# Influence of the nuclear symmetry energy slope on observables of compact
stars with $\Delta$-admixed hypernuclear matter

Vivek Baruah Thapa<EMAIL_ADDRESS>Monika Sinha<EMAIL_ADDRESS>Indian
Institute of Technology Jodhpur, Jodhpur 342037, India

###### Abstract

In this work, we study the effects of nuclear symmetry energy slope on neutron
star dense matter equation of state and its impact on neutron star observables
(mass-radius, tidal response). We construct the equation of state within the
framework of covariant density functional theory implementing coupling schemes
of non-linear and density-dependent models with viability of heavier non-
nucleonic degrees of freedom. The slope of symmetry energy parameter
($L_{\text{sym}}$) is adjusted following density-dependence of isovector meson
coupling to baryons. We find that smaller values of $L_{\text{sym}}$ at
saturation favour early appearance of $\Delta$-resonances in comparison to
hyperons leading to latter’s threshold at higher matter densities. We also
investigate the dependence of $L_{\text{sym}}$ on tidal deformability and
compactness parameter of a $1.4~{}M_{\odot}$ neutron star for different
equation of states and observe similar converging behaviour for larger
$L_{\text{sym}}$ values.

neutron stars; equation of state; hyperons; $\Delta$-resonances; tidal
deformability

##  I Introduction

In stellar evolution, when nuclear fusion stops and the electron degeneracy
pressure can not prevent the gravitational collapse, electrons happen to
combine with protons and produce neutron rich nuclei. Eventually neutrons drip
out of the nuclei, and the matter becomes mainly composed of neutrons with
some admixture of protons and electrons. This remnant of stellar evolution is
called neutron star (NS). Thus, after the type-II supernova explosion the
stellar remnant forms highly dense NS. Naturally, the matter density inside
the NS varies from subnuclear density near surface to supranuclear density
towards the centre. Matter is composed of ions and electrons near the surface,
i.e. in the outer crust region, and in the inner crust region neutron-rich
nuclei and some free neutrons appear. Then in the core, matter is completely
made of free neutrons with comparatively less number of protons and electrons.
With the increase of density towards the centre of the star, exotic components
of matter viz. hyperons, heavier non-strange baryons, Boson condensates, even
deconfined quarks may appear while near surface where density is comparatively
low up to $\approx 2$ times nuclear saturation density ($n_{0}$), the matter
is composed of only nucleons Glendenning (1996); Weber (2017). Hence, the
interior of NS is a good domain to study the dense nuclear matter in bulk with
and without exotic degrees of freedom. In general, we have good knowledge
about finite nuclei at $n_{0}$. Hence, the theoretical idea of uniform
symmetric nuclear matter in bulk even at varied densities above $n_{0}$ is
just the extrapolation and idealization of finite nuclei knowledge. On the
other hand, the observational test of bulk matter properties in varied density
above $n_{0}$ can only be obtained from astrophysical observations of compact
NSs.

One important ingredient for energy density of nucleons inside nuclear matter
is the symmetry energy ($E_{\text{sym}}$). Consequently, nuclear symmetry
energy and its density dependence play a salient role in comprehending dense
matter behaviour Ducoin  _et al._ (2010); Lattimer and Prakash (2016). In case
of finite nuclei, the contribution of symmetry energy to the mass of the
nuclei is small compared to other terms in the semi-empirical mass formula.
Hence, the experimental information regarding $E_{\text{sym}}$ is not very
sound even at $n_{0}$. As mentioned earlier, from the astrophysical properties
of NSs the nuclear matter properties at varied density can be determined. The
nuclear symmetry energy and its variation with density affects substantially
the composition and matter pressure what in consequence affects the NS
properties specially the radius Centelles  _et al._ (2009); Fattoyev  _et al._
(2018). Although from nuclear data we have very little knowledge of
$E_{\text{sym}}$, a good comprehensive idea of the same can now be drawn from
the recent radius measurement of certain NS candidates PSR J$0030+0451$ Miller
_et al._ (2019); Riley  _et al._ (2019); Landry  _et al._ (2020) and PSR
J$0740+6620$ Riley  _et al._ (2021); Miller  _et al._ (2021) from NICER
(Neutron star Interior Composition ExplorER) space mission. The detection of
gravitational wave (GW) emissions from binary NS mergers (GW170817 Abbott  _et
al._ (2017, 2018, 2019), GW190425 Abbott  _et al._ (2020a)) by the LIGO-Virgo
Collaboration (LVC) marked an appreciable breakthrough in the domain of multi-
messenger astronomy. The GW observations set bound on the mutual tidal
deformability ($\tilde{\Lambda}$) which also depends on the matter property
linked with $E_{\text{sym}}$.

Not only that, comprehensive idea regarding symmetry energy behaviour can be
gathered via studying its effects on other NS properties such as maximum mass,
compactness, tidal deformability. The observational estimations of certain
parameters viz. maximum mass and radius of compact objects impose vital
constraints on narrowing down to a unique dense matter equation of state
(EOS). For instance, with the observations of massive compact stars ($M\geq
2~{}M_{\odot}$) such as PSR J$1614-2230$ ($1.908\pm 0.016~{}M_{\odot}$)
Demorest  _et al._ (2010); Arzoumanian  _et al._ (2018), PSR J$0348+0432$
($2.01\pm 0.04~{}M_{\odot}$) Antoniadis  _et al._ (2013), PSR J$0740+6620$
($2.08^{+0.07}_{-0.07}$ $M_{\odot}$ with 68.3% credibility Fonseca  _et al._
(2021)), the soft EOSs tend to be invalid. The mass-radius measurements of PSR
J$0030+0451$ Miller  _et al._ (2019); Riley  _et al._ (2019); Landry  _et al._
(2020) and PSR J$0740+6620$ Riley  _et al._ (2021); Miller  _et al._ (2021)
from NICER space mission provide significant constraint on dense matter EOS.
The latest NICER mass-radius measurement (PSR J$0740+6620$) suggests repulsive
matter behaviour at higher density regimes. Joint analysis of data from NICER
observations for PSR J$0030+0451$ and GW170817 event provides bounds on NS
properties viz. tidal deformability ($\Lambda_{1.4}$) and radius ($R_{1.4}$)
for a $1.4M_{\odot}$ star Jiang  _et al._ (2020).

Many recent studies have been done to constrain the values of $E_{\text{sym}}$
at $n_{0}$ and its slope ($L_{\text{sym}}$) at $n_{0}$ based on data from
various astrophysical observations as well as terrestrial experiments Lattimer
and Steiner (2014); Roca-Maza  _et al._ (2015); Tews  _et al._ (2017). Very
recently an improved value of neutron skin thickness of ${}^{208}\text{Pb}$
was reported in Lead Radius EXperiment-II (PREX-2) to be
$R_{\text{skin}}=R_{n}-R_{p}=(0.283\pm 0.071)$ fm Adhikari  _et al._ (2021).
This evaluates the corresponding symmetry energy and its slope to be
$E_{\text{sym}}=(38.1\pm 4.7)$ MeV and $L_{\text{sym}}=(106\pm 37)$ MeV
respectively at $n_{0}$ with correlation coefficient as 0.978 Reed  _et al._
(2021). These updated values of isospin asymmetry parameters are larger than
the ones ($28.5$ MeV $\leq E_{\text{sym}}(n_{0})\leq 34.9$ MeV; $30.6$ MeV
$\leq L_{\text{sym}}(n_{0})\leq 86.8$ MeV) previously reported in ref.-Oertel
_et al._ (2017) obtained by comparison of experimental data from finite nuclei
and heavy-ion collisions with different microscopic model calculations. In
this work, we explore the influence of nuclear symmetry energy on dense matter
EOS, consequently on NS properties. As a consequence, we attempt to constrain
the $E_{\text{sym}}$ and $L_{\text{sym}}$ from the observational features of
NSs. To do so, we consider covariant density functional (CDF) model
implementing the density-dependence of isovector-vector coupling as introduced
in ref.-Spinella (2017) and incorporating non-linear GM1 Glendenning and
Moszkowski (1991) and density dependent DD-MEX Taninah  _et al._ (2020)
coupling parametrizations. In the CDF scheme, the coupling constants are
chosen in such a way that the model can reproduce the experimental quantities
known at $n_{0}$. Thus the observational properties of NSs which depends on
the EOS parameters i.e. coupling constants will determine $E_{\text{sym}}$ and
$L_{\text{sym}}$. In several previous studies the symmetry energy effects on
dense matter has been considered with the matter composition to be purely
nucleonic Ji  _et al._ (2019); Hu  _et al._ (2020); Wu  _et al._ (2021).
However, due to increasing Fermi energy of nucleons interior to NS, the
appearance of additional degrees of freedom such as hyperons Glendenning and
Moszkowski (1991); Weissenborn  _et al._ (2012); Bonanno and Sedrakian (2012);
Colucci and Sedrakian (2013); Oertel  _et al._ (2015); Tolos  _et al._ (2017);
Raduta  _et al._ (2018); Li  _et al._ (2018); Lopes and Menezes (2021); Lopes
(2021), $\Delta$-resonances Drago  _et al._ (2014); Cai  _et al._ (2015); Zhu
_et al._ (2016); Sahoo  _et al._ (2018); Kolomeitsev  _et al._ (2017); Li  _et
al._ (2018); Li and Sedrakian (2019); Ribes  _et al._ (2019); Li  _et al._
(2020); Thapa  _et al._ (2020, 2021), meson ($\pi,\bar{K},\rho$) condensations
Mannarelli (2019); Haensel and Proszynski (1982); Glendenning and Schaffner-
Bielich (1999); Banik and Bandyopadhyay (2001); Prakash  _et al._ (1997);
Malik  _et al._ (2021); Thapa and Sinha (2020); Thapa  _et al._ (2021) are
inevitable or energetically favorable for massive stars as recently observed
Demorest  _et al._ (2010); Arzoumanian  _et al._ (2018); Antoniadis  _et al._
(2013); Fonseca  _et al._ (2021). Moreover, the appearance of these exotic
matter softens the EOS reducing maximum mass of the stars which is in contrary
to the observations. Hence, the choice of EOS parametrization is limited with
the observational constraints of massive stars. For that we have chosen two
EOS parametrizations within CDF model viz. GM1 with nonlinear model and DD-MEX
with density dependent coupling constant which satisfy the observational
constraints of massive stars. This work will therefore explore the novel
aspects of density-dependent isovector coupling on dense matter EOS with the
onset of heavier strange and non-strange degrees of freedom and study the
symmetry energy slope effects on NS properties.

The paper is organized as follows. In Sec.-II, we briefly describe the CDF
model formalism, its extension to additional heavier degrees of freedom and
coupling parameters for constructing the EOS. The effects of nuclear symmetry
energy on dense matter are shown and discussed in Sec.-III. Sec.IV provides
the summary and conclusions of this work.

Conventions: We implement the natural units $G=\hbar=c=1$ throughout the work.

##  II Formalism

###  II.1 CDF Model

This section briefly describes the CDF model implemented in this work to
construct the dense matter EOS. The dense matter composition considered in
this work are the entire baryon octet ($N\equiv n,p$;
$Y\equiv\Lambda^{0},\Sigma^{\pm,0},\Xi^{-,0}$) and $\Delta$-resonances
($\Delta\equiv\Delta^{+},\Delta^{++},\Delta^{0},\Delta^{-}$) along with
leptons ($l\equiv e^{-},\mu^{-}$). In order to mediate the effective
interactions between the baryons, isoscalar-scalar $\sigma$, isoscalar-vector
$\omega$ and isovector-vector $\rho$-mesons are considered. An additional
hidden strangeness isoscalar-vector $\phi$-meson is also brought into
consideration to describe the hyperon-hyperon repulsive interactions. The
total Lagrangian density is given by Glendenning (1996); Li  _et al._ (2018)

$\displaystyle\mathcal{L}$ $\displaystyle=\sum_{b\equiv
N,Y}\bar{\psi}_{b}(i\gamma_{\mu}D^{\mu}_{(b)}-m^{*}_{b})\psi_{b}+\sum_{l}\bar{\psi}_{l}(i\gamma_{\mu}\partial^{\mu}-m_{l})\psi_{l}$
(1)
$\displaystyle+\sum_{\Delta}\bar{\psi}_{\Delta\nu}(i\gamma_{\mu}D^{\mu}_{(\Delta)}-m^{*}_{\Delta})\psi^{\nu}_{\Delta}+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})$
$\displaystyle-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\boldsymbol{\rho}_{\mu\nu}\cdot\boldsymbol{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\boldsymbol{\rho}_{\mu}\cdot\boldsymbol{\rho}^{\mu}$
$\displaystyle-\frac{1}{4}\phi_{\mu\nu}\phi^{\mu\nu}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\text{U}(\sigma),$

where the baryon octet, lepton Dirac and $\Delta$ baryon Schwinger-Rarita
fields are represented by $\psi_{b},\psi_{l}$ and $\psi_{\Delta}$
respectively. The covariant derivative is given by
$D_{\mu(j)}=\partial_{\mu}+ig_{\omega j}\omega_{\mu}+ig_{\rho
j}\boldsymbol{\tau}_{j3}\cdot\boldsymbol{\rho}_{\mu}+ig_{\phi j}\phi_{\mu}$
with $j$ denoting the baryon particle spectrum. The isospin projection of the
third component of isovector-vector meson field is represented by
$\boldsymbol{\tau}_{j3}$. The scalar self-interaction term, which is present
only in non-linear model, introduced to account for the incompressibility
Glendenning (1996) is given by

$\text{U}(\sigma)=\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}$ (2)

where $g_{2}$, $g_{3}$ are the coefficients of self-interactions.
$\omega_{\mu\nu}$, $\boldsymbol{\rho}_{\mu\nu}$ and $\phi_{\mu\nu}$ are the
anti-symmetric field tensors corresponding to vector meson fields and given by

$\displaystyle\omega_{\mu\nu}$
$\displaystyle=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},$ (3)
$\displaystyle\boldsymbol{\rho}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\boldsymbol{\rho}_{\nu}-\partial_{\nu}\boldsymbol{\rho}_{\mu},$
$\displaystyle\phi_{\mu\nu}$
$\displaystyle=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}.$

The Dirac and Schwinger-Rarita effective masses are respectively given by

$\displaystyle m_{b}^{*}$ $\displaystyle=m_{b}-g_{\sigma b}\sigma,\quad
m_{\Delta}^{*}=m_{\Delta}-g_{\sigma\Delta}\sigma$ (4)

with $m_{b}$, $m_{\Delta}$ representing the bare masses of baryon octet and
$\Delta$-quartet particle spectrum respectively. In mean-field approximation,
the non-vanishing meson fields obtained by solving the Euler-Lagrange
equations are given by

$\displaystyle\sigma$
$\displaystyle=-\frac{1}{m_{\sigma}^{2}}\frac{\partial\text{U}}{\partial\sigma}+\sum_{b}\frac{1}{m_{\sigma}^{2}}g_{\sigma
b}n_{b}^{s}+\sum_{\Delta}\frac{1}{m_{\sigma}^{2}}g_{\sigma\Delta}n_{\Delta}^{s},$
(5) $\displaystyle\omega_{0}$
$\displaystyle=\sum_{b}\frac{1}{m_{\omega}^{2}}g_{\omega
b}n_{b}+\sum_{\Delta}\frac{1}{m_{\omega}^{2}}g_{\omega\Delta}n_{\Delta},$
$\displaystyle\rho_{03}$ $\displaystyle=\sum_{b}\frac{1}{m_{\rho}^{2}}g_{\rho
b}\boldsymbol{\tau}_{b3}n_{b}+\sum_{\Delta}\frac{1}{m_{\rho}^{2}}g_{\rho\Delta}\boldsymbol{\tau}_{\Delta
3}n_{\Delta},$ $\displaystyle\phi_{0}$
$\displaystyle=\sum_{Y}\frac{1}{m_{\phi}^{2}}g_{\phi Y}n_{Y}.$

The scalar and vector (number) densities of the constituent particle spectrum
are given by $n^{s}=\langle\bar{\psi}\psi\rangle$ and
$n=\langle\bar{\psi}\gamma^{0}\psi\rangle$ respectively. The chemical
equilibrium conditions between species in the particle spectrum when
strangeness is not conserved is given by Banik and Bandyopadhyay (2001); Drago
_et al._ (2014)

$\mu_{j}=\mu_{n}-q_{j}\mu_{e}$ (6)

where $q_{j}$ is the charge of $j$th baryon, $\mu_{e}$, $\mu_{n}$ denote the
chemical potential of electron and neutron respectively. The chemical
potential of $j$th baryon is defined as

$\displaystyle\mu_{j}=\sqrt{p_{F_{j}}^{2}+m_{j}^{*2}}+\Sigma^{0}+\Sigma^{r},$
(7)

with $\Sigma^{0}=g_{\omega j}\omega_{0}+g_{\phi j}\phi_{0}+g_{\rho
j}\boldsymbol{\tau}_{j3}\rho_{03}$ and the re-arrangement term necessary to
maintain thermodynamic consistency is given by

$\displaystyle\Sigma^{r}$ $\displaystyle=\sum_{b}\left[\frac{\partial
g_{\omega b}}{\partial n}\omega_{0}n_{b}-\frac{\partial g_{\sigma b}}{\partial
n}\sigma n_{b}^{s}+\frac{\partial g_{\rho b}}{\partial
n}\rho_{03}\boldsymbol{\tau}_{b3}n_{b}\right.$ (8)
$\displaystyle\left.+\frac{\partial g_{\phi b}}{\partial
n}\phi_{0}n_{b}\right]+\sum_{\Delta}(\psi_{b}\longrightarrow\psi_{\Delta}^{\nu}).$

In case of non-linear coupling model for this present work, only the isovector
$\rho$-meson coupling contributes to re-arrangement term.

The dense matter EOS is calculated self-consistently taking into account two
additional constraints, viz. charge neutrality and global baryon number
conservation respectively given by

$\displaystyle\sum_{b}q_{b}n_{b}+\sum_{\Delta}q_{\Delta}n_{\Delta}-n_{e}-n_{\mu}=0,$
(9) $\displaystyle\sum_{b}n_{b}+\sum_{\Delta}n_{\Delta}=n.$

The total energy density is given by

$\displaystyle\begin{aligned}
\varepsilon&=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}\\\
&+\sum_{j\equiv
b,\Delta}\frac{2J_{j}+1}{2\pi^{2}}\left[p_{{F}_{j}}E^{3}_{F_{j}}-\frac{m_{j}^{*2}}{8}\left(p_{{F}_{j}}E_{F_{j}}\right.\right.\\\
&\left.\left.+m_{j}^{*2}\ln\left(\frac{p_{{F}_{j}}+E_{F_{j}}}{m_{j}^{*}}\right)\right)\right]+\frac{1}{\pi^{2}}\sum_{l}\left[p_{{F}_{l}}E^{3}_{F_{l}}\right.\\\
&\left.-\frac{m_{l}^{2}}{8}\left(p_{{F}_{l}}E_{F_{l}}+m_{l}^{2}\ln\left(\frac{p_{{F}_{l}}+E_{F_{l}}}{m_{l}}\right)\right)\right].\end{aligned}$
(10)

The matter pressure is then evaluated following the Gibbs-Duhem relation
defined as

$P=\sum_{j\equiv b,\Delta}\mu_{j}n_{j}+\sum_{l}\mu_{l}n_{l}-\varepsilon.$ (11)

###  II.2 NS structure and properties

The NS properties (mass-radius) are deduced from the Tolman-Oppenheimer-
Volkoff (TOV) equations for non-rotating, spherically symmetric NS
configurations corresponding to constructed EOSs which are given by
Glendenning (1996)

$\displaystyle\frac{dP(r)}{dr}$
$\displaystyle=-\frac{[\varepsilon(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r^{2}[1-2M(r)/r]},$ (12) $\displaystyle\frac{dM(r)}{dr}$
$\displaystyle=4\pi r^{2}\varepsilon(r),$

where, $M(r)$ is the gravitational mass enclosed within radius $r$. Solutions
of TOV equations are obtained on applying the boundary conditions,
$P(R)=M(0)=0$.

Table 1: (i) Parameter values of CDF coupling models considered in this work. Coupling | $g_{\sigma N}$ | $g_{\omega N}$ | $g_{\rho N}$ | $g_{2}$ | $g_{3}$ | $m_{\sigma}$ | $m_{\omega}$ | $m_{\rho}$ | $m_{\phi}$
---|---|---|---|---|---|---|---|---|---
Model |  |  |  | (fm-1) |  | (MeV) | (MeV) | (MeV) | (MeV)
GM1 | 9.5708 | 10.5964 | 8.1957 | 12.2817 | $-8.9780$ | 550 | 783 | 770 | 1019.45
DD-MEX | 10.7067 | 13.3388 | 7.2380 | $-$ | $-$ | 547.3327 | 783 | 763 | 1019.45
(ii) Coefficient values corresponding to eqns.-(19), (21) in DD-MEX coupling
model.
Meson ($i$) | $a_{i}$ | $b_{i}$ | $c_{i}$ | $d_{i}$ |
$\sigma$ | 1.3970 | 1.3350 | 2.0671 | 0.4016 |
$\omega$ | 1.3926 | 1.0191 | 1.6060 | 0.4556 |
$\rho$ | 0.6202 |  |  |  |

The tidal response of compact stars to an external gravitational field is
quantified in terms of dimesionless tidal deformability defined as Hinderer
(2008)

$\Lambda=\frac{2}{3}k_{2}\left(\frac{M}{R}\right)^{-5}$ (13)

where $k_{2}$ is the tidal Love number defined as

$\displaystyle k_{2}=$
$\displaystyle\frac{8C^{5}}{5}(1-2C^{2})[2+2C(y-1)-y]\cdot$ (14)
$\displaystyle\\{2C[6-3y+3C(5y-8)]+4C^{3}[13-11y$
$\displaystyle+C(3y-2)+2C^{2}(1+y)]+3(1-2C^{2})\cdot$
$\displaystyle[2-y+2C(y-1)]\log(1-2C)\\}^{-1}$

and obtained by solving the differential equation Binnington and Poisson
(2009),

$r\frac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,$ (15)

with boundary condition as $y(0)=2$ where the functions are defined as,

$F(r)=\frac{r-4\pi r^{3}[\varepsilon(r)-P(r)]}{r-2M(r)},$ (16) $\displaystyle
Q(r)=$ $\displaystyle\frac{4\pi
r[5\varepsilon(r)+9P(r)+\frac{\varepsilon(r)+P(r)}{\partial
P(r)/\partial\varepsilon(r)}]}{r-2M(r)}$ (17)
$\displaystyle-4\left[\frac{M(r)+4\pi r^{3}P(r)}{r^{2}[1-2M(r)/r]}\right].$

In binary NS mergers, the tidal response is encoded in combined dimensionless
tidal deformability parameter given by Favata (2014)

$\displaystyle\tilde{\Lambda}$
$\displaystyle=\frac{16}{13}\frac{(M_{1}+12M_{2})M_{1}^{4}\Lambda_{1}+(M_{2}+12M_{1})M_{2}^{4}\Lambda_{2}}{(M_{1}+M_{2})^{5}}.$
(18)

where $\Lambda_{1}$, $\Lambda_{2}$ are the dimesionless tidal deformabilities
corresponding to stars with masses $M_{1}$ and $M_{2}$ respectively.

###  II.3 Coupling parameters

As mentioned earlier in Sec.-I, in the present work, we implement GM1
parametrization Glendenning and Moszkowski (1991) with density-dependence of
isovector-vector $\rho$-meson for meson-baryon couplings Spinella (2017) and
DD-MEX parametrization Taninah  _et al._ (2020). In case of GM1
parametrization, the density dependent coupling constant for the isovector
$\rho$-meson is given by

$g_{\rho N}(n)=g_{\rho N}(n_{0})e^{-a_{\rho}(x-1)}$ (19)

where $x=n/n_{0}$, while the coupling constants for $\sigma$, $\omega$-mesons
are considered to be density-independent.

In case of density-dependent model, the isoscalar meson-nucleon couplings are
defined as

$g_{iN}(n)=g_{iN}(n_{0})f_{i}(x)\quad\quad\text{for }i=\sigma,\omega$ (20)

where, the function is given by

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}$ (21)

and the density dependence of isovector-vector $\rho$-meson coupling is given
by eqn.-(19). Table-1 provides the parameter values of GM1 and DD-MEX coupling
parametrizations in nucleonic sector. In the standard GM1 parametrization,
$g_{\rho N}$ is density independent and for the standard DD-MEX
parametrization the coefficient $a_{\rho}$ is given in table-1. For variation
in $L_{\text{sym}}$, it is evaluated by calibrating the coefficient $a_{\rho}$
without altering the other nuclear saturation properties. Table-1 provides the
values of $g_{iN}$ and $g_{\rho N}$ at $n_{0}$. Since the non-strange baryons
do not couple with $\phi$-meson, $g_{\phi N}=g_{\phi\Delta}=0$.

The bare masses of baryons are considered as $m_{N}=939$ MeV,
$m_{\Lambda}=1115.68$ MeV, $m_{\Sigma^{+}}=1189.37$ MeV,
$m_{\Sigma^{0}}=1192.64$ MeV, $m_{\Sigma^{-}}=1197.45$ MeV,
$m_{\Xi^{-}}=1321.71$ MeV, $m_{\Xi^{0}}=1314.86$ MeV and $m_{\Delta}=1232$
MeV. The saturation property parameters viz. $n_{0}$, saturation energy
($E_{0}$), incompressibility ($K_{0}$), $E_{sym}$, $L_{sym}$, curvature of
symmetry energy ($K_{sym}$) and effective nucleonic Dirac mass ($m^{*}_{N}$)
corresponding to these parametrizations are given in table-2.

Table 2: The nuclear properties of the GM1 (A) and DD-MEX (B) CDF models at $n_{0}$. $n_{0}$ | $E_{0}$ | $K_{0}$ | $E_{sym}$ | $L_{sym}$ | $K_{sym}$ | $m^{*}_{N}/m_{N}$
---|---|---|---|---|---|---
(fm-3) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) |
(A) 0.153 | $-16.30$ | 300 | 32.50 | 93.86 | 17.91 | 0.700
(B) 0.152 | $-16.14$ | 267 | 32.27 | 49.58 | $-71.47$ | 0.556

The values of coefficient $a_{\rho}$ adjusted to estimate different values of
$L_{\text{sym}}$ at $n_{0}$ for GM1 and DD-MEX parametrizations are provided
in table-3,

Table 3: $a_{\rho}$ coefficient values for various estimations of $L_{\text{sym}}(n_{0})$ and corresponding $K_{\text{sym}}(n_{0})$ for GM1 and DD-MEX coupling models. $L_{\text{sym}}(n_{0})$ (MeV) | $a_{\rho}$ |  $K_{\text{sym}}(n_{0})$ (MeV)
---|---|---
| GM1 | DD-MEX | GM1 | DD-MEX
35 | 0.5893 | 0.8052 | $-127.13$ | $-34.44$
50 | 0.4390 | 0.6148 | $-129.67$ | $-72.21$
65 | 0.2888 | 0.4242 | $-105.17$ | $-75.72$
85 | 0.0885 | 0.1702 | $-30.43$ | $-27.06$

where the slope and curvature of $E_{\text{sym}}$ at $n_{0}$ are respectively
given by

$\displaystyle L_{\text{sym}}(n_{0})$
$\displaystyle=3n_{0}\left[\frac{\partial E_{\text{sym}}(n)}{\partial
n}\right]_{n=n_{0}},$ (22) $\displaystyle K_{\text{sym}}(n_{0})$
$\displaystyle=9n_{0}^{2}\left[\frac{\partial^{2}E_{\text{sym}}(n)}{\partial
n^{2}}\right]_{n=n_{0}}.$

with the nuclear symmetry energy defined as

$E_{\text{sym}}=\frac{1}{2}\left[\frac{\partial^{2}(\varepsilon/n)}{\partial\alpha^{2}}\right]_{\alpha=0}$
(23)

where $\alpha=(n_{n}-n_{p})/n$ is the asymmetry parameter.

For the hyperonic sector, the vector couplings are implemented according to
SU(6) symmetry and quark counting rule Schaffner  _et al._ (1994). For the
scalar couplings, we consider the optical potential values
$U_{\Lambda}^{N}(n_{0})=-30$ MeV, $U_{\Sigma}^{N}(n_{0})=+30$ MeV and
$U_{\Xi}^{N}(n_{0})=-14$ MeV Gomes  _et al._ (2015); Gal  _et al._ (2016) in
symmetric nuclear matter (SNM). Ref.-Friedman and Gal (2021) recently reported
an attractive optical potential for $\Xi$-hyperons in SNM corresponding to
$U_{\Xi}^{N}(n_{0})\geqslant-20$ MeV. Table-4 provides the scalar meson-
hyperon coupling values at $n_{0}$.

Table 4: Scalar meson-hyperon coupling constants, $R_{\sigma Y}=g_{\sigma Y}/g_{\sigma N}$ (normalized to meson-nucleon coupling) for considered parametrizations in this work. | $\Lambda$ | $\Sigma$ | $\Xi$
---|---|---|---
GM1 | 0.6164 | 0.4033 | 0.3047
DD-MEX | 0.6172 | 0.4734 | 0.3088

For the $\Delta$-resonance sector, we consider the meson-$\Delta$ couplings as
parameters. This is due to scarcity of $\Delta$-nucleon interaction
experimental data. Experimental studies Koch and Ohtsuka (1985); Wehrberger
_et al._ (1989); Nakamura  _et al._ (2010) based on pion-nucleus scattering,
$\Delta$-quartet excitations have reported to constrain meson-$\Delta$
resonance couplings. Recent studies Drago  _et al._ (2014); Kolomeitsev  _et
al._ (2017) have narrowed the $\Delta$-potential ($V_{\Delta}$) in nuclear
medium to be $-30~{}\text{MeV}+V_{N}\leq V_{\Delta}\leq V_{N}$ and $0\leq
R_{\sigma\Delta}-R_{\omega\Delta}\leq 0.2$ with
$R_{\sigma\Delta}=g_{\sigma\Delta}/g_{\sigma N}$,
$R_{\omega\Delta}=g_{\omega\Delta}/g_{\omega N}$. Several works Li  _et al._
(2018); Ribes  _et al._ (2019); Cai  _et al._ (2015); Raduta (2021); Thapa
_et al._ (2021) have considered the vector couplings in the range values
$R_{\omega\Delta}\in[0.6-1.2]$ and $R_{\rho\Delta}\in[0.5-3.0]$. In the
present discussion, we consider $R_{\omega\Delta}=1.10$ and
$R_{\rho\Delta}=1.00$ in the vector coupling sector. And in the scalar
meson-$\Delta$ resonance coupling sector, we consider
$R_{\sigma\Delta}=1.10,1.20$.

##  III Results and Discussion

Figure 1: Nuclear symmetry energy as a function of baryon number density (in
units of $n_{0}$) for, upper panel: GM1 and lower panel: DD-MEX coupling
parametrizations. The shaded regions denote the constraints on density-
dependent symmetry energy from heavy-ion collision data Tsang  _et al._ (2009,
2011). The constraint $38\leqslant E_{\text{sym}}(2n_{0})/\text{MeV}\leqslant
64$ Li  _et al._ (2021) at $68\%$ confidence level obtained via analyses of
data from recent NS observables and heavy-ion collisions is denoted by the
vertical error bars. The solid lines in both the panels represent the original
coupling parametrizations. The other cases with adjusted values of
$L_{\text{sym}}$ at $n_{0}$ are denoted by dot-dot-dashed
($L_{\text{sym}}=35$), dotted ($L_{\text{sym}}=50$), dash-dotted
($L_{\text{sym}}=65$) and dashed ($L_{\text{sym}}=85$) curves respectively.

In this section, we report the numerical results for purely nucleonic (N),
hypernuclear (NY) and $\Delta$-admixed hypernuclear (NY$\Delta$) matter
compositions and investigate the effects of symmetry energy on dense matter
EOS. In order to do so, as mentioned in Sec. I, we implement the density-
dependent modification in isovector $g_{\rho b}$ couplings within the
framework of non-linear GM1 parametrization and consider the density-dependent
coupling schemes with DD-MEX parametrization. We proceed by studying the
effect of variation of $L_{\text{sym}}$ on different properties of matter and
star.

Figure 2: Isovector coupling to nucleons as a function of baryon number
density (in units of $n_{0}$) in case of GM1 (upper panel) and DD-MEX (lower
panel) parametrizations. The different curves represent the same cases as
captioned in fig.-1. Figure 3: Pressure variation as a function of energy
density (EOS) for $T=0$ case with matter compositions as, left panels: pure
nucleonic, middle panels: NY$\Delta$ ($R_{\sigma\Delta}=1.10$) and right
panels: NY$\Delta$ ($R_{\sigma\Delta}=1.20$) for different
$L_{\text{sym}}(n_{0})$ values in upper panels: GM1 and lower panels: DD-MEX
parametrizations. The different curves represent the same cases as captioned
in fig.-1. The matter pressure constraint (vertical line) at $n\sim 2n_{0}$ is
deduced from GW170817 Abbott  _et al._ (2018) event data.

The behaviour of nuclear symmetry energy with varying baryon number density is
plotted in fig.-1 for different values of $L_{\text{sym}}$. In density regime
$n<n_{0}$, cases with higher values of $L_{\text{sym}}$ yield lower values of
$E_{\text{sym}}$ while the vice-versa is observed in case of higher density
regimes ($n>n_{0}$). The experimental constraints on $E_{\text{sym}}(n)$ at
sub-saturation densities shown by the shaded region in the fig.-1 allows EOSs
with $L_{\text{sym}}(n_{0})\geqslant 50$ MeV. The values of $E_{\text{sym}}$
at $n_{0}$ are same for all values of $L_{\text{sym}}$ as it is constrained by
the isovector coupling value at nuclear saturation. This result is consistent
with that of ref.-Wu  _et al._ (2021) found considering NL3 Lalazissis  _et
al._ (1997) parametrization. The constraint on $E_{\text{sym}}(2n_{0})$ is
broader and allows for almost all EOSs corresponding to
$L_{\text{sym}}(n_{0})$ values considered in this work.

Fig.-2 displays the density-dependent nature of isovector couplings in
variation with baryon number density for different values of $L_{\text{sym}}$
in both coupling schemes. In sub-saturation densities, it is observed that
with lower $L_{\text{sym}}$ values, the isovector coupling values are larger.
This behaviour is vice-versa in supra-saturation density regimes. At the
saturation density, $g_{\rho N}$ values are identical owing to eqn.-(19). With
higher values of $L_{\text{sym}}$, the variation of $g_{\rho N}$ with baryon
number density is found to be more steep. The $g_{\rho N}(n)$ coupling values
with lower $L_{\text{sym}}$ approach zero at high density regimes resulting in
similar corresponding $E_{\text{sym}}(n)$ values at those densities.

The EOSs for different NS matter compositions (N, NY$\Delta$) are presented in
fig.-3 for GM1 parametrization in upper panels and for DD-MEX parametrization
in lower panels. The EOSs with modified isovector couplings within non-linear
GM1 model as well as with DD-MEX parametrization are observed to lie well
within bounds of the matter pressure constraint from GW170817 event data
Abbott  _et al._ (2018) shown by the vertical arrows in fig.-3. This is true
for all the matter composition cases. The prominent differences in EOSs are
observed at low density regimes ($n\leqslant 0.4$ fm-3).

Table 5: Threshold densities, $n_{u}$ (in units of $n_{0}$) for hyperons and $\Delta$-quartet in NY and NY$\Delta$ matter with varying $L_{\text{sym}}(n_{0})$ values in GM1 parametrization. Model | $R_{\sigma\Delta}=0$ | $R_{\sigma\Delta}=1.10$ | $R_{\sigma\Delta}=1.20$
---|---|---|---
|  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{\Delta}$ ($n_{0}$) |  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{\Delta}$ ($n_{0}$)
GM1 | 2.25 | 2.25 | 2.89 | 2.29 | 2.11
$L_{\text{sym}}=35$ | 2.54 | 2.68 | 2.22 | 2.95 | 1.87
$L_{\text{sym}}=50$ | 2.49 | 2.57 | 2.27 | 2.84 | 1.90
$L_{\text{sym}}=65$ | 2.42 | 2.43 | 2.35 | 2.66 | 1.94
$L_{\text{sym}}=85$ | 2.30 | 2.30 | 2.61 | 2.39 | 2.04
Figure 4: Solutions of TOV equations corresponding to left panels: pure
N-matter, middle panels: NY$\Delta$ ($R_{\sigma\Delta}=1.10$) and right
panels: NY$\Delta$ ($R_{\sigma\Delta}=1.20$) EOSs displayed in fig.-3 for
upper panels: GM1 and lower panels: DD-MEX parametrizations. The different
curves represent the same cases as captioned in fig.-1. The astrophysical
observable constraints from GW190814 Abbott  _et al._ (2020b), PSR
J$0740+6620$ Riley  _et al._ (2021); Miller  _et al._ (2021), PSR J$0348+0432$
Antoniadis  _et al._ (2013) and PSR J$0030+0451$ Miller  _et al._ (2019);
Riley  _et al._ (2019) are represented by shaded regions. The horizontal lines
represent the joint radius constraints from PSR J$0030+0451$ and GW170817
event data for a typical $1.4~{}M_{\odot}$ NS Jiang  _et al._ (2020); Landry
_et al._ (2020).

Tables-5 and 6 provide the threshold densities of heavier baryons in NY and
NY$\Delta$ matter with GM1 and DD-MEX coupling parametrizations respectively.
It is observed that low values of $L_{\text{sym}}$ shifts the onset of
hyperons to higher density regimes. While the opposite behaviour is seen in
case of $\Delta$-resonances. Higher values of normalized scalar meson-$\Delta$
couplings denote attractive $\Delta$-potentials in SNM which result in early
appearance of $\Delta$-quartet in NS matter. Onset of $\Delta$-resonances
delay the appearance of hyperons in NS matter.

Table 6: Similar to table-5 but with DD-MEX parametrization. Model | $R_{\sigma\Delta}=0$ | $R_{\sigma\Delta}=1.10$ | $R_{\sigma\Delta}=1.20$
---|---|---|---
|  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{\Delta}$ ($n_{0}$) |  $n_{u}^{Y}$ ($n_{0}$) |  $n_{u}^{\Delta}$ ($n_{0}$)
DD-MEX | 2.13 | 2.27 | 1.79 | 2.47 | 1.46
$L_{\text{sym}}=35$ | 2.15 | 2.31 | 1.77 | 2.51 | 1.44
$L_{\text{sym}}=50$ | 2.13 | 2.27 | 1.79 | 2.47 | 1.46
$L_{\text{sym}}=65$ | 2.09 | 2.19 | 1.82 | 2.39 | 1.47
$L_{\text{sym}}=85$ | 2.03 | 2.07 | 1.89 | 2.24 | 1.50

It is observed that onset of heavier baryons softens the EOSs marked by change
in slopes as shown in fig.-3. Now at high matter densities, $g_{\rho b}(n)$
coupling values tend to approach zero resulting in less contribution to EOSs
from $\rho$-meson fields. With increase in attractive $\Delta$-potential, the
onset of $\Delta^{-}$ shifts towards lower density regimes as marked by the
kinks in fig.-3.

The mass-radius relationships obtained by solving TOV equations for non-
rotating, spherically symmetric stars corresponding to the EOSs for N,
NY$\Delta$ ($R_{\sigma\Delta}=1.10,~{}1.20$) matter with GM1 and DD-MEX
parametrizations are displayed in different panels of fig.-4. For the crust
region, Baym-Pethick-Sutherland (BPS) Baym  _et al._ (1971) EOS is implemented
maintaining thermodynamic consistency with modelling crust-core transition
following ref.-Fortin  _et al._ (2016). It can be observed that almost all
EOSs (in N, NY$\Delta$-matter compositions) fit within the limits of recent
astrophysical constraints. However, the joint constraints on radius of a
$1.4~{}M_{\odot}$ NS Jiang  _et al._ (2020); Landry  _et al._ (2020) are
satisfied by EOS models with $L_{\text{sym}}\leq 65$ MeV. Incorporation of
$\Delta$-quartet further softens the EOS at lower density in addition to high
density regimes leading to NS configurations with smaller radii as evident
from middle and right panels of fig.-4. The candidature of the secondary
compact component involved in GW190814 as a NS is still not completely
univocal Sedrakian  _et al._ (2020); Li  _et al._ (2020), hence the maximum
mass constraint from this candidate is not so stringent. The variation of
symmetry energy slope has slight impact on maximum mass NS configurations
owing to the similar values of $M_{\text{max}}$ (refer to table-7). The
softening of EOSs due to inclusion of $\Delta$-resonances is more protruding
in density-dependent scenario. The effect of varying $L_{\text{sym}}$ is least
for pure nucleonic matter and large for NY$\Delta$-matter (with more
attractive $\Delta$-potential). For DD-MEX coupling parametrization, the
maximum mass NS configuration with purely nucleonic matter reaches $\sim
2.55M_{\odot}$ satisfying the mass constraint from GW190814 event. This is
consistent with the results in ref.-Rather  _et al._ (2021).

Figure 5: Particle populations, $n_{i}$ (in units of $n$) as a function of
baryon number density for NY$\Delta$-matter ($R_{\sigma\Delta}=1.20$) with
upper panel: $L_{\text{sym}}(n_{0})=93.86$ MeV (original), lower panel:
$L_{\text{sym}}(n_{0})=50$ MeV cases within non-linear coupling (GM1)
parametrization. Figure 6: Similar to fig.-5 but with upper panel:
$L_{\text{sym}}(n_{0})=49.57$ MeV (original), lower panel:
$L_{\text{sym}}(n_{0})=85$ MeV cases within density-dependent coupling (DD-
MEX) parametrization.

The particle abundances in NY$\Delta$-matter composition
($R_{\sigma\Delta}=1.20$) with GM1 (without density dependent $g_{\rho N}$)
and with $L_{\text{sym}}(n_{0})=50$ MeV coupling parametrization models are
shown in fig.-5. It is observed that in case of density-dependent $g_{\rho
N}(n)$ coupling (lower panel), the onset of hyperons is shifted to higher
densities and early appearance of $\Delta$-resonances is favoured. This
resulted in faster decrease of lepton populations in comparison to constant
$g_{\rho N}(n_{0})$ coupling case (upper panel). In $g_{\rho N}(n_{0})$ case,
the $\Delta^{-},~{}\Xi^{-},~{}e^{-},~{}\mu^{-}$ composition provides the
negative charge to balance the proton charge resulting in appearance of
$\Lambda$-hyperons. At high density regimes, this negative particle
composition leads to onset of $\Delta^{0}$ and $\Xi^{0}$ baryons. While in
case of $g_{\rho N}(n)$ couplings, the onset of $\Delta^{0}$ baryons are due
to charge neutrality condition maintained by $\Delta^{-},~{}e^{-}$ with
protons. Fig.-6 displays the particle populations as a function of baryon
number density similar to fig.-5 but with DD-MEX parametrization for
NY$\Delta$-matter. In this too, it is observed that with decrasing value of
$L_{\text{sym}}$, the onset of hyperons is shifted to higher densities while
early onset of $\Delta$-quartet is favoured. At sub-saturation densities, the
charged particle abundances are enhanced with lower value of $L_{\text{sym}}$.
In case of $L_{\text{sym}}=49.57$ MeV (upper panel), the lepton populations
are seen to decrease at a faster rate with rising matter density in comparison
to $L_{\text{sym}}=85$ MeV (lower panel) case. This is because in the latter
case $\Delta^{-}$ abundances fall short to maintain the charge neutrality
condition with protons demanding lepton populations to stay till higher
density regimes.

Figure 7: Strangeness fraction, $f_{s}$ as a function of baryon number density
for matter composition as, upper panels: NY and lower panels: NY$\Delta$
($R_{\sigma\Delta}=1.20$) with varying $L_{\text{sym}}$ for left panels: GM1
and right panels: DD-MEX parametrizations. The different curves represent the
same cases as captioned in fig.-1. Figure 8: Adiabatic sound velocity (in
units of $c$) as a function of energy density for, upper panels: pure
nucleonic, lower panels: $\Delta$-admixed hypernuclear matter
($R_{\sigma\Delta}=1.20$) with left panels: non-linear (GM1) and right panels:
density-dependent (DD-MEX) coupling models. The different curves represent the
same cases as captioned in fig.-1.

In order to see the effect of varying $L_{\text{sym}}$ on hyperons we plot the
strangeness fraction as a function of baryon number density in fig.-7 which is
defined as Cavagnoli  _et al._ (2011),

$f_{s}=\frac{1}{3}\frac{\sum_{Y}|s_{Y}|n_{Y}}{n}$ (24)

where $s_{Y}$, $n_{Y}$ denote the strangeness and number density of $Y$-th
hyperon respectively. It is seen that $f_{s}$ is sensitive to varying
$L_{\text{sym}}$ and decreases with lowering of $L_{\text{sym}}$ values. The
shifting of hyperon threshold densities to higher densities with lower values
of $L_{\text{sym}}$ (as seen in figs.-5 and 6) is also evident from fig.-7.
The similar strangeness fraction for $L_{\text{sym}}=65,50,35$ MeV cases at
high densities relates with the almost similar values (approaching zero) of
$g_{\rho N}(n)$ (refer to fig.-2). In both the coupling parametrization cases,
similar $f_{s}$ values at high density regimes relates to the almost vanishing
values of $\rho$-meson coupling. The delaying appearance of hyperons into NS
matter with onset of $\Delta$-quartet is also apparently seen in fig.-7.

Due to the utmost dependence of matter pressure explicitly over energy density
in NS matter, it is noteworthy that the EOSs follow the causality condition
(i.e. adiabatic speed velocity, $v_{s}$ to be subluminal) given by $v_{s}<c$.
Fig.-8 displays the adiabatic speed of sound as a function of energy density
for different matter compositions with variation in $L_{\text{sym}}$ values
within non-linear and density-dependent CDF models. It is observed that the
EOSs considered in this work satisfy the causality condition. The effect of
$L_{\text{sym}}$ is more prominent on the lower density regimes. This can be
attributed to the diverse $g_{\rho N}$ coupling values at lower densities.
Lower values of $L_{\text{sym}}$ results in reduced $v_{s}$ at lower matter
densities. Kinks in the lower panels denote onset of heavier baryons in NS
matter.

Table 7: NS properties evaluated from the EOS considering various matter compositions (N, NY, NY$\Delta$) with varying $L_{\text{sym}}(n_{0})$ values with non-linear (GM1) and density-dependent (DD-MEX) coupling schemes. The maximum gravitational mass NS and its corresponding radius are denoted by $M_{\text{max}}$ (in units of $M_{\odot}$), $R$ (in units of km) respectively; central number density, central energy density, central matter pressure are represented by $n_{c}$ (in units of fm-3), $\varepsilon_{c}$ and $P_{c}$ (in units of MeV/fm3). Matter pressures at 2 and 6 times saturation densities are denoted by $P(2n_{0})$, $P(6n_{0})$ respectively. The global properties such as radius, compactness parameter, tidal Love number and tidal deformability for a $1.4M_{\odot}$ NS are given by $R_{1.4}$ (in units of km), $C_{1.4}$, $k_{2(1.4)}$ and $\Lambda_{1.4}$ respectively. Matter | CDF Model | $M_{\text{max}}$ | $R$ | $n_{c}$ | $\varepsilon_{c}$ | $P_{c}$ | $P(2n_{0})$ | $P(6n_{0})$ | $R_{1.4}$ | $C_{1.4}$ | $k_{2(1.4)}$ | $\Lambda_{1.4}$
---|---|---|---|---|---|---|---|---|---|---|---|---
composition |  |  | ($M_{\odot}$) | (km) | ($\text{fm}^{-3}$) | (MeV/fm3) | (MeV/fm3) | (MeV/fm3) | (MeV/fm3) | (km) |  |  |
|  | GM1 | 2.36 | 11.93 | 0.865 | 1116.75 | 500.67 | 30.48 | 574.12 | 13.77 | 0.150 | 0.100 | 882
|  | $L_{\text{sym}}=85$ | 2.33 | 11.77 | 0.888 | 1145.36 | 512.52 | 28.62 | 553.83 | 13.53 | 0.153 | 0.098 | 785
| NL | $L_{\text{sym}}=65$ | 2.31 | 11.51 | 0.917 | 1185.47 | 547.87 | 25.54 | 549.66 | 13.06 | 0.158 | 0.096 | 640
|  | $L_{\text{sym}}=50$ | 2.32 | 11.45 | 0.919 | 1186.87 | 555.04 | 24.22 | 554.15 | 12.79 | 0.162 | 0.096 | 581
Pure |  | $L_{\text{sym}}=35$ | 2.33 | 11.42 | 0.916 | 1180.98 | 553.01 | 23.59 | 556.13 | 12.58 | 0.164 | 0.102 | 568
Nucleonic |  | DD-MEX | 2.56 | 12.33 | 0.776 | 1000.40 | 487.24 | 32.26 | 704.96 | 13.29 | 0.156 | 0.106 | 773
Matter |  | $L_{\text{sym}}=85$ | 2.55 | 12.50 | 0.767 | 988.90 | 469.57 | 34.70 | 698.51 | 13.84 | 0.149 | 0.105 | 939
| DD | $L_{\text{sym}}=65$ | 2.55 | 12.36 | 0.777 | 1003.21 | 486.80 | 32.74 | 703.14 | 13.49 | 0.153 | 0.104 | 821
|  | $L_{\text{sym}}=50$ | 2.56 | 12.33 | 0.776 | 1000.68 | 487.42 | 32.27 | 704.93 | 13.30 | 0.155 | 0.105 | 772
|  | $L_{\text{sym}}=35$ | 2.56 | 12.31 | 0.773 | 995.26 | 484.09 | 32.31 | 705.36 | 13.14 | 0.157 | 0.108 | 748
|  | GM1 | 1.99 | 11.97 | 0.926 | 1126.57 | 317.96 | 30.48 | 312.09 | 13.77 | 0.150 | 0.101 | 882
|  | $L_{\text{sym}}=85$ | 1.98 | 11.72 | 0.964 | 1179.58 | 345.34 | 28.62 | 310.77 | 13.53 | 0.153 | 0.098 | 785
| NL | $L_{\text{sym}}=65$ | 1.98 | 11.41 | 1.001 | 1233.99 | 382.86 | 25.54 | 317.59 | 13.06 | 0.158 | 0.095 | 639
|  | $L_{\text{sym}}=50$ | 2.00 | 11.37 | 0.994 | 1222.21 | 379.64 | 24.22 | 320.41 | 12.79 | 0.162 | 0.096 | 581
Hypernuclear |  | $L_{\text{sym}}=35$ | 2.01 | 11.36 | 0.983 | 1203.98 | 371.28 | 23.59 | 321.37 | 12.58 | 0.164 | 0.102 | 568
Matter |  | DD-MEX | 2.18 | 12.00 | 0.875 | 1082.30 | 362.19 | 32.26 | 395.89 | 13.29 | 0.156 | 0.106 | 777
|  | $L_{\text{sym}}=85$ | 2.16 | 12.14 | 0.876 | 1085.95 | 357.77 | 34.70 | 390.54 | 13.83 | 0.149 | 0.105 | 937
| DD | $L_{\text{sym}}=65$ | 2.17 | 12.02 | 0.882 | 1093.24 | 367.13 | 32.74 | 395.07 | 13.49 | 0.153 | 0.104 | 821
|  | $L_{\text{sym}}=50$ | 2.18 | 12.00 | 0.876 | 1082.86 | 362.48 | 32.27 | 395.88 | 13.30 | 0.155 | 0.105 | 774
|  | $L_{\text{sym}}=35$ | 2.19 | 11.99 | 0.869 | 1071.27 | 356.43 | 32.31 | 396.04 | 13.14 | 0.157 | 0.108 | 748
|  | GM1 | 1.99 | 11.95 | 0.928 | 1130.21 | 320.17 | 30.48 | 312.53 | 13.77 | 0.150 | 0.101 | 882
|  | $L_{\text{sym}}=85$ | 1.97 | 11.63 | 0.980 | 1204.54 | 360.38 | 28.62 | 312.22 | 11.63 | 0.153 | 0.098 | 785
| NL | $L_{\text{sym}}=65$ | 1.97 | 11.19 | 1.045 | 1302.99 | 428.33 | 25.54 | 324.02 | 13.05 | 0.158 | 0.095 | 637
$\Delta$-admixed |  | $L_{\text{sym}}=50$ | 1.99 | 11.13 | 1.042 | 1297.38 | 429.21 | 24.22 | 328.62 | 12.77 | 0.162 | 0.095 | 570
Hypernuclear |  | $L_{\text{sym}}=35$ | 2.00 | 11.11 | 1.031 | 1278.87 | 420.59 | 23.59 | 330.15 | 12.55 | 0.165 | 0.100 | 551
Matter |  | DD-MEX | 2.18 | 11.75 | 0.911 | 1138.96 | 405.08 | 28.03 | 406.19 | 13.19 | 0.157 | 0.102 | 717
($R_{\sigma\Delta}=1.10$) |  | $L_{\text{sym}}=85$ | 2.16 | 11.92 | 0.909 | 1138.40 | 395.67 | 33.15 | 398.31 | 13.83 | 0.149 | 0.104 | 935
| DD | $L_{\text{sym}}=65$ | 2.17 | 11.76 | 0.918 | 1152.14 | 411.28 | 29.43 | 405.19 | 13.43 | 0.154 | 0.102 | 788
|  | $L_{\text{sym}}=50$ | 2.18 | 11.75 | 0.911 | 1139.52 | 405.38 | 28.06 | 406.18 | 13.20 | 0.157 | 0.102 | 720
|  | $L_{\text{sym}}=35$ | 2.19 | 11.74 | 0.904 | 1127.36 | 398.93 | 27.51 | 406.38 | 13.02 | 0.159 | 0.104 | 686
|  | GM1 | 1.99 | 11.78 | 0.962 | 1179.86 | 345.22 | 30.48 | 313.94 | 13.77 | 0.150 | 0.101 | 881
|  | $L_{\text{sym}}=85$ | 1.96 | 11.34 | 1.037 | 1290.93 | 410.47 | 28.62 | 316.04 | 13.50 | 0.153 | 0.097 | 773
| NL | $L_{\text{sym}}=65$ | 1.97 | 10.87 | 1.102 | 1392.46 | 488.48 | 24.65 | 332.45 | 12.82 | 0.161 | 0.089 | 541
$\Delta$-admixed |  | $L_{\text{sym}}=50$ | 1.98 | 10.82 | 1.094 | 1377.87 | 485.28 | 22.32 | 338.37 | 12.43 | 0.166 | 0.086 | 449
Hypernuclear |  | $L_{\text{sym}}=35$ | 2.00 | 10.81 | 1.083 | 1358.24 | 476.19 | 20.88 | 340.33 | 12.18 | 0.169 | 0.083 | 395
Matter |  | DD-MEX | 2.19 | 11.48 | 0.938 | 1177.66 | 439.80 | 18.67 | 416.19 | 12.59 | 0.164 | 0.090 | 505
($R_{\sigma\Delta}=1.20$) |  | $L_{\text{sym}}=85$ | 2.15 | 11.62 | 0.945 | 1193.37 | 436.81 | 24.49 | 405.46 | 13.41 | 0.154 | 0.094 | 720
| DD | $L_{\text{sym}}=65$ | 2.18 | 11.48 | 0.947 | 1194.49 | 447.69 | 20.08 | 414.88 | 12.84 | 0.161 | 0.089 | 565
|  | $L_{\text{sym}}=50$ | 2.19 | 11.48 | 0.938 | 1177.94 | 439.93 | 18.69 | 416.17 | 12.59 | 0.164 | 0.090 | 506
|  | $L_{\text{sym}}=35$ | 2.20 | 11.47 | 0.930 | 1165.22 | 433.33 | 18.30 | 416.43 | 12.43 | 0.166 | 0.093 | 487

Now we move to examine the effect of $L_{\text{sym}}$ on the star properties.
The properties of NS for variation of $L_{\text{sym}}$ along with matter
properties with different EOSs considered in this work have been displayed in
table-7. $L_{\text{sym}}$ has practically no effect on the maximum mass of the
star family. However, $L_{\text{sym}}$ variation has a commendable impact on
the radius of NS configurations. This feature is already clear from the
fig.-4. We have tabulated the comparative values of radius for typical
$1.4~{}M_{\odot}$ mass NS ($R_{1.4}$). With increasing value of
$L_{\text{sym}}$ the $R_{1.4}$ increases. Consequently, the compactness
$C_{1.4}$ decreases. Similar impact is also observed in case of tidal
deformability $\Lambda_{1.4}$ with increase of $L_{\text{sym}}$ softness
decreases. A recent study Dietrich  _et al._ (2020) based on the joint
analysis GW170817 and GW190425 events data reported a radius bound $10.94\leq
R_{1.4}/\text{km}\leq 12.61$ at $90\%$ confidence level. From the EOSs
considered in this work, it can be inferred that to satisfy the said $R_{1.4}$
range, the conditions of $\Delta$-resonances onset into NS matter composition
and $L_{\text{sym}}(n_{0})\leqslant 50$ MeV are favourable. Following the
$69\leq L_{\text{sym}}(n_{0})/\text{MeV}\leq 143$ range deduced from recent
PREX-2 data, it is to be noted that DD-MEX parameterization satisfies the
$\Lambda_{1.4}$ upper bound (GW170817 event) for $L_{\text{sym}}(n_{0})\leq
85$ with $R_{\sigma\Delta}=1.20$.

Figure 9: Upper panel: Compactness parameter, lower panel: tidal deformability
as a function of $L_{\text{sym}}$ considering matter composition to be pure N,
NY$\Delta$ ($R_{\sigma\Delta}$=1.20) with non-linear (GM1) and density-
dependent (DD-MEX) coupling models. The different curves denote the EOS models
as labelled.

The variations of compactness parameter and tidal deformability of
$1.4~{}M_{\odot}$ NS with $L_{\text{sym}}$ considering various matter
compositions are shown in fig.-9. The softness decreases in both
parametrizations following similar trend of convergence towards higher
$L_{\text{sym}}$ values. This relates to the fact that lowering of
$L_{\text{sym}}$ shifts the onset of $\Delta$-quartet to lower density regimes
thus increasing compactness and decreasing tidal deformability. The quadratic
fit of compactness parameter and tidal deformability as a function of
$L_{\text{sym}}$ for different matter compositions with GM1 and DD-MEX
parametrizations is given by,

$C_{1.4}~{}\text{or,}~{}\Lambda_{1.4}=a~{}L_{\text{sym}}^{2}+b~{}L_{\text{sym}}+c,$
(25)

where the coefficient $a$, $b$ and $c$ values are provided in table-8.

Table 8: Coefficient values of the quadratic fits in eqn.-(25). The coefficient of determination, $\mathcal{R}^{2}\sim 0.999$ for all the fits considered in this work. CDF Model | $a$ | $b$ | $c$
---|---|---|---
| GM1 (N) | $-1.83\times 10^{-6}$ | $-2.10\times 10^{-6}$ | 0.1663
$C_{1.4}$ | GM1 (NY$\Delta$) | $-2.01\times 10^{-6}$ | $-6.51\times 10^{-5}$ | 0.1738
| DD-MEX (N) | $-1.53\times 10^{-6}$ | $2.38\times 10^{-5}$ | 0.1581
| DD-MEX (NY$\Delta$) | $-3.31\times 10^{-6}$ | 0.0002 | 0.1646
| GM1 (N) | 0.1023 | $-7.884$ | 719.10
$\Lambda_{1.4}$ | GM1 (NY$\Delta$) | 0.1082 | $-5.617$ | 458.70
| DD-MEX (N) | 0.0623 | $-3.676$ | 800.80
| DD-MEX (NY$\Delta$) | 0.0998 | $-7.339$ | 622.40
Figure 10: Tidal deformability as a function of NS mass for, left panels: pure
nucleonic, right panels: NY$\Delta$ ($R_{\sigma\Delta}=1.20$) matter with
varying $L_{\text{sym}}$ values in upper panels: GM1, lower panels: DD-MEX
parametrizations. The different curves represent the same cases as captioned
in fig.-1. The vertical lines denote the bounds on $\Lambda_{1.4}$ deduced in
refs.-Abbott  _et al._ (2018); Jiang  _et al._ (2020).

Fig.-10 shows the variation of dimensionless tidal deformability with NS mass
corresponding to different values of density-dependent $L_{\text{sym}}$
parameter. As already mentioned, we observe that with higher values of
$L_{\text{sym}}$, tidal deformability parameter value increases as the matter
stiffens. The effects of $L_{\text{sym}}$ is significant only in case of lower
mass stars and for massive stars, these effects are inconsequential. In
addition, the inclusion of heavier non-strange baryons softens the EOS at
lower density regimes consequently decreasing $\Lambda$ or, assembling NS
matter to be more compact. This relates with the results from refs.-Li and
Sedrakian (2019); Raduta (2021).

##  IV Summary and Conclusions

In this work, we discuss the density-dependent symmetry energy effects on
dense matter EOSs with different matter compositions viz. pure nucleonic,
hypernuclear, $\Delta$-admixed hypernuclear within CDF theory framework. We
consider different values of symmetry energy slope $L_{\text{sym}}$ to
introduce variation of symmetry energy with density. The $L_{\text{sym}}$ at
saturation is taken to be within the range of $35-85$ MeV. The NS
configurations evaluated from the EOSs considered in this work within this
range of $L_{\text{sym}}$ satisfy the recent astrophysical observable
constraints obtained from NICER (PSR J$0030+0451$ Miller  _et al._ (2019);
Riley  _et al._ (2019), PSR J$0740+6620$ Riley  _et al._ (2021); Miller  _et
al._ (2021)) and GW Abbott  _et al._ (2017, 2020a) observations.

We find that with smaller values of $L_{\text{sym}}$, the EOS is evaluated to
be softer around density range of $1-2~{}n_{0}$. This is because of the
corresponding lower values of $E_{\text{sym}}$ in the said density regimes.
This results in smaller radii alongside making the matter tidally less
deformable for intermediate mass NSs viz. $1.4~{}M_{\odot}$. Although the
range of $L_{\text{sym}}$ considered in this work is consistent with the
astrophysical observations, the lower values of $L_{\text{sym}}$ are more
favourable for the radius observations from NICER Miller  _et al._ (2019);
Riley  _et al._ (2019, 2021); Miller  _et al._ (2021) as well as estimate of
tidal deformability from GW observations. While at the high density regimes,
the EOS is similar for all $L_{\text{sym}}$ values. This attributes to the
vanishing $\rho$ meson fields due to small (approaching zero) $g_{\rho N}$
coupling values.

Different values of $L_{\text{sym}}$ has also substantial effect on appearance
of exotic component of matter. The lower values of $L_{\text{sym}}$ shift the
threshold density for appearance of hyperons to higher side and favours early
appearance of $\Delta$-quartet particles. The early appearance of $\Delta$
particles is also one of the causes for higher threshold density for
appearance of hyperons with lower values of $L_{\text{sym}}$. The early
appearance of $\Delta$ particles for lower values of $L_{\text{sym}}$ makes
the EOS softer at lower density regime attributing to smaller radius for stars
having mass $\sim 1.4~{}M_{\odot}$. Consequently, the compactness of the stars
increases and the deformability decreases with the decreasing values of
$L_{\text{sym}}$. The different values of $L_{\text{sym}}$ does not
practically affect the maximum mass of the NS family as at higher density
regime, the effect of different values of $L_{\text{sym}}$ on the EOS is
negligible. This relates with the results from refs.-Cavagnoli  _et al._
(2011); Providência and Rabhi (2013). As for higher values of
$L_{\text{sym}}$, the EOSs do not differ much, the values of compactness and
tidal deformability merge at higher values of $L_{\text{sym}}$. Hence the
possibility of exotic matter appearance with lower values of $L_{\text{sym}}$
is most favourable from all astrophysical observations.

However the recent update from nuclear physics sector (PREX-2) suggests higher
values of $L_{\text{sym}}(n_{0})$, which plays a tension with the
astrophysical observables viz. tidal deformability, radius of a canonical
$1.4~{}M_{\odot}$ NS. Considering the viability of inclusion of non-strange
$\Delta$-baryons into NS dense matter EOS is seen to be a reasonable option.
Moreover, different choices of coupling models/parametrizations might or might
not provide a feasible solution to this tension and so may have to be
recalibrated. Further analysis regarding this aspect is beyond the scope of
this work and will be discussed in future studies.

The particle population of protons in $\beta$-equilibrated matter largely
depends on the variation on nuclear symmetry energy with density. This in turn
influences the threshold densities of nucleonic dUrca processes consequently
in cooling processes. Several studies Steiner (2006); Fortin  _et al._ (2016);
Raduta  _et al._ (2018); Raduta (2021) have been done to understand the
effects of $L_{\text{sym}}$ on this aspect. Such analysis with non-nucleonic
matter composition following the recent update in isospin asymmetry parameter,
$L_{\text{sym}}$ range will also be addressed in future works.

###### Acknowledgements.

The authors thank the anonymous referee for the constructive comments which
enhanced the quality of the manuscript.

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# RakutenAI-7B: Extending Large Language Models for Japanese

Rakuten Group, Inc.
Aaron Levine, Connie Huang, Chenguang Wang, Eduardo Batista, Ewa Szymanska,
Hongyi Ding,
Hou Wei Chou, Jean-François Pessiot, Johanes Effendi, Justin Chiu, Kai Torben
Ohlhus,
Karan Chopra, Keiji Shinzato, Koji Murakami, Lee Xiong, Lei Chen, Maki Kubota,
Maksim Tkachenko,
Miroku Lee, Naoki Takahashi, Prathyusha Jwalapuram, Ryutaro Tatsushima,
Saurabh Jain,
Sunil Kumar Yadav, Ting Cai, Wei-Te Chen, Yandi Xia, Yuki Nakayama, Yutaka
Higashiyama  The authors are listed in the alphabetical order by their first
name.

###### Abstract

We introduce RakutenAI-7B, a suite of Japanese-oriented large language models
that achieve the best performance on the Japanese LM Harness benchmarks among
the open 7B models. Along with the foundation model, we release instruction-
and chat-tuned models, RakutenAI-7B-instruct and RakutenAI-7B-chat
respectively, under the Apache 2.0 license.

##  1 Introduction

The shift towards a new “Pre-Train, Prompt, and Predict” paradigm Liu et al.
(2023) in natural language processing (NLP) has resulted in the active
development of large language models (LLMs). These models deliver a unified
solution to a wide variety of NLP tasks, often showcasing a superior
performance over the “Pre-Train and Fine-Tune” paradigm. This new research
trend, however, has primarily been focused on English leaving out languages
such as Japanese.

RakutenAI-7B is a systematic initiative that brings the latest technologies to
the world of Japanese LLMs. RakutenAI-7B achieves the best scores on the
Japanese language understanding benchmarks while maintaining a competitive
performance on the English test sets among similar models such as OpenCalm
Ryosuke (2023), Elyza Sasaki et al. (2023), Youri Zhao et al. (2023), Nekomata
Zhao et al. (2024), and Swallow Okazaki et al. (2024). RakutenAI-7B leverages
the Mistral model architecture Jiang et al. (2023) and is based on Mistral
7B-v0.1 pre-trained checkpoint, exemplifying a successful retrofitting of the
pre-trained model weights. Moreover, we extend Mistral’s vocabulary from 32k
to 48k to offer a better character-per-token rate for Japanese.

RakutenAI-7B achieves a significant milestone for Japanese-oriented LLMs,
while maintaining a competitive English performance. Our aim is to help the
community to create more affordable and efficient Japanese language models
that can be used in a wide variety of applications. We release our models to
the public under the Apache 2.0 License. The models are accessible at
https://huggingface.co/Rakuten/RakutenAI-7B.

In the rest of the report, we describe the key aspects of RakutenAI-7B,
delving into the tokenizer extension, pre-training, fine-tuning, and model
evaluation.

##  2 Technical Details

###  2.1 Tokenization

The Mistral tokenizer often encodes a single Japanese character into multiple
tokens, which has two implications. First, as the language model operates
within a limited context size, it significantly reduces the length of Japanese
text that can be processed by the model. Second, text generation with the low
character-per-token rate tokenizer would spend more GPU cycles to generate the
same text, as opposed to a tokenizer with a higher rate. The problem stems
from the logographic part of Japanese (kanji): Mistral’s limited vocabulary
size is not optimized specifically to parse the kanji characters, which causes
the tokenizer to encode a large proportion of them as a sequence of bytes.
This results in a character-per-token rate that is less than 1. The character-
per-token rate is the ratio between the number of UTF-8 characters and the
number of tokens in an excerpt.

To address this issue, we extend the Mistral tokenizer with 16k additional
tokens specifically selected to improve character-per-token rate on Japanese,
resulting in a vocabulary size of 48k tokens. The extended tokenizer
significantly improves the effective context length in comparison to the
Mistral 7B-v0.1 tokenizer when processing Japanese text.

By improving the tokenization process for Japanese, we can achieve cost
efficient text processing during model training as well as inference.

###  2.2 Foundation Model Training

Previous research studies indicate that the quality of text data used for pre-
training is critical to improving LLM performance Rae et al. (2022); Dodge et
al. (2021); Chowdhery et al. (2022). Alongside these studies, we experiment
with and develop data filtering techniques that allow us to further improve
the quality of available internet-scale datasets for Japanese and English. We
remove any personally identifiable information in the datasets such as phone
numbers, email addresses etc.

The filtering process comprises of multiple stages, including normalization,
deduplication and classification, that helps us to distinguish between high
and low-quality data. This approach ensures that our model is trained on a
curated dataset, enhancing its ability to generate coherent and relevant
outputs. We train the released models on approximately 175 billion tokens of
filtered data.

###  2.3 Model Fine-tuning

We fine-tune our foundation model to create RakutenAI-7B-instruct and
RakutenAI-7B-chat using a mix of open and internally hand-crafted datasets.
This tuning process enhances the model’s capacity to adhere to provided
instructions and prompts for RakutenAI-7B-instruct and enables
RakutenAI-7B-chat to generate conversational responses akin to those of a
virtual assistant, facilitating more natural dialogue interactions.

We fine-tune both the instruction- and chat-tuned models on safety datasets to
mitigate the generation of explicit or offensive context, preventing the
spread of misinformation or harmful biases. However, we advise caution, as
they may still exhibit unintended behaviour. It is crucial to constantly
monitor the model performance to ensure it adheres to ethical and social
standards.

###  2.4 Evaluation

####  2.4.1 Evaluation Settings

We evaluate the performance of our models using the Japanese
111https://github.com/Stability-AI/lm-evaluation-
harness/tree/0fa86429679f521161d5b81a94c0c385e0a0976d and the English
222https://github.com/EleutherAI/lm-evaluation-
harness/tree/b281b0921b636bc36ad05c0b0b0763bd6dd43463 versions of the Language
Model Evaluation Harness (LM-Harness), running the evaluation with same setup
to ensure fair comparison across the models. The list of the Japanese NLP
tasks used for metrics is as follows:

  * •

JCS / JCommonSenseQA Kurihara et al. (2022) is the Japanese version of
CommonSenseQA Talmor et al. (2019), which consists of multiple-choice
questions that evaluate model ability in answering commonsense questions.

  * •

JNLI / Japanese Natural Language Inference Kurihara et al. (2022) is a test
set to evaluate a model’s ability to recognize the inference relation that a
premise sentence has to a hypothesis sentence. There are three inference
relations, namely: entailment, contradiction, and neutral which are presented
to the model in a multiple-choice question format.

  * •

MARC-ja / Multilingual Amazon Review Corpus is a text classification test set
that is based on the MARC dataset Keung et al. (2020). MARC-ja is the Japanese
subset of this dataset, which is a binary classification task for positive and
negative review sentiment.

  * •

JSQuAD / Japanese Stanford Question Answering Dataset Kurihara et al. (2022)
is a Japanese reading comprehension dataset which measures model ability to
answer a question given a passage.

  * •

JAQKET / Japanese Questions on Knowledge of Entities Suzuki et al. (2020) is a
Japanese open-domain question answering dataset which measures model knowledge
on open topics such as TV quiz questions.

  * •

XLSUM-ja Hasan et al. (2021) is a Japanese subset of XLSUM, which is an
abstractive summarisation dataset for 44 languages. We took the test set to
measure model ability to summarise news articles. The hypothesis summary is
evaluated against a reference summary using ROUGE-2 Lin (2004), a metric that
measures n-gram co-occurrences.

  * •

xWino/ xWinograd-ja is the Japanese subset of the Winograd schema challenge
Emelin and Sennrich (2021), which is a pair of sentences that differ in only
one or two contrastive words that are easily disambiguated by a human reader.
This evaluation measures model’s ability to use commonsense knowledge to
disambiguate and debias the ambiguous sentence.

  * •

MGSM / Multilingual Grade School Math Benchmark Shi et al. (2022) contains a
set of grade-school level math problems. We took the Japanese subset to
measure the models’ ability to answer basic mathematical problems that require
multi-step reasoning.

The list of the English NLP tasks used for metrics is as follows:

  * •

ARC / AI2 Reasoning Challenge Clark et al. (2018) is a grade-school science
question test set that is aimed at testing model ability to handle tough
queries that basic algorithms struggle with.

  * •

HellaSwag Zellers et al. (2019) is a test set to benchmark model ability to
perform commonsense reasoning, given questions that require understanding and
reasoning beyond simple pattern matching.

  * •

MMLU / Massive Multitask Language Understanding Hendrycks et al. (2021) is a
test to measure model multitask accuracy. It covers 57 tasks that covers
history, computer science, mathematics, chemistry, and other topics.

  * •

TruthfulQA Lin et al. (2022) measures models’ inclination to replicate common
online falsehoods.

For multiple choice question tasks, the choice with the highest likelihood is
chosen as the model answer. We compute the likelihood for each choice
concatenated with the question (including n-shot question-answer pairs when
provided). The model answer is then compared with the reference answer to
measure the accuracy.

For question-answering test sets such as JSQuAD and JAQKET, we count the model
output as correct if it exactly matches the reference answer (we allow some
variation in the answers by removing punctuation and emoticons, normalizing
whitespaces, etc.).

In Tables 1 and 2, along with the task names we indicate the metrics and the
number of shots for n-shot used for evaluation. acc stands for the perplexity-
based accuracy for multiple-choice questions, em stands for the exact-match
metric, and rouge-2 indicates the ROUGE-2 Ganesan (2018) score.

####  2.4.2 Evaluation Results for Foundation Models

Model Name | 7-Avg. | Avg. | JCS | JNLI | MARC-ja | JSQuAD | Jaqket v2 | XLSum-ja | xWino | MGSM
---|---|---|---|---|---|---|---|---|---|---
| excl. |  | acc | acc | acc | em | em | rouge-2 | acc | acc
| XLSum-ja |  | 3-shots | 3-shots | 3-shots | 2-shots | 1-shot | 1-shot | 0-shot | 5-shots
rakuten-ai-7b | 69.80 | 62.83 | 84.27 | 48.69 | 96.29 | 79.09 | 80.67 | 14.08 | 77.16 | 22.40
nekomata-7b | 66.01 | 58.83 | 85.43 | 40.14 | 96.80 | 76.29 | 71.99 | 8.59 | 73.83 | 17.60
japanese-stablelm-base-gamma-7b | 64.83 | 59.12 | 80.07 | 14.71 | 92.41 | 81.38 | 85.05 | 19.16 | 82.59 | 17.60
youri-7b | 62.71 | 56.90 | 76.94 | 51.11 | 90.96 | 57.45 | 78.09 | 16.27 | 78.00 | 6.40
swallow-7b | 60.86 | 55.18 | 78.91 | 15.16 | 90.27 | 73.28 | 80.24 | 15.41 | 76.96 | 11.20
elyza-japanese-Llama-2-7b | 60.24 | 53.26 | 75.60 | 50.74 | 87.51 | 71.48 | 57.56 | 4.40 | 71.22 | 7.60
elyza-japanese-Llama-2-7b-fast | 58.31 | 51.34 | 71.49 | 45.77 | 86.61 | 70.91 | 64.18 | 2.54 | 61.63 | 7.60
open-calm-7b | 45.27 | 39.67 | 62.65 | 31.92 | 85.37 | 38.05 | 33.42 | 0.45 | 65.07 | 0.40
Table 1: RakutenAI-7B foundation model performance on Japanese LM-Harness metrics in comparison with other models. Our model achieves the highest average score, more than 3 points ahead of the next best model. The models are sorted by 7-Avg. Model Name | Avg. | ARC | HellaSwag | MMLU | TruthfulQA
---|---|---|---|---|---
|  | acc | acc | acc | acc
|  | 25-shots | 10-shots | 5-shots | 6-shots
rakuten-ai-7b | 60.50 | 60.24 | 82.20 | 61.31 | 38.25
japanese-stablelm-base-gamma-7b | 56.08 | 50.60 | 77.43 | 54.99 | 41.30
elyza-japanese-Llama-2-7b | 52.76 | 51.62 | 76.54 | 44.85 | 38.02
elyza-japanese-Llama-2-7b-fast | 52.07 | 51.79 | 75.46 | 44.41 | 36.63
nekomata-7b | 51.97 | 47.35 | 72.78 | 48.38 | 39.38
youri-7b | 50.60 | 49.15 | 75.02 | 42.36 | 35.89
swallow-7b | 49.90 | 47.35 | 72.20 | 39.36 | 40.68
open-calm-7b | 29.87 | 20.56 | 31.01 | 23.73 | 44.16
Table 2: RakutenAI-7B foundation model performance on English LM-Harness
metrics in comparison with other models. Our model achieves the highest
average score, more than 4 points ahead of the next best model.

Tables 1 and 2 report the Japanese and English LM-Harness performance of
RakutenAI-7B. It achieves the best overall performance compared with other
Japanese 7-billion parameter foundation models on both Japanese and English
test sets. The average RakutenAI-7B score for the Japanese LM-Harness
evaluation is 62.83, which is more than 3 points above Japanese-StableLM-Base-
Gamma-7b, the second best model, at 59.12. We also report the average of 7
metrics (7-Avg.) that excludes XLSum-ja, as we find the performance on this
task generally has a low correlation with human judgements. Our model also
showcases the highest performance for 7-Avg. In general, our foundation model
shows a balanced performance across all test sets, with a particularly high
score on MGSM.

In addition, RakutenAI-7B retains a high performance for English, as shown by
its average English LM-Harness score compared with other Japanese models. Our
model achieves an average score of 60.50, while Japanese-StableLM-Base-
Gamma-7b Akiba et al. (2023), which also uses Mistral 7B-v0.1 as the base
model for continual training, lags by more than 4 points compared to
RakutenAI-7B, at 56.08.

####  2.4.3 Evaluation Results for Instruction-Tuned Models

We measure the performance of the instruction-tuned models in the same manner
as the foundation model.

Model Name | 7-Avg. | Avg. | JCS | JNLI | MARC-ja | JSQuAD | Jaqket v2 | XLSum-ja | xWino | MGSM
---|---|---|---|---|---|---|---|---|---|---
| excl. |  | acc | acc | acc | em | em | rouge-2 | acc | acc
| XLSum-ja |  | 3-shots | 3-shots | 3-shots | 2-shots | 1-shot | 1-shot | 0-shot | 5-shots
rakuten-ai-7b-instruct | 77.32 | 68.74 | 93.03 | 90.39 | 96.00 | 80.44 | 81.79 | 8.67 | 75.18 | 24.40
youri-7b-instruction | 73.35 | 66.84 | 86.06 | 70.13 | 97.03 | 82.53 | 79.47 | 21.29 | 79.04 | 19.20
japanese-stablelm-instruct-gamma-7b | 65.46 | 59.98 | 83.82 | 16.97 | 95.68 | 76.20 | 81.87 | 21.58 | 82.06 | 21.60
swallow-7b-instruct | 64.29 | 58.25 | 83.38 | 26.50 | 94.46 | 75.62 | 81.01 | 16.01 | 76.23 | 12.80
elyza-japanese-Llama-2-7b-instruct | 60.04 | 53.19 | 65.15 | 57.44 | 91.51 | 67.29 | 58.51 | 5.20 | 70.80 | 9.60
elyza-japanese-Llama-2-7b-fast-instruct | 57.22 | 50.48 | 70.69 | 36.48 | 92.75 | 68.87 | 62.29 | 3.36 | 59.44 | 10.00
nekomata-7b-instruction | 49.04 | 44.14 | 85.08 | 42.48 | 96.99 | 8.51 | 10.91 | 9.81 | 76.12 | 23.20
Table 3: RakutenAI-7B-instruct model performance on Japanese LM-Harness metrics in comparison with other models. Our model achieves the highest average score, more than 3 points ahead of the next best model. The models are sorted by 7-Avg. Model Name | Avg. | ARC | HellaSwag | MMLU | TruthfulQA
---|---|---|---|---|---
|  | acc | acc | acc | acc
|  | 25-shots | 10-shots | 5-shots | 6-shots
rakuten-ai-7b-instruct | 61.32 | 58.62 | 82.70 | 60.32 | 43.63
japanese-stablelm-instruct-gamma-7b | 55.91 | 50.43 | 77.10 | 54.61 | 41.50
elyza-japanese-Llama-2-7b-fast-instruct | 54.21 | 53.58 | 77.69 | 46.91 | 38.67
elyza-japanese-Llama-2-7b-instruct | 54.07 | 52.05 | 78.33 | 47.09 | 38.83
nekomata-7b-instruction | 52.84 | 50.34 | 73.67 | 48.53 | 38.81
youri-7b-instruction | 52.11 | 48.98 | 75.66 | 45.41 | 38.38
swallow-7b-instruct | 50.32 | 47.61 | 72.27 | 40.77 | 40.62
Table 4: RakutenAI-7B-instruct model performance on English LM-Harness metrics
in comparison with other models. Our model achieves the highest average score,
more than 5 points ahead of the next best model.

In general, we observe improvements in our instruction-tuned model over the
foundation model, which shows the effectiveness of our instruction fine-
tuning. Tables 3 and 4 report the Japanese and English LM-Harness performance
of RakutenAI-7B-instruct in comparison with other Japanese instruction-tuned
models within the 7 billion parameter category. The instruct model achieves an
average score of 68.74, leading by almost 2 points over Youri-7B-instruction,
the second best model (this trend also holds for 7-Avg.). Our instruct model
achieves the best scores on JCS, JNLI, and MGSM.

In addition, RakutenAI-7B-instruct improves its LM-Harness English performance
compared to the foundation model, achieving the best performance as shown by
its average score compared to other previously published Japanese models. It
leads the next best instruct model by more than 5 points.

##  3 Conclusion

Our systematic approach to model development, including data filtering
techniques and curation, ensures that RakutenAI-7B is trained on high-quality
datasets, which results in coherent and high-quality outputs for Japanese and
English. The empirical evaluation results demonstrate the consistently high
performance of RakutenAI-7B across a diverse range of NLP tasks, on average
outperforming other open Japanese models. Moreover, the RakutenAI-7B tokenizer
is more suitable for processing Japanese text, potentially resulting in faster
and cheaper training and inference.

We release the suite of RakutenAI-7B models to researchers, developers, and
industry practitioners alike as part of our continuous effort to foster
innovation and drive positive impact across various domains.

##  4 Acknowledgements

We want to thank Rakuten Group, Inc. for providing the resources to make this
release possible. We would also like to acknowledge the research community,
and the Mistral AI team in particular, for open-sourcing and sharing their
work on resource intensive foundation model research.

##  5 Limitations

The suite of RakutenAI-7B models is capable of generating human-like text on a
wide range of topics. However, like all LLMs, they have limitations and can
produce biased, inaccurate, or unsafe outputs. Please exercise caution and
judgement while interacting with them.

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