#  $\eta$ mesons in hot and dense asymmetric nuclear matter

Rajesh Kumar<EMAIL_ADDRESS>Arvind Kumar<EMAIL_ADDRESS>Department
of Physics, Dr. B R Ambedkar National Institute of Technology Jalandhar,
Jalandhar – 144011,Punjab, India

###### Abstract

We study the $\eta N$ interactions in the hot and dense isospin asymmetric
nuclear matter using two different approaches. In the first approach, the in-
medium mass and optical potential of $\eta$-meson have been calculated in the
chiral SU(3) model, considering the effect of explicit symmetry breaking term
and range terms in the $\eta N$ interaction Lagrangian density. In the second
scenario, the conjunction of chiral perturbation theory and chiral SU(3) model
is employed. In this case, the next-to-leading order $\eta N$ interactions are
evaluated from the chiral perturbation theory (ChPT), and the in-medium
contribution of scalar densities are taken as input from chiral SU(3) model.
We observe a larger negative mass-shift in the ChPT+chiral model approach
compared to the chiral SU(3) model alone as a function of nuclear density.
Moreover, the increase in the asymmetry and temperature cause a decrease in
the magnitude of mass-shift. We have also studied the impact of $\eta N$
scattering length $a^{\eta N}$ on the $\eta$ meson mass $m^{*}_{\eta}$ and
observed that the $m^{*}_{\eta}$ decrease more for increasing the value of
scattering length.

##  I Introduction

The meson-baryons interactions are very imperative topic of research to study
the physics of non-perturbative QCD regime Tolos2020 ; Papazoglou1999 ;
Hayashigaki2000 ; Jenkins1991 ; Kaplan1986 ; Tolos2004 ; Tolos2006 ; Tolos2008
; Cieply2014 ; Zhong2006 ; Waas1997 ; Vogt2007 . The heavy ion-collisions
(HICs) are used to study the strong-interaction physics by colliding high
energy particles. As a byproduct of the collision, the Quark Gluon Plasma
(QGP) appears under the utmost conditions of density and temperature Vogt2007
. Afterward with the expansion of fireball the QGP cools down and changes its
phase to the hadronic matter through hadronization process Vogt2007 . These
two regimes $i.e.$ QGP phase and hadronic phase have different
characterization of the respective medium. For example, in the former phase
quarks and gluons act as a degree of freedom whereas in the latter, mesons and
baryons play this role. In QGP phase the chiral symmetry is followed
($m_{q}\sim 0$) but in hadronic phase it is broken explicitly ($m_{q}\neq 0$)
and spontaneously ($\langle\bar{q}q\rangle\neq 0$) Rapp2010 ; Vogt2007 .
Furthermore, in the hadronic ensemble, the thermodynamics quantities namely
nuclear density (number density of nucleons), isospin asymmetry (number of
neutrons vs the number of protons), and temperature also play a crucial role
to modify the in-medium properties of the mesons and baryons Papazoglou1999 ;
Tolos2020 ; Vogt2007 . The operation of future experimental facilities such as
CBM and PANDA at GSI, Germany, NICA at Dubna, Russia and, J-PARC at Japan may
lead to considerable progress in the understanding of meson-baryons
interactions Vogt2007 ; Rapp2010 .

On the theoretical side, several potential models have been theorized to study
the physics of the non-perturbative regime. Some of these are: Nambu-Jona-
Lasinio (NJL) model Nambu1961 , the Polyakov loop extended NJL (PNJL) model
Fukushima2004 ; Kashiwa2008 ; Ghosh2015 , chiral perturbation theory (ChPT)
Zhong2006 ; Jenkins1991 , coupled channel approach Tolos2020 ; Tolos2004 ;
Tolos2006 ; Tolos2008 ; Hofmann2005 , chiral $SU(3)$ model Papazoglou1999 ;
Mishra2004 ; Mishra2006 ; Kumar2010 ; Kumar2011 ; Kumar2019a ; Kumar2020b ;
Mishra2004a ; Mishra2009 ; Kumar2010 ; Kumar2019 ; Kumar2020 , Quark-Meson
Coupling (QMC) model Guichon1988 ; Hong2001 ; Tsushima1999 ; Sibirtsev1999 ;
Saito1994 ; Panda1997 , Polyakov Quark Meson (PQM) model Chatterjee2012 ;
Schaefer2010 , and QCD sum rules Reinders1981 ; Hayashigaki2000 ; Hilger2009 ;
Reinders1985 ; Klingl1997 ; Klingl1999 , etc. Various effective models are
formulated keeping in view the fundamental QCD properties such as broken scale
invariance and spontaneous and explicit breaking of the chiral symmetry.

For the first time, Haider and Liu anticipated that the $\eta N$ interactions
are attractive and suggested the possibility of $\eta$-meson bound states
Haider1986 ; Liu1986 . The negative mass-shift/optical potential of
$\eta$-meson has attracted researchers to study the possibilities of
$\eta$-mesic nuclei formation Jenkins1991 ; Zhong2006 ; Waas1997 . At nuclear
saturation density, the optical potential of -20 MeV was anticipated in the
chiral coupled channel approach, considering leading order terms Waas1997 .
Using same coupled channel model, Chiang $et.al.$ obtained optical potential
$U_{\eta}$ = -34 MeV in the normal nuclear matter, assuming the $\eta N$
interactions dominated by $N^{*}$(1535) excitation Chiang1991  and anticipated
that the attractive potential can produce $\eta$-meson bound state with light
and heavy nucleus. Using the QMC model, authors of Ref. Tsushima1998  obtained
optical potential -60 MeV at $\rho_{N}=\rho_{0}$. The chiral unitary approach
was also implied to evaluate the $\eta$ potential and it was observed to be
-54 MeV Inoue2002 . A more deep optical potential of -72 MeV was observed in
Ref. Wang2010 . In this article the possibility of a bound state with
$\eta$-meson was also explored.

In Ref. Zhong2006 , using $\eta N$ Lagrangian off-shell terms, at normal
nuclear density, the in-medium mass of $\eta$-meson was found to be (0.84
$\pm$ 0.015)$m_{\eta}$ and the corresponding optical potential was observed as
-(83 $\pm$ 5) MeV. Furthermore, using the relativistic mean-field theory, Song
$et$ $al.$ observed the optical potential by varying the scattering length
Song2008 . Clearly, the values of $\eta$ optical potential predicted in
various studies varies over large range, $i.e.,$ -20 MeV to -85 MeV and hence,
have considerable model dependence. In addition to theoretical attempts, there
are experimental studies to explore the properties of $\eta$ mesons Peng1987 ;
Berg1994 ; Chiavassa1998 ; Martinez1999 ; Averbeck2003 ; Agakishiev2013 . For
example, for different $\eta$ hadron interactions, the $\eta$-meson production
has been studied in Refs. Peng1987 ; Martinez1999 ; Agakishiev2013  and the
transverse momentum spectra of $\eta$-meson is measured in HICs near the free
$N$-$N$ production threshold Agakishiev2013 .

In the current investigation, we present the in-medium mass and optical
potential of the $\eta$-meson in hot and dense asymmetric nuclear matter using
chiral SU(3) model. By incorporating the medium induced nucleon scalar
densities, we calculate the in-medium mass-shift of $\eta$-meson using the
$\eta N$ effective Lagrangian of chiral SU(3) model. Furthermore, as discussed
earlier, the in-medium mass and optical potential of $\eta$-meson have been
studied using the unitary approach of ChPT and relativistic mean-field model
Zhong2006 ; Song2008 . Following this work, as a second part of the current
investigation the effective mass of $\eta$-meson is also evaluated using the
chiral $\eta N$ Lagrangian of chiral perturbation theory Zhong2006 . In this
approach, the nucleon scalar densities are calculated from chiral SU(3) model
and plugged in the dispersion relation of $\eta N$ interactions derived from
ChPT Lagrangian.

The chiral $SU(3)$ model is extensively used to explore the in-medium
properties of the mesons and baryons in the hot and dense hadronic matter
Kumar2010 ; Zschiesche2004 ; Mishra2004 . For example, the model was used to
study the in-medium mass and optical potential of kaons, antikaons and phi
mesons in the nuclear and hyperonic matter Mishra2004 ; Kumar2011 ; Kumar2020b
. Furthermore, in the nuclear and hadronic matter the in-medium mass of spin
0, spin 1 $D$ mesons and quarkonia were calculated using the conjunction of
chiral SU(3) model and QCD sum rules with Kumar2020 ; Kumar2020a ; Kumar2019 ;
Kumar2019a  and without taking the effect of magnetic field Kumar2014 ;
Chhabra2017 ; Chhabra2017a ; Chhabra2018 ; Kumar2010 . The model was extended
to $SU(4)$ and $SU(5)$ sector to evaluate the medium induced properties of
heavy mesons such as $D$ and $B$ Mishra2004a ; Mishra2009 ; Kumar2011 . On the
other hand, the chiral perturbation theory is also a successful theoretical
framework to study the baryon-meson interactions. The in-medium properties of
$K$ meson were first studied by Kaplan and Nelson using chiral perturbation
theory (ChPT) Kaplan1986 . The same theory was applied to study the
$\eta$-nucleon interactions via adding leading order terms in the model
Lagrangian Jenkins1991 . The heavy baryon chiral perturbation theory was also
applied to study the kaon condensation which is an imperative property to
study the neutron star matter Brown1994 ; Lee1995 ; Kaiser1995 . The ChPT
theory was also improved by the introduction of next-to-leading order terms in
the chiral effective Lagrangian. By including these off-shell terms, Zhong
$et.al.$ anticipated appreciable decrease in the in-medium mass of
$\eta$-meson which is favorable for the formation of $\eta$-mesic nuclei
Zhong2006 .

The layout of the present paper is as follows: In the next section, we will
give brief explanation of the formalism used in the present work. In section
II.1.1, we will derive the $\eta N$ interactions in the chiral SU(3) model
whereas, in section II.1.2, $\eta N$ methodology will be given in the unified
approach of chiral perturbation theory and chiral model. In section III, we
will discuss the in-medium effects on the mass of $\eta$-meson, and finally in
section IV, we will present the summary.

##  II  FORMALISM

###  II.1 IN-MEDIUM SCALAR FIELDS IN THE CHIRAL SU(3) MODEL

The Lagrangian density of the chiral SU(3) model is written as

${\cal L}_{\text{chiral}}={\cal L}_{kin}+\sum_{M=S,V}{\cal L}_{NM}+{\cal
L}_{vec}+{\cal L}_{0}+{\cal L}_{SB}.$ (1)

The model preserves the fundamental QCD properties such as the broken scale
invariance and non-linear realization of the chiral symmetry Weinberg1968 ;
Coleman1969 ; Zschiesche1997 ; Bardeen1969 ; Kumar2020 ; Papazoglou1999 ;
Kumar2019 . It is successfully used to explain the nuclear matter, finite
nuclei, neutron star, and hypernuclei Weinberg1968 ; Coleman1969 ;
Zschiesche1997 ; Bardeen1969 ; Kumar2020 ; Papazoglou1999 ; Kumar2019 . In
this model, the nucleons and baryons interact by the exchange of the vector
fields $\omega$ and $\rho$ along with the scalar fields $\sigma$, $\zeta$ and
$\delta$ in the nuclear medium. The vector fields give short-range repulsion
or attraction which depends on the type of meson-nucleon interaction whereas
the scalar fields give attractive contributions to the medium Kumar2020b . The
$\sigma$ field is a non-strange scalar-isoscalar field which represents the
scalar mesons $\sigma$ ($u\bar{d}$) whereas the $\zeta$ field is a strange
scalar-isoscalar field which represent the scalar meson ($s\bar{s}$)
Zakout2000 . Moreover, the scalar-isovector field $\delta$
$\sim(\bar{u}u-\bar{d}d$) is incorporated in the present model to study the
effect of the isospin asymmetric matter. Further, the glueball field, $\chi$
is a hypothetical gluon field that contains gluon particles and is introduced
in the chiral models to incorporate the scale invariance property of QCD
Papazoglou1999 ; Pwang2001 . We have used mean-field approximation to simplify
the model by neglecting the effect of quantum and thermal fluctuations near
phase transitions Kumar2020 ; Reddy2018 .

In Eq.(1), the ${\cal L}_{kin}$ term describes the the kinetic energy term and
the second term ${\cal L}_{NM}$ given by

$\displaystyle{\cal L}_{NM}=-\sum_{i}\bar{\psi_{i}}\left[m_{i}^{*}+g_{\omega
i}\gamma_{0}\omega+g_{\rho i}\gamma_{0}\rho\right]\psi_{i},$ (2)

defines the nucleon-meson interactions with in-medium nucleon mass as

$\displaystyle m_{i}^{*}=-(g_{\sigma i}\sigma+g_{\zeta i}\zeta+g_{\delta
i}\tau_{3}\delta),$ (3)

where $\tau_{3}$ denotes the 3rd component of isospin and $g_{\sigma i}$,
$g_{\zeta i}$ and $g_{\delta i}$ are the coupling constants of $\sigma$,
$\zeta$ and field $\delta$ with nucleons ($i$=$p,n$), respectively. The next
term ${\cal L}_{vec}$ is given by

$\displaystyle{\cal L}_{vec}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}\right)\frac{\chi^{2}}{\chi_{0}^{2}}+g_{4}(\omega^{4}+6\omega^{2}\rho^{2}+\rho^{4}),$
(4)

reproduces the mass of vector mesons through self-interactions. The ${\cal
L}_{0}$ defines the spontaneous chiral symmetry breaking by the equation

$\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}k_{0}\chi^{2}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)+k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)^{2}$
(5) $\displaystyle+$ $\displaystyle
k_{2}\left(\frac{\sigma^{4}}{2}+\frac{\delta^{4}}{2}+3\sigma^{2}\delta^{2}+\zeta^{4}\right)+k_{3}\chi\left(\sigma^{2}-\delta^{2}\right)\zeta$
$\displaystyle-$ $\displaystyle
k_{4}\chi^{4}-\frac{1}{4}\chi^{4}{\rm{ln}}\frac{\chi^{4}}{\chi_{0}^{4}}+\frac{d}{3}\chi^{4}{\rm{ln}}\Bigg{(}\bigg{(}\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\bigg{)}\bigg{(}\frac{\chi}{\chi_{0}}\bigg{)}^{3}\Bigg{)}.$

In this equation, the $\sigma_{0}$, $\zeta_{0}$, $\delta_{0}$ and $\chi_{0}$
denote the vacuum values of $\sigma$, $\zeta$, $\delta$ and $\chi$ scalar
fields, respectively. Also, the parameter $d$=0.064 along with $k_{i}(i=1$ to
$4)$ and other medium parameters are fitted to regenerate the vacuum values of
scalar and vector fields, $\eta$, $\eta^{\prime}$ mesons and the nucleon mass
Papazoglou1999 ; Kumar2010 ; Kumar2019 . In table 1, we have tabulated the
values of various parameters. Furthermore, the last term ${\cal L}_{SB}$ in
Eq.(1) describes the explicit chiral symmetry breaking property and is written
as

$\displaystyle{\cal
L}_{SB}=-\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[m_{\pi}^{2}f_{\pi}\sigma+\big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\big{)}\zeta\right].$
(6)

In the above equation, $m_{\pi}$, $m_{K}$, $f_{\pi}$, and $f_{K}$ symbolize
the masses and decay constants of pions and kaons, respectively.

The non-linear coupled equations of motion of the scalar and vector fields are
deduced by solving the total Lagrangian (Eq.(1)) using the Euler-Lagrange
equations Kumar2019 ; Kumar2019a  and are given as

$\displaystyle
k_{0}\chi^{2}\sigma-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\sigma-2k_{2}\left(\sigma^{3}+3\sigma\delta^{2}\right)-2k_{3}\chi\sigma\zeta$
$\displaystyle-\frac{d}{3}\chi^{4}\bigg{(}\frac{2\sigma}{\sigma^{2}-\delta^{2}}\bigg{)}+\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\pi}^{2}f_{\pi}=\sum
g_{\sigma i}\rho_{i}^{s},$ (7) $\displaystyle
k_{0}\chi^{2}\zeta-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\zeta-4k_{2}\zeta^{3}-k_{3}\chi\left(\sigma^{2}-\delta^{2}\right)$
$\displaystyle-\frac{d}{3}\frac{\chi^{4}}{\zeta}+\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right]=\sum
g_{\zeta i}\rho_{i}^{s},$ (8) $\displaystyle
k_{0}\chi^{2}\delta-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\delta-2k_{2}\left(\delta^{3}+3\sigma^{2}\delta\right)+2k_{3}\chi\delta\zeta$
$\displaystyle+\frac{2}{3}d\chi^{4}\left(\frac{\delta}{\sigma^{2}-\delta^{2}}\right)=\sum
g_{\delta i}\tau_{3}\rho_{i}^{s},$ (9)

$\displaystyle\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\omega}^{2}\omega+g_{4}\left(4{\omega}^{3}+12{\rho}^{2}{\omega}\right)=\sum
g_{\omega i}\rho_{i}^{v},$ (10)

$\displaystyle\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\rho}^{2}\rho+g_{4}\left(4{\rho}^{3}+12{\omega}^{2}{\rho}\right)=\sum
g_{\rho i}\tau_{3}\rho_{i}^{v},$ (11)

and

$\displaystyle
k_{0}\chi\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)-k_{3}\left(\sigma^{2}-\delta^{2}\right)\zeta+\chi^{3}\left[1+{\rm{ln}}\left(\frac{\chi^{4}}{\chi_{0}^{4}}\right)\right]+(4k_{4}-d)\chi^{3}$
$\displaystyle-\frac{4}{3}d\chi^{3}{\rm{ln}}\Bigg{(}\bigg{(}\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\bigg{)}\bigg{(}\frac{\chi}{\chi_{0}}\bigg{)}^{3}\Bigg{)}+\frac{2\chi}{\chi_{0}^{2}}\left[m_{\pi}^{2}f_{\pi}\sigma+\left(\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right]$
$\displaystyle-\frac{\chi}{{{\chi_{0}}^{2}}}(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2})=0,$
(12)

respectively.

In above equations, the $\rho^{s}_{i}$ and $\rho^{v}_{i}$ denote the scalar
and vector densities of $i^{th}$ nucleons ($i=n,p$) Kumar2019 ; Papazoglou1999
and are given as

$\displaystyle\rho_{i}^{v}=\gamma_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\Bigg{(}\frac{1}{1+\exp\left[\beta(E^{\ast}_{i}(k)-\mu^{*}_{i})\right]}-\frac{1}{1+\exp\left[\beta(E^{\ast}_{i}(k)+\mu^{*}_{i})\right]}\Bigg{)},$
(13)

and

$\displaystyle\rho_{i}^{s}=\gamma_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{m_{i}^{*}}{E^{\ast}_{i}(k)}\Bigg{(}\frac{1}{1+\exp\left[\beta(E^{\ast}_{i}(k)-\mu^{*}_{i})\right]}+\frac{1}{1+\exp\left[\beta(E^{\ast}_{i}(k)+\mu^{*}_{i})\right]}\Bigg{)},$
(14)

respectively, where $\beta=\frac{1}{kT}$,
$E^{\ast}_{i}(k)=\sqrt{k^{2}+{m^{*}_{i}}^{2}}$, $\mu^{*}_{i}=\mu_{i}-g_{\omega
i}\omega-g_{\rho i}\tau_{3}\rho$ and $\gamma_{i}$ is the degeneracy factor.
Moreover, the isospin effect on the scalar and vector density is measured by
the definition, $I=-\frac{\Sigma_{i}\tau_{3i}\rho^{v}_{i}}{2\rho_{N}}$. In the
next section, we calculate the medium-modified mass of $\eta$ mesons in hot
asymmetric nuclear matter. The medium modified $\eta$ meson mass is evaluated
from the dispersion relation which is obtained from the $\eta N$ equation of
motion.

$k_{0}$ | $k_{1}$ | $k_{2}$ | $k_{3}$ | $k_{4}$
---|---|---|---|---
2.53 | 1.35 | -4.77 | -2.77 | -0.218
$\sigma_{0}$ (MeV) |  $\zeta_{0}$ (MeV) |  $\chi_{0}$ (MeV) | $d$ |  $\rho_{0}$ ($\text{fm}^{-3}$)
-93.29 | -106.8 | 409.8 | 0.064 | 0.15
$g_{\sigma N}$ | $g_{\zeta N}$ | $g_{\delta N}$ | $g_{\omega N}$ | $g_{\rho N}$
10.56 | -0.46 | 2.48 | 13.35 | 5.48
$m_{\pi}$ (MeV) |  $m_{K}$ (MeV) |  $f_{\pi}$ (MeV) |  $f_{K}$ (MeV) | $g_{4}$
139 | 498 | 93.29 | 122.14 | 79.91
$m_{\sigma}$ (MeV) |  $m_{\zeta}$ (MeV) |  $m_{\delta}$ (MeV) |  $M_{N}$ (MeV) |  $m_{\eta}$ (MeV)
466.5 | 1024.5 | 899.5 | 939 | 574.374
Table 1: Different constants used in the present work Papazoglou1999 .

####  II.1.1 $\eta$N INTERACTIONS IN THE CHIRAL SU(3) MODEL

In the chiral SU(3) model, the $\eta N$ interaction Lagrangian density can be
written as

$\displaystyle\mathcal{L_{\eta}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{\sigma^{\prime}+4\zeta^{\prime}(2f_{K}-f_{\pi})}{\sqrt{2}f^{2}}\right)\partial^{\mu}\eta\partial_{\mu}\eta$
(15) $\displaystyle-$
$\displaystyle\frac{1}{2}\left(m_{\eta}^{2}-\frac{(\sqrt{2}\sigma^{\prime}-4\zeta^{\prime})m^{2}_{\pi}f_{\pi}+8\zeta^{\prime}m^{2}_{K}f_{K}}{\sqrt{2}f^{2}}\right)\eta^{2}$
$\displaystyle+\frac{d^{\prime}}{f^{2}}\left(\frac{\rho^{s}_{p}+\rho^{s}_{n}}{4}\right)\partial^{\mu}\eta\partial_{\mu}\eta,$

The above chiral $\eta N$ Lagrangian consists of three terms.

  * •

First Range Term:

The first term in the chiral Lagrangian describes the first range term
Kumar2011 ; Papazoglou1999  and is obtained from

${\mathcal{L}}_{{\mathrm{1strangeterm}}}=Tr(u_{\mu}Xu^{\mu}X+Xu_{\mu}u^{\mu}X).$
(16)

In the above equation,
$u_{\mu}=-\frac{i}{2}\left[u^{\dagger}(\partial_{\mu}u)-u(\partial_{\mu}u^{\dagger})\right]$
and $u$=$\text{exp}\left[\frac{i}{\sqrt{2}\sigma_{0}}P\gamma_{5}\right]$,
which is expanded up to second order. Here, symbols $X$ and $P$ represent the
scalar and pseudoscalar meson matrices Papazoglou1999 , respectively and are
explicitly given by Eqs. (39) and (40) in the appendix A. Furthermore, the
vacuum values of $\sigma$ and $\zeta$ fields are deduced in terms of pions and
kaons decay constant by solving the axial current of pions and kaons
Papazoglou1999  through relation

$\sigma_{0}=-f_{\pi}\qquad\zeta_{0}=-\frac{1}{\sqrt{2}}(2f_{K}-f_{\pi}).$ (17)

Moreover, in the first term of $\eta N$ Lagrangian
$\sigma^{\prime}(=\sigma-\sigma_{0})$, $\zeta^{\prime}(=\zeta-\zeta_{0})$ and
$\delta^{\prime}(=\delta-\delta_{0})$ define the digression of the expectation
values of fields from their vacuum expectations. Also,
$f$=$\sqrt{f_{\pi}^{2}+2(2f_{K}-f_{\pi})^{2}}$ and $d^{\prime}$=$3d_{1}+d_{2}$
are the constant parameters.

  * •

Mass Term:

Further, the mass term of the chiral model gives the second term of $\eta N$
Lagrangian and is given by

${\cal L}_{SB}=-\frac{1}{2}{\rm
Tr}A_{p}\left(uXu+u^{\dagger}Xu^{\dagger}\right),$ (18)

where $A_{p}$ is a diagonal matrix given in the Eq.(41). The vacuum mass of
$\eta$ meson, $m_{\eta}$ , is extracted from the above term and is given by
the relation

$m_{\eta}=\frac{1}{f}\sqrt{\left(3m_{\pi}^{2}f_{K}m_{K}^{2}+\frac{8f_{K}^{2}m_{K}^{2}}{f_{\pi}^{2}}-\frac{4f_{K}m_{\pi}^{2}}{f_{\pi}}\right)}.$
(19)

Substituting the values of various constants in above $m_{\eta}$ turns out to
be 574.374 MeV which is with an accuracy of 4.9 $\%$ of physical mass $i.e.$
547.862 MeV PDG2020 . The vacuum mass of $\eta$-meson has model dependency
Burakovsky1997  but here in the present work, we are more concerned in the
$\eta$ in-medium mass-shift which is nearly same for both the masses. In Ref.
Burakovsky1997 , using Gell-Mann Okubo mass formula under octet approximation
in the SU(4) meson multiplets, authors calculated the vacuum mass of
$\eta$-meson to be 567 MeV which is with an accuracy of 3.6 $\%$.

  * •

$d^{\prime}$ Term:

The third term ($i.e.$ $d^{\prime}$ term) in the $\eta N$ Lagrangian
originates from the baryon-meson interaction Lagrangian densities Mishra2004a
; Mishra2006

${\cal L}_{d_{1}}^{BM}=\frac{d_{1}}{2}Tr(u_{\mu}u^{\mu})Tr(\bar{B}B),$ (20)

and

${\cal L}_{d_{2}}^{BM}=d_{2}Tr(\bar{B}u_{\mu}u^{\mu}B).$ (21)

In above, $B$ denotes the baryon matrix (see Eq.(42)).

It should be noted that in case of $\eta N$ interactions of Eq. (15), the
terms corresponding to vectorial Weinberg-Tomozawa term vanishes. On the the
other hand, the Weinberg-Tomozawa term plays a crucial role in the
determination of $K(\bar{K})$ and $D(\bar{D})$ in-medium mass Kumar2011 ;
Mishra2009 .

Using the $\eta N$ Lagrangian in the Euler-Lagrange equation for $\eta$ meson,
the equation of motion is evaluated as

$\displaystyle\partial^{\mu}\partial_{\mu}\eta-\left(m_{\eta}^{2}-\frac{(\sqrt{2}\sigma^{\prime}-4\zeta^{\prime})m^{2}_{\pi}f_{\pi}+8\zeta^{\prime}m^{2}_{K}f_{K}}{\sqrt{2}f^{2}}\right)\eta$
$\displaystyle+\frac{2d^{\prime}}{f^{2}}\left(\frac{\rho^{s}_{p}+\rho^{s}_{n}}{4}-\frac{\sigma^{\prime}+4\zeta^{\prime}(2f_{K}-f_{\pi})}{\sqrt{2}}\right)\partial^{\mu}\partial_{\mu}\eta=0.$
(22)

Performing the Fourier transformation on the above equation, the dispersion
relation for $\eta$ meson turns out to be

$-\omega^{2}+{\textbf{k}}^{2}+m_{\eta}^{2}-\Pi^{*}(\omega,|\textbf{k}|)=0.$
(23)

In the above equation, $\Pi^{*}$ denotes the effective self-energy of $\eta$
meson, explicitly given as

$\displaystyle\Pi^{*}(\omega,|\textbf{k}|)$ $\displaystyle=$
$\displaystyle-\frac{(\sqrt{2}\sigma^{\prime}-4\zeta^{\prime})m^{2}_{\pi}f_{\pi}+8\zeta^{\prime}m^{2}_{K}f_{K}}{\sqrt{2}f^{2}}+\frac{2d^{\prime}}{f^{2}}\left(\frac{\rho^{s}_{p}+\rho^{s}_{n}}{4}\right)(\omega^{2}-{\vec{k}}^{2})$
(24) $\displaystyle-$
$\displaystyle\frac{2}{f^{2}}\left(\frac{\sigma^{\prime}+4\zeta^{\prime}(2f_{K}-f_{\pi})}{\sqrt{2}}\right)(\omega^{2}-{\textbf{
k}}^{2}).$

The unknown parameter, $d^{\prime}$ is approximated from the experimental
values of scattering length, $a^{\eta N}$ Zhong2006 . In the chiral model, the
expression of scattering length derived from the scattering amplitude is given
by

$\displaystyle a^{\eta N}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\left(1+\frac{m_{\eta}}{M_{N}}\right)}\Big{[}\Big{(}\frac{d^{\prime}}{\sqrt{2}}-\frac{g_{\sigma
N}}{m^{2}_{\sigma}}+\frac{4(2f_{K}-f_{\pi})g_{\zeta
N}}{m^{2}_{\zeta}}\Big{)}\frac{m_{\eta}^{2}}{\sqrt{2}f^{2}}$ (25)
$\displaystyle+$ $\displaystyle\left(\frac{\sqrt{2}g_{\sigma
N}}{m^{2}_{\sigma}}-\frac{4g_{\zeta
N}}{m^{2}_{\zeta}}\right)\frac{m^{2}_{\pi}f_{\pi}}{2\sqrt{2}f^{2}}+\tau_{3}\frac{2\sqrt{2}g_{\delta
N}}{m^{2}_{\delta}}\frac{m^{2}_{K}f_{K}}{f^{2}}\Big{]}.$

Rearranging the above for $d^{\prime}$ gives

$\displaystyle d^{\prime}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2\pi\left(1+\frac{m_{\eta}}{M_{N}}\right)}\frac{a^{\eta
N}}{m^{2}_{\eta}}+\frac{\sqrt{2}g_{\sigma
N}}{m^{2}_{\sigma}}-\frac{4\sqrt{2}(2f_{K}-f_{\pi})g_{\zeta
N}}{m^{2}_{\zeta}}$ (26) $\displaystyle-$
$\displaystyle\left(\frac{\sqrt{2}g_{\sigma
N}}{m^{2}_{\sigma}}-\frac{4g_{\zeta
N}}{m^{2}_{\zeta}}\right)\frac{m^{2}_{\pi}f_{\pi}}{\sqrt{2}m^{2}_{\eta}}-\tau_{3}\frac{4\sqrt{2}g_{\delta
N}m^{2}_{K}f_{K}}{m^{2}_{\delta}m^{2}_{\eta}}.$

Using the condition, $m_{\eta}^{*}=\omega(|\textbf{k}|$=0) in Eq. (23), we
obtain the effective mass of $\eta$ meson in the nuclear medium. Further, the
momentum dependent optical potentials are defined through the relation
Mishra2008 ; Mishra2009

$U^{*}_{\eta}(\omega,\textbf{
k})=\omega(\textbf{k})-\sqrt{\textbf{k}^{2}+m^{{}^{2}}_{\eta}}.$ (27)

At zero momentum, the above equation gives

$U^{*}_{\eta}=\Delta m_{\eta}^{*}={m_{\eta}^{*}}-m_{\eta}.$ (28)

####  II.1.2  UNIFICATION OF CHIRAL PERTURBATION THEORY (ChPT) AND CHIRAL
MODEL

In this section, we discuss the unified approach of ChPT and chiral model to
compute the in-medium mass of $\eta$ mesons. The ChPT comprises the underlying
chiral symmetry property of QCD and use an effective field theory approach
Zhong2006 . The same theory along with Relativistic mean-field model has been
used to deduce the eta-nucleon interactions in the symmetric nuclear matter
Zhong2006 ; Song2008 . The Lagrangian density defining the meson-baryons
interactions in this theory is given by

$\displaystyle{\mathcal{L}_{\text{ChPT}}}={\mathcal{L}_{P}}+{\mathcal{L}_{PB}},$
(29)

with $P$ representing the pseudoscalar meson multiplet (see Eq.(40)). Up to
second chiral order, the $\mathcal{L}_{P}$ term is defined as Zhong2006 ;
Kaplan1986

$\displaystyle{\mathcal{L}_{P}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}f_{\pi}^{2}\textrm{Tr}\partial^{\mu}\Sigma\partial_{\mu}\Sigma^{\dagger}+\frac{1}{2}f_{\pi}^{2}B_{0}\left\\{\mbox{Tr}M_{q}(\Sigma-1)+\mathrm{h.c.}\right\\},$
(30)

where $\Sigma=\xi^{2}=\exp{(i\sqrt{2}P/f_{\pi})}$ and
$M_{q}=\mbox{diag}\\{m_{q},m_{q},m_{s}\\}$ is the current quark mass matrix.
The Lagrangian term,
$\mathcal{L}_{PB}$=$\mathcal{L}^{L}_{PB}$+$\mathcal{L}^{NL}_{PB}$ describes
the leading and next to leading order contributions Kaplan1986 . Jenkins and
Manohar developed the next to leading order terms using heavy baryon chiral
theory Jenkins1991 . In this Lagrangian, the loop contributions are not
considered as the higher-order corrections get suppressed for the small
momentum scale, $Q^{2}$ Zhong2006 . The different nuclear properties are
studied successfully using $\mathcal{L}^{NL}_{PB}$ Park1993 .

The $\eta N$ Lagrangian is obtained by expanding the Eq.(29) up to the second
order of multiplet $P$ Zhong2006

$\displaystyle\mathcal{L_{\eta N}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{\mu}\eta\partial_{\mu}\eta-\frac{1}{2}\left(m{{}^{\prime}}_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{f_{\pi}^{2}}\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\right)\eta^{2}+\frac{1}{2}\frac{\kappa}{f_{\pi}^{2}}\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\partial^{\mu}\eta\partial_{\mu}\eta.$
(31)

Here, $m{{}^{\prime}}_{\eta}=\sqrt{\frac{2}{3}B_{0}(m_{q}+2m_{s})}$ denotes
the vacuum mass of $\eta$ meson calculated in chiral perturbation theory. In
the mass expression, $B_{0}$ symbolize the relation with the order parameter
of spontaneously broken chiral symmetry and $m_{q(s)}$ denote the mass of
light (strange) quarks Burakovsky1997 . For consistency with the chiral SU(3)
model, we have used the same value of $\eta$ meson vacuum mass $i.e.$
$m{{}^{\prime}}_{\eta}$=$m_{\eta}$= 574.374 MeV in the further calculations of
ChPT. The $\eta N$ sigma term $\Sigma_{\eta\mathrm{N}}$, obtained from
“$a_{i}$” terms of the next-to-leading order chiral Lagrangian density is
given as Zhong2006

$\displaystyle\Sigma_{\eta\mathrm{N}}=-\frac{2}{3}[a_{1}m_{q}+4a_{2}m_{s}+2a_{3}(m_{q}+2m_{s})].$
(32)

The $\Sigma_{\eta\mathrm{N}}$ value is estimated to be 280 $\pm$ 130 MeV from
the different empirical observations of $\Sigma_{\mathrm{KN}}$ term having
value 380 MeV $\pm$ 100 MeV Lyubovitskij2001 ; Dong1996 ; Hatsuda1994 ;
Brown1994 ; Georgi1984 ; Politzer1991 ; Lee1995 ; Zhong2006 .

Also, the parameter $\kappa$ in the last term of the Eq.(31) comprises the
contributions from the “off-shell” $d_{i}$ terms of the next to Leading order
Lagrangian Zhong2006 . In the present work, we determined $\kappa$ using the
expression of $\eta N$ scattering length, $a^{\eta\mathrm{N}}$, calculated
from the ChPT matrix amplitude (on-shell constraints) Zhong2006

$\displaystyle a^{\eta\mathrm{N}}=\frac{1}{4\pi
f_{\pi}^{2}(1+m_{\eta}/M_{\mathrm{N}})}\left(\Sigma_{\eta\mathrm{N}}+\kappa
m_{\eta}^{2}\right),$ (33)

and by re-arranging for $\kappa$ it becomes

$\displaystyle\kappa=4\pi
f_{\pi}^{2}\left(\frac{1}{m_{\eta}^{2}}+\frac{1}{m_{\eta}M_{\mathrm{N}}}\right)a^{\eta\mathrm{N}}-\frac{\Sigma_{\eta\mathrm{N}}}{m_{\eta}^{2}}.$
(34)

We have taken the experimentally determined $a^{\eta\mathrm{N}}$ values $i.e.$
0.91 $\sim$ 1.14 fm in the present investigation Green2005 ; Renard2002 ;
Arndt2005 ; Green1999 ; Zhong2006 . Furthermore, the $\eta N$ equation of
motion has been derived using the interaction Lagrangian (Eq.(31)) in the
Euler Lagrange equation of motion:

$\displaystyle\left(\partial_{\mu}\partial^{\mu}+m_{\eta}^{2}-\frac{\Sigma_{\eta
N}}{2f_{\pi}^{2}}\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle+\frac{\kappa}{2f_{\pi}^{2}}\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle\partial_{\mu}\partial^{\mu}\right)\eta=0.$
(35)

In above,
$\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle\equiv\rho^{s}_{N}$=$\left(\rho^{s}_{p}+\rho^{s}_{n}\right)$
defines the in-medium scalar density of nucleons calculated within the mean-
field chiral SU(3) model (see Eqs.(13) and (14)). The Fourier transformation
of Eq.(35) gives

$\displaystyle-\omega^{2}+\textbf{k}^{2}+m_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}\rho^{s}_{N}+\frac{\kappa}{2f_{\pi}^{2}}\rho^{s}_{N}\left(-\omega^{2}+\textbf{k}^{2}\right)=0,$
(36)

From the above equation, the effective mass
$m_{\eta}^{*}=\omega(|\textbf{k}|$=0) of $\eta$ meson can be written as

$\displaystyle
m_{\eta}^{*}=\sqrt{\left(m_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}\rho^{s}_{N}\right)\Big{/}\left(1+\frac{\kappa}{2f_{\pi}^{2}}\rho^{s}_{N}\right)}.$
(37)

Further, the $\eta$-meson self-energy derived from Eq.(36) is given by

$\displaystyle\Pi^{*}(\omega,\textbf{k})=\Big{(}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}+\frac{\kappa}{2f_{\pi}^{2}}(-\omega^{2}+\textbf{k}^{2})\Big{)}\rho^{s}_{N}.$
(38)

##  III Results and Discussions

In this section, at first we discuss the behavior of in-medium nucleon scalar
densities in the hot asymmetric nuclear matter. Further, we discuss the
effective mass of $\eta$ meson which is derived using the chiral SU(3) model
alone in section III.1 and with the unified approach of ChPT and chiral SU(3)
model in section III.2 . In both approaches, we show the results for range of
scattering length, $a^{\eta_{N}}$=0.91-1.14 fm. Various parameters used in the
present investigation are mentioned in table 1.

In chiral model, the scalar densities of nucleons have been calculated through
Eq. (14). This equation contains the effect of medium modified scalar and
vector fields Papazoglou1999 . The in-medium behavior of these fields is
obtained by solving the coupled equations of motion (Eqs. (7) to (12))
Kumar2020b . In fig. 1, we plot the scalar density of proton and neutron as a
function of number density for finite values of temperature, $T$ and isospin
asymmetry parameter. In symmetric nuclear matter as the contribution of
$\delta$ and $\rho$ field is zero Kumar2019 , we get the same behavior of
neutron and proton scalar densities. The $\delta$ and $\rho$ field changes the
in-medium value of baryon mass $m^{*}_{i}$ and effective chemical potential
$\mu^{*}_{i}$, respectively which further modifies the nucleon scalar density
(see Eq.(14)) Papazoglou1999 . In figure, at $T=0$ the scalar density
increases linearly in the low density regime and becomes non-linear in the
high density regime. When we move from $I$=0 to I$\neq$0 region, we observe a
gradual increase in the neutron scalar density whereas the proton scalar
density decreases. This is because of the non-zero contribution of $\delta$
and $\rho$ field in the isospin asymmetric nuclear matter which changes the
effective mass as well as chemical potential and therefore scalar density
Papazoglou1999 .

Another thermodynamic quantity $i.e.$ temperature is also a main property of
the nuclear medium and in fig. 1 we have shown how the in-medium dynamics
changes under non-zero temperature. The effect of temperature is observed more
in the high density regime as compared to the low density regime. For
symmetric matter in sub-plots (a) and (c), we anticipate appreciable effect of
temperature. Here, for a particular value of nuclear density the value of
scalar densities decrease as a function of temperature. This is because of the
Fermi distribution integral, due to the coupled nature of the equations (Eqs.
(7) to (12)) the value of scalar density in Eq.(14) decreases when we increase
the temperature in integral. On the other hand, in the highly asymmetric
matter $i.e.$ $I$=0.5, for the neutron scalar density the temperature effects
become more appreciable. In addition, we observe a minor contribution to the
proton scalar density for higher temperature values. This is because at finite
temperature the proton condensate ($\bar{p}p$) $i.e.$ proton scalar density
still populates in the medium despite the zero value of proton number density
$\rho_{p}$. The observed behavior of scalar densities in the symmetric nuclear
matter is in agreement with the calculations of the Relativistic mean-field
model Zhong2006 ; Song2008 .

Figure 1: (Color online) The in-medium scalar density of nucleons.

###  III.1 Optical Potential and Mass of $\eta$ Meson in Chiral Model

|  |  $\eta$=0 |  $\eta$=0.5
---|---|---|---
| $a^{\eta N}(\text{fm})$ | T=0 | T=100 | T=0 | T=100
|  | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$
| 0.91 | -46.18 | -132.88 | -37.78 | -120.31 | -44.74 | -123.35 | -37.74 | -116.46
$\Delta m^{*}_{\eta}$ | 1.02 | -54.61 | -146.77 | -45.22 | -133.79 | -52.99 | -136.93 | -45.16 | -129.78
| 1.14 | -63.37 | -160.51 | -52.98 | -147.21 | -61.58 | -150.42 | -52.92 | -143.07
Table 2: Values of in-medium mass-shift of $\eta$-meson for different medium
attributes calculated in chiral model are tabulated (in units of MeV).

In fig. 2, we have illustrated the medium modified mass of $\eta$-meson as a
function of nuclear density for different values of scattering length. In the
same figure, we also show the impact of isospin asymmetry and temperature. For
given value of asymmetry, temperature, scattering length, the in-medium mass
of $\eta$-meson is observed to decrease as a function of nuclear density. The
rate of decrease is linear in the low density regime whereas in the high
density regime it becomes non-linear. This behavior reflects the opposite
variation of nucleon scalar density plotted in fig. 1. This is because the
self-energy of $\eta$-meson (see Eq.(24)) has a direct dependence on the sum
of scalar densities of nucleons. When we change the value of $a^{\eta N}$ from
0.91 to 1.14 fm, we observe a decrement in the effective mass. For example, at
$\rho_{N}$=$\rho_{0}(4\rho_{0})$, $I$=$T$=0, the effective mass of
$\eta$-meson changes from 528 (441) to 512 (423) MeV when we change $a^{\eta
N}$ value from 0.91 to 1.14 fm, respectively. This is due to the $d^{\prime}$
term in Eq.(24). The $d^{\prime}$ term has direct dependence on $a^{\eta N}$
as shown in Eq.(26) and therefore increasing the value of scattering length
cause an increase in the value of $d^{\prime}$. Due to the attractive
contribution of the self-energy part corresponding to $d^{\prime}$ term the
value of effective mass decreases. We also observed the substantial impact of
the temperature on the in-medium mass in the symmetric nuclear matter which
reflects the in-medium behavior of scalar densities. However, in the
asymmetric nuclear matter, we observe the temperature effects on the mass to
be less appreciable which reflects the less contribution of the net scalar
density ($\rho^{s}_{p}+\rho^{s}_{n}$).

The self-energy expression given by Eq.(24) contain three terms (i) first
range term (ii) mass term and (iii) $d^{\prime}$ term. To understand the
contribution of these individual terms, we illustrated the in-medium mass of
$\eta$-meson at zero and non-zero value of asymmetry and temperature in fig. 3
for these different terms. At zero temperature and asymmetry, one can see that
the first range term gives an appreciable repulsive contribution to the
effective mass whereas the mass and $d^{\prime}$ terms give attractive
contributions. We observe the dominant contribution of $d^{\prime}$ term which
in turn gets reflected in the net effective mass. For non-zero temperature and
asymmetry, the variation in the $d^{\prime}$ term becomes less hence we get a
lower value of effective mass. This is due to the effect of scalar density
terms present in the $d^{\prime}$ term (Eq.(24)). For further understanding,
in fig. 2 we plot the $\eta$-meson effective mass as a function of scattering
length $a^{\eta N}$ at $\rho_{N}$=$\rho_{0},4\rho_{0}$. At nuclear saturation
density, we observe a linear decrease of effective mass with the increase in
scattering length. Furthermore, the effective mass decrease more rapidly in
the high density regime. The observed behaviour emphasizes the importance of
scattering length in the $\eta N$ interactions.

The decrease in the in-medium mass leads to a negative mass-shift which
suggests the bound state formation of $\eta$-meson with a nucleus Jenkins1991
; Zhong2006 . To understand the bound state phenomenon, the study of in-medium
optical potential is very imperative. By using the effective mass in Eq.(28),
we plotted the optical potential of $\eta$-meson as a function of momentum
$\lvert\textbf{k}\rvert$ for different values of $\eta N$ scattering length
and other medium parameters in figs. 5, 6 and 7. In fig. 5, at
$\rho_{N}$=$\rho_{0}$ we observe a negative value of the optical potential.
The value of optical potential becomes less negative as we increase the
momentum of the $\eta$-meson. The variation of optical potential reflects the
interplay between the effective mass and momentum. At high values of the
momentum, Eq.(28) gets dominated by momentum and the contribution of effective
mass becomes less.

A similar phenomenon happens in the high density regime. In this region, we
observe appreciable values of optical potential which become less as momentum
increases. Moreover, in the presence of a high density of neutron matter , we
anticipate less effect of temperature which reflects the in-medium behavior of
$\eta$-meson mass. In figs. 6 and 7 we observe a similar trend of optical
potential with $\eta$ momentum. In these figures, we observe a more negative
value of optical potential as we increase the scattering length. As discussed
earlier, the optical potential is directly related to in-medium mass, here it
is illustrated to get a clear idea of negative potential. In the cold
symmetric nuclear matter, at $\rho_{N}$=$\rho_{0}(4\rho_{0})$ we observe
optical potential to be -54.61 (-146.77) MeV for $a^{\eta N}$=1.02 fm, whereas
for $I$=0.5 these values changes to -52.99 (-136.93) MeV. For better
understanding, we have tabulated the in-medium mass-shift of $\eta$-meson at
zero momentum in table 2.

Figure 2: (Color online) In-medium $\eta$ meson mass in the chiral model.
Figure 3: (Color online) Comparison of the different terms of $\eta$-meson
effective mass in chiral model at $a^{\eta N}=1.14$ fm.  Figure 4: (Color
online) The in-medium $\eta$-meson mass as a function of scattering length.
Figure 5: (Color online) The in-medium $\eta$ meson optical potential in
chiral model at $a^{\eta N}$=0.91 fm.  Figure 6: (Color online) The in-medium
$\eta$ meson optical potential in chiral model at $a^{\eta N}$=1.02 fm. Figure
7: (Color online) The in-medium $\eta$ meson optical potential in chiral model
at $a^{\eta N}$=1.14 fm.

###  III.2 In-medium Mass of $\eta$ Meson in Unified Approach of ChPT and
Chiral Model

|  |  $\eta$=0 |  $\eta$=0.5
---|---|---|---
| $a^{\eta N}(\text{fm})$ | T=0 | T=100 | T=0 | T=100
|  | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$ | $\rho_{0}$ | $4\rho_{0}$
| 0.91 | -107.54 | -219.71 | -93.73 | -205.06 | -105.19 | -208.52 | -93.64 | -200.43
$\Delta m^{*}_{\eta}$ | 1.02 | -116.83 | -232.28 | -102.21 | -217.49 | -114.35 | -220.99 | -102.11 | -212.80
| 1.14 | -126.36 | -244.56 | -110.96 | -229.72 | -123.75 | -233.24 | -110.86 | -225.00
Table 3: Values of in-medium mass-shift of $\eta$-meson for different medium
attributes calculated in the ChPT+chiral model are tabulated (in units of
MeV).

In this section, we have used the unified approach of chiral SU(3) model and
chiral perturbation theory to calculate the medium induced mass of
$\eta$-meson. As discussed in the methodology section, the $\eta N$ equation
of motion is obtained from the Lagrangian density of ChPT. Further, the scalar
density of nucleons appearing in the ChPT equation of motion is obtained from
the chiral SU(3) model. In this calculation, we have taken the value of
parameter $\Sigma_{\eta N}$ to be 280 MeV. We have not considered the
contribution of uncertainties in the $\Sigma_{\eta N}$ parameter because of
the less contribution of sigma term in the in-medium mass as compared to
$\kappa$ term which we will see later. The value of in-medium $\eta$ mass-
shift calculated using present unified approach are given in table 3.

In fig. 8, we illustrated the ratio of the in-medium and vacuum mass of the
$\eta$-meson as a function of nuclear density. In this figure, we have also
included the effect of $\eta N$ scattering length, temperature, and medium
isospin asymmetry. Moreover, we compared the results obtained from two
different approaches $i.e.$ (i) chiral model alone (ii) ChPT and chiral model.
Using the second approach, we observed a substantial decrease in the mass of
$\eta$-meson. We observed the same behavior of the in-medium mass with respect
to temperature, asymmetry, and scattering length as was observed in the
situation when the chiral model was used alone. The main difference is, in the
ChPT the $\eta$ meson gets a more net attractive contribution than the chiral
model which is due to the absence of first range term in ChPT Lagrangian. In
fig. 9 we have plotted the contributions of individual terms to the in-medium
mass of $\eta$-meson calculated from the unified approach. The $\eta$-meson
in-medium mass given by Eq.(37) in the ChPT+chiral model approach has two
terms (i) $\Sigma_{\eta N}$ term and (ii) $\kappa$ term. In this figure, we
have shown the individual contribution of these terms with increasing nuclear
density and observed a non-appreciable contribution with $\Sigma_{\eta N}$ but
appreciable with $\kappa$ term. This is because in the $\eta$-meson in-medium
mass expression (see Eq.(37)), the denominator has a positive contribution of
the scalar densities and the increase in scalar density with number density
increases the denominator hence the value of effective mass becomes more
negative. Clearly, there is no first range term with the positive contribution
as was observed in the previous chiral model calculations, and therefore in
the present case we get substantial attractive mass-shift.

The present observations can be compared with the $\eta$-meson effective mass
calculated in the unified approach of ChPT and relativistic mean-field model
of Ref. Zhong2006 . In this article, authors also considered the effect of
scattering length and at nuclear saturation density and $a^{\eta N}$=1.02 fm
they anticipated the effective mass to be 0.84 $m_{\eta}$ whereas we observed
it to be 0.79 $m_{\eta}$. At nuclear saturation density, the effective mass
equal to 0.95 $m_{\eta}$ was obtained within the coupled channel approach with
scattering length $a^{\eta N}$ $\sim$ 0.25 fm Waas1997 . In this non-diagonal
coupled channel approach, there are only leading order contributions and
hence, only a small decrement in the in-medium mass is observed. Also, in the
QMC model at $\rho_{0}$ the in-medium mass of $\eta$-meson having value 0.88
$m_{\eta}$ was observed Tsushima1998 . The obtained values are comparable with
the calculations of the ChPT+chiral model for $a^{\eta N}$ $\sim$ 0.50 fm.

In the cold symmetric nuclear matter, at $\rho_{N}$=$\rho_{0}(4\rho_{0})$ and
$\lvert\textbf{k}\rvert$=0, we observe optical potential to be -116.83
(-232.28) MeV for $a^{\eta N}$=1.02 fm and in the cold isospin asymmetric
nuclear matter the values modifies to -114.35 (-220.99) MeV. Using the
ChPT+chiral model approach, we observed a even deeper optical potential than
evaluated in the relativistic mean-field model+ChPT approach of Ref. Zhong2006
. This is due to the difference in the in-medium scalar densities obtained in
two models. In our approach, we have taken the effect of scalar and vector
fields under the influence of isospin asymmetry, and finite temperature
whereas in the relativistic model approach only cold symmetric medium was
considered. The $\eta$ optical potential was also observed in the different
theoretical observations Waas1997 ; Tsushima1998 ; Cieply2014 ; Inoue2002 ;
Zhong2006 . $U_{\eta}$=-34 MeV was observed by studying the $\eta N$
interactions near the threshold using free space chirally inspired coupled
channel approach by considering the contributions of $N^{*}$(1535) baryon
resonance Cieply2014 . Besides, the optical potential $U_{\eta}$=-54, -60, and
-83 MeV was observed in the chiral unitary approach Inoue2002 , QMC model
Tsushima1998 , and the ChPT Zhong2006 , respectively.

Figure 8: (Color online) Comparison of in-medium $\eta$ meson mass calculated
from chiral model and ChPT.  Figure 9: (Color online) Comparison of different
terms of the effective mass of $\eta$-meson calculated using unification of
ChPT and chiral model at $a^{\eta N}=1.14$ fm.

##  IV SUMMARY

We investigated the in-medium mass of $\eta$-meson in the asymmetric nucear
matter at finite temperature. Under these medium conditions, we studied the
behavior of the $\eta$-meson using two different methodologies. In the first
methodology, using the chiral model alone we calculated the medium modified
mass and optical potential of $\eta$-meson by considering the $\eta N$
interactions up to second order in the Lagrangian and observe a decrease in
the effective mass of $\eta$-meson as a function of density. We find the in-
medium effects to be more appreciable in the high density regime. In the
second, we used the unified approach of chiral perturbation theory (ChPT) and
chiral SU(3) model to study the in-medium attributes of $\eta$-meson. In this
approach, we took the next-to-leading order contributions. We incorporated the
medium effects from the chiral SU(3) model through scalar density which is
plugged in the $\eta N$ equation of motion, which is calculated from the
effective $\eta N$ Lagrangian of ChPT. Using this methodology, we find a
substantial decrease in the mass of $\eta$-meson as a function of nuclear
density. The temperature and asymmetry effects are also studied and found to
be slight repulsive in nature. Also, in the both approaches the mass-shift is
observed to increase with an increase in the value of scattering length. The
decrement on the $\eta$-meson mass leads to a negative mass-shift/optical
potential which further suggests the possibility of $\eta N$ bound states. The
optical potential calculated in the present work will be used in future to
calculate the spectroscopic state of the $\eta$-mesic nuclei Zhong2006 . Also,
the momentum dependent optical potential can be used to study the $\eta$-meson
production rate Peng1987 ; Martinez1999 ; Agakishiev2013  and its momentum
dependence in the asymmetric nuclear medium David2018 ; Chen2017 ; Berg1994 .

## Acknowledgment

One of the authors, (R.K) sincerely acknowledge the support towards this work
from Ministry of Science and Human Resources Development (MHRD), Government of
India via Institute fellowship under National Institute of Technology
Jalandhar.

##  Appendix A EXPLICIT REPRESENTATION OF DIFFERENT MATRICES

Here, we give the matrix representation of meson, baryon and mass matrices
which are used in present calculations Papazoglou1999 .

  * •

The Scalar Meson Matrix, $X$:

$X=\frac{1}{\sqrt{2}}\sigma^{a}\lambda_{a}=\left(\begin{array}[]{ccc}(\delta+\sigma)/\sqrt{2}&\delta^{+}&\kappa^{+}\\\
\delta^{-}&(-\delta+\sigma)/\sqrt{2}&\kappa^{0}\\\
\kappa^{-}&\overline{\kappa^{0}}&\zeta\end{array}\right).$ (39)

  * •

The Pseudoscalar Meson Matrix, $P$:

$P=\frac{1}{\sqrt{2}}\pi_{a}\lambda^{a}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\pi^{0}+{\frac{\eta}{\sqrt{1+2\,{w}^{2}}}}\right)&\pi^{+}&2\,{\frac{K^{+}}{w+1}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\pi^{-}&\frac{1}{\sqrt{2}}\left(-\pi^{0}+{\frac{\eta}{\sqrt{1+2\,{w}^{2}}}}\right)&2\,{\frac{K^{0}}{w+1}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
2\,{\frac{K^{-}}{w+1}}&2\,{\frac{\overline{K}^{0}}{w+1}}&-{\frac{\eta\,\sqrt{2}}{\sqrt{1+2\,{w}^{2}}}}\end{array}\right),$
(40)

where $w=\sqrt{2}\zeta_{0}/\sigma_{0}$.

  * •

The $A_{p}$ Matrix:

$A_{p}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}m_{\pi}^{2}f_{\pi}&0&0\\\
0&m_{\pi}^{2}f_{\pi}&0\\\
0&0&2m_{K}^{2}f_{K}-m_{\pi}^{2}f_{\pi}\end{array}\right).$ (41)

  * •

The Baryon Matrix, $B$:

$B=\frac{1}{\sqrt{2}}b^{a}\lambda_{a}=\left(\begin{array}[]{ccc}\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda^{0}}{\sqrt{6}}&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda^{0}}{\sqrt{6}}&n\\\
\Xi^{-}&\Xi^{0}&-2\frac{\Lambda^{0}}{\sqrt{6}}\end{array}\right).$ (42)

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