Topological charge and topological susceptibility

arXiv:hep-th/0204010v326Apr2002TopologicalchargeandtopologicalsusceptibilityinconnectionwithtranslationandgaugeinvarianceA.M.Kotzinian1,O.Yu.Shevchenko2andA.N.Sissakian3JointInstituteforNuclearResearchDubna,Moscowregion141980,RussiaAbstractItisshownthattheevaluationoftheexpectationvalue(EV)oftopologicalchargedensityoverθ-vacuumisreducedtoinvestigationoftheChern-SimonstermEV.Anequationforthisquantityisestab-lishedandsolved.EVofthetopologicalchargedensityatanarbitraryθoccursequaltozeroand,asaconsequence,topologicalsusceptibilityofbothQCDandpureYang-MillsvacuadefinedinaWicksenseisequaltozero,whereaswhendefinedinaDysonsenseitdiffersfromzerobythequantityproportionaltotherespectivecondensateofthechromomagneticfield.Thus,theusualWitten-Venezianoformulafortheη′mesonmassismodified.1E-mailaddress:[e-mail]2E-mailaddress:[e-mail]3E-mailaddress:[e-mail]1Theeffectsconnectedwiththenontrivialtopologicalconfigurationsofthegaugefieldsattractagreatattentioninmodernphysics.InthisrespecttheQCDtopologicalsusceptibilityχQCD=Zd4xhTq(x)q(0)i(1)isthequantityofaspecialimportancebecauseitentersasakeyobjectinalotofphysicaltasks,inparticular,insuchimportantpuzzlesasafamousU(1)problem[1-5](see[6]forarecentreview)andthe”spincrisis”[7].InEq.(1)q(x)isthetopologicalchargedensityq(x)=g232π2trFaμν(x)˜Fμνa(x)(2)relatedwiththeChern-Simonscurrent,Kμ(x)byq(x)=∂μKμ(x),(3)whereKμ=g232π2ǫμνρσAaνFaρσ−g3fabcAbρAcσ.(4)Itiswellknown,thattopologicalsusceptibilityχQCDisequaltozeroinallordersofperturbationtheoryand,also,thatthisquantityisjustzerointhepresenceofevenonemasslessquark(Crewthertheorem[2]).Inthispapertheconsiderationbasedonthefundamentaltranslationandgaugesymmetrieswillbeperformedwhichwillallowtodrawsomeunex-pectedconclusionsaboutthetopologicalchargeandsusceptibility.Letusproofthefollowingstatement.EVofthetopologicalchargedensity(θ|q(0)|θ)=(1/VT)(θ|Q|θ)overθ-vacuumwithanarbitraryθisequaltozeroifEVofoperatorKμ(0)overθ-vacuumexists,i.e.,|(θ|Kμ(0)|θ)|∞,(5)wheresymbol|θ)denotestheθ-vacuumstatenormalizedtounity:(θ|θ)=1.(6)2Thisstatementdirectlyfollowsfromtranslationinvarianceofθ-vacuum:(θ|q(0)|θ)=(θ|∂μKμ(0)|θ)=−i(θ|[ˆPμ,Kμ(0)]|θ)=−i(Pμθ−Pμθ)(θ|Kμ(0)|θ)=0.(7)Thekeypointhereisthecondition(5)which,aswewillseebelow,inA0=0gaugeisequivalenttothecondition|(θ|WCS(0)|θ)|∞,(8)where4WCS(t)≡Zd3xK0(x)=g232π2Zd3xǫijkAaiFajk−g3fabcAbjAck(9)istheChern-Simonsoperator(see[8]forreview).However,aswewillsee,withintheconventionalformulationofθ-vacuumtheoryratheramazingsit-uationarises.Ontheonehandthecondition(8)isnotsatisfiedduetothegaugenon-invarianceoftheoperatorWCSwithrespecttothe”large”,topologicallynontrivialgaugetransformations.Nevertheless,despiteEVhθ′|WCS(0)|θiismoresingularfunctionthanδ(θ′−θ)atθ′→θ(namely,itbehavesasδ′(θ′−θ)inthislimit),theEVofthetopologicalchargeden-sityoverθ-vacuumisjustzeroagain.Sincewedealwiththegaugeinvariantquantity(EVofthetopologicalcharge)letuschoosetheWeilgaugeA0=0,(10)whichallowstoessentiallysimplifyaconsideration.Choosingtheperiodicboundaryconditionsinthespacedirections(topologyofahypercylinderorientedalongthetimeaxis)onehasZd3x∂iKi(t,~x)=0,(11)andtheexpressionforthetopologicalchargeQ≡Zd4xq(x)=Zd4x∂μKμ(12)4IntheliteratureChern-SimonstermWCS[A]isalsooftencalled”collectivecoordi-nate”(see,forexample,[5])andisdenotedbyX[A].3becomes(see,forexample,[8,9])Q=WCS(t=∞)−WCS(t=−∞).EV(θ|ˆO|θ)ofanarbitraryoperatorˆOoverθ-vacuumisdefinedas(see,forexample[8,9])(θ|ˆO|θ)=hθ|ˆO|θihθ|θi(13)where|θiis,simultaneously,theeigenfunctionofthefullQCDHamiltonianHandoftheunitaryoperatorTνofthelargegaugetransformationswithawindingnumberν:H|θi=Eθ|θi,(14)Tν|θi=e−iθν|θi,(15)i.e,thestate|θiis,uptoaphasemultiplier,gaugeinvariantagainstthelargegaugetransformations.Noticealsothatonthecontraryto(6)thestates|θiarenormalizedashθ′|θi=δ(θ′−θ).(16)sothattheprescription(13)readshθ′|ˆO|θi=(θ|ˆO|θ)δ(θ′−θ),(17)i.e.,(θ|ˆO|θ)isjustthelimitatθ′→θofthemultiplieratδ-functionintheexpressionforhθ′|ˆO|θi.Sinceweareinterestedinthequantity(θ|q(0)|θ)=(VT)−1(θ|Q|θ),(18)wewillkeepthenormalizationfactor(VT)−1.Using(11)andtheHeisenbergequationsoneeasilygets(VT)−1hθ′|Q|θi=(VT)−1Zdtei(Eθ′−Eθ)thθ′|Zd3xq(0,~x)|θi=2π(VT)−1δ(Eθ′−Eθ)hθ′|˙WCS(0)|θ′i=2π(VT)−1δ(Eθ′−Eθ)hθ′|−i[WCS(0),H]|θi=2πi(VT)−1δ(Eθ′−Eθ)(Eθ′−Eθ)hθ′|WCS(0)|θi,(19)4and,thus,thetasknowistoevaluateEVhθ′|WCS(0)|θi.TheremarkablepropertyoftheChern-Simonstermisitstransformationlawundertopologicallynontrivial(oftencalled”large”[8])gaugetransfor-mationsAi→AΩνi=ΩνAiΩ−1ν+∂iΩνΩ−1ν(i=1,2,3;Ω=Ω(~x)),(20)withtopologicalindex(windingnumber)ν.Namely,thequantityWCS[A]isnotgaugeinvariantunder(20)buttransformsasWCS[A]→WCS[AΩν]=WCS[A]+ν,(21)i.e.itonlyshiftsbythewindingnumberνoftherespectivegaugetransfor-mation.ThecompatibilityofthequantumWCS[AΩν]=TνWCS[A]T+ν=WCS(A)+[Tν,WCS(A)]T−1ν(22)andclassical(21)gaugetransformationlawsoftheChern-Simonsterm(9)givesrisetothecommutationlaw5[Tν,WCS(t)]=[Tν,WCS(0)]=νTν.(23)Nowonealreadycanevaluatehθ′|WCS(0)|θi.Indeed,duetotheunitarityoftheoperatorTνandEq.(15)onehashθ′|[Tν,WCS(0)]|θi=e−iνθ′−e−iν