# Effect of isovector scalar meson on equation of state of dense matter within
relativistic mean field model

Virender Thakur<EMAIL_ADDRESS>Department of Physics, Himachal
Pradesh University, Shimla-171005, India  Raj Kumar<EMAIL_ADDRESS>Department of Physics, Himachal Pradesh University, Shimla-171005, India
Pankaj Kumar  Department of Applied Sciences, CGC College of Engineering,
Landran, Mohali 140307, India  Vikesh Kumar  Department of Physics, Himachal
Pradesh University, Shimla-171005, India  Mukul Kumar  Department of Physics,
Himachal Pradesh University, Shimla-171005, India  C. Mondal  Laboratoire de
Physique Corpusculaire, CNRS, ENSICAEN, UMR6534, Université de Caen Normandie,
F-14000, Caen Cedex, France  B.K. Agrawal<EMAIL_ADDRESS>Saha Institute
of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India  Shashi K. Dhiman
<EMAIL_ADDRESS>Department of Physics, Himachal Pradesh University,
Shimla-171005, India  School of Applied Sciences, Himachal Pradesh Technical
University, Hamirpur-177001, India

###### Abstract

The effects of the isovector-scalar $\delta$-meson field on the properties of
finite nuclei, infinite nuclear matter and neutron stars are investigated
within the Relativistic Mean Field (RMF) model which includes non-linear
couplings. Several parameter sets (SRV’s) are generated to asses the influence
of $\delta$-meson on the properties of neutron star. These parametrizations
correspond to different values of coupling constant of $\delta$-meson to the
nucleons with remaining ones calibrated to yield finite nuclei and infinite
nuclear matter properties consistent with the available experimental data. It
is observed that to fit the properties of finite nuclei and infinite nuclear
matter, a stronger coupling between isovector-vector $\rho$ meson and nucleons
is required in the presence of $\delta$ field. Furthermore, the $\delta$-meson
is found to affect the radius of canonical neutron star significantly. The
value of dimensionless tidal deformability, ${\Lambda}$ for the canonical
neutron star also satisfies the constraints from the waveform models analysis
of GW170817 binary neutron star merger event. A covariance analysis is
performed to estimate the statistical uncertainties of the model parameters as
well as correlations among the model parameters and different observables of
interest.

Equation of State; Neutron star

††preprint: APS/123-QED

##  I Introduction

Neutron stars are the densest objects in the observable universe and deep
knowledge of the Equation of State (EoS) of the dense matter in beta-
equilibrium is thus required to understand their behavior. It has been shown
that the dense matter EoS must be treated relativistically [1, 2]. For this
reason, relativistic mean field (RMF) models have been widely used to obtain a
realistic description of the properties of finite nuclei, bulk nuclear matter
or the properties of neutron stars. Currently, many different variants of RMF
models with various couplings are in use to study the finite nuclei and
neutron star properties [3, 4, 5]. Accurate constraints are necessary to
understand the limits of these different types of models. During the last
decade, a wide range of astrophysical observations such as the precise
measurement of massive millisecond pulsars using Shapiro delay technique [6,
7], detection of gravitational wave generated by binary neutron stars in the
GW170817 event by the LIGO-Virgo collaboration [8, 9], or the joint mass
radius measurement of neutron stars using X-ray timing technique by NICER
collaboration [10, 11, 12, 13] started to provide unprecedented new
constraints on the dense matter EoS. They have triggered plethora of
theoretical studies to look at the dense matter EoS from very different
perspectives c.f. Ref. [14] and references therein. First of its kind model
independent measurement of neutron skin thickness $\Delta r_{np}$ of 208Pb
[15] and 48Ca [16] in the Jefferesen lab also inspired theoretical studies to
take a fresher look at the isovector channel of the nuclear interaction [17,
18, 19, 20, 21].

Effective mass of nucleon quantifies the momentum dependence of nuclear force
in the medium. It can be quoted for infinite nuclear matter at the Fermi
surface. It is, however, necessary to realize that the concept of effective
mass is different in non-relativistic [22, 23] and relativistic formalism
[24]. Nevertheless, it plays some crucial roles to determine various finite
nuclear properties e.g. isoscalar giant quadrupole resonance (ISGQR) [25],
nucleon nucleon scattering in optical potentials [26] or even in realizing
various properties of nuclear matter and neutron stars [27, 28]. Recently, a
systematic study was performed using RMF models which assessed the impact of
relativistic (Dirac) effective mass ($M^{*}$) on the properties of neutron
star [29]. The isovector splitting of the effective mass, which measures the
difference between neutron ($M_{n}^{*}$) and proton ($M_{p}^{*}$) effective
mass, can influence greatly as well the physical properties of finite nuclei
such as locating the drip-lines [30] or nucleon-nucleus scattering of
asymmetric systems [26]. Its impact increases manifold in high density
environment which can alter thermal and transport properties of asymmetric
matter [31, 32] or neutrino opacities of neutron star matter [33]. To settle
its value, even at saturation, remains a persisting challenge both
theoretically [34, 35, 36, 37, 38] and experimentally [39, 26, 40].

Appearance of isovector splitting of effective mass to the leading order in
RMF models occurs through the isovector-scalar $\delta$ mesons. It impacts the
proton fraction in neutron stars and hence the cooling process of neutron
stars after formation [41, 42]. It can also influence the global properties of
neutron stars [43, 44, 45, 27, 28, 46]. A systematic study of RMF models with
added freedom in the isospin channel through $\delta$ meson, optimized using
well constrained finite nuclear properties and extrapolating at high density
to understand the properties of neutron stars and in general dense matter EoS,
can enhance our knowledge on the density dependence of the isovector channel
of nuclear interaction.

The present study is aimed towards investigating the effects of $\delta$ meson
on the dense matter EoS within the framework of RMF model. We generate several
parameter sets by varying the coupling strength of $\delta$ meson to the
nucleons such that the low density behaviour of the EOSs remain consistent
with the available finite nuclei data and a few empirical properties of
infinite nuclear matter evaluated at the saturation density. The properties of
neutron stars obtained with these EOSs are them compared to asses the role of
$\delta$ mesons.

The paper is organized as follows. In section II, the theoretical framework
which is used to construct the EoS for neutron stars is discussed. We also
discuss the procedure to optimize the coupling constants and the method to
perform a covariance analysis in the same section. In section III, we present
our results. We summarize and draw our conclusions in section IV.

##  II Formalism

###  II.1 Theoretical model

The Lagrangian density for the RMF model used in the present study based on
different non-linear, self and inter-couplings among isoscalar-scalar
$\sigma$, isoscalar-vector $\omega_{\mu}$, isovector-scalar $\delta$ and
isovector-vector $\rho_{\mu}$ meson fields and nucleonic Dirac field $\Psi$
[47, 48], is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{q}\overline{\Psi}[i\gamma^{\mu}\partial_{\mu}-(M-g_{\sigma}\sigma-
g_{\delta}\delta\cdot\tau)$ (1) $\displaystyle-$
$\displaystyle(g_{\omega}\gamma^{\mu}\omega_{\mu}+\frac{1}{2}g_{\mathbf{\rho}}\gamma^{\mu}\tau\cdot\mathbf{\rho}_{\mu})]\Psi+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})$ $\displaystyle-$
$\displaystyle\frac{\overline{\kappa}}{3!}g_{\sigma}^{3}\sigma^{3}-\frac{\overline{\lambda}}{4!}g_{\sigma}^{4}\sigma^{4}-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}$
$\displaystyle+$ $\displaystyle\frac{1}{4!}\zeta
g_{\omega}^{4}(\omega_{\mu}\omega^{\mu})^{2}-\frac{1}{4}\mathbf{\rho}_{\mu\nu}\mathbf{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\mathbf{\rho}_{\mu}\mathbf{\rho}^{\mu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}(\partial_{\mu}\delta\partial^{\mu}\delta-
m_{\delta}^{2}\delta^{2})+\frac{1}{2}c_{1}g_{\omega}^{2}g_{\rho}^{2}\omega_{\mu}\omega^{\mu}\rho_{\mu}\rho^{\mu}.$

The Dirac effective mass for the nucleons (q) appearing in the Lagrangian
density above is specified as

$M^{*}_{q}=(M-g_{\sigma}\sigma-g_{\delta}\delta\cdot\tau),$ (2)

where, $\tau=1(-1)$ for $q=$ neutron (proton). Following the Euler-Lagrange
formalism one can readily find the expressions for energy density ${\cal E}$
and pressure $P$ as a function of density from Eq. (1) [49].

###  II.2 Optimization and covariance analysis

In the present study, five new relativistic interactions SRV00, SRV01, SRV02,
SRV03, and SRV04 have been generated for the Lagrangian density given by Eq.
(1) to investigate the effect of $\delta$ meson on the properties of finite
nuclei and neutron star matter. Here, SRV00, SRV01, SRV02, SRV03 and SRV04
parametrizations correspond to different value of the coupling of
$\delta$-meson to the nucleon i.e. $g_{\delta}$ = 0.0, 1.0, 2.0, 3.0 and 4.0
respectively. As the effect of $\delta$ meson is predominantly important at
suprasaturation densities, one can a-priori anticipate its insignificant
impact in finite nuclei, which is primarily sensitive to the EoS at
subsaturation densities. This is the reason why we kept fixed the $g_{\delta}$
at aforementioned values optimizing the rest of the parameters in Eq. (1).
This is not far from the strategy recently used by Li et. al. in Ref. [46].
The parameters of the model are obtained by fitting the experimental data [50]
on binding energies ($BE$) and charge rms radii ($r_{ch}$) [51] of some
spherical nuclei 16,24O, 40,48Ca, 56,78Ni, 88Sr,90Zr, 100,116,132Sn and 208Pb.
For the open shell nuclei, the pairing has been included using BCS formalism
with constant pairing gaps [52, 53] that are taken from the nucleon separation
energies of neighboring nuclei [50]. Neutron and proton pairing gaps are
evaluated by using fourth order finite difference mass formula (five point
difference) [54]. The neutron and proton pairing gaps
($\Delta_{n}$,$\Delta_{p}$) in MeV for the open shell nuclei are
${}^{88}Sr$(0.0,1.284), ${}^{90}Zr$(0.0,1.239) and ${}^{116}Sn$(1.189,0.0).
The neutron pairing gap for ${}^{24}O$ practically vanishes since the first
unoccupied orbit 1$d_{3/2}$ is almost 4.5 MeV above the completely filled
2$s_{1/2}$ orbit [55, 56]. The pairing correlation energies for a fix gap
$\Delta$ is calculated by using the paring window of 2$\hbar\omega$, where
$\hbar\omega$ = $45A^{-1/3}$ \- $25A^{-2/3}$ MeV [48]. We also incorporated
the recently measured neutron skin thickness of 208Pb using the parity
violating electron scattering experiment [15] in our fit data.

Table 1:  SRV parameter sets for the Lagrangian of RMF model as given in Eq.(1). The parameter $\overline{\kappa}$ is in fm-1. The values of meson masses $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$ and $m_{\delta}$ are in MeV. The nucleonic mass (M) and meson masses ($m_{\omega}$, $m_{\rho}$ and $m_{\delta}$) are taken as 939, 782.5, 762.468 and 980 MeV, respectively. The values of $\overline{\kappa}$, $\overline{\lambda}$, and ${c_{1}}$ are multiplied by $10^{2}$. ${\bf{Parameters}}$ | SRV00 | SRV01 | SRV02 | SRV03 | SRV04
---|---|---|---|---|---
${\bf g_{\sigma}}$ | 10.3109$\pm$0.1109 | 10.3442$\pm$0.0978 | 10.3344$\pm$0.1135 | 10.3723$\pm$0.0999 | 10.3735$\pm$0.1294
${\bf g_{\omega}}$ | 13.1772$\pm$0.1621 | 13.2508$\pm$0.1379 | 13.2137$\pm$0.1414 | 13.3113$\pm$0.1269 | 13.2984$\pm$0.1815
${\bf g_{\rho}}$ | 10.8834$\pm$1.1730 | 11.2832$\pm$1.0110 | 11.5834$\pm$0.9676 | 12.4834$\pm$1.1274 | 13.1833$\pm$0.8043
${\bf\overline{\kappa}}$ | 1.8509$\pm$0.0500 | 1.8624$\pm$0.0700 | 1.8780$\pm$0.0200 | 1.8242$\pm$0.0500 | 1.8490$\pm$0.0800
${\bf\overline{\lambda}}$ | -0.05151$\pm$0.0700 | -0.05803$\pm$0.0600 | -0.06060$\pm$0.0600 | -0.04934$\pm$0.0800 | -0.05621$\pm$0.0800
${\bf{\zeta}}$ | 0.02116$\pm$0.0017 | 0.02050$\pm$0.0013 | 0.02017$\pm$0.0011 | 0.02177$\pm$0.0012 | 0.02089$\pm$0.0027
${\bf c_{1}}$ | 4.60038$\pm$2.3600 | 4.49219$\pm$1.9200 | 4.20853$\pm$1.5600 | 3.66169$\pm$1.4900 | 3.01068$\pm$0.8800
${\bf m_{\sigma}}$ | 501.9596$\pm$0.9230 | 501.0200$\pm$1.0071 | 501.6638 $\pm$1.3141 | 500.7480$\pm$1.2629 | 501.0215$\pm$1.3663

The optimization of the parameters (p) appearing in the Lagrangian (Eq. 1) is
done by using the simulated annealing method (SAM) [57, 58, 59] by following
$\chi^{2}$ minimization procedure which is given as,

${\chi^{2}}(\textbf{p})=\frac{1}{N_{d}-N_{p}}\sum_{i=1}^{N_{d}}\left(\frac{M_{i}^{exp}-M_{i}^{th}}{\sigma_{i}}\right)^{2},$
(3)

where $N_{d}$ is the number of experimental data points and $N_{p}$ is the
number of fitted parameters. The $\sigma_{i}$ denotes adopted errors [60, 61]
and $M_{i}^{exp}$ and $M_{i}^{th}$ are the experimental and the corresponding
theoretical values, respectively, for a given observable. The minimum value of
${{\chi}}^{2}_{0}$ corresponds to the optimal values $\textbf{p}_{0}$ of the
parameters.

Once the optimized parameter set is obtained, the correlation coefficient
between two quantities Y and Z, can be calculated by covariance analysis [62,
63, 64, 60, 61] as

$\textit{r}_{\scriptstyle{YZ}}=\frac{\overline{\Delta{Y}\Delta{Z}}}{\sqrt{\overline{\Delta{Y^{2}}}\quad\overline{\Delta{Z^{2}}}}},$
(4)

where covariance between Y and Z is expressed as

$\overline{\Delta{Y}\Delta{Z}}=\sum_{\alpha\beta}\left(\frac{\partial{Y}}{\partial{p}_{\alpha}}\right)_{\textbf{p}_{0}}C_{\alpha\beta}^{-1}\left(\frac{\partial{Z}}{\partial{p}_{\beta}}\right)_{\textbf{p}_{0}}.$
(5)

Here, $C_{\alpha\beta}^{-1}$ is an element of inverted curvature matrix given
by

$\textit{C}_{\alpha\beta}=\frac{1}{2}\left(\frac{\partial^{2}\chi^{2}(\textbf{p})}{\partial{p}_{\alpha}\partial{p}_{\beta}}\right)_{\textbf{p}_{0}}.$
(6)

The standard deviation, $\overline{\Delta{Y}^{2}}$, in Y can be computed using
Eq. (5) by substituting Z = Y.

##  III Results and Discussion

We have obtained five different parameter sets corresponding to different
values of $g_{\delta}$ by calibrating the remaining prameters to a suitable
set of on finite nuclei as described earlier. All parametrizations obtained in
the present work give equally good fit to the properties of finite nuclei
which were used for the optimization procedure. In Table 1 we display optimum
values of the model parameters for all the five SRV parameter sets along with
the uncertainties on them computed using Eq. (5). It can be seen that the
parameter $g_{\rho}$ increases with the increase in value of $g_{\delta}$. A
larger value of $g_{\rho}$ is required in the presence of the $\delta$-field
to fit the properties of finite nuclei. As the contribution of the
$\delta$-field is attractive, increased binding due to the $\delta$-field has
to be compensated by the higher value of the repulsion by the $\rho$-field.
The parameter $g_{\rho}$ has its lowest value for SRV00 parametrization
($g_{\delta}$=0). For any finite value of $\delta$-coupling ($g_{\delta}>0$)
i.e. for SRV01, SRV02, SRV03 and SRV04 parametrizations, the strength of
$\rho$-coupling ($g_{\rho}$) increases gradually. The cross-coupling between
the $\omega_{\mu}$ and $\rho_{\mu}$ fields quantified by the term $c_{1}$
decreases slightly from 4.6003 to 3.0107 as the value of coupling constant
$g_{\delta}$ increases from 0.0 to 4.0 corresponding to different SRV
parametrizations.

Table 2:  The values of binding energy (BE) and charge radii ($r_{ch}$) of fitted nuclei along with theoretical errors obtained for different SRV parametrizations. The corresponding experimental values [50, 51] are also listed. The value of neutron skin thickness ($\Delta r_{np}$) for 208Pb is given along with the experimental data [15]. The adopted errors on the observables ($\sigma$) used for optimisation of parameters are also displayed. The value of BE are given in units of MeV and $r_{ch}$ and $\Delta r_{np}$ are in fm.  ${\bf{Nucleus}}$ | ${\bf{Observables}}$ | Exp. | $\sigma$ | SRV00 | SRV01 | SRV02 | SRV03 | SRV04
---|---|---|---|---|---|---|---|---
${}^{16}O$ | BE | 127.62 | 4.0 | 128.67$\pm$0.52 | 128.83$\pm$0.50 | 128.94$\pm$0.55 | 128.90$\pm$0.51 | 128.94$\pm$0.61
| $r_{ch}$ | 2.699 | 0.04 | 2.709$\pm$0.008 | 2.712$\pm$0.023 | 2.711$\pm$0.031 | 2.710$\pm$0.013 | 2.710$\pm$0.023
${}^{24}O$ | BE | 168.96 | 1.0 | 169.95$\pm$0.96 | 169.38$\pm$0.90 | 169.64$\pm$0.89 | 168.90$\pm$1.01 | 169.00$\pm$0.93
${}^{40}Ca$ | BE | 342.04 | 3.0 | 343.40$\pm$0.69 | 343.14$\pm$0.65 | 343.18$\pm$0.53 | 344.41$\pm$0.63 | 344.39$\pm$0.91
| $r_{ch}$ | 3.478 | 0.02 | 3.454$\pm$0.014 | 3.457$\pm$0.007 | 3.455$\pm$0.016 | 3.455$\pm$0.014 | 3.455$\pm$0.017
${}^{48}Ca$ | BE | 415.97 | 1.0 | 415.59$\pm$0.54 | 415.15$\pm$0.52 | 415.37$\pm$0.45 | 415.37$\pm$0.63 | 415.48$\pm$0.52
| $r_{ch}$ | 3.477 | 0.02 | 3.468$\pm$0.016 | 3.469$\pm$0.014 | 3.468$\pm$0.014 | 3.467$\pm$0.014 | 3.467$\pm$0.022
${}^{56}Ni$ | BE | 484.01 | 5.0 | 482.05$\pm$1.28 | 482.39$\pm$1.21 | 482.28$\pm$1.31 | 483.12$\pm$1.48 | 483.22$\pm$1.56
| $r_{ch}$ | 3.750 | 0.02 | 3.712$\pm$0.022 | 3.709$\pm$0.017 | 3.708$\pm$0.14 | 3.707$\pm$0.017 | 3.705$\pm$0.013
${}^{78}Ni$ | BE | 642.56 | 2.0 | 641.39$\pm$1.03 | 640.01$\pm$0.97 | 641.05$\pm$1.01 | 640.21$\pm$1.08 | 640.18$\pm$1.07
${}^{88}Sr$ | BE | 768.42 | 1.0 | 768.24$\pm$0.60 | 767.72$\pm$0.57 | 767.84$\pm$0.58 | 768.39$\pm$0.60 | 768.40$\pm$0.62
| $r_{ch}$ | 4.219 | 0.02 | 4.226$\pm$0.016 | 4.227$\pm$0.014 | 4.227$\pm$0.013 | 4.225$\pm$0.021 | 4.225$\pm$0.0146
${}^{90}Zr$ | BE | 783.81 | 1.0 | 783.92$\pm$0.69 | 783.58$\pm$0.64 | 783.59$\pm$0.68 | 784.37$\pm$0.69 | 784.36$\pm$0.68
| $r_{ch}$ | 4.269 | 0.02 | 4.280$\pm$0.019 | 4.280$\pm$0.019 | 4.280$\pm$0.013 | 4.278$\pm$0.029 | 4.278$\pm$0.015
${}^{100}Sn$ | BE | 825.10 | 2.0 | 826.60$\pm$1.28 | 826.95$\pm$1.24 | 827.56$\pm$1.14 | 827.09$\pm$1.23 | 827.01$\pm$1.66
${}^{116}Sn$ | BE | 988.67 | 2.0 | 988.34$\pm$0.83 | 987.74$\pm$0.73 | 987.72$\pm$2.18 | 988.49$\pm$0.82 | 988.25$\pm$0.88
| $r_{ch}$ | 4.627 | 0.02 | 4.617$\pm$0.016 | 4.618$\pm$0.014 | 4.617$\pm$0.009 | 4.615$\pm$0.012 | 4.615$\pm$0.020
${}^{132}Sn$ | BE | 1100.22 | 1.0 | 1101.38$\pm$0.84 | 1100.58$\pm$0.80 | 1101.48$\pm$0.77 | 1100.59$\pm$0.86 | 1100.70$\pm$0.82
| $r_{ch}$ | 4.709 | 0.02 | 4.721$\pm$0.019 | 4.721$\pm$0.019 | 4.720$\pm$0.010 | 4.719$\pm$0.012 | 4.717$\pm$0.011
${}^{208}Pb$ | BE | 1636.34 | 1.0 | 1636.58$\pm$1.03 | 1635.98$\pm$0.98 | 1636.32$\pm$0.96 | 1636.27$\pm$1.05 | 1636.10$\pm$1.02
| $r_{ch}$ | 5.501 | 0.02 | 5.530$\pm$0.015 | 5.531$\pm$0.014 | 5.529$\pm$0.014 | 5.527$\pm$0.021 | 5.526$\pm$0.011
| $\Delta r_{np}$ | $0.283\pm 0.071$ | 0.071 | 0.222$\pm$0.032 | 0.223$\pm$0.028 | 0.217$\pm$0.026 | 0.215$\pm$0.035 | 0.214$\pm$0.029
Figure 1:  (Color online) Correlation coefficients (absolute values) among the
model parameters for SRV02 parametrization for the Lagrangian given by Eq.
(1).

In Table 2, different observables fitted in the present work, their
experimental values [50, 51], adopted errors $\sigma$ on them [65] along with
the calculated values for different SRV parametrizations are displayed. The
estimated uncertanties are also listed for the fitted observables. The fitted
values of finite nuclei properties are quite close to their experimental
counterparts. The root mean square (rms) errors on the $BE$ are found to be in
the range $1.50-1.86$ MeV, and the ones for $r_{ch}$ are found to be $~{}0.02$
fm for the different parameterizatons. It is quite interesting to observe that
even though $g_{\delta}$ influences the coupling $g_{\rho}$, the isovector
sensitive observable $\Delta r_{np}$ varies only slightly $\sim 0.01$ fm
across the different SRV models obtained in the present work. This observation
is quite similar to the one obtained by Li et. al. [46].

Table 3:  The bulk nuclear matter properties at saturation density for SRV parametrizations are listed: $\rho_{0}$, E/A, K, J, L and $M^{*}/M$ denotes the saturation density, binding energy per nucleon, incompressibility coefficient, symmetry energy, the slope of symmetry energy, and the ratio of effective nucleon mass to the nucleon mass, respectively. ${\bf{Parameters}}$ | SRV00 | SRV01 | SRV02 | SRV03 | SRV04
---|---|---|---|---|---
${\bf{\rho_{0}~{}(fm^{-3})}}$ | 0.149$\pm$0.003 | 0.149$\pm$0.002 | 0.149$\pm$0.002 | 0.149$\pm$0.003 | 0.149$\pm$0.008
${\bf E/A~{}(MeV)}$ | -16.11$\pm$0.06 | -16.11$\pm$0.05 | -16.09$\pm$0.05 | -16.12$\pm$0.06 | -16.11$\pm$0.04
${\bf K~{}(MeV)}$ | 223.94$\pm$8.57 | 221.78$\pm$9.95 | 222.05$\pm$5.47 | 221.72$\pm$10.62 | 221.11$\pm$23.20
${\bf J~{}(MeV)}$ | 33.49$\pm$1.82 | 33.75$\pm$1.77 | 33.31$\pm$1.78 | 33.54$\pm$2.13 | 33.34$\pm$2.08
${\bf L~{}(MeV)}$ | 65.23$\pm$15.37 | 63.82$\pm$13.50 | 61.49$\pm$13.22 | 58.06$\pm$15.93 | 55.31$\pm$13.76
${\bf~{}M^{*}/M}$ | 0.606$\pm$0.013 | 0.0.602$\pm$0.010 | 0.603$\pm$0.005 | 0.601$\pm$0.009 | 0.600$\pm$0.009

In Table 3, we present the results for the properties of Symmetric Nuclear
Matter (SNM) such as binding energy per nucleon (E/A), incompressibility (K),
the ratio of effective mass to the mass of nucleon ($M^{*}/M$) along with
symmetry energy coefficient (J), and its slope (L), all are evaluated at the
saturation density ($\rho_{0}$). We also quote the theoretically calculated
error on them. The results are presented for all five SRV parametrizations.
The value of E/A lies in the range $-16.09$ to $-16.12$ for the five
parametrizations. The value of J and L obtained by our parametrizations are
consistent with the constraints from observational analysis J = 31.6 $\pm$
2.66 MeV and L = 58.9 $\pm$ 16 MeV [66, 67]. The value of K is also in
agreement with the value 240 $\pm$ 20 MeV determined from isoscalar giant
monopole resonance (ISGMR) for 90Zr and 208Pb nuclei [68, 69]. It can also be
seen from Table 3 that the mean values of the slope of of symmetry energy (L)
for SRV parametrizations decrease with the increase in the value of
$g_{\delta}$. The average value of L decreases from 65.23 MeV in SRV00 to
55.31 MeV for SRV04. It can be noted that the isoscalar properties (E/A, K,
$\rho_{0}$ and $M^{*}/M$) are well constrained for all SRV parametrizations.
The only exception is the error on K in case of SRV04, where the error is
almost 10% of its central value. But in the isovector sector, the percentage
error on the slope of symmetry energy (L) are consistently on the larger side
for all SRV parametrizations.

A great deal of importance to perform covariance analysis in theoretical
studies has been pointed out recently [63, 60]. It not only enables one to
quote statistical uncertainties on model parameters or any calculated
observables, but also privedes complementary information about the sensitivity
of the parameters to physical observables, redundancies among fitted
observables or interdependences among model parameters. As our primary
objective is not to establish an ultimate model, for demonstrative purpose, we
will discuss the results of covariance analysis as outlined in Section IIB,
only for the model SRV02. The results for other parameter sets are quite
similar (not shown here). In Fig. 1, the correlation coefficients between
different model parameters appearing in Eq. (1) are outlined for SRV02
parametrization. A strong correlation is found between the several pairs of
model parameters, like, $g_{\sigma}$ and $g_{\omega}$, $g_{\rho}$ and $c_{1}$
and $\overline{\lambda}$ and $\zeta$ with correlation coefficients 0.99, 0.98
and 0.95, respectively. These interdependences mean that if one of these pairs
are fixed at a particular value, the other must attain the precise value as
suggested by their correlation to satisfactorily obtain the fit data. The
results obtained for the correlations among model parameters presented in Fig.
1 are quite similar to the obtained in Refs. [70, 19]. Anticipating the strong
correlation between L and $\Delta r_{np}$ which is shown later (see Fig. 3),
this may be attributed to the large experimental error on the $\Delta r_{np}$
for 208Pb, which also led us choosing a rather large adopted error during
optimization. The theoretical errors on the $\Delta r_{np}$ of 208Pb nucleus
are found to be 0.032, 0.028, 0.026, 0.0353 and 0.029 fm for SRV00, SRV01,
SRV02, SRV03, and SRV04 parametrizations, respectively. These are much smaller
compared to the adopted error (0.071 fm, which is also the experimental error
obtained in Ref. [15]).

Figure 2:  (Color online) Correlation coefficients (absolute values) between
the model parameters and a set of physical observables for SRV02
parametrization (see text for details).

We now display in Fig. 2, the correlation coefficients between the model
parameters apearing in Lagrangian (Eq. 1) and the different properties of
interest corresponding to the SNM and the density dependent symmetry energy as
displayed in Table 3 and the $\Delta r_{np}$ of 208Pb for SRV02. A strong
correlation is observed between the isovector parameter $g_{\rho}$ with the
symmetry energy coefficient (J), its slope (L) and $\Delta r_{np}$ of 208Pb.
The vector mixing parameter $c_{1}$ is also found to have a strong correlation
with the J and L. This strong correlation is anticipated, as $c_{1}$ and
$g_{\rho}$ are strongly correlated to each other, which was observed in Fig.
1. It is also realized from Fig. 2 that bulk properties of SNM like E/A, K,
$\rho_{0}$ and $M^{*}/M$ have strong correlations with isoscalar coupling
parameters $g_{\sigma}$, $g_{\omega}$ and $\overline{\kappa}$. This study is
quite consistent with previous calculations in the literature [70, 56, 20].

Figure 3:  (Color online) Correlation coefficients (absolute values) for few
bulk nuclear matter properties and neutron skin of 208Pb for SRV02
parametrization.

In Fig. 3, we display the correlation coefficients among the different
observables in graphical form, particularly which were also studied in Fig. 2.
In the isoscalar sector the only strong correlation observed is between
binding energy per nucleon (E/A) and incompressibility coefficient (K). The K
also shows some mild correlations with all other observables displayed in the
figure. The symmetry energy J and its slope parameter L are found to be
strongly correlated. As mentioned earlier in the discussion of Table 3, we
observe a strong correlation of the neutron skin thickness of 208Pb with J and
L. These results are also in line with the earlier ones [70].

Figure 4:  (Color online) Effective masses of proton and neutron for a few
values of asymmetry parameter $\alpha$ for SRV04 parametrization.

It is quite important to emphasize that we kept fixed the strength of the
coupling of $\delta$ meson to different values and optimized the rest. This
might be partially responsible to impart a strong correlation among the
isovector sensitive parameters $g_{\rho},c_{1}$ to J, L or $\Delta r_{np}$ of
208Pb (see Fig. 2) to reproduce the fitted data within bounds. It further gets
clarified in the strong correlations among $\Delta r_{np}$ (208Pb), J and L in
Fig. 3. Anticipating the results obtained for neutron stars which are
discussed later, this strong correlation somewhat restricts the behavior of
the matter at high densities in the isovector channel, resulting in a
monotonic increasing in the radius and tidal deformability of 1.4 $M_{\odot}$
(see Table 4) with the increase of the $\delta$ meson coupling $g_{\delta}$. A
full optimization is thus needed with suitable data in the future including
$g_{\delta}$, to understand this behavior further. In the present work we have
included the simplest form of $\delta$-meson coupling ($g_{\delta}$) to the
nucleons. Furthermore, a higher order mixed scalar interactions of $\delta$
meson has a large influence on the symmetry energy and its density dependence.
This enables one to have flexibility to vary the behaviour of EoS at high
density and gives a large influence on the properties of neutron stars [71,
72, 73]. To study the effects of $\delta$-meson on nucleon mass, in Fig. 4,
the effective mass of proton and neutron are plotted as a function of baryon
density for three values of asymmetry parameter $\alpha$= 0, 0.5, 1 ($\alpha$=
$\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$) for SRV04 parametrization,
which has the largest value of $g_{\delta}$ amongst all SRV variants obtained
in the present work. The asymmetry parameter $\alpha$ = 0.0 represents the SNM
and $\alpha$=1 corresponds to the pure neutron matter (PNM). It is clear from
the Eq. (2) that the presence of $\delta$-meson leads to splitting of nucleon
mass. For SNM, there is no splitting of the nucleon mass. In Fig. 4, the solid
(dashed) lines depict the effective mass of proton (neutron) for $\alpha$= 0,
0.5 and 1.0. One can observe from the figure that the effective proton mass is
larger than the neutron effective mass. The splitting of the proton and
neutron effective masses due to the $\delta$-meson can be important in the
highly asymmetric system like a neutron star or supernova environment. At the
center of a neutron star the density can reach $\sim$ 5-6 $\rho_{0}$ and
$\alpha\sim$ 0.7-0.8. One can readily estimate the amount of splitting in the
effective mass in this situation looking at Fig. 4. It can also affect the
transport properties of neutron star matter [74].

Figure 5:  (Color online) Mass-Radius relation of neutron star for SRV parametrizations. Table 4: The properties of nonrotating neutron star for the various EOSs computed with SRV parameter sets are presented along with the theoretical errors on them. MG and Rmax denote the Maximum Gravitational mass and corresponding radius. The values for R1.4 and $\Lambda_{1.4}$ denote radius and dimensionless tidal deformability at 1.4M⊙.  No. | EOS | $M_{G}$ | Rmax | R1.4 | $R_{2.0}$ | ${\bf{\Lambda_{1.4}}}$
---|---|---|---|---|---|---
|  | (M⊙) | (km) | (km) | (km) |
1. | SRV00 | 2.04$\pm$0.03 | 11.48$\pm$0.08 | 12.92$\pm$0.22 | 12.07$\pm$0.31 | 484.16$\pm$62.10
2. | SRV01 | 2.06$\pm$0.02 | 11.55$\pm$0.13 | 12.99$\pm$0.20 | 12.24$\pm$0.21 | 504.95$\pm$58.02
3. | SRV02 | 2.07$\pm$0.02 | 11.67$\pm$0.07 | 13.16$\pm$0.13 | 12.43$\pm$0.14 | 565.52$\pm$60.33
4. | SRV03 | 2.08$\pm$0.02 | 11.81$\pm$0.11 | 13.41$\pm$0.21 | 12.60$\pm$0.16 | 652.60$\pm$52.01
5. | SRV04 | 2.13$\pm$0.06 | 12.11$\pm$0.23 | 13.86$\pm$0.21 | 13.09$\pm$0.33 | 783.96$\pm$70.03

To assess the impact of $\delta$-meson on the global properties of neutron
star, we plot the gravitational mass ($M_{G}$) of non-rotating neutron star as
a function of radius for all SRV parametrizations in Fig. 5. The maximum mass
($M_{max}$) and the corresponding radius ($R_{max}$) of neutron star for all
the models obtained here lie in the range 2.04 - 2.13 $M_{\odot}$ and 11.48 -
12.11 km, respectively. This satisfies the recently measured radius of
PSRJ0740+6620 with $12.45^{+0.65}_{-0.65}$ km by NICER collaboration [12, 13].
The radius of neutron star of 2$M_{\odot}$ is also in accordance with the
observational data of PSRJ0740+6620 by NICER [12, 13]. The maximum mass of the
neutron star attained by various SRV parametrizations supports the constraint
from PSRJ0740+6620 with the mass of 2.08 $\pm$ 0.07 $M_{\odot}$ [75, 76]. It
is observed that the radius ($R_{1.4}$) of neutron star with mass
1.4$M_{\odot}$, can be significantly affected by the presence of
$\delta$-meson as we move from parametrization set SRV00 to SRV04. The value
of $R_{1.4}$ with the inclusion of TM1 crust EoS [77] lies in the range
12.92-13.86 km, which is also in line with the range proposed in Ref. [12,
78]. It is observed that the radius $R_{1.4}$ increases by 7.27 % and the
maximum mass of neutron star changes by 4.4 % from SRV00 to SRV04
parametrizations with the variation of coupling $g_{\delta}$= 0.0 to 4.0. This
change in the neutron star properties may be attributed to the impact of
$\delta$-meson, which affects high-density behavior of asymmetric nuclear
matter.

Tidal deformability imparted by the companion stars on one another in a binary
system can yield remarkable information on the EoS for neutron star [79, 80].
In Fig. 6, we show the results of dimensionless tidal deformability $\Lambda$,
defined as $\Lambda$= (2/3)$k_{2}$$(R/M_{G})^{5}$, where $k_{2}$ is the love
number, as a function of the neutron star mass $M_{G}$ for different SRV
models. The recent constraints on the tidal deformability $\Lambda_{1.4}$ of
1.4$M_{\odot}$ neutron star including GW170817 [8, 81] is also given in the
figure. The value of $\Lambda_{1.4}$ lies in the range 484 - 783 for different
SRV parametrizations, which satisfies the proposed limit are listed in Refs.
[8, 82, 83, 78]. The value of $\Lambda_{1.4}$ increases with the increase in
the value of the coupling $g_{\delta}$ corresponding to the SRV
parametrizations as can be seen from Fig. 6. All these results are summarized
in Table 4. The theoretical erros/uncertainities in neutron star properties
for SRV parametrizations are also mentioned in the table. The neutron star
properties such as $M_{max}$, $R_{max}$, $R_{1.4}$, $R_{2.0}$ are relatively
well constrained for all SRV parametrizations (at $\leq$ 3 $\%$) whereas for
$\Lambda_{1.4}$, the theoretical uncertainities are found to be $\leq$ 10$\%$.

Figure 6:  (Color online) Variation of dimensionless tidal deformability
($\Lambda$) with respect to gravitational mass for SRV parametrizations.

To this end, we may mention that the contribution of $\delta$ meson is
considered only through its linear interaction with nucleons. However,
inclusion of the self interaction of $\delta$ mesons and the mixed
interactions with other mesons may alter the symmetry energy at
suprasaturation densities. This would further enhance the flexibility of EOS
of dense matter and accordingly the properties of neutron stars [71, 72, 73].
Inclusion of the higher order contributions of $\delta$ meson thus enable one
to model the properties of neutron stars in somewhat independent of the
properties of finite nuclei. The optimization of effective Lagrangian that
includes different terms involving $\delta$ meson field requires accurate
knowledge of neutron star properties over a wide range of mass.

##  IV Summary and conclusions

The effect of the isovector-scalar field corresponding to $\delta$-meson in
relativistic mean field theory is investigated. We have generated five sets of
SRV parametrizations SRV00, SRV01, SRV02, SRV03 and SRV04 to explore the
effects of $\delta$-meson on the properties of finite nuclei, infinite nuclear
matter, and neutron stars. A covariance analysis to measure the accuracy of
model predictions is also performed. This also enabled us to carry out a
systematic study of correlations among model parameters and various finite
nuclei and infinite nuclear matter properties of interest. The SRV
parametrizations have been obtained in such a way that they reproduce the
ground state properties of the finite nuclei and infinite nuclear matter
properties quite convincingly. In turn, they satisfy the constraints on mass
and radius of the neutron star and its dimensionless tidal deformability,
$\Lambda$, from recent astrophysical observations [66, 67, 8, 81, 78]. It is
observed that to fit the properties of finite nuclei and infinite nuclear
matter, a stronger coupling between the $\rho$-meson and nucleons ($g_{\rho}$)
is required in the presence $\delta$-meson field. Furthermore, the
$\delta$-meson significantly affects the radius of canonical neutron star. It
is found that the contributions from $\delta$-meson is important and has some
significant effects on the dense matter EoS. The value of $\Lambda_{1.4}$ for
different SRV parametrization is also in line with the constraint obtained
from GW170817 event. It is clear that the isovector splitting of effective
mass of nucleon in the presence of $\delta$ in dense asymmetric matter, like
the scenario present in the core of a neutron star, can be significant. It
remains, however, an open question how to identify in future the signatures of
isovector effective mass splitting from astrophysical observations.

###### Acknowledgements.

Virender Thakur is highly thankful to Himachal Pradesh University and DST-
INSPIRE (Govt. of India) for providing computational facility and financial
assistance (Junior/Senior Research Fellowship under grant number “DST/INSPIRE
Fellowship/2017/IF170302”). C.M. acknowledges partial support from the IN2P3
Master Project “NewMAC”.

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