Beta Function in Supersymmetric Gauge Theories Instantons Versus Traditional Approach

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1、Volume166B,number3PHYSICSLETTERS16January1986THEBETAFUNCTIONINSUPERSYMMETRICGAUGETHEORIES.INSTANTONSVERSUSTRADITIONALAPPROACHV.A.NOVIKOV,M.A.SHIFMAN,A.I.VAINSHTEIN1andV.I.ZAKHAROVInstttuteofTheorettcalandExpertmentalPhystcs,Moscow117259,USSRRecoved15July1985,revis

2、edmanuscriptrecoved4October1985WithinthebackgroundfieldformahsmwediscussvacuumloopsmsupersymmetrlcgaugetheoriesAdirectconnectionisrevealedbetweentheabsence(orpresence)ofhigh-ordercontributionsandinfraredregularlzatlonAsimpleexplanationisgivenwhythemstantonamplitud

3、easexhaustedbyoneloopwhilstmthestandardsupergraphtechniquetheeffectiveactioncontainstermsofallordersmthecouphngconstantWepresentanexactrelationbetweentheGell-Mann-Lowfunctionandtheanomalousd~mens~onsofmattersuperfieldsstemmingfrommstantoncalculus1.Introduction.Ino

4、urpreviouswork[1]wehaveappearafewindependentZfactors.Thenacertainconsideredapplicationsofinstantonstocalculationsrelationcanbederivedbetweentheexactt-functionoftheGell-Mann-Lowfunctioninsupersymmetricandtheanomalousdimensionsofthematterfields.Ingaugetheories.Inbri

5、ef,theresultsreducetothefol-ref.[1]wehaveusedtherelationtoobtaintwo-looplowing.coefficientsofthet-function.Inthegeneralform,as(i)Thevacuum-to-vacuumamplitudeintheinstan-willbeshownbelow,therelationisasfollows:tonexternalfieldisexactlycalculableandisdeter-ct21mined

6、byoneloop.7(¢))fl(a)=--~((ng--½nf)+~ze~ro(ii)IntheorieswithoneindependentZfactor(su-/modesper-symmetricgluodynamicsanditsextensions)onecanextractfromtheaboveamplitudetheexactt-X[1-(cx/4rO(ng-nx)]-1.(2)function.Moreover,thecoefficientsofthet-functionhaveageometrica

7、lmeaning:Herenxisthenumberofthegluinozeromodeswhile7(¢)standsfortheanomalousdimensionofthemat-fl(~)=da0/dlnM0terfieldscz217(¢)=dInZ(¢)/dInM0.-21r(ng--~nf)1-(a/4rr)(ng-nf)'(1)Notice,thatforeachfennionmatterfield7(¢)enterswhereM0istheultravioletcutoff,ngandnfarethet

8、heRHSofeq.(2)multipliedbythenumberofthenumbersofzeromodes,gluonicandfermionic,respec-correspondingzeromodes.tively.Ontheotherhandtheordinarysupergraphte