Physical Unitarity for Massive Non-abelian Gauge T

arXiv:hep-th/0408168v26Dec2004IFUM-800-FTandMPP-2004-86PhysicalUnitarityforMassiveNon-abelianGaugeTheoriesintheLandauGauge:St¨uckelberg&HiggsRuggeroFerraria1andAndreaQuadrib2aPhys.Dept.UniversityofMilan,viaCeloria16,20133Milan,ItalyI.N.F.N.,sezionediMilanobMax-Planck-Institutf¨urPhysik(Werner-Heisenberg-Institut)F¨ohringerRing,6-D80805M¨unchen,GermanyAbstractWediscusstheproblemofunitarityforYang-MillstheoryintheLandaugaugewithamassterm`alaSt¨uckelberg.Weas-sumethatthetheory(non-renormalizable)makessenseinsomesubtractionscheme(inparticulartheSlavnov-Tayloridentitiesshouldberespected!)andwedevotethepapertothestudyofthespaceoftheunphysicalmodes.Wefindthatthetheoryisuni-taryonlyunderthehypothesisthatthe1-PItwo-pointfunctionofthevectormesonshasnopoles(atp2=0).Thisnormaliza-tionconditionmightberathercrucialintheverydefinitionofthetheory.Withalltheseprovisosthetheoryisunitary.Theproofofunitarityisgivenbothinaformthatallowsadirecttranscrip-tionintermsofFeynmanamplitudes(cuttingrules)andintheoperatorialform.Thesameargumentsandconclusionsapplyverbatimtothecaseofnon-abeliangaugetheorieswherethemassofthevectormesonisgeneratedviaHiggsmechanism.Tothebestofourknowledge,thereisnomentionintheliteratureonthenecessaryconditionimpliedbyphysicalunitarity.1E-mailaddress:[e-mail]2E-mailaddress:[e-mail]11IntroductionThequestforaconsistentnon-abeliangaugetheory[1]ofmassivegaugebosonsisasubjectwithalongandvenerablehistory.Todaythepreferredsolution,combiningunitarityandrenormalizability,isstillthespontaneoussymmetrybreakingmechanismbasedontheintroductionoftheHiggsfield[2].Nevertheless,withinthecontextofnon-power-countingrenormalizablemodels,theSt¨uckelbergmechanism[3]hasbeenrepeatedlyadvocated[4,5]asapossiblealternativeforthegenerationofmassivenon-abelianvectorfields.ThispaperisdevotedtothediscussionofsomecrucialpointsintheproofofPhysicalUnitarityforamassivenon-abeliangaugetheoryinthepresenceofamassterm`alaSt¨uckelberg.Suchtermhasbeenoriginallyintroducedinordertohaveagaugeinvarianttheoryformassivephotons.ItcanbeseenastheresultofanoperatorialgaugetransformationonthefieldsintheProcagauge.Thesameprocedurecanbeenvisagedalsointhecaseofnon-abeliangaugetheories[6]-[11].Whileintheabeliancasethetheoryisrenormalizableandmoreovertheproofofunitarityonthephysicalstatesposesnoproblems,thenon-abeliancaseisfarmorecomplicated.Theoriginofthetroublesismainlythetermgeneratedinthemassbytheoperatorialgaugetransformation:ityieldsanon-polynomiallagrangian.Theeffortsmadeinordertoovercomethesedifficultiesisalonglistinthehistoryofquantumfieldtheory.Alongthislineoneofthefirststepstobeaccomplishedisacloseanalysisoftheproblemofunitarity.Infactevenifthetheoryismadefinitebysomesubtractionscheme,physicalunitaritywillalwaysbeonecrucialitemtobeconsidered.Thepresentworkisaimedatfocusingontheconditionsthathavetobemetinordertoguaranteethisimportantproperty.PreviousattemptstoproveunitaritymadeuseofadirectdiagrammaticstudyoftheFeynmanamplitudesandtheyarelimitedtoone-loopintheperturbativeexpansion[11].TheclassicallagrangianforSU(2)isL=−14GaμνGμνa+m2TrAμ+igΩ∂μΩ†2+LM,Gaμν=∂μAaν−∂νAaμ+ǫabcAbμAcν,(1)wheremistheSt¨uckelbergmassandΩisparameterizedintermsofthe2St¨uckelbergfields~φbymΩ=φ01+i~φ·~τ(2)withφ0=pm2−φ2a.ThediscussionisdevotedtotheparticularformulationprovidedbytheLandaugauge-fixingtermSg.f.=Zd4x2Tr(B∂μAμ−¯c∂μD[A]μc).(3)WechosethisgaugebecausewithatransversevectorfieldpropagatortheFeynmanrulesareparticularlysimple.Inparticulartherearenoimportantout-of-diagonaltermsintheconnectedtwo-pointfunctions.InAppendixAweelaborateuponothercovariantgaugesanddemon-stratethatthesubspaceoftheunphysicalmodesincludesalsodipolefields,asitisknowninpower-countingrenormalizabletheories.Thepresenceofafurtherscalarmode,introducedviatheSt¨uckelbergmassterm,requiresarevisitationofthestandardproofofphysicalunitarityinnon-abeliangaugetheories[12],[13],[14],[15].InparticularadetailedstudyoftheFockspaceisnecessaryinordertoidentifytheunphysicalmodes.TheusualmethodbasedonthestudyofthekerneloftheBRSTcharge[14][15]|Physi∈KerQ/ImQ(4)hastobesupportedbyapreliminarystudyoftheFockspaceofthetheory.Inparticularitappearsthattoomanyfieldsdescribeasymptoticallymass-lessunphysicalmodes(thevectorfield,theNakanishi-Lautrupfield[17][18]andtheSt¨uckelbergfield).Aconditionhastobemetinorderthatthedefinitionofphysicalspaceineq.(4)guaranteesphysicalunitarity.TheLandaugaugerequiresthattheconnectedtwo-pointfunctionforthegaugebosonsistransverseWμνab=δabWT(p2)gμν−pμpνp2.(5)Themainresultofthepaperisthefollowing.Ifonecanmakesenseoutofanon-abeliangaugetheorywithaSt¨uckelbergmassterm,thenthephysicalunitarityissatisfiedprovidedonecanimposethenormalizationconditionWT(0)=limp2=0Wφφp2W2Bφ,(6)3whereφistheSt¨uckelbergfield.Theimportanceoftheresultisquitetransparent:iftheconditioncannotbeenforced,unitarity(forthephysicalstates)islost.Thusthisseemstobetheverycrucialconditiontomeetinordertodefinethetheory.Thesamediscussionandthesamecondition(6)isvalidforamassivenon-abeliangaugetheorywherethemassisgeneratedbytheHiggsmecha-nism.Inthi