# Production cross-sections and Radiative Decay widths
of Heavy Quarkonia in magnetized matter

Amruta Mishra<EMAIL_ADDRESS>Ankit Kumar
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Delhi, New Delhi - 110016  S.P. Misra<EMAIL_ADDRESS>Institute of Physics, Bhubaneswar – 751005, India

###### Abstract

We study the production cross-sections and radiative decay widths of heavy
quarkonia (charmonia and bottomonia) in magnetized nuclear matter. The
production cross-sections of the $\psi(3770)$ and $\Upsilon(4S)$, from the
$D\bar{D}$ and $B\bar{B}$ scatterings respectively, are studied from the
medium modifications of the masses and partial decay widths to open charm
(bottom) mesons, of these heavy flavor mesons. Within a chiral effective
model, the masses of the vector and pseudoscalar charmonium (bottomonium)
states are calculated from the medium modification of a dilaton field, $\chi$,
which mimics the gluon condensates of QCD. The effects of the Dirac sea (DS)
and the anomalous magnetic moments (AMMs) of the nucleons are taken into
consideration in the present study. In the presence of a magnetic field, there
is mixing of the pseudoscalar (P) meson and the longitudinal component of the
vector (V) meson (PV mixing), which leads to appreciable modifications of
their masses. The radiative decay widths of the vector (V) heavy quarkonia to
the pseudoscalar (P) mesons ($J/\psi\rightarrow\eta_{c}(1S)\gamma$,
$\psi(2S)\rightarrow\eta_{c}(2S)\gamma$ and
$\psi(1D)\rightarrow\eta_{c}(2S)\gamma$ for the charm sector and
$\Upsilon(NS)\rightarrow\eta_{b}(NS)\gamma$, $N$=1,2,3,4, for the bottom
sector) in the magnetized asymmetric nuclear matter are also investigated in
the present work. The difference in the mass of the transverse component from
the longitudinal component of the vector meson, arising due to PV mixing, is
observed as a double peak structure in the invariant mass spectrum of the
production cross-section of $\psi(3770)$. This is observed to be appreciably
more pronounced as the value of the magnetic field is increased. For the
bottomonium ($\Upsilon(4S)$) production cross-section, the effect of the PV
mixing effect is observed to be marginal, and there is observed to be a
downward shift in the peak position due to the Dirac sea contributions. The
modifications of the production cross-sections as well as the radiative decay
widths of the heavy quarkonia in the magnetized matter should have observable
consequences on the production of these heavy flavour mesons resulting from
ultra-relativistic peripheral heavy ion collision experiments, where the
created magnetic field can be extremely large.

##  I Introduction

The study of in-medium properties of the hadrons is an important topic of
research in high energy physics, due to its relevance in relativistic heavy
ion collision experiments Hosaka . The modifications of the hadrons in the
strongly interacting matter resulting from the high energy nuclear collisions
can affect the experimental observables. In the recent years, there have also
been a lot of studies on the effects of magnetic fields on the properties of
the hadrons, as the magnetic fields produced in peripheral ultra-relativistic
heavy ion collision experiments, e.g., at LHC, CERN and RHIC, BNL are
estimated to be huge tuchin . The strong magnetic fields created in the non-
central heavy ion collisions decrease rapidly after the collison, as the ion
remnants move away from the collision zone. This leads to induced currents
which slows down the decrease of magnetic field by inducing a magnetic field
in the same direction as the external magnetic field, thereby prolonging the
lifetime of the magnetic field. The study of the time evolution of the
magnetic field in the QCD plasma tuchin  is obtained from the solutions of the
Maxwell’s equations, and, assume Ohm’s law, $\vec{J}=\sigma\vec{E}$. However,
the validity of Ohm’s law has been questioned recently in Refs. Z_Wang ;
Igor_Particles_2022_5 , where it was demonstrated that the relaxation time for
the current, $\vec{J}$ to reach the value of Ohm’s law can be much larger than
the lifetime of the external electromagnetic field as well as the formation of
the QGP. This leads to what is termed as the ‘incomplete electromagnetic
response’ of the hot QCD matter Z_Wang ; Igor_Particles_2022_5 . Compared to
the calculations using Ohm’s law Z_Wang ; Igor_Particles_2022_5 , this was
observed to lead to significant suppression in the induced magnetic field (of
the order of $10^{-2}$) for the non-expanding Z_Wang ; Igor_Particles_2022_5
as well as slowly expanding plasma Igor_Particles_2022_5 . On the other hand,
for rapidly expanding plasma, there was observed to be an enhancement of the
electrical conductivity and hence of the induced magnetic field
Igor_Particles_2022_5 . The time evolution of the magnetic field depends
crucially on the electrical conductivity of the medium and is still an open
question. Further investigations are needed on the topic, including obtaining
a proper estimate of the electrical conductivity of the hot QCD matter.
However, since the heavy flavour mesons are created at the early stage when
the magnetic field can still be large, the magnetic field effects on the
properties of the heavy flavour mesons could affect the experimental
observables of these heavy ion collisions.

The heavy quarkonia (charmonia and bottomonia) are bound states of heavy
quark, $Q$ and heavy antiquark, $\bar{Q}$, $Q=c,b$). Studies of these states
in the presence of a gluonic field pes1 ; pes2 ; voloshin  show that the
modification of their masses is proportional to the change in the scalar gluon
condensate in the leading order, treating the $Q$-$\bar{Q}$ separation to be
small as compared to the gluonic fluctuations. Using the leading order
formula, the in-medium masses of the vector charmonium states in nuclear
matter have been studied from the medium change of the scalar gluon
condensate, calculated in a linear density approximation in Ref.leeko . The
study shows that the masses of the excited charmonium states have much larger
shifts as compared to the ground state. Using the QCD sum rule approach, the
modifications of the charmonium states arise due to the medium modifications
of the gluon condensate in the (hot) nuclear medium kimlee ; klingl ;
amarvjpsi_qsr ; jpsi_etac_mag ; moritalee . On the other hand, the
modifications of the masses of the open heavy flavour mesons, which are bound
states of a light ($q=u,d$) quark (antiquark) and a heavy (charm or bottom)
antiquark (quark) arise due to the medium change of the light quark
condensates in the nuclear medium within the framework of QCD sum rule
approach open_heavy_flavour_qsr ; Wang_heavy_mesons ; arvind_heavy_mesons_QSR
. Within the quark meson coupling (QMC) model qmc , the medium modifications
of the masses of the open heavy flavour mesons in the nuclear medium arise
from the modifications of the scalar potentials of the constituent light
quarks (antiquarks) of the open heavy flavour mesons open_heavy_flavour_qmc .

Within a chiral effective model Schechter ; paper3 ; kristof1 , the broken
scale invariance is incorporated in the model by introduction of a scalar
dilaton field, which simulates the gluon condensates of QCD. The in-medium
heavy quarkonium (charmonium and bottomonium) masses are thus obtained from
the medium changes of the scalar dilaton field amarvdmesonTprc ; amarvepja ;
AM_DP_upsilon  using the leading order QCD formula pes1 ; pes2 ; voloshin ;
leeko . The in-medium masses of the open charm (bottom) mesons ($D(\bar{D})$,
$B(\bar{B})$, ${D_{s}}^{\pm})$ within the chiral effective model amdmeson ;
amarindamprc ; amarvdmesonTprc ; amarvepja ; AM_DP_upsilon ; DP_AM_Ds ;
DP_AM_bbar ; DP_AM_Bs  arise from the interactions of these open heavy flavour
mesons with the baryons and scalar mesons. The partial decay widths of the
heavy quarkonium states to the open heavy flavour mesons in (hot) strange
hadronic matter amarvepja , arising from the mass modifications of these
mesons have been studied using a light quark creation model friman , namely
the ${}^{3}P_{0}$ model 3p0_1 ; 3p0_2 ; 3p0_3 ; 3p0_4  as well as using a
field theoretical model for composite hadrons amspmwg ; amspm_upsilon . Within
the chiral effective model, in the presence of a magnetic field, the
contributions of the Landau energy levels to the number densities and the
scalar densities of the charged baryons, and, the lowest Landau level (LLL)
contributions to the masses of the charged mesons are taken into account
dmeson_mag ; bmeson_mag ; charmonium_mag ; charmdecay_mag . In the presence of
a magnetic field, there is mixing of the pseudoscalar meson ($S=0$) and the
longitudinal component of the vector meson ($S=1$) states (PV mixing) leading
to appreciable modifications to their masses at high magnetic fields
charmonium_mag_QSR ; charmonium_mag_lee ; Suzuki_Lee_2017 ;
Alford_Strickland_2013 ; Zhao_Prog_Part_Nucl_Phys_114_2020_103801 ;
Quarkonia_B_Iwasaki_Oka_Suzuki . The heavy quarkonium state in the presence of
a constant magnetic field has been studied as a non-relativistic two body
system of a heavy quark and heavy antiquark bound by a Cornell potential,
using the Schrodinger equation in Refs. Alford_Strickland_2013 ;
Zhao_Prog_Part_Nucl_Phys_114_2020_103801 . In the presence of the magnetic
field, the kinetic momentum is not a conserved quantity due to the breaking of
the translational invariance, instead, the pseudomomentum, ${\bf
K}=\sum_{i=1}^{2}({\bf p}_{i}+q_{i}{\bf A}_{i})$, is conserved
Alford_Strickland_2013 ; Zhao_Prog_Part_Nucl_Phys_114_2020_103801 . The
studies show a strong dependence of the charmonia ($\eta_{c}$ and longitudinal
as well as transverse components of $J/\psi$) energy eigenvalues (which are
defined as ‘masses’) on the kinetic momentum ($\langle P_{kin}\rangle$) for
strong magnetic fields Alford_Strickland_2013 ;
Zhao_Prog_Part_Nucl_Phys_114_2020_103801 . However, for the values of the
magnetic field ($eB\leq 10m_{\pi}^{2}$) considered in the present study, the
dependence on $\langle P_{kin}\rangle$ is observed to be rather marginal upto
$\langle P_{kin}\rangle\sim$ 2 GeV Zhao_Prog_Part_Nucl_Phys_114_2020_103801 .
In the present study of the production cross-sections and radiative decay
widths of the heavy quarkonia, the mixing of the pseudoscalar ($S=0$) and the
vector meson ($S=1$) states for the charm sector are studied using a
phenomenological interaction Lagrangian
($\sim{\tilde{F}}_{\mu\nu}(\partial^{\mu}P)V^{\nu}$) charmonium_mag_lee ;
Quarkonia_B_Iwasaki_Oka_Suzuki , where the coupling parameter for the
interaction is fitted from the observed radiative decay width of $V\rightarrow
P\gamma$. For zero spatial momenta of the pseudoscalar (P) and vector (V)
states, there is mixing of the pseudoscalar meson and the longitudinal
component of the vector meson (PV mixing), which leads to a drop (increase) in
the mass of the pseudoscalar (longitudinal component of the vector meson) in
the presence of an external magnetic field, and the masses of the tranverse
components of the vector mesons remain unaffected. Due to lack of radiative
data for the bottom sector, the PV mixing effect is considered using an
interaction Hamiltonian ($\sim-{\mbox{\boldmath$\mu$}}\cdot{\bf B}$), which
modifies the masses of the pseudoscalar meson and the longitudinal component
of the vector meson due to their mixing Alford_Strickland_2013 ;
Zhao_Prog_Part_Nucl_Phys_114_2020_103801 . These mass modifications due to PV
mixings are considered for the study of the heavy quarkonia radiative decay
widths as well as the production cross-sections. As we shall see later, the
difference in the mass of the transverse component from the longitudinal
component of the vector meson, arising due to PV mixing is observed as a
double peak structure in the invariant mass spectrum of the production cross-
section of $\psi(3770)$, which is observed to be significantly more pronounced
as the strength of the magnetic field is increased. On the other hand, for the
bottomonium ($\Upsilon(4S)$) production cross-section, it is observed that the
PV mixing effects are marginal and there is a downward shift in the peak
position due to the contributions of the magnetized Dirac sea.

Within the chiral effective model, the effects of the Dirac sea (DS) of the
nucleons on the masses of the quarkonia and open heavy flavour mesons in
magnetized nuclear matter have been studied in Refs. Open_bottom_MC ;
Heavy_Quarkonia_masses_MC ; charmdecay_mag_MC  and are observed to be
significant. The effects of these mass modifications on the charmonium partial
decay widths to $D\bar{D}$ mesons, have been studied using the ${}^{3}P_{0}$
model charmdecay_mag_MC , and, the charmonium (bottomonium) decay widths to
$D\bar{D}$ ($B\bar{B}$) using the field theoretic model of composite hadrons
charmdw_mag ; open_charm_mag_AM_SPM ; upslndw_mag ; charmdw_mag_MC . The
effects of PV mixing as well as the Dirac sea (DS) effects of the nucleons are
taken into account for the study of the heavy quarkonia states. The Dirac sea
(DS) effects have been studied for the quark matter using the Nambu-Jona-
Lasinio (NJL) model Preis ; menezes , as well as, for nuclear matter within
Walecka model arghya ; haber  as well as an extended linear sigma model haber
. The effects of the Dirac sea of nucleons are taken into consideration by
summing over the vector and scalar tadpole diagrams for obtaining the nucleon
self-energy arghya  within the chiral effective model Open_bottom_MC ;
Heavy_Quarkonia_masses_MC ; charmdecay_mag_MC ; charmdw_mag_MC . These are
observed to lead to an enhancement of the quark condenstaes (through the
scalar fields) with increase in the magnetic field, an effect called the
‘magnetic catalysis’, for zero density as well as for the baryon density,
$\rho_{B}=\rho_{0}$, the nuclear matter saturation density, both for symmetric
and asymmetric nuclear matter, when the AMMs are not taken into account.
However, the opposite trend is observed for the quark condensates (inverse
magnetic catalysis) for $\rho_{B}=\rho_{0}$, when the AMMs of the nucleons are
taken into account. In peripheral ultra-relativistic heavy ion collisions, the
created matter has very small density. In the present work, we study the
effects of the magnetic field on the production cross-sections of the heavy
quarkonium states, $\psi(3770)$ and $\Upsilon(4S)$, which are the lowest
states which decay to the open charm (bottom) mesons in vacuum. In the
magnetized nuclear matter, these are studied from the spectral functions of
these states, obtained from their masses and the partial decay widths to
$D\bar{D}$ ($B\bar{B}$), taking into consideration the effects of PV mixing as
well as the Dirac sea of the nucleons Heavy_Quarkonia_masses_MC ;
charmdw_mag_MC . The in-medium decay widths of the heavy quarkonia are
calculated within a field theoretic model of composite hadrons charmdw_mag_MC
. The effects of the in-medium masses of the heavy quarkonia on the radiative
decay widths of $V\rightarrow P\gamma$ in the magnetized nuclear matter are
also studied in the present work. The effects of the PV mixing as well as
Dirac sea on the heavy quarkonia production cross-sections as well as
radiative decay widths are observed to be significant, which should affect the
experimental observables, e.g., the yields of the heavy quarkonia and the open
heavy flavour mesons produced in peripheral ultra-relativistic heavy ion
collision experiments.

The outline of the paper is as follows. In section 2, we briefly recapitulate
the chiral efective model used for calculation of the masses of the heavy
quarkonia (charmonia and bottomonia) as well as open charm (bottom) mesons in
magnetized asymmetric nuclear matter. The masses of these states are computed
including the effects of nucleon Dirac sea as well as the mixing of the
pseudoscalar (P) and vector (V) mesons (PV mixing). The in-medium decay widths
of charmonia (bottomonia) to open heavy flavour meson pairs
$D\bar{D}(B\bar{B}$), arising due to the mass modifications of the initial and
final mesons, are obtained using a field theoretical model with composite
hadrons. Sections 3 describes the production cross-sections of the heavy
quarkonium states (the charmonium state, $\psi(3770)$ and the upsilon state,
$\Upsilon(4S)$) arising from the scatterings of $D\bar{D}$ and $B\bar{B}$
respectively. The computation of the radiative decay widths of the vector (V)
heavy quarkonia to pseudoscalar (P) states in the magnetized nuclear matter is
described in Section 4. Section 5 describes the results of the present
investigation of the the production cross-sections and radiative decay wdths
of heavy quarkonia in magnetized nuclear matter. The summary of the findings
of the present work are presented in section 6.

##  II Masses and Decay widths of Heavy Quarkonium states

###  II.1 Masses

The masses of the vector and pseudoscalar heavy quarkonium (charmonium and
bottomonium) states in the magnetized nuclear matter are calculated within a
chiral effective model. The model is based on chiral symmetry and broken scale
invariance of QCD, the latter incorporated into the model through a
logarithmic potential of a scalar dilaton field, $\chi$, which mimics the
gluon condensates of QCD. Studies of the heavy quarkonium state in a gluon
field, assuming the distance between the heavy quark and antiquark to be small
as compared to the scale of gluonic fluctuations pes1 ; pes2 ; voloshin  leads
to the mass shift of the heavy quarkonium state to be proportional to the
change in the gluon condensate in the medium, in the leading order. In the
mean field approximation, the mass shifts of the heavy quarkonia in the
magnetized nuclear matter are calculated within the chiral effective model
from the medium change of $\chi$ amarvdmesonTprc ; amarvepja ; AM_DP_upsilon ,
which is related to the shift in the scalar gluon condensate. The value of the
dilaton field, $\chi$ is obtained from the solution of the coupled equations
of motion of $\chi$ and the scalar (non-strange isoscalar, $\sigma$, strange
isoscalar, $\zeta$ and non-strange isovector $\delta$) fields, for given
values of the baryon density, $\rho_{B}$, the isospin asymmetry parameter,
$\eta=(\rho_{n}-\rho_{p})/(2\rho_{B})$ (with $\rho_{n}$ and $\rho_{p}$ as the
neutron and proton number densities), and, the magnetic field, $B$ (chosen to
be along z-direction). The anomalous magnetic moments (AMMs) of the nucleons
dmeson_mag ; bmeson_mag ; charmonium_mag , and, the contributions due to the
nucleon Dirac sea, are taken into account to study the masses of the
charmonium ($J/\psi$, $\psi^{\prime}\equiv\psi(2S)$,
$\psi^{\prime\prime}\equiv\psi(1D)$, $\eta_{c}\equiv\eta_{c}(1S)$ and
$\eta_{c}^{\prime}\equiv\eta_{c}(2S)$) and the bottomonium ($\Upsilon(1S)$,
$\Upsilon(2S)$, $\Upsilon(3S)$, $\Upsilon(4S)$, $\eta_{b}\equiv\eta_{b}(1S)$,
$\eta_{b}^{\prime}\equiv\eta_{b}(2S)$, $\eta_{b}(3S)$ and $\eta_{b}(4S)$)
states in magnetized (nuclear) matter. These masses have additonal
contributions due to the mixing of pseudoscalar (P) meson and the longitudinal
component of the vector (V) meson (PV mixing) in the presence of a magnetic
field charmonium_mag_QSR ; charmonium_mag_lee ; Suzuki_Lee_2017 ;
Alford_Strickland_2013 ; Zhao_Prog_Part_Nucl_Phys_114_2020_103801 ;
Quarkonia_B_Iwasaki_Oka_Suzuki ; charmdw_mag ; open_charm_mag_AM_SPM . These
PV mixing effects ($J/\psi-\eta_{c}$, $\psi(2S)-\eta_{c}(2S)$,
$\psi(1D)-\eta_{c}(2S)$ for the charmonium states and $\Upsilon(1S)-\eta_{b}$,
$\Upsilon(2S)-\eta_{b}(2S)$, $\Upsilon(3S)-\eta_{b}(3S)$,
$\Upsilon(4S)-\eta_{b}(4S)$ for the bottomonium states), are considered in the
present study of the production cross-sections and radiative decays of heavy
quarkonia.

Figure 1: The effects of Dirac sea (DS) and PV mixing on the masses and decay
widths of $\psi(3770)$ are shown as functions of $eB/{m_{\pi}^{2}}$, for the
symmetric ($\eta$=0) and asymmetric (with $\eta$=0.5) nuclear matter. The
decay width of $\psi(3770)$ to (I) $D^{+}D^{-}$, (II) $D^{0}\bar{D^{0}}$ and
(III) $D^{+}D^{-}$ as well as $D^{0}\bar{D^{0}}$, are shown in panels (c) and
(d) with Dirac sea effects and compared with the results without Dirac sea
effects (shown as dotted lines). Figure 2: The effects of Dirac sea (DS) and
PV mixing on the masses and decay widths of $\Upsilon(4S)$ are shown as
functions of $eB/{m_{\pi}^{2}}$, for the symmetric ($\eta$=0) and asymmetric
(with $\eta$=0.5) nuclear matter. The decay widths of $\Upsilon(4S)$ to (I)
$B^{+}B^{-}$, (II) $B^{0}\bar{B^{0}}$ and (III) $B^{+}B^{-}$ as well as
$B^{0}\bar{B^{0}}$, are shown including the Dirac sea effects and compared
with the results without Dirac sea effects (shown as dotted lines). Figure 3:
Production cross-section of $\psi(3770)$ due to scattering of $D\bar{D}$
mesons for $eB=3m_{\pi}^{2}$, showing the contributions from the longitudinal
(L) and transverse (T) components of $\psi(3770)$. Figure 4: Same as figure 3,
for $eB=8m_{\pi}^{2}$. Figure 5: Production cross-section of $\Upsilon(4S)$
due to scattering of $B\bar{B}$ mesons for $eB=3m_{\pi}^{2}$, showing the
contributions from the longitudinal (L) and transverse (T) components. Figure
6: Same as figure 5, for $eB=8m_{\pi}^{2}$. Figure 7: $\Gamma(V\rightarrow
P\gamma)$ (in MeV) for the charmonium states
($J/\psi\rightarrow\eta_{c}\gamma$,
$\psi^{\prime}\rightarrow\eta^{\prime}_{c}\gamma$ and
$\psi^{\prime\prime}\rightarrow\eta^{\prime}_{c}\gamma$), as functions of
$eB/{m_{\pi}}^{2}$, showing the contributions from the longitudinal (L) and
transverse (T) components, for symmetric and asymmetric nuclear matter. These
are compared with the case when the Dirac sea effects are nor taken into
account (shown as dotted lines).  Figure 8: $\Gamma(V\rightarrow P\gamma)$ in
MeV for the bottomonium states ($\Upsilon(1S)\rightarrow\eta_{b}(1S)\gamma$
and $\Upsilon(2S)\rightarrow\eta_{b}(2S)\gamma$), as functions of
$eB/{m_{\pi}^{2}}$ for symmetric and asymmetric nuclear matter, showing the
individual contributions from the longitudinal (L) and transverse (T)
components. These are compared with the case when the Dirac sea effects are
nor taken into account (shown as dotted lines).  Figure 9: Same as figure 8,
for $\Upsilon(3S)\rightarrow\eta_{b}(3S)\gamma$ and
$\Upsilon(4S)\rightarrow\eta_{b}(4S)\gamma$.

The mass modifications due to PV mixing have been studied using an effective
Lagrangian density of the form charmonium_mag_lee ;
Quarkonia_B_Iwasaki_Oka_Suzuki

${\cal
L}_{PV\gamma}=\frac{g_{PV}}{m_{av}}e{\tilde{F}}_{\mu\nu}(\partial^{\mu}P)V^{\nu},$
(1)

where $m_{\rm{av}}=(m_{V}+m_{P})/2$, $m_{P}$ and $m_{V}$ are the masses of the
pseudoscalar and vector mesons, ${\tilde{F}}_{\mu\nu}$ is the dual
electromagnetic field. In the presence of the PV mixing effects (incorporated
through the phenomenological Lagrangian (1)), the masses of the pseudoscalar
and the longitudinal component of the vector meson are given as
charmonium_mag_lee

$m^{2\
{(PV)}}_{P,V^{||}}=\frac{1}{2}\Bigg{(}M_{+}^{2}+\frac{c_{PV}^{2}}{m_{av}^{2}}\mp\sqrt{M_{-}^{4}+\frac{2c_{PV}^{2}M_{+}^{2}}{m_{av}^{2}}+\frac{c_{PV}^{4}}{m_{av}^{4}}}\Bigg{)},$
(2)

where $M_{\pm}^{2}=m_{V}^{2}\pm m_{P}^{2}$ and $c_{PV}=g_{PV}eB$. The coupling
parameter $g_{PV}$ is fitted from the observed value of the radiative decay
width for $V\rightarrow P\gamma$ for the charmonium sector. However, due to
lack of radiative decay data for the bottomonium states, we estimate the
modifications to the masses of the bottomonium pseudoscalar and vector mesons
upslndw_mag  due to mixing of these states in the presence of a magnetic
field, using the Hamiltonian Alford_Strickland_2013 ;
Zhao_Prog_Part_Nucl_Phys_114_2020_103801 ; Quarkonia_B_Iwasaki_Oka_Suzuki

$H_{\rm{spin-mixing}}=-{\sum_{i=1}^{2}}{\vec{\mu}}_{i}\cdot{\vec{B}},$ (3)

which decribes the interaction of the magnetic moments of the constituent
quark (antiquark) with the external magnetic field. In the above,
${\vec{\mu}}_{i}=g|e|{q_{i}}{\vec{S_{i}}}/(2m_{i})$ is the magnetic moment of
the $i$-th particle, $g$ is the Lande g-factor (taken to be $2(-2)$ for the
quark(antiquark)), $q_{i}$, $\vec{S_{i}}$, $m_{i}$ are the electric charge (in
units of the magnitude of the electronic charge, $|e|$), spin and mass of the
$i$-th particle charmonium_mag_lee ; Quarkonia_B_Iwasaki_Oka_Suzuki . This
interaction leads to the mass of the pseudoscalar (longitudinal component of
the vector meson) to be modified to Alford_Strickland_2013

$\displaystyle{m^{(PV)}_{P(V^{||})}}_{\rm{spin-mixing}}$ $\displaystyle=$
$\displaystyle m_{P(V)}\mp\frac{\Delta
E}{2}\Big{(}(1+\Delta^{2})^{1/2}-1\Big{)}$ (4) $\displaystyle\equiv$
$\displaystyle m_{P(V)}\mp{\Delta M}^{PV},$

where, $\Delta=2g|eB|((q_{1}/m_{1})-(q_{2}/m_{2}))/\Delta E$, and, $\Delta
E=m_{V}-m_{P}$ is the difference in the masses of the vector and pseudoscalar
mesons. In equations (2) and (4), $m_{V}$ and $m_{P}$ refer to the masses
calculated from the medium change of the dilaton field within the chiral
effective model. The mass modifications of the heavy quarkonium states as
discussed above, solely arise due to the coupling of the spin angular momenta
of the constituent of the constituent heavy quark (antiquark) with the
external magnetic field, as decsribed by the Hamiltonian given by equation
(3). These lead to drop (increase) in the mass of the pseudoscalar
(longitudinal component of the vector) heavy quarkonium state, whereas, the
masses of the transverse components of the vector meson remain unaffected due
to the PV mixing.

The PV mixing effects for the masses of the charmonium states charmdw_mag  and
the open charm ($D$ and $\bar{D}$) mesons are incorporated using the
phenomenological Lagrangian interaction given by equation (1) and for the
bottomonium states and open bottom mesons, due to unavailability of radiative
decays data, using the Hamiltonian interaction given by equation (3) for the
study of the heavy quarkonia decay widths. There is observed to be appreciable
mass modifications of the pseudoscalar and the longitudinal component of the
vector charmonium states, due to the PV mixings charmdw_mag . In the present
work, the effects of the Dirac sea contributions are also considered on the
masses of the of the heavy quarkonium as well as open heavy flavour mesons and
their effects on the partial decay widths to open charm (bottom) mesons using
a field theoretic model of composte hadrons.

###  II.2 Decay widths of Heavy Quarkonia

The partial decay widths of the heavy quarkonia to open heavy flavour mesons
in the magnetized nuclear matter are calculated from the in-medium masses of
the initial and final state particles. These have been studied using the
${}^{3}P_{0}$ model for the charmonium sector charmdecay_mag , as well as,
using a field theoretic model of composite hadrons amspmwg ; amspm_upsilon
for both the charmonium and bottomonium sectors charmdw_mag ; upslndw_mag .
The mass modifications of the charmonium states and the open charm mesons are
observed to lead to substantial modification of the partial decay width of
$\psi(1D)\rightarrow D\bar{D}$ charmdw_mag  due to $\psi(1D)-\eta_{c}(2S)$
mixing, as well as, due to $D(\bar{D})-D^{*}(\bar{D}^{*})$ mixing effects
open_charm_mag_AM_SPM . The in-medium masses of the upsilon states and open
bottom mesons, including the PV mixing effects and their effects on the
partial decay width of $\Upsilon(4S)\rightarrow B\bar{B}$ have been studied in
Ref. upslndw_mag .

In this subsection, we briefly describe the field theoretical model of
composite hadrons spm781 ; spm782 ; spmdiffscat , used to study the partial
decay widths of the vector heavy quarkonium states to open heavy flavour
mesons in magnetized (nuclear) matter. As the matter produced in peripheral
ultra-relativistic heavy ion collision experiments (where strong magnetic
fields are created) is extremely dilute, we study the decay widhts of the
charmonium state $\psi(3770)$ and the bottomonium state $\Upsilon(4S)$, which
are the lowest states which decay to $D\bar{D}$ and $B\bar{B}$ in vacuum. The
model used for the calculation of the decay widths describes the hadrons as
comprising of quark (and antiquark) constituents. The constituent quark field
operators of the hadron in motion are constructed from the constituent quark
field operators of the hadron at rest, by a Lorentz boosting. Similar to the
MIT bag model MIT_bag , where the quarks (antiquarks) occupy specific energy
levels inside the hadron, it is assumed in the present model for the composite
hadrons that the quark (antiquark) constituents carry fractions of the mass
(energy) of the hadron at rest (in motion) spm781 ; spm782 . With explicit
constructions of the charmonium (bottomonium) state and the open charm
(bottom) mesons, the decay width of the heavy quarkonium state to open heavy
flavour mesons. is calculated using the light quark antiquark pair creation
term of the free Dirac Hamiltonian for constituent quark field amspmwg . The
relevant part of the quark pair creation term is through the $q\bar{q}(q=u,d)$
creation for decay of the charmonium (bottomonium) state, $\Psi$ ($\Upsilon$),
to the final state, $D\bar{D}$ ($B\bar{B}$). The pair creation term is given
as

${\cal H}_{q^{\dagger}\tilde{q}}({\bf x},t=0)=Q_{q}^{(p)}({\bf
x})^{\dagger}(-i\mbox{\boldmath$\alpha$}\cdot\mbox{\boldmath$\bigtriangledown$}+\beta
M_{q}){\tilde{Q}}_{q}^{(p^{\prime})}({\bf x})$ (5)

where, $M_{q}$ is the constituent mass of the light quark (antiquark). The
subscript $q$ of the field operators in equation (5) refers to the fact that
the light antiquark, $\bar{q}$ and light quark, $q$ are the constituents of
the $D(B)$ and $\bar{D}(\bar{B})$ mesons with momenta ${\bf p}$ and ${\bf
p^{\prime}}$ respectively in the final state of the decay of the charmonium
(bottomonium) state, $\Psi(3770)(\Upsilon(4S))$. The decay width of the
quarkonium state, $M$, for the decay process $M\rightarrow F\bar{F}$, with
$(M,F,\bar{F})\equiv(\psi(3770),D,\bar{D}),(\Upsilon(4S),B,\bar{B}$), is
calculated from the matrix element of the light quark-antiquark pair creation
part of the free Dirac Hamiltonian, between the initial quarkonium state and
the final state mesons for the reaction $M\rightarrow F({\bf p})\bar{F}({\bf
p^{\prime}})$ as given by

$\displaystyle\langle F({\bf p})|\langle{\bar{F}}({\bf
p^{\prime}})|{\int{{\cal H}_{q^{\dagger}\tilde{q}}({\bf x},t=0)d{\bf
x}}}|M^{m}(\vec{0})\rangle=\delta({\bf p}+{\bf p}^{\prime})A_{M}(|{\bf
p}|)p_{m},$ (6)

where, $A_{M}(|{\bf p}|)$ is a polynomial of the magnitude of the momentum,
$|{\bf p}|$ of the outgoing mesons. The decay width is calculated to be

$\displaystyle\Gamma(M\rightarrow F({\bf p}){\bar{F}}(-{\bf
p}))=\gamma_{\psi}^{2}\frac{8\pi^{2}}{3}|{\bf p}|^{3}\frac{p^{0}_{F}(|{\bf
p}|)p^{0}_{\bar{F}}(|{\bf p}|)}{m_{M}}A_{M}(|{\bf p}|)^{2}$ (7)

with $p^{0}_{F({\bar{F}})}(|{\bf p}|)=\big{(}m_{F({\bar{F}})}^{2}+|{\bf
p}|^{2}\big{)}^{{1}/{2}}$, and, $|{\bf p}|$, the magnitude of the momentum of
the outgoing $F(\bar{F})$ meson is given as,

$|{\bf
p}|=\Big{(}\frac{{m_{M}}^{2}}{4}-\frac{{m_{F}}^{2}+{m_{\bar{F}}}^{2}}{2}+\frac{({m_{F}}^{2}-{m_{\bar{F}}}^{2})^{2}}{4{m_{M}}^{2}}\Big{)}^{1/2}.$
(8)

In the above, the masses of the $F(\bar{F})$ and heavy quarkonium state are
the in-medium masses in the magnetized nuclear matter calculated in the chiral
effective model including the effects of the Dirac sea of nucleons, with
additional contributions from lowest Landau levels for the charged open charm
(bottom) mesons, as well as, effects due to PV mixing charmdw_mag_MC . The
parameter, $\gamma_{M}$, in the expression for the quarkonium decay width, is
a measure of the coupling strength for the creation of the light quark
antiquark pair, to produce the $F\bar{F}$ final state. This parameter is
adjusted to reproduce the vacuum decay widths of $\psi(3770)$ to $D^{+}D^{-}$
and $D^{0}\bar{D^{0}}$ amspmwg  for the charm sector and
$\Upsilon(4S)\rightarrow B^{+}B^{-}$ and $\Upsilon(4S)\rightarrow
B^{0}\bar{B^{0}}$ amspm_upsilon  for the bottom sector.

Including the PV mixing effect, the expression for the decay width is modified
to

$\displaystyle\Gamma^{PV}(M$ $\displaystyle\rightarrow$ $\displaystyle F({\bf
p}){\bar{F}}(-{\bf
p}))=\gamma_{M}^{2}\frac{8\pi^{2}}{3}\Bigg{[}\Bigg{(}\frac{2}{3}|{\bf
p}|^{3}\frac{p^{0}_{F}(|{\bf p}|)p^{0}_{\bar{F}}(|{\bf p}|)}{m_{M}}A_{M}(|{\bf
p}|)^{2}\Bigg{)}$ (9) $\displaystyle+$ $\displaystyle\Bigg{(}\frac{1}{3}|{\bf
p}|^{3}\frac{p^{0}_{F}(|{\bf p}|)p^{0}_{\bar{F}}(|{\bf
p}|)}{m_{M}^{PV}}A^{M}(|{\bf p}|)^{2}\Bigg{)}\Big{(}{|{\bf p}|\rightarrow|{\bf
p|}(m_{M}=m_{M}^{PV})}\Big{)}\Bigg{]}.$

In the above, the first term corresponds to the transverse polarizations for
the quarkonium state, $M$, whose masses remain unaffected by the mixing of the
pseudoscalar and vector quarkonium states. The second term in (9) corresponds
to the longitudinal component, whose mass is modified due to the mixing with
the pseudoscalar meson in the presence of the magnetic field.

##  III Production Cross-sections of Heavy Quarkonia

The relativistic Breit Wigner spectral function and the production corss-
section (from scattering of particles $a$ and $b$) of the vector meson $V$ are
given as Elena_16 ; Elena_17 ; Elena_13_1 ; BW_CS_Haglin ; BW_CS_Li

$A_{V}(M)=C\cdot\frac{2}{\pi}\frac{M^{2}\Gamma_{V}^{*}}{(M^{2}-{m_{V}^{*}}^{2})^{2}+(M\Gamma_{V}^{*})^{2}}$
(10)

and

$\sigma(M)=\frac{6\pi^{2}\Gamma_{V}^{*}A_{V}(M)}{q(m_{V}^{*},m_{a}^{*},m_{b}^{*})^{2}}.$
(11)

where $M$ is the invariant mass, $m_{V}^{*}$ and $\Gamma_{V}^{*}$ are the in-
medium mass and decay width of the vector meson, $m_{a}^{*}$ and $m_{b}^{*}$
are the in-medium masses of the scattered particles. $C$ in the expression for
the spectral function given by equation (10) is a constant determined by
normalization condition $\int_{0}^{\infty}A_{V}(M)dM=1.$ In equation (11),
$q(m_{V}^{*},m_{a}^{*},m_{b}^{*})$ is the momentum of the scattering particle
$a(b)$ in the center of mass frame of the vector meson, $V$, and is given as,

$\displaystyle
q(m^{*}_{V},m_{a}^{*},m_{b}^{*})=\frac{1}{2M}\Big{(}\big{[}{m_{V}^{*}}^{2}-(m_{a}^{*}+m_{b}^{*})^{2}\big{]}\cdot\big{[}{m_{V}^{*}}^{2}-(m_{a}^{*}-m_{b}^{*})^{2}\big{]}\Big{)}^{1/2}.$
(12)

The vector meson, $V$ may, however, be created from scattering of (as well as
decay to) particles in different modes, say particles $a_{i}$ and $b_{i}$ with
masses $m^{*}_{a_{i}}$ and $m^{*}_{b_{i}}$ in the channel $i$. The in-medium
decay width of the vector meson is then given as the sum of the decay widths
in these channels, i.e., $\Gamma_{V}^{*}={\sum_{i}}{\Gamma_{V}^{i}}^{*}$. The
production cross-section of the vector meson, accounting for all these
channels is given as

$\sigma(M)=6\pi^{2}\sum_{i}\frac{{\Gamma_{V}^{i}}^{*}}{q({m^{*}_{V}},m^{*}_{a_{i}},m^{*}_{b_{i}})^{2}}A_{V}(M),$
(13)

where $q(m^{*}_{V},m^{*}_{a_{i}},m^{*}_{b_{i}})$ is the center of mass
momentum of the particle $a_{i}$ as well as $b_{i}$ corresponding to the
channel $i$ in the center of mass frame of the vector meson, $V$.

In the present work, we study the production cross-sections of the vector
charmonium (bottomonium) states, $\psi(3770)$ and $\Upsilon(4S)$, which are
the lowest states which decay to open charm ($D\bar{D}$) and open bottom
($B\bar{B}$) mesons. The charmonium vector meson ($V\equiv\psi(3770)$) is
produced through scattering of $D^{+}D^{-}$ and $D^{0}\bar{D^{0}}$ mesons,
and, the bottomonium state ($V\equiv\Upsilon(4S))$) is produced through
scattering of $B^{+}B^{-}$ and $B^{0}\bar{B^{0}}$ mesons. The masses of the
heavy quarkonium states and the open charm ($D$ and $\bar{D}$) and open bottom
($B$ and $\bar{B}$) in the magnetized asymmetric nuclear matter are calculated
within the chiral effective model accounting for the nucleon Dirac sea and the
anomalous magnetic moments of the nucleons. The PV mixing is taken into
account to calculate the masses of the vector quarkonium states, $\psi(3770)$
and $\Upsilon(4S)$, due to $\psi(3770)-\eta_{c}(2S)$ and
$\Upsilon(4S)-\eta_{b}(4S)$ mixings, as well as, for the open charm
$D(\bar{D}$) and open bottom mesons, $B(\bar{B})$ due to
$D-D^{*}(\bar{D}-\bar{D^{*}})$ and $B-B^{*}(\bar{B}-\bar{B^{*}})$ mixings.
Including the PV effect introduces mass difference between the longitudinal
and transverse components of the vector state ($\psi(3770)$ and
$\Upsilon(4S)$), since the longitudinal component undergoes mass modification
due to mixing with the pseudoscalar meson ($\eta_{c}(2S)$ and $\eta_{b}(4S)$),
whereas the transverse component is unaffected due to PV mixing.

In the presence of PV mixing, the production corss-section of the vector
meson, V, given by equation (11), can be written in terms of the contributions
from the longitudinal and transverse components as

$\sigma(M)=6\pi^{2}\Bigg{(}\frac{1}{3}\sum_{i}\frac{{\Gamma_{V}^{*i}}^{L}}{q({{m^{*}_{V}}^{L}},m^{*}_{a_{i}},m^{*}_{b_{i}})^{2}}A_{V}^{L}(M)+\frac{2}{3}\sum_{i}\frac{{\Gamma_{V}^{*i}}^{T}}{q({{m^{*}_{V}}^{T}},m^{*}_{a_{i}},m^{*}_{b_{i}})^{2}}A_{V}^{T}(M)\Bigg{)},$
(14)

where, ${m^{*}_{V}}^{T(L)}$ and ${\Gamma_{V}^{*i}}^{T(L)}$ are the mass and
the decay width (in channel $i$) of the transverse (longitudinal) component of
the vector meson, $A_{V}^{T(L)}(M)$ is the contribution from the transverse
(longitudinal) component to the spectral function, given as,

$A_{V}^{T(L)}(M)=C\cdot\frac{2}{\pi}\frac{M^{2}\Gamma_{V}^{*T(L)}}{\Big{(}M^{2}-{{m^{*T(L)}_{V}}^{2}\Big{)}}^{2}+\Big{(}M\Gamma_{V}^{*T(L)}\Big{)}^{2}},$
(15)

with the constant $C$ determined from the normalization condition

$\int_{0}^{\infty}\Big{(}\frac{1}{3}A_{V}^{L}(M)+\frac{2}{3}A_{V}^{T}(M)\Big{)}dM=1.$
(16)

##  IV Radiative Decay widths of Heavy Quarkonia:

The radiative decay widths of the vector meson, V are obtained from the decay
width as calculated using the interaction Lagrangian given by equation (1).
The parameter, $g_{PV}$ is calculated from the observed decay width of
$V\rightarrow P\gamma$ in vacuum, for the charmonium states. However, due to
lack of experimental data of radiative decays for the upsilon states, we
compute the value of $g_{PV}$ for these states using the expressions of the
masses of the pseudoscalar meson and the longitudinal component of the vector
meson with PV mixing, obtained from the phenomenological Lagrangian given by
(1), in the limit of weak magnetic field. The expressions for the masses
(given by equation (2)), retaining terms upto the second order in $eB$ and
leading order in $\frac{(m_{V}-m_{P})}{(m_{V}+m_{P})}$, reduce to
charmonium_mag_lee

$m^{2\
{(PV)}}_{P,V^{||}}=m^{2}_{P,V}\mp\frac{{(g_{PV}eB)}^{2}}{(m_{V}^{2}-m_{P}^{2})^{2}},$
(17)

where, $m_{V}$ and $m_{P}$ are the masses of the pseudoscalar and vector
mesons as calculated within the chiral effective model. The values obtained
for the parameter $g_{PV}$ for the processes
$\Upsilon(1S)\rightarrow\eta_{b}(1S)\gamma$,
$\Upsilon(2S)\rightarrow\eta_{b}(2S)\gamma$,
$\Upsilon(3S)\rightarrow\eta_{b}(3S)\gamma$, and,
$\Upsilon(4S)\rightarrow\eta_{b}(4S)\gamma$, are 1.1726, 1.2436, 1.2850 and
1.2932, using which the values of radiative decay widths (in eV) in vacuum are
obtained to be 8.70907, 0.4748802, 0.56039 and 0.0078332 respectively.

The radiative decay widths for the vector heavy quarkonium state are
calculated in the magnetized nuclear matter, from the in-medium masses for the
vector and pseudoscalar mesons. In the presence of the PV mixing, the mass of
the longitudinal component of the vector meson is modified due to mixing with
the pseudoscalar meson, whereas, the mass of the transverse component is not
affected by PV mixing. This leads to decays of these components to be
different, in the presence of a magnetic field.

The radiative decay width, $\Gamma(V\rightarrow P+\gamma)$ given as

$\displaystyle\Gamma(V\rightarrow P\gamma)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{12}\frac{g_{PV}^{2}}{\pi
m_{\rm{av}}^{2}}\Bigg{(}{\frac{1}{3}{p^{L}_{cm}}^{3}+\frac{2}{3}{p^{T}_{cm}}^{3}}\Bigg{)}\equiv\frac{1}{3}\Gamma^{L}+\frac{2}{3}\Gamma^{T}$
(18)

where,

$p^{L}_{\rm{cm}}=\frac{{m^{(PV)}_{V^{||}}}^{2}-{m^{(PV)}_{P}}^{2}}{2m^{(PV)}_{V^{||}}}\,\,\,{\rm
and}\;\;p^{T}_{\rm{cm}}=\frac{m_{V}^{2}-{m^{(PV)}_{P}}^{2}}{2m_{V}},$ (19)

are the magnitudes of the center of mass momentum in the final state
corresponding to the longitudinal and transverse components of the decaying
vector meson.

##  V Results and Discussions

In the present study, the effects of the Dirac sea and PV mixing are
investigated on the production cross-sections and radiative decays of the
heavy quarkonia in magnetized nuclear matter. The production cross-sections
are computed from the masses and partial decay widths (to open charm (bottom)
mesons) of these states and the radiative decay widths, $\Gamma(V\rightarrow
P\gamma)$ are calculated from the in-medium masses of the vector ($V$) and
pseudoscalar ($P$) heavy quarkonium states in the magnetized matter. The study
of the effects of magnetic field can be important for observables of
peripheral ultra-relativistic heavy ion collision experiments, e.g., at RHIC,
BNL and LHC, CERN, where the produced magnetic fields are estimated to be
extremely strong. The created matter is extremely dilute in these ultra-
relativistic heavy ion collision experiments. In the present work, we study
the production cross-sections of $\psi(3770)$ and $\Upsilon(4S)$, which are
the lowest states, which decay to $D\bar{D}$ and $B\bar{B}$ respectively, in
vacuum.

The masses of the heavy quarkonium states in magnetized nuclear matter have
been studied within a chiral effective model, including the effects of the
nucleon Dirac sea in Ref. Heavy_Quarkonia_masses_MC . These are obtained from
the medium modification of the scalar dilaton field, $\chi$, which simulates
the gluon condensates of QCD. For given values of the baryon density,
$\rho_{B}$, the isospin asymmetry parameter,
$\eta=(\rho_{n}-\rho_{p})/(2\rho_{B})$ (with $\rho_{n}$ and $\rho_{p}$ as the
neutron and proton number densities), and, the magnetic field, $B$ (chosen to
be along z-direction), the value of the dilaton field is calculated through
the solution of the coupled equations of motion of the dilaton field, $\chi$
and the scalar-isoscalar non-strange ($\sigma$), the scalar-isoscalar strange
($\zeta$), and, scalar-isovector non-strange ($\delta$) fields. The anomalous
magnetic moments (AMMs) of the nucleons are taken into consideration in the
present study. Due to presence of the external magnetic field, there is mixing
of the pseusoscalar (P) meson and the longitudinal component of the vector (V)
meson (PV mixing) which further modifies the masses of these mesons. The
masses of the charmonium states ($J/\psi$, $\psi^{\prime}\equiv\psi(2S)$,
$\psi^{\prime\prime}\equiv\psi(1D)$, $\eta_{c}$ and
$\eta_{c}^{\prime}\equiv\eta_{c}(2S)$), modified due to PV mixing are
calculated using the equation (2), obtained from the phenomenological
Lagrangian density given by equation (1). The parameter, $g_{PV}$ is
calculated from the observed decay width of $V\rightarrow P\gamma$ in vacuum.
For the decay processes $J\psi\rightarrow\eta_{c}\gamma$,
$\psi^{\prime}\rightarrow\eta^{\prime}_{c}\gamma$, and,
$\psi^{{}^{\prime\prime}}\rightarrow\eta^{\prime}_{c}\gamma$, the values of
$g_{PV}$ obtained are 2.094, 3.184 and 7.657 charmdw_mag , consistent with the
vacuum decay widths (in keV) of 92.9, 0.2058 and 24.48 respectively pdg . Due
to lack of data on radiative decay widths for the bottomonium states
($\Upsilon(NS)$ and $\eta_{b}(NS)$, for $N=1,2,3,4$), the masses of these
mesons, as modified due to PV mixing ($\Upsilon(NS)-\eta_{b}(NS)$,
$i=1,2,3,4$), are calculated using the equation (4), obtained from the
interaction Hamiltonian (3). The masses of the charmonium and bottomonium
states have been calculated including the effects of Dirac sea and PV mixing
in Ref. Heavy_Quarkonia_masses_MC . The in-medium decay widths of
$\psi(3770)\rightarrow D\bar{D}$ and $\Upsilon(4S)\rightarrow B\bar{B}$ have
been studied including the effects from the nucleon Dirac sea in Ref.
charmdw_mag_MC . These have been computed from the mass modifications of the
decaying and produced mesons, using a field theoretical model of composite
hadrons, as described in section 2.2. The PV mixing effects are also taken
into account for the study of mass modifications of the open heavy flavour
mesons, $D$, $\bar{D}$, $B$ and $\bar{B}$, arising from $D-D^{*}$,
$\bar{D}-{\bar{D^{*}}}$, $B-B^{*}$ and $\bar{B}-{\bar{B^{*}}}$ mixings for the
study of the partial decay widths of the heavy quarkonium states
charmdw_mag_MC .

The production cross-sections of the states $\psi(3770)$ and $\Upsilon(4S)$
are computed from the in-medium masses Heavy_Quarkonia_masses_MC , and, the
partial decay widths of these states to $D\bar{D}$ and $B\bar{B}$ respectively
charmdw_mag_MC . The DS and PV mixing effects are taken into account for the
calculation of the masses of the decaying and the produced mesons for
computing these partial decay widths. For the sake of completeness, we show
the masses and decay widths of the heavy quarkonia Heavy_Quarkonia_masses_MC ;
charmdw_mag_MC , which are used to calculate their production cross-sections.
In fig. 1, the effects of the Dirac sea as well as PV mixing on the mass, and,
the partial decay width of $\psi(3770)$ to $D\bar{D}$, are shown for both the
symmetric and asymmetric nuclear matter for $\rho_{B}=\rho_{0}$ charmdw_mag_MC
. In the absence of the DS and PV effects, the in-medium mass of the
charmoniun state, $\psi(3770)$ in the magnetized nuclear matter arises due to
the the medium modification of the scalar dilaton field, $\chi$ charmonium_mag
, which mimics the scalar gluon condensates of QCD within the chiral effective
model. As has already been mentioned, within the mean field approximation, the
dilaton field, $\chi$ and the scalar fields, $\sigma$, $\zeta$ and $\delta$
are solved from their equations of motion, accounting for Landau level
contributions for the proton. These are solved for given values of the baryon
density, $\rho_{B}$, isospin asymmetry parameter, $\eta$ and the magnetic
field, $\vec{B}=(0,0,B)$. In Refs. dmeson_mag ; charmonium_mag , the effects
of the anomalous magnetic moments of the nucleons and the isospin asymmetry of
the magnetized nuclear matter on the scalar fields were studied and observed
to be appreciable only for high densities and high values of the magnetic
field. For $\rho_{B}=\rho_{0}$, both for symmetric and asymmetric nuclear
matter, the medium modifications of the scalar fields are marginal and hence
the mass of the charmonium is observed to be insensitive to the variation of
the magnetic field charmonium_mag , as can be seen from figure 1. The PV
mixing effects on the mass of $\psi(3770)$ (due to mixing of the longitudinal
component of $\psi(3770)$ with $\eta_{c}(2S)$), are observed to be large, in
the presence as well as absence of the Dirac sea effects. When the PV mixing
effect is not taken into account, there is observed to be a monotonic drop in
the mass of $\psi(3770)$ with increase in the magnetic field with the
inclusion of Dirac sea effects, whereas, the mass remains almost unchanged in
the absence of nucleon Dirac sea effect. The PV mixing introduces a mass
difference between the longitudinal and transverse components of $\psi(3770)$.
The $\psi(3770)-\eta^{\prime}_{c}$ mixing leads to an increase in the mass of
the longitudinal component of $\psi(3770)$, whereas the transverse component
is unaltered due to the PV mixing. The decay width of $\psi(3770)\rightarrow
D\bar{D}$, which is obtained after averaging over the contributions from the
longitudinal and transverse components, is observed to rise appreciably with
increase in the magnetic field, both for symmetric and asymmetric nuclear
matter charmdw_mag_MC . The PV mixing leading to difference in masses of the
longitudinal and transverse components of the vector state $\psi(3770)$ is
observed as a distinct double peak structure in the production cross-sections
of $\psi(3770)$ plotted in figures 3 and 4 for $eB=3m_{\pi}^{2}$ and
$eB=8m_{\pi}^{2}$ respectively. The separation between the peaks is observed
to be larger for the higher value of the magnetic field, for which the PV
mixing is more appreciable.

In figure 2, the mass and decay width (to open bottom mesons) of
$\Upsilon(4S)$ are plotted as functions of $eB/{m_{\pi}^{2}}$ at
$\rho_{B}=\rho_{0}$ for symmetric and asymmetric nuclear matter, accounting
for the nucleon Dirac sea as well as PV mixing effects charmdw_mag_MC . In the
absence of the DS and PV effects, similar to $\psi(3770)$ mass (shown in
figure 1), the mass of $\Upsilon(4S)$ is also observed to be insensitive to
the variation of the magnetic field. This is a reflection of the fact that for
$\rho_{B}=\rho_{0}$, the change in the dilaton field, $\chi$ is negligible
with variation in the magnetic field charmonium_mag . The effects of PV mixing
are observed to be much smaller on the mass of $\Upsilon(4S)$, as compared to
the effects from the Dirac sea. This is contrary to the modification of the
mass of the charmonium state $\psi(3770)$, where both the PV mixing as well as
the Dirac sea effects have significant contributions to the mass of
$\psi(3770)$. There is observed to be a monotonic drop in the mass of
$\Upsilon(4S)$ with the increase in the magnetic field, when the PV mixing is
not taken into account, but the Dirac sea contributions to the nucleon self
energy are taken into consideration. The mass shift is observed to be much
smaller for the isospin asymmetric nuclear matter, as compared to symmetric
nuclear matter. In the presence of the Dirac sea contributions, the PV mixing
is observed to lead to a smaller drop as compared to when it is ignored for
$\eta$=0.5 case. However, for symmetric nuclear matter, there is observed to
be a positive contribution due to PV mixing upto $eB\sim 7m_{\pi}^{2}$, above
which the contribution turns out to be negative. This is because the effective
mass of the $\Upsilon(4S)$ turns out to be smaller than that of
$\eta_{b}(4S)$, as calculated within the chiral effective model, which makes
$\Delta E$ (and hence $\Delta M^{PV}$) of equation (4) to be negative. As
might be observed from figure 2, There is observed to be an initial increase
in the decay width of $\Upsilon(4S)$ with magnetic field upto $eB\sim
3m_{\pi}^{2}$, followed by a drop with further increase in the magnetic field
upto around $eB\sim 7m_{\pi}^{2}$. At still higher values of the magnetic
field, there is seen to be again a drop. The decay width as calculated from a
field theoretical model of composite hadrons, has the dependence on the center
of mass momentum, $|{\bf p}|$ (given by equation (8)), which depends on the
in-medium masses of the decaying and produced mesons, as a polynomial term
multiplied by an exponential part, which leads to the observed dependence of
the decay width on the magnetic field. The dotted lines show the decay width
when the Dirac sea effects are not taken into account. The values of the mass
as well as the decay width are observed to be quite close for
$eB=3m_{\pi}^{2}$ for both without and with the Dirac sea effects, which is
observed as similar behaviour for the production cross-section of
$\Upsilon(4S)$, as shown in figure 5. For $\eta$=0, the mass of $\Upsilon(4S)$
is observed to have much larger drop as compared to the $\eta$=0.5 case for
$eB=8m_{\pi}^{2}$, which is seen as a higher downward shift of the peak
position in the production cross-section of $\Upsilon(4S)$ in figure 6.

The production cross-section of $\psi(3770)$ is plotted in figures 3 and 4 for
$eB=3m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$ respectively. There is observed to be
a double peak structure of the production cross-section of $\psi(3770)$, which
is is observed to be significantly more pronounced for the higher value of the
magnetic field. This is due to the difference in the masses of the
longitudinal and transverse components of the vector meson (V) due to PV
mixing, for the $\psi(3770)-\eta^{\prime}_{c}$ mixing. The production cross-
section of $\Upsilon(4S)$ is plotted in figures 5 and 6 for $eB=3m_{\pi}^{2}$
and $eB=8m_{\pi}^{2}$ respectively. However, since the PV mixing is much
smaller compared to the Dirac sea effect for the bottomonium sector, the
difference in the longitudinal and transverse contributions to the production
cross-sections for $\Upsilon(4S)$ are observed to be marginal. There is
observed to be a much larger (downward) shift of the peak position due to
Dirac sea contributions, for the higher magnetic field of $eB=8m_{\pi}^{2}$
for the symmetric as compared to asymmetric nuclear matter, as can be seen in
figure 6. The maximun value of the cross-section is observed to drop when the
Dirac sea effects are considered for the higher magnetic field of
$eB=8m_{\pi}^{2}$ and the effect due to PV mixing, leading to different
contributions from the longitudinal and transverse components, is still
observed to be very small.

The PV mixing leads to different contributions from the longitudinal and
transverse components (L and T) for the radiative deay widths ($V\rightarrow
P\gamma$) for the processes. In figure 7, the radiative decay widths of the
charmonium states ($J/\psi\rightarrow\eta_{c}\gamma$,
$\psi^{\prime}\rightarrow\eta^{\prime}_{c}\gamma$, and
$\psi^{\prime\prime}\rightarrow\eta^{\prime}_{c}\gamma$) and in figures 8 and
9, for the bottomonium states (for the processes
$\Upsilon(NS)\rightarrow\eta_{b}(4S)\gamma$, $N=1,2,3,4$), are plotted as
functions of $eB/{m_{\pi}}^{2}$, showing the contributions from the
longitudinal (L) and transverse components. The radiative decay widths of the
heavy quarkonium states in the magnetized nuclear matter are calculated from
the mass modifications of the vector and pseudoscalar states and the total
radiative decay width is obtained by averaging over the contributions from the
transverse and longitudinal components using equation (18). The dependence of
the radiative decay widths on the masses of the vector and pseudoscalar mesons
are through the center of mass momenta, $p_{\rm{cm}}^{L}$ and
$p_{\rm{cm}}^{T}$ for the longitudinal and transverse components of the vector
meson, which are different due to the difference in the masses of these
components arising from the PV mixing. When the DS effects are not taken into
account, the masses of the heavy quarkonium states for both the symmetric and
asymmetric nuclear matter have negligible dependence on the variation of the
magnetic field in the absence of PV mixing effect. This leads to the mass of
the longitudinal component of the vector meson (calculated using equation (2))
also to be very similar in behaviour with change in the magnetic field, for
the symmetric and asymmetric nuclear matter, as can be seen from figures 1 and
2. These lead to the radiative decay widths also to have similar dependence on
the magnetic field for symmetric as well as asymmetric nuclear matter, both
for the longitudinal and transverse components, hence for the total width
which is calculated by averaging over the decay widths of these components, by
using equation (18). In the presence of the Dirac sea contributions, however,
significant drops in the masses of $\psi(3770)$ as well as of $\Upsilon(4S)$
with increase in the magnetic field can be observed from figures 1 and 2, when
the PV mixing is not taken into account. The PV mixing leads to a significant
increase in the mass of the longitudinal component of $\psi(3770)$, whereas,
for $\Upsilon(4S)$, the rise is observed to be rather moderate. For
$\rho_{B}=\rho_{0}$, the values of the masses (in MeV) of the psuodoscalar
meson $\eta_{c}^{\prime}$ (longitudinal component of $\psi(3770)$) are
observed to be 3548.4 (3638.5) at zero magnetic field and 3185.1 (3624.2) for
$eB=10m_{\pi}^{2}$ in symmetric nuclear matter and 3551.4 (3643) at $eB=0$ and
3247.7 (3680.5) for $eB=10m_{\pi}^{2}$ in asymmetric nuclear matter (with
$\eta$=0.5). For the transverse component of $\psi(3770)$, which is unaffected
by PV mixing, the mass (in MeV) is 3638.5 (3402.8) for $\eta$=0 and 3643
(3474.8) for $\eta$=0.5 at $eB=0(10m_{\pi}^{2})$. The value of
$p_{\rm{cm}}^{L}(p_{\rm{cm}}^{T})$ (in MeV) is modified from 88.9 for zero
magnetic field to 412.5 (210.7) for $eB=10m_{\pi}^{2}$ in symmetric ($\eta$=0)
nuclear matter and from 90.4 at $B$=0 to 407.35 (219.63) for
$eB=10m_{\pi}^{2}$ in asymmetric ($\eta$=0.5) nuclear matter. For the same
values of the magnetic fields, the values of the decay widths (in MeV) for the
longitudinal and transverse components, which are proportional to
${p_{\rm{cm}}^{L}}^{3}$ and ${p_{\rm{cm}}^{T}}^{3}$ respectively, (as can be
seen from equation (18)), are modified from 7.7725$\times 10^{-3}$ for $eB=0$
to and 0.86736 (0.115636) at $eB=10m_{\pi}^{2}$ for symmetric nuclear matter
and from 8.15$\times 10^{-3}$ for zero magnetic field to 0.80654 (0.1264) at
$eB=10m_{\pi}^{2}$ for asymmetric ($\eta$=0.5) nuclear matter. As can be seen
from figures 8 and 9, the radiative decay widths of
$\Upsilon(NS)\rightarrow\eta_{b}(NS)\gamma$, $N=1,2,3,4$ have similar
qualitative dependence on the magnetic field for the symmetric and asymmetric
nuclear matter. The modifications due to the DS effect is observed to be
moderate for the radiative decay widths of the charmonium and bottomonium
states investigated in the present work, except for
$\Upsilon(3S)\rightarrow\eta_{b}(3S)\gamma$, where the effect due to the
nucleon Dirac sea contributions is observed to be appreciable at higher values
of the magnetic field, both for the symmetric as well as asymmetric nuclear
matter. In the presence of the Dirac sea contributions, for $\eta=0$, it is
observed that the decay width $\Upsilon(4S)\rightarrow\eta_{b}(4S)\gamma$ is
not possible, for $eB$ larger than $7m_{\pi}^{2}$, as might be seen from
figure 9. This is because the mass of $\eta_{b}(4S)$ (modified due to PV
mixing) becomes larger than the mass of both the transverse (without PV
mixing) and longitudinal component (modified due to the PV mixng)
$\Upsilon(4S)$, because of which the center of mass momenta of radiative
decays (given by equaiton (19)) are no longer positive definite. There is
observed to be increase in the radiative decay widths in both the charm and
bottom sectors. The production cross-sections are observed to be modified
appreciably for the vector charmonium state, $\psi(3770)$, due to Dirac sea
(DS) as well as PV mixing effects. There is observed to be distinct double
peak structure in the invariant mass spectrum of production cross-section of
$\psi(3770)$ due to the PV mixing, which might be observed in dilepton
spectra. For the bottom sector, the DS effect is observed to be the dominant
magnetic field effect. The observed enhancement of the radiative decay widths
at high magnetic fields can affect the production of the vector and
pseudoscalar heavy quarkonium states and the modifications of the production
corss-sections of the states $\psi(3770)$ and $\Upsilon(4S)$ can affect the
yields of the heavy quarkonium states and the open heavy flavour mesons in
peripheral ultra-relativistic heavy ion collision experiments, where the
magnetic fields created can be huge.

##  VI Summary

To summarize, we have studied the production cross-sections and radiative
decay widths of the heavy quarkonium (charmonium and bottomonium) states in
magnetized nuclear matter. The effects of Dirac sea (DS) of nucleons as well
as the PV mixing (mixing of the pseudoscalar meson and the longitudinal
component of the vector meson) are taken into consideration for obtaining the
mass modifications of the heavy quarkonium states as well as open heavy
flavour mesons within a chiral effective model. The anomalous magnetic moments
(AMMs) of the nucleons are considered in the present study. The production
cross-sections of $\psi(3770)$ and $\Upsilon(4S)$ are obtained from the masses
of these mesons, as well as, the partial decay width of
$\psi(3770)$($\Upsilon(4S)$) to $D\bar{D}$($B\bar{B}$), the latter computed
using a field theoretical model of composite hadrons. Contrary to the charm
sector, where both PV mixing and the DS contributions are observed to be
important, it is seen that the PV mixing has much smaller contribution as
compared to the Dirac sea contributions for the bottomonium states. The PV
mixing in the charm sector ($\psi(3770)-\eta_{c}(2S)$ mixing), which modifies
the mass of only the longitudinal component of the vector charmonium state
$\psi(3770)$, is observed to be quite significant leading to appreciable
difference in the masses of the longitudinal and the transverse components of
the state ($\psi(3770)$). This is seen as a double peak in the invariant mass
spectrum of the production cross-section of $\psi(3770)$, which is observed to
be much more pronounced as the magnetic field is increased. If the produced
magnetic field is strong enough, this might be observed in the dilepton
spectra as distinct peaks corresponding to the longitudinal and transverse
components of the state $\psi(3770)$. For the bottom sector, in the presence
of the magnetic field, the Dirac sea (DS) effect is the more dominant effect
as compared to the PV mixing effect. The radiative decay widths of the vector
(V) heavy quarkonium state to pseudoscalar (P) are calculated from the medium
modifications of the masses of these mesons. There is observed to be
appreciable increase in the values of the radiative decay widths in both the
charm and bottom sectors. The modifications of the production cross-sections
and the radiative decay widths of the heavy quarkonia in the magnetized matter
should have observable consequences on the production of heavy quarkonium
states and open heavy flavour mesons, as these are created at the early stage
of the non-central ultra-relativistic heavy ion collision experiments, when
the magnetic field can be still be extremely large.

## Acknowledgements

Amruta Mishra acknowledges financial support from Department of Science and
Technology (DST), Government of India (project no. CRG/2018/002226) and Ankit
Kumar from University Grants Commission (UGC), Government of India (Ref.
1279/(CSIR-UGC NET June 2017)).

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