Rapid chemical equilibration of antihyperons by means of the interplay
between strong annihilation on baryons and the corresponding backreactions of
multi-mesonic (fusion-type) processes in the later, hadronic stage of an
ultrarelativistic heavy ion collision will be discussed. Explicit rate
calculations for a dynamical setup are presented. At maximum SPS energies
yields of each antihyperon specie are obtained which are consistent with
chemical saturated populations of $T \approx 150-160 $ MeV. The proposed
picture supports dynamically the popular chemical freeze-out parameters
extracted within thermal models.Comment: revised version for some typos and two references; 6 pages, 4
  figures; contribution to the Int. Workshop XXX on Gross Properties of Nuclei
  and Nuclear Excitations: Ultrarelativistic Heavy-Ion Collisions, Hirschegg,
  Jan. 13 - 19, 200

Greiner, C.

English

arXiv

Rapid chemical equilibration of antihyperons by means of the interplay between strong annihilation on baryons and the corresponding backreactions of multi-mesonic (fusion-type) processes in the later, hadronic stage of an ultrarelativistic heavy ion collision will be discussed. Explicit rate calculations for a dynamical setup are presented. At maximum SPS energies yields of each antihyperon specie are obtained which are consistent with chemical saturated populations of $T \approx 150-160 $ MeV. The proposed picture supports dynamically the popular chemical freeze-out parameters extracted within thermal models

Greiner, C

CERN Document Server

CHEMICAL EQUILIBRATION OF ANTIHYPERONSC. GREINERInstitut fu¨r Theoretische Physik, Universita¨t Giessen,Heinrich-Bu-Ring 16, D-35392 Giessen, GermanyAbstractRapid chemical equilibration of antihyperons by means of the interplaybetween strong annihilation on baryons and the corresponding backre-actions of multi-mesonic (fusion-type) processes in the later, hadronicstage of an ultrarelativistic heavy ion collision will be discussed. Explicitrate calculations for a dynamical setup are presented. At maximum SPSenergies yields of each antihyperon specie are obtained which are con-sistent with chemical saturated populations of T  150−160 MeV. Theproposed picture supports dynamically the popular chemical freeze-outparameters extracted within thermal models.1 Physics of antihyperon productionStrangeness enhancement has been predicted a long time ago as a diagnosticprobe to prove for the short-time existence of a QGP in the course of a rel-ativistic heavy ion collision. With respect to the high production thresholdsin the various binary hadronic reaction channels, especially the antihyperonswere then advocated as the appropriate candidates [1]. Assuming the existenceof a temporarily present phase of QGP and following statistical coalescenceestimates, the abundant (anti-)strange quarks can easily be redistributed tocombine to strange (anti-)baryons [1], which do then, in return, come close totheir chemical equilibrium values at the onset of the hadronic phase.Indeed, such a behaviour of nearly chemically saturated populations ofthe antihyperons has been experimentally demonstrated with the Pb+Pb ex-periments NA49 and WA97 at CERN-SPS. For this statement, of course, aquantitative, theoretical analysis within a thermal model has to be invokedby tting the thermodynamical parameters to the set of individual (strange)hadronic abundancies [2]. Some phenomenological attempts to explain a moreabundant production of antihyperons within a hadronic transport description[3] do exist like the color rope or the high-dense cluster formation. The un-derlying mechanisms, however, have to be considered as exotic. In particular,a dramatic role of antibaryon annihilation is observed, which, in return, hasto be more than counterbalanced in a rather ad hoc way by these more exoticmechanisms.Be it as it is, a correct incorporation of the strong baryonic annihilationchannels had actually not been achieved within the various microscopic imple-mentations. Recently we have conjectured that a suciently fast redistribu-tions of strange and light quarks into (strange) baryon-antibaryon pairs shouldbe achieved by multi-mesonic fusion-type reactions of the typen1pi + n2K $ Y + p (1)for a moderately dense hadronic system [4]. The beauty of this argument liesin the fact that (at least) these special kind of multi-hadronic reactions haveto be present because of the fundamental principle of detailed balance. As theannihilation of antihyperons on baryons is of dramatic relevance, the multi-mesonic (fusion-like) ‘back-reactions’ involving n1 pions and n2 kaons, where n2counts the number of anti-strange quarks within the antihyperon Y ), must, inprinciple, be taken care of in a dynamical simulation. This reasoning has rstbeen raised by Rapp and Shuryak concerning the production of anti-protons[5], which then triggered our work.It is plausible to assume that the annihilation crosssections are approxi-mately the same as for N p at the same relative momenta, i.e. σBY¯!npi+nY K σ0  50 mb. The equilibration timescale (ΓY¯ )(−1)  1/(σBY¯ vB¯NρB) is to agood approximation proportional to the inverse of the density of baryons andtheir resonances. Adopting an initial density of 1{2 times normal nuclear mat-ter density ρ0 for the initial and thermalized hadronic reball, the antihyperonswill equilibrate on a timescale of 1{3 fm/c! This timescale competes with theexpansion timescale of the late hadronic reball. To be more quantitative ratecalculations for a dynamical setup will be presented in the next section.2 Rate equation results and their implicationsFollowing the concepts of kinetic theory, the microscopic starting point wouldbe a Boltzmann-type equation of the form∂tfY¯ +pEY¯rfY¯ =∑{n1};B12EY¯∫d3pB(2pi)32EB∏{n1,n2}∫d3pi(2pi)32Ei(2pi)4δ4(pY¯ + pB −∑{n1,n2}pi)jhhn1, n2jT jY¯ Biij2(−)fY¯ fB∏{n1,n2}(1 + fi) +∏{n1,n2}fi(1 − fY¯ )(1− fB) ,whereσ{n1}Y¯ B 1‘Flux′∏{n1,n2}∫d3pi(2pi)32Ei(2pi)4δ4pY¯ + pB −∑{n1,n2}pi jhhn1, n2jT jY¯ Biij2corresponds to the annihilation cross section. Assuming vrel σY¯ B(ps) to beroughly constant, which is actually a good approximation for the pp-annihilation,or, invoking a standard coarse grained description of thermally averaged crosssections and distributions, and furthermore taking the distributions in theBoltzmann approximantion, the following master equation for the respectivelyconsidered antihyperon density is obtainedddtρY¯ = −hhσY¯ BvY¯ BiiρY¯ ρB −∑fn1gR(n1,n2)(T, µB, µs)(ρpi)n1(ρK)n2 , (2)where the ‘back-reactions’ of several eectively coalescing pions and kaons areincorporated in the ‘mass-law’ factorR(n,nY )(T, µB, µs) =ρeq.Y¯ ρeq.B(ρeq.pi )n1(ρeq.K )n2pn1 .Here pn1 states the relative probability of the reaction (1) to decay into aspecic number n1 of pions and ρB denotes the total number density of baryonicparticles. R depends only on the temperature and the baryon and strangequark chemical potentials. ΓY¯  hhσY¯ NvY¯ NiiρB gives the eective annihilationrate of the respective antihyperon specie on a baryon.Nonequilibrium inelastic hadronic reactions can explain to a good extentthe overall strangeness production seen experimentally: The major amount ofthe produced kaons at SPS-energies can be understood in terms of still earlyand energetic non-equilibrium interactions [6, 7]. Refering to the master equa-tion (2), one can then take the pions, baryons and kaons to stay approximatelyin thermal equilibrium throughout the later hadronic evolution of the collision,the later being modelled to be an isentropic expansion with xed total entropycontent being specied via the entropy per baryon ratio S/A. (2) becomesddtρY¯ = −ΓY¯{ρY¯ − ρeqY¯}. (3)The ‘eective’ (global or at midrapidity) volume V (t) is parametrized asfunction of time by longitudinal Bjorken expansion and including an acceler-ating radial flow, e.g.Veff (t  t0) = pi (ct)(R0 + v0(t− t0) + 0.5a0(t− t0)2)(4)with R0 = 6.5 fm, v0 = 0.15 c and a0 = 0.05 c2/fm. At starting time t0an initial temperature T0 is chosen. (T0 is set to 190 MeV for the SPS and150MeV for the AGS situation, while the initial energy densities are thenabout 1 GeV/fm3.) From (4) together with the constraint of conserved entropythe temperature and the chemical potentials do follow as function of time. Asa minimal assumption the initial abundancy of antihyperons is set to zero.Equation (3) is solved for each specie.0 2 4 6 8 10 12 140.0000.0020.0040.0060.0080.0100.40.50.60.81.0  Y/Btime [fm/c]Fig. 1: The anti-Λ to baryon number ratio NΛ¯/NB (t) as a function of time for variousimplemented annihiation cross section σeff  λσ0. The entropy per baryon is taken asS/A = 30, t0 = 3 fm/c and T0 = 190 MeV.In Fig. 1 the number of s (normalized to the conserved net baryon number)as a function of time is depicted. The entropy per baryon is chosen as S/A =30 being characteristic to global SPS results. In addition the cross sectionemployed is varied by a constant factor, i.e. σeff  λ σ0. The characteristicsis that rst the antihyperons are dramatically being poulated, and then inthe very late expansion some more are still being annihilated. The results arerather robust against a variation by a factor of 2 in the cross section.In Fig. 2 the number of antihyperons of each specie are now shown as a func-tion of the decreasing temperature T (t) of the hadronic system. For a directcomparison the instantaneous equilibrium abundancy N eq.Y¯(T (t), µB(t), µs(t))/NBis also given. As noted above, after a fast initial population, the individualyields of the antihyperons do overshoot their respective equilibrium numberand then do nally saturate at some slightly smaller value. Moreover, onenotices that the yields eectively do saturate at a number which can be com-pared to an equivalent equilibrium number at a temperature parameter aroundTeff  150− 160 MeV, being strikingly close to the ones obtained within thevarious thermal analyses [2].180 160 140 1200.0000.0050.0100.0150.020180 160 140 1200.00000.00250.00500.00750.01000.0125200 180 160 140 1200.00000.00250.00500.00750.01000.0125200 180 160 140 1200.0000.0010.0020.0030.004eq.eq.eq.eq.Ω /BΞ /BΛ/Bp/B      temperature [MeV]  temperature [MeV]Fig. 2: The antihyperon to baryon number ratio NY¯ /NB (T ) and Neq.Y¯/NB(T ) (dotted line)as a function of the decreasing temperature. Parameters are the same as in Fig. 1.In Fig. 3 the number of anti-s as a function of time is given for vari-ous entropy per baryon ratios. One notices that the nal value in the yieldsignicantly depends on the entropy content, or, in other words, on the bary-ochemical potential. We note that the results at midrapidity from WA97 canbest be reproduced by employing an entropy to baryon ratio S/A = 40, whereone qualitatively expects a larger entropy content.There is also a clear hint at AGS energies of enhanced anti- production[8]. For most central collisions more anti-s are found compared to anti-protons, which is quite puzzling as within a thermal model analysis this ratiois found not to be larger than 1. This enhanced ratio of anti-s compared toanti-protons at AGS energies one can understand in a way that one assumesthat their annihilation cross section on baryons is just slightly smaller thanfor the antiprotons. In Fig. 4 a similar study like that of Fig. 1 is shown for acharacteristic situation at AGS. For smaller, yet not too small eective crosssections the nal yield can here be enlarged by a factor of 2 compared to thecase with a ‘full’ crossection, as the nal reabsorption is not as eective. But,of course, this idea is speculation at present. Also, we remark that the seectively do saturate at an equivalent equilibrium number at a temperatureparameter around Teff  120−130 MeV. Unfortunately, there are no data for at AGS. A detailed measurement of all antihyperons represents an excellentopportunity for future heavy ion facilities at an energy upgraded GSI.0 2 4 6 8 10 12 140.0000.0020.0040.0060.0080.0100.0120.0142025303540  Y/Btime [fm/c]150 140 130 120 110 1000.04.0x10-58.0x10-51.2x10-41.6x10-42.0x10-42.4x10-4 Yeq0.40.20.60.81.0  Y/Btemperature [MeV]Fig. 3: NΛ¯/NB (t) as a function of time forvarious entropy content described via the en-tropy per baryon ratio (S/A = 20 − 40).Other parameters are as in Fig. 1.Fig. 4: NΛ¯/NB (T ) and Neq.Λ¯/NB(T ) asa function of decreasing temperature for acharacteristic AGS situation with an entropycontent of S/A = 12 for various implementedannihiation cross section σeff  λσ0. t0 = 5fm/c and T0 = 150 MeV.References[1] P. Koch, B. Mu¨ller and J. Rafelski, Phys. Rep. 142, 167 (1986); J. Rafel-ski, Phys. Lett. B 262, 333 (1991).[2] P. Braun-Munzinger, I. Heppe and J. Stachel, A. Kera¨nen, E. Suhonenand K. Redlich, PRC 64, 024901 (2001).[3] H. Sorge, Z. Phys. C 67, 479 (1995); Phys. Rev. C 52, 3291 (1995);K. Werner and J. Aichelin, Phys. Lett. B 300, 158 (1993) and Phys. Lett.B 308, 372 (1993); M. Bleicher et al, Phys. Lett. B 485, 133 (2000).[4] C. Greiner and S. Leupold, J. Phys. G 27, L95 (2001); C. Greiner, nucl-th/0011026 and nucl-th/0103079.[5] R. Rapp and E. Shuryak, Phys. Rev. Lett. 86, 2980 (2001); R. Rapp,these proceedings.[6] J. Geiss, W. Cassing and C. Greiner, Nucl. Phys. A 644, 107 (1998).[7] C. Greiner, nucl-th/0112080.[8] L. Ahle et al, (E-802 Collaboration), Phys. Rev. Lett. 81, 2650 (1998);B. Back et al, (E-917 Collaboration), Phys. Rev. Lett. 87, 242301 (2001).

Chemical equilibration of antihyperons

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Chemical Equilibration of Antihyperons

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