RMF models with \sigma-scaled hadron masses and couplings for description of heavy-ion collisions below 2A GeV

RMF models with -scaled hadron masses and couplings for description of heavy-ion collisions below 2A GeV

Konstantin A. Maslov National Research Nuclear University (MEPhI), 115409 Moscow, Russia
Bogoliubov Laboratory for Theoretical Physics, Joliot-Curie street 6, 141980 Dubna, Russia
   Dmitry N. Voskresensky National Research Nuclear University (MEPhI), 115409 Moscow, Russia
Bogoliubov Laboratory for Theoretical Physics, Joliot-Curie street 6, 141980 Dubna, Russia

Within the relativistic mean-field framework with hadron masses and coupling constants dependent on the mean scalar field we study properties of nuclear matter at finite temperatures, baryon densities and isospin asymmetries relevant for heavy-ion collisions at laboratory energies below 2 GeV. Previously constructed (KVORcut-based and MKVOR-based) models for the description of the cold hadron matter, which differ mainly by the density dependence of the nucleon effective mass and symmetry energy, are extended for finite temperatures. The baryon equation of state, which includes nucleons and resonances is supplemented by the contribution of the pion gas described either by the vacuum dispersion relation or with taking into account the -wave pion-baryon interaction. Distribution of the charge between components is found. Thermodynamical characteristics on plane are considered. The energy-density and entropy-density isotherms are constructed and a dynamical trajectory of the hadron system formed in heavy-ion collisions is described in these terms. The effects of taking into account the isobars and the -wave pion-nucleon interaction on pion differential cross sections, pion to proton and ratios are studied. The liquid-gas first-order phase transition is studied within the same models of the equation of state in isospin-symmetric and asymmetric systems. We demonstrate that our models yield thermodynamic characteristics of the phase transition compatible with available experimental results. In addition, we discuss the scaled variance of baryon and electric charge in the phase transition region. Effect of the non-zero surface tension on spatial redistribution of the electric charge is considered for a possible application to heavy-ion collisions at low energies.

1 Introduction

Knowledge of the equation of state (EoS) of cold dense hadronic matter is required for the description of atomic nuclei and neutron stars after minutes-hours since their formation, and EoS of warm and hot hadron matter is required for the description of supernovae and heavy-ion collisions. Nowadays there exists a vast number of EoSs and a large set of experimental and observational constraints, which an appropriate EoS should fulfill Klahn:2006ir (). No one of existing EoSs satisfies all the known constraints. Flexible phenomenological approaches to EoSs are introduced within relativistic mean-field (RMF) models with density dependent couplings, see Typel (); Typel2005 (); Voskresenskaya (); Oertel:2016bki () and refs. therein, and with hadron masses and coupling constants dependent on the mean scalar field Kolomeitsev:2004ff (). The latter model has been generalized to the description of the isospin-symmetric hot hadronic matter including various baryon resonances and bosonic excitations Khvorostukhin:2006ih (); Khvorostukhin:2008xn (); Khvorostukhin:2010aj () and was applied to the description of heavy-ion collisions in a broad range of collision energies. Isospin-asymmetric matter (IAM) was not considered in mentioned works. Bosonic excitations were considered in the ideal gas model.

Recent measurements of masses of the most massive binary pulsars demonstrated that the maximum compact star mass, predicted by an EoS, should exceed . It was found that PSR J1614-2230 has the mass Demorest:2010bx (); Fonseca:2016tux () and PSR J0348+0432 has the mass  Antoniadis:2013pzd (). These measurements rule out many soft EoSs of the purely nucleon matter. However additional degrees of freedom may appear in dense neutron-star interiors, such as hyperons, isobars and meson condensates. This leads to an additional softening of the EoS of the beta-equilibrium matter (BEM) resulting in a decrease of the maximum neutron-star mass. On the other hand, description of the particle flow in heavy-ion collisions requires a rather soft EoS of the isospin-symmetric matter (ISM) Danielewicz:2002pu (). Thereby it is challenging to construct an EoS, which would simultaneously satisfy the maximum neutron-star mass constraint together with the flow constraint.

The working model with hadron masses and coupling constants dependent on the mean scalar field was constructed in Kolomeitsev:2004ff () and labeled in Klahn:2006ir () as KVOR model. It satisfies the flow constraint for ISM and yields the maximum neutron-star mass for BEM, however only if no baryons other than nucleons are included into consideration. In our subsequent works Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu (); Kolomeitsev:2017gli () we constructed RMF models of the cold hadronic matter of arbitrary isospin composition with effective hadron masses and coupling constants dependent on the scalar field with hyperons and resonances taken into account Kolomeitsev:2016ptu (), as well as with the charged condensate Kolomeitsev:2017gli (), which successfully pass the maximum neutron-star mass constraint and the flow constraint simultaneously with many other constraints.

Various characteristics of heavy-ion collisions at energies below a few  GeV have been extensively studied within the expanding fireball framework Siemens:1978pb (); Friedman:1981qm (); Mishustin:1983nv () and then within the ideal hydrodymamics and in various transport models, cf. Arsene:2006vf (); Buss:2011mx () and refs. therein. Densities reached at such collision energies are typically , where is the nuclear saturation density, and temperatures are below the pion mass MeV. Only EoS for ISM has been studied within the expanding fireball framework Siemens:1978pb (); Friedman:1981qm (); Mishustin:1983nv (). Reference Friedman:1981qm () used a variational theory of nuclear matter for a description of nucleons in ISM and Voskresensky:1997vq () used the original Walecka RMF model Walecka:1974qa (), whereas pions were considered with taking into account -wave pion-baryon medium polarization effects. Then Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud () exploited a modified Walecka RMF model of Cubero:1987pr () for the nucleon ISM. A comparison with the data on pion and nucleon spectra available at that time demonstrated advantages of such description. At the freeze-out stage an influence of the effects of the nuclear polarization on the pion spectra was considered within the prompt breakup model Senatorov:1989cg (). However, being extended to describe BEM, such EoSs do not satisfy modern data on the high masses of cold compact stars.

In this work we generalize the models with -scaled hadron masses and couplings developed for the description of the cold BEM in Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu (); Kolomeitsev:2017gli (), now for the case of the ISM and IAM formed in heavy-ion collision reactions for collision energies  GeV, so the reached baryon densities are and temperatures are below . In heavy-ion collisions the strangeness is approximately conserved. Thereby, the hyperon contribution to the thermodynamical values, , where is the hyperon effective mass, can be neglected. Effect of boson , , excitations can be also disregarded for and . The temperature dependence can be then included only in the nucleon and kinetic energy terms and in pion quantities. We use a simplified expanding fireball framework. In a subsequent work we plan to check the validity of our EoSs in actual hydrodynamical calculations. Up to now simulations of heavy-ion collisions have been done within ideal hydrodynamics with various EoSs of isospin symmetric matter with pions treated as particles obeying the vacuum dispersion law, cf. Ivanov:2016xev (). As the first step, in the present work pions will be treated either as ideal gas of the particles obeying the vacuum dispersion law or as the quasiparticle gas with the -wave pion-baryon interactions included. The latter contribute only for IAM. More involved effects of the -wave pion-baryon interaction will be disregarded.

In the heavy-ion collisions at very low collision energies ( MeV) in the expansion stage of the matter at nucleon densities and at low temperatures, MeV, there may occur a spinodal instability during the first-order liquid-gas (LG) phase transition Ropke:1982vzx (). Possible effects of the supercooled gas and superheated liquid phases, as well as the effects of the spinodal instabilities, have been discussed in Ropke:1982vzx (); Schulz:1983pz (); BS (); Panagiotou:1984rb (). The nuclear LG transition phenomenon remained an arena of intense research both on theoretical and experimental sides during subsequent years, cf. Muller:1995ji (); Li:1997px (); Ducoin:2005aa (); Alam:2017krb (). Occurrence of a negative specific heat at constant pressure was reported, as the first experimental evidence of the LG phase transition in heavy-ion collision reactions DAgostino:1999dod (); Schmidt:2000zs (). For a review of this interesting topic see Chomaz:2003dz () and more recent works Skokov:2008zp (); Skokov:2009yu (); Skokov:2010dd (); Voskresensky:2010gf (); Borderie:2018fsi (). Isospin dependence was studied in Colonna:2002ti (); Ducoin:2005aa (). Effects of a finite surface tension were disregarded. Below we apply our models with effective hadron masses and coupling constants dependent on the scalar field also to describe the LG first-order phase transition occurring at a low temperatures and densities. First we assume the surface tension to be zero and then include effects of the non-zero surface tension, which may result in preparation of the pasta-like structures in heavy-ion collisions.

The manuscript is organized as follows. In section 2 we present the RMF model with -scaled hadron masses and couplings generalized for description of the hadron matter at arbitrary isospin composition for , including pion gas. For specificity we further use the KVORcut03-based and MKVOR*-based models of EoS Kolomeitsev:2016ptu () in the region of the baryon densities and temperatures . In Sect. 3 we present results obtained in a simplified expanding fireball model. This simplified description allows us to demonstrate many qualitative and quantitative effects. In Sect. 4 we focus on the description of the region of the LG phase transition first considering ISM and then IAM. Consideration is first performed within the RMF framework and then effects of fluctuations are discussed. Importance of the surface tension effects will be then emphasised. Then in Sect. 5 we formulate our conclusions. For completeness in Appendix 6 we indicate effects of the polarization of the medium due to the -wave pion-baryon interaction, which were disregarded in the present study.

2 RMF models with -scaled hadron masses and couplings. EoS of hadron matter in the region , ,

We use the framework proposed in Kolomeitsev:2004ff () and then developed further in Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu () for and an arbitrary isospin composition and generalized in Khvorostukhin:2006ih (); Khvorostukhin:2008xn (); Khvorostukhin:2010aj () for the case of , but only for ISM. In the present work we focus on the description of matter produced in heavy-ion reactions at collision energies less than few  GeV. Thereby we study the ISM, for which , and the IAM matter, when , where is the total charge of the colliding nuclei and is the corresponding baryon number.

Our model is a generalization on the case of the non-linear Walecka model with effective coupling constants and hadron masses


dependent on the scalar field . Here denotes mesons, for which we use the mean-field solutions of the equations of motion, lists the included baryon species, nucleons and isobars, . We neglect a contribution of hyperons and anti-baryons, which are tiny for collision energies under consideration, for hyperons due to the strangeness conservation Khvorostukhin:2006ih (); Khvorostukhin:2008xn () and for anti-baryons. In absence of the hyperon occupations there is no contribution of the meson mean field. Besides, we include pions , as lightest among pseudo-Goldstone particles. Other pseudo-Goldstone particles and their heavier partners and are not included, since their contributions, , remain tiny for . The quantities and are the dimensionless scaling functions have been fitted in Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu () for the best description of the cold baryon matter.

Using mean-field solutions for meson fields we present the energy density of the hadronic system as Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu ()


where is the baryon spin, is the chemical potential for the given baryon species , is the baryon-charge chemical potential, is the chemical potential of a negative electric charge, is the electric charge of a particle , is the isospin projection of baryon .

For given and eqs. (3) can be solved to find the particle densities and their effective chemical potentials . Then the definitions of the total baryon density and charge density are treated as equations for finding the chemical potentials for given . Here are the pion number densities


where is the dispersion relation of a pion and is its chemical potential. For ISM . The term


is the contribution of species to the energy density.

For any set of baryon concentrations and the temperature equation is solved to find the equilibrium value of the scalar field.

The effective hadron masses and scaling functions of mesons enter the volume part of the thermodynamic quantities only in combinations ,


where the self-interaction potential employed in standard RMF models is included into the definition of the scaling function . The scaling functions of the mass are


and thereby . We suppose that  , . Explicit expressions for the scaling functions are presented in Kolomeitsev:2016ptu ().

The coupling constant ratios are introduced as . The vector-meson coupling constants to s are chosen following the quark SU(6) symmetry:

The coupling constants with the scalar field are deduced from the values of the optical potentials in ISM at the saturation density given by


The value of the potential is poorly constrained by the data. As in Kolomeitsev:2016ptu (), we use as the most realistic estimate. Models including s will be denoted by ”” suffix.

Also we assume that the size of the system under consideration is such that the volume part of the thermodynamic quantities of our interest is much larger than the surface part. Moreover, we first disregard finite-size Coulomb effects compared to the strong-interaction effects. The former effects will be discussed in Sect. 4. Focusing on the description of heavy-ion collisions we do not include lepton terms.

In ref. Maslov:cut () we demonstrated that within an RMF model the EoS becomes stiffer for , if a growth of the scalar field as a function of the density is quenched and the nucleon effective mass becomes weakly dependent on the density for . In Maslov:cut () such a quenching was achieved by adding to the scalar potential a rapidly rising function of at , where is corresponding to . We called it the cut-mechanism. In Ref. Maslov:cut () the cut-mechanism is realized in the sector. We focus now on two models based on KVORcut03 and MKVOR* models proposed in Maslov:2015msa (); Maslov:2015wba (); Kolomeitsev:2016ptu (). These models proved to satisfy many constraints on the hadronic EoS. In neutron-star matter for large densities the hyperons and baryons appear in our models. These two models utilize the cut-mechanisms in the and sectors, respectively. The cut mechanism in sector is implemented in MKVOR-based models in order simultaneously to keep the EoS not too stiff in ISM (to satisfy the flow constraint from heavy-ion collisions Danielewicz:2002pu ()) and to do the EoS as stiff as possible in the BEM to safely fulfill the constraint on the maximum mass of neutron stars. The mean field is coupled to the isospin density that makes the -saturation mechanism very sensitive to the composition of the BEM. Incorporating baryons we use the MKVOR* extension of the MKVOR model Kolomeitsev:2016ptu (), which prevents the effective nucleon mass from vanishing at high density.

Free parameters of the model are fitted to reproduce properties of the cold nuclear matter near the saturation point. These parameters are defined as the coefficients of the Taylor expansion of the energy per nucleon for ,

[MeV] [fm] [MeV] [MeV] [MeV] [MeV] [MeV]
KVORcut03 0.16 275 0.805 32 71 422 -86
MKVOR 0.16 240 0.730 30 41 557 -158
Table 1: Coefficients of the energy expansion (8) for KVORcut03 and MKVOR models.

in terms of small and parameters. The parameters for the MKVOR* and MKVOR models are identical. The properties of the KVORcut03 and MKVOR* models, which we exploit in this work, at the nuclear saturation density are illustrated in Table 1, where we collect coefficients of the expansion of the nucleon binding energy per nucleon near .

For the difference of the neutron and proton chemical potentials we get


The pion quasiparticle spectrum is determined as a solution of the dispersion equation Migdal:1990vm (); Voskresensky:1993ud ()


where is the pion polarization operator in the baryon medium. In this work we will consider pions as an ideal gas of quasiparticles including for IAM only their -wave interaction with baryons, being determined by the so called Weinberg-Tomozawa term. For ISM the -wave interaction is suppressed Baym:1975tm (); Migdal:1978az (); Friedman:2019zhc (); Migdal:1990vm (); Voskresensky:1993ud (). Including only -wave pion-nucleon interactions the retarded pion polarization operator is given by

where MeV is the pion weak decay constant, cf. Migdal:1990vm (); Kolomeitsev:2002gc (), and the spectrum is thereby as follows


Models with pion quasiparticles treated following eq. (11) will be labeled by “” suffix and models with pions described by the vacuum dispersion law we label by “” suffix, respectively. Important role of the -wave pion-baryon interactions has been intensively studied in Baym:1975tm (); Migdal:1978az (); Migdal:1990vm (). This issue will be briefly reviewed in the Appendix.

Figure 1: Left panel: The effective mass scaling function for the models KVORcut03 and MKVOR* for ISM as a function of the baryon density for various temperatures indicated in the legend in MeV. Right panel: The symmetry energy coefficient in models KVORcut03 (dashed line) and MKVOR* (solid line) for ISM. For comparison by dash-dotted line is shown the symmetry energy coefficient obtained in model Ma:2018xjw () with a topology change mimicking the baryon-quark continuity taking place at .

One of the main differences between the KVORcut-based and MKVOR*-based models we consider here is the behavior of the nucleon effective mass with the density. Therefore, on left panel in fig. 1 we show the baryon density dependence of the scaling function in ISM calculated for various temperatures in the models KVORcut03 and MKVOR*. We see that the density dependence of this quantity is significant for , whereas the temperature dependence is moderate for . In the KVORcut03 model  MeV in the density and temperature interval under consideration, whereas in the MKVOR* model we have  MeV.111We should note that here is the mass-coefficient of the mean field rather than the effective mass of excitations, , cf. Khvorostukhin:2006ih (); Khvorostukhin:2008xn (). Thus the effective masses of the mesons remain significantly larger than , and the thermal contribution of the , , , excitations, which is , can be neglected for temperatures and densities we consider here. The curves computed for and models prove to be visually almost not distinguishable for . Therefore below we mainly focus consideration on models. Besides, we note that the dependence of proves to be very weak in the interval of our interest. The curves computed for and for are visually almost not distinguishable.

The temperature dependence of the nucleon energy is and thereby it is essential already for , where is the nucleon Fermi energy ( MeV for , and for the Landau nucleon effective mass ). The energy of the pion ideal gas is and becomes significant for . The contribution of the s to the energy is suppressed compared to the nucleon one as . All these contributions are included in our work. The first-order phase transition to the resonance matter considered in Kolomeitsev:2016ptu () for does not occur for , , for the optical potential MeV that we use in this work. In Khvorostukhin:2008xn () within the model, where pions are treated with the vacuum dispersion law, the effect of the nonzero width was estimated as not significant. Therefore in what follows within our RMF-based model the resonances will be treated as quasiparticles with an effective mass. Concluding this discussion, for temperatures below and for densities of our interest here, the temperature dependence can be included only in the nucleon and isobar quasiparticle contributions and the pion kinetic energy terms, which within the model are described by the free dispersion law.

Our KVORcut03-based and MKVOR*-based models differ also by the density dependence of the symmetry energy coefficient. On the right panel in fig. 1 we show the baryon density dependence of the symmetry energy coefficient derived in our models KVORcut03 and MKVOR* for ISM. The density dependence of the symmetry energy in the KVORcut03-based models, where the cut-mechanism is implemented in the sector Kolomeitsev:2016ptu () for , is rather smooth (dashed line). In the MKVOR*-based models, where the cut-mechanism is used in the sector Kolomeitsev:2016ptu () responsible for the symmetry energy, the dependence of on becomes sharp for (solid line). In Ma:2018xjw () the dramatic change in the density dependence of the nuclear symmetry energy for above some value varied in the interval was associated with the change of the topology mimicking the baryon-quark continuity. By the dash-dotted line in figure is shown the symmetry energy coefficient obtained in model Ma:2018xjw () for . The resulting density dependence is similar to that we obtain in the MKVOR-based models.

In the heated dense nuclear system formed in the collision of nuclei, initial charge per baryon is redistributed among all the involved electrically charged hadrons appeared for , cf. Muller:1995ji (); Li:1997px (), following minimum of the free energy expressed in variables. We use the following decomposition of the system charge:

where is the ratio of the initially fixed number of protons to the fixed total baryon number, is the ratio of the number of protons to that of the baryons inside the thermal system, is the ratio of the total charge of the baryon subsystem to the total baryon density, that includes contribution of charged isobars, and is the ratio of the total charge of the pions to the total baryon density.

Figure 2: The ratio of the excess/deficiency of the positively charged hadrons of various species to the total baryon density at as a function of the temperature for the model KVORcut03 on left panel and for the model MKVOR* on right panel. Solid curves are for , dashed lines for , and dash-dotted ones for . See text for further details.

In fig. 2 we show the charge per baryon for various species for the model KVORcut03 on the left panel and for the model MKVOR* on the right panel. We take as an example relevant to the matter formed in heavy-ion collisions. At zero temperature , since the isobars do not appear in both our models for Kolomeitsev:2016ptu (), which we consider here, and there are no pions for . With an increase of the temperature the abundance of resonances and pions increases. For the charge chemical potential is positive, and there appears an excess of respectively to . The ratio (lines labeled ) increases with the temperature . The pion charge per unit of baryon density proves to be higher for smaller density. Unlike in the neutron-star matter, the subsystem in heavy-ion collisions at remains positively charged. Indeed, . Since the value of remains rather small for , the and abundances remain close to each other. Indeed, as we see in fig. 2 excess of the positively charged resonances (lines labeled ) increases with an increase of in agreement with above estimate. The value (lines labeled by ) slightly increases with increase of for and then it sharply decreases for higher . The reason of a slight increase of for low is that an enhancement of the proton fraction should compensate a small increase with of the negative pion charge to fulfill the charge conservation condition. However, at the isobar concentration increases noticeably and the proton fraction decreases because of the baryon charge conservation. The total charge of the baryon subsystem per baryon is shown in fig. 2 by the lines labeled “”.

For MKVOR* model at for the system is negatively charged. This happens because in the MKVOR* model the symmetry energy coefficient and, correspondingly, are larger at such densities than in the KVORcut03 model, see fig. 1. Therefore in agreement with the estimate given above the density of becomes greater than the sum of densities of and . The negative pion excess contributes less to the charge conservation at than for lower because the number of pions per baryon at a fixed temperature decreases with an increase of the baryon density.

For all the ratios obtained in both of our models are very close to each other. Only for the contribution of to the charge excess computed within the KVORcut03 model proves to be a bit higher than that in MKVOR* model.

Note that in MKVOR* model a transition to the resonance-enriched matter may occur in the dense medium. For and for MeV that we use in this work the s appear by the crossover at , cf. fig. 11 in Kolomeitsev:2016ptu (). With an increase of the temperature in the MKVOR* model s arise for a lower density and the phase transition becomes the transition of the first order. We found that for at the first-order phase transition occurs for . The critical density decreases very smoothly with an increase of the temperature and we get at , thus remains above , i.e, outside the density range we consider in the given work. However we should point out that some papers argue that could be a more attractive. For MeV the first-order phase transition to the resonance matter would occur for already at the density , cf. a discussion in Kolomeitsev:2016ptu (). In this work we use and consider , thereby the first-order phase transition to the reach matter does not occur within our MKVOR* model. In the KVORcut03 model the phase transition to the resonance matter does not occur for all relevant values of at densities and temperatures we are interested in this work.

3 Simplified model for heavy-ion collisions at  GeV

3.1 EoS and the system dynamics

In a heavy-ion collision, nucleons can be subdivided on “participants,” which intensively interact with each other during the collision, and “spectators”, which remain practically unperturbed. Baryons-participants form a nuclear fireball, which first is compressed (for ) and then (for ) is expanded under action of the internal pressure.

The kinetic energy of the projectile nucleus per nucleon (per ) in the laboratory system is related to the kinetic energy per nucleon in the center-of-mass frame as follows


is the nucleon number of the target nucleus. As in Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud (), we assume that at energies less than a few  GeV in the laboratory frame the energy in the center-of-mass frame of the nucleus-nucleus collision, , which corresponds to the nucleons-participants, is spent on the creation of an initially quasi-equilibrium nuclear fireball resting in the center-of-mass frame at the end of the compression stage, for . The energy per baryon, , as a function of the baryon density at fixed has the concave shape in our KVORcut03 and MKVOR* models, it decreases with increase of , gets a minimum at , and then begins to increase, see fig. 3 below. Following Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud () we assume that the initial fireball state is characterized by the temperature , and the baryon density corresponding to the minimum of the energy per baryon, , as a function of the baryon density for ,


The quantity is the binding energy per baryon in a cold nucleus of the nucleon number , . Below we will use values which follow from (2) at ignorance of surface and Coulomb effects, i.e., as would be for very heavy nuclei. Thus we shall take ,  MeV.

The specific entropy is a decreasing function of , see fig. 3 below. Thereby the state of the minimum of the energy on the right branch of (where increases with ) corresponds to the maximum of the entropy on the given isotherm for the states belonging this branch. Moreover, the state also corresponds to the maximum temperature among all available solutions of the eq. (13) for all curves and respectively this state corresponds to the maximum of the stirring of the degrees of freedom possible at assumption of the full stopping of the matter in the center-of-mass frame for .

Figure 3: Energy per baryon (left panel) and total entropy per baryon (right panel) in approximately ISM (, as for Ar+Kcl collisions) for the KVORcut03 EoS as functions of the baryon density in units of fm for different temperatures indicated on the lines in MeV. The bold curves correspond to the set of collision energies in laboratory system shown in GeV by arrows at the right edge of the left panel indicating a position of the corresponding minimum of the . Solid bold curves are presented for the case and two dashed bold curves are shown for the case , as for La+La collisions at and MeV. For comparison by dotted bold curves we show the quantities at the same temperatures, as for the corresponding solid lines, but without the inclusion of  resonances. Thin lines indicate the initial fireball configurations constructed as described in the text. Thin dashed horizontal lines on the right panel denote isoentropic trajectories and small dots are the break up points obtained by fitting of the production differential cross sections, cf. in figs. 6, 7 below.
Figure 4: Same as fig. 3 but for the MKVOR* model.

Note that for a weakly non-equilibrium system moving with the velocity the local pressure can be presented as Ivanov:2013uga () , where is the bulk viscosity and is the quasi-equilibrium pressure depending on the local temperature and density and following a given EoS. Since is a positive-definite quantity, the non-equilibrium correction to the pressure is positive during the compression stage of the nuclear system and it is negative during the expansion stage, see also Voskresensky:2010qu (). So, on the stage of expansion of the fireball in the heavy ion collision the non-equilibrium pressure is in reality smaller than the equilibrium one, , being used in the ideal hydrodynamics. This means that a realistic equilibrium EoS to be used in non-ideal hydrodynamical calculations should be stiffer than the one used to fit experimental data within ideal hydrodynamical simulations. Besides, the entropy in the viscous process increases, whereas it stays constant within the ideal hydrodynamics.

The viscosity effects prove to be rather small at energies less than (1-2) GeV we study in this work. This conclusion is supported by the analyses of heavy-ion collisions performed in an expanding fireball model Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud (), by calculations used ideal hydrodynamics in a broad energy range Ivanov:1991te (); Mishustin:1991sp (); Ivanov:2005yw (); Arsene:2006vf (); Buss:2011mx (), by simulations done within transport codes Bass:1998ca (); Arsene:2006vf () and by estimates of the viscosity Khvorostukhin:2010aj (). In the ideal hydrodynamics dynamical trajectories of the system in heavy-ion collisions are characterized by constant initial values of the entropy per (i.e. by total entropy density per net baryon density of baryons-participants, ). In our RMF approach with the contribution of the ideal pion gas included we neglect the inelastic processes and thereby the entropy is assumed to be conserved. Thus we have


From this relation we obtain the dependence . Note that actually all the results of this model hold locally, and therefore are applicable to the case of either quasi-homogeneous fireball expansion, or inhomogeneous hydrodynamical expansion with and depending on the space point. Just for the illustration purposes, as in Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud (), we will further assume that the expansion is uniform.

We use the KVORcut03 and MKVOR models. The ideal pion gas either is described by the vacuum dispersion law in the model or by the law (11) in the quasiparticle model for IAM. The energy density is given by eq. (2). As we have mentioned, from the relation (13) we unambiguously determine quantities and . The values of the energy per baryon and specific entropy are shown in fig. 3 (left and right) for KVORcut03 model and in fig. 4 (left and right) for MKVOR* model as functions of the density at various temperatures. Being computed with the dispersion law (11) in model, thermodynamic quantities prove to be visually almost not distinguishable from those calculated in model for . Therefore we do not show the curves for model in figs. 3, 4. Values of the temperatures are indicated on lines in MeV. The horizontal arrows on the left panels denote the initial energy per baryon in the laboratory system. Thin lines indicate minima of the energies per baryon, which exist in our models for all and corresponding to  GeV.

With a knowledge of these quantities, the dynamics of the expanding nuclear fireball is determined by the constant value of the entropy per baryon (thin horizontal dashed lines on right panel). Solid bold curves are presented for (as for Ar+KCl collisions). Two dashed bold curves on each figure illustrate the case of 0.8 GeV and 0.246 GeV collisions of La+La . Comparing the bold dashed and solid curves for the collisions with  GeV we see that the effect of the dependence is tiny for . This is because the symmetry energy of asymmetric matter is approximately and within the interval the quantity changes from to 0 and the contribution remains negligible. For the neutron-star matter and thereby the symmetry energy gives significant contribution to the total energy. Also in figs. 3 and 4 by dotted bold lines we show the results for KVORcut03 and MKVOR* models, i.e. without the inclusion of resonances. We see that the contribution of s becomes noticeable for all densities already at  MeV, which roughly corresponds to  MeV. Compared to the case without s, with s included the values and are lower for all , see the thin dotted line connecting minima of the energy. Contrary to that, the initial value of the entropy is larger for the models with s for all .

Figure 5: Left panel: The initial fireball density (upper left panel) and temperature (lower left panel) as functions of for KVORcut03-based and MKVOR*-based models supplemented with ideal pion gas with vacuum dispersion law, with and without included ( and , respectively). The curves visually coincide for both models within each set of included species, see the text for details. Right panel: Mean flow velocity at the system breakup as a function of for the models KVORcut03 (squares) and MKVOR* (triangles). Points correspond to fits of pion spectra in Ne+NaF collisions.

We assume that in the fireball, expanding with the velocity (in reality with ), the thermodynamical quasi-equilibrium is sustained up to a certain rather short breakup stage at which the nucleon and pion mean free paths become compatible with the fireball size, or more precisely, the typical expansion time becomes comparable with the inverse collision frequency Mishustin:1983nv ()). After that the nucleon and pion momentum distributions can be considered as frozen. We assume that the breakup stage is characterized by the baryon density and temperature . First fireball models estimated values of the freeze-out densities in the interval Gosset:1988na (); DasGupta:1981xx (); Nagamiya:1981sd (); Barz:1982ed (); Mishustin:1983nv (). Resonance gas model, cf. Randrup:2006nr (), yields in the whole interval of available collision energies. References Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud () argued that at lowest SIS energies since for higher densities there appears a significant contribution to the scattering amplitude from the exchange by soft pions with momenta , region of larger is usually called the liquid phase of the pion condensate Voskresensky:1993ud (). Besides, the freeze-out densities can be estimated from analysis of the HBT pion interferometry, cf. fig. 2 in Mishra:2007xg (). In figs. 3 and 4 the breakup moments are indicated by small dots. The choice of these points is explained further in the text.

The fireball expansion velocity is supposed to be zero at the initial moment and grows with time, since a part of the energy which is found using condition (14) is transformed to the kinetic energy of particles. Neglecting an energy loss due to the particle radiation in direct reactions and surface radiation during the fireball expansion up to its breakup, from an approximate conservation of the energy we may evaluate the velocity of the collective flow. They are slightly overestimated because of ignoring the mentioned effects. Resulting values of the mean flow velocities at the freeze-out can be estimated as


where are respectively the maximum energy per participant nucleon reachable for a given and the energy per participant nucleon at the breakup. Within the KVORcut03 model we obtain , 0.29, 0.39, and 0.54 for collision energies , 0.4, 0.8, and 2.1 GeV respectively and within the MKVOR* model we get values 0.21, 0.299, 0.40, 0.53, which differ only slightly from those obtained in the KVORcut03 model. Note that the values obtained for  GeV are probably too high and would be smaller, if we included effects of the -wave pion-nucleon interaction, cf. Voskresensky:1993ud (), see also results of a fit of the data for higher energies Adamczyk:2017iwn ().

On the left panel in fig. 5 we show the initial baryon density and temperature, , as functions of the collision energy in the laboratory system within MKVOR*-based models (solid and dash-dotted lines) and KVORcut03-based models (dashed and dotted lines). We see that for all the initial density and temperature for models without s (see dash-dotted and dotted lines) are larger than those with s (see solid and dashed lines). Also for all the values of in the MKVOR* model are higher than those in the KVORcut03 model. However, it is remarkable that the initial temperature dependence on proves to be almost model independent within the same particle set. Thus we see that in our approach to choosing the initial state the dependence of the initial thermodynamic state on the model for the EoS resides in the value of the maximum reachable baryon density, while the maximum fireball temperature depends only weakly on the employed model. On the right panel in fig. 5 we show the mean velocity of the fireball expansion at the breakup, , as a function of the collision energy in the laboratory system evaluated within our models. We see that values evaluated in KVORcut03 and MKVOR* models prove to be approximately the same.

Taking into account the non-zero particle velocities at the breakup leads to a modification of their spectra. Experimental slopes of the spectra, which are determined by effective temperatures at freeze out, for nucleons are a bit higher than those for pions, cf. Nagamiya:1981sd (). The mentioned difference is attributed to the fact that differential cross sections of massive nucleons are more affected by the presence of non-zero mean expansion velocity than differential cross sections of lighter pions, cf. Siemens:1978pb (). Indeed, in a frame moving with the 3-velocity the particle distribution is expressed through that in the rest frame by a replacement , where is the 4-velocity of the frame. For non-relativistic particles and for the transition to the moving reference frame is reduced to the replacement and nucleon distributions are more affected than pion ones since . In relativistic case in presence of the expansion velocity the particle distributions are characterized by the effective temperatures and by shifted momenta. Below we focus on pion distributions and determine the values and fitting the pion distributions. For we may neglect compared to . Owing to this circumstance and taking into account that a slight decrease of in comparison with is partially compensated by the fact that would be a bit higher, if we took into account an increase of the entropy in a realistic viscous expansion of the fireball, as in Voskresensky:1989sn (); Migdal:1990vm (); Voskresensky:1993ud (), we simplifying put .

The momentum-dependent pion free path length proves to be short for pions with momenta up to the breakup stage and thereby such pions radiate from the fireball breakup Voskresensky:1991uv (). Oppositely, pions with momenta have larger mean-free path Voskresensky:1991uv () and radiate from an intermediate stage of the fireball expansion, cf. Voskresensky:1993mw (); Voskresensky:1995wn (). Moreover, a contribution to the pion yield comes from the decay of thermal -resonances at the breakup, or may be a bit later, if typical time of the reaction at , is larger than for thermal pions. Further we determine the values , from the best fit of the differential pion cross sections for the momenta .

3.2 Description of pion differential cross sections

The differential cross section in inclusive processes reads Senatorov:1989cg (); Voskresensky:1989sn ()


where . The first term in the curly brackets is the contribution of thermal pions at the breakup stage, whereas the second term corresponds to the decay occurring at the breakup stage with direct radiation of free pions. For simplicity we consider collisions of nuclei with equal atomic weights , is the volume of the fireball at the breakup, is the geometric factor for inclusive processes. The numeric coefficient corresponds to all the quantities being measured in units of , in particular is measured in , .

We assume that for during the breakup stage, which lasts for , nucleons and pions decouple and for pions, which before breakup stage were described by thermal distributions, can be considered as freely moving particles. In the quasiparticle model the distribution of pions is described by Senatorov:1989cg (); Voskresensky:1989sn ()


The contribution from appears for and distinguishes the from other pion species. The quantity coincides with , if one uses the vacuum dispersion law, and is given by eq. (11) in the model. In the model of the sudden breakup (if the typical time for the pion sub-system breakup is ) the value of is given by Senatorov:1989cg ()