Effective Action of Composite Fields for General G

arXiv:hep-th/9604061v23Jul1996EFFECTIVEACTIONOFCOMPOSITEFIELDSFORGENERALGAUGETHEORIESINBLT–COVARIANTFORMALISMP.M.LAVROV1,S.D.ODINTSOV2DepartmentofMathematicalAnalysis,TomskStatePedagogicalUniversity,Tomsk,634041,RussiaandDept.ECM,Fac.deFisica,UniversidaddeBarcelona,Diagonal647,08028Barcelona,SpainandA.A.RESHETNYAK3QuantumFieldTheoryDepartment,TomskStateUniversity,Tomsk,634050,RussiaAbstractThegaugedependenceoftheeffectiveactionofcompositefieldsforgeneralgaugetheoriesintheframeworkofthequantizationmethodbyBatalin,LavrovandTyutinisstudied.ThecorrespondingWardidentitiesareobtained.Thevariationofcompositefieldseffectiveactionisfoundintermsofnewsetofoperatorsdependingoncompositefield.Thetheoremoftheon-shellgaugefixingindependencefortheeffectiveactionofcompositefieldsinsuchformalismisproven.Briefdiscussionofgravitational-vectorinducedinteractionforMaxwelltheorywithcompositefieldsisgiven.1e-mailaddress:[e-mail]2e-mailaddress:[e-mail]3e-mailaddress:[e-mail]11INTRODUCTIONTheadvancedmethodsofcovariantquantizationforgeneralgaugetheo-riesarebasedeitherontheBRSTsymmetryprinciplerealizedinthewell-knownquantizationschemebyBatalinandVilkovisky[1]orontheextendedBRSTsymmetryprinciplerecentlyrealizedwithinthequantizationmethodbyBatalin,LavrovandTyutin(BLT)[2].ThevariousaspectsandpropertiesofthegaugefieldtheorywithintheBVquantizationhavebeenunderstudyforquitealongtimebynowandmaybeconsideredaswell-knownones(see,forexample,reviews[3,4]).OnthesametimethestudyofpropertiesaswellasvariouspossibilitiesofinterpretationandgeneralizationsofgaugetheoriesintheBLTquantization[2]hasbeenstartedquiterecently[5-16].Followingthelineoftheresearchofrefs.[5-16]presentpaperisdevotedtothestudyofoneofthecentralproblemsarisinginquantumgaugefieldtheorywithintheLa-grangianformalism,i.e.gaugedependenceofgeneratingfunctionalsofGreen’sfunctionsingeneralgaugetheorieswithcompositefields.OurinterestinconsiderationofcompositefieldswithintheBLT-formalismiscausedbytoanumberofreasons.Firstofall,sincethework[17](forareview,see[18])wheretheformalismtostudytheeffectiveaction(EA)forcompositefieldshasbeenintroducedsuchEAisoftenusedtodiscussthedynamicalchiralsymmetrybreakingphenomenonindifferentmodelsusingforexampleSchwinger-Dysonequations.Second,infour-fermionmodels[19]thefermionsformthecompositeboundstateswhichmayplaytheroleofHiggsfieldfordiscussionofdynamicalsymmetrybreakingintheStandardModel(see[20]andreferencestherein).Third,inthemodelsofinflationaryUniversethecomposite2boundstatemayplaytheroleoftheinflaton.Finally,theWilsoneffectiveactionforcompositefieldsmaybeextremelyimportantinrecentstudiesontheexactresultsinSUSYtheories(forareview,see[21]).Thepaperisorganizedasfollows.InSec.2wegiveashortreviewoftheBLTformalism.InSec.3wederivetheWardidentitiesforeffectiveactionofcompositefieldsingeneralgaugetheoriesintheframeworkofthequantizationmethodbyBatalin,LavrovandTyutin.Sec.4isdevotedtothestudyofgaugedependencestructureforcompositefieldsEA.InSec.5theexampleofthiseffectiveactionisconsidered.ConcludingremarksaregiveninSec.6.Inourpaperweusethenotationsofrefs.[2].2BATALIN-LAVROV-TYUTINQUANTIZATIONInthissectionwegiveashortreviewofthemainfeaturesofBLT–quantizationmethodforgeneralgaugetheories.Inordertodothiswestartfromthedefini-tionofgeneralgaugetheories.LetusconsiderthetheoryoffieldsAi(i=1,2,...,n,ε(Ai)=εi)forwhichtheinitialclassicalactionS(A)isinvariantunderthegaugetransformationsδAi=Riα(A)ξα:S,i(A)Riα(A)=0,α=1,2,...,m,0mn,ε(ξα)=ε(α),(1)whereε(ξα)arearbitraryfunctions,andtheRiα(A)aregeneratorsofgaugetransformations.WesupposethesetRiα(A)beingthelinearlyindependent(caseofirreducibletheories)andcomplete.Onecansaythatasaconsequence3oftheconditionofcompletenessthealgebraofgeneratorshasthefollowinggeneralform:Riα,j(A)Rjβ(A)−(−1)εαεβRiβ,j(A)Rjα(A)=−Riγ(A)Fγαβ(A)−S,i(A)Mijαβ(A),(2)whereMijαβsatisfytheconditionsMijαβ=−(−1)εiεjMjiαβ=−(−1)εαεβMijβαThegaugetheorieswhosegeneratorssatisfyEq.(2)arecalledgeneralgaugethe-ories.Asithasalreadybeenmentioned,covariantquantizationofsuchtheories(aswellasreducibleones)intheframeworkofastandardBRSTsymmetryinmodernformhasbeenproposedbyBatalinandVilkovisky[1].ToconstructtheBLT–quantizationschemeitisnecessarytointroducethetotalconfigurationspaceφA.ForirreducibletheoriesthetotalconfigurationspaceφAhasthefollowingformφA=(Ai,Bα,Cαa),ε(φA)=εA.HereCαaisSp(2)–doubletofghost(a=1)andantighost(a=2)fields(Faddeev-Popovfields),Bαareauxiliaryfieldsε(Bα)=εα,ε(Cαa)=εα+1.ForreducibletheoriesthecompletesetoffieldvariablesφAincludesalsopyra-midsoftheghosts,theantighostsandtheLagrangemultiplierswhicharecom-binedintoirreduciblerepresentationsoftheSp(2)–group(formoredetaileddiscussion,see[2]).ForeachfieldφAofthetotalconfigurationspaceoneintroducesthreekindsofantifieldsφ∗Aa,ε(φ∗Aa)=εA+1and¯φA,ε(¯φA)=εA.Theantifieldsφ∗Aa4maybetreatedassourcesofBRSTandantiBRSTtransformations,while¯φAcorrespondstothesourceoftheircombinedtransformation.OnthespaceoffieldsφAandantifieldsφ∗Aaonedefinesoddsymplecticstruc-tures(,)acalledtheextendedantibrackets(F,G)a≡δFδφAδGδφ∗Aa−(F↔G)(−1)(ε(F)+1)(ε(