Relationship between Quark-Meson Coupling Model and Quantum Hadrodynamics

RelationshipbetweenQuark-MesonCouplingModelandQuantum

Hadrodynamics

KoichiSaito

TohokuCollegeofPharmacy,Sendai981-8558,Japan

arXiv:nucl-th/0207053v1 17 Jul 2002(Received)

Usingthequark-mesoncoupling(QMC)model,westudynuclearmatterfromthepointofviewofquarkdegreesoffreedom.Performingare-definitionofthescalarfieldinmatter,wetransformQMCtoaQHD-typemodelwithanon-linearscalarpotential.ThepotentialsobtainedfromQMCarecomparedwiththoseoftherelativisticmean-fieldmodels.

Sinceatpresentrigorousstudiesofquantumchromodynamics(QCD)arelimitedtomattersystemwithhightemperatureandzerobaryondensity,itisimportanttobuildmodelswhichhelptobridgethediscrepancybetweennuclearphenomenologyandQCD.Wehaveproposedarelativisticquarkmodelfornuclearmatterandfinitenuclei,thatconsistsofnon-overlappingnucleonbagsboundbytheself-consistentexchangeofisoscalar,scalar(σ)andvector(ω)mesonsinmean-fieldapproximation(MFA)–thismodeliscalledthequark-mesoncoupling(QMC)model.1)Ontheotherhand,recenttheoreticalstudiesshowthatvariouspropertiesoffinitenucleicanbeverywelldescribedbytherelativisticmean-field(RMF)models,i.e.,Quantumhadrodynamics(QHD).2)Inthisletter,weconsiderrelationshipbetweenQMCandQHDandstudyhowtheinternalstructureofthenucleonshedsitseffectoneffectivenuclearmodels.

Inourpreviousworks,1)theMITbagmodelhasbeenusedtodescribethequarkstructureofthenucleon.Sincetheconfinedquarksinteractwiththescalarfield,σ,

⋆,inQMCisgivenbyafunctionofσinmatter,theeffectivenucleonmass,MN

throughthequarkmodelofthenucleon.(Althoughthequarksalsointeractwiththeωmeson,ithasnoeffectonthenucleonstructureexceptforashiftinthenucleonenergy.1))The(relativistic)constituentquarkmodel(CQM)isanalternativemodelforthenucleon.Recently,ShenandToki3)haveproposedanewversionofQMC–thequarkmean-field(QMF)model,whereCQMisusedtodescribethenucleon.

Inthepresentstudy,aswellasthebagmodel(BM),wewanttousetherela-tivisticCQMwithconfiningpotentials,V(r),ofasquarewell(SW)andaharmonicoscillator(HO)toseethedependenceofthematterpropertiesonthequarkmodel.Itisassumedthatthelight(uord)quarkmass,mq,is300MeVinCQM,whilemq=0MeVinBM.Furthermore,weintroduceaLorentz-vectortypeconfiningpotential,whichisproportionaltoγ0,aswellasthescalarone:

V(r)=(1+βγ0)U(r),

(1)

wherethepotential,U(r),isgivenbySWorHOandβ(0≤β<1)isaparametertocontrolthestrengthoftheLorentz-vectortypepotential.WeassumethattheshapeoftheLorentz-vectortypeconfiningpotentialisthesameasthatofthescalartype

typesetusingPTPTEX.sty

2one.

Letters

InSW,thesolutionforaquarkfield,ψq,canbecalculateda`laBogolioubov.4)ThepotentialisgivenbyU(r)=0forr≤RandMforr>R,whereRistheradiusofthesphericalwellandMistheheightofthepotentialoutsidethewell.Afterfinishingallcalculations,thelimitM→∞istaken.5)ThissystemmaybedescribedbyLagrangiandensity

¯q(iγ·∂−mq)ψqθ(R−r)−1LSW=ψ

(1−β)(E−mq)

todeterminethequarkenergyisgivenby

󰀇

2cr

2

(ctheoscillatorstrength),acondition

c.

(4)

Thec.m.energycanbeevaluatedexactly,asinthenon-relativisticharmonicoscil-lator,anditisjustonethirdofthetotalenergy.7)Thus,thenucleonmassisgiven

byMN=2E−Eg,whereEgdescribesgluonfluctuationcorrections.7)

FortheMITbagmodeltherearemanygoodreviews.5)InBM,wetakemq=0MeVandβ=0.(EveninBMitispossibletoincludetheLorentz-vectortypepotentialusingEqs.(2)and(3).However,ifweusealargeβinBM,itishardtogetgoodvaluesofthenuclearmatterproperties.)

Nowweconsideraniso-symmetricnuclearmatterwithFermimomentumkF,

3/3π2(ρthenuclearmatterdensity).Then,thetotalwhichisgivenbyρB=2kFB

energypernucleon,Etot,canbewrittenas1)

Etot=

4

⋆2(σ)MN

m2

2󰀬+k+σ

2m2ω

ρB,(5)

⋆iscalculatedbythequarkmodel.Theσandωmesonmasses,mandm,whereMNσω

aretakentobe550MeVand783MeV,respectively.Theωfieldisdeterminedbybaryonnumberconservation:ω=gωρB/m2ω(gωistheω-nucleoncouplingconstant),whilethescalarmean-fieldisgivenbyaself-consistencycondition:(∂Etot/∂σ)=0.1)

Letters3

InSW,wesettheradiusofthepotentialtobeR=0.8fmanddetrminezsoastofitthefreenucleonmass,MN(=939MeV).Theparameterβischosentobe0and0.5toexaminetheeffectoftheLorentz-vectortypeconfiningpotential.Wefindthatz=4.396and5.1forβ=0and0.5,respectively.InHO,therearetwoadjustableparameters,candEg.Wedeterminethoseparameterssoastofitthefree

2=0.6nucleonmassandtheroot-mean-square(charge)radiusofthefreeproton:rN

fm2.8)(rNiscalculatedbythequarkwavefunction.)Wefindthatc=1.591fm−3andEg=344.7MeVforthefreenucleon.Innuclearmatter,wekeepcandEgconstantandthequarkenergyEvaries,dependingonthescalarfield.InBM,thebagconstant,B,andtheparameter,z,arefixedtoreproducethefreenucleonmass.AsinSW,wechoosethebagradiusofthefreenucleontobe0.8fm.WefindB1/4=170.3MeVandz=3.273.1)

Nowweareinapositiontode-⋆

TableI.Couplingconstants,MNandK.Theterminethecouplingconstants:theσ-⋆

2,andg2effectivenucleonmass,MN,iscalculatednucleoncouplingconstant,gσω

atρ0.Thenuclearincompressibility,K,is

arefixedtofitthenuclearbindingen-quotedinMeV.TheSWmodelwithβ=

ergy(−15.7MeV)atthesaturationden-0(0.5)isdenotedbySW0(5).

sity(ρ0=0.15fm−3)fornuclearmatter.22⋆

gσgωMN/MNK

Thecouplingconstantsandsomecalcu-latedpropertiesformatterarelistedinTableI.Thepresentquarkmodelscanprovidegoodvaluesofthenuclearin-compressibility,K.

InSWandBMwithmasslessquarks,thequarkscalardensityinthenucleon1)vanishesinthelimitβ→1,whichmeansthattheσmesondoesnotcoupletothenucleon.9)Thisfactimpliesthatasβislargertheσ-nucleoncouplingisweakerinmatter.Thus,wecanconcludethatqualitativelyalargemixtureoftheLorentz-vectortypeconfiningpotentialleadstoaweakscalarmean-fieldandhencealargeeffectivenucleonmassinnuclearmatter.SinceinMFAasmalleffectivenucleonmass(andhenceastrongscalarfield)isfavorabletofitvariouspropertiesoffinitenuclei,2)theconfiningpotentialincludingastrongLorentz-vectortypeonemaynotbesuitablefordescribinganuclearsystem.

ThemaindifferencebetweenQMCandQHDathadroniclevel1)liesinthedependenceofthenucleonmassonthescalarfieldinmatter.Byperformingare-definitionofthescalarfield,theQMCLagrangiandensity1)canbecastintoaformsimilartoaQHD-typemean-fieldmodel,inwhichthenucleonmassdependsonthescalarfieldlinearly,withself-interactionsofthescalarfield.10)InQMC,the

⋆nucleonmassinmatterisgivenbyafunctionofσ,MN,QMC(σ),throughthequark

modelofthenucleon,whileinQHDthemassdependsonascalarfieldlinearly,⋆MN,QHD=MN−g0φ(φisthescalarfieldinaQHD-typemodel).Hence,totransformQMCintoaQHD-typemodel,wecanapplyare-definitionofthescalarfield,

⋆

g0φ(σ)=MN−MN,QMC(σ),

(6)

toQMC,whereg0isaconstantchosensoastonormalizethescalarfieldφinthe

4Letters

⋆limitσ→0:φ(σ)=σ+O(σ2).Thus,g0isgivenbyg0=−(∂MN,QMC/∂σ)σ=0.In

QMC,wefindg0=gσforSWandBM,whileg0=2

2

󰀆

2

d󰀬r[(∇σ)2+m2σσ]=

󰀆

d󰀬r

󰀁

1

2

2m2σσ(φ)

andh(φ)=

󰀈

∂σ

mσ

NotethatinuniformlydistributednuclearmatterthederivativeterminEscldoes

notcontribute.(TheeffectofthistermonthepropertiesoffinitenucleihasbeenstudiedinRef.10).)NowQMCcanbere-formulatedintermsofthenewscalarfield,φ,anditisofthesameformasQHDwiththenon-linearscalarpotential,Us(φ),andthecoupling,h(φ),tothegradientofthescalarfield.(Notethatsincethisre-definitionofthescalarfielddoesnotconcernthevectorinteraction,theenergyoftheωfield(seeEq.(5))isnotmodified.)

TheZimanyi-Moszkowski(ZM)model11)isagoodexample.Byre-definigthescalarfield,ZMcanbeexactlytransformedtoaQHD-typemodelwithanon-linearpotential.SincetheeffectivenucleonmassinZMisgivenby11)

⋆

=MN,ZM

󰀅

∂φ

󰀊

.(8)

MN

1+g′σ

󰀈

andσ(φ)=

φ

2

m2σ

φ

1−4dφ

Letters5

withd=bgσ/aMN.Thissatisfiesthecondition:σ→0inthelimitφ→0.Thenon-linearpotentialisthuscalculated

Us(φ)=

=

m2σ

d

m2σ

󰀊

,

󰀄

φ3+

5

󰀊2

3

φ4+O(gσ),

(14)

MN

wherer=b/a.

TableII.Parametersa,b,κandλ.

a

b

κ(fm−1)

λ

MN

Thestandardformofthenon-linear

scalarpotentialisusuallygivenbyUs(φ)=

1

6φ3+

λ

6Letters

32.52

Us(φ) (fm)1.510.50-0.5-1-1-0.50φ (fm)

-1

-40.51Fig.1.Non-linearscalarpotentialsgenerated

byQMC.ThesolidcurveshowsUs=

m2σ

φ2.The

dashedcurvewithopen(solid)circlesisforG2(NLB),12)whilethedashedonewith-outanymarksisforPM3.13)ThedashedcurvewithcrossesisforZM.11)Thepo-tentialsinTM114)andNL115)arerespec-tivelyshownbythedottedanddot-dashedcurves.

2

ifthepotentialduetotheinternalstructureofthenucleoncouldbeinferredbyanalyzingexperimentaldatainthefuture.