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