Chemical freeze-out conditions and fluctuations of conserved charges in heavy-ion collisions within quantum van der Waals model

The chemical freeze-out parameters in central nucleus-nucleus collisions are extracted consistently from hadron yield data within the quantum van der Waals (QvdW) hadron resonance gas model. The beam energy dependences for skewness and kurtosis of net baryon, net electric, and net strangeness charges are predicted. The QvdW interactions in asymmetric matter, $Q/B \neq 0.5$, between (anti)baryons yield a non-congruent liquid-gas phase transition, together with a nuclear critical point (CP) with critical temperature of $T_c=19.5$ MeV. The nuclear CP yields the collision energy dependence of the skewness and the kurtosis to both deviate significantly from the ideal hadron resonance gas baseline predictions even far away, in $(T,\mu_B)$-plane, from the CP. These predictions can readily be tested by STAR and NA61/SHINE Collaborations at the RHIC BNL and the SPS CERN, respectively, and by HADES at GSI. The results presented here offer a broad opportunity for the search for signals of phase transition in dense hadronic matter at the future NICA and FAIR high intensity facilities.


I. INTRODUCTION
The structure of the phase diagram of strongly interacting matter is one of the most important and still open topics in nuclear and particle physics to date. The known phenomenology of the physics of strong interactions suggests both, short-range repulsion and intermediate-range attraction between nucleons in proximity of nuclear saturation density n 0 = 0.16 fm −3 . This yields a first-order liquid-gas phase transition (LGPT) from a dilute (gaseous) to a dense (liquid) phase of nuclear matter, which smoothens out in the nuclear critical point (CP). In contrast to the hypothetical deconfinement-related CP, the existence of the LGPT and the nuclear CP is better established [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], see [16] for a review.
Theoretical arguments suggest the enhancement of certain fluctuations of conserved quantities in the critical region [17][18][19][20][21][22], namely, the fluctuation of the conserved charges that are related to the so-called order parameter. The signals of the CP in the scaled variance of the charge fluctuations fade-out rather quickly when moving away from the CP [23,24]. On the other hand, the CP signals in fluctuation measures which are related to the higher order moments of charge distributions, namely skewness and kurtosis of charge fluctuations, can be seen even far away from the location of the CP on the phase diagram [25][26][27]. Thus, the observed large deviations of the higher order charge fluctuations from the ideal hadron resonance gas (IHRG) baseline can be taken as a signal for the existence of a CP.
Here we study this issue by employing the quantum van der Waals hadron resonance gas (QvdW-HRG) model, which is the extension of the classical vdW model: The QvdW model was recently generalized to include the grand canonical ensemble (GCE) [28], the effects of relativity and quantum statistics [25], and the full known spectrum of hadrons and resonances [27]. The QvdW-HRG model is a minimal interaction-extension of the IHRG model. It takes into account both, attractive and repulsive, interactions between only baryons and between only anti-baryons. These interactions yield the LGPT and the nuclear CP within the model [25].
The model includes two parameters only, which are fixed by the properties of the nuclear ground state.
The QvdW-HRG at low temperatures is reduced to normal nuclear matter, described by the QvdW model, see Refs. [23,25,26,[28][29][30]. The results for symmetric nuclear matter are similar to the Walecka model results [31]. The QvdW model was applied to describe asymmetric nuclear matter and its non-congruent LGPT in Ref. [24].
The skewness and the kurtosis of baryonic charge fluctuations were calculated within the QvdW-HRG model for central nucleus-nucleus (A+A) collisions along the chemical freeze-out line in Ref. [26]. The present paper extends these results in two directions. First, the chemical freeze-out line is derived consistently for central A+A collisions within the QvdW-HRG model.
Second, both the baryonic and electric charge fluctuations are calculated in T − µ B plane and along the freeze-out line. The electric charge is a more convenient quantity for experimental measurement, compared to baryonic charge, as it does not require the detection of the dominant electrically neutral baryons. The Thermal-FIST [32] package is used for the calculations within the QvdW-HRG model.
The paper is organized as follows. Section II briefly describes the QvdW-HRG model.
Section III discusses the non-congruent LGPT in asymmetric nuclear matter within the QvdW-HRG model and presents chemical freeze-out lines obtained within the QvdW-HRG and IHRG models. Section IV presents the QvdW-HRG and IHRG results on the skewness and the kurtosis of charge fluctuations as functions of the collision energy and in the coordinates of baryochemical potential and temperature. A summary closes the article in Sec. V.

II. THE QUANTUM VAN DER WAALS -HADRON RESONANCE GAS MODEL
The total baryon (B), electric (Q), and strangeness (S) charges of the hot, dense, hadronic system in the GCE are regulated by the corresponding chemical potentials, µ B , µ Q , and µ S .
The chemical potential of the j-th type hadron is µ j = b j µ B + s j µ S + q j µ Q , where b j , s j , and q j are, respectively, the baryonic number, the strangeness, and the electric charge of the hadron of j type. The QvdW model yields the total pressure of the system as a sum of the partial pressures of baryons, anti-baryons, and mesons [27]: The partial pressure of the baryons is given as Here T is the temperature, p id j is the ideal Fermi-Dirac pressure of the baryons of j type, µ B * j and n B are, respectively, the shifted baryonic chemical potential of baryons of j type and the total density of all baryons: The corresponding expressions for pB, µB * j , and nB of the antibaryons are analogous to Eqs.  Tables [33] and which have a confirmed status there.

III. PHASE TRANSITION AND CHEMICAL FREEZE-OUT
The LGPT in the QvdW-HRG model is due to an interplay of the repulsive and the attractive interactions. The mixed phase boundary and the location of the CP in asymmetric nuclear matter are found from the Gibbs equilibrium condition [24]. The QvdW-HRG model with Note that isospin asymmetry alters qualitatively the properties of the PT, rendering it as being a "non-congruent" PT. As a result, the mixed phase in (µ B , T) coordinates can not be presented by a line, but is rather a region of finite width. Moreover, for a non-congruent PT, the location of the CP differs from the location of the temperature endpoint (TEP), the point with the The GCE can be used for all data sets considered, except for the lowest energies at SIS. The exact net strangeness conservation is enforced for the SIS data, i.e., the calculations are done for these low-energy Au+Au collisions within the strangeness canonical ensemble (SCE) [54,55].
Finite resonance widths are treated in the present paper in the framework of energy independent Breit-Wigner scheme [56]. Note, that the energy dependent Breit-Wigner scheme leads to a better description of hadron yields at the LHC [56]. Another possibility is to neglect finite widths of resonances altogether. Here we stick to the energy independent Breit-Wigner scheme so as to preserve consistency with our earlier works regarding the chemical freeze-out conditions in the IHRG model [57] or thermodynamic properties of the QvdW-HRG model [27]. We did verify that differences in the extracted freeze-out parameters obtained within these different schemes are small, with a possible exception of the strangeness saturation factor γ S . A detailed study of finite resonance widths effects on hadron yields for intermediate collision energies will be presented elsewhere.
The fitted freeze-out parameters are µ B , T , volume of the system V , and the strangeness under-saturation parameter γ S (see Ref. [58]). The corresponding IHRG and QvdW-HRG fit results for µ B , T , V , and γ s are presented in Table I and (b). The parameter γ S however cannot be reliably determined (see Ref. [57]), Hence, γ S is not shown in Fig. 3 (a) at SIS. We define uncertainties of the extracted µ B , T , and γ s values following the procedures given in Ref. [59], by multiplying the uncertainties inferred from the χ 2 = χ 2 min + 1 contours by a factor χ 2 min /dof [33]. We adopt a simple thermodynamic parametrization of the chemical freeze-out line, used previously in Ref. [63]. Here we use it to parameterize the extracted T and µ B values. The five newly fitted parameters in Eq. (5), a 1 , a 2 , a 3 , b 1 , b 2 , are presented in Table II for both the IHRG and QvdW-HRG model. Note that the QvdW-HRG parameters a 2 , a 3 , b 1 differ by about 20% from the IHRG fits, but that the b 2 value of the IHRG fit exceeds the QvdW-HRG value by 70%. The parametrization (5) extrapolates the freeze-out line from µ B ≈ 0 region at the highest collision energies down to the nuclear matter region of the phase diagram at the lowest collision energies. The chemical freeze-out line close to the region of the nuclear liquid-gas transition was previously considered in Refs. [64,65] in a context of light cluster formation.
Susceptibilities, χ ch i , are calculated in the GCE from the scaled total pressure by taking the derivatives with respect to the corresponding powers of the chemical potentials over the tem- perature: We have checked that the strangeness suppression effect due to γ s < 1 is rather small in the skewness and kurtosis of the net strangeness fluctuations. Therefore, in calculations of charge fluctuations, γ s is fixed to unity and the GCE is used. Figure 7 shows  In order to reduce the volume fluctuation effects the most central collisions must be selected [68]. Another way to reduce the volume fluctuation effects is to use the so-called strongly intensive quantities [69] as the fluctuation measures. A detailed analysis of the required acceptance and volume fluctuation corrections is however outside the scope of the present paper and will be a subject of the future research.
The baryon number fluctuations are calculated in the present paper with the assumption that all baryons are experimentally detectable. This is not the case in reality. Therefore, a binomial acceptance procedure, see Refs. [21,26,70], is usually applied to account for this inability of the event-by-event measurements of (anti)neutron numbers. This procedure leads to an essential decrease of the observable baryon number fluctuations. In contrast to the baryon number, nearly all electric charges can be experimentally detected. Thus, our results obtained for electric charge fluctuations are more suitable for a comparison with the experimental data.
Finally, an analysis of experimental data should be complemented with dynamical model simulations of heavy-ion collisions, where the effects of baryon-baryon interactions studied here are incorporated. The dynamical models can naturally incorporate the effects related to baryon number conservation and acceptance. Some recent developments in this direction using transport models can be found in Refs. [71,72].
Note that the predicted large differences of the fluctuation measures at the high baryon den-

V. SUMMARY
The quantum van der Waals hadron resonance gas model has been applied to study chemical freeze-out properties in heavy-ion collisions as well as the higher-order fluctuations of net baryon and net charge numbers. The extracted chemical freeze-out parameters exhibit larger uncertainties as compared to the ideal hadron resonance gas model. Similar to the IHRG model, the dependence of T ch on µ ch B in the QvdW-HRG model can be parametrized as a quartic polynomial in µ ch B , with parameters differing quite substantially from the IHRG case. Since both the LGPT and the chemical freeze-out are consistently obtained within a single model, their relative location is clarified.
The beam energy dependences of the skewness and the kurtosis of the baryonic, electric, and strange charge fluctuations has been calculated along the obtained chemical freeze-out curve.