Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion

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1、NuclearPhysicsB277(1986)456-486North-Holland,AmsterdamSOLUTIONOFTHEANOMALYPUZZLEINSUSYGAUGETHEORIESANDTHEWILSONOPERATOREXPANSIONM.A.SHIFMAN*andA.I.VAINSHTEINInstituteofNuclearPhysics,Novosibirsk90,USSRReceived23April1986Thepresentpapercompletesaseriesofworksonflfunctionsandth

2、eanomalyprobleminsupersymmetrictheories.Exactexpressionsfortheflfunctionsareobtainedwithintheframeworkofstandardperturbationtheory.ThekeyobservationisthattheWilsoneffectiveactionSw(~)doesnotcoincidewiththesumofvacuumloopsintheexternalfieldF(/O.Thedifferenceisduetoinfraredeffe

3、cts.Thecoefficient1/g2infrontoftheoperatorW2inSwisrenormalizedonlyatone-looplevel(extensionofthenon-renormalizationtheoremforF-terms).Thisfactresultsintheone-loopformoftheanomalousoperatorequationforthesupercurrent(generalizationoftheAdler-Bardeentheorem).ThefullGell-Mann-Low

4、functionemergesafterpassingtomatrixelementsoftheoperators.Thequantityenteringobservableamplitudesdiffersfrom1/g2by5"ilnZiwherethefactorsZidescriberenormalizationofthefields.(InthissensetheZfactorsofthematterfieldsbecomeobservable.)Wediscusstherelationwithcalculationsoftheinst

5、antontype.1.IntroductionItiswell-knownthattheperturbativeseriesinsupersymmetric(SUSY)models[1]possessmiraculousproperties.Thus,forFtermstheloopcorrectionsareabsentatall(theso-callednon-renormalizationtheorems[2]),whiletheGell-Mann-LowfunctionsinN=2gaugetheoriesareexhaustedbyt

6、hefirstloop[3].InthepresentworkwewilldiscusscalculationoftheeffectiveactioninN=1SUSYgaugetheories,i.e.inparticular,thegaugecouplingconstantrenormalization.Intheliteraturetheterm"effectiveaction"isused,actually,intwodistinctsenses.Accordingtooneprocedurewecalculatevacuumloopsi

7、nexternal(back-ground)fields.ThefunctionaloftheexternalfieldsobtainedinthiswayF(/~)isoftencalled"effectiveaction",althoughmorepreciseisanothername-thegenera-torofone-particle-irreduciblevertices.Wewillsticktothelatterterminology.Thesecondconstructionisacalculationoftheeffecti

8、veactionSw(~t)hlaWilson[4].ThedifferencebetweenSw(/~)andY(#)isduetot