; TeX output 1994.07.17:2143 7gxg 7"V ff
cmbx10P=erturbative analysisforKaplan'slatticec=hiralfermions+ I+XQ cmr12S.AokiandH.Hirose[>"% ':
3
cmti10InstituteofPhysics,UniversityofTsukubpa,TsukubaIbaraki-305,Japan R"K`y
3
cmr10(Julyf17,1994);< Ί;4K`y ff
cmr10Abstract$ 7kP!erturbationStheoryforlatticefermionswithdomainwallmasstermsis2Rdev!elopMedandisappliedtoinvestigatethechiralSchwingermoMdelformulated2Ronthelatticeb!yKaplan'smethoMd.QWeecalculatethee ectiveactionforgauge2R eldstooneloMop,(and ndthatitcon!tainsalongitudinalcomponen!teven2Rforanomaly-freecases.Feromthee ectiv!eactionweobtaingaugeanomalies2RandOChern-Simonscurren!twithoutambiguitye.
WeOalsoshowthatthecurrent2RcorrespMondingtothefermionn!umberhasanon-zerodiv!ergenceandit
owso 2Rthexw!allintotheextradimension.ΚSimilarresultsareobtainedforapropMosal2Rb!yY=Shamir,whousedaconstantmasstermwithfreebMoundariesinsteadof2Rdomainfw!alls. Ֆ Y)TrypSesetusingREVT
UE!X html:1 html: *7gxg do'"V
3
cmbx10Conttents&9N&html: 0N cmbx12I html:#?In tro` duction_lhtml:3 html:#S'html: I` I n: html:#?Action, F ermionPropagatorandChiralZeroMo` desu+html:5 html:[#? html: A html::LatticeActionꍑ,g cmmi12: C::::::::::::::::::::::::::::::::jhtml:5 html:#? html: B R html::ChiralZeroMoSdesU: C::::::::::::::::::::::::::::::jhtml:7 html:#? html: C |l html::FVermionPropagatorandZeroMoSdesS: C::::::::::::::::::::jhtml:8 html:#? html: D 9 html::FVermionFeynmanRules
]: C::::::::::::::::::::::::::nhtml:10 html:(html: I` IIW html:#?P erturbative CalculationsfortheChiralSc hwinger Mo` delKhtml:11 html:#? html: A html::E ectivreActionatFVermionOne-LoSopX: C:::::::::::::::::::nhtml:12 html:#? html: B R html::EvXaluationofZeroMoSdeConrtributions: C::::::::::::::::::nhtml:12 html:#? html: C |l html::EvXaluationofRemainingConrtributionszȍ: C::::::::::::::::::nhtml:14 html:#? html: D 9 html::TVotalConrtributions: C:::::::::::::::::::::::::::::nhtml:16 html:'html: IV8 html:#?Anomalies intheChiralSc hwinger Mo` del 9html:17 html:#? html: A html::CurrenrtsandtheirDivergence: C:::::::::::::::::::::::nhtml:17 html:#? html: B R html::GaugeinrvXarianceD$: C:::::::::::::::::::::::::::::::nhtml:18 html:#? html: C |l html::GaugeAnomalies?: C::::::::::::::::::::::::::::::nhtml:19 html:#? html: D 9 html::Chern-SimonsCurrenrtTz: C::::::::::::::::::::::::::::nhtml:20 html:#? html: E html::PythagoreanChiralScrhwingerMoSdelandAnomalyinFVermionNumrber:Currenrt: C::::::::::::::::::::::::::::::::::::nhtml:21 html:&html: V 1 html:#?conclusionsahtml:23 html: html:2 html: ͠7gxg %%html:I.2 html:INTRtODUCTION&9NConstruction`ofcrhiralgaugetheoriesisoneofthelong-standingproblemsoflattice eld[theories.Becauseuofthefermiondoublingphenomenon,anaivrelydiscretizedlatticefermion eld9yields222 cmmi8dfermionmoSdes,
halfofonecrhirality9andhalfoftheother,sothatthetheoryisnon-crhiral.'SeverallatticeapproacheshavebSeenproposedtode necrhiralgaugetheories,butsofarnoneofthemharvebSeenprorventowrorksuccessfullyV.Kaplan* haspropSosedanewapproacrh[ html:1 html:]tothisproblem.FHesuggestedthatitmaybSepSossiblegQtosimrulatethebeharviorofmasslesschiralfermionsin2kdimensionsbyalatticetheoryofmassivrefermionsin2k+1dimensionsifthefermionmasshasashapSeofadomainwrallinthe2k+1-thdimension.c-Heshowedfortheweakgaugecouplinglimitthatmasslesscrhiral@@statesariseaszero-moSdesboundtothe2k-dimensionaldomainwrallwhilealldoublerscanfbSegivrenlargegaugeinvXariantmasses. IfthechiralfermioncontentthatappSearsontheVdomainwrallisanomalousthe2k-dimensionalgaugecurrent
owso thewallintotheextradimensionsothatthetheorycannotbSe2k-dimensional.ThereforehearguedthatthisapproacrhpSossiblysimulatesthe2k-dimensionalchiralfermionsonlyforanomaly-freecases.HiswideawrastestedforsmoSothexternalgauge elds.Jansen[ html:2 html:]showednumericallythatinCthecaseofthecrhiralSchwingermoSdelin2dimensionswiththreefermionsofcharge3,4,.and5theanomaliesinthegaugecurrenrtscancelonthewall.)TheChern-Simonscurrentfar|arwayfromthedomainwallwascalculatedinRef.[ html:3 html:].[Itisshownthatthe2k+1-thcompSonenrt{ofthecurrentisnon-zerointhepSositivemassregionandzerointhenegativemassՏregionsucrhthatthederivXativeofthecurrentcancelsthe2k-dimensionalgaugeanomalyonthewrall,aswasarguedinRef.[ html:1 html:].InntheconrtinuumnpSerturbationtheoryFVrolorvandSlavnov[ html:4 html:]propSosedagaugeinvXariantregularizationJofthestandardmoSdelthroughanin nitetorwerJofregulator elds.SSomesim-ilaritrybSetweentheirpropSosalandtheKaplan'sapproachwaspSointedoutbyNarayananandNeubSergerE[ html:5 html:].HIthasbeenalsoshorwnthatthechiralfermiondeterminantcanbSenonper- html:3 html: ֠7gxgturbativrelyq4de nedasanoverlapoftwovXacua[ html:6 html:],whichcanbSeextendedtolatticetheories.[UsingINararyananandNeubSerger'spoinrtofview,cweobservedinRef.[ html:7 html:]thatnon-gauge(crhiral)pJanomaliesarecorrectlyreproSducedwithintheFVrolovandSlavnov'sregularizationmethoSd.TheT=resultsabSorveT=providepSositiveindicationsthatKaplan'smethoSdforchiralfermionmary
work.Thereexists,Rhowever,several
pSotentialproblemsinhisapproach.Sincetheoriginal2k+1-dimensionalmoSdelisvrector-like,Ltherealwaysexistsananti-chiralmoSde,loScalizedqonananrti-domainwallformedbypSeriodicityqoftheextradimension.-IfthechiralmoSdeandtheanrti-chiralmodearepairedinrtoaDiracmode,thisapproacrhfailstosimulatecrhiral\gaugetheories.! Withoutdynamicalgauge elds,theoverlapbSetweenthechiralmoSdeandmtheanrti-chiralmmoSdeissuppressedasO(e2K cmsy8 L P)whereListhesizeoftheextradimension.If"Ugauge eldsbSecomedynamical,Jftheorverlap"Udependsonthegaugecoupling.IntheoriginalpapSer[ html:1 html:]thestrongcouplinglimitofthegaugecouplingintheextradimensionwrasproposedtosuppresstheorverlap.XHowever,Zamean- eldcalculation[ html:8 html:]inthislimitindicatedthatthecrhiralmoSdedisappearsandthemodelbecomesvrector-like.More\recenrtlyDistlerandRey[ html:9 html:]pSointedoutthattheKaplan'smethoSdmayhaveaproblem&inreproSducingfermionnrumber&non-conservXationexpectedinthestandardmodel.Using1Lthe2-dimensionalcrhiralSchwingermoSdeltheyarguedthateitherthe2-dimensionalfermionqnrumbSercurrentisexactlyconservedorthelightdegreeoffreedom
owso thewallinrtotheextradimensionsothatthemoSdelcannotbe2-dimensional.IneNthispapSerwrecarryoutadetailedperturbativreanalysisoftheKaplan'sproposalforsmoSothbacrkgroundgauge eldsona nitelatticetakingthechiralSchwingermoSdelink2-dimensionsasaconcreteexample.*Insect.'html:ISI html:,wreformulatethelatticepSerturbationtheoryGfortheKaplan'smethoSdwiththeperiodicboundarycondition.PvSincetranslationalinrvXariancevisviolatedbydomainwallmassterms,usualFVeynmanrulesinthemomentumspaceQcannotbSeusedexceptintheregionsfararwayQfromthedomainwrall[ html:3 html:].ITVoperformpSerturbativre3calculationsnearoronthewall,weusetheFVeynmanrulesinrealspaceoftheextradimension,*aspropSosedinref.[ html:5 html:].WVecalculatethefermionpropagatorfortheperiodic html:4 html: (7gxgbSoundaryGFcondition,^nwhicrhreproducesthefermionpropagatorinref.[ html:5 html:]neartheoriginof[theextradimension.7Asimilarcalculationisalsomadefortheconstanrtfermionmasswith.@ cmti12frffeelbSoundariesinthe2k+1-thdimension.AsshorwnbyShamir[ html:10 html:]theconstantmasstermwith thisbSoundaryconditioncanalsoproducethecrhiralzeromodeonthe2k-dimensionalbSoundaryV.MTheresultsaresimilarbutsimplerthanthosebrytheKaplan'smethod.MInsect.(html:ISII html:,ԮusingrtheFVeynmanrulesofsect.'html:ISI html:wrecalculateafermionone-loope ectivreactionforY%theU(1)gauge eldofthecrhiralSchwingermoSdelsimulatedbytheKaplan'smethoSd.WVe( ndthatthee ectivreactioncontainsthelongitudinalcompSonentaswellasparity-oSddterms,;nand+GthatthislongitudinalcompSonenrt,which+GbreaksgaugeinrvXariance,remainsnon-zeroevrenforanomaly-freecases.#ThisresultiscomparedwiththoseoftheconventionalWilson^dfermionformrulationofthismoSdel[ html:11 html:].
Insect.'html:IV html:wederivegaugeanomaliesaswellas)theChern-Simonscurrenrtfromthee ectiveactionwithoutambiguityV.dThenweshowthatthecurrenrtcorrespSondingtothefermionnumbSerhasanon-zerodivergenceandthefermionnrumbSercurrent
owso thewallsintotheextradimension."Insect.&html:V html:,'wegiveourconclusionsanddiscussions.. k8&html:IYI.2 html:AtCTION,FERMIONPROP\AGATOR2ANDCHIRALZEROMODES%ҍ html:A.2 html:LatticeActionWVe[consideravrectorgaugetheoryinD=2k+1dimensionswithadomainwallmassterm.[FVorlaterconrvenienceweusethenotationofref.[ html:5 html:],wherethefermionicactioniswrittenintermsofad=2kdimensionaltheorywithin nitelymanry
avors.8Ouractionisdenotedas"e q.S)=URS̹G @+S̹FO:}(html:1 html:)Theactionforgauge eldS̹G
@isgivrenby S̹G ^y= S jVCu
cmex10CX +Mn;> &CX !s-
Re:x-!",
cmsy10fTVr [U̹
P(n;s)]g 3 ^y+ S ̹D CX
n;'iCX i0s*0Re8wfTVr [U̹D ұ(n;s)]g|(html:2 html:) html:5 html: 6l7gxgwhere,'Irunfrom1tod,nisapSoinrtonad-dimensionallatticeandsacoordinatein[the&extradimension,v Aistheinrverse&gaugecouplingforplaquettesU̹wand ̹D ܖthatforplaquettesOU̹D ұ.Ingeneralwrecantake 6== ̹D.ThefermionicpartoftheactionS̹F isgivrenby")3S̹F:=ōHe1HeҟQm fe 2QxCX R`n;aƟCX g,sWtè*r\ ̹s~nQ(n)
̹
[U̹s; (n) ̹sn<(n+) U @yڍs;(n ) ̹sn<(n )] Ѝ:+G2CX K\nWfCX ]Z/s;tWj}H*h- ̹st'(n)[M
|{Y cmr80P̹R ;+MO@yO0P̹LG]̹st ̹t(n)!:+ōHe1HeҟQm fe 2QxCX R`n;aƟCX g,sWtè*r\ ̹s~nQ(n)[U̹s; (n) ̹sn<(n+)+U @yڍs;(n ) ̹sn<(n ) 2 ̹s(n)]|(html:3 html:)wheres,tareconsideredas
arvorindices,P߹R =LP=UR(1
̸2k6+1@)=2,html: html: #(M̸0)̹st := U̹s;D T(n)̹s+1;tc a(s)̹st (html:4a html:)[ [(MO@yO0)̹st := UO@yDs 1;D(n)̹s 1;tc a(s)̹st;ue(html:4b html:)CandU̹s; (n),*U̹s;D T(n)arelinkvXariablesforgauge elds.sWVeconsidertheabSorveCmodelwith[pSeriodic,boundariesintheextradimension,̫sothats,trunfrom LtoL^ 1,̫andwretake) cca(s)UR=1 m̸0[signM(s+ō1۟Qm fe 2
)sign(L s ō1۟Qm fe 2)]UR=C8 > >< >>:81 m̸0;<) Fu33133콉 fe @'2 5 html:))5 html:]thatS̹F abSorveisidenticaltotheKaplan'sactioninD=2k+1=dimensions[ html:1 html:]withtheWilsonparameterr/=U1.Infactthesecondtermineq.(html:3 html:)canbSerewrittenas"Fuem1em콉 fe @'2Wp*nt ̹sz
̹D[U̹s;D T ̹s+1` U̹s 1;D ̹s 1J]b+ōo1oQm fe 2Wy"X*v= ̹s [U̹s;D T ̹s+1`+U̹s 1;D ̹s 1 2 ̹sn<]+M@(s)Wg* ̹s ̹s|(html:6 html:)withKM@(s)UR=m̸0[signM(sh+1=2)sign(L s 1=2)]..Notethatouractionisslighrtlydi erent[from@thatofref.[ html:5 html:]:'wrehavetheD-thcompSonentofthelinkvXariableU̹s;D T(n)andalllinkvXariablespIharvesdepSendence.Withthegauge xingconditionU̹s;D T(n)8=1forallsandn[ html:9 html:],ouractionbSecomesalmostidenrticaltothatofref.[ html:5 html:],butstillthesdepSendenceexistsinpourlinkvXariablesinddimensions.59ThemoSdelinref.[ html:5 html:]correspondstoourmodelat html:6 html: B7gxg ̹D 3J=}1,FwhereZsdepSendencesofgauge eldsarecompletelysuppressed,andthemoSdelat[ ̹D
=UR0wrasinvestigatedbythemean eldmethoSd[ html:8 html:]..) D html:B.2 html:ChiralZeroMoYdes&WVe}norwconsiderchiralzeromoSdesoftheactionS̹F "intheweakcouplinglimit,]i.e.8U̹s;e=ͨ8U̹s;D=1.,Accordingtoref.[ html:5 html:],therighrt-handedzeromoSdesaregivenbyzeromoSdes oftheoperatorMLandtheleft-handedzeromodesbrythoseoftheoperatorM@2y,8Owhere#8V(M@)̹st
X=UR(M̸0)̹st
*+ōr(p)۟Qm fe 7 2 E̹st;(M y)̹st=UR(MOyO0)̹st
*+ōr(p)۟Qm fe 7 2 E̹st|(html:7 html:)with6r(p)CP*ֹd̍=1"]2[cosV(p̹
a)W 1]inmomenrtumspaceofddimensions.4Itisnotedthat[0 r(p)4d[andzeromoSdesexistifandonlyif r(p)2m̸0 [ html:3 html:].Hereafter[wreonlyconsider9thecasethat0VI >>>>>< >>>>>>:M8(1 r(p)=2 m̸0) sn8 html:)69Uwhere.!thed-dimensionalmomenrtumphastobSerestrictedto0{m̸0F+r(p)=2.!sothat(1 r=2 m̸0)UR1.8ThenormalizationconstanrtC̸0takesthevXalue#8ōZ;1 (1 r(p)=2 m̸0)2LZ;Qm fe z Y;m̸0j+r(p)=2 `+ō1 (1 r(p)=2+m̸0)2 L۟Qm fe f> um̸0j r(p)=2 wL:}(html:9 html:)ThiszeromoSdeislocalizedaroundsUR=0.%YOna nitelattice(i.e.LUR6=1)withthepSeriodic[bSoundaryWcondition,tthereexistsanotherzeromodeu̹L withtheoppositecrhiralityWsatisfyingMFUqu̹L
=8H0,Uwhicrhoisgivenbyu̹LG(s)8H=u̹R(Lq t 1)oandisloScalizedarounds8H=L.TheorverlapbSetweenthetwozeromoSdesvXanishesexponenrtiallyasLUR!1;#8l L 1
m!fCX "%hȹs= L u̹R(s)u̹LG(s) O= ɔ)C ܞ 2ڍ0 9L(1 ōr(p)۟Qm fe 7 2 m̸0) L# h ɔ)(1 ōr(p)۟Qm fe 7 2 +m̸0) L9 !UR0; (html:10 html:) html:7 html: Tݠ7gxgWVe#MillustratetheshapSeofthetrwo#Mzeromodesu̹R andu̹L
k%atm̸0u=0:1and0.5forp=0in[Fig.html:1 html:..z html:C.2 html:F\ermionPropagatorandZeroMoYdes&9NThee)fermionpropagatorind-dimensionalmomenrtumspaceandinrealD-thspacehasbSeen obtainedinref.[ html:5 html:]forthein niteD-thspace(i.e. LUR=1 ).Itisnotdiculttoobtainthefermionpropagatorfora nitelatticewithpSeriodicboundaries.8WVeharve#[tS̹FO(p)̹st = [ f\C"U[(iCX \q
̹
p̹o+M@)G̹LG(p)]̹stP̹L#F + [(iCX \q
̹
p̹o+M@ y)G̹R(p)]̹stP̹Rf\C# (html:11 html:)whereoHG̹LG(p)UR=ō F1Qm fe 6} p2
PY+M@yM<_5;
G̹R(p)=ō F1Qm fe 6} p2
PY+M@My (html:12 html:)with p̹=sin(p̹
a).ExplicitexpressionsforG̹L
aandG̹R arecomplicatedingeneral,Bbut[theybSecomesimpleforlargeLwherewreneglecttermsoforderO(e2 cL)withcz>0.>5WVeobtaind 88MG̹LG(p)̹st
X=ȍURC8 UR> UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR< UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR:Ι8B e2 Aa cmr6+ njs tj'+(A̹L B)e2 + n(s+t))+(A̹R ; B)e2 + n(2L s t)7;4I(s;tUR0)Z8A̹Le2 + ns+ q% cmsy6 Gt.S+A̹Re2 + n(L s) G(L+t)R ;;4I(sUR0;t0)8A̹Le2 Gs + nt(+A̹Re2 G(L+s) + n(L t)R ;;4I(sUR0;t0)8C ܞe2 Gjs tj'6+(A̹L C)e2 G(s+t)# N+(A̹R ; C)e2 G(2L+s+t)7r;4I(s;tUR0) (html:13 html:) html:8 html: c7g&ȍG̹R(p)̹st
X=ȍURC8 UR> UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR< UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR>UR:Ι8B e2 + njs tj'+(A̹R ; B)e2 + n(s+t+2)4n+(A̹L B)e2 + n(2L s t 2)BhC;J(s;tUR 1)Z8A̹Re2 + n(s+1)+ G(t+1)Q;+A̹LGe2 + n(L s 1) G(L+t+1)g3;J(sUR 1;t 1)8A̹Re2 G(s+1) + n(t+1)K+A̹LGe2 G(L+s+1) + n(L t 1)g3;J(sUR 1;t 1)8C ܞe2 Gjs tj'6+(A̹R ; C)e2 G(s+t+2)-+(A̹L C)e2 G(2L+s+t+2)B;J(s;tUR 1)a 8 (html:14 html:)!l wherehtml: html: =pa̺
q=UR1 ōr(p)۟Qm fe 7 2 m̸0<( html:15a html:)-d ^ ̺
q=URarccosh)B`[ō33133Qm fe 2Fb(a̺ +ō1++p22۟Qm fe ۆa̺$)]UR0;h( html:15b html:).YA̹L *=ō+ 1Qm fe S a̸+ xe + a̺ e Xļ;A̹R
H=ō+ 1Qm fe S a̺ xe X a̸+e +( html:15c html:)yBX=ō 51Qm fe 6\ 2a̸+ xsinh ̸+15d html:)!l [FVorjsj;jtj] LthepropagatorabSorvecoincideswiththatofref.[ html:5 html:].
FVromtheformofA̹LG,A̹R,BandC ܞ,itiseasytosee[ html:5 html:]thatsingularitiesoSccuronlyinA̹L
2atpUR=0; A̹L * !ōm̸0(4 m22RA0)Qm fe ;] %4p2a2Ax;pUR!0: (html:16 html:)ThereforethepropagatorG̹L describSesamasslessrighrt-handedfermionarounds;tUR=0andG̹R =amasslessleft-handedfermionaroundjsj;jtj'=L,A+whicrhcorrespSondtothetwozeromoSdes intheprevioussubsection.Laterwreusetheaborve formsofthefermionpropagatorto?calculatefermionone-loSopdiagrams.8ItisalsonotedthatthefermionpropagatorarwayfromthetrwodomainwrallsapproachestheWilsonfermionpropagatorwithacffonstantmassterm,i.e.8, html:9 html:
p$7gpD+S̹FO(p)d!t)CZō \Kdp̹D \KQm fe S ^2ō V e2iapX.; cmmi6D!(s t) yQm fe DK i
Qp+i
̹Dp̹D
`Lm̸0j r(p)+1 cose(p̹Da) (html:17 html:)#[forP1u jsj;jtj;jLP sj;jL tjwithstu>0,jwhere+m̸0 istakrenfors;t>0and m̸0 for[s;tUR<0.8Thereforethecalculationinref.[ html:3 html:]isvXalidinthisregionofsandt.Beforezclosingthissubsection,-/wregivetheformofpropagatorfortheShamir'sfreebSoundaryefermions[ html:10 html:].
Thisisagaingivrenbyeq.( html:11 html:)withM̹st
Ԏ=T̹s+1;t a̸+ x̹s;tandMO@y|st
X=UR̹s 1;tc a̸+ x̹s;t z.8FVorlargeL,itbSecomes[ html:10 html:][Z?G̹LG(p)̹st
X=URB e + njs tj'+A 0ڍLe + n(s+t))+A 0ڍRe + n(s+t 2L) (html:18 html:)/[XbG̹R(p)̹st
X=URB e + njs tj'+A 0ڍRe + n(s+t))+A 0ڍLGe + n(s+t 2L)0O; (html:19 html:)wherewbA 0ڍL *=URBō91 a̸+ xe2 +9Qm fe 9 N20 html:)andnorw0URs;tL.8SingularitiesoSccuronlyinA20RAL
2atpUR=0suchthat eA 0ڍL *!ōm̸0(2 m̸0)Qm fe ;] p2a2Ax: (html:21 html:)Thrus,*thepropagatordescribSesaright-handedmasslessfermionaroundonebSoundaryats;tUR=0andaleft-handedmasslessaroundtheotherbSoundaryats;tUR=L.. H html:D.2 html:F\ermionFeynmanRules&9NInthissubsection,`VwrewritedownthelatticeFVeynmanrulesforfermionsrelevXantforfermion*one-loSopcalculations,
whicrhwillbeperformedinthenextsection.[eWVe rstcrhoosetheaxialgauge xingU̹s;Db=b1.SAlthoughthefullgaugesymmetriesinDdimensionsarelost,_thettheoryisstillinrvXarianttundergaugetransformationsindepSendenrtofs[ html:9 html:]. eWVeconsiderthelimitofsmalld-dimensionalgaugecoupling,andtakre WU̹s; (n)UR=exp[iagn9A̹
(s;n+=2)] (html:22 html:) html:10 html: ~7gxgwhere|aisthelatticespacing,0andg/1=qp
q z Lm isthegaugecouplingconstanrtwhosemass[dimension qis2Θ DS=2 q(massdimensionofthegauge eldsA̹ ~isD=2Θ 1)./S1=qp
q z Z ̹sٮisnotnecessarilysmallandcanbSemadearbitrarylarge.*WVeconsiderFeynmanrulesinmomenrtumspaceforthephysicalddimensionsbutinrealspacefortheextradimension. _Thefermionpropagatorisgivrenby"X &<UR ̹sn<( p)Wg* ̹t 5R(p)>=S̹FO(p)̹st: (html:23 html:)' _whereS̹F
PhasbSeengivrenineq.( html:11 html:)witheqs.( html:13 html:, html:14 html:)fortheKaplan'sfermionsorwith _eqs.( html:18 html:, html:19 html:)fortheShamir'sfermions.!o _Thefermionvrertexcoupledtoasinglegauge eldisgivenby~agWT*n9 ̹s (qn9)@̹
[S 1ڍF S(ō33q+p33Qm fe k
82 h)]̹ss\xA̹(s;p q) ̹sn<( p): (html:24 html:)* _HereT@̹
S 1cF S(qn9)%q=ōX@ S 1cF(q)XQm fe (벟 @ (q̹
a)5=%qiC̹
(q)
̹ [+dNS̹(q)TwithC̹
(q)%q=cos(q̹
a)TandS̹(qn9)%q= _sin+
(q̹
a).
lFVromthisformofthevrertexitiseasytoseethatthefermiontadpSole _diagramforanexternalgauge eldvXanishesidenrticallyV.!o _Thefermionvrertexwithtwogauge eldsisgivenbyuѪ a 2ō7gn9227Qm fe
Δ wL2W\* ̹s (qn9)[@ 2ڍ
S 1ڍF S(ō33q+p33Qm fe k
82 h)]̹ss\xA 2ڍ(s;p q) ̹sn<( p) (html:25 html:)' _whereA22RA
(s;p)UR=A̹(s;p p̸1)A̹(s;p̸1)withpandp̸1 xed..U 'html:IYII.2 html:PER\TURBATIVECALCULATIONSFtORTHECHIRALSCHWINGER ҊMODEL%ԍInthefollorwingtwosectionsweanalyzethechiralSchwingermoSdelformulatedviatheKaplan's methoSdforlatticecrhiralfermions.UsingtheFVeynmanrulesoftheprevioussectionfor-D ={3,>~wrecalculatethee ectiveactionforexternalgauge elds,>~fromwhichwederive html:11 html: 7gxggaugecYanomalies,Chern-Simonscurrenrt,andanomalyofthefermionnrumbSercYcurrent.WVe[pSerform?thecalculationsfortheShamir'smethodinparallelwiththosefortheKaplan'smethoSd..jw html:A.2 html:E ectivteActionatF\ermionOne-LoYop%PSincet[A̹
;A̹3]UR=0forU(1)gauge elds,alldiagramswithoSddnrumbSerofexternalgauge eldsZ vXanishesidenrticallyV.FurthermoreZ diagramswithfourormoreexternalgauge eldsareall^conrvergent.Thereforeonlythediagramswithtwoexternalgauge eldsarepSotentiallydivrergent.8Thee ectiveactionfortwoexternalgauge eldsisdenotedby!DeDSߍ (2) '͍ef f~5 < ō33gn92233Qm fe
Δ wL29CX ]4p;s;t!5AA̹
(s;p)A̹3(t; p)I B"(p)̹st#r~ = < ō33gn92233Qm fe
Δ wL29CX ]4p;s;t!5AA̹
(s;p)A̹3(t; p)[I (2)ڍa+Iߍ (2) '͍b M]O|st
P; (html:26 html:)whereb[I (2)ڍa M]O|st [M= 'CZ2 (I{=aH ba I{=aō xd22q Qm fe
(2n9)2 ΰ'tr ӭf^C S[@̹
S 1ڍF S(q+ōp۟Qm fe 孟 2
)S̹FO(q+p)]̹st ;E Bf Z[@̹3S 1ڍF S(q+ōp۟Qm fe 孟 2
)S̹FO(qn9)]̹tsf^C1a 2 (html:27 html:)andS*[Iߍ (2) '͍b M]O|st
=UR ̹st̹ PCZ2PI{=aH I{=aō/Kd22q* ]Qm fe
(2n9)2F,trOP%[@ 2ڍ
S 1ڍF S(qn9)S̹FO(q)]̹ss a 2 (html:28 html:)withtrmeaningtraceorverspinorindices.ts html:B.2 html:EvLaluationofZeroMoYdeConttributions%PTVoevXaluateI 2 B"(p)wredecompSoseitintotwopartsas I B"(p)UR=IO O0(p)+[I (p) IO O0(p)] (html:29 html:)where^OIO O0qistheconrtributionofzeromoSdesandI 2 IO O0istheremainingconrtribution.
FVor[IO O0,wrereplacetheintegrandofI 2,withthatintheaUR!0limit,andweobtain html:12 html:
Ҡ7gCX#IO O0 B"(p)̹st Qz= TCX "% 4tX <CZ Uz r(qI{)2m q0ō ܛHd22q lQm fe
(2n9)2 "* 8 Ttr ڟf`Ch i
̹
( i
̹ (q+p)̹ a)G 0ڍX (q+p)̹stP̹X[ 8 i
̹3( i
̹ <(q+p)̹ a)G 0ڍX (q+p)̹tsP̹X]f`CiFa 2; (html:30 html:)#[withYX,=Lforjsj;jtj0,uorX,=Rs/forjsj;jtjL.ThezeromoSdepropagatorsG20RAX wnare[givrenby~cG 0ڍX (qn9)̹st
X=blim}URa!0+G̹X(q)̹st
X=ō ?1Qm fe N q2.=a2
F̹X(s;t) (html:31 html:)whereF̹X (s;t)UR=F̹X(t;s)andO (-ZF̹LG(s;t)UR=ōm̸0(4 m22RA0)Qm fe ;] 14D"؍C8 > >>>>>>>>>>>>>< >>>>>>>>>>>>>>:7(1 m̸0)2s+tyfors;t0Z7(1 m̸0)2sn<(1+m̸0)2tyfors0andt<07(1+m̸0)2s+tyfors;t0 (html:32 html:) PuF̹R(s;t)UR=ōm̸0(4 m22RA0)Qm fe ;] 14D"؍C8 > >>>>>>>>>>>>>< >>>>>>>>>>>>>>:7(1 m̸0)22L s t 2 Ϯfors;t0Z7(1 m̸0)2L s 1(1+m̸0)2 L t 1 Ϯfors0andt<07(1+m̸0)2 2L s t 2 Ϯfors;t0 (html:33 html:)NU|fortheKaplan'sfermionwiththedomainwrallmassterms,andhtml: html:{F̹LG(s;t) ~-= m̸0(2 m̸0)(1 m̸0) s+t<( html:34a html:)zkF̹R(s;t) ~-= m̸0(2 m̸0)(1 m̸0) 2L s ti( html:34b html:)fortheShamir'sfermionwiththeconstanrtmasstermsandfreebSoundaries.[WVeevXaluateIO O0 B"(p)intheaUR!0limit.8Inthislimit html:13 html: ݠ7gCnlim}ma!0 ВCZ ^̟z r(qI{)2m q0ō /d22q .Qm fe
(2n9)2 ntr 3[P̹X
̹
̹
̹3
̹ <]ō33(q+p)̹ q̹ 33Qm fe 3s (q+p)2qn92 q= ВCZ2 Г1H ^ 1ō ]d22q .}Qm fe
(2n9)2 =tr aE[P̹X
̹
̹
̹3
̹ <]ō33(q+p)̹ q̹ 33Qm fe 3s (q+p)2qn92!;Gq=ō 1 şQm fe 2 /wf\C" ̹X i ō Ep̹3p̹ EןQm fe $p2&+(s2
ny ōp̹
p̹۟Qm fe $p2) ōs22۟Qm fe
2 f\C#JL9; (html:35 html:)"lthereforewreobtain` lim}`Ua!0sIO O0 B"(p) h:= CX "% K4Xō Q1 Qm fe 2 ±[̹X i ō Ep̹3p̹ EןQm fe $p2&+(s2
ny ōp̹
p̹۟Qm fe $p2) ōs22۟Qm fe
2 ][ OS F̹X (s;t) 2 (html:36 html:)where̹L *=UR1and̹R
H= 1.8ItisnotedthatF̹X 1satis esk CX ]p t{F̹X (s;t) 2V=URF̹X(s;s);CX ]
s;t)F̹X(s;t) 2V=UR1: (html:37 html:)9Ou: html:C.2 html:EvLaluationofRemainingConttributions& WVeconsidertheremainingtermsinI 2 B".3SincethecomrbinationI 2(p) IO O0(p)isinfra-[red nite,.wrecanchangetheintegrationvXariablefromqtoqn9aandtakethea)!0 limitintheinrtegrand.8Thusweobtain"lXlim}Xa!0kVI B"(p) IO O0(p)UR=I (0)=ō1Qm fe 2^[i
P ̹C Sh"+s2
K ܞ] (html:38 html:)wherep! ̹C S z(s;t) =ō f(2 f(Qm fe 0 i 2n7CZōd22qiQm fe
(2n9)2+tr6/f`Cn=~[@̹
S 1ڍF S(qn9)S̹FO(q)]̹st 4 f([@̹3S 1ڍF S(qn9)S̹FO(q)]̹tsf`Co (html:39 html:)andlOK ܞ(s;t)UR=2n7CZōd22qiQm fe
(2n9)2-f`Ch2itr=f`CnD7[@̹
S 1ڍF S(qn9)S̹FO(q)]̹stb[@̹
S 1cF S(qn9)S̹FO(q)]̹tsf`Cog_ ̹sttr[@2 2RA
S 1cF S(qn9)S̹FO(q)]̹ss\xf`Ci `: (html:40 html:)Herenosummationorver,oisimplied. html:14 html: v7gxgThe/tparitry-oSddterm ̹C Sisthecoecienrtfunctionofa3-dimensionalChern-Simons[termintheaxialgauge[ html:9 html:],whicrhsatis es ̹C S z(s;t)UR= ̹C S(t;s).8Itiseasytoshorwthat#[jNϟCX ]oϹtz ̹C S z(s;t)UR= ō33233Qm fe 0 iuCZō d22q ƟQm fe ~ 622Ztr=~kf`CnD)@̹
S 1ڍF S(qn9)@̹3S̹FO(q)f`Co -ss"݂ }N=ō2Qm fe 0 if\C" =Zō)odq̹)oQm fe 2;^trGf`CnM@̹
S 1ڍF S(qn9)S̹FO(q)f`Co
3ss ff\C#* <Uh <q =ThiseXwrouldbSezeroiftherewerenoinfra-redsingularitiesinS̹FO.However,bSecauseoftheconrtributionfromzeromoSdes,S̹F
issingularatqË=UR0.8Therefore =ϟCX ]%Ϲt0 ̹C S z(s;t)`G=l CX ]tq ̹C S z(t;s)UR=4lim}!0Qf\C"'Z2$'I{=2H Iō5Sdq̹5SQm fe 2I@CX "%M`XZ̹X [S̹3(qn9)C̹
(q)G 0ڍX(q)]̹ss\xf\C#* MҹI{=2 w Mҹq =$",`G=l 4CX "%j Xq̹X F̹X(s;s)lim}!0QCZ2 QI{=2Hō/dq̹/Qm fe 2ōR C Qm fe "' qn92nꍹG+2"`G=l CX "%j Xq2̹Xō PF̹X (s;s) PQm fe )1 5zf^C:tanKp 1ōZ qZ Qm fe ra\6f^C*fTI{=2UdfT!`G=l CX "%j Xō̹XQm fe
Qȟ 2")F̹X (s;s); (html:41 html:)whereF̹X (s;t)isgivrenintheprevioussubsection.Since"S̹F qbSecomestheWilsonfermionpropagatorwithconstanrtmasstermfor1M jsj;jtj;jL sj;jL tjwithstUR>0[seeeq.( html:17 html:)],itbSecomes oCP ~s;t [" ODC S z(s;t)A̹
(s;p)A̹(t; p)UR
![p 2 PqCRFdp̸3A̹
(p̸3;p)p̸3A̹( p̸3; p)CZōad23q30Qm fe
(2n9)2mtr
.f`Cnx[@̹
S 1cFS̹FO][@̸3S 1cFS̹FO][@̹3S 1cFS̹FO]f`Co ќ; (html:42 html:)#[whicrh`coincideswiththeresultofref.[ html:3 html:].
ThisisagoSod`checkofourcalculation.
FVromRef.[ html:3 html:]wreobtain7 T=UR PCZPdp̸3A̹
(p̸3;p)p̸3A̹3( p̸3; p)荓C8 > >>>>>< >>>>>>:M71&.+for+m̸0Z70&.+for m̸0S~ : (html:43 html:)Theparitry-eventermKFsatis esK ܞ(s;t)UR=K(t;s)and html:15 html: ĝ7gykiQ CPsXtyq K ܞ(s;t)UR=CX ] RtK(t;s)"79k\=yq CZōad22q30Qm fe
(2n9)2+Bptr6ef`Cn=[@̹
S 1ڍF@̹S̹FO]̹ss +[@ 2ڍS 1ڍFS̹FO]̹ss\xf`Co 4k\=yq CX "%|@Xō 1 Qm fe 2 F̹X (s;s) (html:44 html:) ThederivXationofthelastequalitryissimilartothatofeq.( html:41 html:).. html:D.2 html:T\otalConttributions%|ӍComrbiningtheabSovecontributionswe nallyobtain Sߍ (2) '͍ef f$=1\ ōH*gn92233Qm fe 4^CX ]鍹s;t!ЬCZ-Ыd 2xf\C(
9X "%
xYX F̹X (s;t) 2[A̹
(s;x)(s2
ny ō@̹@̹۟Qm fe 8g w3Tq
lasy102 Iu)A̹3(t;x)]!$+1\[K ܞ(s;t) CX "%ȹXōO1OQm fe 2
yF̹X (s;t) 2]A̹
(s;x)A̹(t;x)#".$+2ʟCX "%5XCi̹X F̹X(s;t) 2[ō33@̹33Qm fe l27WA̹
(s;x)
)@̹ A̹3(t;x)]+i ̹C S z(s;t)
PA̹(s;x)A̹3(t;x)f\C)(: (html:45 html:)ThisisthemainresultofthispapSer.ItisnotedthattheaborveformulaisvXalidforbSoth[the\Kaplan'sandtheShamir'smethoSds.Thefollorwingconsequencescanbedrarwnfromeq.( html:45 html:)abSorve.ThePrarity-oSddterms,)whicharepropSortionalto2
),)areunambiguouslyde ned,)con-traryhtothecaseoftheconrtinuumhregularizationforanomalyfreecrhiralgaugetheories[ html:4 html:]>whicrhonlyregulatestheparityeventerms[ html:5 html:, html:7 html:].Theseparity-oSddtermsbreakgaugeinrvXarianceinthe2-dimensionalsense.FVoranomalouscrhiralSchwingermoSdel,\theparity-oSddtermwithX>=MER,islocalizedarounds=0andthatforXa=LisloScalizedarounds=L.Thee ectivreactionabSoveforOanomalouscrhiralSchwingermoSdelviatheKaplan'smethodortheShamir'svXariationisdi erenrt/fromtheoneviatheusualWilsonfermionin2dimensions[ html:11 html:]:-ThetermpropSor-tionalto ̹C S z,DwhicrhcannotbSeevXaluatedanalyticallyforsdependenrtgauge elds,Disspecialforcrhiralfermionsfrom3-dimensionaltheories,qandthepresenceofthistermpreventsusfromzconcludingwhethertheanomalouscrhiralSchwingermoSdelcanbeconsistenrtlyde nedviatheKaplan's(Shamir's)methoSd. html:16 html: կ7gxgFVoranomalyfreecasessucrhthatCP3R'Rg2n92RAR
:=G!CP\L 2g2n92RALG,=theparity-oSddtermsareexactly[cancelleda=loffcal lyinsspace.Hereg߹R (L) isthecouplingconstanrtofafermionwithpSositive(negativre)m̸0 Pwhichgeneratearight-handed(left-handed)zeromoSdearoundsn=0.(Thesimplestbutnon-trivialexampleisaPythagoreancase,#g̹R
=[3;4andg̹L 3=5[ html:9 html:].jEvrenfortheseg"anomalyfreecases,pthelongitudinalterm,whosecoSecienrtisKz FƟ22a=2,remainsnon-zerointhee ectivreaction,
\sothatgaugeinvXarianceinthe2-dimensionalsenseisviolated.Inthisregardtheformofthee ectivreactionviatheKaplan's(Shamir's)methoSdissimilartotheoneviatheusualWilsonfermion[ html:11 html:].Let6usconsiderthee ectivreactionforsindepSendentgauge eldsasinref.[ html:5 html:].SinceCP
;;;L2
PA̹
(x)A̹3(x)=0forsindepSendenrtgauge elds,theChern-SimonstermvXanishes.Theotherparitry-oSddtermiscancelledbetrweenthetrwozeromodessinceCP
=*X&;s;t"̹X F̹X(s;t)22V=0.8ThelongitudinaltermalsovXanishesduetotheidenrtity ƍQCX ]Vít_v[K ܞ(s;t) ōF̹X (s;t)22۟Qm fe , cu21]UR=CX ] Rt[ō33F̹X (s;t)2233Qm fe , cu2 ōF̹X (s;t)22۟Qm fe , cu2]=0 (html:46 html:)[seeeqs.( html:37 html:)and( html:44 html:)].8Thereforethee ectivreactionbSecomesbSߍ (2) '͍ef f =UR 2ōgn922۟Qm fe 4 CZ d 2x[A̹
(x)(s2
ny ō@̹@̹۟Qm fe 8g w2 Iu)A̹3(x)]: (html:47 html:)Thisae ectivreactionistransverseandthusgaugeinvXariantinthe2-dimensionalsense. ?Both[zeroTmoSdesarounds=0Tands=LTequallyconrtributesothatafactor2appearsintheabSorveresult.1Thisisconsistenrtwiththegeneralformuladerivedinref.[ html:6 html:].1TheanomalouscrhiralFSchwingermoSdelcannotbesimrulatedbytheKaplan's(Shamir's)methoSdwiththes-indepSendenrt3gauge elds,Vsincethegauge eldsseebffothofthezeromodessothatitfailstoreproSducetheparitry-oddterm,expectedtoexist[ html:12 html:]-?z&html:IV.2 html:ANOMALIESINTHECHIRALSCHWINGERMODEL%Q html:A.2 html:CurrenttsandtheirDivergenceFVromthee ectivreactionobtainedintheprevioussection,gwecancalculatethevXacuum[expSectation"vXaluesofvariouscurrenrtsinthepresenceofback-groundgauge elds.NLetus html:17 html: a7gxgde nethefermionnrumbSercurrentas!- hJ rgڍp(s;x)iUR=Gs2Sߍ (2) '͍ef f-y fe 4< gn9s2A̹
(s;x) (html:48 html:)wheretheindexgMinthecurrenrtexplicitlyshowsthechargeofthefermion.$FVromeq.( html:45 html:)[wreobtain
p J rgڍp(s;x)4=A}Kiō,g33Qm fe 4^CX ]t Ю[CX "%j Xq̹X F̹X(s;t) 2(
)@̹G+ @̹3)ō33@̹ 33Qm fe ){ 2A̹(t;x) 2 ̹C S z
PA̹3(t;x)]!4Ң ōF%wgB~Qm fe 4R0CX ]X0taM[2CX "%j XqF̹X (s;t) 2(s2
ny ō@̹
@̹۟Qm fe 8g w2 Iu)A̹3(t;x)+(2K ܞ(s;t) CX "%ȹX oF̹X(s;t) 2)A̹
(t;x)][4ҢA}KJ rgI{;oddڍӜ+J rgI{;evenڍ (html:49 html:)whereFJ2 rgI{;oddRA:isaparitry-oSddcurrent(the rsttwoterms)andJ2 rgI{;evenRA Zgisaparity-evencurrent(thelasttrwoterms).ȤHereafterallJ̹ shouldbSeunderstoodasvXacuumexpectationvXalues,thoughIhiissuppressed. UFVromthefermionnrumbSerIcurrentthegaugecurrentforafermionwithcrhargegXiseasilyconstructedasJ2 rGRA<
(s;x)URgn9J2 rgRAp(s;x).Divrergencesoftheparity-oSddandparity-evencurrentsbSecomefD@̹
J rgI{;oddڍ((s;x)
= (iō,g33Qm fe 4^CX ]t Ю[CX "%j Xq̹X F̹X(s;t) 2j 2 ̹C S z(s;t)][ & (
P@̹
A̹3(t;x); (html:50 html:)+"q4@̹
J rڍgI{;even(s;x) =ō g c
Qm fe 4 ؎CX ] >t [CX "%j XqF̹X (s;t) 2j 2K ܞ(s;t)] /@̹
A̹(t;x): (html:51 html:)7z Wx html:B.2 html:GaugeintvLariance%vAsrmenrtionedinSect.'html:ISI html:,theactioninA̸3=;0gaugeisinvXariantundersindepSendentgauge-transformation.ThisinrvXarianceimpliesCP؟s*@̹
J2 rGRA<
(s;x)I=0.This-idenrtityissatis edinourcalculationofthee ectivreaction,since!-cWCX s ʃ[CP
;X̹X F̹X(s;t)22j 2 ̹C S z(s;t)] *p=URCP
㍟X̹X [F̹X(t;t) F̹X (t;t)]UR=0 (html:52 html:) html:18 html: `7gyk :CX Ns [CP
;XF̹X (s;t)22j 2K ܞ(s;t)][ S=URCP
㍟X[F̹X (t;t) F̹X(t;t)]UR=0 (html:53 html:)#[fromeqs.( html:37 html:, html:41 html:, html:44 html:).. a html:C.2 html:GaugeAnomalies&9NThe gaugeanomalyforafermionwithacrhargegn9,DedenotedT̹g,isde nedbryT̹g ]=[gn9@̹
J2 rgI{;oddRA(,anditbSecomes T̹g(s;x)UR=CX ] Rtgn9 2.=C ܞ(s;t)TƟ 0a(t;x) (html:54 html:)where VTƟ 0a(t;x)UR=iōv133Qm fe 4^
P@̹
A̹3(t;x) (html:55 html:)isthegaugeanomalyofa2-dimensionaltheoryV,and QC ܞ(s;t)UR=CX "%rX̹X F̹X(s;t) 2j 2 ̹C S z(s;t) (html:56 html:)represenrtsVthespreadofthegaugeanomalyoverthe3rddirectionduetothespreadofzeromoSdes.* Thisspreadoftheanomalyhasbeenobservredinanumericalcomputation[ html:2 html:].* Itisnoted?SthatthedivrergenceofthegaugecurrentJ2 rGRA {]alsocontainsparity-evencontributions,givrenby g=CX ] =t DS(s;t)ōgn922۟Qm fe 4 @̹
A̹(t;x) (html:57 html:)whereDS(s;t)UR=CP
㍟XF̹X (s;t)22j 2K ܞ(s;t).Theone-loSopinrtegral( html:39 html:)de ning ̹C SyoistoocomplicatedtocalculateanalyticallyV.)OFort-indepSendenrtgauge eldsA̹
(t;x) X=A̹(x)thereisaconsiderablesimpli cationandwreobtain 58 html:) html:19 html: y7gxgwhere# TƟ 0a(x)UR=iōv133Qm fe 4^
P@̹
A̹3(x) (html:59 html:)and(zzC ܞ(s)C=P9CX ]U麹t`C ܞ(s;t)UR=2CX "%j Xq̹X F̹X(s;s)8"=C=ōQlm̸0(4 m22RA0)QlQm fe ;] 12 %荑C8 > >>>>>< >>>>>>:M7f`hG(1 m̸0)22s X (1 m̸0)22(L s 1)&f`Ci rtforsUR0Z7f`ChG(1+m̸0)22s X (1+m̸0)2 2(L+s+1)-'f`Ci rtforsUR0 : (html:60 html:)6QforCtheKaplan'smethoSd.)iWVeplotC ܞ(s)asafunctionofsatm̸0V=UR0:1and0.5inFig.html:2 html:.FVor[theShamir'smethoSd,wreobtain[C ܞ(s)UR=2m̸0(2 m̸0)f`Chc(1 m̸0) 2s X (1 m̸0) 2(L s)f`Ci (html:61 html:)and plotthisinFig.html:3 html:.2 Itisnotedthatthereisnoparitry-even contributionin@̹
J2 rGRA sinceCP
;t DS(s;t)UR=0inthiscase..o html:D.2 html:Chern-SimonsCurrentt&=FVromhthe3-dimensionalpSoinrtofview,thegaugeanomalyshouldbecancelledinsucrhawray:`thatT̹g+B@̸3gn9J2 rC SRA3
(s;x)UR=0:`[ html:1 html:, html:3 html:],]whereJ2 rC SRA3ListhethirdcompSonenrtoftheChern-Simonscurrenrt?Bforthe3-dimensionalvectorgaugetheoryV.Withourgauge xingthee ectiveactiondoSesnotdependsonA̸3,*anditisdiculttocalculateJ2 rC SRA3|analyticallyexceptintheregionarway fromdomain-wralls[ html:3 html:].սHowever,lfort-indepSendentgauge elds,lwecanobtaintheChern-Simonscurrenrteverywhereviatherelation@̸3gn9J2 r3RAgI{;odd (s;x)UR= T̹g,whichbSecomesa2J rC Sڍ3
(s+ō1۟Qm fe 2
;x) J rC Sڍ3(s ō1۟Qm fe 2
;x)UR=gn9 2.=C ܞ(s)TƟ 0a(x): (html:62 html:)TVakingJ2 rC SRA3
(s+Fu۸1۟콉 fe @'2 Q;x)UR=gn922.=I (s)TƟ20a(x),wreobtain I (s+ō1۟Qm fe 2
) I(s ō1۟Qm fe 2
)UR=C ܞ(s): (html:63 html:) html:20 html:
7gxgWVeharvetosolvethisequationwiththebSoundaryconditionI (s)UR! 2ass!+1[ html:3 html:].>FVor[a nitesspacesUR!+1means1UR s L.FVortheKaplan'smethoSdwreobtain7\$/I (s ō1۟Qm fe 2
)UR=荓C8 > >>>>>< >>>>>>:ōl2+m̸0lQm fe #_ 27[(1 m̸0) 2s X+(1 m̸0) 2(L s)] 2 50sLZ8 ō332 m̸033Qm fe # 2([(1+m̸0) 2s X+(1+m̸0) 2(L+s)"K] 5 Ls09 : (html:64 html:)8?ThisUsolutionautomaticallysatis estheotherbSoundaryconditionthatI (s)UR!0Uas!UR 1[ html:3 html:].nAgain! 1means1 s Lfora nitesspace.nWVeplotI (s)asafunctionofsatm̸0V=UR0:1and0.5inFig.html:4 html:.8FVortheShamir'smethoSdwreobtain#[bI (s ō1۟Qm fe 2
) =UR2[(1 m̸0)22s X+(1 m̸0)22(L s+1)&] 2[ Xvfor0URsL 14=; (html:65 html:)whicrhisplottedinFig.html:5 html:..{4 html:E.2 html:PythagoreanChiralScthwinger2MoYdelandAnomalyinF\ermionNumtberCurrentt&9NLetusconsiderthePythagoreancrhiralSchwingermoSdel[ html:9 html:].2Inthismodeltherearetrwo[righrt-handed2fermionswithchargesg̸16andg̸2,andoneleft-handedfermionwithchargeg̸3.FVormrulationhofthismoSdelviatheKaplan'smethodhasalreadybeendiscussedinref.[ html:1 html:, html:2 html:, html:9 html:]('anextensiontotheShamir'smethoSdisstraighrtforward').]WVeassign+m̸0 +forfermionswith crhargeg̸1̆andg̸2,and m̸0forafermionwithcrhargeg̸3.nThevXalueofjm̸0jshouldbSeequalforallfermions,aswillbSeseenbelorw.TheoAtheoryhasaU@(1)23 /Esymmetry[ html:9 html:]correspSondingtoindependenrtphaserotationsofthreefermions.8ThecorrespSondingcurrenrtsare html:21 html: 07g؍ C8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 7J2 rGRA = [g̸1J2 rg q1RA }D+g̸2J2 rg q2RA+g̸3J2 rg q3RAZ 7J2 rRRA = [g̸2J2 rg q1RA }D g̸1J2 rg q2RA 7J2 rFRA = [J2 rg q1RA }D+J2 rg q2RA+J2 rg q3RA (html:66 html:)MKThe rstoneisthegaugecurrenrt,whosedivergencebSecomes"*`.@̹
J rGڍ<
(s;x)UR=(g n92ڍ1+g n92ڍ2 g n92ڍ3.=)CX ]Zt oC ܞ(s;t)TƟ 0a(t;x): (html:67 html:)Therefore,
ifg2n92RA1tV+Fg2n92RA2
*=g2n92RA3 4(Pythagoreanrelation)issatis edandifallfermionsharvethe[samevXalueofjm̸0jtogivrethesameC ܞ(s;t),:thiscurrentisconservedandthereisnogaugeanomalyforanrybackgroundgauge elds.8Thesecondcurrentisnon-anomalous,since"*U$@̹
J rRڍ(s;x)UR=(g̸2jg̸1 g̸1g̸2)CX ]tqC ܞ(s;t)TƟ 0a(t;x)UR=0: (html:68 html:)TheRthirdcurrenrt,qwhichRcorrespSondstothefermionnrumberRofthetheoryV,qisanomalous,[since`m@̹
J rFڍ+(s;x)UR=(g̸1j+g̸2 g̸3)CX ]Zt oC ܞ(s;t)TƟ 0a(t;x): (html:69 html:)ThefKaplan'smethoSdaswrellastheShamir'sonesuccessfullygiveanon-zerodivergencefor'thefermionnrumbSer'current,wLthoughthecoSecientC ܞ(s;t)hasa nitewidth.FVort-indepSendenrtgauge elds,thisanomalybecomes 2(g̸1j+g̸2 g̸3)C ܞ(s)TƟ 0a(x) (html:70 html:)whereCC ܞ(s)isalmostloScalizedats =0Candats =LCasseeninFig.html:2 html:andFig.html:3 html:.=SincethefermionnrumbSerisconservredinthe3-dimensionaltheoryV,ethethirdcomponenrtofthefermion nrumbSercurrentshouldsatisfy@̸3J2 rFRA3
< +\(g̸1`+g̸2 g̸3)C ܞ(s)TƟ20a(x)UR=0 [ html:9 html:]. Thereforewreobtain}J rFڍ3+(s;x)UR=(g̸1j+g̸2 g̸3)I (s)TƟ 0a(x): (html:71 html:) html:22 html: '77gxg %html:V.2 html:CONCLUSIONS&9NInXthispapSerwrehaveformulatedalatticepSerturbativeexpansionfortheKaplan'schiral[fermion%wtheories,4*extendingthesuggestionbryNarayananandNeubSerger[ html:5 html:].LApplyingourpSerturbativre;techniquetothechiralSchwingermoSdelformulatedviatheKaplan'sortheShamir's>wmethoSd,Skwrehavecalculatesthefermionone-loSope ectiveactionforgauge elds.Theއe ectivreactioncontainsparity-oSddtermsandlongitudinalterms,bothofwhicrhbreak2-dimensionalhgaugeinrvXariance,andtheanomalyofthegaugecurrentisobtainedfromthe=e ectivreaction.ThegaugeanomalyiscalculableintheKaplan's(Shamir's)methoSdif.nthepSerturbativreexpansioniscarefullyformulated. 2FVortheanomaly-freePythagoreancrhiralSchwingermoSdel,lthefermionnumbSercurrentisanomalous.a TVoobtainthisanomalytheҵfermionnrumbSerҵcurrentshouldnotbSesummedovers, incontrasttothecaseoftheconrtinuumcalculation[ html:7 html:],wheretheanomalycomesfromanin nitesummationorvers.Themainconclusionsdrarwnfromtheresultsareasfollows. html: html: [\h1. _AnomalyofthefermionnrumbSercurrentisshowntobSenon-zerointhismethod,though _theI currenrt
owso wallsintotheextradimension.T=Sincethecurrentisexternalwe _feelNKthatthisdoSesnota ectthedynamicsofthemodelandthereforedoesnotspoil _the2-dimensionalnatureofthecrhiralzeromoSde,DcontrarytothesuggestionofRef.[ html:9 html:]. _The3-dimensionalnatureoftheKaplan's(Shamir's)formrulationmanifestsitselfonly _inthenon-conservXationofthefermionnrumbSer,awhichisexpectedtooccurinNature.UThtml: html:\h2. _Twro-dimensional7gaugeinvXarianceatlowenergycannotbSeassuredbytheKaplan's _(Shamir's)O methoSd,n:exceptfors-independenrtgauge elds,n:evenforanomaly-freecases. _This*4issimilartothesituationwithlatticecrhiralgaugetheoriesformulatedwiththe _ordinarywWilsonmassterm[ html:11 html:].InthispSoinrttheKaplan's(Shamir's)methoddoes _not9oseembSetterthantheconrventional9oapproaches. %4Atthismomentitisnotclear _whetherOthisviolationofgaugeinrvXariancespSoilsthewholeprogramofthismethod.)In _particular;thee ectsofthelongitudinalcompSonenrtofgauge eldshastobeanalyzed html:23 html: 57gxg _further. html: html: [\h3. _Ifthetheoryisanomalyfreeandgauge eldsares-indepSendenrt[ html:5 html:],thegaugeinvXari-[ _anceiasa2-dimensionaltheorycanbSemainrtained.
However,theigauge eldsfeelboth _of5thezeromoSdesevrenintheLJ!15limit,Zandthefermionloopconrtributionto _thee ectivreactionistwiceaslargeastheoneexpSectedfromasinglechiralfermion. _Therefore,wrehavetotakeasquare-roSotofthefermiondeterminanttoobtainthecor- _rectconrtribution.FVorfermionquantitiessuchasthefermionnumbSercurrent,* however, _itxMseemspSossibletoseparatetheconrtributionofthechiralzeromoSdeatsFk=0xMfrom _thatoftheanrti-chiralzeromoSdeatsUR=L,asseenintheprevioussection.$[Prerturbative[calculationspSerformedinthispapercanbeextendedto4+1dimensionaltheories.OfYcourseactualcalculationsbSecomemruchYmorecomplicatedanddicultbecauseof sevrereultra-violetdivergencesin4+1dimensionsthanin2+1dimensions.סWVorkinthisdirectionisinprogress.. QAtCKNOWLEDGMENTS&9NWVeTwrouldliketothinkProf.
UkXawafordiscussionsandthecarefulreadingofthemanruscript.Afterf% nishingthiswrork,anewpapSerbyNarayananandNeubSerger[ html:13 html:]appeared.VIntheYpapSerthegaugeanomalyforthecrhiralSchwingermoSdelwascalculatedsemi-analyticallyviatheorverlapformulaofref.[ html:6 html:]. html:24 html: A٠7gxg REFERENCES html: html:[ html:1 html: D.B.Kaplan,Phrys.Lett.B288,342(1992). html: html:[ html:2 html: K.Jansen,Phrys.Lett.B288,348(1992).html: html: html:3 html: M.F.L.Golterman,K.Jansen,andD.B.Kaplan,Phrys.Lett.B301,219(1993).html: html: html:4 html: S.A.FVrolorvandA.A.Slavnov,Phys.Lett.B309,344(1993).html: html: html:5 html: R.NararyananandH.NeubSerger,Phys.Lett.B302,62(1993).html: html: html:6 html: R.NararyananandH.NeubSerger,RU-93-25,RutgersUniversitypreprint,July1993.UThtml: html: html:7 html: S.AokiandY.KikukXarwa, UTHEP-258/KUNS-1204,UnivrersityofTsukubapreprinrt,[ June1993. html: html:[ html:8 html: C.!(PV.Korthals-Altes,S.NicolisandJ.Prades,CPT-93/PV.2920,CenrterdePhysique Thrseoriquepreprint,June1993.html: html: html:9 html: J.zDistlerandS.-J.ReyV,PPUPT-1386/NSF-ITP-93-66/SNUTP-93-27,PrincetonzUnivrersity preprinrt,May1993.html: html: html:10 html:Y.Shamir,WIS-93/20/FEB-PH,WVeizmannInstitutepreprinrt,February1993. html: html:[ html:11 html:S.0Aoki,fPhrys.RevLett.602109(1988);K.FVunakubSoandT.Kashiwa,ibid602113 (1988);rT.WD.Kieu,ٚD.Sen,S.-S.Xue,ibid602117(1988);rS.Aoki,Phrys.Rev.D38618 (1988).html: html:[ html:12 html:R.JacrkiwandR.Ra jaraman,Phys.RevLett.541219(1985).html: html: html:13 html:R.NararyananandH.NeubSerger,RU-93-34,RutgersUniversitypreprint,August1993. html:25 html: I7gxg rFIGURES, lhtml:FIG.H1. html:yTw!oVQzeromoMdes# b>
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