- Email: [email protected]
Multiparticle dynamics of Pb+Pb collisions at the CERN SPS
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
Multiparticle dynamics of Pb+Pb collisions at the CERN SPS G. Roland a for the NA49 collaboration aInstitut fiir Kernphysik, Universit~it Frankfurt, August-Euler Str. 6, 60486 Frankfurt/Main, Germany We discuss recent results on single- and multiparticle distributions obtained for Pb+Pb collisions at 158 GeV/nucleon at the CERN SPS. By combining information from single particle spectra and Bose-Einstein correlation of like-sign pions with the results of an event-by-event analysis of charged hadron phase space distributions, a consistent picture of a particle source in local thermal equilibrium undergoing a longitudinal and transverse expansion emerges. We discuss further measurements that will allow a quantitative analysis of thermal equilibration based on event-by-event correlation measurements. This will provide vital information on the question whether conditions suitable for the existence of an equilibrated deconfined state of partonic matter, the quark gluon plasma, are reached in the early stages of these collisions.
1. Introduction The ultimate goal of the CERN heavy ion program and similar studies foreseen at future collider facilities is the production and characterisation of an extended volume of equilibrated partonic matter, the quark gluon plasma (QGP) [1]. A phase transition to this state has been predicted by lattice QCD calculations (see [2] for recent reviews) for energy densities above 12 GeV/fm 3 and temperatures above a critical value Te of about 150 to 200 MeV/c. In heavy ion collisions, if a QGP were indeed formed, it would be established in the early stage of the collision, after an initial pre-equilibrium state. Subsequently we expect a re-hadronisation of the partonic matter, followed by hadronic rescattering, chemical freeze-out and finally thermal freezeout. Clearly, experimental information on the dynamics of the source evolution is required to reconstruct the initial conditions of the strongly interacting system and understand whether this picture is realized. Measurements of the total transverse energy produced in central P b + P b collisions at the CERN SPS indicate that energy densities of 2-3 GeV/fm 3 are indeed reached in the early collision stage [3]. In this paper we will use information from singleparticle spectra and two-particle Bose-Einstein correlations to reconstruct typical sizes, time0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. Pll S0920-5632(98)00352-1
scales and expansion velocities for the hadronemitting system. Additional insight is provided by event-by-event measurements of global observables [4]. By analyzing the fluctuations of average transverse momentum of charged hadrons from event to event, we can obtain quantitative information on thermal equilibration in the collision.
2. E x p e r i m e n t a l Setup and Data selection The results shown here are based on a data set of 160.000 central P b + P b collisions collected by the NA49 experiment at the CERN SPS. The upper 4% of the total inelastic cross section were selected by a trigger on the energy deposited in the NA49 forward calorimeter. This corresponds to an impact parameter range of b < 3.5 fm. See [5] for a description of the NA49 experimental setup. For the investigation of multiplicity and momentum space fluctuations, only particles with transverse momentum 0.005 < PT < 1.5 G e V / c and rapidity 4 < y~ < 5.5 with a measured track length of more than 2 m in the NA49 Main T P C s outside the magnetic field were used. Using these cuts, we expect to achieve the overall smallest systematical errors at the present state of the analysis. In this region the momentum resolution is on the order of A p / p = 0.3%, with a reconstruction efficiency greater 95% and a two-track resolution that only affects particles closer than Qinv ~ 8 MeV/c in
262
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
momentum space.
Pb+Pb, NA49 Preliminary >
In this chapter we will present results for singleparticle transverse momentum spectra, two-pion Bose-Einstein correlations and global event-byevent fluctuations obtained for central Pb+Pb collisions by the NA49 experiment. Combining these measurements provides quantitative information on thermalization and collective transverse expansion of the particle source.
3.1. Single-Particle distributions Results on single particle transverse momentum distributions for Pb+Pb collisions at 158 GeV/nucleon have been presented by the NA44 and NA49 collaborations recently [5],[6]. Particular emphasis has been put on systematic studies of the inverse slope parameter T obtained from a fit to the transverse mass spectrum using
1 dn _ C e x p ( r o T ) . T mT dmT
500
,-
3. R e s u l t s
(1)
A compilation of results from NA49 on inverse slopes for hadrons of different mass can be found in Fig. 1. One clearly observes an increase in the inverse slope parameter, for both mesons and baryons, with increasing particle mass. It is particularly interesting to note that protons and phi mesons, which are similar in mass, also show very similar inverse slope parameters of around 300 MeV [7]. Also the measurement by NA49 for the deuterons shows an inverse slope of close to 400 MeV [8]. Clearly an inverse slope of this magnitude, more than two times higher than the limiting 'Hagedorn' temperature, can not be interpreted as a temperature of a hadronic system. A coherent description of the particle spectra in terms of thermodynamical variables needs additional parameters beyond a single common temperature to describe the experimental data. The simplest solution is the introduction of a common transverse expansion velocity/~±, which naturally leads to the observed dependence of inverse slopes on the particle mass [9]. It has been shown in several publications that, with the choice of an ap-
I
I
1
+
d)
400
..+
300
200 100 O0
i
I
0.5
i
I
1
,
I
,
1.5
particle mass [GeV/c 2] Figure 1. Inverse slope parameters for various hadron species in central Pb+Pb collisions.
propriate velocity profile, such an extended model can indeed describe the inverse slope systematics with just two parameters, a common freeze-out temperature Tf and a common flow velocity ~± [10],[11],[12],[13]. It has also been shown however that the minima obtained in the goodness-of-fit parameter as a function of the parameters T / a n d f~± are rather shallow, with a strong correlation between the parameters, leading to a large uncertainty in particular in the determination of ~3± [10]. To resolve this ambiguity independent information on the flow velocity is required. As we will discuss in the next section, such information can be obtained in a quantitative fashion from two-particle Bose-Einstein correlations. The success of the flow analysis in fitting the data however does not prove that the the observed non-thermal velocity field at the final particle freeze-out is of hydrodynamical origin. This is of particular importance for any attempts to extrapolate from the observed particle densities
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
NA35,NA49Preliminary >
300
........ []@ h ........ *' '+'''
250 200 150 i
,
,
,,,,,I
10
102
103
Figure 2. Inverse slope parameters for net protons and negative hadrons as a function of system size measured by the number of participating nucleons.
and flow velocities backwards in time, trying to evaluate the conditions in the early phase of the collision. Several authors have recently argued that the systematic trends observed in the single particle distributions can in fact also be explained in models that explicitely do not assume any kind of thermalization of the particle emitting source [14],[15]. In this models the velocity field observed in the single particle spectra is ascribed to initial state scattering, where successive parton scattering leads to a rotation of the collision axis for multiple collisions. This effectively translates part of the original longitudinal momenta into transverse momentum and results in a broadening of the PT distribution in the reference frame of the primary collision axis. In [14] the initial state scattering is described as a 'random walk' in transverse rapidity. This can be parametrized as particle emission from sources
263
('fireballs') that have a gaussian transverse rapidity distribution, where the width of the distribution is calculated as a function of the number of scatterings the nucleons undergo in the collision. This number increases with the size of the nuclei involved, which in this model explains the observed system size dependence of the inverse slopes (Fig. 2). The parameters of the model are fixed from data on proton+proton collisions and from fits to transverse momentum spectra from proton+nucleus collisions. While the quantitative agreement with the nucleus+nucleus data obtained for the random walk model is not as good as that for fits within a two-parameter hydrodynamical model, one has to keep in mind that the description of the evolution of the transverse momentum spectra provided by the random walk model is essentially parameterfree. In [13] it is argued that present experimental data on transverse momentum spectra can not distinguish between hydrodynamical and initial scattering models. We will describe how additional tests that constrain random walk models can be performed using the event-by-event measurement of the average transverse momentum of charged hadrons [16],[17]. This measurement tests correlations in the particle source at large scales in momentum, i.e. averaged over large regions of phase space. Thermalization of the source in general implies the absence of such correlations. A quantitative measurement can be performed by comparing the correlations observed in nucleus+nucleus reactions with those expected for a superposition of nucleon-induced collisions [18]. 3.2. B o s e - E i n s t e i n correlations Bose-Einstein correlations of identical particles are a well established tool for studying the spacetime evolution of the particle emitting system in high-energy collisions. The theoretical framework for this measurement has been considerably refined recently [19]. Using the dependence of the two-particle correlation function in relative momentum on the average pair momentum, which reflects the correlation between position and momentum of particles in an expanding source, it is
264
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
now possible to extract dynamical parameters like expansion velocities and lifetime from the correlation measurement. Results from the CERN SPS for 32S induced heavy ion collisions have been published by the NA44 and NA35 collaborations. While this unambigously established the existence of a longitudinal velocity field and allowed a first estimate of the total source lifetime, experimental difficulties and the relative weakness of the signal in the small systems available at that time, did not allow a Clear determination of the parameters of the transverse expansion. Recent advances in detector technology have allowed a measurement of two-particle BoseEinstein correlations over a very large acceptance and with sufficient statistics to study the dependence of the observed correlation parameters on rapidity and transverse momentum in fine bins. It has been demonstrated in [11] that a convenient way to parametrize the correlation function is given by the Yano-Koonin-Podgoredsky formalism, where the correlation function is written as C2(Q, K)
=
1 + ~exp[-Q~_R~_ -
Qo)Rll
-
-
+
Here the colliding system is described as a succession of local source elements along the longitudinal axis, where each element is characterized by its space-time extent ( RFl ( k ± , y , , ) , R ± ( k ± , y~,~ ), Ro(kz, y ~ ) ) and longitudinal velocity fl(k±, y~,) and k± and y . , are the average transverse momentum and rapidity of the pion pairs. The parametrization is then fitted to the measured three dimensional correlation function in bins of rapidity and transverse momentum. As we are dealing with central events, we average over the azimuthal angle. The experimental correlation function was corrected for the final state Coulomb interaction using a scheme based on the measured correlation function for unlike-sign pions [20],[21] Further experimental details are described in [20]. In particular the effect of performing the correlation analysis for negative hadrons originating at the main vertex, rather than for positively identified pions, was investigated and found to be small (2 to 6 %) for the extracted radius parameters. The A parameter, which is more strongly affected,
Pb+Pb
-~ h'h" (100 MeV/c < kT < 200 MeVlc) /j//
5
.
,'¢
,~
4
@,-'~ 3
1
2
3
4
5
Y,~
Figure 3. Rapidity of the pion-emitting source element obtained from the YKP fit versus the rapidity of the emitted pion pairs.
will not be discussed here. In Fig. 3 the rapidity of the source elements as obtained from the fit of the correlation function to the data in bins of rapidity, is plotted versus the average pair rapidity of the pion pairs contributing to the correlation function. For an ideal boost-invariant source, the points would fall exactly on the dotted line. We see that for Pb+Pb collisions at SPS energy the data lie close to that scenario, with somewhat larger deviations further away from the CMS rapidity. This is expected due to the finite size of the available longitudinal phasespace. While Fig. 3 demonstrates a somewhat trivial aspect of the dynamics of the particle source, Fig. 4 shows that also for the transverse radius parameter a clear dependence on transverse momentum can be observed. This provides information on the strength of the transverse velocity field, as was shown in [11]. By fitting the source
265
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
NA49 Preliminary T
....
' ....
' ....
' ....
'
.
~- 1 6 0
5
140
41
120 100 800 0.2 0.4 0.6 0.8 1
~VT_ 377,± ?.5,Ce,,v-i,.... 30 0.1 0.2 0.3 0.4 0.5 kT [OeV] Figure 4. Dependence of the transverse correlation length R± on the average transverse momentum k± of the pion pair.
function from [11] to the correlation data, a value of ~ / T f = (3.7 ± 1.5) G e V -1 is determined. Again, the values for the expansion velocity and the freeze-out temperature are correlated, as was the case for the fit to the single particle spectra. But for the two-particle correlations, as is shown in Fig. 5, the valley for the )/2 minimum runs almost perpendicular to that for the single particle spectra. By combining the information from single- and two-particle spectra one can therefore tightly constrain fl± and Tf at the same time. The allowed region in Fig. 5 suggests Tf ~, 120 MeV and and transverse velocity of ~z ~ 0.55. So far the influence of resonance decays has been not taken into account consistently in this analysis.
3.3. Event-by-event analysis As has been shown in the previous chapter, a two-parameter fit of a simple source function
Figure 5. Freeze-out temperature T versus transverse flow velocity ~±. The curves show constraints from fits to single-particle m T spectra and two-particle correlation analysis.
inspired by hydrodynamics can successfully describe the single- and two-particle spectra of P b + P b collisions at the SPS. However the success of the fit does not necessarily imply that the evolution of the source has indeed been governed by hydrodynamics. The considerations that lead to the formulation of initial state scattering models show, that a large part of the observed effect could in fact be caused by non-equilibrium phenomena. Clearly this has to be considered very carefully when using the parameters obtained from the fit to extrapolate backwards in time, trying to estimate the initial conditions of the collision system. Using the data from the NA49 large acceptance spectrometer, it is now possible to obtain previously unavailable information on the thermodynamic properties of the system created in heavyion collisions by studying the size of large-scaie
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
266
Preliminary
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ill,I
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Figure 6. Multiplicity distribution of accepted charged hadrons in the event-by-event fluctuation analysis.
Figure 7. Distribution of event-by-event average transverse momentum and average rapidity for charged hadrons (assuming the pion mass).
fluctuations. For a thermally equilibrated system, the size of these fluctuations is directly related to fundamental properties of the system. The fluctuation measurement can also be used to detect correlations that contradict the assumption of thermal equilibration. In this investigation we will concentrate on the measurement of fluctuations in event-by-event transverse momentum in a large region of phase space detected by the NA49 Main TPCs. The multiplicity distribution for the accepted particles per event is shown in Fig. 6. The distribution of average transverse momentum and average rapdity per event are shown in Fig. 7. These distributions can well be approximated by gaussian fits and do not show evidence for classes of 'anomalous' events above a level of one in 104, which would be compatible with event pile-up in our trigger window. Event-by-event fluctuations in nuclear collisions
are usually dominated by the trivial variation in impact parameter from event to event and the purely statistical variation of the measured quantities. A statistical method that allows us to remove these trivial contributions and to determine the dynamical event-by-event fluctuations has been proposed in [18]. Following this reference we define for every particle i zi = PTi -- PT
where ~-~ is the mean transverse momentum of accepted particles averaged over all events. Using zi we calculate for every event N
Z -~ E z i i=1
where N is the number of particles in the event. With this definition we obtain the following measure to characterise the degree of fluctuation in
G Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
267
the PT distribution from event to event:
Preliminary _
(2)
where {N) and {Z 2} are averages over all events. The variable Opt was orginally proposed in [18], following the observation that particles in nucleon+nucleon collisions are not produced independently [22]. The data show large scale correlations that lead to e.g. a correlation between event multiplicity and the average PT of the particles. This property can be used to probe the dynamics of nucleus+nucleus collisions by measuring to which degree these correlations are reduced or increased when going to nucleus+nucleus collisions. Here, (~PT as a measure of fluctuations has two important properties. For a large system (i.e. a nucleus+nucleus collision) that is a random superposition of many independent elementary systems (i.e. nucleon+nucleon collisions), (I)pr has a constant value that is identical to that of the elementary system. If a P b + P b collision was a simple superposition of nucleon+nucleon collisions, the value of (I)pT would therefore remain constant, independent of the number of participants. If on the other hand the large system consists of particles that have been emitted independently, (~PT assumes a value of zero. (I)pr therefore provides us with a scale on which to characterise fluctuations in nuclear collisions relative to elementary collisions at the same energy. As a representation of the nucleon+nucleon reference point we use V7.02 of the F R I T I O F Monte-Carlo code, where the simulation of rescattering of produced particles was not turned on. In this mode P b + P b collisions are treated as simple superpositions of nucleon+nucleon collisions, which leads to a constant, non-zero value of ~PT" The magnitude of Cpr is determined by the correlation between transverse momentum and multiplicity present in nucleon+nucleon collisions at v/(s) ~ 20GeV/c, which was reported in [22] and is reproduced in the F R I T I O F simulation. The Monte-Carlo events were run through an experimental filter that included a parametrization of the acceptance, momentum and two-track res-
_
5
N+N superposition ( F r i t i o f ) ~ .
T
Pb+Pb d a t a - - ~
iiT
,
,
,
I
,
J
,/I
i
200 400 Participants Figure 8. Dynamical PT fluctuation measure (I)pr for F R I T I O F (circle) and P b + P b data (square symbol). The errors are statistical only (see text).
olution measured for the NA49 apparatus. This filter allows us to treat the simulated events with the same analysis chain as the real data. As shown in Fig. 8 the data point obtained for P b + P b gives a value of (~pr = 0.7 ± 0.5(stat.), which is significantly lower than the value obtained for superimposed nucleon+nucleon collisions of (I)pr = 4.5 ± 0.8(stat.). At the present state of the analysis we estimate a systematical error of ~ 30% on the value obtained for the simulated events, which reflects the uncertainty in the acceptance and two-track resolution as represented by the experimental filter. Based on the present result we conclude that the correlations in the particle production in nucleon+nucleon collisions, which are responsible for the large value of (I)pr in the F R I T I O F simulation, seem to be largely dissolved in the environment of real P b + P b collisions, leading to the small value for (~PT observed in the data.
268
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
This gives a quantitative measure of the approach towards randomized, independent particle emission. Qualitatively this can be explained either as the result of the hadronic rescattering phase of the collision or as a sign of particles not being produced from string excitations as in nucleon+nucleon collisions but from a homogenous thermalized state, which is also in line with models of QGP production.
3.3.1. Event-by-event fluctuations and initial state scattering The possibility to disentangle the effects of hydrodynamical flow and initial state scattering using event-by-event measurements was discussed in [17]. This study was motivated by the fact that particles in the random walk model are not produced independently. They are emitted from sources that, after an initial scattering, are moving in transverse direction. This transverse movement of every individual source is superimposed on the spectrum of particles emitted from that source. The calculation [17] of the effect of initial state scattering has been performed based on the random walk model in [14] and a simple parametrization of proton+proton data at V~ ~ 20GeV/c. The initial state scattering is included by giving the particle sources a Gaussian distribution in transverse rapidity p [14] lAB(P) (X exp(--p2/~B),
(3)
where ¢~AB= 0.44 corresponds to central Pb+Pb collisions. The results of the simulation are summarized in Fig. 9. The dashed lines show the evolution of ~pr with the width of the source transverse rapidity distribution (~AB (i.e. increasing system size). Results are shown for two limiting cases of the longitudinal distribution of particles emitted by the elementary sources (upper curve: p+p parametrization, lower curve: isotropic emission). The solid line in Fig. 9 indicates the reference value of ~pr = 4.5 MeV, for the proton+proton parametrization. For both versions of the random walk model a strong increase in @pr above
the reference line is observed with increasing system size. In contrast, the preliminary analysis of NA49 Pb+Pb data shows the opposite effect, with fluctuations that are decreased in the large system compared to the proton+proton reference.
,..,0.06
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i
i
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0.4
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~AB
Figure 9. Dynamical PT fluctuation measure (hpT for the random walk model as a function of system size ((~AB). Shown are two limiting cases for the random walk model (dashed lines, see text) and the results without initial scattering (solid line).
We conclude that the introduction of the random walk source movement does not wash out the correlations present in proton+proton collisions. Rather the correlation introduced by the transverse source velocity leads to an amplification of large scale fluctuations in transverse momentum. Further investigations of the ¢pT observable will proceed along two lines. We will study PT fluctuations in various microscopic models, which include a realistic description of hadronic rescattering, to differentiate between the possible explanations of the observed effect. More importantly
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
we will also collect further experimental data on (hpT in peripheral nucleus+nucleus collisions as well as nucleon+nucleon and nucleon+nucleus reactions. 4. S u m m a r y and Conclusions
Studying the dependence of the inverse slope of transverse mass distributions for hadrons in central P b + P b collisions, a significant dependence on the particle mass is observed, leading to the assumption of a collective transverse velocity field. This hypothesis is supported by the dependence of the transverse radius parameter on the pair transverse momentum observed in an analysis of two particle Bose-Einstein correlations. A fit with a source function inspired by a hydrodynamical picture of the collision yields a rather low freeze-out temperature T ~ 120 MeV and an expansion velocity of/~± ,~ 0.55c. The validity of the hydrodynamical approach assuming local thermal equilibrium is confirmed by a measurement of event-by-event transverse momentum fluctuations. This measurement shows that in P b + P b collisions large scale correlations that are present in the final state of nucleon+nucleon collisions are reduced. In a simple random walk model we observe an increase in event-by-event fluctuations due to the correlations introduced by initial state scattering, which are not compatible with the data. This lends additional support to a picture of a source in local thermal equilibrium undergoing a transverse expansion. NA49 will follow these results with an extensive experimental program on nucleon+nucleon and nucleon+nucleus collisions that might finally allow to use the initially present correlations to quantitatively trace the approach of the particle source in heavy ion collisions towards thermal equilibrium and understand how the additional velocitylike components of transverse momentum are generated. REFERENCES
1. J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34 (1975) 151, E. V. Shuryak, Phys. Rep. C61 (1980) 71 and C l 1 5 (1984) 151. 995.
269
2. E. Laermann, Nucl. Phys. A610 (1996) 1; E. Laermann, Proceedings of the Hirschegg Workshop on QCD Phase transitions, 1997. 3. T. Alber et al., Phys. Rev. Lett. 75 (1995) 3814 4. R. Stock, Proceedings of a NATO Advanced Research Workshop on the Hot Hadronic Matter: Theory and Experiment, 1994, Divonne, France. 5. P. Jones et al. (NA49 Collaboration), Nucl. Phys. A610 (1996) 188c. 6. N. Xu et al. (NA44 Collaboration), A610 (1996) 175c. 7. V. Friese et al. (NA49 Collaboration), Proceedings of Strangeness in Quark Matter 1997, Thera, Greece, April 1997 8. NA49 spectral paper, LBL draft 9. J. Sollfrank et al., Z. Phys. C52 (1991) 593; E. Schnedermann et al., Phys. Rev. C48 (1993) 2462. 10. B. K~mpfer, preprint FZR-149, hep-ph/9612336 preprint. 11. U. Heinz, Nucl. Phys. A610 (1996) 264 12. U. Wiedemann et al., nucl-th/9611031 preprint. 13. J. Alam, J. Cleymans, K. Redlich, H. Satz, nucl-th/970742 preprint 14. A. Leonidov, M. Nardi, H. Satz, Nucl. Phys. A610 (1996) 124. 15. S. Jeon, J. Kapusta, nucl-th/9703033 preprint 16. G. Roland, Proceedings of the Hirschegg Workshop on QCD Phase transitions, 1997. 17. M. Gaidzicki, A. Leonidov, G. Roland, IKFHENPG 5-97 preprint. 18. M. Galdzicki, St. Mr6wczyfiski, Z. Phys. C54 (1992) 127. 19. S. Pratt, Phys. Rev. D33 (1986) 1314, A. N. Maklin and Y. M. Sinyukov, Z. Phys. C39 (1988) 69; G. Bertsch, Nucl. Phys. A498 (1989) 173; S. Chapman, J. R. Nix and U. Heinz, Phys. Rev. C52 (1995) 2694; S. Pratt, Phys. Rev. D33 (1986) 1314. 20. H. Appelsh~iuser, PhD thesis Frankfurt 1997; S. SchSnfelder, PhD thesis, MPI Munich 1997. 21. T. Alber et al., Z. Phys. C73 (1997) 443. 22. T. Kafka et al., Phys. Rev. C16 (1977) 1261
Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
Multiparticle dynamics of Pb+Pb collisions at the CERN SPS G. Roland a for the NA49 collaboration aInstitut fiir Kernphysik, Universit~it Frankfurt, August-Euler Str. 6, 60486 Frankfurt/Main, Germany We discuss recent results on single- and multiparticle distributions obtained for Pb+Pb collisions at 158 GeV/nucleon at the CERN SPS. By combining information from single particle spectra and Bose-Einstein correlation of like-sign pions with the results of an event-by-event analysis of charged hadron phase space distributions, a consistent picture of a particle source in local thermal equilibrium undergoing a longitudinal and transverse expansion emerges. We discuss further measurements that will allow a quantitative analysis of thermal equilibration based on event-by-event correlation measurements. This will provide vital information on the question whether conditions suitable for the existence of an equilibrated deconfined state of partonic matter, the quark gluon plasma, are reached in the early stages of these collisions.
1. Introduction The ultimate goal of the CERN heavy ion program and similar studies foreseen at future collider facilities is the production and characterisation of an extended volume of equilibrated partonic matter, the quark gluon plasma (QGP) [1]. A phase transition to this state has been predicted by lattice QCD calculations (see [2] for recent reviews) for energy densities above 12 GeV/fm 3 and temperatures above a critical value Te of about 150 to 200 MeV/c. In heavy ion collisions, if a QGP were indeed formed, it would be established in the early stage of the collision, after an initial pre-equilibrium state. Subsequently we expect a re-hadronisation of the partonic matter, followed by hadronic rescattering, chemical freeze-out and finally thermal freezeout. Clearly, experimental information on the dynamics of the source evolution is required to reconstruct the initial conditions of the strongly interacting system and understand whether this picture is realized. Measurements of the total transverse energy produced in central P b + P b collisions at the CERN SPS indicate that energy densities of 2-3 GeV/fm 3 are indeed reached in the early collision stage [3]. In this paper we will use information from singleparticle spectra and two-particle Bose-Einstein correlations to reconstruct typical sizes, time0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. Pll S0920-5632(98)00352-1
scales and expansion velocities for the hadronemitting system. Additional insight is provided by event-by-event measurements of global observables [4]. By analyzing the fluctuations of average transverse momentum of charged hadrons from event to event, we can obtain quantitative information on thermal equilibration in the collision.
2. E x p e r i m e n t a l Setup and Data selection The results shown here are based on a data set of 160.000 central P b + P b collisions collected by the NA49 experiment at the CERN SPS. The upper 4% of the total inelastic cross section were selected by a trigger on the energy deposited in the NA49 forward calorimeter. This corresponds to an impact parameter range of b < 3.5 fm. See [5] for a description of the NA49 experimental setup. For the investigation of multiplicity and momentum space fluctuations, only particles with transverse momentum 0.005 < PT < 1.5 G e V / c and rapidity 4 < y~ < 5.5 with a measured track length of more than 2 m in the NA49 Main T P C s outside the magnetic field were used. Using these cuts, we expect to achieve the overall smallest systematical errors at the present state of the analysis. In this region the momentum resolution is on the order of A p / p = 0.3%, with a reconstruction efficiency greater 95% and a two-track resolution that only affects particles closer than Qinv ~ 8 MeV/c in
262
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
momentum space.
Pb+Pb, NA49 Preliminary >
In this chapter we will present results for singleparticle transverse momentum spectra, two-pion Bose-Einstein correlations and global event-byevent fluctuations obtained for central Pb+Pb collisions by the NA49 experiment. Combining these measurements provides quantitative information on thermalization and collective transverse expansion of the particle source.
3.1. Single-Particle distributions Results on single particle transverse momentum distributions for Pb+Pb collisions at 158 GeV/nucleon have been presented by the NA44 and NA49 collaborations recently [5],[6]. Particular emphasis has been put on systematic studies of the inverse slope parameter T obtained from a fit to the transverse mass spectrum using
1 dn _ C e x p ( r o T ) . T mT dmT
500
,-
3. R e s u l t s
(1)
A compilation of results from NA49 on inverse slopes for hadrons of different mass can be found in Fig. 1. One clearly observes an increase in the inverse slope parameter, for both mesons and baryons, with increasing particle mass. It is particularly interesting to note that protons and phi mesons, which are similar in mass, also show very similar inverse slope parameters of around 300 MeV [7]. Also the measurement by NA49 for the deuterons shows an inverse slope of close to 400 MeV [8]. Clearly an inverse slope of this magnitude, more than two times higher than the limiting 'Hagedorn' temperature, can not be interpreted as a temperature of a hadronic system. A coherent description of the particle spectra in terms of thermodynamical variables needs additional parameters beyond a single common temperature to describe the experimental data. The simplest solution is the introduction of a common transverse expansion velocity/~±, which naturally leads to the observed dependence of inverse slopes on the particle mass [9]. It has been shown in several publications that, with the choice of an ap-
I
I
1
+
d)
400
..+
300
200 100 O0
i
I
0.5
i
I
1
,
I
,
1.5
particle mass [GeV/c 2] Figure 1. Inverse slope parameters for various hadron species in central Pb+Pb collisions.
propriate velocity profile, such an extended model can indeed describe the inverse slope systematics with just two parameters, a common freeze-out temperature Tf and a common flow velocity ~± [10],[11],[12],[13]. It has also been shown however that the minima obtained in the goodness-of-fit parameter as a function of the parameters T / a n d f~± are rather shallow, with a strong correlation between the parameters, leading to a large uncertainty in particular in the determination of ~3± [10]. To resolve this ambiguity independent information on the flow velocity is required. As we will discuss in the next section, such information can be obtained in a quantitative fashion from two-particle Bose-Einstein correlations. The success of the flow analysis in fitting the data however does not prove that the the observed non-thermal velocity field at the final particle freeze-out is of hydrodynamical origin. This is of particular importance for any attempts to extrapolate from the observed particle densities
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
NA35,NA49Preliminary >
300
........ []@ h ........ *' '+'''
250 200 150 i
,
,
,,,,,I
10
102
103
Figure 2. Inverse slope parameters for net protons and negative hadrons as a function of system size measured by the number of participating nucleons.
and flow velocities backwards in time, trying to evaluate the conditions in the early phase of the collision. Several authors have recently argued that the systematic trends observed in the single particle distributions can in fact also be explained in models that explicitely do not assume any kind of thermalization of the particle emitting source [14],[15]. In this models the velocity field observed in the single particle spectra is ascribed to initial state scattering, where successive parton scattering leads to a rotation of the collision axis for multiple collisions. This effectively translates part of the original longitudinal momenta into transverse momentum and results in a broadening of the PT distribution in the reference frame of the primary collision axis. In [14] the initial state scattering is described as a 'random walk' in transverse rapidity. This can be parametrized as particle emission from sources
263
('fireballs') that have a gaussian transverse rapidity distribution, where the width of the distribution is calculated as a function of the number of scatterings the nucleons undergo in the collision. This number increases with the size of the nuclei involved, which in this model explains the observed system size dependence of the inverse slopes (Fig. 2). The parameters of the model are fixed from data on proton+proton collisions and from fits to transverse momentum spectra from proton+nucleus collisions. While the quantitative agreement with the nucleus+nucleus data obtained for the random walk model is not as good as that for fits within a two-parameter hydrodynamical model, one has to keep in mind that the description of the evolution of the transverse momentum spectra provided by the random walk model is essentially parameterfree. In [13] it is argued that present experimental data on transverse momentum spectra can not distinguish between hydrodynamical and initial scattering models. We will describe how additional tests that constrain random walk models can be performed using the event-by-event measurement of the average transverse momentum of charged hadrons [16],[17]. This measurement tests correlations in the particle source at large scales in momentum, i.e. averaged over large regions of phase space. Thermalization of the source in general implies the absence of such correlations. A quantitative measurement can be performed by comparing the correlations observed in nucleus+nucleus reactions with those expected for a superposition of nucleon-induced collisions [18]. 3.2. B o s e - E i n s t e i n correlations Bose-Einstein correlations of identical particles are a well established tool for studying the spacetime evolution of the particle emitting system in high-energy collisions. The theoretical framework for this measurement has been considerably refined recently [19]. Using the dependence of the two-particle correlation function in relative momentum on the average pair momentum, which reflects the correlation between position and momentum of particles in an expanding source, it is
264
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
now possible to extract dynamical parameters like expansion velocities and lifetime from the correlation measurement. Results from the CERN SPS for 32S induced heavy ion collisions have been published by the NA44 and NA35 collaborations. While this unambigously established the existence of a longitudinal velocity field and allowed a first estimate of the total source lifetime, experimental difficulties and the relative weakness of the signal in the small systems available at that time, did not allow a Clear determination of the parameters of the transverse expansion. Recent advances in detector technology have allowed a measurement of two-particle BoseEinstein correlations over a very large acceptance and with sufficient statistics to study the dependence of the observed correlation parameters on rapidity and transverse momentum in fine bins. It has been demonstrated in [11] that a convenient way to parametrize the correlation function is given by the Yano-Koonin-Podgoredsky formalism, where the correlation function is written as C2(Q, K)
=
1 + ~exp[-Q~_R~_ -
Qo)Rll
-
-
+
Here the colliding system is described as a succession of local source elements along the longitudinal axis, where each element is characterized by its space-time extent ( RFl ( k ± , y , , ) , R ± ( k ± , y~,~ ), Ro(kz, y ~ ) ) and longitudinal velocity fl(k±, y~,) and k± and y . , are the average transverse momentum and rapidity of the pion pairs. The parametrization is then fitted to the measured three dimensional correlation function in bins of rapidity and transverse momentum. As we are dealing with central events, we average over the azimuthal angle. The experimental correlation function was corrected for the final state Coulomb interaction using a scheme based on the measured correlation function for unlike-sign pions [20],[21] Further experimental details are described in [20]. In particular the effect of performing the correlation analysis for negative hadrons originating at the main vertex, rather than for positively identified pions, was investigated and found to be small (2 to 6 %) for the extracted radius parameters. The A parameter, which is more strongly affected,
Pb+Pb
-~ h'h" (100 MeV/c < kT < 200 MeVlc) /j//
5
.
,'¢
,~
4
@,-'~ 3
1
2
3
4
5
Y,~
Figure 3. Rapidity of the pion-emitting source element obtained from the YKP fit versus the rapidity of the emitted pion pairs.
will not be discussed here. In Fig. 3 the rapidity of the source elements as obtained from the fit of the correlation function to the data in bins of rapidity, is plotted versus the average pair rapidity of the pion pairs contributing to the correlation function. For an ideal boost-invariant source, the points would fall exactly on the dotted line. We see that for Pb+Pb collisions at SPS energy the data lie close to that scenario, with somewhat larger deviations further away from the CMS rapidity. This is expected due to the finite size of the available longitudinal phasespace. While Fig. 3 demonstrates a somewhat trivial aspect of the dynamics of the particle source, Fig. 4 shows that also for the transverse radius parameter a clear dependence on transverse momentum can be observed. This provides information on the strength of the transverse velocity field, as was shown in [11]. By fitting the source
265
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
NA49 Preliminary T
....
' ....
' ....
' ....
'
.
~- 1 6 0
5
140
41
120 100 800 0.2 0.4 0.6 0.8 1
~VT_ 377,± ?.5,Ce,,v-i,.... 30 0.1 0.2 0.3 0.4 0.5 kT [OeV] Figure 4. Dependence of the transverse correlation length R± on the average transverse momentum k± of the pion pair.
function from [11] to the correlation data, a value of ~ / T f = (3.7 ± 1.5) G e V -1 is determined. Again, the values for the expansion velocity and the freeze-out temperature are correlated, as was the case for the fit to the single particle spectra. But for the two-particle correlations, as is shown in Fig. 5, the valley for the )/2 minimum runs almost perpendicular to that for the single particle spectra. By combining the information from single- and two-particle spectra one can therefore tightly constrain fl± and Tf at the same time. The allowed region in Fig. 5 suggests Tf ~, 120 MeV and and transverse velocity of ~z ~ 0.55. So far the influence of resonance decays has been not taken into account consistently in this analysis.
3.3. Event-by-event analysis As has been shown in the previous chapter, a two-parameter fit of a simple source function
Figure 5. Freeze-out temperature T versus transverse flow velocity ~±. The curves show constraints from fits to single-particle m T spectra and two-particle correlation analysis.
inspired by hydrodynamics can successfully describe the single- and two-particle spectra of P b + P b collisions at the SPS. However the success of the fit does not necessarily imply that the evolution of the source has indeed been governed by hydrodynamics. The considerations that lead to the formulation of initial state scattering models show, that a large part of the observed effect could in fact be caused by non-equilibrium phenomena. Clearly this has to be considered very carefully when using the parameters obtained from the fit to extrapolate backwards in time, trying to estimate the initial conditions of the collision system. Using the data from the NA49 large acceptance spectrometer, it is now possible to obtain previously unavailable information on the thermodynamic properties of the system created in heavyion collisions by studying the size of large-scaie
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
266
Preliminary
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ill,I
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Figure 6. Multiplicity distribution of accepted charged hadrons in the event-by-event fluctuation analysis.
Figure 7. Distribution of event-by-event average transverse momentum and average rapidity for charged hadrons (assuming the pion mass).
fluctuations. For a thermally equilibrated system, the size of these fluctuations is directly related to fundamental properties of the system. The fluctuation measurement can also be used to detect correlations that contradict the assumption of thermal equilibration. In this investigation we will concentrate on the measurement of fluctuations in event-by-event transverse momentum in a large region of phase space detected by the NA49 Main TPCs. The multiplicity distribution for the accepted particles per event is shown in Fig. 6. The distribution of average transverse momentum and average rapdity per event are shown in Fig. 7. These distributions can well be approximated by gaussian fits and do not show evidence for classes of 'anomalous' events above a level of one in 104, which would be compatible with event pile-up in our trigger window. Event-by-event fluctuations in nuclear collisions
are usually dominated by the trivial variation in impact parameter from event to event and the purely statistical variation of the measured quantities. A statistical method that allows us to remove these trivial contributions and to determine the dynamical event-by-event fluctuations has been proposed in [18]. Following this reference we define for every particle i zi = PTi -- PT
where ~-~ is the mean transverse momentum of accepted particles averaged over all events. Using zi we calculate for every event N
Z -~ E z i i=1
where N is the number of particles in the event. With this definition we obtain the following measure to characterise the degree of fluctuation in
G Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
267
the PT distribution from event to event:
Preliminary _
(2)
where {N) and {Z 2} are averages over all events. The variable Opt was orginally proposed in [18], following the observation that particles in nucleon+nucleon collisions are not produced independently [22]. The data show large scale correlations that lead to e.g. a correlation between event multiplicity and the average PT of the particles. This property can be used to probe the dynamics of nucleus+nucleus collisions by measuring to which degree these correlations are reduced or increased when going to nucleus+nucleus collisions. Here, (~PT as a measure of fluctuations has two important properties. For a large system (i.e. a nucleus+nucleus collision) that is a random superposition of many independent elementary systems (i.e. nucleon+nucleon collisions), (I)pr has a constant value that is identical to that of the elementary system. If a P b + P b collision was a simple superposition of nucleon+nucleon collisions, the value of (I)pT would therefore remain constant, independent of the number of participants. If on the other hand the large system consists of particles that have been emitted independently, (~PT assumes a value of zero. (I)pr therefore provides us with a scale on which to characterise fluctuations in nuclear collisions relative to elementary collisions at the same energy. As a representation of the nucleon+nucleon reference point we use V7.02 of the F R I T I O F Monte-Carlo code, where the simulation of rescattering of produced particles was not turned on. In this mode P b + P b collisions are treated as simple superpositions of nucleon+nucleon collisions, which leads to a constant, non-zero value of ~PT" The magnitude of Cpr is determined by the correlation between transverse momentum and multiplicity present in nucleon+nucleon collisions at v/(s) ~ 20GeV/c, which was reported in [22] and is reproduced in the F R I T I O F simulation. The Monte-Carlo events were run through an experimental filter that included a parametrization of the acceptance, momentum and two-track res-
_
5
N+N superposition ( F r i t i o f ) ~ .
T
Pb+Pb d a t a - - ~
iiT
,
,
,
I
,
J
,/I
i
200 400 Participants Figure 8. Dynamical PT fluctuation measure (I)pr for F R I T I O F (circle) and P b + P b data (square symbol). The errors are statistical only (see text).
olution measured for the NA49 apparatus. This filter allows us to treat the simulated events with the same analysis chain as the real data. As shown in Fig. 8 the data point obtained for P b + P b gives a value of (~pr = 0.7 ± 0.5(stat.), which is significantly lower than the value obtained for superimposed nucleon+nucleon collisions of (I)pr = 4.5 ± 0.8(stat.). At the present state of the analysis we estimate a systematical error of ~ 30% on the value obtained for the simulated events, which reflects the uncertainty in the acceptance and two-track resolution as represented by the experimental filter. Based on the present result we conclude that the correlations in the particle production in nucleon+nucleon collisions, which are responsible for the large value of (I)pr in the F R I T I O F simulation, seem to be largely dissolved in the environment of real P b + P b collisions, leading to the small value for (~PT observed in the data.
268
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
This gives a quantitative measure of the approach towards randomized, independent particle emission. Qualitatively this can be explained either as the result of the hadronic rescattering phase of the collision or as a sign of particles not being produced from string excitations as in nucleon+nucleon collisions but from a homogenous thermalized state, which is also in line with models of QGP production.
3.3.1. Event-by-event fluctuations and initial state scattering The possibility to disentangle the effects of hydrodynamical flow and initial state scattering using event-by-event measurements was discussed in [17]. This study was motivated by the fact that particles in the random walk model are not produced independently. They are emitted from sources that, after an initial scattering, are moving in transverse direction. This transverse movement of every individual source is superimposed on the spectrum of particles emitted from that source. The calculation [17] of the effect of initial state scattering has been performed based on the random walk model in [14] and a simple parametrization of proton+proton data at V~ ~ 20GeV/c. The initial state scattering is included by giving the particle sources a Gaussian distribution in transverse rapidity p [14] lAB(P) (X exp(--p2/~B),
(3)
where ¢~AB= 0.44 corresponds to central Pb+Pb collisions. The results of the simulation are summarized in Fig. 9. The dashed lines show the evolution of ~pr with the width of the source transverse rapidity distribution (~AB (i.e. increasing system size). Results are shown for two limiting cases of the longitudinal distribution of particles emitted by the elementary sources (upper curve: p+p parametrization, lower curve: isotropic emission). The solid line in Fig. 9 indicates the reference value of ~pr = 4.5 MeV, for the proton+proton parametrization. For both versions of the random walk model a strong increase in @pr above
the reference line is observed with increasing system size. In contrast, the preliminary analysis of NA49 Pb+Pb data shows the opposite effect, with fluctuations that are decreased in the large system compared to the proton+proton reference.
,..,0.06
! o.os ia.
g g
0.04
g
2
g g g
g g
0.03 0.02 0.01 i
i
i
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0.1
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i
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i
[
~
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i
i
~
[
~
0.3
i
~
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[
i
0.4
i
i
i
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~AB
Figure 9. Dynamical PT fluctuation measure (hpT for the random walk model as a function of system size ((~AB). Shown are two limiting cases for the random walk model (dashed lines, see text) and the results without initial scattering (solid line).
We conclude that the introduction of the random walk source movement does not wash out the correlations present in proton+proton collisions. Rather the correlation introduced by the transverse source velocity leads to an amplification of large scale fluctuations in transverse momentum. Further investigations of the ¢pT observable will proceed along two lines. We will study PT fluctuations in various microscopic models, which include a realistic description of hadronic rescattering, to differentiate between the possible explanations of the observed effect. More importantly
G. Roland~Nuclear Physics B (Proc. Suppl.) 71 (1999) 261-269
we will also collect further experimental data on (hpT in peripheral nucleus+nucleus collisions as well as nucleon+nucleon and nucleon+nucleus reactions. 4. S u m m a r y and Conclusions
Studying the dependence of the inverse slope of transverse mass distributions for hadrons in central P b + P b collisions, a significant dependence on the particle mass is observed, leading to the assumption of a collective transverse velocity field. This hypothesis is supported by the dependence of the transverse radius parameter on the pair transverse momentum observed in an analysis of two particle Bose-Einstein correlations. A fit with a source function inspired by a hydrodynamical picture of the collision yields a rather low freeze-out temperature T ~ 120 MeV and an expansion velocity of/~± ,~ 0.55c. The validity of the hydrodynamical approach assuming local thermal equilibrium is confirmed by a measurement of event-by-event transverse momentum fluctuations. This measurement shows that in P b + P b collisions large scale correlations that are present in the final state of nucleon+nucleon collisions are reduced. In a simple random walk model we observe an increase in event-by-event fluctuations due to the correlations introduced by initial state scattering, which are not compatible with the data. This lends additional support to a picture of a source in local thermal equilibrium undergoing a transverse expansion. NA49 will follow these results with an extensive experimental program on nucleon+nucleon and nucleon+nucleus collisions that might finally allow to use the initially present correlations to quantitatively trace the approach of the particle source in heavy ion collisions towards thermal equilibrium and understand how the additional velocitylike components of transverse momentum are generated. REFERENCES
1. J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34 (1975) 151, E. V. Shuryak, Phys. Rep. C61 (1980) 71 and C l 1 5 (1984) 151. 995.
269
2. E. Laermann, Nucl. Phys. A610 (1996) 1; E. Laermann, Proceedings of the Hirschegg Workshop on QCD Phase transitions, 1997. 3. T. Alber et al., Phys. Rev. Lett. 75 (1995) 3814 4. R. Stock, Proceedings of a NATO Advanced Research Workshop on the Hot Hadronic Matter: Theory and Experiment, 1994, Divonne, France. 5. P. Jones et al. (NA49 Collaboration), Nucl. Phys. A610 (1996) 188c. 6. N. Xu et al. (NA44 Collaboration), A610 (1996) 175c. 7. V. Friese et al. (NA49 Collaboration), Proceedings of Strangeness in Quark Matter 1997, Thera, Greece, April 1997 8. NA49 spectral paper, LBL draft 9. J. Sollfrank et al., Z. Phys. C52 (1991) 593; E. Schnedermann et al., Phys. Rev. C48 (1993) 2462. 10. B. K~mpfer, preprint FZR-149, hep-ph/9612336 preprint. 11. U. Heinz, Nucl. Phys. A610 (1996) 264 12. U. Wiedemann et al., nucl-th/9611031 preprint. 13. J. Alam, J. Cleymans, K. Redlich, H. Satz, nucl-th/970742 preprint 14. A. Leonidov, M. Nardi, H. Satz, Nucl. Phys. A610 (1996) 124. 15. S. Jeon, J. Kapusta, nucl-th/9703033 preprint 16. G. Roland, Proceedings of the Hirschegg Workshop on QCD Phase transitions, 1997. 17. M. Gaidzicki, A. Leonidov, G. Roland, IKFHENPG 5-97 preprint. 18. M. Galdzicki, St. Mr6wczyfiski, Z. Phys. C54 (1992) 127. 19. S. Pratt, Phys. Rev. D33 (1986) 1314, A. N. Maklin and Y. M. Sinyukov, Z. Phys. C39 (1988) 69; G. Bertsch, Nucl. Phys. A498 (1989) 173; S. Chapman, J. R. Nix and U. Heinz, Phys. Rev. C52 (1995) 2694; S. Pratt, Phys. Rev. D33 (1986) 1314. 20. H. Appelsh~iuser, PhD thesis Frankfurt 1997; S. SchSnfelder, PhD thesis, MPI Munich 1997. 21. T. Alber et al., Z. Phys. C73 (1997) 443. 22. T. Kafka et al., Phys. Rev. C16 (1977) 1261