#  $c\bar{b}$ spectrum and decay properties with coupled channel effects

Antony Prakash Monteiro<EMAIL_ADDRESS>Manjunath Bhat  K. B.
Vijaya Kumar  P. G. Department of Physics, St Philomena college Darbe, Puttur
574 202, India  Department of Physics, Mangalore University, Mangalagangothri
P.O., Mangalore - 574199, INDIA

###### Abstract

The mass spectrum of $c\bar{b}$ states has been obtained using the
phenomenological relativistic quark model (RQM) with coupled channel effects.
The Hamiltonian used in the investigation has confinement potential and
confined one gluon exchange potential (COGEP). In the frame work of RQM a
study of M1 and E1 radiative decays of $c\bar{b}$ states has been made. The
weak decay widths in the spectator quark approximation have been estimated. An
overall agreement is obtained with the experimental masses and decay widths.

###### keywords:

relativistic quark model (RQM); radiative decay; confined one gluon exchange
potential (COGEP); $B_{c}$ meson states

††journal: Journal of LaTeX Templates

top=0.75in,left=0.80in,bottom=0.75in,right=0.80in

##  1 INTRODUCTION

The $B_{c}$ meson is a double heavy quark-antiquark bound state and carries
flavours explicitly and provides a good platform for a systematic study of
heavy quark dynamics. $B_{c}$ mesons are predicted by the quark model to be
members of the $J^{P}=0^{-}$ pseudo scalar ground state multiplet [1]. The
first successful observation of $B_{c}$ meson was made by CDF collaboration in
1998 from run I at TEVATRON through the semileptonic decay channel
$B_{c}\rightarrow J/\Psi+l^{+}+\bar{\nu}_{l}$ [2]. They measured the mass of
$B_{c}$ to be $m_{B_{c}}=6400\pm 390\pm 130$ MeV and the life time
$\tau_{B_{c}}=0.46^{+0.18}_{-0.16}\pm 0.03$ ps. The more precise measurement
of mass of $B_{c}$ i.e.,$m_{B_{c}}=6275.6\pm 2.9(\rm stat)\pm 5(\rm syst)$
$\rm MeV$ was done by the CDF collaboration through the exclusive non-leptonic
decay $B_{c}\rightarrow J/\Psi\pi^{+}$[3, 4, 5]. The results of the CDF
collaboration was confirmed by the observations made by the D0 collaboration
[6, 7] at TEVATRON. The LHCb has reported several new observations on $B_{c}$
decays recently. More experimental data on $B_{c}$ meson are expected in near
future from LHCb and TEVATRON.

A suitable theoretical model is required to explain the properties such as
mass spectrum, decays, reaction mechanism and bound state behaviour of mesons
which involve heavy quarks. The properties of the light and heavy mesons were
studied using the phenomenological models. A De Rujula et al [8] proposed
first QCD based model for the study of hadron spectroscopy. The model had a
reasonable success and predicted the masses of charmed mesons and baryons.
Several non-relativistic phenomenological potentials with radial dependencies
for the confinement along with one gluon exchange potential (OGEP) were
examined by Bhaduri et al [9]. The ground state heavy meson spectrum has been
studied by Vijaya Kumar et al [10]. Radiative decay properties of light vector
mesons have been studied by Monteiro et al [11]. Bottomonium spectrum and its
decay properties have been studied in a non relativistic model using OGEP by
Monteiro et al [12]. Bhagyesh et al [13] in their non relativistic model used
Hulthen potential to study the orbitally excited quarkonium states. In these
models the relativistic effects were completely ignored.

There have been many calculations of baryon properties using relativistic
models, like MIT bag models [14, 15], cloudy bag models[16, 17], chiral bag
models [18, 19] etc. Relativistic calculations, where constituent quarks are
confined in a potential, have also been performed [20, 21, 22]. There are
other bag models in literature too. In Budapest bag model the volume energy
term is replaced by a surface energy term [23]. Another model which
effectively contains a surface tension term is the ’SLAC’ bag, developed by
Bardeen et al [24] which begins from a local field theory in which heavy
quarks interact through a neutral scalar field. Ferreria et al [25, 26] used
relativistic quark model to study several properties of low lying hadrons. In
this model both, the linear and quadratic confinement schemes were used.
Bander et al [27] used a relativistic bound state formalism to make
simultaneous study of all meson systems. Isgur et al [28, 29] in their
relativized quark model used a parametrized potential and incorporated
relativistic kinematics to describe all mesons in the same frame work.

In NRQM formalism though the mass spectra of ground state $c\bar{b}$ meson has
been produced successfully, the radiative decay rates, particularly hindered
$M1$ decay rates are significantly influenced by relativistic effects.
Therefore, it is necessary to include these effects for the correct
description of the decays. Radiative decays are the most sensitive to
relativistic effects. Hindered radiative decays are forbidden in the non
relativistic limit due to the orthogonality of initial and final meson wave
functions. They have decay rates of the same order as the allowed ones. In the
relativistic description of mesons an important role is played by the
confining quark-antiquark interaction, particularly its Lorentz structure.
Thus comparison of theoretical predictions with experimental data can provide
valuable information on the form of the confining potential. Hence we use
relativistic quark model formalism to study the properties of $c\bar{b}$ meson
states.

The paper is organized in 4 sections. In sec. 2 we briefly review the
theoretical background for relativistic model, the framework of the coupled-
channel analysis and the relativistic description of radiative decay widths.
In sec. 3 we discuss the results and the conclusions are drawn in sec. 4 with
a comparison to other models.

##  2 THEORETICAL BACKGROUND

###  2.1 The Relativistic Harmonic Model

We investigate properties of $c\bar{b}$ states using confined one gluon
exchange potential in the frame work of relativistic harmonic model (RHM)
[20]. The Hamiltonian used has the confinement potential and a two body
confined one gluon exchange potential(COGEP) [30, 31, 32, 33].

The confinement potential has Lorentz scalar and a vector harmonic oscillator
potential part[34, 35]

$V_{CONF}(r)=\frac{1}{2}\left(1+\gamma_{0}\right)A^{2}r^{2}+M$ (1)

where $\gamma_{0}$ is the Dirac matrix, M is a constant mass and $A^{2}$ is
the confinement strength.

We use the following harmonic oscillator wave equation

$\left(\frac{p^{2}}{E+M}+A^{2}r^{2}\right)\phi=(E-M)\phi$ (2)

the eigenvalue of which is given by

$E^{2}_{N}=M^{2}+(2N+1)\Omega_{N}$ (3)

where $\Omega_{N}$ is the energy dependent oscillator size parameter given by

$\Omega_{N}=A(E_{N}+M)^{1/2}$ (4)

where $\vec{p}$ is the momentum. For the detailed description of RHM see[20,
34, 35].

###  2.2 Confined One Gluon Exchange Potential

In the present existing models for low energy nuclear phenomena the gluon
degrees of freedom have been eliminated from the theoretical frame work and it
is assumed that the gluon exchange can be incorporated into the theory through
OGEP. But in deriving the OGEP[8] the gluon propagators used are similar to
the free photon propagators used in obtaining Fermi-Breit interaction in QED.
Since the confinement of color means the confinement of quarks as well as
gluons, the confined dynamics of gluons should play a decisive role in
determining the hadron spectrum and in the hadron-hadron interaction. The
confinement schemes for quarks and gluons have to be more closely connected to
each other in QCD and the confinement of gluons has to be taken into account.
The COGEP is obtained from the scattering amplitude using confined gluon
propagators[30, 31, 32, 36]. Here $D^{ab}_{\mu\nu}=\partial^{ab}D_{\mu\nu}$
are the gluon propagators in the momentum representation in current
confinement model (CCM)[37]. The CCM was developed for the confinement of
gluons in the spirit of RHM and aims at a unified confinement theory for the
study of quark-gluon bond system in the spirit of RHM for the confinement of
gluons. In the CCM the coupled non-linear terms in the Yang-Mill tensor is
treated as a color gluon super current in analog with Ginzburg-Landu’s theory
of superconductivity. The coupled non-linear terms in the equation of motion
of a gluon are simulated by a self induced color current
$j_{\mu=\theta^{\vartheta}_{\mu}}A_{\nu}$(=$m^{2}A_{\mu}$) or equivalently an
effective mass term for all the gluons with
$m^{2}=c^{4}r^{2}-2c^{2}\delta_{\mu 0}$. The equation of motion is solved
using harmonic oscillator modes in the general Lorentz gauge imposing a
secondary gauge condition termed ‘oscillator gauge’. The two confined gluon
propagators are then obtained in this gauge using the property of the harmonic
oscillator wave functions. The RHM with COGEP has been quite successful in
obtaining the N-N phase shifts and in hadron spectroscopy[32].

The COGEP is obtained from the scattering amplitude [30, 31, 32, 33]

${\cal
M}_{fi}=\frac{g^{2}_{s}}{4\pi}\bar{\psi^{\prime}}_{1}\gamma^{\mu}\psi_{1}D^{ab}_{\mu\nu}(q)\bar{\psi^{\prime}}_{2}\gamma^{\nu}\psi_{2}$
(5)

where $\bar{\psi}=\psi^{\dagger}\gamma_{0}$, $\psi_{1,2}$ are the wave
functions of the quarks in RHM. The $D_{00}(q)~{}\rm and~{}D_{ik}(q)$ are the
zero energy CCM gluon propagators in momentum representation, where
$q=P^{\prime}_{1}-P_{1}=P_{2}-P^{\prime}_{2}$ is the four momentum transfer.
$g^{2}_{s}/4\pi=\alpha_{s}$ is the quark gluon coupling constant. In CCM,
propagators in the momentum representation are given by,

$D_{00}(q)=4\pi D_{0}(q)$ (6)

The $D_{ik}(q)$ are given by,

$D_{ik}(q)=-4\pi\left\\{\delta_{ik}-\frac{a^{\dagger}_{q_{i}}a_{q_{k}}}{a_{q}\cdot
a^{\dagger}_{k}}\right\\}D_{1}(q)$ (7)

where $a_{q}$ and $a^{\dagger}_{q}$ are the creation and destruction operators
in the momentum space.
The scattering amplitude (5) is written as

$\begin{split}{\cal
M}_{fi}&=\frac{g^{2}_{s}}{4\pi}(\psi^{\prime\dagger}_{1}\psi_{1}\psi^{\prime\dagger}_{2}\psi_{2})D_{00}(q)+\\\
&(\psi^{\prime\dagger}_{1}\alpha_{i}\psi_{1})(\psi^{\prime\dagger}_{2}\alpha_{k}\psi_{2})D_{ik}(q)\end{split}$
(8)

We express the 4-component RHM wave function $\psi$ in terms of 2-component
wave function $\phi$ by a similarity transformation.
i.e.

$\displaystyle\psi^{\prime\dagger}_{1}\psi_{1}$
$\displaystyle=\psi^{\prime\dagger}_{1}U^{\prime\dagger}_{1}(U^{\prime\dagger}_{1})^{-1}U^{-1}_{1}U_{1}\psi_{1}$
(9)
$\displaystyle=\phi^{\prime\dagger}_{1}(U^{\prime\dagger}_{1})^{-1}U^{-1}_{1}\phi_{1}$
(10)

where

$N=\sqrt{\frac{2(E+M)}{3E+M}}$ (11)

and

$U=\frac{1}{N\left[1+\frac{p^{2}}{(E+M)^{2}}\right]}\left(\begin{array}[]{cc}1&\frac{{\bf\sigma}\cdot{\bf
p}}{E+M}\\\ -\frac{{\bf\sigma}\cdot{\bf p}}{E+M}&1\end{array}\right)$ (12)

The above expression can be simplified to

$\psi^{\prime\dagger}_{1}\psi_{1}=N^{2}\phi^{\prime\dagger}_{1}\left\\{1+\left[\frac{P^{2}_{1}+q\cdot
P_{1}+i\sigma_{1}\cdot(q\times P_{1})}{(E+M)^{2}}\right]\right\\}\phi_{1}$
(13)

We have,

$\psi^{\prime\dagger}_{2}\psi_{2}=\phi^{\prime\dagger}_{2}(U^{\prime\dagger}_{2})^{-1}U^{-1}_{2}U_{2}\phi_{2}$
(14)

i.e.

$\psi^{\prime\dagger}_{2}\psi_{2}=N^{2}\phi^{\prime\dagger}_{2}\left\\{1+\left[\frac{P^{2}_{2}+q\cdot
P_{2}+i\sigma_{2}\cdot(q\times P_{2})}{(E+M)^{2}}\right]\right\\}\phi_{2}$
(15)

Similarly we can write,

$\displaystyle\psi^{\prime\dagger}_{1}\alpha_{i}\psi_{1}=\frac{N^{2}}{(E+M)}\left[\phi^{\prime\dagger}_{1}\left[2P_{1}+q+i(\sigma_{1}\times
q)\right]\phi_{1}\right]_{i}$ (16)
$\displaystyle\psi^{\prime\dagger}_{2}\alpha_{k}\psi_{2}=\frac{N^{2}}{(E+M)}\left[\phi^{\prime\dagger}_{2}\left[2P_{2}-q-(i\sigma_{2}\times
q)\right]\phi_{2}\right]_{k}$ (17)

Substituting (13), (15), (16) and (17) in (8), the scattering amplitude now
expressed in terms of the two component spinor $\phi$ and the momentum
dependent operator $U$ can be written as,

${\cal
M}_{fi}=4\pi\alpha_{s}N^{4}\phi^{\dagger}_{1}\phi^{\dagger}_{2}\left[U[P_{1},P_{2},q]\right]\phi_{1}\phi_{2}$
(18)

The function $U(P_{1},P_{2},q)$ is the particle interaction operator in the
momentum representation and by taking the Fourier transform of each term in
the scattering amplitude we get the potential operator
$U(\hat{P}_{1},\hat{P}_{2},r)$ in the co-ordinate space. We drop all the
higher order momentum dependent terms in $U(\hat{P}_{1},\hat{P}_{2},r)$ to
obtain the scattering amplitude which is given by

$\begin{split}{\cal
M}_{fi}&=4\pi\alpha_{s}N^{4}\left[1+\frac{1}{(E+M)^{2}}\left[\sigma_{1}\cdot(\nabla\times\hat{P}_{1})-\sigma_{2}\cdot(\nabla\times\hat{P}_{2})\right]\right]D_{0}(\vec{r})+4\pi\alpha_{s}N^{4}\\\
&\times\left[\frac{1}{(E+M)^{2}}\left[2\sigma_{2}\cdot(\nabla\times\hat{P}_{1})-2\sigma_{1}\cdot(\nabla\times\hat{P}_{2})-\nabla^{2}[1-\sigma_{1}\cdot\sigma_{2}]-(\sigma_{1}\cdot\nabla)(\sigma_{2}\cdot\nabla)\right]D_{1}(\vec{r})\right]\end{split}$
(19)

The terms which contribute to the central part of COGEP are,

$D_{0}(\vec{r})$, $\nabla^{2}[\sigma_{1}\cdot\sigma_{2}-1]D_{1}(\vec{r})$ and
$(\sigma_{1}\cdot\nabla)(\sigma_{2}\cdot\nabla)D_{1}(\vec{r})$

In CCM the propagator $D_{1}(\vec{r})$ satisfies the differential equation

$(-\nabla^{2}+c^{4}r^{2})D_{1}(\vec{r})=4\pi\delta^{3}(\vec{r})$ (20)

The term $(\sigma_{1}\cdot\nabla)(\sigma_{2}\cdot\nabla)D_{1}(\vec{r})$, has
angular dependence. But the tensor operator is constructed in such a way that
the average value of tensor operator over the angular variables vanishes. The
averaging over the direction of r gives

$(\sigma_{1}\cdot\nabla)(\sigma_{2}\cdot\nabla)D_{1}(\vec{r})=(1/3)\sigma_{1}\cdot\sigma_{2}[\nabla^{2}D_{1}(\vec{r})]$
(21)

Substituting for $[\nabla^{2}D_{1}(\vec{r})]$, the central part of the COGEP
becomes

$V^{cent}_{COGEP}(\vec{r})=\frac{\alpha_{s}N^{4}}{4}\vec{\lambda_{i}}\cdot\vec{\lambda_{j}}\left[D_{0}(\vec{r})+\frac{1}{(E+M)^{2}}\left[4\pi\delta^{3}(\vec{r})-c^{4}r^{2}D_{1}(\vec{r})\right]\left[1-\frac{2}{3}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\right]\right]$
(22)

where $D_{0}(\vec{r})$ and $D_{1}(\vec{r})$ are the propagators given by

$\displaystyle
D_{0}(\vec{r})=\frac{\Gamma_{1/2}}{4\pi^{3/2}}c(cr)^{-3/2}W_{1/2;-1/4}(c^{2}r^{2})$
(23) $\displaystyle
D_{1}(\vec{r})=\frac{\Gamma_{1/2}}{4\pi^{3/2}}c(cr)^{-3/2}W_{0;-1/4}(c^{2}r^{2})$
(24)

where$\lambda_{i}$ and $\lambda_{j}$ are color matrices,
$\Gamma_{1/2}=\sqrt{\pi}$, W’s are Whittaker functions and $c$(fm-1) is a
constant parameter which gives the range of propagation of gluons and is
fitted in the CCM to obtain the glue-ball spectra and r is the distance from
the confinement center.

The terms which contribute to the spin orbit part of the COGEP are

$[\sigma_{1}\cdot(\nabla\times\hat{P}_{1})-\sigma_{2}\cdot(\nabla\times\hat{P}_{2})]D_{0}(\vec{r})+[2\sigma_{2}\cdot(\nabla\times\hat{P}_{1})-2\sigma_{1}\cdot(\nabla\times\hat{P}_{2})]D_{1}(\vec{r})$
(25)

Operating $\nabla$ on $D_{0}(\vec{r})$ and $D_{1}(\vec{r})$ and defining

$\hat{P}=(\hat{P}_{1}-\hat{P}_{2})/2~{}\rm{and}~{}\hat{P}_{CM}=\hat{P}_{1}+\hat{P}_{2}$

The spin orbit part of COGEP is

$\begin{split}V^{LS}_{12}(\vec{r})=\frac{\alpha_{s}}{4}\frac{N^{4}}{(E+M)^{2}}\frac{\lambda_{1}\cdot\lambda_{2}}{2r}\times\left[[r\times(\hat{P}_{1}-\hat{P}_{2})\cdot(\sigma_{1}+\sigma_{2})](D^{\prime}_{0}(\vec{r})+2D^{\prime}_{1}(\vec{r}))+\right.\\\
\left.[r\times(\hat{P}_{1}+\hat{P}_{2})\cdot(\sigma_{1}-\sigma_{2})](D^{\prime}_{0}(\vec{r})-D^{\prime}_{1}(\vec{r}))\right]\end{split}$
(26)

The spin orbit term has been split into the symmetric
$(\sigma_{1}+\sigma_{2})$ and anti symmetric $(\sigma_{1}-\sigma_{2})$ terms.

The terms which contribute to the tensor part of the COGEP are,

$\left[(\sigma_{1}\cdot\nabla)(\sigma_{2}\cdot\nabla)D_{1}(\vec{r})-(\frac{1}{3}\sigma_{1}\cdot\sigma_{2}[\nabla^{2}D_{1}(\vec{r})])\right]$
(27)

The tensor part of the COGEP is,

$V^{TEN}_{12}(\vec{r})=-\frac{\alpha_{s}}{4}\frac{N^{4}}{(E+M)^{2}}\lambda_{1}\cdot\lambda_{2}\left[\frac{D^{\prime\prime}_{1}(\vec{r})}{3}-\frac{D^{\prime}_{1}(\vec{r})}{3r}\right]S_{12}$
(28)

where

$S_{12}=[3(\sigma_{1}\cdot\hat{r})(\sigma_{2}\cdot\hat{r})-\sigma_{1}\cdot\sigma_{2}]$
(29)

###  2.3 Coupled Channel Effects

In this section we briefly review coupled channel models. For detailed
discussions on coupled channel effects see [38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52].

Current QCD inspired potential models generally neglect the hadron loop
effects (continuum couplings). These couplings lead to two body strong decays
of the meson above threshold and below threshold they give rise to mass shifts
of the bare meson states.

In the coupled channel model, the full hadronic state is given by [46, 48, 49]

$\lvert\psi\rangle=\lvert A\rangle+\sum_{BC}\lvert BC\rangle$ (30)

where we have considered open flavour strong decay $A\to BC$. Here A, B, C
denote mesons.

The wave function $\lvert\psi\rangle$ obeys the equation

$H\lvert\psi\rangle=M\lvert\psi\rangle$ (31)

The Hamiltonian H for this combined system consists of a valence Hamiltonian
$H_{0}$ and an interaction Hamiltonian $H_{I}$ which couples the valence and
continuum sectors.

$H=H_{0}+H_{I}$ (32)

where

$H_{I}=g\int d^{3}x\bar{\psi}\psi$ (33)

The matrix element of the valence-continuum coupling Hamiltonian is given by
[48, 49]

$\langle BC\rvert H_{I}\lvert
A\rangle=h_{fi}\delta(\vec{P}_{A}-\vec{P}_{B}-\vec{P}_{C})$ (34)

where $h_{fi}$ is the decay amplitude.

The mass shift of hadron A due to its continuum coupling to BC can be
expressed in terms of partial wave amplitude ${\cal M}_{LS}$ [46, 49]

$\displaystyle\Delta
M_{A}^{(BC)}=\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}-i\epsilon}\int
d\Omega_{p}|h_{fi}(p)|^{2}$
$\displaystyle\quad=\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}-i\epsilon}\sum_{LS}|{\cal
M}_{LS}|^{2}$ $\Delta M_{A}^{(BC)}={\cal
P}\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}}\sum_{LS}|{\cal
M}_{LS}|^{2}+i\pi\left(\frac{p*E_{B}*E_{C}}{M_{A}}\sum_{LS}|{\cal
M}_{LS}|^{2}\right)|_{E_{B}+E_{C}=M_{A}}$ (35)

The decay amplitude $h_{fi}$ can be combined with relativistic phase space to
give the differential decay rate, which is

$\frac{d\Gamma_{A\to BC}}{d\Omega}=2\pi P\frac{E_{B}E_{C}}{M_{A}}|h_{f}i|^{2}$
(36)

where in the rest frame of A, we have $\vec{P}_{A}=0$ and
$P=|\vec{P}_{B}|=|\vec{P}_{C}|$.

$P=\sqrt{[M^{2}_{A}-(M_{B}+M_{C})^{2}][M^{2}_{A}-(M_{B}-M_{C})^{2}]}/(2M_{A})$
(37)

The total decay rate is given by [46, 49]

$\Gamma_{A\to BC}=2\pi P\frac{E_{B}E_{C}}{M_{A}}\sum_{LS}|{\cal M}_{L}S|^{2}$
(38)

###  2.4 Radiative Decays

Radiative decays are a powerful tool for the study of the quark structure of
mesons, and the calculation of corresponding amplitudes is a subject of the
increasing interest. We consider two types of radiative transitions of the
$B_{c}$ meson:

a) Electric dipole (E1) transitions are those transitions in which the orbital
quantum number is changed ($\Delta L=1$, $\Delta S=0$). E1 transitions do not
change quark spin. Examples of such transitions are $n^{3}S_{1}\rightarrow
n^{\prime}{{}^{3}P_{J}}\gamma(n>n^{\prime})$ and $n^{3}P_{J}\rightarrow
n^{\prime}{{}^{3}S_{1}}\gamma(n\geq n^{\prime})$. The partial widths for
electric dipole (E1) transitions between states ${}^{2S+1}L_{iJ_{i}}$ and
${}^{2S+1}L_{fJ_{f}}$ are given by

$\begin{split}\Gamma_{a\rightarrow
b\gamma}=\frac{4\alpha}{9}\mu^{2}\left(\frac{Q_{c}}{m_{c}}-\frac{Q_{\bar{b}}}{m_{\bar{b}}}\right)^{2}\frac{E_{b}(k_{0})}{m_{a}}k^{3}_{0}\left|\langle
b\rvert r\lvert a\rangle\right|^{2}\\\
\left\\{\begin{array}[]{cl}(2J+1)/3,&{}^{3}S_{1}\rightarrow^{3}P_{J}\\\
1/3,&{}^{3}P_{J}\rightarrow^{3}S_{1}\\\
1/3,&{}^{1}P_{1}\rightarrow^{1}S_{0}\\\
1,&{}^{1}S_{0}\rightarrow^{1}P_{1}\end{array}\right.\end{split}$ (39)

where $k_{0}$ is the energy of the emitted photon,

$k_{0}=\frac{m^{2}_{a}-m^{2}_{b}}{2m_{a}}$ in relativistic model.

$\alpha$ is the fine structure constant. $Q_{c}=2/3$ is the charge of the c
quark and $Q_{\bar{b}}=1/3$ is the charge of the $\bar{b}$ quark in units of
$|e|$, $\mu$ is reduced mass, $m_{\bar{b}}$ and $m_{c}$ are the masses of b
quark and c quar respectively, $m_{a}$ and $m_{b}$ are the masses of initial
and final mesons.

$\mu=\frac{m_{\bar{b}}m_{c}}{m_{\bar{b}}+m_{c}}$

and

$\frac{E_{b}(k_{0})}{m_{a}}=1$ $\langle b\rvert r\lvert
a\rangle=\int^{\infty}_{0}r^{3}R_{b}(r)R_{a}(r)dr$ (40)

is the radial overlap integral which has the dimension of length, with
$R_{a,b}(r)$ being the normalized radial wave functions for the corresponding
states.

b) Magnetic dipole (M1) transitions are those transitions in which the spin of
the quarks is changed ($\Delta S=1,~{}~{}\Delta L=0$) and thus the initial and
final states belong to the same orbital excitation but have different spins.
Examples of such transitions are vector to pseudo scalar
($n~{}^{3}S_{1}\rightarrow n^{\prime}~{}^{1}S_{0}+\gamma$, $n\geq n^{\prime}$)
and pseudo scalar to vector ($n~{}^{1}S_{0}\rightarrow
n^{\prime}~{}^{3}S_{1}+\gamma$, $n>n^{\prime}$) meson decays.
The magnetic dipole amplitudes between $S$-wave states are independent of the
potential model.

The M1 partial decay width between S wave states is [53, 54, 55, 56, 57, 58,
59]

$\begin{split}\Gamma_{a\to b\gamma}=\delta_{L_{a}L_{b}}4\alpha
k^{3}_{0}\frac{E_{b}(k_{0})}{m_{a}}\left(\frac{Q_{c}}{m_{c}}+(-1)^{S_{a}+S_{b}}\frac{Q_{\bar{b}}}{m_{\bar{b}}}\right)^{2}(2S_{a}+1)\times(2S_{b}+1)(2J_{b}+1)\\\
\left\\{\begin{array}[]{ccc}S_{a}&L_{a}&J_{a}\\\
J_{b}&1&S_{b}\end{array}\right\\}^{2}\left\\{\begin{array}[]{ccc}1&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&S_{a}&S_{b}\end{array}\right\\}^{2}\times\left[\int^{\infty}_{0}R_{n_{b}L_{b}}(r)r^{2}j_{0}(kr/2)R_{n_{a}L_{a}}(r)dr\right]^{2}\end{split}$
(41)

where $\int^{\infty}_{0}drR_{n_{b}L_{b}}(r)r^{2}j_{0}(kr/2)R_{n_{a}L_{a}}(r)$
is the overlap integral for unit operator between the coordinate wave
functions of the initial and the final meson states, $j_{0}(kr/2)$ is the
spherical Bessel function. $S_{a}$, $S_{b}$, $L_{a}$, $J_{a}$ and $J_{b}$ are
the spin quantum number, orbital angular momentum and total angular momentum
of initial and final meson states respectively.

###  2.5 Weak Decays

The weak decays of mesons provide information about the underlying quark
dynamics within the system. The weak decays of bound state of a quark and an
anti-quark, which carries heavy flavour c and b - enable us to probe the
validity of the standard model of elementary particle interactions and
determine several parameters of this model. A rough estimate of the $B_{c}$
weak decay widths can be done by treating the $\bar{b}$-quark and $c$-quark
decay independently so that $B_{c}$ decays can be divided into three classes
[60, 61]$\colon$ (i)the $\bar{b}$-quark decay with spectator $c$-quark, (ii)
the $c$-quark decay with spectator $\bar{b}$-quark, and (iii) the annihilation
$B^{+}_{c}\rightarrow l^{+}\nu_{l}$ ($c\bar{s},~{}u\bar{s}$), where
$l=e,~{}\mu,~{}\tau$.

##  3 Results and Discussion

###  3.1 Mass Spectrum of $c\bar{b}$ states with coupled-channel effects

The quark-antiquark wave functions in terms of oscillator wave functions
corresponding to the relative and center of mass coordinates have been
expressed here, which are of the form,

$\Psi_{nlm}(r,\theta,\phi)=N(\frac{r}{b})^{l}~{}L_{n}^{l+\frac{1}{2}}(\frac{r^{2}}{b^{2}})\exp(-\frac{r^{2}}{2b^{2}})Y_{lm}(\theta,\phi)$
(42)

where N is the normalising constant given by

$\displaystyle{\lvert
N\rvert}^{2}={\frac{2n!}{b^{3}\pi^{1/2}}}\frac{2^{[2(n+l)+1]}}{(2n+2l+1)!}(n+l)!$
(43)

$L_{n}^{l+\frac{1}{2}}$ are the associated Laguerre polynomials.

The six parameters are the mass of charm quark $m_{c}$, the mass of beauty
quark $m_{\bar{b}}$, the harmonic oscillator size parameter $b$, the
confinement strength $A^{2}$, the CCM parameter $c$ and the quark-gluon
coupling constant $\alpha_{s}$. The parameters $m_{c},~{}m_{\bar{b}},~{}A^{2}$
are obtained by a $\chi^{2}$ square fit to the available experimental data of
charmonium, bottomonium and $B_{c}$ meson mass spectra. The CCM parameter $c$
is taken from ref ([30, 62, 37]) which was obtained by fitting the iota (1440
MeV)$0^{-+}$ as a digluon glue ball.There are several papers in literature
where the size parameter $b$ is defined [29, 63]. We obtain the value ’b’ by
minimizing the expectation value of the Hamiltonian i.e,
$\frac{\partial\langle\psi\rvert H\lvert\psi\rangle}{\partial b}=0$. We then
tune the parameter $\alpha_{s}$ to reproduce the experimental mass value. In
literature we find different sets of values for $m_{c}$ and $m_{\bar{b}}$,
which are listed in Table 1.

Table 1: mc and m${}_{\bar{b}}$ for various theoretical models (in MeV). Parameter | Ref.[64] | Ref. [65] | Ref. [66] | Ref. [67] | Ref.[68]
---|---|---|---|---|---
$m_{c}$ | 1800 | 1480 | 1480 | 1480 | 1550
$m_{\bar{b}}$ | 5174 | 5180 | 4880 | 4880 | 4880

The values of strong coupling constant $\alpha_{s}$ in literature are listed
in Table 2. The value of strong coupling constant ($\alpha_{s}$=0.3) used in
our model is compatible with the perturbative treatment.

Table 2: $\alpha_{s}$ for various theoretical models. Parameter | Ref. [28] | Ref. [68] | Ref.[69] | Ref. [1] | Ref. [70]
---|---|---|---|---|---
$\alpha_{s}$ | 0.21 | 0.265 | 0.357 | 0.361 | 0.391

We use the following set of parameter values.

$\begin{split}&m_{c}=1525.00\pm 0.37~{}~{}~{}{\rm
MeV};~{}~{}~{}m_{\bar{b}}=4825.00\pm 0.29~{}{\rm MeV};\\\ &b=0.3~{}{\rm
fm};~{}~{}~{}\alpha_{s}=0.3;~{}~{}A^{2}=550.00\pm 0.78~{}{\rm
MeV~{}fm^{-2}};c=1.74~{}{\rm fm^{-1}}\end{split}$ (44)

Table 3: Mass shifts (in MeV). Bare $c\bar{b}$ State | BD | $B_{s}D_{s}$ | $B_{0}D_{0}$ | $B^{*}D$ | $B^{*}_{s}D_{s}$ | $B^{*}D^{*}$ | $B^{*}_{s}D^{*}_{s}$ | Total
---|---|---|---|---|---|---|---|---
$1~{}^{1}S_{0}$ | 0 | 0 | 0 | -5.661 | -5.033 | -10.434 | -9.328 | -30.456
$1~{}^{3}S_{1}$ | -2.046 | -1.805 | -2.052 | -3.955 | -3.496 | -7.293 | -6.488 | -27.135
$1~{}^{3}P_{0}$ | -57.922 | -57.406 | -57.946 | 0 | 0 | -19.088 | -18.932 | -211.294
$1~{}^{1}P_{1}$ | 0 | 0 | 0 | -18.49 | -18.393 | -37.603 | -37.901 | -112.387
$1~{}^{3}P_{1}$ | 0 | 0 | 0 | -38.390 | -38.049 | 0 | 0 | -76.439
$1~{}^{3}P_{2}$ | -40.618 | -40.314 | -40.632 | 0 | 0 | 0 | 0 | -121.557
$2~{}^{1}S_{0}$ | 0 | 0 | 0 | -1.547 | -1.361 | -2.837 | -2.523 | -8.268
$2~{}^{3}S_{1}$ | -0.546 | -0.476 | -0.548 | -1.929 | -1.711 | -1.049 | -0.920 | -7.179
$1~{}^{3}D_{1}$ | -30.675 | -30.312 | -30.682 | -15.326 | -15.146 | -3.077 | -3.044 | -128.262
$1~{}^{1}D_{2}$ | 0 | 0 | 0 | -3.147 | -3.111 | -27.643 | -27.49 | -61.391
$1~{}^{3}D_{2}$ | 0 | 0 | 0 | -27.214 | -27.552 | -69.486 | -68.957 | -193.209
$1~{}^{3}D_{3}$ | -40.753 | -40.359 | -40.772 | -54.308 | -53.783 | -20.835 | -20.606 | -230.663
$2~{}^{3}P_{0}$ | -148.72 | -146.395 | -148.828 | 0 | 0 | -48.589 | -47.903 | -540.435
$2~{}^{1}P_{1}$ | 0 | 0 | 0 | -25.081 | -24.744 | -49.343 | -48.741 | -147.909
$2~{}^{3}P_{1}$ | 0 | 0 | 0 | -98.623 | -97.088 | 0 | 0 | -195.711
$2~{}^{3}P_{2}$ | -79.114 | -77.890 | -79.171 | 0 | 0 | 0 | 0 | -236.175
Table 4: $B_{c}$ meson mass spectrum (in MeV). State |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
$n~{}^{2S+1}L_{J}$ | This work | Ref.[71] | Ref. [72] | Ref. [73] | Ref. [1] | Ref.[68] | Ref.[28] | Ref.[74] | Ref.[75]
$1~{}^{1}S_{0}$ | 6275 | 6247 | 6253 | 6260 | 6264 | 6270 | 6271 | 6280$\pm 30\pm 190$ | 6286
$1~{}^{3}S_{1}$ | 6314 | 6308 | 6317 | 6340 | 6337 | 6332 | 6338 | 6321$\pm 20$ | 6341
$1~{}^{3}P_{0}$ | 6672 | 6689 | 6683 | 6680 | 6700 | 6699 | 6706 | 6727$\pm 30$ | 6701
$1P1$ | 6766 | 6738 | 6717 | 6730 | 6730 | 6734 | 6741 | 6743$\pm 30$ | 6737
$1P1^{\prime}$ | 6828 | 6757 | 6729 | 6740 | 6736 | 6749 | 6750 | 6765$\pm 30$ | 6760
$1~{}^{3}P_{2}$ | 6776 | 6773 | 6743 | 6760 | 6747 | 6762 | 6768 | 6783$\pm 30$ | 6772
$2~{}^{1}S_{0}$ | 6838 | 6853 | 6867 | 6850 | 6856 | 6835 | 6855 | 6960$\pm 80\pm$ | 6882
$2~{}^{3}S_{1}$ | 6850 | 6886 | 6902 | 6900 | 6899 | 6881 | 6887 | 6990$\pm 80$ | 6914
$1~{}^{3}D_{1}$ | 7078 |  | 7008 | 7010 | 7012 | 7072 | 7028 |  | 7019
$1D2$ | 7009 |  | 7001 | 7020 | 7012 | 7077 | 7041 |  | 7028
$1D2^{\prime}$ | 7154 |  | 7016 | 7030 | 7009 | 7079 | 7036 |  | 7028
$1~{}^{3}D_{3}$ | 6980 |  | 7007 | 7040 | 7005 | 7081 | 7045 |  | 7032
$2~{}^{3}P_{0}$ | 6914 |  | 7088 | 7100 | 7108 | 7091 | 7122 |  |
$2P1$ | 7259 |  | 7113 | 7140 | 7135 | 7126 | 7145 |  |
$2P1^{\prime}$ | 7322 |  | 7124 | 7150 | 7142 | 7145 | 7150 |  |
$2~{}^{3}P_{2}$ | 7232 |  | 7134 | 7160 | 7153 | 7156 | 7164 |  |

We evaluate the bare state masses and shifts due to $BD$,$B_{s}D_{s}$,
$B^{0}D^{0}$, $B^{*}D$, $B^{*}_{s}D_{s}$, $B^{*}D^{*}$ and
$B^{*}_{s}D^{*}_{s}$ loops (with $M_{B}=5279.26~{}\rm{MeV}$,
$M_{B_{s}}=5366.77~{}\rm{MeV}$, $M_{B^{0}}=5279.58~{}\rm{MeV}$,
$M_{B^{*}}=5324.6~{}\rm{MeV}$, $M_{B^{*}_{s}}=5415.4~{}\rm{MeV}$,
$M_{D}=1869.61~{}\rm{MeV}$, $M_{D_{s}}=1968.30~{}\rm{MeV}$,
$M_{D_{0}}=1864.84~{}\rm{MeV}$, $M_{D^{*}}=2006.96~{}\rm{MeV}$ and
$M_{D^{*}_{s}}=2112.1$).

For the case of a bound system of quark and antiquark of unequal mass, charge
conjugation parity is no longer a good quantum number so that states with
different total spins but with the same total angular momentum, such as the
${}^{3}P_{1}-^{1}P_{1}$ and ${}^{3}D_{2}-^{1}D_{2}$ pairs, can mix via the
spin orbit interaction or some other mechanism. The $B_{c}$ meson states with
$J=L$ are linear combination of spin triplet $\lvert^{3}L_{J}\rangle$ and spin
singlet $\lvert^{1}L_{J}\rangle$ states which we describe by the following
mixing$\colon$

$\displaystyle\lvert nL^{\prime}\rangle=\lvert
n~{}^{1}L_{J}\rangle\cos\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\sin\theta_{nL}$ (45) $\displaystyle\lvert
nL\rangle=-\lvert n~{}^{1}L_{J}\rangle\sin\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\cos\theta_{nL}$ (46) $~{}J=L=1,2,3,\cdots$

where $\theta_{nL}$ is a mixing angle, and the primed state has the heavier
mass. For $L=J=1$ we have mixing of P states

$\displaystyle\lvert nP^{\prime}\rangle=\lvert
n~{}^{1}P_{1}\rangle\cos\theta_{nP}+\lvert
n~{}^{3}P_{1}\rangle\sin\theta_{nP}$ (47) $\displaystyle\lvert
nP\rangle=-\lvert n~{}^{1}P_{1}\rangle\sin\theta_{nP}+\lvert
n~{}^{3}P_{1}\rangle\cos\theta_{nP}$ (48)

The values of the mixing angles for P states are $\theta_{1P}=0.4^{\circ}$ and
$\theta_{2P}=0.05^{\circ}$

Similarly for $L=J=2$ we have mixing of D states,

$\displaystyle\lvert nD^{\prime}\rangle=\lvert
n~{}^{1}D_{2}\rangle\cos\theta_{nD}+\lvert
n~{}^{3}D_{2}\rangle\sin\theta_{nD}$ (49) $\displaystyle\lvert
nD\rangle=-\lvert n~{}^{1}D_{2}\rangle\sin\theta_{nD}+\lvert
n~{}^{3}D_{2}\rangle\cos\theta_{nD}$ (50)

The value of mixing angle for D states is $\theta_{1D}=0.05^{\circ}$
The calculated masses of the $c\bar{b}$ states are listed in Table 4. Our
calculated mass value for $B_{c}(1S)$ is 6275.851 MeV and for $B^{*}_{c}$(1S)
is 6314.161 MeV. $B^{*}_{c}$(1S) is heavier than $B_{c}$(1S) by 38.193 MeV.
This difference is justified by calculating the ${}^{3}S_{1}-{}^{1}S_{0}$
splitting of the ground state which is given by

$M({}^{3}S_{1})-M({}^{1}S_{0})=\frac{32\pi\alpha_{s}|\psi(0)|^{2}}{9m_{c}m_{b}}$
(51)

The mass of first radial excitation $B_{c}$(2S) is 6838.232 MeV which is
heavier than $B_{c}$(1S) by 562.381 MeV. This value agrees with the
experimental value of $B_{c}$(2S) 6842$\pm$4$\pm$5 MeV[76]. The difference
between the $B^{*}_{c}$(2S) and $B^{*}_{c}$(1S) masses turns out to be 536.412
MeV. Our prediction for masses of orbitally excited $c\bar{b}$ states are in
good agreement with the other model calculations.

###  3.2 Radiative Decays

The calculation of radiative (EM) transitions between the meson states can be
performed from first principles in lattice QCD, but these calculation
techniques are still in their developmental stage. At present, the potential
model approaches provide the detailed predictions that can be compared to
experimental results.

The possible $E1$ decay modes have been listed in Table 5 and the predictions
for E1 decay widths are given. Also our predictions have been compared with
other theoretical models. Most of the predictions for $E1$ transitions are in
qualitative agreement. However, there are some differences in the predictions
due to differences in phase space arising from different mass predictions and
also from the wave function effects. For the transitions involving $P1$ and
$P1^{\prime}$ states which are mixtures of the spin singlet ${}^{1}P_{1}$ and
spin triplet ${}^{3}P_{1}$ states, there exists huge difference between the
different theoretical predictions. These may be due to the different
${}^{3}P_{1}-^{1}P_{1}$ mixing angles predicted by the different models. Wave
function effects also appear in decays from radially excited states to ground
state mesons such as $2~{}^{3}P_{0}\rightarrow 1~{}^{3}S_{1}\gamma$. The
overlap integral for these transitions in our model vanishes and hence we get
decay width for these transitions zero.

The M1 transitions contribute little to the total widths of the 2S levels.
Because it cannot decay by annihilation. Allowed M1 transitions correspond to
triplet-singlet transitions between S-wave states of the same n quantum
number, while hindered M1 transitions are either triplet-singlet or singlet-
triplet transitions between S-wave states of different n quantum numbers.

The possible radiative M1 transition modes are as follows, (i) 2
${}^{3}S_{1}\rightarrow 2^{1}S_{0}+\gamma$, (ii) 2 ${}^{3}S_{1}\rightarrow
1^{1}S_{0}+\gamma$, (iii) 2 ${}^{1}S_{0}\rightarrow 1^{3}S_{1}+\gamma$, (iv)1
${}^{3}S_{1}\rightarrow 1^{1}S_{0}+\gamma$. In the above (ii) and (iii)
represent hindered transitions and (i) and (iv) represent allowed transitions.
In order to calculate decay rates of hindered transitions we need to include
relativistic corrections.There are three main types of corrections:
relativistic modification of the non relativistic wave functions, relativistic
modification of the electromagnetic transition operator, and finite-size
corrections. In addition to these there are additional corrections arising
from the quark anomalous magnetic moment. Corrections to the wave function
that give contributions to the transition amplitude are of two categories:
1) higher order potential corrections, which are distinguished as a) Zero
recoil effect and b) recoil effects of the final state meson and 2)Colour
octet effects. The colour octet effects are not included in potential model
formulation and are not considered so far in radiative transitions.

Table 5: E1 transition rates of $B_{c}$ meson. Transition | k0 | This work | Ref. [68] | Ref. [1] | Ref. [72] | Ref.[75]
---|---|---|---|---|---|---
| MeV | keV | keV | keV | keV | keV
$1^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ | 348.527 | 42.384 | 75.5 | 79.2 | 65.3 | 74.2
$1P1\rightarrow 1^{3}S_{1}\gamma$ | 437.575 | 83.879 | 87.1 | 99.5 | 77.8 | 75.8
$1P1^{\prime}\rightarrow 1^{3}S_{1}\gamma$ | 494.615 | 121.143 | 13.7 | 0.1 | 8.1 | 26.2
$1^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ | 446.371 | 89.04 | 122 | 112.6 | 102.9 | 126
$1P1\rightarrow 1^{1}S_{0}\gamma$ | 473.213 | 106.088 | 18.4 | 0 | 11.6 | 32.5
$1P1^{\prime}\rightarrow 1^{1}S_{0}\gamma$ | 529.934 | 148.992 | 147 | 56.4 | 131.1 | 128
$2^{3}S_{1}\rightarrow 1^{3}P_{0}\gamma$ | 175.953 | 3.635 | 5.53 | 7.8 | 7.7 | 9.6
$2^{3}S_{1}\rightarrow 1P1\gamma$ | 83.181 | 0.384 | 7.65 | 14.5 | 12.8 | 13.3
$2^{3}S_{1}\rightarrow 1P1^{\prime}\gamma$ | 22.416 | 0.00751 | 0.74 | 0 | 1.0 | 2.5
$2^{3}S_{1}\rightarrow 1^{3}P_{2}\gamma$ | 73.879 | 0.269 | 7.59 | 17.7 | 14.8 | 14.5
$2^{1}S_{0}\rightarrow 1P1\gamma$ | 70.979 | 0.238 | 1.05 | 0 | 1.9 | 6.4
$2^{1}S_{0}\rightarrow 1P1^{\prime}\gamma$ | 10.104 | 0.00068 | 4.40 | 5.2 | 15.9 | 13.1
$2^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ | 573.927 | 0 |  | 21.9 | 16.1 |
$2P1\rightarrow 1^{3}S_{1}\gamma$ | 88.815 | 0 |  | 22.1 | 15.3 |
$2P1^{\prime}\rightarrow 1^{3}S_{1}\gamma$ | 938.854 | 0 |  | 2.1 | 2.5 |
$2^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ | 859.837 | 0 |  | 25.8 | 19.2 |
$2P1\rightarrow 1^{1}S_{0}\gamma$ | 917.035 | 0 |  |  | 3.1 |
$2P1^{\prime}\rightarrow 1^{1}S_{0}\gamma$ | 971.788 | 0 |  |  | 20.1 |
$2^{3}P_{0}\rightarrow 2^{3}S_{1}\gamma$ | 63.253 | 0.422 | 34.0 | 41.2 | 25.5 |
$2P1\rightarrow 2^{3}S_{1}\gamma$ | 397.439 | 104.751 | 45.3 | 54.3 | 32.1 |
$2P1^{\prime}\rightarrow 2^{3}S_{1}\gamma$ | 456.65 | 158.896 | 10.4 | 5.4 | 5.9 |
$2^{3}P_{2}\rightarrow 2^{3}S_{1}\gamma$ | 371.628 | 85.639 | 75.3 | 73.8 | 49.4 |
$2P1\rightarrow 2^{1}S_{0}\gamma$ | 409.075 | 114.223 | 13.8 |  | 8.1 |
$2P1^{\prime}\rightarrow 2^{1}S_{0}\gamma$ | 468.191 | 171.244 | 90.5 |  | 58.0 |

The spherical Bessel function $j_{0}(kr/2)$ introduced in equation (41) takes
into account the so called finite-size effect (equivalently, re summing
multipole-expanded magnetic amplitude to all orders). For small $k$,
$j_{0}(kr/2)\rightarrow 1$, so that transitions with $n^{\prime}=n$ have
favoured matrix elements, though the corresponding partial decay widths are
suppressed by smaller $k^{3}$ factors. For large value of photon energy ($k$)
transitions with $n\neq n^{\prime}$ have favoured the matrix element, since
$j_{0}(kr/2)$ becomes very small. $M1$ transition rates are very sensitive to
hyperfine splitting of the levels due to the $k^{3}$ factor in equation (41).

There have been many models which study the radiative decays of $B_{c}$ meson
using non relativistic and relativistic quark models. Eichten and Quigg [1]
calculated the radiative M1 transition rates for the allowed and hindered
transitions. They used the equation (41) in their potential model approach to
determine the M1 transition rates of $B_{c}$ meson. Allowed transition rates
for processes (i) and (iv) were found to be 0.0040 keV and 0.130 keV
respectively. Hindered transition rates for the processes (ii) and (iii) were
0.253 keV and 0.223 keV respectively. Abd El-Hady et al [69] have investigated
the radiative decay properties of $B_{c}$ meson in a Bethe-Salpeter model. The
allowed transition rates for processes (i) and (iii) were found to be 0.0037
keV and 0.0189 keV respectively. The hindered transition rates for the
processes (ii) and (iv) were found to be 0.135 keV and 0.1638 respectively.
Ebert et al [68] have studied these M1 transitions including full relativistic
corrections in their relativistic model. They depend explicitly on the Lorentz
structure of the non relativistic potential. Several sources of uncertainty
make M1 transitions particularly difficult to calculate. The leading-order
results depend explicitly on the constituent quark masses, and corrections
depend on the Lorentz structure of the potential. They estimated the allowed
transition rates to be 0.033 keV and 0.017 keV respectively. For the hindered
transition, decay rates were found to be 0.428 keV and 0.488 keV. Also it is
clear from their calculations that the predicted decay rates for hindered
transitions which are increased by relativistic effects almost by a factor of
3 and they are larger than the rates of allowed M1 transitions by an order of
magnitude.

We have calculated the M1 transition rates for $c\bar{b}$ meson states using
equation (41). The resulting M1 radiative transition rates of these states are
presented in table 6. In this table we give calculated values for decay rates
of M1 radiative transition in comparison with the other relativistic and non
relativistic quark models. We see from these results that the relativistic
effects play a very important role in determining the $B_{c}$ meson M1
transition rates. The relativistic effects reduce the decay rates of allowed
transitions and increase the rates of hindered transitions. The M1 transition
rates calculated in our model agree well with the values predicted by other
theoretical models.

Table 6: M1 transition rates for the $B_{c}$ meson. Transition | $k_{0}(MeV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ |
---|---|---|---|---|---|---|---|---|---
|  | This work | Ref.[69] | Ref.[68] | Ref.[75] | Ref.[70] | Ref.[68] | Ref.[1] |
$1~{}^{3}S_{1}\rightarrow 1^{1}S_{0}\gamma$ | 38.193 | 0.0185 | 0.0189 | 0.033 | 0.059 | 0.060 | 0.073 | 0.135 |
$2~{}^{3}S_{1}\rightarrow 2^{1}S_{0}\gamma$ | 12.329 | 0.0018 | 0.0037 | 0.017 | 0.012 | 0.010 | 0.030 | 0.029 |
$2~{}^{3}S_{1}\rightarrow 1^{1}S_{0}\gamma$ | 550.614 | 0.193 | 0.1357 | 0.428 | 0.122 | 0.098 | 0.141 | 0.123 |
$2~{}^{3}S_{1}\rightarrow 1^{3}S_{1}\gamma$ | 515.410 | 0.123 | 0.1638 | 0.488 | 0.139 | 0.096 | 0.160 | 0.093 |

###  3.3 Weak Decays and Life Time of $B_{c}$ meson

In accordance with the classification given in section 2.5, the total decay
width can be written as the sum over partial widths

$\Gamma(B_{c}\rightarrow X)=\Gamma(b\rightarrow X)+\Gamma(c\rightarrow
X)+\Gamma(ann)$ (52)

In the spectator approximation:

$\Gamma_{1}(\bar{b}\rightarrow
X)=\frac{9G^{2}_{F}|V_{cb}|^{2}m^{5}_{b}}{192\pi^{3}}$ (53)

Calculated value of $\Gamma_{1}(\bar{b}\rightarrow X)$ is $1.041\times
10^{-3}~{}\rm{eV}$ and

$\Gamma_{2}(c\rightarrow
X)=\frac{5G^{2}_{F}|V_{cs}|^{2}m^{5}_{c}}{192\pi^{3}}$ (54)

Calculated value of $\Gamma_{2}(c\rightarrow X)$ is $8.958\times
10^{-4}~{}\rm{eV}$.
In the above expressions $V_{cb}$ and $V_{cs}$ are the elements of the CKM
matrix, $G_{F}=1.16637\times 10^{-5}$ is the Fermi coupling constant, $m_{c}$
and $m_{b}$ are the masses of c and b quarks respectively. The decay widths
are calculated using $|V_{bc}|=0.044$ [77] and $|V_{cs}|=0.975$ [77].

The leptonic partial widths are probe of the compactness of quarkonium system
and provide important information complementary to level spacings. The quark-
antiquark assignments for the vector mesons, as well as the fractional values
for the quark charges, are tested from the values of their leptonic decay
widths. The decay of vector meson into charged leptons proceeds through the
virtual photon $(q\bar{q}\rightarrow l^{+}l^{-})$. The ${}^{3}S_{1}$ and
${}^{3}D_{1}$ states have quantum numbers of a virtual photon, $J^{PC}=1^{--}$
and can annihilate into lepton pairs through one photon. Annihilation widths
such as $c\bar{b}\rightarrow l\nu_{l}$ are given by the expression

$\Gamma_{3}=\frac{G^{2}_{F}}{8\pi}|V_{bc}|^{2}f^{2}_{B_{c}}M_{B_{c}}\sum_{i}m^{2}_{i}\left(1-\frac{m^{2}_{i}}{M^{2}_{B_{c}}}\right)C_{i}$
(55)

Calculated value of $\Gamma_{3}$ is $5.663\times 10^{-6}~{}\rm{eV}$.
Here $m_{i}$ is the mass of the heavier fermion in the given decay channel.
For lepton channels $C_{i}=1$ while for quark channels
$C_{i}=3|V_{q\bar{q}}|^{2}$ and $f_{B_{c}}$ is the pseudo scalar decay
constant for $B_{c}$ meson.

Adding these results we get the total decay width
$\Gamma(\rm{total})=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}=19.428\times
10^{-4}~{}\rm{eV}$ corresponding to a life time of $\tau=0.339~{}\rm{ps}$.

The pseudo scalar decay constant $f_{B_{c}}$ is defined by:

$\langle 0\rvert\bar{b}(x)\gamma^{\mu}c(x)\lvert
B_{c}(k)\rangle=if_{B_{c}}V_{cb}k^{\mu}$ (56)

where $k^{\mu}$ is the four-momentum of the $B_{c}$ meson. In the non
relativistic limit the pseudo scalar decay constant is proportional to the
wave function at the origin and is given by van Royen-Weisskopf formula [78]

$f_{B_{c}}=\sqrt{\frac{12}{M_{B_{c}}}}\psi(0)$ (57)

Here $\psi(0)$ is wavefunction at the origin. The values of decay constant in
various theoretical models are listed in table 7 and in table 8 we compare the
life time of $B_{c}$ meson calculated in our model with other models.

Table 7: Comparison of predictions for the pseudo scalar decay constant of the $B_{c}$ meson ($f_{B_{c}}$). This work | Ref.[66] | Ref. [64] | Ref.[67] | Ref.[74]
---|---|---|---|---
554.125 | 500 | 512 | 479 | 440$\pm$20
Table 8: Comparison of life time of $B_{c}$ meson (in ps). This work | Experiment[77] | Ref.[60] | Ref.[72] | Ref.[79] | Ref. [80]
---|---|---|---|---|---
0.339 | 0.452$\pm 0.033$ | 0.47 | 0.55$\pm 0.15$ | 0.50 | 0.75

###  3.4 Strong Decays

The $c\bar{b}$ states which lie below BD threshold are stable against strong
decays. However, the states which are above the BD threshold undergo two body
strong decays. We have calculated strong decay widths of $c\bar{b}$ states
which lie above the BD threshold using the equation (38). The decay widths are
calculated within the ${}^{3}P_{0}$ pair creation model. The results are
presented in table 9.

Table 9: Strong decay widths of the $B_{c}$ meson. Transition | $\Gamma(MeV)$
---|---
$2~{}^{1}P_{1}\rightarrow B^{*}+D$ | 54.599
$2~{}^{3}P_{1}\rightarrow B^{*}+D$ | 2.145
$2~{}^{3}P_{2}\rightarrow B+D$ | 99.386
$2~{}^{3}P_{2}\rightarrow B^{0}+D^{0}$ | 108.185
$2~{}^{3}P_{2}\rightarrow B^{*}+D$ | 31.247
$1~{}^{3}D_{2}\to B^{*}+D$ | 0.198
$1~{}^{3}D_{2}\to B^{*}_{s}+D_{s}$ | 5.837
$1~{}^{3}D_{2}\to B^{*}+D^{*}$ | 2.123
$1~{}^{3}D_{2}\to B^{*}_{s}+D^{*}_{s}$ | 20.885

##  4 Conclusions

The complete spectrum of $c\bar{b}$ states has been calculated in a
relativistic quark model with coupled channel effects. We have calculated the
meson loop effects on the masses of 1S, 2S, 1P, 2P and 1D $c\bar{b}$ states.
The mass shifts calculated due to these loop effects are large. The ground
state mass of $c\bar{b}$ state calculated in our model matches the
experimental data. When the results for $c\bar{b}$ state mass spectrum are
compared with the previous calculations, it is found that the predictions for
the mass spectrum agree within a few MeV. The differences between the
predictions in most cases do not exceed 30 MeV and the higher orbitally
excited states are 50-100 MeV heavier in our model. The hyperfine splitting of
the ground state vector and pseudo scalar $c\bar{b}$ states in our model is in
good agreement with the prediction made by other theoretical models. The
ground state pseudo scalar $B_{c}$ and vector $B^{*}_{c}$ meson masses lie
within the ranges quoted by Kwong and Rosner in their survey of techniques for
estimating the masses of the $c\bar{b}$ ground state: i.e.,
$6194~{}\rm{MeV}<M_{B_{c}}<6292~{}\rm{MeV}$ and
$6284~{}\rm{MeV}<M_{B^{*}_{c}}<6357~{}\rm{MeV}$.

The difference of quark flavours forbids $B_{c}$ meson from annihilation into
gluons. As a result the excited $B_{c}$ meson states lying below the BD
production threshold (i.e. with $M<M_{D}+M_{B}=7143.1\pm 2.1$ GeV) undergo
radiative transition to ground state which then decays weakly. Radiative
decays are the dominant decay modes of the $B_{c}$ excited states having
widths of about a fraction of MeV. Therefore, it is very important to
determine the masses and the radiative decay widths of $B_{c}$ meson
accurately in order to understand the $B_{c}$ spectrum and distinguishing
exotic states. The radiative E1 and M1 decay rates of $c\bar{b}$ states have
been calculated using spectroscopic parameters obtained from RQM. Most of our
predictions for the E1 decay rates are in good agreement with the other
theoretical calculations. The differences in the prediction for the decay
rates in various theoretical models can be attributed to the differences in
mass predictions, wave function effects and singlet - triplet mixing angels.
The calculated M1 transition rates reasonably agree with the other theoretical
model predictions as listed in table 6. It is clearly seen in this calculation
that the relativistic effects play an important role in determining the
radiative transition rates, since the hindered transition rates are suppressed
due to the wave function orthogonality in the NRQM formalism. The inclusion of
these relativistic effects enhances the hindered transition rates and reduces
the allowed transition rates. It is evident from the table that the hindered
transition rates are larger than the allowed transition rates by an order of
magnitude. Experimental results for the masses of excited states and radiative
decays of $B_{c}$ meson are needed to clarify these predictions. Experimental
results will give us more insight into $B_{c}$ spectrum and will help us to
clarify the hyperfine splitting calculated in different models.

We have done an estimation of weak decay widths in the spectator quark
approximation and calculated the life time of $c\bar{b}$ state. We get about
53% branching ratio for $b$-quark decays, about 46% for $c$-quark decays and
about 1% branching ratio in annihilation channel. The life time of $c\bar{b}$
state predicted in this calculation is listed in table 8 and is found to be in
good agreement with experimental value as well as with other theoretical
predictions. The decay constant of $c\bar{b}$ state ($f_{B_{c}}$) has been
calculated and compared with other model predictions and it is found that the
decay constant is consistent with these predictions. We have calculated two
body strong decay widths of $c\bar{b}$ states in the framework of
${}^{3}P_{0}$ pair creation model.

A simple relativistic model employing COGEP and harmonic-oscillator
confinement potential along with coupled channel effects used in this study is
successful to predict the various properties of $c\bar{b}$ states and this can
shed further light on their non leptonic transitions.

Acknowledgements

One of the authors (APM) is grateful to BRNS, DAE, India for granting the
project and JRF (37(3)/14/21/2014BRNS).

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