Pion distributions from the fluid dynamical model of ultra-relativistic heavy ion collisions

Volume 229, number 4

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PION DISTRIBUTIONS FROM THE FLUID DYNAMICAL MODEL O F U L T R A - R E L A T I V I S T I C HEAVY I O N C O L L I S I O N S Espen F. STAUBO, Alv K. H O L M E , L,'iszl6 P. C S E R N A I Ph),mcs Department. Umver~ty of Bergen..411i,gaten 55. N-5007 Bergen. Norway

Minzhuan G O N G Nattonal Superconducting ('),clotron Lahoratoo'. Mwhtgan State ~:mversity. East Lanstng. MI 48824-1321. US.4

and Daniel S T R O T T M A N lheorettcal Dittsion. Los Alamos ,'rational Laboratory. Los Alamos. ,V.1487545. USA

Rece~,ed 15 June 1989: revised manuscript received I August 1989

Ultra-relativistic heavy ion reactions are described by a three d~menstonal, relativistic, perfect, one-fluid model. Numerical solutions are calculated for heavy ion reactions up to 60 GeV per nucleon. Observables arc evaluated for emitted pions by adding the random thermal motion to the collective flow.

I. Introduction Recent AGS and SPS experiments lead to the conclusion that in central heavy ion collisions hadronic matter is strongly stopped and a significant portion o f the incident energy can be observed in transverse directions. The normalized p s c u d o r a p i d i t y distributions o f charged secondaries are strongly peaked at mid rapidities for 14.6, 60, and 200 G e V / n u c l e o n O + A g / B r collisions and do not show any d e v e l o p m e n t o f a central plateau as shown by the KLM Collaboration [1,21. A similar conclusion was drawn in another emulsion experiment [3 ]. The present data from WA-80 and NA-35 also indicate that the stopping o f baryons is strong at energies up to 200 G e V / n u c l e o n [ 4-71. At the lower energies ( 14.5 and 60 G e V ) there is evidence for complctc stopping in central collisions where no energy is detected a r o u n d zero degrees in calorimctcrs. Thus it is not illogical to attempt a description o f these latter two reactions in terms o f a one-fluid hydrodynamical model in which there is complete stopping [8,9]. One- and two-fluid dynamical calculations have already been performed for high energy nuclear collisions [ 8 - 1 0 ] . In a thrcc-dimensional one-fluid calculation [9] Rentsch et al. concluded that the baryon rapidity distribution is strongly peaked in ultra-relativistic collisions, and the peaks can be as narrow as 1.5 units o f rapidity. The aim o f these calculations and experiments is to d e t e r m i n e under what conditions a q u a r k - g l u o n plasma ( Q G P ) is formed in heavy ion collisions. However, all three dimensional calculations so far have used an equation o f state ( E O S ) without any phase transition [ 11 ] whatsoever! Also, the breakup mechanism o f the fluid cells is important if we intend to describe observable quantities, because the r a n d o m thermal motion o f particles is not negligible. In ref. [9] the first comparisons to experimental data were presented, but their breakup mechanism was not described. The phase transition in baryon-rich m a l t e r has bccn considered only in one dimensional calculations up to 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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now --t. They predict essentially that the transverse flow should decrcase in the prescncc of a phase transition in a given cnergy region.

2. Equation of state We have selected a phenomcnological EOS reproducing the characteristic phase diagram of the Q G P phase transition [ 13 ]. The nuclear part is describcd by the Sicrk-Nix parametrization [ 14 ]. the Q G P part is given by the bag model EOS, and the mixed phase is obtained by Maxwcll construction. Nucleons and pions as well as nonstrange quarks and gluons were considered. A few similar calculations also describc the baryon rich matter in the ultra-relativistic cnergy region [ 15 ]. This EOS has a first order phase transition. The critical temperature at n = 0 is T, = 216 MeV, consistent with recent lattice Q C D results: 7~ = 177-276 MeV [16]. The energy density increases monotonically with T until Tc is reached where the critical energy density is e~.H(n=0) = 280 M e V / f m ~. For comparison we performed two sets o f fluid dynamical calculations with EOS's with and without a phase transition to QGP.

3. Fluid dynamical model Thc equations ofrclativistic perfect fluid dynamics were solved numericly by thc PIC method [ I 0 ] on CRAYXMP computers in Trondheim and Los Alamos. Here we present results for 32S+2°SPb at 14.5 and 60 G e V / nucleon beam energy. The calculational grid was chosen to have 10 cells along the transverse diameter o f t h c ~2S nuclei. Mostly, in the beam direction the fluid cells were 57.6% and 43% of the transverse size to partially balance the Lorentz contraction. The size of the fluid cells at/Tf~h = 14.5 ( 60 ) GeV/nucleon was Az = 0.43 (0.32) fm, A x = ,53:=0.75 fro. The number of marker particles per cell was two in the z and three in the transverse directions, so that the total number of marker particles is 18 per cell. We had 18 728 ( 10 818) marker particles for one half of the configuration space. The numerical solutions satisfied energy and m o m e n t u m conservation to machine accuracy o f the CRAY's. Symmetries in symmetric collisions like A +A and in b = 0 collisions were maintained in the calculations although they were not explicitly enforced. The results of the calculations for equal mass target and projectile were and stayed completely reflection symmetric about the CM point and about the beam (z) axis. Furthermore, in exactly central collisions the transverse momentum, ( p ~ ) , was zero. In the calculations presented here we studied the breakup dependence of the results by choosing several breakup times one after the other. At these times the fluid was frozen out globally everywherc. The phase transition to Q G P contributes to an appreciable increase of the maximum density and in the expansion time. For the purely hadronic EOS at 14.5 (60) GeV/nuclcon the average density reached a maximum of 2.3 (7) no at 2.5 (1.2) fm/c. while this average density dropped below 2no at 4.5 (4.5) fm/c. For the Q G P EOS at the same energies, the average density reached a higher maximum of 4.3 (9) no at 3.5 (1.6) fro~c, while it dropped below 2no at 6.7 (8.5) fm/c. Pure Q G P was formed for a period of about 3.5 (6.5) fm/c with a maximum volume of 13 (40) fm ~. The volume average of the Q G P temperature had a maximum value of 205 (265) MeV. Unfortunately the maximum density is not directly observable, and the estimates on the time extent of a collision [ 17 ] are theoretically biased and accurate to about ( 15-30)% only. This problem is crucial at 60 GeV/nucleon with the Q G P EOS, where the mixed phase did not decay completely even at 15 fm/c. At this time the volume of the expanding matter reached the limits of the maxim um possible calculational grid on a CRAYs~ Most two-dtmenstonalcalculation~ include the fixed Bjorkcn scaling solution longitudinally [ 12], and mathematically the fluid dynamical equations are solved for the radial coordinate only. 352

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XMP. The low density, ~ no, and high temperaturc, ~ 210 MeV, mixed phase domains pcrsistcd for a long time becausc thc prcssure in this statc is cxtrcmely low, ~ 100 McV/fm 3 (see ref. [ 13] ). By this time thc spectator and participant regions wcrc completely separated. At 14.5 GeV/nucleon lhc mixed phase domains decayed smoothly bccausc their temperaturc staycd considerably below 210 McV, and so thc pressure was large enough [ 131 to maintain a rapid expansion.

4. Pion measurables Pion cmission in heavy ion reactions may arise from different mechanisms such as black body radiation, thermal emission, delta decay, ctc. The production mechanisms of mesons arc complicated. They could include for instance ratc equations and non-equilibrium features [ 18.19 ]. However, in a rcccnt analysis Hahn and Glendenning concludcd that the dominant pion production mechanism at high energies is thermal cmission [20]. Thus similar to the baryon distributions we can calculate the pion distributions, assuming a thcrmal volume cmission at the breakup inside each cell. This is consistcnt with our EOS where thermal pions were also considcrcd. We also assume that the pion chemical potcntial vanishes, in accordance with the one-fluid model whcrc local thermodynamical equilibrium is assumed for all particles. At the breakup the local pion momentum distribution at a spacetime point x is assumcd to be a relativistic Bosc distribution:f~(x, p; U '~, T). The pions are emitted in thcir own local rcst framc (LR), and their number is proportional to this volume VrR as well as to 1 "3. At the breakup each fluid cell is moving with thc vclocity U'= y( 1,1~,, ,flj ) = )'l (cosh 0'o), sinh (),o), v, ), where Yo is the rapidity of the fluid cell and the other variables provide the usual decomposition of the four velocity.

5. Pion multiplicities The total pion multiplicity in the calculation is sensitive to thc EOS. For cxamplc at 14.5 GeV/nuclcon the phase transition to Q G P yields a total pion multiplicity (n +, n", rt- ) 60 whilc the hadronic EOS yields about 40 at 10 f m / c breakup time, when the volume average of the density dropped to normal nuclear density in both cascs. Earlicr brcakup times yield a higher pion multiplicity for both EOS's, later breakup times yield a smaller one. At 8 fm/c breakup time the pion multiplicities are 75 (50) for QGP (hadronic) EOS. The higher pion multiplicity in case of Q G P EOS can be understood as a consequence of the reduced collective flow and thus a larger part of the cnergy is thermalized. At 60 GeV/nucleon the phasc transition to Q G P yields a total pion multiplicity of 245. while the hadronic EOS yiclds about 235 at 10 fm/c breakup time. Earlier breakup times yield a higher pion multiplicity for the hadronic EOS, but in the case of the plasma the mixed phase is still dccaying at this time and the pion multiplicity is still increasing. At this time the "'breakup" densities are still 1.5 ( 1.3 ) no for thc Q G P (hadronic) EOS. At this cnergy the decay of the mixed phase still continues at 15 fm/c, and by then the pion multiplicity increases to 300. Duc to competing mechanisms and to the long lifctimc of the mixed phase at higher energics, it is not possible to draw a conclusion regarding how the phase transition affects thc pion multiplicities quantitativcly. Breakup conditions play an essential role in detcrmining pion multiplicities, and these conditions are outside the prcdictive power of fluid dynamics.

6. Pion rapidiq,' distributions Thc contribution of a fluid cell to the final pion rapidity distribution is defined as 353

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dN,,~,,

d3~, -TV~¢,,

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J" d3p d

~6- ~-y [p"U,,L(x.p)] •

(1)

Performing the integrals yields

×

{[

v

I-

,

}

Kk~ ~/2(jhm.)-j~m ~ K~. ~/2(j'hm,) ,

(2)

where 7" is the cell temperature, g, = 3 is the degeneracy of pions, g = ;,,8~ / T = :~,~vi / T. and h = 7~ cosh ( y T-- ), [ cosh (y) - fl~sinh (),) ] / T. Assuming that m,,.g2 << h, and g << h yields

2rrg, Tm2, ~ , ( 2 2 d~-f-- ~- ),V,~,, -(~-~3 ~=,~exp(-jhrn,) l + j h ~ n , + ( j ~

dA,',.~,,,

)

.

Yo ) / (3)

Our final pion rapidity distribution is then obtained by summing up the contributions of this expression from all the fluid cells. For central symmetric collisions the width of the pion rapidity distribution calculated in this model agrees with the prediction of the Landau model and with experiments in the 10-60 GeV/nucleon energy range [I ]. The calculated dN.,/dy for an S+ Pb reaction at 14.5 GeV/nucleon beam energy are shown in fig. I for hadronic and QGP EOS's, at impact parameter b=O.3(Rp+R,) and at breakup times tin:=8.4 and 10.7 fm/c. At 10.7 f m / c the rapidity distribution peaks at y = 0 . 9 (0.8) for QGP (hadronic) EOS. and it is about 1.9 (1.8) units wide. At earlier breakup times the rapidity distributions are wider and peak at higher rapidities for both EOS's. As the "breakup" time increases, more and more target particles will become participants and this leads to a decrease of the peak rapidity cspecially in the hadronic case. The differencc between the two EOS's are larger at the earlier breakup time of 8.4 fm/c. The fact that the distributions peak at relatively low rapidities is a consequence of the one-fluid picture wherein the projectile is completely stopped by the massive targct. The larger pressure in the hadronic case leads to the participation of more target nucleons and this yields a smaller pcak rapidity. Partial transparency of earlier breakup would shift the peak to higher rapidities. For the same reaction at 60 GeV/nucleon, dN,/dy is shown in fig. 2 at breakup times of 7.6 and 10 fm/c. At 10 f m / c the rapidity distributions peak at y = 2 . 0 (1.4) for Q G P (hadronic) EOS, they are about 2.2 (2) units wide, and they have the same amplitude for both EOS's. The smaller pressure in the plasma case leads to a much sharper cut between participants and spectators, and consequently to less participants from the target. This causes the much higher peak rapidity for QGP. At 10 fm/c the peak of rapidity distribution is still strongly increasing, while for hadronic matter it is slightly decreasing. Note the limit of no transverse flow and massless pions ( I/N, ) dN~,.c~Jdy= ~ cosh- 2 ( y - y o ) .

7. Average transverse momentum, at given rapidity At relativistic energics the transverse flow arising from standard compression shocks [21 ] is a relevant observable to determine the features of the EOS and the transport properties of the nuclear matter [22]. Here we also calculated the transverse momentum spectra of the pions [23 ]. The contribution of a fluid cell to the total transverse momentum projected to the reaction plane (x, z) in the CM system is: P~H., = V,~, J d3ppXf,(x, p), and the yield falling into a unit rapidity interval around v is dy 354

- I~, f d2p_ p°lCf~,(x, p) . a

(4)

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b = 0.3

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b

0.3

=

40

dN

Pions

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Pions

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30 75

/

Hadronic : / \~ \ \~ ~' / / ' ~ \

QGP

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,

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'::,~,..~,

Hadronic

_.../ 0. 2

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,, 0

2

0-2

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0

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6

4

Y Fig. 2. The total pion rapidity distribution dN,/dy from the relativistic three dimensional one-fluid dynamical model for the reaction S+ Pb at /:,,h=60 GcV/nucleon. at impact parameter b=0.3 (RI,+ Rt). The breakup times are 7.6 and 10 fm/c, with increasing breakup time the pion multiplicity decreases for hadronic EOS and increases for QGP. so that at 10 fm/c the peak values for QGP and hadronic EOS's are the same. QGP (hadronic) EOS's arc ,nd,catcd by full (dashed) lines.

Fig. I. lhe total pion rapidity distribution dNUdy from the relativistic three dimensional one-fluid dynamical model for the reacuon S+ Pb at EL,0= 14.5 (ieV/nucleon. at impact parameter h=O.3(Rp+R,). The breakup times are 8.4 and 10.7 fm/c. with increasing breakup time the pion muhiplicity decreases for both EOS's. QGP (hadronic) EOS's arc indicated by full (dashed) lines.

A f t e r a s t r a i g h t f o r w a r d calculation we a r r i v e at thc result

=2ncosh(y)m~

dy

,=l*=o

.

"

N/2

3 Vc~. COS(¢R) (2a'h) ~ 2(k+2)

\jh J

K,+~/2(J'hm,,)+ jhm,~

K,+s/2(jhm~)),

(5)

where OR is the a z i m u t h angle o f the velocity o f the fluid cell m e a s u r e d from the reaction plane. T h e average transverse m o m e n t u m per pion at a given rapidity y is then g i v e n by ( p ' / N ~ ) = ( Yc~.dP)~...,/dy) / ( ~c~,dN~.c~u/ dy). T h e average transverse m o m c n t u m o f p i o n s (p±/N.) which is the m a g n i t u d e o f t r a n s v e r s e p r o j e c t i o n o f m o m e n t u m was calculated also:

dP~ ~ . , ~

_

V~,, f d2p, pOp~J ; ( x ,

p)

dy

=2, coshU,) v.o,, ,=, ~=o (2k)!! 2 (2k+3)!!

\j~-}

2 ~

"

(6) 355

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The average p ~ per pion is obtained as it is given above for p ~. The transverse m o m e n t u m (pX) is small due to the large target projectile asymmetry and to thc small impact parameter. For emitted baryons the recoil at the target side is p ~ = 2 3 0 (160) (MeV/c)/nucleon at the maximum for hadronic ( Q G P ) EOS at 14.5 GeV/nucleon and only 40 M c V / c more at 60 GcV/nucleon. At projectile rapidities ( p ~ ) is always below 100 M e V / c except for the hadronic EOS at 60 GeV. Pion ( p ' ) are even smaller and do not exceed 100 MeV/c. The pion transverse momenta, ( P l >, at 14.5 GeV are about 480 M e V / c with both EOS's if the breakup happens at normal nuclear density; this value decreases with increasing breakup time. At 60 GeV/nucleon and at 8 fm/c breakup time the maximum pion transversc momentum, p~, is 670 (860) M e V / c for Q G P (hadronic) EOS at 8 fm/c, and is increasing (decreasing) with increasing breakup time. The maximum o f the average p, for Q G P EOS peaks at 12 f m / c breakup with 720 MeV/c, i.e. again comparable to the value for purely hadronic matter. Another consequence is that pion azimuthal correlations [ 1 ] are unlikely to be observed at ultra-relativistic energies because p, >>p~, so thermal fluctuations and limited statistics will wash out the signs o f the small transverse flow projected to the reaction plane.

8. Pion transverse momentum distributions

Following the same basis lines as presented in ref. [24] the pion transverse m o m e n t u m distribution can be evaluated: dN~..,.~,, yl~,,g,_T ~ [agt(.la)lo(jb)-bK,,(ja)i,(jb)] p_ dpt - (2a-h) 3 ,__z..,

(7)

where a = 7, m_, / 1 ; and b = ; ' l vi p~ /7-. The resulting spectra are shown in figs. 3, 4. There is a clear deviation from the straight exponential shape in all cases. The formation of Q G P decreases both the temperature and the transverse flow so the p~ spectra are steeper. The observable effective temperature or slope parameter is higher than the average temperature of the fluid cells at the same time. Most importantly, the spectrum shows a concave shape which is a consequence o f the collective expansion; the pion mass is small hence the spectrum has a concave shape in the I - 2 GeV range. This shape (also seen in several experiments [25,26] ) is connected to transverse radial flow as pointed out in ref. [25] using a model with idealized cylindrical geometry. That the same shape can be obtained from a one-fluid dynamical calculation shows that the concave shaped p, spectrum is a consequence of the collective flow but it is not necessarily related to complete transparency and the Bjorken scaling expansion [ 12 ]. We obtain with hadronic EOS more concave spectra, indicating a larger transverse flow. This is consistent with the larger average p values for this EOS. The Q G P causes a reduced flow in the transverse direction and a broader width in the longitudinal direction. This can be explained by the fact that the Q G P is much softer, leading to a more compressed and flat shape before the expansion of the matter starts. Then the expansion starts dominantly in the beam direction due to the larger pressure gradients in this direction. At the same time the transverse flow is reduced.

9. Summary In the stopping energy region, which is expected to include the phase transition threshold, one can make use o f a phenomenological EOS if the matter is not tort far from local thermal equilibrium. The one-fluid dynamical model yields a flow patterns depending on the EOS. With a phase transition to Q G P in a S + Pb collision at 60 GeV/nucleon pure plasma was created in a volume which reached 18% of the rest volume o f the projectile. For a period of 2 fm/c the plasma occupied more than 10% o f the initial nuclear volume. At 14.5 GcV/nucleon pure plasma was created in a small volume only (max356

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32S -{" 2°8pb

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6 0 GeV

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10"'

1

,,,,\\ \\

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\\\

10"

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\\\\S\\\ 10 .6

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\\~NNNN \\\ \\\ \\ \\

10"

"XN\\

\\\\ \\%N

10.6

10.8

"\\\~ 10 "g 0

I

1000 2000 p, []VleV/c]

\

3000

Fig. 3. The pion transverse momentum distribution d:V,,I (pj dp~ ) from the relati,,istic three dimensional one-fluid dynamical model using the PIC method (see Clare and Strottman [ 10] ). The reaction ts S+ Pb at A,,t,= 14.5 GeV/nucleon, at impact parameter h = 0.3(R r + R,). The breakup times are the same as tn fig. 1. Two different EOS's were used without (dashcd Itncs) and with (full lines) a phase transition into QGP.

10.7

0

1000

200(3

3000

p []VleV/c] F~g• 4• The pion transverse momentum distrtbution dN,,/ (p~ dp, ) from the relativistic three dimensional one-fluid dynamical model using the PIC method (see Clare and Stronman [ 10 ] ). Thc reaction is S + Pb at E,,t,= 60 GeV/nucleon, at impact parameter b = 0.3 ( Rp + R, ). The break up t imes are the same as m fig. 2. Two different EOS's ~,ere used without (dashed lines) and with (full lines ) a phase transition into QGI'.

imal 6% o f the p r o j e c t i l e ) , and it existed for a very short time. Thus, the m o d i f i c a t i o n o f the flow pattern was caused by the m i x e d phasc only. In a s y m m e t r i c collisions likc S + P b the d i f f e r e n c c in thc average transverse m o m e n t u m , p~, causcd by the EOS is small in nearly central reactions. A l t h o u g h ( p j IN,,) has a strong rapidity d e p e n d e n c e , this is i n f l u e n c e d by the i m p a c t p a r a m e t e r and b r e a k u p times, so a v c r a g i n g o v e r the a b o v e will wash out this structure. T h e m a i n effccts o f the Q G P f o r m a t i o n are increascd longitudinal and r c d u c c d transverse flow which is c o n sistent with the findings o f ref. [ 271 with Bjorken [ 121 type initial c o n d i t i o n s . At 60 G e V / n u c l e o n thcsc effects arc stronger, while at the lower energy thc small a m o u n t o f t h c plasma p r o d u c e d and thc u n c e r t a i n t i e s in b r e a k u p t i m e m a k e it difficult to o b s e r v e a clcar change in the flow experimentally•

Acknowledgement E n l i g h t e n i n g discussions with H.W. Barz, A.S. G o l d h a b e r , H. Schulz and M. T o h y a m a arc gratefully a c k n o w l edged. T h i s w o r k was s u p p o r t e d by the U S N a t i o n a l Science F o u n d a t i o n , thc U S D c p a r t m e n t o f Energy, the San D i e g o S u p e r c o m p u t e r C e n t e r , and the N o r w c g i a n R e s e a r c h C o u n c i l ( N A V F ) . O n e o f the a u t h o r s ( L . P . C s . ) thanks the H u n g a r i a n A c a d e m y o f Sciences and thc C c n t r a l R c s c a r c h l n s t i t u t c for Physics for g r a n t i n g him e x t e n d e d leave. 357

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References [ 1 ] H. yon Gersdorffet al., Phys. Rev. C 39 (1989) 1385. [ 2 ] L.M. Barbler et al.. Phys. Rev. Lett. 60 ( 1988 ) 405. [31P.L Jam, K. Sengupta and G. Singh, Phys. Rev. Left. 59 11987) 2531. [4 ] W~-80 Collab., H.R. Schmidt ct al., in: Proc. 6th Intern. Conf. on I)ltra-relativlstic nuclet,s-nucleus collisions (1987), Z. Phys. C 38 (1988) 109. [ 51 WA-80 ('ollab., F. Plasil et al., in" Proc. Winter Workshop on Nuclear dynamics V ~Sun Valley, ID. Fcbruar's 1988 ), unpublished. [ 6 ] B. Jacak, private commumcation. [ 7 ] E802 Collab, K. ~an Dok et al., in: Proc. Winter Workshop on Nuclear dynamics V (Sun Valley. ID, February 1988 ), unpublished. [8] L.P. Csernai, M. Gong and D. Strottman, in: Proc. Intern. Workshop on Nuclear dynamics at medium and high energy (Bad Honnef). Nucl. Phys. A 495 (1988) 403c. [ 9 ] T. Rcntsch. G. Graebner. J.A. Maruhn, H St6cker and W. Grcincr, Mod. Phys. Lett. 2 ( 1987 ~ 193. [ 10] H. St6ckerand W. Greiner, Phys. Rep, 137 (1986) 277R.B. Clare and D. Strottman, Phys. Rep. 141 ( 1986 ) 177. [ 11 ] L P. Csemai, in: Hadronic matter in collision, eds. P. Carruthers and D. Strottman (World Scientific, Singapore, 1986) p. 432. [ 12 ] J.D. Bjorken, Phys. Roy. D 27 ( 1983 ) 140. [ 13 ] A.K. Holme, E.F. Johansen, L.P. Csernat, E Osnes and I). Strottman, Bergen Sci./Tech. Report 201 / 1989. [ 14] A. Sicrk and J.R. Nix, Phys. Rev. C 22 (1980) 1920. [15] H.W. Barz, L.P. Csemai, B. Kiimpfer and B. Luk~ics, Phys. Roy. D 32 (1985) 115; II.W. Barz et al., Phys. Roy. C 31 ( 1985 ) 268; Phys. Lett. B 143 ( 1984 ) 334; L.P. Csernat and B Luk~ics, Phys. Lett. B 132 (1983) 295; H. St6cker, G. Graebner, J. Maruhn and W. Greiner. Z. I'hys. A 295 (1980) 401; LI. Heinz, P.R Subramantan, H. St(Joker and W. Greiner. J. Phys. G 12 (1986) 1237; K.S. Lee, M.J. Rhoades-Brown and U. Heinz, Phys. Rev. C 37 (1988) 1452. 1463; J. Kuti, B. Luk/tcs, J. Pol6nyt and K. Szlachlinyi, Phys. Lett. B 95 (1980) 75; SA. Chin, Phys. Lctt. B 78 (1978) 552; J. Cleymans. R. Gava, and E. Suhonen, Phys. Rep. 130 (1986) 217; H.W. Barz. B. K/i.mpfer, LP. Cserna~ and B. Lukzics, Nucl. Phys. A 465 (1987) 743. [ 16] F. Karsch, Z. Ph~.s. C 38 (1988) 147. [ 17] T. Humanicet al, Z Phys. C 38 (1988) 79. [ 18 ] HW. Barz. L.P. Csernai and W. Grcincr, Phys. Rev. C 26 ( 1982 ) 740; II.W. Barz, B. K~impfer and B. Luk,'ics, Soy. J. Elcm. Part Nucl. 18 (1987) 1234: H.W. Bar,, and B. Kiimpfcr, Phys Lett B 206 (1988) 399. [ 19 ] H.W. Bar.,, B.L. Fnman, J. Knoll and H Schulz, Nucl. Phys. A 484 ( 1988 ) 66 I. [20] D. Hahn and N.K. Glendenning, I'hys. Roy. C 37 (1988) 1053. [211W. Schcid, H. Mfillerand W. Greiner. Phys. Re~. Lett. 32 (1974) 741; G.F. Chapllne, M.I-I. Johnson, E. Teller and MS. Weiss. Phys. Rev. D 8 ( 1973 ) 4302. [ 22 ] A. Bonasera and L.P. Csernai, Phys. Rev. Lctt. 59 ( 1987 ) 630. [231P. Danielewicz and G. Odyniec. Phys. Lett. B 157 (1985) 146. 124] P.V. Ruuskanen, Acta Phys. Polon. B 18 (19871 551. 125 ] T. Atwater, PS. Freier and J.I. Kapusta, Phys. Lctt. B 199 ( 1987 ) 30. [ 26 ] T. Alcxopulos el al., Phys. Rev. Left. 60 ( 1988 ) 1622. [271M. Kataja et al. Phys. Re','. D 34 (1986) 2755.

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