# Proto-neutron stars with quark cores

Adamu Issifu  0000-0002-2843-835X<EMAIL_ADDRESS>Instituto Tecnológico
de Aeronáutica,
CEP 12.228-900, São José dos Campos, SP, Brazil  Débora P. Menezes
0000-0003-0730-6689<EMAIL_ADDRESS>Departamento de Física, CFM -
Universidade Federal de Santa Catarina;
C.P. 476, CEP 88.040-900, Florianópolis, SC, Brazil.  Zeinab Rezaei
0000-0002-0646-0425<EMAIL_ADDRESS>Department of Physics, School of
Science, Shiraz University, Shiraz 71454, Iran.  Biruni Observatory, School of
Science, Shiraz University, Shiraz 71454, Iran.  Tobias Frederico
0000-0002-5497-5490<EMAIL_ADDRESS>Instituto Tecnológico de Aeronáutica,
CEP 12.228-900, São José dos Campos, SP, Brazil

###### Abstract

This work investigates the evolution of proto-neutron stars (PNSs) from birth
as neutrino-rich objects to maturity as cold-catalyzed neutrino-poor objects
with nucleonic and non-nucleonic degrees of freedom. The focus is on the
star’s core where the nucleons, hyperons, and the $\Delta$-isobars are
expected to dissolve into a "soup" of deconfined quarks, at higher baryon
densities, to establish a possible hadron-quark phase transition. We
separately calculate the nuclear equations of state (EoS) for the hadronic
matter (composed of all the baryon octet and $\Delta$-isobars) and the strange
quark matter (SQM) under the same thermodynamic conditions characteristic of
PNS and proto-strange star (PSS) evolution and construct the hybrid EoS using
Maxwell’s construction. The study allows us to determine the hadron-quark
phase transitions along the evolution lines of the star. We observed a phase
transition from hadronic matter to quark matter (QM) phase when the neutrinos
have completely escaped from the star’s core. The EoSs utilized are
constrained to meet the $2\,\rm M_{\odot}$ threshold in accordance with the
observational data.

##  I Introduction

Core-collapse supernova is a suitable laboratory for probing the emission of
light particles with masses within $\lesssim 100\,\rm MeV$ RAFFELT19901 ;
Raffelt:1987yt . For instance, the observation of neutrino burst in SN 1987A
event Kamiokande-II:1987idp ; Hirata:1988ad ; PhysRevLett.58.1494  is a clear
case in point. This observation set a milestone in particle physics leading to
a further constraint on the properties of neutrinos Kolb:1987qy ;
Raffelt:1990yu . Further analysis of the data led to constraints on particles
such as axions Turner:1987by ; Chang:2018rso ; Lucente:2022vuo , muonic bosons
Caputo:2021rux , dark photons DeRocco:2019njg , gravitons Hannestad:2001jv ,
unparticles Hannestad:2007ys , scalars mixed with Higgs boson Balaji:2022noj ,
and particles escaping from extra dimension Friedland:2007yj , largely
because, the emission of these particles are suspected to affect the time
duration of the neutrino burst. Comparatively, the most explored among these
particles is the quantum chromodynamic (QCD) axions otherwise known as the
axion-like-particles in the literature Bharucha:2022lty ; Lella:2022uwi .
Additionally, the emergence of data from the first authoritative neutron star
(NS) binary merger led to the observation of gravitational wave in the event
GW170817 LIGOScientific:2017vwq ; LIGOScientific:2018cki , which with the
additional data from the Neutron Star Interior Composition Explorer (NICER)
observatory Riley:2019yda ; Miller:2019cac ; Riley:2021pdl ; Miller:2021qha
have lead to strong constraints on cold EoS that governs the structure of NSs.
These constraints clearly define the mass threshold within which neutron stars
can be explored and vital boundaries on the NS radii. Also, they have ruled
out many phenomenological models for EoSs (see e.g. Akmal:1998cf ) that do not
satisfy these new constraints. Consequently, the focus has been directed to
the EoSs that satisfy these constraints. These EoSs are being confronted with
the current observational data to narrow down to more realistic EoSs for
multipurpose applications in the future (See a review of relativistic mean
field models in Menezes:2021jmw , and other forms of EoSs in
CompOSECoreTeam:2022ddl ).

The main ingredients for exploring high energy astrophysical phenomena such as
core-collapse supernovae Burrows:2012ew ; Janka:2016fox , PNSs Prakash:1996xs
; Janka:2006fh  and binary compact star mergers involving at least one NS
Baiotti:2016qnr ; Shibata:2019wef ; Radice:2020ddv  are relationships between
pressure ($P$), energy density ($\varepsilon$), baryon density ($n_{B}$) and
temperature ($T$). The connection between these quantities and the knowledge
of matter composition gives rise to the EoS used in exploring PNSs
Lattimer:2015nhk ; Oertel:2016bki . There is a growing interest in recent
years in these astrophysical phenomena motivating several studies intended to
provide a precise description of matter under extreme conditions of
temperature and density. Owing to the high temperature and density variations
in the stellar core during the supernova explosion and binary merger remnants
Fischer:2013eka ; Perego:2019adq , the determination of their EoSs requires
theoretical and numerical efforts.

Another phenomenon of interest is the appearance of non-nucleonic degrees of
freedom in the core of compact objects. The leading non-nucleonic degree of
freedom theoretically favored to be present in the stellar core has long been
conjectured to be the quark matter Blaschke:2018mqw ; Annala:2019puf . The
major doubt remains to be established, at what stage does the quark core begin
to appear in the stellar matter during its evolution? This is one of the many
questions we will attempt to answer with this contribution. In
PhysRevC.100.015803 , the authors attempt to answer a similar question using a
different model while, in Shao:2011nu , the authors undertake a similar study
considering nucleons and the Gibbs construction of hybrid stars, which differs
from the approach we intend to use in this work. However, in
PhysRevC.100.015803 , they concluded that there are no free quarks in hot
stars within their model framework. The aim is to go beyond the usual zero-
temperature hybrid neutron stars that have been widely investigated in the
literature Issifu:2023ovi ; Karimi:2022kcx ; Cierniak:2021knt  to include
thermodynamic conditions relevant for hot QM Issifu:2023qoo  phase transition
during the stellar evolution.

The deconfinement phase of quarks and gluons is believed to exist at the two
extreme ends on the QCD phase diagram depending on the magnitude of the
temperature and the chemical potential. A smooth crossover is observed in
lattice QCD calculations at high temperature and low chemical potential
($T\gg\mu$) giving rise to a phase transition from a nuclear matter to quark
matter phase Philipsen:2012nu ; Borsanyi:2013bia ; Bellwied:2015rza . On the
other hand, as the temperature decreases and the chemical potential increases
($T\ll\mu$) another phase transition (perhaps of first order Fukushima:2010bq
; Most:2018eaw ) from hadronic to quark matter is observed. The former is
likely to be the case in the early universe and the latter is the phenomenon
believed to occur in the inner core of (proto-) neutron stars Roark:2018uls
or merger remnants. The region between these two extreme ends can be explored
through relativistic heavy-ion collision experiments.

The (proto-) neutron stars are mostly composed of protons and neutrons at low
temperatures and high baryon densities. However, heavy and strangeness-rich
baryons such as the hyperons and even nonstrange $\Delta$-isobars are
theorized to appear deep towards the inner core Marquez:2022gmu . As the
baryon density keeps increasing and the particles get tightly parked, their
dissociation into a deconfined quark matter becomes energetically favorable.
This suggests the possibility of different phases of matter inside the NS:
hadronic matter at low baryon densities and quark matter phase at higher
baryon densities Glendenning:1992vb ; Annala:2019puf .

In this work, we investigate the possibility of a hadron-quark phase
transition in superdense matter during the evolution of NS from birth to
maturity at zero temperature. The main goal is to establish the regime at
which the deconfined quark matter begins to appear in the stellar matter
during the star’s evolution. We broadly considered two regimes: the neutrino-
trapped regime and the neutrino-transparent one after all the neutrinos have
escaped from the stellar core. The quark matter was described with the
density-dependent quark model (DDQM) Backes:2020fyw ; Wen:2005uf  using the
free model parameters determined through Bayesian inference daSilva:2023okq ,
this model was applied in studying PSSs in Issifu:2023qoo . On the other hand,
the hadronic matter is described by the relativistic mean-field approximation
model using the density-dependent couplings adjusted by the density-dependent
meson-nucleon coupling parameterization PhysRevC.71.024312 ; Roca-Maza:2011alv
. This model has also been widely used in studying PNSs in the literature (see
e.g. PhysRevC.100.015803 ; Issifu:2023qyi ; Raduta:2020fdn  and references
therein). In particular, we adopt the EoS calculated in Issifu:2023qyi  to aid
our current study. The two models were studied under similar thermodynamic
conditions characteristic of PNS (PSS) evolution Shao:2011nu ; Prakash:1996xs
and the hadron-quark phase transition investigated through Maxwell’s
construction of hybrid stars Han:2019bub . This approach predicts a sharp
phase transition from hadronic to quark matter phase when chemical and
mechanical equilibrium are established between both matter phases.

The hadronic matter phase is presumed to be composed of nucleons, hyperons,
and $\Delta$-isobars. Their corresponding equations of motion are solved using
an improved and extended parameterization of the baryon-meson couplings with
density-dependent constraints Lopes:2022vjx . The quark matter, on the other
hand, is composed of up ($u$), down ($d$), and strange ($s$) quarks. The
baryon density at which the SQM starts appearing in the stellar matter is also
assumed to be several times greater than the nuclear saturation density
($n_{0}=0.152\,\rm fm^{-3}$). Moreover, our EoSs satisfy the $2\,\rm
M_{\odot}$ threshold of NSs imposed by the discovery of PSR J0740+6620
NANOGrav:2019jur ; Fonseca:2021wxt , PSR J2215+5135 Linares:2018ppq , and PSR
J0952-0607 Romani:2022jhd  pulsars and the detection of the gravitational wave
in event GW170817 by LIGO/Virgo Collaborations LIGOScientific:2017vwq ;
LIGOScientific:2016aoc ; LIGOScientific:2018cki ; LIGOScientific:2018hze .

The paper is organized as follows, in Sec. II we present the model that forms
the basis of the investigations. The section is divided into various
subsections to thoroughly discuss each model and its properties. In Subsec.
II.1, we describe the hadronic model and discuss it in detail, in Subsec.
II.2, we present the general overview of the quark model and separately
discuss the zero-temperature formalism of the model in II.2.1 and the finite
temperature formalism in II.2.2. In Subsec. II.3, we present the properties of
the stellar matter used in the analysis, and in Subsec. II.4, we discuss the
procedure used in constructing the hybrid stars and how to determine the phase
transitions. We present our results in Sec. III and the final remarks in Sec.
IV.

##  II The models

This study adopts separate, hadronic, and quark matter models used to
calculate EoS under thermodynamic conditions characteristic of PNSs and PSSs
evolution, respectively. We use the relativistic model within the mean-field
approximation to describe the hadronic matter with density-dependent couplings
adjusted by the density-dependent meson-nucleon coupling (DDME2)
parameterization PhysRevC.71.024312 ; Roca-Maza:2011alv . On the other hand,
we use the DDQM daSilva:2023okq ; Peng:1999gh ; Xia:2014zaa  to describe the
quark matter. We calculate the EoSs from these two models differently and then
construct the hybrid EoSs by establishing chemical and mechanical equilibrium
between the hadronic and the quark matter phases which serves as a condition
for determining phase transition (this will be elaborated in subsequent
sections) between the two matter phases. We will briefly discuss the hadronic
and quark models separately below, before discussing how the hybrid EoS is
constructed from these two models in the subsequent sections.

###  II.1 The hadronic model

We use the relativistic model within the mean field approximation to study the
hadronic matter comprising the baryon octet, $\Delta$-isobars, leptons, and
mesons that mediate the strong interactions among the hadrons in a Lagrangian
density formalism

$\mathcal{L}_{\rm RMF}=\mathcal{L}_{H}+\mathcal{L}_{\Delta}+\mathcal{L}_{\rm
m}+\mathcal{L}_{\rm L},$ (1)

where the relativistic mean field (RMF) Lagrangian, $\mathcal{L}_{\rm RMF}$
consists of the baryon octet, $\mathcal{L}_{H}$, $\Delta$-isobars,
$\mathcal{L}_{\Delta}$, mesons, $\mathcal{L}_{\rm m}$, and leptons,
$\mathcal{L}_{\rm L}$. A Dirac-like Lagrangian is used to describe the
spin-1/2 baryon octet in the form

$\displaystyle\mathcal{L}_{H}={}$ $\displaystyle\sum_{b\in
H}\bar{\psi}_{b}\Big{[}i\gamma^{\mu}\partial_{\mu}-\gamma^{0}\big{(}g_{\omega
b}\omega_{0}+g_{\phi b}\phi_{0}+g_{\rho b}I_{3b}\rho_{03}\big{)}$
$\displaystyle-\Big{(}m_{b}-g_{\sigma b}\sigma_{0}\Big{)}\Big{]}\psi_{b},$ (2)

where $\sigma$ is a scalar meson, $\omega$ and $\phi$ are vector mesons (with
$\phi$ carrying a hidden strangeness), they are both isoscalars and
$\vec{\rho}$ is a vector-isovector meson. The subscript ‘0’ in the meson
fields in $\mathcal{L}_{H}$ indicates their mean-field version. On the other
hand, the Rarita-Schwinger–type Lagrangian is used to describe the spin-3/2
baryon decuplet in the form

$\displaystyle\mathcal{L}_{\Delta}={}$
$\displaystyle\sum_{d\in\Delta}\bar{\psi}_{d}\Big{[}i\gamma^{\mu}\partial_{\mu}-\gamma^{0}\left(g_{\omega
d}\omega_{0}+g_{\rho d}I_{3d}\rho_{03}\right)$
$\displaystyle-\left(m_{d}-g_{\sigma d}\sigma_{0}\right)\Big{]}\psi_{d},$ (3)

due to the additional vector-valued spinor component relative to the spin-1/2
Dirac spinors. That notwithstanding, it has been demonstrated in
dePaoli:2012eq  that the spin-3/2 model equations of motion can be written in
the same way as the spin-1/2 particles in the regime of the RMF approximation.
Additionally, the mesonic fields that mediate the strong interactions between
the hadronic fields are described by the mean-field Lagrangian density

$\displaystyle\mathcal{L}_{\rm
m}=-\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}+\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2},$
(4)

in which, the $m_{i}$ are the meson masses tabulated in Tab. 1. Finally, the
free leptons are introduced through the Dirac Lagrangian density for free
particles

$\mathcal{L}_{\rm
L}=\sum_{L}\bar{\psi}_{L}\left(i\gamma^{\mu}\partial_{\mu}-m_{L}\right)\psi_{L},$
(5)

where the summation index $L$ runs over all the leptons present. For a cold
catalyzed stellar matter, we consider only electrons ($e$) and muons ($\mu$)
i.e. $L\in(e,\,\mu)$ with a degeneracy of $\gamma_{L}=2$. In the case of fixed
entropy stellar matter, we consider the electron neutrino $\nu_{e}$ in
addition, with $\gamma_{L}=1$, neglecting muons (when $\nu_{e}$’s are trapped
in the stellar matter) and their corresponding neutrinos since muons only
become relevant in the stellar matter when all the electron neutrinos have
escaped from the core of the star consistent with supernova physics
Malfatti:2019tpg ; Prakash:1996xs .

Following the DDME2 parameterization PhysRevC.71.024312 , the meson-baryon
couplings are adjusted by the expression

$g_{ib}(n_{B})=g_{ib}(n_{0})a_{i}\frac{1+b_{i}(\eta+d_{i})^{2}}{1+c_{i}(\eta+d_{i})^{2}},$
(6)

where $i=\sigma,\omega,\phi$ and

$g_{\rho b}(n_{B})=g_{\rho
b}(n_{0})\exp\left[-a_{\rho}\big{(}\eta-1\big{)}\right],$ (7)

with $\eta=n_{B}/n_{0}$. The model parameters are determined by fitting to the
experimental data of bulk nuclear properties at the saturation density
($n_{0}=0.152\,{\rm fm^{-3}}$). The calculated binding energy
($E_{B}=-16.4\,{\rm MeV}$), the compressibility modulus, ($K_{0}=251.9\,{\rm
MeV}$), symmetry energy ($J=32.3\,{\rm MeV}$) and its slope ($L_{0}=51.3\,{\rm
MeV}$) also agree fairly with the recent constraints on bulk symmetric nuclear
matter properties Dutra:2014qga ; Reed:2021nqk ; Lattimer:2023rpe .

As the free parameters of the model are determined through fitting to pure
nucleonic matter, the extension to determine the meson-baryon coupling is done
relative to the nucleon couplings through the ratio $\chi_{ib}=g_{ib}/g_{iN}$
(the couplings for the nucleons at saturation density have been tabulated in
Tab. 1). There are several forms of calculating these couplings using either
SU(3) or SU(6) flavor arguments which are extensively documented in the
literature (see e.g. Glendenning:1991es ; Stone:2019blq ; Lopes:2020rqn ;
RevModPhys.38.215 ; Weissenborn:2011kb ; DOVER1984171 ; Schaffner:1993qj ).
However, we adopt the results presented in Lopes:2022vjx  where the authors
have calculated the meson-hyperon couplings and extended it to the
meson-$\Delta$-isobar couplings in a model-independent manner (see Tab. 2).
The particle potentials used in fitting these couplings are
$U_{\Lambda}=-28\,{\rm MeV}$, $U_{\Sigma}=30$ MeV, $U_{\Xi}=-4$ MeV, and
$U_{\Delta}\approx-98$ MeV. The couplings displayed on Tab. 2 correspond to
$\alpha_{v}=0.5$, the only free parameter of SU(2) and SU(3) groups used in
fixing the baryon-meson couplings as presented in Tab. II of Lopes:2022vjx
which has been recalculated taking into account the isospin projections in
Eqs. (II.1) and (II.1).

The relevant thermodynamic quantities required to calculate the EoS can be
derived from the Lagrangian densities. The baryon density can be determined as

$n_{b}=\gamma_{b}\int\frac{d^{3}k}{(2\pi)^{3}}\left[f_{b\,+}(k)-f_{b\,-}(k)\right]$
(8)

with $\gamma_{b}=2$, the degeneracy of the spin-1/2 baryon octet while $f_{\rm
b\pm}(k)$ is the Fermi–Dirac distribution function given by

$f_{b\pm}(k)=\frac{1}{1+\exp[(E_{b}\mp\mu^{\ast}_{b})/T]}$

where $E_{b}=\sqrt{k^{2}+m_{b}^{*}}$ is the single particle energy with
effective baryon mass $m_{b}^{*}$ and an effective chemical potential
$\mu_{b}^{*}$. We can retrieve the relation for the $\Delta$-isobars by
interchanging $b\leftrightarrow d$, here, the $E_{d}=\sqrt{k^{2}+m_{d}^{*}}$
with $\gamma_{d}=4$. The effective chemical potentials can be expressed as

$\displaystyle\mu_{b,d}^{\ast}$
$\displaystyle=\mu_{b,d}-g_{\omega{b,d}}\omega_{0}-g_{\rho{b,d}}I_{3{b,d}}\rho_{03}-g_{\phi{b}}\phi_{0}-\Sigma^{r},$
(9)

where $\Sigma^{r}$ is the rearrangement term arising from the thermodynamic
consistency of the model. Its explicit form reads:

$\displaystyle\Sigma^{r}={}$ $\displaystyle\sum_{b}\Bigg{[}\frac{\partial
g_{\omega b}}{\partial n_{b}}\omega_{0}n_{b}+\frac{\partial g_{\rho
b}}{\partial n_{b}}\rho_{03}I_{3b}n_{b}+\frac{\partial g_{\phi b}}{\partial
n_{b}}\phi_{0}n_{b}$ $\displaystyle-\frac{\partial g_{\sigma b}}{\partial
n_{b}}\sigma_{0}n_{b}^{s}+b\leftrightarrow d\Bigg{]}.$ (10)

The effective masses are given by

$m_{b}^{\ast}=m_{b}-g_{\sigma b}\sigma_{0},\quad\quad
m_{d}^{\ast}=m_{d}-g_{\sigma d}\sigma_{0},$ (11)

with a scalar density

$n_{b}^{s}=\gamma_{b}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{m^{\ast}_{b}}{E_{b}}\left[f_{b\,+}(k)+f_{b\,-}(k)\right],$
(12)

similar expressions can be obtained for the $\Delta$-isobars by interchanging
$b$ and $d$ with the appropriate degeneracy. It is important to note that the
expressions in Eqs. (8) and (11) can be used to calculate the densities of the
leptons taking into consideration that the leptons have constant mass and
chemical potentials with $\gamma_{L}$ degeneracy. The mean-field equations of
motion for Eqs. (II.1), (II.1) and (4) are

$\displaystyle m^{2}_{\sigma}\sigma_{0}=\sum_{b}g_{\sigma
b}n^{s}_{b}+\sum_{d}g_{\sigma d}n_{d}^{s},$ (13) $\displaystyle
m^{2}_{\omega}\omega_{0}=\sum_{b}g_{\omega b}n_{b}+\sum_{d}g_{\omega d}n_{d},$
(14) $\displaystyle m^{2}_{\phi}\phi_{0}=\sum_{b}g_{\phi b}n_{b},$ (15)
$\displaystyle m^{2}_{\rho}\rho_{03}=\sum_{b}g_{\rho
b}n_{b}I_{3b}+\sum_{d}g_{\rho d}n_{d}I_{3d}.$ (16)

The total energy density and the pressure of the system can be determined
through

$\displaystyle\varepsilon_{B}$
$\displaystyle=\varepsilon_{b}+\varepsilon_{m}+\varepsilon_{d}+\varepsilon_{L},$
(17) $\displaystyle P_{B}$ $\displaystyle=P_{b}+P_{m}+P_{d}+P_{L}+P_{r},$ (18)

where the subscripts denote matter under consideration. The kinetic
contributions from the baryon octet are

$\varepsilon_{b}=\gamma_{b}\int\frac{d^{3}k}{(2\pi)^{3}}E_{b}\left[f_{b+}(k)+f_{b-}(k)\right],$
(19)

$P_{b}=\gamma_{b}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}}{E_{b}}\left[f_{b+}(k)+f_{b-}(k)\right],$
(20)

a similar expression can be used for $\varepsilon_{d}$, $\varepsilon_{L}$,
$P_{d}$ and $P_{L}$ with the exchange of $b$ with $d$ and $L$ with the
appropriate degeneracies. The contribution from the meson fields can be
calculated from the energy-momentum tensor, leading to

$\varepsilon_{m}=\frac{m_{\sigma}^{2}}{2}\sigma_{0}^{2}+\frac{m_{\omega}^{2}}{2}\omega_{0}^{2}+\frac{m_{\phi}^{2}}{2}\phi_{0}^{2}+\frac{m_{\rho}^{2}}{2}\rho_{03}^{2},$
(21)

and

$P_{m}=-\frac{m_{\sigma}^{2}}{2}\sigma_{0}^{2}+\frac{m_{\omega}^{2}}{2}\omega_{0}^{2}+\frac{m_{\phi}^{2}}{2}\phi_{0}^{2}+\frac{m_{\rho}^{2}}{2}\rho_{03}^{2}.$
(22)

The total pressure received a correction

$P_{r}=n_{B}\Sigma^{r},$ (23)

known as the rearrangement self-energy term which occurs as a consequence of
thermodynamic consistency, which can be verified directly from

$P_{B}=n_{B}^{2}\dfrac{\partial}{\partial
n_{B}}\left(\dfrac{\varepsilon_{B}}{n_{B}}\right).$ (24)

From the above thermodynamic quantities, we can determine the free energy
expression $\mathcal{F}_{B}=\varepsilon_{B}-Ts_{B}$, where $T$ is temperature
and $s_{B}=S/n_{B}$ is the entropy per baryon given by

$s_{B}=\frac{\varepsilon_{B}+P_{B}-\sum_{b}\mu_{b}n_{b}-\sum_{d}\mu_{d}n_{d}}{T}.$
(25)

meson($i$) | $m_{i}(\text{MeV})$ | $a_{i}$ | $b_{i}$ | $c_{i}$ | $d_{i}$ | $g_{iN}(n_{0})$
---|---|---|---|---|---|---
$\sigma$ | 550.1238 | 1.3881 | 1.0943 | 1.7057 | 0.4421 | 10.5396
$\omega$ | 783 | 1.3892 | 0.9240 | 1.4620 | 0.4775 | 13.0189
$\rho$ | 763 | 0.5647 | — | — | — | 7.3672
Table 1: DDME2 parameters. b | $\chi_{\omega b}$ | $\chi_{\sigma b}$ | $\chi_{\rho b}$ | $\chi_{\phi b}$
---|---|---|---|---
$\Lambda$ | 0.714 | 0.650 | 0 | -0.808
$\Sigma^{0}$ | 1 | 0.735 | 0 | -0.404
$\Sigma^{-}$, $\Sigma^{+}$ | 1 | 0.735 | 0.5 | -0.404
$\Xi^{-}$, $\Xi^{0}$ | 0.571 | 0.476 | 0 | -0.606
$\Delta^{-}$, $\Delta^{0}$, $\Delta^{+}$, $\Delta^{++}$ | 1.285 | 1.283 | 1 | 0
Table 2: The ratio of the baryon coupling to the corresponding nucleon
coupling for hyperons and $\Delta$s.

###  II.2 The quark Model

The DDQM model is constructed through the QCD Hamiltonian density

$H_{\rm QCD}=H_{k}+\sum_{q=u,\,d,\,s}m_{i0}\bar{q}q+H_{I},$ (26)

where $H_{k}$ is the kinetic term, $q(\bar{q})$ are the quark fields, and
$m_{i0}$ is the bare quark mass together with the equivalent Hamiltonian
density

$H_{\rm eqv}=H_{k}+\sum_{q=u,\,d,\,s}m_{i}\bar{q}q.$ (27)

To maintain the constant energy of the free quark system, the equivalent quark
mass ($m_{i}$) term must take the form

$m_{i}=m_{i0}+m_{I},$ (28)

with $m_{I}$ being associated with the interaction term. To determine $m_{i}$,
we ensure that both $H_{\rm QCD}$ and $H_{\rm eqv}$ reproduce the same
eigenvalues for the same eigenstates, $|n_{B}\rangle$, therefore

$\langle n_{B}|H_{\rm QCD}|n_{B}\rangle=\langle n_{B}|H_{\rm
eqv}|n_{B}\rangle,$ (29)

must be satisfied. Additionally, taking the vacuum expectation values,
$|0\rangle$, of both expressions and taking the difference, yields

$m_{i}=m_{i0}+\dfrac{\langle H_{I}\rangle_{n_{B}}-\langle
H_{I}\rangle_{0}}{\sum_{q}\Big{[}\langle\bar{q}q\rangle_{n_{B}}-\langle\bar{q}q\rangle_{0}\Big{]}}={m_{i0}+m_{I}},$
(30)

with $\langle n_{B}|H_{I}|n_{B}\rangle=\langle H_{I}\rangle_{n_{B}}$, $\langle
n_{B}|q\bar{q}|n_{B}\rangle=\langle\bar{q}q\rangle_{n_{B}}$ and the vacuum
expectation values are expressed in the same way with subscript ‘0’. The
second term in Eq.(30) is precisely the $m_{I}$ term. Since $m_{I}$ depends on
the interaction energy density and the quark condensate, it is flavor
independent. The interaction energy density can be expressed in terms of the
average inter-particle confining potential and the baryon density as

$\langle H_{I}\rangle_{n_{b}}-\langle H_{I}\rangle_{0}=3n_{B}V_{c}(r_{*}),$
(31)

where factor 3, represents the three quark flavors present, $V_{c}(r_{*})$ is
the confining potential and $r_{*}$ is the average inter-particle separation
distance. Also, from the model-independent quark condensate in nuclear matter
PhysRevLett.67.961 ; PhysRevC.45.1881 , we can express the denominator as

$\dfrac{\langle\bar{q}q\rangle_{n_{B}}}{\langle\bar{q}q\rangle_{0}}=1-\dfrac{n_{B}}{n_{B}^{*}},$
(32)

where $n_{B}^{*}$ is a constant associated with the pion mass $M_{\pi}$, its
decay constant $f_{\pi}$, and the pion-nucleon sigma term $\sigma_{N}$ in a
form,

$n^{*}_{B}=\dfrac{M_{\pi}^{2}f_{\pi}^{2}}{\sigma_{N}}.$ (33)

Therefore, the general structure of (30) becomes

$m_{i}=m_{i0}+\dfrac{D}{(n_{B})^{z}},$ (34)

with

$D\sim-\dfrac{3\sigma_{0}n_{B}^{*}}{\sum_{q}\langle\bar{q}q\rangle_{0}},$ (35)

where $\sigma_{0}$ is the proportionality constant of the confining potential
representing the string tension. The last term in (34) is precisely the
$m_{I}$ satisfying the condition

$\lim_{n_{B}\rightarrow 0}m_{I}\rightarrow\infty;\qquad{\text{color
confinement condition}}.$ (36)

The potential $V_{c}(r_{*})=\sigma r^{n}$ also satisfies the same conditions
considering that $r_{*}^{n}\sim 1/n_{B}^{z}$, where $n$ and $z$ are arbitrary
constants that are determined through dimensional analysis. To obtain quark
stars with masses comparable to the observed neutron star (NS) mass threshold
NANOGrav:2019jur ; Fonseca:2021wxt , we employ the famous Cornell potential
PhysRevLett.34.369 ; Bali:2000gf , given by
$V_{c}(r_{*})=-(\beta/r_{*})+\,\sigma r_{*}$, which incorporates both the
confinement and the asymptotic freedom behavior determined from QCD lattice
calculations. Here, an additional term is included in (34), where $\sigma$
represents the string tension and $\beta$ is a constant representing the
deconfinement strength. Hence, the new expression becomes

$m_{i}=m_{i0}+\dfrac{D}{(n_{B})^{z}}+C(n_{B})^{z},$ (37)

the additional term will contribute to repulsive pressure;

$P_{i}\sim(n_{B})^{2}\dfrac{\partial m_{i}(n_{B})}{\partial n_{B}},$ (38)

to balance the attractive pressure generated by the confining term to hold the
star from collapsing as it reaches its maximum possible mass. Hereafter, we
shall choose $z=1/3$ in conformity with the lattice QCD (LQCD) confining
potential, $V_{c}(r_{*})=\sigma r_{*}$ as well as the Cornell potential for
confining heavy quarks, here $n=1$ as well. That notwithstanding, this cubic
root mass scaling law has been widely applied in studying quark stars using
the DDQM by several authors (see e.g. Peng:1999gh ; Xia:2014zaa ;
daSilva:2023okq  and references therein).

To introduce temperature into the system, we follow the LQCD approach, where
the temperature is introduced through the string tension Bicudo:2010hg ;
Kaczmarek:1999mm ; Digal:2003jc . In this case, the string tension decreases
with increasing temperature and breaks at the critical temperature ($T_{C}$)
where the particles become asymptotically free. Thus, the normalized string
tension takes the form

$\dfrac{\sigma(T)}{\sigma_{0}}=\left[1-\dfrac{T^{2}}{T_{C}^{2}}\right]^{\alpha},$
(39)

where $\alpha$ is an arbitrary constant whose value depends on the model under
consideration. At $T=0$, we recover the $\sigma(T=0)=\sigma_{0}$ and it
vanishes at $T=T_{C}$. For the sake of the current work, we will adopt
$\alpha=1/2$ (see e.g. Philipsen:2007rj ; Ilgenfritz:2017kkp ; Bicudo:2010hg
and references therein). Therefore, the full expression for the equivalent
mass term becomes

$m_{i}=m_{i0}+\dfrac{D}{n_{B}^{1/3}}\left[1-\dfrac{T^{2}}{T_{C}^{2}}\right]^{\frac{1}{2}}+Cn_{B}^{1/3}\left[1-\dfrac{T^{2}}{T_{C}^{2}}\right]^{-\frac{1}{2}},$
(40)

similar to the mechanism introduced in Refs. Karsch:2000kv ; Laermann:2003cv .
This approach has been adopted by several authors (see e.g. Wen:2005uf ;
Chen:2021fdj ; Chen:2023rza ; Chu:2017huf ; Issifu:2023qoo ; Issifu:2024zvq )
to introduce temperature into the equivalent mass model arguing that $D$
(confining term) decreases with temperature, on the other hand, $C$ (the
asymptotic freedom term) increases with temperature to stabilize the mass of
the star. The quark masses adopted for our calculations are $m_{u}=2.16\,\rm
MeV$, $m_{d}=4.67\,\rm MeV$, and $m_{s}=93.4\,\rm MeV$, as documented in
ReviewOfParticlePhysics . The free model parameters for the quark model are
$\sqrt{D}=128.4\,\rm MeV$, and $C=0.8$. These values represent a slight
adjustment from the findings reported in daSilva:2023okq , aimed at rendering
the quark stars marginally unstable, preventing the rapid transformation of
hybrid stars into quark stars. The next step is to check for the thermodynamic
consistency of the mass formulas derived above.

####  II.2.1 Zero Temperature Formalism

To ensure that the model is thermodynamically consistent at zero temperature,
we define the thermodynamic free energy ($F$) where the free particle mass is
replaced with $m_{i}(n_{b})$ and the real chemical potential $\mu_{i}$
replaced with an effective chemical potential $\mu^{*}_{i}$ therefore,

$F=\Omega_{0}\left(\\{\mu^{*}_{i}\\},\\{m_{i}\\}\right)+\sum_{i}\mu^{*}_{i}n_{i},$
(41)

where $\Omega_{0}$ is the thermodynamic potential given explicitly by

$\Omega_{0}=-\sum_{i}\frac{\gamma_{i}}{24\pi^{2}}\left[\mu^{*}_{i}\nu_{i}\left(\nu_{i}^{2}-\frac{3}{2}m_{i}^{2}\right)+\frac{3}{2}m_{i}^{4}\ln{\frac{\mu^{*}_{i}+\nu_{i}}{m_{i}}}\right],$
(42)

with $\gamma_{i}=6~{}(3~{}{\rm colors}\times 2~{}{\rm spins})$ the
degeneracies of the quarks. The Fermi momenta in terms of $\mu_{i}^{*}$ is
given as

$\nu_{i}=\sqrt{\mu^{*2}_{i}-m_{i}^{2}}.$ (43)

It follows that the number density of the quarks can be expressed as

$n_{i}=\dfrac{\gamma_{i}}{2\pi^{2}}\int_{0}^{\nu_{i}}p^{2}dp=\dfrac{\gamma_{i}\nu_{i}^{3}}{6\pi^{2}},$
(44)

with $p$ the momenta of the quarks, the $\mu_{i}$ in terms of the
$\mu_{i}^{*}$ is also given as

$\mu_{i}=\mu^{*}_{i}+\sum_{j}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial n_{i}}\equiv\mu_{i}^{*}-\mu_{I},$ (45)

where $\mu_{I}$ is the interacting chemical potential that links $\mu_{i}$ and
$\mu^{*}_{i}$ together. From the above expressions, we can determine the
energy density ($\varepsilon$) for the system of quarks and its pressure ($P$)
through

$\varepsilon=\Omega_{0}-\sum_{i}\mu_{i}^{*}\frac{\partial\Omega_{0}}{\partial\mu_{i}^{*}},$
(46)

and

$P=-\Omega_{0}+\sum_{i,j}\frac{\partial\Omega_{0}}{\partial
m_{j}}n_{i}\frac{\partial m_{j}}{\partial n_{i}}.$ (47)

respectively. The interested reader can check Refs daSilva:2023okq ;
Backes:2020fyw ; Peng:1999gh  for more detailed derivations.

####  II.2.2 Finite Temperature Formalism

At a finite temperature, the system of quark matter (QM) depends on the
$T,\,n_{i}$ and the volume ($V$) of the system, consequently, the
thermodynamic free energy of the system becomes

$\displaystyle F$ $\displaystyle=F(T,V,\\{n_{i}\\},\\{m_{i}\\})$
$\displaystyle=\Omega_{0}(T,V,\\{\mu^{*}_{i}\\},\\{m_{i}\\})+\sum_{i=u,d,s}\mu^{*}_{i}n_{i}.$
(48)

In this expression, the $\Omega_{0}$ is connected to the independent state
variables through

$n_{i}=-\dfrac{\partial}{\partial\mu^{*}_{i}}\Omega_{0}(T,V,\\{\mu^{*}_{i}\\},\\{m_{i}\\}).$
(49)

The explicit expression for the $\Omega_{0}$ and $n_{i}$ in terms of particle
and anti-particle contributions derived from (II.2.2), are

$\displaystyle\Omega^{\pm}_{0}$
$\displaystyle=-\sum_{i}\dfrac{\gamma_{i}T}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\bigg{[}\ln[1+e^{-(\epsilon_{i}-\mu_{i}^{*})/T}]$
$\displaystyle+\ln[1+e^{-(\epsilon_{i}+\mu_{i}^{*})/T}]\bigg{]},$ (50)

and

$\displaystyle
n_{i}^{\pm}=-\dfrac{\partial\Omega_{0}^{\pm}}{\partial\mu^{*}_{i}}$
$\displaystyle=\sum_{i}\dfrac{\gamma_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\bigg{[}\dfrac{1}{1+e^{(\epsilon_{i}-\mu_{i}^{*})/T}}$
$\displaystyle-\dfrac{1}{1+e^{(\epsilon_{i}+\mu_{i}^{*})/T}}\bigg{]},$ (51)

with $\epsilon_{i}=\sqrt{(p^{2}+m_{i}^{2})}$, the single particle energy
respectively. The $\mu_{i}$ is expressed in the same way as (45) with its
temperature dependence introduced through $\Omega_{0}(m_{i},\mu^{*},T)$ and
$m_{i}(n_{b},T)$. The other thermodynamic quantities such as the entropy
density ($S$),

$S=-\dfrac{\partial\Omega_{0}}{\partial T}-\sum_{i}\dfrac{\partial
m_{i}}{\partial T}\dfrac{\partial\Omega_{0}}{\partial m_{i}},$ (52)

the energy density

$\varepsilon=\Omega_{0}+\sum_{i}\mu_{i}^{*}n_{i}-T\dfrac{\partial\Omega_{0}}{\partial
T}-T\sum_{i}\dfrac{\partial m_{i}}{\partial
T}\dfrac{\partial\Omega_{0}}{\partial m_{i}},$ (53)

and the pressure

$\displaystyle
P=-\Omega_{0}+\sum_{i,j}n_{i}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial n_{i}},$ (54)

can all be derived from (II.2.2) in terms of $T$ and $n_{i}$. The volume term
that appears in (II.2.2) does not affect the pressure considering an
infinitely large system of quark matter Wen:2005uf ; Peng:2000ff ; Xia:2014zaa
; Issifu:2023qoo ; Issifu:2024zvq .

###  II.3 Stellar Matter Properties

The hadronic matter comprises nucleons, hyperons, and $\Delta$-isobars,
whereas the quark matter consists of $u$, $d$, and $s$ quarks. In both
scenarios, within the neutrino-trapped region, the prevalent leptons are
electrons and their corresponding electron neutrinos ($\nu_{e}$), as depicted
in the initial and intermediate stages of Fig. 2. The appearance of muons
becomes relevant only after the star has transitioned to a neutrino-
transparent state by supernova physics Malfatti:2019tpg ; Prakash:1996xs ;
Weber:2019xvv . Moreover, the presence of $\tau$-leptons is disregarded due to
their substantial mass. We develop numerical codes to independently compute
the EoS for both hadronic and quark matter phases and their associated
temperature profiles while maintaining fixed values for $S/n_{B}$ and
$Y_{L,e}$. These thermodynamic conditions are selected to accurately describe
the evolution of NSs, drawing from relevant literature on the subject
Raduta:2020fdn ; Oertel:2016xsn ; Sedrakian:2022kgj ; PhysRevC.100.015803 ;
Shao:2011nu ; Chen:2021edy ; Kumari:2021tik . In both scenarios, we examine
$\beta$-equilibrated matter. Consequently, the hadronic matter involves the
utilization of the chemical equilibrium conditions:

$\displaystyle\mu_{\Lambda}=\mu_{\Sigma^{0}}=\mu_{\Xi^{0}}=\mu_{\Delta^{0}}=\mu_{n}=\mu_{B},$
(55)
$\displaystyle\mu_{\Sigma^{-}}=\mu_{\Xi^{-}}=\mu_{\Delta^{-}}=\mu_{B}-\mu_{Q},$
(56) $\displaystyle\mu_{\Sigma^{+}}=\mu_{\Delta^{+}}=\mu_{p}=\mu_{B}+\mu_{Q},$
(57) $\displaystyle\mu_{\Delta^{++}}=\mu_{B}+2\mu_{Q},$ (58)

and similar expressions can be found in Raduta:2021xiz ; Raduta:2020fdn ;
Issifu:2023qyi  where the authors consider hyperons and $\Delta$-isobars in
the stellar matter. The baryon chemical potential is denoted as $\mu_{B}$ and
the charged chemical potential is defined as $\mu_{Q}=\mu_{p}-\mu_{n}$, where
the subscripts correspond to the individual particles present. In the context
of neutrino-trapped matter, $\mu_{Q}$ assumes the following expression

$\mu_{Q}=\mu_{\nu l}-\mu_{l},$ (59)

and in the neutrino-transparent regime, it takes on the following form

$\mu_{Q}=-\mu_{l},$ (60)

where $l$ is the leptons present in that regime. On the other hand, the quark
matter was studied following chemical potential expressions

$\mu^{*}_{d}=\mu_{s}^{*}=\mu_{u}^{*}+\mu_{e}-\mu_{\nu_{e}},\quad\text{neutrino
trapped matter}\quad$ (61)

and

$\mu^{*}_{d}=\mu_{s}^{*}=\mu_{u}^{*}+\mu_{e},\quad\text{neutrino transparent
matter}\quad$ (62)

here, the subscripts $u,\,d,\,s$ represents up, down and strange quarks. Also,
charge neutrality, baryon number conservation, and the lepton number
conservation (in the neutrino-trapped region) were separately ensured for both
the hadronic and the quark matter phases (see the explicit expressions in
Refs. Issifu:2023qyi ; Issifu:2023qoo ). The lepton numbers are conserved
through

$Y_{L,l}=\dfrac{n_{l}+n_{\nu l}}{n_{B}},$ (63)

where $Y_{L,e}=Y_{e}+Y_{\nu e}$ and $Y_{L,\mu}=Y_{\mu}+Y_{\nu_{\mu}}\approx
0$, following the assumptions previously stated based on supernova physics.
The values of $Y_{L,e}$ are primarily influenced by the efficiency of the
electron capture reaction occurring during the initial phases of proto-neutron
star formation. Additionally, $Y_{L,\mu}\approx 0$ indicates the absence of
muons in the stellar matter during the period of neutrino trapping
Prakash:1996xs . The particle distribution in the stellar matter was computed
utilizing

$Y_{i}=\dfrac{n_{i}}{\sum_{i}n_{i}},$ (64)

where the summation index $i$ runs over all the individual baryons and the
quarks in each matter phase.

###  II.4 Construction of the hybrid stars

Figure 1: The graphs depict the pressure as a function of the baryon chemical
potential. Solid lines denote nucleons, dot-dot lines signify nucleons
combined with hyperons, and dash-dot lines represent nucleons in addition to
hyperons and $\Delta$-isobars. The mechanical and chemical equilibrium points,
denoting the phase transition, are identified by the intersection between the
quark matter (depicted in black) and hadronic matter curves and hadronic
matter curves. In the left panel, depicting neutrino-rich stellar matter, the
quark matter curves do not intersect the hadronic matter curves, indicating
the absence of a phase transition at this stage. Conversely, in the right
panel, where the stellar matter is neutrino-poor, both curves intersect,
indicating a phase transition occurring at this stage.

There are generally two ways hybrid stars are constructed in the literature,
the Gibbs construction Shao:2011nu  which involves a mixed phase and a smooth
transition, and the Maxwell construction which shows a sharp first-order phase
transition without a mixed phase – see Ref. Han:2019bub  for an elaborate
discussion of the two scenarios. In the microphysical realm, it has been
demonstrated that these two approaches exhibit no substantial differences
Maruyama:2007ey ; Paoli:2010kc . In the present investigation, we employ the
Maxwell construction, using the two-phase approach wherein the higher density
phase is characterized by quark matter. In comparison, the lower density phase
consists of hadronic matter comprising baryon octet and $\Delta$-isobars
Issifu:2023qyi ; Sedrakian:2022kgj . In this construction, the EoS for the
hadronic phase and the quark matter phase are developed differently and the
hybrid star is constructed using chemical and mechanical equilibrium
conditions i.e.

$P_{H}=P_{Q}=P_{c};\quad\text{mechanical equilibrium}\quad$ (65)

and

$\mu_{H}=\mu_{Q}=\mu_{c};\quad\text{chemical equilibrium}\quad$ (66)

where $P_{H}$ and $\mu_{H}$ are the pressure and the chemical potential of the
hadronic phase and $P_{Q}$ and $\mu_{Q}$ are the pressure and the chemical
potential of the quark phase. The points of equality between the pressures
($P_{c}$) and the chemical potentials ($\mu_{c}$) are referred to as the
critical pressure and critical chemical potential respectively (the
equilibrium points are shown in Fig. 1 with red dots). Marking the point of
the phase transition from the hadronic to the quark matter phase
Issifu:2023ovi ; Lopes:2021jpm .

Even though the existence of hybrid stars was theorized decades ago, the
observation of binary NS merger leading to the detection of the gravitational
wave in event GW170817 (GW) LIGOScientific:2017vwq  and the associated
$\gamma$-ray burst, GRB 170817A (GRB) LIGOScientific:2017zic ;
Goldstein:2017mmi  reinforce the belief of its existence. In these instances,
the relative time difference between GW and GRB, along with the measured speed
of sound of the ejecta from the core of the remnants, offer guidance for
categorizing matter phases within NSs. These observations reinforce the
theoretical plausibility of a quark core, which could give rise to hybrid NSs.
Indeed, the investigation of hybrid stars remains model-dependent, primarily
due to the absence of experimental determination or direct observation by any
observatory as of the current state. However, despite this limitation,
numerous studies exploring the existence of hybrid stars can be found in the
literature (see e.g. Annala:2019puf ; Karimi:2022kcx ; Cierniak:2021knt ;
Contrera:2022tqh ; Alford:2004pf ; Clevinger:2022xzl ).

From the previous sections discussing the hadronic and quark matter models, we
compute the EoS for both matter phases and plotted $P_{H}(\mu_{H})$ and
$P_{Q}(\mu_{Q})$ in Fig.1 to identify chemical and mechanical equilibrium
points, illustrated with red dots in Fig.1. In the right panel, we find that
the $P_{H}(\mu_{H})$ and $P_{Q}(\mu_{Q})$ do not cross each other due to the
absence of phase transition in this regime when neutrinos are present in the
stellar matter, so no equilibrium is established between the two matter
phases. However, in the right panel, the curves cross each other indicating a
phase transition from hadronic to quark matter phase.

##  III Results and Analysis

Stellar properties
---
Particle content |  S$/$nB |  YL,e | $M[{\rm M}_{\odot}]$ | $M_{b}[{\rm M}_{\odot}]$ | $R[{\rm km}]$ | $\varepsilon_{0}[{\rm MeVfm^{-3}}]$ | $n_{c}[n_{0}]$ | $T_{co}[{\rm MeV}]$ | $T_{C}[{\rm MeV}]$ | $\mu_{c}[{\rm MeV}]$ | $P_{c}[{\rm MeVfm^{-3}}]$
| N
---
NH
NH$\Delta$
1 | $0.4$ |  | $2.44$
---
$2.33$
2.33
| $2.81$
---
$2.65$
2.65
| $12.33$
---
$12.38$
12.39
| $1070$
---
$1078$
1072
| 5.26
---
5.36
5.36
| 30.08
---
21.16
21.13
| —
---
—
—
| —
---
—
—
| —
---
—
—
| N
---
NH
NH$\Delta$
2 | $0.2$ |  | $2.49$
---
$2.28$
2.29
| $2.89$
---
$2.61$
2.62
| $12.83$
---
$12.56$
12.52
| $1007$
---
$1070$
1073
| 5.00
---
5.42
5.42
| 71.83
---
46.42
45.41
| —
---
—
—
| —
---
—
—
| —
---
—
—
| N
---
NH
NH$\Delta$
2 | $Y_{\nu e}=0$ |  | $2.43$
---
$2.22$
$2.22$
| $2.83$
---
$2.54$
$2.55$
| $13.66$
---
$13.03$
$12.86$
| $1068$
---
$1084$
$1107$
| 5.59
---
5.56
5.65
| 23.08
---
51.22
49.41
| 62.61
---
48.90
47.30
| 1470
---
1510
1520
| 238
---
280
290
| N
---
NH
NH$\Delta$
$T=0$ | $Y_{\nu e}=0$ |  | $2.26$
---
$2.21$
$2.21$
| $2.67$
---
$2.60$
$2.61$
| $13.04$
---
$12.65$
$12.55$
| $611$
---
$876$
$1021$
| 3.56
---
4.87
5.46
| —
---
—
—
| —
---
—
—
| 1425
---
1476
1486
| 177
---
222
230
Fixed $M_{b}$ and the corresponding stellar matter properties
Particle content |  S$/$nB |  YL,e | $M_{b}[{\rm M}_{\odot}]$ |  M$[{\rm M}_{\odot}]$ | $R[{\rm km}]$ | $\varepsilon_{0}[{\rm MeVfm^{-3}}]$ | $n_{c}[n_{0}]$ | $T_{c}[{\rm MeV}]$ | $T_{C}[{\rm MeV}]$ | $\mu_{c}[{\rm MeV}]$ | $P_{c}[{\rm MeVfm^{-3}}]$
| N
---
NH
NH$\Delta$
1 | $0.4$ |  | $2.45(1.57)$
---
| $2.19(1.48)$
---
$2.18(1.47)$
2.19(1.47)
| $13.65(14.55)$
---
$13.49(14.55)$
13.46(14.545)
| $569.34(348.42)$
---
$631.04(348.38)$
638.30(348.38)
| 3.30(2.19)
---
3.59(2.19)
3.62(2.19)
| 24.21(18.71)
---
20.47(18.26)
20.24(18.15)
| —
---
—
—
| —
---
—
—
| —
---
—
—
| N
---
NH
NH$\Delta$
2 | $0.2$ |  | $2.45(1.57)$
---
| $2.19(1.46)$
---
$2.17(1.48)$
2.18(1.49)
| $14.44(15.57)$
---
$13.71(15.25)$
13.60(15.12)
| $501.81(301.75)$
---
$653.83(329.52)$
659.96(340.48)
| 3.01(1.93)
---
3.72(2.09)
3.76(2.16)
| 57.00(41.93)
---
43.99(38.36)
42.66(36.96)
| —
---
—
—
| —
---
—
—
| —
---
—
—
| N
---
NH
NH$\Delta$
2 | $Y_{\nu e}=0$ |  | $2.45(1.57)$
---
| $2.17(1.48)$
---
$2.16(1.46)$
2.16(1.47)
| $14.50(15.56)$
---
$13.50(15.41)$
13.36(15.027)
| $491.47(305.51)$
---
$720(333.67)$
710(354.47)
| 2.94(1.96)
---
4.08(2.06)
4.02(2.26)
| 54.89(43.27)
---
46.74(38.59)
45.68(38.25)
| —
---
—
—
| —
---
—
—
| —
---
—
—
| N
---
NH
NH$\Delta$
$T=0$ | $Y_{\nu e}=0$ |  | $2.45(1.57)$
---
| $2.10(1.42)$
---
2.10(1.41)
2.10(1.43)
| $13.22(13.22)$
---
13.00(13.21)
12.85(13.14)
| $503.68(343.21)$
---
607.57(341.77)
611.71(363)
| 3.14(2.29)
---
3.66(2.26)
3.69(2.39)
| —
---
—
—
| —
---
—
—
| —
---
—
—
| —
---
—
—
Table 3:  Here, N represents nucleons only, NH represents nucleon plus
hyperons admixture, and NH$\Delta$ represents nucleon plus hyperons plus
$\Delta$-isobars admixture. We show the maximum gravitational mass (Mmax) and
its corresponding radii ($R$), the central energy density ($\varepsilon_{0}$),
the central baryon density $n_{c}$, the core temperature ($T_{co}$) and the
critical temperature ($T_{C}$) for the evolution stages of the PNS considered.
The stars without $P_{c}$ and $\mu_{c}$ values do not undergo a phase
transition to a free quark core. Additionally, we choose two fixed baryonic
masses; ${\rm M_{b}}=1.57\,{\rm M}_{\odot}$ and ${\rm M_{b}}=2.45\,{\rm
M}_{\odot}$ and calculate the other properties of the stellar matter for the
four stages of the stellar evolution. The results are arranged such that the
star with the smaller baryonic mass is in a bracket and the larger one is out
of the bracket i.e. 2.45(1.57).

On Tab. 3, we present the results of the stellar properties determined from
our calculations. Upon examination of the table, we observe that the maximum
stellar masses generally decrease as the degrees of freedom of the stellar
matter increase, as expected. Additionally, from the neutrino-trapped region
where $Y_{L,e}$ is fixed, we note that the maximum stellar mass decreases
slightly as the star deleptonizes, except for stars composed solely of
nucleons, which exhibit no significant changes in mass which may be due to
accretion. However, during deleptonization, the stellar radii increase
alongside higher core temperatures as neutrinos diffuse from the stellar core.
For the neutrino transparent matter, on the other hand, the maximum stellar
mass and radii decrease as the star cools down and shrinks through thermal
emissions. For further clarity, we profiled two stars with fixed baryon masses
$M_{b}=1.57\,\rm M_{\odot}$ and $M_{b}=2.45\,\rm M_{\odot}$ to determine their
corresponding gravitational masses, radii, core temperature, central energy,
and baryon densities and presented them on the table.

Figure 2: The particle fraction for the PNS from birth to maturity as a cold
catalyzed hybrid NS. The upper panels shows the stages when the star is
neutrino-rich and the lower panels when it is neutrino-poor. The proto-neutron
stars undergo a phase transition from a hadronic to a quark matter phase once
all the neutrinos have escaped from the stellar core. They then continue
cooling down to form a cold hybrid NS, as depicted in the bottom panels. The
solid vertical lines in the lower panels show when the phase transition of
each particle begins and the corresponding dot-dot line shows where the phase
transition ends for each particle. The black lines represent the nucleon, the
red lines represent nucleon plus hyperons and the blue line represents the
nucleon plus hyperons plus $\Delta$-isobars.

From the snapshots of the particle distributions depicted in Fig. 2, two fates
await the supernova remnants at each stage of its evolution: it can either
form a black hole or a PNS. However, the formation of a black hole signifies
the culmination of stellar evolution. Therefore, in this study, we presume
that all conditions favored the formation of PNS Prakash:1996xs . In the first
stage ($S/n_{B}=1$ and $Y_{L,e}=0.4$), if the explosion lacks the requisite
strength to deleptonize the mantle of the star, it promptly accreates matter,
leading to the formation of a black hole. In such a scenario, the evolution of
the star is abruptly terminated, and it does not progress to the second stage
($S/n_{B}=2$ and $Y_{L,e}=0.2$). During the transition from the second to the
third stage ($S/n_{B}=2$ and $Y_{\nu_{e}}=0$), the star undergoes
deleptonization via neutrino diffusion. If neutrino loss transpires too
rapidly, it can induce softening of the EoS, thereby increasing the risk of
black hole formation, particularly if the gravitational pressure reaches
sufficiently high levels. In the third stage, once all neutrinos have escaped
from the stellar matter, the likelihood of strangeness-carrying hyperons
emerging in the stellar matter rises. If the formation of strangeness-carrying
hyperonic matter becomes excessive, it can potentially lead to the formation
of a black hole Vidana:2002rg ; Burgio2007 . This phenomenon occurs if there
is an accretion of matter at this stage.

In Fig. 2, we illustrate the particle distribution in the stellar matter
determined using (64). The particle distribution provides valuable insights
into the isospin asymmetry of the stellar matter as the star evolves
Issifu:2023qyi ; Chu:2017huf , highlighting how temperature influences the
emergence of particles during stellar evolution and its consequent impact on
the EoS Raduta:2020fdn . The isospin asymmetry of the quark matter is computed
using

$\delta_{q}=3\dfrac{n_{d}-n_{u}}{n_{d}+n_{u}},$ (67)

where $n_{3}=n_{d}-n_{u}$ represents the isospin density and
$n_{B}=(n_{d}+n_{u})/3$ denotes the baryon density for a two-flavor quark
system. Conversely, the isospin asymmetry of the hadronic matter is determined
by

$\delta_{H}=\dfrac{n_{n}-n_{p}}{n_{n}+n_{p}},$ (68)

where $n_{B}=n_{n}+n_{p}$. Along the panels, it is clear that the ratio of the
particle fraction $Y_{p}/Y_{n}$ decreases along the panel as the star evolves.
This means the isospin asymmetry increases along the evolution lines. However,
the variations in $Y_{u}/Y_{d}$ are not evident from the bottom panels, as the
quark fractions appear to remain constant at higher densities, well beyond
several times $n_{0}$. Nonetheless, as discussed in Issifu:2023qoo , it was
noted that this ratio decreases as PSSs evolve; consequently, the isospin
asymmetry $\delta_{q}$ increases with PSS evolution. Additionally, there is a
delay in the emergence of strangeness-carrying particles, primarily the
$\Lambda$ hyperon, in the low entropy ($S/n_{B}=1$ and $Y_{L,e}=0.4$,
corresponding to low temperature) neutron-rich matter and in neutrino-poor
stellar matter with $T=0$, as illustrated in the first and fourth panels,
respectively. As the matter deleptonizes and $S/n_{B}$ increases ($S/n_{B}=2$
and $Y_{L,e}=0.2$), the strangeness-carrying particles shift further towards
the low-density region. Subsequently, after all the neutrinos have escaped
from the core ($S/n_{B}=2$ and $Y_{\nu_{e}}=0$), all the strangeness-carrying
particles appear either before or at $n_{B}\approx 2n_{0}$, as evident in the
second and third panels. This establishes a relation between the appearance of
strangeness-carrying particles with the increase in temperature and entropy.
Another crucial point to note is that the presence of neutrinos hinders the
appearance of free quarks in stellar matter.

Moreover, the $\Delta^{-}$ heavy non-strange baryon prominently appears in the
stellar matter at each stage along the evolution lines, causing a reduction in
the lepton population. The most prevalent strangeness-carrying hyperon in the
stellar matter across all regimes is the $\Lambda$. At low $S/n_{B}$ (first
panel) and low temperatures (fourth panel), $\Lambda$ and $\Xi$ hyperons
emerge as the dominant strangeness-carrying particles at low $n_{B}$.
Nonetheless, at higher $S/n_{B}$, this phenomenon changes, and $\Lambda$ and
$\Sigma$ strangeness-carrying hyperons dominate at low $n_{B}$, as observed in
the second and third panels. Hence, the temperature favors the production of
$\Sigma$ particles, as evident in the second and third panels. Zooming in on
the bottom panels, we observe a phase transition from hadronic to quark matter
phase. The phase transition begins lowly immediately after the neutrinos have
completely escaped from the stellar core, at the point where the matter is
expected to be maximally heated. This is evident by the wider gray area in the
third panel compared to the fourth one. In the fourth panel, it’s evident that
the quark core expands significantly as the matter cools down to $T=0$, with
comparatively faster phase transitions. We have depicted the onset of phase
transitions with solid vertical lines and the cessation points with a
corresponding dotted vertical line. As the degrees of freedom of the stellar
matter increase, the phase transition slows down.

Figure 3: The figures display the pressure as a function of the energy
density. The right panel shows the EoSs of the stellar matter when the matter
is neutrino-rich with no phase transitions. The right panel shows both a
neutrino-free proto-strange star phase transition and a cold hybrid star EoS.

In Fig. 3, we show the EoSs for various stages of the star’s evolution. In the
left panel, we show the EoSs of both the hadronic and quark phases when
neutrinos are trapped in the stellar core. We determined from Fig. 1 that
there is no chemical and mechanical equilibrium established between both
matter phases at this regime hence, there is no hadron-quark phase transition
here. On the other hand, the curves in the right panel show the EoSs for
neutrino-transparent matter where phase transition from hadronic to quark
matter phase occurs. This is evident by the discontinuity in the energy
density. Generally, we observe that the EoSs become softer as the degrees of
freedom of the stellar matter increase as expected Ribes:2019kno ;
PhysRevC.98.045801 . This property is essential in the determination of the
structure of the corresponding stars since stiffer EoS signifies a higher
maximum gravitational mass of the star and vice versa. In the right panel, it
is noticeable that the phase transition from hadronic to quark matter phase
for the fixed entropy stars ($S/n_{B}=2,\;Y_{\nu_{e}}=0$) happens at higher
energy densities compared to the $T=0$ stars. This is attributed to the slow
nature of the phase transition in the fixed entropy stars compared to the cold
ones as discussed in Fig. 2.

In the neutrino-trapped regime, higher fixed $Y_{L,e}$ results in the increase
of the stiffness of the EoS, hence the first stage of the star’s evolution is
associated with stiffer EoSs compared to the deleptonization phase in the
second stage where the entropy increases and $Y_{L,e}$ decreases. This
correlation can be attributed to the emergence of strangeness-carrying
hyperons in the stellar matter, as depicted in Fig. 2. The increased presence
of hyperons at lower densities results in a softer EoS in this regime. In the
neutrino-transparent matter, where the hybrid stars are formed, the EoSs
become softer as the star cools down from $S/n_{B}=2$, undergoing catalyzation
at $S/n_{B}=0$, and reaches $T=0$ when a ’mature’ NS is formed.

Figure 4: Here we show the gravitational masses of the stars as a function of
radii.

The mass-radius diagram that represents the macroscopic structure of the NSs
is determined by solving the Tolman–Oppenheimer–Volkoff (TOV) equations
PhysRev.55.374 , assuming a hydrostatic spherically symmetric fluid. The
relevant equations to solve are

$\displaystyle\frac{dP(r)}{dr}$
$\displaystyle=-[\varepsilon(r)+P(r)]\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}-2M(r)r},$ (69) $\displaystyle\frac{dM(r)}{dr}$
$\displaystyle=4\pi r^{2}\varepsilon(r).$ (70)

With the radial coordinate $r$, gravitational mass as a function of radius
$M(r)$, pressure $P(r)$, and energy density $\varepsilon(r)$, we utilized
natural units where $G=c=1$. Fig. 4 depicts the solution of these equations.
The baryon mass ($M_{B}$) is computed using

$\dfrac{dM_{B}(r)}{dr}=4\pi m_{n}\dfrac{r^{2}n_{B}(r)}{[1-2M(r)/r]^{1/2}},$
(71)

where $m_{n}=939\,\rm MeV$ is the nucleon mass.

At this juncture, it’s crucial to note that there are two broader approaches
to investigating PNSs: the quasi-static approximation description, applicable
when the stellar matter is isentropic, and the full hydrostatic simulation,
utilized when the stellar matter is out of equilibrium. Thus, the TOV
equations are applicable in the former, not the latter (see e.g.
Raduta:2020fdn ; Oertel:2016xsn ; Sedrakian:2022kgj ; Malfatti:2019tpg ;
Shao:2011nu ; Chen:2021edy ; Kumari:2021tik ). In the quasi-static
approximation description the time derivatives of density, pressure, and the
metric vanish therefore, the average hydrostatics effects that focus on the
evolution of intensive thermodynamic quantities such as the $Y_{L,e}$ and the
$S/n_{B}$ over the Kelvin-Helmholtz timescale is enough for the analysis
Pons:1998mm ; roberts2012new . The boundary conditions at this stage are
$P(R)=P_{\rm surf}$, with $P_{\rm surf}$, the pressure on the surface of the
star, and $R$ is the stellar radius, in the case of ‘mature’ NSs at $T=0$,
$P(R)=0$. In the finite temperature (fixed entropy) scenario, the selection of
$P_{\rm surf}$ is somewhat arbitrary owing to the presence of thermal
pressure. Nevertheless, its value has been found to impact the properties of
the stellar mantle during its initial formation stages. However, studies have
shown that significantly small values of $P_{\rm surf}$ have a negligible
effect on the long-term evolution of the star’s internal structures
RevModPhys.74.1015 ; Gulminelli:2015csa .

The outcomes of the mass-radius diagram derived for pure nucleonic matter,
nucleons plus hyperons, and nucleons plus hyperons plus $\Delta$-isobars
admixed hypernuclear matter all adhere to the $2\,\rm M_{\odot}$ mass
constraint imposed on NSs. As additional degrees of freedom are introduced
into the stellar matter, the maximum masses decrease, in accordance with
expectations. The observed radii also fall within the ranges determined for
the recently observed pulsar from the NICER observatory: PSR J0740+6620
NANOGrav:2019jur ; Fonseca:2021wxt .

Figure 5: The figure shows the velocity of sound squared as a function of the
baryon density. The horizontal gray line is the conformal limit,
$c_{s}^{2}=1/3$, below this line the stars are made up of self-bound quarks,
and above it, we have hadronic stars. The curves increase monotonically with
the baryon density, however, at the point of phase transition the $c_{s}$
drops drastically below the conformal limit.

The square of the sound velocity depicted in Fig. 5 is computed using the
relation

$c_{s}^{2}=\dfrac{\partial P}{\partial\varepsilon},$ (72)

which assists in classifying the particle composition of NSs using conformal
symmetry arguments and the EoS as input. The $c_{s}^{2}$ leverages the
distinct properties exhibited by QCD theory in both perturbative and
nonperturbative regimes to differentiate between particle compositions.
Studies indicate that in the perturbative regime of QCD theory particles
exhibit behavior akin to scale invariance, whereas the opposite behavior is
observed in the nonperturbative regime. This is distinguishable using
$c_{s}^{2}$, since the matter is exactly scale-invariant at $c_{s}^{2}=1/3$.
This value is approached from below at higher density regions
($n_{B}>40n_{0}$) in QM Kurkela:2009gj , where matter is considered to be
scale-invariant. However, in the hadronic matter, $c_{s}^{2}$ rises to values
$c_{s}^{2}\gtrsim 0.5$, clearly violating the scale-invariant limit. The
$c_{s}^{2}$ must also satisfy causality, $c_{s}^{2}\,\leq\,1$, and
thermodynamic stability, $c_{s}^{2}>0$, conditions simultaneously
Annala:2019puf ; Bedaque:2014sqa . We observe from Fig. 5 that the curves for
the neutrino-trapped matter grow monotonically beyond the conformal limit
without decreasing. On the other hand, in neutrino-transparent matter, where
hadron-quark phase transitions take place, the $c_{s}^{2}$ curves rise with
density in the hadronic regime until they reach their maximum. At densities
where free quarks begin to appear in the stellar matter, $c_{s}^{2}$ drops
quickly below the conformal limit, indicating the presence of conformal
invariant matter Fraga:2013qra ; Borsanyi:2010cj . The region of the steep
drop in the curves represents the phase transition region, similar to the
regions marked with gray areas in Fig. 2. The bumps in the $T=0$ curves
(purple) may represent shocks propagating through stacks of different
particles. Comparing with the $n_{B}/n_{0}$ at which new particles appear in
Fig. 2. The $\Delta^{-}$ appears at $\sim 1.8n_{0}$, $\Lambda^{0}$ at $\sim
2.2n_{0}$ and the $\Xi^{-}$ at $\sim 3.3n_{0}$ in the same position the bumps
appear depending on the matter composition. For NH matter, we have two bumps:
the appearance of $\Lambda^{0}$ and $\Xi^{-}$ hyperons at $\sim 2.2n_{0}$ and
$\sim 3.3n_{0}$ respectively. In the NH$\Delta$ matter we have three bumps
first at $\sim 1.8n_{0}$ when $\Delta^{-}$ appears, the second at $\sim
2.2n_{0}$ and the last at $\sim 3.3n_{0}$ when $\Xi^{-}$ appears. Thus, no
bumps are seen when matter is composed of only nucleons.

Figure 6: The graph illustrates the temperature of the stellar matter as a
function of $n_{B}/n_{0}$. It shows a rise in temperature with $n_{B}/n_{0}$,
followed by a sharp decrease at the critical temperature, $T_{C}$ (indicated
by a black star), marking the phase transition from hadronic to quark matter
phase. The core temperatures, $T_{co}$, are also highlighted with purple stars
on the curves, except where they lie outside the critical temperature range.

In Fig. 6, we depict the temperature as a function of baryon density.
Generally, we observe that an increase in the degrees of freedom of the
stellar matter leads to a decrease in the net temperature of the stellar
matter. Ideally, increasing the degrees of freedom increases the entropy of
the stellar matter. Since entropy is kept fixed at each stage, an increase in
the degrees of freedom will lead to a decrease in the net temperature of the
stellar matter. Also, in the neutrino-trapped matter, we observe that the
temperature profile in the first stage when the star is capturing neutrinos,
constitutes the least profiles. When the star begins deleptonizing, the
temperature of the stellar matter increases, as represented by the second
stage in Fig. 6 (for temperature fluctuation in PNSs with exotic baryons see
Issifu:2023qyi ; Sedrakian:2022kgj ; Raduta:2020fdn ). At this point, the
temperature of the stellar matter is relatively higher. In the third stage,
when the quark core begins to form in the stellar matter, the temperature
increases until it reaches a critical point (marked with black stars on the
curves) in the hadronic matter region. Subsequently, it decreases sharply as
free quarks emerge in the stellar matter. The sharp decrease in the
temperature when the quark core starts forming is attributed to the higher
degeneracies of the quarks. The core temperatures are denoted by purple stars
on the curves, except where the core temperature falls outside the critical
temperature range.

Figure 7: The curves represent the adiabatic indices of the hadronic and
hybrid stars as a function of $n_{B}/n_{0}$. A horizontal gray line at
$\Gamma=4/3$ denotes the instability threshold. Curves intersecting this line
are deemed unstable, while those above it are considered stable. The red
curves depicting the PNS in its early stages intersect the instability line at
various densities. Both the N and NH curves cut the instability line at the
low-density region while the NH$\Delta$ curve cuts it at the high-density
region.

In Fig. 7, we analyze the stability of PNSs (with quark cores) using the
adiabatic index as the benchmark. Based on the seminal presentations by
Chandrasekhar Chandrasekhar:1964zza ; PhysRevLett.12.114 , it was established
that the dynamical instability of compact stars can be studied through
variational methods. Mathematically, the adiabatic index ($\Gamma$) is
expressed as

$\Gamma=\dfrac{P+\varepsilon}{P}\Bigg{(}\dfrac{dP}{d\varepsilon}\Bigg{)}_{s_{B}},$
(73)

where the term in the bracket is precisely the speed of sound and
$s_{B}=S/n_{B}$ is the entropy per baryon at which the $\Gamma$ is being
computed. $\Gamma$ is a dimensionless quantity determined through the EoS,
hence, its value depends on the stiffness or otherwise of the corresponding
EoS. A star is considered stable when $\Gamma>4/3$ in its core. Conversely, a
star is deemed unstable when $\Gamma<4/3$, indicating an impending collapse.
Consequently, a critical point exists at $\Gamma=4/3$, serving as the boundary
that distinguishes between the stable and unstable regions glass1983stability
; PhysRevC.50.460 .

From Fig. 7, it is evident that the PNS formed during the initial stage of
stellar evolution when neutrinos are tapped in its core intersects the
instability threshold at various $n_{B}/n_{0}$ values. This indicates that
PNSs formed in this regime are unstable regardless of the composition of the
stellar matter, as highlighted by the red curves. The rest of the stars
satisfy the stability conditions set through $\Gamma$. After the neutrinos
have escaped and the quark core starts forming, the resulting hybrid stars
remain stable as shown in the figure above. However, at the point where the
quark core starts forming, the curves of the stars decrease steadily until
they approach the instability line from above but do not cross it. Thus, the
stars with the quark core are stable as well.

Figure 8: The pressure profile of two stars with fixed baryon masses,
$M_{b}=1.57\,\rm M_{\odot}$(dot-dash lines) and $M_{b}=2.45\,\rm
M_{\odot}$(solid lines). Comparatively, the presence of trapped neutrinos in
the star increases its pressure drastically, this is not immediately obvious
when we consider a group of stars as in Fig. 3 due to the corresponding
increase in energy density.

Subsequently, we profile two stars with fixed baryon mass to determine how the
pressure, temperature, and sound velocity behave with radius in relatively
lighter and heavier stars. It is important to note, that during the stellar
evolution, the baryon density does not vary significantly, as indicated by the
maximum baryon masses in Tab. 3. In the subsequent figures, solid lines
describe a massive star with $M_{b}=2.45\,\rm M_{\odot}$, while dashed-dot
lines describe the corresponding $M_{b}=1.57\,\rm M_{\odot}$ stars. In the
left panel of Fig. 8, we display the pressure profile of the neutrino-trapped
matter, while in the right panel, we illustrate the pressure profile of the
neutrino-transparent matter. Generally, a more massive star is associated with
higher core pressure than a lighter star. However, both exhibit a constant
pressure at the surface of the star. Comparing the two panels, it’s evident
that trapped neutrinos increase the pressure in the core of the star compared
to the neutrino-free ones. Nonetheless, after the neutrinos have escaped from
the star, the surface pressure remains constant for a few more kilometers
before it starts rising steeper towards the core compared to the neutrino-
trapped ones.

Figure 9: The temperature profiles for the two selected stars are depicted on
the graph. Dot-dashed lines distinguish the star with $M_{b}=1.57\,\rm
M_{\odot}$, while the star with $M_{b}=2.45\,\rm M_{\odot}$ is represented by
solid lines.

In Fig. 9, we present the temperature profiles in the core of the two selected
stars. The left panel represents the instance where neutrinos are trapped
inside the star, while the right panel illustrates when the neutrinos have all
escaped from the star’s core, and the stellar matter is maximally heated.
Generally, the temperature is higher in more massive stars than in lighter
ones, as depicted in the figure above. Additionally, the temperature remains
constant for a few kilometers on the star’s surface before it starts
increasing towards the core of the star. The initial stages of star formation
are associated with lower temperature profiles due to higher $Y_{L,e}$, as
expected. As the star begins deleptonizing, the temperature of the stellar
matter increases as depicted in the figure above.

Figure 10: The $c_{s}^{2}$ of two stars with fixed baryon masses,
$M_{b}=1.57\,\rm M_{\odot}$(dot-dash lines) and $M_{b}=2.45\,\rm
M_{\odot}$(solid lines).

In Fig. 10, we illustrate the $c_{s}^{2}$ of the two selected stars as a
function of the star’s radius. This will allow us to determine the composition
of matter that makes up these stars. The left panel represents stars with
neutrinos trapped in their core, which we know to be composed only of hadrons.
Therefore, their $c_{s}^{2}$ are expected to rise monotonously towards the
core, as shown in the graph above. In the right panel, we display the results
of the stars formed with neutrino-transparent matter, with a possible quark
core formation. Generally, the $c_{s}^{2}$ is higher in heavier stars than in
lighter ones. Also, relatively colder stars are associated with higher sound
velocity than their hotter counterparts. We demarcate the conformal boundary
with a horizontal gray line at $c_{s}^{2}=1/3$. In smaller and hotter stars,
such as those with $S/n_{B}=2$ and $Y_{L,e}=0.2$, as well as those with
$S/n_{B}=2$ and $Y_{\nu_{e}}=0$, the $c_{s}^{2}$ falls below the conformal
limit. This might suggest the possibility of free quark phase transitions.
However, upon closer examination of the $n_{c}$ values for
$M_{b}=1.57\,M_{\odot}$ stars, it becomes evident that this behavior is not
indicative of a quark phase transition at this point in time. Instead, it can
be attributed to the mass of the stars under consideration. On the contrary,
the $c_{s}^{2}$ determined for heavier stars with $M_{b}=2.45\,M_{\odot}$
exhibits characteristics typical of hadron stars. As discussed below Fig. 5,
the occurrence of the bumps in $c_{s}^{2}$ observed in the stars at $T=0$,
coincide with the appearance of new degrees of freedom in the star when it is
cold.

##  IV Final remarks

We investigated the evolution of PNSs from their birth to maturity to
determine the conditions under which a quark core begins to form in their
interior, amidst extreme temperatures and densities. The stellar matter
comprises nucleons, nucleons plus hyperons, and nucleons plus hyperons plus
$\Delta$-isobars. For the first time, we used DDQM and RMF approximation with
density-dependant coupling to show that the formation of the quark core during
stellar evolution begins only when all the neutrinos have escaped from the
star. The study demonstrates that the presence of neutrinos in the core of the
PNS hinders the formation of free quarks in its core. However, once all the
neutrinos have escaped from the stellar core, and the stellar matter is
expected to reach maximum heating, the phase transition begins to occur.
Consequently, within the model framework, the phase transition is found to be
highly dependent on the lepton fractions rather than the temperature of the
stellar matter. The observed drastic drop in temperature and sound velocity at
the onset of the quark core can indeed serve as a reliable indicator for
detecting hybrid stars, as our results demonstrate. In Sandin:2007zr ;
Ruester:2005ib , the authors explore the impact of neutrino trapping on the
quark-hadron phase transition in PNSs. They conclude that the possibility of a
quark core appearing in PNSs before deleptonization is completed is highly
unlikely. These conclusions align with our findings. Our study led to the
determination of the critical chemical potential, critical temperature, and
critical pressure at which the hadron-quark phase transition can occur in
neutrino-transparent matter for different compositions of matter, and the
results are displayed in Tab. 3. The specific findings from the study are
itemized below:

  * •

We calculated the $Y_{i}$, and presented our results in Fig. 2 to illustrate
the various particle compositions in the stellar matter, at which density they
appear in the stellar matter, and to pinpoint when the presence of free quarks
become apparent in the stellar matter. We find that there is a gradual phase
transition from hadronic to quark matter phase after the neutrinos have
escaped from the stellar core ($S/n_{B}=2$ and $Y_{\nu_{e}}=0$) within the
baryon density range $3.73\leq n_{B}[n_{0}]\leq 6.28$. After the stellar
matter has cooled down to $T=0$, the phase transition accelerates, and the
quark core expands relatively. This transition occurs within the baryon
density range $3.53\leq n_{B}[n_{0}]\leq 5.47$. Similar studies were conducted
in PhysRevC.100.015803 , where the authors found phase transitions occurring
solely at $T=0$. Additionally, in Shao:2011nu , the authors conducted similar
investigations using Gibbs construction for hybrid stars, considering only
nucleons in their model.

  * •

In Fig. 3, we presented the EoS, which forms the foundation of the entire
study. We observe that in the neutrino-trapped region, the increase in
$Y_{L,e}$ results in the stiffening of the EoS, and vice versa. However, in
the neutrino-transparent matter, a hadron-quark phase transition occurs,
leading to a noticeable discontinuity in the energy density. Here, the EoS
softens as the stellar matter cools from $S/n_{B}=2$ and $Y_{\nu_{e}}=0$ to
$T=0$ through the emission of thermal radiations (see e.g. PhysRevC.100.015803
; Sedrakian:2022kgj ; Raduta:2020fdn ; Issifu:2023ovi  for an insight into
fixed entropy EoSs and hybrid EoS).

  * •

In Fig. 4, we presented the results of the structure of the stars. We find
that the maximum masses are consistent with those obtained for PSR J0740+6620
NANOGrav:2019jur ; Fonseca:2021wxt , PSR J2215+5135 Linares:2018ppq , and PSR
J0952-0607 Romani:2022jhd  pulsars. Additionally, the radius falls within the
bounds of the NICER observed pulsar PSR J0740+6620.

  * •

We distinguished between the hadronic and quark matter phases Annala:2019puf ;
Annala:2023cwx  calculating $c_{s}^{2}$ in Fig. 5. The $c_{s}^{2}$ exhibits a
rise with increasing density in the hadronic regime and then begins to
decrease sharply when the quark core starts to appear, eventually reaching a
value below the conformal limit. Moreover, we profiled two different stars
with $M_{b}=1.57\,M_{\odot}$ and $M_{b}=2.45\,M_{\odot}$ and used the
$c_{s}^{2}$ as a function of star’s radii (Fig.10) and $n_{c}$ (Tab.3) to
determine their particle composition. We found that both stars profiled are
hadron stars. Additionally, we observed that the $c_{s}$ in the core increases
when neutrinos are trapped in the stellar matter. It drops immediately after
all the neutrinos have escaped from the core and increases again when the star
cools down to $T=0$.

  * •

We calculated the variation of temperature with the baryon density and
presented our findings in Fig. 6. Additionally, we studied the temperature
profile in two selected stars and presented our results in Fig. 9. Generally,
the temperature rises with $n_{B}$ and $R$ towards the stellar core. However,
when new degrees of freedom start appearing in the stellar matter, it
significantly reduces the temperature gradient. In Fig. 6, where the
$S/n_{B}=2$ and $Y_{\nu_{e}}=0$ stellar matter possesses the quark core, the
temperature drops drastically as the quark core begins to form in the stellar
matter. This represents a hadron-quark phase transition when the hadronic
matter reaches a critical temperature $T_{C}$. The observed properties of the
temperature variations qualitatively agree with the ones presented in
Issifu:2023qoo ; Raduta:2020fdn ; Issifu:2023ovi .

  * •

We explored the stability of both the hadronic and hybrid NSs using the
adiabatic index as our benchmark for analysis. We found that the neutrino-rich
($S/n_{B}=1$ and $Y_{L,e}=0.4$) PNS is unstable for the various particle
compositions, as illustrated in Fig. 7. The star stabilizes during
deleptonization, transitioning through neutrino transparency to the formation
of a cold-catalyzed hybrid NS. The point of the hadron-quark phase transition
is depicted by a steep drop in the curves, which then becomes horizontally
constant as it approaches the $\Gamma=4/3$ threshold. A detailed discussion on
the derivation of $\Gamma$ and the conditions under which stability can be
determined within its framework is presented in Moustakidis:2016ndw .

  * •

We profiled the pressure variation as a function of the star’s radii of two
NSs with fixed baryon masses as shown in Fig. 9. We observed that the pressure
is relatively constant on the star’s surface for both the lighter and the
heavier stars analyzed, and it begins to rise with radius towards the star’s
core a few kilometers from the surface. The results indicate that the presence
of neutrinos in the star’s core significantly increases its core pressure. As
the star undergoes deleptonization and neutrino diffusion, the pressure begins
to drop and then rises again when the star catalyzes, shrinks, and
compactifies to form a “mature" NS at $T=0$.

From the study, we determined that the presence of neutrino concentration in
the stellar core during its formative stages impedes the dissociation of the
hadrons into SQM in the stellar core. Additionally, neutrino concentration
suppresses temperature rise, increasing pressure and $c_{s}^{2}$ of the
stellar matter toward the star’s core. This results in an increase in the
maximum mass of the neutrino-trapped PNSs relative to neutrino-poor ones.
During the deleptonization stages, the matter heats up, leading to an
expansion in the size of the star and a relative drop in the maximum
gravitational mass, as shown in Tab. 3, in the absence of accretion and the
formation of a black hole. We established the formation of a quark core when
all the neutrinos had escaped from the stellar core. It would be nice to
reproduce the present study with other models to check how model-dependent our
conclusions are.

## Acknowledgements

This work is a part of the project INCT-FNA Proc. No. 464898/2014-5. D.P.M.
was partially supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq/Brazil) under grant 303490-2021-7. A.I. thanks the financial
support from the São Paulo State Research Foundation (FAPESP) Grant No.
2023/09545-1. T. F. thanks the financial support from the Brazilian
Institutions: CNPq (Grant No. 306834/2022-7), Improvement of Higher Education
Personnel CAPES (Finance Code 001) and FAPESP (Grants No. 2017/05660-0 and
2019/07767-1). Z.R. wishes to thank the Shiraz University Research Council.

## References

  * (1) G. G. Raffelt, “Astrophysical methods to constrain axions and other novel particle phenomena,” Physics Reports, vol. 198, no. 1, pp. 1–113, 1990.
  * (2) G. Raffelt and D. Seckel, “Bounds on Exotic Particle Interactions from SN 1987a,” Phys. Rev. Lett., vol. 60, p. 1793, 1988.
  * (3) K. Hirata et al., “Observation of a Neutrino Burst from the Supernova SN 1987a,” Phys. Rev. Lett., vol. 58, pp. 1490–1493, 1987.
  * (4) K. S. Hirata et al., “Observation in the Kamiokande-II Detector of the Neutrino Burst from Supernova SN 1987a,” Phys. Rev. D, vol. 38, pp. 448–458, 1988.
  * (5) R. M. Bionta et al., “Observation of a neutrino burst in coincidence with supernova 1987a in the large magellanic cloud,” Phys. Rev. Lett., vol. 58, pp. 1494–1496, Apr 1987.
  * (6) E. W. Kolb and M. S. Turner, “Supernova SN 1987a and the Secret Interactions of Neutrinos,” Phys. Rev. D, vol. 36, p. 2895, 1987.
  * (7) G. G. Raffelt, “What Have We Learned From SN 1987A?,” Mod. Phys. Lett. A, vol. 5, pp. 2581–2592, 1990.
  * (8) M. S. Turner, “Axions from SN 1987a,” Phys. Rev. Lett., vol. 60, p. 1797, 1988.
  * (9) J. H. Chang, R. Essig, and S. D. McDermott, “Supernova 1987A Constraints on Sub-GeV Dark Sectors, Millicharged Particles, the QCD Axion, and an Axion-like Particle,” JHEP, vol. 09, p. 051, 2018.
  * (10) G. Lucente, L. Mastrototaro, P. Carenza, L. Di Luzio, M. Giannotti, and A. Mirizzi, “Axion signatures from supernova explosions through the nucleon electric-dipole portal,” Phys. Rev. D, vol. 105, no. 12, p. 123020, 2022.
  * (11) A. Caputo, G. Raffelt, and E. Vitagliano, “Muonic boson limits: Supernova redux,” Phys. Rev. D, vol. 105, no. 3, p. 035022, 2022.
  * (12) W. DeRocco, P. W. Graham, D. Kasen, G. Marques-Tavares, and S. Rajendran, “Observable signatures of dark photons from supernovae,” JHEP, vol. 02, p. 171, 2019.
  * (13) S. Hannestad and G. Raffelt, “New supernova limit on large extra dimensions,” Phys. Rev. Lett., vol. 87, p. 051301, 2001.
  * (14) S. Hannestad, G. Raffelt, and Y. Y. Y. Wong, “Unparticle constraints from SN 1987A,” Phys. Rev. D, vol. 76, p. 121701, 2007.
  * (15) S. Balaji, P. S. B. Dev, J. Silk, and Y. Zhang, “Improved stellar limits on a light CP-even scalar,” JCAP, vol. 12, p. 024, 2022.
  * (16) A. Friedland and M. Giannotti, “Astrophysical bounds on photons escaping into extra dimensions,” Phys. Rev. Lett., vol. 100, p. 031602, 2008.
  * (17) A. Bharucha, F. Brümmer, N. Desai, and S. Mutzel, “Axion-like particles as mediators for dark matter: beyond freeze-out,” JHEP, vol. 02, p. 141, 2023.
  * (18) A. Lella, P. Carenza, G. Lucente, M. Giannotti, and A. Mirizzi, “Protoneutron stars as cosmic factories for massive axionlike particles,” Phys. Rev. D, vol. 107, no. 10, p. 103017, 2023.
  * (19) B. P. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett., vol. 119, no. 16, p. 161101, 2017.
  * (20) B. P. Abbott et al., “GW170817: Measurements of neutron star radii and equation of state,” Phys. Rev. Lett., vol. 121, no. 16, p. 161101, 2018.
  * (21) T. E. Riley et al., “A $NICER$ View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation,” Astrophys. J. Lett., vol. 887, no. 1, p. L21, 2019.
  * (22) M. C. Miller et al., “PSR J0030+0451 Mass and Radius from $NICER$ Data and Implications for the Properties of Neutron Star Matter,” Astrophys. J. Lett., vol. 887, no. 1, p. L24, 2019.
  * (23) T. E. Riley et al., “A NICER View of the Massive Pulsar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy,” Astrophys. J. Lett., vol. 918, no. 2, p. L27, 2021.
  * (24) M. C. Miller et al., “The Radius of PSR J0740+6620 from NICER and XMM-Newton Data,” Astrophys. J. Lett., vol. 918, no. 2, p. L28, 2021.
  * (25) A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, “The Equation of state of nucleon matter and neutron star structure,” Phys. Rev. C, vol. 58, pp. 1804–1828, 1998.
  * (26) D. P. Menezes, “A Neutron Star Is Born,” Universe, vol. 7, no. 8, p. 267, 2021.
  * (27) S. Typel et al., “CompOSE Reference Manual,” Eur. Phys. J. A, vol. 58, no. 11, p. 221, 2022.
  * (28) A. Burrows, “Colloquium: Perspectives on core-collapse supernova theory,” Rev. Mod. Phys., vol. 85, p. 245, 2013.
  * (29) H. T. Janka, T. Melson, and A. Summa, “Physics of Core-Collapse Supernovae in Three Dimensions: a Sneak Preview,” Ann. Rev. Nucl. Part. Sci., vol. 66, pp. 341–375, 2016.
  * (30) M. Prakash, I. Bombaci, M. Prakash, P. J. Ellis, J. M. Lattimer, and R. Knorren, “Composition and structure of protoneutron stars,” Phys. Rept., vol. 280, pp. 1–77, 1997.
  * (31) H.-T. Janka, K. Langanke, A. Marek, G. Martinez-Pinedo, and B. Mueller, “Theory of Core-Collapse Supernovae,” Phys. Rept., vol. 442, pp. 38–74, 2007.
  * (32) L. Baiotti and L. Rezzolla, “Binary neutron star mergers: a review of Einstein’s richest laboratory,” Rept. Prog. Phys., vol. 80, no. 9, p. 096901, 2017.
  * (33) M. Shibata and K. Hotokezaka, “Merger and Mass Ejection of Neutron-Star Binaries,” Ann. Rev. Nucl. Part. Sci., vol. 69, pp. 41–64, 2019.
  * (34) D. Radice, S. Bernuzzi, and A. Perego, “The Dynamics of Binary Neutron Star Mergers and GW170817,” Ann. Rev. Nucl. Part. Sci., vol. 70, pp. 95–119, 2020.
  * (35) J. M. Lattimer and M. Prakash, “The Equation of State of Hot, Dense Matter and Neutron Stars,” Phys. Rept., vol. 621, pp. 127–164, 2016.
  * (36) M. Oertel, M. Hempel, T. Klähn, and S. Typel, “Equations of state for supernovae and compact stars,” Rev. Mod. Phys., vol. 89, no. 1, p. 015007, 2017.
  * (37) T. Fischer, M. Hempel, I. Sagert, Y. Suwa, and J. Schaffner-Bielich, “Symmetry energy impact in simulations of core-collapse supernovae,” Eur. Phys. J. A, vol. 50, p. 46, 2014.
  * (38) A. Perego, S. Bernuzzi, and D. Radice, “Thermodynamics conditions of matter in neutron star mergers,” Eur. Phys. J. A, vol. 55, no. 8, p. 124, 2019.
  * (39) D. Blaschke and N. Chamel, “Phases of dense matter in compact stars,” Astrophys. Space Sci. Libr., vol. 457, pp. 337–400, 2018.
  * (40) E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, “Evidence for quark-matter cores in massive neutron stars,” Nature Phys., vol. 16, no. 9, pp. 907–910, 2020.
  * (41) G. Malfatti, M. G. Orsaria, G. A. Contrera, F. Weber, and I. F. Ranea-Sandoval, “Hot quark matter and (proto-) neutron stars,” Phys. Rev. C, vol. 100, p. 015803, Jul 2019.
  * (42) G.-y. Shao, “Evolution of proto-neutron stars with the hadron-quark phase transition,” Phys. Lett. B, vol. 704, pp. 343–346, 2011.
  * (43) A. Issifu, F. M. da Silva, and D. P. Menezes, “Hybrid stars built with density-dependent models,” Mon. Not. Roy. Astron. Soc., vol. 525, no. 4, pp. 5512–5519, 2023.
  * (44) R. Karimi and H. R. Moshfegh, “Hybrid stars within the framework of the Sigma-Omega-Rho model combined with the MIT and NJL models,” Nucl. Phys. A, vol. 1037, p. 122684, 2023.
  * (45) M. Cierniak and D. Blaschke, “Hybrid neutron stars in the mass-radius diagram,” Astron. Nachr., vol. 342, no. 5, pp. 819–825, 2021.
  * (46) A. Issifu, F. M. da Silva, and D. P. Menezes, “Proto-strange quark stars from density-dependent quark mass model,” European Physical Journal C, vol. 84, p. 463, 2024.
  * (47) O. Philipsen, “The QCD equation of state from the lattice,” Prog. Part. Nucl. Phys., vol. 70, pp. 55–107, 2013.
  * (48) S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Szabo, “Full result for the QCD equation of state with 2+1 flavors,” Phys. Lett. B, vol. 730, pp. 99–104, 2014.
  * (49) R. Bellwied, S. Borsanyi, Z. Fodor, J. Günther, S. D. Katz, C. Ratti, and K. K. Szabo, “The QCD phase diagram from analytic continuation,” Phys. Lett. B, vol. 751, pp. 559–564, 2015.
  * (50) K. Fukushima and T. Hatsuda, “The phase diagram of dense QCD,” Rept. Prog. Phys., vol. 74, p. 014001, 2011.
  * (51) E. R. Most, L. J. Papenfort, V. Dexheimer, M. Hanauske, S. Schramm, H. Stöcker, and L. Rezzolla, “Signatures of quark-hadron phase transitions in general-relativistic neutron-star mergers,” Phys. Rev. Lett., vol. 122, no. 6, p. 061101, 2019.
  * (52) J. Roark and V. Dexheimer, “Deconfinement phase transition in proto-neutron-star matter,” Phys. Rev. C, vol. 98, no. 5, p. 055805, 2018.
  * (53) K. D. Marquez, D. P. Menezes, H. Pais, and C. Providência, “$\Delta$ baryons in neutron stars,” Phys. Rev. C, vol. 106, no. 5, p. 055801, 2022.
  * (54) N. K. Glendenning, “First order phase transitions with more than one conserved charge: Consequences for neutron stars,” Phys. Rev. D, vol. 46, pp. 1274–1287, 1992.
  * (55) B. C. Backes, E. Hafemann, I. Marzola, and D. P. Menezes, “Density dependent quark mass model revisited: Thermodynamic consistency, stability windows and stellar properties,” J. Phys. G, vol. 48, no. 5, p. 055104, 2021.
  * (56) X. J. Wen, X. H. Zhong, G. X. Peng, P. N. Shen, and P. Z. Ning, “Thermodynamics with density and temperature dependent particle masses and properties of bulk strange quark matter and strangelets,” Phys. Rev. C, vol. 72, p. 015204, 2005.
  * (57) F. M. da Silva, A. Issifu, L. L. Lopes, L. C. N. Santos, and D. P. Menezes, “Bayesian study of quark models in view of recent astrophysical constraints,” Phys. Rev. D, vol. 109, no. 4, p. 043054, 2024.
  * (58) G. A. Lalazissis, T. Nikšić, D. Vretenar, and P. Ring, “New relativistic mean-field interaction with density-dependent meson-nucleon couplings,” Phys. Rev. C, vol. 71, p. 024312, Feb 2005.
  * (59) X. Roca-Maza, X. Vinas, M. Centelles, P. Ring, and P. Schuck, “Relativistic mean field interaction with density dependent meson-nucleon vertices based on microscopical calculations,” Phys. Rev. C, vol. 84, p. 054309, 2011.  [Erratum: Phys.Rev.C 93, 069905 (2016)].
  * (60) A. Issifu, K. D. Marquez, M. R. Pelicer, and D. P. Menezes, “Exotic baryons in hot neutron stars,” Mon. Not. Roy. Astron. Soc., vol. 522, no. 3, pp. 3263–3270, 2023.
  * (61) A. R. Raduta, M. Oertel, and A. Sedrakian, “Proto-neutron stars with heavy baryons and universal relations,” Mon. Not. Roy. Astron. Soc., vol. 499, no. 1, pp. 914–931, 2020.
  * (62) S. Han, M. A. A. Mamun, S. Lalit, C. Constantinou, and M. Prakash, “Treating quarks within neutron stars,” Phys. Rev. D, vol. 100, no. 10, p. 103022, 2019.
  * (63) L. L. Lopes, K. D. Marquez, and D. P. Menezes, “Baryon coupling scheme in a unified SU(3) and SU(6) symmetry formalism,” Phys. Rev. D, vol. 107, no. 3, p. 036011, 2023.
  * (64) H. T. Cromartie et al., “Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar,” Nature Astron., vol. 4, no. 1, pp. 72–76, 2019.
  * (65) E. Fonseca et al., “Refined Mass and Geometric Measurements of the High-mass PSR J0740+6620,” Astrophys. J. Lett., vol. 915, no. 1, p. L12, 2021.
  * (66) M. Linares, T. Shahbaz, and J. Casares, “Peering into the dark side: Magnesium lines establish a massive neutron star in PSR J2215+5135,” Astrophys. J., vol. 859, no. 1, p. 54, 2018.
  * (67) R. W. Romani, D. Kandel, A. V. Filippenko, T. G. Brink, and W. Zheng, “PSR J0952$-$0607: The Fastest and Heaviest Known Galactic Neutron Star,” Astrophys. J. Lett., vol. 934, no. 2, p. L17, 2022.
  * (68) B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett., vol. 116, no. 6, p. 061102, 2016.
  * (69) B. P. Abbott et al., “Properties of the binary neutron star merger GW170817,” Phys. Rev. X, vol. 9, no. 1, p. 011001, 2019.
  * (70) G. X. Peng, H. C. Chiang, J. J. Yang, L. Li, and B. Liu, “Mass formulas and thermodynamic treatment in the mass density dependent model of strange quark matter,” Phys. Rev. C, vol. 61, p. 015201, 2000.
  * (71) C. J. Xia, G. X. Peng, S. W. Chen, Z. Y. Lu, and J. F. Xu, “Thermodynamic consistency, quark mass scaling, and properties of strange matter,” Phys. Rev. D, vol. 89, no. 10, p. 105027, 2014.
  * (72) M. G. de Paoli, D. P. Menezes, L. B. Castro, and C. C. Barros, Jr, “The Rarita-Schwinger Particles Under de Influence of Strong Magnetic Fields,” J. Phys. G, vol. 40, p. 055007, 2013.
  * (73) G. Malfatti, M. G. Orsaria, G. A. Contrera, F. Weber, and I. F. Ranea-Sandoval, “Hot quark matter and (proto-) neutron stars,” Phys. Rev. C, vol. 100, no. 1, p. 015803, 2019.
  * (74) M. Dutra, O. Lourenço, S. S. Avancini, B. V. Carlson, A. Delfino, D. P. Menezes, C. Providência, S. Typel, and J. R. Stone, “Relativistic Mean-Field Hadronic Models under Nuclear Matter Constraints,” Phys. Rev. C, vol. 90, no. 5, p. 055203, 2014.
  * (75) B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, “Implications of PREX-2 on the Equation of State of Neutron-Rich Matter,” Phys. Rev. Lett., vol. 126, no. 17, p. 172503, 2021.
  * (76) J. M. Lattimer, “Constraints on Nuclear Symmetry Energy Parameters,” Particles, vol. 6, pp. 30–56, 2023.
  * (77) N. K. Glendenning and S. A. Moszkowski, “Reconciliation of neutron star masses and binding of the lambda in hypernuclei,” Phys. Rev. Lett., vol. 67, pp. 2414–2417, 1991.
  * (78) J. R. Stone, V. Dexheimer, P. A. M. Guichon, A. W. Thomas, and S. Typel, “Equation of state of hot dense hyperonic matter in the Quark–Meson-Coupling (QMC-A) model,” Mon. Not. Roy. Astron. Soc., vol. 502, no. 3, pp. 3476–3490, 2021.
  * (79) L. L. Lopes and D. P. Menezes, “Broken SU(6) symmetry and massive hybrid stars,” Nucl. Phys. A, vol. 1009, p. 122171, 2021.
  * (80) A. PAIS, “Dynamical symmetry in particle physics,” Rev. Mod. Phys., vol. 38, pp. 215–255, Apr 1966.
  * (81) S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, “Hyperons and massive neutron stars: the role of hyperon potentials,” Nucl. Phys. A, vol. 881, pp. 62–77, 2012.
  * (82) C. Dover and A. Gal, “Hyperon-nucleus potentials,” Progress in Particle and Nuclear Physics, vol. 12, pp. 171–239, 1984.
  * (83) J. Schaffner, C. B. Dover, A. Gal, C. Greiner, D. J. Millener, and H. Stoecker, “Multiply strange nuclear systems,” Annals Phys., vol. 235, pp. 35–76, 1994.
  * (84) T. D. Cohen, R. J. Furnstahl, and D. K. Griegel, “From qcd sum rules to relativistic nuclear physics,” Phys. Rev. Lett., vol. 67, pp. 961–964, Aug 1991.
  * (85) T. D. Cohen, R. J. Furnstahl, and D. K. Griegel, “Quark and gluon condensates in nuclear matter,” Phys. Rev. C, vol. 45, pp. 1881–1893, Apr 1992.
  * (86) E. Eichten, K. Gottfried, T. Kinoshita, J. Kogut, K. D. Lane, and T. M. Yan, “Spectrum of charmed quark-antiquark bound states,” Phys. Rev. Lett., vol. 34, pp. 369–372, Feb 1975.
  * (87) G. S. Bali, “QCD forces and heavy quark bound states,” Phys. Rept., vol. 343, pp. 1–136, 2001.
  * (88) P. Bicudo, “The QCD string tension curve, the ferromagnetic magnetization, and the quark-antiquark confining potential at finite Temperature,” Phys. Rev. D, vol. 82, p. 034507, 2010.
  * (89) O. Kaczmarek, F. Karsch, E. Laermann, and M. Lutgemeier, “Heavy quark potentials in quenched QCD at high temperature,” Phys. Rev. D, vol. 62, p. 034021, 2000.
  * (90) S. Digal, S. Fortunato, and P. Petreczky, “Heavy quark free energies and screening in SU(2) gauge theory,” Phys. Rev. D, vol. 68, p. 034008, 2003.
  * (91) O. Philipsen, “Lattice QCD at finite temperature and density,” Eur. Phys. J. ST, vol. 152, pp. 29–60, 2007.
  * (92) E.-M. Ilgenfritz, J. M. Pawlowski, A. Rothkopf, and A. Trunin, “Finite temperature gluon spectral functions from $N_{f}=2+1+1$ lattice QCD,” Eur. Phys. J. C, vol. 78, no. 2, p. 127, 2018.
  * (93) F. Karsch, E. Laermann, and A. Peikert, “Quark mass and flavor dependence of the QCD phase transition,” Nucl. Phys. B, vol. 605, pp. 579–599, 2001.
  * (94) E. Laermann and O. Philipsen, “The Status of lattice QCD at finite temperature,” Ann. Rev. Nucl. Part. Sci., vol. 53, pp. 163–198, 2003.
  * (95) H.-M. Chen, C.-J. Xia, and G.-X. Peng, “Strangelets at finite temperature in a baryon density-dependent quark mass model,” Phys. Rev. D, vol. 105, no. 1, p. 014011, 2022.
  * (96) H.-M. Chen, C.-J. Xia, and G.-X. Peng, “Strangelets formation in high energy heavy-ion collisions,” Phys. Rev. D, vol. 109, no. 5, p. 054031, 2024.
  * (97) P.-C. Chu and L.-W. Chen, “Warm asymmetric quark matter and protoquark stars within the confined isospin-density-dependent mass model,” Phys. Rev. D, vol. 96, no. 10, p. 103001, 2017.
  * (98) A. Issifu and T. Frederico, “Hot quark matter and merger remnants,” 1 2024.
  * (99) P. D. Group, “Review of particle physics,” PTEP, vol. 2022, p. 083C01, 2022.
  * (100) G. X. Peng, H. C. Chiang, and P. Z. Ning, “Thermodynamics, strange quark matter, and strange stars,” Phys. Rev. C, vol. 62, p. 025801, 2000.
  * (101) F. Weber, D. Farrell, W. M. Spinella, M. G. Orsaria, G. A. Contrera, and I. Maloney, “Phases of Hadron-Quark Matter in (Proto) Neutron Stars,” Universe, vol. 5, no. 7, p. 169, 2019.
  * (102) M. Oertel, F. Gulminelli, C. Providência, and A. R. Raduta, “Hyperons in neutron stars and supernova cores,” Eur. Phys. J. A, vol. 52, no. 3, p. 50, 2016.
  * (103) A. Sedrakian and A. Harutyunyan, “Delta-resonances and hyperons in proto-neutron stars and merger remnants,” Eur. Phys. J. A, vol. 58, no. 7, p. 137, 2022.
  * (104) H.-M. Chen, C.-J. Xia, and G.-X. Peng, “Strange quark matter and proto-strange stars in a baryon density-dependent quark mass model *,” Chin. Phys. C, vol. 46, no. 5, p. 055102, 2022.
  * (105) M. Kumari and A. Kumar, “Properties of strange quark matter and strange quark stars,” Eur. Phys. J. C, vol. 81, no. 9, p. 791, 2021.
  * (106) A. R. Raduta, “$\Delta$-admixed neutron stars: Spinodal instabilities and dUrca processes,” Phys. Lett. B, vol. 814, p. 136070, 2021.
  * (107) T. Maruyama, S. Chiba, H.-J. Schulze, and T. Tatsumi, “Hadron-quark mixed phase in hyperon stars,” Phys. Rev. D, vol. 76, p. 123015, 2007.
  * (108) M. G. Paoli and D. P. Menezes, “The importance of the mixed phase in hybrid stars built with the Nambu-Jona-Lasinio model,” Eur. Phys. J. A, vol. 46, pp. 413–420, 2010.
  * (109) L. L. Lopes, C. Biesdorf, and D. P. Menezes, “Hypermassive quark cores,” Mon. Not. Roy. Astron. Soc., vol. 512, no. 4, pp. 5110–5121, 2022.
  * (110) B. P. Abbott et al., “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett., vol. 848, no. 2, p. L13, 2017.
  * (111) A. Goldstein et al., “An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A,” Astrophys. J. Lett., vol. 848, no. 2, p. L14, 2017.
  * (112) G. A. Contrera, D. Blaschke, J. P. Carlomagno, A. G. Grunfeld, and S. Liebing, “Quark-nuclear hybrid equation of state for neutron stars under modern observational constraints,” Phys. Rev. C, vol. 105, no. 4, p. 045808, 2022.
  * (113) M. Alford, M. Braby, M. W. Paris, and S. Reddy, “Hybrid stars that masquerade as neutron stars,” Astrophys. J., vol. 629, pp. 969–978, 2005.
  * (114) A. Clevinger, J. Corkish, K. Aryal, and V. Dexheimer, “Hybrid equations of state for neutron stars with hyperons and deltas,” Eur. Phys. J. A, vol. 58, no. 5, p. 96, 2022.
  * (115) I. Vidana, I. Bombaci, A. Polls, and A. Ramos, “Microscopic study of neutrino trapping in hyperon stars,” Astron. Astrophys., vol. 399, pp. 687–693, 2003.
  * (116) G. F. Burgio, M. Baldo, O. E. Nicotra, and H. J. Schulze, “A microscopic equation of state for protoneutron stars,” Astrophys. Space Sci., vol. 308, pp. 387–394, 2007.
  * (117) P. Ribes, A. Ramos, L. Tolos, C. Gonzalez-Boquera, and M. Centelles, “Interplay between $\Delta$ Particles and Hyperons in Neutron Stars,” Astrophys. J., vol. 883, p. 168, 2019.
  * (118) H. S. Sahoo, G. Mitra, R. Mishra, P. K. Panda, and B.-A. Li, “Neutron star matter with $\mathrm{\Delta}$ isobars in a relativistic quark model,” Phys. Rev. C, vol. 98, p. 045801, Oct 2018.
  * (119) J. R. Oppenheimer and G. M. Volkoff, “On massive neutron cores,” Phys. Rev., vol. 55, pp. 374–381, Feb 1939.
  * (120) J. A. Pons, S. Reddy, M. Prakash, J. M. Lattimer, and J. A. Miralles, “Evolution of protoneutron stars,” Astrophys. J., vol. 513, p. 780, 1999.
  * (121) L. F. Roberts, “A new code for proto-neutron star evolution,” The Astrophysical Journal, vol. 755, no. 2, p. 126, 2012.
  * (122) S. E. Woosley, A. Heger, and T. A. Weaver, “The evolution and explosion of massive stars,” Rev. Mod. Phys., vol. 74, pp. 1015–1071, Nov 2002.
  * (123) F. Gulminelli and A. R. Raduta, “Unified treatment of subsaturation stellar matter at zero and finite temperature,” Phys. Rev. C, vol. 92, no. 5, p. 055803, 2015.
  * (124) A. Kurkela, P. Romatschke, and A. Vuorinen, “Cold Quark Matter,” Phys. Rev. D, vol. 81, p. 105021, 2010.
  * (125) P. Bedaque and A. W. Steiner, “Sound velocity bound and neutron stars,” Phys. Rev. Lett., vol. 114, no. 3, p. 031103, 2015.
  * (126) E. S. Fraga, A. Kurkela, and A. Vuorinen, “Interacting quark matter equation of state for compact stars,” Astrophys. J. Lett., vol. 781, no. 2, p. L25, 2014.
  * (127) S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, “The QCD equation of state with dynamical quarks,” JHEP, vol. 11, p. 077, 2010.
  * (128) S. Chandrasekhar, “Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity,” Phys. Rev. Lett., vol. 12, pp. 114–116, 1964.
  * (129) S. Chandrasekhar, “Dynamical instability of gaseous masses approaching the schwarzschild limit in general relativity,” Phys. Rev. Lett., vol. 12, pp. 114–116, Jan 1964.
  * (130) E. Glass and A. Harpaz, “The stability of relativistic gas spheres,” Monthly Notices of the Royal Astronomical Society, vol. 202, no. 1, pp. 159–171, 1983.
  * (131) M. Onsi, H. Przysiezniak, and J. M. Pearson, “Equation of state of homogeneous nuclear matter and the symmetry coefficient,” Phys. Rev. C, vol. 50, pp. 460–468, Jul 1994.
  * (132) F. Sandin and D. Blaschke, “The quark core of protoneutron stars in the phase diagram of quark matter,” Phys. Rev. D, vol. 75, p. 125013, 2007.
  * (133) S. B. Ruester, V. Werth, M. Buballa, I. A. Shovkovy, and D. H. Rischke, “The Phase diagram of neutral quark matter: The Effect of neutrino trapping,” Phys. Rev. D, vol. 73, p. 034025, 2006.
  * (134) E. Annala, T. Gorda, J. Hirvonen, O. Komoltsev, A. Kurkela, J. Nättilä, and A. Vuorinen, “Strongly interacting matter exhibits deconfined behavior in massive neutron stars,” Nature Commun., vol. 14, no. 1, p. 8451, 2023.
  * (135) C. C. Moustakidis, “The stability of relativistic stars and the role of the adiabatic index,” Gen. Rel. Grav., vol. 49, no. 5, p. 68, 2017.