Equilibration in relativistic heavy-ion collisions with relativistic VUU

NUCLEAR PHYSICS A

Nuclear Physics A559 (1993) 603-616 North-Holland

Equilibration in relativistic heavy-ion collisions with relativistic VUU Li Zhuxia, Zhuo Yizhong r, Gu Yingqi, Sun Zemin ofAtomic Energy, Be&g 102413, China

Institute

Yu Ziqiang Dept. of Physics, Nankai Unit,~ersi&,Tianjin, China

M. San0 The institute of Physical and Chemical Research 0UKENI

Wako, Saitama, Japan

Received 19 November 1991 (Revised 18 November 1992)

Abstract The equilibration in relativistic heavy-ion collisions for the systems 160 + *%, 40Ca i-40Ca and 139La+ “39(La is studied with relativistic VUU theory. We have found that the mean field still plays an important role in addition to the collision term in the equilibration process in relativistic heavy-ion collisions at energy around 1 GeV/nucleon. The influence of the medium effect is studied. The light and medium systems do not reach complete equilibrium so that the concept based on it is questionable at the energy domain 1 GeV/nucleon.

1. Introduction High-energy heavy-ion collisions offer a unique possibility to study the properties of nuclear matter far from its ground state. But the interesting information about the dynamics of nuclear reactions and the equation of state of hot and dense matter can only be obtained indirectly through certain theoretical models. Many models are based on the assumption of very fast thermalization of the nuclear matter during nuclear collisions. One of the basic questions that those models must face is “Does the colliding system eventually reach some kind of thermal equilibrium during a certain period of the collision process?” The answer to this question is quite different for different calculations. Correspondence to: Prof. Z.-X. Li, Institute of Atomic Energy, PG. Box 275(18), Beijing 102413, China. ’ Also Institute of Theoretical Physics, Academia Sinica, Beijing, China. 0375-9474/93/$06.~

0 1993 - Elsevier Science Publishers B.V. All rights reserved

604

Z. Li et al. / Equilibration

Recent investigations of FrimHn, Ivanov, Norenberg [1,2] by considering two counter-streaming nuclear matter currents suggested a very fast equilibration in heavy-ion collisions at energy around Elab = 1 GeV/nucleon and then a hydrodynamical model can be used. Similar conclusions were obtained by Randrup [3] based on the relativistic Uehling-Uhlenbeck equation with similar geometry. Cugnon et al. [4] studied the equilibration of the head-on collisions of two 40Ca nuclei at Ebeam = 1 GeV/nucleon by the Monte Carlo cascade model and concluded that complete thermal equilibrium was never reached. Recently Lang et al. [5] investigated the local thermal properties and equilibrium phenomena for systems from two 40Ca nuclei to two 93Nb nuclei at energy 1 GeV/nucleon by the relativistic VUU equation. They found that local equilibrium was reached only when the density significant dropped at the later stage of expansion. So from their study one would draw a conclusion that the assumption of local equilibrium is very questionable and so is the hydrodynamical model. In view of the situation mentioned above it seems to us that the equilibration process might be energy and mass dependent. Furthermore it was predicted that at low energy the mean field plays a dominant role and at intermediate role on the reaction dynamics. Then one would ask to what extent the mean field still plays a role in the nuclear dynamics at relativistic heavy-ion collision (- 1 GeV). So it would be worthwhile to study more systematically the equilibrium process in heavy-ion collisions with a more realistic model. We therefore in this work make a further study of this problem with relativistic VUU for a mass range from two 160 to ‘39La and energy around 1 GeV/nucleon. The brief description of the model will be given in sect. 2. In sect. 3 the results of our calculations are presented. The summary and conclusions will be given in sect. 4.

2. The model Based on the Walecka’s many-body field theory by means of the closed-time path Green function technique and assuming that the Green functions and the self-energy terms are slowly varying functions of the center-of-mass coordinates we have derived the relativistic Boltzmann-Uehling-Uhlenbeck equation which reads as

(1)

Z. Li et al. / Equilibration

605

where

In our derivation not only the Hartree term but also the Fock term of self-energy are involved (superscript H for Hartree term and F for Fock term). For the details of the derivation of eq. (1) we refer the readers to ref. [6]. The detailed expressions of c+(s, t), ZH+(xl, SF+(~, ~1, ZH(xl, X’(X, p), Ft, FF, etc. are given in the appendix. Here the right-hand side of eq. (11, the collision term part, is also derived self-consistently, where the cross section a(.~, cl is computed based on the same lagrangian, so that the medium effects are taken into account automatically. In the present investigation, for simplicity we will not include the Fock terms and in this case the equation is found to be the standard form used in many literatures [8,9]. We assume that the coupling to scalar and vector mesons for both nucleons and deltas are the same following the arguments of refs. [lO,ll]. Hence deltas obey the same equation as nucleons when propagating in the nuclear medium except for the mass difference. An furthermore in the numerical simulations for the collision term we will not carry out a selfconsistent calculation but the Cugnon parametrization 141is used as in many literatures. We take the in-medium cross section to be equal to the free one at the same relative momentum in the c.m. system of two colliding nucleons and then we test the effect of the in-medium cross section by comparing the results with that of introducing the effective cross section with ueeff/grfree= 0.81, 0.64, 0.45, respectively. In the collision term we include elastic and inelastic channels which are reasonable for the energy domain of interest. The calculations based on selfconsistent effective cross sections are in progress.

3. Equilibration

process

In order to study the equilibration in relativistic heavy-ion collisions our calculations are mainly devoted to head-on collisions of the two nuclei 40Ca + 40Ca with Ebeam = 1 GeV/nucleon. For the study of the mass dependence the collisions of two 160 and two 139La are also presented. The effects of the equation of state are tested by introducing different sets of coupling parameters with fixed masses of the scalar and vector meson m, = 550 MeV, M, = 783 MeV, which are commonly assumed in the NN collisions. The coupling constants are taken from ref. [7] namely, (a> set I the linear coupling a-field with parameters of g, = 9.57, g, = 11.67, (bl set II the non-linear coupling p-field with g, = 6.9, g, = 7.54, B = -40.49 fm-‘, C = 383.07, (cl set III the non-linear coupling u field with g, = 9.4, g, =

Z. Li et al. / Equilibration

606

10.95, B = -0.67 fm-‘, C = 40.44 and a soft one from ref. [5], (d) set IV g, = 7.937, g, = 6.696, B = 42.35, C = 157.55. The corresponding effective mass and compressibility are as follows: the effective mass is 0.83 for (b), 0.70 for (c) and 0.85 for (d); the compressibility is 380 MeV for both (b) and (c) and 210 MeV for (d); for the linear coupling case (a) after a slight readjustment in order to fit the binding energy of 160 and 40Ca the effective mass is 0.56 and the compressibility is 540 MeV for this case. 3.2. Head-on collisions of 4oCa + 4oCa at beam energy Ebeam = 1 GeV/nucleon Fig. la, b shows the time evolution of baryon number density in the reaction plane p(x, 0, t> and momentum distribution pP(px, p,) with set I. At the initial time two nuclei are separated both in coordinate space and momentum space. At t = 5 fm/c two nuclei partly overlap each other in the coordinate space. At t = 8 fm/c two nuclei have completely overlapped in coordinate space and reach the maximum density while they are still far from complete overlap in the momentum space. During this stage, the compression stage, the size of the density distribution of the two overlapping nuclei in the z-direction is similar to that of single 40Ca but with wider extent in the x-direction. At t = 15 fm/c, when the system in coordinate space is expanded considerably, two spheres in momentum space become overall overlapped and afterwards the shape of the momentum distribution does not change much. Fig. 2 shows the time evolution of the average collision number per nucleon as well as the density at center of mass for set I, set II and set III, respectively. We find that the time when the momentum distribution reaches the maximum overlap is far later than the end of compression, which is correlated with the time that the average collision number suffered by a nucleon almost reaches the saturation value as is seen by comparing fig. 2 to fig. lb. One can also easily find from fig. 2 that the average collision number and density at center of mass are similar for sets I and III but quite different with set II. Fig. 3 shows the time evolution of the distribution in rapidity space. When t = 5 fm/c there are two peaks located at the initial rapidities of two nuclei. As is easily observed, the small rapidity is quickly populated at t = 8 fm/c and represents the overall longitudinal slowing down. At t = 15 fm/c, following the complete overlapping of two spheres in momentum space, the shape of the rapidity distribution looks like that equilibrium is almost reached in the longitudinal direction. In order to study the role of mean field and the collision term on the equilibration process and the degree of the relaxation reached during the reaction process more quantitatively we calculate the kinetic flow tensor, which is defined in ref. [121 as d4p d3x T” = j~P’pJf(x>

P),

(2)

Z. Li et al. / Equilibration

607

for the quantity, the relative difference of T,,,, and Tperp, R = ( I TX3- T,, I)/(T,, + T,,) can be taken as a measure of the degree of relaxation of the system. Fig. 4 shows the time evolution of the components of kinetic flow tensor T,,, TX3 for the following cases; (a> mean field with set I plus collision term, Cb) nonlinear u-field with set II plus collision term, Cc> nonlinear a-field with set III plus collision term, Cd) soft equation of state (e.0.s.) i.e. set IV plus collision term, (e) mean field only (with set I> i.e. switching off the collision term, (f> collision term only i.e. switching off the mean field. As can be easily seen from fig. 4, the initial

25FM/C

lSFM/C

Fig. la. Time evolution

of baryon

number density p(x, 0, z) for head-on unit of the axes is 10 fm.

collision

of 40Ca+40Ca.

One

608

Z. Li et al. / Equilibration

La-....,....,,...,....

*

8FM/C

SFM/C \

I

8.1 -

..I

d

7 L

@ 1.1 -

1.) -

-4.s -

-

P

,

-II

1

-a.

-1..

1.1,.

-

.

.

.

‘

,

.

.

.

I..

s

1.

X-i&S .

.

,

-&I-

1. 8

.

.

.

La-....,....,....

,

,,.,

4

. r

C

25FM/C

,

. .

-1..

Fig. lb. Time evolution

4.

s

x-i&s

of the momentum distribution p,,(p,, p,) or head-on One unit of the axes is 10 fm-‘.

6

Ca+Ca

k E=l

4 2 0

0

10

----

GeV/n

20

Set

I

Set

II

Set

III

collisions

I.

s

of 40Ca+40Ca.

No. Of collision Density P?,

30

40

t(f m/c)

Fig. 2. Time evolution of the average collision number per nucleon and the density p/p0 at center of mass for head-on collisions of 40Ca + 40Ca. The thin solid curve corresponds to the parameter set I. The thick solid curve corresponds to set II. The dashed curve corresponds to set III.

1.

a

Z. Li et al. / Equilibration

-z.

-1.

cl.

I.

2.

-z.

-1.

609

a

L

z.

Y

Fig. 3. Time evolution

of the distribution

in rapidity

space for head-on

collisions

of ‘“Ca + 4”Ca.

value of Tj3 for the case with the mean field taken into account is much smaller than that without the mean field. This is because when we boost the projectile and target nuclei from their rest system to the rest system of two nuclei the longitudinal momentum p3 of each nucleon is roughly proportional to its energy Cm** +P*)‘/~ in the projectile or target rest system, which is roughly proportional to the effective mass while the initial transverse momentum is not affected. So the smaller the effective mass is, the smaller the initial value of Tj3 and simultaneously the larger the potential energy stored in the initial system. While the initial value of T,, is the same both with and without the mean field.

120 Ca+Ca

E=lGev/n

\

100

___________

_ _ - _ _ _ ---_

60

\ \

_

\,:,,,

_

-

Fig. 4. Time evolution

_

_

_

of T,,, Tj3 for head-on

set I set II set Ill set N Mean

field

Collision

collisions

of 40Ca+ “Ca.

610

Z. Li et al. / Equilibration

Controlled by both the mean field and collision term in cases (a), (b), (c) and (d) following the start of the compression stage from t = 5 to 10 fm/c, two-body collisions quickly thermalize the system and consequently, T33 drops very rapidly and at the same time T,, quickly increases. From t = 10 - 15 fm/c Tj3 does not change very much and T,, increases further and then reaches its final value. Following the expansion of the system the density becomes much lower and both T33 and T,, remain at their constant values. In order to study the role of the collision term and mean field on the equilibration process we also make simulations for cases that only the mean field (case (e>>or only collision term (case (f)) is taken into account. We can easily find from fig. 4 that the final value of relative difference R for case (f) is larger than cases (a), (b), (c) and (d) but less than the case (e), which implies that the mean field still plays a considerable role in addition to the collision term in the equilibration process for heavy-ion collisions at energy around 1 GeV/nucleon. As to the dependence of the equilibration process on the e.o.s. let us make a comparison between case (a)-(d). The final values of T,, of (a)-(d) show a tendency for larger g, to correspond to larger Tll, which can be understood as the larger g, providing a stronger Lorentz-like force leading to a larger deflection of particle to the beam axis. We can also see from fig. 4 that for cases tbl and (d) with similar effective mass and very different compressibility the Tj3 are similar, but for (b) and (c) with very different effective mass and same compressibility the T33 are very different. This indicates that the final value of TX3 strongly depends on the effective mass but only weakly on the compressibility. As is known the effective mass is closely correlated with g, so we can conclude that the dependence on the effective mass is stronger than on the compressibility as far as the dependence of the degree of relaxation reached in HIC at 1 GeV/nucleon is concerned. The effect of the in-medium cross section on the relaxation process is tested by calculating the ratio of T33,11(~~ff)/T33,11(~free) for different a,,. The ratios of are 1.10, 0.95 for geeff/crffree = 0.81; 1.20, 0.89 for ~,,/a,,,, = T33,11(~~eff)/T33.11(~~free) = 0.45, i.e., the degree of equilibrium reached is 0.64 and 1.41, 0.76 for a&a,,,, lowered for small effective cross section. The results imply that the medium effect influences the equilibrium process very strongly in the heavy-ion collisions at energy around 1 GeV/nucleon. 3.2. The mass dependence As mentioned in the Introduction the degree of relaxation in the reaction process may depend on the mass of the system so we will make a systematic study of this aspect. The investigation of the mass dependence of the thermalization process in heavy-ion collisions is performed by carrying out the calculations for ‘“0 + 160 collisions as well as 139La + 139La collisions. Figs. 5 and 6 show the time

Z. Li et al. / Equilibration

611

E=lGev/n

Fig. 5. Time

evolution

of T,,,

Tj3 and

density p/p0 at center ‘60+ ‘ho.

of mass

for head-on

collisions

of

evolution of the same quantities as in fig. 4 and fig. 5 for I60 + I60 and fig. 6 for ‘39La + 139La, respectively. The general feature of the time evolution of the kinetic flow tensor for systems 160 + 160 and ‘39La + ‘39La is similar with 48Ca + 40Ca as discussed in sect 3.1. But the final value of the relative difference of T,, and Tj3 for light systems becomes much larger. For a heavy system we find even more rapid decrease of Tj3 and

280 Laa+La \

240

_________--

set II

----

set

E-lGw/n :J

..200 -

I\ :

N

p I\ :

160

;t

-

‘E

z

F

*\ : :\ ‘y_ -->d=d;;,;~,:~_-~_-_es-_________ /

120

60

TU T-7

,/ 40

I

I

0 0

2&rl/cf0

Fig. 6. Time evolution

1

10

of T,,, TX3 for head-on

4o

collisions

5o

of 13”La + ‘39La.

612

Z. Li et al. / Equilibration

E= 1 Gev/n

0

Fig. 7. Same quantities

as fig. 5 but with no delta production.

increase of T,, at the compression stage. The value of T,, becomes much closer to T33 or even exceeds the value of Tj3 at a late time of the compression stage depending on which coupling constant set is used. But it seems that the complete match of TX3 and T,, is difficult for a finite system since after the compression stage the density becomes quite low. For collisions of two bulks of nuclear matter the complete match of T, 1 and Tj3 can be very possibly reached since the system is homogeneous and the density never changes in coordinate space [l]. 3.4. Role of the delta particles As is well known the delta particles play an important role in heavy-ion collisions around energy E = 1 GeV/nucleon. In order to study the role of delta particles in the equilibration process we make a calculation in which we suppose that the elastic cross section equals the total cross section so that delta productions are switched off (off case). In fig. 7 we show the time evolutions of T,,, T33 and density p/p0 at center of mass for I60 + 160 collision for “off case”, respectively. For simplicity we call the case shown in fig. 5 “on case”. Let us make a comparison between fig. 5 and fig. 7. We can find that the shape of the time evolution of p/p0 at center of mass is very similar for both cases. A much higher final value of T33 is obtained for the “off case”. As for the time evolution of T,,, not only the final value of T,, but also the shape of T,, are similar. The very different effects of delta particle production on T,, and T33 imply

Z. Li et al. / Equilibration

613

that the delta production plays an important role only in slowing down the longitudinal momentum due to the energy carried away by delta production but has almost no effect on the transverse motion.

4. Discussion In summary, in this work we have studied thermalization process and equilibration in the collisions of 160 + 160, 4”Ca + 40Ca and lJ9La + ‘39La at beam energy 1 GeV/nucleon with relativistic VUU theory. We find that the time to reach complete overlap in coordinate space is much earlier than that in momentum space for head-on heavy-ion collisions around beam energy 1 GeV/nucleon. The former is correlated to the time when the system reaches the maximum density, while the latter is correlated to the time when the average collision number reaches the saturation value as well as when the equilibrium in the longitudinal direction is almost reached, when the density in the coordinate space is reduced considerably. After the system reaches complete overlap in momentum space the momentum distribution will not change much, and nor does the average collision number experienced by a nucleon. We have also studied the time evolution of the kinetic flow tensor T,, and T33. Both Tll and T33 reach constant value around time 1.5fm/c, which corresponds to the time when two nuclei become overall overlapped in momentum space. From the comparisons of calculations for various different cases it can be concluded that the mean field still plays an important role in addition to the collision term which is well known in the thermalization process for heavy-ion collision at energy around 1 GeV/nucleon. We have found that the mass dependence of the degree of the equilibrium is quite significant. For the light and medium system like 40Ca f4”Ca the camp 1e te equilibrium is not reached at beam energy around 1 GeV/nucleon. Since real collisions are mostly not head-on our study can be taken as an upper limit of the degree of relaxation reached in the real collision. In this case the concept based on complete equilibrium is questionable. It is also indicated that the medium effect has strong influence on the equilibration process. So the selfconsistent calculations based on eq. (1) seem to be necessary.

Appendix

For readers convenience in the appendix we give the expressions of X”~‘*(x), ZF+L(_x,p), XH(~), ZF(x, p), &, 11, Fr%xI, F-j’(x), which appear in eq. (3). ,ZH+(.r), X’+‘(X, p,>, _XH(~), ZF(.x, p), are Hartree and Fock terms contributed from vector and scalar mesons, respectively. Without introducing confusion we

614

Z. Li et al. / Equilibration

omit the superscript of m* in the following expressions and Z”J‘(X), _ZF,w(.x,p), Z”(X), ZF(x, P> read as

(A.21 (A.3) (A-4)

z,“(x)

z$y

= - gf +dp, 2mZ (27~)~

X,

(A.5)

1 P)

=

-

-

i(p:+m’)

(W3

-z&(x7 P) = -

Pf

/dp,

&

jdP3

(p-p3f2-&f(x’

/&&

p3)7

_,;,"f(+? p3). (A.7)

(p -P3;2

For the collision term we take the notation of De Groot, Van Leeuwen and Van Weert [13]: Zcdl

=

i

’

dP, a(s,

t)Z/(F;-F;)

Here

~(s, t) should take the average

an,(s,

t) in the

center

of mass

do

.

Value

of

(A.8) a,,(S,

t),

a,,,(S,

t),

gnn(S~ l)

and

of two colliding particles. 7 is the Mailer velocity

and FP = f( x, P)f( F; = (1

-f(x,

x, P,)(l p))(l

-f(x7P3))(1 -f(x, -f(X,

Pz))f(X,

p4),

(A.9) (A.lO) (A.ll)

s = (I, +P2)2Y t=(p-p3)2=~(t-4m2)(cos

P3)f(X,

a>)~

O-l),

(A.12)

Z. Li et al. / Equilibration and

B

615

the angle between p and p3. The expressions of a,,(~, t), q,, read as (A.13)

Let t’ = 4m2 - s - t then oi(s, t) can be expressed as

(+l(S, t) =gs”[D*(S,

t) +L),(s,

t’) +c,(sY

(T*(s, t) =g,“[D,(s,

t) +D,(s,

t’) + C,(s3 t, t’)],

$( s, t) =&,2[

Oj( s, t> +D,(s,

t, t’)],

t’) + C,(s,

t, t’)].

(A.14) (A.15) (A.16)

The expressions of s+,LT~~read as (A.17) where a;(s,

t> =s,“[q&

t> +D,(s7

t’)],

(A.18)

fli(s,

t> =8,4[Lfz(s,

t) +&(s,

f’)],

(A.19)

q].

(A.20)

g;;(s, t) =g,2g,z[~&,

t)

D,, I),, D,, C,, C,, C, are defined as

Functions

D,(S, t) = (r-4m”2 8( t - m$

cl(s,

t) = -

D,(s,

t> =

c (s 2

+D,(s>

(A.21) ’

( t2 + st) + 47727s - t) qt-m;)(t’-mf) ’

2s2 + 2st + t2 - 8m2s + 8m4 (A.23)

4(t-mt)2

t> = (s - 2m2)(s 9

(A.22)

’

- 6m2)

2(t -m$)(t’-rnt)

(A.24)

’

(2s + t - 4mz)mz D3(s7

t,

~~(3(s

I)=

f

=

p

_mz)(t

_m;)

(A.25)

’

(t+s)2-2m2s+2m2t 4(t -m”,)(t’-rni)

t2-4m2s-10m2t+24m4 +

4(t -m$)(t’-m2,)

(A.26)

616

Z. Li et al. / Equilibration

This work was supported partly by National Natural Science Foundation of China. One of the authors (Li Zhuxia) acknowledges the hospitality of RIKEN, Japan. Part of this work was done during her visit there.

References [l] M. Cubero, B. Friman, Y. Ivanov, W. Niirenberg, V. Russkikh and M. Schonhofen, Z. Phys. 340 (1991) 385 [2] M. Cubero, M. Schonhofen, B.L. Friman and W. Ndrenberg, Nucl. Phys. A519 (1990) 34% [3] J. Randrup, Nucl. Phys. A314 (1979) 429 [4] J. Cugnon, T. Mizutani and J. Vandermeulen, Nucl. Phys. A359 (1981) 505 [5] A. Lang, B. Blattel, V. Koch, K. Weber, W. Cassing and U. Mosel, to be published in Z. Phys. A [6] Z.-Q. Yu, G.-J. Mao, Y.-Z. Zhuo, Z.-X. Li and Z.-M. Sun, Chin. J. High Energy Phys. Nucl. Phys. 16 (1992) 312 [7] X.-M. Jin, Y.-Z. Zhuo, X.-Z. Zhang and M. Sano, Nucl. Phys. A506 (1990) 655 [8] C.M. Ko, Q. Li and R. Wang, Phys. Rev. Lett. 59 (1987) 1084 [9] H.T. Elze et al., Mod. Phys. Lett. 2 (1987) 451 [lo] B.M. Waldhauser, J. Theis, J.A. Maruhn, H. Stocker and W. Greiner, Phys. Rev. C36 (1987) 1019. 1111 S.I.A. Garpman, N.K. Glendenning and Y.J. Karant, Nucl. Phys. A322 (1979) 382 1121 M. Gyulassy, K.A. Frankel and H. Stocker, Phys. Lett. BllO (1982) 185 [13] S.R. de Groot, W.A. van Leeuwen and Ch.G. van Weert, Relativistic kinetic theory (North-Holland, Amsterdam, 1980)