PHYSICAL REVIEW C, VOLUME 62, 054906

Time dependence of strange baryon freeze-out in relativistic heavy ion collisions

R. Bellwied,1 H. Caines,2 and T. J. Humanic21Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202

2Department of Physics, The Ohio State University, Columbus, Ohio 43210~Received 3 April 2000; published 16 October 2000!

We investigate chemical and thermal freeze-out time dependencies of strange baryon production for CERNSPS heavy ion collisions in the framework of a dynamical hadronic transport code. We show that theL yieldchanges considerably after hadronization in the case of Pb1Pb collisions, whereas for smaller system sizes~e.g., S1S! the direct particle production dominates over production from inelastic rescattering. Chemicalfreeze-out times for strange baryons in Pb1Pb are smaller than for nonstrange baryons, but they are stillsufficiently long for hadronic rescattering to contribute significantly to the finalL yield.

PACS number~s!: 25.75.2q

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Recent model calculations, which were successfullyplied to CERN SPS data, implied that most measured pticle distributions in relativistic heavy ion collisions can bdescribed with statistical models@1#. The underlying kine-matics requires at least thermal equilibration at freeze-All kinetic spectra, such as the transverse momentum andrapidity distributions, can be explained with a single freeout temperature, if one assumes a certain expansion veloafter hadronization. Attempts to describe not only the kinespectra but also the particle abundances and particle rwith a common temperature were successful, but led tdifferent temperature. Whereas the slopes of all momenspectra yield aT around 130 MeV after inclusion of thexpansion velocity, the particle ratios lead to aT of about180 MeV for a common chemical potential. Two questioarise: is the chemical freeze-out decoupled from the therfreeze-out and is the system chemically equilibratedchemical freeze-out? In a systematic study of measuredticle ratios and spectra in the framework of a thermal modHeinz has shown that for all CERN and AGS data the checal freeze-out seems to be decoupled from the therfreeze-out@2#.

Rafelksi has recently postulated that, on the basisstrange particle ratios measured in Pb1Pb at CERN, one hasto assume a phase transition from QGP to hadron gas,that at the time of hadronization the system is not in checal equilibrium@3#. To describe the kinetic spectra and paticle yields this model requires that the chemical compositof the fireball remains unchanged between hadronizationthermal freeze-out. That means all interactions duringtime interval have to be elastic. This model is in agreemwith the fact that all systems frome1e2 collisions up toAAcollisions seem to yield the same chemical freeze-out tperature@1#. Thus, the shapes of the particle spectra michange due to rescattering, but the particle abundancesratios remain constant from hadronization on, a theory whis commonly referred to as ‘‘sudden hadronization.’’ Thprocess explicitly prohibits hadrochemical equilibratithrough rescattering. Only elastic processes can be emplto explain the difference between the temperature calculfrom particle ratios and the temperature calculated fromnetic particle spectra. Thus, the ratios effectively reflecttemperature at hadronization, whereas the kinetic freeze

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temperature describes the actual freeze-out~last elastic finalstate interaction!. The fact that the hadronization temperatuis about 180 MeV, close to the critical temperature, canviewed as an indicator that the system actually crossephase transition. However, the fact that all inelastic scating processes cease at hadronization time is a very stconstraint. Hadronic population ratios are, in this theory,result of the hadronization mechanism and not causedinteractions during the hadronic rescattering phase. The margument employed is that the relaxation times of all relevhadronic channels is well abovet53 fm/c at which timethe temperature of the system has dropped belowT5185MeV which leads to a small probability of chemical equibration after hadronization. Also based on results of HanbBrown–Twiss ~HBT! measurements, Stock argues that ttime between hadronization and thermal freeze-out misimply be too short to develop a significant inelastic crosection contribution@4#. The required time interval for hadronic processes to adjust the strangeness content is longthe chemical rates are small since the production of pairstrange hadrons carries a large energy penalty factorStock’s scenario it takes upward of 3 fm/c to equilibratestrangeness in the hadronic phase, a time span that, hetulates, is not available owing to the rapid expansion preving at hadronization time.

Surprisingly, a partonic cascade approach, which is vdifferent from the model described above, yields resusimilar to a thermal sudden hadronization theory. GeigerEllis @5# have shown that at large incident energies the hronization process can determine the final particle rabased on its combined nonperturbative mechanisms ancan even lead to direct chemical equilibrium. The final mtihadronic state materializes into maximal entropy~equilib-rium! straight out of the partonic phase. Although certaremnants of the initial particle structure functions remain,hadron yield near midrapidity stems mostly from the initparton cascade processes. In this theory, the partonic pexhibits e>2 GeV/fm3 until about 2 fm/c. Hadronizationoccurs after a formation time of about 1 fm/c and a mixedphase of hadrons and partons ends att'20 fm/c, whene isless than 0.2 GeV/fm3. Although the concept of sudden hadronization is lost in this theory, due to the required lomixed phase between partons and hadrons, the resultscomparable to a thermal model without inelastic final st

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R. BELLWIED, H. CAINES, AND T. J. HUMANIC PHYSICAL REVIEW C62 054906

interactions. In addition there are nonequilibrium proceswhich may cause a rapid hadronization of the QGP@6#. Inparticular nucleation theories that assume supercoolingrapidly expanding quark gluon plasma show a hadroprodtion rate comparable to thermal hadronization theories@7#.Spieleset al. nicely documented the similarities and diffeences between a static thermal hadronic source and nonlibrium hadronization of a QGP droplet@8#. While the partonmodels require a mixed phase for the proper kinematicpansion of the system, they do not require any final sinteractions to describe the measured particle abundan

FIG. 1. Particle abundances as a function of time between hronization and kinematic freeze-out.

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Kapusta and Mekjian@9# derived estimates for the dynamicequilibration ~relaxation! times of quark flavors in a quarkgluon gas and deduced predictions for several equilibriabundance ratios. These ratios are in good agreementthe data, which led Stock@4# to postulate that the equilibrated ratios formed in prehadronization reactions dochange during the final state interactions.

The main motivation for this paper is the fact that modynamic transport codes, which in the past were successapplied to describe data from CERN and the AGS, behvery differently from either the parton cascade or the thermsudden hadronization approach. These models employ msured inelastic scattering cross sections for different partspecies to describe the hadronic transport from hadronizato kinetic freeze-out. Many of the relevant particle specare susceptible to number changing interactions, in particinelastic meson interactions lead to sizable contributionsthe hyperon production cross section. Both strangenessation interactions~e.g.,pN→KL) and strangeness exchanginteractions~e.g., KN→pL) should affect strange baryoproduction.

Pratt and Haglin have recently pointed out that numchanging cross sections after hadronization seem to beevant for the description of the pion abundance at CE@10#. They have shown that hadrons interact several timbefore the freeze-out temperatures are reached if onesumes that binary modeling starts atT5160 MeV in achemically and thermally equilibrated system. It is intereing to note that the chemical equilibration times for stranand non strange quarks shown by Stock and Pratt and Haare very comparable. An independent study by Humausing an early version of the transport code which isbasis of this paper, shows that pions interact on averagetimes between hadronization and freeze-out@11#. The timefrom hadronization to freeze-out is quite long~about 15 fm/

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FIG. 2. Time dependence of channel contributions to theL cross section in~a! Pb1Pb and~b! S1S.

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c) in particular for the lightest mesons. If these transpmodels are correct then we should be able to simulatquantitative dynamic evolution of all particle abundancthrough the rescattering phase.

HADRONIC PARTICLE RATIO SIMULATIONS

In the following we attempt to prove that in particular thabundance of the singly strange baryon, theL, is affected bya series of inelastic rescattering processes by the timemal freeze-out is accomplished. We have employed the

FIG. 3. Proof of detailed balance in a nonexpanding systemfixed maximum radius (r 510 fm!.

FIG. 4. Relation between chemical freeze-out time and kimatic freeze-out time for measurableL particles.

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namic transport code described in Ref.@11# to model theinitial state of the system at hadronization and the subseqhadronic rescattering to freeze-out. This code has bshown to well represent the features and dynamic depdences of the transverse mass spectra and HBT observfor CERN SPS Pb1Pb data@11#. In addition to the alreadyimplemented cross sections for pions, kaons, nucleonstheir associated resonances@12#, we have augmented thcode to include elastic and inelastic rescattering ofL bary-ons. Certainly the application of vacuum cross sectionsto be considered an approximation at the relevant partand energy densities. Attempts to parametrize the dendependence of measured interaction cross sections are uway @13#, but for the time being most event generators,particular cascade programs, quite successfully applymeasured cross sections to describe relativistic heavymeasurements. We expect the inelastic and elastic crosstions to increase at higher densities, in which case our calations of the effect of rescattering between hadronizatand thermal freeze-out should be considered a lower lim

The initial conditions in our model are described bycommon temperature and spatial extension. All partichadronize at a proper time of 1 fm/c. A Bjorken-type geom-etry was used to simulate the dynamic evolution of the fiball from hadronization time. No initial radial flow is needefor the calculations to agree with data@11#. Calculations arecarried out assuming initial parameter values and multiplties for each type of particle. In the last stage of the calcution, the freeze-out and decay momenta and space-timesused to produce single-particle and two-particle observasuch as pion, kaon, nucleon, and lambda multiplicities, traverse momentum and rapidity distributions, and two-boscorrelation functions. The values of the initial parametersthe calculation are constrained to give observables whagree with available measured hadronic observables. Fo

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FIG. 5. Chemical freeze-out time for measurableL particles.

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FIG. 6. ~a! Kinetic freeze-out time for the different components of theL spectrum.~b! Comparison between the kinetic freeze-out timdistributions of measurableL ’s, p ’s, and protons.

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calculations studied in the present work, measured obsables were obtained from the NA44@14,15# and NA49@16–18# experiments. Initial resonance multiplicity fractions ataken from the HELIOS experiment@19#.

Figure 1 shows the effect of rescattering on all four revant particle abundances (p, K, nucleons, andL). In thecase of the nucleons, the nucleon resonance productionsubsequent decay leads to no change in the number of nons and is thus considered an elastic process, whereas ipion case, generation from resonance decay is considernumber changing process~e.g., Np→Dp→Npp). Thesame is true for vector meson production (r,f,v) and decayinto pions. Figure 1 shows that the nucleon abundancclose to constant, but the meson and strange baryon adances display a steady increase. One of the large contring elastic reactions ispp→r→pp which is fast and keepsthe number ofr ’s in equilibrium but does not change thpion number. Reactions that change the overall pion numsuch aspp→h8→ppp are generally slower and thus dnot have a large effect on the chemical state formed at hronization. In addition reactions that conserve the net numof strange quarks occur rapidly, e.g.,KN→Lp, but reac-tions that change the number of strange quarks, e.g.,pN→KL, are slow as they require a strange and an antistrahadron to either interact or to be produced jointly.

Figure 2 shows the relative contributions of all relevareaction channels to theL abundance as a function of timfor the Pb1Pb and the S1S systems at the SPS. In the Pcase the directL production at hadronization yields onlaround 50% ('30 L ’s! of the total yield at kinetic freezeout. The additional enhancement can be attributed toL pro-duction well after hadronization. In comparison, in tsmaller S1S system the rescattering contributes only ab10% of the totalL yield. It should be noted that, in eithe

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case, the contribution from the strangeness creating, ethermic reaction (p1N→K1L) is considerably higherthan the one from the simple exothermic strangenesschange reaction (K1N→p1L). We attribute this effect tothe large pion density in the rescattering volume.

The plots also show thatL production and annihilationthrough inelastic rescattering stops after a certain ti~chemical freeze-out!. It turns out, though, that this is not duto our model reaching strangeness chemical equilibriumrather based on the rapid expansion and the resulting droparticle density. After around 30 fm/c the particle density issimply too small to support further inelastic scatterinwhich produce strange baryons. To prove that our moproperly invokes detailed balance in our cross sectionschose a fixed maximum radius of 10 fm and plotted theLproduction and annihilation channels as a function of timFigure 3 shows that, for the case of a nonexpanding systhe totalL yield becomes constant as the annihilation yieequals that of the production at long times. Thus, chemequilibration is possible if the expansion velocity is sufciently small. However, our calculations agree with oththermal model calculations in predicting that the Pb1Pb sys-tem at CERN SPS energies does not reach chemical equrium before freeze-out. Based on Fig. 1 we can concludeall particle ratios that include theL yield will undergo sig-nificant changes during rescattering, a fact which is incopatible with the notion of sudden hadronization. If we forsudden hadronization in our code by simply turning offinelastic rescattering modes, our model calculations leadmuch smaller HBT radius in both thepp andKK channelscompared to the actually measured radii.

Figure 4 shows the relation between kinetic and chemfreeze-out time for everyL measurable in the Pb1Pb systemafter kinetic freeze-out. It is apparent that only about half

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the emittedL ’s show a chemical freeze-out time consistewith the hadronization time, which is required for suddhadronization (tchem50,tkin5any). ManyL ’s ~about 25%!are actually produced in their final rescattering step (tchem5tkin). The one-dimensional projection of the chemicfreeze-out time distribution for theL ’s, as shown in Fig. 5,reveals that the chemical freeze-out occurs fast but it is‘‘sudden.’’ 90% of theL ’s are chemically frozen out afterfm/c, but of those about 40% freeze-out after hadronizatiDetails of the according kinetic freeze-out times are dplayed in Fig. 6. Figure 6~a! shows the difference in kineticfreeze-out time between the direct and the producedL com-ponent. The production after rescattering significantlyhances the average freeze-out time and alters the shapedistribution. But theL freeze-out is still peaked significantlearlier than the pion and proton freeze-out in our model@11#as seen in Fig. 6~b!. The main reason is that the inelastic aelastic cross sections forL induced reactions are mucsmaller than the proton or pion interaction cross sectioThis trend will continue for multistrange baryons andshould lead to a very early decoupling of theV from thefireball @20#. For multistrange baryons the notion of suddhadronization seems thus more sensible.

Figure 7 shows the transverse mass spectrum of thezen outL ’s. The model is in perfect agreement with the dameasured by NA49@21#, as it is for the pions and nucleon

CONCLUSIONS

Based on our calculations we conclude that neither thLyield nor any particle ratio includingL ’s are suitable ther-mometers for the hadronization temperature. These measments may still be indirect QGP indicators in any transpcode description, though, simply through determination

FIG. 7. Transverse mass spectrum forL from our calculations.The extracted slope parameter is seen to agree with data~NA49!.

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the required initial hadronization conditions~initial tempera-ture and energy density!. This is a departure from the original wisdom that only leptonic probes can act as real inditors of the initial conditions.

We have shown that theL abundance is strongly affecteby final state interactions and is not reproducible understrict assumption of sudden hadronization, which requiresinelastic rescattering after hadronization. This does not pclude a very short mixed phase and there is no attempt inpaper to describe the prehadronization phase or hadrontion itself. Our model simply estimates the considerable ctributions between hadronization and thermal freeze-out.probability of inelastic scattering is sufficiently large theven during the rather short time lapse between hadrontion and thermal freeze-out the yields change consideraThus, conclusions drawn from the strange baryon ratiosCERN concerning a QGP phase transition are probablyoversimplification regarding the actual interactions insidefireball.

Our simulations agree with other thermal models thatchemical properties are frozen well before the kinetic spebut we also demonstrate that hadronization and chemfreeze-out mostly do not coincide for strange baryons pduced in Pb1Pb. Thus the hadronization should occur aboT5180 MeV which was deduced from the ratios as the teperature for chemical freeze-out. Indeed, in our calculatthe system hadronizes at 213 MeV, well above the crititemperature. It is also evident that both, theL chemical andthermal freeze-out, occur much earlier than the protonpion freeze-out. Recent detailed simulations of multistranbaryon ratios and spectra, seem to indicate that this edecoupling follows a trend as a function of the strangencontent. This effect can be attributed to differences inrescattering probabilities@22#. In particular the apparent lacof flow in the V transverse momentum spectrum couldcorrelated with the very highV particle ratios@23#. Bothpoint at a uniquely different production mechanism for tV. In Dumitru’s calculations theV ’s undergo on averagetwo collisions after hadronization, whereas the numbercreases to 3.5 and 5 forJ and L, respectively@22#. Thisearly decoupling of theV led van Heckeet al. to postulatethat this is the reason for the lower emission temperaturesV ’s as measured at the SPS@20#. Our calculation indicatesthat even theL thermally freezes out after a few fm/c, due tothe rather small elastic scattering cross section duringrescattering phase. It is our goal to extend our transport cto include the multiply strange baryons to determine mquantitatively the dynamic differences between strangemultistrange baryons.

The complete measurement of strange particle producat RHIC, from the kaon to theV, will be an important ex-ercise. Our SPS study shows that final state interactions hto taken into account in order to properly describe singltrange particle production. Thus, kaon andL measurementsalone might not lead to conclusive proof of the formationa QGP. However, plans for detailed measurementsstrangeness production up to and including theV are wellunderway at RHIC and we are looking forward to this excing new era in relativistic heavy ion physics.

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