- Email: [email protected]
Analytical study of finite temperature lattice f4 theory
16 March 1995 PHYSICS
LETTERS
B
Physics Letters B 347 (1995) 131-135
Analytical study of finite temperature lattice qb4theory * Hong Li a, Tim-lun Chen a,b a Department of Physics, Nankai University, lianjin 300071, People’s Republic of China b CCAST (World Laboratory), PO. Box 8730, Beijing 100080, People’s Republic of China
Received 7 November 1994 Editor: M. Dine
Abstract With the Variational-Cumulant Expansion method, the phase diagram and the external current J as a function of the expectation value of 4 of lattice (b4 theory at finite temperature are given analytically. It is shown that the broken phase at zero temperature might be restored at high temperature.
1. Introduction Symmetry restoration in a quantum field theory at finite temperature is of interest due to its physical relevance to the study of the Higgs sector in the standard electroweak model and in the grand unified theories. Therefore, an investigation of the symmetry restoration by taking the scalar +4 theory on the lattice as a simple model can be revealing. A lot of analytical [ 1,2] and numerical [ 3-51 work concerning this model at zero temperature has been done, which presents strong evidence that 4D &J~theory is “trivial”, meaning that the renormalized coupling constant hR vanishes in the continuum limit. But for 2D and 3D +4 theory, the “triviality” argument is still not definite 16-91. At finite temperature, the recent work mainly concentrates on the perturbative study [lo] and the temperature renormalization group equation method [ 111 in the continuum situation. In order to investigate the “triviality” problem at finite temperature, the lattice +4 model should be studied. Although
there are some finite temperature studies in ( 1 + 1) and (2 + 1) dimensions [ 12-141 with Monte Carlo (MC) simulation under periodic boundary condition, the research in (3 + 1) dimensions has not been done neither numerically nor analytically, probably because of the computer limitations [ 141. Recently, the Variational-Cumulant Expansion (VCE) method has reached a series of successes in the analytical studies of statistical models and lattice gauge theories. So we attempt to investigate the lattice 44 theory at finite temperature in ( 1 + l), (2 + 1) and (3 + 1) dimensions with the VCE method systematically. In the present paper, we give some results about the phase diagram and the external current J as a function of the expectation value of 4.
2. Lattice 4” model at finite temperature The Euclidean given by
*Supported by Doctoral Foundation of Chinese Educational Commission and National Natural Science Foundation of China. 0370-2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDlO370-2693(95)00059-3
Lagrangian
density of q54 model is
132
H. Li. T.L. Chen /Physics Letters B 347 (I995) 131-135
$ww4
+
(1)
where Aa and me are the bare coupling constant and the bare mass, respectively. To consider the finite temperature quantum field theory, we restrict the length of the imaginary time axis by imposing the periodic boundary condition of the field in this direction, which means that 4(x,
P) = $(x,0)
are determined by minimizing the free energy in the cumulant expansion. The trial partition function of the SO system can be calculated as ZI = W(-Fo) =
=
J
[dhl
exp[-Su(&,J)]
10
(7)
where FO is the free energy of the trial system, lo is given by
(2)
where p = l/T, T is the temperature of the system. The discrete version of this action would be
=-
loo n 1 2 cc n=Om=. (n - m) !m!
x J;-mJ,m
IY$(n+m+ d AL
1)) 4
n + m = even
where the scalar field 4~ (n) is defined in the site n and the following dimensionless parameters are also used in Eq. (3)
The partition function of a true system can be obtained via the trial system as Z=edF=
= a d-=&$(X
&(n)
rni = d + irnia*
J
[%I
[d#L]exp(Sc--S-So) < exp(S0 - S) >a
(9)
(4)
0
where a is the lattice spacing, and d is the dimension. All the quantities in Eq. (3) are dimensionless. Thus the partition function of this system is
Z =
J
= exp( -Fo)
hr. = LA &d 4
= n),
(8)
ev(-9
(5)
In general, it is very difficult to calculate the partition function of a certain system because of the nearest neighbor interaction in the action S. According to the VCE method, in order to calculate the partition function, one can introduce a trial action SO which should be close to the action 5’ and the SO system should be easy to calculate. We introduce the same trial action with Ref. [ I] :
where F is the free energy of the system and < . . . >c= & s[ dc$LI(. . .) exp( -&) . According to the VCE method, we obtain the free energy of the true system from Rq. (9) F=Fo-FA
< (So-S)“>, il=l
or Fz
a-2:
< (So-S)”
>c
(11)
n=l
where < . . . >c indicates an average value for the cumulant expansion, and,=a-o
where Jt and 52 are the two variational parameters corresponding to the linear and quadratic terms which
< (So - S)* >,=<
(So - s>* >(I - < So - S >;
< (So - s>s >c=<
(So - s>s >c
- 3 < So - S >c< +2;
(So - S)* >o (12)
H. Li, T.L. Chen / Physics Letters B 347 (1995) 131-135
The expectation value of any physical quantity can also be calculated in the VCE as follows:
0.60
-
0.20
-
133
[d4L]Lexp(-S) J =
m ;1
c Go
sy
< L(S0 -
A
>C
(13)
.
-z "
=,=o < L1>=<
L(So-S)
>C
-7.00
-3.00
-5.00
4
Fig. 1. < (PL > a$ a function of rni for AL = 10.0 in d = 2+ 1(n) and d = 3 + l(b).
=o-~~ =, =< L(So - s>2 >o - 2 < L(So - S) >,< so - s >o
0.0
- < L >o< (So - S)2 >o < L3 >=< =< L(s)
L(So - s)3 >C - sp
-5.0
>o
- 3 < L(So - q2
-10.0
>,< so - s >I)
- 3 < L( so - S) >C< (So - sp - < L >o< (So - sp
>o
-15.0
>o -20.0
(14) They are also functions
_1\ 0.0
XL 5.0
10.0
15.0
20.0
25.0
Fig. 2. Phase diagrams at finite temperature in d = 1 + 1(n), d = 2 + l(b), d = 3 + I(c). The dashed line are the phase
of Ji, Jz, rni and AL.
diagramsatzerotemperatureind=2+l(d),d=3+I(e).
3. Calculations
and results
First,we should determine the variational parameter J1 and Jz as a function of AL and rni by minimizing the free energy F, ( AL, rni, J) as follows:
aFtI,
-0,
dJ1a2F
220,
aJ;
Substituting Eqs. ( 13)) $JL which symmetric 0) phases.
aFIt!
aJ=O 2
a2F, a2Fm
--
aJ; aJ;
-(g$12>_0
(15)
the variational parameters JI and 52 into ( 14)) we can get the expectation value of is the order parameter characterizing the ( < #JL >= 0) and asymmetric ( < r$L > f When making the cumulant expansion of
F,,, and < 4~ >, one should notice the periodic condition in the imaginary time direction. Under this condition, the number of connected graphs increases and the theoretical factors of which are different from those at zero temperature. In the calculations, we take p = 2a and expand the free energy and the < 4~ > up to the third order. In Fig. 1, we give the expectation value of 4~ as a function of rni when AL = 10.0 in (2 + 1) and (3 + 1) dimensions. Obviously, the phase transition is of second order and the transition points are rnk = -4.938 (d = 3) and rnL = -2.237 (d = 4). The phase diagrams of 44 model at finite temperature in (1 + l), (2 + 1) and (3 + 1) dimensions are also given in Fig. 2. The dashed line shows the results at zero temperature from Ref. [ 11. All transitions are
134
H. Li, T.L. Chen /Physics
Table 1 Critxal values mi, in ( 1+ 1) dimension. our
AL
results
Letters B 347 (1995) 131-135
1.00
1
Ref. 171
XL = 10.0
0.75
0.10 0.167 0.25 0.33 0.50
1.91 1.76 1.62 1.46 1.19
1.745 I.558 1.375 I.225 0.900
1
of second order and the critical values m& at finite temperature are much lower than those at zero temperature. For a clear comparison with the MC results, we list several critical values rni, from our results and from Ref. [ 141 in Table 1. Although the periodic condition is used in the spatial direction in Ref. [ 141, it can be rotated to the imaginary time direction. When making a comparison, we change the scale of Ref. [ 141 to our scale with a simple algebraic relation. In order to get the diagram of the external current J as a function of < q5~ >, we introduce a constant external current J in the action
S’=S-
Jxh(n)
0.00
0.25
0.50
Fig. 3. /versus < +L >J form: = -5.4(a), -6.1(d) in d = 1+ 1. I
0.75
-5,62(h),
-5.8(c),
1.00
nbc
d
(16)
n
and change the trial action from SO to Sh:
0.00
+AL~&.)
-
Jc4~(4
Fig. 4. J versus < 4~ >J -5.2(c), -6.1(d) in d =2+
The expectation value of #q. corresponding ternal current J is >J=
JFlTl -aJ
dU
= J(< 4~ >J)
In Figs. 3-5, the diagrams of J (3 + l), (2 + 1) and (1 + 1) definite AL, rn: values are given, correspond to the critical values the only solution of J( < 4~ >)
0.75
for rnt = -4.4(a),
-4.938(b),
I.
to the ex-
(18)
Regarding J as a function of < 4~ > J, we obtain the diagram of J as a function of < q5~ >J. From this function, the effective potential U can be obtained by
d-c 4~ >J
0.50
(17)
n
<4L
0.25
(19) versus < 4~ >J in dimensions at some in which, lines (b) m&. For rni 2 rnZk, = 0 is at < CpL>=
0, while for rn? < mi,, J( < c,b~ >) is zero when < 4~ > has a non-zero value. In Fig. 6, the MC results of J versus < 4~ > J in ( 1 + 1) dimension from Ref. [ 131 are given for a comparison with our results. It should be noticed that a scale change is also made. There are some differences in our results as compared to the MC results, which might be caused by the limited order cumulant expansion in the VCE method and the finite size effects from MC simulations [141.
H. Li, T.L. Chen /Physics Letters B 347 (1995) 131-135
effects make the critical point at which the symmetry restoration occurs move to a lower value of m2,, or further means that the broken phase at zero temperature may be restored at a high temperature, which agrees with the conclusion of Refs. [ 15,161. On the other hand, according to the .I(< 4~ >) relation at different rnivalue in Figs. 3-5, the symmetry restoration phase transition can also be characterized by the appearance of zero solution of equation J( < 4~ >) = 0. So this relation might be used to justify the phase transition. From the previous work of VCE method [ 171, we know that a higher order expansion will give a better result. So we expect to expand the free energy and < $JL > to a higher order to investigate the critical behavior of lattice 44 model at finite temperature. This work is in preparation.
1.00 -
,,
0.25
c
b
d
-
1.do
0.110
2.60
< al. >
Fig. 5. .I versus -2.9(c), -3.2(d)
-2.237(b),
< 4~ >_I for rr$ = -1.8(a), in d = 3+ 1.
135
References
/
, // 0.00
-;; Cl.00
(
(
,
,
,
,
1.00
,
,
2.00
,
,
,
(
,
3.00
< al. > Fig. 6. J versus < qb~ >J for tr$ = -2.25, d = 1 + 1. The dashed line is the MC result.
AL = 0.0556
in
4. Discussions Comparing the phase diagram at finite temperature with that at zero temperature in Fig. 2, we can find that the critical value rn& corresponding to a definite AL value at finite temperature is much lower than that at zero temperature. This means that the thermodynamic
[I] K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75. [2] CM. Wu, Z.K. Zhu et al., Phys. Lea. B 216 (1989) 381. [3] K. Huang, E. Manousakis and J. Polonyi, Phys. Rev. D 35 (1987) 3187. [4] J. Kuti and Yue Shen, Phys. Rev. Lett. 60 (1988) 85. [5] M. Luscher and P Weisz, Nucl. Phys. B 290 ( 1987) 25; B 295 (1988) 65. [6] E Cooper et al., Nucl. Phys. B 210 (1982) 210. [7] B. Freedman, Phys. Lett. B 113 (1982) 481. [8] F. Cooper and B. Freedman, Nucl. Phys. B 239 ( 1984) 459. [9] G. Baker, J. Stat. Phys. 24 (1981) 487. [lo] H. Arther Weldon, Phys. Lett. B 174 (1986) 427, and references therein. [ 1 I] Per Elmfors, Nordita-92/P, preprint, and references therein. [ 121 M.G. do Amaral et al., Phys. Lett. B 165 (1985) 117. [ 131 M.G. do Amaral et al., Z. Phys. C 32 (1986) 609. [ 141 J.C. Ciria and A. Tarancon, Phys. Rev. D 49 (1994) 1020. [ 151 J.R. Li, Introduction to Quark Matter Theory, Ed. Q.L. Tan, Hu Nan Educational press. [ 161 Y. Fujimoto et al., Z. Phys. C 35 ( 1987) 351. [ 171 T.L. Chen and R.J. Liu, Cummun. Theor. Phys. 22 (1994).
LETTERS
B
Physics Letters B 347 (1995) 131-135
Analytical study of finite temperature lattice qb4theory * Hong Li a, Tim-lun Chen a,b a Department of Physics, Nankai University, lianjin 300071, People’s Republic of China b CCAST (World Laboratory), PO. Box 8730, Beijing 100080, People’s Republic of China
Received 7 November 1994 Editor: M. Dine
Abstract With the Variational-Cumulant Expansion method, the phase diagram and the external current J as a function of the expectation value of 4 of lattice (b4 theory at finite temperature are given analytically. It is shown that the broken phase at zero temperature might be restored at high temperature.
1. Introduction Symmetry restoration in a quantum field theory at finite temperature is of interest due to its physical relevance to the study of the Higgs sector in the standard electroweak model and in the grand unified theories. Therefore, an investigation of the symmetry restoration by taking the scalar +4 theory on the lattice as a simple model can be revealing. A lot of analytical [ 1,2] and numerical [ 3-51 work concerning this model at zero temperature has been done, which presents strong evidence that 4D &J~theory is “trivial”, meaning that the renormalized coupling constant hR vanishes in the continuum limit. But for 2D and 3D +4 theory, the “triviality” argument is still not definite 16-91. At finite temperature, the recent work mainly concentrates on the perturbative study [lo] and the temperature renormalization group equation method [ 111 in the continuum situation. In order to investigate the “triviality” problem at finite temperature, the lattice +4 model should be studied. Although
there are some finite temperature studies in ( 1 + 1) and (2 + 1) dimensions [ 12-141 with Monte Carlo (MC) simulation under periodic boundary condition, the research in (3 + 1) dimensions has not been done neither numerically nor analytically, probably because of the computer limitations [ 141. Recently, the Variational-Cumulant Expansion (VCE) method has reached a series of successes in the analytical studies of statistical models and lattice gauge theories. So we attempt to investigate the lattice 44 theory at finite temperature in ( 1 + l), (2 + 1) and (3 + 1) dimensions with the VCE method systematically. In the present paper, we give some results about the phase diagram and the external current J as a function of the expectation value of 4.
2. Lattice 4” model at finite temperature The Euclidean given by
*Supported by Doctoral Foundation of Chinese Educational Commission and National Natural Science Foundation of China. 0370-2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDlO370-2693(95)00059-3
Lagrangian
density of q54 model is
132
H. Li. T.L. Chen /Physics Letters B 347 (I995) 131-135
$ww4
+
(1)
where Aa and me are the bare coupling constant and the bare mass, respectively. To consider the finite temperature quantum field theory, we restrict the length of the imaginary time axis by imposing the periodic boundary condition of the field in this direction, which means that 4(x,
P) = $(x,0)
are determined by minimizing the free energy in the cumulant expansion. The trial partition function of the SO system can be calculated as ZI = W(-Fo) =
=
J
[dhl
exp[-Su(&,J)]
10
(7)
where FO is the free energy of the trial system, lo is given by
(2)
where p = l/T, T is the temperature of the system. The discrete version of this action would be
=-
loo n 1 2 cc n=Om=. (n - m) !m!
x J;-mJ,m
IY$(n+m+ d AL
1)) 4
n + m = even
where the scalar field 4~ (n) is defined in the site n and the following dimensionless parameters are also used in Eq. (3)
The partition function of a true system can be obtained via the trial system as Z=edF=
= a d-=&$(X
&(n)
rni = d + irnia*
J
[%I
[d#L]exp(Sc--S-So) < exp(S0 - S) >a
(9)
(4)
0
where a is the lattice spacing, and d is the dimension. All the quantities in Eq. (3) are dimensionless. Thus the partition function of this system is
Z =
J
= exp( -Fo)
hr. = LA &d 4
= n),
(8)
ev(-9
(5)
In general, it is very difficult to calculate the partition function of a certain system because of the nearest neighbor interaction in the action S. According to the VCE method, in order to calculate the partition function, one can introduce a trial action SO which should be close to the action 5’ and the SO system should be easy to calculate. We introduce the same trial action with Ref. [ I] :
where F is the free energy of the system and < . . . >c= & s[ dc$LI(. . .) exp( -&) . According to the VCE method, we obtain the free energy of the true system from Rq. (9) F=Fo-FA
< (So-S)“>, il=l
or Fz
a-2:
< (So-S)”
>c
(11)
n=l
where < . . . >c indicates an average value for the cumulant expansion, and
where Jt and 52 are the two variational parameters corresponding to the linear and quadratic terms which
< (So - S)* >,=<
(So - s>* >(I - < So - S >;
< (So - s>s >c=<
(So - s>s >c
- 3 < So - S >c< +2
(So - S)* >o (12)
H. Li, T.L. Chen / Physics Letters B 347 (1995) 131-135
The expectation value of any physical quantity can also be calculated in the VCE as follows:
0.60
-
0.20
-
133
[d4L]Lexp(-S) J =
m ;1
c Go
sy
< L(S0 -
A
>C
(13)
.
-z "
L(So-S)
>C
-7.00
-3.00
-5.00
4
Fig. 1. < (PL > a$ a function of rni for AL = 10.0 in d = 2+ 1(n) and d = 3 + l(b).
=
0.0
- < L >o< (So - S)2 >o < L3 >=< =< L(s)
L(So - s)3 >C - sp
-5.0
>o
- 3 < L(So - q2
-10.0
>,< so - s >I)
- 3 < L( so - S) >C< (So - sp - < L >o< (So - sp
>o
-15.0
>o -20.0
(14) They are also functions
_1\ 0.0
XL 5.0
10.0
15.0
20.0
25.0
Fig. 2. Phase diagrams at finite temperature in d = 1 + 1(n), d = 2 + l(b), d = 3 + I(c). The dashed line are the phase
of Ji, Jz, rni and AL.
diagramsatzerotemperatureind=2+l(d),d=3+I(e).
3. Calculations
and results
First,we should determine the variational parameter J1 and Jz as a function of AL and rni by minimizing the free energy F, ( AL, rni, J) as follows:
aFtI,
-0,
dJ1a2F
220,
aJ;
Substituting Eqs. ( 13)) $JL which symmetric 0) phases.
aFIt!
aJ=O 2
a2F, a2Fm
--
aJ; aJ;
-(g$12>_0
(15)
the variational parameters JI and 52 into ( 14)) we can get the expectation value of is the order parameter characterizing the ( < #JL >= 0) and asymmetric ( < r$L > f When making the cumulant expansion of
F,,, and < 4~ >, one should notice the periodic condition in the imaginary time direction. Under this condition, the number of connected graphs increases and the theoretical factors of which are different from those at zero temperature. In the calculations, we take p = 2a and expand the free energy and the < 4~ > up to the third order. In Fig. 1, we give the expectation value of 4~ as a function of rni when AL = 10.0 in (2 + 1) and (3 + 1) dimensions. Obviously, the phase transition is of second order and the transition points are rnk = -4.938 (d = 3) and rnL = -2.237 (d = 4). The phase diagrams of 44 model at finite temperature in (1 + l), (2 + 1) and (3 + 1) dimensions are also given in Fig. 2. The dashed line shows the results at zero temperature from Ref. [ 11. All transitions are
134
H. Li, T.L. Chen /Physics
Table 1 Critxal values mi, in ( 1+ 1) dimension. our
AL
results
Letters B 347 (1995) 131-135
1.00
1
Ref. 171
XL = 10.0
0.75
0.10 0.167 0.25 0.33 0.50
1.91 1.76 1.62 1.46 1.19
1.745 I.558 1.375 I.225 0.900
1
of second order and the critical values m& at finite temperature are much lower than those at zero temperature. For a clear comparison with the MC results, we list several critical values rni, from our results and from Ref. [ 141 in Table 1. Although the periodic condition is used in the spatial direction in Ref. [ 141, it can be rotated to the imaginary time direction. When making a comparison, we change the scale of Ref. [ 141 to our scale with a simple algebraic relation. In order to get the diagram of the external current J as a function of < q5~ >, we introduce a constant external current J in the action
S’=S-
Jxh(n)
0.00
0.25
0.50
Fig. 3. /versus < +L >J form: = -5.4(a), -6.1(d) in d = 1+ 1. I
0.75
-5,62(h),
-5.8(c),
1.00
nbc
d
(16)
n
and change the trial action from SO to Sh:
0.00
+AL~&.)
-
Jc4~(4
Fig. 4. J versus < 4~ >J -5.2(c), -6.1(d) in d =2+
The expectation value of #q. corresponding ternal current J is >J=
JFlTl -aJ
dU
= J(< 4~ >J)
In Figs. 3-5, the diagrams of J (3 + l), (2 + 1) and (1 + 1) definite AL, rn: values are given, correspond to the critical values the only solution of J( < 4~ >)
0.75
for rnt = -4.4(a),
-4.938(b),
I.
to the ex-
(18)
Regarding J as a function of < 4~ > J, we obtain the diagram of J as a function of < q5~ >J. From this function, the effective potential U can be obtained by
d-c 4~ >J
0.50
(17)
n
<4L
0.25
(19) versus < 4~ >J in dimensions at some in which, lines (b) m&. For rni 2 rnZk, = 0 is at < CpL>=
0, while for rn? < mi,, J( < c,b~ >) is zero when < 4~ > has a non-zero value. In Fig. 6, the MC results of J versus < 4~ > J in ( 1 + 1) dimension from Ref. [ 131 are given for a comparison with our results. It should be noticed that a scale change is also made. There are some differences in our results as compared to the MC results, which might be caused by the limited order cumulant expansion in the VCE method and the finite size effects from MC simulations [141.
H. Li, T.L. Chen /Physics Letters B 347 (1995) 131-135
effects make the critical point at which the symmetry restoration occurs move to a lower value of m2,, or further means that the broken phase at zero temperature may be restored at a high temperature, which agrees with the conclusion of Refs. [ 15,161. On the other hand, according to the .I(< 4~ >) relation at different rnivalue in Figs. 3-5, the symmetry restoration phase transition can also be characterized by the appearance of zero solution of equation J( < 4~ >) = 0. So this relation might be used to justify the phase transition. From the previous work of VCE method [ 171, we know that a higher order expansion will give a better result. So we expect to expand the free energy and < $JL > to a higher order to investigate the critical behavior of lattice 44 model at finite temperature. This work is in preparation.
1.00 -
,,
0.25
c
b
d
-
1.do
0.110
2.60
< al. >
Fig. 5. .I versus -2.9(c), -3.2(d)
-2.237(b),
< 4~ >_I for rr$ = -1.8(a), in d = 3+ 1.
135
References
/
, // 0.00
-;; Cl.00
(
(
,
,
,
,
1.00
,
,
2.00
,
,
,
(
,
3.00
< al. > Fig. 6. J versus < qb~ >J for tr$ = -2.25, d = 1 + 1. The dashed line is the MC result.
AL = 0.0556
in
4. Discussions Comparing the phase diagram at finite temperature with that at zero temperature in Fig. 2, we can find that the critical value rn& corresponding to a definite AL value at finite temperature is much lower than that at zero temperature. This means that the thermodynamic
[I] K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75. [2] CM. Wu, Z.K. Zhu et al., Phys. Lea. B 216 (1989) 381. [3] K. Huang, E. Manousakis and J. Polonyi, Phys. Rev. D 35 (1987) 3187. [4] J. Kuti and Yue Shen, Phys. Rev. Lett. 60 (1988) 85. [5] M. Luscher and P Weisz, Nucl. Phys. B 290 ( 1987) 25; B 295 (1988) 65. [6] E Cooper et al., Nucl. Phys. B 210 (1982) 210. [7] B. Freedman, Phys. Lett. B 113 (1982) 481. [8] F. Cooper and B. Freedman, Nucl. Phys. B 239 ( 1984) 459. [9] G. Baker, J. Stat. Phys. 24 (1981) 487. [lo] H. Arther Weldon, Phys. Lett. B 174 (1986) 427, and references therein. [ 1 I] Per Elmfors, Nordita-92/P, preprint, and references therein. [ 121 M.G. do Amaral et al., Phys. Lett. B 165 (1985) 117. [ 131 M.G. do Amaral et al., Z. Phys. C 32 (1986) 609. [ 141 J.C. Ciria and A. Tarancon, Phys. Rev. D 49 (1994) 1020. [ 151 J.R. Li, Introduction to Quark Matter Theory, Ed. Q.L. Tan, Hu Nan Educational press. [ 161 Y. Fujimoto et al., Z. Phys. C 35 ( 1987) 351. [ 171 T.L. Chen and R.J. Liu, Cummun. Theor. Phys. 22 (1994).