A new treatment of the in-medium chiral condensates

arXiv:hep-ph/0309304v1 26 Sep 2003Anewtreatmentofthein-mediumchiralcondensates

G.X.Penga,b,M.Loewec,U.Lombardod,X.J.Wenb

ab

ChinaCenterofAdvancedScienceandTechnology(WorldLaboratory),Beijing100080,ChinaInstituteofHighEnergyPhysics,ChineseAcademyofSciences,Beijing100039,Chinac

FacultaddeFisica,PontificiaUniversidadCat´olicadeChile,Casilla306,Santiago22,Chiled

DipartimentodiFisica,57CorsoItalia,andINFN-LNS,ViaSantaSofia,9500Catania,Italy

Anewformalismtocalculatethein-mediumchiralcondensateispresented.Atlowerdensities,thisapproachleadstoalinearexpression.Ifwedemandacompatibilitywiththefamousmodel-independentresult,thenthepion-nucleonsigmatermshouldbesixtimestheaveragecurrentmassoflightquarks.QCD-likeinteractionsmayslowthedecreasingbehaviorofthecondensatewithincreasingdensities,comparedwiththelinearextrapolation,ifdensitiesarelowerthantwicethenuclearsaturationdensity.Athigherdensities,thecondensatevanishesinevitably.

Thebehaviorofchiralcondensatesinamediumhasbeenaninterestingtopicinnuclearphysics[1].Apopularmethodtocalculatethein-mediumquarkcondensateistheFeynman-Helmannthe-orem.Themaindifficulty,however,istheas-sumptionswehavetomakeonthederivativesofmodelparameterswithrespecttothequarkcur-rentmass.

Tobypassthisdifficulty,wewillapplyasimilarideaasinthestudyofstrangequarkmatter[2–5]bydefininganequivalentmass.Adifferentialequationwhichdeterminestheequivalentmasswillbederived.Atlowerdensities,thenewfor-malismleadstoalineardecreasingcondensate.Acomparisonwiththeresultinnuclearmatterimpliesthatthepion-nucleonsigmatermshouldbesixtimestheaveragecurrentmassoflightquarks.Athigherdensities,itturnsoutthatthedecreasingspeedofthecondensatewithincreas-ingdensitiesislowered,comparedwiththelinearextrapolation.

TheQCDHamiltoniandensitycanbeschemat-icallywrittenas

󰀅

HQCD=Hk+mi0q¯iqi+HI,(1)

i

Thebasicideaofthemass-density-dependent

modelofquarkmatteristhatthesystemenergycanbeexpressedasinanoninteractingsystem,wherethestronginteractionimpliesavariationofthequarkmasseswithdensity.Inordernottoconfusewithothermassconcepts,letuscallsuchadensity-dependentmassasanequivalentmass.Itcanbeseparatedintotwoparts,i.e.,mi=mi0+mI,

(2)

wherethefirsttermisthequarkcurrentmassandthesecondpartisaflavorindependentinteract-ingpart.Therefore,wewillhaveaHamiltoniandensityoftheform

󰀅

miq¯iqi.(3)Heqv=Hk+

i

WerequirethatthetwoHamiltoniandensities

HeqvandHQCDshouldhavethesameexpecta-tionvalueforanystate|Ψ󰀋,i.e.,󰀊Ψ|Heqv|Ψ󰀋=󰀊Ψ|HQCD|Ψ󰀋.

(4)

Applyingthisequalitytothestate|nB󰀋withbaryonnumberdensitynBandtothevacuumstate|0󰀋,wehave

󰀊Heqv󰀋nB−󰀊Heqv󰀋0=󰀊HQCD󰀋nB−󰀊HQCD󰀋0.(5)Hereweuse󰀊A󰀋nB≡󰀊nB|A|nB󰀋and󰀊A󰀋0≡󰀊0|A|0󰀋foranarbitraryoperatorA.

whereHkisthekineticterm,mi0isthecurrentmassofquarkflavori,andHIistheinteractingpartoftheHamiltonian.Thesumgoesoverallflavorsinvolved.

2

Werestrictourselvestosystemswithuni-formlydistributedparticleswherewecanwrite󰀊Ψ|m(nB)¯qq|Ψ󰀋=m(nB)󰀊Ψ|q¯q|Ψ󰀋.AccordinglywecansolveEq.(5)formI,getting

mǫI=

I

2π2

󰀈

kf0

󰀊mi

󰀐,

(8)

whereg=3(colors)×2(spins)=6isthedegener-acyfactor,and

kf=

󰀍18

8

[x󰀇2

󰀎x󰀇2π2

󰀈

kf00

󰀊

kNnB

󰀅

mf

i0F

i

󰀍

f0

2

nB

g

πkf0

3n=

B

󰀅󰀁m󰀍kiFf

i

NF

f

󰀍

∂λ

H(λ)|Ψ󰀋=

∂

󰀊Ψ|foreachflavori.Applying∂mi0

this󰀏d3xHQCD|Ψ󰀋equality,re-spectively,tothestate|nB󰀋(quarkmatterwithbaryonnumberdensitynB)andtothevacuum|0󰀋,oneobtains󰀊q¯iqi󰀋nB−󰀊q

¯∂ǫ

iqi󰀋0=mi

󰀐

[1+∇mI](17)

with∇≡󰀋

i∂/∂mi0.Notethat∇isadifferen-tialoperatorinmassspace.

ComparingthisequationwithEq.(6)wehave∇m)

I=

ǫI/(3nB∇ǫI

Nf

󰀋

i

f

󰀉

kf0

∂m=

mF(kf/m)−

m0

−1,(20)

0

mIf(kf/m)

󰀊q¯q󰀋nB

ǫI

Nf󰀊q¯q󰀋0

m

󰀐

−m0

m=

0

󰀐

ǫI

3N(23)

fn.B

ThismeansmI=ǫI/(3NfnB),i.e.,ǫI/mI=

3NfnB.SubstitutingthisratiointoEq.(21),weget󰀊q¯q󰀋nB

n∗

(24)

withn∗≡−

1

6m,

(25)0

wheremπ≈140MeVisthepionmassandfπ≈

93.2MeVisthepiondecayconstant.

Sincewehavesaidnothingabouttheformoftheinteractingenergydensity,ourresultismodel

3

independent.Recallingthatthereisamodel-dependentresultinnuclearmatter,i.e.,󰀊q¯q󰀋ρ

∗

Mπ2Fπ

2

ρ∗

withρ≡

2

N0nBv(m0,r¯).(27)

Theaverageinter-quarkdistancer¯islinkedto

densitythroughr¯=ξ/n1/3

b.Hereξisageomet-ricalfactorrelatedtothewayinwhichwegroupthequarkstogether.Inwhatfollows,wehavedi-videdthesystemintosubcubicboxes,beingthenξ=1/31/3.WewilltakeN0=2sinceaquarkhasatrendtointeractstronglywithothertwoquarkstoformabaryon.TheconcretevalueofN0aswellasthevalueofξhaveonlyamarginalinfluenceonthedensitybehaviorofthechiralcondensate.

SubstitutingEq.(27)intoEqs.(21)and(22),wehave,respectively,󰀊q¯q󰀋nB

nB

2Nf

m,

(28)

ImF

󰀍kf

NFf

󰀍kf0

2Nv(mf0,nB).(29)

4

IftheparameterNn1/3

0divergesfasterthankforB

atextremelyhigherdensities,wehave

nlim,r¯)=0.

(30)

B→∞

v(m0whichisconsistentwithasymptoticfreedom.TosolveEq.(20),weneedaninitialconditionatm0=m∗0.Letussupposeittobem(m∗0,nB)=m(nB).(31)

Usually,wewillhavev(m0,nB)|m0=m∗0

=v(¯r),(32)wherev(¯r)istheinter-quarkinteractionforthespecialvaluem∗0ofthequarkcurrentmassm0.Eq.(20)isdifficulttosolveanalytically.How-ever,thiscanbedoneatlowerdensities.Let’srewriteEq.(19)asmI=

N0f

Atlower󰀉

v(m0,r¯)

k.(33)

f0

∂m0

densities,theFermimomentumkfissmall,sothefunctionF(x)approachesto1.NAccordingly,fromEq.(29)wegetmI=

0

0

2Nv(¯r)+

1−

1

f

󰀈

mm∗0

󰀁

m0

󰀐󰀂

dm0.(34)

Ingeneral,anexplicitanalyticalsolutionfor

thecondensateisnotavailable,andwehavetoperformnumericalcalculations.Foragiveninter-quarkinteractionv(¯r),wecanfirstsolveEq.(29)toobtaintheinitialconditioninEq.(31)fortheequivalentmass,thensolvethedifferentialEq.(20),andfinallycalculatethequarkcondensatethroughEq.(28).

Therearevariousexpressionsforv(¯r)inlitera-ture,e.g.,theCornellpotential[12],theRichard-sonpotential[13],theso-calledQCDpotentials[14,15],etc.Theyareallflavor-independent.Let’stakeaQCD-likeinteractionoftheformv(¯r)=σr¯−

4

r¯

.

(35)

Thefirsttermσr¯isthelong-rangeconfiningpart.Thesecondtermincorporatesperturbativeef-fects.Tosecondorderinperturbationtheory,onehas[14,15]α4π

lnλ(¯r)

s(¯r)=b20

λ(¯r)

󰀂

(36)

where[16]λ(¯r)≡ln[(¯rΛ

ms,

andb.TheQCDscalepa-rameterisusuallytakentobeΛ

1.00.9 b=100.8 b=20 b=300.7 linearextrapolation0>0.6qqq0.4q<0.30.20.10.00.00.10.20.30.40.5n-3B (fm)Figure1.Densitydependenceofthequarkcon-densateinquarkmatter.A.WhyshouldtheeffectiveFermimo-mentumbeboostedInthisappendix,weshowthattheeffectiveFermimomentumintheequivalentmassap-proachshouldbeboostedtoahighervalue.Westartfromd(VE)=Td(VS)−PdV+µd(Vn),(38)whichisthecombinationofthefirstandsecondlawsofthermodynamics.Herenistheparticlenumberdensity,Eistheenergydensity,andSistheentropydensity.Becausethesystemisuni-formlydistributed,thecorrespondingextensivequantitiesare,respectively,Vn,VE,andVS.µisthechemicalpotential.FromthisexpressionwecangetT=dEdV󰀃󰀃󰀃󰀃=0,(40)S,n5

µ=

dE

f

2π2

󰀈

k0󰀇

16π2

󰀁kf

󰀊m

󰀐󰀂

,(45)

where󰀇theFermimomentumkfsatisfieskf≡∂kf

dkf+

∂E

2π2dkf

+

g∗m

󰀇

3

g∗

n

󰀐1/.(47)

Eq.(47)isthewell-knownexpressionforthenon-interactingsystem.However,inthemass-density-dependentcasewhereinteractionsaretreatednon-pertutbativelybydefininganequiv-alentmass,thequarknumberdensityshouldbegivenbyintegratingoverbothsidesofEq.(46):ng∗k3=

f

󰀈󰀄m2sh−1

(kf/m)

4π2kf−k2f

+m2󰀆mdm(48)Usuallytheequivalentmassisabigquantity,

muchlargerthanthecurrentmass.Therefore,

6

theratiokf/missmallifthedensitiesarenottoohigh.LetusthenexpandtheintegrandofthesecondtermontherighthandsideofEq.(48)withrespecttokf/m,takingthenthelowestor-derterm.Weget:

󰀈3

kfg∗

n=

6π2

Zkf

.(50)

ComparingEqs.(47)and(51),itisobvious

that,forthesamedensity,theFermimomentumoftheinteractingsystemisdifferentfromthatofthenon-interactingcase.WhentakingZ=3/2,Eq.(51)becomesEq.(9).Acknowledgements

ThesupportfromtheNationalNaturalScienceFoundationofChina(19905011and10135030),FONDECYT,Chile,(Proyectos3010059and1010976),theCASpresidentfoundation(E-26),andtheBES-BEPCfund(G6501)aregratefullyacknowledged.REFERENCES

1.See,e.g.,G.E.BrownandM.Rho,Phys.Rep.

363(2002)85;M.C.Birse,J.Phys.G20(1994)1537.

2.G.X.Pengetal.Phys.Rev.C62(2000)

025801;C61(2000)015201;C59(1999)3452;C56(1997)491.

withCbeingaconstant.Tobeconsistentwiththelinearconfinement,theexponentZisequalto11.However,toreproducethepresentlyacceptedvalueforthepion-nucleonterm(about45MeV),Zshouldbeabout3/2.SubstitutingEq.(50)intoEq.(49)thengives

󰀍󰀐1/3

6

kf=.(51)

1−Z/3