hot dense matter in the quark-meson-coupling …hot dense matter in the quark-meson-coupling model...
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Hot Dense Matter in The Quark-Meson-Coupling Model (QMC):
Equation of State and Composition of Proto-Neutron Stars.
J. R. Stone1,2, V. Dexheimer3, P. A. M. Guichon4, and A. W. Thomas5
1Department of Physics (Astro), University of Oxford,
Oxford OX1 3RH, United Kingdom2Department of Physics and Astronomy,
University of Tennessee, Knoxville, TN 37996, USA3 Department of Physics, Kent State University, Kent, OH 44243 USA
4SPhN-IRFU, CEA Saclay, F91191 Gif sur Yvette, France and5ARC Centre of Excellence in Particle Physics at the Terascale and CSSM,
Department of Physics,University of Adelaide, SA 5005 Australia
AbstractWe report the first results of the extension of the QMC model for asymmetric dense matter at
finite temperature. The effects of temperature on particle composition (including the full baryon
octet content) of the core of (proto-)neutron stars, as well as on the equation of state, are studied.
We consider both dense matter in chemical equilibrium and matter in which neutrinos are trapped.
In order to simulate stellar temperature profiles that increase with density and stellar radius, the
entropy per baryon is fixed. Under these conditions, the model predicts that proto-neutron stars
are already born with hyperons present at about the threshold density for their appearance in
cold neutron stars, reaching ∼ 20% of the baryon content in the center of the most massive star
produced.
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I. INTRODUCTION
Properties of young proto-neutron stars (PNS’s) born in core-collapse supernova have
been one of the main topics of interest in observation and theoretical modeling for a long
time [1–3]. More recently, the topic has resurfaced in the context of the possible emission
of detectable gravitational waves (e.g. [4]). It has been generally accepted that, in addition
to the nucleons, heavy baryons exist in the cores of proto-neutron stars subject to weak
interactions. Dexheimer et al. [5] explored the effects of trapped neutrinos and temperature
in stars with hyperons in the SU(3) version of the chiral mean-field (CMF) model. Oertel
et al. [6] studied thermal effects on the Equation of State (EoS) of dense matter with non-
nucleonic degrees of freedom and the possible influence of hyperons on stellar mergers in the
framework of the Relativistic-Mean-Field (RMF) model with a large variety of parameter
sets. Sumiyoshi et al. [7] reported the appearance the hyperons in supernovae appearing
∼0.5–0.7 s after the bounce to trigger a recollapse into a black hole. And, again in the
context of gravitational waves, Sekiguchi et al. [8] and Radice et al. [9] have shown how
neutron star mergers can be influenced by the appearance of hyperons.
Several variants of the QMC model, differing somewhat from the formulation adopted
in this work, have also been applied to model neutron stars and dense matter. They are
listed and briefly discussed in Refs. [10] and [11]. Pertinent to this work, Panda et al. [12]
studied neutrino-free stellar matter and matter with trapped neutrinos at fixed temperatures
and fixed entropies per baryon and compared their results to the outcome of a non-linear
Walecka model. They calculated the hyperon population in the core of a neutron star at
T=0 and 10 MeV and obtained results close to those reported here. However, they did
not study stellar particle population at higher temperatures. Their model predicted an
increase in pressure with density in matter with trapped neutrinos (as compared to matter
without neutrinos) and a shift of the threshold for appearance of strangeness to lower density
(at T=10 MeV when compared to T=0). It was also demonstrated that, in neutrino-free
chemically equilibrated matter, the EoS softens due to the onset of hyperons but stiffens
again when a higher temperature is accounted for.
Models assuming deconfined quarks in addition to hadrons have been extensively reported
in the literature and the hadron-quark phase transition and its consequence for the proto-
neutron stars have been studied (e.g. [13–20]). Very recently Roark et al. [21] explored
2
the role of hyperons and quarks in proton-neutron stars using the CMF model at finite
temperatures and fixed entropy per baryon, with and without neutrino trapping. Hyperons
and quarks were found in the cores of large-mass stars and their interplay and the possibility
of mixtures of phases was taken into account and analyzed. Despite the great variety of
models and approaches, no convergence to a general consensus on the EoS and structure
of high density matter in the core of neutron stars has been achieved as yet [22], although
the presence of hyperons at finite temperatures has been repeatedly predicted in different
formalisms.
In this paper we study hot and dense hadronic matter in the framework of the latest
version of the QMC model [10], extended to finite temperature with the aim to follow
detailed evolution of baryon and lepton populations as a function of temperature and entropy
per baryon and its consequences for the EoS of (proto)neutron stars. The QMC model of
Guichon and collaborators [23–26] was created in order to explore the connection between
nuclear binding and the modification of the structure of a particle embedded in a nuclear
medium. It was shown that, when the quarks in one nucleon interact self-consistently with
the quarks in surrounding nucleons by exchanging a σ meson, the effective mass of the bound
nucleon is no longer linear in the scalar mean field (σ):
M∗N = MN − gσσ +
d
2(gσσ)2, (1)
where the coefficient d is known as the ”scalar polarizability". The appearance of this
term, a natural consequence of the quark structure of the nucleon, is sufficient to lead to
nuclear saturation. The QMC model has been applied successfully to nuclear matter at zero
temperature, predicting the appearance of Λ,Ξ−, and Ξ0 hyperons in the interior of cold
neutron stars [27]. It has also led to impressive results when applied to finite nuclei [22, 28,
29] (for a recent review, see Ref. [10]). The full derivation of the finite temperature formalism
for the QMC model will appear in a separate publication. Nevertheless, the main expressions
relevant to this paper can be found in the supplementary material in QMC-finite-T.pdf at
https://www2.physics.ox.ac.uk/contacts/people/stonej.
3
II. RESULTS
A. Particle composition
Predictions of the QMC model for the particle population distribution of protons, neu-
trons, Λ,Ξ−, and Ξ0 hyperons, electrons, and νe are shown in Figs. 1 - 4 at selected tem-
peratures of 10, 20, 40, 70 MeV, respectively. Only populations higher than 1% of the
total are shown. The range of temperatures was selected to cover both the proto-neutron
stars born in core-collapse supernovae, which are likely to reach temperature of several tens
of MeV [3, 30, 31] and remnants of neutron star mergers, which could reach even higher
temperatures [32]. Charge neutrality was always imposed and two extreme regimes were
considered, one with trapped neutrinos (imposed through a large fixed lepton fraction) and
the other in which neutrinos were allowed to escape (chemical equilibrium), throughout the
whole temperature range. In reality, models suggest that matter is opaque to neutrinos
in core-collapse supernovae and, only after some time (∼1 min), does it start to cool via
URCA type processes, when neutrinos diffuse to the surface and the stars become neutrino
transparent. This time evolution has a significant effect on the distribution of baryonic and
leptonic constituents in the star interior, in dependence on both density and temperature,
but is beyond the scope of the current work.
Careful examination of the left panel of the figures reveals that proto-neutron stars are
most likely born with hyperons present and the threshold baryon density for their appear-
ance increases with decreasing temperature to reach about 0.4-0.6 fm−3 at T=10 MeV. The
chemical equilibrated neutron star scenario (right panel) follows a similar development and,
in addition, predicts the baryon density at which the deleptonization happens to be around
1 fm−3 at T=10 MeV.
The calculations of particle populations performed at fixed temperature, however, may
not be telling the full story because they assume, somewhat schematically, that the whole
neutron star core is at the same temperature. Calculations of the particle population at
fixed entropy per baryon or entropy density per baryon density S/A are more realistic, as
they yield a temperature variation with baryon number density. Figs. 5 and 6 show the
entropy per baryon profile in the fixed temperature case and the temperature profile in the
fixed entropy per baryon case. We present results for the latter case in Figs. 7 - 9, again,
4
up to temperatures relevant for neutron-star mergers. It can be seen that, even in the fixed
entropy per baryon scenario, the hyperon population is always significant.
In particular, the temperature range about 10 - 40 MeV expected for the cores of proto-
neutron stars from core-collapse supernovae corresponds to the range of S/A = 2–3 kB,
being almost constant within 0.4 ≤ nB ≤ 1.0 fm−3 in both the neutrino free and trapped
scenario. Most of the baryon octet is predicted to be present under these conditions and
further cooling of stars should not appreciably change the already developed particle make-
up of the core [33]. Even the higher temperature regime, expected to be reached during
mergers and by the merger remnants, is predicted not to be significantly affected by the
presence of neutrinos. Fig. 5 shows that S/A becomes more density dependent and decreases
more rapidly with increasing density in the chemically equilibrated case. Fig. 6 shows
that temperature becomes more density dependent and grows more rapidly with increasing
density in the chemically equilibrated case.
We note that the hyperon composition in all of the scenarios discussed above does not
predict Σ hyperons to appear at baryon densities below 1.2 fm−3 in cold matter. This finding
is a direct consequence of features which are present in the QMC model [10, 27] and absent
in conventional RMF. In our case, the nucleon-hyperon (N-Y) interactions are not a subject
of choice, but emerge naturally from the formalism. In particular, the hyperfine interaction
which splits the Λ and Σ masses in free space is significantly enhanced in-medium [34],
leading to what is effectively a repulsive three-body force for the Σ hyperons, with no
additional parameter. Of course, it has been shown that the baryon populations proposed
by the QMC model can be obtained in other models for a specific choice of the nucleon-
hyperon N-Y and Y-Y interactions [35, 36]. Gomes et al. [37] explored the effects of many-
body forces simulated by nonlinear self-couplings and meson–meson interaction contributions
to the model Lagrangian and obtained the QMC-like hyperon population as a function of
baryon density for a particular choice of their parameters.
Moreover, the QMC prediction of the absence Σ hyperons is supported by the fact that
no bound Σ-hypernucleus at medium or high mass has been found as yet, despite dedicated
search [38, 39]. The appearance of the Ξ hyperons at rather low densities indicates the exis-
tence of a bound Ξ-hypernucleus. This prediction is in line with recent results of Nakazawa
et al., who reported observation of a bound state of the Ξ−14N system [40]. As shown in
Figs. 1 - 4, the absence of Σ hyperons in cold dense nuclear persists in hot matter.
5
B. The Equation of State
The composition of neutron stars significantly affects their EoS. The QMC EoS is pre-
sented in Fig. 10 for all possible different scenarios. The pressure, as expected, is larger for
larger temperatures, at least until hyperons appear and their enhancement at higher tem-
perature softens the equation of state. This softening causes a “kink” in the EoS at lower
temperatures, which disappears at higher temperature, when the hyperons are present at
all densities. The QMC EoS for nucleon-only matter has been added for illustration of the
inevitable softening of the EoS by the appearance of hyperons. This effect is less pronounced
for the case of neutrino trapping, when there are less hyperons and the overall EoS pressure
is larger. The EoS for three different values of fixed entropy per baryon is illustrated in
Fig. 11, for which similar conclusions can be drawn.
C. Compact objects
As pointed by Lattimer and Swesty [41], in stars with finite temperature/entropy a crust
of high entropy should be used, since the shock wave created during supernova explosions
leaves the outer regions with a much higher entropy than the rest of the star. The crust
remains warmer for a longer time serving as an insulating blanket for the core, delaying
the star from coming to a complete thermal equilibrium with the interstellar medium. In
this case, the crust can be stiff enough to generate massive stars for small central densities,
resulting in large radii.
Fig 12 illustrates the solution of the Tolman-Oppenheimer-Volkoff (TOV) equations for
the EoSs calculated in this work for the cases with trapped neutrinos (left panel) and neutrino
free (right panel), for several values of the entropy per baryon. In both scenarios, the
maximum mass of the object is above the currently observed gravitational mass limit for
neutron stars (considered cold) [42]. Rezzolla et al [43], combining the GW observations
of merging systems of binary neutron stars and quasi-universal relations, set constraints on
the maximum mass that can be attained by nonrotating stellar models of neutron stars,
implying the the maximum mass of a non-rotating neutron star is between 2.01+0.04−0.04 and
2.16+0.17−0.15.The maximum mass of a cold neutron star, predicted in the QMC model, 2.00
M�, lies within these limits. However, in these estimates, no effects of finite temperature
6
were taken into account. The QMC model predicts higher stellar maximum mass for the
cases with larger entropies per baryon and trapped neutrinos (when compared with the cold
beta-equilibrated case). This is consistent with the EoS behavior shown in Fig. 11.
The results of the QMC model shown in this work, consistent with results of other
models [6, 12], predict a substantial presence of hyperons in massive compact objects with
temperatures equal to and above 10 MeV. Previous calculation at T=0 MeV also yielded a
hyperon presence in cold massive neutron star cores [27]. As shown by Roark et al. [21], even
calculations including a phase transition to deconfined quark matter predict that hyperons,
specially the Λ, can be present in different stages of neutron star evolution.
Finally, note that our results re-open the question of the existence of r-modes in rotating
neutron stars [44, 45]. Jones [46, 47] reported that the bulk viscosity of hyperonic matter
in neutron stars would produce a serious damping of the r-modes. Lidblom and Owen [48]
argued that the cooling of the proto-neutron star is too rapid to influence the r-modes.
Damping arising from different phases of quark matter in strange quark stars and hybrid
stars have been also studied [49]. It will be interesting to pursue the connection between
r-modes and the internal composition of neutron stars in the future.
III. SUMMARY
In this work we presented for the first time results for the QMC model for matter at differ-
ent stages of stellar evolution. These included trapped neutrinos and chemically-equilibrated
matter and were shown for constant temperatures up to 70 MeV, using an entropy per baryon
prescription that allows the temperature to increase with baryon density. In all cases, a sub-
stantial amount of hyperons were found in the core of massive neutron stars, which were
predicted to be considerably larger and even more massive right after they are created in
core-collapse supernova explosions. This has to do with the amount of hyperons (which
usually soften the equation of state) being larger when larger temperatures / entropies per
baryon are considered, but being suppressed when trapped neutrinos are included.
The absence of Σ hyperons, already reported in Ref. [10, 27] for the zero-temperature
case, persisted in this work, both with the inclusion of temperature and trapped neutrinos.
This novel result is in agreement with the fact that no bound Σ-hypernucleus has been found
as yet [38, 39]. In the future, we intend to use the QMC model to generate equation of state
7
tables as a function of density, temperature, and lepton fraction in order to study the role
played by hyperons in core-collapse supernova explosions.
Acknowledgments
JRS and PAMG acknowledge a fruitful discussion with Andrew Steiner during the course
of writing the computer code used in this work. This work was supported by the Uni-
versity of Adelaide and the Australian Research Council through the ARC Centre of Ex-
cellence in Particle Physics at the Terascale (CE110001004) and grants DP180100497 and
DP150103101 (AWT). It was also supported by the National Science Foundation under grant
PHY-1748621.
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0.2 0.4 0.6 0.8 1.0nB [fm-3]
0.01
0.10
1.00
n i/nB
protonneutronΛΞ0
Ξ-
electronneutrino
0.2 0.4 0.6 0.8 1.0nB [fm-3]
T = 10 MeV PNS (YL = 0.4)
T = 10 MeV NS(β)
FIG. 1: Composition of dense matter for temperature T=10 MeV. The case of trapped neutrinos
in proto-neutron star matter (left panel) and chemically equilibrated neutron star matter (right
panel) are shown.
12
0.2 0.4 0.6 0.8 1.0nB [fm-3]
0.01
0.10
1.00
n i/nB
protonneutronΛΞ0
Ξ-
electronneutrino
0.2 0.4 0.6 0.8 1.0nB [fm-3]
T = 20 MeV PNS (YL =0.4)
T = 20 MeV NS(β)
FIG. 2: The same as Fig. 1 but for temperature T=20 MeV.
13
0.2 0.4 0.6 0.8 1.0nB [fm-3]
0.01
0.10
1.00
n i/nB
protonneutronΛΞ0
Ξ-
electronneutrino
0.2 0.4 0.6 0.8 1.0nB [fm-3]
T = 40 MeV PNS (YL = 0.4) T = 40 MeV NS(β)
FIG. 3: The same as Fig. 1 but for temperature T=40 MeV.
14
0.2 0.4 0.6 0.8 1.0nB [fm-3]
0.01
0.10
1.00
n i/nB
protonneutronΛΞ0
Ξ-
electronneutrino
0.2 0.4 0.6 0.8 1.0nB [fm-3]
T = 70 MeV PNS (YL = 0.4) T = 70 MeV NS(β)
FIG. 4: The same as Fig. 1 but for temperature T=70 MeV.
15
0 0.2 0.4 0.6 0.8 1nB [fm-3]
0
2
4
6
8
10
12
S/A
[kB]
T = 10 MeVT = 20 MeVT = 40 MeVT = 70 MeV
0 0.2 0.4 0.6 0.8 1nB [fm-3]
PNS (YL = 0.4) NS(β)
FIG. 5: Entropy per baryon as a function of baryon number density at fixed temperatures. The
case of trapped neutrinos in proto-neutron star matter (left panel) and chemically equilibrated
neutron star matter (right panel) are shown.
16
0.0 0.2 0.4 0.6 0.8 1.0nB [fm-3]
0
20
40
60
80
100
T [M
eV]
S/A = 2 kBS/A = 3 kBS/A = 4 kB
0.0 0.2 0.4 0.6 0.8 1.0nB [fm-3]
PNS (YL = 0.4) NS(β)
FIG. 6: Temperature as function of baryon number density at fixed entropy per baryon. The case
of trapped neutrinos in proto-neutron star matter (left panel) and chemically equilibrated neutron
star matter (right panel) are shown.
17
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
0.0
0.1
1.0
n i/nB
protonneutronΛΞ0
Ξ−
electronneutrino
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
NS (β)S/A=2 kB PNS (YL = 0.4) S/A=2 kB
FIG. 7: The same as Fig. 1 but for entropy per baryon S/A=2 kB.
18
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
0.0
0.1
1.0
n i/nB
protonneutronΛΞ0
Ξ−
electronneutrino
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
S/A=3 kB PNS (YL = 0.4) S/A=3 kB NS (β)
FIG. 8: The same as Fig. 1 but for entropy per baryon S/A=3 kB.
19
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
0.0
0.1
1.0
n i/nB
protonneutronΛΞ0
Ξ−
electronneutrino
0.0 0.2 0.4 0.6 0.8 1.0nB [fm
-3]
NS (β)S/A=4 kB PNS (YL = 0.4) S/A=4 kB
FIG. 9: The same as Fig. 1 but for entropy per baryon S/A=4 kB.
20
0.2 0.4 0.6 0.8 1.0nB [fm-3]
0.0
1.0
2.0
3.0
4.0
Pres
sure
[fm
-4]
T = 10 MeVT = 20 MeVT = 40 MeVT = 70 MeVT = 10 MeV nuclT = 20 MeV nuclT = 40 MeV nuclT = 70 MeV nucl
0.2 0.4 0.6 0.8 1.0nB [fm-3]
NS(β)PNS (YL = 0.4)
FIG. 10: Pressure as function of baryon number density at fixed temperatures with trapped
neutrinos (left panel) and in chemical equilibrium (right panel). Results for matter containing only
nucleons and leptons are also shown.
21
0.0 0.2 0.4 0.6 0.8 1.0nb [fm
-3]]
0.0
1.0
2.0
3.0
4.0
Pres
sure
[fm
-4]
S/A = 2 kBS/A = 3 kBS/A = 4 kBS/A = 2 kB npS/A = 3 kB npS/A = 4 kB np
0.0 0.2 0.4 0.6 0.8 1.0nB [fm-3]
S/A = 2 kBS/A = 3 kBS/A = 4 kBS/A = 2 kB npS/A = 3 kB npS/A = 4 kB np
PNS (YL = 0.4) NS(β)
FIG. 11: The same as Fig. 10 but for fixed entropy.
22
10 20 30 40 50R [km]
0.0
0.5
1.0
1.5
2.0
2.5
Mg /[
Mso
lar]
S/A = 2 kBS/A = 3 kBS/A = 4 kB
10 20 30 40 50R [km]
T = 0 MeV
PNS (YL = 0.4] NS(β)
FIG. 12: The gravitational mass - radius diagram is shown for families of stars with fixed entropy
per baryon and trapped neutrinos (left panel) and in chemical equilibrium (right panel). The result
for a cold neutron star is also shown.
23