Color plasma oscillation in strangelets

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

Colorplasmaoscillationinstrangelets

KeiIida

DepartmentofPhysics,UniversityofTokyo,7-3-1Hongo,Bunkyo,Tokyo113-0033,Japan

(February9,2008)

Thedispersionrelationanddampingrateoflongitudinalcolorplasmonsin nitestrangequark

matter(strangelets)areevaluatedinthelimitsofweakcoupling,lowtemperature,andlongwave-

length.ThepropertyoftheQCDvacuumsurroundingastrangeletmakesthefrequencyofthe

plasmonsnearlythesameasthecolorplasmafrequencyofbulkmatter.Theplasmonsaredamped

bytheircouplingwithindividualexcitationsofparticle-holepairsofquarks,ofwhichtheenergy

levelsarediscretizedbytheboundary.Forstrangeletsofmacroscopicsize,thelifetimeoftheplas-

monsisfoundtobeproportionaltothesize,asinthecaseoftheusualplasmaoscillationsinmetal

nanoparticles.

arXiv:nucl-th/9907039v1 9 Jul 1999PACSnumbers:12.38.Mh,25.75.-qI.INTRODUCTIONAtextremelyhightemperatureand/orbaryondensity,quark-gluonplasmas[1]areconsideredtobeenergeticallyfavorableascomparedwithhadronicmatter.Suchextremeconditionspreventusfromcon rmingthepresenceofquark-gluonplasmas,althoughitisexpectedininteriorsofneutronstarsandinultrarelativisticheavy-ioncollisions.Theintriguingpossibilitythatstrangequarkmatter,composedofu,d,andsquarks,mightbethegroundstateofthestronginteractionwaspointedoutbyWitten[2].Ifthisiscorrect, nitestrangequarkmatter,whichisusuallyreferredtoasstrangelets,wouldbeamorestableself-boundsystemthana56Fenucleus.Then,disruptionofneutronstarsandhighlyenergeticheavy-ioncollisionscouldleavebehindmanystrangelets.Thismightallowustodetectquark-gluonplasmasintheformofstrangeletsundervariousexperimentalsituationsTheground-statepropertiesofstrangeletsweredescribedbyFarhiandJa e[6]withintheMITbagmodel; nite-sizee ectssuchassurface,Coulomb,andshelle ectsweretakenintoaccount.Inultrarelativisticheavy-ioncollisions,however,thepossiblyformedstrangeletsshouldbeaccompaniedbythermallyexcitedstates.Amongvariousexci-tations,wetakenoteofspin-and avor-symmetric,longitudinalcolorplasmons[7,8]thatrepresentcollectivecoun-teroscillationsofdi erentcolordegreesoffreedom.Inastrangelet,theseplasmonsareexpectednotonlytoexhibitadispersionrelationdependentonthecolordielectricpropertyofthemedium,butalsotoundergoasize-dependentdampingasinthecaseofanopticallyexcitedmetalclusterornanoparticle.Whenthenanoparticleissmallenoughfortheelectronstobenearlycollisionlessandfortheelectricdipoleradiationtobeoflittlesigni cancebutistoolargefortheshellstructuretotakee ect,thesurfaceplasmonexcitationsaredampedinatimeproportionaltothesizeTheirenergyisresonantlyabsorbedbyindividualexcitationsofparticle-holepairsoftheelectronshavingdiscreteenergylevelsduetothepresenceoftheboundary.Noticethatthequark-gluonplasmainastrangeletisfully

relativisticincontrasttotheelectrongasinananoparticle.Consequently,newfeaturesshouldarisefromspin-orbitinteractionsandfromexcitationsinvolvingantiquarks,bothofwhichmaya ectthelifetimesofthecolorplasmons.Longitudinalcolorplasmaoscillationsinrelativisticanddegeneratebulkmatterofu,d,andsquarkswereinves-tigated[7,8]bycalculatingthecurrent-currentcorrelationfunctionwithintherandom-phaseapproximation(RPA)whichisvalidintheweak-couplingThedispersionlawanddampingfortheseoscillationsareessentiallythesameasthoseexaminedbyJancovici[11]forelectromagneticplasmaoscillationsinarelativisticanddegenerateelec-trongas.Ineithercase,themodepropertiesarecharacterizedbythecorrespondingplasmafrequencyandLandaudampingasinanonrelativisticplasma.Inthispaper,weestimatethelong-wavelength,low-temperaturepropertiesoflongitudinalcoloroscillationmodesinastrangeletofmacroscopicsize,surroundedbytheQCDvacuum,byin-corporatingthediscreteenergylevelsofquarksintothecurrent-currentcorrelationfunctionwithintheRPA.Thevanishingcolordielectricconstantofthevacuumleadstotheabsenceofthesurfacecolorcharge;ifpresent,itwouldmodifythee ectivecolorelectric eldinsidethestrangeletaccordingtoaMie-typeformula.Wethus ndthatthefrequencyofthelongitudinalcolorplasmonsinthestrangeletisalmostthesameasthatinbulkquarkmatter.ThelifetimeoftheplasmonsisdeterminedbythedipoletransitionsfromaquarkstateintheFermiseatoanunoccupiedone.Itisshownthatthequarkparticle-holepairsinvolveddamptheplasmonsatarateinverselyproportionaltothestrangeletsize.BeyondtheRPA,however,theplasmonsarefurtherdampedbycollisionsbetweenquarks.

InSec.II,thelongitudinalpolarizationfunctioniscalculateduptoone-looporder,allowingforthediscretequark

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

statesinsideastrangelet.Theresultingdispersionrelationanddampingrateofthelongitudinalcolorplasmaoscil-lationaredescribedinSecs.IIIandIV,respectively.ConcludingremarksaregiveninSec.V.

II.POLARIZATION

WeconsiderasphericalstrangeletofradiusRandbaryonnumberA,composedofanearlynoninteractinganduniformgasofu,d,andsquarksandembeddedintheQCDvacuum.Thechemicalpotentialµiofquarksof avori,includingtherestmassmi,isassumedtobeindependentofi.Suchindependenceensures avorequilibriumamongthequarks,sincenormallyleptonshaveavanishinglysmallchemicalpotentialinthestrangelet[6].Weworkinunitsinwhichh¯=c=kB=1.Wesetµi≡µ=300MeV,mu=md=0,andms=0–300MeV.ThetemperatureTofthesystemistakentobeintherange0≤T µ.Thezero-temperaturemodelforquarkmatterwillthusbeadoptedbelow.

Letusnowexaminethepropertiesoflongitudinalcoloroscillationmodesinastrangeletbycalculatingthecor-respondingproperpolarizationΠ00intheRPA(one-looporder).AtT=0,Π00canbeexpressedintermsofthe0correlationfunctionofthetimecomponentofthequarkcurrentJα(α=1,...,8)as

Π00(q)=

withi2 200 [Jα(x),Jα(y)] θ(x0 y0),(1)

00¯Jα(x)=λαabψai(x)γψbi(x),

whereq=(ω,q)isthefour-momentumtransfer,V=4πR3/3isthevolumeofthestrangelet,gistherenormalizedcolorcouplingconstant,ψaiisthequark eldofcoloraand avori,λαabaretheSU(3)generators,and ··· denotestheexpectationvalueinthegroundstate.Theright-handsideofEq.(1)isnotsummedoverαandisinvariantunderchangeinα.InthelimitV→∞,thisformofΠ00(q)reducestotheusualformwrittenforbulkuniformmatter.Hereafter,weassumethatthemomentumtransfer|q|isfarsmallerthanR 1.

Theessentialpartofthepresentstudyistoincorporatethediscretequarklevelsintotheone-loopordercalculationsofΠ00.Forsuchcalculations,we rstwritethequark eldintheformofamodeexpansion

0(+)( )0iωkix0ψai(x,x)=[cξκaiΨξκai(x)e iωkix+d Ψ(x)e],(2)ξκaiξκai

ξκ

whereΨξκaiandcξκaiarethespinorandtheannihilationoperatorforthequarkeigenstateofenergyωki=(+)

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

fkκi(r)=(±)k

j m+1Yj+1/2( )√

m 1/2m 1/2j+mYj 1/2( )

and

with√√|j3/2[±2ωkiR(κ±ωkiR)+miR] 1/2,κ=j+1/2,κ(kR)|R0,κ= (j+1/2),B(±)kκi= 0,κ=j+1/2, kRE+mjforκ=±il=j 1/2(kR) j+121i)θ(ωk2iiξ1ξ µi2κ1κ2 ωk µi)× 2 1 θ(ω+iη ωk M(qq)iξ1ξ2κ1κ2(q)+θ(µi1i+ωk2i× 1ω+iη ω1i ωk2i ωk1i)M(¯qq)iξ1ξ2κ1κ2(q) ,k

M(qq)1

iξ1ξ2κ1κ2=

4V| k1,j1,κ1,i,+|r|k2,j2,κ2,i,+ |2 (j1 m1+1)(j1+m1+1)

j2δj1 1,j2δm1,m2θ(κ1κ2)

1

+m21(10)(13)

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

M(¯qq)1

iξ1ξ2κ1κ2=

4V| k1,j1,κ1,i,+|r|k2,j2,κ2,i, |2 (j1 m1+1)(j1+m1+1)

j2δj1 1,j2δm1,m2θ(κ1κ2)

1

+m21

2 1

2 iξ1ξ2κ1κ2ω+iη ωk1i ωk2i

2

2[(ωk1i+ωk2i)2R2+(κ1+κ2)(ωk1i+ωk2i)R]2

(k21 k22)2R

2[(ωk1i+ωk22i)2R+(κ1+κ2)(ωk1i+ωk2i)R]2

(k21 k22)2R

22mi

k2 (ωk1i ωk2i)R+,

1 k22 [2ωk1iR(κ1+ωk1iR)+miR][2ωk2iR(κ2+ωk2iR)+miR]

| k1,j1,κ1=±(j1+1/2),i,+|r|k2,j2=j1+1,κ2=±(j1+3/2),i, |2

= 2k1k2

[2ωκ,k1iR(1+ωk1iR)+miR][2ωk2iR( κ2+ωk2iR)+miR]

| k1,j1,κ1=±(j1+1/2),i,+|r|k2,j2=j1 1,κ2=±(j1 1/2),i, |2

= 2k1k2

[2ω,k1iR(κ1+ωk1iR)+miR][2ωk2iR( κ2+ωk2iR)+miR]

| k1,j1,κ1=±(j1+1/2),i,+|r|k2,j2=j1,κ2= (j1+1/2),i, |2

= 2k1k2Rω+(κk1 κ2)

1i+ωk2i 2(19)(20)(21)

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

Wethus ndthatΠ00(Π00)arisesfromexcitationsofquarkparticle-holepairs,accompaniedbythechangeinthe

(¯qq)(vp)angularmomentumofthescatteredquark j≡j2 j1=±1( j=0);Π00(Π00)arisesfromexcitationsof

quark-antiquarkpairsviascatteringofthequarkinthe lledFermi(Dirac)sea,accompaniedby j=0,±1,onto

(so)(¯qq)(vp)thevacantDirac(Fermi)sea.ThequantitiesΠ00,Π00,andΠ00areassociatedwithe ectsofrelativity:the

spin-orbitinteractions, uctuationsoftheFermiseainvolvingquark-antiquarkpairs,andthesecond-ordervacuumpolarization,respectively.

III.DISPERSIONRELATION(0)(so)

Thedispersionrelationofthelongitudinalcolorplasmaoscillationhasbeenobtainedfromtheone-looppolarizationΠ00givenbyEqs.(13)and(16).Bynotingthecompleteanalogywiththecaseofametalnanoparticleinadielectricmedium[10],sucharelationcanbewrittenas

21+q2

colorplasmafrequencyofbulkquarkmatter[7,8].|ReΠ00|/q2(|ReΠ00+ReΠ00|/q2)issuppressedbyonepower

(0)1/kF,iR(αs)withrespectto|ReΠ00|/q2(unity),whereαs=g2/4πistheQCD nestructureconstant.Wethus

obtainthemodefrequencyω≈ωp.Itsagreementwiththebulkvaluestemsfromabsenceofthesurfacecolorchargedueto vac=0.Thevaluesofωpcalculatedasafunctionofηs=ms/µforαs=0.4havebeenexhibitedinFig.1.Wehavefoundthatωpdependsonmsonlyweakly.

IV.DAMPINGRATE(so)(¯qq)(vp) ig2ni/6µiisthe

WithintheframeworkoftheRPA,thedampingofthelongitudinalcoloroscillationmodeisdeterminedbyitsresonantenergytransferintoindividualexcitationsofparticle-holepairsofthequarkswhoseeigenenergiesaremadediscretebytheboundary.ThecorrespondingdampingratemaybederivedfromthepolarizationΠ00obtainedaboveas[8]

ΓRPA= 3ωp

π4µRq2 h(ν,ηi)

i

2 ln 1 64z(2z+ν)

222π2ν(8z3+16νz2+12ν2z+3ν3)z ηi+π 22.1+ln z2 ηiµR 1+32z(z+ν)

ν 1dzmax(1 ν,ηi) 2(z+ν)2 ηiz2 (26)

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

InFig.2,wehaveshownΓRPA/ωpasafunctionofRforvariousvaluesofηs.Forω≤µ+mi,whichissatis edat

(vp)(¯qq)(0)(so)ω=ωp,ImΠ00cancelsoutImΠ00.Ontheotherhand,thecontributionsofImΠ00andImΠ00,thelatterbeingsuppressedbyonepowerαswithrespecttotheformer,leadtothedampingofwhichtheratebehavesas～c/R

[16].SuchcontributionsmayberegardedasLandaudamping,i.e.,adecayofthecollectivemodebyexcitingasinglerealparticle-holepair.Inthiscase,thesurfaceofthestrangelet,whichactsasaboundaryconditiondeterminingthediscretequarklevelsratherthanasascatterer,rendersΓRPAnonzeroquantummechanicallyandindependentofαsintheweak-couplinglimit.AscanbeobservedfromFig.2,ΓRPAhasonlyaslightdependenceonms.In(0)thenonrelativisticlimit,onlythepartImΠ00remainsuptoO(1/kF,iR),stillyieldingΓRPA∝R 1.WeremarkinpassingthatforastrangeletwithradiusR<～3fm,whichhasabaryonnumberoflessthan～30,higher-orderterms 1in(kF,iR)asignoredheremaywellhaveconsequencetothefrequencyanddampingrateofthecolorplasmons.Wenextconsiderthedampingofthelongitudinalcolorplasmonsduetocollisions[17]betweenquarksintheweak-couplingregime.Thesecollisions,bycolorexchange,disrupttheself-consistent eldsresponsiblefortheplasmonsandaddΓcoltotheRPAresultΓRPA.IfoneattemptstocalculateΓcolinperturbationtheory,onemeetsaninfrareddivergencethatarisesfromthecolormagneticinteractionsmediatedbytransversegluons.Atleastfor nitetemperature[18],suchadivergencecannotberemovedevenbydynamicalscreening[19]ofthemagneticinteractions

[20].ThisindicatesthatcalculationsofΓcoluptoleadingorderinαsrequirenonperturbativefeaturesoftherelativisticquark-gluonplasmawhichmaypossiblydevelopamagneticmassmmagandtherebyprovideane ectivecuto [21].Forathermal,masslessplasmaatzerobaryondensity,wheremmag～αsT,onethusobtainsΓcol～αsln(1/αs)Tfromkinetictheory[17].Thisratearisesfromthecollisionintegralassociatedwithone-gluonexchangeinthebulkplasma; nite-sizecorrectionsofO(1/kF,iR)tosuchabulkresultforΓcolshouldbeofhigherorderinαsthanΓRPA,whichremainsintheabsenceofthecollisionintegral.Atthenonzerostrangequarkmass,baryondensity,andtemperatureofinteresthere,however,thevaluesofmmagandhenceofΓcolareessentiallyunknown.Therefore,allwecanmentionisthatsuchacollisionaldampingortheLandaudampingduetoindividualexcitationsofquarkparticle-holepairs1dominatesthetotaldampingrateΓtot=ΓRPA+Γcol,accordingtowhetherthemeanfreepathofthequarks～Γ colissmallorlargecomparedwithR.Here,itisinstructivetonotethatnodampingoftheplasmonsduetogluonradiationoccursintheabsenceofthesurfacecolorcharge.

V.CONCLUDINGREMARKS

Wehavecalculatedthedispersionrelationanddampingrateofthelongitudinalcolorplasmaoscillationsinastrangeletintheregimesofweakcoupling,lowtemperature,andlongwavelength.ForkF,iR 1,wehavefoundthatthefrequencyofsuchoscillationsisalmostaslargeasthatofbulkcolorplasmonsandthattheLandaudampingyieldsalifetimeproportionaltoR.However,thepresentanalysisisaccompaniedbyseveraluncertainties.Wehavecon nedourselvestothecaseoflow|q|andT.Withincreasing|q|,multipoleplasmonexcitationsshoulddevelop.At nitetemperature,furthermore,gluonicexcitationsintheintermediatestatecontributetotheproperpolarizationΠ00;withintheone-loopapproximation,suchacontributionissimplyaddedtothequark-loopcontributionwithoutinterference[8].Astudyallowingforthoseexcitationshasyettobedone.Shelle ectsinstrangeletsalsoremaintobetakenintoaccount;suche ectsarepredictedtomanifestthemselvesremarkablyforA<～100[6,22].Itisexpectedfromtheanalogywiththecaseofcollectiveexcitationsinnormalnucleithatwhetherstrangeletsareinthevicinityofclosedshellsornotplaysaroleinthepropertiesofthecolorplasmaoscillations.Third,itistobenotedthatthepresentanalysisisrestrictedtotheweak-couplingregime(αs 1).Inthestronglyinteractingnonperturbativeregime(αs～1),morerelevanttoarealsystem,thedampingrateofthecolorplasmonscouldbegreatlymodi ed.Thesenonperturbativee ectsshouldhaveprimarysigni cancetothecollisionaldamping.

Toconclude,webrie yaddresswhattypeoftracesastrangelet,ifformedinheavy-ioncollisionexperiments[23]performedatabeamenergyoforder200GeVpernucleonona xedtarget,couldleaveviaexcitationsofthelongitudinalcolorplasmons.Immediatelyaftercentralnucleus-nucleuscollisionsatsuchenergies,itisexpectedthataquark-gluonplasmawouldbeproducedintheresultingmatterofsu cientlyhighenergydensity.Then,thequark-gluonplasmamightdevelopintostrangeletsviapre-freeze-outevaporationofpions,K+,andK0whichcarryawayentropyandantistrangenessfromthesystem.Thisstrangenessdistillationmechanism,suggestedbyGreineretal.

[24],wouldworkaslongasstrangeletsarestableagainststrongdecay.Notethatthefreezeouttemperature～100–200MeV[25]iscomparablewiththefrequencyωpforrealisticvaluesofαsintherange～0.4–1.Longitudinalcolorplasmaoscillationswouldthusbethermallyexcitedinstrangeletsuntilthetemperaturefullydrops.Onatimescale1oforderΓ tot,theexcitedcolorplasmonswoulddecayintoquarkparticle-holepairs,leadingtoemissionofphotonsandleptonpairsviaelectromagneticinteractions.Theseelectromagneticsignals,havingenergyoforderorlessthan～ωp,mightbeobservedinthehighlyenergeticheavy-ioncollisionsatCERN’sSPSexperiments[23].Inordertosupportthepicturementionedabove,however,estimatesallowingforrelaxationprocessesoccurringinsuchacooledand

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

decompressedsystem,forothermechanismsproducingphotonsandleptonpairs,andfortheuncertaintiesdiscussedintheprecedingparagraph,arerequired.Elucidationofthisproblemisbeyondthescopeofthispaper.

ACKNOWLEDGMENTS

TheauthorthanksG.Baymforusefuldiscussion.HeacknowledgesthehospitalityofAspenCenterforPhysics,wherethisworkwasinitiated.ThisworkwassupportedinpartbyaGrant-in-AidforScienti cResearchprovidedbytheMinistryofEducation,Science,andCultureofJapanthroughGrantsNos.07CE2002and199803687.

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

FIG.1.Colorplasmafrequencyωpasafunctionofηs,calculatedforµ=300MeVandαs=0.4.

The dispersion relation and damping rate of longitudinal color plasmons in finite strange quark matter (strangelets) are evaluated in the limits of weak coupling, low temperature, and long wavelength. The property of the QCD vacuum surrounding a strangelet

FIG.2.DampingratesΓRPAdividedbyωpasafunctionofthestrangeletradiusR,whichhavebeenevaluatedforµ=300MeVandαs=0.4.Thelinesfrombottomtotoparetheresultsobtainedforηs=0,2/3,5/6,1.

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