Evidence for non-hadronic interactions of charm degrees of freedom in heavy-ion collisions at relativistic energies
aa r X i v : . [ nu c l - t h ] M a r Evidence for non-hadronic interactions ofcharm degrees of freedom in heavy-ioncollisions at relativistic energies
O. Linnyk, a , ∗ E. L. Bratkovskaya, a W. Cassing, b a Frankfurt Institute for Advanced Studies,60438 Frankfurt am Main,Germany b Institut f¨ur Theoretische Physik,Universit¨at Giessen,35392 Giessen,Germany
Abstract
Within the Hadron-String Dynamics (HSD) transport approach we study the sup-pression pattern of charmonia at RHIC with respect to centrality and rapidity em-ploying various model concepts such as variants of the ‘comover absorption’ model orthe ‘charmonium melting’ scenario. We find that especially the ratio of the forwardto mid-rapidity nuclear modification factors of J/ Ψ ( R forwardAA ( J/ Ψ) /R midAA ( J/ Ψ))cannot be explained by the interactions with ‘formed’ comoving mesons or by the‘color screening mechanism’ alone. Only when incorporating interactions of the c or¯ c quark with a pre-hadronic medium satisfactory results are obtained. A detailedcomparison to the PHENIX data demonstrates that non-hadronic interactions aremandatory to describe the narrowing of the J/ Ψ rapidity distribution from pp tocentral Au+Au collisions. The Ψ ′ to J/ Ψ ratio is found to be crucial in disentan-gling the different charmonium absorption scenarios especially in the RHIC energyrange. Furthermore, a comparison of the transport calculations to the statisticalmodel of Gorenstein and Gazdzicki as well as the statistical hadronization modelof Andronic et al. shows differences in the energy dependence as well as centralitydependence of the J/ Ψ to pion ratio which may be exploited experimentally to dis-entangle different concepts. We find additionally that the collective flow of charmin the HSD transport appears compatible with the data at SPS energies but sub-stantially underestimates the data at top RHIC energies such that the large ellipticflow v of charm seen experimentally has to be attributed to early interactions ofnon-hadronic degrees of freedom. Key words:
Relativistic heavy-ion collisions, Meson production, Quark-gluonplasma, Charmed mesons, Charmed quarksPACS 25.75.-q, 13.60.Le, 12.38.Mh, 14.40.Lb, 14.65.Dw ∗ corresponding author Email address: [email protected] (O. Linnyk,).
Preprint submitted to Elsevier 28 October 2018
Introduction
An investigation of the formation and suppression dynamics of J/ Ψ, χ c andΨ ′ mesons opens the possibility to address fundamental questions about theproperties of the state of matter at high temperature and density. Up to date,a simultaneous description of the seemingly energy-independent suppression of J/ Ψ together with its narrow rapidity distribution and a strong elliptic flow v of charmed hadrons - as found at the Relativistic-Heavy-Ion-Collider (RHIC)- has presented a challenge to microscopic theories. The large discrepanciesof present studies are striking in view of the success of the hadron-stringtransport theories in describing charmonium data at SPS energies. This haslead to the conjecture that the sizeable difference between the measured yieldsand transport predictions is due to a neglect of the transition from hadronic topartonic matter, e.g. a strongly-coupled Quark-Gluon-Plasma (sQGP). In thepresent work, we report new results on the charmonium nuclear modificationfactor R AA , rapidity distribution, the elliptic flow v of D mesons, the ratios h J/ Ψ i / h π i and Ψ ′ / ( J/ Ψ) for energies from about 20 A · GeV - relevant forthe future Facility-for-Antiproton-and-Ion-Research (FAIR) - up to top RHICenergies.We recall that in the early stage of the nucleus-nucleus collisions the disso-ciation and the regeneration of J/ Ψ by fundamentally different mechanismsare possible: The c ¯ c pairs produced early in the reaction - by gluon-gluon fu-sion in primary nucleon-nucleon interactions - might be completely dissociatedin the dense medium and not be formed as bound states due to color screening.In this model scenario charmonia have to be recreated by some mechanism toyield a finite production cross section of J/ Ψ and Ψ ′ . The c ¯ c pairs might alsobe formed in some pre-hadronic resonance (color-dipole) state that will furtherdevelop to the charmonium eigenstates in vacuum. Such resonance states canbe dissociated in the medium due to interactions with other degrees of freedombut also be recreated by the inverse reaction channels. Independently, char-monia might also be generated in a statistical fashion at the phase boundarybetween the QGP and an interacting hadron gas such that their abundancewould appear in statistical (chemical) equilibrium with the light and strangehadrons [1,2]. In the latter model the charmonium spectra carry no informa-tion on a possible preceeding partonic phase. Indeed, in Ref. [3] a success ofthe statistical hadronization model [4,5] has been put forward. Another alter-native is the model for coalescence of charmonium in the sQGP [6]. For furthervariants or model concepts for charmonium suppression/enhancement we referthe reader to the reviews [7,8]. In this work our aim is to shed some light onvarious model concepts by exploiting relativistic microscopic transport theory.The Hadron-String-Dynamics (HSD) approach [9] provides the space-time ge-ometry of nucleus-nucleus reactions and a rather reliable estimate for the lo-2al energy densities achieved, since the production of secondary particles withlight and single strange quarks/antquarks is described well from SIS to RHICenergies [10]. As we will show in Section 2, the high energy-densities reachedin Au + Au collisions at RHIC clearly indicate that a strongly interacting QGP(sQGP) has been created for a couple of fm/c in the central overlap volume.However, a proper understanding of the transport properties of the partonicphase is still lacking. Presently the effective degrees of freedom in the sQGPare much debated and charmonia are a unique and promising probe that issensitive to the properties of the early (and so far unknown) medium.In the present systematic study we first test the HSD results for charmoniumproduction in p + p and d + Au reactions at RHIC energies in comparisonto the recent data. This is crucial in order to obtain an accurate baseline forthe study of any anomalous suppression of charmonia in nucleus-nucleus col-lisions (see Sections 3 and 4). The interactions of J/ Ψ’s with mesons in thelate stages of the collision (when the energy density falls below a critical valueof about 1 GeV/fm corresponding roughly to the critical energy density for aparton/hadron phase transition) gives a sizable contribution to its anomaloussuppression at all beam energies as demonstrated in Refs. [11,12,13,14,15].Accordingly, this more obvious ‘hadronic’ contribution has to be incorpo-rated when comparing possible models for QGP-induced charmonium sup-pression to experimental data. On the other hand, as known from our studiesin Refs. [11,12] charmonium interactions with the purely hadronic mediumalone (which is modeled rather precisely by HSD) are not sufficient to de-scribe the J/ Ψ suppression pattern at RHIC in detail.Based on the microscopic HSD transport theory, we investigate in particularthe following scenarios for the anomalous absorption of charmonia:(1) the ‘threshold melting’ mechanism;(2) a dissociation by the scattering on hadron-like correlators, i.e. the ‘co-mover’ scenario;(3) additional scattering of charm with pre-hadrons which might be consid-ered as color neutral precursors of hadronic states (cf. Refs. [16,17,18,19]).All implemented scenarios will be described in detail in Section 5. In Section 6we will investigate in particular the effect of the interactions of charm quarksin the pre-hadronic medium on R AA ( y ) of J/ Ψ by comparing our calculationsto RHIC data. To complete our study, we will provide excitation functions forthe J/ Ψ survival probability S ( J/ Ψ) and the ratios B µµ σ ( J/ Ψ) /B ′ µµ σ (Ψ ′ ) inSection 7. Furthermore, by studying the J/ Ψ to π ratio as a function of thenumber of participating nucleons N part , we will test the assumption of char-monium production by statistical hadronization as advocated in Refs. [2,3,20](subsection 7.2). A summary of results as well as a discussion of open problemswill close our study in Section 8. 3 Au+Au, s =200 GeVb=1 fm e ( , , z ) [ G e V / f m ] z [f m ] t i m e [f m / c ] Fig. 1. The energy density ε ( x = 0 , y = 0 , z ; t ) from HSD for a central Au+Aucollision at √ s = 200 GeV. The time t is given in the nucleon-nucleon center-of-masssystem. The HSD transport model - employed in a large variety of π + A , p + A , d + A and A + A reactions - allows to calculate the energy-momentum tensor T µν ( x ) for allspace-time points x and thus the energy density ε ( x ) = T ( x ) in the local restframe. In order to exclude contributions to T µν from noninteracting nucleonsin the initial phase all nucleons without prior interactions are discarded in therapidity intervals [ y tar − . , y tar + 0 .
4] and [ y pro − . , y pro + 0 .
4] where y tar and y pro denote projectile and target rapidity, respectively. Note that the initialrapidity distributions of projectile and target nucleons are smeared out dueto Fermi motion by about ± .
4. Some comments on the choice of the grid inspace-time are in order here: In the actual calculation (for Au+Au collisions)the initial grid has a dimension of 1 fm × × γ cm fm, where γ cm denotesthe Lorentz γ -factor in the nucleon-nucleon center-of-mass system. After thetime of maximum overlap t m of the nuclei the grid-size in beam direction∆ z = 1 /γ cm [fm] is increased linearly in time as ∆ z = ∆ z + a ( t − t m ), wherethe parameter a is chosen in a way to keep the particle number in the localcells of volume ∆ V ( t ) = ∆ x ∆ y ∆ z ( t ) roughly constant during the longitudinalexpansion of the system. In this way local fluctuations of the energy density ε ( x ) due to fluctuations in the particle number are kept low. Furthermore,the time-step is taken as ∆ t = 0 . z ( t ) and increases in time in analogy to4 z ( t ). This choice provides a high resolution in space and time for the initialphase and keeps track of the relevant dynamics throughout the entire collisionhistory.The energy density ε ( r ; t ) – which is identified with the matrix element T ( r ; t )of the energy momentum tensor in the local rest frame at space-time ( r , t ) –becomes very high in a central Au+Au collision at √ s = 200 GeV as shownin Fig.1 ( cf. Fig. 1 of [11] for the corresponding energy density evolution incase of central collisions at top SPS energies). Fig. 1 shows the space-timeevolution of the energy density ε ( x = 0 , y = 0 , z ; t ) for a Au+Au collision at21300 AGeV or √ s = 200 GeV. It is clearly seen that energy densities above16 GeV/fm are reached in the early overlap phase of the reaction and that ε ( x ) drops after about 6 fm/c (starting from contact) below 1 GeV/fm inthe center of the grid. On the other hand the energy density in the region ofthe leading particles - moving almost with the velocity of light - stays above1 GeV/fm due to Lorentz time dilatation since the time t in the transportcalculation is measured in the nucleon-nucleon center-of-mass system. Notethat in the local rest frame of the leading particles the eigentime τ is roughlygiven by τ ≈ t/γ cm . As seen from Fig. 1, the energy density in the local restframe is a rapidly changing function of time in nucleus-nucleus collisions. Fororientation let us recall the relevant time scales (in the cms reference frame):– The c ¯ c formation time τ c ≈ /M ⊥ is about 0.05 fm/c for a transverse massof 4 GeV; the transient time for a central Au+Au collision at √ s = 200 GeVis t r ≈ R A /γ cm ≈ .
13 fm/c. According to standard assumptions, the c ¯ c pairsare produced in the initial hard N N collisions dominantly by gluon fusion inthe time period t r . In fact, the formation time τ c is significantly smaller than t r , which implies the c or ¯ c quarks may interact with the impinging nucleonsof the projectile or target for times t ≤ t r .– Using the Borken estimate for the energy density and employing the time-scale t r = 0 .
13 fm/c, the energy density – after the nuclei have punchedthrough each other – amounts to about 5 / . >
30 GeV/fm as quoted alsoin the HSD calculations in Ref. [11]. Even when adding the c ¯ c formation time,this gives an energy density ∼ / . ≈
28 GeV/fm . So these numbers agreewith transparent and simple estimates (cf. Fig. 1) and illustrate the high initialdensities after c ¯ c production from primary interactions.The energy densities quoted above are considerably different from the Bjorkenestimate τ · ǫ Bj = < E T > dNdη πR T , (1)where < E T > is the average transverse energy per particle, dN/dη the number5
100 200 300 400012345 0 100 200 300 40001234567
PHENIX
HSD
Au+Au, s =200 GeV d E T / d h / . N p a r t [ G e V ] N part PHENIX
HSD
Au+Au, s =200 GeV e B j * t [ G e V / f m / c ] N part Fig. 2. Left part: The energy density E T per pseudorapidity interval dη dividedby the number of participant pairs (0 . N part ) from HSD (solid line) in comparisonto the PHENIX data (dots) [21]. Right part: The Bjorken energy density ε Bj · τ from HSD (solid line) for Au+Au collisions at √ s = 200 GeV in comparison to thePHENIX data (dots) [21]. of particles per unit of pseudorapidity, and τ a formation time parameteroften used as τ = 1 fm/c. Furthermore, πR T denotes the overlap area for thecorresponding centrality. Is is important to point out that the estimate (1) isonly well defined for the product τ ǫ Bj ! The question naturally arises, if thetransport calculations follow the corresponding experimental constraints.To this aim we show dE T /dη (divided by half the number of participants N part )from HSD (l.h.s.) in comparison to the measurements by PHENIX [21]. Ac-cordingly, the Bjorken energy density ǫ Bj – multiplied by the time-scale τ (1)–from HSD is shown additionally in the r.h.s. in comparison to the PHENIXmeasurements as a function of N part . The similarity between the calculatedquantities and the experimental data demonstrates that the space-time evo-lution of the energy-momentum tensor T µν in HSD is sufficiently well undercontrol. We now may step on with the actual investigation of the charmoniumdynamics. In order to examine the dynamics of open charm and charmonium degreesof freedom during the formation and expansion phase of the highly excitedsystem created in a relativistic nucleus-nucleus collision, one has to knowthe number of initially produced particles with c or ¯ c quarks, i.e. D, ¯ D, D ∗ ,¯ D ∗ , D s , ¯ D s , D ∗ s , ¯ D ∗ s , J/ Ψ(1 S ) , Ψ ′ (2 S ) , χ c (1 P ). In this work we follow the previ- The open source code is available from Ref. [22] y HSD PHENIX pp->J/ Y +X, s =200 GeV B mm d s / d y [ nb ] Fig. 3. Cross section for the differential J/ Ψ production in rapidity (times thebranching ratio to di-muons B µµ ) in pp collisions at √ s = 200 GeV. The HSD(input) parametrization (solid line) is compared to the PHENIX data (symbols)from Ref. [23]. ous studies in Refs. [9,11,13,14,24] and fit the total charmonium cross sections( i = χ c , J/ Ψ , Ψ ′ ) from N N collisions as a function of the invariant energy √ s by the expression [12] σ NNi ( s ) = f i a − m i √ s ! α √ sm i ! β θ ( √ s − √ s i ) , (2)where m i denotes the mass of charmonium i while √ s i = m i + 2 m N is thethreshold in vacuum. The parameters in (2) have been fixed to describe the J/ Ψ and Ψ ′ data up to the RHIC energy √ s = 200 GeV ( cf. [11]). We use a = 0 .
16 mb, α = 10, β = 0 . a by ∼ f i are fixed as f χ c = 0 . , f J/ Ψ = 0 . , f Ψ ′ = 0 .
21 in orderto reproduce the experimental ratio B ( χ c → J/ Ψ) σ χ c + B ( χ c → J/ Ψ) σ χ c σ expJ/ Ψ = 0 . ± . pp and πN reactions [25,26] as well as the averaged pp and pA ratio ( B µµ (Ψ ′ ) σ Ψ ′ ) / ( B µµ ( J/ Ψ) σ J/ Ψ ) ≃ . cf. the compilation of exper-imental data in Ref. [27]). The experimentally measured J/ Ψ cross section7ncludes the direct J/ Ψ component ( σ J/ Ψ ) as well as the decays of highercharmonium states χ c and Ψ ′ , i.e. σ expJ/ Ψ = σ J/ Ψ + B ( χ c → J/ Ψ) σ χ c + B (Ψ ′ → J/ Ψ) σ Ψ ′ . (3)Note, we do not distinguish the χ c (1 P ) and χ c (1 P ) states. Instead, we useonly the χ c (1 P ) state (which we denote as χ c ), however, with an increasedbranching ratio for the decay to J/ Ψ in order to include the contribution of χ c (1 P ), i.e. B ( χ c → J/ Ψ) = 0 .
54. Furthermore, we adopt B (Ψ ′ → J/ Ψ) =0 .
557 from Ref. [28].We recall that (as in Refs. [13,14,29,30,31]) the charm degrees of freedom inthe HSD approach are treated perturbatively and that initial hard processes(such as c ¯ c or Drell-Yan production from N N collisions) are ‘pre-calculated’to achieve a scaling of the inclusive cross section with the number of projectileand target nucleons as A P × A T when integrating over impact parameter. Forfixed impact parameter b , the c ¯ c yield then scales with the number of binaryhard collisions N coll ( cf. Fig. 8 in Ref. [13]).In addition to primary hard
N N collisions, the open charm mesons or char-monia may also be generated by secondary meson-baryon ( mB ) reactions.Here we include all secondary collisions of mesons with baryons by assumingthat the open charm cross section (from Section 2 of Ref. [13]) only dependson the invariant energy √ s and not on the explicit meson or baryon state.Furthermore, we take into account all interactions of ‘formed’ mesons – aftera formation time of τ F = 0.8 fm/c (in their rest frame) [32] – with baryonsor diquarks. For the total charmonium cross sections from meson-baryon (or πN ) reactions we use the parametrization (in line with Ref. [33]): σ πNi ( s ) = f i b − m i √ s ! γ (4)with γ = 7 . b = 1 .
24 mb, which describes the existing experimental dataat low √ s reasonably well, as seen in Ref. [11].Apart from the total cross sections for charmonia we also need the differentialdistribution of the produced mesons in the transverse momentum p T and therapidity y (or Feynman x F ) from each individual collision. We recall that x F = p z /p maxz ≈ p z / √ s with p z denoting the longitudinal momentum. Forthe differential distribution in x F from N N and πN collisions we use theansatz from the E672/E706 Collaboration [34] and for the p T distribution apower low parametrization from Ref. [35] which has been fixed by the STAR8
10 20 30 4010 -3 -2 -1 -3 -2 -1 | y | < -> J/ Y +X, s =200 GeV PHENIX HSD / ( p p T ) B ll d s / d y dp T [ nb / ( G e V / c ) ] p T2 [GeV/c] PHENIX HSD1.2 < | y | < -> J/ Y +X, s =200 GeVp T2 [GeV/c] Fig. 4. Differential cross section of J/ Ψ production in pp collisions at √ s = 200 GeVat mid-rapidity ( | y | < .
35, l.h.s.) and at forward rapidity (1 . < | y | < .
2, r.h.s.) vs the transverse momentum squared p T as implemented in HSD (solid line) comparedto the PHENIX data from Ref. [23] (dots). data [36], i.e. dNdx F dp T ∼ (1 − | x F | ) c p T b p T ! c pT , (5)with b p T = 3 . c and c p T = − .
3. The exponent c is given by c = a/ (1 + b/ √ s ) and the parameters a, b are chosen as a NN = 16, b NN = 24 . N N collisions and a πN = 4 . b πN = 10 . πN collisions.The resulting rapidity distribution for J/ Ψ production in pp collisions at √ s =200 GeV is shown in Fig. 3 which is in line with the data from Ref. [23] withinerror bars. We also present the pp → J/ Ψ + X differential cross section in p T at mid-rapidity ( | y | < .
35) and at forward rapidity (averaged in the interval1 . < | y | < .
2) in Fig. 4. The HSD parametrization is compared to therecent measurements of the corresponding quantities by PHENIX [23]. Thetotal cross sections for D + ¯ D production in this study are the same as thosepresented in Ref. [11].The parametrizations of the total and differential cross sections for open charmmesons from pN and πN collisions are taken as in Refs. [13,14], apart from areadjustment of the parameter a NN in order to reproduce the recently mea-sured rapidity distribution of J/ Ψ in p + p reactions at √ s = 200 GeV byPHENIX [23].For D, D ∗ , ¯ D, ¯ D ∗ - meson ( π, η, ρ, ω ) scattering we refer to the calculationsfrom Ref. [37,38] which predict elastic cross sections in the range of 10–20 mbdepending on the size of the form factor employed. As a guideline we use aconstant cross section of 10 mb for elastic scattering with formed mesons andalso baryons, although the latter might be even higher for very low relative9 s =200 GeV R d A y PHENIX HSD
Fig. 5. J/ Ψ production cross section in d + Au collisions relative to that in p + p collisions (see text for the definition of R dA ) in HSD (red stars) as compared to thePHENIX data [39] (full dots). momenta. We will discuss this issue in more detail in Section 5.1. The yield of J/ Ψ in p + A and A + A reactions is modified compared to that in p + p scaled with the number of initial binary scatterings N coll [40,41]. Indeed,the produced c ¯ c can be dissociated or absorbed on either the residual nucleusof the projectile or target or on light co-moving particles (usually on mesonsor, at high energy, on partons) produced in the very early phase. The latterreactions are only important in nucleus-nucleus collisions and not in p + A or d + A as the number of ‘comovers’ created in proton- or deuteron-inducedprocesses is small. In contrast, charmonium absorption on baryons is the lead-ing suppression mechanism in d + A ( p + A ) scattering and is an importantbase-line for the study of the absorption in the hot and dense medium createdin A + A reactions.In order to study the effect of charmonium rescattering on projectile/targetnucleons, we adopt in HSD the following dissociation cross sections of char-monia with baryons independent of the energy: σ c ¯ cB = 4 .
18 mb; (6) σ J/ Ψ B = 4 .
18 mb; σ χ c B = 4 .
18 mb; σ Ψ ′ B = 7 . .
10n (6) the cross section σ c ¯ cB stands for a (color dipole) pre-resonance ( c ¯ c ) -baryon cross section, since the c ¯ c pair produced initially cannot be identifiedwith a particular charmonium due to the uncertainty relation in energy andtime. For the life-time of the pre-resonance c ¯ c pair (in it’s rest frame) a valueof τ c ¯ c = 0.3 fm/c is assumed following Ref. [42]. This time scale correspondsto the mass difference of the Ψ ′ and J/ Ψ.The values for the cross sections σ J/ Ψ N , σ c ¯ cN at RHIC energies are currentlydebated in the literature. On one side, all the data on the J/ Ψ production in p + A at energies √ s ≤
40 GeV were found to be consistent with an energy-independent cross section of the order of 4 − √ s = 200 GeV, e.g. at RHIC, are expected to be smaller [47], since part ofthe suppression might be attributed to other (initial-state) cold-matter effects,such as gluon shadowing [48,49,50], radiative gluon energy loss in the initialstate or multiple gluon rescattering. We recall that ‘shadowing’ is a depletionof low-momentum partons in a nucleon embedded in a nucleus compared tothe population in a free nucleon, which leads to a lowering in the charmoniumproduction cross section. The reasons for depletion, though, are numerous,and models of shadowing vary accordingly. There is, therefore, a considerable(about a factor of 3) uncertainty in the amount of shadowing predicted atRHIC [48,49,50,51,52]. In the analysis of the d + Au data at √ s = 200 GeV, inwhich the maximum estimate for the effect of the shadowing was made [47,50],the additional absorption on baryons allowed by the data was found to lead to σ J/ Ψ N = 1 − σ J/ Ψ N = 3 mb in order to preserve the agreementwith the data of the Fermilab experiment E866. The PHENIX Collaboration[39] finds a breakup cross section of 2 . +1 . − . mb (using EKS shadowing) whichstill overlaps with the CERN value of 4.18 mb (though with large error bars).However, the theoretical uncertainty is still large, since in the works aboveonly an approximate model for baryonic absorption was applied and not amicroscopic transport approach that e.g. also includes secondary productionchannels of charm pairs as described in Section 3.Within HSD we have found the baryoninc absorption cross sections (6) toagree with the data at SPS energies [11]. In Fig. 5 we compare the HSD result(employing the same cross sections (6) for baryonic absorption and neglectingshadowing) for the J/ Ψ production in d + Au collisions at √ s = 200 GeV to theinclusive PHENIX data [39]. The quantity plotted is the nuclear modificationfactor defined as R dA ≡ dN dAuJ/ Ψ /dy h N coll i · dN ppJ/ Ψ /dy , (7)11 R d A N coll |y|< 0.35 R d A s =200 GeV-2.2 < y < -1.2 R d A Fig. 6. The ratio R dA (7) for backward, central and forward rapidity bins as afunction of the number of binary collisions N coll for d + Au at √ s = 200 GeV. Theexperimental data have been taken from Ref. [39]. The HSD results (stars connectedby red dashed lines) show calculations without including low- x gluon shadowing andslightly overestimate R dA in the forward interval 1.2 < y < where dN dAuJ/ Ψ /dy is the J/ Ψ invariant yield in d + A collisions, dN ppJ/ Ψ /dy is the J/ Ψ invariant yield in p + p collisions; h N coll i is the average numberof binary collisions for the same rapidity bin. In our analysis we have used h N coll i = 7 . ± . R dA with rapidity, however, with a tendency to overshoot at forward rapidity.Within error bars we find the values of σ c ¯ cB from (6) to be compatible with12he inclusive RHIC measurement as well as with the lower energy data [44].This finding is also in line with the analysis of the PHENIX Collaboration inRef. [39]In order to shed some further light on the role of shadowing, we compare ourcalculations for R dA in different rapidity bins as a function of the centrality ofthe d + Au collision, which in Fig. 6 is represented by the number of binarycollisions N coll . The latter number is directly taken from the number of binaryhard N N collisions in the transport calculation while the comparison withexperiment is based on a Glauber model analysis of the data similar to thatperformed in Ref. [53]. The actual results displayed in Fig. 6 (stars connectedby dashed lines) and the PHENIX data from Ref. [39] are roughly compatiblefor the rapidity intervals -2.2 < y < -1.2 and | y | < < y < Au + Au collisions: There are ‘cold nuclear mat-ter effects’ such as ‘gluon shadowing’ beyond those incorporated in the trans-port calculations, and especially quantitative statements about any ‘agreementwith data’ might have to be reconsidered. In case of Au + Au reactions theshadowing from projectile/target will show up symmetrically around y = 0and in part contribute to the stronger J/ Ψ suppression at forward/backwardrapidities. Nevertheless, following Granier de Cassagnac [53], an anomaloussuppression of J/ Ψ beyond ‘cold nuclear matter effects’ is clearly present inthe Au + Au data to be investigated below. J/ Ψ suppression It is well known that the baryonic (normal) absorption alone cannot explainthe suppression of charmonia in heavy-ion collisions with increasing centrality[7]. We have implemented in HSD several different mechanism for the addi-tional (anomalous) suppression of charmonia which will be explained in thefollowing Subsections. By comparing the results from these scenarios to eachother and to the available data the mechanism of charmonium interactionswith the medium can be probed. 13 .1 ‘Comover’ suppression (and recombination)
First of all let us stress that the interactions with ‘comoving’ mesons leadnot only to the dissociation of charmonia, but also to their recreation via theinverse recombination process D + ¯ D → c ¯ c + m , where m = { π, ρ, ω, K, ... } . Asalready pointed out before, the J/ Ψ , χ c , Ψ ′ formation cross sections by opencharm mesons or the inverse ‘comover’ dissociation cross sections are not wellknown and the significance of these channels is discussed controversely in theliterature [4,6,54,55,56,57,58]. We here follow the concept of Refs. [14,15] andintroduce a simple 2-body transition model with a single parameter | M | , thatallows to implement the backward reactions uniquely by employing detailedbalance for each individual channel.Since the charmonium-meson dissociation and backward reactions typicallyoccur with low relative momenta (‘comovers’), it is legitimate to write thecross section for the process 1 + 2 → σ → ( s ) = 2 E E E E s | ˜ M i | m + m √ s ! p f p i , (8)where E k denotes the energy of hadron k ( k = 1 , , , √ s are given by p i = ( s − ( m + m ) )( s − ( m − m ) )4 s ,p f = ( s − ( m + m ) )( s − ( m − m ) )4 s , (9)where m k denotes the mass of hadron k . In (8) | ˜ M i | ( i = χ c , J/ Ψ , Ψ ′ ) standsfor the effective matrix element squared, which for the different 2-body chan-nels is taken of the form | ˜ M i | = | M i | for ( π, ρ ) + ( c ¯ c ) i → D + ¯ D (10) | ˜ M i | = 3 | M i | for ( π, ρ ) + ( c ¯ c ) i → D ∗ + ¯ D, D + ¯ D ∗ , D ∗ + ¯ D ∗ | ˜ M i | = 13 | M i | for ( K, K ∗ ) + ( c ¯ c ) i → D s + ¯ D, ¯ D s + D | ˜ M i | = | M i | for ( K, K ∗ ) + ( c ¯ c ) i → D s + ¯ D ∗ , ¯ D s + D ∗ , D ∗ s + ¯ D, ¯ D ∗ s + D, ¯ D ∗ s + D ∗ The relative factors of 3 in (10) are guided by the sum rule studies in [59]which suggest that the cross section is increased whenever a vector meson D ∗ or ¯ D ∗ appears in the final channel while another factor of 1/3 is introduced14or each s or ¯ s quark involved. The factor (( m + m ) / √ s ) in (8) accounts forthe suppression of binary channels with increasing √ s and has been fitted tothe experimental data for the reactions π + N → ρ + N, ω + N, φ + N, K + + Λin Ref. [60].We use the same matrix elements for the dissociation of all charmonium states i ( i = χ c , J/ Ψ , Ψ ′ ) with mesons: | M J/ Ψ | = | M χ c | = | M Ψ ′ | = | M | . (11)We note for completeness that in Ref. [14] the parameter | M | was fixed bycomparison to the J/ Ψ suppression data from the NA38 and NA50 Collabo-rations for S+U and Pb+Pb collisions at 200 and 158 AGeV, respectively. Ina later study [11], however, this parameter has been readjusted in accordancewith the updated value of the cross section (6) of charmonium dissociationon baryons (following the latest NA50 and NA60 analysis [40,43]). The bestfit is obtained for | M | = 0 .
18 fm /GeV ; this value will be employed in ourfollowing studies, too.The advantage of the model introduced in [14,15] is that detailed balance forthe binary reactions can be employed strictly for each individual channel, i.e. σ → ( s ) = σ → ( s ) (2 S + 1)(2 S + 1)(2 S + 1)(2 S + 1) p i p f , (12)and the role of the backward reactions (( c ¯ c ) i +meson formation by D + ¯ D fla-vor exchange) can be explored without introducing any additional parameteronce | M | is fixed. In Eq. (12) the quantities S j denote the spins of the parti-cles, while p i and p f denote the cms momentum squared in the initial and finalchannels, respectively. The uncertainty in the cross sections (12) is of the sameorder of magnitude as that in Lagrangian approaches using e.g. SU (4) flavor symmetry [37,38], since the form factors at the vertices are essentially un-known [59]. It should be pointed out that the ‘comover’ dissociation channelsfor charmonia are described in HSD with the proper individual thresholds foreach channel in contrast to the more schematic ‘comover’ absorption model inRefs. [52,61].The regeneration of charmonia by recombination of D ( D ∗ ) mesons in thehadronic phase was first studied by C.M. Ko and collaborators in [58]. The con-clusion at that time was that this process is unlikely at RHIC energies [55,58,62].On the other hand, it has been shown within HSD [14] that the contributionof the D + ¯ D annihilation to the produced J/ Ψ at RHIC is considerable.Moreover, the equilibrium in the reaction J/ Ψ + m ↔ D ¯ D is reached (i.e.the charmonium recreation is comparable with the dissociation by ‘comov-15ng’ mesons). The reason for such differences is that the pioneering study [58]within the hadron gas model was confined to J/ Ψ reactions with π ’s into twoparticular D ¯ D channels ( D + ¯ D ∗ and D ∗ + ¯ D ∗ ). On the contrary, in Ref. [14]the interactions with all mesons into all possible combinations of D ¯ D stateshave been taken into account. Note that the ρ -meson density at RHIC is largesuch that the channel with the most abundant ρ -meson resonance is domi-nant. Furthermore, in Ref. [14] the feed down from χ c and Ψ ′ decays has beenconsidered. The results of [14] are in accordance with independent studies inRefs. [63,64,65,66]. Later work within the HSD approach [12] has supportedthe conclusions of Ref. [14] and stressed the importance for D ¯ D annihilationin the late (purely hadronic) stages of the collisions. This scenario is based on the idea of sequential dissociation of charmonia withincreasing temperature [67,68,69,70], i.e. of charmonium melting in the QGPdue to color screening as soon as the fireball temperature reaches the dissocia-tion temperatures of ( ≈ T c for J/ Ψ, ≈ T c for excited states, where T c standsfor the critical temperature of the deconfinement phase transition). In the earlyapproaches the temperature of the fireball has been estimated using e.g. theBjorken formula (1). We modify the standard sequential dissociation modelin two aspects: (i) the energy density is calculated locally and microscopicallyinstead of using schematic estimates ( cf. section 2); (ii) the model incorporatesa charmonium regeneration mechanism (by D ¯ D annihilation processes).The ‘threshold scenario’ for charmonium dissociation now is implemented ina straight forward way: whenever the local energy density ε ( x ) is above athreshold value ε j , where the index j stands for J/ Ψ , χ c , Ψ ′ , the charmoniumis fully dissociated to c + ¯ c . The default threshold energy densities adoptedare ε J/ Ψ = 16 GeV/fm , ε χ c = 2 GeV/fm , and ε Ψ ′ = 2 GeV/fm . (13)The dissociation of charmonia is widely studied using lattice QCD (lQCD)[71,72,73,74,75] in order to determine the dissociation temperature (or en-ergy density) via the maximum entropy method. On the other hand one mayuse potential models - reproducing the charmonium excitation spectrum invacuum - to calculate Mott transition temperatures in a hot medium. Bothapproaches have their limitations and the quantitative agreement between thedifferent groups is still unsatisfactory: • (A) Potential models employ the static heavy quark-antiquark pair freeenergy - calculated on the lattice - to obtain the charmonium spectral func-16ions. This leads to the dissociation temperatures [76] T melt ( J/ Ψ) ≤ . T c , T melt ( χ c ) ≤ T c , T melt (Ψ ′ ) ≤ T c . • (B) The maximum entropy method is used to relate the Euclidean thermalcorrelators of charmonia - calculated on the lattice - to the correspondingspectral functions and yields higher dissociation temperatures [71] T melt ( J/ Ψ) = 1 . − T c , T melt ( χ c ) = 1 . − . T c or [72] T melt ( J/ Ψ) ≥ . T c , T melt ( χ c ) = 1 . T c . Our earlier analysis of experimental data at the SPS in the ‘threshold melt-ing’ approach [11] lead us to conclude from the observation of a considerableamount of J/ Ψ in the most central
P b + P b collisions that the assumption ofa melting of J/ Ψ close to T c contradicts the data. Therefore, the values (13)are applied also in the current study. Two more scenarios are implemented in our present HSD simulations thatare closely related to the ‘comover suppression’ and the ‘threshold melting’scenarios outlined in the previous sub-sections. The essential difference is thatthe comoving hadrons (including the D -mesons) exist only at energy densi-ties below some energy density ε cut , which is a free parameter. We employ ε cut = ε c ≈ , which is equal to the critical energy density ε c forthe parton/hadron phase transition. This scenario clearly separates ‘formedhadrons’ from possible pre-hadronic states at higher energy densities. Indeed,it is currently not clear whether D - or D ∗ -mesons survive at energy densitiesabove ε c but hadronic correlators with the quantum numbers of the hadronicstates are likely to persist above the phase transition [77]. One may speculatethat similar correlations (pre-hadrons) survive also in the light quark sectorabove T c such that ‘hadronic comovers’ – with modified spectral functions –might show up also at energy densities above ε c .We recall that the concept of (color neutral) pre-hadrons - explained in moredetail in Refs. [16,18] - has been also used in the hadron electroproductionstudies off nuclei in Refs. [16,17] as well as for high p T hadron suppression[18] or jet suppression at RHIC energies [19]. It has been found that thepre-hadron concept works well for hadron attenuation in nuclei at HERMESenergies [16,17] but underestimates the high p T hadron suppression [18] aswell as the jet attenuation at RHIC energies [19]. Nevertheless, the amount ofattenuation due to such pre-hadronic interactions emerged to be about 50%17f the experimentally observed suppression at RHIC such that their effectmight not simply be discarded. It should be stressed that the concept of pre-hadrons refers to the string breaking mechanism as described in Refs. [16,18]and is independent on the energy density. A detailed study on the space-timeevolution of pre-hadrons and their formation to hadrons for pp collisions hasbeen performed by Gallmeister and Falter in Ref. [78].In line with the investigations in Refs. [18,19] we also study J/ Ψ produc-tion and absorption in Au + Au collisions at √ s = 200 AGeV assuming theabsorption of charmonia on pre-hadrons as well as their regeneration by pre-hadrons. This adds additional interactions of the particles with charm quarks(antiquarks) in the very early phase of the nucleus-nucleus collisions as com-pared to the default HSD approach. Since these pre-hadronic (color-dipole)states represent some new degrees-of-freedom, the interactions of charmedstates with these objects have to be specified separately.For notation we define a pre-hadronic state consisting of a quark-antiquarkpair as pre-meson ˜ m and a state consisting of a diquark-quark pair as pre-baryon ˜ B . The dissociation cross section of a c ¯ c color dipole state with apre-baryon is taken to be of the same order as with a formed baryon, σ dissc ¯ c ˜ B = 5 . , (14)whereas the cross section with a pre-meson follows from the additive quarkmodel as [16,17] σ dissc ¯ c ˜ m = 23 σ dissc ¯ c ˜ B . (15)Elastic cross sections are taken as σ elc ¯ c ˜ B = 1 . , σ elc ¯ c ˜ m = 23 σ elc ¯ c ˜ B . (16)Furthermore, elastic interactions of a charm quark (antiquark) are modeledby the scattering of an unformed D or D ∗ meson on pre-hadrons with onlylight quarks as σ elD ˜ B = 3 . , σ elD ˜ m = 23 σ elD ˜ B . (17)In this way we may incorporate in HSD some dynamics of quark-antiquarkpairs with a medium that has not yet formed the ordinary hadrons. However,it has to be stressed that further explicit partonic degrees of freedom, i.e.gluons and their mutual interactions as well as gluon interactions with quarks18
100 200 300 4000.00.51.0 N part Au+Au, s =200 GeVComover absorption
HSD |y|<0.35 1.2<|y|<2.2 PHENIX, |y|<0.35 PHENIX, 1.2<|y|<2.2 R AA ( J / Y ) Fig. 7. The J/ Ψ nuclear modification factor R AA (18) for Au+Au collisions at √ s = 200 AGeV as a function of the number of participants N part in comparison tothe data from [41] for midrapidity (full circles) and forward rapidity (full triangles).The HSD results for the purely hadronic ‘comover’ scenario are displayed in termsof the lower (green solid) line with open circles for midrapidity J/ Ψ ′ s ( | y | ≤ . . ≤ | y | ≤ . and antiquarks, are not taken into account in the present HSD approach.Therefore, we do not expect to reproduce any details of the measured J/ Ψyield. The study of this particular model situation is motivated first of allby the possibility to assess the conceptual influence of charm scattering onpre-hadrons (in the early reaction phase) on the final rapidity distribution ofthe J/ Ψ’s (see below).
In the transport approach we calculate the J/ Ψ survival probability S J/ Ψ andthe nuclear modification factor R AA as S J/ Ψ = N J/ Ψ fin N J/ Ψ BB , R AA = dN J/ Ψ AA /dyN coll · dN J/ Ψ pp /dy , (18)where N J/ Ψ fin and N J/ Ψ BB denote the final number of J/ Ψ mesons and the numberof J/ Ψ’s produced initially by BB reactions, respectively. Note that N J/ Ψ fin
50 100 150 200 250 300 350 4000.00.51.01.52.02.5
Au+Au, s =200 GeV
HSD R AA (f o r w a r d y ) / R AA ( m i d - y ) N part PHENIX comover absorption threshold melting prehadron interactions
Fig. 8. The ratio of the nuclear modification factors R AA at mid-rapidity ( | y | < . . < | y | < . vs centrality in Au + Au collisions at √ s = 200 GeV. The HSD results in the purely hadronic scenario (‘comover ab-sorption’) are displayed in terms of the blue dashed line (with open circles) and incase of the ‘threshold melting’ scenario in terms of the violet dot-dashed line (withopen squares). The error bars on the theoretical results indicate the statistical un-certainty due to the finite number of Monte-Carlo events in the calculations. Thelower full green dots represent the data of the PHENIX Collaboration [41]. Notethat the data have an additional systematic uncertainty of ± includes the decays from the final χ c . In (18), dN J/ Ψ AA /dy denotes the finalyield of J/ Ψ in AA collisions, dN J/ Ψ pp /dy is the yield in elementary pp reactionswhile N coll is the number of initial binary collisions.The suppression of charmonia by the ‘comover’ dissociation channels withinthe model described in [11] for a matrix element squared | M | = 0.18 fm /GeV has been presented already in Ref. [12] as well as the results for the ‘thresholdmelting scenario’ employing the thresholds ε J/ Ψ = 16 GeV/fm , ε χ c = ε Ψ ′ = 2GeV/fm . Note that the charmonium reformation channels by D + ¯ D channelshad been incorporated, too ( cf. Ref. [14]). Since the PHENIX Collaborationhas released a new data set we compare our calculations with the most recentPHENIX data [41] in Fig. 7 for the J/ Ψ nuclear modification factor R AA (18)for Au+Au collisions at √ s = 200 AGeV as a function of the number of partic-ipants N part for midrapidity (full circles) and forward rapidity (full triangles).The HSD results for the purely hadronic ‘comover’ scenario are displayed interms of the lower (blue solid) line with open circles for midrapidity J/ Ψ ′ s ( | y | ≤ .
35) and in terms of the upper (red dashed) line with open trianglesfor forward rapidity (1 . ≤ | y | ≤ .
100 200 300 4000.00.51.0
Au+Au, s =200 GeVPrehadron interactions
PHENIX |y|<0.35 1.2<|y|<2.2 N part HSD |y|<0.35 1.2<|y|<2.2 R AA ( J / Y ) Fig. 9. The J/ Ψ nuclear modification factor R AA (18) for Au+Au collisions at √ s = 200 AGeV as a function of the number of participants N part in comparisonto the data from [41] for midrapidity (full circles) and forward rapidity (full trian-gles). The HSD results for the hadronic ‘comover’ scenario including additionallypre-hadronic interactions of charm according to (14) - (17) are displayed in terms ofthe upper (green solid) line with open circles for midrapidity J/ Ψ ′ s ( | y | ≤ .
35) andin terms of the lower (orange dashed) line with open triangles for forward rapidity(1 . ≤ | y | ≤ . able at midrapidity ( | y | ≤ .
35) but the even larger suppression at forwardrapidity (seen experimentally) is fully missed (cf. Ref. [12]).The failure of the traditional ’comover absorption’ model as well as ’thresholdmelting’ scenario at the top RHIC energy is most clearly seen in the centralitydependence of the ratio of the nuclear modification factors R AA at forwardrapidity (1 . < | y | < .
2) and at mid-rapidity ( | y | < .
35) as shown in Fig. 8.The HSD results in the purely hadronic scenario (‘comover absorption’) aredisplayed in terms of the blue dashed line (with open circles) and in case ofthe ‘threshold melting’ scenario in terms of the dot-dashed violet line (withopen squares). The error bars on the theoretical results indicate the statisticaluncertainty due to the finite number of Monte-Carlo events in the calcula-tions. The lower full green dots in Fig. 8 represent the corresponding data ofthe PHENIX Collaboration [41] which show a fully different pattern as a func-tion of centrality (here given in terms of the number of participants N part ).The failure of these ’standard’ suppression models at RHIC has lead to theconclusion in Ref. [12] that the hadronic ’comover absorption and recombina-tion’ model is falsified by the PHENIX data and that strong interactions inthe pre-hadronic (or partonic) phase should be necessary to explain the largesuppression at forward rapidities. 21 -4 -3 -2 -1 0 1 2 310 -4 central, 0-20% B mm d N ( J / Y ) / d y -3 -2 -1 0 1 2 310 -5 -4 comover prehadron interactions PHENIX Au+Au, s =200 GeV
HSD semi-central, 20-40%semi-peripheral, 40-60% B mm d N ( J / Y ) / d y y -3 -2 -1 0 1 2 310 -6 -5 y peripheral, 60-90% Fig. 10. The rapidity distribution dN J/ Ψ /dy for different centralities from the stan-dard ‘comover’ model (dashed blue lines) and the ’comover’ model with additionalpre-hadronic interactions of charm according to (14) - (17) (solid red lines). The fulldots show the respective data from the PHENIX Collaboration [41]. The calculatedlines have been smoothed by a spline algorithm. The reactions are Au+Au at √ s = 200 GeV. In this work we follow up the latter idea and incorporate in the ’comoverscenario’ the additional pre-hadronic cross sections (14) - (17) for the earlycharm interactions to have a first glance at the dominant effects. The J/ Ψsuppression pattern in this case is shown in Fig. 9 in comparison to the samedata as in Fig. 7. Now, indeed, the suppression pattern for central and forwardrapidities becomes rather similar to the data within the statistical accuracy ofthe calculations. Indeed, the ratio of R AA at forward rapidity to midrapiditynow follows closely the experimental trend as seen in Fig. 8 by the lower redsolid line.Some further information may be gained from the J/ Ψ rapidity distributionsin Au+Au collisions at RHIC. The latter distribution is shown in Fig. 10 incomparison to the PHENIX data for central collisions (upper l.h.s.), semi-central (upper r.h.s.), semi-peripheral (lower l.h.s.) and peripheral reactions(lower r.h.s.) for the standard ’comover’ scenario (dashed blue lines) and the’comover’ model including additionally pre-hadronic interactions of charm ac-cording to (14) - (17) (solid red lines). Whereas for peripheral reactions these22dditional early interactions practically play no role, the latter lead to a nar-rowing of the J/ Ψ rapidity distribution with the centrality of the collision(roughly in line with the data). In the standard ’comover’ model an oppositetrend is seen: here the interactions of charmonia with formed hadrons producea dip in the rapidity distribution at y ≈ c ¯ c pairs is the same (for the respective central-ity class) and detailed balance is incorporated in the reaction rates we find ansurplus of J/ Ψ at more forward rapidities. The net result is a broadening ofthe J/ Ψ rapidity distribution with centrality opposite to the trend observedin experiment.Summarizing the results displayed in Figs. 7 - 10 we like to point out thatthe hadronic ’comover’ dynamics for charmonium dissociation and recreation- as well as the standard charmonium ’melting’ scenario - do not match thegeneral dependences of the J/ Ψ in rapidity and centrality as seen by thePHENIX Collaboration. In fact, a narrowing of the J/ Ψ rapidity distributioncannot be achieved by comover interactions with formed hadrons since thelatter appear too late in the collision dynamics. Only when including early pre-hadronic interactions with charm a dynamical narrowing of the charmoniumrapidity distribution with centrality can be achieved as demonstrated moreschematically within our pre-hadronic interaction model. Consequently, thePHENIX data on J/ Ψ suppression demonstrate the presence and importantimpact of pre-hadronic or partonic interactions in the early charm dynamics.This finding is line with earlier studies in Refs. [14,18,19] demonstrating thenecessity of non-hadronic degrees of freedom in the early reaction phase forthe elliptic flow v , the suppression of hadrons at high transverse momentum p T and far-side jet suppression in central Au+Au collisions at RHIC energies. In this Section we continue with predictions for future measurements as wellas model comparisons in order to allow for an experimental discriminationbetween the model concepts. Ψ ′ as an independent probe As pointed in Ref. [11] an independent measurement of Ψ ′ will provide furtherinformation on the charm reaction dynamics and final charmonium formation.For instance, a leveling off of the Ψ ′ to J/ Ψ ratio with increasing centrality23
50 100 150 200 250 300 350 4000.00.51.0
Au+Au, s =200 GeV
HSD S ( Y ´) N part Comover QGP threshold
Fig. 11. Survival probability of Ψ ′ in Au + Au reactions at √ s = 200 GeV in the‘threshold melting’ (dashed line with triangles) and ‘comover’ suppression (solid linewith circles) approaches, see text for details. would be a signal for charm chemical equilibration in the medium [3,4,5]. Ad-ditionally, it provides a very clear distinction between the ‘threshold melting’scenario and the ‘comover’ approach. Since detailed predictions for the Ψ ′ to J/ Ψ ratio as a function of centrality have already been presented in Ref. [11]for FAIR and SPS energies we here complement the latter studies by resultsfor the top RHIC energy although the suppression of Ψ ′ mesons has not yetbeen measured at RHIC.In Fig 11, we accordingly present the Ψ ′ survival probability S Ψ ′ defined as S Ψ ′ = N Ψ ′ fin N Ψ ′ BB , (19)for Au + Au at √ s = 200 GeV. In equation (19), N Ψ ′ fin and N Ψ ′ BB denote thenumber of final Ψ ′ mesons and of those produced initially by BB reactions,respectively. One can see from Fig 11 that the ‘threshold melting’ scenario atRHIC predicts an almost complete melting of Ψ ′ , while a hadronic ‘comover’absorption scenario shows a gradual decrease of the number of Ψ ′ with N part .Similar differences between the models have also been found at SPS energies[11] where the presently available data sets clearly favor the ’comover’ model.On the other hand our predictions for the top RHIC energy imply that theΨ ′ signal will be very low for mid-central and central Au+Au collisions suchthat actual measurements will turn out to be very demanding.24
200 400 600 80001234567 HSD Comover QGP threshold Andronic et al
Pb+Pb, E beam =158 A GeV, mid-rapidity < J / Y > / < p > * N coll Pb+Pb, E beam =158 A GeV, 4 p < J / Y > / < p > * N coll Fig. 12. Ratio of the averaged J/ Ψ to π multiplicity for P b + P b at the SPS beamenergy of 158 A · GeV at mid-rapidity (l.h.s.) and in full 4 π acceptance (r.h.s.) as afunction of the number of binary collisions N coll for the different suppression scenar-ios implemented in HSD - the ‘comover’ model (dashed blue line with open circles)and the ‘threshold melting’ scenario (green dot-dashed line with open triangles) - incomparison to the statistical model by Gorenstein and Gazdzicki [2] (r.h.s.; straightorange line) and the statistical hadronization model by Andronic et al. [3] (l.h.s.;solid black line). et al Au+Au, s =200 GeV, mid-rapidity < J / Y > / < p > * N coll HSD Comover QGP threshold Prehadron interactions
Au+Au, s =200 GeV, 4 p < J / Y > / < p > * N coll HSD Comover QGP threshold Prehadron interactions
Fig. 13. Same as Fig. 12 but for Au + Au at the top RHIC energy of √ s = 200 GeV.The red solid line shows additionally the result of the ’comover’ model includingthe pre-hadronic charm interactions (see text). The assumption of statistical hadronization – i.e. of J/ Ψ’s being dominantlyproduced at hadronization in a purely statistically fashion according to avail-able phase space and the number of available c and ¯ c quarks – leads to ascaling of the h J/ Ψ i / h h i ratio with the system size [2], where h h i is the aver-age hadron multiplicity. Since h h i ∼ h π i , we calculate the ratio h J/ Ψ i / h π i inHSD in the different scenarios for charmonium suppression: • ‘threshold melting’ + recombination via D ¯ D → c ¯ c + m including the back-ward reactions c ¯ c + m → D ¯ D , 25 hadronic (‘comover’) absorption: D ¯ D → c ¯ c + m and the backward reactions c ¯ c + m → D ¯ D ; • ‘prehadron interactions’: D ¯ D → c ¯ c + m and the backward reactions c ¯ c + m → D ¯ D as well as early pre-hadronic charm interactions as described in Section6.The results of our calculations are shown in Fig. 12 together with the predic-tion of the statistical model of Gorenstein and Gazdzicki [2] for the full phasespace (straight orange line; r.h.s.) and the statistical hadronization model byAndronic et al. [3,79] for mid-rapidity (solid black line; l.h.s.) for Pb+Pb at158 A · GeV. The centrality dependence here is given by the number of initialbinary collisions N coll . The actual comparison in Fig. 13 indicates that the sta-tistical model by Andronic et al. [3] predicts a sizeably larger J/ Ψ to π ratio atmidrapidity for peripheral and semi-peripheral reactions than the microscopicHSD results for the different scenarios. For central reactions - where an ap-proximate equilibrium is achieved - all scenarios give roughly the same ratio.In full 4 π phase space the HSD results indicate also a slightly higher J/ Ψ to π ratio in the ’comover’ model relative to the ’melting’ scenario but both ratiosonly weakly depend on centrality roughly in line with the statistical modelof Gorenstein and Gazdzicki [2] (orange straight line). Consequently, onlyperipheral reactions of heavy nuclei might be used to disentangle the differentscenarios at top SPS energies at midrapidity (or in full phase space).The situation is different for Au+Au collisions at the top RHIC energy asmay be extracted from Fig. 13 where the J/ Ψ to pion ratio (l.h.s.: at midra-pidity; r.h.s.: for 4 π acceptance) is shown again as a function of N coll . Thestandard ’comover’ model (dashed blue lines) is only shown for reference butis unrealistic according to the analysis in Section 6. We find that the ’comover’model with early pre-hadronic charm interactions (solid red line with stars,l.h.s.) is very close to the statistical hadronization model [3] (solid black line)at midrapidity except for very peripheral collisions. The ’threshold melting’scenario follows the trend in centrality but is down by about 30%. Thus atmidrapidity there is no essential extra potential in differentiating the scenar-ios. Considering the full 4 π acceptance (r.h.s.) we find a practically constant J/ Ψ to pion ratio for N coll >
200 from the HSD calculations as expected fromthe statistical model, however, the early model of Gorenstein and Gazdzicki[2] is down by about a factor of ∼
10 (and may be ruled out by present data).
In this Subsection we present the excitation functions for the J/ Ψ survivalprobability in Au + Au collisions from FAIR to top RHIC energies in the dif-ferent scenarios in order to allow for a further distinction between the different26
00 1000 100000.00.20.40.60.81.0
Central S ( J / Y ) E beam , A GeV Comover
QGP threshold
100 1000 100000.00.20.40.60.81.0
Minimum bias
HSDJ/ Y excitation function S ( J / Y ) E beam , A GeV
100 1000 100000.0000.0040.0080.012
Central B mm ( Y ' ) sY ' / B mm ( J / Y ) s J / Y E beam , A GeV Comover QGP threshold
100 1000 100000.0000.0040.0080.012 B mm ( Y ' ) sY ' / B mm ( J / Y ) s J / Y Minimum bias
HSD Y ' to J/ Y ratio E beam , A GeV Fig. 14. upper part: The excitation function for the J/ Ψ survival probability in the‘QGP threshold melting + hadronic recombination’ scenario (dashed green lineswith triangles) and the ‘comover absorption + recombination’ model (solid redlines with circles) for central (l.h.s.) and minimum bias Au+Au reactions (r.h.s.) asa function of the beam energy. Lower part: The Ψ ′ to J/ Ψ ratio for the same reac-tions as in the upper part of the figure in the ‘QGP threshold melting + hadronicrecombination’ scenario (dashed green lines with triangles) and the ‘comover ab-sorption + recombination’ model (upper solid red lines with circles ). concepts. The results of our HSD calculations are presented in the upper partof Fig. 14 for the ‘QGP threshold melting + hadronic recombination’ scenario(dashed green lines with open triangles) and the ‘comover absorption + re-combination’ model (solid red lines with open circles) for central (l.h.s.) andminimum bias (r.h.s.) Au+Au reactions as a function of the beam energy. Wefind that from FAIR energies of 20 - 40 A · GeV up to top SPS energies of 158 A · GeV there is no significant difference for the J/ Ψ survival probability in case of27
In+In, E beam =160 A GeV
HSD central HSD peripheral NA60 v ( J / Y ) Central / Peripheral
Fig. 15. Elliptic flow v of J/ Ψ’s produced in central and peripheral In + In collisionsat 158 A · GeV beam energy in the hadronic ‘comover’ mode of HSD (open circleand open triangle) compared to the NA60 data [80] represented by black diamonds. central collisions. The differences here show mainly up in the full RHIC energyrange where the ‘QGP threshold melting + hadronic recombination’ scenarioleads to a substantially lower J/ Ψ survival probabilities. In case of minimumbias collisions the ‘comover absorption + recombination’ model (solid lines)leads to a roughly energy independent J/ Ψ survival probability whereas the‘QGP threshold melting + hadronic recombination’ scenario shows lower J/ Ψsurvival probabilities (lower dashed green lines) for laboratory energies above ∼
100 A · GeV due to a larger initial melting of J/ Ψ at high energy density.A clearer distinction between the different concepts is offered by the excitationfunctions for the Ψ ′ to J/ Ψ ratio in Au + Au collisions. The calculated re-sults are shown in the lower part of Fig. 14 for the ‘QGP threshold melting +hadronic recombination’ scenario (dashed green lines with open triangles) andthe ‘comover absorption + recombination’ model (solid red lines) for central(l.h.s.) and minimum bias reactions (r.h.s.). Here the Ψ ′ is already meltingaway in central Au+Au reactions in the ‘QGP threshold melting’ scenarioat bombarding energies above 40 A · GeV whereas a substantial amount of Ψ ′ survives in the ‘comover absorption + recombination’ model. Thus measure-ments of Ψ ′ suppression at the lower SPS or top FAIR energies will clearlydistinguish between the different model concepts.28 .4 Elliptic flow of charm The elliptic flow of particles defined as v ( y, p T ) = * p x − p y p T + y,p T (20)(with p T = p x + p y ) provides additional information on the collective cur-rents and pressure evolution in the early phase of the complex reaction [81]since it is driven by different pressure gradients in case of nonvanishing spatialanisotropy ǫ = < y − x y + x > . Since ǫ decreases fast during the expansion of anoncentral reaction the magnitude of v gives information about the interac-tion strength or interaction rate of the early medium.In Fig. 15 we test the HSD result for v ( J/ Ψ) at SPS in the purely hadronic’comover’ scenario in comparison to the data for v of the NA60 collaborationfor In+In collisions [80]. In central collisions the elliptic flow is practicallyzero both in the calculation as well as in the experiment whereas in peripheralreactions a nonzero flow emerges. The agreement (within error bars) betweenthe theory and the data indicates that in line with the reproduction of the J/ Ψsuppression data [11] the low amount of v does not point towards additionalstrong partonic interactions. Consequently, the present measurements of J/ Ψelliptic flow at SPS energies do not provide further constraints on the modelassumptions.The situation, however, is different for the collective flow of D -mesons at topRHIC energies. In Fig. 16 we show the elliptic flow of D -mesons produced in Au + Au collisions at √ s = 200 GeV as a function of the transverse momen-tum p T in HSD (solid blue line with open circles) compared to the PHENIXdata [82] on v of non-photonic electrons. Here the elliptic flow of D -mesons isclearly underestimated in the standard HSD model (cf. Ref. [35]). Only whenincluding pre-hadronic charm interactions - as described in Section 6 - the el-liptic flow increases (red line with open stars) but still stays clearly below thePHENIX data for p T < p T hadron suppression [18] as well as far-side jet suppression [19] inthe pre-hadronic interaction model. Independently, also the charm collectiveflow points towards strong partonic interactions in the early reaction phasebeyond the pre-hadronic scattering incorporated so far.Since a large fraction of J/ Ψ’s in central Au+Au collisions at RHIC are createdby D − ¯ D recombination, the elliptic flow of J/ Ψ’s obtained from HSD in the29 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5-0.050.000.050.100.15
Au+Au, s =200 GeV
HSD, D-mesons + prehadron interactions PHENIX v ( p T ) p T [GeV/c] Fig. 16. Elliptic flow of D -mesons produced in Au + Au collisions at √ s = 200 GeVas a function of p T from HSD (solid blue line with open circles) in comparison tothe PHENIX data [82] on v of non-photonic electrons. The red line with openstars shows the HSD result for the v of D -mesons when including additionallypre-hadronic charm interactions as described in Section 6. comover (purely hadronic) case is comparatively small, too, and should not bein accord with future experimental data. We consequently discard an explicitrepresentation of the J/ Ψ elliptic flow at RHIC energies since the calculationsshow the v of charmonium to be very close to the D -meson flow within errorbars. Our present study essentially completes the investigations of charm produc-tion, propagation and chemical reactions within the HSD transport approachinitiated more than a decade ago [30,31]. The present systematic investigationextends earlier work to RHIC energies and clearly shows - as advocated before[12] - that the traditional concepts of ‘charmonium melting’ in a QGP stateas well as the hadronic ‘comover absorption and recreation model’ are in se-vere conflict with the data from the PHENIX Collaboration at RHIC energieswhereas both model assumptions work reasonably well at top SPS energies[11].The essential new result of this work is that (at top RHIC energies) we findevidence for strong interactions of charm with the pre-hadronic medium fromcomparison to recent data from the PHENIX Collaboration [41]. In partic-ular, pre-hadronic interactions (of unformed hadrons) with charm lead to30ramatically different rapidity distributions for J/ Ψ’s and consequently to asubstantially modified ratio R forwardAA ( J/ Ψ) to R midAA ( J/ Ψ) compared to earliercalculations/predictions.Further results of the present microscopic transport study may be stated asfollows: • The J/ Ψ suppression in d + Au collisions at √ s = 200 GeV is only roughlycompatible with the charmonium absorption on nuclei as observed at SPSenergies in p + A reactions. We find a clear indication for shadowing effects atforward rapidity, but a conclusive answer about the size of this effect is notpossible due to the statistical error bars in both the experimental data andthe calculations. A proper answer can only be given by future high statisticsdata that allow to fix the scale of shadowing in a model independent way. • The Ψ ′ to J/ Ψ ratio is found to be crucial in disentangling the differentcharmonium absorption scenarios. This result essentially emerges from theearly dissociation of Ψ ′ above the critical energy density ǫ c ≈ inthe ‘QGP melting scenario’ whereas the Ψ ′ in the ‘comover model’ survivesto higher energy densities. • A comparison of the transport calculations to the statistical model of Goren-stein and Gazdzicki [2] (in 4 π acceptance) or the statistical hadronizationmodel of Andronic et al. [3] (at midrapidity) shows differences in the energyas well as centrality dependence of the J/ Ψ to pion ratio, which might beexploited experimentally to discriminate the different concepts. • The collective flow of charm in the HSD transport appears compatible withthe data at SPS energies, but the data are substantially underestimatedat top RHIC energies (cf. Fig. 16). This not only holds for the standardhadronic comover scenario, but also when including interactions of charmwith pre-hadronic states (unformed hadrons). Consequently the large el-liptic flow v of charm seen experimentally has to be attributed to earlyinteractions of non-hadronic degrees of freedom.The open problem - and future challenge - is to incorporate explicit partonicdegrees of freedom in the description of relativistic nucleus-nucleus collisionsand their transition to hadronic states in a microscopic transport approach.On the experimental side, further differential spectra of charmonia and opencharm mesons then will constrain the transport properties of charm in theearly non-hadronic phase of nucleus-nucleus collisions at RHIC (and possiblyat SPS or even FAIR energies). 31 cknowledgements We acknowledge stimulating correspondence with A. Andronic, T. Gunji,D. Kim and J. Skullerud as well as helpful discussions with P. Braun-Munzinger,M. Gorenstein, R. Granier de Cassagnac, K. Redlich, J. Stachel and H. St¨ocker.Furthermore, O. L. and E.L.B. would like to thank the BMBF for financialsupport.
References [1] P. Braun-Munzinger, D. Miskowiec, A. Drees, and C. Lourenco, Eur. Phys. J. C1 , 123 (1998).[2] M. Gazdzicki and M. I. Gorenstein, Phys. Rev. Lett. , 4009 (1999).[3] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Phys. Lett. B652 , 259 (2007).[4] P. Braun-Munzinger and J. Stachel, Phys. Lett.
B490 , 196 (2000).[5] P. Braun-Munzinger and J. Stachel, Nucl. Phys.
A690 , 119 (2001).[6] R. L. Thews, M. Schroedter, and J. Rafelski, Phys. Rev.
C63 , 054905 (2001).[7] C. Lourenco and H. Wohri, Phys. Rept. , 127 (2006).[8] B. Brambilla et al. , CERN Yellow Report, CERN-2005-005 (2005), hep-ph/0412158.[9] W. Cassing and E. L. Bratkovskaya, Phys. Rep. , 65 (1999).[10] H. Weber, E. Bratkovskaya, W. Cassing, and H. St¨ocker, Phys. Rev.
C67 ,014904 (2003).[11] O. Linnyk, E. L. Bratkovskaya, W. Cassing, and H. St¨ocker, Nucl. Phys.
A786 ,183 (2007).[12] O. Linnyk, E. L. Bratkovskaya, W. Cassing, and H. St¨ocker, Phys. Rev.
C76 ,041901 (2007).[13] W. Cassing, E. L. Bratkovskaya, and A. Sibirtsev, Nucl. Phys.
A691 , 753(2001).[14] E. L. Bratkovskaya, W. Cassing, and H. St¨ocker, Phys. Rev.
C67 , 054905(2003).[15] E. L. Bratkovskaya, A. P. Kostyuk, W. Cassing, and H. St¨ocker, Phys. Rev.
C69 , 054903 (2004).
16] T. Falter, K. Gallmeister, W. Cassing, and U. Mosel, Phys. Rev.
C70 , 054609(2004).[17] T. Falter, K. Gallmeister, W. Cassing, and U. Mosel, Phys. Lett.
B594 , 61(2004).[18] W. Cassing, K. Gallmeister, and C. Greiner, Nucl. Phys.
A735 , 277 (2004).[19] K. Gallmeister and W. Cassing, Nucl. Phys.
A748 , 241 (2005).[20] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Nucl. Phys.
A789 , 334 (2007).[21] PHENIX, S. S. Adler et al. , Phys. Rev.
C71 et al. , Phys. Rev. Lett. , 232002 (2007).[24] W. Cassing, E. L. Bratkovskaya, and S. Juchem, Nucl. Phys. A674 , 249 (2000).[25] E705, L. Antoniazzi et al. , Phys. Rev. Lett. , 383 (1993).[26] WA11, Y. Lemoigne et al. , Phys. Lett. B113 , 509 (1982).[27] NA50, B. Alessandro et al. , Phys. Lett.
B553 , 167 (2003).[28] K. Hagiwara et al. , Phys. Rev.
D66 , 010001 (2002), (Review of ParticleProperties).[29] J. Geiss, C. Greiner, E. L. Bratkovskaya, W. Cassing, and U. Mosel, Phys. Lett.
B447 , 31 (1999).[30] W. Cassing and E. L. Bratkovskaya, Nucl. Phys.
A623 , 570 (1997).[31] W. Cassing and C. M. Ko, Phys. Lett.
B396 , 39 (1997).[32] J. Geiss, W. Cassing, and C. Greiner, Nucl. Phys.
A644 , 107 (1998).[33] R. Vogt, Phys. Rep. , 197 (1999).[34] E672/E706, V. Abramov et al. , FERMILAB-Pub-91/62-E, IFVE-91-9, Mar.1991.[35] E. L. Bratkovskaya, W. Cassing, H. St¨ocker, and N. Xu, Phys. Rev.
C71 ,044901 (2005).[36] STAR, A. Tai et al. , J. Phys.
G30 , S809 (2004).[37] Z. Lin and C. M. Ko, Phys. Rev.
C62 , 034903 (2000).[38] Z. Lin and C. M. Ko, J. Phys.
G27 , 617 (2001).[39] PHENIX, A. Adare et al. , Phys. Rev.
C77 , 024912 (2008).[40] NA60, A. Foerster et al. , J. Phys.
G32 , S51 (2006).
41] PHENIX, A. Adare et al. , (2006), nucl-ex/0611020.[42] D. Kharzeev and R. L. Thews, Phys. Rev.
C60 , 041901 (1999).[43] NA50, B. Alessandro et al. , nucl-ex/0612012.[44] NA50, G. Borges et al. , J. Phys.
G32 , S381 (2006).[45] K. Martins, D. Blaschke, and E. Quack, Phys. Rev.
C51 , 2723 (1995).[46] C. Gerschel and J. Hufner, Z. Phys.
C56 , 171 (1992).[47] PHENIX, S. S. Adler et al. , Phys. Rev. Lett. , 012304 (2006).[48] I. C. Arsene, L. Bravina, A. B. Kaidalov, K. Tywoniuk, and E. Zabrodin, (2007),arXiv:0708.3801 [hep-ph].[49] A. Capella and E. G. Ferreiro, (2006), hep-ph/0610313.[50] R. Vogt, Phys. Rev. C71 , 054902 (2005).[51] B. Kopeliovich, A. Tarasov, and J. Hufner, Nucl. Phys.
A696 , 669 (2001).[52] A. Capella et al. , (2007), arXiv:0712.4331 [hep-ph].[53] PHENIX, R. Granier de Cassagnac, J. Phys.
G34 , S 955 (2007).[54] B. M¨uller, Nucl. Phys.
A661 , 272c (1999).[55] P. Braun-Munzinger and K. Redlich, Eur. Phys. J.
C16 , 519 (2000).[56] K. Martins, D. Blaschke, and E. Quack, Phys. Rev.
C51 , 2723 (1995).[57] C. Y. Wong, E. S. Swanson, and T. Barnes, Phys. Rev.
C62 , 045201 (2000).[58] C. M. Ko, B. Zhang, X. N. Wang, and X. F. Zhang, Phys. Lett.
B444 , 237(1998).[59] F. O. Duraes, H. Kim, S. H. Lee, F. S. Navarra, and M. Nielsen, Phys. Rev.
C68 , 035208 (2003).[60] W. Cassing, L. A. Kondratyuk, G. I. Lykasov, and M. V. Rzjanin, Phys. Lett.
B513 , 1 (2001).[61] N. Armesto and A. Capella, Phys. Lett.
B430 , 23 (1998).[62] P. Braun-Munzinger and K. Redlich, Nucl. Phys.
A661 , 546 (1999).[63] L. Grandchamp and R. Rapp, Phys. Lett.
B523 , 60 (2001).[64] L. Grandchamp and R. Rapp, Nucl. Phys.
A709 , 415 (2002).[65] Z.-w. Lin and C. M. Ko, J. Phys.
G27 , 617 (2001).[66] Z.-w. Lin and C. M. Ko, Phys. Rev.
C65 , 034904 (2002).[67] T. Matsui and H. Satz, Phys. Lett.
B178 , 416 (1986).
68] H. Satz, Rep. Progr. Phys. , 1511 (2000).[69] H. Satz, J. Phys. G32 , R25 (2006).[70] F. Karsch, D. Kharzeev, and H. Satz, Phys. Lett.
B637 , 75 (2006).[71] G. Aarts, C. Allton, M. B. Oktay, M. Peardon, and J.-I. Skullerud, (2007),arXiv:0705.2198[hep-lat].[72] S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke, Phys. Rev.
D69 , 094507(2004).[73] P. Petreczky and K. Petrov, Phys. Rev.
D70 , 054503 (2004).[74] RBC-Bielefeld, K. Petrov, PoS
LAT2006 , 144 (2006).[75] O. Kaczmarek, PoS
CPOD07 , 043 (2007).[76] A. Mocsy and P. Petreczky, (2007), arXiv:0706.2183 [hep-ph].[77] H. van Hees and R. Rapp, Phys. Rev.
C71 , 034907 (2005).[78] K. Gallmeister and T. Falter, Phys. Lett.
B630 , 40 (2005).[79] A. Andronic, private communication.[80] NA60, R. Arnaldi et al. , Nucl. Phys.
A783 , 261 (2007).[81] H. St¨ocker, Nucl. Phys.
A750 , 121 (2005).[82] PHENIX, A. Adare et al. , Phys. Rev. Lett. , 172301 (2007)., 172301 (2007).