Physics Letters B 647 (2007) 366–370

www.elsevier.com/locate/physletb

DD correlations as a sensitive probe for thermalization in high energynuclear collisions

X. Zhu a,b,c,∗, M. Bleicher a, S.L. Huang d, K. Schweda e, H. Stöcker a,b, N. Xu d, P. Zhuang c

a Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germanyb Frankfurt Institute for Advanced Studies (FIAS), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany

c Physics Department, Tsinghua University, Beijing 100084, Chinad Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

e Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

Received 26 September 2006; accepted 22 January 2007

Available online 28 February 2007

Editor: W. Haxton

Abstract

We propose to measure azimuthal correlations of heavy-flavor hadrons to address the status of thermalization at the partonic stage of lightquarks and gluons in high-energy nuclear collisions. In particular, we show that hadronic interactions at the late stage cannot significantly disturbthe initial back-to-back azimuthal correlations of DD pairs. Thus, a decrease or the complete absence of these initial correlations does indicatefrequent interactions of heavy-flavor quarks and also light partons in the partonic stage, which are essential for the early thermalization of lightpartons.© 2007 Elsevier B.V. All rights reserved.

PACS: 25.75.-q

1. Introduction

Lattice QCD calculations, at vanishing or finite net-baryondensity, predict a cross-over transition from the deconfinedthermalized partonic matter (the Quark Gluon Plasma, QGP) tohadronic matter at a critical temperature Tc ≈ 150–180 MeV [1].

Measurements of hadron yields in the intermediate and hightransverse momentum (pT) region indicate that dense matteris produced in Au + Au collisions at RHIC [2,3]. The experi-mentally observed large amount of elliptic flow of multi-strangehadrons [4], such as the φ meson and the Ω baryon, suggestthat collective motion develops in the early partonic stage ofthe matter produced in these collisions. A crucial issue to be ad-dressed next is the thermalization status of this partonic matter.

Heavy-flavor (c, b) quarks are particularly excellent tools [5–7] to study the thermalization of the initially created matter. As

* Corresponding author.E-mail address: zhu@fias.uni-frankfurt.de (X. Zhu).

0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.01.072

shown in Fig. 1, their large masses are almost exclusively gen-erated through their coupling to the Higgs field in the electro-weak sector, while masses of light quarks (u, d, s) are dom-inated by spontaneous breaking of chiral symmetry in QCD.This means that in a QGP, where chiral symmetry might be re-stored, light quarks are left with their bare current masses whileheavy-flavor quarks remain heavy. Due to their large masses(� ΛQCD), the heavy quarks can only be pair-created in earlystage pQCD processes. Furthermore, their production cross sec-tions in nuclear collisions are found to scale with the number ofbinary nucleon–nucleon collisions [8,9]. In the subsequent evo-lution of the medium, the number of heavy quarks is conservedbecause the typical temperature of the medium is much smallerthan the heavy quark (c, b) masses, resulting in negligible sec-ondary pair production. In addition, the heavy quarks live muchlonger than the lifetime of the formed high-density medium, de-caying well outside. These heavy quarks (c, b) can participatein collective motion or even kinetically equilibrate if, and onlyif, interactions at the partonic level occur at high frequency.The idea of statistical hadronization of kinetically equilibrated

X. Zhu et al. / Physics Letters B 647 (2007) 366–370 367

Fig. 1. Quark masses in the QCD vacuum and the Higgs vacuum. A large frac-tion of the light quark masses is due to the chiral symmetry breaking in theQCD vacuum. The numerical values were taken from Ref. [12].

charm quarks [10] even predicted significant changes in hiddencharm hadron production [11]. Hence, heavy-flavor quarks arean ideal probe to study early dynamics in high-energy nuclearcollisions.

Recent STAR and PHENIX results on elliptic flow and nu-clear modification factors of non-photonic electrons indicatethat charm quarks might indeed participate in the collective mo-tion of the matter produced in Au + Au collisions at RHIC.To explain the data, a large drag diffusion coefficient in theLangevin equation, describing the propagation of charm quarksin the medium, is found to be necessary [13]. Two- and three-body interactions [14,15] of heavy-quarks and resonant rescat-tering [17] in the partonic stage seem to become important.These investigations suggest that heavy-quarks actively partici-pate in the partonic stage.

In this Letter, we study the change of azimuthal correlationsof D and D meson pairs as a sensitive indicator of frequent oc-currences of partonic scattering. Since heavy-flavor quarks arepair-created by initial scattering processes, such as gg → cc,each quark–antiquark pair is correlated in relative azimuth �φ

due to momentum conservation. In elementary collisions, thesecorrelations survive the fragmentation process to a large extentand hence are observable in the distribution of the relative az-imuth of pairs of D and D mesons. We evaluate by how muchthese correlations should be affected by the early QGP stageand by the hadronic scattering processes in the late hadronicstage.

2. Results and discussions

We start by reviewing the D and D angular correlations inpp collisions. The Monte Carlo event generator PYTHIA [18]reproduces well the experimentally observed correlations of D

Fig. 2. Correlations in relative azimuth, �φ, of DD pairs from pp collisionsat

√s = 200 GeV, as calculated by PYTHIA (v. 6.139) for different ranges of

single D meson pT.

mesons, measured at fixed target energies [19]. Fig. 2 showssuch correlations as calculated by PYTHIA (v. 6.139) for pp

collisions at√

s = 200 GeV. The �φ distribution is peaked at180◦, especially for high pT D mesons, as one would expectfor back-to-back pairs stemming from hard scatterings of par-tons.

To explore how the QCD medium generated in central ultra-relativistic nucleus–nucleus collisions influences the correla-tions of D and D meson pairs, we employ a non-relativisticLangevin approach which describes the random walk of charmquarks in a QGP and was first described in Refs. [5–7],

d �pdt

= −γ (T ) �p + �η,

where �η is a Gaussian noise variable, normalized such that〈ηi(t)ηj (t

′)〉 = α(T )δij δ(t − t ′) with i, j indexing directions.Both the drag coefficient γ and the momentum-space diffusioncoefficient α depend on the local temperature, T . We use thesame parameterization for γ as in Ref. [5], γ (T ) = aT 2, witha = 2 × 10−6 (fm/c)−1 MeV−2 and also neglect a momen-tum dependence of γ . γ and α are related by the fluctuation-dissipation relation in equilibrium, 〈p2

i 〉 = α/2γ . As in Ref. [5],α is calculated from the above equation with 〈p2

i 〉 = 1.33mcT ,with a charm quark mass mc = 1.5 GeV/c2.

For simplicity, the initial conditions for central collisions ofheavy nuclei (Au or Pb) are the same as in Ref. [5]. We assumea plasma in thermal equilibrium occupying a cylinder of fixedradius R, equal to the radius of the colliding nuclei (R = 7 fmin the present calculations), at the time of its full formation τ0.After that, the plasma evolves according to Bjorken’s hydrody-namics, with a temperature dropping like T = T0 (τ0/τ)1/3. Be-fore the time τ0, the plasma is changing rapidly and is not fullyequilibrated. We assume (as done in Ref. [5]) that the plasma

368 X. Zhu et al. / Physics Letters B 647 (2007) 366–370

is already in equilibrium before τ0, with T = T0 and with thecharmed quarks diffusing in the same way as they do after τ0.In order to isolate the effects purely due to parton–parton re-scattering in the outlined medium, we generated c and c quarkswith the same pT and zero longitudinal momentum back-to-back, i.e., with �φ = π at a time of the order of 1/mc �0.1 fm/c, and used a delta function for fragmenting the charmquark into a charmed hadron at the hadronization stage. The ra-dius r where each pair is created is randomly generated from adistribution reflecting the number of binary nucleon–nucleoncollisions taking place at that radius, p(r)dr ∝ (R2 − r2) ×2πr dr . The angular distribution of the initially produced charmquarks in the transverse plane is isotropic.

The evolution of the charm momenta with the Langevinequation is stopped when T reaches the critical temperatureTc = 165 MeV or when the charm quark leaves the QGP vol-ume. Furthermore, to get a first estimation of the QGP effecton the charm quark pairs azimuthal correlation, we omit thepossible contribution of the mixed phase. Fig. 3(a) shows the re-sults for the D meson (charm quark) pairs angular correlationsfor different initial charm quark pT values, for T0 = 300 MeVand τ0 = 0.5 fm/c (typical values for RHIC collisions). We seethat the fastest charm quarks (represented by the pT = 3 GeV/c

red curve) are able to escape from the QGP without sufferingsignificant medium effects, while the slower quarks (see thepT = 0.5 GeV/c black line) have their pair azimuthal corre-lation almost completely smeared out by the interactions in themedium.

Fig. 3(b) shows the corresponding results for T0 = 700 MeVand τ0 = 0.2 fm/c, values representative of LHC energies.Here, the interactions of the charm quarks with the medium are

Fig. 3. Correlations in relative azimuth �φ of DD pairs from Langevin calcula-tions with T0 = 300 MeV (RHIC) and 700 MeV (LHC). The upper part showsthe dependence of the correlations on the initial pT of the c quark; the lowerpart shows the drag coefficient dependence. For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version of thisarticle.

so frequent that only the most energetic charm quarks preservepart of their initial angular correlation; low pT pairs can evenbe completely stopped by the medium. Because the c and thec quarks of a given pair are created together, in the same spacepoint, the pair has a higher escape probability if both quarksescape the medium from the side of the fireball where the thick-ness is smaller. Thus, the �φ distribution is shifted towardssmaller values. Presently, there is only longitudinal flow in thehydro-dynamical model used in the calculations. We expect thatthe strong radial flow will further enhance the same side corre-lations in the case of fully kinetically thermalized charm pairs.Because the two quarks of a pair are created in the same posi-tion in the medium, they will be pushed in the same directionby the radial flow. In fact, this mechanism is a known non-floweffect in the measurement of collective flow [21].

The above results were obtained with a drag coefficient es-timated with perturbative QCD adopting a large coupling con-stant αs = 0.6 [20]. The recent pQCD calculations [16,17] showa factor of 2–3 smaller drag coefficient. Besides, as shown inRef. [17], non-perturbative contributions that arise from quasi-hadronic bound states in the QGP might be important. The pres-ence of these resonances at moderate QGP temperatures wouldresult in a much larger drag coefficient. Since exact values ofthe drag coefficient from lattice QCD calculations do not exist,we show the sensitivity of our calculations when varying the pa-rameter a within reasonable values. Fig. 3(c) and (d) show thedrag coefficient dependence of the cc angular correlations forhigh-energy c quarks: pT = 3 and 10 GeV/c for T0 = 300 and700 MeV, respectively. The cc angular correlations are washedout when a is increased by around a factor of 5 in the first caseand by around a factor of 2 in the second, with respect to thepQCD value.

While the initial correlations of cc pairs are clearly affectedby the formed QCD medium, and sensitive to its temperatureand drag coefficient, it is important to evaluate to which extentthe loss of correlations can be mimicked by hadronic interac-tions. Therefore, we have investigated by how much the cc cor-relations deteriorate due to hadronic scattering in the late stageof high-energy nuclear collisions. For this purpose, we usedthe microscopic hadronic transport approach UrQMD (v. 2.2)[22,23].

This microscopic transport approach is based on the covari-ant propagation of constituent quarks and diquarks accompa-nied by mesonic and baryonic degrees of freedom. It simulatesmultiple interactions of in-going and newly produced particles,the excitation and fragmentation of color strings, and the for-mation and decay of hadronic resonances. A phase transitionto a QGP state is not incorporated into the model dynamics. Inthe latest version of the model, PYTHIA (v. 6.139) was inte-grated to describe the energetic primary elementary collisions[24]. Measured particle yields and spectra are well reproducedby the approach [24].

In this model, D mesons stem from the very early stage high-energy nucleon–nucleon collisions, calculated with PYTHIA.For their propagation in the hadronic medium, we considerelastic scattering of D mesons with all other hadrons. Thehadronic scattering cross-section for D mesons is generally

X. Zhu et al. / Physics Letters B 647 (2007) 366–370 369

Fig. 4. Ratios between the �φ distributions of DD pairs produced in mini-mum bias Au + Au reactions at

√sNN = 200 GeV, before and after hadronic

rescattering, using forward (left panel) and isotropic (right panel) angular dis-tributions.

considered to be small [25] and we take 2 mb in our calcu-lation. To evaluate the sensitivity of the calculations to thisparameter, we have repeated them using a ten times highercross section, 20 mb. The final state angular distribution ofthe elastic scattering between D mesons and other hadrons canaffect the DD correlations. An isotropic angular distributionin the scattering process, motivated by the idea of intermedi-ate resonances formation (essentially D + π ↔ D∗), has thestrongest effect on the charm correlations. For comparison, wehave also used a forward peaked distribution, simulated accord-ing to the distribution p(cos θ) ∝ exp(7 cos θ), where θ is thescattering angle in the center of mass system of the decayingresonance.

The results of the UrQMD calculations are shown in Fig. 4,for minimum bias Au + Au reactions at

√sNN = 200 GeV. To

better illustrate the change in the angular correlations of the DDpairs, we show the ratio between the final distribution, affectedby the evolution with UrQMD, and the initial one, directly ob-tained from PYTHIA (see Fig. 2). For this specific analysis, allDD pairs were selected, irrespective of the pT of the D mesons.We observe that the DD pair correlation, evaluated throughthe relative azimuthal angle distribution, is barely affected byhadronic interactions. Except in the somewhat drastic case ofusing a 20 mb hadronic cross-section and even in this case onlyif we use an isotropic scattering angular distribution (Fig. 4(b))the initial correlations are weakened.

We also find that most scatterings happen at the earliesttimes, when the hadron density is very high. At RHIC ener-gies, the (pure) hadronic stage is presumably shorter than as-sumed in the present calculations. Therefore, the hadronic stageshould not induce visible changes to the measured DD angularcorrelations. Thus, a change of the DD angular correlation inheavy-ion collisions, is dominated by frequent parton–partonscatterings occurring in the QGP phase.

The calculations reported above show that heavy-flavor cor-relations provide a sensitive tool to directly access the status ofequilibration of the partonic medium. A similar picture emergesfrom the observation and suppression of back-to-back corre-lations of unidentified high pT hadrons (jets). Both featuresoriginate from the propagation of partons (heavy quarks or jets)in the medium. However, D mesons (and B mesons) have the

advantage that they can still be identified, even if they lose asignificant fraction of their energy and get kinetically equili-brated.

Compared to the corresponding balance functions [26] ofelectric charge or strangeness, heavy-flavor correlations haveobvious advantages: Heavy quarks and anti-quarks are paircreated in the early stage hard processes only and their sub-sequent annihilations are negligible. Hence, both numbers ofheavy quarks and anti-quarks are conserved separately. On theother hand, only the net-charge or total strangeness is conservedwhile the number of positive (and negative) charges or strangeand anti-strange particles can be changed due to new pair pro-duction and annihilation throughout the whole lifetime of thefireball.

3. Conclusions and outlook

In summary, we argue that the observation of broadened an-gular correlations of heavy-flavor hadron pairs in high-energyheavy-ion collisions would be an indication of thermalizationat the partonic stage (among light quarks and gluons). We haveseen that hadronic interactions at a late stage in the collisionevolution cannot significantly disturb the azimuthal correla-tions of DD pairs. Thus, a visible decrease or the completeabsence of such correlations, would indicate frequent inter-actions of heavy-flavor quarks and other light partons in thepartonic stage, therefore imply early thermalization of lightquarks (which would be the main scattering centers) in nucleus–nucleus collisions at RHIC and LHC.

These measurements require good statistics of events inwhich both D mesons are cleanly reconstructed. A completereconstruction of the D mesons (i.e., of all their decay prod-ucts) in full azimuth is essential to preserve the kinematic in-formation and to optimize the acceptance for detecting corre-lated D meson pairs. Solid experimental measurements in ppand light-ion collisions, at the same energy, are crucial for de-tailed studies of changes in these azimuthal correlations, andshould be performed as a function of pT. Upgrades of the STARand PHENIX detectors with micro-vertex capabilities [27] anddirect open charm reconstruction should make these measure-ments possible.

Note that more than one pair of DD could be produced inhigh-energy nuclear collisions. At mid-rapidity, the total num-ber of charm pairs is in the order of 2–4 and 6–10 at RHIC andLHC, respectively. The multiple production of uncorrelated Dmesons will give an additional background to the measured an-gular correlations. Our tests show that the effect can be removedby the mixed event method. The mixed-event subtracted corre-lation function can be compared with the calculated results. Inaddition, the next to leading order contributions, such as gluonsplitting, to charm production may become more important inhigher energy collisions. These DD pairs are produced with-out a clear back-to-back correlations and will lead to a muchweaker peak at 180 degrees. In order to understand the thermal-ization at RHIC, more theoretical analyses in this direction arecalled for.

370 X. Zhu et al. / Physics Letters B 647 (2007) 366–370

Acknowledgements

After finishing this work, P.-B. Gossiaux informed us abouthis similar unpublished studies. We thank C. Lourenço andH.K. Wöhri for valuable comments and several suggestions.This work has been supported by GSI and BMBF, in part bythe US Department of Energy under Contract No. DE-AC03-76SF00098 and the NSFC Grant No. 10428510.

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