Medium effects on freeze-out of light clusters at NICA energies
G. Roepke, D. Blaschke, Yu. B. Ivanov, Iu. Karpenko, O. V. Rogachevsky, H. H. Wolter
aa r X i v : . [ nu c l - t h ] D ec Medium effects on freeze-out of light clusters at NICA energies
G. R¨opke,
1, 2, ∗ D. Blaschke,
2, 3, 4, † Yu. B. Ivanov,
2, 3, 5, ‡ Iu. Karpenko, § O. V. Rogachevsky, ¶ and H. H. Wolter ∗∗ Institut f¨ur Physik, Universit¨at Rostock, Rostock, Germany National Research Nuclear University (MEPhI), Moscow, Russia Joint Institute for Nuclear Research (JINR), Dubna, Russia Institute of Theoretical Physics, University Wroclaw, Poland National Research Center ”Kurchatov Institute”, Moscow, Russia SUBATECH, Universit´e de Nantes, Nantes, France Fakult¨at f¨ur Physik, Universit¨at M¨unchen, M¨unchen, Germany (Dated: September 21, 2018)
Abstract
We estimate the chemical freeze-out of light nuclear clusters for NICA energies of above 2 AGeV. On the one hand we use results from the low energy domain of about 35 A MeV, wheremedium effects have been shown to be important to explain experimental results. On the highenergy side of LHC energies the statistical model without medium effects has provided results forthe chemical freeze-out. The two approaches extrapolated to NICA energies show a discrepancythat can be attributed to medium effects and that for the deuteron/proton ratio amounts to afactor of about three. These findings underline the importance of a detailed investigation of lightcluster production at NICA energies.
PACS numbers: 21.65.-f, 21.60.Gx, 25.75.-q, 05.30.-dKeywords: Relativistic heavy-ion collisions, Nuclear Matter, Cluster models, QS mechanics ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] . INTRODUCTION The particle production measured in heavy ion collisions (HIC) is of interest to infer theproperties of dense matter. The description of the time evolution of the fireball produced inHIC demands a non-equilibrium approach to describe the time dependence of the distributionfunction of the observed products, which are mainly neutrons, protons, and clusters at lowenergies, but also mesons, hyperons and antiparticles at high energies. Different transportcodes have been developed to describe the time evolution of the fireball, but the formationof bound states (clusters) remains an open problem and semi-empirical assumptions such asthe coalescence model have been applied.In contrast, the composition of dense matter is well investigated in equilibrium. Thechemical freeze-out concept assumes that the system is approximately in equilibrium aslong as collisions are sufficiently frequent to establish the corresponding distributions. Foran expanding fireball this is no longer the case at a critical density, so that the chemicalequilibrium freezes out at the corresponding parameter values for temperature T , baryonnumber density n B , and proton fraction Y p . This concept has been proven to be an appro-priate starting point to describe HIC at moderate laboratory energies E lab = 35 A MeV [1]that have been analyzed in this scheme in Ref. [2], but also up to highest energies providedby heavy-ion collisions at the LHC [3, 4].New facilities such as FAIR and NICA are under construction to investigate the regionbetween these limiting domains, i.e. at temperatures of about 100 MeV and densities ex-ceeding the saturation density. Of interest are the yields of particles and light clusters likeneutrons (n), protons (p), deuterons ( H, d), tritons ( H, t), helions ( He, h), and α -particles( He). We discuss here what can be expected for this intermediate region, using a quantumstatistical (QS) model, as discussed in Ref. [5].The freeze-out concept can only be considered as an approximation to describe disassem-bling matter. It has the advantage that correlations and bound state formation are correctlydescribed within a QS approach. For a non-equilibrium theory, the equilibrium is a limitingcase, and even more, the quasi-equilibrium (generalized Gibbs ensemble) serves to define theboundary conditions for the non-equilibrium evolution, see Ref. [6].As an intermediate step to full dynamical calculations the freeze-out concept has recentlyby combined with hydrodynamical calculations within the framework of the recently devel-2
200 400 600 800 1000 µ B [MeV]10100 T c f [ M e V ] NatowitzCleymansAndronicQS, n B = 0.03 fm -3 FIG. 1: Freeze-out temperature as function of the baryon chemical potential. The fit of Eq.(1)according to Cleymans et al. [10] (green dash-double-dotted line), is compared with the fit of Eq.(3)of Andronic et al. [3] (red dashed line). The star denotes the data point according to Ref. [3] withlowest T . Also shown is the freeze-out temperature derived from HIC at moderate energies [2](black solid line). A calculation in the QS model with fixed freeze-out density n B = 0 .
03 fm − (blue dash-dotted line) is also shown. oped THESEUS event generator [7] to produce more realistic freeze-out conditions. Thesuccess of a local coalescence approach within the three-fluid-hydrodynamical (3FH) model[8] in reproducting the rapidity distributions of light fragments measured by the NA49 col-laboration [9] at SPS energies is encouraging. The local coalescence is in fact the same localthermal model where only the overall normalization is a free parameter. The difference ofthis overall normalization from that predicted by the free-hadronic-gas model may indicatemedium effects in the light-fragment production. II. FREEZE-OUT PARAMETRIZATIONS
The freeze-out concept is surprisingly well appropriate to describe experiments at highenergies [10]. An empirical relation T Cleymans cf has been given to calculate it as function of3he baryon chemical potential µ B , T Cleymans cf GeV = 0 . − . (cid:16) µ B GeV (cid:17) − . (cid:16) µ B GeV (cid:17) (1)with µ B GeV = 1 . . √ s NN / GeV (2)and √ s NN = p m N E lab + 2 m N , m N = 0 .
939 GeV. More recently in Ref. [3], using newdata from LHC, another fit has been proposed: T Andronic cf = 158 . . − ln( √ s NN / GeV) / . , (3)with µ B = 1307 . . √ s NN / GeV (4)The two parametrizations are compared in Fig. 1 and are seen to agree well for T cf > T ≈ µ B ≈
760 MeV). At low temperatures, evidently, the fit of Andronic et al. [3] is notapplicable. Freeze-out parameter values relevant for NICA energies are shown in Tab. I. E lab [GeV] √ s NN [GeV] T Cleymans cf [MeV] µ Cleymans B [MeV] T Andronic cf [MeV] µ Andronic B [MeV]2 2.35 56.38 796.8 52.513 779.763.85 3 79.956 719.07 72.93 701.459.84 4.5 111.8 586.94 107.32 569.47TABLE I: Freeze-out parameter values relevant for NICA energies according to Cleymans et al.[10] and Andronic et al. [3]. Freeze-out conditions are also well investigated experimentally at rather low laboratoryenergies, e.g., by Natowitz et al. [2]. The time evolution of the expanding fireball is deducedfrom the velocity of the emitted particles together with coalescence models. As seen in Fig.1, the fit T Cleymans cf , Eq. (1), meets nicely the low-temperature data of Ref. [2]. However, thelaw of mass action (LMA) [11], also denoted as nuclear statistical equilibrium (NSE), wasfound to be not sufficient to explain the data, and medium effects have to be considered.This was achieved in the QS approach, described in the next section. Alternatively, an4 B [fm -3 ]020406080 T c f [ M e V ] NatowitzQS, n B = 0.03 fm -3 CleymansAndronic
FIG. 2: Freeze-out temperature as function of the baryon number density. Notations as in Fig. 1. excluded volume concept [12] has been used to include medium effects in a semi-empiricalway. An attempt to reproduce the parametrized chemical freeze-out line in the QCD Phasediagram from a kinetic condition has been made in Ref. [13], involving chiral symmetryrestoration and deconfinement.
III. FREEZE-OUT DENSITIES
The determination of the baryon freeze-out density is not simple, as already known fromHIC at moderate energies. In contrast to the freeze-out temperature which is well describedby the yield ratios of different emitted particles, the freeze-out density is very sensitive tothe chemical potential and the considered approximation.Th treat this problem a quantum statistical (QS) approach including light clusters [14]is used here, which includes medium effects of nucleons and light clusters due to Pauliblocking and self energy shifts [14, 15]. The single-nucleon quasiparticle shift was takenaccording to the density-dependent relativistic mean-field approach (DD2-RMF) of Typel[16]. The nucleons i = n, p with rest masses m i are treated as quasiparticles of energy E i ( p ) = p p + ( m i − S i ) + V i , with scalar and vector potentials, S i and V i , which dependon density, proton/neutron asymmetry and temperature. A low density expansion of thesepotentials, which is useful in the present context, is given in the appendix.5
10 20 30 40 50 60 70 80T cf [MeV]10 -6 -5 -4 -3 -2 n i [f m - ] np d t h α FIG. 3: Densities of different constituents as a function of the freeze-out temperature calculated inthe QS model for different choices of the chemical potential. Dashed lines: chemical potential ac-cording to Cleymans et al. [10], stars according to the lowest temperature value given by Andronicet al. [3]. Full lines: chemical potential from the QS model at fixed baryon density n B = 0 . − . The freeze-out densities at low temperatures are determined by the liquid-gas phasetransition. Above the critical temperature of the liquid-gas phase transition of about 12MeV the freeze-out density seems to remain nearly constant. A value n B ≈ .
03 fm − seemsto be reasonable. From Ref. [4] this can be considered to be representative for the freeze-outat the NICA energies. But, as shown there, the freeze-out realistically dependes on densityand temperature.Using different expressions for the baryonic chemical potential at given temperature, thefreeze-out temparature as a function the baryon density is shown in Fig. 2. Also the resultof Natowitz et al. [2] is given.Here, dynamical transport simulations would be helpful to support the interpretation ofthe data with statistical models. The calculations using the coalescence model are of interestas also used in [2]. Also the treatment with the the combined hydrodynamical and statisticalmodel [8] will give further insight. 6
10 20 30 40 50 60 70 80T cf [MeV]00,20,40,60,8 n d / n p fit d/pQS, n B = 0.03 fm -3 CleymansAndronicNatowitz
FIG. 4: Deuteron to proton fraction at freeze-out temperature. Shown is the fit of Eq.5 (orange,dot-double-dash). The meaning and notations of the other curves are as in Fig. 1.
IV. COMPOSITION
The composition of the matter at freeze-out is shown in Fig. 3 as calculated in the QSapproach with different choices of the chemical potential. It is seen that the constant-densityQS calculations matches fairly well with chemical potential taken from the parametrizationof Cleymans et al. [10], as one could have already expected from the results shown in Fig. 1.This should also match with the low-energy results of Natowitz et al. [2].The d/p ratio is shown in Fig. 4. A fit formula [17] d/p = 0 . √ s NN / GeV] − . + 0 . √ s NN ∼ . CONCLUSIONS HICs are non-equilibrium processes. The yields of particles and clusters should ideally bedescribed by a transport approach which includes also the formation, propagation, and colli-sions of bound states. The freeze out approach may be considered as a step in this direction,by including the many-body correlations in a correct way. It describes the situation whererelaxation to equilibrium is fast compared with the time evolution of the thermodynamicalparameters. This way, it serves a an ingredient (source term in the sense of the Zubarevapproach) to solve the dynamical evolution of the non-equilibrium system.The results obtained from the freeze-out concept are surprisingly good. At high energiesas well as at low energies, the applicability of this concept has been demonstated. Here weare interested to combine both limiting cases to obtain results for the intermediate region.In particular, the NICA facility is appropriate to investigate this region.The density is very sensitive to the value of the chemical potential, and the determina-tion of the freeze-out density is a issue of future discussions. Similarly, the composition isalso very sensitive to the thermodynamic parameters as well as the treatment of mediumeffects. Whereas the role of in-medium effects is clearly shown for freeze-out densities atlow temperatures, the influence of in-medium effects at intermediate temperatures such asrelevant for the NICA experiments is under discussion. This concerns in particular the ratioof deuterons to protons. We point out that this may be an important issue of future ex-periments at NICA. We also plan to address this problem within the framework of recentlydeveloped THESEUS event generator [7] that provides more realistic freeze-out conditions.
Acknowledgements
This work was supported by the Russian Science Foundation grant No. 17-12-01427.
A. Appendix: Low-density expansion
A density dependent RMF model was considered in Ref. [16]. The following low-densityexpansions are derived from this model and reproduce the DD-RMF results below the baryondensity n ≤ . − within 0.1 %. Variables are the total baryon density n = n totn + n totp in units of fm − , the asymmetry δ = ( n totn − n totp ) /n , and the temperature T in MeV. These8xpressions update the expansions given in the appendix of Ref. [18]. For the scalar part ofthe DD2-RMF we use the fit S i ( T, n B , Y p ) = (4463 . − . T − . δ + 4 . δ ) n B × c n b + c n B c n b + c n B ,c = 0 . − . T − . δ − . δ ,c = 15 . . T − . δ − . δ ,c = 24 . − . T − . δ + 1 . δ ,c = 114 . . T + 2 . δ + 0 . δ , (6)where δ = (1 − Y p ), and i = n, p . For the vector part one obtains V p ( T, n B , Y p ) = (3403 .
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