# Probing $\tau$ polarization in $B_{c}\to(\eta_{c},J/\psi,D,D^{*})\,l\nu_{l}$

Sonali Patnaik<EMAIL_ADDRESS>School of Physical Sciences,
National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Jatni, 752050, Odisha, India  Lopamudra Nayak
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science
Education and Research, An OCC of Homi Bhabha National Institute, Jatni,
752050, Odisha, India  Rajeev Singh<EMAIL_ADDRESS>Center for
Nuclear Theory, Department of Physics and Astronomy, Stony Brook University,
Stony Brook, New York, 11794-3800, USA  Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

In this research, we focus on exploring lepton flavor universality violating
(LFUV) observables, including longitudinal polarization ($P_{L}$),
$\tau$-polarization ($P_{\tau}$), and forward-backward asymmetry
($\mathcal{A}_{FB}$). Our investigation is conducted within the Relativistic
Independent Quark Model (RIQM), emphasizing the model-dependent analysis of
these observables in the decay channel:
$B_{c}\to\eta_{c}(J/\psi)/D(D^{*}),l\nu_{l}$. With a keen eye on the
experimental aspects of B-hadron decays, the rapid confirmation of these
quantities could signify a significant detection of physics beyond the
Standard Model, opening up new avenues for further research into New Physics.

††preprint:

##  I Introduction

The success of the LHC and the Standard Model (SM) in particle physics has
ushered in a paradigm shift, offering opportunities to explore new questions.
In the elegant framework of the SM, fermions serve as the basic elementary
units, further classified into quarks and leptons. For detailed properties of
elementary particles, refer to Ref. [1]. Leptons, traditionally considered
point-like particles with no substructure, undergo both weak and
electromagnetic interactions, while neutrinos exclusively participate in weak
interactions. The SM treats charged leptons ($e^{-}$, $\mu^{-}$, $\tau^{-}$)
as universal across three generations, accounting for kinematic effects due to
differing masses. Electroweak interaction predictions in the SM exhibit Lepton
Flavor Universality (LFU) [2, 3, 4, 5], experimentally verified in $\mu$
decays [6, 7, 8], $\tau$ decays [1], and Z boson decays [9]. LFU is an
accidental symmetry in the SM, arising from the Yukawa interactions’ strength
between lepton-Higgs, leading to different lepton masses
($m_{\tau}>m_{\mu}>m_{e}$). This implies that physical processes involving
charged leptons should exhibit Lepton Flavor Universality (LFU), an
approximate lepton flavor symmetry among observables like decay rates or
scattering cross-sections. In the Standard Model (SM), LFU is only broken by
charged lepton mass terms. Flavor-changing charged currents (FCCC) transitions
are solely mediated by charged weak bosons $W^{\pm}$, while flavor-changing
neutral current (FCNC) transitions involve neutral weak boson ($Z^{0}$) at the
tree level and virtual $W^{\pm}$ bosons and quarks in loop processes.

However, in current literature, evidence of Lepton Flavor Universality (LFU)
breaking has emerged in $b\to sl^{+}l^{-}$ ($l=e,\mu$) observed by various
collider experiments [10], challenging our fundamental understanding of
physics. This suggests the possible existence of new particles in extensions
to the Standard Model (SM) that violate this symmetry, leading to observable
changes in rates of quark-lepton transitions in $B$-hadron decays. Intriguing
anomalies have surfaced in different decay modes over the past few years [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], with recent theoretical
investigations detailed in Refs. [23, 24, 25, 26, 27, 28, 29].

First-order decays of beauty ($B$) mesons to final $\tau$ lepton states have
been observed in BaBar [30, 31], Belle [11, 32, 33, 34, 35], and LHCb [36, 19,
21, 37, 38, 10, 39, 40] experiments. In all cases, indications of Lepton
Flavor Universality Violation (LFUV) were reported. Recent observations in
leptonic and semileptonic $B$ meson decays ($B\to\tau\nu_{\tau}$ and $B\to
D(D^{*})l\,\nu_{l}$ with $l=e,\,\mu\,{\rm or}\,\tau$) challenge lepton
universality. Any evidence and observation of this violation, along with
clarification of interactions involving new particles, present an open window
to further explore physics Beyond the Standard Model (BSM). This avenue may
provide indirect insights into the nature of dark matter, the origins of
matter-antimatter asymmetry, or the dynamics of the electroweak scale.

This paper presents the decay rates of $B$ mesons into $\tau$ and $\mu$
leptons. The anomalies in charge-current transitions, specifically $B\to
D,D^{*}l{\bar{\nu}_{l}}$ decays, were initially measured by the BaBar
Collaboration in 2012 [30]. In particular, the LFU ratio is defined as:

$\mathcal{R}_{D^{(*)}}=\frac{Br(B\to D^{(*)}\tau{\bar{\nu}_{\tau}})}{Br(B\to
D^{(*)}l{\bar{\nu}_{l}})}\,,\qquad\text{with}\qquad\ell=\mu,e\,,$ (1)

where $D^{(*)}$ refers to $D(D^{*})$ meson and $Br$ indicates the branching
ratio.

The findings reported in Ref. [30] exhibit a 3.4 $\sigma$ discrepancy with the
Standard Model (SM). Subsequent measurements were conducted at BaBar (2013)
[31], Belle (2015) [32], and LHCb (2015) [36], largely confirming the earlier
results. A recent updated measurement of $R_{D}$ and $R_{D^{*}}$ by LHCb
(first joint measurement) [41], based on LHC Run 1 data, aligns with the
previous measurement [42, 43, 44] at a 3 $\sigma$ level (see Table 3 for the
summary). The $\tau$ is reconstructed in $\tau\to\mu\nu{\bar{\nu}}$,
surpassing the 2015 result [36]. Notably, the deviation features observables
consistently higher than SM predictions, implying Lepton Flavor Universality
Violation (LFUV).

In March 2023, LHCb released the results of $\mathcal{R}_{D^{*}}$ based on
$B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays, using the hadronic
$\tau\to\pi^{-}\pi^{+}\pi^{+}(\pi^{0})\nu_{\tau}$ channel. The analysis
utilized pp collision data collected at $\sqrt{s}$ = 13 TeV in 2015 and 2016,
updating a Run 1 measurement. The 2015 and 2016 data, with a luminosity of 2
$fb^{1}$, constitutes a new sample with approximately 1.5 times more $B^{0}$
decays than the previous dataset [45].

Simultaneously, Belle II presented, for the first time, the result of
$\mathcal{R}_{D^{*}}$ using $\bar{B}\to D^{*}l\bar{\nu}_{l}$. This new result
boasts a 40% improvement in statistical precision over the Belle experiment at
the same sample size, and importantly, it aligns with the predictions of the
SM.

$\mathcal{R}_{D^{*}}=0.267^{+0.041}_{-0.039}(stat.)\,^{+0.028}_{-0.033}(syst.)$

The world average is gradually approaching SM predictions [46, 47], and the
significance of the deviation remains above 3 $\sigma$, thanks to reduced
uncertainties. However, persistent anomalies in differential branching
fractions and angular analyses of the muon modes suggest ongoing
discrepancies. These theoretical and experimental measurements will contribute
to a more dedicated parameterization of form factors and their accurate
determination in semileptonic transitions. This motivates an examination of
$\mathcal{R}$ for other $b\to cl\nu_{l}$ semileptonic decay modes. LHCb
recently presented initial results for $\mathcal{R}(J/\psi)$ from $B_{c}\to
J/\psi$ decay i.e

$\mathcal{R}_{J/\psi}=\frac{Br(B_{c}{{}^{+}}\to
J/\psi\tau^{+}\nu_{\tau})}{Br(B_{c}{{}^{+}}\to J/\psi\mu^{+}\nu_{\mu})}\,,$
(2)

with the $\tau^{+}$ decaying leptonically to $\mu^{+}$ $\nu_{\mu}$
${\bar{\nu}_{\tau}}$. In this analysis, a sample of pp collision data
corresponding to 3fb-1 was collected at center-of-mass energies $\sqrt{s}=7$
TeV and $8$ TeV [48]. The $\tau$ lepton was reconstructed, and a global fit
was conducted on the missing mass squared ($q^{2}$) and the decay time of
$B_{c}$. In all cases, the decay branching fractions consistently deviate from
SM predictions, currently in the range of 0.25-0.28 [49, 50, 51]. The observed
values are approximately 2 $\sigma$ lower, as shown in Table 3. The spread in
Standard Model (SM) predictions arises from different modeling approaches for
determining the form factors [50, 52]. This observed anomaly, deviating from
SM predictions, suggests a potential violation of LFU. The $B$-factories
primarily operate on the $\Upsilon(4S)$ resonance for most data collection,
allowing for measurements using other $B_{q}$ species at the LHC.
Additionally, ongoing tree-level LFU tests at LHCb include $R(D^{+})$ and
baryonic observables $R_{{\Lambda_{c}}^{*}}$.

In addition to decay rate ratios, the longitudinal $\tau$-polarization is
crucial for investigation. In $b\to c\tau\nu$ decays, the subsequent decay of
the $\tau$ within the detector enables the measurement of the $\tau$
polarization fraction, given by
$[\Gamma(S_{\tau}=+)-\Gamma(S_{\tau}=-)]/\Gamma$, where $s_{\tau}$ is the
$\tau$ spin projection along a specific polarization axis, and $\Gamma$ is the
total rate. This polarization fraction is dependent on the hadronic final
state and is sensitive to beyond SM contributions, serving as a complementary
probe to the branching ratios or differential distributions of the three-body
final state. The recent Belle Collaboration measurement of $P_{\tau}(D^{*})$
and the fraction of $D^{*}$ longitudinal polarization $F_{L}^{D^{*}}$ for
$B\to D^{*}\tau\nu_{\tau}$ is detailed in [33, 53]

$\displaystyle P_{\tau}(D^{*})$ $\displaystyle=$ $\displaystyle-0.38\pm
0.51\,({\rm stat.})^{0.21}_{-0.16}\,({\rm syst.})\,,$ $\displaystyle
F_{L}(D^{*})$ $\displaystyle=$ $\displaystyle 0.60\pm 0.08\,({\rm
stat.})\pm(0.04)\,({\rm syst.})$ (3)

These results are compatible with the SM predictions [54, 55, 56]

$\displaystyle P_{\tau}(D^{*})$ $\displaystyle=$ $\displaystyle-0.497\pm
0.013\,,$ $\displaystyle F_{L}(D^{*})$ $\displaystyle=$ $\displaystyle
0.441\pm 0.006\quad{\rm or}\quad 0.457\pm 0.010\,.$ (4)

All these measurements unmistakably indicate a deviation from SM predictions.
To address these anomalies and assess the status of NP, various approaches
[54, 51, 56, 57] are being considered. These observables play a crucial role
in constraining and determining the most plausible NP extension of the SM.
Notably, the $\tau$-polarization ($P_{\tau}$), longitudinal polarization
($P_{L}$), and forward-backward asymmetry parameters $\mathcal{A}_{FB}$, along
with kinematic distributions, polarization asymmetry, and other LFUV
parameters, collectively form a comprehensive set for rigorously testing the
SM. This motivates us to present predictions for these observables in the
current study. Meanwhile, Quantum Chromodynamics (QCD), the theory of strong
interactions, is expected to address the quark confinement problem within
hadrons, which is governed by the internal dynamics of constituent quarks and
gluons. However, extracting form factors directly from first principles of QCD
has proven challenging due to the non-abelian and non-perturbative
characteristics of QCD. As an alternative, phenomenological models are
considered to describe the bound state nature of hadrons and their decay
properties. The study of LFUV can be approached theoretically through various
model-dependent and model-independent methods. In the literature, there are
groundbreaking theoretical approaches explaining anomalies in $B$-decays,
involving extensions to SM couplings

  * •

Utilization of heavy quark effective theory (HQET) to parameterize form
factors and derive order-by-order relations in $1/m_{Q}$ and $\alpha_{s}$.

  * •

Application of various quark models and potential models, including the QCD
sum rule, light cone sum rule, Bethe Salpeter approach, and relativistic quark
model approaches, to approximately compute form factors in different kinematic
regimes of $q^{2}$.

  * •

Theoretical calculations based on Lattice QCD (LQCD), presently available for
a limited subset of form factors and kinematic regimes. These developments
enable model-independent predictions on hadronic phenomena, testing the
electroweak theory in the SM.

In summarizing various models outlined above, it’s important to acknowledge
that our references may not be exhaustive. However, whether non-relativistic,
relativistic, QCD-inspired, or purely phenomenological, each model has its
advantages and limitations. A quark potential model succeeds when it
reasonably reproduces available observed data in different hadron sectors.
Regardless of the Lorentz structure of the interaction potential used, a
phenomenological model is considered reliable if it describes constituent-
level dynamics within the hadron core and predicts various hadronic
properties, including decays. However, the parameterization process at the
potential level involves some arbitrariness. In this sense, the potential
model approach is not unique, especially when limited to reproducing
experimental data in a narrow range. Therefore, it is crucial to extend the
applicability of a quark model to a broader range of observed data.

In this context, we present results on $\tau$ polarization in $B_{c}$ decays
within the Relativistic Independent Quark Model (RIQM), as discussed in the
latter part of the article. This paper focuses on recent results in
$B_{c}$-hadron decays, testing the applicability of the RIQM framework in
explaining LFU ratios. Predictions for various angular observables, valuable
for New Physics studies, are also illustrated. With SM Lattice QCD (LQCD)
vector and axial form factors available for $B_{c}\to J/\psi$ decay [58, 59],
we systematically compare SM observables computed with LQCD input in the RIQM.
Despite overall normalization inconsistencies, good agreements are found for
observables based on ratios of distributions, such as the forward-backward
parameter ($\mathcal{A}_{FB}$) and $\tau$-polarization, $P_{\tau}$
asymmetries. Ratios between predictions in $\tau$ and ($e,\mu$) modes, like
$\mathcal{R}_{J/\psi}$, $\mathcal{R}_{\mathcal{A}_{FB}}$, and
$\mathcal{R}_{P_{\tau}}$, also show good agreement.

The article is organized as follows: Section II provides a brief experimental
outlook on LFU and LFUV, covering LHCB, Belle, Atlas, and CMS. Section III
discusses our model framework, studies of $b\to c$ decays, and reports various
results on LFUV observables. Finally, Section IV presents a short summary and
future outlook.

##  II Experimental outlook on LFUV observables

Since the discovery of the $b$ quark, substantial samples of $B$-hadrons have
been generated at colliders like CESR, LEP, or Tevatron. However, prior to the
advent of $B$-factories and the LHC, it was impractical to explore third-
generation LFUV in $B$ mesons, even with their specialized detectors and
larger samples. $B$ meson decay measurements fall into two categories: those
involving FCNC with lepton pairs emitted and those involving FCCC with leptons
and neutrinos emitted. FCNC decays are suppressed at the tree level, requiring
higher-order processes (Fig. 1). Understanding these dynamics necessitates
significant modifications to amplitudes by NP mediators, such as leptoquarks
[60, 61, 62, 63] and $Z^{\prime}$ [64, 65, 66, 67]. FCNC decays have lower
branching ratios ($Br\sim 10^{-6}-10^{-7}$), while FCCC decays have larger
branching ratios (up to a few percent) but are experimentally challenging due
to neutrinos in the final state. LHCb, starting data collection in 2010, has
recorded a massive number of $b\bar{b}$ pairs up to 2020, enabling it to
overcome the challenges of pp collisions [68]. The detector’s design,
influenced by the collinear production of $b$ quarks, facilitates the study of
heavy meson decays, demonstrating good particle identification and excellent
momentum resolution [69]. LFU tests at LHCb reveal differences in electron and
muon hardware trigger efficiencies, and the detector studies various
$B$-hadron species, including $B_{c}$ mesons, despite their low production
rate. Recent neutral-current anomalies in clean ratios at LHCb, notably in
$R_{K}$ and $R_{K^{*}}$, surpass previous measurements and result from tighter
electron identification criteria and improved modeling of misidentified
hadronic backgrounds [10].

The systematic uncertainties in these measurements are currently smaller than
the statistical uncertainties, expected to further decrease with additional
data in Run 3. The ongoing commissioning of the LHCb Upgrade I detector aims
to boost the instantaneous luminosity by a factor of 5, collecting around 50
$fb^{-1}$ during Run 3. The proposed LHCb Upgrade II, planned for installation
during the LHC long shutdown 4, targets an integration of 300 $fb^{-1}$
throughout the HL-LHC era, allowing for a comprehensive exploration of flavor-
physics observables with unprecedented precision.

Simultaneously, the Belle detector, dedicated to heavy meson decays, includes
a superconducting solenoid coil generating a 1.5 T magnetic field. Its
components comprise a silicon vertex detector, a central drift chamber,
aerogel threshold Cherenkov Counters, time-of-flight scintillation counters,
and an electromagnetic calorimeter with Si(Ti) crystals. Strip detectors in
various particle colliders play a crucial role in understanding beam-beam
dynamics and decay vertices of long-lived particles. Silicon strip detectors,
along with layers of gaseous detectors, facilitate straightforward
measurements of charged particle momenta and trajectories. Decays like $B\to
D(D^{*})l\nu_{l}$ have been studied in BaBar [30, 31], Belle [32, 70, 33], and
LHCb [36, 71]. The upgraded Belle detector, Belle II [11], commenced data
collection in 2018, with plans to record over 40 billion $B\bar{B}$ pairs.
Brief discussions on the LFUV prospects for Belle II are available in [5].

The Belle II experiment and the LHCb detector, undergoing upgrades in 2019–21
and 2031, respectively, are anticipated to accumulate data over the next
decade and a half, surpassing current samples by more than an order of
magnitude. Recent Belle II measurements, including $\mathcal{R}_{D^{*}}$ with
a 40% improvement in statistical precision over Belle, align with SM
predictions. With LHC Run 3, Belle II aims to provide an $R_{D}$ measurement
with uncertainties 2 to 3 times smaller than the current world average,
consistent with the SM.

Experiments at ATLAS [72] and CMS [73] actively explore physics beyond the SM
through both direct and indirect methods. Indirect searches focus on rare and
forbidden decays of heavy mesons and baryons, where NP contributions can be
significant. CMS recently detected the excited $B_{c}$ state through
$B_{c}^{+}\pi^{+}\pi^{-}$, providing its mass. While limitations exist
compared to LHCb, ATLAS and CMS complement for decay processes with muons.
FCNC $b\to sll$ decays have been extensively studied by CMS during LHC Run 1,
with ongoing analyses for HL-LHC data, estimating precision on angular
parameters like $p^{\prime}_{5}$. The potential to measure RX ratios (X being
$D(D^{*}),J/\psi$) at ATLAS and CMS is yet to be demonstrated, making future
contributions uncertain. Despite challenges, CMS has conducted angular
analyses for FCNC transitions, paving the way for further exploration during
HL-LHC data taking.

The CMS experiment has released a recent study on the $B_{s}\to\mu^{+}\mu^{-}$
decay rate and lifetime, along with a search for the $B^{0}\to\mu^{+}\mu^{-}$
decay, utilizing extensive data from the 2016-2018 Run 2 of the LHC. The
analysis benefits from both a large data set and advanced machine learning
algorithms, enabling the identification of rare signal events amidst a
challenging background. The precision achieved in the decay rate measurement
surpasses previous experiments, and the observed $B_{s}$ decay rate aligns
closely with theoretical predictions. This result, in comparison with recent
LHCb measurements, reduces overall tension in B-decay anomalies, contributing
significantly to our understanding of these phenomena. The accomplishment is
attributed to the substantial data provided by LHC Run 2, the exceptional
performance of the CMS detector, and the expertise of physicists conducting
this advanced analysis. Continued investigations into these rare decays remain
highly compelling. Having firmly established and precisely measured the
$B_{s}\to\mu^{+}\mu^{-}$ decay, the focus now turns to studying the ultimate
rare transition: the $B^{0}\to\mu^{-}\mu^{+}$ decay. Anticipating large
datasets in the upcoming years of LHC Run 3, we may obtain the initial
insights into this exceptionally rare process, shedding light on the
perplexing anomalies and the physics beyond the standard model.

Table 1: Quarks and meson masses in RIQM framework Leptons | Quarks | Mesons
---|---|---
$e$ | 0.00051 | u | 0.078 | $B_{c}$ | 6.274 | $D$ | 1.869
$\mu$ | 0.10565 | c | 1.492 | $\eta_{c}$ | 2.983 | $D^{*}$ | 2.006
$\tau$ | 1.77686 | b | 4.776 | $J/\psi$ | 3.096 |  |
Table 2: Results of ratios of branching fractions for Semileptonic $B_{c}-$ decays in the ground state Ratio of Branching fractions($R$) (l = $e,\mu$) | RIQM | (CQM)[74] | (RCQM)[52] | (PQCD)[75] | PQCD + LQCD [76]
---|---|---|---|---|---
$R_{\eta_{c}}=\frac{{\cal B}(B_{c}\to\eta_{c}\tau\nu_{\tau})}{{\cal B}(B_{c}\to l\nu_{l})}$ | 0.4325 | 0.25 | 0.27 | 0.34 | 0.31
$R_{J}/\psi=\frac{{\cal B}(B_{c}\to J/\psi\tau\nu_{\tau})}{{\cal B}(B_{c}\to J/\psi l\nu_{l})}$ | 0.21 | 0.23 | 0.236 | 0.29 | 0.27
$R_{D}=\frac{{\cal B}(B_{c}\to D\tau\nu_{\tau})}{{\cal B}(B_{c}\to Dl\nu_{l})}$ | 0.784 | 0.63 | 0.59 | - | -
$R_{D^{*}}=\frac{{\cal B}(B_{c}\to D^{*}\tau\nu_{\tau})}{{\cal B}(B_{c}\to D^{*}l\nu_{l})}$ | 0.9165 | 0.56 | 0.58 | - | -

##  III Tests of LFUV observable in a model dependent formalism for
$B_{c}\to(\eta_{c},J/\psi,D,D^{*})l\nu_{l}$ with $l=e,\mu$, $\tau$

In this section, we introduce the R ratios for $b\to\tau,\mu$ leptons and
explore the longitudinal polarization of leptons in the context of $B_{c}\to
Xl\nu_{l}$, where $X=\eta_{c},J/\psi$, $D(D^{*})$. The corresponding Feynman
diagram is depicted in Fig. 1. The study of $B_{c}$ mesons is intriguing due
to their unique characteristics as the lowest bound state of two heavy open-
flavored quarks (charm and bottom). Positioned between charmonium $(c\bar{c})$
and bottomonium $(b\bar{b})$ in mass and size, the $B_{c}$ family provides a
valuable intermediate for understanding heavy quark interactions. Despite this
potential, the $B_{c}$ meson with two explicit heavy flavors has not been
extensively studied due to limited available data. Unlike the $c\bar{c}$ and
$b\bar{b}$ systems with hidden flavors that decay via strong and
electromagnetic interactions, the $B_{c}$ meson with open flavors decays
solely through weak interactions, residing below the $B\bar{D}$ threshold.
This characteristic results in a comparatively long lifetime and a rich
spectrum of weak decay channels with substantial branching ratios. Notably,
the CMS Collaboration [77] recently detected the excited $B_{c}$ state through
the study of $B_{c}^{+}\pi^{+}\pi^{-}$ based on the entire LHC sample of pp
collisions, providing valuable insights into the $B_{c}$(2S) meson mass,
measured at $6871\pm 1.2\pm 0.8$ MeV. Detection of the ground and excited
state of $B_{c}^{*}$ remains elusive. The prospect of enhanced event
accumulation rates for these undetected states, facilitated by the available
energy and higher luminosity at LHC and at a $Z_{0}$ factory, offers promising
opportunities for detailed studies of $B_{c}$ and $B_{c}^{*}$ counterparts.
The recently observed data, coupled with the anticipated high-statistics
$B_{c}$ events in ongoing and upcoming experiments, motivates a comprehensive
exploration of various decay properties in this sector. Consequently, the
$B_{c}$ meson serves as a unique window into heavy quark dynamics, providing
an independent test of QCD. With these considerations, we present a model-
dependent discussion to address the substantial discrepancies between SM and
BSM physics.

###  III.1 RIQM framework

Figure 1: SM contribution for $B_{c}\to Xl\nu_{l}$ where $X=\eta_{c},J/\psi$ ,
$D(D^{*})$.

Studying exclusive semileptonic decays involving non-perturbative hadronic
matrix elements poses a challenge. Rigorous application of field-theoretic
techniques and QCD from first principles for reliable measurements of
invariant transition amplitudes is currently unattainable. Consequently,
various theoretical approaches resort to phenomenological models to explore
non-perturbative QCD dynamics. Several existing theoretical approaches, such
as the use of HQET to generate relations in $1/m_{Q}$ and $\alpha_{s}$ between
form factors, contribute valuable insights. Quark models, including QCD sum
rule, light cone sum rule, and lattice QCD calculations, offer approximations
for form factors. These approaches play a crucial role in shaping differential
decay amplitudes and enabling sensitive measurements at the forefront of
physics.

In this context, we provide a concise overview of world averages using a
model-dependent approach, specifically adopting the Relativistic Independent
Quark Model (RIQM). The RIQM is grounded in a confining harmonic potential in
the equally mixed scalar-vector form [78, 79, 80]

$U(r)=\frac{1}{2}\left(1+\gamma^{0}\right)\,V(r)\,,$ (5)

where $V(r)=(ar^{2}+V_{0})$. In the given expression, $r$ represents the
relative distance between the quark and antiquark, $\gamma^{0}$ denotes the
time-like Hermitian matrix, and the parameters $a$ and $V_{0}$ are associated
with the potential. These parameter values have been determined in prior
applications of the model through hadron spectroscopy, and their specific
values can be found in [78]

$\displaystyle\left(a,\,V_{0}\right)$ $\displaystyle=$
$\displaystyle\left(0.017166\,{\rm GeV}^{3},\,-\,0.1375\,{\rm GeV}\right)\,.$

The internal dynamics of the constituent quarks are assumed to be described by
a quark Lagrangian density, featuring an appropriate Lorentz structure, given
by

${\cal
L}^{0}_{q}(x)={\bar{\psi}}_{q}(x)\;\left[\frac{i}{2}\,\gamma^{\mu}\partial_{\mu}-m_{q}-U(r)\;\right]\;\psi_{q}(x)\,,$
(6)

leading to the Dirac equation for individual quark

$\left[\gamma^{0}E_{q}-{\vec{\gamma}}.{\vec{p}}-m_{q}-U(r)\right]\psi_{q}(\vec{r})=0\,,$
(7)

with $\psi_{q}(\vec{r})$ representing the four-component Dirac normalized wave
function that can be written as

$\displaystyle\psi_{q}(\vec{r})=\left(\begin{array}[]{c}\psi_{A}(\vec{r})\\\
\psi_{B}(\vec{r})\end{array}\;\right)=\left(\begin{array}[]{c}\psi_{A}(\vec{r})\\\
\left(\frac{{\vec{\sigma}}\cdot{\vec{p}}}{E_{q}+m_{q}}\right)\;\psi_{A}(\vec{r})\end{array}\right)\,.$
(12)

Here, $\psi_{A}(\vec{r})$ and $\psi_{B}(\vec{r})$ are the upper and lower
components of quark wave functions with opposite parity, respectively.

The RIQM is a QCD-inspired phenomenological model, akin to other potential
models. It characterizes the confinement of constituent quarks within hadrons
through an interaction potential with a specified Lorentz structure. The model
aims to derive observable properties of composite hadrons from the constituent
quark dynamics, as a direct derivation from first principles of QCD is
currently unattainable due to inherent complexities. The chosen potential,
represented by the form in Eq. (5), is assumed to depict non-perturbative
multi-gluon interactions. Additionally, residual interactions, such as quark-
pion coupling from chiral symmetry restoration in the SU(2)-sector and one-
gluon exchange at short distances, are treated perturbatively in this model.
The selected confining potential yields a simple and manageable form,
facilitating the analysis of various hadronic properties and providing an
adequate tree-level description for decays involving FCNC. To achieve this
goal, we expand the RIQM framework’s scope to different sectors, demonstrating
it as a viable phenomenological scheme for studying various hadronic
phenomena. We compare our results with other theoretical approaches and
available experimental data. The model’s applicability and reliability have
been previously validated across a broad spectrum of hadronic phenomena,
encompassing radiative, weak radiative, rare radiative [81, 82, 83, 84, 85,
86], leptonic [87], weak leptonic [88], semileptonic [89, 90, 91], radiative
leptonic [92, 93, 94], and non-leptonic [95, 96, 97, 98] decays in both light
and heavy flavor sectors [99, 100, 78]. The invariant matrix element for
$B_{c}\to\eta_{c}(J/\psi)l^{-}\bar{\nu}_{l}$ and $B_{c}\to
D(D^{*})l^{-}\bar{\nu}_{l}$ is expressed in a general form [101]

${\cal M}(p,k,k_{l},k_{\nu})={\frac{\cal
G_{F}}{\sqrt{2}}}V_{bq^{{}^{\prime}}}\,{\cal H}_{\mu}(p,k)\,{\cal
L}^{\mu}(k_{l},k_{\nu})\,,$ (13)

with ${\cal G}_{F}$ being the effective Fermi coupling constant,
$V_{bq^{{}^{\prime}}}$ being the relevant CKM parameter, ${\cal L}^{\mu}$ and
${\cal H}_{\mu}$ are leptonic and hadronic current, respectively. Here,
$p,k,k_{l},k_{\nu}$ denote parent ($B_{c}$) and daughter (X) meson’s, lepton
and neutrino four-momentum, respectively. In the context of decay processes,
the physical occurrence relies on the participating mesons being in their
momentum eigenstates. Thus, a field-theoretic depiction of any decay process
necessitates representing meson bound states through suitable momentum wave
packets, reflecting the momentum and spin distribution between the constituent
quark and antiquark within the meson core. In the RIQM approach, the wave
packet denoting a meson bound state, such as
$|B_{c}(\vec{p},S_{B_{c}})\rangle$ at a specific momentum $\vec{p}$ and spin
$S_{B_{c}}$, is formulated as follows [62, 63, 64, 65, 66, 67, 69]

$\big{|}B_{c}\left(\vec{p},S_{B_{c}}\right)\big{\rangle}=\hat{\Lambda}\left(\vec{p},S_{B_{c}}\right)\big{|}\left(\vec{p_{b}},\lambda_{b}\right);\left(\vec{p_{c}},\lambda_{c}\right)\big{\rangle}\,.$
(14)

In the given expression,
$|(\vec{p_{b}},\lambda_{b});(\vec{p_{c}},\lambda_{c})\rangle$ denotes the Fock
space representation of the unbound quark and antiquark in a color-singlet
configuration with their respective momentum and spin, and
${\hat{\Lambda}}(\vec{p},S_{B_{c}})$ represents an integral operator
encapsulating the bound state characteristics of a meson.

###  III.2 Helicity amplitudes and LFUV observables

By incorporating the weak form factors derived through the covariant expansion
of hadronic amplitudes based on the model dynamics, the angular decay
distribution in the momentum transfer squared $q^{2}$ is obtained. Here,
$q=p-k=k_{l}+k_{\nu}$, and the decay distribution is calculated within the
allowed kinematic range: $0\leq q^{2}\leq(M-m)^{2}$, where M and m represent
the mass of the parent and daughter meson, respectively, [101]

$\frac{d\Gamma}{dq^{2}d\cos\theta}={\frac{{\cal
G}_{F}}{(2\pi)^{3}}}|V_{bq}|^{2}\frac{(q^{2}-m_{l}^{2})^{2}}{8M^{2}q^{2}}|\vec{k}|{\cal
L}^{\mu\sigma}{\cal H}_{\mu\sigma}\,.$ (15)

In the provided equation, ${\cal L}^{\mu\sigma}$ and ${\cal H}_{\mu\sigma}$
represent the lepton and hadron correlation functions, respectively.
Additionally, $m_{l}$ denotes the mass of the charged lepton, and $M$
represents the mass of the parent meson ($B_{c}$). Utilizing the completeness
property, the lepton and hadron tensors in Eq. (15) can be expressed as
follows

$\displaystyle{\cal L}^{\mu\sigma}{\cal H}_{\mu\sigma}$ $\displaystyle=$
$\displaystyle{\cal
L}_{\mu^{\prime}\sigma^{\prime}}g^{\mu^{\prime}\mu}g^{\sigma^{\prime}\sigma}{\cal
H}_{\mu\sigma}\,,$ (16) $\displaystyle=$ $\displaystyle{{\cal
L}_{\mu^{\prime}\sigma^{\prime}}}{\epsilon^{\mu^{{}^{\prime}}}}(m){\epsilon^{\mu^{\dagger}}}(m^{{}^{\prime}}){g_{mm^{{}^{\prime}}}}{\epsilon^{\sigma^{\dagger}}}(n){\epsilon^{\sigma^{{}^{\prime}}}}(n^{{}^{\prime}})g_{nn^{{}^{\prime}}}{\cal
H}_{\mu\sigma}\,,$ $\displaystyle=$
$\displaystyle{L(m,n)}{g_{mm^{\prime}}}{g_{nn^{\prime}}}H({m^{\prime}}{n^{\prime}})\,.$

In the calculation for convenience, the lepton and hadron tensors are
introduced in the space of helicity components as

$\displaystyle L(m,n)$ $\displaystyle=$
$\displaystyle{\epsilon^{\mu}}(m){\epsilon^{\sigma^{\dagger}}}(n){\cal
L}_{\mu\nu}\,,$ $\displaystyle H(m,n)$ $\displaystyle=$
$\displaystyle{\epsilon^{\mu^{\dagger}}}(m){\epsilon^{\sigma}(n)}{\cal
H}_{\mu\nu}\,.$ (17)

Hence, for simplicity, we express physical observables in terms of helicity
components. Accordingly, the helicity form factors are defined using Lorentz
invariant form factors representing the decay amplitudes, and their evaluation
is performed based on the model dynamics. The Lorentz contraction in Equation
(15) is carried out with the helicity amplitudes, as demonstrated in [101].

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

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

The upper and lower signs in Equation (18) are related to the parity-violating
term. Among the seven terms in Equation (18), four terms are recognized as
tilde rates $\tilde{\Gamma}_{i}$, linked to the lepton mass, while the
remaining terms are identified as $\Gamma_{i}$, which are lepton mass-
independent. The relationship between them involves a flip factor
$m_{l}^{2}/(2q^{2})$, expressed as

$\frac{d{\tilde{\Gamma}}_{i}}{dq^{2}}=\frac{m_{l}^{2}}{2q^{2}}\frac{d{\Gamma_{i}}}{dq^{2}}\,.$
(19)

In the limit of vanishing lepton mass, the tilde rates do not contribute to
the decay rate. While they can be disregarded for $e$ and $\mu$ modes, they
are anticipated to have a significant impact on $\tau$-modes. Consequently,
the tilde rates play a crucial role in assessing the lepton mass effects in
the semileptonic decay modes. The differential partial helicity rates
$d\Gamma_{i}/dq^{2}$ are defined by

$\frac{d\Gamma_{i}}{dq^{2}}=\frac{{\cal
G}_{f}^{2}}{(2\pi)^{3}}{|V_{bq^{{}^{\prime}}}|}^{2}\frac{({q^{2}}-{m_{e}^{2}})^{2}}{12{M^{2}}q^{2}}|\vec{k}|H_{i}\,.$
(20)

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

${H_{U}}=Re\left({H_{+}}{H_{+}^{\dagger}}\right)+Re\left({H_{-}}{H_{-}^{\dagger}}\right)\,,$
${H_{L}}=Re\left({H_{0}}{H_{0}^{\dagger}}\right)\,,$
${H_{P}}=Re\left({H_{+}}{H_{+}^{\dagger}}\right)-Re\left({H_{-}}{H_{-}^{\dagger}}\right)\,,$
${H_{S}}=3Re\left({H_{t}}{H_{t}^{\dagger}}\right)\,,$
${H_{SL}}=Re\left({H_{t}}{H_{0}^{\dagger}}\right)\,.$

Here, $H_{U}$, $H_{L}$, $H_{P}$, $H_{S}$, and $H_{SL}$ represent unpolarized-
transverse, longitudinal, parity-odd, scalar, and scalar-longitudinal
interference functions, respectively.

We assume that the helicity amplitudes are real since the available
$q^{2}$-range ($q^{2}\leq(M-m)^{2}$) is below the physical threshold
$q^{2}=(M+m)^{2}$. Therefore, we drop the angular terms multiplied by
coefficients $Im(H_{i}H_{j}^{*}),i\neq j^{*}$. Integrating over $\cos\theta$,
one obtains the differential $q^{2}$ distribution. Finally, integrating over
$q^{2}$, one gets the total decay rate $\Gamma$ as the sum of the partial
decay rates: $\Gamma_{i}$ (where $i=U,L,P$) and $\tilde{\Gamma}_{i}$ (where
$i=U,L,S,SL$). In the numerical analysis section (III.3), we present our
results for the integrated partial helicity rates and the total decay rate in
Table 8 for all the semileptonic transitions investigated in this study.

Moving forward, we explore a significant quantity of interest: the lepton
polarization asymmetry for leptons in the final state. Recently, polarization
measurements of final state $\tau$ leptons have been proposed as a valuable
tool for discerning physics beyond the Standard Model (SM) in various
processes. Examples include determining the longitudinal polarization
($P_{L}$) of leptons in $B_{c}\to\eta_{c}(J/\psi)D(D^{*})$. In addition to the
decay rates and the ratios $\mathcal{R}(X)$, the ($P_{L}$) of the leptons and
the fraction of longitudinal polarization $\mathcal{R}(P_{L})$ are additional
physical observables sensitive to certain types of New Physics (NP) [54, 33,
53]. Given the recent measurements of $P_{\tau}(D^{*})$ and the fraction of
$D^{*}$ longitudinal polarization $F_{L}^{D^{*}}$ by the Belle Collaboration
[33, 53], we extend our analysis to include similar observables for
$\eta_{c}$, $J/\psi$, D, and $D^{*}$ within the RIQM framework. For the decays
$B_{c}\to\eta_{c}(J/\psi),D(D^{*})l\nu_{l}$, we also consider the relevant
longitudinal polarizations and $\tau$ polarizations
($P_{\tau}(\eta_{c}),P_{\tau}(J/\psi),P_{\tau}(D),$ and $P_{\tau}(D^{*})$)
following the definitions used in [54, 57].

The rates $\frac{d\Gamma_{i}}{dq^{2}}$ and
$\frac{d\tilde{\Gamma_{i}}}{dq^{2}}$ correspond to the neutrino-lepton spin
no-flip and flip contributions, respectively. The composition of flip and no-
flip contributions determines the longitudinal polarization $P_{L}$ of the
leptons. Therefore, we define the observable $P_{L}$ as

$P_{L}=\frac{\frac{d\tilde{\Gamma_{i}}}{dq^{2}}^{h=-1}-\frac{d\tilde{\Gamma_{i}}}{dq^{2}}^{h=+1}}{\frac{d\tilde{\Gamma_{i}}}{dq^{2}}^{h=-1}+\frac{d\tilde{\Gamma_{i}}}{dq^{2}}^{h=+1}}\,.$
(21)

Here, $h$ represents helicity and is defined as $h=s\cdot p=\pm 1$, where the
sign corresponds to the case when the spin polarization is parallel (+) or
anti-parallel (-) to the lepton ($e,\mu,\tau$) momentum direction. This is
associated with the usual description of a polarization asymmetry, which is
given in terms of the couplings of helicity amplitudes as mentioned below

$P_{L}=\frac{\tilde{U}+\tilde{L}+\tilde{S}-U-L}{\tilde{U}+\tilde{L}+\tilde{S}+U+L}\,.$
(22)

The predictions for $P_{L}$, $P_{\tau}$, and the fraction
$\mathcal{R}_{P_{L}}$ are provided in Table 4, 5, and 6, respectively.
Additionally, we introduce another parameter for investigation, the forward-
backward asymmetry $\mathcal{A}_{FB}$ of the leptons in the $(l\nu_{l})$
frame. This allows us to predict the forward-backward asymmetry fraction
$\mathcal{R}_{\mathcal{A}_{FB}}$, which is defined as

$\mathcal{A}_{FB}=\frac{3}{4}\Bigg{[}\frac{\pm
P+4\tilde{S}L}{U+\tilde{U}+L+\tilde{L}+\tilde{S}}\Bigg{]}\,.$ (23)

The results are given in Table 7.

Table 3: LFU observables for $B_{c}$-decays to radially excited charmonium states Ratio | RIQM | [102] | [103, 104] | [105] | [106] | [107]
---|---|---|---|---|---|---
$R_{\eta_{c}}(2S)$ | 0.136 | 0.051 | - | 0.068 | 0.74 | 0.028
$R_{\eta_{c}}(3S)$ | 0.021 | 0.0103 | 0.0009 | 0.0013 | 0.030 |
$R_{\psi}(2S)$ | 0.085 | 0.049 | - | 0.069 | - | 0.071
$R_{\psi}(3S)$ | 0.077 | 0.009 | 0.0063 | 0.001 |  |

###  III.3 Numerical analysis and results

Our aim is to investigate LFU ratios, longitudinal polarization,
$\tau$-polarization, ratios of $\tau$-polarization fraction, and ratios of
forward-backward asymmetry fraction within the RIQM framework. To begin, we
gather the input parameters for lepton masses: $m_{e}=0.511\ \text{MeV}$,
$m_{\mu}=105.65\ \text{MeV}$, $m_{\tau}=1776.86\ \text{MeV}$, as well as
$V_{ckm}$ and the lifetime of the $B_{c}$ meson: $V_{B_{c}}=0.041$,
$V_{bu}=0.00382$, $\tau_{B_{c}}=0.510\ \text{ps}$ from the PDG [1]. The
relevant masses for quarks and mesons are determined from the model dynamics
and listed in Table 2. We initially calculate the observable $\mathcal{R}$
within the model framework [101]. Our predicted values for $\mathcal{R}$ in
$B_{c}\to X(nS)l\nu_{l}$, where $X=\eta_{c},J/\psi$, $D(D^{*})$, for both
ground and radially excited states are found to be comparable to other SM
predictions and lattice predictions, as presented in Table 2 and Table 3,
respectively. The deviation of SM predictions of $\mathcal{R}$ from the
experimental data suggests anomalies in semileptonic decays. While our
predictions for LFUV observables for the charm and charmonium higher states,
$R_{D}(2S)$, $R_{D}(3S)$, $R_{D^{*}}(2S)$, and $R_{D^{*}}(3S)$ lack
established model approach data in the literature, they can be valuable for
identifying $B_{c}$ channels in the upcoming Run 3 data at LHCb.

In table 8, we present our results for integrated partial helicity rates
($\Gamma_{i}$, $i=U,L,P$) and tilde rates ($\tilde{\Gamma}_{i}$,
$i=U,L,S,SL$), along with total decay rates. The partial tilde rates in
$e^{-}$-mode are negligible due to limited phase space, while the
corresponding rates in $\tau^{-}$-mode are significant and cannot be
neglected. Our predicted decay rates generally follow the trend of being
smaller in $\tau^{-}$ modes compared to $e^{-}$ modes, consistent with other
model predictions. For CKM-suppressed $B_{c}\to D(D^{*})$ transitions, the
$\tau^{-}$-modes show only marginal reduction compared to their $e^{-}$
counterparts.

Theoretical predictions for the longitudinal polarization of the tau lepton
($P_{\tau}$) in the considered semileptonic
$B_{c}\to(\eta_{c},J/\psi)l\nu_{l}$ decays are available in two approaches:
Perturbative QCD (PQCD) and PQCD + Lattice.

In the “PQCD approach”:

$P_{\tau}(\eta_{c})=0.37\pm 0.01,\quad P_{\tau}(J/\psi)=-0.55\pm 0.01$

In the “PQCD+Lattice” approach: [76]

$P_{\tau}(\eta_{c})=0.36\pm 0.01,\quad P_{\tau}(J/\psi)=-0.53\pm 0.01$

Table 4: Results of longitudinal polarization $(P_{L})$ of final state meson for $B_{c}\to Xl\nu_{l}$ where X is $\eta_{c}$, $J/\psi$, D and $D^{*}$, $l=e,\mu$ and $\tau$ $P_{L}$ Parameters | RIQM Framework
---|---
$B_{c}\to D(e)$, $P^{D}_{e}$ | -0.99
$B_{c}\to D(\mu)$, $P^{D}_{\mu}$ | -0.99
$B_{c}\to D(\tau)$, $P^{D}_{\tau}$ | -0.46
$B_{c}\to D^{*}(e)$, $P^{D^{*}}_{e}$ | -0.99
$B_{c}\to D^{*}(\mu)$, $P^{D^{*}}_{\mu}$ | -0.98
$B_{c}\to D^{*}(\tau)$, $P^{D^{*}}_{\tau}$ | 0.13
$B_{c}\to\eta_{c}(e)$, $P^{\eta_{c}}_{e}$ | -0.99
$B_{c}\to\eta_{c}(\mu)$, $P^{\eta_{c}}_{\mu}$ | -0.97
$B_{c}\to\eta_{c}(\tau)$, $P^{\eta_{c}}_{\tau}$ | 0.28
$B_{c}\to J/\psi(e)$ $P^{J/\psi}_{e}$ | -0.99
$B_{c}\to J/\psi(\mu)$ $P^{J/\psi}_{\mu}$ | -0.98
$B_{c}\to J/\psi(\tau)$ $P^{J/\psi}_{\tau}$ | -0.56

Hence, considering the established theoretical predictions, we present our
results for the longitudinal polarization $(P^{X}_{L})$ of leptons in Table 4
for $\eta_{c}$, $J/\psi$, $D$, and $D^{*}$, where $l=e,\mu$, and $\tau$. The
predictions for semileptonic modes involving $\mu$ leptons exhibit
insignificant differences compared to the electron case. Notably, our results
from the RIQM framework align well with predictions from LQCD and PQCD
approaches, as evident from the comparison.

Table 5: Predictions of Longitudinal $\tau$-polarization ($P_{\tau}$) for $\eta_{c}$, $J/\psi$, $D$ and $D^{*}$ [76] $P_{\tau}$ | RIQM Framework | SM | PQCD | Lattice + PQCD
---|---|---|---|---
$P_{\tau}(\eta_{c})$ | 0.28 | 0.34 | 0.37 | -0.36
$P_{\tau}(J/\psi)$ | -0.56 | -0.51 | -0.55 | -0.53
$P_{\tau}(D)$ | -0.46 | - | - | -
$P_{\tau}(D^{*})$ | 0.13 | - | - | -

Additionally, we present our predictions for the ratios of longitudinal
polarization fractions for $J/\psi$ and $D^{*}$ ($b\to c$), along with the
ratios of forward-backward asymmetry fractions for $\eta_{c}$, $J/\psi$, $D$,
and $D^{*}$. The corresponding results are detailed in Table 6 and 7,
respectively.

$\mathcal{R}_{P_{L}}=\frac{P_{L(\tau)}}{P_{L(\mu(e))}}$
$\mathcal{R}_{\mathcal{A}_{FB}}=\frac{\mathcal{A}_{FB(\tau)}}{\mathcal{A}_{FB(\mu(e))}}$

It’s noteworthy that our RIQM approach provides a comprehensive
parametrization of relevant form factors for semileptonic decay amplitudes
across the entire kinematic range ($0\leq q^{2}\leq q^{2}_{\text{max}}$). This
approach enhances the reliability and accuracy of our predictions compared to
other theoretical models. In various cited models, form factors are initially
calculated with an endpoint normalization at either $q^{2}=0$ (maximum recoil
point) or $q^{2}_{\text{max}}$ (minimum recoil point). Subsequently, using
monopoles, dipoles, and Gaussian ansatz, they are phenomenologically
extrapolated to the entire physical region, introducing uncertainties in the
form factor estimation. To circumvent such uncertainties, our approach avoids
any reliance on such phenomenological ansatz. With the anticipated high
statistics of $B_{c}$-events, expected to yield up to $10^{10}$ events each
year at colliders, the study of semileptonic $B_{c}$ decays to charm and
charmonium states provides an intriguing avenue to explore the new physics
scale further.

Table 6: Results of longitudinal polarization fraction ($\mathcal{R}_{P_{L}}$) for $J/\psi$, and $D^{*}$ $\mathcal{R}_{P_{L}}$ | RIQM Framework
---|---
$\mathcal{R}_{P_{L}}(J/\psi)$ | 0.507 (0.518)
$\mathcal{R}_{P_{L}}(D^{*})$ | 0.347(0.347)
Table 7: Results of forward-backward asymmetry fraction for $\eta_{c}$, $J/\psi$, D and $D^{*}$ Forward-backward asymmetry fraction | RIQM Framework
---|---
$\mathcal{R}_{\mathcal{A}_{FB}}(\eta_{c})$ | $4.15\times 10^{-5}(1.25\times 10^{-8})$
$\mathcal{R}_{\mathcal{A}_{FB}}(J/\psi)$ | 0.507 (0.518)
$\mathcal{R}_{\mathcal{A}_{FB}}(D)$ | -
$\mathcal{R}_{\mathcal{A}_{FB}}(D^{*})$ | 0.347(0.347)

##  IV Summary & future outlook

In conclusion, despite the SM being a highly successful mathematical
framework, it remains incomplete. This paper explores the landscape of LFUV
anomalies in $B$ physics, offering insights from both theoretical and
experimental perspectives. Notably, neutral-current LFU ratios
($R_{K},R_{K^{*}}$) align with SM predictions, considered theoretically clean
observables. Updated results for $R_{D}$ and $R_{D^{*}}$ reveal a significance
level around 3 $\sigma$ deviation, studied within a model-dependent (RIQM)
framework for various LFUV observables.

Table 8: Helicity rates (in $10^{-15}$ GeV) of semileptonic $B_{c}$-meson decays into charmonium and charm-meson state: Decay mode | $U$ | $\tilde{U}$ | $L$ | $\tilde{L}$ | $P$ | $S$ | $\tilde{S}$ | $\tilde{SL}$ | $\Gamma$
---|---|---|---|---|---|---|---|---|---
${B^{-}_{c}}\to\eta_{c}e^{-}\nu_{e}$ |  |  | 4.844 | 4.432$10^{-7}$ |  |  | 15.397$10^{-7}$ | 4.712$10^{-7}$ | 4.844
$B_{c}\to\eta_{c}\tau^{-}\nu_{\tau}$ |  |  | 0.756 | 0.172 |  |  | 1.194 | 0.253 | 2.122
${B^{-}_{c}}\to J/\psi e^{-}\nu_{e}$ | 18.634 | 6.052$\times 10^{-7}$ | 16.283 | 27.813$\times 10^{-7}$ | 8.368 | 1.188 | 66.653$\times 10^{-7}$ | 22.856$\times 10^{-7}$ | 34.918
${B^{-}_{c}}\to J/\psi\tau^{-}\nu_{\tau}$ | 3.823 | 0.846 | 1.922 | 0.437 | 1.704 | 0.614 | 0.307 | 0.197 | 7.336
${B^{-}_{c}}\to De^{-}\nu_{e}$ |  |  | 0.047 | 4.611$\times 10^{-10}$ |  |  | 1.072$\times 10^{-9}$ | 4.038$\times 10^{-10}$ | 0.047
${B^{-}_{c}}\to D\tau^{-}\nu_{\tau}$ |  |  | 0.028 | 0.003 |  |  | 0.007 | 0.0027 | 0.038
${B^{-}_{c}}\to D^{*}e^{-}\nu_{e}$ | 0.2439 | 4$\times 10^{-9}$ | 0.078 | 7.760$\times 10^{-9}$ | 0.169 | 0.081 | 4.092$\times 10^{-8}$ | 3.648$\times 10^{-9}$ | 0.322
${B^{-}_{c}}\to D^{*}\tau^{-}\nu_{\tau}$ | 0.113 | 0.015 | 0.0156 | 0.0021 | 0.092 | 0.046 | 0.151 | 0.0094 | 0.297

The $\tau-$ polarization in $B_{c}\to(\eta_{c},J/\psi,D,D^{*})l\nu_{l}$ is
identified as highly sensitive to New Physics, serving as a potent probe of
the SM. $P_{\tau}$ values align well with LQCD and other established models.
Anticipating the large data samples from upcoming Run 3 at B-factories, we
await machine completion. Predictions of LFUV observables are offered as
valuable probes for NP determination.

Various NP models, including leptoquark, $Z^{\prime}$-boson, composite Higgs
boson, dark matter, and right-handed neutrinos, address the deviation.
Continued confirmation of these results would constitute unambiguous evidence
of NP interpretations. With Run 3 data collection (2022-2025) at the LHC,
event statistics are expected to triple, reducing statistical and systematic
uncertainties. LHCb detector upgrades and Belle II analysis are crucial for
clarifying flavor anomalies. Theoretical reporting on existing models and the
consideration of more NP models for experimental testing are essential. A
comprehensive study on Run 3 and insights into the Future Circular Hadron
Collider (FCC-hh) at CERN are detailed in Ref. [5].

Additionally, there are other anomalies recently observed, strongly indicating
New Physics. The anomalous magnetic moments of the muon and electron
($a_{\mu}$, $a_{e}$) at Fermilab, along with the W-boson mass, potentially
share a common origin with anomalies in $B$ meson decays. The Fermilab (g-2)
Collaboration [108], in conjunction with the old BNL result, confirms a
deviation from the data-driven theory prediction in the SM by about 4.2
$\sigma$. Significantly, each of these flavor anomalies exceeds 3 $\sigma$,
raising the likelihood of at least one anomaly persisting. This could lead to
a new understanding of physics. If the LFUV anomalies persist, there is a high
probability of uncovering remarkable evidence for New Physics in the realm of
flavor physics. This, in turn, would catalyze intense efforts among future
experimentalists and theorists [109].

###### Acknowledgements.

R.S. acknowledges the support of Polish NAWA Bekker program No.
BPN/BEK/2021/1/00342 and Polish NCN Grant No. 2018/30/E/ST2/00432. R.S. also
thank the Institute for Nuclear Theory at the University of Washington for its
kind hospitality and stimulating research environment. This research was
supported in part by the INT’s U.S. Department of Energy grant No. DE-
FG02-00ER41132.

## References

  * [1] Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * [2] HFLAV Collaboration, Y. Amhis et al., “Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016,” Eur. Phys. J. C 77 no. 12, (2017) 895, arXiv:1612.07233 [hep-ex].
  * [3] S. Bifani, S. Descotes-Genon, A. Romero Vidal, and M.-H. Schune, “Review of Lepton Universality tests in $B$ decays,” J. Phys. G 46 no. 2, (2019) 023001, arXiv:1809.06229 [hep-ex].
  * [4] P. Gambino et al., “Challenges in semileptonic $B$ decays,” Eur. Phys. J. C 80 no. 10, (2020) 966, arXiv:2006.07287 [hep-ph].
  * [5] F. U. Bernlochner, M. F. Sevilla, D. J. Robinson, and G. Wormser, “Semitauonic b-hadron decays: A lepton flavor universality laboratory,” Rev. Mod. Phys. 94 no. 1, (2022) 015003, arXiv:2101.08326 [hep-ex].
  * [6] BESIII Collaboration, M. Ablikim et al., “Precision measurements of ${\cal B}[\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi]$ and ${\cal B}[J/\psi\rightarrow l^{+}l^{-}]$,” Phys. Rev. D 88 no. 3, (2013) 032007, arXiv:1307.1189 [hep-ex].
  * [7] NA62 Collaboration, C. Lazzeroni et al., “Precision Measurement of the Ratio of the Charged Kaon Leptonic Decay Rates,” Phys. Lett. B 719 (2013) 326–336, arXiv:1212.4012 [hep-ex].
  * [8] PiENu Collaboration, A. Aguilar-Arevalo et al., “Improved Measurement of the $\pi\rightarrow\textrm{e}\nu$ Branching Ratio,” Phys. Rev. Lett. 115 no. 7, (2015) 071801, arXiv:1506.05845 [hep-ex].
  * [9] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., “Precision electroweak measurements on the $Z$ resonance,” Phys. Rept. 427 (2006) 257–454, arXiv:hep-ex/0509008.
  * [10] LHCb Collaboration, R. Aaij et al., “Test of lepton universality in beauty-quark decays,” Nature Phys. 18 no. 3, (2022) 277–282, arXiv:2103.11769 [hep-ex].
  * [11] Belle Collaboration, J. T. Wei et al., “Measurement of the Differential Branching Fraction and Forward-Backward Asymmetry for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$,” Phys. Rev. Lett. 103 (2009) 171801, arXiv:0904.0770 [hep-ex].
  * [12] BaBar Collaboration, J. P. Lees et al., “Measurement of Branching Fractions and Rate Asymmetries in the Rare Decays $B\rightarrow K^{(*)}l^{+}l^{-}$,” Phys. Rev. D 86 (2012) 032012, arXiv:1204.3933 [hep-ex].
  * [13] LHCb Collaboration, R. Aaij et al., “Differential branching fractions and isospin asymmetries of $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays,” JHEP 06 (2014) 133, arXiv:1403.8044 [hep-ex].
  * [14] LHCb Collaboration, R. Aaij et al., “Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 113 (2014) 151601, arXiv:1406.6482 [hep-ex].
  * [15] LHCb Collaboration, R. Aaij et al., “Differential branching fraction and angular analysis of $\Lambda^{0}_{b}\rightarrow\Lambda\mu^{+}\mu^{-}$ decays,” JHEP 06 (2015) 115, arXiv:1503.07138 [hep-ex]. [Erratum: JHEP 09, 145 (2018)].
  * [16] LHCb Collaboration, R. Aaij et al., “Angular analysis and differential branching fraction of the decay $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$,” JHEP 09 (2015) 179, arXiv:1506.08777 [hep-ex].
  * [17] LHCb Collaboration, R. Aaij et al., “Angular analysis of the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity,” JHEP 02 (2016) 104, arXiv:1512.04442 [hep-ex].
  * [18] Belle Collaboration, S. Wehle et al., “Lepton-Flavor-Dependent Angular Analysis of $B\rightarrow K^{\ast}\ell^{+}\ell^{-}$,” Phys. Rev. Lett. 118 no. 11, (2017) 111801, arXiv:1612.05014 [hep-ex].
  * [19] LHCb Collaboration, R. Aaij et al., “Measurements of the S-wave fraction in $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the $B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ differential branching fraction,” JHEP 11 (2016) 047, arXiv:1606.04731 [hep-ex]. [Erratum: JHEP 04, 142 (2017)].
  * [20] CMS Collaboration, A. M. Sirunyan et al., “Measurement of angular parameters from the decay $\mathrm{B}^{0}\rightarrow\mathrm{K}^{*0}\mu^{+}\mu^{-}$ in proton-proton collisions at $\sqrt{s}=$ 8 TeV,” Phys. Lett. B 781 (2018) 517–541, arXiv:1710.02846 [hep-ex].
  * [21] LHCb Collaboration, R. Aaij et al., “Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays,” JHEP 08 (2017) 055, arXiv:1705.05802 [hep-ex].
  * [22] ATLAS Collaboration, M. Aaboud et al., “Angular analysis of $B^{0}_{d}\rightarrow K^{*}\mu^{+}\mu^{-}$ decays in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” JHEP 10 (2018) 047, arXiv:1805.04000 [hep-ex].
  * [23] J. Bonilla, A. de Giorgi, B. Gavela, L. Merlo, and M. Ramos, “The cost of an ALP solution to the neutral $B$-anomalies,” arXiv:2209.11247 [hep-ph].
  * [24] H. Gedeonová and M. Hudec, “All possible first signals of gauge leptoquark in quark-lepton unification and beyond,” arXiv:2210.00347 [hep-ph].
  * [25] N. Castro and K. Skovpen, “Flavour-changing neutral scalar interactions of the top quark,” arXiv:2210.09641 [hep-ex].
  * [26] S. Descotes-Genon, I. Plakias, and O. Sumensari, “On the impact of meson mixing on $B_{s}\rightarrow\phi ee$ angular observables at low $q^{2}$,” arXiv:2210.11995 [hep-ph].
  * [27] A. J. Buras, “Standard Model Predictions for Rare K and B Decays without New Physics Infection,” arXiv:2209.03968 [hep-ph].
  * [28] A. Datta, A. Hammad, D. Marfatia, L. Mukherjee, and A. Rashed, “Dark photon and dark $Z$ mediated $B$ meson decays,” arXiv:2210.15662 [hep-ph].
  * [29] D. Barbosa, F. Díaz, L. Quintero, A. Flórez, M. Sanchez, A. Gurrola, E. Sheridan, and F. Romeo, “Probing a $\mathrm{Z}^{\prime}$ with non-universal fermion couplings through top quark fusion, decays to bottom quarks, and machine learning techniques,” arXiv:2210.15813 [hep-ph].
  * [30] BaBar Collaboration, J. P. Lees et al., “Evidence for an excess of $\bar{B}\rightarrow D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays,” Phys. Rev. Lett. 109 (2012) 101802, arXiv:1205.5442 [hep-ex].
  * [31] BaBar Collaboration, J. P. Lees et al., “Measurement of an Excess of $\bar{B}\rightarrow D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ Decays and Implications for Charged Higgs Bosons,” Phys. Rev. D 88 no. 7, (2013) 072012, arXiv:1303.0571 [hep-ex].
  * [32] Belle Collaboration, M. Huschle et al., “Measurement of the branching ratio of $\bar{B}\rightarrow D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\rightarrow D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle,” Phys. Rev. D 92 no. 7, (2015) 072014, arXiv:1507.03233 [hep-ex].
  * [33] Belle Collaboration, S. Hirose et al., “Measurement of the $\tau$ lepton polarization and $R(D^{*})$ in the decay $\bar{B}\rightarrow D^{*}\tau^{-}\bar{\nu}_{\tau}$,” Phys. Rev. Lett. 118 no. 21, (2017) 211801, arXiv:1612.00529 [hep-ex].
  * [34] BELLE Collaboration, S. Choudhury et al., “Test of lepton flavor universality and search for lepton flavor violation in $B\rightarrow K\ell\ell$ decays,” JHEP 03 (2021) 105, arXiv:1908.01848 [hep-ex].
  * [35] Belle Collaboration, A. Abdesselam et al., “Test of Lepton-Flavor Universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ Decays at Belle,” Phys. Rev. Lett. 126 no. 16, (2021) 161801, arXiv:1904.02440 [hep-ex].
  * [36] LHCb Collaboration, R. Aaij et al., “Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\rightarrow D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\rightarrow D^{*+}\mu^{-}\bar{\nu}_{\mu})$,” Phys. Rev. Lett. 115 no. 11, (2015) 111803, arXiv:1506.08614 [hep-ex]. [Erratum: Phys.Rev.Lett. 115, 159901 (2015)].
  * [37] LHCb Collaboration, R. Aaij et al., “Test of lepton universality with ${\Lambda}_{b}^{0}\to{pK}^{-}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-}$ decays,” JHEP 05 (2020) 040, arXiv:1912.08139 [hep-ex].
  * [38] LHCb Collaboration, R. Aaij et al., “Tests of lepton universality using $B^{0}\rightarrow K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\rightarrow K^{*+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 128 no. 19, (2022) 191802, arXiv:2110.09501 [hep-ex].
  * [39] LHCb Collaboration, R. Aaij et al. JHEP 11 (2021) 043, arXiv:2107.13428 [hep-ex].
  * [40] LHCb Collaboration, R. Aaij et al., “Branching Fraction Measurements of the Rare $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B^{0}_{s}\rightarrow f_{2}^{\prime}(1525)\mu^{+}\mu^{-}$\- Decays,” Phys. Rev. Lett. 127 no. 15, (2021) 151801, arXiv:2105.14007 [hep-ex].
  * [41] LHCb Collaboration, “First joint measurement of $R(D^{*})$ and $R(D^{0})$ at LHCb,”. https://indico.cern.ch/event/1187939/.
  * [42] HFLAV Collaboration, Y. S. Amhis et al., “Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2018,” Eur. Phys. J. C 81 no. 3, (2021) 226, arXiv:1909.12524 [hep-ex].
  * [43] HFLAV Collaboration, Y. Amhis et al., “Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021,” arXiv:2206.07501 [hep-ex].
  * [44] HFLAV Collaboration, “Average of $R(D)$ and $R(D^{*})$ for end of 2022,”. https://hflav-eos.web.cern.ch/hflav-eos/semi/fall22/html/RDsDsstar/RDRDs.html.
  * [45] LHCb Collaboration, R. Aaij et al., “Test of lepton flavour universality using $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays with hadronic $\tau$ channels,” arXiv:2305.01463 [hep-ex].
  * [46] S. Iguro and R. Watanabe, “Bayesian fit analysis to full distribution data of $\overline{\mathrm{B}}\to{\mathrm{D}}^{\left(\ast\right)}\mathrm{\ell}\overline{\nu}:\left|{\mathrm{V}}_{\mathrm{cb}}\right|$ determination and new physics constraints,” JHEP 08 no. 08, (2020) 006, arXiv:2004.10208 [hep-ph].
  * [47] M. Bordone, M. Jung, and D. van Dyk, “Theory determination of $\bar{B}\to D^{(*)}\ell^{-}\bar{\nu}$ form factors at $\mathcal{O}(1/m_{c}^{2})$,” Eur. Phys. J. C 80 no. 2, (2020) 74, arXiv:1908.09398 [hep-ph].
  * [48] LHCb Collaboration, R. Aaij et al., “Measurement of the ratio of branching fractions $\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\tau^{+}\nu_{\tau})$/$\mathcal{B}(B_{c}^{+}\,\to\,J/\psi\mu^{+}\nu_{\mu})$,” Phys. Rev. Lett. 120 no. 12, (2018) 121801, arXiv:1711.05623 [hep-ex].
  * [49] M. A. Ivanov, J. G. Körner, and P. Santorelli, “Semileptonic and nonleptonic decays of $B_{c}$,” in 18th Conference on High Energy Physics, pp. 251–255.  2007\.  arXiv:hep-ph/0609122.
  * [50] E. Hernandez, J. Nieves, and J. M. Verde-Velasco, “Study of exclusive semileptonic and non-leptonic decays of $B_{c}$ \- in a nonrelativistic quark model,” Phys. Rev. D 74 (2006) 074008, arXiv:hep-ph/0607150.
  * [51] R. Watanabe, “New Physics effect on $B_{c}\to J/\psi\tau\bar{\nu}$ in relation to the $R_{D^{(*)}}$ anomaly,” Phys. Lett. B 776 (2018) 5–9, arXiv:1709.08644 [hep-ph].
  * [52] M. A. Ivanov, J. G. Korner, and P. Santorelli, “Exclusive semileptonic and nonleptonic decays of the $B_{c}$ meson,” Phys. Rev. D 73 (2006) 054024, arXiv:hep-ph/0602050.
  * [53] Belle Collaboration, A. Abdesselam et al., “Measurement of the $D^{\ast-}$ polarization in the decay $B^{0}\to D^{\ast-}\tau^{+}\nu_{\tau}$,” in 10th International Workshop on the CKM Unitarity Triangle.  3, 2019.  arXiv:1903.03102 [hep-ex].
  * [54] M. Tanaka and R. Watanabe, “New physics in the weak interaction of $\bar{B}\to D^{(*)}\tau\bar{\nu}$,” Phys. Rev. D 87 no. 3, (2013) 034028, arXiv:1212.1878 [hep-ph].
  * [55] Z.-R. Huang, Y. Li, C.-D. Lu, M. A. Paracha, and C. Wang, “Footprints of New Physics in $b\to c\tau\nu$ Transitions,” Phys. Rev. D 98 no. 9, (2018) 095018, arXiv:1808.03565 [hep-ph].
  * [56] S. Bhattacharya, S. Nandi, and S. Kumar Patra, “$b\rightarrow c\tau\nu_{\tau}$ Decays: a catalogue to compare, constrain, and correlate new physics effects,” Eur. Phys. J. C 79 no. 3, (2019) 268, arXiv:1805.08222 [hep-ph].
  * [57] J. G. Korner and G. A. Schuler, “Lepton Mass Effects in Semileptonic $B$ Meson Decays,” Phys. Lett. B 231 (1989) 306–311.
  * [58] LATTICE-HPQCD Collaboration, J. Harrison, C. T. H. Davies, and A. Lytle, “$R(J/\psi)$ and $B_{c}^{-}\rightarrow J/\psi\ell^{-}\bar{\nu}_{\ell}$ Lepton Flavor Universality Violating Observables from Lattice QCD,” Phys. Rev. Lett. 125 no. 22, (2020) 222003, arXiv:2007.06956 [hep-lat].
  * [59] HPQCD Collaboration, J. Harrison, C. T. H. Davies, and A. Lytle, “$B_{c}\rightarrow J/\psi$ form factors for the full $q^{2}$ range from lattice QCD,” Phys. Rev. D 102 no. 9, (2020) 094518, arXiv:2007.06957 [hep-lat].
  * [60] G. Hiller and M. Schmaltz, “$R_{K}$ and future $b\rightarrow s\ell\ell$ physics beyond the standard model opportunities,” Phys. Rev. D 90 (2014) 054014, arXiv:1408.1627 [hep-ph].
  * [61] B. Gripaios, M. Nardecchia, and S. A. Renner, “Composite leptoquarks and anomalies in $B$-meson decays,” JHEP 05 (2015) 006, arXiv:1412.1791 [hep-ph].
  * [62] I. de Medeiros Varzielas and G. Hiller, “Clues for flavor from rare lepton and quark decays,” JHEP 06 (2015) 072, arXiv:1503.01084 [hep-ph].
  * [63] R. Barbieri, C. W. Murphy, and F. Senia, “B-decay Anomalies in a Composite Leptoquark Model,” Eur. Phys. J. C 77 no. 1, (2017) 8, arXiv:1611.04930 [hep-ph].
  * [64] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, “Quark flavor transitions in $L_{\mu}-L_{\tau}$ models,” Phys. Rev. D 89 (2014) 095033, arXiv:1403.1269 [hep-ph].
  * [65] A. Crivellin, G. D’Ambrosio, and J. Heeck, “Explaining $h\rightarrow\mu^{\pm}\tau^{\mp}$, $B\rightarrow K^{*}\mu^{+}\mu^{-}$ and $B\rightarrow K\mu^{+}\mu^{-}/B\rightarrow Ke^{+}e^{-}$ in a two-Higgs-doublet model with gauged $L_{\mu}-L_{\tau}$,” Phys. Rev. Lett. 114 (2015) 151801, arXiv:1501.00993 [hep-ph].
  * [66] A. Celis, J. Fuentes-Martin, M. Jung, and H. Serodio, “Family nonuniversal Z’ models with protected flavor-changing interactions,” Phys. Rev. D 92 no. 1, (2015) 015007, arXiv:1505.03079 [hep-ph].
  * [67] A. Falkowski, M. Nardecchia, and R. Ziegler, “Lepton Flavor Non-Universality in B-meson Decays from a U(2) Flavor Model,” JHEP 11 (2015) 173, arXiv:1509.01249 [hep-ph].
  * [68] LHCb Collaboration, A. A. Alves, Jr. et al., “The LHCb Detector at the LHC,” JINST 3 (2008) S08005.
  * [69] LHCb Collaboration, R. Aaij et al., “LHCb Detector Performance,” Int. J. Mod. Phys. A 30 no. 07, (2015) 1530022, arXiv:1412.6352 [hep-ex].
  * [70] Belle Collaboration, A. Abdesselam et al., “Measurement of the branching ratio of $\bar{B}^{0}\rightarrow D^{*+}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}^{0}\rightarrow D^{*+}\ell^{-}\bar{\nu}_{\ell}$ decays with a semileptonic tagging method,” in 51st Rencontres de Moriond on EW Interactions and Unified Theories.  3, 2016.  arXiv:1603.06711 [hep-ex].
  * [71] LHCb Collaboration, R. Aaij et al., “Measurement of the ratio of the $B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau}$ and $B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ branching fractions using three-prong $\tau$-lepton decays,” Phys. Rev. Lett. 120 no. 17, (2018) 171802, arXiv:1708.08856 [hep-ex].
  * [72] ATLAS Collaboration, G. Aad et al., “The ATLAS Experiment at the CERN Large Hadron Collider,” JINST 3 (2008) S08003.
  * [73] CMS Collaboration, S. Chatrchyan et al., “The CMS Experiment at the CERN LHC,” JINST 3 (2008) S08004.
  * [74] A. Issadykov, M. A. Ivanov, and G. Nurbakova, “Semileptonic decays of Bc mesons into charmonium states,” EPJ Web Conf. 158 (2017) 03002, arXiv:1907.13210 [hep-ph].
  * [75] W.-F. Wang, Y.-Y. Fan, and Z.-J. Xiao, “Semileptonic decays $B_{c}\rightarrow(\eta_{c},J/\Psi)l\nu$ in the perturbative QCD approach,” Chin. Phys. C 37 (2013) 093102, arXiv:1212.5903 [hep-ph].
  * [76] X.-Q. Hu, S.-P. Jin, and Z.-J. Xiao, “Semileptonic decays $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l}$ in the ”PQCD + Lattice” approach,” Chin. Phys. C 44 no. 2, (2020) 023104, arXiv:1904.07530 [hep-ph].
  * [77] CMS Collaboration, A. M. Sirunyan et al., “Observation of Two Excited B${}^{+}_{\mathrm{c}}$ States and Measurement of the B${}^{+}_{\mathrm{c}}$(2S) Mass in pp Collisions at $\sqrt{s}=$ 13 TeV,” Phys. Rev. Lett. 122 no. 13, (2019) 132001, arXiv:1902.00571 [hep-ex].
  * [78] S. Patnaik, L. Nayak, P. C. Dash, S. Kar, and N. Barik, “Semileptonic $B_{c}$ meson decays to S-wave charmonium states,” Eur. Phys. J. Plus 135 no. 11, (2020) 936, arXiv:1910.13206 [hep-ph].
  * [79] S. Patnaik and R. Singh, “A Light Shed on Lepton Flavor Universality in B Decays,” Universe 9 no. 3, (2023) 129, arXiv:2211.04348 [hep-ph].
  * [80] S. Patnaik, L. Nayak, and R. Singh, “Assessing Lepton Flavor Universality Violations in Semileptonic Decays,” arXiv:2308.05677 [hep-ph].
  * [81] N. Barik, P. C. Dash, and A. R. Panda, “Radiative decay of mesons in an independent quark potential model,” Phys. Rev. D 46 (1992) 3856–3861.
  * [82] N. Barik and P. C. Dash, “Radiative decay of light and heavy mesons,” Phys. Rev. D 49 (1994) 299–308. [Erratum: Phys.Rev.D 53, 4110 (1996)].
  * [83] M. Priyadarsini, P. C. Dash, S. Kar, S. P. Patra, and N. Barik, “Electromagnetic form factors of heavy flavored vector mesons,” Phys. Rev. D 94 no. 11, (2016) 113011.
  * [84] N. Barik and P. C. Dash, “Radiative weak decays of pseudoscalar mesons in the charm sector,” Mod. Phys. Lett. A 10 (1995) 103–114.
  * [85] N. Barik, S. Kar, and P. C. Dash, “Exclusive rare radiative decays of B mesons,” Phys. Rev. D 57 (1998) 405–412.
  * [86] N. Barik, S. Naimuddin, S. Kar, and P. C. Dash, “Rare radiative decay of the $B_{c}$ meson,” Phys. Rev. D 63 (2001) 014024.
  * [87] N. Barik, P. C. Dash, and A. R. Panda, “Leptonic decay of light vector mesons in an independent quark model,” Phys. Rev. D 47 (1993) 1001–1006. [Erratum: Phys.Rev.D 53, 4110 (1996)].
  * [88] N. Barik and P. C. Dash, “Weak leptonic decay of light and heavy pseudoscalar mesons in an independent quark model,” Phys. Rev. D 47 (1993) 2788–2795.
  * [89] N. Barik and P. C. Dash, “Exclusive semileptonic decay of D and B mesons in the independent quark model,” Phys. Rev. D 53 (1996) 1366–1377.
  * [90] N. Barik, S. K. Tripathy, S. Kar, and P. C. Dash, “Exclusive semileptonic decays of charmed and b flavored mesons,” Phys. Rev. D 56 (1997) 4238–4249.
  * [91] N. Barik, S. Naimuddin, P. C. Dash, and S. Kar, “Semileptonic decays of the Bc meson,” Phys. Rev. D 80 (2009) 074005.
  * [92] N. Barik, S. Naimuddin, P. C. Dash, and S. Kar, “Radiative leptonic decay: $B^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\gamma$ in a relativistic independent quark model,” Phys. Rev. D 77 (2008) 014038.
  * [93] N. Barik, S. Naimuddin, P. C. Dash, and S. Kar, “Radiative leptonic $B_{c}$ decay in the relativistic independent quark model,” Phys. Rev. D 78 (2008) 114030.
  * [94] N. Barik, S. Naimuddin, and P. C. Dash, “Radiative leptonic decay of D/s and D mesons,” Int. J. Mod. Phys. A 24 (2009) 2335–2355.
  * [95] N. Barik, S. Kar, and P. C. Dash, “Exclusive nonleptonic decays of B mesons,” Phys. Rev. D 63 (2001) 114002.
  * [96] N. Barik, S. Naimuddin, P. C. Dash, and S. Kar, “Exclusive nonleptonic $B\rightarrow VV$ decays,” Phys. Rev. D 80 (2009) 014004.
  * [97] S. Naimuddin, S. Kar, M. Priyadarsini, N. Barik, and P. C. Dash, “Nonleptonic two-body Bc-meson decays,” Phys. Rev. D 86 (2012) 094028.
  * [98] S. Kar, P. C. Dash, M. Priyadarsini, S. Naimuddin, and N. Barik, “Nonleptonic $B_{c}\rightarrow VV$ decays,” Phys. Rev. D 88 no. 9, (2013) 094014.
  * [99] S. Patnaik, P. C. Dash, S. Kar, S. Patra, and N. Barik, “Magnetic dipole transitions of $B_{c}$ and $B_{c}^{*}$ mesons in the relativistic independent quark model,” Phys. Rev. D 96 no. 11, (2017) 116010, arXiv:1710.08242 [hep-ph]. [Erratum: Phys.Rev.D 99, 019901 (2019)].
  * [100] S. Patnaik, P. C. Dash, S. Kar, and N. Barik, “Electromagnetic transitions of $(b{\bar{c}})$ bound system,” Phys. Rev. D 97 no. 5, (2018) 056025, arXiv:1801.10503 [hep-ph]. [Erratum: Phys.Rev.D 99, 019902 (2019)].
  * [101] L. Nayak, S. Patnaik, P. C. Dash, S. Kar, and N. Barik, “Lepton mass effects in exclusive semileptonic $B_{c}$-meson decays,” Phys. Rev. D 104 (2021) 036012, arXiv:2106.09463 [hep-ph].
  * [102] T. Zhou, T. Wang, Y. Jiang, L. Huo, and G.-L. Wang, “The weak $B,B_{s}$ and $B_{c}$ decays to radially excited states,” J. Phys. G 48 no. 5, (2021) 055006, arXiv:2006.05704 [hep-ph].
  * [103] D. Ebert, R. N. Faustov, and V. O. Galkin, “Weak decays of the $B_{c}$ meson to charmonium and $D$ mesons in the relativistic quark model,” Phys. Rev. D 68 (2003) 094020, arXiv:hep-ph/0306306.
  * [104] Z. Rui, H. Li, G.-x. Wang, and Y. Xiao, “Semileptonic decays of $B_{c}$ meson to S-wave charmonium states in the perturbative QCD approach,” Eur. Phys. J. C 76 no. 10, (2016) 564, arXiv:1602.08918 [hep-ph].
  * [105] J.-F. Sun, D.-S. Du, and Y.-L. Yang, “Study of $B_{c}\rightarrow J/\psi\pi$, $\eta_{c}\pi$ decays with perturbative QCD approach,” Eur. Phys. J. C 60 (2009) 107–117, arXiv:0808.3619 [hep-ph].
  * [106] Y.-M. Wang and C.-D. Lu, “Weak productions of new charmonium in semi-leptonic decays of $B_{c}$,” Phys. Rev. D 77 (2008) 054003, arXiv:0707.4439 [hep-ph].
  * [107] I. Bediaga and J. H. Munoz, “Production of radially excited charmonium mesons in two-body nonleptonic $B_{c}$ decays,” arXiv:1102.2190 [hep-ph].
  * [108] Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 no. 14, (2021) 141801, arXiv:2104.03281 [hep-ex].
  * [109] ILC International Development Team Collaboration, A. Aryshev et al., “The International Linear Collider: Report to Snowmass 2021,” arXiv:2203.07622 [physics.acc-ph].