#  $\eta$ mesons in hot magnetized nuclear matter

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

###### Abstract

The $\eta N$ interactions are investigated in the hot magnetized asymmetric
nuclear matter using chiral SU(3) model and chiral perturbation theory (ChPT).
In the chiral model, the in-medium properties of $\eta$-meson are calculated
by the medium modified scalar densities under the influence of an external
magnetic field. Further, in the combined approach of chiral model and ChPT,
off-shell contributions of $\eta N$ interactions are evaluated from the ChPT
effective $\eta N$ Lagrangian, and the in-medium effect of scalar densities
are incorporated from the chiral SU(3) model. We observe a significant effect
of magnetic field on the in-medium mass and optical potential of $\eta$ meson.
We observe a deeper mass-shift in the combined approach of ChPT and chiral
model compared to the effect of solo chiral SU(3) model. In both approaches,
no additional mass-shift is observed due to the uncharged nature of $\eta$
mesons in the presence of magnetic field.

##  I Introduction

The investigation of in-medium meson-baryon properties under the effect of
strong external magnetic field is a demanding area of research in the non-
central Heavy-Ion Collisions (HICs) Cho2014 ; Cho2015 ; Gubler2016 ; Reddy2018
; Kharzeev2008 ; Kharzeev2013 ; Fukushima2008 ; Skokov2009 . Besides, the
presence of temperature and isospin asymmetry ,i.e., uneven numbers of
neutrons and protons, lead to significant modifications in the in-medium
properties of hadrons Reddy2018 ; Papazoglou1999 ; Mishra2009 . The strong
magnetic fields (of the order of $eB\sim{15{m^{2}_{\pi}}}$ ($5\times 10^{19}$
gauss) at large hadron collider (LHC), CERN and $eB\sim{2{m^{2}_{\pi}}}$
($6.5\times 10^{18}$ gauss) at relativistic heavy ion collider (RHIC), BNL may
have been produced Kharzeev2008 ; Fukushima2008 ; Skokov2009 . As the remnant
move away from the collision zone, the magnitude of the magnetic field
decreases rapidly. Meanwhile, the decaying magnetic field interacts with
residual matter and as per Lenz’s law, the induced current comes into the
picture. These induced currents further generate the opposite magnetic field
which interacts with the primary magnetic field. This whole process slows down
the decay rate of the primary magnetic field and gives it enough time to leave
imprints on the mesons and hadrons Tuchin2011 ; Tuchin2011a ; Tuchin2013 ;
Marasinghe2011 ; Das2017 ; Reddy2018 ; Cho2015 . The slowing down of
relaxation time is known as chiral magnetic effect Kharzeev2013 ;
Fukushima2008 ; Vilenkin1980 ; Burnier2011 . In HICs, the time evolution of
the magnetic field is still unclear, therefore to have a correct estimate of
the medium’s electrical conductivity and relaxation time, further study is
required Reddy2018 .

In the future experiments namely compressed baryonic matter (CBM) and
antiproton annihilation at Darmstadt (PANDA) at GSI, Germany, Japan proton
accelerator research complex (J-PARC) at Japan, and nuclotron-based ion
collider facility (NICA) at Dubna, Russia, we anticipate significant research
in the in-medium meson-baryons interactions Kumar2019 ; Rapp2010 ; Kumar2020a
. On the theoretical side, various effective models have been built to study
meson-baryon interactions. These models are namely quark-meson coupling (QMC)
model Guichon1988 ; Hong2001 ; Tsushima1999 ; Sibirtsev1999 ; Saito1994 ;
Panda1997 , Polyakov quark meson (PQM) model Chatterjee2012 ; Schaefer2010 ,
coupled channel approach Tolos2004 ; Tolos2006 ; Tolos2008 ; Hofmann2005 ,
chiral SU(3) model Papazoglou1999 ; Mishra2004a ; Mishra2009 ; Kumar2010 ;
Kumar2019 , chiral perturbation theory (ChPT) Zhong2006 ; Jenkins1991 ;
Kumar2020c , QCD sum rules Reinders1981 ; Hayashigaki2000 ; Hilger2009 ;
Reinders1985 ; Klingl1997 ; Klingl1999 , Nambu-Jona-Lasinio (NJL) model
Nambu1961 , and the Polyakov loop extended NJL (PNJL) model Fukushima2004 ;
Kashiwa2008 ; Ghosh2015 . In the present work, we have used two of the above
theoretical approaches i.e. chiral SU(3) model and ChPT.

The $K/\pi/\eta-B$ interactions are much studied in literature using various
theoretical and experimental approaches Jenkins1991 ; Haider1986 ; Liu1986 ;
Kaplan1986 ; Kumar2020c ; Chen2017 ; David2018 ; Inoue2002 . Among these
mesons, the $\eta$-meson is of special interest due to the possibility of
$\eta$-nucleon bound state formation Jenkins1991 ; Zhong2006 ; Waas1997 . The
properties of $\eta$ mesons, Peng1987 ; Berg1994 ; Chiavassa1998 ;
Martinez1999 ; Averbeck2003 ; Agakishiev2013 , for instance, the transverse
momentum spectrum near the threshold of free $N$-$N$ production Agakishiev2013
and the $\eta$-meson production have been studied experimentally Peng1987 ;
Martinez1999 ; Agakishiev2013 . On the theoretical side, Haider and Liu were
the first to observe that the $\eta N$ interactions show attractive behavior
and therefore, the $\eta$-meson can form bound state with nucleons Haider1986
; Liu1986 . Chiang $et.al.$ anticipated an optical potential $U_{\eta}$ = -34
MeV at $\rho_{0}$ using the chiral coupled channel approach Chiang1991 . The
authors also predicted that the negative potential can be used to generate
$\eta$-mesic atom with light/heavy nucleus. At nuclear saturation density
$\rho_{0}$, by incorporating the leading order terms in the coupled channel
approach a mass-shift of -20 MeV was evaluated Waas1997 . In Ref. Wang2010 ,
the optical potential of -72 MeV was anticipated. The authors predicted the
optical potential of -60 MeV at $\rho_{N}=\rho_{0}$ using the QMC model
Tsushima1998 . Using ChPT and relativistic mean-field model at nuclear
saturation density by including the $\eta N$ off-shell terms in the equation
of motion, the authors anticipated optical potential of -83 $\pm$ 5 MeV
Zhong2006 . Furthermore, using the same approach Song $et$ $al.$ obtained the
negative optical potential as a function of $\eta N$ scattering length
Song2008 . The authors studied the $\eta$ production rate and momentum
dependence under the influence of isospin asymmetric HICs in Ref. Chen2017 ,
also the effect of $\eta N$ interactions were studied using intranuclear
cascade model under the effect of distinct medium attributes David2018 .
Recently, using the combined (chiral SU(3) model $+$ ChPT) and solo (chiral
SU(3) model) approach, we derived $\eta N$ equation of motion in the non-
magnetized nuclear matter and observed (-54.61) -116.83 MeV mass-shift at
$\rho_{0}$ with $a^{\eta N}$=1.02 fm Kumar2020c . Evidently, the mass and
optical potential of $\eta$-mesons have the model dependencies and therefore
still need more exploration.

In this article, we investigated the magnetic field effect on the in-medium
mass and optical potential of the $\eta$-meson in the hot asymmetric nuclear
matter. In this work, we extended our previous study in the nuclear medium at
zero magnetic field Kumar2020c . First, we evaluated the in-medium dispersion
relation of $\eta$-meson using the $\eta N$ Lagrangian by the magnetically
induced scalar densities from the chiral SU(3) model Kumar2020c . In the
second approach, we used the scalar density of nucleons calculates using
chiral SU(3) model in the dispersion relation of $\eta$-mesons which is
derived from the chiral effective $\eta N$ Lagrangian of chiral perturbation
theory Zhong2006 .

The chiral perturbation theory is widely used to study the in-medium baryon-
meson interactions. For the first time, the theory was used to investigate the
in-medium properties of kaons Kaplan1986  and later it was modified by adding
leading order terms in the Lagrangian to study the interactions of $\eta$ with
nucleons Jenkins1991 . The theory was also used to study astrophysical objects
such as neutron stars. In the neutron star matter, the heavy baryon ChPT was
applied to study the kaon condensation Brown1994 ; Lee1995 ; Kaiser1995 .
Furthermore, to give correct description of $\eta N$ interactions, the next-
to-leading order terms were introduced in the ChPT Lagrangian. Consequently,
the authors anticipated more deep optical potential of $\eta$-mesons in the
nuclear medium Zhong2006 . The chiral SU(3) model is also widely used to study
the hot and dense hadronic matter Kumar2010 ; Zschiesche2004 ; Mishra2004 .
For instance, the methodology was used in the strange hadronic matter to study
the in-medium properties of kaons and antikaons Mishra2004 . Recently, the
mass and decay width of the $\phi$ meson was also calculated in the strange
hadronic matter by considering the $K\bar{K}$ loop at one-loop level
Kumar2020b . The chiral SU(3) model was extended to charm SU(4) and bottom
SU(5) sector to study the properties of heavy $D$ and $B$ mesons, respectively
Mishra2004a ; Mishra2009 ; Kumar2011 . The chiral model is also successfully
used to anticipate the in-medium properties of baryons and mesons in the
presence of strong external magnetic field. For instance, using the combined
approach of chiral model and QCD sum rules the in-medium mass and decay
constant of scalar, pseudoscalar, vector, and axial-vector $D$ mesons were
calculated with Kumar2020 ; Kumar2020a  and without incorporating the effect
of the external magnetic field Kumar2014 ; Chhabra2017 ; Chhabra2017a ;
Chhabra2018 . Using the same combination, the medium modified properties of
charmonia and bottomonia were studied in the hot magnetized asymmetric nuclear
matter Kumar2019 ; Kumar2019a ; Kumar2010 .

The outline of the present paper is as follows: In the coming section, we will
give a brief explanation of the magnetic field effect in the present work. In
Section II.1.1, we will derive the magnetic induced $\eta N$ interactions in
the chiral SU(3) model whereas, in Section II.1.2, $\eta N$ formalism will be
given in the joint approach of the chiral model and chiral perturbation
theory. In Section III, we will discuss the in-medium effects of strong
magnetic field on the mass of $\eta$-meson, and finally, in Section IV, we
will conclude our investigation.

##  II  Methodology

###  II.1 MAGNETIC FIELD INDUCED SCALAR FIELDS IN THE CHIRAL SU(3) MODEL

The hadronic chiral SU(3) model incorporates the trace anomaly and the non-
linear realization of chiral symmetry Weinberg1968 ; Coleman1969 ;
Zschiesche1997 ; Bardeen1969 ; Kumar2020 ; Papazoglou1999 ; Kumar2019
property of the QCD. In this methodology, the $\eta$-nucleon interactions are
computed by the exchange of scalar ($\sigma$, $\zeta$, $\delta$ and $\chi$)
and vector ($\omega$ and $\rho$) fields. The glueball field $\chi$ is
introduced in the model to preserve the broken scale invariance property of
QCD Kumar2020 . The isospin asymmetry, $I$, of the nuclear matter is
introduced by the addition of the scalar-isovector field $\delta$ and vector-
isovector field $\rho$ Kumar2010 . In the present work, the impact of the
strong magnetic field along $Z$-axis with the vector potential
$A^{\mu}=(0,0,Bx,0)$ is studied by including the magnetic induced Lagrangian
density to the chiral model’s effective Lagrangian density Kumar2019 ;
Reddy2018 . Thus, we write the modified Lagrangian density of chiral model as

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

Individually,

$\displaystyle{\cal L}_{NM}=-\sum_{i}\bar{\psi_{i}}\left[m_{i}^{*}+g_{\omega
i}\gamma_{0}\omega+g_{\rho i}\gamma_{0}\rho\right]\psi_{i},$ (2)
$\displaystyle{\cal L}_{vec}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}\right)\frac{\chi^{2}}{\chi_{0}^{2}}+g_{4}(\omega^{4}+6\omega^{2}\rho^{2}+\rho^{4}),$
(3) $\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}k_{0}\chi^{2}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)+k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)^{2}$
(4) $\displaystyle+$ $\displaystyle
k_{2}\left(\frac{\sigma^{4}}{2}+\frac{\delta^{4}}{2}+3\sigma^{2}\delta^{2}+\zeta^{4}\right)+k_{3}\chi\left(\sigma^{2}-\delta^{2}\right)\zeta$
$\displaystyle-$ $\displaystyle
k_{4}\chi^{4}-\frac{1}{4}\chi^{4}{\rm{ln}}\frac{\chi^{4}}{\chi_{0}^{4}}+\frac{d}{3}\chi^{4}{\rm{ln}}\Bigg{(}\bigg{(}\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\bigg{)}\bigg{(}\frac{\chi}{\chi_{0}}\bigg{)}^{3}\Bigg{)},$

$\displaystyle{\cal
L}_{SB}=-\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[m_{\pi}^{2}f_{\pi}\sigma+\big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\big{)}\zeta\right],$
(5)

and

${\cal
L}_{mag}=-{\bar{\psi_{i}}}q_{i}\gamma_{\mu}A^{\mu}\psi_{i}-\frac{1}{4}\kappa_{i}\mu_{N}{\bar{\psi_{i}}}\sigma^{\mu\nu}F^{\mu\nu}\psi_{i}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}.$
(6)

In Eq. (1), the first term ${\cal L}_{kin}$ denotes the kinetic energy term,
the second term ${\cal L}_{NM}$ given by Eq. (2) describes the nucleon-meson
interaction term. In this equation, the in-medium mass of nucleons is given as
$m_{i}^{*}=-(g_{\sigma i}\sigma+g_{\zeta i}\zeta+g_{\delta i}\tau_{3}\delta)$
where $\tau_{3}$ denotes the $z$th component of isospin quantum number and
$g_{\sigma i}$, $g_{\zeta i}$ and $g_{\delta i}$ denote the coupling strengths
of scalar-isoscalar field $\sigma$, scalar-isoscalar field $\zeta$ and scalar-
isovector field $\delta$ with nucleons ($i$=$p,n$) respectively. The term
${\cal L}_{vec}$ (Eq. (3)) generates the mass of vector mesons through the
interactions with scalar mesons and quartic self-interaction terms, ${\cal
L}_{0}$ represents the spontaneous chiral symmetry breaking where
$\sigma_{0}$, $\zeta_{0}$, $\delta_{0}$ and $\chi_{0}$ symbolize the vacuum
values of the $\sigma$, $\zeta$, $\delta$ and $\chi$ scalar fields,
respectively. To calculate the value of the $d$ parameter in the last term of
Eq. (4), we recall the QCD $\beta$ function at one loop level, for $N_{c}$
colors and $N_{f}$ flavours and is given by Schechter1980 ; Gomm1986

$\beta_{\rm{QCD}}\left(g\right)=-\frac{11N_{c}g^{3}}{48\pi^{2}}+\frac{N_{f}g^{3}}{24\pi^{2}}+O(g^{5}).$
(7)

In the above expression, the first term comes from the gluons self-interaction
(anti-screening) and the second term comes from interactions of quark pairs
(screening). For $N_{c}$=3 and $N_{f}$=3, we estimate the value of $d$ to be
6/33, whereas for $N_{c}$=3 and $N_{f}$=2, the parameter $d$ gives the value
4/33 Schechter1980 ; Gomm1986 ; Kumar2010 . In the present model, we use
$d$=0.064 Kumar2010 , which is fitted along with the other medium parameters
such as $k_{i}(i=1$ to $4)$ to generate the vacuum values of fields
($\sigma_{0}$, $\zeta_{0}$, $\delta_{0}$, $\chi_{0}$, $\omega_{0}$ and
$\rho_{0}$) and the masses of the nucleons and $\eta$, $\eta^{\prime}$ mesons
Papazoglou1999 ; Kumar2010 ; Kumar2019 . The values of fitted medium
parameters are tabulated in Table 1.

Furthermore, the ${\cal L}_{SB}$ in Eq. (5) denotes the explicit chiral
symmetry breaking property. The term ${\cal L}_{mag}$ accounts for the hadrons
interaction with the magnetic field. In Eq. (6), the symbol $\psi_{i}$
represents a wave function of $i$th nucleon and the second term describes the
tensorial interaction with the electromagnetic tensor, $F_{\mu\nu}$. Also, the
symbols $\mu_{N}$ and $k_{i}$ represent the nuclear magneton
($\mu_{N}$=$\frac{e}{2m_{N}}$) and anomalous magnetic moment of $i$th nucleon,
respectively.

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

$\displaystyle
k_{0}\chi^{2}\sigma-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\sigma-2k_{2}\left(\sigma^{3}+3\sigma\delta^{2}\right)-2k_{3}\chi\sigma\zeta$
$\displaystyle-\frac{d}{3}\chi^{4}\bigg{(}\frac{2\sigma}{\sigma^{2}-\delta^{2}}\bigg{)}+\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\pi}^{2}f_{\pi}=\sum
g_{\sigma i}\rho_{i}^{s},$ (8) $\displaystyle
k_{0}\chi^{2}\zeta-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\zeta-4k_{2}\zeta^{3}-k_{3}\chi\left(\sigma^{2}-\delta^{2}\right)$
$\displaystyle-\frac{d}{3}\frac{\chi^{4}}{\zeta}+\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right]=\sum
g_{\zeta i}\rho_{i}^{s},$ (9) $\displaystyle
k_{0}\chi^{2}\delta-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\delta-2k_{2}\left(\delta^{3}+3\sigma^{2}\delta\right)+2k_{3}\chi\delta\zeta$
$\displaystyle+\frac{2}{3}d\chi^{4}\left(\frac{\delta}{\sigma^{2}-\delta^{2}}\right)=\sum
g_{\delta i}\tau_{3}\rho_{i}^{s},$ (10)

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

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

and

$\displaystyle
k_{0}\chi\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)-k_{3}\left(\sigma^{2}-\delta^{2}\right)\zeta+\chi^{3}\left[1+{\rm{ln}}\left(\frac{\chi^{4}}{\chi_{0}^{4}}\right)\right]+(4k_{4}-d)\chi^{3}$
$\displaystyle-\frac{4}{3}d\chi^{3}{\rm{ln}}\Bigg{(}\bigg{(}\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\bigg{)}\bigg{(}\frac{\chi}{\chi_{0}}\bigg{)}^{3}\Bigg{)}+\frac{2\chi}{\chi_{0}^{2}}\left[m_{\pi}^{2}f_{\pi}\sigma+\left(\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right]$
$\displaystyle-\frac{\chi}{{{\chi_{0}}^{2}}}(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2})=0,$
(13)

respectively.

In the above equations, the symbols $m_{\pi}$, $m_{K}$, $f_{\pi}$ and $f_{K}$
represent the masses and decay constants of pions and kaons, respectively. The
isospin effect is measured by the parameter through definition,
$I=-\frac{\Sigma_{i}\tau_{3i}\rho^{v}_{i}}{2\rho_{N}}$. Furthermore,
$\rho^{s}_{i}$ and $\rho^{v}_{i}$ describe the magnetic field induced scalar
and vector densities of $i$th nucleons ($i=n,p$) Kumar2019 ; Broderick2000 ;
Broderick2002 . Due to Landau quantization, the magnetic field interact with
proton and neutron differently. For uncharged neutron the expressions for
scalar and vector densities are given as

$\displaystyle\rho^{s}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{s=\pm
1}\int^{\infty}_{0}k^{n}_{\bot}dk^{n}_{\bot}\left(1-\frac{s\mu_{N}\kappa_{n}B}{\sqrt{m^{*2}_{n}+\left(k^{n}_{\bot}\right)^{2}}}\right)\int^{\infty}_{0}\,dk^{n}_{\parallel}\frac{m^{*}_{n}}{\tilde{E}^{n}_{s}}\left(f^{n}_{k,s}+\bar{f}^{n}_{k,s}\right),$
(14)

and

$\displaystyle\rho^{v}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\sum_{s=\pm
1}\int^{\infty}_{0}k^{n}_{\bot}dk^{n}_{\bot}\int^{\infty}_{0}\,dk^{n}_{\parallel}\left(f^{n}_{k,s}-\bar{f}^{n}_{k,s}\right),$
(15)

respectively. Likewise, the scalar and vector densities for a charged proton
with the effect of Landau quantization are given by Broderick2000 ;
Broderick2002

$\rho^{s}_{p}=\frac{|q_{p}|Bm^{*}_{p}}{2\pi^{2}}\Bigg{[}\sum_{\nu=0}^{\nu_{max}^{(s=1)}}\int^{\infty}_{0}\frac{dk^{p}_{\parallel}}{\sqrt{(k^{p}_{\parallel})^{2}+(\bar{m}_{p})^{2}}}\left(f^{p}_{k,\nu,s}+\bar{f}^{p}_{k,\nu,s}\right)+\sum_{\nu=1}^{\nu_{max}^{(s=-1)}}\int^{\infty}_{0}\frac{dk^{p}_{\parallel}}{\sqrt{(k^{p}_{\parallel})^{2}+(\bar{m}_{p})^{2}}}\left(f^{p}_{k,\nu,s}+\bar{f}^{p}_{k,\nu,s}\right)\Bigg{]},$
(16)

and

$\displaystyle\rho^{v}_{p}=\frac{|q_{p}|B}{2\pi^{2}}\Bigg{[}\sum_{\nu=0}^{\nu_{max}^{(s=1)}}\int^{\infty}_{0}dk^{p}_{\parallel}\left(f^{p}_{k,\nu,s}-\bar{f}^{p}_{k,\nu,s}\right)+\sum_{\nu=1}^{\nu_{max}^{(s=-1)}}\int^{\infty}_{0}dk^{p}_{\parallel}\left(f^{p}_{k,\nu,s}-\bar{f}^{p}_{k,\nu,s}\right)\Bigg{]},$
(17)

respectively.

In the above equations,
$\bar{m}_{p}$=$\sqrt{m^{*2}_{p}+2\nu|q_{p}|B}-s\mu_{N}\kappa_{p}B$ defines the
effective mass of proton where symbol $\nu$ represents the Landau levels. The
effective energy of neutron and proton is given by

$\tilde{E}^{n}_{s}=\sqrt{\left(k^{n}_{\parallel}\right)^{2}+\left(\sqrt{m^{*2}_{n}+\left(k^{n}_{\bot}\right)^{2}}-s\mu_{N}\kappa_{n}B\right)^{2}},$
(18)

and

$\tilde{E}^{p}_{\nu,s}=\sqrt{\left(k^{p}_{\parallel}\right)^{2}+\left(\sqrt{m^{*2}_{p}+2\nu|q_{p}|B}-s\mu_{N}\kappa_{p}B\right)^{2}},$
(19)

respectively. Also, the symbols ${f}^{n}_{k,\nu,s}$, $\bar{f}^{n}_{k,\nu,s}$,
${f}^{p}_{k,s}$ and $\bar{f}^{p}_{k,s}$ define the finite temperature Fermi
distribution functions for nucleon and their antinucleons, and are given as

$\displaystyle f^{n}_{k,s}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\left[\beta(\tilde{E}^{n}_{s}-\mu^{*}_{n})\right]},\qquad\bar{f}^{n}_{k,s}=\frac{1}{1+\exp\left[\beta(\tilde{E}^{n}_{s}+\mu^{*}_{n})\right]}.$
(20) $\displaystyle f^{p}_{k,\nu,s}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\left[\beta(\tilde{E}^{p}_{\nu,s}-\mu^{*}_{p})\right]},\qquad\bar{f}^{p}_{k,\nu,s}=\frac{1}{1+\exp\left[\beta(\tilde{E}^{p}_{\nu,s}+\mu^{*}_{p})\right]}.$
(21)

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

####  II.1.1 $\eta$N INTERACTIONS IN THE MAGNETIZED NUCLEAR MATTER

In this subsection, we evaluate the in-medium mass of $\eta$ mesons via
dispersion relation in hot magnetized asymmetric nuclear matter. The medium
modified $\eta$ meson mass is obtained in terms of scalar and vector fields of
the chiral model which are solved by considering the interactions of nucleons
with $\eta$ mesons in the presence of an external magnetic field. These scalar
and vector fields modify the scalar and vector densities of the nucleons which
in result modifies the self-energy of the $\eta$ mesons.

The $\eta N$ interaction Lagrangian is given as

${\mathcal{L}}_{{\eta N}}={\mathcal{L}}_{{RT}}+{\cal L}_{\eta SB}+{\cal
L}_{d_{1}}^{BM}+{\cal L}_{d_{2}}^{BM}.$ (22)

The individual terms are given in detail as follows

  1. 1.

${\mathcal{L}}_{{RT}}$, The first range term:

The first term in the $\eta N$ Lagrangian comes from the first range term

${\mathcal{L}}_{{\mathrm{1strangeterm}}}=Tr(u_{\mu}Xu^{\mu}X+Xu_{\mu}u^{\mu}X),$
(23)

where
$u_{\mu}=-\frac{i}{2}\left[u^{\dagger}(\partial_{\mu}u)-u(\partial_{\mu}u^{\dagger})\right]$
and $u$=$\text{exp}\left[\frac{i}{\sqrt{2}\sigma_{0}}P\gamma_{5}\right]$. In
the present investigation, we have taken the interactions up to second order.
The $X$ and $P$, represent the scalar and pseudoscalar meson matrices
Zhong2006 , respectively and are explicitly given as

$X=\frac{1}{\sqrt{2}}\sigma^{a}\lambda_{a}=\left(\begin{array}[]{ccc}(\delta+\sigma)/\sqrt{2}&\delta^{+}&\kappa^{+}\\\
\delta^{-}&(-\delta+\sigma)/\sqrt{2}&\kappa^{0}\\\
\kappa^{-}&\overline{\kappa^{0}}&\zeta\end{array}\right),$ (24)

and

$P=\frac{1}{\sqrt{2}}\pi_{a}\lambda^{a}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\pi^{0}+{\frac{\eta}{\sqrt{1+2\,{w}^{2}}}}\right)&\pi^{+}&2\,{\frac{K^{+}}{w+1}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\pi^{-}&\frac{1}{\sqrt{2}}\left(-\pi^{0}+{\frac{\eta}{\sqrt{1+2\,{w}^{2}}}}\right)&2\,{\frac{K^{0}}{w+1}}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
2\,{\frac{K^{-}}{w+1}}&2\,{\frac{\overline{K}^{0}}{w+1}}&-{\frac{\eta\,\sqrt{2}}{\sqrt{1+2\,{w}^{2}}}}\end{array}\right).$
(25)

In Eq. (23), the calculations of the axial current of pions and kaons result
in the following relations

$\sigma_{0}=-f_{\pi}\qquad\zeta_{0}=-\frac{1}{\sqrt{2}}(2f_{K}-f_{\pi}),$ (26)

for the vacuum values of the scalar condensates $\sigma$ and $\zeta$ found in
the linear $\sigma$-model Papazoglou1999 . In the Eq. (25), the re-
normalization factor $w=\sqrt{2}\zeta_{0}/\sigma_{0}$ is incorporated to
obtain the canonical form of the kinetic energy terms Papazoglou1999 . The
matrix $P$ reduces to the matrix normally used in in chiral perturbation
theory Zhong2006  for $w=1$ (i.e. $f_{\pi}$=$f_{K}$). The advantage of $w\neq
1$ is that the SU(3)V breaking effects are accounted in the $P$ matrix for
even at lowest order Zhong2006 .

  2. 2.

${\cal L}_{\eta SB}$, The mass term:

The second term in Eq. (22), represents the scale breaking term of the chiral
model Lagrangian, which is given by

${\cal L}_{SB}=-\frac{1}{2}{\rm
Tr}A_{p}\left(uXu+u^{\dagger}Xu^{\dagger}\right),$ (27)

with $A_{p}$ as a diagonal matrix, given as

$A_{p}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}m_{\pi}^{2}f_{\pi}&0&0\\\
0&m_{\pi}^{2}f_{\pi}&0\\\
0&0&2m_{K}^{2}f_{K}-m_{\pi}^{2}f_{\pi}\end{array}\right).$ (28)

The $\eta$ meson vacuum mass is extracted from the Lagrangian [Eq. (27)] and
given as

$m_{\eta}=\frac{1}{f}\sqrt{\left(3m_{\pi}^{2}f_{K}m_{K}^{2}+\frac{8f_{K}^{2}m_{K}^{2}}{f_{\pi}^{2}}-\frac{4f_{K}m_{\pi}^{2}}{f_{\pi}}\right)}.$
(29)

Using the values of various constants, the value of $m_{\eta}$ turns out to be
574.374 MeV which is with an accuracy of 4.9 $\%$ of experimental mass, i.e.,
547.862 MeV PDG2020 . Moreover, using Gell-Mann Okubo mass formula under octet
approximation, the authors calculated the vacuum mass of $\eta$-meson as 567
MeV which is with an accuracy of 3.6 $\%$ of physical mass Burakovsky1997 . It
has been observed that the vacuum mass of $\eta$-meson has model dependencies
Burakovsky1997  but here in the current scenario, the in-medium mass-shift of
$\eta$-meson is nearly the same for both obtained masses and therefore can be
neglected.

  3. 3.

${\cal L}_{d_{1}}^{BM}+{\cal L}_{d_{2}}^{BM}$, The $d$ terms:

The last term in the Eq. (22) is called another range term which basically
arises from the baryon-meson interaction Lagrangian terms of chiral model
Mishra2004a ; Mishra2006  and are given as

${\cal L}_{d_{1}}^{BM}=\frac{d_{1}}{2}Tr(u_{\mu}u^{\mu})Tr(\bar{B}B),$ (30)

and,

${\cal L}_{d_{2}}^{BM}=d_{2}Tr(\bar{B}u_{\mu}u^{\mu}B).$ (31)

In above, $B$ denotes the baryon matrix, given as

$B=\frac{1}{\sqrt{2}}b^{a}\lambda_{a}=\left(\begin{array}[]{ccc}\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda^{0}}{\sqrt{6}}&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{\Sigma^{0}}{\sqrt{2}}+\frac{\Lambda^{0}}{\sqrt{6}}&n\\\
\Xi^{-}&\Xi^{0}&-2\frac{\Lambda^{0}}{\sqrt{6}}\end{array}\right).$ (32)

The explicit form of above three terms are inserted in Eq. (22) and the
interaction Lagrangian is given as

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

In above, the fields $\sigma^{\prime}(=\sigma-\sigma_{0})$,
$\zeta^{\prime}(=\zeta-\zeta_{0})$ and $\delta^{\prime}(=\delta-\delta_{0})$
are the digression of the expectation values of scalar fields from their
vacuum expectation values, the constant $f$, is equal to
$\sqrt{f_{\pi}^{2}+2(2f_{K}-f_{\pi})^{2}}$ and the parameter
$d^{\prime}$=$3d_{1}+d_{2}$.

At the mean-field level, the equation of motion for the $\eta$ meson field is
simplified to

$\displaystyle\partial^{\mu}\partial_{\mu}\eta+\left(m_{\eta}^{2}-\frac{(\sqrt{2}\sigma^{\prime}-4\zeta^{\prime})m^{2}_{\pi}f_{\pi}+8\zeta^{\prime}m^{2}_{k}f_{k}}{\sqrt{2}f^{2}}\right)\eta$
$\displaystyle+\frac{2d^{\prime}}{f^{2}}\left(\frac{\rho^{s}_{p}+\rho^{s}_{n}}{4}-\frac{\sigma^{\prime}+4\zeta^{\prime}(2f_{K}-f_{\pi})}{\sqrt{2}}\right)\partial^{\mu}\partial_{\mu}\eta=0.$
(34)

Furthermore, the dispersion relation for $\eta$ meson field is obtained by
Fourier transformation of the above equation

$-\omega^{2}+{\vec{k}}^{2}+m_{\eta}^{2}-\Pi^{*}(\omega,|\vec{k}|)=0,$ (35)

where $\Pi^{*}$ symbolize the in-medium self-energy of $\eta$ meson, and it is
explicitly given as

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

In the asymmetric nuclear matter, the in-medium mass of $\eta$ meson is
evaluated by solving Eq. (35) under the condition,
$m_{\eta}^{*}=\omega(|\vec{k}|$=0). The parameter $d^{\prime}$ in the
expression of self energies is estimated from the empirical value of
scattering length $a^{\eta N}$ of $\eta$ meson Zhong2006 , whose expression is
given as

$\displaystyle d^{\prime}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2\pi\left(1+\frac{m_{\eta}}{M_{N}}\right)}\frac{a^{\eta
N}}{m^{2}_{\eta}}+\frac{\sqrt{2}g_{\sigma
N}}{m^{2}_{\sigma}}-\frac{4\sqrt{2}(2f_{K}-f_{\pi})g_{\zeta
N}}{m^{2}_{\zeta}}$ (37) $\displaystyle-$
$\displaystyle\left(\frac{\sqrt{2}g_{\sigma
N}}{m^{2}_{\sigma}}-\frac{4g_{\zeta
N}}{m^{2}_{\zeta}}\right)\frac{m^{2}_{\pi}f_{\pi}}{\sqrt{2}m^{2}_{\eta}}-\tau_{3}\frac{4\sqrt{2}g_{\delta
N}m^{2}_{K}f_{K}}{m^{2}_{\delta}m^{2}_{\eta}},$

where $m_{\sigma}$, $m_{\zeta}$ , $m_{\delta}$ and $m_{N}$ denote the vacuum
masses of the fields $\sigma$, $\zeta$, $\delta$ and nucleons, respectively
and their values are given in Table 1. Using, the in-medium mass of $\eta$
mesons, the optical potential for $\eta$-meson for finite momentum Mishra2008
; Mishra2009  in the nuclear matter is given by

$U^{*}_{\eta}(\omega,\textbf{k})=\omega(\textbf{k})-\sqrt{\textbf{k}^{2}+m^{2}_{\eta}},$
(38)

and for zero momentum, the relation becomes

$U^{*}_{\eta}=\Delta m_{\eta}^{*}={m_{\eta}^{*}}-m_{\eta}.$ (39)

####  II.1.2  FUSION OF CHIRAL PERTURBATION THEORY AND CHIRAL SU(3) MODEL

Chiral Perturbation theory (ChPT) is one of the phenomenological approach to
study the low-energy dynamics of QCD with an effective field theory Lagrangian
based on the underlying chiral symmetry of quantum chromodynamics Zhong2006 .
In this, the SU(3)${}_{\mathrm{L}}\times$SU(3)R Lagrangian describing the
pseudoscalar mesons and baryons interactions is given as

$\displaystyle{\mathcal{L}_{\text{ChPT}}}={\mathcal{L}_{P}}+{\mathcal{L}_{PB}},$
(40)

where the pseudoscalar mesonic term, $\mathcal{L}_{P}$ is taken up to second
chiral order Zhong2006 ; Kaplan1986  and is given by Zhong2006 ,

$\displaystyle{\mathcal{L}_{P}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}f_{\pi}^{2}\textrm{Tr}\partial^{\mu}\Sigma\partial_{\mu}\Sigma^{\dagger}+\frac{1}{2}f_{\pi}^{2}B_{0}\left\\{\mbox{Tr}M_{q}(\Sigma-1)+\mathrm{H.c.}\right\\}.$
(41)

In above equation, $\Sigma=\xi^{2}=\exp{(i\sqrt{2}P/f_{\pi})}$, the symbol
$B_{0}$ represent the connection with order parameter of spontaneously broken
chiral symmetry and $M_{q}=\mbox{diag}\\{m_{q},m_{q},m_{s}\\}$ being the
current quark mass matrix. The second term in the Eq. (40), $\mathcal{L}_{PB}$
defines the leading order and next-to leading order baryon-meson interactions
Kaplan1986 . The off-shell terms are developed by using heavy baryon chiral
theory Jenkins1991 . However, the former theory has additional properties such
as quantum corrections and Lorentz invariance. The properties of the nuclear
system has been described successfully by using the off-shell Lagrangian and
the higher-order terms of this next-to-leading order Lagrangian are also
studied Park1993 . In the present article, we have limited our calculations up
to the small momentum scale, $Q^{2}$ without loop contributions (for s-wave
$\eta{N}$ scattering) because the higher order corrections are suppressed
Zhong2006 .

By using the heavy-baryon approximation and expanding the Eq. (40) up to the
order of $1/f_{\pi}^{2}$, we get the $\eta N$ Lagrangian as

$\displaystyle\mathcal{L_{\eta N}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{\mu}\eta\partial_{\mu}\eta-\frac{1}{2}\left(m{{}^{\prime}}_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{f_{\pi}^{2}}\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\right)\eta^{2}+\frac{1}{2}\frac{\kappa}{f_{\pi}^{2}}\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\partial^{\mu}\eta\partial_{\mu}\eta.$
(42)

In above equation, $m_{\eta}$ represent the mass of $\eta$-meson calculated in
ChPT and is evaluated by relation
$m{{}^{\prime}_{\eta}}^{2}=\frac{2}{3}B_{0}(m_{q}+2m_{s})$. In this mass
relation, $m_{q(s)}$ defines the mass of light (strange) quarks Burakovsky1997
. We have used the same value of $\eta$ meson vacuum mass i.e.
$m{{}^{\prime}}_{\eta}$=$m_{\eta}$= 574.374 MeV in the ChPT+chiral model
calculations for consistency with the chiral SU(3) model. Also, the
$\Sigma_{\eta\mathrm{N}}$, the $\eta N$ sigma term and the $\kappa$ term is
determined by relations

$\displaystyle\Sigma_{\eta\mathrm{N}}=-\frac{2}{3}[a_{1}m_{q}+4a_{2}m_{s}+2a_{3}(m_{q}+2m_{s})],$
(43)

and

$\displaystyle\kappa=4\pi
f_{\pi}^{2}\left(\frac{1}{m_{\eta}^{2}}+\frac{1}{m{{}^{\prime}}_{\eta}M_{\mathrm{N}}}\right)a^{\eta\mathrm{N}}-\frac{\Sigma_{\eta\mathrm{N}}}{m_{\eta}^{2}},$
(44)

respectively. The $a$ terms in the Eq. (43) corresponds to the chiral breaking
effects and are fitted from the parameter $\Sigma_{\mathrm{KN}}=380\pm 100$
MeV, where $\pm 100$ MeV reflects the uncertainty Lyubovitskij2001 ; Dong1996
; Hatsuda1994 ; Brown1994 ; Georgi1984 ; Politzer1991 ; Lee1995 ; Zhong2006 .
The parameter, $\kappa$ is estimated from the $\eta$N scattering length
Zhong2006  with the range of $a^{\eta\mathrm{N}}$ values i.e. 0.91 $\sim$ 1.14
fm, which is assumed from the empirical investigations Green2005 ; Renard2002
; Arndt2005 ; Green1999 ; Zhong2006 .

The equations of motion for $\eta N$ interactions in the unified approach of
chiral SU(3) model and ChPT can be written as by

$\displaystyle\left(\partial_{\mu}\partial^{\mu}+m_{\eta}^{2}-\frac{\Sigma_{\eta
N}}{2f_{\pi}^{2}}\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle+\frac{\kappa}{2f_{\pi}^{2}}\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle\partial_{\mu}\partial^{\mu}\right)\eta=0,$
(45)

where
$\langle\bar{\Psi}_{\mathrm{N}}\Psi_{\mathrm{N}}\rangle\equiv\rho^{s}_{N}$=$\left(\rho^{s}_{p}+\rho^{s}_{n}\right)$
is the magnetic field influenced scalar density of nucleon calculated within
the chiral SU(3) model. The plane wave decomposition of Eq. (45) gives

$\displaystyle-\omega^{2}+\vec{\textbf{k}}^{2}+m_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}\rho^{s}_{N}+\frac{\kappa}{2f_{\pi}^{2}}\rho^{s}_{N}\left(-\omega^{2}+\vec{\textbf{k}}^{2}\right)=0.$
(46)

By solving the above quadratic equation, we get

$\omega=\sqrt{{m_{\eta}^{*}}^{2}+\vec{\textbf{k}}^{2}},$ (47)

and the explicit expression of magnetic field induced mass of $\eta$ meson,
$m_{\eta}^{*}$ is given by

$\displaystyle
m_{\eta}^{*}=\sqrt{\left(m_{\eta}^{2}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}\rho^{s}_{N}\right)\Big{/}\left(1+\frac{\kappa}{2f_{\pi}^{2}}\rho^{s}_{N}\right)}.$
(48)

The last two terms of the Eq. (46) gives the $\eta$-meson self-energy

$\displaystyle\Pi^{*}(\omega,\vec{\textbf{k}})=\Big{(}-\frac{\Sigma_{\eta\mathrm{N}}}{2f_{\pi}^{2}}+\frac{\kappa}{2f_{\pi}^{2}}(-\omega^{2}+\vec{\textbf{k}}^{2})\Big{)}\rho^{s}_{N},$
(49)

where $\omega$ is $\eta$-meson single-particle energy and $\vec{\textbf{k}}$
is the momentum.

##  III Results and Discussions

In this section, we discuss the magnetic field induced optical potential of
$\eta$ meson evaluated using two approaches i.e. (i) chiral SU(3) model in
Section III.1 and (ii) ChPT + chiral SU(3) model in Section III.2. In both
methodologies, we have taken the values of scattering length, $a^{\eta_{N}}$
in the range 0.91-1.14 fm. We start by discussing the in-medium behavior of
nucleon scalar densities under the influence of a strong magnetic field for
different values of nuclear density, isospin asymmetry, and temperature.

In Figure 1, at nuclear saturation density, we illustrate the scalar density
of neutron and proton as a function of temperature for zero and non-zero
values of the magnetic field. In the left (right) column of the figure, we
present the scalar densities for symmetric (anti-symmetric) nuclear matter.
For symmetric nuclear matter and zero magnetic field, we observe the same
behavior of neutron and proton scalar density with temperature. The scalar
densities slowly decrease linearly up to $T\approx$150 MeV and start
increasing for higher values of temperature. These modifications reflect the
interplay between the contributions from higher momenta states and the thermal
distribution functions in the scalar density expressions [see Eqs. (16) and
(14)]. Further, on increasing the magnetic field the proton and neutron scalar
density behave unevenly, for a particular value of temperature, the proton
scalar density increases significantly whereas the neutron scalar density
slightly decreases. The additional effects in proton scalar density are
because of the charged nature of proton, the positively charged proton
interacts with the magnetic field and experiences Landau quantization and
contributions from the anomalous magnetic moment [see Eq. (16)]. Whereas due
to uncharged nature, the neutron only experiences contributions from the
anomalous magnetic moment [see Eq. (14)]. Moreover, the transition temperature
decrease with the increase in the magnetic field which supports the
restoration of chiral symmetry Kumar2020 . On the other side, for asymmetric
nuclear matter [sub-plot (b) and (d)], even for zero value of magnetic field
we find unequal values of proton and nucleon scalar density. In the chiral
SU(3) model, the scalar densities of protons and neutrons are calculated
through Eq. (16) and Eq. (14), respectively and these equations include the
effect of in-medium scalar and vector fields Papazoglou1999 . In asymmetric
nuclear matter the iso-scalar(vector) $\delta(\rho)$ field show non-zero
contributions which eventually leads to unequal values of proton and neutron
densities Kumar2020c . The neutron scalar density modifies significantly in
asymmetric nuclear medium and decreases with the increase in the temperature
whereas the proton scalar density shows zero value up to $T\approx$ 90 MeV and
then increases rapidly. Naively, the value of proton scalar density should be
zero for $I$=0.5 but at higher temperatures, despite $\rho^{v}_{p}$=0, the
proton condensate ($\bar{p}p$) still populates in the nuclear matter. The
inclusion of magnetic field does significant changes in the proton scalar
density whereas the neutron scalar density shows a small decrement with the
increasing magnetic field.

In Figure 2, we plot the scalar densities for the same values of medium
parameters but $\rho_{N}$=4$\rho_{0}$. On the same line, at $eB$=0 and $I$=0,
we observe similar behavior of proton and neutron scalar density. When we move
from zero to non-zero values of magnetic field strength, for a particular
value of temperature we observe the proton scalar density increase appreciably
whereas the neutron scalar density slightly decreases. This is again due to
the additional energy levels in the protons due to the magnetic field
intervention. It is to be noted that the effect of the magnetic field is more
pronounced in the high density regime. Furthermore, in the highest value of
medium asymmetry, as a function of temperature, we observe that the proton
scalar density remains zero up to $T\approx$50 MeV and further increases non-
linearly with the increase in the magnetic field. On the contrary, the neutron
scalar density modifies appreciably in the asymmetric matter, it decreases
with the increase in temperature and magnetic field. In the symmetric nuclear
matter, the observed scalar densities at zero magnetic field are in agreement
with the results of the relativistic mean-field model Zhong2006 ; Song2008 .

Figure 1: (Color online) The in-medium scalar density of nucleons at
$\rho_{0}$.  Figure 2: (Color online) The in-medium scalar density of nucleons
at 4$\rho_{0}$.

###  III.1 Impact of Magnetic Field on the $\eta$ Mesons in Chiral Model

In this section, we present the result and discussion for the in-medium mass
of $\eta$-meson calculated in the chiral SU(3) model under the influence of
the external magnetic field. In Figures 3 and 4, we illustrate the in-medium
$\eta$ mass as a function of temperature for different other parameters such
as isospin asymmetry, magnetic field, and scattering length at
$\rho_{N}$=$\rho_{0}$ and 4$\rho_{0}$, respectively. In Figure 3, for any
value of $I$, $eB$ and $a^{\eta N}$, we observe the in-medium mass gradually
increases with the increase in temperature up to a certain value of
temperature and then it starts decreasing. This behavior represents the
opposite reflection of in-medium scalar densities plotted in Figure 1 as the
expression of $\eta$-meson [see Eq. (48)] has an inverse dependence on the sum
of scalar densities of nucleons. In symmetric nuclear matter, the impact of
the magnetic field leads to a more attractive contribution in the in-medium
$\eta$ mass for a particular value of temperature, scattering length. With the
increase in the magnetic field, we observe the transition point (i.e. the
temperature where in-medium mass starts decreasing as a function of
temperature) moves towards the lower temperature side. As discussed earlier,
the medium modified mass of $\eta$-meson has indirect dependence on the sum of
the nucleon’s scalar densities and therefore it shows opposite behavior to the
scalar densities. However, in the asymmetric nuclear matter, we observe that
the in-medium mass slowly increases for zero value of the magnetic field as
was observed for the symmetric matter. This is because the in-medium mass
depends upon the sum of scalar densities with no additional parameter
dependence. Further, at $I\neq$0, we observe a little change in $\eta$ mass
for the lower temperature region whereas a substantial change in the higher
temperature region concerning the magnetic field. This is because in the
highly asymmetric matter for lower (higher) temperatures, we have a negligible
(substantial) contribution to the proton scalar density. It is to be noted
that due to the uncharged nature of $\eta$-meson it does not couple with
magnetic field directly and therefore does not show any additional
contributions from Landau energy levels as was observed for charged $D$ mesons
Kumar2020 ; Reddy2018 ; Kumar2020a . From fig. 3, we also anticipate the
effect of scattering length. When we change $a^{\eta N}$ from 0.91 to 1.14 fm,
we observe a significant decrement in the effective mass for a particular
value of magnetic field, and temperature. This is because of the parameter
$d^{\prime}$’s direct relationship with the scattering length in Eq. (37). The
$d^{\prime}$ parameter gives attractive contribution to the $\eta$ in-medium
mass through equation of motion [Eq. (35)] and self-energy [Eq. (36)].

In Figure 4, we plot the in-medium mass of $\eta$-meson for same values of
medium parameters but $\rho_{N}$=4$\rho_{0}$. In the left panel, at $eB$=0 and
$I$=0, we observe a similar trend of in-medium mass as a function of
temperature as was observed for $\rho$=$\rho_{0}$. For high nuclear density,
we observe a significant drop in the mass of $\eta$-meson. The drop in
effective mass increase with the increase in the magnetic field and scattering
length. When we move from symmetric nuclear matter to asymmetric nuclear
matter, we observe the cross-over behavior of in-medium mass as a function of
temperature for a particular value of scattering length. This is due to a
similar reason that was discussed for the nuclear saturation density case. The
difference is, here in the high density regime the proton scalar density
populates little (but greater than the $\rho$=$\rho_{0}$ case) in the lower
temperature regime and substantially in the higher temperature regime. Also,
the effect of the magnetic field is more pronounced in the high density
regime.

Furthermore, for a better understanding of in-medium $\eta$-meson mass, in
Figure 5, we plot the individual terms of $\eta$-meson’s self-energy. The
expression of self-energy [Eq. (36)] have three interaction terms: (i) first
range term (ii) mass term and (iii) $d^{\prime}$ term. At nuclear saturation
density, in this figure, we show the contributions of the individual terms as
a function of temperature and asymmetry for $a^{\eta N}$=1.02 fm. At zero
magnetic field and asymmetry, we anticipate that the first range term gives a
significant repulsive contribution to the in-medium mass whereas the mass and
$d^{\prime}$ terms give little and significant attractive contributions,
respectively. For the non-zero magnetic field (asymmetry), the modification in
the $d^{\prime}$ term becomes more (less). This behavior is due to the
presence of nucleon’s scalar density terms in the second term of the self-
energy expression [Eq. (36)]. The $d^{\prime}$ term dependence emphasizes the
importance of scattering length $a^{\eta N}$, in the eta-nucleon interactions.

The optical potential at zero and non zero momentum can be used to study the
$\eta$-mesic nuclei Jenkins1991 ; Kumar2020c ; Zhong2006  and eta-meson
momentum dependence Berg1994 ; Chen2017 ; David2018 . In Figure 6, in
symmetric nuclear matter we plot the optical potential as a function of medium
momentum $\lvert\textbf{k}\rvert$ for various values of magnetic field and
density at $a^{\eta N}$=0.91 fm. In this figure at $\rho_{N}$=$\rho_{0}$, we
observe that the magnitude of optical potential decreases with the increase in
momentum. With the increase in the magnetic field (temperature), we observe
the drop in optical potential become more (less). The behavior of in-medium
optical potential reflects the interplay between the in-medium mass and
momentum which can be understood from the expression given by Eq. (38). At
higher values of the momentum $\lvert\textbf{k}\rvert$, in the optical
potential curve, the contribution of effective mass is suppressed by the
increase in momentum states. Furthermore, in the right panel ,i.e., high
density regime, we anticipate deep optical potential which becomes less as
momentum states increase. In Figures 7 and 8, we find likewise trend of
optical potential with $\eta$ momentum. In these figures, we find a more deep
optical potential with the increasing scattering length. The behavior of
optical potential with scattering length and other medium parameters can be
understood in terms of in-medium mass. For a more clear picture, we listed the
values of in-medium optical potential in chiral SU(3) model at
$\lvert\textbf{k}\rvert$=0 in Table 2.

|  |  $I$=0 |  $I$=0.5
---|---|---|---
| $a^{\eta N}(\text{fm})$ | T=0 | T=100 | T=0 | T=100
|  |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$
| 0.91 | -46.60 | -52 | -38.24 | -46.36 | -45.17 | -42.78 | -38.21 | -38.23
$\Delta m^{*}_{\eta}$ | 1.02 | -55 | -61 | -45.66 | -54.59 | -53.42 | -50.69 | -45.62 | -45.64
| 1.14 | -63.75 | -70.30 | -53.41 | -63.14 | -62 | -59 | -53.36 | -53.38
Table 2: In-medium mass-shift (MeV) of $\eta$-meson with and without taking
the effect of magnetic field at $\rho=\rho_{0}$ and different parameters
calculated in the chiral SU(3) model.

###  III.2 Impact of Magnetic Field on the $\eta$ Mesons in ChPT+Chiral Model

In this section, we evaluate the in-medium mass of $\eta$-meson mass using the
joint approach of chiral SU(3) model and chiral perturbation theory and also
compared it with the results calculated in the chiral SU(3) model alone. As
discussed in the Section II.1.2, the $\eta N$ equation of motion is derived
from the ChPT $\eta N$ Lagrangian density. The magnetic field influenced
scalar density of nucleons in the ChPT self-energy [Eq. (49)] is taken from
the chiral SU(3) model discussed in Section II.1. In the present work, we took
the value of parameter $\Sigma_{\eta N}$ as 280 MeV by neglecting the
uncertainties in the parameter Kumar2020c . We will see later that the
contribution of $\Sigma_{\eta N}$ term is very less as compared to the kappa
term.

In Figures 9 and 10, we plot the mass ratio $m^{*}_{\eta}/m_{\eta}$ with
respect to temperature, scattering length and isospin asymmetry at
$\rho_{N}$=$\rho_{0}$ and 4$\rho_{0}$, respectively. In these figures, we have
also compared the in-medium mass evaluated from the different approaches i.e.
(i) ChPT and chiral model (ii) chiral model alone. In Figure 9, at nuclear
saturation density, using the joint approach, we observe a significant
decrement in the in-medium mass of $\eta$-meson. We find a similar behavior of
the medium modified $\eta$-meson mass concerning the magnetic field, isospin
asymmetry, and scattering length as was found in the observations of the
chiral SU(3) model. The substantial decrement in the joint approach lies in
the fact that there is no term having repulsive contribution term in the ChPT.
The net contribution in ChPT comes from the $\Sigma_{\eta N}$ and $\kappa$
term (both attractive in nature). In Figure 10, at a higher value of nuclear
density, we observe that the trend of mass ratio with temperature remains the
same but here we get more negative mass-shift. The ratio shows similar
behavior concerning other medium parameters and scattering length. To have a
clear understanding, in fig. 11 at $\rho_{N}$=$\rho_{0}$ and $a^{\eta N}$=1.02
fm, we have illustrated the in-medium behavior of individual terms present in
the ChPT self-energy in magnetized asymmetric nuclear matter which contribute
to the in-medium mass of $\eta$-meson through Eq. (48). From this figure, we
observe that the contribution of $\Sigma_{\eta N}$ term is negative but very
less as compared to $\kappa$ term. The $\kappa$ term has a significant
attractive contribution to the in-medium mass because, in the in-medium mass
mathematical relation given by Eq. (48), the numerator has negative
contribution of nucleon scalar density whereas the denominator has a positive
contribution. Therefore, due to this inverse relationship, with the increase
in scalar density the value of effective mass decreases. The values of in-
medium optical potential at zero momentum calculated using ChPT+chiral model
are given in table 3.

To the best of our knowledge, no work has been done to study the effect of
magnetic field on the in-medium mass of $\eta$-mesons. The current results at
zero magnetic field can be compared with the existing literature Zhong2006 ;
Waas1997 ; Tsushima1998 ; Song2008 . In our previous work at zero magnetic
field, we have elaborately compared the results and observed that the findings
of different papers are in agreement for varying values of scattering length
Kumar2020c .

|  |  $I$=0 |  $I$=0.5
---|---|---|---
| $a^{\eta N}(\text{fm})$ | T=0 | T=100 | T=0 | T=100
|  |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$ |  $eB$=0 |  $eB$=5${{m^{2}_{\pi}}}$
| 0.91 | -107.57 | -115.95 | -93.79 | -105.70 | -105.25 | -102.11 | -93.72 | -93.70
$\Delta m^{*}_{\eta}$ | 1.02 | -116.83 | -125.69 | -102.21 | -114.89 | -114.35 | -110.22 | -102.11 | -102.17
| 1.14 | -126.36 | -135.64 | -110.96 | -124.32 | -123.75 | -119.42 | -110.86 | -110.93
Table 3: In-medium mass-shift (MeV) of $\eta$-meson with and without taking
the effect of magnetic field at $\rho=\rho_{0}$ and different parameters
calculated in the ChPT+chiral SU(3) model. Figure 3: (Color online) The in-
medium $\eta$ meson mass in chiral model at $\rho_{0}$ .  Figure 4: (Color
online) The in-medium $\eta$ meson mass in chiral model at 4$\rho_{0}$ .
Figure 5: (Color online) The different terms of in-medium $\eta$ meson mass in
chiral model at $\rho_{0}$ and $a^{\eta N}$=1.02 fm.  Figure 6: (Color online)
The in-medium $\eta$ meson optical potential in chiral model at $a^{\eta
N}$=0.91 fm and $I$=0. Figure 7: (Color online) The in-medium $\eta$ meson
optical potential in chiral model at $a^{\eta N}$=1.02 fm and $I$=0. Figure 8:
(Color online) The in-medium $\eta$ meson optical potential in chiral model at
$a^{\eta N}$=1.14 fm and $I$=0. Figure 9: (Color online) Comparison of in-
medium $\eta$ meson mass at $\rho_{0}$.  Figure 10: (Color online) Comparison
of in-medium $\eta$ meson mass at 4$\rho_{0}$.  Figure 11: (Color online) The
different terms of in-medium $\eta$ meson mass in ChPT+chiral model at
$\rho_{0}$ and $a^{\eta N}$=1.02 fm.

##  IV SUMMARY

To summarize, we studied the effect of an external magnetic field on the in-
medium mass of $\eta$-meson in the hot asymmetric nuclear matter. We studied
the in-medium $\eta N$ interactions using two separate methodologies. In the
first approach, we computed the in-medium mass-shift of $\eta$-meson using the
chiral SU(3) model and observed a decrement in the effective mass as a
function of the magnetic field and nuclear density. We anticipated substantial
medium effects in the regime of high magnetic field and density. In the second
approach, we used the combined method of chiral perturbation theory (ChPT) and
chiral SU(3) model to compute the in-medium properties of $\eta$-meson. In the
latter approach, we introduced the medium effects through the nucleon scalar
density which is calculated in the chiral SU(3) model. Using the joint
approach, we found a substantial decrease in the mass of $\eta$-meson
concerning the magnetic field and nuclear density which is much deeper than
the observations of the first approach. The effects of isospin asymmetry and
temperature are also incorporated and found to be a little repulsive. In both
approaches, we observe a direct dependence of negative mass-shift with
$a^{\eta N}$ scattering length. Furthermore, due to zero charge on the $\eta$
meson, we do not observe Landau quantization therefore no additional energy
levels were discovered. The optical potential at finite momentum can be used
to study the experimental properties such as momentum dependence David2018 ;
Chen2017 ; Berg1994  and $\eta$-meson production rate Peng1987 ; Martinez1999
; Agakishiev2013  in the magnetized nuclear medium. Also, the observed
negative mass-shift can be used to study the possibility of $\eta N$ bound
states formation Jenkins1991 ; Zhong2006 . Also, the magnetic field influenced
optical potential may be used in future experiments to study the in-medium
observables of $\eta$-mesons Rapp2010 ; Vogt2007 .

## Acknowledgment

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

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