation at the $\\cPZ$ resonance is 0.979 $\\pm$ 0.006\\ (EB)\nand 0.993 $\\pm$ 0.011\\ (EE).  To determine the efficiency applicable to\nhigh-energy electrons in the data sample, this correction factor is\napplied to the efficiency found using MC simulation.\nThe efficiency of electron identification increases as a\nfunction of the electron transverse energy until it becomes flat beyond an\n$\\ET$ value of about 45\\GeV. Between 30 and 45 GeV it increases by about 5\\%.\n\n\n\\section{Limits on the Production Cross Section}\n\nThe observed invariant mass spectrum agrees with expectations based on\nstandard model processes, therefore limits are set on the possible\ncontributions from a narrow heavy resonance.\nThe parameter of interest is the ratio of the products of cross sections\nand branching fractions:\n\\begin{equation}\n\\label{eq:rsigma}\nR_\\sigma = \\frac{\\sigma(\\Pp\\Pp\\to \\cPZpr+X\\to\\ell\\ell+X)}\n                {\\sigma(\\Pp\\Pp\\to \\cPZ+X       \\to\\ell\\ell+X)}.\n\\end{equation}\n\nBy focusing on the ratio\n$R_\\sigma$, we eliminate the uncertainty in the integrated\nluminosity, reduce the dependence on experimental\nacceptance, trigger, and offline efficiencies, and generally obtain a more robust result.\n\nFor statistical inference about $R_\\sigma$, we first estimate the Poisson mean\n$\\mu_\\cPZ$ of the number of $\\cPZ\\to\\ell\\ell$ events in the sample\nby counting the number of events in the $\\cPZ$ peak mass region and\ncorrecting for a small ($\\sim 0.4\\%$) background contamination (determined with MC simulation).\nThe uncertainty on $\\mu_\\cPZ$ is about 1\\% (almost all statistical) and contributes negligibly to\nthe uncertainty on $R_\\sigma$.\n\nWe then construct an extended unbinned likelihood function for the\nspectrum of $\\ell\\ell$ invariant mass values $m$ above\n200\\GeV, based on a sum of analytic probability density functions\n(pdfs) for the signal and background shapes.\n\nThe pdf $f_\\mathtt{S}(m|\\Gamma,M,w)$ for the resonance signal is a\nBreit-Wigner of width $\\Gamma$ and mass $M$ convoluted with a Gaussian\nresolution function of width $w$ (section~\\ref{sec:leptonID}). The width $\\Gamma$ is taken to\nbe that of the $\\ZPSSM$ (about 3\\%); as noted below, the high-mass limits\nare insensitive to this width.\nThe Poisson mean of\nthe yield is $\\mu_\\mathtt{S} = R_\\sigma \\cdot \\mu_\\cPZ \\cdot R_\\epsilon$,\nwhere $R_\\epsilon$ is the ratio of selection efficiency times detector\nacceptance for $\\cPZpr$ decay to that of $\\cPZ$ decay; $\\mu_\\mathtt{B}$\ndenotes the Poisson mean of the total background yield.\nA background\npdf $f_\\mathtt{B}$ was chosen and its shape parameters fixed by\nfitting to the simulated Drell--Yan spectrum in the mass range\n$200 < m_{\\ell\\ell} < 2000$\\GeV.\nTwo functional forms for the dependence of $f_\\mathtt{B}$\non shape parameters $\\alpha$ and $\\kappa$ were tried:\n$f_\\mathtt{B}(m|\\alpha,\\kappa) \\sim  \\exp(-\\alpha m^\\kappa)$ and\n$\\sim \\exp(-\\alpha m)m^{-\\kappa}$.  Both\nyielded good  fits and consistent results for both the dimuon and dielectron spectra.\nFor definiteness, this Letter presents results obtained with the latter form.\n\nThe extended likelihood ${\\cal L}$ is then\n\n\\begin{equation}\n\\label{eq:likelihood}\n{\\cal L}({\\boldsymbol m}|R_\\sigma,M,\\Gamma,w,\\alpha,\\kappa,\\mu_\\mathtt{B}) =\n\\frac{\\mu^N e^{-\\mu}}{N!}\\prod_{i=1}^{N}\\left(\n\\frac{\\mu_\\mathtt{S}(R_\\sigma)}{\\mu}f_\\mathtt{S}(m_i|M,\\Gamma,w)+\n\\frac{\\mu_\\mathtt{B}}{\\mu}f_\\mathtt{B}(m_i|\\alpha,\\kappa)\n\\right),\n\\end{equation}\n\nwhere  ${\\boldsymbol m}$ denotes the dataset in which\nthe observables are the invariant mass values of the lepton pairs,\n$m_i$; $N$ denotes the total number of events observed above 200\\GeV;\nand $\\mu=\\mu_\\mathtt{S}+ \\mu_\\mathtt{B}$ is\nthe mean of the Poisson distribution from which $N$ is an observation.\n\nStarting from Eqn.~\\ref{eq:likelihood}, confidence\/credible intervals\nare computed using more than one approach, both frequentist (using\nprofile likelihood ratios) and Bayesian (multiplying ${\\cal L}$ by prior\npdfs including a uniform prior for the signal mean).\nWith no candidate events in the region of small expected background\nabove 465\\GeV, the result is insensitive to\nthe statistical technique, and also with respect to the width of the $\\cPZpr$\nand to changes in systematic uncertainties and their functional forms,\ntaken to be log-normal distributions with fractional uncertainties.\n\nFor $R_\\epsilon$, we assign an uncertainty of 8\\% for the dielectron\nchannel and 3\\% for the dimuon channel. These values reflect our current\nunderstanding of the detector acceptance and reconstruction efficiency\nturn-on at low mass (including PDF uncertainties\nand mass-dependence of $k$-factors), as well as the corresponding values\nat high mass, where\ncosmic ray muons are available to study muon performance but not electron\nperformance.  The uncertainty in the mass scale affects only the mass\nregion below 500\\GeV where there are events in both channels\nextrapolating from the well-calibrated observed resonances.\nFor the dielectron channel, it is set to 1\\% based on linearity studies.\nFor the dimuon channel, it is set to zero, as a sensitivity study showed\nnegligible change in the results up to the maximum misalignment consistent\nwith alignment studies (corresponding to several percent change in momentum scale).\nThe acceptance for $\\GKK$ (spin 2) is higher than for $\\cPZpr$ (spin 1) by\nless than 8\\% over the mass range 0.75--1.1 TeV. This was\nconservatively neglected when calculating the limits.\n\nIn the frequentist calculation, the mean background level\n$\\mu_\\mathtt{B}$\nis the maximum likelihood estimate; in the fully\n Bayesian\ncalculation a prior must be assigned to the mean background\n level,\nbut the result is insensitive to reasonable choices (i.e., for which\nthe likelihood dominates the prior).\n\n\n\nThe upper limits on $R_{\\sigma}$ (Eqn.~\\ref{eq:rsigma}) from the various approaches\nare similar, and we report the Bayesian\nresult (implemented with Markov Chain Monte Carlo in\n{\\sc RooStats}~\\cite{MCMC}) for definiteness.\nFrom the dimuon and dielectron data, we obtain the upper limits on the cross section\nratio $R_{\\sigma}$ at 95\\% confidence level (C.L.) shown in\nFigs.~\\ref{fig:limits}(upper) and (middle), respectively.\n\nIn Fig.~\\ref{fig:limits}, the predicted cross section ratios\nfor $\\ZPSSM$ and $\\ZPPSI$  production are superimposed\ntogether with those\nfor $\\GKK$ production with dimensionless graviton coupling\nto SM fields $k\/\\overline{M}_\\mathrm{Pl}=0.05$ and $0.1$.\nThe leading order cross section predictions for\n$\\ZPSSM$ and $\\ZPPSI$\nfrom \\PYTHIA using {\\sc CTEQ6.1} PDFs are corrected for a mass dependent\n$k$-factor obtained using {\\textsc ZWPRODP}~\\cite{Accomando:2010fz,Hamberg:1990np,\nvanNeerven:1991gh,ZWPROD} to account for NNLO contributions.\nFor the RS graviton model, a constant NLO $k$-factor of 1.6 is\nused~\\cite{Mathews:2005bw}. The uncertainties\ndue to the QCD scale parameter and PDFs are indicated as a band.\nThe NNLO prediction for the $\\cPZ$ production cross\nsection is 0.97$\\pm\\,$0.04~nb~\\cite{Melnikov:2006kv}.\n\n\nPropagating the above-mentioned uncertainties into the\ncomparison of the experimental limits with the predicted cross section\nratios, we exclude\nat 95\\%~C.L. $\\cPZpr$ masses as follows.  From the dimuon only\nanalysis,\nthe $\\ZPSSM$ can be excluded below 1027\\GeV, the\n$\\ZPPSI$ below 792\\GeV, and the\nRS $\\GKK$ below 778 (987)\\GeV for couplings\nof 0.05 (0.1). For the dielectron analysis, the\nproduction of $\\ZPSSM$ and $\\ZPPSI$ bosons is\nexcluded for masses below 958 and 731\\GeV, respectively. The\ncorresponding lower limits on the mass for\nRS $\\GKK$ with\ncouplings of 0.05 (0.10) are 729 (931)\\GeV.\n\n\\subsection{Combined Limits on the Production Cross Section Using\nDimuon and Dielectron Events}\n\nThe above statistical formalism is generalized to combine the results from the\ndimuon and dielectron channels,\nby defining the combined likelihood as the product\nof the likelihoods for the individual channels with $R_\\sigma$ forced to\nbe the same value for both channels. The combined limit is shown in\nFig.~\\ref{fig:limits}~(bottom).\n\nBy combining the two channels, the following\n 95\\% C.L. lower limits on the mass of a $\\cPZpr$ resonance are obtained:\n1140\\GeV for the $\\ZPSSM$, and 887\\GeV for $\\ZPPSI$ models. RS\nKaluza--Klein gravitons are excluded below 855 (1079)\\GeV\n for values of couplings 0.05 (0.10).\nOur observed limits are more restrictive than or comparable to\nthose previously obtained via similar direct searches by the\nTevatron experiments~\\cite{D0_RS,D0_Zp,CDF_RS,CDF_Zp},\nor indirect searches by LEP-II experiments~\\cite{delphi,aleph,opal,l3},\nwith the exception of $\\ZPSSM$, where the value from LEP-II\nis the most restrictive.\n\n\n\nThe distortion of the observed limits at $\\sim$400\\GeV visible in\nFig.~\\ref{fig:limits} is the result of a clustering of several dimuon\nand dielectron events around this mass.  We have tested for the\nstatistical significance of these excesses ($p$-values expressed as\nequivalent $Z$-values, i.e. effective number of Gaussian sigma in a\none-sided test), using the techniques described in \\cite{PTDR2}.\nFor the dimuon sample, the probability of an enhancement at\nleast as large as that at 400 GeV occurring anywhere above 200 GeV in\nthe observed sample size corresponds to $Z<0.2$; for the electron\nsample, it is less. For the combined data sample, the corresponding\nprobability in a joint peak search is equivalent to $Z=1.1$.\n\n\\begin{figure}[htbp!]\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_mumu_40pb_a.pdf}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_ee_35pb_b.pdf}\n\\includegraphics[width=0.65\\textwidth,angle=0]{figures\/zpr_ssm_ratio_mcmc_comb_40pb_c.pdf}\n\\end{center}\n\\caption{\\label{fig:limits} Upper limits as a function\nof resonance mass $M$,\non the production ratio $R_{\\sigma}$ of\n cross section times branching fraction into lepton pairs\n  for $\\ZPSSM$ and $\\GKK$ production and $\\ZPPSI$\n boson production. The limits are shown from (top) the $\\Pgmp\\Pgmm$ final\n state, (middle) the $\\Pe\\Pe$ final state and (bottom) the combined dilepton result.\nShaded yellow and red  bands correspond to the $68\\%$ and  $95\\%$ quantiles \nfor the expected  limits.\nThe predicted cross section ratios are shown as bands, with widths\nindicating the theoretical uncertainties. \n}\n\n\n\\end{figure}\n\nIn the narrow-width approximation, the cross section for the process\n$\\Pp\\Pp\\to \\cPZpr+X\\to\\ell\\ell+X$ can be\nexpressed~\\cite{Carena:2004xs,Accomando:2010fz} in terms of the\nquantity $c_u w_u + c_d w_d$, where $c_u$ and $c_d$ contain the\ninformation from the model-dependent $\\cPZpr$ couplings to fermions\nin the annihilation of charge 2\/3 and charge $-$1\/3 quarks,\nrespectively, and where $w_u$ and $w_d$ contain the information about\nPDFs for the respective annihilation at a given $\\cPZpr$ mass.\n\n\nThe\ntranslation of the experimental limits into the ($c_u$,$c_d$) plane has\nbeen studied in the context of both the narrow-width and finite width\napproximations. The procedures have been shown to give the same\nresults. In Fig.~\\ref{fig:CuCd} the limits on the $\\cPZpr$ mass are\nshown as lines in the $(c_d,c_u)$ plane intersected by curves from\nvarious models which specify $(c_d,c_u)$ as a function of a model\nmixing parameter.\nIn this plane, the thin solid lines labeled by mass are\niso-contours of cross section with constant\n$c_u + (w_d\/w_u)c_d$, where $w_d\/w_u$ is in the range 0.5--0.6 for the\nresults relevant here.  As this linear combination\nincreases or decreases by an order of magnitude, the mass limits\nchange by roughly 500 GeV.\nThe point labeled SM corresponds to the\n$\\ZPSSM$; it lies on the more general curve for the\nGeneralized Sequential Standard Model (GSM) for which the generators\nof the $U(1)_{T_{3L}}$ and $U(1)_Q$ gauge groups are mixed with a mixing\nangle $\\alpha$. Then $\\alpha = -0.072\\pi$ corresponds to the\n$Z^\\prime_{SSM}$ and $\\alpha=0$ and $\\pi\/2$ define the $T_{3L}$ and\n$Q$ benchmarks, respectively, which have larger values of $(c_d,c_u)$\nand hence larger lower bounds on the masses.  Also shown are contours\nfor the E$_6$ model (with $\\chi$, $\\psi$, $\\eta$, $S$, and $N$\ncorresponding to angles 0, 0.5$\\pi$, $-0.29\\pi$, 0.13$\\pi$, and\n0.42$\\pi$, respectively) and Generalized LR models (with $R$, $B-L$,\n$LR$, and $Y$ corresponding to angles 0, 0.5$\\pi$, $-0.13\\pi$, and\n0.25$\\pi$, respectively)~\\cite{Accomando:2010fz} .\n\n\n\\begin{figure}[htbp!]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{figures\/plot_lhc_ee_mm_35_40_all_ALL_cms.pdf}\n\\end{center}\n\\caption{\\label{fig:CuCd}\n95\\% C.L. lower limits on the $\\cPZpr$ mass, represented by the thin continuous lines\nin the $(c_d,c_u)$\nplane.  Curves for three classes of model are shown.  Colours on the\ncurves correspond to different mixing angles of the generators defined\nin each model. For any point on a curve, the mass limit corresponding\nto that value of $(c_d,c_u)$ is given by the intersected contour.}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Eulerian summation of coprimes}\n\nLet be $r \\in \\mathbb{N}$, $\\mathbf{i} \\in \\mathbb{N}^r$ and $n \\in \\mathbb{N}$.  We introduce the sum $\\sum_{\\substack{  p_1 +  p_2 + ... +  p_r =n \\\\   p_1 \\wedge p_2 \\wedge ... \\wedge p_r= 1 }} p_1^{i_1} p_2^{i_2}... p_r^{i_r}$ with the aim to asymptotically estimate it. We denote it $S_n(\\mathbf{i})$, and $j = \\mathbf{1} \\cdot \\mathbf{i} $. Note that $S_n(i)$ is the sum of a power function over the primitive points contained in the surface of the dilates of the $(r-1)$-dimensional standard simplex. It is noteworthy that the number of such points has been computed in \\cite{DezaPournin:primitivepoint} and that more generally a coprime Ehrhart theory is introduced in \\cite{Sanyal:erhart}.   \\\\\n\n\\begin{prop}\\label{eulerian_sum_notation} \n\n\\begin{align}\n    \\sum_{k \\geq 0} k^j \\sum_{n \\geq 1} S_n(\\mathbf{i}) y^{nk} =  \\frac { \\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y)}{(1-y) ^{j+r}} .\n\\end{align} \n\\end{prop}\n\n\\begin{proof}\n\nThis relation relies on the partition $\\mathbb{Z}_+^r =\\bigcup_{k \\geq 0} \\left\\{ kp, p \\in \\mathbb{P}^r_+ \\right\\}$. We write\n\\\\\n\n\\begin{align*}\n    \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) \\left(y^n\\right)^k &= \\sum_{k \\geq 0} \\sum_{n \\geq 1} \\left(\\sum_{\\substack{  p_1 +  ... +  p_r =n \\\\   p_1 \\wedge ... \\wedge p_r= 1 }}(k p_1)^{i_1}y^{kp_1} ... (k p_{r})^{i_{r}}y^{kp_{r}} \\right) \\\\\n&= \\sum_{l \\geq 1} \\left( \\sum_{\\substack{ l_1 +  l_2 + ... +  l_r =l\\\\   l_p \\geq 1 }}l_1^{i_1}y^{l_1} (l_2)^{i_2}y^{l_2}... (l_r)^{i_r}y^{l_r}  \\right) &= \\prod_{p = 1}^{r} \\left( \\sum_{l_p \\geq 1} l_p^{i_p} y^{l_p} \\right)  = \\frac { \\prod_{p = 1}^{r}  A_{i_p}(y)}{(1-y) ^{j+r}}  .\n\\end{align*}\n\n\\end{proof}\n\nThe $n^{th}$ Eulerian polynomial is of degree $n$ except for $A_0(y) = y$ and has no constant term which causes the degree of $\\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y)$ to be no more than $j+r$. We write it as follows: \n\n$$\\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(y) = B_{\\mathbf{i}}(y) = b_{\\mathbf{i},j+r} y^{j+r} + ... b_{\\mathbf{i}, 1} y. $$\n\nDespite this heavy notation, the sums under study in the zonotopal case are usually $S_n(\\mathbf{0})$, $S_n(1, 0,...,0)$ or $S_n(2, 0,...,0)$, making $B_{\\mathbf{i}}(y)$ equal to $y^{d}$, $y^{d+1}$, or $y^{d+2}+ y^{d+1}$.\n\n\\subsection{Mellin transform}\n\nMellin transforms appear naturally in the asymptotic analysis of infinite products (see \\cite{Bodini:polyomino,Flajolet:AC,Gittenberger:hadmi}). In the following, the analysis of zonotopes will require to apply a Mellin transform on sums of coprime numbers. The following proposition is of primary importance in the sequel:\n\\begin{prop}\\label{property2}\nFor $t> 0$, the Mellin transform of $\\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{-n t}$ is \n \\begin{align}\n     \\frac{1}{\\zeta(s-j)} \\left( \\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\sum_{k \\geq 1} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s} \\Gamma(s)\\right) .\n \\end{align} \n \\end{prop}\n\n\\begin{proof}\n\nAs we do throughout the paper, we use the change of variables $y = e^{-t}$ and will always try to write products as sums, to have an expression compatible with the Mellin transform. This leads to:\n\n\\begin{align}\\label{sum_expression}\n \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) \\left(e^{-n t}\\right)^k = \\prod_{\\lambda = 1}^{r}  A_{i_\\lambda}(e^{-t})  \\left( \\sum_{k\\geq 0}  \\binom{ k + j + r - 1}{j + r -1 } e^{-kt} \\right) = \\sum_{n = 0}^{j + r} b_{\\mathbf{i},n}  \\:\\: \\sum_{k\\geq 1}  \\binom{ k + j + r -n - 1}{j + r -1 } e^{-k t} .\n\\end{align}\n\nWe can now make use of the Mellin transform which is the transform that matches the exponential with the Gamma function:  \n\n$$\\forall s, \\Re(s) > 0, \\:\\:\\: \\Gamma(s) = \\int_{0}^{+ \\infty} e^{-t} t^{s - 1} dt.$$\n\nUnder conditions that will be detailed just below, the Mellin transform of a function $f$ the expression is defined as\n\n$$ \\mathcal{M}\\left[f\\left(e^{-t}\\right)\\right](s) = \\int_0^{+\\infty} f\\left(e^{-t}\\right)t^{s-1}dt.$$\n\nFor $\\Re(s) > j+r -1$, the term $ \\left|\\sum_{n\\geq0} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s}\\right|  $ is bounded, which makes the use of Fubini's theorem possible, so the Mellin function of $\\sum_{n\\geq1}S_(i)e^{-nt}$ can be written, on the one hand:\n\n$$ \\int_{0}^{+\\infty} \\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\:\\: \\sum_{k\\geq 1}  \\binom{ k + j + r -n - 1}{j + r -1 } e^{-k t} t^{s -1} d t =\\sum_{n = 0}^{j + r } b_{\\mathbf{i},n}  \\sum_{k \\geq 1} \\frac{\\binom{ k  + j + r - n - 1}{j + r -1 }}{k^s} \\Gamma(s),$$\n\nand on the other hand:\n\n$$\\int_0^{+\\infty} \\sum_{k \\geq 0 } k^{j} \\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{- n k t} t^{s-1} d t= \\zeta(s-j) \\int_0^{+\\infty} \\sum_{n \\geq 1} S_n(\\mathbf{i}) e^{- n t} t^{s-1} dt. $$\n\nThese holomorphic functions being defined and equal on $\\Re(s) > j+r-1$, we can extend it to its meromorphic continuation on $\\mathbb{C}$.\n\\end{proof}\n\nThe Mellin transform is widely used to estimate asymptotic behaviors using its correspondence with the poles of the transformed function, as we will do in next subsection (and as it is done in \\cite{Bodini:polyomino} and \\cite{Bureaux:polygons}).  Property \\ref{property2} make it possible to compute the asymptotic equivalent of any of the sums $\\sum_{n \\geq 1} S_n(\\mathbf{i}) y^n $ near $y =1$. \n\n\n\n\\subsection{Riemann's non trivial zeros challenge, with the example \\texorpdfstring{$\\sum_{ \\protect\\substack{ p < n \\\\  p \\wedge n = 1 }}\\protect p^2$}{TEXT} }\n\nWe use this subsection to detail more specifically the way the Mellin inversion formula is handled, as we will encounter it multiple times. Riemann's $\\zeta$ function intervenes in it, and its non-trivial zeros too. We will use the integral of \\cite{Bureaux:polygons} to manage all the zeros in the critical strip with an integral, but we could also write a sum over all non-trivial zeros. This term is numerically discussed in the Section \\ref{numerical_discu}. \n\n\n\n\\begin{propos}\nWe denote the contour  $\\gamma = \\gamma_{\\text{right}}\\cup \\gamma_{\\text{left}}$, with $\\gamma_\\text{right} = 3 - \\frac{A}{\\log(2 + |t|)} +it$ and $\\gamma_\\text{left} = 2 + \\frac{A}{\\log(2 + |t|)}+it $ as $t$ runs from $-\\infty$ to $+\\infty$, with $A >0$ such that $\\gamma$ surrounds all the zeros in the critical strip of $\\zeta(s-2)$. \\\\\n\nHereafter we will denote $$I_\\text{crit} = \\frac{1}{2i\\pi} \\int_\\gamma  \\mathcal{M}[\\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}](s) t^{-s} d s$$,\\\\\n\nand $$I_\\text{err} =\\frac{1}{2i\\pi} \\int_{\\frac{1}{2} - i\\infty}^{\\frac{1}{2}+i\\infty}  \\mathcal{M}[\\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}](s) t^{-s} d s$$.\\\\\n\nWe have for all $t>0$:\n\n\\begin{align}\n     \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} - \\frac{1}{90 \\zeta'(-1)}\\log(t)  + I_\\text{err}(t).\n\\end{align}\n\n\nWhen $t$ goes to 0,  $I_\\text{err}(t) = O(t^{1\/2})$, $I'_\\text{err}(t) = o(t^{-1})$  and the $k$-th derivative of $I_\\text{crit}$ is $o(t^{-k-1})$.\n\n\\end{propos}\n\n\\begin{proof}\nWe take the general set of sums  $\\left(\\sum_{\\substack{ p < n \\\\  p \\wedge n = 1 }} p^i \\right)_{i \\geq 0}$.\\\\\nAs $p \\wedge n = 1$ being equivalent to $p \\wedge (n-p) = 1$, those sums can be rewritten as $\\sum_{ \\substack{ p +q  = n \\\\  p \\wedge q = 1 }} p^i = S_n(i,0)$.\\\\\nWe fix $i = 2$. Applying properties \\ref{eulerian_sum_notation}  and \\ref{property2}, we have:\n\n\\begin{align*}\n    \\sum_{k \\geq 0 } k^{2} \\sum_{n \\geq 1 } S_n (2, 0) e^{- k n t} &= \\frac{A_0(e^{-t})  A_2 (e^{-t})}{(1-e^{-t})^{4}}\\\\\n    &= (e^{-3 t} + e^{-2t}) \\sum_{k \\geq 0} \\binom{k+3}{3} e^{-k t}. \n\\end{align*}\n\n\nFor $t >0, s >1 $, we have the following Mellin transform: \n\n\\begin{align*}\n    \\mathcal{M} \\left[ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t}\\right] (s) &= \\frac{1}{\\zeta(s-2)}\\left(  \\sum_{k \\geq 1} \\frac{\\binom{k+1}{3} + \\binom{ k }{3}}{k^s}  \\Gamma(s)\\right)\\\\\n    &= \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) .\n\\end{align*}\n\n\nTherefore we have, using  the Mellin inversion formula, for all $c > 4$ (to ensure $\\sup_{s\\in c+ i \\mathbb{R}} \\zeta(s-3) < \\infty$):\n \n $$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{1}{2i\\pi} \\int_{c- i\\infty}^{c+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s. $$\n\nWe use the residue theorem to shift the line of integration to the left as $s= 4$ is a location of a pole of the Mellin transform. It follows that, for all $c_1\\in ]3, 4[$:\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{-4}} + \\frac{1}{2i\\pi} \\int_{c_1- i\\infty}^{c_1+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s.$$\n\nThe remaining poles of the Mellin transform are the double pole at $s = 0$, and simple ones at $s=0$ and at the non-trivial zeros of $\\zeta(s-2)$,  which are in the critic strip $2<\\Re(z)<3$, and in the left side of the complex domain. \\\\\n\nNow consider the contour $\\gamma$ as described in the proposition. The existence of $A$, such that $\\gamma$ surrounds all the zeros of $\\zeta(s-2)$ in the critical strip is a consequence of Theorem 3.8 in \\cite{Titchmarsh:riemann}.\n\n\\begin{align}\\label{riemann_crit_strip}\n    I_\\text{crit}(t) = \\frac{1}{2i\\pi} \\int_\\gamma  \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s.\n\\end{align}\n\nFor any $c_2 \\in ]0,2[$, we have:\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{-4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} +  \\frac{1}{2i\\pi} \\int_{c_2- i\\infty}^{c_2+ i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s. $$\n\nFinally denoting $I_\\text{err} (t) = \\frac{1}{2i \\pi}\\int_{\\frac{1}{2} - i\\infty}^{\\frac{1}{2}+i\\infty} \\frac{2 \\zeta(s-3) - 3 \\zeta(s-2) + \\zeta(s-1)}{6 \\zeta(s-2)}\\Gamma(s) t^{-s} d s$, we have for $t>0$\n\n$$ \\sum_{n \\geq 1 } S_n (2, 0) e^{-  n t} = \\frac{2}{\\zeta(2) t^{4}} + I_\\text{crit}(t) + \\frac{1}{3 t^2} - \\frac{1}{90 \\zeta'(-1)}\\log(t)  + I_\\text{err}(t). $$\n\nThe bounds over the order of $I_\\text{crit}$ and $I_\\text{err}$ are coming from Lemma 2.2 from \\cite{Bureaux:polygons}.\n\\end{proof}\n\n\n\n\n\\subsection{Intermediate sequence of polynomials}\\label{polynome_zono}\n\nAs we make use of the Mellin inversion formula over the logarithm of the generating function, for each $\\delta$ between 1 and $d$, the coefficient $\\binom{k-1}{\\delta-1}$ from Property \\ref{property2} will appear with $S_n(\\underbrace{0,...,0}_{\\delta \\text{ times}})$. In the rest of the paper (as well as in Theorem \\ref{theorem} ), we denote:\n\n$$\\sum_{\\delta = 1}^d   \\binom{d}{\\delta} 2^{\\delta-1} \\binom{{k} - 1}{\\delta-1} = P_d({k}) = p_{d,d-1} {k}^{d-1} + ... + p_{d,1} {k} + p_{d,0} .$$\n\nThe first terms of $\\left( P_d({X})\\right)_{d \\geq 2}$ are: \\\\\n\\begin{align*}\n        P_1({X}) &= 1 \\\\\n        P_2({X}) &= 2 {X}\\\\\n        P_3({X}) &= 2 {X}^2 + 1    \n\\end{align*}\n\nThese polynomials verify the relation $P_{d+2}({X})=  \\frac{2 {X}}{d+1}P_{d+1}({X}) + P_d({X} )$ for $d \\geq 1$. \\\\\n\nTwo properties naturally follow:\n\\begin{itemize}\n    \\item The odd polynomials only contain even powers of $i$ and have a constant term equal to $1$. Similarly, the even polynomials only contain odd powers of $i$. \n        \\item The leading term of $P_d$ is of degree $d-1$ and its coefficient is $\\frac{2^{d-1}}{(d-1)!}$.\n\\end{itemize}\n\nAs each exponent $\\delta$ of $k$ is reflected in the Riemann $\\zeta$ function as $\\zeta(s-\\delta)$ when using Mellin transform, we introduce an operator over complex functions:\n\n\\begin{defn}\\label{definition_operator}\nLet $\\mathrm{M}(\\mathbb{C})$ be the field of meromorphic functions in $\\mathbb{C}$. We define the operator $\\mathbf{\\Pi}_d$ as\n\\begin{align*}\n   \\mathbf{\\Pi}_d \\colon \\mathrm{M}(\\mathbb{C}) &\\to \\mathrm{M}(\\mathbb{C})\\\\\n  \\phi &\\mapsto p_{d,d-1} \\phi(\\cdot \\: - (d-1)) + ... + p_{d,1} \\phi(\\cdot \\: - 1) + p_{d,0} \\phi.\n\\end{align*}\n\\end{defn}\n\nIt is important to note that we can write, for $s\\neq 1$, $\\mathbf{\\Pi}_d (\\zeta)(s) = \\sum_{k \\geq 1} \\frac{P_d(k)}{k^s}$.\n\n\n\\subsection{Asymptotic analysis of the generating function}\n\n\\begin{lem}\\label{lemma_equivalent}\nLet $A$ be a positive number such that $\\gamma = \\gamma_{\\text{right}}\\cup \\gamma_{\\text{left}}$, with $\\gamma_\\text{right} = 1 - \\frac{A}{\\log(2 + |\\theta|)} +it$ and $\\gamma_\\text{left} = \\frac{A}{\\log(2 + |\\theta|)}+it $ as $\\theta$ goes from $-\\infty$ to $+\\infty$, surrounds all the zeros of $\\zeta$ in the critical strip.\\\\\n\nFor $\\theta>0$, as $\\theta \\rightarrow 0$, we have: \n\\begin{align}\n     Zon \\left(e^{-\\theta} \\right) = \\theta^{2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) } e^{2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta  -  \\zeta' \\right)(0) } \\exp\\left( \\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + I_{\\text{crit}}(d, \\theta) + I_\\text{err}(d, \\theta)    \\right)    , \n\\end{align}\n\nwith $$I_\\text{crit}(d,\\theta) = \\frac{1}{2i\\pi}\\int_{\\gamma}\\frac{\\mathbf{\\Pi}_d (\\zeta)(0) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) \\theta^{-s} \\delta s $$,\\\\\n\nand $$I_\\text{err}(d,\\theta) = \\frac{1}{2i\\pi}\\int_{-\\frac{1}{2} - i\\infty}^{- \\frac{1}{2}+ i \\infty}\\frac{\\mathbf{\\Pi}_d (\\zeta)(0) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) \\theta^{-s} \\delta s =O(\\theta^{1\/2})$$.  \\\\\n\n\\end{lem}\n\n\n\\begin{proof}\nWe apply the logarithm to the generating function with the change of variables $x = e^{-\\theta}$, which gives:\\\\\n\\begin{align*}\n    \\log\\left(Zon\\left(e^{-\\theta}\\right)\\right) &=  - \\sum_{\\delta=1}^d \\left(\\binom{d}{\\delta}2^{\\delta-1} \\sum_{\\mathbf{v} \\in \\mathbb{P}^{\\delta}_+} \\log\\left( 1 - e^{- \\mathbf{1}\\cdot \\mathbf{v} \\theta} \\right)\\right)\\\\\n    &=- d \\log(1 - e^{-\\theta}) - \\sum_{\\delta = 2}^d  \\left(  \\binom{d}{\\delta} 2^{\\delta-1} \\sum_{n= 2}^{+\\infty}  \\left( \\sum_{\\substack{ n_{1}+...+ n_{\\delta} = n \\\\  n_{1} \\wedge ... \\wedge n_{\\delta}  = 1   }} 1 \\right)  \\log \\left( 1 - e^{- n \\theta} \\right) \\right).    \n\\end{align*}\n\n\nThe sum $\\sum_{\\substack{ n_{1}+...+ n_{\\delta} = n \\\\  n_{1} \\wedge ... \\wedge n_{\\delta}  = 1   }} 1 $ is of the form of the sums studied in section \\ref{section_eulerset} as being $S_n(\\overbrace{0,...,0}^{\\delta \\text{ times}})$. In a geometrical point of view, it is exactly the number of primitive points . Therefore we compute the Mellin transform accordingly (using logarithmic series), for all $s > d$\\\\\n\n$$ \\mathcal{M}\\left[\\log\\left(Zon\\left(e^{-\\theta}\\right)\\right)\\right](s) = \\sum_{\\delta = 1}^d   \\binom{d}{\\delta} 2^{\\delta-1} \\left(\\sum_{k=1}^{\\infty} \\frac{\\binom{k-1}{\\delta-1}}{k^s} \\right) \\frac{ \\zeta(s+1) \\Gamma(s) }{\\zeta(s)}.$$\n\nWith the notation introduced in definition \\ref{definition_operator}, the transform simplifies to the following form:\\\\\n\\begin{propos} for every $d>1$, we have:\n\\begin{align}\n    \\mathcal{M}\\left[\\log \\left(Zon \\left(e^{-\\theta}\\right)\\right)\\right] (s) =\\frac{\\mathbf{\\Pi}_d (\\zeta(s)) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s).\n\\end{align}\n\\end{propos}\n\n\n\\leavevmode\\\\\n\nWe can extend this function into a meromorphic one on $\\mathbb{C}$, and use successive residue theorems to scan from the right to the left each pole of the Mellin transform, as we did in the previous section. We start with $c> d$ and\\\\\n\n$$ \\log(Zon(e^{-\\theta})) = \\frac{1}{2i\\pi} \\int_{c - i \\infty}^{c + i \\infty} \\frac{p_{d,d-1} \\zeta(s - d +1) + ... + p_{d,1} \\zeta(s - 1) + p_{d,0} \\zeta(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s} \\delta s,$$\n\n\\leavevmode\\\\\n\nwhich leads to \n\n\\begin{align*}\n    \\log(Zon(e^{-\\theta})) =  &\\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + \\frac{1}{2i\\pi} \\int_{\\gamma} \\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s}  ds + 2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta -  \\zeta'\\right) (0)  \\\\\n    &+ 2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) \\log(\\theta)   + \\frac{1}{2i\\pi} \\int_{ -\\frac{1}{2}- i \\infty}^{- \\frac{1}{2} + i \\infty} \\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s)  \\theta^{-s} ds.\n\\end{align*}\n\nWe denote $I_\\text{crit}(d,\\theta)$ and $I_\\text{err}(d,\\theta)$ respectively the first and the second integral above. \n\n\n\\end{proof}\n\n\\begin{cor}\nThe asymptotic equivalent of $ Zon\\left(e^{-\\theta}\\right) $ is:\n\n$$ Zon \\left(e^{-\\theta} \\right) \\sim  \\theta^{2 \\mathbf{\\Pi}_d \\left(\\zeta\\right)(0) } e^{2 \\mathbf{\\Pi}_d \\left(\\log(2\\pi) \\zeta  -  \\zeta' \\right)(0) } \n\\exp\\left( \\sum_{\\delta=1}^{d-1}\\left(p_{d,\\delta}\\frac{\\zeta(\\delta+2) \\Gamma(\\delta+1)}{\\zeta(\\delta+1) \\theta^{\\delta+1}} \\right) + I_{\\text{crit}}(d, \\theta) \\right), \\text{ as } \\theta\\rightarrow 0    .   $$\n\n\\end{cor}\n\n\n\n\\subsection{The saddle-point equation}\n\n\\begin{lem}\\label{sadlle_equation_solution}\nDenoting $\\widetilde{\\theta}_n = \\left( \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d) n} \\right)^{\\frac{1}{d+1}}$, we have: \n\n$$\\left.\\frac{\\partial  \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}}\\right) \\right) }{\\partial \\theta_i} \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}:= \\widetilde{\\theta}_n \\textbf{1} } = - n + o(n) $$\n\n$\\widetilde{\\theta}_n$ is called the asymptotic solution of the saddle-point equation.\n\\end{lem}\n\n\n\\begin{proof}\n\nWe fetch the coefficient of the multivariate series of the generating function by using Cauchy's integral formula and therefore we look for parameters $\\boldsymbol{\\widetilde{\\theta}_n} = (\\widetilde{\\theta}_{n,1},..., \\widetilde{\\theta}_{n,d})$ such that for all $ 1 \\leq i \\leq d$,\n\n\\begin{align}\\label{saddle_equation}\n    \\left.\\frac{\\partial  }{\\partial \\theta_i} \\left( \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}}\\right) \\right) + (n + 1) \\theta_{i}  \\right) \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n} } = 0.\n\\end{align}\n \n\nThe symmetry of the variables in $Zon(\\boldsymbol{\\theta})$  implies $\\theta_{n,i} = \\theta_n $ for $1 \\leq i \\leq d$. The system (\\ref{saddle_equation}) is then restricted to $\\theta_1$ only:\n\n\n\\begin{align*}\n    \\left.\\frac{\\partial}{\\partial \\theta_1} \\log \\left(Zon\\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} &= -  \\sum_{\\delta=1}^d  2^{\\delta-1} \\binom{d-1}{\\delta-1} \\sum_{\\textbf{\\textit{v}}\\in \\mathbb{P}^{\\delta}_+} v_1 \\frac{e^{- \\boldsymbol{<\\widetilde{\\theta}_n}, \\textbf{\\textit{v}}> }}{1 - e^{-<\\boldsymbol{\\widetilde{\\theta}_n}, \\textbf{\\textit{v}}> }}  \\\\\n    &= - \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\sum_{m=\\delta}^{+\\infty} \\:\\:\\:\\underbrace{\\sum_{\\substack{\\textbf{\\textit{v}}\\in \\mathbb{P}^{\\delta}_+ \\\\ <\\textbf{1},\\textbf{\\textit{v}}> = m}} v_1} _ { S_m(1, \\textbf{0}_{\\mathbb{R}^{d-1}}) } \\frac{e^{- m \\theta_n }}{1 - e^{-m \\theta_n }}\n\\end{align*}\n\\leavevmode\\\\\n\nUsing the analysis of the sums $S_m$ in Section \\ref{section_eulerset}, we have the following Mellin transform, for $\\Re(s) > d+1$:\n\n$$\\mathcal{M}\\left[\\left.\\frac{\\partial}{\\partial \\theta_1} \\log \\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} \\right](s) =  - \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\left( \\sum_{k \\geq d} \\frac{\\binom{k}{\\delta}}{k^s} \\right) \\frac{\\zeta(s) \\Gamma(s)}{\\zeta(s-1)}. $$\n\nThis function is a meromorphic function on $\\mathbb{C}$, so we are taking $c\\in \\mathbb{C}$ with $\\Re(c) > d+1$ to use the Mellin inversion formula\n\n\n$$\\left.\\frac{\\partial}{\\partial \\theta_1} \\log\\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} = - \\frac{1}{2 i \\pi} \\int_{c - i\\infty}^{c+ i \\infty} \\sum_{\\delta=1}^d 2^{\\delta-1} \\binom{d-1}{\\delta-1} \\left( \\sum_{k \\geq d} \\frac{\\binom{k}{\\delta}}{k^s} \\right) \\frac{\\zeta(s) \\Gamma(s)}{\\zeta(s-1)} \\theta_n^{-s} d s. $$\n\nWith the residue theorem, we can shift the line of the integral to the left of $\\Re(s) = d+1$. The resulting integral is $o\\left(\\frac{1}{\\theta^{d+1}}\\right)$, which leads to:\n\n$$\\left.\\frac{\\partial}{\\partial \\theta_1} \\log\\left(Zon \\left(\\textbf{\\textit{e}}^{- \\boldsymbol{\\theta}} \\right)\\right) \\right|_{\\substack{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}}} = - \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d) \\theta_n^{d+1}} + o\\left(\\frac{1}{\\theta_n^{d+1}}\\right). $$\n\nTo generalize the notation of \\cite{Bureaux:polygons}, we denote $\\kappa_d = \\frac{2^{d-1} \\zeta(d+1)}{\\zeta(d)}$. Finally get the parameter which is solution of the saddle equation: \n\n\\begin{align}\n    \\theta_n = \\left( \\frac{\\kappa_d}{n} \\right)^{\\frac{1}{d+1}}\n\\end{align}\n\n\\end{proof}\n\n\\subsection{H-admissibility}\n\n\\import{Parties\/Development\/}{4h_admissibility.tex} \n\n\\subsection{Asymptotic number of lattice zonotopes}\n\nAs $Zon$ is H-admissible, and given the solution of the saddle-point equation $\\widetilde{\\theta}_n$, we can write:\n\n$$ z_d(n):=  \\frac{1}{(2i\\pi)^d} \\int_{[0,2\\pi]^d} Zon\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}+ i \\boldsymbol{\\theta}}\\right)e^{(n+1)( d \\theta_n +i <  \\textbf{1},\\boldsymbol{\\theta}>)} d \\boldsymbol{\\theta} \\underset{n \\rightarrow \\infty}{\\sim} \\frac{Zon\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}}\\right)e^{ n d \\theta_n }}{\\sqrt{(2 \\pi)^d \\left|\\textbf{B}\\left( \\textbf{\\textit{e}}^{- \\boldsymbol{ \\widetilde{\\theta}_n}}\\right) \\right|}}, $$\n\nwith the notation from the H-admissibility demonstration ($\\textbf{B} $ is the matrix of the second partial derivatives of the generating function's logarithm). We recall that:\n\n$$ |\\textbf{B}(\\textbf{\\textit{x}}_n)| \\underset{n \\rightarrow \\infty}{\\sim} (d+1) \\left(\\frac{\\kappa_d}{ \\theta_n^{d+2}} \\right)^{d}. $$\n\nUsing $\\kappa_d = \\frac{2^{d-1}\\zeta(d+1)}{\\zeta(d)}$, the contour $\\gamma = \\gamma_\\text{right} \\cup \\gamma_\\text{left}$ with $\\gamma_\\text{right} = 1 - \\frac{A}{\\log(2 + |t|)} +it$ and $\\gamma_\\text{left} = \\frac{A}{\\log(2 + |t|)}+it $, $A>0$ such that $\\gamma$ surrounds the critical strip of the $\\zeta$ function, and the operator $\\mathbf{\\Pi}_d $ defined in Definition~\\ref{definition_operator},  we finally have: \n\n\\begin{align}\\label{final_result}\n     z_d(n) \\underset{n \\rightarrow \\infty}{\\sim}  \\alpha_d n ^{\\beta_d} \\exp \\left( Q_d(n^\\frac{1}{d+1}) + I_{\\text{crit}}(d, t) \\right),\n\\end{align}\n\n\\begin{align*}\n    \\text{with } \n    &\\alpha_d = \\frac{\\kappa_d^{\\frac{d}{2(d+1)} +  \\frac{2}{d+1} \\mathbf{\\Pi}_d (\\zeta)(0) } \\exp \\left( 2\\mathbf{\\Pi}_d \\left(\\log(2\\pi)\\zeta -  \\zeta' \\right)(0) \\right) }{ (2\\pi)^{d\/2 } \\sqrt{d+1}},\\\\\n    &\\beta_d = \\frac{-1}{d+1}\\left(d(d+2)\/2 + 2 \\mathbf{\\Pi}_d \\left(\\zeta\\right) (0) \\right),\\\\\n    &Q_d(X) = (d+1) \\kappa_d^{\\frac{1}{d+1}}  X^d +  \\sum_{\\delta=2}^{d-1} p_{d,\\delta-1}\\frac{\\zeta(\\delta+1) (\\delta-1)!}{\\zeta(\\delta) }\\kappa_d^{-\\frac{\\delta}{d+1}} X^{\\delta}, \\\\\n    &J_{\\text{crit}}(d, n) =  \\left.\\frac{1}{2i\\pi}\\int_{\\gamma}\\frac{\\mathbf{\\Pi}_d (\\zeta)(s) }{\\zeta(s)}\\zeta(s+1)\\Gamma(s) t^{-s} d s \\right|_{t = \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{1}{d+1}}}.\n\\end{align*}\n\nNote that $J_{\\text{crit}}(d, n)$ is just the evaluation of $I_{\\text{crit}}(d, \\theta)$ at the saddle point value  $\\theta= \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{1}{d+1}}.$\n\n\n\n\n\n\\leavevmode\\\\\n\\subsection{Order of size of $J_{crit}$}\nThroughout this paper, we left $J_{crit}$ under its integral form but we could also write it as an infinite sum of residues as we did in the introduction, with $r$ ranging over the non-trivial zeros of $\\zeta$: \n\n$$ \\sum_{r} Res\\left(\\frac{1}{\\zeta(r)}\\right) \\frac{\\mathbf{\\Pi}_d (\\zeta(r)) }{\\zeta(r)}\\zeta(r+1)\\Gamma(r) \\left( \\frac{\\kappa_d}{n} \\right)^{ \\frac{-r}{d+1}} .$$\n\nThis sum is actually invisible for \"computably large\" numbers $n$. The density of poles $r$ is known to be strongly bounded since Selberg in the beginning of the 1940's \\cite{Selberg:Riemann}. Due to this and to the exponential decrease of $\\Gamma$ function, the value of $J_{crit}$ is with a good approximation the first term of the sum. It is composed of the term at the first non-trivial zero, approximately at $r = \\frac{1}{2} + 14.13472i$ and of the term at its conjugate, which gives for respectively $J_{\\text{crit}}(2, n)$,  $J_{\\text{crit}}(3, n)$, and $J_{\\text{crit}}(4, n)$ the following approximations:\n\n\\begin{align}\\label{numeric_I_crit}\n    J_{\\text{crit}}(2, n) &\\approx -1.3579 {\\scriptstyle\\times} 10^{-10}n^{1\/6}\\cos\\left(4.7116\\ln(0.6842 n)\\right) - 1.4236{\\scriptstyle\\times}10^{-9}n^{1\/6}\\sin\\left(4.7116\\ln(0.6842n)\\right),\\nonumber\\\\\n    J_{\\text{crit}}(3, n) &\\approx -1.2325 {\\scriptstyle\\times}10^{-10}n^{1\/8}\\cos\\left(3.5337\\ln(0.2777n)\\right) - 1.2921 {\\scriptstyle\\times}10^{-9}n^{1\/8}\\sin\\left(3.5337\\ln(0.2777n)\\right),\\\\\n    J_{\\text{crit}}(4, n) &\\approx -3.1764{\\scriptstyle\\times}10^{-9}n^{1\/10}\\cos\\left(2.8269\\ln(0.1305n)\\right) - 8.0628{\\scriptstyle\\times}10^{-9}n^{1\/10}\\sin(2.8269\\ln(0.1305n)). \\nonumber\n\\end{align}\n\n\n\n\nWe can see that the size of these numbers mainly relies on $\\Gamma(r)$. The second non-trivial zero of $\\zeta$ is approximately in $r= \\frac{1}{2} + 21,02203 i$, which is smaller by $10^{-4}$ for the first dimensions. \n\n\\begin{comment}\nIt becomes obvious when we compute the graph of the quotient $\\frac{z_2(n)}{\\alpha_2 n ^{\\beta_2} \\exp(Q_2(n^{1\/3}))}$, as follows:\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{images\/zonogones convergence asymptotique.png}\n    \\caption{Quotient of $z_2(n)$ and $\\alpha_2 n ^{\\beta_2} \\exp(Q_2(n^{1\/3}))$. }\n    \\label{fig:my_label}\n\\end{figure}\n\n\\end{comment}\n\n\\subsection{About moving up in dimension}\n\nFor a more down-to-earth analysis of the logarithmic equivalent, i.e. the leading term of the exponential term, we can use the expansion of $\\zeta(d) = 1+ 2^d + o(2^d)$ as $d $ grows large. One can see that: \n\n$$ \\left(\\frac{2^{d-1}\\zeta(d+1)}{\\zeta(d)} \\right)^{\\frac{1}{d+1}} = 2 + O\\left(\\frac{1}{d}\\right), \\:\\:\\:\\:\\:\\:\\:\\:\\text{ as } d \\rightarrow + \\infty.$$\n\nTherefore, when we get that, in higher dimension, the logarithm of $z(n)$ is nearly equivalent to $ 2(d+1) n^{\\frac{d}{d+1}} $.\\\\\n\n\n\n\n\\subsection{Generalization to rectangular boxes}\nOur result is generalizable to rectangular boxes $[0,n_1]\\times...\\times[0,n_d]$ with $n_i = \\alpha_i n$ ($\\alpha$ is a non-zero positive number). In order to lighten the results, and because the proof is very similar to what has been done before,e write the formula here with the sketch of the proof. One can refer to \\cite{Bogatchev:limit} to see detailed generalization in dimension 2. \n\n\\begin{thm}\n\\end{thm}\n\n\\begin{proof}\nThe saddle-point equation differs depending on the coordinate. It brings the following result:\n\n\\begin{lem}\nDenoting $\\boldsymbol{\\widetilde{\\theta}_n} = \\left( \\widetilde{\\theta}_n^i\\right)$ with $\\widetilde{\\theta}_n^i = \\frac{\\prod_k \\alpha_k^{1\/(d+1)}}{\\alpha_i} \\widetilde{\\theta}_n$ for each $i$, $\\boldsymbol{\\widetilde{\\theta}_n}$ is the solution of the saddle equation, i.e. for each $i$ we have\n\n$$\\left.\\frac{\\partial  \\log\\left(Zon\\left(\\boldsymbol{\\theta}\\right)\\right) }{\\partial \\theta_i} \\right|_{\\boldsymbol{\\theta} = \\boldsymbol{\\widetilde{\\theta}_n}} = - \\alpha_i n + o(n) .$$\n\\end{lem}\n\nThen the uni-variate generating function that has to been asymptotically approximate is \n\n$$Zon\\left(e^{-\\frac{t}{\\alpha_1} },..., e^{- \\frac{t}{\\alpha_d} }\\right) $$\n \n  \n\\end{proof}\n\\end{comment} \n\n\\begin{comment}\n\\subsection{On polytopes, zonotopes and probabilities}\n\nAs said in the introduction, this work falls within the framework of counting polytopes in convex bodies. Here, the asymptotic number of zonotopes was established in hypercubes only, so it calls for looking for attempt to extend this result to more general bodies, such as rectangular boxes, cones etc.  \n\nConcerning combinatorial parameters that come with zonotopes, one could wonder if the number of $\\delta$-dimensional faces of a $d$-dimensional zonotope (aggregated into the f-vector) can be studied with this method (i.e. the mean number of $\\delta$-dimensional faces).\n\nFinally, the estimation of zonotopes was addressed using a combinatorial point of view, but as it was mentioned previously, the same equations appear when we lay a Boltzmann-like distribution over sets of generators and compute the estimation of the number of zonotopes through local limit theorem (see \\cite{Bureaux:polygons}). In that case, it is interesting to watch the deformation of the limit shape of the zonotopes and the evolution of the parameters we computed while varying the probability distribution. \\\\  \n\n\\end{comment}\n\n\n\n\n\n\n\\section{Class of zonotopes and generating function}\\label{section_combi}\n\\import{Parties\/Development\/}{1zonotopes_generating_function.tex}\n\\leavevmode\\\\\n\n\\section{Study of sums of coprime numbers}\\label{section_eulerset}\n\\import{Parties\/Development\/}{2eulerian_poly.tex}\n\\leavevmode\\\\\n\n\\section{Analysis of the generating function}\n\\import{Parties\/Development\/}{3analysis.tex}\n\\leavevmode\\\\\n\n\\section{Proof of the Theorem}\\label{proof_main}\n\\import{Parties\/Development\/}{5proof_theorem.tex}\n\\leavevmode\\\\\n\n\\section{Numerical elements about the speed of convergence to the asymptotic regime}\\label{numerical_discu}\n\\import{Parties\/Development\/}{6comments.tex}\n\\leavevmode\\\\\n\n\n\\section{The average diameter of Zonotopes}\n\\import{Parties\/Development\/}{7properties.tex}\n\\leavevmode\\\\\n\n\\section{Acknowledgments}\n\\import{Parties\/Development\/}{8acknowledgment.tex}\n\\leavevmode\\\\\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn quiescent regions irradiated by cosmic-- or X--rays, the oxygen chemistry is initiated by the charge transfer from\nH$^+$ and H$_{3}^{+}$ to atomic oxygen, forming O$^+$ and OH$^+$. \nIn warm environments it can also start\nwith the reaction of atomic oxygen with H$_2$($\\nu$=0) to form OH.\nThis  endothermic reaction (by $\\sim$0.08\\,eV) possesses an activation barrier \nof a few thousand~K and high gas temperatures ($\\gtrsim$400\\,K)\nare needed to produce significant OH abundances (\\textit{e.g.,}~in shocked gas).\nIn  molecular clouds exposed to strong far-ultraviolet (FUV) \n radiation fields, the so--called PDRs,\nthe gas is heated to relatively high temperatures and there are also high\nabundances of FUV-pumped vibrationally excited molecular hydrogen  H$_{2}^{*}$($\\nu$$=$1,2...) \\citep{hol97}. \nThe internal energy available in H$_{2}^{*}$ \ncan be used to overcome the O($^3P$)~+~H$_2$($\\nu$=0) reaction  barrier \\citep[see][and references therein]{agu10}.\nAlthough not well constrained observationally, enhanced OH abundances are   expected in warm~PDRs. \n\nOH is a key intermediary molecule in the FUV-illuminated gas because further reaction of OH with \nH$_2$, C$^+$, O, N or S$^+$  leads to  the formation of H$_2$O, CO$^+$, O$_2$, NO or SO$^+$ respectively. \nBesides, OH is  the product of H$_2$O photodissociation,\nthe main destruction route of water vapour in the gas unshielded against FUV radiation.\nObservations of OH in specific environments thus constrain different chemical routes of the oxygen chemistry.\n\nPrevious observations with  {\\em KAO} and {\\em ISO} \nhave demonstrated that OH is a powerful tracer of the warm neutral gas in shocked gas;\nfrom protostellar outflows and supernova remnants to extragalactic nuclei \\citep[\\textit{e.g.},][]{sto81,mel87,gon04}.\nUnfortunately, the poor angular resolution ($>$1$'$) and sensitivity of the above telescopes  prevented us\nfrom resolving the OH  emission from interstellar PDRs. \n\n\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f1.eps}}\n\\caption{OH (X $^2$$\\Pi$; $\\nu$=0) rotational lines detected with \\textit{Herschel}\/PACS towards the \n$\\alpha_{2000}$$\\simeq$5$^h$35$^m$21.9$^s$, $\\delta_{2000}$$\\simeq$$-$5$^o$25$'$06.7$''$ position\nwhere the higher excitation OH line peak is observed (see Figure~\\ref{fig:OH_maps}).\nThe red dotted lines show the expected wavelength for each $\\Lambda$-doublet\n (see Figure~\\ref{fig_levels} in the appendix~\\ref{sec:appenixB}  for a complete OH rotational energy diagram).\nSmall intensity asymmetries are observed in most OH $\\Lambda$--doublets.\nTransition upper level energies and $A_{ij}$ spontaneous transition probabilities\n are indicated.} \n\\label{fig:OH_lines}\n\\end{figure*}\n\n\nOwing to its proximity  \nand nearly edge-on orientation, \nthe interface region between the Orion Molecular Cloud~1 (OMC1) and the \nH\\,{\\sc ii} region illuminated by the Trapezium cluster, \nthe  Orion Bar,\nis the  prototypical warm PDR \\citep[with a FUV radiation field\n at the ionization front of $\\chi$$\\simeq$2.5$\\times$10$^4$ times the mean interstellar \nfield in Draine units;][]{mar98}.\nThe most commonly accepted scenario is that \nan extended gas component, with mean gas densities $n_H$ of 10$^{4-5}$\\,cm$^{-3}$,\ncauses the chemical stratification seen in the PDR \\citep{hog95}. \nMost of the low-$J$ molecular  line emission arises in this extended ``interclump'' medium \\citep{tie85,sim97,wie09,hab10}.\nIn addition, another component  of  higher density clumps\n was introduced to fit\nthe observed  H$_{2}$, high--$J$~CO, CO$^+$ and other high density and temperature  tracers \n\\citep{bur90,par91,sto95,wer96,you00}.\nOwing their small filling factor this clumpy structure would allow FUV radiation to permeate the region.\nThe presence of dense clumps is  still controversial. \n\nIn this letter we present initial results from a  spectral scan of the Orion Bar\ntaken with  the PACS instrument \\citep{pog10} on board $\\textit{Herschel}$ \\citep{pil10} \nas part of  the ``HEXOS'' key programme \\citep{ber10}. \nPACS observations of OH lines towards young stellar objects have recently been reported\nby Wampfler et al. \\citep{wam10}.\nHere we present the first detection of OH towards this\nprototypical PDR.  \n\n\n\n\n\n\\vspace{-0.2cm}\n\\section{Observations and data reduction}\n\nPACS observations were carried out on 7 and 8 September 2010\nand consist of two spectral scans in Nyquist sampling wavelength\nrange spectroscopy mode.\nThe PACS spectrometer uses photoconductor detectors and  provides 25 spectra  over a 47$''$$\\times$47$''$ field-of-view\nresolved in 5$\\times$5 spatial pixels (``spaxels''), each with a size of $\\sim$9.4$''$$\\times$9.4$''$ on the sky.\nThe measured width of the spectrometer point-spread function (PSF) is relatively constant at\n$\\lambda$$\\lesssim$125\\,$\\mu$m but it  increases above the spaxel size for longer wavelengths. \nThe resolving power varies between $\\lambda$\/$\\Delta$$\\lambda$$\\sim$1000 (R1 grating order) and $\\sim$5000  \n(B3A). \nThe central spaxel was centred at  \n$\\alpha_{2000}$: 5$^h$35$^m$20.61$^s$, $\\delta_{2000}$: -5$^o$25$'$14.0$''$ target  position.\nObservations were carried out in the  ``chop-nodded'' mode with the largest chopper throw of 6~arcmin.\nNominal capacitances (0.14\\,pF) were used.\nThe integration time was 3.2\\,h for the 1342204117 observation (B2B and R1) \n and 2.7\\,h for the 1342204118 observation (B3A and R1). \nThe data were processed with HIPE using a pipeline upgraded with a spectral flatfield\nalgorithm that reduces the spectral fringing seen in the  Nyquist-sampled wavelength spectra of bright sources.\nFigure~\\ref{fig:OH_lines} shows the resulting OH lines towards the OH emission peak and Figure~\\ref{fig_correlations} \nshows the \nintensities measured in each of the 25 spaxels for several lines of OH, CO, CH$^+$, H$_2$O and [N\\,{\\sc{ii}}]. \nIn order to better sample the PSF and obtain accurate line intensities to be reproduced with our models,  \nwe fit the  OH  emission  averaged over several adjacent spaxels in \nSection~\\ref{sec:excitation} (see also appendix~\\ref{sec:appenixA}).\n\n\n\\begin{figure}[h]\n\\vspace{-0.2cm}\n\\includegraphics[width=8.cm, angle=-90]{16977f2.eps}\n\\caption{PACS rotationally excited\n OH  $^{2}\\Pi_{3\/2}$ $J$=7\/2$\\rightarrow$5\/2 lines at $\\sim$84\\,$\\mu$m ($E_u$\/$k$=291\\,K)\noverlaid on the\ndistribution of the CO $J$=6-5 peak brightness temperature (colour image)\nobserved with the CSO telescope at $\\sim$11$''$  resolution \\citep{lis98}.\nWhite contours show the brightest regions of H$_{2}^{*}$ $v$=1-0 $S$(1) emission \\citep{wal00}. \nLower intensity H$_{2}^{*}$ extended emission is\npresent in the entire field \\citep{wer96}.\nViolet stars shows the position of the H$^{13}$CN $J$=1-0  clumps deeper inside the Bar\n\\citep{lis03}. Note the decrease of OH line intensity\nwith distance from the ionization front.}\n\\label{fig:OH_maps}\n\\end{figure}\n\n\n\\vspace{-0.2cm}\n\\section{Results}\n\nOf all the observed OH lines only the ground-state lines at $\\sim$119\\,$\\mu$m  show widespread bright emission \nat all positions. The $\\sim$119\\,$\\mu$m lines mainly arise from the background OMC1 complex\n(the same applies to most ground-state lines of other species).\nFigure~\\ref{fig:OH_maps} shows the spatial distribution of the\nrotationally excited OH $^{2}\\Pi_{3\/2}$ $J$=7\/2$\\rightarrow$5\/2 lines at $\\sim$84\\,$\\mu$m ($E_u$\/$k$=291\\,K)\nsuperimposed over the CO $J$=6-5 peak brightness temperature \\citep[colour image from][]{lis98} \nand over the brightest  H$_{2}^{*}$ $\\nu$=1-0 $S$(1) line emission regions\n\\citep[white contours from][]{wal00}.\nThe emission from the other  OH $\\Lambda$-doublets at $\\sim$79 and $\\sim$163\\,$\\mu$m \n(see Figure~\\ref{fig:OH_maps2} in appendix~\\ref{sec:appenixB}) \ndisplays a similar spatial distribution that follows the ``bar'' morphology   peaking\nnear the H$_{2}^{*}$~$v$=1-0 $S$(1) bright emission region and then decreases with distance from the\nionization front \\citep[note that H$_{2}^{*}$ shows lower level extended\nemission with small-scale structure in the entire observed field;][]{wer96}.\nThe excited OH spatial distribution, however,  does not follow the CO~$J$=6-5 emission maxima, \nwhich approximately trace the gas temperature \nin the extended ``interclump'' component.\n\n\n\n\n\nFigure~\\ref{fig:OH_lines} shows  the detected OH $\\Lambda$-doublets \n(at $\\sim$65, $\\sim$79, $\\sim$84, $\\sim$119 and $\\sim$163\\,$\\mu$m) towards the \nposition where the higher excitation OH lines peak. \nThe total intensity of the observed FIR lines \n is $\\sum$$I$(OH)$\\simeq$5$\\times$10$^{-4}$~erg\\,s$^{-1}$\\,cm$^{-2}$\\,sr$^{-1}$.\n All OH doublets appear in emission, with intensity asymmetry ratios up to 40\\% \n(one line of the $\\Lambda$-doublet is  brighter than the other).\n\n\nNote that the upper energy level of the $^{2}\\Pi_{3\/2}$~$J$=9\/2$\\rightarrow$7\/2 transition \nat $\\sim$65\\,$\\mu$m lies at  E$_u$\/$k$$\\sim$511\\,K.\nThe critical densities ($n_{cr}$) of  the observed  OH transitions are high,\n$n_{cr}$$\\gtrsim$10$^{8}$\\,cm$^{-3}$. For much lower gas densities, and in the presence of strong\nFIR radiation fields,\nmost lines would have been observed in absorption, especially those in the $^{2}\\Pi_{3\/2}$ ladder \\citep{goi02}.\nHence, the observed OH lines  must\narise in an widespread component of warm and dense gas.\n\n\n\nAlthough our PACS observations do not provide a fully sampled map,\n the line emission observed in the 25 spaxels \ncan be used to carry out a first-order analysis on the spatial correlation of  different\n lines (neglecting perfect PSF sampling, line opacity and excitation effects).\nExcept for the OH ground-state lines at $\\sim$119\\,$\\mu$m (that come from the background OMC1 cloud),\nwe find that the rotationally excited OH lines correlate well with the high-$J$ CO and CH$^+$ emission\nbut, as expected, they do not correlate with the ionized gas emission.\nFigure~\\ref{fig_correlations} (\\textit{lower panel}) compares the observed OH~$\\sim$84.597\\,$\\mu$m\nline intensities with those of \nCO $J$=21-20 ($E_u$\/$k$$\\sim$1276\\,K), CH$^+$ $J$=3-2 ($E_u$\/$k$$\\sim$240\\,K)\nand [N\\,{\\sc{ii}}]121.891\\,$\\mu$m  (all observed with a similar PSF)\nand also with the  CO $J$=15-14 ($E_u$\/$k$$\\sim$663\\,K), H$_2$O 3$_{03}$-2$_{12}$ ($E_u$\/$k$$\\sim$163\\,K) and\nOH~$\\sim$163.397\\,$\\mu$m lines (\\textit{upper panel}).\nThis simple analysis suggests\nthat the excited OH, high-$J$ CO and CH$^+$ $J$=3-2 lines  arise from the same gas  component.\nIt also shows that the emission from different excited OH lines is well correlated,\nwhile the OH and H$_2$O  emission is not (within the PSF sampling caveats).\n\n\n\n\\vspace{-0.2cm}\n\\section{OH column density determination}\n\\label{sec:excitation}\n\nDetermining the OH level populations is no  trivial excitation problem.\nIn addition to relatively strong asymmetries in the collisional \nrate coefficients\\footnote{We used  \ncollision rate coefficients of OH with $para$- and $ortho$-H$_2$  \nfrom  Offer \\& van Dishoeck \\citep{off92} and Offer et al. \\citep{off94}.\nStrong differences in the intensity of\neach OH $\\Lambda$-doublet component due to  asymmetries in the collisional\nrates between OH and $para$-H$_2$ were predicted\n (\\textit{e.g.,}~$I$(84.597)\/$I$(84.420)$>$$I$(119.441)\/$I$(119.234)$>$1).\nAsymmetries are significantly reduced when collisions with $ortho$-H$_2$ are included\n (\\textit{i.e.,}~in the warm gas) and when FIR radiative pumping plays a role.\nWe assume that the H$_2$ $ortho$-to-$para$ ratio is thermalized to the gas temperature, \n\\textit{e.g.,}~$\\sim$1.6 at 100\\,K and $\\sim$2.9 at 200\\,K.} between\neach $\\Lambda$-doubling component (Offer \\& Van Dishoeck 1992),\nradiative and opacity effects (pumping by the ambient IR radiation field and line trapping) can play a significant role\nif the  gas density is much lower than~$n_{cr}$.\nHere we use a nonlocal\nand non-LTE code that treats both the OH line \nand continuum radiative transfer \\citep[see appendix in][]{goi06}.\nThe continuum measured by PACS and SPIRE in the region (H. Arab et al. in prep.)\n can be approximately reproduced by a modified blackbody with a colour temperature of $\\sim$55\\,K and a\nopacity dependence of $\\sim$0.05(100\/$\\lambda$)$^{1.75}$ \\citep[see][]{lis98}.\nOur calculations include thermal, turbulent, and\nopacity line broadening with a turbulent velocity dispersion of\n$\\sigma$=1.0\\,km\\,s$^{-1}$ and FWHM=2.355$\\times\\sigma$\n\\citep[see \\textit{e.g.}, the  linewidths measured by][]{hog95}. \nA grid of single-component models for different $N$(OH), gas temperatures, densities and beam filling factors  \nwere run. \nThe best fit model was found by minimizing the ``$\\chi^2$-value\" (see the appendix~\\ref{sec:appenixA} for its definition).\n\n  \n\nFrom the excitation models we conclude that the\nhigh $I$(65.279)\/$I$(84.596)$\\simeq$0.3 and $I$(65.132)\/$I$(84.420)$\\simeq$0.5  intensity ratios in the $^{2}\\Pi_{3\/2}$ ladder\ncan only be reproduced if the gas is dense, at least $n_H$=$n$(H)+2$n$(H$_2$) of a few 10$^6$\\,cm$^{-3}$.\nIn addition, no model is able to produce even a crude fit to the data if one assumes \nthe average molecular gas temperature  (T$_k$$\\simeq$85\\,K)  in the lower density ``interclump'' medium \\citep{hog95}.\nIn the $^{2}\\Pi_{1\/2}$ ladder, the intensity of the $\\sim$163\\,$\\mu$m lines is sensitive to the \nFIR radiation field in the region through the absorption of FIR dust continuum photons in the\nOH\\,$\\sim$34 and $\\sim$53\\,$\\mu$m cross-ladder transitions ($^{2}\\Pi_{3\/2}$-$^{2}\\Pi_{1\/2}$ $J$=3\/2$\\rightarrow$5\/2\nand $J$=3\/2$\\rightarrow$3\/2 respectively).   \nHowever, the $\\sim$163\\,$\\mu$m lines in the Orion Bar are not particularly strong and the OH~$\\sim$34 and $\\sim$53\\,$\\mu$m lines\nare not present in the ISO spectra,  \nthus FIR pumping does not dominate the OH excitation. \nAll in all, the best model is found for a source of high density ($n_H$$\\lesssim$10$^7$~cm$^{-3}$)\nand warm gas temperatures (T$_{k}$=160-220\\,K).  \nThis temperature lies in between\nthe  $\\sim$600\\,K derived in the H$_2$ emitting regions \\citep{all05}\nand the $\\sim$150\\,K derived from  NH$_3$ lines \\citep{bat03}  \n near the ridge of dense and cooler H$^{13}$CN clumps \\citep[T$_k$$\\simeq$50\\,K;][]{lis03}.  \nIn our simple model, the best fit is obtained for a source of small filling factor ($\\eta\\simeq$10$\\%$)\nwith a source-averaged OH column density of $\\gtrsim$10$^{15}$\\,cm$^{-2}$ (see~appendix~\\ref{sec:appenixA}).\n\n\\begin{figure} [t]\n\\vspace{-0.0cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f3.eps}}\n\\caption{Line intensity spatial correlations between the  OH \n$^{2}\\Pi_{3\/2}$ $J$=7\/2$^-$$\\rightarrow$5\/2$^+$  line at 84.597\\,$\\mu$m \nand lines from other species.\nIntensities are in units of 10$^{-5}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$\\,sr$^{-1}$.}\n\\label{fig_correlations}\n\\end{figure}\n\n\n\\vspace{-0.2cm}\n\\section{OH chemistry and line emission origin}\n\\label{sec:PDRmods}\n\nWe  used the Meudon PDR code \\citep{lpt06,goi07} to estimate the OH column \ndensity in a slab of gas at  different densities ($n_H$ from 5$\\times$10$^4$ to 10$^7$\\,cm$^{-3}$). \nThe adopted FUV radiation field, $\\chi$=10$^4$, roughly corresponds to the attenuation\nof the FUV field at the ionization front  by a column density of $N_H$$\\simeq$10$^{21}$\\,cm$^{-2}$ in\na $n_H$$\\simeq$10$^4$\\,cm$^{-3}$  medium. This attenuation is equivalent to a spatial length\nof $\\sim$10$^{17}$\\,cm, consistent with the observed  decrease of excited\n OH emission with projected distance from the ionization front (see Figure~\\ref{fig:OH_maps}).\nGiven the high gas temperature, FUV field and moderate grain temperature \n(T$_{gr}$$\\sim$70-100\\,K) in the regions traced by  \nFIR OH, CO and CH$^+$ lines, we neglected molecule freeze-out and ice desorption\n\\citep[which are important deeper inside at A$_V$$\\gtrsim$3;][]{hol09}.\nIn our models   OH is a surface tracer that reaches its  peak abundance at A$_V$$\\lesssim$1\n(Figure~\\ref{fig:pdr_mod}) where OH formation is driven by the endothermic reaction O($^3P$)~+~H$_2$~$\\rightarrow$~OH~+~H, \nslightly enhanced by the O($^3P$)~+~H$_{2}^{*}$ reaction \n\\citep[included in our models; see][]{agu10}.\nGas temperatures around $\\sim$1000-500\\,K are predicted near the slab surface at A$_V$=0.01\nand around $\\sim$100\\,K at A$_V$$\\gtrsim$1. In these H\/H$_2$ transition layers where the OH abundance peaks,\nthe electron density is still high ($\\lesssim$[C$^+$\/H]\\,$n_H$) and hydrogen is not  fully molecular, \nwith  $f$(H$_2$)=2$n$(H$_2$)\/[$n$(H)+2$n$(H$_2$)]$\\simeq$0.5.\nIn general, the higher the gas temperature  where enough H$_2$ has formed, \nthe higher  the predicted OH abundance.\n\n\nIn the A$_V$$\\lesssim$1 layers,  OH destruction is dominated by photodissociation \n(OH~+~$h\\nu$~$\\rightarrow$~O~+~H)\nand to a  lesser extent, by reactions of OH with H$_2$ to form H$_2$O \n(only when the gas temperature and density are very high).\nWater vapour photodissociation (H$_2$O~+~$h\\nu$~$\\rightarrow$~OH~+~H) in the surface layers limits the H$_2$O formation\n  and leads to OH\/H$_2$O abundance ratios ($>$1), much higher than those expected in equally warm regions without \nenhanced FUV radiation fields (\\textit{e.g.} in C--shocks).\nThe lack of apparent correlation between the excited OH and H$_2$O 3$_{03}$-2$_{12}$ lines \n(see Figure~\\ref{fig_correlations}) and the\nabsence of high excitation H$_2$O lines in the PACS spectra (only weak H$_2$O 2$_{21}$-2$_{12}$, 2$_{12}$-1$_{01}$ \nand 3$_{03}$-2$_{12}$ lines are clearly detected) suggests\nthat the bulk of OH and H$_2$O column densities arise from different  depths.\n\nAs the temperature decreases inwards, the gas-phase production of OH also decreases.\nThe  spatial correlation between excited OH, CH$^+$ $J$=3-2  and high-$J$ CO lines is a good \nsignature of their common origin in the warm gas at low $A_V$ depths. \n\n\n\nOur PDR models predict OH column densities in the range  $\\sim$10$^{12}$\\,cm$^{-2}$ to $\\sim$10$^{15}$\\,cm$^{-2}$\nat  A$_V$$\\lesssim$1\nfor gas densities between $n_H$=5$\\times$10$^4$ and 10$^7$\\,cm$^{-3}$ respectively.\nHence, even if we take into account possible inclination \neffects, high density and temperature models \nproduce OH columns closer to the values derived in Section~4 just from excitation\nconsiderations (note that a precise determination of the\ngas density would require knowing the collisional rate coefficients of OH with H atoms and electrons). \nThe OH abundance in these dense surface layers is of the order of $\\approx$10$^{-6}$ with respect to H nuclei.\nHowever, optical depths of A$_V$$\\lesssim$1 correspond to spatial scales of only $\\sim$10$^{15}$\\,cm\n\\citep[\\textit{i.e.}, much smaller than the H$^{13}$CN clumps detected by][deeper\ninside the cloud]{lis03},\nbut we detect extended OH emission over $\\sim$10$^{17}$\\,cm scales. Therefore, we  have to conclude that \nthe observed OH emission arises from a small filling factor ensemble of unresolved structures \nthat are exposed to FUV radiation (overdense clumps or filaments). \nNote that owing to the lower grain temperature compared to the gas, \nthe expected FIR continuum emission from these clumps will still be below the continuum\nlevels observed by \\textit{Herschel}\/PACS.\n\nThe minimum size of the dense clumps is $\\sim$10$^{15}$\\,cm (from OH photochemical models)\nwith a maximun size of $\\sim$10$^{16}$\\,cm (from the inferred beam dilution factor). Both correspond\nto $\\lesssim$0.2$''$-2$''$ at the distance of Orion.\nAs an example, H$_2$ photoevaporating clumps of $\\sim$10$^{16}$\\,cm size have been unambiguously resolved \ntowards  S106 PDR \\cite{noe05}. However, higher angular resolution observations \n(\\textit{e.g.,}~with 8-10m  telescopes) are needed\nto resolve smaller H$_2$ clumps from the H$_2$ interclump emission in the Orion Bar.\n\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.00cm}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{16977f4.eps}}\n\\caption{Gas-phase PDR models of a FUV-illuminated slab ($\\chi$=10$^4$) of gas with\n different densities:\n$n_H$=10$^7$, 10$^6$, 10$^5$ and 5$\\times$10$^4$\\,cm$^{-3}$. \nOH column densities are shown as continuous curves (left axis) \nwhile OH abundances, $n$(OH)\/$n_H$, are shown as dashed curves (right axes).}\n\\label{fig:pdr_mod}\n\\end{figure}\n\n\n\n\nIf the observed FIR OH line emission does not arise from such a high density gas component, a different non-thermal \nexcitation mechanism able to populate the  OH $^{2}\\Pi_{3\/2}$ $J$=9\/2 and 7\/2 levels would be needed.\nTwo alternative scenarios can be explored, at least qualitatively.\nFirst, OH molecules produced by H$_2$O photodissociation are expected to form mostly in the ground electronic\nand vibrational state but in unusually high energy $J$$>$70\/2 levels \\citep[a few thousands~K!;][]{van00}. \nNevertheless, they   \n will cascade down radiatively to  lower energy rotational ladders extremely rapidly.\nAlthough very excited suprathermal OH $J$$\\simeq$70\/2-15\/2 lines have been reported\ntowards the HH\\,211 outflow\nand were interpreted as H$_2$O photodissociation \\citep{tap08},  we\ndid not find any of them in the SWS or in the IRS spectra of the Orion Bar\n(taken and processed by us from the ISO and \\textit{Spitzer} basic calibrated data archives). \nBesides, even if photodissociation is the main H$_2$O destruction mechanism in the A$_V$$\\lesssim$1 warm layers, \nthis  is not the main production pathway of OH (the O($^3P$)~+~H$_2$ reaction dominates).\n\n\nSecond,  experiments and quantum calculations  suggest that reaction of O($^3P$) atoms with H$_{2}^{*}$($\\nu$=1)  \ncan produce significant OH in the $\\nu$=1 vibrationally excited state, with OH($\\nu$=1)\/OH($\\nu$=0) \npopulation ratios $\\gtrsim$1 for moderate kinetic energies\n\\citep[$\\gtrsim$0.05-0.1\\,eV; \\textit{e.g.,}][]{bal04}.\nComplementarily, absorption of near-- and mid--IR photons (from the continuum or from bright overlapping H$_{2}^{*}$ \nand ionic lines) can also pump OH to the $\\nu$=1 state.\nIn both cases, subsequent de-excitation  through the $\\nu$=1-0 rotation--vibration band \nat $\\sim$2.80\\,$\\mu$m\nwould  populate the  OH($\\nu$=0) rotationally excited levels  that we observe with PACS.\nHowever, the OH $\\nu$=1-0 band at $\\sim$2.80\\,$\\mu$m is not present in the ISO\/SWS spectra.\nEven assuming that the O($^3P$)+H$_{2}^{*}$($\\nu$=1) reaction only forms OH($\\nu$=1),\nsignificant gas column densities at very hot temperatures (T$_k$$\\sim$2000\\,K) will be needed to match\nthe observations if  $n_H$$\\simeq$10$^{4-5}$\\,cm$^{-3}$.\nStudying the OH vibrationally-pumping mechanism quantitatively is beyond the scope of this work \nbut these chemical and pumping  effects could contribute to the excitation of  FIR OH lines\nin lower density gas.\n\nDifferent scenarios for the origin and nature of photoevaporating clumps have been proposed\n\\citep{gor02}, but without a more precise determination of their\nsizes and densities it is hard to conclude on any of them. Subarcsec resolution observations of OH gas-phase products, \n\\textit{e.g.,}~direct observation of CO$^+$ or SO$^+$ clumps with \\textit{ALMA},\nwill help us to assay the clumpy nature of the Orion Bar in the near future.\n \n\n\n\n\\begin{acknowledgements}\nPACS has been developed by\na consortium of institutes led by MPE (Germany)\nand including UVIE (Austria); KU Leuven, CSL,\nIMEC (Belgium); CEA, LAM (France); MPIA (Germany); \nINAF-IFSI\/OAA\/OAP\/OAT, LENS, SISSA (Italy); IAC (Spain). \nWe thank Darek Lis and Malcolm Walmsley for providing us their CO $J$=6-5 and H$_{2}^{*}$~1-0~S(1) maps.\nWe also thank Emilie Habart, Bart Vandenbussch and Pierre Royer  for useful discussions and  the referee \nfor his\/her constructive report.\nWe thank the Spanish MICINN for funding support\nthrough grants AYA2006-14876, AYA2009-07304 and CSD2009-00038. \nJRG is supported by a Ram\\'on y Cajal research contract.\n We acknowledge the use of the LAMDA data base \\citep{sch05}. \n\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe concept of a fuzzy set was introduced by Zadeh \\cite{zadeh}, in 1965.\nSince its inception, the theory has developed in many directions and found\napplications in a wide variety of fields. There has been a rapid growth in\nthe interest of fuzzy set theory and its applications from the past several\nyears. Many researchers published high-quality research articles on fuzzy\nsets in a variety of international journals. The study of fuzzy set in\nalgebraic structure has been started in the definitive paper of Rosenfeld\n1971 \\cite{Rosen}. Fuzzy subgroup and its important properties were defined\nand established by Rosenfeld \\cite{Rosen}. In 1981, Kuroki introduced the\nconcept of fuzzy ideals and fuzzy bi-ideals in semigroups in his paper \\cit\n{Kuroki}.\n\nThere are several kinds of fuzzy set extensions in the fuzzy set theory, for\nexample, intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets,\netc. Bipolar-valued fuzzy set is another extension of fuzzy set whose\nmembership degree range is different from the above extensions. Lee \\cit\n{Lee} introduced the notion of bipolar-valued fuzzy sets. Bipolar-valued\nfuzzy sets are an extension of fuzzy sets whose membership degree range is\nenlarged from the interval $[0,1]$ to $[-1,1]$. In a bipolar-valued fuzzy\nset, the membership degree $0$ indicate that elements are irrelevant to the\ncorresponding property, the membership degrees on $(0,1]$ assign that\nelements somewhat satisfy the property, and the membership degrees on \n[-1,0) $ assign that elements somewhat satisfy the implicit counter-property \n\\cite{Lee, Lee2}.\n\nAkram et al. \\cite{Akram} introduced the concept of bipolar fuzzy\nK-algebras. In \\cite{Jun}, Jun and park applied the notion of bipolar-valued\nfuzzy sets to BCH-algebras. They introduced the concept of bipolar fuzzy\nsubalgebras and bipolar fuzzy ideals of a BCH-algebra. Lee \\cite{KLee2}\napplied the notion of bipolar fuzzy subalgebras and bipolar fuzzy ideals of\nBCK\/BCI-algebras. Also some results on bipolar-valued fuzzy BCK\/BCI-algebras\nare introduced by Saeid in \\cite{Saeid}.\n\nThis paper concerns the relationship between bipolar-valued fuzzy sets and\nleft almost semigroups. The left almost semigroup abbreviated as an\nLA-semigroup, was first introduced by Kazim and Naseerudin \\cite{Kazim}.\nThey generalized some useful results of semigroup theory. They introduced\nbraces on the left of the ternary commutative law $abc=cba,$ to get a new\npseudo associative law, that is $(ab)c=(cb)a,$ and named it as left\ninvertive law. An LA-semigroup is the midway structure between a commutative\nsemigroup and a groupoid. Despite the fact, the structure is non-associative\nand non-commutative. It nevertheless possesses many interesting properties\nwhich we usually find in commutative and associative algebraic structures.\nMushtaq and Yusuf produced useful results \\cite{1979}, on locally\nassociative LA-semigroups in 1979. In this structure they defined powers of\nan element and congruences using these powers. They constructed quotient\nLA-semigroups using these congruences.\\ It is a useful non-associative\nstructure with wide applications in theory of flocks.\n\nIn this paper, we have introduced the notion of bipolar-valued fuzzy\nLA-subsemigroups and bipolar-valued fuzzy left (right, bi-, interior) ideals\nin LA-semigroups.\n\n\\section{\\textbf{Preliminaries and basic definitions}}\n\n\\textbf{Definition 2.1. }\\cite{Kazim} A groupoid $\\left( S,\\cdot \\right) $\\\nis called an LA-semigroup$,$ if it satisfies left invertive law \n\\begin{equation*}\n\\left( a\\cdot b\\right) \\cdot c=\\left( c\\cdot b\\right) \\cdot a,\\text{ \\ for\nall }a,b,c\\in S\\text{.}\n\\end{equation*}\n\n\\textbf{Example 2.1 }\\cite{1978} Let $\\left( \n\\mathbb{Z}\n,+\\right) $\\ denote the commutative group of integers under addition. Define\na binary operation \\textquotedblleft $\\ast $\\textquotedblright\\ in \n\\mathbb{Z}\n$\\ as follows$:\n\\begin{equation*}\na\\ast b=b-a,\\text{ \\ for all }a,b\\in \n\\mathbb{Z}\n\\text{.}\n\\end{equation*\nWhere \\textquotedblleft $-$\\textquotedblright\\ denotes the ordinary\nsubtraction of integers. Then $\\left( \n\\mathbb{Z}\n,\\ast \\right) $\\ is an LA-semigroup.\n\n\\textbf{Example 2.2 }\\cite{1978} Define a binary operation \\textquotedblleft \n$\\ast $\\textquotedblright\\ in \n\\mathbb{R}\n$\\ as follows$:\n\\begin{equation*}\na\\ast b=b\\div a,\\text{ \\ for all }a,b\\in \n\\mathbb{R}\n\\text{.}\n\\end{equation*\nThen $\\left( \n\\mathbb{R}\n,\\ast \\right) $\\ is an LA-semigroup.\n\n\\begin{lemma}\n\\label{l1}\\cite{1979} If $S$ is an LA-semigroup with left identity $e$, then \n$a(bc)=b(ac)$ for all $a,b,c\\in S.$\n\\end{lemma}\n\nLet $S$ be an LA-semigroup. A nonempty subset $A$ of $S$ is called an\nLA-subsemigroup of $S$ if $ab\\in A$ for all $a,b\\in A$. A nonempty subset $L$\nof $S$ is called a left ideal of $S$ if $SL\\subseteq L$ and a nonempty\nsubset $R$ of $S$ is called a right ideal of $S$ if $RS\\subseteq R$. A\nnonempty subset $I$ of $S$ is called an ideal of $S$ if $I$ is both a left\nand a right ideal of $S$. A subset $A$ of $S$ is called an interior ideal of \n$S$ if $(SA)S\\subseteq A$. An LA-subsemigroup $A$ of $S$ is called a\nbi-ideal of $S$ if $(AS)A\\subseteq A$.\n\nIn an LA-semigroup the medial law holds\n\\begin{equation*}\n(ab)(cd)=(ac)(bd),\\text{ \\ for all }a,b,c,d\\in S.\n\\end{equation*}\n\nIn an LA-semigroup $S$ with left identity, the paramedial law holds\n\\begin{equation*}\n(ab)(cd)=(dc)(ba),\\text{ \\ for all }a,b,c,d\\in S.\n\\end{equation*}\n\nNow we will recall the concept of bipolar-valued fuzzy sets.\n\n\\textbf{Definition 2.2 }\\cite{Lee2} Let $X$\\ be a nonempty set. A\nbipolar-valued fuzzy subset (BVF-subset, in short) $B$\\ of $X$\\ is an object\nhaving the for\n\\begin{equation*}\nB=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu _{B}^{-}(x)\\right\\rangle :x\\in\nX\\right\\} .\n\\end{equation*\nWhere $\\mu _{B}^{+}:X\\rightarrow \\lbrack 0,1]$\\ and $\\mu\n_{B}^{-}:X\\rightarrow \\lbrack -1,0]$.\n\nThe positive membership degree $\\mu _{B}^{+}(x)$ denotes the satisfaction\ndegree of an element $x$ to the property corresponding to a bipolar-valued\nfuzzy set $B=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu\n_{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} $, and the negative membership\ndegree $\\mu _{B}^{-}(x)$ denotes the satisfaction degree of $x$ to some\nimplicit counter property of $B=\\left\\{ \\left\\langle x,\\mu _{B}^{+}(x),\\mu\n_{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} $. For the sake of simplicity, we\nshall use the symbol $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nfor the bipolar-valued fuzzy set $B=\\left\\{ \\left\\langle x,\\mu\n_{B}^{+}(x),\\mu _{B}^{-}(x)\\right\\rangle :x\\in X\\right\\} .$\n\n\\textbf{Definition 2.3 }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ and $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle $ be two BVF-subsets of a nonempty set $X$. Then\nthe product of two BVF-subsets is denoted by $B_{1}\\circ B_{2}$ and defined\nas: \n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\left( x\\right) \n&=&\\left\\{ \n\\begin{array}{l}\n\\tbigvee_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( z\\right) \\right\\} ,\\text{ if }x=yz\\text{ for some }y,z\\in\nS \\\\ \n\\text{ }0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ otherwise.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{array\n\\right.  \\\\\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\left( x\\right) \n&=&\\left\\{ \n\\begin{array}{l}\n\\tbigwedge_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( z\\right) \\right\\} ,\\text{ if }x=yz\\text{ for some }y,z\\in\nS \\\\ \n\\text{ }0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ otherwise.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{array\n\\right. \n\\end{eqnarray*}\n\nNote that an LA-semigroup $S$ can be considered as a BVF-subset of itself\nand le\n\\begin{eqnarray*}\n\\Gamma &=&\\left\\langle \\mathcal{S}_{\\Gamma }^{+}(x),\\mathcal{S}_{\\Gamma\n}^{-}(x)\\right\\rangle \\\\\n&=&\\left\\{ \\left\\langle x,\\mathcal{S}_{\\Gamma }^{+}(x),\\mathcal{S}_{\\Gamma\n}^{-}(x)\\right\\rangle :\\mathcal{S}_{\\Gamma }^{+}(x)=1\\text{ and }\\mathcal{S\n_{\\Gamma }^{-}(x)=-1,\\text{ for all }x\\text{ in }S\\right\\}\n\\end{eqnarray*\nbe a BVF-subset and $\\Gamma =\\left\\langle \\mathcal{S}_{\\Gamma }^{+}(x)\n\\mathcal{S}_{\\Gamma }^{-}(x)\\right\\rangle $ will be carried out in\noperations with a BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ such that $\\mathcal{S}_{\\Gamma }^{+}$ and $\\mathcal{\n}_{\\Gamma }^{-}$ will be used in collaboration with $\\mu _{B}^{+}$ and $\\mu\n_{B}^{-}$ respectively.\n\nLet $BVF(S)$ denote the set of all BVF-subsets of an LA-semigroup $S.$\n\n\\begin{proposition}\n\\label{P1}Let $S$ be an LA-semigroup, then the set $(BVF(S),\\circ )$ is an\nLA-semigroup.\n\\end{proposition}\n\n\\textbf{Proof. }Clearly $BVF(S)$ is closed. Let $B_{1}=\\left\\langle \\mu\n_{B_{1}}^{+},\\mu _{B_{1}}^{-}\\right\\rangle ,$ $B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ and $B_{3}=\\left\\langle \\mu\n_{B_{3}}^{+},\\mu _{B_{3}}^{-}\\right\\rangle $ be in $BVF(S)$. Let $x$ be any\nelement of $S$ such that $x\\neq yz$ for some $y,z\\in S$. Then we hav\n\\begin{equation*}\n\\left( \\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) (x)=0=\\left( \\left( \\mu _{B_{3}}^{+}\\circ \\mu\n_{B_{2}}^{+}\\right) \\circ \\mu _{B_{1}}^{+}\\right) (x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) (x)=0=\\left( \\left( \\mu _{B_{3}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\right) \\circ \\mu _{B_{1}}^{-}\\right) (x).\n\\end{equation*\nLet $x$ be any element of $S$ such that $x=yz$ for some $y,z\\in S$. Then we\nhav\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) (x) &=&{\\bigvee }_{x=yz}\\left\\{ \\left( \\mu\n_{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (y)\\wedge \\mu\n_{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=yz}\\left\\{ \\left( {\\bigvee }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{+}(p)\\wedge \\mu _{B_{2}}^{+}(q)\\right\\} \\right) \\wedge \\mu\n_{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=yz}\\text{ }{\\bigvee }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{+}(p)\\wedge \\mu _{B_{2}}^{+}(q)\\wedge \\mu _{B_{3}}^{+}(z)\\right\\}\n\\\\\n&=&{\\bigvee }_{x=(pq)z}\\left\\{ \\mu _{B_{1}}^{+}(p)\\wedge \\mu\n_{B_{2}}^{+}(q)\\wedge \\mu _{B_{3}}^{+}(z)\\right\\} \\\\\n&=&{\\bigvee }_{x=(zq)p}\\left\\{ \\mu _{B_{3}}^{+}(z)\\wedge \\mu\n_{B_{2}}^{+}(q)\\wedge \\mu _{B_{1}}^{+}(p)\\right\\} \\\\\n&=&{\\bigvee }_{x=sp}\\left\\{ \\left( {\\bigvee }_{s=zq}\\left\\{ \\mu\n_{B_{3}}^{+}(z)\\wedge \\mu _{B_{2}}^{+}(q)\\right\\} \\right) \\wedge \\mu\n_{B_{1}}^{+}(p)\\right\\} \\\\\n&=&{\\bigvee }_{x=sp}\\left\\{ \\left( \\mu _{B_{3}}^{+}\\circ \\mu\n_{B_{2}}^{+}\\right) (s)\\wedge \\mu _{B_{1}}^{+}(p)\\right\\} \\\\\n&=&\\left( \\left( \\mu _{B_{3}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\mu\n_{B_{1}}^{+}\\right) (x).\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) (x) &=&{\\bigwedge }_{x=yz}\\left\\{ \\left( \\mu\n_{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (y)\\vee \\mu _{B_{3}}^{-}(z)\\right\\}\n\\\\\n&=&{\\bigwedge }_{x=yz}\\left\\{ \\left( {\\bigwedge }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{-}(p)\\vee \\mu _{B_{2}}^{-}(q)\\right\\} \\right) \\vee \\mu\n_{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=yz}\\text{ }{\\bigwedge }_{y=pq}\\left\\{ \\mu\n_{B_{1}}^{-}(p)\\vee \\mu _{B_{2}}^{-}(q)\\vee \\mu _{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=(pq)z}\\left\\{ \\mu _{B_{1}}^{-}(p)\\vee \\mu\n_{B_{2}}^{-}(q)\\vee \\mu _{B_{3}}^{-}(z)\\right\\} \\\\\n&=&{\\bigwedge }_{x=(zq)p}\\left\\{ \\mu _{B_{3}}^{-}(z)\\vee \\mu\n_{B_{2}}^{-}(q)\\vee \\mu _{B_{1}}^{-}(p)\\right\\} \\\\\n&=&{\\bigwedge }_{x=sp}\\left\\{ \\left( {\\bigwedge }_{s=zq}\\left\\{ \\mu\n_{B_{3}}^{-}(z)\\vee \\mu _{B_{2}}^{-}(q)\\right\\} \\right) \\vee \\mu\n_{B_{1}}^{-}(p)\\right\\} \\\\\n&=&{\\bigwedge }_{x=sp}\\left\\{ \\left( \\mu _{B_{3}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\right) (s)\\vee \\mu _{B_{1}}^{-}(p)\\right\\} \\\\\n&=&\\left( \\left( \\mu _{B_{3}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\mu\n_{B_{1}}^{-}\\right) (x).\n\\end{eqnarray*}\n\nHence $(BVF(S),\\circ )$ is an LA-semigroup. $\\ \\ \\Box $\n\n\\begin{corollary}\n\\label{C1}If $S$ is an LA-semigroup, then the medial law holds in $BVF(S)$.\n\\end{corollary}\n\n\\textbf{Proof. }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $, $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle ,$ $B_{3}=\\left\\langle \\mu _{B_{3}}^{+},\\mu\n_{B_{3}}^{-}\\right\\rangle $ and $B_{4}=\\left\\langle \\mu _{B_{4}}^{+},\\mu\n_{B_{4}}^{-}\\right\\rangle $ be in $BVF(S)$. By successive use of left\ninvertive la\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) \\circ \\left( \\mu\n_{B_{3}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) &=&\\left( \\left( \\mu\n_{B_{3}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) \\circ \\mu _{B_{2}}^{+}\\right)\n\\circ \\mu _{B_{1}}^{+} \\\\\n&=&\\left( \\left( \\mu _{B_{2}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) \\circ \\mu\n_{B_{3}}^{+}\\right) \\circ \\mu _{B_{1}}^{+} \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{3}}^{+}\\right) \\circ \\left( \\mu\n_{B_{2}}^{+}\\circ \\mu _{B_{4}}^{+}\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) \\circ \\left( \\mu\n_{B_{3}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) &=&\\left( \\left( \\mu\n_{B_{3}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) \\circ \\mu _{B_{2}}^{-}\\right)\n\\circ \\mu _{B_{1}}^{-} \\\\\n&=&\\left( \\left( \\mu _{B_{2}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) \\circ \\mu\n_{B_{3}}^{-}\\right) \\circ \\mu _{B_{1}}^{-} \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{3}}^{-}\\right) \\circ \\left( \\mu\n_{B_{2}}^{-}\\circ \\mu _{B_{4}}^{-}\\right) .\n\\end{eqnarray*\nHence this shows that the medial law holds in $BVF(S)$. $\\ \\ \\Box $\n\n\\section{\\textbf{Bipolar-valued fuzzy ideals in LA-semigroup}}\n\n\\textbf{Definition 3.1 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy LA-subsemigroup of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( x\\right) \\wedge \\mu\n_{B}^{+}\\left( y\\right) \\text{\\ \\ and \\ \\ }\\mu _{B}^{-}\\left( xy\\right) \\leq\n\\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\n\\textbf{Definition 3.2 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy left ideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\text{ \\ \\ and\n\\ \\ }\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\n\\textbf{Definition 3.3 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a bipolar-valued\nfuzzy right ideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( x\\right) \\text{ \\ \\ and\n\\ }\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( x\\right)\n\\end{equation*\nfor all $x,y\\in S$.\n\nA BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of an\nLA-semigroup $S$ is called a BVF-ideal or BVF-two-sided ideal of $S$ if \nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is both BVF-left and\nBVF-right ideal of $S$.\n\n\\textbf{Example 3.1} Let $S=\\{a,b,c,d\\}$, the binary operation \"$\\cdot $\" on \n$S$ be defined as follows\n\\begin{equation*}\n\\begin{tabular}{l|llll}\n$\\cdot $ & $a$ & $b$ & $c$ & $d$ \\\\ \\hline\n$a$ & $b$ & $d$ & $c$ & $a$ \\\\ \n$b$ & $a$ & $b$ & $c$ & $d$ \\\\ \n$c$ & $c$ & $c$ & $c$ & $c$ \\\\ \n$d$ & $d$ & $a$ & $c$ & $b\n\\end{tabular\n\\end{equation*\nClearly, $S$ is an LA-semigroup. But $S$ is not a semigroup because \nd=d\\cdot (b\\cdot a)\\neq (d\\cdot b)\\cdot a=b.$ Now we define BVF-subset a\n\\begin{equation*}\nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle =\\left\\langle \\left( \n\\frac{a}{0.2},\\frac{b}{0.2},\\frac{c}{0.7},\\frac{d}{0.2}\\right) ,\\text{ \n\\left( \\frac{a}{-0.5},\\frac{b}{-0.5},\\frac{c}{-0.8},\\frac{d}{-0.5}\\right)\n\\right\\rangle .\n\\end{equation*\nClearly $B$ is a BVF-ideal of $S.$\n\n\\begin{proposition}\n\\label{BP2}Every BVF-left (BVF-right) ideal $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is a bipolar-valued fuzzy\nLA-subsemigroup of $S$.\n\\end{proposition}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-left ideal of $S$ and for any $x,y\\in S$,\n\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\geq \\mu\n_{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( y\\right) \\text{.}\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right) \\leq \\mu\n_{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right) .\n\\end{equation*\nHence $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nbipolar-valued fuzzy LA-subsemigroup of $S$. The other case can be prove in\na similar way. $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L56}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S.$ Then\n\\end{lemma}\n\n(1) $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-LA-subsemigroup of $S$\\ if and only if $\\mu _{B}^{+}\\circ \\mu\n_{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu _{B}^{-}\\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$\n\n(2) $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a BVF-left\n(resp. BVF-right) ideal of $S$\\ if and only if $\\mathcal{S}_{\\Gamma\n}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mathcal{S}_{\\Gamma\n}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$ (resp. $\\mu _{B}^{+}\\circ \n\\mathcal{S}_{\\Gamma }^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\supseteq \\mu _{B}^{-}$)$.$\n\n\\textbf{Proof. }(1)\\textbf{\\ }Let $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ be a BVF-LA-subsemigroup of $S$ and $x\\in S.$ If \n\\left( \\mu _{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) =0$ and $\\left(\n\\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) =0,$ then $\\left( \\mu\n_{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) \\leq \\mu _{B}^{+}\\left(\nx\\right) $ and $\\left( \\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right)\n\\geq \\mu _{B}^{-}\\left( x\\right) .$ Otherwise\n\\begin{equation*}\n\\left( \\mu _{B}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) ={\\bigvee \n_{x=yz}\\left\\{ \\mu _{B}^{+}\\left( y\\right) \\wedge \\mu _{B}^{+}\\left(\nz\\right) \\right\\} \\leq {\\bigvee }_{x=yz}\\mu _{B}^{+}\\left( yz\\right) =\\mu\n_{B}^{+}\\left( x\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\mu _{B}^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) ={\\bigwedge \n_{x=yz}\\left\\{ \\mu _{B}^{-}\\left( y\\right) \\vee \\mu _{B}^{-}\\left( z\\right)\n\\right\\} \\geq {\\bigwedge }_{x=yz}\\mu _{B}^{-}\\left( yz\\right) =\\mu\n_{B}^{-}\\left( x\\right) .\n\\end{equation*\nThus $\\mu _{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mu\n_{B}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, let $\\mu _{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$, $\\mu\n_{B}^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$ and $x,y\\in S,$ the\n\\begin{equation*}\n\\mu _{B}^{+}\\left( xy\\right) \\geq \\left( \\mu _{B}^{+}\\circ \\mu\n_{B}^{+}\\right) \\left( xy\\right) ={\\bigvee }_{xy=ab}\\left\\{ \\mu\n_{B}^{+}\\left( a\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\geq \\mu\n_{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( y\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( xy\\right) \\leq \\left( \\mu _{B}^{-}\\circ \\mu\n_{B}^{-}\\right) \\left( xy\\right) ={\\bigwedge }_{xy=ab}\\left\\{ \\mu\n_{B}^{-}\\left( a\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\leq \\mu\n_{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( y\\right) .\n\\end{equation*\nSo $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-LA-subsemigroup of $S.$\n\n(2) Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-left ideal of $S$ and $x\\in S.$ If $\\left( \\mathcal{S}_{\\Gamma\n}^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) =0$ and $\\left( \\mathcal{S\n_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) =0,$ then $\\left( \n\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) \\leq \\mu\n_{B}^{+}\\left( x\\right) $ and $\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ \\mu\n_{B}^{-}\\right) \\left( x\\right) \\geq \\mu _{B}^{-}\\left( x\\right) .$\nOtherwise\n\\begin{eqnarray*}\n\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\left( x\\right) &=\n{\\bigvee }_{x=ab}\\left\\{ \\mathcal{S}_{\\Gamma }^{+}\\left( a\\right) \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} ={\\bigvee }_{x=ab}\\left\\{ 1\\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}\\mu _{B}^{+}\\left( b\\right) \\leq {\\bigvee }_{x=ab}\\mu\n_{B}^{+}\\left( ab\\right) =\\mu _{B}^{+}\\left( x\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\right) \\left( x\\right) &=\n{\\bigwedge }_{x=ab}\\left\\{ \\mathcal{S}_{\\Gamma }^{-}\\left( a\\right) \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} ={\\bigwedge }_{x=ab}\\left\\{ -1\\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}\\mu _{B}^{-}\\left( b\\right) \\geq {\\bigwedge \n_{x=ab}\\mu _{B}^{-}\\left( ab\\right) =\\mu _{B}^{-}\\left( x\\right) .\n\\end{eqnarray*\nThus $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and \n$\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, let $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu\n_{B}^{+}$, $\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu\n_{B}^{-} $ and $x,y\\in S,$ then \n\\begin{eqnarray*}\n\\mu _{B}^{+}\\left( xy\\right) &\\geq &\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ\n\\mu _{B}^{+}\\right) \\left( xy\\right) ={\\bigvee }_{xy=ab}\\left\\{ \\mathcal{S\n_{\\Gamma }^{+}\\left( a\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&\\geq &\\mathcal{S}_{\\Gamma }^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left(\ny\\right) =1\\wedge \\mu _{B}^{+}\\left( y\\right) =\\mu _{B}^{+}\\left( y\\right) .\n\\end{eqnarray*\nAnd \n\\begin{eqnarray*}\n\\mu _{B}^{-}\\left( xy\\right) &\\leq &\\left( \\mathcal{S}_{\\Gamma }^{-}\\circ\n\\mu _{B}^{-}\\right) \\left( xy\\right) ={\\bigwedge }_{xy=ab}\\left\\{ \\mathcal{S\n_{\\Gamma }^{-}\\left( a\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&\\leq &\\mathcal{S}_{\\Gamma }^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left(\ny\\right) =-1\\vee \\mu _{B}^{-}\\left( y\\right) =\\mu _{B}^{-}\\left( y\\right) .\n\\end{eqnarray*\nThus $\\mu _{B}^{+}\\left( xy\\right) \\geq \\mu _{B}^{+}\\left( y\\right) $ and \n\\mu _{B}^{-}\\left( xy\\right) \\leq \\mu _{B}^{-}\\left( y\\right) .$\\ Thus \nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a BVF-left ideal\nof $S.$ The second case can be seen in a similar way. $\\ \\ \\Box $\n\nLet $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu _{B_{1}}^{-}\\right\\rangle $ and \n$B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ be two\nBVF-subsets of an LA-semigroup $S.$ The symbol $B_{1}\\cap B_{2}$ will mean\nthe following\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) (x)=\\mu\n_{B_{1}}^{+}(x)\\wedge \\mu _{B_{2}}^{+}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) (x)=\\mu\n_{B_{1}}^{-}(x)\\vee \\mu _{B_{2}}^{-}(x),\\text{ for all }x\\in S.\n\\end{equation*\nThe symbol $A\\cup B$ will mean the following\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\cup \\mu _{B_{2}}^{+}\\right) (x)=\\mu\n_{B_{1}}^{+}(x)\\vee \\mu _{B_{2}}^{+}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\cap \\mu _{B_{2}}^{-}\\right) (x)=\\mu\n_{B_{1}}^{-}(x)\\wedge \\mu _{B_{2}}^{-}(x),\\text{ for all }x\\in S.\n\\end{equation*}\n\n\\begin{theorem}\nLet $S$ be an LA-semigroup and $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ be a BVF-right ideal of $S$ and \nB_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $ be a\nBVF-left ideal of $S,$ then $B_{1}\\circ B_{2}\\subseteq B_{1}\\cap B_{2}.$\n\\end{theorem}\n\n\\textbf{Proof. }Let for any $x,y,z\\in S,$ if $x\\neq yz,$ then we hav\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (x)=0\\leq \\mu\n_{B_{1}}^{+}(x)\\wedge \\mu _{B_{2}}^{+}(x)=\\left( \\mu _{B_{1}}^{+}\\cap \\mu\n_{B_{2}}^{+}\\right) (x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (x)=0\\geq \\mu\n_{B_{1}}^{-}(x)\\vee \\mu _{B_{2}}^{-}(x)=\\left( \\mu _{B_{1}}^{-}\\cup \\mu\n_{B_{2}}^{-}\\right) (x).\n\\end{equation*\nOtherwis\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\right) (x) &=&{\\bigvee \n_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nz\\right) \\right\\}  \\\\\n&\\leq &{\\bigvee }_{x=yz}\\left\\{ \\mu _{B_{1}}^{+}\\left( yz\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( yz\\right) \\right\\}  \\\\\n&=&\\left\\{ \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nx\\right) \\right\\}  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) (x).\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\circ \\mu _{B_{2}}^{-}\\right) (x) &=&{\\bigvee \n_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nz\\right) \\right\\}  \\\\\n&\\geq &{\\bigvee }_{x=yz}\\left\\{ \\mu _{B_{1}}^{-}\\left( yz\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( yz\\right) \\right\\}  \\\\\n&=&\\left\\{ \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nx\\right) \\right\\}  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) (x).\n\\end{eqnarray*\nThus we get $\\mu _{B_{1}}^{+}\\circ \\mu _{B_{2}}^{+}\\subseteq \\mu\n_{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}$ and $\\mu _{B_{1}}^{-}\\circ \\mu\n_{B_{2}}^{-}\\supseteq \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}.$ Hence \nB_{1}\\circ B_{2}\\subseteq B_{1}\\cap B_{2}.$ $\\ \\ \\Box $\n\n\\begin{proposition}\n\\label{P101}\\textit{Let }$B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $\\textit{\\ and }$B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $\\textit{\\ be two\nBVF-LA-subsemigroups of }$S$\\textit{. Then }$B_{1}\\cap B_{2}$\\textit{\\ is\nalso a BVF-LA-subsemigroup of }$S$\\textit{.\\ }\n\\end{proposition}\n\n\\textbf{Proof.\\ }Let $B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $ and $B_{2}=\\left\\langle \\mu _{B_{2}}^{+},\\mu\n_{B_{2}}^{-}\\right\\rangle $ be two \\textit{BVF-LA-subsemigroups} of $S$. Let \n$x,y\\in S$. The\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( xy\\right)  &=&\\mu\n_{B_{1}}^{+}\\left( xy\\right) \\wedge \\mu _{B_{2}}^{+}\\left( xy\\right)  \\\\\n&\\geq &\\left( \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{1}}^{+}\\left(\ny\\right) \\right) \\wedge \\left( \\mu _{B_{2}}^{+}\\left( x\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\left( x\\right) \\wedge \\mu _{B_{2}}^{+}\\left(\nx\\right) \\right) \\wedge \\left( \\mu _{B_{1}}^{+}\\left( y\\right) \\wedge \\mu\n_{B_{2}}^{+}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( x\\right)\n\\wedge \\left( \\mu _{B_{1}}^{+}\\cap \\mu _{B_{2}}^{+}\\right) \\left( y\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( xy\\right)  &=&\\mu\n_{B_{1}}^{-}\\left( xy\\right) \\vee \\mu _{B_{2}}^{-}\\left( xy\\right)  \\\\\n&\\leq &\\left( \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{1}}^{-}\\left(\ny\\right) \\right) \\vee \\left( \\mu _{B_{2}}^{-}\\left( x\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\left( x\\right) \\vee \\mu _{B_{2}}^{-}\\left(\nx\\right) \\right) \\vee \\left( \\mu _{B_{1}}^{-}\\left( y\\right) \\vee \\mu\n_{B_{2}}^{-}\\left( y\\right) \\right)  \\\\\n&=&\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( x\\right) \\vee\n\\left( \\mu _{B_{1}}^{-}\\cup \\mu _{B_{2}}^{-}\\right) \\left( y\\right) .\n\\end{eqnarray*\nThus $B_{1}\\cap B_{2}$\\textit{\\ is also a bipolar-valued fuzzy\nLA-subsemigroup of }$S$\\textit{.} $\\ \\ \\Box $\n\n\\begin{proposition}\n\\textit{Let }$B_{1}=\\left\\langle \\mu _{B_{1}}^{+},\\mu\n_{B_{1}}^{-}\\right\\rangle $\\textit{\\ and }$B_{2}=\\left\\langle \\mu\n_{B_{2}}^{+},\\mu _{B_{2}}^{-}\\right\\rangle $\\textit{\\ be two BVF-left (resp.\nBVF-right, BVF-two-sided) ideal of }$S$\\textit{. Then }$B_{1}\\cap B_{2}\n\\textit{\\ is also a BVF-left (resp. BVF-right, BVF-two-sided) ideal of }$S.$\n\\end{proposition}\n\n\\textbf{Proof.\\ }The proof is similar to the proof of Proposition \\ref{P101\n. $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L2}In an LA-semigroup $S$ with left identity, for every BVF-left\nideal $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of $S$, we\nhave $\\Gamma \\circ B=B.$ Where $\\Gamma =\\left\\langle \\mathcal{S}_{\\Gamma\n}^{+}(x),\\mathcal{S}_{\\Gamma }^{-}(x)\\right\\rangle .$\n\\end{lemma}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-left ideal of $S.$ It is sufficient to show that $\\mathcal{S\n_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\mathcal{S\n_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}$. Now $x=ex$, for all \n$x$ in $S$, as $e$ is left identity in $S$. S\n\\begin{equation*}\n(\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+})(x)={\\bigvee }_{x=yz}\\left\\{ \n\\mathcal{S}_{\\Gamma }^{+}(y)\\wedge \\mu _{B}^{+}(z)\\right\\} \\geq \\mathcal{S\n_{\\Gamma }^{+}(e)\\wedge \\mu _{B}^{+}(x)=1\\wedge \\mu _{B}^{+}(x)=\\mu\n_{B}^{+}(x).\n\\end{equation*\nAn\n\\begin{equation*}\n(\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-})(x)={\\bigwedge }_{x=yz}\\left\\{ \n\\mathcal{S}_{\\Gamma }^{-}(y)\\vee \\mu _{B}^{-}(z)\\right\\} \\leq \\mathcal{S\n_{\\Gamma }^{-}(e)\\vee \\mu _{B}^{-}(x)=-1\\vee \\mu _{B}^{-}(x)=\\mu _{B}^{-}(x).\n\\end{equation*\nThus $\\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\supseteq \\mu _{B}^{+}$ and \n$\\mathcal{S}_{\\Gamma }^{-}\\circ \\mu _{B}^{-}\\subseteq \\mu _{B}^{-}.$ Hence \n\\Gamma \\circ B=B.$ $\\ \\ \\Box $\n\n\\textbf{Definition 3.4 }Let $S$ be an LA-semigroup and let $\\emptyset \\neq\nA\\subseteq S.$ Then bipolar-valued fuzzy characteristic function $\\chi\n_{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi _{A}}^{-}\\right\\rangle $\nof $A$ is defined a\n\\begin{equation*}\n\\mu _{\\chi _{A}}^{+}=\\left\\{ \n\\begin{array}{c}\n1\\text{ \\ \\ if }x\\in A \\\\ \n0\\text{ \\ \\ if }x\\notin \n\\end{array\n\\right. \\text{ \\ \\ and \\ }\\mu _{\\chi _{A}}^{-}=\\left\\{ \n\\begin{array}{c}\n-1\\text{ \\ \\ if }x\\in A \\\\ \n\\text{ \\ }0\\text{ \\ \\ if }x\\notin A\n\\end{array\n\\right.\n\\end{equation*}\n\n\\begin{theorem}\n\\label{T119}Let $A$ be a nonempty subset of an LA-semigroup $S$. Then $A$ is\nan LA-subsemigroup of $S$ if and only if $\\chi _{A}$ is a\nBVF-LA-subsemigroup of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }Let $A$ be an LA-subsemigroup of $S$. For any $x,y\\in S,$ we\nhave the following cases:\n\nCase $\\left( 1\\right) :$ If $x,y\\in A$, then $xy\\in A$. Since $A$ is an\nLA-subsemigroup of $S$. Then $\\mu _{\\chi _{A}}^{+}\\left( xy\\right) =1,$ $\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) =1$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right)\n=1$. Therefore \n\\begin{equation*}\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) =\\mu _{\\chi _{A}}^{+}\\left( x\\right)\n\\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd $\\mu _{\\chi _{A}}^{-}\\left( xy\\right) =-1,$ $\\mu _{\\chi _{A}}^{-}\\left(\nx\\right) =-1$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1$. Therefore \n\\begin{equation*}\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) =\\mu _{\\chi _{A}}^{-}\\left( x\\right)\n\\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nCase $\\left( 2\\right) :$ If $x,y\\notin A$, then $\\mu _{\\chi _{A}}^{+}\\left(\nx\\right) =0$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right) =0$. So \\ \\ \\ \\ \\ \\ \\ \n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu _{\\chi\n_{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd $\\mu _{\\chi _{A}}^{-}\\left( x\\right) =0$ and $\\mu _{\\chi _{A}}^{-}\\left(\ny\\right) =0$. So \n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*}\n\nCase $\\left( 3\\right) :$ If $x\\in A$ or $y\\in A$. If $x\\in A$ and $y\\notin A\n, then $\\mu _{\\chi _{A}}^{+}\\left( x\\right) =1$ and $\\mu _{\\chi\n_{A}}^{+}\\left( y\\right) =0$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nNow if $x\\notin A$ and $y\\in A$, then $\\mu _{\\chi _{A}}^{+}\\left( x\\right) =0\n$ and $\\mu _{\\chi _{A}}^{+}\\left( y\\right) =1$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) \\geq 0=\\mu\n_{\\chi _{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) .\n\\end{equation*\nAnd if $x\\in A$ or $y\\in A$. If $x\\in A$ and $y\\notin A$, then $\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) =-1$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =0$.\nS\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nNow if If $x\\notin A$ and $y\\in A$, then $\\mu _{\\chi _{A}}^{-}\\left(\nx\\right) =0$ and $\\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1$. S\n\\begin{equation*}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) \\leq 0=\\mu\n_{\\chi _{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) .\n\\end{equation*\nHence $\\chi _{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi\n_{A}}^{-}\\right\\rangle $ is a BVF-LA-subsemigroup of $S$.\n\nConversely, suppose $\\chi _{A}=\\left\\langle \\mu _{\\chi _{A}}^{+},\\mu _{\\chi\n_{A}}^{-}\\right\\rangle $ is a BVF-LA-subsemigroup of $S$ and let $x,y\\in A$.\nThen we have \n\\begin{eqnarray*}\n\\text{ \\ \\ }\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &\\geq &\\mu _{\\chi\n_{A}}^{+}\\left( x\\right) \\wedge \\mu _{\\chi _{A}}^{+}\\left( y\\right) =1\\wedge\n1=1 \\\\\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &\\geq &1\\text{ but }\\mu _{\\chi\n_{A}}^{+}\\left( xy\\right) \\leq 1 \\\\\n\\mu _{\\chi _{A}}^{+}\\left( xy\\right) &=&1\n\\end{eqnarray*\nAnd \n\\begin{eqnarray*}\n\\text{ \\ \\ }\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &\\leq &\\mu _{\\chi\n_{A}}^{-}\\left( x\\right) \\vee \\mu _{\\chi _{A}}^{-}\\left( y\\right) =-1\\vee\n-1=-1 \\\\\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &\\leq &-1\\text{ but }\\mu _{\\chi\n_{A}}^{-}\\left( xy\\right) \\geq -1 \\\\\n\\mu _{\\chi _{A}}^{-}\\left( xy\\right) &=&-1\n\\end{eqnarray*\nHence $xy\\in A$. Therefore $A$ is an LA-subsemigroup of $S.$ $\\ \\ \\Box $\n\n\\begin{theorem}\nLet $A$ be a nonempty subset of an LA-semigroup $S$. Then $A$ is a left\n(resp. right) ideal of $S$ if and only if $\\chi _{A}$ is a BVF-left (resp.\nBVF-right) ideal of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }The proof of this theorem is similar to Theorem \\ref{T119}. \n\\ \\ \\Box $\n\n\\textbf{Definition 3.5 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a\\textbf{\\ \nBVF-generalized bi-ideal\\textbf{\\ }of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(y\n\\text{ \\ and \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}(x)\\vee \\mu\n_{B}^{-}(y),\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\textbf{Definition 3.6 }A BVF-LA-subsemigroup $B=\\left\\langle \\mu\n_{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called \n\\textbf{\\ }BVF-bi-ideal\\textbf{\\ }of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(y\n\\text{ \\ and \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}(x)\\vee \\mu\n_{B}^{-}(y),\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{L57}A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is a BVF-generalized bi-ideal\nof $S$ if and only if $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma\n}^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nbe a BVF-generalized bi-ideal of an LA-semigroup $S$ and $x\\in S.$ If \n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) =0$ and $\\left( \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\right) (x)=0,$ the\n\\begin{equation*}\n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) =0\\leq \\mu _{B}^{+}\\left( x\\right) \\text{ \\\nand \\ }\\left( \\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right)\n\\circ \\mu _{B}^{-}\\right) (x)=0\\geq \\mu _{B}^{-}(x).\n\\end{equation*\nOtherwis\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\right) \\left( x\\right) &=&{\\bigvee }_{x=ab}\\left\\{ \\left( \\mu\n_{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\left( a\\right) \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}\\left\\{ {\\bigvee }_{a=mn}\\left\\{ \\mu _{B}^{+}\\left(\nm\\right) \\wedge \\mathcal{S}_{\\Gamma }^{+}\\left( n\\right) \\right\\} \\wedge \\mu\n_{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}{\\bigvee }_{a=mn}\\left\\{ \\left( \\mu _{B}^{+}\\left(\nm\\right) \\wedge 1\\right) \\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigvee }_{x=ab}{\\bigvee }_{a=mn}\\left\\{ \\mu _{B}^{+}\\left( m\\right)\n\\wedge \\mu _{B}^{+}\\left( b\\right) \\right\\} \\\\\n&\\leq &\\mu _{B}^{+}\\left( x\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\left( \\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu\n_{B}^{-}\\right) \\left( x\\right) &=&{\\bigwedge }_{x=ab}\\left\\{ \\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\left( a\\right) \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}\\left\\{ {\\bigwedge }_{a=mn}\\left\\{ \\mu _{B}^{-}\\left(\nm\\right) \\vee \\mathcal{S}_{\\Gamma }^{-}\\left( n\\right) \\right\\} \\vee \\mu\n_{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}{\\bigwedge }_{a=mn}\\left\\{ \\left( \\mu _{B}^{-}\\left(\nm\\right) \\vee -1\\right) \\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&=&{\\bigwedge }_{x=ab}{\\bigwedge }_{a=mn}\\left\\{ \\mu _{B}^{-}\\left( m\\right)\n\\vee \\mu _{B}^{-}\\left( b\\right) \\right\\} \\\\\n&\\geq &\\mu _{B}^{-}\\left( x\\right) .\n\\end{eqnarray*\nThus $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu\n_{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu _{B}^{-}\\circ \\mathcal{S\n_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq \\mu _{B}^{-}.$\\newline\nConversely, assume that $\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma\n}^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mu\n_{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\supseteq\n\\mu _{B}^{-}.$ Let $x,y,z\\in S,$ the\n\\begin{eqnarray*}\n\\mu _{B}^{+}\\left( (xy)z\\right) &\\geq &\\left( \\left( \\mu _{B}^{+}\\circ \n\\mathcal{S}_{\\Gamma }^{+}\\right) \\circ \\mu _{B}^{+}\\right) \\left(\n(xy)z\\right) ={\\bigvee }_{(xy)z=cd}\\left\\{ \\left( \\mu _{B}^{+}\\circ \\mathcal\nS}_{\\Gamma }^{+}\\right) \\left( c\\right) \\wedge \\mu _{B}^{+}\\left( d\\right)\n\\right\\} \\\\\n&\\geq &\\left( \\mu _{B}^{+}\\circ \\mathcal{S}_{\\Gamma }^{+}\\right) \\left(\nxy\\right) \\wedge \\mu _{B}^{+}\\left( z\\right) =\\left\\{ {\\bigvee \n_{xy=pq}\\left\\{ \\mu _{B}^{+}\\left( p\\right) \\wedge \\mathcal{S}_{\\Gamma\n}^{+}\\left( q\\right) \\right\\} \\right\\} \\wedge \\mu _{B}^{+}\\left( z\\right) \\\\\n&\\geq &\\left\\{ \\mu _{B}^{+}\\left( x\\right) \\wedge \\mathcal{S}_{\\Gamma\n}^{+}\\left( y\\right) \\right\\} \\wedge \\mu _{B}^{+}\\left( z\\right) =\\left\\{\n\\mu _{B}^{+}\\left( x\\right) \\wedge 1\\right\\} \\wedge \\mu _{B}^{+}\\left(\nz\\right) \\\\\n&=&\\mu _{B}^{+}\\left( x\\right) \\wedge \\mu _{B}^{+}\\left( z\\right) .\n\\end{eqnarray*\nAn\n\\begin{eqnarray*}\n\\mu _{B}^{-}\\left( (xy)z\\right) &\\leq &\\left( \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu _{B}^{-}\\right) \\left(\n(xy)z\\right) ={\\bigwedge }_{(xy)z=cd}\\left\\{ \\left( \\mu _{B}^{-}\\circ \n\\mathcal{S}_{\\Gamma }^{-}\\right) \\left( c\\right) \\vee \\mu _{B}^{-}\\left(\nd\\right) \\right\\} \\\\\n&\\leq &\\left( \\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\left(\nxy\\right) \\vee \\mu _{B}^{-}\\left( z\\right) =\\left\\{ {\\bigwedge \n_{xy=pq}\\left\\{ \\mu _{B}^{-}\\left( p\\right) \\vee \\mathcal{S}_{\\Gamma\n}^{-}\\left( q\\right) \\right\\} \\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) \\\\\n&\\leq &\\left\\{ \\mu _{B}^{-}\\left( x\\right) \\vee \\mathcal{S}_{\\Gamma\n}^{-}\\left( y\\right) \\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) =\\left\\{ \\mu\n_{B}^{-}\\left( x\\right) \\vee -1\\right\\} \\vee \\mu _{B}^{-}\\left( z\\right) \\\\\n&=&\\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( z\\right) .\n\\end{eqnarray*\nThus $\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}\\left( x\\right)\n\\wedge \\mu _{B}^{+}\\left( z\\right) $ and $\\mu _{B}^{-}\\left( (xy)z\\right)\n\\leq \\mu _{B}^{-}\\left( x\\right) \\vee \\mu _{B}^{-}\\left( z\\right) ,$ which\nimplies that $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is a\nBVF-generalized bi-ideal of $S.$ $\\ \\ \\Box $\n\n\\begin{lemma}\n\\label{L30}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S$ then $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-bi-ideal of $S$\\ if and only if $\\mu\n_{B}^{+}\\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+},$ $\\mu _{B}^{-}\\circ \\mu\n_{B}^{-}\\supseteq \\mu _{B}^{-},$ $\\left( \\mu _{B}^{+}\\circ \\mathcal{S\n_{\\Gamma }^{+}\\right) \\circ \\mu _{B}^{+}\\subseteq \\mu _{B}^{+}$ and $\\left(\n\\mu _{B}^{-}\\circ \\mathcal{S}_{\\Gamma }^{-}\\right) \\circ \\mu\n_{B}^{-}\\supseteq \\mu _{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }Follows from Lemma \\ref{L56}(1) and Lemma \\ref{L57}. $\\ \\\n\\Box $\n\n\\textbf{Definition 3.7 }A BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of an LA-semigroup $S$ is called a BVF-interior\nideal of $S$ i\n\\begin{equation*}\n\\mu _{B}^{+}\\left( (xy)z\\right) \\geq \\mu _{B}^{+}\\left( y\\right) \\text{ \\\nand \\ }\\mu _{B}^{-}\\left( (xy)z\\right) \\leq \\mu _{B}^{-}\\left( y\\right) \n\\text{ for all }x,y,z\\in S.\n\\end{equation*}\n\n\\begin{lemma}\n\\label{L31}Let $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a\nBVF-subset of an LA-semigroup $S$ then $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-interior ideal of $S$\\ if and only if \n\\left( \\mathcal{S}_{\\Gamma }^{+}\\circ \\mu _{B}^{+}\\right) \\circ \\mathcal{S\n_{\\Gamma }^{+}\\subseteq \\mu _{B}^{+}$ and $\\left( \\mathcal{S}_{\\Gamma\n}^{-}\\circ \\mu _{B}^{-}\\right) \\circ \\mathcal{S}_{\\Gamma }^{-}\\supseteq \\mu\n_{B}^{-}.$\n\\end{lemma}\n\n\\textbf{Proof. }The proof of this lemma is similar to the proof of Lemma \\re\n{L57}$.$ $\\ \\ \\Box $\n\n\\begin{remark}\nEvery BVF-ideal is a BVF-interior ideal of an LA-semigroup $S,$ but the\nconverse is not true.\n\\end{remark}\n\n\\textbf{Example 3.2 }Let $S=\\{a,b,c,d\\}$, the binary operation \"$\\cdot $\" on \n$S$ be defined as follows\n\\begin{equation*}\n\\begin{tabular}{l|llll}\n$\\cdot $ & $a$ & $b$ & $c$ & $d$ \\\\ \\hline\n$a$ & $c$ & $c$ & $c$ & $d$ \\\\ \n$b$ & $d$ & $d$ & $c$ & $c$ \\\\ \n$c$ & $d$ & $d$ & $d$ & $d$ \\\\ \n$d$ & $d$ & $d$ & $d$ & $d\n\\end{tabular\n\\end{equation*\nClearly, $S$ is an LA-semigroup. But $S$ is not a semigroup because \nc=a\\cdot (a\\cdot b)\\neq (a\\cdot a)\\cdot b=d.$ Now we define BVF-subset a\n\\begin{equation*}\nB=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle =\\left\\langle \\left( \n\\frac{a}{0.5},\\frac{b}{0.3},\\frac{c}{0.1},\\frac{d}{0.8}\\right) ,\\text{ \n\\left( \\frac{a}{-0.7},\\frac{b}{-0.4},\\frac{c}{-0.2},\\frac{d}{-0.9}\\right)\n\\right\\rangle .\n\\end{equation*\nIt can be verified that $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is a BVF-interior ideal of $S.$ But, sinc\n\\begin{equation*}\n\\mu _{B}^{+}\\left( b\\cdot c\\right) =\\mu _{B}^{+}\\left( c\\right) =0.1<0.3=\\mu\n_{B}^{+}\\left( b\\right) .\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}\\left( b\\cdot c\\right) =\\mu _{B}^{-}\\left( c\\right)\n=-0.2>-0.4=\\mu _{B}^{-}\\left( b\\right) .\n\\end{equation*\nThus $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ is not a\nBVF-right ideal of $S,$ that is, $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is not a BVF-two-sided ideal of $S.$\n\n\\begin{proposition}\nEvery BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ of\nan LA-semigroup $S$ with left identity is a BVF-right ideal if and only if\nit is a BVF-interior ideal.\n\\end{proposition}\n\n\\textbf{Proof. }Let every BVF-subset $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ of $S$ is a BVF-right ideal. For $x$, $a$ and $y$ of \n$S$, conside\n\\begin{equation*}\n\\mu _{B}^{+}((xa)y)\\geq \\mu _{B}^{+}(xa)=\\mu _{B}^{+}((ex)a)=\\mu\n_{B}^{+}((ax)e)\\geq \\mu _{B}^{+}(ax)\\geq \\mu _{B}^{+}(a).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xa)y)\\leq \\mu _{B}^{-}(xa)=\\mu _{B}^{-}((ex)a)=\\mu\n_{B}^{-}((ax)e)\\leq \\mu _{B}^{-}(ax)\\leq \\mu _{B}^{-}(a).\n\\end{equation*\nWhich implies that $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $\nis a BVF-interior ideal. Conversely, for any $x$ and $y$ in $S$ we have\n\\begin{equation*}\n\\mu _{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x)\\text{ \\ and \\ }\\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x).\n\\end{equation*\nHence required. $\\ \\ \\Box $\n\n\\begin{theorem}\nLet $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle $ be a BVF-left\nideal of an LA-semigroup $S$ with left identity, then $B=\\left\\langle \\mu\n_{B}^{+},\\mu _{B}^{-}\\right\\rangle $ being BVF-interior ideal is a\nBVF-bi-ideal of $S$.\n\\end{theorem}\n\n\\textbf{Proof. }Since $B=\\left\\langle \\mu _{B}^{+},\\mu _{B}^{-}\\right\\rangle \n$ is an BVF-left ideal in $S$, so $\\mu _{B}^{+}(xy)\\geq \\mu _{B}^{+}(y)$ and \n$\\mu _{B}^{-}(xy)\\leq \\mu _{B}^{-}(y)$ for all $x$ and $y$ in $S$. As $e$ is\nleft identity in $S$. So\n\\begin{equation*}\n\\mu _{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x)\\text{ \\ and \\ }\\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x),\n\\end{equation*}\nwhich implies that $\\mu _{B}^{+}(xy)\\geq \\mu _{B}^{+}(x)\\wedge \\mu\n_{B}^{+}(y)$ and $\\mu _{B}^{-}(xy)\\leq \\mu _{B}^{-}(x)\\vee \\mu _{B}^{-}(y)$\nfor all $x$ and $y$ in $S$. Thus $B=\\left\\langle \\mu _{B}^{+},\\mu\n_{B}^{-}\\right\\rangle $ is an BVF-LA-subsemigroup of $S$. For any $x,y$ and \nz$ in $S$, we ge\n\\begin{equation*}\n\\mu _{B}^{+}((xy)z)=\\mu _{B}^{+}((x(ey))z)=\\mu _{B}^{+}((e(xy))z)\\geq \\mu\n_{B}^{+}(xy)=\\mu _{B}^{+}((ex)y)\\geq \\mu _{B}^{+}(x).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xy)z)=\\mu _{B}^{-}((x(ey))z)=\\mu _{B}^{-}((e(xy))z)\\leq \\mu\n_{B}^{-}(xy)=\\mu _{B}^{-}((ex)y)\\leq \\mu _{B}^{-}(x).\n\\end{equation*\nAls\n\\begin{equation*}\n\\mu _{B}^{+}((xy)z)=\\mu _{B}^{+}((zy)x)=\\mu _{B}^{+}((z(ey))x)=\\mu\n_{B}^{+}((e(zy))x)\\geq \\mu _{B}^{+}(zy)=\\mu _{B}^{+}((ez)y)\\geq \\mu\n_{B}^{+}(z).\n\\end{equation*\nAn\n\\begin{equation*}\n\\mu _{B}^{-}((xy)z)=\\mu _{B}^{-}((zy)x)=\\mu _{B}^{-}((z(ey))x)=\\mu\n_{B}^{-}((e(zy))x)\\leq \\mu _{B}^{-}(zy)=\\mu _{B}^{-}((ez)y)\\leq \\mu\n_{B}^{-}(z).\n\\end{equation*\nHence $\\mu _{B}^{+}((xy)z)\\geq \\mu _{B}^{+}(x)\\wedge \\mu _{B}^{+}(z)$ and \n\\mu _{B}^{-}((xy)z)\\leq \\mu _{B}^{-}(x)\\vee \\mu _{B}^{-}(z)$ for all $x,y$\nand $z$ in $S$. $\\ \\ \\Box $\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIn 1954 Wick and Cutkosky noticed that a certain class of ladder-type Feynman integrals with massive propagators features a massive dual conformal symmetry \\cite{Wick:1954eu,Cutkosky:1954ru}. \nWhile most of the formal insights into quantum field theory inspired by the AdS\/CFT correspondence are limited to massless situations, this massive dual conformal symmetry is naturally realized in the context of this duality~\\cite{Alday:2009zm,Caron-Huot:2014gia}. In particular, the extended dual conformal symmetry limits the variables that certain massive Feynman integrals can depend on and thus simplifies their computation. In the present letter we argue that for large classes of Feynman integrals, this massive dual conformal symmetry is in fact only the zeroth level of an infinite dimensional massive Yangian algebra. In addition to limiting the number of variables, this new symmetry strongly constrains the functional form of the integrals. While these symmetry properties naturally extend the observations on the integrability of massless Feynman integrals \\cite{Chicherin:2017cns,Chicherin:2017frs,Loebbert:2019vcj}, to the knowledge of the authors this is the first occurence of quantum integrability in massive quantum field theory in $D>2$ spacetime dimensions. \n\nFor \\emph{massless} $\\mathcal{N}=4$ super Yang--Mills (SYM) theory it was recently argued that planar integrability is preserved in a certain double scaling limit, which (in the simplest case) results in the so-called bi-scalar fishnet theory \\cite{Gurdogan:2015csr}. Here, individual (massless) Feynman integrals inherit the Yangian symmetry that underlies the integrability of the prototypical examples of the AdS\/CFT duality \\cite{Chicherin:2017cns,Chicherin:2017frs}. A similar starting point, i.e.\\ an \\emph{integrable massive} avatar of $\\mathcal{N}=4$ super Yang--Mills theory, is not known. We thus investigate massive Feynman integrals directly, i.e.\\ we consider the properties of functions of the type\n\\begin{equation}\n\\includegraphicsbox{FigTwoStars.pdf}\n\\quad\n=\n\\int \\frac{\\mathrm{d}^D x_0 \\mathrm{d}^D x_{\\bar 0}}\n{\n\\hat x_{01}^{2a_1}\n\\hat x_{02}^{2a_2}\nx_{0\\bar 0}^{2b_0}\n\\hat x_{\\bar 03}^{2a_3}\n\\hat x_{\\bar 04}^{2a_4}\n},\n\\end{equation}\nwhere $x_{jk}^\\mu=x_j^\\mu-x_k^\\mu$ and $\\hat x_{jk}^2=x_{jk}^2+(m_j-m_k)^2$. Here the dashed internal propagator is massless, i.e.\\ $m_0=m_{\\bar 0}=0$, while the other propagators are massive.\nThe $x$-variables denote dualized momenta (dotted green diagram) related via $p^{\\mu}_j=x^{\\mu}_j-x^{\\mu}_{j+1}$  \\footnote{Note that the $p_j^2$ are unconstrained and the $m_j$ are generic; we have $x^{\\mu}_j=x^{\\mu}_{1}-\\sum_{k < j} p^{\\mu}_k $ and $x_{n+1}^\\mu=x_{1}^\\mu$.}.\nOur findings suggest that all Feynman graphs, which are cut from regular tilings of the plane and have massive propagators on the boundary, feature a massive $D$-dimensional Yangian symmetry. \nWe will demonstrate the usefulness of this Yangian for bootstrapping massive Feynman integrals. Finally, we will show that, when translated to momentum space, the non-local Yangian symmetry can be interpreted as a massive generalization of momentum space conformal symmetry. \nThis suggests to interpret this novel symmetry within the AdS\/CFT correspondence.\n\n\\section{Massive Yangian}\n\nMassive dual conformal symmetry is realized in the form of partial differential equations obeyed by coordinate space Feynman integrals. That is, the integrals are annihilated by the tensor product action of the level-zero dual conformal generators\n$\n\\gen{J}^a = \\sum_{j=1}^n \\gen{J}_{j}^a, \n$\nwhere $\\gen{J}_j^a$\ndenotes one of \nthe following densities  acting on $x_j$:\n\\begin{align}\n\\gen{P}^{\\hat \\mu}_j &= -i \\partial_{x_{j}}^{\\hat \\mu}, \n\\qquad\\qquad\n\\gen{L}_j^{\\hat \\mu\\hat \\nu} = i x_j^{\\hat \\mu} \\partial_{x_{j}}^{\\hat \\nu} - ix^{\\hat \\nu}_j \\partial_{x_{j}}^{\\hat \\mu}, \n\\nonumber\n\\\\\n\\gen{D}_j &= -i \\brk!{x_{j\\mu} \\partial_{x_j}^\\mu + m_j \\partial_{m_j} + \\Delta_j},\n\\label{eqn:massdualconfrep}\n\\\\\n \\gen{K}^{\\hat \\mu}_j &= -2ix_j^{\\hat \\mu}\\brk!{x_{j\\nu}  \\partial_{x_j}^\\nu  + m_j\\partial_{m_j} + \\Delta_j} +i (x^2_j + m^2_j)\\partial_{x_{j}}^{\\hat \\mu}.\\notag\n\\end{align}\nThese can be understood as massless generators in $D+1$ dimensions with $x_j^{D+1}=m_j$. Only the components $\\hat \\mu=1,\\dots,D$ of the generators correspond to symmetries.\nHere we work with the Euclidean metric and the index $\\hat{\\mu}$ runs from 1~to~$D+1$, while $\\mu$ runs from 1~to~$D$.\n\nThe massive Yangian is spanned by the above level-zero Lie algebra generators and the bi-local level-one generators \ndefined as\n\\begin{equation}\n\\label{eq:DefLev1}\n \\gen{\\widehat J}^a=\\sfrac{1}{2} f^a{}_{bc}\\sum_{j<k=1}^n \\gen{J}_{j}^c \\gen{J}_{k}^b+ \\sum_{j=1}^n \\Eval_j \\gen{J}_{j}^a,\n\\end{equation}\nwhere $ f^a{}_{bc}$ denotes the Lie algebra structure constants.\nThe so-called evaluation parameters $\\Eval_j$ depend on the considered Feynman integral and will be specified below.\nThe level-one momentum generator for instance reads \n\\begin{align}\n\\label{eq:DefPhat}\n\\gen{\\widehat P}^{\\hat\\mu}\n=\\sfrac{i}{2} \\sum_{j,k=1}^n \n\\text{sign}(k-j)\\brk!{\\gen{P}_j^{\\hat\\mu} \\gen{D}_k + \\gen{P}_{j\\nu} \\gen{L}_k^{\\hat\\mu \\nu} }\n + \\sum_{j=1}^n \\Eval_j \\gen{P}_j^{\\hat \\mu}.\n\\end{align}\nHere we do not sum the internal index $\\nu$ over $D+1$ dimensions but rather define this contribution separately, \ne.g.\\\n$\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}=\\sfrac{i}{2}\\sum_{j,k}\\text{sign}(k-j)\\gen{P}_{j,D+1} \\gen{L}_k^{\\hat\\mu,D+1}$.\nNote that the whole Yangian algebra is spanned by the above level-one momentum generator \\eqref{eq:DefPhat} and the level-zero generators~\\eqref{eqn:massdualconfrep}. \n\n\\section{One and Two Loops }\n\n\nFirst, we consider a generic scalar $n$-point Feynman integral at one-loop order with massive propagators. Propagator powers $a_j$ and the spacetime dimension $D$ are arbitrary:\n\\begin{align}\nI_n = \\int \\text{d}^D x_0 \\ \\rho_n\n=\\includegraphicsbox{FigNStarPropWeights.pdf}\\,.\n\\label{eq:nGonIntegral}\n\\end{align}\nHere $\\rho_n\n=\n\\prod_{j=1}^n (x_{0j}^2 + m_j^2)^{-a_j}$ and we use the notation $x_{jk}^\\mu=x_j^\\mu-x_k^\\mu$.\nThese integrals have been expressed in terms of hypergeometric functions by Davydychev~\\cite{Davydychev:1990cq}.\nSurprisingly, we find that the integrand is invariant under the massive level-one generators acting on legs $1$ to $n$ defined in \\eqref{eq:DefLev1}, \ni.e.\\ we have $\\gen{\\widehat J}^a \\rho_n = 0$ which implies \n\\begin{align}\n\\gen{\\widehat J}^a I_n = 0.\n\\label{eq:Lvl1Symm}\n\\end{align}\nThe scaling dimensions in \\eqref{eqn:massdualconfrep} take values $\\Delta_j=a_j$ and the evaluation parameters entering the level-one generators \\eqref{eq:DefLev1} can be written in the compact form\n\\begin{align}\n\\Eval_j\n=\\sfrac{1}{2} \\sum_{k=1}^n \\text{sign}(k-j) \\,a_k.\n\\label{eq:EvalsNGons}\n\\end{align}\nWe emphasize that this symmetry also holds for cases where an arbitrary subset of the masses is set to zero~\n\\footnote{In these cases one acts on massive legs with the generators of the massive symmetry and on massless legs with the usual dual-conformal generators obtained by dropping the mass terms in \\protect\\eqref{eqn:massdualconfrep}.}.\nMoreover, the above level-one symmetry holds even if we do not impose conformal level-zero symmetry, i.e.\\ the constraint $\\sum_{j=1}^n a_j=D$ on the propagator powers. Hence, the level-one symmetry can be used to constrain dual conformal and non-dual-conformal integrals.\nFinally we should mention that also the generators as $\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}$ given below \\eqref{eq:DefPhat}, which can be interpreted as contributions to an internal sum over $D+1$ dimensions, define a separate symmetry of the above one-loop integrands \\footnote{Switching on the internal mass $m_0$, only the combination $\\gen{\\widehat P}^{\\hat \\mu}+\\gen{\\widehat P}^{\\hat \\mu}_{\\text{extra}}$ is a symmetry.}.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|l|c|c|c|}\\hline\nLoops&Graphs&Conformal&Not Conformal\n\\\\\\hline\nOne&$n$-gons&Full Yangian \\& $\\levo{J}_{\\text{extra}}^a$ & All $\\levo{J}^a$  \\& $\\levo{J}_{\\text{extra}}^a$ \n\\\\\nTwo&$l$-$r$-gons&Full Yangian&  $\\levo{P}^\\mu$\n\\\\\nAll&Tilings&Full Yangian&  $\\levo{P}^\\mu$\n\\\\\\hline\n\\end{tabular}\n\\caption{Overview of non-local symmetries. We speak of a (dual) conformal integral, if at each $x$-space vertex the propagator powers $a_j$ sum up to the spacetime dimension $D$.}\n\\label{tab:allsyms}\n\\end{table}\n\nWhile \\eqref{eq:Lvl1Symm} represents the central symmetry equation that extends to higher loop integrals, see \\Tabref{tab:allsyms}, at one-loop order even stronger invariance statements can be formulated:\nFirstly, the one-loop invariance even holds at the integrand level. Secondly, due to the total permutation symmetry, the $n$-gon integrand is even invariant under the two-point generator density:\n\\begin{equation}\n\\label{eq:TwoPointSymNgon}\n\\gen{\\widehat J}^a_{jk}\\rho_n=0,\n\\qquad\n\\gen{\\widehat J}^a_{jk}=\\sfrac{1}{2} f^a{}_{bc} \\gen{J}_{j}^c \\gen{J}_{k}^b+ \\Eval_j \\gen{J}_{j}^a+ \\Eval_k \\gen{J}_{k}^a,\n\\end{equation}\nfor $j,k\\in\\{1,\\dots,n\\}$ and with  \n$\\Eval_j= a_k\/2$ and\n$\\Eval_k=- a_j\/2$.\n\nAt two loops we find that $n$-point integrals (not integrands) built from an $l+1$-gon and an $r+1$-gon with $n=l+r$ connected by a massless internal propagator are Yangian invariant:\n\\begin{equation}\n\\includegraphicsbox{FiglrGon.pdf}\n\\end{equation}\nUsing the permutation symmetries of this class of diagrams as well as their level-zero symmetry and the two-point level-one symmetry \\eqref{eq:TwoPointSymNgon} of the one-loop integrals, it is not difficult to prove this two-loop invariance. \n\n\n\\section{Higher Loops}\n\\label{sec:HigherLoops}\n\nBeyond the analytic proofs for one- and two-loop integrals, we have numerically tested the invariance of higher loop diagrams for various examples, cf.\\ \\Tabref{tab:ExampleGraphs}.\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{tabular}{|c | c | c||c|}\\hline\n\\multicolumn{3}{|c||}{Non-local Symmetry}&{No Sym.}\n\\\\\\hline\nTriangle&Square&Hexagon&Irregular\\\\\n\\hline\n\\includegraphicsbox[scale=.7]{FigTwoHexagons.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigBrokenWindowDiagram.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigThreeStars.pdf}\n&\n\\includegraphicsbox[scale=.6]{FigThreePentagons.pdf}\n\\\\\n\\includegraphicsbox[scale=.7]{FigThreeHexagons.pdf}\n&\n\\includegraphicsbox[scale=.7]{FigWindowDiagram.pdf}\n&\n\\raisebox{1mm}{\\includegraphicsbox[scale=.7]{FigSixStars.pdf}}\n&\n\\includegraphicsbox[scale=.7]{FigIrreg4.pdf}\n\\\\\\hline\n\\end{tabular}\n\\caption{\nExamples of integrals up to 6 loops and 12 legs explicitly tested for level-one momentum invariance by acting on the Feynman parametrization. Graphs cut from regular tilings are invariant if propagators on the boundary of the graph are massive or massless (solid), while internal propagators have to be massless (dashed). Graphs not cut from regular tilings have no level-one symmetry as described here.}\n\\label{tab:ExampleGraphs}\n\\end{table}\nTo be precise, we have checked for the presence of the symmetry by acting on the respective Feynman parametrization and numerical integration of the result.\nOn this basis we conjecture that all planar Feynman graphs, which are cut along a closed contour from one of the three regular tilings of the plane, have massive Yangian symmetry if all internal propagators are massless. The external \npropagators can be massive or massless.\nHere an $x$-space propagator is called external if it is connected to an external point of the diagram. The integration can be evaluated in generic spacetime dimension $D$ and propagator powers may take generic values $a_j$.\nFor full Yangian invariance we require the conformal constraint\n$D=\\sum_{j\\in \\text{vertex}} a_j$ at each integration vertex.\nIf this constraint is not satisfied, we still have level-one momentum symmetry $\\levo{P}^\\mu$, which corresponds to a massive generalization of momentum space conformal symmetry that we introduce below.\nIn the massless limit this matches the observations of \\cite{Chicherin:2017cns,Chicherin:2017frs,Loebbert:2019vcj} on the integrability of massless Feynman integrals in $D=3$, 4 and 6 spacetime dimensions.\n\nThe values of the evaluation parameters $\\Eval_j$ entering \\eqref{eq:DefPhat} can be read off from the Feynman graph as follows: \ni)~Choose an arbitrary leg $1$ with $s_1=\\sfrac{1}{2}\\sum_{k=2}^n a_k$, ii)~walk clockwise along the external boundary of the ($x$-space) graph and pick up the respective propagator weights, iii)~here an external propagator with power $a_j$ contributes $-{a_j}\/{2}$ while an internal propagator with weight $b_k$ contributes $-b_k + {D}\/{2}$. The evaluation parameter $\\Eval_{j+1}$ is obtained from $\\Eval_j$ by adding the respective terms, e.g.\\\n\\begin{equation}\n\\label{eq:EvalRules}\n\\includegraphicsbox{FigEvalRulesExternalProp.pdf}\n\\qquad\n\\includegraphicsbox{FigEvalRulesInternalProp.pdf}.\n\\end{equation}\n\\section{Yangian Bootstrap}\n\n\n\nLet us now employ the above non-local symmetry to bootstrap Feynman integrals with massive propagators. This discussion extends the algorithm of \\cite{Loebbert:2019vcj} to the massive case.\nWe start with some examples containing conformal vertices obeying the condition $\\sum_j a_j=D$. Notably, after solving these integrals, we can obtain an infinite class of integrals by acting with $r$ mass derivatives $\\partial_{m_k}$ which yields an integral with propagator weights $\\tilde a_j$ with $\\sum_j \\tilde a_j=D+r$.\n\n\\paragraph{Conformal, 2 points, 2 masses, 1 loop:}\n\nAs a simple starting example consider the two-point integral \n(see e.g.\\ \\cite{Boos:1990rg,Davydychev:1990cq,Kniehl:2011ym} for the non-conformal case)\n\\begin{align}\nI_2^{m_1m_2} = \n\\includegraphicsbox{FigConfTwoPointsTwoMasses.pdf}\n=\\frac{ (1-v^2)^{\\beta\/2}}{m_1^{a_1} m_2^{a_2}}\\phi(v),\n\\end{align}\nwritten in terms of a function $\\phi$ of the conformally invariant variable\n$v \n=(m_1^2+m_2^2+x_{12}^2)\/2m_1m_2$.\nFor convenience we set\n$2\\alpha=a_1-a_2-1$\nand \n$2\\beta=-a_1-a_2+1$.\nActing on $I_2$ with the level-one momentum generator yields the associated Legendre differential equation \\footnote{Note that this equation arises from the differential operator $\\levo{ P}^{\\hat \\mu}_{\\text{extra}}$, while the rest of $\\levo{P}$ trivially annihilates $I_2$. This is a generic feature for two-point invariants.}:\n\\begin{equation}\n\\brk[s]*{\\alpha(\\alpha+1)+\\sfrac{\\beta^2}{v^2-1}}\\phi-2v\\phi'+(1-v^2)\\phi''=0.\n\\end{equation}\nThe general solution to the above differential equation is easily found, e.g.\\ in Mathematica, to be a linear combination of the associated Legendre functions of the first and second kind, i.e.  $P_\\alpha^\\beta$ and $Q_\\alpha^\\beta$. The coefficients of these functions can be fixed using numerical input for the integral, which yields\n\\begin{equation}\n\\phi(v) = 2^{\\beta}\\pi^{1-\\beta} P_\\alpha^\\beta(v).\n\\end{equation}\nWe have thus completely constrained the integral.\n\\paragraph{Conformal, 3 points, 3 masses, 1 loop:}\nConsider next the one-loop three-point integral with all propagators massive and the conformal constraint $D=a_1+a_2+a_3$ (see e.g.\\ \\cite{Nickel:1978ds,Davydychev:2017bbl,Bourjaily:2019exo} for the case with $a_j=1$):\n\\begin{equation}\n\\label{eq:I3start}\nI_3^{m_1m_2m_3}=\n\\includegraphicsbox{FigStarPropWeights.pdf}=\\frac{\\phi_3(u,v,w)}{m_1^{a_1}m_2^{a_2}m_3^{a_3}}.\n\\end{equation}\nHere we have chosen the conformal variables $u=u_{12}$, $v=u_{13}$ and $w=u_{23}$, where\n$u_{jk}=\n-\\hat x_{jk}^2\/4m_jm_k$.\nMaking a series ansatz in $u$, $v$ and $w$, the Yangian PDEs translate into recurrence equations for the series coefficients. \nThese equations can straightforwardly be solved, which yields Srivastava's triple hypergeometric function $H_{C}$, a generalization of Appell's hypergeometric function $F_1$, see e.g.\\ \\cite{srivastava2015note}:\n\\begin{equation}\n\\label{eq:SrivastavaHC}\nH_{C}(u,v,w)\n=\n\\sum_{k,l,n=0}^\\infty \\frac{(a_1)_{k+l}(a_2)_{k+n}(a_3)_{l+n}}{(\\gamma)_{k+l+n}}\\frac{u^k}{k!}\\frac{v^l}{l!}\\frac{w^n}{n!},\n\\end{equation}\nwith $\\gamma=D\/2+1\/2$ and \nthe Pochhammer symbol\n$(a)_k= \\Gamma_{a+k}\/\\Gamma_a.$\nThis series converges for\n\\begin{equation}\n\\label{eq:ConvergenceHC}\n \\abs{u}+\\abs{v}+\\abs{w}-2\\sqrt{(1-\\abs{u})(1-\\abs{v})(1-\\abs{w})}<2,\n\\end{equation}\nand in this region it is the only converging solution of 29 formal series solutions to the Yangian equations. Numerical comparison to the Feynman parameter representation of \n\\eqref{eq:I3start} yields the overall constant such that\n\\begin{equation}\nI_3^{m_1m_2m_3}\n=\n\\frac{\\pi^{D\/2}\\Gamma_{D\/2}}{\\Gamma_D m_1^{a_1}m_2^{a_2}m_3^{a_3}}\nH_{C}(u,v,w).\n\\end{equation}\nTo the knowledge of the authors this result has not been given elsewhere in the literature.\n\n\n\n\n\\paragraph{Non-Conformal, 3 points, 1 mass, 1 loop:}\n\nLet us finally demonstrate the usefulness of the level-one symmetry \nfor an integral that has no massive dual conformal (level-zero) symmetry. We write the triangle with massive leg $1$ as\n\\begin{equation}\nI_3^{m_100} = \n\\includegraphicsbox{FigStarPropWeights1Fat.pdf}\n=m_1^{D-2a_1-2a_2-2a_3} \\psi_3\\left(u,v,w\\right),\n\\end{equation}\nwith\n$u=u_{12}$,\n$v=u_{13}$ and\n$w=-u_{23}$,\nand $u_{jk}=-x_{jk}^2\/m_1^2$.\nWe act with the level-one PDEs on the series ansatz\n\\begin{equation}\n\\label{eq:SeriesAnsatz3pt1Mass}\nG_{xyz} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} \n\t= \\sum_{k,l,n}\n\t\tf_{kln} \\,\\frac{u^k}{k!} \\frac{v^l}{l!} \\frac{w^n}{n!},\n\\end{equation}\nwhere the sum runs over $k\\in x+\\mathbb{Z}$, $l\\in y+\\mathbb{Z}$, $n\\in z+\\mathbb{Z}$ for some $(x,y,z)\\in \\mathbb{R}^3$.\nThe resulting recurrence equations can straightforwardly be solved which results in\n\\begin{align}\nf_{kln}\n=\n \\frac{(\\alpha_1)_{k+l+n}(\\alpha_2)_{k+n}(\\alpha_3)_{l+n}}{(\\gamma_1)_{k+l+n}(\\gamma_2)_{n}},\n\\label{eq:FundSol3GonNotConf1Mass}\n\\end{align}\nwith the new parameters\n\\begin{equation}\n\\label{eq:params3pt1MassNonConf} \n\\alpha_1 = a_1 + a_2 + a_3-\\sfrac{D}{2}   , \\quad\n\\alpha_2 =  a_2 , \\quad\n\\alpha_3 =   a_3,  \\quad\n\\gamma_1 = \\sfrac{D}{2},\n\\end{equation}\nand the abbreviation\n$\\gamma_2 = 1 - \\gamma_1 + \\alpha_2 +\\alpha_3$.\nWe find 36 combinations of $(x,y,z)$ in \\eqref{eq:SeriesAnsatz3pt1Mass} for which the series terminates after rescaling, which\nwe take as a necessary criterion for convergence. \nHowever, only for the two choices $(0,0,0)$ and \n$(0,0,1-\\gamma_2)$ of these 36 combinations, $u$, $v$ and $w$ are the effective series variables.\nIn fact, the ansatz combining these two functions, \ni.e.\\ \n\\footnote{\nNote the following shift identity for $A=G_{001-\\gamma_2} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} \n$\nand \n$B=G_{000}^{\\alpha_1-\\gamma_2+1,\\alpha_2-\\gamma_2+1,\\alpha_3-\\gamma_2+1,\\gamma_1-\\gamma_2+1,2-\\gamma_2}$:\n\\begin{equation*}\nA\n=\nw^{1-\\gamma_2} \\frac{\\Gamma_{\\gamma_1}\\Gamma_{\\gamma_2}\\Gamma_{1+\\alpha_1-\\gamma_2}\\Gamma_{1+\\alpha_2-\\gamma_2}\\Gamma_{1+\\alpha_3-\\gamma_2}}{\\Gamma_{\\alpha_1}\\Gamma_{\\alpha_2}\\Gamma_{\\alpha_3}\\Gamma_{2-\\gamma_2}\\Gamma_{1+\\gamma_1-\\gamma_2}} B\n\\end{equation*}\n}\n\\begin{align}\n\\psi_3&=\\pi^\\frac{D}{2} \\brk*{ c_1 \\,\n\t\tG_{000} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2}\n\t\t+ c_2  \\,\n\t\tG_{001-\\gamma_2} ^{\\alpha_1 \\alpha_2 \\alpha_3\\gamma_1\\gamma_2} }, \t\n\\end{align}\nmaps to the solution given in \\cite{Boos:1990rg}, if the constants \n$c_1$ and $c_2$ are chosen as\n$\nc_1=\\Gamma_{\\alpha_1}\\Gamma_{1-\\gamma_2}\/\\Gamma_{\\alpha_1-\\gamma_2+1}\\Gamma_{\\gamma_1}\n$\nand\n\\begin{equation*}\nc_2=\\frac{\\Gamma_{\\alpha_1}\\Gamma_{\\gamma_1-\\alpha_2}\\Gamma_{\\gamma_1-\\alpha_3}\\Gamma_{\\gamma_2-1}\\Gamma_{2-\\gamma_2}}\n{\\Gamma_{\\gamma_1}\\Gamma_{\\gamma_2}\\Gamma_{1+\\alpha_1-\\gamma_2}\\Gamma_{1+\\alpha_2-\\gamma_2}\\Gamma_{1+\\alpha_3-\\gamma_2}}.\n\\end{equation*}\nIn this last example we have thus successfully employed the \nlevel-one  symmetry to constrain the functional form of an integral without conformal level-zero symmetry. This suggests an alternative interpretation of this non-local symmetry.\n\\section{Massive Momentum Space Symmetry}\n\\label{sec:MassMomSpace}\n\nIn the massless case, the Yangian symmetry of amplitudes in planar $\\mathcal{N}=4$ SYM theory \\cite{Drummond:2009fd} as well as in the fishnet theory \\cite{Chicherin:2017cns} can be understood as the closure of a dual (region momentum space) and an ordinary (momentum space) conformal symmetry.\nIt is thus a natural question whether the discovered massive Yangian can be understood as the closure of the massive dual conformal symmetry generated by \\eqref{eqn:massdualconfrep} and some massive generalization of momentum space conformal symmetry. To address this question we translate the level-one generator $\\levo{P}^\\mu$ of \\eqref{eq:DefPhat} given in terms of region momenta $x_j$ into momenta $p_j$ defined via\n$p^{\\mu}_j=x^{\\mu}_j-x^{\\mu}_{j+1}$. The masses, on the contrary, stay untouched because the mass dependence of the integrals cannot be phrased in terms of dual masses $m_j-m_{j+1}$ only. By making the respective substitutions and using the chain rule, one finds \n\\begin{equation}\n\\label{eq:PhatToK}\n\\levo{P}^\\mu=\\sfrac{i}{2} \\gen{\\bar K}^\\mu \\, ,\n\\end{equation} \nwhich holds up to terms that vanish when the integral is expressed in terms of the first $n-1$ momenta by using momentum conservation. Here, the special conformal operator $\\gen{\\bar K}^\\mu$ forms part of the following massive generalization of the momentum space conformal generators~\\footnote{Here we could also use a local version of the dilatation generator density acting only on the mass $m_j$. In the tensor product this makes no difference.}:\n\\begin{align}\n\\gen{\\bar P}_{j}^\\mu &= p_j^\\mu \\, , \n\\hspace{1cm} \\gen{\\bar L}_{j}^{\\mu\\nu} =  p_j^\\mu \\partial_{p_j}^{\\nu} -  p_j^\\nu \\partial_{p_j}^{\\mu},\n\\nonumber\\\\\n\\gen{\\bar D}_{j} &= p_{j\\nu} \\partial^{\\nu}_{p_j}+ \\sfrac{m_j \\partial_{m_j}+m_{j+1} \\partial_{m_{j+1}}}{2} + \\bar{\\Delta}_j,\n\\label{eqn:MomConfMassRep}\n\\\\\n\\gen{\\bar K}_{j}^\\mu \n&= p_j^\\mu \\partial_{p_j}^2\n-2\\brk[s]2{p_{j\\nu} \\partial_{p_j}^\\nu+\\sfrac{m_j \\partial_{m_j}+m_{j+1} \\partial_{m_{j+1}}}{2} + \\bar{\\Delta}_j }\\partial_{p_j}^{\\mu}.\n\\nonumber \n\\end{align}\nThese generators thus furnish symmetries of the above integrals expressed in momentum space and they obey the ordinary conformal algebra. We have $\\bar \\Delta_j=\n\\sfrac{1}{2}\\Delta_{j}+ \\sfrac{1}{2}\\Delta_{j+1} +  \\Eval_j - \\Eval_{j+1}$ with the evaluation parameters $s_j$ determined by the rules \\eqref{eq:EvalRules}\n\\footnote{For the one-loop $n$-gons for instance, we thus find $\\bar \\Delta_j = a_j+a_{j+1}$.}.\nWe note that the operator $\\gen{\\bar K}_{j}^\\mu$ includes a mildly non-local contribution acting on the masses of nearest-neighboring legs $j$ and $j+1$ of the Feynman graph.\nIn the massless limit, the above translation \\eqref{eq:PhatToK} of the Yangian level-one generator to momentum space also makes connection to the momentum space Ward identities that were recently studied for conformal correlation functions, see e.g.\\ \\cite{Coriano:2013jba,Bzowski:2013sza,Coriano:2020ccb}.\n\n\\section{Conclusions and Outlook}\n\nThe main results presented in this letter are summarized as follows:\nAn infinite number of specific massive Feynman integrals features a highly constraining Yangian symmetry --- a hallmark of integrability.\nThe observed Yangian algebra can be understood as the closure of a novel momentum space conformal symmetry and the known massive dual conformal symmetry; this strongly suggests an embedding of these findings into a larger picture within the AdS\/CFT duality.\nOur finding shows that the massive IR regulator proposed in \\cite{Alday:2009zm} does not only preserve dual conformal symmetry but also the Yangian symmetry of Feynman integrals.\n\n\nThere are plenty of further directions worth being investigated.\nFirstly, it would be important to formally prove our conjecture of higher loop Yangian and massive momentum space conformal symmetry. This may be possible along the lines of the massless case \\cite{Chicherin:2017cns,Chicherin:2017frs}, but it could be tricky to find a proof that applies directly to all cases considered in this letter (e.g.\\ to generic spacetime dimensions $D$). \nSecondly, the applicability of Yangian symmetry to massive Feynman integrals and generic (double) $n$-gons allows us to refine the algorithm of \\cite{Loebbert:2019vcj} on many more examples whose complexity interpolates between the simplest massless cases with $2$ and $9$ variables, respectively. This may finally open the door to computing e.g.\\ the massless hexagon or double box integral. \nMoreover, some of the integrals discussed above have a geometric interpretation, see e.g.\\ \\cite{Nandan:2013ip,Davydychev:2017bbl,Herrmann:2019upk,Bourjaily:2019exo}. It would be fascinating to understand the considered symmetries in this context. \nAnother highly interesting task is to seek for a massive generalization of the Basso--Dixon formula for massless fishnet four-point integrals \\cite{Basso:2017jwq}, whose finding was motivated by integrability.\nFinally, in the massless situation integrability took its way from AdS\/CFT via the fishnet theory to Feynman integrals \\cite{Gurdogan:2015csr,Chicherin:2017cns,Chicherin:2017frs}. Here we have found similar properties for the massive building blocks of quantum field theory. Tracing back these symmetries to an origin within the AdS\/CFT duality\nis very suggestive and may open the door to understanding integrability in massive phases of AdS\/CFT.\nIn this direction we note that in $D=4$ for instance, the massive square fishnet diagrams discussed above can be considered as planar off-shell amplitudes generated by the Lagrangian\n\\begin{align}\n\\mathcal{L}=\n&N_\\text{c} \\text{tr}\n\\brk*{\n-\\partial_\\mu\\bar X \\partial^\\mu X -\\partial_\\mu\\bar Z\\partial^\\mu Z+\\xi^2 \\bar X\\bar Z X Z\n}\n\\\\\n&- N_\\text{c}\\brk*{m_j-m_k}^2 \\bar Z^j{}_k Z^k{}_j\n-N_\\text{c}\\brk*{ m_j-m_k}^2 \\bar X^j{}_kX^k{}_j.\\nonumber\n\\end{align}\nHere, $N_\\text{c}-n$ of the masses $m_j$ are taken to be zero, such that graphs with massless propagators in the bulk of the diagram are dominant in the planar limit~\\footnote{Note that the mass terms generated by spontaneous symmetry breaking in the fishnet theory have product form instead of the difference form \\cite{Karananas:2019fox,Loebbert:2020tje}.}. \nIt would be interesting to test this Lagrangian for Yangian symmetry with the methods of~\\cite{Beisert:2017pnr}.\n\n\n\\medskip \n\n\\acknowledgments{\nFL and DM are very grateful to Till Bargheer and Jan Plefka for initial collaborations on similar ideas in the context of amplitudes in $\\mathcal{N}=4$ SYM theory and for helpful discussions. \nWe thank Niklas Beisert, Lance Dixon and Jan Plefka\nfor helpful comments on the manuscript.\n The work of FL is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)--Projektnummer 363895012. JM is  supported  by  the  International  Max  Planck  Research  School  for  Mathematical  and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory. DM was supported by DFF-FNU through grant number DFF-FNU 4002-00037. The work of HM has been supported by the grant no.\\ 615203 of the European Research Council under the FP7 and by the Swiss National Science Foundation by the NCCR SwissMAP. \n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}