Compact stars within an asy-soft quark-meson-coupling model

Abstract

We investigate compact star properties within the quark meson coupling model
(QMC) with a soft symmetry energy density dependence at large densities. In
particular, the hyperon content and the mass/radius curves for the families of
stars obtained within the model are discussed. The hyperon-meson couplings are
chosen according to experimental values of the hyperon nuclear matter
potentials, and possible uncertainties are considered. It is shown that a
softer symmetry energy gives rise to stars with less hyperons, smaller radii
and larger masses. Hyperon-meson couplings may also have a strong effect on the
mass of the star.

Full text

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arXiv:1110.4708v1 [nucl-th] 21 Oct 2011
Modified quark-meson-coupling model: softer symmetry energy and related compact
star properties
Prafulla K. Panda,1,2Alexandre M. S. Santos,1,3D´ ebora P. Menezes,4and Constan¸ ca Providˆ encia1
1Centro de F´ ısica Computacional - Departamento de F´ ısica,
Universidade de Coimbra - P-3004 - 516 - Coimbra - Portugal
2Department of Physics, C.V. Raman College of Engneering, Vidya Nagar, Bhubaneswar-752054, India
3Universidade Federal de Santa Catarina - Campus Curitibanos,
Caixa Postal 101 - 89.520-000 - Curitibanos-SC - Brazil
4Departamento de F´ ısica, CFM, Universidade Federal de Santa Catarina,
CP 476, CEP 88.040-900 Florian´ opolis - SC - Brazil
We propose a new version of the quark meson coupling model (QMC) with a softer symmetry
energy density dependence at large densities, by adding a nonlinear coupling bewtween the isoscalar
and isovector vector-meson. The instabilities of asymmetric nuclear matter are studied. Implications
of the new parametrization of the QMC model on the compact star properties are investigated. In
particular, the crust-core transition properties, the hyperon content and the mass/radius curves
for the families of stars obtained within the models are discussed. The hyperons coupling mesons
are chosen according to experimental values of the hyperon nuclear matter potentials, and possible
uncertainties are considered. It is shown that a softer symmetry energy gives rise to stars with less
hyperons, smaller radius and larger masses. Hyperon-meson couplings may also have a strong effect
on the mass of the star.
PACS numbers: 21.65.-f, 21.30.-x, 95.30.Tg
I. INTRODUCTION
In the last years important efforts have been done to
determine the density dependence of the symmetry en-
ergy of asymmetric nuclear matter (see the reviews [1–3]
and references therein). Correlations between different
quantities in bulk matter and finite nuclei have been es-
tablished. For instance, the correlation between the slope
of the pressure of neutron matter at ρ = 0.1 fm−3and
the neutron skin thickness of208Pb [4, 5], or the corre-
lation between the crust-core transition density and the
neutron skin thickness of208Pb [6] are well determined.
Presently, there also exist different experimental mea-
surements that constrain the saturation properties of the
symmetry energy [7].
The quark-meson-coupling (QMC) model [8, 9] is an
effective nuclear model that takes into account the inter-
nal structure of the nucleon explicitly. Within the QMC
model, matter at low densities and temperatures is a sys-
tem of nucleons interacting through meson fields, with
quarks and gluons confined within MIT bags [10]. For
matter at very high density or temperature, one expects
that baryons and mesons dissolve and that the entire
system of quarks and gluons becomes confined within a
single, big, MIT bag. Within QMC it is possible to de-
scribe in a consist way both nucleons and hyperons [11].
The energy of the baryonic MIT bag is identified with
the mass of the baryon and is obtained selfconsistently
from the calculation. It is important to stress that within
the QMC model the coupling of the hyperon to the σ-
meson is fixed at the level of the saturation properties of
the equation of state (EOS). Hypernuclei properties [12–
15] will then allow the determination of the coupling of
hyperons to the isoscalar-vector meson without any am-
biguity except for the uncertainty on the experimental
hypernuclei data.
In the present study we consider an extension of the
QMC that takes into account a meson nonlinear term
involving the ω and ρ mesons.
isovector channel of the QMC equation of state, namely
the density dependence of the symmetry energy [6, 16].
Moreover, choosing the coupling constant adequately it is
possible to correct the too stiff behavior of the symmetry
energy at large densities in the QMC model.
In [17] the instabilities in asymmetric nuclear matter
(ANM) have been investigated within the QMC model
with and without the isovector-scalar δ-meson. The δ-
meson was introduced to include in the isovector channel
of the EOS between the scalar δ and the vector ρ mesons,
the same symmetry already existing in the isoscalar chan-
nel between the σ and ω mesons [18]. The presence of
the δ meson gave rise to an even stiffer symmetry energy.
Once the ω − ρ coupling term is introduced, a softer
symmetry energy is obtained and we investigate, with
this new parameterizations of the QMC, the thermody-
namical instabilities. In particular, we also discuss the
distillation effect within the modified QMC model and
determine the density and proton fraction at the inner
edge of the crust of a compact star. The results are then
compared with the predictions from other models and
approaches.
Stellar matter within the modified QMC model is also
studied. We want to investigate how the symmetry en-
ergy affects the radius, mass and hyperon content of a
compact star. This was recently done within the NLWM
with the nonlinear ωρ term and it was shown that while
the mass was not affected the radius of the star decreases
when the slope of the symmetry energy decreases [19].
This term affects the

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However, the QMC model has a softer EOS and gener-
ally predicts smaller hyperon fractions in stellar matter.
This could affect the behavior of the star properties and
its relation with the slope of the symmetry energy.
The paper is organized as follows: in section II an ex-
tension of the QMC model to include the ω −ρ coupling
is discussed, in section III we make a short review of
the calculation of the spinodal surface, in section IV re-
sults are presented and discussed and some conclusions
are drawn in the last section.
II. THE QUARK-MESON COUPLING MODEL
In what follows we present a review of the QMC model
and its generalization to include the isoscalar-isovector
ω − ρ coupling.
In the QMC model, the nucleon in nuclear medium is
assumed to be a static spherical MIT bag in which quarks
interact with the scalar (σ) and vector (ω, ρ) fields, and
those are treated as classical fields in the mean field ap-
proximation (MFA) [8, 9]. The quark field, ψqi, inside
the bag then satisfies the equation of motion:
[i/ ∂ − (m0
+1
q− gq
2gq
σ) − gq
ωω γ0
ρτzρ03γ0
?
ψqi(x) = 0 ,q = u,d(1)
where m0
denote the quark-meson coupling constants. The nor-
malized ground state for a quark in the bag is given by
qis the current quark mass, and gq
σ, gq
ωand gq
ρ
ψqi(r,t) = Nqiexp(−iǫqit/Ri)
×
iβqi? σ · ˆ rj1i(xqir/Ri)
?
j0i(xqir/Ri)
?
χq
√4π
, (2)
where
ǫqi= Ωqi+Ri
?
gq
ωω +1
2gq
ρτzρ03
?
; βqi=
?
Ωqi− Rim∗
Ωqi+ Rim∗
q
q
,
(3)
with the normalization factor given by
N−2
qi
= 2R3
ij2
0(xq)?Ωq(Ωq− 1) + RNm∗
?
the bag radius of nucleon i and χq is the quark spinor.
The bag eigenvalue for nucleon i, xqi, is determined by
the boundary condition at the bag surface
q/2??
q− gq
x2
q, (4)
where Ωqi≡
x2
qi+ (RNm∗
q)2, m∗
q= m0
σσ, Ri is
j0i(xqi) = βqij1i(xqi) .(5)
The energy of a static bag describing nucleon i consisting
of three quarks in ground state is expressed as
Ebag
i
=
?
q
nqΩqi
Ri
−Zi
Ri
+4
3π R3
iBN ,(6)
where Zi is a parameter which accounts for zero-point
motion of nucleon i and BN is the bag constant. The
effective mass of a nucleon bag at rest is taken to be M∗
Ebag
i
. The equilibrium condition for the bag is obtained
by minimizing the effective mass, M∗
bag radius
i=
iwith respect to the
dM∗
dR∗
i
i
= 0, i = p,n,(7)
and the unknowns Zi= 3.986992 and B1/4
MeV are obtained for Ri= 0.6 fm. Furthermore, the de-
sired values of EN≡ ǫ/ρ−M = −15.7 MeV at saturation
ρ = ρ0= 0.15 fm−3, are achieved by setting gq
gω= 8.954.
N
= 211.30306
σ= 5.981,
A.QMC with coupled ω − ρ fields
The consideration of a coupling between the isoscalar
and isovector fields is carried out much like in the manner
it was performed in [6]. A note is however on demand:
we start out from quarks, which find themselves confined
in a bag [10], and the boundary conditions for achieving
confinement hold the same.
couplings) and fittings (to the symmetry energy) are done
otherwise for hadronic matter. The total energy density
of the nuclear matter then reads
ε =1
2m2
1
π2
0
The relevant changes (of
σσ2+1
2m2
ωω2
0+1
2m2
ρρ2
03+ g2
ωg2
ρΛb2
0ω2
0
+
?
N
?kN
k2dk?k2+ M∗2
N
?1/2,(8)
where the ω − ρ coupling is introduced. The free energy
density is given by
F = ε − µpρp− µnρn,(9)
where the chemical potentials are given by
µp=
?
k2
p+ M∗
p
2+ gωρ +gρ
2ρ03,(10)
µn=
?
k2
p+ M∗
n
2+ gωρ −gρ
2ρ03.(11)
The vector mean field ω0 and ρ03 are determined
through
ω0=gω(ρp+ ρn)
m2
ω
,ρ03=gρ(ρp− ρn)
m2
ρ
,(12)
where gω= 3gq
is fixed by
ωand gρ= gq
ρ. Finally, the mean fields σ0
∂ε
∂σ= 0.(13)
In order to set the model parameters, we fix the free
space bag properties. They are obtained by fitting the

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TABLE I. Nuclear matter properties of the models used in the present work. All quantities are taken at saturation, where
B/A=15.7 MeV, the compressibility K0 = 290 MeV, and Q0 = −456.76 MeV. ρt and Ypt are the density and proton fraction
at the crust-core transition estimated from the thermodynamical spinodal section.
ModelΛv
gρ
Esym
(MeV) (MeV) (MeV) (MeV) (fm−3)
8.8606 33.7093.59
QMCωρ 0.01 8.9837 33.02
0.02 9.1122 32.43
0.03 9.2463 31.88
0.05 9.5335 30.87
0.10 10.3869 27.78
NL3 [23]0 8.9480 37.34
TW[24]0 7.3220 32.76
LKsym
Kτ
ρt
Yp,t
QMC0 -42.93 -457.37 0.0756 0.0223
-98.29 -474.66 0.0786 0.0241
77.35 -138.38 -480.92 0.0807 0.0250
70.55 -166.38 -478.80 0.0814 0.0218
59.03 -195.70 -457.13 0.0887 0.0289
39.04 -182.65 -355.53 0.0982 0.0332
118.30 100.50-698.4
55.30 -124.70 -508.1
84.99
0.065
0.084
0.021
.0.038
nucleon mass and enforcing the stability condition for the
bag in free space.
We consider the inclusion of the ω − ρ mixing and de-
termine the values of the couplings gρ and Λv so that
Esym= 23.27 MeV at ρ = 0.1 fm−3(kF ∼ 1.14 fm−1).
The increase of Λv requires a larger gρ coupling. The
parameters gρand Λ are listed in Table I. The coupling
gω = 8.9541 for all the QMC models considered. We
take the standard values for the meson masses, namely
mσ= 550 MeV, mω= 783 MeV, and mρ= 770 MeV.
III. STABILITY CONDITIONS
The stability conditions for asymmetric nuclear mat-
ter, keeping constant volume and temperature are ob-
tained from the free energy density F, imposing that this
function is a convex function of the densities ρpand ρn,
i.e. the symmetric matrix with elements
Fij=
?
∂2F
∂ρi∂ρj
?
T
, (14)
is positive [20–22]. This is equivalent to imposing
∂µp
∂ρp
> 0,
∂(µp,µn)
∂(ρp,ρn)> 0,(15)
where we have used µi=
∂F
∂ρi
???
T,ρj?=i.
The two eigenvalues of the stability matrix are given
by [20]
λ±=1
2
?
Tr(F) ±
?
Tr(F)2− 4Det(F)
?
,(16)
and the eigenvectors δρ±by
δρ±
δρ±
i
j
=λ±− Fjj
Fji
,i,j = p,n.
The largest eigenvalue is always positive whereas the
other can take on negative values.We are interested
in the latter, as it defines the spinodal surface, which is
determined by the values of T, ρ, and the proton fraction
yp= ρp/ρ, for which the smallest eigenvalue of Fij be-
comes negative. The associated eigenvector defines the
instability direction of the system, in isospin space.
It is well known that in asymmetric nuclear matter, the
spinodal instabilities cannot be separately classified as
mechanical or chemical instabilities [22]. In fact, the two
conditions that give rise to the instability of the system
are coupled so that the instability appears as an admix-
ture of nucleon density and concentration fluctuations.
In the following we study the direction of instability and
the spinodal for the different models considered.
IV.RESULTS AND DISCUSSIONS
A.Model properties
We now compare the symmetry energy and its slope
and curvature for all models (Fig. 1 and Table I). The
symmetry energy in the present relativistic mean field
models is given by
Esym=
kF2
6ǫF2+ρ
2
g2
4m2
ρ
ρ
(17)
where ǫF=
ons, and M∗
The symmetry energy slope L(ρ) is defined by
?P2
F+ M∗2
0is the Fermi energy of the nucle-
0is their effective mass in symmetric matter.
L = 3ρ0∂Esym/∂ρ.(18)
The curvature parameter of the symmetry energy
Ksym= 9ρ2
0∂2Esym/∂ρ2
(19)
(Fig. 1c)) is also of interest because it distinguishes be-
tween different parametrizations. Of particular interest
is the quantity
Kτ= Ksym− 6L − (Q0/K0)L,

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0
10
20
30
40
50
60
0 0.05 0.1 0.15 0.2 0.25
εsym (MeV)
ρ (fm-3)
(a)
QMC
QMC-Λv=0.01
QMC-Λv=0.03
QMC-Λv=0.05
QMC-Λv=0.1
0
50
100
150
200
250
0 0.05 0.1
ρ (fm-3)
0.15 0.2 0.25
L (MeV)
(b)
QMC
QMC-Λv=0.01
QMC-Λv=0.03
QMC-Λv=0.05
QMC-Λv=0.1
-500
-400
-300
-200
-100
0
0 0.05 0.1
ρ (fm-3)
0.15 0.2 0.25
Ksym (MeV)
(c)
QMC
QMC-Λv= 0.1
QMC-Λv= 0.05
QMC-Λv= 0.03
QMC-Λv= 0.01
FIG. 1. (Color online) Symmetry energy (a), its slope param-
eter L = 3ρ0E′
of density for QMC and QMCωρ for different values of the
coupling Λv.
sym(b), and curvature Ksym (c) as a function
where K = 9ρ2
Q = 27ρ3
isoscalar properties take the values K0= 290 MeV and
Q0 = −456.76 MeV, and they are the same for all the
QMC models considered in the present work. Kτ can
be directly extracted from measurements of the isotopic
dependence of the giant monopole resonance (GMR) [3].
Recent measurements of the GMR on even-A Sn isotopes
give a quite stringent value of Kτ= −550 ± 100 MeV.
In Table I, the saturation properties of nuclear matter
0∂2(E/ρ)/∂ρ2is the incompressibility, and
0∂3(E/ρ)/∂ρ3its derivative. At saturation these
for all the QMC models considered in the present work as
well as the corresponding properties for two well known
relativistic nuclear models NL3 [23] and TW [24], which
are considered for comparison. The symmetry energy
of the QMCωρ model with Λv = 0.1 seems to be a bit
low. The slope L and the parameter Kτ of the models
are well within the experimental constraints coming from
different sources [7].
The symmetry energy within QMC shows a rather lin-
ear behavior with density.
NLWM models. Once the nonlinear ωρ term is incor-
porated. an extra symmetry energy density dependence
is included. As a result, the symmetry energy of the
QMCωρ (Fig. 1(a)) becomes softer at higher densi-
ties. The slope parameter L, plotted in Fig. 1(b), con-
firms that fact, with smaller values at the larger densities
shown. However, below ρ = 0.1 fm−3the symmetry en-
ergy is larger in the models with a larger Λvand this has
important effects on the properties of the crust-core tran-
sition as it will be discussed latter. The density depen-
dence of the three properties ǫsym, L, and Ksymbehavior
shown in Fig. 1 is in accordance with the dependence ex-
pected from relativistic nuclear models (see Fig. 2 of ref.
[25]). The curves for the symmetry energy cross by con-
struction at 0.1 fm−3. However, this feature is also true
for most of the symmetry energy of other parameteri-
zations which cross each other at ρ = 0.1 − 0.12 fm−3.
Equilibrium properties of nuclei constrain the symmetry
energy in this range of densities. The crossing of the slope
at ρ = 0.05 − 0.06 fm−3as seen in Fig. 1(b) is again a
result of the same constraints as discussed in [25].
This is a feature of many
B. Instabilities
In the present subsection we discuss the results for
the instability region at subsaturation densities obtained
within QMC and QMCωρ models. For comparison we in-
clude results for the relativistic nuclear models NL3 [23]
and TW [24]. In [17] the thermodynamical instabilities of
nuclear matter as described by the QMC including the δ-
meson were studied and compared with NL3 and TW. It
was shown that the spinodal surface obtained with QMC
appeared in between the NL3 and TW spinodal surfaces.
On the other hand the inclusion of the δ-meson reduces
the spinodal region at large asymmetries with respect to
QMC due to the slightly smaller values of the symmetry
energy at subsaturation densities. This occurs because
the inclusion of the δ-meson increases the symmetry en-
ergy slope. However, the inclusion of the ωρ term reduces
the slope L so we may expect that this term gives rise to
an opposite effect.
In Fig. 2 we plot the spinodal curves for np matter.
As referred before, they are defined by the points, for a
given temperature, density and isospin asymmetry, that
make the curvature matrix of the free energy vanish. We
confirm that the QMCωρ presents larger instability re-
gions than QMC at large isospin asymmetries. The larger

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0
0.02
0.04
0.06
0.08
0.1
0 0.02 0.04
ρn ( fm-3 )
0.06 0.08 0.1
ρp ( fm-3 )
QMC -Λv=0.00
Λv=0.01
Λv=0.02
Λv=0.03
Λv=0.05
Λv=0.1
NL3
TW
FIG. 2. (Color online) Spinodal (thermodynamical instability
border) for QMC, QMCωρ, NL3 [23] and TW [24].
the magnitude of the coupling Λv the smaller the slope
L and the larger the instability region. The same behav-
ior was obtained in [16] for NL3ωρ. In [25] the effect of
the slope L on the spinodal surface at large asymmetries
was discussed and it was shown that larger values of L
gave rise to smaller spinodal regions at large asymme-
tries, where matter is closer to neutron matter. Neutron
matter pressure is essentially proportional to the slope L
and, therefore, a larger L corresponds to a harder EOS.
The crust-core transition densities are shown in Table
I and we can see that they increase with the increase
of the ω − ρ coupling. QMC gives a higher transition
density than the one obtained within NL3 and TW in-
terpolates in between the QMCω−ρ transition densities,
as expected due to the values of L.
The nuclear liquid-gas coexistence phase is character-
ized by different isospin contents for each phase, i.e., the
clusterized regions are more isospin symmetric than the
surrounding nuclear gas, the so-called isospin distillation
[26, 27]. The extension of the distillation effect is model
dependent and it has been shown that NL3 and other
NLWM parameterizations lead to larger distillation ef-
fects than the density dependent hadron models [28–30].
In Fig. 3 we show the ratio of the proton versus the neu-
tron density fluctuations corresponding to the unstable
mode. This ratio defines the direction of the instability
of the system. We show the results for different proton
fractions Yp= 0.3, 0.1, 0.05, for the sake of studying the
effectiveness of the models in restoring the symmetry in
the liquid phase. One can see that the effect of the ω−ρ
coupling is to decrease the δρp/δρnratio. This same ten-
dency was obtained in [16] for NL3ωρ, although in this
last work a dynamical calculation was performed. Com-
paring with the result reported in [28] we conclude that:
a) QMC behaves differently from NLWM models such
as NL3 and TM1. For these models the ratio of the pro-
ton versus the neutron density fluctuations increases with
the density, while for QMC after a maximum obtained at
ρ ∼ 0.02 fm−3, this ratio decreases and more strongly if
Λv is large; b) QMC presents a behavior similar to the
one of relativistic models with density dependent cou-
plings such as TW, however the decrease of the distilla-
tion effect with density is not so strong [28, 31], even for
the largest Λv coupling we have considered. Moreover,
the ratio of the proton versus the neutron density fluc-
tuations reaches larger values for larger proton fractions,
as already seen in other relativistic and non-relativistic
models [28, 30, 31].
0.5
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1
δρp
-/δρn
-
ρ (fm-3 )
(a)
yp=0.05
QMC
Λv = 0.01
Λv = 0.02
Λv = 0.03
Λv = 0.05
Λv = 0.1
0.5
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1
δρp
-/δρn
-
ρ (fm-3 )
(b)
yp=0.1
QMC
Λv = 0.01
Λv = 0.02
Λv = 0.03
Λv = 0.05
Λv = 0.1
0.5
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1
δρp
-/δρn
-
ρ (fm-3 )
yp=0.3
(c)
QMC
Λv = 0.01
Λv = 0.02
Λv = 0.03
Λv = 0.05
Λv = 0.1
FIG. 3. (Color online) Direction of instability (eigenvector for
negative eigenvalue λ−) for yp=0.05 (a), 0.1 (b) and 0.3 (c).