Bound states in quantum field theory, scalar field

arXiv:hep-ph/9907483v124Jul1999BOUNDSTATESINQUANTUMFIELDTHEORY,SCALARFIELDSG.V.EfimovBogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,141980Dubna,RussiaThemainaimofthispaperistodemonstratethemethodcalled”theBosonizationofNonlocalCurrents”(BNC),usedforcalculationsofboundstatesinaquarkmodel,withinthesimplestrelativisticquantumfieldmodeloftwoscalarfieldswiththeYukawatypeinteraction:L=Φ+(2−M2)Φ+12φ(2−m2)φ+gφΦ+Φ.AsecondaimistoclarifytherelationbetweenBNCandtwowidelyusedmethods,employedinrecentparticlephysicstocalculateboundstatesofinteractingparticles,basedonthenonrelativisticSchr¨odingerequation(theS-method),andtherelativisticBethe-Salpeterequation(theBS-method),andtodeterminetheconditionsonparametersofaquantumfieldmodeldictatingadefinitemethodtobeapplied.Itisshownthatallthesemethodscanbeappliedonlyintheweakcouplingregime(theeffectivedimensionlesscouplingconstantλshouldbelessthan1).Thebasicparameterseparatingtherelativisticandnonrelativisticpicturesisξ=mM,namely,ξ≪1withλ≪1leadstothepotentialpicture,i.e.,theboundstateisdescribedbythenonrelativisticSchr¨odingerequation.Forξ≥1andλ≤1theSchr¨odingerpotentialpictureisnotvalidandtheBethe-SalpeterequationortheBNCshouldbeused,wheretheBNCmethodhasaslightlybitwiderregionofapplicability.PACSnumber(s):03.65.Ge,03.65.Db,03.70.+k,11.10.St,11.15.TkI.INTRODUCTION.Greateffortshavebeenmadetounderstandhowboundstatesariseintheformalismofquantumfieldtheoryandtoworkouteffectivemethodstocalculateallcharacteristicsoftheseboundstates,especiallytheirmasses.Unfortunately,weobservethatthereisnowell-defineduniquemethod,liketheSchr¨odingerequationinnonrelativisticquantummechanics,whichcanbeusedpracticallyforanyproblemsofnonrelativisticquantumphysics.WecanconcludethatQFToftodayisnotwellsuitedtodescribeboundstateproblem,(see,forexample,[1]).Theanalysisofaboundstateissimplestwhentheconstituentparticlescanbeconsideredtobenonrelativistic,i.e.,whentheytravelatspeedsconsiderablylessthanc.Thephysicalevidentcriteriontotellthataboundstateisnonrelativisticisforthebindingenergytobesmallcomparedtotherestenergiesoftheconstituents.Thetheoreticalcriterionisthatthecouplingconstantsshouldtobeweakandmassesofintermediateparticles(photoninQED,mesonsinnuclearphysics,gluonsinQCD)shouldbesmallincomparisonwithmassesofconstituents.Thebestexampleishydrogen-likesystems,whichcanbeconsiderednonrelativistic,andtheexperimentswithgreataccuracywererequiredtodevelopthemethodstocalculatenextrelativisticcorrections(see,forexample,[1–3]).Thesituationinnuclearandparticlephysicsiscompletelydifferent.First,thecouplingconstantsarenotsmallanymore.Second,innuclearphysics,althoughthebindingenergyisrelativelysmallincomparisonwithnucleonmasses,themassesofintermediatemesonsrealizingthestrongnuclearinteractionarenotsmall.Inparticular,themostadequatedescriptionofthedeuteroncanbedonebytheBethe-Salpeterequationwherethecontributionofalllightmesonsshouldbetakenintoaccount(see,forexample,[4]).Inparticlephysics,onlyhadronsmadeoutofheavyquarkscanbeconsideredbythenonrelativisticpotentialmeth-odsalthoughthedecaysofheavyhadronsintolightonesrequirerelativisticmethodstodescribethesetransformations.Themostfamiliarlight-quarkstatesareintrinsicallyrelativistic,sothattheyrequirepurerelativisticmethods.Be-sides,theyareconstitutedatdistanceswheretheconfinementphenomenonshouldbetakenintoaccount.Inaddition,onemayaskwhetherthefreeDiracequationapplicabletodescribethelightquarksinthisregion?Therefore,usingthenonrelativisticSchr¨odingerequationtodescribethelight-quarksystemsbyfittingparametersofpotentials(see,forexample,[5,6])canbeconsideredheroicattempttounderstandlightmesonphysicsbyunsuitablemethodsinaveryroughtheoreticalapproximation.Inrelativisticquantumfieldtheory,boundstatesareidentifiedbytheoccurrenceofpolesofcorrespondingampli-tudesorGreenfunctionswithappropriatequantumnumbers.Thesepoleshaveanonperturbativecharacter,sothattheycanariseasaresultofanonperturbativerearrangementofseriesoveracouplingconstant.TheinvestigationofnonperturbativepropertieswasdonebyestablishingintegralequationsamongamplitudesandGreenfunctions,using1thespecificstructureofaLagrangian.Oneshouldsaythattheseequations,havingabsolutelygeneralform,inrealitycanbeusedwhenthekernelscontaincontributionsofthelowestFeynmandiagramsonly.Itimpliesthatinsomesensethecouplingconstantshouldbesmallenough.TheBethe-Salpeterequationisthemostimportantintegralequationofthistypeanditiswidelyused,especiallyforcalculationofrelativisticcorrectionsinhydrogen-likesystemsanddeuteronphysics(see,forexample,[1,3,7,9]).Nowwewouldliketopayattentiontotheso-calledZ2=0approach(see,forexample,[8,10])whichisnotusedsowidely,althoughthebosonizationofQCDintroducingbilocalboson-typefieldswasdevelopedin[11].Theobjectionandprejudiceagainstthismethodarebasedonthepersuasionthatinlocalquantumfieldtheoryweshouldhavelocalinteractiononly.Forexample,thepionasaquark-antiquarkboundstateisrepresentedbythelocaltermπ(¯qγ5q).AsaresulttherenormalizationconstantZ2forthepioncontainstheultravioletdivergenceandthereforetheconditionZ2=0makesnopr