# Spectral properties of $\omega$, $\rho$ and $A_{1}$ mesons in hot magnetized
matter: effects of (inverse) magnetic catalysis

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

###### Abstract

In-medium masses of the light vector $\omega$, $\rho$ and axial-vector $A_{1}$
mesons are studied in the magnetized nuclear matter, accounting for the
effects of (inverse) magnetic catalysis at finite temperature. The in-medium
partial decay widths for the $A_{1}\rightarrow\rho\pi$ channels are studied
from the in-medium masses of the initial and the final state particles, by
applying a phenomenological Lagrangian to account for the $A_{1}\rho\pi$
interaction vertices. The masses are calculated within the QCD sum rule
framework, with the medium effects coming through the light quark
($\sim\langle\bar{q}q\rangle$) and the scalar gluon condensates ($\sim\langle
G^{2}\rangle$), as well as the light four-quark condensate
($\sim\langle\bar{q}q\rangle^{2}$). The condensates are calculated within the
chiral $SU(3)$ model in terms of the medium modified scalar fields: isoscalar
$\sigma$, $\zeta$, isovector $\delta$ and the dilaton field $\chi$. The
effects of magnetic fields are incorporated through the magnetized Dirac sea
contribution as well as the Landau energy levels of protons and anomalous
magnetic moments (AMMs) of the nucleons at finite temperature nuclear matter.
The effects of temperature are included through the Fermi distribution
functions in the number ($\rho_{p,n}$) and scalar ($\rho^{s}_{p,n}$) densities
of nucleons within the chiral effective model framework. The incorporation of
the magnetic field through the Dirac sea of nucleons lead to an enhancement
(reduction) of the light quark condensates with magnetic field, give rise to
the phenomenon of magnetic (inverse) catalysis. The effects of (inverse)
magnetic catalysis at finite temperature nuclear matter are studied on the
spectral functions and production cross-sections of the neutral $\rho$ and
$A_{1}$ mesons. This may affect the production of the light vector and axial-
vector mesons in the peripheral heavy-ion collision experiments, where
estimated magnetic field is very large at the early stages of collisions with
very high temperature.

##  I Introduction

The study of the in-medium spectral properties (masses, decay widths etc.) of
hadrons under extreme conditions of density and/or temperature, has become a
very important topic of research in the strong interaction physics. The study
is of great relevance in the context of high energy heavy-ion collision
experiments. In the peripheral ultra relativistic heavy-ion collision
experiments, very large magnetic fields have been estimated, for e.g., at RHIC
in BNL, LHC in CERN kharzeev -tuchin . Thus, the in-medium study of hadrons in
the presence of an external strong magnetic field attracts a lot of research
interest in this area. The large number difference between the neutrons and
protons of the heavy colliding nuclei leads to the incorporation of the
effects of isospin asymmetry in the study of hadronic properties.

In-medium properties of the light vector mesons may affect the low mass
dilepton production in the heavy-ion collision experiments rap . The heavy
leptonic decay of $\tau\rightarrow\nu_{\tau}+X$, indicates the coupling of the
hadronic state $X$ to an axial-vector current. Its decay fraction to the three
pion states through the intermediate state of $\rho\pi$ with respect to the
total width shows that, the dominant decay mode of the $A_{1}$ meson is the
partial ($s$-wave) $\rho\pi$ mode tao . The study of the in-medium spectral
properties of the $A_{1}$ meson is important in the context of partial
restoration of chiral symmetry and to provide valuable information to further
experimental studies of the axial-vector meson state. Its coupling to the
$\pi$ meson $f_{\pi}$, is defined in the usual way $\langle
0|\bar{u}\gamma_{\mu}\gamma_{5}d|\pi\rangle=if_{\pi}P_{\mu}$ B196 . The
fundamental symmetry of Quantum chromodynamics (QCD) is the chiral symmetry,
which is spontaneously broken at low densities and low temperatures due to the
non-zero expectation values of the chiral condensates. The formation of the
quark and gluon condensates in QCD vacuum leads to the generation of hadron
masses. The spontaneous chiral symmetry breaking effect induces mass
splittings between the opposite parity states in the hadron spectra, e.g.,
$\pi$-$\sigma,$ $\rho$-$A_{1}$.
The condensates are expected to change with temperature and/or density, and
also with the magnetic field. The values tend to decrease with baryon density
in the strange hadronic matter at different values of strangeness $f_{s}=0,\
0.3,\ 0.5$ and isospin asymmetry parameter $\eta=0,\ 0.5$ 91 . The phenomena
of enhancement (reduction) of the light quark condensates with magnetic field
is called magnetic (inverse) catalysis Shovkovy ; elia ; kharzeevmc . In the
literature, there are few studies related to the effects of (inverse) magnetic
catalysis on the hadron properties in magnetized nuclear matter. There are
several studies in the context of quark matter with a background magnetic
field using the framework of Nambu-Jona-Lasinio model Preis ; menezes ; ammc ;
lemmer ; guinjl . The effects of the magnetized Dirac sea have been studied on
the nuclear matter phase transition using the Walecka model and an extended
linear sigma model haber . In ref. arghya , the effects of (inverse) magnetic
catalysis have been studied using the weak-field expansion of the fermion
propagator incorporating its anomalous magnetic moment, to evaluate the
nucleonic one-loop self energy functions in the hot magnetized nuclear matter.
At very high temperatures and high densities, the chiral symmetric phase is
expected to be (partially) restored, which can be inferred from its observable
consequence on the hadron spectrum. The dilepton data, as measured in the
relativistic heavy-ion collisions, provided evidence on the in-medium spectral
changes of $\rho$ meson, while there is very little access to the experimental
evidence on the spectral changes of $A_{1}$ meson. Therefore, to investigate
on the chiral symmetry restoration from the in-medium spectral changes of the
$\rho$ meson hohler , a theoretical investigation of the same is required on
the $A_{1}$ meson spectrum. Towards this aim, sum rules serve as good non-
perturbative tool to connect the hadronic spectral properties directly to the
QCD vacuum condensates. The medium modifications of the spectral properties of
hadrons may serve as good evidence to the in-medium nature of QCD vacuum
condensates.

There have been QCD sum rule studies on the spectral density functions induced
by the axial-vector current B196 ; shifman448 . The spectral functions of
$\rho$ and $A_{1}$ mesons have been analyzed in vacuum using the hadronic
models and have been tested further with QCD sum rules rapp . The Weinberg-
type sum rules weinberg  for $\rho$ and $A_{1}$ mesons (in the exact chiral
limit), have been studied at zero and finite temperature 49 . In ref.
hatsudab394 , finite temperature ($T\neq$ 0) study of $\rho$, $\omega$ and
$A_{1}$ mesons have been performed by using Borel sum rule. The effects of
temperature have been incorporated through the thermal average of the local
operators in the operator product expansion, which leads to the non-vanishing
values of Lorentz non-scalar operators that would have otherwise been zero at
$T=0$. The contributions of the scalar four-quark condensates are observed to
be significant on these meson properties. In ref. leupold , the Breit-Wigner
parametrization for the $\rho,A_{1}$ spectral functions have been used and the
constraints of QCD sum rules have been investigated on their masses and decay
widths, in vacuum and at finite nuclear matter density. The coupling of $\pi$
meson to the axial-vector current has been considered by adding a
$\delta$-function peak at $m_{\pi}$ to the correlator of axial-vector channel.
In ref. kwon , the parity-mixing ansatz including the finite widths of
$\rho,A_{1}$ spectra has been considered at finite temperature in the context
of finite energy sum rules. The mass of the $A_{1}$ meson is reported to drop
with rising temperature, indicating the expected tendency of $\rho$ -$A_{1}$
mass degeneracy nearby the critical temperature $T_{c}$ of chiral symmetry
restoration. The temperature dependencies of the meson-nucleon coupling
constants for $\rho$ sahin  and $A_{1}$ sahin2 , have been investigated using
the soft-wall AdS/QCD model with thermal dilaton field. An interaction
Lagrangian of the $\rho(A_{1})$-nucleon interaction along with the thermal
dilaton field system has been constructed in the bulk of space-time, to form
the integral representation of the $g_{A_{1}NN}(g_{\rho NN})$ coupling. The
mesons and nucleons profile functions are applied to the model. The AdS/CFT
correspondence states the equivalence of two different physical theories of
gravity (in the bulk of AdS space-time) and a quantum field theory (on the
boundary of this space-time). There is a strong-weak duality which is applied
to describe the low-energy phenomena of QCD (theory of strong interactions),
and is called AdS/QCD. There are two approaches in AdS/QCD used to study the
hadron properties in nuclear medium, which is expected to create in the
relativistic heavy ion collision experiments. The modified thermal soft-wall
model has been used to incorporate the effects of temperature through the
thermal dilaton field, which is also related to the chiral condensate.

In ref. 100 , the effects of magnetic field have been studied on the in-medium
masses of the light vector mesons ($\rho,\omega,\phi$) in magnetized nuclear
matter using the QCD sum rule method. In-medium masses are obtained by
incorporating the medium effects through the scalar fields
$\sigma,\zeta,\delta,\chi$, calculated within the chiral $SU(3)$ model. In the
absence of an external magnetic field, masses of these mesons have been
studied in the strange hadronic matter using the sum rule approach 91 , in
terms of the light quark (up to the scalar four quark condensates) and the
scalar gluon condensates calculated within the chiral model framework. In ref.
100 , contribution of an external magnetic field has been incorporated through
the Landau quantization of protons and the anomalous magnetic moments of the
nucleons. In our present study, contributions of the magnetized Dirac sea are
incorporated to study the effects of (inverse) magnetic catalysis on the
masses of $\rho$, $\omega$ and $A_{1}$ mesons as well as the spectral function
and production cross-sections of $\rho^{0},A_{1}^{0}$ mesons in hot and dense
nuclear matter, in an external magnetic field.
The present paper is organized as follows: in sec.II, the chiral effective
model is briefly discussed to find the in-medium light quark and the scalar
gluon condensates. In sec.III, QCD sum rule approach is presented to calculate
the masses of the light vector and axial-vector mesons under study. Sec.IV,
describes the phenomenological Lagrangian formulation to find the hadronic
decays of the $A_{1}$ and $\rho$ mesons. Sec.V contains the formalism of the
Breit-Wigner spectral functions and production cross-sections of the neutral
$\rho$ and $A_{1}$ mesons incorporating the medium effects through their
respective mass and decay widths. The results of the present investigation are
discussed in sec.VI. Finally, sec.VII, summarizes the findings of the present
work.

##  II The chiral effective Model

The in-medium masses of the light vector and axial-vector mesons within the
QCD sum rule approach are obtained through the light quark and scalar gluon
condensates, which are calculated using a chiral effective model framework.
The effective chiral model is based on the non-linear realization of chiral
$SU(3)_{L}\times SU(3)_{R}$ symmetry coleman ; weinberg1 ; bardeen  and the
broken scale-invariance of QCD papa ; 69 ; zschi . A scale-invariance breaking
logarithmic potential in the scalar dilaton field $\chi$ sech ; ellis  within
the effective Lagrangian simulates QCD gluon condensate. The general
Lagrangian density of the model contains the kinetic energies of the baryons
and mesons, along with the baryon-meson (both spin-0 and spin-1) interactions.
Some of which generate the mass of the baryons (due to their couplings with
the scalar mesons fields). The meson-meson interactions give rise to their
mass and dynamics in the effective model Lagrangian. The spontaneous symmetry
breaking and explicit symmetry breaking of QCD are incorporated within the
chiral $SU(3)$ model Lagrangian accordingly papa . The scale-symmetry breaking
logarithmic potential simulates the scalar gluon condensates in the model. In
the magnetized nuclear medium, the magnetic field contributions are coming
through the Landau energy levels of protons and anomalous magnetic moments of
the nucleons in the Fermi sea p97 ; am98 ; am981 ; prakash ; brod ; wei ; mao
.

In some earlier works on the in-medium vector meson properties using Quantum
Hadrodynamics model, the dominant contribution to the mass shifts are
originated from the vacuum polarization effects in the baryon (nucleon)
sector. This significant drop can not be obtained while considering the mean-
field approximation only. The in-medium properties of vector mesons and their
observable effects on the low mass dilepton spectra have been studied
extensively in the literature accounting for the quantum correction (through
relativistic Hartree approximations) effects within the Walecka model. It is
therefore interesting in the context of modern day experimental facilities at
RHIC, LHC, where at the peripheral heavy ion collisions ultra strong transient
electromagnetic field is expected to be produced at the very early stages of
the collisions, which eventually decay with time in the produced medium.
Depending on the collision geometry, the effect of an uniform magnetic field
background in a less dense medium can be considered and the quantum
corrections in the nucleonic sector (as in the low density matter the possible
constituents can be protons and neutrons) should be taken into account.

The propagation of a particle through an external medium at finite density
and/or temperature lead to the modifications to its field theoretical
propagator. The same is also true when the particle is going through an
external magnetic background. In the formalism of the relativistic Hartree
approximation (RHA), the effective interaction of a fermion e.g., nucleon in a
medium is taking into account through the one-loop self-energy function
derived using the Feynman tadpole diagrams. The diagrams are corresponding to
the scalar and vector mesons interactions with the nucleons. The tadpole
diagrams, in turn, correspond to the second-order contributions to the various
Green’s functions, e.g., fermion propagators. In the non-central ultra-
relativistic heavy ion collisions, there have been estimation of very strong
electromagnetic fields at the very early stages of the collisions. The
strength of the produced fields may depend on the various collision parameters
and collision geometry. However, the magnetic field at the freeze-out hyper-
surface (FOHS) is estimated to be of the order of $0.07$ Ge$V^{2}$ which is
around $4m_{\pi}^{2}$ fohs . In order to obtain a more complete in-medium
effects, the effect of finite temperature is also accounted for in the present
study. As it will be discussed further in the results and discussion section,
the solutions of the scalar fields within the present model incorporating the
magnetized Dirac sea contributions, are obtained up to a magnetic field
strength of 3.9$m_{\pi}^{2}$ in the vacuum and almost around 9$m_{\pi}^{2}$ at
the nuclear matter saturation density $\rho_{0}$, which is nearby the field
strength at the FOHS. In this process, the one-loop baryons i.e., protons and
neutrons are completely dressed in the sense that their effective interacting
propagator is considered at the given medium while computing the loop
integrals. The effective interacting propagator of fermion is obtained from
the perturbative expansion in the Dirac equation in presence of an external
magnetic field with finite anomalous magnetic moments (AMMs) of the nucleons.
The weak-field expansion is considered in the present study by keeping up to
the second order contributions in magnetic field and AMMs of the Dirac sea of
nucleons. The method of dimensional regularization is applied to obtain the
simplified form of the divergent integrals involved in the one-loop self
energy functions. The ultraviolet divergence resulting from the pure vacuum
term is neglected using the mean-field approximation. However, the other
divergent contribution which is extracted from the pole of the gamma function
is treated using the $\overline{MS}$ renormalization scheme. In ref.arghya ,
the above procedure has been implemented to study the vacuum to nuclear matter
phase transition at finite temperature nuclear matter within the Walecka model
by considering $\sigma-N$ interaction in the one-loop scalar self energy
function of nucleons. In our present work, the formalism is extended to apply
in the context of chiral $SU(3)$ model, which incorporates the $\sigma-N$,
$\zeta-N$ and $\delta-N$ interactions in the scalar one-loop self energy
functions. The contribution of the anomalous magnetic moments of the Dirac sea
of nucleons is considered in the weak field expansion of the magnetized
fermion propagator. Thus, the contribution of the magnetized Dirac sea is
incorporated through the summation of the nucleonic tadpole diagrams
corresponding to their interactions with the scalar meson fields $\sigma$,
$\zeta$ and $\delta$ in the chiral $SU(3)$ model Lagrangian.

$\displaystyle\mathcal{L}_{BX}$
$\displaystyle=\sum_{i=p,n}\bar{\psi}_{i}(g_{\sigma i}\sigma+g_{\zeta
i}\zeta+g_{\delta i}\delta)\psi_{i}.$ (1)

Where, $\sigma$, $\zeta$ are the non-strange and strange scalar-isoscalar
fields, respectively and $\delta$ is the scalar-isovector field. The coupling
parameters $g_{\sigma i},\ g_{\zeta i},\ g_{\delta i}$ corresponding to the
$i^{th};\ i=p,n$, baryon are fitted from the empirically known masses of the
baryons. The one-loop self energy functions of the Dirac sea of nucleons are
thus evaluated using the weak-field expansion of nucleonic propagators (up to
second order in magnetic field). The effects of AMMs of the Dirac sea of
nucleons are also accounted for in this study. The weak-field approximation of
fermion propagator leads to a reliable solutions of the self-consistent scalar
fields equations in the magnetized nuclear matter, till the magnetic field
strength of $0.04$ Ge$V^{2}$ or around 2-3 $m_{\pi}^{2}$ within the Walecka
model arghya . In the present work’s chiral $SU(3)$ model, the similar field
expansion of the propagators of Dirac nucleons is considered to find the
magnetized vacuum corrections in the nuclear matter context. The self-
consistent scalar fields equations for $\sigma$, $\zeta$, $\delta$ and $\chi$
including both Fermi and Dirac sea effects in the magnetic nuclear matter give
rise to a feasible solution up to a magnetic field strength which is
consistent with that of the freeze-out hypersurface for hadronic cooling zone.
As there is a finite chance of the in-medium hadron properties to be affected
by a low-strength magnetic field in this regime. In this sense the weak-field
expansion is considerable for a qualitative estimate of the hadron properties
in a magnetized matter. Beside this the degree of divergence involved in the
loop integral would also be out of control if higher order terms in the
propagator expansion were taken into account. The nucleonic self energy
functions contribute to the scalar densities of nucleons chrmmc ; dmc ; bmc ,
which then give impact to the scalar fields coupled equations of motion. The
scalar meson fields are treated as classical, whereas the nucleons as quantum
fields in the evaluation of the magnetized Dirac sea contribution. The scale-
invariance breaking phenomena of QCD leads to the trace anomaly of QCD, i.e.,
non-zero value for the trace of the energy-momentum tensor in QCD, which in
the limit of finite light quark masses $m_{i}(i=u,d,s)$ become 100 ; cohen

$\langle T^{\mu}_{\mu}\rangle=\sum_{i=u,d,s}\langle
m_{i}\bar{q}_{i}q_{i}\rangle+\langle\frac{\beta_{QCD}}{2g}G_{\mu\nu}^{a}G^{a\mu\nu}\rangle.$
(2)

In eq.(2), $G_{\mu\nu}^{a}$ is the gluon field strength tensor of QCD. The
trace of the energy momentum tensor within the model can be obtained from the
Lagrangian density terms containing the field $\chi$ 91 ; arvind82 ; heide  as

$\langle\theta_{\mu}^{\mu}\rangle=\chi\frac{\partial\mathcal{L}}{\partial\chi}-4\mathcal{L}=-(1-d)\chi^{4}$
(3)

The first term in eq.(2) corresponds to the explicit chiral symmetry breaking
term in QCD

$\mathcal{L}_{SB}^{QCD}=-Tr[diag(m_{u}\bar{u}u,m_{d}\bar{d}d,m_{s}\bar{s}s)],$
(4)

which in the chiral $SU(3)$ model under the mean-field approximation is
written as 91

$\mathcal{L}_{SB}=Tr\left[diag\left(-\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma+\delta),-\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma-\delta),\left(\sqrt{2}m_{k}^{2}f_{k}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right)\right]$
(5)

Comparing eqs. (4) and (5), the light quark condensates are related to the
scalar fields $\sigma$, $\zeta$ and $\delta$ as

$\displaystyle m_{u}\langle\bar{u}u\rangle$
$\displaystyle=\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma+\delta),$ (6)
$\displaystyle m_{d}\langle\bar{d}d\rangle$
$\displaystyle=\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma-\delta),$ (7)
$\displaystyle m_{s}\langle\bar{s}s\rangle$
$\displaystyle=\left(\sqrt{2}m_{k}^{2}f_{k}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta$
(8)

From eqs. (2) and (3), one obtains the scalar gluon condensate as

$\sum_{i=u,d,s}m_{i}\langle\bar{q}_{i}q_{i}\rangle-\frac{9}{8}\left\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\right\rangle=-(1-d)\chi^{4}$
(9)

In eq.(2),
$\beta_{QCD}(g)=-\frac{N_{c}g^{3}}{48\pi^{2}}(11-\frac{2}{N_{c}}N_{f})=-\frac{9\alpha_{s}g}{4\pi}$
is the QCD $\beta$ function at the one-loop level with $N_{f}=3$ flavors,
$N_{c}=3$ colors of quarks and $\alpha_{s}=\frac{g^{2}}{4\pi}$; by using the
expressions for $m_{i}\langle\bar{q_{i}}q_{i}\rangle$ $(i=u,d,s)$ from eqs.
(6)-(8), the scalar gluon condensate is given by

$\left\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\right\rangle=\frac{8}{9}\left[(1-d)\chi^{4}+\left(m_{\pi}^{2}f_{\pi}\sigma+\left(\sqrt{2}m_{k}^{2}f_{k}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\right)\zeta\right)\right]$
(10)

The coupled equations of motion of the scalar fields $\sigma$, $\zeta$,
$\delta$ and $\chi$ obtained from the chiral $SU(3)$ model Lagrangian are
solved in the magnetized (asymmetric) nuclear matter at finite temperature,
accounting for the effects of the Dirac sea under the mean-field
approximation. In the coupled equations of motion arvind82 , the number and
scalar densities of nucleons ($\rho_{i},\rho^{s}_{i};i=p,n$) incorporate the
effects of density, temperature and magnetic fields as given below for the
charged protons and neutral neutrons in hot magnetized nuclear matter,

$\displaystyle\rho_{p}$ $\displaystyle=\frac{|eB|}{2\pi^{2}}\sum_{\nu,\ s=\pm
1}\int_{0}^{\infty}dk_{||}\left(\frac{1}{e^{(E_{p}^{*}-\mu_{p}^{*})/T}+1}-\frac{1}{e^{(E_{p}^{*}+\mu_{p}^{*})/T}+1}\right),$
(11) $\rho_{p}^{s}=\frac{|eB|m^{*}_{p}}{2\pi^{2}}\sum_{\nu,\ s=\pm
1}\int_{0}^{\infty}\frac{dk_{||}}{\sqrt{k_{||}^{2}+(\sqrt{m^{*2}_{p}+2\nu|eB|}+s\Delta_{p})^{2}}}\left(\frac{1}{e^{(E_{p}^{*}-\mu_{p}^{*})/T}+1}+\frac{1}{e^{(E_{p}^{*}+\mu_{p}^{*})/T}+1}\right)+\Delta\rho^{s}_{p},$
(12)

$\rho_{n}=\frac{1}{2\pi^{2}}\sum_{s=\pm
1}\int_{0}^{\infty}dk_{\perp}k_{\perp}\int_{0}^{\infty}dk_{||}\left(\frac{1}{e^{(E_{n}^{*}-\mu_{n}^{*})/T}+1}-\frac{1}{e^{(E_{n}^{*}+\mu_{n}^{*})/T}+1}\right),$
(13)
$\rho_{n}^{s}=\frac{1}{2\pi^{2}}\sum_{s}\int_{0}^{\infty}dk_{\perp}k_{\perp}\left(1+\frac{s\Delta_{n}}{\sqrt{m_{n}^{*2}+k_{\perp}^{2}}}\right)\int_{0}^{\infty}dk_{||}\frac{m^{*}_{n}}{\sqrt{k_{||}^{2}+(\sqrt{m^{*2}_{n}+k_{\perp}^{2}}+s\Delta_{n})^{2}}}\\\
\times\left(\frac{1}{e^{(E_{n}^{*}-\mu_{n}^{*})/T}+1}+\frac{1}{e^{(E_{n}^{*}+\mu_{n}^{*})/T}+1}\right)+\Delta\rho^{s}_{n}.$
(14)

In the above expressions, the term related to the anomalous magnetic moments
of protons and neutrons is $\Delta_{i(i=p,n)}=-\frac{\kappa_{i}\mu_{N}B}{2}$,
with $\kappa_{p}=3.5856$ and $\kappa_{n}=-3.8263$, are the gyromagnetic ratio
corresponding to the anomalous magnetic moments (AMMs) of the proton and the
neutron, respectively wei ; mao . The energy spectra of protons and neutrons
in the magnetized nuclear matter are
$E^{*}_{p}=\big{[}k_{||}^{2}+(\sqrt{m^{*2}_{p}+2\nu|eB|}+s\Delta_{p})^{2}\big{]}^{1/2}$,
and
$E^{*}_{n}=\big{[}k_{||}^{2}+(\sqrt{m^{*2}_{n}+k_{\perp}^{2}}+s\Delta_{n})^{2}\big{]}^{1/2}$;
$\mu^{*}_{i}$ denote the effective chemical potential of protons and neutrons
($i=p,n$) at finite temperature matter am81 ; am91 . In equations (12) and
(14), the additional contribution $\Delta\rho^{s}_{i};\ ({i=p,n})$ due to the
magnetized Dirac sea effects can be related to the vacuum self-energy
corrections in a background magnetic field as

$-\frac{g_{Si}^{2}}{m_{S}^{2}}\Delta\rho^{s}_{N}\Big{|}_{S=\sigma,\zeta,\delta}=\Sigma^{s}_{vacuum}=\frac{g_{Si}^{2}}{4\pi^{2}m_{S}^{2}}\Bigg{[}\frac{|eB|^{2}}{3m^{*}_{i}}+\Big{(}\Delta_{p}^{2}m^{*}_{p}+\Delta_{n}^{2}m^{*}_{n}-|eB|\Delta_{p}\Big{)}\left\\{\frac{1}{2}+2ln\left(\frac{m^{*}_{i}}{m_{i}}\right)\right\\}\Bigg{]}$
(15)

In equation (15), the corresponding electric charge for proton should be used
which is zero for neutron. In the magnetized nuclear matter, the number and
scalar densities of charged fermions (protons) have the contributions of the
Landau energy levels of protons. The number and scalar densities of both
protons and neutrons incorporate the effects of the anomalous magnetic moments
(AMMs) of the nucleons from the magnetized Fermi sea of nucleons p97 ; am98 ;
wei ; mao ; rabhi . The scalar densities of the nucleons incorporate an
additional contribution ($\Delta\rho_{i}^{s};i=p,n$) due to the magnetized
Dirac sea, which is obtained by summing over the nucleonic tadpole diagrams
corresponding to the scalar mesons ($\sigma$, $\zeta$ and $\delta$) and
nucleons interactions within the chiral effective model. The Fermi
distribution functions give rise to the effect of temperature on the solutions
of the scalar fields ($\sigma$, $\zeta$, $\delta$ and $\chi$) through
$\rho_{i}$ and $\rho^{s}_{i};\ i=p,n$ rabhi . The scalar fields equations of
motion are thus solved self-consistently at the given values of the baryon
density $\rho_{B}=\rho_{p}+\rho_{n}$, isospin asymmetry parameter
$\eta=\frac{\rho_{n}-\rho_{p}}{2\rho_{B}}$, temperature $T$, and magnetic
fields $|eB|$, taken into account the important effects of the magnetic field
through Dirac sea.

##  III QCD sum rule framework

In the present section, the masses of the light vector mesons $\omega$, $\rho$
and the axial-vector meson $A_{1}$ are calculated using the QCD sum rule
(QCDSR) approach. In-medium masses of these mesons are investigated in the
magnetized nuclear matter at finite temperature, accounting for the effects of
(inverse) magnetic catalysis through the magnetized Dirac sea. The in-medium
masses are obtained in terms of the light quark condensates (up to the scalar
four-quark condensate) and the scalar gluon condensate, which are calculated
within the chiral $SU(3)$ model. The time-ordered current-current correlator
is given by B196 ; shifman448

$\Pi_{\mu\nu}(q)=i\int d^{4}x\ e^{iqx}\left<T[J_{\mu}(x),J_{\nu}(0)]\right>.$
(16)

Where ${}^{\prime}T^{\prime}$ denotes the time-ordered product, and the symbol
${}^{\prime}\langle\rangle^{\prime}$ indicates the in-medium expectation value
of the T-ordered product of currents. The current quark-bilinears of all the
three axial-vector meson states (with $J^{P}=1^{+}$) are defined as

$J^{(A_{1}^{+})}_{\mu}=\bar{d}\gamma_{\mu}\gamma_{5}u;\quad
J^{(A_{1}^{-})}_{\mu}=\bar{u}\gamma_{\mu}\gamma_{5}d;\quad
J^{(A_{1}^{0})}_{\mu}=\frac{1}{2}(\bar{u}\gamma_{\mu}\gamma_{5}u-\bar{d}\gamma_{\mu}\gamma_{5}d)$
(17)

For the vector currents with $J^{P}=1^{-}$ they are given by

$J^{(\rho^{+})}_{\mu}=\bar{d}\gamma_{\mu}u;\quad
J^{(\rho^{-})}_{\mu}=\bar{u}\gamma_{\mu}d;\quad J^{(\omega,\
\rho^{0})}_{\mu}=\frac{1}{2}(\bar{u}\gamma_{\mu}u\pm\bar{d}\gamma_{\mu}d)$
(18)

The correlation function is written out in the following tensor structure
shifman448 ; leupold ; k1 ; urban

$\Pi_{\mu\nu}(q)=q_{\mu}q_{\nu}R(q^{2})-g_{\mu\nu}K(q^{2})$ (19)

The above expression is valid for mesons at rest. For the conserved vector
currents, $K(q^{2})=q^{2}R(q^{2})$. As for the non-conserved axial-vector
current this relation no longer holds and $R(q^{2})$ has contributions from
pseudoscalar mesons k1 ; shifman448 ; urban . In principle, QCD sum rule can
be carried out with either $R(q^{2})$ or $K(q^{2})$. Further studies in this
work are based on $R(q^{2})$ as the Borel transform with the other quantity is
rather unstable and there will be no new information obtained by working with
$K(q^{2})$ along with $R(q^{2})$ for the $A_{1}$ current leupold .
Among the two representations of QCDSR, the real part of the correlator
$R(q^{2})$ on the phenomenological side is related to its imaginary part via a
dispersion relation 91 ; k1

$R_{phen.}(q^{2})=\frac{1}{\pi}\int^{\infty}_{0}ds\
\frac{ImR^{phen.}(s)}{(s-q^{2})}.$ (20)

Where $ImR^{phen.}(s)$ is called the spectral density, parametrized in terms
of the hadronic resonance plus perturbative continuum. On the other
representation, the real part of $R(q^{2})$ is expressed in the large space
like region $(Q^{2}=-q^{2})>>1$ $GeV^{2}$ by the Wilson’s operator product
expansion (OPE), given as hatsudab394 ; klingl

$R_{OPE}(q^{2}=-Q^{2})=\left(-c_{0}\ln\left(\frac{Q^{2}}{\mu^{2}}\right)+\frac{c_{1}}{Q^{2}}+\frac{c_{2}}{Q^{4}}+\frac{c_{3}}{Q^{6}}+...\right)$
(21)

In the above equation the operators up to dimension-6 are considered and the
scale $\mu$ has been chosen as 1 GeV 91 ; hatsudab394 ; klingl . The first
term in the OPE is a contribution from perturbative QCD. The coefficients
$c_{i}$ $(i=1,2,3)$ of the subsequent terms contain QCD non-perturbative
effects in terms of the light quark and scalar gluon condensates and some
parameters from the QCD Lagrangian. The condensates are affected in presence
of a medium. The effects of magnetic field through the Dirac sea lead to the
significant changes in the condensates with magnetic field, which are of main
concern of study on the in-medium spectral properties of light mesons. The
coefficient $c_{3}$ is being associated with the light four-quark condensate
and it is different corresponding to the different current quark-bilinears of
the charged and neutral mesons with the same $J^{PC}$ quantum numbers. In
Eq.(21), the $c_{i}$ $(i=0,1,2,3)$ coefficients for the light axial-vector
meson $A_{1}$ are given by hatsudab394 ; leupold ; k1 ; F1

$c_{0}=\frac{1}{8\pi^{2}}\left(1+\frac{\alpha_{s}}{\pi}\right);\quad
c_{1}=-\frac{3}{8\pi^{2}}(m_{u}^{2}+m_{d}^{2});\quad
c_{2}=\frac{1}{24}\langle\frac{\alpha_{s}}{\pi}G^{\mu\nu}G_{\mu\nu}\rangle-\frac{1}{2}(m_{u}\langle\bar{u}u\rangle+m_{d}\langle\bar{d}d\rangle),$

and

$c_{3}^{(A_{1}^{0})}=\pi\alpha_{s}\times\frac{88}{81}\kappa_{1}\left(\langle\bar{u}u\rangle^{2}+\langle\bar{d}d\rangle^{2}\right).$

Here we study the case of neutral $A_{1}$ meson only. To simplify the
expression for the scalar four-quark condensates factorization technique is
adopted. The parameter $\kappa_{1}$ introduces the deviation from the exact
factorization which is one in the vacuum saturation assumption shifman448 ;
zos . The value of the running coupling constant is $\alpha_{s}=0.35$ at
$\mu=1$ GeV scale. For the (neutral) vector meson states $\rho$ and $\omega$,
the Wilson coefficients are already given in refs.91 ; 100 , which will be
used here to study the important effects of (inverse) magnetic catalysis on
their masses at finite temperature nuclear matter. In-medium masses of the
charged $\rho$ mesons are also calculated to study the in-medium hadronic
decay widths of $A_{1}\rightarrow\rho^{\pm}\pi^{\mp}$, have the corresponding
coefficients $c_{i};i=0,1,2$, similar to the neutral partner $\rho_{0}$.
However, $c_{3}$ term is different than that of neutral $\rho$ in the
following way,

$c_{3}^{\rho^{\pm}}=\pi\alpha_{s}\times\kappa_{2}\left[\frac{16}{81}\left(\langle\bar{u}u\rangle^{2}+\langle\bar{d}d\rangle^{2}\right)-\frac{16}{9}\langle\bar{u}u\rangle\langle\bar{d}d\rangle\right]$

In practice, the parametrization of the current-current correlator is given
for the energy region of the lowest hadronic resonance, usually there is no
model which can be valid for arbitrary high energies. To achieve a larger
suppression on the high energy part of the hadronic spectral distribution,
Borel transform is applied. After applying the Borel transform, the
phenomenological side of Eq.(20) is connected to the OPE side in Eq.(21) as

$\frac{1}{\pi}\int^{\infty}_{0}ds\ e^{-s/M^{2}}\
ImR^{phen.}(s)=\left[c_{0}M^{2}+c_{1}+\frac{c_{2}}{M^{2}}+\frac{c_{3}}{2M^{4}}\right]$
(22)

Where, $M$ is called Borel mass. The exponential function on the l.h.s
enhances the contribution of the low energy resonance by suppressing the high
energy continuum part at large $s$. Higher dimensional operators in $R_{OPE}$
(on the r.h.s) are suppressed by an additional factor of $1/(n-1)!$, leading
to the better convergence of the operator product expansion side.

The hadronic spectral density function $ImR^{phen.}(s)$ separates into a
resonance part, $R_{res}(s)$ (for $s\leq s_{0}$) and a perturbative continuum
(for $s>s_{0}$) 91 ; leupold ; klingl

$\frac{ImR^{phen.}(s)}{\pi}=R_{res}(s)\ \Theta(s_{0}-s)+c_{0}\
\Theta(s-s_{0})$ (23)

Where, $s_{0}$ is the threshold between the low energy resonance region and
the high energy continuum part, latter being calculated in perturbative QCD.

Next, the finite energy sum rules (FESRs) are derived by inserting Eq.(23)
into the l.h.s of Eq.(22). The exponential function then expanded in powers of
$\frac{s}{M^{2}}$ for $s\leq s_{0}$ and $M>\sqrt{s_{0}}$. Comparing the
different powers of $1/M^{2}$ on both sides of Eq.(22), the finite energy sum
rules in vacuum can be written as

$\int^{s_{0}}_{0}ds\ R_{res}(s)=(c_{0}s_{0}+c_{1})$ (24) $\int^{s_{0}}_{0}ds\
sR_{res}(s)=\left(\frac{c_{0}s_{0}^{2}}{2}-c_{2}\right)$ (25)
$\int^{s_{0}}_{0}ds\
s^{2}R_{res}(s)=\left(\frac{c_{0}s_{0}^{3}}{3}+c_{3}\right)$ (26)

The spectral parametrization $R_{A}^{phen.}(s)$, used for the axial-vector
current is given by

$R_{A}^{phen.}(s)=F_{A}\delta(s-m_{A}^{2})+c_{0}\Theta(s-s_{0}^{A})+f_{\pi}^{2}\delta(s-m_{\pi}^{2})$
(27)

Where, $\frac{ImR^{phen.}(s)}{\pi}=R^{phen.}(s)$. In the axial-vector meson
channel, apart from the $A_{1}$ resonance term there is a contribution from
the pseudoscalar meson resonance, due to its coupling to the axial-vector
current B196 ; shifman448 ; leupold . Similar ansatz is used for the vector
meson channel except for the pseudoscalar meson pole 91 ; klingl ; hatsudab394

$R_{V}^{phen.}(s)=F_{V}\delta(s-m_{V}^{2})+c_{0}\Theta(s-s_{0}^{V})$ (28)

In the nuclear medium, for mesons at rest, the meson-nucleon scattering effect
is incorporated through the Landau damping term $\rho_{sc}$ in the spectral
function as given by,

$\int^{\infty}_{0}ds\ e^{-s/M^{2}}\
\frac{ImR^{phen.}(s)}{\pi}+\rho_{sc}=\left[c_{0}M^{2}+c_{1}+\frac{c^{\prime}_{2}}{M^{2}}+\frac{c^{\prime}_{3}}{2M^{4}}\right]$
(29)

Where the primed symbols denote the corresponding in-medium quantities. The
damping term originates due to the absorption of a space-like meson by an on-
shell nucleon leupold ; flower . The contributions for the vector and axial-
vector meson channels are taken according to the formalism based on the
scattering amplitudes with the nucleons of the respective polarization states
of vector and axial-vector correlators flower . The coefficient in the scalar
four-quark condensate $\kappa_{j}$ for the vector and axial-vector states, are
determined by solving their respective vacuum FESRs. Thus, the in-medium
resonance parameters of the vector and axial-vector spectral functions, i.e.,
their corresponding mass $m^{\prime}$, strength $F^{\prime}$ and threshold
energy $s^{\prime}_{0}$, are solved from the in-medium FESRs as derived from
Eq.(29).

##  IV In-medium Hadronic Decay widths

An effective Lagrangian involving the axial-vector (a), vector (v) and
pseudoscalar ($\phi$) mesons vertices, is used in the present investigation to
calculate the in-medium partial decay width of $A_{1}$ meson going to $\rho$
and $\pi$ mesons states. The effects of (inverse) magnetic catalysis on the
in-medium hadronic decay widths of $A_{1}\rightarrow\rho\pi$ are obtained
through the in-medium mass of the initial state $A_{1}$ and final state $\rho$
mesons particles, calculated using QCD sum rule approach. The effects of
temperature in the magnetized nuclear matter is considered in the present
study, including the effects of magnetized Dirac sea contribution. The
phenomenological Lagrangian used for the $av\phi$ interaction

$\mathcal{L}_{{av\phi}}=i\tilde{f}\langle a_{\mu\nu}[v^{\mu\nu},\phi]\rangle.$
(30)

Where, the $SU(3)$ multiplet of the mesons are considered; The symbol $<>$
denotes the trace of the product of matrices and $i$ in front of $\tilde{f}$
is to make the Lagrangian hermitian. The spin-1 meson fields in Eq.(30) are
treated as anti-symmetric tensor fields. The channel of neutral axial-vector
meson $A_{1}^{0}\rightarrow\rho^{\pm}\pi^{\mp}$ is studied in the hot
magnetized matter, accounting for the effects of (inverse) magnetic catalysis.
The amplitude for the process is calculated using the above Lagrangian.

The coupling parameters in the interaction vertices are fitted from the
observed decay modes of $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ family of axial-
vector mesons with their respective branching ratio roca70 ; pdg . The value
of the fitted parameter $\tilde{f}$ in Ref. roca70 , is used here to find the
in-medium partial decay widths of the $A_{1}^{0}$ meson in an external medium.
For tree level calculations $(\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu})$
can be used instead of the tensor $X_{\mu\nu}$ of $X=a,v$ fields.
Normalization leads to eckerb321

$\langle
0|X_{\mu\nu}|X;p,\epsilon\rangle=\frac{i}{m_{X}}[p_{\mu}\epsilon_{\nu}(X)-p_{\nu}\epsilon_{\mu}(X)].$
(31)

$m_{X}$ is the mass of the $X$ meson. The decay width of
$A_{1}\rightarrow\rho\pi$ channel is given by

$\Gamma_{A_{1}\rightarrow\rho\pi}=\frac{q}{8\pi
m_{A_{1}}^{2}}\overline{|\mathcal{M}|^{2}}$ (32)

Where, $q$ is the momentum of the final state particles in the rest frame of
$A_{1}$ meson

$q(m_{A_{1}},m_{\rho},m_{\pi})=\frac{1}{2m_{A_{1}}}\Big{(}[m_{A_{1}}^{2}-(m_{\rho}+m_{\pi})^{2}]\times[m_{A_{1}}^{2}-(m_{\rho}-m_{\pi})^{2}]\Big{)}^{1/2}$
(33)

The interaction terms corresponding to the $A_{1}\rho\pi$ vertices as given by
Lagrangian (30), are

$\mathcal{L}_{A_{1}\rho\pi}=\sqrt{2}i\tilde{f}\Big{[}A_{1}^{0}\left(\rho^{-}\pi^{+}-\rho^{+}\pi^{-}\right)+A_{1}^{+}\left(\rho^{+}\pi^{0}-\rho^{0}\pi^{+}\right)+A_{1}^{-}\left(\rho^{0}\pi^{-}-\rho^{-}\pi^{0}\right)\Big{]}$
(34)

These terms result from the parity and charge conjugation symmetry of the
Lagrangian density. The amplitude obtained from the interaction vertices is
thus given as

$\mathcal{M}=\frac{-2\lambda_{av\phi}}{m_{A_{1}}m_{\rho}}\left(p^{\prime}.p\quad\epsilon^{\prime}.\epsilon-\epsilon^{\prime}.p\quad\epsilon.p^{\prime}\right)$
(35)

With $p^{\prime}$, $\epsilon^{\prime}$ and $p$, $\epsilon$ are the four-
momenta and polarization vectors of the $A_{1}$ and $\rho$ mesons,
respectively, and $\lambda_{av\phi}=\sqrt{2}i\tilde{f}$. Using the
normalization of the fields [eq.(31)], the gauge invariant amplitude is
obtained. The decay width of $A_{1}\rightarrow\rho\pi$ is therefore written as

$\Gamma_{A_{1}\rightarrow\rho\pi}=\frac{|\lambda_{av\phi}|^{2}}{2\pi
m_{A_{1}}^{2}}q\left(1+\frac{2q^{2}}{3m_{\rho}^{2}}\right)$ (36)

To consider the decay of the neutral light vector meson $\rho^{0}$ into two
pseudoscalar mesons $\pi^{+}\pi^{-}$, an effective vector-pseudoscalar
interaction Lagrangian is taken into account with the tree-level coupling
constant $\tilde{g}$ fitted experimentally by the observed decay width of
$\rho\rightarrow\pi^{+}\pi^{-}$. The tree level amplitude leads to the
following decay width klingl2

$\Gamma_{\rho\rightarrow\pi^{+}\pi^{-}}=\frac{\tilde{g}^{2}}{48\pi}m_{\rho}\left(1-\frac{4m_{\pi}^{2}}{m_{\rho}^{2}}\right)^{3/2}$
(37)

##  V Spectral Function and Production Cross-sections

The normalized form of the relativistic Breit-Wigner spectral function for the
light vector and axial-vector mesons are considered as amspec ; elena

$A(M)=\frac{2}{\pi}\frac{M^{2}\Gamma^{*}}{(M^{2}-m^{*2})^{2}-M^{2}\Gamma^{*2}}.$
(38)

Where, $M$ is the invariant mass $m^{*}$ and $\Gamma^{*}$ are the in-medium
mass and decay widths of the corresponding vector and axial-vector meson
states. The relativistic Breit-Wigner cross-section for productions of neutral
$\rho$ meson from the scattering of $\pi^{+}$ and $\pi^{-}$ mesons and $A_{1}$
in the $\rho^{+}\pi^{-}$ and $\rho^{-}\pi^{+}$ channels, in a hot magnetized
nuclear matter is given by

$\sigma(M)=\frac{6\pi^{2}\Gamma^{*}A(M)}{q(m^{*},m_{1},m_{2})^{2}}.$ (39)

Here, $q$ is the momentum of the scattering particles in the center-of-mass
frame of the produced particle amspec . In eq.(39), $m_{1}$, $m_{2}$ are the
masses of the scattering particles. The spectral functions and production
cross-sections of $\rho^{0}$ and $A_{1}^{0}$ mesons are studied in the present
work by incorporating the effects of the magnetized Dirac sea to the in-medium
masses as described in sections (II) and (III). This effect leads to the
phenomena of (inverse) magnetic catalysis at vacuum and at nuclear matter
saturation density $\rho_{0}$, depending on the finite or zero values of the
anomalous magnetic moments of the nucleons. The effects are studied both at
zero and finite temperature nuclear matter in a background magnetic field for
baryon density $\rho_{B}=0$ and $\rho_{0}$. As the density of the medium in a
strong magnetic field in the peripheral ultra-relativistic heavy ion collision
is typically very low. In-medium masses are calculated within the QCD sum rule
approach by incorporating the medium effects through the light quark (up to
the scalar four quark) condensates and the scalar gluon condensates, which are
obtained from the chiral effective model. In-medium decay widths of the
processes $A_{1}^{0}\rightarrow\rho^{\pm}\pi^{\mp}$ and
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ are computed using the appropriate form of
the effective interaction Lagrangian involving the parent and daughter
particles as illustrated in section (IV).

##  VI Results and Discussions

In the present work, the in-medium spectral properties of mass and decay
widths of the light vector mesons $\omega$, $\rho$ and the light axial-vector
meson $A_{1}$ are investigated in the asymmetric nuclear medium at finite
temperature, in presence of an external magnetic field incorporating the
effects of the magnetized Dirac sea. Using the in-medium mass and decay widths
of the mesons, the spectral functions as well as the production cross-sections
of the neutral $\rho$ and $A_{1}$ mesons are studied from their respective
scattering modes (also decay modes) of $\rho\rightarrow\pi^{+}\pi^{-}$ and
$A_{1}\rightarrow\rho^{\pm}\pi^{\mp}$. The masses are computed using the QCD
sum rule (QCDSR) approach in terms of the in-medium values of the light quark
condensates, the scalar gluon condensates, and the parameters of the QCD
Lagrangian, namely the current quark masses $m_{u}$, $m_{d}$, the QCD running
coupling constant $\alpha_{s}$. In-medium values of the QCD condensates are
obtained from that of the scalar isoscalar fields $-$ non-strange $\sigma$,
strange $\zeta$, the scalar isovector field $\delta$ and the scalar dilaton
field $\chi$, within the chiral $SU(3)$ model [as given by eqs.(6)-(10)]. In
the chiral $SU(3)_{L}\times SU(3)_{R}$ model, the meson fields are treated to
be classical and the nucleons as the quantum fields in obtaining the one-loop
self energy functions of the nucleons with the magnetized fermionic propagator
in the weak-field limit. In this way the effects of the Dirac sea at finite
magnetic field are taken into account. The effects of temperature are
incorporated through the Fermi distribution functions in the number and scalar
densities of protons and neutrons in magnetized matter, eqs.(11)-(14). The
coupled equations of motion of the scalar fields are then solved at the given
values of baryon density $\rho_{B}$, isospin asymmetry
$\eta\big{(}=\frac{\rho_{n}-\rho_{p}}{2\rho_{B}}\ \big{)}$, temperature $T$
and magnetic field $|eB|$, through the number ($\rho_{p,n}$) and scalar
densities ($\rho^{s}_{p,n}$) of protons and neutrons. Furthermore, the energy
levels of the charged nucleons, i.e., protons are modified by the magnetic
field, giving rise to the Landau energy levels. There is another effect coming
from the magnetic field due to the anomalous magnetic moments (AMMs) of the
nucleons in both the Fermi and Dirac sea of nucleons. The contribution of an
external magnetic field on the Dirac sea are obtained through summation over
the nucleonic tadpole diagrams due to their interactions with the scalar meson
fields $\sigma$, $\zeta$ and $\delta$ in the chiral effective model
Lagrangian. The fermionic propagator are taken up to second order expansion in
magnetic field $|eB|$ in the calculation of the one loop self energy functions
of the nucleons, along with the non-zero anomalous magnetic moments of the
nucleons. Thus, the Dirac sea contributes to the scalar densities of the
nucleons. In ref.chrmmc , the effects of magnetized Dirac sea have been
studied on the in-medium mass of the lowest lying states of heavy quarkonia at
finite density and zero temperature. In the present investigation, we have
studied the effects of the magnetized Dirac sea on the in-medium spectral
properties of the light vector and axial-vector mesons in asymmetric nuclear
matter at finite temperature.

The light quark condensates ($\langle\bar{q}q\rangle$; $q=u,d$) and the scalar
gluon condensate
$(\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\rangle)$, thus
obtained are used to calculate the in-medium mass and other spectral
parameters of the light axial-vector and vector mesons, by solving their
respective finite energy sum rules. The vacuum masses of the light mesons are
taken as per the particle data group values of $m_{\rho}=770$ MeV,
$m_{\omega}=783$ MeV and $m_{A_{1}}=1230$ MeV, pdg . Using these values the
vacuum finite energy sum rules for $\rho$, $\omega$ and $A_{1}$ are solved to
obtain the respective coefficients $\kappa_{j}$ of the scalar four quark
condensates. The values obtained are: $\kappa_{1}=-2.204$ for $A_{1}^{0}$;
$\kappa_{2}=-8.804$ for $\rho^{\pm}$, $\kappa=7.237$ and $\kappa=7.789$ for
$\rho^{0}$ and $\omega$ mesons respectively. In the QCD sum rule study of the
light vector mesons at strange hadronic matter 91 , the values of
$\kappa_{j}=7.236,7.788,-1.21$ were obtained for $j=\rho^{0},\omega,\phi$
mesons respectively, by solving their corresponding vacuum FESRs. For the case
of $A_{1}$ meson there is an extra pion pole contribution in its spectral
density function eq.(27). The pion decay constant is $f_{\pi}=93.3$ MeV and
the pion mass $m_{\pi}=139$ MeV, remain fixed as in the context of chiral
effective model Lagrangian papa . The vacuum values of the perturbative
continuum threshold $s_{0}$ and the resonance strength $F$ (both in $GeV^{2}$)
are thus found to be $1.266$ and $2.114$ (for $\rho^{0}$ state), $2.268$ and
$2.755$ (for $A_{1}^{0}$ state) and $1.3046$ and $0.2419$ (for $\omega$
state), respectively. The nuclear matter saturation density of $\rho_{0}=0.15\
fm^{-3}$ is used in the present study.

Figure 1:  In-medium masses of the $\omega$ and $\rho^{0}$ mesons are plotted
as functions of $|eB|/m_{\pi}^{2}$ at $\rho_{0}$ and $\eta=0$ for $T=0,\ 50,\
100$ MeV. The effects of the magnetized Dirac sea are shown on the masses.
Effects of the nucleons’ AMMs are considered and compared to the case with
zero anomalous magnetic moment. Figure 2:  In-medium masses of the $\rho$ and
$A_{1}$ mesons are plotted as functions of temperature T (in MeV) for
$|eB|/m_{\pi}^{2}=2,\ 9$ at $\rho_{0}$ and $\eta=0$. The effects of the
magnetized Dirac sea are incorporated with non-zero nucleons’ AMMs.

Fig.1 illustrates the in-medium masses of the (a) $\omega$ and (b) $\rho$
mesons as functions of magnetic field $|eB|$ (in units of $m_{\pi}^{2}$), in
the symmetric nuclear matter ($\eta=0$) at $\rho_{0}$ for zero as well as
finite temperature $T=0,\ 50,\ 100$ MeV. The anomalous magnetic moments of the
nucleons are considered in the magnetized Dirac sea effects and comparison is
shown to the case when the AMMs are taken to be zero. The behavior obtained
can be explained from the underlying trend of variation of the light quark and
the scalar gluon condensates with changing magnetic field at different medium
conditions. At $\rho_{0}$, for zero as well as finite temperatures the values
of the light quark condensates, the scalar gluon condensate decrease with
increasing magnetic field for non zero AMMs of the nucleons, an effect called
inverse magnetic catalysis. The opposite behavior is obtained when nucleonic
AMM is taken to be zero, lead to magnetic catalysis. This is observed for both
symmetric and asymmetric nuclear matter, for e.g., for pure neutron rich
matter with $\eta=0.5$. At $\rho_{B}=0$, there is no additional contribution
from the protons Landau energy levels, any effect due to the magnetic field is
obtained through the magnetized Dirac sea contribution only. At zero density,
the light quark condensates increase with magnetic field for both finite and
zero anomalous magnetic moments of the Dirac sea of nucleons, leading to the
effect known as magnetic catalysis. There is observed to be considerable
amount of difference in the behavior of the light quark condensates for finite
values of the AMMs of the Dirac sea of nucleons as compared to the zero AMM
case.

In fig.2, the mass of (a) $\rho$ and (b) $A_{1}$ mesons are plotted as
functions of temperature (in MeV) at the nuclear matter saturation density
$\rho_{0}$, $\eta=0$. The masses of both $\rho$, $A_{1}$ mesons tend to rise
with temperature by very small amount at the magnetic field strength of
$2m_{\pi}^{2}$. However, at the field strength of $9m_{\pi}^{2}$ (plotted as
long dashed line), the masses initially increase with temperature with a
sudden drop after 100 MeV till 150 MeV. The initial rise in mass with
temperature may be due to the contribution from the higher momenta region in
the three-momentum integral of $\rho_{p,n}$ and $\rho^{s}_{p,n}$, within the
context of chiral $SU(3)$ model at finite density and temperature. However, at
high enough temperatures the condensates tend to decrease as it approach to
the critical temperature of chiral symmetry restoration. In fig.2 the effects
of inverse magnetic catalysis on the masses are observed at finite temperature
and density, as it can be inferred from the fact that the overall mass
decreases from $2m_{\pi}^{2}$ to $9m_{\pi}^{2}$, at any fixed temperature. At
high magnetic fields, the condensates decrease at high temperature region
towards the chiral symmetry restoration, an effect induced by the phenomenon
of inverse magnetic catalysis in hot magnetized nuclear matter.

In the present work, we have investigated the effects of the magnetized Dirac
sea at finite (but low) density and temperature (below critical temperature of
chiral transition) medium over a range of magnetic field from $|eB|=0$ to
$9m_{\pi}^{2}$. The variation is shown in figs.1-2 until the behavior of the
condensates starts fluctuating with respect to the change in magnetic field.
As we have applied a weak-field expansion of the nucleonic propagator up to
the second order in magnetic field, it gives reliable results till
$|eB|=3.9m_{\pi}^{2}$ at $\rho_{B}=0$, and $|eB|=9m_{\pi}^{2}$ at
$\rho_{B}=\rho_{0}$, $\eta=0,0.5$ for non-vanishing AMMs of the Dirac sea of
nucleons at $T=0$ to 150 MeV. Now, this range gives considerably smooth
variation of the condensates with magnetic field, leading to the phenomena of
magnetic (inverse) catalysis at the specific conditions of baryon density and
AMMs of the Dirac sea of nucleons. Considerably the region of valid field
strength is consistent with that of the hadronic freeze-out hypersurface
(FOHS), where the light mesons e.g., pseudoscalar $\pi$, vector $\rho$ mesons
etc., are profusely produced. At FOHS, the estimated magnetic field is as low
as $4m_{\pi}^{2}$ fohs , thus the weak-field expansion is well applicable in
this regime. In ref.fohs , it has been studied that the impacts of magnetic
field on the charged light mesons e.g., $\pi^{\pm}$, $\rho^{\pm}$, should have
significant observable consequences at the freeze-out hypersurface. The amount
of particle production and particle ratio at FOHS can be used to estimate the
field strength, lifetime (still a major open question to search for) of the
strong and transient magnetic field which is expected to be produced at the
very early stages of the non-central heavy ion collision experiments at RHIC,
LHC. Thus, the evolution and magnitude of the produced magnetic field at FOHS
can be estimated from the ratio of the production number of charged particles
(e.g., $\rho^{\pm}/\pi^{\pm}$). As their production are strongly dependent on
their masses having strong magnetic field dependencies, this may lead to their
modified yields at freeze-out. The study of magnetic field effects of
(inverse) magnetic catalysis on the light vector and axial-vector mesons,
should have considerable amount of observable impacts, e.g., in the production
of different light mesons species at the stage of freeze-out in the non-
central heavy ion collision experiments. However, at finite density and finite
temperature medium, the validity range is somewhat extended covering the
region at the freeze-out hypersurface.

Hadron mass e.g., that of the mesons is, in general, not a directly
experimentally observable quantity. However, from the in-medium mass and decay
widths the spectral function can be constructed. In general, the spectral
function then receives the peak shift, width broadening, appearance of new
peaks etc., due to the various in-medium effects in the relativistic heavy ion
collisions. The in-medium masses of the $\omega$, $\rho$ and $A_{1}$ mesons at
different values of temperature and magnetic fields, as shown in figs.1-2,
give an impact of the effects of magnetic (inverse) catalysis on the in-medium
properties of light hadrons. The masses are calculated using the QCD sum rule
approach as described in section III, in terms of the in-medium light quark
condensates (up to scalar four quark) and the scalar gluon condensate. The
light quark condensates increase (decrease) with magnetic field due to the
phenomena of magnetic (inverse) catalysis, which is observed to be an
important effect in the present analysis of light meson properties in a
background magnetic field.

In the chiral $SU(3)$ model the critical temperature of chiral phase
transition or a crossover is obtained around $T_{c}=150$ MeV. This phase
transition is of purely hadronic nature as the model involves only hadronic
degrees of freedom. In ref.borsanyi , the transition temperature is found to
be around $T_{c}=155$ MeV from the inflection point of chiral condensate using
the lattice QCD simulation at zero baryon chemical potential $\mu_{B}=0$. From
an estimate of the chiral condensate at the pseudo-critical temperature it has
been observed that its value is about half of that in vacuum, which is also
close to the one estimated at $2\rho_{0}$ in the cold nuclear matter holt . In
the presence of a magnetic field background at $\mu_{B}=0$, the magnetic field
variation of the chiral condensates can be obtained from the first principle
calculations of lattice QCD simulations. The lattice results indicate that the
critical temperature increases monotonically with increasing magnetic field.
The enhancement of the QCD light quark condensates with the magnetic field is
named as magnetic catalysis (MC), while its opposite behavior is known as
inverse magnetic catalysis (IMC). It has been illustrated that the effect of
IMC is attributed to the dominance of the sea contribution over the valence
term of the quark condensate, e.g., in the context of Nambu-Jona-Lasinio (NJL)
model, quark-meson models, etc. The values of the light quark condensates tend
to decrease with increasing temperature and/or density of the surrounding
medium. Consequently, the enhancement in the critical temperature corresponds
to the increment in the quark condensate. However, some lattice results show
IMC at specific medium conditions, correspondingly the critical temperature
shows opposite behavior compared to the case in magnetic catalysis. In our
present study, at zero baryon density $\rho_{B}=0$, effects of the magnetized
Dirac sea lead to the phenomenon of magnetic catalysis for both non-zero AMM
(till $3.9m_{\pi}^{2}$) and zero AMM of the Dirac sea of nucleons below
critical temperature. At $\rho_{B}=\rho_{0}$, inverse magnetic catalysis is
observed for non-vanishing nucleonic AMMs, which turn out to be magnetic
catalysis for zero AMM, both at zero and finite temperatures. Notably, the
current study concerns with the behavior of the light vector and axial-vector
mesons in a magnetized hot nuclear matter, accounting for the effects of
(inverse) magnetic catalysis [coming from magnetized Dirac sea] below the
pseudo-critical temperature of chiral phase transition.

The in-medium partial decay width of $A_{1}\rightarrow\rho\pi$ process is
calculated from eq.(36), by using the experimentally fitted parameters in the
Lagrangian, masses of the initial and final states particles. The mass of the
neutral and charged pions to be used here are $134.97$ MeV and $139.57$ MeV
respectively pdg . The vacuum decay widths for
$A_{1}^{0}\rightarrow\rho^{\pm}\pi^{\mp}$ channels are calculated to be
$202.58$ MeV for $\tilde{f}=1540$ MeV, obtained from experimentally fitted
values of the relevant decay modes as mentioned in table.(4) of ref.roca70 .
The value is very close to the central average value of $210$ MeV
(corresponding to the $50\%$ branching ratio) for $A_{1}\rightarrow\rho\pi$
channel. The phenomenological interaction Lagrangian contains no term
involving $A_{1}^{0}\rho^{0}\pi^{0}$ vertex, thus there is no decay for
$A_{1}^{0}\rightarrow\rho^{0}\pi^{0}$ mode. The authors of CLEO Collaboration
tao  have been reported a branching ratio of about $60\%$ for the intermediate
$\rho\pi$ ($S$-wave) state of $A_{1}^{-}$ meson decaying into
$\pi^{-}\pi^{0}\pi^{0}$ relative to the total
$A_{1}^{-}\rightarrow\pi^{-}\pi^{0}\pi^{0}$ decay. The works of chen91  showed
$\rho\pi$ ($S$-wave) to be the dominant decay channel for the ground state of
$A_{1}$ meson in its class of axial-vector mesons with quantum numbers
$(I^{G}(J^{PC})=1^{-}(1^{++}))$. They have analyzed the mass spectra of all
the possible light axial-vector meson states by Regge trajectory analysis and
have applied a light quark pair creation model micu  to calculate their OZI-
allowed strong decays.

(a)  (b)

Figure 3:  (a) Spectral function $A(M)$ (in Ge$V^{-1}$), (b) production cross-
section $\sigma$ (in mb) of the neutral $A_{1}$ meson are plotted as functions
of the invariant mass $M$ (in GeV) at $\rho_{B}=\rho_{0}$, in the symmetric
nuclear matter ($\eta=0$), for $|eB|/m_{\pi}^{2}=2,6,8$ at $T=50$ MeV. The
vacuum case with $\rho_{B}=0$, $T=0$ also $|eB|=0$ is shown for comparison as
the dashed line.

In fig.3, (a) the Breit-Wigner spectral functions $A(M)$ (in units of
$GeV^{-1}$), and (b) the production cross-sections $\sigma$ (in units of $mb$)
of $A_{1}^{0}$ are plotted as functions of the invariant mass $M$ (in units of
GeV) in the symmetric nuclear matter at $\rho_{0}$ for $T=50$ MeV and various
magnetic field strengths. The production cross-section of $A_{1}^{0}$ due to
the scattering (and decay to) $\rho^{\pm}\pi^{\mp}$ channels are considered
here. The hadronic decay widths of $A_{1}^{0}\rightarrow\rho^{\pm}\pi^{\pm}$
are computed using a phenomenological effective interaction vertex of
$A_{1}\rho\pi$ as illustrated in Sec. IV. Effects of the magnetized Dirac sea
along with the Landau energy levels contributions of protons and anomalous
magnetic moments of the nucleons in hot and dense matter, are incorporated in
the solutions of the scalar fields $\sigma$, $\zeta$, $\delta$ and $\chi$
within the framework of chiral effective model. The solutions are obtained at
finite temperature (well below the critical temperature of chiral phase
transition) and nuclear matter saturation density as well as zero baryon
density in strong magnetic fields, which occurs at low density matter. As the
$A_{1}$ mass changes very slightly with magnetic field as compared to the
light vector meson $\rho$, there is almost no shift in the peak position of
the spectral function at $\rho_{0}$ ($\eta=0$) with magnetic field. However,
there is distinct difference in the peak position with change in baryon
density $\rho_{B}$ from 0 to $\rho_{0}$. As mass decreases with density the
peak shifted towards the low invariant mass region on the left in plot (a).
In-medium decay widths increase with density, and also separately with
magnetic field (for finite AMMs) at $\rho_{0}$. This is due to the fact that
for non-zero anomalous magnetic moments of the nucleons, Dirac sea
contribution in the hot magnetized matter leads to the effect of inverse
magnetic catalysis at $\rho_{0}$ and $T=0,\ 50,\ 100,150$ MeV in symmetric as
well as asymmetric nuclear matter. As a consequence, the light quark
condensates, the scalar gluon condensate and hence the masses decrease with
magnetic field in the QCDSR approach. Therefore, from eq. (36), in-medium
partial decay widths of $A_{1}^{0}\rightarrow\rho\pi$ increase with magnetic
field. In our study the masses of the charged $\rho^{\pm}$ mesons are taken to
be equal and the difference from the neutral partner $\rho^{0}$ is obtained
from the scalar four quark condensate in the $c_{3}$ coefficient of the OPE
side of QCDSR. In ref. k1 , the operators with non-zero spin dimension are
also considered and due to the charge symmetry breaking effect, odd-
dimensional operators in the OPE side lead to the non-degenerate mass of
$K_{1}^{+}$ and $K_{1}^{-}$. The twist-2 operators are not considered here and
the masses of the oppositely charged states are taken to be degenerate. An
important thing to note here and in the subsequent plots is that the Dirac sea
contribution in the magnetized matter is accounted for with non-zero anomalous
magnetic moments of the nucleons. Fig.4 represents the plots of the (a)
spectral functions and (b) productions cross-sections of $A_{1}^{0}$ at
different temperatures $T=0,\ 50,\ 150$ MeV for a particular magnetic field of
$2m_{\pi}^{2}$ in the symmetric nuclear matter at $\rho_{0}$. The vacuum case
with $\rho_{B}=0$, $T=0$ and $|eB|=0$ is shown in the plots for comparison. It
is observed that the production cross-section is increasing slightly with
temperature as the decay width of $A_{1}\rightarrow\rho\pi$ process decreases
and there is very small increment in the mass of the initial state particle of
$A_{1}^{0}$ with $T$, at a fixed value of density $\rho_{0}$, and magnetic
field strength. Although this amount of change in the spectral properties of
$A_{1}$ meson is very less as compared to the change observed with respect to
the variation in density and magnetic field including the Dirac sea effects.

(a)  (b)

Figure 4:  (a) Spectral function $A(M)$ (in Ge$V^{-1}$), (b) production cross-
section $\sigma$ (in mb) of the neutral $A_{1}$ meson are plotted as functions
of the invariant mass $M$ (in GeV) at $\rho_{B}=\rho_{0}$, in the symmetric
nuclear matter ($\eta=0$), for $|eB|=2m_{\pi}^{2}$ at $T=0,\ 50$ and 150 MeV.
The vacuum case with $\rho_{B}=0$, $T=0$ also $|eB|=0$ is shown for comparison
as the dashed line. Effects of the magnetized Dirac sea with non-zero
nucleonic AMMs are incorporated here.

(a)  (b)

Figure 5:  (a) Spectral function (b) production cross-section of the
$A_{1}^{0}$ meson at $\rho_{0}$, in the asymmetric nuclear matter $\eta=0.5$,
for $|eB|=2m_{\pi}^{2}$ at $T=0,\ 50,\ 100$ and 150 MeV. Effects of the
magnetized Dirac sea is incorporated here for finite anomalous magnetic
moments.

(a)  (b)

Figure 6:  (a) Spectral function $A(M)$ (in Ge$V^{-1}$), (b) production cross-
section $\sigma$ (in mb) of $A_{1}^{0}$ meson are plotted as functions of the
invariant mass $M$ (in GeV) at $\rho_{B}=0$, $T=50$ MeV for
$|eB|/m_{\pi}^{2}=2,\ 3,\ 3.5,\ 3.9$. The vacuum case with $\rho_{B}=0$, $T=0$
also $|eB|=0$ is shown for comparison as the dashed line. Effects of the
magnetized Dirac sea with non-zero nucleonic AMMs are incorporated here.

(a)  (b)

Figure 7:  (a) Spectral function $A(M)$ (in Ge$V^{-1}$), (b) production cross-
section $\sigma$ (in mb) of the neutral $\rho$ meson are plotted as functions
of the invariant mass $M$ (in GeV) at $\rho_{0}$, $\eta=0$, and $T=50$ MeV for
$|eB|/m_{\pi}^{2}=2,\ 8,\ 9$. The vacuum case with $\rho_{B}=0$, $T=0$ also
$|eB|=0$ is shown for comparison as the solid line. Effects of the magnetized
Dirac sea with non-zero nucleonic AMMs are incorporated here.

Fig.5 shows the similar plots as fig.4 for pure neutron rich matter with
isospin asymmetry parameter $\eta=0.5$ at $\rho_{0}$. The production cross-
section is observed to increase very slightly with temperature at fixed
magnetic field strength of $2m_{\pi}^{2}$, accounting for the effects of the
magnetized Dirac sea with finite AMMs.

In fig.6, similar plots of $A_{1}^{0}$ meson are shown at $\rho_{B}=0$, $T=50$
MeV, and for different values of magnetic field $|eB|/m_{\pi}^{2}=2,\ 3,\ 3.5$
and 3.9. The mass of $A_{1}^{0}$ meson rise with magnetic field at zero
density and at any fixed temperature, due to the effects of magnetic catalysis
for both non-zero and zero AMMs of the Dirac sea of nucleons. Hence the decay
width in the $A_{1}\rightarrow\rho\pi$ channel decreases with $|eB|$ at a
fixed temperature. This leads to an increment in $\sigma$ with magnetic field
at $\rho_{B}=0$ and $T=50$ MeV. Thus, in the magnetized vacuum when there
remains only the contribution of the Dirac sea, the production of $A_{1}^{0}$
meson is found to be increased due to the important phenomena of magnetic
catalysis for both zero and non-zero nucleonic AMMs and at any temperature. In
fig.7, (a) the in-medium Breit Wigner spectral function $A(M)$, and (b) the
production cross-section $\sigma$ are plotted as functions of the invariant
mass $M$ for the neutral light vector meson $\rho$, at $\rho_{B}=\rho_{0}$,
$\eta=0,$ and $T=50$ MeV for $|eB|/m_{\pi}^{2}=2,\ 8,\ 9$. The pure vacuum
case in solid line is shown to emphasize on the in-medium spectral properties
of $\rho^{0}$ meson due to the effects of inverse magnetic catalysis in hot
and dense matter. The contributions of non-zero anomalous magnetic moments of
the Dirac sea of nucleons are taken into account. Similar behavior are shown
in fig.8 for the asymmetric magnetized nuclear matter at $\eta=0.5$. The
production of $\rho^{0}$ mesons from the scattering of $\pi^{+}\pi^{-}$ mesons
and also its decay to $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ channel is
considered in the calculation of Breit Wigner spectral function from eq.(38)
and cross-section from eq.(39). The mass of $\rho$ meson decreases with
magnetic field at the nuclear matter saturation density, for both $\eta=0,\
0.5$, and finite AMMs of the nucleons accounting for the effects of magnetized
Dirac sea, as it is obvious from plot (b) in fig.(1). The contributions of
Fermi distribution functions in the number and scalar densities of nucleons
incorporate the effects of temperature in the current study. The decreasing
nature in mass with $|eB|$ for non-zero AMMs of the nucleons, is obtained for
different values of temperatures $T=0,\ 50,\ 100$ and 150 MeV. From eq.(37)
the in-medium decay width of $\rho\rightarrow\pi^{+}\pi^{-}$ process is
calculated using an effective Lagrangian of $\rho\pi$ interaction with the
experimentally fitted coupling parameter $\tilde{g}=6.05$ klingl2 . The
modified decay width is obtained from the in-medium mass of the parent
particle $\rho$ meson with no medium changes in $\pi$ mesons masses in our
present study, as stated before. The decay width is seen to decrease with
magnetic field at the given condition from eq.(37). Hence, the production
cross-section of $\rho\rightarrow\pi^{+}\pi^{-}$ process increases with $|eB|$
due to the phenomena of inverse magnetic catalysis at $\rho_{0}$ and for
finite AMMs of the nucleons. However, unlike the case of $A_{1}$ meson, the
mass of $\rho$ changes considerably with magnetic field in this process and
the shift in the peak position is significantly visible towards the low
invariant mass region of $\rho$ meson. In ref.chernodub , the minimal
effective mass squared of the free charged, spin-0 pions $\pi^{\pm}$ received
a positive contribution due to the lowest Landau energy level contribution in
an external magnetic field background, which is also known as point-article
correction. It has been used to study the onset of $\pi^{\pm}$ stability
regions leading to the longer lifetime of $\rho$ mesons by closing the main
strong decay channels of $\rho\rightarrow\pi^{+}\pi^{-}$.

(a)  (b)

Figure 8:  (a) Spectral function (b) production cross-section of $\rho^{0}$
meson are plotted as functions of the invariant mass $M$ (in GeV) at
$\rho_{0}$, $\eta=0.5$, and $T=50$ MeV for $|eB|/m_{\pi}^{2}=2,\ 8,\ 9$. Rest
of the description is similar as in fig.7.

(a)  (b)

Figure 9:  (a) Spectral function (b) production cross-section of $\rho^{0}$
meson are plotted as functions of the invariant mass $M$ (in GeV) at
$\rho_{0}$, $\eta=0$, and $T=0,\ 50,\ 100,\ 150$ MeV for $|eB|=2m_{\pi}^{2}$.
Effect of magnetized Dirac sea is incorporated with finite nucleonic AMMs.

(a)  (b)

Figure 10: Similar as in fig.9 for asymmetric nuclear matter at $\eta=0.5$.

(a)  (b)

Figure 11:  (a) Spectral function (b) production cross-section of $\rho^{0}$
meson are plotted as functions of the invariant mass $M$ (in GeV) at
$\rho_{0}$, $\eta=0$, and $T=0,\ 50,\ 100,\ 150$ MeV for $|eB|=9m_{\pi}^{2}$.
Effects of inverse magnetic catalysis is incorporated.

(a)  (b)

Figure 12: (a) Spectral function (b) production cross-section of $\rho^{0}$
meson are plotted as functions of the invariant mass $M$ (in GeV) at
$\rho_{B}=0$, and $T=50$ MeV for $|eB|/m_{\pi}^{2}=$ $2,\ 3,\ 3.5$, and 3.9.
The pure vacuum case is shown for comparison as the dotted line. Effects of
the magnetized Dirac sea are incorporated with finite nucleonic AMMs.

(a)  (b)

Figure 13: (a) Spectral function (b) production cross-section of $\rho^{0}$
meson are plotted as functions of the invariant mass $M$ (in GeV) at
$\rho_{0}$, $\eta=0$ and $T=0$ MeV. The effects of the magnetized Dirac sea
for the case of finite anomalous magnetic moments and zero AMM are shown for
comparison for $|eB|=2m_{\pi}^{2}$ and $8m_{\pi}^{2}$. Effects of the
magnetized Dirac sea are incorporated with finite nucleonic AMMs.

(a)  (b)

Figure 14: Same as in fig.13 for the asymmetric nuclear matter $\eta=0.5$.

Figs. 9 and 10 exhibit the plots of (a) $A(M)$ and (b) $\sigma$ of $\rho^{0}$
meson with variation in $M$ at the given values of density $\rho_{0}$,
magnetic field $|eB|=2m_{\pi}^{2}$, for four temperatures $T=0,\ 50,\ 100,\
150$ MeV in the symmetric ($\eta=0$) and asymmetric ($\eta=0.5$) nuclear
matter, respectively. The effects of the magnetized Dirac sea with finite AMMs
are incorporated in this behavior. At the given conditions, in-medium mass of
$\rho$ meson increases very slightly with temperature at lower magnitude of
magnetic fields e.g., at $2m_{\pi}^{2}$, leading to the increasing decay width
of $\rho\rightarrow\pi^{+}\pi^{-}$ channel by eq.(37). As a result, the
production cross-section of the neutral $\rho$ meson decreases slowly with
temperature in plot (b) at the nuclear matter saturation density. The peak
position of the spectral function also shifts gradually towards the high
invariant mass region with temperature in plot (a).

In fig.11, similar plots as fig.9 are shown at the magnetic field strength of
$9m_{\pi}^{2}$. Here the peak position of the spectral function in plot (a)
shifts slightly towards left at high enough temperature of $T=100$ to 150 MeV,
and the production cross-section also increases slightly due to the similar
variation of temperature. This behavior is obtained because of the lowering of
light quark condensates at high temperature and high magnetic field,
accounting for the effects of inverse magnetic catalysis.

Fig.12 contains the spectral behavior of $\rho^{0}$ meson similar to the
previous plots at zero baryon density $\rho_{B}=0$, $T=0$, for
$|eB|/m_{\pi}^{2}=2,\ 3,\ 3.5,\ 3.9$. The phenomena of magnetic catalysis at
$\rho_{B}=0$, for both finite and vanishing AMMs of nucleons, mass of $\rho$
meson increases with magnetic field accompanied by increased decay widths of
$\rho\rightarrow\pi^{+}\pi^{-}$ process. Hence, the production cross section
$\sigma$ decreases with magnetic field in plot (b), and the peak of the
spectral function in (a) is shifted towards the high $M$ region on the right.

In figs. 13 and 14, the effect of the magnetized Dirac sea with finite
nucleonic AMMs, is compared to the case when nucleons’ AMMs are taken to be
zero at the nuclear matter saturation density $\rho_{0}$, at zero temperature,
for symmetric ($\eta=0$) and asymmetric ($\eta=0.5$) nuclear matter,
respectively. For non-zero AMMs, the Dirac sea contribution in the magnetized
nuclear matter leads to the phenomena of inverse magnetic catalysis and the
opposite effect of magnetic catalysis is obtained when the AMM is taken to be
zero. Therefore, the mass of $\rho$ meson is seen to be increasing
(decreasing) with magnetic field due to the phenomena of magnetic (inverse)
catalysis at $\rho_{0}$ for both the conditions of $\eta=0$ and 0.5. In plot
(a) of figs. 13-14, the peak position of the spectral function is therefore
shifted towards the low $M$ region (on the left) for the variation of magnetic
field strength from $|eB|=2$ to $8m_{\pi}^{2}$ for finite anomalous magnetic
moments of the nucleons, which then shifts towards the high $M$ region (on the
right) for the case of without AMMs of nucleons. The variation of $\rho$ meson
mass in this way leads to the increasing (decreasing) decay width of
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ process due to the effects of magnetic
(inverse) catalysis at $\rho_{0}$ without (with) anomalous magnetic moments of
the nucleons in the magnetized Dirac sea contribution. As a consequence, the
production cross-section of $\rho$ meson decreases (increases) with magnetic
field in plot (b), corresponding to the case of zero (non-zero) anomalous
magnetic moments of nucleons.

The vacuum spectral functions of the vector and axial-vector meson states are
related to some physical processes e.g., scattering. The vectorial spectral
function can be obtained from the $e^{+}e^{-}$ annihilation into an even
number of pions. While the axial-vector one can be extracted from data on
$\tau$ decay into a $\nu_{\tau}$ and an odd-number of pions. Their vacuum
spectral functions are different which gives one of the experimental
signatures that the chiral symmetry is spontaneously broken. The presence of
pions in the thermal heat bath can lead to the mixing between the vacuum
vector and axial-vector spectral functions via a model independent ”mixing
theorem”. To the lowest order in temperature there is obtained no change in
the spectral shape itself. Only a temperature dependent coupling between the
vector and axial-vector correlators can be obtained. However, as an
experimental signal it is instructive to observe the consequences on the
dilepton spectra or, the $l^{+}l^{-}$ invariant mass spectra. Dileptons or,
$l^{+}l^{-};\ l=e,\mu$ pairs are provided to be an important tool to measure
the medium modifications of the vector mesons e.g., $\rho,\omega,\phi$ in
nuclear media, as their final-state interactions are very negligible. Both the
mixed vector and axial-vector correlator are indistinguishable down to 1 GeV,
which is lower than its vacuum position at around 1.5 GeV. Hence the effects
of finite temperature lead to the lowering of the duality threshold from its
vacuum value of 1.5 GeV down to 1 GeV in medium. Therefore, the direct
signature of chiral restoration is the degeneracy between vector and axial-
vector spectral functions. As the vector current directly couples to the
virtual photons which further decay into $l^{+}l^{-}$, the vector spectral
function is directly related to the physically measurable observable. The
broadening or melting of the resonances in the medium that flattens the $\rho$
meson resonance structure and generates low-mass dilepton enhancement below
rho mass. The spontaneous breaking of chiral symmetry (SBCS) is inherently a
soft phenomena occurring at scales of less than or around 1 GeV, for this the
low mass dilepton spectra is considered to be one of the most promising
observables to detect changes in the chiral properties. E.g., change in the
condensates at finite temperature and/or density, also with magnetic field in
the context of non-central heavy ion collisions. At the hadronic level, SBCS
is related to the splitting of opposite parity states in chiral multiples
e.g., $\pi-\sigma$, $\rho-A_{1}$ etc. Thus, to extract the signatures of (or,
the approach to) chiral symmetry restoration from dilepton spectra dominated
by the $\rho$ channel, one is lead to the theoretical study of the in-medium
spectral properties of $A_{1}$ meson as well. Phenomenologically it is also
rather well established that the low mass dilepton enhancement is a common
feature observed in every experiments, from the low energy SIS experiments
with center-of-mass (square-root) $\sqrt{s}=2.25$ GeV sis  via SPS
($\sqrt{s}=8.8,17.3$ GeV) sps , which is again obtained via the RHIC beam
energy scan ranging up to its maximum energy ($\sqrt{s}=19.6,27,39,62,200$
GeV) rhic . It shows a broad $\rho$ meson spectral function due to the thermal
radiation mostly of hadronic origin. Thus, utilizing the $\rho$ meson spectral
function which has direct connection to the observed dilepton spectra, the
spectral function of its chiral partner $A_{1}$ can be evaluated using the in-
medium values of the condensates from a chiral effective model and using the
constraints of QCD Sum rules. It is almost impossible to measure $A_{1}$
spectral function via dilepton emissions, as the neutral $A_{1}$ current can
only couple to dileptons via weak interaction through $Z^{0}$ boson hayano .
However, one point to be noted here is that the chirally symmetric condensates
do not need to be vanished at the critical point of the chiral phase
transition. Hence, it is not straightforward to relate the spectral
modifications of vector mesons to the chiral symmetry restoration.

The amount of shift in the spectral peak position of $\rho^{0}$ meson, which
has the direct connection to the dilepton pairs are determined in the present
work incorporating the effects of the magnetized Dirac sea at finite
temperature and density matter in a background magnetic field. The amount of
shift in the peak position of the invariant mass spectra of $\rho^{0}$ meson
spectral function $A(M)$ (in Ge$V^{-1}$) from its vacuum position of $M=0.77$
GeV is given in the symmetric nuclear matter $\eta=0$, at $\rho_{0}$ and
$T=50$ MeV, for $|eB|=2,8,9m_{\pi}^{2}$ respectively as (in GeV) -0.138,
-0.17, -0.208, respectively, which are towards the low-invariant mass region
on the left side as shown in fig.7 (a). Now the shifts with varying
temperature at a fixed magnetic field strength of $2m_{\pi}^{2}$ and
$T=50,100,150$ MeV are (in GeV) 0.138, 0.129, 0.122, respectively, as shown in
fig.9 (a) (towards higher $M$ region from its vacuum position). The similar
shifts obtained for $9m_{\pi}^{2}$ as 0.208, 0.178, 0.187 GeV at
$T=50,100,150$ MeV, respectively, as in fig.11 (a). The spectral changes in
the vacuum are obtained by incorporating the effects of the magnetized Dirac
sea as an external effect. At $\rho_{B}=0$, $T=50$ MeV, the peak position is
shifted towards right from its vacuum position by an amount of 0.012, 0.033,
and 0.094 GeV for $|eB|=2,3,3.9m_{\pi}^{2}$, respectively, in fig.12 (a), as a
consequence of the phenomena of magnetic catalysis originated due to the
magnetized Dirac sea of nucleons. The data of spectral peak shifts given above
are obtained while taking into account the finite anomalous magnetic moments
of the nucleons at $\rho_{B}=0,\rho_{0}$, for various temperatures and
magnetic field strengths. The lowering of the peak position with increasing
magnetic field at $\rho_{0}$ ($\eta=0$) is due to the effect of inverse
magnetic catalysis, while that at vacuum due to catalysis, in both cases AMMs
are non-zero. The effects of temperature is somewhat noticeable at higher
magnetic field strength e.g., at $9m_{\pi}^{2}$ for finite nucleonic AMMs. The
shifting happen in the opposite direction as one goes from $T=100$ to 150 MeV
with $|eB|$ changing from 2 to $9m_{\pi}^{2}$. At higher $eB$, the effects of
inverse magnetic catalysis lead the peak position to shift towards the low-
mass region with increasing temperature, i.e., an apparent tendency towards
the chiral symmetry restoration.

In the peripheral ultra-relativistic heavy ion collision experiments e.g., at
RHIC with $\sqrt{s}=200$ GeV in Au+Au collisions and at LHC with
$\sqrt{s}=2.76$ TeV in Pb+Pb collisions, the early stage time evolution of the
produced electromagnetic fields (at the origin) have been studied for the
central as well as non-central collisions deng . The contributions of the
electrically charged participants, spectators, and remnant particles are taken
into account. At a later stage of the collisions, e.g., at around $1$ fm/c,
the contributions are expected to come predominantly from the remnants as they
move at a slower rate than the spectators from the collision zone. Studies
have been discussed that the remnants may lead to the slowing down of the
decay process of transverse electromagnetic (EM) fields. Although there might
have been enhanced longitudinal components of EM fields due to the
fluctuations of the remnants from the equilibrium position. For central
collisions, any non-vanishing field is generated by the position fluctuations
of the charged particles, which cause a sizable amount of fields in the
transverse directions at the initial time which decay rapidly (to a large
extent for higher center-of-mass energies). On the contrary, the off-central
collisions lead to a dominant field component perpendicular to the reaction
plane coming due to the motion of the spectators. However, the dominance of
remnants over the spectators at a later time gives rise to the equal
contribution to the magnitude of the transverse components of the magnetic
field. From an estimate of the time-evolution of EM fields before and after
collisions at the center-of-mass energy of RHIC, there can be obtained very
small but a sizable component of the transverse fields at or around 1 fm/c
deng . Therefore, as studied in our present work the effects of the magnetic
field on the mesons in a magnetized matter are coming out to be significant
when incorporating the effects of the magnetized Dirac sea into the
computation of mesons spectral properties. For the $A_{1}$ meson there is
obtained a noticeable amount of modifications in the production cross
sections, whereas there is obtained distinct peak shifts of $\rho$ meson
spectral function with changing magnetic field strength at various other
medium conditions as described in this section.

##  VII Summary

In the summary, we have investigated the in-medium spectral properties of the
light vector $\rho$, $\omega$ and the light axial-vector $A_{1}$ mesons in the
isospin asymmetric magnetized nuclear matter, accounting for the effects of
(inverse) magnetic catalysis at zero and finite temperatures. The in-medium
masses of $\rho$, $\omega$ and $A_{1}$ mesons are calculated within the QCD
sum rule framework, using the Borel transform followed by the finite energy
sum rules. The values of the other spectral parameters $F,\ s_{0}$, are
modified by the medium effects. Masses are obtained in terms of the in-medium
values of the light quark (up to the scalar four-quark condensate) and the
scalar gluon condensates. The condensates are further obtained within the
chiral effective model in terms of the scalar isoscalar fields non-strange
$\sigma$, strange $\zeta$, scalar isovector $\delta$ and the scalar dilaton
field $\chi$, at the given values of density, isospin asymmetry, magnetic
field and temperature. The effects of the magnetized Dirac sea is incorporated
through one-loop self energy functions of the nucleons due to their
interactions with the scalar mesons $\sigma$, $\zeta$, and $\delta$ within the
chiral $SU(3)$ model framework. However, the effects of finite anomalous
magnetic moments of the Dirac sea of nucleons are incorporated to the self
energy part by computing the weak field expansion (up to second order in
$|\textbf{eB}|$) of the fermionic propagators. Effects of Landau energy levels
of protons and anomalous magnetic moments of the Fermi sea of nucleons are
also incorporated in the study of magnetized nuclear matter at finite
temperature. Temperature effects are taken into account by introducing the
Fermi distribution functions to the number ($\rho_{p,n}$) and scalar
($\rho^{s}_{p,n}$) densities of nucleons in solving the coupled equations of
motion of the four scalar fields $\sigma$, $\zeta$, $\delta$ and $\chi$. The
hadronic decay widths of $A_{1}\rightarrow\rho\pi$ channel is determined using
an effective interaction Lagrangian of $A_{1}\rho\pi$ vertex from the in-
medium masses of the $A_{1}^{0}$ and $\rho^{\pm}$ states by keeping the pion
$\pi^{\pm}$ mass fixed. The in-medium decay widths in the main decay channel
of $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ is also determined using an effective
Lagrangian of $\rho\pi$ interaction. In both cases the coupling parameter is
fitted from the experimentally observed decay widths (or, decay branching
ratio of several modes) to calculate the medium modified strong decay widths
of neutral $\rho$ and $A_{1}$ mesons. From the in-medium masses and decay
widths thus obtained, the spectral functions and production cross-sections of
the neutral $A_{1}$ and $\rho$ mesons are investigated at different values of
density, isospin asymmetry, temperature and magnetic field in addition to the
important effects of (inverse) magnetic catalysis. The (decrement) increment
in the values of the light quark condensates with magnetic field give rise to
the phenomenon of (inverse) magnetic catalysis for (non-zero) zero anomalous
magnetic moments of the nucleons at the nuclear matter saturation density
$\rho_{0}$, for both $\eta=0$ and 0.5, and at $T=0,\ 50,\ 100,$ and 150 MeV of
temperatures. The contribution of magnetic field at finite density matter is
significant through the magnetized Dirac sea along with the additional
important effects due to the non-zero anomalous magnetic moments of the Dirac
sea of nucleons. There are observed to be non-trivial effects of temperature
on the in-medium mass, decay widths, hence on the spectral function and
production cross-sections of the light vector and axial-vector mesons. The
masses of $\rho$ and $A_{1}$ mesons increase slightly with temperature at
$\rho_{0}$ for finite nucleonic AMMs and magnetic field strength of
$2m_{\pi}^{2}$, which show opposite behavior at high temperature region for
$9m_{\pi}^{2}$, due to the effects of inverse magnetic catalysis. The
productions of the light vector and axial-vector mesons are thus modified
considerably due to the effects of (inverse) magnetic catalysis in a
magnetized nuclear matter considered here at finite temperature case. These
results may affect the experimental observables in the peripheral ultra
relativistic heavy-ion collision experiments, where huge magnetic fields are
expected to be produced at the initial stage of collisions with very high
temperature.

###### Acknowledgements.

Amruta Mishra acknowledges financial support from Department of Science and
Technology (DST), Government of India (project no. CRG/2018/002226). Pallabi
Parui wants to acknowledge Sourodeep De for many useful suggestions and a
thoughtful discussion.

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