# Impacts of symmetry energy slope on the oscillation frequencies of neutron
stars with short-range correlation and admixed dark matter

Bin Hong1<EMAIL_ADDRESS>ZhongZhou Ren1<EMAIL_ADDRESS>Chen Wu2
XueLing Mu3 1 School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China  2 Shanghai Advanced Research Institute, Chinese
Academy of Sciences, Shanghai 201210, China  3 College of Electronic
Information and Electrical Engineering, Chengdu University, Chendu 610106,
China

###### Abstract

Oscillation modes of compact stars, in general, can serve as a fingerprint in
determining the equation of state (EOS) of dense matter. In this study, we
examine the impact of symmetry energy slope ($L$) on the oscillation
frequencies of neutron stars (NSs) with nucleon-nucleon short range
correlation (SRC) and admixed dark matter (DM) for the first time within the
relativistic mean-field theory. By adjusting the $L$, we revise the EOS and
coupling parameters in light of the SRC and DM effects, and construct the new
sets. The results reveal that NSs containing SRC and DM inside are more likely
to satisfy the observational constraints, and we find that smaller $L$
exhibits larger fundamental non-radial and radial frequencies, and that the
effect on Large Separation (LG) is also mainly concentrated in the low-mass
region. Moreover, we update the linear relationship between the non-radial
frequency and mean density, and we further give empirical relations between
non-radial and radial frequencies and tidal deformability at different $L$ for
1.4$M_{\odot}$ and 2$M_{\odot}$. These findings will enable us to more
effectively confine the NS EOSs, in turn, also provide a strategy to place
constraints on the $L$.

Keywords: Neutron stars; Nuclear astrophysics; Dark matter

††preprint: APS/123-QED

##  I Introduction

As a class of very dense objects in astronomy, the description of matter at
high density inside NSs has attracted a lot of attention from nuclear physics
(Oertel  _et al._ , 2017), particle physics, and astrophysics (Jiang  _et al._
, 2020; Luo  _et al._ , 2022). However, given the non-perturbative nature of
nuclear forces, we cannot derive the EOS directly from quantum chromodynamics
(QCD). As a result, it is more common to construct NS EOSs from the
microscopic first principles, such as $\chi EFT$ (Epelbaum  _et al._ , 2009;
Hammer  _et al._ , 2013; Holt  _et al._ , 2013; Tews  _et al._ , 2013; Lynn
_et al._ , 2016), or from the self-consistent phenomenological models, typical
approaches like the Skyrme-Hartree-Fock (Stone and Reinhard, 2007; Rikovska
Stone  _et al._ , 2003; Dutra  _et al._ , 2012) and Gogny-Hartree-Fock (Loan
_et al._ , 2011; Gonzalez-Boquera  _et al._ , 2017), or from parameterization
models (Read  _et al._ , 2009; Luo  _et al._ , 2022). These models are also
strongly constrained by multi-messenger observations, including the recent
series of discoveries of twice-solar-mass ($2M_{\odot}$) NSs (Demorest  _et
al._ , 2010; Fonseca  _et al._ , 2016; Arzoumanian  _et al._ , 2018;
Antoniadis  _et al._ , 2013; Cromartie  _et al._ , 2020; Fonseca  _et al._ ,
2021), the tidal deformability extracted from the binary NS merger event
GW170817 (Abbott  _et al._ , 2019; Fasano  _et al._ , 2019), the simultaneous
mass and radius measurements of the isolated PSR J0030$+$0451 by NICER
(Neutron Star Interior Composition Explorer) (Miller  _et al._ , 2019; Riley
_et al._ , 2019) and the gravitational wave event GW190814 from the
coalescence of a stellar-mass black hole and a mysterious compact star (Abbott
_et al._ , 2020). To some extent, these observations undoubtedly disprove
certain EOSs, making it imperative to establish plausible ones that can
describe the low density properties while meeting the observational
constraints.

Nuclear microscopic scale nucleon-nucleon SRC (Arrington  _et al._ , 2012;
degli Atti, 2015; Hen  _et al._ , 2017) and cosmic macroscopic size DM have
proven to be relatively challenging issues. Several theoretical works have
demonstrated that, in contrast to the free Fermi gas, the SRC originating from
the strongly repulsive core of nuclear force and its tensor part leads to an
appreciable depletion below the Fermi surface, with some nucleons occupying
regions above the Fermi surface and giving rise to a high-momentum tail, as
has been verified by experiments like $(e,e^{\prime}p)$ (Rohe  _et al._ ,
2004) and $(e,e^{\prime}NN)$ (Onderwater  _et al._ , 1998; Starink  _et al._ ,
2000). Furthermore, the SRC effects also play an important role in nuclear
physics (Panda  _et al._ , 2006; Sammarruca, 2014; Cai and Li, 2015; Hen  _et
al._ , 2015a; Wang  _et al._ , 2017; Yang  _et al._ , 2019), such as allowing
us to have a better understanding of the EMC effect (Weinstein  _et al._ ,
2011; Hen  _et al._ , 2012) and to explain the neutrino oscillation
measurements (Fields  _et al._ , 2013; Fiorentini  _et al._ , 2013) as well as
density-dependent behavior of nuclear symmetry energy (Hen  _et al._ , 2015a).
When it is extended to study NSs, tidal deformability (Souza  _et al._ ,
2020a; Hong  _et al._ , 2022), mass-radius relations (Li  _et al._ , 2019; Lu
_et al._ , 2021; Lourenço  _et al._ , 2022) and cooling efficiency will all be
affected (Souza  _et al._ , 2020b). For DM, many observations such as
gravitational lensing, galaxy rotation curves, velocity dispersions, galaxy
clusters and cosmic microwave background have predicted its existence (for a
review, see (Bertone  _et al._ , 2005; Bertone and Hooper, 2018; Buen-Abad
_et al._ , 2022)). Although there are many candidates for DM, the origin and
properties remain a mystery, moreover, DM does not interact directly with
normal matter, but it has a more pronounced gravitational effect on dense
objects like NSs (Kouvaris and Tinyakov, 2011), and play a important role in
determining NS mass-radius relation (Quddus  _et al._ , 2020; Lourenço  _et
al._ , 2022) and tidal deformability (Das  _et al._ , 2021, 2020). Despite
this, there is still little ongoing research to incorporate nucleon-nucleon
SRC into the admixed DM NSs and to further examine their effects.

The symmetry energy and its slope play a crucial role in determining the
equation of state of pure neutron matter, which involves determining the
neutron skin thickness in neutron-rich matter, the dynamics of heavy-ion
collisions, and also affects the structure and properties of NSs, helping us
understand isospin asymmetry physics under extreme conditions. Given that $L$
characterizes the density dependence of the symmetry energy, it not only
determines the behavior of high-density asymmetry nuclear matter but also
directly affects the supernova, and even the production of heavy elements in
nucleosynthesis. Moreover, its value remains large uncertain and still
requires theoretical and experimental studies. Therefore, we choose the
symmetry energy slope $L$ to study its effect on the properties of NSs under
the background of short-range correlations and dark matter admixed. In the
work (Kong  _et al._ , 2017) the authors used two different methods to deduce
its range around $L=64.29\pm 11.84$ MeV and $L=53.85\pm 10.29$ MeV,
respectively, while the latest study employs Bayesian analysis to estimate $L$
to be around $L=70^{+21}_{-18}$ MeV (Tang  _et al._ , 2021). Recently, a more
accurate model-independent experiment PREX-2 reported very thick 208Pb neutron
skin thickness ($R_{\text{skin }}^{208}$) (Adhikari  _et al._ , 2021), and
based on this result with taking into account the strong correlation between
$R_{\text{skin }}^{208}$ and $L$, the work (Reed  _et al._ , 2021) extracted a
$L$ up to $(106\pm 37)$ MeV, a value that seriously challenges our
understanding of neutron-rich matter near the nuclear saturation density and
will further aggravate the uncertainty in $L$. Additionally, as the
oscillation frequency is highly dependent on the NS internal structure, it is
possible to understand the internal physics by analyzing GW signals from
compact star oscillations in addition to signals from binary NS mergers. The
presence of any non-axisymmetric perturbation in NSs caused by dynamical
instabilities, such as starquakes, magnetic reconfiguration, and rotating
(Jones, 2002), is known to cause non-radial oscillations and thus emission of
gravitational waves. Furthermore, depending on the restoring force, they can
be classified as fundamental $f$-modes, gravity $g$-modes, pure time
$w$-modes, pressure $p$-modes, and rotational $r$-modes (Kokkotas and Schutz,
1992; Lau  _et al._ , 2010; Kokkotas and Schmidt, 1999). It is well known that
non-radial oscillations of NSs are also accompanied by radial oscillations
(Passamonti  _et al._ , 2007), and although radial oscillations do not produce
gravitational radiation directly, they can couple and further amplify it
(Passamonti  _et al._ , 2006, 2007). Many excellent works have been carried
out to investigate the radial and non-radial oscillations, for example, by
studying the difference in the oscillation frequency caused by the different
components inside a NS, it is possible to identify whether it is a hadronic,
hybrid or quark star (Vaeth and Chanmugam, 1992; Vásquez Flores and Lugones,
2010; Flores  _et al._ , 2017; Ranea-Sandoval  _et al._ , 2018; Pereira  _et
al._ , 2018; Sun  _et al._ , 2021; Jaikumar  _et al._ , 2021; Lau and Yagi,
2021). Moreover, numerical relativistic simulations show that binary NSs
merger to form a hypermassive NS may experience strong radial oscillations and
radiate GWs at kHz (Bauswein  _et al._ , 2014; Takami  _et al._ , 2014), and
the identification of quasi-periodic oscillations in giant flares caused by NS
torsional oscillations can also help us to effectively constrain the NS crust
properties (Sotani  _et al._ , 2012a, b, 2018, 2019).

Given the fact that the $L$ is still theoretically uncertain, and that the NS
oscillation frequency is expected to detect in the near future with the
upgrade of the observational devices (Perkins  _et al._ , 2021; Punturo  _et
al._ , 2010; Dwyer  _et al._ , 2015), so it is necessary to establish a link
between them, which will not only assist us in establishing a bridge between
theory and observation to better constrain the EOSs, but also to infer the
theoretical value of $L$ from frequency. In light of this, we introduce the
nucleon-nucleon SRC in the interior of admixed DM neutron stars and rebuilt
the EOSs that not only satisfy the properties of nuclear matter at saturation
density, but also satisfy the constraints from multi-messenger observation.
Based on this, we examine the impact of $L$ on the NS radial and non-radial
oscillations, build their relationship and predict their frequency ranges.

This article is organized as follows: Section II provides a basic introduction
to theoretical models of SRC and DM in NSs. Section III discusses in detail
the parameter construction that meets the constraints from saturation nuclear
behaviors and multi-messenger observations. Section IV is devoted to
investigate the implication of $L$ on the NS oscillation frequency, and
Section V provides a brief summary and remark. Finally, Appendix shows a
comprehensive derivation of relativistic nucleon coupling parameters by
incorporating the SRC effect.

##  II Relativistic mean field theory with Short-Range correlation and Dark
Matter Admixed

In this paper, we adopt relativistic mean field theory (RMF) with
$\sigma\omega\rho$ model. Based on this framework, we absorb the SRC and
introduce the DM to give the SRC-DM-revised RMF. In the mean-field
approximation, the energy density and pressure are expressed as follows:

$\displaystyle\mathcal{E}=\sum_{N}\langle\psi_{N}^{+}i\dot{\psi}_{N}\rangle+\sum_{l}\langle\psi_{l}^{+}i\dot{\psi}_{l}\rangle+\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}-\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}$
$\displaystyle-\frac{1}{2}m_{\varrho}^{2}\varrho_{0}^{2}-\Lambda_{\omega}\left(g_{\varrho
N}\varrho_{0}\right)^{2}\left(g_{\omega
N}\omega_{0}\right)^{2}+\frac{1}{3}g_{2}\sigma_{0}^{3}+\frac{1}{4}g_{3}\sigma_{0}^{4}$

and

$\displaystyle
P=\frac{1}{3}\sum_{N}\langle\psi_{N}^{+}(-i\alpha\cdot\nabla)\psi_{N}\rangle+\frac{1}{3}\sum_{l}\langle\psi_{l}^{+}(-i\alpha\cdot\nabla)\psi_{l}\rangle$
$\displaystyle-\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\varrho}^{2}\varrho_{0}^{2}+\Lambda_{\omega}\left(g_{\varrho
N}\varrho_{0}\right)^{2}\left(g_{\omega N}\omega_{0}\right)^{2}$
$\displaystyle-\frac{1}{3}g_{2}\sigma_{0}^{3}-\frac{1}{4}g_{3}\sigma_{0}^{4}$
(2)

where $N$ and $l$ denote the nucleon $N(n,p)$ and lepton $l(e,\mu)$ degrees of
freedom, and $m_{\sigma}$,$m_{\omega}$,$m_{\varrho}$ correspond to the mass of
$\sigma,\omega,\rho$. Coupling parameters $g_{\sigma N},g_{\omega
N},g_{\varrho N},g_{2},g_{3},\Lambda_{\omega}$ need to be further revised by
incorporating SRC and DM and the detailed analytical derivation have been
attached in the Appendix. For DM component, here we adopt the lightest
neutralino as the Fermi DM candidate with a mass of $m_{\chi}$=200 GeV.
Moreover, DM and nucleons do not interact directly but through coupled Higgs
fields, with the following Lagrangian form (Panotopoulos and Lopes, 2017a):

$\displaystyle\mathcal{L}_{\mathrm{DM}}=$
$\displaystyle\bar{\chi}\left[i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(3)
$\displaystyle-\frac{1}{2}M_{h}^{2}h^{2}+\sum_{N}f\frac{m_{N}}{v}\bar{\psi}_{N}h\psi_{N},$

where $\chi$, $h$ denote the DM and Higgs field, and $y$ is the coupling
strength between DM and Higgs field which usually takes values between 0.001
and 0.1, here we adopt the generally used value of 0.07 (Murakami and Wells,
2001; Panotopoulos and Lopes, 2017a). $M_{h}$ refers to the mass of the Higgs
boson, which is 125 GeV. $f\frac{m_{N}}{v}$ is the effective Yukawa coupling
strength between nucleon and Higgs field, where $v$ is the Higgs vacuum
expectation value with $v=246$ GeV and $f$ is the Higgs-nucleon formation
factor. Based on the lattice computations, we choose the optimal value of $f$
as 0.3, which is not only consistent with theoretical result (Cline  _et al._
, 2013) but also satisfies the experimental constraints of PandaX-II (Tan  _et
al._ , 2016), PandaX-4T (Meng  _et al._ , 2021) for the DM-nucleon scattering
cross section.

Finally, we assume that the DM number density is about 1000 times lower than
the number density of nucleons, and if we take the typical nuclear saturation
number density $n_{0}=0.16$ fm-3, the calculated DM Fermi energy
$k_{F}^{\mathrm{DM}}$ is about 0.033 GeV. Applying the mean field
approximation, the effective mass of nucleons and DM and its scalar density
are given as,

$\displaystyle
M_{N}^{\star}=m_{N}-g_{\sigma}\sigma_{0}-\frac{fm_{N}}{v}h_{0},$ (4)
$\displaystyle M_{\chi}^{\star}=m_{\chi}-yh_{0},$ (5)
$\displaystyle\rho_{s}^{\mathrm{DM}}=\frac{M_{\chi}^{*}}{\pi^{2}}\int_{0}^{k_{F}^{\mathrm{DM}}}\frac{k^{2}\mathrm{d}k}{\left(k^{2}+M_{\chi}^{*2}\right)^{1/2}},$
(6)

while the energy density and pressure of dark matter are

$\displaystyle\mathcal{E}_{\mathrm{DM}}=\frac{1}{\pi^{2}}\int_{0}^{k_{F}^{\mathrm{DM}}}k^{2}\mathrm{d}k\sqrt{k^{2}+\left(M_{\chi}^{\star}\right)^{2}}+\frac{1}{2}M_{h}^{2}h_{0}^{2},$
(7)

and

$\displaystyle
P_{\mathrm{DM}}=\frac{1}{3\pi^{2}}\int_{0}^{k_{F}^{\mathrm{DM}}}\frac{k^{4}\mathrm{d}k}{\sqrt{k^{2}+\left(M_{\chi}^{\star}\right)^{2}}}-\frac{1}{2}M_{h}^{2}h_{0}^{2}.$
(8)

Both $\langle\psi_{N}^{+}i\dot{\psi_{N}}\rangle$ and
$\langle\psi_{N}^{+}(-i\alpha\cdot\nabla)\psi_{N}\rangle$ in the Eqs. (1) and
(2) depend directly on the momentum distribution of the nucleon in the mean-
field approximation, and therefore also on the SRC. Recent theoretical and
experimental studies indicate that the short-range correlation produces a
single nucleon momentum distribution that approximately satisfies the $\sim
1/k^{4}$ distribution near the Fermi surface (Wang  _et al._ , 2017; Hen  _et
al._ , 2015a; Cai and Li, 2015). The $1/k^{4}$ functional form has been shown
to provide a good fit to experimental data on nucleon momentum distributions,
and it is insensitive to the specific details of the nucleon-nucleon
interaction and the nuclear wave function, making it a versatile tool in the
study of nucleon momentum distributions. The form can be applied in a range of
contexts, including electron scattering experiments (Rohe  _et al._ , 2004;
Onderwater  _et al._ , 1998; Starink  _et al._ , 2000). Although the
parameterized model mentioned above can well characterize the high-momentum
tail caused by nucleon short-range correlations and is consistent with
experimental results, there is still one thing to point out that the
correlation percentage between nucleons and the form of high-momentum
distribution caused by SRCs still depend on the model, and different models
also have their own advantages in characterizing SRC, for example, the Green’s
function method Dickhoff and Van Neck (2008) provides a powerful framework for
describing nucleon SRCs in nuclear systems, and can help to shed light on the
physics underlying these phenomena. In contrast, the $1/k^{4}$ form has the
advantage of portraying a high-momentum tail on the one hand in its simple
form, and on the other hand, it can qualitatively give similar high-momentum
tail calculated using microscopic self-consistent Green’s function (SCGF)
theory (Rios  _et al._ , 2014; Rios, 2020). Moreover, because of its
simplicity, the $1/k^{4}$ form has also been widely used in understanding
nuclear matter properties (Wang  _et al._ , 2017; Sammarruca, 2014; Cai and
Li, 2015; Hen  _et al._ , 2015a; Wang  _et al._ , 2017; Guo  _et al._ , 2021)
and NS prosperties (Souza  _et al._ , 2020a; Hong  _et al._ , 2022; Li  _et
al._ , 2019; Lu  _et al._ , 2021; Lourenço  _et al._ , 2022).

For asymmetry nuclear matter, based on the microscopic theoretical model
predicting an approximate linear correlation between isospin-asymmetry and
single-nucleon distribution, we can express the nucleon momentum distribution
in the following form:

$f(k)_{p}=\left\\{\begin{array}[]{lll}C_{1}^{(p)},&k\leq k_{F}^{(p)}\\\
\frac{C_{2}^{(p)}(1-\beta)}{k^{4}},&k_{F}^{(p)}<k\leq\lambda k_{F}^{(p)}\\\
0,&k>\lambda k_{F}^{(p)}\\\ \end{array}\right.$ (9)

$f(k)_{n}=\left\\{\begin{array}[]{ll}C_{1}^{(n)},&k\leq k_{F}^{(n)}\\\
\frac{C_{2}^{(n)}(1+\beta)}{k^{4}},&k_{F}^{(n)}<k\leq\lambda k_{F}^{(n)}\\\
0,&k>\lambda k_{F}^{(n)}\end{array}\right.$ (10)

where $\beta$ is the isospin-asymmetry, and $k_{F}$ and $\lambda$ are the
Fermi momentum and high-momentum cutoff, respectively. According to
experiments with proton-knockout reactions utilizing megaelectron-energy
electron beams, around $20\%\sim 25\%$ of the nucleons have high momentum
distribution (Subedi  _et al._ , 2008). Furthermore, the deuteron momentum
distribution suggests that the cutoff $\lambda$ should be around $2.75\pm
0.25$ (Hen  _et al._ , 2015b, a). In the next discussion, we take the cutoff
to be 2.75, while taking high-momentum ratio to be typically 25%. Combining
the normalization conditions, we can fix the values of $C_{1}^{(n,p)}$,
$C_{2}^{(n,p)}$, and they are $C_{1}^{(p)}=1-0.25(1-\beta)$,
$C_{2}^{(p)}=0.25k_{F}^{(p)4}/(3-3/\lambda)$, $C_{1}^{(n)}=1-0.25(1+\beta)$
and $C_{2}^{(n)}=0.25k_{F}^{(n)4}/(3-3/\lambda)$ respectively. After
incorporating results into Eqs. (1) and (2), the SRC-revised energy density
and pressure are derived as follows:

$\displaystyle\mathcal{E}_{\mathrm{SRC}}=\sum_{N}\frac{1}{\pi^{2}}\int_{0}^{k_{N}}\mathrm{~{}d}kk^{2}\sqrt{k^{2}+m^{*2}}+\sum_{l}\frac{1}{\pi^{2}}\int_{0}^{k_{l}}\mathrm{d}kk^{2}\sqrt{k^{2}+m_{l}^{2}}+\sum_{N}\frac{1}{3\pi^{2}}k_{N}^{3}\left(g_{\omega
N}\omega_{0}+g_{\varrho N}I_{3N}\varrho_{0}\right)$
$\displaystyle-\frac{0.25(1-\beta)}{\pi^{2}}\int_{0}^{k_{F}^{(n)}}\mathrm{d}kk^{2}\left(g_{\omega
n}\omega_{0}+g_{\varrho
n}I_{3n}\varrho_{0}+\sqrt{k^{2}+m^{*2}}\right)-\frac{0.25(1+\beta)}{\pi^{2}}\int_{0}^{k_{F}^{(p)}}\mathrm{d}kk^{2}\left(g_{\omega
p}\omega_{0}+g_{\varrho p}I_{3p}\varrho_{0}+\sqrt{k^{2}+m^{*2}}\right)$
$\displaystyle+\frac{C_{2}^{(n)}(1-\beta)}{\pi^{2}}\int_{k_{F}^{(n)}}^{\lambda
k_{F}^{(n)}}\mathrm{d}kk^{2}\frac{\left(g_{\omega n}\omega_{0}+g_{\varrho
n}I_{3n}\varrho_{0}+\sqrt{k^{2}+m^{*2}}\right)}{k^{4}}+\frac{C_{2}^{(p)}(1+\beta)}{\pi^{2}}\int_{k_{F}^{(p)}}^{\lambda
k_{F}^{(p)}}\mathrm{d}kk^{2}\frac{\left(g_{\omega p}\omega_{0}+g_{\varrho
p}I_{3p}\varrho_{0}+\sqrt{k^{2}+m^{*2}}\right)}{k^{4}}$
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}-\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}-\frac{1}{2}m_{\varrho}^{2}\varrho_{0}^{2}-\Lambda_{\omega}\left(g_{\varrho
N}\varrho_{0}\right)^{2}\left(g_{\omega
N}\omega_{0}\right)^{2}+\frac{1}{3}g_{2}\sigma_{0}^{3}+\frac{1}{4}g_{3}\sigma_{0}^{4}$

and

$\displaystyle
P_{\mathrm{SRC}}=\frac{1}{3}\sum_{N}\frac{1}{\pi^{2}}\int_{0}^{k_{N}}\mathrm{~{}d}k\frac{k^{4}}{\sqrt{k^{2}+m^{*2}}}+\frac{1}{3}\sum_{l}\frac{1}{\pi^{2}}\int_{0}^{k_{l}}\mathrm{d}k\frac{k^{4}}{\sqrt{k^{2}+m_{l}^{2}}}-\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\varrho}^{2}\varrho_{0}^{2}+\Lambda_{\omega}\left(g_{\varrho
N}\varrho_{0}\right)^{2}\left(g_{\omega N}\omega_{0}\right)^{2}$
$\displaystyle-\frac{1}{3}g_{2}\sigma_{0}^{3}-\frac{1}{4}g_{3}\sigma_{0}^{4}-\frac{1}{3}\frac{0.25(1-\beta)}{\pi^{2}}\int_{0}^{k_{F}^{(n)}}\mathrm{d}k\frac{k^{4}}{\sqrt{k^{2}+m^{*2}}}+\frac{C_{2}^{(n)}(1-\beta)}{3\pi^{2}}\int_{k_{F}^{(n)}}^{\lambda
k_{F}^{(n)}}\mathrm{d}k\frac{k^{4}}{k^{4}\sqrt{k^{2}+m^{*2}}}$
$\displaystyle-\frac{1}{3}\frac{0.25(1+\beta)}{\pi^{2}}\int_{0}^{k_{F}^{(p)}}\mathrm{d}k\frac{k^{4}}{\sqrt{k^{2}+m^{*2}}}+\frac{C_{2}^{(p)}(1+\beta)}{3\pi^{2}}\int_{k_{F}^{(p)}}^{\lambda
k_{F}^{(p)}}\mathrm{d}k\frac{k^{4}}{k^{4}\sqrt{k^{2}+m^{*2}}}$ (12)

Then the total energy density and pressure, revised jointly by admixing DM as
well as SRC, are expressed as

$\displaystyle\mathcal{E}=\mathcal{E}_{\mathrm{DM}}+\mathcal{E}_{\mathrm{SRC}},~{}~{}~{}P=P_{\mathrm{DM}}+P_{\mathrm{SRC}}.$
(13)

As a part of standard NS calculation scheme, in the NS outer crust, where the
density is around $6.3\times 10^{-12}\mathrm{fm}^{-3}\leqslant n\leqslant
2.46\times 10^{-4}\mathrm{fm}^{-3}$, we employ the Baym-Pethick-Sutherland
(BPS) EOS (Baym  _et al._ , 1971). For NS inner crust area with a density of
$2.46\times 10^{-4}\mathrm{fm}^{-3}\leqslant n\leqslant n_{t}$, we adopt the
polytropic parametrized EOSs of $P=a+b\varepsilon^{4/3}$(Cai and Li, 2016; Xu
_et al._ , 2009; Carriere  _et al._ , 2003), where $a$ and $b$ are related to
the core-crust transition $n_{t}$. In this study, we examine the possible weak
interaction between DM and ordinary baryonic matter, which occurs via the
exchange of Higgs bosons. When solving the hydrostatic Tolman-Oppenheimer-
Volkoff (TOV) equation, they resemble the degrees of freedom of hyperons
inside NSs, hence we can regard the mixed system of DM and baryonic matter as
a single-fluid system. Adopting the static spherically symmetric space-time
background given by

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (14)

with solving the Einstein equation, we can derive the Tolman-Oppenheimer-
Volkoff (TOV) equation (Oppenheimer and Volkoff, 1939)

$\frac{\mathrm{d}p}{\mathrm{d}r}=-\frac{(p+\epsilon)\left(M+4\pi
r^{3}p\right)}{r(r-2M)},\\\ $ (15) $\mathrm{d}M=4\pi
r^{2}\epsilon\mathrm{d}r,$ (16)

in which the corresponding metric functions express as

$e^{\lambda(r)}=(1-2m/r)^{-1},$ (17)
$\nu(r)=2\int_{r}^{\infty}dr^{\prime}\frac{e^{\lambda\left(r^{\prime}\right)}}{r^{\prime
2}}\left(m+4\pi r^{\prime 3}p\right).$ (18)

Then with the beta equilibrium and charge conservation condition, the NS mass
and radius can be determined.

Figure 1: The NS mass-radius relationships are calculated from four widely
used relativistic parameter sets, each of which considers four cases,
corresponding to the original version (black solid line), the SRC-revised
version (red solid line), the DM-revised version (black dashed line), and both
the SRC and DM-revised version (red dashed line). The three horizontal light-
colored bars indicate the three massive NSs MSP J0740+6620, PSR J0348+0432 and
PSR J1614-2230, the middle elliptical shaded region is the constraints given
by GW170817 and NICER, the vertical long shaded region is the mass-radius
constraint given by NICER in conjunction with XMM-Newton.

##  III The construction of parameters

To better visualize the implications of SRC and DM on NSs, we choose four
extensively used parameter sets to characterize the NS mass-radius relation,
namely NL3 (Lalazissis  _et al._ , 1997), FSUGold (Todd-Rutel and Piekarewicz,
2005), GM1 (Glendenning and Moszkowski, 1991) and NL$\rho$ (Liu  _et al._ ,
2002), as these sets can provide a good description of the properties of
ground state nuclei and can be extrapolated to study standard NS matter.
Moreover, as we mentioned before after the introduction of SRC in the mean
field, in addition to the revisions of energy density and pressure, the
coupling parameters between nucleons also need to be revised accordingly. The
revised results are shown in Fig.1, where the black solid line indicates the
mass-radius curve obtained from the original version, while the red solid line
indicates the result of SRC effect, which makes the EOSs stiffer due to the
additional correlation effect, being consistent with the conclusions from
(Souza  _et al._ , 2020a; Lu  _et al._ , 2021). The black dashed line
indicates the case of admixed DM, and it can be found that, compared to the
original parameters, the introduction of DM substantially softens the EOS. To
further compare with the astronomical observations, the constraints from
different groups are attached to the figure, where the three horizontal light-
colored bars indicate the three massive NSs (Demorest  _et al._ , 2010;
Arzoumanian  _et al._ , 2018; Antoniadis  _et al._ , 2013; Cromartie  _et al._
, 2020; Fonseca  _et al._ , 2021), the middle elliptical shaded region
indicates the constraints given by GW170817 (Abbott  _et al._ , 2017, 2018;
Annala  _et al._ , 2018; Most  _et al._ , 2018; De  _et al._ , 2018) and NICER
(Miller  _et al._ , 2019; Riley  _et al._ , 2019; Miller  _et al._ , 2021),
the vertical long shaded region indicates the mass-radius constraint given by
NICER in conjunction with XMM-Newton through Bayesian analysis (Riley  _et
al._ , 2021; Miller  _et al._ , 2021), in which the blue and gray region
represent the 75% and 90% confidence intervals, respectively. Unfortunately,
considering the SRC alone would largely increase the radius of NSs and thus
make it easier to deviate from the astronomical constraints, as in the case of
parameter NL3 and FSUGold, while considering admixed DM alone would make the
EOSs excessively soft, as in the case of parameter NL$\rho$. If the effects of
both SRC and DM are considered inside the NSs, as shown by the red dotted
line, their complementary mechanism makes the EOS curves easier to satisfy the
observational constraints. Although we cannot draw an absolute conclusion from
this due to the model dependence of the parameters, all four parameter sets
support our conclusion of having both SRC and DM effects in the NS interior.
Moreover, the SRC and admixed DM exhibit similar effects on the mass-radius
curves for all four sets, and in view of this, here we have chosen only GM1
for the next discussion. The values are shown in Table I, where the first
column is the original version for GM1 and the second column is the SRC and DM
revised version (GM1+SD). By adjusting the range of $L$ while maintaining the
other saturation properties, we investigate the impact of $L$ on the modified
version. Such an approach keep the isoscalar nature unchanged and can satisfy
the properties at the saturation density, and has been used in many works
(Souza  _et al._ , 2020a; Hornick  _et al._ , 2018). We extract $L$ located
around 50$\sim$90 MeV within a relatively credible interval based on previous
experimental and theoretical studies, and its corresponding modified versions
(GM1+SD+L) are shown in the second half of the Table I.

Table 1:  The original relativistic parameter set GM1 and its associated revised version, GM1+SD denotes the set revised by incorporating SRC and DM, and GM1+SD+L denotes the new parameters based on the GM1+SD by adjusting the isovector $L$ while keeping the isoscalar properties (saturation density $n_{0}$, incompressibility coefficient $K$, nucleon effective mass $m^{*}/m$, binding energy per nucleon $-B/A$ ) unchanged.  | GM1 | GM1+SD | GM1+SD+L50 | GM1+SD+L60 | GM1+SD+L70 | GM1+SD+L80 | GM1+SD+L90
---|---|---|---|---|---|---|---
$m_{N}(\textrm{MeV})$ | 939 | 939 | 939 | 939 | 939 | 939 | 939
$m_{\sigma}(\textrm{MeV})$ | 512 | 512 | 512 | 512 | 512 | 512 | 512
$m_{\omega}(\textrm{MeV})$ | 783 | 783 | 783 | 783 | 783 | 783 | 783
$m_{\varrho}(\textrm{MeV})$ | 770 | 770 | 770 | 770 | 770 | 770 | 770
$n_{0}(\textrm{fm}^{-3})$ | 0.153 | 0.153 | 0.153 | 0.153 | 0.153 | 0.153 | 0.153
$K(\textrm{MeV})$ | 300 | 300 | 300 | 300 | 300 | 300 | 300
$m^{*}/m$ | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70
$-B/A(\textrm{MeV})$ | 16.3 | 16.3 | 16.3 | 16.3 | 16.3 | 16.3 | 16.3
$E_{\textrm{sym}}(\textrm{MeV})$ | 32.5 | 32.5 | 32.5 | 32.5 | 32.5 | 32.5 | 32.5
$L(\textrm{MeV})$ | 94 | 94 | 50 | 60 | 70 | 80 | 90
$g_{\sigma}$ | 8.910 | 9.321 | 9.321 | 9.321 | 9.321 | 9.321 | 9.321
$g_{\omega}$ | 10.610 | 9.645 | 9.645 | 9.645 | 9.645 | 9.645 | 9.645
$g_{\varrho}$ | 8.196 | 13.566 | 14.934 | 14.587 | 14.263 | 13.960 | 13.675
$b$ | 0.0029 | 0.0029 | 0.004 | 0.004 | 0.004 | 0.004 | 0.004
$c$ | -0.001 | -0.001 | -0.003 | -0.003 | -0.003 | -0.003 | -0.003
$\Lambda_{\omega}$ | 0 | 0.002 | 0.011 | 0.009 | 0.007 | 0.005 | 0.003

As a next step, we need to decide a key question: whether the sets constructed
by different $L$ are able to describe empirical properties at low densities
while also fulfilling constraints from multi-messenger observations? To this
end, we have calculated the binding energy per nucleon (E/A) for symmetry
nuclear matter (SNM) and pure neutron matter (PNM), as shown in Fig.2. The
different color curves in the upper part indicate different $L$ for PNM, in
which below the saturation number density $n_{0}$ (corresponding to the black
vertical dashed line), the smaller $L$ gives a relatively large E/A, while the
opposite is true above $n_{0}$, and these results are in agreement with (Wu
_et al._ , 2021). Furthermore, in order to constrain the PNM behavior at
subsaturation density, we adopt the $\chi EFT$ given in (Tews  _et al._ ,
2013), which presents next-to-next-to-next-to-leading order (N3LO) in the
chiral expansion based on potentials developed by Epelbaum, Gl$\ddot{o}$ckle
and Meißner (EGM). As shown in the light green area (the lower right corner
corresponds to the enlarged version), our revised parameters are largely in
compliance with the constraints. Furthermore, it can be seen that $\chi EFT$
is more supportive of smaller $L$. The blue line shows the E/A for SNM, which,
unlike PNM, is clearly independent of a specific $L$ and has a minimum value
at $n_{0}$, corresponding to the most stable ground state, of about -16 MeV.

Figure 2: The energy per nucleon (E/A) as a function of number density in pure
neutron matter (PNM) and symmetry nuclear matter (SNM). The light green area
gives constraints from $\chi EFT$.  Figure 3: The symmetry energy as a
function of baryon number density for different $L$s.

Fig.3 shows the symmetry energy $E_{\mathrm{sym}}$ as a function of density
for different $L$. As can be seen, $E_{\mathrm{sym}}$ with a smaller $L$ have
a lower (higher) value than those with a larger $L$ at $n>n_{0}$ ($n<n_{0}$),
similar behavior is observed in (Hu  _et al._ , 2020; Wu  _et al._ , 2021).
Fig.4 gives the pressure of SNM, and we includes the experimental constraints
obtained from collective flow data in heavy-ion collisions which are often
used to constrain the EOS of SNM (Huang  _et al._ , 2020; Cavagnoli  _et al._
, 2011). The enclosed area represents experimental data according to
(Danielewicz  _et al._ , 2002), and our result is in line with this
constraint.

Figure 4: Pressure as a function of baryon number density for different $L$s
in SNM.

Apart from that, reasonable parameters also need to satisfy the constraints
from multi-messenger observations, including being able to generate massive
NSs, meeting the mass-radius constraint derived from NICER+XMM-Newton, and
satisfying the tidal deformability range given by GW170817. Fig.5 depicts the
mass-radiu relation, with different colors representing different $L$s and
with shaded regions showing the same constraints as in Fig.1. The influence
trend of the symmetry energy slope $L$ on the mass-radius curve can be
observed to be similar, where a larger $L$ can support a harder EOS and hence
give a larger radius. In addition, the range of $L$ we have selected, from the
lowest 50 MeV to the highest 90 MeV, can yield to the observational
constraints in both low-mass and massive NS regions. Specially, at low-mass of
$1.4M_{\odot}$, the radius varies from 11.5 km ($L$ = 50) to 12.5 km ($L$ =
90), and at massive mass of $2M_{\odot}$, the radius varies from 11.87 km ($L$
= 50) to 12.16 km ($L$ = 90). In addition to the above, the tidal
deformability-mass relations at different $L$, along with respective tidal
deformability of two NSs in GW170817 are given in Fig.6, where the vertical
line in the left panel indicates the tidal constraint inferred from GW170817,
the solid dot in the right panel represents the $1.4M_{\odot}$ NSs, and the
gray dashed lines indicate the confidence intervals of 50% and 90%,
respectively. As can be seen in Fig.6, a smaller $L$ gives a relatively
smaller tidal deformability, which can be inspired by Fig.4, where a smaller
$L$ yields a smaller radius, so a smaller radius makes it harder for a star to
distort in the tidal field, resulting in a smaller tidal value. More
importantly, we find that the tidal deformability obtained from the five sets
after absorbing SRC and DM all fall within the range given by the GW170817
event.

Figure 5: The mass-radius relations for different $L$s, with the horizontal
lines indicating the three massive NSs MSP J0740+6620, PSR J0348+0432 and PSR
J1614-2230. The shaded areas indicate the constraints from GW170817 as well as
NICER and XMM-Newton, respectively.  Figure 6: Left: Relationship between
tidal deformability and mass, with different colors indicating different $L$s.
Vertical line shows constraint from GW170817. Right: The relationship between
$\Lambda_{1}$ and $\Lambda_{2}$ in GW170817 event calculated with different
$L$, with solid dots indicating the $1.4M_{\odot}$ NSs. The gray dash-lines
indicate the 50% and 90% confidence intervals.

As a whole, after taking into account the SRC and admixed DM in NS interior,
the five parameter sets constructed by varying $L$ based on the GM1 can not
only characterize the behavior of nuclear matter at low densities (Figs.2-4),
but also meet the requirements inferred from recent multi-messenger
observations (Figs.5-6). Next, using these parameter sets, we investigate the
effect of $L$ on the currently undetectable NS oscillation frequencies.

##  IV Non-radial oscillation and radial oscillation

###  IV.1 Non-radial oscillation

Thorne and Campollataro (Thorne and Campolattaro, 1967) are credited with
being the ones who first suggested discussing non-radial modes within the
context of general relativity. In our work, we consider a non-rotating NS with
an ideal fluid interior, using the Cowling approximation (Cowling, 1941;
McDermott  _et al._ , 1983; Torres-Forné  _et al._ , 2017), which ignores the
space-time metric perturbation but retains density perturbations caused by
fluid oscillations (Kokkotas and Schmidt, 1999). Recent research indicates
that the difference between the $f$-mode calculated using the Cowling
approximation approach and the complete linearized equations of general
relativity is less than 20%, the $p$-mode error is about 10% (Yoshida and
Kojima, 1997), and the $g$-mode error is only a few percent (Sotani  _et al._
, 2001). This shows the practicality of Cowling approximation (Flores and
Lugones, 2014), and based on this, the fluid perturbations consist of a
spherical harmonic function $Y_{lm}(\theta,\phi)$ and a time-dependent part
eiωt, resulting in the following Lagrangian fluid displacements associated
with infinitesimal oscillatory perturbations:

$\displaystyle\zeta^{\mathrm{i}}=$
$\displaystyle{\left[\mathrm{e}^{-\Lambda(r)}W(r),-V(r)\partial_{\theta},-V(r)\sin^{-2}\theta\partial_{\phi}\right]r^{-2}}$
(19)
$\displaystyle\times\mathrm{Y}_{lm}(\theta,\phi)\mathrm{e}^{\mathrm{i}\omega
t},$

where $W(r)$ and $V(r)$ satisfy

$\displaystyle\frac{\mathrm{d}W(r)}{\mathrm{d}r}=\frac{\mathrm{d}\varepsilon}{\mathrm{d}p}\left[\omega^{2}r^{2}\mathrm{e}^{\Lambda(r)-2\phi(r)}V(r)+\frac{\mathrm{d}\Phi(r)}{\mathrm{d}r}W(r)\right]$
$\displaystyle-l(l+1)\mathrm{e}^{\Lambda(r)}V(r),$ (20)
$\displaystyle\frac{\mathrm{d}V(r)}{\mathrm{d}r}=2\frac{\mathrm{d}\Phi(r)}{\mathrm{d}r}V(r)-\frac{1}{r^{2}}\mathrm{e}^{\Lambda(r)}W(r),$

with

$\displaystyle\frac{\mathrm{d}\Phi(r)}{\mathrm{d}r}=\frac{-1}{\varepsilon(r)+p(r)}\frac{\mathrm{d}p}{\mathrm{~{}d}r}.$
(21)

the above equations can be viewed as the eigenvalue equations of $\omega$ for
suitable boundary conditions. Inside NS($r=0$), $W(r)$ and $V(r)$ exhibit the
following approximate behaviors:

$\displaystyle W(r)=Ar^{l+1},\quad V(r)=-\frac{A}{l}r^{l},$ (22)

with A being an arbitrary constant. The outer boundary condition is the
pressure will disappear at the NS surface

$\displaystyle\omega^{2}\mathrm{e}^{\Lambda(R)-2\Phi(R)}V(R)+\left.\frac{1}{R^{2}}\frac{\mathrm{d}\Phi(r)}{\mathrm{d}r}\right|_{r=R}W(R)=0.$
(23)

In this paper, we determine the typical non-radial bar mode instability for
quadrupole oscillations ($l$ = 2) (Passamonti  _et al._ , 2013) using the
above revised parameter sets, and show the $f$-mode frequencies under
different $L$s in Fig.7. If we take a closer look at the region roughly
between $1.4M_{\odot}$ and $2M_{\odot}$, a smaller $L$ is more likely to
excite $f$-mode oscillations, giving a larger frequency. The curve starts to
bend after approaching maximum mass, after which it corresponds to the
unstable NS. In the stable region, the frequency increases with mass, for
$1.4M_{\odot}$, different $L$s give frequencies in the range of 2.10 kHz-2.19
kHz, and for $2M_{\odot}$, in the range of 2.35 kHz-2.39 kHz. If future
gravitational wave detectors like the Cosmic Explorer or Einstein Telescope
(Perkins  _et al._ , 2021; Punturo  _et al._ , 2010; Dwyer  _et al._ , 2015)
are able to pick up the frequency in this range, we may have a good reason to
speculate about the approximate mass of this NS.

Figure 7: The $f$-mode frequency of non-radial oscillations as a function of
NS mass. We determine the typical non-radial bar mode instability for
quadrupole oscillations ($l$ = 2)  Figure 8: Left: Linear relationship between
$f$-mode frequency and $\sqrt{\frac{\bar{M}}{\bar{R}^{3}}}$ with different
$L$. Right: The relationship between $\omega M$ and compactness $M/R$.

In GW asteroseismology, we can relate the oscillation frequency and damping
timescale to NS bulk properties like mass, radius, and tidal deformability,
and then establish an empirical relationship between them (Doneva  _et al._ ,
2013; Benhar  _et al._ , 2004; Passamonti  _et al._ , 2013; Gaertig  _et al._
, 2011; Pradhan and Chatterjee, 2021; Wen  _et al._ , 2019). The original
study was proposed by Andersson and Kokkotas (Andersson and Kokkotas, 1996,
1998), who suggested that the stellar dynamical timescale is related to its
mean density and established a empirical relationship as

$\displaystyle f(\mathrm{kHz})=a+b\sqrt{\frac{\bar{M}}{\bar{R}^{3}}},$ (24)

where the dimensionless parameters $\bar{M}=M/(1.4M_{\odot})$ and
$\bar{R}=R/(10\mathrm{km})$. A further evaluation of this relationship was
carried out by taking into account the rotation effect (Doneva  _et al._ ,
2013) and exotic matter (Benhar  _et al._ , 2004) in the NS. However, these
studies are not supported by the current results of multi-messenger astronomy
and need further adjustment. Our goal is to re-calibrate this empirical
relationship by considering SRC and admixed DM inside NSs, with satisfying the
behaviours near saturation density and multi-messenger observations. The
specific results are shown on the left in Fig.8, where the different scatter
points indicate the results given by different $L$s and the solid green line
is the result of linear fitting. We present the fitting result in Table 2, as
a comparison, we also present the results by other researchers. It is found
that incorporating SRC and DM within NSs still maintains a good linear
relationship, and the discrepancies are very small compared to other models,
further illustrating the model-independent nature. Another similar scenario is
that of a excellent linear correlation between $\omega M$ and $M/R$, which has
been used to analyze $g$-mode, $p$-mode, and $f$-mode, and the fitting is
shown in the right of Fig.8 with $\omega M=208.687(M/R)-7.257$, which shows a
stronger linear correlation than the case on the left. These empirical
relations can be used to easily decode NS internal information once the
$f$-mode frequencies can be detected. On the one hand, the $f$-mode frequency
is capable of not only providing information on the NS density, but also
inferring the compactness ($M/R$) which further correlates to NS tidal
deformability. On the other hand, as the NICER detector are upgraded to
simultaneously give accurate masses and radii in the foreseeable future, we
can in turn be able to predict its $f$-mode frequency well. As mentioned,
since the compactness controls the tidal deformability directly, we can also
relate the $f$-mode frequency to the tidal deformability by extrapolating the
relationship between $f$-mode and compactness. For this purpose we plot
$\Lambda_{1.4}$ and $\Lambda_{2.0}$ with the corresponding $f$-mode
frequencies in Fig.9, where the dots in different shades indicate different
$L$s. It can be found that the larger $L$ is, the larger corresponding radius
is (see Fig.5 for details), and the NS is more likely to undergo a deformation
resulting in a larger tidal deformability (see Fig.6), but a larger radius is
more conducive to NS stability and thus more difficult to vibrate(see Fig.7),
so the larger $L$, the larger tidal deformability and the smaller $f$-mode
frequency. Furthermore, the frequency and tidal deformability show a good
linear correlation, and we have fitted with $f_{0}=-0.001\Lambda_{1.4}+2.386$
and $f_{0}=-0.001\Lambda_{2.0}+2.719$, and the range of $\Lambda_{1.4}$ and
$\Lambda_{2.0}$ extracted from different $L$ both satisfy the constraints from
GW170817 and theoretical calculations (De  _et al._ , 2018). This connection
not only explains the influence of $L$ on frequency and tidal deformability,
but it also predicts the observations. Together with the empirical relation
shown in Fig.8, the constraints on the EOSs will become increasingly rigorous.

Figure 9: Left and right show the linear relationship between the $f$-mode frequency and $\Lambda_{1.4}$ and $\Lambda_{2.0}$ respectively, where the different scatter points indicate the results given by different $L$s and the solid green line indicates the linear fitting.  Table 2:  The numerical fitting results data from Fig.8. The variables $a$ and $b$ stand for the intercept and slope in Eq.(20), respectively, and the first row, which highlights the data of this work. The results from other groups are also provided for comparison in the final four lines.  CASES | a(kHz) |  | b(kHz)
---|---|---|---
SRC and admixed DM | 1.0581 |  | 1.4507
Andersson and Kokkotas | 0.78 |  | 1.6350
(Andersson and Kokkotas, 1996) |  |  |
Omar Benhar et al. | 0.79 |  | 1.5000
(Benhar  _et al._ , 2004) |  |  |
Daniela D. Doneva et al. | 1.5620 |  | 1.1510
(Doneva  _et al._ , 2013) |  |  |
Bikram Keshari Pradhan et al. | 1.0750 |  | 1.4120
(Pradhan and Chatterjee, 2021) |  |  |
Figure 10: Left: The radial $F_{0}$-mode frequency as a function of NS mass
for different $L$s. Right: Relationship between radial $F_{0}$-mode frequency
and $\sqrt{\frac{\bar{M}}{\bar{R}^{3}}}$ for different $L$s.  Figure 11:  The
perturbations $\xi$ and $P$ of a NS with 1.4$M_{\odot}$ (upper panel) or
2$M_{\odot}$ (down panel) as a function of the dimensionless radial parameter
$r/R$ for different overtone numbers $n$=1,2,3,4. Different symmetry energy
slopes are denoted by thin lines in each overtone group.

###  IV.2 Radial oscillation

It has been shown that radial oscillations of NSs, accompanied by non-radial
oscillations(Passamonti  _et al._ , 2007), are a special class of non-radial
oscillations, which in astroseismology correspond to spherical harmonic
functions with zero orbital angular momentum. A seminal paper by Chandrasekhar
(Chandrasekhar, 1964) was the first to discuss radial perturbations of compact
stars and in the past 60 years, different work has been done to study the NSs’
radial oscillations under the general relativity (Glass and Lindblom, 1983;
Chanmugam, 1977; Passamonti  _et al._ , 2006, 2007; Di Clemente  _et al._ ,
2020; Sagun  _et al._ , 2020; Sun  _et al._ , 2021). Some studies have pointed
out that radial oscillation can be used to distinguish between traditional NSs
and strange stars (Vaeth and Chanmugam, 1992) as well as hybrid stars (Pereira
_et al._ , 2018; Sun  _et al._ , 2021), and can also often be used to describe
the NS stability. Despite the fact that radial oscillations cannot directly
produce gravitational radiation, they can couple and amplify gravitational
wave signals (Passamonti  _et al._ , 2006, 2007). As described in Ref.(Gondek
_et al._ , 1997), the differential equations that characterize infinitesimal
radial oscillations can be expressed as:

$\displaystyle\frac{\mathrm{d}\xi}{\mathrm{d}r}=-\frac{1}{r}\left(3\xi+\frac{\Delta
p}{\Gamma
p}\right)-\frac{\mathrm{d}p}{\mathrm{~{}d}r}\frac{\xi}{(p+\varepsilon)},$ (25)
$\displaystyle\frac{\mathrm{d}\Delta p}{\mathrm{~{}d}r}=$
$\displaystyle\xi\left\\{\omega^{2}\mathrm{e}^{\lambda-v}(p+\varepsilon)r-4\frac{\mathrm{d}p}{\mathrm{~{}d}r}\right\\}+\xi\left\\{\left(\frac{\mathrm{d}p}{\mathrm{~{}d}r}\right)^{2}\frac{r}{(p+\varepsilon)}\right.$
(26)
$\displaystyle\left.-8\pi\mathrm{e}^{\lambda}(p+\varepsilon)pr\right\\}+\Delta
p\left\\{\frac{\mathrm{d}p}{\mathrm{~{}d}r}\frac{1}{(p+\varepsilon)}\right.$
$\displaystyle\left.-4\pi(p+\varepsilon)r\mathrm{e}^{\lambda}\right\\}$

where $\omega$ is the radial eigenfrequency, $\xi$($\equiv\triangle r/r$) and
$\triangle p$ characterize the radial displacement and Lagrangian perturbation
of pressure respectively, both of which rely on the harmonic time form of
$e^{i\omega t}$. $\Gamma$ stands for the relativistic adiabatic index:

$\displaystyle\Gamma=\left(1+\frac{\varepsilon}{p}\right)\frac{dp}{d\varepsilon}$
(27)

Numerically solving above equations requires two boundary conditions, one is
that in the NS interior ($r$ = 0) the term associated with $1/r$ must be
finite (Gondek  _et al._ , 1997), so we have

$(\Delta p)_{\text{center }}=-3(\xi\Gamma p)_{\text{center }}$ (28)

with $\xi(0)=1$ for the normalization condition and the second one is that the
pressure perturbation near NS surface should become negligible, i.e. $\Delta
p\rightarrow 0$ for $r\rightarrow R$. Taking a given NS EOS as an input, the
above oscillation equations are Sturm-Liouville boundary value problems (Vaeth
and Chanmugam, 1992) and using the numerical shooting method we can get a
series of so-called eigenvalue frequencies
$\omega^{2}_{1}<\omega^{2}_{2}<...<\omega^{2}_{n}<...$, with $n$ being the
number of nodes.

Table 3:  Eleven sets of high-mode frequencies $\nu_{n}$ under different $L$s. The two sets of data in each column represent the 1.4$M_{\odot}$ and 2$M_{\odot}$ counterparts respectively  Overtone number |  |  $L$=50 MeV |  |  $L$=60 MeV |  |  $L$=70 MeV |  |  $L$=80 MeV |  |  $L$=90 MeV
---|---|---|---|---|---|---|---|---|---|---
$n$=1 |  | 3.095 (2.142) |  | 3.031 (2.113) |  | 2.952 (2.074) |  | 2.861 (2.031) |  | 2.777 (1.991)
$n$=2 |  | 7.885 (7.086) |  | 7.705 (7.031) |  | 7.498 (6.952) |  | 7.284 (6.842) |  | 7.090 (6.735)
$n$=3 |  | 12.141 (11.129) |  | 11.839 (11.029) |  | 11.507 (10.895) |  | 11.183 (10.715) |  | 10.888 (10.545)
$n$=4 |  | 16.288 (15.025) |  | 15.871 (14.877) |  | 15.417 (14.687) |  | 14.991 (14.441) |  | 14.594 (14.213)
$n$=5 |  | 20.395 (18.868) |  | 19.866 (18.671) |  | 19.294 (18.425) |  | 18.765 (18.114) |  | 18.256 (17.824)
$n$=6 |  | 24.482 (22.683) |  | 23.844 (22.438) |  | 23.153 (22.137) |  | 22.522 (21.768) |  | 21.906 (21.415)
$n$=7 |  | 28.558 (26.484) |  | 27.814 (26.192) |  | 26.999 (25.835) |  | 26.261 (25.401) |  | 25.516 (24.994)
$n$=8 |  | 32.626 (30.275) |  | 31.777 (29.936) |  | 30.829 (29.526) |  | 29.978 (29.031) |  | 29.066 (28.565)
$n$=9 |  | 36.688 (34.061) |  | 35.734 (33.676) |  | 34.638 (33.212) |  | 33.661 (32.657) |  | 32.521 (32.131)
$n$=10 |  | 40.746 (37.842) |  | 39.684 (37.412) |  | 38.417 (36.894) |  | 37.392 (36.280) |  | 35.815 (35.694)
$n$=11 |  | 44.798 (41.620) |  | 43.628 (41.145) |  | 42.158 (40.575) |  | 40.872 (39.901) |  | 38.895 (39.253)

Our next step is to examine how $L$ influences radial oscillations in terms of
fundamental radial frequencies ($F_{0}$) and consecutive frequency differences
(so-called Large Separation)$\triangle\nu_{n}=(\omega_{n}-\omega_{n-1})/2\pi$
(Lopes, I. P., 2001; Panotopoulos and Lopes, 2017b). The fundamental radial
frequencies $F_{0}$($n=0$) as a function of NS mass are given on the left of
Fig.10. The smaller $L$ exhibit larger radial frequency, similar to the result
from non-radial one, and the effect of $L$ on the frequency is more pronounced
in low mass regions, where the frequency drops from 3.11 kHz ($L=50$ MeV) to
2.76 kHz ($L=90$ MeV) for 1.4$M_{\odot}$ and from 2.16 kHz to 1.99 kHz for
2$M_{\odot}$. Furthermore, unlike non-radial oscillations, the radial
frequency decreases more rapidly at massive NS regions, and tends to zero when
the mass indefinitely approaches the maximum stable mass (MSM). This is
because when a NS reaches its MSM, any small radial perturbation does not
cause the star to oscillate. Due to this fact, the linear relationship between
the radial frequency and the mean density is no longer satisfied as it was for
the non-radial one (see the right of Fig.10). Moreover, we can apply radial
frequency equivalently to determine the star equilibrium condition, where a
stable NS condition requires $\partial M/\partial n_{c}\geq 0$ in the mass-
radius curve, in which $\partial M/\partial n_{c}=0$ gives the MSM,
corresponding to the point in figure where the frequency is zero. This is
important because the MSM obtained from the mass-radius curve can only be
given by a theoretical model calculation, whereas the radial oscillation
frequency allows the MSM to be determined from an observed perspective.
Combined with the results from non-radial oscillations together, we can infer
whether the NS has a MSM, as radial oscillations are less likely to occur near
the MSM (see Fig.10), while non-radial oscillations are more likely to produce
high frequencies (see Fig.7), which means that if a small radial frequency be
detected along with a high non-radial frequency, the NS is closer to its MSM.
This will be possible in the near future with the upgrade of high-sensitivity
GW detectors, and from this point of view, NS astroseismology could provide at
least one new way to determine the NS MSM.

Figure 12: Consecutive frequency differences (so-called Large Separation)
$\triangle\nu=(\omega_{n}-\omega_{n-1})/2\pi$ (in kHz) as a function of $\nu$.
Left and right panel consider 1.4$M_{\odot}$ and 2$M_{\odot}$ respectively, in
which different color dots represent different $L$s.  Figure 13: Left and
right show the linear relationship between the radial $F_{0}$-mode frequency
and $\Lambda_{1.4}$ and $\Lambda_{2.0}$ respectively, where the different
scatter points indicate the results given by different $L$s and the solid
green line is the linear fitting.

For high-mode oscillations with overtone numbers $n>0$, we examined the
implications of $L$ for their eigenfunctions. Fig.11 shows the first four
high-mode eigenfunctions, with $\xi$ shown in the left part and $P$ in the
right, and 1.4$M_{\odot}$ and 2$M_{\odot}$ correspond to upper and down panel,
respectively. The intersection of eigenfunction curve with dash-line
represents the overtone number, for example, $n$=1 indicates that there is
only one node. Furthermore, as the overtone numbers increase, the curve
fluctuations become more prominent and reflect a higher frequency. In the NS
interior, the different symmetry energy slopes (corresponding to different
lines under each overtone curve) have little impact on the high-mode
oscillation. It is essential to emphasize that the $F_{0}$-mode still remains
the most promising frequency band for detection since the larger overtone
numbers $n$ produce higher the frequencies, which would far exceed the
sensitivity of current detection devices, making it still difficult to detect.

The consecutive frequency differences or Large Separation (LS) can also be
extracted from the high-mode eigenfunctions, which is a measurable quantity
that has been used to identify the DM component(Panotopoulos and Lopes,
2017b). To further examine the possible implications from $L$, we calculated
11 consecutive high-mode oscillation frequencies listed in Table 3, and plot
the results in Fig.12, with the left and right parts showing 1.4$M_{\odot}$
and 2$M_{\odot}$ cases, respectively. It can be seen that the LS in both cases
is roughly around 4 kHz to 5 kHz, which is consistent with the study of
strange quark stars admixed DM (Panotopoulos and Lopes, 2017b). Moreover, LS
decreases as $L$ increases, where for 1.4$M_{\odot}$ the decrease in LG is
about 0.8 kHz or more as $L$ grows from 50 MeV to 90 MeV, while for
2$M_{\odot}$ LG decreases by approximately 0.4 kHz, which implies that the
effect of $L$ on LG is more pronounced for low-mass NS than for massive NS, as
also implied from the previous discussion on this point (see Figs.5-6).
Finally, as discussed in section 4.1, the non-radial oscillations give a very
strong linear correlation between frequency and tidal deformability for
different $L$s, and as an analogy we also give results for the radial ones,
apart from the different frequency values, 1.4$M_{\odot}$ and 2$M_{\odot}$
also satisfy a fairly good linear relationship, with the fitted results being
$F_{0}=-0.004\Lambda_{1.4}+3.769$ and $F_{0}=-0.062\Lambda_{2.0}+3.032$ shown
in Fig.13 respectively. These results together with the non-radial
oscillations will be more helpful in constraining EOSs for NSs.

##  V Summary

In this study, we absorb the nucleon-nucleon SRC and admixed DM effect in NS
interior, adopting the widely accepted SRC model and assuming the DM component
to be one thousandth of the conventional nucleon number density. As a result
of incorporating SRC and DM, we re-derive the pressure and energy density at
the relativistic mean field theory and revise the corresponding nucleon
coupling parameters. It is found that the simultaneous introduction of SRC and
DM enables NS to better satisfy the observational constraints, and on this
basis a typical set of parameters GM1 is selected to evaluate the impact of
$L$ on the oscillation frequency of NSs with incorporating SRC and DM. For the
purpose of verifying the feasibility of sets constructed from different $L$s,
we calculated its behavior at low and high densities and discovered that it
not only fulfills the properties near the saturation nuclear matter density
well, but also meets the astronomical constraints. The following are the
important physical findings of this study.

1\. For non-radial oscillations, absorbing SRC and DM, the smaller $L$ is more
likely to excite $f$-mode for stable NSs, thus giving larger frequencies.

2\. After absorbing SRC and DM, we updated the empirical relationship between
frequency and mean density with $f=1.0581+1.4507\sqrt{\bar{M}/\bar{R}^{3}}$,
and the relationship between $\omega M$ and the compactness parameter with
$\omega M=208.687(M/R)-7.257$, these relations can be used to decode NS
internal information from the $f$-mode frequencies.

3\. Our results indicate that a larger $L$ results in a larger tidal
deformability and a smaller non-radial frequency. Furthermore, the frequency
and tidal deformability show a good linear correlation, corresponding to
$f_{0}=-0.001\Lambda_{1.4}+2.386$ and $f_{0}=-0.001\Lambda_{2.0}+2.719$ for
1.4$M_{\odot}$ and 2$M_{\odot}$.

4\. For fundamental radial oscillations, the smaller $L$ exhibit larger radial
frequency, and the effect of $L$ on the frequency is more pronounced in low
mass regions, where the frequency drops from 3.11 kHz to 2.76 kHz for
1.4$M_{\odot}$ and from 2.16 kHz to 1.99 kHz for 2$M_{\odot}$ as $L$ changes
from 50 MeV to 90 MeV. Moreover, with the features of non-radial frequencies,
we can provide a methodology for estimating the maximum stable mass of NS.

5\. Finally, we evaluate the effect of $L$ on the high-mode eigenfunction as
well as LG, and find that the $L$ has little impact on the high-mode
oscillation and the effect on LG is more pronounced for low-mass NS than for
massive NS. We also found that the frequency of radial oscillations also
satisfies a strong linear relationship with tidal deformability, for which we
fitted this relationship at 1.4$M_{\odot}$ and 2$M_{\odot}$, corresponding to
$F_{0}=-0.004\Lambda_{1.4}+3.769$ and $F_{0}=-0.062\Lambda_{2.0}+3.032$,
respectively.

The above findings establish a link between $L$ and NS oscillation frequency,
which allows us to understand how the $L$ affects the frequency and on other
NS properties, with the continuous improvement of observational equipments, it
is anticipated that this band frequency will be observed in the future, which
in turn can be used to constrain $L$.

###### Acknowledgements.

Bin thanks the anonymous referee for his or her kind guidance. This work is
supported by the National Natural Science Foundation of China ( Grants
No.12035011, No.11975167, No.11961141003, No.11905103, No.11761161001,
No.11375086), and by the National Key R&D Program of China (Contracts No.
2018YFA0404403 and No.2016YFE0129300).

## Data Availability Statement

This manuscript has no associated data or the data will not be deposited.
[Authors’ comment: All results are shown in figures or table and are clearly
given. The experimental data we cited can be obtained from references.]

## appendix

In this study within SRC-revised RMFT, in addition to the energy density and
pressure that need to be revised, the six nucleon coupling parameters of
$g_{\sigma N},~{}g_{\omega N},~{}g_{\rho N},~{}b,~{}c,\Lambda_{\omega}$ also
need to be revised, which rely on the saturation number density $n_{0}$,
binding energy per nucleon $B/A$, incompressibility coefficient $K$ and
nucleon effective mass $m^{*}$. Next we will re-derive their connection by
considering SRC effect. Given that the sum of binding energy per nucleon $B/A$
and nucleon mass $m_{B}$ determine energy per nucleon $E/A$, i.e.
$E/A=B/A+m_{B}$, thus we can yield

$\displaystyle\frac{B}{A}+m_{B}=(\frac{g_{\omega}}{m_{\omega}})^{2}n_{0}+C_{1}\sqrt{k_{F}^{2}+m^{*2}}$
$\displaystyle+\frac{C_{2}\lambda^{3}\sqrt{(\lambda
k_{F})^{2}+m^{*2}}}{(\lambda
k_{F})^{4}}-\frac{C_{2}\sqrt{k_{F}^{2}+m^{*2}}}{k_{F}^{4}}$
$\displaystyle+\frac{1}{k_{F}^{2}}\frac{\partial C_{2}}{\partial
k_{F}}\int_{k_{F}}^{\lambda
k_{F}}\frac{k^{2}\sqrt{k^{2}+m^{*2}}}{k^{4}}\mathrm{d}k,$ (29)

where $m^{*}$ stands for the nucleon effective mass, and since the isospin-
asymmetry $\beta=0$ for saturation density, we get
$C_{1}^{(n)}=C_{1}^{(p)}=C_{1}$, $C_{2}^{(n)}=C_{2}^{(p)}=C_{2}$,
$k_{F}^{(n)}=k_{F}^{(p)}=k_{F}$. As described in RMFT, the isoscalar-scalar
meson $\sigma$ meet

$\displaystyle\left(\frac{m_{\sigma}}{g_{\sigma}}\right)^{2}\left(m_{B}-m^{*}\right)+m_{B}\left(m_{B}-m^{*}\right)^{2}b+c\left(m_{B}-m^{*}\right)^{3}$
$\displaystyle=2\Bigg{[}\frac{C_{1}}{\pi^{2}}\int_{0}^{k_{F}}\frac{m^{*}k^{2}}{\sqrt{k^{2}+m^{*2}}}\mathrm{~{}d}k$
$\displaystyle+\frac{C_{2}}{\pi^{2}}\int_{k_{F}}^{\lambda
k_{F}}\frac{m^{*}k^{2}}{k^{4}\sqrt{k^{2}+m^{*2}}}\mathrm{~{}d}k\Bigg{]},$ (30)

and energy density satisfies as

$\displaystyle\frac{1}{2}\left(m_{B}-m^{*}\right)^{2}\left(\frac{m_{\sigma}}{g_{\sigma}}\right)^{2}+\frac{1}{3}m_{B}\left(m_{B}-m^{*}\right)^{3}b$
$\displaystyle+\frac{1}{4}\left(m_{B}-m^{*}\right)^{4}c+\frac{1}{2}\left(\frac{g_{\omega}}{m_{\omega}}\right)^{2}n_{0}^{2}$
$\displaystyle=\left(B/A+m_{B}\right)n_{0}-2\times\Bigg{[}\frac{C_{1}}{\pi^{2}}\int_{0}^{k_{F}}k^{2}\sqrt{k^{2}+m^{*2}}\mathrm{d}k$
$\displaystyle+\frac{C_{2}}{\pi^{2}}\int_{k_{F}}^{\lambda
k_{F}}\frac{k^{2}\sqrt{k^{2}+m^{*2}}}{k^{4}}\mathrm{d}k\Bigg{]}.$ (31)

At saturation density, the incompressibility coefficient reads as

$\displaystyle
K=(\frac{g_{\omega}}{m_{\omega}})^{2}\cdot\frac{6k_{F}^{3}}{\pi^{2}}+\Bigg{(}AA+BB+CC\Bigg{)}^{{}^{\prime}k}3k_{F}$
$\displaystyle-\Bigg{(}AA+BB+CC\Bigg{)}^{{}^{\prime}m*}\frac{6k_{F}\left\langle\bar{\psi}\psi\right\rangle^{{}^{\prime}k}}{\Bigg{[}(\frac{m_{\sigma}}{g_{\sigma}})^{2}+2\left\langle\bar{\psi}\psi\right\rangle^{{}^{\prime}m*}-\frac{d^{2}U}{d\sigma_{0}^{2}}\frac{1}{g_{\sigma}^{2}}\Bigg{]}},$

where prime refers to the partial derivative of momentum $k$ and nucleon
effective mass $m^{*}$, and $AA$,$BB$,$CC$ and
$\left\langle\bar{\psi}\psi\right\rangle$ read as

$\displaystyle AA$ $\displaystyle=C_{1}\sqrt{k_{F}^{2}+m^{*2}},$ (33)
$\displaystyle BB$ $\displaystyle=\frac{C_{2}\lambda^{3}\sqrt{(\lambda
k_{F})^{2}+m^{*2}}}{(\lambda
k_{F})^{4}}-\frac{C_{2}\sqrt{k_{F}^{2}+m^{*2}}}{k_{F}^{4}},$ (34)
$\displaystyle CC$
$\displaystyle=\frac{C^{\prime}_{2}}{k_{F}^{2}}\int_{k_{F}}^{\lambda
k_{F}}\frac{k^{2}\sqrt{k^{2}+m^{*2}}}{k^{4}}\mathrm{d}k,$ (35)
$\displaystyle\left\langle\bar{\psi}\psi\right\rangle$
$\displaystyle=\frac{2C_{1}}{\pi^{2}}\int_{0}^{k_{F}}\frac{m^{*}k^{2}}{\sqrt{k^{2}+m^{*2}}}\mathrm{~{}d}k+\frac{2C_{2}}{\pi^{2}}\int_{k_{F}}^{\lambda
k_{F}}\frac{m^{*}k^{2}}{k^{4}\sqrt{k^{2}+m^{*2}}}\mathrm{~{}d}k.$

Moreover, we can correlate the isovector coupling parameters $g_{\varrho}$ and
$\Lambda_{\omega}$ with symmetry energy $E_{\text{sym}}$ and its slope $L$,
for symmetry energy at saturation density, it can be divided into two parts
and reads as

$\displaystyle
E_{\mathrm{sym}}(n_{0})=\frac{1}{2}\frac{\partial^{2}(E/A)}{\partial\beta^{2}}\Bigg{|}_{\beta=0}=\frac{1}{2}\left[\frac{\partial^{2}(\epsilon/n)}{\partial\beta^{2}}\right]_{\beta=0}$
$\displaystyle=E_{1\mathrm{sym}}(n_{0})+E_{2\mathrm{sym}}(n_{0}),$ (37)

where two parts correspond to

$\displaystyle
E_{1\mathrm{sym}}(n_{0})=\frac{k_{0}^{3}}{12\pi^{2}}\frac{g_{\varrho}^{2}}{m_{\varrho}^{2}+2\Lambda_{\omega}\left(g_{\omega}\omega_{0}\right)^{2}g_{\varrho}^{2}}$
(38)

and

$\displaystyle
E_{2\mathrm{sym}}(n_{0})=\frac{k_{0}^{2}}{8\sqrt{k_{0}^{2}+m^{*2}}}$
$\displaystyle+\frac{1}{4}\sqrt{k_{0}^{2}+m^{*2}}+\Bigg{[}(\frac{C^{\prime}_{2}}{2k_{0}^{2}})^{\prime}\cdot\frac{k_{0}}{3}-\frac{C^{\prime}_{2}}{k_{0}^{2}}\Bigg{]}\int_{k_{0}}^{\lambda
k_{0}}\mathrm{~{}d}k\frac{k^{2}\sqrt{k^{2}+m^{*2}}}{k^{4}}$
$\displaystyle+\bigg{(}\frac{C^{\prime}_{2}k_{0}}{3}-C_{2}\bigg{)}\bigg{(}\frac{\lambda^{3}\sqrt{\left(\lambda
k_{0}\right)^{2}+m^{*2}}}{(\lambda
k_{0})^{4}}-\frac{\sqrt{k_{0}^{2}+m^{*2}}}{k_{0}^{4}}\bigg{)}$
$\displaystyle+\frac{k_{0}C_{2}}{6}\bigg{(}\frac{\lambda^{3}\sqrt{\left(\lambda
k_{0}\right)^{2}+m^{*2}}}{(\lambda
k_{0})^{4}}-\frac{\sqrt{k_{0}^{2}+m^{*2}}}{k_{0}^{4}}\bigg{)}^{\prime}.$ (39)

While, for symmetry energy slope $L$, it also can be divided into two parts
accordingly

$\displaystyle L(n_{0})=$ $\displaystyle 3n_{0}\frac{\partial
E_{\mathrm{sym}}}{\partial n}\Bigg{|}_{n=n_{0}}=L_{1}(n_{0})+L_{2}(n_{0}),$
(40)

where $L_{1,2}$ represent as

$\displaystyle
L_{1}(n_{0})=3E_{3\mathrm{sym}}\Bigg{[}1-32\frac{g_{\omega}^{2}}{m_{\omega}^{2}}\times
g_{\omega}\omega\Lambda_{\omega}\times E_{3\mathrm{sym}}\Bigg{]}$ (41)

and

$\displaystyle L_{2}(n_{0})=k(E_{2\mathrm{sym}})^{{}^{\prime}k}$
$\displaystyle-\frac{2k(E_{2\mathrm{sym}})^{{}^{\prime}m*}\langle\bar{\psi}\psi\rangle^{\prime
k}}{\Bigg{[}(\frac{m_{\sigma}}{g_{\sigma}})^{2}+2\langle\bar{\psi}\psi\rangle^{\prime
m*}+2bm(m-m^{*})+3c(m-m^{*})^{2}\Bigg{]}}.$

At this point, it is clear that the six nucleon coupling parameters under the
SRC-revised RMFT are successfully associated with six saturation nuclear
parameters, and can be quantitatively solved.

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