On the occurrence of mass in field theory

arXiv:hep-th/0008229v921May2003ONTHEOCCURRENCEOFMASSINFIELDTHEORYGiampieroEspositoINFN,SezionediNapoli,ComplessoUniversitariodiMonteS.Angelo,ViaCintia,Edi-ficioN’,80126Napoli,Italy;e-mail:[e-mail]isiche,Universit`adiNapoli“FedericoII”,ComplessoUniversi-tariodiMonteS.Angelo,ViaCintia,EdificioN’,80126Napoli,ItalyReceived19April2002,revised10June2002ThispaperprovesthatitispossibletobuildaLagrangianforquantumelectrodynamicswhichmakesitexplicitthatthephotonmassiseventuallysettozerointhephysicalpartonobservationalground.Gaugeindependenceisachieveduponconsideringthejointef-fectofgauge-averagingtermandghostfields.ItremainspossibletoobtainacountertermLagrangianwheretheonlynon-gauge-invarianttermisproportionaltothesquareddiver-genceofthepotential,whilethephotonpropagatorinmomentumspacefallsofflikek−2atlargekwhichindeedagreeswithperturbativerenormalizability.TheresultingradiativecorrectionstotheCoulombpotentialinQEDarealsoshowntobegauge-independent.Theexperienceacquiredwithquantumelectrodynamicsisusedtoinvestigatepropertiesandproblemsoftheextensionofsuchideastonon-Abeliangaugetheories.Keywords:quantumelectrodynamics,pathintegrals,perturbativerenormalization.11.INTRODUCTIONAkeytaskoftheoreticalphysicshasbeenalwaysthedescriptionofawidevarietyofnaturalphenomenawithinaunifiedconceptualframework,wheretheycanallbederivedfromafewbasicprincipleswhichhavebeencarefullytestedagainstobservation.Thedevelopmentoflocalornon-localfieldtheories,theinvestigationofperturbativeandnonperturbativeproperties,andtheconstructionofgaugetheoriesoffundamentalinteractionsprovidegoodexamplesofhowsuchataskcanbeaccomplished.Moreover,whenacommonlyacceptedmodelremainsunprovenforalongtime,thetheoreticalphysicisthastoperformacarefulassessmentoftheideasleadingtosuchaprediction,andhe/sheisexpectedtofindeitheranindependentwaytoconfirmit,oranalternativewaytounderstandthephenomenon.Withinthisframework,itistheaimofourpapertoreconsideralongstandingprob-leminparticlephysicsandfieldtheory,i.e.thegenerationofmassingaugetheoriesoffundamentalinteractions.AlthoughtheHiggsmechanismprovidesawellunderstoodthe-oreticalmodelforthegenerationofmass,(1)theanalysisofalternativemodelsappearsnecessaryforatleastafundamentalreason:noconclusiveevidenceontheexistenceoftheHiggsfieldisavailableasyet.Atpresentonecanonlysaythat,fromtheprecisionmeasurementsofthemassoftheWbosonandtheeffectiveleptonicweakmixingangleattheZ-bosonresonance,onefindsa95percentconfidencelevelupperboundontheHiggs-bosonmassgivenbyMH188GeV.(2)Forexample,intheWeinberg–Salammodel,(3−5)theLagrangiandensityL(hereafterweomittheword“density”forsimplicity)containsfivetermsdescribinggaugebosons,thecouplingofgaugebosonstoscalars,thecouplingof2gaugebosonstoleft-handedandright-handedfermions,andthegauge-invariantinterac-tionamongscalarsandfermions,respectively.Inparticular,thecouplingofgaugebosonstoscalarsisdescribedbythetermLGB−S=(Dμφ)†Dμφ−V(φ†φ),(1.1)whereφisaHiggsfieldandthegauge-covariantderivativereadsDμ≡∂μ+ig3Xk=1Wkμτk+ig′W0μτ0.(1.2)Withastandardnotation,WkμaretheSU(2)gaugefieldswithassociatedgeneratorsτk,whileW0μistheU(1)gaugefieldwithgeneratorτ0=121001.Intheunitarygauge,theHiggsfieldisexpressedbythe“columnvector”φ=0eρ,andafterwritingthetransformation(θwbeingtheWeinbergangle)W3μW0μ=cosθwsinθw−sinθwcosθwZμAμ,(1.3)thekineticterminEq.(1.1)readseventually(Dμφ)†Dμφ=g24W1μWμ1+W2μWμ2eρ2+g24ZμZμeρ2cos2θw.(1.4)Thus,thevectormesonsW+,W−andZarefoundtohavesquaremasses12g2eρ2,12g2eρ2and12cos2θwg2eρ2,respectively.FromtheknownexperimentalvalueoftheWeinbergangle,onethenfindsattreelevelmassesmWandmZoforder80GeVand90GeV,respectively.3Nevertheless,sincetheHiggsfieldremainsunobserved,weareledtoaskourselveswhetherthefundamentalprinciplesofquantumfieldtheorymakeitpossibletofittheexperimentaldatawithouthavingtoassumetheexistenceofaHiggsfield.Motivatedbythisoutstandingproblem,Secs.2and3studyanewclassofgauge-averagingfunctionalsinthepathintegralforbosonicgaugetheories,andotheroriginalresultsarederivedinSecs.4–8,whicharedevotedtophotonpropagatorsinquantumelectrodynamics;perturbativerenormalizationofaQEDmodelwherethemassofthephotonissettozeroonlyonobservationalgroundatalaterstage;radiativecorrectionsinQED;masstermsforvectormesonsinnon-Abeliangaugetheory.ConcludingremarksandopenproblemsarepresentedinSec.9.2.GAUGE-AVERAGINGFUNCTIONALSANDGAUGE-FIELDOPERATORSAtthisstage,thefundamentalpointinourinvestigationistheneedtorecallawellknownpropertyofallgaugetheories:sinceaninvariancegroupispresent,theoperatorobtainedfromsecondfunctionalderivativesS,ijoftheclassicalactionSisnotinvertible.ToobtainaninvertibleoperatoronfielddisturbancesonehastoaddtoS,ijatermob-tainedfromthegeneratorsofinfinitesimalgaugetransformationsandtheiradjoints.(6)Inthecorrespondingquantumtheory,thecounterpartofthisconstructionistheadditionofagauge-averaging(alsocalled,morefrequently,gauge-breakingorgauge-fixing)termtothe4originalLagrangianL.(7)TheresultingLagrangianleadstowelldefinedfunctionaldeter-minantsintheone-loopsemiclassicaltheoryandispartofthepath