# Open Strange Mesons in (magnetized) nuclear matter

Ankit Kumar<EMAIL_ADDRESS>Amruta Mishra
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Delhi, Hauz Khas, New Delhi - 110016

###### Abstract

We investigate the mass modifications of open strange mesons (vector $K^{*}$
and axial vector $K_{1}$) in (magnetized) isospin asymmetric nuclear matter
using quantum chromodynamics sum rule (QCDSR) approach. The in-medium decay
widths of $K^{*}$ $\rightarrow$ $K\pi$ and $K_{1}$ $\rightarrow$ $K^{*}\pi$
are studied from the mass modifications of $K_{1}$, $K^{*}$ and $K$ mesons,
using a light quark-antiquark pair creation model, namely the ${}^{3}P_{0}$
model. The in-medium decay width for $K_{1}$ $\rightarrow$ $K^{*}\pi$ is
compared with the decay widths calculated using a phenomenological Lagrangian.
The effects of magnetic fields are also studied on the mass and the partial
decay width of the vector $K^{*}$ meson decaying to $K\pi$. Within the QCD sum
rule approach, the medium effects on the masses of the open strange mesons are
calculated from the light quark condensates and the gluon condensates in the
hadronic medium. The quark condensates are calculated from the medium
modifications of the scalar fields ($\sigma$, $\zeta$, and $\delta$) in the
mean field approximation within a chiral $SU(3)$ model, while the scalar gluon
condensate is obtained from the medium modification of a scalar dilaton field
($\chi$), which is introduced within the model to imitate the scale invariance
breaking of QCD.

##  I Introduction

The investigation of various properties of hadrons Hosaka_PPNP96_88_2017  has
become an important and emerging topic of research interest in high energy
physics due of its relevance to relativistic heavy ion collision (HIC)
experiments. The medium produced in these heavy ion colliders has high density
and/or high temperature, which can affect the experimental observables due to
the medium modifications of the produced hadrons. It is important to study the
effects of isospin asymmetry as the colliding heavy ions have more number of
neutrons as compared to the number of protons. The study of light vector
mesons is important due to its relevance to observables e.g., the dilepton
spectra in the HIC experiments. The dileptons are promising observables for
the study the properties of hadrons in dense nuclear matter as their
interaction with the hadronic environment is negligible, and, they give
information about all stages of the evolution of the strongly interacting
created in heavy ion collision experiments. The in-medium masses of light
vector mesons ($\rho$, $\omega$, and $\phi$) have been studied in strange
hadronic matter AM_PRC91_035201_2015  and isospin asymmetric magnetized
nuclear medium AM_PRC100_015207_2019  within the QCD sum rule approach, using
the quark and gluon condensates as calculated within a chiral $SU(3)$ model.
The study of strange mesons has been the center of attention due to their
significance in studying the yield and spectra of these mesons, produced in
HIC experiments Hartnack_PR510_119_2012  as well as in the study of certain
astronomical bodies where strange matter could exist in the interior of the
neutron stars Tolos_PPNP112_103770_2020 ; Kaplan_PLB175_57_1986 . The in-
medium masses of the open pseudoscalar mesons, e.g., the kaons and antikaons
have been studied in strange hadronic matter using a chiral effective model
AM_PRC70_044904_2004  and the effects of temperature have also been
incorporated. The effects from baryonic Dirac sea are also investigated in
AM_PRC70_044904_2004  and the results for in-medium masses are compared to
that obtained from chiral perturbation theory (ChPT). The energies and optical
potentials for kaons and antikaons have been studied in isospin asymmetric
nuclear (hyperonic) matter within a chiral effective model
AM_PRC78_024901_2008 ; AM_PRC74_064904_2006 ; AM_EPJA41_205_2009 . The isospin
asymmetry effects are observed to be significant at high densities. These can
have observable consequences in the heavy-ion beam collisions at Compressed
Baryonic Matter (CBM) experiment at future Facility for Antiproton and Ion
Research (FAIR). Moreover, the low energy scattering of antikaons with
nucleons ($N$) have been studied in the framework of coupled channel approach
Oset_NPA635_99_1998 ; Ramos_NPA671_481_2000  and the $\Lambda(1405)$ is
reproduced dynamically just below the $\bar{K}N$ threshold due to coupling of
$\bar{K}N$ channel to $\pi\Sigma$ channel in $I=0$ Koch_PLB337_7_1994 .

In the present work, we study the in-medium masses as well as the decay widths
of vector $K^{*}$ (decaying to $K\pi$) and axial vector $K_{1}$ (to
$K^{*}\pi$) mesons, in (magnetized) isospin asymmetric dense nuclear medium.
These strange vector and axial-vector mesons are the chiral partners of each
other and are considered as obvious systems to study the chiral symmetry
breaking effects and its restoration in the medium. In Ref.
Gubler_PLB767_336_2017 , the properties of $f_{1}(1285)$ and $\omega$ mesons
are studied in QCDSR approach to probe the chiral symmetry restoration in
nuclear environment. The in-medium masses for the light vector mesons ($\rho$,
$\omega$, and $\phi$) have been calculated within QCD sum rule approach
Hatsuda_PRC46_R34_1992  from the medium modifications of quark condensates and
scalar gluon condensates. The lowest Charmonium states $J/\psi$ and $\eta_{c}$
have been studied in isospin asymmetric hot nuclear matter
AK_PRC82_045207_2010  within QCD sum rule approach, and the effects of medium
density are found to be the dominant effects. Moreover, within the QCD sum
rule approach, the masses of $1S$ and $1P$ states of heavy quarkonium
(charmonium and bottomonium) have been studied in isospin asymmetric nuclear
matter including the effects of strong magnetic fields
Parui_PRD106_114033_2022 . The mass modification for heavy quarkonium states,
within QCD sum rule approach, arises from the medium modifications of scalar
gluon condensates and twist-2 gluon condensates. This work includes the
effects from Dirac sea through summing over the nucleonic tadpole diagrams and
leads to a decrease in the values of light quark condensates with magnetic
field, an effect known as inverse magnetic catalysis, when the anomalous
magnetic moments (AMMs) of nucleons are taken into account. The decay width of
channel $K^{*}$ $\rightarrow$ $K\pi$ is studied using the ${}^{3}P_{0}$ model,
from the in-medium masses of vector $K^{*}$ and pseudoscalar $K$ meson, which
are calculated within QCD sum rule approach and chiral $SU(3)$ model,
respectively. The open flavor mesons decay through the creation of a
$\bar{q}q$ pair which is produced with vacuum quantum numbers
($J^{PC}=0^{++}$) corresponding to a ${}^{3}P_{0}$ state
Ackleh_PRD54_6811_1996 ; Micu_NPB10_521_1969 . The ${}^{3}P_{0}$ model has
been used extensively in the literature to study the decays of various mesons
Yaouanc_PRD8_2223_1973 ; Barnes_PRD55_4157_1997 ; Friman_PLB548_153_2002 .
This model indicates the importance of taking into account the internal
structures of hadrons as it has explained the experimentally observed
suppression of $\psi(4040)$ decay mode to $\bar{D}D$ and ($\bar{D^{*}}D$ \+
$\bar{D}D^{*}$) in comparison with $\bar{D^{*}}D^{*}$ decay mode
Yaouanc_PLB71_397_1977 .

The masses of strange mesons ($K$, $K^{*}$, and $\phi$) have been investigated
in the presence of strong magnetic fields in Ref. AM_IJMPE30_2150014_2021  due
to $\phi-\eta^{\prime}$ and $K^{*}-K$ mixing, and also the decay widths for
$\phi\rightarrow K\bar{K}$, $K^{*}\rightarrow K\pi$ are studied in a field
theoretic model for composite hadrons from the mass modifications of the
mesons. The in-medium spectral functions and production cross-sections for the
strange mesons ($K^{*},\bar{K}^{*}$, and $\phi$), in strange hadronic medium,
have also been studied from the in-medium masses and decay widths for these
mesons AM_EPJA57_98_2021 . The effects of medium density as well as
strangeness on the production cross-sections of $K^{*},\bar{K}^{*}$, and
$\phi$ from the $K\pi$, $\bar{K}\pi$, and $\bar{K}K$ scattering respectively,
have been found to be quite appreciable when compared to the vacuum
conditions. The properties of vector $\bar{K^{*}}$ meson in the nuclear matter
are investigated in Ref. Tolos_PRC82_045210_2010  using a unitary approach in
coupled channels. The strange vector $K^{*}$($\bar{K^{*}}$) mesons are
produced mainly at later stages from the $K\pi(\bar{K}\pi$) scattering and the
contribution from direct hadronization from the quark gluon plasma (QGP) state
is quite small, as calculated within Parton–Hadron-String-Dynamics (PHSD)
transport model Ilner_PRC95_014903_2017 ; Ilner_PRC99_024914_2019 . Although
the study of medium density effects might find relevance in future experiments
at the GSI Facility for Antiproton and Ion Research (FAIR) and Nuclotron-based
Ion Collider facility (NICA) Kumar_EPJC79_403_2019 ; Rapp_PNP65_209_2010 ,
where matter having large baryon density will be produced. The axial vector
$K_{1}$ meson masses have been analyzed in the QCD sum rule analysis
Song_PLB792_160_2019  in the nuclear matter and the decay widths for
$K_{1}\rightarrow K\pi\pi$ channel have been studied using the ${}^{3}P_{0}$
model Tayduganov_PRD85_074011_2012 .

In QCD sum rule approach AM_PRC91_035201_2015 ; Hatsuda_PRC46_R34_1992 ;
Hatsuda_PRC52_3364_1995 , we expand the current-current correlation function
for the corresponding meson using operator product expansion (OPE) in terms of
local operators and their coefficients. The central idea of this approach is
to relate the spectral density of this correlation function with the OPE
expression via a dispersion relation, for large space-like regions. The medium
modifications in masses arise due to the medium modifications of light quark
condensates and gluon condensates within the QCD sum rule approach. These
light quark condensates are related to scalar fields ($\sigma$, $\zeta$, and
$\delta$) of the medium by comparing the explicit chiral symmetry breaking
term of QCD Lagrangian to the corresponding Lagrangian term in the chiral
$SU(3)$ model Papazoglou_PRC59_411_1999 ; Zschiesche_PRC63_025211_2001 ; while
the gluon condensates are related to the scalar dilaton field ($\chi$) of the
medium. The chiral effective Lagrangian is written such that it includes
various symmetries of low energy QCD and the symmetry breaking effects. The
coupled equations of motion for various scalar fields ($\sigma$, $\zeta$,
$\delta$, and $\chi$) are solved within the chiral $SU(3)$ model including
various medium effects.

The estimation of strong magnetic field production in the peripheral HIC
experiments Tuchin_AHEP2013_490495_2013 ; Skokov_IJMPA24_5925_2009  have grown
immense interest in the study of the magnetic field effects on the produced
medium. The estimated magnetic field strength at Relativistic Heavy Ion
Collider (RHIC) is $\sim$ $2m_{\pi}^{2}$ and at Large Hadron Collider (LHC) is
$\sim$ $15m_{\pi}^{2}$, calculated considering Lienard-Wiechert potential in
numerical simulations within a microscopic transport model
Skokov_IJMPA24_5925_2009 . The study of strong magnetic field effects on
produced medium is also important due to novel interesting quantum effects
like chiral magnetic effect Fukushima_PRD78_074033_2008 , magnetic catalysis
Kharzeev_Springer_2013 ; Shovkovy_LNP871_13_2013  and inverse magnetic
catalysis Kharzeev_Springer_2013 ; Preis_LNP871_49_2013  as well as in the
study of neutron stars and magnetars where large magnetic fields are estimated
to exist. The charged mesons have contribution due to Landau level
quantization in the presence of magnetic field and the effects of PV mixing,
considered through effective Lagrangian vertex Gubler_PRD93_054026 ;
Cho_PRD91_045025_2015 ; AM_PRC102_045204_2020 ; AM_IJMPE30_2150064_2021  and
spin-magnetic field interaction term Alford_PRD88_105017_2013 ;
Yoshida_PRD94_074043_2016 ; Machado_PRD88_034009_2013 , are also investigated
in this study. In Ref. AM_PRC102_045204_2020 , the PV mixing between vector
and pseudoscalar charmonium states is considered through the effective
Lagrangian vertex in the presence of strong magnetic fields and the decay
width of vector charmonium state $\psi(3770)$ to $\bar{D}D$ is also studied
within a field theoretic model of composite hadrons. Furthermore, the PV
mixing between open charm mesons ($D^{*}$ and $D$) is also studied in Ref.
AM_IJMPE30_2150064_2021  alongwith the contribution due to Landau levels for
charged mesons, and the decay widths ($D^{*}\rightarrow D\pi$) and
($\psi(3770)\rightarrow\bar{D}D$) are also studied using a field theoretic
model. In field theoretic model of composite hadrons, the hadronic states like
charmonium ($\psi(3770)$) state, open charm mesons ($D^{*},\bar{D},D$) and
pion ($\pi$) states are constructed explicitly from constituent quark fields
assuming harmonic oscillator wave functions for these states, and the matrix
element for the decay is then calculated from the light quark-antiquark pair
creation term of the free Dirac Hamiltonian density. However, the produced
medium density will be very small in these peripheral collision experiments.
Therefore, to understand the behavior at these conditions of high magnetic
field and low medium density, we also study the effects of high magnetic
fields on the properties of vector $K^{*}$ meson. Hence, this present work
might have relevance in lower energy central collisions, where produced medium
has high density, as well as in high energy peripheral collision experiments
where large magnetic fields are produced but the produced medium has low
density.

The present work is organized in the following manner: In section II, we
briefly describe the chiral $SU(3)$ model to compute the medium modifications
of the quark condensates and scalar gluon condensates from the medium
modifications of the scalar fields ($\sigma$, $\zeta$, $\delta$, and $\chi$).
In section III, we discuss the QCD sum rule approach, which is used to study
the in-medium masses of these open strange mesons in isospin asymmetric
nuclear medium. We also discuss the effects of strong magnetic field on the
masses (and hence the decay widths) of the open strange mesons. In section IV,
we discuss the in-medium masses and decay width of $K^{*}$ meson within a
phenomenological model. The decay width of $K_{1}\rightarrow K^{*}\pi$ is also
studied within a phenomenological Lagrangian approach. In section V, we
briefly discuss the ${}^{3}P_{0}$ model, which will be further used to
calculate the in-medium decay width of vector $K^{*}$ and axial vector $K_{1}$
mesons. In section VI, we discuss and analyze the results obtained and compare
them with earlier work to emphasize on the relevance of this work. In section
VII, we summarize the results obtained in the present work.

##  II The Hadronic Chiral $SU(3)$ Model

We make use of an effective chiral $SU(3)$ model Papazoglou_PRC59_411_1999 ;
Zschiesche_PRC63_025211_2001 ; AM_PRC69_024903_2004 ; Weinberg_PR166_1568_1968
; Coleman_PR177_2239_1969 ; Bardeen_PR177_2389_1969  to calculate the quark
condensates and scalar gluon condensates in the nuclear matter, which will
further be used in the QCD sum rule approach to calculate the in-medium masses
of vector $K^{*}$ and axial vector $K_{1}$ mesons. The effective chiral model
is formulated on the basis of nonlinear realization of chiral symmetry of QCD
and its scale invariance breaking AM_PRC69_024903_2004 ;
Weinberg_PR166_1568_1968 ; Coleman_PR177_2239_1969 ; Bardeen_PR177_2389_1969 .
The broken scale invariance of QCD symmetry leads to trace anomaly of QCD,
$\theta_{\mu}^{\mu}=\big{<}\frac{\beta_{QCD}}{2g}G_{\mu\nu}^{a}G^{a\mu\nu}\big{>}$,
where $G^{a}_{\mu\nu}$ is the gluon field strength tensor. At the tree level,
this is introduced in the effective Lagrangian density through a logarithmic
scale breaking term given by Schechter_PRD21_3393_1980 ,

$\displaystyle\mathcal{L}_{\rm scale\,breaking}=-\frac{1}{4}\chi^{4}{\rm
ln}\left(\frac{\chi^{4}}{\chi_{0}^{4}}\right)+\frac{d}{3}\chi^{4}{\rm
ln}\left(\Big{(}\frac{\sigma^{2}\zeta}{\sigma_{0}^{2}\zeta_{0}}\Big{)}\Big{(}\frac{\chi}{\chi_{0}}\Big{)}^{3}\right)$
(1)

and the total Lagrangian density for the dilaton field ($\chi$) is given by,

$\displaystyle\mathcal{L}_{\chi}=\frac{1}{2}\big{(}\partial_{\mu}\chi\big{)}\big{(}\partial^{\mu}\chi\big{)}-k_{4}\chi^{4}+\mathcal{L}_{\rm
scale\,breaking}$ (2)

where first term is the kinematic term and the second term is introduced to
ensure the vacuum expectation value (VEV) for scalar dilaton field. Then the
energy momentum tensor in chiral model is given by,

$\displaystyle
T_{\mu\nu}=\big{(}\partial_{\mu}\chi\big{)}\left(\frac{\partial\mathcal{L}_{\chi}}{\partial\big{(}\partial^{\nu}\chi\big{)}}\right)-g_{\mu\nu}\mathcal{L}_{\chi}$
(3)

On the other hand, the energy momentum tensor in QCD Lee_P72_97_2009 ;
Morita_PRC77_064904_2008 , accounting for the current quark masses, is written
as,

$\displaystyle
T_{\mu\nu}=-ST\big{(}G^{a}_{\mu\sigma}G^{a\sigma}_{\nu}\big{)}+\frac{g_{\mu\nu}}{4}\Big{(}\sum_{i}m_{i}\big{<}\bar{q_{i}}q_{i}\big{>}+\Big{<}\frac{\beta_{QCD}}{2g}G_{\sigma\kappa}^{a}G^{a\sigma\kappa}\Big{>}\Big{)}$
(4)

where the first term represents the symmetric traceless part and the second
term contains the trace part. After multiplying eq. (3) and (4) by
$g^{\mu\nu}$, we get the trace ($T_{\mu}^{\mu}$) of the energy momentum tensor
in chiral model and QCD respectively, and the expression for scalar gluon
condensate is then given as,

$\displaystyle\Big{<}\frac{\alpha_{s}}{\pi}G^{a}_{\mu\nu}G^{a\mu\nu}\Big{>}=\frac{8}{9}\Big{[}\big{(}1-d\big{)}\chi^{4}+\sum_{i}m_{i}\big{<}\bar{q_{i}}q_{i}\big{>}\Big{]}$
(5)

where the one-loop QCD $\beta$-function is given by,

$\beta_{QCD}(g)=-\frac{11N_{c}g^{3}}{48\pi^{2}}\Big{(}1-\frac{2N_{f}}{11N_{c}}\Big{)}+\mathcal{O}\big{(}g^{5}\big{)}$
(6)

with $N_{c}=3$, and $N_{f}=3$, are the number of colors and quark flavors
respectively and the strong coupling constant of QCD, $\alpha_{s}=g^{2}/4\pi$.
Thus, the scalar gluon condensate is introduced in the chiral $SU(3)$ model
through a scalar dilaton field ($\chi$). The scalar gluon condensate has
additional contribution due to finite non-zero quark masses $m_{i}(i=u,d,s)$.
The non-zero quark condensates $\big{<}\bar{q}_{i}q_{i}\big{>}$ are introduced
in the QCD vacuum by the spontaneous breaking of chiral symmetry by the ground
state. The finite quark mass term ($m_{i}\big{<}\bar{q_{i}}q_{i}\big{>}$) of
equation (5) is given in terms of scalar fields ($\sigma$, $\zeta,$ and
$\delta$) by comparing the explicit chiral symmetry breaking term of the
chiral model, after mean field approximation,

$\mathcal{L}_{SB}={\rm Tr}\left[{\rm
diag}\,\Big{(}-\frac{1}{2}m_{\pi}^{2}f_{\pi}\big{(}\sigma+\delta\big{)},-\frac{1}{2}m_{\pi}^{2}f_{\pi}\big{(}\sigma-\delta\big{)},\Big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\Big{)}\zeta\Big{)}\right]$
(7)

with the corresponding Lagrangian density term of the QCD, which is written
as,

$\mathcal{L}_{SB}^{QCD}=-{\rm Tr}\left[{\rm
diag}\,\big{(}m_{u}\bar{u}u,m_{d}\bar{d}d,m_{s}\bar{s}s\big{)}\right]$ (8)

Within the chiral model, the coupled equations of motion for the scalar fields
($\sigma$, $\zeta$, $\delta$ and $\chi$), derived from the chiral Lagrangian
density, are given as,

$\displaystyle
k_{0}\chi^{2}\sigma-4k_{1}\big{(}\sigma^{2}+\zeta^{2}+\delta^{2}\big{)}\sigma-2k_{2}\big{(}\sigma^{3}+3\sigma\delta^{2}\big{)}-2k_{3}\chi\sigma\zeta$
$\displaystyle-\frac{d}{3}\chi^{4}\left(\frac{2\sigma}{\sigma^{2}-\delta^{2}}\right)+\left(\frac{\chi}{\chi_{0}}\right)^{2}m_{\pi}^{2}f_{\pi}-\sum_{i}g_{\sigma
i}\rho_{i}^{s}=0$ (9) $\displaystyle
k_{0}\chi^{2}\zeta-4k_{1}\big{(}\sigma^{2}+\zeta^{2}+\delta^{2}\big{)}\zeta-4k_{2}\zeta^{3}-k_{3}\chi\big{(}\sigma^{2}-\delta^{2}\big{)}-\frac{d}{3}\frac{\chi^{4}}{\zeta}$
$\displaystyle+\left(\frac{\chi}{\chi_{0}}\right)^{2}\left[\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right]-\sum_{i}g_{\zeta
i}\rho_{i}^{s}=0$ (10) $\displaystyle
k_{0}\chi^{2}\delta-4k_{1}\big{(}\sigma^{2}+\zeta^{2}+\delta^{2}\big{)}\delta-2k_{2}\big{(}\delta^{3}+3\sigma^{2}\delta\big{)}+2k_{3}\chi\delta\zeta$
$\displaystyle+\frac{2}{3}d\chi^{4}\left(\frac{\delta}{\sigma^{2}-\delta^{2}}\right)-\sum_{i}g_{\delta
i}\rho_{i}^{s}=0$ (11) $\displaystyle
k_{0}\chi\big{(}\sigma^{2}+\zeta^{2}+\delta^{2}\big{)}-k_{3}\big{(}\sigma^{2}-\delta^{2}\big{)}\zeta+\chi^{3}\left[1+{\rm
ln}\Big{(}\frac{\chi^{4}}{\chi_{0}^{4}}\Big{)}\right]+$
$\displaystyle\big{(}4k_{4}-d\big{)}\chi^{3}-\frac{4}{3}d\chi^{3}{\rm
ln}\left(\Big{(}\frac{(\sigma^{2}-\delta^{2})\zeta}{\sigma_{0}^{2}\zeta_{0}}\Big{)}\Big{(}\frac{\chi}{\chi_{0}}\Big{)}^{3}\right)+\bigg{(}\frac{2\chi}{\chi_{0}^{2}}\bigg{)}$
$\displaystyle\left[\Big{(}m_{\pi}^{2}f_{\pi}\sigma+\big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\big{)}\zeta\Big{)}\right]=0$
(12)

The medium effects due to the baryon density, isospin asymmetry, and magnetic
field, are incorporated into the model through the scalar fields, which depend
on the scalar densities ($\rho_{i}^{s}$) of the baryons. The coupled equations
of motion, given by (9,10,11,12), are solved to find the medium dependent
values of the scalar fields, from which we obtain the scalar gluon condensates
$\big{<}\frac{\alpha_{s}}{\pi}G^{a}_{\mu\nu}G^{a\mu\nu}\big{>}$ and the quark
condensates ($\big{<}\bar{u}u\big{>}$, $\big{<}\bar{d}d\big{>}$, and
$\big{<}\bar{s}s\big{>}$) in the nuclear medium.

##  III QCD Sum Rule Approach

In this section, we will discuss the QCD sum rule method AM_PRC91_035201_2015
; Song_PLB792_160_2019 ; Klingl_NPA624_527_197 , which is used to calculate
the in-medium masses through the medium modifications of the quark and gluon
condensates, calculated within the chiral model. The current-current
correlation function, written in terms of the time-ordered product of two
currents, for the meson $V$ is given by,

$\Pi^{V}_{\mu\nu}(q)=i\int
d^{4}x\,e^{iq.x}\big{<}0\big{|}T\big{[}j^{V}_{\mu}(x),j^{V}_{\nu}(0)\big{]}\big{|}0\big{>}$
(13)

where the currents for vector $K^{*}$ meson are given as
$j^{K^{*+}}_{\mu}=\bar{s}\gamma_{\mu}u$ and
$j^{K^{*0}}_{\mu}=\bar{s}\gamma_{\mu}d$; while the currents for the axial
vector $K_{1}$ meson are given by
$j^{K_{1}^{+}}_{\mu}=\bar{s}\gamma_{\mu}\gamma_{5}u$ and
$j^{K_{1}^{0}}_{\mu}=\bar{s}\gamma_{\mu}\gamma_{5}d$. We write the transverse
tensor structure for the correlation function as a sum of two independent
functions as Song_PLB792_160_2019 ; Leupold_PRC64_015202_2001 ,

$\Pi^{V}_{\mu\nu}(q)=-g_{\mu\nu}\Pi_{1}(q^{2})+q_{\mu}q_{\nu}\Pi_{2}(q^{2})$
(14)

For the conserved vector current $j^{K^{*}}_{\mu}$, these two functions are
related as $\Pi_{1}(q^{2})=q^{2}\Pi_{2}(q^{2})$. As the axial current
$j^{K_{1}}_{\mu}$ is not conserved, this relation does not hold true for axial
current. We can make use of either $\Pi_{1}(q^{2})$ or $\Pi_{2}(q^{2})$ to
carry out QCDSR, but $\Pi_{2}(q^{2})$ will have contributions from the
pseudoscalar mesons which require further investigation of the in-medium
properties of pseudoscalar mesons. Therefore we will make use of
$\Pi_{1}(q^{2})$ in this work throughout. The main idea of QCDSR is to relate
the spectral density of the correlator function $\Pi^{V}(q^{2}=-Q^{2})$ on the
phenomenological side via a dispersion relation with the QCD operator product
expansion (OPE) side. The correlator function, on the phenomenological side,
can be written as,

$12\pi^{2}\widetilde{\Pi}^{V}_{\rm phen}(Q^{2})=\int ds\frac{R^{V}_{\rm
phen}(s)}{s+Q^{2}}$ (15)

where $\widetilde{\Pi}^{V}(Q^{2})=\Pi^{V}(Q^{2})/Q^{2}$ and the spectral
density $R^{V}_{\rm phen}(s)$ is related to the imaginary part of the
correlator as $R^{V}_{\rm phen}(s)=12\pi Im(\Pi^{V}_{\rm phen}(s))$. For
enhancing the contribution of the pole, we make use of Borel transformation
Shifman_NPB147_385_1979 ; Reinders_PR127_1_1985  and we get,

$12\pi^{2}\,\widetilde{\Pi}^{V}(M^{2})=\int dse^{-\frac{s}{M^{2}}}R^{V}_{\rm
phen}(s)$ (16)

For large space-like regions, $Q^{2}=-q^{2}>>1$ GeV2, the asymptotic freedom
in QCD allows for series expansion of correlation function in terms of
operator product expansion (OPE) as Klingl_NPA624_527_197 ;
Kwon_PRC81_065203_2010 ,

$12\pi^{2}\,\widetilde{\Pi}^{V}(q^{2}=-Q^{2})=d_{V}\bigg{[}-c_{0}^{{V}}{\rm
ln}\frac{Q^{2}}{\mu^{2}}+\frac{c_{1}^{V}}{Q^{2}}+\frac{c_{2}^{V}}{Q^{4}}+\frac{c_{3}^{V}}{Q^{6}}+...\bigg{]}$
(17)

where $d_{K^{*},K_{1}}=3$ and the scale $\mu$ is taken to be 1 GeV here. The
first term is the leading perturbative QCD term and subsequent higher order
terms, containing the non-perturbative effects of QCD, are suppressed by
powers of $1/Q^{2}$. The coefficients $c_{i}^{\prime}s$ ($i=2,3...$) are
related to quark condensates and scalar gluon condensates. For the $K^{*+}$
meson Song_PLB792_160_2019 ; Shifman_NPB147_385_1979 ; Shifman_NPB147_448_1979
, these coefficients are given by,

$c_{0}^{K^{*}}=1+\frac{\alpha_{s}(Q^{2})}{\pi},\,\,c_{1}^{K^{*}}=-3(m_{u}^{2}+m_{s}^{2})$
(18)
$c_{2}^{K^{*}}=\frac{\pi^{2}}{3}\Big{<}\frac{\alpha_{s}}{\pi}G^{\mu\nu}G_{\mu\nu}\Big{>}+\frac{16\pi^{2}}{d_{K^{*}}}\big{<}m_{u}\bar{s}s+m_{s}\bar{u}u\big{>}-\frac{4\pi^{2}}{d_{K^{*}}}\big{<}m_{u}\bar{u}u+m_{s}\bar{s}s\big{>}$
(19) $\displaystyle
c_{3}^{K^{*}}=-8\pi^{3}\Bigg{[}2\big{<}\alpha_{s}(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}s)(\bar{s}\gamma^{\mu}\gamma^{5}\lambda^{a}u)\big{>}+\frac{2}{9}\Big{<}(\bar{u}\gamma_{\mu}\lambda^{a}u+\bar{s}\gamma_{\mu}\lambda^{a}s)\times$
$\displaystyle\Big{(}\sum_{q=u,d,s}\bar{q}\gamma^{\mu}\lambda^{a}q\Big{)}\Big{>}\Bigg{]}=-8\pi^{3}\alpha_{s}\kappa_{n}\bigg{[}\frac{32}{9}\big{<}\bar{u}u\big{>}\big{<}\bar{s}s\big{>}+\frac{32}{81}\big{(}\big{<}\bar{u}u\big{>}^{2}+\big{<}\bar{s}s\big{>}^{2}\big{)}\bigg{]}$
(20)

where $\alpha_{s}(Q^{2})=4\pi/\big{[}b\,{\rm
ln}(Q^{2}/\Lambda^{2}_{QCD})\big{]}$ is the running coupling constant of QCD
and $(m_{u},m_{s})$ are current quark masses for up and strange quark. The QCD
scale is taken to be $\Lambda_{QCD}=140$ MeV, with $b=11-(2/3)N_{f}$ with
$N_{f}=3$ as the number of quark flavors. To evaluate the four quark operators
of coefficient $c_{3}^{K^{*}}$, we have used the factorization method
Shifman_NPB147_448_1979 ,

$\big{<}(\bar{q_{i}}\gamma_{\mu}\gamma_{5}\lambda^{a}q_{j})^{2}\big{>}=-\big{<}(\bar{q_{i}}\gamma_{\mu}\lambda^{a}q_{j})^{2}\big{>}=\delta_{ij}\frac{16}{9}\kappa_{n}\big{<}\bar{q_{i}}q_{i}\big{>}^{2}$
(21)

where $q_{i}=(u,d,s)$ and the parameter $\kappa_{n}$ is introduced to
parameterize the deviation from the exact factorization with $`n$’
corresponding to different mesons. For the neutral $K^{*0}$ meson, the up
($u$) quark flavor is replaced by the down ($d$) quark flavor. The
coefficients for the strange axial vector meson $K_{1}^{+}$ are given by
Song_PLB792_160_2019 ,

$c_{0}^{K_{1}}=1+\frac{\alpha_{s}(Q^{2})}{\pi},\,\,c_{1}^{K_{1}}=-3(m_{u}^{2}+m_{s}^{2})$
(22)
$c_{2}^{K_{1}}=\frac{\pi^{2}}{3}\Big{<}\frac{\alpha_{s}}{\pi}G^{\mu\nu}G_{\mu\nu}\Big{>}-\frac{16\pi^{2}}{d_{K_{1}}}\big{<}m_{u}\bar{s}s+m_{s}\bar{u}u\big{>}+\frac{4\pi^{2}}{d_{K_{1}}}\big{<}m_{u}\bar{u}u+m_{s}\bar{s}s\big{>}$
(23) $\displaystyle
c_{3}^{K_{1}}=-8\pi^{3}\Bigg{[}2\big{<}\alpha_{s}(\bar{u}\gamma_{\mu}\lambda^{a}s)(\bar{s}\gamma^{\mu}\lambda^{a}u)\big{>}+\frac{2}{9}\Big{<}(\bar{u}\gamma_{\mu}\lambda^{a}u+\bar{s}\gamma_{\mu}\lambda^{a}s)\times$
$\displaystyle\Big{(}\sum_{q=u,d,s}\bar{q}\gamma^{\mu}\lambda^{a}q\Big{)}\Big{>}\Bigg{]}=-8\pi^{3}\alpha_{s}\kappa_{n}\bigg{[}-\frac{32}{9}\big{<}\bar{u}u\big{>}\big{<}\bar{s}s\big{>}+\frac{32}{81}\big{(}\big{<}\bar{u}u\big{>}^{2}+\big{<}\bar{s}s\big{>}^{2}\big{)}\bigg{]}$
(24)

Thus the difference in correlator function for the vector and axial vector is
proportional to chiral symmetry breaking dimension-4
(¡$m_{q_{i}}\bar{q_{i}}q_{i}$¿) and dimension-6
($\big{<}\bar{q_{i}}q_{i}\big{>}\big{<}\bar{q_{j}}q_{j}\big{>}$) operators.
After doing the Borel transformation of equation (17) for improved
convergence, we get,

$12\pi^{2}\,{\widetilde{\Pi}^{V}(M^{2}})=d_{V}\bigg{[}c_{0}^{V}M^{2}+c_{1}^{V}+\frac{c_{2}^{V}}{M^{2}}+\frac{c_{3}^{V}}{2M^{4}}\bigg{]}$
(25)

We assume that the spectral density has a resonance part $R^{V({\rm
res})}_{\rm phen}(s)$ and a perturbative continuum that contains all higher
energy states, which are separated by an energy scale $\sqrt{s_{0}^{V}}$ as
Klingl_NPA624_527_197 ; Kwon_PRC81_065203_2010 ,

$R^{V}_{\rm phen}(s)=R^{{V}({\rm res})}_{\rm
phen}(s)\,\Theta(s_{0}^{V}-s)+d_{V}c_{0}^{V}\,\Theta(s-s_{0}^{V})$ (26)

As the rapid crossover between the resonance and continuum part is not
realistic, the scale $s_{0}^{V}$ is taken as the average scale characterizing
the smooth transition from low-lying resonance region to high-energy continuum
part. Due to larger exponential suppression of the correlator function through
utilizing the Borel transform, the more detailed description of the crossover
and continuum region becomes insignificant Leupold_PRC64_015202_2001 . The
correlator function is matched from equations (16) and (25) to get,

$\int ds\,e^{-\frac{s}{M^{2}}}R^{V}_{\rm
phen}(s)=d_{V}\bigg{[}c_{0}^{V}M^{2}+c_{1}^{V}+\frac{c_{2}^{V}}{M^{2}}+\frac{c_{3}^{V}}{2M^{4}}\bigg{]}$
(27)

We expand the exponential expression, for $M>\sqrt{s_{0}^{V}},$ of equation
(27) in powers of $\frac{s}{M^{2}}$ for $s<s_{0}^{V}$ and get the finite
energy sum rules (FESRs) AM_PRC91_035201_2015 ; Klingl_NPA624_527_197  in
vacuum, by comparing the coefficients of various powers of $1/M^{2}$. Using
simple ansatz for the vector $K^{*}$ meson spectral function $R^{K^{*}}_{\rm
phen}(s)$ as AM_PRC91_035201_2015 ; Klingl_NPA624_527_197 ;
Kwon_PRC81_065203_2010 ,

$R^{K^{*}}_{\rm
phen}(s)=F_{K^{*}}\,\delta(s-m_{K^{*}}^{2})+d_{K^{*}}c_{0}^{K^{*}}\,\Theta(s-s_{0}^{K^{*}})$
(28)

the finite energy sum rules (FESRs) for the vector $K^{*}$ meson in vacuum are
given as,

$F_{K^{*}}=d_{K^{*}}(c_{0}^{K^{*}}s_{0}^{{K^{*}}}+c_{1}^{K^{*}})$ (29)
$F_{K^{*}}\,m_{K^{*}}^{2}=d_{K^{*}}\Big{(}\frac{(s_{0}^{{K^{*}}})^{2}c_{0}^{K^{*}}}{2}-c_{2}^{{K^{*}}}\Big{)}$
(30)
$F_{K^{*}}\,m_{K^{*}}^{4}=d_{K^{*}}\Big{(}\frac{(s_{0}^{K^{*}})^{3}c_{0}^{K^{*}}}{3}+c_{3}^{K^{*}}\Big{)}$
(31)

These equations are solved by putting in the vacuum masses and vacuum values
of condensates to find the delineation scale $s_{0}^{K^{*}}$, the overlap
strength $F_{K^{*}}$ between the current and lowest-lying resonance, and the
coefficient $c_{3}^{K^{*}}$ for vector meson $K^{*+}$ and $K^{*0}$ separately.
The parameter $\kappa_{n}$ can be fixed from the value of coefficient
$c_{3}^{K^{*}}$.

However, in the presence of the nuclear medium, the quark and gluon
condensates are also modified due to medium modifications of quark and gluon
condensates, and these medium effects are incorporated in the FESRs through
the medium-modified coefficients $c_{2}^{*}$ and $c_{3}^{*}$. Then the finite
energy sum rules for the $K^{*}$ meson in the nuclear medium are given by,

$F_{K^{*}}^{*}=d_{K^{*}}(c_{0}^{K^{*}}s_{0}^{*{K^{*}}}+c_{1}^{K^{*}})$ (32)
$F_{K^{*}}^{*}\,m_{K^{*}}^{*2}=d_{K^{*}}\Big{(}\frac{(s_{0}^{*{K^{*}}})^{2}c_{0}^{K^{*}}}{2}-c_{2}^{*{K^{*}}}\Big{)}$
(33)
$F_{K^{*}}^{*}\,m_{K^{*}}^{*4}=d_{K^{*}}\Big{(}\frac{(s_{0}^{*{K^{*}}})^{3}c_{0}^{K^{*}}}{3}+c_{3}^{*{K^{*}}}\Big{)}$
(34)

These FESRs are solved to find the in-medium scale $s_{0}^{*K^{*}}$, overlap
strength $F^{*}_{K^{*}}$, and mass $m_{K^{*}}^{*}$ of the vector meson.

However, for the strange axial vector meson channel, the spectral density
$R^{K_{1}}_{\rm phen}(s)$ will have a contribution from the pseudoscalar $K$
meson as well as from the axial vector $K_{1}$ meson resonance and we
parameterize the spectral density for strange axial vector $K_{1}$ meson as,

$R^{K_{1}}_{\rm
phen}(s)=f_{K}^{2}\,\delta({s-m_{K}^{2}})+F_{K_{1}}\,\delta(s-m_{K_{1}}^{2})+d_{K_{1}}c_{0}^{K_{1}}\,\Theta(s-s_{0}^{K_{1}})$
(35)

with $f_{K}$ and $m_{K}$ being the kaon decay constant and kaon mass
respectively. A similar parameterization scheme have been used for non-strange
axial vector $A_{1}$ meson which gets additional contribution to the spectral
density due to pseudoscalar pion ($\pi$) Leupold_PRC64_015202_2001 ;
Shifman_NPB147_448_1979 ; Parui_arXiv_2209_02455 . The FESRs for the strange
axial vector meson in the nuclear medium are given as,

$F_{K_{1}}^{*}=d_{K_{1}}(c_{0}^{K_{1}}s_{0}^{*{K_{1}}}+c_{1}^{K_{1}})-f_{K}^{2}$
(36)
$F_{K_{1}}^{*}\,m_{K_{1}}^{*2}=d_{K_{1}}\Big{(}\frac{(s_{0}^{*{K_{1}}})^{2}c_{0}^{K_{1}}}{2}-c_{2}^{*{K_{1}}}\Big{)}-f_{K}^{2}m_{K}^{2}$
(37)
$F_{K_{1}}^{*}\,m_{K_{1}}^{*4}=d_{K_{1}}\Big{(}\frac{(s_{0}^{*{K_{1}}})^{3}c_{0}^{K_{1}}}{3}+c_{3}^{*{K_{1}}}\Big{)}-f_{K}^{2}m_{K}^{4}$
(38)

The above FESRs are solved simultaneously to find the in-medium masses for the
axial vector $K_{1}$ mesons.

###  III.1 Effects of strong magnetic fields on $K^{*}$ meson masses

(1) Whenever a charged particle moves in an external magnetic field, the
Lorentz force comes into play and the particle’s momenta perpendicular to the
direction of magnetic field, are discretized to certain levels characterized
by an integral label $(n)$ called Landau levels; while the particle’s momenta
in the direction of magnetic field remains unaffected. Therefore the energy
levels of charged pseudoscalar (spin-0) and vector (spin-1) mesons, in the
presence of magnetic field, are discretized to different Landau levels
Gubler_PRD93_054026  and are given by,

$m_{P}(eB)=\sqrt{m_{P}^{2}+(2n+1)|eB|+p_{z}^{2}}$ (39)
$m_{V}(eB)=\sqrt{m_{V}^{2}+(2n+1)|eB|+p_{z}^{2}+gS_{z}|eB|}$ (40)

where $m_{P}$ and $m_{V}$ are the masses in the absence of magnetic field. The
integer ‘$n$’ specifies the Landau levels, $p_{z}$ is the continuous momentum
in the z-direction, g is the Lande g-factor, and $S_{z}$ is the spin quantum
number along the direction of magnetic field. The internal structure of the
mesons is not considered while writing the above expressions
Chernodub_LNP871_143_2013 . In the present discussion, we will consider only
the lowest Landau level (LLL), $n=0$, contribution at zero momentum in the
z-direction ($p_{z}=0$). The contribution from the higher Landau levels
($n\geq 1$) is more important in the weak magnetic field case as these will be
very close to the lowest Landau level and hence can not be treated as the
continuum part. The three polarization states ($S_{z}=+1,0,-1$) of charged
vector $K^{*+}$ meson have different Landau contributions to the masses, and
pseudoscalar $K$ meson mass also gets modified, which are given by,

$m_{P}(eB)=\sqrt{m_{P}^{2}+eB}$ (41)
$m_{V^{\perp}_{+1}}(eB)=\sqrt{m_{V}^{2}+3\,eB},\;m_{V^{||}}(eB)=\sqrt{m_{V}^{2}+eB},\;m_{V^{\perp}_{-1}}(eB)=\sqrt{m_{V}^{2}-eB}$
(42)

(2) The presence of external magnetic field can induce mixing among some spin
states due to breaking of a part of spatial rotation symmetry. Because to
this, only the mesonic spin states oriented along the direction of the
magnetic field can persist as a good quantum number and there is a mixing
between the pseudoscalar meson (($S,S_{z})=(0,0$)) and longitudinal part
(($S,S_{z})=(1,0$)) of vector meson. The PV mixing effect can be incorporated
through an effective interaction vertex, ensuring the Lorentz invariance, in
the Lagrangian as Cho_PRD91_045025_2015 ; AM_PRC102_045204_2020 ;
AM_IJMPE30_2150064_2021 ,

$\mathcal{L}_{\gamma PV}=\frac{g_{PV}}{m_{\rm
avg}}e\tilde{F}_{\mu\nu}(\partial^{\mu}P)V^{\nu}$ (43)

where $m_{\rm avg}$ is the average of masses of pseudoscalar ($m_{P}$) and
longitudinal part of vector meson ($m_{V^{||}})$. Here $P$ and $V^{\mu}$ are
the pseudoscalar and vector meson fields and $\tilde{F}_{\mu\nu}$ is the dual
field strength tensor of QCD. Here, we considered the magnetic field in
z-direction so that the only non-zero components of dual field strength tensor
are $\tilde{F}_{03}=-\tilde{F}_{30}=B$ and the mesons are considered to be at
rest. We fit the dimensionless coupling parameter $g_{PV}$ from the
experimentally observed radiative decay width $\Gamma(V\rightarrow
P\gamma)$Gubler_PRD93_054026 ; AM_PRC102_045204_2020 ; AM_IJMPE30_2150064_2021
. From the equations of motion, obtained from the effective Lagrangian
containing the kinetic and interaction term, we find that only the
longitudinal part $V^{||}$ is mixed with pseudoscalar P and there is no mixing
for the transverse parts $V^{\perp}$ of vector meson. The energy eigenvalues
for the physical state are given by Gubler_PRD93_054026 ;
Cho_PRD91_045025_2015 ; AM_PRC102_045204_2020 ; AM_IJMPE30_2150064_2021 ,

$m_{V^{||},P}^{PV}=\sqrt{\frac{1}{2}\Bigg{(}M_{+}^{2}+\frac{c_{PV}^{2}}{m_{\rm
avg}^{2}}\pm\sqrt{M_{-}^{4}+\frac{2c_{PV}^{2}M_{+}^{2}}{m_{\rm
avg}^{2}}+\frac{c_{PV}^{4}}{m_{\rm avg}^{4}}}\Bigg{)}}$ (44)

where $M_{+}^{2}=m_{V}^{2}+m_{P}^{2}$, $M_{-}^{2}=m_{V}^{2}-m_{P}^{2}$, and
$c_{PV}=g_{PV}\,eB$. The effects of PV mixing, on charged meson masses, can be
studied separately with and without Landau contributions on the masses, while
the neutral mesons do not have contribution due to Landau quantization.

(3) Furthermore, the PV mixing effects can also be studied through the spin-
magnetic field interaction ($-\mu_{i}.B$) term. The external magnetic field
with quantum numbers ($J^{P}=1^{+}$) can induce mixing between a pseudoscalar
($J^{P}=0^{-}$) and vector meson ($J^{P}=1^{-}$). The spin singlet state
$\big{|}00\big{>}$ is mixed with longitudinal component $\big{|}10\big{>}$ of
spin triplet state,

$\big{(}-\mu_{i}.B\big{)}\big{|}00\big{>}=-\frac{gB}{4}\Big{(}\frac{q_{1}}{m_{1}}-\frac{q_{2}}{m_{2}}\Big{)}\big{|}10\big{>}$
(45)

where the magnetic moment of the $i$th particle (with $i=1,2$ corresponding to
quark, antiquark in this work) is $\mu_{i}=gq_{i}\sigma_{i}/4m_{i}$, where
$\sigma_{i}$ are the Pauli spin matrices. The Lande factor $g$ is taken to be
2 and ($q_{i},m_{i}$) are the charges and constituent masses of
quark/antiquark respectively. Since the $\big{|}10\big{>}$ and
$\big{|}00\big{>}$ states are not pure eigenstates of interaction Hamiltonian
term ($\mathcal{H}_{s}=-\mu_{i}.B$), we will consider a two dimensional
eigensystem, for the $\big{|}10\big{>}$ and $\big{|}00\big{>}$ states, to
determine the spin mixing effect Alford_PRD88_105017_2013 . Then effective
physical mass eigenvalues are given by,

$m_{V^{||},P}^{\rm eff}=m_{V^{||},P}\pm\frac{\Delta
E}{2}\big{(}\sqrt{1+\chi_{sB}^{2}}-1\big{)}$ (46)

where $\Delta E=m_{V^{||}}-m_{P}$, and $\chi_{sB}=\big{(}\frac{2g|eB|}{\Delta
E}\big{)}\Big{(}\frac{q_{1}}{m_{1}}-\frac{q_{2}}{m_{2}}$). For the charged
meson, the effects of Landau levels are also considered here in
$m_{V^{||},P}$. The transverse polarized states, $\big{|}1+1\big{>}$ and
$\big{|}1-1\big{>}$, do not get mixed with any other states
Yoshida_PRD94_074043_2016 ; Machado_PRD88_034009_2013 ;
Iwasaki_EPJA57_222_2021 . The energy eigenvalue equation for transverse
polarized states is given by,

$\big{(}-\mu_{i}.B\big{)}\big{|}1\pm
1\big{>}=\mp\frac{gB}{4}\Big{(}\frac{q_{1}}{m_{1}}+\frac{q_{2}}{m_{2}}\Big{)}\big{|}1\pm
1\big{>}$ (47)

This contribution vanishes for meson states, like charmonium and bottomonium,
which have same quark-antiquark as constituents due to $q_{1}=-q_{2}$ and
$m_{1}=m_{2}$. The quark masses taken here are the constituent quark masses.

##  IV Decay Width of $K^{*}\rightarrow K\pi$ and $K_{1}\rightarrow K^{*}\pi$
in a Phenomenological Model

(1) We now discuss the medium modifications of masses and decay widths of
$K^{*}$ meson from the self energy at one-loop level. The interaction term for
a vector meson V decaying to two pseudoscalar mesons $P_{1}$ and $P_{2}$ can
be written as Klingl_ZPA356_193_1996 ,

$\mathcal{L}_{V\bar{P_{1}}P_{2}}=igV^{\mu}[\bar{P_{1}}(\partial_{\mu}P_{2})-(\partial_{\mu}\bar{P_{1}})P_{2}]$
(48)

The parameter ‘$g$’ is known as the coupling strength of the decay channel.
This interaction term of the effective Lagrangian generates a hadronic
current, which couples with the $K^{*}$ meson field to produce the self
energy, which is given by,

$\Pi^{\mu\nu}(p)=-ig_{K^{*}}^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{K}^{2}+i\epsilon}\Big{[}\frac{q^{\mu}q^{\nu}}{(q-p)^{2}-m_{\pi}^{2}+i\epsilon}\Big{]}$
(49)

where $p$ is the $4$ -momenta carried by vector $K^{*}$ meson and the $4$
-momentas of intermediate $K$ and $\pi$ are $q$ and ($q-p$) respectively, and
we have integrated over the internal loop momenta.

Figure 1:  $K^{*}$ meson self energy diagram at one loop level

Writing $\Pi_{K^{*}}(p^{2})=\frac{1}{3}g_{\mu\nu}\Pi^{\mu\nu}(p)$ and
$\mathring{m}_{K^{*}}$ as the bare $K^{*}$ meson mass, the physical mass can
be written as Cobos-Martinez_PLB771_113_2017 ,

$m_{K^{*}}^{2}=\mathring{m}_{K^{*}}^{2}+Re\,\Pi_{K^{*}}(p^{2}=m_{K^{*}}^{2})$
(50)

The decay width, at resonance, is related to the imaginary part of the
propagator as,

$\Gamma(K^{*}\rightarrow
K\pi)=-Im\,\Pi_{K^{*}}(p^{2}=m_{K^{*}}^{2})/m_{K^{*}}$ (51)

The imaginary part of the propagator is calculated by the standard Cutkosky
rule Itzykson_QFT_1980  and is given by,

$Im\,\Pi_{K^{*}}(p^{2})=-\frac{g_{K^{*}}^{2}m_{K^{*}}^{2}}{64\pi}\Big{[}1-\frac{(m_{K}-m_{\pi})^{2}}{m_{K^{*}}^{2}}\Big{]}^{3/2}\Big{[}1-\frac{(m_{K}+m_{\pi})^{2}}{m_{K^{*}}^{2}}\Big{]}^{3/2}$
(52)

The scalar part of the in-medium self-energy, in the rest frame of $K^{*}$,
can be written as,

$i\Pi_{K^{*}}(p)=-\frac{1}{3}g_{K^{*}}^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\vec{q^{2}}D_{K}(q)D_{\pi}(q-p)$
(53)

where $D_{K}(q)=(q^{2}-m^{2}_{K}+i\epsilon)^{-1}$ and
$D_{\pi}(q)=\big{(}(q-p)^{2}-m^{2}_{\pi}+i\epsilon\big{)}^{-1}$ are the kaon
and pion propagators, respectively. The real part of the self energy is
written as,

$Re\,\Pi_{K^{*}}(p)=-\frac{1}{3}g_{K^{*}}^{2}\mathcal{P}\int\frac{d^{3}q}{(2\pi)^{3}}\vec{q^{2}}\frac{(E_{K}+E_{\pi})}{(2E_{K}E_{\pi})\big{(}(E_{K}+E_{\pi})^{2}-m_{K^{*}}^{2}\big{)}}$
(54)

Here $\mathcal{P}$ denotes the principal value of the integral and the
energies are given by $E_{K}=\sqrt{\vec{q^{2}}+m_{K}^{2}}$ and
$E_{\pi}=\sqrt{\vec{q^{2}}+m_{\pi}^{2}}$. The integral (54) is divergent and
it needs to be regularized to avoid singularities. We use a phenomenological
form factor approach with a cutoff parameter $\Lambda_{c}$
Leinweber_PRD64_094502_2001 ; Krein_PLB697_136_2011 , and the integral then
becomes,

$\displaystyle
Re\,\Pi_{K^{*}}(p)=-\frac{1}{3}g_{K^{*}}^{2}\mathcal{P}\int_{0}^{\Lambda_{c}}\frac{d^{3}q}{(2\pi)^{3}}\vec{q^{2}}$
$\displaystyle\Big{(}\frac{\Lambda_{c}^{2}+m_{K^{*}}^{2}}{\Lambda_{c}^{2}+4E_{K}^{2}}\Big{)}^{2}\Big{(}\frac{\Lambda_{c}^{2}+m_{K^{*}}^{2}}{\Lambda_{c}^{2}+4E_{\pi}^{2}}\Big{)}^{2}$
$\displaystyle\frac{(E_{K}+E_{\pi})}{(2E_{K}E_{\pi})\big{(}(E_{K}+E_{\pi})^{2}-m_{K^{*}}^{2}\big{)}}$
(55)

where the quantities, called vertex form factors, are given by,

$u_{K,\pi}(q^{2})=\Bigg{(}\frac{\Lambda_{c}^{2}+m_{K^{*}}^{2}}{\Lambda_{c}^{2}+4E_{K,\pi}^{2}}\Bigg{)}^{2}$
(56)

The coupling parameter $g_{K^{*}}$ is determined separately, from the
empirical decay width of the $K^{*}$ meson in vacuum, for each particular
value of cut-off parameter $\Lambda_{c}$. Then we fix the bare mass of the
$K^{*}$ meson by the relation (50), from the vacuum mass of $K^{*}$ meson. In
a simple picture, the cutoff parameter is related to the overlap region of the
parent and daughter particles at the vertex and depends on the size of their
wave functions as done in Krein_PLB697_136_2011 ; Lee_PRC67_038202_2003 . So
we also calculate the vertex form factors using the ${}^{3}P_{0}$ model for
quark-antiquark pair creation with Gaussian wave functions for the mesons.
Then the root mean square radii from the two form factors is compared to get
an estimate of the cutoff mass $\Lambda_{c}$ as done in Ref.
Krein_PLB697_136_2011  for the $J/\psi$ meson. A similar work has been done
for the $\phi$ meson in Cobos-Martinez_PLB771_113_2017 . To include the
uncertainty in the estimated value, we take the range of $\Lambda_{c}$ from
$1$ to $4$ GeV.

(2) Further, we also use a phenomenological approach for the study of
$K_{1}\rightarrow K^{*}\pi$ decay. The interaction Lagrangian is constructed
from the anti-symmetric tensor fields for the axial-vector and vector meson.
The $SU(3)$ matrices associated with the tensor fields for the two axial
vector nonets $a_{1}$ and $b_{1}$, and vector meson nonet are, respectively
given as

$A_{\mu\nu}$ =
${\begin{pmatrix}\frac{1}{\sqrt{2}}a_{1}^{0}+\frac{1}{\sqrt{2}}f_{1}(1285)&a_{1}^{+}&K_{1A}^{+}\\\
a_{1}^{-}&-\frac{1}{\sqrt{2}}a_{1}^{0}+\frac{1}{\sqrt{2}}f_{1}(1285)&K_{1A}^{0}\\\
K_{1A}^{-}&\bar{K_{1A}^{0}}&f_{1}(1420)\end{pmatrix}}_{\mu\nu}$

$B_{\mu\nu}$ =
${\begin{pmatrix}\frac{1}{\sqrt{2}}b_{1}^{0}+\frac{1}{\sqrt{2}}h_{1}(1170)&b_{1}^{+}&K_{1B}^{+}\\\
b_{1}^{-}&-\frac{1}{\sqrt{2}}b_{1}^{0}+\frac{1}{\sqrt{2}}h_{1}(1170)&K_{1B}^{0}\\\
K_{1B}^{-}&\bar{K_{1B}^{0}}&h_{1}(1380)\end{pmatrix}}_{\mu\nu}$

$V_{\mu\nu}$ =
${\begin{pmatrix}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega^{0}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega^{0}&K^{*0}\\\
K^{*-}&\bar{K^{*0}}&\phi\end{pmatrix}}_{\mu\nu}$

The mesons corresponding to two nonets, $a_{1}$ and $b_{1}$ above, corresponds
to $J^{PC}=1^{++}$ and $1^{+-}$ states. The interaction vertex, ensuring the
$SU(3)$ invariance, charge conjugation (C), and parity (P) conservation, for
the decay of an axial vector meson A (and B) to a vector meson V and a
pseudoscalar meson P is written as Roca_PRD70_094006_2004 ,

$\mathcal{L}_{AVP}=i\tilde{F}\big{<}A_{\mu\nu}[V^{\mu\nu},P]\big{>}$ (57)
$\mathcal{L}_{BVP}=\tilde{D}\big{<}B_{\mu\nu}\\{V^{\mu\nu},P\\}\big{>}$ (58)

where $P$ is the usual $SU(3)$ matrix for the pseudoscalar meson nonet
considering the standard $\eta-\eta^{{}^{\prime}}$ mixing. The free
parameters, $\tilde{F}$ and $\tilde{D}$, are fitted globally from the
available data of various decays and branching ratios of the members of
$a_{1}$ and $b_{1}$ nonets in Ref. Roca_PRD70_094006_2004  and the mixing
angle is taken to be $\theta_{K_{1}}$ $\sim$ $62^{\circ}$. The symbol
$\big{<}...\big{>}$ represents the $SU(3)$ trace and factor $`i$’ ensures that
the Lagrangian is hermitian. The decay width for the $K_{1}\rightarrow
K^{*}\pi$ decay calculated within the phenomenological approach is given as,

$\Gamma_{K_{1}\rightarrow K^{*}\pi}=\frac{q}{8\pi
M_{K_{1}}^{2}}{\mathcal{|M|}^{2}}$ (59)

where the momentum $q$ of the final state particles, in the rest frame of the
parent particle $K_{1}$, is given by,

$q(M_{K_{1}},M_{K^{*}},M_{\pi})=\frac{1}{2M_{K_{1}}}\sqrt{\big{(}M_{K_{1}}^{2}-(M_{K^{*}}+M_{\pi})^{2}\big{)}\big{(}M_{K_{1}}^{2}-(M_{K^{*}}-M_{\pi})^{2}\big{)}}$
(60)

and the matrix element is given by,

$\mathcal{M}=\frac{-2\lambda_{AVP}}{M_{K_{1}}M_{K^{*}}}(p^{\prime}.p\,\epsilon^{\prime}.\epsilon-\epsilon^{\prime}.p\,\epsilon.p^{\prime})$
(61)

with $p^{\prime},\epsilon^{\prime}$ and $p,\epsilon$ being the momentum and
polarization vector of the axial-vector $K_{1}$ and vector $K^{*}$ meson,
respectively. The decay width then becomes,

$\Gamma_{K_{1}\rightarrow K^{*}\pi}=\frac{|\lambda_{AVP}|^{2}}{2\pi
M_{K_{1}}^{2}}q\Big{(}1+\frac{2}{3}\frac{q^{2}}{M_{K^{*}}^{2}}\Big{)}$ (62)

The coefficients $\lambda_{AVP}$ of the interaction vertex are given as
$\pm\frac{1}{\sqrt{2}}({\rm cos}\,\theta\,\tilde{D}+{\rm
sin}\,\theta\,\tilde{F})$, $\pm({\rm cos}\,\theta\,\tilde{D}+{\rm
sin}\,\theta\,\tilde{F})$, $\pm({\rm cos}\,\theta\,\tilde{D}+{\rm
sin}\,\theta\,\tilde{F})$, and $+\frac{1}{\sqrt{2}}({\rm
cos}\,\theta\,\tilde{D}+{\rm sin}\,\theta\,\tilde{F})$ for the
$K_{1}^{\pm}\rightarrow K^{*\pm}\pi^{0}$, $K_{1}^{\pm}\rightarrow
K^{*0}\pi^{\pm}$, $K_{1}^{0}\rightarrow K^{*\pm}\pi^{\mp}$, and
$K_{1}^{0}\rightarrow K^{*0}\pi^{0}$ decays respectively, with
$\theta\equiv\theta_{K_{1}}$.

##  V Decay Widths of $K^{*}\rightarrow K\pi$ and $K_{1}\rightarrow K^{*}\pi$
within the ${}^{3}P_{0}$ model

(1) First, we study the decay width of vector $K^{*}$ meson to two
pseudoscalar mesons which are pion and kaon, using the ${}^{3}P_{0}$ model
Barnes_PRD55_4157_1997 ; Friman_PLB548_153_2002 . This model was first
introduced by L. Micu to calculate the decay rates of various meson resonances
Micu_NPB10_521_1969  and then extended further by A. Le Yaouanc and others to
explain strong decay amplitudes of mesons and baryons Yaouanc_PRD8_2223_1973 .
In this model, a light quark-antiquark pair is assumed to be produced in the
${}^{3}P_{0}$ state having vacuum quantum numbers $J^{PC}=0^{++}$. The quark
(antiquark) of this produced pair combines with the antiquark (quark) of the
parent meson, which is assumed to be at rest initially, to give the final
state mesons. The matrix element for the general decay $A\rightarrow BC$ in
the ${}^{3}P_{0}$ model is given as,

$\mathcal{M}_{A\rightarrow
BC}=\big{<}A\big{|}\gamma\,[\bar{q_{s}}q_{s}]^{{}^{3}P_{0}}\big{|}BC\big{>}$
(63)

where $\gamma$ is the coupling strength which is related to the probability of
production of a quark antiquark pair in the ${}^{3}P_{0}$ state. The wave
functions for the produced $\bar{q}q$ pair are chosen to be simple harmonic
oscillator (SHO) wave functions to calculate the strong decay amplitude. In
the present case of a vector $K^{*}$ meson decaying to two pseudoscalar
mesons, the matrix element for the $1({}^{3}S_{1})\rightarrow
1({}^{1}S_{0})+1({}^{1}S_{0})$ channel with proper flavor factor $I_{f}$ is
given by,

$\mathcal{M}_{K^{*}\rightarrow K\pi}=\frac{\gamma_{K^{*}}}{\pi^{1/4}\beta_{\rm
avg}^{1/2}}\bigg{[}\frac{-2^{4}}{3^{1/2}}\frac{r^{3/2}(1+r^{2})x}{(1+2r^{2})^{5/2}}\bigg{]}{\rm
exp}\Big{(}\frac{-x^{2}}{4(1+2r^{2})}\Big{)}I_{f}$ (64)

where the ratio $r=\frac{\beta_{K^{*}}}{\beta_{\rm avg}}$, with
$\beta_{K^{*}}$ being the harmonic oscillator potential strength for the
$K^{*}$ meson and $\beta_{\rm avg}$ as the average of harmonic oscillator
potential strength of two daughter mesons. Taking $r\neq 1$ i.e.
$\beta_{K^{*}}\neq\beta_{\rm avg}$ allows to account for the different sizes
of daughter and parent mesons. The factor $I_{f}$ gives the flavor weight
factor contributions from the two Feynman decay diagrams, called $d_{1}$ and
$d_{2}$ diagrams, of the parent meson Barnes_PRD55_4157_1997  and is taken to
be $1/(2\sqrt{2})$ for the above decay. The quantity ‘$x$’ is the scaled
momentum, given by $p_{K}/\beta_{\rm avg}$, carried by the daughter meson. The
decay width is then given by,

$\Gamma({K^{*}}\rightarrow
K\pi)=2\pi\frac{p_{K}E_{K}E_{\pi}}{M_{K^{*}}}\sum_{LS}\Big{|}\mathcal{M}_{LS}\Big{|}^{2}$
(65)

where $p_{K}$ is the 3-momentum, given by,

$p_{K}=\sqrt{\frac{M_{K^{*}}^{2}}{4}-\frac{M_{K}^{2}+M_{\pi}^{2}}{2}+\Big{(}\frac{M_{K}^{2}-M_{\pi}^{2}}{2M_{K^{*}}}\Big{)}^{2}}$
(66)

and the energies are given by $E_{K}=\sqrt{p_{K}^{2}+m_{K}^{2}}$ and
$E_{\pi}=\sqrt{p_{\pi}^{2}+m_{\pi}^{2}}$. As the decaying meson is assumed to
be at rest, momentum conservation gives $p_{K}=p_{\pi}$. The masses
$M_{K^{*}}$ and $M_{K}$ are the in-medium masses, from which we calculate the
in-medium decay width. In this work, we do not take into account the medium
modification of the pion mass.

(2) The axial vector $K_{1}(1270)$ meson is not a pure $1{}^{3}P_{1}$
($J^{PC}=1^{++}$) or $1{}^{1}P_{1}$ ($J^{PC}=1^{+-}$) state, but the
physically observed $K_{1}(1270)$ and $K_{1}(1400)$ are a mixture of the two
non mass eigenstates, $\big{|}K_{1A}\big{>}$ and $\big{|}K_{1B}\big{>}$, of
the two strange axial vector nonets $1{}^{3}P_{1}$ and $1{}^{1}P_{1}$,
respectively Tayduganov_PRD85_074011_2012 ; Suzuki_PRD47_1252_1993 ,

$\big{|}K_{1}(1270)\big{>}={\rm sin}\,\theta_{K_{1}}\big{|}K_{1A}\big{>}+{\rm
cos}\,\theta_{K_{1}}\big{|}K_{1B}\big{>}$ (67) $\big{|}K_{1}(1400)\big{>}={\rm
cos}\,\theta_{K_{1}}\big{|}K_{1A}\big{>}-{\rm
sin}\,\theta_{K_{1}}\big{|}K_{1B}\big{>}$ (68)

where $\theta_{K_{1}}$ is the mixing angle. The matrix element for the axial
vector meson decay channel ($K_{1}\rightarrow K^{*}\pi$) is given as,

$\mathcal{M}_{LS}=\frac{\gamma_{K_{1}}}{\pi^{1/4}\beta_{\rm
avg}^{1/2}}\mathcal{P}_{LS}(x,r)\,{\rm
exp}\Big{(}\frac{-x^{2}}{4(1+2r^{2})}\Big{)}I_{f}$ (69)

where the polynomials for various decays are given as Barnes_PRD55_4157_1997 ;
Friman_PLB548_153_2002 ,

$\mathcal{P}_{01}^{\big{(}1{}^{3}P_{1}\rightarrow
1{}^{3}S_{1}+1{}^{1}S_{0}\big{)}}=\Bigg{(}2^{5}\Big{(}\frac{r}{1+2r^{2}}\Big{)}^{5/2}\Big{(}1-\frac{1+r^{2}}{3(1+2r^{2})}x^{2}\Big{)}\Bigg{)}$
(70)

$\mathcal{P}_{21}^{\big{(}1{}^{3}P_{1}\rightarrow
1{}^{3}S_{1}+1{}^{1}S_{0}\big{)}}=-\sqrt{\frac{5}{6}}\Bigg{(}\frac{1}{\sqrt{15}}\frac{r^{5/2}2^{5}(1+r^{2})}{(1+2r^{2})^{7/2}}x^{2}\Bigg{)}$
(71)

$\mathcal{P}_{01}^{\big{(}1{}^{1}P_{1}\rightarrow
1{}^{3}S_{1}+1{}^{1}S_{0}\big{)}}=-\frac{1}{\sqrt{2}}\Bigg{(}2^{5}\Big{(}\frac{r}{1+2r^{2}}\Big{)}^{5/2}\Big{(}1-\frac{1+r^{2}}{3(1+2r^{2})}x^{2}\Big{)}\Bigg{)}$
(72)

$\mathcal{P}_{21}^{\big{(}1{}^{1}P_{1}\rightarrow
1{}^{3}S_{1}+1{}^{1}S_{0}\big{)}}=-\sqrt{\frac{5}{3}}\Bigg{(}\frac{1}{\sqrt{15}}\frac{r^{5/2}2^{5}(1+r^{2})}{(1+2r^{2})^{7/2}}x^{2}\Bigg{)}$
(73)

with $r=\frac{\beta_{K_{1}}}{\beta_{\rm avg}}$, and $\beta_{\rm avg}$ being
the average of harmonic oscillator potential strength of the daughter $K^{*}$
and $\pi$ mesons. The flavor factor $I_{f}$ is scaled together with the
coupling constant $\gamma_{K_{1}}$ in this work. The decay width for
$K_{1}\rightarrow K^{*}\pi$ decay is then evaluated from equation (65) after
changing the corresponding variables for this decay.

##  VI Results and Discussion

Now we discuss the medium modifications of the $K^{*}$ and $K_{1}$ meson
masses and decay widths in nuclear matter including the effects of isospin
asymmetry, and the effects of the magnetic field on the vector $K^{*}$ meson
will also be discussed. Isospin asymmetry corresponds to the fact that the
number of neutrons in heavy ion collision experiments is more than the number
of protons inside the colliding heavy ions. Isospin asymmetry factor ($\eta$)
gives the amount of isospin asymmetry in the medium. For example, $\eta=0.5$
corresponds to a medium consisting of neutrons only. We investigate the in-
medium masses and decay widths at different values ($\rho_{0}$, $2\rho_{0}$,
and $4\rho_{0}$) of medium density where nuclear matter saturation density,
$\rho_{0}$ = $0.15\,{\rm fm}^{-3}$.

First, the medium modifications of scalar fields ($\sigma$, $\zeta$, $\delta$,
and $\chi$) are calculated using a chiral effective model by solving the
coupled equations of motion for these fields. Then these modifications are
included in scalar gluon condensates,
$\big{<}\frac{\alpha_{s}}{\pi}G^{a}_{\mu\nu}G^{a\mu\nu}\big{>}$ and light
quark condensates
($\big{<}\bar{u}u\big{>},\big{<}\bar{d}d\big{>},\big{<}\bar{s}s\big{>}$).
Finally, we calculate the in-medium masses using QCD sum rule (QCDSR) approach
from the medium modified scalar quark and gluon condensates, and then in-
medium decay widths are calculated from the in-medium masses, using the
${}^{3}P_{0}$ model. The in-medium masses and decay widths for the $K^{*}$
meson are also calculated from the in-medium self energy of the $K^{*}$ meson
at the one-loop level as discussed in section IV.

The values of constant parameters in sections II and III, taken in this work,
are given as $\alpha_{s}(Q^{2}=1$ GeV${}^{2})=0.3551$, $d=0.064018$,
$m_{\pi}$=139 MeV, pion decay constant $f_{\pi}=93.3$ MeV, $m_{K}$= 498 MeV,
kaon decay constant $f_{K}=122.143$ MeV. The vacuum values of the scalar
fields are $\sigma_{0}=-93.3$ MeV, $\zeta_{0}=-106.6$ MeV, and $\chi_{0}$=
409.77 MeV. The vacuum mass values of $K^{*+}$, $K^{*0}$, and $K_{1}$ mesons
are taken to be 891.67, 895.55, and 1253 MeV respectively
AM_IJMPE30_2150014_2021 ; Zyla_PDG_2020 . The current quark masses, used in
QCD sum rule approach, for the light quarks are taken as $m_{u}=4$ MeV,
$m_{d}=7$ MeV, and $m_{s}=150$ MeV. The isospin asymmetry parameter, in the
medium consisting of nucleons only, is given by
$\eta=\frac{\rho_{n}-\rho_{p}}{2\rho_{B}}$, where ($\rho_{n},\rho_{p}$) are
the vector number density of (neutrons, protons) respectively with $\rho_{B}$
as total baryon density. The coefficient $\kappa_{n}$ for $K^{*}$ and $K_{1}$
mesons is found from the vacuum FESRs with the values of vacuum masses and
values of the scalar quark and gluon condensates. The obtained values are
0.73586, 2.30955, 22.53633, and 17.77003 for the $K^{*+}$, $K^{*0}$,
$K_{1}^{+}$, and $K_{1}^{0}$ mesons, respectively.

Figure 2: Masses of vector $K^{*}$ meson and axial vector $K_{1}$ meson
plotted as a function of nuclear medium density ($\rho_{B}$) in terms of
nuclear matter saturation density $\rho_{0}$ for various values of isospin
asymmetry parameter $\eta$, within QCD sum rule approach at zero magnetic
field.

###  VI.1 In-medium Masses using QCD Sum Rule Approach

In figure (2), the masses of $K^{*}$ and $K_{1}$ mesons are plotted as a
function of nuclear medium density in terms of nuclear matter saturation
density (${\rho_{0}}$) for various isospin asymmetry parameters $(\eta)$,
calculated within the QCD sum rule approach. The mass values are observed to
decrease monotonically as the medium density increases. This is because of the
fact that the chiral condensates $\big{<}\bar{q}q\big{>}$ and the gluon
condensates $\big{<}\frac{\alpha_{s}}{\pi}G^{a}_{\mu\nu}G^{a\mu\nu}\big{>}$
decrease in magnitude as a function of medium density. The non-strange quark
condensates ($\big{<}\bar{u}u\big{>}$, $\big{<}\bar{d}d\big{>}$) decrease more
in nuclear matter as compared to strange quark condensate
$\big{<}\bar{s}s\big{>}$ and gluon condensates AM_PRC100_015207_2019 . There
is observed to be a smaller mass drop in isospin asymmetric nuclear matter
($\eta\neq 0$) as compared to isospin symmetric matter ($\eta=0$) for the
charged $K^{*+}$ and $K_{1}^{+}$ meson. The mass modifications for the neutral
$K^{*0}$ and $K_{1}^{0}$ mesons are observed to be smaller as compared to the
corresponding charged mesons and the mass drop is slightly higher in $\eta\neq
0$ matter, for the neutral mesons, as compared to isospin symmetric matter.
Also, there is observed to be a sharper mass drop for smaller values of
nuclear medium density up to around 2$\rho_{0}$. As nuclear medium density
increases, the drop in strange quark condensates decreases even more which
results in smaller changes in $K^{*}$ and $K_{1}$ meson mass at higher
densities. However, the isospin asymmetry effects are observed to be larger at
higher medium density. The masses for these open strange mesons are given in
table I and II as calculated within the framework of QCD sum rule approach.

Table I: Masses of Vector $K^{*}$ Meson (QCD Sum Rule Approach)
mass of $K^{*+}$ (MeV) mass of $K^{*0}$ (MeV) density $\eta=0$ $\eta=0.5$
$\eta=0$ $\eta=0.5$ $\rho_{0}$ 759.67 776.05 789.28 782.79 2$\rho_{0}$ 665.18
703.53 722.43 713.72 4$\rho_{0}$ 562.15 627.55 658.54 649.02

Table II: Masses of Axial Vector $K_{1}$ Meson (QCD Sum Rule Approach)
mass of $K_{1}^{+}$ (MeV) mass of $K_{1}^{0}$ (MeV) density $\eta=0$
$\eta=0.5$ $\eta=0$ $\eta=0.5$ $\rho_{0}$ 1138.45 1150.91 1148.44 1143.18
2$\rho_{0}$ 1059.59 1089.87 1088.04 1081.76 4$\rho_{0}$ 967.85 1024.71 1033.85
1027.84

Figure 3: Masses of vector ($K^{*}$) and pseudoscalar (K) meson plotted as a
function of $eB/m_{\pi}^{2}$ at zero baryon density. Here ‘LLL’ represents the
lowest Landau level contribution, ‘PV’ represents the PV mixing considered
through effective Lagrangian, ‘spin’ represents the PV mixing considered
through spin-magnetic field interaction, and ‘Z’ represents the anomalous
Zeeman splitting. The three polarization states of vector $K^{*}$ meson
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$K^{*\perp}_{+1},K^{*||},K^{*\perp}_{-1}$. Figure 4: Masses of vector
($K^{*}$) and pseudoscalar (K) meson plotted as a function of $eB/m_{\pi}^{2}$
at baryon density $\rho_{B}$= $\rho_{0}$ in isospin symmetric ($\eta=0$)
nuclear matter. Here ‘LLL’ represents the lowest Landau level contribution,
‘PV’ represents the PV mixing considered through effective Lagrangian, ‘spin’
represents the PV mixing considered through spin-magnetic field interaction,
and ‘Z’ represents the anomalous Zeeman splitting. The three polarization
states of vector $K^{*}$ meson
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$K^{*\perp}_{+1},K^{*||},K^{*\perp}_{-1}$. Figure 5: Masses of vector
($K^{*}$) and pseudoscalar (K) meson plotted as a function of $eB/m_{\pi}^{2}$
at baryon density $\rho_{B}$= $\rho_{0}$ in isospin asymmetric ($\eta=0.5$)
nuclear matter. Here ‘LLL’ represents the lowest Landau level contribution,
‘PV’ represents the PV mixing considered through effective Lagrangian, ‘spin’
represents the PV mixing considered through spin-magnetic field interaction,
and ‘Z’ represents the anomalous Zeeman splitting. The three polarization
states of vector $K^{*}$ meson
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$K^{*\perp}_{+1},K^{*||},K^{*\perp}_{-1}$.

###  VI.2 Effects of Magnetic Field on $K^{*}$ Meson Masses

We now discuss the effects of the magnetic field on the vector ($K^{*}$) and
pseudoscalar ($K$) meson masses. First, due to lowest Landau level (LLL)
contribution, the masses of the electrically charged mesons will be modified
as given by equations (41) and (42). The charge neutral mesons will not
undergo Landau quantization. We study the effects of the magnetic field at
zero medium density as well as at medium density $\rho_{B}=\rho_{0}$.
Furthermore, due to PV mixing effects incorporated through effective
interaction term, the masses of the longitudinal part of vector meson
$K^{*||}$ and pseudoscalar meson, both charged and neutral, are further
modified as given by equation (44). The effects of PV mixing considered
through the effective interaction term are labelled as ‘PV’ in figures (3),
(4), and (5) and we conclude that the PV mixing effects alone are small. Due
to PV mixing, there is a level repulsion between vector ($K^{*||}$) and
pseudoscalar $(K)$ meson, and the mass of vector increases, while the mass of
pseudoscalar $(K)$ decreases as a function of magnetic field. The effect of PV
mixing increases even more at higher magnetic field values.

The effects of PV mixing incorporated through spin-magnetic field interaction,
labelled as ‘spin’, are also shown in figures (3), (4), and (5) for medium
density $\rho_{B}=0$, $\rho_{0}(\eta=0)$, and $\rho_{0}(\eta=0.5)$,
respectively. The values are plotted after including the combined effects of
lowest Landau level contribution and spin-magnetic field interaction term
(shown as label ‘LLL + spin’) for charged pseudoscalar $K$ meson and
longitudinal polarization $K^{*||}$ of vector meson. The masses of transverse
polarized states are also modified, due to spin-magnetic field interaction, in
the presence of magnetic field due to unequal quark masses and/or charges for
the $K^{*}$ meson. This is known as anomalous Zeeman splitting, shown as ‘Z’,
which is due to the non-vanishing intrinsic magnetic moment of the meson. This
splitting contributes to both charged as well as neutral $K^{*}$ meson masses.
The constituent quark masses are taken as $m_{u,d}=330$ MeV and $m_{s}=480$
MeV. The mass of vector $K^{*||}$ meson increases and that of pseudoscalar
$(K)$ meson decreases when we consider only the spin mixing effects. For the
charged vector $\big{|}1+1\big{>}$ state, there is a decrease in mass due to
the spin interaction term, while the $\big{|}1-1\big{>}$ state has a positive
mass shift. However, for the neutral vector $\big{|}1+1\big{>}$
($\big{|}1-1\big{>}$) state, there is observed to be an increase (decrease) in
the masses due to the spin interaction term which is due to the polarity of
constituent quark/antiquark charges. The effects of medium density and isospin
asymmetry on the pseudoscalar $K$ meson mass are calculated within a chiral
$SU(3)$ model AM_EPJA45_169_2010 ; AM_EPJA55_107_2019 . The effects of
magnetic field are observed to be quite appreciable and should be visible in
the mass spectra of these particles and other observables in heavy ion
collision experiments. The effects of PV mixing, considered through the spin-
magnetic field interaction (-$\mu_{i}.B$) term, have also been studied on the
masses of open bottom mesons and upsilon states in magnetized nuclear matter
which, in turn, affects the decay width of $\Upsilon(4S)$ to $\bar{B}B$ pair
appreciably AM_IJMPE31_2250060_2022 .

###  VI.3 Masses and Decay Widths of $K^{*}$ meson within a phenomenological
model

The in-medium mass of $K^{*}$ meson is calculated from the medium modified
self-energy at one loop level. The effects of the medium are incorporated
through in-medium $K$ meson masses, which are calculated in chiral $SU(3)$
model AM_EPJA45_169_2010 ; AM_EPJA55_107_2019 . The coupling parameter
$g_{K^{*}}$ is fixed for various decay channels, from the vacuum decay width
given by equation (51). Then we fix the bare mass of $K^{*}$ meson using
equation (50) for a particular value of cut-off parameter $\Lambda_{c}$ by
taking vacuum mass values for mesons. After fixing the bare mass
$\mathring{m}_{K^{*}}$ and coupling parameter $g_{K^{*}}$, we can calculate
the in-medium masses for $K^{*}$ meson from equation (50) by inserting in-
medium $K$ meson masses. The mass shifts for $K^{*+}$ (from $K^{+}\pi^{0}$ and
$K^{0}\pi^{+}$ loop) and $K^{*0}$ (from $K^{0}\pi^{0}$ and $K^{+}\pi^{-}$
loop) mesons in the nuclear medium at zero magnetic fields, for various values
of cut-off parameter, are given in table III(a) and III(b) respectively. We
observe a positive mass shift of $27.8$ MeV for the charged $K^{*+}$ meson for
isospin symmetric nuclear matter at density $\rho_{B}$ = $\rho_{0}$ for cut-
off parameter $\Lambda_{c}$ = $1000$ MeV. A positive self energy indicates
that the nuclear mean field provides repulsion to $K^{*+}$ meson. This mass
shift can be compared with a mass shift of around $+40$ MeV in self-consistent
scattering amplitude calculations and a mass modification of $+50$ MeV, at
$\rho_{0}$, in the low density T$\rho$ approximation
Cabrera_JPCS503_012017_2014 .

Table III(a): Vector $K^{*+}$ Meson Mass shifts (MeV) from $K\pi$ loop for
$eB$ = 0 $\rho_{B}=\rho_{0}$ $\rho_{B}=4\rho_{0}$ $\eta=0$ $\eta=0.5$ $\eta=0$
$\eta=0.5$ $\Lambda_{c}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$
$K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$
$(K\pi)^{+}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$ 1000 18.0 9.8 27.8
11.9 11.8 23.7 5.3 6.0 11.3 3.6 8.7 12.3 2000 10.3 4.2 14.5 5.9 6.1 12.0 4.3
12.0 16.3 1.6 7.6 9.2 3000 5.5 2.6 8.1 6.0 5.0 11.0 9.2 2.2 11.4 3.9 10.9 14.8
4000 7.1 8.1 15.2 7.1 8.0 15.1 11.8 8.9 20.7 6.3 11.0 17.3

Table III(b): Vector $K^{*0}$ Meson Mass shifts (MeV) from $K\pi$ loop for
$eB$ = 0
$\rho_{B}=\rho_{0}$ $\rho_{B}=4\rho_{0}$ $\eta=0$ $\eta=0.5$ $\eta=0$
$\eta=0.5$ $\Lambda_{c}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$
$K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$
$(K\pi)^{0}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$ 1000 1.5 15.6 17.1 3.4
16.9 20.3 1.8 19.5 21.3 0.7 7.7 8.4 2000 0.8 7.7 8.5 3.8 0.6 4.4 2.1 3.1 5.2
0.6 3.7 4.3 3000 2.2 0.4 2.6 1.8 1.7 3.5 6.1 2.3 8.4 3.6 5.4 9.0 4000 6.5 2.0
8.5 4.6 12.7 17.3 4.0 5.4 9.4 5.6 4.5 10.1

The partial decay widths of various decay channels of vector $K^{*+}$ and
$K^{*0}$ mesons to pseudoscalar kaon and pion are also given in Tables III(c)
and III(d), where the effects of medium density and isospin asymmetry are
studied. These are calculated using equation (51) by considering the in-medium
masses of $K^{*}$ as calculated from the self-energy loop and $K$ meson masses
calculated within the chiral model AM_EPJA55_107_2019 . As the increase in $K$
meson masses is more as compared to vector $K^{*}$ meson masses, the in-medium
decay width is seen to decrease when compared to its vacuum value. We also
observe that the decay widths have very small dependency on the cut-off
parameter $\Lambda_{c}$ which is an encouraging result. The wide range of cut-
off parameter value were taken to account for the rough estimate made by
comparing the vertex form factor $u(q^{2})$ of 56 with the vertex form factor
calculated in the ${}^{3}P_{0}$ model.

Table III(c): Vector $K^{*+}$ Meson Decay Width from $K\pi$ loop for $eB$ = 0
$\rho_{B}$ = $\rho_{0}$ $\rho_{B}$ = $4\rho_{0}$ $\eta=0$ $\eta=0.5$ $\eta=0$
$\eta=0.5$ $\Lambda_{c}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$
$K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$
$(K\pi)^{+}$ $K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $(K\pi)^{+}$ 1000 15.6 29.6 45.2
15.4 29.1 44.5 12.2 24.4 36.6 13.6 16.7 30.3 2000 15.0 28.7 43.7 14.9 28.1
43.0 12.1 25.4 37.5 13.5 16.5 30.0 3000 14.6 28.4 43.0 14.9 27.9 42.8 12.5
23.8 36.3 13.7 17.0 30.7 4000 14.7 29.3 44.0 15.0 28.4 43.4 12.7 24.9 37.6
13.8 17.0 30.8

Table III(d): Vector $K^{*0}$ Meson Decay Width from $K\pi$ loop for $eB$ = 0
$\rho_{B}=\rho_{0}$ $\rho_{B}=4\rho_{0}$ $\eta=0$ $\eta=0.5$ $\eta=0$
$\eta=0.5$ $\Lambda_{c}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$
$K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$
$(K\pi)^{0}$ $K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $(K\pi)^{0}$ 1000 13.3 28.5 41.8
13.1 29.2 42.3 11.3 24.7 35.9 7.5 25.8 33.3 2000 13.3 27.3 40.6 13.1 26.7 39.8
11.3 22.2 33.5 7.5 25.2 32.6 3000 13.4 26.1 39.5 12.9 26.9 39.8 11.6 22.1 33.7
7.7 25.4 33.1 4000 13.7 26.4 40.1 13.2 28.6 41.7 11.4 22.6 34.0 7.8 25.3 33.1

We also compute the effects of strong magnetic field on vector $K^{*}$ masses,
through the self energy loop, from the pseudoscalar $K$ meson masses
calculated after including the effects of Landau quantization ( for charged
mesons only) and spin mixing (through $-\mu_{i}.B$ term) in the presence of
magnetic field. These are tabulated in III(e) and III(f) at various values of
cut-off parameter $\Lambda_{c}$. We observe that the modifications in $K^{*}$
meson masses due to self energy loop are small when compared with the direct
effects of magnetic field like Landau quantization and spin-magnetic field
interaction.

Table III(e): Vector $K^{*+}$ Meson Masses from $K\pi$ loop at $\rho_{B}=0$
$K^{+}\pi^{0}$ $K^{0}\pi^{+}$ $eB/m_{\pi}^{2}$ $\Lambda_{c}$ 1000 2000 3000
4000 1000 2000 3000 4000 0 891.67 891.67 891.67 891.67 891.67 891.67 891.67
891.67 2 887.81 894.24 894.72 892.58 889.75 904.00 894.77 922.53 4 893.73
889.92 894.03 899.75 900.90 886.30 894.24 893.08 6 891.73 891.70 889.53 890.44
897.61 894.75 895.77 893.40 8 894.41 894.14 899.05 895.98 890.769 892.27 891.2
893.14 10 895.75 895.60 898.77 895.94 895.51 898.78 888.47 880.38

Table III(f): Vector $K^{*0}$ Meson Masses from $K\pi$ loop at $\rho_{B}=0$
$K^{0}\pi^{0}$ $K^{+}\pi^{-}$ $eB/m_{\pi}^{2}$ $\Lambda_{c}$ 1000 2000 3000
4000 1000 2000 3000 4000 0 895.55 895.55 895.55 895.55 895.55 895.55 895.55
895.55 2 883.93 890.73 896.40 899.95 897.67 902.91 898.48 901.17 4 893.47
896.23 896.75 905.33 903.26 905.13 894.39 895.69 6 890.78 891.09 901.23 903.09
899.86 894.35 897.74 896.41 8 890.47 888.51 897.80 898.71 904.42 901.33 902.63
904.85 10 893.10 891.30 893.76 899.33 905.43 904.15 905.23 903.89

Figure 6: Decay width of vector $K^{*}$ meson plotted as a function of
relative baryon density ($\rho_{B}/\rho_{0}$) for various values of isospin
asymmetry parameter $\eta$ for zero magnetic field case.

###  VI.4 In-medium Decay Width of Vector $K^{*}$ Meson within the
${}^{3}P_{0}$ model

We study the in-medium decay width of vector $K^{*}$ meson to two pseudoscalar
mesons (kaon and pion), using the ${}^{3}P_{0}$ model, from the mass
modifications of vector $K^{*}$ meson and pseudoscalar kaon ($K$). The mass
modifications for the pseudoscalar kaon have been studied using the chiral
$SU(3)$ model AM_EPJA55_107_2019 . The vacuum masses for pseudoscalar mesons
are taken to be $m_{K^{+}}=493.677$ MeV, $m_{K^{0}}=497.611$ MeV,
$m_{\pi^{\pm}}=139.57039$ MeV, $m_{\pi^{0}}=134.9768$ MeV Zyla_PDG_2020 . By
putting the vacuum values of decay widths $\Gamma(K^{*}\rightarrow K\pi)$ and
vacuum masses for various decay channels, we find the coupling strength
$\gamma$ related to the strength of ${}^{3}P_{0}$ vertex of each channel
individually.

The coupling strength parameter $\gamma$ for the decays $(K^{*+}\rightarrow
K^{+}\pi^{0})$, $(K^{*+}\rightarrow K^{0}\pi^{+})$, $(K^{*0}\rightarrow
K^{0}\pi^{0})$, $(K^{*0}\rightarrow K^{+}\pi^{-})$ comes out to be
$0.1242963$, $0.1781849$, $0.1198312$, $0.1673111$ MeV, respectively for
vacuum decay widths to be 16.98, 33.77, 15.87, and 31.31 MeV, respectively. We
assume spherical harmonic oscillator potential for the wave functions of
vector $K^{*}$ as well as for pseudoscalar $K$ and $\pi$ mesons. The harmonic
oscillator strength parameter for pion, fitted from its charge radius squared
value (0.4 ${\rm fm}^{2}$), is ($211$ MeV)-1 SPM_PRD18_1673_1978 ;
AM_IJMPE24_1550053_2015 . Then we find the harmonic oscillator strength
parameter for the $K$ meson by assuming the ratio $\beta_{K}/\beta_{\pi}$ to
be the same as the ratio of their charge radii, $(r_{\rm ch})_{\pi}/(r_{\rm
ch})_{K}$. The charge radius for $K$ and $K^{*}$ mesons are taken to be
$\big{(}(r_{\rm ch})_{K}=0.56\,{\rm fm}\big{)}$ and $\big{(}(r_{\rm
ch})_{K^{*}}=0.74\,{\rm fm}\big{)}$, which gives $\beta_{K}$ = ($238.3$ MeV)-1
and $\beta_{K^{*}}$= ($184.84$ MeV)-1 AM_EPJA57_98_2021 . By taking into
account the mass modifications of vector $K^{*}$ meson calculated within the
QCD sum rule approach and of pseudoscalar $K$ meson using the chiral $SU(3)$
model, we calculate the in-medium decay width for the decay $(K^{*}\rightarrow
K\pi)$ for various sub-channels by using the ${}^{3}P_{0}$ model.

The decay width for the $K^{*}$ meson for various decay channels is plotted in
figure (6) as a function of relative medium density ($\rho_{B}/\rho_{0}$) for
different isospin asymmetry parameter ($\eta=0,0.5$) and is also given in
table IV. We observed that the decay width for each decay channel decreases as
the medium density is increased. This is because the mass for $K^{*}$ meson
decreases as a function of density, while the mass of pseudoscalar kaon (both
$K^{+}$ and $K^{0}$) increases with density as calculated within the chiral
$SU(3)$ model. The effect of isospin asymmetry on the decay width is also more
pronounced at higher density. The drop in decay width for the channels
$(K^{*+}\rightarrow K^{+}\pi^{0})$ and $(K^{*+}\rightarrow K^{0}\pi^{+})$ is
observed to be more for isospin symmetric case ($\eta=0$) as compared to
isospin asymmetric case ($\eta\neq 0$), while for the $(K^{*0}\rightarrow
K^{+}\pi^{-})$ and $(K^{*0}\rightarrow K^{0}\pi^{0})$ decay channels, the drop
is observed to be more for isospin asymmetric case ($\eta\neq 0$). This is due
to the fact that as the medium density is increased, the mass of charged
$K^{+}$ meson is modified less for isospin asymmetric matter as compared to
isospin symmetric matter, while neutral $K^{0}$ meson mass is modified greatly
for isospin asymmetric matter AM_EPJA55_107_2019 .

Table IV(a): Decay Width of Vector $K^{*+}$ Meson
$\Gamma(K^{*+}\rightarrow K^{+}\pi^{0})$ (MeV) $\Gamma(K^{*+}\rightarrow
K^{0}\pi^{+})$ (MeV) density $\eta=0$ $\eta=0.5$ $\eta=0$ $\eta=0.5$
$\rho_{0}$ 2.98 4.11 5.48 6.90 2$\rho_{0}$ 0.10 0.73 0.05 0.23

Table IV(b): Decay Width of Vector $K^{*0}$ Meson
$\Gamma(K^{*0}\rightarrow K^{0}\pi^{0})$ (MeV) $\Gamma(K^{*0}\rightarrow
K^{+}\pi^{-})$(MeV) density $\eta=0$ $\eta=0.5$ $\eta=0$ $\eta=0.5$ $\rho_{0}$
4.25 3.63 8.45 7.99 2$\rho_{0}$ 0.91 0.30 1.76 1.69

Figure 7: Partial decay widths of vector $K^{*}$ meson plotted as a function
of $eB/m_{\pi}^{2}$ at zero baryon density. The effects of Landau quantization
and PV mixing, considered through spin-magnetic field interaction term, are
included. The decay widths for different polarized states
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$\perp(+1)$, $|\,|$, and $\perp(-1)$, respectively. Figure 8: Partial decay
widths of vector $K^{*}$ meson plotted as a function of $eB/m_{\pi}^{2}$ at
baryon density equal to nuclear matter saturation density ($\rho_{0}$) with
$\eta=0$. The effects of Landau quantization and PV mixing, considered through
spin-magnetic field interaction term, are included. The decay widths for
different polarized states
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$\perp(+1)$, $|\,|$, and $\perp(-1)$, respectively. Figure 9: Partial decay
width of vector $K^{*}$ meson plotted as a function of $eB/m_{\pi}^{2}$ at
baryon density equal to nuclear matter saturation density ($\rho_{0}$) with
$\eta=0.5$. The effects of Landau quantization and PV mixing, considered
through spin-magnetic field interaction term, are included. The decay widths
for different polarized states
($\big{|}1\,+1\big{>},\big{|}1\,0\big{>},\big{|}1\,-1\big{>}$) are written as
$\perp(+1)$, $|\,|$, and $\perp(-1)$, respectively. Figure 10: Decay width of
axial vector $K_{1}$ meson, plotted as a function of relative baryon density
$(\rho_{B}/\rho_{0})$, calculated within a phenomenological Lagrangian
approach.  Figure 11: Decay width of axial vector $K_{1}$ meson, plotted as a
function of relative baryon density $(\rho_{B}/\rho_{0})$, calculated within
the ${}^{3}P_{0}$ model.

Furthermore, we have studied the effects of strong magnetic field on the decay
widths of each decay channel by considering the effects from each polarization
state individually. In Fig. (7), the partial decay widths of vector $K^{*}$
meson are plotted as a function of $eB/m_{\pi}^{2}$ at zero baryon density
after including the effects of Landau quantization and PV mixing (computed by
using the spin mixing ($-\mu_{i}.B$) term). The qualitative behavior of decay
width, as a function of magnetic field, reflects the medium dependence of the
masses of mesons involved in that decay channel. As the neutral $K^{*0}$
vector meson does not undergo Landau quantization, the decay width
modifications for its transverse polarized states arise due to medium
modifications of vector meson masses due to Zeeman splitting and pseudoscalar
meson mass due to PV mixing. On the other hand, the decay width modification
for the longitudinal component of $K^{*0}$ arises due to medium modifications
of masses of longitudinal part of vector meson and pseudoscalar meson due to
PV mixing. In Figs. (8) and (9), the effects of magnetic field on the partial
decay widths of $K^{*}$ are shown at nuclear medium density equal to nuclear
matter saturation density $(\rho_{0})$, in isospin symmetric $(\eta=0)$ and
asymmetric (with $\eta=0.5)$ matter, respectively.

###  VI.5 In-medium Decay Width of Axial Vector $K_{1}$ Meson

The two physical strange axial vector mesons $K_{1}(1270)$ and $K_{1}(1400)$
are the admixture of two strange members, $K_{1A}$ and $K_{1B}$, of two axial
vector nonets. The larger branching ratio of $K_{1}(1400)$ to $K^{*}\pi$ as
compared to $\rho K$ decay channel, and of $K_{1}(1270)$ to $\rho K$ as
compared to $K^{*}\pi$ decay mode, indicates that there is a large mixing
between $K_{1A}$ and $K_{1B}$ to give the physical mesons
Roca_PRD70_094006_2004 . We have studied the in-medium $K_{1}\rightarrow
K^{*}\pi$ decay width from the medium modifications of the $K_{1}$ and $K^{*}$
meson masses, calculated within the QCD sum rule approach. In a
phenomenological approach, the decay width is given by equation (62). The
parameters $\tilde{F}$ and $\tilde{D}$ are fitted in Roca_PRD70_094006_2004
from the various observed decay channels of $K_{1}(1270)$ and $K_{1}(1400)$
mesons as $1400$ MeV and $-1250$ MeV respectively. This fitting leads to the
vacuum partial decay widths for the $K_{1}^{+}\rightarrow K^{*+}\pi^{0}$,
$K_{1}^{+}\rightarrow K^{*0}\pi^{+}$, $K_{1}^{0}\rightarrow K^{*+}\pi^{-}$,
and $K_{1}^{0}\rightarrow K^{*0}\pi^{0}$ decays as $6.2789,12.3173,12.4761$,
and $6.1998$ MeV, respectively. The in-medium decay width is observed to
increase as the medium density is increased as shown in figure (10). This is
due to a larger relative decrease in the $K^{*}$ meson mass as compared to the
relative mass decrease for the $K_{1}$ meson within the QCDSR approach. As the
density of the nuclear medium increases, we observed a significant increase of
the effects of isospin asymmetric matter.

Moreover, the $K_{1}$ meson decay width is also studied using the
${}^{3}P_{0}$ model. The properties of strange axial vector mesons are still
not very clearGuo_NPR36_125_2019  and we take the vacuum partial decay widths
to be the same as calculated within the phenomenological approach. The decay
widths of an axial vector ($A_{1}\rightarrow\rho\pi$) are also studied in a
phenomenological approach in reference Parui_arXiv_2209_02455 . The main idea
here is to analyze the qualitative behavior of the strange axial-vector
mesons. In figure (11), we have plotted the partial decay widths in nuclear
medium calculated in ${}^{3}P_{0}$ model. The variation trends in the decay
width are caused by the interplay between the mass modifications of the
involved $K^{*}(K^{*+}$ or $K^{*0})$ and $K_{1}(K_{1}^{+}$ or $K_{1}^{0})$
mesons. There is observed to be a smaller variation as a function of medium
density when compared with the results of the phenomenological approach.

##  VII Summary

In the present work, we have studied the in-medium masses of vector $K^{*}$
meson and axial vector $K_{1}$ meson in isospin asymmetric nuclear matter,
using the QCD sum rule approach. The self-energy loop of $K^{*}$ meson is
evaluated at one loop level from the medium modifications of the kaon masses.
In the presence of strong magnetic field, the effects of Landau quantization
and PV mixing (through effective interaction vertex as well as spin-magnetic
field interaction term) are also included for the $K^{*}$ meson and are found
to be quite significant. The decay widths of various decay channels for the
$K^{*}\rightarrow K\pi$ decay are also studied by using the ${}^{3}P_{0}$
model. Also, the partial decay widths for the $K_{1}\rightarrow K^{*}\pi$
decay are analyzed within a phenomenological approach as well as within the
${}^{3}P_{0}$ model. In QCDSR, the mass modifications in the medium occur due
to medium modifications of the light quark condensates and scalar gluon
condensates, which are calculated using a chiral effective Lagrangian approach
within a chiral effective model. The effects of density are found to be more
pronounced than the effects of isospin asymmetry of the nuclear medium. The
isospin asymmetry effects are observed to be more pronounced at higher medium
density. The effects of PV mixing are observed to be larger at larger magnetic
fields. The effects of medium density and/or magnetic field on decay widths
are a reflection of medium modifications of the masses of mesons involved.
This present analysis of the in-medium properties of open strange particles
might find relevance in heavy-ion collision experiments in the Relativistic
Heavy Ion Collider (RHIC) low-energy scan programme and the High Acceptance
DiElectron Spectrometer (HADES) Collaboration at GSI, Darmstadt.

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