# Mass spectra and decays of ground and orbitally excited $c\bar{b}$ states in
non relativistic quark model

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

###### Abstract

The complete spectrum of $c\bar{b}$ states is obtained in a phenomenological
non relativistic quark model(NRQM), which consists of a confinement potential
and one gluon exchange potential (OGEP) as effective quark - antiquark
potential. We make predictions for the radiative decay (E1 and M1) widths and
weak decay widths of $c\bar{b}$ states in the framework of NRQM formalism.

###### keywords:

Mesons;Phenomenological quark models;Non relativistic quark
models;Leptonic;semileptonic;radiative decays of mesons

PACS: 14.40.-n;12.3.-x,12.39.-Jh,13.20.-v

##  1 INTRODUCTION

The $B_{c}$ state is the only bound system that consists of two heavy quarks
of different flavours that offers a sound laboratory opportunity to observe
both QCD and weak interaction. The ground state $B_{c}(1S)$ was first observed
in the CDF and DO experiments (Tevatron) in two decay modes $\colon$
$B_{c}\rightarrow J/\psi l\nu$ and $B_{c}\rightarrow J/\psi\pi$ [1, 2, 3, 4,
5, 6, 7], also observed by LHC experiments in the various decay modes of
$c\bar{b}$ states such as $\colon$ $B_{c}\rightarrow J/\psi\pi$(LHCb, CMS,
ATLAS), $B_{c}\rightarrow J/\psi\pi\pi\pi$ (LHCb and CMS),
$B^{+}_{c}\rightarrow\psi(2S)\pi^{+}$, $B^{+}_{c}\rightarrow J/\psi
D^{+}_{s}$, $B^{+}_{c}\rightarrow J/\psi D^{*+}_{s}$, $B^{+}_{c}\rightarrow
J/\psi K^{+}$, $B^{+}_{c}\rightarrow J/\psi 3\pi^{+}2\pi^{-}$ and
$B^{+}_{c}\rightarrow J/\psi K^{+}K^{-}\pi^{+}\pi^{-}$ (LHCb) [8, 9, 10, 11,
12, 13]. Recently the ATLAS collaboration at the LHC has observed radially
excited $c\bar{b}$ state (i.e, $B_{c}(2S)$) through the decay channel
$B^{\pm}_{c}(2S)\rightarrow B^{\pm}_{c}(1S)\pi^{+}\pi^{-}$ [14]. $B_{c}$
mesons are predicted by the quark model to be members of the $J^{P}=0^{-}$
pseudo scalar ground state multiplet [15]. The vector $B^{*}_{c}(1S)$ meson is
the triplet state of $B_{c}(1S)$ which has not been observed by means of
experiments to date.

The discovery of $B_{c}$ state has made the families of well investigated
charmonium and bottomonium complete and has given a new insight into the study
of heavy quark dynamics. Since the charmed bottom meson $c\bar{b}$ is an
intermediate state of the $c\bar{c}$ mesons and $b\bar{b}$ mesons, its
analysis could give detailed information on the balance between the
perturbative and non perturbative effects. The investigation of masses of
$c\bar{b}$ states gives us an opportunity to obtain information on the nature
of the strong interaction thereby it throws up an interesting issue and a
tantalising problem. Since the flavour asymmetry of $c\bar{b}$ state unlike in
symmetric quarkonium, forbids the annihilation of $c\bar{b}$ state into
gluons, the ground state of $c\bar{b}$ state below the BD threshold, can only
decay through weak interaction that provides an ideal platform to study weak
decays and provides new methods for calculating the CKM matrix. Some decay
channels of $c\bar{b}$ states show that bound state effects are significant in
$c\bar{b}$ decays. The lighter c quark has a greater decay rate (65$\%$) than
the heavier b quark (25$\%$). The pseudo scalar $c\bar{b}$ state decays only
weakly. The $c\bar{b}$ is unique in that either one of its quarks can decay,
leaving the other as a spectator or both the quarks may involve in its weak
decay. Thus $B_{c}$ meson can serve as a great laboratory for QCD sum rules,
Heavy Quark Effective Theory, lattice QCD and potential models.

This work uses NRQM formalism to study both mass spectra and decay properties
of $c\bar{b}$ states. The NRQM formalism is found to provide systematic
treatment of the perturbative and non perturbative components of QCD at the
hadronic scale [16, 17, 18, 19, 20, 21]. The masses of the $c\bar{b}$ spectrum
can be predicted using NRQM whose parameters are tuned to reproduce the
spectra of the observed charmonium and bottomonium states [22, 23, 24, 15, 25,
26]. There are a good number of theoretical models that study leptonic,
semileptonic and hadronic decay channels of $c\bar{b}$ state [27, 28, 29, 30,
31, 32, 33, 34, 35, 36].

In the last two decades, the lattice QCD Monte Carlo calculations have emerged
as a reliable non-perturbative method to study hadron spectra. For $q\bar{q}$
systems it has been shown unambiguously that the ground state potential is
$V_{q\bar{q}}=-\frac{A_{q\bar{q}}}{r}+\sigma_{q\bar{q}}r+C_{q\bar{q}}$, with
inter quark distance r [37, 38, 39], which is consistent with the standard
NRQM potential of Coulombic + OGEP + linear confinement[40]. Also, from
lattice QCD, the effect of the gluonic excitation in the three quark system
has been investigated. It has been shown that for low-lying hadrons with
excitation energy below 1 GeV, the effect of the gluonic excitations is
negligible and hence quark degrees plays a dominant role in low-lying hadrons
and hence resolves the absence of gluonic excitation modes in low-lying hadron
spectra[41]. Also, the static three-quark potential has been studied in detail
using SU(3) lattice QCD. The detailed analyses lattice QCD data of the 3q
potential support the Y-ansatz[38].

In the recent work on lattice QCD, an investigation to estimate the masses of
B,$B_{s}$ and $B_{c}$ has been carried out. The work takes into account of the
effect of the u, d and s sea quarks. The highly improved staggered quark
(HISQ) action for u/d,s and c quarks and non-relativistic QCD is used for b
quarks. Using HISQ formalism, the mass difference and decay constant ratios
between B,Bs and Bc mesons. The mass obtained for $B_{c}$ mesons is 6280(10)
GeV. which is a significant improvement on an earlier value of Bc mesons
obtained from lattice QCD which is a strong test of lattice QCD to the B
physics[42].

Our basic aim is to develop a consistent model which could reproduce both the
spectra and the decay widths with the same set of parameters. The parameters
in our work, are fixed from the mass spectroscopy and the same set of a
parameters have been used to obtain the decay widths. From our analysis we
infer that our model has the right prediction both for the mass spectrum and
decay widths with the same set of parameters. It should be noted that
obtaining the mass spectrum alone in accordance with the experimental results
doesnot guarantee the validity of a model for describing hadronic
interactions. Different potentials can reproduce the same spectra. Hence, in a
given model, one must be able to calculate other observables like the decay
constants, leptonic decay widths, the radiative decay widths, etc. Heavy
quarkonium decays provide a deeper insight on the exact nature of the inter
quark forces and decay mechanisms. For example, the leptonic decay widths are
a probe of the quarkonium system provide important information complementary
to level spacings. In the existing quark models, The OGEP has its origin in
the exchange of a single gluon which belongs to an octet representation of the
$SU(3)_{c}$. The OGEP is obtained from the QCD Lagrangian in the non-
relativistic by retaining terms to the order of $1/c^{2}$. This procedure is
similar to the derivation of the Fermi-Breit interaction in quantum
electrodynamics[43]. In deriving OGEP,the gluon propagtaors used are similar
to the free photon propagators used in obtaining Ferm-Breit interaction in
QED. Since the confinement of color means confinement of quarks as well as
gluons, the confined dynamics of gluons should play a decisive role in
determining the spectroscopy of the mesonic states. But, Fermi-Breit
interaction which gives rise to the splitting for singlet and triplet states
are treated as perturbation. But, the OGEP is attractive for singlet states
and repulsive for triplet states, hence, naïve perturbative treatment of OGEP
is incorrect. This leads to further renormalization of strength of interaction
for a better fit[44]. Also, the most prominent flaw of non-relativistic
potential models is the neglect of gluon dynamics[45, 46, 47, 16]. Hence, it
is required to obtain the mass spectrum by diagonalizing the Hamiltonian
matrix.

The paper is organized in 4 sections. In sec. 2 we briefly review the
theoretical background for non relativistic model, the description of
radiative and weak decay widths. In sec. 3 we discuss the results and the
conclusions are drawn in sec. 5 with a comparison to other models.

##  2 THEORETICAL BACKGROUND

###  2.1 The Hamiltonian

Essentially, in all phenomenological non relativistic QCD based quark models,
the Hamiltonian for the quark system consists of the kinetic energy, the two-
body confinement potential and OGEP. In most of these works, it is assumed
that the principal binding forces of hadrons are the long range quark
confining forces. These should be independent of quark spins and quark masses,
depending just on the spatial separations of the constituent quarks. In
addition there exist short range forces depending on the quark spins and
masses. The effective short range force stems from the one gluon exchange
mechanism. The exchange of gluons can provide binding between quarks in a
hadron. The Hamiltonian employed in our model [48, 49, 50] includes kinetic
energy part, confinement potential and one gluon potential (OGEP) [45].

$H=K+V_{CONF}+V_{OGEP}$ (1)

The kinetic energy part (K) is the sum of the kinetic energies including the
rest mass minus the kinetic energy of the center of mass motion (CM) of the
total system, i.e.,

$\displaystyle K$ $\displaystyle=$
$\displaystyle\left[\sum_{i=1}^{2}M_{i}+\frac{P^{2}_{i}}{2M_{i}}\right]-K_{\mathrm{cm}},$
(2)

with $M_{i}$ and $P_{i}$ as the mass and momentum of the $i{\rm th}$ quark,
respectively. $K_{CM}$ is the kinetic energy of the centre of mass motion.

The confinement potential must come ultimately from a non-perturbative
treatment of QCD, whereas the residual interaction OGEP is based on
perturbation theory. In phenomenological quark models the confinement
potential is assumed to be harmonic oscillator potential ($V\sim r^{2}$) or
logarithmic potential ($V\sim ln(r)$) or linear potential ($V\sim r$). For our
model we have chosen the linear potential which represents the non
perturbative effect of QCD that confines quarks within the color singlet
system [49, 50].

$V_{CONF}(\vec{r}_{ij})=-a_{c}r_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}$
(3)

where $a_{c}$ is the confinement strength and $\lambda_{i}$ and $\lambda_{j}$
are the generators of the color SU(3) group for the $i{\rm th}$ and $j{\rm
th}$ quarks.

It should be noted that the two body confinement potential[51], has symmetric
and antisymmetric terms.

$V^{conf}_{so}=\sum_{i>j}-\frac{1}{4m^{2}r_{ij}}\frac{dV^{conf}}{dr_{ij}}\\{(r_{ij}\times
p_{ij})\cdot(\sigma_{i}+\sigma_{j})+[r_{ij}\times(p_{i}+p_{j})/2]\cdot(\sigma_{i}-\sigma_{j})\\}$
(4)

If one includes both the symmetric and antisymmetric terms the two terms of
$V^{conf}_{ij}$ cancel each other and give an almost vanishing contribution to
the single baryon/meson spectra. Also, it should be noted that the symmetric
term of the $V^{conf}_{ij}$ has opposite sign to the one in $V^{OGEP}_{ij}$
and hence cancels the contribution from the symmetric term of $V^{conf}_{ij}$
in single baryon/meson system. The antisymetric term is Galilei non-invariant.
It should be noted that the spin orbit term in $V^{conf}_{ij}$ comes from the
relativistic effects. The above two body potential is having problems with the
long range attractive color Van der Waals force[52, 53, 54, 55] when it is
used for the two hadron system. Since the attraction between color singlet
hadrons comes from the virtual excitation of the color octet dipole state of
each hadron, this problem is evaded by restricting the model space to describe
hadrons. Also, if the two body confinement mechanism is extended to two
nucleon system, it essentially gives a zero contribution to the spin-orbit
force of the N-N potential[56]. Since the mechanism which we take for
confinement, namely two body confinement, is purely phenomenological it is
advisable to leave out the spin-orbit term due to confinement.

We consider a purely linear confinement potential in our calculation. The
spin-orbit splittings calculated in our model[57] and in [58] for heavy quark
system such as bottomonium and charmonium suggests a scalar confinement. W.
Lucha et al.[58], Bhagyesh et al.[59, 60] and Bhagyesh and K. B. Vijaya Kumar
[61] used a mixture of scalar and vector confinement potential to explain the
mass spectra and decays of heavy quarkonia. In our calculation we use a one
gluon exchange potential and a purely linear confinemnt potential. The one
gluon exchange potential is of purely vector nature and the confinement
potential is of purely scalar nature. The combination $V(r)=V_{OGEP}+V_{CONF}$
is a mixture of Lorentz scalar and Lorentz vector nature.

The one gluon exchange potential is given by

$V_{OGEP}=-\frac{4}{3}\frac{\alpha_{s}}{r}+V_{SD}(r)$ (5)

where the spin dependent potential $V_{SD}$ is introduced as an additional
term to the potential to take into the account the spin-orbit and spin-spin
interactions, causing the splitting of the $nL$ levels (n is the principal
quantum number, L is the orbital momentum), so it has the form [62, 63, 24]

$\begin{split}V_{SD}(r)=&\left(\frac{L\cdot S_{c}}{2m^{2}_{c}}+\frac{L\cdot
S_{b}}{2m^{2}_{b}}\right)\left(-\frac{dV(r)}{rdr}+\frac{8}{3}\alpha_{s}\frac{1}{r^{3}}\right)+\frac{4}{3}\alpha_{s}\frac{1}{m_{c}m_{b}}\frac{L\cdot
S}{r^{3}}+\frac{4}{3}\alpha_{s}\frac{2}{3m_{c}m_{b}}S_{c}\cdot
S_{b}4\pi\delta(r)\\\
&+\frac{4}{3}\alpha_{s}\frac{1}{3m_{c}m_{b}}\left[3(S_{c}\cdot n)(S_{b}\cdot
n)-S_{c}\cdot S_{b}\right]\frac{1}{r^{3}}\end{split}$ (6)

The central part of the two-body potential due to OGEP is [45],

$V_{OGEP}(\vec{r}_{ij})=\frac{\alpha_{s}}{4}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\left[\frac{1}{r_{ij}}-\frac{\pi}{M_{i}M_{j}}\left(\frac{M_{i}}{M_{j}}+\frac{M_{j}}{M_{i}}+\frac{2}{3}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\right)\delta(\vec{r}_{ij})\right]$
(7)

where the first term represents the residual Coulomb energy and the second
term is the chromo-magnetic interaction leading to the hyperfine splitting.
$\sigma_{i}$ is the Pauli spin operator and $\alpha_{s}$ is the quark-gluon
coupling constant.

The non$-$central part of OGEP has two terms, namely the spin$-$orbit
interaction $V^{SO}_{OGEP}(\vec{r})$ and tensor term
$V^{ten}_{OGEP}(\vec{r})$. The spin-orbit interaction of OGEP is given by,

$V^{SO}_{OGEP}(\vec{r})=-\frac{\alpha_{s}}{4}{\bf\lambda_{i}\cdot\lambda_{j}}\left[\frac{3}{8M_{i}M_{j}}\frac{1}{r^{3}}(\vec{r}\times\vec{p})\cdot({\bf\sigma_{i}}+{\bf\sigma}_{j})\right]$
(8)

where the relative angular momentum is defined as usual in terms of relative
position $\vec{r}$ and the relative momentum $\vec{p}$. Unlike the tensor
force, the spin$-$orbit force does not mix states of different $\vec{L}$,
since $L^{2}$ commutes with $\vec{L}\cdot\vec{S}$, $\vec{L}$ is still a
constant of motion, but $L_{z}$ is not.

We use the following tensor term [64, 65]

$V^{ten}_{OGEP}(\vec{r})=-\frac{\alpha_{s}}{4}{\bf\lambda_{i}\cdot\lambda_{j}}\left[\frac{1}{4M_{i}M_{j}}\frac{1}{r^{3}}\right]\hat{S}_{ij}$
(9)

where,

$\hat{S}_{ij}=[3(\vec{\sigma}_{i}\cdot\hat{r})(\vec{\sigma}_{j}\cdot\hat{r})-\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}]$
(10)

The tensor potential is a scalar which is obtained by contracting two second
rank tensors. Here, $\hat{r}=\hat{r}_{i}-\hat{r}_{j}$ is the unit vector in
the direction of $\vec{r}$. In the presence of the tensor interaction,
$\vec{L}$ is no longer a good quantum number.

The only short-range force between the quarks is spin dependent which comes
from OGEP. The spin-orbit couplings give rise to long range forces according
to lattice QCD, since the interaction energy between two distant static quarks
in lattice QCD involves a power-series expansion in inverse powers of the
gluon quark coupling constant[45]. In QCD the Fermi-Breit corrections are
large for mesons and baryons. For attractive potentials for the Dirac delta
function $\delta(r)$, the wave equations have no physically acceptable
solutions and lead to collapse both for the quark-antiquark and three quark
system. Hence, there are models which have introduced a cut-off or smearing
function to weaken the singularity. Though, the finite range function is
calculable, it has to be parameterized[66]. But it has been shown that the
short range spin interactions itself is strong enough to support a bound state
of a single $q\bar{q}$ pair. The tensor and spin-orbit potentials (eqs 7 and
8) are of long range, in contrast to the “zero range”
$\delta(r)\sigma_{1}\cdot\sigma_{2}$ central potential (eq. 6)[67]. But the
actual range of the $\delta(r)\sigma_{1}\cdot\sigma_{2}$ is still unkown and
hence is required to do a non-perturbative calculation ( or a larger
basis)[68]. Also there are other effects like suppression of the spin orbit
potential from OGEP from the spin orbit potential arising from the confinement
potential (they have opposite sign).

###  2.2 Radiative Decays

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 ($\Delta L=1$, $\Delta S=0$). 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 strength of the electric
dipole transitions is governed by the size of the radiator and the charges of
the constituent quarks. The E1 partial decay width is given by [69],

$\Gamma_{a\rightarrow
b}=\frac{4\alpha}{9}\mu^{2}\left(\frac{Q_{c}}{m_{c}}-\frac{Q_{\bar{b}}}{m_{\bar{b}}}\right)^{2}\frac{E_{b}(k_{0})}{m_{a}}k^{3}_{0}\left|\langle
R_{b}\rvert r\lvert
R_{a}\rangle\right|^{2}\left\\{\begin{array}[]{cl}(2J+1)/3,&{}^{3}S_{1}\rightarrow^{3}P_{J}\\\
1/3,&{}^{3}P_{J}\rightarrow^{3}S_{1}\\\
1/3,&{}^{1}P_{1}\rightarrow^{1}S_{0}\\\
1,&{}^{1}S_{0}\rightarrow^{1}P_{1}\end{array}\right.$ (11)

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

$k_{0}=m_{a}-m_{b}$ in non relativistic limit.

$\alpha$ 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 $\mu=\frac{m_{b}m_{c}}{m_{b}+m_{c}}$ and
$\frac{E_{b}(k_{0})}{m_{a}}=1$ in non relativistic limit.

$\langle R_{b}\rvert r\lvert
R_{a}\rangle=\int^{\infty}_{0}r^{3}R_{b}(r)R_{a}(r)dr$ (12)

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 M1 partial decay width between S wave states is [69, 70]

$\begin{split}\Gamma=\delta_{L_{a}L_{b}}4\alpha
k^{3}_{0}\frac{E_{b}(k_{0})}{m_{a}}\left(\frac{Q_{c}}{m_{c}}+(-1)^{S_{a}+S_{b}}\frac{Q_{b}}{m_{b}}\right)^{2}(2S_{a}+1)\times(2S_{b}+1)(2J_{b}+1)\\\
\left\\{\begin{array}[]{ccc}S_{a}&L_{a}&J_{a}\\\
J_{b}&1&S_{b}\end{array}\right\\}^{2}\left\\{\begin{array}[]{ccc}1&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&S_{a}&S_{b}\end{array}\right\\}^{2}\times\left[\int^{\infty}_{0}R_{n_{b}L_{b}}(r)r^{2}R_{n_{a}L_{a}}(r)dr\right]^{2}\end{split}$
(13)

which can be further simplified to

$\Gamma_{M1}(a\to
b+\gamma)=\frac{16}{3}\alpha\mu^{2}_{eff}k^{3}_{0}(2J_{b}+1)\\\
\left[\int^{\infty}_{0}R_{n_{b}L_{b}}(r)r^{2}R_{n_{a}L_{a}}(r)dr\right]^{2}$
(14)

where $\int^{\infty}_{0}drR_{n_{b}L_{b}}(r)r^{2}R_{n_{a}L_{a}}(r)$ is the
overlap integral for unit operator between the coordinate wave functions of
the initial and the final meson states, $m_{c}$ and $m_{b}$ are the masses of
the charm and bottom quarks and
$\mu^{2}_{eff}=\frac{m_{b}Q_{c}-m_{c}Q_{\bar{b}}}{4m_{c}m_{b}}$. $S_{a}$,
$S_{b}$, $L_{a}$, $J_{a}$ and $J_{b}$ are the spin quantum number, orbital
angular momentum and total angular momentum quantum numbers of initial and
final meson states respectively.
The M1 transitions contribute little to the total widths of the 2S levels,
since 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
allowed M1 transitions are essentially $1~{}^{3}S_{1}\rightarrow
1~{}^{1}S_{0}$ and $2~{}^{3}S_{1}\rightarrow 1~{}^{1}S_{0}$.

###  2.3 Weak Decays

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 decays independently so that
$B_{c}$ decays can be divided into three classes [71, 72]$\colon$ (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 Spectra

mc and mb for various theoretical models (in GeV). Parameter Ref.[73] Ref.
[74] Ref. [75] Ref. [76] Ref.[77] $m_{c}$ 1.8 1.48 1.48 1.48 1.55 $m_{b}$
5.174 5.18 4.88 4.88 4.88

The quark-anti quark 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,

$\Psi_{nlm}(r,\theta,\phi)=N(\frac{r}{b})^{l}~{}L_{n}^{l+\frac{1}{2}}(\frac{r^{2}}{b^{2}})\exp(-\frac{r^{2}}{2b^{2}})Y_{lm}(\theta,\phi)$
(15)

where N is the normalising constant given by

$\displaystyle{\lvert
N\rvert}^{2}={\frac{2n!}{b^{3}\pi^{1/2}}}\frac{2^{[2(n+l)+1]}}{(2n+2l+1)!}(n+l)!$
(16)

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

The wave function used in this calculation (eqn.15), is the standard form of
the harmonic oscillator wave functions which has been extensively used in
earlier works in atomic, and nuclear physics. The wave function is normalized
and the normalization constant is given in eqn.16. The main advantage of using
the harmonic oscillator wave function is that it allows the separation of the
motion of the center of mass and has been extensively used to classify the
spectra of baryons and mesons [78, 79] and extending to nucleon-nucleon
interaction is straight forward [51, 80, 81]. If the basic states are the
harmonic oscillator wave functions, then it is straightforward to evaluate the
matrix elements of few body systems such as mesons or baryons. Since the basic
states are the products of the harmonic oscillator wave functions they can be
chosen in a manner that allows the product wave functions to be expanded as a
finite sum of the corresponding products for any other set of Jacobi
coordinates. It is advantageous to use the Gaussian form since in the
annihilation of quark-anti quark into lepton pairs, the amplitude of the
emission or absorption processes depend essentially on the overlap of initial
and final hadrons and hence the overlap depends only on the intermediate
distance region of the spatial wave functions which can extend up to 0.5 fm.
This intermediate region can be described by potentials that are similar in
this region and hence harmonic oscillator wave functions are expected to
reproduce emission and absorption processes quite well.

The four parameters in our model are the mass of charm quark $m_{c}$, the mass
of bottom quark $m_{b}$, the harmonic oscillator size parameter $b$ and the
quark-gluon coupling constant $\alpha_{s}$. There are several papers in
literature where the size parameter $b$ is defined [67, 82]. The value of $b$
is fixed by minimizing the expectation value of the Hamiltonian for the vector
meson. The confinement strength $a_{c}$ is fixed by the stability condition
for variation of mass of the vector meson against the size parameter $b$. To
fit $\alpha_{s}$ , $m_{b}$ and $m_{c}$, we start with a set of reasonable
values and diagonalize the matrix for $B_{c}$ meson. Then we tune these
parameters to obtain an agreement with the experimental value for the mass of
$B_{c}$ meson. In literature we find different sets of values for $m_{c}$ and
$m_{b}$, which are listed in table 3.1.

The values of strong coupling constant $\alpha_{s}$ in literature are listed
in table 3.1. The value of strong coupling constant ($\alpha_{s}$=0.3) used is
compatible with the perturbative treatment.
We use the following set of parameter values.

$\begin{split}m_{c}=1480.0~{}~{}~{}{\rm MeV};~{}~{}~{}m_{b}=4750.0~{}{\rm
MeV};\\\ b=0.350~{}{\rm fm};~{}~{}~{}\alpha_{s}=0.300;~{}~{}a_{c}=145~{}{\rm
MeV~{}fm^{-1}};\end{split}$ (17)

$\alpha_{s}$ for various theoretical models. Parameter Ref. [17] Ref. [77]
Ref.[83] Ref. [15] Ref. [84] $\alpha_{s}$ 0.21 0.265 0.357 0.361 0.391

The mass spectrum has been obtained by diagonalizing the Hamiltonian in a
large basis of $15\times 15$ matrix which has not been carried out in other
existing models which is a new ingredient in our model. The calculation
clearly indicates that masses for both pseudo scalar and vector mesons
converge to the experimental values when the diagonalization is carried out in
a larger basis. In our earlier work also, we had come to the similar
conclusion while investigating light meson spectrum[48, 85]. The
diagonalization of the Hamiltonian matrix in a larger basis leads to the
lowering of the masses and justifies the perturbative technique to calculate
the mass spectrum. The calculation clearly indicates that when diagonalization
is carried out in a larger basis convergence is achieved both for pseudo-
scalar mesons and vector mesons to the respective experimental values.

For the case of a bound system of quark and anti-quark of unequal mass, charge
conjugation parity is no longer a good quantum number so that the 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$\colon$

$\displaystyle\lvert nL^{\prime}\rangle=\lvert
n~{}^{1}L_{J}\rangle\cos\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\sin\theta_{nL}$ (18) $\displaystyle\lvert
nL\rangle=-\lvert n~{}^{1}L_{J}\rangle\sin\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\cos\theta_{nL}$ (19) $~{}J=L=1,2,3,\cdots$

Table 1: : (continued) State |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
$n~{}^{2S+1}L_{J}$ | This work | Ref.[86] | Ref. [24] | Ref. [87] | Ref. [15] | Ref.[77] | Ref.[17] | Ref.[88] | Ref.[89]
$1~{}^{1}S_{0}$ | 6.275 | 6.247 | 6.253 | 6.260 | 6.264 | 6.270 | 6.271 | 6.280$\pm 30\pm 190$ | 6.286
$1~{}^{3}S_{1}$ | 6.357 | 6.308 | 6.317 | 6.340 | 6.337 | 6.332 | 6.338 | 6.321$\pm 20$ | 6.341
$1~{}^{3}P_{0}$ | 6.638 | 6.689 | 6.683 | 6.680 | 6.700 | 6.699 | 6.706 | 6.727$\pm 30$ | 6.701
$1P$ | 6.686 | 6.738 | 6.717 | 6.730 | 6.730 | 6.734 | 6.741 | 6.743$\pm 30$ | 6.737
$1P^{\prime}$ | 6.734 | 6.757 | 6.729 | 6.740 | 6.736 | 6.749 | 6.750 | 6.765$\pm 30$ | 6.760
$1~{}^{3}P_{2}$ | 6.737 | 6.773 | 6.743 | 6.760 | 6.747 | 6.762 | 6.768 | 6.783$\pm 30$ | 6.772
$2~{}^{1}S_{0}$ | 6.862 | 6.853 | 6.867 | 6.850 | 6.856 | 6.835 | 6.855 | 6.960$\pm 80\pm$ | 6.882
$2~{}^{3}S_{1}$ | 6.897 | 6.886 | 6.902 | 6.900 | 6.899 | 6.881 | 6.887 | 6.990$\pm 80$ | 6.914
$1~{}^{3}D_{1}$ | 6.973 |  | 7.008 | 7.010 | 7.012 | 7.072 | 7.028 |  | 7.019
$1D$ | 6.974 |  | 7.001 | 7.020 | 7.012 | 7.077 | 7.041 |  | 7.028
$1D^{\prime}$ | 7.003 |  | 7.016 | 7.030 | 7.009 | 7.079 | 7.036 |  | 7.028
$1~{}^{3}D_{3}$ | 7.004 |  | 7.007 | 7.040 | 7.005 | 7.081 | 7.045 |  | 7.032
$2~{}^{3}P_{0}$ | 7.084 |  | 7.088 | 7.100 | 7.108 | 7.091 | 7.122 |  |
$2P$ | 7.137 |  | 7.113 | 7.140 | 7.135 | 7.126 | 7.145 |  |
$2P^{\prime}$ | 7.173 |  | 7.124 | 7.150 | 7.142 | 7.145 | 7.150 |  |
$2~{}^{3}P_{2}$ | 7.175 |  | 7.134 | 7.160 | 7.153 | 7.156 | 7.164 |  |
$3~{}^{1}S_{0}$ | 7.308 |  |  | 7.240 | 7.244 | 7.193 | 7.250 |  |
$3~{}^{3}S_{1}$ | 7.333 |  |  | 7.280 | 7.280 | 7.235 | 7.272 |  |
$2~{}^{3}D_{1}$ | 7.377 |  |  |  |  |  |  |  |
$2D$ | 7.385 |  |  |  |  |  |  |  |
$2D^{\prime}$ | 7.408 |  |  |  |  |  |  |  |
$2~{}^{3}D_{3}$ | 7.410 |  |  |  |  |  |  |  |
$3~{}^{3}P_{0}$ | 7.492 |  |  |  |  |  |  |  |
$3P$ | 7.546 |  |  |  |  |  |  |  |
$3P^{\prime}$ | 7.572 |  |  |  |  |  |  |  |
$3~{}^{3}P_{2}$ | 7.575 |  |  |  |  |  |  |  |
$4~{}^{1}S_{0}$ | 7.713 |  |  |  | 7.562 |  |  |  |
$4~{}^{3}S_{1}$ | 7.734 |  |  |  | 7.594 |  |  |  |
$3~{}^{3}D_{1}$ | 7.761 |  |  |  |  |  |  |  |
$3D$ | 7.781 |  |  |  |  |  |  |  |
$3D^{\prime}$ | 7.783 |  |  |  |  |  |  |  |
$3~{}^{3}D_{3}$ | 7.796 |  |  |  |  |  |  |  |
$4~{}^{3}P_{0}$ | 7.970 |  |  |  |  |  |  |  |
$4P$ | 7.943 |  |  |  |  |  |  |  |
$4P^{\prime}$ | 7.942 |  |  |  |  |  |  |  |
$4~{}^{3}P_{2}$ | 7.970 |  |  |  |  |  |  |  |
$5~{}^{1}S_{0}$ | 8.097 |  |  |  |  |  |  |  |
$5~{}^{3}S_{1}$ | 8.115 |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
State |  |  |  |  |  |  |  |  |
$n~{}^{2S+1}L_{J}$ | This work |  | $n~{}^{2S+1}L_{J}$ | This work |  | $n~{}^{2S+1}L_{J}$ | This work |  |
$4~{}^{3}D_{1}$ | 8.132 |  | $9~{}^{1}S_{0}$ | 9.543 |  | $12P^{\prime}$ | 10.778 |  |
$4D$ | 8.154 |  | $9~{}^{3}S_{1}$ | 9.553 |  | $12~{}^{3}P_{2}$ | 10.787 |  |
$4D^{\prime}$ | 8.155 |  | $8~{}^{3}D_{1}$ | 9.552 |  | $13~{}^{1}S_{0}$ | 11.015 |  |
$4~{}^{3}D_{3}$ | 8.168 |  | $8D$ | 9.573 |  | $13~{}^{3}S_{1}$ | 11.020 |  |
$5~{}^{3}P_{0}$ | 8.254 |  | $8D^{\prime}$ | 9.572 |  | $12~{}^{3}D_{1}$ | 10.964 |  |
$5P$ | 8.314 |  | $8~{}^{3}D_{3}$ | 9.586 |  | $12D$ | 10.972 |  |
$5P^{\prime}$ | 8.312 |  | $9~{}^{3}P_{0}$ | 9.695 |  | $12D^{\prime}$ | 10.971 |  |
$5~{}^{3}P_{2}$ | 8.337 |  | $9P$ | 9.646 |  | $12~{}^{3}D_{3}$ | 10.977 |  |
$6~{}^{1}S_{0}$ | 8.469 |  | $9P^{\prime}$ | 9.731 |  | $13~{}^{3}P_{0}$ | 11.192 |  |
$6~{}^{3}S_{1}$ | 8.484 |  | $9~{}^{3}P_{2}$ | 9.749 |  | $13P$ | 11.016 |  |
$5~{}^{3}D_{1}$ | 8.494 |  | $10~{}^{1}S_{0}$ | 9.896 |  | $13P^{\prime}$ | 11.203 |  |
$5D$ | 8.516 |  | $10~{}^{3}S_{1}$ | 9.905 |  | $13~{}^{3}P_{2}$ | 11.208 |  |
$5D^{\prime}$ | 8.517 |  | $9~{}^{3}D_{1}$ | 9.893 |  | $14~{}^{1}S_{0}$ | 11.458 |  |
$5~{}^{3}D_{3}$ | 8.530 |  | $9D$ | 9.912 |  | $14~{}^{3}S_{1}$ | 11.462 |  |
$6~{}^{3}P_{0}$ | 8.621 |  | $9D^{\prime}$ | 9.912 |  | $13~{}^{3}D_{1}$ | 11.356 |  |
$6P$ | 8.675 |  | $9~{}^{3}D_{3}$ | 9.925 |  | $13D$ | 11.361 |  |
$6P^{\prime}$ | 8.674 |  | $10~{}^{3}P_{0}$ | 10.051 |  | $13D^{\prime}$ | 11.361 |  |
$6~{}^{3}P_{2}$ | 8.696 |  | $10P$ | 9.855 |  | $13~{}^{3}D_{3}$ | 11.365 |  |
$7~{}^{1}S_{0}$ | 8.832 |  | $10P^{\prime}$ | 10.081 |  | $14~{}^{3}P_{0}$ | 11.623 |  |
$7~{}^{3}S_{1}$ | 8.846 |  | $10~{}^{3}P_{2}$ | 10.097 |  | $14P$ | 11.544 |  |
$6~{}^{3}D_{1}$ | 8.851 |  | $11~{}^{1}S_{0}$ | 10.253 |  | $14P^{\prime}$ | 11.629 |  |
$6D$ | 8.872 |  | $11~{}^{3}S_{1}$ | 10.261 |  | $14~{}^{3}P_{2}$ | 11.632 |  |
$6D^{\prime}$ | 8.873 |  | $10~{}^{3}D_{1}$ | 10.228 |  | $15~{}^{1}S_{0}$ | 12.005 |  |
$6~{}^{3}D_{3}$ | 8.886 |  | $10D$ | 10.243 |  | $15~{}^{3}S_{1}$ | 12.008 |  |
$7~{}^{3}P_{0}$ | 8.981 |  | $10D^{\prime}$ | 10.242 |  | $14~{}^{3}D_{1}$ | 11.801 |  |
$7P$ | 9.027 |  | $10~{}^{3}D_{3}$ | 10.253 |  | $14D$ | 11.804 |  |
$7P^{\prime}$ | 9.029 |  | $11~{}^{3}P_{0}$ | 10.403 |  | $14D^{\prime}$ | 11.804 |  |
$7~{}^{3}P_{2}$ | 9.051 |  | $11P$ | 10.172 |  | $14~{}^{3}D_{3}$ | 11.806 |  |
$8~{}^{1}S_{0}$ | 9.189 |  | $11P^{\prime}$ | 10.428 |  | $15~{}^{3}P_{0}$ | 12.175 |  |
$8~{}^{3}S_{1}$ | 9.201 |  | $11~{}^{3}P_{2}$ | 10.441 |  | $15P$ | 11.917 |  |
$7~{}^{3}D_{1}$ | 9.203 |  | $12~{}^{1}S_{0}$ | 10.621 |  | $15P^{\prime}$ | 12.179 |  |
$7D$ | 9.224 |  | $12~{}^{3}S_{1}$ | 10.628 |  | $15~{}^{3}P_{2}$ | 12.180 |  |
$7D^{\prime}$ | 9.224 |  | $11~{}^{3}D_{1}$ | 10.604 |  | $15~{}^{3}D_{1}$ | 12.350 |  |
$7~{}^{3}D_{3}$ | 9.238 |  | $11D$ | 10.617 |  | $15D$ | 12.351 |  |
$8~{}^{3}P_{0}$ | 9.339 |  | $11D^{\prime}$ | 10.616 |  | $15D^{\prime}$ | 12.351 |  |
$8P$ | 9.366 |  | $11~{}^{3}D_{3}$ | 10.625 |  | $15~{}^{3}D_{3}$ | 12.352 |  |
$8P^{\prime}$ | 9.381 |  | $12~{}^{3}P_{0}$ | 10.762 |  |  |  |  |
$8~{}^{3}P_{2}$ | 9.401 |  | $12P$ | 10.561 |  |  |  |  |

where $\theta_{nL}$ is a mixing angle, and the primed state has the heavier
mass. The values of the mixing angle for P states are
$\theta_{1P}=0.2^{\circ}$, $\theta_{2P}=0.10^{\circ}$ and
$\theta_{3P}=0.05^{\circ}$

Similarly for $L=J=2$ we have mixing of D states and the values of mixing
angles for D states are $\theta_{1D}=0.20^{\circ}$ and
$\theta_{2D}=0.05^{\circ}$.
The calculated masses of the $c\bar{b}$ states after diagonalization are
listed in Table LABEL:spectrum. Our calculated mass value for $B_{c}(1S)$ is
6275.75 MeV which agrees withe experimental value 6.275 GeV[90] and for
$B^{*}_{c}$(1S) is 6357.27 MeV. $B^{*}_{c}$(1S) is heavier than $B_{c}$(1S) by
81.52 MeV. This difference is justified by calculating the
${}^{3}S_{1}-{}^{1}S_{0}$ splitting of the ground state which is given by

$M({}^{3}S_{1})-M({}^{1}S_{0})=\frac{32\pi\alpha_{s}|\psi(0)|^{2}}{9m_{c}m_{b}}$
(20)

The mass of first radial excitation $B_{c}$(2S) is 6862.88 MeV which is
heavier than $B_{c}$(1S) by 587.13 MeV. This value agrees with the
experimental value of $B_{c}$(2S) 6842$\pm$4$\pm$5 [14]. The difference
between the $B^{*}_{c}$(2S) and $B^{*}_{c}$(1S) masses turns out to be 540.14
MeV. Our prediction for masses of orbitally excited $c\bar{b}$ states are in
good agreement with the other model calculations. Some of the states (i.e.,
$2^{3}P_{0}$, 2P1,$2P1^{\prime}$,$2^{3}P_{2}$) are 50-100 MeV heavier in our
model.

Vijande et al. have derived Coulomb strength
$A_{3Q}\approx\frac{1}{2}A_{Q\bar{Q}}$ and confinement strength $B_{3Q}\approx
B_{Q\bar{Q}}$ from the triply baryon spectra calculated in Lattice QCD [91,
92, 93]. They obtained a nice fit of the nonperturbative QCD results with the
Cornell like potential

$V^{3Q}(r)=-A\sum_{i<j}\frac{1}{|\vec{r}_{i}-\vec{r}_{j}|}+B\sum_{i<j}|+C$
(21)

with Coulomb strength $A_{3Q}=0.1875$ and confinement strength
$B_{3Q}=0.1374~{}\rm{GeV}^{2}$. The Coulomb strength and the confinement
strength calculated in our model is $A_{Q\bar{Q}}=0.4$ and
$B_{Q\bar{Q}}=0.1526$ GeV2 (i.e. coefficients of Coulomb and linear
potentials) respectively. Using these values we come to the conclusion that
$\frac{A_{3Q}}{A_{Q\bar{Q}}}\approx\frac{1}{2}$ and $B_{3Q}\approx
B_{Q\bar{Q}}$. Vijande et al. obtained
$\frac{A_{3Q}}{A_{Q\bar{Q}}}<\frac{1}{2}$ slightly different from
$\frac{1}{2}$ as the one gluon exchange result and our result [92].

###  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 development stage. At present, the potential
model approaches provide the detailed predictions that can be compared to
experimental results. In our non relativistic model we consider the Magnetic
dipole (M1) transitions and Electric dipole (E1) transitions of $B_{c}$ meson.

We have listed the possible $E1$ decay modes in table 3.3 and given the
predictions for E1 decay widths. Also we have compared our predictions 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. This is due to the orthogonality
condition for wave functions.

The possible radiative M1 transition modes are (1) 2 ${}^{3}S_{1}\rightarrow
2^{1}S_{0}+\gamma$, (2) 2 ${}^{3}S_{1}\rightarrow 1^{1}S_{0}+\gamma$, (3) 2
${}^{1}S_{0}\rightarrow 1^{3}S_{1}+\gamma$ and (4)1 ${}^{3}S_{1}\rightarrow
1^{1}S_{0}+\gamma$.

In the above (2) and (3) represent hindered transitions and (1) and (4)
represent allowed transitions.

The resulting M1 radiative transition rates of these states are presented in
table 3.3. In this table we give the calculated M1 decay widths for allowed
transitions (n${}^{3}S_{1}\rightarrow n^{\prime}{{}^{1}S_{0}}+\gamma$,
$n=n^{\prime}$ ) and we compare the decay widths with other non relativistic
quark models [15, 84, 89]. The hindered transitions are strongly suppressed in
the non relativistic limit due to the orthogonality of the initial and final
state wave functions. By adding relativistic effects to the wave function the
hindered M1 transition rates can be enhanced.

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

E1 transition rates of $B_{c}$ meson. Transition k0 $\Gamma(i\rightarrow
f+\gamma)$ Ref. [77] Ref. [15] Ref. [24] Ref.[89] MeV keV keV keV keV keV
$1^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ 281.22 30.67 75.5 79.2 65.3 74.2
$1P\rightarrow 1^{3}S_{1}\gamma$ 329.71 49.438 87.1 99.5 77.8 75.8
$1P^{\prime}\rightarrow 1^{3}S_{1}\gamma$ 377.72 74.331 13.7 0.1 8.1 26.2
$1^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ 380.55 112.754 122 112.6 102.9 126
$1P\rightarrow 1^{1}S_{0}\gamma$ 411.23 31.974 18.4 0 11.6 32.5
$1P^{\prime}\rightarrow 1^{1}S_{0}\gamma$ 459.24 44.531 147 56.4 131.1 128
$2^{3}S_{1}\rightarrow 1^{3}P_{0}\gamma$ 258.92 7.98 5.53 7.8 7.7 9.6
$2^{3}S_{1}\rightarrow 1P\gamma$ 210.43 8.568 7.65 14.5 12.8 13.3
$2^{3}S_{1}\rightarrow 1P^{\prime}\gamma$ 162.42 3.939 0.74 0 1.0 2.5
$2^{3}S_{1}\rightarrow 1^{3}P_{2}\gamma$ 159.59 6.228 7.59 17.7 14.8 14.5
$2^{1}S_{0}\rightarrow 1P\gamma$ 175.9 5.004 1.05 0 1.9 6.4
$2^{1}S_{0}\rightarrow 1P^{\prime}\gamma$ 127.89 1.923 4.40 5.2 15.9 13.1
$2^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ 726.95 0 21.9 16.1 $2P\rightarrow
1^{3}S_{1}\gamma$ 797.33 0 22.1 15.3 $2P^{\prime}\rightarrow 1^{3}S_{1}\gamma$
816.04 0 2.1 2.5 $2^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ 818.37 0 25.8 19.2
$2P\rightarrow 1^{1}S_{0}\gamma$ 861.56 0 3.1 $2P^{\prime}\rightarrow
1^{1}S_{0}\gamma$ 897.56 0 20.1 $2^{3}P_{0}\rightarrow 2^{3}S_{1}\gamma$
186.81 14.987 34.0 41.2 25.5 $2P\rightarrow 2^{3}S_{1}\gamma$ 239.9 31.739
45.3 54.3 32.1 $2P^{\prime}\rightarrow 2^{3}S_{1}\gamma$ 275.9 48.280 10.4 5.4
5.9 $2^{3}P_{2}\rightarrow 2^{3}S_{1}\gamma$ 278.23 49.513 75.3 73.8 49.4
$2P\rightarrow 2^{1}S_{0}\gamma$ 274.43 47.512 13.8 8.1
$2P^{\prime}\rightarrow 2^{1}S_{0}\gamma$ 310.43 68.770 90.5 58.0

M1 transition rates for the $B_{c}$ meson. Transition $k_{0}$ Ref. [89] Ref.
[84] Ref. [77] Ref.[15] This work $\Gamma(keV)$ $\Gamma(keV)$ $\Gamma(keV)$
$\Gamma(keV)$ $\Gamma(keV)$ $\Gamma(keV)$ $1~{}^{3}S_{1}\rightarrow
1^{1}S_{0}\gamma$ 81.52 0.190 0.060 0.073 0.135 0.0581
$2~{}^{3}S_{1}\rightarrow 2^{1}S_{0}\gamma$ 34.53 0.043 0.010 0.030 0.029
0.00173

Table 2: Comparison of life time of $B_{c}$ meson (in ps). This work | Experiment[90] | Ref.[71] | Ref.[24] | Ref.[94] | Ref. [95]
---|---|---|---|---|---
0.379 | 0.452$\pm 0.033$ | 0.47 | 0.55$\pm 0.15$ | 0.50 | 0.75
Table 3: Comparison of predictions for the pseudo scalar decay constant of the $B_{c}$ meson. Parameter | Ref.[75] | Ref. [73] | Ref.[76] | Ref.[88] | This work
---|---|---|---|---|---
$f_{B_{c}}$ | 500 | 512 | 479 | 440$\pm$20 | 439.735

In accordance with the classification given in section 2.3, the total decay
width can be written as the sum over partial widths

$\Gamma(B_{c}\rightarrow X)=\Gamma_{1}(\bar{b}\rightarrow
X)+\Gamma_{2}(c\rightarrow X)+\Gamma_{3}(ann)$ (22)

In the spectator approximation:

$\Gamma_{1}(\bar{b}\rightarrow
X)=\frac{9G^{2}_{F}|V_{cb}|^{2}m^{5}_{b}}{192\pi^{3}}$ (23)

Calculated value of $\Gamma_{1}(\bar{b}\rightarrow X)$ is $9.628\times
10^{-4}~{}\rm{eV}$ and

$\Gamma_{2}(c\rightarrow
X)=\frac{5G^{2}_{F}|V_{cs}|^{2}m^{5}_{c}}{192\pi^{3}}$ (24)

where $V_{cb}$ and $V_{cs}$ are the elements of the CKM matrix. The decay
widths are calculated using $|V_{bc}|=0.044$ [90] and $|V_{cs}|=0.975$ [90].
Calculated value of $\Gamma_{2}(c\rightarrow X)$ is $7.712\times
10^{-4}~{}\rm{eV}$.

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

$\Gamma_{3}(ann)=\frac{G^{2}_{F}}{8\pi}|V_{bc}|^{2}f^{2}_{B_{c}}M_{B_{c}}\sum_{i}m^{2}_{i}\left(1-\frac{m^{2}_{i}}{M^{2}_{B_{c}}}\right)C_{i}$
(25)

where $m_{i}$ is the mass of the heavier fermion in the given decay channel.
For lepton channels $C_{i}=1$ while for quark channels
$C_{i}=3|V_{q\bar{q}}|^{2}$.
Calculated value of $\Gamma_{3}$ is $3.56\times 10^{-6}~{}\rm{eV}$
Adding these results we get the total decay width
$\Gamma(\rm{total})=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}=18.104\times
10^{-4}~{}\rm{eV}$ corresponding to a life time of $\tau=0.364~{}\rm{ps}$.

The pseudo scalar decay constant $f_{B_{c}}$ is defined by [15]

$\langle 0\rvert\bar{b}(x)\gamma^{\mu}\gamma_{5}c(x)\lvert
B_{c}(k)\rangle=if_{B_{c}}V_{cb}k^{\mu}$ (26)

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 [96]

$f_{B_{c}}=\sqrt{\frac{12}{M_{B_{c}}}}\psi(0)$ (27)

The value of decay constant in the non relativistic potential model is listed
in table 3.

##  4 Theoretical uncertainties in the predictions of the model

The goal of the present work is to obtain a reliable estimate of the masses of
the $c\bar{b}$ states and the decay widths and understand the uncertainties in
the calculation in the frame work of non-relativistic quark models. The main
reason for using the non-relativistic model is, it gives a good account of the
spectra and it is possible to separate the center-of-mass motion and the
relativistic correction can be incorporated by $v/c$ expansion. For $c\bar{b}$
system quark velocities are sizeable and hence needs significant corrections,
but are still very small to make any significant changes in the spectra or
decay widths and $v/c$ corrections are very small and the non-relativistic
predictions radiative decays are very accurate[97]. The non-relativistic quark
models have many other basic features of the QCD and has met with great
success in predicting the spectra and decay widths of hadrons as it allows
direct calculations of the relevant matrix elements for each hadron.

The standard way of estimating the uncertainties in any model is to vary
different parameters in the model.

Also it is known that using a larger harmonic oscillator basis increases the
wave function at the origin since the higher order states mixing into the wave
function can probe the short distance of the potential[98, 85]. Hence there is
a slight theoretical uncertainty in the value of the oscillator size parameter
(b). Also, there is theoretical uncertainty in the form of the wave function
$\psi(0)$ at the origin. The $\psi(0)$ is relatively flat for linear and
harmonic oscillator potentials, but it raises sharply for the Columbic
potential. Ultimately, the form of the wave function at the origin has to be
settled by lattice QCD calculations which will be the most reliable results as
they are calculations from the first principles. But, for the orbitally
excited states, the wave function at the origin vanishes in the non
relativistic limit and hence the lowest order approximation vanishes. In
calculating the masses and decay widths of quarkonium there are ambiguities in
the value of $\alpha_{s}$. These ambiguities are both theoretical and
experimental. In theory there is uncertainty in the scale $\mu$ to be
considered in computing $\alpha_{s}(\mu^{2})$. On the other hand in
experimental side, there is uncertainty in the scale parameter $\Lambda$ of
the QCD.

Another source of theoretical uncertainty is the masses of the charm ($m_{c}$)
and bottom quark ($m_{b}$). For the charm quark, masses of $m_{c}$ used in
literature are 1 GeV, 1.5 GeV and 1.8 GeV. But it should be noted that the
spectra of the quarkonium are not very sensitive to the mass of the quark, but
$|\psi(0)|^{2}$ is strongly dependent on $m_{c}$. But, since we are making use
of the non-relativistic model both for spectrum and decay the choices for the
masses of $m_{c}$ are minimum. The $m_{c}$ is constrained by the rate of the
M1 transition $J/\psi\to\eta_{c}\gamma$. This branching ratio restricts
$m_{c}\sim$ 2 GeV. Our choice for $m_{c}$ is 1.48 GeV which is the value
quoted in PDG. For the b quark, the mass is again taken from PDG. For the b
quark, for a fixed energy eigen value the dependence on $|\psi(0)|^{2}$ is
minimal as the variation is little because of smaller uncertainty of the
$m_{b}$.

##  5 Conclusions

The study highlights the mass spectra of $c\bar{b}$ meson in a non
relativistic quark model. 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 50-80 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 Penin et
al. They predicted the hyperfine splitting of the vector and pseudo scalar
$B_{c}(1S)$ mesons to be $M(B^{*}_{c})-M(B_{c})=50\pm 17(th)\rm{MeV}$. 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.,
$6.194~{}\rm{GeV}<M_{B_{c}}<6.292~{}\rm{GeV}$ and
$6.284~{}\rm{GeV}<M_{B^{*}_{c}}<6.357~{}\rm{GeV}$.

Radiative decays are the dominant decay modes of the $c\bar{b}$ excited states
having widths of about a fraction of MeV. In order to understand the
$c\bar{b}$ spectrum and distinguishing exotic states, it is very essential
that the masses and the radiative decay widths of $c\bar{b}$ states are
accurately determined. The calculated M1 transition rates reasonably agree
with the other theoretical model predictions as listed in table 3.3. It is
clearly seen in this calculation that the relativistic effects play an
important role in determining the M1 radiative transition rates, since the
hindered transition rates are zero due to the wave function orthogonality in
the NRQM formalism. The inclusion of these relativistic effects may enhance
the hindered transition rates and reduce the allowed transition rates.

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 differences in
mass predictions, wave function effects and singlet - triplet mixing angels.
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 42% for $c$-quark decays and
about 5% branching ratio in annihilation channel. The life time of $B_{c}$
meson predicted in in othe theoreticla model is listed in table 2. The life
time of $c\bar{b}$ state predicted in this calculation 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.

A simple non relativistic model employing OGEP and linear confinement
potential 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
transition rates.

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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