Anomaly Cancellation in 2+1 dimensions in the pres

arXiv:hep-th/9311050v19Nov1993CU-TP-615OCTOBER-93hep-th/9311050Anomalycancellationin2+1dimensionsinthepresenceofadomainwallmassShaileshChandrasekharan†DepartmentofPhysicsColumbiaUniversity,NewYork,N.Y.10027AbstractAFermionin2+1dimensions,withamassfunctionwhichdependsononespatialcoordinateandpassesthroughazero(adomainwallmass),isconsidered.Inthismodel,originallyproposedbyCallanandHarvey,thegaugevariationoftheeffectivegaugeactionmainlyconsistsoftwoterms.OnecomesfromtheinducedChern-Simonstermandtheotherfromthechiralfermions,boundtothe1+1dimensionalwall,andtheyareexpectedtocanceleachother.Thoughthereexistargumentsinfavourofthis,basedonthepossibleformsoftheeffectiveactionvalidfarfromthewallandsomefactsabouttheoriesofchiralfermionsin1+1dimensions,acompletecalculationislacking.Inthispaperwepresentanexplicitcalculationofthiscancellationatoneloopvalidevenclosetothewall.Weshowthat,integratingoutthe“massive”modesofthetheorydoesproducetheChern-Simonsterm,asappreciatedpreviously.Inadditionweshowthatitgeneratesatermthatsoftensthehighenergybehaviourofthe1+1dimensionaleffectivechiraltheorytherebyresolvinganambiguitypresentinageneral1+1dimensionaltheory.——————————–†email:[e-mail]1IntroductionItwasunderstoodsometimeagothatthereexistintimateconnectionsbetweentheChern-Simonsterminanodddimensionalspace-timeandthechiralanomalyinonelowerdimension.Aftersuchaconnectionwasunderstood,CallanandHarvey[1]proposedamodelinwhichtheconnectionwasphysicallyrealised.Theyconsideredathreedimensionalfermionwithadomainwallmass(amasstermthatdependsononespacecoordinate,passesthroughzeroattheoriginandgoestoaconstantwithoppositesignsatplusandminusinfinity)coupledtoagaugetheory.Sincetherearenoanomaliesinthecontinuoussymmetriesinodddimension,thetheorymustbegaugeinvariant.However,inthetheorywithadomainwallmass,onecanshow,aswewillseelater,thatthereexisteffectivelytwo-dimensionalmasslesschiralfermionsattachedtothedomainwall.Theresultingtwodimensionalchiraltheoryshouldhaveananomalyinthegaugecurrent.However,sincethewholetheoryhasnoanomalytheremustbeyetanothercontributiontothecurrentinthewholetheorywhichwillcancelthechiralanomaly.Itwasfoundthatthereindeedexistcurrentsoneithersideofthewallwhichflowintoorawayfromthewalldependingonthesignoftheanomalyonthewall.ThiscurrentcanbeapproximatelycalculatedawayfromthewallusingmethodsofGoldstoneandWilczek[2].WewillcallthesecurrentsGoldstone-Wilczekcurrents.However,whenoneinvestigatestheGoldstone-Wilczekcurrentsflowingfromthethirddimensionintothewallandthusaccountsforthechargeappearanceonthewall(thepersononthewallconsidersthechargeappearingasananomaly),oneen-countersdifficulties.Inanabeliantheory,forexamplethechargeappearingonthewallistwiceasmuchasthatpredictedfromananomalyinanexclusively1+1di-mensionaltheory[4].Inanon-abeliantheorytheproblemismoreevident.Here,theanomalyinthetwodimensionalchiraltheoryisnecessarilygaugenon-covariant.Thisnon-covariantformisrequiredbytheWess-Zumino[3]consistencyconditionsobeyedbytheusualdefinitionofthecurrent.Hencetheanomalyinthiscurrentisalsore-ferredtoasaconsistentanomaly.OntheotherhandtheGoldstone-Wilczekcurrentaccountsfortheanomalyonthewallwhichisgaugecovariantinitsform.Thisformoftheanomaly,whichisgaugecovariantinitsformisalsoreferredtoasthecovariantanomaly.ThusthisGoldstone-Wilczekcurrentalonecannotcompletelycancelthe1consistentanomalyonthewall,asoneisgaugecovariantandtheotherisnot.WhenBardeenandZuminodiscussconsistentandcovariantanomalies[5]theyshowhowanadditionofanextratermtotheconsistentcurrentcanmaketheanomalycovariantinitsform.Thusitseemsthattheremustexistanextrapieceofcurrentonthewall,thatarisesnaturallyandwhichmakestheanomalyintheeffective1+1dimensionaltheorycovariantinitsform.Thistermcannotbeobtainedfromthelagrangianofanexclusively1+1dimensionaltheoryastheconsistencyconditionswouldnotallowitspresence.Ontheotherhandinourmodelthisextrapieceofthecurrentcanbeinducedbytheeffectsoftheextradimension.ThisproblemwasaddressedbyNaculich[4]inwhichhesuggestshowaparticularformoftheChern-Simonsterminthe2+1dimensionaleffectiveaction(producedwhenyouintegrateoutthemassivefermionmodesofthetheory)caninducethisextrapiece.Infact,thisparticularformoftheChern-SimonstermwasoriginallysuggestedbyCallanandHarvey[1].HoweverthereisnocompletederivationforthisChern-Simonstermintheeffectiveaction,whichisvalidarbitrarilyclosetothewall.Thisisnotsatisfactorybecause,thecalculationssuggestedtoextracttheeffectiveChern-Simonstermarevalidonlyfarfromthewall,ontheotherhandtheactualquestions,whichareatissuehere,arerelatedtotermsinducedonthewall.Alsoonewonders,ifthereisoneextraeffectonthewall,otherthanthatofasimple1+1dimensionalmasslesschiraltheory,theremaybeothersthatarehiddenandunclear,untilacompletecalculationoftheeffectiveactionisdone.Amorecleanandsimpleviewofthisanomalycancellationcomesfromconsideringaneffectiveactionintermsofthegaugefieldsafterintegratingoutthefermions.Thiseffectiveactionmustbegaugeinvariant.ThiswastheoriginalwayCallanandHarvey[1]analysedtheproblem.Intheirpaper[1]theyarguethatintegratingoutthemassivef