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1、NuclearPhysicsB221(1983)381-408©North-HollandPublishingCompanyTHEAXIALANOMALYANDTHELATTICEDIRACSEAJ.AMBJORN,J.GREENSITEandC.PETERSON*Nordita,Blegdamsvej17,DK-2100Copenhagen0,DenmarkReceived28January1983Inthehamiltonianformulationoffermionscoupledtoexter
2、nalgaugefields,theaxialanomalyhasasimplephysicalinterpretationintermsoflevelshiftingatthetopoftheDiracsea.WeapplythisformalismtolatticeQEDin1+1and3+1dimensions,inordertostudyhowthelatticeregulationandsmallfermionmassaffectthepicture.Forasimplechoiceofth
3、eEandBfields,itispossibletoaccuratelyfollowthetimeevolutionofthe(lattice)DiracseaofWilsonfermions.WefindthattheWilsonrtermplaysarole,withrespecttotheanomaly,whichiscloselyanalogoustopoint-~plittinginthecontinuum.ThisroleisjeopardizedwhenevertheWilsonmas
4、sscaler/aiscomparabletootherphysicalmassscalesintheproblem,asisthecase,forexample,instrongcouplingcalculations.Remarkably,wefindthathadronicscalesofonlyafewlatticespacingsareprobablysufficienttoguaranteetheproperanomalystructure.1.IntroductionOneofthemo
5、stintriguingeffectsinquantumfieldtheoryistheaxialanomaly,inwhichtransitionsthatshouldbeforbiddenbytheaxialsymmetryofthelagrangian(suchas7r°~2y)areneverthelessinducedsomehowbyquantumeffects.Theoriginofthiseffectisratherobscureintheeuclideanformulation,wh
6、ereitappearstodependontheshort-distancestructureofthetheory.Forexample,thedecay7r°-~2yisaconsequenceoflineardivergencesinthewell-knowntrianglediagram[1],whichsuggeststhatthedecayisanultravioleteffect.Ontheotherhand,fromtheQCDpointofview,itseemspuzzlingt
7、hattheannihilationofapairofquarksinapionintotwophotonsshoulddependcruciallyondynamicsfarbelowthehadronicscale.ThisobscurityiscompoundedwhenonestudiestheanomalyonthelatticewithWilsonfermions[2].Inthiscasethedivergenceoftheaxialcurrentisjust51"--F~_ieaA(x
8、)l[ix+atzA"l"(X)=2m°~xYsO~+--g'xYSOXa~aa(d~xyse+h.c.),(1.1)andthe"anomalous"divergenceoff5isdueentirelytotheWilsonrterm,whichbreakstheaxialsymmetrystronglyandexplicitly.ItwasshownbyKarstenandSmit[3]thattheWilsonrtermpreciselyrepr
