Solving the Bethe-Salpeter Equation in Minkowski S

SolvingtheBethe-SalpeterEquationinMinkowskiSpaceScalarTheoriesandBeyondarXiv:hep-ph/9601364v130Jan1996ADP-96-1/T206,hep-ph/9601364TalkgivenatJointJapan-AustraliaWorkshoponQuarks,HadronsandNuclei,Adelaide,SouthAustralia,November15-24,1995(Toappearintheconferenceproceedings)SolvingtheBethe-SalpeterEquationinMinkowskiSpace:ScalarTheoriesandBeyondK.Kusaka,K.M.SimpsonandA.G.WilliamsDepartmentofPhysicsandMathematicalPhysics,UniversityofAdelaide,S.Aust5005,AustraliaAbstractTheBethe-Salpeterequation(BSE)forboundstatesinscalartheoriesisre-formulatedandsolvedintermsofageneralizedspectralrepresentationdirectlyinMinkowskispace.Thisdi?ersfromtheconventionalapproach,wheretheBSEissolvedinEuclideanspaceafteraWickrotation.Forallbutthelowest-order(i.e.,ladder)approximationtothescatteringkernel,thenaiveWickrotationisinvalid.OurapproachgeneratesthevertexfunctionandBethe-Salpeteramplitudefortheentireallowedrangeofmomenta,whereasthesecannotbedirectlyobtainedfromtheEuclideanspacesolution.OurmethodisquitegeneralandcanbeappliedevenincaseswheretheWickrotationisnotpossible.E-mail:kkusaka,ksimpson,awilliam@(BSE)[1]describesthe2-bodycomponentofbound-statestructurerelativisticallyandinthelanguageofQuantumFieldTheory(foranextensivereview,seeRef.[2];Ref.[3]isanexhaustivelistofBSEliteraturepriorto1988).Ithasapplicationsin,forexample,calculationofelectromagneticformfactorsof2-bodyboundstatesandrelativistic2-bodyboundstatespectraandwavefunctions.BSEshavebeensolvedanalyticallyforseparablekernelsandforscatteringkernelsintheladderapproximation.SolutionsforBSEshavealsobeenobtainedforQCD-basedmodelsofmesonstructureinEuclideanspace;thesesolutionsmustbeanalyticallycon-tinuedbacktoMinkowskispace.ItisimportanttonotethatanalyticalcontinuationbacktoMinkowskispacefromtheEuclideanspacesolutionisquitedi?cultevenforthesimplecaseofconstituentsinteractingviasimpleparticleexchangeintheladderapproximationtothescatteringkernel.Inparticular,BSamplitudeswithtimelikemomentacannotbeunambiguouslyobtainedfromtheEuclideanspacesolutionwithoutsolvingfurthersin-gularintegralequations.Furthermore,foranyBSEwithanon-ladderscatteringkerneland/orwithdressedpropagatorsfortheconstituentparticles,theproperimplementa-tionofthisprocedure(knownastheWickrotation[4])itselfishighlynon-trivial.ForthesetworeasonsthedirectsolutionoftheMinkowskispaceBSEispreferable.Hereweoutlinesuchamethodforscalartheories,basedonthePerturbationTheoreticIntegralRepresentation(PTIR)ofNakanishi[5].ThePTIRisageneralisationofthespectralrepresentationfor2-pointGreen’sfunc-tionston-pointfunctions.Ageneraln-pointfunctionmaybewrittenasanintegraloveraweightdistribution,whichcontainscontributionsfromgraphsatallordersinpertur-bationtheory.Asanygraphwithn?xedexternallegscanbewritteninPTIRform,thismustalsobetrueofanysumofsuchgraphs.HencethePTIRforaparticularrenor-malisedn-pointfunctionisanintegralrepresentationofthecorrespondingin?nitesumofFeynmangraphswithn?xedexternallegs.Thescalar-scalarBSEhasbeensolvednumericallyintheladderapproximationaftertheWickrotationbyLindenandMitter[6].HerewepresentMinkowskispacesolutionstotheladderBSE,whichwillactasacheckofourimplementationoftheapproachtobeusedhere.OurnumericalsolutionsareobtainedbyusingthePTIRtotransformtheequationfortheproperbound-statevertex,whichisanintegralequationinvolvingcomplexdistributions,intoarealintegralequation.Thisequationmaythenbesolvednumericallyforanarbitraryscatteringkernel[11].Wewillrestrictourconsiderationofexplicitnumericalsolutionstotheladderapproximation,althoughtheapproachisacompletelygeneralone.Calculationsfornon-ladderkernelsareunderwayandtheseresultswillbepresentedelsewhere[12].Asaspeci?cexampleofascalartheorytowhichourformalismmaybeapplied,considertheφ2σmodel,whichhasaLagrangiandensityL=12(?μσ?μσ?m2σσ2)?gφ2σ,(1)wheregistheφ-σcouplingconstant.2FormalismandPTIRTheBethe-Salpeterequationinmomentumspaceforascalar-scalarboundstatewithscalarexchangeisΦ(p,P)=?D(p1)D(p2)d4q2=η2,andsohenceforththesevaluesoftheηiwillbeused.Thisnotationisused,forexample,byItzyksonandZuber[7].ThequantitiesD(pi)arethepropagatorsforthescalarconstituents.Wewillusefreepropagatorshereforsimplicity,althoughwecouldincludearbitrarynonperturbativepropagatorsbymakinguseoftheirspectralrepresentation[8]:D(q2)=?1α?q2?i?,(3)where(m+μ)istheinvariantmassofthe?rstthresholdinD.Itisrelativelystraight-forwardtogeneralisethediscussionbelowtoincludeρφ(α)=0.Wenowrede?nethescatteringkernelKsuchthatK(p,q;P)=iI(p,q;P),andrewritethemomentaoftheconstituentsintermsoftherelativemomentumpandthebound-statemomentumP.TheBethe-SalpeterequationbecomesΦ(p,P)=D(P2?p)d4q[α?(q2+zq·P+P22(m+μ)2+1+zP22,1?zP22+1+zisgivenbythes-wavevertexmultipliedbytheappropriatesolidharmonic[9].Thesolidharmonicisanlthorderpolynomialofitsarguments,andcanbewrittenasYml(p)=|p|lYml(?p),(7)withYml(?p)beingtheordinarysphericalharmonicforangularmomentumquantumnum-berslandmandwhere?p≡p/|p|isaunitvector.Wehaveintroducedadummyparametern,whichwill