Table Of ContentDD Correlations asa SensitiveProbe forThermalizationinHighEnergy Nuclear Collisions
X. Zhu1,2,3, M. Bleicher1, S.L. Huang4, K. Schweda5, H. Sto¨cker1,2, N. Xu4, P. Zhuang3
1 Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universita¨t,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
2 FrankfurtInstituteforAdvancedStudies(FIAS),Max-von-Laue-Str.1,D-60438FrankfurtamMain,Germany
3 Physics Department, Tsinghua University, Beijing 100084, China
4 NuclearScience Division, LawrenceBerkeleyNationalLaboratory, Berkeley, CA94720, USAand
5 PhysikalischesInstitut,Universita¨t Heidelberg, Philosophenweg 12, D-69120Heidelberg, Germany
(Dated:February2,2008)
Weproposetomeasureazimuthalcorrelationsofheavy-flavorhadronstoaddressthestatusofthermalization
atthepartonicstageoflightquarksandgluonsinhigh-energynuclearcollisions. Inparticular,weshowthat
hadronicinteractionsatthelatestagecannotsignificantlydisturbtheinitialback-to-backazimuthalcorrelations
of DD pairs. Thus, a decrease or the complete absence of these initial correlations does indicate frequent
7
interactionsofheavy-flavorquarksandalsolightpartonsinthepartonicstage,whichareessentialfortheearly
0
thermalizationoflightpartons.
0
2
PACSnumbers:25.75.-q
n
a
J Introduction frequency. The idea of statistical hadronization of kineti-
3 cally equilibrated charm quarks [10] even predicted signifi-
LatticeQCDcalculations,atvanishingorfinitenet-baryon
2 cantchangesinhiddencharmhadronproduction[11].Hence,
density, predict a cross-over transition from the decon-
2 fined thermalized partonic matter (the Quark Gluon Plasma, heavy-flavorquarksareanidealprobetostudyearlydynamics
inhigh-energynuclearcollisions.
v QGP) to hadronicmatter at a critical temperatureT 150–
c
8 180MeV[1]. ≈
7
1 Measurementsofhadronyieldsintheintermediateandhigh t
Higgs Vacuum
4 transverse momentum(pT) region indicate that dense matter 110055 Electroweak symmetry breaking
0 is produced in Au+Au collisions at RHIC [2, 3]. The ex- )
V
6 perimentallyobservedlargeamountofellipticflowofmulti- e
0 strange hadrons[4], such as the f meson and the W baryon, (M 110044 b
h/ suggest that collective motion developsin the early partonic ss c
p a
stageofthematterproducedinthesecollisions. Acrucialis- m
p- sue to be addressed next is the thermalization status of this k 110033
r s
e partonicmatter. ua
:h Heavy-flavor (c, b) quarks are particularly excellent s q 110022
v g
tools[5, 6, 7]to studythethermalizationoftheinitiallycre- g
Xi atedmatter. AsshowninFig.1,theirlargemassesarealmost Hi 1100 d QCD Vacuum
r exclusivelygeneratedthroughtheircouplingtotheHiggsfield c c symmetry breaking
a u
in the electro-weak sector, while masses of light quarks (u,
d, s) are dominated by spontaneous breaking of chiral sym- 11
22 33 44 55
11 1100 1100 1100 1100 1100
metryinQCD.ThismeansthatinaQGP,wherechiralsym-
Total quark mass (MeV)
metry mightbe restored, lightquarksare left with their bare
currentmasseswhileheavy-flavorquarksremainheavy. Due
to their large masses ( L ), the heavy quarks can only FIG. 1: (Color online) Quark masses in the QCD vacuum and the
≫ QCD Higgsvacuum. Alargefractionofthelightquarkmassesisdueto
bepair-createdinearlystagepQCDprocesses. Furthermore,
thechiral symmetry breaking inthe QCD vacuum. Thenumerical
theirproductioncrosssectionsinnuclearcollisionsarefound valuesweretakenfromRef.[12].
to scale with the number of binary nucleon-nucleon colli-
.
sions [8, 9]. In the subsequent evolution of the medium,
thenumberofheavyquarksisconservedbecausethetypical Recent STAR and PHENIX results on elliptic flow and
temperature of the medium is much smaller than the heavy nuclear modification factors of non-photonic electrons indi-
quark (c, b) masses, resulting in negligible secondary pair cate that charm quarks might indeed participate in the col-
production. In addition, the heavy quarks live much longer lectivemotionofthematterproducedinAu+Aucollisionsat
than the lifetime of the formed high-density medium, de- RHIC. To explain the data, a large drag diffusioncoefficient
caying well outside. These heavy quarks (c, b) can partic- intheLangevinequation,describingthepropagationofcharm
ipate in collective motion or even kinetically equilibrate if, quarksinthemedium,isfoundtobenecessary[13].Two-and
and only if, interactions at the partonic level occur at high three-bodyinteractions[14,15]ofheavy-quarksandresonant
2
rescattering[17]inthepartonicstageseemtobecomeimpor- 6,7],
tant. Theseinvestigationssuggestthatheavy-quarksactively
d~p
participateinthepartonicstage. = g (T)~p+~h , (1)
dt −
In this paper, we study the change of azimuthal correla-
tions of D and D meson pairs as a sensitive indicatorof fre- where~h is a Gaussian noise variable, normalized such that
quentoccurrencesof partonic scattering. Since heavy-flavor h i(t)h j(t′) =a (T)d ijd (t t′)with i,j indexingdirections.
quarksarepair-createdbyinitialscatteringprocesses,suchas Bhoth the driag coefficient g−and the momentum-space diffu-
gg cc,eachquark-antiquarkpairiscorrelatedinrelativeaz- sion coefficient a depend on the local temperature, T. We
imu→thDf duetomomentumconservation. Inelementarycol- use the same parameterization for g as in Ref. [5], g (T) =
lisions,thesecorrelationssurvivethefragmentationprocessto aT2, with a=2 10−6 (fm/c)−1 MeV−2 and also neglect a
a largeextentandhenceareobservableinthe distributionof momentum depen·dence of g . g and a are related by the
tahteebreylahtoivwemazuicmhutthheosefcpoarirrseloaftioDnsanshdoDuldmbeesoanfsfe.cWtedebevyatlhue- AfluscitnuaRtieofn.-[d5i]s,siapaitsiocnalrceullaattieodnfirnomequthileibarbiuomve,ehqpu2iaiti=ona w/2itgh.
early QGP stage and by the hadronicscattering processes in hp2ii=1.33mcT,withacharmquarkmassmc=1.5GeV/c2.
thelatehadronicstage. For simplicity, the initial conditions for central collisions
of heavy nuclei (Au or Pb) are the same as in Ref. [5]. We
ResultsandDiscussions
assume a plasma in thermal equilibrium occupying a cylin-
We startbyreviewingtheD andD angularcorrelationsin deroffixedradiusR, equalto theradiusof thecollidingnu-
ppcollisions.TheMonteCarloeventgeneratorPYTHIA[18] clei (R=7 fm in the present calculations), at the time of its
reproduces well the experimentally observed correlations of fullformationt . Afterthat,theplasmaevolvesaccordingto
0
D mesons, measured at fixed target energies [19]. Figure 2 Bjorken’s hydrodynamics, with a temperature dropping like
showssuchcorrelationsascalculatedbyPYTHIA(v.6.139) T =T (t /t )1/3. Beforethetimet ,theplasmaischanging
0 0 0
for pp collisions at √s = 200 GeV. The Df distribution is rapidlyand is notfully equilibrated. We assume (as done in
peakedat180◦,especiallyforhighpTDmesons,asonewould Ref. [5]) thatthe plasma is alreadyin equilibriumbeforet 0,
expectforback-to-backpairsstemmingfromhardscatterings withT=T andwiththecharmedquarksdiffusinginthesame
0
ofpartons. wayastheydoaftert . Inorderto isolatetheeffectspurely
0
duetoparton-partonre-scatteringintheoutlinedmedium,we
generated c and c quarks with the same p and zero longi-
T
tudinal momentum back-to-back, i.e. with Df =p at a time
p + p collisions (Pythia 6.1) of the order of 1/m 0.1 fm/c, and used a delta function
c
≃
2 forfragmentingthecharmquarkintoacharmedhadronatthe
p > 4 GeV/c hadronizationstage. The radiusr where each pair is created
1 T
-) 1.5 < p < 4 GeV/c israndomlygeneratedfromadistributionreflectingthenum-
d 1.5 T ber of binary nucleon-nucleoncollisions taking place at that
(ra pT < 1.5 GeV/c rthadeiiunsi,tipa(llry)dprro(cid:181)du(cRe2d−chr2a)rm2p qrudarr.kTshineathnegutrlaanrsdviesrtrsiebuptliaonneoisf
All
fd 1 isotropic.
/
N The evolution of the charm momenta with the Langevin
d equation is stopped when T reaches the critical temperature
0.5
T =165MeVorwhenthecharmquarkleavestheQGPvol-
c
ume. Furthermore, to get a first estimation of the QGP ef-
fectonthecharmquarkpairsazimuthalcorrelation,weomit
0
the possible contribution of the mixed phase. Figure 3(a)
0 0.5 1 1.5 2 2.5 3 shows the results for the D meson (charm quark) pairs an-
Df
(rad.) gularcorrelationsfordifferentinitialcharmquark p values,
T
forT =300MeVandt =0.5fm/c(typicalvaluesforRHIC
0 0
collisions). Weseethatthefastestcharmquarks(represented
FIG.2: (Color online) Correlationsinrelativeazimuth, Df , ofDD
by the p =3 GeV/c red curve) are able to escape from the
pairsfromppcollisionsat√s=200GeV,ascalculatedbyPYTHIA T
(v.6.139)fordifferentrangesofsingleDmesonp . QGP withoutsufferingsignificantmediumeffects, while the
T
slowerquarks(seethe p =0.5GeV/cblackline)havetheir
T
pairazimuthalcorrelationalmostcompletelysmearedoutby
ToexplorehowtheQCDmediumgeneratedincentralultra- theinteractionsinthemedium.
relativistic nucleus-nucleus collisions influences the correla- Figure 3(b) shows the corresponding results for T =
0
tions of D and D meson pairs, we employ a non-relativistic 700 MeV and t =0.2 fm/c, values representative of LHC
0
Langevin approach which describes the random walk of energies. Here,theinteractionsofthecharmquarkswiththe
charm quarks in a QGP and was first described in Refs. [5, medium are so frequent that only the most energetic charm
3
quarks preserve part of their initial angular correlation; low Whiletheinitialcorrelationsofccpairsareclearlyaffected
p pairscanevenbecompletelystoppedbythemedium. Be- bytheformedQCDmedium,andsensitivetoitstemperature
T
cause the c and the c quarks of a given pair are created to- and drag coefficient, it is importantto evaluate to which ex-
gether, in the same space point, the pair has a higherescape tentthelossofcorrelationscanbemimickedbyhadronicin-
probability if both quarks escape the medium from the side teractions. Therefore,wehaveinvestigatedbyhowmuchthe
of the fireball where the thickness is smaller. Thus, the Df cc correlations deteriorate due to hadronic scattering in the
distributionisshiftedtowardssmallervalues. Presently,there latestageofhigh-energynuclearcollisions. Forthispurpose,
isonlylongitudinalflow inthe hydro-dynamicalmodelused weusedthemicroscopichadronictransportapproachUrQMD
inthecalculations. Weexpectthatthestrongradialflowwill v.2.2[22,23].
furtherenhancethesamesidecorrelationsinthecaseoffully Thismicroscopictransportapproachisbasedonthecovari-
kineticallythermalizedcharmpairs. Becausethetwo quarks antpropagationofconstituentquarksanddiquarksaccompa-
ofapairarecreatedinthesamepositioninthemedium,they nied by mesonic and baryonic degrees of freedom. It sim-
will be pushed in the same direction by the radial flow. In ulates multiple interactions of in-going and newly produced
fact, this mechanism is a knownnon-flow effectin the mea- particles, the excitation and fragmentation of color strings,
surementofcollectiveflow[21]. andtheformationanddecayofhadronicresonances.Aphase
Theaboveresultswereobtainedwithadragcoefficientesti- transitiontoaQGPstateisnotincorporatedintothemodeldy-
matedwithperturbativeQCDadoptingalargecouplingcon- namics.Inthelatestversionofthemodel,PYTHIA(v.6.139)
stant a =0.6 [20]. The recent pQCD calculations [16, 17] was integrated to describe the energetic primary elementary
s
show a factor of 2-3 smaller drag coefficient. Besides, as collisions[24]. Measuredparticleyieldsandspectraarewell
shown in Ref. [17], non-perturbativecontributionsthat arise reproducedbytheapproach[24].
fromquasi-hadronicboundstatesintheQGPmightbeimpor- In this model, D mesons stem from the very early
tant. ThepresenceoftheseresonancesatmoderateQGPtem- stagehigh-energynucleon-nucleoncollisions,calculatedwith
peratureswouldresultinamuchlargerdragcoefficient.Since PYTHIA. For their propagationin the hadronicmedium, we
exactvaluesofthedragcoefficientfromlatticeQCDcalcula- considerelasticscatteringofDmesonswithallotherhadrons.
tionsdonotexist,weshowthesensitivityofourcalculations Thehadronicscatteringcross-sectionforD mesonsisgener-
whenvaryingtheparameterawithinreasonablevalues. Fig- allyconsideredtobesmall[25]andwetake2mbinourcal-
ures3(c)and(d)showthedragcoefficientdependenceofthe culation. Toevaluatethesensitivityofthecalculationstothis
ccangularcorrelationsforhigh-energycquarks: p =3and parameter, we have repeated them using a ten times higher
T
10 GeV/c for T =300 and 700 MeV, respectively. The cc cross section, 20 mb. The final state angular distribution of
0
angular correlations are washed out when a is increased by theelasticscatteringbetweenDmesonsandotherhadronscan
aroundafactorof5inthefirstcaseandbyaroundafactorof affectthe DD correlations. An isotropic angulardistribution
2inthesecond,withrespecttothepQCDvalue. in the scattering process, motivated by the idea of interme-
diate resonances formation (essentially D+p D ), has the
∗
↔
strongest effect on the charm correlations. For comparison,
(a) T = 300 MeV (b) T = 700 MeV we have also used a forward peaked distribution, simulated
2 0 0
pT (GeV/c) pT (GeV/c) accordingtothedistribution p(cosq )(cid:181) exp(7cosq ),whereq
0.5 1
1.5 1 3 isthescatteringangleinthecenterofmasssystemofthede-
2 6
1 3 10 cayingresonance.
TheresultsoftheUrQMDcalculationsareshowninFig.4,
-1d.) 0.5 for minimum bias Au+Au reactions at √sNN=200 GeV. To
a 0 better illustrate the change in the angular correlationsof the
(r DD pairs, we show the ratio between the final distribution,
fd (c) T = 300 MeV (d) T = 700 MeV
N/ 2 0 0 affected by the evolution with UrQMD, and the initial one,
d a 1x 10-6 (pT = 3 GeV/c) a 1x 10-6 (pT = 10 GeV/c) directly obtained from PYTHIA (see Fig. 2). For this spe-
1.5 2 2
6 3 cific analysis, all DD pairs were selected, irrespective of the
1 12 4 p oftheDmesons. WeobservethattheDDpaircorrelation,
20 6 T
0.5 evaluatedthroughtherelativeazimuthalangledistribution,is
barelyaffectedbyhadronicinteractions. Exceptinthesome-
0
whatdrasticcaseofusinga20mbhadroniccross-sectionand
0 0.5 1Df 1(r.5ad.2) 2.5 3 0 0.5 1Df 1(r.5ad.2) 2.5 3 eveninthiscaseonlyifweuseanisotropicscatteringangular
distribution(Fig.4(b))theinitialcorrelationsareweakened.
FIG. 3: (Color online) Correlations in relative azimuth Df of DD We also find that most scatterings happen at the earliest
pairs from Langevin calculations with T0 =300 MeV (RHIC) and times, when the hadron density is very high. At RHIC en-
700MeV(LHC).Theupperpartshowsthedependenceofthecorre- ergies, the (pure) hadronic stage is presumably shorter than
lationsontheinitialpTofthecquark;thelowerpartshowsthedrag assumedin the presentcalculations. Therefore,the hadronic
coefficientdependence.
stage shouldnotinducevisible changesto the measuredDD
4
angularcorrelations. Thus, a changeof the DD angularcor- reconstructionof the D mesons (i.e. of all their decay prod-
relation in heavy-ion collisions, is dominated by frequent ucts)infullazimuthisessentialtopreservethekinematicin-
parton-partonscatteringsoccurringintheQGPphase. formationandtooptimizetheacceptancefordetectingcorre-
latedDmesonpairs. Solidexperimentalmeasurementsinpp
and light-ion collisions, at the same energy, are crucial for
2.5 (a) forward (b) isotropic
detailed studies of changes in these azimuthal correlations,
p
|p+ 2 and should be performed as a function of pT. Upgrades of
fd theSTARandPHENIXdetectorswithmicro-vertexcapabil-
N/ 1.5
d ities [27] and direct open charm reconstructionshould make
m.b. /Au+Au 1 thesemeasurementspossible.
| Note that morethan one pair of DD couldbe producedin
fN/d 0.5 s D = 20 (mb) s D = 20 (mb) high-energynuclearcollisions.Atmid-rapidity,thetotalnum-
d 0 s D = 2 (mb) s D = 2 (mb) berofcharmpairsisintheorderof2-4and6-10atRHICand
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Df (rad.) Df (rad.) LHC, respectively. Themultipleproductionof un-correlated
Dmesonswillgiveanadditionalbackgroundtothemeasured
FIG.4:RatiosbetweentheDf distributionsofDDpairsproducedin angular correlations. Our tests show that the effect can be
minimumbiasAu+Aureactionsat√s =200GeV,beforeandafter removedby the mixed event method. The mixed-eventsub-
NN
hadronicrescattering,usingforward(leftpanel)andisotropic(right tracted correlationfunctioncan be comparedwith the calcu-
panel)angulardistributions.
lated results. In addition, the nextto leadingordercontribu-
tions, such as gluon splitting, to charm production may be-
Thecalculationsreportedaboveshowthatheavy-flavorcor- comemoreimportantin higherenergycollisions. TheseDD
relations provide a sensitive tool to directly access the sta- pairs are producedwithout a clear back-to-backcorrelations
tus of equilibration of the partonic medium. A similar pic- andwillleadtoamuchweakerpeakat180degrees. Inorder
tureemergesfromtheobservationandsuppressionofback-to- to understand the thermalization at RHIC, more theoretical
backcorrelationsofunidentifiedhigh pT hadrons(jets). Both analysesinthisdirectionarecalledfor.
features originate from the propagation of partons (heavy
Acknowledgements After finishing this work, P.-B. Gossi-
quarks or jets) in the medium. However, D mesons (and B
aux informed us about his similar unpublished studies. We
mesons) have the advantage that they can still be identified,
thank C. Lourenc¸o and H.K. Wo¨hri for valuable comments
eveniftheyloseasignificantfractionoftheirenergyandget
and several suggestions. This work has been supported by
kineticallyequilibrated.
GSIandBMBF,inpartbytheU.S.DepartmentofEnergyun-
Compared to the corresponding balance functions [26] of derContractNo.DE-AC03-76SF00098andtheNSFCGrants
electricchargeorstrangeness,heavy-flavorcorrelationshave No. 10428510.
obvious advantages: Heavy quarks and anti-quarks are pair
created in the early stage hard processes only and their sub-
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