# Hot quark matter and merger remnants

Adamu Issifu  0000-0002-2843-835X<EMAIL_ADDRESS>Instituto Tecnológico
de Aeronáutica,
CEP 12.228-900, São José dos Campos, SP, Brazil  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 hot quark matter under the thermodynamic conditions
characteristic of a binary neutron star (BNS) merger remnants. We use the
density-dependent quark mass model (DDQM) to access the microscopic nuclear
equation of state (EoS). The strange quark matter (SQM) is studied at finite
temperature and entropy in the presence of electrons and muons and their
neutrinos to simulate the BNS merger conditions. We observe that as the
entropy of the SQM increases, the merger remnant becomes more massive, and
increases in size whereas the neutrino population also increases. In the fixed
temperature case, on the other hand, we observe that the entropy spreads from
the surface towards the center of the remnant. We determine the particle
distribution in the remnants’ core, the remnant’s structure, the temperature
profile, sound velocity, and the polytropic index and discuss their effects.
The strange quark (SQ) remnants satisfy the $2\,{\rm M_{\odot}}$ mass
constraint associated with neutron stars (NS).

##  I Introduction

The merger of two NSs leading to the detection of a gravitational wave in
event GW170817 (GW) [1] results in the formation of massive remnants and
subsequent matter ejection in the interstellar medium [2, 3]. The remnant
comprises a central object with a massive disk around it. The characteristics
of the disk and the central object depend on the masses of the merging stars,
the magnetic field, NS spin, and their corresponding EoSs [4, 5, 6]. The
gravitational wave was observed from the inspiral phase by Advanced LIGO [7]
and Advanced Virgo [8] interferometers. This observation paved a new path into
the microphysics of NSs, offering the possibility to probe matter dynamics at
extreme temperature and density conditions inside these remnants. The
constraint on the tidal effect of the coalescing bodies and the maximum NS
mass and radii has enabled nuclear astrophysicists to impose a strict
constraint on the EoS for cold NSs [9]. The detection also coincided with the
observation of a $\gamma$-ray burst, GRB 170817A (GRB) [10, 11] which helps in
classifying the binary contained matter using the relative time delay between
the GW and GRB and the measured speed of sound [10].

Currently, gravitational waves have been observed, and future gravitational
wave observatories are expected to observe more neutron star mergers with
higher confidence levels. In this light, numerical simulations for the BNS
merger are important to interpret the data and further constrain the EoS for
NSs. Temperatures in the postmerger phase are expected to reach as high as
$\sim 50-100$ MeV, making it suitable for finite temperature studies [12].
Even though the current gravitational wave detectors have not reported on a
postmerger signal yet [1, 13, 14], however, it is expected that future
detectors such as the Einstein Telescope [15] and Cosmic Explorer [16]
detectors with improved sensitivities and high-frequency features may detect
BNS postmerger signals. The current sensitivity upgrades on the existing
instruments also promise a better quality BNS detection with a better sky
location [17].

It has been predicted through numerical relativity simulations that the final
product of BNS mergers is a prompt formation of a black hole through
gravitational collapse or the formation of a remnant NS [18]. In the case of
NS formation, the baryon density ($n_{b}$) of a typical $\beta$-equilibrated
matter of the individual NSs of about $n_{b}\sim 2-3n_{0}$ ($n_{0}$ is the
nuclear saturation density) increases to about $n_{b}\sim 5-6n_{0}$ with a
high remnant temperature [19, 20, 21]. A typical temperature in the postmerger
remnant core at the initial stages ranges from 20 to 60 MeV [22]. This
temperature further increases in the contact layer at the early stages
shooting up to about 150 MeV [22, 23]. These extreme conditions are expected
to be reached on shorter time scales due to the violent nature of the event,
in milliseconds, under conditions relatively different from the formation of
proto-NSs (PNS) through supernova remnants [24, 25]. Also, higher isospin
asymmetries are expected in BNS merger events due to an excess of neutrons
over protons in close binary systems NS-NS, and NS-BH [26, 27, 28], contrary
to PNSs [29, 30].

Since Ivanenko and Kurdgelaideze hypothesized the existence of strange quark
stars (SQSs) in 1965 [31, 32] and together with 1984 Witten’s conjecture [33],
namely, that SQM is likely to be the true ground state of hadronic matter, QSs
have been intensively studied ( see e.g. [34, 35, 36, 37, 38]). Nowadays there
is sufficient information about compact objects through the electromagnetic
spectrum and gravitational wave detection. Analysis of these observational
data has led to constraints on the macroscopic properties such as the mass,
radii, and tidal deformability of compact astrophysical objects [1, 39, 9].
Recent data obtained from the Neutron Star Interior Composition Explorer
(NICER) led to the detection of two compact objects PSR J0030+0451 and PSR
J0740+6620, which further constrain the composition of stellar matter [40, 41,
42, 43]. However, these pulsars’ exact matter composition is still a research
subject. Thus, these observations in recent years have led to robust
restrictions on the EoS ruling out most phenomenological quark models.
Additionally, it has been established theoretically that, at a high enough
baryon density (in the inner core of a compact object) the nucleon Fermi
energy is high enough to allow possible new degrees of freedom such as exotic
baryons [44, 45, 46, 47] and deconfined QM [48, 49]. Since SQM has been
conjectured to be the true ground state of QCD [33, 35, 50, 51] it was
understood that hadronic matter can transform into QM, transitioning to QSs
instead of NSs. As discussed in [52, 53, 54, 55, 56, 57, 58], proto-strange
QSs with bare or dynamically irrelevant nucleonic matter crust can be formed
through BNS merger. That notwithstanding, there is a strong argument against
the presence of a crust on QSs [59, 60, 61].

In this paper, we employ the DDQM under thermodynamic conditions
characteristic of the BNS merger remnant, assuming that the remnant is a NS
composed of self-bound free QM. The DDQM is characterized by two free
parameters D (a dimensionful constant representing the confining strength) and
C (a dimensionless constant representing the deconfinement strength) given
through the cubic-root law [62]

$m_{i}=m_{i0}+\dfrac{D}{n_{b}^{1/3}}+Cn_{b}^{1/3},$ (1)

with $m_{i}$ the equivalent of a dressed quark mass, $m_{i0}$ the current
quark mass and $n_{b}$ the baryon density. The last two terms in (1) can be
associated to a Cornell-like potential for confining heavy quarks [63, 64,
65], $V_{c}(r_{*})=-\beta/r_{*}+\sigma r_{*}$, where $\sigma$ and $\beta$ are
constants that can be determined through fitting to lattice QCD (lQCD) data.
It comprises a Coulomb like term dominant at short distances with strength
$\beta$ from asymptotic freedom and a linear confining part with strength
$\sigma$, which is also associated with the QCD string tension. In the above
expression, the average inter-particle separation distance $r_{*}$ can be
associated with the baryon density through $r_{*}\sim 1/n_{b}^{1/3}$. We
choose $\sqrt{D}=127.40\,{\rm MeV}$ and $C=0.8$ reached in [34] through
Bayesian analysis of the DDQM at $T=0\,{\rm MeV}$ to serve as a benchmark for
our study. These parameters produce a stable QS according to Bodmer and
Witten’s conjecture which hypothesizes that the binding energy of SQM must be
less than the binding energy of 56Fe, i.e. $(\varepsilon/n_{b})_{\rm SQM}\leq
930{\rm MeV}$. In the same vein, the two-flavor quark matter (2QM ) must
satisfy $(\varepsilon/n_{b})_{\rm 2QM}>930{\rm MeV}$ [33, 35, 66] i.e., the
binding energy should be larger for 2QM for $u$ and $d$ quarks to be more
tightly bound otherwise, they would dissociate into free $u$ and $d$ quarks.
Also, these choices lead to SQS that satisfies the maximum mass constraint of
pulsars PSR J0952-0607 [67], PSR J0740+6620 [43] and PSR J0030+0451 [40], and
others that satisfy the 2M⊙ constraint. We fix the entropy per baryon,
$S/n_{b}$ of the remnant matter, and calculate the temperature profile,
particle distributions, speed of sound, polytropic index, and the structure of
the star. On the other hand, for comparison, we consider a remnant matter at a
uniform temperature and study the entropy distribution from the disk to the
core of the remnant and the other properties of the matter investigated at
fixed entropy per baryon.

The paper is organized as follows: In Sec. II, we present the model that
serves as the basis for this investigation. This section is divided into four
subsections, in Subsec. II.1 we present the thermodynamic treatment of the
model, in Subsec. II.2 we present the foundation of the mass scaling and its
scaling with temperature, in Subsec. II.3 we present the properties of the SQM
under investigation, and finally, we present the thermodynamic conditions for
BNS merger in Subsec. II.4. We present our findings and analyze them in Sec.
III and the final remarks in Sec. IV.

##  II The model

The variation of particle self-energies depending on the media in which they
are propagating is well-known in literature [68, 69]. In the simplest case,
the mass is usually referred to as the effective mass to distinguish it from
the ‘bare’ mass. When models are expressed in terms of temperature and
chemical potentials they are referred to as the quasiparticle models [70] (see
e.g. [71, 72, 73, 74, 75, 76]).

Now we focus our attention on DDQM, which is used in our study. Indeed, the
DDQM is a quasiparticle model originally formulated to investigate the
confining characteristics of QM [77, 78] and was later on extended to
calculate the EoS [79, 80, 81, 82], SQM viscosity and r-modes dissipation
[83], diquark properties [84] and compact astrophysical objects [34, 25, 71,
73]. The application of this model to study compact astrophysical objects has
largely been successful in fulfilling the 2${\rm M}_{\odot}$ mass and radii
constraints of NSs, making it a viable QS model. That notwithstanding, it is
associated with two known limitations, namely: the equivalent mass scaling and
thermodynamic consistency. However, these issues have been tackled by several
authors [71, 85, 82, 86] with some promising outcomes.

###  II.1 Finite temperature formalism of DDQM

The free energy density of the hot SQM system can be expressed as

$\displaystyle F$ $\displaystyle=F(T,V,\\{n_{i}\\},\\{m_{i}\\})$
$\displaystyle=\Omega_{0}(T,V,\\{\mu^{*}_{i}\\},\\{m_{i}\\})+\sum_{i=u,d,s}\mu^{*}_{i}n_{i},$
(2)

where $\mu^{*}_{i}$ is the effective chemical potential for the $i$th
particle, $m_{i}\rightarrow m_{i}(n_{b},T)$ is the density and temperature
dependent particle mass, and $\Omega_{0}$ is the thermodynamic potential
density similar to the non-interacting particle system, which will be used as
an intermediate quantity at this point. The thermodynamic potential density
$\Omega$ will be determined and presented at the end of this subsection as a
function of $\Omega_{0}$.

The expression for $F$ has the same form as the well-known expression for the
free fermionic system with the particle mass replaced by the equivalent
particle mass $m_{i}(n_{b},T)$ and the real chemical potential $\mu_{i}$
replaced by $\mu^{*}_{i}$ for thermodynamic consistency. We connect
$\mu_{i}^{*}$ and $\Omega_{0}$ to the independent state variables; $T,\;n_{i}$
and the volume $V$ through

$n_{i}=-\dfrac{\partial}{\partial\mu^{*}_{i}}\Omega_{0}(T,V,\\{\mu^{*}_{i}\\},\\{m_{i}\\}),$
(3)

this expression is well-known in thermodynamics with $\mu_{i}^{*}$ in place of
$\mu_{i}$ to ensure thermodynamic consistency. To determine the other
thermodynamic quantities, we take a derivative of (3), which yields

$dF=d\Omega_{0}+\sum_{i}n_{i}d\mu^{*}_{i}+\sum_{i}\mu^{*}_{i}dn_{i},$ (4)

here, $\Omega_{0}$ can be expressed in terms of the dependent variables as

$d\Omega_{0}=\dfrac{\partial\Omega_{0}}{\partial
T}dT+\sum_{i}\dfrac{\partial\Omega_{0}}{\partial\mu^{*}_{i}}d\mu^{*}_{i}+\sum_{i}\dfrac{\partial\Omega_{0}}{\partial
m_{i}}dm_{i}+\dfrac{\partial\Omega_{0}}{\partial V}dV,$ (5)

and the equivalent mass as

$dm_{i}=\dfrac{\partial m_{i}}{\partial T}dT+\sum_{j}\dfrac{\partial
m_{i}}{\partial n_{j}}dn_{j}.$ (6)

Combining (4), (5) and (6) and regrouping terms, we find that:

$\displaystyle dF$ $\displaystyle=\left(\dfrac{\partial\Omega_{0}}{\partial
T}+\sum_{i}\dfrac{\partial\Omega_{0}}{\partial m_{i}}\dfrac{\partial
m_{i}}{\partial T}\right)dT$
$\displaystyle+\sum_{i}\left(\mu_{i}^{*}+\sum_{j}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial
n_{i}}\right)dn_{i}+\dfrac{\partial\Omega_{0}}{\partial V}dV.$ (7)

Comparing the above result to the standard expression for the thermodynamic
free energy density;

$dF=-SdT+\sum_{i}\mu_{i}dn_{i}+\left(-P-F+\sum_{i}\mu_{i}n_{i}\right)\dfrac{dV}{V},$
(8)

where $S$ is the entropy density and $P$ is the pressure of the system, we
identify:

$S=-\dfrac{\partial\Omega_{0}}{\partial T}-\sum_{i}\dfrac{\partial
m_{i}}{\partial T}\dfrac{\partial\Omega_{0}}{\partial m_{i}},$ (9)

and

$P=-F+\sum_{i}\mu_{i}n_{i}-V\dfrac{\partial\Omega_{0}}{\partial V},$ (10)

where the real chemical potential is given in terms of the effective one as

$\mu_{i}=\mu^{*}_{i}+\sum_{j}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial n_{i}}\equiv\mu_{i}^{*}-\mu_{I},$ (11)

where $\mu_{I}$ is the chemical potential that carries the interaction. For an
infinitely large system of SQM, Eq. (II.1) is independent of the volume of the
system and hence (10) takes the form:

$\displaystyle P$ $\displaystyle=-F+\sum_{i}\mu_{i}n_{i}$
$\displaystyle=-\Omega_{0}+\sum_{i,j}n_{i}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial n_{i}},$ (12)

in the second step we have substituted (II.1) in place of $F$ and (11) in
place of $\mu_{i}$ to obtain a result in terms of $\Omega_{0}$ and
$\mu^{*}_{i}$. Similar expressions were derived in [72, 87, 85] for DDQM using
a thermodynamic consistent approach, and in [82] using the general ensemble
theory. Additionally, the last term in (10) only persists if the finite size
effect of the system cannot be overlooked whether the particle masses are
fixed or not. The energy density of the system can then be determined from the
fundamental relation with the Helmholtz free energy, namely
$\varepsilon=F-TS$, by substituting (II.1) and (9), resulting in:

$\varepsilon=\Omega_{0}+\sum_{i}\mu_{i}^{*}n_{i}-T\dfrac{\partial\Omega_{0}}{\partial
T}-T\sum_{i}\dfrac{\partial m_{i}}{\partial
T}\dfrac{\partial\Omega_{0}}{\partial m_{i}}.$ (13)

Finally, the real thermodynamic potential density $\Omega$ of the system can
be determined as

$\Omega=F-\sum_{i}\mu_{i}n_{i}=\Omega_{0}-\sum_{i,j}n_{i}\dfrac{\partial\Omega_{0}}{\partial
m_{j}}\dfrac{\partial m_{j}}{\partial n_{i}}.$ (14)

It should be noted that the $\Omega_{0}$ is composed of both particle and
anti-particle, therefore, the full contribution will be
$\Omega_{0}^{\pm}=\Omega_{0}^{-}+\Omega^{+}_{0}$ with $\Omega^{-}_{0}$ and
$\Omega^{+}_{0}$ representing particle and antiparticle contributions
respectively. Consequently, the explicit expression for the $\Omega_{0}^{\pm}$
is given as

$\displaystyle\Omega^{\pm}_{0}$
$\displaystyle=-\sum_{i}\dfrac{g_{i}T}{2\pi^{2}}\int
p^{2}dp\Big{[}\ln[1+e^{-(\epsilon_{i}-\mu_{i}^{*})/T}]$
$\displaystyle+\ln[1+e^{-(\epsilon_{i}+\mu_{i}^{*})/T}]\Big{]},$ (15)

where $i=u,\;d,\;s,\;e,\;\mu,\;\nu_{\mu}\;\text{and}\;\nu_{e}$ represents the
quarks and leptons present in the system and $g_{i}$ the degeneracy factor.
The number density of each fermion species is also given by two terms,
$n_{i}^{\pm}=n^{+}_{i}-n_{i}^{-}$ thus,

$\displaystyle
n_{i}^{\pm}=-\dfrac{\partial\Omega_{0}^{\pm}}{\partial\mu^{*}_{i}}$
$\displaystyle=\sum_{i}\dfrac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\Big{[}\dfrac{1}{1+e^{(\epsilon_{i}-\mu_{i}^{*})/T}}-\dfrac{1}{1+e^{(\epsilon_{i}+\mu_{i}^{*})/T}}\Big{]}.$
(16)

The derivatives in $S,\;\mu_{i}^{*},\;P,\text{and},\;\varepsilon$ can be
written explicitly as

$\displaystyle\dfrac{\partial\Omega_{0}^{\pm}}{\partial m_{i}}=$
$\displaystyle\sum_{i}\dfrac{g_{i}m_{i}}{2\pi^{2}}\int_{0}^{\infty}\Big{[}\dfrac{1}{e^{(\epsilon_{i}-\mu^{*}_{i})/T}+1}$
$\displaystyle+\dfrac{1}{e^{(\epsilon_{i}+\mu^{*}_{i})/T}+1}\Big{]}\dfrac{p^{2}dp}{\epsilon_{i}},$
(17)

and

$\displaystyle\dfrac{\partial\Omega_{0}^{\pm}}{\partial T}$
$\displaystyle=-\dfrac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}\Big{(}\ln\left[1+e^{-(\epsilon_{i}-\mu^{*}_{i})/T}\right]$
$\displaystyle+\dfrac{(\epsilon_{i}-\mu_{i}^{*})/T}{1+e^{-(\epsilon_{i}-\mu^{*}_{i})/T}}+\ln\left[1+e^{-(\epsilon_{i}+\mu^{*}_{i})/T}\right]$
$\displaystyle+\dfrac{(\epsilon_{i}+\mu_{i}^{*})/T}{1+e^{-(\epsilon_{i}+\mu^{*}_{i})/T}}\Big{)}p^{2}dp,$
(18)

with $\epsilon_{i}=\sqrt{p^{2}+m_{i}^{2}}$.

###  II.2 Mass scaling with temperature

We will adopt the cubic-root equivalent mass scaling formula originally
derived for $T=0$ [62, 87] and was extended for a finite temperature treatment
[72, 85]. The derivation of quark mass scaling and its thermodynamic
consistency have been elaborated in [72, 62]. We do not intend to repeat the
derivation here, but we will highlight the parts fundamental to this work and
focus more on how we intend to introduce temperature into the system. We start
with the QCD Hamiltonian density

$H_{\rm QCD}=H_{k}+\sum_{q=u,\,d,\,S}m_{i0}\bar{q}q+H_{I},$ (19)

with $H_{k}$ the kinetic term and $H_{I}$ the interaction term. We now
introduce an equivalent Hamiltonian density with the interacting term absorbed
into an equivalent mass $m_{i}$ in the form

$H_{\rm eqv}=H_{k}+\sum_{q=u,\,d,\,s}m_{i}\bar{q}q,$ (20)

hence,

$m_{i}=m_{0i}+m_{I},$ (21)

this means to keep the energy of the system of free quarks constant, its mass
must be $m_{i}$. To determine $m_{i}$, we impose the condition that $H_{\rm
eqv}$ and $H_{\rm QCD}$ must have the same eigenvalues for the same
eigenstates, $|n_{b},T\rangle$ i.e.,

$\langle n_{b},T|H_{\rm QCD}|n_{b},T\rangle=\langle n_{b},T|H_{\rm
eqv}|n_{b},T\rangle,$ (22)

applying the same equality to the vacuum state $|0\rangle$ and taking the
difference we obtain

$m_{i}=m_{i0}+\dfrac{\langle H_{I}\rangle_{n_{b},T}-\langle
H_{I}\rangle_{0}}{\sum_{q}\Big{[}\langle\bar{q}q\rangle_{n_{b},T}-\langle\bar{q}q\rangle_{0}\Big{]}},$
(23)

therefore $m_{I}$ can be identified with,

$m_{I}=\dfrac{\langle H_{I}\rangle_{n_{b},T}-\langle
H_{I}\rangle_{0}}{\sum_{q}\Big{[}\langle\bar{q}q\rangle_{n_{b},T}-\langle\bar{q}q\rangle_{0}\Big{]}}.$
(24)

We remark that the interacting part of the equivalent quark mass, $m_{I}$,
depends on the interaction energy density, and the relative quark condensates
(the denominator of the above expression) which are assumed to be flavor
independent. For color confinement and asymptotic freedom, $m_{I}$ must
satisfy

$\lim_{n_{b}\rightarrow 0}m_{I}\rightarrow\infty;\qquad{\text{color
confinement condition}}\qquad$ (25)

and

$\lim_{n_{b}\rightarrow\infty}m_{I}\rightarrow 0;\qquad{\text{asymptotic
freedom condition.}}\qquad$ (26)

The proposed structure of the equivalent mass in terms of $n_{b}$ is

$m_{i}=m_{i0}+\dfrac{D}{n_{b}^{z}},$ (27)

consequently, $z>0$ satisfies the conditions (25) and (26). Assuming the
quarks are confined, the interaction energy density depends on the particle
density, while the linear confining potential, $V_{c}(r_{*},T)$, is a function
of the average inter-particle separation distance $r_{*}$ and $T$. Explicitly,

$\langle H_{I}\rangle_{n_{b},T}-\langle
H_{I}\rangle_{0}=3n_{b}V_{c}(r_{*},T),$ (28)

where factor 3 is due to the number of quark flavors participating in the
strong interaction. The average quark-quark distance is related to $n_{b}$ by

$r_{*}\sim\dfrac{1}{n_{b}^{1/3}},$ (29)

we have adopted the cubic-root law where $z$ was determined to be $1/3$ in
[62, 87, 72]. The confining potential is well-known in LQCD to be proportional
to $r_{*}$ in the form $V_{c}(r_{*},T)=\sigma(T)r_{*}$, where $\sigma$ is the
proportionality constant known as the string tension [88, 89, 90], with this
expression, (26) and (25) are satisfied;

$\lim_{n_{b}\rightarrow\infty}V_{c}(r_{*},T)\rightarrow 0,$ (30)

which expresses the asymptotic freedom and, on the other hand, the low-density
limit:

$\lim_{n_{b}\rightarrow 0}V_{c}(r_{*},T)\rightarrow\infty,$ (31)

represents color confinement.

The potential $V_{c}$ has been widely investigated as a function of
temperature in the literature – see, for example, Refs. [91, 92, 93] –, it was
found in all cases that the string tension decreases with temperature and
vanishes at $T=T_{C}$ ($T_{C}$ is the critical temperature) i.e.,
$\sigma(T=T_{C})=0$. Hence, the normalized $\sigma(T)$ can generally be
expressed as

$\dfrac{\sigma(T)}{\sigma_{0}}=\left[1-\dfrac{T^{2}}{T_{C}^{2}}\right]^{\gamma},$
(32)

where $\gamma$ is a constant usually determined in the model framework and
$\sigma_{0}$ is the string tension determined at $T=0$. We adopt the case in
which $\gamma=1$ as applied in [94] to a density and temperature-dependent
quark mass in the framework of the MIT bag model. Also, a phenomenological
study of quark and glueball confinements in [95] arrived at a similar outcome.
The denominator of (24) is compared to the chiral condensate in the QCD vacuum
in a simplified form. Even though the general expression must include density
and temperature dependence, for simplicity without loss of generality, we will
consider only density-dependent and use the well-known model-independent quark
condensate in nuclear matter [96, 97],

$\dfrac{\langle\bar{q}q\rangle_{n_{b}}}{\langle\bar{q}q\rangle_{0}}=1-\dfrac{n_{b}}{n_{b}^{*}},$
(33)

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

$n^{*}_{b}=\dfrac{M_{\pi}^{2}f_{\pi}^{2}}{\sigma_{N}}.$ (34)

Substituting (28) and (33) into (24) and rescaling D as

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

we get

$m_{I}=\dfrac{D}{n_{b}^{1/3}}\left[1-\dfrac{T^{2}}{T^{2}_{C}}\right].$ (36)

therefore, the full expression in (23) becomes

$m_{i}=m_{i0}+\dfrac{D}{n_{b}^{1/3}}\left[1-\dfrac{T^{2}}{T^{2}_{C}}\right],$
(37)

in this expression, the interacting part vanishes when $T=T_{C}$, similar to
the QCD string-breaking scenario however, the current quark mass permanently
breaks the chiral symmetry.

Comparing the above expression to the well-known Cornell potential
$V(r_{*})=-\beta/r_{*}+\sigma r_{*}$, which incorporates both the confinement
and the asymptotically free behavior of the QM, one can introduce a
deconfinement parameter proportional to $n_{b}$, looking at the Coulomb part
of the Cornell potential, thus, $z<0$ in (27). The Cornell potential has been
widely explored in phenomenological and lQCD and it has been proven to show
characteristics consistent with the behavior of QCD matter. This additional
term is necessary to balance the attractive pressure generated by the linear
confining term to support the star to attain the maximum required mass for NSs
without collapsing. In this case, the full expression becomes

$m_{i}=m_{i0}+\dfrac{D}{n_{b}^{1/3}}\left[1-\dfrac{T^{2}}{T^{2}_{C}}\right]+Cn_{b}^{1/3}\left[1-\dfrac{T^{2}}{T^{2}_{C}}\right]^{-1}.$
(38)

From (1) the confining strength is represented by $D$ and the Coulomb-like
strength is represented by $C$ and their values can be determined through
stability analysis. In quenched lQCD the $T_{C}$ is determined to be around
$T_{C}\sim 270\,{\rm MeV}$ [98, 99, 100, 101].

###  II.3 Properties of the SQM

The SQM is, as usual, composed of up ($u$), down ($d$), and strange ($s$)
quarks with leptons (electrons $e$ and muons $\mu$ with their corresponding
neutrinos $\nu_{e}$ and $\nu_{\mu}$ respectively) in $\beta$\- equilibrium.
Due to the weak interactions, such as $d,s\leftrightarrow u+e+\bar{\nu}_{e}$,
$s+u\leftrightarrow u+d$, etc., the $\beta$-equilibrated matter are required
to satisfy the chemical equilibrium condition,

$\mu_{d}^{*}=\mu_{s}^{*}=\mu_{u}^{*}+\mu_{l}-\mu_{\nu_{l}},$ (39)

where the subscripts represent the particles present,
$\mu_{l}=\\{\mu_{e}\;,\mu_{\mu}\\}$ and
$\mu_{\nu_{l}}=\\{\mu_{\nu_{e}},\;\mu_{\nu_{\mu}}\\}$. It is important to
mention here that the real chemical potentials $\mu_{i}$ satisfy the same
chemical equilibrium relation as the effective ones above (see Ref.[71]).
Charge neutrality of the system requires that

$\dfrac{2}{3}n_{\mu}-\dfrac{1}{3}n_{d}-\dfrac{1}{3}n_{s}-n_{e}-n_{\mu}=0,$
(40)

here, $n_{i}$ is the particle number density and the subscript $i$ represents
the identity of the particle present. The baryon number is also conserved
through

$n_{b}=\dfrac{1}{3}\big{(}n_{u}+n_{d}+n_{s}\big{)}=\dfrac{1}{3}\sum_{i}n_{i}.$
(41)

At finite temperature where neutrinos are trapped in the SQM, we fix
separately, the lepton numbers; $Y_{L,e}=Y_{e}+Y_{\nu_{e}}$ and
$Y_{L,\mu}=Y_{\mu}+Y_{\nu_{\mu}}$, with $Y_{L,e}=(n_{e}+n_{\nu_{e}})/n_{b}$
and $Y_{L,\mu}=(n_{\mu}+n_{\nu_{\mu}})/n_{b}$ for electron and muon families
respectively. The thermodynamic conditions are based on the astrophysical
phenomena under consideration. It is known that neutrinos are trapped in the
stellar matter when their mean-free path is less than the size of the system
(typically the radius of the stellar object) [102, 103]. Also, the electron
and muon neutrino spheres are not identical hence, the trapping regime might
be related to a family of leptons. Therefore we fix the lepton numbers
separately for each family of leptons and ignore the $\tau$leptons, assuming
they are too heavy to be relevant.

The quark flavors have degeneracy of $g_{i}=6$ (3 colors $\times$ 2 spins),
the $e$, the $\mu$ have a degeneracy of $g_{i}=2$ and the neutrinos have
degeneracy of $g_{i}=1$. The current quark masses for $u,\;d$, and $s$ used
are 2.16 MeV, 4.67 MeV, and 93.4 MeV respectively as reported in the PDG
[104]. We choose a critical temperature, $T_{C}=270$MeV, the strength of the
linear confinement, $\sqrt{D}=127.40$MeV, and the dimensionless single gluon-
exchange interaction strength, $C=0.8$. The values of C and D were taken from
[34] determined at $T=0$ using Bayesian study and stability analysis to serve
as a benchmark for our analysis. The EoS and the other remnant properties were
determined by either fixing $T$ and calculating $S/n_{b}$ or fixing $S/n_{b}$
and calculating the temperature profile in addition to the other properties.

###  II.4 Thermodynamic conditions for the BNS merger remnants

The thermodynamic conditions such as the lepton number concentration, entropy
per baryon, and temperature were derived from BNS simulations results [12,
105, 2, 106, 107, 106]. The degrees of freedom considered in BNS merger
simulations are nucleons, nuclei, electrons, and positrons. Aside from that,
some include hyperons, quarks, and/or neutrinos [108, 20, 109, 110, 111]. On
the other hand, most state-of-the-art simulations of binary NS mergers do not
include muons, even though the exotic temperature and density conditions in
the BNS merger event favor their generation. A recent simulation of BNS
postmerger remnants [112] considering muons shows that their presence
significantly affects the EoS. In this work, we include muons and their
neutrinos in our numerical codes to study their effect on the EoS,
particularly, we pay attention to the production of $\nu_{e}$ and $\nu_{\mu}$
as the merger remnant evolves, assuming it does not form a black hole
promptly.

Muons are known to play a vital role in the microphysics of cold NSs and are
expected to be significant in mergers where the thermodynamic conditions
rather favor their production. The two NSs that participated in the merger
event were originally in a quasi-circular orbit until the emission of GWs
caused their merger – see a review on this subject in [2]. In the merging
process, the two stars are compressed a few times $n_{0}$ ($n_{0}=0.152\,$fm-3
is the nuclear saturation density [113]) and heated substantially resulting in
supersonic velocities and shock wave formation. This provokes a sharp rise in
temperature and local entropy production. Considering that the BNS merger
ejecta comes from the crust and the outer core of two NSs, the initial lepton
number is expected to be low. It should be noted that cold NSs are associated
with low lepton fractions. The stellar matter of the two cold NSs have lepton
fractions in the range of $0.00\leq Y_{L,l}\leq 0.2$ [23, 114]. We chose
$Y_{L,e}=Y_{L,\mu}=0.1$ as the working assumption for our analysis similar to
the study of a merger event in hadronic stars in [115, 103].

The cores of the NSs are not significantly shocked at merger so, the entropy
per baryon of the remnant core is estimated to be $S/n_{b}\lesssim 2k_{B}$
($k_{B}$ is the Boltzmann constant), here, we chose $S/n_{b}=1\,k_{B}$ for our
analysis. When there is a prompt black hole (BH) formation, the apparent
horizon removes the high-density part leaving a relatively cold disk with a
slightly higher lepton fraction, $Y_{L,e}=Y_{L,\mu}=0.25$. At this point, the
entropy of the remnant also increases and the evolution process terminates
here, therefore, we chose $S/n_{b}=1.5\,k_{B}$ for our analysis. When the
action of the inspiral arms on the innermost part of the disk increases the
entropy of the remnant and the temperature also increases accordingly. The
estimated entropy at this stage is around $2<S/n_{b}[k_{B}]\lesssim 10$
however, we chose a lower limit of $S/n_{b}=3\,k_{B}$ and an upper limit of
$S/n_{b}=6\,k_{B}$ for our analysis. In [116] the authors used the
thermodynamic conditions of BNS merger and heavy ion collision to probe their
similarities and differences. In this study, the entropy flow from these two
events played a key role. Additionally, we considered the QM at uniform
temperature with a lower limit of $T=5\,{\rm MeV}$ (neutrino trapping
temperature) and an upper limit of $T=50\,{\rm MeV}$ to study the $S/n_{b}$
variation with $n_{b}$. Although there is a wide spread of $T$ and $S/n_{b}$
at the initial stages of the remnant’s evolution, these thermodynamic
conditions become homogeneous in the remnant matter at the latter stages of
its evolution.

Moreover, matter under extreme conditions of temperature and densities above
the $n_{0}$ are also formed through core-collapse supernovae (CCSN)
explosions, aside from the BNS merger remnant, leading to the formation of
PNSs that go through several evolution stages to form an NS or a BH. The
difference between the CCSN remnant and BNS merger remnant is that, the latter
involves a higher isospin asymmetry due to excess neutron over protons in NS-
NS or NS-BH merger events (see [107] for a review). In the case of CCSN the
core of the massive star traps neutrinos (particularly $\nu_{e}$) which heats
it, and expands it before the explosion occurs. This process increases the
proton fraction at the initial stage of the star’s birth before
deleptonization sets in after the core bounce causing a reduction in the
proton fraction in the PNS as it cools down to a cold NS [29, 117]. In the
case of a BNS merger, the proton fraction in the two cold NSs that initially
take part in the merger is small. There has been an enormous effort in the
past few years in theory, laboratory experiments, and astrophysical
observations to constrain the thermodynamic properties and the chemical
composition of stellar matter under the conditions of CCSN and NS mergers.
This is necessary because of the link between the macroscopic remnant
structure during the evolution of such stellar remnants and the fundamental
interactions between their constituents at the microscopic level. These
motivate the study of such objects since they contribute to the effort in
creating general-purpose EoSs and also challenge our understanding of nature
on both scales.

##  III Results and analysis

S$/$n${}_{B}[k_{B}]$; YL,l | M${}_{max}[{\rm M}_{\odot}]$ | $R[{\rm km}]$ | $\varepsilon_{0}[{MeVfm^{-3}}]$ |  $n$ [$fm^{-3}$] |  $T_{c}$ [MeV]
---|---|---|---|---|---
$1;\;Y_{L,e}=Y_{L,\mu}=0.1$ | 2.25 | 14.30 | 685 | 0.59 | 9.02
$1.5;\;Y_{L,e}=Y_{L,\mu}=0.25$ | 2.34 | 14.78 | 635 | 0.52 | 12.17
$3;\;Y_{L,e}=Y_{L,\mu}=0.1$ | 2.33 | 14.69 | 634 | 0.54 | 26.71
$6;\;Y_{L,e}=Y_{L,\mu}=0.1$ | 2.67 | 16.45 | 498 | 0.41 | 51.30
T = 5; $Y_{L,e}=Y_{L,\mu}=0.1$ | 2.25 | 14.36 | 662 | 0.59 | —
T = 50; $Y_{L,e}=Y_{L,\mu}=0.1$ | 2.57 | 16.92 | 483 | 0.41 | —
Table 1: BNS remnant properties.

In Table 1 we show the lepton fraction $Y_{L,l}$, the maximum mass ${\rm
M_{max}}$, its radii $R$, entropy per baryon $S/n_{b}$, core temperature
$T_{c}$, and the central energy and baryon densities, $\varepsilon_{0}$ and
$n$, respectively. We observe that the ${\rm M_{max}}$ and $R$ increase by
increasing $S/n_{b}$, while $\varepsilon_{0}$ and $n$ decrease causing a rise
in the core temperature of the NS remnant matter. In the case of an increase
in $Y_{L,l}$ and $S/n_{b}$ (condition for prompt formation of a black hole)
the ${\rm M_{max}}$, $R$ and $T_{c}$ increases with a decrease in
$\varepsilon_{0}$ and $n_{0}$. For uniform temperature matter, a hotter QM has
higher ${\rm M_{max}}$ and $R$ while $\varepsilon_{0}$ and $n_{0}$ decrease,
as usual. Naturally, the ratio $\varepsilon_{0}/n$ increases as $S/n_{B}$
rises and $T$ increases. It has been established (see [6, 12]) that a typical
density in BNS merger remnant is between $2n_{0}$ and $6n_{0}$, this agrees
with the values of $n$ on Table 1 which ranges between $2.7n_{0}$ and
$4n_{0}$. Following the tabulated results for fixed $S/n_{b}$ and $T$, hotter
remnants accrete more mass and increase in size.

Figure 1: In this diagram we show the $Y_{i}$ as a function of $n_{b}/n_{0}$.
In the upper panel, we choose two values of $T$; a lower limit of $T=5\,{\rm
MeV}$ at which neutrinos are trapped in the remnant matter, and an upper limit
$T=50\,{\rm MeV}$, using $Y_{L,e}=Y_{L,\mu}=0.1$ and their corresponding
$S/n_{b}$ distribution for each particle in the bottom panel.

In Fig. 1, we show the particle distributions, assuming that the NS remnant
has a uniform temperature using the relation

$Y_{i}=\dfrac{n_{i}}{n_{u}+n_{d}+n_{s}},$ (42)

where $i$ represents the individual particles present. Comparing the particle
distributions in the upper panels, we observe that at low $T$ we have a higher
$d$-population relative to $u$ and delayed appearance of the $s$-quark at low
$n_{b}$ regions. The $e$’s and $\mu$’s show a marginal increase in the hotter
remnant matter. In both cases, the particles do not show any significant
changes at higher densities beyond $n_{b}\sim 2.5n_{0}$. Also, higher $T$
produces more $\nu_{e}$ population while $\nu_{\mu}$ only increases marginally
at higher $n_{b}$ and rather shows a decrease at low $n_{b}$ with an early
appearance of the $s$-quark and $\mu$ in the remnant matter.

These effects at higher $T$ causes an increase in the ${\rm M_{max}}$ and R as
can be seen on Table 1. The isospin asymmetry (or $u-d$ quark asymmetry),

$\delta=3\dfrac{n_{d}-n_{u}}{n_{d}+n_{u}},$ (43)

with $n_{3}=n_{d}-n_{u}$, the isospin density and $n_{b}=(n_{d}+n_{u})/3$ the
baryon density for two flavor quark systems. The $\delta$ is higher in the low
$T$ remnant matter due to excess of $d$-quark relative to $u$ at densities
below $n_{b}\sim 2.7n_{0}$. In the lower panel, we show the $S/n_{b}$
contribution of each particle to the overall $S/n_{b}$. We observe that
temperature increase significantly increases the entropy of the particles, as
expected, from low $n_{b}/n_{0}$ and steadily falls towards the center of the
remnant. The $d$-quark shows a higher $S/n_{b}$ contribution while $\nu_{e}$
shows the least contribution in both cases.

Figure 2: Net entropy of the particles per baryon at different fixed
temperatures as a function of baryon density. Solid line for $T=50\,$MeV and
dot-dashed line for $T=5\,$MeV.

In Fig. 2, we show the net $S/n_{b}$ as a function of $n_{b}/n_{0}$. We
observe a higher $S/n_{b}$ in hotter QM, as expected. The $S/n_{b}$ is higher
towards the surface (low $n_{b}$) of the remnant matter where the disk is
formed and decreases toward the center (high $n_{b}$). The BNS merger is
associated with net angular momentum and is gravitationally bound, however,
thermodynamics in the local rest frame of the system remains relevant. In the
absence of physical viscosity, the matter flow is isentropic, hence $S/n_{b}$
can serve as a meaningful tool for analysis. It has been found in [118] that
entropy rise is associated with neutrino absorption with an initial value in
the disk estimated at $S/n_{b}\sim 5-10k_{B}$ and rise to $S/n_{b}\sim
15-20k_{B}$ in the later stages of the remnant evolution. We can deduce that,
in the model framework, the $\nu_{e}$ contributes highly to the rise in the
entropy compared to the $\nu_{\mu}$. The muon neutrinos have higher chemical
potentials compared to the electrons, this means the $\Omega_{0}$ becomes more
negative and $S$ becomes more positive for the same T, see Eqs. (9) and (II.1)
together.

Figure 3: We show the $Y_{i}$ for fixed entropy BNS remnants. The first panel
(top left) shows the initial condition at merger $t\sim 0$ ($t$ is time), and
the second panel (top right) shows the $Y_{i}$ in the remnant disk assuming
there was prompt collapse of the remnants to a black hole. The bottom panels
show two stages after the merger $t>0$.

In Fig. 3, we show four snapshots of the evolution stages of the postmerger NS
remnant. The first stage (top left) is the $Y_{i}$ in the core of the remnant
at the merger, $t\sim 0$. At this stage, the innermost part of the merging NS
cores $n_{b}>n_{0}$ does not experience significant shocks so the $S/n_{b}$ in
the central part of the remnant remains low, it is estimated to be around
$1\lesssim S/n_{b}[k_{B}]\lesssim 2$. This stage is characterized by a small
temperature profile in the core, mainly generated by matter compression of
degenerate nuclear matter above $n_{0}$. At this stage, we use the initial
conditions $S/n_{b}=1$ and $Y_{L,l}=Y_{L,e}=Y_{L,\mu}=0.1$ for the analysis,
because at $t\sim 0$, the lepton fractions in the cores of the two NSs is
frozen to the initial neutrino-less ($\nu$-less) cold weak equilibrium value,
which is estimated to be in the range $0.0\lesssim Y_{L,l}\lesssim 0.2$ (this
estimate was initially made for only electrons but we generalize it to include
muons) towards the higher baryon density regions. The second panel (top right)
is based on the assumption of prompt formation of a black hole. In this case,
the thermodynamic conditions change drastically. The formation of an apparent
horizon removes all the high-density part of the remnants leaving a cold disk
with an estimated temperature of about $T\lesssim 10\,{\rm MeV}$, with a
significant reprocessed lepton fraction of about $Y_{L,l}\,\sim\,0.25$. At
this stage the disk entropy is slightly higher than when massive NS is present
at the center, hence, we used $S/n_{b}=1.5k_{B}$ as our working assumption.
Comparing the $Y_{i}$ in the two panels we can see that the $Y_{i}$ in the
right panel has less isospin asymmetry due to excess $u$-quark relative to
$d$. Also, there is more $e$, $\mu$, $\nu_{\mu}$ and $\nu_{e}$ content due to
higher $Y_{L,l}$. At some dynamical timescales, a few milliseconds after the
merger, if the remnant does not form a black hole promptly, the inspiral arms’
activities at the disk’s central part increase the entropy of the remnant
matter, creating a strong correlation between the matter density and entropy.
The highest temperature of the stream is expected at this stage, however, they
may rapidly decrease because of fluid expansion and neutrino diffusion. The
lepton fraction is reset to the initial condition due to excess lepton
antineutrino emissions and absorption. The entropy at this stage is also
estimated to be between $2\leq S/n_{b}[k_{B}]\lesssim 10$. In the bottom
panels, we set lower and upper limits for $S/n_{b}$ to study the particle
distribution as the $S/n_{b}$ increases through the evolution of the merger
remnant. Comparing the first panel (bottom left) and the second panel (bottom
right) we observe that the NS remnant absorbs more neutrinos as the $S/n_{b}$
increases, particularly the $\nu_{e}$ is significantly enhanced while
$\nu_{\mu}$ only increases marginally. The isospin asymmetry also decreases as
$S/n_{b}$ increases. Comparing the $Y_{i}$ for the initial condition at merger
(top right panel) and the two panels below following the evolution of the NS
remnant, assuming it did not form a black hole promptly, the fraction of
$\nu_{e}$ significantly increases and the isospin asymmetry decreases along
the evolution lines. It is worth pointing out that, through the evolution
stages, only a fraction of $\nu_{e}$ show significant variation at all
densities as the NS remnant evolves. Additionally, higher $S/n_{b}$
facilitates the emergence of $\mu$ and $s$-quarks but their effects can be
seen at relatively low $n_{b}$ and does not show any significant variation
along the evolution stages at $n_{b}\approx 1.6n_{0}$ and above.

Figure 4: The EoSs for hot quark matter.

Fig. 4 shows the EoS of both the fixed entropy NS remnants and the uniform
temperature ones. We observe that, in the case of $T=50{\rm MeV}$, the EoS is
stiffer at low $\varepsilon$ and begins to soften at the higher $\varepsilon$
compared to the $S/n_{b}=6k_{B}$ case. This effect can be associated with the
increase of the strength of the term proportional to $n_{b}^{1/3}$ in the
equivalent quark mass as $T$ increases. Comparing the maximum masses and radii
data on Table 1, we observe that the fixed entropy NS remnant is more compact
than the uniform temperature one. Also, comparing their neutrino content,
Figs. 1 (top panel, right) and 3 (bottom panel, right), we can see that the
$S/n_{b}=6k_{B}$ remnant has visibly high $\nu_{e}$ content, particularly
towards the center of the NS remnant.

Interestingly, as we see in Fig. 4, the initial conditions for uniformly
heated remnant matter at $T=5{\rm MeV}$ coincides with the initial condition
set for the fixed entropy remnant matter, even though it has a relatively
higher core temperature of $T_{c}\sim 9{\rm MeV}$. Their EoSs exactly overlap
each other with no distinguishable differences. Comparing their particle
fractions, Figs. 1 (top panel, left) and 3 (top panel, left), they have almost
the same $\nu_{e}$ content and early appearance of the $s$-quark and $\mu$,
their $\delta$ are also comparably the same. Generally, hotter remnant matter
has stiffer EoS than their relatively colder counterparts, similarly, higher
$S/n_{b}$ remnant matter has stiffer EoS than relatively lower $S/n_{b}$ ones
since higher values of $S/n_{b}$ means higher $T$. In the case of
$S/n_{b}=1.5k_{B}$ an increase in the $Y_{L,l}$ stiffens the EoS as well. We
can deduce from the particle distribution in the NS remnant matter that a
delayed appearance of $\mu,\;\nu_{e}$ and $s$-quark rather softens the EoS and
leads to lower maximum masses and radii. Likewise, a higher $\nu_{e}$ content
in the NS remnant matter leads to a stiffer EoS. Moreover, stiffer EoS are
associated with low central energy and baryon densities as shown on Table 1.

Figure 5: The temperature profiles of the fixed entropy NS remnant matter as a
function of the $n_{b}$. The red stars in the curves represent the position of
the core temperature.

In Fig. 5, we show the temperature variations in the NS remnant matter as a
function of $n_{b}/n_{0}$. At the contact interface of the two NSs, matter
from their crust and the outer cores slips out and mix causing instability.
The cores originally behind these contact interfaces begin to fuse over time,
at this point, a series of compressions and expansions occur while the
remnants bounce severally. Hence, temperature and density increase immediately
after the merger due to these compressions, oscillations and bounce dynamics
[12, 119]. Besides, the high sound velocity of the nuclear matter
($c_{s}\gtrsim 0.2c$) and nuclear supra-densities ($n_{b}\gtrsim n_{0}$)
prevent the generation of hydrodynamics shocks in the cores of the two
coalescing NSs, thus, the temperature remains low (estimated to be $T\gtrsim
10{\rm MeV}$) in the core at merger.

In Fig. 5, we observe that the temperature profile is at its least at the
initial merging phase when $S/n_{b}=1$ ($T_{c}\sim 9{\rm MeV}$) compared to
the other stages when $S/n_{b}$ was higher (temperature profile for $t\sim
5$ms postmerger has been studied in [106] as a function of $n_{b}$). When the
remnant promptly forms a black hole, the $S/n_{b}$ increases, and the maximum
density decreases, as can be seen in Fig 3 for $S/n_{b}=1.5k_{B}$, which
corresponds to a higher temperature ($T_{c}\sim 12{\rm MeV}$) and lower
central density relative to the first stage. The evolution process terminates
when the remnant forms a black hole promptly. Furthermore, when the remnant
continues evolving as an NS, the cores of the coalescing NSs continue to fuse,
causing compression and shear dissipation, increasing the entropy and as a
result, the temperature. We choose a lower limit of $S/n_{b}=3k_{B}$ and an
upper limit of $S/n_{b}=6k_{B}$ to obtain a fair idea of the temperature
variations at this stage of the remnant evolution, we found $T_{c}\sim 27{\rm
MeV}$ and $T_{c}\sim 51{\rm MeV}$ respectively (higher temperatures in the
range of $T\sim 70-110{\rm MeV}$ were determined in [119] and $T\sim 158{\rm
MeV}$ reported in [112]). Comparing Figs. 5 and 3 we can deduce that higher
$S/n_{b}$ are associated with higher $\nu_{e}$ production and an early
appearance of $\mu$ and $s$-quark in the remnant matter.

Figure 6: The mass-radius diagram of hot NS remnant. The red dots represent
the precise point of the maximum mass and radii of the remnant star.

In Fig. 6, we determine the structure of the NS remnant by calculating the
mass-radius (M-R) diagram using the Tolman–Oppenheimer–Volkoff equation (TOV)
[120] given by the expressions

$\displaystyle\frac{dP(r)}{dr}$
$\displaystyle=-[\varepsilon(r)+P(r)]\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}-2M(r)r},$ (44) $\displaystyle\frac{dM(r)}{dr}$
$\displaystyle=4\pi r^{2}\varepsilon(r),$ (45)

with $r$ the radial coordinate, $M(r)$ the gravitational mass, $P(r)$ the
pressure, and $\varepsilon(r)$ the energy density, here, we use the natural
units ($G=\hbar=c=1$). The initial condition is two equal-mass NSs with a
combined mass estimated to be $2.74^{+0.04}_{-0.01}M_{\odot}$ and individual
masses between 1.17 and 1.6 M⊙ [1, 121, 40], neglecting the effect of spin in
the analysis based on the assumption that most systems are practically not
rotational at merger [122]. Generally, NSs have a known mass threshold of
about 2M⊙ [42, 41] whereas stellar black hole masses observed in binaries are
greater than the GW170817 components [123, 124]. We compare our results with
the measured mass and radii of the millisecond pulsar PSR J0740+6620
determined through the observation and analyses of the NICER X-ray observatory
data. The estimated measured masses and radii were 2.07${}^{+0.07}_{-0.07}{\rm
M}_{\odot}$ and 2.08${}^{+0.09}_{-0.09}{\rm M}_{\odot}$ at 68% confidence
level (CL); and radii 12.39${}^{+1.30}_{-0.98}{\rm km}$ [43] and
13.71${}^{+2.61}_{-1.50}{\rm km}$ [125] determined by different groups. It is
worth mentioning that the mass of this pulsar was also independently
determined to be 2.08${}^{+0.07}_{-0.07}{\rm M}_{\odot}$ through Shapiro delay
effect [41]. The mass and radii of PSR J0030+0451 determined through fitting
to the NICER data led to two predictions (at 68% CL);
$M=1.34^{+0.15}_{-0.16}{\rm M}_{\odot}$, $R=12.71^{+1.14}_{-1.19}{\rm km}$
[40] and $M=1.44^{+0.15}_{-0.14}{\rm M}_{\odot}$, $R=13.02^{+1.24}_{-1.06}{\rm
km}$ [126].

From our results, the M-R diagram of the initial conditions for fixed $T$
($T=5{\rm MeV}$) and fixed $S/n_{b}=1k_{B}$ overlap, similar to their Mmax’s
on Table 1, and are also marked with red dots on the curves. On the other
hand, the M-R diagram of $T=50{\rm MeV}$ NS remnant matter has the largest
size with relatively small Mmax compared to the $S/n_{b}=6k_{B}$ NS remnant
matter with the highest Mmax even though it’s core temperature is higher than
50 MeV. Hence, fixed $S/n_{b}$ NS remnants appear more compact than fixed $T$
NS remnants. Additionally, we observed that the mass and radius of the remnant
increase with an increasing $S/n_{b}$. In the case of prompt black hole
formation, the remnant mass also increases with increasing lepton number and
$S/n_{b}$. Analysing Figs. 1 and 3 together with the M-R diagram, we can
deduce that fewer neutrino content (particularly $\nu_{e}$) and early
appearance of the $\mu$ and $s$-quark in the remnant matter reduces its
pressure thereby softening the EoS and consequently, reducing the maximum mass
and radii. On the other hand, the existence of NSs with a larger mass and
radius when increasing $T$ or $S/n_{b}$ are indeed a consequence of the
enhancement of the strength of the term proportional to $n_{b}^{1/3}$ in the
equivalent quark mass when $T$ increases towards $T_{C}$.

Figure 7: The sound velocity through the NS remnant matter as a function of
$n_{b}/n_{0}$. The horizontal gray line represents the conformal limit,
$c_{s}^{2}=1/3$, above this limit matter is considered to be in a hadronic
state, and below it, matter is considered asymptotically free.

In Fig. 7, we show the result of the square of sound velocity, $c_{s}^{2}$ as
a function of the $n_{b}/n_{0}$. The $c_{s}^{2}$ is determined using the EoS
as inputs from the expression

$c_{s}^{2}=\dfrac{\partial P}{\partial\varepsilon},$ (46)

where $c_{s}$ is measured in the unit of constant speed of light ($c$). The
$c_{s}$ helps to distinguish between the properties of strongly interacting
matter using conformal symmetry arguments and more importantly measure the
stiffness of the EoS. The confined matter (hadron) phase is not conformally
symmetric due to chiral symmetry breaking, but the deconfined QM phase, on the
other hand, is conformally symmetric due to asymptotic freedom. The matter is
classified as exactly symmetric under conformal transformation when
$c_{s}^{2}=1/3$, in highly dense free QM ($n_{b}>40n_{0}$) this limit, is
approached from below [127] due to quark deconfinement, asymptotic freedom,
and small quark masses. The $c_{s}$ is estimated to be $c_{s}^{2}\gtrsim 0.5$
in hadronic matter, it must also satisfy causality constraint $c_{s}^{2}\leq
1$, and thermodynamic stability condition $c_{s}^{2}>0$ for all kinds of
matter compositions [128]. It is an important parameter for determining quark
core in massive NSs (i.e. hybrid NSs) [38, 49] and for determining the
behavior of QM that form QSs [34], whether the quarks behave as being confined
or free. The results above show that the $c_{s}$ is higher in high $S/n_{b}$
remnant matter, in general. Hence, more massive remnants are associated with
high $c_{s}$ than less massive ones. However, in the case of larger sized and
less compact remnants (we are comparing $T=50{\rm MeV}$ and $S/n_{b}=6k_{B}$)
the $c_{s}$ is relatively higher at low $n_{b}$ and falls slightly as $n_{b}$
increases. The small tail seen in the $S/n_{b}=6k_{B}$ curve at low $n_{b}$ is
due to the different behavior of the $u$ and $d$-quarks at this entropy (see
Fig. 3), in that region. They show a sharp rise at low $n_{b}$ and start
decreasing steadily relative to their behavior in other stages of the remnant
evolution. Comparing the $c_{s}$ curves to the particle distributions in Figs.
1 and 3, we can infer that the $\nu_{e}$ has a significant impact on the
$c_{s}$ in the remnant matter. The early appearance of $\nu_{e}$ in the
remnant shows higher $c_{s}$ at low $n_{b}$ and a higher rise of $\nu_{e}$
toward the center of the remnant results in higher $c_{s}$ at higher $n_{b}$.

Figure 8: The polytropic index as a function of $n_{b}/n_{0}$. The horizontal
gray line is the conformal limit $\gamma=1$.

In Fig. 8, we show the polytropic index as a function of $n_{b}/n_{0}$
determined through the relation

$\gamma=\dfrac{\partial\ln P}{\partial\ln\varepsilon},$ (47)

using the EoS as input. The $\gamma$ is also used to classify the inner
composition and the forms of matter in the NS remnant matter. The application
of the $\gamma$ is similar to the $c_{s}$ discussed above. They constitute the
two most important parameters for determining the classes of matter that form
an NS or exist in the core of the NS (if the existence of a hybrid NS is
assumed [38]) using the EoS. A conformally symmetric matter is scale-
independent and does not involve any dimensionful parameter, therefore, $P$
and $\varepsilon$ are expected to be proportional given rise to $\gamma=1$. On
the contrary, the value of $\gamma$ in the hadronic matter, for instance, is
estimated to be $\gamma\approx 2.5$ around and above the $n_{0}$ in chiral
effective theories, setting a threshold for classifying hadrons and free quark
matter [127, 129]. This is expected because the ground state of the QCD theory
does not have its usual approximate chiral symmetry at low to intermediate
baryon densities [130]. From Fig. 8, we observe that $\gamma$ decreases as
$n_{b}$ increases and attains its minimum value $\gamma\approx 1.1$ at the
high-density region. This value is slightly above the conformal limit (the
horizontal gray line), however, it is in good agreement with conformal matter
classification, comparing this results to the new limit set in [38] after
examining a wide range of EoSs, estimating it at $\gamma\leq 1.75$.
Additionally, we observe that heavier remnants approach the $\gamma=1$
threshold faster than less massive remnants.

##  IV Final remarks

We studied the BNS postmerger remnants in the presence of $e$ and $\mu$ and
their corresponding neutrinos, assuming that the merger remnant is a neutron
star composed of self-bound free quarks, using DDQM. Generally, the aftermath
of the merger has three different outcomes; 1) it may form a stable NS, 2)
form a black hole, or 3) a supramassive rotationally supported NS, which will
later collapse to form a black hole due to angular momentum losses. The
possibility of any of these three scenarios occurring depends on the masses
and the EoSs of the NS binaries participating in the merger event. In each
scenario, the event will affect the GW signal and its electromagnetic
counterpart (see Refs. [131, 132, 119] for more detailed discussions).
Assuming a stable NS was formed in the framework of this study, a few seconds
after the merger, the remnant will go through deleptonization, neutrino
diffusion, and thermal radiations to form a stable strange quark star (SQS) at
$T=0$, several years later. In that case, the $M_{\rm max}=2.18{\rm
M}_{\odot}$ and the radius $R=13.86{\rm km}$ with dimensionless tidal
deformability parameter $\Lambda_{1.4}=2163$, as shown on Table II of [34].
Hence, as the remnant cools down it shrinks, and its mass and radius decrease
as compared to the hotter counterparts. That notwithstanding, the model
strongly violates the $\Lambda_{1.4}$ extracted from the GW170817 event with
an upper bound of $\Lambda_{1.4}\leq 800$. However, this seems to be a problem
with most of the phenomenological quark models used to study SQSs; a similar
extreme violation of the $\Lambda_{1.4}$ was found in [133] using a confining
quark model and in [73] using quasiparticle model. These violations may be a
result of higher radii beyond the $\sim 13{\rm km}$ threshold generated by
these models to achieve the $2$M⊙ minimum threshold as projected in [134]. The
violation suggests that the two NS binaries that were involved in the merger
event that led to the observation of the GW170817 might not be quark stars.

The main findings of our investigation are:

  * •

In the upper panels of Fig. 1 and in Fig. 3 we observed a significant increase
in $\nu_{e}$ as the merger remnant evolves with increasing $T$ and $S/n_{b}$.
The $\nu_{e}$’s shift more towards the low $n_{b}$ region and their population
also increases significantly toward the center of the remnant matter along the
evolution lines. The $\nu_{\mu}$ population drops marginally at low $n_{b}$
and does not show any significant increase at higher $n_{b}$ along the
evolution lines. The $\delta$ of the remnant matter also decreases with
increasing $T$ and $S/n_{b}$ due to excess $u$ over $d$-quarks along the
evolution lines. As expected, an increase in $T$ and $S/n_{b}$ facilitates the
early appearance of $s$-quark in the remnant matter, a similar effect was
observed with the $\mu$’s. Particle distribution under conditions of merger
involving hadronic matter can be found in [115, 135], their results
qualitatively agree with our findings considering that neutrons are rich in
$u$, protons are rich in $d$ and hyperons contain $s$-quarks.

  * •

In the lower panels of Fig. 1, we found that massless particles like $\nu_{e}$
and $\nu_{\mu}$ contribute less to the overall $S/n_{b}$ of the system.
Generally, an increase in $T$ mutually increases the $S/n_{b}$ of each
particle in the system thereby increasing the net $S/n_{b}$ as shown in Fig.
2. Additionally, the $S/n_{b}$ decreases with increasing $n_{b}$ among the
individual particles, a similar effect is observed in the net system of
particles in Fig. 2. This reinforces the point that the $S/n_{b}$ spreads from
the disk towards the center of the remnants (higher $S/n_{b}$ in the time
interval of $\sim 40{\rm ms}$ postmerger have been reported in [118]).

  * •

In Fig. 4, we found that a delayed appearance of the $s$-quark and $\mu$ in
the remnant matter coupled with a large $\delta$ soften the EoS resulting into
lower Mmax and $R$. On the other hand, a shift of the $\nu_{e}$ towards the
low $n_{b}$ region increases the pressure in the remnant matter and stiffens
the EoS leading to higher Mmax and $R$. The $\nu_{\mu}$ constituent does not
significantly affect the EoS along the evolution lines however, its marginal
decrease at low $n_{b}$ and marginal increase at higher $n_{b}$ coincides with
the stiffening of the EoS as the remnant evolves. Generally, we observed that
the remnants’ EoS becomes stiffer as $T$ and $S/n_{b}$ increase, as a result
of the $n_{b}^{1/3}$ dependence of the equivalent quark mass and the
enhancement of this term by driving $T$ to the critical deconfinement
temperature.

  * •

We observed a connection between temperature rise, neutrino production, and
the emergence of different particles in the remnant matter. The matter with
the highest temperature profile in Fig. 5 corresponds to the matter with the
highest neutrino content, particularly $\nu_{e}$, early appearance of $s$ and
$\mu$, and less $\delta$. We find that the temperature increases with $n_{b}$
and higher values of $S/n_{b}$, reaching core temperatures ranging between
$T_{c}\sim 9-51{\rm MeV}$ along the evolution lines considered. These values
are within the lower limits reported in Refs. [112, 12, 116] for postmerger
remnant matter. Temperature increases in the remnant reduce both the central
energy and baryon densities. It should be noted that temperatures in quark
matter are generally lower than a corresponding hadronic matter investigated
under the same conditions due to a higher degeneracy in quark matter.

  * •

The EoSs calculated and analyzed produced non-rotating spherically symmetric
static compact objects with maximum masses of between $2.25-2.67$M⊙, which is
fairly in agreement with the estimated combined masses of the NS binaries that
were involved in the GW170817 event [1] and the secondary component in the
GW190814 event with maximum mass of $2.59^{+0.08}_{-0.09}$M⊙ [136]. The
maximum masses also satisfy the threshold set by PSR J0740+6620, and the
remnant radii determined are within the upper limits of the radius determined
for PSR J0740+6620 by [125] through the analyses of the NICER observatory data
(see the result in Fig. 6 and Table 1). Since the maximum masses obtained here
are higher than the ones obtained for PSR J0740+6620, even though their radii
correspond to the upper limit, we can infer that the NS remnants analyzed here
are more compact. Besides, the results satisfy the maximum NS mass constraints
set through more massive PSR J2215+5135 [137] and PSR J0952-0607 [67] observed
pulsars.

  * •

Finally, we analyzed the composition of the remnant matter through the
conformal symmetry arguments using the EoS as input. The results obtained for
$c_{s}$ and $\gamma$ are presented in Figs. 7 and 8 respectively. The graphs
show $c_{s}^{2}<1/3$ and $\gamma\gtrsim 1.1$, in conformity with conformal
matter classification detailed in [38] through a wide range of EoSs analyzed.

Consequently, we have established that the BNS merger remnant can be an SQS
that satisfies the 2M⊙ constraint of NSs. The $\mu$ populations in the SQS
remnant showed a significant variation as the remnant evolved, influencing the
EoS temperature profile, and the macroscopic structure of the SQS remnant. On
the other hand, we found from our calculations that the presence of the
$\nu_{\mu}$ increases the maximum mass of the NS remnants by approximately
$\sim 0.94$% and radii by $\sim 0.75$%, drops the central energy density by
$\sim 0.086$%, the baryon density by $\sim 0.41$% and almost $\sim 0.92$% drop
in the core temperature as well. Also, the presence of $\nu_{e}$ in the
remnant matter significantly influences the EoS along the evolution lines as
compared to the $\nu_{\mu}$.

##  V Acknowledgements

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: National Council for Scientific and Technological
Development 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).This work is a part of the project Instituto Nacional de Ciência
e Tecnologia - Física Nuclear e Aplicações Proc. No. 464898/2014-5.

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