# Heavy Magnetic Neutron Stars

Ishfaq A. Rather Department of Physics
Aligarh Muslim University
Aligarh, 202002, India  Usuf Rahaman  Department of Physics
Aligarh Muslim University
Aligarh, 202002, India  V. Dexheimer  Department of Physics,
Kent State University, Kent OH 44242 USA  A. A. Usmani  Department of Physics
Aligarh Muslim University
Aligarh, 202002, India  S. K. Patra  Institute of Physics, Bhubaneswar 751005,
India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar,
Mumbai 400094, India

###### Abstract

We systematically study the properties of pure nucleonic and hyperonic
magnetic stars using a density-dependent relativistic mean field (DD-RMF)
equations of state. We explore several parameter sets and hyperon coupling
schemes within the DD-RMF formalism. We focus on sets that are in better
agreement with nuclear and other astrophysical data, while generating heavy
neutron stars. Magnetic field effects are included in the matter equation of
state and in general relativity solutions, which in addition fulfill Maxwell’s
equations. We find that pure nucleonic matter, even without magnetic field
effects, generates neutron stars that satisfy the potential GW190814 mass
constraint; however, this is not the case for hyperonic matter, which instead
only satisfies the more conservative 2.1 M☉ constraint. In the presence of
strong but still somehow realistic internal magnetic fields $\approx 10^{17}$
G, the stellar charged particle population re-leptonizes and de-hyperonizes.
As a consequence, magnetic fields stiffen hyperonic equations of state and
generate more massive neutron stars, which can satisfy the possible GW190814
mass constraint but present a large deformation with respect to spherical
symmetry.

Equation of State – Magnetic fields – Neutron stars

##  1 Introduction

Neutron stars (NSs) are considered to be the densest objects that are formed
as a result of the gravitational collapse of supernovae. Their interior covers
a wide range of densities (reaching $\sim$ 10 times the normal nuclear
density) and, hence, invites the possibility of exotic degrees of freedom like
quarks, hyperons, and kaons. They also provide a distinctive environment to
investigate many unknowns in physics and astrophysics. The properties of the
interior dense matter affect NS macroscopic properties, such as mass, radius,
and the amount by which it can be deformed.

The most precise measurement of a NS mass is 1.4$M_{\sun}$, that of the Hulse-
Taylor pulsar (Hulse & Taylor, 1975). Recently, higher masses of binary
neutron-star (BNS) systems like millisecond pulsar (MSP) PSR J1614-2230 (M=
1.97 $\pm$ 0.04 $M_{\sun}$) (Demorest et al., 2010), PSR J0348+0432 (M= 2.01
$\pm$ 0.04 $M_{\sun}$) (Antoniadis & Freire et al., 2013), and PSR J0740+6620
(M= 2.04${}^{+0.10}_{-0.09}$ $M_{\sun}$) (Cromartie & Fonseca et al., 2019)
have been measured with high precision. The gravitational wave detection by
LIGO and Virgo Collaborations (LVC) of a BNS merger GW170817 event (Abbott et
al., 2017, 2018) further provided a precise measurement of the NS mass,
1.16-1.60 $M_{\sun}$ for low spin priors and a maximum mass approaching 1.9
$M_{\sun}$ for high spin priors (Abbott et al., 2019), which allowed one to
study dense matter properties at extreme conditions due to the measurement of
a new property of NSs, the tidal deformability. The latter provided a new
constraint on the NS EoS and its variation with the radius approximated by the
relation $\Lambda\propto R^{5}$ (Hinderer et al., 2010; Kumar et al., 2017)
resulting in a strong constraint on the nuclear EoS at intermediate density.

Very recently, a new gravitational wave event reported by LVC as GW190814
(Abbott et al., 2020a) was observed with a 22.2-24.3 $M_{\sun}$ black hole and
2.50-2.67 $M_{\sun}$ secondary component. The secondary component of GW190814
attracted a lot of attention as it contained no measurable tidal deformability
signatures. As the mass of the secondary component of GW190814 lies in the
lower region of the mass-gap (2.5$M_{\sun}$ $<$ M $<$ 5$M_{\sun}$), it raised
the question whether it is a light black hole or a very massive NS. To
understand this object, many interesting theories have been proposed recently
regarding its nature as the most massive NS, lightest black hole, or fastest
pulsar (Godzieba et al., 2021; Most et al., 2020; Zhang & Li, 2020; Tsokaros
et al., 2020; Fattoyev et al., 2020; Lim et al., 2020a; Tews et al., 2021;
Rather et al., 2021c; Sedrakian et al., 2020a; Beniamini et al., 2020; Biswas
et al., 2020; Roupas et al., 2020; Zhou et al., 2021; Wu et al., 2021;
Khadkikar et al., 2021; Bombaci et al., 2020; Goncalves & Lazzari, 2020;
Christian & Schaffner-Bielich, 2021a; Demircik et al., 2021a; Li et al., 2020;
Riahi & Kalantari, 2021; Dhiman et al., 2007; Shahrbaf et al., 2020; Thapa &
Sinha, 2020; Huang et al., 2020a; Lim et al., 2020b; Tan et al., 2020; Gupta
et al., 2020; Dexheimer et al., 2021).

Different values of NS radii have been obtained from the the analysis of X-ray
spectra emitted by NSs (Özel et al., 2010; Lattimer & Steiner, 2014; Lattimer
& Prakash, 2016; Özel & Freire, 2016). Lattimer & Prakash determined the range
for radius at the NS canonical mass 10.7 $\leq R_{1.4}$ $\leq$ 13.1 km from
the properties of symmetric matter and nuclear experiments (Lattimer &
Prakash, 2016). The recent measurement of the radius for the NS canonical mass
inferred from PSR J0030+0451 by the Neutron Star Interior Composition Explorer
(NICER) (Gendreau et al., 2016) is R=13.02${}_{-1.06}^{+1.24}$ km for
M=1.44${}_{-0.14}^{+0.15}$ $M_{\sun}$ (Miller et al., 2019) or
R=12.71${}_{-1.19}^{+1.14}$ km for M=1.34${}_{-0.16}^{+0.15}$ $M_{\sun}$
(Riley et al., 2019), obtained using a Bayesian inference approach. In Ref.
(Fattoyev et al., 2018), an upper limit on the NS radius for 1.4 $M_{\sun}$,
$R_{1.4}$ $\leq$ 13.76 km was deduced. The measurements from NICER provide an
estimate of the canonical mass NS radius within a precision of 1 km. For a
canonical mass NS with small radius, a softening of the pressure of neutron
matter at intermediate density is required, which results in a small value of
the nuclear symmetry energy around saturation density $\rho_{0}$ (Lattimer &
Prakash, 2007; Özel & Freire, 2016). However, the requirement of the
2$M_{\sun}$ NS doesn’t allow the pressure to be reduced too much. The small
radius and large mass constraints are satisfied simultaneously by very few
models that provide an additional accurate description of finite nuclei
properties (Chen & Piekarewicz, 2015; Jiang et al., 2015; Guichon et al.,
2018).

At densities of a few times normal nuclear density, the composition of matter
inside NS is not known. With increasing density, the appearance of exotic
degrees of freedom inside NSs is possible and has been studied over the past
decade. In particular, the appearance of hyperonic matter under NS inner core
conditions is energetically favored (Ambartsumyan & Saakyan, 1960;
Glendenning, 1985). The onset of hyperons reduces the pressure, leading to a
softer EoS as they open new channels filling their Fermi sea, which decreases
the maximum mass of NSs by about 0.5 $M_{\sun}$ (Glendenning, 1982; Chatterjee
& Vidaña, 2016). Several works has been performed recently considering
hyperonic matter in NSs (Li & Sedrakian, 2019a; Li et al., 2018b; Li &
Sedrakian, 2019b; Sedrakian et al., 2020a; Fortin et al., 2017; Li et al.,
2018a; Baldo et al., 2000; Tolos et al., 2017; Colucci & Sedrakian, 2014;
Gusakov et al., 2014; Bhowmick et al., 2014; Oertel et al., 2015; Chatterjee &
Vidaña, 2016; Lopes & Menezes, 2018; Isaka et al., 2017; Spinella & Weber,
2019; Guichon et al., 2018; Vidaña, 2018; Fortin et al., 2020; Tolos &
Fabbietti, 2020; Gomes et al., 2015; Dexheimer & Schramm, 2008; Fortin et al.,
2021).

Heavy NSs are expected to contain exotic matter in their interior, even if
they are rotating fast (Avancini et al., 2018; Dexheimer et al., 2021). Very
massive and/or fast rotating stars could be the result of accretion, or even a
previous stellar merger, both of which have been shown to enhance stellar
magnetic fields (Pons & Viganò, 2019; Bransgrove et al., 2017; Yakhshiev et
al., 2019; Aguirre, 2020; Zhong & Dai, 2020; Most et al., 2019; Ciolfi et al.,
2019; Giacomazzo et al., 2015; Xue et al., 2019; Raynaud et al., 2020; Sur &
Haskell, 2021). With exceptionally high density, a magnetic field reaching
$\approx$ 109 to 1018 G (Bocquet et al., 1995; Cardall et al., 2001;
Chatterjee et al., 2015) is attainable in massive NS centers. Among various
classes of compact stars available, Anomalous X-ray pulsars (AXPs) and the
Soft gamma repeaters (SGRs), usually called magnetars (Mereghetti et al.,
2015; Kaspi & Beloborodov, 2017), are considered to have surface magnetic
field within the order 1014-1015 G (Harding & Lai, 2006; Turolla et al.,
2015). Fast radio bursts (FRBs) have also shown evidence of magnetars
(Margalit et al., 2020; Beniamini et al., 2020; Beloborodov, 2020; Levin et
al., 2020; Zanazzi & Lai, 2020).

In the interior of the magnetars, the magnetic field cannot be measured
directly, and hence only estimated using theoretical models. The absolute
largest value of magnetic field a star can possess in the interior (usually
taken as an upper bound for the magnetic field) can be estimated using the
relativistic version of the virial theorem Bonazzola et al. (1993), for which
the negative total energy implies that the magnitude of the gravitational
potential energy must be greater than the magnetic field energy, which results
in magnetic field $\approx$ 1018 G. As discussed in Refs. Bocquet et al.
(1995); Cardall et al. (2001), assuming a poloidal configuration, the maximum
allowed central magnetic ﬁeld that still fulfills Einstein’s and Maxwell’s
equations is around a few times 1018 G, the exact value being dependent on the
equation of state. Similar limits exist in purely toroidal configurations Pili
et al. (2017). As discussed in Refs. Dexheimer et al. (2017b); Pili et al.
(2017), the magnitude of magnetic field does not increase much beyond one
order of magnitude within a star, regardless of the equation of state or
magnetic field configuration. This implies that, regardless of its
feasibility, the star would have to present $\geq 10\times^{16}$ G on the
surface in order to reach a magnetic field $\geq 10\times^{17}$ G or higher in
the center. Most of the magnetar observations inferred magnetic fields of
$\sim 10\times^{15}$ G or below, the only exception being data from the source
4U 0142+61 for slow phase modulations in hard x-ray pulsations (interpreted as
free precession) that suggests magnetic fields of the order of $10\times^{16}$
G Makishima et al. (2014). Ref. Dall’Osso et al. (2018) provides
$10\times^{16}$ G as a low bound for the same source. To be strongly
structurally changed by magnetic fields, either more sources have not yet been
discovered because their magnetic fields do not fit in the magnetar model
(used to infer magnetic fields through period-period derivative diagrams),
their magnetic fields decay quickly due to misalignment of rotation and
magnetic axes Lander & Jones (2018); Kuzur & Mallick (2019), or these stars
are not stable, being formed for example in previous mergers Price & Rosswog
(2006); Giacomazzo et al. (2015); Most et al. (2019). It should be mentioned
that magnetic fields $\geq 10\times^{15}$ G could be produced by three-
dimensional simulations of supernova explosions with main requirement to
produce a strong field dynamo being sufficient angular momentum in the
progenitor star Raynaud et al. (2020). Previous works have studied the effect
of the magnetic field on the NS Equation of state (EoS) and on the stellar
properties like mass and radius (Chakrabarty et al., 1997; Broderick et al.,
2000; Rabhi et al., 2008b; Mallick & Schramm, 2014; Bandyopadhyay et al.,
1997; Aguirre, 2019; Gomes et al., 2017; Dexheimer et al., 2012; Pili et al.,
2017; Felipe et al., 2008; Paulucci et al., 2013; Casali et al., 2014; Rabhi
et al., 2008a). The presence of strong magnetic fields deviates neutron-star
structures from spherical symmetry very strongly and, hence, the spherically
symmetric Tolman–Oppenheimer–Volkoff (TOV) equations can no longer be applied
for studying their macroscopic structure (Chatterjee et al., 2015; Gomes et
al., 2019).

In the present work, we employ recently proposed density-dependent
Relativistic Mean-Field (DD-RMF) parameter sets to study the properties of
NSs. By reproducing different hyperon-hyperon optical potentials, the values
of the hyperon couplings are obtained, using several different coupling
schemes, which are then used to model hyperons in our calculations. The
presence of hyperons lowers NS maximum masses (to a value still larger than 2
$M_{\sun}$) by softening the EoS, although the radius at the NS canonical mass
remains insensitive to them. We further analyze the effect of magnetic fields
on nucleonic and hyperonic matter by employing a realistic chemical potential-
dependent magnetic field (Dexheimer et al., 2017a). The macroscopic stellar
matter properties for the magnetic EoSs are obtained using the publicly
available Language Objet pour la RElativité NumériquE (LORENE) library
(LORENE, -; Bocquet et al., 1995; Bonazzola et al., 1993), which solves the
coupled Einstein-Maxwell field equations in order to determine stable magnetic
star configurations. In this case, NSs become more massive, especially the
ones containing hyperons in their interior.

The main motivation behind our work is to demonstrate that a possible
measurement of a neutron star mass $\sim 2.5$ M⊙ does not necessarily rule out
exotic degrees of freedom in its interior, as several works in the literature
claim. In this regard, despite the fact that the GW190814 data has been
available for about a year Abbott et al. (2020b), our conclusions differ
significantly from all other works on the subject that have been published.
While some argue that if the secondary object in GW190814 is a neutron star,
it either has no exotic degrees of freedom Li et al. (2020); Sedrakian et al.
(2020b) or, if it does, it rotates extremely fast Dexheimer et al. (2021);
Rather et al. (2021a) or uses an unusual approach to model deconfined quark
matter Christian & Schaffner-Bielich (2021b); Roupas et al. (2021); Gonçalves
& Lazzari (2020); Bombaci et al. (2021); Demircik et al. (2021b); Cao et al.
(2020), We investigate the possibility of a star with a strong magnetic field
inside. We investigate how different particle populations and nuclear
interactions affect microscopic and macroscopic stellar properties, and we
illustrate how, if the secondary object in GW190814 is a massive neutron star,
we can learn about the dense matter inside them.

The paper is organized as follows: in section 2, the density-dependent RMF
model is presented and the inclusion of the magnetic field for the beta-
equilibrium EoS is discussed. The various DD-RMF parameter sets, nuclear
matter properties, and hyperon couplings are discussed in section 3. The
nucleonic and hyperonic EoSs are described in subsection 4.1, along with
spherical solutions for star matter properties. Subsection 4.2 deals with the
neutron and hyperon EoS with effects of magnetic fields, together with the
discussion of their stellar properties. Subsection 4.3 presents additional
results produced assuming different hyperon couplings. We summarize our
results and outline an outlook for the future in sec. 5. Finally, the
equations of motion and the expressions for the energy and baryon density in
the presence of magnetic fields are shown in Appendix A.

##  2 Theory and Formalism

The Lagrangian density is usually the starting point in RMF theory: the
baryons interact through meson exchanges, including the scalar-isoscalar sigma
$\sigma$, the vector-isoscalar $\omega$, and the vector-isovector $\rho$. The
Lagrangian density contains contributions from the baryonic octet, leptons and
the mesons, followed by the interactions between them. Within the DD-RMF
model, the coupling constants are density-dependent (Brockmann & Toki, 1992),
as shown in the following.

The DD-RMF Lagrangian density is written as (Huang et al., 2020b)

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

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

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

The Lagrangian density for the pure electromagnetic part is written as

$\mathcal{L}_{f}=-\frac{1}{16\pi}F_{\mu\nu}F^{\mu\nu},$ (3)

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

$\mathcal{L}=\mathcal{L}_{m}+\mathcal{L}_{f}.$ (4)

The density dependent coupling constants in the DD-RMF model are parametrized
by the relation

$g_{iB}(\rho_{B})=g_{iB}(\rho_{0})f_{i}(x),$ (5)

for $i=\sigma,\omega$. The function $f_{i}(x)$, where $x=\rho_{B}/\rho_{0}$
with $\rho_{0}$ as the nuclear matter saturation density, is defined as

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}.$ (6)

For the $\rho$ meson, the density-dependent coupling constant is given by an
exponential relation as

$g_{\rho}(\rho_{B})=g_{\rho}(\rho_{0})exp[-a_{\rho}(x-1)].$ (7)

This parametrization has eight parameters, which are reduced to three using
the constraint conditions from refs (Typel & Wolter, 1999; Rather et al.,
2021c) to reproduce symmetric and asymmetric nuclear matter properties.

For the present case that includes all baryons from the octet, the NS chemical
equilibrium condition between different particles are

$\displaystyle\mu_{n}$ $\displaystyle=\mu_{\Sigma^{0}}=\mu_{\Xi^{0}},$
$\displaystyle\mu_{p}$ $\displaystyle=\mu_{\Sigma^{+}}=\mu_{n}-\mu_{e},$
$\displaystyle\mu_{\Sigma^{-}}$ $\displaystyle=\mu_{\Xi^{-}}=\mu_{n}+\mu_{e},$
$\displaystyle\mu_{\mu}$ $\displaystyle=\mu_{e}.$ (8)

The charge neutrality condition follows as

$\rho_{p}+\rho_{\Sigma^{+}}=\rho_{e}+\rho_{\mu^{-}}+\rho_{\Sigma^{-}}+\rho_{\Xi^{-}}.$
(9)

The expressions for the energy density and pressure in presence of a magnetic
field can be obtained by solving the energy-momentum tensor relation

$T^{\mu\nu}=T_{m}^{\mu\nu}+T_{l}^{\mu\nu},$ (10)

where (Huang et al., 2010; Khalilov, 2002)

$\displaystyle T_{m}^{\mu\nu}$
$\displaystyle=\mathcal{E}_{m}u^{\mu}u^{\nu}-P(g^{\mu\nu}-u^{\mu}u^{\nu})$
$\displaystyle+\mathcal{M}B\Bigg{(}g^{\mu\nu}-u^{\mu}u^{\nu}+\frac{B^{\mu}B^{\nu}}{B^{2}}\Bigg{)},$
$\displaystyle T_{l}^{\mu\nu}$
$\displaystyle=\frac{B^{2}}{4\pi}\Bigg{(}u^{\mu}u^{\nu}-\frac{1}{2}g^{\mu\nu}\Bigg{)}-\frac{B^{\mu}B^{\nu}}{4\pi}.$
(11)

Here, $\mathcal{M}$ is the magnetization per unit volume and
$B^{\mu}B_{\mu}=-B^{2}$. For the nuclear matter in presence of a magnetic
field, the single particle energy of all charged baryons and leptons is
quantized in the direction perpendicular to the magnetic field.

For a uniform magnetic field locally pointing in the $z$-direction,
$B=B\hat{z}$, the total energy of a charged particle becomes

$E_{cb}=\sqrt{k_{z}^{2}+M^{*2}+2\nu|q|B},$ (12)

The quantity
$\nu=\Big{(}n+\frac{1}{2}-\frac{1}{2}\frac{q}{|q|}\sigma_{z}\Big{)}=0,1,2,...$
indicates the Landau levels of fermions with electric charge $q$. $n$ is the
orbital angular momentum quantum number and $\sigma_{z}$ is the Pauli matrix.
The Fermi momentum of all baryons charged $k_{F,\nu}^{cb}$ and leptons
$k_{F,\nu}^{l}$ with Fermi energies $E_{F}^{cb}$ and $E_{F}^{l}$,
respectively, are given as (Broderick et al., 2000)

$\displaystyle k_{F,\nu}^{cb}$
$\displaystyle=\sqrt{(E_{F}^{cb})^{2}-M_{cb}^{*2}-2\nu|q_{cb}|B},$
$\displaystyle k_{F,\nu}^{l}$
$\displaystyle=\sqrt{(E_{F}^{l})^{2}-m_{l}^{2}-2\nu|q_{l}|B}.$ (13)

The highest value of $\nu$ is obtained with sum over Landau levels under the
condition that the Fermi momentum of each particle is positive

$\displaystyle\nu_{max}$
$\displaystyle=\Bigg{[}\frac{(E_{F}^{cb})^{2}-M_{cb}^{*2}}{2|q_{cb}|B}\Bigg{]},$
$\displaystyle\nu_{max}$
$\displaystyle=\Bigg{[}\frac{(E_{F}^{l})^{2}-m_{l}^{2}}{2|q_{l}|B}\Bigg{]},$
(14)

for charged baryons $cb$ and leptons, respectively.

The expressions for the matter energy density and baryon density obtained in
presence of magnetic field are shown in Appendix A. From Eq. 2, the total
energy density is

$\mathcal{E}=\mathcal{E}_{m}+\frac{B^{2}}{8\pi}.$ (15)

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

$\displaystyle P_{\perp}$
$\displaystyle=P_{m}-\mathcal{M}B+\frac{B^{2}}{8\pi},$ $\displaystyle
P_{\parallel}$ $\displaystyle=P_{m}-\frac{B^{2}}{8\pi},$ (16)

where the magnetization is calculated as

$\mathcal{M}=\partial P_{m}/\partial B.$ (17)

##  3 Parameter Sets

For the present study, we employ two recently proposed density dependent DD-
MEX (Taninah et al., 2020), and DD-LZ1 (Wei et al., 2020) parameter sets.
These include the necessary tensor couplings of the vector mesons to nucleons
and were obtained by fitting the ground state properties of finite nuclei,
which considered parametric corrections and shell evaluations of mesons.
Additionally, widely used parameter sets DD-ME1 (Nikšić et al., 2002) and DD-
ME2 (Lalazissis et al., 2005) are also used.

Table 1 displays the nucleon and meson masses and the coupling constants
between nucleon and mesons for the given parameter sets. The parameters
$a,b,c,d$ for $\sigma$, $\omega$, and $\rho$ mesons are also shown.

Table 1: Nucleon and meson masses and different coupling constants for various DD-RMF parameter sets. Nuclear matter properties: density, binding energy ($E/A$), incompressibility ($K_{0}$), symmetry energy ($J$), slope parameter ($L$), and nucleon effective masses at saturation density are also shown. | DD-LZ1 | DD-ME1 | DD-ME2 | DD-MEX
---|---|---|---|---
$m_{n}$ |  938.9000 |  939.0000 |  939.0000 |  939.0000
$m_{p}$ |  938.9000 |  939.0000 |  939.0000 |  939.0000
$m_{\sigma}$ |  538.6192 |  549.5255 |  550.1238 |  547.3327
$m_{\omega}$ |  783.0000 |  783.0000 |  783.0000 |  783.0000
$m_{\rho}$ |  769.0000 |  763.0000 |  763.0000 |  763.0000
$g_{\sigma}(\rho_{0})$ |  12.0014 |  10.4434 |  10.5396 |  10.7067
$g_{\omega}(\rho_{0})$ |  14.2925 |  12.8939 |  13.0189 |  13.3388
$g_{\rho}(\rho_{0})$ |  15.1509 |  7.6106 |  7.3672 |  7.2380
$a_{\sigma}$ |  1.0627 |  1.3854 |  1.3881 |  1.3970
$b_{\sigma}$ |  1.7636 |  0.9781 |  1.0943 |  1.3350
$c_{\sigma}$ |  2.3089 |  1.5342 |  1.7057 |  2.0671
$d_{\sigma}$ |  0.3799 |  0.4661 |  0.4421 |  0.4016
$a_{\omega}$ |  1.0592 |  1.3879 |  1.3892 |  1.3926
$b_{\omega}$ |  0.4183 |  0.8525 |  0.9240 |  1.0191
$c_{\omega}$ |  0.5386 |  1.3566 |  1.4620 |  1.6060
$d_{\omega}$ |  0.7866 |  0.4957 |  0.4775 |  0.4556
$a_{\rho}$ |  0.7761 |  0.5008 |  0.5647 |  0.6202
$\rho_{0}(fm^{-3})$ |  0.158 |  0.152 |  0.152 |  0.152
$E/A$(MeV) |  -16.126 |  -16.668 |  -16.233 |  -16.140
$K_{0}$(MeV) |  231.237 |  243.881 |  251.306 |  267.059
$J$(MeV) |  32.016 |  33.060 |  32.310 |  32.269
$L$(MeV) |  42.467 |  55.428 |  51.265 |  49.692
$M_{n}^{*}/M$ |  0.558 |  0.578 |  0.572 |  0.556
$M_{p}^{*}/M$ |  0.558 |  0.578 |  0.572 |  0.556

Some important nuclear matter properties for the DD-LZ1, DD-ME1, DD-ME2, and
DD-MEX parameter sets are also shown in table (1). The symmetry energy
parameter $J$ for the listed parameter sets is compatible with the values
$J=31.6\pm 2.66$ MeV (Li & Han, 2013). The symmetry energy slope parameter $L$
also satisfies standard constraints $L=59.57\pm 10.06$MeV (Zhang et al., 2020;
Danielewicz & Lee, 2014). A very recent publication by the PREX collaboration
reported a value for the neutron skin thickness of 208 Pb of $R_{skin}$ =
(0.283 $\pm$ 0.071) fm (Adhikari et al., 2021). Such a value is significantly
larger than the $R_{skin}$ = 0.20 fm predicted by DD-ME1. Given the strong
correlation between $R_{skin}$ and $L$, symmetry energy slope parameter values
as large as $L$ $\approx$ 100 MeV have been proposed (Reed et al., 2021).
Although our parametrizations reproduce a lower symmetry energy and slope,
they are still in excellent agreement with a large amount of constraints, as
shown in Fig. 1 from Ref. (Li et al., 2021). The currently accepted value of
incompressibility determined from the isoscalar giant monopole resonance
(ISGMR) lies in the range $K_{0}=240\pm 20$MeV. The $K_{0}$ value for all the
given parameter sets is within this range, except for the DD-MEX, which
predicts a value little higher.

The density-dependent coupling constants of the hyperons to the vector mesons
are determined from the SU(6) symmetry as (Banik & Bandyopadhyay, 2001;
Schaffner & Mishustin, 1996; Banik et al., 2014; Tolos et al., 2016)

$\displaystyle\frac{1}{2}g_{\omega\Lambda}$
$\displaystyle=\frac{1}{2}g_{\omega\Sigma}=g_{\omega\Xi}=\frac{1}{3}g_{\omega
N},$ $\displaystyle\frac{1}{2}g_{\rho\Sigma}$
$\displaystyle=g_{\rho\Xi}=g_{\rho N},g_{\rho\Lambda}=0.$ (18)

These couplings are calculated by fitting the hyperon optical potential
obtained from the experimental data. See Refs. (Inoue, 2019a, b) for current
lattice calculations and experimental data for these potentials.

The hyperon coupling to the $\sigma$ field is obtained, so as to reproduce the
hyperon potential in symmetric nuclear matter (SNM) derived from the
hypernuclear observables (Banik et al., 2014; Klevansky, 1992)

$U_{\Lambda}^{N}(\rho)=g_{\omega\Lambda}\omega_{0}+\sum_{R}-g_{\sigma\Lambda}\sigma_{0}.$
(19)

For the present study, we reproduce the following optical potentials for the
hyperons (Thapa et al., 2020)

$\displaystyle U_{\Lambda}^{N}(\rho_{0})$ $\displaystyle=-30~{}\mathrm{MeV},$
$\displaystyle U_{\Sigma}^{N}(\rho_{0})$ $\displaystyle=+30~{}\mathrm{MeV},$
$\displaystyle U_{\Xi}^{N}(\rho_{0})$ $\displaystyle=-14~{}\mathrm{MeV}.$ (20)

These potentials correspond to the value of the density-dependent scalar
couplings $g_{\sigma\Lambda}/g_{\sigma N}=0.6105$, $g_{\sigma\Xi}/g_{\sigma
N}=0.3024$, and $g_{\sigma\Sigma}/g_{\sigma N}=0.4426$.

Figure 1:  Dense matter Equation of State for the DD-LZ1, DD-ME1, DD-ME2, and
DD-MEX parameter sets. The solid lines represent pure nucleonic matter, while
the dashed lines represent hyperonic matter including the entire baryon octet.

##  4 Results and Discussions

###  4.1 Nucleonic and Hyperonic neutron stars

The EoS of pure nucleonic (solid lines) and hyperonic matter (dashed lines)
under chemical-equilibrium and charge neutrality conditions for several DD-RMF
parameter sets are displayed in figure 1. For pure nucleonic matter, the DD-
MEX parameter set produces a stiff EoS at low density region, while the DD-ME2
set produces a stiff EoS in the high density regime. The hyperons start to
appear in the density range $\approx$ 300-400 MeV/fm3 for all parameter sets.
The onset of hyperonization softens the EoS (reduction in the pressure) due to
the hyperons replacing the neutrons and opening new channels to distribute the
Fermi energy. To build a unified EoS, the Baym-Pethick-Sutherland (BPS) EoS
(Baym et al., 1971) has been used in the outer crust region. For the inner
crust part, the EoS for non-uniform matter is generated by using DD-ME2
parameter set in a Thomas-Fermi approximation (Avancini et al., 2009; Pais &
Providência, 2016; Rather et al., 2021b). All the DD-RMF parameter sets
considered are very similar in the outer crust density regime.

With the EoSs obtained for pure nucleonic and hyperonic matter, stellar matter
properties like mass and radius are obtained by solving the Tolmann-
Oppenheimer-Volkov (TOV) coupled differential equations (Tolman, 1939;
Oppenheimer & Volkoff, 1939) for a static isotropic spherically symmetric star

$\frac{dP(r)}{dr}=-\frac{[\mathcal{E}(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r^{2}(1-2M(r)/r)},$ (21)

and

$\frac{dM(r)}{dr}=4\pi r^{2}\mathcal{E}(r),$ (22)

where $M(r)$ contained in the radius $r$ represents the gravitational mass for
a specific EoS and a given choice of central energy density $\mathcal{E}_{c}$.

The tidal deformability $\lambda$ is defined as (Hinderer et al., 2010; Kumar
et al., 2017)

$\lambda=-\frac{Q_{ij}}{\mathcal{E}_{ij}}=\frac{2}{3}k_{2}R^{5},$ (23)

where, $Q_{ij}$ represents the mass quadrupole moment of a star and
$\mathcal{E}_{ij}$ corresponds to its external tidal field. The dimensionless
tidal deformability $\Lambda$ is related to the compactness parameter $C=M/R$
as

$\Lambda=\frac{\lambda}{M^{5}}=\frac{2k_{2}}{3C^{5}},$ (24)

where $k_{2}$ is the second love number. To estimate the love number along
with the evaluation of the TOV equations, the function $y=y(R)$ as defined in
ref (Hinderer, 2008) is computed with the initial condition of $y(0)=2$ for
the second love number. This value can be computed from the following first-
order differential equations (Kumar et al., 2017; Hinderer, 2008)

$r\frac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,$ (25)

where,

$\displaystyle F(r)$ $\displaystyle=\frac{r-4\pi
r^{3}[\mathcal{E}(r)-P(r)]}{r-2M(r)},$ (26) $\displaystyle Q(r)$
$\displaystyle=\frac{4\pi
r\Big{(}5\mathcal{E}(r)+9P(r)+\frac{\mathcal{E}(r)+P(r)}{\partial
P(r)/\partial\mathcal{E}(r)}-\frac{6}{4\pi r^{2}}\Big{)}}{r-2M(r)}$
$\displaystyle-4\Bigg{[}\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}(1-2M(r)/r)}\Bigg{]}^{2}.$ (27)

Once the above equations are solved for $y=y(R)$ with $R$ being the radius of
the spherical star in isolation, the second tidal love number is obtained from
the following expression (Hinderer et al., 2010)

$\displaystyle k_{2}$
$\displaystyle=\frac{8}{5}(1-2C)^{2}[2C(y-1)]\Bigl{\\{}2C(4(y+1)C^{4}+(6y-4)C^{3}$
$\displaystyle+(26-22y)C^{2}+3(5y-8)C-3y+6)$
$\displaystyle-3(1-2C)^{2}(2C(y-1)-y+2)\log\Big{(}\frac{1}{1-2C}\Big{)}\Bigr{\\}}^{-1}.$
(28)

Given the initial boundary conditions for the NS at the center $P(0)=P_{c}$,
$M(0)=0$, and $y(0)=2$ with $P_{c}$ as the central pressure, to the surface of
the star $P(R)=0$, $M(R)=M$, and $r(R)=R$, the above equations are solved for
a given central density to determine the macroscopic properties of a NS.

Figure 2:  Mass-Radius relation for pure nucleonic (solid lines) and hyperonic
(dashed lines) stars using several DD-RMF parameters. The colored areas show
recent constraints inferred from GW190814 and from the massive pulsars MSP
J0740+6620 and PSR J0348+0432 (Abbott et al., 2020a; Cromartie & Fonseca et
al., 2019; Antoniadis & Freire et al., 2013). The constraints on the mass-
radius limits inferred from NICER observations (Miller et al., 2019; Riley et
al., 2019) are also shown.

Figure 2 shows the mass-radius relation for pure nucleonic and hyperonic
matter for the parameter sets DD-ME1, DD-ME2, DD-LZ1, and DD-MEX. The shaded
areas represent the recent constraints on the NS maximum mass inferred from
GW190814 (M=2.50-2.67 $M_{\sun}$), the massive pulsars MSP J0740+6620
(M=2.14${}_{-0.09}^{+0.10}$ $M_{\sun}$), and PSR J0348+0432 (M=2.01$\pm 0.04$
$M_{\sun}$). The constraints on the radius limits around the NS canonical mass
inferred from PSR J0030+0451 by NICER, R=13.02${}_{-1.06}^{+1.24}$ km at
M=1.44${}_{-0.14}^{+0.15}$ $M_{\sun}$ (Miller et al., 2019) and
R=12.71${}_{-1.19}^{+1.14}$ km at M=1.34${}_{-0.16}^{+0.15}$ $M_{\sun}$ (Riley
et al., 2019), are also shown. For pure nucleonic matter, the DD-LZ1 and DD-
MEX parameter sets reach a maximum mass of 2.55 and 2.57 $M_{\sun}$, with a
radius of 12.30 and 12.46 km, respectively, indicating the possibility of
GW190814 secondary component being a very massive NS. The hyperonic
counterparts of the given DD-RMF parameter sets, which soften the EoS, produce
a maximum mass of 2.18 and 2.13$M_{\sun}$ with a radius of 12.24 and 12.07 km
respectively. For the DD-ME1 and DD-ME2 parameter sets, the NS maximum mass
decreases from 2.45 and 2.48 to 1.98 and 2.01 $M_{\sun}$ respectively, while
the respective radius decreases by $\approx$ 0.5 km. The radius at the NS
canonical mass, R1.4, remains insensitive to the appearance of hyperons. The
hyperonic configurations satisfy the maximum mass limit from the massive
pulsars MSP J0740+6620 and PSR J0348+0432, but are inconsistent with the
GW190814 potential constraint (see discussion in Sedrakian et al. (2020a)).
All of the nucleonic and hyperonic configurations satisfy the mass-radius
limits inferred from NICER.

Figure 3:  Dimensionless tidal deformability variation with the NS mass for
nucleonic (dashed lines) and hyperonic (dot dashed lines) stars using DD-LZ1,
DD-ME1, DD-ME2, and DD-MEX parameter sets. The orange dotted line represents
the upper limit on the dimensionless tidal deformability set by measurement
from GW170817 (Abbott et al., 2017).

The variation of the dimensionless tidal deformability with the NS mass for
the nucleonic and hyperonic stars based on the DD-RMF parametrizations are
shown in figure 3. The dimensionless tidal deformability of pure nucleonic as
well as hyperonic matter for all the parameter sets lies well below the upper
limit of $\Lambda_{1.4}$=800 obtained from the gravitational wave event
GW170817 (Abbott et al., 2017). The shift in the tidal deformability for the
hyperonic matter is seen as the mass increases. Table 2 displays the different
properties of neutron and hyperonic stars obtained with different DD-RMF
parameter sets. A subsequent analysis by the LVC suggest a much smaller upper
limit of 580 on the tidal deformability, which is smaller than all the values
displayed in Table 2 Abbott et al. (2018). However, this value corresponds to
the 50% confidence region, the 90% confidence region extracted from the same
data includes the values we reproduce.

Table 2:  Table 2. Properties of pure nucleonic and hyperonic NSs for
different DD-RMF parameter sets, including maximum mass,
m respective radius, and radius and dimensionless tidal deformability of a 1.4
$M_{\sun}$ star.

9.7cm  Nucleonic star Hyperonic star Mmax ($M_{\sun}$) R (km) R1.4 (km)
$\Lambda_{1.4}$ Mmax ($M_{\sun}$) R (km) R1.4 (km) $\Lambda_{1.4}$ DD-ME1
2.449 11.981 12.898 689.342 1.983 11.515 12.898 689.342 DD-ME2 2.483 12.017
12.973 733.149 2.013 11.674 12.973 733.149 DD-LZ1 2.555 12.297 13.069 728.351
2.130 12.067 13.069 728.351 DD-MEX 2.575 12.465 13.168 791.483 2.183 12.238
13.168 791.483

###  4.2 Magnetic stars

In order to study the effects of magnetic fields on our microscopic
description of matter, we employ a chemical potential-dependent magnetic field
derived from the solutions of the Einstein-Maxwell’s equations. The quadratic
relation between the magnetic field and the chemical potential depends on the
magnetic dipole moment and is given by (Dexheimer et al., 2017a)

$B^{*}(\mu_{B})=\frac{(a+b\mu_{B}+c\mu_{B}^{2})}{B_{c}^{2}}\mu,$ (29)

with $\mu_{B}$ being the baryon chemical potential in MeV and $\mu$ the dipole
magnetic moment in units of Am2 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 $a=-0.0769$ G2/(Am2), $b=1.20\times 10^{-3}$ G2/(Am2 MeV) and
$c=-3.46\times 10^{-7}$ 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 $M_{\sun}$.

Figure 4 displays the magnetic field profile as a function of baryon density
for a 2.2 $M_{\sun}$ baryonic mass star obtained for the DD-MEX EoS. The
magnetic field effect on the DD-ME2 EoS has already been calculated (Thapa et
al., 2020) and the DD-ME1 EoS predicts similar behavior. In Ref. (Thapa et
al., 2020), spherically symmetric TOV equations were used for central magnetic
fields $\approx$ 1018 G with a density-dependent and universal profile for
magnetic field. Since the DD-MEX EoS predicts a heavier NS than other
parameter sets, we choose this parameter set to study magnetic effects and
verify whether this model predicts the possibility of the GW190814 secondary
component being a hyperonic magnetar. In fig. 4, the dashed curves represent a
NS without hyperons and dot-dashed curves represent a NS with hyperons. It is
clear that the magnetic field produced by the NS without hyperons is larger.
This illustrates that the transition from $\mu_{B}$ to $\rho_{B}$ is model and
particle population dependent. For magnetic dipole moment greater than 1031
Am2, the magnetic field produced at large densities is larger than 1017 G,
which is strong enough to cause a large deformation in the NS structure. The
values of the magnetic field produced at the surface and at large densities
using different values of the magnetic dipole moment for NSs with and without
hyperons are shown in table 3. For a magnetic dipole moment 1032 Am2, the
magnetic field produced at large densities is greater than 4$\times$ 1017 for
both cases.

Figure 4:  Magnetic field profile as a function of baryon density for the DD-MEX EoS with different values of magnetic dipole moment. The dashed lines represent the profile for NS without hyperons, while the dot-dashed lines represent hyperonic stars. Figure 5:  Variation of matter pressure in transverse direction vs energy density for the DD-MEX parameter set without and with magnetic field effects at different values of magnetic dipole moment. Left panel depicts the EoSs without hyperons and right panel depicts the EoS with hyperons. The insets in each panel show the variation in the pressure at higher value of the energy density for different magnetic moments.  Table 3: Magnetic field at low densities Bs (corresponding to stellar surfaces) and at high values of densities Bc calculated for the DD-MEX EoS with and without hyperons. |  Nucleonic star |  Hyperonic star
---|---|---
$\mu$ (Am2) |  Bs (G) |  Bc (G) |  Bs (G) |  Bc (G)
5$\times$1030 |  1.01$\times$1015 |  2.59$\times$1016 |  6.65$\times$1015 |  1.96$\times$1016
5$\times$1031 |  8.98$\times$1016 |  2.28$\times$1017 |  5.83$\times$1016 |  1.89$\times$1017
1032 |  1.79$\times$1017 |  4.55$\times$1017 |  1.12$\times$1017 |  3.77$\times$1017
Figure 6: Particle fraction of the baryons and leptons as a function of
normalized baryon density for the DD-MEX model without magnetic field (a) and
with magnetic field with different magnetic dipole moments b)
$\mu$=5$\times$1030 Am2, c) $\mu$=5$\times$1031 Am2, and d)
$\mu$=1$\times$1032 Am2.

Figure 5 shows the variation of the transverse pressure vs the energy density
for the DD-MEX parameter set with and without hyperons. The solid line
represents the variation in the pressure without including magnetic field
($\mu$=0), while the other lines represent the variation obtained with the
magnetic field at magnetic dipole moments $\mu$=5$\times$1030 Am2,
5$\times$1031 Am2 and 1030 Am2. The insets show the pressure at higher values
of the energy density. It is clear that the change in the pressure at a given
value of energy density is larger for a NS with hyperons as compared to the NS
without hyperons, which implies that the EoS with hyperons becomes stiffer
than without hyperons (when compared to the $B$= 0 case) in the presence of
strong magnetic field. The reason for such behavior will be discussed in the
following. The magnetic field produced at the magnetic dipole moment
5$\times$1030 Am2 is of the order of 1016 G at the center, which is small
enough to be indistinguishable from the zero magnetic case. For higher
magnetic dipole moments, the magnetic field produced $\approx$ 4 $\times$ 1017
G is strong enough to increase the matter pressure to higher values, thus
producing a distinguishable effect.

Figure 6 shows the particle fractions as a function of baryon density for beta
stable NS matter obtained using the DD-MEX EoS. Figure 6 panel (a) displays
the fractions in the absence of magnetic field, while panels (b), (c), and (d)
depict the particle fractions in the presence of magnetic fields with fixed
magnetic dipole moments $\mu$=5$\times$1030 Am2, $\mu$=5$\times$1031 Am2, and
$\mu$=5$\times$1032 Am2, respectively. Clearly, in all the cases, the
$\Lambda$ particle is the dominant hyperonic component, which starts appearing
in the density range 2 - 3 $\rho_{0}$ (DHapo et al., 2010; Vidaña, 2013;
Rather et al., 2018; Sen et al., 2018). The neutral $\Xi$ hyperon appears at a
density $\approx$ 5.5 $\rho_{0}$ for $B$=0, which remains unaltered with the
inclusion of the magnetic field. As expected, the charged particles are more
strongly affected by the magnetic field and an increase in their population is
seen with the increase in the magnetic field strength. For $B$=0 (and all
other cases), the $e^{-}$ and $\mu^{-}$ population is large at low densities,
which suppresses the appearance of $\Xi^{-}$ hyperons. With an increase in the
magnetic dipole moment, the magnetic field strength increases, which shifts
the appearance of $\Xi^{-}$ hyperon from $\approx$ 2.5 to around $\approx$
3.5$\rho_{0}$. At this threshold, the density of the negatively charged
leptons $e^{-}$ and $\mu^{-}$ starts to drop, as the charge neutrality
condition from (2) allows the $\Xi^{-}$ hyperon to take over. Similarly, the
appearance of the $\Lambda$ hyperon accelerates the disappearance of neutrons,
as both are neutral particles. Overall, we see that the addition of magnetic
field increases the population of leptons (re-leptonizes) and correspondingly
decreases the population of hyperons (de-hyperonizes), which renders the EoS
stiffer (Tolos et al., 2016).

Because of the repulsive nature of the $\Sigma$ potential, the formation of
$\Sigma^{0}$ and $\Sigma^{-}$ is totally suppressed for the densities
considered in the present work. The absence of $\Sigma$ hyperons is supported
by the fact that no bound $\Sigma^{-}$ hypernuclei has been found yet, despite
several searches (Harada & Hirabayashi, 2006, 2015) and this has been
reproduced in several works (Glendenning, 2001; Thapa et al., 2020; Guichon et
al., 2018). The inclusion of strong magnetic fields do not change this
feature. The properties of nucleonic and hyperonic stars at different values
of the magnetic dipole moment, which correspond to different magnetic field
values at the surface and the center, are shown in table 4.

The effect of magnetic field on the mass-radius relation of a NS with and
without hyperons is displayed in figure 7. These calculations are performed
for the DD-MEX EoS using the LORENE library (LORENE, -). Different values of
magnetic dipole moment are used to obtain different values of the magnetic
field at the stellar surface and at the center. As can be seen, the NS maximum
mass without hyperons increases from 2.575 $M_{\sun}$ for $B$=0 to 2.711
$M_{\sun}$ for $\mu$=1032 Am2. The corresponding radius changes from 12.465 to
13.474 km. The radius at 1.4$M_{\sun}$ increases by around 1 km. With the
hyperons included, the mass increases from 2.183 to 2.463 $M_{\sun}$ when
magnetic field effects are included with $\mu$=1032 Am2. The variation
obtained in the mass-radius is larger for hyperonic stars than for the pure
nucleonic stars due to the additional effect of de-hyperonization that takes
place due to the magnetic field. The de-hyperonization results in the
enhancement of the matter pressure $P_{m}$ for a given $\mathcal{E}_{m}$.

Figure 7:  Relation between mass and circumferential radius for a NS without
magnetic field and with magnetic field effects considering different magnetic
dipole moments without hyperons (left panel) and with hyperons (right panel)
using the DD-MEX parameter set. The colored areas show the recent constraints
inferred from GW190814, MSP J0740+6620, and PSR J0348+0432 (Abbott et al.,
2020a; Cromartie & Fonseca et al., 2019; Antoniadis & Freire et al., 2013).
The constraints on the mass-radius limits inferred from NICER (Miller et al.,
2019; Riley et al., 2019) are also shown. Figure 8:  Dimensionless tidal
deformability ($\Lambda$) as a function of NS mass for EoS without magnetic
field (solid line) and with magnetic field effects considering different
magnetic dipole moments. Left panel shows the results obtained for NS without
hyperons, while right panel shows the results for NS with hyperons. The orange
dotted line represents the upper limit on the dimensionless tidal deformabilty
set by measurement from GW170817 (Abbott et al., 2017).

For higher magnetic fields produced at magnetic dipole moments $\mu$=5$\times$
1031 and $\mu$=1032 Am2, the effect of the magnetic field is seen to be very
large at smaller stellar masses. For low magnetic fields, pure nucleonic stars
still satisfy the possible maximum mass constraint from the GW190814 event,
implying the possibility that of its secondary component to be a magnetar. The
radius constraints inferred from NICER are satisfied by both nucleonic as well
as hyperonic stars with low magnetic fields. For central magnetic fields
$\approx$ 7 $\times$ 1016 G, the EoS obtained for pure nucleonic matter
satisfies the radius constraints from NICER measurement. For hyperonic matter,
a lower magnetic field $\approx$ 4 $\times$ 1016 G produces a hyperonic star
with radius that satisfies NICER constraints. This is due to the fact that for
magnetic fields less than 1017 G, the deformation produced in the stellar
structure is negligible and, hence, the variation in radius is too small when
compared to the non-magnetic case. Note that the surface magnetic field of the
stars observed by NICER is expected to be $~{}\approx 10^{8}$ G (Miller et
al., 2019).

Figure 8 shows the variation in the dimensionless tidal deformability as a
function of gravitational mass with magnetic field effects and considering
different values of magnetic dipole moment. The results for EoS with and
without hyperons are shown. For pure nucleonic stars, the tidal deformability
increases to a value $\Lambda_{1.4}\approx$ 1500 for a central magnetic field
of 4.55$\times$ 1017 G produced fixing the magnetic dipole moment to
$\mu$=1032 Am2, thus violating the constraint on the dimensionless tidal
deformability from GW170817, which provides an upper limit of 800 on
$\Lambda_{1.4}$ at 90% confidence (Abbott et al., 2017). Seen in table 4, a
small magnetic field produces a NS with tidal deformability larger than the
upper limit from GW170817. This confirms that the BNS merger event GW170817
did not consist of magnetars.

Table 4:  Table 4. Stellar properties: Maximum mass ($M_{max}$), maximum mass
radius ($R_{max}$), canonical mass radius ($R_{1.4}$), and
m dimensionless tidal deformability ($\Lambda_{1.4}$) of pure nucleonic and
hyperonic star for the DD-MEX EoS without magnetic field
m and with magnetic field effects for different values of the magnetic dipole
moment.

9.5cm  $\mu$ (Am2) Nucleonic star Hyperonic star Mmax ($M_{\sun}$) R (km) R1.4
(km) $\Lambda_{1.4}$ Mmax ($M_{\sun}$) R (km) R1.4 (km) $\Lambda_{1.4}$ 0
2.575 12.465 13.168 791.483 2.183 12.238 13.168 791.483 5$\times$ 1030 2.580
12.536 13.235 802.801 2.224 12.506 13.168 791.483 5$\times$ 1031 2.632 13.024
14.507 1175.35 2.325 13.269 14.112 998.882 1032 2.711 13.474 15.027 1517.09
2.463 13.894 15.105 1559.194

For hyperonic stars, the value at 1.4$M_{\sun}$ remains unchanged when
considering a magnetic dipole moment of $\mu$=5$\times$ 1030 Am2. As the
magnetic dipole moment increases, stronger magnetic fields increase the
stellar radius, which allows the tidal deformability $\Lambda_{1.4}$ to reach
a value around 1550. For a nucleonic star and a hyperonic star with
dimensionless tidal deformability well within the limit of GW170817 at 90%
confidence, a magnetic field with a maximum value of $\approx$ 2 $\times$ 1016
G is required. See Ref (Biswas & Bose, 2019) for a discussion of the tidal
deformability of deformed stars and Ref. (Zhu et al., 2020) for a discussion
on how very large magnetic fields generate new high-order corrections to the
deformability that can modify the gravitational-wave evolution of magnetars.
Note that magnetic NSs can also be a source of gravitational waves due to
their asymmetric nature (Sur & Haskell, 2021; Haskell et al., 2008; Gomes et
al., 2019; Bonazzola & Gourgoulhon, 1996; Frieben & Rezzolla, 2012).

###  4.3 Additional hyperon couplings

To investigate how different hyperon couplings and hyperon potentials affect
the results we presented so far, we use a more general symmetry group SU(3) to
determine the coupling constants of all baryons and mesons (Lopes & Menezes,
2014, 2021). For the $\omega$ meson, we have

$\frac{g_{\omega\Lambda}}{g_{\omega
N}}=\frac{4+2\alpha_{v}}{5+4\alpha_{v}},\frac{g_{\omega\Sigma}}{g_{\omega
N}}=\frac{8-2\alpha_{v}}{5+4\alpha_{v}},\frac{g_{\omega\Xi}}{g_{\omega
N}}=\frac{5-2\alpha_{v}}{5+4\alpha_{v}}.$ (30)

For the $\rho$ meson, we have

$\frac{g_{\rho\Sigma}}{g_{\rho N}}=2\alpha_{v},\frac{g_{\rho\Xi}}{g_{\rho
N}}=2\alpha_{v}-1,\frac{g_{\rho\Lambda}}{g_{\rho N}}=0.$ (31)

The hyperon-scalar meson coupling constants are fixed so as to reproduce the
following optical potentials (Glendenning & Moszkowski, 1991; Dover & Gal,
1984; Schaffner et al., 1994; Schaffner-Bielich & Gal, 2000)

$\begin{gathered}U_{\Lambda}=-28~{}\mathrm{MeV},\\\
U_{\Sigma}=+30~{}\mathrm{MeV},\\\ U_{\Xi}=-18~{}\mathrm{MeV}.\end{gathered}$
(32)

The coupling constants at different values of the parameter $\alpha_{v}$ are
displayed in table 5. For $\alpha_{v}$ = 1.0, the hybrid SU(6) group is
recovered.

Table 5: Hyperon meson coupling constants for different values of the hyperon coupling parameter $\alpha_{v}$.  |  $\alpha_{v}$ = 1.0 |  $\alpha_{v}$ = 0.75 |  $\alpha_{v}$ = 0.50 |  $\alpha_{v}$ = 0.25
---|---|---|---|---
$g_{\omega\Lambda}/g_{\omega N}$ |  0.667 |  0.687 |  0.714 |  0.75
$g_{\omega\Sigma}/g_{\omega N}$ |  0.667 |  0.812 |  1.00 |  1.25
$g_{\omega\Xi}/g_{\omega N}$ |  0.333 |  0.437 |  0.571 |  0.75
$g_{\rho\Sigma}/g_{\rho N}$ |  2.0 |  1.5 |  1.0 |  0.5
$g_{\rho\Xi}/g_{\rho N}$ |  1.0 |  0.5 |  0.0 |  -0.5
$g_{\sigma\Lambda}/g_{\sigma N}$ |  0.610 |  0.626 |  0.653 |  0.729
$g_{\sigma\Sigma}/g_{\sigma N}$ |  0.403 |  0.514 |  0.658 |  0.850
$g_{\sigma\Xi}/g_{\sigma N}$ |  0.318 |  0.398 |  0.500 |  0.638
Figure 9:  EoS for the DD-MEX parameter set at different values of the
parameter $\alpha_{v}$.

The particle population of hyperons and other particles depend upon the
hyperon meson coupling constants, which vary with the parameter $\alpha_{v}$.
Changing the value of $\alpha_{v}$ from 1.0 to 0.25 suppresses strange
particles. The suppression of $\Xi^{-}$ increases the lepton fraction at lower
value of $\alpha_{v}$. The neutrons and protons are the most populated
particles at $\alpha_{v}$=0.25. Figure 9 displays the EoS obtained using
different values of the parameter $\alpha_{v}$. As the $\alpha_{v}$ value
decreases from 1.0 to 0.25, the stiffness of the EoS increases due to the
increase in the value of $\omega$ meson couplings.

Figure 10 displays the mass-radius relation for hyperonic stars without
magnetic field effects at different values of $\alpha_{v}$. Lowering the value
of $\alpha_{v}$ stiffens the EoS, which increases the maximum mass to 2.283
$M_{\sun}$ for $\alpha_{v}$=0.25 (from 2.183 for $\alpha_{v}$=1.0). It is to
be mentioned that the stellar properties of hyperonic stars obtained at
$\alpha_{v}$ =1.0 almost resemble to that obtained from the hyperon potentials
given by (3). The change in the hyperon potentials alters the values of sigma
meson couplings by a small fraction and, hence, the changes obtained in the
particle population and stellar properties are negligible.

Figure 10:  Mass-Radius for a NS with hyperons at different values of the
parameter $\alpha_{v}$ using the DD-MEX parameter set. The shaded regions show
the recent constraints inferred from GW190814, MSP J0740+6620, and PSR
J0348+0432 (Cromartie & Fonseca et al., 2019; Antoniadis & Freire et al.,
2013). The green overlaid arrows show constraints on the mass-radius limits
inferred from NICER (Miller et al., 2019; Riley et al., 2019). Figure 11:
Relation between mass and circumferential radius for a hyperonic star with
magnetic field effects considering different magnetic dipole moments at
different hyperon couplings $\alpha_{v}$=0.75 (left panel), $\alpha_{v}$=0.50
(middle panel) and $\alpha_{v}$=0.25 (right panel) using the DD-MEX parameter
set. The colored areas show the recent constraints inferred from GW190814, MSP
J0740+6620, and PSR J0348+0432 (Abbott et al., 2020a; Cromartie & Fonseca et
al., 2019; Antoniadis & Freire et al., 2013). The constraints on the mass-
radius limits inferred from NICER (Miller et al., 2019; Riley et al., 2019)
are also shown.

With the addition of magnetic field effects, the results of different hyperon
couplings on hyperonic EoSs is determined. Figure 11 displays the relation
between mass and radius for a hyperonic star at different $\alpha_{v}$ values
without magnetic field and with magnetic field effects considered at different
values of magnetic dipole moment. For $\alpha_{v}$=0.75, the maximum mass
increases to a value 2.437 $M_{\sun}$ for a magnetic dipole moment $\mu$=1032
Am2, which corresponds to a central magnetic field of 3.77 $\times$ 1017 G.
Similarly, for $\alpha_{v}$=0.50 and 0.25, the maximum mass reaches a value
2.480 and 2.556$M_{\sun}$, respectively, at $\mu$=1032 Am2. This implies that
the secondary component of GW190814 could be a hyperonic magnetar. For the
present, hyperon couplings with a magnetic dipole moment $\mu$=5 $\times$ 1031
Am2, which corresponds to a central magnetic field greater than 1017 G, the
deviation in the hyperonic star radius at canonical mass is very large, around
1.5 km, as compared to that obtained from previous couplings. But with even
stronger magnetic fields, the deviation obtained is less in the present case.
This implies that a change in hyperon couplings effects the stellar
properties, especially the radius at canonical mass at strong central magnetic
field.

Figure 12:  Dimensionless tidal deformability as a function of NS mass for
EoSs without magnetic field (solid line) and with magnetic field effects
considering different magnetic dipole moments. Different panels show the
results obtained for hyperonic EoSs at different values of the parameter
$\alpha_{v}$. The orange dotted line represents the upper limit on the
dimensionless tidal deformability set by measurement from GW170817 (Abbott et
al., 2017). Table 6:  Table 6. Stellar properties: Maximum mass ($M_{max}$),
corresponding radius ($R_{max}$), canonical mass radius ($R_{1.4}$), and
dimensionless m tidal deformability ($\Lambda_{1.4}$) of hyperonic star for
the DD-MEX EoS at different values of the parameter $\alpha_{v}$ without
magnetic field and m with magnetic field with different values of the magnetic
dipole moment.

9.7cm  $\mu$ (Am2) $\alpha_{v}$=0.75 $\alpha_{v}$=0.50 $\alpha_{v}$=0.25 Mmax
($M_{\sun}$) R (km) R1.4 (km) $\Lambda_{1.4}$ Mmax ($M_{\sun}$) R (km) R1.4
(km) $\Lambda_{1.4}$ Mmax ($M_{\sun}$) R (km) R1.4 (km) $\Lambda_{1.4}$ 0
2.217 12.335 13.168 791.483 2.239 12.335 13.168 791.483 2.283 12.453 13.168
791.483 5$\times$ 1030 2.240 12.543 13.213 815.398 2.271 12.632 13.224 869.281
2.316 12.652 13.237 914.012 5$\times$ 1031 2.349 13.522 14.741 1338.714 2.366
13.708 14.821 1397.389 2.407 13.563 14.938 1425.185 1032 2.437 13.995 14.992
1510.783 2.480 14.224 15.069 1615.288 2.556 14.469 15.403 1879.880

Figure 12 displays the variation in the dimensionless tidal deformability of a
hyperonic star with magnetic field effects considered at different values of
magnetic dipole moment. For small values of the magnetic dipole moment at
different values of $\alpha_{v}$, the $\Lambda$-M curve follows similar
pattern as for the previous hyperon couplings. Since the variation in radius
at canonical mass are large for the present couplings, the tidal
deformability, $\Lambda_{1.4}$ obtained is also large. For $\alpha_{v}$=0.25,
the dimensionless tidal deformability reaches a value of $\approx$ 1900 for
magnetic dipole moment 1032 Am2. The properties of hyperonic stars using
different hyperon coupling values according to the value of $\alpha_{v}$ with
and without magnetic field effects considered at different magnetic dipole
moments, are shown in table 6.

##  5 Summary and Conclusion

We modeled massive nucleonic and hyperonic stars that fulfill the constraints
set by the observation of the possibly most massive neutron star (NS) ever
detected (in the secondary object of the gravitational wave event GW190814)
using a density-dependent relativistic mean field model (DD-RMF). This simple,
yet powerful formalism provides the freedom necessary to fulfill
simultaneously nuclear and astrophysical constraints. The results obtained
from the TOV equations show that the hyperons soften the EoS, lowering the
maximum mass of NS to around 2.2$M_{\sun}$, thereby satisfying more
conservative massive constraints from the astrophysical observations. The
radius of the NS canonical mass is seen to be insensitive to the appearance of
hyperons. Both the nucleonic and the hyperonic stars satisfy the constraints
from mass-radius limits inferred from NICER observations and tidal
deformability constraints from the LIGO and VIRGO collaborations.

We also studied the effects of strong magnetic fields on DD-RMF nucleonic and
hyperonic EoSs. We investigated the EoS and particle populations using a
realistic chemical potential-dependent magnetic field that was developed by
solving Einstein-Maxwell equations. For very strong magnetic fields, the
spherical symmetric solutions obtained by solving the TOV equations lead to an
overestimation of the mass and the radius and, hence, cannot be used for
determining stellar properties. For this reason, we used the LORENE library to
determine stellar properties of magnetic NSs. For low values of the magnetic
dipole moment, implying lower strengths of magnetic fields, the EoS resembles
the non-magnetic one. For higher magnetic dipole moments, the EoS stiffens at
higher energy density. The amount of stiffness is larger in case of hyperonic
EoSs than for the case of pure nucleonic ones. As the magnetic field strength
is increased, the particle fractions of leptons ($e^{-}$ and $\mu^{-}$)
increases at higher densities and the appearance of charged hyperons
($\Xi^{-}$) is delayed.

The stiffening in the EoS caused by the changes in population described above
increases the maximum mass of magnetic stars. For a small dipole moment of
5$\times$ 1030 Am2, the nucleonic and hyperonic mass-radius profiles resemble
the non-magnetic case due to the lower magnetic field produced. For higher
magnetic dipole moments, although the variation in the maximum mass is small,
a variation of about 1 km is seen in the radius of the NS canonical mass. For
hyperonic stars, the maximum mass increases by $\approx$ 0.3 $M_{\sun}$,
allowing them to satisfy the GW190814 possible mass constraint for a dipole
moment $\mu\geq 10^{32}$ Am2 when different coupling schemes are considered.
For strong magnetic fields, the different coupling schemes for hyperons
predict increases in radius of $\approx 0.2-1$ km. Thus, we see that different
hyperon couplings and different hyperon potentials populate stellar matter
differently and, hence, can change stellar properties significantly.

The change in the dimensionless tidal deformability for NS was studied. It was
found that, for higher value of the magnetic dipole moment, the tidal
deformability of the canonical mass surpasses the upper limit of 800 set by
the measurement from GW1701817 at 90% confidence, which is consistent with the
acknowledgement of such objects possessing weak magnetic fields. No such
measurement of tidal deformability is available for GW190814, as it has no
tidal signatures. But if in the future, measurements of tidal deformability of
massive NSs such as the one measured for the secondary object in GW190814 are
$\leq$ 800, this would imply that object could not be a hyperonic magnetar.

In the future, we will investigate the effect of considering more exotic
particles in our modelling of magnetic NSs. These include, kaons,
$\Delta$-resonances and deconfined quark matter (with and without mixtures of
phases). We also intend to analyze combined effects of magnetic fields and
fast rotation on stars containing exotic matter, as both of these aspects
could be important in young stars (Lasky et al., 2019; Yu et al., 2019; Lasky
et al., 2017).

AAU acknowledges the Inter-University Centre for Astronomy and Astrophysics,
Pune, India for support via an associateship and for hospitality. VD thanks
Valentina Sarti and Nicolas Yunes for useful discussions and acknowledges
support from the National Science Foundation under grant PHY-1748621 and
PHAROS (COST Action CA 16214).

## 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 (2) is written as

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

where, $M_{B}^{*}=M_{B}-g_{\sigma}(\rho_{B})\sigma$ is the effective mass of
baryons and $\sum_{R}$ is the rearrangment term due to the density dependence
of the coupling constants

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

The equation 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 scalar density $\rho_{s}$, baryon density $\rho_{B}$, and isovector
densities $\rho_{s3}$, and $\rho_{3}$, are given as

$\displaystyle\rho_{s}$ $\displaystyle=\sum_{B}\bar{\psi}\psi,$
$\displaystyle\rho_{B}$ $\displaystyle=\sum_{B}\psi^{\dagger}\psi,$
$\displaystyle\rho_{s3}$ $\displaystyle=\sum_{B}\bar{\psi}\tau_{3}\psi,$
$\displaystyle\rho_{3}$
$\displaystyle=\sum_{\alpha}\psi^{\dagger}\tau_{3}\psi=\rho_{p}-\rho_{n}.$
(A.4)

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

$\displaystyle\mathcal{E}_{m}$
$\displaystyle=\sum_{B}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{B}}d^{3}kE_{B}^{*}(k)+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{2}m_{\omega}^{2}\omega^{2}-\frac{1}{2}m_{\rho}^{2}\rho^{2}$
$\displaystyle+g_{\omega}(\rho_{B})\omega\rho_{B}+\frac{g_{\rho}(\rho_{B})}{2}\rho\rho_{3},$
$\displaystyle P_{m}$
$\displaystyle=\sum_{B}\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{B}}d^{3}k\frac{k^{2}}{E_{B}^{*}(k)}-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho^{2}-\rho_{B}\sum_{R}(\rho_{B}),$
(A.5)

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

In the presence of magnetic field, the scalar and vector density for charged
baryons $cb$, uncharged baryons $ub$, and leptons follow as (Broderick et al.,
2000)

$\displaystyle\rho_{s}^{cb}$
$\displaystyle=\frac{|q_{cb}|BM_{cb}^{*2}}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}ln\Bigg{(}\frac{k_{F,\nu}^{cb}+E_{F}^{cb}}{\sqrt{M_{cb}^{*2}+2\nu|q_{cb}|B}}\Bigg{)},$
$\displaystyle\rho_{s}^{ub}$
$\displaystyle=\frac{M_{ub}^{*2}}{2\pi^{2}}\Bigg{[}E_{F}^{ub}k_{F}^{ub}-M_{ub}^{*2}ln\Bigg{(}\frac{k_{F,\nu}^{ub}+E_{F}^{ub}}{M_{ub}}\Bigg{)}\Bigg{]},$
$\displaystyle\rho^{cb}$
$\displaystyle=\frac{|q_{cb}|B}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}k_{F,\nu}^{cb},$
$\displaystyle\rho^{ub}$ $\displaystyle=\frac{(k_{F}^{ub})^{3}}{3\pi^{2}},$
$\displaystyle\rho_{l}$
$\displaystyle=\frac{|q_{l}|B}{2\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}k_{F,\nu}^{l},$
(A.6)

where $r_{\nu}$ is the Landau degeneracy of $\nu$ level. The expressions for
the baryon and lepton energy densities in the presence of magnetic field
become

$\displaystyle\mathcal{E}_{cb}$
$\displaystyle=\frac{|q_{cb}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}$
$\displaystyle\times\Bigg{[}k_{F,\nu}^{cb}E_{F}^{cb}+(M_{cb}^{*2}+2\nu|q_{cb}|B)ln\Bigg{(}\frac{k_{F,\nu}^{cb}+E_{F}^{cb}}{\sqrt{M_{cb}^{*2}+2\nu|q_{cb}|B}}\Bigg{)}\Bigg{]},$
$\displaystyle\mathcal{E}_{ub}$
$\displaystyle=\frac{1}{8\pi^{2}}\Bigg{[}k_{F}^{ub}(E_{F}^{ub})^{3}+(k_{F}^{ub})^{3}E_{F}^{ub}-M_{ub}^{*4}ln\Bigg{(}\frac{k_{F}^{ub}+E_{F}^{ub}}{M_{ub}^{*}}\Bigg{)}\Bigg{]},$
$\displaystyle\mathcal{E}_{l}$
$\displaystyle=\frac{|q_{l}|B}{4\pi^{2}}\sum_{\nu=0}^{\nu_{max}}r_{\nu}$
$\displaystyle\times\Bigg{[}k_{F,\nu}^{l}E_{F}^{l}+(m_{l}^{2}+2\nu|q_{l}|B)ln\Bigg{(}\frac{k_{F,\nu}^{l}+E_{F}^{l}}{\sqrt{m_{l}^{2}+2\nu|q_{l}|B}}\Bigg{)}\Bigg{]}.$
(A.7)

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