[1,2]Rajeev Singh 1]School of Physical Sciences, National Institute of Science
Education and Research, An OCC of Homi Bhabha National Institute, Jatni,
752050, Odisha, India 2]Center for Nuclear Theory, Department of Physics and
Astronomy, Stony Brook University, Stony Brook, 11794-3800, New York, USA

# Assessing Lepton Flavor Universality Violations in Semileptonic Decays

Sonali Patnaik<EMAIL_ADDRESS>Lopamudra Nayak
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

In light of recent measurements suggesting potential lepton flavor
universality violations in semileptonic decays at collider experiments, this
article provides a concise study of tree- and loop-level $B$-hadron
semileptonic decays, $b\to cl\nu_{l}$ and $b\to sl^{+}l^{-}$. We provide
predictions for lepton flavor violating observables, $\mathcal{R}_{J/\psi}$
and $\mathcal{R}_{\eta_{c}}$, across the entire $q^{2}$ range. Our study
employs the Relativistic Independent Quark Model (RIQM), highlighting a model-
dependent approach to these observables. We compare our model’s predictions
with existing lattice predictions, demonstrating the strong applicability of
the RIQM framework in describing $B_{c}$ decays. Additionally, we reassess
global averages for $\mathcal{R}_{D(D^{*})}$ and $\mathcal{R}_{K(K^{*})}$ in
semileptonic transitions. With the upcoming experimental upgrades and the
anticipated Run 3 data on $B_{c}$ meson decays, rapid confirmation of these
quantities could indicate significant evidence of physics beyond the Standard
Model, thereby opening new pathways for understanding the complex flavor
dynamics in $B$ meson decays.

###### keywords:

BSM Physics, $B_{c}$ meson decays, LFUV, Anomalies, Semi-leptonic transitions

##  1 Introduction

The Standard Model (SM) of particle physics identifies the basic elements as
fermions, including quarks and leptons, as detailed in Ref. [1]. Leptons,
considered as point-like particles without substructure, partake in weak and
electromagnetic interactions, while neutrinos are involved solely in weak
interactions. The SM recognizes these charged leptons ($e^{-}$, $\mu^{-}$,
$\tau^{-}$) as universal across generations, except for variations due to
their differing masses, a concept validated in multiple decay experiments [2,
3, 4, 1, 5]. The Lepton Flavor Universality (LFU), an accidental symmetry in
the SM, results from varying Yukawa interactions between lepton-Higgs leading
to different lepton masses ($m_{\tau}>m_{\mu}>m_{e}$) [6, 7, 8, 9]. However,
recent studies suggest potential LFU breaking in $b\to sl^{+}l^{-}$
$(l=e,\mu)$, challenging our fundamental understanding of physics and implying
the possible existence of new particles that violate this symmetry [10]. These
could alter the rates of quark-lepton transitions in the $B$-hadron decays of
the SM [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29]. Anomalies hinting at LFU violation have also been observed in the
first-order decays of beauty ($B$) mesons to final $\tau$ lepton states [30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 10, 42, 43].

Additionally, leptonic and semileptonic decays of $B$ mesons appear to contest
lepton universality [32, 44, 45, 46, 47, 48, 49, 50, 38, 51, 39, 52].
Detecting such violations and elucidating the interactions of new particles
could pave the way for investigating physics beyond the SM (BSM), potentially
unveiling insights into dark matter, matter-antimatter asymmetry, and the
dynamics of the electroweak scale.

This paper reviews anomalies in semileptonic transitions, focusing on
differences in $B$ decay rates to $\tau$ and $\mu$ leptons, stemming from the
notable $\mu$-$\tau$ mass disparity. The first charge-current anomalies were
observed in $B\to D,D^{*}l{\bar{\nu}_{l}}$ decays by the BaBar Collaboration
[30], defining the LFU ratio as

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

where $D^{(*)}$ denotes $D(D^{*})$ meson, and $Br$ represents the branching
ratio. The results deviated from the SM by 3.4 $\sigma$ [30] and were later
affirmed by measurements at BaBar [31], Belle [33], and LHCb [37]. Recently,
LHCb provided an updated measurement of $\mathcal{R}_{D}$ and
$\mathcal{R}_{D^{*}}$, based on LHC Run 1 data, aligning with previous
findings [53, 54, 55, 56]. This $\tau$ reconstruction in
$\tau\to\mu\nu{\bar{\nu}}$ supersedes the 2015 result [37], with measured
observables exceeding SM predictions, implying Lepton Flavor Universality
Violation (LFUV). While recent averages edge closer to SM predictions [57,
58], the deviation significance remains above 3 $\sigma$ due to reduced
uncertainties. These theoretical and experimental measurements promote further
exploration of form factors in semileptonic transitions. The study of
$\mathcal{R}$ in various semileptonic decay modes prompted LHCb’s first
results for $\mathcal{R}(J/\psi)$ from $B_{c}\to J/\psi$ decay, defined as

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

with $\tau^{+}$ decaying leptonically to $\mu^{+}$ $\nu_{\mu}$
${\bar{\nu}_{\tau}}$. This analysis utilized a 3fb-1 pp collision data sample
at $\sqrt{s}=7$ and $8$ TeV [59]. The $\tau$ lepton was reconstructed, and a
global fit performed, revealing consistent deviation from SM predictions,
currently between 0.25-0.28 [60, 61, 62], roughly 2 $\sigma$ lower. These
deviations hint at LFU violation.

Further LFU tests are underway at LHCb for other $B_{q}$ species, including
$\mathcal{R}(D^{+})$ and $\mathcal{R}_{{\Lambda_{c}}^{*}}$ [61, 63]. LFU
measurements in $b\to sl^{+}l^{-}$ transitions present anomalies [46, 39, 64,
40, 35, 10, 41], and LFUV observables

${\mathcal{R}}_{K^{(*)}}=\frac{Br(B\to K^{(*)}\mu^{+}\mu^{-})}{Br(B\to
K^{(*)}e^{+}e^{-})}\,,\quad{\cal
R}_{\phi}=\frac{Br(B_{s}\to\phi\mu^{+}\mu^{-})}{Br(B_{s}\to\phi
e^{+}e^{-})}\,,$ (3)

display deviations from SM predictions, significant to 3.1$\sigma$, with a
combined uncertainty of 5% [65, 66]. LHCb’s recent tests on muon-electron
universality in $B^{+}\to Kl^{+}l^{-}$ and $B^{0}\to K^{0}l^{+}l^{-}$ decays,
using $B$ meson data from Run 1 and 2 (total luminosity: 9 fb-1), concur with
SM predictions [67]. Updated measurements for $\mathcal{R}_{K}$ and
$\mathcal{R}_{K^{*}}$ supersede previous ones [10] and align with SM values.
We reassess these transitions while constructing observables and analyzing
their SM tension. Table 1 contains the summary of experimental world averages
versus SM predictions.

Table 1: Summary of experimental measurements versus SM predictions [68]

LFU Parameters | LHCb measurements | SM prediction | Deviation.
---|---|---|---
${\mathcal{R}}_{D}^{LHCb2022}$ | 0.441 $\pm$ 0.060 $\pm$ 0.066 [69] | 0.298 $\pm$ 0.004 [54] | 2.16 $\sigma$
${\mathcal{R}}_{D^{*}}^{LHCb2023}$ | 0.257 $\pm$ 0.012 $\pm$ 0.014 [56] | 0.254 $\pm$ 0.005 [54] | 1 $\sigma$
${\mathcal{R}}_{J/\psi}$ | 0.71 $\pm$ 0.17 $\pm$ 0.181 [59] | 0.283 $\pm$ 0.048 [62] | 2 $\sigma$
${\mathcal{R}}_{K}^{*}$ | 1.027 $\pm$ 0.072 $\pm$ 0.027 [59] | 1.00 $\pm$ 0.0.01 [65] | 0.2 $\sigma$
${\mathcal{R}}_{K}$ | 0.949$\pm$ 0.042$\pm$ 0.022 [59] | 1.00 $\pm$ 0.01 [65] | 0.2 $\sigma$

Quantum Chromodynamics (QCD) addresses quark confinement inside hadrons and
characterizes decay transition rates through Lorentz invariant form factors.
These factors represent information about hadrons as bound systems. Yet, QCD
complexities from non-abelian and non-perturbative properties hinder the
extraction of these factors. Instead, phenomenological models are employed to
describe the bound state nature of hadrons and decay properties. Various
theoretical methods, such as heavy quark effective theory (HQET), quark
models, and Lattice QCD (LQCD), aim to explain $B$-decay anomalies [70, 71,
72, 73, 74, 75]. These methods have strengths and weaknesses. Regardless of
their Lorentz interaction potential structure, a quark potential model’s
success is determined by its ability to reproduce observed data across various
hadron sectors. The model’s reliability is based on its accurate depiction of
constituent-level dynamics inside the hadron core and predictions of various
hadronic properties, including decays. The downside is that potential models
are not unique and may only reproduce experimental data within a limited
range. Hence, expanding a quark model’s applicability to a broader range of
observed data is crucial.

In this work, we present our findings on $B_{c}$-decay anomalies within the
Relativistic Independent Quark Model (RIQM) framework, aiming to test the
model’s ability to explain LFU ratios. Given the availability of SM LQCD
vector and axial form factors for the $B_{c}\to J/\psi$ decay, which have
motivated us to further undertake this study and therefore we present a
systematic comparison of our results with LQCD inputs. This paper is organised
as follows: Section 2 presents an experimental outlook on LFU. Sections 3 and
4 discuss model-dependent studies of $b\to c$ and a brief review on $b\to s$
decays, respectively. The final section 5, offers our conclusions and future
directions.

##  2 Experimental outlook on Lepton Flavor Universality

The discovery of the $b$ quark [76] paved the way for the production of
$B$-hadrons at colliders such as CESR, LEP, and Tevatron. Comprehensive
studies on third-generation LFUV in $B$ mesons only became viable with the
advent of the $B$-factories and the LHC [77]. The examination of $B$ meson
decay is split into two: decays that involve FCNC (flavor-changing neutral
current) ($b$-quark to $s$-quark transition with lepton pair emission) and
FCCC (flavor-changing charge current) decays (transition from a $b$-quark to a
$c$-quark, with leptons and neutrinos emission). New physics (NP) mediators
like leptoquarks [78, 79, 80, 81] and $Z^{\prime}$ [82, 83, 84, 85]
significantly modify their amplitudes, facilitating their study. Since 2010,
LHCb has collected a large data set of $b\bar{b}$ pairs, compensating for the
challenging pp collision environment [77]. The detector design optimizes heavy
meson decay studies, with excellent identification and momentum resolution
capabilities [86]. The LHCb platform also allows for LFU tests on all
$B$-hadron species produced at the LHC, including those with low production
rates like $B_{c}$ mesons [87]. The Belle detector is also instrumental in
heavy meson decay studies and is equipped with various high-precision
components. Alongside the upgraded Belle II [32, 9], they have already made
significant strides in the examination of $B\to D(D^{*})l\nu_{l}$ decays [30,
31, 33, 88, 34, 37, 89]. Belle II and the upgraded LHCb detector are set to
continue data collection for the next 15 years, anticipated to multiply
current data samples. Measurements from Belle II and LHCb are expected to
illuminate current flavor anomalies and align with SM predictions [90, 91,
92]. Experiments at ATLAS [93] and CMS [94] also contribute significantly to
our understanding of physics beyond the SM. Both ATLAS and CMS, with the CMS
having recently made a significant detection of an excited $B_{c}$ state [95],
are limited compared to LHCb in flavor physics capabilities due to certain
requirements and limitations but provide complementary contributions to muon
decay processes.

The rare decay of $B_{s}\to\mu^{+}\mu^{-}$ has been studied extensively by CMS
using data from LHC Run 2. This study revealed a clear signal of $B_{s}$ meson
decaying to a muon-antimuon pair, the decay rate of which closely matches
theoretical predictions. This result, alongside future studies of extremely
rare transitions like $B^{0}\to\mu^{-}\mu^{+}$ decay expected to be
facilitated by data from LHC Run 3, could prove instrumental in understanding
puzzling anomalies and the very physics beyond the standard model.

##  3 Tests of LFU ratios in $B_{c}\to Xl\nu_{l}$ where $X$ is $\eta_{c}$,
$J/\psi$, $D$ and $D^{*}$ and $l=e,\mu,\tau$

In this section, we explore the ratios of $b\to\tau,\mu$ leptons due to their
mass difference influencing decay amplitudes. Our focus is on the SM tree-
level depiction of $B_{c}\to Xl\nu_{l}$, where $X=\eta_{c},J/\psi$ ,
$D(D^{*})$, as shown in Fig. 1. We examine the $B_{c}$ meson for its unique
traits as the only bound state of two heavy open flavored quarks (charm and
bottom). Its position between charmonium $(c\bar{c})$ and bottomonium
$(b\bar{b})$ makes it intriguing but not well-studied due to limited data. It
decays exclusively via weak interaction as it resides below the $B\bar{D}$
threshold, yielding a long life span and multiple weak decay channels with
notable branching ratios.

Figure 1: SM contribution for $B_{c}\to Xl\nu_{l}$ where $X=\eta_{c},J/\psi$ ,
$D(D^{*})$ [68].

Recent evidence of an excited $B_{c}$ state was found by the CMS Collaboration
[95] through the study of $B_{c}^{+}\to\pi^{+}\pi^{-}$ with a total integrated
luminosity of 143 fb-1 at $\sqrt{s}=$ 13 TeV, determining a $B_{c}$(2S) meson
mass of $6871\pm 1.2\pm 0.8$ MeV. Detection of ground and excited states of
$B_{c}^{*}$ remains elusive. However, the energy and higher luminosity at LHC
and $Z_{0}$ factory should bolster the accumulation rate of these undetected
states, promoting a deeper understanding of $B_{c}$ and $B_{c}^{*}$
counterparts. This emerging data and the anticipated surge of $B_{c}$ events
in ongoing and forthcoming experiments underpin the importance of
investigating decay properties in this area. Thus, the $B_{c}$-meson offers a
unique view of heavy quark dynamics and a distinct test for QCD. In light of
these points, we offer a model-dependent discussion to reconcile the
disparities between SM and BSM physics.

###  3.1 RIQM framework

Studying exclusive semileptonic decays involving non-perturbative hadronic
matrix elements poses challenges due to current limitations in rigorously
applying QCD principles for reliable invariant transition amplitude
measurements. As a result, various theoretical approaches utilize
phenomenological models to investigate non-perturbative QCD dynamics [96, 61,
60, 97]. HQET’s usefulness lies in relating form factors via $1/m_{Q}$ and
$\alpha_{s}$, and several quark models like QCD Sum rule, light cone sum rule,
and lattice QCD are employed to approximate these form factors. Form factor
parametrization insights shape differential decay amplitudes and provide
measurements on NP scales. We thus present an overview of global averages in a
model-dependent approach. We used the RIQM model, based on confining harmonic
potential in an equally mixed scalar-vector form [98, 99]

$U(r)=\frac{1}{2}\left(1+\gamma^{0}\right)\,V(r)\,.$ (4)

In the potential $V(r)=(ar^{2}+V_{0})$, $r$ is the quark-antiquark relative
distance, $\gamma^{0}$ is the time-like Hermitian matrix, and $a$ and $V_{0}$
are potential parameters. The values of the potential parameters is listed in
3.3 which are set from prior model application. The internal dynamics of
quarks are represented by a quark Lagrangian density of suitable Lorentz
structure

${\cal
L}^{0}_{q}(x)={\bar{\psi}}_{q}(x)\;\left[\frac{i}{2}\,\gamma^{\mu}\partial_{\mu}-m_{q}-U(r)\;\right]\;\psi_{q}(x)\,.$
(5)

This leads to the individual quark’s Dirac equation

$\left[\gamma^{0}E_{q}-{\vec{\gamma}}.{\vec{p}}-m_{q}-U(r)\right]\psi_{q}(\vec{r})=0\,,$
(6)

where $\psi_{q}(\vec{r})$ is the Dirac normalized wave function, expressible
in a two-component form

$\displaystyle\psi_{q}(\vec{r})=\left(\begin{array}[]{c}\psi_{A}(\vec{r})\\\
\psi_{B}(\vec{r})\end{array}\;\right)=\left(\begin{array}[]{c}\psi_{A}(\vec{r})\\\
\left(\frac{{\vec{\sigma}}\cdot{\vec{p}}}{E_{q}+m_{q}}\right)\;\psi_{A}(\vec{r})\end{array}\right)\,.$
(11)

In this form, $\psi_{A}(\vec{r})$ and $\psi_{B}(\vec{r})$ represent the quark
wave function’s upper and lower components of opposite parity, respectively.

The RIQM, like other potential models, is a QCD-inspired model that determines
the confinement of quarks in a hadron through an interaction potential with
appropriate Lorentz structure. Theoretical derivations of observable
properties of composite hadrons from their constituent dynamics should be
feasible but have proven challenging due to inherent QCD complexities. As
such, numerous effective quark potential models have been used as
phenomenological means to describe these dynamics. The model assumes a non-
perturbative multi-gluon interaction represented by the potential form, with
residual interactions arising from quark-pion coupling and short-distance one-
gluon exchange treated perturbatively. The confining potential $U(r)$, as
outlined in Eq. (4), provides a simple yet effective framework for analyzing
various hadronic properties, including FCCC-involved decays. We aim to broaden
the applicability of the RIQM, showcasing its suitability as an alternative
phenomenological scheme for examining various hadronic phenomena, and
comparing results with other theories and available experimental data. The
model’s efficacy has been verified across a range of hadronic phenomena,
including various types of hadron decays [100, 101, 102, 103, 104, 105, 106,
107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 98]. 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 general form as [120]

${\cal M}(p,k,k_{l},k_{\nu})={\frac{\cal
G_{F}}{\sqrt{2}}}V_{bq^{{}^{\prime}}}\,{\cal H}_{\mu}(p,k)\,{\cal
L}^{\mu}(k_{l},k_{\nu})\,.$ (12)

The effective Fermi coupling constant is represented by ${\cal G}_{F}$, with
$V_{bq^{{}^{\prime}}}$ as the corresponding CKM parameter. ${\cal L}^{\mu}$
and ${\cal H}_{\mu}$ denote the leptonic and hadronic currents, respectively.
The four-momentum of parent ($B_{c}$) and daughter (X) mesons, lepton, and
neutrino are symbolized by $p,k,k_{l},k_{\nu}$ respectively. It’s important to
note that decay processes occur when the involved mesons are in their momentum
eigenstates. Thus, any field-theoretic decay process representation
necessitates the depiction of meson bound states via suitable momentum wave
packets, which illustrate the momentum and spin distribution between the
constituent quark and antiquark within the meson core.

In the RIQM method, a wave packet representing a meson bound state, such as
$|B_{c}(\vec{p},S_{B_{c}})\rangle$, possessing a definite momentum $\vec{p}$
and spin $S_{B_{c}}$, is formulated as follows [118, 119, 98, 102, 100, 107,
106, 101, 103, 104, 108, 109, 105, 114, 112, 111, 110]

$\big{|}B_{c}\left(\vec{p},S_{B_{c}}\right)\big{\rangle}=\hat{\Lambda}\left(\vec{p},S_{B_{c}}\right)\big{|}\left(\vec{p_{b}},\lambda_{b}\right);\left(\vec{p_{c}},\lambda_{c}\right)\big{\rangle}\,.$
(13)

Here, $|(\vec{p_{b}},\lambda_{b});(\vec{p_{c}},\lambda_{c})\rangle$ denotes
the Fock space representation of unbound quark and antiquark in a color-
singlet setup, each with its momentum and spin. Meanwhile,
${\hat{\Lambda}}(\vec{p},S_{B_{c}})$ signifies an integral operator reflecting
the meson’s bound state properties.

###  3.2 Helicity amplitudes and observables

Using the model’s dynamics, the weak form factors are integrated and the
angular decay distribution in $q^{2}$ (where $q=p-k=k_{l}+k_{\nu}$) is
expressed as [120]

$\frac{d\Gamma}{dq^{2}d\cos\theta}={\frac{{\cal
G}_{F}}{(2\pi)^{3}}}|V_{bq}|^{2}\frac{(q^{2}-m_{l}^{2})^{2}}{8M^{2}q^{2}}|\vec{k}|{\cal
L}^{\mu\sigma}{\cal H}_{\mu\sigma}\,.$ (14)

Here, ${\cal L}^{\mu\sigma}$ and ${\cal H}_{\mu\sigma}$ represent lepton and
hadron correlation functions, respectively. The mass of the charged lepton is
denoted as $m_{l}$, and $M$ is the mass of the parent ($B_{c}$) meson. The
completeness property allows rewriting the lepton and hadron tensors in Eq.
(14) as

$\displaystyle{\cal L}^{\mu\sigma}{\cal H}_{\mu\sigma}$ $\displaystyle=$
$\displaystyle{\cal
L}_{\mu^{\prime}\sigma^{\prime}}g^{\mu^{\prime}\mu}g^{\sigma^{\prime}\sigma}{\cal
H}_{\mu\sigma}\,,$ (15) $\displaystyle=$ $\displaystyle{{\cal
L}_{\mu^{\prime}\sigma^{\prime}}}{\epsilon^{\mu^{{}^{\prime}}}}(m){\epsilon^{\mu^{\dagger}}}(m^{{}^{\prime}}){g_{mm^{{}^{\prime}}}}{\epsilon^{\sigma^{\dagger}}}(n){\epsilon^{\sigma^{{}^{\prime}}}}(n^{{}^{\prime}})g_{nn^{{}^{\prime}}}{\cal
H}_{\mu\sigma}\,,$ $\displaystyle=$
$\displaystyle{L(m,n)}{g_{mm^{\prime}}}{g_{nn^{\prime}}}H({m^{\prime}}{n^{\prime}})\,.$

The lepton and hadron tensors, defined in terms of helicity components, are as
follows

$\displaystyle L(m,n)={\epsilon^{\mu}}(m){\epsilon^{\sigma^{\dagger}}}(n){\cal
L}_{\mu\nu}\,,\qquad
H(m,n)={\epsilon^{\mu^{\dagger}}}(m){\epsilon^{\sigma}(n)}{\cal
H}_{\mu\nu}\,.$ (16)

Expressing physical observables on a helicity basis simplifies the
calculation. Hence, the helicity form factors are written as Lorentz invariant
form factors that depict decay amplitudes. The Lorentz contraction in Eq. (14)
can thus be performed with the helicity amplitudes, as illustrated in [120].
In this study, we don’t account for the azimuthal $\chi$ distribution of the
lepton pair. Therefore, by integrating over the lepton tensor’s azimuthal
angle dependence, we obtain the differential partial helicity rates
$(d\Gamma_{i}/dq^{2})$

$\frac{d\Gamma_{i}}{dq^{2}}=\frac{{\cal
G}_{f}^{2}}{(2\pi)^{3}}{|V_{bq^{{}^{\prime}}}|}^{2}\frac{({q^{2}}-{m_{l}^{2}})^{2}}{12{M^{2}}q^{2}}|\vec{k}|H_{i}\,.$
(17)

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

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

Table 2: Helicity rates (in $10^{-15}$ GeV) of semileptonic $B_{c}$-meson decays into charmonium and charm-meson state [68] 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

In Table 2, we present the computed integrated partial helicity rates
$\Gamma_{i}(i=U,L,P)$, $\tilde{\Gamma}_{i}(i=U,L,S,SL)$, and the total decay
rates for the respective channels under study. We find the partial tilde rates
for the $e^{-}$-mode to be insignificant and thus dismissable. However, the
rates in the $\tau^{-}$-mode are comparable to the tilde rates and cannot be
overlooked. Taking individual helicity rates into account, we provide decay
rates predictions for each process in both $e^{-}$ and $\tau^{-}$ modes. Like
all model predictions, our $\tau^{-}$ mode decay rates are typically smaller
than those in the $e^{-}$ modes. For instance, our $B_{c}\to J/\psi$
transition decay rate prediction in $\tau^{-}$-mode is around 5 times smaller
than its $e^{-}$ mode counterpart. For the $B_{c}\to\eta_{c}$ transition, the
suppression factor is about 2. The $\tau^{-}$-modes for the CKM suppressed
$B_{c}\to D(D^{*})$ transitions are only slightly lower than their
$e^{-}$-mode counterparts.

###  3.3 Numerical analysis and results

Our aim is to examine LFU ratios, using the RIQM framework. We commence by
assembling the input parameters used in this study. The relevant quark masses,
the RIQ model parameters $(a,V_{0})$, and quark binding energy
${}^{\prime}E_{q}^{\prime}$ as in [107, 109, 119, 118] are determined from the
model dynamics. The meson masses as, the $V_{\rm CKM}$, and the lifetime of
the $B_{c}$ meson along with their uncertainities are taken from the PDG [1]

$\displaystyle(a,V_{0})$ $\displaystyle=$ $\displaystyle(0.017166\ {\rm
GeV}^{3},-0.1375\ {\rm GeV}){\rm GeV}$ $\displaystyle m_{b}$
$\displaystyle\approx$ $\displaystyle 4.78\,{\rm GeV},\quad m_{c}\approx
1.5\,{\rm GeV},\quad m_{u}\approx 0.079\,{\rm GeV},\quad m_{e}\approx
0.51\,{\rm MeV},$ $\displaystyle m_{\tau}$ $\displaystyle\approx$
$\displaystyle 1.78\,{\rm GeV},\quad E_{b}\approx 4.77\,{\rm GeV}\quad
E_{c}\approx 1.58\,{\rm GeV}\quad\ E_{u}\approx 0.47\,{\rm GeV},$
$\displaystyle m_{J/\psi}$ $\displaystyle\approx$ $\displaystyle 3096.9\pm
0.006\,{\rm MeV},\quad\ m_{D}\approx 1864.84\pm 0.05\,{\rm MeV},\quad
m_{B_{c}}\approx 6274.47\pm 0.32\,{\rm MeV},$ $\displaystyle m_{\eta_{c}}$
$\displaystyle\approx$ $\displaystyle 2983.9\pm 0.04\,{\rm MeV},\quad
m_{D^{*}}\approx 2006.85\pm 0.05\,{\rm MeV},$ $\displaystyle V_{cb}$
$\displaystyle\approx$ $\displaystyle 0.0408\pm 0.0014,\quad\quad
V_{bu}\approx 0.00382\pm 0.00020,\quad\quad\tau_{B_{c}}\approx 0.51\pm
0.009\,{\rm ps}\,.$ (18)

We compute the observable ${\cal R}$ using the model outlined in [118, 119].
The mechanisms behind the decay process are well comprehended within the
context of a suitable phenomenological model (RIQ), achieved by comparing its
predictions for observable phenomena with those from conventional theoretical
methods. A quark potential model succeeds when it reasonably reproduces
available observed data in different hadron sectors. Regardless of the Lorentz
structure of the interacting 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 scenarios, we have also adopted a
potential model framework in the present analysis, which has been extensively
used to describe a wide range of hadronic phenomena,

Our calculated ${\cal R}$ for $B_{c}\to X(nS)l\nu_{l}$, where
$X=\eta_{c},J/\psi$, $D,D^{*}$, in both ground and radially excited states,
aligns well with other SM predictions as shown in Tables 3 and 4. The mismatch
between SM predictions of ${\mathcal{R}}$ and experimental data underscores
anomalies in semileptonic decays and limitations of our RIQM in incorporating
NP constraints.

To quote here, in our current analysis, we have only focused on uncertainities
stemming from meson masses, $V_{CKM}$ and lifetime of $B_{c}$. We have not
incorporated uncertainities from the model framework in initial and final
meson states, quark masses, corressponding binding energies used in the
calculation. It may be pointed out that, we do not have any adjustable free
parameters in this model. The potential parameters ($a$, $V_{0}$), quark
masses ($m_{q}$), corressponding binding energies ($E_{q}$) etc., have already
been fixed from hadron spectroscopy in the basic level application of this
model in reproducing the hyperfine splitting of baryons and mesons. We use the
same set of fixed parameters to describe wide ranging hadronic phenomenas as
pointed out earlier. As such we have no free parameters at our disposal to
show any kind of uncertainties from our model framework. Therefore the
dominant theoretical errors stem solely from the uncertainties of the input
parameters $|V_{\rm cb}|=(42.2\pm 0.8)\times 10^{-3}$, $|V_{\rm ub}|=(3.94\pm
0.36)\times 10^{-3}$, $m_{\tau}=1776.86\pm 0.12$ MeV. The uncertainties in our
predictions are concurrently within a range of $\pm 5\%$ of their central
values. As a result, we have only reported the central values of our
predictions in this work.

Given the lack of predictions from recognized models in existing literature,
our predictions for LFUV observables for the higher charm and charmonium
states, ${\cal R}_{D}(2S)$, ${\cal R}_{D}(3S)$, ${\cal R}_{D^{*}}(2S)$, and
${\cal R}_{D^{*}}(3S)$, may be instrumental in identifying the $B_{c}$
channels in the forthcoming Run 3 data at LHCb.

Table 3: Ratios of branching fractions for Semileptonic $B_{c}-$ decays in the
ground state [68]

Ratio of Branching fractions (${\cal R}$) ($l=e,\mu$) | RIQM | (CQM)[121] | RCQM[63] | PQCD[122] | LQCD [123, 124]
---|---|---|---|---|---
${\cal R}_{\eta_{c}}=\frac{{\cal B}(B_{c}\to\eta_{c}\tau\nu_{\tau})}{{\cal B}(B_{c}\to l\nu_{l})}$ | 0.4325 | 0.25 | 0.27 | 0.34 | -
${\cal R}_{J/\psi}=\frac{{\cal B}(B_{c}\to J/\psi\tau\nu_{\tau})}{{\cal B}(B_{c}\to J/\psi l\nu_{l})}$ | 0.21 | 0.23 | 0.236 | 0.29 | 0.27
${\cal R}_{D}=\frac{{\cal B}(B_{c}\to D\tau\nu_{\tau})}{{\cal B}(B_{c}\to Dl\nu_{l})}$ | 0.784 | 0.63 | 0.59 | - | 0.682
${\cal R}_{D^{*}}=\frac{{\cal B}(B_{c}\to D^{*}\tau\nu_{\tau})}{{\cal B}(B_{c}\to D^{*}l\nu_{l})}$ | 0.9165 | 0.56 | 0.58 | - | -

Table 4: LFU observables for $B_{c}$-decays to radially excited charmonium
states [68]

Ratio | RIQM | [BSA][125] | [RQM][126, 127] | [PQCD][128] | [LCQCD][129] | [NRISGW2][130]
---|---|---|---|---|---|---
${\cal R}_{\eta_{c}}(2S)$ | 0.136 | 0.051 | - | 0.068 | 0.74 | 0.028
${\cal R}_{\eta_{c}}(3S)$ | 0.021 | 0.0103 | 0.0009 | 0.0013 | 0.030 |
${\cal R}_{\psi}(2S)$ | 0.085 | 0.049 | - | 0.069 | - | 0.071
${\cal R}_{\psi}(3S)$ | 0.077 | 0.009 | 0.0063 | 0.001 |  |

Our RIQM approach evaluates parametrization of relevant form factors for
semileptonic decay amplitudes across the complete kinematic range ($0\leq
q^{2}\leq q^{2}_{max}$), enhancing reliability and accuracy over other models.
These models often calculate form factors with endpoint normalization at
either $q^{2}=0$ or $q^{2}$ = $q^{2}_{max}$, then extrapolate them
phenomenologically using monopoles, dipoles, and Gaussian ansatz to the entire
physical region, potentially compromising reliability. To avoid such
uncertainties, we refrain from using such phenomenological ansatz. Given the
projected high statistics of $B_{c}$-events, expected to deliver up to
$10^{10}$ events annually at colliders, semileptonic $B_{c}$ decays to charm
and charmonium states offer an intriguing avenue to further explore emerging
scales in physics.

##  4 Tests of LFU violation in $b\to sl^{+}l^{-}$ transitions

Figure 2: (a) SM contribution for $b\to sl^{+}l^{-}$ transition including W, Z
and $\gamma$ via loop level. (b) NP contribution involving a leptoquark (LQ
coupling directly to quarks and leptons) [68].

In the SM, quark flavor transitions are mediated solely by the charged weak
bosons $W^{\pm}$. Hence, FCNC transitions involving same charge quarks aren’t
directly enabled by the neutral weak boson $Z^{0}$, but instead occur through
rare loop processes involving virtual $W^{\pm}$ and extra virtual quarks.
These transitions, depicted in penguin- and box-like Feynman diagrams, are
accurately predicted by the SM. New particles can either partake in the loops
or contribute additional tree-level diagrams, see Fig 2. Decays regulated by
$b\to sl^{+}l^{-}$ transitions, which are suppressed, are excellent
environments to probe for new physics due to potentially sizeable effects
compared to competing SM processes.

Recently, significant deviations from SM predictions have been reported in LFU
measurements related to these transitions. The authors of Ref. [10]
demonstrated evidence of LFU violation with a 3.1$\sigma$ significance and
around 5% combined statistical and systematic uncertainty. Considerable
interest has been drawn to decays at the quark level, like $b\to sl^{+}l^{-}$,
due to significant lepton universality violations. For the first time, these
violations were reported by LHCb [46, 64, 10], with the result

${\cal R}_{K}=0.745({\rm stat})\pm 0.036({\rm syst})\,,$ (19)

deviating from the SM prediction ${\cal R}_{K}=1.0004$ [65] with a
significance of 2.6 $\sigma$. Recent findings by LHCb have confirmed that
${\cal R}_{K}$ and ${\cal R}_{K^{*}}$ measurements align with SM predictions,
showing no anomalies. The reported values are [67]

$\displaystyle{\cal R}_{K}$ $\displaystyle=0.994({\rm stat})\pm 0.029({\rm
syst})\,,$ $\displaystyle{\cal R}_{K^{*}}$ $\displaystyle=0.927({\rm stat})\pm
0.036({\rm syst})\,,$ (20) $\displaystyle{\cal R}_{K}$
$\displaystyle=0.949({\rm stat})\pm 0.022({\rm syst})\,,$ $\displaystyle{\cal
R}_{K^{*}}$ $\displaystyle=1.027({\rm stat})\pm 0.027({\rm syst})\,.$ (21)

The measured ${\cal R}_{K}$ and ${\cal R}_{K^{*}}$ values for $q^{2}$
intervals: $0.1<q^{2}<1.1~{}{\rm GeV}^{2}/c^{4}$ (low $q^{2}$ region) and
$1.1<q^{2}<6.0~{}{\rm GeV}^{2}/c^{4}$ (central $q^{2}$) supersede earlier
measurements [10], and are consistent with the SM. Data from all pp collisions
recorded by the LHCb detector from 2011-2018 were used, equivalent to
integrated luminosities of 1.0, 2.0, and 6.0 $fb^{-1}$ at 7, 8, and 13 TeV
center-of-mass energies. Systematic uncertainties, significantly less than the
statistical uncertainties, are projected to further reduce with additional
data.

In the SM, decays with negligible electron-muon mass difference can be
precisely predicted as they are exempted from hadronic uncertainties and only
occur in loop order. Their suppression can be explained by SM’s fundamental
symmetries. Theoretical models [66] extensively examine universality effects,
including QED-radiative corrections in the Monte Carlo framework, effectively
negating non-perturbative QCD effects. Their work, using the meson effective
theory’s analytic results, identified ${\cal R}_{K}$ as a reliable SM
prediction. Such corrections offer insight into the most sensitive moments and
encourage experimental investigation.

Since lepton universality violation observables can be influenced by the charm
loop due to NP and SM contribution interference, an unbiased NP picture was
investigated in [131] using charming penguins to solve charm loop amplitudes.
Further details, including a detailed numerical comparison for neutral mode
$\bar{B^{0}}\to\bar{K^{0}}l^{+}l^{-}$, which is relevant for LFU ratios, are
in Ref. [132]. However, Ref. [133] used an updated global fit of muon Wilson
Coefficients to interpret anomalies. The significant 5.6 $\sigma$ discrepancy
motivates the proposition of various NP models. The LHCb collaboration
reported new measurements for ${\cal R}_{{K^{0}_{s}}}$ and ${\cal
R}_{K^{*+}}$, and updated measurements for certain
$B_{s}\to\phi\mu^{+}\mu^{-}$ observables that deviate from the SM by a 3.6
$\sigma$ level [41, 43, 42]. An effective field theory approach offered a
solid data fit [134]. Further analysis of $b\to sl^{+}l^{-}$ decay transitions
may provide insight into the observed matter-antimatter disparity in the
universe. A study focusing on CP-violating angular observable with a complex
phase offers a unique insight into potential NP structures in this transition
[135].

Glashow, Guadagnoli, and Lane (GGL) proposed an explanation for the ${\cal
R}_{K}$ anomaly, demonstrating that a NP model can explain both ${\cal R}_{K}$
and ${\cal R}_{D^{(*)}}$ anomalies using an effective field theory approach
[136]. Under the assumption of NP predominantly coupling to the third
generation and having a scale far beyond the weak scale, they identified two
types of fully gauge-invariant NP operators, incorporating both neutral-
current and charged-current interactions. Similar unified anomaly theories in
the framework of effective theory are discussed in [137, 138, 139].

##  5 Conclusion & future outlook

Although the SM has been a potent mathematical framework, it’s not without its
limitations. This paper examines the recent landscape of LFUV anomalies in $B$
physics from both theoretical and experimental perspectives, which could
unveil BSM physics. Notably, LFU ratios related to neutral-current (${\cal
R}_{K},{\cal R}_{K^{*}}$) align with SM predictions and are considered
theoretically clean observables. Recently updated results for ${\cal R}_{D}$
and ${\cal R}_{D^{*}}$ show around 3 $\sigma$ deviation, which has been
explored in a model-dependent (RIQM) framework. NP models including leptoquark
and others involving $Z^{\prime}$-boson [140], composite Higgs boson [141],
dark matter [142], right-handed neutrinos [143] have been proposed to explain
these discrepancies. If validated, these findings would provide clear evidence
of NP. The 2022-2025 LHC Run 3 is projected to triple data collections over
three years, leading to improved event statistics, reduced uncertainties, and
unprecedented precision for flavor measurements. Several upgrades to the LHCb
detector will lower background from charged and neutral tracks and increase
accessibility to electronic and tauonic modes. The Belle II analysis will also
be instrumental in independently clarifying flavor anomalies.

From a theoretical perspective, we should thoroughly report results from
current models and encourage development of more NP models to explain these
anomalies. A detailed examination of the upcoming Run 3 and its future
implications has been reported in Ref. [9], which also discusses the Future
Circular Hadron Collider (FCC-hh) at CERN’s potential for direct observation
of NP mediators up to the multi-TeV range.

Recent observations have indicated other anomalies suggestive of NP, including
the anomalous magnetic moment of the muon and electron ($a_{\mu}$, $a_{e}$),
and the mass of the W-boson. These could share an origin with anomalies in $B$
meson decays. The Fermilab (g-2) Collaboration’s confirmation [144] of the old
BNL result has increased the discrepancy from the data-driven SM prediction to
about 4.2 $\sigma$. Intriguingly, each of these flavor anomalies exceeds a 3
$\sigma$ significance. The survival of at least one of these anomalies could
herald a new era in our understanding of physics. If the LFUV anomalies
persist, they may soon provide compelling evidence of NP within flavor
physics, prompting intensive investigation from future experimentalists and
theorists [145, 146]. Acknowledgments We thank Sanjay Swain and Sudhir Vempati
for fruitful discussions. 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.

Data Availability Statement: No Data associated in the manuscript.

## References

  *   * Workman et al. [2022] Workman, R.L., et al.: Review of Particle Physics. PTEP 2022, 083–01 (2022) https://doi.org/10.1093/ptep/ptac097
  * Ablikim et al. [2013] Ablikim, M., 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(3), 032007 (2013) https://doi.org/10.1103/PhysRevD.88.032007 arXiv:1307.1189 [hep-ex]
  * Lazzeroni et al. [2013] Lazzeroni, C., et al.: Precision Measurement of the Ratio of the Charged Kaon Leptonic Decay Rates. Phys. Lett. B 719, 326–336 (2013) https://doi.org/10.1016/j.physletb.2013.01.037 arXiv:1212.4012 [hep-ex]
  * Aguilar-Arevalo et al. [2015] Aguilar-Arevalo, A., et al.: Improved Measurement of the $\pi\rightarrow\textrm{e}\nu$ Branching Ratio. Phys. Rev. Lett. 115(7), 071801 (2015) https://doi.org/10.1103/PhysRevLett.115.071801 arXiv:1506.05845 [hep-ex]
  * Schael et al. [2006] Schael, S., et al.: Precision electroweak measurements on the $Z$ resonance. Phys. Rept. 427, 257–454 (2006) https://doi.org/10.1016/j.physrep.2005.12.006 arXiv:hep-ex/0509008
  * Amhis et al. [2017] Amhis, Y., et al.: Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016. Eur. Phys. J. C 77(12), 895 (2017) https://doi.org/10.1140/epjc/s10052-017-5058-4 arXiv:1612.07233 [hep-ex]
  * Bifani et al. [2019] Bifani, S., Descotes-Genon, S., Romero Vidal, A., Schune, M.-H.: Review of Lepton Universality tests in $B$ decays. J. Phys. G 46(2), 023001 (2019) https://doi.org/%****␣revised.bbl␣Line␣150␣****10.1088/1361-6471/aaf5de arXiv:1809.06229 [hep-ex]
  * Gambino et al. [2020] Gambino, P., et al.: Challenges in semileptonic $B$ decays. Eur. Phys. J. C 80(10), 966 (2020) https://doi.org/10.1140/epjc/s10052-020-08490-x arXiv:2006.07287 [hep-ph]
  * Bernlochner et al. [2022] Bernlochner, F.U., Sevilla, M.F., Robinson, D.J., Wormser, G.: Semitauonic b-hadron decays: A lepton flavor universality laboratory. Rev. Mod. Phys. 94(1), 015003 (2022) https://doi.org/10.1103/RevModPhys.94.015003 arXiv:2101.08326 [hep-ex]
  * Aaij et al. [2022] Aaij, R., et al.: Test of lepton universality in beauty-quark decays. Nature Phys. 18(3), 277–282 (2022) https://doi.org/%****␣revised.bbl␣Line␣200␣****10.1038/s41567-021-01478-8 arXiv:2103.11769 [hep-ex]
  * Bonilla et al. [2023] Bonilla, J., Giorgi, A., Gavela, B., Merlo, L., Ramos, M.: The cost of an ALP solution to the neutral B-anomalies. JHEP 02, 138 (2023) https://doi.org/10.1007/JHEP02(2023)138 arXiv:2209.11247 [hep-ph]
  * Gedeonová and Hudec [2023] Gedeonová, H., Hudec, M.: All possible first signals of gauge leptoquark in quark-lepton unification and beyond. Phys. Rev. D 107(9), 095029 (2023) https://doi.org/10.1103/PhysRevD.107.095029 arXiv:2210.00347 [hep-ph]
  * Castro and Skovpen [2022] Castro, N.F., Skovpen, K.: Flavour-Changing Neutral Scalar Interactions of the Top Quark. Universe 8(11), 609 (2022) https://doi.org/10.3390/universe8110609 arXiv:2210.09641 [hep-ex]
  * Descotes-Genon et al. [2023] Descotes-Genon, S., Plakias, I., Sumensari, O.: On the impact of meson mixing on Bs→ $\phi$ee angular observables at low q2. JHEP 02, 096 (2023) https://doi.org/10.1007/JHEP02(2023)096 arXiv:2210.11995 [hep-ph]
  * Buras [2023] Buras, A.J.: Standard Model predictions for rare K and B decays without new physics infection. Eur. Phys. J. C 83(1), 66 (2023) https://doi.org/10.1140/epjc/s10052-023-11222-6 arXiv:2209.03968 [hep-ph]
  * Datta et al. [2023] Datta, A., Hammad, A., Marfatia, D., Mukherjee, L., Rashed, A.: Dark photon and dark Z mediated B meson decays. JHEP 03, 108 (2023) https://doi.org/10.1007/JHEP03(2023)108 arXiv:2210.15662 [hep-ph]
  * Barbosa et al. [2023] Barbosa, D., Díaz, F., Quintero, L., Flórez, A., Sanchez, M., Gurrola, A., Sheridan, E., Romeo, F.: Probing a $\textrm{Z}^{\prime}$ with non-universal fermion couplings through top quark fusion, decays to bottom quarks, and machine learning techniques. Eur. Phys. J. C 83(5), 413 (2023) https://doi.org/10.1140/epjc/s10052-023-11506-x arXiv:2210.15813 [hep-ph]
  * Dutta and Bhol [2017] Dutta, R., Bhol, A.: $B_{c}\to(J/\psi,\,\eta_{c})\tau\nu$ semileptonic decays within the standard model and beyond. Phys. Rev. D 96(7), 076001 (2017) https://doi.org/10.1103/PhysRevD.96.076001 arXiv:1701.08598 [hep-ph]
  * Dutta [2019a] Dutta, R.: Predictions of $B_{c}\to(D,\,D^{\ast})\tau\,\nu$ decay observables in the standard model. J. Phys. G 46(3), 035008 (2019) https://doi.org/10.1088/1361-6471/ab0059 arXiv:1809.08561 [hep-ph]
  * Dutta [2019b] Dutta, R.: Model independent analysis of new physics effects on $B_{c}\to(D_{s},\,D^{\ast}_{s})\,\mu^{+}\mu^{-}$ decay observables. Phys. Rev. D 100(7), 075025 (2019) https://doi.org/10.1103/PhysRevD.100.075025 arXiv:1906.02412 [hep-ph]
  * Mohapatra et al. [2022] Mohapatra, M.K., Rajeev, N., Dutta, R.: Combined analysis of Bc→Ds(*)$\mu$+$\mu$\- and Bc→Ds(*)$\nu$$\nu$¯ decays within Z’ and leptoquark new physics models. Phys. Rev. D 105(11), 115022 (2022) https://doi.org/%****␣revised.bbl␣Line␣400␣****10.1103/PhysRevD.105.115022 arXiv:2108.10106 [hep-ph]
  * Dutta et al. [2013] Dutta, R., Bhol, A., Giri, A.K.: Effective theory approach to new physics in b → u and b → c leptonic and semileptonic decays. Phys. Rev. D 88(11), 114023 (2013) https://doi.org/10.1103/PhysRevD.88.114023 arXiv:1307.6653 [hep-ph]
  * Rajeev et al. [2021] Rajeev, N., Sahoo, N., Dutta, R.: Angular analysis of $B_{s}\,\to\,f_{2}^{\prime}\,(1525)\,(\to K^{+}\,K^{-})\,\mu^{+}\,\mu^{-}$ decays as a probe to lepton flavor universality violation. Phys. Rev. D 103(9), 095007 (2021) https://doi.org/10.1103/PhysRevD.103.095007 arXiv:2009.06213 [hep-ph]
  * Rajeev and Dutta [2022] Rajeev, N., Dutta, R.: Consequences of b→s$\mu$+$\mu$\- anomalies on B→K(*)$\nu$$\nu$¯, Bs→($\eta$,$\eta$’)$\nu$$\nu$¯ and Bs→$\phi$$\nu$$\nu$¯ decay observables. Phys. Rev. D 105(11), 115028 (2022) https://doi.org/10.1103/PhysRevD.105.115028 arXiv:2112.11682 [hep-ph]
  * Das and Dutta [2022] Das, N., Dutta, R.: New physics footprints in the angular distribution of Bs→Ds*(→Ds$\gamma$, Ds$\pi$)$\tau$$\nu$ decays. Phys. Rev. D 105(5), 055027 (2022) https://doi.org/10.1103/PhysRevD.105.055027 arXiv:2110.05526 [hep-ph]
  * Das et al. [2023] Das, N., Adhikary, A., Dutta, R.: Revisiting $b\to c\tau\nu$ anomalies with charged Higgs boson (2023) arXiv:2305.17766 [hep-ph]
  * Gao et al. [2022] Gao, J., Huber, T., Ji, Y., Wang, C., Wang, Y.-M., Wei, Y.-B.: B → D$\ell$$\nu$ℓ form factors beyond leading power and extraction of —Vcb— and R(D). JHEP 05, 024 (2022) https://doi.org/10.1007/JHEP05(2022)024 arXiv:2112.12674 [hep-ph]
  * Cui et al. [2023] Cui, B.-Y., Huang, Y.-K., Wang, Y.-M., Zhao, X.-C.: Shedding New Light on $\mathcal{R}(D^{(*)}_{(s)})$ and $|V_{cb}|$ from Semileptonic $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ Decays (2023) arXiv:2301.12391 [hep-ph]
  * Huang et al. [2018] Huang, Z.-R., Li, Y., Lu, C.-D., Paracha, M.A., Wang, C.: Footprints of New Physics in $b\to c\tau\nu$ Transitions. Phys. Rev. D 98(9), 095018 (2018) https://doi.org/10.1103/PhysRevD.98.095018 arXiv:1808.03565 [hep-ph]
  * Lees et al. [2012] Lees, J.P., et al.: Evidence for an excess of $\bar{B}\rightarrow D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays. Phys. Rev. Lett. 109, 101802 (2012) https://doi.org/10.1103/PhysRevLett.109.101802 arXiv:1205.5442 [hep-ex]
  * Lees et al. [2013] Lees, J.P., 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(7), 072012 (2013) https://doi.org/10.1103/PhysRevD.88.072012 arXiv:1303.0571 [hep-ex]
  * Wei et al. [2009] Wei, J.-T., et al.: Measurement of the Differential Branching Fraction and Forward-Backward Asymmetry for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$. Phys. Rev. Lett. 103, 171801 (2009) https://doi.org/10.1103/PhysRevLett.103.171801 arXiv:0904.0770 [hep-ex]
  * Huschle et al. [2015] Huschle, M., 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(7), 072014 (2015) https://doi.org/10.1103/PhysRevD.92.072014 arXiv:1507.03233 [hep-ex]
  * Hirose et al. [2017] Hirose, S., 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(21), 211801 (2017) https://doi.org/10.1103/PhysRevLett.118.211801 arXiv:1612.00529 [hep-ex]
  * Choudhury et al. [2021] Choudhury, S., et al.: Test of lepton flavor universality and search for lepton flavor violation in $B\rightarrow K\ell\ell$ decays. JHEP 03, 105 (2021) https://doi.org/10.1007/JHEP03(2021)105 arXiv:1908.01848 [hep-ex]
  * Abdesselam et al. [2021] Abdesselam, A., et al.: Test of Lepton-Flavor Universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ Decays at Belle. Phys. Rev. Lett. 126(16), 161801 (2021) https://doi.org/10.1103/PhysRevLett.126.161801 arXiv:1904.02440 [hep-ex]
  * Aaij et al. [2015] Aaij, R., 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(11), 111803 (2015) https://doi.org/10.1103/PhysRevLett.115.111803 arXiv:1506.08614 [hep-ex]. [Erratum: Phys.Rev.Lett. 115, 159901 (2015)]
  * Aaij et al. [2016] Aaij, R., 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, 047 (2016) https://doi.org/10.1007/JHEP11(2016)047 arXiv:1606.04731 [hep-ex]. [Erratum: JHEP 04, 142 (2017)]
  * Aaij et al. [2017] Aaij, R., et al.: Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays. JHEP 08, 055 (2017) https://doi.org/10.1007/JHEP08(2017)055 arXiv:1705.05802 [hep-ex]
  * Aaij et al. [2020] Aaij, R., et al.: Test of lepton universality with ${\Lambda}_{b}^{0}\to{pK}^{-}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-}$ decays. JHEP 05, 040 (2020) https://doi.org/10.1007/JHEP05(2020)040 arXiv:1912.08139 [hep-ex]
  * Aaij et al. [2022] Aaij, R., 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(19), 191802 (2022) https://doi.org/10.1103/PhysRevLett.128.191802 arXiv:2110.09501 [hep-ex]
  * Aaij et al. [2021a] Aaij, R., et al.: Angular analysis of the rare decay. JHEP 11, 043 (2021) https://doi.org/10.1007/JHEP11(2021)043 arXiv:2107.13428 [hep-ex]
  * Aaij et al. [2021b] Aaij, R., 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(15), 151801 (2021) https://doi.org/10.1103/PhysRevLett.127.151801 arXiv:2105.14007 [hep-ex]
  * Lees et al. [2012] Lees, J.P., et al.: Measurement of Branching Fractions and Rate Asymmetries in the Rare Decays $B\rightarrow K^{(*)}l^{+}l^{-}$. Phys. Rev. D 86, 032012 (2012) https://doi.org/10.1103/PhysRevD.86.032012 arXiv:1204.3933 [hep-ex]
  * Aaij et al. [2014a] Aaij, R., et al.: Differential branching fractions and isospin asymmetries of $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ decays. JHEP 06, 133 (2014) https://doi.org/10.1007/JHEP06(2014)133 arXiv:1403.8044 [hep-ex]
  * Aaij et al. [2014b] Aaij, R., et al.: Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays. Phys. Rev. Lett. 113, 151601 (2014) https://doi.org/10.1103/PhysRevLett.113.151601 arXiv:1406.6482 [hep-ex]
  * Aaij et al. [2015a] Aaij, R., et al.: Differential branching fraction and angular analysis of $\Lambda^{0}_{b}\rightarrow\Lambda\mu^{+}\mu^{-}$ decays. JHEP 06, 115 (2015) https://doi.org/10.1007/JHEP06(2015)115 arXiv:1503.07138 [hep-ex]. [Erratum: JHEP 09, 145 (2018)]
  * Aaij et al. [2015b] Aaij, R., et al.: Angular analysis and differential branching fraction of the decay $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$. JHEP 09, 179 (2015) https://doi.org/10.1007/JHEP09(2015)179 arXiv:1506.08777 [hep-ex]
  * Aaij et al. [2016] Aaij, R., et al.: Angular analysis of the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity. JHEP 02, 104 (2016) https://doi.org/10.1007/JHEP02(2016)104 arXiv:1512.04442 [hep-ex]
  * Wehle et al. [2017] Wehle, S., et al.: Lepton-Flavor-Dependent Angular Analysis of $B\rightarrow K^{\ast}\ell^{+}\ell^{-}$. Phys. Rev. Lett. 118(11), 111801 (2017) https://doi.org/10.1103/PhysRevLett.118.111801 arXiv:1612.05014 [hep-ex]
  * Sirunyan et al. [2018] Sirunyan, A.M., 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, 517–541 (2018) https://doi.org/10.1016/j.physletb.2018.04.030 arXiv:1710.02846 [hep-ex]
  * Aaboud et al. [2018] Aaboud, M., 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, 047 (2018) https://doi.org/10.1007/JHEP10(2018)047 arXiv:1805.04000 [hep-ex]
  * Amhis et al. [2021] Amhis, Y.S., et al.: Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2018\. Eur. Phys. J. C 81(3), 226 (2021) https://doi.org/10.1140/epjc/s10052-020-8156-7 arXiv:1909.12524 [hep-ex]
  * Amhis et al. [2023] Amhis, Y.S., et al.: Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2021. Phys. Rev. D 107(5), 052008 (2023) https://doi.org/10.1103/PhysRevD.107.052008 arXiv:2206.07501 [hep-ex]
  * [55] Average of $R(D)$ and $R(D^{*})$ for end of 2022
  * Aaij et al. [2023] Aaij, R., et al.: Test of lepton flavour universality using $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays with hadronic $\tau$ channels (2023) arXiv:2305.01463 [hep-ex]
  * Iguro and Watanabe [2020] Iguro, S., Watanabe, R.: 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(08), 006 (2020) https://doi.org/10.1007/JHEP08(2020)006 arXiv:2004.10208 [hep-ph]
  * Bordone et al. [2020] Bordone, M., Jung, M., Dyk, D.: Theory determination of $\bar{B}\to D^{(*)}\ell^{-}\bar{\nu}$ form factors at $\mathcal{O}(1/m_{c}^{2})$. Eur. Phys. J. C 80(2), 74 (2020) https://doi.org/10.1140/epjc/s10052-020-7616-4 arXiv:1908.09398 [hep-ph]
  * Aaij et al. [2018] Aaij, R., 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(12), 121801 (2018) https://doi.org/10.1103/PhysRevLett.120.121801 arXiv:1711.05623 [hep-ex]
  * Ivanov, Mikhail A. and Körner, Jürgen G. and Santorelli, Pietro [2007] Ivanov, Mikhail A. and Körner, Jürgen G. and Santorelli, Pietro: Semileptonic and nonleptonic decays of $B_{c}$, pp. 251–255 (2007). https://doi.org/10.1007/978-88-470-0530-3_43
  * Hernandez et al. [2006] Hernandez, E., Nieves, J., Verde-Velasco, J.M.: Study of exclusive semileptonic and non-leptonic decays of $B_{c}$ \- in a nonrelativistic quark model. Phys. Rev. D 74, 074008 (2006) https://doi.org/10.1103/PhysRevD.74.074008 arXiv:hep-ph/0607150
  * Watanabe [2018] Watanabe, R.: New Physics effect on $B_{c}\to J/\psi\tau\bar{\nu}$ in relation to the $R_{D^{(*)}}$ anomaly. Phys. Lett. B 776, 5–9 (2018) https://doi.org/10.1016/j.physletb.2017.11.016 arXiv:1709.08644 [hep-ph]
  * Ivanov et al. [2006] Ivanov, M.A., Korner, J.G., Santorelli, P.: Exclusive semileptonic and nonleptonic decays of the $B_{c}$ meson. Phys. Rev. D 73, 054024 (2006) https://doi.org/10.1103/PhysRevD.73.054024 arXiv:hep-ph/0602050
  * Aaij et al. [2019] Aaij, R., et al.: Search for lepton-universality violation in $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays. Phys. Rev. Lett. 122(19), 191801 (2019) https://doi.org/10.1103/PhysRevLett.122.191801 arXiv:1903.09252 [hep-ex]
  * Bordone et al. [2016] Bordone, M., Isidori, G., Pattori, A.: On the Standard Model predictions for $R_{K}$ and $R_{K^{*}}$. Eur. Phys. J. C 76(8), 440 (2016) https://doi.org/10.1140/epjc/s10052-016-4274-7 arXiv:1605.07633 [hep-ph]
  * Isidori et al. [2020] Isidori, G., Nabeebaccus, S., Zwicky, R.: QED corrections in $\bar{B}\rightarrow\bar{K}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-}$ at the double-differential level. JHEP 12, 104 (2020) https://doi.org/10.1007/JHEP12(2020)104 arXiv:2009.00929 [hep-ph]
  * Aaij et al. [2023] Aaij, R., et al.: Test of Lepton Universality in $b\rightarrow s\ell^{+}\ell^{-}$\- Decays. Phys. Rev. Lett. 131(5), 051803 (2023) https://doi.org/10.1103/PhysRevLett.131.051803 arXiv:2212.09152 [hep-ex]
  * Patnaik and Singh [2023] Patnaik, S., Singh, R.: A Light Shed on Lepton Flavor Universality in B Decays. Universe 9(3), 129 (2023) https://doi.org/10.3390/universe9030129 arXiv:2211.04348 [hep-ph]
  * Iguro et al. [2022] Iguro, S., Kitahara, T., Watanabe, R.: Global fit to $b\to c\tau\nu$ anomalies 2022 mid-autumn (2022) arXiv:2210.10751 [hep-ph]
  * Crivellin et al. [2017] Crivellin, A., Müller, D., Ota, T. JHEP 09, 040 (2017) https://doi.org/10.1007/JHEP09(2017)040 arXiv:1703.09226 [hep-ph]
  * Crivellin et al. [2019] Crivellin, A., Greub, C., Müller, D., Saturnino, F.: Importance of Loop Effects in Explaining the Accumulated Evidence for New Physics in B Decays with a Vector Leptoquark. Phys. Rev. Lett. 122(1), 011805 (2019) https://doi.org/10.1103/PhysRevLett.122.011805 arXiv:1807.02068 [hep-ph]
  * Crivellin et al. [2020] Crivellin, A., Müller, D., Saturnino, F.: Flavor Phenomenology of the Leptoquark Singlet-Triplet Model. JHEP 06, 020 (2020) https://doi.org/10.1007/JHEP06(2020)020 arXiv:1912.04224 [hep-ph]
  * Blanke and Crivellin [2018] Blanke, M., Crivellin, A.: $B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background. Phys. Rev. Lett. 121(1), 011801 (2018) https://doi.org/10.1103/PhysRevLett.121.011801 arXiv:1801.07256 [hep-ph]
  * Calibbi et al. [2018] Calibbi, L., Crivellin, A., Li, T.: Model of vector leptoquarks in view of the $B$-physics anomalies. Phys. Rev. D 98(11), 115002 (2018) https://doi.org/10.1103/PhysRevD.98.115002 arXiv:1709.00692 [hep-ph]
  * Calibbi et al. [2015] Calibbi, L., Crivellin, A., Ota, T.: Effective Field Theory Approach to $b\rightarrow s\ell\ell^{(^{\prime})}$, $B\rightarrow K^{(*)}\nu\bar{\nu}$ and $B\rightarrow D^{(*)}\tau\nu$ with Third Generation Couplings. Phys. Rev. Lett. 115, 181801 (2015) https://doi.org/10.1103/PhysRevLett.115.181801 arXiv:1506.02661 [hep-ph]
  * Herb et al. [1977] Herb, S.W., et al.: Observation of a Dimuon Resonance at 9.5-GeV in 400-GeV Proton-Nucleus Collisions. Phys. Rev. Lett. 39, 252–255 (1977) https://doi.org/10.1103/PhysRevLett.39.252
  * Alves et al. [2008] Alves, A.A. Jr., et al.: The LHCb Detector at the LHC. JINST 3, 08005 (2008) https://doi.org/10.1088/1748-0221/3/08/S08005
  * Hiller and Schmaltz [2014] Hiller, G., Schmaltz, M.: $R_{K}$ and future $b\rightarrow s\ell\ell$ physics beyond the standard model opportunities. Phys. Rev. D 90, 054014 (2014) https://doi.org/10.1103/PhysRevD.90.054014 arXiv:1408.1627 [hep-ph]
  * Gripaios et al. [2015] Gripaios, B., Nardecchia, M., Renner, S.A.: Composite leptoquarks and anomalies in $B$-meson decays. JHEP 05, 006 (2015) https://doi.org/10.1007/JHEP05(2015)006 arXiv:1412.1791 [hep-ph]
  * de Medeiros Varzielas and Hiller [2015] Medeiros Varzielas, I., Hiller, G.: Clues for flavor from rare lepton and quark decays. JHEP 06, 072 (2015) https://doi.org/10.1007/JHEP06(2015)072 arXiv:1503.01084 [hep-ph]
  * Barbieri et al. [2017] Barbieri, R., Murphy, C.W., Senia, F.: B-decay Anomalies in a Composite Leptoquark Model. Eur. Phys. J. C 77(1), 8 (2017) https://doi.org/10.1140/epjc/s10052-016-4578-7 arXiv:1611.04930 [hep-ph]
  * Altmannshofer et al. [2014] Altmannshofer, W., Gori, S., Pospelov, M., Yavin, I.: Quark flavor transitions in $L_{\mu}-L_{\tau}$ models. Phys. Rev. D 89, 095033 (2014) https://doi.org/10.1103/PhysRevD.89.095033 arXiv:1403.1269 [hep-ph]
  * Crivellin et al. [2015] Crivellin, A., D’Ambrosio, G., Heeck, J.: 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, 151801 (2015) https://doi.org/10.1103/PhysRevLett.114.151801 arXiv:1501.00993 [hep-ph]
  * Celis et al. [2015] Celis, A., Fuentes-Martin, J., Jung, M., Serodio, H.: Family nonuniversal Z’ models with protected flavor-changing interactions. Phys. Rev. D 92(1), 015007 (2015) https://doi.org/10.1103/PhysRevD.92.015007 arXiv:1505.03079 [hep-ph]
  * Falkowski et al. [2015] Falkowski, A., Nardecchia, M., Ziegler, R.: Lepton Flavor Non-Universality in B-meson Decays from a U(2) Flavor Model. JHEP 11, 173 (2015) https://doi.org/10.1007/JHEP11(2015)173 arXiv:1509.01249 [hep-ph]
  * Aaij et al. [2015a] Aaij, R., et al.: LHCb Detector Performance. Int. J. Mod. Phys. A 30(07), 1530022 (2015) https://doi.org/%****␣revised.bbl␣Line␣1475␣****10.1142/S0217751X15300227 arXiv:1412.6352 [hep-ex]
  * Aaij et al. [2015b] Aaij, R., et al.: Measurement of $B_{c}^{+}$ production in proton-proton collisions at $\sqrt{s}=8$ TeV. Phys. Rev. Lett. 114, 132001 (2015) https://doi.org/10.1103/PhysRevLett.114.132001 arXiv:1411.2943 [hep-ex]
  * Abdesselam et al. [2016] Abdesselam, A., 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. (2016)
  * Aaij et al. [2018] Aaij, R., 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(17), 171802 (2018) https://doi.org/10.1103/PhysRevLett.120.171802 arXiv:1708.08856 [hep-ex]
  * Albrecht et al. [2017] Albrecht, J., Bernlochner, F., Kenzie, M., Reichert, S., Straub, D., Tully, A.: Future prospects for exploring present day anomalies in flavour physics measurements with Belle II and LHCb (2017) arXiv:1709.10308 [hep-ph]
  * Abudinén et al. [2020] Abudinén, F., et al.: First flavor tagging calibration using 2019 Belle II data (2020) arXiv:2008.02707 [hep-ex]
  * Aggarwal et al. [2022] Aggarwal, L., et al.: Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond (2022) arXiv:2207.06307 [hep-ex]
  * Aad et al. [2008] Aad, G., et al.: The ATLAS Experiment at the CERN Large Hadron Collider. JINST 3, 08003 (2008) https://doi.org/10.1088/1748-0221/3/08/S08003
  * Chatrchyan et al. [2008] Chatrchyan, S., et al.: The CMS Experiment at the CERN LHC. JINST 3, 08004 (2008) https://doi.org/10.1088/1748-0221/3/08/S08004
  * Sirunyan et al. [2019] Sirunyan, A.M., 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(13), 132001 (2019) https://doi.org/10.1103/PhysRevLett.122.132001 arXiv:1902.00571 [hep-ex]
  * Abd El-Hady et al. [2000] Abd El-Hady, A., Munoz, J.H., Vary, J.P.: Semileptonic and nonleptonic B(c) decays. Phys. Rev. D 62, 014019 (2000) https://doi.org/10.1103/PhysRevD.62.014019 arXiv:hep-ph/9909406
  * Colquhoun et al. [2015] Colquhoun, B., Davies, C.T.H., Dowdall, R.J., Kettle, J., Koponen, J., Lepage, G.P., Lytle, A.T.: B-meson decay constants: a more complete picture from full lattice QCD. Phys. Rev. D 91(11), 114509 (2015) https://doi.org/10.1103/PhysRevD.91.114509 arXiv:1503.05762 [hep-lat]
  * Patnaik et al. [2020] Patnaik, S., Nayak, L., Dash, P.C., Kar, S., Barik, N.: Semileptonic $B_{c}$ meson decays to S-wave charmonium states. Eur. Phys. J. Plus 135(11), 936 (2020) https://doi.org/10.1140/epjp/s13360-020-00898-4 arXiv:1910.13206 [hep-ph]
  * Patnaik et al. [2023] Patnaik, S., Nayak, L., Sadangi, P., Swain, S., Singh, R.: Study of angular observables in exclusive $B_{c}$ decays (2023) arXiv:2312.17114 [hep-ph]
  * Barik et al. [1992] Barik, N., Dash, P.C., Panda, A.R.: Radiative decay of mesons in an independent quark potential model. Phys. Rev. D 46, 3856–3861 (1992) https://doi.org/10.1103/PhysRevD.46.3856
  * Barik and Dash [1994] Barik, N., Dash, P.C.: Radiative decay of light and heavy mesons. Phys. Rev. D 49, 299–308 (1994) https://doi.org/10.1103/PhysRevD.49.299 . [Erratum: Phys.Rev.D 53, 4110 (1996)]
  * Priyadarsini et al. [2016] Priyadarsini, M., Dash, P.C., Kar, S., Patra, S.P., Barik, N.: Electromagnetic form factors of heavy flavored vector mesons. Phys. Rev. D 94(11), 113011 (2016) https://doi.org/10.1103/PhysRevD.94.113011
  * Barik and Dash [1995] Barik, N., Dash, P.C.: Radiative weak decays of pseudoscalar mesons in the charm sector. Mod. Phys. Lett. A 10, 103–114 (1995) https://doi.org/10.1142/S0217732395000119
  * Barik et al. [1998] Barik, N., Kar, S., Dash, P.C.: Exclusive rare radiative decays of B mesons. Phys. Rev. D 57, 405–412 (1998) https://doi.org/10.1103/PhysRevD.57.405
  * Barik et al. [2001] Barik, N., Naimuddin, S., Kar, S., Dash, P.C.: Rare radiative decay of the $B_{c}$ meson. Phys. Rev. D 63, 014024 (2001) https://doi.org/10.1103/PhysRevD.63.014024
  * Barik et al. [1993] Barik, N., Dash, P.C., Panda, A.R.: Leptonic decay of light vector mesons in an independent quark model. Phys. Rev. D 47, 1001–1006 (1993) https://doi.org/10.1103/PhysRevD.53.4110 . [Erratum: Phys.Rev.D 53, 4110 (1996)]
  * Barik and Dash [1993] Barik, N., Dash, P.C.: Weak leptonic decay of light and heavy pseudoscalar mesons in an independent quark model. Phys. Rev. D 47, 2788–2795 (1993) https://doi.org/%****␣revised.bbl␣Line␣1800␣****10.1103/PhysRevD.47.2788
  * Barik and Dash [1996] Barik, N., Dash, P.C.: Exclusive semileptonic decay of D and B mesons in the independent quark model. Phys. Rev. D 53, 1366–1377 (1996) https://doi.org/10.1103/PhysRevD.53.1366
  * Barik et al. [1997] Barik, N., Tripathy, S.K., Kar, S., Dash, P.C.: Exclusive semileptonic decays of charmed and b flavored mesons. Phys. Rev. D 56, 4238–4249 (1997) https://doi.org/10.1103/PhysRevD.56.4238
  * Barik et al. [2009] Barik, N., Naimuddin, S., Dash, P.C., Kar, S.: Semileptonic decays of the Bc meson. Phys. Rev. D 80, 074005 (2009) https://doi.org/10.1103/PhysRevD.80.074005
  * Barik et al. [2008a] Barik, N., Naimuddin, S., Dash, P.C., Kar, S.: Radiative leptonic decay: $B^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\gamma$ in a relativistic independent quark model. Phys. Rev. D 77, 014038 (2008) https://doi.org/10.1103/PhysRevD.77.014038
  * Barik et al. [2008b] Barik, N., Naimuddin, S., Dash, P.C., Kar, S.: Radiative leptonic $B_{c}$ decay in the relativistic independent quark model. Phys. Rev. D 78, 114030 (2008) https://doi.org/10.1103/PhysRevD.78.114030
  * Barik et al. [2009] Barik, N., Naimuddin, S., Dash, P.C.: Radiative leptonic decay of D/s and D mesons. Int. J. Mod. Phys. A 24, 2335–2355 (2009) https://doi.org/10.1142/S0217751X09043262
  * Barik et al. [2001] Barik, N., Kar, S., Dash, P.C.: Exclusive nonleptonic decays of B mesons. Phys. Rev. D 63, 114002 (2001) https://doi.org/10.1103/PhysRevD.63.114002
  * Barik et al. [2009] Barik, N., Naimuddin, S., Dash, P.C., Kar, S.: Exclusive nonleptonic $B\rightarrow VV$ decays. Phys. Rev. D 80, 014004 (2009) https://doi.org/10.1103/PhysRevD.80.014004
  * Naimuddin et al. [2012] Naimuddin, S., Kar, S., Priyadarsini, M., Barik, N., Dash, P.C.: Nonleptonic two-body Bc-meson decays. Phys. Rev. D 86, 094028 (2012) https://doi.org/10.1103/PhysRevD.86.094028
  * Kar et al. [2013] Kar, S., Dash, P.C., Priyadarsini, M., Naimuddin, S., Barik, N.: Nonleptonic $B_{c}\rightarrow VV$ decays. Phys. Rev. D 88(9), 094014 (2013) https://doi.org/10.1103/PhysRevD.88.094014
  * Patnaik et al. [2017] Patnaik, S., Dash, P.C., Kar, S., Patra, S., Barik, N.: Magnetic dipole transitions of $B_{c}$ and $B_{c}^{*}$ mesons in the relativistic independent quark model. Phys. Rev. D 96(11), 116010 (2017) https://doi.org/10.1103/PhysRevD.96.116010 arXiv:1710.08242 [hep-ph]. [Erratum: Phys.Rev.D 99, 019901 (2019)]
  * Patnaik et al. [2018] Patnaik, S., Dash, P.C., Kar, S., Barik, N.: Electromagnetic transitions of $(b{\bar{c}})$ bound system. Phys. Rev. D 97(5), 056025 (2018) https://doi.org/10.1103/PhysRevD.97.056025 arXiv:1801.10503 [hep-ph]. [Erratum: Phys.Rev.D 99, 019902 (2019)]
  * Nayak et al. [2021] Nayak, L., Patnaik, S., Dash, P.C., Kar, S., Barik, N.: Lepton mass effects in exclusive semileptonic $B_{c}$-meson decays. Phys. Rev. D 104, 036012 (2021) https://doi.org/10.1103/PhysRevD.104.036012 arXiv:2106.09463 [hep-ph]
  * Issadykov et al. [2017] Issadykov, A., Ivanov, M.A., Nurbakova, G.: Semileptonic decays of Bc mesons into charmonium states. EPJ Web Conf. 158, 03002 (2017) https://doi.org/10.1051/epjconf/201715803002 arXiv:1907.13210 [hep-ph]
  * Wang et al. [2013] Wang, W.-F., Fan, Y.-Y., Xiao, Z.-J.: Semileptonic decays $B_{c}\rightarrow(\eta_{c},J/\Psi)l\nu$ in the perturbative QCD approach. Chin. Phys. C 37, 093102 (2013) https://doi.org/10.1088/1674-1137/37/9/093102 arXiv:1212.5903 [hep-ph]
  * Hu et al. [2020] Hu, X.-Q., Jin, S.-P., Xiao, Z.-J.: Semileptonic decays $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l}$ in the ”PQCD + Lattice” approach. Chin. Phys. C 44(2), 023104 (2020) https://doi.org/10.1088/1674-1137/44/2/023104 arXiv:1904.07530 [hep-ph]
  * Cooper et al. [2022] Cooper, L.J., Davies, C.T.H., Wingate, M.: Form factors for the processes $B^{+}_{c}\to D^{0}\ell^{+}\nu_{\ell}$ and $B^{+}_{c}\to D^{+}_{s}\ell^{+}\ell^{+}(\nu\bar{\nu})$ from lattice QCD. Phys. Rev. D 105(1), 014503 (2022) https://doi.org/10.1103/PhysRevD.105.014503 arXiv:2108.11242 [hep-lat]
  * Zhou et al. [2021] Zhou, T., Wang, T., Jiang, Y., Huo, L., Wang, G.-L.: The weak $B,B_{s}$ and $B_{c}$ decays to radially excited states. J. Phys. G 48(5), 055006 (2021) https://doi.org/10.1088/1361-6471/abc3a6 arXiv:2006.05704 [hep-ph]
  * Ebert et al. [2003] Ebert, D., Faustov, R.N., Galkin, V.O.: Weak decays of the $B_{c}$ meson to charmonium and $D$ mesons in the relativistic quark model. Phys. Rev. D 68, 094020 (2003) https://doi.org/10.1103/PhysRevD.68.094020 arXiv:hep-ph/0306306
  * Ebert et al. [2010] Ebert, D., Faustov, R.N., Galkin, V.O.: Semileptonic and Nonleptonic Decays of $B_{c}$ Mesons to Orbitally Excited Heavy Mesons in the Relativistic Quark Model. Phys. Rev. D 82, 034019 (2010) https://doi.org/10.1103/PhysRevD.82.034019 arXiv:1007.1369 [hep-ph]
  * Rui et al. [2016] Rui, Z., Li, H., Wang, G.-x., Xiao, Y.: Semileptonic decays of b_c b c meson to s-wave charmonium states in the perturbative qcd approach. The European Physical Journal C 76, 1–11 (2016)
  * Wang and Lu [2008] Wang, Y.-M., Lu, C.-D.: Weak productions of new charmonium in semi-leptonic decays of $B_{c}$. Phys. Rev. D 77, 054003 (2008) https://doi.org/10.1103/PhysRevD.77.054003 arXiv:0707.4439 [hep-ph]
  * Bediaga and Munoz [2011] Bediaga, I., Munoz, J.H.: Production of radially excited charmonium mesons in two-body nonleptonic $B_{c}$ decays (2011) arXiv:1102.2190 [hep-ph]
  * Ciuchini et al. [2021] Ciuchini, M., Fedele, M., Franco, E., Paul, A., Silvestrini, L., Valli, M.: New Physics without bias: Charming Penguins and Lepton Universality Violation in $b\rightarrow s\ell^{+}\ell^{-}$ decays (2021) arXiv:2110.10126 [hep-ph]
  * Isidori et al. [2022] Isidori, G., Lancierini, D., Nabeebaccus, S., Zwicky, R.: QED in $\bar{B}\rightarrow\bar{K}l^{+}l^{-}$ LFU ratios: theory versus experiment, a Monte Carlo study. JHEP 10, 146 (2022) https://doi.org/%****␣revised.bbl␣Line␣2225␣****10.1007/JHEP10(2022)146 arXiv:2205.08635 [hep-ph]
  * Altmannshofer and Stangl [2021] Altmannshofer, W., Stangl, P.: New physics in rare B decays after Moriond 2021. Eur. Phys. J. C 81(10), 952 (2021) https://doi.org/10.1140/epjc/s10052-021-09725-1 arXiv:2103.13370 [hep-ph]
  * Alok et al. [2022] Alok, A.K., Singh Chundawat, N.R., Gangal, S., Kumar, D.: A global analysis of $b\rightarrow s\ell\ell$ data in heavy and light $Z^{\prime}$ models. Eur. Phys. J. C 82(10), 967 (2022) https://doi.org/10.1140/epjc/s10052-022-10816-w arXiv:2203.13217 [hep-ph]
  * Singh Chundawat [2023] Singh Chundawat, N.R.: $CP$ violation in $b\to s\ell\ell$: a model independent analysis. Phys. Rev. D 107, 075014 (2023) https://doi.org/10.1103/PhysRevD.107.075014 arXiv:2207.10613 [hep-ph]
  * Bhattacharya et al. [2015] Bhattacharya, B., Datta, A., London, D., Shivashankara, S.: Simultaneous Explanation of the $R_{K}$ and $R(D^{(*)})$ Puzzles. Phys. Lett. B 742, 370–374 (2015) https://doi.org/10.1016/j.physletb.2015.02.011 arXiv:1412.7164 [hep-ph]
  * Bhattacharya et al. [2017] Bhattacharya, B., Datta, A., Guévin, J.-P., London, D., Watanabe, R.: Simultaneous Explanation of the $R_{K}$ and $R_{D^{(*)}}$ Puzzles: a Model Analysis. JHEP 01, 015 (2017) https://doi.org/10.1007/JHEP01(2017)015 arXiv:1609.09078 [hep-ph]
  * Kumar et al. [2019] Kumar, J., London, D., Watanabe, R.: Combined Explanations of the $b\to s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}$ Anomalies: a General Model Analysis. Phys. Rev. D 99(1), 015007 (2019) https://doi.org/10.1103/PhysRevD.99.015007 arXiv:1806.07403 [hep-ph]
  * Bhupal Dev et al. [2020] Bhupal Dev, P.S., Mohanta, R., Patra, S., Sahoo, S.: Unified explanation of flavor anomalies, radiative neutrino masses, and ANITA anomalous events in a vector leptoquark model. Phys. Rev. D 102(9), 095012 (2020) https://doi.org/10.1103/PhysRevD.102.095012 arXiv:2004.09464 [hep-ph]
  * Mahata et al. [2022] Mahata, S., Biswas, S., Sahoo, S.: Study of $B\rightarrow{K}_{0}^{*}\left(1430\right){l}^{+}{l}^{-}$ decays in non-universal ${Z}^{{\prime}}$ models and in a model-independent way. Eur. Phys. J. Plus 137(11), 1220 (2022) https://doi.org/10.1140/epjp/s13360-022-03258-6
  * Marzocca [2018] Marzocca, D.: Addressing the B-physics anomalies in a fundamental Composite Higgs Model. JHEP 07, 121 (2018) https://doi.org/10.1007/JHEP07(2018)121 arXiv:1803.10972 [hep-ph]
  * Barman et al. [2019] Barman, B., Borah, D., Mukherjee, L., Nandi, S.: Correlating the anomalous results in $b\rightarrow s$ decays with inert Higgs doublet dark matter and muon $(g-2)$. Phys. Rev. D 100(11), 115010 (2019) https://doi.org/10.1103/PhysRevD.100.115010 arXiv:1808.06639 [hep-ph]
  * Mandal et al. [2020] Mandal, R., Murgui, C., Peñuelas, A., Pich, A.: The role of right-handed neutrinos in $b\rightarrow c\tau\bar{\nu}$ anomalies. JHEP 08(08), 022 (2020) https://doi.org/%****␣revised.bbl␣Line␣2425␣****10.1007/JHEP08(2020)022 arXiv:2004.06726 [hep-ph]
  * Abi et al. [2021] Abi, B., et al.: Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. Phys. Rev. Lett. 126(14), 141801 (2021) https://doi.org/10.1103/PhysRevLett.126.141801 arXiv:2104.03281 [hep-ex]
  * Aryshev et al. [2022] Aryshev, A., et al.: The International Linear Collider: Report to Snowmass 2021 (2022) arXiv:2203.07622 [physics.acc-ph]
  * Abir et al. [2023] Abir, R., et al.: The case for an EIC Theory Alliance: Theoretical Challenges of the EIC (2023) arXiv:2305.14572 [hep-ph]