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e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Siksha ${}^{\prime}O^{\prime}$
Anusandhan Deemed to be University, Bhubaneswar-751030, India 22institutetext:
Department of Physics, Maharaja Sriram Chandra Bhanja Deo University,
Baripada-757003, India 33institutetext: Department of Physics, Utkal
University, Bhubaneswar-751004, India

# The nonleptonic decays of $b$-flavored mesons to $S$-wave charmonium and
charm meson states

Kalpalata Dashe1,addr1  Lopamudra Nayake2,addr1  P. C. Dashaddr1  Rabinarayan
Pandaaddr1  Susmita Karaddr2  N. Barikaddr3

###### Abstract

The detection of radially excited heavy meson
states in recent years and measurement of heavy meson decays, particularly
$B_{c}^{+}\to J/\psi D_{s}^{+}$ and $B_{c}^{+}\to J/\psi D_{s}^{*+}$, by the
LHCb and ATLAS Collaborations, have aroused a lot of theoretical interest in
the nonleptonic decays of $b$-flavored mesons. In this paper, we study the
exclusive two-body nonleptonic $\bar{B}^{0}$, $\bar{B_{s}^{0}}$, $B^{-}$ and
$B_{c}^{-}$-meson decays to two vector meson ($V_{1}(nS)V_{2}$) states.
Assuming the factorization hypothesis, we calculate the weak-decay form
factors from the overlapping integrals of meson wave functions, in the
framework of the relativistic independent quark (RIQ) model. We find a few
dominant decay modes: $B^{-}\to D^{*0}\rho^{-}$, $\bar{B^{0}}\to
D^{*+}\rho^{-}$, $\bar{B_{s}^{0}}\to D_{s}^{*+}\rho^{-}$, $B^{-}\to J/\psi
K^{*-}$ and $B_{c}^{-}\to J/\psi D_{s}^{*-}$ with predicted branching
fractions of 1.54, 1.42, 1.17, 0.53 and 0.52 (in $\%$), which are
experimentally accessible. The predicted branching fractions for corresponding
decay modes to excited ($2S$) states, obtained in the order ${\cal
O}(10^{-3}-10^{-4})$ lie within the detection accuracy of the current
experiments at LHCb and Tevatron. The sizeable $CP$-odd fractions predicted
for $B_{c}^{-}$-meson decay to two charmful states: $D^{*0}D^{*-}_{(s)}$ and
$\bar{D}^{*0}D^{*-}_{(s)}$ indicate significant $CP$-violation hinting at the
so-called new physics beyond the standard model.

††journal: Eur. Phys. J. C

##  1 Introduction

The experimental probes over the last two decades in the $b$-flavored heavy
meson ($B,B_{s},B_{c}$) sector have led to the discovery of many excited
states which include the radially excited charmonium $\psi(2S)$ and
$\eta_{c}(2S)$ states by the Belle Collaborations A1 . In the heavy-light
meson sector, a number of charm meson states such as $D_{s1}^{*}(2710)^{\pm}$
A2  by Belle, $D(2550)^{0}$ A3 , and $D_{J}(2580)^{0}$ A4  by the LHCb,
$D^{*}(2640)^{\pm}$ A5  by Delphi and $D^{*}(2650)^{0}$ A4  and
$D_{1}^{*}(2680)^{0}$ A6  by LHCb have also been discovered, which can be
identified as $D_{s}^{*}(2S)^{\pm}$ A7 , $D(2S)^{0}$ A8 , $D^{*}(2S)^{\pm}$ A8
and $D^{*}(2S)^{0}$ state, respectively. Recently LHCb discovered
$D_{1}^{*}(2600)$ A3  which might be the same object as $D^{*}(2650)^{0}$ and
$D_{1}^{*}(2650)$. The current RUN-II at Tevatron, RUN-III at CERN LHC, and
the $e^{+}e^{-}$ collider activities at Belle-II are all designed to boost the
measurement scenario in heavy flavor physics. Specially designed detectors at
BTeV and LHCb, dedicated to enhancing the event accumulation rates, are
expected to yield high statistics $b$-flavored ($B,B_{s}$) events and the
$B_{c}$-events, in particular, at the rate $\sim 10^{10}$ events per annum;
providing a fascinating area of research in $B$-physics. The measurement of
$B,B_{s}$-meson decays by BaBar, Belle, and LHCb Collaborations A9 ; A10 ; A11
and recent measurement of $B_{c}$-meson decays: $B_{c}\to J/\psi D_{s}$ and
$B_{c}\to J/\psi D_{s}^{*}$, performed by LHCb A12  and ATLAS A13
Collaborations have aroused a great deal of theoretical interest in
nonleptonic decays of heavy-flavored mesons. The study of nonleptonic decays
of heavy mesons is important as it helps in probing the interplay of QCD and
electroweak interactions, determining the Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements, testing predictions of the standard model (SM), and exploring
new physics beyond SM.

The nonleptonic decays of $b$-flavored mesons have been widely studied in
different theoretical and phenomenological model approaches (see the
classified bibliography of Ref.14). Most of these earlier studies refer to the
$b$-flavored $(B,B_{s},B_{c})$-meson decays to ground states of charmonium and
charm mesons. A number of theoretical attempts have also been made in this
sector, yielding predictions on decay modes to radially excited states. A few
noteworthy among them include studies based on the non-relativistic potential
model (NRPM) using the Bathe-Salpeter equation A15 ; A16 , relativistic
constituent quark model (RCQM) based on the Bathe-Salpeter formalism A17 ,
improved Bathe-Salpeter (IB
S) approach A18 ; A19 , relativistic quark model (RQM) A20 ; A21 ; A22 ; A23 ,
ISGW2 quark model A24 , covariant confined quark model (CCQM) A25 ,
relativistic quark model (RQM) using quasi-potential approach A26 , QCD
relativistic potential model (QCDRPM)A27 , QCD factorization A28 ,
perturbative QCD (pQCD) A29 , covariant light-front quark model (CLFQM) A30 ,
light-cone sum rule (LCSR) A31  and the heavy quark sum rules (HQS) A32  etc.
The recent predictions of branching fraction for $B_{c}\to J/\psi
D_{s}^{*}(2S)\simeq 1.75\times 10^{-2}$, $B_{c}\to\psi(2S)\\\ D^{*}_{s}\simeq
2.72\times 10^{-3}$ based on improved Bathe-Salpeter approach A18 ; A19 ,
$B_{s}\to D^{*}_{s}(2S)D^{*}_{s}\simeq 1.7\times 10^{-3}$ in RQM A26 ,
$B_{c}^{-}\to\psi(2S)\rho^{-}\simeq 1.1\times 10^{-3}$,
$B_{c}\to\psi(2S)D^{*}_{s}\simeq 1.2\times 10^{-3}$ in ISGW2 quark model A24 ,
$B_{c}^{+}\to D^{*+}\bar{D}^{*0}\simeq 5.14\times 10^{-3}$ in QCDRPM A27  etc.
are accessible in the LHCb experiment. The branching fractions for decay
modes: $B_{c}^{-}\to\psi(2S)\\\ \rho^{-}$ A16 , $B_{s}\to D_{s}^{*}\rho^{-}$
and $B_{c}^{-}\to\psi(3S)\rho^{-}$A18 , etc. predicted in the range $\sim
10^{-4}$, lie within the detection accuracy of the current experiments and can
be measured in near future.

The analysis of nonleptonic decays is notoriously non-trivial as it is
strongly influenced by confining color forces and it involves matrix elements
of local four-quark operators in the non-perturbative QCD approach, the
mechanism of which is not yet clear in the SM framework. Ignoring the weak
annihilation contribution, the transition amplitudes can be conveniently
described in the so-called naive factorization approximation A17 ; A20 ; A21 ;
A22 ; A23 ; A27 ; A33 ; A34 ; A35 ; A36 ; A37 ; A38 ; A39 ; A40 ; A41 ; A42 ;
A43 ; A44 ; A45 ; A46 ; A47 ; A48 ; A49 ; A50 ; A51 ; A52 ; A53 ; A54 ; A55 ,
which works reasonably well in the nonleptonic $b$-flavored meson decays,
where the quark-gluon sea is suppressed in the heavy quarkonium A47 ; A48 ;
A49 ; A50 . In this approach, the transition matrix element of local four-
quark operators is factorized into two single current matrix elements. One of
the factorized amplitudes, in which the decaying parent meson is connected to
one final meson state, can be covariantly expanded in terms of Lorentz
invariant weak form factors as in the case of semileptonic decays. The other
factorized amplitude, where the vacuum is connected to the second final meson
state, can be parametrized in terms of meson decay constants that describe the
leptonic decays. The description of the nonleptonic decay process is thus
reduced to the calculation of weak decay form factors in the framework of a
suitable phenomenological model.

Bjorken’s intuitive argument on color transparency in his pioneering work A51
, theoretical development based on the QCD approach in the $\frac{1}{N_{c}}$
limit A52  and the heavy quark effective theory (HQET) A53 , etc. justify the
naive factorization approximation, where strong-interaction effects such as
the final-state interaction, rescattering of the final state hadrons and the
renormalization-point dependence of amplitudes are shown to be marginal A56 .
Discovery of excited charmonium and charm meson states and prediction of
$b$-flavored meson decays to the ground and radially excited states by
different theoretical approaches, inspired our group to predict energetic
nonleptonic $B$ and $B_{c}\to PP,PV,VP$ decays A33 ; A34  as well as their
decays to two vector meson ($VV$) ground states A35 ; A36 , within the
framework of our relativistic independent quark (RIQ) model. Here $P$($V$)
refers to a pseudoscalar (vector) meson state. The nonleptonic $B_{c}$-decays
to an $S$-wave charmonium and a light or charm meson state A55  have also been
predicted by our group in good comparison with the experiment and other SM
predictions.

Note here that the approach based on naive factorization approximation may be
justified in the analysis of energetic nonleptonic decays of $B_{F}\to PP,PV$
type, with the quark flavor $F\to d,u,s,c$; where the strong interaction
effects such as the final state interaction, rescattering of final state meson
as well as the renormalization point dependence of factorized amplitude have
been shown to be marginal A57 . Such an approach, however, may not hold up
well in the description of $B_{F}\to V_{1}(nS)V_{2}$ decays, where both the
final state mesons being heavy, are expected to be in the region close to zero
recoil point. Here also both the longitudinal and transverse polarization
components contribute to decay amplitude which can be measured experimentally.
From the naive counting rules, the longitudinal polarization fraction in this
sector is expected to dominate over transverse components which can be checked
as well. The nonleptonic $B_{F}$-decays to two charmful vector meson states
are of special interest as they provide valuable information which is
different from cases with light meson productions. For example, the evaluation
of $CP$-asymmetries in $B^{-}_{c}$ decays:
$B^{-}_{c}\to\bar{D}^{*0}D_{(s)}^{*-}$ and $B^{-}_{c}\to D^{*0}D_{(s)}^{*-}$
provides an important clue in testing the SM predictions and exploring new
physics beyond SM.

In this paper we would like to extend the applicability of our RIQ model to
study, within factorization approximation, the nonleptonic $B_{F}\to
V_{1}(nS)V_{2}$ decays to $S$\- wave charmonium and charm vector meson states
$(nS)$ along with a light or a heavy-light meson state, where $n=1,2,3$. We
ignore the decay channels involving higher $(4S)$ charmonium and charm meson
states since their properties are still not understood well. We adopt here the
general formalism used in Ref. A55 . In the present study we consider the
contribution of the current-current operators A58  only in calculating the
tree-level diagram, expected to be dominant in $B_{F}\to V_{1}(nS)V_{2}$
decays. The contribution of the penguin diagram may be significant in the
evaluation of $CP$-violation and search for new physics beyond SM, but its
contribution to these decay amplitudes is considered less significant. In
fact, the QCD and electroweak penguin operators’ contribution have been shown
A59 ; A60 ; A61 ; A62  negligible compared to the contribution of current-
current operators in these decays due to serious suppression of CKM matrix
elements. The Wilson’s coefficients of penguin operators being very small,
their contribution to decay amplitude is only relevant in rare decays, where
the tree-level contribution is either strongly CKM-suppressed as in
$\bar{B}\to\bar{K}^{*}\pi$ or matrix elements of current-current operators do
not contribute at all as in the case of rare decays:
$\bar{B}\to\bar{K}^{*}\gamma$ and $\bar{B^{0}}\to\bar{K}^{0}\phi$ A58 .

The rest of the paper is organized as follows. In the following section, we
present a general remark on the factorization approximation and discussed the
factorized amplitudes of the nonleptonic decay. In Section 3, we obtain the
model expressions for invariant weak-decay form factors and the factorized
transition amplitudes. Section 4 is devoted to the numerical results and
discussion and Section 5 encompasses our brief summary and conclusion. A brief
review of the model conventions, wave-packet representation of the meson state
and momentum probability amplitudes of the constituent quarks inside the meson
bound-state are given in the Appendix.

##  2 Factorization approximation and nonleptonic transition amplitude

The transition amplitude for two-body nonleptonic transition: $B_{F}\to
V_{1}(nS)V_{2}$ can be written as

$A(B_{F}\to V_{1}(nS)V_{2})=\langle V_{1}(nS)V_{2}|{\cal
H}|B_{F}\rangle={\frac{G_{F}}{\sqrt{2}}}\sum_{i}\lambda_{i}C_{i}(\mu){\langle{\cal
O}\rangle}_{i},$

(1)

where $G_{F}$ is the Fermi Coupling constant, $\lambda_{i}$ the CKM factor,
$C_{i}$ is the Wilson coefficients and ${\langle{\cal O}\rangle}_{i}$ is the
matrix element of local four-quark operators. In the factorization
approximation, the matrix element of the local four-quark operator is
factorized into two single-particle matrix elements of quark current as

${\langle V_{1}(nS)V_{2}|{\cal H}|B_{F}\rangle}_{i}={\langle
V_{2}|J^{\mu}|0\rangle}{\langle
V_{1}(nS)|J_{\mu}|B_{F}\rangle}+{(V_{1}(nS)\leftrightarrow V_{2})},$

(2)

where $J_{\mu}\equiv V_{\mu}-A_{\mu}$ is the vector-axial vector current.

The difficulty inherent in such an approach is that Wilson’s coefficients
$C_{i}(\mu)$, which include the short distance QCD effect between $\mu=m_{N}$
and $\mu=m_{b}$ are $\mu$ scale and renormalization scheme dependent, whereas
${\langle{\cal O}\rangle}_{i}$ are $\mu$ scale and renormalization scheme
independent. As a result, physical amplitude depends on the $\mu$ scale.
However, in the naive factorization approach, the long-distance effects are
disentangled from the short-distance effect assuming that the matrix element
${\langle{\cal O}\rangle}$ at the $\mu$ scale contains nonfactorizable
contributions. This results in the cancellation of the $\mu$ dependence and
scheme dependence of $C_{i}(\mu)$.

We neglect here the so-called $W$ exchange and annihilation diagram, since in
the limit $M_{W}\to\infty$ they are connected by Fiertz transformation and
doubly suppressed by a kinematic factor of order
$(\frac{m_{i}^{2}}{M_{W}^{2}})$ A52 . We also discard the color octet current
which emerges after the Fiertz transformation of color singlet operators.
Clearly, these currents violate factorization since they cannot provide
transitions to vacuum states. Taking into account the Fiertz reordered
contribution, the relevant coefficients are not $C_{1}(\mu)$ and $C_{2}(\mu)$
but the combination

$a_{1,2}(\mu)=c_{1,2}(\mu)+\frac{1}{N_{c}}c_{2,1}(\mu).$ (3)

The factorization approximation, in general, works well in the description of
two-body nonleptonic decays of heavy mesons in the limit of a large number of
colors. Assuming a large $N_{c}$ limit to fix the QCD coefficients
$a_{1}\approx c_{1}$ and $a_{2}\approx c_{2}$ at $\mu\approx m_{b}^{2}$,
nonleptonic decays of heavy mesons have been analyzed in Refs. A15 ; A24 ; A63
; A64 ; A65 ; A66 ; A67 .

The hadronic matrix element of the weak current $J_{\mu}$ are covariantly
expanded in terms of weak form factors as

$\displaystyle{\langle V_{1}(\vec{k})|A_{\mu}|B_{F}(\vec{p})\rangle}$
$\displaystyle=f(q^{2})\epsilon_{\mu}^{*}+a_{+}(q^{2})(\epsilon^{*}.p)(p+k)_{\mu}$
(4) $\displaystyle+a_{-}(q^{2})(\epsilon^{*}.p)(p-k)_{\mu},$

$\langle
V_{1}(\vec{k})|V_{\mu}|B_{F}(\vec{p})\rangle=ig(q^{2})\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}(p+k)^{\rho}(p-k)^{\sigma},$

(5)

where $\epsilon^{*}$ is the polarization of the vector meson $V_{1}$. $p$ and
$k$ represent the four-momentum of the parent meson $B_{F}$ and daughter meson
$V_{1}$, respectively. With the four-momentum transfer
$q=p-k\equiv(E,0,0,|\vec{q}|)$ and mass $m_{V_{1}}$, the polarization of the
daughter meson $V_{1}$ is taken in the form

$\epsilon_{\mu}^{\pm}\equiv\frac{1}{\sqrt{2}}(0,\mp 1,-i,0),\
\epsilon_{\mu}^{L}\equiv\frac{1}{m_{V_{1}}}(|\vec{q}|,0,0,E).$ (6)

The matrix element of the current $J^{\mu}$ between vacuum and vector-meson
$V_{2}$ in the final state can be parametrized in terms of meson decay
constant $f_{V_{2}}$ as

${\langle V_{2}|J^{\mu}|0\rangle}=\epsilon_{V_{2}}^{*\mu}f_{V_{2}}m_{V_{2}}.$
(7)

In the factorization approach, the nonleptonic transition amplitude can be
calculated from one of the three possible tree-level diagrams shown in Fig. 1.
The color-favored transitions, shown in quark level diagram in Fig. 1(a),
represent “class I” transitions which are characterized by external emission
of $W$-boson. In these transitions, the factorized amplitude coupled to the
QCD factor $a_{1}$ only give the nonvanishing contribution. On the other hand,
color-suppressed transitions shown in the diagram in Fig. 1(b) representing
“class II” transitions are characterized by internal $W$ emission. In such
transitions, the nonvanishing contribution to the decay rate comes from
factorized amplitude proportional to the QCD factor $a_{2}$. Figure 1(c),
however, represents “class III” transitions which are due to both color-
favored and color-suppressed diagrams. In such decays, the factorized
amplitudes corresponding to $a_{1}$ and $a_{2}$ contribute coherently to give
the transition amplitude.

For the color-favored general type tree-level transition $B_{F}\to
V_{1}(nS)V_{2}$ pertaining to “class I” transitions, the decay rate can be
written as A35 ; A36 ; A46

$\Gamma={\frac{{G_{F}}^{2}}{16\pi}}{a_{1}^{2}(\mu)}|V_{bq^{{}^{\prime}}}V_{q_{i}\bar{q}_{j}}|^{2}{\frac{|\vec{k|}}{M^{2}}}{|\cal
A|}^{2},$ (8)

Figure 1:  Quark-level diagram of the nonleptonic decay of a meson $B_{F}\to
V_{1}(nS)V_{2}$.

where $M$ and $\vec{k}$ represent the parent-meson mass and three-momentum of
the recoiled daughter meson $V_{1}$, respectively, in the parent-meson rest
frame. ${|\cal A|}^{2}$ is the sum of the polarized amplitude squared with
${\cal A}_{j}\equiv{\langle V_{2}|J^{\mu}|0\rangle}{\langle
V_{1}|J_{\mu}|B_{F}\rangle}$, such that

${|\cal A|}^{2}=\sum_{j}{|{\cal A}_{j}|^{2}}.$ (9)

We use the notation $j=\pm$, $\mp$ or $ll$, where the first and second labels
denote the helicity of the $V_{1}$ and $V_{2}$ meson, respectively. From the
polarized amplitudes expressed in terms of the weak form factors $f$, $g$ and
$a_{+}$ and the decay constant $f_{V_{2}}$ shown in Eqs.(4)–(7), it is
straightforward to find expressions for the positive, negative, and
longitudinal polarizations, respectively, of the daughter meson $V_{1}$ as

$\displaystyle{\cal A}_{\pm}$ $\displaystyle=$ $\displaystyle-
f_{V_{2}}m_{V_{2}}\\{{f(q^{2})+2g(q^{2})|\vec{k}|M}\\},$ $\displaystyle{\cal
A}_{\mp}$ $\displaystyle=$ $\displaystyle-
f_{V_{2}}m_{V_{2}}\\{{f(q^{2})-2g(q^{2})|\vec{k}|M}\\},$ $\displaystyle{\cal
A}_{ll}$
$\displaystyle=\frac{f_{V_{2}}}{m_{V_{1}}}\Big{[}f(q^{2})\Big{\\{}|\vec{k}|^{2}+\frac{1}{4M^{2}}(M^{2}+m_{V_{1}}^{2}-m_{V_{2}}^{2})$
(10)
$\displaystyle\times(M^{2}+m_{V_{2}}^{2}-m_{V_{1}}^{2})\Big{\\}}+2a_{+}(q^{2})|\vec{k}|^{2}M^{2}\Big{]},$

where

$|\vec{k}|=\Big{[}\Big{(}\frac{M^{2}+m_{V_{1}}^{2}-m_{V_{2}}^{2}}{2M}\Big{)}^{2}-m_{V_{1}}^{2}\Big{]}^{1/2}.$
(11)

The decay widths and branching fractions for $B_{F}\to V_{1}(nS)V_{2}$ decays
can be predicted from Eq.(8) using the expressions in Eqs. (9–11) for the
polarized amplitudes in terms of the weak form factors derivable in the
framework of the RIQ model.

##  3 Transition amplitude and weak form factors

As discussed in the preceding section, the nonleptonic transition amplitude
for the process $B_{F}\to V_{1}(nS)V_{2}$ can be calculated from the tree-
level diagram shown in Fig.1. The class-I type decay modes, depicted in Fig
1(a), are induced by the $b$-quark transition to the daughter quark
$q^{{}^{\prime}}$ with the emission of $W$-boson. The daughter quark
$q^{{}^{\prime}}$ and the antiquark $\bar{q}$ of the decaying parent meson
state $|B_{F}(\vec{p},S_{B_{F}})\rangle$ hadronize to form a vector meson
state $|V_{1}(\vec{k},S_{V_{1}})\rangle$. The externally emitted $W$-boson
first decays to a quark-antiquark pair ($q_{i}$ ${\bar{q}_{j}}$), which
subsequently hadronizes to other vector meson state
$|V_{2}(\vec{q},S_{V_{2}})\rangle$.

The decay process, in fact, occurs physically in the momentum eigenstate of
participating mesons. Therefore, a field-theoretic description of a decay
process demands meson-bound states to be represented by appropriate momentum
wave packets reflecting momentum and spin distribution between the quark
constituents in the meson core. A brief discussion of the wave-packet
representation of meson bound state in the RIQ model is given in the Appendix.
Using the wave-packet representation (A.9-A.11) of participating meson states,
the residual dynamics responsible for the decay process can, therefore, be
described at the constituent level by the otherwise unbound quark and
antiquark using the usual Feynman technique. The constituent-level $S$-matrix
element $S_{fi}^{b\to q^{{}^{\prime}}{q_{i}}{\bar{q}_{j}}}$ obtained from the
appropriate Feynman diagram when operated upon by the bag-like operator
${\hat{\Lambda}}(\vec{p},S_{B_{F}})$ in the wave packet representation can
give rise to the mesonic-level $S$-matrix element in the form

$S_{fi}^{B_{F}\to V_{1}(nS)V_{2}}\to\hat{\Lambda}(\vec{p},S_{B_{F}})\
S_{fi}^{b\to q^{{}^{\prime}}q_{i}{\bar{q}_{j}}}.$ (12)

Using the wave packet representation of the parent and daughter meson state,
$|B_{F}(\vec{p},S_{B_{F}})\rangle$ and $|V_{1}(\vec{k},S_{V_{1}})\rangle$,
respectively, we calculate the Feynman Diagram Fig. 1(a) and obtain the
$S$-matrix element in the parent meson rest frame in the general form:

$S_{fi}=(2\pi)^{4}\ {\delta^{(4)}(M-q-k)}\ (-i{\cal
M}_{fi}){\frac{1}{\sqrt{V2M}}}\prod_{f}{\frac{1}{\sqrt{V2E_{f}}}}.$

(13)

where the invariant transition amplitude ${\cal M}_{fi}$ is obtained in the
form:

${\cal
M}_{fi}=\frac{G_{F}}{\sqrt{2}}{V_{bq^{{}^{\prime}}}V_{q_{i}\bar{q}_{j}}}a_{1}{\cal
A},$ (14)

with ${\cal A}=h^{\mu}H_{\mu}$,
$h^{\mu}=\epsilon_{V_{2}}^{*\mu}(\vec{q},\lambda_{2})f_{V_{2}}m_{V_{2}}$ and

$\displaystyle H_{\mu}=$
$\displaystyle\frac{1}{\sqrt{N_{B_{F}}(0)N_{V_{1}}(\vec{k})}}\int
d^{3}\vec{p}_{b}$ (15) $\displaystyle\frac{{\cal
G}_{B_{F}}(\vec{p}_{b},-\vec{p}_{b})\ {\cal
G}_{V_{1}}(\vec{p}_{b}+\vec{k},-\vec{p}_{b})}{\sqrt{E_{b}(\vec{p}_{b})E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})}}$
$\displaystyle\times\sqrt{[E_{b}(\vec{p}_{b})+E_{\bar{q}}(-\vec{p}_{b})][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+E_{\bar{q}}(-\vec{p}_{b})]}$
$\displaystyle\langle S_{V_{1}}|J_{\mu}(0)|S_{B_{F}}\rangle.$

The terms $E_{b}(\vec{p}_{b})$ and $E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})$
in (15) stand for the energy of the non-spectator quark of the parent and
daughter meson, $\vec{p}_{b}$ and $\vec{k}$ represent three momentum of the
non-spectator constituent quark $b$ and the daughter meson $V_{1}$,
respectively and $q=p-k$ is the four-momentum transfer. Finally, $\langle
S_{V}|J_{\mu}|S_{B_{F}}\rangle$ is the symbolical representation of the spin
matrix elements of the effective vector-axial vector current; which can be
written in the explicit form:

$\displaystyle\langle S_{V_{1}}|J_{\mu}(0)|S_{B_{F}}\rangle$
$\displaystyle=\sum_{{\lambda_{b}},{\lambda_{q^{{}^{\prime}}}},{\lambda_{\bar{q}}}}\zeta_{b\bar{q}}^{B_{F}}(\lambda_{b},\lambda_{\bar{q}})\
\zeta_{q^{{}^{\prime}}\bar{q}}^{V_{1}}(\lambda_{q^{{}^{\prime}}},\lambda_{\bar{q}})$
(16)
$\displaystyle\bar{u}_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k},\lambda_{q^{{}^{\prime}}})\
\gamma_{\mu}(1-\gamma_{5})\ u_{b}(\vec{p}_{b},\lambda_{b}).$

Here, $u_{i}$ stands for free Dirac
spinor.$\zeta^{B_{F}}(\lambda_{b},\lambda_{\bar{q}})$ and
$\zeta^{V_{1}}(\lambda_{q^{{}^{\prime}}},\\\ \lambda_{\bar{q}})$ are the
appropriate $SU(6)$ spin flavor coefficients corresponding to the parent and
daughter meson, respectively.

It may be pointed out here that, in our description of the decay process,
$B_{F}\to V_{1}(nS)V_{2}$ the three momentum conservation is ensured
explicitly via $\delta^{(3)}(\vec{p}_{b}+\vec{p}_{\bar{q}}-\vec{p})$ and
$\delta^{(3)}(\vec{p}_{i}\\\ +\vec{p}_{j}-\vec{k})$ in the participating meson
states. However, energy conservation in such a scheme is not ensured so
explicitly. This is in fact a typical problem in all potential model
descriptions of mesons as bound states of valence quarks and antiquarks
interacting via some instantaneous potential. This problem has been addressed
in the previous analysis in this model in the context of radiative leptonic
decays of heavy flavored meson $B$, $B_{c}$, $D$, $D_{s}$ A68 ; A69 ; A70  and
also in the QCD relativistic quark model A71 ; A72 , where the effective
momentum distribution function ${\cal
G}_{B_{F}}(\vec{p}_{b},\vec{p}_{\bar{q}})$ that embodies bound-state
characteristics of the meson ensures energy conservation in an average sense
satisfying $E_{M}=\langle
B_{F}(\vec{p},S_{B_{F}})|[E_{b}(\vec{p}_{b})+E_{\bar{q}}(\vec{p}_{\bar{q}})]|B_{F}(\vec{p},S_{B_{F}})\rangle$.
In view of this, we take the energy conservation constraint
$M=E_{b}(\vec{p}_{b})+E_{\bar{q}}(-\vec{p}_{b})$ in the parent meson rest
frame. This along with the three momentum conservation via appropriate
$\delta^{(3)}(\vec{p}_{b}+\vec{p}_{\bar{q}}-\vec{p})$ in the meson state
ensures the required four-momentum conservation ${\delta^{(4)}(p-k-q)}$ at the
mesonic level, which is pulled out of the quark-level integration to obtain
the $S$-matrix element in the standard form (13). This has been discussed
elaborately in earlier works A33 ; A34 ; A35 ; A36 ; A55 .

Using usual spin algebra the spacelike and timelike components of the spin
matrix elements $\langle S_{V_{1}}|J_{\mu}(0)|S_{B_{F}}(0)\rangle$
corresponding to vector and axial vector current are obtained in the form

$\langle
S_{V_{1}}(\vec{k},\hat{\epsilon}_{V_{1}}^{*})|V_{0}|S_{B_{F}}(0)\rangle=0,$
(17)

$\langle
S_{V_{1}}(\vec{k},\hat{\epsilon}_{V_{1}}^{*})|V_{i}|S_{B_{F}}(0)\rangle=\frac{i[E_{b}(\vec{p}_{b})+m_{b}]}{\sqrt{[E_{b}(\vec{p}_{b})+m_{b}][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+m_{q^{{}^{\prime}}}]}}(\hat{\epsilon}_{V_{1}}^{*}\times\vec{k})_{i}$,

(18)

$\langle
S_{V_{1}}(\vec{k},\hat{\epsilon}_{V_{1}}^{*})|A_{i}|S_{B_{F}}(0)\rangle=\frac{[E_{b}(\vec{p}_{b})+m_{b}][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+m_{q^{{}^{\prime}}}]-(\vec{p}_{b}^{2}/3)}{\sqrt{[E_{b}(\vec{p}_{b})+m_{b}][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+m_{q^{{}^{\prime}}}]}}(\hat{\epsilon}_{V_{1}}^{*})_{i}$,

(19)

$\langle
S_{V_{1}}(\vec{k},\hat{\epsilon}_{V_{1}}^{*})|A_{0}|S_{B_{F}}(0)\rangle=\frac{[E_{b}(\vec{p}_{b})+m_{b}]}{\sqrt{[E_{b}(\vec{p}_{b})+m_{b}][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+m_{q^{{}^{\prime}}}]}}(\hat{\epsilon}_{V_{1}}^{*}.\vec{k})$.

(20)

Here $E_{b}(\vec{p}_{b})=\sqrt{\vec{p}_{b}^{2}+m_{b}^{2}}$ and
$E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})=\sqrt{(\vec{p}_{b}+\vec{k})^{2}+m_{q^{{}^{\prime}}}^{2}}$
are, respectively, the energy of the non-spectator quark $b$ and daughter
quark $q^{{}^{\prime}}$. The spacelike component of the hadronic matrix
element $H_{\mu}$ obtained from Eq.(15) via Eqs. (18) and (19) are compared
with the corresponding expressions from Eqs. (4) and (5), which lead to the
model expressions of the weak form factors $g(q^{2})$ and $f(q^{2})$ in the
form:

$\displaystyle g(q^{2})$ $\displaystyle=$ $\displaystyle-\frac{1}{2M}\int
d\vec{p}_{b}\ {\cal Q}(\vec{p}_{b})\ [E_{b}(\vec{p}_{b})+m_{b}],$ (21)
$\displaystyle f(q^{2})$ $\displaystyle=$ $\displaystyle-\int
d\vec{p}_{b}{\cal R}(\vec{p}_{b}),$ (22)

where

$\displaystyle{\cal Q}(\vec{p}_{b})$ $\displaystyle=\frac{{\cal
G}_{B_{F}}(\vec{p}_{b},-\vec{p}_{b}){\cal
G}_{V_{1}}(\vec{p}_{b}+\vec{k},-\vec{p}_{b})}{\sqrt{N_{B_{F}}(0)N_{V_{1}}(\vec{k})}}$
(23)
$\displaystyle\frac{\sqrt{[E_{b}(\vec{p}_{b})+E_{\bar{q}}(-\vec{p}_{b}))][E_{q^{{}^{\prime}}}(\vec{p}_{b}+\vec{k})+E_{\bar{q}}(-\vec{p}_{b})]}}{\sqrt{E_{b}(\vec{p}_{b})E_{q^{{}^{\prime}}}(\vec{p}_{b})[E_{b}(\vec{p}_{b})+m_{b}][E_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}})+m_{q^{{}^{\prime}}}]}},$
${\cal R}(\vec{p}_{b})={\cal
Q}(\vec{p}_{b})\Big{[}(E_{b}(\vec{p}_{b})+m_{b})(E_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}})+m_{q^{{}^{\prime}}})-\frac{\vec{p}_{b}^{2}}{3}\Big{]}.$
(24)

The timelike component of hadronic amplitude obtained from Eq. (15) via Eq.
(20), when compared with the corresponding expression from Eq. (4) yields an
expression of the form factor $a_{+}(q^{2})$ in the form:

$\displaystyle a_{+}(q^{2})$
$\displaystyle=\frac{1}{2M^{2}}\Big{[}E_{V_{1}}\int d\vec{p}_{b}\ {\cal
Q}(\vec{p}_{b})\ (E_{b}(\vec{p}_{b})+m_{b})-$ (25) $\displaystyle\int
d\vec{p}_{b}\ {\cal R}(\vec{p}_{b})\Big{]}.$

Then it is straightforward to get the model expression for the polarized
amplitude squared $|{\cal A}_{j}|^{2}$ using Eqs.(21-25). Summing over
possible polarization states and integrating over the final-state particle
momenta, the decay width $\Gamma(B_{F}\to V_{1}(nS)V_{2})$ is obtained in the
parent-meson rest frame from the generic expression

$\displaystyle\Gamma(B_{F}\to V_{1}(nS)V_{2})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int\frac{d\vec{k}\
d\vec{q}}{2M2E_{V_{1}}2E_{V_{2}}}$ (26)
$\displaystyle\delta^{(4)}(p-k-q)\times\sum|{\cal M}_{fi}|^{2}.$

The two-body nonleptonic decay $(B_{F}\to V_{1}(nS)V_{2})$, described so far
in this section refers to the color-favored “class I” decays involving
external emission of $W$-boson. Similarly, class II and III type $B_{F}\to
V_{1}(nS)V_{2}$ decays can be calculated from the corresponding Feynman
diagrams shown in Figs. 1(b) and 1(c), respectively. The model expressions for
relevant form factors and decay rates for such decays (class II and class III)
can be obtained by suitable replacement of appropriate flavor degree of
freedom, quark masses, quark binding energies, QCD factors $a_{1},a_{2}$, and
the meson decay constants.

##  4 Numerical results and Discussion

In this section, we present our numerical results in comparison with other
model predictions and the available experimental data. For numerical
calculation, we use the model parameters $(a,V_{0})$, quark mass $m_{q}$ and
quark binding energies $E_{q}$, which have been fixed from hadron spectroscopy
by fitting the data of heavy and heavy-light flavored mesons in their ground
state as A73 ; A74 ; A75

$\displaystyle(a,V_{0})$ $\displaystyle=$ $\displaystyle(0.017166\
GeV^{3},-0.1375\ GeV),$ $\displaystyle(m_{b},m_{c})$ $\displaystyle=$
$\displaystyle(4.77659,1.49276)\ GeV,$ $\displaystyle(m_{s},m_{u}=m_{d})$
$\displaystyle=$ $\displaystyle(0.31575,0.07875)\ GeV,$
$\displaystyle(E_{b},E_{c})$ $\displaystyle=$ $\displaystyle(4.76633,1.57951)\
GeV,$ $\displaystyle(E_{s},E_{u}=E_{d})$ $\displaystyle=$
$\displaystyle(0.591,0.47125)\ GeV.$ (27)

The description of the decay process involving radially excited meson states,
the constituent quarks in the meson-bound states are expected to have higher
binding energies compared to their ground-state binding energies. For this, we
solve the cubic equation representing the binding energy condition (A.5) for
respective constituent quarks $(c,s,u=d)$ in radially excited $2S$ and $3S$
states of the $(\bar{c}c)$, $(\bar{c}u)$, $(\bar{c}d)$ systems as

$\displaystyle(E_{c},E_{s},E_{u}=E_{d})_{2S}$ $\displaystyle=$
$\displaystyle(1.97015,1.07737,0.96221)GeV,$
$\displaystyle(E_{c},E_{s},E_{u}=E_{d})_{3S}$ $\displaystyle=$
$\displaystyle(2.22478,1.40043,1.29359)GeV.$ (28)

Using the above input parameters (27), wide-ranging hadronic phenomena have
been described within the framework of the RIQ model, which includes the two
body nonleptonic decays of $B$ and $B_{c}$ mesons to ground state mesons in
the charmonium, charm, strange and non-strange light flavor sectors A33 ; A34
; A35 ; A36 . For CKM parameters and the lifetime of decaying mesons, we take
their respective central values from the Particle Data Group (2022) A82 :

$\displaystyle(V_{bc},V_{bu})$ $\displaystyle=$
$\displaystyle(0.0408,0.00382),$ $\displaystyle(V_{cs},V_{cd})$
$\displaystyle=$ $\displaystyle(0.975,0.221),$ $\displaystyle(V_{us},V_{ud})$
$\displaystyle=$ $\displaystyle(0.2243,0.97373),$ (29)

and

$\displaystyle(\tau_{B_{c}^{-}},\tau_{B^{-}})$ $\displaystyle=$
$\displaystyle(0.510,1.638)\ ps,$
$\displaystyle(\tau_{\bar{B^{0}}},\tau_{\bar{B_{s}^{0}}})$ $\displaystyle=$
$\displaystyle(1.519,1.5672)\ ps,$ (30)

respectively. For the mass and decay constant of the participating mesons,
considered as phenomenological inputs in the numerical calculation, we take
the central values of the available observed data from Ref. A5 ; A6 ; A82 . In
the absence of the observed data on the mass of excited ($2S$ and $3S$)
charmed and strange-charmed mesons and the meson decay constants, we take the
corresponding predicted data from established theoretical approaches A83 ; A84
; A85 ; A86 . Accordingly, the updated meson masses and decay constants used
in the present study are listed in Table-1.

Table 1: The masses and decay constants of mesons. Particle |  |  Mass(MeV) |  Decay constant(MeV) |
---|---|---|---|---
$\rho^{-}$ |  | 775.11 A82  |  208.5A84  |
$K^{*\pm}$ |  | 891.67 A82  |  217 A85  |
$D^{*0}(1S)$ |  | 2006.8 A82  |  339 A86  |
$D^{*\pm}(1S)$ |  | 2010.2 A82  |  341 A86  |
$D_{s}^{*\pm}(1S)$ |  | 2112.2 A82  |  375 A86  |
$J/\psi(1S)$ |  |  3096.9 A82  |  459 A86  |
$B_{c}^{-}$ |  | 6274.47A82  |  |
$\bar{B}^{0}$ |  | 5279.6 A82  |  |
$\bar{B}_{s}^{0}$ |  | 5415.4 A82  |  |
$B^{-}$ |  | 5279.3 A82  |  |
$D^{*\pm}(2S)$ |  | 2637 A5  |  290 A86  |
$D^{*0}(2S)$ |  | 2681 A6  |  289 A86  |
$D_{s}^{*\pm}(2S)$ |  | 2732 A83  |  312 A86  |
$\psi(2S)$ |  | 3686.1 A82  |  364 A86  |
$D^{*0}(3S)$ |  | 3080 A18  |  |
$D^{*\pm}(3S)$ |  | 3110 A83  |  |
$D_{s}^{*\pm}(3S)$ |  | 3193 A83  |  |
$\psi(3S)$ |  | 4039.1 A82  |  319 A86  |

It may be mentioned here that, in the prediction of nonleptonic decay,
uncertainties mostly creep into the calculation through input parameters:
potential parameter $(a,V_{0})$, quark mass $(m_{q})$ and quark binding energy
$(E_{q})$, CKM parameters, meson decay constants and QCD coefficients
$(a_{1},a_{2})$ etc. As mentioned above, the potential parameters, quark
masses and quark binding energies (27,28) have already been fixed at the
static level application of RIQ model by fitting the mass spectra of heavy and
heavy-light mesons. In order to avoid uncertainty in our model predictions, we
take the central values of the CKM parameter as well as the observed value of
the decay constants. As such we do not have the liberty to use any free
parameter in our calculation which could be fine-tuned from time to time to
predict any hadronic phenomena. In that sense, we perform almost a parameter-
free calculation in our studies. As regards the QCD coefficients: ($a_{1}$,
$a_{2}$), different sets of data for decays induced by the $b$-quark
transition at the quark level, are used in the literature. For example,
Colangelo and De Fazio, in Ref. A27  use QCD coefficients, Set 1:
$(a_{1}^{b},a_{2}^{b})=(1.12,-0.26)$, as fixed in Refs. A87 ; A88 . In most
earlier calculations, the authors use a different set of QCD coefficients, Set
2: $(a_{1}^{b},a_{2}^{b})=(1.14,-0.2)$, fixed by Buras et al. A52  in the
mid-1980s, whereas Dubnicka et al. A89  use a different set of numerical
value, i.e., Set 3: $(a_{1}^{b},a_{2}^{b})=(0.93,-0.27)$. We use all three
sets of the Wilson coefficients in our calculation.

Before using the above input parameters in our numerical analysis, it is
pertinent to elaborate a bit on the energy conservation ansatz mentioned in
Sec.3. The present analysis based on the energy conservation constraints
$M=E_{b}(\vec{p}_{b})+E_{q^{{}^{\prime}}}(-\vec{p}_{b})$ in the parent meson
rest frame might lead to spurious kinematic singularities at the quark-level
integration appearing in the decay amplitude. This problem has already been
addressed previously in their QCD relativistic quark model approach A71 ; A72
and later by our group in the study of radiative leptonic decays of heavy and
heavy-light flavored meson sector A68 ; A69 ; A70 , by assigning a running
mass $m_{b}$ to the non-spectator quark that satisfies the relation:

$\displaystyle m_{b}^{2}(|\vec{p}_{b}|)$ $\displaystyle=$ $\displaystyle
M^{2}-m_{\bar{q}}^{2}-2M\sqrt{|\vec{p}_{b}|^{2}+m_{\bar{q}}^{2}},$

as an outcome of the energy conservation ansatz, while retaining definite mass
$m_{\bar{q}}$ of the spectator quark $\bar{q}$. This leads to an upper bound
on the quark momentum $|\vec{p}_{b}|<\frac{M^{2}-m_{\bar{q}}^{2}}{2M}$ in
order to retain $m_{b}^{2}(|\vec{p}_{b}|)$ the positive definite. The upper
limit $|\vec{p}_{b}|_{max}$ would have no other bearing to seriously affect
the calculation which is apparent from the shape of the radial quark momentum
distribution $|\vec{p}_{b}|{\cal G}(\vec{p}_{b},-\vec{p}_{b})$. In fact, the
quark momentum distribution obtained in this model A68 ; A69 ; A70  is similar
to the prediction of the QCD relativistic quark model analysis A71 ; A72 . The
rms value of the active quark momentum
$\sqrt{\langle|\vec{p}_{b}^{2}|\rangle}$, where
$\langle|\vec{p}_{b}^{2}|\rangle=\langle
B_{F}(0)|\vec{p}_{b}^{2}|B_{F}(0)\rangle$, the expectation value of the
binding energies of the active quark $b$, and spectator $\bar{q}$ and the sum
of the binding energy of quark and antiquark pair $\langle
E_{b}(\vec{p}_{b}^{2})\rangle$, $\langle
E_{\bar{q}}(|-\vec{p}_{b}^{2}|)\rangle$ and
$\langle[E_{b}(\vec{p}_{b}^{2})+E_{\bar{q}}(|-\vec{p}_{b}^{2}|)]\rangle$,
respectively, calculated in the framework of RIQ model, are presented in Table
2.

Table 2: The rms values of quark momentum, expectation values of the momentum
of quark and antiquark, and the sum of the momentum of quark and antiquark in
the meson states.

Meson state | $\sqrt{\langle|\vec{p}_{b}^{2}|\rangle}$ | $\langle E_{b}(\vec{p}_{b}^{2})\rangle$ | $\langle E_{\bar{q}}(|-\vec{p}_{b}^{2}|)\rangle$ | $\langle[E_{b}(\vec{p}_{b}^{2})+E_{\bar{q}}(|-\vec{p}_{b}^{2}|)]\rangle$ | Observed meson
---|---|---|---|---|---
$|X(0)\rangle$ | (GeV) | (GeV) |  (GeV) |  (GeV) | mass (GeV)
$|B_{u}(0)\rangle$ | 0.51 | 4.799 |  0.480 |  5.279 |  5.27925
$|B_{c}(0)\rangle$ | 0.66 | 4.657 |  1.629 |  6.286 |  6.27447
$|D(0)\rangle$ | 0.4506 | 1.4418 |  0.4275 |  1.8693 |  1.86965
$|D_{s}(0)\rangle$ | 0.4736 | 1.4165 |  0.5517 |  1.9682 |  1.96835

It is noteworthy to discuss four important aspects of our present approach.
(1) The rms value of the quark momentum in the meson-bound state is much less
than the corresponding upper bound $|\vec{p}_{b}|_{max}$, as expected. (2) The
average energies of a constituent quark of the same flavor in different meson-
bound states do not exactly match. This is because the kinematics and binding
energy conditions for constituent quarks due to the color forces involved are
different from one meson-bound state to other. The constituent quarks in the
meson-bound state are considered to be free particles of definite momenta,
each associated with its momentum probability amplitude derivable in this
model via momentum space projection of the respective quark eigenmodes. On the
other hand, the energies shown in Eq. (27,28), which are the energy
eigenvalues of the corresponding bound quarks with no definite momenta of
their own, are obtained from respective quark orbitals by solving the Dirac
equation in this model. This makes a marginal difference between the energy
eigenvalues (27,28) and the average energy of constituent quarks shown in
Table 2. (3) The expectation values of the sum of the energy of a constituent
quark and antiquark in the meson-bound state are obtained in good agreement
with the corresponding observed meson masses as shown in Table 2. These
important aspects of our results lend credence to our energy conservation
ansatz in an average sense through the effective momentum distribution
function like ${\cal G}_{B_{F}}(\vec{p}_{b},-\vec{p}_{b})$ in the meson-bound
state $|B_{F}(0)\rangle$. This ansatz along with three momentum conservation
in the meson-bound state (A.1) ensures the required energy-momentum
conservation in our description of several decay processes pointed out
earlier. In the absence of any rigorous field theoretic description of the
meson-bound states, invoking such an ansatz is no doubt a reasonable approach
for a constituent-level description of hadronic phenomena. (4) Finally, in a
self-consistent dynamic approach, we extract the form factors from the
overlapping integrals of meson wave functions, where the $q^{2}$ dependence of
the decay amplitude is automatically encoded. This is in contrast to some
model approaches cited in the literature where the form factors are determined
only at one kinematic point, i.e., either at $q^{2}\to 0$ or $q^{2}\to
q^{2}_{max}$, and then extrapolated to the entire kinematic range using some
phenomenological ansatz (mainly dipole or Gaussian form).

The invariant form factors $g(q^{2})$, $f(q^{2})$ and $a_{+}(q^{2})$, which
represent the decay amplitudes, are found to have different dimensions. In
order to study their $q^{2}$-dependence over the allowed kinematic range, they
need to be treated on equal footing by casting them in the dimensionless form
:

$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle(M+m_{V_{1}})g(q^{2}),$ $\displaystyle A_{1}(q^{2})$
$\displaystyle=$ $\displaystyle(M+m_{V_{1}})^{-1}f(q^{2}),$ $\displaystyle
A_{2}(q^{2})$ $\displaystyle=$ $\displaystyle-(M+m_{V_{1}})a_{+}(q^{2}).$ (31)

One may naively expect the form factor in the dimensionless form to satisfy
the heavy quark-symmetry (HQS) relation:

$V(q^{2})\simeq A_{2}(q^{2})\simeq\tilde{A_{1}}(q^{2}),$ (32)

as an outcome of the heavy quark effective theory (HQET), where

$\tilde{A_{1}}(q^{2})=\Big{[}1-\frac{q^{2}}{(M+m_{V_{1}})^{2}}\Big{]}^{-1}A_{1}(q^{2}).$
(33)

Figure 2: $q^{2}$-dependence of form factors in $B_{c}^{-}\to\psi(nS)$ type
decays.

Figure 3: $q^{2}$-dependence of form factors in $B_{c}^{-}\to D^{*-}(nS)$ type
decays.

Figure 4: $q^{2}$-dependence of form factors in $B_{c}^{-}\to D_{s}^{*-}(nS)$
type decays.

Figure 5:  $q^{2}$-dependence of form factors in $B^{-}\to\psi(nS)$ type
decays.

Figure 6:  $q^{2}$-dependence of form factors in $B^{-}\to D^{*0}(nS)$ type
decays.

Figure 7:  $q^{2}$-dependence of form factors in $\bar{B^{0}}\to D^{*+}(nS)$
type decays.

Figure 8:  $q^{2}$-dependence of form factors in $\bar{B_{s}^{0}}\to D_{s}^{*+}(nS)$ type decays. Table 3:  Predicted values of form factors for decays to 1S state. Decay mode | value at | $V(q^{2})$ | $A_{1}(q^{2})$ | $A_{2}(q^{2})$
---|---|---|---|---
${B_{c}}^{-}\to J/\psi$ | $q^{2}=0$ | 0.482 | 0.125 | 0.0095
| $q^{2}=q^{2}_{max}$ | 2.121 | 0.452 | 0.0006
${B_{c}}^{-}\to D_{s}^{*-}$ | $q^{2}=0$ | 0.077 | 0.016 | 0.0192
| $q^{2}=q^{2}_{max}$ | 4.584 | 0.466 | 0.8853
${B_{c}}^{-}\to{D}^{*-}$ | $q^{2}=0$ | 0.033 | 0.006 | 0.0105
| $q^{2}=q^{2}_{max}$ | 6.527 | 0.403 | 1.5552
$B^{-}\to J/\psi$ | $q^{2}=0$ | 0.727 | 0.198 | 0.0003
| $q^{2}=q^{2}_{max}$ | 2.049 | 0.485 | 0.0097
$B^{-}\to{D}^{0*}$ | $q^{2}=0$ | 0.578 | 0.307 | 0.0954
| $q^{2}=q^{2}_{max}$ | 1.447 | 0.605 | 0.2885
$\bar{B^{0}}\to D^{*+}$ | $q^{2}=0$ | 0.578 | 0.307 | 0.0953
| $q^{2}=q^{2}_{max}$ | 1.448 | 0.604 | 0.2877
$\bar{B_{s}^{0}}\to D_{s}^{*+}$ | $q^{2}=0$ | 0.523 | 0.254 | 0.0725
| $q^{2}=q^{2}_{max}$ | 1.514 | 0.584 | 0.2674
Table 4:  Predicted values of form factors for decays to 2S state. Decay mode | value at | $V(q^{2})$ | $A_{1}(q^{2})$ | $A_{2}(q^{2})$
---|---|---|---|---
${B_{c}}^{-}\to\psi(2S)$ | $q^{2}=0$ | 0.245 | 0.048 | 0.022
| $q^{2}=q^{2}_{max}$ | 1.829 | 0.314 | 0.155
${B_{c}}^{-}\to D_{s}^{*-}(2S)$ | $q^{2}=0$ | 0.066 | 0.010 | 0.019
| $q^{2}=q^{2}_{max}$ | 3.961 | 0.325 | 1.087
${B_{c}}^{-}\to{D}^{*-}(2S)$ | $q^{2}=0$ | 0.046 | 0.006 | 0.015
| $q^{2}=q^{2}_{max}$ | 5.789 | 0.300 | 1.869
$B^{-}\to\psi(2S)$ | $q^{2}=0$ | 0.415 | 0.086 | 0.037
| $q^{2}=q^{2}_{max}$ | 1.460 | 0.273 | 0.164
$B^{-}\to{D}^{0*}(2S)$ | $q^{2}=0$ | 0.273 | 0.094 | 0.019
| $q^{2}=q^{2}_{max}$ | 1.231 | 0.357 | 0.097
$\bar{B^{0}}\to D^{*+}(2S)$ | $q^{2}=0$ | 0.265 | 0.094 | 0.020
| $q^{2}=q^{2}_{max}$ | 1.224 | 0.363 | 0.106
$\bar{B_{s}^{0}}\to D_{s}^{*+}(2S)$ | $q^{2}=0$ | 0.252 | 0.084 | 0.014
| $q^{2}=q^{2}_{max}$ | 1.287 | 0.360 | 0.090

With the input parameters (27,28), we first study the $q^{2}$-dependence of
the dimensionless form factors in their allowed kinematic ranges. Our
predictions do not entirely agree with HQS relation. This is due to the well-
known fact that the heavy flavor symmetry is not strictly applicable in the
heavy meson sector, particularly when two heavy constituent quarks are
involved. The $q^{2}$-dependence of form factors for
$B_{c}^{-},B^{-},\bar{B^{0}},\bar{B_{s}^{0}}$-decays to $S$-wave ($1S,2S,3S$)
charmonium and charm meson states in the present model is shown in the Figs.
(2-8). We find that the departure from HQS relation is more and more
pronounced in the decays to higher and higher excited states. This is due to
different kinematics and four-momentum transfer involved in different decay
modes.

Table 5:  Predicted values of form factors for decays to 3S state. Decay mode | value at | $V(q^{2})$ | $A_{1}(q^{2})$ | $A_{2}(q^{2})$
---|---|---|---|---
${B_{c}}^{-}\to\psi(3S)$ | $q^{2}=0$ | 0.1121 | 0.0198 | 0.0015
| $q^{2}=q^{2}_{max}$ | 1.5727 | 0.2383 | 0.0015
${B_{c}}^{-}\to D_{s}^{*-}(3S)$ | $q^{2}=0$ | 0.0286 | 0.0036 | 0.0014
| $q^{2}=q^{2}_{max}$ | 3.6723 | 0.2490 | 0.0014
${B_{c}}^{-}\to{D}^{*-}(3S)$ | $q^{2}=0$ | 0.0213 | 0.0022 | 0.0014
| $q^{2}=q^{2}_{max}$ | 5.3214 | 0.2283 | 0.0014
$B^{-}\to\psi(3S)$ | $q^{2}=0$ | 0.2570 | 0.0465 | 0.0014
| $q^{2}=q^{2}_{max}$ | 1.2710 | 0.2084 | 0.0014
$B^{-}\to{D}^{0*}(3S)$ | $q^{2}=0$ | 0.1177 | 0.0337 | 0.0034
| $q^{2}=q^{2}_{max}$ | 1.1429 | 0.2742 | 0.0020
$\bar{B^{0}}\to D^{*+}(3S)$ | $q^{2}=0$ | 0.1203 | 0.0339 | 0.0029
| $q^{2}=q^{2}_{max}$ | 1.1469 | 0.2716 | 0.0044
$\bar{B_{s}^{0}}\to D_{s}^{*+}(3S)$ | $q^{2}=0$ | 0.1143 | 0.0307 | 0.0009
| $q^{2}=q^{2}_{max}$ | 1.1360 | 0.2569 | 0.0176

Figure 9:  Overlap of momentum distribution amplitudes of the initial and
final meson states.

Figure 10:  Overlap of momentum distribution amplitudes of the initial and
final meson states.

Our predicted form factors at $q^{2}\to 0$ (maximum recoil) and $q^{2}\to
q^{2}_{max}$ (minimum recoil) point for the transition to $1S$, $2S$ and $3S$
charmonium and charm meson states are shown, respectively in Table (3,4,5).
Before predicting the physical quantities of interest: decay width and
branching fraction etc., it is interesting to go for a qualitative assessment
of transition probabilities for transition to different $S$-wave states. For
this, we study the radial quark momentum distribution amplitude
$|\vec{p}_{q}|{\cal G}_{B_{F}}(\vec{p}_{q},-\vec{p}_{q})$ of the parent and
daughter mesons over the physical range of respective quark momentum
$\vec{p}_{q}$, for each such decay mode. From the plot shown in Figs. (9-10)
the overlap region between the momentum distribution profile of the parent and
daughter meson in the transition to the $1S$ state is found to be maximum and
it decreases for transitions to higher excited $2S$ and $3S$ states. Since the
invariant form factors are evaluated from overlapping integrals of
participating meson wave function, it is expected that the contribution of the
form factors to decay width/ branching fractions should be in decreasing order
of magnitude, going from decays to $1S$ state to higher excited ($2S$ and
$3S$) states.

Table 6: Decay widths in units of $10^{-15}$ GeV in terms of Wilson
coefficients $a_{1}$ and $a_{2}$.

$B_{c}^{-}\to J/\psi X^{-}$ | Width | $B_{c}^{-}\to\psi(2S)X^{-}$ | Width | $B_{c}^{-}\to\psi(3S)X^{-}$ | Width
---|---|---|---|---|---
$\rho^{-}$ | 1.8565$a_{1}^{2}$ | $\rho^{-}$ | 0.2076$a_{1}^{2}$ | $\rho^{-}$ | 0.0266$a_{1}^{2}$
$K^{*-}$ | 0.1123$a_{1}^{2}$ | $K^{*-}$ | 0.0127$a_{1}^{2}$ | $K^{*-}$ | 0.0016$a_{1}^{2}$
$D_{s}^{*-}$ | $(3.124a_{1}+3.529a_{2})^{2}$ | $D_{s}^{*-}$ | $(1.344a_{1}+3.6702a_{2})^{2}$ | $D_{s}^{*-}$ | $(0.942a_{1}+3.502a_{2})^{2}$
$D^{*-}$ | $(0.758a_{1}+0.359a_{2})^{2}$ | $D^{*-}$ | $(0.331a_{1}+0.842a_{2})^{2}$ | $D^{*-}$ | $(0.191a_{1}+0.847a_{2})^{2}$
$\bar{B_{s}^{0}}\to D_{s}^{*+}(1S)X^{-}$ | Width | $\bar{B_{s}^{0}}\to D_{s}^{*+}(2S)X^{-}$ | Width | $\bar{B_{s}^{0}}\to D_{s}^{*+}(3S)X^{-}$ | Width
$\rho^{-}$ | 3.929$a_{1}^{2}$ | $\rho^{-}$ | 0.3502$a_{1}^{2}$ | $\rho^{-}$ | 0.0350$a_{1}^{2}$
$K^{*-}$ |  0.2319$a_{1}^{2}$ | $K^{*-}$ | 0.0211$a_{1}^{2}$ | $K^{*-}$ | 0.0021$a_{1}^{2}$
$\bar{B^{0}}\to D^{*+}(1S)X^{-}$ | Width | $\bar{B^{0}}\to D^{*+}(2S)X^{-}$ | Width | $\bar{B^{0}}\to D^{*+}(3S)X^{-}$ | Width
$\rho^{-}$ | 4.918$a_{1}^{2}$ | $\rho^{-}$ | 0.386$a_{1}^{2}$ | $\rho^{-}$ | 0.0367$a_{1}^{2}$
$K^{*-}$ |  0.2885$a_{1}^{2}$ | $K^{*-}$ |  0.0232$a_{1}^{2}$ | $K^{*-}$ | 0.0022$a_{1}^{2}$
$B^{-}\to D^{*0}(1S)X^{-}$ | Width | $B^{-}\to D^{*0}(2S)X^{-}$ | Width | $B^{-}\to D^{*0}(3S)X^{-}$ | Width
$\rho^{-}$ | 4.931$a_{1}^{2}$ | $\rho^{-}$ | 0.3787$a_{1}^{2}$ | $\rho^{-}$ | 0.0371$a_{1}^{2}$
$K^{*-}$ | 0.289$a_{1}^{2}$ | $K^{*-}$ | 0.0228$a_{1}^{2}$ | $K^{*-}$ | 0.0023$a_{1}^{2}$
$B^{-}\to J/\psi X^{-}$ | Width | $B^{-}\to\psi(2S)X^{-}$ | Width | $B^{-}\to\psi(3S)X^{-}$ | Width
$\rho^{-}$ | 1.032$a_{2}^{2}$ | $\rho^{-}$ | 0.892$a_{2}^{2}$ | $\rho^{-}$ |  0.628$a_{2}^{2}$
$K^{*-}$ | 31.29$a_{2}^{2}$ | $K^{*-}$ |  20.912$a_{2}^{2}$ | $K^{*-}$ | 12.22$a_{2}^{2}$

Table 7:  Branching fractions in $\%$ for values of Wilson coefficients
$a_{1}$ and $a_{2}$.

| Branching Fraction |  | A20  | A23  | A25  | A49  | A89  |
---|---|---|---|---|---|---|---|---
$B_{c}^{-}\to J/\psi X^{-}$ |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93 |  |  |  |  |  |  |
|  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27 |  |  |  |  |  |  |
$\rho^{-}$ | 0.1804 | 0.186 | 0.124 |  |  0.16 | 0.49 |  | 0.40 |  |
$K^{*-}$ | 0.0109 | 0.011 |  0.0075 |  |  0.01 | 0.028 |  $0.0109\pm 0.0033$ | 0.022 |  |
$D_{s}^{*-}$ | 0.5159 |  0.6317 |  0.2953 |  |  | 0.97 | $0.412\pm 0.123$ | 0.67 | 0.78 |
$D^{*-}$ | 0.0442 |  0.0486 |  0.0286 |  |  | 0.045 |  $0.0182\pm 0.0055$ | 0.028 | 0.031 |
$\bar{B_{s}^{0}}\to D_{s}^{*+}(1S)X^{-}$ |  | Branching Fraction |  | A18  |  |  |  |  |  | A82
$\rho^{-}$ | 1.1734 | 1.2157 |  0.8091 | $0.726^{+0.076}_{-0.071}$ |  |  |  |  |  | $0.96\pm 0.21$
$K^{*-}$ | 0.0692 | 0.0717 |  0.0477 |  $0.0688^{+0.0067}_{-0.0064}$ |  |  |  |  |  |
$\bar{B^{0}}\to D^{*+}(1S)X^{-}$ |  | Branching Fraction |  | A18  |  |  |  |  |  | A82
$\rho^{-}$ | 1.4238 | 1.4751 |  0.9817 | $0.873^{+0.078}_{-0.073}$ |  |  |  |  |  | $0.68\pm 0.09$
$K^{*-}$ | 0.0834 | 0.0864 |  0.0575 |  $0.0758^{+0.0064}_{-0.0059}$ |  |  |  |  |  | $0.033\pm 0.006$
$B^{-}\to D^{*0}(1S)X^{-}$ |  | Branching Fraction |  | A18  |  |  |  |  |  | A82
$\rho^{-}$ | 1.5391 | 1.5946 |  1.0612 | $0.873^{+0.079}_{-0.073}$ |  |  |  |  |  | $0.98\pm 0.17$
$K^{*-}$ | 0.0901 | 0.0934 |  0.0621 |  $0.0846^{+0.0070}_{-0.0064}$ |  |  |  |  |  | $0.081\pm 0.014$
$B^{-}\to J/\psi X^{-}$ |  | Branching Fraction |  |  |  |  |  |  |  | A82
$\rho^{-}$ | 0.0173 | 0.0102 | 0.0187 |  |  |  |  |  |  | $0.0041\pm 0.0005$
$K^{*-}$ | 0.5264 | 0.3115 | 0.5677 |  |  |  |  |  |  | $0.143\pm 0.008$

We calculate the decay width from the expression (26) via (14) and (9-10) and
our predicted decay widths $\Gamma(B_{F}\to V_{1}(nS)V_{2})$ for general
values of QCD coefficients $(a_{1},a_{2})$ of the operator product expansion
are listed in Table 6 to facilitate a comparison with other dynamical model
predictions. Our predicted branching fractions (BFs) for
$B_{c}^{-},\bar{B_{s}^{0}},B^{-},\bar{B^{0}}$-decays to $1S$, $2S$ and $3S$
charmonium and charm meson states, are listed in Table (7,8,9), respectively,
in reasonable agreement with available experimental data A82  and other model
predictions. Our results for BFs of decays to $1S$, $2S$ and $3S$ states
corresponding to 3 sets of QCD parameters are listed in the second column of
each table. As expected, our predicted branching fractions are obtained in the
hierarchy:

$\displaystyle{\cal B}(B_{F}\to V_{1}(3S)V_{2})$ $\displaystyle<{\cal
B}(B_{F}\to V_{1}(2S)V_{2})$ $\displaystyle<{\cal B}(B_{F}\to
V_{1}(1S)V_{2}).$

Table 8:  Branching fractions in $10^{-4}$ for values of Wilson coefficients
$a_{1}$ and $a_{2}$.

| Branching Fraction | A15  | A18  |  A20  | A25  |
---|---|---|---|---|---|---
$B_{c}^{-}\to\psi(2S)X^{-}$ |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93 |  |  |  |  |
|  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27 |  |  |  |  |
$\rho^{-}$ | 2.01 | 2.09 | 1.39 | 5.69 | $4.04^{+0.21}_{-0.17}$ |  1.8 |  |
$K^{*-}$ |  0.124 |  0.128 |  0.085 |  0.304 |  $0.2822^{+0.0156}_{-0.0126}$ |  1.0 |  $0.439\pm 0.071$ |
$D_{s}^{*-}$ | 2.35 | 4.92 | 0.52 |  |  |  | $8.85\pm 2.54$ |
$D^{*-}$ |  0.178 | 0.33 | 0.05 |  |  |  |  $0.432\pm 0.117$ |
$\bar{B_{s}^{0}}\to D_{s}^{*+}(2S)X^{-}$ |  | Branching Fraction |  |  | A18  |  |  A26  |
$\rho^{-}$ |  10.45 |  10.83 | 7.21 |  |  $0.475^{+0.164}_{-0.121}$ |  |  22 |
$K^{*-}$ | 0.63 | 0.65 | 0.43 |  |  $0.0332^{+0.0122}_{-0.0090}$ |  |  1.2 |
$\bar{B^{0}}\to D^{*+}(2S)X^{-}$ |  | Branching Fraction |  |  | A18  |  |  |
$\rho^{-}$ |  11.17 |  11.57 |  7.706 |  |  $0.267^{+0.081}_{-0.061}$ |  |  |
$K^{*-}$ | 0.67 | 0.69 | 0.46 |  |  $0.0162^{+0.0050}_{-0.0038}$ |  |  |
$B^{-}\to D^{*0}(2S)X^{-}$ |  | Branching Fraction |  |  | A18  |  |  |
$\rho^{-}$ |  11.81 |  12.24 | 8.14 |  |  $0.287^{+0.088}_{-0.066}$ |  |  |
$K^{*-}$ | 0.70 | 0.72 | 0.48 |  |  $0.0173^{+0.0054}_{-0.0041}$ |  |  |
$B^{-}\to\psi(2S)X^{-}$ |  | Branching Fraction |  |  |  |  | A82  |
$\rho^{-}$ | 1.49 | 0.88 | 1.61 |  |  |  |  |
$K^{*-}$ |  35.18 |  20.81 |  37.93 |  |  |  | $6.7\pm 1.4$ |

Table 9:  Branching fractions in $10^{-5}$ for values of Wilson coefficients
$a_{1}$ and $a_{2}$.

| Branching Fraction | A18
---|---|---
$B_{c}^{-}\to\psi(3S)X^{-}$ |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93 |
|  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27 |
$\rho^{-}$ | 2.5 | 2.6 | 1.7 | $3.35^{+0.78}_{-0.68}$
$K^{*-}$ |  0.16 |  0.17 |  0.11 |  $0.229^{+0.055}_{-0.045}$
$D_{s}^{*-}$ | 1.6 | 10.8 |  0.37 |
$D^{*-}$ |  0.003 |  0.181 |  0.202 |
$\bar{B_{s}^{0}}\to D_{s}^{*+}(3S)X^{-}$ |  | Branching Fraction |  | A18
$\rho^{-}$ | 10.4 |  10.85 | 7.2 | $3.55^{+0.88}_{-0.69}$
$K^{*-}$ | 0.65 | 0.67 |  0.44 |  $0.251^{+0.061}_{-0.048}$
$\bar{B^{0}}\to D^{*+}(3S)X^{-}$ |  | Branching Fraction |  | A18
$\rho^{-}$ | 10.6 |  11.03 | 7.34 | $1.03^{+0.44}_{-1.04}$
$K^{*-}$ | 0.66 | 0.68 | 0.45 |  $0.0709^{+0.0310}_{-0.0719}$
$B^{-}\to D^{*0}(3S)X^{-}$ |  | Branching Fraction |  | A18
$\rho^{-}$ | 11.5 | 12 | 7.9 | $1.11^{+0.50}_{-1.12}$
$K^{*-}$ | 0.72 |  0.74 |  0.49 |  $0.0753^{+0.0347}_{-0.0766}$
$B^{-}\to\psi(3S)X^{-}$ |  | Branching Fraction |  |
$\rho^{-}$ | 10.5 | 6.2 | 11.3 |
$K^{*-}$ |  205.5 |  121.6 |  221.6 |

Our results for transitions to $2S$ and $3S$ states are obtained two and three
orders of magnitude down compared to those obtained for transition to $1S$
state. The node structure of the $2S$ wave function is responsible for small
BFs. Since there is no node for the initial wave function, the contribution
from the positive and negative parts of the final wave function cancel each
other out yielding small BFs. In the case of the transition to $3S$ states,
there are even more serious cancellations; leading to still smaller BFs. As
expected, the tighter phase space and the $q^{2}$-dependence of the form
factors typical to the decay mode lead to smaller BFs for transitions to
higher excited $2S$ and still smaller for the transition of $3S$ states.

The BFs of the decay modes, considered in the present study, are obtained in a
wide range of $\sim 10^{-2}-10^{-6}$. For nonleptonic $B_{F}$-meson decays to
1S, 2S, and 3S charmonium and charm meson states, BFs range from $\sim
10^{-2}-10^{-3}$, $\sim 10^{-4}-10^{-5}$ and $\sim 10^{-5}-10^{-6}$,
respectively. The dominant decay modes: $B_{c}^{-}\to J/\psi D_{s}^{*-}$,
$\bar{B_{s}^{0}}\to D_{s}^{*+}\rho^{-}$, $\bar{B^{0}}\to D^{*+}\rho^{-}$,
$B^{-}\to D^{*0}\rho^{-}$ and $B^{-}\to J/\psi K^{*-}$ to $1S$ charmonium and
charm meson states are found to have BFs of 0.52, 1.17, 1.42, 1.54 and 0.53,
respectively in the order ${\cal O}(\sim 10^{-2})$ which should be
experimentally accessible. For such decays to corresponding $2S$ modes, the
predicted BFs, of 2.35, 10.45, 11.17, 11.81 and 35.18 in the order of
magnitude ${\cal O}(\sim 10^{-4})$, lie within the detection accuracy of
current experiments. The neutral $B$-meson decay in the present study is found
to have smaller BFs than those of charged $B$-meson decays, as expected. This
may be due to the spectator interaction effects of $d$ and $u$ quarks.

Table 10: Decay widths in units of $10^{-15}$ GeV and branching fractions in
$\%$ for values of Wilson coefficient $a_{1}$ and $a_{2}$.

|  | Branching Fraction | A82
---|---|---|---
Decay modes | Decay width |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93 |
|  |  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27 |
$\bar{B^{0}_{s}}\to D_{s}^{*+}(1S)D^{*-}$ | 0.8116$a_{1}^{2}$ | 0.2423 | 0.2510 | 0.1670 |
$\bar{B^{0}_{s}}\to D_{s}^{*+}(2S)D^{*-}$ | 0.1052$a_{1}^{2}$ | 0.0313 | 0.0324 | 0.0216 |
$\bar{B^{0}_{s}}\to D_{s}^{*+}(3S)D^{*-}$ | 0.0304$a_{1}^{2}$ | 0.0090 | 0.0094 | 0.0062 |
$\bar{B^{0}}\to D^{*+}(1S)D^{*-}$ | 1.1406$a_{1}^{2}$ | 0.2631 | 0.2726 | 0.1814 |
$\bar{B^{0}}\to D^{*+}(2S)D^{*-}$ | 0.1077$a_{1}^{2}$ | 0.0311 | 0.0323 | 0.0215 |
$\bar{B^{0}}\to D^{*+}(3S)D^{*-}$ | 0.0329$a_{1}^{2}$ | 0.0095 | 0.0098 | 0.0065 |
$\bar{B^{0}}\to D^{*+}(1S)D_{s}^{*-}$ | 21.924$a_{1}^{2}$ | 6.3460 | 6.5740 | 4.3750 | $0.8\pm 0.11$
$\bar{B^{0}}\to D^{*+}(2S)D_{s}^{*-}$ | 2.7331$a_{1}^{2}$ | 0.7911 | 0.8196 | 0.5454 |
$\bar{B^{0}}\to D^{*+}(3S)D_{s}^{*-}$ | 1.1440$a_{1}^{2}$ | 0.3312 | 0.3432 | 0.2284 |
$B^{-}\to D^{*0}(1S)D^{*-}$ | 0.9118$a_{1}^{2}$ | 0.2846 | 0.2948 | 0.1962 |  $0.081\pm 0.017$
$B^{-}\to D^{*0}(2S)D^{*-}$ | 0.1072$a_{1}^{2}$ | 0.0334 | 0.0346 | 0.0230 |
$B^{-}\to D^{*0}(3S)D^{*-}$ | 0.0305$a_{1}^{2}$ | 0.0095 | 0.0098 | 0.0065 |
$B^{-}\to D^{*0}(1S)D_{s}^{*-}$ | 21.984$a_{1}^{2}$ | 6.8625 | 7.1098 | 4.7316 | $1.71\pm 0.24$
$B^{-}\to D^{*0}(2S)D_{s}^{*-}$ | 2.7074$a_{1}^{2}$ | 0.8451 | 0.8756 | 0.5827 |
$B^{-}\to D^{*0}(3S)D_{s}^{*-}$ | 1.0567$a_{1}^{2}$ | 0.3298 | 0.3417 | 0.2274 |

As discussed earlier, the nonleptonic $B_{F}$-decays to two charmful vector
meson ($V_{1}V_{2}$) states are of special interest as they help to evaluate
$CP$-asymmetry factors, which provide the clue for testing SM predictions and
exploring possible new physics beyond SM. The predicted decay widths (in
$10^{-15}$ GeV) for general values of QCD parameters ($a_{1},a_{2}$) for
$B^{-}$, $\bar{B^{0}}$ and $\bar{B_{s}^{0}}$ decays to two charmful
($1S,2S,3S$) states along with their BFs, obtained in order of magnitude $\sim
10^{-2}-10^{-4}$, are shown in Table 10. However, the predicted BFs for
$B_{c}^{-}$-decays: $B_{c}^{-}\to D^{*-}_{(s)}D^{*0}$ and
$B_{c}^{-}\to\bar{D}^{*0}D^{*-}_{(s)}$, obtained in the order of magnitude
${\cal O}(10^{-6})$, are shown in Table 11 and 12, which cannot be measured in
current experiments. Our predicted BFs for $\bar{B^{0}}$, $B^{-}$, $B_{c}^{-}$
decays to two charmful states, however, are found somewhat underestimated
compared to predictions to Ref. A23  and available experimental data A82 .
The relative size of BFs for nonleptonic decays is broadly estimated from a
power counting of QCD factors: ($a_{1}$, $a_{2}$) in the Wolfenstein
parameterization A90 . Accordingly, class I decays determined by $a_{1}$ are
found to have comparatively large BFs as shown in Table. (7,8,9,10). On the
other hand, class II decay modes, determined by $a_{2}$, are found to have
relatively small BFs Table. (7,8,9,11), as expected, except for decay modes:
$B^{-}\to J/\psi K^{*-}$ and $B^{-}\to\psi(2S)K^{*-}$, characterized by a
product of CKM factors: $V_{bc}V_{cs}$, which have BFs $\sim 0.31\%$ and
$0.12\%$, respectively. These modes should be measured at high luminosity
hadron colliders. In class III decay modes that are characterized by Pauli
interference, the BFs are determined by the relative value of $a_{1}$ with
respect to $a_{2}$. Considering positive values of $a_{1}=1.12$ and negative
value of $a_{2}=-0.26$ in Set 1, for example, which leads to destructive
interference, the decay modes are suppressed compared to the case where
interference is switched off. However, at a qualitative level, where the ratio
$\frac{a_{2}}{a_{1}}$, a function of running coupling constant $\alpha_{S}$
evaluated at the factorization scale, is shown to be positive in the case of
$b$-flavored meson decays corresponding to small coupling A32 . The
experimental data also favor constructive interference of the color-favored
and color-suppressed $b$-flavored meson decay modes. Considering positive
value of $a_{2}^{b}=0.26$, our predicted BFs of class III decay modes:
$B_{c}^{-}\to J/\psi D_{s}^{*-}$ and $B_{c}^{-}\to\psi(2S)D_{s}^{*-}$ find
enhancement by a factor of $\sim 3$ and $\sim 20$, respectively, over that
obtained with $a_{2}=-0.26$. For decay modes to $3S$ states, the enhancement
is still more significant.

Table 11: Decay widths in units of $10^{-15}$ GeV and branching fractions in
$10^{-6}$ for values of Wilson coefficient $a_{1}$ and $a_{2}$.

|  | Branching Fraction |  A23
---|---|---|---
Decay modes | Decay width |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93 |
|  |  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27 |
${B^{-}_{c}}\to D_{s}^{*-}(1S)D^{*0}$ | 9.6099${a_{2}^{2}}$ | 0.5033 | 0.2978 | 0.5428 |  1.6
${B^{-}_{c}}\to D_{s}^{*-}(2S)D^{*0}$ | 2.2700${a_{2}^{2}}$ | 0.1189 | 0.0703 | 0.1282 |
${B^{-}_{c}}\to D_{s}^{*-}(3S)D^{*0}$ | 0.3628${a_{2}^{2}}$ | 0.0190 | 0.0112 | 0.0204 |
${B^{-}_{c}}\to D^{*-}(1S)D^{*0}$ | 0.1440${a_{2}^{2}}$ | 0.0075 | 0.0044 | 0.0081 |  21
${B^{-}_{c}}\to D^{*-}(2S)D^{*0}$ | 0.0664${a_{2}^{2}}$ | 0.0034 | 0.0020 | 0.0037 |
${B^{-}_{c}}\to D^{*-}(3S)D^{*0}$ | 0.0106${a_{2}^{2}}$ | 0.0005 | 0.0003 | 0.0006 |

Table 12:  Decay widths in units of $10^{-15}$ GeV and branching fractions of
order $10^{-6}$ for general values of the Wilson coefficients $a_{1}$ and
$a_{2}$.

|  | Branching Fraction
---|---|---
Decay mode |  Decay width |  $a_{1}$=1.12 |  $a_{1}$=1.14 |  $a_{1}$=0.93
|  |  $a_{2}$=-0.26 |  $a_{2}$=-0.20 |  $a_{2}$=-0.27
${B_{c}}^{-}\to\bar{D}^{*0}(1S)D_{s}^{*-}$ | $(0.0648a_{1}+0.0979a_{2})^{2}$ | 1.72 | 2.28 | 0.887
${B_{c}}^{-}\to\bar{D}^{*0}(2S)D_{s}^{*-}$ | $(0.0438a_{1}+0.1450a_{2})^{2}$ | 0.1007 | 0.3416 | 0.002
${B_{c}}^{-}\to\bar{D}^{*0}(1S)D^{*-}$ | $(0.0120a_{1}+0.0119a_{2})^{2}$ | 0.082 | 0.098 | 0.0489
${B_{c}}^{-}\to\bar{D}^{*0}(2S)D^{*-}$ | $(0.0049a_{1}+0.0212a_{2})^{2}$ | 0.4463$\times 10^{-6}$ | 0.0014 | 0.0010

In the spirit of the experimental data favoring a constructive interference of
the color-favored and color-suppressed $b$-flavored meson decays, the effect
of Pauli interference inducing enhancement of BFs can be further probed by
casting the decay width ($\Gamma$) in the form:
$\Gamma=\Gamma_{0}+\Delta\Gamma$, where
$\Gamma_{0}=x_{1}^{2}a_{1}^{2}+x_{2}^{2}a_{2}^{2}$,
$\Delta\Gamma=2x_{1}x_{2}a_{1}a_{2}$ and then evaluating
$\frac{\Delta\Gamma}{\Gamma_{0}}$ in each case as done in A33 ; A34 ; A35 ;
A36 ; A37 ; A38 ; A39 ; A40 ; A55 . We find that the absolute values of
$\frac{\Delta\Gamma}{\Gamma_{0}}$ for $B_{c}^{-}\to J/\psi
D_{s}^{*-},B_{c}^{-}\to\psi(2S)D_{s}^{*-})$, $B_{c}^{-}\to J/\psi D^{*-}$ and
$B_{c}^{-}\to\psi(2S)D^{*-}$ (in $\%$) are $\sim 49$, $\sim 90$, $\sim 22$,
and $\sim 88$, respectively. For $B_{c}$-decays to two charmful states:
$B_{c}^{-}\to\bar{D}^{*0}D_{s}^{*-}$,
$B_{c}^{-}\to\bar{D}^{*0}(2S)D_{s}^{*-}$, $B_{c}^{-}\to\bar{D}^{*0}D^{*-}$,
$B_{c}^{-}\to\bar{D}^{*0}(2S)D^{*-})$, the enhancement in $\%$ is found to be
$\sim 63$, $\sim 96$, $\sim 44$, and $\sim 100$, respectively. This indicates
that interference is more significant in $B_{c}^{-}$-decays to two charmful
states: $B_{c}^{-}\to\bar{D}^{*0}(1S,2S)D^{*-}_{(s)}$ compared to other decay
modes. This is particularly important since such decay modes have been
proposed A91 ; A92 ; A93 ; A94  for extraction of the CKM angle $\gamma$
through amplitude relation.

Table 13: Predicted longitudinal fraction ($R_{L}$) and $CP$-odd fraction ($R_{\perp}$).  | Longitudinal |
---|---|---
| polarization |  $CP$ -odd
Decay modes | fraction($R_{L}$) |  fraction($R_{\perp}$)
${B^{-}_{c}}\to D^{*0}(1S)D_{s}^{*-}$ | 0.796 | 0.133
${B^{-}_{c}}\to D^{*0}(2S)D_{s}^{*-}$ | 0.688 | 0.189
${B^{-}_{c}}\to D^{*0}(3S)D_{s}^{*-}$ | 0.603 | 0.202
${B^{-}_{c}}\to D^{*0}(1S)D^{*-}$ | 0.797 | 0.156
${B^{-}_{c}}\to D^{*0}(2S)D^{*-}$ | 0.690 | 0.224
${B^{-}_{c}}\to D^{*0}(3S)D^{*-}$ | 0.606 | 0.248
${B^{-}_{c}}\to\bar{D}^{*0}(1S)D_{s}^{*-}$ | 0.786 | 0.148
${B^{-}_{c}}\to\bar{D}^{*0}(2S)D_{s}^{*-}$ | 0.640 | 0.234
${B^{-}_{c}}\to\bar{D}^{*0}(1S)D^{*-}$ | 0.797 | 0.156
${B^{-}_{c}}\to\bar{D}^{*0}(2S)D^{*-}$ | 0.638 | 0.276
Table 14: Predicted longitudinal fraction ($R_{L}$) and $CP$-odd fraction ($R_{\perp}$) of $\bar{B^{0}_{s}}$,$\bar{B^{0}}$,$B^{-}$.  | Longitudinal |
---|---|---
| polarization |  $CP$ -odd
Decay modes | fraction($R_{L}$) |  fraction($R_{\perp}$)
$\bar{B^{0}_{s}}\to D_{s}^{*+}(1S)D^{*-}$ | 0.677 | 0.0627
$\bar{B^{0}_{s}}\to D_{s}^{*+}(2S)D^{*-}$ | 0.524 | 0.0597
$\bar{B^{0}_{s}}\to D_{s}^{*+}(3S)D^{*-}$ | 0.396 | 0.0291
$\bar{B^{0}}\to D^{*+}(1S)D^{*-}$ | 0.675 | 0.0606
$\bar{B^{0}}\to D^{*+}(2S)D^{*-}$ | 0.514 | 0.0571
$\bar{B^{0}}\to D^{*+}(3S)D^{*-}$ | 0.381 | 0.0230
$\bar{B^{0}}\to D^{*+}(1S)D_{s}^{*-}$ | 0.646 | 0.0629
$\bar{B^{0}}\to D^{*+}(2S)D_{s}^{*-}$ | 0.482 | 0.0537
$\bar{B^{0}}\to D^{*+}(3S)D_{s}^{*-}$ | 0.350 | 0.0094
$B^{-}\to D^{*0}(1S)D^{*-}$ | 0.675 | 0.0605
$B^{-}\to D^{*0}(2S)D^{*-}$ | 0.502 | 0.0554
$B^{-}\to D^{*0}(3S)D^{*-}$ | 0.390 | 0.0263
$B^{-}\to D^{*0}(1S)D_{s}^{*-}$ | 0.647 | 0.0629
$B^{-}\to D^{*0}(2S)D_{s}^{*-}$ | 0.470 | 0.0514
$B^{-}\to D^{*0}(3S)D_{s}^{*-}$ | 0.358 | 0.0137

Another area of interest is $CP$-violation in nonleptonic decay of
$b$-flavored mesons to two charmful states. The evaluation of the $CP$-odd
fraction $R_{\perp}$ indeed indicates the degree of $CP$-violation in a decay
process. We predict the longitudinal polarization fraction $R_{L}$ and
$CP$-odd fraction $R_{\perp}$ from their expressions in terms of positive,
negative and longitudinal polarization (10) as:

$\displaystyle R_{L}$ $\displaystyle=$ $\displaystyle\frac{|{\cal
A}_{ll}|^{2}}{[|{\cal A}_{\pm}|^{2}+|{\cal A}_{\mp}|^{2}+|{\cal
A}_{ll}|^{2}]},$ $\displaystyle R_{\perp}$ $\displaystyle=$
$\displaystyle\frac{|{\cal A}_{\pm}-{\cal A}_{\mp}|^{2}}{2[|{\cal
A}_{\pm}|^{2}+|{\cal A}_{\mp}|^{2}+|{\cal A}_{ll}|^{2}]}.$ (34)

Our predicted $R_{L}$ and $R_{\perp}$ for $(B_{c}^{-}\to
D^{*0}(nS)D^{*-}_{(s)})$ and ($\bar{B^{0}}\to D^{*+}(nS)D^{*-}_{(s)}$,
$B^{-}\to D^{*0}(nS)D^{*-}_{(s)}$, $\bar{B_{s}^{0}}\to D^{*+}_{s}(nS)D^{*-}$)
are shown in Table 13 and Table 14, respectively. In all decay modes to two
charmful states considered here, the longitudinal polarization
fractions$(R_{L})$ dominate over the transverse polarization fraction
$(R_{\perp})$. However the $CP$-odd fractions $(R_{\perp})$ in nonleptonic
$B_{c}$-decays to two charmful states are obtained here one order of magnitude
higher than that in such other $b$-flavored meson decays; which indicates that
$CP$-violation is more pronounced in $B_{c}^{-}\to D^{*0}D^{*-}_{(s)}$ decays
compared to that obtained in corresponding decays of ($\bar{B^{0}}$,
$\bar{B_{s}^{0}}$ and $B^{-}$) mesons. It is also found that $R_{\perp}$ in
$B_{c}^{-}$-sector gets enhanced going from decays to ground state to
corresponding decays to higher excited states ($2S$ and $3S$). Apart from
significant $CP$-odd fraction $R_{\perp}$ predicted in $B_{c}^{-}$-decays to
two charmful states, its noticeable enhancement is also predicted for
$B_{c}^{-}$ and $B^{-}$ decays to charmonium states:
$B_{c}^{-}\to\psi(nS)D^{*-}$, $B_{c}^{-}\to\psi(nS)D_{s}^{*-}$,
$B^{-}\to\psi(nS)\rho^{-}$ and $B^{-}\to\psi(nS)K^{*-}$ as shown in Table 15.

For color-favored $B_{c}^{-}\to D^{*}(D^{*-},D_{s}^{*-})$ decays, the effect
arising due to the short-distance non-spectator contribution is shown to be
marginal A95 . However, the long-distance (LD) nonfactorizable contributions
from rescattering effects, final-state interactions, etc., may not be
negligible. If a significant LD effect exists, one expects a large $CP$-odd
fraction in these decays. The predicted longitudinal and transverse helicity
amplitudes and the form factor $g(q^{2})$ yield $R_{\perp}$ values for
different $B_{F}\to V_{1}(nS)V_{2}$ decays, which are shown in Table 15. In
particular, the predicted $R_{\perp}$ values for the transitions with two
charmful final states indicate nonvanishing LD contributions, which lead to
CP-violation in $B_{c}^{-}\to D^{*0}D^{*-}_{(s)}$ and
$\bar{D}^{*0}D^{*-}_{(s)}$ decays.

Table 15: Predicted longitudinal fraction ($R_{L}$) and $CP$-odd fraction ($R_{\perp}$) of state. | Longitudinal |
---|---|---
| polarization |  $CP$ -odd
Decay modes | fraction($R_{L}$) |  fraction($R_{\perp}$)
${B^{-}_{c}}\to J/\psi\rho^{-}$ | 0.932 | 0.018
${B^{-}_{c}}\to\psi(2S)\rho^{-}$ | 0.891 | 0.024
${B^{-}_{c}}\to\psi(3S)\rho^{-}$ | 0.853 | 0.026
${B^{-}_{c}}\to J/\psi K^{*-}$ | 0.911 | 0.023
${B^{-}_{c}}\to\psi(2S)K^{*-}$ | 0.860 | 0.030
${B^{-}_{c}}\to\psi(3S)K^{*-}$ | 0.813 | 0.032
${B^{-}_{c}}\to J/\psi D_{s}^{*-}$ | 0.573 | 0.176
${B^{-}_{c}}\to\psi(2S)D_{s}^{*-}$ | 0.405 | 0.353
${B^{-}_{c}}\to\psi(3S)D_{s}^{*-}$ | 0.336 | 0.137
${B^{-}_{c}}\to J/\psi D^{*-}$ | 0.568 | 0.152
${B^{-}_{c}}\to\psi(2S)D^{*-}$ | 0.379 | 0.246
${B^{-}_{c}}\to\psi(3S)D^{*-}$ | 0.447 | 0.415
${B^{-}}\to J/\psi\rho^{-}$ | 0.757 | 0.159
${B^{-}}\to\psi(2S)\rho^{-}$ | 0.565 | 0.276
${B^{-}}\to\psi(3S)\rho^{-}$ | 0.421 | 0.345
${B^{-}}\to J/\psi K^{*-}$ | 0.728 | 0.141
${B^{-}}\to\psi(2S)K^{*-}$ | 0.549 | 0.200
${B^{-}}\to\psi(3S)K^{*-}$ | 0.426 | 0.201
${B^{-}}\to D^{*0}(1S)\rho^{-}$ | 0.945 | 0.013
${B^{-}}\to D^{*0}(2S)\rho^{-}$ | 0.895 | 0.021
${B^{-}}\to D^{*0}(3S)\rho^{-}$ | 0.849 | 0.024
${B^{-}}\to D^{*0}(1S)K^{*-}$ | 0.928 | 0.017
${B^{-}}\to D^{*0}(2S)K^{*-}$ | 0.865 | 0.027
${B^{-}}\to D^{*0}(3S)K^{*-}$ | 0.807 | 0.030
$\bar{B^{0}}\to D^{*+}(1S)\rho^{-}$ | 0.945 | 0.013
$\bar{B^{0}}\to D^{*+}(2S)\rho^{-}$ | 0.899 | 0.020
$\bar{B^{0}}\to D^{*+}(3S)\rho^{-}$ | 0.845 | 0.025
$\bar{B^{0}}\to D^{*+}(1S)K^{*-}$ | 0.928 | 0.017
$\bar{B^{0}}\to D^{*+}(2S)K^{*-}$ | 0.870 | 0.026
$\bar{B^{0}}\to D^{*+}(3S)K^{*-}$ | 0.802 | 0.031
$\bar{B^{0}_{s}}\to D_{s}^{+*}(1S)\rho^{-}$ | 0.945 | 0.013
$\bar{B^{0}_{s}}\to D_{s}^{+*}(2S)\rho^{-}$ | 0.902 | 0.021
$\bar{B^{0}_{s}}\to D_{s}^{+*}(3S)\rho^{-}$ | 0.852 | 0.025
$\bar{B^{0}_{s}}\to D_{s}^{+*}(1S)K^{*-}$ | 0.928 | 0.017
$\bar{B^{0}_{s}}\to D_{s}^{+*}(2S)K^{*-}$ | 0.873 | 0.026
$\bar{B^{0}_{s}}\to D_{s}^{+*}(3S)K^{*-}$ | 0.811 | 0.031

##  5 Summary and conclusion

In this work, we study the exclusive two-body nonleptonic decays of
$b$-flavored ($\bar{B^{0}}$, $\bar{B_{s}^{0}}$, $B^{-}$ and $B_{c}^{-}$)
mesons to $S$-wave charmonium and charm meson $1^{-}$ states, in the framework
of relativistic independent quark (RIQ) model. The weak decay form factors
representing decay amplitude and their $q^{2}$-dependence are extracted from
the overlapping integrals of the meson wave functions obtainable in the RIQ
model. The predicted branching fractions for different decay modes are
obtained in a wide range, from ${\cal O}(10^{-6})$ for $B_{c}^{-}$ decays to
two charmful states to as high as $\sim 1.54\%$, $\sim 1.42\%$ and $\sim
1.17\%$ for $B^{-}\to D^{*0}\rho^{-}$, $\bar{B^{0}}\to D^{*+}\rho^{-}$ and
$\bar{B_{s}^{0}}\to D_{s}^{*+}\rho^{-}$, respectively. Our results are in
general agreement with the available experimental data and other SM
predictions. The decay modes with predicted branching fractions in the order:
${\cal O}(10^{-2})$, which include the $B_{c}$-meson decays to $1S$ charmonium
and charm meson states as well as $\bar{B_{s}^{0}}$, $\bar{B^{0}}$, $B^{-}$
decays to two charmful mesons in their ground state, should be experimentally
accessible. The decay modes to $2S$ and $3S$ charmonium states such as
$B_{c}^{-}\to\psi(2S)\rho^{-},\psi(2S)D_{s}^{*-}$, $\bar{B_{s}^{0}}\to
D_{s}^{*+}(2S)\rho^{-}$, $\bar{B^{0}}\to D^{*+}(2S)\rho^{-}$, $B^{-}\to
D^{*0}(2S)\rho^{-}$, $B^{-}\to\psi(2S)K^{*-}$, $B^{-}\to\psi(2S)\rho^{-}$ and
$\bar{B_{s}^{0}}\to D_{s}^{*+}(3S)\rho^{-}$, $\bar{B^{0}}\to
D^{*+}(3S)\rho^{-}$, $B^{-}\to D^{*0}(3S)\rho^{-}$, $B^{-}\to\psi(3S)K^{*-}$,
$B^{-}\to\psi(3S)\rho^{-}$ with predicted branching fractions upto $\sim
10^{-4}$ may be accesible at high luminosity hadron colliders in near future.
Other decay modes and especially $B_{c}$-decay to two charmful states with
predicted branching factions in the order ${\cal O}(10^{-6})$ can not reach
the detection ability of the current experiments. As expected, our predicted
branching fractions are obtained in the hierarchy:

$\displaystyle{\cal B}(B_{F}\to V_{1}(3S)V_{2})$ $\displaystyle<{\cal
B}(B_{F}\to V_{1}(2S)V_{2})$ $\displaystyle<{\cal B}(B_{F}\to
V_{1}(1S)V_{2}).$

This is due to i) the nodal structure of the participating daughter mesons in
their excited states, ii) tighter phase space and iii) typical
$q^{2}$-dependence of the weak decay form factors for decay modes to higher
excited ($2S$ and $3S$) states in comparison to that for the corresponding
decay modes to the corresponding ground $(1S)$ state.

The relative size of branching fractions is broadly estimated from a power
counting of QCD factors: ($a_{1},a_{2}$) in the Wolfenstein parametrization.
The class I decay modes characterized by $a_{1}$ are found to have large
branching fractions, as expected; compared to those obtained for class II
decays which are determined by $a_{2}$. The branching fractions of class III
decays characterized by Pauli interference for $B_{c}$-decays to two charmful
states in particular, obtained in the order of magnitude ${\cal O}(10^{-6})$
can not be measured in current experiments.

In view of experimental data favoring a constructive interference of the
color-favored and color-suppressed $b$\-
flavored meson decays, the effect of Pauli interference is studied in
different decay modes, by evaluating the enhancement factor in each such decay
mode. For ${B^{-}_{c}}\to\bar{D}^{*0}D_{s}^{*-}$,
${B^{-}_{c}}\to\bar{D}^{*0}(2S)D_{s}^{*-}$, ${B^{-}_{c}}\to\bar{D}^{*0}D^{*-}$
and ${B^{-}_{c}}\to\bar{D}^{*0}(2S)D^{*-}$, the enhancement (in $\%$) is found
to be $\sim 63$, $\sim 96$, $\sim 44$ and $\sim 100$, respectively. This shows
that the Pauli interference is more significant in $B_{c}^{-}$-decays to two
charmful states: ${B^{-}_{c}}\to\bar{D}^{*0}(1S,2S)D_{(s)}^{*-}$ compared to
other decay modes. This is particularly important since such decay modes have
been proposed for extracting the CKM angle $\gamma$ through amplitude
relations.

We predict the longitudinal polarization fraction ($R_{L}$) and transverse
polarization ($CP$-odd) fraction ($R_{\perp}$). We find that predicted $R_{L}$
dominates in all decay modes to two charmful states; considered in the present
study. The $CP$-odd fraction ($R_{\perp}$) in nonleptonic $B_{c}^{-}$-decays
to two charmful states are obtained one order magnitude higher than that in
such other $b$-flavored meson decays; which indicates the $CP$\- violation is
more significant in $B_{c}^{-}\to D^{*0}D_{(s)}^{*-}$ compared to that
obtained in ($\bar{B^{0}}$, $\bar{B_{s}^{0}}$ and $B^{-}$)-meson decays. For
color-favored $B_{c}^{-}\to D^{*}(D^{*-},D_{s}^{*-})$ decays, the effect
arising due to the short-distance non-spectator contribution is shown to be
marginal. However, the long-distance (LD) nonfactorizable contributions from
rescattering effects, final-state interactions, etc., may not be negligible.
If a significant LD effect exists, one expects a large $CP$-odd fraction in
these decays. The predicted longitudinal and transverse helicity amplitudes
and the form factor $g(q^{2})$ yield $R_{\perp}$ values for different
$B_{F}\to V_{1}(nS)V_{2}$ decays. In particular, the predicted $R_{\perp}$
values for the transitions with two charmful final states indicate
nonvanishing LD contributions, which lead to a significant CP-violation in
$B_{c}^{-}\to D^{*0}D^{*-}_{(s)}$ and $\bar{D}^{*0}D^{*-}_{(s)}$ decays.

In conclusion, the present analysis shows that the factorization approximation
works reasonably well in describing the exclusive nonleptonic $B_{F}\to
V_{1}(nS)V_{2}$ decays in the framework of the RIQ model.

##  Appendix A Quark orbitals and wave packet representation of the meson
state in RIQ model framework

In the RIQ model, a meson is picturized as a color-singlet assembly of a quark
and an antiquark independently confined by an effective and average flavor-
independent potential in the form:
$U(r)=\frac{1}{2}(1+\gamma^{0})(ar^{2}+V_{0})$, where ($a$, $V_{0}$) are the
potential parameters. It is believed that the zeroth-order quark dynamics
generated by the phenomenological confining potential $U(r)$ taken in equally
mixed scalar-vector harmonic form can provide an adequate tree-level
description of the decay process being analyzed in this work. With the
interaction potential $U(r)$ put into the zeroth-order quark lagrangian
density, the ensuing Dirac equation admits a static solution of positive and
negative energy as:

$\displaystyle\psi^{(+)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{ig_{\xi}(r)}{r}\\\
\frac{{\vec{\sigma}}.{\hat{r}}f_{\xi}(r)}{r}\end{array}\;\right){{\chi}_{l,j,m_{j}}}(\hat{r}),$
(37) $\displaystyle\psi^{(-)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{i({\vec{\sigma}}.{\hat{r}})f_{\xi}(r)}{r}\\\
\frac{g_{\xi}(r)}{r}\end{array}\;\right)\
{\tilde{\chi}}_{l,j,m_{j}}(\hat{r}),$ (40)

where, $\xi=(nlj)$ represents a set of Dirac quantum numbers specifying the
eigenmodes. $\chi_{ljm_{j}}(\hat{r})$ and ${\tilde{\chi}}_{ljm_{j}}(\hat{r})$
are the spin angular parts given by,

$\displaystyle\chi_{ljm_{j}}(\hat{r})$ $\displaystyle=$
$\displaystyle\sum_{m_{l},m_{s}}<lm_{l}\;{1\over{2}}m_{s}|jm_{j}>Y_{l}^{m_{l}}(\hat{r})\chi^{m_{s}}_{\frac{1}{2}},$
$\displaystyle{\tilde{\chi}}_{ljm_{j}}(\hat{r})$ $\displaystyle=$
$\displaystyle(-1)^{j+m_{j}-l}\chi_{lj-m_{j}}(\hat{r}).$ (41)

With the quark binding energy $E_{q}$ and quark mass $m_{q}$ written in the
form $E_{q}^{\prime}=(E_{q}-V_{0}/2)$, $m_{q}^{\prime}=(m_{q}+V_{0}/2)$ and
$\omega_{q}=E_{q}^{\prime}+m_{q}^{\prime}$, one can obtain solutions to the
resulting radial equation for $g_{\xi}(r)$ and $f_{\xi}(r)$ in the form:

$\displaystyle g_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}\Big{(}\frac{r}{r_{nl}}\Big{)}^{l+1}\exp(-r^{2}/2r^{2}_{nl})L_{n-1}^{l+1/2}(r^{2}/r^{2}_{nl}),$
$\displaystyle f_{nl}$ $\displaystyle=$
$\displaystyle\frac{N_{nl}}{r_{nl}\omega_{q}}\Big{(}\frac{r}{r_{nl}}\Big{)}^{l}\exp(-r^{2}/2r^{2}_{nl})$
(42) $\displaystyle\times$
$\displaystyle\left[\Big{(}n+l-\frac{1}{2}\Big{)}L_{n-1}^{l-1/2}(r^{2}/r^{2}_{nl})+nL_{n}^{l-1/2}(r^{2}/r^{2}_{nl})\right],$

where, $r_{nl}=(a\omega_{q})^{-1/4}$ is a state independent length parameter,
$N_{nl}$ is an overall normalization constant given by

$N^{2}_{nl}=\frac{4\Gamma(n)}{\Gamma(n+l+1/2)}\frac{(\omega_{nl}/r_{nl})}{(3E_{q}^{\prime}+m_{q}^{\prime})},$
(43)

and $L_{n-1}^{l+1/2}(r^{2}/r_{nl}^{2})$ etc. are associated Laguerre
polynomials. The radial solutions yield an independent quark bound-state
condition in the form of a cubic equation:

$\sqrt{(\omega_{q}/a)}(E_{q}^{\prime}-m_{q}^{\prime})=(4n+2l-1).$ (44)

From the solution of the cubic equation (A.5), the zeroth-order binding
energies of the confined quark and antiquark are obtained for all possible
eigenmodes.

In the relativistic independent particle picture of this model, the
relativistic constituent quark and antiquark are thought to move independently
inside the meson bound-state $|B_{F}(\vec{p},S_{B_{F}})\rangle$ with their
momentum $\vec{p}_{b}$ and ${\vec{p}}_{\bar{q}}$, respectively. In order to
study the decay process which takes place in the momentum eigenstates of
participating mesons, we Fourier transform the quark orbitals (A.1) to
momentum space and obtain the momentum probability amplitude of the quark and
antiquark of participating mesons in the following forms:
For ground state mesons:($n=1$,$l=0$),

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}\omega_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}(E_{p_{b}}+E_{b})$
$\displaystyle\times\exp{\Big{(}-{{\vec{p_{b}}}^{2}\over{4\alpha_{b}}}\Big{)}},$
$\displaystyle{\tilde{G}}_{\bar{q}}(\vec{p}_{\bar{q}})$
$\displaystyle=-{{i\pi{\cal
N}_{\bar{q}}}\over{2\alpha_{\bar{q}}\omega_{\bar{q}}}}\sqrt{{(E_{p_{\bar{q}}}+m_{\bar{q}})}\over{E_{p_{\bar{q}}}}}(E_{p_{\bar{q}}}+E_{\bar{q}})$
(45)
$\displaystyle\times\exp{\Big{(}-{{\vec{p}}_{\bar{q}}^{2}\over{4\alpha_{\bar{q}}}}\Big{)}}.$

For the excited meson state:($n=2$,$l=0$),

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}\omega_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}(E_{p_{b}}+E_{b})$
$\displaystyle\times\Big{(}\frac{\vec{p_{b}}^{2}}{2\alpha_{b}}-\frac{3}{2}\Big{)}\exp{\Big{(}-{{\vec{p_{b}}}^{2}\over{4\alpha_{b}}}\Big{)}},$
$\displaystyle{\tilde{G}}_{\bar{q}}(\vec{p}_{\bar{q}})$
$\displaystyle=-{{i\pi{\cal
N}_{\bar{q}}}\over{2\alpha_{\bar{q}}\omega_{\bar{q}}}}\sqrt{{(E_{p_{\bar{q}}}+m_{\bar{q}})}\over{E_{p_{\bar{q}}}}}(E_{p_{\bar{q}}}+E_{\bar{q}})$
(46)
$\displaystyle\times\Big{(}\frac{\vec{p}_{\bar{q}}^{2}}{2\alpha_{\bar{q}}}-\frac{3}{2}\Big{)}\exp{\Big{(}-{{\vec{p}}_{\bar{q}}^{2}\over{4\alpha_{\bar{q}}}}\Big{)}}.$

For the excited meson state:($n=3$,$l=0$),

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}\omega_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}(E_{p_{b}}+E_{b})$
$\displaystyle\times\Big{(}\frac{{\vec{p}_{b}}^{4}}{8\alpha_{b}^{2}}-\frac{5\vec{p_{b}}^{2}}{4\alpha_{b}}+\frac{15}{8}\Big{)}\exp{\Big{(}-{{\vec{p_{b}}}^{2}\over{4\alpha_{b}}}\Big{)}},$
$\displaystyle{\tilde{G}}_{\bar{q}}(\vec{p}_{\bar{q}})$
$\displaystyle=-{{i\pi{\cal
N}_{\bar{q}}}\over{2\alpha_{\bar{q}}\omega_{\bar{q}}}}\sqrt{{(E_{p_{\bar{q}}}+m_{\bar{q}})}\over{E_{p_{\bar{q}}}}}(E_{p_{\bar{q}}}+E_{\bar{q}})$
(47)
$\displaystyle\times\Big{(}\frac{{\vec{p}_{\bar{q}}}^{4}}{8\alpha_{\bar{q}}^{2}}-\frac{5\vec{p}_{\bar{q}}^{2}}{4\alpha_{\bar{q}}}+\frac{15}{8}\Big{)}\exp{(-{{\vec{p}}_{\bar{q}}^{2}\over{4\alpha_{\bar{q}}}})}.$

With the momentum probability amplitudes of quark constituents, we construct
an effective momentum distribution function for the meson state in the form
${\cal G}_{B_{F}}(\vec{p}_{b},\ \vec{p}_{\bar{q}})=\\\
\sqrt{G_{b}(\vec{p}_{b})G_{\bar{q}}(\vec{p}_{\bar{q}})}$ in the light of an
ansatz of Margolis and Mendel in their bag model description of the meson
bound state A96 .

Here the effective momentum distribution function ${\cal G}_{B_{F}}\\\
(\vec{p}_{b},\ \vec{p}_{\bar{q}})$, which in fact, embodies the bound-state
characteristics of $|B_{F}(\vec{p},S_{B_{F}})\rangle$, defines the meson bound
state at definite momentum $\vec{p}$ and spin projection $S_{B_{F}}$ in the
form:

$\displaystyle|B_{F}(\vec{p},S_{B_{F}})\rangle$
$\displaystyle={\hat{\Lambda}}(\vec{p},S_{B_{F}})|(\vec{p}_{b},\lambda_{b});(\vec{p}_{\bar{q}},\lambda_{\bar{q}})\rangle$
(48)
$\displaystyle={\hat{\Lambda}}(\vec{p},S_{B_{F}})\hat{b}^{\dagger}_{b}(\vec{p}_{b},\lambda_{b})\hat{\tilde{b}}^{\dagger}_{\bar{q}}(\vec{p}_{\bar{q}},\lambda_{\bar{q}})|0\rangle,$

where
$|(\vec{p}_{b},\lambda_{b});(\vec{p}_{\bar{q}},\lambda_{\bar{q}})\rangle$ is
the Fock-space representation of unbound quark and antiquark in a color-
singlet configuration with respective momentum and spin:
$(\vec{p}_{b},\lambda_{b})$ and $(\vec{p}_{\bar{q}},\lambda_{\bar{q}})$.
$\hat{b}^{\dagger}_{b}(\vec{p}_{b},\lambda_{b})$ and
$\hat{\tilde{b}}^{\dagger}_{\bar{q}}(\vec{p}_{\bar{q}},\lambda_{\bar{q}})$ are
the quark and antiquark creation operators and
${\hat{\Lambda}}(\vec{p},S_{B_{F}})$ is a bag-like integral operator taken in
the form:

$\displaystyle{\hat{\Lambda}}(\vec{p},S_{B_{F}})$
$\displaystyle=\frac{\sqrt{3}}{\sqrt{N_{B_{F}}(\vec{p})}}\sum_{\lambda_{b},\lambda_{\bar{q}}}\zeta_{b,\bar{q}}^{B_{F}}\int
d^{3}\vec{p}_{b}\ d^{3}\vec{p}_{\bar{q}}$ (49)
$\displaystyle\delta^{(3)}(\vec{p}_{b}+\vec{p}_{\bar{q}}-\vec{p})\ {\cal
G}_{B_{F}}(\vec{p}_{b},\vec{p}_{\bar{q}}).$

Here, $\sqrt{3}$ is the effective color factor and $\zeta_{b,\bar{q}}^{B_{F}}$
is the $SU(6)$ spin-flavor coefficients for the $B_{F}$-meson state. Imposing
the normalization condition in the form $\langle
B_{F}(\vec{p})|B_{F}(\vec{p}^{{}^{\prime}})\rangle=\\\
\delta^{(3)}{(\vec{p}-\vec{p}^{{}^{\prime}})}$, the meson state normalization
$N_{B_{F}}(\vec{p})$ is obtainable in an integral form

$N_{B_{F}}(\vec{p})=\int d^{3}\vec{p}_{b}\ |{\cal G}_{B_{F}}(\vec{p}_{b},\
\vec{p}_{\bar{q}})|^{2}.$ (50)

###### Acknowledgements.

The library and computational facilities provided by authorities of Siksha ’O’
Anusandhan Deemed to be University, Bhubaneswar, 751030, India are duly
acknowledged.

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