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NuclearPhysicsB277(1986)456-486North-Holland,AmsterdamSOLUTIONOFTHEANOMALYPUZZLEINSUSYGAUGETHEORIESANDTHEWILSONOPERATOREXPANSIONM.A.SHIFMAN*andA.I.VAINSHTEINInstituteofNuclearPhysics,Novosibirsk90,USSRReceived23April1986Thepresentpapercompletesaseriesofworksonflfunctionsandtheanomalyprobleminsupersymmetrictheories.Exactexpressionsfortheflfunctionsareobtainedwithintheframeworkofstandardperturbationtheory.ThekeyobservationisthattheWilsoneffectiveactionSw(~)doesnotcoincidewiththesumofvacuumloopsintheexternalfieldF(/O.Thedifferenceisduetoinfraredeffects.Thecoefficient1/g2infrontoftheoperatorW2inSwisrenormalizedonlyatone-looplevel(extensionofthenon-renormalizationtheoremforF-terms).Thisfactresultsintheone-loopformoftheanomalousoperatorequationforthesupercurrent(generalizationoftheAdler-Bardeentheorem).ThefullGell-Mann-Lowfunctionemergesafterpassingtomatrixelementsoftheoperators.Thequantityenteringobservableamplitudesdiffersfrom1/g2by5"ilnZiwherethefactorsZidescriberenormalizationofthefields.(InthissensetheZfactorsofthematterfieldsbecomeobservable.)Wediscusstherelationwithcalculationsoftheinstantontype.1.IntroductionItiswell-knownthattheperturbativeseriesinsupersymmetric(SUSY)models[1]possessmiraculousproperties.Thus,forFtermstheloopcorrectionsareabsentatall(theso-callednon-renormalizationtheorems[2]),whiletheGell-Mann-LowfunctionsinN=2gaugetheoriesareexhaustedbythefirstloop[3].InthepresentworkwewilldiscusscalculationoftheeffectiveactioninN=1SUSYgaugetheories,i.e.inparticular,thegaugecouplingconstantrenormalization.Intheliteraturetheterm"effectiveaction"isused,actually,intwodistinctsenses.Accordingtooneprocedurewecalculatevacuumloopsinexternal(back-ground)fields.ThefunctionaloftheexternalfieldsobtainedinthiswayF(/~)isoftencalled"effectiveaction",althoughmorepreciseisanothername-thegenera-torofone-particle-irreduciblevertices.Wewillsticktothelatterterminology.ThesecondconstructionisacalculationoftheeffectiveactionSw(~t)hlaWilson[4].ThedifferencebetweenSw(/~)andY(#)isduetothefactthatinthevacuumloopsforSwwekeeponlythecontributionofvirtualmomentap>/~.TheactionSw(~t)*ITEP,Moscow117259.0550-3213/86/$03.50@ElsevierSciencePublishersB.V.(North-HollandPhysicsPublishingDivision) M.A.Shifman,A.LVainshtein/Anomalypuzzle457isthenormalactionwithrespecttothelow-frequencyfields.ThesubscriptWintroducedaboveemphasizesthedistinctionbetweenthetwonotations.Thus,withintheframeworkoftheWilsonprocedurewedealwiththenormaloperatorproductexpansion.InordertopassfromSwtoFonemusttakematrixelementsofexp(iSw(/~))eir(~)=(eiSw(~)).Letusemphasizethatthedifferencebetweenthetwodefinitionsisduetothecontributionoftheinfrareddomainp>m.Themaximalvalueof/~isequaltoMo,theultravioletcutoffparameter.Atthispointtheaction(4)isjusttheoriginalSQEDactionandthecoefficients1/eZ(Mo),Z(Mo)arebareparameters.Forarbitrary/~thecoefficientfunctionsaredeterminedbynormalgraphsofperturbationtheoryconstructedstartingfromtheoriginalactionwiththefollowingconstraint-andthisisthemostcrucialpointforus-thattheintegrationdomainovermomentakinallloopsislimitedbythecondition/~"-7]'>+soz7-,(22)wherefortheZfactorwehaveZ(-~)=1-(1+½f)a°lnM°rr/~(23)NoticethatZdependsonthegaugeofthephotonfieldwhosepropagatorischosenintheform~=e2(_g~.+~--~-)--~'k~k~~1InthisapproximationF(/t)superficiallycoincideswithSw(/~)sincethephotonicmatrixelementof(Z-1)(~rp)*~,cpistobetakenintoaccountonlyinthetwo-looporder.Letusproceednowtoatwo-loopanalysis.Inthetwo-loopapproximationthecoefficientinfrontofFainSwisdeterminedbythediagramoffig.1.Letussingle 464M.A.Shifman,A.I.Vainshtein/AnomalypuzzleFig.1.Thetwo-loopcontributiontoSw(~)inscalarelectrodynamics.Thesolidline-thescalarparticlepropagatorintheexternalfield;wavyline-thephotonpropagator.Fig.2.Bycuttingoffthephotonlineinfig.1wearriveatthephotonpolarizationoperator/-/~,,.WeareinterestedinthecoefficientinfrontofF~aF~intheoperatorexpansionforH,,.outintegrationoverthevirtualphotonandperformitattheveryend.Then,beforethislastintegration,calculationofSw(/~)isequivalenttoacalculationofthephotonpolarizationoperatorinoneloop(fig.2).Morestrictly,oneneedstofindonlyonetermintheoperatorexpansionforH,.,namelyCF.BF.~.Thecoefficientofthistermisfiniteandwell-defined.Thenthelastintegrationoverthephotonmomentumkwillyieldalogarithmicintegralofthetypefd4k/k4whichcanbesimplycutofffromaboveatM0andfrombelowat/~.(Forfurtherdetailsseeref.[27].)Specifically,inthexrepresentation~(=)eff=-2IfMId4xi~.~,(x)(F2)()X(24);-~.(53)Hereallquantitiesarematricesinthecolourspace,(p~,).b=i(8.bov+sf:~bAc"""y]~=_GL-ab__facbGcTosimplifytheexpressionswehavescrappedthesubscript"ext"fortheexternalfields.Theformalismweexploitisexplicitlygaugeinvariantwithrespecttotheexternalfield.Asfarasthequantumfieldgaugeisconcerned,eq.(53)impliestheFeynmangauge,.~ggf..=-~(~,a,)1a2.Needlesstosay,thefinalanswershouldbeindependentofthequantumfieldgauge.Wewillreturntothisissuelater.Now,letusexpandthepropagatorineq.(53)inpowersofG/P2.ThezerothordertermsinGdropsoutbecauseofcontractionwith%~r8.Thesecond-orderterm111containstwoG'scontractedoveroneindex;hencethereisnowayofgettingtheonlystructure,q~,G,q3G,#~,determiningthelongitudinalpartofKz.Thus,weareleftwiththelinearinGterm:11=-8g2Trco,our(xlPr~Gz¢-~lx).(54) M.A.Shifman,A.I.Vainshtein/Anomalypuzzle475Insteadofdirectcomputationof(54)(whichpresentsnodifficulties,though)onecancomparethisexpressionwiththematrixelementofthespinoraxialcurrenta,,whoseanomalyiswell-known.Forthespinorcurrent1(at`)=-½ig2Trco,o~Trspin(x17t`Y5--Ix)1=-½ig2Trco,ourTrspin(XlYt`Y5~p2_~G,~#o,~Ix>"Asinthepreviouscaseonecanconvinceoneselfthatinthelongitudinalpartof(at`)onlythelinearinoGtermsurvives,11(55)Asimpleinspectionofeqs.(54)and(55)showsthepresenceinjustthesamepoleasin(at,>,(qJq2)GG,butwithanadditionalfactor4.Sincea'a<°3~at'>-4~r(G;~G~)ext'(561wegetfor~-<0t`K/.t>=~,-t`vt`t,lex,~"{'-wherewehaveaddedtheunittermfromtheclassicalpart.Eqs.(56)and(57)implyfortheGcomponentofW2(seeeq.(49)):T(G)a~T(G)a(TrW21G)=W2aextf~1+--(58)=Ic+2~-)"Now,keepinginmindsupersymmetry,weseethattheone-looppieceofrelation(46)isreproduced.Whatremainstobedoneistodemonstrateindependenceofthequantumfieldgauge.Inarbitrarygaugethepropagatorofthefielda~hastheform-@t,~(x,y)=(x[(P2g~,_2G~,~-~Pt`P~)-l[x>.(59) 476M.A.Shifman,A.LVainshtein/AnomalypuzzleOnecanreadilycheckthefollowingoperatorequalityPt,(p2g~.v-2G~++)=P2Pv-1-i~vayv.(60)Iftheexternalfieldisassumedtosatisfytheequationsofmotion,~vGy~=O,asrequiredintheexternalfieldmethod,thesecondtermonther.h.s,dropsout,andtheGreenfunctionisrepresentableintheclosedform:1]~1N.+(x,y)=(xlez-2G.~+1-,~P,,-~-iP,,Iy).ReturningnowtothematrixelementofK.(seeeq.(53))wewritethe~dependentpartasfollows2~~1(61)Formally,duetogaugeinvariancewithrespecttotheexternalfield,(x]p-4e~lx)mustbeproportionalto-@,Gv,,whichiszero.However,inthekinematicsconsidered(k12=k22=0)theexpression(xlP4pvlx)isill-definedintheinfrared(itcontainsl/k2).Forregularizationonecanintroduceaninfraredmassmtothequantumfielda~.Inotherwords,inallpropagatorsP2~(p2_m2)1.Itisassumedthatm2<>Mv).Withinthistechniquetherearenocorrectionsofsecondandhigherordersin0,a~.TheassertionasitstandsreferstotheamplitudeswithexternalmomentapintheintervalMF>>p>>Mv,i.e.itbearstheoperatorcharacterinourlanguage*.Inref.[20]wehaveextendedthetwo-limittechniquetosupersymmetryandfoundthatthesituationwith0~wasjustthesameaswitha~a~.Inotherwords,0~isexhaustedbyoneloopinthetwo-limitsense.Theanswer,however,didnotsatisfyussinceofmostinterestistheone-limitregularization(evolutiontop<0).Ind=4-e,apartfromW2^,thereexitsanotheroperatorgaugeinvariantwithrespecttotheexternalfield,I'I',whereI"isaconnection,andthedoublecaretaccordingto[21,22]denotesprojectiononthe"additional"edimensions.Theanswerforthetwo-loopdiagramobtainedin[21,22]reducestotheoperator(C/e)fdaxd40~2,andnottotheoperator(C/e)fd4xd20W2appearingintheone-loopgraph.ThentheauthorshaveusedthefactthatinSRDR~,2~,2=_eWZ.Accordingtothepicturedevelopedherethesolutionoftheanomalyproblemdoesnotrequireintroductionoftwoaxialcurrents,twooperatorsGG,etc.More-over,thetwocurrentsconsideredinref.[22]actuallydiffernotbyanultravioletconstant,butbyaninfraredsingularnon-localexpression.Asamanifestation,thedifferenceatAB-a~SSfromref.[22]couldnotbewritteninthelimite~0.Inourlanguagethesituationiseasilyexplainable:inessence,thetwo-loopcomputationofref.[22]isacomputationofthematrixelementoffd20W2withintheSRDRprocedure.Thematrixelementiscompletelysaturatedintheinfrareddomain.AsregardstotheissueofdifferentschemesfortheoperatorGG,themainpointisnotthedistinctionbetweenGGinthedifferentschemesbutthedistinctionbetweentheoperatoranditsmatrixelement.Thelatterisfixedunambiguously.Hereitwillbeinordertoexplaintowhichrenormalizationschemetheflfunctionsquotedineqs.(1),(2),(9)refer.OurdefinitionisclosetotheMOMscheme.Specifically,wefixthegaugecoupling[g2(/Q]forsomeexternalfieldmomentump-/zandexpressg2(~)intermsofthebarechargeandtheultravioletcut-off.Thus,wegetarelationbetweengoandM0.Nosubtractionsaremadeatintermediatestages.Thelatterpointseemstoexplainthedisagreementbetweenourthree-loopcoefficientineq.(2)andthatfoundin[41].Inconclusion,letusmentionthepaper[7]whichpresentstwoperturbativederivationsofeq.(1).Onederivationwasbasedonaninfraredregularizationina 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