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Temperature-Dependent Model

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Temperature-Dependent Modelmodel,Model,tem

    Temperature-Dependent Model

    Commun.Theor.Phys.(Beijing,China)47(2007)PP.78_84

    @InternationalAcademicPublishersVo1.47,No.1,January15,2007

    PhaseStructureinaQuarkMassDensity?and?Temperature.DependentModel WENXinJian,,2PENGGuang-Xiong,

    andSHENPengNian

    InstituteofHighEnergyPhysics,theChineseAcademyofSciences,Beijing100049,China GraduateUniversityoftheChineseAcademyofSciences,Beijing100049,China (ReceivedMarch3,2006;RevisedMay19,2006)

    Abstractephasediagramofbulkquarkmatterinequilibriumwithafinitehadronicgasisstudied.D.erent

    frompreviousinvestigations,wetreatthequarkphasewlththequarkmassdensity-and

    temperature-dependentmodel

    totakethestrongquarkinteractionintoaccount,whilethehadronphaseistreatedbyhardcorerepulsionfactor.It

    isfoundthatthephasediagraminthismodelislinseveralaspectsldifferentfromthoseintheconventionalMITbag

    model,especiallyathightemperature.Thenewphasediagramalsohasstrongeffectsonthemassradiusrelationof

    compacthybridstars.

    PACSnumbers:12.38.Mh,21.65.+fj25.75.Nq,26.60.+c

    Keywords:phasediagram,quarkmatter,hadronmatter

    1Introduction

    Strangequarkmatter(SQM)hasbeenaninterest.

    ingtopicsinceitwasproposedtobeapossibleground

    stateofthestronginteraction.Liu,Shaw,andGreiner

    eta1.havearguedthatstrangelets,lumpsofSQM,can

    actasaunique0r.unambiguoussignatureforquark-gluon plasmaformation,theymighthavebeencreatedinthe ultrarelativisticheavyioncollisions.[2-4]Astronomically, SQMalsoplaysimportantroles,e.g.,itcanbeadom

    inantpartinthecoreofdensestars.5,6Jacandidateof

    darkmatter.landapossibleexplanationfortheGZK cut.off.etc.Forarecentreviewonthecontextoftheex- perimentalsearchesforquark-gluonplasma,seeR|ef.f81; forageneralreviewonSQMincompactstars,seeRfef.91;

    andforrecentstudiesonsuperconductivity,seeRefs.f101 and[11_

    ThespecialfeatureofQCDinteractionsistimquark confinement.Usually,oneaddstothesystemthermody- namicpotentialanextraconstant.Thisisthefamous bagmechanism.Manyinvestigationshavebeencarried outwithintheframeworkofthebagmechanism.7,1215]

    Recently,Akimuraeta1.appliedmoleculardynamicssim ulationtoinvestigatingbaryonquarkphasetransition.6J

    wheretheseparationofstrangeness,thesurfaceand Coulombeffects,thefinitesizeeffect,theinfluenceof symmetryenergy,andthekaoncondensationhavebeen studied.Basedontheconservedstrangenessthephase structurehasbeenpresentedbyLeeandHeinzeta1.17J

    IntheisentropicexpansionfromQGPtohadronicmat

    ter,properthermodynamicconditionsforstrangeletsfor

    mationhavebeenpresented.Greinereta1.18gavethe

    distillationofstrangequarkmatterduringthephasetran

    sitionattheconditionofzeroinitialstrangeness.Applica- bly,Burgioeta1.foundthemaximummassinarelatively

narrowinterval(1.4/14o,1.7Mo)forneutronstars.[19

    Anotherapproachistomakethequarkmassesdensity- dependent.Whenthequarknumberdensityapproaches zero,quarkmassesbecomesolargethatthevacuum cannotsupportanyseparatequark.Thisisthequark massdensitydependentmodel(QMDD)orquarkmass

    dependentmodel(QMDTD)at densityandtemperature

    finitetemperature.Inrecentyears,thismodelhasbeen appliedtothedescriptionofquarkmatterandstrangelets successfully.206lForthelatestprogressinthismode1. seeRef.I27I.

    Inthepresentpaper,weadoptthelatestversionof theQMDTDmodelwithanewquarkmassscalingtoin vestigatetheQGPhadronphasediagram.Detailsonthe

    phasediagramandisentropicexpansionaredisCussed.Fi nally,basedontheequationofstatefromthephasedia- gram,wecalculatethepropertiesofhybridstarsandfind thatthestrangenessaffectsthemassradiusrelations.

    ThePaperproceedsasfollows.InSec.2,wein- troducethethermodynamictreatmentwiththedensity. andtemperaturedependentmassinquarkphaseandthe

    Hagedornfactorin1ladronicphaserespectively,andgive theequationofstate.Thephaseequilibriumcondition ipresentedinSec.3.InSec.4wegivethenumerical resultsaboutthephasetransitionwiththequark.density- and,temperature-dependentmassincomparisonwithbag mode1.Naturally,weplottheisentropicexpansionpro- cesswiththeconservedcharges(baryonandstrangeness number).Inthenextsectionwecalculatethemassradius

    relationofhybridstarswithnonzerostrangeness.Ashort

conclusionisshownfinally.

    2EquationofStatewithDensity-Dependent

    Masses

    2.1QuarkPhase

    Followingthepreviouspapers.,weconsiderthe

    bulkquarkmatterasamixtureofthreeflavors,i.e. up,down,andstrangequarks,andtheirantiquarks.In QMDTD,thetotalthermodynamicpotentialdensityis writtenas

    =

    ?i(T,{,TD,i),(1)

    TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNos.103

    75074,90203004,10135030,and10475089

    tEmailaddress:wen~@ihep.ac.ca

    No.1PhaseStructureinaQuarkMassDensity?andTemperature-DependentModel79 wherethesummationindexigoesoveru,d,sflavors.At finitetemperatures,wetreattheanti-quarksasawhole withquarks.Sothethermodynamicpotentialdensityfor flavoriis

    

    27r2n]

    +ln[1+eT])p.dp

    Inthispaper,wedescribethequarkmasswith

    quarkmassdensityandtemperaturedependentmodel,

    i.e.mi=m(nb,T),nb=?ni/3,wherenbandnide-

    notethebaryonandquarknumberdensitiesrespectively. Thismeansthatquarkandantiquarkmassesvarywiththe stateparametersinthenuclearenvironment.Sothequark interaction,tosomeextent,hasbeenincluded,thoughwe

    adoptthenoninteractionstatisticformulas.Wecandi. videthequarkmassintotwoparts.i.e.currentmassand interactingterm,mimio+mz,wheremziscommon fo?difierentflavorquarks.0urquarkmassscalingcanbe 一啪+一唧(A)]

    Hereisthecriticaltemperature,andtheconstant A=LambertW(S1?1.60581199632.Themassscal

    inginEq.(3)hasbeenderivedatzerotemperaturein Ref.122I,andatfinitetemperaturesinRef.271.Itiscon,

    sistentwiththermodynamicsnaturallyandautomatically. Theparticlenumberdensityforeachquarkflavorcall bederivedbythefollowingexpression,

    0Q

    n

    .o#i

    wheretheupindexQisadoptedtodistinguishquark phasefromthehadronphase.Itwillbeimpliedinthe followingsections.

    The

    ,

    p,

    ressurederivationwasinducedintheprevious paper.[28]

    PQ=QO~Oml(5)

    Accordingly,theentropycanbederivedfromessential thermodynamiclawsandisexpressedas

    

    .Omi.Owti

    ThepartialderivativesOmi/OnbandOml/OTareeasily obtainedfromEq.(3),

Onb=

    [一唧(AT)]j(7)n+A,J'.

    

    Omq

    OT[+p(AT)l?一一lpYnI^_=In(【A.TJ"/

    Inliterature,thereisanotherparametrizationforthe quarkmasses,i.e.,

    m,(T)=B0+6(,

    whereaandbareconstantsandgiveninRefs. [29and

    [3O].WhenT0,thisnlassformulacanresultinunrea~ sonablethermodynamicsresults.27]Sointhispaperwe

    investigateitonlyforacomparisonpurpose. 2.2FinitedronieMatter

    Herewestartourtreatmentabouthadronicphaseas aweaklyinteractingmixtureofnonstrangehadrons7r,,

    N,A(1232)andstrangehadronsK+,K,A,?,,Q,and

    theiranti.particles.Thisboundstateparticlespectrumis sufficientforthestudyofthehadronicgas(HG)phase.In ordertoincludethefiniteeigenvolumeofpointlikeha&on weresorttotheHagedorncorrectionfactor.17,31

    Thehadronicchemicalpotentialcanbedescribedby quarkchemicalpotentialaccordingtothequarkcompo

    nentofdistincthadrons,ie

    =?(n:n;),

    q

    (10)

    whereidenotesthehadronicspeciesandqthequarkfla- vor.(n;i)isthenetnumberofthequarkqforthe -

thbaryon.

    Thetotalenergy,andthesystempressure,andtile baryonnumberdensityforha&onphaseareasfollows: =

    ?,

    n=

    ?n

    (11)

    (12)

    Intheaboveequationstheenergy,pressure,andparticle numberforthetthpointhadroncanbeenachievedac

    cordingtoBose-EinsteinandFermi-Diracstatistics. The

    parametergimeansthedegeneracyglspinxisospin. Herethefunctionsofidealhadrongasexcludingrepulsive interactionsbetweenhadrons[32]are

    .

    ptg-

    ~/0..

    p:—三一

    671-2

    p(一

    

    

    2zr2

    p4dp

    gi(e(gi-m)/T-4-11'

    p,)/Te(一士1'

    SPit=-4-g

    i

    .

    {?n[ec?口T

    

    ep(黾一皿t)/T1J

    (14)

    (15)

    (16)

    (17)

    wheret=,andtheuppersignisforferrnions

    andthelowerforbosons.Thenthephysicalpressure

    ,

    baryonnumberdensity,andentropydensityreadbyHage.

    dorncorrectionfactor:

    pH:

    nbH=

    SH:

    

    1+~vtgp/4B?,'

    1+gp./4B

    1

    1+gp./4B

    ?6n,

    (18)

    (19)

    ?,(20)

    where6isthebaryonnumberofthei-thha&on.Please

    note,anoverheadbarlikethatin,,mimeansthecorre

    spondingquantitiesforhadronorinhadronphase,rather

thanaverage.

    ?

    {l??

    WENXin-Jian,PENGGuang-Xiong,andSHENPeng-NianVl01.47 3PhaseDiagramatFiniteTemperatures

    Inthissection,weconsidertheisolatedsystemconsist ingofthehadronicgasphaseandquarkphases.Inthe equilibriouscoexistenceoftwophases,theeigenquan. titiesmustsatisfyGibbsequilibriumlaws,i.e,mechanic equilibriumpQGP=pHG,

    chemicalequilibriamGP:

    .

    ,

    andthermodynamicequilibriumTQGP=THG.In

    theearlyuniverseexpansiontransitionafterabigbang, theweakequilibriumisallowedbyweakinteraction.On thecontrary,inheavy.ioncollisions.aflavorequilibrium duetoweakinteractionispreventedfortheshortcollision timeofstronginteractingscale.Sointhecompactsystem thetotalstrangeness,aswellasthetotalbaryonnumber, areconstant.1JTheflavorsymmetryissupposeddueto thestronginteractioninthewholecollisionprocess. Thefixedstrangenessfractionis

    =

    

    '(21)

    Weassumethatthesystemcontainspositivenumberof strangequarksandthestrangenessfractionisintherange 0<3tofinditsinfluenceonthermodynamicphase

diagram..

    Becausethequarkmassisdensity.dependent.when n-y0thequarkmasswillbecomeinfiniteandthesys. temcanhardlygetthemechanicequilibriumpQGP= PHG.S0wedefinearatioofthehadronphasevolumeto thetotalvolume,iea=V/t.Whenaapproaches

    160

    80,,

    ,

    O

    120

    O

    1,i.e.,thehadronicphaseisthedominatepartinthe system,thenwegettheboundarybetweenmixedphase andh~lrongas.ThistrickissimilartothewaybyHee(

    inRef.[i5].Inthequarkphaseandhadronphasethe baryondensitiesarerespectivelydefinedas=NIVq}

    n=N0/VH.Becausetheparticlescannotescapefrom thesurfaceofthesystem,thetotalbaryonnumbercon. serves.Hereweassumecommonlightquarkschemical potentials"=d.BasedontheGibbsconditionsmen. tionedaboveandtheconservedbaryondensityandthe conservedstrangenessdensity,theequationgroupforthe mixedphasecanbeexpressedas

    P(,,.,)=P(T,,.),(22)

    ;0=n(,,.,)(1a)+(,,.)a,(23)

    t

    .

    o

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