The equilibration of hot and dense nuclear matter produced in the central
region in central Au+Au collisions at $\sqrt{s}=200$ AGeV is studied within the
microscopic transport model UrQMD. The pressure here becomes isotropic at $t
\approx 5$ fm/c. Within the next 15 fm/c the expansion of the matter proceeds
almost isentropically with the entropy per baryon ratio $S/A \approx 150$.
During this period the equation of state in the $(P,\epsilon)$-plane has a very
simple form, $P=0.15 \epsilon$. Comparison with the statistical model (SM) of
an ideal hadron gas reveals that the time of $\approx 20$ fm/$c$ may be too
short to attain the fully equilibrated state. Particularly, the fractions of
resonances are overpopulated in contrast to the SM values. The creation of such
a long-lived resonance-rich state slows down the relaxation to chemical
equilibrium and can be detected experimentally.Comment: Talk at the conference Strangeness'2000, to be published in J. of
  Phys. 

A Faessler

Bleicher M

Blättel B

Bravina L V

C Fuchs

Cleymans J

E E Zabrodin

Fermi E

H Stöcker

L V Bravina

Landau L D

Letessier J

M Bleicher

M Brandstetter

M I Gorenstein

S A Bass

S Soff

Shuryak E

W Greiner

English

arXiv

The equilibration of hot and dense nuclear matter produced in the central region in central Au+Au collisions at square root s = 200A GeV is studied within the microscopic transport model UrQMD. The pressure here becomes isotropic at t approx 5 fm/c. Within the next 15 fm/c the expansion of the matter proceeds almost isentropically with the entropy per baryon ratio S/A approx 150. During this period the equation of state in the (P, epsilon)-plane has a very simple form, P = 0.15 epsilon. Comparison with the statistical model (SM) of an ideal hadron gas reveals that the time of approx 20 fm/c may be too short to attain the fully equilibrated state. Particularly, the fractions of resonances are overpopulated in contrast to the SM values. The creation of such a long-lived resonance-rich state slows down the relaxation to chemical equilibrium and can be detected experimentally

Bravina, Larissa

Zabrodin, Eugene E.

Bleicher, Marcus

Bass, Steffen A.

Brandstetter, Mathias

Fäßler, Amand

Fuchs, Christian

Greiner, Walter

Gorenstein, Mark I.

Soff, Sven

Stöcker, Horst

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0010088 v1   26 Oct 2000Transition to resonance-rich matter in heavy ioncollisions at RHIC energiesL V Bravina†‡, E E Zabrodin†‡, M Bleicher§, S A Bass ,M Brandstetter¶, A Faessler†, C Fuchs†, W Greiner¶,M I Gorenstein+, S Soﬀ∗, H St¨ ocker¶† Institut f¨ ur Theoretische Physik, Universit¨ at T¨ ubingen, T¨ ubingen, Germany‡ Institute for Nuclear Physics, Moscow State University, Moscow, Russia§ Nuclear Science Division, Lawrence Berkeley Laboratory, USA  National Superconducting Cyclotron Lab, Michigan State University, USA¶ Institut f¨ ur Theoretische Physik, Universit¨ at Frankfurt, Frankfurt, Germany+ Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine∗ Gesellschaft f¨ ur Schwerionenforschung, Darmstadt, GermanyAbstract. The equilibration of hot and dense nuclear matter produced in thecentral region in central Au+Au collisions at√s = 200A GeV is studied withinthe microscopic transport model UrQMD. The pressure here becomes isotropicat t ≈ 5 fm/c. Within the next 15 fm/c the expansion of the matter proceedsalmost isentropically with the entropy per baryon ratio S/A ≈ 150. During thisperiod the equation of state in the (P,ε)-plane has a very simple form, P = 0.15ε.Comparison with the statistical model (SM) of an ideal hadron gas reveals thatthe time of ≈ 20 fm/c may be too short to attain the fully equilibrated state.Particularly, the fractions of resonances are overpopulated in contrast to the SMvalues. The creation of such a long-lived resonance-rich state slows down therelaxation to chemical equilibrium and can be detected experimentally.1. IntroductionThe assumption that strongly interacting matter, produced in nucleus-nucleuscollisions at high energy, can reach the state of local equilibrium (LE) [1,2] is oneof the most important topics in the relativistic heavy ion programme [3]. The degreeof equilibration can be checked by ﬁtting the measured particle yields and transversemomentum spectra to that of the thermal model in order to extract the conditionsof the ﬁreball at the chemical and thermal freeze-out [4–8]. Here the equilibriumparticle abundances, which correspond to a certain temperature T, baryon chemicalpotential µB, and strangeness chemical potential µS, can be determined. However,the analysis is complicated, e.g., by the presence of collective ﬂow and the non-homogeneity of the baryon charge distribution in the reaction volume. The volumeof the ﬁreball is also taken as a free parameter in the thermal model. Various non-equilibrium microscopic transport models have been applied to verify the appearanceof at least local equilibrium in the course of heavy ion collisions at relativistic andultra-relativistic energies [9–14]. To reduce the number of unknown parametersand simplify the analysis it has been suggested [15–17] to examine the equilibriumconditions in the central cell of relativistic heavy ion collisions. For this purpose themicroscopic transport model UrQMD [18] is employed. Previous studies at energiesResonance-rich matter in HICs at RHIC 2from 10.7A GeV (AGS) to 160A GeV (SPS) [15,16] have shown that the cubic cellwith volume V = 125 fm3 is well suited for the analysis. The aim of the present paperis to study the relaxation of hot nuclear matter, simulated within the microscopicmodel, in the central cell in Au+Au interactions at√s = 200A GeV.2. Relaxation to thermal and chemical equilibriumFirst, the kinetic equilibrium has to be veriﬁed. The collective ﬂow in the cell shouldbe isotropic and small, so it cannot signiﬁcantly distort the momentum distributionsof particles. Microscopic simulations show that the longitudinal ﬂow rapidly dropsand converges to the developing transverse ﬂow [16, 17]. Isotropy of the velocitydistributions results in the pressure isotropy. Pressure in longitudinal direction in thecell, calculated according to the virial theorem, is compared in ﬁgure 1(a) with thetransverse pressure. The time of convergence of longitudinal pressure to the transverseone decreases from 10 fm/c to 5 fm/c with rising incident energy from AGS to RHIC,respectively. Thus, the kinetic equilibrium in the central cell in Au+Au collisions atRHIC energy is reached at t ≈ 5 fm/c.10-310-210-11100 10 1101022 200501001500 10000501001502002500 2000Figure 1. (a) The longitudinal (solid lines) and the transverse (dashed lines)diagonal components of the pressure tensor P in the central cell of heavy ioncollisions at AGS, SPS, and RHIC energies compared to the SM results (dottedlines). (b) Time evolution of the entropy per baryon ratio, s/ρB = S/A, in thecentral cell calculated with the UrQMD (solid lines) and the SM (dashed lines).(c) The evolution of pressure P and baryon density ε in the central cell. (d) Thesame as (c) but for the (T,µB)-plane. Solid symbols correspond to the stage ofkinetic equilibrium, open symbols indicate the pre-equilibrium stage. The hatchedarea shows the expected region of the quark-hadron phase transition.To verify that the matter in the cell is in thermal and chemical equilibrium onehas to compare the snapshot of hadron yields and energy spectra in the cell withthe equilibrium spectra of hadrons obtained in the statistical model (SM) of an idealResonance-rich matter in HICs at RHIC 3hadron gas. The description of the SM employed in our calculations can be foundelsewhere [19]. The values of energy density ε, baryon density ρB, and strangenessdensity ρS, extracted from the microscopic calculations in the cell, are used as an inputto the system of nonlinear equations of the SM [19] to determine the temperature T,baryon chemical potential µB, and strangeness chemical potential µS. This enablesone to calculate the particle yields NSMi , total energy ESMi , pressure P SM, and entropydensity s. The thermal and chemical equilibrium is assumed to occur in the cell whenthe microscopic spectra of hadrons become close to the spectra predicted by the SM.At the isotropic stage the total microscopic pressure is close to the grand canonicalpressure, P SM, as shown in ﬁgure 1(a). Therefore, the time t = 5 fm/c is chosen asa starting point for the comparison with the SM. The ﬁnal time of the calculations,deﬁned from the conventional freeze-out conditions ε ≈ 0.1 GeV/fm3 or ρtot ≈ 0.5ρ0[16], corresponds to t ≈ 21 fm/c. The hadron-string matter in the central cell expandsnearly isentropically, see ﬁgure 1(b), with s/ρB ≡ S/A ≈ 12 (AGS), 32 (SPS), and 150(RHIC). The results of the simulations at AGS and SPS energies are intriguingly closeto the entropy per baryon values extracted from the thermal model ﬁts to experimentaldata, namely, (S/A)AGS ≈ 14 and (S/A)SPS ≈ 36 [7]. It is most interesting to comparethe predicted value (s/ρB)RHIC = 150 − 170 to the upcoming RHIC data.The evolution of the pressure with the energy density is depicted in ﬁgure 1(c).The pressure drops linearly with the decreasing energy density for all three energiesin question. Thus, the equation of state in the (P,ε)-plane has a very simple form:P = 0.12ε at AGS, and P = 0.15ε at SPS and RHIC, i.e. the ratio P/ε in the centralcell is saturated already at SPS energies.Figure 1(d) presents the evolution of the EOS in the (T,µB)-plane. In accordancewith general estimates [7,20] the baryon chemical potential at RHIC energies is smallwhile the temperatures are well above the anticipated temperature of the QCD phasetransition, T ≈ 150 MeV. Note that the evolution of temperature as a function ofε/ρB and µB is mainly determined by the change of energy per baryon, which dropsto ≈ 80−90% of its initial value, but not by changing of the baryon chemical potential.The slopes of energy spectra of particles in the cell are steeper compared to thepredictions of the statistical model [17]. This means that the apparent temperaturesof hadrons, especially pions, are lower than the temperature given by the SM. Sincethe energy density ε is the same in both models, one might expect that the fractionsof mesons and resonances in the UrQMD cell are overpopulated. These extra-particlesconsume signiﬁcant part of the total energy and eﬀectively “cool” the hadron cocktail.Figure 2 depicts the yields of the main hadron species in the cell within the timeinterval 5 fm/c ≤ t ≤ 19 fm/c. It is interesting that microscopic spectra of pions, whichare underestimated by the SM in the central cell at lower energies [16], converge tothe SM predictions at t ≈ 15 fm/c. Also, the statistical model overestimates yields ofnucleons, lambdas, and kaons, while the yields of both baryon and meson resonancesare reproduced quite well. Since the baryon number and the strangeness are conservedin strong interactions, where is the rest of the hypercharge, Y = B + S, in the UrQMDcalculations in the cell? As a matter of fact, the total yields of baryons and antibaryonsin the SM are larger than those of the UrQMD. For instance, at t = 10 fm/c the valuesof partial baryon and antibaryon density in the microscopic model are RmicB = 0.05fm−3 and Rmic¯ B = 0.02 fm−3, respectively, while the SM predicts the values RSMB = 0.08fm−3 and RSM¯ B = 0.05 fm−3. The hadron-resonance-string matter in the cell is not inchemical equilibrium; that is why the density of antibaryons is 2-3 times smaller thanthe equilibrium values. Similarly, the SM predicts signiﬁcantly larger abundances ofResonance-rich matter in HICs at RHIC 410-210-111010210-210-111010211010210-111010-11105 10 1510-11105 10 15Figure 2. The yields of main hadron species in the central cell of Au+Aucollisions at√s = 200A GeV as a function of time as obtained in the modelUrQMD (circles) together with the predictions of the SM (stars).both strange baryons and strange antibaryons at 5 ≤ t ≤ 11 fm/c in the cell, whilethe densities of strange mesons are nearly the same.To check the relevance of the central cell results for a larger reaction volume atRHIC energies the rapidity distributions of baryon resonances are presented in ﬁgure 3(other global observables have been studied in [21]). Here only those resonances thatdecay directly into groundstate hadrons (no ﬁnal state interactions) are accountedfor. These distributions are remarkably ﬂat within the interval |y| ≤ 3.5. More than80% of the baryon non-strange resonances are ∆’s (1232). The fraction of resonancesreconstructible from the experimental data is quite large and, therefore, the formationof resonance-abundant state can be traced experimentally.0 1 2 3 4 5 6y110100dN/dyAu+Au @ 200AGeV, b=3fm****+N*Figure 3. The rapidity distributions of baryon resonances in Au+Au (√s =200A GeV) interactions with the impact parameter b = 3 fm.Resonance-rich matter in HICs at RHIC 53. ConclusionsIn summary, the following conclusions can be drawn. The expansion of matter incentral Au+Au collisions at√s = 200A GeV proceeds with constant entropy perbaryon ratio in the central cell, S/A = s/ρB ∼ = 150. Since the S/A ratios for thecentral cell in A+A collisions, calculated at AGS and SPS energies, are very close tothe ratios extracted from the analysis of the experimental data, the expected valueof the entropy per baryon ratio at RHIC lies within the range 150 ≤ s/ρB ≤ 170.The microscopic pressure in the cell is close to the SM pressure. It shows a lineardependence on the energy density in the cell, P = 0.15ε. The obtained result isin accord with the EOS P ∼ = 0.2ε, derived for an ideal gas of hadrons and hadronresonances [22]. The temperature T SM in the cell at RHIC energies is shown to benearly independent on the baryon chemical potential µB ≈ 50 MeV.The comparison of the energy spectra and yields of hadrons with those of theSM shows that the times t ≈ 20fm/c may be too short to reach full thermal andchemical equilibrium. Particularly, the deceleration of the relaxation to equilibrium isattributed to the creation of the long-lived resonance-abundant matter. The yields ofresonances are in accord with the SM spectra from the very early times t ≈ 5 fm/c,while the densities of strange and non-strange baryons and their antiparticles arelower than the equilibrium values. (However, the ﬁtting temperature of the thermalmodel is higher than the temperatures of hadron species in the cell.) According tomicroscopic calculations, the resonance-rich matter survives until the thermal freeze-out. It remains a challenging task to verify the formation of long-lived resonance-abundant matter in heavy ion collisions at√s = 200A GeV experimentally.References[1] Fermi E 1950 Prog. Theor. Phys. 5 570[2] Landau L D 1953 Izv. Akad. Nauk SSSR 17 51[3] Proc. Int. Conf. Quark Matter’99 1999 Nucl. Phys. A 661 1c[4] Braun-Munzinger P, Heppe I and Stachel J 1999 Phys. Lett. B 465 15[5] Becattini F, Gazdzicki M and Sollfrank J 1998 Eur. Phys. J. C 5 143[6] Letessier J and Rafelski J 1999 J. Phys. G: Nucl. Part. Phys. 25 451Letessier J and Rafelski J 1999 Phys. Rev. C 59 947[7] Cleymans J and Redlich K 1999 Phys. Rev. C 60 054908Cleymans J and Redlich K 1998 Phys. Rev. Lett. 81 5284[8] Yen G D and Gorenstein M I 1999 Phys. Rev. C 59 2788[9] Geiger K and Kapusta J I 1993 Phys. Rev. D 47 4905[10] Bl¨ attel B, Koch V and Mosel U 1993 Rep. Prog. Phys. 56 1[11] Bass S A et al 1998 Phys. Rev. Lett. 81 4092[12] Bleicher M et al 1998 Phys. Lett. B 435 9[13] Sollfrank J et al 1999 Phys. Rev. C 59 1637[14] Cassing W, Bratkovskaya E L and Juchem S 2000 Nucl. Phys. A 674 249[15] Bravina L V et al 1998 Phys. Lett. B 434 379Bravina L V et al 1999 J. Phys. G: Nucl. Part. Phys. 25 351[16] Bravina L V et al 1999 Phys. Rev. C 60 024904Bravina L V et al 1999 Nucl. Phys. A 661 600c[17] Bravina L V et al 2000 (to be submitted)[18] Bass S A et al 1998 Prog. Part. Nucl. Phys. 41 255Bleicher M et al 1999 J. Phys. G: Nucl. Part. Phys. 25 1859[19] Belkacem M et al 1998 Phys. Rev. C 58 1727Bratkovskaya E L et al 2000 Nucl. Phys. A 675 661[20] Braun-Munzinger P and Stachel J 1998 Nucl. Phys. A 638 3c[21] Bleicher M et al 2000 Phys. Rev. C 62 024904[22] Shuryak E 1973 Sov. J. Nucl. Phys. 16 395

Transition to resonance-rich matter in heavy ion collisions at RHIC energies

Abstract

Similar works

Full text

Available Versions

Hochschulschriftenserver - Universität Frankfurt am Main

Crossref

GSI Repository