The initial gluon multiplicity in heavy ion collisions

AlexKrasnitz

UCEH,UniversidadedoAlgarve,CampusdeGambelas,P-8000Faro,Portugal.

RajuVenugopalan

PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA.

February1,2008

Abstract

Theinitialgluonmultiplicityperunitareaperunitrapidity,dN/L2/dη,inhighenergynuclearcollisions,isequaltofN(g2µL)(g2µ)2/g2,withµ2proportionaltothegluondensityperunitareaofthecollidingnuclei.ForanSU(2)gaugetheory,wecomputefN(g2µL)=0.14±0.01forawiderangeing2µL.ExtrapolatingtoSU(3),wepredictdN/L2/dηforvaluesofg2µLintherangerelevanttotheRelativisticHeavyIonColliderandtheLargeHadronCollider.Wecomputetheinitialgluontransversemomentumdistribution,dN/L2/d2k⊥,andshowittobewellbehavedatlowk⊥.

Atopicofconsiderablecurrentinterestisthepossibilityofforminganequilibratedplasmaofquarksandgluons,aquark–gluonplasma(QGP),inveryhighenergynuclearcollisions.ExperimentalsignaturesofsuchaplasmamayprovideinsightintothenatureoftheQCDphasediagramatfinitetemperatureandbaryondensity[1].

Equallyinteresting,istheinformationthatheavyioncollisionsmayprovideaboutthedistributionsofpartonsinthewavefunctionsofthenucleibeforethecollision.Atveryhighenergies,thegrowthofpartondistributionsinthenuclearwavefunctionsatu-rates,formingastateofmattersometimescalledaColorGlassCondensate(CGC)[2].ThecondensateischaracterizedbyabulkmomentumscaleQs.IfQs≫ΛQCD,thepropertiesofthiscondensate,albeitnon–perturbative,canbestudiedinweakcoupling.Thepartonsthatcomprisethiscondensatearefreedinacollision.Sincethescaleofthecondensate,Qs,istheonlyscaleintheproblem,theinitialmultiplicityandenergydistributionsofproducedgluonsatcentralrapiditiesaredeterminedbythisscalealone.Wewillbrieflydiscusslatertherelationofthesequantititiestophysicalobservables.Theabovestatementsmaybequantifiedinaclassicaleffectivefieldtheoryapproach(EFT)tohighenergyscattering[4].TheEFTandp⊥≫ΛQCD,(x∼p⊥/

√isclassicalbecause,atcentralrapidities,wherex≪1decreasingxgivingrisetolargeoccupationnumbers.Briefly,theEFTseparatespartonsinahadron(ornucleus)intostatic,highxvalenceandhardgluesources,and“wee”smallxfields.Foralargenucleusintheinfinitemomentumframe,thehardsourceswithcolorchargedensityρ,arerandomlydistributedinthetransverseplanewiththedistribution

P([ρ])=exp−

󰀁

1

IntheclassicalEFT,thisdivergenceislogarithmic.Thenumberdistributionshavetheform

nk⊥∝

1

k⊥

󰀋4

ln

󰀁

k⊥

2

󰀃󰀅

k

|π(k)|+ω(k)|φ(k)|

222

󰀈

,(3)

whereφ(k)isthekthmomentumcomponentofthefield,π(k)isitsconjugatemomentum,andω(k)isthecorrespondingeigenfrequency.Theaverageparticlenumberofthek-th

3

modeisthen

N(k)=ω(k)󰀑|φ(k)|󰀒=

2

󰀆

8

√

therighthandsideof(6)willdiverge.TheexpressioninEq.(4)canthenstillbefor-mallydeterminedintheCoulombgaugebutitsinterpretationasaparticlenumberisproblematic.Thissituationdeservesspecialattentionandwillbediscussedindetailelsewhere.Wedidnotobserveanyeffectsofmetastabiltywithintherangeofparame-tersofthecurrentnumericalstudy.Inparticular,weverifiedtheconvergenceofEq.(6)withrespecttotheupperlimitofintegration.

Wenowpresentourresultsusingboththetechniquesdiscussed.Webeginwiththenumberdistribution,whichcanonlybecomputedinCoulombgauge.Wehaveverified,forg2µL=35.35,thatintherangeofvaluesofg2µaconsideredherethesystemisclosetothecontinuumlimit.Thisisconsistentwithourearlieranalysisofthelatticespacingdependenceofamoreultraviolet-sensitivequantity,theenergydensity[15].InFig.1a,

weplotthegluonnumberdistribution,n(k⊥)≡dN/L2/dk2

⊥=N(k⊥)/(2π)2versusk⊥forfixedg2µL=35.5,butfordifferentvaluesofthelatticespacingg2µa.Forlargek⊥,thefinestlattice(g2µa=0.138)agreeswellwiththelatticeperturbationtheory(LPTh)analogueofEq.(2).Atsmallerk⊥,thedistributionissofter,andconvergestoaconstantvalue.InFig.1b,weplotthegluondistributionintheinfrared,atfixedg2µa,fordifferentg2µL(148.5and297).Wenoticethatthesedistributionsarenearlyuniversalandindependentofg2µL!Also,theconvergenceofthedistributiontoaconstantvalueismoreclearlyvisibleinFig.1b.

Whenk⊥≤g2µ,non–perturbativeeffectsqualitativelyaltertheperturbativenum-berdistribution,renderingitfiniteintheinfrared.Unfortunately,sincetheseeffectsarelarge,ananalyticalunderstandingofthebehavioratlowk⊥islacking.Ourre-sults,despitebeinguniversal,arenotsimplyfitbyanyofthephysicallymotivatedparametrizationswehaveconsidered.Fromourpreviousdiscussion,theformula

1dη

=1

g2µLg2µa

fN(cooling)fN(Coulomb)fN(res.)×103

35.36.276

.116±.001.127±.002

14±270.71.276

.119±.001.125±.0027.8±0.2106.10.207.127±.001.135±0.0018.9±0.2148.5.29

.138±.001142±.0015.6±0.1212.1.41

.146±.001.145±.0017.12±0.08297.0.29

.151±.001.153±.0014.83±.04

Table1:ValuesoffNvsg2µL,forfixedg2µa,plottedinFig.2.fN(res.)isdefinedinRef.[18].

WecancompareourresultsforthenumberdistributiontotheonepredictedbyA.H.Mueller[17].IntermsofQsandR,wecanre–writeEq.7as

1

dη

=cN

2Nc−1

4π2αS

Q2s.

AlargenumberofmodelsofparticleproductioninnuclearcollisionsatRHICand

LHCenergiescanbefoundintheliterature.AnicerecentsummaryoftheirvariouspredictionsandrelevantreferencescanbefoundinthecompilationofRef.[21].NaivelyextrapolatingourresultstoSU(3),wefindforAu-AucentralcollisionsatRHICenergies,dN/dη∼950forfN=0.132±0.006(g2µL≈120—wetakethemeanofthe106and148coolingpoint).Similarly,forLHCenergiesfN=0.148±0.002(g2µL∼255—themeanofthe212and297coolingpoint),onefindsdN/dη∼4300.Inparticular,comparingourpredictionswiththoseofpQCDbasedmodels[19],wefindournumberstobeinroughagreement.However,ifweincludeaKfactorlikemanyofthesemodelsdo,ournumberswillberoughlyafactorof2larger.

Muellerestimatesthenon–perturbativecoefficientcNtobeoforderunity.IfwetakefN=0.14±0.01,asisthecaseformuchoftherangestudied,wefindcN=1.29±0.09,anumberoforderunityaspredictedbyMueller.Despitethiscloseagreement,thetransversemomentumdistributions,showninFig.1aand1b,anddiscussedearlier,look

2

quitedifferentfromMueller’sguessofθ(Q2s−k⊥).Theθ–functiondistributionwasonlyaroughguesstorepresentaqualitativechangeofthedistributionsatk⊥∼Qs.

ThereisconsiderableuncertainityinthevalueofQsbecausethegluondensitiesattherelevantxandQ2areill–known.SincethemultiplicitydependsquadraticallyonQs,apredictionofthesameisperforceunreliable.Distinguishingbetweendifferentmodelswillthereforerequire,atthe√veryleast,testingtheirpredictionsforthescalingofmultiplicitieswithAandwith

ln(s/s0)),wheres0isaconstant.Inthelattercase[23],itisclaimedthata

goodfittothemultiplicityfromexistinghighenergyhadronscatteringdataisobtained.DatafromRHICwillhelpconstraintheenergydependenceofQs.

6

Thereadershouldalsonotethatourrelationsarederivedonlyfortheinitialpartonmultiplicitydistributionsatcentralrapidities.Theseprovidetheinitialconditionsforthesubsequentevolutionofthesystem,whichcanbeinvestigatedinatransportap-proach[17,24].Therateofchemicalequilibration,anduncertainitiesduetohadroniza-tionalsohavetobetakenintoaccountinpredictionsofobservablessuchaschargedhadronmultiplicities.Conversely,theseobservablesmayhelpconstrainthesaturationscaleQs,andinformusabouttheveryearlieststagesofnuclearcollisons.

Insummary,wehavederivedanon–perturbativerelationbetweenthemultiplicityofproducedpartonsandthesaturationscaleofpartondistributionsinhighenergynuclearcollisions.Wehavecomputednumberdistributionswhichhavethepredictedperturbativebehaviorintheultraviolet,andarefiniteintheinfrared.Atpresent,inourapproach,weareonlyabletomakequalitative“ball-park”predictions.However,wehavedevelopedaframeworkinwhichthesecanbequantifiedandextendedinaconsistentmannertostudyalargenumberoffinalstateobservablesinheavyioncollisions.

Acknowledgments

WewouldliketothankLarryMcLerranandAlMuellerforveryusefuldiscussions.R.V.’sresearchwassupportedbyDOEContractNo.DE-AC02-98CH10886.TheauthorsacknowledgesupportfromthePortugueseFCT,undergrantsCERN/P/FIS/1203/98andCERN/P/FIS/15196/1999.

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8

+23+23+23+23+23+23+23+32+3233+2+2+2+2+2+2++2+2+++2+222

222

2222

22222

22222

0.05

++0.043

3+3

0.03

b

n(k⊥)

0.02

0.01

0

+3+3+33++33++3+33++3+33+3++33++3+33++3+3+3

0

0.2

0.4

0.6

0.8

1

Figure2:ThefunctionfN,definedinEq.(7)asafunctionofg2µL,obtainedbytherelaxationmethod(plusses)andbytheCoulombgaugefixing(diamonds).Thevaluesofg2µaare0.276forg2µL=35.35andg2µL=70.8;0.29forg2µL=148.5andg2µL=297;and0.414forg2µL=212.

g2µ

k⊥

Figure3:Gluondispersionrelationω(k⊥)obtainedfromEq.(4),forthevalues70.8(diamonds),148.5(plusses),and297(squares),withthevaluesofg2µaasinFigure2.

10