example of a gauge field theory

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1、Ref.TH.1571-CERNEXAMPLEOFAGAUGEFIELDTHEORYG.'tHooftandM.Veltman*)CERN—GenevaABSTRACTThetechniquesandresultsofapreviouspaperaredemons-tratedonasimplenon-Abelianmodel.TobepublishedintheProceedingsoftheMarseilleConference,June1972*)OnleavefromtheUniversityofUtrecht,Netherlands.Ref.TH.1571-CERN

2、17October19721.IntroductionThecombinatoricsofrenormalizablemassivevectorfieldtheoriesisquitecomplicated,mainlybecausesomanyverticeshavetobeincludedintheconsiderations.Awayaroundthisdifficultyistheuseofcompositelinesandverygeneralidentitiesbetweendiagrams,andinapreviouspublication1)wehaveuse

3、dsuchmethodstogiveageneralcombinatorialproofoftheequivalenceoftheS-matrixindifferentgauges.Alsorenormalizationproblemswerestudiedinthisway.Inthesecircumstancesanexplicitexamplemaybeverywelcome,andalsoinspiremoreconfidenceinthemethod.Inthisnotewewillconsidersuchanexampleinfulldetail.Thecasew

4、etakeiscase2ofref.2,resemblingmostcloselypuremassiveYang-Millsfields.Inthatexamplethereare,asphysicalparticles,threemassivevectormesonsofequalmassforminganisospintripletandonemassivescalarparticleofisospin0.ThereareseveralsetsofFeynmanrulesallgivingrisetothesameS-matrix.Oneofthegaugesisthep

5、hysicalgaugewheretheonlyinternallinesarethosethatcorrespondtophysicalparticles,anothergaugeshowsthatthetheoryisrenormalizable.Equivalenceofgaugesimpliesthatwehaveaunitaryrenormalizabletheory.Inthisnotewewillmoreorlessfollowthegeneraltreatmentofref.1.Theideaistogiveanexplicitexampleofthevari

6、ousequations.Therenormalizationprocedurewillbecarriedthroughuptooneclosedloop.Themodelcontainsthreeparameters:thevectormesonmass,themassofthe(physical)Higgs-Kibbleparticleandthecouplingconstant.Theseconstantsmustberenormalized,howeverinagaugeindependentway.3.MassivevectorfieldsConsideramass

7、lessYang-Millsisospin1tripletofvectormesonscoupledtoanisospinfieldK(2.1)(2.2)(2.3)ThisLagrangianisinvariant,tofirstorderinthefunctionsforthereplacement(2.4)(2.5)Since(2.5)isaninfinitesimalrotationinI-spinspacethelastandlastbutonetermof(2.1)areobviouslyin