Probing chemical freeze-out criteria in relativistic nuclear collisions with coarse grained transport simulations
EEPJ manuscript No. (will be inserted by the editor)
Probing chemical freeze-out criteria in relativistic nuclearcollisions with coarse grained transport simulations
Tom Reichert , , Gabriele Inghirami , , and Marcus Bleicher , , , Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Helmholtz Research Academy Hesse for FAIR, Campus Frankfurt, Max-von-Laue-Str. 12, 60438 Frankfurt, Germany University of Jyv¨askyl¨a, Department of Physics, P.O. Box 35, FI-40014 University of Jyv¨askyl¨a, Finland Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt , Germany John von Neumann-Institut f¨ur Computing, Forschungszentrum J¨ulich, 52425 J¨ulich, GermanyReceived: July 14, 2020/ Revised version:
Abstract
We introduce a novel approach based on elastic and inelastic scattering rates to extract thehyper-surface of the chemical freeze-out from a hadronic transport model in the energy range fromE lab = 1 .
23 AGeV to √ s NN = 62 . (cid:104) E (cid:105) / (cid:104) N (cid:105) ≈ s/T = 7 and n B + n ¯B ≈ .
12 fm − are limited to higher collision energies. PACS.
The collision of heavy ions in today’s largest particle ac-celerators provides an excellent tool to explore nuclear andsub-nuclear matter under extreme conditions as they oc-cur e.g. in neutron stars, around black holes or in theearly universe. Matter created under these conditions sus-tains tremendous temperatures, pressures and densities involumes on the order of ∼ over timescales of10 − s.Since separating quarks creates new quark-anti-quarkpairs from the vacuum in order to bind to color neutralhadrons, a direct measurement of the inner degrees of free-dom of strongly interacting matter in heavy ion collisionsis, unfortunately, still impossible. In contrast to QED, thisphenomenon called confinement forbids perturbative cal-culations at small momenta. Access to the early and inter-mediate stage of a collision can be gained via electromag-netic probes e.g. with real photons and virtual photonsin the di-lepton channel [1,2,3], via the study of hadronsemitted at the chemical freeze-out [4,5], or by flow observ-ables, like v , v , v , · · · .Typically, the final state of a heavy ion collision in-volves two parts: 1.) the chemical freeze-out and 2.) thekinetic (or thermal) freeze-out. At the chemical freeze-out inelastic flavour changing reactions cease (particle yieldsget fixed) and at the kinetic freeze-out also elastic reac-tions cease (particle 4-momenta get fixed) and the systemdecouples. This behavior is also reflected in the scatter-ing rates as shown in [6] where the chemical freeze-out isestimated to occur at τ chem = 6 fm. It has been shown,that the kinetic freeze-out proceeds in a more continuousfashion [7] than being an instantaneous process. Assum-ing thermal equilibrium, particle spectra or ratios can befitted with statistical and blastwave models to extract ki-netic or chemical freeze-out properties like the tempera-ture T and the baryo-chemical potential µ B . Thermal fitsof particle ratios resulting from CERN/SPS [8,9,10,11],BNL/AGS [12,13,14] and GSI/SIS [15,16] measurementsled to a unified description of the chemical freeze-out linevia a hadronic gas model at a constant energy per particleof (cid:104) E (cid:105) / (cid:104) N (cid:105) = 1 GeV [17]. Besides this, other quantitieswere proposed to define the chemical freeze-out, e.g. a con-stant total baryon density n B + n ¯B = 0 .
12 fm − [18] or aconstant entropy per temperature s/T = 7 [19].The present investigation introduces a novel approachto determine the chemical freeze-out hyper-surface directlyfrom a combined UrQMD/coarse-grained framework. Thefull time evolution of Au+Au collisions in the GSI/FAIRand RHIC BES-II energy regime is analyzed for this pur- a r X i v : . [ nu c l - t h ] J u l T. Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations pose and every final state pion is traced back to the space-time point of its creation. These space-time points definethe hyper-surface of the chemical freeze-out of the pions.We focus on pions, because they are the most abundanthadrons and they are produced in sufficient amount at allinvestigated energies. To obtain the thermodynamic pa-rameters ( T , µ B ) on this hyper-surface, the UrQMD datais coarse grained and supplemented by a Hadron Reso-nance Gas EoS [20]. The energy dependence of the calcu-lated thermodynamic variables is then investigated to testdifferent suggested criteria for the chemical freeze-out. The present study uses the Ultra-relativistic QuantumMolecular Dynamics (UrQMD) [21,22] transport model incascade mode. UrQMD is used to compute both the bulkevolution of the system and to determine the space-timecoordinates of the chemical freeze-out of the hadrons. Werecall that UrQMD is a hadron cascade model that simu-lates the dynamical evolution of heavy ion collision eventsby following the propagation of the individual hadrons,modeling their interactions via the excitation of color flux-tubes (strings) and by further elastic and inelastic scat-terings. A transition to a deconfined stage is not explicitlyincluded in the cascade mode employed here.
The UrQMD coarse-graining approach [7,23,24,25,26,27]consists in computing the temperature and the baryonchemical potential from the average energy-momentumtensor and net baryon current of the hadrons formed in alarge set of heavy ion collision events with the same colli-sion energy and centrality. The computation is done in thecells of a fixed spatial grid at constant intervals of time.In the present study, the cells are four-cubes with spatialsides of length ∆x = ∆y = ∆z = 1 fm and ∆t = 0 .
25 fmlength in time direction. First, we evaluate the net-baryonfour current j µ B as j µ B ( t, r ) = 1∆ V (cid:42) N h ∈ ∆V (cid:88) i =1 B i p µi p i (cid:43) , (1)and the energy momentum tensor T µν as T µν ( t, r ) = 1 ∆V (cid:42) N h ∈ ∆V (cid:88) i =1 p µi p νi p i (cid:43) , (2)in which ∆V is the volume of the cell, B i is the baryonnumber and p µi stands for the µ component of the fourmomentum of the hadron i . The sums run over all hadrons N h in the cell. We adopt the Eckart’s frame definition [28]and we obtain the fluid four velocity u µ from j µ B as u µ = j µ B (cid:112) j ν B j B ν = ( γ, γ v ) , (3) in which γ is the Lorentz factor and v the fluid velocityin natural units ( c = (cid:126) = 1). By a Lorentz transformationof the net-baryon current and of the energy momentumtensor in the Local Rest Frame (LRF) of the fluid, wecompute the baryon density ρ B and the energy density ε as: ρ B = j , LRF , ε = T . (4)Often the chemical freeze-out occurs in cells with an an-isotropy between the pressure in the parallel ( P (cid:107) ) and inthe transverse direction ( P ⊥ ), with respect to the beamaxis. To take into account this condition, we rescale ε [29,30,24,31] as: ε corr = ε/r ( χ ) , (5)where χ = ( P ⊥ /P (cid:107) ) / and r ( χ ) = χ − / (cid:20) χ Artanh √ − χ √ − χ (cid:21) , if χ < χ − / (cid:20) χ Artanh √ χ − √ χ − (cid:21) , if χ > . (6)This corrections was also applied in all our previous stud-ies [24,25,26,27]. The final step in the coarse graining pro-cedure consists in associating to each cell of the coarsegrained grid the temperature T ( ε corr , ρ B ) and the baryonchemical potential µ B ( ε corr , ρ B ) through the interpolationof a tabulated Hadron Resonance Gas EoS [20]. To test this novel way of tracking down the chemical freeze-out coordinates in a full transport simulation we focus onpions. The reason for this is twofold: a) pions are veryabundant hadrons in the investigated energy regime andb) pions are stable particles under strong interactions,both facts simplify the reconstruction of the chemical freeze-out coordinates with high accuracy. When does a π freeze-out chemically? Typically, pions are either produced di-rectly in a string decay (dominant at higher energies) orvia N + N → N + ∆ , and subsequently ∆ → N + π reactions. Of course the ∆ can be replaced by other res-onances. In addition cascades like ∆ → N + ρ , and sub-sequently ρ → ππ are possible. Not all pions producedinitially make it to the final state due to absorption pro-cesses, e.g. π + N → N ∗ → K + Λ . To extract the space-time point of the production of a finally observed π , wefollow all observed pions backwards through the evolutionuntil we reach their point of production. This defines thechemical freeze-out coordinates ( t, r ) for each individualpion. The results are obtained by analyzing central Au+Au col-lisions calculated with the UrQMD model. The simula-tions are used as input to extract the chemical freeze-outcoordinates and calculate the fields ( T ( t, r ) , µ B ( t, r )) with . Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations 3 the coarse-graining procedure. Central events are selectedvia an impact parameter cut at b max = 3 . | z | ≤ Let us start with the distribution of the chemical freeze-out times at a specific energy to first analyze the influenceof the different reconstruction scenarios. For this purposewe show the contributions to the full chemical freeze-outdistribution at √ s NN = 19 . π distribution is shown as a solid black line. Itconsists of pions that are created hidden in carriers (e.g. ∆ , ρ , etc.), shown as red lines and those created directly aspions, shown as blue line. In general the full distributionshows two local maxima: one large maximum centered at ≈ ≈
10 fm. Thefirst peak consists mainly of hadrons created in string exci-tation reactions, while the second bump consists of decaysof resonances and secondary strings. To better interpret t [fm] d N / d t [ f m ] Au+Au (UrQMD/cg) s NN = 19.6 GeVb 3.4 fm|z| 5 fmAll created Hidden created in decayHidden created in stringVisible created in string Figure 1. [Color online] Contributions to the full chemicalfreeze-out distribution of π having their last interaction in | z | ≤ π are shown in red and visible π are shownin blue, while dashed lines represent decays and dotted linesstand for string reactions. the appearing structure, we now show in Fig. 2 the chemi-cal freeze-out distributions of π mesons reconstructed withthe presented algorithm at | z | ≤ lab = 1 .
23 AGeV (yellow), E lab = 4 . lab = 10 . √ s NN = 7.7 GeV (cyan), √ s NN = 9.1 GeV (blue), √ s NN = 11.5 GeV (magenta), √ s NN = 14.5 GeV (purple), √ s NN = 19.6 GeV (pink), √ s NN = 27 GeV (red), √ s NN = 39 GeV (brown) and √ s NN = 62.4 GeV (black) in central Au+Au collisions( b ≤ . ≈ ∆ or the ρ . This findingvalidates therefore that the conceptual idea of an instan-taneous chemical freeze-out is to some extent reasonablewhile the kinetic freeze-out studied in e.g. [7] is shown tobe a more continuous and spread-out process. The aver- t [fm] d N / d t [ f m ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm E lab = 1.23 AGeVE lab = 4.0 AGeVE lab = 10.8 AGeV s NN = 7.7 GeV s NN = 9.1 GeV s NN = 11.5 GeV s NN = 14.5 GeV s NN = 19.6 GeV s NN = 27.0 GeV s NN = 39.0 GeV s NN = 62.4 GeV Figure 2. [Color online] Distribution of the chemical freeze-out times of π ’s reconstructed with the presented algorithm at | z | ≤ lab = 1 .
23 AGeV (yellow), E lab = 4 . lab = 10 . √ s NN = 7.7 GeV(cyan), √ s NN = 9.1 GeV (blue), √ s NN = 11.5 GeV (magenta), √ s NN = 14.5 GeV (purple), √ s NN = 19.6 GeV (pink), √ s NN =27 GeV (red), √ s NN = 39 GeV (brown) and √ s NN = 62.4 GeV(black) in central Au+Au collisions ( b ≤ . age chemical freeze-out times τ chem can be found in Tab.1 stating that the chemical freeze-out in the RHIC BES-IIenergy regime occurs at ≈ T. Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations
Now that the freeze-out four-coordinates are determined,the question arises how the coarse-grained temperaturesand the baryon chemical potentials will be distributed.For this we show in Fig. 3 the distribution of the coarse-grained chemical freeze-out temperatures of pions with | z | ≤ lab = 1 .
23 AGeV(yellow), E lab = 4 . lab = 10 . √ s NN = 7.7 GeV (cyan), √ s NN = 9.1 GeV(blue), √ s NN = 11.5 GeV (magenta), √ s NN = 14.5 GeV(purple), √ s NN = 19.6 GeV (pink), √ s NN = 27 GeV (red), √ s NN = 39 GeV (brown) and at √ s NN = 62.4 GeV (black)from UrQMD as well as we show the distributions of thebaryon chemical potential at the chemical freeze-out inFig. 4. We observe two major features in the distribu- T [MeV] / N d N / d T [ M e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm E lab =1.23 AGeVE lab =4.0 AGeVE lab =10.8 AGeV s NN =7.7 GeV s NN =9.1 GeV s NN =11.5 GeV s NN =14.5 GeV s NN =19.6 GeV s NN =27.0 GeV s NN =39.0 GeV s NN =62.4 GeV Figure 3. [Color online] Distribution of the chemical freeze-out temperatures of π ’s reconstructed with the presented al-gorithm at | z | ≤ lab = 1 .
23 AGeV (yellow), E lab =4 . lab = 10 . √ s NN =7.7 GeV (cyan), √ s NN = 9.1 GeV (blue), √ s NN = 11.5 GeV(magenta), √ s NN = 14.5 GeV (purple), √ s NN = 19.6 GeV(pink), √ s NN = 27 GeV (red), √ s NN = 39 GeV (brown)and √ s NN = 62.4 GeV (black) in central Au+Au collisions( b ≤ . tions of temperatures extracted from the reconstructedhypersurface: a) with increasing beam energy, the tem-perature distributions merge into one peak at ≈
150 MeV(in line with expectations from a thermal model analy-sis) while the typical widths of the temperature spread isFWHM ≈
50 MeV, different from the expectations from athermal model analysis with a single temperature. b) Thedistributions of the baryon chemical potentials at chemi-cal freeze-out is very narrow at each energy. As expected [MeV] / N d N / d [ M e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm E lab =1.23 AGeVE lab =4.0 AGeVE lab =10.8 AGeV s NN =7.7 GeV s NN =9.1 GeV s NN =11.5 GeV s NN =14.5 GeV s NN =19.6 GeV s NN =27.0 GeV s NN =39.0 GeV s NN =62.4 GeV Figure 4. [Color online] Distribution of the baryon chemicalpotential at the chemical freeze-out of π ’s reconstructed withthe presented algorithm at | z | ≤ lab = 1 .
23 AGeV(yellow), E lab = 4 . lab = 10 . √ s NN = 7.7 GeV (cyan), √ s NN = 9.1 GeV (blue), √ s NN = 11.5 GeV (magenta), √ s NN = 14.5 GeV (purple), √ s NN = 19.6 GeV (pink), √ s NN = 27 GeV (red), √ s NN =39 GeV (brown) and √ s NN = 62.4 GeV (black) in centralAu+Au collisions ( b ≤ . the chemical potential decreases with increasing energy.In contrast to the temperatures we do not observe a sat-uration of µ B , but a continuous decrease. In this section the collision energy dependence of the tem-perature and the baryon chemical potential are investi-gated on the chemical freeze-out hyper surface. We com-pare the chemical freeze-out values to kinetic freeze-outvalues. The kinetic freeze-out is defined as the point of theparticles last interaction and can also easily be extractedfrom the presented algorithm. We show in Fig. 5 the av-erage temperature at the kinetic freeze-out (blue circles)and the average temperature at the chemical freeze-out(red circles) as a function of the collision energy. Firstly,it is noticeable and important that T chem > T kin over thewhole energy range. This implies that indeed the chem-ical and the kinetic freeze-out processes happen sequen-tially. The energy dependence of the absolute difference ∆T = T chem − T kin between the two freeze-out temper-atures stays according to the present analysis approxi-mately constant (20 ± ≈
150 MeVwhile the kinetic freeze-out temperature saturates at ≈
130 MeV. Fig. 6 shows the average baryon chemical poten- . Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations 5 s NN [GeV] T [ M e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm UrQMD/cg (chem)UrQMD/cg (kin)
Figure 5. [Color online] Average temperature at the chemicalfreeze-out (red circles) and at the kinetic freeze-out (blue cir-cles) extracted from the coarse-graining procedure with a HRGEoS in central Au+Au collisions ( b ≤ . tials at the chemical and the kinetic freeze-out. Both quan-tities drop rapidly with increasing energy, lining up withthe vanishing chemical potential inferred from observedbaryon-charge symmetric matter formed at top RHIC andLHC energies. After the analysis of the time distributions and the energydependence of the chemical freeze-out quantities (temper-atures and baryon chemical potentials), we will now relateour results to the phase structure of QCD in the phasediagram of nuclear matter. Fig. 7 shows the calculatedaverage temperature (cid:104) T (cid:105) and the average chemical poten-tial (cid:104) µ B (cid:105) as red circles for the chemical freeze-out and asblue circles for the kinetic freeze-out. Fits to the exper-imental data points obtained at RHIC [32,33,34,35,36],CERN/SPS [37,38], BNL/AGS [37,38], GSI/SIS [39,40,41] and the recent HADES point [42] are shown as greenstars (summarized in [43,44]).We observe in general that our calculated chemicalfreeze-out line matches the estimates from thermal modelfits to the experimental data surprisingly well. This resultis remarkable because it is ex-ante not expected that apurely hadronic transport model, which does neither in-volve the concept of the chemical freeze-out explicitly nora phase-transition to a deconfined phase, should reproduce( T , µ B ) combinations near the data points obtained fromthermal model fits. This raises a question: Why does anon-equilibrium transport model without phase-transition s NN [GeV] B [ M e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fmUrQMD/cg (chem)UrQMD/cg (kin)
Figure 6. [Color online] Average baryon chemical potential atthe chemical freeze-out (red circles) and at the kinetic freeze-out (blue circles) extracted from the coarse-graining procedurewith a HRG EoS in central Au+Au collisions ( b ≤ . and chemical break-up reproduce thermal (i.e. equilib-rium) quantities such precise? A hint towards the answercan be found the scattering rates. It is well known frome.g. chemistry that the onset of equilibrium can be charac-terized by the ratio of the expansion rate to the scatteringrate. This means to maintain equilibrium, the (inelastic)scattering rate Γ must be much larger than the expan-sion rate θ . Where θ/Γ = Kn , can be interpreted as theKnudsen number Kn . Typically, Γ ∼ f i f j σ ij , with f i be-ing the phase space density of species i and σ ij being the(inelastic) interaction cross section, and θ = ∂ µ u µ , the di-vergence of the 4-velocity field [45]. I.e., if the scatteringrate is larger than the expansion rate then enough equi-librating reactions happen to keep the system in equilib-rium. If otherwise the scattering rate is smaller than theexpansion rate then the system does not have enough timeto adjust its properties and the state becomes frozen forthe remaining time of the expansion. While in our analy-sis, the competition between scattering rate and expansionrate is modelled by microscopic dynamics, it can also beused to calculate the chemical and kinetic freeze-out in aself consistent way in expanding gases and fluids, see e.g.[45,46].Therefore, the temperatures and the baryon chemicalpotentials at chemical and kinetic freeze-out as calculatedhere emerge from the local interplay of the elastic andinelastic collision rates with the local expansion rates ofthe system. Thus, the chemical and kinetic freeze-out linesare not related to a phase-transition or a cross over. T. Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations B [MeV] T [ M e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm UrQMD/cg (chem)UrQMD/cg (kin)Data RHIC/SPS/AGS/SIS (fits)
Figure 7. [Color online] Phase diagram of the chemical freeze-out calculated with the UrQMD/cg model (red circles) and ofthe kinetic freeze-out (blue circles). Also shown are thermalmodel fits to RHIC [32,33,34,35,36], SPS [37,38], AGS [37,38], SIS data [39,40,41] and the recent HADES point [42] asgreen stars.
Our analysis can be further deepened by the investiga-tion of quantities which have been proposed to classifythe chemical freeze-out line. The systematic investigationof the chemical freeze-out in experiments and theory led tothe proposal of several quantities which apparently seemto function as criteria for the chemical freeze-out to occur:1. In Ref. [17] the average energy per particle (cid:104) E (cid:105) / (cid:104) N (cid:105) was proposed to stay constant along the chemical freeze-out line with (cid:104) E (cid:105) / (cid:104) N (cid:105) = 1 GeV.2. Secondly, an entropy based criterion, s/T = 7 [19] wassuggested for the meson dominated energy regime.3. It was further proposed that the total baryon numbercan also serve as an estimation for the freeze-out lineby demanding n B + n ¯B = 0 .
12 fm − [18].These suggestions can be directly accessed and tested inthe coarse-graining method employed here. With increasing collision energy a major part of the ad-ditional energy is used to produce new (heavier) parti-cles than to increase the momentum of the existing parti-cles [47] leading to the limiting Hagedorn temperature forhadrons. In fact, the study of hadronic abundances at SIS,AGS, SPS, RHIC and LHC energies has shown that the chemical freeze-out line can be characterized by a constant (cid:104) E (cid:105) / (cid:104) N (cid:105) of ≈ (cid:104) E (cid:105) / (cid:104) N (cid:105) = 0.96- 1.08 GeV [43,48]. In [38] using a statistical model withtwo thermal sources (TSM) it was predicted that the av-erage energy per particle as a function of √ s NN may notbe constant, but depend on the resonance share of theproduced matter and suggested an increase of (cid:104) E (cid:105) / (cid:104) N (cid:105) to1.1 - 1.2 GeV for √ s NN ≤
10 GeV.We show in Fig. 8 the calculated (cid:104) E (cid:105) / (cid:104) N (cid:105) ratio onthe chemical and kinetic freeze-out surface arising fromthe coarse-graining method in dependence of the collisionenergy. The results at the chemical freeze-out are shownas red circles and the results for the kinetic freeze-outare shown as blue circles. First of all, the computed av-erage energy per particle at the chemical freeze-out hy-per surface varies between 0.95 and 1.2 and is thereforevery close to the phenomenological 1 GeV/ particle . Inline with [38,48], we observe a slight energy dependenceof the ratio with an increase towards 1.2 at the baryondominated energy region. When studying this ratio onthe kinetic freeze-out hyper surface, it is surprising thatthe same quantity computed here yields only a slightlylower value of (cid:104) E (cid:105) / (cid:104) N (cid:105) ≈ .
95 GeV. This can be ex-plained when examining how (cid:104) E (cid:105) / (cid:104) N (cid:105) varies with timewhich has been done e.g. in [6]. The system is character-ized by a plateau in the energy per particle over the dura-tion of the chemical equilibration. Thus we can concludethat the chemical freeze-out line is indeed characterizedby (cid:104) E (cid:105) / (cid:104) N (cid:105) ≈ particle . Towards lower energies,we predict an increase by about 20 % in line with [38,48]. In thermodynamics, the entropy density is the determin-ing quantity that characterizes when (chemical) equilib-rium sets in and is thus a prime candidate to constrain thechemical freeze-out. For an ideal gas of massless hadronswith zero net-baryon density the dimensionless term s/T is equivalent to the number of active degrees of freedom.E.g. a pion gas with s/T = 3 would have exactly 3 de-grees of freedom. In reality, however, pions are not theonly degrees of freedom in the hadron gas, and the en-tropy has contributions from all mesons and baryons, thus s/T is a proxy for the effective degrees of freedom. Thisrelation was extensively studied with the conclusion thatchemical equilibration is characterized by a constant en-tropy density per cubic temperature of 7 (equivalent to7 effective degrees of freedom) by thermal model studies[19]. Therein it was further shown that the decompositionof s/T = 7 into hadronic and mesonic contributions re-veals an interchange in the dominant contribution around √ s NN = 7 . . Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations 7 Table 1.
For each energy (column 1) the average chemical freeze-out time τ chem of all pions (column 2), the average chemicalfreeze-out temperature T chem (column 3), the average baryon chemical potential at the chemical freeze-out µ chemB (column 4)and the freeze-out criteria evaluated at the chemical freeze-out (cid:104) E (cid:105) / (cid:104) N (cid:105) (column 5), s/T (column 6) and n B + n ¯B (column 7)are shown. √ s NN [GeV] (cid:104) τ chem (cid:105) [fm] (cid:104) T chem (cid:105) [MeV] (cid:104) µ chemB (cid:105) [MeV] (cid:104) E (cid:105) / (cid:104) N (cid:105) [GeV] s/T n B + n ¯B [1/fm ]2.4 13.7 67.3 837.5 1.088 31.478 0.3343.3 10.3 108.8 702.1 1.194 13.626 0.3474.9 8.1 132.2 541.5 1.166 9.248 0.3037.7 6.9 145.6 395.1 1.092 7.350 0.2339.1 6.7 147.8 355.8 1.068 6.999 0.20911.5 6.7 149.5 310.1 1.038 6.607 0.17814.5 6.8 150.2 272.4 1.015 6.321 0.15319.6 7.1 150.7 231.6 0.995 6.100 0.12927.0 7.5 151.3 195.5 0.987 6.047 0.11139.0 8.0 152.3 161.1 0.992 6.151 0.09662.4 8.4 154.7 123.2 1.018 6.449 0.081 s NN [GeV] E / N [ G e V ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm UrQMD/cg (chem)UrQMD/cg (kin)Stat. model values
Figure 8. [Color online] The (cid:104) E (cid:105) / (cid:104) N (cid:105) at the chemical freeze-out (red circles) and at the kinetic freeze-out (blue circles) ex-tracted from the coarse-graining procedure with a HRG EoS incentral in central Au+Au collisions ( b ≤ . The calculated entropy density s/T is shown in Fig.9. The red circles show the values obtained in the presentanalysis at the chemical freeze-out and the blue circlescorrespond to the values at the kinetic freeze-out, both independence of the collision energy. The filled circles rep-resent the meson dominated region and the empty circlesshow the baryon dominated region. To avoid numericalproblems related to the temperature extraction in cellswith high baryon density and low energy density (i.e. cold cells), we include only cells with T ≥
30 MeV in the aver-aging. Extracting the chemical freeze-out from UrQMD, s NN [GeV] s / T Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fmUrQMD/cg (chem)UrQMD/cg (kin)s/T = 6 7 Figure 9. [Color online] The average entropy density s/T atthe chemical freeze-out (red circles) and at the kinetic freeze-out (blue circles) extracted from the coarse-graining procedurewith a HRG EoS in central in central Au+Au collisions ( b ≤ . we find that between 7.7 GeV and 62.4 GeV s/T remainsat values between 6 and 7 lining up with suggested val-ues of Ref. [19]. Below the √ s NN = 8 GeV, the entropyper cubic temperature rises rapidly as expected from [19].For the kinetic freeze-out, we find that s/T saturates be-tween 4 − T. Reichert et al. : Probing chemical f.o. criteria in rel. nuclear collisions with coarse grained transport simulations the baryon dominated region. In conclusion, the s/T cri-terion motivated by the number of degrees of freedom isfound to be in line with the previously obtained value of 7[19] in the whole energy range and above √ s NN = 8 GeV. Finally, we explore the suggested baryon-anti-baryon cri-terion with coarse grained UrQMD. Fig. 10 shows the cal-culated n B + n ¯B in dependence of the collision energy. Theresults for the chemical freeze-out are shown as red cir-cles, while the results for the kinetic freeze-out are shownas blue circles. The results for the total baryon and anti-baryon density at the chemical freeze-out start at a den-sity of 0 .
35 fm − at the low energies and drop rapidly to0 .
15 fm − at √ s NN = 20 GeV. This increase at low ener-gies is qualitatively in line with expectations in [43]. Theresults at the kinetic freeze-out follow the same trend, butthe decrease is more moderate starting at 0 .
15 fm − andending at 0 .
05 fm − . Generally, the baryon density cri-terion shows a stronger energy dependence than the pre-viously discussed criteria for the chemical freeze-out line.Nevertheless, towards higher energies ( √ s NN ≥
20 GeV)also this criterion can be applied to characterize the chem-ical freeze-out line. s NN [GeV] n B + n B [ f m ] Au+Au (UrQMD/cg)b 3.4 fm|z| 5 fm UrQMD/cg (chem)UrQMD/cg (kin)0.12 fm criterion Figure 10. [Color online] The average n B + n ¯B at the chemicalfreeze-out (red circles) and at the kinetic freeze-out (blue cir-cles) extracted from a coarse-graining procedure with a HRGEoS in central in central Au+Au collisions ( b ≤ . In this article we have developed a novel approach todetermine the chemical freeze-out hyper-surface directlyfrom a microscopic simulation. The UrQMD transportmodel was employed to simulate the underlying eventsand the microscopic evolution of the system. Using a newalgorithm, we traced final state particles back to theiroriginal creation space-time coordinate through sequentialdecay and re-formation chains allowing to reconstruct thechemical freeze-out hyper-surface using a coarse-grainingmethod. We found that the average chemical break-uptime remains constant at ≈ √ s NN = 7 . s/T = 7and n B + n ¯B ≈ .
12 fm − are good proxies to character-ize the chemical freeze-out curve at high collision energies( √ s NN ≥
20 GeV). The originally suggested Cleymans-Redlich criterion (cid:104) E (cid:105) / (cid:104) N (cid:105) = 1 GeV, however, providesthe best estimate for the chemical freeze-out line over allenergies. Acknowledgments
The authors thank Paula Hillmann, Michael Wondrak,Vincent Gaebel, Michel Bonne, Alexander Elz and HarriNiemi for fruitful discussions. G. Inghirami is supportedby the Academy of Finland, Project no. 297058. This workwas supported by Deutscher Akademischer Austauschdi-enst (DAAD), Helmholtz Forschungsakademie Hessen (HFHF)and in the framework of COST Action CA15213 (THOR).The computational resources were provided by the Centerfor Scientific Computing (CSC) of the Goethe-UniversityFrankfurt.
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