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e1e-mail<EMAIL_ADDRESS>e2e-mail:
<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>e4e-mail:
<EMAIL_ADDRESS>
11institutetext: Department of Physics, BITS-Pilani, K. K. Birla Goa Campus,
Goa 403726, India 22institutetext: CFisUC, Department of Physics, University
of Coimbra, 3004-516 Coimbra, Portugal

# High-Density behavior of symmetry energy and speed of sound in the dense
matter within an effective chiral model

Prashant Thakure1,addr1  N. K. Patrae2,addr1  T. K. Jhae3,addr1  Tuhin Malik
e4,addr2

(Received: date / Accepted: date)

###### Abstract

With an effective chiral model, we investigate how the mesonic cross couplings
$\sigma-\rho$ and $\omega-\rho$ affect the density content of the symmetry
energy and its higher-order slope parameters. Earlier mentioned cross-
couplings are crucial to controlling the density content of symmetry energy.
For this purpose, we did a case study for different values of the symmetry
energy $J_{1}$, defined at density 0.1 fm-3 in the range (23.4 - 25.2) for a
fixed value of the slope of the symmetry energy $L_{0}=60$ MeV at saturation
density and investigate its effect on the higher-order coefficients and their
influence on the underlying equation of state. We found that the model with
$J_{1}=24.6$ MeV is more favorable with the pure neutron matter (PNM)
constraints obtained from $\chi$EFT calculations. In addition, we show that
all of our models predict a monotonically increasing speed of sound up to four
times the saturation density. The speed of sound decreases/saturates above
that point and approaches the conformal limit approximately $\sqrt{1/3}~{}c$
at the center of the maximum mass star.

##  1 Introduction

Constraining the equation of state (EOS) of dense nuclear matter, supported by
astrophysical observations Lattimer and Prakash (2001); Watts  _et al._
(2016); Özel and Freire (2016); Oertel  _et al._ (2017); Vidaña (2018);
Bombaci and Logoteta (2018), and terrestrial experiments Danielewicz  _et al._
(2002); Baran  _et al._ (2005); Li  _et al._ (2008); Tsang  _et al._ (2019);
Trautmann and Wolter (2012); López and Terrazas Porras (2017); Giuliani  _et
al._ (2014); Garg and Colò (2018); Ono (2019) remains one of the primary focus
in dense matter studies for decades. A dense matter EOS can consist of several
phases or compositions, including hyperons, quarks, superconducting matter,
and colored superconducting matter. The density-dependence of symmetry energy
plays an important role in regulating the particle content in the underlying
EOS, particularly in studies of highly asymmetric matter such as those
speculated in the core of neutron stars. It is to be noted that the symmetry
energy slope parameters can be equivalently expressed as a series expansion in
matter density. Here, in addition to $J_{0}$, the symmetry energy defined at
nuclear saturation density ($\rho_{0}\sim 0.153~{}$fm-3), its slope parameter
$L_{0}$, the curvature parameter $K_{\rm sym,0}$, and the skewness parameter
$Q_{\rm sym,0}$, as well as other higher-order terms may dictate the behavior
of a particular EOS at higher densities. Based on the current knowledge of
nuclear masses and giant dipole polarizability, the symmetry energy has been
constrained $J_{0}\simeq 32.25\pm 2.5$ MeV Roca-Maza  _et al._ (2013, 2015);
Viñas  _et al._ (2014); Mondal  _et al._ (2016) and it’s slope parameter
$L_{0}\simeq 58.9\pm 16$ MeV Li  _et al._ (2015); Trippa  _et al._ (2008);
Möller  _et al._ (2012); Tsang  _et al._ (2012); Lattimer and Lim (2013);
Oertel  _et al._ (2017) at nuclear saturation density. However, there are very
few known theoretical constraints on $K_{\rm sym,0}$ and constraints on
$Q_{\rm sym,0}$ Lattimer and Lim (2013); Lattimer (2023); Malik  _et al._
(2022) and therefore, one of our objectives here is to analyze their
interdependence with a model based on microphysics. Numerous relativistic
mean-field models (RMF) Walecka (1974); Serot and Walecka (1986); Gambhir  _et
al._ (1990); Patra and Praharaj (1991); Serot (2002) are known to have been
successfully applied theoretically to extract finite nuclear properties across
the periodic table as well as in applications of nuclear matter. Broadly, in
these theories, nuclear interactions have been expressed in terms of the
scalar meson $\sigma$, the vector mesons $\omega$ and the iso-vector meson
$\rho$ and their cross-couplings and the respective fields being expressed in
mean-field where quantum fluctuations are ignored. The formalism seems to be
more so valid when the density or the source terms are large Serot and Walecka
(1986); Todd-Rutel and Piekarewicz (2005); Agrawal (2010); Kumar  _et al._
(2017).

Analogously, models based on chiral symmetry, introduced by Gell-Mann $\&$
Levy Gell-Mann and Levy (1960) were subsequently applied to nuclear matter
studies Lee and Wick (1974); however, had limited applications in the finite
nuclear domain Ogawa  _et al._ (2004); Chanfray and Ericson (2007). One of the
major drawbacks of the model is that at short distances or high-momentum
transfer, the interactions become weaker Thomas (1984). The vacuum jumps to a
chirally restored abnormal vacuum (Lee-Wick vacuum) Lee and Wick (1974); Lee
and Margulies (1975). The problem Thomas  _et al._ (2004) may be overcome by
including logarithmic terms of the scalar field in chiral potentials Furnstahl
and Serot (1993); Serot (2002); Heide  _et al._ (1994); Mishustin  _et al._
(1993); Papazoglou  _et al._ (1999) which intercept the normal vacuum from
collapsing which may help in describing the finite nuclear properties Schramm
(2002); Tsubakihara and Ohnishi (2007); Tsubakihara  _et al._ (2010).
Subsequent inclusion of the dynamically generated mass of the vector meson in
the model Boguta (1983); Sahu  _et al._ (1993) reported an unrealistic high
nuclear incompressibility ($K$) value, which was taken care of later by
introducing higher-order terms of scalar meson field and applied to nuclear
matter studies at both low Sahu  _et al._ (2004) and high densities and
neutron stars Jha and Mishra (2008); Jha  _et al._ (2006, 2010). Lately, in
order to improve the density content of the symmetry energy, mesonic cross-
couplings were incorporated Malik  _et al._ (2017) and applied to study
magnetized neutron stars Patra  _et al._ (2020). Moreover, the higher-order
interactions in the chiral fields are desirable as it is known to mimic the
three-body forces, which may play a vital role in dense matter studies.

In the present work, we focus on the aspects of the isospin part of nuclear
interaction and analyze its dependence on the in-medium nuclear interactions
and the resulting equation of state (EOS), particularly on the density
dependence of the nuclear symmetry energy Danielewicz  _et al._ (2002).
Although the nuclear symmetry energy at normal matter density is fairly well
known, however, they are poorly known or known with large uncertainties at
supra-normal densities Li  _et al._ (1998). Our primary objective is to study
the behavior of different order slope parameters of the nuclear symmetry
energy up to larger densities. For this, we employ the effective chiral model
with mesonic cross interaction such as $\sigma-\rho$ and $\omega-\rho$. There
are three parameters in our model regulating the density content of symmetry
energy. We systematically study the correlations and the effect of those
parameters on different slope parameters of the nuclear symmetry energy.

It was often shown by several authors that the speed of sound has exhibited a
maximum at an energy density of approximately 500 MeV fm-3 or three times the
saturation density when the EOS developed by physics-agnostic perturbative QCD
calculations is conditioned at high densities Kurkela (2022); Gorda  _et al._
(2022). A similar phenomenon, however, does not occur when only astronomical
restrictions are applied. Accordingly, we also investigate the behavior of
sound up to higher densities.

The paper is organized as follows. In Section 2, we present the details of the
hadronic model. The results of our calculations are discussed in Section 3.
Section 4 contains the summary and conclusions of the present work.

##  2 The equation of state

The effective chiral lagrangian is as given in Eq. (1), where the hadronic
degrees of freedom are the pseudo-scalar meson $\pi$, the scalar meson
$\sigma$, the vector meson $\omega$ and the isovector $\rho-$meson Malik  _et
al._ (2017).

$\displaystyle{\cal{L}}$
$\displaystyle=\bar{\psi}_{B}\Big{[}\left(i\gamma_{\mu}\partial^{\mu}-g_{\omega}\gamma_{\mu}\omega^{\mu}-\frac{1}{2}g_{\rho}\vec{\rho_{\mu}}.\vec{\tau}\gamma^{\mu}\right)-g_{\sigma}(\sigma+i\gamma_{5}\vec{\tau}.\vec{\pi})\Big{]}\psi_{B}$
(1)
$\displaystyle+\frac{1}{2}\left(\partial_{\mu}\vec{\pi}.\partial^{\mu}\vec{\pi}+\partial_{\mu}\sigma\partial^{\mu}\sigma\right)-\frac{\lambda}{4}\left(x^{2}-x_{0}^{2}\right)^{2}-\frac{\lambda
b}{6m^{2}}(x^{2}-x_{0}^{2})^{3}$ $\displaystyle-\frac{\lambda
c}{8m^{4}}(x^{2}-x_{0}^{2})^{4}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}g_{\omega}^{2}x^{2}\left(\omega_{\mu}\omega^{\mu}\right)-\frac{1}{4}\vec{R_{\mu\nu}}.\vec{R^{\mu\nu}}$
$\displaystyle+\frac{1}{2}{m_{\rho}^{\prime}}^{2}\vec{\rho_{\mu}}.\vec{\rho^{\mu}}+\eta_{1}\left(\frac{1}{2}g_{\rho}^{2}x^{2}\vec{\rho_{\mu}}.\vec{\rho^{\mu}}\right)+\eta_{2}\left(\frac{1}{2}g_{\rho}^{2}\vec{\rho_{\mu}}.\vec{\rho^{\mu}}\omega_{\mu}\omega^{\mu}\right).$

In the lagrangian above, $\psi_{B}$ is the nucleon iso-spin doublet
interacting with the mesons. The $\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and
$\frac{1}{4}\vec{R_{\mu\nu}}\vec{R^{\mu\nu}}$ are kinetic terms for $\omega$
and $\rho$ respectively, where
$F_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$ and
$\vec{R_{\mu\nu}}=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}$. The
coupling strength for higher order scalar fields are $b$ and $c$,
respectively, and $\gamma^{\mu}$ and $\tau$ are the Dirac matrices and Pauli
matrices, respectively. The interaction terms and higher order terms are given
in terms of the chiral invariant field $x^{2}=(\pi^{2}+\sigma^{2})$. The last
two terms contain mesonic cross-coupling between $\rho$-$\sigma$ and
$\rho$-$\omega$ whose coupling strengths are $\eta_{1}$ and $\eta_{2}$,
respectively. The vacuum expectation value of the scalar field is $x_{0}$ and
the mass of the nucleon ($m$), the scalar ($m_{\sigma}$), and the vector meson
mass ($m_{\omega}$), are related to $x_{0}$ through

$\displaystyle
m=g_{\sigma}x_{0},~{}~{}m_{\sigma}=\sqrt{2\lambda}x_{0},~{}~{}m_{\omega}=g_{\omega}x_{0}\
,$ (2)

where, $\lambda=\frac{(m_{\sigma}^{2}-m_{\pi}^{2})}{2f_{\pi}^{2}}$, where and
$f_{\pi}$ is the pion decay constant. In the mean-field treatment, we ignore
the pion field and also set $m_{\pi}=0$. The derivation of the equation of
motion of the meson fields and its EOS $(\varepsilon~{}\&~{}p)$ for the
present model can be found in Ref. Malik  _et al._ (2017). In terms of
$Y=x/x_{0}=m^{*}/m$ and the corresponding couplings $C_{\sigma}\equiv
g_{\sigma}^{2}/m_{\sigma}^{2}$ for scalar, $C_{\omega}\equiv
g_{\omega}^{2}/m_{\omega}^{2}$ for vector and $C_{\rho}\equiv
g_{\rho}^{2}/m_{\rho}^{2}$ for the iso-vector the energy density $(\epsilon)$
and pressure $(p)$ of the present model for a given baryon density can be
calculated as,

$\displaystyle\epsilon$
$\displaystyle=\frac{1}{\pi^{2}}\sum_{i=n,p}\int_{0}^{k_{F}^{i}}k^{2}\sqrt{k^{2}+m^{*}{{}^{2}}}dk+\frac{m^{2}}{8C_{\sigma}}(1-Y^{2})^{2}$
(3)
$\displaystyle-\frac{b}{12C_{\sigma}C_{\omega}}(1-Y^{2})^{3}+\frac{c}{16m^{2}C_{\sigma}C_{\omega}^{2}}(1-Y^{2})^{4}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}Y^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\Big{[}1-\eta_{1}(1-Y^{2})(C_{\rho}/C_{\omega})+3\eta_{2}C_{\rho}\omega_{0}^{2}\Big{]}(\rho_{3}^{0})^{2},$
$\displaystyle p$
$\displaystyle=\frac{1}{3\pi^{2}}\sum_{i=n,p}\int_{0}^{k_{F}^{i}}\frac{k^{4}}{\sqrt{k^{2}+m^{*}{{}^{2}}}}dk-\frac{m^{2}}{8C_{\sigma}}(1-Y^{2})^{2}$
(4)
$\displaystyle+\frac{b}{12C_{\sigma}C_{\omega}}(1-Y^{2})^{3}-\frac{c}{16m^{2}C_{\sigma}C_{\omega}^{2}}(1-Y^{2})^{4}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}Y^{2}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\Big{[}1-\eta_{1}(1-Y^{2})(C_{\rho}/C_{\omega})+\eta_{2}C_{\rho}\omega_{0}^{2}\Big{]}(\rho_{3}^{0})^{2}.$

In the equation above, $k_{F}^{i}$ is the fermi momentum of a nucleon and
$\gamma$ is the spin degeneracy factor ($\gamma=4$ for Symmetric Nuclear
Matter (SNM)). The parameters $C_{\sigma}$ and $C_{\omega}$ can be obtained by
solving the field equations self-consistently tuned to satisfy standard
nuclear matter saturation properties. The detailed calculation of the same can
be found in our previous work, where it was evident that the couplings are
very sensitive to the value of the effective mass Jha and Mishra (2008).
Accordingly, the symmetry energy $S(\rho)$ in the present model using cross
interactions (Eq. (1)) is given as,

$\displaystyle S(\rho)$ $\displaystyle=$
$\displaystyle\frac{k_{F}^{2}}{6\sqrt{k_{F}^{2}+m^{*}{{}^{2}}}}+\frac{C_{\rho}k_{F}^{3}}{12\pi^{2}(m^{*}_{\rho}/m_{\rho})^{2}}+\frac{\eta_{2}C_{\rho}^{2}\omega_{0}^{2}k_{F}^{3}}{6\pi^{2}(m^{*}_{\rho}/m_{\rho})^{4}}$
(5)
$\displaystyle-\frac{2\eta_{2}C_{\rho}^{2}C_{\omega}k_{F}^{9}}{27\pi^{6}m_{\omega}^{2}Y^{4}(m^{*}_{\rho}/m_{\rho})^{4}},$

where,
${m^{*}_{\rho}}^{2}=m_{\rho}^{2}\Big{[}1-\eta_{1}(1-Y^{2})(C_{\rho}/C_{\omega})+\eta_{2}C_{\rho}\omega_{0}^{2}\Big{]}$
and $k_{F}=(3\pi^{2}\rho/2)^{1/3}$. The three coupling parameters $C_{\rho}$,
$\eta_{1}$ and $\eta_{2}$ are fixed numerically for given values of symmetry
energy at 0.1 fm-3 $J_{1}$, symmetry energy $J_{0}$ and its slope parameter
$L_{0}$ at saturation density. In general, the symmetry energy can be
expressed in Taylor series expansion around saturation density $(\rho_{0})$ as
Dong  _et al._ (2015)

$\displaystyle S(\rho)$
$\displaystyle=J_{0}+L_{0}(\frac{\rho-\rho_{0}}{3\rho_{0}})+\frac{1}{2}K_{\rm
sym,0}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{2}$ (6)
$\displaystyle+\frac{1}{6}Q_{\rm
sym,0}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{3}+\frac{1}{24}Z_{\rm
sym,0}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{4}$
$\displaystyle+\mathcal{O}(\frac{\rho-\rho_{0}}{3\rho_{0}})^{5},$

where $J_{0}$ is the symmetry energy coefficient, its slope parameter $L_{0}$,
symmetry energy curvature parameter $K_{\rm sym,0}$ and the third order
density derivatives of symmetry energy $Q_{\rm sym,0}$, all defined at
$\rho_{0}$ Lattimer and Lim (2013). The density dependence of all those
parameters can be defined as,

$\displaystyle L$ $\displaystyle=$ $\displaystyle 3\rho_{0}\frac{\partial
S(\rho)}{\partial\rho},$ (7) $\displaystyle K_{\rm sym}$ $\displaystyle=$
$\displaystyle 9\rho_{0}^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}},$ (8)
$\displaystyle Q_{\rm sym}$ $\displaystyle=$ $\displaystyle
27\rho_{0}^{3}\frac{\partial^{3}S(\rho)}{\partial\rho^{3}},$ (9)
$\displaystyle Z_{\rm sym}$ $\displaystyle=$ $\displaystyle
81\rho_{0}^{4}\frac{\partial^{4}S(\rho)}{\partial\rho^{4}}.$ (10)

The nuclear incompressibility of asymmetric matter can be elaborated in terms
of isospin asymmetry $\delta=\frac{(\rho_{n}-\rho_{p})}{\rho}$ at $\rho_{0}$
as $K(\delta)=K+K_{\tau}\delta^{2}+\mathcal{O}(\delta^{4})$ and $K_{\tau}$ is
given by Chen  _et al._ (2009)

$\displaystyle K_{\tau}=K_{sym}-6L_{0}-\frac{Q_{0}L_{0}}{K},$ (11)

Where $K$ is the nuclear matter incompressibility
($9\rho^{2}\frac{\delta^{2}(\epsilon/\rho)}{\delta\rho^{2}}|_{\rho_{0}}$) and
$Q_{0}$ is the skewness parameter
($27\rho^{3}\frac{\delta^{3}(\epsilon/\rho)}{\delta\rho^{3}}|_{\rho_{0}}$ ).

##  3 Result and Discussion

In this section, we present the results for the density-dependence of symmetry
energy, its slope, and curvature parameters over densities obtained for
different cross-coupling parameters. For the present investigation, the EOS
for the symmetric nuclear matter is kept unchanged by fixing the parameters
$C_{\sigma}$, $C_{\omega}$, $B$ and $C$ at $\rho_{0}$ as obtained in Ref.
Malik  _et al._ (2017), with the nuclear saturation properties, such as
binding energy, nuclear incompressibility, and nucleon effective mass and are
enlisted in Table 1. The values are well within the acceptable ranges
prescribed in the literature. However, the effective mass $m^{*}/m=0.87$ is
relatively larger than some of the well-known relativistic mean field models,
such as NL3 or IUFSU. It is to be noted that the magnitude of the effective
mass is very important for reproducing the spin-orbital splittings of finite
nuclei. If the effective nucleon mass is larger, the spin-orbital splitting
will be decreased. Furthermore, the $\sigma$-meson mass is also smaller
compared to other RMF models. In one of the previous worksJha and Mishra
(2008), the correlation of nucleon effective mass and the corresponding mass
of the $\sigma-$meson was analyzed. The authors found that with larger the
nucleon effective mass in the model, the smaller the mass of the scalar meson
to satisfy the nuclear matter saturation properties. Another rare feature of
the present model is the behavior of the nucleon effective mass with density.
In our case, the effective mass goes down with density but again increases
after $\sim~{}3\rho_{0}$ Jha  _et al._ (2006); Jha and Mishra (2008). This
behavior is attributed to the fact that repulsive forces start to dominate as
the matter density increases.

Table 1: Model parameters for the symmetric nuclear matter calculated at energy per nucleon ($E_{0}$), nuclear incompressibility ($K$), and effective nucleon mass ($m^{*}/m$) at the saturation density $\rho_{0}$=0.153 fm-3. $C_{\sigma}=g_{\sigma}^{2}/m_{\sigma}^{2}$ and $C_{\omega}=g_{\omega}^{2}/m_{\omega}^{2}$ are the scalar and vector meson coupling parameters, respectively. The higher-order self-couplings of scalar field parameters are $B=b/m^{2}$ and $C=c/m^{4}$, respectively, where $m$ is the mass of the nucleon. The masses of nucleon, $\omega$ meson and $\sigma$ meson are 939 MeV, 783 MeV and 444.6 MeV respectively. $C_{\sigma}$ | $C_{\omega}$ | $B$ | $C$ | $E_{0}$ | K | $m^{*}/m$
---|---|---|---|---|---|---
($fm^{2}$) | ($fm^{2}$) | ($fm^{2}$) | ($fm^{4}$) | (MeV) | MeV |
7.221 | 1.670 | -6.435 | 0.001 | -16.0 | 231 | 0.87
Table 2:  The slope parameter of symmetry energy for all the models is kept
fixed $L_{0}=60$MeV. The values of the various symmetry energy
parameters($J_{0}$, $J_{1}$,$K_{\rm sym,0}$, $Q_{\rm sym,0}$, $Z_{\rm sym,0}$,
and $K_{\tau}$) as shown in Eqs. [7-11] are listed according to the model
coupling schemes as discussed in the text.

TPJM | $J_{1}$ | $J_{0}$ | $C_{\rho}$ | $\eta_{1}$ | $\eta_{2}$ | $K_{\rm sym,0}$ | $Q_{\rm sym,0}$ | $Z_{\rm sym,0}$ | $K_{\tau}$
---|---|---|---|---|---|---|---|---|---
| (MeV) | (MeV) | (fm2) |  |  | (MeV) | (MeV) | (MeV) | (MeV)
1 | 23.4 | 31.5 | 7.45 | -0.32 | 5.20 | -153.74 | 124.97 | 703.76 | -427.04
32.0 | 6.78 | -0.15 | 6.99 | -191.31 | 97.22 | 2918.98 | -461.85
32.5 | 6.35 | -0.06 | 6.48 | -192.33 | -37.344 | 3435.22 | -482.22
2 | 24.0 | 31.5 | 12.56 | -0.80 | 1.19 | -79.19 | 302.59 | -2209.27 | -348.94
32.0 | 8.48 | -0.41 | 4.55 | -146.75 | 182.84 | 19.97 | -419.05
32.5 | 7.14 | -0.17 | 6.81 | -195.20 | 125.73 | 2750.89 | -466.51
3 | 24.6 | 31.5 | 13.13 | -0.82 | 1.05 | -76.66 | 310.06 | -2624.05 | -346.07
32.0 | 13.32 | -0.79 | 1.63 | -87.44 | 302.36 | -2343.15 | -353.06
32.5 | 10.23 | -0.55 | 3.24 | -124.97 | 242.90 | -1621.37 | -398.85
4 | 25.2 | 33 | 12.75 | -0.68 | 2.54 | -110.01 | 296.53 | -2509.99 | -377.39
34 | 7.01 | -0.02 | 8.15 | -247.63 | 167.07 | 6205.68 | -520.32

Figure 1:  (Color Online) The pressure is plotted against density for (left)
SNM, and (right) PNM is plotted. The results for PNM EOS obtained for
different classes of our model based on $J_{1}$ values are displayed and
compared with a few RMF models such as NL3, IUFSU, BSP, and BKA22. The shaded
regions on both panels are the constraints from Havey Ion Collision (HIC)
experiments Danielewicz  _et al._ (2002).

For density-dependent of symmetry energy, we have three parameters, namely
$C_{\rho}$, $\eta_{1}$, and $\eta_{2}$. We tune these three parameters to
obtain different values of $J_{1}$ and $J_{0}$, but a fixed value of slope
$L_{0}=60$ MeV, which is reasonably good for the central value obtained in
different analyses Malik  _et al._ (2017). In Table 2, we present the
parameter sets $C_{\rho}$, $\eta_{1}$ and $\eta_{2}$ along with $K_{\rm
sym,0},Q_{\rm sym,0},Z_{\rm sym,0}$ for different $J_{0}$, $J_{1}$. The
symmetry energy slope parameter $L_{0}$ for all those models is kept fixed at
60 MeV. In the following, we present the density behavior of those symmetry
energy parameters (Eq.(7$-$10) ) for different $J_{1}$ and $J_{0}$, and for a
fixed value of $L_{0}$. The value of $J_{1}$ in our present work lies in the
range (23.4$-$25.2) MeV obtained in Ref. Trippa  _et al._ (2008); Roca-Maza
_et al._ (2013) by analyzing the experimental data on isovector giant
resonances.

The pressure for both symmetric nuclear matter (left) and pure neutron matter
(right) are plotted in Fig. 1 as a function of density. A comparison with the
constraints obtained from heavy-ion collision flow (HIC) experiments and a few
well-known theoretical models is shown. It is to be noted that the HIC PNM
pressure is calculated theoretically with two extreme parameterizations, the
Asy-soft (weakest) and Asy-stiff (strongest) of symmetry energy Danielewicz
_et al._ (2002). Here, we find that our model prediction is in good agreement
with the band of the HIC data in the entire range, very similar to other
models such as IUFSU Fattoyev  _et al._ (2010), BSP Agrawal  _et al._ (2012),
and BKA22 Agrawal (2010). It is to be noted that the NL3 has both a stiff SNM
and a PNM content as its nuclear matter incompressibility K is 270 MeV, and
the slope of the symmetry energy $L_{0}$ is 120 MeV at saturation density
$\rho_{0}$. The figure also reflects the same as it is out of the HIC
constraints for both SNM and PNM. The models employed in the present study
based on different classes of $J_{1}$ don’t show any substantial difference in
the pressure. It is interesting to note that all of our models have different
symmetry energy behavior or sufficient wide dispersion in symmetry energy at
high density, but it dissolves in pressure. It is because of many degeneracies
of proton fraction or symmetry energy for a given pressureImam  _et al._
(2022); Mondal and Gulminelli (2022).

Figure 2:  (Color Online) Energy per neutron as a function of density
($\rho_{n}$) is shown for different classes of our models based on $J_{1}$
values. Few selected RMF models such as NL3, IUFSU, BSP and BKA22 are also
compared Gezerlis and Carlson (2010); Hebeler  _et al._ (2013). Figure 3:
(Color Online) Symmetry energy ($S(\rho)$) as a function of
density(${\rho_{b}}$) is plotted for different classes of our models based on
$J_{1}$ values along with a few selected RMF models such as NL3, IUFSU, BSP,
and BKA22. The constraint of symmetry energy from the isobaric analog states
(IAS) is also displayed Danielewicz and Lee (2014).

The Chiral EFT describes the hierarchy of two-, three-, and weaker higher-body
forces and provides theoretical uncertainty estimates.. Microscopic
calculations, which are derived from chiral effective field theory, help us to
constrain the properties of neutron-rich matter up to nuclear saturation
density to a high degree. In Figure 2, we show the energy per particle for PNM
for different classes of our models based on $J_{1}$ values. We compare the
pure neutron matter energy obtained from $\chi$EFT calculations in yellow
hatched Hebeler  _et al._ (2013) area. We also plot this constraint with twice
the uncertainty in the yellow-shaded region. As we know, this PNM constraint
strongly influences symmetry energy up to supra-saturation density and is a
stringent constraint for most of the RMF models. For comparison, we also show
the PNM energy obtained from a few well-known RMF models, such as NL3, IUFSU,
BSP, and BKA22. As we can see, our models for different $J_{1}$ are in
agreement with the $\chi$EFT constraints from $\approx$ 0.4 $\rho_{0}$ density
similar to others. However, the prediction of $J_{1}=24.6$ MeV is the closest
to $\chi$EFT PNM constraints compared to all the models throughout the density
range shown. The mesonic cross couplings, therefore, seem to be instrumental
in dictating and regulating the EOS at both high and low densities,
particularly with the symmetry energy aspects of the matter.

Figure 4:  (Color Online) The slope parameter $L$ (Eq. 7) as a function of
density ($\rho_{b}$) is shown. Different bands of $L$ corresponding to
different values of $J_{1}$ are displayed (see Table 2 ).

The density content of the symmetry energy as a function of baryon density is
plotted in Fig. 3. Additionally, we compare the symmetry energy constraint
provided by the isobaric analog states (IAS) Danielewicz and Lee (2014). It
can be seen from the figure that all of our model’s predictions agree with IAS
well. We also compare the density-dependent symmetry energy derived from a few
RMF models, namely NL3, IUFSU, BSP, and BKA22. All of our models for different
values of $J_{1}$ show softer symmetry energy with a spread above two times
saturation density compared to other RMF models shown in the figure. It is to
be noted for all of our models, the slope of the symmetry energy at saturation
is fixed to $L_{0}$ = 60 MeV. The values of $L_{0}$ are 120.65, 49.26, 50, and
78.79 for the models NL3, IUFSU, BSP, and BKA22, respectively. We find that
the model with $J_{1}$ = 25.2 MeV represents the softest symmetry energy among
others.

Figure 5:  (Color Online) The $J_{0}-L_{0}$ phase space is plotted. Present
model calculations have been shown with various other theoretical approaches
such as microscopic, the Skyrme, the non-linear Walecka model, and the
density-dependent mesonic couplings model (DDM). The different experimental
constraints such as heavy-ion collisions (HIC) Tsang  _et al._ (2009),
isobaric analog states (IAS) Danielewicz  _et al._ (2017), giant dipole
resonances (GDR) Trippa  _et al._ (2008) those obtained from fitting
astrophysical M–R observationsSteiner  _et al._ (2010); Silich and Tenorio-
Tagle (2013), electric dipole polarizability Roca-Maza  _et al._ (2015) and
neutron-skin thicknesses of Sn isotopes Chen  _et al._ (2010) are also
displayed.

In Figure 4, we plot the density-dependence of the slope parameter $L$ as a
function of baryon density for the models with different $J_{1}$ values,
corresponding to the symmetry energy at saturation density, $J_{0}$ in the
range (31.5 $-$ 34) MeV (see Table 2). The values agree to $J_{0}\simeq
32.25\pm 2.5$ MeV as obtained from Refs. Roca-Maza  _et al._ (2013, 2015);
Viñas  _et al._ (2014); Mondal  _et al._ (2016). The value of $L_{0}=60$ MeV
is fixed for all the cases and is the median value which is obtained from
recent analyses of isospin diffusion and the astrophysical dataChen  _et al._
(2010); Lattimer and Lim (2013); Oertel  _et al._ (2017). Our models have a
fixed value of $L_{0}$ at saturation density, which can be inferred from the
plot, where they all converge to the same point at $\rho_{0}$. However, we
find that the model with a lower value of $L$ below saturation density results
in a step up in values after saturation density. In Fig. 5, We plot
$J_{0}-L_{0}$ phase space for all of our models and compare them with
available empirical/experimental constraints. Our model results are comparable
with other theoretical models as well as various terrestrial experimental
constraints and astrophysical observations. Compatibility with the overlap
region of the $J_{0}-L_{0}$ phase space is of interest as it agrees to a
diverse set of constraints available. We find that our model predictions lie
in this overlap region, which again speaks of its prominence in the current
model.

Figure 6:  (Color Online) The curvature parameter $K_{\rm sym}$ (Eq. 8) as a
function of density ($\rho_{b}$) is shown. The results of our models with
different bands of $K_{\rm sym}$ corresponding to symmetry energy ($J_{1}$) is
displayed.

In Fig. 6, we plot the density dependence of symmetry energy curvature
parameter $K_{\rm sym}$ against baryon number density for all of our different
models corresponding to different symmetry energy($J_{1}$). We also compare
the range of $K_{\rm sym}$ at saturation density $K_{\rm sym,0}\in[-177,71]$
MeV obtained from RMF within the Bayesian Inference approach with the minimal
constraint imposed Malik  _et al._ (2022, 2023). All of our models show good
agreement with the value at saturation density except the model with
$J_{1}=25.2$ MeV. The density dependence of curvature parameter $K_{\rm sym}$,
in general, shows an increasing trend over density, contrary to the behavior
of $L$.

Figure 7:  (Color Online) The skewness parameter $Q_{\rm sym}$(Eq.9) is
plotted as a function of density ($\rho_{b}$). Different bands of $Q_{\rm
sym}$ corresponding to symmetry energy $J_{1}$ are displayed.

We plot the density dependence of the symmetry energy skewness parameter
$Q_{\rm sym}$ in Figure 7 against baryon number density for all models
corresponding to different symmetry energy ($J_{1}$). Also, we compare the
range of $Q_{\rm sym,0}\in[-88,1736]$ at saturation density obtained from the
microscopic model in RMF treatment with the minimal constraint imposed Malik
_et al._ (2022, 2023). The values of $Q_{\rm sym,0}$ do also agree to the
recently obtained range (-607.65 $-$ 783.21) MeV in Ref.Dutra  _et al._
(2014); Patra  _et al._ (2022). At saturation density, this parameter may have
a wide range, but all of our models fall within the constraints. The density
dependence of the skewness parameter $Q_{\rm sym}$ shows a decreasing trend
and becomes saturated at higher densities. However, all the models show
nonlinearity around saturation density except for the model with $J_{1}=24.6$
MeV. The nonlinearity is greater for models with $J_{1}$ equal to 25 MeV. This
is because we have fixed the value of slope parameter $L_{0}$ for all models.
The density dependence of the symmetry energy kurtosis parameter $Z_{\rm sym}$
against scaled baryon number density in Fig. 8 for all models corresponding to
different symmetry energy ($J_{1}$). We put a band of $Z_{\rm
sym,0}\in[-19290,236]$ obtained in Ref Malik  _et al._ (2022, 2023). Except
for the model with $J_{1}=25.2$, all of our models show good agreement with
the value at saturation density. The figure illustrates that the kurtosis
parameter $Z_{\rm sym}$ increases over density and becomes saturated at a
higher density above 2 $\times\rho_{0}$.

Figure 8:  (Color Online) The higher-order parameter $Z_{\rm sym}$ (Eq. 10) as
a function of density ($\rho_{b}$). Different bands of $Z_{\rm sym}$
corresponding to symmetry energy $J_{1}$ are displayed.

As mentioned earlier, the nuclear matter incompressibility $K$ and the
$K_{\tau}$ are crucial in the determination of a reliable EOS as they control
the stiffness of the symmetric and asymmetric parts of the EOS separately. The
influence of parameters $L_{0}$ and $K_{\rm sym,0}$ predominantly determine
the value of $K_{\tau}$ and is related to the incompressibility of asymmetric
nuclear matter. It is to be noted that various theoretical approaches fail to
converge on a particular range of nuclear incompressibility (K), and there is
also disagreement with the experimental data that is available. For example,
in the non-relativistic domain, random phase calculations (RPA) calculations
indicate $K$ in the range (210 - 220) MeV Blaizot  _et al._ (1995); Blaizot
(1980), microscopic Gogny effective interactions limit the value to 231$\pm$5
MeV Youngblood  _et al._ (1999). Recently a range of (240$\pm$20) MeV was
reported using various experimental data on isoscalar giant monopole
resonances Shlomo  _et al._ (2006). Combined together, the mean value would
come to be $\sim 230$ MeV. Similarly, the constraints on $K_{\tau}$ ranges
from $-840$ to $-350$ MeV Stone  _et al._ (2014); Pearson  _et al._ (2010). In
Fig. 9, we have shown the range on the $K$-$K_{\tau}$ phase space for our
model (marked with a red triangle) along with several other well-known models
and experimental constraints. All our models predict the incompressibility
($K$) value at 231 MeV, and we have only modified the symmetry energy
component, which affects $K_{\tau}$ only. It can be seen that our model
predictions seem to agree with the majority of the theoretical approaches.

Figure 9:  (Color Online) The values of $K_{\tau}$ and $K$ of our model
compared with different theoretical models Sagawa  _et al._ (2007); Colo  _et
al._ (2014). The horizontal and vertical dashed lines represent the empirical
ranges of $K_{\tau}$ and $K$, respectively.  Figure 10:  (Color Online) The
speed of sound $c_{s}^{2}$(in $c^{2}$) as a function of baryon density is
shown. The color lines correspond to a few selected RMF models, such as NL3,
BSR2, and FSU2. For comparison, we displayed the speed of the sound domain
from Ref.Kurkela (2022) obtained with pQCD, and astrophysical constraints
(shaded light green and light purple region ).

We extend our analysis to study the speed of sound of beta-equilibrated charge
neutral matter inside the star. The behavior of $c_{s}$ at such a high density
is important because it controls the stiffness of the equation of state. In
Fig. 10, we plot the square of the speed of sound as a function of density for
all models mentioned in Table 2. We observe that all of our models give a
monotonically increasing speed of sound up to four times saturation density,
and above this density, it gets saturated and tends to decrease towards
$\approx\sqrt{1/3}$c. However, in all of our models, the rate of decrease is
minimal (in the third decimal place only). Generally, in most microscopic
models within RMF treatment, it commonly shows a monotonically increasing
trend and gets saturated at a value higher than 0.4. For a better comparison,
we show the same obtained from other RMF models: FSU2 Chen and Piekarewicz
(2014), BSR2 Agrawal (2010), and NL3 Lalazissis  _et al._ (1997). We also
compare the results obtained in Ref. Kurkela (2022); Gorda  _et al._ (2022),
where the authors calculated the possible domain of speed of sound by imposing
astrophysical and QCD constraints. It was shown that a knee-like feature forms
at densities of approximately 3$\rho_{0}$, which is the critical density when
QCD softens the equation of state. The speed of sound, which is obtained from
the equation state, exhibits the effects of this property in the form of a
peak-like structure. The speed of sound obtained with our parameter set falls
in the overlap region of Astro and Astro+ QCD constraints (Mass-Radius-Tidal
deformability). It can be seen from the figure that the NL3 and BSR2 do not
satisfy the constraints imposed by astro+pQCD (shaded light green), whereas
they are within the region obtained with astro constraints (shaded light
purple) Kurkela (2022); Gorda  _et al._ (2022). It is to be noted that all of
our models for different values of $J_{1}$ noticeably agree with those
constraints.

##  4 Conclusions

Studies of dense matter physics are complicated because of the unknown
knowledge of nuclear forces. In dense matter studies, chiral symmetry models
are promising to replicate three-body forces, which may play a vital role Lee
and Wick (1974); Ogawa  _et al._ (2004); Chanfray and Ericson (2007). On the
other hand, the particle content in the underlying EOS is significantly
influenced by the density-dependent symmetry energy. Several phases or
compositions, such as hyperons, quarks, superconducting matter, and colored
super conditions, can be suitable candidates for a dense matter EOS. The value
of symmetry energy at saturation density is quite known from nuclear physics
experimental or empirical knowledge, however, is not well known at higher
densities. In this work, we have done a case study employing the existing
chiral model Malik  _et al._ (2017) to investigate the density-dependent
behavior of different slope parameters of the symmetry energy. We also
investigate the behavior of the stiffness parameter of EOS: the square of the
speed of sound with different values of $J_{1}$.

In the present work, we bring out the essence of the density behavior of
several higher-order symmetry energy parameters in the presence of various
mesonic cross-couplings. At both high and low densities, mesonic cross
couplings appear to be crucial in determining and controlling the EOS,
especially when considering the symmetry energy features of the matter. For
models with varying $J_{1}$ in the range (23.4-25.2) MeV for a fixed value of
$L_{0}=60$ MeV, we found that model with $J_{1}=24.6$ shows better performance
over all the empirical constraints of nuclear symmetry energy shown in this
article. The incompressibility of nuclear matter, due to their independent
influence over the symmetric and asymmetric components of the EOS, $K$, and
the $K_{\tau}$, are essential in determining a reliable EOS.

The behavior of the square of the speed of sound $c_{s}^{2}$ at such a high
density is important because it controls the stiffness of the equation of
state. Our model predictions for the square of the speed of sound fall within
the overlap region of Astro and QCD constraints (Mass-Radius-Tidal
deformability) obtained in Ref. Kurkela (2022); Gorda  _et al._ (2022). We
find that up to four times the saturation density, the models predict a
monotonically increasing sound speed, and thereafter, it saturates and trends
toward the value of asymptotic limit $\sqrt{1/3}$c.

It is interesting to note that the square of the sound speed at high density
is similar in all of our models, but the symmetry energy at high density has
spread and therefore, has uncertainty in composition. In order to fully
understand this, a model-independent statistical future calculation is
required, which is beyond the scope of this study.

##  5 Acknowledgements

N.K.P. would like to acknowledge the Department of Science and Technology
(DST), India, for the support DST/INSPIRE Fellowship/2019/IF190058.

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