Pauli-blocking-Effect-in-a-Quark-Model

arXiv:hep-ph/0209247v121Sep2002Pauli-blockingEffectinaQuarkModelSachikoTakeuchiJapanCollegeofSocialWork,Kiyose,JapanKiyotakaShimizuDepartmentofPhysics,SophiaUniversity,Chiyoda-ku,Tokyo,Japan(Dated:February1,2008)Pauli-Blockingeffectonthekinetictermisinvestigatedbyemployingthequarkclustermodel.Theeffectcanbeunderstoodbythechangeofthedegreesofthemixingbetweentheincomingwaveandthe0ℓstateoftheinter-clusterwavefunction,whichcanbeexpressedbyapotentialwhichishighlynonlocal.Welookintothepropertiesofthiseffectbycomparingequivalentlocalpotentials.InthechannelwherethePauli-blockingeffectissmall,theon-shellequivalentlocalpotentialsimulatesthenonlocalpotentialwellevenfortheoff-shellbehavior.Ontheotherhand,theoff-shellbehaviorisverydifferentfromtheoriginalonewheretheeffectislarge.Thisoff-shellbehavior,however,canwellbesimulatedbyconsideringthenonlocalmatrixelementsonlybetweenthe0sstateandtheotherstates.Theenergydependentpotentialsarealsoconstructedandfoundtobehelpfultounderstandtheenergydependenceoftheeffect.PACSnumbers:12.39.Jh,13.75.Ev,03.65.Nk,02.30.Zz21.60.Gx,Keywords:Pauli-blocking,Constituentquarkclustermodel,Inversescattering,EquivalentlocalpotentialI.INTRODUCTIONTheclusterstructurecanoftenbeseeninmany-bodysystems.Oneofthefamousexamplesis8Be,wherethe4Hestructurehasanimportantrole[1].Itisalsoknown,thatthequarkclustermodel,wherethebaryoniscon-sideredasaclusterofthreevalencequarks,reproducessuccessfullymanyofthefeaturesofthetwo-baryonsys-tems[2,3,4,5,6].Inthiswork,wefocusourattentiononthePauli-Blockingeffectonthekineticterm.Wetakethequarkclustermodelasanexamplethoughthedefini-tion,theprocedure,andthebasicfeaturediscussedherecanbeappliedtootherclustersystems.ThePauli-Blockingeffectonthekinetictermshouldalwaysbetakenintoaccountregardlessoftheinterac-tionbetweentheconstituentfermions.Inthissense,thiscanbeconsideredasa‘pure’Pauli-Blockingeffectonaconcerningsystem.Actually,aswasreportedinref.[7],inthechannelwherethePauli-blockingeffectplaysanimportantrole,theeffectcanstillbeseeninthephaseshiftsafterthequarkinteractionsareintroduced.Thelow-energypropertiesofthequarkPauli-blockingeffecthasbeenconsideredtocauseastrongrepulsionbe-tweentwobaryonsincertainchannels.It,however,alsoproducesanattractiveforceforahigherenergyregionaswewilldiscussindetailinthispaper.ThePauli-blockingeffectamongthreeormoreclustersmaybecomeimportant[8].Here,however,weconcentrateonthetwo-clustersystems.Inthispaper,weinvestigatethePauli-blockingeffectbyconstructinganonlocalpotentialandbylookingintothephaseshiftsandthewavefunctionsofthesystemswiththispotential.Then,wefurtherinvestigateitsprop-ertiesbycomparingfourkindsoflocalpotentials:1)theonewhichgivesthesamescatteringphaseshiftasthatoftheoriginalnonlocalpotential,2)theonewhichgivesthesameBornphaseshift,3)theenergy-dependentpoten-tialobtainedfromthewavefunction,and4)theenergy-dependentpotentialobtainedbyusingtheWKBmethod.Thecontentsofthisarticleisasfollows.InsectionII,wedefinethenonlocalpotentialwhichexpressesthePauli-blockingeffectonthenonrelativistickineticterm.Forthispurpose,weemploythequarkclustermodelwithoutanyinteractionbetweenquarks.Wewillshowthattheeffectcanbetakenintoaccountbyintroducinganonlocaltwo-baryonpotentialwhichdoesnotdependontheenergy.InsectionIII,weexplainlocalpotentialswhichexpressapartofthefeaturesofthenonlocalpotential.Insec-tionIIIA,wementionthatonecanconstructauniquepotentialwhichhasthesameon-shellbehaviorasthatofagivennonlocalpotential.InsectionIIIB,weconstructalocalpotentialwhichreproducesthesameBornphaseshiftasthatofthenonlocalpotential.Forthislocalpo-tential,theanalyticformcanbeobtained.Inthenexttwosubsections,twokindsoftheenergy-dependentlocalpotentialsarediscussed.Wementionthatapotentialcanbeobtainedbyasimpledivisionbythewavefunc-tioninsectionIIIC.Theenergy-dependentpotentialcanalsobederivedwiththehelptheWKBmethod,whichisdiscussedinsectionIIID.TheresultsareshowninsectionIV.InsectionIVA,weshowthatthePauli-Blockingeffectcanbeexpressedintermsofanonlocalpotential,whichisseparable-like.Itisfoundthattheeffectdoesnotcauseasimpleat-tractionorrepulsion.Thephaseshiftchangesitssignattheenergywhichcorrespondstothekineticenergyoftherelative0ℓstate.AsseeninsectionIVB,thelocalpotentialwhichgivesthesameon-shellbehaviorinthestronglyprohibitedchannelisfoundtohaveadeepat-tractivepocketattheveryshortrange.InthechannelwherethePauli-blockingeffectissmall,orinthehigher2partial-wavechannels,thisequivalent-localpotentialap-proachseemsvalidevenfortheoff-shellbehavior;thewavefunctionisalsosimilartothatoftheoriginalnon-localpotential.Wefindthatthenonlocaltreatmentofthe0s-1scomponentisessentialtoreproducetheoff-shellbehaviorinthestronglyprohibitedchannel,whichisdiscussedinsectionIVC.InsectionIVD,theenergydependenceofthepoten-tialsisinvestigated.Theenergy-dependentpotentialsobtainedbythedivisionbythewavefunctionhavesin-gularitiesatthenodepointsofthewavefunction.They,however,arehelpfultounderstandtheenergydepen-denceoftheshortrangeregion.TheenergydependentpotentialsgivenbytheWKBmethodchangesconsider-ablyastheenergyincreases.Thepotent