August 28, 2024

$B_{c}$ to A Transition Form Factors and Semileptonic Decays

in Self-consistent Covariant Light-front Approach

Avijit<EMAIL_ADDRESS>Thejus Mary
<EMAIL_ADDRESS>Neelesh<EMAIL_ADDRESS>Rohit
Dhir†444Corresponding author<EMAIL_ADDRESS>
†Department of Physics and Nanotechnology,

SRM Institute of Science and Technology, Kattankulathur 603203, India.

‡ Paradigm of Science Cultivation and Ingenious, Kangra 176032, India.

Abstract

We present a comprehensive analysis of the semileptonic weak decays of $B_{c}$
meson decaying to axial-vector mesons for bottom-conserving and bottom-
changing decay modes. We employ self-consistent covariant light-front quark
model to obtain the weak transition form factors. The self-consistent
covariant light-front quark model uses type-II correspondence to eliminate
inconsistencies in the traditional type-I covariant light-front quark model.
We confirm that type-II correspondence produces self-consistent form factors
involving $B_{c}$ to axial-vector meson transitions free from zero-mode
contributions. We also study the $q^{2}$ dependence of the form factors in
weak hadronic currents for the whole accessible kinematic range $0\leq
q^{2}\leq q^{2}_{max}$ for both bottom-conserving as well as bottom-changing
transitions. We thoroughly investigate the effects of self-consistency between
type-I and type-II schemes using a comparative analysis. We predict the
branching ratios of the semileptonic weak decays by adopting the factorization
hypothesis. We evaluate the lepton mass effect on these branching ratios and
various other important physical observables such as forward-backward
asymmetries, lepton-side convexity parameter, the asymmetry parameter, and
longitudinal polarization asymmetries and fractions. Finally, we obtain the
lepton-flavor-universality ratios for various decays.

##  I Introduction

The CDF Collaboration discovered $B_{c}$ meson ground state at Fermilab in
1998 CDF:1998axz ; CDF:1998ihx . Thereafter, considerable experimental
progress has been made in $B_{c}$ meson studies, including spectroscopy,
lifetime, decay channels, etc. In the past few decades, the study of the
$B_{c}$ meson decays has garnered significant interest because of its unique
characteristic of containing two distinct heavy open flavors. This flavor
asymmetry in $B_{c}$ prevents it from strong (electromagnetic) annihilation
into gluons (photons), however, it can decay weakly. As a result, the $B_{c}$
state is much more stable than heavy quarkonia, with a relatively longer life
span of $0.5134\pm 0.011\pm 0.0057$ ps LHCb:2014glo . The LHCb reported latest
precise measurement of $B_{c}$ meson mass as $6274.47\pm 0.27\pm 0.17$ MeV
LHCb:2020ayi . Thus, the $B_{c}$ system is an exceptional tool for exploring
decay properties in both strong and weak interactions, as well as probing
heavy quark dynamics.

The modern B factories and collaborations such as LHCb, CMS, ATLAS, and CDF
have explored and planned Berezhnoy:2019yei ; CDF:2016hra ; Chen:2018obq
experimental study of $B_{c}$ meson, where the substantial volume of data on
$B_{c}$ will yield valuable insights into the understanding of heavy flavor
dynamics. In this regard, significant advancements have been achieved through
the measurement of the relative $B_{c}/B$ production fraction,
$B_{c}^{+}-B_{c}^{-}$ production asymmetry, and other experimental
investigations on the properties and decays of $B_{c}$ system LHCb:2012ihf ;
LHCb:2013xlg ; LHCb:2013vrl ; LHCb:2013hwj ; Berezhnoy:2019yei ; LHCb:2019bem
; CMS:2019uhm ; LHCb:2017lpu ; ATLAS:2015jep ; LHCb:2015azd ; LHCb:2014mvo ;
ATLAS:2014lga ; LHCb:2014rck ; ATLAS:2019jpi ; LHCb:2019tea ; CDF:2005yjh ;
CDF:2007umr ; D0:2008bqs ; D0:2008thm ; LHCb:2016vni ; LHCb:2016djy ;
LHCb:2017ogk ; LHCb:2017rqe ; LHCb:2022ioi ; LHCb:2021tdf . In addition, LHCb
has investigated $B_{c}^{+}$ decays into two charm mesons LHCb:2021azb ,
giving an upper bound for their branching ratios. More recently, the ATLAS
Collaboration reported the ratio of branching fractions, Br$(B_{c}^{+}\to
J/\psi D_{s}^{*+})$/Br$(B_{c}^{+}\to J/\psi D_{s}^{+})=1.93\pm 0.26$. They
also measured the transverse polarization fraction for $B_{c}^{+}\to J/\psi
D_{s}^{*+}$ ATLAS:2022aiy . Furthermore, the measured branching ratio of
$B_{c}^{+}$ decaying to a p-wave meson has been reported in
$B_{c}^{+}\rightarrow\chi_{c0}(\to K^{+}K^{-})\pi^{+}$ decay mode LHCb:2016utz
. While a number of observations have been reported for bottom-changing
transitions of $B_{c}$, the $B_{c}^{+}\rightarrow B_{s}^{0}\pi^{+}$ mode is
the only bottom-conserving transition seen experimentally. Moreover, the
$B_{c}^{+}\rightarrow B_{s}^{0}\pi^{+}$ decay is expected to have a large
branching ratio due to the Cabibbo-Kobayashi-Maskawa (CKM) favored, $c\to s$,
transition LHCb:2013xlg . Recently, LHCb reported the first direct measurement
of the ratio of branching fractions of $B_{c}^{+}\rightarrow B_{s}^{0}\pi^{+}$
and $B_{c}^{+}\rightarrow J/\psi\pi^{+}$ LHCb:2022htj . Among the semileptonic
decays, only $B_{c}^{+}\to J/\psi(1S)\mu^{+}\nu_{\mu}$ and $B_{c}^{+}\to
J/\psi(1S)\tau^{+}\nu_{\tau}$ were seen by CDF and LHCb collaborations
LHCb:2017vlu ; LHCb:2014rck , however, their branching ratios are yet to be
measured.

Concurrently, the investigation of the semileptonic decays of $B_{(s)}$ mesons
into excited charm mesons such as $D_{(s)}^{**}$ offers a valuable insight
into the properties and behavior of these excited states Chen:2016spr ;
Atoui:2013ksa ; Gan:2014jxa ; Huang:2004et ; Zhao:2006at . Similarly,
semileptonic $B_{c}$ decay into $B^{**}$ and $B_{s}^{**}$ mesons can
significantly contribute to understanding the spectroscopy and mixing
phenomena of $B_{(s)}$ excitations CMS:2017hfy ; LHCb:2016dxl ; D0:2017qqm ;
Chen:2016spr ; ParticleDataGroup:2022pth ; ATLAS:2018udc ; D0:2016mwd ;
Wang:2016wkj ; Chen:2016mqt . These studies encourage experimental efforts for
the semileptonic decay of $B_{c}$ to orbitally excited (axial-vector ($A$),
scalar ($S$), and tensor ($T$)) mesons. Due to the simple theoretical
description via tree level processes in the standard model (SM), the analysis
of semileptonic weak decays of $B_{c}$ mesons holds significant importance in
experimental studies. These decays act as a probe to strong and weak
interactions interference, phases of the CKM matrix elements, CP-violation
within and beyond the SM, and examine the lepton flavor universality (LFU) in
the electroweak interactions.

Theoretically, numerical estimates of semileptonic $B_{c}$ decays involving
axial-vectors have been reported based on a variety of models, namely, the
covariant light-front quark model (CLFQM), relativistic quark model (RQM),
nonrelativistic quark model (NRQM), relativistic constituent quark model
(RCQM), light cone QCD sum rules (LCQSR), QCD motivated relativistic quark
model (QRQM), perturbative QCD (pQCD), Bethe-Salpeter (BS) approach, three-
point QCD sum rules (QCDSR) Zhang:2023ypl ; Wang:2009mi ; Ivanov:2005fd ;
Ivanov:2006ni ; Hernandez:2006gt ; Akan:2022vtf ; Ebert:2010zu ;
Colangelo:2022awx ; Rui:2018kqr ; Shi:2016gqt ; Wang:2011jt ; Azizi:2009ny ;
Khosravi:2015tea . Thus far, the focus of such theoretical studies has been on
bottom-changing and charm-conserving decays. However, comprehensive
investigations involving bottom-conserving and bottom-changing semileptonic
decays of $B_{c}$ (except for decays to charmonia) are scarce.

Exclusive semileptonic decays can be characterized by dependence on weak
transition form factors. Knowledge of these form factors of heavy mesons is
crucial for precise phenomenological predictions of helicity amplitude,
branching ratios, and other observables in various decay processes. In this
work, we calculate the axial-vector meson emitting transition form factors of
$B_{c}$ for bottom-conserving and bottom-changing transitions in type-II CLFQM
Choi:2013mda . It should be noted that type-II CLFQM is termed as self-
consistent CLFQM on account of the challenges associated with type-I
correspondence Jaus:1999zv ; Choi:2013mda ; Cheng:2003sm . Qin Chang et al.
Chang:2018zjq ; Chang:2019mmh ; Chang:2020wvs have investigated the issues of
self-consistency and covariance in the CLFQM within type-I scheme (hereafter
termed as type-I CLFQM). It has been found that, analogous to the decay
constants of vector (V) and axial-vector mesons, the form factors are also
likely to encounter problems in terms of consistency, particularly with regard
to the $B^{(i)}_{j}$ functions (for details see Chang:2019mmh ). The authors
have examined the form factors for pseudoscalar (P) to vector transitions,
focusing on both the type-I and type-II correspondence schemes in the CLFQM;
however, they have restricted their study to bottom-changing transitions only.
Moreover, the effects of self-consistency has not been attempted on form
factors for axial-vector meson emitting transitions. Furthermore, it has been
demonstrated in Chang:2019mmh that the self-consistency issue arises in form
factors, $a_{-}(q^{2})$ and $f(q^{2})$, due to the involvement of
$B^{(i)}_{j}$ functions in the type-I scheme. However, this problem can be
addressed by adopting the type-II correspondence scheme, as the contributions
associated with $B^{(i)}_{j}$ functions numerically diminish when the meson
mass $M\to M_{0}$, where $M_{0}$ is the kinetic invariant mass. The zero-mode
contributions to these form factors are present in theory for both the type-I
as well as type-II schemes; however, they are numerically negligible in the
type-II scheme. Additionally, the self-consistency and covariance challenges
in the CLFQM can be understood as stemming from the contribution of
$B^{(i)}_{j}$ functions. It is important to note that, similar to vector
mesons, axial-vector states also encounter self-consistency issues. Therefore,
we incorporate the suggested improvements by using type-II CLFQM for
comprehensive analysis of $B_{c}\to A$ form factors111$A$ represents both
types of axial-vector mesons involved in transition form factors and
semileptonic decays.. Furthermore, we believe that the visible impact of such
effects can only be weighed by studying weak decays. Thus, we also present an
analysis of semileptonic $B_{c}\to A\ell\nu_{\ell}$ weak decays in the current
work, for both bottom-conserving and bottom-changing modes. We study the self-
consistency effects in corresponding form factors involving axial-vector
mesons, i.e., $c_{-}(q^{2})$ and $l(q^{2})$222The form factors $l(q^{2})$,
$q(q^{2})$, $c_{+}(q^{2})$, and $c_{-}(q^{2})$ are related to CLF form factors
$A(q^{2})$, $V_{0}(q^{2})$, $V_{1}(q^{2})$, $V_{2}(q^{2})$, and $V_{3}(q^{2})$
through the Eq. (11), as shown in Sec. II.. In contrast to the vector meson
system, it is noteworthy that self-consistency effects have a substantial
numerical influence on $B_{c}\to A$ transitions and subsequently lead to
prominent impact on the decay rates. We further explored the implication of
$q^{2}$ dependence for $B_{c}\to A$ form factors over the entire momentum
range. We found that one type of $q^{2}$ parameterization is not sufficient to
describe the momentum dependence of form factors in the broad kinematical
region. We use a two-parameter fit to describe the behavior of $B_{c}\to
B_{1}/B_{s1}$ form factors in smaller $q^{2}$ range, while a three-parameter
fit is preferred in $B_{c}\to D_{1}/\chi_{c1}$ transition, where large $q^{2}$
range is available. We also plotted the type-II $B_{c}\to A$ form factor to
analyze the behavior of the form factors with respect to available $q^{2}$. We
further use these form factors to predict the helicity amplitudes and,
consequently, the branching ratios of the semileptonic decays. In addition, we
also calculated various physical observables important for experimental as
well as theoretical understanding, such as $\langle A_{FB}\rangle$, $\langle
C^{\ell}_{F}\rangle$, $\langle P^{\ell}_{L}\rangle$, $\langle F_{L}\rangle$,
and $\langle\alpha^{*}\rangle$ and plot their $q^{2}$ dependence. Further, our
investigation includes the calculation of lepton flavor universality (LFU)
ratios pertaining to $B_{c}\to B_{1}/B_{s1}/D_{1}/\chi_{c1}$ semileptonic
decay modes.

The paper is organized as follows. In Sec. II, we describe the decay amplitude
and matrix elements for $B_{c}$ meson decays. We discuss the $q^{2}$
dependence of the form factors and its implications in Sec. III. The form
factors are calculated using the methodology given in Sec. IV. Further, in
Sec. V, we provide the expressions for decay rates, helicity amplitudes, and
other observables. Following this, we present our numerical results and
discussions in Sec. VI. Finally, we summarize our findings in Sec. VII. The
details of the mixing between two axial-vector mesons and the complete
expressions for the decay form factors are provided in Appendices A and B,
respectively.

##  II Decay amplitude and matrix elements

The semileptonic decay amplitude, $\mathcal{M}\left(B_{c}\rightarrow
A\ell\nu_{\ell}\right)$, can be expressed as Richman:1995wm

$\mathcal{M}\left(B_{c}\rightarrow
A\ell\nu_{\ell}\right)=\frac{G_{F}}{\sqrt{2}}\left|V_{q_{1}q_{2}}\right|\mathrm{L}^{\mu}\mathrm{H}_{\mu},$
(1)

where $V_{q_{1}q_{2}}$ is the CKM matrix element for $q_{1}\rightarrow q_{2}$
transition, namely, $c\rightarrow d(s)$ and $b\rightarrow u(c)$; $G_{F}$ is
the Fermi coupling constant, $\ell$ denotes the lepton flavor, and
$\mathrm{L}^{\mu}$ is the lepton current defined by

$\mathrm{L}^{\mu}=\bar{u}_{\ell}\gamma^{\mu}\left(\mathbb{1}-\gamma_{5}\right)v_{\nu},$
(2)

where $u_{\ell}$, $v_{\nu}$ are Dirac spinors. $\mathrm{H}_{\mu}$ is the
hadronic current given in terms of the matrix elements between $B_{c}$ and
axial-vector meson states as

$\mathrm{H}_{\mu}=\langle
A\left|\gamma^{\mu}\left(\mathbb{1}-\gamma_{5}\right)\right|B_{c}\rangle.$ (3)

It may be noted that the axial-vector mesons are of two types,
${}^{3}P_{1}\leavevmode\nobreak\ (J^{PC}=1^{++}$) and
${}^{1}P_{1}\leavevmode\nobreak\ (J^{PC}=1^{+-})$, and they can mix. Their
spectroscopy and mixing schemes are given in Appendix A. The matrix elements
of $B_{c}\to A$ transition decompose into Cheng:2003sm

$\displaystyle\langle
A(p^{{\prime}{\prime}},\varepsilon^{*})|\bar{q}_{2}\gamma^{\mu}q_{1}|B_{c}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
i\left\\{l(q^{2})\varepsilon^{*}+\varepsilon^{*}\cdot
P[P^{\mu}c_{+}(q^{2})+q^{\mu}c_{-}(q^{2})]\right\\},$ (4)
$\displaystyle\langle
A(p^{{\prime}{\prime}},\varepsilon^{*})|\bar{q}_{2}\gamma^{\mu}\gamma_{5}q_{1}|B_{c}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle-
q_{(}q^{2})\epsilon^{\mu\nu\rho\sigma}\varepsilon^{*}_{\nu}P^{\rho}q^{\sigma},$
(5)

where the four-momentum transfer, $q^{\mu}=p^{{\prime}}-p^{{\prime}{\prime}}$,
and $P=p^{\prime}+p^{{\prime}{\prime}}$. Further, in terms of conventional
Bauer-Stech-Wirbel (BSW)-type form factors, the above matrix elements can be
expressed as Cheng:2003sm

$\displaystyle\langle
A(p^{{\prime}{\prime}},\varepsilon^{*})|\bar{q}_{2}\gamma^{\mu}q_{1}|B_{c}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle-\Bigg{\\{}2M_{A}V_{0}(q^{2})\frac{\varepsilon^{*}\cdot
q}{q^{2}}\
q^{\mu}+(M_{B_{c}}-M_{A})V_{1}(q^{2})\left(\varepsilon^{*\mu}-\frac{\varepsilon^{*}\cdot
q}{q^{2}}\ q^{\mu}\right)$ (8) $\displaystyle-
V_{2}(q^{2})\frac{\varepsilon^{*}\cdot
q}{M_{B_{c}}-M_{A}}\left[p^{{\prime}\mu}+p^{{\prime}{\prime}\mu}-\frac{M_{B_{c}}^{2}-M_{A}^{2}}{q^{2}}\
q^{\mu}\right]\Bigg{\\}},$ $\displaystyle\langle
A(p^{{\prime}{\prime}},\varepsilon^{*})|\bar{q}_{2}\gamma^{\mu}\gamma_{5}q_{1}|B_{c}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{2iA(q^{2})}{M_{B_{c}}-M_{A}}\epsilon^{\mu\nu\rho\sigma}\varepsilon^{*}_{\nu}p^{\prime}_{\rho}p^{{\prime}{\prime}}_{\sigma},$
(9)

where $M_{B_{c}}(M_{A})$ represents the mass of initial $B_{c}$ (final axial-
vector $A$) meson and the form factors satisfy the relations,

$\begin{gathered}V_{3}(0)=V_{0}(0),\\\
V_{3}(q^{2})=\frac{M_{B_{c}}-M_{A}}{2M_{A}}V_{1}(q^{2})-\frac{M_{B_{c}}+M_{A}}{2M_{A}}V_{2}(q^{2}).\end{gathered}$
(10)

These BSW-type form factors are related to
$l(q^{2}),c_{+}(q^{2}),c_{-}(q^{2})$, and $q(q^{2})$ form factors in CLFQM by
the following equations:

$\begin{gathered}A^{PA}(q^{2})=-(M_{B_{c}}-M_{A})q(q^{2}),\quad
V_{1}^{PA}(q^{2})=-\frac{l(q^{2})}{M_{B_{c}}-M_{A}},\\\
V_{2}^{PA}(q^{2})=(M_{B_{c}}-M_{A})c_{+}(q^{2}),\quad
V_{3}^{PA}(q^{2})-V_{0}^{PA}(q^{2})=\frac{q^{2}}{2M_{A}}c_{-}(q^{2}).\end{gathered}$
(11)

The expressions for $l(q^{2}),c_{+}(q^{2}),c_{-}(q^{2})$, and $q(q^{2})$ form
factors are discussed in Appendix B. The study of semileptonic $B_{c}$ decays
involving axial-vector meson transitions presents certain challenges,
particularly regarding the $q^{2}$ behavior of form factors. Therefore, a
careful analysis of these form factors is necessary to gain a comprehensive
understanding of the underlying decay processes. In the next section, we
describe the parameterization used to determine the physical form factors for
$q^{2}>0$.

##  III Squared momentum dependence of the form factors

The aforementioned transition form factors that appear in hadronic matrix
elements receive contributions from momentum transfer ($q^{2}$) and are
expressed as a function of the same. To establish a coherent theoretical
framework for $B_{c}$ decays to axial-vector mesons, it is crucial to
accurately ascertain the $q^{2}$ dependence of the decay amplitudes across the
full range of kinematics. The conventional dependence of the form factor on
$q^{2}$ is typically expressed as

$F(q^{2})=\frac{F(0)}{(1-\frac{q^{2}}{M_{p}^{2}})},$ (12)

where a pole structure arises from vector meson dominance (VMD) Wirbel:1985ji
, and is commonly referred to as the BSW-type monopole approximation. Here,
the pole mass, $M_{p}$, is the mass of the lowest-lying $q_{1}\bar{q_{2}}$
meson with the vector quantum numbers. It is anticipated that this will yield
the most significant contribution to the form factor for the transition
$q_{1}\rightarrow q_{2}$. However, it has been observed that fit to the
experimental data does not reproduce the expected vector meson masses
ParticleDataGroup:2022pth ; HeavyFlavorAveragingGroup:2022wzx ; Hill:2006ub .
Further, it has been argued that the monopole form alone cannot be sufficient
to explain the $q^{2}$ behavior, and higher resonance contributions to the
form factor are expected Burdman:1992hd . Furthermore, it was argued that the
validity of the nearest pole dominance assumption depends on the specific
system under consideration. In some cases, multiple resonances may contribute
significantly to the form factor, and neglecting the contributions of higher-
mass poles may lead to inaccuracies in the description of the physical process
Richman:1995wm ; Burdman:1996kr . We wish to remark that it is difficult to
anticipate the non-perturbative physics that governs the $q^{2}$ dependence;
therefore, the most accurate way to express $q^{2}$ dependence is typically
not evident. The models make reasonable assumptions only for certain ranges of
$q^{2}$. Therefore, the assumptions should only be seen as reasonable
approximations in the models. The most generic assumption to parameterize
$q^{2}$ involves the use of a simple pole and the summation of effective poles
HeavyFlavorAveragingGroup:2022wzx , which involves multiple undetermined
expansion parameters to be determined from the experiment. On the other hand,
the simple pole is derived as the lowest-order approximation when the
effective poles are disregarded. It may be noted that the assumptions inherent
in the models should be considered reasonably accurate, particularly when the
range of $q^{2}$ is not excessively large. In the context of bottom-conserving
decays of the $B_{c}$ meson, where the range of momentum transfer is $0\leq
q^{2}\leq 0.30$ GeV2, the Becirevic-Kaidalov (BK) Becirevic:1999kt
parameterization can be employed to yield a reliable squared momentum
dependence Melikhov:2000yu ; Melikhov:2001zv :

$\displaystyle{F}(q^{2})=\frac{{F}(0)}{\left(1-q^{2}/M_{p}^{2}\right)\left(1-a(q^{2}/M_{p}^{2})\right)}.$
(13)

In Eq. (13), the parameter a represents the deviation from the single
resonance contribution in the $c\rightarrow d(s)$ transition. The parameter
$a$ is also referred to as the slope parameter. The physical interpretation of
this parameter is that it represents an effective pole other than $M_{p}$ and
captures the effects of higher-order resonances
HeavyFlavorAveragingGroup:2022wzx . Typically, the BK parameterization is
identified as a three-parameter fit, where $F(0)$, a, and $M_{p}$ are
determined from available experimental data. To ensure the reliability of our
calculations and the proper selection of quark-model parameters, we set the
pole mass $M_{p}$ to the mass of the nearest pole according to the values
given in Table 1 and describe it as a two-parameter fit Melikhov:2000yu ;
Melikhov:2001zv .

Table 1: Pole masses for the transition form factor parameterization. $J^{P}$ | Form factor | Pole masses (GeV)
---|---|---
$b\bar{u}$ | $b\bar{c}$ | $c\bar{u}$ | $c\bar{s}$
$1^{+}$ | $A(q^{2})$ | 5.72 | 6.87 | 2.41 | 2.46
$0^{+}$ | $V_{0}(q^{2})$ | 5.70 | 6.84 | 2.34 | 2.32
$1^{-}$ | $V_{1}(q^{2}),V_{2}(q^{2})$ | 5.32 | 6.47 | 2.01 | 2.11

Consequently, the magnitude of $a/{M_{p}}^{2}$ term will determine how
effectively sub-threshold resonances contribute to the form factors in the
available $q^{2}$ range. The proposed parameterization in Eq. (13) for
$B_{c}\to B_{(s)1}$ transitions can be justified by the fact that the
$q^{2}\leq 0.30$ GeV2. Furthermore, this $q^{2}$ parameterization provides
reliable predictions within a small $q^{2}$ region. By employing this
parameterization, we can simplify the analysis and accurately compute the
transition form factors in the small $q^{2}$ region.

It may be noted that compared to $P\to V(A)$ transitions, $P\to P$ transitions
are simpler from a first-principles point of view. Studying the transitions
involving vector or orbitally excited particles is challenging because they
have multiple invariant form factors, as described in Eqs. (8) and (9). In
addition, the form factors’ analytic structure is complicated due to the
unstable nature of the excited mesons and may be influenced by sub-threshold
contributions Hill:2005ju ; Hill:2006ub ; Melikhov:2001zv . Furthermore, the
inclusion of higher-order poles becomes especially relevant when the final
meson is in a radially or orbitally excited state. For instance, the momentum
dependence of $A(q^{2})$ is governed by the mass of the nearest resonance with
$J^{P}=1^{+}$ pole, while $V_{0}(q^{2})$ and $V_{i}(q^{2})$ (for $i=1,2$)
include contributions from $J^{P}=0^{+}$ and $J^{P}=1^{-}$ pole, respectively
Bauer:1988fx ; Gubernari:2022hrq .

It is not surprising that the available $q^{2}$ in bottom-changing decays is
significantly larger than that of the bottom-conserving modes, as the entire
momentum is carried by the bottom and bottom-charmed resonances. While the
nearest pole dominance described above can be a useful approximation, its
applicability should be carefully assessed on a case-by-case basis,
considering the properties of the system, available $q^{2}$ range, and
experimental data. Moreover, as we approach the $q^{2}_{max}$, the
contribution of valence states to the form factor starts to decrease and
deviates from the parameterization based on the $q^{2}$ defined by Eq. (13).
In the vicinity of $q^{2}_{max}$, the dominant contribution comes from non-
valence states associated with higher poles Heger:2021gxt . Thus, in bottom-
changing transitions with a substantial momentum transfer with
$q^{2}_{max}\simeq 15$ GeV2, the inclusion of additional higher-order
contributions is necessary to accurately predict the dynamics of the physical
decay processes. This can be achieved by assuming Melikhov:2000yu

$\displaystyle{F}(q^{2})=\frac{{F}(0)}{\left(1-q^{2}/M_{p}^{2}\right)\left(1-a(q^{2}/M_{p}^{2})+b(q^{4}/M_{p}^{4})\right)}\,.$
(14)

The terms involving the slope parameters a and b introduce effective poles,
similar to the description provided for Eq. (13). For meson (bottom-changing)
transitions described by Eq. (14), the masses of the lowest resonances are
expected to be closer to the bottom mass. As before, we set the pole masses in
Eq. (14) equal to the physical states given in Table 1 and describe it as a
three-parameter fit. The phenomenological accuracy and reliability of $q^{2}$
dependence, given by Eq. (14), have been extensively discussed in Refs.
Melikhov:2000yu ; Melikhov:2001zv . As indicated in Ref. Becirevic:2014kaa ,
it was observed that when the value of $q^{2}$ is high, the form factors
cannot be fully described by only considering the initial few physical poles.
In order to explain experimental data effectively, an expansion of the form
factors using a series of poles is necessary.

We wish to remark that various other theoretical works have used the following
$q^{2}$ structure

$\displaystyle{F}(q^{2})=\frac{F(0)}{1-a(q^{2}/M_{B_{c}}^{2})+b(q^{4}/M_{B_{c}}^{4})},$
(15)

with the parent pole mass Cheng:2003sm . Eq. (15) is expected to be valid for
the region that lies farther away from the physical decay region. More
appropriately for $A(q^{2})$ and $V_{0}(q^{2})$ form factors where axial-
vector and scalar resonances may contribute. Further, Eq. (15) can be regarded
as an approximation of Eq. (14) with modified slope parameters.

To determine the form factors for $B_{c}\to A^{({\prime})}$ we employ self-
consistent CLFQM (with type-II correspondence), details of which are discussed
in Sec. IV.

##  IV Self-consistency in covariant light-front approach

As already stated, we focus on the proposed self-consistency in the CLFQM, for
which details can be found in Refs. Jaus:1999zv ; Jaus:2002sv ; Cheng:2003sm ;
Choi:2013mda ; Chang:2018zjq ; Chang:2019mmh ; Chang:2020wvs . Therefore, we
summarize the formulation of the theoretical framework and adopt the notations
of Chang:2019mmh . For a meson transition, we define the momentum of the
initial and final mesons as $p^{\prime}$ and $p^{{\prime}{\prime}}$,
respectively. Further, the meson momenta can be expressed in terms of the off-
shell quark ($q_{1}$), and an antiquark ($\bar{q}_{2}$), momenta, i.e,
$p^{{\prime}({\prime}{\prime})}=k_{1}^{{\prime}({\prime}{\prime})}+k_{2}$,
where $k_{1}^{{\prime}({\prime}{\prime})}$ and $k_{2}$ represent quarks’
momenta. We denote $m_{1}^{{\prime}({\prime}{\prime})}$ and $m_{2}$ as the
masses of the incoming (outgoing) quark and the spectator quark, respectively.
In addition, quarks’ momenta can be expressed in terms of the internal light-
front (LF) variables $(x,\mathbf{k_{\bot}})$ as

$\displaystyle
k_{1}^{+}=xp^{+}\,,\quad\,\mathbf{k}_{1\bot}=x\mathbf{p}_{\bot}+\mathbf{k}_{\bot}\,,\qquad
k_{2}^{+}=\bar{x}p^{+}\,,\quad\,\mathbf{k}_{2\bot}=\bar{x}\mathbf{p}_{\bot}-\mathbf{k}_{\bot}\,,$
(16)

with $x+\bar{x}=1$. The meson momentum is defined as
$p=(p^{+},p^{-},\mathbf{p}_{\bot})$ with $p^{\pm}=p^{0}\pm p^{z}$, such that
$p^{2}=p^{+}p^{-}-\mathbf{p}^{2}_{\bot}$, while the transverse quark and meson
momenta are given by $\mathbf{k}_{\bot}=(k^{x}\,,k^{y})$ and
$\mathbf{p}_{\bot}=(p^{x}\,,p^{y})$, respectively. Conventionally, a meson
bound-state ($q_{1},\bar{q}_{2}$) can be represented as

$\displaystyle|M(p,^{2S+1}L_{J},J_{z})\rangle=\int\\{d^{3}\tilde{k}_{1}\\}\\{d^{3}\tilde{k}_{2}\\}\leavevmode\nobreak\
2(2\pi)^{3}\delta^{3}({\tilde{p}-\tilde{k}_{1}-\tilde{k}_{2}})\leavevmode\nobreak\
$
$\displaystyle\times\sum_{h_{1},h_{2}}\Psi^{JJ_{z}}_{LS}(\tilde{k}_{1},\tilde{k}_{2},h_{1},h_{2})|q_{1}(k_{1},h_{1})\bar{q}_{2}(k_{2},h_{2})\rangle,$
(17)

where L and J are angular orbital and total spin quantum numbers, respectively
Cheng:2003sm . Further, $\tilde{p}=(p^{+},\mathbf{p_{\bot}})$, and
$\tilde{k}_{1,2}=(k_{1,2}^{+},\mathbf{k}_{1,2\bot})$ represent the on-mass-
shell LF momenta, and $\\{d^{3}\tilde{k}\\}\equiv{dk^{+}d^{2}k_{\bot}\over
2(2\pi)^{3}}$. The wave function
$\Psi^{JJ_{z}}_{LS}(\tilde{k}_{1},\tilde{k}_{2},h_{1},h_{2})$, which describes
the distribution of momentum in space for ${}^{2S+1}L_{J}$ meson, can be
written as

$\Psi^{JJ_{z}}_{LS}(\tilde{k}_{1},\tilde{k}_{2},h_{1},h_{2})=\frac{1}{\sqrt{N}_{c}}\langle
LS;L_{z}S_{z}|LS;JJ_{z}\rangle
R^{SS_{z}}_{h_{1}h_{2}}(x,\mathbf{k}_{\bot})\leavevmode\nobreak\
\psi_{LL_{z}}(x,\mathbf{k}_{\bot}).$ (18)

The radial wave function $\psi_{LL_{z}}(x,\mathbf{k}_{\bot})$ characterizes
how the constituent quarks’ momenta are distributed in a bound state that
possesses orbital angular momentum $L$, and the corresponding Clebsch-Gordan
coefficient is represented by $\langle LS;L_{z}S_{z}|LS;JJ_{z}\rangle$. The
spin-orbital LF wave function, $R^{SS_{Z}}_{h_{1}h_{2}}$, represents definite
spin state ($S,S_{Z}$) corresponding to the LF helicity ($h_{1},h_{2}$)
eigenstates. For more details on the normalization procedure see Cheng:2003sm
. The radial wave function is usually assumed to be of the Gaussian-type

$\displaystyle\psi_{s}(x,\mathbf{k}_{\bot})=$ $\displaystyle
4\frac{\pi^{\frac{3}{4}}}{\beta^{\frac{3}{2}}}\sqrt{\frac{\partial
k_{z}}{\partial
x}}\exp\left[-\frac{k_{z}^{2}+\mathbf{k}_{\bot}^{2}}{2\beta^{2}}\right]\,$
(19)

and

$\displaystyle\psi_{p}(x,\mathbf{k}_{\bot})=$
$\displaystyle\frac{\sqrt{2}}{\beta}\psi_{s}(x,\mathbf{k}_{\bot})\,,$ (20)

for s-wave and p-wave mesons, respectively. The shape parameter, $\beta$, in
Eq. (19), which describes the momentum distribution, is expected to be of
order $\Lambda_{QCD}$ Verma:2011yw . The relative momentum in the
$z$-direction, $k_{z}$, is expressed as,

$\displaystyle
k_{z}=(x-\frac{1}{2})M_{0}+\frac{m_{2}^{2}-m_{1}^{2}}{2M_{0}}\,,$ (21)

and

$\displaystyle\frac{\partial k_{z}}{\partial
x}=\frac{M_{0}}{4x(1-x)}[1-(\frac{m^{2}_{1}-m^{2}_{2}}{M^{2}_{0}})^{2}]$ (22)

Choi:2021mni , where the kinetic invariant mass squared is given by
Chang:2019mmh ; Chang:2020wvs

$\displaystyle
M_{0}^{2}=\frac{m_{1}^{2}+\mathbf{k}_{\bot}^{2}}{x}+\frac{m_{2}^{2}+\mathbf{k}_{\bot}^{2}}{\bar{x}}\,,$
(23)

with
$\mathbf{k}_{\bot}^{{\prime}{\prime}}=\mathbf{k}^{{\prime}}_{\bot}-\bar{x}\mathbf{q}_{\bot}$.
For the meson transition, one can take a convenient Drell-Yan-West frame,
characterized by $q^{+}=0$ Chang:2019mmh . The implication of $q^{+}=0$ is
such that the form factors are known only for space-like momentum transfer,
i.e., $q^{2}=-\mathbf{q}_{\bot}^{2}\leqslant 0$. One shall extrapolate $q^{2}$
to the time-like region ($q^{2}>0$) to obtain these form factors. We follow
Chang:2019mmh ; Chang:2020wvs to calculate $M^{\prime}\to
M^{{\prime}{\prime}}$ transition matrix element given by

$\displaystyle{\cal B}(q^{2})=N_{c}\int\frac{{\rm
d}^{4}k_{1}^{{\prime}}}{(2\pi)^{4}}\frac{H_{M^{{\prime}}}H_{M^{{\prime}{\prime}}}}{N_{1}^{{\prime}}\,N_{1}^{{\prime}{\prime}}\,N_{2}}iS_{\cal
B}\,.$ (24)

Thereafter, a conversion of the covariant formulation to LF is achieved by
implementing the following substitutions, as given in Refs. Jaus:1999zv ;
Cheng:2003sm ,

$\displaystyle N_{1}\to\hat{N}_{1}=x\left(M^{2}-M_{0}^{2}\right),$ (25)

and

$\displaystyle\chi_{M}=H_{M}/N_{1}\to h_{M}/\hat{N}_{1}\,,\qquad D_{M,{\rm
con}}\to D_{M,{\rm LF}}\,,\qquad$ (26)

which is referred to as the type-I correspondence scheme between the covariant
BS and LF approaches Jaus:1999zv ; Choi:2013mda ; Cheng:2003sm . The LF forms
of the vertex function, $\chi_{M}$, for P, $A_{A}$, and $A_{B}$ mesons are
given by

$\displaystyle\chi_{P}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N_{c}}}\sqrt{\frac{\bar{x}}{x}}\frac{\psi_{s}}{\hat{M}^{\prime}_{0}},$
$\displaystyle\chi^{\prime}_{{}^{A_{A}}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N_{c}}}\sqrt{\frac{\bar{x}}{x}}\frac{\hat{M}^{{\prime}2}_{0}}{2\sqrt{2}M^{\prime}_{0}}\frac{\psi_{p}}{\hat{M}^{{\prime}}_{0}},$
$\displaystyle\chi^{\prime}_{{}^{A_{B}}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N_{c}}}\sqrt{\frac{\bar{x}}{x}}\frac{\psi_{p}}{\hat{M}^{\prime}_{0}},$
$\displaystyle D_{{}^{A_{A}}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\hat{M}^{{\prime}2}_{0}}{m^{\prime}_{1}-m_{2}},\quad
D^{\prime}_{{}^{A_{B}}}=2\,.$ (27)

with $\hat{M}_{0}^{2}\equiv M_{0}^{2}-(m_{1}^{{\prime}}-m_{2})^{2}$. It may be
noted that type-I correspondence suffers from the problems of self-consistency
and covariance, which have been thoroughly investigated in Cheng:2003sm ;
Choi:2013mda . It has been argued that these issues are connected with zero-
mode contributions in type-I correspondence. Furthermore, it has been pointed
out that, in addition to vector and axial-vector decay constants, the $P\to V$
meson transition form factors have self-consistency problems Choi:2013mda .
Due to the inconsistency, the form factors and decay constants extracted from
the longitudinal and transverse polarization states are not equal within
type-I CLFQM Chang:2019mmh . This inconsistency arises because of the
contributions from $B^{(i)}_{j}$ functions. Likewise, $P\to V$ transition form
factors, these issues are expected to affect $P\to A$ transition form factors
as well. These issues can be resolved with type-II correspondence after
replacing the meson mass ($M$) with the kinetic invariant mass ($M_{0}$). The
effect of type-II correspondence is such that the contributions from
$B^{(i)}_{j}$ functions numerically vanish. In $P\to A$ transitions, the
$l(q^{2})$ and $c_{-}(q^{2})$ form factors receive zero-mode contributions for
the longitudinal polarization states through $B^{(2)}_{1}$ and $B^{(3)}_{3}$
functions, which are absent in the transverse polarization states. The $M\to
M_{0}$ transformation eliminates the unnecessary spurious contribution and
shows self-consistency in $P\to A$ transition form factors, i.e., their
numerical evaluation is free from zero-mode contributions.

It is crucial to recognize that the type-II correspondence relies on the fact
that while results from the BS model provide information about binding energy
($M^{2}-M_{0}^{2}$), LF results are derived based on a zero binding energy
($M\to M_{0}$) Choi:2021qza . Therefore, Choi and Ji Choi:2013mda have
proposed a generalized type-II correspondence scheme given as

$\sqrt{2N_{c}}\frac{\chi_{V}(x,k_{\bot})}{1-x}\to\frac{\psi_{V}(x,k_{\bot})}{\sqrt{x(1-x)}\hat{M}_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\ M\to M_{0}.$ (28)

The type-II correspondence scheme effectively resolves the issues concerning
self-consistency and covariance in axial-vector mesons form factors $l(q^{2})$
and $c_{-}(q^{2})$. After some simplifications (see Chang:2019mmh ), Eq. (24)
can be written into a more convenient LF form

$\displaystyle\hat{{\cal B}}=N_{c}\int\frac{{\rm d}x{\rm
d}^{2}\mathbf{k}_{\bot}^{{\prime}}}{2(2\pi)^{3}}\frac{h_{M^{{\prime}}}h_{M^{{\prime}{\prime}}}}{\bar{x}\hat{N}_{1}^{{\prime}}\,\hat{N}_{1}^{{\prime}{\prime}}\,}\hat{S}_{\cal
B}\,.$ (29)

In Refs. Jaus:1999zv ; Cheng:2003sm , it has been observed that the quantity
${\cal B}$ is subjected to additional unwanted spurious contributions that are
eliminated by zero-mode contributions.

It has been established that the $\omega$ dependence defined by
$\omega^{\mu}=(0,2,\mathbf{0}_{\bot})$, mainly arising through $C^{(i)}_{j}$
functions, can be eliminated by incorporating appropriate zero-mode
contributions Jaus:1999zv . However, there exist remnant $\omega$ dependencies
related to the $B^{(i)}_{j}$ functions, which are independent of zero-mode
contributions. Such remnant contributions violate the covariance of the LF
matrix element in the type-I scheme. Therefore, one can conclude that the
self-consistency and covariance originate for the same reason. The authors in
Chang:2019mmh have shown that both these problems can be simultaneously
resolved by employing type-II correspondence. Thus, we proceed by using the
proposed type-II correspondence of the CLFQM to obtain the form factors,

$\displaystyle[{F}(q^{2})]=N_{c}\int\frac{{\rm d}x\,{\rm d}^{2}{\bf
k}_{\bot}^{{\prime}}}{2(2\pi)^{3}}\frac{\chi^{{\prime}}_{P}\chi_{A_{A},A_{B}}^{{\prime}{\prime}}}{\bar{x}}\,\widetilde{F}(x,{\bf
k}_{\bot}^{{\prime}},q^{2})\,,$ (30)

where the function $\widetilde{F}(x,{\bf
k}_{\bot}^{{\prime}},q^{2})=\widetilde{q}(x,{\bf
k}_{\bot}^{{\prime}},q^{2}),\widetilde{l}(x,{\bf k}_{\bot}^{{\prime}},q^{2})$,
and $\widetilde{c}_{\pm}(x,{\bf k}_{\bot}^{{\prime}},q^{2})$ are defined in
Appendix B. The numerical discussion of form factors and their $q^{2}$
behavior is discussed in the next section.

##  V Decay rate and helicity amplitudes

The branching ratio of semileptonic $B_{c}$ decays involving axial-vector
mesons can be written as333We define our expressions for $\ell^{+}$.

$\mathcal{B}(B_{c}\to
A\ell\nu_{\ell})=\frac{\tau_{B_{c}}}{\hbar}\int_{m_{\ell}^{2}}^{q^{2}_{max}}\left(\frac{d\Gamma(B_{c}\to
A\ell\nu_{\ell})}{dq^{2}}\right)\,dq^{2},$ (31)

where $\tau_{B_{c}}$ denote $B_{c}$ meson lifetime. The differential decay
rate for $B_{c}\to A\ell\nu_{\ell}$ is Hernandez:2006gt ; Ivanov:2005fd

$\frac{d\Gamma(B_{c}\to
A\ell\nu_{\ell})}{dq^{2}}=\frac{G_{F}^{2}}{(2\pi)^{3}}|V_{q_{1}q_{2}}|^{2}\frac{\lambda^{1/2}q^{2}}{24M_{B_{c}}^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}{\cal
H}_{\rm Total},$ (32)

where $m_{\ell}$ is the lepton mass. The 3-momentum distribution of the hadron
in the $B_{c}$ rest frame is given by the Källén function
$\lambda\equiv\lambda(M_{B_{c}}^{2},M_{A}^{2},q^{2})=(M_{B_{c}}^{2}+M_{A}^{2}-q^{2})^{2}-4M_{B_{c}}^{2}M_{A}^{2}$.
The total helicity amplitude ${\cal H}_{\rm Total}$ is expressed in terms of
helicity components ${\cal H}_{i}(i=U,L,P,S,SL$) as,

${\cal H}_{\rm Total}=({\cal H}_{U}+{\cal
H}_{L})\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}{\cal
H}_{S}.$ (33)

Further, the unpolarized–transverse ($U$), longitudinal ($L$), parity–odd
($P$), scalar ($S$) and scalar– longitudinal interference ($SL$) helicity
components can be expressed in terms of four helicity amplitudes $H_{i}$
($i=+,-,0,t$) as Ivanov:2006ni

${\cal H}_{U}=|H_{+}|^{2}+|H_{-}|^{2},\quad{\cal H}_{L}=|H_{0}|^{2},\quad{\cal
H}_{P}=|H_{+}|^{2}-|H_{-}|^{2},\\\ {\cal H}_{S}=|H_{t}|^{2},\quad{\cal
H}_{SL}=Re(H_{0}H_{t}^{\dagger}).$ (34)

The helicity amplitudes are related to hadronic transition form factors as
given below:

$\displaystyle H_{\pm}(q^{2})$ $\displaystyle=$
$\displaystyle-(M_{B_{c}}-M_{A})V_{1}(q^{2})\pm\frac{\lambda^{1/2}}{M_{B_{c}}-M_{A}}A(q^{2}),$
(35) $\displaystyle H_{0}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2M_{A}\sqrt{q^{2}}}\left[-(M_{B_{c}}-M_{A})(M_{B_{c}}^{2}-M_{A}^{2}-q^{2})V_{1}(q^{2})+\frac{\lambda}{M_{B_{c}}-M_{A}}V_{2}(q^{2})\right],$
(36) $\displaystyle H_{t}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{1/2}}{\sqrt{q^{2}}}V_{0}(q^{2}).$ (37)

In addition to the branching ratio, we also study multiple experimental
observables for a comprehensive understanding of semileptonic $B_{c}$ decays
as given below:

  * 1.

Forward-backward asymmetry $(A_{FB})$: It is defined as the difference between
the number of particles produced in the forward and backward directions. The
expression for $A_{FB}(q^{2})$ is given as Hernandez:2006gt ; Ivanov:2005fd ;
Becirevic:2016hea

$A_{FB}(q^{2})=\frac{3}{4}\frac{{\cal H}_{P}+2\frac{m_{\ell}^{2}}{q^{2}}{\cal
H}_{SL}}{{\cal H}_{\rm Total}}.$ (38)

  * 2.

Lepton-side convexity parameter $(C_{F}^{\ell})$: The convexity parameter
gives the angular decay distribution over $\cos\theta$ dependence and is
defined as Ivanov:2019nqd ; Zhang:2020dla

$C^{\ell}_{F}(q^{2})=\frac{3}{4}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)\frac{{\cal
H}_{U}-2{\cal H}_{L}}{{\cal H}_{\rm Total}}.$ (39)

  * 3.

Polarization asymmetry $(P_{L(T)}^{\ell})$: The longitudinal lepton-
polarization asymmetry corresponding to the charge of the final state lepton
is given by Ivanov:2019nqd ; Zhang:2020dla

$P_{L}^{\ell}(q^{2})=\frac{({\cal H}_{U}+{\cal
H}_{L})\left(1-\frac{m_{\ell}^{2}}{2q^{2}}\right)-\frac{3m_{\ell}^{2}}{2q^{2}}{\cal
H}_{S}}{{\cal H}_{\rm Total}}.$ (40)

  * 4.

Longitudinal (transverse) polarization fraction $(F_{L(T)})$: $F_{L}$ for the
final axial-vector meson can be expressed as

$F_{L}(q^{2})=\frac{{\cal
H}_{L}\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}{\cal
H}_{S}}{{\cal H}_{\rm Total}},$ (41)

such that, $F_{T}(q^{2})=1-F_{L}(q^{2})$ Ivanov:2019nqd .

  * 5.

Asymmetry parameter $(\alpha^{*})$: The transverse (longitudinal) composition
of the final state mesons in the decay process are governed by asymmetry
parameter Ivanov:2005fd , i.e.,

$\alpha^{*}(q^{2})=\frac{{\cal H}_{U}+\bar{\cal H}_{U}-2({\cal
H}_{L}+\bar{\cal H}_{L}+3\bar{\cal H}_{S})}{{\cal H}_{U}+\bar{\cal
H}_{U}+2({\cal H}_{L}+\bar{\cal H}_{L}+3\bar{\cal H}_{S})},$ (42)

where $\bar{\cal H}_{i}=\frac{m^{2}_{\ell}}{2q^{2}}{\cal H}_{i}$.

  * 6.

Lepton flavor universality ratio ($R_{A}$): In the SM, the semileptonic decay
modes involving different lepton flavors are expected to be approximately
equal due to the lepton flavor universality, which treats all leptons equally.
The LFU ratio, $R_{A}$, is defined by Wang:2022yyn ,

$R_{A}=\frac{\int_{m_{\ell^{{\prime}}}^{2}}^{q^{2}_{max}}\left(\frac{d\Gamma(B_{c}\to
A\ell^{{\prime}}\nu_{\ell^{{\prime}}})}{dq^{2}}\right)\,dq^{2}}{\int_{m_{\ell}^{2}}^{q^{2}_{max}}\left(\frac{d\Gamma(B_{c}\to
A\ell\nu_{\ell})}{dq^{2}}\right)\,dq^{2}},$ (43)

where $\ell^{{\prime}}$ and $\ell$ represents heavy and light leptons,
respectively. This ratio can be used as a test for the SM predictions.

We wish to emphasize that $A_{FB}(q^{2})$, $C^{\ell}_{F}(q^{2})$,
$P^{\ell}_{L}(q^{2})$, $F^{\ell}_{L}(q^{2})$, and $\alpha^{*}(q^{2})$
observables discussed above are independent. We present the expectation values
of all the aforementioned physical observables by individually integrating the
numerator and denominator over $q^{2}$ inclusive of kinematical factor
$\lambda^{1/2}q^{2}(1-m_{\ell}^{2}/q^{2})^{2}$.

##  VI Numerical results and discussion

In this work, we investigated the $B_{c}\to A_{A}(A_{B})$ transition form
factors in the self-consistent CLF approach, $A_{A}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }A_{B}$ are defined in Appendix A. Consequently, we
predicted the branching ratios of semileptonic weak decays of $B_{c}$
involving $A$ meson in the final state for bottom-conserving and bottom-
changing decay modes. We studied the form factors over their entire $q^{2}$
range and performed a comprehensive analysis of the momentum dependence of
these form factors. We have calculated the form factors using self-consistent
(type-II) CLFQM in the $q^{2}<0$ region and, thereafter, extrapolated them to
time-like region with $q^{2}>0$ using the parameterizations defined in Sec.
III. We employed two different parameterizations for $q^{2}$ dependence based
on the available range of momentum transfer, i.e., for bottom-conserving given
by Eq. (13) and for bottom-changing we used Eq. (14). The self-consistency
proposed in the type-II CLFQM approach Chang:2019mmh confirms that the full
and valence contribution results are equivalent for $P\to V$ transitions
because they are not affected by zero-mode contributions arising from
$B^{(i)}_{j}$ functions. The issue of self-consistency arises not only in
vector mesons but also in axial-vector mesons Chang:2018zjq ; Chang:2019mmh .
Therefore, we conducted a thorough analysis of the self-consistency problem by
examining the $B_{c}\to A_{A}(A_{B})$ form factors, considering both bottom-
conserving and bottom-changing transitions. We calculated the $B_{c}\to
A_{A}(A_{B})$ transition form factors ($A(q^{2})$, $V_{0}(q^{2})$,
$V_{1}(q^{2})$, and $V_{2}(q^{2})$) using the numerical inputs, i.e.,
constituent quark masses and $\beta$, derived from Gaussian-type light-front
wave functions Verma:2011yw ; Chang:2019mmh as listed below. We use

$\begin{gathered}m_{u}=m_{d}=0.27\pm 0.04\leavevmode\nobreak\ \mathrm{GeV},\
m_{s}=0.48\pm 0.05\leavevmode\nobreak\ \mathrm{GeV},\\\ m_{c}=1.6\pm
0.2\leavevmode\nobreak\ \mathrm{GeV},\text{ and }m_{b}=4.8\pm
0.2\leavevmode\nobreak\ \mathrm{GeV}\\\ \end{gathered}$ (44)

as quark masses, and the $\beta$’s (in $\mathrm{GeV}$) are given as
Verma:2011yw ,

$\begin{gathered}\beta_{B_{c}}=0.920\pm 0.092,\ \beta_{D_{1}}=0.389\pm
0.039,\\\ \beta_{\chi_{c1}}=0.420\pm 0.042,\ \beta_{B_{1}}=0.500\pm 0.050,\\\
\text{ and }\beta_{B_{s1}}=0.550\pm 0.055,\end{gathered}$ (45)

except for $\beta_{B_{c}}$, which has been fixed by averaging from the extreme
range available in the literature Verma:2011yw ; Shi:2016gqt ; Chang:2019mmh .
It is worth pointing out that $\beta_{B_{c}}$ is reasonably close to the
recent calculations Chang:2018zjq ; Chang:2019mmh . It should be noted that we
have allowed large uncertainties in $\beta$ parameters as compared to previous
works due to the lack of experimental information. In addition, to account for
flavor dependence, we allowed variation in initial and final states $\beta$
parameters to observe its impact on form factors and branching ratios. The CKM
matrix elements are sourced from Ref. ParticleDataGroup:2022pth . The axial-
vector meson masses used in the numerical calculation are given in Appendix A
and the lepton masses are given as $m_{e}=0.511$ MeV, $m_{\mu}=105.66$ MeV,
and $m_{\tau}=1776.86$ MeV ParticleDataGroup:2022pth . The uncertainties
present in the masses of mesons have been disregarded due to their
significantly smaller magnitude compared with the uncertainties in both quark
masses and shape parameters. An important aspect of our investigation is the
evaluation of transition form factors using the self-consistent CLFQM and
employing the appropriate $q^{2}$ dependence formulation described by Eqs.
(13) and (14). The results for the type-II approach are presented in the
respective columns of Tables 2 and 3, which are comprehensively elucidated in
the subsequent sections. To facilitate comparison, we have computed the type-I
form factors444Note that we used the $B_{c}$ pole mass to comply with Eq.
(15). In addition, we used Eqs. (13) and (14) for the type-I scheme to
estimate the exact magnitude of self-consistency effects, as shown in Tables 2
and 3. utilizing the $q^{2}$ dependence formulation given by Eq. (15). This
allows us to contrast the results obtained from the type-I and type-II
approaches within the CLFQM. A comprehensive comparison between these two
types is presented in Tables 2 and 3, providing valuable insights into the
discrepancies associated with $q^{2}$ formulation and zero-mode contributions.
Furthermore, it should be emphasized that the form factors listed in Tables 2
and 3 will also be affected by the mixing of $A_{A}$ and $A_{B}$ axial-vector
states, which will be thoroughly examined in the forthcoming analysis of
semileptonic decays. After evaluating the form factors, we continue to
calculate the semileptonic branching ratios where we employed all the helicity
amplitudes. In addition, we computed the expectation values of significant
physical observables, namely $A_{FB}$, $C_{F}^{\ell}$, $P_{L}^{\ell}$,
$F_{L}$, and $\alpha^{*}$. These observables provide valuable insights into
the decay processes and have the potential for experimental verification in
future observations.

###  VI.1  Analysis of transition form factors

We studied the form factors of $B_{c}\rightarrow A_{A}(A_{B})$ in bottom-
conserving as well as bottom-changing transitions in the self-consistent type-
II CLF approach. As mentioned before, the $q^{2}$ dependence of the form
factors is described using the parameterization given by Eq. (13) and Eq. (14)
for bottom-conserving and bottom-changing transitions, respectively. By
incorporating the type-II correspondence, we computed the form factors
numerically. The extension of the type-II scheme from $P\to V$ to $B_{c}\to A$
transitions has demonstrated the self-consistency in our approach. We observed
that the longitudinal and transverse form factors $c_{-}(q^{2})(V_{0}(q^{2}))$
and $l(q^{2})(V_{1}(q^{2}))$ in type-II CLFQM result in equivalent numerical
values, which confirms their self-consistency. The obtained results are
presented in Tables 2 and 3. We list our observations below:

|
---|---
|

Figure 1: $q^{2}$ dependence of $B_{c}\rightarrow B_{1}$ form factors.

  * i.

The observed $q^{2}$ range for bottom-conserving $B_{c}\rightarrow B_{(s)1}$
transitions is limited to a narrow interval of $0\lesssim q^{2}\lesssim 0.30$
GeV2, indicating the small magnitude of available $q^{2}$ range. Consequently,
it is anticipated that these form factors will demonstrate minimal variations
within the permissible $q^{2}$ range, as described by Eq. (13). Furthermore,
Table 2 provides the numerical values of the form factors $A(0)$, $V_{0}(0)$,
$V_{1}(0)$, and $V_{2}(0)$ for $B_{c}\to A_{A}(A_{B})$ transitions without
considering any mixing between the $A_{A}$ and $A_{B}$ states. However, these
transition form factors will exhibit mixing when they appear in the decay
processes.

  * ii.

To observe the variation of form factors with respect to $q^{2}$, we plotted
all the bottom-conserving $B_{c}\to B_{(s)1}$ transition form factors, as
shown in Figures 1 and 2. As expected we observe that these form factors show
negligible variation with respect to $q^{2}$. In addition, to examine the
effect of mixing on the magnitude and variation of $q^{2}$ in the transition
form factors involving $A_{A}$ and $A_{B}$ states, we have also plotted the
mixed transition form factors $B_{c}\to A^{({\prime})}$. It is worth
mentioning that the magnitude of form factor $V_{1}(q^{2})$ is larger for both
mixed and unmixed transitions as compared to other form factors.

|
---|---
|

Figure 2: $q^{2}$ dependence of $B_{c}\rightarrow B_{s1}$ form factors.

  * iii.

Also, we notice that numerical values of all the bottom-conserving transition
form factors are positive except for $V_{2}^{B_{c}B_{(s)1A}}(0)$. In addition,
the numerical values of the slope parameter a are negative for most of the
form factors. The sign and magnitude of the slope parameter555The mixing among
the slope parameters corresponding to $A_{A}$-type and $A_{B}$-type axial-
vector mesons have been considered. signifies how sharply the form factor will
vary with respect to allowed $q^{2}$. Interestingly, the slope parameter has
magnitude less than one for most of the form factors except for
$V_{2}^{B_{c}B_{s1B}}(q^{2})$, which indicates that the form factors do not
receive significant contributions from higher-order resonances beyond the
decay region. One may suspect the influence of the spectator quark mass on
these transition form factors, however, lattice QCD calculations have recently
shown that the bottom-conserving form factors do not depend upon the spectator
quark mass Cooper:2020wnj . To investigate the impact of varying quark masses
and $\beta$ parameters on the form factors, we evaluate these form factors by
considering uncertainties in the quark masses and $\beta$ parameters given by
(44) and (45), respectively. The corresponding results are given in Table 2.
Furthermore, we observed that the numerical values of the form factors
associated with the $B_{c}\to A_{B}$ transitions are more prominent than
$B_{c}\to A_{A}$ form factors.

  * iv.

It may be noted that the form factors for bottom-conserving transitions at
$q^{2}=0$ differ marginally compared to those observed at their maximum
$q^{2}$, i.e., $q^{2}_{max}$. This equivalence can be attributed to the
limited range of available $q^{2}$. This observation suggests that a
straightforward parameterization, as described in Eq. (13), is sufficient to
explain these form factors. To simultaneously emphasize the effect of type-II
correspondence and the sensitivity of the form factors with respect to the
choice of the formulation of $q^{2}$ dependence, we also investigate the form
factors in the type-I scheme Cheng:2003sm for most commonly used $q^{2}$
dependency given by Eq. (15). The numerical results for type-I CLFQM are
presented in columns $5-8$ of Table 2. We found that the numerical values of
all the form factors for the type-I scheme are less than one apart from
$V_{1}(0)$, which take numerical values greater than $10$ for all the
transitions using Eq. (15) and Eq. (13). In addition to $V_{1}(0)$ form
factor, $V_{0}(0)$ form factor show a substantial change in the numerical
values. Moreover, the slope parameter, $a$, exceeds unity and its magnitude
increases as we approach small numerical values of form factors at $q^{2}=0$.
Additionally, the slope parameter $b\gtrsim 1$ is observed for the form
factors pertaining to $B_{c}\to A_{B}$ transitions, with an exceedingly high
value noted specifically for $V_{2}^{B_{c}B_{(s)1B}}(0)$.

  * v.

Dynamically, the contributions to the branching ratios of semileptonic decays
mainly come from the terms containing form factors $V_{1}(q^{2})$,
$V_{2}(q^{2})$, and $A(q^{2})$, while the impact of the term with form factor
$V_{0}(q^{2})$ is negligible. It is important to note that in vector meson
emitting decays of the $B_{c}$ meson, the contributions of the $A_{2}(q^{2})$
form factor is usually disregarded because of its small magnitude resulting
from division by $(M_{B_{c}}+M_{V})$ in the amplitude. However, such
contributions cannot be overlooked in $B_{c}\to A$ semileptonic decays, where
it is divided by $(M_{B_{c}}-M_{A})$ as shown in Eq. (35). In general, large
numerical values for $V_{1}(q^{2})$ form factors cause substantial amplitudes,
and therefore, large branching ratios involving these transition form factors.
On the other hand, for the type-I scheme, the branching ratio $B_{c}\to
A(A^{\prime})$ semileptonic decays are expected to be underestimated
(overestimated) due to exceedingly large values of the $V_{1}(q^{2})$ form
factors and their mixing.

  * vi.

We wish to point out that the numerical values of the form factors in type-I
correspondence suffer from a two-fold problem. First, it has large values of
slope parameters for most of the form factors; second, the form factor
$V_{1}(0)$ attains an excessively large numerical value. One may expect that
consideration of appropriate pole masses may resolve the issues, however, the
problem persists. We observed that the inclusion of transition pole masses, as
listed in Table 1 for type-I scheme with Eq. (13), leads to smaller values of
the slope parameters, yet the numerical values of the $V_{1}(0)$ ($V_{0}(0)$)
transition form factors remain large. Most recently, Li et al. Li:2023wgq have
obtained these transition form factors using different $q^{2}$
parameterization, which suffer from a similar issue where the form factors at
$q^{2}=0$ are small but the slope parameters take exceedingly large numerical
values. Thus, we conclude that it is very difficult to accurately predict the
behavior of the form factors within type-I correspondence. The same can be
observed from the numerical results of CLFQM type-I analysis Shi:2016gqt .
Furthermore, we found that such dubious behavior can be attributed to zero-
mode contributions in the type-I scheme, which affects the form factors
$V_{0}(q^{2})$ and $V_{1}(q^{2})$. We also observe that the self-consistency
effects are predominantly large for bottom-conserving decays. The efficient
resolution for the above stated problems can be reached by employing type-II
correspondence, as the zero-mode contributions appearing through $B^{(i)}_{j}$
functions numerically vanish when $M\to M_{0}$ Choi:2013mda . This has been
thoroughly discussed in Sec. IV.

  * vii.

For bottom-changing transitions, particularly in $B_{c}\rightarrow D_{1}$ form
factors, the $q^{2}$ range is expected to be significantly broader with
$0<q^{2}<15$ GeV2. In contrast to bottom-conserving transitions, this
extensive range provides an opportunity to analyze how the dependence on
$q^{2}$ affects the form factors involved in these transitions. For the
reasons stated in Sec. III, we adopt $q^{2}$ parameterization provided in Eq.
(14) for which the resulting numerical values of the form factors and slope
parameters are listed in columns $2-5$ of Table 3. Similar to bottom-
conserving transitions, we have plotted mixed and unmixed transition form
factors involving axial-vector charm mesons as shown in Figure 3. Throughout
the accessible range of $q^{2}$, we observe a consistent pattern in the
behavior of all the form factors, with slight deviations for some of the mixed
form factors. All unmixed form factors have positive numerical values except
for $V_{2}(0)$, which deviates from this trend by assuming negative values for
$B_{c}\to D_{1B}$. It may be noted that $A^{B_{c}D_{1B}}(0)$ form factor
acquires a small numerical value, indicating that its magnitude is relatively
small and approaches zero at $q^{2}_{max}$. Further, these form factors show
significantly different $q^{2}$ behavior after mixing. Since the contribution
to decay width corresponding to $V_{0}(q^{2})$ form factor is negligible, the
variation of $V_{1}(q^{2})$ form factor with respect to $q^{2}$ have
significant implications and is a matter of special interest in semileptonic
decay analysis.

  * viii.

In contrast to bottom-conserving decays, the bottom-changing decays are
characterized by two slope parameters denoted by $a$ and $b$, as described in
Eq. (14). Both $a$ and $b$ acquire positive values greater than one, with few
exceptions, and $a<b$ for all type-II transitions. A similar trend can be
observed for type-I results using $q^{2}$ dependence given by Eq. (15), as
shown in column $6-9$ of Table 3. It is worth mentioning that even though the
central values of form factors $A(0)$ and $V_{2}(0)$ are similar between
type-I and type-II schemes, their respective slope parameters differ
significantly with larger magnitudes observed exclusively for parameter $b$.
This shows that the form factors with $q^{2}$ dependence given by Eq. (15) are
more sharply varying. Moreover, the utilization of $q^{2}$ dependence
expressed by Eq. (14) (with transition pole masses) in the type-I scheme
addresses this issue effectively, as observed in bottom conserving
transitions. However, it does not significantly impact the values of form
factors at $q^{2}=0$. It is worth noting that the type-I scheme has a
negligible effect on the central values of all form factors for both Eq. (14)
and Eq. (15), whereas the type-II scheme shows a small but reasonable
variation in $V_{0}$ and $V_{1}$. Another characteristic feature of zero-mode
transitions can be observed in the variation of affected form factors
throughout the large $q^{2}$ range, i.e., they show considerable change in
numerical values at $q^{2}_{max}$ for transition pole masses in type-I, as
could also be seen in the results of Ref. Li:2023wgq . Furthermore, the form
factor $A^{B_{c}D_{1B}}(0)$ is negligibly small and therefore possesses
exceptionally large slope parameters for both schemes, as shown in Table 3. As
a result, the mixed form factors $A^{B_{c}D_{1}^{({\prime})}}(q^{2})$ approach
negligible values at large $q^{2}$ as shown in Figure 3. It is interesting to
note that after mixing the form factors corresponding to $B_{c}\to D_{1}$
transitions become negative except for $V_{0}(q^{2})$ transition.

|
---|---
|

Figure 3: $q^{2}$ dependence of $B_{c}\to D_{1}$ form factors.

  * ix.

In the case of charmonium states, it may be noted that the transition form
factors $B_{c}\to\chi_{c1}^{({\prime})}$ are free from mixing effects666As
mentioned in the Appendix A, we are denoting $h_{c}$ as
$\chi_{c1}^{{\prime}}$.. The $q^{2}$ dependence of these transition form
factors is shown in Figure 4. We observe that the form factors
$V_{1}^{B_{c}\chi_{c1}}(q^{2})$ and $V_{2}^{B_{c}\chi_{c1}^{\prime}}(q^{2})$
decrease with increasing $q^{2}$. However, the form factors
$V_{0}^{B_{c}\chi_{c1}^{\prime}}(q^{2})$ and
$V_{1}^{B_{c}\chi_{c1}^{\prime}}(q^{2})$ change almost linearly with respect
to $q^{2}$. Furthermore, we observe that $B_{c}\to\chi_{c1}^{({\prime})}$ form
factors are least affected by the problem of self-consistency as compared to
other transitions, as can be observed from Table 3.

  * x.

In general, a comparison of the numerical values of the form factor between
type-I and type-II correspondence reveals that the effect of self-consistency
and covariance leads to significant changes in the numerical values of
$V_{0}(q^{2})$ and $V_{1}(q^{2})$. Such deviations between the two schemes are
expected to be decisive for the study of weak semileptonic decays. We also
observe that the effects of self-consistency on bottom-changing transition
form factors are smaller than those of the bottom-conserving transition form
factors. As a result, the quest of self-consistency in the decays is of utmost
importance to make reliable predictions given the strong dependence on
$V_{1}(q^{2})$ form factor.

|
---|---
|

Figure 4: $q^{2}$ dependence of $B_{c}\to\chi_{c1}^{({\prime})}$ form factors.

We have employed the type-II correspondence to axial-vector emitting
transition for both bottom-changing and bottom-conserving decays. Finally, we
ensure that on the application of type-II correspondence to axial-vector
emitting transition form factors for both bottom-changing and bottom-
conserving $B_{c}\to A$ transitions are self-consistent, i.e., zero-mode
contributions vanish numerically. We now proceed to calculate branching ratios
of semileptonic $B_{c}\to A^{({\prime})}\ell\nu_{\ell}$ decays involving
$B_{c}\to A^{({\prime})}$ transition form factors.

###  VI.2 Semileptonic decays

In this subsection, we predict the branching ratios of the semileptonic weak
decays of $B_{c}$ meson to the bottom and charm axial-vector mesons in the
final state, as shown in Table 4. To evaluate these branching ratios, we
utilize the decay rate formulae given by Eqs. (31) and (32) and the numerical
values of the form factors are listed in Tables 2 and 3. In addition, we
calculate the expectation values of the observables, such as the forward-
backward asymmetry $\langle A_{\rm FB}\rangle$, lepton-side convexity
parameter $\langle C_{F}^{\ell}\rangle$, polarization parameters ($\langle
P_{L}^{\ell}\rangle,\leavevmode\nobreak\ \langle F_{L}\rangle$), and asymmetry
parameter $\langle\alpha^{*}\rangle$, which are of significant experimental
importance. Predictions corresponding to these observables are given in the
respective columns of Table 5.

####  VI.2.1 Bottom-conserving decay modes

The bottom-conserving semileptonic decay modes of $B_{c}$ mesons, which
involve the bottom mesons in the final states, undergo kinematic suppression
due to their heavier masses. These semileptonic decay processes provide an
excellent opportunity to observe the effects of form factors on the branching
ratios and, therefore, to test the theoretical models. As mentioned earlier,
the branching ratios are affected by the numerical values of the form factor
as well as their signs. In addition, kinematic and CKM factors play a
significant role in determining their magnitude. We analyzed $B_{c}\to
B_{(s)1}$ semileptonic transitions using the self-consistent CLF approach. We
list our major findings as follows:

  * i.

Despite the kinematic suppression of bottom-conserving modes, our predictions
indicate that the branching ratios for these decay channels are of the order
of $\mathcal{O}(10^{-4})$ to $\mathcal{O}(10^{-6})$. Among them, the
semileptonic decays $B_{c}^{+}\to B_{s1}^{0{\prime}}e^{+}\nu_{e}$ and
$B_{c}^{+}\to B_{s1}^{0{\prime}}\mu^{+}\nu_{\mu}$ are most dominant with
branching ratios $({1.04}^{+0.36+1.03}_{-0.00-0.08})\times 10^{-4}$ and
$({8.37}^{+1.89+6.12}_{-0.00-0.98})\times 10^{-5}$, respectively. In these
CKM-favored decays, the dominant CKM factor ($V_{cs}$) overcomes the kinematic
suppression and leads to significant branching ratios similar to those
observed in $b\to u(c)$ transitions. However, the branching ratios of the
$B_{c}\to B_{1}^{({\prime})}\ell\nu_{\ell}$ decays are smaller due to lower
value of the associated CKM factor ($V_{cd}$). $B_{c}^{+}\to
B_{1}^{0{\prime}}e^{+}\nu_{e}$ and $B_{c}^{+}\to
B_{1}^{0{\prime}}\mu^{+}\nu_{\mu}$, have the larger branching ratios given by
$({2.12}^{+0.22+0.45}_{-0.04-0.32})\times 10^{-5}$ and
$({1.77}^{+0.15+0.37}_{-0.03-0.29})\times 10^{-5}$, respectively.

  * ii.

In addition, we observe that the branching ratios of $B_{c}$ decays to primed
axial-vector mesons are roughly larger by an order of magnitude than those of
decays involving unprimed axial-vector mesons. This difference between their
branching ratios can be attributed mainly to the combined effect of axial-
vector meson mixing and the numerical values of the form factors. Therefore,
experimental determination of the mixing angle can provide valuable insights
into these decay channels. We wish to point out that we have also estimated
the uncertainties in the branching ratios corresponding to errors in form
factors. The uncertainties in the branching ratios corresponding to form
factors because of quark masses and $\beta$ parameters are dealt with
independently. It is interesting to note that despite the nearly symmetric
uncertainties in the form factors, the uncertainties in the branching ratios
are highly asymmetric, particularly for bottom-conserving decays. Although the
impact of uncertainties on branching ratios is mostly incremental, we noticed
that variations in quark masses result in changes opposite to those caused by
$\beta$ uncertainties. However, for the sake of uniformity in the
presentation, we have given the respective mass uncertainties similar to those
of $\beta$. As previously mentioned, we anticipated significant uncertainties
due to the inclusion of a wide range of values for $\beta$ parameter, in
addition to variation in quark masses. Furthermore, the branching ratios show
greater sensitivity to variation due to uncertainties corresponding to $\beta$
parameter. The uncertainties in $\beta$ (quark mass) lead to a maximum change
in branching ratios roughly by a factor of three (two). Taking into account
both types of uncertainties, we notice that the maximum increase (by a factor
of four) in the branching ratios is observed for the decay $B_{c}\to
B_{s1}e\nu_{e}$. On the other hand, uncertainties for $B_{c}\to
B_{1}^{({\prime})}\ell\nu_{\ell}$ are relatively smaller with a maximum
increase (approximately by a factor of two) observed for $B_{c}\to
B_{1}e\nu_{e}$ and $B_{c}\to B_{1}^{({\prime})}\mu\nu_{\mu}$. This further
highlights the sensitivity of branching ratios to form factors, slope
parameters, and mixing777Note that uncertainty in slope parameters play a
significant role in overall errors in the branching ratios.. We wish to point
out that we observe large uncertainties mainly for $B_{c}\to
B^{({\prime})}_{s1}\ell\nu_{\ell}$ decays where the branching ratio of
$B_{c}\to B_{s1}^{{\prime}}\mu\nu_{\mu}$ changes by an order of magnitude due
to uncertainties. Such an expanded range of uncertainties will provide a
reasonable domain for experimental searches.

  * iii.

In order to assess the impact of zero-mode contributions (self-consistency),
we ignored the mixing and found that the branching ratios of these decays in
type-I scheme for the $q^{2}$ formulation described by Eq. (13) as well as Eq.
(15) are larger by $\mathcal{O}(10^{1}\sim 10^{2})$ as compared to type-II
results without mixing. These large variations can also be observed in the
presence of mixing, as shown in Table 4, indicating that similar to form
factors the self-consistency substantially affects the branching ratios of
bottom-conserving decays. We emphasize that large self-consistency effects
cannot be ignored for bottom-conserving transitions.

  * iv.

The effect of the lepton mass in two types of axial-vector transitions is
noteworthy. It should be asserted that the difference in mass between the
electron and muon has a significant impact on the branching ratios as well as
other parameters of these transitions. Interestingly, the branching ratios in
these modes display an opposite trend compared to the lepton mass. In the
decays $B_{c}^{+}\to
B_{1}^{0({\prime})}/B_{s1}^{0({\prime})}\ell^{+}\nu_{\ell}$ (as shown in Table
4), the branching ratios decrease approximately by $10\%\sim 24\%$ with
increasing lepton mass. Furthermore, the effect of lepton mass is more
pronounced in $B_{c}\to B_{s1}^{({\prime})}\ell\nu_{\ell}$ decays.
Consequently, this observation highlights the importance of considering the
lepton mass difference when studying these decays and provides valuable
insights into the contribution of lepton masses to the branching ratios.
Similar remarks can also be made for other observables.

  * v.

We also calculated the forward-backward asymmetry, $A_{FB}$, for bottom-
conserving transitions, as shown in column $2$ of Table 5. As mentioned
before, we have followed Hernandez:2006gt ; Ivanov:2005fd ; Becirevic:2016hea
for $A_{FB}$ analyses in p-wave emitting semileptonic decays of $B_{c}$. We
observe that $A_{FB}({B_{c}\to B_{(s)1}^{{\prime}}\ell\nu_{\ell}})$ is
opposite in sign compared to $A_{FB}({B_{c}\to B_{(s)1}\ell\nu_{\ell}})$. The
positive values of $A_{FB}({B_{c}\to B_{(s)1}e\nu_{e}})$ reflect the dominance
of the parity violating helicity structure function, $\mathcal{H}_{P}$, with a
larger contribution from $H_{+}$ amplitude. Further, the effect of lepton mass
can be observed for $A_{FB}({B_{c}\to B_{(s)1}\mu\nu_{\mu}})$, which increases
due to a larger, i.e., $\sim 39\%(74\%)$ contribution from scalar-longitudinal
interference ($\mathcal{H}_{SL}$). The $\mathcal{H}_{SL}$ contributions are
negligible for electron in the final state. For decays involving primed axial-
vector mesons, the predominant contribution occurs from $\mathcal{H}_{P}$ with
larger magnitude for $H_{-}$ amplitude. In addition, $\mathcal{H}_{SL}$
contributions are lesser for such decays, i.e.,
$25\%\text{\leavevmode\nobreak\ and\leavevmode\nobreak\ }37\%$ for $B_{c}\to
B_{1}^{{\prime}}\mu\nu_{\mu}$ and $B_{c}\to B_{s1}^{{\prime}}\mu\nu_{\mu}$,
respectively. The uncertainties in $A_{FB}$ are also given corresponding to
the errors in the form factors. The uncertainties in the $A_{FB}$ involving
primed states behave in a manner similar to that of branching ratios, however,
they behave in the opposite way for unprimed partners. Although individual
uncertainties corresponding to mass and $\beta$ parameters do not result in
sign flip but $A_{FB}(B_{c}\to B_{s1}^{{\prime}}\mu\nu_{\mu})$ goes to zero
with summed errors.

  * vi.

The bottom-conserving semileptonic decay modes have negative lepton-side
convexity parameter, $C_{F}^{\ell}$. For $B_{c}\to B_{s1}e\nu_{e}$ and
$B_{c}\to B_{s1}\mu\nu_{\mu}$, the magnitude of $C_{F}^{\ell}$ is very small
compared to their primed partners. It is worth noting that as the mass of the
lepton increases, the numerical magnitude of the observable decreases. Note
that we have ignored the error due to quark masses and $\beta$ in
$C_{F}^{\ell}$ and other observables.

  * vii.

The longitudinal polarization fraction of the final state meson,
$F_{L}^{\ell}$, for the bottom-conserving semileptonic modes ranges from
$40\%$ to $50\%$, except for $B_{c}\to B_{s1}e\nu_{e}$. It increases with the
lepton mass for unprimed axial-vector, while stays roughly constant for prime
axial-vector mesons.

  * viii.

The asymmetry parameter, $\alpha^{*}$, in bottom-conserving mode is
consistently negative for all decays, indicating the dominance of the
longitudinal character of the daughter meson. This dominance reduces for
decays involving $B_{c}\to B_{s1}$ transitions. In fact, for $B_{c}\to
B_{s1}e\nu_{e}$ and $B_{c}\to B_{s1}\mu\nu_{\mu}$ transitions, the
longitudinal character is marginally greater than the transverse, i.e., $53\%$
and $57\%$ respectively.

  * ix

We have plotted the $q^{2}$ variation of the branching ratios , $A_{FB}$,
$C^{\ell}_{F}$, $P^{\ell}_{F}$, $F_{L}$, and $\alpha^{*}$, respectively, as
shown in Figures 5, 6, 7, and 8. The plots show comparative variation of the
said observables with respect to lepton mass.

####  VI.2.2 Bottom-changing decay modes

In this subsection, we focus on bottom-changing decays, i.e., $B_{c}$ meson
decaying to the charm meson final state. One of the most interesting features
of bottom-changing decays is that, apart from decays involving $e^{+}\nu_{e}$
and $\mu^{+}\nu_{\mu}$ lepton pairs, they include $\tau^{+}\nu_{\tau}$ pair in
the final state. Kinematically, these semileptonic decays are more favorable
than bottom-conserving decays. However, they are governed by $b\to u(c)$ quark
transitions that are CKM suppressed. A few kinematically enhanced bottom-
changing semileptonic $B_{c}$ decays have the most dominant branching ratios
despite of the CKM suppression. We have analyzed and listed our major findings
on $B_{c}\to D_{1}/\chi_{c1}$ semileptonic transitions as follows:

  * i.

The order of branching ratios for these semileptonic decay channels ranges
from $\mathcal{O}(10^{-3})$ to $\mathcal{O}(10^{-6})$, as shown in Table 4.
The predominant decays have the following branching ratios:
Br$(B_{c}^{+}\to\chi_{c1}^{{\prime}}e^{+}\nu_{e})=({1.09}^{+0.05+0.35}_{-0.05-0.33})\times
10^{-3}$ and
Br$(B_{c}^{+}\to\chi_{c1}^{{\prime}}\mu^{+}\nu_{\mu})=({1.08}^{+0.04+0.35}_{-0.04-0.33})\times
10^{-3}$. As stated before, despite of numerical suppression by $V_{cb}$, the
kinematic factors contribute significantly to the enhancement of branching
ratios of $B_{c}^{+}\to\chi_{c1}^{({\prime})}\ell^{+}\nu_{\ell}$ decays.
Furthermore, the CKM suppressed $B_{c}^{+}\to
D_{1}^{({\prime})}\ell^{+}\nu_{\ell}$ decays, involving $b\to u$ transition,
have the branching ratios of the order $\mathcal{O}(10^{-5})$ to
$\mathcal{O}(10^{-6})$, which are comparable to bottom-conserving decays.

  * ii.

As observed in bottom-conserving decays, the branching ratios of transitions
involving primed axial-vectors in the final state are roughly higher by an
order of magnitude than those of decays with unprimed axial-vector mesons,
except for $B_{c}^{+}\to\chi_{c1}^{({\prime})}\tau^{+}\nu_{\tau}$ decays. The
branching ratio of $B_{c}^{+}\to\chi_{c1}^{{\prime}}\tau^{+}\nu_{\tau}$ decay
is enhanced by $\sim 35\%$ as compared to
$B_{c}^{+}\to\chi_{c1}\tau^{+}\nu_{\tau}$ decay. As observed before, these
branching ratios decrease with increasing lepton mass,
$B_{c}^{+}\to\chi_{c1}^{({\prime})}\ell^{+}\nu_{\ell}$ decays are most
influenced by the lepton mass effect. It may be noted that the decays
involving the $\tau$ lepton have the lowest branching ratios among all the
decays because of the significantly larger mass of the $\tau$ lepton.

  * iii.

Likewise, for bottom-conserving decays, the branching ratios of bottom-
changing decays have uncertainties as large as $\sim 140\%$ inclusive of both
types of uncertainties. Moreover, uncertainties in these decays are less
asymmetric when compared to those of bottom-conserving decays. In contrast to
bottom-conserving decays, where uncertainties mostly cause an increase in
branching ratios, uncertainties in bottom-changing decays lead to noticeable
changes through gradual increases or decreases in the branching ratios of
these decays. For $B_{c}^{+}\to\chi_{c1}\ell^{+}\nu_{\ell}$ decays, the
uncertainties are less than $\sim 40\%$ and are nearly symmetric. As pointed
out earlier, the self-consistency effects are expected to be smaller in
semileptonic decays involving $B_{c}\to D_{1}^{({\prime})}$ transitions,
wherein we observe that the branching ratios of unmixed states are differently
affected by zero-mode contributions. In comparison to the type-II scheme, the
branching ratios of unmixed $B_{c}\to D_{1}^{({\prime})}\ell^{+}\nu_{\ell}$
decays experience changes ranging from approximately $60\%$ to $100\%$ for
$q^{2}$ dependence using Eq. (15) and $26\%$ to $92\%$ with Eq. (14) in type-I
CLFQM, where the branching ratios for $B_{c}\to D_{1}\ell^{+}\nu_{\ell}$ are
varying at least by $90\%$. In addition, we find that these effects are more
prominent in the presence of mixing. Therefore, such effects for
$D_{1}^{({\prime})}$ emitting decays can be significant and cannot be ignored.
Nevertheless, the self-consistency has a minimal effect on the branching
ratios of semileptonic decays to $\chi_{c1}^{({\prime})}$ states, with
variation of $5\%-40\%$.

  * iv.

The numerical values of $A_{FB}$ parameters for bottom-changing transitions
are listed in column $2$ of Table 5. Unlike bottom-conserving decays, $A_{FB}$
for bottom-changing decays are consistently negative in numerical values
because of the predominant contribution from $\mathcal{H}_{P}$ with larger
magnitude for $H_{-}$ helicity amplitude. The exception is evident in the
decays that involve the pairing of a $\tau$ lepton and a primed axial-vector
particle in the final state. In all bottom-changing $B_{c}\to
D_{1}(\chi_{c1})e\nu_{e}$ decays, $\mathcal{H}_{SL}$ contributions are
negligible due to the large $q^{2}$ value for electrons in the final state.
Further, as we increase the mass of the lepton, the $A_{FB}({B_{c}\to
D_{1}\tau\nu_{\tau}})$ receives approximately $54\%$ contribution from
$\mathcal{H}_{SL}$, however, this contribution increases to (from being
negligible) $\sim 10\%$ for $B_{c}\to\chi_{c1}\tau\nu_{\tau}$. In addition, we
observe a sign flip in $\mathcal{H}_{SL}$ with $\tau$ lepton mass for decays
including $D_{1}$. The positive values of $A_{FB}({B_{c}\to
D_{1}^{{\prime}}(\chi_{c1}^{{\prime}})\tau\nu_{\tau}})$ signify the dominance
of $\mathcal{H}_{SL}$, with negligible contributions from $\mathcal{H}_{P}$.
On comparison with Hernandez:2006gt ; Ivanov:2005fd , we found that our
results of $A_{FB}$ for both $B_{c}\to\chi_{c1}\ell\nu_{\ell}$ (apart from
$\tau$) and $B_{c}\to\chi_{c1}\tau\nu_{\tau}$ decays are consistent with
Hernandez:2006gt , however, they exhibit opposite signs compared to those of
Ref. Ivanov:2005fd . Furthermore, the uncertainties corresponding to $\beta$
values show exceptional variation in $A_{FB}(B_{c}\to D_{1}\ell\nu_{\ell})$
making them more negative with large numerical values. Such exceptional
variation in $A_{FB}$ highlights the importance of interference between
$\mathcal{H}_{P}$ and $\mathcal{H}_{SL}$ contributions and mixing in the form
factors. However, uncertainties in masses behave symmetrically manner with
negligibly small variations in some cases. For
$A_{FB}({B_{c}\to\chi_{c1}^{{\prime}}\mu\nu_{\mu}})$, we observe a possible
sign flip corresponding to large $\beta$ error.

  * v.

The numerical values of $C_{F}^{\ell}$ in bottom-changing semileptonic mode
have both signs, i.e., primed-type decays have negative values with larger
magnitudes compared to the unprimed-type. The lepton mass has a substantial
effect on their numerical values, as can be seen for the decays involving
$\tau$ lepton. Unlike bottom-conserving decays, the longitudinal polarization
fraction ($F_{L}^{\ell}$) for axial-vector mesons lies between $20\%$ and
$30\%$, on the other hand the primed states fractions vary from $60\%$ to
$90\%$.

  * vi.

Interestingly, the asymmetry parameter $\alpha^{*}$ for all bottom-changing
transitions involving primed axial-vector mesons in the final state are
dominant in longitudinal character, whereas unprimed states favor transverse
asymmetry, as shown in Table 5. The unprimed states for $B_{c}\to
D_{1}\ell\nu_{\ell}$ decays are marginally transverse in nature, this behavior
reduces further for decays involving $\tau$ leptons. In addition to the lepton
mass, the mixing effect plays a significant role in these decays. In contrast,
the asymmetry in $B_{c}\to\chi_{c1}\ell\nu_{\ell}$ decays is of transverse
nature with a negligible effect of the lepton mass.

  * vii.

It is noteworthy to mention that most of the observables for bottom-conserving
as well as bottom-changing semileptonic decays show a dominant longitudinal
behavior, with few exceptions, especially for heavier leptons. The plots for
all the observables, including branching ratios showing variation with $q^{2}$
and mass, are given in Figures 9, 10, 11, and 12.

  * viii.

Recently, some experiments involving B meson decay have shown deviations in
${R}_{D^{*}}$ from the SM expectation. While the measured numerical value is
${R}_{D^{*}}^{exp}=0.281\pm 0.018\pm 0.024$ BaBar:2012obs ; BaBar:2013mob ;
Belle:2015qfa ; Belle:2016ure ; Abdesselam:2016xqt ; LHCb:2015gmp ;
LHCb:2023zxo , the average theoretical value is ${R}_{D^{*}}^{SM}=0.254\pm
0.005$ HeavyFlavorAveragingGroup:2022wzx . Thus, the study of semileptonic
$B_{c}$ decays involving the heavy to heavy $B_{c}\to A$ transition may also
suffer from similar anomalies. Motivated by these intriguing measurements, we
obtain the LFU ratios for axial-vector meson emitting decays of $B_{c}$ meson
by considering the masses of the leptons:
$R_{D_{1}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
D_{1}^{0}\tau^{+}\nu_{\tau})}{\mathcal{B}(B_{c}^{+}\rightarrow
D_{1}^{0}e^{+}\nu_{e})}$ = $0.50^{+0.22+0.40}_{-0.29-0.70}$ ;
$R_{D_{1}^{{\prime}}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
D_{1}^{0{\prime}}\tau^{+}\nu_{\tau})}{\mathcal{B}(B_{c}^{+}\rightarrow
D_{1}^{0{\prime}}e^{+}\nu_{e})}$ = $0.44^{+0.26+0.28}_{-0.32-0.50}$;
$R_{\chi_{c1}}$ =
$\frac{\mathcal{B}(B_{c}^{+}\rightarrow\chi_{c1}\tau^{+}\nu_{\tau})}{\mathcal{B}(B_{c}^{+}\rightarrow\chi_{c1}e^{+}\nu_{e})}$
= $0.11^{+0.01+0.03}_{-0.01-0.03}$; $R_{\chi_{c1}^{{\prime}}}$ =
$\frac{\mathcal{B}(B_{c}^{+}\rightarrow\chi_{c1}^{{\prime}}\tau^{+}\nu_{\tau})}{\mathcal{B}(B_{c}^{+}\rightarrow\chi_{c1}^{{\prime}}e^{+}\nu_{e})}$
= $0.08^{+0.01+0.04}_{-0.01-0.04}$;
Similarly, $c\to d/s$ transitions yield,
$R_{B_{1}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
B_{1}^{0}\mu^{+}\nu_{\mu})}{\mathcal{B}(B_{c}^{+}\rightarrow
B_{1}^{0}e^{+}\nu_{e})}$ = $0.90^{+0.00+0.00}_{-0.60-0.70}$ ;
$R_{B_{1}^{{\prime}}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
B_{1}^{0{\prime}}\mu^{+}\nu_{\mu})}{\mathcal{B}(B_{c}^{+}\rightarrow
B_{1}^{0{\prime}}e^{+}\nu_{e})}$ = $0.84^{+0.02+0.19}_{-0.11-0.25}$;
$R_{B_{s1}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
B_{s1}^{0}\mu^{+}\nu_{\mu})}{\mathcal{B}(B_{c}^{+}\rightarrow
B_{s1}^{0}e^{+}\nu_{e})}$ = $0.88^{+0.00+0.00}_{-0.80-2.10}$;
$R_{B_{s1}^{{\prime}}}$ = $\frac{\mathcal{B}(B_{c}^{+}\rightarrow
B_{s1}^{0{\prime}}\mu^{+}\nu_{\mu})}{\mathcal{B}(B_{c}^{+}\rightarrow
B_{s1}^{0{\prime}}e^{+}\nu_{e})}$ = $0.80^{+0.00+0.11}_{-0.33-1.00}$.

Finally, we compare our predictions for branching ratios with those of
existing theoretical works from various models. As mentioned before, there has
been limited research on bottom-conserving semileptonic decays to p-wave
mesons until recently. The results for bottom-conserving and changing decays
from other models, namely, RQM Ebert:2010zu , pQCD Rui:2018kqr , BS
Wang:2011jt , NRQM Hernandez:2006gt , RQM Ivanov:2005fd , RCQM Ivanov:2006ni ,
QCDSR Azizi:2009ny , CLFQM Wang:2009mi , modified Godfrey-Isgur model
(MGI)-CLFQM Li:2023wgq , and CLFQM Shi:2016gqt are given in Tables 6 and 7,
respectively. We observe that our predictions for $B_{c}\to
B_{(s)1}\ell\nu_{\ell}$ compare well with those of RQM Ebert:2010zu . However,
our predictions for $B_{c}\to B_{(s)1}^{{\prime}}\ell\nu_{\ell}$ are larger
than those of RQM Ebert:2010zu , although they are consistent within the
uncertainties. When compared to the type-I CLFQM Shi:2016gqt , we consistently
observe smaller magnitudes for our results, while maintaining similar decay
orders involving primed axial-vector mesons in the final state. In contrast,
our predictions agree well with Li et al. Li:2023wgq (MGI with type-I CLFQM)
for $B_{c}\to B_{1}^{({\prime})}\ell\nu_{\ell}$ decays, while there are
noticeable discrepancies in the case of $B_{c}\to
B_{s1}^{({\prime})}\ell\nu_{\ell}$ decays. It should be noted that besides the
type-I correspondence, differences exist between their approaches and ours
regarding the formulation of $q^{2}$ dependence as well as other parameters
such as quark masses and $\beta$ parameters. For bottom-changing $B_{c}\to
D_{1}^{({\prime})}\ell\nu_{\ell}$ decays, our results are smaller than RQM
Ebert:2010zu and show an opposite trend with respect to order of magnitude as
compared to Li:2023wgq . Moreover, our predictions for $B_{c}\to
D_{1}^{({\prime})}e\nu_{e}$ decays are smaller by more than an order of
magnitude in comparison to the predictions of QCDSR Khosravi:2015tea .
Additionally, when considering charmonium decays, the branching ratios of
$B_{c}\rightarrow\chi_{c1}^{({\prime})}\ell\nu_{\ell}$ decays from various
models including ours demonstrate a reasonably consistent pattern among
results obtained for $e\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}\mu$ leptons. However, all these models present a scattered range of
branching ratios for the case of $\tau$ lepton, which could be of immense
experimental significance for near future experiments. Additionally, we also
compared our predictions for LFU ratios with available results from other
models. We observe that the CLFQM predictions of $R_{\chi_{c1}}=0.11$ and
$R_{\chi_{c1}^{{\prime}}}=0.07$ Wang:2009mi and relativistic constituent quark
model predictions of $R_{\chi_{c1}}$ = $0.12$ and $R_{\chi_{c1}^{{\prime}}}$ =
$0.08$ Ivanov:2005fd agree well with our results.

##  VII Summary

In the present work, we have done an in-depth analysis of the self-consistency
of the CLFQM in $B_{c}\to A^{({\prime})}$ transition form factors and
semileptonic $B_{c}\to A^{({\prime})}\ell\nu_{\ell}$ decays. The self-
consistent type-II correspondence incorporates the resolution of existing
inconsistencies associated with zero-mode contributions and violation of
covariance of matrix element in type-I CLFQM. These issues have been studied
for bottom-changing transition form factors involving vector mesons
Chang:2019mmh . However, problems related to self-consistency were also
expected to be present in axial-vector meson emitting transitions, which has
been comprehensively examined in the current work for both bottom-conserving
and bottom-changing transition form factors. In addition, to examine the
impact of self-consistency via type-II correspondence, we have done a thorough
analysis of semileptonic weak decays involving $B_{c}\to A^{({\prime})}$
transitions. Aforementioned, in the type-I scheme Jaus:1999zv ; Cheng:2003sm ;
Choi:2013mda , it is important to note that the form factors
$A(q^{2})(q(q^{2}))$ and $V_{2}(q^{2})(c_{+}(q^{2}))$ are self-consistent,
i.e., they are free from zero-mode contributions. However, the challenges
regarding self-consistency and strict covariance still exist in
$V_{0}(q^{2})(c_{-}(q^{2}))$ and $V_{1}(q^{2})(l(q^{2}))$ form factors in the
CLFQM Cheng:2003sm ; Choi:2013mda ; Chang:2018zjq ; Jaus:1999zv . These form
factors are inconsistent due to non-vanishing $\omega$-dependent spurious
contributions associated with $B^{(i)}_{j}$ functions and violated covariance
Chang:2019mmh . The type-II correspondence suggests that both of these issues
share a common source, and it resolves them simultaneously. In the present
work, we also done a comparative study of form factors and branching ratios
between type-I and type-II CLFQM to measure the impact of inconsistencies. In
addition to the prediction of the branching ratios of $B_{c}\to
A^{({\prime})}\ell\nu_{\ell}$ decays in type-II CLFQM, we have calculated the
expectation values of physical observables such as $\langle A_{\rm
FB}\rangle,\langle C_{F}^{\ell}\rangle,\langle
P_{L}^{\ell}\rangle,\leavevmode\nobreak\ \langle F_{L}\rangle$, and
$\langle\alpha^{*}\rangle$. Further, we have plotted the $q^{2}$ variation of
form factors, branching ratios, and rest of the observables for different
generations of leptons. We list our important findings as follows:

  * •

We observed that the form factors $V_{0}(q^{2})(c_{-}(q^{2}))$ and
$V_{1}(q^{2})(l(q^{2}))$ in type-I CLFQM, acquire zero-mode contributions
through $B$-functions, i.e., $B_{1}^{(2)}$ and $B_{3}^{(3)}$. Consequently,
the form factors corresponding to longitudinal and transverse states yield
different numerical values leading to inconsistency in type-I scheme. As
expected, application of type-II correspondence resolves such issues. We
confirm that though zero-mode contributions associated with $B$-functions
exist formally in form factor relations but they vanish numerically in type-II
correspondence.

  * •

Numerically speaking, the effect of zero-mode contributions are overwhelming
in bottom-conserving $V_{0}(q^{2})$ and $V_{1}(q^{2})$ transition form
factors. These contributions are visibly prominent in $V_{1}(q^{2})$ form
factor. It should be emphasized that $V_{1}^{B_{c}B_{(s)1}}(q^{2})$ form
factors have profound ramifications on the branching ratios of semileptonic
$B_{c}\to A^{({\prime})}\ell\nu_{\ell}$ decays. The contributions of
$V_{1}^{B_{c}B_{(s)1}}(q^{2})$ form factors to these decays become even more
important given the fact that contributions from
$V_{0}^{B_{c}B_{(s)1}}(q^{2})$ form factors are negligible.

  * •

We observe that there exists a two-fold problem in type-I form factors
irrespective of $q^{2}$ dependence formulation. The fitting parameters, either
form factors at $q^{2}=0$ or slope parameters acquire the large numerical
values. The choice of pole masses play significant role in lowering the slope
parameters, however, the form factors still stay large due to zero-mode
contributions. Similar observations can be made from the numerical values of
the form factors in other analyses in type-I scheme Li:2023wgq ; Shi:2016gqt .
All the above stated issues can be resolved in type-II CLFQM.

  * •

In comparative study of type-I and type-II correspondence, we also observe a
sequence in type-I inconsistencies in different flavor transition form
factors. The form factors for $B_{c}\to B_{(s)1}$ transitions are more
seriously affected by zero-mode contributions as compared to $B_{c}\to D_{1}$
transitions. On the other hand, the effect is least in case of
$B_{c}\to\chi_{c1}$ transitions.

  * •

The analyses of semileptonic $B_{c}\to A^{({\prime})}\ell\nu_{\ell}$ decays
pose more serious challenges in addition to inconsistency, i.e., corresponding
to $q^{2}$ dependence of the form factors, and the mixing of final states. The
branching ratios of all the decays involving primed axial-vector meson states
are larger as compared to decays including their unprimed partner. The
numerical effects of self-consistency are profound in bottom-conserving decays
as compared to bottom-changing decays. Interestingly, the
$B_{c}\to\chi_{c1}^{({\prime})}\ell\nu_{\ell}$ decays are affected minimally
due to self-consistency. Further, the branching ratios of self-consistent
type-II CLFQM differ by at least an order of magnitude as compared to type-I
CLFQM. This effect can even be observed in the absence of mixing. It is worth
noting that the branching ratios of $B_{c}\to
B_{(s)1}^{({\prime})}\ell\nu_{\ell}$ decays are comparable to those of
$B_{c}\to D_{1}^{({\prime})}\ell\nu_{\ell}$ decays.

  * •

Finally, we have analyzed multiple physical observables, like, $\langle A_{\rm
FB}\rangle,\langle C_{F}^{\ell}\rangle,\langle
P_{L}^{\ell}\rangle,\leavevmode\nobreak\ \langle F_{L}\rangle$, and
$\langle\alpha^{*}\rangle$. In addition, we also estimated LFU within the SM
through the lepton mass effects in semileptonic decays. It should be
emphasized that the mass of $e$ and $\mu$ has considerable effect on the
bottom-conserving decays. In contrast, for bottom-changing decays the effect
is prominent only when $\tau$ lepton is involved. Subsequently, we obtained
LFU ratios that would be useful to highlight anomalies in comparison to future
experiments.

The numerical results obtained from CLFQM and the phenomenological analysis
offer crucial insights into the underlying physics and can guide future
experimental investigations to validate and refine our understanding of these
semileptonic decay processes.

## Acknowledgment

The authors gratefully acknowledge the financial support by the Department of
Science and Technology (SERB:CRG/2018/002796), New Delhi.

## Appendix

##  Appendix A ${}^{3}P_{1}$ and ${}^{1}P_{1}$ mixing

Based on C-parity, there exist two types of spectroscopic axial-vector mesons,
i.e., ${}^{3}P_{1}\leavevmode\nobreak\ (J^{PC}=1^{++}$) and
${}^{1}P_{1}\leavevmode\nobreak\ (J^{PC}=1^{+-})$ and they behave well with
respect to the quark model $q_{1}\bar{q_{2}}$ assignments. We denote
${}^{3}P_{1}$ and ${}^{1}P_{1}$ as $A_{A}$ and $A_{B}$, respectively. Due to
the absence of a quantum number that prohibits the mixing of strange and
charmed states, they are expressed as a mixture of $A_{A}$ and $A_{B}$.
However, due to their opposite C-parity, the nonstrange and uncharmed mesons
$A_{A}$ and $A_{B}$ cannot undergo mixing. The following non-strange and
uncharmed mesons have so far been observed, experimentally
ParticleDataGroup:2022pth , i.e.,

  1. i.

$A_{A}$ multiplet: isovector $a_{1}(1.230)$ and two isoscalars $f_{1}(1.282)$,
$f_{1}(1.426)$, $f{{}^{{\prime}}}_{1}(1.518)$, $a_{1}(1.655)$, and
$\chi_{c1}(3.511)$, where the numerical values given in the parentheses
represent the masses (in GeV) of the individual mesons.

  2. ii.

$A_{B}$ multiplet: isovector $b_{1}(1.230)$ and two isoscalars $h_{1}(1.166)$,
$h{{}^{{\prime}}}_{1}(1.416)$, $h_{1}(1.595)$, and $h_{c1}(3.525)$. C-parity
of $h^{{{}^{{\prime}}}}_{1}(1.416)$, and spin and parity of the
$h_{c1}(3.525)$888 We denote $h_{c}(3.525)$ as $\chi_{c1}^{{\prime}}(3.525)$.
remains to be confirmed.

In the current work, we are focusing on charm and bottom mesons, therefore we
consider the mixing of these states as follows999The form factors and slope
parameters will exhibit mixing as per (46), which has been duly considered in
our numerical calculations.:

$\begin{array}[]{l}{A=A_{A}\sin\theta+A_{B}\cos\theta,}\\\
{A^{\prime}=A_{A}\cos\theta-A_{B}\sin\theta,}\end{array}$ (46)

where $\theta$ is the mixing angle between $A_{A}$ and $A_{B}$, while
$A^{({\prime})}$ denotes physical states. The charmed physical states
$D_{1}(2.422)$ and ${D}^{\prime}_{1}(2.427)$ mesons are then given as,

$\begin{array}[]{l}{D_{1}(2.422)=D_{1A}\sin\theta_{D_{1}}+D_{1B}\cos\theta_{D_{1}},}\\\
{{D}^{\prime}_{1}(2.427)=D_{1A}\cos\theta_{D_{1}}-D_{1B}\sin\theta_{D_{1}},}\end{array}$
(47)

where the mixing angle $\theta_{D_{1}}=-58^{\circ}$ Godfrey:2015dva ;
Chen:2016spr ; Wang:2022tuy . Note that the measured mass for
${D}^{\prime}_{1}$ state is $2.412$ $\mathrm{GeV}$ LHCb:2019juy , which has
been used for calculation in our work. Similarly, for b-flavored mesons we use
the following:

$\begin{array}[]{l}{B_{1}(5.710)=B_{1A}\sin\theta_{B_{1}}+B_{1B}\cos\theta_{B_{1}},}\\\
{B^{{\prime}}_{1}(5.726)=B_{1A}\cos\theta_{B_{1}}-B_{1B}\sin\theta_{B_{1}};}\end{array}$
(48)

and

$\begin{array}[]{l}{B_{s1}(5.820)=B_{s1A}\sin\theta_{B_{s1}}+B_{s1B}\cos\theta_{B_{s1}},}\\\
{B^{{\prime}}_{s1}(5.830)=B_{s1A}\cos\theta_{B_{s1}}-B_{s1B}\sin\theta_{B_{s1}}.}\end{array}$
(49)

Based on previous studies that have reported various mixing angles for the
bottom states, we adopt $\theta_{B_{1}}=-55^{\circ}$ and
$\theta_{B_{s1}}=-55^{\circ}$ in our analysis Chen:2016spr ; li:2021hss ;
Li:2018eqc .

##  Appendix B Form factors of $B_{c}\to A$ transitions

Using the framework of the CLFQM, the $P\rightarrow A$ form factors can be
written as

$\displaystyle[{F}(q^{2})]=N_{c}\int\frac{{\rm d}x\,{\rm d}^{2}{\bf
k}_{\bot}^{{\prime}}}{2(2\pi)^{3}}\frac{\chi^{{\prime}}_{P}\chi_{A_{A},A_{B}}^{{\prime}{\prime}}}{\bar{x}}\,\widetilde{F}(x,{\bf
k}_{\bot}^{{\prime}},q^{2})\,,$ (50)

with $\widetilde{F}=\widetilde{q},\widetilde{l}$, and $\widetilde{c}_{\pm}$
for longitudinal polarization state are given as

$\displaystyle\widetilde{q}(x,{\bf k}_{\bot}^{{\prime}},q^{2})=$
$\displaystyle-2\left\\{\bar{x}m_{1}^{{\prime}}+xm_{2}+(m_{1}^{{\prime}}+m_{1}^{{\prime}{\prime}})\frac{{\bf
k}_{\bot}^{{\prime}}\cdot{\bf
q}_{\bot}}{q^{2}}+\frac{2}{D^{{\prime}{\prime}}_{{{}^{A_{A}}}}}\left[{{\bf
k}_{\bot}^{{\prime}}}^{2}+\frac{({\bf k}_{\bot}^{{\prime}}\cdot{\bf
q}_{\bot})^{2}}{q^{2}}\right]\right\\}\,,$ (51)
$\displaystyle\widetilde{c}_{+}(x,{\bf k}_{\bot}^{{\prime}},q^{2})=$
$\displaystyle
2\bigg{\\{}(-m_{1}^{{\prime}{\prime}}-2xm_{1}^{{\prime}}+m_{1}^{{\prime}}+2xm_{2})\,\frac{{\bf
k}_{\bot}^{{\prime}}\cdot{\bf q}_{\bot}}{{\bf
q}_{\bot}^{2}}+(x-\bar{x})(\bar{x}m_{1}^{{\prime}}+xm_{2})$
$\displaystyle+\frac{2}{D^{{\prime}{\prime}}_{{{}^{A_{A}}}}}\frac{{\bf
k}_{\bot}^{{\prime}{\prime}}\cdot{\bf q}_{\bot}}{\bar{x}{\bf
q}_{\bot}^{2}}\left[{\bf k}_{\bot}^{{\prime}}\cdot{\bf
k}_{\bot}^{{\prime}{\prime}}+(xm_{2}+\bar{x}m_{1}^{{\prime}{\prime}})(xm_{2}+\bar{x}m_{1}^{{\prime}})\right]\bigg{\\}}\,,$
(52) $\displaystyle\widetilde{l}(x,{\bf k}_{\bot}^{{\prime}},q^{2})=$
$\displaystyle-2\Bigg{\\{}-(m_{1}^{{\prime}}-m_{1}^{{\prime}{\prime}})^{2}(m_{1}^{{\prime}}-m_{2})+(xm_{2}-\bar{x}m_{1}^{{\prime}}){M^{{\prime}}}^{2}+(xm_{2}+\bar{x}m_{1}^{{\prime}}){M^{{\prime}{\prime}}}^{2}$
$\displaystyle-x(m_{2}-m_{1}^{{\prime}})({M_{0}^{{\prime}}}^{2}+{M_{0}^{{\prime}{\prime}}}^{2})-2xm_{1}^{{\prime}{\prime}}{M_{0}^{{\prime}}}^{2}-4\left(m_{1}^{{\prime}}-m_{2}\right)\left({{\bf
k}_{\bot}^{{\prime}}}^{2}+\frac{({\bf k}_{\bot}^{{\prime}}\cdot{\bf
q}_{\bot})^{2}}{q^{2}}\right)$ $\displaystyle-
m_{2}q^{2}-(m_{1}^{{\prime}}-m_{1}^{{\prime}{\prime}})(q^{2}+q\cdot
P)\frac{{\bf k}_{\bot}^{{\prime}}\cdot{\bf
q}_{\bot}}{q^{2}}{+4(m_{1}^{{\prime}}-m_{2})B_{1}^{(2)}}$
$\displaystyle+\frac{2}{D^{{\prime}{\prime}}_{{{}^{A_{A}}}}}\bigg{[}\left({{\bf
k}_{\bot}^{{\prime}}}^{2}+\frac{({\bf k}_{\bot}^{{\prime}}\cdot{\bf
q}_{\bot})^{2}}{q^{2}}\right)\bigg{(}(x-\bar{x}){M^{{\prime}}}^{2}+{M^{{\prime}{\prime}}}^{2}-2(m_{1}^{{\prime}}+m_{1}^{{\prime}{\prime}})(m_{1}^{{\prime}}-m_{2})$
$\displaystyle+2x{M_{0}^{{\prime}}}^{2}-q^{2}-2(q^{2}+q\cdot P)\frac{{\bf
k}_{\bot}^{{\prime}}\cdot{\bf q}_{\bot}}{q^{2}}\bigg{)}$
$\displaystyle-\left({M^{{\prime}}}^{2}+{M^{{\prime}{\prime}}}^{2}-q^{2}+2(m_{1}^{{\prime}}-m_{2})(-m_{1}^{{\prime}{\prime}}+m_{2})\right)B_{1}^{(2)}+2B_{3}^{(3)}\bigg{]}\Bigg{\\}}\,,$
(53) $\displaystyle\widetilde{c}_{-}(x,{\bf k}_{\bot}^{{\prime}},q^{2})=$
$\displaystyle-2\Bigg{\\{}(3-2x)(\bar{x}m_{1}^{{\prime}}+xm_{2})-\left[(6x-7)m_{1}^{{\prime}}+(4-6x)m_{2}-m_{1}^{{\prime}{\prime}}\right]\frac{{\bf
k}_{\bot}^{{\prime}}\cdot{\bf q}_{\bot}}{q^{2}}$
$\displaystyle+4(m_{1}^{{\prime}}-m_{2})\left[2\left(\frac{{\bf
k}_{\bot}^{{\prime}}\cdot{\bf q}_{\bot}}{q^{2}}\right)^{2}+\frac{{{\bf
k}_{\bot}^{{\prime}}}^{2}}{q^{2}}\right]{-4\frac{(m_{1}^{{\prime}}-m_{2})}{q^{2}}B_{1}^{(2)}}$
$\displaystyle+\frac{1}{D^{{\prime}{\prime}}_{{{}^{A_{A}}}}}\bigg{[}-2\left({M^{{\prime}}}^{2}+{M^{{\prime}{\prime}}}^{2}-q^{2}+2(m_{1}^{{\prime}}-m_{2})(-m_{1}^{{\prime}{\prime}}+m_{2})\right)(A_{3}^{(2)}+A_{4}^{(2)}-A_{2}^{(1)})$
$\displaystyle+\left(2{M^{{\prime}}}^{2}-q^{2}-\hat{N}_{1}^{{\prime}}+\hat{N}_{1}^{{\prime}{\prime}}-2(m_{1}^{{\prime}}-m_{2})^{2}+(m_{1}^{{\prime}}-m_{1}^{{\prime}{\prime}})^{2}\right)\left(A_{1}^{(1)}+A_{2}^{(1)}-1\right)$
$\displaystyle+2Z_{2}\left(2A_{4}^{(2)}-3A_{2}^{(1)}+1\right)+2\frac{q\cdot
P}{q^{2}}\left(4A_{2}^{(1)}A_{1}^{(2)}-3A_{1}^{(2)}\right)$
$\displaystyle{+\frac{2}{q^{2}}\left(\left({M^{{\prime}}}^{2}+{M^{{\prime}{\prime}}}^{2}-q^{2}+2(m_{1}^{{\prime}}-m_{2})(-m_{1}^{{\prime}{\prime}}+m_{2})\right)B_{1}^{(2)}-2B_{3}^{(3)}\right)}\bigg{]}\Bigg{\\}}\,.$
(54)

To obtain the results for the case of transverse polarization state, one can
derive them from these formulas by excluding the terms that are related to the
$B^{(i)}_{j}$ functions. The analytic expressions for $P\rightarrow A$
transition form factors are obtained from that of $P\rightarrow V$ ones
Cheng:2003sm ; Chang:2019mmh by using the following transformations:

$\displaystyle F^{A_{A},A_{B}}(q^{2})=[\mathcal{F}(q^{2})]_{full}\text{ with
}\left(m_{1}^{{\prime}{\prime}}\rightarrow-
m_{1}^{{\prime}{\prime}},\leavevmode\nobreak\
\chi_{V}^{{\prime}{\prime}}\to\chi_{A_{A},A_{B}}^{{\prime}{\prime}},D^{{\prime}{\prime}}_{V}\to
D^{{\prime}{\prime}}_{{{}^{A_{A},A_{B}}}}\right)$ (55)

It should be noted that, in the $B_{c}\to A_{B}$ form factors, only the terms
$1/D^{{\prime}{\prime}}_{{{}^{A_{A}}}}$ are retained by replacing
$D^{{\prime}{\prime}}_{{{}^{A_{A}}}}\to D^{{\prime}{\prime}}_{{{}^{A_{B}}}}$.
One must exercise caution when considering the replacement of
$m_{1}^{\prime\prime}\to-m_{1}^{\prime\prime}$ and ensure that it is not
applied indiscriminately to $m_{1}^{\prime\prime}$ in both $D^{\prime\prime}$
and $\chi^{\prime\prime}$. The corresponding definitions of the functions
needed to investigate the form factors are provided below Jaus:1999zv ;
Cheng:2003sm .

$\displaystyle Z_{2}$
$\displaystyle=\hat{N}_{1}^{{\prime}}+m_{1}^{{\prime}2}-m_{2}^{2}+(\bar{x}-x){M^{{\prime}}}^{2}+(q^{2}+q\cdot
P)\frac{k_{1\bot}^{{\prime}}\cdot q_{\bot}}{q^{2}}\,$ $\displaystyle
A_{1}^{(1)}$ $\displaystyle=\frac{x}{2}\,,\qquad
A_{2}^{(1)}=\frac{x}{2}-\frac{k_{1\bot}^{{\prime}}\cdot
q_{\bot}}{q^{2}}\,,\qquad$ $\displaystyle A_{1}^{(2)}$
$\displaystyle=-k_{1\bot}^{{\prime}2}-\frac{(k_{1\bot}^{{\prime}}\cdot
q_{\bot})^{2}}{q^{2}}\,,$ $\displaystyle A_{2}^{(2)}$
$\displaystyle=(A_{1}^{(1)})^{2}\,,\qquad
A_{3}^{(2)}=A_{1}^{(1)}A_{2}^{(1)}\,,\qquad A_{4}^{(2)}=(A_{2}^{(1)})^{2}\,,$
$\displaystyle B_{1}^{(2)}$
$\displaystyle=\frac{x}{2}Z_{2}-A_{1}^{(2)}\,,\qquad
B_{3}^{(3)}=B_{1}^{(2)}Z_{2}+\left(P^{2}-\frac{(q\cdot
P)^{2}}{q^{2}}\right)A_{1}^{(1)}A_{1}^{(2)}.$ (56)

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Table 2: Bottom-conserving $B_{c}\rightarrow A$ transition form factors in the CLFQM. Note that Type-I represents the form factors calculated using Eq. (15), while Type-I* in parenthesis uses Eq. (13) and Eq. (14) with transition pole masses listed in Table 1. The first and second errors correspond to the quark masses and $\beta$ parameters, respectively. F | Type-II | Type-I
---|---|---
(Type-I*)
$F(0)$ | a | $F(q^{2}_{max})$ | $F(0)$ | a | b | $F(q^{2}_{max})$
$A^{B_{c}B_{1A}}$ |  ${0.10}^{+0.00+0.00}_{-0.00-0.00}$101010We have rounded the errors upto second decimal places, however, in numerical calculations upto three significant digits are considered. | ${-0.01}^{+0.06+0.01}_{-0.05-0.01}$ | ${0.10}^{+0.00+0.00}_{-0.00-0.00}$ | $0.10$ | $6.46$ | $0.22$ | $0.10$
($0.10$) |  ($-0.01$) |  |  ($0.10$)
$V_{0}^{B_{c}B_{1A}}$ | ${0.08}^{+0.00+0.01}_{-0.00-0.01}$ | ${0.10}^{+0.19+0.05}_{-0.12-0.04}$ | ${0.08}^{+0.01+0.01}_{-0.00-0.01}$ | $0.52$ | $4.07$ | $0.04$ | $0.54$
($0.55$) |  ($-0.14$) |  |  ($0.57$)
$V_{1}^{B_{c}B_{1A}}$ | ${1.84}^{+0.11+0.16}_{-0.09-0.17}$ | ${-0.33}^{+0.01+0.00}_{-0.00-0.00}$ | ${1.95}^{+0.11+0.17}_{-0.09-0.18}$ | $10.65$ | $2.34$ | $-0.49$ | $10.85$
($11.96$) |  ($-0.21$) |  |  ($12.75$)
$V_{2}^{B_{c}B_{1A}}$ | ${-0.02}^{+0.02+0.02}_{-0.00-0.00}$ | ${-0.64}^{+0.22+0.17}_{-0.03-0.00}$ | ${-0.02}^{+0.03+0.02}_{-0.00-0.00}$ | $-0.004$ | $-20.96$ | $65.11$ | $-0.003$
($-0.01$) |  ($-0.52$) |  |  ($-0.01$)
$A^{B_{c}B_{1B}}$ | ${0.11}^{+0.01+0.00}_{-0.01-0.00}$ | ${0.08}^{+0.05+0.05}_{-0.04-0.04}$ | ${0.11}^{+0.02+0.00}_{-0.01-0.00}$ | $0.10$ | $6.61$ | $6.25$ | $0.11$
($0.11$) |  ($0.09$) |  |  ($0.11$)
$V_{0}^{B_{c}B_{1B}}$ | ${0.22}^{+0.00+0.03}_{-0.00-0.03}$ | ${0.14}^{+0.12+0.02}_{-0.08-0.02}$ | ${0.24}^{+0.00+0.03}_{-0.00-0.03}$ | $0.72$ | $1.68$ | $3.18$ | $0.73$
($0.78$) |  ($-0.22$) |  |  ($0.82$)
$V_{1}^{B_{c}B_{1B}}$ | ${4.63}^{+0.08+0.47}_{-0.09-0.49}$ | ${-0.22}^{+0.00+0.00}_{-0.00-0.00}$ | ${4.92}^{+0.09+0.50}_{-0.10-0.52}$ | $14.21$ | $4.13$ | $0.91$ | $14.68$
($15.52$) |  ($-0.14$) |  |  ($16.62$)
$V_{2}^{B_{c}B_{1B}}$ | ${0.005}^{+0.00+0.00}_{-0.03-0.03}$ | ${-0.13}^{+0.00+0.00}_{-1.22-0.93}$ | ${0.01}^{+0.00+0.00}_{-0.03-0.03}$ | $-0.004$ | $-117.25$ | $431.74$ | $-0.002$
($0.003$) |  ($-0.23$) |  |  ($0.003$)
$A^{B_{c}B_{s1A}}$ | ${0.07}^{+0.00+0.00}_{-0.00-0.00}$ | ${-0.06}^{+0.05+0.00}_{-0.04-0.00}$ | ${0.08}^{+0.00+0.00}_{-0.00-0.00}$ | $0.07$ | $5.44$ | $0.22$ | $0.07$
($0.07$) |  ($-0.05$) |  |  ($0.08$)
$V_{0}^{B_{c}B_{s1A}}$ | ${0.07}^{+0.00+0.01}_{-0.00-0.01}$ | ${0.04}^{+0.17+0.05}_{-0.11-0.04}$ | ${0.07}^{+0.00+0.01}_{-0.00-0.01}$ | $0.56$ | $3.23$ | $0.37$ | $0.57$
($0.59$) |  ($-0.18$) |  |  ($0.61$)
$V_{1}^{B_{c}B_{s1A}}$ | ${1.57}^{+0.13+0.16}_{-0.12-0.18}$ | ${-0.39}^{+0.00+0.00}_{-0.00-0.00}$ | ${1.61}^{+0.13+0.17}_{-0.12-0.18}$ | $13.92$ | $2.17$ | $-0.46$ | $14.08$
($15.37$) |  ($-0.22$) |  |  ($15.95$)
$V_{2}^{B_{c}B_{s1A}}$ | ${-0.01}^{+0.01+0.01}_{-0.02-0.01}$ | ${-0.74}^{+0.26+0.14}_{-0.52-0.21}$ | ${-0.01}^{+0.01+0.01}_{-0.02-0.01}$ | $-0.01$ | $-30.90$ | $88.13$ | $-0.01$
($-0.04$) |  ($-1.20$) |  |  ($-0.04$)
$A^{B_{c}B_{s1B}}$ | ${0.08}^{+0.01+0.00}_{-0.01-0.00}$ | ${0.07}^{+0.05+0.04}_{-0.04-0.04}$ | ${0.09}^{+0.01+0.00}_{-0.01-0.00}$ | $0.08$ | $5.85$ | $5.03$ | $0.08$
($0.08$) |  ($0.07$) |  |  ($0.09$)
$V_{0}^{B_{c}B_{s1B}}$ | ${0.24}^{+0.00+0.03}_{-0.00-0.03}$ | ${0.10}^{+0.11+0.02}_{-0.08-0.02}$ | ${0.25}^{+0.00+0.03}_{-0.00-0.03}$ | $0.72$ | $2.43$ | $2.10$ | $0.73$
($0.77$) |  ($-0.19$) |  |  ($0.80$)
$V_{1}^{B_{c}B_{s1B}}$ | ${5.65}^{+0.12+0.58}_{-0.11-0.60}$ | ${-0.25}^{+0.00+0.00}_{-0.00-0.00}$ | ${5.85}^{+0.12+0.60}_{-0.12-0.62}$ | $17.44$ | $3.60$ | $0.68$ | $17.76$
($18.87$) |  ($-0.16$) |  |  ($19.62$)
$V_{2}^{B_{c}B_{s1B}}$ | ${0.06}^{+0.00+0.00}_{-0.13-0.19}$ | ${-5.17}^{+7.15+10.00}_{-0.00-0.00}$ | ${0.05}^{+0.00+0.00}_{-0.13-0.23}$ | $-0.03$ | $-44.22$ | $494.41$ | $-0.03$
($0.05$) |  ($-5.51$) |  |  ($0.05$)
Table 3: Bottom-changing $B_{c}\rightarrow A$ transition form factors in the CLFQM. F | Type-II | Type-I
---|---|---
(Type-I*)
$F(0)$ | a | b | $F(q^{2}_{max})$ | $F(0)$ | a | b | $F(q^{2}_{max})$
$A^{B_{c}D_{1A}}$ | ${0.12}^{+0.03+0.03}_{-0.03-0.03}$ | ${1.95}^{+0.04+0.15}_{-0.08-0.22}$ | ${3.63}^{+1.14+2.36}_{-1.00-1.32}$ | ${0.25}^{+0.15+0.15}_{-0.10-0.13}$ | $0.12$ | $2.85$ | $12.66$ | $0.07$
($0.12$) |  ($1.95$) |  ($3.63$) |  ($0.25$)
$V_{0}^{B_{c}D_{1A}}$ | ${0.08}^{+0.00+0.02}_{-0.00-0.02}$ | ${1.57}^{+0.15+0.25}_{-0.20-0.22}$ | ${2.08}^{+0.91+1.32}_{-0.77-0.76}$ | ${0.21}^{+0.01+0.08}_{-0.03-0.08}$ | $0.04$ | $2.90$ | $14.25$ | $0.02$
($0.04$) |  ($2.07$) |  ($4.11$) |  ($0.09$)
$V_{1}^{B_{c}D_{1A}}$ | ${0.30}^{+0.05+0.08}_{-0.05-0.08}$ | ${0.64}^{+0.18+0.31}_{-0.21-0.28}$ | ${0.92}^{+0.35+0.64}_{-0.26-0.30}$ | ${0.69}^{+0.08+0.13}_{-0.11-0.19}$ | $0.26$ | $2.06$ | $4.91$ | $0.28$
($0.26$) |  ($0.69$) |  ($0.93$) |  ($0.61$)
$V_{2}^{B_{c}D_{1A}}$ | ${0.09}^{+0.02+0.02}_{-0.02-0.02}$ | ${1.49}^{+0.08+0.17}_{-0.13-0.22}$ | ${2.24}^{+0.77+1.56}_{-0.65-0.84}$ | ${0.23}^{+0.11+0.11}_{-0.08-0.11}$ | $0.09$ | $2.78$ | $11.43$ | $0.06$
($0.09$) |  ($1.49$) |  ($2.24$) |  ($0.23$)
$A^{B_{c}D_{1B}}$ | ${0.001}^{+0.00+0.03}_{-0.00-0.00}$ | ${-59.96}^{+0.64+62}_{-0.00-2.44}$ | ${208}^{+0.00+8.01}_{-2.17-204}$ | ${0.00}^{+0.00+0.07}_{-0.00-0.00}$ | $0.001$ | $-92.70$ | $402.04$ | $0.000$
($0.001$) |  ($-59.96$) |  ($208.29$) |  ($0.000$)
$V_{0}^{B_{c}D_{1B}}$ | ${0.20}^{+0.04+0.07}_{-0.04-0.06}$ | ${1.93}^{+0.02+0.18}_{-0.05-0.23}$ | ${3.23}^{+0.92+2.17}_{-0.85-1.21}$ | ${0.46}^{+0.24+0.30}_{-0.16-0.25}$ | $0.16$ | $-4.94$ | $45.80$ | $0.02$
($0.17$) |  ($-2.33$) |  ($18.32$) |  ($0.05$)
$V_{1}^{B_{c}D_{1B}}$ | ${0.11}^{+0.03+0.04}_{-0.03-0.04}$ | ${1.50}^{+0.08+0.19}_{-0.12-0.23}$ | ${2.12}^{+0.75+1.47}_{-0.65-0.81}$ | ${0.28}^{+0.15+0.18}_{-0.11-0.15}$ | $0.004$ | $-92.61$ | $401.38$ | $0.000$
($0.005$) |  ($-50.04$) |  ($149.40$) |  ($0.000$)
$V_{2}^{B_{c}D_{1B}}$ | ${-0.06}^{+0.01+0.02}_{-0.01-0.02}$ | ${1.90}^{+0.05+0.00}_{-0.07-0.08}$ | ${4.82}^{+1.01+2.69}_{-0.96-1.56}$ | ${-0.10}^{+0.03+0.05}_{-0.05-0.08}$ | $-0.06$ | $2.59$ | $22.00$ | $-0.02$
($-0.06$) |  ($1.90$) |  ($4.82$) |  ($-0.10$)
$A^{B_{c}\chi_{c1}}$ | ${0.21}^{+0.01+0.02}_{-0.01-0.03}$ | ${2.19}^{+0.08+0.22}_{-0.10-0.23}$ | ${4.28}^{+0.94+2.11}_{-0.91-1.34}$ | ${0.33}^{+0.01+0.03}_{-0.02-0.05}$ | $0.21$ | $2.39$ | $6.29$ | $0.27$
($0.21$) |  ($2.19$) |  ($4.28$) |  ($0.33$)
$V_{0}^{B_{c}\chi_{c1}}$ | ${0.07}^{+0.00+0.01}_{-0.01-0.01}$ | ${0.98}^{+0.09+0.14}_{-0.06-0.12}$ | ${1.47}^{+0.42+0.69}_{-0.42-0.46}$ | ${0.09}^{+0.01+0.01}_{-0.01-0.01}$ | $0.07$ | $2.26$ | $9.98$ | $0.08$
($0.07$) |  ($2.26$) |  ($7.38$) |  ($0.10$)
$V_{1}^{B_{c}\chi_{c1}}$ | ${0.47}^{+0.01+0.05}_{-0.01-0.05}$ | ${-1.20}^{+0.09+0.26}_{-0.10-0.22}$ | ${2.28}^{+0.09+0.18}_{-0.09-0.06}$ | ${0.45}^{+0.00+0.03}_{-0.00-0.04}$ | $0.48$ | $-0.01$ | $2.23$ | $0.44$
($0.48$) |  ($-0.81$) |  ($1.98$) |  ($0.49$)
$V_{2}^{B_{c}\chi_{c1}}$ | ${0.08}^{+0.01+0.01}_{-0.01-0.01}$ | ${0.79}^{+0.15+0.27}_{-0.17-0.25}$ | ${2.13}^{+0.35+0.93}_{-0.31-0.55}$ | ${0.11}^{+0.01+0.00}_{-0.01-0.01}$ | $0.09$ | $1.48$ | $3.84$ | $0.10$
($0.08$) |  ($0.79$) |  ($2.13$) |  ($0.11$)
$A^{B_{c}\chi_{c1}^{\prime}}$ | ${0.04}^{+0.01+0.01}_{-0.01-0.03}$ | ${2.69}^{+0.00+0.00}_{-0.01-97.49}$ | ${7.04}^{+1.08+471}_{-1.10-2.09}$ | ${0.06}^{+0.02+0.01}_{-0.01-0.06}$ | $0.04$ | $2.62$ | $9.66$ | $0.04$
($0.04$) |  ($2.69$) |  ($7.04$) |  ($0.06$)
$V_{0}^{B_{c}\chi_{c1}^{\prime}}$ | ${0.35}^{+0.00+0.05}_{-0.00-0.05}$ | ${2.55}^{+0.04+0.22}_{-0.06-0.24}$ | ${4.78}^{+1.10+2.50}_{-1.05-1.56}$ | ${0.58}^{+0.02+0.09}_{-0.02-0.11}$ | $0.33$ | $2.51$ | $8.71$ | $0.39$
($0.33$) |  ($2.47$) |  ($6.13$) |  ($0.51$)
$V_{1}^{B_{c}\chi_{c1}^{\prime}}$ | ${0.30}^{+0.02+0.06}_{-0.02-0.06}$ | ${1.58}^{+0.12+0.25}_{-0.15-0.24}$ | ${2.38}^{+0.63+1.31}_{-0.58-0.80}$ | ${0.46}^{+0.03+0.08}_{-0.03-0.09}$ | $0.25$ | $2.28$ | $5.04$ | $0.33$
($0.25$) |  ($1.66$) |  ($2.42$) |  ($0.39$)
$V_{2}^{B_{c}\chi_{c1}^{\prime}}$ | ${-0.16}^{+0.01+0.02}_{-0.00-0.00}$ | ${2.40}^{+0.04+0.09}_{-0.03-0.14}$ | ${6.02}^{+0.76+2.56}_{-0.75-1.66}$ | ${-0.26}^{+0.00+0.05}_{-0.00-0.04}$ | $-0.16$ | $2.64$ | $10.98$ | $-0.18$
($-0.16$) |  ($2.40$) |  ($6.02$) |  ($-0.26$)
Table 4: Branching ratios of $B_{c}^{+}\to A^{({\prime})}\ell^{+}\nu_{\ell}$ decays. Type-I and Type-I* have the same formulation as stated in the caption of Table 2. Decays | Type-II | Type-I (Type-I*)
---|---|---
$B_{c}^{+}\to B_{1}^{0}e^{+}\nu_{e}$ | $({1.14}^{+0.56+0.68}_{-0.00-0.00})\times 10^{-6}$ |  ${3.56}\times 10^{-7}$ (${9.80}\times 10^{-7}$)
$B_{c}^{+}\to B_{1}^{0{\prime}}e^{+}\nu_{e}$ | $({2.12}^{+0.22+0.45}_{-0.04-0.32})\times 10^{-5}$ |  ${2.59}\times 10^{-4}$ (${3.21}\times 10^{-4}$)
$B_{c}^{+}\to B_{1}^{0}\mu^{+}\nu_{\mu}$ | $({1.03}^{+0.40+0.53}_{-0.00-0.00})\times 10^{-6}$ |  ${2.88}\times 10^{-7}$ (${7.72}\times 10^{-7}$)
$B_{c}^{+}\to B_{1}^{0{\prime}}\mu^{+}\nu_{\mu}$ | $({1.77}^{+0.15+0.37}_{-0.03-0.29})\times 10^{-5}$ |  ${2.18}\times 10^{-4}$ (${2.70}\times 10^{-4}$)
$B_{c}^{+}\to B_{s1}^{0}e^{+}\nu_{e}$ | $({1.13}^{+0.93+2.21}_{-0.00-0.00})\times 10^{-5}$ |  ${7.25}\times 10^{-6}$ (${2.48}\times 10^{-5}$)
$B_{c}^{+}\to B_{s1}^{0{\prime}}e^{+}\nu_{e}$ | $({1.04}^{+0.36+1.03}_{-0.00-0.08})\times 10^{-4}$ |  ${1.84}\times 10^{-3}$ (${2.13}\times 10^{-3}$)
$B_{c}^{+}\to B_{s1}^{0}\mu^{+}\nu_{\mu}$ | $({1.00}^{+0.48+1.31}_{-0.00-0.00})\times 10^{-5}$ |  ${5.62}\times 10^{-6}$ (${1.60}\times 10^{-5}$)
$B_{c}^{+}\to B_{s1}^{0{\prime}}\mu^{+}\nu_{\mu}$ | $({8.37}^{+1.89+6.12}_{-0.00-0.98})\times 10^{-5}$ |  ${1.41}\times 10^{-3}$ (${1.64}\times 10^{-3}$)
$B_{c}^{-}\to{D}_{1}^{0}e^{+}\nu_{e}$ | $({7.00}^{+2.77+6.89}_{-2.14-3.65})\times 10^{-6}$ |  ${2.26}\times 10^{-6}$ (${3.19}\times 10^{-6}$)
$B_{c}^{+}\to{D}_{1}^{0{\prime}}e^{+}\nu_{e}$ | $({3.93}^{+1.89+3.31}_{-1.55-2.27})\times 10^{-5}$ |  ${2.64}\times 10^{-7}$ (${3.75}\times 10^{-7}$)
$B_{c}^{+}\to{D}_{1}^{0}\mu^{+}\nu_{\mu}$ | $({6.98}^{+2.76+6.88}_{-2.14-3.65})\times 10^{-6}$ |  ${2.24}\times 10^{-6}$ (${3.17}\times 10^{-6}$)
$B_{c}^{+}\to{D}_{1}^{0{\prime}}\mu^{+}\nu_{\mu}$ | $({3.92}^{+1.89+3.30}_{-1.55-2.27})\times 10^{-5}$ |  ${2.90}\times 10^{-7}$ (${4.08}\times 10^{-7}$)
$B_{c}^{+}\to{D}_{1}^{0}\tau^{+}\nu_{\tau}$ | $({3.49}^{+1.47+3.27}_{-1.13-1.87})\times 10^{-6}$ |  ${2.33}\times 10^{-7}$ (${5.10}\times 10^{-7}$)
$B_{c}^{+}\to{D}_{1}^{0{\prime}}\tau^{+}\nu_{\tau}$ | $({1.74}^{+0.96+1.52}_{-0.75-1.08})\times 10^{-5}$ |  ${5.50}\times 10^{-8}$ (${1.77}\times 10^{-7}$)
$B_{c}^{+}\to\chi_{c1}e^{+}\nu_{e}$ | $({5.25}^{+0.17+1.01}_{-0.17-1.09})\times 10^{-4}$ |  ${5.07}\times 10^{-4}$ (${5.59}\times 10^{-4}$)
$B_{c}^{+}\to\chi_{c1}^{{\prime}}e^{+}\nu_{e}$ | $({1.09}^{+0.05+0.35}_{-0.05-0.33})\times 10^{-3}$ |  ${8.38}\times 10^{-4}$ (${9.29}\times 10^{-4}$)
$B_{c}^{+}\to\chi_{c1}\mu^{+}\nu_{\mu}$ | $({5.22}^{+0.17+1.00}_{-0.17-1.08})\times 10^{-4}$ |  ${5.03}\times 10^{-4}$ (${5.55}\times 10^{-4}$)
$B_{c}^{+}\to\chi_{c1}^{{\prime}}\mu^{+}\nu_{\mu}$ | $({1.08}^{+0.04+0.35}_{-0.04-0.33})\times 10^{-3}$ |  ${8.30}\times 10^{-4}$ (${9.22}\times 10^{-4}$)
$B_{c}^{+}\to\chi_{c1}\tau^{+}\nu_{\tau}$ | $({5.79}^{+0.19+1.02}_{-0.20-1.16})\times 10^{-5}$ |  ${5.38}\times 10^{-5}$ (${6.33}\times 10^{-5}$)
$B_{c}^{+}\to\chi_{c1}^{{\prime}}\tau^{+}\nu_{\tau}$ | $({8.82}^{+0.64+2.98}_{-0.61-2.85})\times 10^{-5}$ |  ${5.44}\times 10^{-5}$ (${7.11}\times 10^{-5}$)
Table 5: Predictions of physical observables in semileptonic $B_{c}^{+}\to A^{({\prime})}\ell^{+}\nu_{\ell}$ decays. Decays | $\langle A_{FB}\rangle$ | $\langle C_{F}^{\ell}\rangle$ | $\langle P_{L}^{\ell}\rangle$ | $\langle F_{L}\rangle$ | $\langle\alpha^{*}\rangle$
---|---|---|---|---|---
$B_{c}^{+}\to B_{1}^{0}e^{+}\nu_{e}$ | ${0.16}^{+0.00+0.00}_{-0.08-0.04}$ | $-0.25$ | $1.00$ | $0.44$ | $-0.23$
$B_{c}^{+}\to B_{1}^{0{\prime}}e^{+}\nu_{e}$ | ${-0.24}^{+0.02+0.02}_{-0.00-0.01}$ | $-0.45$ | $1.00$ | $0.53$ | $-0.39$
$B_{c}^{+}\to B_{1}^{0}\mu^{+}\nu_{\mu}$ | ${0.21}^{+0.03+0.00}_{-0.07-0.03}$ | $-0.14$ | $0.77$ | $0.46$ | $-0.27$