Interacting hadron resonance gas meets lattice QCD

We present, in the framework of the interacting hadron resonance gas, an evaluation of thermodynamical quantities. The interaction is modelled via a correction for the finite size of the hadrons. We investigate the sensitivity of the model calculations on the radius of the hadrons, which is a parameter of the model. Our calculations for thermodynamical quantities as energy and entropy densities and pressure are confronted with predictions using the lattice Quantum Chromodynamics (QCD) formalism.


Introduction
One of the major goals of ultrarelativistic nuclear collision studies is to obtain information on the QCD phase diagram [1]. A promising approach is the investigation of hadron production. Hadron yields measured in central heavy ion collisions from AGS up to RHIC energies can be described very well [2,3,4,5,6,7,8,9] within a hadro-chemical equilibrium (also called hadron resonance gas or statistical) model. The main result of these investigations is that the extracted temperature values rise rather sharply from low energies on towards a center-of-mass energy per colliding nucleon pair √ s N N ≃10 GeV and reach, for higher collision energies, constant values near T =160-165 MeV, while the baryochemical potential µ b decreases monotonically as a function of energy. The Hagedorn limiting temperature [10] behavior suggests a connection to the phase boundary between the hadronic world and the deconfined phase. It was, indeed, argued [11] that the quark-hadron deconfinement phase transition drives the equilibration dynamically, at least for SPS energies and above. For lower energies, the quarkyonic state of matter [12] could complement this picture. The conjecture of the triple point [13] between hadronic, deconfined and quarky-onic matter was put forward in this context. In a recent study [14] it is argued, however, that the chemical frezee-out region at large µ b is not close to the phase boundary.
Theoretical investigations of the QCD phase diagram are an important priority for ongoing research. While effective field theories need to be employed to model QCD in the strongly interacting regime for finite µ b [15,16,17], QCD calculations on the lattice are an increasingly reliable approach for µ b ≃ 0. Employing calculations of QCD on lattice all groups predict indeed a steep increase of thermodynamical quantitites near a critical temperature for deconfinement, T c . Until recently, values for T c between 151 MeV [18] and 192 MeV [19] were obtained. New results on larger lattices and with quark masses approaching the physical values [20,21] lead to better agreement on the T c value in the range 155-160 MeV, while important details of lattice QCD calculations continue to be addressed. In this context, the hadron resonance gas (HRG) is used by lattice QCD groups as reference for their calculations in the hadronic sector [22,23]. This follows earlier ideas of [24] and involves modelling of quark mass dependence in the hadron resonance gas model to account for the finite lattice spacing [22,23,24]. Conversely, lattice data are used to constrain effective models based on hadronic resonances [25] to describe hadronic matter near T c .
The details of the hadron resonance gas (or statistical) model are important too, as we have recently shown in [9], where we demonstrated that the completeness of the hadron spectrum involved in calculations is important for a precise description of data in nucleusnucleus collisions. The aim of this paper is to confront our HRG model calculations with lattice QCD predictions for thermodynamical observables in the hadronic sector. In particular, we investigate the effect of the excluded volume correction employed in the model to approximate a short-range repulsive hadron-hadron interaction. Such excluded volume corrections were first introduced in [26], albeit not yet in a thermodynamically consistent way. A thermodynamically consistent approach was first developed in [27] and will be the basis for our investigations.

Model description
We restrict ourselves here to the basic features and essential results of the statistical model approach. A complete survey of the assumptions and results, as well as of the relevant references, is available in ref. [28].
The basic quantity required to compute the thermal composition of hadron yields and the thermodynamical quantities is the partition function Z(T, V ). In the grand canonical (GC) ensemble, the partition function for a particle species i in the limit of large volume takes the following form (k = = c = 1): from which the particle density n i , the partial pressure P i , the energy density ε i and the entropy density s i are then calculated according to: where g i = (2J i + 1) denotes the spin degeneracy factor, T is the temperature and E i = p 2 + m 2 i is the total energy. The (+) sign corresponds to fermions and (-) corresponds to bosons. For the hadron species i of baryon number B i , third component of the isospin I 3i , strangeness S i , and charm C i , the chemical potential is The chemical potentials related to baryon number (µ b ), isospin (µ I 3 ), strangeness (µ C ) and charm (µ C ) ensure the (on average) conservation, in the collision, of the respective quantum numbers: i) isospin: V cons i n i I 3i = I tot 3 , with V cons = N B / i n i B i ; ii) strangeness: i n i S i = 0; iii) charm: i n i C i = 0. The (net) baryon number N B and the total isospin I tot 3 of the system are input values which need to be specified according to the colliding nuclei and rapidity interval studied. Taking into account the conservation laws (i-iii), the freeze-out temperature T , the baryochemical potential µ b and the fireball volume at chemical freeze-out V are the parameters of the model, which are obtained from fits to experimentally measured hadron yields. The following hadrons (number of species, not counting g i ) are included in the calculations: i) mesons: non-strange (123), strange (32), charm (40), bottom (28); ii) baryons: non-strange (48), strange (48), charm (32), bottom (14). The corresponding anti-particles are of course also included. Their characteristics, including a rather complete set of decay channels (all strong and electromagnetic decays), are implemented according to the 2008 PDG compilation 1 [29], with hadron masses reaching 3 GeV. We use vacuum masses for all hadrons.
Usually, whenever thermal fits are performed, the finite widths of resonances are taken into account in the density calculation by an additional integration, over the particle mass, with a Breit-Wigner distribution as a weight [8]. For the range of temperatures investigated in this work the effect of the finite resonance widths is small and to save computing time we have not employed the additional integration in the present calculations except where stated otherwise.
When comparing thermodynamical quantities computed within the framework of the hadron resonance gas model with results obtained using lattice QCD methods one has to decide how to incorporate interactions among the hadrons. One approach is to use results obtained by the authors of [30,31,32] where two-body collisions are taken into account through scattering phase shifts. Here the interaction measure (the 2nd virial coefficient) is related to the derivative of the phase shifts with respect to energy. To compute the thermodynamics of the interacting hadron resonance gas in this way one would need knowledge of the energy dependence of all phase shifts. At first glance this seems quite impractical. An interesting result was, in this context, obtained in [33]. These authors show by explicit construction that, for simple systems such as gases of pions, pions and nucleons, and pions, kaons, and nucleons, the equation of state of the interacting system is obtained by adding the relevant resonances, the ρ and f 0 (980) mesons, the ∆ baryon, the K * meson, to the list of particles and by computing the partition function of the enlarged gas assuming no interactions.
This interesting result has led some authors [22,23] to argue that the thermodynamics of the interacting hadron resonance gas is well approximated, via the Dashen, Ma and Bernstein theorem [31,32], by that of the non-interacting case, provided that all states (resonances) are included in the partition function. It is one of the goals of this paper to address the accuracy of this approximation within the framework of our interacting HRG model. Even at the formal level, there are points to be considered. First, the η, ω, η ′ , φ and a 0 mesons cannot be treated like this [33]. Second, the baryon-baryon interaction is largely repulsive, with no known resonance structure, 2 see, e.g. fig. 8 of [33]. More importantly, the approach of [31,32] is, as also discussed there, a low density approach, relevant for dilute systems. At temperatures near T c , the temperature of the phase boundary between hadron gas and quark-gluon plasma, the hadron resonance gas is not dilute anymore. As will be shown below, overall densities exceed 0.5 fm −3 and total baryon densities are close to normal nuclear matter densities of 0.15 fm −3 . This implies that for the whole range of baryon chemical potentials considered here the baryon densities near T c are close to or exceed the critical value worked out in [33] above which the virial expansion breaks down. In this environment also the concept of asymptotic states needed for the S-matrix approach of [30,31,32] is ill defined.
We therefore explore in the following, in addition to the 'free' hadron resonance gas, also the thermodynamic properties of a hadron resonance gas in which short-distance repulsion is explicitely taken into account using the thermodynamically consistent excluded volume approach developed in [27]. In essence this amounts to a Van-der-Waals construction. This is implemented according to [27,34] in an iterative procedure for the total pressure as: where P id.gas = i P id.gas i (T,μ i ) and for each particle i the chemical potential at a given iteration is recalculated as:μ This approach yields the following formulae for the particle densities n i , the total energy density ε and the total entropy density s expressed as a function of the respective quantities in the ideal gas, which are given in (2)(3)(4)(5): where V eigen,i = 4 · 4πR 3 i /3 is the eigenvolume of a hadron 3 with radius R i . We checked numerically that thermodynamical consistency, expressed by ε = T s − P + i µ i n i , is well fulfilled by our calculations, explicitly confirming the consistency of the procedure [27] used for the excluded volume correction.
For the radius parameter R i , governing the excluded volume calculation, we follow the earlier arguments in [4]. There it was argued that, for baryons, the radius is given by the hard-core repulsive interaction as extracted from nucleon-nucleon scattering [35], giving a radius of about 0.3 fm. Values for other baryons should be similar. For mesons, in the absence of detailed information on their interactions at short distance, we assign the same radius value, based on the similarity of the meson charge radii compared to baryons and on the energy dependence of the pion-nucleon phase shifts [36]. For illustration, we have included the case of R=0 for mesons, although we believe that the physical case is for mesons with repulsive core radius R comparable to baryons.
For the baryon-antibaryon system there are likely no short-range repulsive interactions, because of annihilation processes, which are by construction included in the hadron resonance gas at equilibrium. We have modelled the absence of short-range repulsion in a schematic way, introducing a correction factor k anni for the (anti)baryon eigenvolume based on the expression: where n baryons is the density of baryons and n antibaryons the density of antibaryons. As shown by the results presented in the following, the absence of short-range repulsion in the baryon-antibaryon system leads to only a small correction, since mesons dominate at small µ b and since there are very few antibaryons at large µ b .
For the rest of the paper we show results of our calculations for radii in the range of 0.3±0.05 fm, along with the cases discussed above and contrast those with the case of no interaction (R i =0). The value of 0.3 fm, common for mesons and baryons, was also used whenever we performed thermal fits to hadron abundancies [4,6,8,9]. In the description of hadron yields with the statistical model, the excluded volume correction leads to a larger volume parameter, while the fit temperature and baryochemical potential are unchanged compared to the case of fitting hadron ratios [8], for the case of identical R i for mesons and baryons. The implication of a pion radius different from all other hadrons for the description of data has been studied by Yen et al. [34].
In Fig. 1 we present the temperature dependence of the hadron densities calculated with our model, with and without interactions modelled via excluded-volume corrections, as well as with the cases of R meson =0 and of absence of short-range interactions for baryonantibaryon pairs. This illustrates our remarks above, namely that, while at low temperatures (low densities) there is no difference between the case of excluded volume correction and the case of free hadron resonance gas, the difference becomes appreciable for T above 130-140 MeV, which is significantly below the value for the critical (crossover) temperature T c ≃160 MeV. Near T c , the hadron resonance gas becomes manifestly dense, with the mean distance between hadrons getting significantly smaller than twice the hadron radius. All approximations appropriate for the dilute gas, discussed above, break down and the non-interacting hadron gas is not a suitable approach anymore.  As pointed out earlier [27,33], a possible problem of the hadron gas model with excluded volume corrections is acausal behavior (speed of sound larger than velocity of light). As we demonstrate in Fig. 3, our model is not plagued by such a behavior (for the case µ b ≃0 considered here). The adiabatic speed of sound, c s , is calculated as: It exhibits a shallow minimum as a function of T for the case of excluded volume corrections. Our calculations are compared to lattice QCD calculations [37]. In general, a good agreement between our and the lattice result is observed. We note that our calculations predict a shallow minimum for T around 140-150 MeV, while the lattice values indicate a more pronounced dip and exhibit a speed of sound value smaller than the hadron resonance gas with interactions. It would be interesting to see if the corresponding low temperature part of the equation of state (EoS) would lead to changes in hydrodynamic calculations, where generally the EoS shown in Fig. 6 or ref. [22] is used.

Hadron resonance gas and lattice QCD results
In the following we compute, in the HRG model, thermodynamical quantities with and without excluded volume corrections and compare the results to predictions from lattice QCD. In Fig. 4 we show the temperature dependence of energy density, pressure, and entropy density, each normalized to appropriate powers of the temperature.
The case without interactions (no excluded volume correction) has the expected strong dependence on temperature. As noted early on by Hagedorn [10], a limiting temperature, also called "Hagedorn temperature", of T H ≃ 200 MeV arises for calculations of thermodynamical quantities within the HRG model if one assumes a hadron mass spectrum which increases exponentially with particle mass. Such exponential behavior is consistent with the present knowledge of hadron resonances [29] up to 2.0-2.5 GeV in mass. At this temperature all thermodynamical quantities for the HRG without excluded volume corrections diverge. We note in passing that all thermal model calculations without excluded volume corrections become meaningless for temperature values close to T H . In the course of investigations reported in [9] we realized that this implies a practical limitation to temperatures below 175 MeV as all calculations for higher temperatures become very sensitive to details of the mass spectrum for masses larger than 3 GeV.
For the case of calculations employing finite hadron volume corrections the Hagedorn infinities are tamed. This was already noted by Hagedorn [10] who was the first to introduce excluded volume corrections [26]. Our findings substantiate this and imply that the Hagedorn limiting temperature is an artifact of the usage of the free hadron resonance gas description at temperatures where the implicit approximations for dilute systems are manifestly inappropriate.
For temperatures below 120 MeV the HRG model results with and without excluded volume correction almost coincide, see Fig. 4. For larger temperatures, the HRG with interactions yields, in our view, a realistic description of the hadronic phase. Therefore, in the confined regime, lattice QCD calculations of thermodynamical variables should give results in agreement with the interacting HRG. The expectation is that, as soon as effects of deconfinement become important in the lattice QCD results, they should increasingly exceed the HRG values. In Fig. 4 the most recent predictions of lattice QCD are compared to the HRG results. Indeed, below T =150 MeV good agreement between results of lattice QCD [37] and the interacting HRG is found. On the other hand, effects of the onset of deconfinement [20,21] are apparent for T in excess of 150 MeV.
In Fig. 5 we compare our calculations for the trace anomaly ε − 3P (normalized to T 4 ) to LQCD results [37,38]. We see an agreement between LQCD data and our calculations for the interacting hadron gas only up to T =140 MeV for the data of Borsányi et al. [37]. For the (preliminary) data of the HotQCD collaboration [38] we have used for illustration the set for hisq action with N t =6, but we note that other available sets are in agreement with those within the errors [38].
We turn now to the energy dependence of the thermodynamical quantities at chemical freeze-out in central nucleus-nucleus collisions [9]. The degree of stopping of the colliding nuclei, which is energy dependent, brings some uncertainty in the choice of N B and I tot 3 . As we study central collisions of heavy nuclei (Au or Pb) and focus on data at midrapidity, we have chosen N B = 400 · µ b /938 MeV and I tot   Fig. 4) are compared to LQCD calculations of Borsányi et al. [37] and HotQCD collaboration [38] (preliminary results).
In Fig. 6 we show the thermodynamical quantities as a function of collision energy for chemical freeze-out in central nucleus-nucleus collisions. The trends seen in Fig. 6 reflect primarily the sharp increase of the temperature at chemical freeze-out (determined from fits of experimental data up to √ s N N =200 GeV [9]) followed by a saturation above √ s N N ≃10 GeV. The characteristic energy dependence of the baryon density, exhibiting a maximum around 8 GeV, is determined by the increase of T combined with the strong decrease of µ b with energy, as discussed in [9]. The effect of interactions leads to up to 30% reduction of the thermodynamical quantities at chemical freeze-out in nucleus-nucleus collisions.

Summary
We have presented an evaluation of thermodynamical quantities in the framework of the interacting hadron gas model, incorporating all known hadrons with masses reaching 3 GeV. A Van-der-Waals-type interaction is modelled via an excluded volume correction. Thermodynamic consistency is ensured by construction and the model exhibits proper causal behavior.
The resulting values for the thermodynamical quantities increase, already for temperatures of 130-150 MeV, i.e. significantly below T c , much less steeply than in case of a free hadron resonance gas. Near T c the free hadron resonance gas calculations already show signs of the Hagedorn divergence. Comparisons of lattice QCD results with the free hadron resonance gas in this temperature regime are therefore in our view problematic. Our  results imply the need to consider the hadron resonance gas with interactions, beyond the usual implementations based on the Dashen, Ma and Bernstein theorem.
On the other hand, lattice QCD simulations start to be precise enough to reproduce the complete hadron gas at low temperatures. The apparent rise of the lattice QCD results above the HRG results is a clear indication of the onset of deconfinement not contained in the latter. In our view, the lattice results show genuine quark and gluon degrees of freedom in the vicinity of the (crossover) transition. In this temperature range, the lattice results produce thermodynamical quantities well above our predictions for the interacting hadron resonance gas. Our findings also imply that the Hagedorn limiting temperature is an artifact of the usage of the free hadron resonance gas description at temperatures where the gas becomes manifestly dense.