The physical properties of asymmetric nuclear matter are studied in the Extended Nambou-Jona-Lasinio (ENJL) model formulated directly in the nucleon degrees of freedom. It results that the density dependence of the nuclear symmetry energy and its related quantities are basically in good agreement with data of recent analyses

Phat, Tran Huu

Tam, Dinh Thanh

Tuan Anh, Nguyen

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 22, No. 2 (2012), pp. 183-192
NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF
NUCLEAR MATTER
TRAN HUU PHAT
Vietnam Atomic Energy Commission
and
Dong Do University
NGUYEN TUAN ANH
Electric Power University
DINH THANH TAM
Institute of Nuclear Science and Technique
and
Tay Bac University, Sonla, Vietnam.
Abstract. The physical properties of asymmetric nuclear matter are studied in the Extended
Nambou-Jona-Lasinio (ENJL) model formulated directly in the nucleon degrees of freedom. It
results that the density dependence of the nuclear symmetry energy and its related quantities are
basically in good agreement with data of recent analyses.
I. INTRODUCTION
In recent years many radioactive beam facilities have been established in the world,
such as FAIR/GSI in Germany, SPIRAL2/GANIL in France, RIB/RIKEN in Japan and so
on. Experimental realizations using radioactive beams with large neutron or proton excess
have created a breakthrough for exploring the role of isospin degree of freedom in modern
nuclear physics. It was shown that most observed phenomena are mainly determined
by the in-medium nucleon-nucleon potentials and the isospin-dependent nuclear equation
of state (EOS), especially the density dependence of the nuclear symmetry energy. As
was known, the latter quantity plays a crucial role for understanding not only a lot of
important issues in nuclear physiscs [1-3], but also many critical problems in astrophysics
[4, 5]. There have been so far a great deal of investigations on this subject based on both
nonrelativistic [6-9] and relativistic [10-17] nuclear theories. The present paper is devoted
to the study of the isospin-dependent equation of state, in particular, the nuclear symmetry
energy and its related quantities. Our starting point is the Extended Nambu-Jona-Lasinio
(ENJL) model, which is able to remove some drawbacks of both nonrelativistic theories
(where the causality is violated at high densities) and relativistic theories (where the chiral
symmetry is absent). In our previous article [18] the chiral phase transition and liquid-gas
phase transition at subsaturation density were scrutinized for symmetric nuclear matter
184 NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF NUCLEAR MATTER
by means of the Lagrangian of the ENJL model given as
LNJL = ψ¯(i∂ˆ −m0)ψ +
Gs
2
[(ψ¯ψ)2 + (ψ¯iγ5~τψ)
2]− Gv
2
(ψ¯γµψ)
2
+
Gsv
2
[(ψ¯ψ)2 + (ψ¯iγ5~τψ)
2](ψ¯γµψ)
2.
Now LNJL is generalized to the case of asymmetric nuclear matter by incorporating rho
meson into consideration, namely,
L= ψ¯(i∂ˆ−m0+γ0µ)ψ + Gs
2
[(ψ¯ψ)2+(ψ¯iγ5~τψ)
2]− Gv
2
[(ψ¯γµψ)2+(ψ¯γ5γ
µψ)2]
+
Gsv
2
[(ψ¯ψ)2+(ψ¯iγ5~τψ)
2][(ψ¯γµψ)2+(ψ¯γ5γ
µψ)2]−Gr
2
[(ψ¯γµ
~τ
2
ψ)2+(ψ¯γ5γ
µ~τ
2
ψ)2],(1)
where µ = diag(µp, µn), µp,n = µB ± µI/2; m0 is the “bare” mass of the nucleon, ~τ is the
isospin Pauli matrices, and Gs, Gv, Gsv, and Gr are coupling constants.
In the mean-field approximation we replace
(ψ¯Γiψ)
2 = 2ψ¯Γiψ[ψ¯Γiψ]− [ψ¯Γiψ]2,
(ψ¯Γiψψ¯Γjψ)
2 = ([ψ¯Γiψ]ψ¯Γjψ)
2 + (ψ¯Γiψ[ψ¯Γjψ])
2 − ([ψ¯Γiψ][ψ¯Γjψ])2
= [ψ¯Γiψ]
2(2ψ¯Γjψ[ψ¯Γjψ]− [ψ¯Γjψ]2) + (2ψ¯Γiψ[ψ¯Γiψ]− [ψ¯Γiψ]2)[ψ¯Γjψ]2
−[ψ¯Γiψ]2[ψ¯Γjψ]2, (2)
with Γ = {1, iγ5~τ , γµ, γ5γµ}, and the averaging at finite density and temperature denoted
by angular brackets.
Eqs. (2) combines with the bosonization
σ = ψ¯ψ, ~pi = ψ¯iγ5~τψ, ωµ = ψ¯γµψ, φµ = ψ¯γ5γµψ, ~%µ = ψ¯γµ
~τ
2
ψ, ~χµ = ψ¯γ5γµ
~τ
2
ψ,
yielding
L = ψ¯(i∂ˆ −m0 + γ0µ)ψ + [Gs +Gsv(ω2 + φ2)]ψ¯(σ + iγ5~τ ~pi)ψ
−[Gv −Gsv(σ2 + pi2)]ψ¯γµ(ωµ + γ5φµ)ψ −Grψ¯γµ
~τ
2
(~%µ + γ5~χµ)ψ
−Gs
2
(σ2 + pi2) +
Gv
2
(ω2 + φ2) +
Gr
2
(%2 + χ2)− 3Gsv
2
(σ2 + pi2)(ω2 + φ2).(3)
In (3) we impose µI = 0 since we are not interested to the pion condensation in
what follows.
This paper is structured as follows. Section II deals with the equations of state
and the expression for nuclear symmetry energy . In Section III are presented the results
of numerical computation for nuclear symmetry energy and isospin-dependent EOS. The
conclusion and discussion are given in Section IV.
II. EQUATIONS OF STATE
Assume the sigma, pion, omega and rho fields develop the ground state expectation
values
〈σ〉 = u, 〈pii〉 = 0, 〈ωµ〉 = ρBδ0µ, 〈%µi〉 = ρI δµ0δi3, (4)
TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM 185
in cold nuclear matter. Inserting (4) into (3) leads to
LMFT = ψ¯
(
i∂ˆ −M∗ + γ0µ∗
)
ψ − U(ρB , u), (5)
where
M∗ = m0 − G˜su, G˜s = Gs +Gsvρ2B = Gs[1 + ξ(ρB/ρ0)2], ξ = ρ20Gsv/Gs, (6)
µ∗p,n = µ
∗
B
± µ∗
I
/2, µ∗
B
= µB − Σv = µB − [Gv −Gsvu2]ρB , (7)
µ∗
I
= −GrρI , (8)
U(ρB , u) =
1
2
[
G˜su
2 − 2ΣvρB +Gvρ2B −Grρ2I
]
. (9)
The solution M∗ of Eq. (7) is the effective mass of nucleon, which reduces to the
nucleon mass in vacuum.
Based on (5) the grand partition function Z is established
Z =
∫
Dψ¯DψDσD~piDωµ exp
∫ β
0
dτ
∫
V
d3x iLMFT .
Integrating out the nucleon degrees of freedom we arrive at
Z = exp
(
− iV U
T
)
detS−1,
in which V is the volume of system and
detS−1(k; ρB , u) = (k0 − E−−)(k0 + E−+)(k0 − E+−)(k0 + E++), (10)
with
E±∓ = E
±
k ∓ µ∗B , E±k = Ek ± µ∗I/2, Ek =
√
k2 +M∗2.
Then the effective potential Ω is derived
Ω(ρB , u) = i
T
V
lnZ = U(ρB , u) + iTr lnS
−1(ρB , u)
= U(ρB , u) + 2
∫
d3k
(2pi)3
[
2Ek + T ln(n
−
−n
−
+) + T ln(n
+
−n
+
+)
]
, (11)
with n±∓ being the Fermi distribution function, n
±
∓ =
[
eE
±
∓/T + 1
]−1
.
The pressure P is defined by means of (11)
P = −Ωtaken at minimum.
The energy density is obtained by the Legendre transform of P ,
E(ρB , u) = Ω(ρB , u) + Tς + µBρB + µIρI
=
1
2
[
G˜su
2+Gvρ
2
B
+Grρ
2
I
]
+2
∫
d3k
(2pi)3
Ek
[
(n−−+n
−
+−1)+(n+−+n++−1)
]
, (12)
186 NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF NUCLEAR MATTER
with the entropy density defined by
ς = −2
∫
d3k
(2pi)3
[
n−− lnn
−
− + (1− n−−) ln(1− n−−) + n−+ lnn−+ + (1− n−+) ln(1− n−+)
+n+− lnn
+
− + (1− n+−) ln(1− n+−) + n++ lnn++ + (1− n++) ln(1− n++)
]
.
The ground state of nuclear matter is determined by the minimum conditions
∂Ω
∂σ
= 0,
∂Ω
∂ω0
= 0,
∂Ω
∂%03
= 0,
or
u = 2
∫
d3k
(2pi)3
M∗
Ek
[
(n−−+n
−
+−1) + (n+−+n++−1)
]
, (13)
ρB = 2
∫
d3k
(2pi)3
[
(n−−−n−+) + (n+−−n++)
]
, (14)
ρI =
∫
d3k
(2pi)3
[
(n−−+n
−
+−1)− (n+−+n++−1)
]
, (15)
which are usually called the gap equations.
In terms of the baryon density (14) the expression for P is rewritten as
P = −(m0−M
∗)2
2G˜s
−Gv
2
ρ2
B
+
Gr
2
ρ2
I
+(µB−µ∗B)ρB−2
∫
d3k
(2pi)3
[
2Ek+T ln(n
−
−n
−
+)+T ln(n
+
−n
+
+)
]
, (16)
and the energy density takes the form
E = (m0 −M
∗)2
2G˜s
+
Gv
2
ρ2
B
+
Gr
2
ρ2
I
+2
∫
d3k
(2pi)3
Ek
[
(n−−+n
−
+−1)+(n+−+n++−1)
]
, (17)
Starting from (16) and (17) we have that
(1) The binding energy per nucleon
Ebin = −MN + E/ρB (18)
(2) The nuclear symmetry energy
Esym =
ρB
8
∂2E
∂ρ2
I
∣∣∣∣
ρ
I
=0
. (19)
(3) The compressibility
K(ρB , α) = 9
∂P
∂ρB
, (20)
where α is isospin asymmetry, α = (ρn − ρp)/ρB .
TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM 187
It obvious that all the thermodynamic properties of our system are governed by the
equations of state (16) and (17). However, in this article we restrict ourselves to consider
only cold nuclear matter, T = 0, in which case Eqs. (13) - (20) are simplified to
u = − 1
pi2
[ ∫ Λ
k
Fp
k2dk
M∗
Ek
+
∫ Λ
k
Fn
k2dk
M∗
Ek
]
, (21)
ρB =
1
3pi2
[
k3
Fp
+ k3
Fn
]
, (22)
ρI =
1
6pi2
[
k3
Fp
− k3
Fn
]
, (23)
and
E = (m0 −M
∗)2
2G˜s
+
Gv
2
ρ2
B
+
Gr
2
ρ2
I
− 1
pi2
[ ∫ Λ
k
Fp
k2dkEk +
∫ Λ
k
Fn
k2dkEk
]
. (24)
III. NUMERICAL COMPUTATIONS AND RESULTS
In order to proceed to the numerical computation the in-vacuum mass of nucleon
is chosen to be MN = 939 MeV. Five parameters m0, Gs, Gv, ξ, and Λ were determined
together with the in-medium mass of nucleon M∗ and the incompressibility K0 in Ref. [18]
for symmetric nuclear matter (Gr = 0). Their values are listed in Table 1.
Table 1. Values of parameters and physical quantities.
Λ (MeV) Gs (fm
2) Gv/Gs m0(MeV) ξ M
∗/MN K0(MeV)
400 8.507 0.933 41.264 0.032 0.684 285.91
As to fixing Gr let us employ the expansion of nuclear symmetry energy around ρ0
Esym = a4 +
L
3
(
ρB − ρ0
ρ0
)
+
Ksym
18
(
ρB − ρ0
ρ0
)2
+ ...
with a4 being the bulk symmetry parameter of the Weiszaecker mass formula. Experi-
mental data provide a4 = 30− 35 MeV; L and Ksym are related to slope and curvature of
NSE at saturation density ρ0
L = 3ρ0
(
∂Esym
∂ρB
)
ρB=ρ0
, (25a)
Ksym = 9ρ
2
0
(
∂2Esym
∂ρ2B
)
ρB=ρ0
(25b)
Then Gρ is fitted to give a4 = 32 MeV, its value is Gr = 0.417Gs. Thus, all of the
model parameters are fixed, they allow us to calcualate the isospin-dependent EOSs of
asymmetric nuclear matter.
188 NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF NUCLEAR MATTER
In Fig. 1 we show the dependence of binding energy Ebin(ρB , α) on density ρB/ρ0
and α. For comparison with the results of the relativistic Brueckner-Hatree-Fock approach
with and without the momentum-dependent self-energies [11] given in Fig. 2 we plot in
Fig. 3 the density dependence of Ebin(ρB , α) at several values of α. From these figures we
deduce that in our model the asymmetric matter is less stiff and the isospin dependence
of saturation density is strong enough.
0
1
2ΡBΡ0 0.0
0.5
1.0
Α
0
50
100
Ebin
Fig. 1. The dependence of
binding energy on nuclear
density and isospin asym-
metry α.
 
Fig. 2. The equation of state of asym-
metric nuclear matter from the rela-
tivistic Brueckner-Hartree-Fock calcu-
lations. Taken from Ref. [11].
Α =
0
Α
=
0.2Α
=
0.4
Α
=
0.6
Α
=
0.8
Α
=
1
0.0 0.5 1.0 1.5 2.0 2.5
0
20
40
60
ΡBΡ0
E b
in
HM
eV
L
Fig. 3. The equation of state
of asymmetric nuclear matter
derived from our chiral model.
0 1 2 3 4
0
20
40
60
80
100
120
140
ΡBΡ0
E s
ym
HM
eV
L
Fig. 4. The ρB/ρ0 dependence of Esym (solid
line), E1 = 32(ρB/ρ0)
0.7 (dotted line), and
E2 = 32(ρB/ρ0)
1.1 (dashed line).
Now the density dependence of nuclear symmetry energy is concerned. It is known
that besides the constraints extracted from astrophysical observation [19] we get significant
progress in constraining the behavior of NSE around and below the nuclear saturation
TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM 189
density from various experiments [20]. However, the predictions for behavior of NSE at
supra-saturation densities are quite divergent. In this regard, the prediction made by our
model could provide some insight into this issue. Taking into account Eqs. (13), (14),
(15) and (19) altogether and the implementing the numerical computation with the aid
of MATHEMATICA [21] we arrive at Fig. 4 describing the density dependence of nuclear
symmetry energy. Here, we also plot the graphs of the functions E1 = 32(ρB/ρ0)
0.7 and
E2 = 32(ρB/ρ0)
1.1. It is evident that
- Around and below the nuclear saturation densities our predicted graph is close
to the Brueckner-Hartree-Fock prediction [22].
- E1 < Esym < E2 for ρB < 2ρ0 and ρB > 3ρ0,
- Esym > E2 > E1 for 2ρ0 < ρB < 3ρ0.
This behavior of Esym at supra-saturation densities is basically in agreement with the
analysis of Refs. [23-25].
Next the theory is highlighted by studying the behaviors of the pressure P and
effective nucleon mass M∗ at high density. In Fig. 5 we show the ρB/ρ0 dependence of
pressure P at different isospin asymmetry and in Fig. 6 is shown the high-density behavior
of the pressure for neutron matter. The shaded region in Fig. 5 (Fig. 6) denotes the
constraint on high-density behavior of the pressure corresponding to symmetric nuclear
matter (neutron matter) derived from the simulations of flow data in heavy-ion collision
experiments [2]. The density dependence of M∗ is represented in Fig. 7 which manifests
the chiral restoration at high density.
Symmetric Nuclear Matter
Α= 0
Α= 0.2
Α= 0.4
Α= 0.6
Α= 0.8
Α= 1
Expt.
0 1 2 3 4 5
1
5
10
50
100
500
ΡBΡ0
P
HM
eV
fm
-
3 L
Fig. 5. The EoS of cold asymmetric nuclear matter at high baryon density at several
isospin asymmetry α. The shaded area means a constraint on the behavior of the pressure
of symmetric nuclear matter consistent with the experimental flow data [2].
Finally let us turn to the determination of several interesting quantities.
190 NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF NUCLEAR MATTER
Neutron Matter
Expt.Asy_soft
Expt.Asy_stiff
0 1 2 3 4 5
1
5
10
50
100
500
ΡBΡ0
P
HM
eV
fm
-
3 L
Fig. 6. The EoS and constraint (shaded region) on the high-density behavior of the
EoS for neutron matter [2].
Α = 0
Α = 0.2
Α = 0.4
Α = 0.6
Α = 0.8
Α = 1
0 1 2 3 4 5
0
200
400
600
800
1000
ΡBΡ0
M
*
HM
eV
L
Fig. 7. The density dependence of effective nucleon mass M∗.
(1) Eqs. (25) provide immediately the values of slope and curvature of NSE: L =
96.732 MeV and Ksym = −347.786 MeV. It is clear that L is consistent with
the constraint 46 MeV ≤ L ≤ 111 MeV obtained from heavy-ion data and the
analyses of other models [24, 26] and Ksym is also within the analyses based on
the in-medium NN cross section in the IBUU04 model [27]: Ksym = −500± 50
MeV.
TRAN HUU PHAT, NGUYEN TUAN ANH, AND DINH THANH TAM 191
(2) At the nuclear saturation density ρ0 and around α = 0, the isobaric incom-
pressibility of asymmetric nuclear matter can be expressed to the second-order
of alpha as
K(α) = K0 +Kasyα
2,
here, Kasy is the isospin-dependent part
Kasy = Ksym − 6L
characterizing the density dependence of NSE. The information on this quan-
tity can be extracted experimentally by measuring the giant monopole reso-
nance of neutron-rich nuclei. Our calculation yields Kasy = −928.18 MeV.
This value is different from the recently experimental measurements [28] that
give Kasy = −550± 100 MeV.
(3) The symmetry pressure and the shift of nuclear saturation density with asym-
metry at lowest order in alpha are obtained, respectively
Psym(ρ0) = ρ0L/3 = 5.482 MeV/fm
3,
∆ρ0 = −3ρ0L
K0
= −0.173 fm−3.
The calculated values of the model parameters and physical quantities are listed in
Table 2 and 3
Table 2. Values of the model parameters.
Λ(MeV) Gs(fm
2) Gv/Gs Gr/Gs m0(MeV) ξ
400 8.507 0.933 0.417 41.264 0.032
Table 3. Values of the model physical quantities.
M∗/MN K0(MeV) a4(MeV) L(MeV) Ksym(MeV) Kasy(MeV) Psym(MeV/fm
3) ∆ρ0(fm
−3)
0.684 285.91 32 96.732 -347.786 -928.181 5.482 -0.173
IV. CONCLUSION AND DISCUSSION
Developing the previous work [18] we have carried out in this article a more realistic
study of nuclear asymmetric matter, where it is found that
- Most calculated physical quantities listed in Table 2 are in agreement with the
recent analyses derived from various models as well as experimental constraints.
192 NUCLEAR SYMMETRY ENERGY IN CHIRAL MODEL OF NUCLEAR MATTER
- Most calculated physical quantities, in particular, the quantities K0, Ksym,
and L, listed in the Table 3 are in agreement with experimental constraints as
well as recent analyses of various models except for Kasy, where the theoretical
value much differs from the experimental one. The formulae Kasy = Ksym −
6L indicates that the experimental measurement for Kasy is probably not in
accordance with those obtained for Ksym and L.
Experimental realizations at high energies in radioactive beam facilities are expected
to create a good chance to explore the high density-behavior of NSE and other EoSs.
ACKNOWLEDGMENTS
D. T. Tam would like to express his sincere thanks to Institute for Nuclear Science
and Technique for the hospitality extended to him during his PhD study.
This paper is financially supported by the Vietnam National Foundation for Science
and Technology Development(NAFOSTED).
REFERENCES
[1] V. Baran, M. Colonna, V. Greco, and Di Toro, Phys. Rep. 410 (2005) 335 and references herein.
[2] P. Danielewicz, R. Lacey, and W. G. Lynch, Science 298 (2002) 1592.
[3] B. A. Li , C. M. Ko, and W. Bauer, Int. J. Mod. Phys. E7 (1998) 147.
[4] A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys. Rep. 411 (2005) 325 and references
herein.
[5] C. H. Lee, Phys. Rep. 275 (1996) 255.
[6] Dao T. Khoa, W. von Oertzen, and A. A. Ogloblin, Nucl. Phys. A602 (1996) 98.
[7] I. Bombaci and U. Lombardo, Phys. Rev. C44 (1991) 1892.
[8] R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C38 (1988) 1010.
[9] M. Lopez-Quelle et al., Nucl. Phys. A483 (1988) 479.
[10] H. Muther, M. Prakash, and T. L. Ainsworth, Phys. Lett. B199 (1987) 469.
[11] H. Huber, F. Weber, and M. K. Weigel, Phys. Rev. C50 (1994) R1287.
[12] B. Ter Haar and R. Malfliet, Phys. Rev. C50 (1994) 31.
[13] Tran Huu Phat, Nguyen Tuan Anh and Nguyen Van Long, Phys. Rev. C77 (2008) 054321.
[14] K. Sumiyoshi, H. Toki , and R. Brochmann, Phys. Lett. B276 (1992) 393.
[15] C. J. Horowitz and B. D. Serot, Nucl. Phys. A464 (1987) 613.
[16] D. Hirata et al., Phys. Rev. C44 (1991) 1467.
[17] Y. Sugahara and H. Toki, Nucl. Phys. A579 (1994) 557.
[18] Tran Huu Phat, Nguyen Tuan Anh and Dinh Thanh Tam, Phys. Rev. C84 (2011) 024321.
[19] A. W. Steiner, J. M. Lattimer, and E. F. Brown, Astrophys. J. 722 (2010) 33.
[20] W. G. Lynch, talk given at the 2nd International Symposium on Nuclear Symmetry Energy (NuSYM
11), June 17-22, 2011, Massachusetts, USA.
[21] S. Wolfram, The Mathematica Book, Wolfram Media, 2003.
[22] W. Zuo,, A. Lejeune, U. Lombardo, and J. F. Mathiot, Eur. Phys. J. A14 (2002 ) 469.
[23] B. A. Li, L. W. Chen, C. M. Ko, and A. W. Steiner, Arxiv: nucl-th/0601028.
[24] B. A. Li and L. W. Chen, Phys. Rev. C72 (2005) 064611.
[25] B. A. Li, Nucl. Phys. A681(2001) 434c .
[26] L. W. Chen, C. M. Ko, and B. A. Li, Phys. Rev. Lett. 94 (2005) 032701.
[27] C. B. Das, S. Das Gupta, C. Gale, and B. A. Li, Phys. Rev. C67 (2003) 034611.
[28] B. A. Li et al., Phys. Rev. Lett. 99 (2007) 162503.
Received 20 October 2011.


Nuclear Symmetry Energy in Chiral Model of Nuclear Matter

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Nuclear Symmetry Energy in Chiral Model of Nuclear Matter

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