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arXiv:1110.4708v1 [nuclth] 21 Oct 2011
Modified quarkmesoncoupling 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  P3004  516  Coimbra  Portugal
2Department of Physics, C.V. Raman College of Engneering, Vidya Nagar, Bhubaneswar752054, India
3Universidade Federal de Santa Catarina  Campus Curitibanos,
Caixa Postal 101  89.520000  CuritibanosSC  Brazil
4Departamento de F´ ısica, CFM, Universidade Federal de Santa Catarina,
CP 476, CEP 88.040900 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 vectormeson. 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 crustcore 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. Hyperonmeson 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 crustcore 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 quarkmesoncoupling (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 isoscalarvector 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 isovectorscalar δmeson. 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 QUARKMESON COUPLING MODEL
In what follows we present a review of the QMC model
and its generalization to include the isoscalarisovector
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 quarkmeson 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 zeropoint
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 crustcore 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]08.9480 37.34
TW[24]07.3220 32.76
LKsym
Kτ
ρt
Yp,t
QMC042.93 457.37 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 (fm3)
(a)
QMC
QMCΛv=0.01
QMCΛv=0.03
QMCΛv=0.05
QMCΛv=0.1
0
50
100
150
200
250
0 0.05 0.1
ρ (fm3)
0.15 0.2 0.25
L (MeV)
(b)
QMC
QMCΛv=0.01
QMCΛv=0.03
QMCΛv=0.05
QMCΛv=0.1
500
400
300
200
100
0
0 0.05 0.1
ρ (fm3)
0.15 0.2 0.25
Ksym (MeV)
(c)
QMC
QMCΛv= 0.1
QMCΛv= 0.05
QMCΛv= 0.03
QMCΛv= 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 evenA 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 crustcore 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 δmeson 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 δmeson 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 ( fm3 )
0.06 0.08 0.1
ρp ( fm3 )
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 crustcore 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 liquidgas 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 socalled 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 nonrelativistic
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 (fm3 )
(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 (fm3 )
(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 (fm3 )
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).
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C.Neutron stars
We now turn to stellar properties as obtained in the
present approach. The composition of the stellar matter
is determined by the requirements of the charge neutral
ity and chemical equilibrium under the weak processes
B1→ B2+ l + ¯ νl;
where B1 and B2 are baryons, l is a lepton. The EOS
which depends on the chemical potentials is now modi
fied according to [11], so that the lowest eight baryons are
taken into account. As we restrict ourselves to zero tem
perature, no trapped neutrinos are considered, but the
electrons and muons are considered so that charge neu
trality and βequilibrium can be enforced. The hyperon
couplings are not relevant to the ground state properties
of nuclear matter, but information about them can be
available from the levels in hypernuclei [12, 32–36]:
B2+ l → B1+ νl
(20)
gσB= xσBgσ, gωB= xωBgω, gρB= xρBgρ
and xσB, xωBand xρBare equal to 1 for the nucleons and
acquire different values in different parameterizations for
the other baryons. Note that the squark is unaffected
by the sigma and omega mesons i.e. gs
In QMC the couplings of the hyperons to the σmeson
do not need to be fixed because the effective masses of
the hyperons are determined selfconsistently at the bag
level. Only the xωB and xρB have to be fixed. We ob
tain xωBfrom the hyperon potentials in nuclear matter,
UB= −(M∗
be 28 MeV, 30 MeV and 18 MeV respectively. We find
that xωΛ= 0.743, xωΣ= 1.04 and xωΞ= 0.346. xρB= 1
is fixed for all the baryons. However, while the binding
of the Λ to symmetric nuclear matter is well settled ex
perimentally [13], the binding values of the Σ−and Ξ−
still have a lot of uncertainties [15]. We, therefore, test
the effect of the coupling to the cascade and show results
also for VΞ = −10 and 0 MeV. In fact, measurements
from the production of Ξ in the12C(K−,K+)12
compatible with a shallow attractive potential VΞ∼ −14
MeV [14]. We obtain xωΞ= 0.3989 for VΞ= −10 MeV
and xωΞ= 0.4643 for VΞ= 0 MeV.
The resulting EOS are displayed in Fig. 4a for the
QMC, QMCωρ for different values of the coupling pa
rameter and QMC with protons and neutrons only. We
also include the empirical EOS obtained by Steiner et
al. from a heterogeneous set of seven neutron stars with
welldetermined distances [37]. These results however,
should still be considered with care because there are
many uncertainties involved. We conclude that the agree
ment of the theoretical EOS with the empirical one when
hyperons are included in the calculation is defined by
the hyperonmeson interaction and the Λv coupling, or,
equivalently, by the symmetry energy.
EOS agrees with the constraints. However, the inclu
sion of hyperons with the hyperon couplings obtained for
σ= gs
ω= 0 .
B− MB) + xωBgωω0, for B = Λ, Σ and Ξ to
ΞBe are
The QMC pn
0.1
1
1 1.5 2 2.5 3 3.5 4 4.5 5
P ( fm4 )
ε ( fm4 )
a)
Steiner 2010
VΞ=18,Λv=0.00
0.01
0.1
VΞ=10, Λv=0.1
VΞ= 0, Λv=0.1
pn
1
1.2
1.4
1.6
1.8
2
2.2
11 11.5 12 12.5
R (km )
13 13.5
QMC
pn
b)
14
Ξ
Ξ
M(M
=0.1, V =−10
v
V =−18, =0.0
)
= 0
v
=0.05
=0.1
Λ
Λ
FIG. 4.
results for QMC and QMCωρ. The EOS for VΞ = −10 MeV
and 0 MeV where only obtained with Λv = 0.1.
(Color online) a) Equation of state and b) TOV
the hyperon nuclear potentials taking VΛ = −28 MeV,
VΣ= 30 MeV and VΞ= −18 MeV makes the EOS too
soft. Increasing Λv makes the EOS harder bringing the
EOS closer to the constraints defined by the empirical
EOS. This is easily understood with the help of Fig. 5.
Increasing Λvgives rise to a softer pn EOS at high den
sities and, therefore, hinders the onset of hyperons. So
the larger Λvthe smaller the hyperon fraction in the star
and the harder the EOS.
The effect of a less attractive VΞpotential is also clear:
the EOS becomes harder because the onset of hyperons
occurs at larger densities as shown in Fig. 5. We con
clude that any mechanism than hinders the formation of
hyperons makes the EOS harder.
The EOS enters as input to the TolmanVolkoff
Oppenheimer [38] equations, which generate the macro
scopic stellar quantities. The obtained mass/radius curve
for stars with a mass larger than 1M⊙ and the cor
responding properties of maximum mass stars are then
shown, respectively, in Fig. 4b) and Table II.
First let us discuss the effect of the symmetry energy
and the hyperon couplings on the mass/radius curve. A
Page 7
7
TABLE II. Stellar properties obtained with the QMC model
and different values of the parameter Λv and the Ξmeson
coupling. pn stands for nucleonic matter with no hyperons
included.
Λv
VΞ(MeV) Mmax(M⊙) Mb(M⊙) R(km) ε0(fm−4)
0.0181.776
0.01 181.836
0.03181.871
0.05181.880
0.1181.888
0.1101.928
0.101.969
0 pn2.131
2.006
2.096
2.152
2.170
2.182
2.243
2.301
2.492
12.657
12.496
12.458
12.415
12.345
12.292
12.218
11.623
4.620
4.837
4.892
4.945
5.004
5.113
5.182
5.986
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7
Strangeness fraction
ρ/ρ0
V =−18,
Ξ
V =−10,
Ξ
V = 0,
Ξ
v
Λ =0.0
0.05
0.1
Λ =0.1
Λ =0.1
FIG. 5. (Color online) Strangeness fraction as a function of
density QMCωρ. For VΞ = −10 and 0 MeV we have taken
Λv = 0.1, For VΞ = −18 MeV, we show results for Λv =
0, 0.05, 0.1.
larger Λv gives rise to a softer EOS and, therefore, a
smaller radius. It is seen that when going from Λv= 0
to 0.1 the radius of a star with a mass M = 1−1.5M⊙de
creases by ∼ 0.3 Km. This effect was discussed recently
in [19] where NLWM models were discussed. In this pa
per it was shown a clear correlation between L and star
radius. However, within the models discussed in [19] the
maximum mass was shown not to depend on L, while in
the framework of the QMC model there is a clear effect
of almost 0.1M⊙ if Λv increases from 0 to 0.1. This is
mainly due to the smaller strangeness fraction inside the
star.
The reduction of the attractiveness of VΞhas a similar
effect on the maximum mass of the star, i.e., the mass
increases ∼ 0.2M⊙if VΞincreases from 18 to 0 MeV.
We conclude that there is still quite a large uncertainty
on the coupling of hyperons to nuclear matter and there
fore, there is still room for a very massive star such as
the recently measured pulsar J16142230 with a mass
M = 1.97 ± 0.04 [39], even including hyperons in the
EOS. This, however, is a particularly massive star. Most
of the known pulsars [40] can be obtained by the present
models.
V. SUMMARY AND DISCUSSION
We have proposed a modified QMC model which in
cludes a nonlinear ωρ in the same fashion as it has been
proposed in a hadron framework [6].
the nucleons are described as nonoverlapping bags. The
new contribution allows the softening of the symmetry
energy at large densities. In normal NLWM or QMC
without this term the symmetry energy increases almost
linearly with the baryonic density giving rise to very hard
stellar matter EOS. The inclusion of this term, done for
the first time within QMC in the present study, remedies
this problem and brings down the slope of the symme
try energy at saturation density to values closer to the
experimental predictions (see [7] for a compilation of all
constraints on L). This allows taking advantage of the
already known good properties of the QMC together with
a symmetry energy not too hard.
In QMC model
We have studied subsaturation nuclear instabilities for
both symmetric and asymmetric matter within the QMC
model, with and without the inclusion of a nonlinear ωρ
coupling. We have also shown that the behavior of the
models get closer to the properties of nuclear relativistic
models with density dependent couplings such as TW.
Namely, the distillation effect in non homogeneous mat
ter does not increase with density as in NL3, but de
creases as in TW.
We have also discussed the effect of the new EOS on
the stellar properties. Hyperons were included in the
EOS. For the hyperon couplings we took advantage of
the fact that QMC predicts the hyperon effective masses
without being necessary to fix the hyperonσ couplings.
We have used information from hypernuclei to fix the
hyperonω coupling and the hyperonρ coupling was con
sidered equal to the one of the nucleon. Since there is a
large uncertainty on hypernuclei with Σ and Ξ, we have
also tested the effect of increasing the potential VΞ so
that it becomes less attractive. It was shown that both
the symmetry energy and the hyperon couplings have a
strong effect on the mass and radius of the star. A softer
symmetry energy gives rise to smaller stars. Also the hy
peron fraction is affected: softer symmetry energy corre
sponds to a smaller hyperon fraction as already discussed
in [19]. However, within QMC the density dependence of
the symmetry energy has also an effect on the maximum
star mass, effect not observed in [19].
It was also shown that the hyperon nuclear interac
tion defines the amount of strangeness in the star, and,
therefore, has a strong influence on the maximum mass
allowed. Even including hyperons in the QMC EOS we
could explain the mass of the pulsar J16142230 if the
cascade nuclear potential is set to be very little attrac
tive. More data on hypernuclei is needed to constrain the
hyperonmeson couplings.
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8
ACKNOWLEDGMENTS
This work was partially supported by the Capes/FCT
232/09 bilateral collaboration, by CNPq and n.
FAPESC/1373/20100 (Brazil), by FCT and FEDER
(Portugal) under the projects CERN/FP/83505/2008
and PTDC/FIS/113292/2009,and by Compstar, an ESF
Research Networking Programme.
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