Quantization of spontaneously broken gauge theory

arXiv:hep-th/9712053v326Jun1998HD–THEP–97–29,SOGANG–HEP223/97QuantizationofspontaneouslybrokengaugetheorybasedontheBFT–BFVFormalismYong-WanKim∗andYoung-JaiPark†DepartmentofPhysicsandBasicScienceResearchInstituteSogangUniversity,C.P.O.Box1142,Seoul100-611,KoreaandInstitutf¨urTheoretischePhysikUniversit¨atHeidelbergPhilosophenweg16,D-69120Heidelberg,GermanyAbstractWequantizethespontaneouslybrokenabelianU(1)HiggsmodelbyusingtheimprovedBFTandBFVformalisms.WehaveconstructedtheBFTphys-icalfields,andobtainthefirstclassobservablesincludingtheHamiltonianintermsofthesefields.Wehavealsoexplicitlyshownthatthereareexactforminvariancesbetweenthesecondclassandfirstclassquantities.Then,accord-ingtotheBFVformalism,wehavederivedthecorrespondingLagrangianhavingU(1)gaugesymmetry.WealsodiscussattheclassicallevelhowoneeasilygetsthefirstclassLagrangianfromthesymmetry-brokensecondclassLagrangian.PACS:11.10.Ef,11.15.Ex,11.15.-qTypesetusingREVTEX∗e-mail:[e-mail]†e-mail:[e-mail]damentaltheoriesofmodernphysicscanbeconsideredasdescriptionsofdynamicalsystemssubjectedtoconstraints.ThefoundationsforHamiltonianquantizationoftheseconstrainedsystemshavebeenestablishedbyDirac[1].Byrequiringthestrongimplementationofsecondclassconstraints,however,thismethodimpliesDiracbrackets,whosenon-canonicalstructuremayposeseriousproblemsonoperatorlevel.Thismakesitdesirabletoembedthesecondclasstheoryintoafirstclassoneinwhichthecommutatorrelationsremaincanonical.AnexampleisprovidedbytheHiggsmodelwithspontaneoussymmetrybreakdown[2]whosequantizationisusuallycarriedoutinthesocalled“unitary”gauge.Asiswellknown,inthisgaugethemodelisapurelysecondclasssystemcharacterizedbytwosetsofthesecondclassconstraints[3,4].Therequiredstrongimplementationoftheseconstraintsleadstonon-polynomialfielddependentDiracbrackets.Asmentionedabove,onecancircumventtheproblemsassociatedwiththisnon-polynomialdependencebyturningthissystemintoafirstclassonewithausualPoissonbracketstructureinanextendedphasespaceandimplementingthefirstclassconstraintsonthephysicalstates.AsystematicprocedureforachievingthishasbeengivenbyBatalinandFradkin(BF)[5]inthecanonicalformalism,andappliedtovariousmodelsobtainingtheWess-Zumino(WZ)action[6].Inparticular,thisanalysisexplicitlycarriedoutfortheaboveHiggsmodel[4].However,itisalreadyprovedthattheconstructionofthefirstclassHamiltonianintheBFframeworkisnon-trivialevenintheabeliancasebecauseofthefielddependenceontheconstraintalgebra.InthiscasetheonlyweaklyinvolutivefirstclassHamiltonianisobtainedafterthefifthiteration,andthusitdoesnotappearparticularlytobesuitedfortreatingnon-abeliancases.Amoresystematicandtransparentapproachforthisiterativeprocedure,calledBatalin–Fradkin–Tyutin(BFT)formalismwhencombinedwiththeBFone,hasbeendevelopedbyBatalinandTyutin[7].Thisprocedurehasbeenappliedtoseveralinterestingmodels[8,9],wheretheiterativeprocessisterminatedaftertwosteps.Ingeneral,ithasbeen,however,stilldifficulttoapplythisBFTformalismtothenonabeliancase[10].Ontheotherhand,wehaverecentlyimprovedtheBFTformalismbyintroducingthenovelconceptoftheBFT2physicalfieldsconstructedintheextendedphasespace[11]inordertoconstructthestronglyinvolutiveobservablesincludingtheHamiltonian.ThismodifiedversionoftheBFTmethodhasbeensuccessivelyappliedforonlyfindingthefirstclassHamiltonianofseveralnontrivialnonabelianmodels[12–14].Inparticular,theoriginofthesecondclassconstraintsoftheHiggsmodelcomesfromthespontaneoussymmetrybreakingeffects,whilethatofnon-abelianProcamodel[12,13]duetotheexistenceoftheexplicitlysymmetrybrokenmassterm.Therefore,itisveryinterestingtoanalyzetheHiggsmodelhavingthedifferentorigin,whichisphenomenologicallyimportant.InthisletterweshallrevisitthespontaneouslybrokenabelianU(1)HiggsmodelbyfollowingtheconstructiveprocedurebasedontheimprovedBFTversion[11].Insection2,weconvertthesecondclassconstraintsintoafirstclassones,andconstructinsection3theBFTphysicalfieldsintheextendedphasespacecorrespondingtotheoriginalfieldsintheusualphasespace,followingtheimprovedBFTformalism.WethensystematicallyobtainallobservablescontainingthefirstclassHamiltonianasfunctionalsoftheBFTphysicalfieldsshowingtheforminvariancesbetweenthesecondclassandfirstclassquantities.Insection4,throughthestandardpathintegralquantizationestablishedbyBatalin,FradkinandVilkovsky(BFV)[16,17],wederivethegaugeinvariantLagrangian.Insection5,wesuggestanovelpathattheclassicallevelhowonecaneasilygetthisLagrangianfromtheoriginalsecondclassonebysimplyreplacingtheoriginalfieldswiththeBFTonesthroughanadditionalrelationwithoutfollowingusuallycomplicatedpathintegralquantization.Thisnewmethodwillbepossibletoanalyzetherealisticnon-abelianHiggsmodels.Wesummarizeinsection6.II.BFTCONSTRUCTIONOFFIRSTCLASSCONSTRAINTSConsidertheabelianU(1)Higgsmodelintheunitarygauge[3,4],Lu=−14FμνFμν+12g2(ρ+v)2AμAμ+12∂μρ∂μρ+V(ρ),(1)wherethesubscript“u”standsfortheunitarygauge,theHiggspotentialisV(ρ)=12μ2(ρ+v)2−λ4(ρ+v)4,andthefieldstrengthtensorFμν=∂μAν−∂νAμ.Themomentacanonically3conjugatetoA0,Aiandρaregivenbyπ0=0,πi=Fi0,andπρ=˙ρ,resp