# Exclusive semileptonic $B_{c}$-meson decays to radially excited charmonium
and charm meson states.

Lopamudra Nayak1111email<EMAIL_ADDRESS>P. C. Dash1, Susmita
Kar2, N. Barik3 1 Department of Physics, Siksha ${}^{\prime}O^{\prime}$
Anusandhan (Deemed to be University), Bhubaneswar-751030, India
2 Department of Physics, Maharaja Sriram Chandra Bhanja Deo University,
Baripada-757003, India
3 Department of Physics, Utkal University, Bhubaneswar-751004, India.

###### Abstract

In this work, we study exclusive semileptonic $B_{c}$-meson decays to radially
excited charmonium and charm meson states in the framework of the relativistic
independent quark(RIQ) model based on an average flavor independent confining
potential in equally mixed scalar-vector harmonic form. We calculate the
invariant form factors representing decay amplitudes from the overlapping
integrals of meson wave functions derivable in the RIQ model. To evaluate the
lepton mass effects in the semileptonic decays, we study the physical
quantities of interest for the decay processes in their $e^{-}$ and $\tau^{-}$
mode separately. We predict the decay rates/ branching fractions, forward-
backward asymmetry, and the asymmetry parameter in reasonable agreement with
other model predictions, which can hopefully be tested in future experiments
at the Tevatron and LHC. We find that the decay channels
$B_{c}\to\eta_{c}/\psi(nS)$ have branching ratios $\sim(10^{-2}-10^{-4})$ and
that for $B_{c}\to D^{*}(nS)$ in their $e^{-}$ mode is $10^{-4}$, which are
within the detection accuracy of current experiments. Our predictions on the
observable ${\cal R}$, which corresponds to the ratio of branching fractions,
for $B_{c}$ decays to radially excited charmonium states are comparable to
other standard models (SM) predictions, which highlight the failure of the
lepton flavor universality hinting at new physics beyond SM. In the absence of
predicted data from established model approaches in the literature for $B_{c}$
decays to radially excited charm meson states, our predictions can be useful
to identify the $B_{c}$-channels characterized by clear experimental
signature.

##  I Introduction

Ever since the discovery of $B_{c}$ meson two decades ago at Fermilab by the
CDF Collaboration [1], with its mass $m_{B_{c}}=6.40\pm 0.39\pm 0.13$ GeV and
the lifetime $\tau_{B_{c}}=0.46\pm^{0.18}_{0.16}\pm 0.003\ ps$, several
experimental studies have been carried out in $B_{c}$ sector. Later on, the
CDF Collaboration [2, 3] gave a more precise and upgraded value of mass
$m_{B_{c}}=6275.6\pm 2.9(Stat.)\pm 5(Syst.)\ MeV/c^{2}$ which was confirmed
subsequently by the D0 Collaboration at Tevatron [4]. The LHCb Collaboration
also reported several new observations on $B_{c}$-decays [5] and a more
precise measurement of the $B_{c}$ lifetime using semileptonic $B_{c}\to
J/\psi\mu\nu_{\mu}X$ decays [6], where X denotes any possible additional
particle in the final state. The current Run II at Tevatron [7, 8], upcoming
Run III at the CERN LHC, and the $e^{+}e^{-}$ experiment at BELLE-II are
designed to boost heavy flavor physics measurement scenario. The dedicated
detectors at $B\ TeV$ and LHCb, designed to enhance the event accumulation
rates, are expected to provide high statistics at the rate $\sim 10^{10}$
$B_{c}$-events per year. Therefore, experimental as well as theoretical
studies on $B_{c}$ meson and its various decay channels will be of great
interest in many aspects.
Among the $b$-flavored mesons, the $B_{c}$ meson has attracted a great deal of
attention due to its outstanding features. (1) It is the only lowest bound-
state of two heavy constituent quarks, a bottom quark (antiquark) and a charm
antiquark (quark) of different flavors and charges. The other heavy quark in
the Standard Model, the top quark, cannot form a hadron because it’s too short
a lifetime to be hadronized. (2) As far as the bound-state characteristics are
concerned, $B_{c}$ meson is quick similar to the $J^{PC}=0^{-+}$ states
$\eta_{c}$ and $\eta_{b}$ in the charmonium ($c\bar{c}$\- bound state) and the
bottomonium ($b\bar{b}$\- bound state) sector. However, the $\eta_{c}$ and
$\eta_{b}$ with hidden (implicit) flavors decay strongly and
electromagnetically whereas the $B_{c}$ meson with open (explicit) flavors
decays weakly since it lies below the $B\bar{D}$ threshold. (3) Although
$B_{c}$ meson lies intermediate in mass and size between the charmonium and
bottomonium family, where the heavy quark interactions are understood rather
well, many aspects in the $B_{c}$-sector remain obscure. (4) Both the
constituents of $B_{c}$ being heavy, they can decay individually yielding very
rich weak $B_{c}$-decay channels with sizeable branching fractions and a long
$B_{c}$ lifetime. The investigation of $B_{c}$-decay modes, therefore, can be
used in determining the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements:
$V_{cb},\ V_{ub},\ V_{cs},\ V_{cd}$, leptonic decay constant, and the origin
of $CP$ and $T$ violation. All these features make $B_{c}$ meson an ideal
system for studying the heavy quark dynamics, and weak decay phenomenology,
thus offering a fascinating area for future research.
One of the most engrossing puzzles in flavor physics in recent years has been
the observed deviations of the semileptonic $B_{c}$-decay observables from the
corresponding SM predictions. Recently the LHCb Collaboration has reported
their measurements of observables ${\cal R}_{J/\psi}$: the ratio of
semileptonic branching fractions [9] as

${\cal R}_{J/\psi}=\frac{{\cal B}(B_{c}\to J/\psi\tau\nu_{\tau})}{{\cal
B}(B_{c}\to J/\psi\mu\nu_{\mu})}=0.71\pm 0.17(stat)\pm 0.18(syst),$ (1)

which lies within $2\sigma$ above the range of existing SM predictions [10,
11, 12, 13, 14]. A series of measurements of identical decay modes $B\to
D^{*}l\nu_{l}$ by BaBar [15, 16], BELLE [17, 18, 19], and LHCb [20, 21] also
hint at the deviation of observables from the corresponding SM predictions.
The average of such measurements of ${\cal R}_{D}$ and ${\cal R}_{D^{*}}$
performed by the HFLAV Collaboration [22] are

$\displaystyle{\cal R}_{D}=\frac{{\cal B}(B\to D\tau\nu_{\tau})}{{\cal B}(B\to
D\mu\nu_{\mu})}=0.407\pm 0.39\pm 0.024$ $\displaystyle{\cal R}_{D}=\frac{{\cal
B}(B\to D^{*}\tau\nu_{\tau})}{{\cal B}(B\to D^{*}\mu\nu_{\mu})}=0.304\pm
0.013\pm 0.007,$

which mark significant deviations from the corresponding SM predictions,
${\cal R}_{D}\simeq 0.299\pm 0.011$ from lattice calculation[23, 24] and
${\cal R}_{D^{*}}\simeq 0.252\pm 0.003$ from model calculation [25],
respectively. The deviation of observables ${\cal R}$ over SM predictions
besides shedding light on the unresolved puzzle in the $b$-flavored sector
hints at the failure of the lepton flavor universality (LFU) invoking new
physics (NP) beyond SM via enhanced weak couplings to third-generation of
charged leptons and quarks in electroweak interactions.
Though many experimental studies on $B_{c}$-decays are cited in the
literature, most of them pertain to the decays to final mesons in their ground
state only. Now a days radially excited $\psi(2S)$ and $\eta_{c}(2S)$ states
in the heavy flavored charmonium sector and many charmonium-like states
including $D^{0}(2S)$ and $D^{*0}(2S)$ states have been detected. The
detection of the radially excited charmonium and charm meson states and the
measured observables in the semileptonic $B_{c}$-decays are of special
interest as it is easier to identify them in the experiment.
Many theoretical studies on semileptonic $B_{c}$-decays to charmonium and
charm meson states have been cited in the literature under different
theoretical approaches. A few noteworthy among them are the potential
model[26], non-relativistic quark model (NRQM) [13], the Baur-Stech-Wirbel
(BSW) approach [27], QCD sum rules (QCDSR) [11, 28, 29, 30, 31, 32], non-
relativistic QCD (NRQCD) [33, 34, 35, 36, 37], the Bethe-Salpeter Equation[38,
39], Bethe-Salpeter relativistic quark model [40], improved Bethe-Salpeter
method [41], relativistic quark model (RQM) [12, 42, 43, 44, 45], relativistic
constituent quark model (RQCM) on light front [46, 47], Covariant confined
quark model (CCQM)[48, 49, 50, 51], Covariant light-front quark model
(CLQM)[52], QCD potential model (QCDPM) [53, 54, 55], perturbative QCD (pQCD)
[56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66], Lattice QCD [67], Light-cone QCD
sum rules [68, 69], and the light-front quark model (LFQM) [70]. The detection
of the semileptonic $B_{c}$-decay modes to radially excited charmonium and
charm meson states, the deviation of observables ${\cal R}$ from the SM
predictions, and predictions of branching fractions by different model
approaches provide necessary motivation to study these decay modes in the RIQ
model framework.
The RIQ model, developed by our group and applied extensively in the
description of wide-ranging hadronic phenomena including the static properties
of hadrons [71] and several decay properties [72] such as the radiative; weak
radiative, rare radiative; leptonic, weak leptonic, radiative leptonic; and
non-leptonic decays, has also been applied in predicting the semileptonic
decays [73, 74, 75, 76] of heavy flavored mesons. The participating mesons in
the description of hadronic phenomena [71, 72, 73, 74, 75, 76] are considered
in their ground state only.
It may be noted here that our analysis based on the vanishing lepton mass
limit may be considered reasonable in the description of semileptonic decay
modes [73, 74, 75, 76], where only the 3-vector (or space component) hadronic
current form factors contribute to the decay amplitude. The scalar-(or time
component) hadronic current form factor is not accessible in such a
description. However, such an approximation is not applicable to describe the
semitaunic decay modes, where both the space and time component hadronic
current form factors contribute to the decay amplitudes. In our recent
analysis of exclusive semileptonic $B_{c}$ meson decays to the charmonium and
charm meson in their ground states, we predict the lepton mass effect by
evaluating the $B_{c}$ decays into both the electronic and taunic modes[77].
Inspired by the detection of several semileptonic $B_{c}$ decay modes to
$S$-wave charmonium and charmonium-like meson states, our predictions on
magnetic dipole and electromagnetic transitions of $B_{c}$\- and $B_{c}^{*}$
in their ground as well as radially excited ($2S$ and $3S$) states[78],
followed by our recent prediction on exclusive nonleptonic $B_{c}$ meson
decays to radially excited charmonium and charmonium like states[79], we would
like to extend the applicability of the RIQ model to investigate the lepton
mass effects in the exclusive semileptonic $B_{c}$-meson decays to the
charmonium and charm mesons in their radially excited $(nS)$ states, where
$n=2,3$. Our purpose here is to predict the branching fractions, related
observables ${\cal R}$ in comparison to those obtained in our previous
analysis[77] and thus provide a ready reference to the existing and
forthcoming experiments in this sector. We ignore the decay channels involving
higher $4S$ final states since their properties are still not understood well.
At the quark level, the semileptonic decay modes:
$B_{c}\to\eta_{c}(J/\psi)l\nu_{l}$ and $B_{C}\to D(D^{*})l\nu_{l}$ are induced
by $b\to cl\nu_{l}$ and $b\to ul\nu_{l}$ transitions, respectively, which
appear at the tree level in the SM. The kinematic range of $q^{2}$ for
$B_{c}\to\eta_{c}(J/\psi)l\nu_{l}$ is $0\leq q^{2}\leq 10\ GeV^{2}$, and that
for $B_{c}\to D(D^{*})l\nu_{l}$ is $0\leq q^{2}\leq 18\ GeV^{2}$. In the
$B_{c}$-meson rest frame, the maximum recoil, momenta of the final state
charmonium ($\eta_{c},\ J/\psi$) and charm ($D,\ D^{*}$) meson states are
estimated to be in the same range of magnitude as their masses. With this
kinematic constraint, it is interesting to analyze here the decay modes
$B_{c}\to\eta_{c}(J/\psi)l\nu_{l}$ and $B_{c}\to D(D^{*})l\nu_{l}$ separately.
Due to their simple theoretical description via a tree-level diagram in the
SM, the effects of the strong interaction can be separated from the effects of
the weak interaction into a set of Lorentz-invariant form factors. The study
of semileptonic decays, therefore, reduces to a calculation of relevant weak
form factors in a suitable phenomenological model framework. We would like to
evaluate the transition form factors parameterizing the decay amplitudes from
the overlapping integrals of the meson wave functions obtainable in the RIQ
model framework.

It is worthwhile to note here following few points: (1) In some of the
theoretical approaches like the pQCD and QCD sum rules, for example, the form
factors are determined first, with an endpoint normalization at minimum
$q^{2}$ (maximum recoil point) or maximum $q^{2}$ (minimum recoil point) and
then extrapolated to the entire physical region using some monopole/ dipole
/Gaussian ansatz which makes the form factor estimation less reliable. In our
RIQ model, the relevant form factors are evaluated in the full kinematic range
of momentum transfer squared $q^{2}$, which makes the predictions of physical
observables more accurate. (2) Since the decay modes considered here involve
both the light and heavy leptons, we would evaluate both the vector- and
scalar components of hadronic form factors relevant for possible semileptonic
$B_{c}$-meson decay modes to radially excited charmonium and charm meson
states. We intend to study the lepton mass effects in such decay modes and
predict the observables ${\cal R}$ in comparison with other SM predictions.
(3) We shall update some input hadronic parameters to be considered in our
calculation according to the Particle Data Group [80].

The outline of the paper is as follows. In Sec. II we briefly discuss the
general formalism and kinematics for semileptonic $B_{c}$-meson decays.
Section III describes the extraction of the transition form factors from the
invariant transition amplitudes in the framework of the RIQ model. Section IV
is devoted to the numerical analysis of the form factors, decay rates,
branching fractions, ratios of branching fractions, and a comparison of our
results with the results of other theoretical approaches. In Sec. V we briefly
summarize our results. In the Appendix, we briefly describe the RIQ model
framework, phenomenological interaction potential used, and the wave packet
representation of the participating meson states.

##  II General formalism and kinematics

The general formalism and kinematics for exclusive semileptonic
$B_{c}$-decays: $B_{c}\to\eta_{c}/J\psi l\nu_{l}$ and $B_{c}\to
D/D^{*}l\nu_{l}$ are described elaborately in our recent study [77], from
which we quote here only a few important expressions related to leptonic and
hadronic parts of the invariant decay amplitude:

${\cal M}(p,k,k_{l},k_{\nu})={\frac{\cal
G_{F}}{\sqrt{2}}}V_{bq^{{}^{\prime}}}{\cal H}_{\mu}(p,k){\cal
L}^{\mu}(k_{l},k_{\nu})$ (2)

where ${\cal L}^{\mu}$ and ${\cal H}_{\mu}$ are leptonic and hadronic parts,
respectively:

$\displaystyle{\cal L}^{\mu}(k_{l},k_{\nu})=$
$\displaystyle\bar{u}(\vec{k_{l}})\gamma^{\mu}(1-\gamma_{5})v(\vec{k_{\nu}})$
$\displaystyle{\cal H}_{\mu}(p,k)=$ $\displaystyle\langle
X(\vec{k},S_{X})|{J^{h}_{\mu}(0)}|B_{c}(\vec{p},S_{B_{c}})\rangle$ (3)

Here $J^{h}_{\mu}=V_{\mu}-A_{\mu}$ is the vector-axial vector current;
$q^{{}^{\prime}}=c,u$. We take $(p,k)$ as four momenta of the parent($B_{c}$)
and daughter($X$) meson with their respective spin state: $S_{B_{c}}$ and
$S_{X}$ and mass: M and m. $k_{l}$ and $k_{\nu}$ are the four momenta of
lepton and lepton neutrino, respectively. $q=p-k=k_{l}+k_{\nu}$ represents the
four-momentum transfer.

The hadronic amplitudes are covariantly expanded in terms of Lorentz invariant
form factors. For $(0^{-}\to 0^{-})$ type transitions, the expansion is

${\cal
H}_{\mu}(B_{c}\to(\bar{c}c/\bar{u}c)_{S=0})=(p+k)_{\mu}F_{+}(q^{2})+q_{\mu}F_{-}(q^{2}),$
(4)

and for $(0^{-}\to 1^{-})$ type transitions, it is expanded as

$\displaystyle{\cal H}_{\mu}(B_{c}\to(\bar{c}c/\bar{u}c)_{S=1})=$
$\displaystyle{\frac{1}{(M+m)}}\epsilon^{\sigma^{\dagger}}\Big{\\{}g_{\mu\sigma}(p+k)qA_{0}(q^{2})$
(5)
$\displaystyle+(p+k)_{\mu}(p+k)_{\sigma}A_{+}(q^{2})+q_{\mu}(p+k)_{\sigma}A_{-}(q^{2})$
$\displaystyle+i\epsilon_{\mu\sigma\alpha\beta}(p+k)^{\alpha}q^{\beta}V(q^{2})\Big{\\}}$

The angular decay distribution differential in the momentum transfer squared
$q^{2}$ is [49, 77]

$\frac{d\Gamma}{dq^{2}dcos\theta}={\frac{{\cal
G}_{F}}{(2\pi)^{3}}}|V_{bq}|^{2}\frac{(q^{2}-m_{l}^{2})^{2}}{8M^{2}q^{2}}|\vec{k}|{\cal
L}^{\mu\sigma}{\cal H}_{\mu\sigma}$ (6)

Here ${\cal L}^{\mu\sigma}$ and ${\cal H}_{\mu\sigma}$ are the lepton and
hadron tensor, respectively; $m_{l}$ is the mass of charged lepton. In our
normalization, the lepton tensor ${\cal L}^{\mu\sigma}$ is found to be

${\cal
L}^{\mu\sigma}_{\mp}=8\Bigg{\\{}k^{\mu}_{l}k^{\sigma}_{\nu}+k^{\sigma}_{l}k^{\mu}_{\nu}-g^{\mu\sigma}\Bigg{(}{\frac{q^{2}-m^{2}_{l}}{2}}\Bigg{)}\pm
i{\epsilon^{\mu\sigma\alpha\beta}}k_{l_{\alpha}}k_{\nu_{\beta}}\Bigg{\\}}$ (7)

It is convenient to express physical observables such as the hadron tensor on
a helicity basis in which the helicity form factors are obtained in terms of
the Lorentz invariant form factors representing the decay amplitudes. While
doing the Lorentz contraction in Eq.(5) with the helicity amplitudes, one
considers four covariant helicity projections $\epsilon^{\mu}(m)$ with
$m=+,-,0$ and $m=t$ for spin 1 and spin 0 part, respectively, of the $W_{off-
shell}$ involved in the decay process. In our attempt to study the lepton mass
effects in the semileptonic decay modes, we need to consider the time
component of the polarization $\epsilon^{\mu}(m=t)$ in addition to its other
three components with $m=+,-,0$. Using orthogonality and the completeness
relations satisfied by helicity projections, the Lorentz contraction in Eq.(5)
can be written as

$\displaystyle{\cal L}^{\mu\sigma}{\cal H}_{\mu\sigma}=$ $\displaystyle{\cal
L}_{\mu^{\prime}\sigma^{\prime}}g^{\mu^{\prime}\mu}g^{\sigma^{\prime}\sigma}{\cal
H}_{\mu\sigma}$ $\displaystyle=$
$\displaystyle{L(m,n)}{g_{mm^{\prime}}}{g_{nn^{\prime}}}H({m^{\prime}}{n^{\prime}})$
(8)

with $g_{mn}=dia(+,-,-,-)$. Here the lepton and hadron tensors are introduced
in the space of helicity components as:

$\displaystyle L(m,n)={\epsilon^{\mu}}(m){\epsilon^{\sigma^{\dagger}}}(n){\cal
L}_{\mu\nu}$ $\displaystyle
H(m,n)={\epsilon^{\mu^{\dagger}}}(m){\epsilon^{\sigma}(n)}{\cal H}_{\mu\nu}$
(9)

For the sake of convenience, we consider two frames of reference: i) the
$\bar{l}\nu$ or $l\bar{\nu}$ center-of-mass frame and ii) the parent
$B_{c}$-meson rest frame. We evaluate lepton tensor $L(m,n)$ in the
$\bar{l}\nu$ or $l\bar{\nu}$ c.m. frame and hadron tensor $H(m,n)$ in the
$B_{c}$-rest frame.

The space-and-time components of the four momenta: $p^{\mu},k^{\mu},q^{\mu}$,
and that of the polarization vectors: $\epsilon^{\mu}(\pm)$,
$\epsilon^{\mu}(0)$, and $\epsilon^{\mu}(t)$ are specified [77] in the
$B_{c}$-rest frame. Then, it is straightforward to obtain the helicity
components of the hadronic tensors in terms of Lorentz invariant form factors.
For $B_{c}\to(\bar{c}c/\bar{u}c)_{s=0}$ transition, they are obtained as

$\displaystyle{H_{t}}=$
$\displaystyle\frac{1}{\sqrt{q^{2}}}\biggl{\\{}(p+k).(p-k){F_{+}}+{q^{2}}{F_{-}}\biggr{\\}}$
$\displaystyle{H_{\pm}}=$ $\displaystyle 0$ (10) $\displaystyle{H_{0}}=$
$\displaystyle\frac{2M|\vec{k}|}{\sqrt{q^{2}}}{F_{+}}$

and for $B_{c}\to(\bar{c}c/\bar{u}c)_{s=1}$ transitions, the space-and-time
components of the projection operator can also be specified after which the
relations between helicity form factors and invariant form factors are
obtained in the form

$\displaystyle{H_{t}}=$
$\displaystyle\frac{1}{(M+m)}\frac{M{|\vec{k}|}}{m\sqrt{q^{2}}}\\{(p+k){.}q\
({-A_{0}}+{A_{+}})+{{q^{2}}{A_{-}}}\\}$ $\displaystyle{H_{\pm}}=$
$\displaystyle\frac{1}{(M+m)}\big{\\{}-(p+k){.}q\ {A_{0}}\mp
2M|\vec{k}|V\big{\\}}$ (11) $\displaystyle{H_{0}}=$
$\displaystyle\frac{1}{(M+m)}\frac{1}{2m\sqrt{q^{2}}}\Big{\\{}-(p+k){.}q({M^{2}}-{m^{2}}-{q^{2}}){A_{0}}+4{M^{2}}|\vec{k}|^{2}{A_{+}}\Big{\\}}$

In the $(l\bar{\nu})$-c.m. frame, the space-and-time components of the
relevant four momenta $q^{\mu},k_{\nu}^{\mu},k_{l}^{\mu}$
$({\vec{k}}_{l}+{\vec{k}}_{\nu}=\vec{q}=0)$ along with those of the
polarization vectors are specified as in Ref. [77]. Then using the expressions
for space and time components of polarization vectors and that of the lepton
tensor in Eq.(6), it is straightforward to evaluate the helicity
representation $L(m,n)$ of the lepton tensor.

In the present analysis, we do not consider the azimuthal $\chi$ distribution
of the lepton pair and therefore integrate over the azimuthal angle dependence
of the lepton tensor. The differential $(q^{2},\cos\theta)$ distribution is
then obtained in the form:

$\displaystyle\frac{d\Gamma}{d{q^{2}}\cos\theta}=$
$\displaystyle\frac{3}{8}(1+\cos^{2}\theta)\frac{d\Gamma_{U}}{dq^{2}}+\frac{3}{4}\sin^{2}\theta.\frac{d\Gamma_{L}}{dq^{2}}\mp\frac{3}{4}\cos\theta\frac{d\Gamma_{P}}{dq^{2}}+$
(12)
$\displaystyle\frac{3}{4}\sin^{2}\theta\frac{d{\tilde{\Gamma}_{U}}}{dq^{2}}+\frac{3}{2}\cos^{2}\theta\frac{d{\tilde{\Gamma}_{L}}}{dq^{2}}+\frac{1}{2}\frac{d{\tilde{\Gamma}}_{S}}{dq^{2}}+3\cos\theta\frac{d{\tilde{\Gamma}}_{SL}}{dq^{2}}$

The upper and lower signs associated with the parity-violating term in Eq.(11)
refer to two cases: $l^{-}\bar{\nu}$ and $l^{+}{\nu}$, respectively. Out of
seven terms in the r.h.s of Eq.(11), four terms identified as tilde rates
$\tilde{\Gamma}_{i}$ are linked with the lepton mass, and other terms
identified as $\Gamma_{i}$ are lepton mass-independent. Both are related via a
flip factor $\frac{m_{l}^{2}}{2q^{2}}$ as:

$\frac{d{\tilde{\Gamma}}_{i}}{dq^{2}}=\frac{m_{l}^{2}}{2q^{2}}\frac{d{\Gamma_{i}}}{dq^{2}}$
(13)

The tilde rates do not contribute to the decay rate in the vanishing lepton
mass limit. They can be neglected for $e$ and $\mu$ modes but they are
expected to provide a sizeable contribution to the $\tau$-modes. Therefore the
tilde rates are crucial in evaluating the lepton mass effects in the
semileptonic decay modes. The differential partial helicity rates
$\frac{d\Gamma_{i}}{dq^{2}}$ are defined by

$\frac{d\Gamma_{i}}{dq^{2}}=\frac{{\cal
G}_{f}^{2}}{(2\pi)^{3}}{|V_{bq^{{}^{\prime}}}|}^{2}\frac{({q^{2}}-{m_{e}^{2}})^{2}}{12{m_{1}^{2}}q^{2}}|\vec{k}|H_{i}$
(14)

Here $H_{i}(i=U,L,P,S,SL)$ represents a standard set of helicity structure
function given by linear combinations of helicity components of hadron tensor
$H(m,n)=H_{m}H_{n}^{\dagger}$:

$\displaystyle{H_{U}}=$ $\displaystyle
Re({H_{+}}{H_{+}^{\dagger}})+Re({H_{-}}{H_{-}^{\dagger}})\ \ \ \ \
:Unpolarized-transversed$ $\displaystyle{H_{L}}=$ $\displaystyle
Re({H_{0}}{H_{0}^{\dagger}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
:Longitudinal$ $\displaystyle{H_{P}}=$ $\displaystyle
Re({H_{+}}{H_{+}^{\dagger}})-Re({H_{-}}{H_{-}^{\dagger}})\ \ \ \ :Parity-odd$
$\displaystyle{H_{S}}=$ $\displaystyle 3Re({H_{t}}{H_{t}^{\dagger}})\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ :Scalar$ $\displaystyle{H_{SL}}=$
$\displaystyle Re({H_{t}}{H_{0}^{\dagger}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ :Scalar-Longitudinal\ Interference$

Here we assume throughout that the helicity amplitudes are real since the
available $q^{2}$\- range: ($q^{2}\leq(M-m)^{2}$) is below the physical
threshold $q^{2}=(M+m)^{2}$. Therefore we drop the angular terms that are
multiplied by coefficients $Im(H_{i}H_{j}^{*}),i\neq j^{*}$. Then integrating
over $\cos\theta$ one gets the differential $q^{2}$ distribution and finally
integrating over $q^{2}$, one obtains the total decay rate $\Gamma$ as the sum
of the partial decay rates : $\Gamma_{i}=(i=U,L,P)$ and
$\tilde{\Gamma}_{i}(i=U,L,S,SL)$.

A quantity of interest is the forward-backward asymmetry $A_{FB}$ of the
lepton in the $(l\bar{\nu})$ c.m. frame which is defined as

$A_{FB}=\frac{3}{4}\biggl{\\{}\frac{\pm
P+4\tilde{SL}}{U+\tilde{U}+L+\tilde{L}+\tilde{S}}\biggr{\\}}$ (15)

Another quantity of interest is the asymmetry parameter $\alpha^{*}$ which is
defined by rewriting Eq.(11) in terms of its $\cos^{2}\theta^{*}$ dependence
i.e $d\Gamma\propto 1+\alpha^{*}\cos^{2}\theta^{*}$. The asymmetry parameter
$\alpha^{*}$ determines the transverse and longitudinal composition of the
final state vector meson is given by:

$\displaystyle\alpha^{*}=\frac{U+\tilde{U}-2(L+\tilde{L}+\tilde{S})}{U+\tilde{U}+2(L+\tilde{L}+\tilde{S})}$
(16)

We list our predictions on helicity rates $\Gamma_{i}$, $\tilde{\Gamma}_{i}$,
$A_{FB}$, and $\alpha^{*}$ in Sec. IV.

##  III Transition amplitude and weak decay form factors

###  III.1 Transition amplitude

Since the decay process physically occurs in the momentum eigenstate of the
participating mesons, its field-theoretic description requires meson bound-
states to be represented by appropriate momentum wave packets with suitable
momentum and the spin distribution between the constituent quark-antiquark
pair in the corresponding meson core. In the relativistic independent particle
picture of the RIQ model, the constituent quark antiquark are assumed to be in
the state of independent and relativistic motion inside the meson bound-state:
$|B_{c}(\vec{p}_{b},\vec{p}_{c})\rangle$, for example, with the momentum
$\vec{p}_{b}$ and $\vec{p}_{c}$, respectively. In our previous analysis of
wide-ranging hadronic decay properties including our recent study in Ref.
[77], we have extracted the momentum probability amplitude of the constituent
quark and antiquark in the participating mesons in their ground states only,
as shown in the Appendix, by taking the momentum projections of corresponding
quark orbitals, derivable in the RIQ model. In the present context, one needs
to extract equivalent model expressions for the momentum probability amplitude
of constituent quark and antiquark inside the charmonium and charm mesons in
their radially excited $2S$ and $3S$ states. For this, the quark orbitals for
radially excited meson states are from the general expression (A1) shown in
the Appendix after specifying appropriate Dirac quantum numbers and associated
Laguerre polynomial. Thereafter the momentum projection of the specified quark
orbitals yields the momentum probability amplitude of the quark and antiquark
in the $S$-wave ($2S$ and $3S$) charmonium and charm meson states,
respectively, in the form

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}{(E_{p_{b}}+E_{b})\over{(E_{b}+m_{b})}}({\vec{p_{b}}^{2}\over{2\alpha_{b}}}-{3\over
2})\exp{(-{{\vec{p}_{b}}^{2}\over{4\alpha_{b}}})}$
$\displaystyle{\tilde{G}}_{c}(\vec{p}_{c})$ $\displaystyle={{i\pi{\cal
N}_{c}}\over{2\alpha_{c}}}\sqrt{{(E_{p_{c}}+m_{c})}\over{E_{p_{c}}}}{(E_{p_{c}}+E_{c})\over{(E_{c}+m_{c})}}({\vec{p_{c}}^{2}\over{2\alpha_{c}}}-{3\over
2})\exp{(-{{\vec{p}_{c}}^{2}\over{4\alpha_{c}}})}$ (17)

and

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}{(E_{p_{b}}+E_{b})\over{(E_{b}+m_{b})}}({\vec{p_{b}}^{4}\over{8{\alpha_{b}}^{2}}}-{{5{\vec{p}_{b}}^{2}}\over{4\alpha_{b}}}+{{15\over
8}})\exp{(-{{\vec{p}_{b}}^{2}\over{4\alpha_{b}}})}$
$\displaystyle{\tilde{G}}_{c}(\vec{p}_{c})$ $\displaystyle={{i\pi{\cal
N}_{c}}\over{2\alpha_{c}}}\sqrt{{(E_{p_{c}}+m_{c})}\over{E_{p_{c}}}}{(E_{p_{c}}+E_{c})\over{(E_{c}+m_{c})}}({\vec{p_{c}}^{4}\over{8{\alpha_{c}}^{2}}}-{{5{\vec{p}_{c}}^{2}}\over{4\alpha_{c}}}+{{15\over
8}})\exp{(-{{\vec{p}_{c}}^{2}\over{4\alpha_{c}}})}$ (18)

The binding energy of constituent quark and antiquark in the radially excited
meson states are also obtained by solving the appropriate cubic equations for
the binding energy condition (A5) shown in the Appendix. After specifying the
momentum probability amplitudes of constituent quarks and their binding
energies, the radially excited meson states, involved in the semileptonic
$B_{c}$-meson decays, are duly constructed which can be used to obtain the
$S$-matrix elements for the decay process.
The internal dynamics at the constituent level, responsible for physically
observable meson-level decay processes, are described by an otherwise unbound
quark and antiquark using the usual Feynman technique. The constituent level
$S$-matrix element $S_{fi}^{b\to c,u}$, obtained from the Feynman diagram,
when operated upon by the bag like operator $\hat{\Lambda}$ used in the RIQ
model [72, 73, 74, 75, 76, 77] yields meson-level $S$-matrix in the form

$\hat{\Lambda}S_{fi}^{b\to c/u}\longrightarrow
S_{fi}^{{B_{c}}\to(\bar{c}c/\bar{u}c)system}$ (19)

Figure 1: Leading order diagram of semileptonic decay of $B_{c}$ meson.

Using the wave packet representation of the participating meson states, the
$S$-matrix element for the decay process is obtained in the standard form [77]

$S_{fi}=(2\pi)^{4}\delta^{(4)}(p-k-k_{l}-k_{\nu})(-{\cal
M}_{fi})\frac{1}{\sqrt{(2\pi)^{3}2E_{B_{c}}}}\Pi_{f}\frac{1}{\sqrt{2E_{f}(2\pi)^{3}}}$
(20)

where ${\cal M}$ is the invariant matrix element involving the hadronic ${\cal
H}_{\mu}$ and leptonic ${\cal L}$ amplitude. In the $B_{c}$-rest frame the
hadronic amplitude ${\cal H}_{\mu}$ is

${\cal
H}_{\mu}=\sqrt{\frac{4ME_{k}}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int\frac{d^{3}p_{b}}{\sqrt{2E_{p_{b}}2E_{k+p_{b}}}}{\cal
G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})\langle
S_{X}|J_{\mu}^{h}(0)|S_{B_{c}}\rangle$ (21)

where $E_{p_{b}}$ and $E_{p_{b}+k}$ stand for the energy of the non-spectator
quark of the parent and daughter meson, respectively, and $\langle
S_{X}|J_{\mu}^{h}(0)|S_{B_{c}}\rangle$ represents symbolically the spin matrix
elements of vector-axial vector current.

###  III.2 Weak decay form factors

For $0^{-}\to 0^{-}$ transitions, the axial vector current does not
contribute. The spin matrix elements corresponding to the non-vanishing vector
current parts are obtained in the form:

$\displaystyle\langle
S_{X}(\vec{k})|V_{0}|S_{B_{c}}(0)\rangle=\frac{(E_{p_{b}}+m_{b})(E_{p_{c/u}}+m_{c/u})+|\vec{p_{b}}|^{2}}{\sqrt{(E_{p_{b}}+m_{b})(E_{p_{c/u}}+m_{c/u})}}$
(22) $\displaystyle\langle
S_{X}(\vec{k})|V_{i}|S_{B_{c}}(0)\rangle=\frac{(E_{p_{b}}+m_{b})k_{i}}{\sqrt{(E_{p_{b}}+m_{b})(E_{p_{b}+k}+m_{c/u})}}$
(23)

With the above spin matrix elements, the expressions for hadronic amplitudes
(20) are compared with corresponding expressions in Eq.(3) yielding the form
factors $f_{+}$ and $f_{-}$ in the form:

$\displaystyle f_{\pm}(q^{2})=$
$\displaystyle\frac{1}{2M}\sqrt{\frac{ME_{k}}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int
d\vec{p_{b}}{\cal G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})$ (24)
$\displaystyle\times\frac{(E_{o_{b}}+m_{b})(E_{p_{c/u}}+m_{c/u})+|\vec{p_{b}}|^{2}\pm(E_{p_{b}}+m_{b})(M\mp
E_{k})}{E_{p_{b}}E_{p_{c/u}}(E_{p_{b}}+m_{b})(E_{p_{c/u}}+m_{c/u})}$

For $(0^{-}\to 1^{-})$ transitions, the spin matrix elements corresponding to
the vector and axial-vector current are found separately as:

$\displaystyle\langle
S_{X}(\vec{k},\hat{\epsilon^{*}})|V_{0}|S_{B_{c}}(0)\rangle=$ $\displaystyle
0$ (25) $\displaystyle\langle
S_{X}(\vec{k},\hat{\epsilon^{*}})|V_{i}|S_{B_{c}}(0)\rangle=$
$\displaystyle\frac{i(E_{p_{b}}+m_{b})(\hat{\epsilon}^{*}\times\vec{k})_{i}}{\sqrt{(E_{p_{b}}+m_{b})(E_{p_{b}+k}+m_{c/u})}}$
(26) $\displaystyle\langle
S_{X}(\vec{k},\hat{\epsilon^{*}})|A_{i}|S_{B_{c}}(0)\rangle=$
$\displaystyle\frac{(E_{p_{b}}+m_{b})(E_{p_{b}+k}+m_{c/u})-\frac{|\vec{p_{b}}|^{2}}{3}}{\sqrt{(E_{p_{b}}+m_{b})(E_{p_{b}+k}+m_{c/u})}}$
(27) $\displaystyle\langle
S_{X}(\vec{k},\hat{\epsilon^{*}})|A_{0}|S_{B_{c}}(0)\rangle=$
$\displaystyle\frac{-(E_{p_{b}}+m_{b})(\hat{\epsilon}^{*}.\vec{k})}{\sqrt{(E_{p_{b}}+m_{b})(E_{p_{b}+k}+m_{c/u})}}$
(28)

With the spin matrix elements (24-27), the expressions for hadronic amplitudes
(20) are compared with corresponding expressions in Eq.(4). The model
expressions of the form factors: $V(q^{2}),A_{0}(q^{2}),A_{+}(q^{2})$ and
$A_{-}(q^{2})$ are obtained in the form:

$\displaystyle V(q^{2})=$
$\displaystyle\frac{M+m}{2M}\sqrt{\frac{ME_{k}}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int
d\vec{p_{b}}{\cal G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})$ (29)
$\displaystyle\times\sqrt{\frac{(E_{p_{b}}+m_{b})}{E_{p_{b}}E_{p_{c/u}}(E_{p_{c/u}}+m_{c/u})}}$
$\displaystyle A_{0}(q^{2})=$
$\displaystyle\frac{1}{(M-m)}\sqrt{\frac{Mm}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int
d\vec{p_{b}}{\cal G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})$ (30)
$\displaystyle\times\frac{(E_{p_{b}}+m_{b})(E^{0}_{p_{c/u}}+m_{c/u})-\frac{|\vec{p_{b}}|^{2}}{3}}{\sqrt{E_{p_{b}}E_{p_{c/u}}(E_{p_{b}}+m_{b})(E_{p_{c/u}}+m_{c/u})}}$

with $E^{0}_{p_{c/u}}=\sqrt{|\vec{p}_{c/u}|^{2}+m^{2}_{c/u}}$ and

$A_{\pm}(q^{2})=\frac{-E_{k}(M+m)}{2M(M+2E_{k})}\big{[}T\mp\frac{3(M\mp
E_{k})}{(E_{k}^{2}-m^{2})}\big{\\{}I-A_{0}(M-m)\big{\\}}\big{]}$ (31)

where $T=J-(\frac{M-m}{E_{k}})A_{0}$

$\displaystyle J=$
$\displaystyle\sqrt{\frac{ME_{k}}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int
d\vec{p_{b}}{\cal G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})$ (32)
$\displaystyle\times\sqrt{\frac{(E_{p_{b}}+m_{b})}{E_{p_{b}}E_{p_{c/u}}(E_{p_{c/u}}+m_{c/u})}}$
$\displaystyle I=$
$\displaystyle\sqrt{\frac{ME_{k}}{N_{B_{c}}(0)N_{X}(\vec{k})}}\int
d\vec{p_{b}}{\cal G}_{B_{c}}(\vec{p_{b}},-\vec{p_{b}}){\cal
G}_{X}(\vec{k}+\vec{p_{b}},-\vec{p_{b}})$ (33)
$\displaystyle\times\biggl{\\{}\frac{(E_{p_{b}}+m_{b})(E^{0}_{p_{c/u}}+m_{c/u})-\frac{|\vec{p}_{b}|^{2}}{3}}{\sqrt{E_{p_{b}}E^{0}_{p_{c/u}}(E_{p_{b}}+m_{b})(E^{0}_{p_{c/u}}+m_{c/u})}}\biggr{\\}}$

With the relevant form factors thus obtained in terms of model quantities, the
helicity amplitudes and hence the decay rates for semileptonic $B_{c}$-meson
decays to radially excited $(2S$ and $3S)$ charmonium and charm meson states
are evaluated and our predictions are listed in the next section.

##  IV Numerical results and discussion

In this section, we present our numerical results on the semileptonic $B_{c}$
decay modes to $S$-wave charmonium and charm meson states. For numerical
calculation, we need the model parameters $(a,\ V_{0})$, quark masses $m_{q}$,
and quark binding energies $E_{q}$, which have already been fixed from hadron
spectroscopy by fitting the data of heavy and heavy-light flavored mesons in
their ground states as

$\displaystyle(a,\ V_{0})=$ $\displaystyle(0.017166\ GeV^{3},-0.1375\ GeV)$
$\displaystyle(m_{b},\ m_{c},\ m_{u})=$
$\displaystyle(4.77659,1.49276,0.07875)GeV$ $\displaystyle(E_{b},\ E_{c},\
E_{u})=$ $\displaystyle(4.76633,1.57951,0.47125)GeV$ (34)

With this set of input parameters, wide-ranging hadronic phenomena have been
described earlier [72, 73, 74, 75, 76, 77] in the framework of the RIQ model.
However, for a description of the decay process, in the present context,
involving radially excited final meson states, the constituent quarks in the
meson bound states are expected to have higher binding energies compared to
the binding energies in their ground states. We solve the cubic equation
representing the binding energy condition (A5) for respective constituent
quarks and obtain the binding energies of the quarks $(c,u)$ in the radially
excited $2S$ and $3S$ states of the $(\bar{c}c)$ and $(\bar{u}c)$ systems as:

$(E_{c},E_{u})_{2S}=(1.97015,0.96221)GeV$ (35)

and

$(E_{c},E_{u})_{3S}=(2.22478,1.29356)GeV$ (36)

For CKM-parameters and $B_{c}$-lifetime, we take its central values from the
Particle Data Group (2020) [80] as:

$\displaystyle(V_{bc},V_{bu})=$ $\displaystyle(0.041,0.00382)$
$\displaystyle\tau_{B_{c}}=$ $\displaystyle 0.510\ ps$ (37)

For the masses of participating mesons, taken as phenomenological inputs in
the present calculation, we take their central values of the available
observed data from [80, 81, 82, 83]. In the absence of observed data in the
charmonium and charm meson sector, we take the corresponding predicted values
from established theoretical approaches [84, 85, 86]. Accordingly, the updated
meson masses used in the present study are listed in Table I.

Table 1: The masses of the participating mesons Particle | Mass(MeV) | Reference
---|---|---
$B_{c}$ | 6274.47 | [80]
$\eta_{c}(2S)$ | 3637 | [80]
$\psi(2S)$ | 3686.1 | [80]
$D^{0}(2S)$ | 2518 | [81]
$D^{0*}(2S)$ | 2681.1 | [82]
$\psi(3S)$ | 4039.1 | [83]
$\eta_{c}(3S)$ | 4007 | [84]
$D^{0}(3S)$ | 3068 | [85]
$D^{0*}(3S)$ | 3110 | [86]

With these input parameters, the Lorentz invariant form factors: $(F_{+},\
F_{-};\ A_{0},\ A_{+},\ A_{-},\ V)$ representing decay amplitudes can be
calculated from the overlapping integral of participating meson wave
functions. We first study the $q^{2}-$dependence of the invariant form factors
in the allowed kinematic range. Since our main purpose is to study the
leptonic mass effect in semileptonic decay of $B_{c}$ meson to radially
excited $(2S$ and $3S)$ charmonium and charm meson states, we plot
$q^{2}$-dependence of the form factors in Figs. 2 and 3 for decays in their
$e^{-},\mu^{-}$ and $\tau^{-}$-modes. We find that the behavior of form
factors in $e^{-}$ and $\mu^{-}$-mode overlap in the entire kinematic range:
$0\leq q^{2}\leq q^{2}_{max}$. This is because of an insignificant change in
the phase space boundary going from $e^{-}$ to $\mu^{-}$ mode. As one can see
here the maximal lepton energy shift $\frac{m^{2}_{\mu}-m^{2}_{e}}{2M}$ is
invisible at the usual scale of the plot. On the other hand for the decays in
the $\tau^{-}$ mode, the relevant form factors behave differently throughout
the accessible kinematic range of $q^{2}_{min}\leq q^{2}\leq q^{2}_{max}$,
where $q^{2}_{min}$ is +ve and away from $q^{2}\to 0$. The $\tau-$phase space,
as compared to the $e^{-}$ and $\mu^{-}$ cases is considerably reduced and
shifted to a large $q^{2}-$ region. In the present analysis, we shall
therefore consider decays in their $e^{-}$ and $\tau^{-}$ modes only for
evaluating the lepton mass effects on the semileptonic $B_{c}$ meson decays.

Figure 2: The $q^{2}$-dependence of invariant form factors for
$B_{c}\to\eta_{c}(2S)/\psi(2S)$ and $B_{c}\to\eta_{c}(3S)/\psi^{*}(3S)$
decays.

Figure 3: The $q^{2}$-dependence of invariant form factors for $B_{c}\to
D(2S)/D^{*}(2S)$ and $B_{c}\to D(3S)/D^{*}(3S)$ decays.

As mentioned earlier it is convenient to calculate decay amplitudes in the
helicity basis in which the partial helicity rate and total decay rates are
expressed in terms of helicity form factors. On this basis, the expressions
for the relevant helicity form factors have also been obtained [77] in terms
of invariant form factors. Using the helicity form factors it is
straightforward to obtain partial helicity rates:$\frac{d\Gamma_{i}}{dq^{2}}$
(without flip) and $\frac{d\tilde{\Gamma}_{i}}{dq^{2}}$(with flip) for all
four decay processes considered here in their $e^{-}$ and $\tau^{-}$ modes.
Before calculating the helicity rates we study the $q^{2}$-dependence of the
helicity form factors, partial helicity rates, and $q^{2}$-spectra of
semileptonic decay rates, separately in their $e^{-}$ and $\tau^{-}$ modes.
For this, we first rescale the helicity form factors according to

$h_{j}=A(q^{2})H_{j},\ \ \ \ j=0,+,-\ \ \ (for\ no\ flip\ case)$ (38)

and for flip cases

$\displaystyle\tilde{h}_{j}=$
$\displaystyle\sqrt{\frac{m_{l}^{2}}{2q^{2}}}A(q^{2})H_{j}\ ,\ \ \ \ \
j=0,+,-$ $\displaystyle\tilde{h}_{t}=$
$\displaystyle\sqrt{\frac{m_{l}^{2}}{2q^{2}}}\sqrt{3}A(q^{2})H_{t}$ (39)

where

$A(q^{2})=\frac{{\cal
G}_{F}}{4M}\big{(}\frac{q^{2}-m_{l}^{2}}{q^{2}}\big{)}\sqrt{\frac{|\vec{k}|q}{6\pi^{3}}}\
|V_{b,c/u}|$ (40)

and $\sqrt{\frac{m_{l}^{2}}{2q^{2}}}$: denotes a flip factor. In terms of the
rescaled helicity form factors, the angle integrated differential $q^{2}$-rate
is expressed as:

$\frac{d\Gamma}{dq^{2}}=\sum_{0,+,-}|h_{j}|^{2}+\sum_{t,0,+,-}|\tilde{h}_{j}|^{2}$
(41)

Figure 4: Reduced helicity amplitudes $h_{j}$ and $\tilde{h_{j}}$,
$(j=t,+,-,0)$ as functions of $q^{2}$ for $B_{c}\to\eta_{c}(2S)/D(2S)$ and
$B_{c}\to\eta_{c}(3S)/D(3S)$ decays.

Figure 5: Reduced helicity amplitudes $h_{j}$ and $\tilde{h_{j}}$,
$(j=t,+,-,0)$ as functions of $q^{2}$ for semileptonic decay rates for
$B_{c}\to\psi(2S)/D^{*}(2S)$ and $B_{c}\to\psi(3S)/D^{*}(3S)$ decays.

In Fig. 4 we plot the $q^{2}-$dependence of the rescaled helicity form factors
$h_{j}$ and $\tilde{h}_{j}$ for $B_{c}\to\eta_{c}$ and $B_{c}\to D$ decays,
respectively in their $e^{-}$ and $\tau^{-}$modes. We find in both the decay
processes, the longitudinal no-flip amplitude $h_{0}$ is reduced in the low
$q^{2}$ region going from $e^{-}$ to $\tau^{-}$ mode. This reduction is due to
the threshold like factor; $\frac{q^{2}-m^{2}_{l}}{q^{2}}$ appearing in the
rescaled helicity amplitude. The longitudinal flip amplitude $\tilde{h}_{0}$
is further reduced by the flip factor $\sqrt{\frac{m_{l}^{2}}{2q^{2}}}$. The
large value of the scalar flip amplitude $\tilde{h}_{t}$ is attributed to
time-like (scalar-)current contribution which proceeds via an orbital
$S-$wave, where there is no pseudo-threshold factor to tamper the enhancement
at large $q^{2}$ resulting from the time-like form factor in the helicity
amplitude.

In Fig. 5 we plot the $q^{2}$-dependence of the rescaled helicity amplitudes
for $B_{c}\to\psi(2S,3S)$ and $B_{c}\to D^{*}(2S,3S)$ transitions in their
$e^{-}$ and $\tau^{-}$ mode. We find a significant reduction of the
longitudinal no-flip amplitude $h_{0}$ in these decay modes. Contrary to
$B_{c}\to\eta_{c}(2S,3S)/D(2S,3S)$ decays we find all the flip amplitudes are
generally small compared to corresponding no-flip amplitudes. This is because
of the partial wave structure of the scalar-current contribution. The effect
of time like (scalar)-current on semileptonic decay in the $\tau^{-}$ mode is
depicted in Figs. 4 and 5. It may be noted here that the contribution of
invariant form factors: $F_{-}(q^{2})$ and $A_{-}(q^{2})$ is dominant to
determine the shape of the plot of flip helicity component $\tilde{h}_{t}$
over the allowed kinematic range of $q^{2}$. In
$B_{c}\to\eta_{c}(2S,3S)/D(2S,3S)$ decay modes the contribution of
$F_{-}(q^{2})$ to the timelike helicity part is destructive for which the
corresponding rescaled helicity amplitude $\tilde{h}_{t}$ increases when
$F_{-}(q^{2})$ is switched off as shown in Fig.4. Since $|\tilde{h}_{t}|^{2}$
contributes dominantly to the decay rate, an accurate determination of
$B_{c}\to\eta_{c}(2S,3S)$ and $B_{c}\to D(2S,3S)$ decays in their
$\tau^{-}$-mode would help extract information on the sign and magnitude of
the scalar-invariant form factors. On the other hand, the contribution of
scalar form factor $A_{-}(q^{2})$ in $B_{c}\to\psi(2S,3S)/D^{*}(2S,3S)$ decays
is constructive for which $\tilde{h}_{t}$ decreases when $A_{-}(q^{2})$ is
switched off; as shown in Fig.5.

Figure 6: Partial helicity rates $\frac{d\Gamma_{i}}{dq^{2}}$ and
$\frac{d\tilde{\Gamma_{i}}}{dq^{2}}$ as functions of $q^{2}$ for semileptonic
decay rates for $B_{c}\to\eta_{c}(2S)/D(2S)$ and $B_{c}\to\eta_{c}(3S)/D(3S)$
decays.

In Fig. 6 we plot the $q^{2}$-spectra for different helicity components in
$B_{c}\to\eta_{c}(e^{-},\tau^{-})$ and $B_{c}\to D(e^{-},\tau^{-})$ decay
modes. As in the case of semileptonic $B_{c}$ decays to ground state
charmonium and charm mesons [77], we also find here a considerable reduction
of the longitudinal no-flip contribution in the $\tau^{-}$ modes compared to
their corresponding contribution in $e^{-}$ modes. Apart from the contribution
of the longitudinal no-flip part, one can see here a sizeable contribution
coming from the scalar flip components over the allowed kinematic range.

Figure 7: Partial helicity rates $\frac{d\Gamma_{i}}{dq^{2}}$ and
$\frac{d\tilde{\Gamma_{i}}}{dq^{2}}$ as functions of $q^{2}$ for semileptonic
decay rates for $B_{c}\to\psi(2S)/D^{*}(2S)$ and $B_{c}\to\psi(3S)/D^{*}(3S)$
decays.

Fig. 7 depicts the $q^{2}$-spectra for different helicity components in
$B_{c}\to\psi(e^{-},\tau^{-})$ and $B_{c}\to D^{*}(e^{-},\tau^{-})$ modes. We
also find here that the spin-flip parts are negligible compared to no-flip
parts as seen earlier [77] in the study of semileptonic $B_{c}$-decays to
ground state charmonium and charm mesons. Apart from the contribution of
longitudinal no-flip part, the dominant contribution also comes here from the
transverse no-flip parts. The helicity rates are found reduced uniformly going
from $e^{-}$ to $\tau^{-}$ modes, as expected. A significant contribution of
the time like scalar-current via $\tilde{S}(\tau)$ is obtained in the high
$q^{2}$-region as shown in Fig. 7.

Figure 8: $q^{2}$-spectrum of semileptonic decay rates
$B_{c}\to\eta_{c}(2S)/\psi(2S)$ and $B_{c}\to D(2S)/D^{*}(2S)$ decays.

Figure 9: $q^{2}$-spectrum of semileptonic decay rates for
$B_{c}\to\eta_{c}(3S)/\psi(3S)$ and $B_{c}\to D(3S)/D^{*}(3S)$ decays.

In Figs. 8 and 9 we plot the total $q^{2}$-spectra: $\frac{d\Gamma}{dq^{2}}$
for $B_{c}\to\eta_{c}(\psi)$ and $B_{c}\to D(D^{*})$ decays, respectively, in
their $e^{-}$ and $\tau^{-}$ modes. For $B_{c}\to\eta_{c}(2S)$ decay in
$e^{-}$ mode, the $q^{2}$-spectra increases rapidly at $q^{2}\to 0$ to a peak
and then flattens with a slow rise to another peak near $q^{2}\simeq 2.5\
GeV^{2}$. Thereafter it decreases slowly to zero value at $q^{2}\simeq 7\
GeV^{2}$. In the $\tau^{-}$-mode, the spectra, which begin at a positive
$q^{2}\simeq 3.3\ GeV^{2}$ away from $q^{2}\to 0$, increase slowly to a peak
at $q^{2}\simeq 5.5\ GeV^{2}$ and then decrease almost at the same rate to
zero at $q^{2}\simeq 7\ GeV^{2}$. The $\tau^{-}$-mode spectra in these decays
is found almost below $e^{-}$-mode spectra throughout only to overcome
slightly above the $e^{-}$-spectra towards the end of the kinematic range:
$q^{2}\geq 6\ GeV^{2}$.

A similar trend of $q^{2}$-spectra is observed for $B_{c}\to\eta_{c}(3S)$
decays in their $e^{-}$ and $\tau^{-}$ modes in the kinematic range of $0\leq
q^{2}\leq 5\ GeV^{2}$ and $3.3\leq q^{2}\leq 5\ GeV^{2}$, respectively. For
$B_{c}\to D(2S)$ decays, the $e^{-}$ mode spectra originating at $q^{2}\to 0$
rise slowly to a peak at about $q^{2}\simeq 11.25\ GeV^{2}$ and then decrease
to zero at $q^{2}\simeq 13.3\ GeV^{2}$. On the other hand, the $\tau^{-}$-mode
spectra for these decays starts at $q^{2}\simeq 3\ GeV^{2}$, slowly rise to a
peak at $q^{2}\simeq 11.25\ GeV^{2}$, and fall to zero thereafter at
$q^{2}\simeq 13.8\ GeV^{2}$ lying all along within its $e^{-}$-mode spectra.
For $B_{c}\to D(3S)$ decays the $e^{-}$ mode spectra rise fast near
$q^{2}\simeq 0$, then rise moderately to a peak at $q^{2}\simeq 6.75\ GeV^{2}$
and thereafter fall to zero at $q^{2}\simeq 10.25\ GeV^{2}$. Its $\tau$-mode
spectra originate at about $q^{2}\simeq 3\ GeV^{2}$ away from $q^{2}\to 0$ and
then rise to the peak at $q^{2}\simeq 7.5\ GeV^{2}$ after which it decreases
to zero at $q^{2}=10.25\ GeV^{2}$, lying all along within its $e^{-}$-mode
spectra.

For $B_{c}\to\psi(2S)$ decays we find a sharp rise of $e^{-}$mode spectra to a
peak at $q^{2}\to 0$ which rises thereafter to a prominent shoulder art
$q^{2}\simeq 3\ GeV^{2}$ beyond which it decreases in the region $3\leq
q^{2}\leq 5.5\ GeV^{2}$. Then the spectra flatten up to $q^{2}\simeq 6\
GeV^{2}$ followed by its sharp fall to zero at about $6.7\ GeV^{2}$. Its
$\tau^{-}$-mode spectra originating at about $q^{2}\simeq 3.2\ GeV^{2}$
attains a slow linear rise up to $q^{2}\simeq 6\ GeV^{2}$; after which it
slows down a bit only to hook up to a very high value at $q^{2}\simeq 6.7\
GeV^{2}$. On the other hand for $B_{c}\to\psi(3S)$ decays in $e^{-}$ mode, the
spectra, which begin with a sharper rise to a peak at $q^{2}\to 0$, flatten to
attain a peak at $q^{2}\simeq 2.3\ GeV^{2}$ and then decrease rapidly to zero
at $q^{2}\simeq 5\ GeV^{2}$. In its $\tau^{-}$-mode, the spectra begin only at
$q^{2}\simeq 3.25\ GeV^{2}$ and increase slowly up to $q^{2}\simeq 4.5\
GeV^{2}$ after which it attains a steep rise to a high value at about
$q^{2}\simeq 4.75\ GeV^{2}$.

For $B_{c}\to D^{*}(2S)$ decays, the $e^{-}$ mode spectra attain a step rise
to a peak at $q^{2}\to 0$. Then the spectra increase moderately up to
$q^{2}\simeq 5\ GeV^{2}$ beyond which it increases rapidly to develop a sharp
shoulder only to fall rapidly to a very small value near $q^{2}\simeq 13\
GeV^{2}$. Its $\tau^{-}$ mode spectra appear identical in shape through out
the kinematic range lying all along within the $e^{-}$-spectra as expected
except at $q^{2}\simeq 13\ GeV^{2}$ where it attains a steep rise to a very
high value at $q^{2}\simeq 13\ GeV^{2}$. For $B_{c}\to D^{*}(3S)$ decays the
$e^{-}$-mode spectra rise swiftly at $q^{2}\to 0$ to a peak and then rises
moderately till it attains a peak at about $q^{2}\simeq 7\ GeV^{2}$.
Thereafter it decreases to a very low value at $q^{2}\simeq 10\ GeV^{2}$. Its
$\tau^{-}$-mode spectra on the other hand begins also at about $q^{2}\simeq 3\
GeV^{2}$ away from $q^{2}\to 0$ and rises afterward till it overtakes its
$e^{-}$-mode spectra at about $q^{2}\simeq 9\ GeV^{2}$. After overtaking the
$e^{-}$-mode spectra, it attains a steep rise to a very high value. It may be
noted here that in $B_{c}\to\psi(2S,3S)$ and $B_{c}\to D^{*}(2S,3S)$ decays,
the shapes of the $\tau^{-}$-mode spectra are significantly reduced compared
to that of the $e^{-}$-mode spectra. Towards the end of the kinematic region,
the respective $\tau^{-}$-mode spectra are found to overtake the $e^{-}$-mode
spectra contrary to the naive phase space expectations.

Before evaluating the physical quantities of interest it is interesting to
study the behavior of radial quark momentum distribution amplitude function of
the decaying $B_{c}$ meson state together with those of the final $S$-wave
charmonium and charm meson states. The behavior of momentum distribution
amplitudes is shown in Fig. 10. One can see that the overlap regions between
the momentum distribution amplitude function of the initial $B_{c}$ meson and
final charmonium and charm meson states are obtained in the decreasing order
from $1S$ to higher $2S$ and $3S$ final states. It is expected that
contribution to decay rate should also be in decreasing order from maximum to
a minimum as on goes from $1S$ to higher $2S$ and $3S$ final states.

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

In Tables II and III we list our results for the integrated partial helicity
rates: $\Gamma_{i}(i=U,L,P)$ and $\tilde{\Gamma}_{i}(i=U,L,S,SL)$ as well as
the total decay rates of semileptonic $B_{c}$ decays to $2S$ and $3S$
charmonium and charm meson states, respectively. The partial tilde decay rates
for each decay process in the $e^{-}$ mode are found tiny as expected from
Eq.(12) and can therefore be neglected. But corresponding rates in
$\tau^{-}$-mode, obtained comparable to the tilde rates, can hardly be
neglected. From the contributions of individual helicity rates, we predict the
decay rates for each process, considered in the present study, in its $e^{-}$
as well as $\tau^{-}$ mode. Like all other model predictions, our predicted
decay rates in $\tau^{-}$-modes are found, in general, smaller than that in
$e^{-}$-modes. For $B_{c}\to\psi(2S)$ and $B_{c}\to\psi(3S)$ transitions, our
predicted decay rate in $\tau^{-}$ mode is suppressed by a factor of $\sim 12$
and $\sim 13$, respectively. However, for $B_{c}\to\eta_{c}(2S)$ and
$B_{c}\to\eta_{c}(3S)$ transitions the suppression is only by a factor of
$\sim 2$ and $\sim 2.5$, respectively compared to their respective decays in
the $e^{-}$ mode. For the CKM suppressed transitions: $B_{c}\to D(2S)$ and
$B_{c}\to D^{*}(2S)$, the $\tau^{-}$ modes are suppressed by a factor of $\sim
2$ whereas for $B_{c}\to D(3S)$ and $B_{c}\to D^{*}(3S)$ the suppression is by
a factor of $\sim 2.5$ over corresponding $e^{-}$ modes.

Table 2: Helicity rates(in $10^{-15}$ GeV) of semileptonic $B_{c}$-meson decays into $2S$ charmonium and charm-meson states: Decay mode | $U$ | $\tilde{U}$ | $L$ | $\tilde{L}$ | $P$ | $S$ | $\tilde{S}$ | $\tilde{SL}$ | $\Gamma$
---|---|---|---|---|---|---|---|---|---
${B^{-}_{c}}\to\eta_{c}(2S)e^{-}\nu_{e}$ |  |  | 2.877 | 5.275$10^{-7}$ |  |  | 17.57$10^{-7}$ | 5.51$10^{-7}$ | 2.877
$B_{c}\to\eta_{c}(2S)\tau^{-}\nu_{\tau}$ |  |  | 0.098 | 0.031 |  |  | 0.266 | 0.0508 | 0.396
${B^{-}_{c}}\to\psi(2S)e^{-}\nu_{e}$ | 9.236 | 4.430$\times 10^{-7}$ | 8.08 | 18.73$\times 10^{-7}$ | 5.685 | 3.166 | 47.04$\times 10^{-7}$ | 16.63$\times 10^{-7}$ | 17.317
${B^{-}_{c}}\to\psi(2S)\tau^{-}\nu_{\tau}$ | 0.688 | 0.206 | 0.170 | 0.050 | 0.392 | 0.0309 | 0.360 | 0.007 | 1.476
${B^{-}_{c}}\to D(2S)e^{-}\nu_{e}$ |  |  | 0.042 | 8.86$\times 10^{-10}$ |  |  | 24.73$\times 10^{-10}$ | 8.51$\times 10^{-10}$ | 0.042
${B^{-}_{c}}\to D(2S)\tau^{-}\nu_{\tau}$ |  |  | 0.017 | 0.0027 |  |  | 0.0077 | 0.0026 | 0.027
${B^{-}_{c}}\to D^{*}(2S)e^{-}\nu_{e}$ | 0.199 | 4.317$\times 10^{-9}$ | 0.070 | 10.17$\times 10^{-9}$ | 0.164 | 0.024 | 2.734$\times 10^{-8}$ | 7.187$\times 10^{-9}$ | 0.270
${B^{-}_{c}}\to D^{*}(2S)\tau^{-}\nu_{\tau}$ | 0.069 | 0.011 | 0.014 | 0.0026 | 0.064 | 0.0093 | 0.0146 | 0.0028 | 0.113
Table 3: Helicity rates(in $10^{-15}$ GeV) of semileptonic $B_{c}$-meson decays into $3S$ charmonium and charm-meson states: Decay mode | $U$ | $\tilde{U}$ | $L$ | $\tilde{L}$ | $P$ | $S$ | $\tilde{S}$ | $\tilde{SL}$ | $\Gamma$
---|---|---|---|---|---|---|---|---|---
${B^{-}_{c}}\to\eta_{c}(3S)e^{-}\nu_{e}$ |  |  | 1.838 | 6.25$\times 10^{-7}$ |  |  | 20.12$\times 10^{-7}$ | 6.45$\times 10^{-7}$ | 1.83
$B_{c}\to\eta_{c}(3S)\tau^{-}\nu_{\tau}$ |  |  | 0.007 | 0.002 |  |  | 0.032 | 0.005 | 0.042
${B^{-}_{c}}\to\psi(3S)e^{-}\nu_{e}$ | 4.678 | 3.09$\times 10^{-7}$ | 3.821 | 15.43$\times 10^{-7}$ | 1.767 | 0.347 | 44.04$\times 10^{-7}$ | 13.442$\times 10^{-7}$ | 8.499
${B^{-}_{c}}\to\psi(3S)\tau^{-}\nu_{\tau}$ | 0.09 | 0.032 | 0.019 | 0.0072 | 0.021 | 0.188 | 0.51 | 0.026 | 0.659
${B^{-}_{c}}\to D(3S)e^{-}\nu_{e}$ |  |  | 0.036 | 16.17$\times 10^{-10}$ |  |  | 47.82$\times 10^{-10}$ | 16.03$\times 10^{-10}$ | 0.036
${B^{-}_{c}}\to D(3S)\tau^{-}\nu_{\tau}$ |  |  | 0.0075 | 0.0017 |  |  | 0.005 | 0.0017 | 0.014
${B^{-}_{c}}\to D^{*}(3S)e^{-}\nu_{e}$ | 0.109 | 3.391$\times 10^{-9}$ | 0.05 | 8.091$\times 10^{-9}$ | 0.087 | 0.0206 | 2.326$\times 10^{-8}$ | 5.532$\times 10^{-9}$ | 0.160
${B^{-}_{c}}\to D^{*}(3S)\tau^{-}\nu_{\tau}$ | 0.023 | 0.0051 | 0.0059 | 0.0014 | 0.0202 | 0.008 | 0.026 | 0.0024 | 0.0623

From the predicted decay rates shown in Tables II and III and $B_{c}$-meson
lifetime $\tau_{B_{c}}=0.510\ ps$, we predict the branching fractions (BF) for
semileptonic $B_{c}\to\eta_{c}(\psi)$ and $B_{c}\to D(D^{*})$ decays in their
$e^{-}$ and $\tau^{-}$ modes. Our results for such decays to radially excited
$2S$ and $3S$ final meson states are shown in Table IV and V, respectively.
Considering our results for semileptonic $B_{c}$ decays to charmonium and
charm meson ground states [77], we find that our predicted branching fractions
for decays to $1S$, $2S$, and $3S$ final states are obtained in the hierarchy:

${\cal B}(B_{c}\to X(3S)l\nu_{l})<{\cal B}(B_{c}\to X(2S)l\nu_{l})<{\cal
B}(B_{c}\to X(1S)l\nu_{l})$ (42)

Our results for $B_{c}\to\eta_{c}(2S)$ and $B_{c}\to\eta_{c}(3S)$ decays in
their $e^{-}$ mode are obtained $\sim 1.5$ and $\sim 2.5$ times down and in
their $\tau^{-}$, mode one and two orders of magnitude down respectively,
compared to that for the decays to ground state charmonium. However, our
predicted branching fractions for $B_{c}\to\psi(2S)$ and $B_{c}\to\psi(3S)$
decays in $e^{-}$ modes are $\sim 2$ and $\sim 4$ times down whereas, in
$\tau$-mode, they are $1/2$ and $1$ order of magnitude down, respectively
compared to those obtained for decays to ground state charmonium $(J/\psi)$.
The node structure of the $2S$ wave function is responsible for small
branching fractions. In the calculation of the overlap integral of meson wave
functions, as there is no node for the initial wave functions, the positive
and negative parts of the final wave function give contributions cancel each
other out yielding a small branching fraction. About 3S final states, there
are even more severe cancellations, which leads to still smaller branching
fractions. As expected the tighter phase space and weaker $q^{2}$-dependence
of form factors also lead to smaller branching fractions for transitions to
higher excited $2S$ and $3S$ final states. Our predictions on branching
fraction for decays to $2S$ final charmonium states in $e$ and $\tau$ modes
are in reasonable agreement with those of Refs. [26, 56, 68] and about $1$
order of magnitude higher than the predictions in Refs.[39, 40, 41]. Although
our predictions for decays to $3S$ charmonium states in their $\tau$-modes are
found over-estimated compared to other model predictions shown in Table. IV,
our predictions for corresponding decay in $e^{-}$-mode agree with those of
[41, 56]. As in all other model predictions we find our prediction for
$B_{c}\to\psi(2S)$ as well as $B_{c}\to\psi(3S)$ is the largest of all decay
modes considered in the present analysis.

Table 4: Branching fractions(in %) of semileptonic $B_{c}$ decays into $2S$ and $3S$ charmonium states: Decay mode | This work | [26] | [39] | [40] | [41] | [42, 43] | [53] | [56] | [68] | [87]
---|---|---|---|---|---|---|---|---|---|---
$B_{c}\to\eta_{c}(2S)e\nu$ | 0.22 | 0.056 | 0.046 | 0.07 | 0.0496 | 0.03 | 0.02 | 0.77 | 0.11 | 0.046
$B_{c}\to\eta_{c}(2S)\tau\nu$ | 0.03 | - | - | - | 0.0025 | - | - | 0.053 | 0.0081 | 0.0013
$B_{c}\to\psi(2S)e\nu$ | 1.341 | 0.112 | 0.014 | 0.1 | 0.081 | 0.03 | 0.12 | 1.2 | - | 0.21
$B_{c}\to\psi(2S)\tau\nu$ | 0.114 | - | - | - | 0.0408 | - | - | 0.084 | - | 0.015
$B_{c}\to\eta_{c}(3S)e\nu$ | 0.14 | - | - | - | 0.00414 | 5.5$\times 10^{-4}$ | - | 0.14 | 1.9$\times 10^{-2}$ | -
$B_{c}\to\eta_{c}(3S)\tau\nu$ | 0.003 | - | - | - | 0.0043$\times 10^{-2}$ | 5.0$\times 10^{-7}$ | - | 1.9$\times 10^{-4}$ | 5.7$\times 10^{-4}$ | -
$B_{c}\to\psi(3S)e\nu$ | 0.658 | - | - | - | 0.0109 | 5.7$\times 10^{-4}$ | - | 3.6$\times 10^{-2}$ | - | -
$B_{c}\to\psi(3S)\tau\nu$ | 0.0511 | - | - | - | 0.010$\times 10^{-2}$ | 3.6$\times 10^{-6}$ | - | 3.8$\times 10^{-5}$ | - | -

Our results for $B_{c}\to D(2S)/D^{*}(2S)$ and $B_{c}\to D(3S)/D^{*}(3S)$
transitions are shown in Table V. We find the branching fractions for decays
in their $e^{-}$ as well as $\tau^{-}$ modes in the decreasing order as one
goes from $1S$ to higher $2S$ and $3S$ final charm meson states as expected.
For $B_{c}\to D$ decays in their $e^{-}$ as well as $\tau^{-}$ mode, the
predicted branching fractions decrease marginally. However, for $B_{c}\to
D^{*}$ decays in $e^{-}$ mode our results decreases marginally from $1S$ to
higher $2S$ and $3S$ states, and in $\tau^{-}$ mode our predictions for $2S$
and $3S$ final states decrease $\sim 3$ and $\sim 5$ times, respectively, as
compared to that for $1S$ final state.

Table 5: Branching fractions(in %) of semileptonic $B_{c}$ decays into $2S$ and $3S$ charm meson states: Decay |  $B_{c}\to D(2S)$ |  $B_{c}\to D^{*}(2S)$ |  $B_{c}\to D(3S)$ |  $B_{c}\to D^{*}(3S)$
---|---|---|---|---
Mode | $e^{-}$ | $\tau^{-}$ | $e^{-}$ | $\tau^{-}$ | $e^{-}$ | $\tau^{-}$ | $e^{-}$ | $\tau^{-}$
This work | 0.0032 | 0.0021 | 0.020 | 0.0087 | 0.0028 | 0.0013 | 0.012 | 0.0048
Table 6: Forward-backward asymmetry $A_{FB}$ and the asymmetry parameter $\alpha^{*}$ for semileptonic $B_{c}$-decays to $2S$ charmonium and charmed meson states. Decay process | $A_{FB}(l^{-})$ | $A_{FB}(l^{+})$ | $\alpha^{*}$
---|---|---|---
$B_{c}\to\eta_{c}(2S)e\nu$ | 5.75$\times 10^{-7}$ | 5.75$\times 10^{-7}$ |
$B_{c}\to\eta_{c}(2S)\tau\nu$ | 0.384 | 0.384 |
$B_{c}\to\psi(2S)e\nu$ | 0.246 | -0.246 | -0.272
$B_{c}\to\psi(2S)\tau\nu$ | 0.183 | -0.214 | -0.130
$B_{c}\to D(2S)e\nu$ | 6.051$\times 10^{-8}$ | 6.051$\times 10^{-8}$ | -
$B_{c}\to D(2S)\tau\nu$ | 0.287 | 0.287 |
$B_{c}\to D^{*}(2S)e\nu$ | 0.457 | 0.457 | 0.172
$B_{c}\to D^{*}(2S)\tau\nu$ | 0.348 | -0.5003 | 0.116
Table 7: Forward-backward asymmetry $A_{FB}$ and the asymmetry parameter $\alpha^{*}$ for semileptonic $B_{c}$-decays to $3S$ charmonium and charmed meson states. Decay process | $A_{FB}(l^{-})$ | $A_{FB}(l^{+})$ | $\alpha^{*}$
---|---|---|---
$B_{c}\to\eta_{c}(3S)e\nu$ | 1.05$\times 10^{-6}$ | 1.05$\times 10^{-6}$ |
$B_{c}\to\eta_{c}(3S)\tau\nu$ | 0.367 | 0.367 |
$B_{c}\to\psi(3S)e\nu$ | 0.155 | -0.155 | -0.240
$B_{c}\to\psi(3S)\tau\nu$ | -0.095 | -0.144 | -0.794
$B_{c}\to D(3S)e\nu$ | 1.329$\times 10^{-7}$ | 1.329$\times 10^{-7}$ | -
$B_{c}\to D(3S)\tau\nu$ | 0.355 | 0.355 |
$B_{c}\to D^{*}(3S)e\nu$ | 0.4064 | -0.4064 | 0.038
$B_{c}\to D^{*}(3S)\tau\nu$ | 0.123 | -0.362 | -0.404

Using appropriate helicity rates for different decay modes, we evaluate two
quantities of interest: the forward-backward asymmetry $A_{FB}$ $(14)$ and
asymmetry parameter $\alpha^{*}$ $(15)$. Our predicted $A_{FB}$ and
$\alpha^{*}$ for decays to $2S$ and $3S$ final meson states are shown in Table
VI and VII, respectively. For decays to spin 0 state, $A_{FB}$, which is
proportional to the helicity amplitude $\tilde{SL}$, is found tiny in $e^{-}$
mode but non-negligible in $\tau^{-}$ mode. Looking at the expression for
$A_{FB}\ (14)$ it is easy to relate corresponding parameters in the $e^{-}$
and $e^{+}$ mode in the form: $A_{FB}(e^{-})=-A_{FB}(e^{+})$ and
$A_{FB}(\tau^{-})\neq-A_{FB}(\tau^{+})$ for decays to spin 1 state. To
determine the transverse and longitudinal components of the final state vector
mesons in $B_{c}\to\psi$ and $B_{c}\to D^{*}$ transitions in their $e^{-}$ and
$\tau^{-}$ modes, we calculate the asymmetry parameter $\alpha^{*}$. One can
see from Tables II and III that the transverse and longitudinal pieces in
$B_{c}\to\psi(2S)$ and $B_{c}\to\psi(3S)$ decays are found almost equal in
$e^{-}$ mode. But the transverse component dominates over longitudinal parts
in their $\tau^{-}$ mode by a factor of $\sim 4$. In the $B_{c}\to D^{*}(2S)$
and $B_{c}\to D^{*}(3S)$ decays, the transverse components dominate over
longitudinal parts by a factor $\sim 2$ in $e^{-}$ mode. However, in the
$\tau^{-}$ mode, transverse components dominate over longitudinal parts by a
factor of $\sim 5$ and $\sim 4$ for $B_{c}\to D^{*}(2S)$ and $B_{c}\to
D^{*}(3S)$, respectively. The asymmetry parameter is found close to $-27\%$ in
$e^{-}$ mode and $-13\%$ in $\tau^{-}$ mode for $B_{c}\to\psi(2S)$ decays. For
$B_{c}\to D^{*}(2S)$ decays it is found close to $17\%$ and $12\%$ in $e^{-}$
and $\tau^{-}$ mode, respectively. In $B_{c}\to\psi(3S)$ decays, $\alpha^{*}$
is obtained close to $-24\%$ and $-79\%$ in their $e^{-}$ and $\tau^{-}$ mode,
respectively, whereas in $B_{c}\to D^{*}(3S)$ decays, it is found to be $\sim
4\%$ in $e^{-}$ mode and $-40\%$ in $\tau^{-}$ mode. This is because the
transverse components of helicity amplitudes provide significant contributions
compared to the tiny contributions of the longitudinal and scalar parts for
decays in $e^{-}$ mode. On the other hand, the scalar flip part $\tilde{S}$ of
the helicity amplitude dominates over the rest part and contributes
destructively in the decays in their $\tau^{-}$ modes yielding a highly
suppressed asymmetry parameter $\alpha^{*}$ as low as $-79\%$ and $-40\%$ in
$B_{c}\to\psi(3S)$ and $B_{c}\to D^{*}(3S)$ decays, respectively.

Finally, we evaluate the observable: ${\cal R}=\frac{{\cal B}(B_{c}\to
X(nS)l\nu_{l})}{{\cal B}(B_{c}\to X(nS)\tau\nu_{\tau})}$ and our results are
listed in Table VIII in comparison with other model predictions. The CKM
matrix elements do not contribute to the ratio ${\cal R}$. The uncertainties
due to the model calculation and CKM parameter etc. are cancelled in
calculating the observable ${\cal R}$. Therefore, contrary to other
observables, the ratio ${\cal R}$, in which the production of $B_{c}$ meson is
cancelled totally, provide an essential test on the applicability of any
phenomenological model in the description of decay processes. Our predicted
${\cal R}$ for $B_{c}$ decays to charmonium states are comparable to other
standard model predictions as shown in Table 8. However, in the absence of
predicted data from established model approaches in the literature for $B_{c}$
decays to charm meson states, our predictions: ${\cal R}_{D}(2S)=1.523,\ {\cal
R}_{D}(3S)=2.153,\ {\cal R}_{D^{*}}(2S)=2.298$ and ${\cal R}_{D^{*}}(3S)=2.5$
can be useful to identify the $B_{c}$-channels characterized by clear
experimental signature. The departure of the SM predictions on ${\cal R}$ from
the observed data would highlight the puzzle in flavor physics and failure of
lepton flavor universality(LFU) hinting at the new physics beyond SM.

Table 8: Ratios of branching fractions for semileptonic $B_{c}$-decays to radially excited charmonium states Ratio | This work | [41] | [42, 43] | [56] | [68] | [87]
---|---|---|---|---|---|---
${\cal R}_{\eta_{c}}(2S)$ | 7.33 | 18.4 | - | 14.5 | 1.35 | 35.38
${\cal R}_{\eta_{c}}(3S)$ | 46.67 | 96.24 | 1.1$\times 10^{3}$ | 7.36$\times 10^{2}$ | 33.33 |
${\cal R}_{\psi}(2S)$ | 11.76 | 1.98 | - | 14.3 | - | 14
${\cal R}_{\psi}(3S)$ | 12.876 | 109 | 158.33 | 947.4 |  |

##  V Summary and conclusion

In this paper, we study the exclusive $B_{c}$-meson decays into radially
excited $2S$ and $3S$ charmonium and charm meson states in the framework of
the RIQ model based on an average flavor-independent confining potential in an
equally mixed scalar-vector harmonic form. The invariant form factors
representing decay amplitudes are extracted from the overlapping integral of
meson wave functions derivable from the model dynamics. Since our main
objective here is to evaluate the lepton mass effects on the semileptonic
$B_{c}$-meson decays, we analyze $B_{c}\to\eta_{c}/\psi(nS)l\nu$ and $B_{c}\to
D/D^{*}(nS)l\nu$ in their $e^{-}$ and $\tau^{-}$ modes separately. For this,
we study the $q^{2}$-dependence of relevant physical quantities such as the
invariant form factors, helicity amplitudes, partial helicity rates, and the
total $q^{2}$-spectra for all decay processes analyzed in the present work.
Before evaluating the decay rates we also study the behavior of radial quark
momentum distribution amplitude function of the decaying $B_{c}$ meson state
together with those of the final $S$-wave charmonium and charm meson states.
From the overlapping regions, one could assess the relative contribution to
the decay rate in each decay mode.

Considering contribution from different partial helicity rates:
$\frac{d\Gamma_{i}}{dq^{2}}\ (i=U,L,P)$ and
$\frac{d\tilde{\Gamma}_{i}}{dq^{2}}\ (i=U,L,S,SL)$, the total $q^{2}$-spectrum
$\frac{d\Gamma}{dq^{2}}$ is obtained for each decay process, from which we
predict the decay rates/branching fractions of the semileptonic $B_{c}$-meson
decays to their $e^{-}$ as well as $\tau^{-}$-modes. Our predictions on the
decay rates/ branching fractions for decay processes are found in overall
agreement with other SM predictions. We find that the decay rate for
$B_{c}\to\psi(2S)$ as well as $B_{c}\to\psi(3S)$ transition is the largest of
all. As expected, the decay rates for $B_{c}\to\eta_{c}/\psi(nS)$ decay
induced by $b\to c$ transition at the quark level dominate over those for
$B_{c}\to D/D^{*}(nS)$ decays induced by the quark level $b\to u$ transition
in their respective $e^{-}$ as well as $\tau^{-}$ modes. Like all other model
prediction our predicted branching ratios for semileptonic $B_{c}$ decays to
charmonium and charm meson ground [77] as well as $2S$ and $3S$ final states
are obtained in the hierarchy:

${\cal B}(B_{c}\to X(3S)l\nu_{l})<{\cal B}(B_{c}\to X(2S)l\nu_{l})<{\cal
B}(B_{c}\to X(1S)l\nu_{l})$ (43)

We also find that the $\tau^{-}$ modes for all decays are suppressed in
comparison with their corresponding $e^{-}$ modes.

Using appropriate helicity rates we evaluate two quantities of
interest:(1)Forward-backward asymmetry ‘$A_{FB}$’ and the asymmetry parameter
‘$\alpha^{*}$’. $A_{FB}$ in the present analysis is found negligible for
transitions into spin 0 state in their $e^{-}$-mode but non-negligible in
$\tau^{-}$-mode, as expected. For transition into spin 1 state, we also find
$A_{FB}(e^{-})=-A_{FB}(e^{+})$ and $A_{FB}(\tau^{-})\neq-A_{FB}(\tau^{+})$. We
evaluate the asymmetry parameter $\alpha^{*}$, which determines the transverse
and longitudinal components of the final vector meson state in
$B_{c}\to\psi(nS)$ and $B_{c}\to D^{*}(nS)$ transitions. We find the asymmetry
parameter close to $-27\%$ in $e^{-}$ mode and $-13\%$ in $\tau^{-}$ mode for
$B_{c}\to\psi(2S)$ decays. For $B_{c}\to D^{*}(2S)$ decays it is found close
to $17\%$ and $12\%$ in $e^{-}$ and $\tau^{-}$ mode, respectively. However,
for $B_{c}\to\psi(3S)$ decays, $\alpha^{*}$ is obtained close to $-24\%$ and
$-79\%$ in their $e^{-}$ and $\tau^{-}$ mode, respectively, whereas in
$B_{c}\to D^{*}(3S)$ decays, it is found to be $\sim 4\%$ in $e^{-}$ mode and
$-40\%$ in $\tau^{-}$ mode. This is because the transverse components of
helicity amplitudes provide significant contribution compared to the tiny
contributions of the longitudinal and scalar parts for decays in $e^{-}$ mode.
On the other hand, the scalar flip part $\tilde{S}$ of the helicity amplitude
dominates over the rest part and contributes destructively in the decays in
their $\tau^{-}$ modes yielding a highly suppressed asymmetry parameter
$\alpha^{*}$ as low as $-79\%$ and $-40\%$ in $B_{c}\to\psi(3S)$ and $B_{c}\to
D^{*}(3S)$ decays, respectively.
Finally, we evaluate the observable ${\cal R}$, which corresponds to the
ratios of branching fractions for transitions in $e^{-}$ modes to the
corresponding value in $\tau^{-}$ modes. Our predicted observable ${\cal R}$
for $B_{c}\to\eta_{c}(nS)$ and $B_{c}\to\psi(nS)$ is found comparable to other
SM predictions. The departure of the SM predictions of ${\cal R}$ from the
experimental data would highlight the puzzle in flavor physics and the failure
of lepton flavor universality hinting at new physics beyond SM. However, in
the absence of predicted data from established model approaches in the
literature for $B_{c}$ decays to charm meson states, our predictions: ${\cal
R}_{D}(2S),\ {\cal R}_{D}(3S),\ {\cal R}_{D^{*}}(2S)$ and ${\cal
R}_{D^{*}}(3S)$ can be useful to identify the $B_{c}$-channels characterized
by clear experimental signature. Given the possibility of high statistics
$B_{c}$-events expected to yield up to $10^{10}$ events per year at the
Tevatron and LHC, semileptonic $B_{c}$-meson decays into charmonium and charm
mesons in their ground as well as excited states offer a fascinating area for
future research.

###### Acknowledgements.

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

##  Appendix A CONSTITUENT QUARK ORBITALS AND MOMENTUM PROBABILITY AMPLITUDES

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:

$\displaystyle\psi^{(+)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{ig_{\xi}(r)}{r}\\\
\frac{{\vec{\sigma}}.{\hat{r}}f_{\xi}(r)}{r}\end{array}\;\right)U_{\xi}(\hat{r})$
(46) $\displaystyle\psi^{(-)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{i({\vec{\sigma}}.{\hat{r}})f_{\xi}(r)}{r}\\\
\frac{g_{\xi}(r)}{r}\end{array}\;\right){\tilde{U}}_{\xi}(\hat{r})$ (49)

where, $\xi=(nlj)$ represents a set of Dirac quantum numbers specifying the
eigen-modes; $U_{\xi}(\hat{r})$ and ${\tilde{U}}_{\xi}(\hat{r})$ are the spin
angular parts given by,

$\displaystyle U_{ljm}(\hat{r})$ $\displaystyle=$
$\displaystyle\sum_{m_{l},m_{s}}<lm_{l}\;{1\over{2}}m_{s}|jm>Y_{l}^{m_{l}}(\hat{r})\chi^{m_{s}}_{\frac{1}{2}}$
$\displaystyle{\tilde{U}}_{ljm}(\hat{r})$ $\displaystyle=$
$\displaystyle(-1)^{j+m-l}U_{lj-m}(\hat{r})$ (50)

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:

$\displaystyle g_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}(\frac{r}{r_{nl}})^{l+l}\exp(-r^{2}/2r^{2}_{nl})L_{n-1}^{l+1/2}(r^{2}/r^{2}_{nl})$
$\displaystyle f_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}(\frac{r}{r_{nl}})^{l}\exp(-r^{2}/2r^{2}_{nl})$ (51)
$\displaystyle\times$
$\displaystyle\left[(n+l-\frac{1}{2})L_{n-1}^{l-1/2}(r^{2}/r^{2}_{nl})+nL_{n}^{l-1/2}(r^{2}/r^{2}_{nl})\right]$

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

$N^{2}_{nl}=\frac{4\Gamma(n)}{\Gamma(n+l+1/2)}\frac{(\omega_{nl}/r_{nl})}{(3E_{q}^{\prime}+m_{q}^{\prime})}$
(52)

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

$\sqrt{(\omega_{q}/a)}(E_{q}^{\prime}-m_{q}^{\prime})=(4n+2l-1)$ (53)

The solution of the cubic equation provides the zeroth-order binding energies
of the confined quark and antiquark for all possible eigenmodes.

In the relativistic independent particle picture of this model, the
constituent quark and antiquark are thought to move independently inside the
$B_{c}$-meson bound state with momentum $\vec{p}_{b}$ and $\vec{p}_{c}$,
respectively. Their momentum probability amplitudes are obtained in this model
via momentum projection of respective quark orbitals (A1) in the following
forms:

For ground state mesons:($n=1$,$l=0$)

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

In terms of individual momentum probability amplitudes: $G_{b}(\vec{p_{b}})$
and $G_{c}(\vec{p_{c}})$ of the constituent quarks, ${\cal
G}_{B_{c}}(\vec{p_{b}},\vec{p}-\vec{p_{b}})$, the effective momentum
distribution function is taken in the form:

${\cal
G}_{B_{c}}(\vec{p_{b}},\vec{p}-\vec{p_{b}})=\sqrt{G_{b}(\vec{p_{b}})G_{c}(\vec{p}-\vec{p_{b}})}$
(55)

in the straightforward extension of the ansatz of Margolis and Mendel in their
bag model description [88].

In the RIQ model, the wavepacket representing a meson bound state
$|B_{c}(\vec{p},S_{B_{c}})\rangle$, for example, at a definite momentum
$\vec{p}$, and spin $S_{B_{c}}$ is taken in the form[72, 73, 74, 75, 76, 77,
78, 79]

$|B_{c}(\vec{p},S_{B_{c}})\rangle=\hat{\Lambda}(\vec{p},S_{B_{c}})|(\vec{p_{b}},\lambda_{b});(\vec{p_{c}},\lambda_{c})\rangle$
(56)

where $|(\vec{p_{b}},\lambda_{b});(\vec{p_{c}},\lambda_{c})\rangle$ is the
Fock space representation of the unbound quark and antiquark in a color-
singlet configuration with their respective momentum and spin:
$(\vec{p_{b}},\lambda_{b})$ and $(\vec{p_{c}},\lambda_{c})$. Here
$\hat{b_{b}}^{\dagger}(\vec{p_{b}},\lambda_{b})$ and
$\hat{b_{c}}^{\dagger}(\vec{p_{c}},\lambda_{c})$ denote the quark-antiquark
creation operator, respectively, and $\hat{\Lambda}(\vec{p},S_{B_{c}})$ is a
bag like integral operator in the form:

$\hat{\Lambda}(\vec{p},S_{B_{c}})=\frac{\sqrt{3}}{\sqrt{N(\vec{p})}}\sum_{\delta_{b}\delta_{c}}\zeta_{b_{c}}^{B_{c}}(\lambda_{b},\lambda_{c})\int
d^{3}p_{b}d^{3}p_{c}\delta^{(3)}(\vec{p_{b}}+\vec{p_{c}}+\vec{p}){\cal
G}_{B_{c}}(\vec{p_{b}},\vec{p_{c}})$ (57)

Here $\sqrt{3}$ is the effective color
factor,$\zeta_{b_{c}}^{B_{c}}(\lambda_{b},\lambda_{c})$ is the $SU(6)$ spin-
flavor coefficients for $B_{c}$-meson and $N(\vec{p})$ is the meson-state
normalization obtained in an integral form:

$N(\vec{p})=\int d^{3}\vec{p_{b}}|{\cal
G}_{B_{c}}(\vec{p_{b}},\vec{p}-\vec{p_{b}})|^{2}$ (58)

by imposing the normalization condition
$\langle{B_{c}}(\vec{p})|{B_{c}}(\vec{p}^{{}^{\prime}})\rangle=\delta^{3}(\vec{p}-\vec{p}^{{}^{\prime}})$.

The binding energy of constituent quark and antiquark in the parent and
daughter meson in their ground as well as radially excited states are obtained
by solving respective cubic equations representing appropriate bound state
conditions(A5).

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