Introduction

The development of the hadron resonance gas model (HRGM) with the multicomponent hardcore repulsion between the constituents [1,2,3,4,5,6], i.e. with several hard-core radii of hadrons, converted the so-called thermal model into a powerful and convenient tool of the heavy ion physics phenomenology, but also it led to a few real breakthroughs in our understanding of the chemical freeze-out (CFO) process. Indeed, using just a few extra parameters compared to the traditional HRGM [7], which employs a single hard-core radius for baryons R b and the one R m for the mesons, it was possible to reach an unprecedented accuracy in the description of hadronic yields measured in the central nuclear collisions from the low AGS BNL collision energy ( √ s N N = 2.7 GeV) to the highest RHIC one ( √ s N N = 200 GeV) with a quality χ 2 /dof 1.15 [3,4,5] (if one includes into the fitting the hard-core radius of pions R π and kaons R K ) or with χ 2 /dof 0.96 [6] (if one includes into the fitting the hard-core radius of Λ-(anti-)hyperons R Λ in addition to R π and R K ). The high accuracy achieved by the HRGM with multicomponent hard-core repulsion allowed us not only to elucidate the characteristics of the CFO of A+A collisions, but also to resolve several long-standing problems of the CFO process [2,3,4,5,6,8,9,10,11,12]: (i) in Refs. [2,3,6] it was shown that the so-called (anti-)Λ puzzle [7] is the result of oversimplifying assumptions; (ii) in Refs. [2,3] it was found a natural solution to the Strangeness Horn [13] description puzzle which troubled the heavy ion community for a decade; (iii) the concept of separate CFOs of strange and non-strange hadrons was independently suggested in Refs. [4,14]. Moreover, the high quality of data description allowed us to find out several new irregularities of thermodynamic quantities at the CFO which helped us to formulate new and promising signals of two phase transitions [5,8,9,10,11,12] that are expected to exist in strongly interacting matter [15,16,17].

One should, however, remember that the multicomponent versions of the HRGM [1,2,3,4,5,6] based on the popular Van der Waals (VdW) approximation to the hard-core repulsion, i.e. which use the classical second virial coefficients, are rather complicated and take a lot of CPU time, since for N different hard-core radii for each iteration of the experimental data fitting it is necessary to solve the system of (N + 1) transcendental equations containing a few hundreds of double integrals. Hence, the application of the multicomponent HRGM based on VdW approximation to cases of N 1 is somewhat problematic [18,19]. However, an entirely new and efficient approach to deal with the multicomponent hard-core repulsion in the grand canonical ensemble for large values of N was invented in Ref. [20].

This novel approach is based on the induced surface tension (IST) concept [20]. It has two principal advantages over the other multicomponent versions of the HRGM: (i) the number of equations which should be solved is two only and does not depend on N , and (ii) as shown in [18,19,21,22] it allows one to go far beyond the usual VdW approximation and to take into account not only the second, but the third and even the fourth virial coefficients of the classical hard spheres. In Refs. [18,19] it was recently shown that, in contrast to the oversimplified version of the HRGM like the one used in [23], there is no proton yield puzzle neither at ALICE energy √ s N N = 2.76 TeV, nor at RHIC energies of collisions.

Our next step was to extend the IST equation of state (EoS) to the description of the mixtures of the hadrons with light nuclear clusters, i.e. the deuterons (d), helium-3 ( 3 He), helium-4 ( 4 He) and hyper-triton ( 3 Λ H) and their antiparticles, and to apply the developed EoS to the simultaneous description of the STAR √ s N N = 200 GeV data on the nuclear multiplicities [24,25,26], the ALICE √ s N N = 2.76 TeV data on light nuclear cluster yields [27,28,29] and the hadronic multiplicities measured at these collision energies. To our great surprise not only the quantum, but also the classical second virial coefficients of such nuclei and hadrons were never discussed in the literature. Therefore, we had to resolve this problem first. Since the HRGM with the classical second virial coefficients of hadrons with the hard-core repulsion, i.e. with the excluded volumes, is rather successful, we extended this approach to the classical second virial coefficients of hadrons and light nuclear clusters [30,31,32,33].

In this work we summarize our very recent results [32,33] obtained on the description of the STAR √ s N N = 200 GeV data on the nuclear multiplicities [24,25,26] and the ALICE √ s N N = 2.76 TeV data on light nuclear cluster yields [27,28,29], and discuss some findings which were not reported previously, in particular, the problematic hyper-triton ratios (PHTR)

Λ H/ 3 He • p/Λ and S 3 measured by the STAR and ALICE Collaborations which were not described until now either by the HRGM or by the coalescence model [34].

HRGM for the mixture of hadrons and light nuclear clusters

The HRGM based on the IST EoS has the following hard-core radii of pions R π =0.15 fm, kaons R K =0.395 fm, Λ-(anti-)hyperons R Λ =0.085 fm, other baryons R b =0.365 fm and other mesons R m =0.42 fm [18,19,11] which only slightly differ from the our previous results found within the VdW approximation [4,6]. Since all the details of the IST EoS and the fitting procedure of the hadronic data are well documented in Refs. [18,19,11], here we do not discuss them.

To account for the classical excluded volumes of light nuclear clusters and hadrons we use two approaches worked out in [30,31,32] with one exception, namely we consider the hypertriton (HTR) differently as it is suggested in [33]. Both of these approaches employ the classical excluded volumes of light nuclei of A ∈ {2, 3, 4} baryonic constituents and hadron h [32,33]

The equations above can be found from the fact that all light nuclear clusters analyzed here are roomy clusters. The mean distances among the baryons inside of such clusters are rather large [32,33] and, hence, it is possible to freely translate any hadron h with the hard-core radius R h around each constituent of a nucleus without touching any other constituent of this nucleus. The first approach is the ISTΛ EoS and it uses exactly the excluded volumes ( 1) and ( 2). It is rigorously derived using a self-consistent treatment of classical excluded volumes of light nuclear clusters and hadrons [32] with the help of the methods developed in [21,22]. In contrast, the IST EoS which employs (1) for the HRT is called the IST EoS.

The second approach is approximate and complementary to the exact one. It is based on an approximate, but the rather accurate treatment of the equivalent hard-core radius of roomy nuclear cluster and pions which are the dominating component of the HRG at the energy range of our interest. In the latter approach one can find an effective hard-core radius of nuclei of A baryons as R A A 1/3 R b , since the hard-core radius of pions is very small and, hence, it generates a negligible correction to R A [30,31,32]. Since the hard-core radius of light nuclear clusters defined in this way is similar to the expression of the Bag Model [35], it is called the BMR EoS. A more accurate expression for the HTR hard-core radius R HT R 2 1/3 R b is derived in [33] and such a model is called the BMRΛ EoS. The main reason to compare the results of these two approaches is that, despite the difference in the equations, they should reproduce the data with the same quality by construction. Hence, finding the region of parameters which provide a similar quality of the data description one can remove the ambiguity in choosing the appropriate CFO parameters by analyzing the wide and shallow minima χ 2

A of light nuclei. Following our ideology outlined in [30], we verify two different scenarios of the CFO of nuclei clusters, namely a single CFO together with the hadrons and their separate CFO from the hadrons. The major reason for such an analysis is that the mechanisms of the hadron production and production of nuclei in collisions can be rather different. One can clearly see from the left panels of Figs 1 and 2 that the minimum of the light nuclear clusters χ 2 A (T A ) as a function of their CFO temperature T A is located far away from the minimum of χ 2 h (T h ) of hadrons as the function of the hadronic CFO temperature T h . The total χ 2 tot (V ) is defined as

cti v el y, t h e m e a n s q u ar e d d e vi ati o n f or t h e r ati os a n d f or t h e yi el ds, w hil e V is t h e C F O v ol u m e of n u cl ei a n d ρ k is t h e p arti cl e n u m b er d e nsit y of t h e k -t h s ort of p arti cl es. A c o m bi n e d fit of p arti cl e yi el ds a n d r ati os is di ct at e d b y t h e a v ail a bl e d at a a n d b y n u m eri c al c o n v e ni e n c e. It is i nt er esti n g t h at f or t h e v a nis hi n g h ar d-c or e r a dii of all n u cl ei t h e mi ni m u m of χ 2 A (T ) is cl os e t o t h e mi ni m u m of χ 2 h (T ) f or t h e S T A R d at a, b ut still it is f ar a w a y f or t h e A LI C E o n e (s e e t h e s h ort d as h e d c ur v es i n t h e l eft p a n els of Fi gs 1 a n d 2).

Fr o m Fi g.