Temperatures and chemical potentials at kinetic freeze-out in relativistic heavy ion collisions from coarse grained transport simulations

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 3.0 https://creativecommons.org/licenses/by-nc-nd/3.0/ Temperatures and chemical potentials at kinetic freeze-out in relativistic heavy ion collisions from coarse grained transport simulations © 2019 IOP Publishing Ltd. Accepted version (Final draft) Inghirami, Gabriele; Hillmann, Paula Christine; Tomášik, Boris; Bleicher, Marcus Inghirami, G., Hillmann, P. C., Tomášik, B., & Bleicher, M. (2020). Temperatures and chemical potentials at kinetic freeze-out in relativistic heavy ion collisions from coarse grained transport simulations. Journal of Physics G: Nuclear and Particle Physics, 47(2), Article 025104. https://doi.org/10.1088/1361-6471/ab53f4 2020 Journal of Physics G: Nuclear and Particle Physics ACCEPTED MANUSCRIPT Temperatures and chemical potentials at kinetic freeze-out in relativistic heavy ion collisions from coarse grained transport simulations To cite this article before publication: Gabriele Inghirami et al 2019 J. Phys. G: Nucl. Part. Phys. in press https://doi.org/10.1088/13616471/ab53f4 Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © 2019 IOP Publishing Ltd. 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Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address 130.234.162.106 on 07/11/2019 at 08:06 Temperatures and chemical potentials at kinetic freeze-out in relativistic heavy ion collisions from coarse grained transport simulations Gabriele Inghirami1,2, Paula Hillmann3,4,5,6, Boris Tom´aˇsik7,8, and Marcus Bleicher3,4,5,6 1University of Jyv¨askyl¨a, Department of Physics, P.O. Box 35, FI-40014 University of Jyv¨askyl¨a, Finland 2Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland 3Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany 4Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 5GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germany 6John von Neumann Institute for Computing, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany 7Fakulta pr´ırodn´ych vied, Univerzita Mateja Bela, 97401 Bansk´a Bystrica, Slovakia 8Fakulta jadern´a a fyzik´alnˇe inˇzen´yrsk´a, ˇ Cesk´e vysok´e uˇcen´ı technick´e v Praze, 11519 Prague, Czechia (Dated: October 29, 2019) 1 Page 1 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript Abstract Using the UrQMD/coarse graining approach we explore the kinetic freeze-out stage in central Au + Au collisions at various energies. These studies allow us to obtain detailed information on the thermodynamic properties (e.g. temperature and chemical potential) of the system during the kinetic decoupling stage. We explore five relevant collision energies in detail, ranging from √sNN = 2.4 GeV (GSI-SIS) to √sNN = 200 GeV (RHIC). By adopting a standard Hadron Resonance Gas equation of state, we determine the average temperature hTiand the average baryon chemical potential hµBion the space-time hyper-surface of last interaction. The results highlight the nature of the kinetic freeze-out as a continuous process. This differential decoupling is an important aspect often missed when summarizing data as single points in the phase diagram as e.g. done in Blast-Wave fits. We compare the key properties of the system derived by using our approach with other models and we briefly review similarities and differences. 2 Page 2 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript I. INTRODUCTION Heavy ion collisions at ultra-relativistic energies have provided strong evidences [1–5] for a novel phase of Quantum-Chromo-Dynamic (QCD) matter. This novel state of deconfined matter [6, 7] is called the (strongly interacting) Quark-Gluon Plasma (QGP). A large variety of approaches [8–20] have been developed to study the properties of this QCD-medium, allowing to test in detail our understanding of the laws of nature at the subatomic scale. Unfortunately, the tiny dimensions of the QGP system under investigation and its extremely fast evolution make it inaccessible to direct measurements. Therefore, one is constrained - even with the most advanced experimental apparatus - to the detection of hadrons and their momentum distributions at distances many orders of magnitudes larger than the typical size of the colliding ions. Dynamical modelling, however, opens a key hole to explore the intriguing and exciting phenomena happening in the early stages of the collision. Nevertheless, it is clear that this indirect view relies on the quality of the model to consistently and accurately reconstruct the relevant dynamics and phases of the collision from hadron formation to their detection. In this work we want to explore the systems properties during the decoupling stage of the evolution. A similar analysis was e.g. done in [21] in a more ab-initio fashion, however with less realistic initial conditions and only with a schematic expansion and more phenomenologically in [22]. Thus, we focus on the last stage of a heavy ion collision event, the so called kinetic or thermal freeze-out [23, 24], when the hadrons stop to interact with each other and their momentum distribution does not change anymore. This condition is different from the so-called chemical freeze-out [25–27], which, instead, refers to the ceasing of the inelastic scatterings and the stabilization of the abundances of the hadronic species. Although single freeze-out models have been proposed [28–30] and some models estimate [31] a chemical freeze-out temperature Tch close to the kinetic freeze-out temperature Tkin, the two phenomena are conceptually different [32, 33]. Tch is tightly connected with the QGP phase transition [34, 35], it depends on the collision energy [36], but not on the collision centrality class [37] and it has common features in different systems well explained by statistical thermal hadronization models [38–42]. On the other hand, Tkin is more related to the dynamics of the system [23]. The results delivered by the recent versions of the multi-source thermal model [43] and, even more, by the Blast-Wave model [44] heavily depend on the 3 Page 3 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript assumptions about the kinematic properties of the system. In particular, in the Blast-Wave model the kinetic freeze-out temperature, the baryon chemical potential and the transverse velocity are parameters obtained by a fit to a certain phase-space density distribution of hadrons [45–49]. The Blast-Wave model can be quite sophisticated [50] and may take into account the anisotropy of the system [51], but, in any case, it is an approach based on the direct evaluation of macroscopic quantities fitted to experimental data. In this paper we adopt a different perspective. We exploit a microscopic description of the system given by the numerical transport code UrQMD [52, 53] and then we associate to the kinetic freeze-out condition the corresponding macroscopic quantities by using a coarsegraining approach [54, 55]. We define the kinetic freeze-out microscopically as the time and the position in space of the last interaction of a hadron, including not only scatterings, but also decays by strong interaction. Therefore, within the present framework, the freezeout coordinates are given by the dynamics and cross sections of the UrQMD simulation. To relate these freeze-out coordinates to the thermal properties at this space-time point, we compute in a second step the average net-baryon current, the energy density and the net-baryon density, by using a coarse graining procedure. Finally, we employ the Equation of State (EoS) to associate to these quantities the corresponding temperature and baryon chemical potential. For first studies in this respect see e.g. [56, 57]. The current limits of the chosen approach do not compromise the main goal of this study: highlighting the nature of the kinetic freeze-out as a continuous, dynamical process, by exploring the distribution of the kinetic freeze-out parameters at different collision energies. We focus on Au+Au reactions in the 0 −5% centrality class and extract temperature and baryon chemical potentials at midrapidity as a function of transverse momentum and as a function of rapidity. We focus on the most abundant hadron species and postpone a detailed analysis of the difference between hadron species to future follow-up studies. The structure of this article is as follows. In Section II we explain the UrQMD model, the coarse graining approach and the extraction procedure in details. In Section III we present our results on (T,µB) values for different collision energies, fluctuations of the decoupling temperatures and chemical potentials and on the transverse momentum and rapidity dependence of kinetic freeze-out parameters. In Section IV, we summarize the main findings of this study, we review its present limitations and we hint at possible further developments in future works. 4 Page 4 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript II. DESCRIPTION OF THE MODEL The present approach is based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) [52, 53] transport model. UrQMD is employed for two different purposes: to compute the time evolution of the average properties of the system, by exploiting a coarsegraining method, and to determine the space-time coordinates of the kinetic freeze-out coordinates of the hadrons. UrQMD itself is a hadron cascade model that simulates the dynamics of a heavy ion collision based on the covariant propagation of hadrons. Interactions are modelled via the excitation of color flux-tubes (strings) and by further elastic and inelastic interactions of the hadrons. For details, the reader is referred to [52, 53]. The UrQMD coarse-graining method was developed in Refs. [55, 58–61] and used successfully to explore and predict dilepton and photon production from GSI-SIS to RHIC energies as well as to provide underlying events for heavy quark studies. Here we employ the same approach and shortly summarize the main ingredients. In the coarse-graining method one reconstructs thermal parameters based on the approximation of the hadronic distribution function f(x,p, t) as f(x,p, t) = *X h δ(3) (x−xh(t)) δ(3) (p−ph(t))+,(1) by performing averages over the total ensemble of hadrons produced in a large set of heavy ion collision events having the same √sNN energy. These averages are done at each space point at fixed times (with respect to the UrQMD computational frame). The spatial grid for the coarse-graining procedure has a typical resolution of 0.8 fm, except for collisions at √sNN = 200 GeV, for which we use a resolution of 1 fm to slightly reduce memory and disk space usage (see Table I). More precisely, we evaluate the net-baryon four current jµ Bas jµ B(x, t) = 1 ∆V*Nh∈∆V X i=1 Bi pµ i p0 i+,(2) and the energy momentum tensor Tµν as Tµν(x, t) = 1 ∆V*Nh∈∆V X i=1 pµ ipν i p0 i+,(3) 5 Page 5 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript in which ∆Vstands for the cell volume, Biand pµ ifor the baryon number and the µ component of the four momentum of the hadron i, respectively, and the sums are done over all hadrons Nh. Adopting the Eckart’s frame definition [62], we obtain the fluid four velocity uµfrom jµ Bas uµ=jµ B qjξ BjBξ = (γ, γ~v),(4) with uµuµ= 1, γthe Lorentz factor and vthe fluid velocity in natural units (c=~= 1). The baryon density ρBand the energy density ε, as measured in the Local Rest Frame (LRF) of the fluid, can be obtained by a Lorentz transformation of the net-baryon current and of the energy momentum tensor as ρB=j0 B,LRF, ε =T00 LRF.(5) The temperature T(ε, ρB) and the baryon chemical potential µB(ε, ρB) are obtained by interpolation from a tabulated Hadron Resonance Gas EoS [63], having consistently the same degrees of freedom as UrQMD. We accept a coarse-grained cell only, if it contains at least 100 particles (summing over all events), so to reduce the statistical fluctuations. Typically, we are able to determine the corresponding medium average bulk properties for more than 95% of the kinetically frozen-out hadrons, except at √sNN = 200 GeV, where we drop at ≈85%. In line with previous studies, we rescale ρBand εbefore the interpolation step by a correction factor to compensate for the anisotropy of the system along the beam direction [55, 64, 65]. However, this correction term is predominantly active only in the initial stages of the collision and not at the late times at which most of the kinetic freeze-out events happen. We verified for selected cases that its influence on the final results is negligible. The knowledge of the bulk properties of the system obtained by the coarse graining approach at different space-time points is then associated to the microscopic freeze-out distribution in space and time as given by UrQMD. III. RESULTS We simulate central Au+Au collisions with impact parameter b= 0 −3.4 fm, roughly corresponding to the 0 −5% centrality class [66, 67], from √sNN = 2.4,GeV to √sNN = 200 GeV, covering a range of energies relevant for the HADES [68] at GSI, NA49 [69] at 6 Page 6 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript CERN and RHIC/BES [70] at BNL experiments. In Table I we provide the details for the coarse graining simulations. The lists of the kinetic freeze-out points have been obtained by running 104UrQMD events for each collision energy until 200 fm/c. We consider the most abundant and significant hadron species, i.e. pions, kaons, protons, neutrons, lambdas and their antiparticles, including the feed-down of resonance decays. √sNN (GeV) Nev. ∆t(fm/c) ∆x(fm) Nx,y Nztmax (fm/c) 2.4 1.8·1060.5 0.8 70 200 80 4.5 7.4·1050.5 0.8 80 250 90 7.7 6.6·1050.5 0.8 80 250 90 19.6 3.6·1050.5 0.8 86 276 100 200 6.4·1040.5 1.0 200 402 200 Table I. List of the main parameters used in the UrQMD/coarse-graining numerical simulations. We report the values of the collision center-of-mass energy √sNN (GeV), the number of events Nev., the time resolution ∆t(fm/c), the spatial resolution ∆x(fm), the number of cells along in the transverse plane (Nx,y) and in longitudinal direction (Nz), and the time tmax (fm) after the collision at which we stop the simulations. A. Freeze-out time distributions, temperature and baryo-chemical potential variations on the decoupling hyper-surface To set the stage, we begin with the decoupling-time distribution defining the kinetic freeze-out. The studies focus on central rapidities (|y|<0.2) and the time tis defined in the center-of-mass frame starting from the beginning of the collision. Fig. 1 shows the decoupling probability (i.e. the normalized time distribution of the decoupling distribution) of the hadrons in central Au+Au reactions from 2.4 GeV to 200 GeV. The peak of the decoupling time is typically between 10 and 25 fm/c. The duration of the decoupling stage lasts typically 15−20 fm/c (FWHM) (resulting in a damping rate Γ(tmax) = 30 −40 MeV) indicating that the kinetic freeze-out happens within a quite broad interval of time. It is interesting to note that the results are in line with the Kadanoff-Baym equation based analysis by Knoll [21]. The position of the emission peak is governed to first approximation by two effects: I) the 7 Page 7 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 3 Au+Au, b<3.4fm, UrQMD/cg <µB/T> pT [GeV] ECM=2.4 GeV ECM=4.5 GeV ECM=7.7 GeV ECM=19.6 GeV ECM=200 GeV Figure 6. (Color online) Average of the ratio hµB/Tiat kinetic freeze-out as a function of transverse momentum pTat midrapidity (|y|<0.2) for central Au+Au reaction at center-of-mass energies of √sNN = 2.4,4.5,7.7,19.6,200 GeV (full line, short dashed line, dashed line, long dashed-dotted line, dotted dashed line). with respect to transverse momentum spectra at midrapidity (|y|<0.05), then with respect to the pseudorapidity, using a non-boost invariant Blast-Wave model, up to a maximum value which depends on the beam rapidity, which, of course, in turn depends on the collision energy. We refer to Ref. [50] for the details. The data of the kinetic freeze-out at √sNN = 7.7 and 19.6 GeV come from Ref. [48] and they have been obtained by a simultaneous fit with a Blast-Wave model with |y|<0.1 of π±(0.5< pT<1.3 GeV), K±(0.24 < pT<1.4 GeV), pand ¯p(0.4< pT<1.3 GeV). The authors of this study excluded other particle species to avoid the consequent implicit assumption that all hadrons share the same kinetic freeze-out temperature. The authors also imposed limits on the transverse momentum selected for the fits. On the low pTend, this restriction was motivated by the issues with the resonance decays, while, on the high pTend, the hydrodynamic models underlying the Blast-Wave model is not adequate to describe hard processes [48]. The data at √sNN = 200 GeV are taken from Ref. [46] and also refer to a fit with a Blast-Wave model of π±,K±,pand ¯pat midrapidity (|y|<0.1), without considering the pion spectra for pT<0.5 GeV. Our 14 Page 14 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 20 40 60 80 100 120 -4 -3 -2 -1 0 1 2 3 4 Au+Au, b<3.4fm, UrQMD/cg <T> [MeV] y (rapidity) ECM=2.4 GeV ECM=4.5 GeV ECM=7.7 GeV ECM=19.6 GeV ECM=200 GeV Figure 7. (Color online) Average kinetic freeze-out temperature hTias a function of rapidity yfor central Au+Au reaction at center-of-mass energies of √sNN = 2.4,4.5,7.7,19.6,200 GeV (full line, short dashed line, dashed line, long dashed-dotted line, dotted dashed line). model of kinetic freeze-out incorporates the resonance feed-down, therefore their contribution should be quantitatively assessed for a detailed comparison between the two models, which, nevertheless, is out of the scope of the present work. Given the small abundance of hadrons at high pTwith respect to those at low pT, we are less concerned by a possible bias introduced by them. The data regarding the chemical freeze-out temperature between √sNN = 2.7 and 4.3 GeV are taken from Ref. [74], while at the remaining reaction energies they are from Ref. [48]. Let us first compare our kinetic freeze-out temperatures with the chemical freezeout temperatures from the Statistical Model fits. We observe that at low collision energies (√sNN = 7 GeV), kinetic and chemical freeze-out are only separated by a small temperature difference on the order of 5-10 MeV, nicely consistent with a very short duration of the expansion phase. At energies above √sNN = 7 GeV the chemical freeze-out temperature is substantially above the kinetic decoupling temperature (∆T > 40−50 MeV). This indicates a rather strong expansion flow of the system from chemical to kinetic freeze-out at high energies. If we compare the kinetic freeze-out temperature from the present study to the kinetic freeze-out temperatures obtained from Blast-Wave fits, we observe that, apart from 15 Page 15 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 -4 -3 -2 -1 0 1 2 3 4 Au+Au, b<3.4fm, UrQMD/cg <µB> [MeV] y (rapidity) ECM=2.4 GeV ECM=4.5 GeV ECM=7.7 GeV ECM=19.6 GeV ECM=200 GeV Figure 8. (Color online) Average baryo-chemical potential hµBiat kinetic freeze-out as a function of rapidity yfor central Au+Au reaction at center-of-mass energies of √sNN = 2.4,4.5,7.7,19.6,200 GeV (full line, short dashed line, dashed line, long dashed-dotted line, dotted dashed line). the point at √sNN = 19.6,GeV, there is a tension between the results of the two approaches, with the Blast-Wave fits suggesting a substantially lower kinetic freeze-out temperature than obtained in the present study, in particular at high collision energy. We relate this difference to the hadronic dynamics that leads to weaker transverse expansion than observed in the data. For the present investigation we employ UrQMD without a hydrodynamic/QGP stage to avoid to introduce an additional parameter, the “particlization” temperature [75], whose proper exploration would require a rather strong computational effort, made heavier by the longer time needed to run UrMQD in hydbrid mode compared to cascade mode. At low collision energy, the discrepancy might be due to the exclusion of the hadrons with low pTin the Blast-Wave fits. The inadequacy of our chosen EoS to describe a system out of chemical equilibrium might introduce a bias, as well. Further investigations to understand the differences between our results and those coming from the Blast-Wave model will be addressed in a future study, probably including a careful evaluation of the bias introduced in the fits by the selection of the pTintervals and the adoption of a different EoS. 16 Page 16 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 2 4 6 8 10 12 14 16 -4 -3 -2 -1 0 1 2 3 4 Au+Au, b<3.4fm, UrQMD/cg <µB/T> y (rapidity) ECM=2.4 GeV ECM=4.5 GeV ECM=7.7 GeV ECM=19.6 GeV ECM=200 GeV Figure 9. (Color online) Average of the ratio hµB/Tiat kinetic freeze-out as a function of rapidity yfor central Au+Au reaction at center-of-mass energies of √sNN = 2.4,4.5,7.7,19.6,200 GeV (full line, short dashed line, dashed line, long dashed-dotted line, dotted dashed line). Under this perspective, one should not forget that the representation of the kinetic freezeout as a single point in the phase diagram is indeed a convenient way to summarize its key properties, but, at the same time, it is also an oversimplification. For example, Fig. 12 shows the density of the kinetic freeze-out parameters in the (T, µB) plane for central Au+Au reactions at √sNN = 19.6 GeV. One clearly observes that different parts of the system decouple at different (T, µB) points. In addition a correlation between hµBiand hTiis present. Such a spread in parameter space is at the moment not included in the present Blast-Wave fits and might yield different results than in the standard Blast-Wave approach. IV. SUMMARY AND CONCLUSIONS In this work we studied the kinetic freeze-out process with the UrQMD/coarse-graining approach [52, 53, 55]. First, we performed a large series of UrQMD simulations to compute the average temperature and baryon chemical potential of the system during its evolution on a coarse-grained grid. Then we determined the time and the position of the points of the last interaction of the most abundant hadron species. These space-time points of last 17 Page 17 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 20 40 60 80 100 120 140 0 100 200 300 400 500 600 700 800 2.4 GeV 4.5 GeV 7.7 GeV 19.6 GeV 200 GeV Au+Au, b<3.4fm, UrQMD/cg <T> [Mev] <µB> [MeV] Figure 10. (Color online) Kinetic freeze-out temperature with respect to the baryon chemical potential in Au+Au reactions at different center-of-mass energies in the rapidity range |y|<0.2. interactions, which include both scatterings and decays by strong interaction, are what we defined as kinetic freeze-out hyper-surface. Afterwards, we associated to these last interaction points the corresponding values of the coarse-grained cell in which they were located. We focused on Au+Au collisions in the centrality class 0 −5%, i.e. with a Glauber model impact parameter b < 3.4 fm, considering five reaction energies: √sNN = 2.4,4.5,7.7,19.6 and 200 GeV. We evaluated the probability distributions for particle emission in time and the probability for the emission at a given temperature and chemical potential. In general, we found that these distributions are rather broad. These results are consistent with the concept of kinetic freeze-out as continuous process which happens at very different space-time points, due to the complex dynamics of the system and due to the different (and energy dependent) cross sections of the hadrons. With increasing collision energy, the average freeze-out times tend to decrease, the average freeze-out temperatures become higher and the average baryon chemical potential decreases. We evaluated also how the average baryon chemical potential, the average temperature and the average of the ratio between the two vary with respect to the transverse momentum and to the rapidity. We found that these average values are 18 Page 18 of 24AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript 0 20 40 60 80 100 120 140 160 180 1 10 100 T [MeV] √sNN [GeV] Tkin UrQMD/cg Tkin Blast Wave Tchem Stat. Model Figure 11. (Color online) Comparison between the average kinetic freeze-out temperature determined in the present study, the kinetic freeze-out temperature obtained from Blast-Wave model fits (Refs. [46, 48, 50]) and the chemical freeze-out temperatures obtained from the Statistical Hadronization model fits (Ref. [48], table VIII, GCER, and Ref. [74]), with respect to the reaction energy. The calculation and the data refer to central Au+Au reactions. We converted the reaction energies of fixed target experiments from Elab to √sNN . essentially independent of rapidity and transverse momentum. Finally, we presented the set of the average temperature and baryon chemical potential points at kinetic freeze-out at the various collision energies under investigation, comparing our results with those coming from Blast-Wave model fits for the kinetic freeze-out temperature and from the Statistical Hadronization model for the chemical freeze-out temperature. We found that the kinetic freeze-out points in the (T, µB) plane follow a regular pattern, from higher to lower baryon chemical potential and from lower to higher temperature as the reaction energy grows, similar to the curve described by the chemical freeze-out points, albeit with different values. We observe some deviations between the results from the BlastWave fits, in which the kinetic freeze-out temperature at high collision energy seems to slightly decrease, which might be interesting to investigate more into depth in future studies. Moreover, we found also a disagreement in the low collision energy region which might 19 Page 19 of 24 AUTHOR SUBMITTED MANUSCRIPT - JPhysG-103004.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Accepted Manuscript Figure 12. (Color online) Profile of the kinetic freeze-out temperature and baryon chemical potential at |y|<0.2 for Au+Au collisions at √sNN = 19.6 GeV. be important to understand, giving the rapid variation of the key thermodynamical properties of the system in that reaction energy region, slightly below the bottom end of the BES-II program [76] and in the range of the upcoming FAIR [77] and NICA [78] facilities. We concluded by showing a density plot of the freeze-out parameters at √sNN = 19.6 GeV to provide the evidence that the common representation of the kinetic freeze-out as a single, well defined point in the phase diagram hides its real nature as a continuous process across many different thermodynamical conditions. ACKNOWLEDGMENTS We gratefully acknowledge Stephan Endres for providing the coarse-graining numerical code which served as a basis for the present work. We sincerely thank the Referees for helping us in improving the quality of the manuscript. G. 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