A Andronic

P Braun-Munzinger

K Redlich

J Stachel

Heavy quark(onium) at LHC: the statistical hadronization case

We discuss the production of charmonium in nuclear collisions within the framework of the statistical hadronization model. We demonstrate that the model reproduces very well the availble data at RHIC. We provide predictions for the LHC energy where, dependently on the charm production cross section, a dramatically different behaviour of charmonium production as a function of centrality might be expected. We discuss also the case in elementary collisions, where clearly the statistical model does not reproduce the measurements.

Introduction

Charmonium production is considered, since the original proposal more than 20 years ago about its suppression in a Quark-Gluon Plasma (QGP) [1], an important probe to determine the degree of deconfinement reached in the fireball produced in ultrarelativistic nucleus-nucleus collisions. In the original scenario of J/ψ suppression via Debye screening [1] it is assumed that the charmonia are rapidly formed in initial hard collisions but are subsequently destroyed in the QGP (see recent summaries in [2,3]).

In recent publications [4] we have demonstrated that the data on J/ψ and ψ production in nucleus-nucleus collisions at the SPS ( √ s NN ≈ 17 GeV) and RHIC ( √ s NN =200 GeV) energies can be well described within the statistical hadronization model proposed in [6] and have provided predictions for the LHC energy ( √ s NN =5.5

TeV) and for energies close to threshold ( √ s NN ≈ 6 GeV). Because the mass of heavy quarks exceeds the transition temperature of the QCD phase transition by a factor of almost 8, heavy flavor hadron production cannot be described in a purely thermal approach. It was, however, realized in [6] that charmonium and charmed hadron production can be well described by assuming that all charm quarks are produced in initial, hard collisions while charmed hadron and charmonium production takes place exclusively at the phase boundary with statistical weights calculated in a thermal approach. For a recent review of this statistical hadronization approach see [5]. An important element is thermal equilibration, at least near the critical temperature, T c , which we believe can be achieved efficiently for charm only in the QGP. An important question is whether the statistical behavior is a unique feature of high energy nucleus-nucleus collisions or whether it is also encountered in elementary collisions, where finite size effects will obscure a possible phase transition. In early analyses (see ref. [7]), it is indeed argued that hadron production in e + e -and pp is thermal in nature. Furthermore, such analyses of hadron multiplicities (for recent results see [8,9,10] and refs. therein) yield also temperature values in the range of 160-170 MeV.

To establish the uniqueness of the J/ψ probe for the diagnosis of a QGP in nucleusnucleus collisions it is important to understand whether similar thermal features as observed in nucleus-nucleus collisions are also at work in heavy-flavor hadron production in elementary collisions. We have recently demonstrated [11] that there are crucial differences in elementary collisions compared to nucleus-nucleus.

The statistical model in elementary collisions

For the study of hadron production in e + e -collisions we employ the canonical statistical model described in [8,11] (see also [9,12]). For the present study, we perform calculations for two cases: i) a 2-jet initial state which carries the quantum numbers of the 5 flavors, with the relative abundance of the five flavors in one jet and corresponding antiflavor in the other jet taken form the measurements at the Z 0 resonance quoted in [13]. These relative abundances (17.6% for uū and cc and 21.6% for d d, ss and b b) are thus external input values, unrelated with the thermal model. ii) a purely thermal ansatz , i.e. a 2-jet initial state characterized by vanishing quantum numbers in each jet.

In Fig. 1 we show a comparison of data [13] and model prediction for charmed and bottom hadron yields in e + e -annihilations at √ s=91 GeV. For the model we have used the parameter set: T =170 MeV, V =16 fm 3 and γ s =0.66, which represents the best fit of multiplicities of hadrons with lighter quarks [8].

We first note that the calculation employing the 5-flavor scheme is in very good agreement with the data, as demonstrated by the good χ 2 per degree of freedom between the model and the data (excluding the Υ and prompt J/ψ) of 21.7/16 (34/18 when including all species) (see also ref. [9]). Despite this overall agreement, the exceptions are significant: the Υ meson yield is underpredicted by the model by 17 orders of magnitude, while the prompt J/ψ yield [START_REF] Acciarri | L3 coll[END_REF] is underpredicted by almost 2 orders of magnitude. Obviously, the production of quarkonia is expected to be strongly suppressed in the statistical model. The disagreement is a consequence of the separate hadronization of the c and c quarks. The measured prompt J/ψ production in Z 0 decays (into hadrons) is about 3×10 -4 [START_REF] Acciarri | L3 coll[END_REF]. The thermal model predicts a prompt yield for J/ψ of 4.1×10 -6 (1.6×10 -7 for ψ and 4.3×10 -7 for χ c1 ), identically for the two calculation schemes. Whenever the model seems to describe the yields of charmonia the measured yields are

Multiplicities of hadrons with charm and bottom

quarks in e + e -collisions compared to the thermal model calculations for two cases: i) the 5flavor jet scheme (thick lines) and ii) no (net) flavor jet scheme (thin lines with diamonds for the charm sector and triangles for bottom). Note, for case ii) the large factors used to scale the model calculations to fit in the plotting range. The data are from the compilation published by the Particle Data Group (PDG) [13]. The prompt J/ψ measurement J/ψ pr is from the L3 experiment [START_REF] Acciarri | L3 coll[END_REF].

dominated by the feed down from bottom hadrons and the agreement only reflects the agreement seen for the open bottom hadrons and their branching ratios to charmonia, properly considered in the model.

The calculation employing a purely thermal ansatz underpredicts all the measurements by many oders of magnitude, while for the light quark sector the differences between calculations with a pure thermal model and with the 5-flavor quarkantiquark scheme were found to be small [8]. The strangeness suppression factor, which for the present results only enters in the calculation of the yields of D s and B s mesons, appears to have no counterpart in the heavy quark sector. This reflects the fact that a negligible number of c and b quarks are formed in the fragmentation process. In this case, the thermal weights describe the distribution of the initial quarks into heavy flavor hadrons. Chemical equilibration is not required in this process.

For the case of hadron production in elementary hadronic collisions we employ the canonical realization of the thermal model [15,4,16]. For the description of the relative production cross sections of heavy flavored hadrons, the energy dependence of the temperature parameter is the only model input, which is taken in a parametrized form from the fits of (u,d,s)-hadron abundancies in central nucleus-nucleus collisions [17]. For c.m. energies beyond 10 GeV per nucleon pair in nucleus-nucleus collisions a limiting temperature T lim =164±5 MeV is reached. Recent fits of hadron yields in pp collisions [10] give very similar values, independent of anergy.

In Fig. 2 we show the model comparison to data for the relative production cross section of ψ and J/ψ charmonia ‡. The measurements in pA and pp(p) collisions are above the model values by about a factor 4 (corresponding to 10 experimental standard [19]; the points for elementary collisions are from the PHENIX experiment at RHIC [20] and from the CDF experiment at Tevatron [22] (see text); the data point for Pb+Pb collisions at the SPS energy is from the NA50 experiment [START_REF] Alessandro | NA50 coll[END_REF].

The average value of the pA and pp(p) measurements with the corresponding error (see text) is represented by the shaded box. The band denotes statistical model calculations for the temperature parametrization from heavy-ion fits [17] (T lim =164 MeV) with ±5 MeV error.

deviations; the average value of the measurements is 0.137±0.009, with a χ 2 per degree of freedom of 0.88). The relative production cross sections of charmonium states, as are observed in all measurements in hadronic collisions cannot be described in the thermal approach. The temperature needed to explain the data would be 300 MeV, well above the Hagedorn limiting temperature, which is about 200 MeV. This is in sharp contrast to the (only currently existing) measurement in central nucleus-nucleus collisions, performed at the SPS by the NA50 experiment [START_REF] Alessandro | NA50 coll[END_REF], which is well described. We note that the pA data exhibit a constant ψ /J/ψ production ratio as a function of energy. In the model, the value is determined only by the temperature and this is reflected in the slight decrease of the ratio towards low energies. A constant value, also up to the LHC energies, is predicted beyond √ s NN 20 GeV.

The measurements reported in Fig. 2 demonstrate that the relative production cross section ψ /J/ψ is identical in pA and in pp(p) collisions, implying no visible influence of the cold nuclear medium. Note that the ratio for the Tevatron energy was derived from the CDF measurements of J/ψ [21] and ψ [22] and is for transverse momentum p t >1. 25 GeV/c (we have extrapolated the ψ measurement from 2 GeV/c down to 1.25 GeV/c).

The statistical hadronization model in nucleus-nucleus collisions

The model has the following input parameters: i) characteristics at chemical freeze-out: temperature, T , baryochemical potential, µ b , and volume corresponding to one unit of rapidity V ∆y=1 , extracted from thermal fits of non-charmed hadrons [17]; ii) the charm production cross section in pp collisions, taken either from NLO pQCD calculations [23,24] or from experiment [25], used to calculate the number of directly produced cc pairs N dir cc . The heavy quark (charm or bottom) balance equation [6] (including canonical suppression factors [15,4], I 1 /I 0 ):

N dir cc = 1 2 g c N th oc I 1 (g c N th oc ) I 0 (g c N th oc ) + g 2 c N th cc . (1)
is used to determine the fugacity factor g c (I n are modified Bessel functions; N th oc and N th cc are the numbers of open and hidden charm hadrons, respectively, in the volume V ∆y=1 , computed from their grand-canonical densities).

A comprehensive set of model predictions as a function of energy, from the charm production threshold up to the LHC, is presented in Fig. 3 for central Au-Au collisions (N part =350). The left panel shows our predictions for the energy dependence of midrapidity yields for various open and hidden charm hadrons. The most striking behavior is observed for the production of Λ + c baryons: their yield relative to other charmed hadrons rises significantly towards lower energies. In our approach this is caused by the increase in baryochemical potential towards lower energies (coupled with the charm neutrality condition). A similar behavior is seen for the Ξ + c baryon. The relative production yields of D-mesons depend on their quark content and depend on energy only around threshold. These results emphasize the importance of measuring, at low energies, in addition to D-mesons, also the yield of charmed baryons to get a complete measure of the total charm production cross section. One of the motivations for the study of charm production at low energies was the expectation [26,27] to provide, by a measurement of D-meson production near threshold, information on their possible in-medium modification near the phase boundary. However, the cross section σ cc is governed by the mass of the charm quark m c ≈ 1.3 GeV, which is much larger than any soft QCD scale such as Λ QCD .

Therefore we expect no medium effects on this quantity. The much later formed Dmesons, or other charmed hadrons, may well change their mass in the hot medium. Whatever the medium effects may be, they can, because of the charm conservation, σ cc = 1 2 (σ D + σ Λc + ...) + (σ ηc + σ J/ψ + ...), expressed in Eq. ( 1), in first order only lead to a redistribution of charm quarks [4]. In contrast, as a consequence of in-medium masses of open charm hadrons, the yields of charmonia can vary by up to 40-50% and this is more prominent at threshold energies [4].

In the right panel of Fig. 3 we show the energy dependence of ratios of mid-rapidity yields of open and hidden charm hadrons and pions. The relative production of charmed hadrons to pions exhibits a monotonic dependence on energy, unlike equivalent ratios involving strange hadrons [17]. The different bahavior compared to the strangeness sector is mainly due to the strong canonical suppression of open charm at low energies (determined in turn by strong decrease of the charm production cross section at low energies). A non-monotonic behavior is observed in the production ratio of J/ψ to D 0 , determined by the interplay of the strong canonical suppression of open charm at low energies and the gradual onset as a function of energy of the quadratic term in Eq. ( 1). AA for two centrality classes. The data from the PHENIX experiment [28] (symbols with errors) are compared to calculations (lines, see text). The shaded area corresponds to calculations for the lower limit of the charm cross section as measured by PHENIX [25], with our shadowing scenario.

In Fig. 4 we present the rapidity dependence of the nuclear modification factor R J/ψ AA (we use experimental data for the pp reference). While earlier [4] we have compared data [28] to our model predictions for the pQCD charm production cross section [23] (dashed line in Fig. 4), we use here as alternative the charm cross section as measured by PHENIX in pp collisions [25] and consider in addition shadowing for Au-Au collsions as extracted from recent dAu data [29]; we assume that the deviation of R J/ψ dAu from unity is entirely due to shadowing. Also in this case, our model describes the observed suppression and its rapidity dependence. The maximum of R J/ψ AA at midrapidity is in our model due to the enhanced generation of charmonium around mid-rapidity, determined by the rapidity dependence of the charm production cross section. In this sense, the above result constitutes strong evidence for the statistical generation of J/ψ at chemical freeze-out. The centrality dependence of R J/ψ AA at midrapidity is shown in the left panel of Fig. 5. Our calculations approach the value in pp collisions around N part =50, which corresponds to an assumed minimal volume for the creation of QGP of 400 fm 3 [4]. The model reproduces very well the decreasing trend versus centrality seen in the RHIC data [28]. Note that, in our model, the centrality dependence of the nuclear modification factor arises entirely as a consequence of the still rather moderate rapidity density of initially produced charm quark pairs at RHIC (dN cc /dy=1.6 for central collisions, using the FONLL charm production cross section [23]) leading to a marked centrality dependence of the canonical suppression factor for open charm hadrons. At the much higher LHC energy the charm production cross section (including shadowing in PbPb collisions [24]) is expected to be about an order of magnitude larger. As a result, the opposite trend as a function of centrality is predicted, with R J/ψ AA exceeding unity for central collisions. A significantly larger enhancement of about a factor of 2 is obtained if the charm production cross section is two times larger than presently assumed, as seen in the right panel of Fig. 5, where we show the J/ψ production relative to the number of initially produced cc pairs.

Conclusions

We have shown that the statistical hadronization model describes well the measured decrease with centrality and the rapidity dependence of R J/ψ AA at RHIC energy.

Importantly, well described in central heavy-ion collisions at SPS energy, where we have the only measurement to date, is the ratio ψ /J/ψ. In contrast, in elementary, e + e -and in pp and p(π)-nucleus, collisions the charmonium measurements cannot be explained within the statistical model. This underlines the relevance of the statistical features in nucleus-nucleus collisions. Extrapolation to LHC energy leads, contrary to the observations at RHIC, to R J/ψ AA increasing with collision centrality. If observed, such a behavior will provide a dramatic proof of charmonium as the ultimate observable to delineate the phase boundary of QCD matter. K.R. acknowledges support from the Alexander von Humboldt Foundation.

  Figure 1.Multiplicities of hadrons with charm and bottom quarks in e + e -collisions compared to the thermal model calculations for two cases: i) the 5flavor jet scheme (thick lines) and ii) no (net) flavor jet scheme (thin lines with diamonds for the charm sector and triangles for bottom). Note, for case ii) the large factors used to scale the model calculations to fit in the plotting range. The data are from the compilation published by the Particle Data Group (PDG)[13]. The prompt J/ψ measurement J/ψ pr is from the L3 experiment[START_REF] Acciarri | L3 coll[END_REF].

Figure 2 .

 2 Figure2. Production cross section of ψ relative to J/ψ. The data for pA collisions are from the compilation by Maltoni et al.[19]; the points for elementary collisions are from the PHENIX experiment at RHIC[20] and from the CDF experiment at Tevatron[22] (see text); the data point for Pb+Pb collisions at the SPS energy is from the NA50 experiment[START_REF] Alessandro | NA50 coll[END_REF].The average value of the pA and pp(p) measurements with the corresponding error (see text) is represented by the shaded box. The band denotes statistical model calculations for the temperature parametrization from heavy-ion fits[17] (T lim =164 MeV) with ±5 MeV error.

Figure 3 .

 3 Figure 3. Left panel: energy dependence of the production yield at midrapidity for open and hidden charm hadrons (with the energy dependence of dσ cc /dy as in[4]). Right panel: ratios calculated from midrapidity yields.

Figure 4 .

 4 Figure 4. Rapidity dependence of R J/ψ

Figure 5 .

 5 Figure 5. Centrality dependence of R J/ψ AA for RHIC and LHC energies (left panel) and of the J/ψ rapidity density at LHC relative to the number of initially produced cc pairs (right panel, curves labelled by the dσ cc /dy) at midrapidity.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 15 February 2010

‡ The charm production cross section, which is an important model input parameter for the calculations of absolute yields, cancels out for this ratio.


J Noronha-Hostler

Carsten Greiner
email: carsten.greiner@th.physik.uni-frankfurt.de

Igor Shovkovy
email: igor.shovkovy@asu.edu

Thermalization through Hagedorn states -the importance of multiparticle collisions

Quick chemical equilibration times of hadrons within a hadron gas are explained dynamically using Hagedorn states, which drive particles into equilibrium close to the critical temperature. Within this scheme master equations are employed for the chemical equilibration of various hadronic particles like (strange) baryon and antibaryons. A comparison of the Hagedorn model to recent lattice results is made and it is found that for both Tc =176 MeV and Tc=196 MeV, the hadrons can reach chemical equilibrium almost immediately, well before the chemical freeze-out temperatures found in thermal fits for a hadron gas without Hagedorn states.

Introduction and Motivation

As two heavy ions collide color neutral clusters are formed within which the number of particles per cluster increase. The clusters become so dense and begin to overlap such that it impossible to distinguish quarks from one cluster from that in another i.e. a percolation transition. The critical density for this is about ≈ 1 GeV /f m 3 . Following the phase transition into Quark Gluon Plasma the interactions are dominated by quarks and gluons. Through gluon fusions, strange quarks can easily be reproduced. Eventually the QGP cools back into hadrons where the particle yields and ratios are measured.

If one only considers binary collisions, which react too slowly for strange particles to reach chemical equilibrium within the hadron gas phase, then it is clear that strange particle yields can only be explained through gluon fusion within QGP [1] and that the hadrons must exist QGP already in full chemical equilibrium [2]. However, multimesonic collisions nπ ↔ X X have been demonstrated to reach chemical equilibration for varoius (strange) antibaryons quickly at SPS [3], although they are still not enough to explain the particle yields of exotic antibaryons at the higher energies at RHIC [4,5]. In order to circumvent such longer time scales ∼ 10 fm/c for a situation of a nearly baryon-free system with nearly as much antibaryons as baryons, it was then suggested by Braun-Munzinger, Stachel and Wetterich [6] that near Tc there exists an extra large particle density overpopulated with pions and kaons, which then drive the baryons/anti-baryons into equilibrium by exactly such multi-mesonic collisions. But it is not clear how and why this overpopulation of pions and kaons should appear, and how the subsequent population of (anti-)baryons would follow in accordance with a standard statistical hadron model: According to the mass action law the overpopulated matter of pions will result in an overpopulation of (anti-)baryons. For such a large number of (anti-)baryons it is difficult to get rid of them quickly enough in order to reach standard hadron equilibrium values before the chemical freeze-out [7].

Rather, understanding the rapid chemical equilibration is possible using Hagedorn states, heavy resonances that drive similar and more multi-hadronic reactions close to T c , as shown in [7,8,9,10]. Close to T c the matter is then a strongly interacting mixture of standard hadrons and such resonances. Using the Hagedorn states as potential and highly unstable catalysts, the standard hadrons can be populated reactions:

nπ ↔ HS ↔ n π + X X (1)
where X X can be substituted with pp , K K , Λ Λ, or Ω Ω. The large masses of the decaying Hagedorn states open up the phase space for multi-particle decays.

In this note we will compare the particle ratios obtained by using reactions driven by Hagedorn states and those of the experiments at RHIC. We find that both strange and non-strange particles match the experimental data well within the error bars. Furthermore, we are able to make estimates for the chemical equilibration time and find that they are very short, which implies that the hadrons can easily reach chemical equilibrium within an expanding, hadronic fireball and that hadrons do not need to be "born" into chemical equilibrium [8,9,10]. Hagedorn states thus provide a microscopic basis for understanding hadronisation of deconfined matter to all hadronic particles.

Before starting with the details, we emphasize that Hagedorn states have become quite popular to understand the physics of strongly interacting matter close to the critical temperature: Hagedorn states have been shown to contribute to the physical description of a hadron gas close to T c . The inclusion of Hagedorn states leads to a low η/s in the hadron gas phase [11], which nears the string theory bound η/s = 1/(4π) . Calculations of the trace anomaly including Hagedorn states also fits recent lattice results well and correctly describe the minimum of the speed of sound squared, c 2 s , near the phase transition found on the lattice [11]. Estimates for the bulk viscosity including Hagedorn states in the hadron gas phase indicate that the bulk viscosity, ζ/s, increases near T c [11]. We also remark here that Hagedorn states can also explain the phase transition above the critical temperature and, depending on the intrinsic parameters, the order of the phase transition [12]. Finally, it has been shown [13] that Hagedorn states provide a better fit within a thermal model to the hadron yield particle ratios. Additionally, Hagedorn states provide a mechanism to relate T c and T chem , which then leads to the suggestion that a lower critical temperature could possibly be preferred, according to the thermal fits [13].

Thermalization through Hagedorn states -the importance of multiparticle collisions 3

Setup

The basis of the Hagedorn spectrum is that there is an exponential mass increase along with a prefactor i.e. the mass spectrum has the form: f (m) ≈ exp m/T H [14]. The exponential mass spectrum drives open the phase space, which allows for multi-mesonic decays to dominate close to T c (we assume T H ≈ T c ). We use the form

ρ = M M 0 A [m 2 + m 2 r ] 5 4 e m T H dm. (2)
where M 0 = 2 GeV and m 2 r = 0.5 GeV. We consider two different 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 [10]. Furthermore, we need to take into account the repulsive interactions and, thus, use volume corrections [10,11,18], which ensure that the our model is thermodynamically consistent. Note that B is a free parameter based upon the idea of the MIT bag constant.

We need to consider the back reactions of multiple particles combining to form a Hagedorn state in order to preserve detailed balance. The rate equations for the Hagedorn resonances N i , pions N π , and the X X pair N X X , respectively, are given by Ṅi = Γ i,π N eq i n

B i,n N π N eq π n -N i + Γ i,X X N eq i N π N eq π n i,x N X X N eq X X 2 -N i Ṅπ = i Γ i,π N i n i -N eq i n B i,n n N π N eq π n + i Γ i,X X n i,x N i -N eq i N π N eq π n i,x N X X N eq X X 2 ṄX X = i Γ i,X X N i -N eq i N π N eq π n i,x N X X N eq X X 2 . ( 3
)
The decay widths for the i th resonance are Γ i,π and Γ i,X X , the branching ratio is B i,n (see below), and the average number of pions that each resonance will decay into is n i . The equilibrium values N eq are both temperature and chemical potential dependent. However, here we set µ b = 0. Additionally, a discrete spectrum of Hagedorn states is considered, which is separated into mass bins of 100 MeV. The branching ratios, B i,n , are the probability that the i th Hagedorn state will decay into n pions where n B i,n = 1 must always hold. We assume the branching ratios follow a Gaussian distribution for the reaction HS ↔ nπ

B i,n ≈ 1 σ i √ 2π e - (n-n i ) 2 2σ 2 i , (4)
which has its peak centered at n i ≈ 3 -34 and the width of the distribution is σ 2 i ≈ 0.8 -510 (see [10]). For the average number of pions when a X X pair is present, we again refer to the micro-canonical model in [7,19] and find

n i,x = 2.7 1.9 (0.3 + 0.4m i ) ≈ 2 -7. ( 5
)
where m i is in GeV. In this paper we do not consider a distribution but rather only the average number of pions when a X X pair is present. We assume that (6) Γ i is a linear fit extrapolated from the data in [20]. It is then seperated into two parts, one for the reaction HS ↔ nπ i.e. Γ i,π and one for the reaction HS ↔ nπ + X X i.e. Γ i,X X . The decay width Γ i,X X is found my multiplying X , which is the average X that a Hagedorn state of mass m will decay into, that is found from both microcanonical [19,7] and canonical models [10]. The large masses open up the phase space for such more special multi-particle decays. A detailed explanation is found in [10].

The equilibrium values are found using a statistical model [21], which includes 104 light or strange particles from the the PDG [20]. As in [21], we also consider the effects of feeding for pions, protons, kaons, and lambdas. Additionally, throughout this paper our initial conditions are the various fugacities at t 0 (at the point of the phase transition into the hadron gas phase) α ≡ λ π (t 0 ) , β i ≡ λ i (t 0 ) , and φ ≡ λ X X (t 0 ) which are chosen by holding the contribution to the total entropy from the Hagedorn states and pions constant i.e. s Had (T 0 , α)V (t 0 ) + s HS (T 0 , β i )V (t 0 ) = s Had+HS (T 0 )V (t 0 ) = const and the corresponding initial condition configurations we choose later can be seen in Tab. 1 (for further discussion see [10]).

Results: Expansion

In order to include the cooling of the fireball we need to find a relationship between the temperature and the time, i.e., T (t). To do this we apply a Bjorken expansion for which the total entropy is held constant

const. = s(T )V (t) ∼ S π N π dN π dy dy. (7)
where s(T ) is the entropy density of the hadron gas with volume corrections. The total number of pions in the 5% most central collisions, dNπ dy , can be found from experimental results in [22]. Thus, our total pion number is i N π i = 0.5 -0.5 dNπ dy dy = 874. While for a gas of non-interacting Bose gas of massless pions S π /N π = 3.6, we do have a mass for a our pions, so we must adjust S π /N π accordingly. In [23] it was shown that when the pions have a mass the ratio changes and, therefore, the entropy per pion is close to S π /N π ≈ 5.5, which is what we use here.

The effective volume at mid-rapidity can be parametrized as a function of time. We do this by using a Bjorken expansion and including accelerating radial flow. The volume term is then

V (t) = π ct r 0 + v 0 (t -t 0 ) + 1 2 a 0 (t -t 0 ) 2 2 (8)
where the initial radius is r 0 (t 0 ) = 7.1 fm for T H = 196 and the corresponding t

(196) 0 ≈ 2f m/c.
For T H = 176 we allow for a longer expansion before the hadron gas phase is reached and, thus, calculate the appropriate t (176) 0 from the expansion starting at T H = 196, which is t (176) 0 ≈ 4f m/c. We use v 0 = 0.5 and a 0 = 0.025 (see [10]). Because the volume expansion depends on the entropy and the Hagedorn resonances contribute strongly to the entropy only close to the critical temperature (see [10]), the effects of the Hagedorn states must be taken into account with calculating the total particle yields otherwise the yields do not increase with the temperature (see [10] for further discussion). Therefore, one has to include the potential contribution of the Hagedorn resonances to the pions as in the case of standard hadronic resonances, e.g. a ρ-meson decays dominantly into two pions and, thus, accounts for them by a factor two. Including Hagedorn states, we arrive at our effective number of pions and X X pairs

Ñπ,X X = N π + i N i [(1 -X i ) n i + X i n i,x ] ÑX X = N X X + i N i X i (9)
because Hagedorn states also contribute strongly to the X X pairs close to T c .

Along with the expansion we also must solve these rate equations, Eq. (3), numerically . We start with various initial conditions as seen in table II and the initial temperature is the respective critical temperature and we stop at T = 110 MeV. The black solid line in each graph is the chemical equilibrium abundances and the colored lines are the various initial conditions listed in Tab. 1. In order to save space, the results are only the expansion of the Λ's and Ω's results are only shown for T H = 196 MeV. However, T H = 176 MeV and further results are shown and discussed in [10]. However, the end particle ratios are all shown in Fig. 2. Note that in all the following figures the effective numbers are shown so that the contribution of the Hagedorn states is included. We can also observe the affects of the expansion on the Λ Λ pairs as seen in Fig. 1 for the reaction nπ ↔ HS ↔ nπ + Λ Λ. They reach the experimental values almost immediately. One can see that the chemical equilibration time does depend slightly on our choice of β i , i.e., a larger β i means a quicker chemical equilibration time. The one exception is for an underpopulation (β i < 1) of Hagedorn states, which reaches chemical equilibrium by T = 170 for T H = 196 MeV. Additionally, when the Λ Λ pairs start at about half their chemical equilibrium values, it only helps the Λ Λ pairs to reach equilibrium at a slightly higher temperature (on the order of a couple of MeV).

α = λ π (t 0 ) β i = λ i (t 0 ) φ = λ X X (t 0 ) IC 1 1 1 0 IC 2 1 1 0.5 IC 3 1.1 0.
We used our model to investigate the possibility of Ω's produced through [7]:

HS ↔ Ω Ω + X HS (sssq q q) ↔ Ω + B + X HS B (sss) ↔ Ω + X. ( 10
)
the first we implemented into our model to obtain Fig. 1 where we are impressively able to populate the Ω Ω's. However, this scenario is slightly more delicate because Γ i by 50%, the total production of Ω is not sufficient up to 25 %, to meet the experimental yield (the other ratios are not significantly affected by the change in Γ i ).

A summary graph of all our results is shown in Fig. 2. The gray error bars cover the range of error for the experimental data points from STAR and PHENIX. The points show the range in values for the initial conditions at a final expansion point with a temperature T = 110 MeV. We see in our graph that our freezeout results match the experiments well and the initial conditions have little effect on the ratios, which implies that information from the QGP regarding multiplicities is washed out due to the rapid dynamics of Hagedorn states. A smaller β i slows the equilibrium time slightly. However, as seen in Fig. 2 it still fits within the experimental values. Furthermore, in [8] we showed the the initial condition play almost no roll whatsoever in K/π + and (B + B)/π + . Thus, strengthening our argument that the dynamics are washed out following the QGP.

Conclusion

To conclude we have found that the Hagedorn states provide a mechanism for quick chemical equilibration times. Our model gives chemical equilibration times on the order of ∆τ ≈ 1 -3 f m c . Furthermore, the particle ratios obtained from decays of Hagedorn states match the experimental values at RHIC very well, which leads to the conclusion that hadrons do not need to be born in chemical equilibrium. Rather a scenario of hadrons that reach chemical freeze-out shortly after the critical temperature due to multi-mesonic reactions driven by Hagedorn states, is entirely plausible. We have shown that both strange (Λ's and K's) and non-strange (π's and p's) hadrons can reach chemical equilibration by T = 160 MeV. Thus, it would be interesting to implement Hagedorn states into a transport approach such as UrQMD [24] Moreoever, even multi-strange baryons such as Ω's can reach chemical equilibrium in such a scenario. However, unlike the other hadrons, the final abundancy of Ωs is slightly dependent on the decay width of the Hagedorn states (they cannot reach full chemical equilibrium if the decay width is 50% of its current value) and, thus, should be further investigated using strange and/or baryonic Hagedorn states [7]. Although they are quite exotic, it is still possible that they might occur. Still, our work indicates that the population and repopulation of potential Hagedorn states close to phase boundary can be the key source for a dynamical understanding of generating and chemically equilibrating the Thermalization through Hagedorn states -the importance of multiparticle collisions 8 standard and measured hadrons. Hagedorn states thus can provide a microscopic basis for understanding hadronisation of deconfined matter.

Figure 1 .

 1 Figure 1. Results for Λ's and Ω's with various initial conditions for T H = 196 MeV.

Figure 2 .

 2 Figure 2. Plot of the various ratios including all initial conditions defined in Tab. 1. The points show the ratios at T = 110 MeV for the various initial conditions (circles are for T H = 176 MeV and diamonds are for T H = 196 MeV). The experimental results for STAR and PHENIX are shown by the brown error bars.

Table 1 .

 1 Initial condition configurations.

			5	0
	IC 4	0.95	1.2	0

n i,x = n i,p = n i,k = n i,Λ = n i,Ω for when a kaon anti-kaon pair, Λ Λ, or Ω Ω pair is present.The decays widths are defined as follows (see[10]):Γ i = 0.15m i -0.0584 = 250 MeV to 1800 MeV Γ i,pp = 3 MeV to 1000 MeV Γ i,K K = 50 MeV to 1700 MeV Γ i,Λ Λ = 3 MeV to 250 MeV Γ i,Ω Ω = .01 MeV to 4 MeV Γ i,π = Γ i -Γ i,X X .

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. I.A.S. thanks the members of the Institut für Theoretische Physik of Johann Wolfgang Goethe-Universität for their hospitality during the final stages of this work. The work of I.A.S. was supported in part by the start-up funds from the Arizona State University.


I Kraus

J Cleymans

H Oeschler

K Redlich

Prospects for Strangeness Production in p -p Collisions at LHC

Keywords: numbers: 12.40.Ee, 25.75.Dw Statistical model, Strangeness undersaturation, Particle production

Prospects for strangeness production in p-p collisions at the Large Hadron Collider (LHC) are discussed within the statistical model. Firstly, the system size and the energy dependence of the model parameters are extracted from existing data and extrapolated to LHC energy. Particular attention is paid to demonstrate that the chemical decoupling temperature is independent of the system size. In the energy regime investigated so far, strangeness production in p -p interactions is strongly influenced by the canonical suppression effects. At LHC energies, this influence might be reduced. Particle ratios with particular sensitivity to canonical effects are indicated.

Secondly, the relation between the strangeness production and the charged-particle multiplicity in p -p interactions is investigated. In this context the multiplicity dependence studied at Tevatron is of particular interest. There, the trend in relative strangeness production known from centrality dependent heavy-ion collisions is not seen in multiplicity selected p -p interactions. However, the conclusion from the Tevatron measurements is based on rather limited data samples with low statistics and number of observables. We argue, that there is an absolute need at LHC to measure strangeness production in events with different multiplicities to possibly disentangle relations and differences between particle production in p -p and heavy-ion collisions.

Introduction

For more than half a century the statistical model has been used to describe particle production in high-energy collisions [1]. In the past it has evolved into a very useful and successful model describing a large variety of data. In particular, hadron yields in central heavy-ion collisions have been described in a very systematic and appealing way unmatched by any other model [2]. The statistical model has also provided a very useful framework for the centrality and system-size dependence of particle production [3,4]. The applicability of the model in small systems like p -p and e + -e -annihilation has been the subject of several recent publications [5,6]. In the following, we present predictions of the statistical model for particle productions at the LHC energy. Any deviation of data from our predictions allows to disentangle a new physics phenomena in high energy p -p collisions.

The thermal parameters of central heavy-ion collisions, given by the temperature T and the baryon chemical potential µ B , appear to fall on a common chemical freeze-out curve in the T-µ B plane. This observation allows to extrapolate the model parameters to the LHC energy regime and to predict particle ratios in heavy-ion collisions. To verify the validity of this method for p -p interactions we investigate the system-size dependence of the model parameters. In Section 2 we focus on differences emerging in the chemical freeze-out temperatures in Refs. [3] and [4] as recently studied in Ref. [7].

The canonical suppression of the strange particle phase-space in p -p collisions was found to be essential to quantify the SPS and RHIC data. Its evolution to LHC energy is necessary to correctly describe the strangeness sector. The energy dependence of the canonical effect is discussed in detail in Section 3. Here, we propose to distinguish between early and late strangeness production from experimental data based on the strength of the canonical suppression.

As a first step towards higher collision energies, we consider Tevatron data in Section 4. We argue that, the strangeness production and in particular its multiplicity dependence might suggest that the production is established at an early stage. However, as we discuss in the Outlook, this observation is based on a rather weak ground and only experiments at LHC energies might provide definite conclusions.

System size and centrality dependence

The trends of temperature and baryon chemical potential when varying the number of participating nucleons from central heavy ion to p-p collisions were studied at SPS [3,4] and RHIC [8]. The same value of µ B is found in phase-space integrated data, while midrapidity data show a decreasing number of net baryons as the multiplicity decreases. This, together with smaller µ B at higher energy, can be explained by weaker stopping due to higher transparency. On the other hand, the chemical freeze-out temperature was found to be multiplicity-independent at RHIC and also in our analysis at SPS [3]. These results are in contrast with findings from Ref. [4] where the temperature was

〉 - π 〈 , 〉 - h 〈 - h - π (GeV/c) lab p 0 200 400 0 0.1 (d) 〉 - π 〈 / 〉 - K 〈 Figure 1.
Charged kaon yields (left panels), negatively charged hadron h -and pions π -(c), and K -/π -ratios (d) in p -p collisions as a function of laboratory momentum. The charged kaon yields, K + (diamonds) and K -(crosses), are from Ref. [12].

The lines are fits to data. The SPS yields from Ref. [10] (circle) and from Ref. [START_REF] Baechler | NA49[END_REF] (triangle) are also shown. The negatively charged hadrons are from Ref. [13].

shown to be higher in smaller systems. Such behavior has dramatic consequences, e.g., it indicates that one can probe in p -p interactions QCD matter beyond the freeze-out line established in heavy-ion collisions [9]. Below we discuss what is the origin of such different predictions.

The largest difference in extracted model parameters from Refs. [3] and [4] appears in p -p collisions. The main reason is that the above analyses are based on different data sets and in some cases also on different numerical values for particle yields. The largest differences are observed in the charged kaon yields as illustrated in Fig. 1. Some data appear below the line which has been obtained from an interpolation between different lab momenta. The use of those data in the statistical model fit imply a stronger suppression of the strange-particle phase-space (smaller γ S ). This has to be compensated by higher T in order to describe various strange-particle yields like e.g. hyperons and K 0 S mesons. This is illustrated in Fig. 2 where the left panel shows the fit to the data set of Ref. [5] while in the right panel the charged kaon yields are replaced by values used in Ref. [4]. The charged kaons in Fig. 2(a) are taken from the parametrisation proposed in Ref. [10]. Figure 3(a) shows that these data are consistent with preliminary data obtained for yields of multi-strange baryons [14].

The analysis in Ref. [4] includes the φ meson. Based on measured data we have shown in Ref. [3] that the φ meson behaves like particle with strangeness between 1 and 2 in its multiplicity dependence. Therefore, this particle cannot be correctly treated in the statistical-model analysis. When including the φ meson in the analysis, we are unable to find a set of parameters that describes all particle yields simultaneously. In addition, the fit results in large χ 2 along with the values of thermal parameters that leave a reasonable range as demonstrated in Fig. 3 meson result in a very high temperature that exceeds that obtained in central heavy-ion collisions at the corresponding energy.

From the above we conclude, that at a given energy the chemical freeze-out temperature is system-size independent both at RHIC and at the SPS. In addition, the magnitude and the centrality dependence of the φ meson is obviously inconsistent with the statistical model systematics. Consequently, including this particle in the fit can lead to misleading conclusions on the model parameters and their qualitative properties.

π p/ + π / + K - π / - K / p Λ Λ / - Ξ - Ξ / - Ω - /K - Ω - π / - Ω - π p/ + π / + K - π / - K / p Λ Λ / - Ξ - Ξ / - Ω - /K - Ω - π / - Ω Ratio - 6 10

Extrapolation to LHC

Based on the discussion above, we assume that the multiplicity-independence of the temperature found at SPS and RHIC is also valid at LHC. In addition, to extrapolate T and µ B to p -p interactions at the LHC we use the same parametrisation of their energy dependence as it was found in the systematics of heavy-ion collisions. The baryon chemical potential at midrapidity decreases towards smaller systems, however, at the LHC µ B is so small that any variation with system size can be neglected. The canonical suppression is implemented in the following way. Strangeness production is assumed to be correlated inside chemically equilibrated clusters that may be smaller than the fireball. The size of these subvolumes (clusters) quantifies the strength of the canonical suppression. The correlation length R C extracted from SPS and RHIC data, as illustrated in Fig. 4 left, does not allow to quantify its energy dependence. We consider two extreme scenarios: (i) Small subvolumes independent of the incident energy which can be interpreted as being driven by the initial size of the proton or the range of strong interactions. Thus strangeness production would be determined at an early stage of the reaction. (ii) Subvolumes that increase with energy. In this case the strangeness production in p -p collisions per charged particle would be multiplicity dependent, reflecting a late determination of the strangeness content. In such a scenario, at a given energy, one expects less strangeness suppression in high multiplicity events. Consequently, high-multiplicity events in p -p interactions could appear like heavy-ion collisions in respect to the observed particle ratios.

As seen in Fig. 4 left, our actual knowledge from the SPS and RHIC energies does not allow for a definite extrapolation of the correlation volume towards the LHC energy.

The radius R C of the canonical correlation volume can appear in the range between 1 and 2 fm. Figure 4 (right) shows the sensitivity of different particle ratios to the value of the correlation radius. From this figure it is clear, that measurements of the Ω/π and Ω/K ratios are best be used to extract the strangeness correlation radius that quantifies canonical suppression effects and the production dynamics of strange particles.

Strangeness production at Tevatron

Before LHC results become available, Tevatron data provide already some insight into strangeness production at higher energies. There, mainly kaons and Λ hyperons were measured as strangeness carriers. The kaon and pion yields were found to increase smoothly and similarly with energy. Thus, the K/π ratio seems to saturate with energy [15]. This hints towards saturating correlation length since the temperature is almost unchanged with collision energy beyond the SPS. When studying the multiplicity dependence at fixed energy, we can rule out a possible counterbalance between T and R C . Indeed, in pp interactions the p/π and K/π ratios were found to be independent of the charged-particle multiplicities. These data indicate, that neither the temperature nor the cluster size exhibits variations with multiplicity. This interpretation is also supported by the multiplicity-dependent Λ/p ratio measured by the E735 and the CDF collaborations [15,16].

After the above qualitative discussion, we show in Fig. 5 a quantitative comparison of Tevatron data to the statistical model. The measured K/π ratio points to significant canonical suppression. The cluster size is small with none or weak energy dependence when comparing to RHIC and SPS results. This supports the interpretation that strangeness production is determined at an early stage and is quantified by the size of the colliding proton or the range of strong interactions. Consequently, we expect strange/non-strange particle ratios to saturate towards LHC energies with only small variations due to decreasing µ B . Due to the large errors the Λ/p ratio contributes weakly to constrain the strength of the canonical suppression.

Summary and Outlook

We have shown that the statistical model results are indeed very sensitive to the data selection. We have traced back the apparent variation of the chemical freezeout temperature at SPS to kaon yields that are not in line with earlier measurements at lower and higher beam momenta. We conclude that the chemical freeze-out temperature exhibits no system-size nor multiplicity dependence in the energy range under study. Thus, we confirm a similar observation made at RHIC.

The strangeness correlation length describes the strength of the canonical suppression. Its energy and multiplicity dependence can give insight into the question whether the relative strangeness production is defined at an early or late stage of the R (fm) reaction. At present, the extrapolation of the correlation length extracted from SPS and RHIC data towards LHC energies is not unique due to rather large uncertainties. The results from the Tevatron point towards small clusters and an early stage of strangeness production. However, this conclusion is based on a rather incomplete and low-statistics data.

0 1 2 3 )
- π + + π ) / ( - + K + (K 0 0.1 0.2 R (fm) 0 1 2
To quantify strangeness production and its energy dependence data with much better statistics are needed to reduce errors. Also there is a lack of multi-strange baryons, and the multiplicity dependence of strangeness production at fixed energy in p -p collisions requires more complete data. Further, the E735 and CDF data cover a charged particle densities up to 25 and 35, respectively. This compares to very peripheral heavy-ion interactions, e.g. it corresponds to Cu-Cu collisions at RHIC where the 40% most central events were left out.

At LHC we expect to have soon good statistics of events with a charged-particle density up to 100 or more in p -p interactions corresponding to semi-central Cu-Cu collisions. It is clear that after a long period of successful running with p -p at the LHC we will be able to investigate the similarities between p -p and heavy-ion interactions. Such comparison requires data with high statistics on many particle species and particularly on multi-strange baryons. We are convinced that the LHC will provide a very deep insight into strangeness production dynamics in p -p and in heavy-ion collisions and it will allow us to verify or strengthen our conclusions drawn from the existing data.

Figure 2 .Figure 3 .

 23 Figure 2. The χ 2 scan in the (T-γ S )-plane. Starting from its minimum, χ 2 increases by 2 for each contour line. Fit to (a) data set of Ref.[5] (data set A1 in[7]) and (b) same data set but charged kaons as used in Ref.[4] (data set A4 in[7]), in the canonical model. The minima are indicated by the crosses.

Figure 4 .

 4 Figure 4. Left panel: The cluster radius R C as a function of energy, extracted from the analysis of SPS data (triangle, square) and RHIC data (star)[5]. The lines illustrate possible evolutions toward LHC energies as discussed in the text. Right panel: Predictions for various particle ratios using different values of R C[5].

Figure 5 .

 5 Figure 5. The statistical model results for the K/π (left panel) and the Λ/p (right panel) ratio obtained with the chemical freeze-out parameters expected at Tevatron (T = 170 MeV and the initial quantum numbers B = S = Q = 0) as a function of the radius R of a spherical volume. Experimental results from Ref.[15] are indicated by horizontal bands. Derived ranges of the correlation length are displayed by vertical bands; for the Λ/p ratio only a lower limit can be extracted (large values that correspond to vanishing canonical suppression are consistent with the Λ/p data, too), the dark vertical band shows the range extracted from the K/π ratio. The gray curve in the left panel indicates the uncertainties due to a mismatch between feed-down corrections in data and model calculations.

We acknowledge the support from the German BMBF. K.R. acknowledges supports of the Polish Ministry of Science MEN and the Alexander von Humboldt Foundation. The financial support of the DFG-NRF and the South Africa-Poland scientific collaborations are also gratefully acknowledged.


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

C Sturm

S Schramm

H Stöcker

Strange Perspectives at FAIR

come   L'archive ouverte pluridisciplinaire

The FAIR Project

The Facility for Antiproton and Ion Research, FAIR [START_REF]FAIR Baseline Technical Report[END_REF][START_REF] Henning | FAIR: Recent developments and status[END_REF][START_REF] Stöcker | FAIR: Challenges Overcome and Still to be Met[END_REF], will provide an extensive range of particle beams from protons and antiprotons to ion beams of all chemical elements up to the heaviest one, uranium, with in many respects world record intensities. As a joint effort of 16 countries ‡ the new facility builds, and substantially expands, on the present accelerator system at GSI, both in its research goals and its technical possibilities. Compared to the present GSI facility, an increase of a factor of 100 in primary beam intensities, and up to a factor of 10000 in secondary radioactive beam intensities, will be a technical property of the new facility.

The start version of FAIR, the so called Modularized Start Version includes a basic accelerator as well as three experimental modules as it is colored illustrated in figure 1. The superconducting synchrotron SIS100 with a circumference of 1100 meters and a magnetic rigidity of 100 Tm is at the heart of the FAIR accelerator facility. Following an upgrade for high intensities, the existing GSI accelerators UNILAC and SIS18 will serve as an injector. Adjacent to the SIS100 synchrotron are two storage-cooler rings and experiment stations, including a superconducting nuclear fragment separator (Super-FRS) and an antiproton production target. By delivering first beams FAIR will supply rare isotope beams (RIBs) and antiproton beams with excellent intensity and quality. Not shown is the additional experimental area above ground for the APPA community (module 1).

Because of the long lead times for civil-construction planning civil work for the first buildings of FAIR will start in the fourth quarter of 2010 earliest. First beams will be delivered in 2015/2016. Moreover, after the start phase the facility will be upgraded by experimental storage rings enhancing capabilities of secondary beams and by the SIS300 providing particle energies 20-fold higher compared to those achieved so far at GSI as additional funds become available.

The Experimental Programme of FAIR

The main thrust of FAIR research focuses on the structure and evolution of matter on both a microscopic and on a cosmic scale. The approved FAIR research programme embraces 14 experiments, which form the four scientific pillars of FAIR and offers a large variety of unprecedented forefront research in hadron, nuclear, atomic and plasma physics as well as applied sciences. Already today, over 2500 scientists and engineers are involved in the design and preparation of the FAIR experiments. They are organized in the experimental collaborations APPA, CBM, NuSTAR, and PANDA.

APPA -Atomic Physics, Plasma Physics and Applications

Atomic physics with highly charged ions [START_REF] Th | Atomic physics with highly-charged ions at the future FAIR facility: A status report[END_REF] will concentrate on two central research themes: (i) the correlated electron dynamics in strong, ultra-short electromagnetic fields including the production of electron-positron pairs and (ii) fundamental interactions between electrons and heavy nuclei in particular the interactions described by Quantum Electrodynamics, QED. Here bound-state QED in critical and supercritical fields is the focus of the research programme. In addition, atomic physics techniques will be used to determine properties of stable and unstable nuclei and to perform tests of predictions of fundamental theories besides QED.

For Plasma physics the availability of high-energy, high-intensity ion-beams enables the investigation of High Energy Density Matter in regimes of temperature, density and pressure not accessible so far [START_REF] Lomonosov | Prospects of High-Energy Density Matter Research at the Future FAIR Facility at Darmstadt[END_REF]. It will allow probing new areas in the phase diagram and long-standing open questions of basic equation of state research can be addressed. The biological effectiveness of high energy and high intensity beams was never studied in the past. It will afford to investigate the radiation damage induced by cosmic rays and protection issues for the Moon and Mars missions. Furthermore, the intense ion-matter interactions with projectiles of energies above 1 GeV/u will endorse systematic studies of material modifications.

CBM/HADES -Compressed Baryonic Matter

Violent collisions between heavy nuclei promise insight into an unusual state in nature, that of highly compressed nuclear matter. In addition to its relevance for understanding fundamental aspects of the strong interaction, this form of matter may exist in various so far unexplored phases in the interior of neutron stars and in the core of supernovae. HADES [START_REF] Agakishiev | The High-Acceptance Dielectron Spectrometer HADES[END_REF][START_REF] Stroth | Di-electron emission from resonance matter[END_REF][START_REF]Future perspectives at SIS-100 with HADES-at-FAIR[END_REF] and CBM [START_REF] Senger | Compressed baryonic matter: Experiments at GSI and FAIR[END_REF][START_REF] Senger | Probing dense baryonic matter[END_REF] at SIS100/300 will explore the QCD phase diagram in the region of very high baryon densities and moderate temperatures. This approach includes the study of the nuclear matter equation-of-state, the search for new forms of matter, the search for the predicted first order phase transition between hadronic and partonic matter, the QCD critical endpoint, and the chiral phase transition, which is related to the origin of hadron masses. It is intended to perform comprehensive measurements of hadrons, electrons, muons and photons created in collisions of heavy nuclei proton-nucleus, and proton-proton collisions at different beam energies. Most of the rare probes like lepton pairs, multi-strange hyperons and charm will be measured for the first time in the FAIR energy range.

NuSTAR -Nuclear Structure, Astrophysics and Reactions

The main scientific thrusts in the study of nuclei far from stability are aimed at three areas of research: (i) the structure of nuclei, the quantal many-body systems built by protons and neutrons and governed by the strong force, towards the limits of stability, where nuclei become unbound, (ii) nuclear astrophysics delineating the detailed paths of element formation in stars and explosive nucleosynthesis that involve short-lived nuclei, (iii) and the study of fundamental interactions and symmetries exploiting the properties of specific radioactive nuclei.

The central part of the NuSTAR programme at FAIR [START_REF] Krücken | The Nustar Facility At Fair[END_REF]12] is the high acceptance Super-FRS with its multi-stage separation that will provide high intensity mono-isotopic radioactive ion beams of bare and highly-ionized exotic nuclei at and close to the driplines. This separator, in conjunction with high intensity primary beams with energies up to 1.5 AGeV, is the keystone for a competitive NuSTAR physics programme. This opens the unique opportunity to study the evolution of nuclear structure into the yet unexplored territory of the nuclear chart and to determine the properties of many short-lived nuclei which are produced in explosive astrophysical events and crucially influence their dynamics and associated nucleosynthesis processes.

PANDA -AntiProton ANnihilation in DArmstadt

The big challenge in hadron physics is to achieve a quantitative understanding of strongly interacting complex systems at the level of quarks and gluons. In pp-annihilation, particles with gluonic degrees of freedom as well as particle-antiparticle pairs are copiously produced, allowing spectroscopic studies with unprecedented statistics and precision. The PANDA experiment at FAIR [START_REF] Fohl | The Panda Detector At The Future Fair Laboratory[END_REF][START_REF] Lange For The | The Panda Experiment: Hadron Physics With Antiprotons At Fair[END_REF] will bring new fundamental knowledge in hadron physics by pushing the precision barrier towards new limits. The charmonium (cc) spectroscopy will take advantage by precision measurements of mass, width, decay branches of all charmonium states. Particular emphasis is placed on mesons with open and hidden charm, which extends ongoing studies in the light quark sector to heavy quarks, and adds information on contributions of the gluon dynamics to hadron masses. The search for exotic hadronic matter such as hybrid mesons or heavy glueballs gains enormously by precise scanning of resonance curves of narrow states as well. Recently, this field has attracted much attention with the surprise observation at electron-positron colliders of the new X, Y and Z states with masses around 4 GeV. These heavy particles show very unusual properties, whose theoretical interpretation is entirely open. Additionally the precision gamma-ray spectroscopy of single and double hypernuclei will allow extracting information on their structure and on the hyperonnucleon and hyperon-hyperon interaction.

Strangeness physics at FAIR

Massive nucleus-nucleus collisions provide an abundant source of strangeness: more than 50 Hyperons and about 30 Anti-Kaons are produced in central collisions of lead nuclei at the CERN-SPS low energy programme and before that at the AGS. With its high luminosity beam the FAIR facility and the CBM experiment offer the opportunity to do high precision measurements of bulk observables related to strangeness production (e.g. ratios, spectra and fluctuations). In addition it will open the door to explore the 3rd dimension of the nuclear chart, as hypernuclei and other exotic forms of bound objects containing strangeness will be produced abundantly at the CBM, PANDA and NuSTAR experiments.

Exotic hypermatter

These forms of metastable or even stable objects with strangeness have been proposed long ago (see [15]) as collapsed states of matter, either consisting of baryons or quarks. For example the H di-baryon (a six quark state) was predicted by Jaffe [16]. The (non-)observation of multi-quark bags, e.g. strangelets and (strange) di-baryons is still one of the open problems of intermediate and high energy physics.

Recent calculations within an effective chiral mean field model for infinite matter [17,18] predict that stable and metastable strange matter can be created when the net strangeness of the bulk matter becomes larger than f s > 0.4. In this simple estimate we set the strength of the hyperon coupling to the strange scalar ζ, whose mean-field value corresponds to the scalar strange-quark condensate, to a value of n s g 2 (n s is the number of strange quarks in the hyperon), rescaling the other couplings to keep the correct vacuum masses of the particles. The full line in figure 2 (left) shows the corresponding values of lowest energy of a system of free nucleons and hyperons for a given f s (net strangeness per baryon). Beyond a value of g 2 ≈ 2 one obtains a bound state for fixed f s . For even higher values of the coupling strength absolutely stable hypermatter at f s ≈ 1.2 is obtained. It was shown, within a hadronic transport model (UrQMD) [19], that strangeness can be separated in phase space due to the non-equilibrium nature of the early phase of a heavy ion collision. Furthermore, in equilibrated fireballs, strange and anti-strange quarks separate by a distillation process. While strangeness is produced exclusively in pairs in the quark gluon plasma phase, hadronic production involves mostly the associated production of a K + and Λ hyperon, which leads to a chemical imbalance if both phases coexist. To regain chemical phase equilibrium the quark phase of the fireball will be loaded with net strangeness.

In addition the emission of Kaons can further enrich the produced fireball with strange quarks. For our calculations we apply an equation of state that follows from an effective chiral SU(3) f Lagrangian, including the full sets of mesonic and baryonic octets.

Quark and gluonic degrees of freedom are included to model the deconfinement phase transition [20]. Both, chiral symmetry restoration and deconfinement, are described as a smooth crossover in the model. As a result we observe a wide region of phase coexistence in the phase diagram, which gives room for a possible distillation scenario.

Figure 2 (right) shows the time evolution (in the center of mass frame of the collision) of the strangeness per baryon fraction for a fireball that is created at a Pb+Pb collision at E lab = 40A GeV. The two vertical lines set the limits to where one would expect a chemical freeze out to occur. Beyond this time, the system decouples and should not be treated as being in chemical equilibrium. The solid lines depict the strangeness fraction in the different phases, while the fireball has no net strange charge. This reflects the original strangeness distillation process as proposed in [21]. In our model this process will lead to a strangeness fraction of f s ≈ 0.2 of the remaining quark droplet, at freeze out.

Due to early emission of Kaons the strangeness fraction can even increase more during the course of the fireballs evolution, as Kaons are more abundant at hight baryon densities than Antikaons [22]. The fireball itself becomes loaded with net strangeness. In combination with the distillation process, the quark part of the system can obtain a strangeness fraction of up to f s ≈ 0.5. This could in turn result in the formation of a (meta-)stable cluster of quarks or hadrons.

Hypernuclei

Conventional hypernuclei are known to exist already for a long time [24,25,26]. At the future FAIR facility several experiments will perform ambitious research programmes in hypernuclear physics. At the PANDA experiment for example, precision gamma-ray spectroscopy of single and double hypernuclei will be performed to investigate their structure, the hyperon-nucleon and hyperon-hyperon interaction as well as the weak decays of these objects.

As the FAIR accelerator offers an unprecedented beam intensity for heavy ion experiments, hypernuclei and multi-hypernuclear objects will be produced abundantly in the CBM experiment [20]. Here the strange matter clusters are created from a hot and dense fireball. In the following we will discuss the production of the single Λ-Hypernucleus from thermal equilibrium. More precisely we will calculate the following ratio R H

R H = 3 Λ H/ 3 He • p/Λ (1)
for collisions of Pb+Pb at a wide range of beam energies. This ratio is especially interesting, as it does not depend on the chemical potential of the particles (as fugacities cancel), and any canonical correction factors for strangeness are canceled.

First we calculate the ratio in a simple thermal model, where we assume a single freeze out temperature for every beam energy. In our calculation we adapted the freeze out curve which was found in a thermal model analysis of Andronic et. al. [27]. The resulting excitation function is shown in figure 3 as dotted line. It is obvious that the temperature dependence of the proposed ratio is (almost) negligible. Please note that, for simplicity, we have neglected the contribution of the feed down from higher resonances to the Λ and proton yields. This contribution will be discussed later. To get a more sophisticated picture of thermal production we also calculate R H from a hybrid model simulation of heavy ion collisions. In this model a full 3+1 dimensional ideal hydrodynamic evolution is embedded in the UrQMD transport code. At the transition hypersurface from hydrodynamics to transport all particles are thermally produced according to the Cooper Frye prescription [28].

The solid line in figure 3 shows results on R H from the hybrid model, extracted at the transition surface from hydrodynamics to transport. When final state interactions (FSI), like resonance decays and production, are explicitly included in the transport phase, the result changes significantly (dashed line). Nevertheless it still does not depend on the beam energy. Note that we neglected contributions to 3 Λ H/ 3 He from FSI (e.g. via coalescence of decay products) which should again increase R H .

Comparing our different results to data taken at the AGS [23], a large discrepancy can be seen, while preliminary results from RHIC experiment are in line with predictions from thermal production (R H ≈ 1) [29]. This discrepancy can be explained by the non equilibrium (not canonical) nature of strangeness production at AGS energies. The FAIR facility therefore offers the opportunity to study the production of such hypernuclei over a very interesting range of beam energies. It may prove to be a unique tool to study the process of strangeness equilibration which might be closely related to the formation of a QGP [30,31,32].

Summary

FAIR will allow forefront research addressing (1) the investigation of the properties and the role of the strong (nuclear) force in shaping basic building blocks of the visible world around us and of its role in the evolution of the universe, (2) tests of symmetries and predictions of the standard model and search for physics beyond in the electro-weak sector and in the domain of the strong interaction, (3) the properties of matter under extreme conditions, both at the subatomic as well as at the macroscopic scale of matter and (4) applications of high-intensity, high-quality ion and antiproton beams in research areas that provide the basis for, or directly address, issues of applied sciences and technology. Due to the high luminosity which exceeds current facilities by up to a factor of 10000, experiments will be feasible that could not be done elsewhere. This opens the unique opportunity to study the evolution of nuclear structure into the yet unexplored 3rd dimension of the nuclear chart and to determine the properties of many short-lived hypernuclei and metastable exotic multi-hypernuclear objects. FAIR will expand the knowledge in the various scientific fields beyond current frontiers. Moreover, there exists strong cross-topical synergies which can be exploited and promise novel insights.

Figure 1 .

 1 Figure 1. On the left hand side the existing GSI facility is shown. Colored displayed is the so called Modularized Start Version of FAIR including module 0 -3 . Coloring: the 100 Tm super conducting synchrotron SIS100 (module 0) -green; the experimental area for CBM/HADES (module 1) -red; the NuSTAR facility including the Super-FRS (module 2)yellow; The Antiproton facility including the PANDA experiment (module 3)orange.Not shown is the additional experimental area above ground for the APPA community (module 1).

Figure 2 .

 2 Figure 2. Left: Binding energy per particle as a function of strangeness per baryon f s for different values of the hyperon interaction with the scalar field ζ (g 2 ). Right: Time evolution of the strangeness per baryon fraction for the 2 coexisting phases (hadrons or quarks) in a heavy ion collision at E lab = 40A GeV. The solid line corresponds to results where the fireball has vanishing net strangeness while the dashed line shows results where emission of Kaons is included.

Figure 3 .

 3 Figure 3. Excitation function of the ratio R H (see text) for different model calculations compared to data [23].

‡ In alphabetical order: Austria, China, Finland, France, Germany, Greece, India, Italy, Poland, Romania, Russia, Slovenia, Slovakia, Spain, Sweden, and the United Kingdom

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).


T S Biró
email: tsbiro@rmki.kfki.hu

K Ürmössy

Z Schram

Thermodynamics of Composition Rules

Keywords: numbers: 24.10.Pa, 25.75.Nq, 12.38.Gc

HAL is

Introduction

Classical thermodynamics can be distilled into generally valid rules, into the so called four laws (from zeroth to third) of thermodynamics stating universal behavior of any system irrespective to the details of microscopic mechanisms -or the relativistic covariance properties of the underlying dynamics. Exactly this universality makes models based on this minimum of information so attractive for application to physical systems emerging after a rather complex dynamic evolution; among others to describe hadron spectra stemming from a fireball in high-energy heavy-ion collisions or, to an increasing degree, from more elementary hadronic reactions [1,2,3,4,[START_REF] Cleymans | Near-thermal equilibration with Tsallis distribution in heavy ion collisions[END_REF].

As a new development of recent decades a strong wish has emerged to generalize classical thermodynamics to be valid for the non-extensive systems, where the classical requirement of having only short range correlations (and henceforth only short range interaction) in near-equilibrium systems is not fulfilled. Such systems are numerous in nature, like gravitating systems, charged plasmas, turbulent fluids, quantum entangled systems, nearly Bose-condensed atoms, and -important for our understanding of the very hadronization process in high-energy collisions -quark matter near to the critical point or near to the edge of mechanical stability. In non-extensive systems one or even all of the extensive variables, like entropy, internal energy, particle species number, volume, cannot be treated as such, because the correlation between large subsystems of thermodynamic bodies is not restricted to a thin layer near to their interface. To the contrary, corrections to the independent subsystem approximation are so important, that they do not vanish in the thermodynamic limit.

A novel approach to deal with such systems is still under vigorous investigation. Suggestions based on to use of an altered entropy formula instead of the Boltzmann-Gibbs-Shannon one trace back to the Hungarian mathematician Alfred Renyi, who has proved that there is a more general, extensive entropy formula than the classical one using the logarithm of probability [START_REF] Renyi | [END_REF][START_REF] Renyi | Probability Theory[END_REF]. An equivalent, but non-extensive entropy formula has been suggested in relation with informatics [START_REF] Daroczy | On measures of Information and their Characterization[END_REF] and later in condensed matter physics by Constantino Tsallis [START_REF] Tsallis | [END_REF]10,11,12]. Discussions about the advantages and disadvantages of one or the other formula triggered ramified investigations of more formal nature: as a most important feature the Lesche-stability of the Renyi and Tsallis entropies was investigated [13,[START_REF] Van | On the continuity and Lesche stability of Tsallis and Renyi entropies and q-expectation values[END_REF]. Now it seems that these formulas can be used equivalently in statistical models and the physical meaning of the parameter q in them is related to the degree of non-extensivity in the system under investigation.

Recently we have worked out a formal and simple approach to non-extensivity formulas based on the infinite repetition of composition rules generalizing the addition [START_REF] Biro | [END_REF]. This procedure reveals that there is a way to treat statistical models with nonextensive properties mathematically clean. In the framework of the thermodynamics of abstract composition rules not only a general recipe is given for producing entropy formulas to each universal class of composition rules, but also suggestive generalizations of both the canonical equilibrium and the kinetic approaches are established [16,17,18,19].

In this paper we review the basic formalism about the behaviour of repeated composition rules and the description of non-extensivity. We present then arguments about hadron p T spectra occurring in relativistic heavy-ion collisions showing nonextensive behaviour which does not seem to vanish with increasing participant number [START_REF] Chinellato | A Non-extensive Equilibrium Analysis of Pion pt Spectra at RHIC[END_REF]. The use of this parameter expands the range of validity of statistical model fits to p T spectra dramatically. Also theoretical ideas will be discussed how relative momentum-dependent QCD interactions among partons may generate a non-additive rule for the individual kinetic energies causing this way a non-extensive statistical canonical distribution of hadron energies with a power-law tail. Finally, taking these hints seriously, we report about first attempts to obtain a non-extensive equation of state for SU(2) gluon matter in the framework of a numerical lattice gauge theory simulation.

Composition rules and non-extensive formulas

The classical Boltzmann-Gibbs-Shannon entropy formula,

S = f ln 1 f , (1)
has the property that it is additive for independent probabilities

S [f 1 • f 2 ] = S [f 1 ] + S [f 2 ] . (2)
The canonical energy distribution, f = e -βE /Z, shows the back side of this coin: it is multiplicative by the addition of subsystem energies:

f (E 1 + E 2 ) = f (E 1 ) • f (E 2 ).
In such systems the absolute temperature can be obtained as satisfying the relation β = 1/T = ∂S/∂E from the canonical requirement of maximizing the total entropy at fixed total energy: S -βE = max. That the entropy is a useful concept is mainly reflected in the fact, that β = 1/T occurs as an integrating factor rendering dS to be a total differential (in contrast to the heat and mechanical work):

dS = 1 T (dE + pdV -µdN + . . .) . (3)
Several physical mechanisms have since been suggested as micro-dynamical models to achieve and maintain a canonical equilibrium state [START_REF] Kodama | [END_REF]. The Langevin equation utilizes an additive white (Gaussian) noise for the momentum, the Fokker-Planck equation describes the linear diffusion process for the distribution of elementary momenta and Boltzmann's Stosszahlansatz incorporates the trend towards the irreversible growth of entropy. All these mechanisms have already been generalized to deal with more entangled situations: the Langevin equation with multiplicative noise leads to a power-law tailed stationary distribution in kinetic energy [22], the anomalous (nonlinear) diffusion can establish the more general Levy or the t-Student distribution instead of the Gaussian, and the Stosszahlansatz also can be generalized exactly by manipulating the product assumption (or by manipulating the energy addition assumption valid for a separated pair of particles in a micro-collision) [16,19]. What is common in all these approaches can be understood as "deforming" the usual formulas using the logarithm and the exponential functions: in a way by keeping the same structure exactly these functions are replaced by their "deformed logarithm" and "deformed exponential" versions.

Starting with an abstract composition rule generalizing the simple addition,

x ⊕ y = h(x, y), (4)
its infinite repetitive use, x n = h(x n-1 , ∆y n ), starting with zero (x 0 = 0)and composing infinitesimally small quantities, -which nevertheless sum up to a finite value, y = n ∆y n , -one can prove that the resulting composed quantity, x = lim n→∞

x n satisfies the scaling differential equation

dx dy = ∂ ∂y h(x, y) y=0 + . ( 5
)
Its solution, y = L(x) is the formal logarithm which maps the composition rule to the simple addition:

x(y 1 + y 2 ) = L -1 (L(x(y 1 )) + L(x(y 2 ))) (6)
is the asymptotic composition rule in the thermodynamic (infinite repetition) limit. Such asymptotic rules are associative and are attractors among all abstract composition rules (for details see [START_REF] Biro | [END_REF]19]).

A scaling parameter, a, can be used to interpolate between the simple addition and the asymptotic rule by introducing L a (x) = L(ax)/a. Now L 0 (x) = x, L 1 (x) = L(x) and the deformed exponential function is given by e a (x) = exp(L a (x)). Its inverse function is the deformed logarithm ln a (x) = L -1 a (ln(x)). Entropy composition formulas then satisfy the rule L(σ(ab)) = L(σ(a)) + L(σ(b)) and hence σ(p) = ln a (1/p). Considering now a generalized canonical problem a non-additive entropy is maximized while a non-additive total energy is kept fixed:

Y (S 1 ) + Y (S 2 ) = Y (S 12 ) = max. X(E 1 ) + X(E 2 ) = X(E 12 ) = const. ( 7
)
This construction also satisfies the zeroth law of thermodynamics because the requirement of canonical equilibrium factorizes to quantities containing data of one or the other system respectively [18]. This defines an absolute temperature as

β = Y (S 1 ) X (E 1 ) S 1 (E 1 ) = Y (S 2 ) X (E 2 ) S 2 (E 2 ). (8)
The relation between entropy, energy and temperature, however, is more complex than in the classical case. Examples for different composition rules and emerging entropy formulas can be constructed, for a list see the reference [19].

Non-extensive hadron spectra

A remarkable composition rule is defined by h(x, y) = x + y + G(xy), where the asymptotic rule becomes ϕ(x, y) = x + y + axy with a = G (0). This is exactly the rule underlying the Tsallis and Renyi entropies. We have suggested in [START_REF] Biro | [END_REF] that in the high kinetic energy kinematics, the individual kinetic energy composition rule may be deformed due to the momentum dependence of the interaction energy,

h(E 1 , E 2 ) = E 1 + E 2 + U (Q 2 ) with Q 2 = 2E 1 E 2 (1 -cos θ)
containing the product of individual energies. This leads to a cut power-law asymptotic distribution in the non-extensive canonical state

f eq = 1 Z (1 + 2U (0)E) -β/2U (0) . (9)
Unfortunately, U (0) is an infrared property and is not readily available from perturbative QCD. By considering power-law tails in hadron spectra the phenomenological fits of this parameter is a possible starting point for further investigations.

In the framework of a quark coalescence picture we have analyzed RHIC mesonic and baryonic p T spectra and fitted them in a blast wave picture as possibly composed from a quark matter with q ≈ 1.2 [23,24]. Here we would like to demonstrate that the descriptive power of such a fit increases dramatically by including the non-extensivity parameter. In figure 1 any thermal model, also its universality (dependence on hadron masses would directly deny the possibility of a state based on the stationarity energy distributions). We note that applying power-law tailed distributions the energy density and pressure is higher at the same temperature than in the classical case. The energy density per degree of freedom for an ideal boson gas using the deformed exponential function with parameter a is given by

e = T 4 2π 2 ∞ n=1 3! n(n -a)(n -2a)(n -3a) . (10)
Consequently aiming the same energy per particle values (which is determined by the experimental facts to be around E/N ≈ 1 GeV in all relativistic heavy-ion experiments) the fitted values for the temperature for q = 1 + aT > 1 are smaller than for q = 1 (a = 0, the traditional case). For massless Boltzmann particles one easily obtains 3T = E/(N + aE) < E/N for a > 0.

Non-extensive lattice SU(2) equation of state

Motivated by the mathematical challenge and encouraged by the phenomenological success we raise the question: may it be that already the quark matter, preceding the hadronic fireball spectra, was in a non-extensive asymptotic state? The Monte Carlo method simulating lattice gauge field dynamics under non-extensive thermodynamic asymptotic conditions is based on the superstatistic interpretation [25,26], where the power-law tailed Tsallis-Pareto distribution is convoluted from an Euler-Gamma distribution for the inverse temperature [17]. This in fact also has been interpreted as an effect of temperature fluctuations due to deficient equilibration in the fireball [27,28].

Since c c Γ(c) ∞ 0 dt t c-1 e -ct e -tβE = 1 + βE c -c , (11)
and the inverse temperature β = 1/T = Na t on the lattice is defined by its Euclidean time periodicity, we supplement the well-known lattice Metropolis method by sampling the asymmetry parameter (the ratio of timelike and spacelike lattice spacings), t = a t /a s , from an Euler Gamma distribution with the parameter c. This distribution stretches over non-negative values only and its width is 1/ √ c. We renew the parameter t after each lattice sweep of throwing new link variables.

In figure 2 we present the equipartition between the chromoelectric and chromomagnetic plaquette average values corrected by the corresponding asymmetry factors: at equilibrates to b/t. A quick equilibration during the Monte Carlo process can be inspected for all different values of c (1024, 32 and 13.5). Raw data towards obtaining the equation of state are collected from numerical simulations in figures 3 and 4. The action difference between chromoelectric and chromomagnetic contributions on an asymmetric lattice is the basis of obtaining the scaled energy density e/T 4 , while the action sum for the interaction measure (e-3p)/T 4 . Our results show already the main trends: smaller c values -corresponding to bigger temperature fluctuations -increase the pressure and energy density. In fact, although numerical uncertainties are still large, a tendency can be observed for thickening the area under the action sum curve related to (e -3p)/T 4 . Of course further analysis has to be done and the renormalization of the physical temperature scale has to be carried out based on the static quark -antiquark potential analysis of Wilson Loop expectation values. The preliminary results presented here we regard as a first step into a promising direction.

In summarizing we have reviewed the thermodynamics of composition rules as a mathematical formulation for dealing with a non-extensive thermodynamic description, first of all formulas for entropy, energy and canonical distribution. We see the main reason for occurring power-law tailed distributions in certain parton cascade simulations and phenomenological fits to fireball hadron and quark matter spectra in the momentumdependent interaction term in QCD. This idea, of course, has to be solicited by further research. Finally we reported first results attempting to obtain a non-extensive equation of state for gluon matter in the framework of a modified Monte Carlo method in lattice simulation. Although first results look promising in this area, there is still a long way to go before the expected deviations from the classical thermodynamical picture can be pointed out by such numerical calculations.

Figure 1 .

 1 Figure 1. Scaling variable for RHIC p T distributions with transverse flow as a blast wave.

Figure 2 .Figure 3 .

 23 Figure 2. Equipartition of chromelectric and chromomagnetic contributions during the combined Euler-Gamma deviate and Metropolis method on a 10 4 su2 lattice at 4/g 2 = 2.10.

Figure 4 .

 4 Figure 4. Action sum (a + b) as a function of the inverse coupling 4/g 2 on an asymmetric 10 3 × 2 lattice normalized to the symmetric lattice. These are the raw data for the interaction measure, (e -3p)/T 4 vs T/T c plot.

Acknowledgments

This work has been supported by the Hungarian National Science Fund OTKA (K49466, K68108 and K46925). Part of this work has been done within the framework of a joint research funded by the DFG and the Hungarian Academy of Sciences. Z.S. and T.S.B. are grateful to the Institue für Theoretische Physik, Johann Wolfgang Goethe Universität, Frankfurt for their hospitality.