††thanks: Mailing address

# Study of $B_{c}$ ${\to}$ $B^{({\ast})}P$, $BV$ Decays with QCD Factorization

Junfeng Sun  College of Physics and Information Engineering, Henan Normal
University, Xinxiang 453007, China  Theoretical Physics Center for Science
Facilities (TPCSF), Institute of High Energy Physics, Chinese Academy of
Sciences (IHEP, CAS)  Yueling Yang  College of Physics and Information
Engineering, Henan Normal University, Xinxiang 453007, China  Wenjie Du
College of Physics and Information Engineering, Henan Normal University,
Xinxiang 453007, China  Huilan Ma  College of Physics and Information
Engineering, Henan Normal University, Xinxiang 453007, China

###### Abstract

The $B_{c}$ ${\to}$ $B^{({\ast})}_{q}P$, $B_{q}V$ decays are studied with the
QCD factorization approach (where $P$ and $V$ denote pseudoscalar and vector
mesons, respectively; $q$ $=$ $u$, $d$ and $s$). Considering the contributions
of both current-current and penguin operators, the amplitudes of branching
ratios are estimated at the leading approximation. We find that the
contributions of the penguin operators are very small due to the serious
suppression by the CKM elements. The most promising decay modes are $B_{c}$
${\to}$ $B_{s}^{({\ast})}{\pi}$, $B_{s}{\rho}$, which might be easily detected
at hadron colliders.

###### pacs:

12.39.St 13.25.Hw

##  I Introduction

The $B_{c}$ meson is one of the unique “double heavy-flavored” binding system
in the standard model (SM). The study of the $B_{c}$ meson has received a
great interest, due to its special properties: (1) The $B_{c}$ meson carries
open flavors. We can study the two heavy flavors of both $b$ and $c$ quarks
simultaneously with the $B_{c}$ meson. (2) The $B_{c}$ meson can serve as a
great laboratory for potential models, QCD sum rules, Heavy Quark Effective
Theory (HQET), lattice QCD, etc. (3) The $B_{c}$ meson has rich decay
channels, because of its sufficiently large mass and that the $b$ and $c$
quarks can decay individually. The $B_{c}$ meson decays may provide windows
for testing the predictions of the SM and can shed light on new physics beyond
SM.

The $B_{c}$ mesons are too massive to access at the $B$-factories near
${\Upsilon}(4S)$. They can be produced in significant numbers at hadron
colliders. The $B_{c}$ meson has been firstly discovered by the CDF
Collaboration cdf98 . Recently the CDF and D0 Collaborations announced some
accurate measurements cdf07 ; d008  with part of their available data. Much
more $B_{c}$ mesons and detailed information about their decay properties are
expected at the Large Hadron Collider (LHC) which is scheduled to run in this
year. It is estimated that one could expect around $5$ ${\times}$ $10^{10}$
$B_{c}$ events per year at LHC 0412158 ; pan67p1559  due to the relatively
large production cross section prd71p074012  plus the huge luminosity ${\cal
L}$ $=$ $10^{34}\,{\rm cm}^{-2}{\rm s}^{-1}$ and high center-of-mass energy
$\sqrt{s}$ $=$ $14$ TeV lhc . There seems to exist a real possibility to study
not only some $B_{c}$ rare decays, but also $CP$ violation and polarization
asymmetries. The study of the $B_{c}$ meson will highlight the advantages of
$B$ physics at hadron colliders.

The $B_{c}$ meson is stable for strong interaction because it lies below the
threshold of the $B$-$D$ mesons. The electromagnetic interaction cannot
transform the $B_{c}$ meson into other hadrons containing both $b$ and $c$
heavy quarks, because the $B_{c}$ meson itself is the ground state. The
$B_{c}$ meson decays via weak interaction only, which can be divided into
three classes: (1) the $b$ quark decay ($b$ ${\to}$ $c$, $u$) with $c$ quark
as a spectator, (2) the $c$ quark decay ($c$ ${\to}$ $s$, $d$) with $b$ quark
as a spectator, and (3) the weak annihilation channels. In the $B_{c}$ meson,
both heavy quark can decay weakly, resulting in its much shorter lifetime than
other $b$-flavored mesons, i.e. ${\tau}_{B_{c}}$ ${\lesssim}$
$\frac{1}{3}{\tau}_{B_{q}}$ (where $q$ $=$ $u$, $d$, and $s$) pdg2006 . Rates
of the Class (1) and (2) are competitive in magnitude. The Cabibbo-Kobayashi-
Maskawa (CKM) ckm  matrix elements ${|}V_{cb}{|}$ ${\ll}$ ${|}V_{cs}{|}$, that
is in favor of the $c$-quark decay greatly, whereas the phase space factor
$m_{b}^{5}$ ${\gg}$ $m_{c}^{5}$ compensates the CKM matrix elements a lot for
the two flavors ijmpa21p777 . In fact, the dominant contributions to the
$B_{c}$ lifetime comes from the $c$-quark decays [Class (2)] (${\approx}$
$70\%$), while the $b$-quark decay [Class (1)] and weak annihilation [Class
(3)] are expected to add about $20\%$ and $10\%$, respectively 0412158 .

The $B_{c}$ meson decays have been widely studied in the literature due to
some of its outstanding features. (1) The pure leptonic $B_{c}$ decays belong
to the Class (3), which are free from strong interaction in final states and
can be used to measure the decay constant $f_{B_{c}}$ and the CKM elements
${|}V_{cb}{|}$, but they are not fully reconstructed due to the missing
neutrino. (2) The semileptonic $B_{c}$ decays provide an excellent laboratory
to measure the CKM elements ${|}V_{cb}{|}$, ${|}V_{ub}{|}$, ${|}V_{cs}{|}$,
${|}V_{cd}{|}$ and form factors for transitions of $B_{c}$ ${\to}$ $b$\- and
$c$-flavored mesons. The first signal of $B_{c}$ is observed via this mode
cdf98 . The most difficult theoretical work at present is how to evaluate the
hadronic matrix elements properly and accurately. (3) The nonleptonic $B_{c}$
decays are the most complicated due to the participation of the strong
interaction, which complicate the extraction of parameters in SM, but they
also provide great opportunities to study perturbative and non-perturbative
QCD, final state interactions, etc.

The earlier nonleptonic decays of $B_{c}$ meson has been studied in 0412158 ;
pan67p1559 ; 07070919 ; prd75p097304 ; prd73p054024 ; epjc45p711 ;
prd70p074022 ; prd68p094020 ; epjc32p29 ; jpg28p595 ; jpg28p2241 ;
prd65p114007 ; pan64p1860 ; pan64p2027 ; prd61p034012 ; prd62p057503 ;
prd62p014019 ; cpl18p498 ; epjc9p557 ; epjc5p705 ; prd56p4133 ; plb387p187 ;
pan60p1729 ; 9605451 ; 9504319 ; prd49p3399 ; prd46p3836 ; plb286p160 ;
prd39p1342 . While $c$-quark decays take the lion’s share of the $B_{c}$
lifetime, the study on the Class (2) has not received enough attention. This
can be explained by the fact that on the one hand the available data on the
$B_{c}$ meson is very few, on the other hand it is assumed that the long
distance effects and final state interferences might be quite huge and that
the Class (2) decays were hard to detect experimentally. Accompanied by the
LHC being about to run, the future copious data require more accurate
theoretical predictions from now on. In this paper, we shall concentrate on
the $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$ (here $P$ and $V$ denote
pseudoscalar and vector mesons, respectively; $q$ $=$ $u$, $d$ and $s$) decays
in Class (2) with QCD factorization approach. Now let us outline a few reasons
and arguments below.

  1. 1.

From the experimental view

     * •

The initial and final $b$-flavored mesons, i.e. the $B_{c}$ and
$B_{q}^{({\ast})}$, all have a long lifetime due to their decays via the weak
interaction. Considered the relativistic boost kinematically due to their
large momentum obtained from huge center-of-mass energy, their information
would be easily recorded by the multipurpose detectors sitting at the hadron
colliders interaction regions (see lhc  for details).

     * •

Although it is perceived that the hadron collider environment is “messy” with
high backgrounds, the $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$ decays are
measurable due to the “clean” final states. Since the $B_{c}$ meson carries
charge, the final $B_{q}^{({\ast})}$ meson is tagged explicitly by the initial
$B_{c}$ meson. The other light meson in the final state could also be
identified effectively by the conservation low of both momentum and energy,
because the dedicated detectors at LHC has excellent performance on trigger,
time resolution, particle identification and so on (see lhc  for details).

  2. 2.

From the phenomenological view

     * •

With very high statistics, we can carefully test the various theoretical
models, precisely determine the CKM elements, and meticulously search for the
signals of new physics. This requires more accurate theoretical predictions.
In this paper, we shall study the $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$,
$B_{q}V$ decays with QCD factorization approach, including the contributions
of both current-current and penguin operators.

     * •

In the rest frame of the $B_{c}$ meson, the velocity
${\beta}_{B_{q}^{({\ast})}}$ of the $B_{q}^{({\ast})}$ meson is very small due
to its large mass, not exceeding $0.18$. The ratio of velocity
${\beta}_{P,V}/{\beta}_{B_{q}^{({\ast})}}$ ${\gtrsim}$ $5.5$, which is very
different from that in the two-body $D$ meson decays where the ratio of
velocities of final states is close to one. This may indicate that the final
state interferences for $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$ decays
might not be so strong as that in $D$ mesons. If it holds true, it will
benefit us in determining the CKM elements $V_{cs}$ and $V_{cd}$, the $B_{c}$
${\to}$ $B_{q}^{({\ast})}$ transition form factors, etc. In this paper, we
shall neglect the effects of final state interferences for the moment.

This paper is organized as follows: In Sec. II, the theoretical framework is
discussed. To estimate the amplitude of the branching ratios, the master QCD
factorization (QCDF) formula are applied to the $B_{c}$ ${\to}$
$B_{q}^{({\ast})}P$, $B_{q}V$ decays at the leading approximation. Section III
is devoted to the numerical results. Finally, we summarize in Sec. IV.

##  II Theoretical framework

###  II.1 The effective Hamiltonian

Using the operator product expansion and renormalization group (RG) equation,
the low energy effective Hamiltonian relevant to the $B_{c}$ ${\to}$
$B_{q}^{({\ast})}P$, $B_{q}V$ decays can be written as

${\cal H}_{\rm
eff}=\frac{G_{F}}{\sqrt{2}}\Big{\\{}\\!\\!\sum\limits_{q_{1}=d,s\atop
q_{2}=d,s}\\!\\!\\!V_{uq_{{}_{1}}}V_{cq_{{}_{2}}}^{\ast}\Big{[}C_{1}({\mu})Q_{1}+C_{2}({\mu})Q_{2}\Big{]}+\\!\\!\\!\sum\limits_{q=d,s\atop
i=3,\ldots,10}\\!\\!\\!\\!\\!V_{uq}V_{cq}^{\ast}C_{i}({\mu})Q_{i}\Big{\\}}+\hbox{H.c.},$
(1)

where $V_{uq_{{}_{\\!1}}}V_{cq_{{}_{2}}}^{\ast}$ is the CKM factor. The cases
$q$ $=$ $d$ and $q$ $=$ $s$ can be treated separately and have the same Wilson
coefficients $C_{i}({\mu})$. The expressions of the local operators are

$\displaystyle
Q_{1}=(\bar{u}_{\alpha}q_{\\!1\,{\alpha}})_{V-A}(\bar{q}_{2{\beta}}c_{\beta})_{V-A},\
\ \ \ \ \ \ \ \ \ \ \ \ \
Q_{2}=(\bar{u}_{\alpha}q_{1{\beta}})_{V-A}(\bar{q}_{2{\beta}}c_{\alpha})_{V-A}$
(2) $\displaystyle
Q_{3}=({\bar{u}}_{\alpha}c_{\alpha})_{V-A}\sum\limits_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\beta})_{V-A},\
\ \ \ \ \ \ \ \ \ \ \
Q_{4}=({\bar{u}}_{\alpha}c_{\beta})_{V-A}\sum\limits_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A},$
(3) $\displaystyle
Q_{5}=({\bar{u}}_{\alpha}c_{\alpha})_{V-A}\sum\limits_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\beta})_{V+A},\
\ \ \ \ \ \ \ \ \ \ \
Q_{6}=({\bar{u}}_{\alpha}c_{\beta})_{V-A}\sum\limits_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A},$
(4) $\displaystyle
Q_{7}=\frac{3}{2}({\bar{u}}_{\alpha}c_{\alpha})_{V-A}\sum\limits_{q^{\prime}}e_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\beta})_{V+A},\
\ \ \ \ \
Q_{8}=\frac{3}{2}({\bar{u}}_{\alpha}c_{\beta})_{V-A}\sum\limits_{q^{\prime}}e_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A},$
(5) $\displaystyle
Q_{9}=\frac{3}{2}({\bar{u}}_{\alpha}c_{\alpha})_{V-A}\sum\limits_{q^{\prime}}e_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\beta})_{V-A},\
\ \ \ \ \
Q_{10}=\frac{3}{2}({\bar{u}}_{\alpha}c_{\beta})_{V-A}\sum\limits_{q^{\prime}}e_{q^{\prime}}({\bar{q}}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A},$
(6)

where the summation over the repeated color indices (${\alpha}$ and ${\beta}$)
is understood. The Dirac current $(\bar{q}_{1}q_{2})_{V{\pm}A}$ $=$
$\bar{q}_{1}{\gamma}(1{\pm}{\gamma}_{5})q_{2}$. $q^{\prime}$ denotes all the
active quarks at scale ${\mu}$ $=$ ${\cal O}(m_{c})$, i.e. $q^{\prime}$ $=$
$u$, $d$, $s$, $c$. $e_{q^{\prime}}$ denotes the electric change of the
corresponding quark $q^{\prime}$ in the unit of ${|}e{|}$, which reflects the
electroweak origin of $Q_{7}$, ${\cdots}$, $Q_{10}$. The current-current
operators ($Q_{1}$, $Q_{2}$), QCD penguin operators ($Q_{3}$, ${\cdots}$,
$Q_{6}$), and electroweak penguin operators ($Q_{7}$, ${\cdots}$, $Q_{10}$)
form a complete basis set under QCD and QED renormalization rmp68p1125 .

The effective coupling constants — Wilson coefficients $C_{i}({\mu})$ — are
calculated in perturbative theory at a high scale ${\mu}$ ${\sim}$ $m_{W}$ and
evolved down to a characteristic scale ${\mu}$ ${\sim}$ $m_{c}$ using the RG
equations. The Wilson coefficient functions are given by rmp68p1125

$\vec{C}({\mu})=U_{4}({\mu},{\mu}_{b})M({\mu}_{b})U_{5}({\mu}_{b},{\mu}_{W})\vec{C}({\mu}_{W})$
(7)

Here $U_{f}({\mu}_{f},{\mu}_{i})$ is the RG evolution matrix for $f$ active
flavors, which includes the RG-improved perturbative contribution from the
initial scale ${\mu}_{i}$ down to the final scale ${\mu}_{f}$. The $M({\mu})$
is the $10{\times}10$ quark-threshold matching matrix. The corresponding
formula and expressions can be found in Ref. rmp68p1125 . The Wilson
coefficients $C_{i}({\mu})$ have been evaluated to the next-to-leading order
(NLO). Their numerical values in the naive dimensional regularization (NDR)
scheme are listed in Table 1.

###  II.2 Hadronic matrix elements within the QCDF framework

For the weak decays of hadrons, the short-distance effects are well-known and
can be calculated in perturbation theory. However, the nonperturbative long-
distance effects responsible for the hadronization from quarks to hadrons
still remain obscure in several aspects. But to calculate the exclusive weak
decays of the $B_{c}$ meson, one needs to evaluate the hadronic matrix
elements, i.e., the weak current operator sandwiched between the initial state
of the $B_{c}$ meson and the concerned final states, which is the most
difficult theoretical work at present. Phenomenologically, these hadronic
matrix elements are usually parameterized into the product of the decay
constants and the transition form factors based on the argument of color
transparency and the naive factorization scheme (NF) bsw . A few years ago,
Beneke, Buchalla, Neubert, and Sachrajda suggested a QCDF formula to compute
the hadronic matrix elements in the heavy quark limit, combining the hard
scattering approach with power counting in $1/m_{Q}$ 9905312  (here $m_{Q}$ is
the mass of heavy quark). At leading order in the power series of heavy quark
mass expansion, the hadronic matrix elements can be factorized into “non-
factorizable” corrections dominated by hard gluon exchange and universal non-
perturbative part parameterized by the form factors and meson’s light cone
distribution amplitudes. This promising approach has been applied to exclusive
two-body nonleptonic $B_{u}$, $B_{d}$, $B_{s}$ decays 0108141 ; prd68p054003 ;
npb657p333 . It is found that with appropriate parameters, most of the QCDF’s
predictions are in agreement with the present experimental data. In this
paper, we would like to apply the QCDF approach to the $B_{c}$ ${\to}$
$B^{({\ast})}_{q}P$, $B_{q}V$ decays.

In the heavy quark limit $m_{c}$ $\gg$ ${\Lambda}_{QCD}$, up to power
corrections of order of the ${\Lambda}_{QCD}/m_{c}$, using the master QCDF
formula, the hadronic matrix elements for the $B_{c}$ ${\to}$
$B^{({\ast})}_{q}M$ decays ($M$ $=$ $P$ or $V$) can be written as 9905312

${\langle}B^{({\ast})}_{q}M{|}O_{i}{|}B_{c}{\rangle}\ =\
F^{B_{c}{\to}B^{({\ast})}_{q}}{\int}dz\ H(z){\Phi}_{M}(z)$ (8)

where $F^{B_{c}{\to}B^{({\ast})}_{q}}$ is the transition form factor and
${\Phi}_{M}(z)$ is the distribution amplitudes for the meson of $M$, which are
assumed to be nonperturbative and dominated by the soft contributions. The
hard-scattering kernels $H(z)$ can be calculated in the perturbative theory.
For details about the QCDF formula Eq.(8), please refer to Ref.9905312 .

To estimate the branching ratios approximately and to have a sense of the
order of amplitudes, we shall adopt a rough approximation, i.e. at the leading
order of ${\alpha}_{s}$. Within this approximation, the hard-scattering kernel
functions become very simple, $H(z)$ $=$ $1$. That is to say the long-distance
interactions between $M$ and $B_{c}$-$B^{({\ast})}_{q}$ system could be
neglected. So the integral of ${\Phi}_{M}(z)$ reduces to the normalization
condition for the distribution amplitudes. Furthermore, according to the
arguments of QCDF 9905312 , the hard interactions with the spectator are power
suppressed in the heavy quark limit. Therefore it is not surprisingly to
reproduce the result of NF.

In our paper, the annihilation amplitudes are neglected due to some reasons.
(1) According to the power counting arguments of QCDF 9905312 , compared with
the leading order contributions to the hard scattering kernel, the
contributions from annihilation topologies are power suppressed. (2) The
annihilation amplitudes are suppressed by the CKM elements. For the decay
modes concerned, the CKM factors in the non-annihilation amplitudes are
$V_{ud}V_{cs}^{\ast}$ ${\sim}$ $1$, $V_{us}V_{cs}^{\ast}$ ${\sim}$
${\lambda}$, $V_{ud}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}$ and
$V_{us}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}^{2}$, while the annihilation
amplitudes are proportional to the CKM factors of $V_{cb}V_{ub}^{\ast}$
${\sim}$ ${\lambda}^{5}$.

The explicit expressions of decay amplitudes for $B_{c}$ ${\to}$
$B^{({\ast})}_{q}P$, $B_{q}V$ decays are collected in appendix A. In our
paper, we define

$\displaystyle a_{i}$ $\displaystyle{\equiv}$ $\displaystyle
C_{i}+\frac{1}{N_{c}}C_{i+1}\ \ \ \ \ \ \ (i={\rm odd})$ (9) $\displaystyle
a_{i}$ $\displaystyle{\equiv}$ $\displaystyle C_{i}+\frac{1}{N_{c}}C_{i-1}\ \
\ \ \ \ \ (i={\rm even})$ (10)

where $i$ runs from $1$ to $10$, $C_{i}$ are the Wilson coefficients. $N_{c}$
$=$ $3$ is the color number.

##  III Numerical results and discussions

Within the QCDF approach, the decay amplitudes depend on many input parameters
including the CKM matrix elements, decay constants, form factors, etc. These
parameters are discussed and specified below.

###  III.1 The CKM matrix elements

We will use the Wolfenstein parameterization. Phenomenologically, it is a
popular approximation of the CKM matrix in which each elements is expanded as
a power series in the small parameter ${\lambda}$. Up to ${\cal
O}({\lambda}^{6})$, the CKM elements can be written as rmp68p1125

$\displaystyle V_{ud}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2}{\lambda}^{2}-\frac{1}{8}{\lambda}^{4}+{\cal O}({\lambda}^{6})$
(11) $\displaystyle V_{us}$ $\displaystyle=$ $\displaystyle{\lambda}+{\cal
O}({\lambda}^{6})$ (12) $\displaystyle V_{ub}$ $\displaystyle=$ $\displaystyle
A{\lambda}^{3}({\rho}-i{\eta})$ (13) $\displaystyle V_{cd}$ $\displaystyle=$
$\displaystyle-{\lambda}+A^{2}{\lambda}^{5}\Big{[}\frac{1}{2}-({\rho}+i{\eta})\Big{]}+{\cal
O}({\lambda}^{6})$ (14) $\displaystyle V_{cs}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2}{\lambda}^{2}-\frac{1}{8}{\lambda}^{4}-\frac{1}{2}A^{2}{\lambda}^{4}+{\cal
O}({\lambda}^{6})$ (15) $\displaystyle V_{cb}$ $\displaystyle=$ $\displaystyle
A{\lambda}^{2}+{\cal O}({\lambda}^{6})$ (16) $\displaystyle V_{td}$
$\displaystyle=$ $\displaystyle
A{\lambda}^{3}\Big{[}1-({\rho}+i{\eta})\Big{(}1-\frac{1}{2}{\lambda}^{2}\Big{)}\Big{]}+{\cal
O}({\lambda}^{6})$ (17) $\displaystyle V_{ts}$ $\displaystyle=$
$\displaystyle-A{\lambda}^{2}+A{\lambda}^{4}\Big{[}\frac{1}{2}-({\rho}+i{\eta})\Big{]}+{\cal
O}({\lambda}^{6})$ (18) $\displaystyle V_{tb}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2}A^{2}{\lambda}^{4}+{\cal O}({\lambda}^{6})$ (19)

The global fit for the four independent Wolfenstein parameters gives pdg2006

$A=0.818^{+0.007}_{-0.017},\ \ \ \ {\lambda}=0.2272{\pm}0.0010,\ \ \ \
\bar{\rho}=0.221^{+0.064}_{-0.028},\ \ \ \ \bar{\eta}=0.340^{+0.017}_{-0.045}$
(20)

where the relationship between (${\rho}$, ${\eta}$) and ($\bar{\rho}$,
$\bar{\eta}$) is pdg2006

${\rho}+i{\eta}=\frac{\sqrt{1-A^{2}{\lambda}^{4}}(\bar{\rho}+i\bar{\eta})}{\sqrt{1-{\lambda}^{4}}[1-A^{2}{\lambda}^{4}(\bar{\rho}+i\bar{\eta})]}$
(21)

If not stated otherwise, we shall use their central values for illustration.

###  III.2 Decay constants and form factors

In principle, information about the decay constants and transition form
factors of mesons can be obtained from experiments and/or theoretical
estimations. Now we specify these parameters. The decay constants $f_{P}$ and
$f_{V}$ corresponding to the pseudoscalar and vector mesons respectively, are
defined by

${\langle}P(q){|}({\bar{q}}_{1}q_{2})_{V-A}{|}0{\rangle}=-if_{P}q^{\mu},\ \ \
\ \ \ \
{\langle}V(q,{\epsilon}){|}({\bar{q}}_{1}q_{2})_{V-A}{|}0{\rangle}=f_{V}m_{V}{\epsilon}^{{\ast}{\mu}},$
(22)

where ${\epsilon}^{{\ast}}$ is the polarization vector of the vector meson
$V$. In this paper, we assume ideal mixing between ${\omega}$ and ${\phi}$
mesons, i.e. ${\omega}$ = $(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and ${\phi}$ =
$s\bar{s}$. In fact, the $B_{c}$ ${\to}$ $B_{u}^{({\ast})}{\phi}$ decays are
not possible, because the $B_{c}$ meson lies below the threshold of the
$B_{u}^{({\ast})}{\phi}$ system. As to the ${\eta}$ and ${\eta}^{\prime}$
mesons, we take the convention in Ref. prd58p114006 , adopting the Feldmann-
Kroll-Stech mixing scheme. Neglecting the possible compositions of both
${\eta}_{c}$ $=$ $c\bar{c}$ and glueball $gg$, the ${\eta}$ and
${\eta}^{\prime}$ are expressed as linear combinations of orthogonal states
${\eta}_{q}$ and ${\eta}_{s}$ with the flavor structure $q\bar{q}$ $=$
$(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and $s\bar{s}$, respectively, i.e.

$\left(\begin{array}[]{c}{\eta}\\\
{{\eta}^{\prime}}\end{array}\right)=\left(\begin{array}[]{lr}{\cos}{\phi}&-{\sin}{\phi}\\\
{\sin}{\phi}&{\cos}{\phi}\end{array}\right)\left(\begin{array}[]{c}{\eta}_{q}\\\
{\eta}_{s}\end{array}\right)$ (23)

where ${\phi}$ $=$ $(39.3{\pm}1.0)^{\circ}$ prd58p114006  is the
${\eta}$-${\eta}^{\prime}$ mixing angle. So the decay constants related to the
${\eta}$ and ${\eta}^{\prime}$ mesons can be defined by

$\left(\begin{array}[]{cc}f^{q}_{\eta}&f^{s}_{\eta}\\\
f^{q}_{{\eta}^{\prime}}&f^{s}_{{\eta}^{\prime}}\end{array}\right)=\left(\begin{array}[]{cc}{\cos}{\phi}&-{\sin}{\phi}\\\
{\sin}{\phi}&{\cos}{\phi}\end{array}\right)\left(\begin{array}[]{cc}f_{q}&0\\\
0&f_{s}\end{array}\right)$ (24)
${\langle}0{|}\bar{q}{\gamma}_{\mu}{\gamma}_{5}q{|}{\eta}^{(\prime)}(p){\rangle}=if^{q}_{{\eta}^{(\prime)}}p_{\mu},\
\ \ \ \ \ \
{\langle}0{|}\bar{s}{\gamma}_{\mu}{\gamma}_{5}s{|}{\eta}^{(\prime)}(p){\rangle}=if^{s}_{{\eta}^{(\prime)}}p_{\mu}.$
(25)

The matrix elements of the pseudoscalar densities are defined by prd58p094009

$\frac{{\langle}0{|}\bar{u}{\gamma}_{5}u{|}{\eta}^{(\prime)}{\rangle}}{{\langle}0{|}\bar{s}{\gamma}_{5}s{|}{\eta}^{(\prime)}{\rangle}}=\frac{f^{u}_{{\eta}^{(\prime)}}}{f^{s}_{{\eta}^{(\prime)}}},\
\ \ \ \ \ \ \ \ \
{\langle}0{|}\bar{s}{\gamma}_{5}s{|}{\eta}^{(\prime)}{\rangle}=-i\frac{m_{{\eta}^{(\prime)}}^{2}}{2m_{s}}(f^{s}_{{\eta}^{(\prime)}}-f^{u}_{{\eta}^{(\prime)}}),$
(26)

The numerical values of the decay constants are collected in Table 2. If not
stated otherwise, we shall take their central values for illustration.

The transition form factors are defined as bsw

${\langle}P(k){|}({\bar{q}}_{3}q_{4})_{V-A}{|}B(p){\rangle}=(p+k)^{\mu}F_{1}^{B{\to}P}(q^{2})+\frac{m_{B}^{2}-m_{P}^{2}}{q^{2}}q^{\mu}\Big{[}F_{0}^{B{\to}P}(q^{2})-F_{1}^{B{\to}P}(q^{2})\Big{]}$
(27)
$\displaystyle{\langle}V(k,{\epsilon}){|}({\bar{q}}_{3}q_{4})_{V-A}{|}B(p){\rangle}$
$\displaystyle=$ $\displaystyle
i\frac{{\epsilon}^{\ast}{\cdot}p}{q^{2}}q_{\mu}2m_{V}A_{0}^{B{\to}V}(q^{2})+i{\epsilon}^{\ast}_{\mu}(m_{B}+m_{V})A_{1}^{B{\to}V}(q^{2})$
(28) $\displaystyle-$ $\displaystyle
i\frac{{\epsilon}^{\ast}{\cdot}p}{m_{B}+m_{V}}(p+k)_{\mu}A_{2}^{B{\to}V}(q^{2})-i\frac{{\epsilon}^{\ast}{\cdot}p}{q^{2}}q_{\mu}2m_{V}A_{3}^{B{\to}V}(q^{2})$
$\displaystyle+$
$\displaystyle{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}{\epsilon}^{{\ast}{\nu}}p^{\alpha}k^{\beta}\frac{2V^{B{\to}V}(q^{2})}{m_{B}+m_{V}}$

where $F_{0,1}$, $V$ and $A_{0,1,2,3}$ are the transition form factors, $q$ =
$p$ $-$ $k$. In order to cancel the poles at $q^{2}$ $=$ $0$, we must impose
the condition

$\displaystyle F_{0}^{B{\to}P}(0)=F_{1}^{B{\to}P}(0),\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ A_{0}^{B{\to}V}(0)=A_{3}^{B{\to}V}(0),$ (29) $\displaystyle
2m_{V}A_{3}^{B{\to}V}(0)=(m_{B}+m_{V})A_{1}^{B{\to}V}(0)-(m_{B}-m_{V})A_{2}^{B{\to}V}(0).$
(30)

In our paper, only the $B_{c}$ ${\to}$ $B^{({\ast})}_{q}$ transition form
factors appear in the amplitudes within the “spectator” model where the
spectator is the $b$-quark for the concerned processes. Their numerical values
are collected in Table 3. From the numbers in Table 3, we can see clearly
that, due to the properties of nonperturbative QCD, there are large
uncertainties about the form factors with different theoretical treatments.
Here, we notice the fact that the velocity of the final state
$B_{q}^{({\ast})}$ meson is very small in the rest frame of the initial
$B_{c}$ meson, as that mentioned in Sec. I. It is commonly assumed that the
velocities of the $b$-quark in the rest frame of the $b$-flavored mesons
should be close to zero. The $B_{q}^{({\ast})}$ meson is neither fast nor
small. By intuition, the overlap between the initial and final states should
be huge, close to unity, as that argued in prd39p1342 . So for illustration
and simplification, we will take the same value for the transition form
factors, i.e. $F_{1,0}(0)$ $=$ $A_{0}(0)$ $=$ $0.8$.

###  III.3 Quark masses

In the decay amplitudes, there exist the “chirally enhanced” factors which are
associated with the hadronic matrix elements of the scalar and pseudoscalar
densities, for example, $R_{c1}$ in Eq.(39). These factors are formally of
order the ${\Lambda}_{\rm QCD}/m_{c}$, power suppressed in the heavy quark
limit, but numerically close to unity because the mass of the $c$ quark is not
infinity in practice. The current quark masses in the denominator appear
through the equations of motions and are renormalization scale dependent.
Their values are pdg2006

$\begin{array}[]{lll}m_{u}(2\,{\rm GeV})=3{\pm}1~{}{\rm
MeV},&{}{}{}{}{}&m_{d}(2\,{\rm GeV})=6.0{\pm}1.5~{}{\rm MeV},\\\ m_{s}(2\,{\rm
GeV})=103{\pm}20~{}{\rm MeV},&&m_{c}(m_{c})=1.24{\pm}0.09~{}{\rm
GeV}.\end{array}$ (31)

Using the renormalization group equation of the running quark mass rmp68p1125
,

$m({\mu})=m({\mu}_{0})\Big{[}\frac{{\alpha}_{s}({\mu})}{{\alpha}_{s}({\mu}_{0})}\Big{]}^{\frac{{\gamma}_{m}^{(0)}}{2{\beta}_{0}}}\Big{\\{}1+\Big{(}\frac{{\gamma}_{m}^{(1)}}{2{\beta}_{0}}-\frac{{\gamma}_{m}^{(0)}{\beta}_{1}}{2{\beta}_{0}^{2}}\Big{)}\frac{{\alpha}_{s}({\mu})-{\alpha}_{s}({\mu}_{0})}{4{\pi}}\Big{\\}}$
(32)

their corresponding values at a characteristic scale ${\mu}$ ${\sim}$ $m_{c}$
can be obtained.

###  III.4 Numerical results and discussions

The numerical results are listed in Table 4, where ${\cal B}r^{T}$ corresponds
to the contributions of the current-current operators only, ${\cal
B}r^{T+P_{s}}$ corresponds to the contributions of both current-current and
QCD penguin operators, ${\cal B}r^{T+P_{s}+P_{\rm e}}$ corresponds to the
contributions of both current-current and penguin operators, i.e. $Q_{1}$,
${\cdots}$, $Q_{10}$.

Here, we would like to point out that these numbers are just the qualitative
estimations on the order of amplitudes, because many of the subtleties and
details, such as final state interactions, the renormalization scale
dependence, the transition form factors, the strong phases, and so on, all
deserve the dedicated researches but are not considered here.

From the numbers in Table 4, we can see

  * •

The contributions of both QCD and electroweak penguin operators are very small
for $B_{c}$ ${\to}$ $B^{({\ast})}P$, $BV$ decays, compared with those of the
current-current operators. This is very different from that of the $B_{u,d,s}$
meson decays. The reason is that the contributions of penguin operators are
seriously suppressed by the CKM elements. The CKM elements corresponding to
different topologies for $c$-quark decay in the $B_{c}$ meson are listed
below.

tree topologies | penguin topologies | annihilation topologies
---|---|---
$V_{ud}V_{cs}^{\ast}$ ${\sim}$ $1$, $V_{us}V_{cs}^{\ast}$ ${\sim}$ $+{\lambda}$ |  $V_{ud}V_{cd}^{\ast}$ $+$ $V_{us}V_{cs}^{\ast}$ ${\sim}$ ${\lambda}^{5}$ |  $V_{cb}V_{ub}^{\ast}$ ${\sim}$ ${\lambda}^{5}$
$V_{us}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}^{2}$, $V_{ud}V_{cd}^{\ast}$ ${\sim}$ $-{\lambda}$ |  |

So, for the $B_{c}$ ${\to}$ $B^{({\ast})}P$, $BV$ decays, the effects of new
physics contributed via the penguin topologies might be tiny and not
detectable even with large statistics, due to the serious suppression by the
CKM elements.

  * •

There are clear hierarchy of amplitudes of the branching ratios. According the
CKM elements and the coefficients of $a_{1,2}$, these decay modes are divided
into different cases listed below.

cases | processes | coefficients | the CKM elements | order of branching ratios
---|---|---|---|---
case 1a |  $c$ ${\to}$ $s$ | $a_{1}$ |  $V_{ud}V_{cs}^{\ast}$ ${\sim}$ $1$ |  ${\sim}$ $10^{-2}$
case 1b |  $c$ ${\to}$ $s$ | $a_{1}$ |  $V_{us}V_{cs}^{\ast}$ ${\sim}$ ${\lambda}$ |  ${\sim}$ $10^{-3}$
|  $c$ ${\to}$ $d$ | $a_{1}$ |  $V_{ud}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}$ |  ${\sim}$ $10^{-3}$
case 1c |  $c$ ${\to}$ $d$ | $a_{1}$ |  $V_{us}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}^{2}$ |  ${\sim}$ $10^{-4}$
case 2a |  $c$ ${\to}$ $u$ | $a_{2}$ |  $V_{ud}V_{cs}^{\ast}$ ${\sim}$ $1$ |  ${\sim}$ $10^{-5}$
case 2b |  $c$ ${\to}$ $u$ | $a_{2}$ |  $V_{us}V_{cs}^{\ast}$, $V_{ud}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}$ |  ${\sim}$ $10^{-6}$ — $10^{-7}$
case 2c |  $c$ ${\to}$ $u$ | $a_{2}$ |  $V_{us}V_{cd}^{\ast}$ ${\sim}$ ${\lambda}^{2}$ |  ${\sim}$ $10^{-8}$

The decay modes determined by $a_{1}$ have comparatively large branching
ratios, which should be detectable experimentally, especially the CKM favored
decay modes, such as $B_{c}$ ${\to}$ $B_{s}^{({\ast})}{\pi}$, $B_{s}{\rho}$,
might be the promising decay modes to be measured in hadron colliders. Due to
the great branching ratios of the decay modes determined by $a_{1}$, the
$B_{c}$ mesons can be used as a source of the $B_{s}$ mesons if the $B_{c}$ is
produced copiously, as that stated in Ref.ijmpa21p777 . The decay modes
determined by $a_{2}$ have comparatively small branching ratios, which are
hard to detect experimentally, especially the CKM suppressed decay modes, such
as $B_{c}$ ${\to}$ $B_{u}^{({\ast})}K^{0}$, $B_{u}K^{{\ast}0}$, their
branching ratios are too tiny to be measured.

  * •

Although the $B_{c}$ ${\to}$ $B_{u}^{({\ast})}{\eta}^{\prime}$ decays belong
to the case 2b modes, their branching ratios are abnormally small, order of
$10^{-8}$. This can be explained by the fact that on one hand the physical
space phase available is too small, on the other hand there are large
destructive interactions between $f^{u}_{{\eta}^{\prime}}a_{2}$ and
$f^{s}_{{\eta}^{\prime}}a_{2}$ due to the serious cancellation between the CKM
elements $V_{ud}V_{cd}^{\ast}$ and $V_{us}V_{cs}^{\ast}$.

  * •

The relations among the $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$ decay
mode become very simple since the effects of penguin topologies is too tiny to
be considered. We can use these relations to determine and overconstrain some
parameters, such as the CKM elements, the form factors, etc. In addition, in
estimating and measuring these parameters, the ratios of the branching ratios
can be used to cancel and/or reduce largely theoretical uncertainties and
experimental errors. For example

$\displaystyle\frac{{\cal B}r(B^{+}_{c}{\to}B^{0}_{d}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}K^{+})}{\approx}\frac{V_{ud}V_{cd}^{\ast}}{V_{us}V_{cd}^{\ast}}\frac{f_{\pi}}{f_{K}}{\approx}\frac{V_{ud}V_{cs}^{\ast}}{V_{us}V_{cs}^{\ast}}\frac{f_{\pi}}{f_{K}}{\approx}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}K^{+})}$ (33) $\displaystyle\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}{\pi}^{+})}{\approx}\frac{V_{cd}^{\ast}}{V_{cs}^{\ast}}\frac{F_{0}^{B_{c}{\to}B_{d}}(m_{B_{c}}^{2}-m_{B_{d}}^{2})}{F_{0}^{B_{c}{\to}B_{s}}(m_{B_{c}}^{2}-m_{B_{s}}^{2})}{\approx}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}K^{+})}{{\cal B}r(B^{+}_{c}{\to}B^{0}_{d}K^{+})}$
(34) $\displaystyle\frac{{\cal B}r(B^{+}_{c}{\to}B^{0}_{d}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}{\rho}^{+})}{\approx}\frac{f_{\pi}F_{0}^{B_{c}{\to}B_{d}}}{f_{\rho}F_{1}^{B_{c}{\to}B_{d}}}~{}~{}~{}~{}~{}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}K^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}K^{{\ast}+})}{\approx}\frac{f_{K}F_{0}^{B_{c}{\to}B_{d}}}{f_{K^{\ast}}F_{1}^{B_{c}{\to}B_{d}}}$
(35) $\displaystyle\frac{{\cal B}r(B^{+}_{c}{\to}B^{0}_{d}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{{\ast}0}_{d}{\pi}^{+})}{\approx}\frac{F_{0}^{B_{c}{\to}B_{d}}}{A_{0}^{B_{c}{\to}B^{\ast}_{d}}}{\approx}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{d}K^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{{\ast}0}_{d}K^{+})}$ (36) $\displaystyle\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}{\rho}^{+})}{\approx}\frac{f_{\pi}F_{0}^{B_{c}{\to}B_{s}}}{f_{\rho}F_{1}^{B_{c}{\to}B_{s}}}~{}~{}~{}~{}~{}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}K^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}K^{{\ast}+})}{\approx}\frac{f_{K}F_{0}^{B_{c}{\to}B_{s}}}{f_{K^{\ast}}F_{1}^{B_{c}{\to}B_{s}}}$
(37) $\displaystyle\frac{{\cal B}r(B^{+}_{c}{\to}B^{0}_{s}{\pi}^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{{\ast}0}_{s}{\pi}^{+})}{\approx}\frac{F_{0}^{B_{c}{\to}B_{d}}}{A_{0}^{B_{c}{\to}B^{\ast}_{d}}}{\approx}\frac{{\cal
B}r(B^{+}_{c}{\to}B^{0}_{s}K^{+})}{{\cal
B}r(B^{+}_{c}{\to}B^{{\ast}0}_{s}K^{+})}$ (38)

##  IV Summary and Conclusion

In prospects of the huge statistics of the $B_{c}$ mesons at the hadron
colliders, accurate and thorough studies of the $B_{c}$ physics will be
accessible very soon. In this paper, we study the two-body nonleptonic
$c$-quark decays in the $B_{c}$ mesons, i.e. $B_{c}$ ${\to}$
$B_{q}^{({\ast})}P$, $B_{q}V$ decays within the QCDF approach for the leading
approximation, and estimate their branching ratios. We find that the
contributions of the penguin operators are very small to the decay amplitudes
due to the serious suppression by the CKM elements. The decay modes determined
by $a_{1}$ have comparatively large branching ratios. The most promising decay
modes are $B_{c}$ ${\to}$ $B_{s}^{({\ast})}{\pi}$, $B_{s}{\rho}$, which might
be easily detected at the hadron colliders.

##  Appendix A Amplitudes for $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$
decays

###  A.1 $c$ ${\to}$ $d$ processes

$\displaystyle{\cal A}(B_{c}^{+}{\to}B_{d}^{0}{\pi}^{+})$ $\displaystyle=$
$\displaystyle-i\frac{G_{F}}{\sqrt{2}}f_{\pi}F_{0}^{B_{c}{\to}B_{d}^{0}}\left(m_{B_{c}}^{2}-m_{B_{d}}^{2}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{1}$
(39)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}+R_{c1}\big{(}a_{6}-\frac{1}{2}a_{8}\big{)}\Big{]}\Big{\\}}$

where $R_{c1}\ =\
\displaystyle\frac{2m_{{\pi}^{+}}^{2}}{(m_{d}+m_{u})(m_{c}-m_{d})}$.

${\cal
A}(B_{c}^{+}{\to}B_{d}^{0}K^{+})=-i\frac{G_{F}}{\sqrt{2}}f_{K}F_{0}^{B_{c}{\to}B_{d}^{0}}\left(m_{B_{c}}^{2}-m_{B_{d}}^{2}\right)V_{us}V_{cd}^{\ast}a_{1}$
(40) $\displaystyle{\cal A}(B_{c}^{+}{\to}B_{d}^{0}{\rho}^{+})$
$\displaystyle=$
$\displaystyle\sqrt{2}G_{F}f_{{\rho}}F_{1}^{B_{c}{\to}B_{d}^{0}}m_{{\rho}^{+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{1}$
(41)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}\Big{]}\Big{\\}}$
${\cal
A}(B_{c}^{+}{\to}B_{d}^{0}K^{{\ast}+})=\sqrt{2}G_{F}f_{K^{\ast}}F_{1}^{B_{c}{\to}B_{d}^{0}}m_{K^{{\ast}+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{us}V_{cd}^{\ast}a_{1}$
(42) $\displaystyle{\cal A}(B_{c}^{+}{\to}B_{d}^{{\ast}0}{\pi}^{+})$
$\displaystyle=$
$\displaystyle\sqrt{2}G_{F}f_{\pi}A_{0}^{B_{c}{\to}B_{d}^{{\ast}0}}m_{B_{d}^{{\ast}}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{1}$
(43)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}+Q_{c1}\big{(}a_{6}-\frac{1}{2}a_{8}\big{)}\Big{]}\Big{\\}}$

where $Q_{c1}\ =\
\displaystyle\frac{-2m_{{\pi}^{+}}^{2}}{(m_{d}+m_{u})(m_{c}+m_{d})}$.

${\cal
A}(B_{c}^{+}{\to}B_{d}^{{\ast}0}K^{+})=\sqrt{2}G_{F}f_{K}A_{0}^{B_{c}{\to}B_{d}^{{\ast}0}}m_{B_{d}^{{\ast}}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{us}V_{cd}^{\ast}a_{1}$
(44)

###  A.2 $c$ ${\to}$ $s$ processes

${\cal A}(B_{c}^{+}{\to}B_{s}^{0}{\pi}^{+})\ =\
-i\frac{G_{F}}{\sqrt{2}}f_{\pi}F_{0}^{B_{c}{\to}B_{s}^{0}}\left(m_{B_{c}}^{2}-m_{B_{s}}^{2}\right)V_{ud}V_{cs}^{\ast}a_{1}$
(45) $\displaystyle{\cal A}(B_{c}^{+}{\to}B_{s}^{0}K^{+})$ $\displaystyle=$
$\displaystyle-i\frac{G_{F}}{\sqrt{2}}f_{K}F_{0}^{B_{c}{\to}B_{s}^{0}}\left(m_{B_{c}}^{2}-m_{B_{s}}^{2}\right)\Big{\\{}V_{us}V_{cs}^{\ast}a_{1}$
(46)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}+R_{c2}\big{(}a_{6}-\frac{1}{2}a_{8}\big{)}\Big{]}\Big{\\}}$

where $R_{c2}\ =\
\displaystyle\frac{2m_{K^{+}}^{2}}{(m_{s}+m_{u})(m_{c}-m_{s})}$

${\cal A}(B_{c}^{+}{\to}B_{s}^{0}{\rho}^{+})\ =\
\sqrt{2}G_{F}f_{{\rho}}F_{1}^{B_{c}{\to}B_{s}^{0}}m_{{\rho}^{+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{ud}V_{cs}^{\ast}a_{1}$
(47) $\displaystyle{\cal A}(B_{c}^{+}{\to}B_{s}^{0}K^{{\ast}+})$
$\displaystyle=$
$\displaystyle\sqrt{2}G_{F}f_{K^{{\ast}}}F_{1}^{B_{c}{\to}B_{s}^{0}}m_{K^{{\ast}+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{us}V_{cs}^{\ast}a_{1}$
(48)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}\Big{]}\Big{\\}}$
${\cal
A}(B_{c}^{+}{\to}B_{s}^{{\ast}0}{\pi}^{+})=\sqrt{2}G_{F}f_{\pi}A_{0}^{B_{c}{\to}B_{s}^{{\ast}0}}m_{B_{s}^{{\ast}}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{ud}V_{cs}^{\ast}a_{1}$
(49) $\displaystyle{\cal A}(B_{c}^{+}{\to}B_{s}^{{\ast}0}K^{+})$
$\displaystyle=$
$\displaystyle\sqrt{2}G_{F}f_{K}A_{0}^{B_{c}{\to}B_{s}^{{\ast}0}}m_{B_{s}^{{\ast}}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{us}V_{cs}^{\ast}a_{1}$
(50)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}-\frac{1}{2}a_{10}+Q_{c2}\big{(}a_{6}-\frac{1}{2}a_{8}\big{)}\Big{]}\Big{\\}}$

where $Q_{c2}\ =\
\displaystyle\frac{-2m_{K^{+}}^{2}}{(m_{s}+m_{u})(m_{c}+m_{s})}$

###  A.3 $c$ ${\to}$ $u$ processes

$\displaystyle{\cal
A}(B_{c}^{+}{\to}B_{u}^{+}{\pi}^{0})=-i\frac{G_{F}}{2}f_{\pi}F_{0}^{B_{c}{\to}B_{u}^{+}}\left(m_{B_{c}}^{2}-m_{B_{u}}^{2}\right)\Big{\\{}-V_{ud}V_{cd}^{\ast}a_{2}$
(51)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}+a_{10}-\frac{3}{2}\big{(}a_{7}-a_{9}\big{)}+R_{c3}\big{(}a_{6}+a_{8}\big{)}\Big{]}\Big{\\}}$

where $R_{c3}\ =\
\displaystyle\frac{2m_{{\pi}^{0}}^{2}}{(m_{d}+m_{u})(m_{c}-m_{u})}$

${\cal
A}(B_{c}^{+}{\to}B_{u}^{+}{\overline{K}}^{0})=-i\frac{G_{F}}{\sqrt{2}}f_{K}F_{0}^{B_{c}{\to}B_{u}^{+}}\left(m_{B_{c}}^{2}-m_{B_{u}}^{2}\right)V_{ud}V_{cs}^{\ast}a_{2}$
(52) ${\cal
A}(B_{c}^{+}{\to}B_{u}^{+}K^{0})=-i\frac{G_{F}}{\sqrt{2}}f_{K}F_{0}^{B_{c}{\to}B_{u}^{+}}\left(m_{B_{c}}^{2}-m_{B_{u}}^{2}\right)V_{us}V_{cd}^{\ast}a_{2}$
(53) $\displaystyle{\cal
A}(B_{c}^{+}{\to}B_{u}^{+}{\eta}^{(\prime)})=-i\frac{G_{F}}{\sqrt{2}}f_{{\eta}^{(\prime)}}^{u}F_{0}^{B_{c}{\to}B_{u}^{+}}\left(m_{B_{c}}^{2}-m_{B_{u}}^{2}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{2}$
(54)
$\displaystyle+\frac{f^{s}_{{\eta}^{(\prime)}}}{f^{u}_{{\eta}^{(\prime)}}}V_{us}V_{cs}^{\ast}a_{2}+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}\frac{f^{s}_{{\eta}^{(\prime)}}}{f^{u}_{{\eta}^{(\prime)}}}\big{\\{}a_{3}-a_{5}+\frac{1}{2}\Big{(}a_{7}-a_{9}\Big{)}\big{\\}}$
$\displaystyle+2\Big{(}a_{3}-a_{5}\Big{)}+a_{4}+a_{10}-\frac{1}{2}\Big{(}a_{7}-a_{9}\Big{)}+\left(1-\frac{f^{u}_{{\eta}^{(\prime)}}}{f^{s}_{{\eta}^{(\prime)}}}\right)R_{c4}^{(\prime)}\Big{(}a_{6}+a_{8}\Big{)}\Big{]}\Big{\\}}$

where $R_{c4}^{(\prime)}\ =\
\displaystyle\frac{2m_{{{\eta}^{(\prime)}}}^{2}}{(m_{s}+m_{s})(m_{c}-m_{u})}$

$\displaystyle{\cal A}(B_{c}^{+}{\to}B_{u}^{+}{\rho}^{0})$ $\displaystyle=$
$\displaystyle
G_{F}f_{{\rho}}F_{1}^{B_{c}{\to}B_{u}^{+}}m_{{\rho}^{0}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}-V_{ud}V_{cd}^{\ast}a_{2}$
(55)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}+a_{10}+\frac{3}{2}\big{(}a_{7}+a_{9}\big{)}\Big{]}\Big{\\}}$
$\displaystyle{\cal
A}(B_{c}^{+}{\to}B_{u}^{+}{\omega})=G_{F}f_{\omega}F_{1}^{B_{c}{\to}B_{u}^{+}}m_{\omega}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{2}$
(56)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}2\big{(}a_{3}+a_{5}\big{)}+a_{4}+a_{10}+\frac{1}{2}\big{(}a_{7}+a_{9}\big{)}\Big{]}\Big{\\}}$
${\cal
A}(B_{c}^{+}{\to}B_{u}^{+}{\overline{K}}^{{\ast}0})={\sqrt{2}}G_{F}f_{{\overline{K}}^{{\ast}}}F_{1}^{B_{c}{\to}B_{u}^{+}}m_{{\overline{K}}^{{\ast}0}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{ud}V_{cs}^{\ast}a_{2}$
(57) ${\cal
A}(B_{c}^{+}{\to}B_{u}^{+}K^{{\ast}0})={\sqrt{2}}G_{F}f_{K^{{\ast}0}}F_{1}^{B_{c}{\to}B_{u}^{+}}m_{K^{{\ast}0}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{us}V_{cd}^{\ast}a_{2}$
(58) $\displaystyle{\cal
A}(B_{c}^{+}{\to}B_{u}^{{\ast}+}{\pi}^{0})=G_{F}f_{\pi}A_{0}^{B_{c}{\to}B_{u}^{{\ast}+}}m_{B_{u}^{{\ast}}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}-V_{ud}V_{cd}^{\ast}a_{2}$
(59)
$\displaystyle+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}a_{4}+a_{10}-\frac{3}{2}\big{(}a_{7}-a_{9}\big{)}+Q_{c3}\big{(}a_{6}+a_{8}\big{)}\Big{]}\Big{\\}}$

where $Q_{c3}\ =\
\displaystyle\frac{-2m_{{\pi}^{0}}^{2}}{(m_{d}+m_{u})(m_{c}+m_{u})}$

${\cal
A}(B_{c}^{+}{\to}B_{u}^{{\ast}+}{\overline{K}}^{0})={\sqrt{2}}G_{F}f_{K}A_{0}^{B_{c}{\to}B_{u}^{{\ast}+}}m_{B_{u}^{{\ast}+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{ud}V_{cs}^{\ast}a_{2}$
(60) ${\cal
A}(B_{c}^{+}{\to}B_{u}^{{\ast}+}K^{0})={\sqrt{2}}G_{F}f_{K}A_{0}^{B_{c}{\to}B_{u}^{{\ast}+}}m_{B_{u}^{{\ast}+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)V_{us}V_{cd}^{\ast}a_{2}$
(61) $\displaystyle{\cal
A}(B_{c}^{+}{\to}B_{u}^{{\ast}+}{{\eta}^{(\prime)}})={\sqrt{2}}G_{F}f_{{\eta}^{(\prime)}}^{u}A_{0}^{B_{c}{\to}B_{u}^{{\ast}+}}m_{B_{u}^{{\ast}+}}\left({\varepsilon}{\cdot}p_{{}_{B_{c}}}\right)\Big{\\{}V_{ud}V_{cd}^{\ast}a_{2}$
(62)
$\displaystyle+\frac{f^{s}_{{\eta}^{(\prime)}}}{f^{u}_{{\eta}^{(\prime)}}}V_{us}V_{cs}^{\ast}a_{2}+\Big{(}V_{ud}V_{cd}^{\ast}+V_{us}V_{cs}^{\ast}\Big{)}\Big{[}\frac{f^{s}_{{\eta}^{(\prime)}}}{f^{u}_{{\eta}^{(\prime)}}}\big{\\{}a_{3}-a_{5}+\frac{1}{2}\Big{(}a_{7}-a_{9}\Big{)}\big{\\}}$
$\displaystyle+2\Big{(}a_{3}-a_{5}\Big{)}+a_{4}+a_{10}-\frac{1}{2}\Big{(}a_{7}-a_{9}\Big{)}+\left(1-\frac{f^{u}_{{\eta}^{(\prime)}}}{f^{s}_{{\eta}^{(\prime)}}}\right)Q_{c4}^{({\prime})}\Big{(}a_{6}+a_{8}\Big{)}\Big{]}\Big{\\}}$

where $Q_{c4}^{({\prime})}\ =\
\displaystyle\frac{-2m_{{{\eta}^{(\prime)}}}^{2}}{(m_{s}+m_{s})(m_{c}+m_{u})}$

## Acknowledgments

This work is supported in part both by National Natural Science Foundation of
China (under Grant No. 10647119, 10710146) and by Natural Science Foundation
of Henan Province, China. We would like to thank Prof. Dongsheng Du, Dr.
Deshan Yang, Prof. Caidian Lü and Prof. Zhizhong Xing for valuable
discussions.

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Table 1: The NLO Wilson coefficients $C_{i}(\mu)$ in the NDR scheme. The input parameters are pdg2006 : ${\alpha}_{s}(m_{Z})$ $=$ $0.1176$, ${\alpha}_{em}(m_{W})$ $=$ $1/128$, $m_{W}$ $=$ $80.403$ GeV, ${\Lambda}_{\rm QCD}^{(f=5)}$ $=$ $220.9$ MeV, ${\Lambda}_{\rm QCD}^{(f=4)}$ $=$ $317.2$ MeV. | ${\mu}=m_{b}$ |  ${\mu}=2.0$ GeV |  ${\mu}=1.5$ GeV | ${\mu}=m_{c}$
---|---|---|---|---
$C_{1}$ | $1.0849$ | $1.1497$ | $1.1883$ | $1.2215$
$C_{2}$ | $-0.1902$ | $-0.3077$ | $-0.3717$ | $-0.4241$
$C_{3}$ | $0.0148$ | $0.0238$ | $0.0296$ | $0.0349$
$C_{4}$ | $-0.0362$ | $-0.0542$ | $-0.0652$ | $-0.0747$
$C_{5}$ | $0.0088$ | $0.0105$ | $0.0107$ | $0.0102$
$C_{6}$ | $-0.0422$ | $-0.0703$ | $-0.0896$ | $-0.1078$
$C_{7}/{\alpha}_{em}$ | $-0.0007$ | $-0.0164$ | $-0.0186$ | $-0.0181$
$C_{8}/{\alpha}_{em}$ | $0.0565$ | $0.0964$ | $0.1235$ | $0.1493$
$C_{9}/{\alpha}_{em}$ | $-1.3039$ | $-1.3966$ | $-1.4473$ | $-1.4901$
$C_{10}/{\alpha}_{em}$ | $0.2700$ | $0.4144$ | $0.4964$ | $0.5656$
Table 2: values of the decay constant (in the unit of MeV) $f_{\pi}$ | $f_{K}$ | $f_{q}$ | $f_{s}$ | $f_{\rho}$ | $f_{\omega}$ | $f_{K^{\ast}}$
---|---|---|---|---|---|---
$131$ pdg2006  |  $160$ pdg2006  |  $(1.07{\pm}0.02)f_{\pi}$ prd58p114006  |  $(1.34{\pm}0.06)f_{\pi}$ prd58p114006  |  $205{\pm}9$ prd71p014029  |  $195{\pm}3$ prd71p014029  |  $217{\pm}5$ prd71p014029
Table 3: Values of transition form factors Ref. | $F^{B_{c}{\to}B_{u,d}}_{0}(0)$ | $F^{B_{c}{\to}B_{s}}_{0}(0)$ | $A^{B_{c}{\to}B_{u,d}^{\ast}}_{0}(0)$ | $A^{B_{c}{\to}B_{s}^{\ast}}_{0}(0)$
---|---|---|---|---
prd39p1342  111The form factors increase with the increasing parameter ${\omega}$ $=$ $0.4$ ${\sim}$ $1.0$ GeV that determines the average transverse quark momentum. The authors of prd39p1342  prefer $F^{B_{c}{\to}B_{u}}_{0}(0)$ $=$ $0.831$, $F^{B_{c}{\to}B_{s}}_{0}(0)$ $=$ $0.859$, $A^{B_{c}{\to}B_{u}^{\ast}}_{0}(0)$ $=$ $0.869$ and $A^{B_{c}{\to}B_{s}^{\ast}}_{0}(0)$ $=$ $0.842$ with the corresponding parameter ${\omega}$ $=$ $0.8$ GeV. | $0.320{\sim}0.910$ | $0.340{\sim}0.925$ | $0.349{\sim}0.916$ | $0.432{\sim}0.931$
zpc57p43  222The definitions of the transition form factors in zpc57p43  are different from ours in Eq.(24) and Eq.(25). The relationship is  $F_{1}^{B_{c}{\to}P}=F_{+},\ \ \ \ \ \ \ \ A_{0}^{B_{c}{\to}V}=\frac{F_{0}^{A}}{2m_{V}}+\frac{m_{{}_{B_{c}}}^{2}-m_{V}^{2}}{2m_{V}}F_{+}^{A}.$ (39) with the values of $F_{+}$ $=$ $0.3{\pm}0.1$ ($0.30{\pm}0.05$), $F_{0}^{A}$ $=$ $4.0{\pm}1.0$ ($4.5{\pm}0.5$) ${\rm GeV}^{-1}$and $F_{+}^{A}$ $=$ $-0.02{\pm}0.01$ ($-0.03{\pm}0.02$) ${\rm GeV}^{-1}$ for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition zpc57p43 . | $0.3{\pm}0.1$ | $0.30{\pm}0.05$ | $0.35{\pm}0.09$ | $0.39{\pm}0.05$
9504319  333Using the relationship of Eq.(29) and Eq.(30) with the input $A_{1}$ $=$ $0.52$, $A_{2}$ $=$ $-2.79$ 9504319 . | — | $0.61$ | — | $0.79$
9810339  444For parameter ${\omega}$ $=$ $0.4$, $0.5$ GeV. | — | $0.403{\sim}0.617$ | — | $0.433{\sim}0.641$
jpg26p1079  555Using the relationship of Eq.(39) with the input $F_{+}$ $=$ $0.4504$ ($0.5917$), $F_{0}^{A}$ $=$ $3.383$ ($5.506$) ${\rm GeV}^{-1}$ and $F_{+}^{A}$ $=$ $-0.0463$ ($-0.0673$) ${\rm GeV}^{-1}$ for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition jpg26p1079 . | $0.4504$ | $0.5917$ | $0.2691$ | $0.4451$
prd63p074010  666Using the relationship of Eq.(29) and Eq.(30) with the input $A_{1}$ $=$ $0.27$ ($0.33$) and $A_{2}$ $=$ $-0.60$ ($-0.40$) for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition prd63p074010 . | $-0.58$ | $-0.61$ | $0.35$ | $0.39$
mpla16p1439  777Using the relationship of Eq.(29) and Eq.(30) with the input $A_{1}$ $=$ $0.28$ and $A_{2}$ $=$ $0.49$ mpla16p1439 . | — | $0.297$ | — | $0.263$
pan64p1860  888Using the relationship of Eq.(39) with the input $F_{+}$ $=$ $1.27$ ($1.3$), $F_{0}^{A}$ $=$ $9.8$ ($8.1$) ${\rm GeV}^{-1}$ and $F_{+}^{A}$ $=$ $0.35$ ($0.2$) ${\rm GeV}^{-1}$ for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition in the framework of QCD sum rules pan64p1860 . | $1.27$ | $1.3$ | $1.29$ | $0.94$
pan64p1860  999Using the relationship of Eq.(39) with the input $F_{+}$ $=$ $1.38$ ($1.1$), $F_{0}^{A}$ $=$ $9.4$ ($8.2$) ${\rm GeV}^{-1}$ and $F_{+}^{A}$ $=$ $0.36$ ($0.3$) ${\rm GeV}^{-1}$ for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition in the framework of potential model pan64p1860 . | $1.38$ | $1.1$ | $1.26$ | $1.04$
epjc32p29  | $0.39$ | $0.50$ | $0.20$ | $0.35$
epjc47p413  | — | — | $0.23{\pm}0.03$ | —
epjc51p833  101010Using the relationship of Eq.(29) and Eq.(30) with the input $A_{1}$ $=$ $0.90$ ($1.01$) and $A_{2}$ $=$ $7.9$ ($9.04$) for $B_{c}$ ${\to}$ $B_{u,d}^{({\ast})}$ ($B_{s}^{({\ast})}$) transition epjc51p833 . | $0.90$ | $1.02$ | $0.27$ | $0.36$
Table 4: The branching ratios for $B_{c}$ ${\to}$ $B_{q}^{({\ast})}P$, $B_{q}V$. ${\cal B}r^{T}$ corresponds to the contributions of the operators $Q_{1}$ and $Q_{2}$. ${\cal B}r^{T+P_{s}}$ corresponds to the contributions of operators $Q_{1}$ ${\sim}$ $Q_{6}$. ${\cal B}r^{T+P_{s}+P_{\rm e}}$ corresponds to the contributions of $Q_{1}$ ${\sim}$ $Q_{10}$. modes | case | ${\cal B}r^{T}$ | ${\cal B}r^{T+P_{s}}$ | ${\cal B}r^{T+P_{s}+P_{\rm e}}$ | $\frac{{\cal B}r^{T+P_{\rm s}}-{\cal B}r^{T}}{{\cal B}r^{T}}$ | $\frac{{\cal B}r^{T+P_{\rm s}+P_{\rm e}}-{\cal B}r^{T}}{{\cal B}r^{T}}$
---|---|---|---|---|---|---
$B_{c}^{+}$ ${\to}$ $B_{s}^{0}{\pi}^{+}$ | case 1a | $5.3089{\times}10^{-2}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{s}^{0}{\rho}^{+}$ | case 1a | $6.2652{\times}10^{-2}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{s}^{{\ast}0}{\pi}^{+}$ | case 1a | $4.5916{\times}10^{-2}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{s}^{0}{K}^{+}$ | case 1b | $3.6746{\times}10^{-3}$ | $3.6759{\times}10^{-3}$ | $3.6759{\times}10^{-3}$ | $3.4{\times}10^{-4}$ | $3.4{\times}10^{-4}$
$B_{c}^{+}$ ${\to}$ $B_{s}^{0}{K}^{{\ast}+}$ | case 1b | $1.6450{\times}10^{-3}$ | $1.6451{\times}10^{-3}$ | $1.6451{\times}10^{-3}$ | $5.0{\times}10^{-5}$ | $5.0{\times}10^{-5}$
$B_{c}^{+}$ ${\to}$ $B_{s}^{{\ast}0}{K}^{+}$ | case 1b | $2.9772{\times}10^{-3}$ | $2.9766{\times}10^{-3}$ | $2.9766{\times}10^{-3}$ | $-1.9{\times}10^{-4}$ | $-1.9{\times}10^{-4}$
$B_{c}^{+}$ ${\to}$ $B_{d}^{0}{\pi}^{+}$ | case 1b | $3.7283{\times}10^{-3}$ | $3.7272{\times}10^{-3}$ | $3.7272{\times}10^{-3}$ | $-3.0{\times}10^{-4}$ | $-3.0{\times}10^{-4}$
$B_{c}^{+}$ ${\to}$ $B_{d}^{0}{\rho}^{+}$ | case 1b | $5.2745{\times}10^{-3}$ | $5.2742{\times}10^{-3}$ | $5.2742{\times}10^{-3}$ | $-5.0{\times}10^{-5}$ | $-5.0{\times}10^{-5}$
$B_{c}^{+}$ ${\to}$ $B_{d}^{{\ast}0}{\pi}^{+}$ | case 1b | $3.2682{\times}10^{-3}$ | $3.2688{\times}10^{-3}$ | $3.2688{\times}10^{-3}$ | $1.9{\times}10^{-4}$ | $1.9{\times}10^{-4}$
$B_{c}^{+}$ ${\to}$ $B_{d}^{0}{K}^{+}$ | case 1c | $2.6616{\times}10^{-4}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{d}^{0}{K}^{{\ast}+}$ | case 1c | $2.2583{\times}10^{-4}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{d}^{{\ast}0}{K}^{+}$ | case 1c | $2.2075{\times}10^{-4}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\overline{K}}^{0}$ | case 2a | $2.2067{\times}10^{-5}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\overline{K}}^{{\ast}0}$ | case 2a | $1.8434{\times}10^{-5}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{{\ast}+}{\overline{K}}^{0}$ | case 2a | $1.8261{\times}10^{-5}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\eta}$ | case 2b | $1.5991{\times}10^{-6}$ | $1.6122{\times}10^{-6}$ | $1.6125{\times}10^{-6}$ | $8.2{\times}10^{-3}$ | $8.4{\times}10^{-3}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{{\ast}+}{\eta}$ | case 2b | $1.3042{\times}10^{-6}$ | $1.2960{\times}10^{-6}$ | $1.2964{\times}10^{-6}$ | $-6.3{\times}10^{-3}$ | $-6.0{\times}10^{-3}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\pi}^{0}$ | case 2b | $4.5968{\times}10^{-7}$ | $4.5161{\times}10^{-7}$ | $4.5134{\times}10^{-7}$ | $-1.8{\times}10^{-2}$ | $-1.8{\times}10^{-2}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\rho}^{0}$ | case 2b | $6.5030{\times}10^{-7}$ | $6.4823{\times}10^{-7}$ | $6.4776{\times}10^{-7}$ | $-3.2{\times}10^{-3}$ | $-3.9{\times}10^{-3}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\omega}$ | case 2b | $5.7921{\times}10^{-7}$ | $5.8199{\times}10^{-7}$ | $5.8212{\times}10^{-7}$ | $4.8{\times}10^{-3}$ | $5.0{\times}10^{-3}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{{\ast}+}{\pi}^{0}$ | case 2b | $4.0262{\times}10^{-7}$ | $4.0722{\times}10^{-7}$ | $4.0685{\times}10^{-7}$ | $1.1{\times}10^{-2}$ | $1.0{\times}10^{-2}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{\eta}^{\prime}$ | case 2d | $8.8676{\times}10^{-8}$ | $8.7700{\times}10^{-8}$ | $8.7738{\times}10^{-8}$ | $-1.1{\times}10^{-2}$ | $-1.1{\times}10^{-2}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{{\ast}+}{\eta}^{\prime}$ | case 2d | $1.7401{\times}10^{-8}$ | $1.7728{\times}10^{-8}$ | $1.7731{\times}10^{-8}$ | $1.9{\times}10^{-2}$ | $1.9{\times}10^{-2}$
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{K}^{0}$ | case 2c | $6.5428{\times}10^{-8}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{+}{K}^{{\ast}0}$ | case 2c | $5.4658{\times}10^{-8}$ | — | — | — | —
$B_{c}^{+}$ ${\to}$ $B_{u}^{{\ast}+}{K}^{0}$ | case 2c | $5.4143{\times}10^{-8}$ | — | — | — | —