IFUP-TH 2696 Topological Susceptibility at zero an

IFUP-TH26/96TopologicalSusceptibilityatzeroandniteTinSU(3)Yang-MillstheoryB.Alles,M.D'EliaandA.DiGiacomoDipartimentodiFisicadell'UniversitaandINFN,PiazzaTorricelli2,56126-Pisa,ItalyAbstractWedeterminethetopologicalsusceptibilityatT=0inpureSU(3)gaugetheoryanditsbehaviouratniteTacrossthedeconningtransition.Weuseanimprovedtopologicalchargedensityoperator.dropssharplybyoneorderofmagnitudeatthedeconningtemperatureTc.PartiallysupportedbyECContractCHEX-CT92-0051andbyMURST.1I.INTRODUCTIONTheflavoursingletaxialcurrentj5=γ5γ(1.1)isnotconservedinQCDbecauseofthetriangleanomaly[1]@j5(x)=2NfQ(x):(1.2)Ineq.(1.2)Q(x)isthetopologicalchargedensity,denedasQ(x)=g2642Fa(x)Fa(x):(1.3)AsaconsequencethecorrespondingUA(1)isnotasymmetry[1].Thenonsingletpartnersj5a=γ5γa(1.4)areconserved,andthecorrespondingsymmetryisspontaneouslybroken,thepseudoscalaroctetbeingtheGoldstoneparticles.IfUA(1)wereasymmetry,eitherparitydoubletsshouldexist,or,incaseofspontaneousbreaking,theinequalitym0p3mshouldhold[2].Neitherofthesepredictionsistrueinnature,andthishasbeenknownastheUA(1)problemformanyyears,beforetheadventofQCD.HoweverUA(1)isasymmetryatleadingorderintheexpansionin1Nc[3],(Ncisthenumberofcolours).ThereareargumentsthattheleadingapproximationinthatexpansiondescribesthemainphysicsofQCD[4,5].Theanomalyisnonleading,andcanbeviewedasaperturbation.Oneofitseectsistodisplacem0fromzero,whichcorrespondstotheGoldstoneparticleintheleadingorderapproximation,byanamountwhichisrelatedtothetopologicalsusceptibilityofthevacuumattheleadingorder.Thepredictionis[6,7]2Nff2=m2+m20−2m2K:(1.5)2ThetopologicalsusceptibilityisdenedasZd4xh0jT(Q(x)Q(0))j0i:(1.6)Leadingorderimpliesabsenceoffermionsandinthelanguageofthelatticethisisknownasquenchedapproximation.Latticeistheidealtooltocomputefromrstprinciples.hasinfactbeendetermined[8]andisconsistentwiththepredictionofref.[6].Anadditionalhintinfavourofitistheindicationthatthe0massishigherinsectorswithhighertopologicalcharge[9].AquestionthenarisesnaturallywhethertheUA(1)symmetryisrestoredinquenchedQCDatthesametemperatureatwhichSUA(3)isrestored,i.e.atTc260MeV[10].Manymodels[11]predictthebehaviouroftheUA(1)chiralsymmetryatTc.AquitegeneralexpectationisthatthetopologicalsusceptibilityshoulddropatTc[12],sinceDebyescreeninginhibitstunnelingbetweenstatesofdierentchiralityanddampsthedensityofinstantons.AttemptshavebeenmadeafewyearsagotostudythebehaviourofthroughTc[13,14].Thestatusisdiscussedinref.[14].Thedicultiesgobacktothedenitionofatopologicalchargeonthelattice.Thecorrectwaytodeneit,accordingtothecommonlyacceptedprescriptionsofeldtheory,istointroduceonthelatticealocaloperatorQL(x)forthetopologicalchargedensitywhichtendstothecontinuumoperatorasthelatticespacinggoestozero.QLprovidesaregularizedversionofQ(x).Ingoingtothecontinuumlimitaproperrenormalizationmustbeperformed,likeinanyotherregularizationscheme.AspecicfeatureofQListhatonthelatticeitisnotthedivergenceofacurrent,likeinthecontinuum,andhenceitrenormalizesmultiplicatively:thismeansthatthelatticetopologicalchargeofacongurationcanbenoninteger[15].InformulaeQL=Z()Qa4+O(a6):(1.7)Asusual,6=g20.ThetopologicalsusceptibilitycanbedenedonthelatticeasLhXxQL(x)QL(0)i:(1.8)3ThestandardrulesofrenormalizationthengiveL=Z()2a4+M()+O(a6);(1.9)whereM()isanadditiverenormalizationcontainingmixingsofLtootheroperatorswiththesamequantumnumbersandlowerorequaldimensions[16].InformulaeM()=B()a4G2+P():(1.10)ThetermsproportionaltoP()andB()arerespectivelythemixingstotheidentityoperatorandtothedensityofactionG2hg242FaFai.TheadditiverenormalizationcomesfromthesingularitiesoftheproductQ(x)Q(0)asx!0andmustberemovedtobeconsistentwiththeprescriptionusedtoderiveeq.(1.5)[17].ThedenitionofQLisnotunique:innitelymanyoperatorscanbedenedwhichobeyeq.(1.7)butdierbytermsoforderO(a6).ThesimplestdenitionofQLis[18]QL(x)=−12924X=1~Tr((x)(x)):(1.11)Here~isthestandardLevi-Civitatensorforpositivedirectionswhilefornegativeonestherelation~=−~−holds.istheplaquetteinthe−plane.WiththisdenitionZ'0:18andthemixingMislargecomparedtothesignalinthescalingregion[19].ZandMcanbecomputednon-perturbatively[20{22].Althoughtheeld-theoreticmethodiscorrectinprinciple,itisunpleasantthatmostofthesignalisduetolatticearti-facts,whichhavethentoberemoved.Moreoveratthetimeofref.[14]thenon-perturbativedeterminationofZandMwasnotknown.Analternativemethodtodetermineisthesocalledcoolingtechnique[13]:theideaistofreezequantumfluctuationsbyalocalalgorithmwhichcoolsthelinksoneaftertheother.Themodesrelevantatadistancedarefrozenafteranumberofstepsn,whichisproportionaltod2,likeinadiusionprocess.Mostoftheinstantonsareexpectedtohaveasizeoftheorderofthecorrelationlength.Afterafewcoolingsteps,theeliminationoflocalfluctuationswillsuppressthemixingMandmakeZ'1,butthenumberofinstantonswill4bepreserved,sothatL'a4.InfactaplateauisreachedinQLafterafewcoolingsteps,whereQLisaninteger,whichstandsmanyfurthersteps[13].AtT=0andbelowTcthemethodworksverywellandagreesw