The phase structure of the linear sigma model with constituent quarks and the electric neutrality is systematically studied in the mean field theory. It results that in the chiral limit, as function of $T$ and chemical potential \$\mu\$, the pion condensate undergoes a first-order phase transition. In the physical world, the phase diagram of chiral condensate exhibits a first-order phase transition, which ends at a critical end point (CEP) for $\alpha  0.3$, where $\alpha$ is the fraction of $\pi^+$  contribution to the condensate

Phat, Tran Huu

Thu, Nguyen Van

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 22, No. 2 (2012), pp. 103-110
PHASE STRUCTURE OF LINEAR SIGMA MODEL WITH
NEUTRALITY CONSTRAINT (II)
TRAN HUU PHAT
Vietnam Atomic Energy Commission
and
Dong Do University
NGUYEN VAN THU
Institute of Nuclear Science and Technique
and
Department of Physics, Hanoi University of Education II
Abstract. The phase structure of the linear sigma model with constituent quarks and the electric
neutrality is systematically studied in the mean field theory. It results that in the chiral limit,
as function of T and chemical potential µ, the pion condensate undergoes a first-order phase
transition. In the physical world, the phase diagram of chiral condensate exhibits a first-order
phase transition, which ends at a critical end point (CEP) for α < 0.3, where α is the fraction of
pi+ contribution to the condensate.
I. INTRODUCTION
It is known that the study of the phase structure of quantum chromodynamics
(QCD) turns out to be a very hot subject attracted more and more attention, both ex-
perimentally and theoretically [1]. Many aspects of in-medium effects such as the chiral
restoration at high baryon density, the in-medium properties of hadrons, the pion con-
densation in dense matter and so on are of special interest. The high energy heavy-ion
collisions are presently the powerful machinery to generate hot and dense hadronic mat-
ter, and, therefore, they create a good chance for exploring the phase structure of QCD
at extreme conditions. It is commonly accepted that the chiral restoration phase transi-
tion accompanies the confinement - deconfinement phase transition at the same critical
temperature. The recent experimental data of RHIC, collected for high-temperature and
low-baryon density region, provide clear signals on the confinement-deconfinement phase
transition. This region is also one of the research problems to be carried out at LHC. The
experimental research in high-baryon density region is planned to be implemented in the
future heavy-ion collision experiments at RHIC, FAIR and NICA.
The first principle calculations of lattice QCD at finite baryon density is rather
problematic due to the sign puzzle. In this respect, one has to invoke different effective
models to address this issue.
We know that heavy nuclei can be considered as nuclear matter with finite isospin
density due to Coulomb interactions and the isospin-dependent quantities of isotopically
104 TRAN HUU PHAT AND NGUYEN VAN THU
asymmetric nuclear matter, such as the in-medium nucleon-nucleon potential, the equa-
tions of state and, in particular, the nuclear symmetry energy plays very important role
in understanding a lot of interesting astrophysical problems [2]- [4], dynamics of heavy-ion
reactions at intermediate energies [5]- [8], the structure of neutron-rich nuclei, and the
nuclei close to the drip-line [9]- [14]. Hence, dealing with QCD at finite isospin chemical
potential is of special significance, too. The lattice-QCD (LQCD) simulations [15,16] at fi-
nite isospin chemical potential, finite temperature and low baryonic density were smoothly
implemented and conjectured that the charged pions condense at high isospin density and
low temperature, and also suggested that the transition to a Bose-Einstein condensed state
coincides with the deconfinement phase transition to quark-gluon plasma at low isospin
chemical potential. The pion condensation and the related phase diagrams of two-flavor
QCD have been investigated by means of various QCD effective models: chiral pertur-
bation theory [17]- [19], Nambu-Jona-Lasinio (NJL)-type models [20]- [23], linear sigma
model [24]. We are aware that, among theories developed so far, the LQCD is the unique
one based on first principle and the Polyakov-loop extended (PNJL) model is an extended
version of the NJL model which respects approximately both mechanisms: chiral symme-
try breaking and confinement. In this regard, perhaps, they provide a guideline for all
studies of the phase structure.
Nowadays, the holographic QCD [25] emerges as a promised theory that is hopeful
to explain satisfactorily all phenomena related to the non- perturbative region of QCD.
In the present paper the effect of the neutrality condition on the phase structure of
the linear sigma model with constituent quarks is studied. We will closely follow Ref. [26]
to consider the system given by the Lagrangian
L = q
[
iγµ∂µ −mq − g(σ + iγ
5~τ~π)
]
q + µqγ0q + LLSM ,
LLSM =
1
2
[∂ασ∂
ασ + ∂α~π∂
α~π] + 2iµI(π1∂0π2 − π2∂0π1)− U,
U =
m2
2
(σ2 + ~π2)−
µ2I
2
(π21 + π
2
2) +
λ2
4
(σ2 + ~π2)2 − ǫfpim
2
piσ, (1)
where q, σ and ~π are respectively the field operators of quarks, sigma meson and pion; mpi
and fpi are respectively pion mass and pion decay constant in vacuum; µ and µI denote
the baryon and isospin chemical potentials, mq is current quark mass and ǫ = 0, 1.
The paper is structured as follows. In Section II the basic formulae and phase
diagram of the system (1) are considered, respectively, in the chiral limit and the physical
world. The conclusion and discussion are given in Section III.
II. THE BASIC FORMULAE AND PHASE DIAGRAM
In our previous works [27] we found the gap equations
[m2 + λ2(u2 + v2)]u− ǫfpim
2
pi +
∂Ωqq
∂u
= 0, (2)
[−µ2I +m
2 + λ2(u2 + v2)]v +
∂Ωqq
∂v
= 0, (3)
PHASE STRUCTURE OF LINEAR SIGMA MODEL WITH NEUTRALITY CONSTRAINT (II) 105
and proved that the quark densities fulfill the equality
ρu + ρd = ρ,
in which ∂Ωqq/∂u, ∂Ωqq/∂v and ρ are given in Eqs. (9), (10) and (12) of [27].
Next we consider the electric neutrality. The bulk matter of stars is electrically
neutral and the color neutrality is automatically satisfied in our case. So, the electric
neutrality has to be incorporated into consideration.
In addition, matter must be stable under the weak processes like
d→ u + e− + ν˜e.
The chemical potential of neutrinos is zero if we assume that the neutrinos can leave the
system, then the chemical equilibrium requires
µu − µd = −µe.
in which µe is the electron chemical potential. The chemical equilibrium related to the
presence of charged pion condensate imposes the condition
µu − µd = µI , (4a)
µe = −µI . (4b)
Let us remember that, in addition to Eq.(4a), we already have another relation
µu + µd = µ,
which together with (4a) lead to
ρu + ρd = ρ, ρu − ρd = ρI . (5)
The electric neutrality means that the total charge density in condensed phase has to be
zero, namely,
Q =
∑
B=u,d
qBρB + αρI − (1− α)ρI + ρe+ = 0, (6)
here qB is the charge of quark B, 0 ≤ α ≤ 1 means the portion of π
+ condensate in the
charged pion condensate, (1 − α) is the portion of π− condensate in the charged pion
condensate and ρe+ denotes the positron density and
ρI = µIv
2.
Taking into account (4b) Eq. (6) is rewritten∑
B=u,d
qBρB = (1− 2α)ρI +
(
∂Ωe
∂µe
) ∣∣∣∣
µe=−µI
, (7)
where Ωe is the effective potential of electron
Ωe = −2
∫
d3~p
(2π)3
{
Ep + T ln
[
1 + e−β(Ep−µe)
]
+ T ln
[
1 + e−β(Ep+µe)
]}
,
Ep =
√
~p2 +m2e,
106 TRAN HUU PHAT AND NGUYEN VAN THU
derived from the Lagrangian
Le = ψe(iγ
µ∂µ + eγ
0µe −me)ψe,
and ψe, e andme denote respectively the electron field, the electron charge and the electron
mass.
Inserting (5) into (7) we arrive at the neutrality constraint which determines µ as a
function of µI and vice versa. This implies that, from now on, our system depends upon
only two parameters, for example T and µ. In the remainder, we neglect the electron mass
for simplicity.
Now we consider how the above calculated phase diagrams change under the neu-
trality constraint. In order to proceed to the numerical computation the values of the
model parameters are usually chosen as mq = 5.5 MeV, fpi = 93 MeV, mpi = 138 MeV,
sigma mass mσ = 500 MeV and the coupling constant g is determined to be g = 3.3.
We are ready to implement the numerical study in two different cases with the aid of
Mathematica [28].
II.1. The chiral limit ǫ = 0
In this case the chiral condensate vanishes and solving simultaneously the gap equa-
tion (3) and the neutrality equation (7) provides in Fig.1(a) the evolution of the pion
condensate versus T at µ = 100 MeV and α = 0 (solid line), 1/4 (dashed line), 1/2 (dot-
ted line). All lines have the same meeting point at T = 96.7 MeV. The pion condensation
is a first-order phase transition. In Fig.1(b) we show the evolution of the pion condensate
versus µ at T = 0 and α = 0 (solid line), 1/4 (dashed line) , 1/2 (dotted line). All lines
have the same meeting point at µ = 205.3 MeV. Evidently, the phase transition is the
first-order everywhere. The corresponding phase diagram in the (T, µ)-plane is plotted in
Fig.2. The NJL model calculations [12] claim that a first-order line ends at a critical point
and the transition is second-order all the way to µ = 0.
II.2. The physical world ǫ = 1
1- We first consider the case mpi < µI where both chiral and pion condensates are
non-vanishing. Simultaneously calculating the gap equations (2), (3) and the neutrality
equation (7) it follows that
- In Fig.3(a) is shown the graphs representing the evolutions of pion condensate
against T at µI = 200 MeV and α = 0 (solid line), 0.25 (dashed line), 0.3 (dotted line), all
lines have the same meeting point at T = 52.5 MeV. Fig.3(b) displays the µ dependence of
the pion condensate at T = 50 MeV and α = 0 (solid line), 0.25 (dashed line), 0.3 (dotted
line), all lines have the same meeting point at µ = 234 MeV. Evidently, in both cases
a first-order phase transition occurs everywhere. The analyses based on NJL model [22]
assert that no charged pion condenses in the physical world.
- Making use of the effective potential Ω(M) we show in Fig.4 the phase diagram of
chiral condensate taken at several values of α. From the bottom to top the lines correspond
to α = 0, 0.25, 0.3. The solid line (dashed line) denotes the first-order (smooth) phase
transition. M (T= 347 MeV, µ = 662 MeV) and N (T = 373.7 MeV, µ = 1232 MeV) are
the second-order critical endpoints. For α = 0.3 the transition is smooth everywhere. The
PHASE STRUCTURE OF LINEAR SIGMA MODEL WITH NEUTRALITY CONSTRAINT (II) 107
0 20 40 60 80 100 120
0.0
0.2
0.4
0.6
0.8
1.0
T@MeVD
v

f Π
(a)
0 50 100 150 200 250 300
0.0
0.2
0.4
0.6
0.8
1.0
Μ@MeVD
v

f Π
(b)
Fig. 1. The pion condensate as a function of T (Fig.1(a)) at µ = 100 MeV and
of µ at T = 50 MeV (Fig.1(b)) in the chiral limit for neutral matter. The solid,
dashed and dotted lines correspond to α = 0, 1/4, 1/2.
v = 0
v > 0
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
Μ@MeVD
T@
M
eV
D
Fig. 2. The phase diagram of pion condensate in the (T, µ)-plane for neutral
matter in chiral limit. In order to have some comparison, we add to the figure the
dashed line representing the phase diagram of pion condensate in the (T, µ)-plane
at µI = 232.576 MeV when the neutrality condition is not used.
present scenario is quite different from what was derived in Ref. [22] which claims that
the chiral condensate decreases with increasing T and µ, but never vanishes.
2- In the case mpi > µI the pion condensate vanishes and we receive in Figs.5(a)
and 5(b) the evolution of the chiral condensate against T for µI = 0 (solid line), 100
108 TRAN HUU PHAT AND NGUYEN VAN THU
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
T@MeVD
v

f Π
(a)
0 50 100 150 200 250
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Μ@MeVD
v

f Π
(b)
Fig. 3. The pion condensate as a function of T (Fig.3(a)) at µI = 200 MeV and
of µ (Fig.3(b)) at T = 50 MeV in the physical world for neutral matter. The solid,
dashed and dotted lines correspond to α = 0, 0.25, 0.3.
u > 0
u = 0
M N
0 500 1000 1500 2000
0
500
1000
1500
2000
Μ@MeVD
T@
M
eV
D
Fig. 4. The phase diagram of chiral condensate in (T, µ)-plane for neutral mat-
ter in the physical world. From the bottom to top the graphs correspond to
α = 0, 0.25, 0.3. The solid line (dashed line) denotes first-order (smooth) phase
transition. M (662, 346.8) MeV and N(1232, 373,6) MeV are second order critical
end points. For α = 0.3 only smooth phase transition occurs.
MeV(dashed line) and the phase diagram in the (T, µI)-plane. They exhibit the smooth
transition everywhere.
PHASE STRUCTURE OF LINEAR SIGMA MODEL WITH NEUTRALITY CONSTRAINT (II) 109
0 50 100 150 200 250 300
0.0
0.2
0.4
0.6
0.8
1.0
T@MeVD
u

f Π
(a)
u > 0
u = 0
0 20 40 60 80 100 120
260
280
300
320
340
ΜI@MeVD
T@
M
eV
D
(b)
Fig. 5. In the region µI < mpi: Fig.5(a) shows T dependence of u for neutral
matter (solid and dashed lines correspond to µI = 0, 100 MeV) and Fig.5(b)
shows the phase diagram of chiral condensate in (T, µI) plane for neutral matter.
III. CONCLUSION AND DISCUSSION
In this paper the phase structure of the linear sigma model with constituent quarks
was investigated systematically within the neutrality condition and the main results are
in order.
a- The neutrality constraint was established.
b- The phase diagrams of the chiral condensate depends strongly on the ratio be-
tween the π+ and π− contributions to the condensate. Meanwhile, the phase diagram of
pion condensate does not depend on α.
c- The phase diagram of the chiral condensate in the (T, µ)-plane exhibits a first
order phase transition which ends at a CEP for α < 0.3.
It is necessarily pointed out that the electric neutrality condition makes basic
changes in the phase diagram of pion and chiral condensate.
ACKNOWLEDGMENT
N.V.Thu expresses his sincere thanks to the Vietnam Atomic Energy Commission
and the Institute of 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] M. A. Stephanov, Prog. Theor. Phys. Suppl. 153 (2004) 139; J. Wambach, Proceedings of International
Workshop on Hot and Cold Baryonic Matter. Budapest, Aug. 15-20, 2010, Hungaria; D. H. Rischke,
Proc. Part. Nucl. Phys. 52 (2004) 197.
[2] A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys. Rept. 411 (2005) 325.
110 TRAN HUU PHAT AND NGUYEN VAN THU
[3] C. H. Hee, Phys. Rept. 275 (1996) 255.
[4] I. Bombaci, in Isospin Physics in Heavy-Ion Collisions, edited by B. A. Li and Udo Schroeder, Nova
Science Publishers, Inc., New York, 2001, pp.35-81.
[5] V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rept. 410 (2005) 335.
[6] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rept. 464 (2008) 113.
[7] P. Danielewicz and Jenny Lee, Int. J. Mod. Phys. E18 (2009) 892.
[8] B. A. Li, C. M. Ko, and W. Bauer, Int. J. Mod. Phys. E7 (1998) 147.
[9] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647 .
[10] S. Typel and B. A. Brown, Phys. Rev. C64 (2001) 027302.
[11] F. Hofmann, C. M. Keil, and H. Lenske, Phys. Rev. C 64, 034314 (2001).
[12] J. R. Stone, J. C. Miller, R. Koncewicz, P. D. Stevenson, and M. R. Strayer, Phys. Rev. C 68, 034324
(2003).
[13] A. W. Steiner and B. A. Li, Phys. Rev. C72 041601 (R) (2005).
[14] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).
[15] J. B. Kogut and D. K. Sinclair, Phys. Rev. D64 (2001) 034508; ibid. D66 (2002) 034505. ibid. D70
(2004) 094501.
[16] S. Gupta, hep-lat/02005; P. de Forcrand, M. A. Stephanov, and U. Wenger, PoSLAT 2007, 237 (2007).
[17] D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86 (2001) 592.
[18] K. Splittorff, D. T. Son, and M. A. Stephanov, Phys. Rev. D64 (2001) 016003.
[19] K. Splittorff, D. Toublan, and J. J. Verbaarshot, Nucl. Phys. B639 (2002) 524.
[20] A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli, Phys. Rev. D69 (2004) 096004; D. Ebert and
K. G. Klimenko, J. Phys. G Part. Nucl. Phys. 32 (2006) 599; D. Ebert and K. G. Klimenko, Eur.
Phys. J. C46 (2006) 771; S. Lawley, W. Bentz, and A. W. Thomas, Phys. Lett. B632 (2006) 495; L.
He, Phys. Rev. D82 (2010) 096003.
[21] L. He, M. Jin, and P. Zhuang, Phys. Rev. D71 (2005) 116001.
[22] J. O. Andersen and L. Kyllingstad, J. Phys. G. Part. Nucl. Phys. 37 (2010) 015003.
[23] T. Sasaki, Y. Sakai, H. Kouno, and M. Yahiro, Phys. Rev. D82 (2010) 116004.
[24] J. O. Andersen, Phys. Rev. D75 (2007) 065011 ; J. O. Andersen and T. Brauner, Phys. Rev. D78
(2008) 014030.
[25] T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113 (2005) 843; ibid. 114 (2006) 1083.
[26] D. K. Campell, R. F. Dashen, and J. T. Manassah, Phys. Rev. D12 (1975) 979; ibid. D12 (1975)
1010.
[27] Tran Huu Phat and Nguyen Van Thu, Comm. Phys. 22 (2012) 15-31.
[28] S. Wolfram, The Mathematica Book, 5th edition, Wolfram Media/Cambridge University Press, Cam-
bridge , UK, 2003.
[29] H. Abuki, R. Anglani, R. Gatto, M. Pellicoro, and M. Ruggieri, arxiv: 0809.2658.
[30] J. W. Harris and B. Muller, Ann. Rev. Nucl. Part. Sci. 46 (1996) 71.
[31] V. Bernard, Ulf- G. Meissner, and I. Zahed, Phys. Rev. D36 (1987) 819.
[32] M. Buballa, Phys. Rept. 407 (2005) 205.
[33] M. Askawa and K. Yazaki, Nucl. Phys. A504 (1989) 668.
[34] T. Hatsuda and T. Kunihiro, Phys. Rept. 247 (1994) 221.
[35] M. A. Stephanov, Int. J. Mod. Phys. A20 (2005) 4387; hep-ph/0402115.
[36] R. V. Gavai and S. Gupta, Phys. Rev. D68 (2003) 034506 .
Received 14 October 2011.


Phase structure of linear sigma model with neutrality constraint (II)

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Phase structure of linear sigma model with neutrality constraint (II)

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