Charmonium production in heavy-ion collisions is investigated within a
kinetic theory framework incorporating in-medium properties of open- and
hidden-charm states in line with recent QCD lattice calculations. A
continuously decreasing open-charm threshold across the phase boundary of
hadronic and quark-gluon matter is found to have important implications for the
equilibrium abundance of charmonium states. The survival of $J/\psi$ resonance
states above the transition temperature enables their recreation also in the
Quark-Gluon Plasma. Including effects of chemical and thermal off-equilibrium,
we compare our model results to available experimental data at CERN-SPS and
BNL-RHIC energies. In particular, earlier found discrepancies in the
$\psi'/\psi$ ratio can be resolved.Comment: 4 pages RevTex including 4 eps-figures. v2: Minor modifications and
  clarifications, typos corrected, Fig. 4 update

E. V. Shuryak

Gerald E. Brown

Loïc Grandchamp

R. Rapp

Ralf Rapp

English

arXiv

Grandchamp, Lo��c

Rapp, Ralf

Brown, Gerald E.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-ph/0306077v2  9 Oct 2003In-Medium Effects on Charmonium Production in Heavy-Ion CollisionsLo¨ıc Grandchamp1,2, Ralf Rapp3,4 and Gerald E. Brown11Department of Physics, SUNY Stony Brook, NY 11794-38002IPN Lyon, 69622 Villeurbanne Cedex, France3NORDITA, Blegdamsvej 17, 2100 Copenhagen, Denmark4Cyclotron Institute and Physics Department,Texas A&M University, College Station, TX 77843(Dated: March 24, 2018)Charmonium production in heavy-ion collisions is investigated within a kinetic theory frameworkincorporating in-medium properties of open- and hidden-charm states in line with recent QCD latticecalculations. A continuously decreasing open-charm threshold across the phase boundary of hadronicand quark-gluon matter is found to have important implications for the equilibrium abundance ofcharmonium states. The survival of J/ψ resonance states above the transition temperature enablestheir recreation also in the Quark-Gluon Plasma. Including effects of chemical and thermal off-equilibrium, we compare our model results to available experimental data at CERN-SPS and BNL-RHIC energies. In particular, earlier found discrepancies in the ψ′/ψ ratio can be resolved.The production systematics of heavy-quark boundstates in (ultra-) relativistic collisions of heavy nuclei (A-A) is believed to encode valuable information on the hotand dense strong-interaction matter formed in these re-actions [1]. Based on the notion that charm quark pairs(cc¯) are exclusively created in primordial (hard) nucleon-nucleon (N -N) collisions, it has been suggested [2] that asuppression of observed J/ψ mesons in sufficiently centraland/or energetic A-A reactions signals the formation ofa deconfined medium (Quark-Gluon Plasma=QGP), astightly bound cc¯ states are conceivably robust in hadronicmatter. While theoretical [3] and (indirect) experimen-tal evidence [4] supports N -N collision-scaling of totalcharm production, it has recently been realized [5, 6, 7]that coalescence of c and c¯ quarks can induce significantregeneration of charmonium states in later stages of A-A collisions, especially if several pairs are present (e.g.,Ncc¯=10-20 in central Au-Au at RHIC). This is a directconsequence of (elastic) c-quark reinteractions, facilitat-ing the backward direction of charmonium dissociationreactions, J/ψ +X1 ⇋ X2 + c+ c¯ (D + D¯).Recent lattice computations of Quantum Chromody-namics (QCD) have revealed important information oncharm(onium) properties at finite temperature T , mostnotably: (i) an in-medium reduction of the open-charmthreshold which is surprisingly continuous even across thephase transition region [8], and, (ii) the survival of J/ψand ηc states as resonances in the QGP phase [9, 10],with essentially unmodified masses. Note that the use ofcharmonium spectral functions, in connection with ap-propriate asymptotic states, incorporates both (static)screening and (dynamical) dissociation mechanisms.In this letter, we propose an approach that implementscharm properties inferred from lattice QCD as micro-scopic in-medium effects into a kinetic rate equation. Itenables a comprehensive treatment of charmonium disso-ciation and regeneration across the phase transition, con-necting hadronic and QGP phases in a continuous way,not present in previous calculations. E.g., in the kineticapproach of Ref. [6], J/ψ’s were only considered in theQGP phase using their vacuum binding energy, whereasrecent transport calculations [11, 12] do not invoke thenotion of a (equilibrium) phase transition. In addition,in-medium open-charm masses will significantly affectthe equilibrium levels of charmonium, as employed in sta-tistical models [5, 7]. We thus conceptually improve ourearlier constructed “two-component” model [13], whichcombined statistical production at hadronization withsuppression in hadronic and QGP phases. Employinga schematic thermal fireball model, the rate equationsare applied to heavy-ion reactions at SPS and RHIC.We begin by evaluating in-medium masses of charmstates. Charmonium masses appear to be essentiallyunaffected at finite T , even above Tc (critical tempera-ture) [9, 10], which we assume from now on. UnquenchedQCD lattice studies of the free energy, FQQ¯(r, T ), of aheavy-quark pair [8], however, exhibit a plateau valuefor r → ∞ which gradually decreases with increasingT . In the hadron gas (HG), this can be interpreted asa reduction of D-meson masses (supported by QCD sumrules [14]) being driven by a reduced constituent light-quark mass,m∗q , due to (partial) chiral symmetry restora-tion [15]. We implement this also for other open-charmhadrons by employing a Nambu-Jona-Lasinio model cal-culation for m∗q at finite T and quark chemical poten-tial µq [16], neglecting heavy-light quark interaction andkinetic energies. For small µq, the typical light-quarkmass reduction amounts to ∆mq(Tc)≃140 MeV. To as-sess open-charm states in the QGP, a key element is theessentially continuous behavior of the open-charm thresh-old through Tc. Within a quark description above Tc, wepropose to identify this with an in-medium charm-quarkmass, m∗c≃1.6-1.7 GeV [17]. In fact, m∗c(Tc) is fixed suchthat the total open-charm density smoothly matches thehadronic side. The difference to the bare charm-quarkmass (mc≃1.3 GeV) is naturally attributed to a thermal2correlation energy of heavy quarks in the QGP.Next, we determine the thermal widths (or inverse life-times), Γψ=(τψ)−1, of charmonium states ψ. They arerelated to (inelastic) dissociation cross sections, σdissψi , byΓψ(T ) =∑i∫d3k(2pi)3f i(ωk;T ) σdissψi vrel , (1)where i runs over all matter constituents with ther-mal distributions f i. Within the QGP, parton-induced “quasifree” break-up, i + ψ → i + c + c¯(i=g,q,q¯), is utilized [13]. For small binding energies,EB=2m∗c -mψ∼<ΛQCD, the latter (with αs≃0.25) has beenshown [13] to dominate standard gluo-dissociation [18],g + ψ → c+ c¯. In addition, essentially unbound charmo-nium resonances such as ψ′ and χ-states can be treatedon an equal footing. The thermal J/ψ width turns out tobe ∼100 MeV at T≃250 MeV, reminiscent to (quenched)lattice results [10] under comparable conditions.For J/ψ’s in the HG, we compute inelastic cross sec-tions with pi- [19] and ρ-mesons within a flavor-SU(4)effective lagrangian formalism [20, 21], including in-medi-um charm hadron masses [22] as described above. Thefree χ and ψ′ cross sections are approximated by geo-metric scaling [23], supported by quark-model calcula-tions [24]. If the DD¯ threshold moves below the char-monium mass, we additionally evaluate direct decays,ψ → DD¯, accounting for wavefunction-node effects ac-cording to Ref. [25]. Despite the increase in availablephase space due to reduced masses, the charmonium life-times in the HG remain longer than in the QGP.In-medium charm properties have important conse-quences for charmonia in heavy-ion reactions. To illus-trate this point, consider the J/ψ number in thermalequilibrium, Neqψ =VFBneqψ (VFB: 3-volume), with thedensityneqψ (T, γc) = 3γ2c∫d3q(2pi)3fψ(mψ, T ) . (2)Here, we have included chemical off-equilibrium effectsvia the charm-quark fugacity γc which is adjusted to thetotal number, Ncc¯, of cc¯ pairs in the system via [5, 7]Ncc¯ =12γcNopI1(γcNop)I0(γcNop)+ VFB∑ηc,J/ψ,...nψ(T, γc) (3)where Nop=VFBnop(m∗c,D;T ) denotes the totalequilibrium-number of open-charm states in the thermalfireball. This procedure resides on the expectation thatessentially all charm-quarks are created in primordial N -N collisions, i.e., Ncc¯ does not chemically equilibrate [3].Thermal equilibration of charm quarks, however, isconceivable [27], implying a hierarchy of relaxationtimes as τchemc,c¯ ≫ τFB ≫ τ thermc,c¯ . The resulting J/ψequilibrium abundances, Neqψ , are shown in Fig. 1 under0.1500.2000.2500.3000.350T [GeV]10-510-410-310-210-1NJ/Ψeqmc = 1.5 GeVmc = 1.8 GeVin-med. Dvac. DRHIC-200SPS-17Central A-AFIG. 1: Equilibrium J/ψ abundances (with mψ(T ) ≡ mvacψ )in an isotropic, adiabatically expanding system at fixed Ncc¯for SPS and RHIC conditions for two values of the charm-quark mass above Tc, and for free and in-medium charmedhadron masses below Tc.conditions resembling central Pb-Pb (Au-Au) collisionsat SPS (RHIC). One observes a large sensitivity to the(in-medium) open-charm masses. At fixed Ncc¯, largervalues for m∗c (or m∗D) imply a thermal suppressionof open-charm states so that an increasing numberof anti-/charm quarks is redistributed into charmonia(cc¯ states). Enforcing continuity on the open-charmspectrum across Tc then has the interesting consequencethat, due to the volume increase in the hadronic phase(reducing γc), the equilibrium charmonium level on theQGP side is significantly larger than on the HG side [17],i.e., J/ψ formation is favored in the QGP.Also note that, due to a constant Ncc¯, the equilibriumJ/ψ numbers in the hadronic phase grow with decreas-ing temperature. Thus, if J/ψ’s equilibrate close to Tc,subsequent hadronic reactions will tend to increase theirabundance (which has indeed been found in transportcalculations at RHIC energies [11, 12]), quite contrary tothe commonly assumed hadronic dissociation.To model the time dependence, Nψ(τ), of charmoniain heavy-ion collisions, we utilize the dissociation (forma-tion) time τψ, Eq. (1), and equilibrium density, Eq. (2),within a kinetic rate equation,dNψdτ= − 1τψ[Nψ −Neqψ]. (4)The underlying temperature and volume evolution, T (τ)and VFB(τ), are taken from a (schematic) thermal fire-ball expansion which is consistent with observed hadro-chemistry and radial flow characteristics (as employedbefore for thermal dilepton production [28]).The simple form of the gain term in Eq. (4) resideson the assumption that the surrounding light and open-charm constituents (quarks or hadrons) are in thermalequilibrium. Following Ref. [23], we relax this assump-tion for c-quarks by introducing a thermal relaxation3time correction, R = 1 − exp(− ∫ dτ/τ thermc ), which re-duces Neqψ in the early phases. Varying τthermc within afactor of 2, the thermal yield is affected by ±10%.Another correction concerns the effective volume overwhich the charm quantum number is conserved (figuringinto Eq. (3) via the Bessel functions). Clearly, if only fewcc¯ pairs are present, their pointlike primordial productionimplies that they cannot explore the entire fireball vol-ume in the early stages[37]. This problem is well-knownfrom strange particle production at fixed target energies,where a phenomenological “correlation volume”, V0, hasbeen introduced to localize strangeness conservation [29].We adopt the same procedure here for local charm con-servation by replacing VFB(τ) in the argument of theBessel functions in Eq. (3) with V0(τ)=4pi(r0+〈vc〉τ)3/3.r0≃1.2 fm represents a minimal radius characterizing therange of strong interactions, and 〈vc〉≃0.5c is the aver-age relative speed of produced c and c¯ quarks as inferredfrom experimental D-meson p⊥-distributions [30]. Wechecked that our results are not very sensitive to the pa-rameterization for V0 (e.g., varying 〈vc〉 within a factorof 2 requires a change in the effective coupling constantαs by ±30%, to fit SPS data).The rate Eq. (4) is readily integrated over the fireballevolution of the collision for J/ψ, ψ′ and χ’s, once ini-tial conditions are specified. In an A-A reaction at givencentrality, Ncc¯ is determined by the collision-scaled N -N cross section. The initial charmonium numbers arealso assumed to follow hard production (i.e., empiricalfractions of Ncc¯ observed in N -N collisions), subject to(pre-equilibrium) nuclear absorption with a recently up-dated absorption cross section σabs=4.4 mb [31]. Typicalvalues for the fireball thermalization time range from 13(RHIC-200) to 1 fm/c (SPS).We first confront our approach to NA50/NA38 datain√sNN=17.3 GeV Pb-Pb at CERN-SPS. Fig. 2 dis-plays the ratio of J/ψ→µµ to Drell-Yan dimuons as afunction of centrality. The agreement between model(solid line) and data is fair for semi-/central collisions(at ET>100 GeV, the data can be reproduced by ac-counting for transverse energy fluctuations and losses inthe minimum bias analysis at impact parameters closeto zero [32], cf. Ref. [23]). Since the initial J/ψ num-ber is well above the equilibrium level (cf. lower arrow inFig. 1), J/ψ regeneration (the gain term in Eq. (4)) isvery moderate (dot-dashed line). Therefore, in line withour previous findings [13], J/ψ suppression is the maineffect at SPS energies.In peripheral collisions, the suppression appears to beslightly overestimated. We believe this discrepancy toreside in limitations of our fireball description. In par-ticular, thermalization is expected to be delayed (and/orincomplete) at large impact parameters due to less en-ergetic initial conditions. This is also borne out of hy-drodynamic models, which, e.g., reproduce the observedelliptic flow for mid-central collisions, but overestimate it0 20 40 60 80 100 120ET [GeV]05101520253035Bµµσ(J/Ψ)/σ(DY)2.9-4.5Pb-Pb, NA50suppression + regenerationsuppression onlyregenerationvariable τ0FIG. 2: Centrality dependence of J/ψ/Drell-Yan dimuonsat SPS; NA50 data [31] are compared to our results with(solid line) and without (dashed line) charmonium regenera-tion (represented by the dot-dashed line). The dotted line in-cludes longer thermalization times τ0 in peripheral collisions.0 50 100 150 200 250 300 350 400 450Npart00.0050.010.0150.02Bµµ(Ψ’)σΨ’/Bµµ(J/Ψ)σJ/ΨPb-Pb,NA50mDvacmD*average pp and pAthermal ratioFIG. 3: Centrality dependence of the ψ′/ψ data [34] at SPScompared to our results with (full line) and without (dashedline) in-medium reduced D-meson masses.for peripheral ones. A suitable increase of the equilibra-tion time by up to a factor of ∼3 [33] indeed improvesthe agreement at small ET , cf. dotted line in Fig. 2.Another important observable is the ψ′/ψ ratio. InRef. [23], the ψ′ dissociation rates were too small by afactor of ∼5 to account for NA50 data [34]. With in-medium D-meson masses, however, Γhadψ′ increases sub-stantially, primarily due to the opening of the ψ′ → DD¯decay channel. As a result, the ψ′/ψ data are reasonablywell described, cf. Fig. 3.We finally examine the impact of in-medium modi-fications at RHIC, cf. Fig. 4. Since reduced D-mesonmasses entail a lower J/ψ equilibrium level (cf. Fig. 1),the regeneration of J/ψ’s is somewhat less pronouncedthan the statistical production with free hadron massesin the 2-component model [23, 35]. Nevertheless, incentral Au-Au collisions, regenerated J/ψ’s still exceed40 50 100 150 200 250 300 350Npart00.050.10.150.20.250.3J/Ψ BeedN/dy(y=0) / Ncoll [x10-5 ]PHENIX directsecondarysumFIG. 4: J/ψ yield per binary N-N collision versus partici-pant number in√sNN=200 GeV Au-Au collisions. Prelim-inary PHENIX data [36] are compared to our model calcu-lations; dot-dashed line: thermal regeneration; dashed line:suppressed primordial production; band: total J/ψ yieldswith different values for in-medium open-charm masses.the suppressed primordial contribution, with the totalyield (solid curve) in line with first PHENIX data [36].The uncertainty in the charm-hadron mass reductionis illustrated by the band in Fig. 4, corresponding to80 MeV<∆mq(Tc)<250 MeV with accordingly matchedcharm-quark masses in the QGP. In any case, most ofthe regeneration occurs above Tc.To summarize, we have proposed a conceptually im-proved model of charmonium production in heavy-ioncollisions. In-medium modifications of both open-charmand charmonium states have been modeled in accordancewith recent finite-T QCD lattice calculations. The ap-parent reduction of the open-charm threshold with in-creasing T has been linked to (partial) chiral symme-try restoration in the SU(2)-sector via decreasing con-stituent light-quark masses in charmed hadrons. Thecontinuity of the open-charm threshold across the phasetransition has been encoded in thermal charm-quarkmasses in the QGP phase, together with T -independentcharmonium masses and J/ψ resonance states survivingabove Tc. Upon inclusion of chemical and thermal off-equilibrium effects, we solved kinetic rate equations forthe time evolution of charmonia in heavy-ion reactions.The results for J/ψ centrality dependences compare wellwith data from SPS and RHIC. The earlier predictedtransition from suppressed production at SPS to pre-dominant (thermal) regeneration at RHIC persists, withboth mechanisms residing on QGP formation. A muchimproved description of the ψ′/ψ ratio at SPS emergeddue to hadronic in-medium effects, accelerating ψ′ disso-ciation. Forthcoming measurements by NA60 (SPS) andPHENIX (RHIC) will be of great importance to scruti-nize the proposed approach.Acknowledgments: We thank U. Heinz and E.V. Shu-ryak for valuable discussions. L.G. is grateful for the hos-pitality at NORDITA and the Niels-Bohr Institute wherepart of this work was done within a Rosenfeld fellowship.This work was supported in part by the US Departmentof Energy under Grant No. DE-FG02-88ER40388.[1] R. Vogt, Phys. Rep. 310, 197 (1999).[2] T. Matsui, H. Satz, Phys. Lett. B178, 416 (1986).[3] P. Levai, B. Mu¨ller, X.-N. Wang, Phys. Rev. C51, 3321(1995).[4] PHENIX Collaboration (K. Adcox et al.), Phys. Rev.Lett. 88, 192303 (2002); Nucl. Phys. A715, 695 (2003).[5] P. Braun-Munzinger, J. Stachel, Phys. Lett. B490, 196(2000).[6] R.L. Thews, M. Schroedter, J. Rafelski, Phys. Rev. C63,054905 (2001).[7] M.I. Gorenstein et al., Phys. Lett. B509, 277 (2001).[8] F. Karsch, E. Laermann, A. Peikert, Nucl. Phys. B605,579 (2001).[9] S. Datta et al., hep-lat/0208012.[10] T. Umeda, K. Nomura, H. Matsufuru, hep-lat/0211003.[11] B. Zhang et al., Phys. Rev. C65, 054909 (2002).[12] E.L. Bratkovskaya, W. Cassing, H. Sto¨cker, Phys. Rev.C67, 054905 (2003).[13] L. Grandchamp, R. Rapp, Phys. Lett. B523, 60 (2001).[14] A. Hayashigaki, Phys. Lett. B487, 96 (2000).[15] S. Digal, P. Petreczky, H. Satz, Phys. Lett. B514, 57(2001).[16] V. Bernard, U.-G. Meissner, I. Zahed, Phys. Rev. D36,819 (1987).[17] R. Rapp, L. Grandchamp, hep-ph/0305143.[18] E.V. Shuryak, Sov. J. Nucl. Phys. 28, 408 (1978).[19] F.O. Dura˜es et al., nucl-th/0211092.[20] Z. Lin, C.M. Ko, Phys. Rev. C62, 034903 (2000).[21] K. Haglin, C. Gale, Phys. Rev. C63, 065201 (2001).[22] A. Sibirtsev et al., Phys. Lett. B484, 23 (2000).[23] L. Grandchamp, R. Rapp, Nucl. Phys.A709, 415 (2002).[24] C.-Y. Wong, E.S. Swanson, T. Barnes, Phys. Rev. C65,104903 (2001).[25] B. Friman, S.H. Lee, T. Song, Phys. Lett. B548, 153(2002).[26] D. Blaschke et al., hep-ph/0210265.[27] B. Svetitsky, Phys. Rev. D37, 2482 (1988).[28] R. Rapp, J. Wambach, Adv. Nucl. Phys. 25, 1 (2000).[29] S. Hamieh, K. Redlich, A. Tounsi, Phys. Lett. B486, 61(2000).[30] Beatrice (WA92) Collaboration (M. Adamovich et al.),Nucl. Phys. B495, 3 (1997).[31] NA50 Collaboration (L. Ramello et al.), Nucl. Phys.A715, 243 (2003).[32] A. Capella, A.B. Kaidalov, D. Sousa, Phys. Rev. C65,054908 (2002).[33] P. Kolb, private communication.[34] NA50 Collaboration (L. Ramello et al.), Nucl. Phys.A638, 261 (1998).[35] L. Grandchamp, R. Rapp, Nucl. Phys.A715, 545 (2003).[36] PHENIX Collaboration (S.S. Adler et al.),nucl-ex/0305030.[37] We thank U. Heinz for pointing this out to us, includinguseful discussions on the correlation volume.

In-Medium Effects on Charmonium Production in Heavy-Ion Collisions

Abstract

Similar works

Full text

Available Versions

OAKTrust Digital Repository (Texas A&M Univ)

HAL-IN2P3

HAL: Hyper Article en Ligne

Copenhagen University Research Information System

Texas A&M University

Crossref