# Impact of finite volume on kaon, antikaon, and $\phi$ meson masses and decay
width in asymmetric strange hadronic matter

Zeeshan Ahmad<EMAIL_ADDRESS>Department of Physics, Dr. B R Ambedkar
National Institute of Technology Jalandhar, Jalandhar – 144008, Punjab, India
Nisha Chahal<EMAIL_ADDRESS>Department of Physics, Dr. B R Ambedkar
National Institute of Technology Jalandhar, Jalandhar – 144008, Punjab, India
Arvind Kumar<EMAIL_ADDRESS>Department of Physics, Dr. B R Ambedkar
National Institute of Technology Jalandhar, Jalandhar – 144008, Punjab, India
Suneel Dutt<EMAIL_ADDRESS>Department of Physics, Dr. B R Ambedkar National
Institute of Technology Jalandhar, Jalandhar – 144008, Punjab, India

###### Abstract

In the present work, we investigate the impact of finite volume on the in-
medium properties of kaons ($K^{+}$, $K^{0}$) and antikaons ($K^{-}$,
$\bar{K^{0}}$), and $\phi$ mesons in the isospin asymmetric strange hadronic
medium at finite density and temperature. We use the chiral SU(3) hadronic
mean-field model, which accounts for the interactions between baryons through
the exchange of scalar ($\sigma,\zeta,\delta$) and vector ($\omega$, $\rho$,
$\phi$) fields. To investigate the effects of finite volume, we apply the
multiple reflection expansion (MRE) technique for calculations of the density
of states. The non-strange scalar field $\sigma$ shows significant variation
in an asymmetric medium, while the strange scalar field $\zeta$ shows good
dependency in the strange medium. We use the medium-modified masses of kaons
and antikaons calculated using the chiral SU(3) model to obtain the masses and
decay width of $\phi$ mesons in finite volume hadronic medium. To obtain the
masses and decay widths of $\phi$ mesons, an effective Lagrangian approach
with $\phi$$K$$\bar{K}$ interactions at one-loop level is used in the present
work. We obtain the effective masses and decay widths in the finite volume
matter, for the spherical geometry of the medium with Neumann and Dirichlet
boundary conditions as well as for the cubic geometry. The finite volume
effects are found to be appreciable at high baryon densities.

##  I Introduction

The study of the in-medium properties of hadrons at finite temperatures and
densities continues to receive significant attention due to their significant
role in understanding the strong interaction physics of the quantum
chromodynamics (QCD) phase diagram [1, 2, 3, 4, 5, 6, 10, 8, 7, 9, 11]. At low
temperatures and baryon chemical potential, colorless hadrons appear as
essential constituents in the QCD [12]. However, as the temperature and/or
baryon density of the medium is increased, a phase transition occurs, leading
to the deconfinement of hadrons into quarks and gluons, and quark-gluon plasma
(QGP) phase may form [13, 14, 15, 16, 17, 18, 19, 20]. Lattice QCD uses a
grid-based simulation technique to explore the phase diagram of QCD at low
baryonic densities, shedding light on the behavior of quarks and gluons in
strongly interacting medium [21]. At the zero value of baryon chemical
potential, lattice calculations estimate the temperature for the chiral
crossover from hadronic to QGP phase at around $T_{c}\approx 156$ MeV [22, 23,
24, 13]. Though the lattice simulations have been successful at the chemical
potentials values close to vicinity of zero, but due to the sign problem,
these techniques are not applied to study the phase transition at high baryon
chemical potential values [25, 26].

In the experimental facilities of heavy-ion collisions (HICs) such as Large
Hadron Collider (LHC) at CERN and Relativistic Heavy-Ion Collider (RHIC) at
BNL, USA, hot and dense matter at high temperatures and low baryon chemical
potential has been produced [27], whereas the future Facility for Antiproton
and Ion Research (FAIR) project at GSI-Germany, Nuclotron-based Ion Collider
fAcility (NICA) at Dubna, and the Japan Proton Accelerator Research Complex
(J-PARC) aim to examine the QCD phase diagram at moderate temperatures and
high baryon densities [28]. In the high-energy HICs, when two isospin
asymmetric nuclei collide, the QGP fireball, which may be produced initially,
further transforms into a hadronic phase with the decrease in temperature. The
size of the fireball produced in these experiments is not of infinite extent,
but has a finite size which depends upon several factors such as the
collisions centrality, size of colliding nuclei, and center of mass energy
[29, 30, 31].

Numerous studies have been conducted to estimate the critical endpoint and for
understanding the thermodynamics of strongly interacting matter, particularly
focusing on the impact of finite volume of QCD matter [32, 33, 34, 35].
Studies have shown that the location of the critical endpoint and the behavior
different thermodynamic quantities is notably influenced by consideration of
the finite volume of the medium. According to Polyakov linear sigma model
(PLSM) [36, 37], with decreasing system volume, the quark-hadron phase
boundary is shifted towards higher values of $\mu$ and $T$ [38], while the
critical endpoint shifts towards higher $\mu$ and lower $T$ [34, 39]. In
theoretical computations, the volume of strongly interacting matter can be
conceptualized with various sizes and shapes. Extensive studies have focused
on examining the consequences of a finite volume on the QCD phase diagram
using a cubic box [40, 41, 42, 43]. However, some studies have considered the
more realistic spherical shape [44, 45]. The multiple reflection expansion
(MRE) method, which accounts for surface and curvature effects, refines the
computation of the density of states by including essential corrections for
small systems [46, 47]. The effects of finite size have been explored through
different approaches, such as chiral perturbation theory [48, 49], the quark-
meson model [50, 51, 52, 53], the Dyson-Schwinger equations employing anti-
periodic boundary conditions [54], the Polyakov loop extended Nambu–Jona-
Lasinio (NJL) model [33, 55], and various non-perturbative renormalization
group techniques [56]. The finite size effects on the properties of strongly
interacting matter have been explored using NJL [40, 35] and relativistic mean
field model [57, 58], applying anti-periodic boundary conditions for the
fermions in the imaginary time coordinate, imposed via Kubo-Martin Schwinger
conditions. In the spatial coordinate directions, no restrictions on the
boundary conditions were imposed, and choice of these led to significant
observations on the masses of mesons. A comprehensive review of finite size
effects is available in Ref. [59]. The above-listed studies using different
non-perturbative techniques have mostly considered the effects of finite size
on the quark matter phase. However, the impact of finite size effects on the
physical properties, which are calculated in hot and dense hadronic fireball
has not been studied much. In Ref. [60], the transport coefficients of
hadronic matter within the framework of the Hadron Resonance Gas (HRG) model
have been studied employing lower momentum cut-off.

The in-medium masses and decay width of different mesons at finite baryon
density and temperatures have been studied using different approaches [62, 19,
63, 64, 65, 66, 67, 68, 69, 61, 70, 71, 72, 74, 73, 75]. The modifications of
optical potentials play a significant role in understanding various
experimental observables [76, 77, 78]. The investigations of the in-medium
characteristics of kaons and antikaons were initiated by Kaplan and Nelson
[79, 80]. The importance of $K$ and $\bar{K}$ isospin doublets in heavy-ion
collisions has led to extensive theoretical [81, 82, 83, 84, 85, 86, 87, 88,
65] and experimental [89, 90, 91] research. The low-energy antikaon–proton
scattering data was successfully explained in terms of s-wave coupled SU(3)
channels in [92, 93], one of the first interesting attempts in this area.
Moreover, the characteristics of $K$ and $\bar{K}$ mesons are investigated in
isospin asymmetric nuclear [94] and strange matter [81] using the chiral SU(3)
model [95]. In the various non-perturbative approaches described above to
calculate the in-medium masses or decay width of different mesons, the dense
hadronic medium is assumed to be of infinite extent. It will be of interest
and also important to examine the impact of finite volume of the hadronic
phase on the optical potentials of different mesons. In the present work, we
investigate the in-medium masses of pseudoscalar kaons,
$K\left(K^{+},K^{0}\right)$ and antikaons
$\bar{K}\left(K^{-},\bar{K^{0}}\right)$ in a finite volume isospin asymmetric
hot and dense medium composed of nucleons and hyperons. Further, using these
medium modified masses of $K$ and $\bar{K}$ mesons, we shall also study the
in-medium masses and decay width of $\phi$ mesons in finite volume matter.

The motivation to investigate the in-medium properties of $\phi$ mesons is as
follows. Light vector mesons ($\omega$, $\rho$, and $\phi$) play a crucial
role in dilepton formation in HICs, prompting interest in both theoretical and
experimental investigations [96, 3, 97, 98, 99, 100, 101, 102]. Because
dileptons interact weakly with baryons and mesons, these are regarded as a
promising observable to study the properties of hadronic medium [103, 104,
105, 106, 107]. In the case of $\phi$ mesons, despite being predominantly
composed of strange quarks, they interact strongly with nuclei containing up
and down quarks as well as with strange baryons and mesons [66, 108, 109, 99].
Since almost 85% of the $\phi$ meson decays to kaons hence, in-medium
properties of kaons and antikaons play a crucial role in the determination of
the properties of $\phi$ mesons. For $\phi$ mesons, at normal nuclear matter
density, the KEK-E325 collaboration [110] observed an increase in the decay
width by 3.6 times with a decrease in mass by 3.4%. SPring8 facility [111],
revealed a substantial in-medium $\phi$N cross-section in comparison to
KEK-E325 collaboration, leading to a decay width of 35 MeV. These results are
consistent with other experiments [112, 113]. In an effort to shed light on
the discrepancies, the CLAS collaboration at JLab conducted new measurements
of nuclear transparency ratios. They estimated in-medium widths to be in the
range of 22–100 MeV [114], consistent with the measurement from SPring8 [111].
The ANKE-COSY collaboration [115] measured $\phi$ meson production from
proton-induced reactions on various nuclear targets and suggested an in-medium
$\phi$ width of approximately 50 MeV. Many theoretical studies have predicted
a downward shift in the in-medium $\phi$ meson mass along with a broadening of
its decay width. Hatsuda and Lee used the QCD sum rule approach to calculate
the in-medium $\phi$ meson mass and predicted a decrease in the range of
1.5%–3% at normal nuclear matter density [62, 116]. In Ref. [117], authors,
taking into account the renormalization of the $K$ and $\bar{K}$ mesons
through a coupled channel approach in the medium, forecast a decay width of 22
MeV for $\phi$ mesons. Meanwhile, the work of Ref. [118] indicates a negative
mass shift of less than 1% and a decay width of 45 MeV at normal nuclear
matter density. In Ref. [119], authors combined a model based on chiral
effective field theory and finite-energy QCD sum rules and observed a downward
mass shift of less than 2% and a significant widening of the width to 45 MeV.
Furthermore, in Ref. [109], building upon previous work [62, 116], the authors
reported a negative mass shift of 3.4% and a substantial decay width of 70 MeV
at normal nuclear matter density.

As stated before, in the current work we shall use an effective chiral SU(3)
hadronic mean field model [95] to investigate the in-medium properties of
kaons and antikaons as well as vector $\phi$ mesons in a finite volume strange
hadronic medium. To incorporate the finite volume effects, we shall follow the
approach of the MRE method, incorporating surface and curvature effects, to
evaluate the density of states, as discussed in detail in the later part of
the manuscript. This article is organized as follows. In Sec. II.1, we briefly
describe the framework of the chiral SU(3) model, incorporating corrections
for finite-size effects. The effective masses of the $K$ and $\bar{K}$ mesons
and the $\phi$ meson mass and decay width in the finite volume hadronic medium
are computed in Secs. II.2 and II.3, respectively. Results of present
investigations are discussed in Sec. III. Finally, a summary is presented in
Sec. IV.

##  II Methodology

###  II.1 Hadronic Chiral SU(3) Model

To study the impact of finite volume on the in-medium masses of kaons and
antikaons, the chiral SU(3) hadronic mean-field model based on the nonlinear
realization of chiral symmetry is used in the present work [95, 120, 121,
122]. In this model, the interactions between baryons are described via the
exchange of the scalar fields $\sigma,\zeta$ and $\delta$ and vector fields
$\omega,\rho$ and $\phi$. The non-strange scalar meson, $\sigma$, is composed
of light $u$ and $d$ quarks, while the strange scalar meson $\zeta$ contains
strange $s$ quark [123]. The scalar iso-vector field $\delta$ and vector iso-
vector field $\rho$ contribute when the medium has finite isospin asymmetry.
Furthermore, the scalar dilaton field $\chi$ representing the glueball field
is introduced in the model to incorporate the broken scale invariance property
of QCD [95, 124].

The Lagrangian density for hadronic chiral SU(3) effective mean-field model is
expressed as [95]

$\displaystyle\mathcal{L}_{\text{chiral}}$
$\displaystyle=\mathcal{L}_{\text{kin}}+\sum_{M=S,V,A,Y}\mathcal{L}_{\text{BM}}+\mathcal{L}_{\text{vec}}+\mathcal{L}_{0}+\mathcal{L}_{\text{SB}}.$
(1)

In Eq.(1), $\mathcal{L}_{\text{kin}}$ represents the kinetic energy term for
baryons and mesons and is given by

$\displaystyle\mathcal{L}_{\text{kin }}$
$\displaystyle=i\operatorname{Tr}\bar{B}\gamma_{\mu}D^{\mu}B+\frac{1}{2}\operatorname{Tr}D_{\mu}SD^{\mu}S+\operatorname{Tr}\left(u_{\mu}Su^{\mu}S+Su_{\mu}u^{\mu}S\right)\quad+\frac{1}{2}\operatorname{Tr}D_{\mu}YD^{\mu}Y$
$\displaystyle+$ $\displaystyle\frac{1}{2}D_{\mu}\chi
D^{\mu}\chi-\frac{1}{4}\operatorname{Tr}\left(V_{\mu\nu}V^{\mu\nu}\right)-\frac{1}{4}\operatorname{Tr}\left(\mathcal{A}_{\mu\nu}\mathcal{A}^{\mu\nu}\right).$
(2)

In the above equation, the first term represents the kinetic energy term for
the baryon octet, $B$. This term also incorporates the interactions of baryons
with pseudoscalar mesons through the covariant derivative $D_{\mu}$ defined as
$D_{\mu}B=\partial_{\mu}B+i\left[\Gamma_{\mu},B\right]$, with
$\Gamma_{\mu}=-\frac{i}{4}\left[u^{\dagger}\partial_{\mu}u-\partial_{\mu}u^{\dagger}u+\right.$
$\left.u\partial_{\mu}u^{\dagger}-\partial_{\mu}uu^{\dagger}\right]$. Here,
$u=\exp\left[\frac{i}{\sigma_{0}}\pi^{a}\lambda^{a}\gamma_{5}\right]$ is the
unitary transformation operator through which pseudoscalar mesons enter into
the calculations. As discussed in Sec. II.2, this kinetic term defined in
terms of covariant derivative contributes towards the interactions of kaons
and antikaons with the nucleons and hyperons in the strange medium (popularly
known as Weinberg Tomozawa term). The second and third terms of Eq. (2) give
the kinetic terms for the scalar and pseudoscalar octet mesons, respectively.
The fourth and fifth terms represent the kinetic terms for the pseudoscalar
singlet $Y$ and the dilaton field $\chi$, respectively. The last two terms of
Eq. (2) defined in terms of field tensors, $V_{\mu\nu}$ and
$\mathcal{A}_{\mu\nu}$, represent the kinetic terms of spin-1 vector and axial
vector mesons, respectively.

In Eq.(1), the second term, ${\cal{L}}_{BM}$, gives the interactions of
baryons with scalar and vector mesons, which in the mean-field approximation
is written as

$\displaystyle\mathcal{L}_{BM}=-\sum_{i}\overline{\psi_{i}}\left[g_{\omega
i}\gamma_{0}\omega+g_{\rho i}\gamma_{0}\tau_{3}\rho+g_{\phi
i}\gamma_{0}\phi+m_{i}^{*}\right]\psi_{i}.$ (3)

In the above, $m_{i}^{*}$ is the effective mass of baryons defined in terms of
scalar fields $\sigma,\zeta$ and $\delta$ and is given by

$\displaystyle m_{i}^{*}=-\left(g_{\sigma i}\sigma+g_{\zeta i}\zeta+g_{\delta
i}\tau_{3}\delta\right).$ (4)

In Eq. (4), symbols $g_{\sigma i}$, $g_{\zeta i}$ and $g_{\delta i}$ denote
the coupling strengths associated with baryons $(i=n,p,\Lambda,\Sigma,\Xi)$
interacting with the fields $\sigma$, $\zeta$, and $\delta$. The term
$\mathcal{L}_{\text{vec }}$ of Eq. (1) gives self interactions of vector
mesons, $\mathcal{L}_{\text{0}}$ gives the self interaction of scalar mesons
$\sigma,\zeta$ and $\delta$ and also the interactions for dilaton field
$\chi$. The interaction Lagrangian densities for these terms are given as

$\displaystyle\mathcal{L}_{\text{vec }}$
$\displaystyle=\frac{1}{2}\frac{\chi^{2}}{\chi_{0}^{2}}\Big{(}m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}+m_{\phi}^{2}\phi^{2}\Big{)}+g_{4}(\omega^{4}+2\phi^{4}+6\omega^{2}\rho^{2}+\rho^{4}),$
(5)

and

$\displaystyle\mathcal{L}_{\text{0}}$
$\displaystyle=-\frac{1}{2}k_{0}\chi^{2}(\sigma^{2}+\zeta^{2}+\delta^{2})+k_{1}(\sigma^{2}+\zeta^{2}+\delta^{2})^{2}+k_{2}(\frac{\sigma^{4}}{2}+\frac{\delta^{4}}{2}+\zeta^{4}+3\sigma^{2}\delta^{2})$
$\displaystyle+k_{3}\chi(\sigma^{2}-\delta^{2})\zeta-
k_{4}\chi^{4}-\frac{1}{4}\chi^{4}\ln\frac{\chi^{4}}{\chi_{0}^{4}}+\frac{d}{3}\chi^{4}\ln\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}},$
(6)

respectively. In Eq. (6), the last two terms account for scale-breaking
effects introduced in the chiral SU(3) model through the dilation field
$\chi$. The term, $\mathcal{L}_{SB}$, of Eq. (1) is the explicit symmetry-
breaking term and is given by

$\displaystyle\mathcal{L}_{SB}$
$\displaystyle=-\left[m_{\pi}^{2}f_{\pi}\sigma+\left(\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right].$
(7)

The thermodynamic potential, $\Omega$, of the grand canonical ensemble per
unit volume $V$, at a given temperature and chemical potential, is expressed
as

$\begin{array}[]{r}\frac{\Omega}{V}=-\frac{T\gamma_{i}}{(2\pi)^{3}}\sum_{i}\int
d^{3}k\left\\{\ln\left(e^{-\beta\left[E_{i}^{*}(\mathbf{k})-\mu_{i}^{*}\right]}+1\right)\right.\\\
\left.+\ln\left(e^{-\beta\left[E_{i}^{*}(\mathbf{k})+\mu_{i}^{*}\right]}+1\right)\right\\}-\mathcal{L}_{\text{vec
}}-\mathcal{L}_{0}-\mathcal{L}_{\mathrm{SB}}-\mathcal{V}_{vac},\end{array}$
(8)

where $E_{i}^{*}(\mathbf{k})=\sqrt{\mathbf{k}^{2}+m_{i}^{*2}}$ and
$\mu_{i}^{*}=\mu_{i}-g_{\omega i}\omega-g_{\rho i}\tau_{3}\rho$ $-g_{\phi
i}\phi$ are the single-particle effective energy and effective chemical
potential, respectively, at temperature $T$. The spin degeneracy factor is
represented by $\gamma_{i}$ and factor, $\beta=\frac{1}{kT}$. In addition, the
vacuum potential energy, $\mathcal{V}_{\text{vac}}$, is subtracted from Eq.
(8) to achieve vanishing vacuum energy. The equations of motion for the non-
strange scalar $\sigma$, the strange scalar $\zeta$, the scalar iso-vector
$\delta$, the vector $\omega$, the vector-isovector $\rho$, strange-vector
$\phi$, and the scalar dilaton $\chi$ fields are derived by minimizing the
thermodynamic potential and are written as

$\displaystyle\frac{\partial(\Omega/V)}{\partial\sigma}=k_{0}\chi^{2}\sigma-2k_{2}\left(\sigma^{3}+3\sigma\delta^{2}\right)-2k_{3}\sigma\zeta\chi-4k_{1}\left(\sigma^{2}+\delta^{2}+\zeta^{2}\right)\sigma$
$\displaystyle-\frac{d}{3}\chi^{4}\left(\frac{2\sigma}{\sigma^{2}-\delta^{2}}\right)+\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\pi}^{2}f_{\pi}-\sum_{i}g_{\sigma
i}\rho_{i}^{s}=0,$ (9)
$\displaystyle\frac{\partial(\Omega/V)}{\partial\zeta}=k_{0}\chi^{2}\zeta-4k_{2}\zeta^{3}+k_{3}\chi\left(\delta^{2}-\sigma^{2}\right)-4k_{1}\left(\sigma^{2}+\delta^{2}+\zeta^{2}\right)\zeta$
$\displaystyle-\frac{d}{3}\frac{\chi^{4}}{\zeta}+\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right]-\sum_{i}g_{\zeta
i}\rho_{i}^{s}=0,$ (10)
$\displaystyle\frac{\partial(\Omega/V)}{\partial\delta}=k_{0}\chi^{2}\delta+2k_{3}\chi\delta\zeta-2k_{2}\left(\delta^{3}+3\sigma^{2}\delta\right)-4k_{1}\left(\sigma^{2}+\delta^{2}+\zeta^{2}\right)\delta$
$\displaystyle+\frac{2}{3}d\chi^{4}\left(\frac{\delta}{\sigma^{2}-\delta^{2}}\right)-\sum_{i}g_{\delta
i}\tau_{3}\rho_{i}^{s}=0,$ (11)
$\displaystyle\frac{\partial(\Omega/V)}{\partial\omega}=\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\omega}^{2}\omega+g_{4}\left(12\rho^{2}\omega+4\omega^{3}\right)-\sum_{i}g_{\omega
i}\rho_{i}^{v}=0,$ (12)
$\displaystyle\frac{\partial(\Omega/V)}{\partial\rho}=\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\rho}^{2}\rho+g_{4}\left(12\omega^{2}\rho+4\rho^{3}\right)-\sum_{i}g_{\rho
i}\tau_{3}\rho_{i}^{v}=0,$ (13)
$\displaystyle\frac{\partial(\Omega/V)}{\partial\phi}=\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\phi}^{2}\phi+8g_{4}\phi^{3}-\sum_{i}g_{\phi
i}\rho_{i}^{v}=0,$ (14)

and

$\displaystyle\frac{\partial(\Omega/V)}{\partial\chi}=k_{0}\chi\left(\sigma^{2}+\delta^{2}+\zeta^{2}\right)+\left(4k_{4}-d\right)\chi^{3}+k_{3}\left(\delta^{2}-\sigma^{2}\right)\zeta+\chi^{3}\left[1+\ln\left(\frac{\chi^{4}}{\chi_{0}^{4}}\right)\right]$
$\displaystyle-\frac{4}{3}d\chi^{3}\ln\left(\left(\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\right)\left(\frac{\chi}{\chi_{0}}\right)^{3}\right)+\frac{2\chi}{\chi_{0}^{2}}\left[m_{\pi}^{2}f_{\pi}\sigma+\left(\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right]$
$\displaystyle-\frac{\chi}{\chi_{0}^{2}}\left(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}+m_{\phi}^{2}\phi^{2}\right)=0,$
(15)

respectively. The scalar and vector densities of the $i$th baryon appearing in
the above equation are defined as

$\displaystyle\rho_{i}^{s}=\gamma_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{m_{i}^{*}}{E_{i}^{*}(\mathbf{k})}\left(n_{i}(\beta)+\bar{n}_{i}(\beta)\right),$
(16)

and

$\displaystyle\rho_{i}^{v}=\gamma_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\left(n_{i}(\beta)-\bar{n}_{i}(\beta)\right),$
(17)

with

$\displaystyle
n_{i}(\beta)=\frac{1}{\exp\left[\beta\left(E_{i}^{*}(\mathbf{k})-\mu_{i}^{*}\right)\right]+1},\quad\bar{n}_{i}(\beta)=\frac{1}{\exp\left[\beta\left(E_{i}^{*}(\mathbf{k})+\mu_{i}^{*}\right)\right]+1},$
(18)

as the finite temperature distribution functions of baryons and antibaryons,
respectively. In the present work, we shall investigate the in-medium
properties of pseudoscalar and vector mesons at finite isospin asymmetry and
finite strangeness fraction of the medium. The isospin asymmetry is introduced
through the parameter $\eta=\frac{-\sum_{i}I_{3i}\rho_{i}^{v}}{\rho_{B}}$,
whereas for the finite strangeness fraction the definition
$f_{s}=\frac{\sum_{i}|s_{i}|\rho_{i}^{v}}{\rho_{B}}$ is used. The symbols
$I_{3i}$, $|s_{i}|$, and $\rho_{B}$ represent the isospin quantum number
(third component), the number of strange quarks, and the total baryonic
density, respectively.

As discussed previously, in the current work, the medium modification of kaons
and antikaons and the vector $\phi$ mesons will be studied in finite volume
strange hadronic matter at finite temperature. To achieve this, the coupled
system of equations, given by Eqs. (9)- (15) for the scalar and vector fields
are solved, incorporating finite volume corrections through the MRE method
[46, 47, 125, 126]. Under the MRE method, the expression for the density of
states, for a medium with finite volume and spherical geometry, having radius
$R$, is written as [127, 128, 129]

$\rho_{MRE}=\frac{k^{2}}{2\pi^{2}}\left[1+\frac{6\pi^{2}}{kR}f_{S}\left(\frac{k}{m_{i}^{*}}\right)+\frac{12\pi^{2}}{(kR)^{2}}f_{C}\left(\frac{k}{m_{i}^{*}}\right)\right].$
(19)

In the above equation, $f_{S}$ and $f_{C}$ are the surface and curvature
contributions to the density of states and are given by

$\displaystyle
f_{S}\left(\frac{k}{m_{i}^{*}}\right)=-\frac{1}{8\pi}\left[1-\frac{2}{\pi}\arctan\frac{k}{m_{i}^{*}}\right],$
(20)

and

$\displaystyle
f_{C}\left(\frac{k}{m_{i}^{*}}\right)=\frac{1}{12\pi^{2}}\left[1-\frac{3k}{2m_{i}^{*}}\left(\frac{\pi}{2}-\arctan\frac{k}{m_{i}^{*}}\right)\right],$
(21)

respectively. Also, as defined earlier, $m_{i}^{*}$ is the effective mass of
baryons in the finite volume hadronic medium. In literature, the finite volume
effects with surface and curvature contributions have been investigated for
quark matter only, with $m_{i}^{*}$ being the constituent quark mass [47, 127,
130]. In some of these studies, $m_{i}^{*}$ is being replaced with the
parameter $\alpha$ defining the penetration of wave function from finite
volume quark matter to hadronic phase and its values are taken as,
$\alpha=\infty$, enforcing Dirichlet boundary conditions and $\alpha=0$ in the
case when Neumann boundary conditions are implemented [125]. For investigating
the properties of finite volume hadronic medium in the present work, we
restrict to these two values for $\alpha$, i.e., $\alpha=\infty$ and $0$.

Figure 1: In the above figure, the variation of density of states,
$\rho_{MRE}$, as a function of momentum, $k$, is shown for various values
effective masses of baryons, $m_{i}^{*}$, in the finite volume hadronic
medium. Results are compared with the case when the density of states is
calculated without the MRE method.

As is apparent from Eq. (19), due to quadratic nature in $k$, the density of
the state has two roots. Between these roots, there can be density of states
that have negative values and, thus, have no physical significance. An
infrared (IR) cut-off, $\Lambda_{IR}$, is introduced in the volume integral
over momentum states appearing in the definitions of scalar and vector
densities of baryons defined through Eqs. (16) and (17), to eliminate negative
values of density of states through the following prescription [47, 46]

$\displaystyle\int_{0}^{\infty}\frac{d^{3}k}{(2\pi)^{3}}\cdots\rightarrow\int_{\Lambda_{IR}}^{\infty}\frac{d^{3}k}{(2\pi)^{3}}\rho_{MRE}\cdots,$
(22)

where $\Lambda_{IR}$ is the largest solution to, $\rho_{MRE}(k)=0$, with
respect to momentum $k$. With $\alpha=\infty$, the Dirichlet boundary
conditions, the density of states will become [131, 130]

$\rho_{MRE}=\frac{k^{2}}{2\pi^{2}}\left[1-\frac{3\pi}{4kR}+\frac{1}{(kR)^{2}}\right],$
(23)

with $\Lambda_{IR}=1.8/R$. On the other side, implementing the Neumann
boundary conditions, i.e., $\alpha=0$, the expression for the density of
states is given by

$\displaystyle\rho_{MRE}=\frac{k^{2}}{2\pi^{2}}\left[1-\frac{1}{2(kR)^{2}}\right],$
(24)

which leads to $\Lambda_{IR}=1/\sqrt{2}R$. The above discussion for the
density of states in the MRE method consider the finite volume in a spherical
geometry, which takes into account surface and curvature effects. As outlined
in Ref. [128], considering cubic geometry for finite volume matter, the
density of states in the MRE method can be written as

$\rho_{MRE}=\frac{k^{2}}{2\pi^{2}}\left(1-\frac{3}{4kR}\right),$ (25)

with IR cut-off conditions $\Lambda_{IR}=0.75/R$, where now $R$ being the
length of cubic volume. For a better understanding of above discussed
different cases, in Fig. (1) densities of states, $\rho_{MRE}$, calculated
using Eq. (19) is plotted as a function of momentum, $k$, for various values
of effective mass, $m_{i}^{*}$, of baryons in the hadronic medium with finite
volume. All results are plotted for system size $R=2$ fm and compared with the
situation when densities of states are calculated without using the MRE
method, in which case they do not become negative for any value of $k$. As can
be seen from this figure, the value of $k$ at which the density of states
become negative decrease with a decrease in $m_{i}^{*}$. Thus, in the
Dirichlet case, i.e., when $\alpha=m_{i}^{*}=\infty$, the value of $k$ and
hence, the lower cut-off $\Lambda_{IR}$ will have largest value compared to
other cases and maximum states are excluded (as they have negative value) from
the integral in Eq. (22).

###  II.2 Kaon and antikaon in the hadronic chiral SU(3) model

To obtain the medium modified masses of kaons and antikaons in the finite
volume strange hadronic matter within the chiral SU(3) mean field model, the
Lagrangian density describing the interactions of kaons and antikaons with the
baryon octet and the scalar mesons $\sigma,\zeta$ and $\delta$ is written as
[81]

$\displaystyle\mathcal{L}_{KB}=$
$\displaystyle-\frac{i}{4f_{K}^{2}}\left[\left(2\bar{p}\gamma^{\mu}p+\bar{n}\gamma^{\mu}n-\overline{\Sigma^{-}}\gamma^{\mu}\Sigma^{-}+\overline{\Sigma^{+}}\gamma^{\mu}\Sigma^{+}-2\bar{\Xi}^{-}\gamma^{\mu}\Xi^{-}-\overline{\Xi^{0}}\gamma^{\mu}\Xi^{0}\right)\right.$
$\displaystyle\quad\times\left(K^{-}\left(\partial_{\mu}K^{+}\right)-\left(\partial_{\mu}K^{-}\right)K^{+}\right)$
$\displaystyle\quad+\left(\bar{p}\gamma^{\mu}p+2\bar{n}\gamma^{\mu}n+\overline{\Sigma^{-}}\gamma^{\mu}\Sigma^{-}-\overline{\Sigma^{+}}\gamma^{\mu}\Sigma^{+}-\overline{\Xi^{-}}\gamma^{\mu}\Xi^{-}-2\overline{\Xi^{0}}\gamma^{\mu}\Xi^{0}\right)$
$\displaystyle\quad\times\left(\overline{K^{0}}\left(\partial_{\mu}K^{0}\right)-\left(\partial_{\mu}\overline{K^{0}}\right)K^{0}\right)\bigg{]}$
$\displaystyle+\frac{m_{K}^{2}}{2f_{K}}\left[(\sigma+\sqrt{2}\zeta+\delta)\left(K^{+}K^{-}\right)+(\sigma+\sqrt{2}\zeta-\delta)\left(K^{0}\bar{K}^{0}\right)\right]$
$\displaystyle-\frac{1}{f_{K}}\left[(\sigma+\sqrt{2}\zeta+\delta)\left(\partial_{\mu}K^{+}\right)\left(\partial^{\mu}K^{-}\right)+(\sigma+\sqrt{2}\zeta-\delta)\left(\partial_{\mu}K^{0}\right)\left(\partial^{\mu}\overline{K^{0}}\right)\right]$
$\displaystyle+\frac{d_{1}}{2f_{K}^{2}}\left(\bar{p}p+\bar{n}n+\bar{\Lambda}^{0}\Lambda^{0}+\bar{\Sigma}^{-}\Sigma^{+}+\bar{\Sigma}^{0}\Sigma^{0}+\bar{\Sigma}^{-}\Sigma^{-}+\bar{\Xi}^{-}\Xi^{-}+\bar{\Xi}^{0}\Xi^{0}\right)$
$\displaystyle\quad\times\left(\left(\partial_{\mu}K^{+}\right)\left(\partial^{\mu}K^{-}\right)+\left(\partial_{\mu}K^{0}\right)\left(\partial^{\mu}\bar{K}^{0}\right)\right)$
$\displaystyle+\frac{d_{2}}{2f_{K}^{2}}\left[\left(\bar{p}p+\frac{5}{6}\Lambda^{0}\Lambda^{0}+\frac{1}{2}\bar{\Sigma}^{0}\Sigma^{0}+\bar{\Sigma}^{+}\Sigma^{+}+\bar{\Xi}^{-}\Xi^{-}+\bar{\Xi}^{0}\Xi^{0}\right)\right.$
$\displaystyle\quad\times\left(\partial_{\mu}K^{+}\right)\left(\partial^{\mu}K^{-}\right)$
$\displaystyle\quad+\left(\bar{n}n+\frac{5}{6}\bar{\Lambda}^{0}\Lambda^{0}+\frac{1}{2}\bar{\Sigma}^{0}\Sigma^{0}+\overline{\Sigma^{-}}\Sigma^{-}+\bar{\Xi}^{-}\Xi^{-}+\bar{\Xi}^{0}\Xi^{0}\right)$
$\displaystyle\quad\times\left(\partial_{\mu}K^{0}\right)\left(\partial^{\mu}\bar{K}^{0}\right)\bigg{]}.$
(26)

In the above equation, $K^{+}$ and $K^{0}$ belong to kaon, $K$, isospin
doublet, whereas, $K^{-}$ and $\bar{K}^{0}$ are the members of antikaon
$\bar{K}$ doublet. In Eq. (26), the first term (first four lines in the square
bracket) is the vectorial Weinberg Tomozawa interaction term, obtained from
the first term of Eq. (2). The second term, which generates an attractive
interaction for kaons and antikaons as a function of the density of the
medium, arises from the explicit symmetry-breaking term of the model [95, 70].
In the framework of the chiral SU(3) model, the third term of Eq. (26) is
derived from the kinetic part of the pseudoscalar mesons (third term of Eq.
(2)) [70, 94]. Fourth and fifth terms in Eq. (26), named as range terms, arise
from the Lagrangian densities [67, 70]

$\mathcal{L}_{\left(d_{1}\right)}^{BM}=d_{1}\operatorname{Tr}\left(u_{\mu}u^{\mu}\bar{B}B\right),$
(27)

and

$\mathcal{L}_{\left(d_{2}\right)}^{BM}=d_{2}\operatorname{Tr}\left(\bar{B}u_{\mu}u^{\mu}B\right).$
(28)

From the interaction Lagrangian density given by Eq.(26), equation of motion
for kaons and antikaons are derived using Euler’s Lagrange equation of motion,
whose Fourier transformation gives the dispersion relation,

$\vec{k}^{2}-\omega^{2}+m_{K(\bar{K})}^{2}=\Pi_{K(\bar{K})}^{*}(\omega,|\vec{k}|,\rho),$
(29)

where $\Pi_{K(\bar{K})}^{*}(\omega,|\vec{k}|,\rho)$ denotes the kaon
(antikaon) self-energy in the medium. Also, $m_{K(\bar{K})}$ in Eq. (29)
represents the vacuum mass of kaon (antikaon). Specifically, for the kaon
doublet, $K$ $(K^{+},{K^{0}})$, self energy is given as

$\displaystyle\Pi_{K}^{*}(\omega,|\vec{k}|,\rho)=$
$\displaystyle-\frac{1}{4f_{K}^{2}}\left[3\left(\rho_{p}^{v}+\rho_{n}^{v}\right)\pm\left(\rho_{p}^{v}-\rho_{n}^{v}\right)\pm
2\left(\rho_{\Sigma^{+}}^{v}-\rho_{\Sigma^{-}}^{v}\right)-3\left(\rho_{\Xi^{-}}^{v}+\rho_{\Xi^{0}}^{v}\right)\pm\left(\rho_{\Xi^{-}}^{v}-\rho_{\Xi^{0}}^{v}\right)\right]\omega$
$\displaystyle+\frac{m_{K}^{2}}{2f_{K}}\left(\sigma^{\prime}+\sqrt{2}\zeta^{\prime}\pm\delta^{\prime}\right)$
$\displaystyle+\left[-\frac{1}{f_{K}}\left(\sigma^{\prime}+\sqrt{2}\zeta^{\prime}\pm\delta^{\prime}\right)+\frac{d_{1}}{2f_{K}^{2}}\left(\rho_{p}^{s}+\rho_{n}^{s}+\rho_{\Lambda^{0}}^{s}+\rho_{\Sigma^{+}}^{s}+\rho^{s}_{\Sigma^{0}}+\rho_{\Sigma^{-}}^{s}+\rho^{s}_{\Xi^{-}}+\rho^{s}_{\Xi^{0}}\right)\right.$
$\displaystyle\quad+\frac{d_{2}}{4f_{K}^{2}}\left(\left(\rho^{s}_{p}+\rho^{s}_{n}\right)\pm\left(\rho^{s}_{p}-\rho^{s}_{n}\right)+\rho^{s}_{\Sigma^{0}}+\frac{5}{3}\rho^{s}_{\Lambda^{0}}+\left(\rho^{s}_{\Sigma^{+}}+\rho^{s}_{\Sigma^{-}}\right)\pm\left(\rho_{\Sigma^{+}}^{s}-\rho_{\Sigma^{-}}^{s}\right)\right.$
$\displaystyle\quad\left.\left.+2\rho_{\Xi^{-}}^{s}+2\rho_{\Xi^{0}}^{s}\right)\right]\left(\omega^{2}-\vec{k}^{2}\right),$
(30)

where contributions to $K^{+}$ and $K^{0}$ are denoted by the $\pm$ signs,
respectively. Also, $\sigma^{\prime}$ $=\sigma-\sigma_{0}$, $\zeta^{\prime}$
$=\zeta-\zeta_{0}$, and $\delta^{\prime}$ $=\delta-\delta_{0}$ are the
variation of $\sigma$, $\zeta$ and $\delta$ fields from their expectation
values in vacuum. A non-zero value for $\delta$ will break the isospin
symmetry in the medium (in vacuum, $\delta_{0}=0$).

Likewise, the expression for the self-energy of antikaon isospin doublet,
$\bar{K}$ $(K^{-},\bar{K^{0}})$, is written as

$\displaystyle\Pi_{\bar{K}}^{*}(\omega,|\vec{k}|,\rho)=$
$\displaystyle\frac{1}{4f_{K}^{2}}\left[3\left(\rho_{p}^{v}+\rho_{n}^{v}\right)\pm\left(\rho_{p}^{v}-\rho_{n}^{v}\right)\pm
2\left(\rho_{\Sigma^{+}}^{v}-\rho_{\Sigma^{-}}^{v}\right)-3\left(\rho_{\Xi^{-}}^{v}+\rho_{\Xi^{0}}^{v}\right)\pm\left(\rho_{\Xi^{-}}^{v}-\rho_{\Xi^{0}}^{v}\right)\right]\omega$
$\displaystyle+\frac{m_{K}^{2}}{2f_{K}}\left(\sigma^{\prime}+\sqrt{2}\zeta^{\prime}\pm\delta^{\prime}\right)$
$\displaystyle+\left[-\frac{1}{f_{K}}\left(\sigma^{\prime}+\sqrt{2}\zeta^{\prime}\pm\delta^{\prime}\right)+\frac{d_{1}}{2f_{K}^{2}}\left(\rho^{s}_{p}+\rho^{s}_{n}+\rho_{\Lambda^{0}}^{s}+\rho_{\Sigma^{+}}^{s}+\rho^{s}_{\Sigma^{0}}+\rho_{\Sigma^{-}}^{s}+\rho^{s}_{\Xi^{-}}+\rho^{s}_{\Xi^{0}}\right)\right.$
$\displaystyle\quad+\frac{d_{2}}{4f_{K}^{2}}\left(\left(\rho_{p}^{s}+\rho^{s}_{n}\right)\pm\left(\rho_{p}^{s}-\rho^{s}_{n}\right)+\rho^{s}_{\Sigma^{0}}+\frac{5}{3}\rho^{s}_{\Lambda^{0}}+\left(\rho^{s}_{\Sigma^{+}}+\rho^{s}_{\Sigma^{-}}\right)\pm\left(\rho_{\Sigma^{+}}^{s}-\rho_{\Sigma^{-}}^{s}\right)\right.$
$\displaystyle\quad\left.\left.+2\rho_{\Xi^{-}}^{s}+2\rho_{\Xi^{0}}^{s}\right)\right]\left(\omega^{2}-\vec{k}^{2}\right),$
(31)

where $K^{-}$and $\bar{K}^{0}$ are denoted by the $\pm$ signs, respectively.
The effective mass of the $K$ ($\bar{K}$) meson in the strange hadronic medium
with finite volume is determined by solving Eq. (29) with the constraint
$m^{*}_{K(\bar{K})}=\omega(|\mathbf{k}|=0)$. The finite volume effects in the
in-medium masses of kaons and antikaons are introduced through the
modification of various quantities appearing in the self-energy terms, which
are calculated in the finite volume hadronic medium as described in the
previous section, i.e, in terms of scalar fields $\sigma,\zeta$, and $\delta$,
the scalar densities $\rho_{i}^{s}$ and vector densities $\rho_{i}^{v}$ of the
baryons. The values of parameters $d_{1}$ and $d_{2}$ used in the expressions
of self-energy are chosen as ${2.56}/{m_{K}}$ and ${0.73}/{m_{K}}$
respectively, fitted using empirical data on $KN$ scattering length [70, 132].

###  II.3 $\phi$ meson self energy

The effective masses of kaons and antikaons calculated in the finite volume
hadronic medium are used as input in the present section to calculate the in-
medium masses and decay width of $\phi$ mesons. To calculate the $\phi$ meson
self-energy in the hadronic medium, we consider the decay process
$\phi\rightarrow K\bar{K}$ at the one-loop (as shown in Figure 2), for which
the interaction Lagrangian density is written as [99, 133]

$\mathcal{L}_{\text{int
}}=ig_{\phi}\phi^{\mu}\left[\bar{K}\left(\partial_{\mu}K\right)-\left(\partial_{\mu}\bar{K}\right)K\right].$
(32)

In the above equation, $g_{\phi}$ represents the coupling constant.

Figure 2: The one loop level interaction of $\phi K\bar{K}$ [99].

Since the contributions of $\phi\phi$K$\bar{K}$ interactions to the in-medium
masses and decay width are smaller than those of $\phi$K$\bar{K}$
interactions, hence these interactions are not considered in the current work
[99]. In the rest frame of $\phi$ mesons, the scalar component of the self-
energy, corresponding to the loop diagram as shown in Fig. (2), can be
expressed as

$i\Pi_{\phi}^{*}(p)=-\frac{8}{3}g_{\phi}^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\vec{q}^{2}D_{K}(q)D_{\bar{K}}(q-p),$
(33)

where, $D_{K}(q)=\left(-m_{K}^{*^{2}}+q^{2}+i\epsilon\right)^{-1}$ and
$D_{\bar{K}}(q-p)=\left(-m_{\bar{K}}^{*^{2}}+(q-p)^{2}+i\epsilon\right)^{-1}$
represent the propagators for kaons and antikaons, respectively. Here,
$p=\left(p^{0}=m_{\phi}^{*},\overrightarrow{0}\right)$ denotes the four-
momentum vector of the $\phi$ meson, with $m_{\phi}^{*}$ as the in-medium
$\phi$ meson mass. Also,
$m_{K}^{*}\left(=\frac{m_{K^{0}}^{*}+m_{K^{+}}^{*}}{2}\right)$ and
$m_{\bar{K}}^{*}\left(=\frac{m_{\bar{K}^{0}}^{*}+m_{K^{-}}^{*}}{2}\right)$,
are the averaged kaons and antikaon masses, respectively. The values of
$m_{K^{+}}^{*},m_{K^{0}}^{*},m_{K^{-}}^{*}$, and $m_{\bar{K}^{0}}^{*}$ are
computed using Eq. (29) in the finite volume strange hadronic medium, at
finite temperature and for finite isospin asymmetry. The in-medium mass of the
$\phi$ meson is calculated from the real part of $\Pi_{\phi}^{*}(p)$, as
determined by the following relation

$m_{\phi}^{*^{2}}=\left(m_{\phi}^{0}\right)^{2}+\operatorname{Re}\Pi_{\phi}^{*}\left(m_{\phi}^{*^{2}}\right),$
(34)

where $m_{\phi}^{0}$ is the bare mass of the $\phi$ meson. The real part of
the self-energy can be written [99, 102] as

$\operatorname{Re}\Pi_{\phi}^{*}=-\frac{4}{3}g_{\phi}^{2}\mathcal{P}\int\frac{d^{3}q}{(2\pi)^{3}}\vec{q}^{2}\frac{\left(E_{K}^{*}+E_{\bar{K}}^{*}v\right)}{E_{K}^{*}E_{\bar{K}}^{*}\left(\left(E_{K}^{*}+E_{\bar{K}}^{*}\right)^{2}-m_{\phi}^{*^{2}}\right)},$
(35)

where $\mathcal{P}$ denotes the principal value of the integral given in Eq.
(33). Also, $E_{K}^{*}=$ $\left(\vec{q}^{\,2}+m_{K}^{*^{2}}\right)^{1/2}$, and
$E_{\bar{K}}^{*}=\left(\vec{q}^{\,2}+m_{\bar{K}}^{*^{2}}\right)^{1/2}$. To
circumvent singularities, the integral in Eq. (35) is regularized by
incorporating a phenomenological form factor with the cutoff parameter
$\Lambda_{c}$, i.e., [134]

$\displaystyle\operatorname{Re}\Pi_{\phi}^{*}=-\frac{4}{3}g_{\phi}^{2}\mathcal{P}\int_{0}^{\Lambda_{c}}\frac{d^{3}q}{(2\pi)^{3}}\vec{q}^{2}\left(\frac{\Lambda_{c}^{2}+m_{\phi}^{*^{2}}}{\Lambda_{c}^{2}+4E_{K}^{*^{2}}}\right)^{4}\frac{\left(E_{K}^{*}+E_{\bar{K}}^{*}\right)}{E_{K}^{*}E_{\bar{K}}^{*}\left(\left(E_{K}^{*}+E_{\bar{K}}^{*}\right)^{2}-m_{\phi}^{*^{2}}\right)}.$
(36)

The decay width of the $\phi$ meson is determined by extracting the imaginary
component of its self-energy, denoted as $\operatorname{Im}\Pi_{\phi}^{*}$,
and is expressed in terms of in-medium masses of $\phi$, $K$, and $\bar{K}$
mesons [135], i.e.,

$\displaystyle\Gamma_{\phi}^{*}=\frac{g_{\phi}^{2}}{24\pi}\frac{1}{m_{\phi}^{*^{5}}}\left[\left(m_{\phi}^{*^{2}}-\left(m_{K}^{*}+m_{\bar{K}}^{*}\right)^{2}\right)\right.\left.\times\left(m_{\phi}^{*^{2}}-\left(m_{K}^{*}-m_{\bar{K}}^{*}\right)^{2}\right)\right]^{3/2}.$
(37)

In the current study, the value of coupling constant $g_{\phi}$ is taken to be
$4.539$, fitted to the empirical width of the $\phi$ meson in vacuum.

##  III Results and discussion

In this section, we present the results on the in-medium masses of kaons and
antikaons alongwith the masses and decay width of $\phi$ mesons in a strange
hadronic medium with finite volume. As discussed earlier, within the chiral
SU(3) model, the in-medium masses of baryons are calculated through the medium
modified scalar fields $\sigma,\zeta$ and $\delta$, which further modify the
properties of kaons and antikaons. In Sec. III.1, we shall discuss the impact
of finite volume isospin asymmetric strange hadronic medium at finite
temperature on the scalar fields $\sigma,\zeta$ and $\delta$ as well as the
dilaton field $\chi$. The in-medium properties of kaons and antikaons are
presented in Sec. III.2, and finally, Sec. III.3 is dedicated to the
discussion on the results for the in-medium masses and decay widths of $\phi$
mesons. The parameters of the chiral SU(3) model used in the present study are
listed in table 1.

Table 1: Various parameters used in the present calculations [61] $g_{\sigma n(p)}$ | $g_{\zeta n(p)}$ | $g_{\delta n(p)}$ | $g_{\omega n(p)}$ | $g_{\rho n(p)}$ | $g_{\sigma\Lambda}$ | $g_{\zeta\Lambda}$ | $g_{\delta\Lambda}$ | $g_{\sigma\Sigma}$ | $g_{\zeta\Sigma}$
---|---|---|---|---|---|---|---|---|---
10.6 | -0.46 | 2.48 | 13.32 | 5.48 | 5.31 | 5.8 | 0 | 6.13 | 5.8
$g_{\delta\Sigma}$ | $g_{\delta\Sigma^{0}}$ | $g_{\sigma\Xi}$ | $g_{\zeta\Xi}$ | $g_{\delta\Xi}$ | $g_{\omega\Lambda}$ | $g_{\omega\Sigma}$ | $g_{\rho\Sigma}$ | $g_{\rho\Sigma^{0}}$ | $g_{\omega\Xi}$
6.79 | 0 | 3.68 | 9.14 | 2.36 | $\frac{2}{3}g_{\omega N}$ | $\frac{2}{3}g_{\omega N}$ | $\frac{2}{3}g_{\omega N}$ | 0 | $\frac{1}{3}g_{\omega N}$
$g_{\rho\Lambda}$ | $g_{\rho\Xi}$ | $g_{\phi\Lambda}$ | $g_{\phi\Sigma}$ | $g_{\phi\Xi}$ | $\sigma_{0}(\mathrm{MeV})$ | $\zeta_{0}(\mathrm{MeV})$ | $\chi_{0}(\mathrm{MeV})$ | $d$ | $\rho_{0}\left(\mathrm{fm}^{-3}\right)$
0 | $\frac{1}{3}g_{\omega N}$ | $-\frac{\sqrt{2}}{3}g_{\omega N}$ | $-\frac{\sqrt{2}}{3}g_{\omega N}$ | $-\frac{2\sqrt{2}}{3}g_{\omega N}$ | -93.29 | -106.8 | 409.7 | 0.064 | 0.15
$m_{\pi}(\mathrm{MeV})$ | $m_{K}(\mathrm{MeV})$ | $f_{\pi}(\mathrm{MeV})$ | $f_{K}(\mathrm{MeV})$ | $g_{4}$ | $k_{0}$ | $k_{1}$ | $k_{2}$ | $k_{3}$ | $k_{4}$
139 | 498 | 93.29 | 122.14 | 79.91 | 2.53 | 1.35 | -4.77 | -2.77 | -0.218
$m_{\omega}(\mathrm{MeV})$ | $m_{\rho}(\mathrm{MeV})$ | $m_{\phi}(\mathrm{MeV})$ |  |  |  |  |  |  |
780.56 | 761.06 | 1019 |  |  |  |  |  |  |

###  III.1 Scalar fields of the chiral SU(3) hadronic model in finite volume
matter

The density and temperature dependence of the scalar and vector fields are
obtained by solving the coupled system of equations defined through Eqs.
(9)-(15). In Figs. (3) and (4), we plot the non-strange scalar field $\sigma$
and the strange scalar field $\zeta$ with respect to baryonic density
$\rho_{B}$ (in units of nuclear saturation density, $\rho_{0}$) in the finite
volume of hadronic medium. The results are shown for nuclear ($f_{s}=0$) and
strange hadronic matter ($f_{s}=0.5$) at temperatures $T=100$ and $150$ MeV
and isospin asymmetry parameters, $\eta=0$ and $0.3$. To understand the impact
of finite volume, we have performed the calculations for $R=2$ fm and compared
the results at $R=\infty$, which corresponds to infinite volume hadronic
matter. For $R=2$ fm (finite volume), we have shown the results for spherical
geometry of the finite volume hadronic matter, with MRE method used to
calculate the density of states under Neumann and Dirichlet boundary
conditions (shown as case (I) and (IV) in the figures). In the figures, the
results are also shown for cubic geometry, for cubic length, $R=2$ fm, and
considering the density of states calculated using Eq. (25) (shown as case II,
with Dirichlet boundary condition) and without MRE method (case III). In case
III, the finite volume of the medium is taken into account by simply applying
the lower momentum cut-off, $k=\frac{\pi}{R}$. This procedure of lower
momentum cut-off without using spherical and curvature effects has been used
to the study of finite volume quark matter in various work [136, 60]. In the
current study, we have not used the periodic and anti-periodic boundary
conditions with a discrete sum over the momentum states for the consideration
of finite volume effect [40]. As has been demonstrated in Ref. [137], the
discrete sum can be transformed to a continuous integral, and the use of lower
momentum cut-off in calculations of finite volume effects is justified.

For a given temperature, $T$, isospin asymmetry, $\eta$, and strangeness
fraction, $f_{s}$, of the medium, the magnitude of scalar fields $\sigma$ and
$\zeta$ decreases with an increase in the density of the medium, both for
infinite ($R=\infty$) and finite ($R=2$ fm) volume hadronic medium. In the
symmetric nuclear medium, at temperature $T=100$ MeV and $R=\infty$, the
magnitude of scalar fields $\sigma$ and $\zeta$ decreases by $63.6$% and
$12.9$%, as $\rho_{B}$ is increased from zero to $4\rho_{0}$. As can be seen
from Figs. (3) and (4), for a given density of the medium, decreasing the
value of $R$ from $\infty$ to $2$ fm leads to an increase in the magnitude of
scalar fields, i.e., less drop in the magnitude of scalar fields is observed.
Since the mass of baryon is directly proportional to the in-medium values of
scalar fields (see Eq. (4)), the effective mass of baryons will increase with
a decrease in the volume of the nuclear or strange hadronic medium, at the
fixed value of baryon density. From Figs. (3) and (4), we also observe that on
moving from infinite to finite volume case, the maximum change in the scalar
fields (an increase of magnitude) is observed in the situation where finite
volume calculations are performed for the spherical geometry of the medium
with Dirichlet boundary conditions in the MRE method. As the symmetric nuclear
medium, at $T=100$ MeV, is squeezed to a finite volume, with $R=2$ fm, values
of percentage drop in the magnitude of scalar field $\sigma$, for an increase
in $\rho_{B}$ from zero to $4\rho_{0}$, are observed to be $63.2$%, $62.5$%,
$60.9$% and $60.2$ %, for cases I to IV, respectively. For the strange scalar
field $\zeta$, these values of percentage drop at $R=2$ fm are observed to be
$12.8$%, $12.7$%, $12.6$% and $12.4$ %. One can compare these values with the
earlier quoted results for infinite nuclear matter, i.e., $R=\infty$. In the
present calculations, we are interested in the study of hadronic matter at
non-zero baryon density and finite temperature. The first term of Eq.(8)
represents the contribution of the medium with finite baryon density to the
total thermodynamic potential. This is in line with the calculations of
various relativistic mean field models employed to study the properties of the
hadronic medium at finite baryon density [66]. The vacuum term, which is
usually taken into account in the NJL/PNJL [46] and some of quark models [138]
and which require regularization techniques to avoid divergence, is not added
in the thermodynamic potential of Eq.(8). In the study of finite volume quark
matter properties using the models which consider the contribution of vacuum
term to the thermodynamic potential, the effective masses of quarks are
observed to decrease with decrease in the volume of system [128]. On the other
side, in Refs. [42] quark matter properties were investigated using the quark
models, neglecting the vacuum term and an increase in the effective mass of
quarks is observed for a decrease in the volume .

An increase in the isospin asymmetry of the medium from zero to finite value
causes less decrease in the magnitude of scalar fields $\sigma$ and $\zeta$.
The scalar field $\sigma$ which is proportional to light quark condensates is
observed to affect more compared to the scalar field $\zeta$, which is
proportional to the strange quark condensate. The impact of isospin asymmetry
is observed to be more in the infinite volume matter compared to the finite
volume case. For example, increasing the value of isospin asymmetry parameter
$\eta$ from zero to 0.3, magnitude of scalar field $\sigma$ increases by
$1.63$% and $1.44$% at $R=\infty$ and $2$ fm (case IV), respectively, at
baryon density $\rho_{B}=4\rho_{0}$ and temperature, $T=100$ MeV. The finite
strangeness fraction of the medium impacts more the strange scalar field
$\zeta$ compared to the non-strange scalar field $\sigma$. For a given density
and temperature of the medium, increasing the strangeness fraction, $f_{s}$,
from $0$ to $0.5$ causes an increase in the magnitude of the scalar field
$\sigma$ and a decrease in $\zeta$. In isospin symmetric medium, for $T=100$
MeV and $\rho_{B}=4\rho_{0}$, as $f_{s}$ is increased from $0$ to $0.5$, at
$R=\infty$ (2 fm (case IV)), the magnitude of scalar field $\sigma$ increases
by $2.8$% ($1.1$%), whereas, the magnitude of $\zeta$ field decreases by
$1.38$% ($1.33$%). As can be seen, the impact of strangeness is observed to be
higher in the infinite strange matter.

Figure 3:  The variation of scalar field $\sigma$ is shown with respect to the
baryon density $\rho_{B}$ (in units of nuclear saturation density $\rho_{0}$),
at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and $150$ MeV [in
subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are for Neumann and
Dirichlet boundary conditions, and MRE denotes multiple reflection expansion.
Figure 4:  The variation of the scalar field $\zeta$ is shown with respect to
the baryon density $\rho_{B}$ (in units of nuclear saturation density
$\rho_{0}$), at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and
$150$ MeV [in subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are
for Neumann and Dirichlet boundary conditions, and MRE denotes multiple
reflection expansion.

In Figs. (5) and (6), the variation of the scalar isovector field $\delta$ and
the scalar dilaton field $\chi$ is shown, respectively. The scalar isovector
field $\delta$ contributes most significantly in the asymmetric nuclear and
hyperonic medium. The dilaton field $\chi$ is observed to have small
variations as a function of density of the medium which is the reason that
this field is treated as a frozen glue ball in some studies [95]. Decreasing
the value of system size from $R=\infty$ to $2$ fm enhances the value of the
dilaton field, $\chi$, in the medium, i.e., smaller drop is observed in finite
volume in comparison to the infinite volume case.

Figure 5:  The variation of scalar field $\delta$ is shown with respect to
baryon density $\rho_{B}$ (in units of nuclear saturation density $\rho_{0}$),
at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and $150$ MeV [in
subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are for Neumann and
Dirichlet boundary conditions, and MRE denotes multiple reflection expansion.
Figure 6:  The variation of scalar dilaton field $\chi$ is shown with respect
to baryon density $\rho_{B}$ (in units of nuclear saturation density
$\rho_{0}$), at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and
$150$ MeV [in subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are
for Neumann and Dirichlet boundary conditions, and MRE denotes multiple
reflection expansion.

###  III.2 Kaon and antikaon in finite volume matter

In this section, we discuss the medium modification of kaon ($K^{+},K^{0}$)
and antikaon ($K^{-},\bar{K^{0}}$) masses in the strange hadronic matter with
finite volume. The in-medium masses of $K$ and $\bar{K}$ mesons are calculated
using the medium modified values of vector and scalar densities of baryons as
well as the scalar fields $\sigma,\zeta$ and $\delta$, in the expressions of
self energies, as given in Eqs. (30) and (31). Figures (7) and (8) show the
medium modified masses of $K^{+}$ and $K^{0}$ mesons with respect to the
baryonic density $\rho_{B}$ (in units of nuclear saturation density,
$\rho_{0}$) at temperatures $T=100$ and 150 MeV, for the system sizes
$R=\infty$ and $2$ fm. In the nuclear medium, both for $R=\infty$ and $2$ fm,
the in-medium masses of kaons, $K^{+}$ and $K^{0}$, increase as a function of
baryon density, $\rho_{B}$, of the nuclear medium, i.e., both members of
$K$-isospin doublet feel repulsive interactions. On the otherside, in the case
of antikaon doublet $\bar{K}$, the in-medium masses of $K^{-}$ and
$\bar{K^{0}}$, plotted in Figs. (9) and (10), decrease as a function of
density of the medium. As can be seen from the interaction Lagrangian density
given by Eq.(26), the medium modification of kaons and antikaons within the
chiral SU(3) hadronic mean field model occurs through the contribution of
Weinberg Tomozawa term, the explicit symmetry breaking term and three range
terms (1st range term stemming from kinetic term of pseudoscalar mesons and
the other two from $d_{1}$ and $d_{2}$ terms, respectively). In Fig. (11),
contributions of individual terms to the effective masses of kaons and
antikaons are shown for finite volume matter, $R=2$ fm (Dirichlet boundary
conditions with MRE case) and are compared with the results calculated in the
infinite volume matter, i.e., $R=\infty$. The results are shown for
temperature, $T=100$ MeV and isospin asymmetry, $\eta=0.3$. The left panel
shows the results for the nuclear medium ($f_{s}=0)$, whereas on the right
side, strangeness fraction $f_{s}$ is fixed at 0.5. In the nuclear medium, as
a function of the baryon density $\rho_{B}$, the Weinberg Tomozawa term gives
repulsive contributions to the in-medium masses of kaons and attractive to
antikaons, leading to an increase and a decrease in their in-medium masses,
respectively. The explicit symmetry-breaking term of the model provides
attractive contributions to both, kaons and antikaons, thereby reducing their
in-medium masses as a function of $\rho_{B}$, for both $R=\infty$ and $2$ fm.
The first range term gives repulsive contributions to both, kaons and
antikaons, whereas the $d_{1}$ and $d_{2}$ terms contribute to the attractive
interactions for the in-medium modification in the masses of kaons and
antikaons, (the term $d_{2}$ is less attractive compared to the $d_{1}$ terms,
for the $K^{+}$ and $K^{-}$ mesons, as we can notice in Fig. (11)). As a
combined effect, the repulsive interaction for $K^{+}$ and $K^{0}$ mesons,
mainly from the WT term and the first range term, dominate over the attractive
interactions provided by the explicit symmetry breaking term, the $d_{1}$ and
$d_{2}$ terms. This results in a net increase in their in-medium masses as a
function of the density of the nuclear medium. The opposite is true for the
antikaon doublet, i.e., the attractive interactions provided by the WT term,
explicit symmetry breaking term, and the $d_{1}$ and $d_{2}$ terms for
antikaons dominate over the repulsive interactions from the first range term.
This causes a net decrease in their in-medium masses with increasing density
of the nuclear matter.

For a given density of the nuclear medium, when the value of $R$ is decreased
from $\infty$ to $2$ fm, the in-medium mass of kaons and antikaons increases.
For example, in symmetric nuclear matter, for baryon density $\rho_{B}$ =
$4\rho_{0}$ and temperature $T=100$ MeV, at $R=\infty$ (2 fm, Dirichlet
boundary conditions with spherical geometry and MRE method), the effective
masses of $K^{+}$, $K^{0}$, $K^{-}$ and $\bar{K^{0}}$ are observed to be
$575.56$ $(591.06)$, $578.88$ ($594.47$), $317.96$ $(326.00)$ and $321.28$
$(329.40)$ MeV, respectively. Looking at the behavior of individual terms as a
function of $R$, (see Fig. (11)), effective mass increases with a decrease in
$R$ for all terms except for the first range term, for which the effective
mass decrease with decreasing $R$. For a given density of the nuclear matter,
an increase of the isospin asymmetry parameter from zero to finite value
causes the mass splitting between the members of a given isospin doublet of
kaons and antikaons. At baryon density $\rho_{B}=4\rho_{0}$, as $\eta$ is
changed from zero to $0.3$, in infinite nuclear matter ($R=\infty$), the mass
of $K^{+}$ meson decreases by $21.16$ MeV, whereas the effective mass of
$K^{0}$ increases by $24.89$ MeV. In case of antikaon doublet $\bar{K}$, the
effective mass of $K^{-}$ ($\bar{K^{0}}$) increases (decreases) by $25.6$
($21.85$) MeV, as $\eta$ is changed from $0$ to 0.3. When the medium is
confined to finite volume with $R=2$ fm (case IV), the values of mass-shift,
as a function of isospin asymmetry (changing $\eta$ from $0$ to $0.3$), for
$K^{+}$, $K^{0}$, $K^{-}$ and $\bar{K^{0}}$ mesons are observed to be
$-23.28$, $26.66$, $25.73$ and $-22.06$ MeV, respectively. From the above, we
observe that the magnitude of mass splitting between the members of a given
isospin doublet increases with decease in the system size, i.e., changing the
volume of medium from infinite to finite value.

Now we discuss the impact of strangeness fraction, $f_{s}$, on the effective
masses of kaons and antikaons in the infinite and finite strange hadronic
medium. As can be seen from Figs. (7) and (8), the effective masses of $K^{+}$
and $K^{0}$ mesons first increase upto certain density beyond which a decrease
is observed with further increase in the baryon density, $\rho_{B}$, of the
strange medium, except for the mass of $K^{0}$ mesons at $\eta=0.3,f_{s}=0.5$
in the finite volume case. As discussed earlier, the finite isospin asymmetry
of the medium causes mass splitting between the isospin partners $K^{+}$ and
$K^{0}$ mesons, and the effective mass of $K^{+}$ decreases whereas the mass
of $K^{0}$ increases with $\eta$. Thus, at finite strangeness fraction $f_{s}$
and isospin asymmetry $\eta=0.3$, the baryon density upto which the mass of
$K^{+}$ mesons increases, decrease to a lower value compared to the symmetric
matter ($\eta=0$) case. On the otherside, increase of isospin asymmetry
decrease the impact of strangeness fraction on the masses of $K^{0}$ mesons,
i.e., less decrease in the mass of $K^{0}$ mesons is observed due to
strangeness fraction $f_{s}$, with increase of $\eta$ from zero to finite
value [as can be seen on comparing the subplots (g) and (h) with subplots (e)
and (f) of Fig. (8)]. As can be seen from Fig. (11), on transition from non-
strange to a strange medium, for the case of kaons, the strength of repulsive
contributions of Weinberg term decreases and the attractive contributions of
explicit symmetry breaking term and the $d_{1}$ term increase, resulting in
overall decrease in the effective mass of $K^{+}$ and $K^{0}$ mesons at higher
values of baryon density of strange medium. Consideration of the finite volume
of the strange medium decreases the impact of attractive interactions, i.e.,
in the strange medium also, the effective mass of kaons and antikaons
increases on changing $R$ from $\infty$ to $2$ fm. An increase of strangeness
fraction causes a lesser drop in the masses of $K^{-}$ and $\bar{K^{0}}$
mesons, i.e., the effective masses of antikaons increase with the strangeness
fraction of the medium (see Figs. (9) and (10)). To quote in terms of numbers,
as the strangeness fraction $f_{s}$ changes from $0$ to $0.5$, at baryon
density $4\rho_{0}$ and temperature $T=100$ MeV, the mass shift for $K^{+}$,
$K^{0}$, $K^{-}$ and $\bar{K^{0}}$ are observed to be $-92.39$ ($-95.93$),
$-95.71$ ($-90.15$), $32.35$ ($33.21$) and $35.15$ ($35.75$) MeV,
respectively, at $R=\infty$ ($2$ fm, (case IV)). As can be seen from Figs. (7)
to (10), among the different cases at finite volume, the effective masses of
kaons and antikaons modify most significantly in the case when the density of
states is calculated using the MRE method and Dirichlet boundary conditions
are implemented, whereas the effects of Neumann boundary conditions cause the
least change. In Ref. [46], the effect of finite volume on the masses of
pseudoscalar mesons was studied using the PNJL model and the masses of kaons
are observed to increase with a decrease in the volume of the system.

Figure 7: The in-medium masses of $K^{+}$ mesons are plotted with respect to
the baryon density $\rho_{B}$ (in units of nuclear saturation density
$\rho_{0}$), at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and
$150$ MeV [in subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are
for Neumann and Dirichlet boundary conditions, and MRE denotes multiple
reflection expansion.  Figure 8: The in-medium masses of $K^{0}$ mesons are
plotted with respect to the baryon density $\rho_{B}$ (in units of nuclear
saturation density $\rho_{0}$), at temperatures, $T=100$ [in subplots (a),
(c), (e) and (g)] and $150$ MeV [in subplots (b), (d), (f) and (h)]. In the
above, NBC and DBC are for Neumann and Dirichlet boundary conditions, and MRE
denotes multiple reflection expansion.  Figure 9: The in-medium masses of
$K^{-}$ mesons are plotted as a function of the baryon density $\rho_{B}$ (in
units of nuclear saturation density $\rho_{0}$), at temperatures, $T=100$ [in
subplots (a), (c), (e) and (g)] and $150$ MeV [in subplots (b), (d), (f) and
(h)]. In the above, NBC and DBC are for Neumann and Dirichlet boundary
conditions, and MRE denotes multiple reflection expansion.  Figure 10: The in-
medium masses of $\bar{K^{0}}$ mesons are plotted as a function of the baryon
density $\rho_{B}$ (in units of nuclear saturation density $\rho_{0}$) , at
temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and $150$ MeV [in
subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are for Neumann and
Dirichlet boundary conditions, and MRE denotes multiple reflection expansion.
Figure 11: The contributions of the individual terms of interaction Lagrangian
to the in-medium masses of $K$ and $\bar{K}$ mesons are plotted as a function
of baryon density $\rho_{B}$ (in units of nuclear saturation density
$\rho_{0}$) in asymmetry ($\eta=0.3$) nuclear ($f_{s}=0$) [in subplots (a),
(c), (e) and (g)] and strange ($f_{s}=0.5$) [in subplots (b), (d), (f) and
(h)] matter, at $T=100$ MeV. In figure, esb and WT denote explicit symmetry
breaking and Weinberg Tomozawa, respectively.

###  III.3 Medium modified mass and decay width of $\Phi$ meson

In this subsection, we shall discuss the impact of medium modified masses of
kaons and antikaons on the masses and decay widths of $\phi$ mesons in finite
volume nuclear and strange hadronic matter. As discussed in Sec. II.3, the
self-energy of $\phi$ mesons is calculated at one loop level for the process
$\phi\rightarrow K\bar{K}$. The effective mass of $\phi$ mesons in the finite
volume strange hadronic matter is calculated using Eq. (34), which depends
upon the real part of the self-energy of $\phi$ mesons, given by Eq. (36). The
medium modified masses of kaons and antikaons calculated in the finite volume
strange hadronic matter are plugged into this equation to obtain the effective
mass of $\phi$ mesons. In Fig. (12), we show the variation of the effective
mass, $m_{\phi}^{*}$, of the $\phi$ mesons in the nuclear and strange hadronic
medium for the infinite ($R=\infty$) and finite ($R=2$ fm) volume of hadronic
matter, considering the cut-off parameter $\Lambda_{c}$= 3 GeV, to regularize
the loop integral of the real part of self-energy. We observe that the
effective mass of $\phi$ mesons first increase slightly and then decreases
with the increase in the baryonic density of the medium, for both $R=\infty$
and $2$ fm. For a given density, a decrease in the volume of the medium causes
an increase in the mass of these mesons. For example, in the nuclear matter at
baryon density $\rho_{B}=4\rho_{0}$ and temperature $T=100$ MeV, the effective
masses of $\phi$ mesons are observed to be $1003.0$ ($1003.7$) and $1007.4$
($1008.0$) MeV, at $R=\infty$ and $2$ fm, case IV, for isospin asymmetry
$\eta=0(0.3)$. In the strange hadronic medium, with $f_{s}=0.5$, these in-
medium masses of $\phi$ mesons change to $991.67$ ($992.19$) and $995.13$
($995.73$) MeV, at $R=\infty$ and $2$ fm , for asymmetry parameter
$\eta=0(0.3)$. As the volume of the medium is changed from infinite to a
finite value, the percentage increase in the effective mass of $\phi$ mesons
is observed to be more in the nuclear medium compared to the strange hadronic
medium. Also, as was the case previously, the impact of finite volume is
observed to be more with the spherical geometry and under Dirichlet boundary
conditions. In Fig.(12), results are shown for the fixed value of the cut-off
parameter, $\Lambda_{c}$. For given density and temperature of the medium, an
increase in the value of cut-off parameter $\Lambda_{c}$ causes an increase in
the effective mass of these mesons [102]. Also, an increase in the temperature
of the nuclear and strange hadronic medium will cause less drop in the mass of
$\phi$ mesons.

In Refs. [139, 99], masses and decay widths of $\phi$ mesons were calculated
in the symmetric infinite nuclear matter through the mass modification of
kaons and antikaons appearing in the loop integral. In these studies, the
effective mass of kaons is calculated using the QMC model, and both kaons and
antikaons were considered to have equal mass in the nuclear medium. However,
as discussed in the previous section, within the chiral SU(3) model, kaons and
antikaons are modified differently in the hadronic medium. This causes
significantly different mass modifications of $\phi$ mesons in the present
calculations within infinite and finite nuclear matter compared to the Ref.
[139, 99]. A very modest decline in the in-medium mass of the $\phi$ meson in
the asymmetric strange matter at zero temperature was noticed in Ref. [96]
through the use of QCD sum rules [140] and the unification of the chiral SU(3)
model.

Figure 12: The in-medium masses of $\phi$ mesons are plotted as a function of
the baryon density $\rho_{B}$ (in units of nuclear saturation density
$\rho_{0}$) , at temperatures, $T=100$ [in subplots (a), (c), (e) and (g)] and
$150$ MeV [in subplots (b), (d), (f) and (h)]. In the above, NBC and DBC are
for Neumann and Dirichlet boundary conditions, and MRE denotes multiple
reflection expansion.

In Fig. (13), the medium modified decay widths, $\Gamma_{\phi}^{\star}$, of
the $\phi$ mesons, decaying to $K$$\bar{K}$ pairs and calculated using Eq.
(37), are plotted as a function of baryon density $\rho_{B}$, for
$\Lambda_{c}$ = 3 GeV. As can be seen from the figure, the $\phi$ meson decay
width increases with increase in the baryonic density, $\rho_{B}$, in both
infinite and finite hadronic medium. However, for a given density of the
medium, as the value of system size, $R$, decreases to a finite value, due to
the increase of effective mass of $K$ and $\bar{K}$ mesons, the decay width
decreases, i.e., less phase space is available for the decay process
$\phi\rightarrow K\bar{K}$, as the volume of the hadronic medium becomes
finite. At baryon density $4\rho_{0}$, the decay widths of $\phi$ mesons are
observed to be $22.20$ ($21.29$) and $16.58$ ($15.80$) MeV at $R=\infty$ and
$2$ fm, case IV, for $\eta=0(0.3)$. Moving to the strange matter
($f_{s}=0.5$), decay widths of $\phi$ mesons for $\eta=0(0.3)$ are observed to
be $39.09$ ($38.15$) and $32.98$ ($31.99$) MeV at $R=\infty$ and $2$ fm,
respectively. Although the masses and decay width of $\phi$ mesons have been
studied in the infinite nuclear matter [62, 116, 117, 119, 118], the current
study is the first attempt to study the impact of finite volume.

Figure 13: The in-medium decay widths of $\phi$ mesons are plotted as a
function of the baryon density $\rho_{B}$ (in units of nuclear saturation
density $\rho_{0}$) , at temperatures, $T=100$ [in subplots (a), (c), (e) and
(g)] and $150$ MeV [in subplots (b), (d), (f) and (h)]. In the above, NBC and
DBC are for Neumann and Dirichlet boundary conditions, and MRE denotes
multiple reflection expansion.

##  IV Summary

To summarize, in the present work, we investigated the mass modifications of
kaons and antikaons in the isospin asymmetric strange hadronic matter confined
to the finite volume using the chiral SU(3) hadronic mean field model. The
finite volume effects are introduced in the present work through the
calculations of the density of states using the MRE method and applying the
lower momentum cut-off in the expressions of scalar and vector densities of
baryons in terms of system size $R$. The magnitude of the scalar fields
$\sigma$ and $\zeta$, which modify the properties of baryons within the chiral
SU(3) model, are observed to increase as the volume of the medium is
considered as finite,i.e., $R$ is changed from infinite to a finite value.
This increases the effective mass of kaons and antikaons with a decrease of
$R$. While calculating the in-medium modifications of kaons, antikaons, and
$\phi$ meson properties in the finite volume of medium, calculations are done
considering (i) Neumann boundary conditions in the MRE method and spherical
geometry, (ii) Dirichlet boundary condition with MRE method and considering
spherical and cubic geometry and (iii) applying only the lower momentum cut-
off, $\pi/R$, in the expressions of scalar and vector densities of baryons
without calculating the density of states using MRE method. Among these
different cases, the most significant modifications occur when calculations
are done considering the spherical geometry of the finite volume of the medium
and applying Dirichlet boundary conditions.

Employing the in-medium masses of $K$ and $\bar{K}$ mesons calculated using
the chiral SU(3) model, the masses and decay width of the $\phi$ mesons are
also calculated in the finite volume nuclear and hyperonic medium. The
effective masses of $\phi$ mesons are observed to increase, whereas the decay
width decreases as the value of $R$ is changed from infinite to finite value.
The impact of finite volume, using periodic-antiperiodic boundary conditions,
on the in-medium properties of hadrons will be investigated in the future
work. Further, the impact of coupled channel calculations, considering finite
volume, on the in-medium masses of different mesons will also be investigated
in the future. Present studies of $\phi$ mesons will also be of relevance in
understanding the observables from the experiments, for example, J-PARC E16,
dedicated to investigate the $\phi$ meson mass shift from the measurement of
di-electron spectra in $p-A$ collisions [141]. Additionally, a proposal has
been made at J-Lab to investigate the interaction of helium nuclei with $\phi$
mesons after the 12 GeV upgrade [142] and in-medium mass-shift and decay width
will be useful to explore the possibility of $\phi$ mesic nuclei in such
experiments.

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