- Email: [email protected]
On a four-nucleon model of nuclear matter
ELSEVIER
Nuclear
Physics
A722
(2003)
548~~552~ www.elsevier.com/locate/npe
On a four-nucleon
model of nuclear
matter
Tran Huu Phata, Nguyen Tuan Anhb and Le Viet HoaC aVietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam. bInstitute for Nuclear Science and Technique, P.O.Box 5T-160, Hoang Quoc Viet, Hanoi, Vietnam. “Department of Physics, Hanoi University of Education II, Hanoi, Vietnam The nuclear matter saturation mechanism based on a four-nucleon interaction is proposed. The model describes rather well nuclear matter properties in the Hartree-Fock (HF) approximation. 1. INTRODUCTION
The successof nuclear physics in satisfactorily explaining low energy nuclear phenomena leads to a strong belief that nucleons and mesons are appropriate degrees of freedom. The relativistic treatment of nuclear many-body systems introduced not long ago by Walecka [I] is the QHD-I model based on nucleons, vector and scalar mesons. At the mean-field level QHD-I describes the nuclear binding and saturation at the normal density po as a result of the interplay between a large scalar and a large vector meson condensates. Later on, the QHD-II model [a], which in’corporated charge vector p meson and pseudoscalar ?r meson into consideration, was developed by Serot and applied to finite nuclei. At present, QHD turned out to be a quite successful tool for the study of many nuclear properties: binding energies and nuclear charge radii are reproduced to within a few percent and the density distributions of doubly magic sphkrical nuclei are in the excellent agreement with electron scattering data. The recent investigation [3] shows that most of the ground state properties of a large number of spherical and deformed nuclei over the entire range of the periodical table can be very well reproduced by relativistic mean field theory (RMF). The relativistic Hartree-Fock formalism is an important extension of RMIF. However, it has not been possible to construct a reliable approximation scheme for these approachs. This is because the loop expansion is not useful to QHD at least up to two-loop order, and, moreover, the perturbation theory and other approximation schemes, such as loop expansion and the random phase approximation, are not valid due to the extremely large self-energies of sigma and omega mesons. Recently, Guichon [4] proposed a quark-meson coupling (&MC) model of nuclear matter, in which the appropriate degrees of freedom are quarks and mesons. In this theory nuclear matter consists of non-overlapping MIT bags interacting through the self-consistent 0375-9474/03/$ - see front matter doi:lO.l016/SO375-9474(03)01425-8
0 2003 Elsevier
Science
B.V:
All rights
reserved.
7: Huu Phat et al. /Nuclear
Physics A722 (2003) 548c-552~
549c
exchange of mesons in the mean-field approximation, and the mesons are directly coupled to the quarks. It was shown that the bulk properties of nuclear matter can be satisfactorily described [5]. The &MC model has been extended fruitfully to study finite nuclei [6]. Surprisingly, the theory seems to provide a semi-quantitative explanation of the Okamoto-Nolen-Schiffer anomaly when quark mass differences are included. In the Skyrme model extended to the nuclear phenomena [7] only the meson degrees of freedom are relevant. In the present paper a four-nucleon model of nuclear matter is presented which consists of only nucleon degrees of freedom. Through the direct interaction between nucleons, two types of nucleon condensates, (iVfV) and {flr,fV), are formed in the nuclear medium. They give the main contribution to the nuclear saturation. ln Section II we formulate the model and derive the Cornwall-Jackiw-Tomboulis (CJT) effective potential. Section III deals with numerical calculations. The conclusion and discussion are given in Section IV. 2. EFFECTIVE
POTENTIAL
FOR
COMPOSITE
OPERATORS
Let us first write down the lagrangian density of the model
c = 7J(iS- n/r)+ + $(a$;2 - $($-i,+)‘,
(1)
where + is nucleon field operator, M nucleon mass, gs and gU nucleon coupling constants. Bosonization of (1) leads to c = ?j[iS - n/r + g&r - g&J;l?J - 3(u2 - w”),
(2)
in which u = gs$+, wP = y,qy,$. It is clear that (2) has the structure of the QHD-I lagrangian density without the kinetic terms for sigma and omega mesons. According to [8], all external sources vanish in the ground state, the expression for the CJT effective potential V[ii, L3,S, C, OPU] has then the form 4
v
=
+2+p+1
1 ($4
d
i 2 -4
d4p tr{[ln C;‘C(p) (27r)”
-
i -2a, i^ &$ +;A
tr[ln S;‘(p)S(p)
- S-l(p; f?, i;r)S(p) + l]
- C<‘C(p) + l] + [In Di$Y”(p)
trP(27)r8hk -p)S(k)C(k -
-YZ?-fi
s (27T)”(2n)4
tr[y’“S(p)P(p,
Ic - p)S(k)D,,(k
- 0&,0”“(p)
PII
- p)],
where iSi’ = i - n/l, is-i(g,G) = iS:i + gso - g,G, iC,i = -1, iD~,i, are the propagators of nucleon and mesons, respectively, and B = (F[upq
=
co71st.,
6, = (Flw,jF)
+ 11)
= gpv; S, C, D,,
= 6,ow = const.
are mean values of the field operators, ]Fj is the ground state of nucleon matter.
55oc
7: Huu Phat et al. /Nuclear Physics A722 (2003) 54&z-552~
The configuration
of the ground
state requires
bV -= F = {a,G}, 6F ‘I 6V -= G = {S, C, D,.w>, SG ” (4) is the gap equation for nucleon condensate and (5) is the Schwinger-Dyson equation for propagator G. Inserting (3) into (4) we obtain the gap equations for 8 and W,
(4) (5) (SD)
4
IF= -gs ___ J ($4 tGYp)l= cm,
trhd(p)l = sum,
(6)
where ps and ,LIB are the scalar and baryon density, respectively. The SD equations (3) into (5) accordingly, propagators S, C and D,, are obtained by substituting is-l(k)
= iS,l(k)
for
- C(k),
(7)
+k)C(P +k)l J--!--N?J)rs(P,P +&WY0 + J-(;$hp”sb)r”h~ + ~P,vb + fill, = -1 - &(k), IT,(k)= ig, & tr[s(P)rshP +k)S(P +k)l, (8) J +~)S(P +~)I,($ J4 tr[7,Ap)rvb,p C(k) = -gsa
- G7,
(&)
i9,
C’(k) iD,;‘(k)
= -gpv
- qw(~)>
qw(~)
= -Gl?J
(gI
C, II, and II,,, are self-energies of nucleon, sigma and omega meson respectively. Next let us derive the formula for energy density in the bare vertex approximation, l?,(p, p) = gs and lYP(p, 4) = -gVyW. As suggested in [Q], we confine our consideration to contribution from the density-dependent part of the nucleon propagator, which is dominant at low density, S(k)
= (k + M*(k))G,(k),
GD(k)
= - &%
- E*(k)P(~oP(b
where E*(k) = (.@ + IW*~(~))‘/~. F& this purpose, let us follow given by the SD equation (7), which is rewritten as follows CT’(k)
= iS,l(k)
- C(k)
= To[ICO+ Co(k)]
where C,, CO and C, are three components leads to the definition of effective quantities k; = ki[l + C,(k)],
k,* = ko + C,(k),
[IO] and start with
- yZ[l + C,(k)]
of Dirac
decomposition W(k)
- [iId + C,(k)], of C(k).
= n/r + C,(k).
Substituting S from (10) into (3), we obtain the expressions part of effective potential V and the energy density of nuclear
- lh(lO) S(k) (lla) Eq. (lla)
(lib)
for the density-dependent matter,
kF
Y &=V+E, P2& &=s
o J
E*(P)
(12)
T. Huu Phat et al. /Nuclear Physics A722 (2003) 548c-5.52~ 3. NUMERICAL
CALCULATION
IN HARTREE-FOCK
551c
APPROXIMATION
In this section let us restrict to the approximation, in which all interacting meson propagators are replaced by their free ones. Furthermore, the vacuum polarization is not taken into account. This is the HF approximation. The numerical calculation is the following: We solve the integral equation, W(/C,) = M + Cs(lcF), and then substitute lW*(k.~) into the binding energy, &in = --&I + &/ps. Two parameters gs and gv are adjusted to reproduce the nuclear saturation point. The values of gs and gv can be determined uniquely by the condition that the binding energy &bz, = -15.8 MeV is minimum at the normal density /cF ’ - 1.42 fin-‘. Their values are fixed to be gf = 229.5/n/1’ and g,” = 0.5853~~:. In Fig 1, we show the saturation curve for nuclear matter with these values of gs and gv. As one can see from Fig 1, the neutron matter is not bound. In Fig 2, the density dependence of n/r*/&’ is plotted. For a comparison, results given by different models are listed in Table 1. One can see that our results are closer to those of NSD.
I 0
0 I
I
2
I
k, [;d]
Figure
Table
1. The
saturation
kF [ fm-I]
curve
of nuclear
matter
Figure
2. Density
dependence
of W/n/I.
1
Coupling constants, nuclear matter QHD, NSD, QMC and our model. and m, = 783 iVIeV.
Model
incompressibilities, effective nucleon masses obtained in four models: The masses used in calculations are n/f = 939 MeV, m., = 550 MeV,
QHD-I NSD [lo]
10.455
625
0.52
8.714
10.678
572
0.53
QMC
5.84
6.29
293
0.78
8.873
9.664
547
0.56
210 SC 30 [ll]
0.6 [12]
Our
Exp.
model
gum,
K (MeV)
gsma 9.159
552c 4. CONCLUSION
T. Huu Phut et al. /Nuckur AND
Physics A722 (2003) 548c-SS2c
DISCUSSION
In the present work we have presented results of the four-nucleon model for nuclear matter, in which mesons emerge as bound states of nucleons in nuclear medium. Within the framework of the CJT formalism, higher order contributions are naturally included and the contributions from ring diagrams for mesons are explicitly treated in the energy density. We reproduce, in the Hartree-Fock approximation, the nuclear matter properties, which are as good as those obtained in QHD and nuclear Schwinger-Dyson (NSD) equations [lo]. Compared to other models, our theory contains a minimal number of parameters (gs, g, and &f), because the nuclear matter is built up from nucleons only and meson degrees of freedom play no role. In the spirit of the unified field theory of Heisenberg [13] it is possible that only nucleon degrees of freedom are relevant for an effective model of nuclear phenomena and, hopefully, this fact has a close relation to the problem raised by Bertsh et. al. [14]. However, it is worth to mention that lacking chiral symmetry is a serious shortcoming of our model. Unfortunately, its generalization to cover chiral symmetry does not yield saturation point and the nuclear matter turns out to be unstable [15, 161. In general, the present theory needs to be extended to study finite nuclei. This is the objective of our further study. This work is supported by Vietnam Atomic Energy Commission and National Science Council of Vietnam. REFERENCES 1. J. D. Walecka, Ann. Phys. 83 (1974) 491. 2. B. D. Serot and J. D. Walecka, Phys. Lett. B87 (1997) 172. 3. Y. K. Gambhir, P. Ring and A. Thimet, Ann. Phys. 198 (1990) 132. 4. P. A. M. Guichon, Phys. Lett. BZOO (1988) 235. 5. K. Saito and A. W. Thomas, Phys. Lett. B327 (1994) 9; Phys. Rev. C52 (1995) 2789. 6. K. Saito, K. Tsushima and A. W. Thomas, Nucl. Phys. A609 (1996) 339. 7. Baryons as Skyrm Solitions, Ed. G. Holzwarth, World Scientific, Singapore, 1993; G. Holzwarth, Nucl. Phys. A672 (2000) 167; G. Holzwarth and R:Machleidt, Phys. Rev. C55 (1997) 1088; R. A. Leese, N. S. Manton and-B. J. Schroers, Nucl. Phys. B442 (1995) 228. 8. J. Cormwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. 9. M. P. Allendes and B. D. Serot, Phys. Rev. C45 (1992) 2975; G. Krein, N. Nielsen, R. D. Puff and L. Wilets, Phys. Rev. C4’7 (1992) 2485; M. Prakash, P. T. Ellis and J. I. Kapusta, Phys. Rev. C45 (1992) 2518. 10. M. Nakano, A. Hasegawa, H. Kouno and K. Koide, Phys. Rev. C49 (1994) 3061; C49 (1994) 3076. 11. J. P.’ Blaizot, Phys. R.ep. 640 (1980) 171. 12. H. A Bethe, Ann. Rev. Nucl. Sci. 21 (1957) 269. 13. W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269. 14. G. F. Bertsch, L. Franfurt and M. Strikman, Science 259 (1993) 773. 15. M. Buballa, Nucl. Phys. A611 (1996) 393; iv. Buballa and M. Oertel, Nucl. Phys. A642 (1998) 39c. 16. Tran Huu Phat, Nguyen Tuan Anh and Le Viet Hoa, in preparation.
Nuclear
Physics
A722
(2003)
548~~552~ www.elsevier.com/locate/npe
On a four-nucleon
model of nuclear
matter
Tran Huu Phata, Nguyen Tuan Anhb and Le Viet HoaC aVietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam. bInstitute for Nuclear Science and Technique, P.O.Box 5T-160, Hoang Quoc Viet, Hanoi, Vietnam. “Department of Physics, Hanoi University of Education II, Hanoi, Vietnam The nuclear matter saturation mechanism based on a four-nucleon interaction is proposed. The model describes rather well nuclear matter properties in the Hartree-Fock (HF) approximation. 1. INTRODUCTION
The successof nuclear physics in satisfactorily explaining low energy nuclear phenomena leads to a strong belief that nucleons and mesons are appropriate degrees of freedom. The relativistic treatment of nuclear many-body systems introduced not long ago by Walecka [I] is the QHD-I model based on nucleons, vector and scalar mesons. At the mean-field level QHD-I describes the nuclear binding and saturation at the normal density po as a result of the interplay between a large scalar and a large vector meson condensates. Later on, the QHD-II model [a], which in’corporated charge vector p meson and pseudoscalar ?r meson into consideration, was developed by Serot and applied to finite nuclei. At present, QHD turned out to be a quite successful tool for the study of many nuclear properties: binding energies and nuclear charge radii are reproduced to within a few percent and the density distributions of doubly magic sphkrical nuclei are in the excellent agreement with electron scattering data. The recent investigation [3] shows that most of the ground state properties of a large number of spherical and deformed nuclei over the entire range of the periodical table can be very well reproduced by relativistic mean field theory (RMF). The relativistic Hartree-Fock formalism is an important extension of RMIF. However, it has not been possible to construct a reliable approximation scheme for these approachs. This is because the loop expansion is not useful to QHD at least up to two-loop order, and, moreover, the perturbation theory and other approximation schemes, such as loop expansion and the random phase approximation, are not valid due to the extremely large self-energies of sigma and omega mesons. Recently, Guichon [4] proposed a quark-meson coupling (&MC) model of nuclear matter, in which the appropriate degrees of freedom are quarks and mesons. In this theory nuclear matter consists of non-overlapping MIT bags interacting through the self-consistent 0375-9474/03/$ - see front matter doi:lO.l016/SO375-9474(03)01425-8
0 2003 Elsevier
Science
B.V:
All rights
reserved.
7: Huu Phat et al. /Nuclear
Physics A722 (2003) 548c-552~
549c
exchange of mesons in the mean-field approximation, and the mesons are directly coupled to the quarks. It was shown that the bulk properties of nuclear matter can be satisfactorily described [5]. The &MC model has been extended fruitfully to study finite nuclei [6]. Surprisingly, the theory seems to provide a semi-quantitative explanation of the Okamoto-Nolen-Schiffer anomaly when quark mass differences are included. In the Skyrme model extended to the nuclear phenomena [7] only the meson degrees of freedom are relevant. In the present paper a four-nucleon model of nuclear matter is presented which consists of only nucleon degrees of freedom. Through the direct interaction between nucleons, two types of nucleon condensates, (iVfV) and {flr,fV), are formed in the nuclear medium. They give the main contribution to the nuclear saturation. ln Section II we formulate the model and derive the Cornwall-Jackiw-Tomboulis (CJT) effective potential. Section III deals with numerical calculations. The conclusion and discussion are given in Section IV. 2. EFFECTIVE
POTENTIAL
FOR
COMPOSITE
OPERATORS
Let us first write down the lagrangian density of the model
c = 7J(iS- n/r)+ + $(a$;2 - $($-i,+)‘,
(1)
where + is nucleon field operator, M nucleon mass, gs and gU nucleon coupling constants. Bosonization of (1) leads to c = ?j[iS - n/r + g&r - g&J;l?J - 3(u2 - w”),
(2)
in which u = gs$+, wP = y,qy,$. It is clear that (2) has the structure of the QHD-I lagrangian density without the kinetic terms for sigma and omega mesons. According to [8], all external sources vanish in the ground state, the expression for the CJT effective potential V[ii, L3,S, C, OPU] has then the form 4
v
=
+2+p+1
1 ($4
d
i 2 -4
d4p tr{[ln C;‘C(p) (27r)”
-
i -2a, i^ &$ +;A
tr[ln S;‘(p)S(p)
- S-l(p; f?, i;r)S(p) + l]
- C<‘C(p) + l] + [In Di$Y”(p)
trP(27)r8hk -p)S(k)C(k -
-YZ?-fi
s (27T)”(2n)4
tr[y’“S(p)P(p,
Ic - p)S(k)D,,(k
- 0&,0”“(p)
PII
- p)],
where iSi’ = i - n/l, is-i(g,G) = iS:i + gso - g,G, iC,i = -1, iD~,i, are the propagators of nucleon and mesons, respectively, and B = (F[upq
=
co71st.,
6, = (Flw,jF)
+ 11)
= gpv; S, C, D,,
= 6,ow = const.
are mean values of the field operators, ]Fj is the ground state of nucleon matter.
55oc
7: Huu Phat et al. /Nuclear Physics A722 (2003) 54&z-552~
The configuration
of the ground
state requires
bV -= F = {a,G}, 6F ‘I 6V -= G = {S, C, D,.w>, SG ” (4) is the gap equation for nucleon condensate and (5) is the Schwinger-Dyson equation for propagator G. Inserting (3) into (4) we obtain the gap equations for 8 and W,
(4) (5) (SD)
4
IF= -gs ___ J ($4 tGYp)l= cm,
trhd(p)l = sum,
(6)
where ps and ,LIB are the scalar and baryon density, respectively. The SD equations (3) into (5) accordingly, propagators S, C and D,, are obtained by substituting is-l(k)
= iS,l(k)
for
- C(k),
(7)
+k)C(P +k)l J--!--N?J)rs(P,P +&WY0 + J-(;$hp”sb)r”h~ + ~P,vb + fill, = -1 - &(k), IT,(k)= ig, & tr[s(P)rshP +k)S(P +k)l, (8) J +~)S(P +~)I,($ J4 tr[7,Ap)rvb,p C(k) = -gsa
- G7,
(&)
i9,
C’(k) iD,;‘(k)
= -gpv
- qw(~)>
qw(~)
= -Gl?J
(gI
C, II, and II,,, are self-energies of nucleon, sigma and omega meson respectively. Next let us derive the formula for energy density in the bare vertex approximation, l?,(p, p) = gs and lYP(p, 4) = -gVyW. As suggested in [Q], we confine our consideration to contribution from the density-dependent part of the nucleon propagator, which is dominant at low density, S(k)
= (k + M*(k))G,(k),
GD(k)
= - &%
- E*(k)P(~oP(b
where E*(k) = (.@ + IW*~(~))‘/~. F& this purpose, let us follow given by the SD equation (7), which is rewritten as follows CT’(k)
= iS,l(k)
- C(k)
= To[ICO+ Co(k)]
where C,, CO and C, are three components leads to the definition of effective quantities k; = ki[l + C,(k)],
k,* = ko + C,(k),
[IO] and start with
- yZ[l + C,(k)]
of Dirac
decomposition W(k)
- [iId + C,(k)], of C(k).
= n/r + C,(k).
Substituting S from (10) into (3), we obtain the expressions part of effective potential V and the energy density of nuclear
- lh(lO) S(k) (lla) Eq. (lla)
(lib)
for the density-dependent matter,
kF
Y &=V+E, P2& &=s
o J
E*(P)
(12)
T. Huu Phat et al. /Nuclear Physics A722 (2003) 548c-5.52~ 3. NUMERICAL
CALCULATION
IN HARTREE-FOCK
551c
APPROXIMATION
In this section let us restrict to the approximation, in which all interacting meson propagators are replaced by their free ones. Furthermore, the vacuum polarization is not taken into account. This is the HF approximation. The numerical calculation is the following: We solve the integral equation, W(/C,) = M + Cs(lcF), and then substitute lW*(k.~) into the binding energy, &in = --&I + &/ps. Two parameters gs and gv are adjusted to reproduce the nuclear saturation point. The values of gs and gv can be determined uniquely by the condition that the binding energy &bz, = -15.8 MeV is minimum at the normal density /cF ’ - 1.42 fin-‘. Their values are fixed to be gf = 229.5/n/1’ and g,” = 0.5853~~:. In Fig 1, we show the saturation curve for nuclear matter with these values of gs and gv. As one can see from Fig 1, the neutron matter is not bound. In Fig 2, the density dependence of n/r*/&’ is plotted. For a comparison, results given by different models are listed in Table 1. One can see that our results are closer to those of NSD.
I 0
0 I
I
2
I
k, [;d]
Figure
Table
1. The
saturation
kF [ fm-I]
curve
of nuclear
matter
Figure
2. Density
dependence
of W/n/I.
1
Coupling constants, nuclear matter QHD, NSD, QMC and our model. and m, = 783 iVIeV.
Model
incompressibilities, effective nucleon masses obtained in four models: The masses used in calculations are n/f = 939 MeV, m., = 550 MeV,
QHD-I NSD [lo]
10.455
625
0.52
8.714
10.678
572
0.53
QMC
5.84
6.29
293
0.78
8.873
9.664
547
0.56
210 SC 30 [ll]
0.6 [12]
Our
Exp.
model
gum,
K (MeV)
gsma 9.159
552c 4. CONCLUSION
T. Huu Phut et al. /Nuckur AND
Physics A722 (2003) 548c-SS2c
DISCUSSION
In the present work we have presented results of the four-nucleon model for nuclear matter, in which mesons emerge as bound states of nucleons in nuclear medium. Within the framework of the CJT formalism, higher order contributions are naturally included and the contributions from ring diagrams for mesons are explicitly treated in the energy density. We reproduce, in the Hartree-Fock approximation, the nuclear matter properties, which are as good as those obtained in QHD and nuclear Schwinger-Dyson (NSD) equations [lo]. Compared to other models, our theory contains a minimal number of parameters (gs, g, and &f), because the nuclear matter is built up from nucleons only and meson degrees of freedom play no role. In the spirit of the unified field theory of Heisenberg [13] it is possible that only nucleon degrees of freedom are relevant for an effective model of nuclear phenomena and, hopefully, this fact has a close relation to the problem raised by Bertsh et. al. [14]. However, it is worth to mention that lacking chiral symmetry is a serious shortcoming of our model. Unfortunately, its generalization to cover chiral symmetry does not yield saturation point and the nuclear matter turns out to be unstable [15, 161. In general, the present theory needs to be extended to study finite nuclei. This is the objective of our further study. This work is supported by Vietnam Atomic Energy Commission and National Science Council of Vietnam. REFERENCES 1. J. D. Walecka, Ann. Phys. 83 (1974) 491. 2. B. D. Serot and J. D. Walecka, Phys. Lett. B87 (1997) 172. 3. Y. K. Gambhir, P. Ring and A. Thimet, Ann. Phys. 198 (1990) 132. 4. P. A. M. Guichon, Phys. Lett. BZOO (1988) 235. 5. K. Saito and A. W. Thomas, Phys. Lett. B327 (1994) 9; Phys. Rev. C52 (1995) 2789. 6. K. Saito, K. Tsushima and A. W. Thomas, Nucl. Phys. A609 (1996) 339. 7. Baryons as Skyrm Solitions, Ed. G. Holzwarth, World Scientific, Singapore, 1993; G. Holzwarth, Nucl. Phys. A672 (2000) 167; G. Holzwarth and R:Machleidt, Phys. Rev. C55 (1997) 1088; R. A. Leese, N. S. Manton and-B. J. Schroers, Nucl. Phys. B442 (1995) 228. 8. J. Cormwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. 9. M. P. Allendes and B. D. Serot, Phys. Rev. C45 (1992) 2975; G. Krein, N. Nielsen, R. D. Puff and L. Wilets, Phys. Rev. C4’7 (1992) 2485; M. Prakash, P. T. Ellis and J. I. Kapusta, Phys. Rev. C45 (1992) 2518. 10. M. Nakano, A. Hasegawa, H. Kouno and K. Koide, Phys. Rev. C49 (1994) 3061; C49 (1994) 3076. 11. J. P.’ Blaizot, Phys. R.ep. 640 (1980) 171. 12. H. A Bethe, Ann. Rev. Nucl. Sci. 21 (1957) 269. 13. W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269. 14. G. F. Bertsch, L. Franfurt and M. Strikman, Science 259 (1993) 773. 15. M. Buballa, Nucl. Phys. A611 (1996) 393; iv. Buballa and M. Oertel, Nucl. Phys. A642 (1998) 39c. 16. Tran Huu Phat, Nguyen Tuan Anh and Le Viet Hoa, in preparation.