Particles Associated with Omega Produced at Intermediate p_T

atIntermediatepT

CharlesB.Chiu1andRudolphC.Hwa2

1

arXiv:0704.2616v1 [nucl-th] 19 Apr 2007CenterforParticlePhysicsandDepartmentofPhysics

UniversityofTexasatAustin,Austin,TX78712,USA

2

InstituteofTheoreticalScienceandDepartmentofPhysicsUniversityofOregon,Eugene,OR97403-5203,USA

Abstract

ThedualobservationoftheΩproductionincentralAu-AucollisionshavingbothanexponentialpTdistributionandalsoassociatedparticlesabovethebackgroundhasbeenreferredtoastheΩpuzzle.Wegiveaquantitativedescriptionofhowthatpuzzlecanbeunderstoodintermsofphantomjets,whereonlyridgeswithoutpeaksareproducedtogiverisetoboththeΩtriggeranditsassociatedparticles.Intheframeworkofrecombinationofthermalpartonsweareabletoreproduceboththe∆φdistributionandthetrigger-momentumdependenceoftheyieldoftheassociatedparticles.Wemakepredictionsonotherobservablesthatcanbecheckedbyfurtheranalysesofthedata.

1

1Introduction

RecentdataontheproductionofφandΩattheRelativisticHeavyionCollider(RHIC)atintermediatepThaverevealedimportantpropertiesofthesquarksinthedensemediumcreatedbythecollisionofAunucleiat200GeV[1,2].BothφandΩexhibitexponentialbehaviorintheirpTdistributionsuptopT≈4.5GeV/c.SuchpropertiesoftheφandΩspectrahavebeenreproducedin[3]asconsequencesofrecombinationofthermalpartons.Theimplicationisthatbeforehadronizationthedensesystemconsistsofthermalizedpartonsthatincludethesquarks.Thatisthemaincharacteristicthatquark-gluonplasma(QGP)shouldpossess.TheproductionofφandΩprovidesaclearwindowthroughwhichonecanobservethethermalsourcewithoutthecontaminationofshowerpartonsduetohardscattering,whichissuppressedforsquarks.SinceQGPisadescriptionofthebulkmedium,onecanreasonablyaskwhatthepropertiesareattheedgeofthatmedium,namely,whenthepartonickTis>1GeV/c.RecentdatafromSTARoncorrelationthatusesΩastriggerintheintermediatepTregionhasrevealedinterestingpropertiesoftheassociatedparticles[4].ThedualfeaturesoftheexponentialspectrumofΩandtheexistenceofassociatedparticlesabovebackgroundhavebeenreferredtoastheΩpuzzle[5].Thispaperisaimedatprovidingaquantitativeresolutionofthatpuzzle.

ParticlesproducedatintermediateandhighpTinheavy-ioncollisionsareassociatedwithjetsduetohardscatteringofpartons.Themediumeffectonhadronizationhasbeensuccessfullydescribedintherecombinationmodelintermsoftherecombinationofthermalandshowerpartons[6].Thesingle-particlepTdistributionsofπ,Kandparewellrepro-duced,showingpower-lawdeviationfromtheexponentialbehavioratlowpTthatisdueto

2

therecombinationofthermalpartons.ThehadronizationmechanismthatworksforthoseparticlesthathavelightquarksasconstituentsdoesnotworkfortheproductionofΩthatcontainsonlysquarks.Strangeshowerpartonsaresuppressedwhatevertheinitiatinghardpartonmaybe[3].ForthatreasonthepTdistributionofΩisessentiallyexponentialuptothehighestmeasuredvalueofnearly6GeV/c[7],ashasbeenwellreproducedbytherecombinationofthreethermalsquarks[3].

SinceshowersquarkdoesnotparticipateintheformationofΩ,itisnaturaltoconcludethatjetsplaynoroleandthattheproductionofΩisnotaccompaniedbyanyassociatedparticles,whichareusuallypresentineventstriggeredbylessstrangehadrons.Thedistri-butionsofassociatedparticlesinthelattercasehavealsobeensatisfactorilydescribedbytherecombinationofthermalandshowerpartonsinpT[8],aswellasin∆ηand∆φ[9].Withoutjetproductionthereshouldbenojetstructurecharacterizedbyassociatedparticlesdistinguishablefromthebackground.

ThediscoverythatΩtriggerisaccompaniedbyassociatedparticlesabovethebackgroundhasbeenastounding[4,10,11].Howcanatriggerparticleformedbythermalpartonshavepartnersabovethebackground,sincetheyarepresumablyalsoofthermalorigin?ThatistheΩpuzzle.

2TheRidge

Conventionaljetstructureconsistsofapeakaboveapedestal,whenthetriggermomentum

assoc

)arenottoohigh.STARhasadoptedthe(ptrigT)andthoseoftheassociatedparticles(pT

terminologyJet(J)forpeakandridge(R)forpedestal.TheJtoRratiooftheiryields

3

assoc

dependsonptrigandcentrality[12].Forptrig>4GeV/candpassoc>2.3GeV/cinT,pTTT

centralAu-Aucollisionsat200GeV,ithasbeenshownthatJ/R∼1.However,atlowervaluesofpassoctheJ/Rratioissmaller,becomingaslowas0.1−0.15atpassoc∼1.2GeV/cTT[13].ForΛ-triggeredeventsthatratioisevenlower(<0.1).NodataforthatratioyetexistforΩ-triggeredevents,butonemayanticipateittobefurtherloweraccordingtothetrendofincreasingstrangeness.Ifthatistrue,thentheparticlesassociatedwithΩwillnotexhibitanysignificantpeakinthe∆ηdistribution,whichiswhereJandRareusuallyseenwithnon-strangetriggers.Onthebasisofthestudydonein[3],whereΩisshowntooriginatefromthermalsources,wecanassertthattheJetcomponentshouldbeabsent,andthatonlytheridgewillbeobserved.

Thenotionofphantomjetwassuggestedin[5]todescribethephenomenoninwhichajetisproducedwithouttheJetpartbeingobserved.Theridgethatisobservedprovidestheevidencethatthereisanunderlyingjet.Suchascenariohasnotbeenconfirmedbydata,butisourconjectureasasolutiontotheΩpuzzle,thequantitativedescriptionofwhichwillbegiveninthefollowingtwosections.AconfirmationwouldbeinevidencewhentheΩ-triggered∆ηdistributionexhibitsaridgewithoutapeak.Atthispointthestatisticsinthedatafor2.5Aphantomjetisseenasanordinaryjetifthetriggerisanon-strangeparticle.Itmaybeinitiatedbyagluonoralightquark,whichcangeneratenon-strangeshowerpartonsthathadronizeasπ,K,porΛ.IfptrigTisnottoohigh,thehardscatteringmayoccurintheinterior

4

<

ofthebulkmedium.Thescatteredhardpartontraversesthemediumandlosesenergyonitswayouttothesurface.Theenergydepositedinthemediumenhancesthethermalmotionofthepartonsinthevicinityofthetrajectory.Thosepartonshadronizeandformtheridgethatisabovethebackground.Suchasequenceofprocessesisconventional,andhasbeendescribedsuccessfullyin[9]thatgivesboththe∆ηand∆φdistributions.However,theΩ-triggeredeventsarenotconventional,eventhoughtheunderlyingjetisconventionalasdescribedabove.Thereasonisthatthesquarksaresuppressedintheshower.WithoutthesshowerpartonsΩcannotbeformedasadirectconsequenceofhardscattering.Itcan,however,beformedindirectlyfromtheridgethatisthermalandhasamplesupplyofsquarks.EventstriggeredbyΩproducedthatwayarerareathighptrigT,sincethethermalspectrumisexponential.Thustheptrigthresholdissetlow,like2.5GeV/c.TheTcorrespondingpassocthresholdisevenlower,like1.5GeV/c,inordertohaveenoughstatisticsTtoproduceameaningfulassociatedparticledistribution.Whenpassocissolow,theridgeTcomponentdominatesoverthepeaksotheJ/Rratioisverysmall.Theordinaryjetthusandlowerpassoc.becomesaphantomjetundertheconditionoflowptrigTT

IftheridgecansupplysquarkstoformtheΩtrigger,itcansurelysupplylightquarkstoproduceotherlower-masschargedhadrons.Thosearetheassociatedparticlesdetected[4].ThustheridgeofthephantomjetisthekeytothesolutionoftheΩpuzzle.TheΩhasapTdistributionthatisexponential,ontheonehand,becauseitisformedbythethermalpartonsintheridge,andithasassociatedparticles,ontheotherhand,becausetheridgeisabovethebackground.

5

3TheBackground

Ouraimnowistodescribethe∆φdistributionoftheassociatedparticlesineventstriggeredbyΩ[4].Thesignalislessthan4%ofthebackgroundheight.Thusinorderforustoreproducequantitativelythesignal,itisnecessarytoshowfirstthatthebackgroundcanbeaccuratelyobtainedinourformalism.Thereisav2oscillationduetoellipticflow,resultingin[4]

dNbg

dp1dη1

andthetwo-particledistributionby

N(1,2)=

dNΩ,2

(2)

d∆ηd∆φ

=

󰀇pd

pc

dp1

󰀇p1

pa

dp2

󰀇1

−1

dη1N(1,2)

wherepa=1.5,pc=2.5andpd=4.5allinunitsofGeV/c.ForthebackgrounddistributionN(1,2)isfactorizable,i.e.,

Nbg(1,2)=N(1)Nbg(2).

(5)

ItfollowsthatontherighthandsideofEq.(4)theη1integrationsinthenumeratorandthedenominatoraretriviallycancelled,leaving

dNbg

󰀇pd

pc

dp1N(p1)

,where

Nbg

(pp1

dNbg

1,∆η,∆φ)=

󰀂

pdp2

a

p1

dNbg

dp2d∆ηd∆φ

≡

󰀂

pdp2

a

dq

=C−q/T0qe0,

where

C0=23.2(GeV/c)−1,

T0=0.317GeV/c.7

(6)

(9)

(10)

Therecombinationofthosepartonstoformthermalpionsisgivenby[6]

dNπ

thp2

󰀂

p0

dqT0(q)T0(p−q)=

C0

2dp=

4

2d∆φ

3

(C0T0)2h(p1,T0),

(13)

where

h(pp1,T0)≡f

󰀅

a

T,T00

󰀈

,

(14)

with

f(x,T0)=(1+b)(1+x)+cT0(2+2x+x2)e−x.

(15)

Tocarryouttheintegrationsoverp1inEq.(6)weneedthesingle-particledistributionfortheproductionofΩ,whichcanbedeterminedinthesamewayasforprotonbyrecombination[3,6],exceptthatitissimplerwhenonlythermalsquarkscontribute.Theresultis[3]

dNΩ

p31

27wherep10=(p21+MΩ2

)

1/2

and

Ts=0.33GeV/c.

(17)

ThenormalizationfactorCΩisimmaterial,sinceitappearsinboththenumeratorandthedenominatorofEq.(6),sotheycanceleachother.

Integratingbothoverp1and∆ηinEq.(6),wefinallyhave

dNbg

3

(C0T0)2󰀈h(p1,T0)󰀉,

(18)

where

󰀈h(p1,T0)󰀉=

󰀂

pdpdpdNΩ

p1

a

c

dp≡f

(19)

1

󰀅

T,T0

0

󰀈󰀄

.ThelasttermontherighthandsideofEq.(19)isgivenby

󰀃

f(

p1

T03

s

−xI(xs,ys)

(x2+µ21/2,(20)

0s)

ewiththevariableofintegratonx=p1/Ts.Thelimitsofintegrationarex0s=pc/T0sandy0s=pd/T0s,whereT0s=T0Ts/(T0+Ts).Theothersymbolsarez=(T0s/T0)x,τ0=cT0andµ0s=mΩ/T0s.Thedenominatorofthefirstfactorisgivenby

I(xs,ys)=

󰀂

ys

xdx

x3

s

partonsgiveninEq.(10),wecannowcomputethebackgroundheightfortheassociatedparticlesandobtain

dNbg

ThebasicdifferencebetweentheridgeandthebackgroundisthatfortheformerwehavenewparametersCandTfortheenhancedthermalpartons,whileforthelatterC0andT0aregiveninEq.(10).Thereisalsoa∆φdependencewhichisforcedtovanishat∆φ=±1duetothesubtractionprocedureusedinthedataanalysis.Wewrite

C(∆φ)=C0H(∆φ)=C0H0exp−(∆φ)/2σ,

󰀆

2

2

󰀉

(23)

wherethevaluesofH0andσarediscussedbelow.Sinceassociatedparticlesintheridgeareunidentifiedchargedhadrons,justasinthebackground,weobtainbythermalrecombinationasdoneintheprecedingsection

dNR

3

(C0T)H(∆φ)󰀈h(p1,T)󰀉−

2

dNbg

thenear-sideyieldforvarioustriggerhadronsthatallexhibitsimilardependenceonptrigT.IncontrasttodNR/d∆φwhichinvolvesintegrationoverp1,theyieldY(p1)involvesintegrationover∆φinstead.Ifwedenote

thentheyieldis

H

˜=󰀂

1

−1

d∆φH(∆φ),(25)

Y(p1)=

4

roughlyinaccordwiththedatanotonlyforΩtrigger,butalsoforΛandΞtriggers,shownalsoinFigs.1and2.AlthoughtheJ/RratioforΛisnotzero,beinglessthan0.1wouldstillimplythattheridgeisdominant,soitsassociatedparticleswouldarisemainlyfromtheridge,withshowerpartonsplayingaminimalrole.ThecasewithΞtriggerwouldbeevenclosertothatoftheΩtrigger.

Assumingthattherangeoftheridgein∆ηisfrom−2to+2,aswehavedoneinEq.(24)inaccordancetothepresentexperimentalcut,wecancalculatethetotalnumberofchargedparticlesintheridgecorrespondingtoanintervalIinthe∆φdistribution

RNI

=

󰀂

I/2

−I/2

d∆φ

dNR

∆η

=

RNI

3

2C0

󰀆

ˆI󰀈h(p1,T)󰀉−T2I󰀈h(p1,T0)󰀉,TH0

2

󰀉

(29)

I=1,are

ˆI=󰀇I/2d∆φH(∆φ).Thepredictedridgeheightsforthetwocasesconsidered,forwhereH−I/2

(a)

dNR

d∆η

=0.054.

(30)

Ifthedataontheridgein∆ηturnouttohaveawidthdifferentfrom4,thentheheightwillalsobedifferentaccordingly.Subjecttothatqualification,aheightof0.06isourfirst-orderprediction.Itisofinteresttonotethattheridgeheightenvisionedhereisroughlythesame<6GeV/candastheoneobservedin[14]forunidentifiedchargedtriggerwith413

5Conclusion

WehavepresentedaquantitativedescriptionofhowtheΩpuzzlecanberesolved.WehaveshownthattheparticlesassociatedwithΩcanbeunderstoodasproductsofrecombinationofthermalpartonsintheridge.Althoughthenon-perturbativedynamicalprocessthatleadstotheformationoftheridgecannotbecalculated,thereareaspectsofthedatathatneedcoordinatedexplanationinaspecifichadronizationscheme.WehavereproducedinFigs.1and2boththe∆φdistributionandtheptrigdependenceoftheyieldoftheassociatedTparticles.OneparameterH0isusedtofittheheightofdNR/d∆φ,whichdependsontheexactvalueofH0sosensitively(towithin1%inthemarginoferror)thatitisbeyondthescopeofanydynamicaltheorytopredict.Whatwehavelearnedfromthisphenomenologicalstudyisthattheparticlesinthepeakobservedinthe∆φdistributionareallfromtheridge,contrarytotheusualidentificationofpeakswithJets.Phantomjetshavenopeaks.Ourfindingimpliesthatthereshouldnotbeapeakinthe∆ηdistribution,stilltobeproducedbyfurtheranalysisofthedata.Whenthatdistributionbecomesavailableandshowsonlyaridge,thenoursolutionoftheΩpuzzlewillfinallybeconsideredconfirmed.

Intheframeworkinwhichwehavecalculatedthe∆φdistributionoftheassociatedparticles,itisimpliedthatthepTdistributionofthepionswillhaveaninverseslopeT=0.33GeVfor1.5WhilewehavefocusedourattentioninthispaperontheΩ-triggeredevents,itisnaturaltopredictthatexactlythesamefeatureswillbeobservedfortheparticlesassociatedwith

14

theproductionofφunderthesameconditions.BothφandΩareproducedfromthesameenhancedthermalsource,sotheirassociatedparticlesshouldbothbeformedfromtheridgesofsimilarcharacteristics.

Acknowledgment

WearegratefultoJanaBielcikovaforhervaluablehelpandtogetherwithBettyAbelevwethankthemfortheircooperativecommunication.Thisworkwassupported,inpart,bytheU.S.DepartmentofEnergyunderGrantNo.DE-FG02-92ER40972.

References

[1]B.I.Abelevetal.(STARCollaboration),nucl-ex/0703033.

[2]J.Adamsetal.(STARCollaboration),Phys.Rev.Lett.92,182301(2004).[3]R.C.HwaandC.B.Yang,Phys.Rev.C(toappear),nucl-th/06020243.[4]J.Bielcikova(forSTARCollaboration),nucl-ex/0701047.[5]R.C.Hwa,nucl-th/0701018.

[6]R.C.HwaandC.B.Yang,Phys.Rev.C70,024905(2004).

[7]J.Adamsetal.(STARCollaboration),Nucl.Phys.A757,102(2005).

[8]R.C.HwaandC.B.Yang,Phys.Rev.C70,054902(2004);R.C.HwaandZ.Tan,

Phys.Rev.C72,057902(2005).

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[9]C.B.ChiuandR.C.Hwa,Phys.Rev.C72,034903(2005).[10]L.Ruanetal.(forSTARCollaboration),nucl-ex/0701070.[11]O.Barranikova,plenarytalkatQuarkMatter2006.

[12]J.Putschke,paralleltalkatQuarkMatter2006,nucl-ex/0701074.[13]J.Bielcikova,talkgivenatHardProbes2006,nucl-ex/0612028.

[14]J.Adamsetal.(STARCollaboration),Phys.Rev.Lett.95,152301(2005).

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dNR+bg/d∆φ7.27 (a)(b)bgΩΞΛ6.8−2−10∆φ12Figure1:(Coloronline)CalculatedassociatedparticledistributionsineventstriggeredbyΩfor(a)solid(red)line,H0=0.795,and(b)dashed(blue)line,H0=0.790.Thedatapointsarefrom[4]forthreehyperontriggers.Thedashed-dottedlineisthebackground.

17

Nearside yield/trigger0.50.40.30.20.102(a)(b)ΩΞΛ3ptrig (GeV/c)T45Figure2:(Coloronline)TheyieldoftheassociatedparticlesonthenearsideforΩtriggerasafunctionofthetriggermomentum.Thesolid,dashedlinesanddatapointsareasinFig.1.

18