J Steinheimer
email: steinheimer@th.physik.uni-frankfurt.de

H Petersen

M Mitrovski

M Bleicher

Strangeness production at SPS energies in a (3+1)-dimensional Boltzmann+hydrodynamics approach

Results on strange particle production at energies from √ s = 2 -200 A GeV from a (3+1)dimensional Boltzmann+hydrodynamics hybrid model are presented and compared to results from the Ultra-relativistic Quantum Molecular Dynamics transport approach and existing experimental data. The hybrid model with an intermediate hydrodynamical evolution for the hot and dense stage of the collision allows us to explore the centrality and system size dependence of (multi-strange) hyperon production in A+A collisions. The transition from a system which is not fully thermalized, like in C+C or Si+Si collisions, to supposedly fully thermalized systems as e.g. created in Pb+Pb reactions is investigated. The hyperon yields are enhanced in the hybrid approach due to the local equilibrium assumption and therefore in line with the experimental data. The excitation functions of strange to nonstrange particle yields (e.g. K + /π + ) calculated in the hybrid approach are able to reproduce the famous 'horn' structure at SPS energies with a purely hadronic equation of state in contrast to the pure transport approach, which failed to explain the experimental data.

Strangeness production at SPS energies in a (3+1)dimensional Boltzmann+hydrodynamics approach

Introduction

One of the main motivations to study high energy heavy ion collisions is the creation of a new deconfined phase of strongly interacting matter, the so called Quark-Gluon Plasma (QGP) [1,2]. At CERN-SPS energies evidence for the creation of this new state of matter is claimed, e.g., with observations of an enhanced K/π ratio ('horn') and the step in the mean transverse mass excitation function for various hadron species [3]. Especially the low energy (high µ B ) program at SPS showed a culmination of exciting results. Therefore, the production of hadrons containing strange quarks is a very interesting topic to study in this energy regime (at the FAIR facility and at critRHIC), as they have to be newly created and their production mechanism is different comparing for example the QGP to a hadron gas. Whether a QGP phase is necessary to reproduce the experimentally measured yields, is not yet clear [4,5,6,7,8]. We address this question by using a relativistic transport approach, namely the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) (version 2.3) [9,10,11], and compare the results to a hybrid approach ‡ in which a hydrodynamic evolution is embedded into the intermediate stage of the model dynamics [12]. The advantage of this hybrid approach, compared to a pure hydrodynamic simulation, is that it allows for the direct comparison between non-equilibrium and locally equilibrated dynamics and accounts for a realistic decoupling of the particles with separate chemical and kinetic freeze-outs. Note that we do not aim to understand the dynamics of the strangeness equilibration process itself, but we are able to compare strangeness production in a full non-equilibrium approach with a dynamical approach involving local thermal and chemical equilibrium.

The model

In the hydrodynamic+transport hybrid model UrQMD v3.3, we use Boltzmann dynamics with hadron-hadron interactions, string excitation and fragmentation to calculate the initial state of a heavy ion collision for the hydrodynamical evolution. Hence we account for non-equilibrium effects in the very early stage of the collision and event-by-event fluctuations are naturally included [13]. Nucleons that did not interact are propagated during the whole evolution in the hadronic cascade, whereas the other particles are mapped to the hydrodynamical grid. The full (3 + 1) dimensional hydrodynamic evolution proceeds employing the SHASTA algorithm [14,15]. For the present comparison we restrict ourselves to a hadron gas equation of state which includes the same degrees of freedom as in the cascade part of UrQMD [16]. The transition from the hydrodynamic phase to the Boltzmann dynamics takes place when the energy density of every cell in a transverse slice drops below 4 (or 5) times the ground state energy density 0 ≈ 145 MeV/fm 3 . This gradual transition procedure provides a reasonable approximation to an iso-eigentime criterion and has Compared are two hybrid model calculations (black lines), UrQMD 2.3 (grey lines) and results from experiments [21,[START_REF] Afanasiev | The NA49 Collaboration[END_REF][START_REF] Ahle | E866 Collaboration and E917[END_REF]24,25,[START_REF] Klay | E-0895[END_REF]3,27]. The two black lines represent results for different freeze out criteria 4 0 (solid line) and 5 0 (dashed line).

been applied successfully in Refs. [17,18]. The hydrodynamic fields are then mapped to particle degrees of freedom via the Cooper-Frye equation on an isochronous hypersurface for each individual slice. For the production of strange baryons the corresponding chemical potentials µ B and µ S are taken into account, where µ S is chosen in a way that net strangeness vanishes locally. The subsequent rescatterings and final decays are calculated in the cascade part of the UrQMD model. For a more detailed description and results of the hybrid model we refer to the references [12,19].

Results

First we compare particle ratios from the two frameworks i.e., the pure transport calculation (grey line) and the hybrid approach (black line). In Fig. 1 the excitation functions of different particle ratios (at mid-rapidity), including strange baryons, are shown for central Au+Au/Pb+Pb collisions for center of mass energies of √ s = 2 -200 AGeV. We also compare two different freeze-out criteria; 4 0 (solid line) and 5 0 (dashed line). Within the hybrid model calculation the production of strange particles is enhanced, due to the establishment of local thermal equilibrium. This leads to an improved description of multi-strange particle ratios at energies above √ s ≈ 5 AGeV, while at lower energies the non-equilibrium approach seems to provide a better description of the data. Especially the 'horn' structure in the K + /π + ratio is reproduced without the explicit inclusion of a phase transition. As in thermal model studies we contribute it to a transition from a baryon dominated to a meson dominated regime in the phase diagram, reached at SPS energies. Note that the transport model is not able to reproduce the 'horn' structure in the K + to π + ratio. This was explained by an over-estimation of the pion yields rather than the actual K + production rate [20].

Centrality dependence

As a next step we investigate the centrality dependence of strange baryon production in Pb + Pb reactions (for details see [35]). The centrality and system size dependence of hyperon production at E lab = 40A and 158A GeV at mid rapidity (|y| ≤ 0.5) is shown in Fig. 2 (left) and compared to data [29,21,30,31,32,33,34]. The dashed line represents calculations with standard UrQMD and the solid line depicts predictions from transport + hydro simulations. The standard transport approach describes the data only in very peripheral collisions quite well. Moving to more central Pb + Pb collisions, it is evident that the production of strange hadrons, from a system in local equilibrium, is in much better agreement with the measured data than obtained in the binary collision prescription (see also [28]).

Especially for multistrange hyperons it is visible that the production via independent strings and resonance excitations is not sufficient to explain the experimentally measured yields.

Instead of varying the centrality of the collision to explore the applicability limit of the hybrid model in terms of collision volume, we can also perform calculations for smaller systems. Figure 2 (right) represents the rapidity spectra of Λ and Λ for central C + C and Si + Si collisions at 158A GeV measured by the NA49 Collaboration [29,30] compared to results from the hybrid model (solid line) and the standard UrQMD (dashed line).

Even if the number of participants in central Si + Si collisions is similar to peripheral Pb + Pb collisions (33.5-43.5 % most central), the geometry and the starting time for the hydrodynamic evolution are different, as both depend on the radius of the nucleus:

t start = 2R 2m N /E lab .
It is evident that standard UrQMD (dashed lines) describes the data better than the transport+hydrodynamics approach (solid lines) for such small systems. The limited system size does not allow for the same degree of equilibration in C + C and Si+Si interactions as in the case of Pb + Pb collisions for all shown centralities. This indicates that up to Si + Si reactions, equilibration cannot be achieved and the system is merely a (non-)trivial convolution of elementary baryon-baryon (and in smaller parts mesonbaryon) interactions.

Conclusion

We presented results on strange hadron production in an hydrodynamic+transport hybrid approach. The assumption of local thermal equilibration of strangeness leads to an improvement of the description of strange particle ratios, at energies above √ s ≈ 5 AGeV, when compared to the pure transport results. At lower energies the assumption of full local equilibration of strangeness seems questionable. In this context, the 'horn' structure in the K + /π + ratio excitation function can be reproduced by our approach, which does not explicitly include a phase transition. Our results on the system size and centrality dependence of strange particle production show that strangeness equilibration is only reached in Pb + Pb collision (at energies above √ s ≈ 5 AGeV) at least in all shown centrality classes. Compared to Pb + Pb collisions the system seems not fully equilibrated in C + C and Si + Si collisions and therefore the Boltzmann approach provides a better description of the measured rapidity spectra. The applicability of our hybrid approach therefore is limited to energies above 5 AGeV, at least for observables concerning strange matter. We conclude that the assumption of local equilibration alone allows us to explain a major part of the strangeness enhancement in central collisions of massive nuclei which does not necessarily invoke the formation of a QGP. For peripheral reactions or small collision systems a microscopic description with binary interactions between hadrons is favored. Furthermore, the possibility of creating an equilibrated fireball, which can undergo a phase transition, is reduced in such small systems because the creation of a QGP would be observed in higher strangeness production.

Figure 1 .

 1 Figure 1. Excitation functions for kaon to pion ratios (left) and strange baryon to pion ratios (right) from most central heavy ion collisions at energies of E c.m. = 2-200 AGeV.Compared are two hybrid model calculations (black lines), UrQMD 2.3 (grey lines) and results from experiments[21,[START_REF] Afanasiev | The NA49 Collaboration[END_REF][START_REF] Ahle | E866 Collaboration and E917[END_REF]24,25,[START_REF] Klay | E-0895[END_REF]3,27]. The two black lines represent results for different freeze out criteria 4 0 (solid line) and 5 0 (dashed line).

Figure 2 .

 2 Figure 2. Left: Centrality dependence of the (anti-) hyperon yields at mid rapidity (|y| ≤ 0.5) at E lab = 40A GeV (left) and E lab = 158A GeV (right) measured by NA49 (squares)[29,21,30,31] and NA57 (circles)[32,33, 34]. The solid line represents calculations with UrQMD+hydro and the dashed line represents calculations with UrQMD[35]. Right: The rapidity spectra of Λ, Λ and Ξ -produced in central C + C and Si + Si collisions at E lab = 158A GeV[29,30]. The closed symbols indicate measured points: open points are reflected with respect to mid rapidity. The solid line represents calculations with UrQMD + hydro and the dashed line represents calculations with UrQMD[35].

‡ This new hybrid approach is available for download at http://urqmd.org/ as UrQMD v3.3.

Acknowledgments

This work was supported by HGS-hire and the Hessian LOEWE initiative through the Helmholtz International center for FAIR (HIC for FAIR). The computational resources were provided by the Frankfurt Center for Scientific Computing (CSC). H.P. was supported by the Feodor Lynen program of the Alexander von Humboldt foundation. This work was supported in part by U.S. department of Energy grant DE-FG02-05ER41367.


O Linnyk

E L Bratkovskaya

W Cassing

Strangeness production within Parton-Hadron-String Dynamics (PHSD)

Keywords: Quark-gluon plasma, Relativistic heavy-ion collisions, Strangeness PACS: 12.38.Mh, 12.38.Aw, 25.75.-q

Strangeness production within partonhadronstring dynamics (PHSD)

The nature of confinement and the phase transition from a partonic system of quarks, antiquarks and gluons -a quark-gluon plasma (QGP) -to interacting hadrons, as occurring in relativistic nucleus-nucleus collisions, is a central topic of modern high energy physics. In the present work, the dynamical evolution of the heavy ion collision is described by the PHSD transport approach [1] incorporating the off-shell propagation of the partonic quasi-particles according to [2] as well as the transition to resonant hadronic states (or strings). Here we employ the PHSD approach -described in [1,3] -to strangeness production in nucleus-nucleus collisions at moderate relativistic energies, i.e. at SPS energies up to 160 A•GeV.

The PHSD approach

A consistent dynamical approach -valid also for strongly interacting systems -can be formulated on the basis of the Kadanoff-Baym equations [START_REF] Kadanoff | Quantum Statistical Mechanics[END_REF][START_REF] Cassing | [END_REF] or off-shell transport equations in phase-space representation, respectively [2,[START_REF] Cassing | [END_REF]. In the Kadanoff-Baym theory the field quanta are described in terms of propagators with complex selfenergies. Whereas the real part of the selfenergies can be related to mean-field potentials, the imaginary parts provide information about the lifetime and/or reaction rates of time-like 'particles' [6]. Once the proper (complex) selfenergies of the degrees of freedom are known, the time evolution of the system is fully governed by off-shell transport equations (as described in Refs. [2,[START_REF] Cassing | [END_REF]).

The PHSD approach is a microscopic covariant transport model that incorporates effective partonic as well as hadronic degrees of freedom and involves a dynamical description of the hadronization process from partonic to hadronic matter [1]. Whereas the hadronic part is essentially equivalent to the conventional HSD approach [7] the partonic dynamics is based on the Dynamical QuasiParticle Model (DQPM) [8] which describes QCD properties in terms of single-particle Green's functions (in the sense of a two-particle irreducible approach) and leads to effective strongly interacting partonic quasiparticles with broad spectral functions as degrees of freedom.

The off-shell parton dynamics also allows for a solution of the hadronization problem: the hadronization occurs by quark-antiquark fusion or 3 quark/3 antiquark recombination which is described by covariant transition rates as introduced in Ref. [3,9], obeying flavor current-conservation, color neutrality as well as energy-momentum conservation. Since the dynamical quarks become very massive close to T c , the formed resonant 'pre-hadronic' color-dipole states (q q or qqq) are of high invariant mass, too, and sequentially decay to the ground state meson and baryon octets increasing the total entropy. This solves the entropy problem in hadronization in a natural way [10].

Partonic energy fractions

We start with a consideration of energy partitions in order to map out the fraction of partonic energy in time for relativistic nucleus-nucleus collisions. The question emerges to what extent the partonic phase shows up as a function of centrality. To this aim we study the partonic energy E p (b) relative to the energy converted from the kinetic motion of the impinging nuclei at impact parameter b, i.e.

R p (b) = E p (b)/(E B (t = 0, b) -E B (t → ∞, b))
, which is displayed in Fig. 1 for impact parameter b from 1 fm to 13 fm in steps of 2 fm. The system is Pb+Pb at 158 A•GeV. One observes that even at very central collisions the partonic energy fraction R p only reaches about 40% and decreases to about 20% (in the peak) for very peripheral reactions. More striking is the fact that the duration of the partonic phase shrinks substantially when going from central to peripheral reactions. Thus the 'popular picture' that a partonic phase is reached in the overlap region of the nuclei at top SPS energies is by far not substantiated by the PHSD calculations. This has been been pointed out before in Refs. [11]. In conclusion, very central collisions show a sizeable fraction of space-time regions of partonic nature, but also of the hadronic (or string-like) corona. At first sight this looks discouraging, but one should concentrate on observables with a special sensitivity to the partonic phase, e.g. strangeness [3], dileptons [12], charm [13], jets [14], etc.

We continue with the ratio R p as a function of bombarding energy concentrating here on the FAIR and full SPS energy regime but considering only central collisions (b = 1 fm). The results are displayed in Fig. 2 for central Pb+Pb collisions at 160, 80, 40, and 20 A•GeV as a function of time and demonstrate that the average duration of the partonic phase does not change very much with bombarding energy, however, the partonic volume shrinks by about a factor of three when stepping down in bombarding energy from 160 to 20 A•GeV. Thus, according to our PHSD calculations there should be a QGP also at lower SPS and at FAIR energies but its space-time volume is significantly smaller than that for the hadronic phase. As one of the consequences, our preliminary findings within PHSD do not indicate a pronounced "horn" [15,16] at 25-30 AGeV (cf. HSD restults [17]). The latter phenomenon might be attributed to a first order phase transition at the lower SPS energy, while the current version of PHSD incorporates only a cross over between the hadronic and partonic phases as expected for low quark chemical potential.

Particle spectra in comparison to experiment

It is of interest, how the PHSD approach compares to the HSD [7] model (without explicit partonic degrees of freedom) as well as to experimental data. In Fig. 3 to the data of the NA49 Collaboration [15]. Here the slope of the π -spectra is only slightly enhanced in PHSD relative to HSD which demonstrates that the pion transverse motion shows no sizeable sensitivity to the partonic phase. However, the K ± transverse mass spectra are substantially hardened with respect to the HSD calculations at all bombarding energies -i.e. PHSD is more in line with the data -and thus suggest that partonic effects are better visible in the strangeness-degrees of freedom.

The hardening of the kaon spectra can be traced back to parton-parton scattering as well as a larger collective acceleration of the partons in the transverse direction due to the presence of repulsive vector fields for the partons. The enhancement of the spectral slope for kaons and antikaons in PHSD due to collective partonic flow shows up much clearer for the kaons due to their significantly larger mass (relative to pions). We recall that in Refs. [17] the underestimation of the K ± slope by HSD (and also UrQMD) had been suggested to be a signature for missing partonic degrees of freedom; our present PHSD calculations support this early suggestion.

The strange antibaryon sector is of further interest since here the HSD calculations have always underestimated the yield [18]. In this respect we compare in Fig. 4 (top part) the multiplicities of (Λ + Σ 0 )/N wound (l.h.s.) and ( Λ + Σ0 )/N wound (r.h.s.) as functions of the number of wounded nucleons N wound for Pb+Pb collisions at 158 A•GeV at mid-rapidity from PHSD and HSD to the experimental data from the NA57 Collaboration [19] and the NA49 Collaboration [20]. Whereas the HSD and PHSD calculations both give a reasonable description of the Λ+Σ 0 yield of the NA49 Collaboration, both models underestimate the NA57 data (open triangles) by about 30%. An even larger discrepancy in the data from the NA49 and NA57 Collaborations is seen for ( Λ + Σ0 )/N wound (r.h.s.); here the PHSD calculations give results which are in between the NA49 data (solid dots) and the NA57 data (open triangles). We see that HSD underestimates the ( Λ + Σ0 ) midrapidity yield at all centralities. This observation points towards a partonic origin but needs further examination. The latter result suggests that the partonic phase does not show up explicitly in an enhanced production of strangeness (or in particular strange mesons and baryons) but leads to a different redistribution of antistrange quarks between mesons and antibaryons. To examine this issue in more detail we show in Fig. 4 (bottom part) the multiplicities of Ξ -baryons (l.h.s.) and Ξ+ antibaryons (r.h.s.) -devided by N wound -as a function of the number of wounded nucleons for Pb+Pb collisions at 158 A•GeV at mid-rapidity from PHSD and HSD in comparison to the experimental data from the NA57 Collaboration [19] and the NA49 Collaboration [21,20]. The situation is very similar to the case of the strange baryons and antibaryons before: we find no sizeable differences in the double strange baryons from HSD and PHSDin a good agreement with the NA49 data -but observe a large enhancement in the double strange antibaryons for PHSD relative to HSD.

Summary

The PHSD approach has been applied to nucleus-nucleus collisions from 20 to 160 A•GeV in order to explore the space-time regions of 'partonic matter'. We have found that even central collisions at the top SPS energy of ∼160 A•GeV show a large fraction of non-partonic matter. It is also found that though the partonic phase has only a very low impact on rapidity distributions of hadrons [3], it has a sizeable influence on the transverse-mass distribution of final kaons due to the repulsive partonic mean fields and parton interactions. On the other hand, the most pronounced effect of the partonic phase is seen on the production of multi-strange antibaryons due to a slightly enhanced ss pair production in the partonic phase from massive time-like gluon decay and a more abundant formation of strange antibaryons in the hadronization process. We also mention that partonic production channels for dileptons appear to be visible in the µ + µ -spectra from In+In collisions at 158 A•GeV in the intermediate invariant mass range [12].

Fig. 1 .Fig. 2 .

 12 Fig. 1. The partonic energy fraction as a function of time for impact parameters b from 1 fm to 13 fm in steps of 2 fm for Pb+Pb at 158 A•GeV.

2 ]Fig. 3 .

 23 Fig.3. The π -, K + and K -transverse mass spectra for central Pb+Pb collisions at 40, 80 and 158 A•GeV from PHSD (thick solid lines) in comparison to the distributions from HSD (thin solid lines) and the experimental data from the NA49 Collaboration[15].

Fig. 4 .

 4 Fig.[START_REF] Kadanoff | Quantum Statistical Mechanics[END_REF]. Top: The multiplicities of (Λ + Σ 0 )/N wound (l.h.s.) and ( Λ + Σ0 )/N wound (r.h.s.) as a function of the number of wounded nucleons for Pb+Pb collisions at 158 A•GeV at mid-rapidity from PHSD (blue solid lines) and HSD (red dashed-dotted lines) in comparison to the experimental data from the NA57 Collaboration[19] (open triangles) and the NA49 Collaboration[20] (solid dots). The calculations have an error of 5-10% due to limited statistics. Bottom: The multiplicities of Ξ -/N wound (l.h.s.) and Ξ+ /N wound (r.h.s.) vs N wound for Pb+Pb collisions at 158 A•GeV at mid-rapidity. Line coding as above.

Acknowledgments

Work supported in part by the "HIC for FAIR" framework of the "LOEWE" program.


S Vogel

J Aichelin

M Bleicher

Resonances as a possible observable of hot and dense nuclear matter

Keywords: numbers: 24.10.Lx, 25.75.-q, 25.75.Dw

One of the most fundamental questions in the field of relativistic heavy ion physics is how to reach and explore densities which are needed to cross the chiral and/or the deconfinement phase transition. In this analysis we investigate the information we can gather by analyzing baryonic and mesonic resonances on the hot and dense phase in such nuclear reactions. The decay products of these resonances carry information on the resonances properties at the space time point of their decay. We especially investigate the percentage of reconstructable resonances as a function of density for heavy ion collisions in the energy range between E lab = 30 AGeV and √ s = 200 AGeV, the energy domain between the future FAIR facility and the present RHIC collider.

Resonances have been investigated as probes for the interior of heavy ion reactions for a long time. Their properties have been analyzed in various experiments ranging from low energy collisions [1,2] through intermediate [START_REF] Afanasev | NA49[END_REF]4] to high energy heavy ion collisions [5][6][7]. In general one distinguishes between leptonic and hadronic decay channels. While the hadronic decay channels have the advantage of larger branching ratios, the leptonic decays have the advantage that the decay particles do not undergo final state interactions. It has been shown recently [11] that dileptons might not be favorable as an observable for the high density phase of heavy ion collisions. Thus we will focus on the hadronic decay channels and investigate their possibility to measure the hot and dense phase of nuclear reactions.

The present Relativistic Heavy Ion Collider (RHIC) at Brookhaven and the upcoming Facility for Antiproton and Ion Research (FAIR, for a recent status on the project we refer to [8] and articles in these proceedings) provide an excellent research environment for probing resonances in matter. At the RHIC experiments it has been observed [9] that less resonances are measured than expected from statistical model calculations [10]. Stable hadrons however follow the prediction of this model. This suggests the conclusion that after chemical freeze-out, when the chemical composition of the final state is determined, hadrons still undergo collisions and therefore some of the resonances cannot be identified by the invariant mass of the decay products.

At FAIR the leptonic as well as the hadronic decay channel can be measured. While the leptonic channel is usually regarded as the 'cleaner' channel recent calculations [11] have shown that the dilepton channel might not probe the dense phase as it was expected before. In light of this new development it is worthwhile to evaluate the density-profile and the space-time-evolution of resonances which can be reconstructed in the hadronic decay channels. Although those channels suffer from the drawback of final state interaction of the decay products, their large branching ratios might make them better suited for the investigation of the high density phase of heavy ion collisions compared to leptonic decay channels.

For our calculations we utilize the UrQMD(v2.3) model, a non-equilibrium transport approach, which relies on the covariant Boltzmann equation. All cross sections are calculated by the principle of detailed balance and the additive quark model or are fitted to available data. UrQMD does not include any explicit in-medium modifications for vector mesons or effects to describe the restoration of chiral symmetry. The model allows to study the full space time evolution of all hadrons, resonances and their decay products in hadron-hadron or nucleus-nucleus collisions. This permits to explore the emission patterns of resonances in detail and to gain insight into their origins and decay channels. For previous studies of resonances within this model see [11][12][13][14][15][16][17]. For further details about the UrQMD model the reader is referred to [18,19].

The experimental reconstruction of resonances is challenging. One often applied technique is to reconstruct the invariant mass spectrum for single events. Then, an invariant mass distribution of mixed events is generated (here, the particle pairs are uncorrelated by definition). The mixed event distribution is subtracted from the invariant mass spectrum of the single (correlated) events. As a result one obtains the mass distributions and yields (after all experimental corrections) of the resonances by fitting the resulting distribution with a suitable function (usually a Confidential: not for distribution. Submitted to IOP Publishing for peer review 20 January 2010 Breit-Wigner function peaked around the pole mass of the respective resonance).

If the resonance spectral function changes in the hadronic medium this is in principle visible in the difference spectrum between true and mixed events. However, if a daughter particle (re-)scatters before reaching the detector the signal for the experimental reconstruction is blurred or even lost. Especially for strongly interacting decay products this effect can be sizeable. It is therefore difficult to judge whether a deviation from an expected Breit-Wigner distribution is due to an initial deformation or an increase of the initial width or due to the momentum dependence of the rescattering cross section of the daughter particles.

What makes this analysis even tougher is the fact that the resonances decay over a wide range of densities and therefore only an average value is measured. If this average value is dominated by resonance decays at low density the information from the high density phase is blurred and thus the resonance analysis may offer only a limited view on the high density phase of the heavy ion collision.

Within UrQMD we apply a different technique for the extraction of resonances. We follow the individual decay products of each decaying resonance (the daughter particles). If the daughter particles do not rescatter in the further evolution of the system, the resonance is counted as "reconstructable". The advantage of this method is that it allows to trace back the origin of each individual resonance to study their spatial and temporal emission pattern. Because UrQMD follows the space time evolution of all particles it is possible to link production and decay point of each individual resonance. This method also allows to explore the reconstruction efficiency in different decay branches. Note however, that this method is restricted to theoretical models and not applicable in experimental analyses.

In order to calculate at which density the resonance decays we have to determine the baryonic density. The baryon density is calculated locally at the position of the resonance in the rest frame of the baryon current (Eckart frame) as ρ B = j 0 with j µ = (ρ B , 0). Details on the calculation of the baryon density are discussed in [11]. One should note that the method chosen is insensitive to the parameters of the density calculation if chosen within reasonable bounds. A variation of the Gaussian widths by 50% resulted in no sizable difference in the obtained results. In all figures we present the density in units of ground state density, where a value of 0.16 1/fm 3 is assumed. All analyses presented in this work are done for central (impact parameter b≤3.4fm) Au+Au collisions at either FAIR energies of E lab = 30 AGeV or top RHIC energies of √ s = 200 AGeV. In the following we discuss the density dependence of the probability that a resonance can be reconstructed. Naively, one would expect that the higher the densities the more the rescattering effect becomes dominant. Therefore it is unlikely that a resonance which decays at high density is reconstructable. The view on the low density zone is expected to remain unblurred but is less interesting because it resembles that observed in elementary collisions. Depicted in Fig. 1 left (right) is the probability that a baryon resonance -shown are ∆, Σ * (1385) and Λ * (1520) baryon resonances (ρ, ω, K * 0 and Φ mesons) -which was produced at a certain density can be reconstructed experimentally. One observes a clear peak at very low density and a steady decrease towards higher density. This means that resonances that are produced at rather low density have a high probability to be detected and as the density increases the chance to reconstruct the resonances decreases. This is nothing unexpected. However, this trend stops at roughly 2 ρ 0 . At higher densities the chance to reconstruct a resonance saturates or even increases slightly again. This increase, which we discuss later in detail, is caused by resonances which picked up very high transverse momenta and leave the interaction zone quickly. This results in a decay in a region with less hadronic activity and a higher chance to be reconstructed. Whereas the form of the curves is qualitatively similar for the different hadrons the absolute value of the fraction of reconstructable resonances is rather different. It can be understood in terms of lifetimes of the resonances and in terms of the rescattering cross sections of the decay products.

Due to the large cross section of pions in nuclear matter (usually undergoing N + π → ∆ or π + π → ρ reactions) the probability to detect a high density ∆ resonance or a ρ meson is rather small compared to the probability to detect a high density Φ meson, since the Φ meson itself has a small cross section in nuclear matter and a long lifetime of ∼ 40 fm/c and the hadronic decay products (mostly kaons and antikaons) have a smaller cross sections compared to the pions from the decay of a ρ meson. Similarly, the long lifetime of the Λ increases their possibility to be reconstructed. As mentioned earlier, the saturation or slight increase of the reconstruction probability as a function of density has its origin in the possibility that resonances with a large p T can escape quickly from the reaction zone which is rather small initially.

Fig. 2 shows for various resonances the probability that an experimentally reconstructable resonance has been created at a density ρ. The integral over all densities is normalized to unity. One observes that most of those resonances are produced at very low densities, which is especially true for the mesonic resonances.

Reconstructable baryon resonances stem from slightly higher baryon densities, however most are still produced at rather low densities (with a peak at roughly 0.1 ground state density). So the detection of resonances produced at densities above ground state densities using hadronic decay channels seems not too encouraging. However, as we discuss next, a loophole might exist.

Let us illustrate this further with two examples which are representative for all investigated particles. Fig. 3 depicts the average transverse momentum of ∆ baryon resonances (left) and ρ meson resonances (right) as a function of baryon density. Lines show reconstructable resonances, symbols show all decayed resonances. The striking feature is the different average transverse momentum between all resonances and those which are reconstructable. The higher the average transverse momentum, the larger is the chance that the resonance can be reconstructed. The < p T > of reconstructable ∆/ρ resonances is about 200 MeV higher than for all resonances. Resonances with a large p T can leave the high density zone rather fast and move with a velocity of about < p T > /m outwards.

Another interesting feature in Fig. 3 is the difference between the √ s=200 AGeV and E lab =30 AGeV curves. While the E lab =30 AGeV data shows a decrease of < p T > as a function of the baryon density, the √ s=200 AGeV data show an increase. At √ s=200 AGeV the initial collisions (which happen at high baryon density) are more energetic and give the particles a high transverse momentum, subsequent rescattering decreases p T . For the E lab =30 AGeV collisions the situation is opposite. Initially the particle p T is small and the rescattering increases the p T due to transverse expansion. Fig. 4 shows the p T dependence of the reconstruction probability in detail. It shows the transverse momentum spectra for all (full symbols) and reconstructable resonances (open symbols) for two different energies. The numbers stated in the three shaded areas (p T < 1 GeV, 1 GeV < p T < 2 GeV, p T > 2 GeV ) are the percentages of reconstructable resonances created at a density higher than 2ρ 0 . One observes that at low transverse momentum the percentage of reconstructable resonances is low and increases when going to higher transverse momenta, i.e. that with increasing p T the chance to reconstruct a resonance produced at high baryon density increases. This effect is more pronounced in RHIC collisions at √ s = 200 AGeV than in collisions with an energy of E lab = 30 AGeV.

One might raise the question whether resonances with high transverse momentum are actually sensitive to the very hot and dense quark gluon phase. Although there is no partonic phase build in the approach we are using, there have been calculated estimates recently [22]. There it has been shown that the formation time of resonances are well within the estimated lifetime of the quark-gluon-plasma. In conclusion, we have discussed that the view on the high density zone may not be as restricted as usually assumed when analyzing hadronic resonances.

We argued that resonances detected with high transverse momentum are sensitive to higher densities. It will be interesting to explore if the properties of these resonances are different from the bulk emitted at low densities. The exploration of high p T resonances might therefore open a new keyhole at the upcoming CBM experiment at FAIR or the low energy program at RHIC to gain information on the high density zone and to observe eventual changes of resonance properties in the medium.

FIG. 1 :

 1 FIG. 1: (Color online) Fraction of reconstructable baryon resonances (top) and meson resonances (bottom) as a function of baryon density at the point of production.

FIG. 2 :

 2 FIG. 2: (Color online) Probability distribution of baryon density at the production vertex for various reconstructable resonances in central (b ≤ 3.4 fm) Au+Au collisions at 30 AGeV (top figure) and 200 AGeV (bottom figure) as a function of baryon density. One observes that most resonances which can be reconstructed in the hadronic decay channel originate from low baryon density.

  FIG. 3: (Color online) Average transverse momentum of reconstructable (line) or all (symbol) ∆ baryons as a function of baryon density for two different energies (left figure). Average transverse momentum of reconstructable (line) or all (symbol) ρ mesons as a function of baryon density for two different energies (right figure).

FIG. 4 :

 4 FIG. 4: (Color online) Transverse momentum spectra for all and reconstructable resonances for central (b≤3.4 fm) Au+Au collision at E lab = 30 AGeV (left) and √ s = 200 AGeV beam energy. Full circles depict the spectrum for all decayed resonances (included in the analysis are ∆, Λ, Σ baryons, as well as ρ, ω, K * 0 and ω mesons), open circles for reconstructable resonances. The numbers indicate the percentage of reconstructable resonances stemming from density region with ρ/ρ0 > 2.

This work was (financially) supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse. The computational resources have been provided by the Center for the Scientific Computing (CSC) at Frankfurt. This work was supported by GSI, DAAD (PROCOPE) and BMBF.


Hélène Ricaud
email: h.ricaud@gsi.de

Alexander Kalweit

Antonin Maire

Study of strange particle production in pp collisions with the ALICE detector

ALICE is well suited for strange particles production studies since it has very good reconstruction capabilities in the low transverse momentum (p t ) region and it also allows to extend the identification up to quite high p t . Charged strange mesons (K + , K -,) are reconstructed via energy loss measurements whereas neutral strange mesons (K 0 s ) and strange hyperons (Λ, Ξ, Ω) are identified via vertex reconstruction. All these particles carry important information: first, the measurement of production yields and the particle ratio within the statistical models can help to understand the medium created and secondly the dynamics at intermediate p t investigated via the baryon over meson ratio (Λ/K 0 s ) allows a better understanding of the hadronization mechanisms and of the underlying event processes. We present these two aspects of the strange particles analysis in pp collisions using simulated data.

Introduction

Over the past few years, strange particle production has been largely studied and it has been proven to be a good tool to investigate the Quark Gluon Plasma (QGP) which can be created in high energy density conditions such as relativistic heavy-ion collisions. Strange particle analysis will be important at LHC to characterize the medium created in the future Pb-Pb collisions but it can also give very useful information to understand pp collisions. These elementary collisions are expected to be a baseline for heavy-ion physics but they can also be interesting in themselves as we will show.

In the second section, we describe the ALICE detector and its particle identification abilities (see also [1]). The third section will be dedicated to two aspects of the physics we can extract from strange particle analysis in pp collisions. The first one is the chemical composition analysis done within statistical models. These models have been very successful in describing hadron yields in central heavy-ion collisions. We present here their applicability in small systems like elementary collisions and their predictions at the top LHC energy for pp collisions. We focus then on the dynamics at intermediate p t via the measurement of the ratio Λ/K 0 s versus p t .

The ALICE detector and the particle reconstruction

The design of the ALICE detector is ideal for particle reconstruction in the soft physics sector. The reconstruction of strange particles is performed inside the so called central barrel of ALICE which is embedded in a large solenoidal magnet and covers polar angles from 45 • to 135 • over the full azimuth. Tracking in the central barrel is mainly divided into the Inner Tracking System (ITS) and the Time Projection Chamber (TPC), the main tracking detector of ALICE.

The particle identification (PID) capabilities of ALICE are remarkable and cover a large part of the phase space by using all known experimental techniques. Several detectors can be used for PID independently or in combination. In addition to tracking the TPC can serve as a PID detector in the non-relativistic and relativistic rise regions of the Bethe-Bloch parametrization. via the energy loss measurements (dE/dx). The four outer layers of ITS have analog PID capabilities via dE/dx measurement in the non-relativistic region. This technique can be used for K 0 s , Λ, Ξ and Ω analysis but also secondary vertex topology identification such as from K 0 s and Λ can help extend PID capabilities to higher p t .

The ALICE TPC provides excellent particle identification in the low p t region which allows us to extract charged kaons spectra for p t < 1.5 GeV/c by deconvolution of several Gaussians in each p t interval. It will be possible to obtain K + and K -spectra at higher p t with the help of the Time of Flight detector and the PID in the relativistic rise with the TPC. We present here in Fig. 1 the dE/dx histogram obtained from high statistics cosmics data. The K + and K -spectra from simulated data are presented in Fig. 2. The particle ratios such as K + /π + , K + /K -, p/π + will be studied as a function of p t and multiplicity in pp collisions. The multi-strange particles, Ξ and Ω, are reconstucted with the help of the ITS and TPC detectors via their decay products. We estimate that, at √ s = 10 TeV, we will need 100 M minimum bias events to obtain Ξ spectra up to p t = 7 GeV/c with good uncertainties. We could however have access to spectra up to p t = 4 GeV/c with 10% of statistical uncertainties for each point with 2 M minimum bias events.

K 0 s and Λ are also reconstructed topologically in the TPC and ITS. In order to reduce the background and purify the sample, it is possible to use the TPC PID to better identify the decay products but we have shown that we can extract the signal even without this help. The signal extraction is also straightforward without any further selection optimization. It is however impossible to reconstruct K 0 s and Λ below p t = 0.2 GeV/c and p t = 0.5 GeV/c respectively. Our analysis showed that the comfortable situation to extract p t spectra will be to have 1 M events but it will be feasible to reach p t = 3.0 GeV/c in the Λ spectrum only with 200 K events.

Physics at LHC with strange particles

Perspectives of statistical model analysis at LHC

Particle production calculated within the statistical models is quantified by several thermal parameters: temperature, volume and a set of chemical potentials related to charge conservation. In pp collisions, strangeness conservation is treated canonically. However, SPS data have shown a stronger suppression of strange particles yields than expected in the canonical model [2]. As a consequence, an additional suppression effect has to be added. This can be done by using the γ S fugacity factor to take into account deviations of strange particles abundance from their chemical equilibrium distribution value [3] or by using the R c parameter to control the radius of the sub-volume within which strangeness is conserved exactly. In the following, we will focus on the second option. A statistical model that quantifies the extra strangeness suppression by this strangeness correlation volume has been formulated in [4,5,6].

The extrapolation of the parameters of the statistical models from SPS to LHC is quite straightforward for temperature and chemical potentials. No temperature variation is expected between RHIC and LHC whereas the baryonic potential μ B should decrease. However, the value of the R c parameter remains largely uncertain at LHC [7], either it saturates and therefore does not vary with the collision energy or it keeps increasing. Strange particles and above all multi-strange particle ratios are shown to be very sensitive to the value of R c . For large values of this parameter, we clearly observe that particles ratios are very close to the Grand-Canonical description used for large systems like heavy-ion collisions [7].

Baryon over meson ratio

In RHIC central heavy-ion collisions, quark coalescence has been suggested as a possible hadronization mechanism which explains qualitatively the amplitude of the mid-rapidity baryon/meson ratio at intermediate p t . In pp collisions, quark coalescence is clearly not favoured due to the low phase space density. The interest of strange particles such as Λ and K 0 s in this analysis is due to their topological reconstruction techniques that allow an identification over a large p t range.

Baryon over meson ratio in elementary collisions: Λ/K 0 s at UA1

In pp collisions at RHIC, the Λ/K 0 s ratio showed a maximum amplitude of 0.6, well below the one obtained in central heavy-ion collisions. This value is consistent with the one obtained at HERA in the √ s = 319 GeV e+p collisions where the hadronization is modeled by fragmentation [8]. However as it is shown in Fig. 3, the Λ/K 0 s ratio in UA1 p + p collisions at √ s = 630 GeV is close to unity. The baryon over meson ratio reaches even higher values at higher energy. We cannot a priori invoke coalescence in pp collisions to explain this amplitude but on the other hand none of the two models, PYTHIA and EPOS, can reproduce the experimental data as shown in Fig. 3.

Baryon over meson ratio: predictions of PYTHIA and EPOS at LHC

In order to understand the mechanisms that take place in pp collisions and that lead to such a high amplitude of the baryon over meson ratio, we have studied two different models, PYTHIA and EPOS, and their predictions at LHC energies. Using PYTHIA [9], a model based on the Lund fragmentation, requires to extrapolate all the parameters from TEVATRON to LHC. To do so the so-called ATLAS tune has been used by ALICE [10]. This set of parameters leads to a very low Λ/K 0 s ratio at √ s = 14 TeV, considerably lower than the UA1 data, but in line with the PYTHIA predictions for the UA1 energies (see Fig. 4 and [11]). With the increase of the collision energy, the probability of multiple-parton interactions is likely to increase. We have therefore modified the ATLAS set of parameters in order to better describe the so-called underlying event, which includes all phenomena other than the hard scattering process, such as the multiple parton interactions. The values of the parameter PARP(90) that controls the multiple interaction rate (see Fig. 4) have been chosen according to PYTHIA authors' suggestions. Increasing the multiple interaction rate helps to increase the baryon over meson ratio so it seems to go in the right direction but a factor 2 at least is still missing (see Fig. 4). We have then investigated EPOS, a multiple interaction model based on Pomeron exchanges [12]. The specificity of EPOS is that it considers the possibility that collective phenomena take place in pp collisions (mini-plasma). As we have seen on Fig. 3, this mini-plasma option has no effect at √ s = 630 GeV. However, the effect is dramatic at √ s = 14 TeV, it leads to a very strong increase of the baryon over meson ratio (see Fig. 5). The high amplitude of the ratio obtained with the mini-plasma option is due to two combined effects: a stronger increase of the Λ yield than of the K 0 s one and a change in the shape of the p t spectra: it is strongly shifted to higher p t in the case of the baryon. If LHC data show such a high amplitude of the Λ/K 0 s ratio, it could be interpreted, according to EPOS, as the manifestation of collective phenomena in pp collisions.

Conclusion

We have discussed several aspects of the anticipated strange particle analysis in pp collisions at LHC energies. Strange particles are an interesting tool to investigate the nature of the system created in pp collisions. They could reveal, at the LHC energies, that the canonical suppression might diminish, which means strange particle yields could reach the value obtained in heavy-ion collisions. Strange particle ratios could also prove the presence of collective phenomena. Both aspects that have been presented, production yield and dynamics at intermediate p t , come to the same conclusion: they could lead us to revise our view of pp collisions at very high energy, which would raise a lot of questions concerning the use of pp collisions as a baseline but would certainly open a perspective on new physics in pp collisions.

Figure 1 .Figure 2 .

 12 Figure 1. Energy loss in the TPC versus momentum measured with cosmics.

Figure 3 .

 3 Figure 3. Λ/K 0 s in p+p collisions at √ s = 630 GeV at mid-rapidity (UA1 experiment) compared to PYTHIA and EPOS simulations.

Figure 4 .Figure 5 .

 45 Figure 4. Predictions of PYTHIA 6.214 for pp collisions at √ s = 14 TeV.


L Turko
email: lturko@ift.uni.wroc.pl

D Prorok
email: prorok@ift.uni.wroc.pl

D Blaschke
email: blaschke@ift.uni.wroc.pl

Charmonium suppression at RHIC and SPS: an unified approach

Keywords: numbers: 14.40.Gx, 24.10.Pa, 25.75.-q

A kinetic equation approach reproduces an anomalous J/ψ suppression at RHIC and SPS by absorption in a hadron resonance gas. The rapidity dependence of the PHENIX data is also reproduced as a geometric effect due to a longer absorption path for J/ψ production at forward rapidity.

Introduction

Recent STAR and PHENIX experiments [1,2,3] brought up two unexpected results. First, at forward rapidity J/ψ is more suppressed than at midrapidity. Second, the dependence of J/ψ suppression at midrapidity on the number of participants N part [3,4] coincides with the one observed at SPS Pb-Pb collisions by the NA50 experiment [5]. Two different mechanisms to explain those RHIC data are usually considered in the literature: recombination [6,7,8,9] or statistical coalescence [10,11,12,13] and nuclear effects [14,15].

We present a step towards a general description of J/ψ absorption in the framework of a statistical analysis. The formalism allows [16,[START_REF] Prorok | AIP Conf. Proc[END_REF] for a unified description of the data from the NA38 and NA50 experiments at CERN and from the PHENIX experiment at RHIC. Both PHENIX results, concerning rapidity and N part dependencies of J/ψ suppression in Au-Au and Cu-Cu collisions are simultaneously reproduced. More details are given in [START_REF] Prorok | [END_REF].

The J/ψ absorption is caused here by the effective hadronic medium consisting of a multi-component non-interacting hadron resonance gas (HRG). This medium does not exclude a suppression scenario based on the existence of the quark-gluon plasma (QGP) phase [START_REF] Matsui | [END_REF]. The recent lattice QCD calculations [20] locate hadronic systems considered here, with energy densities in the range 2.5 -5.4 GeV/fm 3 , slightly above the expected HRG-QGP transitions temperature. This is still not the region of asymptotically free quarks and gluons. All partons are strongly correlated and J/ψ evolution in the medium consists of collisions with free and bound states in the Confidential: not for distribution. Submitted to IOP Publishing for peer review 22 March 2010 surrounding medium. Detailed calculations, based on Green function technique, show the equivalence of descriptions by means of hadronic or partonic degrees of freedom [21,22].

All hadrons from the lowest up to 2 GeV mass are taken into account as constituents of the matter. The gas is in thermal equilibrium and expands longitudinally and transversally according to relativistic hydrodynamics [23]. J/ψ suppression is the result of inelastic scattering on constituents of the HRG and on nucleons in the cold nuclear matter (CNM) of the colliding ions. Both sources of J/ψ suppression, namely absorption in CNM and HRG, are considered simultaneously.

The model applied here is a straightforward generalization of the model of Refs. [16,24] to include the non-zero rapidity case. J/ψ suppression is the result of a J/ψ final state absorption in a confined medium through interactions of the type

J/ψ + h -→ D + D + X , (1)
where h denotes a hadron, D is a charm meson and X stands for a particle which assure conservation laws (charge, baryon number, strangeness). Charmonium states can be absorbed first in nuclear matter and soon after in the hadron gas. In our phenomenological analysis we assume universal cross sections for baryons, with the appropriate thresholds for their dissociation reactions but energy-independent, σ J/ψN = 4 mb. This choice corresponds to the absorption cross section σ abs = 4.2 ± 0.5 mb for J/ψ in cold nuclear matter measured in p-A collisions by the NA50 experiment [25], shown to be compatible [15,26] with recent d-Au data from the PHENIX experiment [27]. The absorption cross section for meson impact is taken as σ m = 2σ J/ψN /3, according to quark counting rules.

In order to justify the magnitude of hadronic absorption cross sections employed in the present work, a strong medium dependence is required which could for instance stem from spectral broadening of light mesons at the chiral restoration transition (mesonic Mott effect) [28,29].

An A-A collision at impact parameter b generates an almond shaped overlap region S ef f . The time t = 0 corresponds to the moment of the maximal overlap of the nuclei. After about half of the time the nuclei need to cross each other, matter appears in the central region. Soon thereafter matter thermalizes and this moment, t 0 , is estimated at about 1 fm. In the real situation the width of the matter at t 0 is also of the order of 1 fm. Then matter starts to expand and cool according to relativistic hydrodynamics (for details see [16,[START_REF] Prorok | AIP Conf. Proc[END_REF]23,24]) until reaching the freeze-out temperature T f.o. and subsequently freely streaming towards the detectors.

A cc pair is created in a hard nucleon-nucleon collision during the passage of the colliding nuclei through each other. It will evolve to a charmonium eigenstate during the formation time which is of the order of t 0 while experiencing cold nuclear matter (CNM) effects. These are due to nuclear modification of the initial state gluon distributions and partial extinction while passing through the colliding nuclei and will be appropriately parametrized using data from pA collisions. We are interested here to describe the "anomalous suppression" effect by subsequent final state interactions with the expanding HRG formed between the receding nuclei. Since these nuclei which border the HRG on both sides in the longitudinal direction move almost with the speed of light, even J/ψs which contribute to the particle spectra measured in the forward (backward) rapidity region can not escape the hadronic medium. Due to the different production process, the J/ψ velocities can be considered as independent from the velocity distribution of the HRG medium. This results in considerable impact velocities, in particular for forward (backward) produced J/ψs traversing counterpropagating HRG flow.

The transverse expansion starts in the form of a rarefaction wave moving inward S ef f . The evolution is decomposed into a longitudinal expansion inside a slice bordered by the front of the rarefaction wave and the transverse expansion. The rarefaction wave moves radially inward with the sound velocity c s . Since the temperature (and hadron gas density) decreases rapidly outside the wave, we shall ignore possible J/ψ scattering there. We denote by t esc the moment of crossing the border of the rarefaction front.

Inside the region bordered by the front of the rarefaction wave the hydrodynamic evolution ceases when the freeze-out temperature is reached, here we take T f.o. = 150 MeV what is suggested by the statistical model analysis of the PHENIX data [30,31]. So J/ψ is subject to absorption for the time t final = min{ t esc , t f.o. } where t esc is the mean time until it escapes the hadron gas region and t f.o. is the time when it passes the kinetic freeze-out surface for its absorption reactions. The rapidity dependence of t final is crucial for the description of J/ψ RHIC absorption processes. This is an increased functions of rapidity as is illustrated by figure 1-Left.

J/ψ absorption

The J/ψ absorption processes in CNM (nuclear absorption -NA) and HRG take place subsequently. Due to this separation in time the J/ψ suppression factor for given rapidity in a heavy-ion collision with impact parameter b assumes the form

R AA (y, b) = S NA (y, b) • S HRG (y, b) , (2)
where S NA (y, b) and S HRG (y, b) are J/ψ suppression factors in CNM and HRG, respectively. For S NA we employ the usual approximation

S NA ∼ = exp {-σ ψN ρ 0 L } , (3)
where ρ 0 = 0.16 fm -3 is the nuclear matter density and L is the mean absorption path length of the J/ψ through the colliding nuclei obtained from the Glauber model. Within this approximate expression, S NA does not depend on rapidity.

The HRG suppression factor S HRG (y, b) is defined as

S HRG (y, b) = d 2 p T d 3 r F( r, y, p T , t)| t=∞ d 2 p T d 3 r F( r, y, p T , t)| t=t0 , (4)
where the initial time t 0 = 1 fm/c denotes the moment of the thermalization of the created hadronic matter and the beginning of the hydrodynamical expansion. The rapidity-momentum J/ψ distribution results from the kinematical change of variables d 3 p → d 2 p T dy and is given by F( r, y, p T , t) = M T cosh(y) ×f ( r, p, t), where f ( r, p, t) is the formal solution of a kinetic equation (for details see [16,24,[START_REF] Prorok | [END_REF]).

For the description of the evolution of the matter, relativistic hydrodynamics is employed. For n B = 0 and a uniform initial temperature distribution with a sharp edge at the border established by the nuclear surfaces, the full solution of the (3+1)dimensional hydrodynamic equations is known [23]. The radial velocity is zero inside the region bordered by the rarefaction wave and v z ( s, z, t) = z/t, the four-velocity u ν ( r, t) simplifies to

u ν ( r, t) = 1 τ (t, 0, 0, z) . (5)
The temperature also does not depend on the radial coordinate s within the region bordered by the rarefaction wave,

T ( r, t) = T (z, t) = T (0, τ) = T 0 • τ -a (6)
and the power a is the sound velocity squared at T 0 , a = c 2 s (T 0 ) [16,[START_REF] Prorok | AIP Conf. Proc[END_REF]. It turned out that T (τ ) has the form given by ( 6) with a varying in the range a = 0.148 -0.156 for 0 in the range 0 = 5.5 -0.5 GeV/fm 3 . For a J/ψ which is at s 0 ∈ S ef f at the moment t 0 and has the velocity v = v T + v L = p T /E + p L /E we denote by t esc the moment of crossing the border of the rarefaction front. Left: An example of t final in a function of rapidity for p T = 2 GeV and the 0 -20% centrality bin for Au-Au (solid) and Cu-Cu (dashed) collisions. More details are given in [START_REF] Prorok | [END_REF]. Right: J/ψ (dashed and dot-dashed lines) inside hadronic gas absorption region and the time evolution of the rarefaction front radius s at different z (continuous lines with labels denoting z in fm). Numerical values correspond to Au-Au collisions. More details are given in [START_REF] Prorok | [END_REF] The continuous lines in figure 1-Right, labeled by different values of the longitudinal variable z, show the time evolution of the rarefaction front radius s at the position z. The dashed and dash-dotted lines in figure 1-Right are (s, t) trajectories of J/ψ particles originating from the same point in the region S ef f , but with different values of transverse velocity v T . The dot-dashed line corresponds here to the lower value of v T .

The final expression for S HRG (y, b) reads

S HRG (y, b) = 1 dp T M T g 0 (p T ) dp T M T g 0 (p T ) × exp - t final t0 dt l i=1 d 3 q (2π) 3 f i ( q, t)σ i v rel,i p ν q ν i EE i , (7)
where t final = min{ t esc , t f.o. }, t esc (b, v) is an average time of leaving the hadron medium (i.e. crossing the rarefaction wave front) by J/ψ with the velocity v produced in a collision at impact parameter b and t f.o. is the freeze-out moment resulting from the longitudinal Bjorken expansion. Equation ( 7) is in fact an approximation of the more involved exact formula. The approximation means that the average of integrals over trajectories has been replaced by the integration over the averaged trajectory, similarly as in (3). Our preliminary results of estimations of the exact formula suggest that the approximation influences only the normalization and in such a way that this can be compensated by the rescaling of the absorption cross-section to the lower values.

Results

In figure 2 and3 we show the J/ψ nuclear modification factor R AA for different centralities versus rapidity in Au-Au and Cu-Cu collisions vs. PHENIX data. There is an overall agreement with the data. The main new effect of the PHENIX data, that J/ψ suppression is stronger at forward rapidity than at midrapidity is shown in figure 4 for the N part dependence. It arises in the present approach as a result of the geometry of the kinetic freezeout surface for the J/ψ which extends much more in the longitudinal direction for sufficiently long times, t ≥ R A -b/2. This entails a longer absorption path for J/ψ production at forward rapidity.

Note that the present model simultaneously accounts for the anomalous J/ψ suppression in the centrality dependence of the NA50/NA60 experiments at CERN, see [16,[START_REF] Prorok | AIP Conf. Proc[END_REF].

The present model may be seen as a preparatory step towards a unified description of charmonium suppression kinetics in SPS and RHIC experiments within a quantum statistical approach to the sQGP to be developed. The geometrical effect on the rapidity dependence decribed in the present work shall be a part of such a description.

  Figure 1.Left: An example of t final in a function of rapidity for p T = 2 GeV and the 0 -20% centrality bin for Au-Au (solid) and Cu-Cu (dashed) collisions. More details are given in[START_REF] Prorok | [END_REF]. Right: J/ψ (dashed and dot-dashed lines) inside hadronic gas absorption region and the time evolution of the rarefaction front radius s at different z (continuous lines with labels denoting z in fm). Numerical values correspond to Au-Au collisions. More details are given in[START_REF] Prorok | [END_REF]

Figure 2 .

 2 Figure 2. J/ψ nuclear modification factor versus rapidity in Au-Au collisions for constant σ J/ψN = 4 mb. PHENIX data are from [1]. Errors shown are the quadratic sum of statistical and uncorrelated systematic uncertainties.

Figure 3 .

 3 Figure 3. Same as Fig. 2 but for Cu-Cu collisions with PHENIX data from [2].

Figure 4 .

 4 Figure 4. J/ψ nuclear modification factor versus centrality in Au-Au (left) and Cu-Cu (right) collisions for constant σ J/ψN = 4 mb. Lines are predictions of the model for midrapidity (solid) and forward rapidity (dashed). PHENIX data are from [1, 2]. Errors shown are the quadratic sum of statistical and uncorrelated systematic uncertainties.

Acknowledgments

We thank Susumu Oda for providing us with the J/ψ rapidity data for various centralities of Cu-Cu collisions and Krzysztof Redlich for valuable discussions. This work was supported in part by the Polish Ministry of Science and Higher Education.


T Hilger
email: t.hilger@fzd.de

R Schulze

B Kämpfer

QCD sum rules for D mesons in dense and hot nuclear matter

Keywords: numbers: 21.65.Jk, 12.40.Yx, 14.40.Lb, 24.85.+p

Open charm mesons (pseudo-scalar and scalar as well as axial-vector and vector) propagating or resting in nuclear matter display an enhanced sensitivity to the chiral condensate. This offers new prospects to seek for signals of chiral restoration, in particular in pA and pA reactions as envisaged in first-round experiments by the CBM and PANDA collaborations at FAIR. Weinberg type sum rules for charming chiral partners are presented, and the distinct in-medium modifications of open-charm mesons are discussed. We also address the gluon condensates near T c and their impact on QCD sum rules.

Introduction

The chiral condensate seems to be of utmost importance as candidate of an order parameter of chiral symmetry which is spontaneously and explicitly broken in nature (vacuum). The approach to restored chiral symmetry in a dense and/or hot medium may be signaled by a modified hadron spectrum. In particular, the spectral strengths of hadrons are expected to become modified when embedded in strongly interacting matter. While for vector mesons and strange mesons fairly exhaustive studies have been performed in this context, charm degrees of freedom require further investigations. The charmed hadron spectroscopy in pA and pA collisions will be addressed by two major experimental set-up's of FAIR, CBM and PANDA. With this motivation we report in section 2 our results of a QCD sum rule analysis of spectral changes of pseudo-scalar open charm D mesons in cold nuclear matter.

In section 3 we consider differences of spectral strengths of chiral partners thus exposing the role of the chiral condensate also for charm. Since, within the QCD sum rule approach which we employ here, the temperature and density dependence of QCD condensates is important, we present in section 4 a first explorative study of two gluon condensates near the deconfinement temperature but at finite net baryon density.

QCD sum rules for D and D mesons

QCD sum rules offer the unique possibility to relate (non-perturbative) QCD parameters and hadronic observables. To be specific, the current-current correlation function

Π(q) = i d 4 x e iqx T j(x)j † (0) , (1)
with currents j(x) (with the quark structure i d(x)γ 5 c(x) for the pseudo-scalar D mesons, for instance) is related to the integrated spectral density

Π(q) = 1 π +∞ -∞ dω ∆Π(ω, | q |) ω -q 0 , (2)
where ∆Π = 1 2i lim

→0 [Π(ω + i ) -Π(ω -i )]
in turn is related to hadronic observables. We are here interested in an exploration of spectral changes. Quite general, one may consider the moments

S n (M) ≡ ω + 0 ω - 0 dω ω n ∆Π(ω)e -ω 2 /M 2 (3)
which allow to define the quantities ∆m ≡ 1 2

S 1 S 2 -S 0 S 3 S 2 1 -S 0 S 2 , (4)
m + m -≡ - S 2 2 -S 1 S 3 S 2 1 -S 0 S 2 , (5)
m 2 ≡ ∆m 2 + m + m -, (6)
which may be interpreted, e.g. by exploiting the pole + continuum ansatz, as mass splitting (actually, ∆m is half of the splitting) and mass shift of D and D. In line with a similar pattern in the strange meson sector, one expects shifts of spectral strengths by an ambient nuclear medium as exhibited schematically in figure 1. An evaluation of the above equations leads to the results displayed in figure 2 (for details and a discussion of different parameter sets cf. [1,2]). These results point to a negative mass splitting

(color online)

The D-D mass splitting ∆m from Eq. ( 4) and the center of gravity m from Eq. ( 6) as a function of nuclear matter density n. The colored region indicates the confidence range.

D + -D -(left panel), while a safe average shift of D + + D -cannot be extracted from the present approach due to uncertainties of input parameters (see right panel of figure 2). Similar considerations apply for B and D s mesons [1].

A discussion of finite width effects always meets the problem of additional parameters which have to be fixed by the same set of equations. Hence, one only is able to give the mass of the particle (i.e. the center of gravity of the spectral function) as a function of its width. In the case of D mesons the widths for particle (+) and anti-particle (-) enter, and the mass-width correlation of the particle is locked with the width of the antiparticle as well. Employing the following Breit-Wigner ansatz for the spectral function

∆Π(ω) = F + π ωΓ + (ω 2 -m 2 + ) 2 + ω 2 Γ 2 + Θ(ω)+ F - π ωΓ - (ω 2 -m 2 -) 2 + ω 2 Γ 2 - Θ(-ω)(7)
and determining m ± , Γ ± such that they fulfill Eqs. ( 4) and (6). For ∆m = -30 MeV and m = 1.915 GeV one obtains the results depicted in figure 3. In a first step, the integration limits in Eq. ( 3) have been extended to ±∞ without including the continuum contribution, to avoid the influence of possible threshold effects.

l V -A, light-light V -A, heavy-light P -S, heavy-light -1 F 2 π 0 0 8 m Q qq 2m Q qq 1 O 6 8m 3 Q qq + 4m Q ∆ -2m 3 Q qq + m Q qgσGq -m Q ∆ 2 8m 5 Q qq - 4m 3 Q qgσGq - 12 ∆ + O 7 -2m 5 Q qq + 3m 3 Q qgσGq - 3 ∆ + O 7 Table 1.
A few selected moments of the spectral differences dss l ∆Π V (P ) -∆Π A(S) for chiral partners. O i denotes a condensate or a combination of condensates of mass dimension i and ∆ stands for a medium-specific combination of condensates, i.e. it vanishes at zero density and temperature.

Weinberg type sum rules

Exploiting the Operator Product Expansion one can evaluate in the spirit of (1) the differences of spectral moments for vector and axial-vector currents, as has been pioneered by Weinberg [3], and applied to a hot medium by Shuryak and Kapusta [4]. Our results for the differences of vector (V) -axial-vector (A) and the scalar (S)pseudo-scalar (P) correlators in case of mesons with one light quark, m 1 Λ QCD , and one heavy quark, m Q = m 2 Λ QCD , are summarized in table 1. The employed currents are j V(A) µ = q 1 (γ 5 )γ µ q 2 , j S(P) = (i)q 1 (γ 5 )q 2 . In table 1, also the first spectral difference moments for mesons consisting of two light quarks, m 1,2 Λ QCD are listed for V -A. In particular, one recognizes for heavy-light mesons that the differences between chiral partners are driven by the chiral condensate qq in conjunction with the heavy quark mass m Q . Furthermore, in case of light-light vector mesons the first condensate which qualifies as an order parameter is a certain combination of four-quark condensates for the 1st moment in next-to-leading order of the strong coupling α s , whereas the chiral condensate can only occur in lowest order of α s and in the 0th moment of the spectral differences if one of the quarks is heavy. In this respect, one may identify such heavylight quark bound states (hadrons) as useful probes for chiral restoration to be linked with the density dependence of the chiral condensate qq n .

Condensates near T c

According to the consideration in [5] one may relate the gluon condensate at finite temperature to the QCD trace-anomaly for N f = 3 and the equation of state:

α s π G 2 T = α s π G 2 0 - 8 9 (e -3p) , (8)
α s π (uG) 2 - G 2 4 T = - 3 4
α s π (e + p) , (9)
u is the medium 4-velocity, whereas contributions from light quarks to (8) have been omitted in a first step, as we focus on the continuation to finite densities. The energy density e and the pressure p are accessible by QCD lattice evaluations [6] and may be extrapolated to finite baryon densities by employing the Rossendorf quasiparticle model [7] which is here adjusted to [6]. Both condensates (8) and (9) are depicted in figure 4 in the region near T c at quark chemical potential µ q = 0 (left panel) and µ q = 270 MeV (right panel); energy density and pressure have been employed from the quasiparticle model which allows for a thermodynamic consistent extrapolation to finite densities. The curves for the condensate (8) are flattened with increasing lattice temporal extend N τ ; [5] is reproduced by the p4 action [6] for N τ = 6. On the other hand, the condensate (9) seems not to be affected by such a choice. At finite densities, the gluon condensate for αs π G 2 T drops significantly due to the non-vanishing chemical potential. The symmetric and traceless gluon condensate αs π (uG) 2 -G 2

4

T is much less influenced by density effects. These effects have to be studied in detail for the J/ψ, which depends essentially on the considered gluon condensates [5].

As mentioned above, finite current quark masses modify the trace of the energy momentum tensor to [8] (N f = 3)

T µ µ = - 9α s 8π G a µν G µν a + m u ūu + m d dd + m s ss + . . . , (10)
where higher order α s terms have been neglected. (The contributions of heavy quarks can be absorbed into the first term by using the heavy-quark mass-expansion.) A precise analysis would hence also have to include the density and temperature dependence of the quark condensates. However, as the chiral condensate is supposed to vanish above T c due to the restoration of chiral symmetry, it is justified to neglect their influence on the gluon condensates in ( 8) and (9). Furthermore, if light quark contributions can be neglected in specific QCD sum rule evaluations, it is consistent with neglecting such terms in ( 8) and (9), too. This offers the avenue towards the evaluation of QCD sum rules near/above the deconfinement transition at non-zero baryon densities.

Summary

In summary we analyzed in-medium modifications of D mesons in cold nuclear matter and find good prospects for an experimental verification, as the spectral strengths of D and D may be displaced by about 60 MeV at saturation density. The key here is the amplification of the chiral condensate by the heavy charm-quark mass. We also had a glimpse on an extension of the celebrated Weinberg and Kapusta-Shuryak sum rules for chiral partners and considered, furthermore, the determination of the gluon condensate via the QCD trace anomaly in the deconfinement region at non-zero baryon density.

Figure 1 .

 1 Figure 1. Expected spectral changes of D mesons in a cold nuclear medium.

Figure 3 .

 3 Figure 3. Mass-width correlations for m + (left panel) and m -(right panel) for different assumptions of widths of the respective other particle. ∆m = -30 MeV, m = 1.915 GeV and F + = F -are assumed.In a first step, the integration limits in Eq. (3) have been extended to ±∞ without including the continuum contribution, to avoid the influence of possible threshold effects.

Figure 4 .

 4 Figure 4. Temperature dependence of the gluon condensates (8) (upper curves) and (9) (lower curves) at µ q = 0 (left panel) and µ q = 270 MeV (right panel, same line and color code as in left panel).

Acknowledgments

The work is supported by BMBF Grant 06DR9059D and GSI-FE.


J Noronha-Hostler

Jorge Noronha

Carsten Greiner

Particle Ratios and the QCD Critical Temperature

We show how the measured particle ratios at RHIC can be used to provide non-trivial information about the critical temperature of the QCD phase transition. This is obtained by including the effects of highly massive Hagedorn resonances on statistical models, which are used to describe hadronic yields. Hagedorn states are relevant close to T c and have been shown to decrease η/s to the KSS limit and allow for quick chemical equilibrium times in dynamical calculations of hadrons. The inclusion of Hagedorn states creates a dependence of the thermal fits on the Hagedorn temperature, T H , which is assumed to be equal to T c , and leads to an overall improvement of thermal fits. We find that for Au+Au collisions at RHIC at √ s N N = 200 GeV the best square fit measure, χ 2 , occurs at T c ∼ 176 MeV and produces a chemical freeze-out temperature of 170.4 MeV and a baryon chemical potential of 27.8 MeV.

Introduction

Hagedorn states, heavy resonances that follow an exponential mass spectrum, have been recently shown to play an important role close to the QCD critical temperature, T c . They have been used to find a low η/s in the hadron gas phase [1], which nears the string theory bound η/s = 1/(4π) [2]. Calculations of the trace anomaly including Hagedorn states also fit recent lattice results well and correctly describe the minimum c 2 s near the phase transition found on the lattice [1]. Furthermore, estimates for the bulk viscosity including Hagedorn states in the hadron gas phase indicate that ζ/s increases near T c [1].

Recent calculations using Hagedorn states to drive hadrons into chemical equilibrium indicate that hadrons do not need to be born in equilibrium at RHIC [3,4,5,6]. This comes about because Hagedorn states allow for very short chemical equilibration times, which were previously not seen using only binary collisions. Previously, it was assumed that anti-baryons and anti-hyperons were already in chemical equilibrium following the phase transition from QGP to the hadron gas phase because gluon fusion more efficiently produces strange quarks. However, reactions including ‡ E-mail: hostler@th.physik.uni-frankfurt.de § E-mail: noronha@phys.columbia.edu E-mail: carsten.greiner@th.physik.uni-frankfurt.de

Hagedorn states may also reproduce anti-baryons and anti-hyperons within the lifetime of a fireball. Because of the importance of Hagedorn states close to T c it is possible that they would have an effect on thermal fits, which are computed within statistical models to reproduce hadron yield ratios in heavy ion collisions [7,8,9,10,11,12]. For Au+Au collisions at RHIC at √ s N N = 200 GeV, specifically, estimates for the chemical freezeout temperature and baryon chemical potential range from T ch = 155 -169 MeV and µ = 20 -30 MeV [10,11,12]. Moreover, because Hagedorn states are dependent on the limiting Hagedorn temperature T H = T c , a relationship between the chemical freeze-out temperature and the critical temperature can be found by including Hagedorn states in thermal fits [13]. This uniquely gives us the ability to distinguish between different critical temperature regions depending on the quality of the fit obtained using the statistical model. Thus, in this paper we explore the possibility of including Hagedorn states into thermal fits.

Setup

In this paper we use a grand-canonical model to describe the particle densities from which we can calculate the corresponding ratios as described in detail in [8]. Hagedorn states are included in our hadron resonance gas model via the exponentially increasing density of states [14]

ρ(M ) = M M 0 A [m 2 + m 2 r ] 5 4 e m T H dm, (1)
where M 0 = 2 GeV and m 2 r = 0.5 GeV. We consider two different lattice results for T c : T c = 196 MeV [15,16] (the corresponding fit to the trace anomaly is then A = 0.5GeV 3/2 , M = 12 GeV, and B = (340M eV ) 4 ), which uses an almost physical pion mass, and T c = 176 MeV [17] (the corresponding fit to the energy density leads to A = 0.1GeV 3/2 , M = 12 GeV, and B = (300M eV ) 4 ). Both are shown and discussed in [6]. Furthermore, we take into account repulsive interactions using volume corrections [1,6,18] that are thermodynamically consistent. Note that B is a free parameter based upon the idea of the MIT bag constant. Also, we include all the known hadrons below with mass below 2 GeV in our analysis. Note that B is a free parameter based upon the idea of the MIT bag constant. Note that due to the volume corrections T H > T c for a discussion on this see [6,19].

In our model we do not just consider the direct number of hadrons but also the indirect number that comes from other resonances. For example, for pions we consider also the contribution from resonances such as ρ's, ω's, and etc. The number of indirect hadrons can be calculated from the branching ratios for each individual species in the particle data book [20]. Moreover, there is also a contribution from the Hagedorn states to the total number of pions, kaons, and so on as described in [4,6]. Thus the total number of "effective" pions can be described by

Ñπ = N π + i N i n i (2)
whereas the total number of "effective" protons, kaons, or lambdas (generalized as X) can be described by

ÑX = N X + i N i X i ( 3
)
where X is the average number of X = p's, K's, or Λ's. Here N is the total number of each species and n i is the average number of pions that each Hagedorn state decays into. To determine X we use the multiplicities in Fig. 2 of Ref. [3] from the microcanonical model in [21] such that p = 0.058 m i -0.10

K + = 0.075 m i + 0.047 Λ = 0.04 m i -0.07. (4)
Clearly, they are all dependent on the mass of the i th Hagedorn state.

In order to get an idea of the quality of the thermal fits, we define χ 2 as

χ 2 = i R exp i -R therm i 2 σ 2 i ( 5
)
where R therm i is our ratio of hadron yields calculated within our thermal model whereas R exp i is the experimentally measured value of the hadron yield with its corresponding error σ 2 i . Then, µ b and T ch are varried until the minimum χ 2 is found. We use the experimental data from both STAR [22] and PHENIX [23] at mid-rapidity for Au+Au collisions at RHIC at √ s N N =200 GeV. Specifically, we observe the ratios: π -/π + , p/p, K -/K + , K + /π + , p/π + , and (Λ + Λ)/π + . All of which are calculated by STAR [22]. However, only π -/π + , p/p, K -/K + , K + /π + , p/π + are given by PHENIX. Because there is such a significant difference between p/π + from PHENIX and STAR we choose only the value from STAR so that we can compare are results to [11] where they also exclude p/π + from PHENIX. It should be noted that Ref. [11] includes more ratios than we do such as multi-strange particles and resonances, which are not included in this paper. This is because the purpose of this paper is not to confirm their results, which have already been confirmed in [12], but rather to compare thermal fits that include the contribution of Hagedorn states and those that exclude them.

Results

We show the thermal fits for a hadron gas excluding Hagedorn states in Fig. 1, where T ch = 160.4 MeV, µ b = 22.9 MeV, and χ 2 = 21.2. This is almost identical to [11] where χ 2 = 20.9, which is shown in Fig. 2. The χ 2 is actually slightly smaller than in Fig. 1. When we consider the lattice results from BMW, which are at the lower end of the The difference in the χ 2 's for BMW and RBC-Bielefeld collaboration is directly related to the contribution of Hagedorn states in the model. Because the RBC-Bielefeld critical temperature region is significantly higher than its corresponding chemical freezeout temperature the contribution of the Hagedorn states is minimal ∼ 4-11% whereas the contribution from the Hagedorn states is 30 -40% for T H = 176 MeV (see [13]). We find that the inclusion of Hagedorn states should not only provide a better fit but they also affect the chemical freeze-out temperature and the baryonic chemical potential. The more mesonic Hagedorn states are present the larger µ b becomes. Furthermore, our fits also have higher T ch 's than seen in the fit without the effects of Hagedorn states.

T ch = 160.

Conclusion

Our results for thermal fits without Hagedorn states concur well with other thermal fit models [11] where the chemical freeze-out temperature (T ch = 160.4 MeV) is almost identical and the baryonic chemical potential (µ b = 22.9 MeV) is only slightly larger. The thermal fit with the known particles in the particle data group provides a decent fit with χ 2 = 21.2. However, the inclusion of Hagedorn states provides an even better fit to the experimental data. In fact, we find χ 2 = 17.8, T ch = 172.6 MeV, and µ b = 39.7 MeV for the BMW collaboration while for the RBC-Bielefeld collaboration we obtained χ 2 = 20.9, T ch = 165.9 MeV, and µ b = 20.9 MeV. This provides further evidence [1,4,5,6] that Hagedorn states should be included in a description of hadronic matter near T c . Since the chemical freeze-out temperature was found to increase from 160 MeV to roughly 165 MeV (RBC-Bielefeld) or 172 MeV (BMW) when including Hagedorn states, this exemplifies the degree of uncertainty in extracting chemical freezeout thermodynamical parameters by means of such thermal analyzes.

  5 and µ b = 20 MeV. The inclusion of Hagedorn states is our primary interest. Starting with the fit for the RBC-Bielefeld collaboration, we obtain T ch = 165.9 MeV, µ b = 25.3 MeV, and

Figure 1 .

 1 Figure 1. Thermal fits for Au+Au collisions at RHIC at √ s N N = 200 GeV without Hagedorn states with mass above 2 GeV.

Figure 2 .

 2 Figure 2. Thermal fits including Hagedorn states with mass above 2 GeV for Au+Au collisions at RHIC at √ s N N = 200 GeV.

Acknowledgements

This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landes-Offensive zur Entwicklung Wissenschaftlich-ökonomischer Exzellenz) launched by the State of Hesse. J.N. was supported by the US-DOE Nuclear Science Grant No. DE-FG02-93ER40764.