# Magnetic-Field Induced Deformation in Hybrid Stars

Ishfaq A. Rather1<EMAIL_ADDRESS>Asloob A. Rather2,3 V.
Dexheimer4 Ilídio Lopes1 A. A. Usmani2 S. K. Patra5,6 1Centro de Astrofísica e
Gravitação-CENTRA, Instituto Superior Técnico-IST, Universidade de Lisboa-UL,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal  2Department of Physics, Aligarh
Muslim University, Aligarh 202002, India  3Department of School Education,
Govt. of Union Territory of Jammu and Kashmir, India  4 Department of Physics,
Kent State University, Kent OH 44242 USA  5Institute of Physics, Bhubaneswar
751005, India  6 Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, India

###### Abstract

The effects of strong magnetic fields on the deconfinement phase transition
expected to take place in the interior of massive neutron stars is studied in
detail for the first time. For hadronic matter, the very general density-
dependent relativistic mean-field (DD-RMF) model is employed, while the
simple, but effective Vector-Enhanced Bag model (vBag) model is used to study
quark matter. Magnetic-field effects are incorporated into the matter equation
of state and in the general-relativity solutions, which also satisfy Maxwell’s
equations. We find that, for large values of magnetic dipole moment, the
maximum mass, canonical-mass radius, and dimensionless tidal deformability
obtained for stars using spherically-symmetric TOV equations and axisymmetric
solutions attained through the LORENE library differ considerably. The
deviations depend on the stiffness of the equation of state and on the star
mass being analyzed. This points to the fact that, unlike what was assumed
previously in the literature, magnetic field thresholds for the correct
assumption of isotropic stars and the proper use of TOV equations depend on
the matter composition and interactions.

##  I Introduction

Matter under extreme densities, temperatures, and magnetic fields is among the
most popular and fascinating current research topics. Neutron stars (NSs)
provide the perfect environment to study physics under extreme conditions.
Recent advances in NS observation have yielded intriguing conclusions
regarding their maximum mass, canonical mass ($1.4$ M⊙), radius, tidal
deformability, and so on. However, there is still a large uncertainty
concerning the equation of state (EoS) of dense matter. The density inside NSs
reaches several times nuclear saturation density ($\rho_{0}=10^{14}$
gcm${}^{-3}=0.15$ fm-3), making it difficult to determine their inner
structure. As interior densities approach $10^{15}$ g cm-3, theoretical models
describing cold and dense matter, calibrated around $\rho_{0}$ for isospin-
symmetric nuclear matter (SNM), must be extrapolated in both density and
isospin asymmetry. To put these ideas to the test, the structure of NSs is
computed starting from the energy–momentum tensor, solving equations for
hydrostatic equilibrium, and comparing them to astrophysical observations. One
constraint that all NS models must meet is the ability to explain the highest
measured NS mass, which according to recent astrophysical data is above $2$ M⊙
[1, 2].

The observation of gravitational waves emitted by NS mergers, as well as their
electromagnetic counterparts, has marked a major breakthrough in multi-
messenger astronomy. The LIGO/VIRGO detection of the binary NS merger event
GW170817 [3, 4] ushered in a revolution, which constrained the maximum mass
and the tidal deformability of NSs and, hence, the EoS. In the following
LIGO/VIRGO run, GW190425 was observed [5] in addition to two black hole-NS
mergers, GW200105 and GW200115 [6]. More recently, the gravitational wave
event GW190814 was observed [7] involving a black hole with mass $22.2-24.3$
M⊙ and a secondary object with mass $2.50-2.67$ M⊙. This gravitational-wave
event received a lot of attention due to the nature of its secondary component
being either an extremely massive NS or the lightest black hole encountered
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29].

From an electromagnetic observation perspective, the extremely precise mass
detections of the pulsars PSR J1614-2230 [30] and PSR J0348+0432 [1] with $M$
$\approx$ $2$ M⊙ have also ushered in a new era of NS research. Since then,
the mass measurement of PSR J1614-2230 was corrected [31], and a new high-mass
pulsar, PSR J0740+6620 with $M>2$ M⊙, was identified [32, 2]. The existence of
these massive NSs rules out a number of soft EoSs, in which the expected
maximum mass of the compact object is less than the observed maximum mass.
Meanwhile, it has become long accepted in the nuclear physics community that
hyperons emerge in dense medium under NS conditions at a few times $\rho_{0}$
[33, 34, 35]. However, the inclusion of hyperons softens the EoS enough to
rule out the occurrence of $2$ M⊙ NSs in many models, unless the EoS is stiff
enough to prevent that, which is related to predicting stellar radii that are
large. The NASA Neutron Star Interior Composition Explorer (NICER) measured
the canonical neutron-star radii to be small, in the range $11.52-14.26$ km
[36, 37] from the analysis of PSR J0030+0451. More recent constraints on the
NS radius from NICER measurements of the massive pulsar J0740+6620 provide the
wide range $11.41-16.30$ km [38, 39]. But, more constraining, canonical
neutron-star radii were estimated from the tidal deformability measured in the
event GW170817 [4] to be in the range $9.1-13.3$ km. Together, all of these
give rise to the so-called hyperon puzzle, which is a severe challenge in
nuclear physics [40]. A possible way out of the puzzle is to consider
deconfinement to quark matter, as discussed in the following.

NSs are not just extremely dense objects, but they are also known to present
extremely strong magnetic fields. Combining a simple magnetic dipole model and
known pulsar spin-down data, surface magnetic fields $\sim 10^{12}-10^{13}$ G
are obtained [41]. But a few X-ray isolated NSs and rotating radio transients
present even stronger magnetic fields. Soft gamma repeaters (SGRs) and
anomalous X-ray pulsars (AXPs) present the strongest magnetic fields found in
NSs with surface values of the order 1014-1015 G [42, 43, 44, 45]. The similar
features of these two groups of objects have resulted in the development of a
unified magnetar model to explain their observed traits. Note that fast radio
bursts (FRBs) also show magnetar behavior [46, 47, 48, 49, 50]. But, more
interestingly, data from the source 4U 0142+61 for slow phase modulations in
hard x-ray pulsations (interpreted as free precession) suggests magnetic
fields of the order of $10^{16}$ G [51, 52] inside this pulsar. Magnetic
fields of the order of $10^{16}$ G or higher are also expected to be produced
in NS mergers [53, 54, 55, 56, 57].

Because the maximal magnetic field in the interior of magnetars cannot be
measured directly, it is commonly predicted using the virial theorem - most
estimates lead to a theoretical maximum of the order of $10^{18}$ G [58]. A
strong magnetic field can have several implications for NSs, such as modifying
the EoS due to Landau quantization of the constituent charged particles [59,
60], changing the the energy–momentum tensor, and breaking of stellar
spherical symmetry. Several studies have shown the effect of the magnetic
field on the NS EoS and on stellar properties [61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 74, 79, 80, 81]. Most of the previous
studies that include magnetic field effects on the EoS use isotropic
Tolman–Oppenheimer–Volkoff (TOV) equations to determine the relevant stellar
properties [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92]. However, it should be
emphasized that for strong magnetic fields, below the threshold at which
Landau quantization effects on the EoS become non-negligible, the deviations
from spherical symmetry can already be considerable [93]. Strong magnetic
fields can substantially deviate NS configurations from spherical symmetry
and, in this case, spherically symmetric TOV equations can no longer be used
to describe their macroscopic structure.

Concerning the EoS, in a density regime that ab-initio methods fail to
describe, relativistic mean-field (RMF) models have been successful in
characterizing both finite and infinite nuclear matter [94]. The primary
mechanism in this case consists of hadrons interacting through a mean field of
mesons. The use of different mesons such as $\sigma$, $\omega$, $\rho$, and
$\delta$ have improved the predictions of SNM properties and constrained them
well inside the expected regime [95, 96, 97, 98, 97]. The density-dependent
RMF (DD-RMF) model replaces the self- and cross-coupling of various mesons in
the classic RMF model with density-dependent coupling constants [99].
Parameter sets like DD-ME1 [100] and DD-ME2 [101] generate very massive NSs
with a $2.3-2.5$ M⊙ maximum mass. Several new DD-RMF parameter sets such as
DD-LZ1 [102] and DD-MEX [103] also produce stiff EoSs. Note that the density-
dependent functional DD2 EoS [104, 105, 106] with excluded volume correction
[107], which accounts for the finite size of nucleons caused by the repulsive
attraction between their internal quarks caused by Pauli blocking effects, is
widely used in simulations of e.g. binary NS mergers [108, 109, 110, 111] or
supernovae [112, 113].

The idea of deconfined quark matter in the interior of NSs have been studied
since the eighties [114]. A NS with hadronic matter in the core followed by a
phase transition to quark matter at least a couple of times saturation density
is referred to as hybrid star. To study quark matter in NS cores, several
different models have been used. The simple MIT Bag model [115, 116, 117] was
the first realistic description proposed to study pure quark and hybrid stars.
The Nambu-Jona-Lasinio (NJL) model, that originally described the mass gap in
the Dirac spectrum of nucleons (in analogy to the BCS mechanism for
superconductivity) [118, 119], was modified to describe interacting quark
matter [120, 121, 122]. Somehow between these, the modified version of the Bag
model, Vector-Enhanced Bag model (vBag) [123], was introduced as an effective
model to study astrophysical processes. This model is preferred over the
simple bag model because it takes into consideration repulsive vector
interactions as well as Dynamic Chiral Symmetry Breaking (D$\chi$SB). The
repulsive vector interaction and the introduction of a mixed-phase allow one
to reproduce hybrid stars with masses larger than the $2$ M⊙ limit.

An alternative for a sharp first-order phase transition emerges when one
considers that the surface tension of quark matter is not infinite. This so-
called non-congruent phase transition gives rise to a mixture of phases in
which quantities such as electric charge neutrality are fulfilled globally
[124]. In astrophysics, this is referred to as a Gibbs construction and
generates a phase in which hadrons “melt” into their constituent quarks
gradually over kilometers inside NSs. See Ref. [125] and references therein
for an extended review of the topic. A more thorough treatment of the mixed
phase including contributions from surface and Coulomb effects requires
accurate knowledge of the surface tension between two phases, which is still
unknown, having an estimated value ranging from 10-100 MeV fm-2 [126, 127,
128, 129, 130]. The Gibbs construction produces results that are somehow
similar to those obtained with the lower value of the above-mentioned surface
tension range, and so the contribution from surface and Coulomb effects can be
ignored (when calculating properties such as stellar masses and radii) [131,
132]. In this work, we study hybrid stars by means of a Gibbs construction
that gives rise to a mixed phase connecting the hadronic phase described by
the DD-RMF model and the quark phase described by the vBag model.

When the magnetic field effects in hadronic and quark phases are compared, it
is found that changes in the population of charged particles in the hadronic
phase are larger than for uncharged particles, noting that all quarks have
electric charge [72, 82, 86]. Also, a difference due to particle masses is
observed, baryons having a substantially larger mass than quarks and leptons
[133]. These two characteristics imply that magnetic field effects should be
more pronounced in quark matter. On the other hand, it has been shown that
changes in the amount of (baryonic) charge fraction affects more strongly the
hadronic phase than the quark phase [134]

To understand better how these effects affect macroscopic stellar properties,
we use our EoS with quark deconfinement in the publicly available Language
Objet pour la RElativité NumériquE (LORENE) library [135, 136, 137] for the
first time to investigate in detail quark deconfinement. LORENE solves the
coupled Einstein-Maxwell field equations in order to determine stable
anisotropic magnetic-star configurations. However, to highlight and calculate
the deviation from spherical symmetry produced in stellar properties, we also
analyze results from TOV solutions.

Our paper is organized as follows: the hadronic and quark matter EoSs,
together with the contribution from photons, are discussed in Sec. II. The
mixed phase region and the deconfinement phase transition properties are
discussed in Sec. III. The equations governing macroscopic NS properties such
as mass and radius are discussed in Sec. IV. In Sec. V, the magnetic field
profile employed to study magnetic effects on NS EoSs is discussed. Results
for magnetic field effects on the EoS with quark deconfinement and hybrid
stars are explained and discussed in Sec. VI. Finally, the summary and
concluding remarks are presented in Sec. VII.

##  II Equation of State

###  II.1 Hadronic matter

The most basic and simple RMF Lagrangian involves the scalar-isoscalar
$\sigma$ and vector-isoscalar $\omega$ mesons without higher-order
interactions [138], resulting in a huge SNM incompressibility $K_{0}$ [94]. A
nonlinear self-coupling of the $\sigma$ field was then incorporated by Boguta
and Bodmer [95], which reduced the value of SNM incompressibility to realistic
levels. The contribution from the vector-isovector $\rho$ and scalar-isovector
$\delta$ mesons was later introduced to study isospin asymmetry effects.

The various non-linear meson coupling terms present in the RMF models are
replaced by the density-dependent nucleon-meson coupling constants in the DD-
RMF [99, 100, 101, 102, 103, 104], Density-Dependent Relativistic Hartree [99]
and also in the Density-Dependent Relativistic Hartree-Fock (DD-RHF) [139,
140, 141] models. Their couplings allow for a consistent study of NSs and
yield results that are comparable with other more complicated models. The
contribution of the rearrangement term self energies to DD-RMF field equations
is the most significant difference from normal RMF models and ensures
thermodynamical consistency.

The hadronic matter DD-RMF matter Lagrangian density (with a free Fermi gas of
leptons) is given by:

$\displaystyle\mathcal{L}_{H}$
$\displaystyle=\sum_{B}\bar{\psi}_{b}\Biggl{\\{}\gamma_{\mu}\Bigg{(}iD^{\mu}-g_{\omega}(\rho_{B})\omega_{\mu}-\frac{1}{2}g_{\rho}(\rho_{B})\rho_{\mu}\tau\Bigg{)}$
$\displaystyle-\Bigg{(}M_{B}-g_{\sigma}(\rho_{B})\sigma\Bigg{)}\Biggr{\\}}\psi_{B}+\frac{1}{2}\Bigg{(}\partial^{\mu}\sigma\partial_{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2}\Bigg{)}$
$\displaystyle-\frac{1}{4}W^{\mu\nu}W_{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}R^{\mu\nu}R_{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\rho_{\mu}\rho^{\mu}$
$\displaystyle+\sum_{l}\bar{\psi}_{l}(i\gamma_{\mu}D^{\mu}-m_{l})\psi_{l}\ ,$
(1)

where $B$ sums over the baryon octet
($n,p,\Lambda,\Sigma^{+},\Sigma^{0},\Sigma^{-},\Xi^{0},\Xi^{-}$) and $l$ over
leptons $e^{-}$ and $\mu^{-}$. $\psi_{B}$ and $\psi_{l}$ represent the baryon
and lepton Dirac fields and $M_{B}$ and $m_{l}$ the baryon and lepton masses,
respectively. The mesonic tensor fields and covariant derivatives are defined
as

$\displaystyle W^{\mu\nu}$
$\displaystyle=\partial^{\mu}W^{\nu}-\partial^{\nu}W^{\mu},$ $\displaystyle
R^{\mu\nu}$ $\displaystyle=\partial^{\mu}R^{\nu}-\partial^{\nu}R^{\mu},$
$\displaystyle D^{\mu}$ $\displaystyle=\partial^{\mu}+iQA^{\mu},$ (2)

where $W_{\mu\nu}$ and $R_{\mu\nu}$ are the antisymmetric tensor fields of
$\omega$ and $\rho$ vector mesons. $\sigma$ is a scalar meson. $A^{\mu}$ is
the photon field that couples to baryons and leptons with electric charge $Q$.

The isoscalar density-dependent coupling constants for the DD-RMF parameter
set are written as a function of baryon (number) density $\rho_{B}$

$g_{i}(\rho_{B})=g_{i}(\rho_{0})f_{i}(x)\ ,$ (3)

where the function $f_{i}(x)$ is given by

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}},\
i=\sigma,\omega\ ,$ (4)

with $x=\rho_{B}/\rho_{0}$. Additional constraints for the function
$f_{i}(1)=1$,$f^{{}^{\prime\prime}}_{\sigma}(1)=f^{{}^{\prime\prime}}_{\omega}(1)$,
$f^{{}^{\prime\prime}}_{i}(0)=0$ reduce the number of free parameters from
eight to three in the Eq. 4. Among them, the first two constraints are

$a_{i}=\frac{1+c_{i}(1+d_{i})^{2}}{1+b_{i}(1+d_{i})^{2}},\ 3c_{i}d_{i}^{2}=1\
.$ (5)

For the isovector $\rho$ and $\delta$ mesons, the density-dependent coupling
constants are given by an exponential dependence

$g_{i}(\rho_{B})=g_{i}(\rho_{0})\exp[-a_{i}(x-1)]\ .$ (6)

The coupling constants of the nucleons to the $\sigma$ and $\omega$ mesons at
saturation are determined from fitting saturation density and binding energy
for SNM. The coupling constant of the nucleons to the $\rho$ meson at
saturation is fitted by reproducing the empirical saturation properties of
nuclear matter, such as the symmetry energy. The coupling constants of the
hyperons to the vector mesons at saturation are determined from SU(6) symmetry

$\displaystyle\frac{1}{2}g_{\omega\Lambda}$
$\displaystyle=\frac{1}{2}g_{\omega\Sigma}=g_{\omega\Xi}=\frac{1}{3}g_{\omega
N},$ $\displaystyle\frac{1}{2}g_{\rho\Sigma}$
$\displaystyle=g_{\rho\Xi}=g_{\rho N},\ \ g_{\rho\Lambda}=0\ .$ (7)

The coupling constants of the hyperons to the $\sigma$ meson at saturation are
determined from fitting the $\Lambda$ hyperon optical potential for SNM

$U_{\Lambda}^{N}(\rho)=g_{\omega\Lambda}\omega_{0}+\sum_{R}-g_{\sigma\Lambda}\sigma_{0}\
,$ (8)

to results obtained from lattice calculations [142, 143], reproducing the
following potentials: $U_{\Lambda}^{N}(\rho_{0})=-30$ MeV,
$U_{\Sigma}^{N}(\rho_{0})=+30$ MeV, and $U_{\Xi}^{N}(\rho_{0})=-14$ MeV. They
correspond to the following coupling values at saturation:
$g_{\sigma\Lambda}/g_{\sigma N}=0.6105$, $g_{\sigma\Xi}/g_{\sigma N}=0.3024$,
and $g_{\sigma\Sigma}/g_{\sigma N}=0.4426$.

For a uniform magnetic field locally pointing in the $z$-direction,
$B=B\hat{z}$, the Fermi energy of a charged baryons and leptons, respectively,
becomes [62]

$E_{B}^{*}=\sqrt{k_{zB}^{2}+M_{B}^{*2}+2\nu|Q_{B}|B}\ ,$ (9)
$E_{l}=\sqrt{k_{zl}^{2}+m_{l}^{2}+2\nu|Q_{l}|B}\ ,$ (10)

where the baryon effective mass is
$M_{B}^{*}=M_{B}-g_{\sigma}(\rho_{B})\sigma$. The quantity
$\nu=\Big{(}n+\frac{1}{2}-\frac{1}{2}\frac{q}{|q|}\sigma_{z}\Big{)}=0,1,2,...$
indicates the Landau levels, $n$ is the orbital angular momentum quantum
number, and $\sigma_{z}$ the Pauli matrix. The highest value of $\nu$ is
obtained under the condition that the Fermi momentum of each particle is real.
This gives us

$\displaystyle\nu_{max}$
$\displaystyle=\Bigg{\lfloor}\frac{E_{B}^{*2}-M_{B}^{*2}}{2|Q_{B}|B}\Bigg{\rfloor},$
$\displaystyle\nu_{max}$
$\displaystyle=\Bigg{\lfloor}\frac{E_{l}^{2}-m_{l}^{2}}{2|Q_{l}|B}\Bigg{\rfloor}\
,$ (11)

for charged baryons and leptons, respectively.

With all baryons from the octet and leptons included, the NS chemical
equilibrium conditions between different particles is

$\displaystyle\mu_{B}=\mu_{n}=\mu_{\Sigma^{0}}=\mu_{\Xi^{0}},$
$\displaystyle\mu_{p}=\mu_{\Sigma^{+}}=\mu_{n}-\mu_{e},$
$\displaystyle\mu_{\Sigma^{-}}=\mu_{\Xi^{-}}=\mu_{n}+\mu_{e},$
$\displaystyle\mu_{\mu}=\mu_{e}\ .$ (12)

The (electric) charge neutrality condition follows as

$\rho_{p}+\rho_{\Sigma^{+}}=\rho_{e}+\rho_{\mu^{-}}+\rho_{\Sigma^{-}}+\rho_{\Xi^{-}}\
.$ (13)

The equations of motion and EoS in the presence of a magnetic field are
discussed in Appendix (A).

###  II.2 Quark matter

The MIT bag model has been extensively used to describe quark matter in NSs.
In the original description [117, 115, 116], quarks are considered to be free
inside a bag and thermodynamic properties are simply derived from a free Fermi
gas model. The vector-enhanced bag (vBag) model [123], a modified version of
the bag model, was introduced as an effective model for studying astrophysical
processes. It is preferred over the simple bag model because it takes into
consideration dynamic chiral symmetry breaking (D$\chi$SB) as well as
repulsive vector interactions. The repulsive vector interaction is important
as it permits hybrid stars to achieve the $2$ M⊙ maximum mass limit [144, 12,
145, 146, 8, 147] and, hence, satisfy the mass constraints from pulsars like
PSR J1614-2230 [30], PSR 0348+0432 [1], and PSR J0740+6620 [32].

The Lagrangian density for the vBag model (with a free Fermi gas of leptons)
reads

$\displaystyle\mathcal{L}$
$\displaystyle=\sum_{f}[\psi_{f}(i\gamma_{\mu}\partial_{\mu}-m_{f}-B)\psi_{f}]\Theta_{H}$
$\displaystyle-
G_{V}\sum_{f}(\bar{\psi_{f}}\gamma_{\mu}\psi_{f})^{2}+\sum_{l}\psi_{l}\gamma_{\mu}(i\partial_{\mu}-m_{l})\psi_{l}\
,$ (14)

where $u$, $d$, and $s$ quarks and $e^{-}$ and $\mu^{-}$ leptons are denoted
by subscript $f$ and $l$, respectively. $B$ denotes the bag constant and
$\Theta_{H}$ is the Heaviside step function which allows for a
confinement/deconfinement of the bag [117]. The vector interaction is
introduced via the coupling of vector-isoscalar meson to the quarks with
coupling constant $G_{V}$. Quarks $u$, $d$, and $s$ with mass $m_{u}$ =
$m_{d}$ = 5 MeV and $m_{s}$ = 100 MeV are considered.

The total energy density and pressure are

$\mathcal{E}_{Q}=\sum_{l}\mathcal{E}_{l}+\sum_{f=u,d,s}\mathcal{E}_{\rm{vBag},f}-B_{dc}\
,$ (15) $P_{Q}=\sum_{l}\mathcal{E}_{l}+\sum_{f=u,d,s}P_{\rm{vBag},f}+B_{dc}\
,$ (16)

where, $B_{dc}$ represents the deconfined bag constant, which lowers the
energy per particle and, thus, favoring stable strange matter. The energy
density and pressure of a single quark flavor are defined as

$\mathcal{E}_{\rm{vBag},f}=\mathcal{E}_{f}(\mu_{f}^{*})+\frac{1}{2}K_{\nu}n_{f}^{2}(\mu_{f}^{*})+B_{\chi,f}\
,$ (17)
$P_{\rm{vBag},f}=P_{f}(\mu_{f}^{*})+\frac{1}{2}K_{\nu}n_{f}^{2}(\mu_{f}^{*})-B_{\chi,f}\
.$ (18)

The expressions for the free quark matter energy density and pressure in the
presence of the magnetic field are shown in the Appendix A. The coupling
constant parameter $K_{\nu}$ results from the vector interactions and controls
the stiffness of matter [148]. In the present study, the coupling constant
parameter $K_{\nu}$ is fixed at $4$ GeV-2 for three flavor configuration. The
bag constant for a single quark flavor is denoted by $B_{\chi,f}$.

The effective chemical potential $\mu_{f}^{*}$ of the system and the quark
density is defined as

$\mu_{f}^{*}=\mu_{f}-K_{\nu}n_{f}(\mu_{F}^{*})\ .$ (19)
$n_{f}(\mu_{f})=n_{f}(\mu^{*})$ (20)

In order for the phase transition to occur at the same chemical potential
$\mu_{B}=\mu_{u}+2\mu_{d}$ for all flavors, an effective bag constant
$B_{\rm{eff}}$ is defined in the vBag model as

$B_{\rm{eff}}=\sum_{f=u,d,s}B_{\chi,f}-B_{dc}\ .$ (21)

Two different values of effective bag constant are used in this work
$B_{eff}^{1/4}=130$ and $160$ MeV and, in the final discussion, $180$ MeV is
also considered.

In presence of a magnetic field, the transverse component of the momentum of
all quarks is quantized into landau levels

$E_{f}=\sqrt{k_{z,f}^{2}+m_{f}^{2}+2\nu|Q_{f}|B}\ ,$ (22)

For quark matter, we obtain the highest value of Landau levels, similar to Eq.
11 as

$\nu_{max}=\Bigg{\lfloor}\frac{E_{f}^{2}-m_{f}^{2}}{2|Q_{f}|B}\Bigg{\rfloor}\
,$ (23)

The charge neutrality and chemical-equilibrium conditions for the quark matter
are

$\frac{2}{3}\rho_{u}-\frac{1}{2}(\rho_{d}+\rho_{s})-\rho_{e}-\rho_{\mu}=0,$
(24) $\displaystyle\mu_{s}=\mu_{d}=\mu_{u}+\mu_{e}$ (25)
$\displaystyle\mu_{\mu}=\mu_{e}.$ (26)

###  II.3 Photons

For the pure electromagnetic part, the Lagrangian density is written as

$\mathcal{L}_{\gamma}=-\frac{1}{16\pi}F_{\mu\nu}F^{\mu\nu}\ ,$ (27)

where, $F_{\mu\nu}$ is the electromagnetic field tensor,
$F_{\mu\nu}$=$\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Hence, the total
Lagrangian density in the presence of a magnetic field is

$\mathcal{L}=\mathcal{L}_{m}+\mathcal{L}_{\gamma}\ ,$ (28)

where “m” stands for matter, including the hadronic, quark, and mixed phases.

For the energy density and pressure in the presence of a magnetic field $B$,
the expressions can be obtained by solving the energy-momentum tensor relation

$T^{\mu\nu}=T_{m}^{\mu\nu}+T_{\gamma}^{\mu\nu},$ (29)

where

$\displaystyle T_{m}^{\mu\nu}$
$\displaystyle=\mathcal{E}_{m}u^{\mu}u^{\nu}-P(g^{\mu\nu}-u^{\mu}u^{\nu})$
$\displaystyle+\mathcal{M}B\Bigg{(}g^{\mu\nu}-u^{\mu}u^{\nu}+\frac{B^{\mu}B^{\nu}}{B^{2}}\Bigg{)},$
$\displaystyle T_{\gamma}^{\mu\nu}$
$\displaystyle=\frac{B^{2}}{4\pi}\Bigg{(}u^{\mu}u^{\nu}-\frac{1}{2}g^{\mu\nu}\Bigg{)}-\frac{B^{\mu}B^{\nu}}{4\pi}\
.$ (30)

Here, $\mathcal{M}$ is the magnetization per unit volume and
$B^{\mu}B_{\mu}=-B^{2}$ where $B^{\mu}$ =
$\epsilon^{\mu\nu\alpha\beta}F_{\nu\alpha}u_{\beta}/2$ with
$\epsilon^{\mu\nu\alpha\beta}$ being the totally antisymmetric Levi-Civita
tensor, and $B$ = $\mathrm{|\textbf{B}|}$. For matter in presence of a
magnetic field, the single particle energies of all charged baryons, quarks,
and leptons are quantized in the direction perpendicular to the magnetic
field.

The expressions for the matter energy density, pressure, and baryon density
obtained in the presence of magnetic field are shown in Appendix A. From Eq.
II.3, the total energy density is

$\mathcal{E}=\mathcal{E}_{m}+\frac{B^{2}}{8\pi}\ .$ (31)

The total pressure in the perpendicular and the parallel directions to the
local magnetic field are

$\displaystyle P_{\perp}$
$\displaystyle=P_{m}-\mathcal{M}B+\frac{B^{2}}{8\pi},$ $\displaystyle
P_{\parallel}$ $\displaystyle=P_{m}-\frac{B^{2}}{8\pi}\ ,$ (32)

where the magnetization is calculated as

$\mathcal{M}=\partial P_{m}/\partial B\ .$ (33)

When we use TOV solutions for macroscopic stellar properties, we consider only
the pressure in perpendicular direction to determine the maximum possible NS
mass obtained with magnetic field. When we use solutions from LORENE, we do
not account for magnetic field effects in the EOS, as this was shown not to
modify significantly stellar masses and radii [149, 150].

##  III Mixed Phase

In this work we assume that the deconfinement phase transition is of first-
order and that the quark-matter surface tension is low enough for a mixture of
phases to appear. By means of a Gibbs construction [124], charge neutrality is
achieved globally, while hadronic matter is positively charged and quark
matter is negatively charged. Within chemical equilibrium, the stiffness of
the phases, determine the extension of the mixed phase in density.

The expressions for the chemical potential and pressure within the mixed-phase
are defined as

$\mu_{B,H}=\mu_{B,Q};\ \ \mu_{e,H}=\mu_{e,Q}\ ,$ (34)

and

$P_{H}(\mu_{B},\mu_{e})=P_{Q}(\mu_{B},\mu_{e})=P_{MP}\ ,$ (35)

where the subscripts $H$, $Q$, and $MP$ represent the hadronic phase, quark
phase, and the mixed phase, respectively, all containing lepton contributions.
From the global charge conservation of the electric and baryon charges, we
have

$\displaystyle\chi\rho_{Q}+(1-\chi)\rho_{H},$
$\displaystyle\chi\rho_{B_{Q}}+(1-\chi)\rho_{B_{H}}=\rho_{B_{MP}}\ .$ (36)

Here, $\chi$ is the quark phase volume fraction given by
$\chi=V_{Q}/(V_{Q}+V_{H})$ and $(1-\chi)$ is the hadronic phase volume
fraction. The charge densities in the hadronic and quark phases are
represented by $\rho_{Q}$ and $\rho_{H}$.

The total energy density of the mixed phase then reads

$\mathcal{E}_{MP}=\chi\mathcal{E}_{Q}+(1-\chi)\mathcal{E}_{H}\ ,$ (37)

The mixed phase is characterized by a value of $\chi$ which varies from 0, the
onset of mixed phase to 1, the onset of pure quark phase. The equations above
determine the properties of the mixed-phase and, combined with the equations
for hadronic and quark phases, allow us to calculate macroscopic properties
for hybrid stars.

##  IV Neutron-Star Structure

The properties of spherically static NSs are obtained using the well known
Tolman Oppenheimer Volkoff coupled differential equations given by [151, 152]

$\frac{dP(r)}{dr}=-\frac{[\mathcal{E}(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r^{2}(1-2M(r)/r)}\ ,$ (38)

and

$\frac{dM(r)}{dr}=4\pi r^{2}\mathcal{E}(r)\ ,$ (39)

where $M(r)$ represents the gravitational mass inside radius $r$. The boundary
conditions $P(0)=P_{c}$ and $M(0)=0$ at the center and $P(R)=0$ at the surface
allow one to solve the differential equations and determine the properties of
a NS for each given central pressure $P_{c}$.

For strong magnetic fields, the spherical symmetric solutions obtained by
solving the TOV equations lead to an overestimation of the mass and the radius
(when the pressure in the local perpendicular direction to the magnetic field
is applied in all directions) and, hence, cannot be used for determining
stellar properties. For this reason, we use the LORENE library [135, 136,
137], which solves the Einstein-field Maxwell’s equations with an axisymmetric
deformation, to determine stellar properties of magnetic NSs.

By assuming a poloidal magnetic field distribution, the numerical code solves
the coupled Einstein-Maxwell field equations to identify stable and stationary
magnetised configurations of stars. The Maximal-Slicing-Quasi-Isotropic (MSQI)
metric is employed for polar spherical symmetry, which allows stars to deform
by letting the metric potentials rely on the radial $r$ and angular
coordinates $\theta$ with respect to the magnetic axis. By employing this
approach, the field is produced self-consistently by a macroscopic current,
which is a function of the stellar radius, angle $\theta$ with respect to
symmetry axis, and dipole magnetic moment $\mu$ for each equation of state.
This allows one to control the strength of the magnetic field throughout the
star either by the magnetic dipole moment $\mu$ or the dimensionless current
function $f_{0}$. In this work, we vary the former.

Table 1: Magnetic field profile values at low and high densities calculated for EoSs with different effective bag constants and dipole magnetic moments.  |  $B^{1/4}_{\rm{eff}}=130$ MeV |  $B^{1/4}_{\rm{eff}}=160$ MeV
---|---|---
$\mu$ (Am2) |  $B_{\rm{low}}$ |  $B_{\rm{high}}$ |  $B_{\rm{low}}$ |  $B_{\rm{high}}$
5$\times$1031 |  $1.1\times 10^{17}$ |  $3\times 10^{17}$ |  $1.1\times 10^{17}$ |  $3.2\times 10^{17}$
$2\times 10^{32}$ |  $4.5\times 10^{17}$ |  $1.2\times 10^{18}$ |  $4.5\times 10^{17}$ |  $1.3\times 10^{18}$

##  V Magnetic Field

We employ a chemical-potential dependent magnetic field, fitted from the
solutions of the Einstein-Maxwell’s equations, to study the effects of
magnetic fields on our microscopic description of matter with a phase
transition. The relation between the magnetic field and the chemical potential
depends on the magnetic dipole moment and is given by the relation [153]

$B^{*}(\mu_{B})=\frac{(a+b\mu_{B}+c\mu_{B}^{2})}{B_{c}^{2}}\mu\ ,$ (40)

where $\mu_{B}$ is the baryon chemical potential in MeV and $\mu$ is the
dipole magnetic moment in units of Am2, so as to produce $B^{*}$ in units of
the electron critical field $B_{c}=4.414\times 10^{13}$ G. The coefficients
$a$, $b$, and $c$ taken as $a=-0.786$ G2/(Am2), $b=1.24\times 10^{-3}$ G2/(Am2
MeV) and $c=-3.51\times 10^{-7}$ G2/(Am2 MeV2) are obtained from a fit for the
magnetic field in the polar direction of a star with a baryon mass of $2.2$
M⊙.

Figure 1: Magnetic field profile as a function of baryon density for the EoSs
calculated with different values of effective bag constant. Different values
of magnetic dipole moment are shown.

Fig. 1 shows the magnetic field profile as a function of baryon density
obtained for two EoSs with quark deconfinement and mixed phase, one with a
larger value of effective bag constant (dashed lines) and one with a smaller
value (full lines). They differ because the effective bag constant affects how
baryon density and chemical potential relate, with the larger value
reproducing a softer quark matter EoS and lower density $\rho_{B}=\partial
P/\partial\mu_{B}$ (at a given $\mu_{B}$ or $B^{*}$) along the mixed phase for
a given value of dipole magnetic moment. Regardless, at large densities, a
dipole magnetic moment approximately determines the strength of the magnetic
field reproduced (see Tab. 1), reaching either $~{}10^{17}$ or $~{}10^{18}$,
depending on the dipole magnetic moment. At low densities, the magnetic field
is less than one order of magnitude lower than its maximum value, showing a
large contrast with ad hoc exponential profiles [65, 154, 155].

We discuss how high the density can become inside stars depending on dipole
magnetic moment and EoS in the next section.

##  VI Results and Discussions

Figure 2: Equation of state (transverse pressure vs. energy density) for different values of dipole magnetic moment. The left (right) panel shows the EoS with effective bag constant $B^{1/4}_{\rm{eff}}$ = 130 MeV ($B^{1/4}_{\rm{eff}}$ = 160 MeV). The hadronic, mixed, and quark phases are identified, respectively, by HP, MP, and QP.  Table 2: Normalized baryon density phase boundaries: the beginning and end of mixed phase for the EoSs calculated with different effective bag constants and dipole magnetic moments.  |  $B^{1/4}_{\rm{eff}}=130$ MeV |  $B^{1/4}_{\rm{eff}}=160$ MeV
---|---|---
$\mu$ (Am2) |  $\rho_{\rm{beg}}/\rho_{0}$ |  $\rho_{\rm{end}}/\rho_{0}$ |  $\rho_{\rm{beg}}/\rho_{0}$ |  $\rho_{\rm{end}}/\rho_{0}$
$0$ |  $2.32$ |  $5.63$ |  $3.04$ |  $6.27$
$5\times 10^{31}$ |  $2.35$ |  $5.78$ |  $3.08$ |  $6.38$
$2\times 10^{32}$ |  $2.47$ |  $6.07$ |  $3.21$ |  $6.75$

Figure 3: Normalized particle fraction of baryons, leptons, and quarks as a
function of normalized baryon density for effective bag constant
$B^{1/4}_{\rm{eff}}=130$ MeV without magnetic field (a) and with magnetic
field with different magnetic dipole moments, $\mu$=5$\times$1031 Am2 (b) and
$\mu$=2$\times$1032 Am2 (c).

Fig. 2 displays NS EoSs with quark deconfinement in the presence of the
magnetic field profiles previously discussed for different values of the
magnetic dipole moment. The left panel shows EoSs with effective bag constant
$B^{1/4}_{\rm{eff}}=130$ MeV, while the right panel shows EoSs with
$B^{1/4}_{\rm{eff}}=160$ MeV. The solid black lines shows the EoS at $\mu=0$
Am2, which correspond to the zero magnetic field case. All curves include
contribution from matter only (no pure photon contribution). As already
mentioned, a larger value of effective bag constant reproduces a softer quark
matter and mixed phase EoS (lower pressure). As a consequence, the onset of
the mixed and pure quark phases occurs at higher energy densities and
densities for the larger value of the effective bag constant [82, 84]. These
statements do not change in the presence of magnetic fields.

For the magnetic dipole moment of $\mu=5\times 10^{31}$ Am2, which corresponds
to a high-density magnetic field of $\sim 10^{17}$ G, for the lower effective
bag constant the hadronic phase in the low density region is not affected by
the magnetic field, while the pure quark phase in the high density is slightly
stiffer than the EoS without magnetic field. The mixed phase region extends to
higher densities. Similar behavior is observed for the EoSs with higher
effective bag constant. For the magnetic dipole moment of $\mu=2\times
10^{32}$ Am2, which corresponds to a high-density magnetic field of $\sim
10^{18}$ G, both the hadronic phase and pure quark phases become stiffer. The
mixed phase region is even broader (than with lower $\mu$). The pure quark
phase appears at a very large densities and, hence, will occupy a small part
(if any) of NSs. For the larger effective bag constant, the onset of mixed
phase and pure quark phase is strongly affected. This is because at the
densities at which it occurs the magnetic field is much stronger. Thus, we see
that the transitions to the mixed phase and to the pure quark phase are
affected by the inclusion of the magnetic field and depends upon its strength.
See Tab. 2 for the exact density of the phase boundaries.

Figure 4: Same as Fig. 3 but with effective bag constant $B^{1/4}_{\rm{eff}}$
= 160 MeV.

Fig. 3 shows the normalized particle population of baryons, leptons, and
quarks without and with magnetic field effects with different values of the
magnetic dipole moment. These plots correspond to the EoSs obtained with
effective bag constant $B^{1/4}_{\rm{eff}}=130$ MeV. For $\mu=0$ Am2, the
hadrons disappear in mixed phase and quarks appear smoothly. Even though
hyperons are included in the calculations, they are partially suppressed by
the appearance of quark phase, a small amount of $\Lambda$ hyperons appear at
density around $2.2$ $\rho_{0}$, just before the hadron-quark mixed phase
starts, and a small amount of $\Sigma^{-}$ appear in the mixed phase. The
density of leptons, $e^{-}$ and $\mu^{-}$, is significant in the hadron phase
but vanishes in the mixed phase. Since the quarks ($d$ and $s$) are negatively
charged, there is no necessity of leptons in the quark phase to maintain beta
equilibrium and charge neutrality conditions. For the lower value of the
magnetic dipole moment (when compared with $\mu=0$), oscillations appear in
the particle population of charged particles, which arise due to the Landau
levels. With the increasing density, charged particles (especially leptons)
are enhanced, as seen in the Fig. 3. As already discussed, the density at
which the mixed phase appears and the density range of the mixed phase
increase with larger values of $\mu$. The hyperons are suppressed due to
increase in the proton density [156]. These effects are enhanced in magnitude
for the larger value of the magnetic dipole moment, as seen in Fig. 3.

Fig. 4 also shows the populations of baryons, leptons, and quarks but now with
a larger effective bag constant $B^{1/4}_{\rm{eff}}$ = 160 MeV. Similar to
Fig. 3, some of the hyperons appear just before the onset of mixed phase and
some after, all being suppressed for larger dipole magnetic moments. As the
magnetic dipole moment increases, the lepton fraction is enhanced to the point
that a small number of leptons extends into the pure quark phase. For higher
values of the magnetic dipole moment, the mixed phase region extends up to
7$\rho_{0}$. In the mixed phase, the abundance of $u$-quarks is found to be
enhanced, while those of $d$ and $s$ quarks remain practically same.

Fig. 5 shows the mass-radius diagram for the hybrid stars obtained for
effective bag constant $B^{1/4}_{\rm{eff}}=130$ MeV without magnetic field and
with magnetic field calculated for different values of the magnetic dipole
moment. The left panel shows the results obtained solving Einstein and
Maxwell’s equations with an axisymmetric deformation (LORENE library), while
the right panel displays the results using solutions from spherically
symmetric TOV equations. For $\mu=B=0$ they coincide. Without magnetic field,
the maximum mass produced is $2.13$ M⊙ at radius of 12.6 km. The radius of the
canonical mass, $1.4$ M⊙ is around 13 km, which satisfies the radius
constraints from PSR J0030+0451 by NICER. The recent NICER constraint on the
radius of the $2$ M⊙ pulsar J0740+6620 NS is also satisfied. Maximum mass
constraints and constraints from LIGO/VIRGO are also satisfied.

With increasing magnetic dipole moment, the mass and radius of entire stellar
sequences increase. For $\mu=5\times 10^{31}$ Am2, which corresponds to a
magnetic field of 3 $\times$ 1017 G in the center of the maximum-mass star,
the radius of low/intermediate-mass stars obtained using the LORENE library
differs significantly from the radius obtained from TOV, $0.6$ km for the
canonical mass when compared with $0.1$ km for the maximum mass. Note that the
radius shown in the left panel of Fig. 5 is the equatorial one, and under the
influence of strong poloidal magnetic fields, stars become oblate. For $\mu$ =
2 $\times$ 1032 Am2, which corresponds to a magnetic field of 1.2 $\times$
1018 G in the center of the maximum-mass star, both the radius and mass of all
stars from the sequence obtained using LORENE differ significantly from the
values obtained from TOV. The difference is $0.7$ km for the canonical mass
and $-0.2$ km for the maximum mass (with the minus meaning that now TOV gives
the larger radius). In the case of TOV, because of the required spherical
symmetry, the absence of the Lorentz force does not enlarge the equator of
less massive stars that possess a more soft EoS, instead the excess magnetic
energy that would deform the star is added to the mass, which becomes
unphysically large.

Fig. 6 also displays the mass radius profiles for hybrid stars but with
effective bag constant $B^{1/4}_{\rm{eff}}$ = 160 MeV. For $\mu$ = 0 Am2, the
maximum mass obtained lowers to $2.02$ M⊙ with corresponding radius of $12.5$
km (when compared with Fig. 5). The radius at the canonical mass is $12.68$
km. Our results are still in agreement with NICER, LIGO/VIRGO and mass
constraints. Using LORENE, as the magnetic dipole moment increases, the
maximum mass increases more than for the previous effective bag constant
considered. The radius of the whole sequence also increases more. The
solutions from TOV, do not depend as much on the effective bag constant, and
in this case are farther from reproducing low/intermediate mass results from
LORENE. A difference of around $0.4$ km is observed in the radius at the
canonical mass and $-0.3$ for the maximum mass with measurements from LORENE
and TOV. This implies that the difference between the two approaches depends
on the EoS (particle composition and interactions) and mass of the star we are
calculating the deformation for.

Figure 5: Mass-radius diagram for hybrid stars with effective bag constant
$B^{1/4}_{\rm{eff}}$ = 130 shown for different values of magnetic dipole
moment. Left panel shows results obtained using the LORENE library, while
right panel shows solutions from TOV. Recent constraints on mass and radius
are also shown [7, 30, 1, 32, 36, 37, 38, 39].  Figure 6:  Same as Fig. 5, but
with $B^{1/4}_{\rm{eff}}$ = 160 MeV.  Figure 7: Same as Fig. 5, but with
$B^{1/4}_{\rm{eff}}$ = 180 MeV.

Table 3 shows the maximum mass, corresponding radius, canonical mass’ radius,
and canonical mass’ dimensionless tidal deformability obtained at different
values of the magnetic dipole moment using both LORENE library and TOV
solutions. Without magnetic field, the dimensionless tidal deformability for
the canonical mass satisfies the constraints from GW170817,
$\tilde{\Lambda}<800$ ($B^{1/4}_{\rm{eff}}=160$ MeV) and the recently observed
GW190814 data, $\Lambda_{1.4}=616_{-158}^{+273}$ ($B^{1/4}_{\rm{eff}}=130$ and
$160$ MeV). This constraint from GW190814 data is obtained by considering its
secondary component to a supermassive NS [7].

To further verify our results, we calculate the deconfinement phase transition
using an effective bag constant of $B^{1/4}_{\rm{eff}}=180$ MeV. Both the
publicly available LORENE and spherically symmetric TOV equations are used to
produce the mass radius profiles and analyze the differences in the stellar
properties. Fig. 7 displays the mass-radius diagram for 180 MeV effective bag
constant for different values of the magnetic dipole moment. For $\mu=0$ Am2
case, the maximum mass and the canonical mass radius for the hybrid star
configuration obtained is $1.86$ M⊙ and $11.94$ km, respectively. The
canonical radius is $12.07$ km. In this case, the maximum mass is not high
enough to fulfill modern mass constraints. For larger dipole magnetic moments,
there is a large increase in mass (than the other effective bag constants)
using Lorene and the radii, particularly for low/intermediate mass stars,
becomes very large. This reinforces the results we discussed above.

Table 3: Maximum mass, maximum mass’ radius, canonical mass’ radius, and canonical mass’ dimensionless tidal deformability of hybrid stars obtained with different effective bag constants and magnetic dipole moments.  | using LORENE and $B^{1/4}_{\rm{eff}}=130$ MeV
---|---
$\mu$ (Am2) |  $M_{max}$ |  $R_{max}$ |  $R_{1.4}$ |  $\Lambda_{1.4}$
0 |  2.13 |  12.64 |  13.01 |  879.54
5$\times$1031 |  2.16 |  12.67 |  13.73 |  1223.98
2$\times$1032 |  2.20 |  12.78 |  13.96 |  1276.95
| using TOV and $B^{1/4}_{\rm{eff}}=130$ MeV
0 |  2.13 |  12.64 |  13.01 |  879.54
5$\times$1031 |  2.18 |  12.66 |  13.13 |  956.02
2$\times$1032 |  2.33 |  12.98 |  13.26 |  1236.42
| using LORENE and $B^{1/4}_{\rm{eff}}=160$ MeV
0 |  2.02 |  12.53 |  12.68 |  700.94
5$\times$1031 |  2.10 |  12.64 |  13.46 |  946.38
2$\times$1032 |  2.19 |  12.56 |  13.86 |  1268.99
| using TOV and $B^{1/4}_{\rm{eff}}=160$ MeV
0 |  2.02 |  12.53 |  12.68 |  700.94
5$\times$1031 |  2.15 |  12.56 |  12.97 |  923.13
2$\times$1032 |  2.29 |  12.89 |  13.39 |  1194.26
| using LORENE and $B^{1/4}_{\rm{eff}}=180$ MeV
0 |  1.86 |  11.94 |  12.07 |  472.26
5$\times$1031 |  1.94 |  11.87 |  12.47 |  567.22
2$\times$1032 |  2.08 |  12.07 |  13.27 |  976.83
| using TOV and $B^{1/4}_{\rm{eff}}=180$ MeV
0 |  1.86 |  11.94 |  12.07 |  472.26
5$\times$1031 |  2.01 |  11.82 |  12.10 |  532.06
2$\times$1032 |  2.19 |  12.19 |  12.92 |  872.23

##  VII Summary and Conclusion

We study the effect of strong magnetic fields on the phase transition between
baryons and quarks in the core of neutron stars (NSs). For this purpose, we
use the widely known density-dependent relativistic mean-field (DD-RMF) model
with the DD-MEX parameter set, which offers enough flexibility to meet both
nuclear and astrophysical constraints. The hyperon couplings are determined
from SU(6) symmetry. For the quark phase, we use the Vector-Enhanced Bag model
(vBag), with the construction of a mixed-phase, which allows to describe
strange hybrid NSs that achieve the $2$ M⊙ limit. Several values of the
effective bag constant, $B^{1/4}_{\rm{eff}}=130,160,180$ MeV, are considered.

To study magnetic field effects on the equation of state (EoS), a physically-
motivated profile is applied, which depends quadratically on the chemical
potential. As a consequence, the profile as a function of density depends on
the EoS around the phase transition, and it is different for different
effective bag constants. At large densities, the magnetic field approximately
only depends on the magnetic dipole moment, reaching $10^{18}$ G for the
largest magnetic dipole moment studied, $\mu=2\times 10^{32}$ Am2. Overall,
the equation of state stiffens in the presence of magnetic fields, but the
effects are stronger in the mixed phase, which is also wider (in density and
energy density) and takes place at larger densities. The effects are stronger
for larger magnetic dipole moments and effective bag constants. The hyperons,
which appear around the onset of mixed phase are suppressed by the magnetic
field (due to an increased proton density). Oscillations appear in the
particle population of charged particles, which arise due to the Landau
levels. Overall, the population of charged particles (especially leptons) are
enhanced. The population of $u$-quark is enhanced, when compared with the
other quarks.

The LORENE library is used to calculate stellar properties of NSs that present
strong magnetic fields and are, therefore, not spherically symmetric. However
in order to quantify the anisotropy, we also use solutions from the
spherically symmetric TOV equations, and discuss the difference. For $\mu=B=0$
they of course coincide. In this case, the maximum mass produced with
effective bag constants $B^{1/4}_{\rm{eff}}=130$ and $160$ MeV, $2.13$ and
$2.02$ M⊙, respectively, satisfy the mass and radius constraints from various
measurements.

Using the LORENE library, the poloidal magnetic fields turns NSs oblate,
increasing the equatorial radius. While the mass and radius of all stars of a
family are modified by the magnetic field, this is not the case for TOV, in
which case just massive stars become larger and their masses become
unphysically large. This qualitative behavior is independent of the effective
bag constant. Quantitatively, the effects of magnetic field affect more the
mass and radii of the entire family of stars described by lower effective bag
constant (softer EoS) using LORENE. Using TOV, the effects of magnetic field
don’t depend as much on the effect bag constant. This means that, as discussed
in detail for the first time in this paper, the difference between anisotropic
and isotropic general relativity solutions depends on the EoS and mass of the
star we are calculating the deformation for.

Interestingly, for some fixed stellar masses, the difference in radius between
LORENE and TOV is larger for lower values of $\mu$ and $B$. Overall, the
canonical radius is always smaller for TOV, and the difference increases with
effective bag constant. For the radius of the maximum-mass star, TOV can have
smaller (lower $\mu$) or larger (larger $\mu$) radius than LORENE. The maximum
mass of the sequence is always too large for TOV, the difference increasing
with $\mu$ and effective bag constant.

Future extensions of this work include adding a crust (see Ref. [157] for a
discussion of magnetic field effects on the neutron-star crust), the effects
of temperature, fast rotation and anomalous magnetic moment (AMM) of baryons,
as well as exploring different parametrizations of the models that allow for
smoother phase transitions. We also intend to use our EoSs in numerical
relativity simulations of merging binary NSs, as these are the key to learning
about the physical processes involved in events such as GW170817. Magnetic
fields, in particular, are known to play an important role in post-merger
evolution, fueling relativistic jets, and shaping electromagnetic counterpart
signals [53, 54, 55, 56, 57].

##  VIII Acknowledgment

I. A. R. and V. D. are thankful to T. Klähn and H. Pais for discussion on the
quark matter and magnetic EoS. A. A. U. acknowledges the Inter-University
Centre for Astronomy and Astrophysics, Pune, India for support via an
associateship and for hospitality. V. D. acknowledges support from the
National Science Foundation under grants PHY1748621, MUSES OAC-2103680, and
NP3M PHY-2116686. I. L. and I. A. R. also acknowledge the Fundação para a
Ciência e Tecnologia (FCT), Portugal, for the financial support to the Center
for Astrophysics and Gravitation (CENTRA/IST/ULisboa) through the Grant
Project No. UIDB/00099/2020 and Grant No. PTDC/FIS-AST/28920/2017.

## Appendix A

Within the relativistic density-dependent mean field (DD-RMF) model, the
equation of motion for baryons obtained by applying the Euler Lagrange
equations to the Lagrangian density (Eq. II.1) is written as

$\displaystyle\sum_{B}\Bigg{[}i\gamma_{\mu}D^{\mu}-\gamma^{0}\Bigg{(}g_{\omega}(\rho_{B})\omega$
$\displaystyle+\frac{1}{2}g_{\rho}(\rho_{B})\rho\tau_{3}+\sum_{R}(\rho_{B})\Bigg{)}$
$\displaystyle-M_{B}^{*}\Bigg{]}\psi_{B}=0\ ,$ (A.1)

where the rearrangement term due to the density dependence of the coupling
constants is

$\sum_{R}(\rho_{B})=-\frac{\partial
g_{\sigma}}{\partial\rho_{B}}\sigma\rho_{s}+\frac{\partial
g_{\omega}}{\partial\rho_{B}}\omega\rho_{B}+\frac{1}{2}\frac{\partial
g_{\rho}}{\partial\rho_{B}}\rho\rho_{3}\ .$ (A.2)

The equations of motion for meson fields are

$\displaystyle m_{\sigma}^{2}\sigma$
$\displaystyle=g_{\sigma}(\rho_{B})\rho_{s}\ ,$ $\displaystyle
m_{\omega}^{2}\omega$ $\displaystyle=g_{\omega}(\rho_{B})\rho_{B}\ ,$
$\displaystyle m_{\rho}^{2}\rho$
$\displaystyle=\frac{g_{\rho}(\rho_{B})}{2}\rho_{3}\ ,$ (A.3)

where the total scalar density $\rho_{s}$, baryon (vector) density $\rho_{B}$,
and isovector density $\rho_{3}$ are given as

$\displaystyle\rho_{s}$
$\displaystyle=\sum_{B}\bar{\psi}\psi=\sum_{B}\frac{1}{2\pi^{2}}\left(k_{B}M^{*}E^{*}-{M^{*}}^{3}\ln{\frac{k_{B}+E^{*}}{M^{*}}}\right)\
,$ $\displaystyle\rho_{B}$
$\displaystyle=\sum_{B}\psi^{\dagger}\psi=\sum_{B}\frac{k_{B}^{3}}{3\pi^{2}}\
,$ $\displaystyle\rho_{3}$
$\displaystyle=\sum_{B}\psi^{\dagger}\tau_{3}\psi=\rho_{p}-\rho_{n}+2\rho_{\Sigma^{+}}-2\rho_{\Sigma^{-}}-3\rho_{\Xi^{0}}-\rho_{\Xi^{0}}\
\ ,$ $\displaystyle\rho_{l}$
$\displaystyle=\sum_{l}\psi^{\dagger}\psi=\sum_{l}\frac{k_{l}^{3}}{3\pi^{2}}\
.$ (A.4)

With the above equations, the expression for the energy density and pressure
are

$\displaystyle\mathcal{E}_{m}$
$\displaystyle=\sum_{B}\frac{1}{\pi^{2}}\Bigg{[}\Bigg{(}\frac{1}{4}k_{B}^{3}+\frac{1}{8}(M_{B}^{*})^{2}k_{B}\Bigg{)}E_{B}^{*}-\frac{1}{8}(M_{B}^{*})^{4}$
$\displaystyle\ln\frac{k_{B}+E_{B}^{*}}{M_{B}^{*}}\Bigg{]}+\sum_{l}\frac{1}{\pi^{2}}\Bigg{[}\Bigg{(}\frac{1}{4}k_{l}^{3}+\frac{1}{8}(m_{l})^{2}k_{l}\Bigg{)}E_{l}$
$\displaystyle-\frac{1}{8}(m_{l}^{*})^{4}\ln\frac{k_{l}+E_{l}}{m_{l}}\Bigg{]}+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho^{2}+g_{\omega}(\rho_{B})\omega\rho_{B}+\frac{g_{\rho}(\rho_{B})}{2}\rho\rho_{3}\
,$ $\displaystyle P_{m}$
$\displaystyle=\sum_{B}\frac{1}{3\pi^{2}}\Bigg{[}\Bigg{(}\frac{1}{4}k_{B}^{3}-\frac{3}{8}(M_{B}^{*})^{2}k_{B}\Bigg{)}E_{B}^{*}+\frac{3}{8}(M_{B}^{*})^{4}$
$\displaystyle\ln\frac{k_{B}+E_{B}^{*}}{M_{B}^{*}}\Bigg{]}+\sum_{l}\frac{1}{3\pi^{2}}\Bigg{[}\Bigg{(}\frac{1}{4}k_{l}^{3}-\frac{3}{8}(m_{l})^{2}k_{l}\Bigg{)}E_{l}$
$\displaystyle+\frac{3}{8}(m_{l})^{4}\ln\frac{k_{l}+E_{l}}{m_{l}}\Bigg{]}-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho^{2}-\rho_{B}\sum_{R}(\rho_{B})\ ,$
(A.5)

where, $E_{B}^{*}=\sqrt{k_{B}^{2}+M_{B}^{*2}}$ and
$E_{l}=\sqrt{k_{l}^{2}+m_{l}^{2}}$. The rearrangement term
$\sum_{R}(\rho_{B})$ contributes to the pressure only.

The expressions for the total charged baryon, uncharged baryon, and lepton
energy densities in the presence of magnetic field become

$\displaystyle\mathcal{E}_{cB}$
$\displaystyle=\frac{|q_{cB}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}\times\Bigg{[}k_{F}^{cB}E_{F}^{cB}+$
$\displaystyle(M_{cB}^{*2}+2\nu|q_{cB}|B)\ln\Bigg{(}\frac{k_{F}^{cB}+E_{F}^{cB}}{\sqrt{M_{cB}^{*2}+2\nu|q_{cB}|B}}\Bigg{)}\Bigg{]}\
,$
$\displaystyle\mathcal{E}_{uB}=\frac{1}{8\pi^{2}}\Bigg{[}k_{F}^{uB}(E_{F}^{uB})^{3}$
$\displaystyle+(k_{F}^{uB})^{3}E_{F}^{uB}$ $\displaystyle-
M_{uB}^{*4}\ln\Bigg{(}\frac{k_{F}^{uB}+E_{F}^{uB}}{M_{uB}^{*}}\Bigg{)}\Bigg{]}\
,$
$\displaystyle\mathcal{E}_{l}=\frac{|q_{l}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}$
$\displaystyle\times\Bigg{[}k_{F}^{l}E_{F}^{l}+(m_{l}^{2}+2\nu|q_{l}|B)\ln\Bigg{(}\frac{k_{F}^{l}+E_{F}^{l}}{\sqrt{m_{l}^{2}+2\nu|q_{l}|B}}\Bigg{)}\Bigg{]}\
.$ (A.6)

The scalar and vector density for a given charged baryon $cB$, uncharged
baryon $uB$, and lepton follow as [62]

$\displaystyle\rho_{s}^{cB}$
$\displaystyle=\frac{|q_{cB}|BM_{cB}^{*2}}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}\ln\Bigg{(}\frac{k_{F}^{cB}+E_{F}^{cB}}{\sqrt{M_{cB}^{*2}+2\nu|q_{cB}|B}}\Bigg{)},$
$\displaystyle\rho_{s}^{uB}$
$\displaystyle=\frac{M_{uB}^{*2}}{2\pi^{2}}\Bigg{[}E_{F}^{uB}k_{F}^{uB}-M_{uB}^{*2}\ln\Bigg{(}\frac{k_{F}^{uB}+E_{F}^{uB}}{M_{uB}}\Bigg{)}\Bigg{]}\
,$ $\displaystyle\rho_{cB}$
$\displaystyle=\frac{|q_{cB}|B}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}k_{F}^{cB},$
$\displaystyle\rho_{uB}$ $\displaystyle=\frac{(k_{F}^{uB})^{3}}{3\pi^{2}},$
$\displaystyle\rho_{l}$
$\displaystyle=\frac{|q_{l}|B}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}k_{F}^{l}\
,$ (A.7)

where $r_{\nu}$ is the Landau degeneracy of $\nu$ level. For hadronic matter,
the spin degeneracy is 2 for all Landau levels, except for the ground state,
$\nu=0$, in which case it is 1. $k_{F}^{cB,uB,l}$ represents the Fermi
momentum of charged baryons, uncharged baryons, and leptons, respectively.

The pressure in the presence of a magnetic field is obtained from the energy
density as

$P_{m}=\sum_{i}\mu_{i}\rho_{i}^{v}-\mathcal{E}_{m}=\mu_{n}\sum_{b}\rho_{b}^{v}-\mathcal{E}_{m}\
,$ (A.8)

where the last equality is obtained using charge neutrality and
$\beta$-equilibrium conditions.

For quark matter, the presence of the magnetic field in the energy density and
pressure modifies the first term of Eq. (17) as

$\displaystyle\mathcal{E}_{f}$
$\displaystyle=\sum_{i=u,d,s,e}\Biggl{\\{}\frac{|q_{i}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu^{i}_{max}}g_{i}\times\Bigg{[}\mu_{i}\sqrt{\mu_{i}^{2}-(M_{\nu}^{i})^{2}}-$
$\displaystyle(M_{\nu}^{i})^{2}\ln\Bigg{(}\frac{\mu_{i}+\sqrt{\mu_{i}^{2}+(M_{\nu}^{i})^{2}}}{(M_{\nu}^{i})^{2}}\Bigg{)}\Bigg{]}\Biggr{\\}}\
,$ $\displaystyle P_{f}$
$\displaystyle=\sum_{i=u,d,s,e}\Biggl{\\{}\frac{|q_{i}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu^{i}_{max}}g_{i}\times\Bigg{[}\mu_{i}\sqrt{\mu_{i}^{2}-(M_{\nu}^{i})^{2}}-$
$\displaystyle(M_{\nu}^{i})^{2}\ln\Bigg{(}\frac{\mu_{i}+\sqrt{\mu_{i}^{2}-(M_{\nu}^{i})^{2}}}{(M_{\nu}^{i})^{2}}\Bigg{)}\Bigg{]}\Biggr{\\}}\
.$ (A.9)

The quark density in presence of strong magnetic field is

$\displaystyle\rho_{q}$
$\displaystyle=\sum_{i=u,d,s}\Biggl{\\{}\frac{|q_{i}|B}{2\pi^{2}}\sum_{\nu=0}^{\nu^{i}_{max}}g_{i}\sqrt{\mu_{i}^{2}-(M_{\nu}^{i})^{2}}\Biggr{\\}}\
,$ (A.10)

where $M_{\nu}^{i}=\sqrt{m_{i}^{2}+2\nu|q_{i}|B}$. The factor $g_{i}$
represents the degeneracy of $i$th particle, which is six for quarks and two
for leptons, except for the zeroth Landau level, in which case it is three and
one, respectively. The baryon density $\rho_{B}$ is defined as
$\rho_{B}=\sum_{q}\rho_{q}/3$ and

$\displaystyle\rho_{3}$
$\displaystyle=\sum_{q}\psi^{\dagger}\tau_{3}\psi=\rho_{u}-\rho_{d}\ .$ (A.11)

## References

  * Antoniadis and Freire et al. [2013] J. Antoniadis and P. C. C. Freire et al., Science 340 (2013), 10.1126/science.1233232.
  * Fonseca  _et al._ [2021] E. Fonseca _et al._ , Astrophys. Jour. Lett. 915, L12 (2021).
  * Abbott  _et al._ [2017] B. P. Abbott, R. Abbott, _et al._ , Phys. Rev. Lett. 119, 161101 (2017).
  * Abbott  _et al._ [2018] B. P. Abbott, R. Abbott, _et al._ , Phys. Rev. Lett. 121, 161101 (2018).
  * Abbott  _et al._ [2020a] B. P. Abbott _et al._ , The Astrophys. Jour. Lett. 892, L3 (2020a).
  * Abbott  _et al._ [2021] R. Abbott _et al._ , The Astrophys. Jour. Lett. 915, L5 (2021).
  * Abbott  _et al._ [2020b] R. Abbott, T. D. Abbott, _et al._ , Astrophys. Jour. 896, L44 (2020b).
  * Dexheimer  _et al._ [2021a] V. Dexheimer, R. O. Gomes, T. Klähn, S. Han, and M. Salinas, Phys. Rev. C 103, 025808 (2021a).
  * Tan  _et al._ [2020] H. Tan, J. Noronha-Hostler, and N. Yunes, Phys. Rev. Lett. 125, 261104 (2020).
  * Fishbach  _et al._ [2020] M. Fishbach, R. Essick, and D. E. Holz, Astrophys. J. 899, L8 (2020).
  * Rather  _et al._ [2021a] I. A. Rather, U. Rahaman, M. Imran, H. C. Das, A. A. Usmani, and S. K. Patra, Phys. Rev. C 103, 055814 (2021a).
  * Rather  _et al._ [2021b] I. A. Rather, U. Rahaman, V. Dexheimer, A. A. Usmani, and S. K. Patra, The Astrophysical J. 917, 46 (2021b).
  * Das  _et al._ [2021] H. C. Das, A. Kumar, and S. K. Patra, Phys. Rev. D 104, 063028 (2021).
  * Godzieba  _et al._ [2021] D. A. Godzieba, D. Radice, and S. Bernuzzi, Astrophys. J. 908, 122 (2021).
  * Most  _et al._ [2020] E. R. Most, L. J. Papenfort, L. R. Weih, and L. Rezzolla, MNRAS 499, L82–L86 (2020).
  * Zhang and Li [2020] N.-B. Zhang and B.-A. Li, Astrophys. Jour. 902, 38 (2020).
  * Tsokaros  _et al._ [2020] A. Tsokaros, M. Ruiz, and S. L. Shapiro, Astrophys. J. 905, 48 (2020).
  * Fattoyev  _et al._ [2020] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, and B. Reed, Phys. Rev. C 102, 065805 (2020).
  * Lim  _et al._ [2021] Y. Lim, A. Bhattacharya, J. W. Holt, and D. Pati, Phys. Rev. C 104, L032802 (2021).
  * Tews  _et al._ [2021] I. Tews, P. T. H. Pang, T. Dietrich, M. W. Coughlin, S. Antier, M. Bulla, J. Heinzel, and L. Issa, Astrophys. J. Lett. 908, L1 (2021).
  * Sedrakian  _et al._ [2020] A. Sedrakian, F. Weber, and J. J. Li, Phys. Rev. D 102, 041301 (2020).
  * Huang  _et al._ [2020] K. Huang, J. Hu, Y. Zhang, and H. Shen, Astrophys. Jour. 904, 39 (2020).
  * Cao  _et al._ [2020] Z. Cao, L.-W. Chen, P.-C. Chu, and Y. Zhou, arXiv e-prints , arXiv:2009.00942 (2020), arXiv:2009.00942 [astro-ph.HE] .
  * Demircik  _et al._ [2021] T. Demircik, C. Ecker, and M. Järvinen, Astrophys. J. Lett. 907, L37 (2021).
  * Bombaci  _et al._ [2021] I. Bombaci, A. Drago, D. Logoteta, G. Pagliara, and I. Vidaña, Phys. Rev. Lett. 126, 162702 (2021).
  * Li  _et al._ [2020] J. J. Li, A. Sedrakian, and F. Weber, Phys. Lett. B 810, 135812 (2020).
  * Roupas  _et al._ [2021] Z. Roupas, G. Panotopoulos, and I. Lopes, Phys. Rev. D 103, 083015 (2021).
  * Blaschke and Cierniak [2021] D. Blaschke and M. Cierniak, Astronomische Nachrichten 342, 227–233 (2021).
  * Nathanail  _et al._ [2021] A. Nathanail, E. R. Most, and L. Rezzolla, The Astrophys. Jour. Lett. 908, L28 (2021).
  * Demorest  _et al._ [2010] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, Nature 467, 1081–1083 (2010).
  * Arzoumanian  _et al._ [2018] Z. Arzoumanian _et al._ , The Astrophys. Jour. Suppl. Ser. 235, 37 (2018).
  * Cromartie and Fonseca et al. [2019] H. T. Cromartie and E. Fonseca et al., Nature Astron. 4, 72–76 (2019).
  * Ambartsumyan and Saakyan [1960] V. A. Ambartsumyan and G. S. Saakyan, Sov. Astron. 4, 187 (1960).
  * Glendenning [1982] N. K. Glendenning, Phys. Lett. B 114, 392–396 (1982).
  * Glendenning [1985] N. K. Glendenning, Astrophys. Jour. 293, 470–493 (1985).
  * Miller et al. [2019] M. C. Miller et al., Astrophys. J. 887, L24 (2019).
  * Riley et al. [2019] T. E. Riley et al., Astrophys. J. 887, L21 (2019).
  * Miller  _et al._ [2021] M. C. Miller _et al._ , The Astrophys. Jour. Lett. 918, L28 (2021).
  * Riley  _et al._ [2021] T. E. Riley _et al._ , Astrophys. J. Lett. 918, L27 (2021).
  * Chamel  _et al._ [2013] N. Chamel, P. Haensel, J. L. Zdunik, and A. F. Fantina, Int. J. Mod. Phys. E 22, 1330018 (2013).
  * Popov  _et al._ [2006] S. B. Popov, R. Turolla, and A. Possenti, MNRAS 369, L23–L26 (2006).
  * Mereghetti  _et al._ [2015] S. Mereghetti, J. A. Pons, and A. Melatos, Space Sci. Rev. 191, 315–338 (2015).
  * Kaspi and Beloborodov [2017] V. M. Kaspi and A. M. Beloborodov, Ann. Rev. Astron. Astrophys. 55, 261–301 (2017).
  * Harding and Lai [2006] A. K. Harding and D. Lai, Rep. Prog. Phys. 69, 2631–2708 (2006).
  * Turolla  _et al._ [2015] R. Turolla, S. Zane, and A. L. Watts, Rep. Prog. Phys. 78, 116901 (2015).
  * Margalit  _et al._ [2020] B. Margalit, P. Beniamini, N. Sridhar, and B. D. Metzger, Astrophys. Jour. 899, L27 (2020).
  * Beniamini  _et al._ [2020] P. Beniamini, Z. Wadiasingh, and B. D. Metzger, MNRAS 496, 3390–3401 (2020).
  * Beloborodov [2020] A. M. Beloborodov, Astrophys. Jour. 896, 142 (2020).
  * Levin  _et al._ [2020] Y. Levin, A. M. Beloborodov, and A. Bransgrove, Astrophys. Jour. 895, L30 (2020).
  * Zanazzi and Lai [2020] J. J. Zanazzi and D. Lai, Astrophys. Jour. 892, L15 (2020).
  * Makishima  _et al._ [2014] K. Makishima, T. Enoto, J. S. Hiraga, T. Nakano, K. Nakazawa, S. Sakurai, M. Sasano, and H. Murakami, Phys. Rev. Lett. 112, 171102 (2014).
  * Dall’Osso  _et al._ [2018] S. Dall’Osso, L. Stella, and C. Palomba, Mon. Not. Roy. Astron. Soc. 480, 1353–1362 (2018).
  * Ciolfi and Rezzolla [2013] R. Ciolfi and L. Rezzolla, Mon. Not. Roy. Astron. Soc. 435, L43–L47 (2013).
  * Giacomazzo  _et al._ [2015] B. Giacomazzo, J. Zrake, P. Duffell, A. I. MacFadyen, and R. Perna, Astrophys. J. 809, 39 (2015).
  * Ciolfi [2020] R. Ciolfi, General Relativity and Gravitation 52, 59 (2020).
  * Palenzuela  _et al._ [2021] C. Palenzuela, R. Aguilera-Miret, F. Carrasco, R. Ciolfi, J. V. Kalinani, W. Kastaun, B. Miñano, and D. Viganò, (2021), arXiv:2112.08413 [gr-qc] .
  * Ruiz  _et al._ [2021] M. Ruiz, A. Tsokaros, and S. L. Shapiro, Phys. Rev. D 104, 124049 (2021).
  * Lai and Shapiro [1991] D. Lai and S. L. Shapiro, Astrophys. Jour. 383, 745 (1991).
  * Landau and Lifshitz [1965] L. D. Landau and E. M. Lifshitz, “Quantum mechanics. Non-relativistic theory,” London-New York: Pergamon Press. Ltd. 1965. XIII, 616 p. 100 s. (1965). (1965).
  * Strickland  _et al._ [2012] M. Strickland, V. Dexheimer, and D. P. Menezes, Phys. Rev. D 86, 125032 (2012).
  * Chakrabarty  _et al._ [1997] S. Chakrabarty, D. Bandyopadhyay, and S. Pal, Phys. Rev. Lett. 78, 2898–2901 (1997).
  * Broderick  _et al._ [2000] A. Broderick, M. Prakash, and J. M. Lattimer, Astrophys. Jour. 537, 351–367 (2000).
  * Rabhi  _et al._ [2008a] A. Rabhi, C. Providência, and J. D. Providência, Jour. Phys. G: Nucl. and Part. Phys. 35, 125201 (2008a).
  * Mallick and Schramm [2014] R. Mallick and S. Schramm, Phys. Rev. C 89, 045805 (2014).
  * Bandyopadhyay  _et al._ [1997] D. Bandyopadhyay, S. Chakrabarty, and S. Pal, Phys. Rev. Lett. 79, 2176–2179 (1997).
  * Aguirre [2019] R. M. Aguirre, Phys. Rev. C 100, 065203 (2019).
  * Gomes  _et al._ [2017] R. O. Gomes, B. Franzon, V. Dexheimer, and S. Schramm, Astrophys. J. 850, 20 (2017).
  * Dexheimer  _et al._ [2012a] V. Dexheimer, R. Negreiros, and S. Schramm, Eur. Phys. J. A 48, 189 (2012a).
  * Pili  _et al._ [2017] A. G. Pili, N. Bucciantini, and L. Del Zanna, Mon. Not. Roy. Astron. Soc. 470, 2469–2493 (2017).
  * Felipe  _et al._ [2008] R. G. Felipe, A. P. Martinez, H. P. Rojas, and M. Orsaria, Phys. Rev. C 77, 015807 (2008).
  * Paulucci  _et al._ [2013] L. Paulucci, E. J. Ferrer, V. de la Incera, and J. E. Horvath, IAU Symp. 291, 465–467 (2013).
  * Casali  _et al._ [2014] R. H. Casali, L. B. Castro, and D. P. Menezes, Phys. Rev. C 89, 015805 (2014).
  * Rabhi  _et al._ [2008b] A. Rabhi, C. Providencia, and J. Da Providencia, J. Phys. G 35, 125201 (2008b).
  * Sotani and Tatsumi [2015] H. Sotani and T. Tatsumi, MNRAS 447, 3155–3161 (2015).
  * Dexheimer  _et al._ [2021b] V. Dexheimer, K. D. Marquez, and D. P. Menezes, Eur. Phys. Jour. A 57, 216 (2021b).
  * Marquez  _et al._ [2022] K. D. Marquez, M. R. Pelicer, S. Ghosh, J. Peterson, D. Chatterjee, V. Dexheimer, and D. P. Menezes, “Exploring the effects of delta baryons in magnetars,”  (2022).
  * Gomes, R. O.  _et al._ [2019] Gomes, R. O., Pais, H., Dexheimer, V., Providência, C., and Schramm, S., A&A 627, A61 (2019).
  * Fogaça  _et al._ [2016] D. A. Fogaça, S. M. Sanches, T. F. Motta, and F. S. Navarra, Phys. Rev. C 94, 055805 (2016).
  * Tolos  _et al._ [2016] L. Tolos, M. Centelles, and A. Ramos, Astrophys. Jour. 834, 3 (2016).
  * Ferrer and Hackebill [2019] E. J. Ferrer and A. Hackebill, Phys. Rev. C 99, 065803 (2019).
  * Pelicer and Menezes [2022] M. R. Pelicer and D. P. Menezes, (2022), arXiv:2207.03878 [nucl-th] .
  * Rabhi  _et al._ [2009] A. Rabhi, H. Pais, P. K. Panda, and C. Providência, Jour. Phys. G: Nucl. Part. Phys. 36, 115204 (2009).
  * Menezes  _et al._ [2009] D. P. Menezes, M. B. Pinto, S. S. Avancini, and C. Providência, Phys. Rev. C 80, 065805 (2009).
  * Menezes and Providência [2003] D. P. Menezes and C. Providência, Phys. Rev. C 68, 035804 (2003).
  * Ferrer  _et al._ [2010] E. J. Ferrer, V. de la Incera, J. P. Keith, I. Portillo, and P. L. Springsteen, Phys. Rev. C 82, 065802 (2010).
  * Dexheimer  _et al._ [2014] V. Dexheimer, D. P. Menezes, and M. Strickland, Jour. Phys. G Nucl. Phys. 41, 015203 (2014).
  * Chatterjee  _et al._ [2015a] D. Chatterjee, T. Elghozi, J. Novak, and M. Oertel, Mon. Not. Roy. Astron. Soc. 447, 3785 (2015a).
  * Dexheimer  _et al._ [2021] V. Dexheimer, K. D. Marquez, and D. P. Menezes, Eur. Phys. Jour. A 57, 216 (2021).
  * Chatterjee  _et al._ [2019] D. Chatterjee, J. Novak, and M. Oertel, Phys. Rev. C 99, 055811 (2019).
  * Thapa  _et al._ [2020] V. B. Thapa, M. Sinha, J. J. Li, and A. Sedrakian, Part. 3, 660–675 (2020).
  * Terrero  _et al._ [2021] D. A. Terrero, S. L. Pérez, D. M. Paret, A. P. Martínez, and G. Q. Angulo, Phys. Rev. C 103, 045807 (2021).
  * Malik  _et al._ [2018] T. Malik, D. Sen, T. K. Jha, and H. Mishra, “Impact of magnetic field on neutron star properties,”  (2018).
  * Gomes  _et al._ [2019] R. O. Gomes, H. Pais, V. Dexheimer, C. Providência, and S. Schramm, Astron. Astrophys. 627, A61 (2019).
  * Walecka [1974] J. D. Walecka, Ann. Phys. 83, 491 (1974).
  * Boguta and Bodmer [1977] J. Boguta and A. Bodmer, Nucl. Phys. A 292, 413 – 428 (1977).
  * Serot [1979] B. D. Serot, Phys. Lett. B 86, 146 – 150 (1979).
  * Sugahara and Toki [1994] Y. Sugahara and H. Toki, Nucl. Phys. A 579, 557 – 572 (1994).
  * Horowitz and Piekarewicz [2001] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86, 5647–5650 (2001).
  * Brockmann and Toki [1992] R. Brockmann and H. Toki, Phys. Rev. Lett. 68, 3408–3411 (1992).
  * Nikšić  _et al._ [2002] T. Nikšić, D. Vretenar, P. Finelli, and P. Ring, Phys. Rev. C 66, 024306 (2002).
  * Lalazissis  _et al._ [2005] G. A. Lalazissis, T. Nikšić, D. Vretenar, and P. Ring, Phys. Rev. C 71, 024312 (2005).
  * Wei  _et al._ [2020] B. Wei, Q. Zhao, Z.-H. Wang, J. Geng, B.-Y. Sun, Y.-F. Niu, and W.-H. Long, Ch. Phys. C 44, 074107 (2020).
  * Taninah  _et al._ [2020] A. Taninah, S. Agbemava, A. Afanasjev, and P. Ring, Phys. Lett. B 800, 135065 (2020).
  * Typel  _et al._ [2010] S. Typel, G. Ropke, T. Klahn, D. Blaschke, and H. H. Wolter, Phys. Rev. C 81, 015803 (2010).
  * Typel and Wolter [1999] S. Typel and H. Wolter, Nucl. Phys. A 656, 331–364 (1999).
  * Banik  _et al._ [2014] S. Banik, M. Hempel, and D. Bandyopadhyay, Astrophys. J. Suppl. 214, 22 (2014).
  * Typel [2016] S. Typel, Eur. Phys. J. A 52, 16 (2016).
  * Sekiguchi  _et al._ [2016] Y. Sekiguchi, K. Kiuchi, K. Kyutoku, M. Shibata, and K. Taniguchi, Phys. Rev. D 93, 124046 (2016).
  * Radice  _et al._ [2018] D. Radice, A. Perego, K. Hotokezaka, S. A. Fromm, S. Bernuzzi, and L. F. Roberts, The Astrophysical Journal 869, 130 (2018).
  * Lehner  _et al._ [2016] L. Lehner, S. L. Liebling, C. Palenzuela, O. L. Caballero, E. O’Connor, M. Anderson, and D. Neilsen, Classical and Quantum Gravity 33, 184002 (2016).
  * Tootle  _et al._ [2022] S. Tootle, C. Ecker, K. Topolski, T. Demircik, M. Järvinen, and L. Rezzolla, “Quark formation and phenomenology in binary neutron-star mergers using V-QCD,” (2022), arXiv:2205.05691 [astro-ph.HE] .
  * Fischer [2021] T. Fischer, Eur. Phys. J. A 57, 270 (2021).
  * Jakobus  _et al._ [2022] P. Jakobus, B. Mueller, A. Heger, A. Motornenko, J. Steinheimer, and H. Stoecker, (2022), arXiv:2204.10397 [astro-ph.HE] .
  * Witten [1984] E. Witten, Phys. Rev. D 30, 272 (1984).
  * Chodos  _et al._ [1974] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974).
  * Freedman and McLerran [1978] B. Freedman and L. McLerran, Phys. Rev. D 17, 1109 (1978).
  * Farhi and Jaffe [1984a] E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984a).
  * Nambu and Jona-Lasinio [1961a] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345–358 (1961a).
  * Nambu and Jona-Lasinio [1961b] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246–254 (1961b).
  * Kleinert [1976] H. Kleinert, in  _14th International School of Subnuclear Physics: Understanding the Fundamental Constitutents of Matter_ (1976).
  * Volkov [1984] M. K. Volkov, Annals Phys. 157, 282–303 (1984).
  * Hatsuda and Kunihiro [1984] T. Hatsuda and T. Kunihiro, Phys. Lett. B 145, 7–10 (1984).
  * Klähn and Fischer [2015] T. Klähn and T. Fischer, Astrophys. J. 810, 134 (2015).
  * Glendenning [1992] N. K. Glendenning, Phys. Rev. D 46, 1274–1287 (1992).
  * Hempel  _et al._ [2013] M. Hempel, V. Dexheimer, S. Schramm, and I. Iosilevskiy, Phys. Rev. C 88, 014906 (2013).
  * Farhi and Jaffe [1984b] E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984b).
  * Kajantie  _et al._ [1991] K. Kajantie, L. Karkkainen, and K. Rummukainen, Nucl. Phys. B 357, 693–712 (1991).
  * Alford and Reddy [2003] M. Alford and S. Reddy, Phys. Rev. D 67, 074024 (2003), arXiv:nucl-th/0211046 .
  * Lugones  _et al._ [2013] G. Lugones, A. G. Grunfeld, and M. Al Ajmi, Phys. Rev. C 88, 045803 (2013), arXiv:1308.1452 [hep-ph] .
  * Lugones and Grunfeld [2017] G. Lugones and A. G. Grunfeld, Phys. Rev. C 95, 015804 (2017), arXiv:1610.05875 [nucl-th] .
  * Maruyama  _et al._ [2007] T. Maruyama, S. Chiba, H.-J. Schulze, and T. Tatsumi, Phys. Rev. D 76, 123015 (2007).
  * Voskresensky  _et al._ [2002] D. Voskresensky, M. Yasuhira, and T. Tatsumi, Phys. Lett. B 541, 93–100 (2002).
  * Peterson  _et al._ [2021] J. Peterson, V. Dexheimer, R. Negreiros, and B. G. Castanheira, Astrophys. J. 921, 1 (2021).
  * Aryal  _et al._ [2020] K. Aryal, C. Constantinou, R. L. S. Farias, and V. Dexheimer, Phys. Rev. D 102, 076016 (2020).
  * [135] LORENE, “Langage objet pour la relativité numérique,” http://www.lorene.obspm.fr (-).
  * Bocquet  _et al._ [1995] M. Bocquet, S. Bonazzola, E. Gourgoulhon, and J. Novak, Astron. Astrophys. 301, 757 (1995).
  * Bonazzola  _et al._ [1993] S. Bonazzola, E. Gourgoulhon, M. Salgado, and J. A. Marck, Astron. Astrophys. 278, 421–443 (1993).
  * Horowitz and Serot [1981] C. Horowitz and B. D. Serot, Nucl. Phys. A 368, 503 – 528 (1981).
  * Bouyssy  _et al._ [1987] A. Bouyssy, J.-F. Mathiot, N. Van Giai, and S. Marcos, Phys. Rev. C 36, 380–401 (1987).
  * Brockmann [1978] R. Brockmann, Phys. Rev. C 18, 1510–1524 (1978).
  * Long  _et al._ [2006] W.-H. Long, N. Van Giai, and J. Meng, Physics Letters B 640, 150 – 154 (2006).
  * Inoue [2019a] T. Inoue (HAL QCD), JPS Conf. Proc. 26, 023018 (2019a).
  * Inoue [2019b] T. Inoue (HAL QCD), AIP Conf. Proc. 2130, 020002 (2019b).
  * Cierniak  _et al._ [2018] M. Cierniak, T. Klähn, T. Fischer, and N. Bastian, Universe 4, 30 (2018).
  * Rather  _et al._ [2021c] I. A. Rather, A. A. Usmani, and S. K. Patra, Jour. Phys. G: Nucl. and Part. Phys. 48, 085201 (2021c).
  * Lopes  _et al._ [2021] L. L. Lopes, C. Biesdorf, and D. e. P. Menezes, Phys. Scripta 96, 065303 (2021).
  * Kumar  _et al._ [2022] A. Kumar, V. B. Thapa, and M. Sinha, Mon. Not. Roy. Astron. Soc. 513, 3788–3797 (2022), arXiv:2204.11034 [astro-ph.HE] .
  * Wei  _et al._ [2019] W. Wei, B. Irving, M. Salinas, T. Klähn, and P. Jaikumar, Astrophys. J. 887, 151 (2019).
  * Chatterjee  _et al._ [2015b] D. Chatterjee, T. Elghozi, J. Novak, and M. Oertel, MNRAS 447, 3785–3796 (2015b).
  * Franzon  _et al._ [2016] B. Franzon, V. Dexheimer, and S. Schramm, Mon. Not. Roy. Astron. Soc. 456, 2937–2945 (2016).
  * Tolman [1939] R. C. Tolman, Phys. Rev. 55, 364–373 (1939).
  * Oppenheimer and Volkoff [1939] J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374–381 (1939).
  * Dexheimer  _et al._ [2017] V. Dexheimer, B. Franzon, R. Gomes, R. Farias, S. Avancini, and S. Schramm, Phys. Lett. B 773, 487–491 (2017).
  * Dexheimer  _et al._ [2012b] V. Dexheimer, R. Negreiros, and S. Schramm, Eur. Phys. Jour. A 48, 189 (2012b).
  * Lopes and Menezes [2015] L. L. Lopes and D. Menezes, JCAP 08, 002 (2015).
  * Broderick  _et al._ [2002] A. E. Broderick, M. Prakash, and J. M. Lattimer, Phys. Lett. B 531, 167–174 (2002).
  * Wang  _et al._ [2022] X. Wang, J. Li, J. Fang, H. Pais, and C. Providência, Phys. Rev. D 105, 063004 (2022).