Production of pionic modes in relativistic heavy-ion collisions

Nuclear Physics A505 (1989) 823-834 North-Holland, Amsterdam

PRODUCTION

OF

PIONIC

MODES

IN RELATIVISTIC

HEAVY-ION

COLLISIONS* G.E.

BROWN

Department of Physics, State University of New York, Stony Brook, NY 11794, USA E. OSET’,

M. VICENTE

VACAS’

Institute of Theoretical Physics, University of Regensburg,

and

W. WEISE’

D-8400 Regensburg, Fed. Rep. Germany

Received 28 March 1989 (Revised 3 July 1989) Production of pionic modes in heavy-ion collisions of energy E/nucleon -1 GeV is substantially enhanced by interactions of the pions with the medium. These interactions provide a momentum-dependent index of refraction for the pions, making Cherenkov radiation of them possible for nucleons traversing such media. Schematic and realistic calculations of these processes are presented.

Abstract:

1. Introduction Studies of pionic many-body degrees of freedom in high-energy heavy-ion collisions were initiated by Migdal ‘) and by Gyulassi and Greiner ‘). The importance of pionic interactions with the nuclear medium was pointed out by Mishustin et al. 3, who suggested that at energies - 1 GeV/nucleon in relativistic heavy-ion collisions, pions could carry an appreciable part of the entropy. The in medium lowering of the pion energies makes it possible for the pions to carry much more entropy than if their energy obeyed the free particle relation o = dk* + rnt . Friedman et al. “) worked out the thermodynamics of the interacting pions. Based on this work,

Ainsworth

et al. ‘) found

that at twice nuclear

density

p = 2p,, the number

of pions in the medium was increased by a factor - 10 by their interactions. Bertsch et al. “) showed that medium corrections increase the production of pions, roughly doubling the inelastic cross section for p = 2p,, at energy E/A = 800 MeV. If so, stopping showing In sect. increased We shall

is assured. These authors found a singularity in the clothed pion propagator that there is the possibility of Cherenkov production of pions in the medium. 2 we shall work out this mechanism, and show that this does give an stopping. Furthermore, it increases in importance with increasing energy. first introduce a schematic calculation to demonstrate the basic physical

in part by USDOE grant no. DE-FG02-88ER40388. address: Dept. of Physics, University of Valencia, Spain. ’ On leave from University of Valencia; supported by Ministerio de Education y Ciencia. 3 Supported in part by BMFT grant MEP 0234 REA and by DFG grant We 655/9-2. l

Supported

’ Permanent

0375-9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

824

process medium. important

G.E. Brown et al. / Production of pionic modes

which couples

a projectile

nucleon

Later on we give results absorptive

damping

to a low-energy

of a more complete

mechanisms

2. Basic developments

pionic calculation

for the pionic

mode in a nuclear which includes

modes.

and a schematic model

Consider the interaction of a projectile nucleon with a piece of nuclear matter. We begin from the nucleon self-energy E involving a dressed pion in the medium, as shown in fig. 1. Here the projectile nucleon with four-momentum k = (E(k), k) and kinetic energy a(k) = E(k) - M passes through nuclear matter and transfers a four-momentum q to the pionic mode. Keeping only Im E, which puts the particles in the intermediate state on shell, we can evaluate the effective cross section per nucleon for “pisobar” production. (We consider first the in-medium normal modes composed of the coupled pion and A-isobar-nucleon hole and call these a pisobar. 7)): a(k, P) = -

2 Im x(k, P) PU

(I)

’

where p is the density and v = Ik]/E is the velocity of the projectile (relative to the medium). In the actual nuclear collision, the propagating nucleon should be thought of as part of the projectile nucleus in the center-of-mass system of the two colliding ions, the system in which the total baryon current is zero. We should point out that the formula (1) is valid for scattering of an external particle in a system close to equilibrium. In the actual nucleus-nucleus collision, pion production takes place most actively during the non-equilibrium phase. As noted in ref. *), a self-consistent treatment incorporating the decay rate of the non-equilibrium system tends to reduce pion production rates. Using pointlike pseudo-vector E becomes (with f= 1):

pion-nucleon

coupling

to start with, the self-energy

(2)

a(k)-qo-E(k-q)+ia+F(k)-qO-E(k-q)-i8

k

k-q k Fig. 1. Nucleon

self-energy

I(k)

D

from emission

9

and absorption

of a pionic

mode.

G.E. Brown et al. / Production

where

n(p) = 1 for Ip( s kF, the Fermi

of pionic modes

momentum.

825

Here we use

E(k-4)=$g, with the effective

nucleon

(3)

mass M*.

The primary point of interest is the appearance of the (complex) pion self-energy n( q; p) which summarizes the pion-nuclear many-body interactions. Pionic modes are then determined by the singularities of the dressed pion propagator [qi-q2VI’,-n(q)]-‘. The integration over q. in eq. (2) is carried out by performing a Wick rotation so as to pick up the contribution from the nucleon pole at I - e(k - q) + 23. This puts the intermediate nucleon on shell. Renormalization at the nucleon quasiparticle pole, which would come in higher order, is neglected here. One obtains

w =

E(k)_&(k_q).

(5)

We will continue to evaluate (4) within a simplified model of the pion self-energy II in which the pion couples to the leading A-hole excitations, as described in refs. 4-7). In our schematic calculation we first neglect the coupling to the nucleon particlehole continuum. The latter will be quite temperature dependent, since temperatures* T 2 50 MeV are obtained in GeV heavy-ion collisions, and the temperature dependence complicates their treatment ‘). Furthermore large local field corrections ‘) (i.e. repulsive spin-isospin interactions gh,a, * u27 * ~~ with gf*i,.,- 0.7-0.9) tend to inhibit the build up of coherence in the nuclear particle-hole sector. We put off dealing with the particle-hole branch until later. At this stage we keep only the pisobar branch from the pionic dispersion relation in ref. “); in other words, we use the spectrum w(q) = Jq2+ mf,, + Re 17 of ref. 4), but we shall employ 19) gkd = 0.4 local field correction. rather than the g&, = 0.6 used there for the nucleon-isobar The dispersion relation for w(q) of ref. “) is repeated here as fig. 2. Let us now return to the integral (4). We drop the Pauli exclusion factor which is not important at large projectile momenta \kl. One observes that the integrand -q21m17(o,q)/[(w2-W2(q))2+(Im17(w,q))2] develops its maximum at the pisobar pole, i.e. where w = I--(k-q) equals w(q) =Jq2+ mt+Re n(o, q). In this region we can then write

-Im~(k)--3($)2~&(&)[o_w~]2+$2~ * Quoted temperatures for energies E/A - 1 GeV are often much higher, do not include the thermostatic effects of pisobar production.

(6) but calculations

giving them

G.E. Brown et al. / Production of pionic modes

826

1

8

0

Fig. 2. Dispersion

relation

with (q( < 2k under

I

w(q) for the pion-isobar-hole

mode

q from Bertsch

et al. “).

the integral

I

I

2 3 1 PION MOMENTUM q[m,l

L

(pisobar)

as a function

of momentum

and

_ImWw(q), 4) 4q) .

r=

(7)

Of course, r is q-dependent, but in our first schematic calculation, we shall take a narrow width approximation ($ < m,). One sees in eq. (6) how the spectrum w(q) of pisobars enters, with lorentzian weighting function and the Bose normalization factor [2w( q)]-‘. In the effective-mass approximation,

The integral over the angle 0, where cos 19= k^*4, is easily carried out in the small r approximation. The step function @(e(k) - e( k - q)) in eq. (4) limits the values of cos 0 to the interval [q/2k, I]. One finds:

kq/M”-q2/2M*-w(q)

+ arctan

(9)

p

This result is readily interpretable. arctan’s in eq. (9) equals rr if

In the narrow

k M*’

-w(q)+ q

- 4 2M”’

width approximation,

the sum of

(10)

otherwise zero. Hence for sufficiently large projectile momentum k and high density, the pisobars can be excited as collective modes of the system. Once this is possible,

827

G.E. Brown et al. / Production of pionic modes

one expects

that the excitation

to the stopping cross section. as given in fig. 2, integrating Actually,

the integration

function

f9(a(k) - s(k -4)).

of these normal

modes

will contribute

substantially

We choose M*/M = 0.7 and use the spectrum o(q) up to qmax = 3m,. This yields o = 128 mb at p = 2p,.

in eq. (4) brings In addition

in a natural

cutoff in q-space

via the theta

there is a cutoff at the n-NA vertex which

cuts down pisobar production at large q. In order to compare with Bertsch et al. 6), local field corrections at the pisobar-nucleon vertices which gives a factor ‘)

we introduce

F=(l+gh,&-*, where

U, is the isobar-hole u

_8fi

A

propagator

*AP 9 m2, oi-co2’

For &A = 0.4, p = 2p, and of 4, leaving

(11)

multiplied

by the TNA coupling

strength:

withfA=2,wA=2m,.

w = m,, this cuts the cross section

a=32mb.

down

by a factor

(12)

This replaces the part of the free inelastic NN cross section which comes from spin-longitudinal interactions and which amounts to roughly one half of a,,(inelas800 MeV. The transverse part is expected to remain relatively tic) = 20 mb at Elab = unaffected, so we arrive at a total inelastic cross section a(inelastic) ~42 mb in medium. This means a mean free path of only about 0.7 fm for p = 2p,, just from the production of pionic modes, without bringing in the quasi-elastic two-body cross section. Carrying out the calculation along the above lines for energy E = 1 GeV/nucleon gives a(inelastic) = 60 mb, the inelastic cross section increasing rapidly with energy. From the condition (10) it is seen that pisobar formation is efficient for the stopping because relatively large momenta can be transferred due to w(q) being kept small by medium dependent effects. Of course, for q - 4m,, as can be seen from fig. 2, w(q) begins to rise more steeply, shutting the process off. From comparison of the results for 1 GeV with those for 800 MeV/nucleon, it is seen that pisobar formation gives an inelastic cross section which rises rapidly with energy. It is easy to see, going over to the relativistic region, k > M that eq. (10) is replaced by (13) This condition expect pisobar

is easy to satisfy up to where w(q) starts rising steeply, so one would production to strongly increase in the multi-GeV region.

G.E. Brown er al. / Production of pionic modes

828

Of course,

pisobar

by the momenta our approximations

q

formation in deriving

be good in the ultra-relativistic by processes

is a “soft”

process,

less than the cutoff at the pisobar

which transfer

the dispersion region.

relation

However,

a lot of momenta

limited

in momentum

production

for o(q)

after the initial

(usually

-f

transfer

vertex. Furthermore, versus

q,

“hard”

of the incident

may not

scattering nucleon

momenta) our Cherenkov process will be very effective in the secondary collisions in the medium. We now return to a discussion of the particle-hole continuum, which was neglected in calculating our pisobar dispersion relation o(q). Often, the role of particle-hole excitations can be neglected, because in the static limit (i.e. q2/2M
3. Realistic calculation A number of additional ingredients, well known as being important in the pion-nuclear many-body problem lo) will now be systematically incorporated: (i) The coupling to the nucleon-hole continuum is added in the lowest order pion self-energy: n(w, where

U, =4(f/mt)4(w,

4) = -q2[ UN(W, 4) + K(w,

cl)17

q) and 4 is the nucleon-hole

(14)

Lindhard

function.

The

imaginary part of U, then accounts for the contributions to Im 2 from the quasi-free nucleon knockout processes. (ii) The A-hole states, described by U,, decay not only via the A + Nn- channel (which is partly blocked by the Pauli principle), but couple strongly to the twonucleon-two-hole nuclei.

continuum.

This is the driving

The sum of all these effects implies

mechanism

a large A-hole

for pion absorption

in

decay width r in

In practice we will use the width which incorporates Pauli blocking and absorptive 2p2h channels as specified in ref. ‘l), including energy and density dependence, which correctly reproduces rr-nuclear absorption cross sections. (iii) While the spin-longitudinal interactions of the type V/a, . ia, ’ &r, * r2 are the driving ones in pionic modes, the spin-tranverse interactions proportional to V,(a, x 6) - (u2 x 4)~~ . 7* are also important (with u’s and 7’s replaced by transition spins S and isospins T when N + A vertices are involved). We use a rr + p exchange

G.E. Brown et al. / Production of pionic modes

model

plus local field corrections

proportional

829

to g’, so that

-q2cAq2)

Ve(w,4) = q2_m;+ie+g’(42) 9

(164

-42qh?2) + g’(q2)*

vt(w, 4) = c, q2-m2p

We use C, = 2. The form factors are parametrized as Fi( q2) = AT/ (A: - q2), with q2= m2- q2, and we leave room for a q2-dependent local field correction with g’( q2) = g’/ (1 - q2/h2). Spin-longitudinal and spin-transverse modes do not mix in nuclear matter, so that they enter separately eq. (4) as follows:

in the self-energy

2 which now replaces

ImZ(k)=3($)2Iml~~l-n(k-p)l8(w) Ve(w,4) 1+ V/(W,(I)U(%

x[

2 K(w, 4) (17) v,(w,q)U(w, q)1 O=E(k)-F(k-4)’

d+l+

with U = U, + U,. We use different values of g’ for nucleons and A’s, so that 1 + VU and V are actually matrices in the coupled NN, NA, AN and AA channels. (iv) Finally, we take into account processes in which the intermediate nucleon in fig. 1 is replaced by a A-isobar, including its many-body decay channels. One expects A-excitation of the projectile nucleon to be no less efficient as a stopping mechanism than A-hole production in the target. The corresponding additional piece of E has the nucleon propagator in eq. (2) replaced by the A-propagator. The results of this calculation are now presented, at Elab = 800 MeV/A, for two different sets of parameters: Set I (gkN= 0.55, gha = gbn = 0.35; A,, = 1.0 GeV, AP = 1.1 GeV, A + 00) is consistent with empirical values of the free NN cross section aNN(free)

= -i

y p=o

(18)

for both the NN + NN and NN + NNa channels at energies up to Elab = 1 GeV. It represents, however, a case in which a pion condensate is formed at densities as low as p-2p,. Set II (gAN = gh = gba = 0.7; A, = 1.2 GeV, AP = A = 1.0 GeV) has increased repulsion (i.e. larger g’ values) in the spin-isospin interaction and avoids pion condensation in our density regime (p -c 2.5~~). These two scenarios can be considered as opposite extremes. We shall nevertheless find that the results for the effective cross section (1) turn out to be qualitatively similar for sets I and II; the conclusions will be quite insensitive to variations over a wide range of input parameters.

830

G.E. Brown et al. / Production

Total effective

cross sections

of pionic modes

o for sets I and II are presented

in figs. 3a, b. In

both cases the spin-longitudinal parts are almost doubled from their free values of about 25 mb to around 50 mb at p = 2p,. The spin-transverse part of u is almost constant increasing

as a function density;

of density

for set I. For set II it decreases

this is due to the large repulsive

g’ term which

smoothly dominates

with the

spin-transverse interaction, and which is slightly quenched as the density grows. In either case, the transverse interaction does not undergo massive changes with increasing density. In contrast, the spin-longitudinal interaction is driven by the attractive pion exchange potential which is larger in magnitude than the repulsive g’ term in most of the available phase space. This attraction is enhanced with increasing density, so that the longitudinal effective cross section grows. In order to further analyse the content of the effective cross section, it is instructive to examine the mechanisms by which energy is transferred during the collision between the beam and the interacting many-body system. This information is in the imaginary part of d2X/dq dw, plotted as a function of momentum transfer q = 141 and energy transfer o, at a given projectile energy. We show typical examples of this quantity for two different densities in figs. 4a, b. One observes the different domains as they contribute to the response of the compressed many-body system to the colliding projectile. The peaks in this landscape represent dominant modes of excitation. At small energy and momentum transfers, the response is primarily due to quasifree NN scattering, as visualized by the pronounced quasielastic peak and its neighbourhood. A second peak appears at larger transferred energy in the A excitation region. At low density, the basic energy loss mechanisms are the

L

o[mbl -

(4 1

0

I

2

1

1

P/P, Fig. 3. Effective cross sections per nucleon as a function of density for incident nucleons with E&A = 800 MeV, calculated in the full scheme described in sect. 3. Spin-longitudinal pionic parts (vL) and transverse parts (q) are shown separately (g= a,++). Figures (a) and (b) correspond to input parameter sets I and II, respectively, as given in sect. 3.

G.E. Brown et al. / Production of pionic modes

831

Fig. 4. Differential nucleon self-energy d Im E/do dq as a function of the energy-momentum transfer (w, q) which goes into the excitation of the pionic or transverse modes. The figures show the sum of all contributions for input set II as described in sect. 3, at an incident projectile energy E,,,/A = 800 MeV. Subfigures a and b correspond to densities p = OSp, and p = 2p,, respectively, where p0 = 0.17 fm-j.

quasifree NN reaction and the pion production process NN + NNv. As the density increases, the peak structures merge with the continuum of two-nucleon-two-hole etc. channels which contribute prominently to the large total inelasticity. Next, consider the differential cross section du/dw derived by integrating -(~/VP) d(Im E)/dq dw over the momentum transfer q = 1~1 in the kinematically allowed region. We study the interesting spin-longitudinal part of this quantity separately, the one shown in fig. 1. Typical results at densities p = 0 and p = 2p, are presented in figs. 5a, b. These figures describe the way in which the incoming energy 800 MeV distributes into different channels, namely pion production, nucleon-hole and two-nucleon-two-hole excitations. Evidently, in the free (p + 0) case (fig. 5a), quasifree NN scattering and pion production (r) are the only relevant mechanisms; pion production is mainly due to the NN + NA + NNr process. At density p = 2p,

832

0.6

0.8 w fGeVl

0.8 w IGeVl Fig. 5. Partial spin-Iongitudinal effective cross sections deJdw as a function of energy transfer w, at incident energy l&,/A = 800 MeV. (a) free cross section (P = 01 from quasielastic NN --yNN and pion production NN-2 NNrr processes fsf. (bf Effective cross seetiom at density p = 2p, for various channels: quasifree (nucleon-hoie excitation) processes, two-nucleon-two-hole excitation j2N2h) and pion production (a). Also shown is the pion production rate in the absence of 2N2h couplings (T, fabs = 0). 0

0.2

0.4

06

the scenario changes (see fig. 5b). The quasifree cross section which corresponds in leading order to nucleon-hole excitations increases substantially at low energy transfers due to effects of higher order in density. Fions are also produced at larger energy transfers, but their production rate (D-) is strongly reduced as a consequence of the competition between r-production and two-nucleon two-hole (2N2h) excitations. In the absence of these absorptive interactions (Tabs = 0), the pion production rate would not differ much from the one in free space. However, the opening of 2NZh continuum channels reduces this rate by roughly a factor of three at p = 2.~~. The relatively large 2N2h contribution to the effective cross section can be understood as follows. Its leading part is generated by a term proportional to p2 in the imaginary part of the nucleon self-energy E(p). The two-nucleon term in u = -(2/pu) Im 3 therefore grows linearly with density and represents a substantial part of the inelastic cross section at p = 2p0. The present calculation includes inelastic processes only through INfh, 2N2h and GTproduction. Other channels such as a(2NZh) etc. for

G.E. Brown et al. / Production of pionic modes

which the self energy

2 grows approximately

so much

the total

in changing

effective

as p2, could

cross

section,

inelasticity among the different final states. Altogether, this analysis suggests that colliding very efficiently energy

in dense nuclear

by producing

interaction

pionic

833

also be important,

but in re-distributing

high energy nucleons

matter. They can lose a large fraction

modes

which decay at once (i.e. within

time scales of 1O-23 s) through

emission

not the

are stopped of their initial typical

strong

of nucleons.

4. Conclusions We have shown that stopping

through

production

of pionic modes is substantially

increased by medium effects in relativistic heavy-ion collisions. At Bevalac energies of 800 MeV - 2 GeV we expect that this stopping is sufficient to quickly equilibrate the colliding material. There are strong empirical indications that this happIens 12). In higher energy heavy-ion collisions, the stopping in secondary collisions twill be greatly increased through these processes. Although many pions will be produced in the medium at Bevalac conditions, because of the strong interactions, the pions will equilibrate, and only a few of these will emerge at the freeze-out temperature. The latter will be relatively low, since the pisobars act as thermostats; they are easy to produce because of their low energies. The situation is, therefore, somewhat paradoxical: Very many pions are produced in medium, but few pions emerge; the actually observed particles are mostly secondary nucleons. It should be, however, possible to look into the interior of the overlapping nuclei and see the vast numbers of pisobars. Gale and Kapusta 13) proposed dilepton observation as a clue to what goes on. Colliding pions going through real or virtual p mesons radiate lepton pairs. The invariant mass of the latter is equal to that of the colliding pions. If this is greatly lowered by medium interactions, then this should appear in striking fashion in the invariant mass spectrum of dileptons. Details of this have been worked out by Xia et al. 14) who show an order of magnitude enhancement in the spectrum at invariant mass -2m, for 40Ca+40Ca

at =2.1 GeV/nucleon

and p = 2p,.

(These

authors

used g;*I* =0.6,

somewhat larger than what we suggest, so they should have somewhat less enhancement). Preliminary Bevalac experiments looking for lepton pairs 15) can be interpreted as showing some enhancement in 4oCa + 40Ca collisions at 1 GeV/nucleon. Experiments are planned for Nb + Nb at this energy, triggering on high multiplicities, or equivalently, central collisions. These should show effects of medium enhancement more clearly because of the higher densities formed in central collisions. Effects of the pisobars on the equation of state (EOS) were treated in ref. 4), but this work is still in the exploratory stage. Bertsch er al. 16) showed that with the increased stopping, the sideways flow is somewhat increased over that from a soft EOS plus momentum dependence (which gave sideways flow equivalent to that from a stiff EOS). The latter fits the data well, so a compensating softening of the EOS from pionic mode effects is needed. Presumably the system will live longer

G.E. Brown et al. / Production

834

because

of the diminished

is small in magnitude. correlations

pressure;

also the group velocity

It will be interesting

16) which measure

of pionic modes

to measure

formed

of the pisobars bound”

pion

chiefly the time that the system stays together.

The work of Xia et al. 14) shows (with a relatively pisobars

&o(q)/aq

the “outward

in 2.1 GeV/nucleon

collisions

large gfJd) that the number

is comparable

to the number

of of

nucleons in the system. The interesting possibility exists that at even higher energies, most of the nucleons are converted into A’s which are in the A-hole component of the pionic mode. In that case, the system should behave as a nonviscous Bose gas, nonviscous because the interactions are strong. (Of course, the pisobar decay width will also increase at higher energies). In fact, essentially the s-wave interaction between pionic modes was invoked by Ainsworth et al. “) in order to bring the mass of the a-meson down with increasing density, and the p-wave one was used by Brown is) to bring the p-meson mass down. These attractive interactions should play a role in the many-body behavior of the pisobar gas. One of us (G.E.B.) would like to thank George Bertsch and Volker Koch, who discovered the possibility of Cherenkov radiation in joint work of ref. “). We are grateful to Ulrich Heinz for a critical reading of the manuscript and helpful comments. References 1) A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107 2) M. Gyulassy and W. Greiner, Ann. of Phys. 109 (1977) 485; M. Gyulassy, Proc. Top. Conf. on heavy ion collisions, Falls Creek Falls, Tenn. (1977), p. 457 3) I.M. Mishustin, F. Myhrer and P.J. Siemens, Phys. Lett. B95 (1980) 361 4) B. Friedman, V.R. Pandharipande and Q.N. Usmani, Nucl. Phys. A372 (1981) 483 5) T.L. Ainsworth, E. Baron, G.E. Brown, J. Cooperstein and M. Prakash, Nucl. Phys. A464 (1987) 740 6) G. Bertsch, G.E. Brown, V. Koch and B. Li, Nucl. Phys. A490 (1988) 745 7) G.E. Brown and V. Koch, Proc. Texas A. and M. Symposium on hot nuclei, College Station, Texas, 7-10 Dec. 1987 (World Scientific, Singapore) to be published 8) G.E. Brown, E. Osnes and M. Rho, Phys. Lett. B163 (1985) 41; SO. Backman, G.E. Brown and J.A. Niskanen, Phys. Reports 124 (1985) 1 9) G.E. Brown and W. Weise, Phys. Reports 22 (1975) 280 10) E. Oset, H. Toki and W. Weise, Phys. Reports 83 (1982) 281; T.E.O. Ericson and W. Weise, Pions and nuclei (Clarendon, Oxford, 1988) 11) E. Oset, L.L. Salcedo and D. Strottman, Phys. Lett. B165 (1985) 13; E. Oset and L.L. Salcedo, Nucl. Phys. A468 (1987) 631 12) H. Strobele et al., in Proc. 7th high energy ion study, GSI, 1984; see also G.E. Brown, Proc. Conf. on nucleus-nucleus collisions, Saint Malo, June 6-l 1, 1988, ed. C. Detraz et al., Nucl. Phys. A488 (1988) 689~ 13) C. Gale and J. Kapusta, Phys. Rev. C35 (1987) 2107 14) L.H. Xia, C.M. Ko, L. Ciong and J.Q. Wu, Nucl. Phys. A485 (1988) 721 15) G. Roche et al., Proc. Conf. on nucleus-nucleus collisions, Saint Malo, June 6-11, (1988), Nucl. Phys. A488 (1988) 477~ 16) G. Bertsch, M. Gong and M. Tohyama, Phys. Rev. C37 (1988) 1896 17) T.L. Ainsworth, G.E. Brown, M. Prakash and W. Weise, Phys. Lett. B200 (1988) 413 18) G.E. Brown, Prog. Theor. Phys., Suppl. No. 91 (1987) 85 19) A. Arima, Progr. Part. Nucl. Phys. 11 (1984) 53